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THESIS

,7 J)

This is to certify that the

dissertation entitled
Coherent and Chaotic Phenomena in Nuclear Matter
presented by
Alexander Volya
has been accepted towards fulfillment

of the requirements for

Ph.D. degree in Physics

00'

Q l flajor professdt

 

Date July 14, 2000

MSU is an Affirmative Action/Equal Opportunity Inxiilulian 0712771

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University

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COHERENT AND CHAOTIC PHENOMENA IN NUCLEAR
MATTER

By

Alexander Volya

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the Degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

2000

ABSTRACT

COHERENT AND CHAOTIC PHENOMENA IN NUCLEAR MATTER

By

Alexander Volya

The interplay between coherence and chaos is often the key question in under-
standing the behavior of a complex many—body system. The research work presented
here considers several topics from nuclear physics aiming to pose questions and search
for answers in this unconquered area of many-body physics.

The first topic in this dissertation deals with pairing, which is responsible for su-
perconductivity and superfluidity in macroscopic Fermi systems. The smallness of
the nuclear system reduces the sharpness of this phase transition, making macro-
scopic methods inadequate. This fact calls for the development of other theoretical
techniques to which this research work is dedicated.

Chaos, intrinsic to many-body systems, often complicates many solutions to nu-
clear physics problems. Nevertheless, as it is shown in the second research project,
chaotic features can play a regularizing role and can be successfully used for extrap-
olating results of large—scale numerical work.

Pionic fusion is the process of pion production in the coalescence of two nuclei
extremely close to the energetic threshold so that the coherent action of all nucle-
ons is required for it to take place. This amazing reaction has been experimentally
observed. The suggested theoretical treatment of pionic fusion successfully describes
experimental results and can be used for future predictions.

In—medium pions may be considerably softened, and the reduction of the effective

pion energy may open an unusual channel of photon “decay” into a pion and another

photon inside the nucleus. The possibility of using this reaction as a tool for probing
the pionic mode, was investigated in detail.

Higher energies and densities make pionic features more pronounced. Squeezed
states forming a pion condensate can be created in a heavy ion collision or in reactions
with cosmic rays. The dynamical evolution of parametrically excited quantum pion
fields is studied in the last chapter. The possibility of the appearance of an oriented
chiral condensate is discussed along with the resulting charge fluctuations and particle

multiplicities.

ACKNOWLEDGMENTS

I owe my deepest gratitude to my advisor Scott Pratt, whose careful guidance, pa-
tience and cheerful humor made my advances possible. Vladimir Zelevinsky played
an essential role in my graduate career. His help and supervision in my research work
were indispensable. I highly appreciate help and professional advice from W. Bauer,
B.A. Brown, P. Danielewicz, M. Dykman and W. Repko. I am indebted to V. F.
Dmitriev, K. Haglin, M. Horoi for joining my efforts in research work presented in
this dissertation. I must thank G. Bertseh, D.A. Brown, F.M.Izrailev, M. Mostovoy,
D. Mulhall and A. Sakharuk for all the constructive and motivating discussions with
regard to my work. The atmosphere in the theory group and at the entire National
Superconducting Cyclotron Lab is very stimulating, it was a privilege to work there.
Finally, I would like to thank Shari Conroy for guiding me through the mazes of

random paperwork and administrative chores that come with graduate school.

iv

CONTENTS

LIST OF FIGURES vii

1 Introduction 1

2 Particle number conserving description of pairing 6
2.1 Experimental evidence of paired collective

states in nuclei .............................. 6

2.2 Pairing forces ............................... 10

2.3 Solutions to the pairing problem ..................... 15

2.3.1 Seniority scheme and degenerate model ............. 15

2.3.2 BCS theory in the case of nuclei ................. 19

2.3.3 Exact solutions .......................... 22

2.4 Particle conserving methods ....................... 22

3 Chaotic wave functions and exponential convergence of low-lying

energy eigenvalues 36
3.1 Introduction ................................ 36
3.2 Examples of exponential convergence .................. 37
3.3 Solvable models .............................. 46
3.4 Conclusion ................................. 51
4 Modeling pionic fusion 52
4.1 Introduction ................................ 52
4.2 Description of the model ......................... 54
4.2.1 The transition amplitude ..................... 54
4.2.2 Nuclear wave functions ...................... 55

4.3 Fusion reactions A + A —> 2A + 7r .................... 58
4.3.1 Charged pion production ..................... 59
4.3.2 Neutral pion production ..................... 61

4.4 Low pion momentum approximation .................. 61
4.5 Application of the model and results .................. 64
4.5.1 The reaction p + p ——> d + 7r+ ................... 64
4.5.2 The reaction 3He + 3He —> 6Li + 7r+ .............. 66
4.5.3 The reaction 12C + 12C ——> 24"Mg + 7r0 .............. 67
4.5.4 Calculations for heavy nuclei ................... 70

4.6 Conclusions ................................ 73

4.7 Appendix ................................. 74
4.7.1 Harmonic oscillator wave functions and overlaps ....... 74
4.7.2 Calculational details of the A + A —+ 2A + 7r+ reaction . . . . 76
4.7.3 Toward a complete analytical answer, the reaction A + A —>

2A —i— 7r0 .............................. 78

5 Exploring the nuclear pion dispersion relation through the anoma-
lous coupling 7 —> y’rro 81
5.1 Introduction ............................... 81
5.2 The pion dispersion relation ....................... 83
5.3 Pion production through the anomalous coupling ........... 86
5.4 Calculating background processes .................... 90
5.5 Conclusions ................................ 97
6 Multiple pion production from an oriented chiral condensate 99
6.1 Introduction ................................ 99
6.2 The linear sigma model .......................... 102
6.3 From classical to quantum solutions ................... 105
6.3.1 Parametric excitation of a harmonic oscillator ......... 105
6.3.2 Infinite number of mixed modes ................. 110
6.4 Application to chiral condensates .................... 122
6.4.1 Separable modes, space-independent effective masses ..... 122
6.4.2 Time- and space-dependent perturbations ........... 127
6.4.3 The spherical square well ..................... 134
6.5 Summary and conclusions ........................ 139
LIST OF REFERENCES 143

vi

LIST OF FIGURES

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

Observed fluctuation of the gap on the background of single particle
energy. ..................................
Moments of inertia for different nuclei are compared to the rigid body
value (dashed line). 1: even-even systems, 2: odd Z, 3: odd N, 4:
odd-odd systems. ............................
Gap size as a function of angular momentum, obtained with exact
numerical calculations. .........................
Paired states of a two particle system. Calculation with a 6-function-
like potential is on the left; and on the right the energy levels of two
neutrons in the 99/2 shell in 21OPb. Each level is labeled with the
corresponding value of the angular momentum. ............
Plot of two particle angular correlation function %Pl(cos(612)) ver-
sus angle (912. The dashed line represents the case of l = 10 and the
solid line corresponds to the l = 5 situation. It may be inferred from
this that the correlation angle is approximately equal to l/l (in radi—
ans). ...................................
Plot of the gap as a function of particle number for three different level
spectra. The pairing constant was chosen at G = 0.4M eV ......
Particle number fluctuations for three different level spectra. The pair
coupling constant is G = 0.4 (M 6V) .................
Energy of the even—N ground state in the model 3, the BCS result
according to Eq. (2.45), dashed curves, particle conserving solutions,
dotted curves, in comparison with the exact results, solid line; the
coupling strength is G = 5, upper part, and G = 1, lower part.

Deviation of the chemical potential from the exact solution as a func-
tion of the particle number for the particle conserving solution, solid

line, and the corrected BCS, dashed line; the coupling constant G = 1 .

Energy gap as a function of the particle number, upper panel; energy
of the paired system relative to energy of the system with no pairing,
G = 0, vs. N; lower panel. On both plots the solid line shows the
particle conserving solution, whereas the dashed curve shows the BCS
result according to Eq. (2.45). .....................

vii

10

12

14

20

21

30

31

33

2.11 Quasiparticle energies 6,, for 102‘13’28n isotopes; the results for the par-
ticle conserving algorithm, solid lines with circles marking the valence
neutron number, and the BCS result, dashed line with calculated points
denoted by crosses. ...........................

3.1 The density plot of matrix elements of a typical shell-model Hamilto-
nian matrix. This particular plot corresponds to the system with a
quadrupole-quadrupole residual interaction. The band-like structure
is clearly seen along the main diagonal which goes from the lower left
to the upper right corners of the plot. .................

3.2 Left panel: energies of the lowest states 1/21‘ and 3/21‘ in 518C as a
function of the matrix truncation n, shell model calculation (points)
and fits A exp(—7n) (solid lines); right panel: energy of the yrast state
2+ in 48Cr, real scale (t0p) and logarithmic scale (bottom). .....

3.3 Summary of the results obtained in [??], that clearly indicate the ex—
ponential nature of the strength function tails. ............

3.4 Energy deviations for the ground state of random matrices of dimension
2000 as a function of the progressive matrix truncation n (diamonds):
the GOE—like banded matrix of width b 2 0.293N, chosen in such
a way that half of the matrix elements vanish, approximately in the
same proportion as in typical shell model cases, left panel; the full GOE
matrix, right panel. Solid lines show a fit A exp(—yn) with A :- 1.18
and ’y = 3.8 X 10’3, left, and A = 1.96, y = 1.53 X 10’3, right. Note
the absence of the horizontal asymptotics in the case of the full matrix.

3.5 Convergence of the ground state energy for a banded random ma-
trix with the exponential cut-off of matrix elements Vkl ~ exp(—|k —
l|/b), b : 0.293N, shown by diamonds; the solid line shows the fit
const- exp(—2n/b)/n. ..........................

3.6 Convergence of the ground state energy in the tight-binding model
of a finite one-dimensional lattice with A as a hopping parameter, left
panels, and for a shifted harmonic oscillator with the Hamiltonian H :
ala+A(a.—i-al), right panels. The upper parts Show the energy deviation
AEn = E001) - E0(oo) as a function of the truncated dimension 71
(solid lines for A = 1 and A = 2); dotted lines show the predicted
analytical convergence of the models, A7r2/n2 (left) and A2n/n! (right).
The lower parts characterize the convergence rate An —-> A by plotting
An E AEnnZ/WZ, left, and An = (AEnnl)1/2", right. ..........

3.7 The exponential convergence rate 7 is shown as a function of the lim-
iting perturbation A in the case of a tri-diagonal matrix. The critical
point at A = 0.5 is indicated. ......................

3.8 Convergence region of five—diagonal matrices shown in the two-parameter
space of the off-diagonal matrix strengths A1 and A2. .........

4.1 The pionic fusion of two nuclei in the sudden approximation is illus-
trated. ..................................

43

44

46

50

4.2

4.3

4.4

4.5

4.6

4.7

5.1
5.2

5.3
5.4

One of the amplitudes of the total fusion process: an initial proton
from the nth orbit produces a 7r+ and ends at the lth final neutron
single-particle orbit. E, is the remaining overlap of a proton system
with the nth initial single-particle state missing. G; is the neutron
overlap with no lth state in the final system. .............
Reaction cross sections for 3He + 3He —-> 6Li + 7r+. The left panel
shows the transition to the ground state and the right panel to the
first excited state of 6Li at 2.18 MeV. The solid lines represent the total
cross section, dashed and dotted lines are p and s-waves, respectively.
Differential cross section of the reaction 3He + 3He —-> 6Li + 7r+. On
the left panel the solid line represents the transition to the ground state
of 6Li and the dashed line to the first excited state; the corresponding
absolute values of pion momentum are 96 and 90 MeV/c, respectively.
The right panel displays the experimentally observed values [2] of the
differential cross section of the transition to the ground state (squares)
and to the first excited state (circles) of 6Li. ..............
The reaction cross section for 12C + 12C —> 24Mg + no with oscillator
parameters '0 = 104 MeV/c and w = 119 MeV/c as a function of pion
momentum. ................................
The total cross section of 12C + 12C ——> 24Mg + 7r0 as a function of
the model parameters 1) and w. The calculation is done for a pion
momentum 41 MeV/c which corresponds to the total energy of about
6 MeV above threshold. .........................
The general behavior of the pionic fusion cross section A+A —> 2A+7r
versus the mass number of initial nucleus A. The plotted value 6 is
related to a total cross section as 0 2 6 (k/mfl)3. Calculations in the
low-momentum limit (filled diamonds) show that cross sections fall by
several orders of magnitude in this mass range, but remain in the pico-
barn region for nuclei as large as oxygen. The highest and the lowest
cross section found within the shell model configurations are repre-
sented by error bars. Experimental measurements are displayed (open
circles) and compared to calculations (filled circles) which were per-
formed for the finite pion momenta corresponding to the experiments.

Delta-hole contributions to the pion self-energy. ............
The pion dispersion relation is shown for both the vacuum case (solid
line) and with an effective coupling ngA=20 and g’=0.8 (dashed line)
in Figure 5.1. The thin straight line shows the space-like-to—time-like
boundary. .................................
Pion production diagram. ........................
Neutral pion production through the anomalous coupling 7 —-> 'y' + 7r”.
The on—shell energy is 225 MeV, the momentum transfer |k.,,| is 275
MeV/c, and the pion’s width is 50 MeV. Cross sections are shown for
three incoming photon energies: .500 MeV (solid line), 1.0 GeV (long
dashes) and 2.0 GeV (short dashes). ..................

ix

67

68

69

70

72
84

85
86

5.5

5.6

5.7

5.8

5.9

6.1

6.2

6.3

6.4

6.5

Neutral pion production through the anomalous coupling 7 —> 7’ + 7r0.
On-shell energies are 225 MeV (solid line), 250 MeV (long dashes) and
275 MeV (short dashes). The momentum transfer |k,,| is 275 MeV/c,
and the pion width is 50 MeV. The incoming photon has an energy of
500 MeV. .................................
Feynman diagrams for the background Compton-like processes 7 +
N —> 7’ + N’. ...............................
The contribution from ”y + N —> ”y’ + N’ assuming the outgoing nu-
cleon has a width of 25 MeV. This component is negligible for energy
transfers greater than 200 MeV. ....................
Matrix elements for calculating background processes where a pion,
photon and excited nucleon are in the final state. These diagrams
can be thought of as processes where a delta-hole (which subsequently
decays) is created. ............................
The cross section for the background process where a charged pion is
created is shown in the upper panel. By requiring the created pion to
be neutral, the background is greatly reduced as shown in the lower
panel. Polarization projections are shown for a,- = 0, of = 0 (solid
1ine),a,- = 0, or) = 7r/2 (short dashes), a, = 7r/2, ozf = 0 (long dashes),
a, = 7r/2, of = 7r/2 (dot-dashes). ...................

Illustration of the “quench” process. Panel (a) shows the normal 7r — 0
vacuum at low temperature. Panel (b) shows the restoration of chiral
symmetry when the system is heated in the nuclear collision, and panel
(c) demonstrates the “quench”, the sudden restoration of the vacuum
to its normal low temperature shape. In the process (a)-(b)-(c) the
expectation values of (7r) and (a) change which is illustratively shown
by the classical motion of a small ball. The curvature in the 7r direction
is associated with the pion mass and the evolution of this quantity is
schematically shown in panel ((1). ....................
Illustration showing the scattering from the quantum problem of scat-
tering. Mathematically this is the same problem as the problem of
parametric excitation of classical harmonic oscillator. .........
The particle number distribution for neutral pions (solid line), charged
pions (long dashes), and for all pions (short dashed line). All curves are
normalized to unity; note that the total number of pions and number
of neutral pions both are always even, while any integer number of

charged pions appearing in pairs is allowed. The parameter p = 0.999.

The probability ’P( f) that a given neutral pion fraction f is observed.
The three curves display cases of one condensate mode (solid line), two

energy-degenerate modes (dotted line), and three modes (dashed line).

Eckart (IIII : 1 (GeV)2, T = 0.5 fm/c) and rectangular (III! 2 1
(GeV)2, T = 1 fm/c) perturbations are shown on the right and left
panels, respectively. ...........................

89

91

93

94

96

104

107

117

119

6.6

6.7

6.8

6.9

6.10

The average number of particles produced in the mode k as a function
of k for square potential (solid line) and Eckart potential (dashed line).
The parameters of the perturbation are chosen so that H = 1 GeV2
for both models, and T is 0.5 fm/c and 1.0 fm/c for Eckart and square
potentials, respectively. .........................
The schematic representation of the perturbation H(:r), with one con-
densate bound state. ...........................
The behavior of the dimensionless variables I (‘1)0, I (1) / f2 and I (‘1)f2+
I (1) /Q is shown in dashed, dotted and solid lines, respectively, as a
function of the potential depth W, left panel, and potential size R,
right panel. A fixed size of 1 fm was used for the left panel and a fixed
depth x/I7 =1 GeV was used for the plot in the right panel. .....
The average number of particles produced as a function of the depth of
the potential field W is shown in the left panel, the size R was fixed
at 1 fm. The right panel shows the number of particles produced as a
function of size R given a fixed depth V : 1 GeVg. The time length
of the perturbation is set at T :1 fm/c. ................
The left side shows the average number of pions of a particular type
as a function of V , the depth of the perturbation. Curves displayed as
solid, dotted and dashed lines correspond to the values of the radius R
of 0.5, 1, and 2 fm, respectively. Plotted on the right hand side is the
number of pions versus the radius R for values of W of 0.5, 1 and 2
GeV as solid, dotted and dashed lines, respectively. ..........

xi

 

125

128

136

137

Chapter 1

Introduction

The complexity and chaos of the real world are not surprising to the modern day
physicist or mathematician. It’s long been known that most systems in nature are not
integrable, furthermore the mathematical description of complex reality often results
in an unstable solution. The order and coherence that we see in life is puzzling.
Paradoxes such as the infinite growth of entropy or time reversibility in physical laws
but not in nature, and concepts such as the anthropic principle are just a few of the
issues that arose from our agelong preoccupation with this puzzle. There is a hope
that a valuable insight to this may be gained from our experience and knowledge of
small systems. The existence of coherent many-body effects such as hydrodynamical
flow, sound waves, oscillations, rotations and phase transitions on the background of
a completely chaotic and nonintegrable dynamics is known and has been reasonably
well studied in the past century. Nuclear physics is one of those exciting sciences that
spans a large variety of topics. Nuclear science straddles all regimes in physics, low
and high energy, classical and quantum, micro and macroscopic, regular and chaotic.
It is an excellent arena to unify our understanding of these diverse limits, and to
understand the coexistence of things coherent and chaotic. This thesis is written as

a “book of stories”, or chapters, each of which can be read separately and presents

separate research, but at the same time all these chapters are strongly related by the
amazing fact that these “stories” describe the same nuclear systems.

For most intents and purposes at low energies nuclei can be described as protons
and neutrons that interact with some phenomenologically inspired forces. These
collections of quantum “balls”, mesoscopic in size, are perfect to explore the question
of how the macro world is related to the micro world? From the beginning of the
20th century there has been clear experimental evidence that nuclei are somewhat
similar to classical liquid drOps. The success of the Weizacker mass formula which
incorporates simple concepts of the volume energy, surface tension and Coulomb
energy of a charged drOp is a clear indication of this. However, this provides only a
rough picture. Real-life quantum many-body effects are readily apparent.

One of the most important forces that remains after all bulk interactions, such as
the mean field, are taken into account is the short-range pairing interaction. In fact
at low excitation energies, i.e., at low temperatures, the nucleons are all in a special
collective quantum state, so that the whole nucleus is a superfluid liquid drop, to
use the condensed matter physics terminology. The experimental evidence for this
along with the current theoretical understanding of nuclear pairing is discussed in
chapter 2. Pairing in mesoscopic systems is theoretically challenging. Exact solutions
are already too complicated to find and macroscopic methods are too approximate
because they rely on the assumption of an infinite system. The last section of chapter
2 describes a method for treating the general pairing problem in a finite Fermi-system
without violating particle number conservation.

Chaos is yet another interesting aspect of nuclear systems. The question of how
the regular and collective nuclear features appear and exist in dynamical chaos is still
open. Chaoticity in the nucleus is also conceptually important because there is hope

to trace the path of classical many—body chaos into the quantum world. Signatures

of quantum chaos such as found in the level spacing and level density distribution
are widely discussed and debated in the scientific community. In the work that is
presented in chapter 3, we study how to utilize chaotic features in order to help solve
the nuclear many—body problem. In this study we suggest that low-lying eigenvalues
of realistic quantum many-body hamiltonians, given, as in the nuclear shell model,
by large matrices, can be calculated by first diagonalizing small truncated matrices
then exponentially extrapolating results, instead of performing full diagonalization.
We consider a number of realistic and model examples where numerical data confirms
this conjecture. Based on these results we argue that exponential convergence in an
appropriate basis may be a generic feature of complicated (“chaotic”) systems where
the wave functions are localized in this basis.

Studies of nuclear systems are burdened by more fundamental considerations such
as the choice of appropriate degrees of freedom. At low energies experience shows
that protons and neutrons give a satisfactory set of variables, nevertheless it is not
always true. Extreme collectivity may be the key for gathering enough energy to
excite other modes such as pions. Recently observed rare heavy ion fusion processes,
where the entire available energy is carried away by a single pion, is an example of
such an unusual coherent behavior in nuclear reactions. Theoretical understanding
of this process is challenging, because the usual methods used for heavy ion reac-
tions must incorporate the precise features of the nuclear structure. In chapter 4 we
calculate these cross sections in the sudden overlap approximation, modeling the ini-
tial and final nuclei as moving harmonic oscillator potentials. This allows for a fully
quantum-mechanical treatment, exact conservation of linear and angular momenta
and fulfillment of the Pauli principle. Our results are in satisfactory agreement with
data. We also discuss general trends of the process and the mass number dependence.

The constitution of nuclear matter and the interplay between different degrees of

freedom is an intriguing question. For pions, the lightest species of mesons, depending
on their coupling to nuclear matter, theory would predict a variety of possible inter-
esting phenomena such as pion condensates or the existence of super-dense nuclei.
Pion behavior in nuclear matter, discussed in chapter 5, is an investigation of the
possibility of measuring pion dispersion through anomalous coupling in the reaction
’y —> y’rro. If the pionic mode is softened in nuclear matter due to mixing with the
delta-hole state, in—medium photon “decay” into pions becomes possible. This reac-
tion, if observed, would give a tremendous contribution to our knowledge about pions
in nuclei. Unfortunately, our calculations suggest that competing backgrounds are
larger than the pionic channel of interest. We were therefore only able to conclude
that the use of such a measurement to explore the pion dispersion relation might be
untenable, although difficult exclusive measurements might allow one to conquer the
background.

With higher energy, the number of different particle species grows considerably
and non-nucleonic contributions become important, if not dominant. Pions in heavy
ion reactions become not only the essential part of the overall picture, but also provide
important detection probes for looking at different states of matter. The formation
of coherent pion states, if not possible in normal matter, is still predicted at high
energies and densities or in quark—gluon-plasma. The final chapter considers the “ori-
ented” chiral condensate. It may be produced in high energy nuclear collisions. Unlike
disoriented chiral condensates, previously considered in the literature i.e. a coherent
state of pions with a disoriented in isospin source, we consider a squeezed state of pi-
ons produced in the frame of an effective field theory with time— and space-dependent
pion masses. We discuss the general properties of the solution, identifying condensate
modes and determining the resulting pion distributions. The implementation of the

dynamics in the form of a sudden perturbation allows us to look for exact solutions.

In the region of condensation, a dramatic increase in pion production and charge fluc-
tuations is demonstrated, which can serve as experimental evidence for the formation
of a chiral condensate.

The main material of the dissertation was published and presented at various
meetings: chapter 3 [1, 2], chapter 4 [3, 4, 5], chapter 5 [6, 7, 8], and chapter 6

[9, 10, 11].

Chapter 2

Particle number conserving

description of pairing

2.1 Experimental evidence of paired collective

states in nuclei

Pairing correlations play an important role in nuclear structure, determining an es-
sential contribution to binding energy, odd-even effects, single—particle occupancies,
quasiparticle excitation spectrum, radiation and beta-decay probabilities, transfer re—
action amplitudes, and low-lying collective modes and moments of inertia [12, 13, 14].
The revival of interest in pairing correlations is related to studies of nuclei far from sta-
bility and predictions of exotic pairing modes [15]. Metal clusters, organic molecules
and Fullerenes give another example of finite Fermi systems with possibilities of pair—
ing correlations of superconducting type [16].

The experimental evidence of a paired superconducting state in nuclei is very rich.
Direct observation of the gap and its variations with the size of the nucleus could be

Obtained simply by plotting the average energy shift between odd and even nuclei,

 

N=8

45
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126

   

 
    
  

 

   

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N

Figure 2.1: Observed fluctuation of the gap on the background of single particle

energy.

which in the degenerate model approximation is directly related to the energy gap
(see Eq.(2.11)). In Figure 2.1 obtained from [17] the energy difference of odd to even
system is plotted, which is just a single particle excitation energy plus the gap. The
behavior of the gap is easy to observe on the background of extra single particle
energy.

Another interesting demonstration of nuclear superconductivity is in the rotational
bands of nuclei. This example is important because of its analogy to the condensed
matter systems where rotation causes the destruction of a superfluid state. Further,
even some analogy could be drawn here to a magnetic field, or current flow as de—
structors of superconductivity. Every rotating system with angular momentum J
possesses a term in its Hamiltonian h2J2/2I, where I is a moment of inertia defined
by a particular configuration of contributing particles. If I is assumed to be constant
the energy spectrum of rotation forms a band th (J + 1)2 / 21 with J (J + 1) being the

eigenvalue of the operator J 2. Observation of this rotational band allows experimental

determination of the moment of inertia. Theoretically, the moment of inertia could
be obtained in the adiabatic approximation of a very slow rotation, cranking model,
Ref.[18, 19, 20, 21]. The major idea of this model, in the context of superconducting
states, is that particles form pairs not exactly of time conjugate orbits, i.e. spin pro-
jections of m and —m for the same j, but with some extra coherent mixture of pairing
between states with m and —m i 1 which creates a total rotational momentum along
the axis perpendicular to the quantization axis. The amount of this mixing plays the
perturbative role in this theory. The observed situation is again similar to the one in
superfluids, where under the conditions of flow the pairing starts to shift from p and
—p paired momenta to p + k / 2 and —p + k / 2. The value of the moment of inertia in

the cranking model is expressed by the following formula

 

, _ 2
[m : 7‘2 (“AUX UA'UA) .3: , 2 2.1
7. AZ; EA‘l'EA’ [(J )A,Al ( i

where the (jx),\,N is a momentum matrix element and the numerator forms a coherence
factor, that could be substituted by occupation numbers using for example BCS

results, see Eq.(2.14).
(“AUX — UA’UA)2 = ”A0 — fix) + "NU — 71A)

With this formula one could show that for a perfect Fermi gas in a deformed container,
the moment of inertia is close to that of a rigid body. Introduction of superconductiv-
ity causes the system to behave more fluid-like. The reason for this is that the energy
denominator in (2.1) increases because of the influence of the gap; and because the
coherence factor in the numerator is now a smaller number due to a smeared occu-
pation in contrast to the steep jump in a Fermi gas. The decrease of strength due to
the coherence factor could be observed in all single-particle nuclear transitions. This
fluid-like behavior is characterized by a significant reduction of the moment of iner-

tia. The systematics of the moment of inertia for different nuclei is presented in Fig

8

 

I ' l E l I I “I

i

we .. A be“ e

ExPerimental ._

’- f
MU—-- ,r’” I 2
‘ 3 points
x 4

f20 - '"

:00

3, MeV'1

80

c

60 --

 

40...

20 ~— -'

 

 

 

:’ a i E I
I50 1‘55 f60 {65 I 70 I75 160 I85 I90

Figure 2.2: Moments of inertia for different nuclei are compared to the rigid body

value (dashed line). 1: even-even systems, 2: odd Z, 3: odd N, 4: odd-odd systems.

 

 

 

 

 

2" iiiridiiri llil
15*- —~
3‘— 10—* ‘4
E:
5.... __..
K
0 ilLllilliillixl
01234567891011121314

00+ 1 i)”

Figure 2.3: Gap size as a function of angular momentum, obtained with exact nu-

merical calculations.

2.2 obtained from [17]. The rotational angular momentum is created by a coherent
shift of all pairs. This shifting from the time conjugated pairing in turn leads to a
reduction of the gap, which in nicely demonstrated [22] in Fig.2.3. This situation
resembles a phase transition when it becomes energetically favorable to break pairs

and re-align them for a higher total angular momentum.

2.2 Pairing forces

While the exact mechanisms of pairing remain ambiguous and widely discussed, their

responsibility in creating microscopic effects is undeniable, as has been shown in the

10

previous section. Historically, pairing forces were first considered in nuclear physics,
although their effects were not well understood until the deveIOpment of BCS theory
[24]. The mean field approximation used for the nuclear physics many-body problem
leads to a shell model, a model where single particle energy levels could be grouped
into shells that have a relatively high energy gap in between. It is not unreasonable
to assume that some residual pairing interactions between nucleons still exist on the
background of the mean field. Experiments show that the typical pairing force is a
short range one, and that it is characterized by energies much smaller then a gap
between shells. A good model of this force could be obtained by considering a 6-

function as a pairing potential [25]

v = max. — x.) = —&6<r1 — unease) — coswgiiael — a).

T1T2
The two—particle state can be written as the eigenstate of their total angular momen-

tum

J,M . .
lJvM) : Z Cj1,m1,j2,m2l.717ml>l32im2>v (22)

7”1,7712

where
l9} 7”) = leX‘i/zlmRnW")

is a single-particle wave function in the shell model. With some analytical work one
could then compute the schematic level diagram of < J, M |V|J, M >, i.e. the energy
of different paired states of a two particle system. This is shown in Figure 2.4 on
the left. For comparison, the right side of the same picture displays experimentally
determined energy levels of 210P!) which has one pair of neutrons in the outer shell.
Two identical particles in the same j-shell could couple only to an even total angular
momentum.

This similarity between calculated and measured level structures, along with the
fact that this picture is observed in basically all systems with one Cooper pair out-

side the main core, confirms that the residual nucleon-nucleon interaction is a short

11

210

 

 

 

 

 

 

 

 

 

 

 

Calculation with S-function Levels of Pb
as a potential of pairing
l
19 — - 1.4 - J
________ l
6 8 [
4 . 6 [
4 l
1.4 1 2 fl ’
..>_ >- o.9 ~ J
o r a)
£2 E 2
"E .E
3
a 0.9 ~ _ 3 i
S 5
:0: C
9 Lu
<
0.4 ~ ~
0.4 — -
0 o
-01 -01

 

 

 

 

 

 

Figure 2.4: Paired states of a two particle system. Calculation with a (5-function—like
potential is on the left; and on the right the energy levels of two neutrons in the
gg/g shell in 210P1). Each level is labeled with the corresponding value of the angular

momentum.

12

range one, for which a 6—function approximation was not a bad choice. Secondly, the
effects of pairing could be only considered on unfilled shells in nuclei, just like in a
Cooper model one electron pair is studied at the Fermi level. From the above con-
sidered case with a 6—function potential it is clear that most of the pairing strength
would occur in those pairs that have the greatest single particle overlap, i.e., between
time-conjugated states. This feature results in a preference of coupling to a total
zero angular momentum, since I j, m) and [j, —m) are time conjugate, and leads to a
considerably lower energy of the 0+ ground state compared to the rest of the states.
The analogy for an electron gas where coupling is strongest for the states with p and
—p could be emphasized at this point.

To complete this discussion about pairing forces in nuclei it would be interesting
to mention the correlation length of time-conjugate pairs which, in contrast to usual

superconductors, is normally much larger than the nuclear radius R.

hpf '
% __6’I‘_’:7_1_2- % RAl/B
{ mA

Another important parameter, correlation angle, is often introduced. Analytically it
_1 l—m.

could be obtained with the use of GR £1er :2 Sim/“1% and the formula for addition of

Legendre polynomials applied to Eq. 2.2 which leads to:

\/2l+1

|0,0 >= 12(i«)(--1)l 4W

B(cos 612)

as a relative wave function. Its value is plotted in Fig. 2.5. Consequently, the
above results lead to a correlation angle of the order l/l, which favors a short range

interaction.

13

1.0 . u . i . T

 

 

---- l=10

 

 

 

(2l+1)1/2P,(cos(0))/41t

 

 

 

 

0.0 0.2 0.4 0.6 0.8
angle 0,, (rad)

Figure 2.5: Plot of two particle angular correlation function %B(COS(012)) versus
angle 612. The dashed line represents the case of l = 10 and the solid line corresponds
to the l = 5 situation. It may be inferred from this that the correlation angle is

approximately equal to 1/l (in radians).

14

2.3 Solutions to the pairing problem

2.3.1 Seniority scheme and degenerate model

The dramatic difference of a nuclear case as compared to a usual superconductor is
that the number of states for the particles near the Fermi energy is quite limited.
The shells in nuclei normally consist of several dozen states, thus despite the relative
simplicity of the problem the finite size effects may not be ignored. The discussion
presented here follows the works [23] and [26]. Let us denote the single-particle phase
space volume as O, which for a given j is just 2 j + 1. In this space we use an arbitrary
basis, denoting by A and A the two time-conjugate states. With this terminology a

two particle paired state could obtained by a creation operator:

P+_1Q + +

where one should not forget an important property of time conjugated states: a; =

—a:’, and that summation goes over all space I). Normalization of these states could

be done as follows:

1 Q 1 9 Q
<0lPP+l0> = Z Z<0|GWAGIICL$|0> = ; Z<6AA’5J\XI — 6076i») = 5 (2-3)
AA’ AA’

Conventionally the notation [N ; s) is used for the systems with pairing, where N is
the total number of particles and s is the seniority, a quantity that reflects the amount

of unpaired particles. In this form one could write:

[20 = \fglfilm

Operators P+ and P are to be considered as creation and annihilation of a spin-zero

quasiboson. The commutation relation for these bosons is easy to obtain in the form:

mpflzg— an

15

where N is a number operator. It is seen from here that the right side of Eq.(2.4)
almost remains constant, in which case it is just a bosonic commutation relation, with
some small N -dependent correction. This correction reflects the fact that quasibosons
are almost but not quite bosons, and there exists a residual effect of Pauli blocking
due to their fermionic components. In general, normalization of states like IN, 0) may

be done using Eq.(2.4). As an example N = 2 gives:

<0|P2(P+)2|0> = <0|P(lP»P+l + P+P>P+io> = 2%? — 1)

and for arbitrary N the normalization is:

N!(Q/2)!
((9/2) — N)!

 

<OlPN<P+>Nl0> = (2.5)

Finally, since the physical interpretation of a seniority is in existence of several un-
paired nucleons, the non zero value of 3 just reduces the effective phase-space by
blocking the creation of possible pairs. For example, the state with s = 1 would be

normalized as:
N!(§2/2 - 1)!
((9/2)-1- N)!

 

(0|aAPN(P+)Naf{|0) = (2.6)

which follows directly from Eq.(2.5) by substitution O —> O — 2 because one pair
state is blocked. Caution should be applied here in that two unpaired particles in
time conjugated states reduce the available space only by 2 and not by 4. The
described above seniority scheme was first introduced by G. Racah and provides a
convenient basis for dealing with pairing correlations.

The usual next step of discussion in this direction would be an introduction of BCS
and solution of its equations for the approximate ground state. However, the litera-
ture often omits that there exists another exactly solvable model of superconductivity
which is intriguing enough to be considered at this step. The degenerate model was
apparently first studied in 1949 but was not completely understood before the arrival

of other microscopic theories of superconductivity. The pairing energy is normally

16

either smaller or of the same order of magnitude than the separation between single
particle levels on the last unfilled shell. In some cases, especially for spherical nuclei,
there are large degeneracies and in certain occasions the approximation of the shell
as one highly degenerate single—particle level could be adopted. In the case of the
harmonic oscillator model with no spin-orbital coupling, the shells are in fact com-
pletely degenerate and therefore this model is precise. Thus the degenerate model
makes a reasonable assumption that at some distance from the filled core there is one
degenerate single particle level at the energy 6, filled with N fermions. Secondly, we
assume that only particles in time-conjugate orbits interact, and that the energy of
this interaction is substituted by some mean value: —G for all pairs. Experimental
data give the value of G at about 23/A MeV. The Hamiltonian for the degenerate

model is :

H = eN—GP+P

The diagonalization of this Hamiltonian could be nicely performed by introducing a

quasispin. One may notice that Operators P+, P and N form a ladder structure:
[P, N] : 2P, [P+,N] = -2P+
so it is possible to define the quasispin L as:
+ 1
L- : P, L+ : P , L2 : 5(N — —) (2.7)

This new vector operator obeys S U (2) symmetry and therefore has all the properties
of the angular momentum operator. The integrals of motion are as usual: LZ which
obviously is a constant that depends only on the number of particles and the volume

of space Eq.(2.7), and the total square of quasispin:
L2 :2 L: — L, + P+P (2.8)

The eigenvalues of L2 are L(L + 1) where L is an integer or half-integer. For a

17

particular nucleus the value of L2 is fixed but L could vary in the limits of :
(2.9)

In this picture the state of the system could be determined by the eigenvalues of L2
and L2 and with the use of (2.8) the eigenstate energy of the Hamiltonian is:

E(L,N) :eN+-:3(N— 52’— —2) —GL(L+1) (2.10)

It would be useful to replace the eigenvalue L with a more physical measure, namely
seniority. From Eq.(2.9) it follows that L takes N values up to the maximum 9/4
that have increments of 1 / 2, while at the same time seniority ranges from 0 to N and
has steps of 1. At the maximum value L, the ground state of Eq.(2.10), the seniority
is zero and one can therefore conclude that:

8:2(g—L)

and the energy spectrum becomes:
G
E(N,s) =€N+Z(N—s)(N+s—Q—2)

Examination of this solution shows that the energy gap, i.e. the energy to break one
pair which corresponds to s = 2, is

9
2A: —
02

and in general the excitation energy of the state with seniority s is:

E(N, s) — E(N,0) = $(n + 2 — 8)

If the nucleus has an odd number of nucleons, then the lowest possible seniority is 1
which just reflects the fact that one particle is always unpaired. The energy required
for the even system to add one particle is

E(N+1,1)—E(N,O)=e+%]—v—

18

This energy could be compared with the energy of two consecutive even systems:

E(N+2,0)—E(N,0):2e+GN+-C329

The resulting conclusion is that on average the ground state of the odd system is

shifted by one half of the gap above the even system.

E(N, 0) + E(N + 2,0)
2

 

E(N + 1) = + A (2.11)

2.3.2 BCS theory in the case of nuclei

The general approach of BCS theory as applied to the nuclei is quite standard. The

ground state of the system is generated from the Hamiltonian

1
H’zH—nNzZe’AaIaA‘l’Z ZGAA’PIpX, €’=€—,u (2.12)
A AA’

by minimizing the energy through the probing wave function:

[0) : HAUL), — vypjflvacuum). (2.13)

After this is done one arrives at the following solutions:

2 1 6’, 2 1 c’A

 

where e) = \/(6A — ,u)2 + Ari; AA and ,u. are the energy gap and chemical potential
respectively. Two additional equations are important here:

A),

1
46A

AAZZGAA’ NZZTL)‘. (2.15)
A’ A

The first is a usual gap equation, and the second one preserves the average number of
particles in this grand canonical ensemble. As was mentioned before, the number of
levels in a nuclear shell is quite small compared to the almost continuum spectrum in
solid state physics. For this reason it is not always good to substitute the summation

in the gap equation (2.15) by integration over some band around the Fermi surface.

19

 

40 i . 1 r ' v r r ' ' v i

 

 

3.0 P // \\ 7
/ \
/ \
/ \
A / \
% // ’’’’’ ’ _ \ \\
E / / / \ T ‘\~\ \
a. 20 - / / ~ ~
go // \ \
8 ///' \\
* / \\
1.0 ~ ~
——— one level
—-— two levels
— Sn shell model

0.0 i i l l I i 1 L l
0.0 8.0 16.0

particle number

 

24.0 32.0

Figure 2.6: Plot of the gap as a function of particle number for three different level

spectra. The pairing constant was chosen at G = 0.4M eV

A good practical example is the case of nuclei exclusively possessing a filling neutron
shell from N :- 50 to N = 82; assuming that the core of 50 neutrons is completely
inert. Certainly the gap is zero for any magic nucleus, but for the systems with
N between 50 and 82 the gap depends on the single-particle level spectrum. In
Figure 2.6 the numerical calculations for the BCS gap are shown as a function of
a particle number in the external shell for three different level spectra: of just one
level (equivalent to the degenerate model), two equally-degenerate levels, and for the
actual distribution of levels within the shell model of tin isotopes.

The BCS is accurate for large systems and becomes exact in the asymptotic limit
[27]. The major drawback of the BCS is the violation of particle number conserva-
tion, which gives rise to deviations from the exact solutions for small systems. Since
this treatment uses a grand canonical ensemble one can study the fluctuation of the

particle number arising from particle non-conservation brought by the wave function

20

3.0 . l . r a l

 

 

 

 

/// \\\
/ \
/ \
/ \
/ _. -

(I) / T‘ " \. \
5 20 ~ ///' \‘\. ./’/ \.\ -
CE ' / ’ V \\
c ‘\
'5 /
(0
:3 /
*6
2
u.
.92
O
"E 1.0 -
(U
o.

——— one level

—-— two levels

—— Sn shell model

0.0 1 l . l . l I
0.0 8.0 16.0 24.0 32.0

particle number

Figure 2.7: Particle number fluctuations for three different level spectra. The pair

coupling constant is G = 0.4 (Ii/16V)

(2.13), given by the formula

 

AN2 2 ZTLA(1— 71.))
A

This result is plotted in the Fig. 2.7.

Various ideas were suggested to correct this deficiency, such as the direct particle-
projection technique [28], number-projection mean-field methods [29, 30], statistical
description [31], and taking into account the residual parts of the Hamiltonian in the
random phase approximation [32]. These methods have found only a limited number
of practical applications. For some approaches the obtained results did not manifest
the desired accuracy, whereas for other methods the complications turn out to be

almost on the same scale as for the exact solution by diagonalization.

21

2.3.3 Exact solutions

The Richardson method, described in the series of papers [33, 34, 35, 36], provides
a formally exact way of solving the pairing Hamiltonian with a constant effective
pairing force. This method reduces the large-scale diagonalization of a many-body
Hamiltonian in truncated Hilbert space to a set of coupled equations of a dimension
equal to the number of valence particles.

Recently, exact solutions have been approached by introducing sephisticated math—
ematical tools such as infinite-dimensional algebras [37]. Nevertheless the numerical
complications currently limit the scope of applicability and the need for a good ap-
proximate theory still persists, especially because it can provide us with a convenient

basis for calculating the effects of other parts of the residual interaction.

2.4 Particle conserving methods

The goal of our research in this area was to investigate a particle—conserving varia—
tional approximate solution that is formulated in the form of a recurrence relation in
the number of particles N. For each step it is required to solve equations for only
two variables, energy gap and chemical potential, as a function of the exact particle
number, and thus even for large N the numerical procedure is quite fast. By making
additional approximations this solution can be reduced to the BCS. The idea of the
method goes back to older papers [38, 39] and to the work of ST. Belyaev and V .G.
Zelevinsky [40] where the set of exact operator equations of motion was formulated by
introducing the gauge angle conjugate to the particle number as a collective variable.
In discrete space the corresponding equations are of a recurrent type. This method
was applied to the so—called pairing rotations (a systematic change of pair separation

energy). Below we construct an algorithm for the solution of recurrence relations

22

derived from operator equations of motion with exact particle number conservation
at each stage. In the well known degenerate model the solution coincides with the
exact one. In a model with equidistant single-particle levels, our particle—conserving
solution is compared to the exact solution and the BCS solution. A remarkable im-
provement over the BCS is observed. Finally, tin isotopes are considered as a realistic
example.

Again, we consider the pairing Hamiltonian of the general form

H = Zeualau — 211-: CW, 1)],er (2.16)

acting in the truncated fermion space (“shell”) of $2 = 2,, 1 orbitals IV). For sim-
plicity we assume the time reversal invariant single-particle spectrum 6,, = 6,; and
real pair scattering matrix elements GW 2 GM; here and below the tilde denotes
time conjugation of the single-particle orbitals IV) defined with the phase condition
[13) = —|V) (the fermionic nature of particles is responsible for the negative sign).

The pair annihilation and creation operators are introduced as

Pu : PD : (”Ll/(1’17) PI. : p11; 2 (1130.: ' (2'17)

The amplitudes GWI are invariant under the tilde operation; the factor 1/4 in Eq.
(2.16) is due to the fact that in complete summation both subscripts V and V’ run
over the entire single—particle space.

A sequence of ground states of systems with even particle number N and corre-
sponding energies E(N) will be denoted by IN). It is assumed that the spectra of
adjacent odd nuclei start with energies E (N i 1; V) of the states [N j: 1; V) of senior-
ity 1 containing one unpaired nucleon with quantum numbers V. The single-particle

transitions between the even and odd systems are described by the amplitudes

MN) = (N —1;I7lau|N>, 11,,(N) = (N +1;V|aI,IN), (2-18)

23

where again N is even. We also define the energy gaps AV(N) : AD(N) as a function

of particle number through the pair-transfer matrix elements between the even ground

states,
13.00 = g <N — 2| Z, a...) Pu'lNi- (2.19)
The exact operator equations of motion for the single-particle operators (1,, and

a: are
MWM=6a+§ZGWeW, (mm

[a[, H] = —6,,a[, — é: Guy/193a,; = —(6,, — G,,,,) a], — g: G,,,,/ a919,], . (2.21)
Instead of introducing the condensate of the pairs with an uncertain particle number,
we make a physically similar splitting of nonlinear terms in the equations of motion
that does not violate the conservation law and keeps the N-dependence intact. Our
variational approximation can be formulated as a truncation of the full Hilbert space
to that spanned by the seniority 0 even ground states and seniority 1 states in odd-N
nuclei. Thus, the step beyond BCS is only in that particle number is treated exactly.
The matrix elements in the equations of motion are disentangled by neglecting the

seniority 2 admixtures in even nuclei. For example,

 

<N — 1.14;; G...» alpw N) z
m>uweW—26L2§zawnmn=Amnww>a, (mm

In this approximation the equations of motion (2.20) and (2.21) give
[E(N) — E(N — 1; V) — 6,,] v,,(N) — A,,(N) u,;(N — 2) = 0, (2.23)
[E(N — 2) — E(N — 1; V) + 6,, — G,,,,] u,,(N —- 2) — A:(N)v,;(N) = 0. (2.24)
It is convenient to introduce, instead of E (N —— 2) and E (N — 1; V), the chemical

potential MN) and quasiparticle energies 6,,(N) according to

 

E(N — 2) = E(N) — 2,1(N), E(N — 1; V) = E(N) — MN) + 6,,(N) — 02 , (2.25)

24

Then we obtain the following simple set of equations:
[6,,(N) + €,,(N)]v,,(N) + AV(N)u,;(N — 2) 2 0, (2.26)

AIXNNJDUV) + l€V(N) _ €V(N)]’U.,,(1V — 2) Z 0, (2'27)

where N is even, and we use the modified single-particle energies

 

— MN). (2.28)

With time reversal invariance, we have 21,; 2 11,, and 2),; 2 21,, . The nontrivial solution
of the set (2.26, 2.27) determines the BCS-like spectrum of quasiparticle excitations

for each particle number N ,

 

6,, = (fgguv) + IA,,(N)|2. (2.29)

On the same level of our variational approximation

<Nl Zaia.lN> 2 N = Z Ivu<Nil2 . (2.30)

and the fermionic anticommutation relations [a,,, a,[]+ 2 1 result in
(2),,(N)|2 + [11,,(N)|2 = 1. (2.31)

Finally Eqs. (2.27, 2.31) result in the recursion relation connecting adjacent even

nuclei,
Viz/(NW
[6,,(N) ‘- EV(N)l

Taking into account Eq. (2.30) and summing over all Q single-particle states, we

 

|v,,(N — 2)]’2 :1— 2 [21,,(N)I2 . (2.32)

obtain

_ mun):
Q " N + 2 ‘ P adv) — awn

The gap defined in Eq. (2.19) is subject to the self-consistency condition

 

211(sz . (2.33)

1 , 1
AV(N) : _§ 2: GVV’UV’(N)u1/’(N - 2) Z 5 Z GVV’

 

ANN) IUV'(N)I2 (2.34)

V(N)-€V(N) '

25

The pairing problem formulated in this manner allows a recursive solution in both
directions, starting from an empty shell or from a completely filled shell. For example,
going down from the completely filled shell with the occupation numbers Iv,,(§2) |2 2 1
by solving Eqs. (2.33) and (2.15), the values of 11(9) and Aum) can be found which
in turn determine the next set of [12,,(Q — 2)]2 via Eq. (2.32). Similar equations in
terms of the amplitudes u,,(N) are suitable for the recursive solution climbing up from
an empty shell. The particle-hole symmetry with respect to the interchange of 'u, and
”U with the additional change of sign of the chemical potential and the appropriate
level reordering states the equivalence of both directions in the recursive solution and
provides a good check. Below we give a number of examples.

1. The BCS limit means that the N -dependence is ignored in Eq. (2.32) assuming
[2

that I’m/(N)? = [MN — 2)]2 = MN)

 

and similarly for u,,. This allows us to solve

the equations in a standard way,

- 2 1 6,,(N)
1),,(N)[ 22(1—6V(N))’

 

 

 

 

u,,(N)[2 = 3 (1+ iii/7i) . (2.35)

There is some ambiguity in the choice of the argument N in equations (2.35). Usually
N 2 N is assumed, which makes a better agreement for the less than half-filled shell,
but with the assumptions made there it should be no difference if N 2 N — 2. The
presence of a difference is related to the particle-number nonconservation in BCS
theory. Generally, in the region of good applicability of BCS, N >> 1, the exact
choice of the argument N has no effect. It turns out that BCS works best with the
interpolating choice of N 2 N — 1. This will be confirmed below for particular cases.
Eqs. (2.15) and (2.35) lead to a conventional form of the BCS gap equation, that

agrees with Eq. 2.15 from the previous section
V,

1 A/
AV : _ UV]; - 2.
2 Z a 26 ( 36)

2. The degenerate model [41] assumes that all single—particle levels have the

same energy 6, and the coupling strength G,,,,r 2 G is independent of V, V’. As it was

26

discussed before this model is exactly solvable in terms of the pseudospin SU(2) group
and thus provides a good testing ground. We assume that e 2 G / 2 and therefore
5,,(N) 2 -—n(N) according to (2.28). This makes the summation in Eqs. (2.33) and

(2.15) trivial:

 

 

 

A N 2
Q—N+22 l ( )l 2 , (2.37)
(6W) - E(N»
G N
1 2 — . 2.38
2 e<N> — :(N) ( )
These equations can be easily solved to give
G
A(N) = §\/N(Q — N + 2), (2.39)
G Q
N 2 — — — — .
M) 2(N 2 1), (240)
and the occupancies are independent of V,
2 _ N 2 _ N
|v(N)l Q . lu(N)l -— 1 Q . (2.41)

These results coincide with the exact solution.
In the BCS solution for the degenerate model, the occupation probabilities co-

incide with the exact values (2.41), whereas the gap and the chemical potential are

given by
A(N) = % Mo — N), (2.42)
MN) = g:— (N — g) (2.43)

respectively. As discussed before, N is defined approximately within the values of N
and N — 2. The considerable improvement to BCS with a small number of particles is
reached if N is chosen as an average value, N 2 N — 1 . With this choice, the chemical
potential becomes exact, and the square of the gap is shifted by a constant from its
exact value.

The ground state energy of the system is yet another sensitive test for the validity

of any approximation. In the recursive algorithm following the definition in Eq.

27

(2.25), energy of even-N ground states is given as a discrete integral of the chemical

potential,

E(N) = 214k). (244)

k:2,4,...N

This result has to agree with the standard evaluation of the ground state expectation

value of the Hamiltonian with the intermediate states of seniority 0 and 1,

E(N) = Z |v,,(N)|2 — E(gl. (2.45)

For the degenerate model the particle conserving solution is exact and therefore it is
not surprising that Eqs. (2.44) and (2.45) agree exactly providing the exact value of

energy

E(N) = —% N(o — N). (2.46)

In general for all models we have tested, very good agreement was observed between
both methods for calculating E (N) within the particle conserving recursive algorithm.
Still, the first way that utilizes Eq. (2.44) is preferable as it guarantees exact values
at the boundaries (empty and completely filled shell).

The BCS solution does not ensure agreement between the different methods of
calculating E (N) and the exact value. In the degenerate model, the evaluation of
energy according Eq. (2.44) gives

E(N) = —%N(Q—N)+—f:—N(1+N—N), (2.47)

while the second method, Eq. (2.45), leads to

E(N)2—%N(Q—N)+GTN+%(N—N)(Q—N—N). (2.48)

Finally, these are to be compared to the exact solution in Eq. (2.46). The chemical
potential method gives an exact result for BCS corrected by the choice of N 2 N — 1 ,
but the results of the second method always differ considerably from the exact answer.

The fact that the ground state energy found in BCS could be smaller than the actual

28

value points to the non—variational nature of BCS (or rather super—variational since
the classes of states with different N are mixed in the trial function) that can result in
uncontrollable errors. The proposed recursive solution is variational, and the resulting
ground state energy is always greater than the exact value.

3. An equidistant model. Here we consider a model of six double-fold Kramers-
degenerate levels (the total space capacity is Q 2 12) with equidistant energies of
e 2 0,1,. . .5. The coupling amplitudes GWI are chosen to be independent of 1/ and
u’. The choice of this model was made almost at random, and the results are typical
for a sufficiently small system, where exact diagonalization is possible, and the error
due to particle non-conservation in BCS might be significant.

For comparison, we first consider ground state energies obtained in the particle-
conserving solution, BCS, and by exact numerical diagonalization. In both approx-
imate solutions the ground state energy may be found by two different means, Eqs.
(2.44) or (2.45), that may not necessarily agree. This would signal poor quality of the
approximation. For the exact numerical solution, the ground state energy is given by
the lowest eigenvalue of the Hamiltonian matrix. In the largest case of the half-filled
shell, N 2 6, the matrix has a dimension 924. The numerical results are given in
Fig. 2.8 for strong pairing, G 2 5, upper panel, and for G 2 1, lower panel. In both
cases, the results exhibit a similar trend. The four curves that correspond to exact
solution, two ways of calculating energy with the particle conserving algorithm and
the BCS result, corrected by the choice N 2 N — 1 with Eq. (2.44), agree to the
degree they can hardly be distinguished. A separate dashed line that represents a
BCS calculation through the gap equation (2.45) is considerably shifted upwards. For
the recursive algorithm the largest errors occur at the edges of the occupation. This
is also clear from a direct analysis of Eqs. (2.23) and (2.24).

The quality of approximations can be also seen from Fig. 2.9, where the devia-

29

40 I l I I T 1 I l

 

E(N), G=5

 

 

 

 

exact ’.
20 f """""" particle conserving , ~

1

 

 

 

E(N), G

 

 

 

 

2 4 6 8 10 12

Figure 2.8: Energy of the even-N ground state in the model 3, the BCS result ac—
cording to Eq. (2.45), dashed Curves, particle conserving solutions, dotted curves, in

comparison with the exact results, solid line; the coupling strength is G 2 5, upper

part, and G 2 1, lower part.

30

 

0.2 . 1 ' 1 . v ‘ 1

particle conserving
— -— - BCS

 

 

 

 

 

0.15 ~ ~

 

 

 

 

Figure 2.9: Deviation of the chemical potential from the exact solution as a function of
the particle number for the particle conserving solution, solid line, and the corrected

BCS, dashed line; the coupling constant C 2 1.

tion of the chemical potential is shown for the particle conserving solution and the
corrected BCS.

4. A realistic example. The chain of tin isotopes, 100‘132Sn, provides a good place
to apply the pairing problem. The outer major neutron shell in these nuclei contains
five single-particle j-levels of total capacity 9 2 32. The energies 6,, of these levels,
see [42], slightly depend on the number of particles due to the presence of other parts
of the residual interaction. For the pairing problem, we assume them to be constant
and equal to €(d5/2) 2 —10.371 MeV, 6(g7/2) 2 —9.925 MeV, 6(81/2) 2 —8.773
MeV, 6(d3/2) 2 —8.122 MeV, and E(hu/Q) 2 —7.612 MeV. The pairing strength was
taken as a constant G' 2 0.2 MeV. In Fig. 2.10 the upper and lower panels display,
respectively, the dependence of the gap and ground state energy (relative to the case

with no pairing) as a function of the particle number in even—N systems. The BCS

31

with N 2 N + 1 is in a good agreement with the particle conserving solution for
the ground state energy (“h-method” of Eq. (2.44)). A similarly good agreement is
observed for the occupancies of single—particle orbitals calculated for the 116Sn isotope.
However, the BCS and our results differ considerably for the calculation of the gap,
upper part of Fig. 2.10, and for the values of 6,,(N), Fig. 2.11, that represent the
spectrum of broken pairs.

As a conclusion, in this work we discuss and test by a number of models the
variational method of solving the pairing problem with the exact conservation of par-
ticle number. Being a generalization of the conventional BCS approach, the method
reduces to the recursive solution of equations for the energy gap and the chemical po-
tential as functions of the total particle number. The results are in a good agreement
with the exact solutions in all considered cases. The BCS value of the ground state
energy in even-N systems can be improved by the interpolating the particle number
and using for calculation the integration of the chemical potential instead of the ex-
pectation value of the Hamiltonian. However, the recursive approach discussed here
has an advantage of providing more precise values of the energy gap and quasiparticle
energies.

The practical merits of the recursive solution are in its relative computational
simplicity and broad applicability. There are no restrictions for the type of single-
particle spectrum or pairing matrix elements; extension to other pairing modes and / or
the presence of time reversal non-invariant forces is also possible. The method can be
incorporated in the self-consistent scheme of the HFB approach taking into account
on equal footing non-pairing components of the residual interaction. We expect the
importance of exact particle number conservation to increase for the description of
soft nuclei where the static mean field is unstable, high order effects play a crucial

role, and the nuclear spectra reveal a stronger N -dependence. It would be interesting

32

 

 

 

 

 

 

1.6 l . . . . . . .
1.4 f —
/"\ 1.2 T F
% .
2 1 ~ ~
<1 0.8 - 2
0.6 ~ ~
0.4 _ ,
9 —2.ol 3
O)
a x
a; —4.0 — ~
0.
a —60~ -
”fl
6: _8.0 1 l . I . 1 i
a o 10 20 30 4o
U—l

Figure 2.10: Energy gap as a function of the particle number, upper panel; energy
of the paired system relative to energy of the system with no pairing, G 2 0, vs.
N ; lower panel. On both plots the solid line shows the particle conserving solution,

whereas the dashed curve shows the BCS result according to Eq. (2.45).

33

 

 

 

 

1 f 1 1 l
/
hll/Z / ’
\
\ /
3 * d \ / / I —
V2 \ \y ,X ’
\\ , X,
\ X x
P X\\ \\ )(I/ /)(I
S1/2 K x‘ X x’ X/
\ \ \ x
2 y\\ x‘ \K / 1’ IX
v 2 F x\ \x \ I, // /)<’ T
> \ \ x
Q) X \ é //
\x< I;
194$ 1"
\ ‘5‘ ‘ ’x’x
g7/2x“*_ __x.vx’ \x’}‘(,¢—1>—x'\:
1 ~ .. ,2» Mr ,
d w’"
5/2
4
O 4 +. I . . 1 . 1 I 1 . 1 . . I

Figure 2.11: Quasiparticle energies 6,, for 102—1328n isotopes; the results for the particle

conserving algorithm, solid lines with circles marking the valence neutron number,

and the BCS result, dashed line with calculated points denoted by crosses.

34

to proceed beyond the present BCS—like approximation and include explicitly the
states of higher seniority and discuss odd particle systems. Other extensions of the
particle—conserving method to problems of isovector pairing and pairing in weakly
bound systems with the presence of the continuum are plausible and will be the

subject of my future work.

35

 

Chapter 3

Chaotic wave functions and
exponential convergence of

low-lying energy eigenvalues

3. 1 Introduction

Statistical prOperties of complex quantum systems have been studied extensively from
various viewpoints. The seminal papers by Wigner [43] and Dyson [44] developed the
random matrix theory (RMT [45]) where the systems are considered as members
of a statistical ensemble, and all Hamiltonians of a given global symmetry appear
with certain probabilities. The canonical Gaussian ensembles [46, 47] correspond to
systems with complicated dynamics when, in almost all bases connected by the trans—
formations preserving global symmetry, the components of generic eigenfunctions are
uniformly distributed on a unit sphere in multidimensional Hilbert space. On a local
scale, Gaussian ensembles predict specific correlations and fluctuations of spectral
properties which are in agreement with empirical data for atoms, nuclei [48], quan-

tum dots [49] and resonators (microwave [50] and acoustic [51] experiments). These

36

spectral features are considered usually as signatures of quantum chaos [47, 52, 53].

Recently, the detailed studies of highly excited states in realistic atomic [54] and
nuclear [55] calculations demonstrated that such many-body systems are close to the
RMT limit although they reveal some deviations, partly due to the presence of the
mean field [56], and coherent components [57] of the residual interaction [46, 58]. In
complex atoms and nuclei, precise experimental information exists, as a rule, about
low-lying states only. Effective residual interactions , such as the Wildenthal-Brown
interaction [59] for the sd—shell model turned out to be successful well beyond the input
used for their original fit. This justifies the use of such interactions for studying generic
complicated states in the region of high level density. The shell model approach
requires large-scale diagonalization even if one is interested in the low-lying states
only. The dimensions of matrices increase dramatically with the number of valence
nucleons, which makes the full diagonalization impractical, even after projecting out
correct angular momentum and isospin states. This problem is avoided in the Monte
Carlo shell model method [60], but, apart from the so-called sign problem [61], which
requires the introduction of an extrapolation when working with realistic interactions,
this method is better suited for calculating thermal properties or strength functions
than spectroscopic characteristics. In order to consider individual levels, one needs
to supplement the Monte Carlo sampling with some variational procedure including
an additional “stochastic” diagonalization ([62] or the QMOD approach [63]). An
important step towards larger dimensions in the standard shell model is made with

the development of the DUPSM code [64].

3.2 Examples of exponential convergence

Here we suggest a simple approach which allows one, while calculating energies of

low lying states, to reduce large dimensions of matrices under study by orders of

37

magnitude, maintaining high precision in the results. The approach is based on the
statistical properties of complicated many-body states [54, 55]. Because of the strong
residual interaction and “geometric Chaoticity” [55, 65] of the angular momentum
coupling, the eigenstates are extremely complex superpositions of independent par-
ticle Slater determinants. However, in contrast to the limiting case of the Gaussian
orthogonal ensemble (GOE), the stationary wave functions are not fully delocalized
in shell model space. Due to the inherently self—consistent nature of the residual in-
teraction (even if it is extracted in a semiempirical manner), its strength does not
exceed the typical spacings between single-particle levels which are determined by the
mean field, i.e., by the same original forces. Together with the fact that the two—body
forces cannot couple very distinct configurations, this leads to a band-like structure
of the Hamiltonian matrices in the shell model basis Fig 3.1.

The theory of banded random matrices did not reach the same degree of com-
pleteness as that of canonical Gaussian ensembles. Nevertheless, both mathematical
[66] and numerical [67] arguments favor the localization of the eigenstates in Hilbert
space, similar to the coordinate localization of electronic states in disordered solids.
The generic many-body states in complex atoms or nuclei have a typical localiza-
tion width [55, 67, 68]. Inversely, the simple shell model configurations are packets
of the eigenstates. Their strength function is fragmented over the range of energies
characterized by the spreading width I‘ which is nearly constant along the spectrum
because the coupling matrix elements between the complicated states are small just
as it is needed to compensate small level spacings in the region of high level density
[69, 70, 71]. The qualitative arguments are confirmed by more general theory [57]
as well as by detailed numerical calculations for atoms [54] and nuclei [55, 72]. The
nuclear case is close to the strong coupling limit [73, 74, 55] where the typical width,

associated with the energy uncertainty, can be estimated [7 2] as I‘ z 26 in terms of

38

 

2505
200L
150~
100}

50-

 

 

 

 

Figure 3.1: The density plot of matrix elements of a typical shell—model Hamiltonian
matrix. This particular plot corresponds to the system with a quadrupole-quadrupole
residual interaction. The band—like structure is clearly seen along the main diagonal

which goes from the lower left to the upper right corners of the plot.

39

the energy dispersion of a simple configuration Ik),

0:: (leglk) - |<l€|H|k>|2 = Z lszIQ- (3-1)

Me

Here HM are the off-diagonal elements of the residual interaction between the basis
states so that the calculation of (3.1) does not require any diagonalization. The
dispersions a), of different simple states fluctuate weakly [55] and in our estimate of
F they are substituted by the appropriate mean value 6 which can be found by the
methods of statistical spectroscopy [75].

The practical method of truncating large shell model matrices was suggested in
[76]. The shell model states are grouped into partitions (sets of states belonging to
the same particle configuration). Since the states separated in energy by an interval
broader than F are not significantly mixed with the studied state, we truncated the
matrix retaining only the partitions whose statistical centroids E 2 (W are
closer than 30. The spin—isospin projection and the elimination of the center-of-mass
admixtures can be done within the truncated subspace only. In order to keep the
correct shell model structure, the partitions should be included as a whole. As shown
in [76], this method allows for the calculation of low-lying energies with sufficient
precision in large shell—model spaces. The truncated eigenvectors overlap with the
exact ones on the level of better than 90%.

Going beyond the simple truncation, we consider the convergence of level energies
to the exact values as a function of the increasing dimension n of the diagonalized
matrix. As an example Fig. 3.2 displays the energies of the yrast 2+ state in 48Or
(the p f-shell model dimension 1922106 in the m-scheme; JT-dimension 27229) and
of the lowest 1/2‘ and 3/2‘ states in the 518C nucleus (JT-dimensions 13016 and
24474, respectively) calculated with the FPD6 interaction [77] for dimensions ranging
from n 2 2000 to the full dimension N. The smallest 77, lead to a deviation within a

few hundred keV for 518C and about 1.5 MeV for 4801 As 72. increases, the energies

40

 

 

0.12 I I I I I I '97.0 ' ' ' l '

 

 

 

 

 

 

 

 

 

 

 

 

 

o 48C12+O
0 ,0 _ ( 5130 (1/2) 1 j .q
S ' 1, _97 5[ I, 3.0 exp(-0.00016 n)-99.22 .
a) \ - ‘ ‘1
2 0.08 _ [o 0.26 exp( 0.000497 n) o _ "x.
1.0 “ S _
D 0.06 P \o _ g 98.0 1 L;
1\_ I l =
8 0.04 " ;\ ‘ -98.5 - w
I ‘
Lu 0.02 - \°\ -
52% -99.0 ~
,__. . "1" . . 1 . . . . . .1... . . .1 5 “0"i:‘?-—-—~~—.~.<—-»..l-2....o _, ,(
0 4000 3000 12000 0 10000 20000 30000
0.l 6 \o r I I r ] 0 1 I 1
.\ [ 48C! 210
- 1 515c(3/2-) 1 -—-- ~ ~
g 0 ,2 _ ‘1‘ [ 3.0 exp(-0.00016 n) 2
2 ' \ 0.23 exp(-0.00025 n) o s‘ 1 “s2,
.4 \ a) X\‘ I]
‘0 h i I '2 i ,
I'— \ s—I
0, 0.08 ~ 1, - m j
!— ‘\ N
o _ °\ . oi
'—' \ o
\ 0.1
Lt 0.04 - X. ~ +
i \0 LLI ‘x
.. \Q‘Q .1
\Of‘o"‘o wow-.9 or i l
.. . .1. .... l' .. . 'Oirr-:-~—..... 0. 0.0] l l I 4 1
0 10000 20000 0 10000 20000 30000
n n

Figure 3.2: Left panel: energies of the lowest states 1 /21_ and 3 / 21— in 518C as a function
of the matrix truncation n, shell model calculation (points) and fits A exp(—yn) (solid
lines); right panel: energy of the yrast state 2+ in 48Cr, real scale (top) and logarithmic

scale (bottom).

converge very fast and monotonically to the exact values. The convergence is almost
purely exponential, E(n) 2 E00 + Aexp(—*yn); the rate of convergence 7 z 6/N for
Sc and y % 4/N for Or. A similar exponential convergence, with slightly varying
parameters (apparently lower 7 for strongly collective states as in Or), is observed in
all studied cases for low-lying states. The study of the convergence rate as a function
of excitation energy will be delayed for further research. The exponential convergence
of eigenvalues would be extremely helpful for shell model practitioners. It would make
almost redundant the full large-scale diagonalization if one is interested in the low-

lying states only. Instead, the calculations for several increasing dimensions (still

41

far from the full value and therefore easily tractable) would end in determining the
exponential parameters and simply extrapolating to the exact result. At present,
the rigorous mathematical theory of convergence is absent, and we limit ourselves by
qualitative arguments and plausible conjectures.

The convergence under consecutive truncations is determined by the type of the
matrix and by the original unperturbed basis which orders the basis vectors in a
certain way. The ordering is done almost uniquely in the spherical shell model where
the mean field is fixed and all many-body states are organized into partitions. For the
lowest levels, the admixtures of highly excited states outside of the starting truncation

correspond to the wings of the strength function
Fk(E) = Z l<a|k>|25(E — Ea), (3-2)

which describes the fragmentation of the state |k) over eigenstates la) . As confirmed
by atomic [54] and nuclear [55, 72] studies, the strength function has on average a
universal shape. This shape evolves from the standard Breit-Wigner function [17]
for the “weak damping” case to the Gaussian form at strong damping (semicircle in
the RMT limit of uniform spectra [57, 66] which is not reached in practical cases).
Correspondingly, the dependence of the spreading width on the strength of the resid-
ual interaction changes from quadratic in the standard golden rule [17] to linear
[74, 57, 72]. The remote wings of the strength function have their own energy be-
havior [54, 72]. With high accuracy they can be described [72] by an exponential
function of energy Fig. 3.3. This is a clear manifestation of the localization of the
eigenfunctions typical for the banded Hamiltonian structures [78, 66]. The strength
function is simply a Fourier image of the decay amplitude of the original state |k) .

eiEt _
F),(E) = / 271 dt(k|e"H‘|k). (3.3)

 

In this limiting regime of distant tails the t —> O decay region has a Breit-Wigner type

42

 

.._ Bren-Wigner
Gaussian
.._ exponential

 

 

 

     

 

 

lllLlllllllllllll
-50 -40 -30 —20 --10 0 10 20 30 4O 50

E — Ek(MeV)

Figure 3.3: Summary of the results obtained in [.77], that clearly indicate the expo-

nential nature of the strength function tails.

dependence on time which follows from the Taylor expansion

P(t) = [we-“1%) z 1— 0,3)? 21/(1+ 031:2), (3.4)

|2
where 0% was defined by the Eq. 3.1. This leads to an exponential form of the
strength function tails. The exponentially weak mixing should lead to exponentially
small energy shifts and to the corresponding convergence of the eigenvalues.

We can expect the exponential behavior to be generic for large matrices of quasi-
banded form with the off-diagonal elements of approximately the same order of mag-
nitude along the spectrum. This conjecture can be checked by generating random
matrices with the desired properties and diagonalizing them in a sequence of progres-
sive truncations (the matrices are first ordered according to their diagonal elements).
Since there is no “vertical” structure in such random matrices, we do not have here

physical arguments concerning the optimal truncation sequence. We show in Fig. 3.4,

left, a typical result for the banded GOE matrix with the band width b 2 0.293N

43

 

 

 

 

 

 

 

 

1-2 I I I I I I I I I -0-4 I I I I 1 fir I I I
o GOE 2000 O 06 a Full GOE 2000 o
- . ... \
1 r\ — b\
.03 r. b. _
0.8 ~ \ ~ \o\
:2 ° ‘1 ” ° ‘
e. X X
.— 0.6 - Q — Lu -1.2 — ~
:1 ‘29 \.
*0 -1.4 ~ ”\v —
0.4 '- "‘ \\K
x -16 ~ "2
b . .g 2
0.2 - N00 ‘ Q‘Q.
00 '1.8 '- Q\°\ 'l
W 0‘.
0 1 I 4 1 l l I $909900> _ 1 1 I l l 1 I I 1 <32
0 200 400 600 800100012001400160018002000 0 200 400 600 800100012001400160018002000
n 11

Figure 3.4: Energy deviations for the ground state of random matrices of dimension
2000 as a function of the progressive matrix truncation n (diamonds): the GOE-like
banded matrix of width b 2 0.293N, chosen in such a way that half of the matrix
elements vanish, approximately in the same proportion as in typical shell model cases,
left panel; the full GOE matrix, right panel. Solid lines show a fit Aexp(—yn) with
A 2 1.18 and *y 2 3.8 X 10‘3, left, and A 2 1.96, 7 2 1.53 X 10‘3, right. Note the

absence of the horizontal asymptotics in the case of the full matrix.

which clearly demonstrates the exponential convergence. The full GOE matrix, Fig.
3.4, right, converges more slowly in the absolute sense and does not saturate. This is
a simple consequence of the fact that the ground state of a GOE matrix is repelled by
the higher states to the edge of the semicircle (—2 with the GOE definition accepted
here and in [55]). This process is driven by the off—diagonal elements; all of them
on average have the same magnitude. Their number and, thence, the dispersion 0,
Eq. (3.1), are greater in the full GOE. Therefore, the distance from the unperturbed
position is also greater in this case (the diagonal and off-diagonal contributions add
in quadrature). In all studied cases with the width b changing from 0.1N to the
full GOE, the convergence is exponential and the exponent y is approximately scaled
inversely proportional to b.

In realistic cases there is also a leading sequence of regular diagonal elements. A

44

similar banded matrix example, considered in [54], goes back to Wigner [43]. The ma-
trix consists of the equidistant diagonal with the spacing D and random off-diagonal
matrix elements Vkl within the band [It — l I g b. At the relatively weak interaction,
9 E (V2) /D2 < 1, the main contribution in the perturbation series for the admixture
run 2 0,2, of a very remote state ln), n >> 1, to the wave function of a low-lying state
[0) is given by the summation of long “straight” paths in Hilbert space connecting
the initial state with the final one through various intermediate stops. Because of the
random character of the off-diagonal interaction, the mean value of 112,, is determined
by the squares of the contributions of these paths (no interference). In the approxi-
mation of a weakly changing level density, this can be approximately written as an

integral equation of the random—walk type,

"(I)" : _g§fn + '22 Z fn—kwk (35)
n n k

where the factor 1/n2 comes from the energy denominators, and fn shows the behav-
ior of typical squared off-diagonal matrix elements V30 as a function of the distance
n from the diagonal. With the sharp band boundary [54], the weights 10,, decrease
quickly, essentially as exp(—nln n) N (n!)"1. With the smooth cut-off, the conver-
gence is nearly exponential. Thus, for the exponential cut-off of matrix elements,
Vk) ~ exp(—|k — ll/b), we have fn = exp(—2n/b), and Eq. (3.5) allows a simple
solution 112,, 2 Aexp(—2n/ 12)/722. Therefore, the contributions to energy of the state
[0) should converge N ann 2 DA exp(—2n/b)/n. Fig. 3.5 illustrates this con-
sideration by an example of the numerical diagonalization of a random matrix with
the equidistant diagonal and the exponential cut-off. One may note that the rate
of convergence is similar to that in a shell—model calculation, see above, where the
effective width of the band is close to b a: N / 4 [55]. However, the method of Eq. (3.5)
becomes invalid in the case of strong interaction when the contributions to the per-

turbation series of additional loops in Hilbert space cannot be neglected. The range

45

0.045 I I I I I I I

 

0.04 -

0.035 - -

0.03 - -

Lu 0.025 - -

0.02 - _

0.015 - _

 

0.01 — If 3 “”"“‘""**<*2 0

VA} (,2

 

 

 

0.005 I l l l l l I
200 300 400 500 600 700 800 900 1000

Figure 3.5: Convergence of the ground state energy for a banded random matrix with
the exponential cut-off of matrix elements V1,) ~ exp(-|k — ll/b), b 2 0.293N, shown

by diamonds; the solid line shows the fit const- exp(—2n/b)/n.

of convergence is seen from the expression for the constant in the above solution,

A 2 g/[l — ng=l k"2], which determines the critical value 90 2 6/22.

3.3 Solvable models

It is interesting to test the character of the convergence in simple solvable models.
A harmonic oscillator, shifted from the equilibrium position by a constant force,
H 2 did + /\(a. + al), lowers its energy by A2. The exact ground state is a coherent
combination of unperturbed states In). In accordance with the composition of the

coherent state, the convergence of the energies in the unperturbed basis of the original

46

oscillator is N A2n/nl. This is clearly seen in the left panel of Fig. 3.6. The fast
convergence is due to the constant level density along the main diagonal while the

1/2. In the case

perturbation has matrix elements growing only prOportional to n
of a quartic anharmonic oscillator, the exponential convergence is modulated with
oscillations.

Another example displays the case of slow convergence. The tight-binding model
of a finite one-dimensional lattice has degenerate levels in each of N identical wells.
The amplitude of hopping between adjacent wells is labeled as v. The eigenstates
of the model are delocalized standing Bloch waves with energies within the band,
Eq 2 2'1) cos cpq, 0)., 2 7Iq/(N +1), q 2 1, 2, . . . , N. It is straightforward to to see that
the truncation in the site basis corresponds to the convergence ~ 1/722, see Fig. 3.6,
right.

Tridiagonal matrices with the entries Hm, E 6,, and Hn_1,n 2 .,,,,,_1 E n
smoothly depending on n can be analyzed in a general way using the recurrence

relation for the secular determinants Dn(E) of the matrix (H — E) truncated at the

nth step,
Dn(E) = (6n. - Ean—1(E)— Vuan—AEI (3-6)

The consecutive approximations for Em”) and E0“), 77. >> 1, to a low-lying eigenvalue
E are the roots of Dn_1(E("‘1)) 2 0 and Dn(E(")) 2 0, respectively. For 6n >> E(")
(with a slight modification, the method works also for initially degenerate matrices
with 6,, 2const), the asymptotic behavior of the energy increments An 2 E (n) —E ("—1)
follows from (3.6) as

AnAn—Z V2
2 n E An. 3.7
(An + An.--1)(An-l + Ari—2) énen—l ( )

 

 

The exponential convergence corresponds to An —> /\ 75 0 at large n. (proportional
increase of diagonal and off-diagonal elements). Then the increment ratio fin 2

An/An,1 also goes to a constant limit 5 2 (1/2A)[1 — (1 — 4A2)1/2] which restricts the

47

10

 

 

 

 

 

 

 

 

 

 

 

 

r
1
I
r7 k=l
. 1

 

 

1.5 — 1.5 ~ -
(<C
1.0 1.0 _ /2 ,2,
0.5 a 1 . 0.5 . . 1 . . . .
0 50 100 150 0.0 10.0 20.0 30.0 40.0
n n

Figure 3.6: Convergence of the ground state energy in the tight-binding model of a
finite one-dimensional lattice with /\ as a hOpping parameter, left panels, and for a
shifted harmonic oscillator with the Hamiltonian H 2 did + Ma + al), right panels.
The upper parts show the energy deviation AEn 2 E0(n) — E0(oo) as a function of
the truncated dimension 12 (solid lines for /\ 2 1 and A 2 2); dotted lines show the
predicted analytical convergence of the models, Aa2/n2 (left) and A2n/n! (right). The
lower parts characterize the convergence rate An —+ A by plotting An E AEnn2 /71’2,

left, and An 2 (AEnn!)1/2", right.

48

10

 

  
     

Critical point
2 ‘ x205 ‘

 

 

0 I 1 I m . 1 . L I I .
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

A=Vn/8n, n->infinity

 

Figure 3.7: The exponential convergence rate 7 is shown as a function of the limiting
perturbation A in the case of a tri—diagonal matrix. The critical point at A 2 0.5 is

indicated.

exponential convergence region to A2 < 1 / 4. The existence of the constant limit is
still compatible with an additional pre-exponential factor weakly dependent on n; at
large 72 corresponding fits are usually indistinguishable. An explicitly solvable (by the
Bogoliubov transformation) model of the harmonic oscillator with the perturbation
A(a2 + al2) agrees completely with this estimate. The case A2 2 1 / 4 corresponds here
to the degeneracy of the oscillator with zero frequency, and at A2 > 1/4 the spectrum
is inverted. In general, A 2 1/2 corresponds to a critical point as shown in Fig. 3.7;
the convergence here is described by a power law being exponential outside of this
region.

The main difference of the tight-binding case from the oscillator model is the

degeneracy of the unperturbed levels (absence of the leading diagonal) which results

49

 

divergence

 

 

 

 

&
Exponential
0'4 P convergence l
0.2 s W
0 l l J l
0 0 2 0.4 O 6 O 8 1

Figure 3.8: Convergence region of five-diagonal matrices shown in the two-parameter

space of the off-diagonal matrix strengths A1 and A2 .

in delocalized wave functions for eigenstates. Because of the degeneracy, the analog
of Eq. (3.7) contains, instead of Ag, the ratio v2 / E (“Em—1) which is just equal to the
critical value 1 / 4 in the limit of large n. The situation is similar for the spin chains
with the nearest neighbor interaction where finite size effects on the ground state

energy were repeatedly studied [79] and corrections also go as n”.

We expect the
presence of disorder (random positions of the original site levels), which leads to the
localization of the eigenfunctions, to be accompanied by the transition to exponential
convergence of the eigenvalues. Numerical studies of five-diagonal matrices show a
similar trend in Fig. 3.8. Depending on the limiting, relative to diagonal, strength
of the off—diagonal elements A1 (nearest to diagonal) and A2, there is a region of

exponential convergence separated by the phase boundary from the divergent region,

where the lowest eigenvalue is unbound.

50

3.4 Conclusion

In conclusion, we discussed the convergence of the low-lying eigenvalues of large ma-
trices describing realistic many-body Hamiltonians of the shell-model type. We gave
arguments in favor of the conjecture that the exact diagonalization of relatively small
matrices, truncated on the grounds of physical partitions and generic spreading widths
of simple configurations, provides a starting approximation which can be extrapolated
to the exact result with the aid of a simple exponential continuation. The arguments
are based on the generic features of quantum chaotic many-body dynamics, simple

models and the results of numerical analysis.

51

Chapter 4

Modeling pionic fusion

4.1 Introduction

Nuclear fusion reactions which produce a pion are often referred to as pionic fu-
sion. Pion production has been observed [80, 81, 82] at energies approaching absolute
threshold, where the entire available energy is converted into the pion, demonstrating
an amazing collective behavior of nucleon systems. However, it remains quite difficult
to incorporate the observed collectivity into existing theoretical models. A variety of
studies [83, 84, 85, 86] have dealt with subthreshold pion production in heavy ion col-
lisions, where the energy per nucleon is below the energy threshold of the elementary
single-particle reaction N N —> N N + 7r. Most models, such as those featuring pion
bremsstrahlung mechanisms [87, 88], quantum molecular dynamics approaches[89],
perturbative calculations using Boltzmann-Nordheim-Vlasov equations [90] or nuclear
structure functions [91], provide a good picture at energies starting from E /A z 30
MeV up to the single-particle threshold E /A 2 280 MeV in the laboratory frame.
In the present work motivated by the experimental results of [80, 81, 82] our aim is
to consider even lower energies and study the behavior of the cross section of fusion

reaction in the region down to 210 MeV above the absolute threshold. This neces-

52

 

 

 

 

 

 

 

A+A

Figure 4.1: The pionic fusion of two nuclei in the sudden approximation is illustrated.

sitates a careful consideration of limitations on the reaction given by conservation
laws and the Pauli exclusion principle which govern the behavior of the cross section
in this extreme situation. The statistical approach uSed in most existing models at
higher energy has to be substituted by low energy many-body structure physics.
Our model, that is described in Sect. 4.2, considers the cross section in the Born
approximation, assuming that pion production occurs through coupling to a single
nucleon. All possible further rescatterings of the pion are expected to significantly
reduce the probability of the reaction and are ignored as higher-order processes. A
schematic picture of the reaction is shown in Fig. 4.1 demonstrating the pionic fusion
of two nuclei A and A’ . Many—body nuclear mean-field parameters are assumed to
be constant and to suddenly change from the initial to the final values. This will be
referred to as the sudden approximation. The three-dimensional harmonic oscillator
shell model is used to describe the structure of the incoming and outgoing nuclei.
This allows analytical calculation of all necessary overlap amplitudes. The stationary

wave functions are constructed as Slater determinants projected onto good angular

53

momentum. Taking into account the center-of-mass motion we preserve linear mo-
mentum. Sect. 4.3 shows the implementation of the model for the case of pionic
fusion of two identical nuclei. In Sect. 4.4 we present a low pion momentum approx-
imation, for which more general results could be derived. The parallel discussion of
mathematical details is given in the appendices. The application of this model to
experimentally observed pionic fusion reactions shows surprisingly good agreement
with data. The comparison is presented in Sect. 4.5 along with some predictions for

heavy nuclei.

4.2 Description of the model

4.2.1 The transition amplitude

We approach the problem of pionic fusion as a stationary scattering problem. We
consider the reaction cross section to be given by the Fermi golden rule in terms of
the transition amplitude (F [H II), where I and F refer to the initial and final states,
respectively, of the whole system including the emitted pion. The density of final pion
states is given by ngdk d9/(27T)3 with k as a momentum of the pion produced; Q
is a solid angle in the center of mass (CM) frame, and V stands for the quantization
volume. In all further calculations the pion is assumed to be fully relativistic whereas
nucleons obey non—relativistic quantum mechanics. We use a set of natural units with

h 2 c 2 1. In this framework the differential cross section can be written as

k . .
do w m 2|(F|H|I)|2V2d0 , (4.1)

: 2pn(27r)

where m is a nucleon mass, pn is the CM momentum per nucleon in the initial state

and w 2 Mk2 + m3, is pion energy.

On the single-nucleon level one can use a phenomenological Hamiltonian density

54

for the pion-nucleon interaction [92],
— .. a A1 -_. _. A2 - ... ;.
7i 2 grbyflrbn + 47rgz—rb7r ' mp -|— 47rT—n31Dr-7r X 7rd) . (4.2)

A number of studies have been performed analyzing this form of the interaction
within the context of chiral perturbation theory [93]. The first term, often called
in the literature the impulse or Born term, is responsible for single—pion production
in a p—wave. We neglect the second and the third s-wave terms which require an
additional interaction to absorb the extra pion created in the four-point vertex. We
believe that due to the difficulty of recombining the nucleons into an appropriate
final state the second and third terms become increasingly unimportant for larger
nuclei. It has also been experimentally observed that in pionic fusion reactions the
pion is predominantly produced in a p-wave [80, 81]. Reduction of the first term in

the Hamiltonian to a non-relativistic case gives an interaction of the form

5'7k

with the coupling 9 appropriately defined according to isospin. Separation of the

quantized pion field,

 

1 . .
V (age—’1“x + ake+‘k'x), (4.4)

7r :r 2
( l 2.: m
in the matrix element of Eq. (4.1) reduces the transition amplitude to the following

form

1 1 .
<F]H]I>:\/Tw_V%<fl Z Qk'ge—Zk'xlw , (4-5)

nucleons

Where If) and [1) are final and initial states of the nucleon system.

4.2.2 Nuclear wave functions

We will approximate a state of a nuclear system with an antisymmetric combina-

tion built upon single-particle (s.p.) states. We take these states from the harmonic

55

oscillator shell model, which allows for the analytic calculation of corresponding over-
laps. The approach however can be extended to any single—particle basis. Each of
the single-particle states can be characterized by the number of excitation quanta in
three Cartesian directions, the nucleon spin and isospin projections. The locations
of the centers of the harmonic oscillator potentials for all separate nuclei have to be
introduced as additional parameters to the wave function. The importance of these
parameters in projecting a nucleon wave function onto a state with correct total mo-
mentum for every nucleus participating in the process is discussed below. Following

these assumptions we will write the wave function of a nucleon system as follows

(A+1)th s.p. state

 

I (C71, 81, t1; EYE, 527 t2; - - - fl); [Bi/1+1, 8A+1,tA+f; (Ag/1+2, SA+27 tA+23 - - - ; 1"» - (4-6)

\ J
r

Y
nucleus A nucleus A’

In this example we assume that the system consists of two nuclei A and A’ with
the centers of their respective harmonic oscillator potentials at r and r’. The single-
particle orbitals are numbered from 1 to A for the first nucleus and from A + 1 up to
the total number of nucleons A f 2 A + A’ for the second one. Labels 62 2 (am, cry, a2)
are Cartesian quantum numbers of single—particle states, while 3 and t are the spin
and isospin projections. Protons and neutrons can be considered separately as well
as different spin projections of the nucleons, reducing the wave function of the state
to a product of four components. If the described separation is performed and the
resulting part of the wave function contains only single-particle states with the same
values of either 5 or t, then the corresponding index is omitted in writing. We use
a standard form for the one-dimensional harmonic oscillator wave functions centered

at r in the coordinate representation:

(am; 7),) = 477%ch Hamid — r))e—“2($—T)2/2 . (4.7)

The parameter 1) is defined for a single oscillator as v 2 me. These parameters

characterize the mean field potentials for every incoming or outgoing nucleus. The

56

 

 

function Hn(:r) is the nth order Hermite polynomial of the variable 11:. The discussion
of the overlap integrals such as ((a', r’)v/|(oz, 7),), and the general form of the results
are presented in Appendix 4.7.1.

A simple projecting technique was used to construct wave functions as eigenstates
of the momentum operators that correspond to the total momenta of each individual

nucleus,

l(&1,517t1; - - JP); (&A+1,SA+1,tA+1; . . .;p')) :

+00 - I I
N‘1// d3rd3r'|(d’1,sl,t1;...;r),(o'ZA+1,sA+1,tA+1;...;r'))el’(p'r+p 'r) .(4.8)
—00

It is easy to check that

A
—iZV.I(&1,...;p>,(&AH,...;p'>>=p|<crl,...;p),(&l+.,...;p')> (4.9)
j=1

and
A+A’
‘i Z Vilfii» - - JP), (CY/1+1, - . JP,» 2 P, [(511, - - JP), (CY/1+1, - - JP,» - (4-10)
j2A+l
In the above example the situation with two-nuclei state is shown, which is appro-
priate for describing the initial state in pionic fusion. The final state containing just
one fused nucleus is constructed analogously.
Due to the finite range of the interaction, the overall normalization N of the
state (4.8) that contains several moving nuclei, is just a product of normalizations for

each of the constituent nuclei individually. It is useful to write the CM coordinates

separately from the relative coordinates of the nucleons

[(521, C72; . . .d’A;r)v) = [(CYCM = (0,0,0);r)v\/Z) W464) . (4.11)

The relative coordinate wave function Wm) can be complicated but the CM part
for the ground state nucleus is simply represented by the unphysical ground state

oscillation of the center of mass in the effective harmonic potential with the parameter

57

m/A. This is removed by a projection (4.8) onto the correct momentum state. The

normalization integral can be expressed as

N2 : f/d3T d3T,<(&CMi r’)v\/Al(&CMi FLA/X) (wrellwrel>eip(r—ri)

_ /d3,’./d3?”16—A(r—r’)2v2/4eip(r—r’) _ (3)3/2 ‘/€3—1)2/A212 (4 12)
.._. — 02A . .

A different method of calculating the normalization along with further justification
of this form for the CM part of the wave function is discussed in Appendix 4.7.2. We

also note here that with a slight modification of Eq. (4.12) the orthogonality of the

nucleon wave functions can be shown

<(&1i&2i - - ' &Ai p,)v](&1i ($2; - - ~ 62A; p)v> I N26p,p’ -

4.3 Fusion reactions A + A —> 2A + 7r

For the remainder of the paper we will assume A to be the mass number of each of the
initial nuclei with proton—neutron composition (Z ,N ) and w the oscillator parameter.
The entire initial state is characterized by a set of the single—particle quantum numbers
{62,-}. The fusion product has 2A 2 A f nucleons, the oscillator parameter 2}, and the
final state quantum numbers {flj}. The collision is considered in the CM reference
frame; therefore we use p and —p to denote the momenta of the incoming nuclei and
k for a final pion momentum with corresponding p f 2 —k as the total momentum of
the recoil nucleus. The integration of the wave functions leading to correct momenta,
Eq. (4.8), is performed at a final stage so initially overlaps are calculated as functions
of r , r’ and R, the locations of the centers of the two initial nuclei and the final

nucleus, respectively.

58

 

 

@ 1——>— }neutrons

 

Figure 4.2: One of the amplitudes of the total fusion process: an initial proton from
the nth orbit produces a 7r+ and ends at the lth final neutron single-particle orbit. Fn
is the remaining overlap of a proton system with the nth initial single-particle state

missing. G; is the neutron overlap with no lth state in the final system.

4.3.1 Charged pion production

We begin with the case of 7r+ production where one of the initial protons interacts
with the pion field producing a neutron and a real on-shell pion. With the assumption
that the pion was produced in a single-particle interaction, the total amplitude of the
process becomes a sum over all possible amplitudes shown in Fig. 4.2, with the pion
vertex connecting any of the initial protons to any of the final state neutrons with
the correct relative sign to preserve antisymmetry. Suppose the interacting proton
in the single-particle state n produced a neutron in the state 1 of the final nucleus.
In the initial state we sum over the occupied orbitals of the first and the second
colliding nucleus, for n 3 Z and for Z < n f 2Z, respectively. We use the notations

01(7", 7", R) for the neutron overlap

Gr:((3151;---gl—1,§z—1;fll+1,§1+1;---;R]](&1731;---;I')(5ZA+178A+13---;I")>a

no lth s.p. state

(4.13)

59

Fn(r, 7", R) for the proton overlap

= ((3131; - - -;R)][551,81; - - . ;&n—1,3n-1;C—¥'n+113n+1; - - )1), (4-14)

no nth s.p. state

 

and H,” for a single-particle matrix element

((51.5;RH95' ke’ik'xlfim 8n;r)) n S Z
nl : _, _ (4.15)
((flz,§);R)|gE-kc‘1k'x|(d’n,sn;r’)) n > Z
Finally, following Eq.(4.5), the total amplitude can be expressed in terms of the

following sum:

1
MNf\/2:J VZ—m

The determinants of the matrices are constructed from a product of the single-

<F|H|1>—

 

—/ / d3 (131' (1311321)"“FnGlH (‘1‘ r’> (Pf R(4.16)

particle overlaps of size (2N) x (2N) for the neutrons (0)) and (2Z — 1) x (2Z — 1)
for the protons (E). The Gaussian nature of the single-particle overlaps allows one
to separate all exponential factors that govern the general trend of the cross section
leaving only some polynomials of a general form that carry spin, isospin and Pauli
blocking information. These mathematical manipulations are discussed in some detail
in Appendix 4.7.2. Here we present a final expression for the square of the transition
amplitude

l<F|H|1>l2

2 2 2 3/2 k2

_ 'V2 2—vA cum—77]
in which the exponential factor g, effective oscillator parameter 77 and reduced ampli-

tude M+ are introduced as follows

 

 

2p2 k2 k2('11)2 — 112)
E —— — — 4.18
.5 exp ( Av'2 "U2 + 2112 2A’02(’u2 + 1112) ( )
21211)
E ._ 4.1
77 '02 + w2 ’ ( 9)
and
M+ a P(k, meek/M + Q(k, mark/M. (4.20)

60

 

 

Here, P and Q are dimensionless polynomials of p and k, the total CM momentum of
the initial nuclei and the final pion momentum. The polynomials are to be determined
using particular configurations of the initial and final nuclei. They are also functions
of v and to which determine the appropriate momentum scale. If the two colliding
nuclei have the same initial shell model state then P(k, p) 2 iQ(k, —p) (the phase
difference given by 31: sign for even or odd Z, respectively, is due to imposed Pauli
antisymmetry, see Appendix 4.7.2). The procedure of analytically calculating P and
Q involves finding the determinants of the matrices constructed from polynomials that
result from integrating a product of Hermite polynomials of the form ((5, R)|(a, r));
and performing the integrational Fourier-type conversion, Eqs. (4.8). This process is
discussed in Appendix 4.7.3. The size of the matrices is determined by the number

of nucleons of the same spin-isospin type.

4.3.2 Neutral pion production

The case of no production can be considered in a similar fashion. A neutral pion can
be produced either by one of the protons or by one of the neutrons, which couple
with a negative relative sign. Compared to charged pions the coupling is larger by a

factor x/2. The final amplitude can then be expressed, similarly to Eq. (4.17), as

2 3/2 2
2_ 9 277 [kl é 6(A—1) 2
[(F]H[I)| — ———V2 (T112) —w 2 77 [A40] . (4.21)

Here the reduced amplitude can be split into a proton and a neutron part:

Mo = Pp(k,p)ep'k/A“2+Qp(k,p)e'p'k/A”2—Pn(k,p)ep‘k/A“2-Qn(k,p)e‘p'k/A”2 - (422)

4.4 Low pion momentum approximation

Due to the specific form of the polynomials discussed above, further simplifications

can be made for the case of no production. Near the absolute threshold, the pion

61

momentum [k] is small compared to all other momentum parameters [p], v and w,

and can be ignored in polynomials. Then

11,, = <(B’..31;R).,Ia - ke-ik'xl(&..sn;r)w>

—k2 k . 2 2
————.— _ 2 (RD +rw ) . (423)
2(1)2 + 2122) ('U2 +1122)

 

z (3...; - klsn> «Emma... r>w> exp(

With this approximation, the interaction part is factorized into exponents as shown
in the expression above. Therefore the total pionic fusion amplitude is a product of
a pure fusion amplitude and the expression that arises from the operator 6 - k acting
on the nucleons. For a given type of the initial and final nucleon, the sum of a single-
particle matrix element multiplied by the corresponding overlap of the remaining
particles reduces to a sum of matrix elements multiplied by the corresponding minor
which is related to a determinant of a full matrix. It is shown in Appendix 4.7.3
that the polynomials can be expressed in an analytical form if all inner harmonic
oscillator shells are completely filled without any gaps in all participating nuclei.
This restriction allows any type of particle-hole excitations within the outer unfilled
shell.

The total differential cross section for neutral pion production close to absolute

threshold is given by the form:

 

(10 92Ak3 < 27V )3/2 776(A—1)+Qf+§21 (fill) QrQi (2122/41)? x

if). : (27r)22pm A212 Av

, [ 2
202/2qu (2]) m) ((1:10 —1)l!(qy —1)ll

Here the integers (1,, j 2 513,11, 2 are introduced as differences between numbers of

2 1 ~ 2
715 [M] . (4.24)

 

 

quanta in final and initial systems for three Cartesian directions; Q,- and Q; are total

numbers of quanta in initial and final systems. These values are defined as

qr: Z 31'— Z 01', 91': Z (04+04y+012), 91: Z (fi$+18y+162)'

nucleons nucleons nucleons nucleons

62

The spin and radial parts of the wave function are completely decoupled in our non-
relativistic description of the nucleon system. This allows one to introduce the matrix

element used in Eq. (4.24)

~ 1

Arzmflfl X:251M% (4%)

nucleons

where f and f are the spin-isospin parts of nucleon wave function of the initial and
final systems, respectively. This matrix element could be directly computed for every
particular nuclear configuration, but for a large number of states, degenerate within

the harmonic oscillator model, it is useful to use an approximation for the average

~

M2fia—A—M+M) Mm)

 

The Cartesian directions of the harmonic oscillator quantization axes are chosen in
such a way that the z axis coincides with the beam direction, though the spin is
quantized along the k axis, simplifying the action of (3' - k which is used to obtain Eq.
(4.27). Integers Z], Z j, NT and N j are mean numbers of particles for each spin-isospin
combination with respect to our axis of spin quantization. The polynomials T "(113),

defined in Eq. (4.36) of Appendix 4.7.1, can be approximated as

2q2/2T ip 2 x ip 2 qz . (4.28)
q. Anvw Anvw

This approximation is valid in the limit that the arguments become large and allows

 

 

for a better quantitative understanding of the behavior of the cross section. The value

of the argument is almost independent of the mass number A at threshold energy:

2

z 6 .
Anvw

 

P

In Eq. (4.24) only the lowest order term in the pion momentum is retained
resulting in a p—wave cross section (exponents with k are also ignored). The equation

has only one numerical parameter 7, the origin of which is discussed in Appendix

63

4.7.3. This parameter is a product of four factors, one for every spin / isospin nucleon
species. Each factor depends on the number of particles of corresponding type and on
their distribution within the highest harmonic oscillator shell for both initial and final
nuclei. Numerically, y ranges from 1 to 10 for light nuclei. The cross section can be
zero if some symmetries are not preserved (spin, isospin, oscillator symmetry) as well
as by virtue of Eq. (4.39) in Appendix 4.7.1 if creation of the final system requires an

odd number of quanta relative to the initial system in any of the transverse directions.

4.5 Application of the model and results

4.5.1 The reaction p +p ——> d + 71+

The first and the simplest example to calculate is the two-nucleon fusion reaction
1) + p —+ d + 7r+. This example serves here only for illustrative purpose as we do not
include pion rescattering due to the full interaction given by Hamiltonian density of
Eq. (4.2) which is important for this elementary process. Moreover, the deuteron can
hardly be approximated with the harmonic oscillator shell model. The polynomials
P and Q, Eq. (4.20), in this case do not depend on p being equal to the matrix
element of (f - k/[k] evaluated between the spinors of initial interacting proton and
final neutron. Here, P and Q correspond to the choice of the first or second initial
proton to produce a pion, respectively.

Dominant partial wave channels are summarized in the following table along with
our results for their reduced amplitudes. The table was constructed by separation of
initial singlet and triplet states of the N N system. The table shows partial waves of

the 7r —— at system printed in the left column that yield the dominant contributions to

64

the amplitudes which are shown in the right column.

 

pion N N state amplitude
s—wave 3P1 2 \/2 sinh(£°—§4?TS€)
(4.29)
p—wave 150

2 cosh(—P———k 420280)
p—wave 1D;;
As can be seen from the table above, this cross section is predominantly p—wave in
nature at low pion energies. The s-wave contribution that comes [92] from rescattering

of the pion due to the interaction (4.2) was not included. The total cross section

averaged over spin projections in the initial state and summed over final states is

do 921133 41024-192 2kpcos6
—2 ——-——— h —— —1 . 4.
d9 2mp\/2—7rv3 exp ( 2222 3 cos '02 ( 30)

The obtained p—wave cross section behaves at low energies as

 

 

0(pp —> (171+) : a (k/m,)3 , (4.31)
where

2 2x/27rg2m75/2 _2mm /,,2

a2 m3/2v3 e . (4.32)
Choosing the oscillator parameter 1) 2 216 MeV/c reproduces the experimental

value[94], 46 z 0.42 fm2. For this case the fusion is sensitive to the tail of the
wave function in momentum space. Since the wave function of a. deuteron is ex-
tremely non—Gaussian with a long tail in coordinate space, choosing v to reproduce
the deuteron’s r.m.s. charge radius would result in a grossly underpredicted cross
section. For the fusion of heavier ions, the incoming nuclei are moving at a slower
velocity and their momenta per nucleon are similar to characteristic momentum scales
of the wave functions.

The oscillator parameter 1) can be best obtained by matching the harmonic os-
cillator type deuteron wave function used here to its experimentally known behavior
[94]. The choice of this parameter between 180 and 220 fm would lead to the values

of 46 in the range of 0.06 to 0.48 fmz.

65

4.5.2 The reaction 3He + 3He —> 6Li + 7r+

As a next step, we apply the model to the experimentally studied pionic fusion reac-
tion 3He + 3He —> 6Li + 7r+, where even first excited states of the 6Li nucleus have
been resolved [81]. This reaction involves heavier nuclei so that the process of pion
rescattering becomes less important as discussed above. The polynomials P and Q for
Eq. (4.20) can be constructed in a direct way considering the shell model structure
of all nuclei involved in the reaction. The ground 1+ and first excited 3+ states of 6Li
were constructed within the [03/2 j-subshell. In Fig. 4.3, the total cross section for this
reaction is calculated for fusion into the ground state (left panel) and the first excited
state (right panel). The contributions of the s-wave and p—wave to the cross section
are plotted together. We choose a value v 2 118.91 MeV/c for 6Li as it corresponds
to the oscillator frequency of 15.06 MeV, the parameter of the MK3W model [95].
The initial parameter w 2 112.7 MeV/c was chosen by assuming the r.m.s. size 2.14
fm of 3He. In Fig. 4.4 we show the differential cross section for this fusion reaction
going into the ground state of 6Li (solid line) and the first excited state (dashed line).
The beam energy is assumed to be fixed so that the corresponding absolute values of
the pion momentum are 96 and 90 MeV/c, respectively.

Comparison with the experiment [81] in which pionic fusion resolves the few lowest
levels of 6Li shows that we obtain a reasonable ratio of the cross sections. However,
we underpredict the magnitude by approximately 40%, compared to the estimated
experimental value of 111 i 11 nb for the ground state transition. We note that the
result is extremely sensitive to parameters of the shell model, and their choice in the
harmonic oscillator approximation is quite uncertain for light nuclei. For example, a
variation of the final oscillator frequency within 10% range of the used value would
lead to values of the cross section between about 20 and 140 nb. Using more realistic

non-Gaussian wave functions might significantly improve the model. We might also

66

 

120.0
100.0 *-

80.0 P

G (nb)

60.0 —
40.0 -

20.0 -

 

 

 

 

 

Figure 4.3: Reaction cross sections for 3He + 3He —> 6Li + 7r+. The left panel shows
the transition to the ground state and the right panel to the first excited state of 6Li
at 2.18 MeV. The solid lines represent the total cross section, dashed and dotted lines

are p and s-waves, respectively.

be underestimating the cross section due to inherent limitations of the approach. For
instance, we do not consider a gradual change of the nuclear mean field in the process

of fusion substituting it with the sudden approximation.

4.5.3 The reaction 12C + 12C —> 24Mg + no

Here we apply our approach to the cross section of the 12C + 12C ——> 24Mg + 7T0 reac-
tion. This process, along with its isospin analog 12C + 12C —> 24Na + 7r+, represents
those few heavy ion pionic fusion reactions for which experimental data. exist [80].
The application of the developed theory does not present a great difficulty except for
the fact that the cross section is quite dependent on the structure of the initial and
final states of the interacting nuclei. Within the harmonic oscillator picture we have

approximately 3 X 108 different combinations of interacting states that correspond

67

 

 

25 " T 50

20

_L
01

—L
O

dO/dQCM (nb/sr)

 

 

 

 

 

 

 

 

 

1 1 1 1 O 1 1 1 l
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

cosz(6CM‘)

Figure 4.4: Differential cross section of the reaction 3He + 3He —> 6Li + 7r+. On
the left panel the solid line represents the transition to the ground state of 6Li and
the dashed line to the first excited state; the corresponding absolute values of pion
momentum are 96 and 90 MeV/c, respectively. The right panel displays the exper-
imentally observed values [2] of the differential cross section of the transition to the

ground state (squares) and to the first excited state (circles) of 6Li.

68

 

60.0 1 1 . 1 1 r 1 1

40.0 ~

6 (pb)

20.0 -

 

 

 

0.0 l 1 l 1 l 1 l 1
0.0 20.0 40.0 60.0 80.0 100.0
k1: (MeV)

 

Figure 4.5: The reaction cross section for 12C + 12C —> 24Mg + n0 with oscillator

parameters 12 2 104 MeV/c and w 2 119 MeV/c as a function of pion momentum.

to the same energy. Angular momentum and isospin conservation constraints reduce
this number by several orders of magnitude. Additional shell model interactions have
to be introduced to build up a realistic nuclear state for each of the nuclei and reduce
this large number of states, that are degenerate in our model, to the ones of interest.
Based on this argument we will present here the Monte-Carlo averaged cross section,
where we average over random Cartesian states. In the following Fig. 4.5 we display
the total reaction cross section as a function of pion momentum. We use here the
oscillator parameters 2) 2 104 MeV/c and w 2 119 MeV/c which are estimated by
various theoretical models [96, 97]. The experimentally estimated cross section for
this reaction is 208 :l: 38 pb which was observed for pion momentum of 41 MeV/c
[80]. In this example we again underestimate the cross section. To see the sensi—
tivity of our results we present in Fig. 4.6 the dependence of the cross section on

oscillator parameters for pion energy at about 6 MeV (momentum 41 MeV/c). This

69

    
 
  

 
 
   

D
....~
W» ---
~90” ...:~:-'-'—"I::

  

Figure 4.6: The total cross section of 12C + 12C —> 24Mg + 7r0 as a function of the
model parameters 11 and w. The calculation is done for a pion momentum 41 MeV/c

which corresponds to the total energy of about 6 MeV above threshold.

figure indicates that a reasonable variation of parameters could cause a change in the
answer by an order of magnitude. We emphasize again that in our calculations we
did not project the participating nuclei onto appropriate shell-model states. Such a
projection would require additional nuclear structure input. Given that the existing
experimental data do not clearly resolve the structure of the final state this seems
sufficient. As a conclusion, within all the limitations discussed above, the agreement
between the introduced theory and the experimental results of this rare process seems

to be remarkable.

4.5.4 Calculations for heavy nuclei

In this section we apply the low—momentum approximation for the cross section de-
scribed by Eq. (4.24) to several reactions, with the goal of understanding the general

dependence with respect to the mass of the incoming nuclei. In order to calculate the

70

cross section, one needs the harmonic oscillator parameter 1) which can be estimated

from the experimentally determined r.m.s. radii of the nuclei[99],

1 1 1 3
Tim... : Z 2; <T12> = :4' Xi: 52‘ (a1 + g) - (4.33)

In order to calculate the cross section, one needs to know the incoming energy of
the nuclei as well as the energy of the outgoing pion. Calculations of the cross sections
were performed for incoming nuclei 9Be, 12C, 16O and 20Ne with correSponding fusion
products 18O, 24Mg, 328 and 40Ca in the limit of low pion momentum. In this limit

the cross section is proportional to the cube of the pion momentum,
a = 6(33/mi). (4.34)

Values of 6 are displayed as a function of the mass number of the incoming nuclei
in Figure 4.7. The shell model configurations are again randomly chosen from the
available set of Cartesian states that conserve isospin and parity. Average values
are represented by filled diamonds while the states with the highest and lowest cross
sections are represented by the boundaries of the error bars. The large error bars
demonstrate the wide fluctuation in strengths for individual states. However, despite
the fluctuations, it is clear that the overall trend is of a decreasing cross section with
increasing mass.

Also shown in Figure 4.7 are experimental measurements represented by open
circles for the pp, 3He3He and 12C12C cases discussed previously. The corresponding
calculations, which were performed for the experimentally measured pion momenta
rather than in the low-momentum limit are also displayed with closed circles. One sees
that the cross sections fall by several orders of magnitude, but the measurements are
still feasible throughout the wide range of masses. Calculations could be performed
for heavier nuclei, but for larger masses the Coulomb barrier becomes important, and

shuts off the possibility of fusion for masses larger than 20.

71

 

 

 

 

1o . I . T . . . .
o
o
10"5 ~ 9 —
«iii/10‘10 — —
1b
10‘15 l —
10-20 J 1 l r 1
o 5 1o 15 20

Figure 4.7: The general behavior of the pionic fusion cross section A + A ——> 2A +
7T versus the mass number of initial nucleus A. The plotted value (3 is related to
a total cross section as a = r} (Ir/mm)? Calculations in the low-momentum limit
(filled diamonds) show that cross sections fall by several orders of magnitude in this
mass range, but remain in the picobarn region for nuclei as large as oxygen. The
highest and the lowest cross section found within the shell model configurations are
represented by error bars. Experimental measurements are displayed (open circles)
and compared to calculations (filled circles) which were performed for the finite pion

momenta corresponding to the experiments.

72

4.6 Conclusions

Near threshold meson production represents a unique area of heavy ion reactions.
In this area the reactions underline the pronounced features of quantum many-body
physics. Most theoretical approaches to understanding and predicting these phenom-
ena lose their validity in such an extreme regime. In this paper we have proposed a
simple model to study the processes of deep subthreshold pion production. The pi-
onic fusion cross section was obtained in a Born approximation with respect to pion
production and in the sudden approximation for the nuclear rearrangement. The
participating nuclei were described by the harmonic oscillator shell model in moving
oscillator potentials. The advantage of the method is that it allows one to incorporate
energy, momentum, spin and isospin conservation laws precisely and respect the Pauli
principle at all steps of the calculation. Further aspects of nuclear structure could be
additionally taken into account. At threshold energies these constraints pose the most
powerful restriction on the reaction and cannot be ignored as is done in statistical
and kinetic models which are reasonable at higher energy. The obvious disadvantage
of the model is that the sudden approximation does not consider the slow changes
of the nuclear mean field in the process of interaction. For the future developments
it seems feasible to incorporate the time dependence and solve the equations for the
evolution of the nuclear mean field.

The nearly analytical form of overlaps greatly simplified the calculations for this
study. We used a spherically symmetric nuclear mean field but in some cases this sym-
metry prohibits the transition and this would require a consideration of deformations,
i.e., different oscillator parameters for different directions. The above mentioned limi-
tations are reflected by the difficulty in determining the parameters of the model, and
lead to about an order of magnitude ambiguity in the results for s-d shell nuclei. More

realistic single-particle wave functions could be incorporated into the model. Some of

73

the exponential factors in Eq. (4.18) arise directly from the Fourier transformation
of the Gaussian tails in harmonic oscillator wave functions and could be substituted
with modified factors that would reflect a more realistic behavior.

We would like to stress here again that pionic fusion is a very rare process pre-
senting a tiny fraction of the total cross section. The agreement that was observed
between calculations and experimental data for the cross sections ranging from 10"4
to 10‘9 barns is remarkable. Within the limits of the low pion momentum approxi-
mation in the class of the reactions A + A ——> 2A + 7r, we were able to obtain a general
formula, Eq. (4.24), for the cross sections. The proposed techniques can certainly
be applied in the same manner to other pion production reactions. The processes of

electrofission [100] present another interesting avenue to exercise this technique.

4.7 Appendix

4.7 .1 Harmonic oscillator wave functions and overlaps

Work with harmonic oscillator wave functions often involves integrations of the expres-
sions constructed of polynomials and Gaussian weights. Thus, the following integral

is useful (a > 0),

+00 2 ‘ 7T 2 ”Lb
In —a:1: +zbzc d = _ —b /4a —n/2T 4
/00 :r e a: (/ a6 a n (2V5) , ( 35)

where Tn(.r) is a sum arising from the binomial expansion,

 

 

" nl(j—1)!l _-
Tn(a:) : , _ . 3:” J . (4.36
j=£§4...3!(n _ Jlm/Z )

This expression can be used for the evaluation of any integral encountered in this

work. There are two important limiting cases for the sum T ”(33), a: —> 0 and :1: >> 1:

95—}; if n is even
lim Tn(:c) = lim Tn(:z:) = 2"”2 :13" . (4.37)
“H" 0 if n is odd H”

74

The Gaussian-Fourier integration of Eq. (4.35) is a transformation on the space of

polynomials defined on the basis
a?” —> [513”](19) = T7100) (4-38)

The following two-dimensional integrals often appear in our calculations,

——1—71/2(n —1)!l if n is even

2 2 7m
f/(a: — fine-“(x +y )da: dy = , (4.39)
0 if n is odd

n/2 ,-
_ n -a($2+y2) ip(:v-y) _ 1(2) —P2/2‘1 ( LP )
a: e e dcrd — e Tn . 4.40
/ /< y) y a a 7.. < >

The basic block of the calculations is the overlap of two one-dimensional harmonic

 

oscillator wave functions with different oscillator parameters, shifted locations of the
centers and a possible additional factor 6”” that enters the single—particle interaction
integral from Eq.(4.5). This type of integral, the generalized Debye-Waller factor, can

be written in a factorized form:

<ffliT)vle_ikx|(a;r')w) = 771/2 x

—k2 (r — 7")2v2w2 z'k(r'u2 + r’wz)
_ _ PC. I'_ ,k;, , 4.41
p (2(2)2 + 1112) 2(2)2 + 2112) (’02 + w?) B ((r T) U 112) ( )

 

 

where 77 is given by Eq. (4.19) and P,,m(r, k; v, w) is a dimensionless polynomial of
'r and k of the highest power 7?. + m with coefficients dependent on w and v. The

following are examples of these polynomials for the smallest values of n and m:

P000", k;'v,w) = 1,
\f2— (ik+rv2) w

212-1-1122
2mm (112 +1112 — (k — 2'7"?)2) (k +irw2))

(v2 + w2)2

P01(T7kivaw) = P10(-7", k;w,v) : —

 

P110", k; 22,10) :

 

The technique of obtaining these expressions is simple though tedious. An important
situation k = 0 would correspond to the overlap of two wave functions without pion

production, in this case we will not write k as an argument. It can be shown that [98]

kzz’lzj ilj! (—1)J_lvj‘lwi”k (7‘7)

kll! (i—k)!(j—l)! \/§

 

z'+j—k—l
P,- -(7"; 21,112) = ) Pk1,(0; 12,111) (4.42)

k+l=0,2,4...

75

and
k!
Pkl(0ivaw) = V— l! —73(/:+i€)//22 (77) , (4-43)

with P5 being the associated Legendre polynomials. A bit simpler case is

1U! jmin(i,j) k i+j_2k 2k

 

 

Any three-dimensional overlap is reduced to the one-dimensional form of Eq. (4.41)

in a direct way
((16) R)l(C—\f, r» : ((flxaR ”Karin?” - (4°45)

Similarly we introduce

<<§»r>le“’k*l<&,r'>> = 773/2 x

 

 

—k2 (r — r ’)2v2w2 —z'k(rv2 + r’w2)
-———— — -- Pu '—- k' 4.46
exp (2012 +w2) 2(7)? + 202) (v2 + M) [30((1’ r)’ ’v’w) ( l
where
P~&((r' — r), k, 21,211) = P530150" —7‘x,k$;v, w) . (4.47)
$21,2,3

4.7 .2 Calculational details of the A + A a 2A + 7r+ reaction

Overlaps of many-body nucleon wave functions can be expressed in our approximation

by a determinant of single-particle overlaps:

«BURHWMI‘» «BuRlanfl'»
<(B,,...,/§,;R)|(a,,...;r)(...;r')) =

 

 

((5n;R)|(551;r)> ((3n;R)l((Yn;r’)>

Eq. (4.46) allows one to take identical exponential multipliers in each row outside
the determinant as a common factor in all calculations leaving only the matrix of
polynomials P352 to be evaluated. A simple example of this is the calculation of the

normalization:

((521, 072 . . .a’A; r),,|(c'x'1, 522 . . . 52A; r'),> = ||P(r — r'; v, v)||e-A<r-r’>2v2/4 ape-r”.

(4.49)

76

In this expression ||P(r—r’; v, 2)) H is a determinant of a matrix with the entries P67, 5]..
As discussed in Sect 4.2, this overlap is equal to that of the CM wave functions of
two harmonic oscillators located at r and r’. For a nucleon system in the lowest
state (in terms of harmonic oscillator shell excitations), the CM wave function is
the harmonic oscillator wave function of the ground state |(0,r)<). We obtain an

interesting mathematical fact
||P(r—r';v,v)|| =P00(r—r';§,c) : 1. (4.50)

Comparison of the exponents in Eq. (4.49) and Eq. (4.46) gives the value of the
oscillator parameter for the center-of-mass oscillation as c : m/A.

With the same strategy, one can approach the calculation of the reaction A+A —>
2A+7r+ extracting all exponential factors. Corresponding values of the overlaps FmG)

and Hm may be rewritten, defining new polynomials fn, g, and h"):

. _ {exp (-7va [Z(R- 1‘)2 + (Z -1)(R— 1")2l /4) fn n S Z,
Fn : 773(2Z 1)/2 X
exp (—77sz [(Z —1)(R — r)2 + Z(R — r’)2] /4) fn n > Z;
(4.51)

G. = 773”)” exp (-N77vw [(R — W + (R — 1")2l/4l 95’

(R — r)2v2’w2 + k2 + Qik - (R112 + r102)
2(2)2 + 102)

 

Hnlan/Zexp <— ) hn) (ifn>Z, rfir').

It is useful to notice here that all the polynomials are functions of distances be-
tween the nuclei (r — R) and (r’ — R) that we will denote as x and y respectively.
Considering integration in Eq. (4.16) over variables x, y and R we observe from Eq.

(4.51) that the oscillating phase has the form

 

—z'k - (R1)2 + r102) , ,
exp ( (“2+7“) +zp(r—r') —zpf.R>
ik - x2112

v2 + 102

=exp<ip~(x—y)— —z'(k+pf)-R>,

and integration over R gives a momentum preserving 6-function that requires k =

—pf. For convenience we split the sum in Eq. (4.16) over 72 3 Z and Z < n g 2Z

77

and substitute F, G and H from Eq. (4.51)

 

1 V 1 2 2 2 2 2
(FIHiI) : / _773A e—Avwn(z +31 )/4 e—k /2(v +w )
Nix/W \/2wV 2m

2 6—ik-xnw/2v Z: (—1)i+jfzgjhzj + e—ik-ynw/Z’U Z (—1)i+jfigjhij €—ip~(x-—y)d3$d3y .
252,3“ i>Z, j

(4.52)

The terms 2 figjhw- are again some polynomials of x and y proportional to [kl and

containing parameters 2) and w. The final integration can be performed with the help

of Eq. (4.35), the corresponding parameters a and b being
azAnvw/4 , bzip—knw/ZU . (4.53)

As a result, we arrive at the formula (4.17) with polynomials

1 - 1 ,p+k77w/2v z'p
P k = — —1 2+] 7; 'hi' — a a
( ,p) lkl Lg; ) f9] 3] ( z x/Anvw «Anvw

—-z'p z' p — knw/Qv (4 54)
y/AU’U’LU ’ «Anvw ’ '

where the first argument is the transformation of elements of vector x and the second

 

 

 

1 . .
QUCaP) = m [Z (_1)z+]figjhij

i>Z,j

that of vector y. From here it is also seen that if before transformation there existed a

symmetry between x and y, i.e. the nuclei were in an identical state, then P (k, —p) =

iQ(k, 1))-

4.7.3 Toward a complete analytical answer, the reaction A +

A——>2A+7r0

As it was pointed out in the main text, the amplitude of the pionic process is ap-
proximately proportional to the amplitude of the fusion reaction. One can study the
properties of the determinants arising in a fusion reaction in a quite general way,

separately considering the four types of particles distinguished by spin and isospin in

78

the reaction of fusion of the type A + A —+ 2A. This leads to the following form of a

single-particle overlap matrix

 

' ((31;R)|(591;r)> ' .. ---<(51;R)|(&A;r’)>
2A< 5 5 . (4.55)
((54; R)|(071; 1)) ' '° ”4(3):; R)|(&A; 1"»
first Anucleons second A nucleons

Without loss of generality, R can be set to zero. A second important feature
is that in the nuclei under consideration all inner shells are filled. Therefore, the
resulting determinant is a function of the nucleon number A and extra parameters
arising from different ways to distribute the particles in the outer shells.

It is interesting to present the exact result for the onesdimensional case where the
problem is uniquely defined. We consider two oscillators with single-particle states
from 0 till A -— 1 overlapping with one larger oscillator with occupied states from 0

up to 2A — 1, see Eqs. (4.41) and (4.42),

 

 

 

 

P00020710) POA—lnyflU)

P2A—10(37;’U,w) P2A—1A—1(yivaw)

_ (—1)A(w(a: — 9))A “(214)! ”(M-1M : E(wm _ y))q ”Q (4 56)
(A—1)A A M ‘ 7 '
2 2 VA! (Hz-:1 2., ) '7
f—llA vA! (A—1)A/2 (A (21)!)

7 A) = 2 . . 4.57
( (2A)! £11 2! ( )

The result is just a single term which depends only on the distance between the two
initial oscillator locations raised to the power equal to the difference in total number
of quanta between initial and final systems, (1 = A2. The term 77 = 222112/(112 + 1112)
comes in the power of total number of quanta in the final nucleus, Q = (2A — 1)A.
This remains true only for Fermi systems in the ground state, i.e. if there are no
gaps in the harmonic oscillator single-particle level occupation. The situation for a

three-dimensional oscillator is similar. The required polynomial is still given by one

79

term that has a form of the product

1
30732: _ yx)(h:($y _ yyyzy (513; _ yz)(Izw(I:z:+qi/+Qz 779; a (4.58)

where integers qx, qy and qz are differences of the number of quanta between the
final and initial systems in the .73, y and .2 directions, respectively. A specific three-
dimensional complication arises from the following aspect. The lowest energy state is,
in general, degenerate as for non-magic nuclei one has the freedom of placing several
particles into (n + 1)(n + 2) / 2 degenerate levels of the n-th shell. The numerical
parameter 7 depends in this case also on the way the particles are placed in the
outer shell of each nucleus. The harmonic oscillator symmetries in the problem often
prohibit the transition.

The polynomials in Eq. (4.52) acquire a form of a product of four components,
each of the form of Eq. (4.58) for each type of nucleons, times the sum of terms
(5' - k) acting on every pair of interacting nucleon species. Using the integrals from
Eq. (4.39) and writing the action of (5 - k) between initial and final spin parts of the
wave function as a matrix element If! we arrive at the expression for the polynomial

in Eq. (4.22)

1 2 , 2 2 (Q:r+(Iy+Qz)/2
0),.) 2 _ (M) 779! x

PQIJII/vQZ (k : 7 AU2

((44 —1)!!(q, —1)!!2”§-qu (rm/Aim» M . (4.59)

In the above expression we have redefined 7 as a product of 7’s for all four types of

 

nucleons.

80

Chapter 5

Exploring the nuclear pion
dispersion relation through the

anomalous coupling 7 ——> #770

5. 1 Introduction

Pions are expected to interact strongly with. nuclear matter due to the mixture of
pionic and delta-hole states. The mixture is especially strong when the momentum of
the pionic mode approaches 300 MeV/c. In this momentum range the energy of the

1/2, crosses the energy of the delta-hole mode, (162 + Mg”? —

pionic mode, (k2 + mi)
M N, and the energy of the pionic branch should lower due to level repulsion. This
topic drew a great deal of attention in the late 19705 when it was thought that the
pionic mode might be pushed below zero energy at sufficient nuclear density, which
would result in pion condensation. An extensive review of pionic excitations and
condensation in nuclear matter was published by Migdal [101], and recently, kaon

condensation has been the subject of several investigations [102]. Since such novel

behavior requires higher densities, the discussions are often in the context of neutron

81

 

 

stars [103], although relativistic heavy ion collisions were also once proposed as a
mechanism for producing sufficient density for condensation.

Despite theoretical efforts in this area, experimental evidence of large in-medium
corrections to the pion dispersion relation is sparse. The most promising informa-
tion is from recent charge-exchange measurements from light ions scattered off heavy
nuclei. The cross-sections appear enhanced for channels where pion-exchange is ex-
pected to dominate. This enhancement is consistent with the lowering of the energy
of the pionic branch, which reduces the amount by which the exchanged pion is off-
shell[104]. The interpretation of this experiment suffers only from the fact that the
probe is hadronic and must traverse the surface of the heavy nucleus before interact-
ing. Heavy-ion collisions, which can produce matter at three to four times nuclear
density, were expected to create environments where the dispersion relation was ex-
tremely distorted. However, experimental signals, such as measuring pion spectra[105]
or dilepton pairs[106, 107], of pionic properties in the interior of these regions proved
difficult to extract.

In this work we study the inelastic scattering of photons off heavy nuclei as a
means to create the pionic excitations within the medium. The anomalous coupling
of a neutral pion to two photons, which is responsible for the decay of the no, can
be used to excite a pionic mode in a heavy nucleus. The photon provides a clean
probe for entering and exiting the interior of the nucleus. Unfortunately, this has
the same drawback as the charge—exchange experiment mentioned in the previous
paragraph —— only space—like excitations can be investigated. However, in-medium
effects are expected to lower the energy of the pionic branch, perhaps to the point
that it crosses into the space-like region, |/;,,| > w”. Thus, the reaction 7 —+ *y’ + 7r0,
which is kinematically forbidden in free space, is allowed in the nuclear medium.

Furthermore, the measurements of such a branch would provide direct evidence of

82

in-medium correction to the pion dispersion relation.

This chapter is organized into five parts. The next section briefly reviews the
in-medium corrections to the pion dispersion relation. The following section shows
the contribution to the cross section from the anomalous coupling to the 7r“, outlines
the procedure one must follow to map out the pion dispersion relation, and presents
a discussion of how gauge-invariance constrains the cross—section to disappear at the
space-like-to—time-like boundary. Due to this constraint, the contribution from the
anomalous coupling is reduced in the region of interest, which allows background
processes to pose a major problem. The bulk of the work is comprised of estimates for
background processes which are presented in section 5.4, where non-pionic delta‘hole
channels are shown to provide most of the background. Given the possibility of using
free-electron lasers to supply high-energy photons with 100% polarization, we also
discuss the use of polarization measurements to eliminate background. We conclude
that the only chance to defeat the background is to perform a difficult exclusive

measurement, which would require a more sophisticated theoretical treatment.

5.2 The pion dispersion relation

The delta-hole and nucleon-hole both contribute to the in-medium correction to the
self-energy of the pion in nuclear matter. The coupling to the nucleon-hole raises
the energy of the pionic mode, while the coupling to the delta-hole state significantly
lowers the energy, especially for momenta approaching 300 MeV/c. Numerous the-
oretical works in this direction [105] have focused on the correction to the pionic
mode in perturbative pictures[108, 101]. Sophisticated treatments of the pion self en-
ergy were performed by Xia, Siemens and Soyeur[109], and Korpa and Malfliet[110].
There, self—consistent corrections to the delta’s self energy and width were included.

In this section we present a simple model to illustrate the modifications of the

83

 

kn
H(k,,) : 1—9’:(k«f/k,2 7 kan) 29+ 0

19+k7r

Figure 5.1: Delta-hole contributions to the pion self-energy.

pion dispersion relation in nuclear matter. We will forgo such lengthy calculations,
as we wish to discuss how to experimentally observe the resulting dispersion relation,
rather than how to better calculate in-medium corrections.

Self energy corrections can be written diagrammatically as shown in Figure 5.1.
Vertex renormalization due to the effective four-point interaction between deltas and
nucleons is incorporated through the phenomenological constant g’. Nucleon-hole
contributions can also be considered but are much less important for E7, approaching
300 MeV/c.

Assuming an interaction Lagrangian density of the form,

am 2 meafl + h.c.) , (5.1)

we obtain the self-energy correction for the pion propagator shown in Figure 5.1.

The delta propagator is assumed to have a Rarita—Schwinger form [111] for all
the calculations in this section, in which the width is inserted by substitution 114A —>
MA — ilk/2 [112].

I/ 2 II/ 1 l/ 2 I, l/
GZ(G)=—— -g" +-7“7 + —2q’q -

g _ MA 3 3 M "7” — WI") (5-2)

M ((1
The nucleon propagator in the presence of nucleons filled to the Fermi momentum
pf has a correction given by:

2'

fl—MN‘l'fE

 

GNU?) = + 24(1) — MN)6(p(2) — E2(P))@(Pf — lfill (5-3)

84

 

350 4

300

250

E,c (MeV)

 

 

 

 

O A A A L 100 i A I i 200
k,[ (MeV/c)

Figure 5.2: The pion dispersion relation is shown for both the vacuum case (solid
line) and with an effective coupling ngA=2.O and g’:0.8 (dashed line) in Figure 5.1.

The thin straight line shows the space—like-to-time—like boundary.

With the obtained self-energy term a dispersion relation could be inferred by find-
ing a pole of the corresponding “dressed” pion propagator. Using these equations
and assuming the coupling strengths, gmN = 2.0 and g’ = 0.8, we obtain the pion
dispersion relation presented in Figure 5.2.

The value of the effective in-medium coupling constant is not precisely known
although vertex corrections have been extensively studied. Many parameters used
in more sophisticated analyses[109, 110] are still rather uncertain. For instance the
delta’s mass and width as well as the nucleon’s mass in matter are somewhat uncer-

tain, as are the coupling constants gflNA and g’.

85

7.- (wi. ’61) 7f (wf, (ff)

Figure 5.3: Pion production diagram.

5.3 Pion production through the anomalous cou-

pling

We consider production of a neutral pion through an effective photon “decay” inside
nuclear matter as shown in Figure 5.3.

This process can only occur if the resulting pion is space-like,
k: = (ki — kf)2 = —2kz’ ° kif = —2w,-wf(1 — COS 0) S 0, (5.4)

where k,- and kf are the incoming and outgoing momenta of the photon, and the
outgoing photon leaves at an angle 6. An interaction that has a gauge invariant and
parity conserving form is £77,, 2 01/ (47r f,)F “VF WW0, where a is the electromagnetic
fine coupling constant, and the pion decay constant is f7r = 93 MeV. The transi-
tion amplitude THf = (f,7r|£,,,,]i) can be expressed in terms of the momenta and

polarizations of the incoming and outgoing photons.

 

a C!
= 27:3.” (1‘53. — 4.2) sma- — a».

where k,- and [cf are the initial and final momenta of the photon and x,- and Xf

are the corresponding polarizations. In the last line of Eq. (5.5) the transition

86

amplitude is written in terms of the momentum of the pion, and the incoming and
outgoing polarization angles, a, and of, as measured relative to the reaction plane.
The principle difficulty in obtaining the goals of this study comes from the fact that
the amplitude vanishes at the space-like-to—time-like boundary where the square of a
pion four-momentum is zero, which is precisely the kinematic region of interest. This
is a direct consequence of gauge symmetries involved in coupling two photons to a
pseudoscalar. This result may be problematic since we investigate the area that is
close to the space-like-to-time-like boundary, and therefore the cross sections obtained
are quite small. The rate of this “decay” can be expressed in terms of the matrix

element 7' [113]:

1 d313, dBk,
2w,- 2wf(27r)3 2w7,(27r)3 '

 

 

dB 2 (27f)464(l€f — p7; — ki)]7',j_,f[2 (5.6)

We will express our answer in terms of the energy and momentum transfer, w7r :-
w, — w; and If, E If, — 13f respectively. Substituting a Lorentzian form in place of the
energy preserving 6 -function, allows the incorporation of a finite width to the pionic

state
1 2w2f‘
7r (w2 — E2)2 + w2F2 '

 

6(w — E) —> (5.7)

In general the complicated form of the in-medium pion self-energy makes the above
parameterization somewhat arbitrary. Nevertheless Eq. (5.7) follows from the choice
of —w,,I‘ being equal to the imaginary part of the self-energy. For our numerical
calculations presented below we used a fixed value of the pion width F = 50 MeV,
which in the region of interest agrees well with the calculations presented in Sec. 5.2
and with the results of more sophisticated calculations [109, 110]. The decay rate

into a pionic mode of energy w7r and momentum It, can be expressed as:

0312 _ a2 |1’,|(E§,—wg)2
dual/2,.) (2702f? 2w?

wfif‘
a); — Efr)2 + (.0317? '

 

(5.8)

 

sin2(a, — af)(

To estimate the pion production cross section in the physical reaction of inelastic

87

 

 

 

 

10 ~
”> i
<1)
‘2 1
..o
3‘ 1
=2? 5 -
I:
8
E
b
..O x
// \
F // \
// \ 1
,,,,, \
.......... \\
O -_-.—.:". ------ 1- """" :- 1 ‘5)
O 100 200 300

Figure 5.4: Neutral pion production through the anomalous coupling 7 —> 7’ + 7r0.
The on-shell energy is 225 MeV, the momentum transfer [k7,] is 275 MeV/c, and the
pion’s width is 50 MeV. Cross sections are shown for three incoming photon energies:

500 MeV (solid line), 1.0 GeV (long dashes) and 2.0 GeV (short dashes).

photon scattering from a heavy nucleus one must multiply the result in Eq. (5.8) by
the nuclear volume. Examples of cross sections for the *y —> ’y’ + 7r° process calculated
for a photon scattering off a Pb nucleus for different incoming photon energies are
shown in Figure 5.4. In the calculations the on—shell energy of the pion is assumed
to be 225 MeV and momentum transfer [Ex] is fixed at 275 MeV/c. Figure 5.5 shows
the dependence of the peak location and height with respect to the on-shell energy.
If the on—shell energy is not more then 25 MeV less than [137,], the peak is probably
too small to be observed.

The fact that the cross section is inversely proportional to the incoming photon
energy suggests the use of lower energies for a greater signal. Lower energies also

allow a more confident prediction of the background as very massive and not well

88

 

 

 

 

.4
O
I

m
T r
n

4:.
\
/

do/dmndkn (ub/GeVz)

N
\

 

 

 

O 100 200 300

Figure 5.5: Neutral pion production through the anomalous coupling 7 —> 7’ + 7r0.
On-shell energies are 225 MeV (solid line), 250 MeV (long dashes) and 275 MeV
(short dashes). The momentum transfer [k7,] is 275 MeV/c, and the pion width is 50

MeV. The incoming photon has an energy of 500 MeV.

89

understood nucleonic resonances do not contribute. Incoming photons with energies
between 400 and 600 MeV are satisfactory for our purposes given the above consid-
erations and the experimental ease with which they can be created. Lower-energy
photons would be difficult to deal with since the final photon could be confused with
those from nuclear processes such as giant-dipole decays.

For a 5 mm (a radiation length) lead target, one would need on the order of 1014
photons to investigate the peak in the region of interest. [This estimate was ascertained
by requiring 100,000 final—state photons to correspond to pion momenta between 225
and 275 MeV and pionic energies within 75 MeV of the momentum. This number
of photons is within the realm of current experimental constraints, although the
elimination of background to be discussed in the next section will push the viability
of these measurements. The role of the width is important, as for vanishing widths the
shape of the differential cross section in Eq. (5.6) becomes a sharp spike which would
be more easily observed. A confident calculation of the width is not trivial, since
the mixture of the pion and the delta-hole might represent a significant contribution.
The off-shell behavior of the W077’ vertex superimposes an additional effect. From the
calculations in Ref. [114] it follows that in the the region of interest k3, = 42?,— MN]? >
—0.03 GeV2 , the reduction of the coupling strength is smooth and is on the level of

one percent, which is negligible.

5.4 Calculating background processes

We consider two processes that might overwhelm the *y —> *y’ + 7rO reaction, namely
ordinary Compton scattering, that could also proceed through an intermediate A
resonance, see Figure 5.6, and reactions that produce a A-hole (which decays into
7rN) as a final state, see Figure 5.8. These reactions are the same order in a as

y —> 7’ + no. One might expect these background processes to be smaller than the

90

Pi Pf Pi P!

P P
71' 7f 71‘ 7!
Pi 1’! Pi Pf
A A
7i 7] ”71' '7!

Figure 5.6: Feynman diagrams for the background Compton—like processes 7 + N —>

7’+N’.

simple 7n) production process since they are further off-shell in the kinematic region
we are exploring. However, even though 7 —> 7’ + 770 might be nearly on—shell, a
gauge constraint forces the matrix element to zero at the k3, = 0 boundary and allows
other channels to compete. We also report on our investigation of using polariza-
tion measurements to project the signal from the background. Unfortunately, our
estimate of the background processes is done without the benefit of an experimental
measurement of relevant processes in vacuum, e.g. 7 + p ——> p’ + 7’ + 7r”. Certainly
such measurements are possible and would greatly increase the confidence with which
we present the background.

First, we discuss our estimate of normal Compton scattering as illustrated in
Figure 5.6. We consider the intermediate state to be either a delta or a proton. The
diagrams with the delta as an intermediate state provide the dominant contribution.
The photon is assumed to interact with the baryon via the following couplings [115,

116,117[

5m}? : BTW—”MAMA

91

G — G _
DYNA = e —1—A“*y“751/JFW + -—:8”A”751/2FW + h.c. (5.9)

We observed that the contribution from the second term proportional to 02 is small,
and we neglected it in our analyses.The value of G 1 = 2.63 has been determined from
experiments[118, 119]. The regular Compton process was also small, and thus did
not warrant including more sophisticated coupling, e.g. through magnetic moments.

The result for the cross section is shown in Figure 5.7. Again, we have assumed
that the photon lost a momentum var] = 275 MeV/c. The form would be a sharp
peak at low energy if it were not for our replacing the final delta function in the cross-
section by a Lorentzian, giving the nucleon an effective width of 25 MeV. Since an
on-shell nucleon with 275 MeV/c of momentum has only 50 MeV of energy, there is
little contribution in the kinematic region of interest, where ca7r approaches [[37,]. Since
the kinematics were effectively smeared by the Fermi motion, changing the nucleon’s
width had little effect.

The primary contribution to the background derives from production of a delta-
hole in the final state. Since this is precisely the process that mixes with the pion due
to its kinematic proximity to the pionic mode, it is not surprising. Even though the
delta-hole should be 50 MeV in energy higher than the pionic branch of the dispersion
relation, its contribution is not constrained to go to zero when HEW] equals w”. This
lack of a constraint derives from the fact that a spin 3 / 2 delta and a spin 1 / 2 nucleon-
hole do not necessarily form a pseudo—vector, and couple to 8%, but can also couple
to J = 2. Furthermore, the delta will decay into a nucleon and a pion. If the pion
is charged, it can radiate photons readily since it is light and moves quickly. We find
that only by gating on the presence of the 710 and by requiring the target nucleus
to be left in it’s ground state, can one confidently translate the measurement into
information regarding the nuclear pion dispersion relation.

The diagrams used for calculating the contributions for a delta-hole being in the

92

 

O) on
I I

A
I

N
1 .

do/dwndkn (ub/GeVZ)

 

 

 

50 250

1&0
(x),t (MeV)

Figure 5.7: The contribution from 7 + N —> ”y’ + N’ assuming the outgoing nucleon
has a width of 25 MeV. This component is negligible for energy transfers greater than

200 MeV.

final state are shown in Figure 5.8. In order to maintain gauge invariance, the decay
of the delta into pions is included. This decay is also crucial as Bremsstrahlung off
the light charged pion is important. The coupling of the electromagnetic field to the
delta is accomplished via minimal substitution.

One might also consider a diagram where the photon decays into a second photon
and a pion, with the pion interacting and exciting the nucleon hole. However, this
excitation is part of the signal rather than the background as it represents a truly
pionic excitation that would disappear at the space-like-to—time—like boundary.

The Lagrangian for the delta that results in the Rarita—Schwinger form of the

propagator is:

£4 = "—301 {(i X9 — MAlgafi — (Vaiafl + 13.174) + 7012' X975 + Mammal A3 (5-10)

93

 

 

 

Figure 5.8: Matrix elements for calculating background processes where a pion, pho-
ton and excited nucleon are in the final state. These diagrams can be thought of as

processes where a delta-hole (which subsequently decays) is created.

94

The associated coupling of a delta to a photon, through minimal substitution, is:

£AA7 = 63—“ {9..47" — 9315 - 95%. + 107“14}N34u (5-11)

Due to the derivative nature of the 7TNA coupling shown in Eq. (5.1), minimal

substitution requires a four point coupling of the photon to the WNA vertex.

 

LWNA : iegWNA AuAudnr + h.c. (5.12)
m

71'

Minimal substitution from the interaction term in Eq. (5.9) also results in a four-point
coupling, which we will neglect since G2 is set to zero.

The 20 terms necessary for creating a nucleon, a photon and a pion in the final
state are shown in Figure 5.8. A fixed Width was used for the delta, which although
is not realistic, simplified self-consistency checks regarding gauge invariance and the
Ward-Takahashi identity. Since the delta-hole is not far off—shell, the answer should
not vary greatly by incorporating an energy and density dependent width.

Transition elements were calculated for specific linear polarizations, a,- and 01f of
the incoming and outgoing photons. The scattering plane of the photons defines the
angles. For our signal, 7 —> myy’, the dependence is proportional to sin2(a,- — af)
as shown in Eq. (5.5). Since the final states illustrated in Figure 5.8 are three-
body states with large widths from the delta decay, the final three particles could be
assumed to be on—shell. Numerical integrations were performed over all final-state
variables except for the photon polarizations and the energy and momentum lost by
the photon, a), and [EN].

The cross sections for the background are shown in Figure 5.9. The upper graph
shows the background for charged pions, while the lower graph presents the back-
ground for the case where a neutral pion is created. The creation of a charged pion
overwhelms the signal by more than an order of magnitude. This strength comes

from the radiation off the light, fast-moving, charged particle.

95

 

300 xwmrrefij.

 

 

1 I;—
200

NA P

> .

CD

CD 100

B .

:3.

v

8!:

w: 0 i

[=2

.54

"E 1
b 15.0 ” [l‘

I
"d I/ -
,/’/
r— I —<
10.0 ’z/

 

 

 

 

Figure 5.9: The cross section for the background process where a charged pion is
created is shown in the upper panel. By requiring the created pion to be neutral, the
background is greatly reduced as shown in the lower panel. Polarization projections
are shown for 01,; = 0, 01] = 0 (solid line),a,— = 0, a; = 7r/2 (short dashes), 01,- =

7r/2, af : 0 (long dashes), or,- = 71/2, of = 7r/2 (dot-dashes).

96

 

Polarization does not significantly ameliorate the background as shown in Figure
5.9. The signal is proportional to sin2(a,- -— of) which has the same shape as part of
the spin-2 delta-hole state.

By gating on neutral pions, one can reduce the background by an order of mag-
nitude as shown by comparing the upper and lower panels of Figure 5.9. Although
this background is now comparable to the signal, it is still possible that a charged
pion could be created and undergo a charge exchange with the medium, resulting in
a neutral pion.

The intractability of the background derives from the gauge invariance constraint
that requires the cross section to disappear at the space-like-to-time—like boundary
for pseudoscalar channels. Since the delta-hole state can couple in other ways than
pseudoscalar modes, the background becomes problematic. We briefly discuss the idea

of using an exclusive measurement to eliminate the background in the conclusion.

5.5 Conclusions

The subject of in-medium hadron masses has been historically inconclusive. The
pion’s in-medium properties are unique in that the dispersion relation may dip down
into the space-like region at normal nuclear density. This permits the consideration
of simple scattering experiments, such as the process proposed here, to investigate
pionic modes.

Unfortunately, background processes where a delta-hole is produced in the final
state with quantum numbers different than the pion overwhelms the pseudoscalar
mode of interest here. Even polarization measurements were not adequate to focus in
on the channel of interest. The only means to defeat such a problem is to eliminate
the delta-hole channel (that which does not have the quantum numbers of a pion)

from consideration. The delta-hole channel decays into a pion and a nucleon hole.

97

By requiring the nucleus to remain in it’s original ground state, one could in principle
eliminate the background. However, such a requirement would also eliminate much of
the signal as a pionic mode can also decay by exciting the nucleus. Before proposing
such a difficult exclusive measurement, one should perform an optical model calcula-
tion to realistically model the escape of the pion from the center of the nucleus. That
is beyond the scope of the current work, where our aim is to investigate the inclusive

measurement.

98

Chapter 6

Multiple pion production from an

oriented chiral condensate

6. 1 Introduction

Pions have always played an important role in heavy ion reactions. With their rela-
tively strong coupling to nuclei and very small mass, pions can be produced in large
quantities and can carry important detectable information about the state of nuclear
matter during the reaction. Large fluctuations in the ratio of charged to neutral pions
have been observed in cosmic ray experiments [120] and have recently been a subject
of many discussions. The major interest in this direction is motivated by an almost
decade-old idea [121, 122, 123] that at high enough energies the pion-sigma ground
state symmetry breaking condensate can be destroyed. In the latter process of sud—
den cooling there is a chance of the formation of disoriented chiral domains, given
that the terms responsible for the explicit breaking of chiral symmetry are small.
The formation of this disoriented chiral condensate (DCC) is presumably responsible
for the observed discrepancy in the ratio of pion species. Further enthusiasm about

these ideas is generated by the fact that it may be possible to achieve an environment

99

capable of producing disoriented chiral domains in relativistic heavy ion collisions. In
that case the large fluctuations in pion types can be a signal of a DCC.

There exist a number of theoretical works investigating the production and devel-
opment of DCCs in heavy ion collisions. The assumption of a random equiprobable
orientation of the pionic isovector leads [122] to the probability of observing a fraction

of neutral pions f :- N,,o/(N,,+ + N,— + N,,o) being
(6.1)

This result is consistent with the slowly varying classical pionic field as a solution of
the non-linear a-model [124]. It can also be obtained as a limiting large N distribution
in the coherent single-mode pion production by an isoscalar operator [125, 126, 127]

(51.51)]V/2 : (aga; + up; + an)” .

The formation and the nature of domain structures is an important question in itself
[128, 129]. In the case of many small domains or in the absence of a DCC, one would
expect a Gaussian distribution P (f) , following from the central limit theorem. More
complicated methods of quantum and classical field theory applied to this problem
with different assumptions of formation, evolution [130, 131] and dissociation of DCCs
[132, 133] lead to different results. Pionic mode mixing and final state interaction may
also greatly change the observed forms of these probability distributions [127, 124,
134].

Most of the existing theoretical works consider pion production from DCC domains
formed in the dynamical process of symmetry restoration and later sudden relaxation
into a non-symmetric vacuum which is some times called a quench [135]. The word
“disoriented” also describes the theoretical foundation of the DCC approach which
is based on introducing an effective disoriented current-type term in the Lagrangian.
The advantage of this formalism is in the simplicity of the solution, as a current-

type interaction can always be solved with a coherent state formalism [136]. In

100

contrast to that the linear sigma model, a well—established effective theory based on
chiral symmetry, naturally leads to a quadratic pion field term in the Lagrangian.
The goal of the present work is to consider an “oriented” chiral condensate with a
chirally symmetric Lagrangian that to the lowest order is quadratic in the pion field.
Throughout all this paper we are going to deal with the wave equation for pions in

the form
827?

at?

 

— V27? + m§fi(x, t) 7? = 0. (6.2)

This is the equation of motion deduced from a quadratic Lagrangian. Equation
(6.2) contains an effective mass mefir(x,t) which is due to a mean field from non-
pionic degrees of freedom. This term produces a parametric excitation of the pionic
field, and, if 7713,, < 0, may lead to amplification of low momentum modes and the
formation of a chiral condensate. The chiral condensate in this picture is “oriented”;
as opposed to the usual picture of a DCC. The scalar mean field (effective mass)
introduced here does not break the symmetry properties of pion fields in any way.
The quantum problems formulated by equations of the form (6.2) address a variety
of physical issues, and can be encountered in different areas of physics. In general,
such non-stationary quantum problems cannot be solved exactly. Even studies with
approximate methods like perturbation theory, impulse approximations, adiabatic
expansions and many others often require sophisticated approaches. We will address
those rare occasions when exact solutions may be obtained. The significance of the
analytical solution should not be underestimated. The exact results could exhibit the
unperturbative features of the solutions, like phase transitions and condensates, as
well as point the way to a good approximate theory.

The paper is structured as follows. We start with a short introduction to the
linear sigma model and show that effectively it leads to Eq. (6.2) for a pion field.

In the next section we discuss an exact solution of Eq. (6.2). First, we show that

101

from the classical solution a quantum solution can be built exactly. Then, based on
the general form of the evolution matrix, we obtain and discuss possible forms of the
multiplicity distribution. The bulk of the paper is concentrated in Sec. 6.4. where we
further analyze the properties of the solutions to Eq. (6.2). By considering a simpler
model of space-independent but variable-in-time effective mass we clearly define the
exponentially growing “condensate” momentum modes. Depending on the number
and strength of the condensate modes we obtain different distributions for a particle
number and distributions over pion species. We emphasize the limits when Eq. (6.1)
is recovered or when 73( f) becomes Gaussian. In the following part of Sec. 6.4 we
consider a full field equation with a time- and space-dependent mass parameter and
address the exactly solvable case of a mass parameter abruptly changing and returning
to normal. We show that in this picture the condensate modes are still identifiable
and they still have a characteristic momentum distribution. Final summary and

conclusions are given in Sec. 6.5.

6.2 The linear sigma model

Below we give a short review of the linear sigma model, point out the origin of Eq.
(6.2) in the context of pion fields and deliberate the possible form of the effective mass
term. We assume the usual linear sigma model Lagrangian of the pion and sigma
fields [137]

1 I /\
5L3. : 5 ((9,,7?) ' (8“7?) + 5 (8,,0) (6"0) — 4 ((7?2 + 02) — v3)2 + 60, (6.3)

where explicit symmetry breaking is introduced by the parameter 6. The sigma field
has a non-zero vacuum expectation value f7, that is set by the Goldberger-Treiman
relation and is related to the above mentioned symmetry breaking parameter as

6

)‘fn '

 

2 .2
710wa—

102

 

The Lagrangian of Eq. (6.3) produces the following masses of the pion and sigma
mesons
m2:—, m222Af2+-§-.
7r f7r 0 7r 7f
Equations of motion for each isospin component of the pion field 7r, have the form

827% _,
~5t—2— — VZWT + A ((7r2 + 02) — v3) 7r, 2 0. (6.4)
In the mean field approximation non-pionic degrees of freedom, like the sigma field,
are approximated by their expectation value. Then the dynamics of the pionic field

can be viewed as a field equation of type (6.2), with a time- and space-dependent

mass parameter

mgffix, t) = A ((7'r'2 + 02) — 22(2)) .

As in the application to chiral condensates, we would like to solve Eq. (6.4) in
the formalism of quantum field theory with reasonable expectation values (7?2 + 02).
We do not solve the problem self-consistently because meg(x, t) , being generated by
other degrees of freedom, is placed in by hand as a mean field. Before and long after
the reaction the fields are in their ground state so that (NT) 2 0 and (a) =2 f7r , which
is equivalent to

2

m fi(x,t 2 —oo) : mefl(x,t = +00) 2 m2

7, .
During the reaction, the behavior of the effective mass is unknown, and in our case
would be an input to the model. When chiral symmetry is restored the expectation of
the a field tends to zero, while the effective squared pion mass is positive. A sudden
return of the effective potential to its vacuum form (quenching) can strand the a
mean field near zero with a large negative value of the effective pion mass squared,
which in the extreme limit would reach mgfl» : —Av3 z —m§ / 2. A negative value of
mgff would lead to an exponential growth of the low momentum pion modes and long

range correlations, i.e., the creation of a chiral condensate [138], see Fig. 6.1.

103

 

  
 

 

ea are ”QM:
d C
Evolution ( ) ( )

l
2 x symm. restored
/

time

 

 

“ridden cceéirig
a 2.. vaeanm germs»

    

Figure 6.1: Illustration of the “quench” process. Panel (a) shows the normal 7r — a
vacuum at low temperature. Panel (b) shows the restoration of chiral symmetry when
the system is heated in the nuclear collision, and panel (c) demonstrates the “quench”,
the sudden restoration of the vacuum to its normal low temperature shape. In the
process (a)—(b)—(c) the expectation values of (7r) and (a) change which is illustratively
shown by the classical motion of a small ball. The curvature in the 7r direction is
associated with the pion mass and the evolution of this quantity is schematically

shown in panel (d).

104

 

6.3 From classical to quantum solutions

The generic form of the Lagrangian density of the field 2,0 that has field terms up to

the second order in its potential energy part is

t = g (M) (8%) — M. t) w — émfix, t) w? (6.5)

To clarify the approach, we consider here a bosonic isoscalar field 1b. This does not
limit the consideration and will be generalized later. It is known that the linear
terms J (x, t), often called a current or force, can always be removed from considera—
tion [139]. The time-dependent current term is responsible for the creation of coherent
states with a Poissonian distribution of particles. Intensive studies of a DCC pro-
duced by linear-type coupling of the pions to the disoriented mean field have been
performed [134]. Unfortunately, many problems cannot be easily reduced to this
linear approximation. As mentioned in the Introduction, for the chiral condensate
this would require the introduction of a symmetry breaking isovector current f(x, t),
whereas introducing a scalar effective mass (a quadratic term in the Lagrangian)
preserves intrinsic symmetries.

There is one important feature of the problem with a quadratic perturbation in
the Lagrangian. The equations of motion for the field, like Eq. (6.2), are linear
for the Lagrangian (6.5); therefore they are identical for the classical fields and the
corresponding operators in the Heisenberg picture. Thus, an exact classical solution

is related to the solution to the quantized version of the problem. The step from

classical to quantum treatment will be considered next.

6.3.1 Parametric excitation of a harmonic oscillator

The parametric excitation of a quantum harmonic oscillator has been extensively

considered in the literature starting from refs. [140, 141, 142]. This corresponds to

105

our problem in the case of no spatial dependence of the effective mass. The presence

of translational symmetry in this case leads to the conservation of linear momentum.

The quantum number of momentum, k, can be used to label the normal modes.

Each mode is just a simple oscillator with the time-dependent frequency. The simple

Fourier transformation from :1: to k transforms the Hamiltonian, that corresponds to
r

the Lagrangian in Eq. (6.0), into a sum over modes (for simplicity we keep one-

dimensional notations)
1 .
H=§/m%Maw+vwah+mgmwah)=
1 ~ 2
5 2 (wk + W 4);? + midtwf) - (6.6)
k
With this reduction we arrive at the problem of the independent development of a

large number of quantum oscillators. With the notation w2(t) = m§fi(t) + k2 , we have

a classical equation of motion for each mode k
713,, + wimp. = 0. (6.7)

After the quantization, the wk and the corresponding momenta become operators.
Assuming a particular normal mode It below we omit this subscript.

Even for the problem of a simple oscillator, the relation between the classical and
quantum solution is quite subtle; not to mention that classically Eq. (6.7) is analogous
to a standard problem in quantum mechanics of scattering from a potential —w2 that
is known to have only a limited number of cases with exact analytical solutions, see
the illustration in Fig 6.2.

Classically, the S—matrix would be defined if we assume that a solution of Eq.

6.7 with the as mptotics in the remote past correspondin to the frequenc w- ,
y g y
2/)(t) : e‘w“ at t —-> —oo, (6.8)

has evolved with time to a final general form of the solution with the frequency w+ ,

z/J(t) = w___ (u 6‘1“” + ”0* Hm”) at t —> +00. (6.9)
“1+

106

 

 

 

 

 

Figure 6.2: Illustration showing the scattering from the quantum problem of scatter-
ing. Mathematically this is the same problem as the problem of parametric excitation

of classical harmonic oscillator.

It is clear that the complex conjugate to Eq. (6.8) will develop into a corresponding
complex conjugate version of Eq. (6.9). Furthermore, the unitarity, i.e. conservation

of probability applied to Eq. (6.7), results in a restriction posed on u and v
lul'2 — |v|2 = 1. (6.10)

In order to analyze the quantum version of the problem we use the language of
secondary quantization, introducing creation and annihilation operators for a single
harmonic oscillator. The time dependence of a quantized field coordinate 1/1(t) in the

Heisenberg representation as t —> —oo is

1 . .
'r/1(t) : W (ae'w‘t + alew“) , (6.11)

where the operators a and aJf do not depend on time and define the “in” state of the

 

system. We will define the “out” state using Operators b and bl as

1 . .
w) = (be—Wth + bl em“) at t —> +00. (6.12)

\/2w_+_

Being supported by the fact that the field 7,0(t), even in its quantum form, satisfies

 

Eq. (6.7), and with our assumptions about the classical solution (6.8, 6.9) we can

107

relate operators a and al with b and bl ,
b : (ua+val), l)T :- (1L*al+v*a) , (6.13)

with the same parameters u and 'u as in the classical solution. As seen (6.13) the “in”
and “out” states are related by a Bogoliubov transformation. The condition that the

commutation relation
lb, 12*] = (lulg — lvl2)[a,a*] = 1
is preserved coincides with Eq. (6.10) .
It is possible to build an S-matrix corresponding to the transformation in Eq.
(6.13), however the solution is very technical, and of limited use. We can use Eq.

(6.13) directly to answer simple but relevant questions. The probability amplitude to

have 77. final quanta if there was vacuum in the initial state is given by
Cn : a<n|3l0>a : a<nl0>bv

where the S-matrix is a transition matrix from the “in” state to the “out” state.

Then
00
|0>b : Z Cnln>aa
n=0
and recursion relation for the coefficients Cn

v n + 1
C71 : _ —
+2 11, n + 2

 

C... (6.14)
may be found from the definition of the vacuum and Eq. (6.13):
b|0)b = 2C7, (rm/filn— 1)a+v\/n+1|n+1)a) = 0.
n=0

Finally, normalized to unity, the transition probability from a vacuum to a state with

2n bosons is expressed by

(2n)!

lC2nl“Z E 13(2"): mp" 1—p,

108

 

where the parameter p = [v/u]2 represents the reflection probability for Eq. (6.7) if it
is viewed as a Schrédinger equation for scattering at zero energy off a potential —w2 .
Conservation of probability, Eq. (6.10), limits the values of p to the region ,0 E [0, 1) .

The distribution of particles produced from vacuum Eq. (6.15) peaks at zero and the

average number of particles produced is

n = {E 2nP(2n) = _1_p_. (6.16)

n=0 — ,0
It is seen from the above expression that the number of particles diverges as p ap-
proaches one. This will be a region of interest in our later studies of chiral condensates.
A rigorous and detailed discussion of all the features in the transition from classical
solutions of Eq. (6.7) to a problem of a quantum oscillator with a time-dependent
frequency can be found in [140, 141, 142]. The corresponding Schréidinger equation
for the wave function of the quantum oscillator is (h = 1)

,8<1>(:z:,t) _ 182<I>(a:,t) 1 2 2
27 — EW—l—éw (t).’L‘ (DUI/3t), (6.17)

where coordinate :6 originates from the field “coordinate” 1/2, Eq. (6.7). Let some
function (Mt) be a solution to classical Eq. (6.7) with w(t z —oo) = w_ and w(t =
+00) : 62+. This function (Mt) can be written in terms of the real part r(t) and phase

7(25) so that 1M1?) 2: r(t) em” . With a direct substitution it can be shown that

 

(P(z,t) = :(t) exp (2%) x(y,T) (6.18)

is a solution to the quantum equation (6.17). Here a rescaled length y and time T are
introduced as y = :c/r(t), T = 7(t)/w_. The function x(:r, t) is a standard solution

of the Schrodinger equation for the oscillator with a constant frequency w- ,

3X(Zl,7) _ 182X(y,’r) 1 2
37 — g @312 +5w—y X(y,T)- (6-19)

7;—

The initial conditions for the solution x(y, T) should be set at t —> —oo by the known

conditions on @($, t) . Equation (6.15) can be generalized utilizing this more general

109

 

technique. It can be shown that the probability for a transition from the state with

m quanta into the final state of n quanta is given by
m____1n( m, n) ('1— le— nl/2(\/_
max( m, n) pP] (m+")/ —p)

where P; are the associated Legendre polynomials and n and m have to be of the

2

, (6.20)

 

same parity. The established bridge between classical and quantum solutions is a
remarkable achievement, but unfortunately analysis of Eq. (6.18) does not present a

pleasant task.

6.3.2 Infinite number of mixed modes
General canonical transformation

The situation becomes more complicated if we return to the general field equation
given by the Lagrangian density of Eq. (6.5), and assume that the modes cannot
be separated. We keep assuming J (:r,t) = 0 which is relevant to our particular
problem but in general this factor can be included back in the discussion without much
complication. The problem of the system of coupled oscillators has been discussed
in great detail in [143]. The transformation analogous to Eq. (6.13) now takes an
N -dimensional symplectic form, where N is the number of coupled modes [144],
N N

6,. .—_ H21 (ukk/akl +vkkraL) , b}; = I; (6,3,6; +vgk,a,,,,) , (6.21)

It is convenient to regard the operators ak as components of an N -dimensional vector,

and the numbers um, and 2);, k, as N X N matrices. Eq. (6.21) in matrix notation

takes the form

b=S"1aS=ua+val, bl=S_1alS=u*al-l-v*a. (6.22)

110

 

Using the properties of the matrices u and 2) shown below, it is straightforward to

check that the inverse transformation is given by
a =szS‘1 = all) — val, a]f = SbJ’S—1 = uTb] — vlb, (6.23)

where u*, u] and uT have their usual meanings of complex conjugate, hermitian
conjugate and transpose matrices, respectively. Further, we will also use an inversion
denoted as u‘l. Similarly to the one-mode example of Eq. (6.10), matrices u and
v are subject to conditions that arise from the fact that the commutation relations

have to be preserved. It follows from Eq. (6.22) that
ual—vol = l, uvT—vuT20, (6.24)
and from the inverse transformation, Eq. (6.23)

ulu—UTU’“ = 1, ulv—vTu*:0. (6.25)

Transitions between coherent states

In the case of many mixed modes it is still possible to obtain recursion relations similar
to the single-mode situation of Eq. (6.14) but they become difficult to analyze. Next
we will discuss the approach found in [143] with a different technique that uses the
transitions between coherent states. We define a single—mode coherent state In) in

the usual way as

 

. OO CY"
la) 2 e-IaV/Z m). (6.26)
In mathematics sums as in Eq. (6.26) are often called generating functions. It is

convenient to introduce states with a different normalization [145] as
Ha) : emf/210.) = em" |0). (6.27)

When applied to the states (6.27), the creation operation is equivalent to the deriva-
tive,

aina> = 3H6. (6228)

111

 

 

We continue to use the notation la) for a multidimensional coherent state, which
should be interpreted as a product of the single—mode states. Here a is a vector and
the derivative 69/80 is understood as a gradient in N-dimensional space of modes.
We are interested in finding the matrix elements of the evolution matrix S that
implements the unitary transformation between initial and final states of the system.
The evolution of the initial state is quite complicated and unless v : 0 the coherent
state does not stay coherent as in general it evolves into a so—called “squeezed” state
[146, 147]. Nevertheless it is possible to obtain an analytic expression for the matrix
elements of the evolution operator S between coherent states. To proceed in this
direction we will consider the action of the matrix S on the creation and annihilation
operators given by Eqs. (6.22, 6.23), that actually serve here as the definition of the
evolution matrix. By acting on the complex conjugate form of the first equation in
Eq. (6.22) with (6 HS from the left and with I] oz) from the right, and utilizing Eq.

(6.28) we arrive at the following differential equation

. (9
(6* — u*a—a — v’koz) (fillSHa) = 0. (6.29)

In a similar manner from the first equation of Eq. (6.23) we obtain

 

6
(a — UT 86* + 6T 5*) (6mm) 2 0. (6.30)

The solution to the differential equations (629,6.30) determines the transition ampli-

tude up to a normalization constant C(u, v):

(BIISHCI) : C(Ua’U) exp (whim—101 + éfl]v(u*)—1fl* — $0571 (u*)—1v*cv) .
(6.31)

This solution can be checked directly by substitution into Eqs. (629,630) and utiliz-
ing the observation which follows from Eqs. (6.24, 6.25) that matrices v (u”‘)‘1 and

(u*)“1 71* are both symmetric. The normalization constant may be obtained using the

112

completeness of coherent states [144],
— 4
C(u,v) : (det(uul)) 1/ . (6.32)

The evolution matrix given in the form of Eq. (6.31) can be transformed via a Taylor

expansion in the particle number basis using the definitions in Eqs. (6.26, 6.27)

N am), N ”I

<6IISIIa>= <nkISIn2> k 0" . (6.33)
{ngfni} 1:61 W H

 

 

In general it is quite complicated to give a finite expression for the coefficients of the
Taylor series arising from the multi-variable Gaussian. Expansions of two-variable
Gaussians are known to be of the form of Legendre polynomials which give rise to
Eq. (6.20). Nevertheless, the algorithm for the expansion is straightforward. First,
the exponent should be expanded in terms of its argument and then each term can
be expanded into a final sum with a generalized binomial expansion.

By considering a transition from the vacuum state we set all terms with o: to zero
in Eq. (6.31). In this case everything is completely determined by the matrix 2) (u*)‘1 .
For the question of multiplicity distributions in the one-mode case, we note that the
relative phase between a and U was of no importance, a single parameter p determined
everything. We will further see that a similar picture holds in the general case, and
only one matrix 222)] is needed to find the particle distributions. With the help of
Eqs. (6.24, 6.25) it can be seen that the matrices uul and Up] can be diagonalized
simultaneously and that the eigenvalues of v'vl(uul)(‘1) = v (u*)‘1('u (u*)*1)l form a

set of parameters pk : ['uk/u,,|2 which are different for each mode.

Multiplicity distributions

Despite the complicated form of a general expression (6.31) one may calculate the
moments of multiplicity distributions in a straightforward manner. Let the initial

state It) be characterized by a diagonal density matrix 71°, a set of numbers of quanta

113

n° in each mode It on the diagonal. The avera e number of quanta in a final state
I: 8

| f) is determined by
m = (flab/elf) = (ilblbklil'

With the above assumptions and Eqs. (6.22, 6.23), the density matrix 7‘). of a final
state is

71 : urf°ul + mfovl + up]. (6.34)

Throughout the rest of the work we will concentrate on the situation of a particular
interest when particles are created from the vacuum and the first two terms in Eq.
(6.34) are identically zero. The average total number of particles in this case may be

expressed in a simple matrix form
fitotal : Z nk : T‘I'(UT'U), (6.35)
k

which is consistent with Eq. (6.16). Higher moments of the particle distributions can
be calculated in the same manner. Unfortunately, the calculation of an arbitrary mo-
ment requires path integration techniques while using Wick’s theorem in the normal

ordering of operators. Low order moments can be directly calculated, for example
Kim] 2 fifoml + 2fitotal + 2Tr ((vvl)2) , (6.36)

firm,“ : 630%, + 6fifoml + 4mm] + 6mm.“ Tr ((1)szr )2) + 12Tr ((mfr )2) + 8Tr ((1)21)r )3) .

(6.37)
Equation (6.36) can be identified as a super-Poissonian distribution of particle pairs.
It follows from the above expressions that particle production is determined by the
matrix vvl which is related in a simple way to ual, Eq. (6.10). As was mentioned
before, the hermitian matrix vol can always be diagonalized with diagonal elements
being average numbers of particles 77, in each eigenmode q. This diagonalization

allows us to view particle production as a production from independent modes. A

114

connection can be established to the one-oscillator case, previously considered, by
defining a set of parameters p as pg 2 7771/ (1 +Wq’) , that are the eigenvalues of voluul .
Considering the number of particles produced in each eigenmode we can restrict
ourselves by the modes with pq —> 1, that dominate the particle production. We
will later refer to these modes as condensate modes. In the following section, it will
be shown that the distinct physical feature of these modes is that they produce an
exponentially large number of particles.

Next we consider a number of condensate modes with equal parameters pq. For
one mode the answer is in Eq. (6.15). For the case with several condensate modes one
needs to know the distribution for the total sum of the particles, which is given by the
convolution of the corresponding probabilities. As for the Fourier transformation, a
convolution of several distributions results in a product of their generating functions.

The generating function for the distribution (6.15) is

~ 00 1—p
P(y) = 2: “My” = —-
n20 l-py

We obtain total probabilities of particle production for any combination of species
or any number of modes convoluted together, as a Fourier expansion of a product
of corresponding generating functions. Suppose we are looking for a distribution

P)(2n) that gives a probability to observe 2n particles appearing in l single-mode

 

distributions (6.15). The Taylor expansion of (1 — py)‘l/2 gives
(l+2(n—-1))ll n

where the double factorial should be understood appropriately for odd and even l .
According to the central limit theorem, only the distributions corresponding to a

small number of condensate modes have a shape that is very different from a Gaus-

sian, see Figure 6.3 below. If parameters {pq} for the condensate modes are very

different then only the important modes that produce many particles can be isolated

115

reducing the problem back to the case of several almost degenerate modes. The
method of generating functions is also convenient in discussions of distributions over
species. For example, the distribution for the total number of pions, regardless of the
isospin projection, can be obtained as a distribution of the sum of three species, via
convolution. Alternatively, one can think of the total condensate modes for pions as
just a sum of numbers of condensate modes for all species.

As an example we apply Eq. (6.38) for one mode and three pion species. The

probability of having 27). neutral pions is

P°<2n> = eon) = 595%?” —p>1/2p". (6.39)

the probability for observing 27?. charged pions is
P+(n) = P-(n) : PChargedQn) = P2(2n) = (1 — p)p”, (6.40)

and finally the probability of having total 277. pions is

(272+ 1)!

Pt°t(2n) : P3(2n) = 22710102

(1 - p)”2 p"- (641)

We note the convolution of three distributions in Eq. (6.39) that give rise to Eq.
(6.41) produces a peaked curve with maximum at particle number

3p—2
l—p'

 

”max =

The probability distributions (6.39, 6.40, 6.41) are shown in Fig. 6.3 with solid line,
long dashed line and short dashed line, respectively. The figure displays a critical
situation when parameter p is 0.999. This corresponds closely to the first mode in
the example of a square perturbation considered in the next section.

Now we can return to the original questions: What is the probability to observe
a certain fraction of neutral pions given the total number of pions detected? Is it
possible to use this distribution for the detection of chiral condensate? In the one-

mode approximation, given a total number of pions produced 277.), the normalized

116

 

   

Probability

I
b

10

1 fi—V-“hV‘I.-- 1

 

 

-5

 

1 0 ‘ 1 ‘ ‘ ‘
0 2000 4000 6000
Particle number n

Figure 6.3: The particle number distribution for neutral pions (solid line), charged
pions (long dashes), and for all pions (short dashed line). All curves are normalized
to unity; note that the total number of pions and number of neutral pions both are

always even, while any integer number of charged pions appearing in pairs is allowed.

The parameter p = 0.999.

117

 

probability of observing 27?. neutral particles is

”P (f : 17:) : P0(2n) PCha’ged(2nt — 2n) _ (2n)! (ntl)2 22727—6

n, Ptot(2n,) " (272,)! H

 

 

(6.42)

Finally, if both n and n.) are large, which is a good approximation in almost all regions,
Stirling’s formula may be used, and one finds

1
(2724 + 1)\/7,

which coincides with Eq. (6.1) if the normalization over m + 1 discrete points between

 

7’(f) = (6.43)

zero and one is switched to the integral over f E [0, 1].

Unfortunately, this result does not hold for all situations when several modes are
participating together. The single-mode result (6.43) may be invalid when the largest
eigenvalue of the matrix 212)] is exactly or almost degenerate. Physically, this may be
due to some symmetry for example. One- and two-mode cases have their probability
peaked at zero, this is no longer true for the convolution of three or more modes, Fig.
6.3. Eq. (6.38) can be applied to give a result for any number of modes j participating
in the condensate, assuming they all have equal strength p. The distribution of neutral

pions is given by

 

703' (f 2 E) : 13,-(2n) P2j(2m — 271) (6.44)

774 P3j(2’flt) . .
With a Stirling’s formula and the assumption n >> j , Eq. (6.44) can be simplified to

(37/2- 1)!
(JV? —1)!(J'— 1)

For comparison, the probability distributions for one, two, and three modes in the

 

72,-( f) = ,fm‘l (1 — f)“- (645)

condensate are shown in Fig. 6.4. The distributions of Eq. (6.45) have maxima,
average and widths as follows

_ 4
—27j+18’

j—3 — ~— —2
max:__.——7 :13: 2—
f 3J_4 f / f f

where the first equation can be applied with the restriction j > 2. It follows from

the central limit theorem that for a large number of modes one should expect a

118

 

 

 

Y ' V
p
I

 

Probability P(f)

—L

 

 

 

O i 1 . 1 . 1 .
O 0.2 0.4 0.6 0.8

neutral pion ratio f

Figure 6.4: The probability 73( f ) that a given neutral pion fraction f is observed. The

three curves display cases of one condensate mode (solid line), two energy-degenerate

modes (dotted line), and three modes (dashed line).

119

 

Gaussian distribution that tends to a 6-function. Thus, as the number of modes in
the condensate grows, there is a fast transition from a 1/\/7 type behavior to a sharp

peak at f =1/3.

Dynamics of matrices u and 2)

As some final remarks about the link between classical and quantum solutions to the
wave equation arising from the Lagrangian in Eq. (6.5), we would like to discuss the
technical question of constructing matrices u and 2}. Unfortunately, Eq. (6.2) has no
general analytic solution even classically. The Green’s function formalism reduces the
problem to a Fredholm type integral equation. The exact solution can be obtained
only in special cases, for a separable kernel [139], or in the sudden perturbation limit
to be considered in the next section. However, numerical studies of Eq. (6.2) seem
to offer a great chance of success.

It would be good to obtain equations for u and 1) that are still exact but written in
the form convenient for numerical work. Let us quantize the field at every intermediate

stage with bare particles [139] so that

t : b t zkx—zwkt th —zka:+zwkt 646
we.) 2. —,ka(k<>e + me ). < >

where bk(t) is now a time-dependent annihilation operator. Eq. (6.46) is written in
an interaction representation, the explicit time dependence of the free field is given by
the exponents, whereas creation and annihilation operators absorb the remaining non-
. o u u 2 ‘2 1/2
tr1v1al t1me dependence. In Eq. (6.46) the var1able w), = (k + mcfl-(t —> —oo))
is the initial time-independent frequency, thus there is no problem if at any point in
time the effective mass goes through zero. As time goes to infinity was, t) becomes
the final “out” state, with operators b and bl defined as before. For one-dimensional

case L is the quantization length. For further simplicity we to denote

mifi(33,t) 2 might —-> —00) + H(a:,t), (6.47)

120

this allows the separation of the interaction Hamiltonian. The perturbed Hamiltonian

from Eq. (6.6) expressed in terms of b(t) and bl(t) is

Him“) I Z {Qkk’(t) (but) be“) + (£211) +

kk’

where the matrices f2 and A are determined as follows:

 

1 . _ H(—k’—k t) .
o ,t : ———§R H k—k’ t “W W A ,t = ’ “WWW.
(6.49)

In the above expression H(k, t) is a Fourier image of II(a:, t) , determined as
H(k,t) = II*(—k,t) = /n(:c,t)eik$dx,

and ER denotes a real part of the expression. Utilizing the Hamiltonian equation of

motion in the interaction picture,

6%b(t) = [6(t),HW-(t)] = Qb(t) + Abl(t),

it is possible to show that if b(t) is defined through the initial operators a and al as
I)(t) : u(t)a. + 22(t)al, (6.50)

where u and 'u must satisfy the matrix equations

d

rat-Mt) = Q(t)u(t) + A(t)v*(t), (6.51)
6%110) : Q(t)v(t) + A(t)u*(t). (6.52)

At infinitely large times when the perturbation II goes to zero, the right hand sides
of Eqs. (6.51, 6.52) vanish and u and 2) become time-independent. In order to obtain
matrices u and v for Eq. (6.22) one has to solve the above equations with initial

conditions u = 1 and v = 0.

121

6.4 Application to chiral condensates

6.4.1 Separable modes, Space-independent effective masses

To illustrate the machinery developed in the previous section we start with simple
cases when the classical solution is known analytically. The first example is the
space-independent field H(x,t) E II(t), see Eq. (6.47), i.e. the situation when the
perturbation is uniform in a box to which the entire pion field is confined. The
wave vector k is a good quantum number. Particles get produced independently in
each mode labeled with k, and production is determined by the classical reflection
probability pk : [12k]2 / lukl2, Eq. (6.15). The distribution of particles for a single
mode, Eq. (6.15), is a decaying function that has a maximum at zero. Its behavior

can be approximated with Stirling’s formula as

P(2n) ~ 5%,

where P(2n) denotes the probability of creating 2n particles from the vacuum. The
distribution in Eq. (6.15) for a value of p = 0.999 is shown in Fig. 6.3 as a solid line;
some additional discussion is given below. The average number of particles produced
in Eq. (6.16) is generally quite small unless we are in the condensate region when

p —> 1 . With the assumption of independent modes Eq. (6.4) reduces to
—d27rT/dt2 — H(t)7r, = (m?r + k2)7rT. (6.53)

Eq. (6.53) is written in a form of the Schrfidinger equation for scattering from a
potential barrier of height —II at “energy” m3, + k2 . The parameter p we are looking
for, which links classical and quantum pictures, is the reflection coefficient for this
scattering process.

Let the effective pion mass change and then return back to normal in a step func-

tion manner. This corresponds to the scattering off a rectangular potential barrier.

122

The situation when the tunneling is involved, is relevant to the case of a low mo-
mentum mode being amplified as the index of reflection is rapidly increasing. We
assume that the perturbation Il(t) has a non-zero value II only for the time interval

75 E [0, T] . The reflection probability for this scattering potential is

lvkl2

pk :1+]’Uk]2,

and the average number fik of particles created in the mode k is given by

2

, (654)

 

H2
4(m,2r + k2) [771,2r + k2 + III

 

fik : lvkl2 :

sin (T\/m,2, + k2 + H)

 

 

Another form of the perturbation II(t) for which an exact analytical solution exists

is the Eckart potential [148]

II

H“) : cosh2(t/T)’

where II = II(0) is the minimum value of II(t) and T is the time scale of perturbation.

The resulting form of the reflection probability is

 

1+Cos(7r\/4IIT2 + 1)

pk : cosh (27rT(/m,2r + k2] + cos (7r\/4IIT2 +1)

Both forms of the perturbation ll(t) with parameters that we use below for our

 

 

numerical estimates are plotted in Fig. 6.5, right side shows the Eckart potential,
rectangular barrier is on the left. Fig. 6.6 shows the average number of particles
produced in the two models described above. For our estimate we made the following
choice of parameters. The effective mass drops to the value of mgfi = —m3/2,
correspondingly the parameter II is chosen at —1 GeV2 for both models. The time
scale, given by parameter T, is 1 fm/c and 0.5 fm/c for the square barrier and the
Eckart potential, respectively. It is important to notice that even though the graph
is plotted over a continuous variable k , our finite size spatial box allows only discrete

values of momentum in each direction. With the expected interaction region size of 2

123

 

1.5 I I l I I I I I I I

 

 

 

 

 

 

 

t (fm/C) t (fm/c)

Figure 6.5: Eckart (|II| = 1 (GeV)2, T = 0.5 fm/c) and rectangular ([11] = 1 (GeV)2,

T : 1 fm/c) perturbations are shown on the right and left panels, respectively.

fm, the lowest momentum in a cubic well of size L is [k] z \/3 7r / L, numerically this is
about 540 MeV. According to Fig. 6.6, both models predict around several hundreds
of particles. The next higher-lying modes, have significantly smaller numbers of
particles. The picture presented shows that one can only hope to have very few
modes that actually form the condensate, as the number of particles falls drastically
for higher momenta. This is consistent with the argument that in order to get a
noticeable condensate one should have the energy of the mode dipping below zero.
The square barrier model provides a simple estimate for the number of particles

produced if the mode just touches zero,

_ HT2 mgr? 6
TI, 2 — z z
4 8 ’

 

 

where the sigma mass m0 is taken 1.4 GeV and the interaction time T is around 1
fm/c .
For more complicated perturbations it is possible to use the WKB approximation

in order to determine the reflection coefficient of classical Eq. (6.7) for tunneling.

124

 

 

 

 

—5 . 1 . l . 1 \ ‘ .
O 500 1 000 1 500 2000
lkl (MeV)

 

Figure 6.6: The average number of particles produced in the mode k as a function of
k for square potential (solid line) and Eckart potential (dashed line). The parameters
of the perturbation are chosen so that II = 1 GeV2 for both models, and T is 0.5

fm/c and 1.0 fm/c for Eckart and square potentials, respectively.

125

With the notation

«s = 2 exp Um 'w(t)19(-w2(t>) dt) ’

‘00
where the Heaviside theta-function limits the integration to the region of negative
690?), we have the average number of particles given in the semiclassical limit of

€>>1by
2
6:3.

4 (6.55)

Concerning the total particle distribution one has to add all particles from all
modes and consider the distribution as a superposition. Based on the results of
the above examples and Sec. 2 we can conclude the following. About a thousand
neutral pions with momentum 540 MeV/c are produced by the lowest mode and
are distributed according to P1(2n), Eq. (6.38). The distribution over species is
771 N 1/\/7, Eq. (6.45). This is the single-mode result described by Eqs. (6.39,
6.40, 6.41) and shown in Fig. 6.3. Geometry is crucial here, as it determines the
energies and degeneracies of other modes that may or may not compete with the
lowest mode(s). Higher momentum modes in the box do not “condense” if, in the
scattering picture, their energy is higher than the barrier and reflection is negligible.
These modes, even jointly, may on average produce just several particles. The case
of the non-condensate particle production will be considered with a better model in
the next subsection.

These simple examples are still far from realistic. One of the major failures is
that pions from the square box are not real pions and therefore all excitations that
we obtain need to be projected onto final pion states given by the plane waves of
the entire space. This projection will produce a momentum spread of the outgoing
states, that will carry the characteristics of each condensate mode. Nevertheless, these
models produced reasonable results and what is more important they have identified

the physics of the process.

126

6.4.2 Time— and space-dependent perturbations
General solution for a step-like temporal perturbation

Here we will solve the perturbed Klein-Gordon equation for the pions in a more

realistic case, where the effective pion mass is both space- and time-dependent,
— — V271“ + (m?r + H(x, t)) 7r = 0. (6.56)

Despite the fact that this dependence is put into the model by hand, the resulting
features can be quite general. In order to keep our solutions analytic we choose the

perturbation II(x, t) as a step function in time

II(x,t) : II(x) for t6 [0,T]

0 otherwise

This choice will allow us to solve Eq. (6.56) classically and to construct matrices u
and 2) that control the evolution of a plane wave. For most of the discussion the form
of II(x) is left as general, but in the last part for the numerical results we take it as
a spherical square well.

The wave function 7r(x,t) can be found in each of the time regions, and the
solutions should be smoothly matched keeping 7I(x, t) and the derivative 871x, t)/8t
continuous. Introducing a separation of variables as 7r(x, t) : X(x) T(t) we obtain
equations of motion for X and T in all three time regions. In the perturbed region

t E [0,T] , Eq. (6.56) becomes

 

—v2X(x) + n(x)2c(x) = exec), (6.57)
82W) 2 _
at? + W 7'(t) _ 0,

where the dispersion relation is
W2 2 m3, + 8. (6.58)

127

 

 

II(x) ll

' continuum,”

 

 

 

 

bound state

 

Figure 6.7: The schematic representation of the perturbation H(:r), with one conden—

sate bound state.

The unperturbed form of the free space wave equations at t < 0 or t > T is

—V2X°(x) = [k]2 X°(x), (6.59)
82T°(t)

(9752 + wiT°(t) = 0,

with the usual relation

61,3 2 m?r + lklg.

Eq. (6.59) has simple plane wave eigensolutions that we denote as |k) with positive
eigenvalues [k]2 . The perturbed Eq. (6.57) with negative H may have negative energy
bound states as well as the usual continuum states with positive energies, see Fig. 6.7.
For such a bound state, W2, Eq. (6.58), becomes negative and W = if? is imaginary.
We denote the eigenstates of Eq. (6.57) as |Ic) and corresponding energy as 8(a).
Here K is a set of quantum numbers labeling the eigenstates. We also assume that

both |k) and [61) are properly normalized and form complete sets,
<k|k'> = 6.1.666) = 6...,1 = Z |k) <k| = Z In) (6.
k K.

128

To determine the Bogoliubov transformation matrices u and v we consider the evo-

lution of the wave function
|7r(t)) : e‘iwkt |k), fort < 0,
through an intermediate stage t E [0, T]

[7r(t)) : Z (akne_iWNt + bkfleiw’d) Ira),

K

into a final state

6) . .
[6(0) = Z w—“ (..., + 6.... 6....) M
k’ k’

Continuity of the wave function and its derivative with respect to time allows the

determination of the unknown coefficients a)“, and bk), as well as the matrix elements

of interest vk kl and uk 14 . The result is, both for real and imaginary WK, ,

U) I 87:14)le I
Ukk’ = [/—k Z<k|H)(H|k> ><
wk 2 n

 

 

 

 

 

 

wk . wk Wm) .

[(1—l— wkl) cos (WK, T) 7. (W). + wk, 8111 (W,, T) , (6.60)
and
—iw IT
vkk, = Z: 6 2" ;(k'|n)(nlk) x
[(1 wk) cos(W T) '(wk WK) sin(W T)] (6 61)
— - . n — z — —— ,6 . .
wk, WK wk:

In order to see whether the condensate was formed we first address the question of the
number of produced pions. Possible bound states in the solutions of Eq. (6.57) should
be carefully treated. As seen below, bound states in Eq. (6.57) with the energy 8
below —m§, produce an exponential growth of the particle number with time which
is our starting criterion for the search of the condensate. The average number of

particles may be then expressed using Eq. (6.35) as

1
fl : 2 [Wk]? : Z Z {14”, cos (WET) cos (WK/T) + B”, sin(WKT)sin(VV,:,T)} ,
k,k’ K K.’
(6.62)

129

A”, = 2 (117,919), — 6,, ) ,

[0,1,2 W W*, _ 2
B,,,,— _ ‘ '1 6,... I( 1,) W,.,W*,
(W: W; (W; W.) + .. ) ’

Where we assumed that the states are chosen in such a way that the amplitudes (kl/4.)

 

   

are real, and we introduced the notations

$123.22: K.(]k) wk( (k|)I6,

k

and

_ _ 1
If. .9 = If. E.) = Z (mm—<54).
k wk

Depending on a particular level, W may be real or imaginary if in Eq. (6.58) the
energy 5 is greater or less than mi; nevertheless Eq. (6.62) works for both cases.

In practice it is convenient to express the overlap of (kl/c) using Eqs. (6.57, 6.59)
for the states [16) and |k). Multiplying Eq. (6.57) by A.” and Eq. (6.59) by X and

subtracting results we obtain
(6 — k2)X 26° = II(x)X X° + V(XVX° — X°VX). (6.63)
Integration of Eq. (6.63) over all space produces a useful result

(kl/*7) =

 

1 3 o
g _ k2 /d mum/ta” . (6.64)

For the continuous spectrum, this expression contains an additional term with 6(8 —
k2) .

Due to large oscillations of the trigonometric factors in Eqs. (6.61) and (6.62),
one should expect considerable particle production only in situations with imaginary
W . This observation makes it natural to separate the sum in Eq. (6.62) into several
contributions depending on the intermediate state IR) . There are exponentially rising
terms that involve transitions through the states with imaginary W; it is important
that the bound states are discrete and that their number is finite. Another contribu-

tion is of all intermediate states that lie in the continuum. This second contribution is

130

always present for any perturbation even with no bound states. With a non-zero pion
mass there maybe a number of discrete states that have real W but these states have
negligible contributions compared to the states from the continuum. In the remaining
part of this subsection we will mainly draw attention to the first two cases which we

call “condensate” and “non-condensate” pion production.

Condensate pion production.

In the following picture we assume that there is one mode [K0) with negative energy
so that Wm0 = if), and in the summation in Eq. (6.61) the term that involves this
condensate state [H0) is dominant. Thus, the distribution of particles and all other
properties are the same as in the single-mode example, i.e. as for the parametrically
excited single oscillator. According to Eqs. (6.35), (6.36), (6.37) and (6.24), the traces
of vol and other higher powers of this matrix completely determine all moments of
the particle distribution and thus the distribution itself. Therefore, it is sufficient to
consider the eigenvalues of a hermitian matrix 211)].

Under the assumption that only one term in the sum in Eq. (6.61) is important,

we can express the matrix element of vvl as

1 LL) U)!

T ] k * ] k r.

U?) /——a——+b./w w1+c ——-+C —, 6.60

{ }k,k ,——I k k I ( )

where the introduced coefficients are

1 _

a z 4 (1(1) cosh2(QT) + 921(71)sinh2(QT)) , (6.66)

b = i (1(‘1)cosh2(QT) + 1(1) sinh2(nT)/Q2) , (6.67)
1 . _

c = 4 (Z[I(1)/Q + 1< 1>12] — 1) , (6.68)

and

 

We wrote an element of the matrix vvl in the form of Eq. (6.65) in order to show
that it consists of only four factorizable terms. In accordance with a general theory of
factorizable kernels, the only non—zero eigenvalues of the matrix 21121 are the eigenvalues

of the following 4 x 4 matrix

(011(4) b c (21(4))

a (21(1) c1“) c*
(6.69)
61H) 1) 0 MM)

 

 

\ a bl“) cl“) 6* )
Having the second two rows the same as the first two in the matrix (6.69) reduces

the actual number of non-zero eigenvalues to two, and the secular equation for the

eigenvalues )1

1
A2 — /\Tr(vvl) + E(1(1) 1H) — 1)2 = 0. (6.70)

The trace of the matrix UN in this expression is the same as the average number of

particles produced by one condensate state [K0) ,

_ 2 - 2 12
fl: Tr(vv]) = ———COSh (QT) (1(‘1)1(1)) + M (1(4)2 Q2 + I( ) > —1

 

 

2 4 Q? 2 l
(6.71)
The result (6.71) can be also obtained directly from Eq. (6.62).
In the limit of exp(2f2T) >> 1, Eq. (6.71) becomes
629T [(1) '2
— = [We — . . 2
n 16 ( + Q ) (6 7 )

This approximation is relevant to our problem as we wish to determine the exponen-
tially growing condensate modes and therefore we choose a physical environment for
which this is true. The limitation exp(2§2T) >> 1 is then a criteria for the environment
of a chiral condensate. As seen from Eq. (6.70), of the remaining two eigenvalues of
the matrix vvl one becomes exponentially large and the other one goes to zero. This

is important because with only one nonzero eigenvalue we recover the single-mode

132

situation, discussed in the previous sections which leads to the distribution of par-
ticles of the form of Eq. (6.39) and a 1/\/T form for the distribution over the pion
species.

As a concluding statement we stress here that in principle just making the ap—
proximation exp(2f2 T) >> 1 allows us to take one term in the sum of Eqs. (6.60,
6.61) from the very beginning, and the same approximation was used again in the
end to purify the single mode. Therefore the condition exp(29 T) >> 1 must be a
clear indication of a condensate in the bound mode with relative energy (2. If there
are several bound condensate states with close energies 52 we would again recover the

system of several modes that was discussed before.

Non-condensate pion production

To make the picture complete we have to estimate the number of particles that are
produced from all the modes not involved in the condensate. The low-lying negative
energy states that have energies above the pion mass do not make a considerable
contribution. They may be estimated by the same method that was used for a
condensate states just having oscillating exponents instead of exponential growth in
Eq. (6.72), 2
1
a N I16 (I(’1)Q + %) . (6.73)

This source does not produce many pions, as will be seen from the numerical example
in the next section.

The contribution to the number of particles from the continuum is of a greater
interest. First, as the number of states is infinite, unlike the previous case of bound
states, we may have a significant contribution. Secondly, it is an important practical
question because the continuum states always produce pions even if the condensate is

absent. It follows from Sec. 6.3 that the charge distribution of pions from many modes

133

of almost equal strength is Gaussian. To perform a particle number estimate we again
address Eq. (6.71). It is reasonable to assume that oscillating terms corresponding
to different arguments W,c and WK: average out to zero and the contributing terms
are those that are in phase, corresponding to the same K. For these terms we take
sin2(WT) z cos2(WT) m 1/2. Applying these approximations to Eq. (6.71) we
obtain
2

g 2 I<k12>12(w:;$: 2) . (6.74)

n,k

 

Ti
As seen from the structure of this equation, the arbitrary 6-contribution in Eq. (6.64)

does not influence the result for 72’.

6.4.3 The spherical square well

The spherical square well potential of finite depth is the simplest spatial perturbation

H(x) that has an analytical solution. We assume that

—V for [x] <R
H(x) :-

0 otherwise

The rotational symmetry makes angular momentum a good quantum number, and the
remaining radial dependence can be expressed in terms of spherical Bessel functions.
Due to the largest exponential enhancement the deepest level is expected to produce
most of the contribution. Thus, we concentrate our attention on an s-wave bound
ground state that will be a dominant condensate state in the process. By virtue of
symmetry, all pions produced from this state will have a spherically symmetrical 3-
wave spatial distribution. Instead of plane waves it is convenient to quantize spherical
waves in a large sphere of radius L. This leads to the substitution of the former
basis |k) by ]k,l,m) where k = [k] and l and m are the orbital momentum and
its projection, respectively. Needed overlaps between perturbed and non—perturbed

states can be computed using the one-dimensional version of Eq. (6.64) where the

134

right hand side is obtained by integrating Eq. (6.63) from zero up to the size R giving

 

 

 

 

 

V (X(r) 8/82X°(r) — X°(r) 0/(97‘X(r))
(kl/6) = (6.75)
6 — k2 E — k2 + V 7:12
In particular, for a bound (8 < 0) S-state [6.0) Eq. ( 6.75) gives
_ 2&2 acot(aR) sin(kR) - k cos(kR) a
(Kolk’ 0’ 0) — \/L sin(ozR) (k2 — 072) (k2 + a2 cot2(07R)) \/ozR — tan(cvR) (676)

where a = (/V — [8 | . This overlap is normalized to one as a sum over all momenta
k: = 7m / L , where n is a nonnegative integer. Converting this sum into an integral over
all positive .16 will remove the quantization radius L. The eigenenergies for s-states

are given by the equation

cvcot(ozR) + V? = 0.

Within these results the matrices v and u can be evaluated via Eqs. (6.60, 6.61) and
all the theory described above can be applied in a straightforward way.

As a realistic physical picture, we take the depth of the spherical well V = 1 GeV2,
radius R = 1 fm, and a lifetime of the condensate T = 1 fm/c. Then the lowest level
is at a depth of 8 =(858 MeV)2 which corresponds to Q = W]: 847 MeV.
These assumptions are probably exaggerated as the average number of pions of one
particular type in this case is fl z 2500.

Figure 6.8 displays the behavior of the dimensionless variables I(_1)Q, I (1)/ Q and
their sum 1(‘1)Q + I (I) / 9 shown in dashed, dotted and solid lines, respectively, as a
function of the potential depth, left panel, and the potential size, right panel. The
singularity of I (1) / Q at threshold corresponds to Q approaching zero and does not
have any physical significance because the approximation exp(QT) >> 1 no longer
holds. In general both plots are dominated by the behavior of Q for the chosen
parameters, whereas the integrals 1(4) and I (1) are weakly influenced by the form of

H(x).

135

 

 

 

 

 

 

 

 

 

4 - 4
9‘
Ci.
r—4 2 _ 2
O 1 l . l . l 1 1 L . I I l . O
O 500 1000 1500 0 1 2 3 4

v“2 (MeV) R (fm)

Figure 6.8: The behavior of the dimensionless variables 1(‘1)Q, I (1)/ Q and I (‘UQ +
I (U/ Q is shown in dashed, dotted and solid lines, respectively, as a function of the
potential depth W, left panel, and potential size R, right panel. A fixed size of 1

fm was used for the left panel and a fixed depth x/l7 =1 GeV was used for the plot

in the right panel.

136

 

 

I C'.

 

 

 

 

 

 

 

1 1 1 1 1 1 1 1 1 1 1 1 1
o 500 1000 1500 0 1 2 3 4
v“2 (MeV) R (fm)

Figure 6.9: The average number of particles produced as a function of the depth of
the potential field W is shown in the left panel, the size R was fixed at 1 fm. The
right panel shows the number of particles produced as a function of size R given a

fixed depth V = 1 GeV2 . The time length of the perturbation is set at T =1 fm/c.

137

Fig. 6.9 shows the average number of particles as a function of the well size, right
panel, and depth of the perturbation, left panel. The average number of particles
grows approximately exponentially with the potential depth (Fig. 6.9, left panel).
This is simply related to the fact that in a deep well the ground state energy grows
almost linearly with the depth having 9 z W. By making the potential wider
the ground state approaches the bottom of the well, limiting Q to a constant. This
restrains the growth of particles shown on the right panel of Fig. 6.9. One should
bear in mind that the conditions considered are quite extreme and were used here to
emphasize the character of the condensate. Practically the time scale may be shorter
and perturbation weaker, leading to a more pre-condensate picture with much fewer
pions. The total energy available in a heavy ion reaction may provide a guideline to
what the perturbation H is and the realistic number of mesons produced are.

As a final part of this analysis we consider non-condensate pion production, which
may be the main mechanism in most practical situations. Eq. (6.74) with the ad—
ditional help of Eq. (6.75) results in the following form for s—wave pions from the

continuum

 

fi_ 2V__2 ‘WT/Oood “[000 dk< (ksin( a(R) cos(kR) — a cos(aR) sin(kR))2

, 6.77
— k2)2w,,w,, (wk + W,,)2 ( )

where the parameter a is defined as or = W, W1, 2 W , and wk 2
W are the total energies of the corresponding modes. As expected, the number
of particles produced with no condensate involved is quite small. The left panel in
Fig. 6.10 shows the average number of non—condensate pions as a function of the
potential depth for different spatial sizes R 20.5, 1, 2 fm. The right panel of the same
figure displays the dependence on the size R for various values of V . The important
conclusion here is that the number of non-condensate pions ranges from a few up
to maybe a dozen for extreme cases. This number is completely negligible in the

presence of a strong condensate that produces hundreds of mesons. However, present

138

 

 

1|:10- (

 

 

 

 

 

 

 

O 1 ’ 1 1 1 1 1
0 500 1000 1500 0
v”2 (MeV)

 

Figure 6.10: The left side shows the average number of pions of a particular type as a
function of V, the depth of the perturbation. Curves displayed as solid, dotted and
dashed lines correspond to the values of the radius R of 0.5, 1, and 2 fm, respectively.
Plotted on the right hand side is the number of pions versus the radius R for values

of W of 0.5, 1 and 2 GeV as solid, dotted and dashed lines, respectively.

experiments may just barely reach the point of the phase transition, and therefore the
fraction of non-condensate pions is considerable, if not dominant. Moreover, other

(conventional) mechanisms of pion production have to be taken into account.

6.5 Summary and conclusions

Our prime objective in this work was to study a mechanism of pion production in
heavy ion and hadronic collisions related to the creation of a chiral condensate and to

explore how pion distributions can signal the presence of a condensate. We studied

139

 

meson production by imposing a pion dispersion relation specific to the medium, i.e.
with a space- and time-dependent effective mass.

In general, the problem of parametric excitation of the field quanta presents an
interesting question as it is encountered in many branches of physics from condensed
matter to high energy physics. The problem also exhibits a vast variety of solutions
ranging from adiabatic to phase transitions and condensates.

We have conducted an extensive study of quantum field equations of a general
form of Eq. (6.2). In our picture the parametric excitation of the field quanta is
carried out by the externally given Space- and time-dependent mass term. This
term in the Lagrangian is quadratic in the field, and the states produced are often
called “squeezed” [146, 147, 149, 150]. Some analogy can be drawn here from well
studied linear current type terms that produce coherent states. However, the essential
difference between coherent and squeezed states arises due to the fact that the latter
correspond to the pairwise generation of quanta. In the case of pseudoscalar pions,
this means that charge, isospin and parity are exactly preserved. We have emphasized
the fact that the quantum solution can be built from the classical solution, and this
important link was established via a canonical Bogoliubov transformation.

With our interest lying in the direction of chiral condensate we focused our atten-
tion on the potential of Eq. (6.2) to form a condensate with fast particle production
and large correlations when the effective mass goes through zero. Along with a gen-
eral formalism that can be used for numerical studies we have analytically solved the
problem when the effective mass experiences sudden abrupt changes. Consideration of
this particular temporal perturbation allowed us to clearly separate the exponentially
rising collective pion condensate modes for any given spatial form of the perturba-
tion in the effective mass. This produced conditions where the condensate and its

signatures can be seen.

140

We have identified two basic channels of pion production. The first involves only
a few discrete condensate modes and a large associated pion population. The second
leads to the production of far fewer (non-condensate) particles with a broad phase
space distribution. Mathematically these channels can be identified as production of
mesons from bound and continuum states of a Schrédinger equation with a potential
of the form of the perturbation itself, see Fig. 6.7. The bound states of negative
energy are responsible for the characteristic features of the condensate.

N umerically, the number of non-condensed pions ranges from a few up to a dozen,
and as expected is not very sensitive to the choice of the spatial and temporal form
of the perturbation in the effective mass. In contrast, the condensate modes have
an exponential sensitivity to the input parameters. As the abrupt changes in the
effective pion mass grow in strength, a critical point is reached with the appearance
of the condensate mode. The pOpulation of this mode increases dramatically from
zero to thousands with a further slight change in the mass parameter. Due to this
hyper-sensitivity of the number of condensed pions to the perturbation it is practically
impossible to predict the effect quantitatively without specifying precisely the scenario
of the process.

However, our results predict the number of non-condensate pions, thereby impos—
ing a lower limit on the statistics needed to unambiguously detect the chiral con-
densate. Furthermore, we have shown that the condensate pions have a specific
momentum distribution due to their common collective condensate mode. We have
also shown that although the distribution over species starts from the famous 1/\/7
form for one mode it quickly becomes Gaussian with the appearance of successive
modes. Therefore the presence of several modes complicates the detection of a chiral
condensate. The number of modes present increases with energy. In addition, the

mass parameter can be strongly perturbed in more than one region, each increasing

141

the number of modes. Tunneling and chaotic dynamics in the resulting multi-well
potential might lead to another class of interesting problems.

This work can be extended in several directions. With the formalism presented
here, large numerical studies of the effective pionic field in a hot medium can be
conducted involving realistic and even self-consistent forms of the perturbation given
by the 0 field. Constraints on the perturbation of the mass parameter should be
related more rigorously to observables. Further analysis of our results applied to
phase transitions, zero—mass particle production, energy transfer and many other field
theory problems would definitely be fruitful. We feel that this work may provide a step
forward in the study and classification of field theories with parametric excitations

and possibly clarify the nature of the produced squeezed states.

142

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