————‘—
——_

QHERAL DYNAMICS CALCULATIONS
0F SENGLE PEON PRGDUCTRON EN
PEGN NUCLEGN EfiELASTIE SCATTERiNG

Thesis for the Degree of Ph. D‘
WCHEGAN STATE UNIVERSlTY
WELLIAM FREDERICK LONG
1969

  
 

L
Michigan State
University

"""" r—‘fl‘L 1
IBRARY ~

   

 

  

This is to certify that the

thesis entitled
CHIRAL DYNAMICS CALCULATIONS
OF SINGLE PION PRODUCTION IN
PION-NUCLEON INELASTIC SCATTERING

presented by

WILLIAM FREDE RI CK LON G

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Physics

9/4//6:;;709/’ i

Major p\feT»sor

Date June 6, 1969

0-169

 
   
   
 

HMS 8 SDNS' :
anoxam RYWC. , '

L: '1
"I t»

    
   

 

 

 

ABSTRACT

CHIRAL DYNAMICS CALCULATIONS
OF SINGLE PION PRODUCTION
IN PION-NUCLEON INELASTIC SCATTERING

BY

William Frederick Long

The current algebra program initiated by Murray
Cell-Mann has been incorporated by dynamical models
by several people. Weinberg and Schwinger, among others,
have constructed so-called "chiral dynamics" Lagrangians
which describe the interaction between nucleons and
"soft" pions. A common difficulty of these formulations
is an ambiguity in the pi-pi interaction. The most
straightforward way of eliminating this ambiguity would
be measurement of pi-pi scattering lengths, but that
is a very difficult experiment. Olsson and Turner have
attempted to resolve the difficulty by calculating the
threshold cross section for the process T!" P—i TI " ‘1‘ n
in which process the disputed pi-pi interaction strongly
contributes. But near threshold the breaking of isospin
symmetry is reflected in large differences in phase
space volumes, depending on which mass of the supposedly
degenerate isomultiplets is used. For this reason, it
is desirable to extend the cross section calculation off

threshold to where the isospin symmetry incorporated in

William Frederick Long
the model is a more realistic approximation and conclusions
about the validity of different models can be based on
a larger set of experimental data. Furthermore, such
a calculation would give some idea of the maximum
energies for which the model, derived for soft pions,
could be applied. I

The calculation was done for five different charge
channels retaining all Feynman diagrams and computing
the integrals necessary to obtain total and differential
cross sections by means of a Monte Carlo method to
within four (4) per cent. The results indicate that the
best fit to low energy total cross sections was given
by a model in which chiral symmetry was broken by a
term which transforms as a rank two chiral tensor.

For incident pion energies much greater than 300 Mev

none of the chiral dynamics models employed fit the total
cross sections well. Comparatively little relevant

data exists for differential cross sections, but what
there is indicates poor agreement between experiment

and chiral dynamics predictions.

CHIRAL DYNAMICS CALCULATIONS
OF SINGLE PION PRODUCTION

IN PION-NUCLEON INELASTIC SCATTERING

BY

William Frederick Long

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Physics

1969

 

ACKNOWLEDGEMENTS

‘I would like to thank Professor J. S. Kovacs
for his help in selecting and solving this problem.

I would also like to thank Dr. H. Zing Ma for his
assistance in the use of the FAKE program and Dr. Jonas
Holdeman for use of his quadrature program.

And finally I would like to thank my wife for her

help in preparation of the manuscript and her patience.

ii

TABLE OF CONTENTS

SECTION
I. INTRODUCTION ....... z .......................
II. CURRENTS AND GAUGE TRANSFORMATIONS ... ......
III. CONSTRUCTION OF CHIRAL DYNAMICS

LAGRANGIANS .............
IV. ,PION PRODUCTION PROCESS .......... ..........
V. FEYNMAN RULES FOR CHIRAL DYNAMICS ..........

VI. MATRIX ELEMENTS FOR 1T1- N ""’1T+ TV + N
VII. NUMERICAL CALCULATIONS .....................
VIII. RESULTS ........................ ........ ....
Ix. CONCLUSIONS .. ..............................
APPENDIX A. GOLDBERGER TREIMAN RELATION ..... .......
APPENDIX B. WICK-DYSON REDUCTION TECHNIQUES ........
APPENDIX C. DERIVATIVE COUPLINGS .... ............. ..
APPENDIX D. MONTE CARLO CALCULATIONS ...............
APPENDIX E. PION SCATTERING LENGTHS ...... ... .......
REFERENCES ................................. .........

iii

PAGE

16

25

51

57

65

75

81

97

Table

LIST OF TABLES

Page
Experimental and theoretical total cross
sections for 11'“ -+fi"'fl"’ n . All
cross sections In millibarns ................. 84
Experimental and theoretical total cross
sections for U“ P" T7" IT ‘7) . All
cross sections in millibarns ................. 85

Trial run of Monte Carlo program with

qua,P£3°}.,oiz,pp= 001193,. ...... 129

Trial run of Monte Carlo program with

f(%£,Pijfi.tt°(-1>P})=Oml 61: ...... 129

Trial run of Monte Carlo program with

(a, a; .. -.. )'= among;
f?!) CI-Ci-FI‘ [an/““J‘... 132

 

iv

10.
ll.
12.

13.

14.

15.

LIST OF FIGURES

Diagram of beta decay process (1.2) ........

Diagram of beta decay with unrenormalized
vertex ........... .... ........ .. ...........

Diagram of beta decay with renormalized
vertex 000...... OOOOOOOOOOOOOOOOOOOOOOOOOOO

Vertex dependent on ‘71‘ ..................
Diagram contributing to process (4.1) .....
Diagram of Figure 5 with factors given

by Feynman rules for chiral dynamics
encircled next to appropriate

topological parts..... ...... . ..............

The five diagrams contributing to
the process (6.1) .............. ...........

Total cross sections for 11": --9‘n'° 1'1" P ..

Total cross sections for 11" p-—>fi"n‘ n .
Total cross sections for Tl" Pfin°n°n .
Total cross sections for Tr"P—) rr+n°P .
Total cross sections for IT‘"p-—)TT“'TTIH
Differential cross sections for
U‘P-‘>U°n‘p at T11 =4SO MeV....
Differential cross sections for
“'PfiTTITT‘V'I at Tu- =290 MeV....

Differential cross sections for

11.1-p._, “1'”? n at T“ = 357 MeV .....

Page

52

53

62

67
89
9O
91
92

93

94

95

96

Figure Page

16. Diagrams for elastic scattering of two
fermions in ps(pv) theory .. ................ 114

17. Diagram for pi—pi scattering ...... ....... .. 134

Vi

NOTAT I ON

Conventions for metric and gamma matrices, as well
as most of the other notation) has been taken from
Relativistic Quantum Fields by James Bjorken and Sidney
Drell. (See especially appendix A, p. 377.) Some

specifics of notation are given below.

Metric and Four—Vectors
A covariant notation is used with metric matrix

?q(3 defined by

file": 1: l

a... ‘= 9.22: 9331‘l

C}q43 1: C) 5 ar=# G!
The summation convention is used, repeated Latin letters
indicating a sum over three indices, repeated Greek
letters indicating a sum over four indices. Vectors
are denoted by a symbol with an arrow, e.g. 7;? , and
four vectors are defined by

rK‘r‘: Cm°, 5?),

“K.‘== Cytrfi!afifics==i:rx<a,"azj)

Products of two four-vectors are of the form

,qtkj':: (Viq’fir.(‘= 1°(afljyat'r ;;r‘ 1;:

vii

We define

a 5 .3—« 99's Q...
'r 306* ) 30"?

Integrals

We use the notation
foa3m rfoqm, c£IX1 09/713

form ‘7] cent, (9»... cameo/...

Operators
Operators are denoted by capital letters. Some

operators frequently encountered are:

Q

~‘

I
X

general operator

 

isospin operator

generator of axial vector current

Currents
Currents are denoted by script letters. Some

currents often used are:

A
i
ll

general current
q
(MK) := hadron vector current

hadron axial current

QC)
as
3
V
II

viii

Particle Field Operators

Particle field operators are denoted by Greek

letters, except for photon fields. Field operators

used are:
‘9 (m)
TCM)
¢(rx)
WAR...)

3(a)

general field

general particle field

general boson field

general fermion field and its adjoint

pion field

TCM),T(#\) = nucleon field and its adjoint

Aqtfh) = photon field

Constants

Constants are written in various ways, as the

examples below show.

¢O3}§§

02A #1.,
8.1:
em =

general boson mass
general fermion mass
pion mass

nucleon mass

weak coupling constant

strong coupling constant, taken such that
2 _

ratio of axial current COUpling constant
to vector current COUpling constant

Kronecker delta

antisymmetric tensor

Functional dependence and subscripts are frequently

suppressed to Simplify notation.

ix

SECTION I

INTRODUCTION

Recent interest in the application of the algebra
of currents to particle physics grew out of studies
of the weak interaction. It was found that the effective
Lagrangian density for the weak interaction, JL.V¢» ,
could be written as a weak current, 9x408), coupled

to itself, i.e.

lw(m)="%_: gwcr<fldrgz C M)

where (3' is a coupling constant which is the same

(1.1)

for all weak interactions. The weak current may be written
as the sum of a leptonic part, ‘5‘ C at) , and a hadronic
part, 950*). Each of these may be broken up into the
difference between a vector current invariant under

spatial inversion, and an axial vector current which

changes sign under spatial inversion. For the leptons,
)1Cm)=&.(m)'—}5(m)

and for the hadrons,

99.0%) ’- 9(M) ‘95 (M).

In the case of beta decay, for example,
n-—)P+e‘+'{3, (1.2)

the lepton currents take the form

#qCM) = Yak“) W? 1"; ((7‘) )

._ (1.3)
isqérx) =\K:(m) Yqu 71AM).

Assuming small momentum transfer, the hadron currents

take the form

gxmhicqu to.) ,

(1.4)
95.,(rx) = 2—3- Yp(m)\(9' Y5 Yr, 1M)
9v

The quantity 3A /9V is the ratio of the axial
vector coupling constant to the vector coupling constant.
It is measured experimentally to be 1.18 i 0.02. If we
use the currents given by (1.3) and (1.4) with the
Lagrangian (1.1) we get a four fermion interaction which
is not renormalizable, but which gives satisfactory
agreement with experiment if used naively in perturbation
theory to lowest order in (3’ .

Two aspects of the interaction (1.1) and the structure
of the weak currents are particularly interesting. First
of all, the coupling constant (3‘ is the same whether
the interaction is leptonic, semi-leptonic, or non-
leptonic. Second, the hadronic and leptonic weak currents

are constructed from their respective vector and axial

3
vector currents in a remarkably similar fashion. In the
leptonic case, the weak current assumes a symmetric
"vector current — axial vector current" form, and in
the hadronic case the weak current deviates from this
form only because of the factor aA/av in the axial
vector current, and fim/QV is tantalizingly close to
unity. Taken together, these two observations mean that
leptonic and hadronic vector currents couple in the same
way in the weak interaction, and leptonic and hadronic
axial vector currents couple in nearly the same way.
Let's consider first the implications of this for the
vector current.

The identical coupling of leptonic and hadronic
vector currents comes as a surprise because we would
expect a current of the strongly interacting particles
to be modified by the pion clouds which surround such
particles. In terms of Feynman diagrams, this means
that as far as the vector current is concerned, the
interaction of Figure 1 consists entirely of the diagrams
of Figure 2 with no contribution by diagrams like

Figure 3.

 

 

Figure 1. Diagram of beta decay process (1.2).

 

Figure’2.

 

 

Figure 3. Diagram of beta decay with renormalized vertex.

In other words, the vector current is unrenormalized
by the strong interaction.

A very similar situation exists in the case of the
electromagnetic interaction. The interaction Lagrangian

density for the electromagnetic interaction may be written

ole-10»): X EQ:CM)A¢V Cm)

(1.5)

where A-(CM) is the photon field operator and 3:3" ("4)
is the four-vector electromagnetic current of either a
hadron or a lepton. Here again we have a current partic-
ipating in a direct way in an interaction, and here again
we have a coupling constant, this time the fundamental
unit of charge, 2 , which is the same if the current in
the interaction is associated with leptons or with
strongly interacting particles. The fact that electric
charge is unrenormalized by the strong interaction
indicates that the strong interaction Hamiltonian commutes

with the charge operator' CD which generates the

5
electromagnetic current, and therefore the electro—
magnetic current is conserved.* This is eXpressed by
the equation
EQd:::n<:OK) =_ (:>.
Eng"

Returning to the discussion of the vector current

 

part of the weak current, we see that application of
arguments similar to those employed for the electromagnetic
current imply that 9(a) is generated by some Operator
which commutes with the strong interaction Hamiltonian,

and

qucm12 O.

. ' (1.6
3m” - )

We'd like to be able to identify the operator generating

 

the vector current. A clue comes from rewriting the
hadronic vector current for beta decay in the following

way:

9.: 4?qu “V..==§?Y«(’C’.+a’ta)i’

(1.7)

where \I’ is the eight component spinor

) . . . . .
and the (I: s. are the Pauli matrices. Now the isospin

current for nucleons may be written

E§3:;(:nk)‘= fi_:;;(n\)‘VCq';E? €1’<,0() .

(1.8)

 

*The formalism relating gauge transformation and currents
is reviewed in section II.

6
Comparing (1.7) and (1.8), we see that the hadronic vector
current for beta decay equals the "plus" component of the
iSOSpin current. Since, moreover, the isospin operator
a

'1‘ commutes with the strong interaction Hamiltonian

and the isospin current satisfies the conservation law
g“
a!
30$

we postulate the identity of the vector current and
...

isospin current, and identify .1. as the generator
of both. The identification of the vector current as a
component of the isospin current together with equation
(1.6) is called the "conserved vector current hypothesis",
which alphabetizes to CVC. A consequence of the conserved
vector current hypothesis is that form factors of the
nucleon weak current and the nucleon isospin current
are, because of the Wigner-Eckart theorem, proportional
to one another, the proportionality constant being just
a Clebsch-Gordon coefficient. This has been verified
experimentally in analyses of the beta spectra of decays
of B12 and N12 into C12. 1

The case of the axial vector current of strongly
interacting particles is somewhat more complicated. The
success of CVC makes it tempting to postulate that the
axial current is also conserved, but two facts militate
against this hypothesis. First, the axial current con-

tribution to the hadronic weak current enters not with a

factor of unity but a factor an/av = 1.18 i 0.02.

7
This is close enough to one to suggest that the renormal—
ization of the axial current is slight, but far enough
from one to indicate that the renormalization may be not
ignored. Second, a conserved axial vector hypothesis
.1
3.3.5.921): O (19)
3M"

would forbid the process
1'? 6/14, + v. (1.10)

This decay is governed by the matrix element

<1 9;(m)ITTCCf)7 (1.11)

where l :> is the vacuum state and Irrcrar) :7 is a
state with one pion of four—momentum (i' . If (1.9) holds,

then

<la" 9;:(0‘H 77(117 =‘S.31<19;(M)IWCC‘.)) (1.12)
= 0

so that the matrix element (1.11) vanishes and the decay
amplitude is zero. Since the pion decay process (1.10)
does take place, equation (1.9) cannot hold. However,
(1.12) contains an important clue to the resolution of
the dilemma. If the pion is at rest when it decays,

(1.11) becomes
fl<l§§<mHfl7=O

where l/QL is pion mass. We see that if /6& = 0,

8
(1.9) could hold without forbidding pion decay by the
process (1.10). Therefore we say that the axial symmetry
of the strong interaction is broken by the pion mass.
The simplest way of accounting for the symmetry
breaking is to write the divergence of the axial current

as follows:

go
3.95 = constant x fl x some pseudoscalar Operator

This has the correct parity and reduces to (1.9) in the
limit as‘/°& "fi> 0. The simplest pseudoscalar operator
we could choose would simply be the pion field Operator,
d d
a . The Operator @ is an isotriplet, so we

incorporate the axial vector current in an isotriplet

and write

...—3

d
aqgs km) =constant x/(A. x @Lm) (1.13)

It was shown by Gell-Mann and Levy2 that the constants
in (1.13) must be chosen such that
d
’301 95C“): ...-[:1
‘2 av

and that it is possible to use this formula to derive

1....)
M @V’“), (1.14)

1f

by a field theoretic approach a relation between the
axial vector form factors and the pion decay rate which
was originally derived from dispersion theory by

*
Goldberger and Treiman. ’3 Here, (a, is the strong

 

*The Goldberger—Treiman relation is discussed in more
detail in Appendix A.

9
coupling constant and P" is the nucleon mass.
Equation (1.14) forms the "partially conserved axial
vector current hypothesis", which alphabetizes to PCAC.
However, PCAC by itself gives no clue about the nature
of the operator generating the axial vector current.

In an early attempt to discover the group properties
of the generator of the axial vector current, Sell—Mann
and Levy proposed three Lagrangian models incorporating
PCAC and CVC;4 a gradient coupling model, the sigma
model, and a variant of the sigma model called the
non-linear model. All these models were unsatisfactory
by the criteria applied, the first and third models being
unrenormalizable and the second requiring the existence
of the never-to-be-discovered sigma meson. Despite the
drawbacks of these models, there were some useful results
of this line of inquiry. The most important result was
the Lie algebra of the generators of the vector current
and the axial vector current for the sigma model and the
non-linear model. If we let .35. be the Operator which
generates the axial vector current, these two models

yield the commutators

‘[ .T:a , -T:& J (:(S adhz‘wfie
[Ta )Xb] = Ce wk. A; (1.15)

[Xe-)Xb] 3 C64.“ T;

If we define two new triplets of Operators by

10

F6.“ (1‘:+—>'<.)/2

_a .a _, (1.16)
L.=CT-x)/?_
routine algebra reduces (1.15) to the symmetric forms
[R«,Rb)= (:5;ch
(1.17)

1: L-Gt) 1.6r] :: C Euxbn. L.¢

[ l_.a ) FIsd] = (3, .
Equation (1.17) shows that the Operators F?. and .E:
generate two independent SU(2) groups, and together they
generate the symmetry group SU(2)R ® SU(2)‘_ with
subscripts referring to the operator generating each
SU(2) algebra. This group is called chiral SU(2)(::)SU(2)
because of the similarity of the forms in (1.16) to the
right and left hand chirality operators of field theory.

It turned out that (1.15) was correct as far as
it went, but it remained to be explained how to incorporate
strange particles into the theory. Clues to the solution
of the problem were furnished by the success of SU(3) as
a symmetry group of the strong interaction and by the
Cabbibo theory of the weak interactions which placed the
weak currents in SU(3) multiplets. These discoveries
set the stage for Gell-Mann's current algebra hypothesis.
Gell-Mann's hypothesis made the logical extension

of chiral SU(2) ® SU(2) to chiral SU(3) ® SU(3).
This may be done by changing the 6:01,, )5 of equation
(1.17) to the structure constants of SU(3), conventionally

denoted lab: I and allowing Q , 6' , and C. to run

11

over the eight indices labeling SU(3) Operators. Equation

(1.17) becomes
E FQG) f?br] : <: fLaMH: F1g
E ‘_.a )1_6r] : C: fiat! L—c.
[ La” R6] 3 O _ (1.18)

(

The operators are related to the eight generators of

vector currents, fi; , and the eight generators of axial

vector currents, F5“- , by

f?£L :' C Fae i- Fisa1)I/:l-
L_.o.‘= ( Fig ‘ fié<l) //:L .

(1.19)

And finally, (1.15) becomes

.[FQ’FL)=‘.I"6‘E (1.20)
[ Era.) FESb]
[ F2§a1)F135:]

H

5. f'klhk Elie

£1.11". F;

N

Gell—Mann went one step further than this, however. If
r—-

we denote vector and axial vector currents by d) (m) and

¢_.

dfi5(}‘)respectively, Cell-Mann postulates the following

equal time commutation relations for the octet charge

densities:
O

(§:(m),’a~1°(7)]mo.,a= rake-3),... E m)
fa: (...) ,a5,(.,)1,... ,«- = 4 2.36.2 4),... 5;: (m)
[Escafisicpnaf = ..3 (aghm aim).

12
Since Fa=§o€3m 3":(m) ,and
Fsagfd3m 3:5: (M) , (1.21) integrates to (1.20)
and is a stronger hypothesis than (1.20). Equation
(1.21) is referred to as the current algebra hypothesis
and it forms the foundation of most recent work in current
algebra, along with PCAC and some version of PCAC for
strange particles.

One interesting aspect of the current algebra
hypothesis is that it is free of specific assumptions
about the dynamics of the strong interaction, and
therefore freed of the limitations inherent in the
perturbation eXpansions of quantum field theory. Yet,
the hypothesis is not sterile and permits calculation of
many interesting physical quantities through manipulations
of matrix elements of the currents. One particularly
interesting class of calculations is that of so-called
"soft pion processes". These calculations express the
matrix element for a process in which one or more pions
of small four-momentum are emitted or absorbed in terms
of certain equal time commutators of currents and the
matrix element of the same process without soft pions.
Soft pion calculations have been applied to ”<13

7’8 pi-pi and pion—

10

decay,6 multiple pion production,
nucleon scattering lengths,9 pion photoproduction,
pion production in pion-nucleon interaction near

threshold,11 and many other problems.

13

But the model independence of the current algebra
.hypothesis was eventually compromised by the introduction
Of specific Lagrangian densities which were vastly easier
to work with, if somewhat less elegant. The first of
these "chiral dynamics" Lagrangians was introduced by
Weinberg.12 His model was developed by transforming the
sigma model in such a way that it would reproduce the
results of current algebra if used to lowest order in
the coupling constant, <3 . The problem of renormal-
izing higher order contributions of perturbation theory
was overcome by fiat: one just ignored terms leading
to divergences. Weinberg's model included only pions and
nucleons. Soon Schwingerl3 introduced a Lagrangian
model for pions and nucleons which also satisfied the
restrictions imposed by the current algebra hypothesis,
but which differed from Weinberg's model. Since then the
chiral dynamics approach has enjoyed increasing popularity.
Several different papers have been published on different
methods of constructing pion-nucleon Lagrangians,l4 and
many authors have published applications involving
extensions to other particle fields.15

As useful as chiral dynamics Lagrangians have been
found to be, they still cannot be considered a true field
theory for the strong interaction in the sense that (1.5)
forms the basis of quantum electrodynamics. The reason for

this has already been alluded to, namely, we avoid using

these Lagrangians to more than lowest order in the coupling

l4
constants, though higher order terms are manifestly
divergent and therefore non-negligible. The algorithm
for using chiral dynamics Lagrangians reduces to this:
Use the Lagrangian with the ordinary Feynman—Dyson rules
to lowest order in (Q. , but include only contributions
from tree diagrams. Tree diagrams are diagrams which
contain no loops and no internal momenta to be integrated
over.

Two main points of View are held about chiral dynamics
theories. They are exemplified by the attitudes of
Schwinger and Weinberg, two of the first to participate
in the Lagrangian revival. Schwinger feels that
Lagrangians represent a suitable way to investigate
strong interaction phenomenology and to remove the weak
(interaction orientation of current algebra. His rebuttal
to anyone who objects to the chiral dynamics algorithm
explained above is,

It is not meaningful to question the use

of coupling terms 'in lowest order'. That is

the nature of a numerical effective Lagrange

function, which gives a direct description of

the phenomena.1 .

Weinberg, on the other hand, is less sanguine about such
procedures, and clings to the primacy of current algebra.
He says,
Opinions differ as to whether any fun-
damental significance resides in the

Lagrangians which have been used... [I]

myself remain uneasy at using a symmetry on

the phenomenological level, when it is not

clear how any fundamental Lagrangian could
give rise to the supposed symmetry of

15

phenomena. From this point of View, chirality

is in good shape because we have current

algebra to underwrite it...17

But no matter what philOSOphy is adOpted, the
utility of chiral dynamics in practical calculations
cannot be denied. Besides, strong precedent exists for
the use Of a Lagrangian with shaky underpinnings in the
case of the weak interaction Lagrangian (1.1). Had
investigators had too many scruples about employing such

an unrenormalizable Lagrangian, current algebra might

never have been discovered.

SECTION II

CURRENTS AND GAUGE TRANSFORMATIONS

If a system may be described by a Lagrangian
density L , which is a function of n fields ?m(m)
and their derivatives 3., ‘9“(m), the equations of
motion of the system are given by the usual Euler—

Lagrange equations

9.5. .. i 9 L =O;n.=l.~-,m(2.1)
38.0») an." acaexcmxad“)

We are interested in the effect of a small change in

each field ?m(m) , of the form

3.021)") f..(«) - c'e AM VAC/x) (2.2)

where 6 is some small constant and )\na is a
constant matrix or, at most, a function of boson fields.
Equation (2.2) is called a gauge transformation. Using
the chain rule, along with (2.1) and (2.2), we find for
8 L , the change in the Lagrangian, after all the

have undergone gauge transformations,

8 11H‘1<3);l.18 ‘3n. 4- 69111. (S (éDq'th.)
Eb'ffix 39(fgarafn)

a-c'eAMiLL 3_______L ]‘€a+[ #3 9.31
fid"€>(€hr‘al) 2§Li§q”fh) EDaC'

16

3

17

$1 =.- E. l Fifi—4.... /\M YoCm)](2.3)
3m“ acaq mm» ‘

Let us now define the four vector current generated
by the gauge transformation (2.2) as
" . L .
9(M)':“ ..‘2 l Am‘ea((1\))
3(3 (mm/am") (2.4)

and define its "charge" as

QC()= d3“ 9.0(m).

an space (2.5)
If <1: is invariant under the transformation (2.2),
3 1 = 0 and from (2.3) we see that go‘C“) satisfies
the conservation equation
a... 9%...) = O.
(2.6)

Furthermore, for a conserved current, C2. is a constant

of motion since
dd

9g =fJ3m ESQ—(0‘): -foersm V \ (m)
. t

a t H space

~f$ .135 ”C3700 = o

u

.1
where we've used Gauss' Law and assumed that S)<:fl\) has
finite extent and therefore vanishes on a surface ii at
infinity.

In canonical field theory, we write that the operator

TTACM) conjugate to the Operator ‘en < M) iS given bY

fin (at) = <9 L .
a ‘9nCd) (2.7)

Using (2.4), (2.5), and (2.7), we get for C2

 

18
~31
O=M$~<l~< 3;.) A... )6.
Q = "(In-13¢ ’ITme) /\m ”(a Cm). (2.8)

We shall need to know the commutator of the operator C:
with the field Operator faméx). Assume first that

we want to know

[0(fi), @PCR‘fi‘U (2.9)

where ¢P Cd) is the pth boson field. The usual
quantization procedures require the following equal-

times commutators:

[$aém), e.c1,)]..-,.=
E 17.9.), n. (17))...» ,.=

( 1mm), ¢.(.,)],.,.,,,= -.' s... a

00

1
U

C fi?-—"'). (2.10)

Applying (2.8) and (2.10) in the evaluation of (2.9),

we get

tom, ¢P<z,:)]
=[-(MilanesAMexgx),23,451.12)?
. -.‘ [.437 [ 17.41330, (3913,01 ./\M‘,Dz‘.<;,t)
= - a fat-K, 2-; gms3ca-s;)3.\wn.(;,z)

. [Q<t),¢p(fi€,f)] = --/\‘M 0Q(R,t. (2.11)

l9

Assume instead we wanted the commutator

(Oct). r...<s:.t)l

where K()pnr Cm) is the V th component of the pth

(2.12)

fermion field. This time, quantization procedures yield

the anticommutators

S ‘rz.M,-VC5K) sf»? )CSC%)}U«.=17o
iY; rm +:1-3,(3C:)3«o 4..

i 1:,chrx),‘13:,?¢3Cc7f)3r~.-c1.

H

O
O

H

 

(2.13)
The operator conjugate to WA)” is
f
a I. 2 (a. Y“ ) 0’ )
’9 Y“) of

so the charge operator becomes
+ ...-I .
Q(t\:S.C03‘?Yfl)’VCC;)t)/\M *a,«(.c7,t ).
(2.14)

Evaluating (2.12) using (2.13) and (2.14), we find

[QC() ‘1’ ,Y'Cm,t)]
‘[fo?3 4+“,4Cc4‘p) /\M YM—CC)‘. f) )Kflp,Y(f-;:,t)]
[<93 ‘1. /\T/+M[YA,e-e(ci,fi)i‘(’a, «(4,19, YF‘V «’93
‘2 Ya “4(6): f)‘i’ra ,Y‘CM,t)K+)a ,ad 4,1“)
+i T3,.Cg,t) fissure} 11.2; e)
"‘2 +p,~r C17),fi)\i’m.,(f§? 1‘.)Ya,er\.a‘,t) ]

‘3 g O‘3k? }\rua‘;‘ 8’3 (’1:“L7‘>§”V3:jlai\r;:63’:!]

20
(OR) , ‘1’,..,Y(.7:,0]=-....,\rm~nm,(5.“t ). (2.15)

Since (2.15) is in the same form as (2.11), we may write

for any field VFW?!) '

[QCt)) cha.t)]=~/\P¢raca‘}t). (2.16)

~

Now let us define an Operator 6" which generates
the gauge transformation (2.2). If we transform some

field ~€n(m), G- is defined by

2““ \encm)a“‘G= face» “A... race)

But, since 5 is small,
2"“; Heat...) p.""‘°g(l+¢eG)ve..am)(Luge)
: ‘en-Cm) 4‘ (..Q [G, vnQM)]

and so
[G 3 “ad/«)1 “ "/\M “6007.)

By comparison with (2.16), we see that effectively <¢P = Cl,
and hence the charge (:1 is the generator of the gauge
transformation. It is important to realize that (2.2)
will give a different form of %K M) for every different
Lagrangian, but in every case the charge (:2 obtained
by integrating %o (M) will generate the gauge trans-
formation which we started with.

Since CQ generates the transformation of all

operators, we may apply it to the Lagrangian density.

21

Hence, under the transformation (2.2),

:Z:.—a) Lz;"_. ‘2 ciicl‘JL'JZ-<.E.C|
§ 1+L€[Q, Z]

The change in the Lagrangian induced by the gauge trans-
formation is » ‘
.
E>.1: 2 .2: ".l: ‘ a £.l.<:), 4:.1.
But from (2.3) and (2.4),

‘§.2:1= 6i 29a! §9f*§.n‘) )
(I [Q(t)) 1):;«r)°r\m)o

Equation (2.17) is a convenient way of finding the

so
(2.17)

divergence of a current without actually constructing
the current or the equations of motion of the system.
What we usually know from experiments is that some
quantum number Cfi. , such as charge, strangeness, or
baryon number is conserved, hence some charge operator
(2! commutes with the Hamiltonian. Next we try to

deduce the commutator

(Git), {afradi

for whatever fields fB‘CFx) are involved. Then we
construct the Lagrangian from the fields in such a way

that

[Q(t),1(“en<m),3q Ken/...»? 0.

22
Y )

If Cu. is not conserved, but we know 3% 9 0‘ , the
divergence of the current, it is trivial to use (2.17)
to generalize this procedure.

One common situation is when we have a Lagrangian
invariant under transformations generated by a set of
operators i Q a 3 which form a Lie algebra with

structure constants Ci. aJJ-r. .'
I: (:lca, (3)6rl‘2 i fi}.a1h;<:)c .

Particle fields @QCm) are composed of matrices
involving one or more fields ‘QAC m) , boson fields
being composed of one field, fermion fields of a spinor
with four fields, etc. If then the particle field
operators are tensors of the n-dimensional representation

of the Lie group of the operators (SICK . then

[QM ®6Cm)]= - (Ana Chan)

(2.13)
where /\a is the at th matrix of the n x n repre-
sentation of the operator Qa . Equation (2.2)
generalizes to

¢a(m)-* (1340,0- ( 6.6( Air)“ (P (m)
=CPaCrn) + (.6213- [Q5 (I)... (00]
(2.19)

65 being the 6‘ th member of a set of small numbers.
An illustration of the formalism discussed above is
the case of the isospin gauge transformation generated by
—-—§

the operator .1. which satisfies the Lie algebra

23

[ _T;a ) -T‘b-:l ‘=. C ‘Scalhz _T:;

The nucleons form an isodoublet, and since the 2 x 2

representation of the isospin generator la. is

(ta)6t ‘ (1% )61. , (2.18) gives
[Ts ) K{Jo-2‘ : - £32.)”; Kl);

'2.

The pions form an isotriplet, and the 3 x 3 representation

of .T:a is

(:tha,)&gy“ " C fzcaxh:

(2.20)

so (2.18) gives

[TQ\@CI‘]=~:QW fig.

4 (2.21)
If in (2.19) we let 6(7' €§@)c , where é is a
unit vector in isospin space, (2.19), (2.20), and (2.21)
give for the isotopic gauge transformations of nucleon
and pion fields

i’<a>—+~1’(a)-—;aa ’8. (Si/(...)

300—9 $(m)“€ C JUNK 3) , (2.22)
If we take for our Lagrangian density the free field
Lagrangian

.... ... _. _. ...

L....=T(a2~r1)i+a<a.da*2~ “((52-3)

the current generated is
531..) = a ?(a)v.,i¢i’cm) + 3(a) x3. (8(a).

Since lfv-ee is an isotOpic scalar, (2.17) shows

24
d and
that a qum) = 0. If we went on to evaluate

d ——-h
T: 5,303,.“ 9°C“ ) and used the second quanti-
zations (2.10) and (2.13) to evaluate [Ta )Tlr] ,

we would recover the Lie algebra I. Ta ) Tb] : (Quake-r:

which we started with.

SECTION III

CONSTRUCTION OF CHIRAL DYNAMICS LAGRANGIANS

Since Weinberg's first paper on chiral dynamics,
several different methods have been put forward for
construction of chiral dynamics Lagrangians. The method
we shall use, that of covariant derivatives, was
developed in a somewhat later paper by Weinberg.18
The discussion here follows closely the discussion in
Weinberg's paper.

We shall restrict ourselves to Lagrangians con-
taining only pion and nucleon fields, hence the symmetry
group we'll be concerned with is chiral SU(2) CS) SU(2).
The program will be to construct the gauge transformation
of chiral SU(2) (:) SU(2), then to construct a chiral
invariant Lagrangian, and finally to add to this chiral
invariant Lagrangian a symmetry breaking term which
reproduces PCAC.

As discussed in Section I, the generators of vector
‘ A
and axial vector currents, 1. anui >< respectively,

satisfy the Lie algebra

[ ‘r;.,-T2»] :‘ a Etna: -TZ
I 1::, ><6r1 : ‘: Gena; 77“C
l: Xe )Xb] = (Gal: 1: ' (3.1)

25

26
...—3
We have already written down the commutator of ‘T’ with
the nucleon field Operator and the pion field operator
in equations (2.20) and (2.21), and used them to derive
the gauge transformation (2.22) which generates the vector
current. We could immediately write down the gauge
_—3

transformation generated by )( in the same way as we've
done for :1: , if we knew the commutators [ )( ) ér(ra) ]

...-l .
and [ X , ‘i’ (m) ]. However, attempts to make physical

d
fields tensors of the operator )( have not been suc-
*

cessful, so we have to use other means of evaluating
these commutators.

First, let's consider the transformation of the

pion field. We wish to evaluate the commutator

[XQ‘ éé. ]. Define a function Kalb- ( Q ) by

"6 fqb(3)=[xa, Os]. (3.2)
Our problem now reduces to evaluating fiWC ) . We
do this by deriving a pair of simultaneous differential
equations for /°I46- ( a) .
We will need two identities to derive these differen~

tial equations. One is the well—known Jacobi identity

[A,[B,C)]’[B,[A,C]]+[[A)B],C].(3.3)

The other identity is

[QQ)/\(‘en)]: 95 [(1%) 38"] (3.4)
E§‘€V\
where Qq is some operator and I’M/Y“) is some

 

 

*The sigma model was one such attempt.

27
general function of the set of n fields f {“3 .
To prove (3.4) , expand A (VA) in a Taylor's series
around some set of constant fields 5 A very close

to YD. . Our commutator becomes

[QM A<Ya>1
= [0a, ms). 3%....) (a- we].

Because A< gm) and 3 A I g are constants,

this becomes

 

[Qa,lr.(‘ea)]=_3__l [Games].

But g“ ’5’ Yr). , so we arrive at (3.4).
Apply the Jacobi identity (3.3) in the following

form:

(T..(x.,(a.))=[x.,( T..@.11+HT~><61»@«1-

Using (2.21), (3.1), and (3.2), this becomes

[Tan go‘( 3)]=cewwa3)+tem/c¢t(3)
( )

3.5
Using (2.21) and (3.4), (3.5) reduces to

QéQECE) eases 9.3.an(3)6.¢4 + ’aeng)=a1&-
$64 I ‘

This is our first differential equation for fatC @ )

Now apply the Jacobi identity in the form
[ x.,[x(, (8‘11-(x.,[x., <3.]]=[[2<.,><.],(B.].

Using (3.1) and (3.2) with this identity, we obtain

28

[Ka,;h(3)]‘[)(b,fu(¥)1= ‘Coélabd Golan 62-

(3.7)
The tensor identity

€¢aem= SaeSK~3wz 5cm

(3.8)
simplifies the right hand side of (3.7), and using (3.2)
and (3.4) to simplify the left hand side, we get
3§e(§)fw(§)_bégsC3)faC3)=3dr «Mart...
334 a <34 (3.9)

This is our second differential equation for d—kc 3.)
We must now solve equation (3.6) and (3.9) for rh(§1 .
We note that (3.5) indicates fécc 3‘) is an isotopic
—-h
tensor. Since both Y and g have odd parity,
[ Xq, @6' ] must have even parity, and (3.2) indicates
{RC 3) must also have even parity. Thus we take as

a solution for 5.6:

1.643% Sec/.C3111- @ér @c OJC an)

1 “‘2
where [-C a ) and 7‘ 6 )are functions to be determined.

(3.10)

If we substitute (3.10) into (3.6) we get no restrictions
‘on [- and C? . This is not the case if we substitute

(3.10) into (3.9). From the chain rule, for a general

A
1
function A( 3 ), we get

321; e. 2 <34 A'
95.9 (3.11)

a...

where a prime denotes differentiation with respect to ‘8

Using the form (3.10) and the identity (3.11),

29
éliéés itamfl'=: ésalf 43‘,f‘?’ 't Skye <54},’
'SécQ +25h§a(%l+3lfif’)
+26e®eéccg1+ 2%,).

From this,
a 5‘ ad~3 H=( -2 5-231 ’(Saefiré (3..
52% fi‘ (7 H 3‘“ ’ H

Substituting this in our second differential equation

(3.9) gives

(54-20422 33406.... (Sealer a... 6. as... (a...
which is satisfied.if? 6- and i} are related by

I
$11: I +':2 .
1 I "
§-2 & (3.12)
Equation (3.12) gives the only relationship between

A1 a
&( a ) and ?,< ‘3 I). This means the commutator

[2(a) ($61

——.
I
is not unique, since f( a ) may be chosen to be any

(3.13)

isoscalar function of the pion field, and each different
I. / A
choice of /.( a ) gives a different function b—alrx é)
which is proportional to the commutator (3.13). What we
‘1) . .

do when we choose {C <8 15 to choose what isovector
. , . . . . / / Ef‘-)

quantity we 11 call the pion field. Changing a, x.

is equivalent to redefining the pion field by

...—3 3‘ é AI “‘1

9 . a 6@Cé )where @C 6 ) is some isoscalar
. . . . / .3“)

function determined by the new chOice of f \

Once we have settled on a definition, we can calculate

30
(3.13) and determine the axial gauge transformation for
the pion field as we've defined it.
. . , -*1
We shall restrict ourselves to the form of f C 6 )

19

used by Weinberg and Schwinger,20 namely

{(3‘)=~_LCI~A° <3“)
‘2)\ (3.14)

where >\ is a constant which we'll determine later.
1
Using (3.12) we find the function ?( a ) corresponding

4

2
to our choice (3.14) for {C 6 > to be simply
..1
3&6 )~ __ /\‘ (3.15)
Our result for the commutator (3.13) is then
. 1A: 1
)\ (3.16)

We now turn our attention to the nucleon field
and the commutator [ .Xq ,‘I’b- ] which determines its
axial gauge transformation. Define a function U'a.‘< é ) by
—A
‘:;K“‘)ff;‘] : Lfiapr‘é')1.i;3?)ccfl KTZJ.
’1 (3.17)
Our problem is now reduced to evaluation of 0:146 C 6 3
and again we solve it by employing the Jacobi identity
twice in order to obtain a pair of differential equations

for U'ab(§‘),
The first form of the Jacobi identity we use is
[11,[xc,1’.]l=[X/.,[T.,T.)] dings]. it).
(3.18)

Using (2.20), (3.1), (3.4), and (3.17), equation (3.18)

31

yields our first differential equation

ail. 6.1.9. <8, = £17,460”; + (roe... 60.13.30.
a QA’ (3.19)

The second Jacobi identity we use is

EXMLx.,i’.1]-[x.,lx.,i’.1]=[[><..><.1.it].

(3.20)
Using (3.4) and (3.17), the left hand side of (3.20)

becomes (suppressing matrix indices on ‘i2 and ’t3 )

fx.,[x3,‘1’]]~[><.,[ x..‘1’1]
-[Xq,U—a-c]32.:zfl‘[X6-,qulgz\f’
'r'[ 1;}; , ’E§;£.} (J‘émfl Lraq: ‘1’.

From (2.20) and (3.1), the right hand side of (3.20) becomes
[[XQ)X6']) ‘1’]: C&W[TC)T]:‘Q\Q1m:c-T

Equating right and left hand sides, we obtain

[x., we]?! -[x.,u..]

N
5-.“
2

+Jwva£[’t_¥,’§’_c]=~gew ”it" (3.21)
2

Using (3.2), (3.4), and the commutation rule
1,4 "5—‘6; 2’
[1”? 4.2.1.“.

equation (3.21) becomes

av“ ad- U38 6192 ‘qucvo-cQagdgi-QW
a $4 6) ¢QQ (3.22)

This is our second differential equation for U'oJrC <8 ) .

32
We see that (3.19), our first differential equation
for URL-( 5) , has the same form as equation (3.6)
..D
for l—mlr( é ) . Since (3.6) just arose because [.afi—C a.)

a

is an isotopic tensor, U‘aué-C @ ) must also be an
——8
isotOpic tensor. Since X has odd parity, (3.17)
shows that qufir must have odd parity, so we take as
our solution the form
(3)- 5‘)
‘Jnels ‘ <£<L61. ibc, LP (
(3.23)
1 0 ~ I o a
where U‘( 8‘ >15 a scalar to be determined. Substitution
of (3.23) into (3.19) gives no more information about
'1
U( a. ) , but substitution into (3.22) does.
Substituting (3.23) in (3.22) and simplifying the

results using (3.8) and (3.11), we get
6.1L. €6-ufl go..- €cuaJ éb'JXiU'j‘i-z U’[/+3;]}

‘f:2<E<~6¢ xiv: =' 530, qgnrifithuxg bqu‘b éiauéc.
(3.24)

Using the identity
“12
fl 6w “T‘Eaficéc‘pn‘tebu‘ga @;+Gm@¢. <34

(3.25)
2
to eliminate the term $4 @9 6. M J from

(3.24), we thereby obtain
{BJEQM £3..- eame <3Hxiu1;+2a~’[jj + 331~J1
A: 1
+€qg[201+ @ U‘- I] = O.

(3.26)

To satisfy (3.26) we must have

33

‘2

u-a),+QU-; (bl-pal? )‘U=O (3.27)
244-2522Lr—I=O. (MB)

The latter equation has the solutions

U‘= #W (01 1V7)?

(3. 29)

 

A long but straightforward calculation shows that

U‘ = (f 1V&1+ @‘ )‘l is also a solution of (3.27).

Substituting (3.29) back into (3.23), we have
a a “)
(3.30)
From this and (3.17) we see that once the pion field
7‘)
is defined, i.e. iC Q is chosen, the nucleon trans-
formation is also determined.* ’
- 1
In particular, let's go back to the form of f( g )

we've chosen in (3.14). This gives for (J' the two

JC¢2)=;\JE1

The first of these yields a non—linear gauge transfor—

solutions

mation with unpleasant properties near where '1: =r0,
so we take the second solution. With this solution our
commutator [Xq ) “1’6. ] becomes

[X‘.i’]=~%(?x5)‘l’.

(3.31)

 

*It is interesting to note that the above discussion may
be generalized from the nucleon field ‘f to a general
field 4 by simply eplacing Tia/'1 by the matrix
representation of appropriate to § .

E

34
With the commutators (3.16) and (3.31), it is easy
to construct the axial gauge transformation using (2.19).
Taking E6. ’ E C «fl-)6- where .8. is a unit vector

in isospin Space, the axial gauge transformation is

T(m)"’ TCM) " <.€/\ f [ 3(a)) xii] T’Cm)

 

12
B<m)‘9 ECM) - 63. {Al )_|--){1 ECm)' 30-4)]
’ 1%

" g): [ .6. EUR” 5(a). (3-32)

Now that we have the gauge transformations generating
vector and axial vector currents, we wish to construct
a Lagrangian invariant under these transformations, to
which we'll later add a symmetry breaking term. Because
the vector current is conserved, we know our Lagrangian
must be an isoscalar. But we must further limit our
Lagrangian to isoscalars which are invariant under the
axial gauge transformation. From (3.17) and (2.20)
we can show that any isoscalar function of the nucleon
field ‘1’ must also be invariant under the axial gauge
transformation. To show this, consider come isoscalar
function of the nucleon field AC ‘1’) . Because it
is an isoscalar . ___.

L T. AC 10] = 0.

But from (2.20) and (3.4)

2‘- 73.) A
[T.,A<‘1’)1 (:3: ..o 1.4%:

and so

(2:).2 “Kc as . o.
€9‘f2_ (3.33)

35

l

The commutator of AC ‘1’) and the generator X1 is,

using (3.4) and (3.17),

[XL AC?)]= vh(§)wp?.og~5.
:2

But using (3.33), this just becomes

['2,A<i’)]=0,

so A(q’ ) is invariant under axial gauge transformations

(2

h

as asserted.

The form of the commutator (3.2) makes it impossible
to go through a parallel argument for isoscalar functions
of the pion field. However, it is possitle to construct
a function proportional to the derivative of the pion
field, for which isoscalars are also chiral invariant,
just as was the case for .i’ . Define this function to be

.1
[)-( §§QL=‘C§Lq5C7)§ ) 23¢ éblS.
(3.34)
The quantity Dar Q is called a covariant derivative.
We assure that isoscalar functions of the covariant
derivative are chiral invariants by choosing J «Jr such

A

that Dar @ transforms like KP , i.e.

[x°1 Di ‘5‘] 3 “c°Uab(¥)€6cJ Der <54 (3°35)
[Ta , D-( @c] = C. €ox40 Der @J (3.36)

where we've let (Tb-)ca ._....a, _. {E 61.0.

The first step in obtaining JMC Q ) is to sub-

stitute (3.34) into (3.36). We shall need the identity

36

HQ... 9., men] = a. (Q... eon], (3.37)

To derive this, use the usual laws governing commutators

to obtain

(Q... 9.. 10.60) = 9.10. 364.0] + IQ.a 9.1m“)

and note that [Qanaq ] = 0 since Qq and 3.,
operate in different spaces. {Using (3.34), (3.36), and
(3.37), we obtain
fT.,D.@.1=[TZ,J¢;>.<Z>.1
= &e.6. c.’ saw ( 8e @J)+[T;,o?e(.](9e <53)
3 L. 6w 00.2. C a? @192). (3.38)

A little rearrangement of (3.38) gives

[12,004.]: c'eqMoUwO+ c‘ Sadr-flour,

(3.39)
By this time we recognize that (3.39) indicates Cpsab‘ is

an isotopic tensor. This suggests that it may be eXpanded

like 5-0.4- , i.e.
coacéhaaazce‘). e. e. dr(31),(3.40)

d ...—.2

where 4C 62) and WC @ )are scalar functions.
We determine C9 and (.J‘ much as we did i. and 3’ ,
that is we form differential equations from (3.35) and

1 '~*z
(3.36) and use them to deduce C.“ 3 ) and vU'C @ ).
If we do this we find, as we've come to eXpect, that (3.36)

“'i “*1

gives no further information about JC Q )and bxf( é ).
All restrictions on these functions are obtained by

substituting (3.34) into (3.35).

37

Using (3. 2), (3. 34), (3.35), and (3.37), we get

[Xq,Ja]3-( @z‘afla 3 0:9 962.
'7 "LUalrEGu-I oVan. 3v®a

For this to be satisfied,

[Xa,ch.6(= icgoc9%_54; —- 5 U3“ €WoJOQ‘”(3.41)

Straightforward, but tedious, manipulations using (3.2),
(3.4), (3.11), and (3.40) give for the left hand side

of (3.41)

[quQoeh-diéh @%(2H'+131?0Q')
+3“ @banJ‘ t 2% a8;
+ gin <56- gh;<:2 ’-+

Equally straightforward, but even more tedious manipula-
tions using (3.10), (3.11), and (3.40) give for the

first term on the right of (3. 41)

L oQer 33,33245812 5321/. +8.25 (gag-r5?!)
+5MJ 34+ 6 536 (2.2? .24 may).
Finally, use (3.8), (3.23), and (3.40) to obtain for
the final term on the right hand Side Of (3.41)
‘t’ U‘Qngt-AOQ&6 =~c1f 80.6- 504 oQar
~3... QMJwJ. 33w). e. <23. e. M}.
V (3.44)
Substituting (3.42), (3.43), and (3.44) hack into (3.41)

and doing some rearranging we finally obtain

38

I

362. §q(’2 ’+'2 513,0? + 74:0.
+3M5GCfl+2%’+éu+§W)
+§dQCCW+afl+gW—cbf)

+5.. @6@c(2i~r'+2$1w' «Pawnee;
+wa.’+2$2ur3'-W3 2 O.

(3.45)
For (3.45) to hold, (.9 and Ld' must satisfy the four

coupled differential equations

:2jkfl’sh:2 Egczicg' 4-C1c2 :<:> (3.46)
W+Qofl¢’+o&r+ $2M =0 (3.47)

A

furi- wdtézyar~obr=0 (3.48)
:Z.f~w-’~f 2! ii C’~UP;;+"ESC3AAI'+Dc£&?.' .
i +‘2w1.’+'2 Q‘W?’*W= . (3'49)

Incredibly enough, a closed solution exists to this
intractable looking set of equations. But since we're

- - . (3‘)
only gOing to be interested in the form of f

chosen in (3.14), we will only obtain the solution for

a“ $2)and w‘C gz)where

f 1: —-('!-— )<2 QS“) ) f}::<dr : —-/>\
1/\

Direct substitution of these forms in (3.48) shows that

(3.50)

4“

W( 5 )3 O. (3.51)

Substitution of (3.50) in (3.46) yields the differential

equation

(1+ )3 $‘>&’+ >34=o

39

which has as its solution

&C52)5f (I + /\1 52):“ (3,52)

Solutions (3.51) along with (3.50) satisfy our remaining
two equations, (3.47) and (3.49). For our covariant

derivative we have, using (3.34), (3.40), (3.51), and (3.52),

Def @ °f'(l +>\ é ) <91 @.
For convenience we choose the prOportionality constant

such that
Dvgzaaré‘l‘

SO
—.

D45

Much the same procedure is necessary to construct

([+- A!2 guy-t3, .55

H

(3.53)

the covariant derivative of the nucleon field’[)q-\{?

It must satisfy the equations

[7:1, D431]: ‘ ”gag?

(3.54)

‘

[ ?<<x ) [D-rkf’:]‘= L/BdE 7%}. [>q'1e . (3.55)

For the form of E)? kf’we take

[)=4\f,=: Ely \f’ 4— c: F“L;( ES) C ébcr 15c3)\35

...-A
where me C Q ) is a 2 x 2 matrix function to be determined.

(3.56)

Substituting (3.56) in (3.55) and simplifying the result

using (3.2), (3.4), (3.17), and (3.37),

40
Excquksz' 1%!» 7‘30- (3., @c)\f+U.a1r:C:/rkavi))
965.; 2
d P1C.C£ fl 523) 1? P1; ;) :n:<:;)w éil)fii)
+ Jab Z} + 313:0

Mfd(3w@)¥= 1,131,6- ’C’G- Dari}
'Lflixb 'Lgb- éDarLi’ +~ ;:(Jfigqyy :ifp #4:; :)q’<5 Ki)

After some rearrangement this becomes
('U¢[1J,nc}=auw:9+md)d+am face
2 9%: Q 64 ago“

Let us take as a form of the solution to (3. 57)

MC5)= €aa€a’_t_y 523(5)

(3.8) and (3.58) along with the identities

‘Ecuéz é5ér 15: 1: (fl E§;‘ &§i)o. 3
I? );@]=C€afig’%’c

the left hand side of (3.57) becomes

‘ ’Z’I -
k U'oJrl: =1") Mr. ‘ Qafit 5g,- éfl 3:! If},
“2
Use of straightforward differentiation and the chain rule

gives for the terms on the right hand side of (3.57)

 

QU‘QJ, E: —Ea&¢9’§£ltf‘l€afidé6- 5g/Z9'VI
am. 1 1 2
am My: some "cu +€o£ééa®5/Q‘
3":2/ ”f (by 2 fl

+2€war5a 5&?_%0(fl’+ 5 w)

~Q€ajnflé 565 ”2:22 £3,
+€étoqé “@6’2529? W

41
Using these last four equations in (3.57), after some

simplification we arrive at
‘em5&@JU}+SMC/ ’J‘LI
+eauc5 56C2%+’2)’/ +3‘5’9'7)
*28W 565; cf’fl} 0-

(3.59)
Comparison with the identity (3.25) shows (3.59) is

satisfied only if

493‘ ‘(zr ‘7 ~42): -27.}‘135 1333:? (3. 6-60).

From the first part of (3.60),

“l
“ U'C/ + U’ 55 > (3.61)
Using {'3 " ((‘A 6 z)/{)\ and U' 3 ‘A, (3.61)

gives

3:2/\‘<l +28 5‘)“

and the nucleon covariant derivative becomes, using

(3.56) and (3.58),
[)qfi? :.;)#ff>‘r 6.255(I-+.>3'§;1)f'?§’.§§>‘;;q‘ag ‘1’.

We now have three covariant quantities from which

we may construct our chiral symmetric Lagrangian, namely

(3.62)
Any isotopic scalar function of these quantities will

be a chiral invariant. But we are not free to choose just

42
any isoscalar, because we want the Lagrangian to contain

the usual free field Lagrangian

1F,..=?<.a~ M>‘I’+ j; 5 a
.. /<..5_:. £5 95 (3.63)

so our Lagrangian will take the form

1‘ : Lfree ‘1' II. (3°64)

In addition,¢1:-I should contain a term corresponding
to the usual ps(ps) or ps(pv) coupling.

We can arrive at a Lagrangian of form (3.64) by
writing (3.63) with the following substitutions to make

it chiral invariant:

But we immediately encounter difficulty with the pion
A
mass term since there is no covariant analogue of g
This is not too surprising since we showed in Section I
that chiral symmetry breaking was intimately connected
with the non-zero mass of the pion. Thus, no chiral
symmetric Lagrangian may be constructed for pions with
finite mass and we shall have to pretend, for the time
being, that./‘k = 0. Thus
.— -‘ A U" A
L‘frce ~= “1’0. 25 *MH’+%_&(@-3 9&5
M‘0
is the free field Lagrangian which belongs in (3.64).

Now we may proceed with the recipe outlined above and write

43

Lchxml symmafr.‘c2" ‘1‘?)(il _. ‘M
__/\2 TC! +}\ <3 >~ IT; @ x
i(l + /\ 32 ) 1 9q§ 3'

if.
if

may

However, we still haven't fulfilled our second require-
ment, which was that the Lagrangian contain the usual

pion-nucleon coupling terms

c'gYYs’t’ 6‘? or £33,? Y5? $7512

The pseudoscalar coupling involves the same problem
with the pion field which the pion mass term had. So
we're forced to use the pseudovector coupling. Adding

on this gradient coupling term, our chiral symmetric
Lagrangian becomes

LCkm-al Symmefr‘ocd 7‘ Ygfa" M )j
~A‘Y(I+A 6 )“’C’ $2.55 @‘1’
+£(HA‘3‘Y7‘9-ré J’é
+ ‘PYs(l+—/\ @1)"?°3§Y
2%

(3.65)
To the chiral symmetric Lagrangian (3.65) we shall

now add a symmetry breaking term IS. B. so that our

final Lagrangian will have the form

I 3 Lakfrqi $~7mmefra.< ‘f‘ ”Z- 3.3.

We shall see that we recover our pion mass terms and fix

the heretofore undetermined constant .>\ by constructing

44
this symmetry breaking term in such a way that PCAC
is satisfied.
From (1.14) we have for the PCAC condition
-—.‘I
a-v 9310‘): £1 at: #1 @Cm 3
9) av

In appendix B it is shown that this form is not unique

(1.14)

and in fact the general form of PCAC is
c—A _.

aqggaggfi/A‘z@[l+§am($l)ml

where fawn}, is a set of arbitrary constants. From

3.66)

(2.17) we see that the divergence of the axial current

is given by
«—a ..3
Q. [X ) ls.B.]=C>*-r 9°; .

Combining (3.66) and (3.67), we obtain

(I. X) 15.8.]‘11 %fl‘g[l+zam<51)ml
3' v

(3.67)

(3.68)
Equation (3.68) is the equation 15.3. must satisfy.
Because the set iQM3 is arbitrary, the solution of
(3.68) is not unique and many different models may be
used as solutions.
The simplest solution is just to restore the pion

mass term, that is, take

.1 - ‘1 ’
5.3. = “/14 § (3.69)
’2.

From (3.16),
[K51]

SO

..s .a -a1 “*
(.‘[ X)‘ 1 @1] =AAiL/Y‘ (”Aw-é )@ (3.71)

H

(I. "( I +/\1 52) a (3.70)

MI}

45

This satisfies (3.68) if we take

,}\'1= .
:2 €ga£b (3.72)
We shall call the model defined by (3.69) the minimal
coupling model, for want of a better name.

We can see from (3.70) and (3.71) that any model

we wish to use must be of the form
00
‘1 '1 1. ha
I - M 3‘ 2. (y (5‘ )
... -" ....- m _ )
5.8. 2 + mu
where 56'an may be any set of numbers. In the minimal
coupling model we've taken 6v“ = 0 for all n. But another
alternative is to take‘arn = 0 for all n. This puts

PCAC in the simplest form, that of equation (1.14). To

Obtain 15. B, in this model we must solve
. -‘ .. I @-
«tx.1....1~-"-”—3$M -

‘3’ v
Using (3.4), (3.11), and (3.16), equation (3.73) becomes

(3.73)

the differential equation

( l 1- A1 $1) 1:33. 1: ‘- Aifli
'2

which has the solution
' 1
L543." -/,3"N2£m(l +20 5 ).

But /\ = so the final form of the solution is
1% as): ,

1“" “373% 21)”£"[‘*(2% 3:) a I

‘ 21 (3.74)
This is the model proposed by Schwinger.

The final solution of (3.68) which we shall consider

will be Weinberg's which consists of making <21‘5-Bn an

46
C N)
SU(2) ® SU(2) tensor. If t-ra -- -Y‘ designates a

traceless symmetric chiral SU(2) ® SU(2) tensor of rank

N, we let
_ (N)
LS.B, ‘ 1N = oo---o
In order to derive a form for dz; we must first make

”I
a short digression into the properties of SU(2) ® SU(2)
tensors. Let U4 be such a tensor of the fundamental

four dimensional representation where or runs over the

indices 1, 2, 3, 0. Equation (2.18) tells us that

EXQ) 114] = ~ (Maxwfi HQ
[Tq , Ex] 2 " < to.)'r(3 Ea (3.75)

where Ma and to. are the 4 x 4 representations of .Kq

and 1:. and therefore satisfy

[ta;tb]= (:6.th
[ tq)/7(b]= dew/7‘5.
[“q3/XG] 2 (:éaoé: t4.

A specific form of the 4 x 4 representation is

(fade; ‘2 -- C.‘ W

(taM-o‘.’ (flak/3“" (ta)... 2 o
(Man-o =‘(r7fia)oér= L'Sauo-
<r><a>6z 3 (Ma)oo “' O.

(’1‘)

(3.76)
From (3.76) we see that the first three components of U—r

form an isotopic triplet, and U9 is an isotopic singlet.

In the sigma model the triplet was identified with the

47
qu—I
pion field @Cm)and the zeroth component with the
sigma field.

Let us temporarily restrict ourselves to

15.3. = l. = to . From (3.75) and (3.76)

EKG) to] = (fa
[,Xq, t&] = 7 c. ‘80.;- to. (3.77)
From (3.77),

[x.,[x.,1.]]=smt.=3t. -3.z

(3 78)
To generalize (3.78), note that a rank N tensor is

constructed from the product of N rank 1 tensors. Hence

Ex.” .53)..]=Nt.f. .T‘itx.,t.)
=<'Nto a.
1: (:I\J ta (SJ).

(3.79)
where the last step uses the symmetric property of't-flin-Vfi
For the generalization of (3.78), using (3.77) and

(3.79) gives

[xb,[ X...Z~]]= 1X3, a Nd")... )

‘ (04) (hi)
- -— bd(:hJ-|) t,xb”3”¢3 +~ P4 ESQA,'t<,...o»
(3.80)
(N)
Since tqe ..v is traceless
(N) __ (N) \N)
t-r-ro---o ‘ tmo- o *‘tOOOm 0:0)

so setting at = b in (3.80),

48

[XME x..1~]]=-N(N~nt‘“" 3mm”

mqowuo‘+ o-o
s) N(N-()+3N3t.‘..‘:',’ = N(l\l+2)LN

We derive from the double commutator

[Xa,[qulN]]=Nw(N+—2)¢LN (3.81)-

a differential equation for .J:_ N using (3.4), (3.11),

(3.14), and (3.15). This equation is ’
(1+ >3 6‘)‘ 3‘1L+i(|+x‘$‘)(3+/<@‘)ZN
+NCN+2)>\21N=O

(3.82)

William Sollfrey22 has shown that, in accord with the

condition that .2; N contain the usual pion mass term

7. "A 7-
../¢_4~ C? , (3.82) has the general solution
‘2

IN = 3w»“ ,, (l+>:§z)m}n[2(N+l)Cam">\5),
HNCNQDG 1(N1—l))\ if

For rank one and rank two tensors this reduces to

all: ‘43? (l + )3 3‘)" 51

1“?~ "
Jl‘2’='— ’%§2 C I +- /A\ i5 ) ‘1 Q5

where constant terms have been omitted since they have

 

no physical effect. Using Sollfrey's solution or solving

(3.82) by series as Weinberg actually did, we obtain

the useful solution

IN = “/giiéz +/_¢‘fN(N+2)+2]>\2(-3I)1
'2 (O

(3.83)
+...

49
Weinberg chose as his hypothesis that the symmetry
breaking term transforms as a rank one SU(2)(:) SU(2)

tensor, that is

I _ 2 “‘2 ‘l '41.
5.13. - -' l +( @ J ®
where, as usual, )\ 32.3. 05:; . This will be referred

to as Weinberg'sz3 model.

We see that the minimal coupling hypothesis that

7. A“.
15.3. = 5/62: ®
and Schwinger's hypothesis __A
29*.qugq.'= [:1 €¥43 //J:z ‘5
‘3 °IV

are dependent on the definition of . implicit in our
1
choice of 6- C 6 ) . Weinberg justifiably objects to
these hypotheses for that reason. His own hypothesis
as incorporated in (3.81) has no reference to the form
of [( é . However, in order to derive the differen-
tial equation form of (3.81) from which .1: N is actually
-*1

obtained, we do have to employ a Specific form of fc @ ),
so it appears that the dependence of Weinberg's model
on the definition of the pion field is merely camou-
flaged. It would seem that the only way to remedy the
ambiguity in the model is to compare the model's pre~
dictions with experiments.

To sum up the results of this section, our chiral

dynamics Lagrangian takes the form

50

1= ‘H as m i ) warm—1m}
xififlvs‘fi-a} a 3- (if (33) “‘83? wk)?
3"

+ii|+(§%)13¥)€3fl} é)... ‘65 31-153
(3.85)

The term 15.3. breaks the chiral symmetry of the rest
of the Lagrangian, and the exact form of IS. B.depends
on specific assumptions about the model. Specific
models we'll consider are these:

1. Minimal Coupling Model
—-‘1

15.3. =-/f_§ @

2. Schwinger's Model

1.5.5. = ‘/%1(%)-1(%X)H2h[{+(2%)( 3)?)
' ~33+2w3>c3~m <3“)

3. Weinberg's Model

1... " ‘31 1+(3)‘(3a)‘$‘]"%‘

“3‘ 3<-?a>‘3( S'3‘)C<‘5““..)..

4. Tensor Rank Two Model
-1.-1

1.... 49-31143? 3:) 3“] 3
—; ..3- 2 ‘1 _‘1L 1

3‘ ‘3 + <33) <31) (.3 W

S. Tensor Rank Three Model

15.3, 3 613‘:
:_/%‘@+_l_ fifl37)1(¢11)4--

[<3

‘3 .-

SECTION IV

‘PION PRODUCTION PROCESS

The process
Tr+N‘——)TT*TT+N (4.1)

has long been of interest to theorists. The first
papers which attempted to calculate theoretical cross
sections for it appeared in the mid-fifties. These
first calculations were based on the static model with
pseudovector coupling and yielded total cross sections
an order of magnitude too small.24 This model was later

modified by consideration of final state interactions25

26 The model in this form

and pi-pi scattering effects.
was fairly successful, but was still unable to explain
final state mass distributions. Another approach was
the (3 exchange model.27 This model proved inadequate
below 1 BeV because it favored the wrong isospin channel
and forbade N* production. More recently, Olsson and
Yodh obtained a very good fit to total and differential
cross sections employing a phenomenological model with
seven free parameters.28

The process (4.1) is of interest in the study of
chiral dynamics because it may be used to discriminate

among the various Lagrangians discussed in the previous

section. To see why this is true, consider the general

51

52

form of the symmetry breaking term,
1 = __ ‘1 $1. 00 - m )QM( )2m( -" )m

5.3. A}? ié‘C J)‘ (2% a @ 71m
where $71,“; is a set of model dependent parameters.
In order to make an experimental determination of ‘y1fh ,
we need to look at processes involving vertices at which
2n pion lines intersect. In particular, if we want ‘W1. ,

we need a process with Feynman diagrams containing

*
the vertex of Figure 4.

Figure 4. Vertex dependent on.'71‘.

The obvious candidate for such a process would be simply
pi-pi scattering which at low energies would be domin—

ated by the diagram
.3? CH3.
3. ,7’

'9)“ 1;" ,,

It is easy to calculate low energy parameters for pi-pi

 

*Dashed lines are used for pions, solid lines for nucleons.

53
scattering using the chiral dynamics,* but unfortu—
nately it is difficult to perform the experiments
necessary for comparison.
Another diagram containing the four pion vertex

in question may be formed by simply attaching a nucleon

line to Figure 4. The resultant diagram is
66; Cal—7

- ~-”>' "?‘~.‘ Caz

"A

0
Q

 

> A 3~
P" m.

Figure 5. Diagram contributing to process (4.1)

which contributes to the pion production process (4.1).
As luck would have it, this diagram usually dominates
the process at low energies and is very sensitive to
the choice of h, , in this region.

The utility of chiral dynamics models in pion

production processes was first noted by Olsson and

Turner.29 Their calculation included, in addition to

the diagram of Figure 5, the contact term
3' 7.4)
\ Q I
‘. ,? 7-
.) I, 1’ a.

\ v
Q ' I

) .~1’ ) fi .
p. m-

*See appendix E for a calculation of s-wave scattering
lengths.

 

 

54

Other terms generated by the Lagrangian contained one
or more nucleon pole terms, and were therefore neglected
as being relatively small. Actual cross sections were
calculated by approximating all final state momenta to
be zero in the transition amplitude m so that the
cross section was essentially I 771‘ 2times phase Space.
This approximation circumvented a laborious integration
over final state phase space, as well as greatly simpli-
fying spin averaging m , but it restricted the calcu-
lation to total cross sections very near threshold which
made conclusions drawn from the calculation very
dependent on the accuracy of a handful of low energy
experimental data. Olsson and Turner concluded from
their calculation that, of the two, Weinberg's model
agreed with the data better than Schwinger's model.

The procedure employed by Olsson and Turner may
be objected to on two main grounds. First of all,
even at the lowest energies for which experimental cross
sections exist (incident pion lab energy T-fl‘ = 210 MeV
versus threshold energy of about 180 MeV), approximating
the final state momenta to be zero causes an error of
20% or so, and the discrepancy could only increase if
the method were applied to higher energies. Secondly,
it is not really clear that we should restrict our
attention to cross sections very near threshold. We
have built into our mouel isospin invariance. We know

that isospin symmetry is not quite good since proton

55

and neutron masses differ by about 0.1% and there is
about 3% difference in the masses of the charged and neutral
pions. These small differences made no difference in
high energy calculations, but near threshold small
differences in final state masses may cause enormous
differences in the magnitude of the phase space integral
since the rest masses of the particles take up a large
fraction of the energy available. In fact, at threshold
the error becomes infinite! Even at -r}T = 210 MeV,

the choice of masses within the pion multiplet can cause
a difference of about 85% in the size of the phase space
integral. So the problem is, we claimed when we made
isospin a good quantum number that within isomultiplets
such as the pions or nucleons, all masses are equal for
all practical purposes. however, in the kinematic
region we are considering, phase space integrals, and
therefore cross sections, are strongly dependent on just
which of the masses we choose from the supposedly degenerate
isomultiplets. hence, the breaking of isospin symmetry
which we neglected when forming our models turns out to
have an important effect. Fortunately, the effect of
this symmetry breaking on phase space diminishes rapidly
with increasing energy. At ‘17" = 300 Mev, for example,
phase space only varies about 20% with different choices
of mass within isomultiplets, and this is within the
experimental errors in this energy region. But it is

clear that it is dangerous to make any judgments on the

56

relative merits of different models without examining
a range of energies above threshold.

The work done for this thesis is an attempt to make
a more accurate determination of cross sections predicted
by chiral dynamics Lagrangians under consideration by
retaining the momentum dependence of the final state within
the amplitude and performing the necessary integrations
by Monte Carlo methods. Because we wished to do this
calculation for energies above threshold, it was no longer
possible to neglect diagrams with nucleon poles, so all
diagrams generated by the Lagrangians were retained
in forming the transition amplitude. It was hoped that
by comparing theoretical and experimental total cross
sections over a range of energies we could get an idea
which of the competing models was most satisfactory,
and incidentally find out at what energies the exact form
of the symmetry breaking assumptions are important.
Differential cross sections were also calculated in order
to gain further insight into the structure of the invariant

amplitudes obtained from the models.

SECTION V

FEYNMAN RULES FOR CHIRAL DYNAMICS

The chiral dynamics algorithm tells us to apply the
usual Wick-Dyson reduction methods to the chiral dynamics
Lagrangian and retain only tree diagrams of lowest order
in the coupling constant 3} . But in actual practice
it is best to use the Wick-Dyson reduction methods only
enough to be able to deduce the Feynman rules and then to
construct transition amplitudes directly from the diagrams?
The procedure is to draw all the tree diagrams for the
process of interest and associate with the nth graph
an amplitudem( m) which is the product of factors
associated with the various topological elements of the
graph. The final invariant amplitude m is just the

711<(h)' . .
sum of all these 5. The folloWing rules speCify
factors which are independent of the form of the inter"
action Lagrangian:

1. For each internal nucleon line (called a nucleon
prOpagator) of momentum [Q , there is a factor

if 1 ... i'( p: + NH
ld‘m*é£ P1_M1+£€*

where ‘1, is a 2 x 2 unit matrix in the space of the

 

*For a review of the reduction methods of quantum field
theory, see appendix B.

57

58

1:' matrices, and G: is a small number which is allowed
to vanish after all integrations over internal momenta
are completed.*

2. For each internal pion line (pion propagator)
of momentum Cb, there is a factor

a. gal.-
CBF‘ ~/(4,2 + (a

where 83.1, is a Kronecker delta connecting isospin indices
at the vertices joined by the internal pion line and
is used as with the internal nucleon line.

3. Define two column matrices

7% =(c'a) > 7L3 ‘4?)-

These matrices operate in the space of the ’27’ matrices
and serve to keep track of nucleon isospin. The rule
is that for an external nucleon line of momentum f3 ,
spin A1 , and isospin z-component T3: Ml“, there is a
spinor and isospin factor as follows:

incoming nucleon: X W U~ ( P: 0. )

outgoing nucleon: XL J. C P) 4)

incoming anti-nucleon: 7S .. U! (f C P: ’1)

T _.

outgoing anti-nucleon: “our U'( P) 0)
Here, “CF; 11) andU‘CP , 0.) are particle and anti-
particle spinors, respectively, and QCI’J J O ) and
‘7‘ < P1 O. > are their conjugates.

4. Define three unit vectors in isospin space by

 

*Because there will be no such integrations in tree diagrams,
we can omit this é, altogether in chiral dynamics cal—
culations.

59

'9» '9w9u)
a +
u u
2 sm—
r\ r\ m
V:
P.
O
u

0

These vectors keep track of the pion isospin. The rule
is that for an external pion line with T2 = t and isospin
index a. , there is a factor as follows:
A
incoming pion: C Qt)“
A A,‘
outgoing pion: C @-t )0. 3 C @t q

The following rules for vertices apply only to the
F)

Lagrangian (3.85). Let us adopt the notation that :2:
means to sum over all permutations of unrepeated indices

in whatever expression follows it, e.g.

Eil‘:é§au£r =: g’cJ? +- Elétg )

2 P6.“ (3); = ém($ )c. +€WC§)Q,

For a pion line of momentum % , define Q to be—C‘. if
the line enters a vertexfi'cbif the line leaves a vertex.
If a pion line is labelled by isospin index Cl , its
momentum is designated atlg. These capital indices

go in tandem with small indices when they stand to the

P
right of :2: , e.g.

f:
E Swag-B ‘-= AMT: 4%th
With this notation in mind, the factors for the three

kinds of vertices generated by chiral dynamics Lagrangians

are as follows: 1

60

1.. I], ’ i 2 ”11‘ l Pg‘On lsnas

 

‘ ‘ "q 32’*‘*'! :7 11~a
var-Tex: ‘(" I) (it) (.31: }A
ZPSWH'BFIA} It’nVs QR

m Kroner. ker- ole )rqg

 

11”! 3 2 m + 2 P‘oon l.ne')

ver‘l’ex \“1)m(i%)2m+2 3:1)2'fl‘t-1

KZP 8°58” Tt€rvyc Q5

m KY‘OhecKer ate.) +33

 

“ ’ ’/' :2 rvw -+- :2 F>fora i.}ficas’
q ’»z’

.' m 1m
VeY‘Ic-x'. 'Cc‘lfn (1%): (a)

Fig 2.
W ,1
m 1" KV‘Onecker
“”35 +Cm+DQRQSJ
2

 

 

61
These somewhat forbidding general forms become much
simpler for special cases. For out purposes we shall

actuall only need four vertices, and they are

 

I Q . _ ‘.
4 3 var-fax. i% 2"qu <16,

.16: var-Tex (‘33): 3%) 2:. 5m (fi-V‘fiay

: :62; ‘3'; var-Tex. 2(%)3 (1)2
'5‘5...’ JEQX, 2'ch 503.3 +5M’Z‘Z-Ys a,“
361; ”C2. \fsqn]

. I/C 0F! var-Tex 41c(.%) (1y);
..g‘.> ’:::"‘ ) “[SGJI'SCJ()1 fl:+ )
d \‘. I 7332‘ $.91
4'qu +5“ 5614(71 fl: +31% 61.42.)
{-80.095/ (72, /u +3133 $91)}
The first of these is just the familiar vertex of ps(pv)
coupling. The latter three vertices are unique to the
chiral dynamics theory, and the last vertex contains

the model dependent parameter n. . The values of Th.

for the different models we'll consider are given in the

table below.

62

MODEL Tl ‘
Minimal Coupling O
Schwinger Model 1/2
Weinberg Model (Tensor Rank 1) l
Tensor Rank 2 2

Tensor Rank 3 17/5

As an example of now to use the rules enumerated
above, let's evaluate the amplitude corresponding to
Figure 5. Figure 5 is redrawn in Figure 6 with each
nucleon line labeled by its four momentum, each pion
line labeled by its four momentum and isospin index,
and the factors given by the Feynman rules encircled

next to the appropriate parts of the graph.

_.-- ...—”n.-." h
.m- -.....— _._.--..-_.

   

+§af36cfl (Vb/u.) +c‘.,h a“ 3.1)
+SOJ36-{CY‘h/IA +q..~1.z 63... 52)]

 

 

 

Figure 6. Diagram of Figure 5 with factors given by
Feynman rules for chiral dynamics encircled next to ap-
propriate topological parts.

63
To form the invariant amplitude 77? for this diagram,
multiply the circled factors, reading along the nucleon
line to keep matrix multiplications in the right order.

The result is

W} -: ix; &(P£241)}{2'?i :Z‘iYslfl}iAX;u(p.}a;)3
‘i19$:diwz)q3i<@f).3i<réi).3
"i ‘16 (2%)71 3i): [§¢$a£(n,/m‘+cl_,h
‘ct,-<7_,)+$M 55d C71,/u‘+<},k—<3.'cpl
+ 8...; 26. (m/A‘a-oharag 1013

Using various kinematic identities and Dirac algebra,
this may be simplified still further. For a specific
charge channel use the appropriate forms of 7‘6 , XL' ,

a A A
fig, ¢' , and $1 and the correct isotopic factors result
from a simple matrix multiplication.

After the total‘??? is obtained for a given process
by summing the 771("‘)'s for all contributing diagrams,
cross sections are obtained by integrating over all un—

observed final state kinematic variables in the formula

 

 

Jo = ‘ (..L ) ...L. ) I777?
[55-33] 2a).. 2%,». a.

x (1935‘. Jakm 3,,(2TT) 5q(p.+p1~:h;)s
2a.). ( 2:1)3 2w,“ ('21?) ..

 

(5.1)

 

A

J _-- ...
where (4)“: \lk‘d- ”“1 , and U‘.‘ and U'iare velocities
of the incident particles. The incident particles have
four momenta f). and P1 and the n final state particles

have four momenta h, , ’12, k3, ...,l‘m. lmnts the

f
spin average of 77) )7}. The factor 5 is obtained

from S a T'l' _L

C ”1;!

where/VI; is the number of identical final state particles

 

of the ith type.* The factor 5 is only included in
calculations of total cross sections, not differential
cross sections. For fermions of mass r'r'n , the factors

--L which appear in (f.l) are replaced by C2? .
29.: E.

 

’By identical particles we mean particles and experimentalist
cannot distinguish. Thus aTT“ and a T!" are not identical
particles even though they are members of the same iso-
multiplet.

SECTION VI

MATRIX ELEMENTS FOR TT + N —5 1T + ”IT + N

Let us now restrict our discussion to the pion

production process

I
“(q-C) +N<PC) a Trl(fil)+fi1(?1)+ N(P{)(61)
The quantities in parenthesis are the four momenta of
the particles. Letting (.405) '5 “C F‘.) 0;) and
“(f)5 U- C ’0’. , A 5.) , the invariant amplitude

for (6.1) may be written in the form

777 = c2<i)(W1-Xq..+‘f¢‘_1 +Zd‘.z¢‘,,)‘(su(£2)).

(6.

Here W, X , Y , and Z are functions of the

kinematics. Other forms such as (1(f) fl—C V5 (A (C):
a(;)q,.-q_.¢p vs a“)! or mpc [Ag-[Ades ace)
which may appear in the course of applying the Feynman
rules to the diagrams may always be brought into the form
(6.2) by using the identities of Dirac algebra.*

Before evaluating cross sections, we must spin

average IWI1=MTW . If we write

’01-: (RC/J r' on“)
where of course P: ( W? X4: *Tfi-‘z +Zfizfi‘7Y5:
and define P: v0 I"? Yo

*See Drell and Bjorken, Relativistic Quantum Mechanics,
appendix A.

, the usual methods yield

 

 

65

66

for (m l1 , the Spin average of 'Ml?
T75? i: :2} Z’m'm . We wear—“Mas?

Now the usefulness of reducing 771 to the form (6.2)

becomes clear, because expressions of the form

Trance ¢.¢.1--- ¢~m

become very difficult to evaluate for large n. If we

apply our spin averaging theorem to (6.2), we arrive at

I’an‘= Thai lleprg‘pé.)-M1]p 1
+ I Xlzf2CP€°&'XPI~9~I)‘W1CP‘Fi)"; ”.1
+ I Y ‘21. 2(p6fi1)(p’71)‘/¢:(p6pfl‘M-L [‘1 1
+ I 21‘ I ac...-axp,o,.)(o,.q.)-2,:(pgq.xp,?\
Q 2M1CPc°fil) 91/3,.) ‘flQCPdplfl‘fl‘qm I
+2MRnM/*X [Cpécfl ‘(pf$u)]
+‘2F1Ra W*Y[(p.qa\‘(r15.qnfl
+1Rs. V‘Zf ((2.: p5)C%.3.1) ‘C p.’qz7(pi cpl
+(p.°cb..)(pfic’1\‘ M1 (:r‘cp”)
+1312 X*Y[( .: 2N q.)‘(r2¢.' (Cinch
+chog,)(gtqql "' Mféc}.c’_1)1
+2NR1. X*Z[2CPCQA\C%.Q1)‘M‘I(fl/.93..)
"79.1 ((349431
+QHRuY*Z[ M‘(p..'q.,) + M1 (Pffi-|\
-2(Pf6}1)(°}'3‘1)]3

 

(6.3)

67
At threshold, this rather involved expression takes
the simple form ‘1
I Hm...“ = ( E '5 ‘ fl) lw+ >9.“wa *2/03l :
11"]

where of course W , X , Y , and Z are calculated
at threshold. It is easy to see that at threshold [7"7'1‘
is real and positive, as it must be.

Chiral dynamics Lagrangians generate five general

types of tree diagrams for the process (6.1). These

are shown in Figure 7.

 

 

 

 

 

. ..I‘o .0 I
o . e
g ' Q I
. p . o
. ¥ '4- x
o
0
“ . / ’0 '
. C O ' U o
. O l g ’ O I
‘ 2.. Z ‘ 'l ‘ A L A
I r I

Figure 7. The five diagrams contributing to the
process (6.1)

For different labeling of the pion lines, these five
general types of diagrams yield altogether fourteen
different Feynman graphs. These graphs along with the
amplitudes they generate upon application of the
Feynman rules of the previous section are tabulated

on pp. 69-74. For specific charge channels, the isospin

68
factors cause about half these diagrams to vanish
identically. Note that if we include only diagrams
(e1) through (e6), we obtain the results of pseudo—

vector coupling theory.

 

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72

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mom .... (2%)3 Exit? ‘31?" @f’l’dcle

xa(;)§2r’1[._ 2,3,3“ .51]

 

 

 

 

 

 

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73

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SECTION VII

NUMERICAL CALCULATIONS

 

- . . mm“
The preVious section showed how to obtain

for processes of the form

TTLCQI'N-NCPJ) ‘9 11‘. (“I”) + fi1tq23+ N (pf)

(6.1)

 

All we have to do now is substitute IMF in (5.1) and
integrate over the apprOpriate quantities. For the
process (6.1), we note that in center of mass
-‘ " - “A -‘ ° —‘. _ _'—‘.
= fi , U"; = %: , and '0‘ - 7,; and
so _. ' dfi = M ,where E’- EJi-OJI'
"J7“‘J3' EL lagil

is total center of mass energy. Using this and the

 

appropriate normalization factors for initial and final

nucleons, (5.1) becomes

 

x {37%. 7173 1:7?"

“SC-L- ",CIJ :3002‘916.)$ (db..+-°‘.1 +13%)

(7.1)
Equation (7.1) is what we must integrate. However, this
integration is no trivial matter, mainly because of the
number of variables to be integrated over. Even after the

75

76

delta function is eliminated there still remain five
variables. Analytic forms might be possible if the
quantity in curly brackets took on a simple form.
But a glance at the amplitudes pp. 69-74 should be
adequate to crush any hopes that this might be the
case for chiral dynamics. 80 it seems clear that we
must resort to numerical methods. However, even the
usual numerical methods like Simpson's rule or Weddle's
rule fail us because of the number of variables to
integrate over. Ordinary quadratures become impractical
for more than two independent variables. So we'll have
to employ a method often used for numerical integration
over large numbers of variables, the Monte Carlo method.

The basic idea of Monte Carlo quadrature methods
is that instead of making a systematic sampling of the
integrand over a grid of the independent variables, a
random sample over the grid of independent variables is
taken and statistical criteria are used to decide when
the sample is large enough for the accuracy desired.
The difference between the two methods is rather like
the differenCe between an election and a Gallup poll.
The drawback of the Monte Carlo method is that it severely
limits accuracy, and must be used only on slowly varying
functions. Its great advantage is that bad as it is, it
doesn't get worse as the number of independent variables

increases.

77
Let us define a covariant phase space for three
particles with masses /L&, , ‘/042 , and P1 and total

center of mass energy E , by

fisc'E:3,A&I)/°L1’ P1)
={{{ 09,: "£3?“ O‘gEFME'wrw-I‘Ep 83(§.¢a‘,‘1+f35)

.__ _. - _‘ .1
wherew.= 43+M1'w13H:+/4:’ and E}.— J Pia-M .

We can prove that

6(5) = ans) El EIMHIIIIMJIIII

where CUE.) is the average of the quantity
i _L. .m: Wm“ ‘3 3
8(‘210f E Iii-c!

over phase space. It turns out that we can reduce

calculation of f3( ES/A.)/(A.1 , M ) to a

 

 

(7.3)

Simpson's rule quadrature, and thus we can obtain values
of the phase space integral with great accuracy. It is
in calculation of a( E) that the Monte Carlo approach
becomes necessary. The method used employs subroutines
of a computer program called FAKE which was developed for
use by high energy experimentalists. FAKE generates
random events uniformly distributed in covariant phase
space, i.e. if we consider n particles of masse54~?1, ,
m2, ..., mm in the center of mass with total mass

energy E , FAKE will generate random sets of vectors
-—I

_—I
ll, , k2, ..., Tim for which

*See appendix D for a proof of (7.2) and a more thorough
discussion of several points mentioned in this section,
expecially the Monte Carlo technique.

 

78

 

ZKII=O 2:.J7:I-+m.-=E I

To( evaluate a(E), we have FAKE generate a three
particle state i G}. , (31.1, Pl. 3 and a two particle
state fag) '3‘} . These states are generated
independently of one another and are oriented randomly
with respect to one another. The four—momenta]: l’
C}. , a: , and a" are now substituted into

(7.3). This process is repeated many times and the
resultant values for (7.3) averaged until statistical
criteria tell us that our average is sufficiently close
to the true average. This resultant average is desig-
nated 0t( E ) ,

A similar technique is used to obtain differential
cross sections. Let's orient a coordinate system with
its z-axis parallel to K where 7': is one of the vectors
a: , i7. , or F5."- . Let the polar and aximuthal angles

A
of fid be 9 and ¢ in this coordinate system. We can

 

prove that
<96(E) ._. C((E, I9 6.) P (EM. ,M1,M\
09.0. “an. ‘1’". (7.4)

wherea(: ,6: 60) is the average of (7.3) over

that part of phase space where 6? is constrained to be
*

equal to can . Program FAKE is used as above, but with

the random events restricted to that region of covariant

 

*See appendix D for proof.

79
phase space where

jgf‘iaéc :: C471 €90 .
( 73H Eiffel

Doubtless, ways of applying the Monte Carlo method
to (7.1) exist besides application of (7.2) and (7.3).
But aside from practical considerations, formulae (7.2)
and (7.3) have the advantage of separating the calculation
of the phase space PS ( E 1 MI >M1. ) M) and the model
dependent factor C“ E ) or CH E ) 9) . As we've
discussed, the phase space factor is very sensitive to
the exact choices of fl. , M1 , and M , so in its
calculation we use eXperimental values of the three
masses involved. The factors 0((5) and C(( E ) 6)
are much less sensitive to the mass values used, and so,
in the spirit of the isospin symmetry built in our
Lagrangians, we use some average mass for the pion mass
l/Lg and nucleon mass rv1 . To be exact, we take

1‘A~:"4A-£‘+/‘4¢‘r/¢‘2 ’ 'v1 : [V16 i- P1LE

3 ’2.

though this choice is pretty arbitrary. We see that

 

we've taken the iSOSpin symmetry breaking into account
solely in the phase integral, but it is in this integral
where it's most important.

In obtaining the results of the next section Monte
Carlo errors were kept equal to or less than 4%. This
means that there is a 68% probability that they are within

8% of the right answer, and a 99.95% probability they are

80

within 12% of the right answer. The Monte Carlo errors
are always smaller than the experimental errors fOr the
cross sections, and a great deal smaller than the

experimental errors below ‘TET = 300 MeV or so, a region

of special interest.

SECTION VIII

RESULTS

Theoretical calculations of total and differential
cross sections were carried out using the five different
chiral dynamics models listed on page 50. Cross sections
were calculated with an error of 4% or less. The results
were compared with the data compilation of Olsson and

Yodh.30

Experimental points and theoretical curves are
shown in Figures 8-15.
Data for total cross sections was available for

five different charge channels which are

n‘p —'—'9Tl'°TT‘p
TT" P ‘——-%> TT*TT‘n
IT‘p———5 TT°TT°h
TT*'r->—7"* TT*TT°P
11‘“ p-—-‘9 Trl‘rr*h.

Certain features are common to the predictions for all
these processes. In each case we have generally good
agreement between chiral dynamics predictions and
experimental data for energies below 300 MeV. Above this
region a discrepancy develops between theory and exper-

iment, but around 900 MeV the gap between the two again

81

82
narrows. The disagreement above 300 MeV is probably due
to the effects of resonances which the models cannot
take into account. The agreement around 900 MeV must
be considered a coincidence which stems from the fact
that the theoretical model gives cross sections which
increase without bound, whereas the experimental cross
section begins to decrease around 1 BeV. Since the
theoretical cross sections are initially smaller, the
two curves are bound to intersect. But in the region
below 300 MeV, Chang has shown that resonances make only
small contributions to the total amplitude,31 so it is
in this energy region where we would expect the chiral
dynamics models to be successful. It is interesting to
note that though the chiral dynamics models we're dealing
with are essentially chiral symmetric versions of the
ps(pv) model, the ps(pv) model gives cross sections
which are vastly different from the chiral dynamics pre-
dictions and the experimental data. This discrepancy
is at its worst in the energy region where we expect the
greatest validity of the chiral dynamics approach.

In the case of the processes 7T- P -—§ "on. P
and W1. P ___§n-+ "0 P we see that all five chiral
dynamics models give good fits to the data, even above
the low energy region, whereas the ps(pv) model is much
too small. This gives support to the chiral dynamics
approach, but is of no help in discriminating among the

models. For the process "I P ---5 no ”O h no

83
data exists below 374 MeV, so it would be futile to try
to draw any conclusions from it.

Two experimental cross sections exist for the
process IT" P -9 1T+ [1" h at moderately low energies
and these are tabulated in Table l. The experimental
cross section at 300 MeV is rather poorly determined.
None of the models are within the error bars, but the
tensor rank 3 model and--surprisingly--the ps(pv) model
come closest with the tensor rank 2 model somewhat farther
away. But the models of Weinberg and Schwinger and the
minimal coupling model disagree badly with the data.
The data at 357 MeV is somewhat above the region where
we expect goOd results from chiral dynamics, but the
data here seems to confirm the above conclusions about
the different models.

A great deal of total cross section data exists
for the process TI" p—HH IT" 71 , much of it at low
energies. Some of the better determined experimental
points are tabulated in Table 2, along with the corre-
Sponding theoretical predictions. None of the models
fits the cross sections at 210 MeV or 222 MeV, although
Weinberg's model and the tensor rank 2 model come close.
From 233 MeV to 290 MeV, however, the tensor rank 2
model fits the data quite well. Weinberg's model is
consistently too small in this region, and the tensor
rank 3 model is generally too large. The remaining

two chiral dynamics models, Schwinger's model and the

84

 

 

 

 

 

 

 

 

 

 

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85

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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000.0 000.0 000.0 000.0 000.0 000.0 000.00000.0 000
000.0 00.0 00.0 00.0 00.0 00.0 00.0H 00.0 000
000.0. 000.0 000.0 000.0 000.0 000.0 000.00000.0 000
0000.0 000.0 000.0 000.0 000.0 000.0 000.0H000.0 000
0000.0 000.0 000.0 000.0 000.0 000.0 000.00000.0 000
Hmooz Hoooz Hove:
00002 0:000:00 0muoz . 0mvoz 0 0:00 0 xcmm 0>ozv
A>mvmm HMEHCHZ Hmmc03nom mnmnc0m3 Homcma Homcma ucmfiflummxm has
.mcumnfiaawa a0 mcowuomm mmouo 00¢
.Cpc «to... lvhuom mCOHuomm mmouo Hmuou HMUHumHomnu 6cm amucmawummxm .m magma

 

86
minimal coupling model, are much too small throughout
this energy region, and the ps(pv) model is completely
negligible.

What little data there is for differential cross
sections exists in the form of histograms of the angular
distribution of the various final state particles.

These histograms are exhibited along with the theoretical
predictions in Figures 13 through 15. Unfortunately,
statistical fluctuations in the histograms sometimes

make it difficult to determine the angular dependency
clearly. The eXperimental ordinate which was number of
events has been converted to microbarns per ste~

radian for easy comparison of theoretical and exper-
imental numbers.

Differential cross sections have been measured
for TI‘P —-)Tl° n“ P at 450 MeV. This energy is rather
high, but the angular dependence of the theoretical
predictions is at least plausible. At 357 MeV there is
differential cross section data for VIP ‘-’ WIIT" V1
The experimental data exhibits unmistakable angular
dependencies for this process which are not matched
very well by any of the chiral dynamics models, though
the ps(pv) model gives a fairly good fit. But 357 MeV
is still a bit above the region where we can expect.
chiral dynamics models to be valid, so we should refrain

from making any judgments on the basis of these results.

87

Once again it appears we shall have to depend on
data for the process "'P -—>IT* “‘ n , for which
experimental differential cross sections have been mea-
sured at 290 MeV. The fact that the experimental total
cross section data leaps from 0.28 i 0.09 millibarns
at 288 MeV to 0.61 i 0.13 millibarns at 290 MeV, indi*
cates that the experimental situation is in some doubt,
so we would probably be well advised to concentrate
mainly on the angular dependence of the cross sections,
experimental and theoretical, rather than their actual
magnitudes. Comparing theory and experiment for
£3 /JIL11-+ , we see that just about any of the
chiral dynamics models exhibit the right angular depen-
dence. In the case of 46/411 11'“ , none of the
chiral dynamics models predict the large backward
scattering, but in general they exhibit the prOper
dependence for positive values of “I 911" . But it
is for the cross section €96 /J-n-n that the most
unmistakable angular dependence exists, and it is here
that chiral dynamics has its most clearcut failure.
While the experimental data shows strong forward peaking,
the chiral dynamics models predict backward peaking, or
at best, isotropy. The reason for this prediction is
clear upon examination of the Feynman diagram below
which makes the dominant contribution to the invariant

amplitude.

88

 

Conservation of momentum at the vertex gives [—36+‘£ 3' F& .
For low energies the exchanged pion will carry off only
small momentum so that fic 90’ F?“ 8. By definition,
(m 9n 2 5:1 ' E“; f: 31:45.°E_‘§'
liallfifl liallfi‘d
But in center of mass, at...“ =‘fi; and (:6), 5n ’3! -- 1 1

hence this diagram will favor negative values of 601 8h

”(FWD-’7‘ rp) (mb)

89

    

mmmu. COUPLING
scnwmesa MODEL
'- " wemaeae MODEL

 

 

 

 

TENSOR RANK 2
TENSOR RANK 3
PS (PV)
.5 —
.2 -
| .—
.05 —
02 -
.O I 1 1 1 1 1 1 1
200 300 400 500 600 700 800 900
. T7 (MeV)

Figure 8. Total cross sections for TT‘ P——> TT°TT‘ P

0‘ (7r -p —or*r‘n I (mb)

90

 

 

 

 

 

I0. F
5. — é Q
2. —
TENSDR RANK 3—
. . _TENSOR RANK 2
/ WEINBERG MODEL
5 _ A / SCHWINGER MODEL
MINIMAL COUPLING
PS(PV)
.2 -
.l -
.05 -
.02 -
.01 1 l l l l 1
200 300 400 500 600 700 800
T1(MOV)

Figure 9. Total cross sections for TT‘P-5 WIN' n

0hr -p-*7r°7r°n) (mb)

 

TENSOR RANK 3————
2. - TENSOR RANK 2

// / WEINBERG MODEL

/ / SCHWINGER MODEL
f/ MINIMAL COUPLING

 

 

 

 

/ P3(PV)
.5 -
.2 0
I - »
.05 -
.02 .—
.OI I I I I I I I
200 300 400 500 600 700 800 900
T, (MeV)

Figure 10.

Total cross sections for IT I P ‘9 TT° no N

a’ (7r+p-' 7r*#°p) (mb)

 

 

 

 

 

IO — 0 §
' O
Q 0
5 - §
2. 0
0.
I . 0 5 A
MINIMAL COUPLING
SCHWINGER MODEL /
‘5 *— WEINBERG MODEL TENSOR RANK 2
TENSOR RANK 3
PS (Pv)
2 I
O
I __ .
.05 -
.02 -
.0I I L 4 I I I J
200 300 400 500 600 700 800 900
T1r (MeV)

Figure 11. Total cross sections for “HIP-“"TIYTOP

93

 

MINIMAL COUPLING

2. _. scuwINGER MODEL fl
WEINBERG MODEL

 

    

 

TENSOR RANK 3

 

ah" p-§ 7’7”) (mb)

 

 

 

TENSOR RANKZ
5— // J
PS(PV) %
.2 —
O
| I-
.05 0
T
.02 -
.0l I I I I I I
200 300 400 500 600 700 800 900

T: (MeV)

Figure 12. Total cross sections for W+p -’ TTITTIFI

(r ’p-Or'r'pII/z b/srr I

(10'
dflp

(r " p-Ozr’r-p [Cub/sir)

dd
(10"

'p-Or'r'pX/tb/slr)

(7

d0
d0:

ISO
0°C

94

 

 

 

 

 

I60
ISO-
I40-
I30
IZO
IIO-
I00-
90
DOI-
70-
60*-
50-
4O
3O
20
ID

I

l

I

 

-I.O
200
C

 

T7 = 450 MOV

 

 

 

 

MINIMAL
COUPLING

SCHWINGER
MODEL

 

 

 

 

 

 

 

 

 

 

l TENSOR RANK 2
TENSOR RANK 3

WEINDERG MOML

 

P8(PV)
I I I I I

I J
-.8 -.6 -.4 -.2 .O .2 .4 .5 .8 LC
COG 9p

 

T1 = 450 MeV

 

 

 

3
o
I

:33
000
III

 

       
    
   
 

 

 

 

 

_J MINIMAL COUPLING

SCHWINGER MODEL
WEINDERG MODEL

 

 

 

 

TENSOR RANK 3
P8 (PVI

 

 

I;
o
I

55355
00000
IIIII

 

| '0 I. .
loo—J MINIMAL COUPLING —1

 

T7 = 450 MeV

 

 

 

 

 

 

 

 

 

SCHWINOER MODEL—
WEINDERG MODEL—

 

 

 

 

 

 

 

 

Figure 13. Differential cross sections for

n-P-)h°n.Pat T1" = 450 MeV.

da'
d8

d0”
dflr

(1r

(r'p+ 7*7r-n )(pb/sfr)

fir " p-P 7* 7'0 ”/4 b/sfr)

"p-P mix-n )(pb/sfr)

n
—|'0
OO

95

I20—

IIO-
IOO T1r -29OM6V

 

 

CD
0
I

 

 

 

~10
00
II

 

 

0’
O
I

 

who
COO

 

 

 

 

 

SCHWINGER MODEL
MINIMAL COUPLING

TENSOR RANK 3

_ r—TENSOR RANK 2

_ WEINBERG MODEL
.\

-I.0 -.8 -.6 -.4 -.2 .O .2 .4 .6 .8 LC "5”")
cos 9n

 

 

 

 

 

 

 

 

 

IOO -

90 - T7 =290M8V
BO _
7O -

 
   
  
 

 

5° TENSOR RANK 3
4O
30
20
I O

4.01:3 -.6 -.4 -.2 .O .2 .4 .6 .8 IO

MINIMAL COUPLING Ps (PV) COG 97*
sCNwINGER MODEL

 

TENSOR RANK 2 ‘--'
wEINDERG MODEL

 

 

 

 

 

 

 

 

 

7o _ T7 =29° MeV — TENSOR RANK 3

 

 

—— TENSOR RANK 2

 

..——-- WEINBERG MODEL
...—— SCHWINGER MODEL
|~ MINIMAL COUPLING

7 d I“ :\
-I.0 -.8 -.6 -.4 -.2 .O .2 .4 .6 .6 LO ”5””

 

 

  

 

 

Figure 14. Differential cross sections for

TT‘P—NT’ TT‘n at T" = 290 MeV.

d_CT.
dan

dO'

dflr

(7r+p-O r*r*n )(p b/str)

7*p-nrf'r'n )( b/sfr)
,u

I00

96

T17: 357 MOV

 

I

90
80
7O -

50
4O
3O
20 -

low

 

MINIMAL COUPLING
I SCHWINGER MODEL
WEINBERG MODEL

TENSOR RANK 2

 

- TENSOR RANK 3

 

0—— Ps (PV)

 

 

 

-I.O

I20
I I0
IOO
90
60
7O
60
50
4O -
3O *-

 

-.8 -.6 -.4 -.2 .O 2 .4 .6 ,8 IO

— MINIMAL COUPLING

T7 = 357 MeV

-- SCHWINGER MODEL

— WEINBERG MODEL

 

—TENSOR RANK 2
\ PG (PH

 

 

aoIf-r

IO

 

-— TENSOR RANK

 

 

 

l 1 l J I l l l

 

-|.O

l
-.8 -.6 -.4 -.2 .0 .2 .4 .6 .8 LC
cos 97+

Figure 15. Differential cross sections for

Tr’p-éTT’TI’h at Tn = 357 MeV.

SECTION IX

CONCLUSIONS

It is difficult to know what to say about the
various chiral dynamics models or the chiral dynamics
method in general on the basis of the pion production
predictions discussed in section VIII. Total cross
section data for different charge channels does not
unequivocally favor any of the models, and the diff-
erential cross section data seems to damn them all.

If anything is clear, though, it is that the
evidence Olsson and Turner found for Weinberg's model
is not convincing on closer examination. While they
were correct in asserting that Weinberg's model works
better than Schwinger's model in reproducing low

energy total cross sections for TI-P"%TI+TI‘ h I

 

more careful calculations show that Weinberg's model is

not much better than the tensor rank 2 model at the

lowest energies, and is inferior to it throughout a range

of higher energies. Because of this and because of the

relatively good agreement the tensor rank 2 model gives

‘-
for total cross sections of the process 1T+P "5 77* IT “I

0
not to mention the processes IT‘- P‘TITITT P and

‘IT‘P—D IT°TT" P where all the models give good results,

97

98
the tensor rank two model appears to be the single model
giving the best all round description of total cross
sections.

But though the chiral dynamics models seem fairly
adequate to eXplain total cross sections, they are unable
to account for differential cross sections which are more
dependent on the detailed form of the invariant amplitude. E
This suggests that the models we've used can only be
applied safely near threshold, where the small final

state momenta make the exact structure of the invariant

 

amplitude unimportant. This is a severe limitation.

The obvious first step in attempting to make the
model more realistic above threshold would be the in-
clusion of resonances, especially the N* resonance which
is known to play an important part in the pion production
process. Another possibility would be the inclusion of
final state interactions. Only if these things suCceeded
in improving differential cross sections at low energies
would it be reasonable to extend the calculation to
higher energies.

Finally, we might hope for more low energy data to
help determine the validity of the conclusions drawn

above.

 

APPENDICES

APPENDIX A

GOLDBERGER TREIMAN RELATION

In investigations of the symmetry groups of the
strong interaction, the usual goal has been to find
unbroken symmetries. In fact, we have always had to
settle for symmetries which are only approximate.

But such broken symmetries may be just as useful as
unbroken symmetries if one understands why they are
broken. For chiral SU(2)® SU(2) , information about'
the symmetry breaking comes from the PCAC hypothesis.

To understand the raison d'etre of this hypothesis and

 

its limitations, it is necessary to understand how it
is used in derivation of the Goldberger-Treiman relation.

The PCAC hypothesis may be written
9°, 9-5008) ‘2 C/btq @Cm)

where (l. is a proportionality constant. The technique

(A.l)

used in deriving the Goldberger-Treiman relation is to

take the matrix element of (A.l) between nucleon states

to evaluatEI C: , and then between a one pion state and

a vacuum state to relate <:; to the pion decay amplitude.
First take the matrix element of the "plus" com~

ponent of (A.l) between a neutron and a proton state:

99

100

<PCFF)I;‘Y§‘5‘I’ ' 6+ I ”CF... ) >
=<P(pp)l 3" (35‘3. ... 9‘1’IIncpnp

‘<PCPP)IC/"1 <3 ¢+ In (Pry)?

 

(A.2)

The matrix element of the left hand side becomes, taking

into account the form factors FA<fif) and FF (6,?)

 

 

<P(P'I’Ha“< 9;: +4. (51:)IVICF0)>
IFS:
= 6@:‘ PAIE(ppI[Y-IY5F ECO; )+q..,Ys 0(0‘I1uCPAI
(1103 VT 1

 

-.-. 2V”:ECQZI+Q1FEC%1I]CCPE)CW5 092..)
C1rr)3 VT
where ark? PFT- P: and we've used the identity

¢“(P)=MU.CP). Now take the limit as fi.—'5 O to

obtain

40°<Pcm>mgé£ +5 9;?) I n q...»
V3:

=2I‘1 ECO) aCF-IP)Q.YS“CP0)_
V2.
But FA(O)= g4: , so finally

(I) (1.)

4.:O<P(PP”30 (95¢: ~I-<- 95<)Ih<{2 )>

-.=. 2MC9‘A/1V)“CPP) 9 YIS “(PT)-
CZTT)3 \l-i

 

 

(A.3)

 

101
Now evaluate the matrix element of the pion field.

If we define

“9"...ch a (CI .701) 3ch

then

<p<pp)lC/X‘ 6+.3Ihcpn)>

 

= SAi‘: <. PCP?” 3+' @317 ‘hCP-n)>
M‘-q.‘

= EA fiflKNNn-(i) anph‘Ys-Otcrzn)
M1”? C2W)3

‘2
where KNNIT (q, )is the form factor of the pion-
nucleon vertex. Again take the limit of <7. “'60,

and obtain
d

#530 <p<pP)Ic/¢A‘ III.- <13 109-...»

3 H CfiKNnna» (TI-(r39) 9V5 “(F203
(22")3

Now KNNIT (0”1) is normalized such that KNNDCJI: l,

but we assume KNNI‘I‘ (O) 94’ KNNW (M13 and so

4:20 <PCFPII C/IA:1 (BIN a In ((10))

2: EC? QCFPII' Y5 MCpAI.
(Zn-)3

(A.4)

102
Using (A.3) and (A.4) in (A.2),

:2 I4 IfC}/\I<jlvr) CZ-C}afa) C; V15 C4.< f3r~3
(Zn)? VE—

'-'- \/_i C? CCCFIP) C \G—ufl’qn)
(211)

from which we obtain
C=£1 ,
2 3‘5
and (A.l) becomes
or “‘ —_0
;)_ sgngIkfl) : I:1 fl~‘/AJ;1 g5 ('7‘)‘
3.2—; (A.5)

Now take the matrix element of the "minus" com-

ponent of (A.5) between a one pion state and a vacuum
state.

<I3«§Sq' é—ITTCG}.I>
= <I a°’(9,‘5”., ~49‘SJHWCOII7
. V? i...
= 2% 2€M1<I@0' @IWCfiI) (A6)
the pion decay amplitude F}? (Gil) I»!
<1 9:- IWCfi.)>-‘6.Mq‘q F11- (3:).

Vkizn”)3 1L<qur
Evaluating the left hand side of (A.6), then

< Ia"< 92.1.3 -.‘ géi’wmo,»
U f V

Define

2L
2 0; °'<I( é” ~é921-3)Irr( I>
OI i an... Cr
:- LEI/0'.2 PRC/41). (A.7)
‘r'l JC‘znPQw:

103
For Mthe right hand side of (A. 6),
M3<I<§ {InCc},\>
9? M a.
GI WM

Substituting (A.7) and (A.8) in (A.6), we get

VIA" F7704") 2 :1 A M“ 0
fichnP 20., v 3;— JIznP 202..tr

which reduces to

 

 

F:fi'(/LC1) z: ‘Ycig Ogigi -
3' 0,01 (A.9)
Equation (A. 9) is the Goldberger-Treiman relation. The
key assumption in its derivation is that KNNW(0‘? )
is a slowly enough varying function that
KNNR(O)2 KNNWCM1)= l. The most important thing
for our purposes is that none of the matrix elements

would be altered if instead of (A. 5) we'd used

9 95:10:02 M 513/“ #1 ¢CMI[I+ZQM( (205))!“ ]
<7, 67v 040.

I
where the Gu03»are arbitrary constants. This is true
...D —52 I»

because matrix elements of QC @ )3 m 3| , will
always vanish for the states used in the derivation.
Thus PCAC tells us what the divergence of the axial
vector current is to lowest order in the pion field, but
additional postulates must be made to determine higher

order terms.

APPENDIX B

WICK-DYSON REDUCTION TECHNIQUES

For actual calculations of scattering amplitudes,
the well-known Wick-Dyson technique is generally much
too cumbersome. It is useful, however, in approaching
an unfamiliar Lagrangian because it takes care of
symmetrizations, normalization factors, etc. quite
mechanically. Because of this and because its use is
necessary in obtaining the Feynman rules for a given
linteraction Lagrangian, the salient results of pertur-
bation theory are summarized here.

Our object is to obtain the matrix element of the
operator :5 between the initial and final states of
the system in question. To do this, we employ the

matrix expansion
82. -,...—-II
"PIWI(M.)-~ WICmmIE

(8.1)
where P indicates a Dyson chronological product and «1(a)
is the interaction Hamiltonian density. The Dyson

chronological product is defined

104

105

PfAMIDCITIE-z g A0069), II- 0.>«,.
BL?) AC“), I; k1,.) >0“.

The generalization of this definition for any number of

operators is obvious. It is clear from the formula

" 3%..

that if the interaction Lagrangian density contains

 

no derivatives, 4K 1' (at) may be replaced by -.Z 1: (a)
in (8.1). It is not clear that this replacement may
be made if 1]: (m) does contain derivatives, but

*
in fact it can. So we may write (B.l) as
an

fS-ztvnuo £%;:‘§10.§Rc9‘*rvc, "'¢d?‘4(7Inn
xPierm,)'°'lr(/Xm)3. (B.2)

If we are dealing with a theory involving fermion fields,
our formalism must use the Wick chronological product
instead of the Dyson chronological product. The Wick

product is defined by
TfACm.)13(m1)"-3=C~|IFPfA(m)D(m-.I“S

where f: is the number of interchanges of pairs of

fermion fields necessary to change EA (0“) B («1.) . . .3
to chronological order. Since physical theories seem

to always require Hamiltonians bilinear in fermion

 

*See appendix C.

 

106
Operators, the Wick chronological product of any
physical Hamiltonian will involve the interchange of
even numbers of fermions. So the Wick product in (B.2)
may be replaced by a 'T' product and the ES matrix
expansion takes the form

3:: £5"'§dq”‘r'° <9qu

n1=cI ,~1I
ATfllcm.)I--11 (00)}. 0.3)

The form (3.3) is useful because of a theorem due to
Wick relating the Wick chronological product of a set
of Operators and the normal product of the set of oper-

ators is defined by
NiAB---L§=(-I)F§QR-~-W}
where Q R " ' \A/ are the operators AB ’ " I—

reordered so that all annihilation Operators stand to the
right of all creation operators, and (3 is the number
of interchanges of fermion Operators. Let us further

define the contraction of two operators la» and E3 , by
AB = < I N<ABII >,.
L__J
Wick's theorem may now be written
TSABCmeszg

= NIABCD'”WX\I’Z! H +0 , l
. Ali/33C D \,/X‘{ 23. 2.0:...1'EJ0ZEL

a II ofhe Ir double

‘I' Ni LBCLB ' ' ' WXYZX+COHIFQCII.O'D Terms
4.

(3.4)

 

107

The contraction YCM) he (. L7) relates to a propagator
L——___—l
connecting the points 44‘ and if . The contraction
over boson fields is given by
¢(MI¢CI1)-CAFCm-¢7I= <9 R .2 I _
"""'—""' ‘1 ‘2. 1. .
(2") k -/m +t€

where a?! is the boson mass. The contraction over

fermion fields is given by

‘I’raCnO 830(7) = -_“I?.(C:1)Ya(m) =6 31:0... C 0‘ “7)

6—0 l

where for the matrix SP (mm?) , we have
5000-0) 10211.2 2.0.0.0“
(211)” [ti-”1+0:
Gel-{h £.&<mfl7) I
(211)” I0 aft-I- (a

A being the fermion mass. Contractions over unlike
fields vanish. We may assume that the contraction of
a derivative is the derivative of the contraction, e.g.
‘€(m)§_‘?¢1 I31§[Q(M)€L‘1)].* The various terms in
the Wick expansion 03y be placed in correspondence with
Feynman diagrams. In applying the expansion, terms with
equal time commutators are omitted, as well as terms
corresponding to unconnected diagrams.

All we lack at this point in order to have all the
equipment necessary to find < i. I % I C. > is the

second quantization of the particle fields we'll use.

 

*See appendii C.

108
In our case this means the nucleon and pion fields. The

nucleon fields fC’X) and ngM) have the quantization

“fléé Ff: 7‘.” E0 .
xIMF, ., ...) 06., 0.; 7’"... 0%, 0-009., 0w]

(3.5)
--- 3
0.00050. w;—

HI-é' (3,4 quCppIR T‘O?(PJ/1,‘(.J)U{P,a)fl~ fl]

 

(3.6)
Notation for spinors, isospin factors, etc. was established
in section III. The creation and annihilation operators
have the properties:

OJYP, D)w)creates a nucleon of four momentum fa ,
with spin of z-componentSa 7" A , and
isospin of z-component—Fi '3 (41‘.

64(0) A, W) annihilates a fermion of momentum ’3 ,
with spin of z-componentS; = A , and
isospin of z-component T}: “r.

(fl (F, 0., (0f) creates an anti-fermion of four momentum [’3 ,
with spin of z—component 52 = 0. , and
isospin of z-component Ti ‘ (if.

CQCP) 4, W") annihilates an anti-fermion of four momentum
f3. , with spin of ascomponent 5‘ =0. ,

and isospin of z-component T2 2 Dr .

Im-

109

These operators have the anti-commutators

.' .. . 3 .. _.,
i6<p,a,ur), (71-(p’, 4',W)3‘§M'§w S (prp)
iOQ(P,0,w-),OQT(P', o'Jw')3=§,gaf$wu/%3Lr3“f3’)

all other anti-commutators vanishing. If we let Latin
letters designate isospin state and Greek letters spinor

components, the relevant propagator is

IAKSCM) :17tprCI-7) z: c: 3M 31-23? (m *7).

 

...—Q
The pion field 0 (’7‘) has quantization

"" = A ‘3
9(0‘) (‘2‘:0" @tj’fifi:

{0300-0 2%.. adopt) rm]
(B.7)

A
where the unit vectors ét are defined in section III.
The creation and annihilation Operators have the prop-
erties:
+ C ) . .
0. 0‘. {6 creates a pion With four momentum C} and
isospin of z-component 11313 t
0L C Cr) {5) annihilates a pion with four momentum CZ.
and isospin of z-component T3.“ C

These operators have the commutator

[q Capt) ) q+(0,',t')I=$001 83(5;0a;’)

110

O u A
all other commutators vanishing. Let and .r1 be
unit vectors in isospin space. The contraction we'll

need is then
A A

25ch115620): 5 <20"). AFC/x07)

We are now equipped to find the matrix elements

 

of 5 for any nucleon—pion Lagrangian by means of a
mechanical and foolproof procedure. First insert the
interaction Lagrangian IICmIin the 3 matrix eXpansion
(3.3). Replace *(m),q;(.rx), andacm)by (B.5) I
through (B.7). Expand the Wick chronological product

by means of Wick's theorem. Now sandwich the ES
operator between a bra (fl and a ket I f. > repre-

senting final and initial states. For the process
II'(<=}£)+ P(p;,aa) "5 "+013 +TT“Cc,,-.,)+ ”(PI-I4?)

for example, (fl and I C > are written

 

IC> = I n-(q.c..)) PCPJJ 00.17 '4' Qf(?..',“ I) 6+(F03/1qt»)

(II ‘~‘ < IT‘qu),IT“Cq.zI,h(rll.,/J.‘JI
=<IQCqUI)QCfi_1,’I) 6CPI'I af’_i)

From this point on, just simplify the resultant expression
by use of the commutation relations of the creation and
annihilation operators.

The matrix element St... 5 <fI S I C > is related

to the invariant amplitude 77Z by

111

8/0.: (2U)L%5HCP.+P1~~ZIZJ )m

 

 

[ I I ... |
2MP. 2dr): 2%). me

where we scatter from an initial state of two bosons
of four momenta f3. and {31,to a state of n bosons of
four momenta k. , . - - , hm. For fermions of mass
,1. , 5'23 is replaced by 4" . The method of

going from the amplitude to cross sections is discussed

in section VII.

APPENDIX c

DERIVATIVE COUPLINGS

Most modern books on field theory slight the tOpic
of derivative couplings, probably because they are not
renormalizable and do not fit easily in the usual ca-

nonical formalism. As an illustration of the inade-

 

quacy of the canonical formalism, let us Specialize I
our considerations to a ps(pv) interaction Lagrangian
density
IICM) = # YCm) Y5 Yq9¢¢(m) YCm).

1AA‘ (C.1)
Here IL is a unitless coupling constant. For simpli-
city, we've omitted the complications introduced by
isosPin. Now the canonical formalism tells us that the
interaction Hamiltonian density corresponding to the
above Lagrangian is obtained from the formula

IN[;r 1' EELEELEL Eb ":Z:r
The result when this isEZpSI:ed to (C.1) is

KICK): # E208)“: V VQXmI ‘I’Cm)
/u.

(C.2)

Now the Lagrangian of (C.1) is Lorentz covariant, as it

must be, but the Hamiltonian obtained from it and given

112

113

by (C.2) is not covariant. If employed as it stands,
in Wick-Dyson reductions (C.2) will lead to formulae
for the invariant amplitude which are not, in fact,
invariant. The reason for this paradox lies in the
canonical formalism. The formalism is itself not com-
pletely covariant because it treats time and Space I
components of four vectors on different footings.

To remedy this difficulty, field theory must be

i
formulated in terms of spacelike surfaces, as Dyson33

or Umezawa34 do. Such a procedure greatly complicates

 

the formalism and will not be discussed here. But the
result of applying this formalism to our ps(pv)
Lagrangian is to show the interaction Hamiltonian must

be written in the covariant form

7f 1(rx ) = ~£ :PC’K)YSY#;H’¢CM) kI’Cm)
‘I‘ J3: (t)1[ kFCraavs Vina; KI’Cflt’)]1

(C.3)
where r~W-r is the normal to a family of spacelike

surfaces.3s P.J. Matthews36

has shown that we can use this
Hamiltonian in perturbation theory omitting the term
dependent on My so that even for derivative couplings

we may take I)<( 1; ’- ‘ 1.! provided we also assume the
contraction of a derivative equals the derivative of a
contraction.

Matthews proved this for any process generated by

the Hamiltonian (C.3). For clarity, let's restrict our

114

attention to fermion-fermion elastic scattering

F, Cr)", ) 0 F; (0.0.) *2 F3CPl-I)+F‘1 (fl/5‘)

Our Hamiltonian generates three diagrams for this process

which are illustrated in Figure 16.

PCI I Pin (2.:- \ [2‘1

 

 

 

 

 

; ' I'
I : I
I3€1l Fifi. {10%; [afI
F‘.‘ FI-z

Figure 16. Diagrams for elastic scattering of two
fermions in ps(pv) theory.

Taking matrix elements of the 5 matrix

<0], I S I e.
__ < I ,; .XJIMTMII Y‘piqmwva/(mnl

-I- (____" 2C) 3&40 JQ?T[( -)£ )IQIVsY §___¢Cm)+(m)

“(fi)CI:(u}.)Ys\/aa¢ WHflWQflgfl‘Ic)

9113!.) ., J. 1 __ .
=<k|ifli y&mN[2(fi) (Y<fl)Y506k+/(fl€»]
...J. 4
2 ffwwe u, so? 3%?
.NRJH $(mxvsv*r(m)rcv)wva‘f(g)]3 Ia>.

(C.4)

 

We know how to evaluate the normal products in this
expression, but it is not clear how to evaluate the
contraction. To evaluate it, we start from the definition

agmagcwle 39cm) ages?) I).
3?? 3‘16 30$? év’ra)

 

 

(C.5)

Let's define a function €(3)by

2(3):.E' ‘°' 3°>°
-| {“W' f}o<0.

With this function we may write a Dyson chronological

product as

HAMBG,» = [ WU ACM) Bo?)
l - Ecru- ) BC )AC .
+[ 2 j] ‘7 0‘)

With this identity, (C.5) becomes

3Q5Cnd 3¢C¥) : ago») Q29?)
@3qu «9:79 <‘(fi am... , Euro 3

 

+£C;‘fi)[ $3) 33:3?)1) l >

116

Dam) 9mg) - a. _ <li¢cna (m 81>
Drier anc-y 9. amqatza ‘7

+£(ovf3)91____<‘[¢(0\),¢Ct7nl>

 

 

‘:Z éDrThwégafis
(C.6) fa
It was possible to bring the derivatives out in front '
of the vacuum ket because the derivative doesn't operate :
in occupation number space. Using the identities of E
Drell and Bjorken,37 f

<1E¢Cm),¢(c7)]l7= L'ACm-c?)
( ' i¢<mh¢€vfil> = A.(m~'~7)

’and (C.6) becomes

92m.) egg?) :1. SzAJCrn-g)
arc-r <9 ma 2 34433;;

+é_€£§;_‘g.> 31‘4““ L4).
' E)l7K-r59‘1rfil

 

 

(C.7)

Using the identity
2 A F (or?) = -¢'A,<m-L1)+2cm—«,)Acm-«7)

where AFCm-v) = ¢Cf70 (2561’, to eliminate '
Aflflqflfrom (C.7) , we get

117

<3¢<o<> 3204): a 3‘ AFCm—g)

 

 

 

 

 

 

30:? 1‘7" advaflfi
- 3i ACm-L?) D‘ECnc-ML)
2 a/X-r 317:3
+'EEEEQZSE3):§‘3<J""§J.+ EDiicrn-g?) SL53J:0‘.£?{3 r5
'ZBnK-r 29 L113 2) b713. 23 nflor ‘

Now we must evaluate the expression in curly brackets.

 

First find the derivative of €(m " t7.) . Because

{(nr ‘9’)only depends on the time component of ’7‘“? ,

9£<“‘$) = “'0 3£CM-<1) .
BM-r 7 ado '

 

Because of the definition of ECO! -'~?),
EL§UCnK-§1) :: C)

3M0 if Mo-‘évo.

But for any positive number Q ,

(,a'fa.
f c0“. DECovg.) =£Ca)-E<~c.) =2
°-°' a ”(o '

hence

3 ZC/Kttgk) .___.. 2 nga- (.10)
30:0

and

Di (“‘1 = 2 (jar 3Cr>‘on‘-/)¢o)-
a M...
(C.8)

Using (C.8) and the fact that A(’))‘ ‘AC~’)), we get

118

BECm-fi) 3ACm-q)=23qOSCMO'V}O)34(d‘9)
3 0(9' 3 ‘16—, 3 “jar

.-.—- 2 ...}.ch ,x[ 3A(C4,~m1]
(2' L1 29 L743 (wa=u,o

(C.9)

 

But [.£)Z§LC£1(~:5)

3 («yrs ]”‘o=‘1°

and [ 34(54,~r7\)] z: -- $3CC;--va“\
3‘10 «o‘va
as may be shown using the explicit expression
A(})=~c'5 093k (L‘W) “£61241
C21T§3120Jng

=ojfb=l)1,3 5

 

So in general,
3:30.31.) 2 -3mos3cqv‘i
3 (qr:
and combining (C.8) and (C.9) .
aan-?)3A(m~4)=’27qo €130 3*(fX‘v).
3 ’7“? a ‘73 (c.10)

Likewise

SiCM‘j19aCmy)—; Qaaro (“7.903% (“‘(7).

r

3 Urn 30w (c.11)
Finally, using (C.8),

4cm “1) 9" 2: (My) g 3.79.119.-.» 36;. “(7:0).
E§CKur£9£1w3 - 29 f7(%’

119

But in general

£LC’1b3£§§%’Sfox-cz) :: “:S<37€-c;) :ééééfifi)

and so

ACm- )313(““1): @oSC o'VKJQACer )
0’ BMqat/{n ‘7 L1 Son-(4

and our result is
A COO-(2),) 91 aCr'ij'J = ‘KZO’VOOIGO 34(or— 11’)
309(an (C.12)

Substituting (C.10), (C.11), and (C.12) in (C.7) and

simplifying, we arrive at

3950)!) QCDCEQ .3 c." azdem-jg)- (9)703903V0‘v).

 

 

adcr g a/X-rs 3
.__ ma *7!
Generalizing from tne ”normal" spacelike surface implicit

in the identities used to derive (C.9), (C.10), and (C.11),

our final expression for the contraction in question is

990x) egg.) .... a 91AF(“"QJ _. gmmaiVawfi.
;)(§:¥ E>§1fi3 23v7<q';>b?fis

 

(C.13)
Now, at long last, we can return to our expression for
the 3 matrix, (C.4). Substituting (C.13) into (C.4),

we get

 

gust <7 = 415 up“. NH-(fl‘flanM)?

.3}: jja‘imoe‘i‘r ( c. 9145‘.- («33 .. L'mqmmqux-V)

art-(«3mm

. N[(fi)1?(m\vgv.‘1’<my?&,)vs Ya *Cv’R i c >

 

= w H SSW 4“» fifiié’ys’

" N [ (ff ‘l’ClesY. flail?) Yarrow} 1 .- >.

We see that terms dependent on Mar have cancelled one
another. In terms of the diagrams of Figure 16, terms
in the prOpagators of the first two diagrams have just
cancelled the last diagram. We could have obtained
the same result by assuming
1.
sow) sung)... a [¢C’*3¢.C‘7n
3 (79f @473 ark-(9‘79 .

 

(C.14)

WI <m>= “ 11(m\

(C.15)
Neither (C.14) nor (C.15) is actually correct, but
as long as we are using perturbation theory we may pretend
they are. It is a case where two wrongs do make a right.
Though we've only dealt with ps(pv) coupling, we may

assume (C.14) and (C.15) to hold quite generally.

APPENDIX D

MON TE CARLO CALCULATI ONS

Integration by means of Monte Carlo methods rests

on two theorems which are here stated without proof.38

THEOREM I. (The Strong Law of Large Numbers)
If (3 Cm, , -- - ,mm)is a probability density

function on a region R such that p<f7<. a --',n¢~.) >/ O,

I
“

and {RPCM;,,’Km) CMp-MJQ/flfi: l, and 6~ is

the integral
7“::S’R}(’7‘.."')’x’“)9(mn”')0"“)oen“ ”.9011.”
A

and 6. N is a random number defined by
A N
N '7: .1... Z l—Cm‘d‘, )mmr')
N w
the set of numbers (My; 3 -~ ) ’Xmg) being the 5th
set of values chosen at random from R according to the

probability density P<mx , 3 Mm) , then
/\ ...-

lfm 2

N->0° J'N 5‘ '

THEOREM II. (Central Limit Theorem)
—. 4 “
.For large N the probability that 6~$$fNé J.+ 3 ,

is independent of the exact nature of me, ,u-J Mm)

121

122
and (DC/x, , )MM) , but depends only on N and

c~ .- 1
the variance (3,": 1 ‘ , in factm
I ?1

ProbabilityifiN< 1....3}: .5]?% 7:921
[2%

+ terms of order
In other words, the probability that I IN- i‘é C5
is 68%, the probability that I 2:“- il<1 6 is 95%, and
the probability that l ‘N" ZI<3 6 is 99. 95%.

The advantage of the algorithm indicated by Theorem I
is that it is independent of the dimensionality of the inte—
gral. The purpose of Theorem II is to show what size sample
need be used in applying Theorem I in order to obtain a
given required accuracy. In applying Theorem II, one
must be a bit cautious because for actual applications
involving finite samples, the predictions of Theorem II
are usually Optimistic and should only be considered

suggestive. In applying Theorem II we approximate <5
A

by V N where

VNsz—I. N (3'3“ 07:1)

and
A 1
il— Th2 N[>;.(‘;H<;y 3 ”<"‘C)1
i 8|
N
A
é‘ NZ f(mlg.)"‘)mms.),
i at
Our method for evaluating an integral to within a
/\
relative error 3.. is to calculate éN for larger

and larger values of N until \' \A/N < i #N ,
and then 6. :: f" , and probably] ‘. "’ J. N (i & .

The integrals we must perform in evaluating cross

123
sections contain Dirac delta functions in the argument,
so we need to find a way of evaluating integrals over
delta functions. Let's consider the very general case

where the integral is

(:1? LY}? [C(rn‘l)” ~,n‘nq3‘i}<'le )---)4fi(nn)
as T3; é;[:<}4{(.nx.). M”‘n~¥]‘£ux (ibflnn

(13.1) » 5

 

Here there are I. delta functions with arguments containg

integrand is broken up into the product of a general function
fCrx' ‘ ) m m ) and a positive definite function
3, (IX. , - - - ) (Wm) . Let's define a probability

density to be

OCmir‘W’fim) = £7 )CmH'” )lmm)
)Q 11F €51: i?l‘.(’7‘t) '“')‘fiK"")-l

(D.2)

where ‘\/' is a normalization constant such that

V: KR 7(0‘.)'“, mm)n5[7“(m"”"n‘m)1

“at

.< cJZnK‘ .- . cfilvsn.

(D-3)
Rewriting our integral C1? as
0?:SRfCrKH-nwxm3pCrx',...)n(m3V
x 094, oQ/Km

(D.4)

124
we see that it is in a form where we may apply the

strong law of large numbers. Doing so gets us

DJ

w v. 3:...[75- .... chm...,... , mu]

(D.5)
where C m‘m,...)mmm\ is the rmth n--tuplet
chosen according to the probability density (D.2). What
this probability density tells us is to generate sets
of points (Mgr-m, - -- , Ohm...) which satisfy

?rg(n(.,~-,f7€m)=o3 h3l,"',l_.

and which are distributed according to the (usually

 

unnormalized) probability function 3 (m |) -~ - , (Km).

Now let's apply these results to the problem of finding
cross sections. First consider total cross sections.
To find a total cross section we must evaluate an inte~

gral of general fOrm

oCE) = 555 09—33? 293:1 21:35.
"/~(C}d .pc‘ , a.” CI-"Pi')8( E (.2 ~01. EH3 (9]. war-1113;)
The function I'CQ’C.’PJ)fi-t)9»1 , P.J.) will be the

square of the transition amplitude times some kinematic

factors. The integral is in the form of (D.l), so we

apply (D. 4) and (D. 5) to obtain

QDCEE) -'iE?S<:E:)/¢1Q /¢A1q r1)
lam - .
K N—ooo N _LkZaTJ‘CCt-Lh,Pc.kicl—lh,7.lkipfkfl

where

125

003

I?év(E: )I“|)/44HL)P1) =‘5;S‘;J cflrE; -Zi;£}J
..sas- m.) ”a. E/ >33Cc“;,+a;-.+f5r3—flr

The sets (flan P h)fi.gn5fi.mn )P-é‘lk) are chosen

from the probability distribution Q “1E5. such that

 

they satisfy
é%;c:1'F3<f‘=‘afs f'E;:1.+‘ri.F.‘= C)
Omit Eg- -.- w‘+w1+E‘. = E.-
In other words, we averagei. ((34, P“) Cl" )Cf1JP‘J')0ver

points randomly selected from covariant phase Space.

If we denote this average by a( E) , that is

Cl(E)= = ,3me J- 2. lv(ci.c.Rafl;h3fi.lh)a(.1k)PI-k)

NRII

then our theorem is

6(5) = P5(E3,a,,MMf1)aCE).

Now let's consider the problem of evaluating differen-

tial cross sections. Noting that

iffcmmxlmmz 'LCms)=f/.(m)3(m-mo)oQ/x,

we write our differential cross section integral as
22.: mm a: sa: as?
a”! (Cb-°':P430h1?1)PI-) §(¢-¢.)§(cm.6«an6’.)
x8(E"C\J‘." (..JI- El')83(€"+5;"+fif)'

Applying (D.2), (D.3), and (D.4) as before

 

126

<駀5(:E:)I _- c9 13; ( EZIB/L‘ 3/L¢1) ‘ W) ‘
:2 n: mm .2 n. m...

,. Hi?» ZM: «an ha....1.n.p,r.)]

 

o n / . u o
The p01nts in phase space \ 9",“ {15h} a‘“)()l'1h3P/l’h)
now are picked at random from that region of phase space

where fl = [10, i.e. ¢ = $0 and 661-9 = 001. 9o
If we define aCE ~(‘.)by

C((Eyflo) )-l.m :Lif(c1,;h,{zgn.)c}_m,c}zh,[lfn)

hflfiao[\|

our theorem becomes

dew.)
dfl

 

 

:CflyI?g<:Ed/AA )/L414P1)l (3((E:)JCLO)
11311, 4 IL “in"
In particular, set up a coordinate system with the z-axis

-—_I _.I
along the vector k where k is one of the vectors

—. .... -—A

Ol..,.a.1 , or '21.. Let 6 and 53 be the polar
and azimuthal angles of a?i.in this system. A little
reflection will show a(E, no) independent of (be

and so

C52605) 2 imEj/QUMmM) CL(E,qn(9°).
can. 51341.0 .911. 11.25:...

 

'Use of the program FAKE in generating random events for
use with these theorems is discussed in section V.

In evaluating cross sections by the method described
above, it is necessary to calculate the phase Space inte~

gral

127
J3 , 3 1 3
fasvlwvnhflr a? 23'. 4%?
x$(E-w,-w1—Ei)$3(§?.+fifa+fé‘fl
dPsggiflo)/W1LM) .
an.

Actual integration over the delta functions gives

FSCESMI3/413M) =f 023,3]. €9¢$2

-é-‘ [Sq-2(é:+/u:)ft+(flf‘/A:) ]
PI’ + P1

and its derivative

(D.6)

 

where 52: E24- M1+2EJFF+MI . From
(D.6). it is clear that
fg’IZSCEEbs/L‘n) /°‘1.,'vl) =r 1:5(:E:3 /°‘a)/b(1; P1) _
(Ln. ~ ' ‘in‘
So our formula for differential cross sections becomes
L6<E) = Q:(E)w‘-693FS(ES/‘uxuzyr1\.
onL

11311., “t 11"

 

So for both total and differential cross sections we only
need to find ES (E 3%. ,Ah.) m ). Going through the

rest of the integration of (D.6) gives

Eda-”...,MMM) .-.- arrlfopm 4,2

.[ 421‘] 5" - 2<M3+M:)E*+c,«7-w:)‘]

 

 

 

 

P1 + M 2 (D.7)
where Punch: =/[ E1~(A.+A‘*M) ][E1‘(M-*M1‘M)11.
1E:

The form (D.7) contains only one integration variable ,

and so is amenable to any of the many variants of Simpson's

 

128
rule. Because the integrand is a slowly varying function,

PS( E3MI)/”~1 ) M) may be evaluated quite accurately
by numerical methods.

It is not easy to find trial integrands with which
to check the technique outlined above because even fairly.
simple functions are difficult to integrate over phase
space. Nevertheless, two trial functions were discovered

and tested. The first of these is the function
a .

/(qa.p:3°f..<}a,pfl=(java-1') 5 “1291‘;-
~ ”aphid
For this function

C((E) 3 3
{m19p,$(E-u. ‘00..- Efléflifl *figi'qz) g"; :45}

_1L

j5(E-~.-w.-E,)é’ffi+*B‘i-“P‘ 5-5-13?” 3 ’2:

 

(menu: can 991.4(007- 9m) a J_
T 3

{meflft'l OQCWGF")

So for this case, Q(E) = 1/3 independent of energy.

Results of a series of trials are shown in Table 3.
'2 1

If<we take )= .4 A‘- 3.001. 9 I

i CF"P"C""C12’P"IOPHO;I

we should still get a( E): 1/3. Trials for this

integrand are shown in Table 4. From the combined
results of Table 3 and Table 4, we see that the Monte
Carlo result is within the predicted error 80% of the time

versus the 68% we'd expect, within twice the predicted

 

129

Table 3. Trial run of Monte Carlo program with

[-(%€)P£jfi-UG}1;P[-)= “1169‘.

 

 

 

 

 

 

 

 

 

Tr: Calculated Predicted Actual
(MeV) a( E) error(%) error(%)
210 0.3311 : 2.8 -0.6

288 0.3311 1 2'8 ~0.6

377 0.3542 1 2.8 6.3

466 1 0.3293 1 3.0 -1.2
905 0.3358 4;;2’8 0.7

 

 

 

 

Table 4. Trial run of Monte Carlo program with

i‘fiemcsaufimm) =m‘9w

 

 

 

 

 

 

 

Tr! Calculated Predicted Actual
(MeV) a C E) error (9:) error (25)
210 0.3324 1 3.4 -0.2
288 0.3196 1 2.7 -4.1
377 0.3424 w i 3.0 2.7

466 0.3409 t 3.2 2,3

905 0.3377 : 2.7 1.3

 

 

 

 

 

130

error 90% of the time versus the 95% we'd expect, and
within three times the predicted error 100% of the time
versus the 99.95% we'd eXpect. Moreover, the calculated
answer is greater than the exact answer half the time

and less than the exact answer half the time, as we'd
expect.

The other trial function was
f(q.£,P.'jfi-I>CI-13PI-) : a), w“ a -
#4th
If we define pCE) andCJs C E) by
(3(5) 2' j S (E-‘01‘U1‘E}.)33(€|+§3*Pp °
"¢of%fid clae}1.¢‘:73f >
(35(E)‘= MnMsfl $(E‘U.-w1-El_) 53(€‘+§-3+F§fi
(2"‘”h‘E56‘ ..ciae,..ar!5,.iafflaf

 

then for the function defined above,

C((EZ) .. (3(5)
95(5)

In Table 5 we tabulate the results of trial runs for this

 

integrand. For comparison,(5 ( E) and (~35 ( E- ) were
calculated by a Simpson's rule method to within 0.1%

or less. For this case the predicted error was smaller
than the actual error every time. On the other hand,
only one of the Monte Carlo answers was larger than the
correct answer, though we'd expect this to happen four or
five times. This might be because of some lystematic

error in calculation of F" E) and (35 < E ) or

131
some slight preference in the choice of random events
by FAKE. At any rate, for such small errors the dis-
crepancy is not significant. The general level of ac-
curacy is particularly impressive for the smaller values
of -r}T . The overall accuracy of the Monte Carlo.
results, especially in the low energy region, comes from
the fact that there is not much phase space to sample
over at these energies, particularly near threshold.
This is a happy coincidence for the calculation of chiral
dynamics models since it is the region near threshold which

is of most interest.

132

Table 5. Trial run of Monte Carlo program with

chpmiiorucm/AH': “‘01 EL

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

/¢&./¢~1_Pq
T11- (J ( E) Calculated Predicted Actual
(MeV) _____ error(%) error(%)
(35(5) C(( E)
210 1.200 1.200 0.1 0.0
722 1.263 1.263 0.1 0.0
233 1.321 1.319 0.2 0.2
310 1.746 1.742 0.3 0.2
371 2.103 2.091 0.5 0.6
377 2.138 2.137 0.5 0.5
454 2.613 2.611 0.5 0.1
590 3.512 3.526 1.1 -0.4
800 5.033 5.003 1.1 0.6
905 5.847 5.837 0.7 0.2

 

 

 

 

APPENDIX E

PION SCATTERING LENGTHS

The most natural place to compare the different
versions of chiral dynamics is in their predictions of
low energy pi-pi scattering parameters, especially the
s-wave scattering lengths. Scattering length GK is ,
defined in terms of the scattering. amplitude f(6,¢)by

a. = 1"" f. ( 6, ¢)
qr-QC)
‘ 0 0
where G} is the momentum in center of mass of one of
the particles. .For identical particles, like pions,

the differential cross section is given by

3% .. lluem) + fur-9,915.11)!

1.

 

But in terms of the «invariant amplitude m , the differ-

ential cross section is

‘2
as = I W72!
Zn. Q‘czn)‘ no"
so the scattering length is related to ”l by

Clan _l;"‘ [<777' 11‘s”.
‘1‘", 3211700. (E.1)

 

the phase factor Of being arbitrary. In our case

it is chosen to match the convention of gflrsey and Chang.

Chiral dynamics Lagrangians give the diagram- of

Figure 17 for pi-pi scattering.

133

134
q"
*3“. o, v
g:>f’f4
7‘14 ‘uf’f-D

Figure 17. Diagram for pi-pi scattering.

From mthe Feynman rule
:441?‘ )2 ( *3) r“...

m 11*?

5.39 9 9: 9.1 [Th/c3 + (3461.: + 01.5 7.0)] .

+ 9.. 9; 9' 91ffluu - (a-Aq.B+C}-c¢}-0”
+9..“ 9:! 96v 9:: [Th/0:. - (Ophfio +$B$c ”3

Go to the Cartesian representation of the vectors ‘,

 

A A A
$5, , Q‘, and $4 and take the threshold limit of (1‘,

3.3, 2‘, and $0 to obtain

(2mm :1)[Salr5a£ ()1 42),“.
.wskCn. 42),. + 3136.101 am]

From (E. M1),

°~" fi'fi M(%v)( )1-[5..4.8an. 2)
+22056: “L?” Sam 8M 01.91143.»

We wish to separate the various isotopic parts
of (3.2). Consider some sort of operator in isospin space,
Q , which operates on isotopic triplets. We may

expand Q in terms of projection Operators

=,,§_ITT-.><T T2!

13‘:r ' as

0: o.P° +Q.P'+Q.,P?

If we take a matrix element of Q between Cartesian
states IQ)<> and 1636‘), we get

«,4! Qla,c7 mars“ + BSwaavc slash.

It can be shown that

 

135
(:34, T54ak‘r E3 +~CL
Q1: B‘C-
Q1 = B+C.
If we apply this result to the scattering length
0L ,

the scattering lengths for different isospin channels

w '52:?- ”HEW-3i?

6a.: O 1 1 H
“1 ‘(’%’,—},3)’“(€%) (Sn-‘1).

If good experimental numbers existed for Clo,

are

 

(E.3)

 

0t. , andq'a. , it would now be easy to determine 71, , 1
and thereby choose one of the chiral dynamics models

over the others. Unfortunately, pi-pi scattering exper—

iments are difficult to perform, and all the available

results can do is indicate that the expressions of

(E.3) are plausible.

REFERENCES

 

.L.‘ *h't

ll.
12.
l3.

14.

15.

REFERENCES

Y. K. Lee, L. W. Mo, and C. S. Wu, Phys. Rev.
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Cell-Mann and Levy, op. cit. :44“
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