HEAVY MESONS AND THEIR EFFECTS ON THE
NN AND “N ELASTIC SCATTERING AMPLITUDES
Thesis Io! the Dawn §I Ph. D.
MICHIGAN STATE UNIVERSITY
George C. Summerfield
1962

_ IIIIIIIIWIIIIIIIIIIIIIIIWWII!'

1293 01743 0129

This is to certify that the
thesis entitled

HEAVY MESONS AND THEIR EFFECTS ON THE NN AND
N ELASTIC SCATTERING AMPLITUDES

presented by

George C. Summerfield

has been accepted towards fulfillment
of the requirements for

Ph D 439,66 in Physics

Major professor

Date 5/2— {//6 2‘
/ f

 

0-169

LIBRARY

Michigan State
University

 

NN AND NM EL STIC SCATTERING AK?LITCZZS

p.
.3 C? S L:““arfield

tzd t0

Submitt
Richigaa State University
In partial fulfillment of the requiremen nts
or the degree of

DOCTOR 0? W ILOSOPHY

I‘

Department or

'0
’T

s and Astronomy

U7
0

\r.‘
0"».
IQ

C/“JIQ/

z?2wbg

«r: V
ALFJ/J

NN AND HE E;£STSC LCilztn

neutral,

These are evaluate:

theory.

ten percent of the p

nucleons, and low

are not significan‘

pseudoscalar coup

are developed for

I
I

amplitudes for one

.—~."~‘a'f\J

{ I)

f' ’1' '\ l‘ 'f‘
s.) O J\.s".v

I» J

---,\\\!:\ A'A‘

. . 5., ,—
l I .—"—~

I . .
‘1 o I ..__

I
l
l
J

at fiVDLITHfiT“
st». .1 u Vex—u

=‘roa “ C'r“°r£°elo
vvlblw v. dusty Il

~ “0 A“ I" i '_
If” IUlchOHJ ére 'CVCIODV SOY

OUTCG O

m
O

The fourtn order terms are shown to be aoout

- J—Aw,,.- .C r. J. a a
ole eras tor neutral, scalar
:nn '7 2.:- °-‘: "hr: 'fii' A’oc‘“.o“
energsoa,ll Ltc COdp xng Coustcuts

arger taan the pion-nucleon

raga, spsnor nucleons. mesons thn

zero strangeness are considered and it is shown that

there are ten that
field, conserving

these mesons are c

t r‘ ,1 I "" A 1 ,. - ‘,,.‘ -Z'
a, r, *, cfl‘ G. Inv pots tor.s o.
-, I" > fi"" , w" ’- . J- " .|I,‘. ‘— r ‘ F ~ ‘ . a
s-ezm;ncd an Lne “x an‘ .. c.as-xc
O O

scattering emolitccas, o;

t

proton-proton and restro.

scattering cross sections

pole terms. This 3 ves

in terms of twelve coup}

L.

pun.

.-l J— , .f-‘w-n ‘
TIC: ROULrC/‘x‘l‘nebc . or”.

CHAPTER
I.
II.
III.
IV.

V.

VI.

INTRODUCTiCE
THE BASlS 3? Di
CAUSALITY
NEUTRAL, SC LAR

l.
2.
3.

4.

CHARGED,

l.

PIONS
].
2.

Uplt I Lilli

j

L7l\:l-l.fll~RIT‘{o o o o o o o

I
L.

lIJCLEONSO o o o o o

I”...-.- __._O
{\liJUJJQLlCSOOOOOQOQoocoo.

ouble ion ncleti
Dete“:fiet2on of Lhe Sca
moi?:ude...............
SF;IOR XUCLECNS......

he Scattering amplitude
Crossing Relations......

ion

Cross Sections.........

A.l“\lD I’lEa:b"-!’Y £.ESO:§IS Q 0 O 0 I O O

G-ParlKEV/OOOOOOOOOOCOOOO

G-Parity of Baryon-An

StatESOOOOOQOOOOOQOOO

Quantum Numbers of “

I
H r
and

Mesons

O
I)

U)

r11-

0

H

(D

Relations

.00.

SPEASION RELATIONS.

tibaryon

1.;esons O O O O C O

the Nucleon-Nucleon

Scattering Amplitude...........

fx)
1:

I 0
Lu

[‘x)
\\I

\lJ
U‘t

C)\ U1 U1 U' 1 \JJ
H KC} "4 .—4 u‘)

(‘M
[Q

0\
\J

J\
to

\J
N

\l
KC)

II-‘h -~_\,

. Pertarba-ion lntegrals..........

O\
O
m
(1
..)
(l
l
(‘I
N
(n
(L
C
:3
rt.
(3)
”'1
(”l‘
‘<
0

\II

CD
U‘I

\! U'l

)

V.)

U;
U.)

\f:
U:

N

\‘c'

\f, -
Cl.)

FIGURE
1.

A Function,

Two Brancn Points

(.3

Contours f

Amplituce..
Four-Moment
Nucleon-Huclaon

(’3
O

Singular't:e

Meson Exchange Di

Contour Defining
Closed Loo;
Fourth Order Box

Singtiarities in

Hitn‘

I
ane

Two Po es and

invariant Scattering

Fourth Order

:0

0
rs
(U
’U
23'

TheorYOOOOOOOOOOOOOOOOOOOOOCOOOOOO

Crossed Box Diagram

OOOOOOOOOOOOOOO

I

eaction Ad-B"

Pertu

oOOOOOOOOOOOOOOOOOOOOOOOOo.

f‘ I ,
C'r

\r) (t. (o u» m N

\C)

V. ,.

k0

U)

Ct)

‘
-

U I

“\I

\f‘)

Nucleon-Xu

Intermedic

G-Parity o
Nesons Wit
Meson-Wucl
Observed L
Meson Pole

Residues,

LIST OF TABLES

cleon Channels..........

3—\ “ A'l-'
Lg; btd»CSooooooooooooooo

~3-
lql

for t 3 O, and m
a

. t
attering................
f i-N States With J<.2..
n J<.2, T<.2............
eon Interactions........
esons...................

Terms..................

3

o\_y‘ij00.000.000.000000000

ions in N-N and fl-N

A CKN OIIVIJED GLIENT S

The author expresses his thanks to
Prof. D. B. Lichtenberg for suggesting this
problem and for his continuing guidance and

encouragement throughout the course of the work.

viii

Presently, no satisfactory theory of the
fundamental interaction bet ween nucleons is availabie.
Calculations have been made using phenomenological
potentials and meson theories, but none of these can

be cla sifI

I
O
a”
U)
0
in
r-I
”—
(D
"i
‘3
U)
0
‘1
O
0
U)
to
(C
0
f I
O
”i
L

quantitative predictions

to any desired accuracy. any potential theory serves

23
If)
0
IS“
(0
:3
(D
(D
"3
0-
O
.«3
( I
L'I
1
(II

primarily as a par“rotcii:atio
insight into the funza ental nature of the interaction
process. Meson the rise attempt to explain the natu.e
of the interaction. However, the role of the meson is
still not well determined, since meson theory has yet
to give a quantitative result for the short range part
of the interaction.

The usual pi-meson theory assumes that the
nuclear force arises from a pion field in the same way

the electromagnetic force arises from a photon field.

 

l. For an extensive survey of the theories of t
nuclear interaction, See H.J. Moravcsil< and H.P. Noyes,
Ann. Rev. Nuclear Sci. ll '

2. A discussion of tr’ .e pion theory of nuclear .orces
can be found in J.D. Jackson, ”The Physics of Elementary
Particles," Princeton Univ. Press, Princeton i;55.

The long range nucleon-nucleon force is given by the
exchange of a single virtual pion between two nucleons.
The pion is virtual in that it does not conserve energy
and can only exist for a finite length of time by the
Heisenberg uncertainty relation. The pion carries
momentum between the two nucleons and, thus, gives rise
to a force of finite range equal to the inverse mass
of the pion exchanged.3 The long range interaction
predicted by single pion exchange is well verified ex-
perimentally.

Since the range of the interaction equals the
inverse mass of the particle exchanged, single pion
exchange should predict accurately the long range part
of the interaction. However, the short razge part
should depend on the exchange of more than one pion.
effects of the exchange of more than one meson have have
been satisfactorfly calculated, and, considering
complexity of the problem, it is unlikely that they wil
be using usual field theoretic techniques.

Since the calculation of the short range
nucleon-nucleon interaction has not been made, it is
not certain that the pion is fundamental in the interaction.
Sakuraih suggests that vector mesons heavier than the
pion are the fundamental particles in the nuclear inter-

action, and the pion is a bound state of an antinucleon-

 

N

3. We use units such that ”h = C. =-

l, I (19

O\

O).

h. J. Sakurai, Ann. Phys.

nucleon pair under the influence of the heavy meson
field.

The existence of heavy me sons has been
confirmed by experiment. If these heavy mesons
contribute to the nucleon-nucleon interaction, their
effects would be felt chiefly at short range, and
would not contribute appreciably to the long range
part of the interaction. Also, the heavy mesons need
not be observed as free particles. They could decay
rapidly into pions if their masses are greater tnan the
mass of two pions and their quantum numbers allow the
decay. We discuss these points in Chapter VI. hitn
the recent discoveries of the two and three pion
resonances,5 this heavy meson theory of nuclear forces
becomes more attractive.

It may be immaterial which of the mesons are
considered elementary and which are considered composites
of other particles. Supporting this viewpoint, hishijima6
has shown the equivalence of elementary and composite

particle theories for local, renormalizable field theories.

 

5. The resonances are T'JT 33”, reported by A. R. Erwin,
et al., Phys. Rev. Letters 6, 628 (196]); bJ-e 37?
lpygorted by B. C. Maglic, et al., Phys. Rev. Letters 1,
(l96l); li—N‘n’ reported by A. Pevsner, et al. ,
Phys. Rev. Letters 1, 421 (1961); and Y- ~> EST" repor
by R. Barloutaud, et al., Phys. Rev. Letters §J ll’

A

I::

6. K. Nishijima, Phys. Rev. ll, 995 (1958).

b,

’ Similar considerations apply to the anti-
nucleon-nucleon interaction, with the added complication
that annihilation into pions or heavy mesons is possible.

Regardless of whether further progress in
understanding the nucleon-nucleon and nucleon-anti-
nucleon interactions can be made within the framework
of field theory, basic objections have been raised to
the use of field theory to explain fundamental processes.
As long ago as l91+3,6Heisenberg7 suggested that the
failure of the concept of the continuum for short time
and space intervals makes even the definition of a

8

Hamiltonian impossible. More recently, Landau and
Chew9 expressed the opinion that field theory is
incapable of completely explaining elementary phenomena.
Our discussion is based within the frameworki
of dispersiOn relations, and the form of-a dispersion
relation is independent of which particles are chosen

.10

as elementary. Chew and Frautschi claim that it is

possible to use dispersion relations to completely

determine all strong interactions. If this claim is

 

7. w. Heisenberg, z. Physik 1.2.0.. 513 and 673 (1943).

8. L. D. Landau, Proceedings Ninth Ann. Inter, Conf.
High Energy Phys., Kiev, l959 -

9. G. F. Chew, "5- Matrix Theory of Strong Interactions, "
W. A. Benjamin Inc., New York, l96l.

12] 6G)F. Chew and S. C. Frautschi, Phys. Rev. 1;}, l#78
9 l

U‘l

correct, the choice of elementary particles is immat-
erial and field theory is unnecessary. However, the

claim has not been substantiated.

THE BASlS CF DISPERSICN RELATIONS.

Even in t.1 e absence of a com .plete theory,

a
v
.-

ects infons

£12
I

dispersion relations can predict some e

nucleon-nucleon and antinucleon-

(i)

and heavy mesons on th

nucleon interactions.
Before discussing the use of dispersion

relations for the nuc iaar interaction, we must des

cribe the interaction. for elastic scattering of equa

s

mass particles, the differential cross section is

related to the T-matrix as foil
/' - r a
Ggiteflzip) 1T E with->3 (I)

h‘here E' is the center of mass energy of each particle.

The T-matrix is related to the S-matrix as follows:

“ii. mam-9 <2)

0"“

<i|$l£> = S“ Jr (3.27) 'i

Where Pi and P, are the initial and final total
03"
four-momenta. The S-mat rix is de fined in terms of the

incident and scattered wave functions in the next chapter.

We use the invariant scattc ring amplitude introduced by

l . . , . . .
Heller. - For react: ns Vitn two inCident and two

 

l. C. M¢ller, Det. Kol. Dansxe Vidensk. Se lsk., Mat.-Fys.

\J

Medd. 22, NR] (l945).

scattered neutral, spin ess particles, the invariant

amplitude A is related to the T-matrix as follows:

 

(qlqlelq3q4)-=OS , [:5 C A (3)

The initial and final fou ~momenta are q1, q2 and

q3, Q4. Labeling the in tiai and final particles “,8
and C,D, we write the scat“ering represented by equations
(1), (2), and (3)

A-r 8 ~—%

D)
m

C)
a-"
l
L'.)
A
[—e.
\l

The scattering amplitude for this reaction is a fenction
of two independent variables, e.g. the total energy and
the scattering angle. However, it is more convenient to
choose the invariants s, the square of the total center

of mass energy, and t, the negative square of the momentum

transfer:

[\D

.- ‘.

\ -

t = ( - C I
ql '3

In the formalism of diSpersion relations,
A(s,t) is continued into the complex s-plaie, while t
is kept fixed. The continued function may have singul-
arities in the séplane. if the location of these singul-

arities can be determined, an integral representation for

A(s,t) can be written using the Cauchy formula.
For example, consider a function A(s,t),

not necessarily a scattering amplitude, which has two

simple poles and two branch points in the s-plane, as

shown in Figure l. The location of the singular ‘ ies

U"
.1

may depend on the second varianle, L. The anch points

are at s}(t) and 52(t), and the poles are at s3(t) and

54(t).

Applying the Cauchy formula for the contour

Np v

shown in Figure l, we obtain an integral representation

for A(s,t).

(. fittES'S) l I
.. _—.—— \r- ~—------’ OHS
Hls’fl at’lTi 5) 3-8 (6)

\
C(t)
Letting the border of the contour extend to
infinity and assuming the contribution of the integral

.. 2 ., .
around the border vanishes, we are left With only tne

pole terms and the integrals along the cuts. Defining

A(s+—é) t) and A(s-& ) t) as the values of A(s,t)
on different sides of the cuts, we rewrite equation (6)

for Figure l as:

 

 

 

 

l
R(s,-It) "' 83/(53"' ‘3) + l}i‘i'/i{5:}_s) ’T’ . “
“{qu‘ .22.: ,J'f'iw (/)
-—-'-v 843' flslé.-i3-RiS'-éi-il y l .845 Fiji-6;: meta-,1)
am 9— s Wt s‘- s
s, 53*

 

2. See Chapter Ill for a discussion of this point.

\O

 

 

 

 

 

 

 

 

 

 

A Function A(s,t) With TWO Poles and Two Branch Points.

10

R3 and Rh are residues, and the numerators of the
integrands are the discontinuities of A(s,t) across

the cuts:

Disc. A(s,t): A(s+€ , t) - A(s-e,t) (8).

Double dispersion relations are obtained
by applying the.same considerations to Disc. A(s,t)
as a continued function in the complex t-plane.3

Our use of dispersion relations depends
entirely on our ability to determine the singularities
of A(s,t). The determination of the singularities of
the scattering amplitude constitutes the major part of
this work. We indicate how the singularities are
located and how residues and discontinuities are
determined from causality and unitarity in Chapter iii.
In Chapter iv and V, we discuss the double dispersion
relations for neutral,scalar, and charged, spinor
nucleons. In Chapter Vi, we discuss the properties of

pions and heavy mesons and how these mesons affect the

dispersion relations.

 

3. see Chapter iV.

CRAPTER lll

CAUSALiTY AXD UNlTARlTY

We now Siow how to determine the singular-
ities of the scattering amplitude using causality and
unitarity. The use of causality has a rather long
history in the theory of scattering. In 1926 Kronigl
and Kramers2 used causality to write an integral
equation relating the dispersive and absorptive parts
of the refractive index for the scattering of light.

Causality was used by Karplus and Ruderman,3

4

r\ particle theory. A

{M

° ‘ ,..-..,.. m;—
In Elan-pant.

\

and Goldberger

rigorous proof that causality implies a Kramers-Kronig

type of integral equation for the scattering amplitude

in field theory was given by Bogohtixm and Symanzik.O
The mathematical details of the field theoretk:

proofs obscure the physical content of the theory.

 

l. R. Kronig, J. Opt. soc. Am. 1;, Shb (l926).

. H.A. Kramers, Atti. cong. intern. fis. Como ;, 545
l927).
. R. Karplus and H.A. Ruderman, Phys. Rev. fig, 771 (1955).

. M.L. Goldberger, Phys. Rev. 21, 508 (l955), and fig,
79 (1955).

N.N. Bogoliubov, Report of the international Conf.
Theor. Phys., Seattle, lSSb.

6. K. Symanzik, Phys. Rev. lOS, 743 (1957).

 

in order to maintain a clear insight into the physical
nature of the theory, we avoid the f'eld theoretic
analysis. Our discussion is based on physically reason-
able arguments using causality and unitarity. This
approach sacrifices some of the rigor and completeness
of the more formal arguments.

We limit the scattering to two particle

.1
Q
L)
'l
('0
U)
(u
1')
N
(0

initial and final states -..i ' d by equation (4),
Chapter II.

Following the usual analysis,7 we consider
that the interaction induces a transition between non-
interacting stationary states. These are the initial
state of particles A and B incoming to the region of

interaction with total energy E, and the final state of

T}

r
«‘1‘,

outgoing particles C aha '. The wave functions of these
i . . , 5

states are 1V in(E)> and l ‘/out(E)> . The

scattering is represented by an operator S(E) connecting

the "in” and “out” wave functions.

l

1351 out(E)> __ 5(5) i“/in(t)> (l)

The relation between the matrix elements of S(E) and the

cross section is given in equation (l), Chapter ll.

 

mentaw

7. For example see J. Hamilton, “The Theory of

Ele
Particles,” Clarendon Press, London, l959, Page 2%3

13
8

This is the S-matrix introduced by Wheeler and used

by Heisenberg.9

The Fourier transforms of the quantities

in equation (l) obey the following relation.

H," outta): _| Vt 3g‘:’,,,lt'i at’ (2)
“CG
Equation (2) states that, for an incident wave packet,
the outgoing wave at a time t is a linear superposition
of contributions from the incoming wave at times t'.
Causality dazzanes that H’in(tl)) contributes
to l+out(t)>only if t‘ comes before t. Thus, the

scattering operator vanishes for t' greater than t.
S(t-t'): O, for t<t‘. (3)

We use the Fourier representation of 5(E),
oo

’

SW) = 3 SW) 6‘ Mali: <4)

-9
to extend the definition of 5(E) into the complex E-plane.

Equation (3) limits ’t’ to positive values in equation
(h). Thus, if E has a positive imaginary part, the

integral has an exponential damping factor and is well

*

8. J.A. Wheeier, Phys. Rev. fig, llO7 (1937).

9. W. Heisenberg, op. cit.

defined and finite. Then, 5(5) is analytic in the
upper half E-plane. On the rea E axis, there is no
exponential damping and 8(5) may not be analytic.

We use the results of causality together

0 derive the analytic properties of

(’1‘

with unitarity
the scattering amplitude.Unitarity is a statement of
the conservation of probability in the scattering.

the initial state is normalized to unity,
<\ljin(E-) l‘i)in(t)>= 1, (5)
the final state must also be normalized to unity.
<Huout(E) l Kljout(E) :l. (6)

Using the definition of S(E) in equation (6), we get the

unitarity property.

<Yin(s) l sls l kl}in(t)>---‘<“ii/n(t)l‘Viiutyel. (7)

S(E)r S(E): l (8)

S(E) is not the most convenient quantity to
use in dispersion theory. A more convenient quantity is

the invariant amplitude, related to the‘l-operator by a

I5

function of the energy given in the definition of A

(equation (3), Chapter ll).

A: C(E-)l. (9)
l'is related to S by
s: l-il. ‘ ' (10)

l,is related to the T—matrix introduced in equation (1),

Chapter ll by:

L".
If

<é \lep = «this la-axilrm

T and S obviously have the same singularities. T(E) is
~ 9'“

analytic in the upper half C-plane. The unitarity of S

I”

leads to the following condition on I;

sTseul- iii-)(I-il) e i,
(II)

We make one further convenient change.
Instead of the energy E, we use the square of the energy

5 as a dynamical variable. Since T(E) is analytic in the

upper half E-plane 3(3) is analytic on the sheet of 5
corresponding to the upper half E-plane, excluding the
real 5 axis.

' 0

K
lat“

[Tl
ll

a'L?
szmfle “ (u)

0<Q<W

Unless C(E) introduces new singularities,10 A(s) is also
analytic in the complex s-plane, except possibly on the
real 5 axis.

With the analytic properties of A(s) estab-
lished, we use the Cauchy formula to write an integral

representation for A(s).

A(s) :: m Li<l : (I3)
CuJ-i-CL
The contours CU and CL are the two semi-circles shown
in Figure 2. If A(s) vanishes for large 5, we can
extend the borders of the contours to infinity, and the
contributions from the borders vanish, leaving only the

integrals along the real axis in equation (l3).

 

IO. C(E) actually removes an undesirable singularity
at 5:0.

A(S) -‘-‘ "T" 6‘1“” 'I‘ ~Ni

ELTifl. ‘3 3-3 “I

Combining the two intrgrals gives:

 

CC
I 1” fl( ‘ 'I I.
.- --—- \ (’1 fl' 3+(C)_H(S~(é) '
We assume that:
A*(s)== A(s*). (15)

This assumption comes from the perturbation theory ex-
pressions for A(s). For certain perturbation theory

‘ diagrams it does not hold. (See the Appendix). It is true
in perturbation theory for nucleon-nucleon and antinucleon-
nucleon scattering, and we apply the condition in equation
(14).

(Ci , Fi(s'+i6)- Himgg

._'__ __
A(s): ATTL \oxS S._ S

..__‘ (00:14.)
am I“ s‘-S

-63

Ms) :

Unitarity (equation (ll) ) gives an expression

for lm A(s).

Im A(s) :: C(E) Im l(s) ,
im A(s):.- isle) (t—I’Wa. (17)
lm A(s): -— CLE)ITE/EL

 

 

 

 

 

m/ C U
r
s-plane
\\
\
e I} > > #___
6 I :< ‘<
I
i
fiSURE 2

Contours for the Invariant Scattering Amplitude.

19

Taking matrix elements between initial and final

states, we get:

 

1m ems):- LBIITIW (l8).

Summin over a complete set of intermediate states ives
. 9

 

CIE
lm<elfile>=~§ ikhlITIanIIIQ (19)

Equation (IS) shows that Im <I3>IFII°L> fails
to vanish only at energies for which there is an allowed
intermediate state In) , since I has only energy conserv-
ing matrix elements. If In) is a single particle state
with the same quantum numbers as ICK> and IfB) ,
Im ((BIHICK) does not vanish at an energy equal to the
mass of the particle in In) . If In) is a multiparticle'
state with the same quantum numbers as ICK> and la)» ,
Im<p|fild> does not vanish at all energies greater than
the sum of the masses of the particles in In) . Of
course, In) could be either Id) or Ia) .

Using equation (l9), we can write equation (l6)
as:

80

h I‘fld>'
Rn S’J‘S' :EJQ‘II‘IS

((3II5II"\>"’§\I Mt-s 4" ’75?

 

s‘—S (20)

{fifihJHIL

”3
[AA
by

The sum on the pole terms is over all allowed single
particle intermediate states. The lower limit on the
integral is the square of the mass of the particles in
the multiparticle intermediate state with the smallest
total mass, and ijs the residue at the pole.

We now have rules connecting the analytic
properties of the scattering amplitude and the allowed
intermediate states.

I. If there is an allowed single particle
intermediate state, the scattering amplitude has a pole

at at energy equal to the mass of the particle.

2. If there is an allowed multiparticle
intermediate state, the scattering amplitude has a branch
point at an energy equal to the sum of the masses of the

particles.

3. The residues at the poles and the discont-

inuities across the cuts are given by equation (19).

To obtain equation (20), we assumed that
A(s) vanished for large real 5. If A(s) is constant the
borders of the contours CU and CL in equation (l3) cannot
be extended to infinity. However, if 50 is not equal to

any of the M‘s, equation (13) applies to A(s)/(s— so)

ZI

and the contours can be extended to infinity for this

function.
I i 9(5.) PI(So)
_L.LA -‘-1 ""cXS'T—"Trt ———-— ,
s- :0 '- aflTl (SAKS—So) 3- So (2:)
CUtCL
'"5tead of equation (20). this gives:

<MHisll°l>_ «undone Rn
. fS'So - 3" Sofi— + g (“l-S)(Mx-§°3

 

 

DC,-

I “ ,Imdilills‘no :
+ ’Il' 8‘“ (s'—S)(s’-s°\
{native-ll

IL)
[\)

 

Multiplying equation (22) by (s— 50) gives:
I

(when) ~<rlells.llo\>= z “(fig-g Rio)
in

l

a

00
. .3..- \
+_I__ (4g Immmslmht, 32s,)
’rr 1
{n‘+u).+lu}

Equation (23) is the subtracted form of the
dispersion relation. If the scattering amplitude goes
asymptotically, as the nth power of s, n+-l subtractions
will give a valid dispersion relation. Notice that in
equation (23) there is one new parameter, A(so). Each
subtraction adds one new parameter, since no two sub-

tractions can be made at the same value of so.

6

22

Consider neutron-proton elastic scattering.

n+ pen-1r P

The deuteron is an allowed 'ntermediate state, giving

rise to a pole at s : H32. Since the mass of the deuteron
is less than the sum of the masses of the neutron and
proton, the pole is below the start of the physical region
of s,((r\"il'\n\\55400 ). The lowest mass multi-
particle intermediate state is the neutron-proton state.
The branch out starts at s: anI' MP) 2 and extends to
plus infinity.H

Now consider antineutron-proton elastic scatter-

ing.

i3+tnérfi+r>.

+
TheTaneson is an allowed intermediate state,

. . .,a . .
giVing a pole at s: hTT . Tne lowest mass multi-

particle state is the two-pion state. The branch out

I] Here,

a . . .
starts at s = 4M“, and extends to plus infinity.
the cut starts well below the physical threshold, while
in neutron-proton scattering the out started at the

physical threshold. This difference makes antinucleon-

 

II. There will be higher mass allowed intermediate

states and they will contribute to the discontinuity

across the cut by equation (19), when s is greater than

the square of the SUm of the masses in the intermediate state.

23

nucleon scattering more difficult to analyze than
nucleon-nucleon scattering.

‘The results presented here were shown to be
valid under the assumptions that A*(s):= A(s*) (equation
(l5) ),12 and that C(E) introduces no singularities in
A beyond those in I. For particular scatterings, of
certain particles, these conditions do not hold and
there are perturbation theory graphs which have differ-
ent singularities than those given by rules I. and 2.
However, these rules are valid for the graphs involved
in nucleon-nucleon scattering.

Thusfar, we have not justified fully our use
of unitarity. Unitarity is well defined for physical
values of s, but it is certainly not obvious that
unitarity applies for non-physical values of s. In the
next Chapter, we show that the scattering amplitude for
negative values of 5 represents a different physical
process.13 For 5 between zero and the physical threshold,
»the meaning of unitarity is still not clear and our use
of it must be justified. Mandelstamlh has shown that unit-
arity can be unambiguously extended to this region

consistent with the continuation of A(s).

 

12. The failure of A*(s)== A(s*) can lead to anomalous
thresholds and complexsingularities. (See the AppendixL.

l3. See Chapter IV. _
l4. S. Mandelstam, Phys. Rev. Letters.fl, 8h (1960).

CHAPTER IV
NEUTRAL, SCALAR NUCLEONS

l. Kinematics
Before deriving the dispersion relations,
we discuss the kinematics of the two body problem. Con-

sider the reaction represented by:
A-k B —A' C%- D. (l)

The kinematics are specified by the four-momenta of the
initial and final particles. Figure 3 is the diagram

of the four-momenta.

Time
Component -

qA QB

 

 

Space Component

FIGURE 3

Four-Momenta of the Reaction A+B -¥C-l-D .

25

The metric is defined such that:

' -+
q2= - q 2+ 52 = NZ. (2)

The statement of energy and momentum conservation in

the reaction is:

= Cl - (3
qA+ qB C-I- qD )

We choose the same variables as in equation

(5), Chapter II and equation (l2), Chapter III.

2 2
s = (CiA+ QB) = (qc+ <20)

t = (qA-qglz = (q3- CIC) (1:)

2 2
U = (qA‘ QC) :3 (CIB -' CID) '

In a two body scattering, there are only two indepen~

dent variables. Therefore, there must be a relation

between s,t, and u. Equations (2) and (3) imply that

2
s+t+ u : (inquz—i (qA- qD)2+ (qA- QC)

_ 2 2 2 2' .-
s + t+ u- BMA + MB + MC + MD—ZqA’(C-IC‘)'CID" 0.3) (a)

sli-ti-u : MA2+MBZ+ MC2+MDZ .

26

For nucleon-nucleon and antinucleon-nucleon scatter-
ing:

MA: MB : MC : MD 2: M.

We give the connection between 5, t, u on;
the center of mass energy and scattering angle for this
case. Kibble1 has derived the connection between
s, t, u and the energy and scattering angle for arb-
bitrary masses, but the results are more complicated.

The center of mass conditions for equal

mass particles are:

- i i
:q - Cl D=(<)lZ+MZ\/°’ (6)

Using equations (6), the following expressions for s,

t, and u derive.

' 2
s-.-. 43,4. 38) + (qu+ qOB)2-.- 1452+ M2)

2 .' 2 -2

-(qA- q0) + (qu- qu) =45, (l-cose) (7)
u:-- (qA- qc) + (qOA-qoc)‘: 'Zq (”COSa)’

9 -+ ~§

cose- a o - .
"qA. .0” qA qc'

where 32

l. T.W.B. Kibble, Phys. Rev. liZ, l159 (196$.

27

From equations (7), the physical range of s, t, and

, u is:

(8)

i:
[A
fi

b
O

2. Crossing Relations
Consider the three reactions represented

by Figure #. Each reaction is called a channel.

   

l.N+N4N+N H.N+N-+N+N HI.N+fi+inN

FIGURE 4

Nucleon-Nucleon Channels.

Crossing relations for neutral scalar
particles state that the same function of s, t, and
u, A(s,t,u), represents the scattering amplitudes of
all three of these reactions.2 (See the Appendix).
However, the connections between 5, t, and u and the
center of mass energies and scattering angles, and the
physical regions of s, t, and u are different for each
channel. Table I shows the connections between 5, t,
and u and the center of mass energies and scattering_
angles, and the physical regions of s, t, and u for
each channel.

A(s,t,u) represents A[(s,t) When 5, t, and
u are in the physical region for channel I, A,l(t,s)
when s, t, u are in the physical region for channel II,
and Alll(u’t) when 5, t, and u are in the physical
region for channel lll. Al’ All’ and Alll are the

physical scattering amplitudes in channels I, II, and

 

2. H. Lehmann, K. Symanzik and W. Zimmermann,Nuovo
Cimento 13 205 (l955), and 6, 3l9 (1957); and

S. Gasiorowicz, Fort. der Phys. §, 665 (l960); and
M.L. Goldberger, Y. Nambu, and R. Oehme, Ann. Phys. g,,

226 (1957).

29

 

23'
:1

El PHYS 1cm. REGlCN

 

 

 

 

i
CHANNEL EENYER or NASS VARIABLES ilOF S, t, and u
l l.
l N+N4N+N‘i “whiz-yr?) ” Lmzss<°°
tz-z'qz (l—cosB!) -s+l+M2.<.t 6.0
Liz-2E??- (l + c059,) ~5+LiM2$ u s O
I
ll N+N+N+N s=~2E§2(1- c056”) -t+t+MZss s o
t== A(E'Z-l-MZ) AMZs t<°0

 

u=“232 (1+ c058”) ~t+L+M a us. 0

 

- -— -z>
Ill N+N+N+N s::--2q2 (1+ coselll) ~u+l+M2ss 5 O
ts—Z'qz (l- cosGHl) ~u+l+M2$t 5 0
U: 4(324- M2) 4:425 u<°o

 

 

 

 

30

Since we are going to relate allowed inter-

mediate states to Singularities in the scattering

amplitude, we Show in Table 2 the pertinent, allowed

intermedie-s states for each of the three channels.

TABLE 2

Intermediate States

 

MULTIPARTICLE INTER-

 

 

 

 

ONE PAR ICLE
CHANNEL INTERSEDIATE MEDIATE STATE WITH
SMALLEST TOTAL MASS
l (D) deuteron* NN -Nucleon-Nucleon
II M - Heson** TT TT - Pion-Pion**
m M - r-«zeson-kv': T1 T1 - Pion-Pion**

 

*The deuteron state is only present for neutron-proton
scattering. We include it in our discussion inside
brackets. In the particular cases of neutron-neutron
and proton-proton scattering, the contents of the

brackets should be taken as zero.

**There are different mesons possible with the quantum
numbers of some nucleon-antinucleon state. Only five
have been observed and of these the pion has the lowest

mass (See Chapter VI).

\U
.._4

By the arguments of Chapter III, Al(s,t) has

no singularities in the complex s-plane for t real and neg-

 

ative. Therefore, we can apply equation (16), Chapter Ill.
00
l I
Al(s,t): 2': gold Ln H'BLQ (9)
H $'_S

—00
When 5‘ is greater than 4M2, lm A|(s',t) is the
imaginary part of the scattering amplitude in channel I.
When 5' is negative, since sl+t+u'= hMZ, and since t‘
is negative, u‘ is greater than 4M2. For negative 5',
lm Al(s',t) is the imaginary part of the scattering
amplitude in channel Ill. This second statement follows
from the crossing relations. Thus, lm A|(s‘,t) is the

sum of contributions from channels I and Ill.
lm Al(s‘,t):: Ai(s‘,t)+tA3(u‘,t).

A1 and A are related to the imaginary parts of T-matrix

3 .
elements by equation (3), Chapter II. For example, A1 is

I
A t Z 6 n ’3‘ :
1(5, ) (I ququchqu) 'm‘<quB'T|'chS>

where the T-matrix is related to the S-matrix by:
<il Sl i) = 3,, , emit (”'03, me fill)

The unitarity of S,

§<ilsiln><nlSlf> = 6;,

32

gives the following expression for lm <i)Tl§> .
(et’ll‘VI‘

mm rm: ,_ 2‘; Sikia-PnlahllanlTlp
Thus, A1 and A3 are:
(AWY) , , 0,, ‘i . ,. ' T
PM“): :— §llbioa°goggoggwl S(Tigris-inkmdr.Iridium.)
' ' (Io)
(AflY‘B a '/ bi 1' 2' -
R3kuit3: T N \‘L EcamoszoccEw); 8 (CER+%L-?VI\ <3A1¢flul®<flinnnagp>

In terms of Al and A , equation (9) is:

3
oo _ .4 ‘x n
. , 1,145.) _._ __I_ . Must)

p

Equations (l0) and Table 3 Show that A1(s‘,t) is non-
zero at 5' equal to the square of the deuteron mass,

s‘:.MD (for neutron-proton scattering only), and.for
s' greater than the square of the mass of two nucleons,
5'13 4M2; and that A9(u',t) is non-zeroat u‘ equal to
the squares of the masses of the mesons discussed in
Chapter VI, u':Vh?, and for u‘ greater than the square
of the mass of two pions, u'ZfikMi . Since sL+t+-u'=’hM2,
A3(u't) fails to vanish for S'::#M2-vn2-t and
S‘éihM2-4M%-t. Using these results, we write

equation (ll) as:

 

 

 

 

)Kn H (35f)
R ‘ l ‘ 1 I
H1)330:) a.“ 3+ 2n Sridrfll-‘miv-T) + E 8‘13 $'-S
“5's 3.
-00 [in
R (u‘rcl
\ I _§

+n‘-4Hi-T

)~'

33

The sum on m extends to 311 single particle
intermediate states. Unitarity (equation (10) )

gives the residues RD and Rm and the discontinuities

 

A s',t and A u',t . For exam le, the term Rm -
1( ) 3( ) p s.- (“MI-mid)

comes from the single particle contribution of the

meson, m, to equation (10) for A3(u',t).

 

Rm 2 8(31\? Emuvtofltoi motmob)?‘ Sq ($fl+%t‘. P M)x

s --(l-lMZ--n1--t)::1r
5-5? (‘6; Til—l [mel’mlTl'ls'g‘bD

'2
B)

'2
D)

I
where s': -
. (qA q
I
: (qA— q

: (qA'+ qc )

Since the sum is over a continuum of intermediate states

of four-momentum Pm,

—: + AL x
g:: SJ Pm ékhyi'q“)

 

(LTfl’

and

Rm
s-(HMz-mZF t)

:55 (‘L%oR%oB%OC%oD) “Build" ‘7“ 55M ”Tilia‘LM'fifl‘UlTIhh}

3h

' I
Since u': (qA+q ')2 and u'+ t-l 5‘: M42, the s'
C

integration gives:

Rws (“antiwar a; z: IT’li%i*t£l><w3+%£)lT I m,»

where: qA‘+qc',._._ qB'+ ‘qD'

2
(qA.+ qc.) '-‘-

. 2
l l_
(qA-qu,s

I I).
(qA -qD )- t.

This expression for Rm is the square of the renormalized

meson-nucleon coupling constant.
2
Rngm '

Similarly, we can obtain a dispersion relation

for A'l(t,s) in t for 5 real and negative,

 

on 4thrlE-S
x' l Fl . l
R m'm Rmt Z‘t-(an‘M‘d lfiLSdtm HAL s +1184" 7L15 (13)

          

4% -oo
and for,A,ll(u,t) in u for t real and negative,
R -t
° l+ arid“ LM’r +1r 'CLE‘M (M)
u-

35

' Single dispersion relations have been
obtained by Go1dberger3, Capps and Takedall for pion~
nucleon scattering and by Goldberger, Nambu and Oehmes
for nucleonenucleon scattering.

Since we have not considered the asymptotic
behavior of the scattering amplitudes, “subtractions”,
described in Chapter 111, may be necessary in

equations (12), (13), and (1%).

3. Double Dispersion Relations
Single dispersion relations have been
used for pion-nucleon forward scattering by Dav con and
Goldberger6 to exclude one of the two sets of phcse
shifts that fitted the scattering data; and by Haber-

8

Schaim7 and Gilbert to determine the pion-nucleon
coupling constant. It is difficult to extract more
information than this from equations (12), (13), and
(1h), since they do not include the momentum transfer
dependence of the scattering amplitude. The momentum

transfer dependence is given in double dispersion relations.

b

M.L. Goldberger, Phys. Rev. 99, 979 (1955).

R.H. Capps and G. Takeda, Phys. Rev. 103, 1337 (1956).

26 (1957).

. W.C. Davidon and M.L. Goldberger, Phys.Rev. 104,
119 (1956).

. U. Haber-Schaim, Phys. Rev. 10%, 1113 (1956).

3
z,
5. M.L. Goldberger, Y. Nambu 8 R. Oehme, Ann. Phys. 3,
2
6
l

7
8. W. Gilbert, Phys. Rev. lg§, 1078 (1957).

36

Since A(s,t,u) must satisfy equations (12),

2

(13), and (lk), it has poles at (s=MDZ), t: m and

uzinz, and branch points at 5::4M2, t = QMTTZ and
u: kMTTZ. In equations (12), (13), and (14), one of
the variables must be real and negative, so these might
not be all the singularities of A(s,t,u).

Mandelsta‘a9 assumed that the only singularitkas
are those required by the single dispersion relations.
Figure 5 shows the location of these singularities in the
real 5, real t plane. Figure 5 shows only the pion
intermediate states for channels 11 and 111, but remember
that the intermediate states in these channels should
include all the mesons discussed in Chapter VI.

If the only singularities are as shown in
Figure 5, A(s,t,u) has singularities only for 5 real
when t is constant and for t real when s is constant.
Therefore, equation (16), Chapter III holds for A(s,t,u)

in both 5 and t.

 

 

00
. 1 I ills‘iie-i ‘1- siiet '
F,(s,t,u)::-fi \ds J 4111 R1 1 11:1 (15)
.oo 5 ”S
his! “IMF {in K46 mille’ll‘hwl' ”(Sil‘It"*I”') (16.)
.00 it. t

 

‘ 9. s. Mandelstam, Phys. Rev. 112, 1344 (1958).

37

 

FIGURE E

Singularities of A(s,t,u)'

Poles ————

Branch Pts. W

38

Applying equation (16) in equation (l5), we get:

 

‘50 oo
(S,tu. *‘JS alt 'P,(' (. ._ -:_ z: u)- 95- £e,'t ieufl) (113321,, t"+ieu')+ His‘de (4.“)
H 3:? 1%"—-—s)(t t) (’7)
-00 .00

The integrand in equation (17) fails to

vanish only for (s‘zzMDZ), t'=.m2, u'zinz, and in the

three cross hatched regions in Figure 5. Actually from
perturbation theory the integrand vanishes in a smaller

region than these. (See the Appendix).
2
s 7 11M , t 7 £11le
5 7 4.142,. u > 11:12

2 , 2
t >1hM , u‘7 4%
Defining the spectral functions as:

_ Flis-ie.t+ie,u)-Eli-{Egt-iém- Rlslié,t+iL-Ju]+HUNefi-(e,u)
5’ 11.11-

I 4_ .
3: 13,91) __MS- 1.6111“, u+iel~ FMS" (Elf; u-ia) - HLSHQJJUHCQJ- Fl($+(e+,-t}u-fg

[i—

f it 011" H(S,t-l&,u+ie)—Rl'>,‘t—ie,u-le) --Fl(s,t+ie,u+ie) +F1(s,t+£e, u-ie)
, -
4.

we get the Mandelstamirepresentation from equation (17).

39

 

 

 

. 0° N o
(if: 0.?— I l (3.1.)
RLS u : {in + ._....1 , {.4 __l- I ’3 4....
’t’ ) (Hp-S) m nil-t T g}, mL-ulqr‘ :15 4: (slant-t)
1'1" 411
co 50 b0 q” ll (19)
r "3' l ' 1
~1- ' 1 ‘--._J_CLM l n 1 fishfll')
+114 45 lWKS'-Sllw-u1 +1? 8‘“ MHL'LHW-“l
a. 3‘ ‘ x ,_
4'“ 4H“ Lll‘lfi (“‘13
The assumptions made in obtaining the
10

Mandelstam representation are too stringent. Eden
has shown the validity of the Mandelstam representation
for the following conditions.

a. 1n the real 5, real t plane the singular-

ities of A(s,t,u) are those shown in Figure 5.

b. A(s,t,u) is analytic when one of the
variables is real and positive and the other two are

complex and satisfy s1-t1-u = AMZ.

The most general conditions for the validity of the

Mandelstam representation have not been determined.

#. Determination of the Scattering Amplitude
Attempts have been made to use the complete
two dimensional Mandelstam representation and unitarity

1] However, most applications

for pion-pion scattering.
reduce the Mandelstam representation to a single dis-

persion relation.

 

10. R. J. Eden, Phys. Rev. 120, lSlR (l960).

11. See G.F. Chew, “S-Matrix Theory of Strong Interactions”,
H.A. Benjamin 1nc., New York, 1961.

40

These single dispersion relations do not fully
utilize the implications of the Mandelstam representation,
but they do give more information than equations (12),
(13), and (1%). The difference between single dispersion
relations obtained from the Mandelstam representation
and equations (12), (13), and (1#) lies in the assumption
that the only singularities of A(s,t,u) are those required
by equations (12), (13), and (14).

Assuming these are the only singularities,
we can remove the restrictions that one of the variables
be negative. For example, equation (13) is valid
in the physical region for channel 1, where previously
5 must have been negative. This is the single dispersion
relation that Cini and Fubini12 use to analize nucleon-
nucleon scattering. We sketch their analysis below.

Making the change of variables:

U:L1M2- S—t
Ulng'MZ— S—t'
-du'.-.- dt.‘

 

12. M. Cini and S. Fubini, Ann. Phys. 19, 352 (1960).

41

in the second pole term and the second integral in

equation (13), we get:

1311‘:th) 21:11:13; * .L 11113.1( 35.1.1.2)1— 8‘3“.“ 4‘1le (20)

Unitarity gives expressions for A2 and A3 similar to

equation (10).

11,11,113 -‘ 111 21 (151,801,513? 3411131194111 (‘13 ‘BDITTI H><V11T1 116113)
(21 1

11,111,114? 2111.11..1..1.11 $11.11 1119111111 11x11 1111111)

The sums are over multiparticle intermediate states,
since single particle intermediate states have been
split off in the pole terms. Cini and Fubini consider
only the two pion intermediate states and approximate
the rest of the sum by a finite series of Legendre
polynomials. They do not consider the posSifility of
heavy mesons.

Ignoring the heavy mesons, this approximation
is reasonable, since intermediate states only contribute
when the energy is greater than the mass of the particles
in the intermediate state. Thus, the two pion state
contributes for t or u greater than 4Mfl-, while the

three pion state contributes for t or u greater than 9Mfl"

42

Noticing the symmetry of equation (20) under the inter-
change of t and u and applying the Cini-Fubini approx-

imation, we get:

R1s,t,ul= film-iii? 3:31—4:13 +1te 9) 17.1 (M 1111.191

where s, t, and u are in the physical rexgion for

(channel 1.

s -.-. 1432+ 112)
t =-232(1- cose)
u=-232(1+cose).
3%fl(t‘,s) is the two pion contribution to A2(t',s).

a. 1
1111‘” 2111-1111 1;.1..‘a;.1"é (int-11,- -?1.1<1',1’,1T1?1,;> 91.113.111.11

“We express the T-matrix in terms of the

scattering amplitude for antinucleon-nucleon annihilation

into two pions:

 

1 l
("Milk-H“ m“ > z 111. 11,1. 11.31 11.51 1111'“- 133111113111; )

Biis the scattering amplitude for the annihilation reaction.

N+ii 9T1 + 11.

These results give the following expression for A(s,t,u).

43

 

g; 41 1,9 ,, 1 3‘11'1‘11'1511‘1’1P') ‘1' .
2 —:f- + '72-231LE) a t1rr a D 11[ - -
R\5,t,u) Mn‘t {1‘1 8 tL-L 2:31} a. 4P0" Po}, (S (‘1); 111111131)
Riki (22)
L

+ (13H u) +2; Wnfiime)

14 and K1 must be determined from experiment. Hopefully,

they are small.

Cini and Fubini write the Mandelstam represent-
ation for B1 , bringing in the crossed channels
771+ N—+TT r N. Then they reduce the Mandelstam re-
presentation for B to a single dispersion relation and
evaluate the single dispersion relation approximately,
including only the two pion state in the unitarity express-
ion. Thus, the dispersion relation for E5 contains the
scattering amplitudes for ‘ri+"Ti-111'+ TT and
11+Tl—)TT'+ TT.

Following the Cini-Fubini approach, one must
simultaneously determine the scattering amplitudes for

three different reactions.
TT+1T——>’1‘1’+’1T
‘W+Nafl+N U”

N+N~+N+N

44

Chew and Handelstan13 use equation (20) to de-
termine the singularities of the partial wave amplitude
in the complex 32 -plane. This method also involves
the simultaneous determination of the scattering
amplitudes for the reactions in equations (23), unless

only pole terms are considered.

Since we consider the effects of ten mesons,
the large number of different reactions to be considered
in the methods mentioned above prohibits their use. We
deal directly with the Mandelstam representation, but
only consider the pole terms‘explicitly, and attempt to
find a convenient parameterization for the integrals.

The spectral functions have been determined
in fourth order perturbation theory for neutral scalar

nucleons from the diagrams shown below (See the Appendix).

0- 31

 

 

 

 

 

 

 

 

 

 

 

 

 

. ‘ D
For 3 (s,t) N '1 a N r
'3‘ N N __
N >, ., b N L
0‘13. q13
qc k 90
r W 4,—
For :3(s,u) g a N
N N _
N K *- efléflv N X
r v rfi
9A QB
13. G.F. Chew and s. Mandelstam, Phys. Rev. 119,
467, (1960.

4S

 

 

 

 

 

 

 

 

 

 

 

 

/’ e
C qB
N a N
N N
N k b N L
I W r
q q
For S°(t,u) A D
x3
q q
D >_ ’~_,»~\_,, _ ;: B
N a N
N N
N b N
__ y y,
0A . qc

a and b are any two mesons.
We express the contributions of these diagrams
to the spectral functions in the following way.

Q
‘ ? (s,t) is the contribution of the first diagram

to ,&(s,t).

ab
Y (s,u) is the contribution of the second diagram

to fi3(s,u).
1.110
?(t,u) is the contribution of the third diagram

to fi3(t,u).

1x10
? (u,t) is the contribution of the fourth diagram

to Uta).
'1 uh ,

The form of all of the § 5 is the same. Thus, we can

write:

Tablvifl

LMW 1‘“ 1”“ 1x21111111-

 

and substitute the appropriate variables for x and
y to obtain the contribution of anyone of the diagrams.
An approximate expression for Iab(x,y) is given
in the following.
With the spectral functions approximated

in fourth order perturbation theory, equation (19) is:

 

 

D 1., l _l___
Hightfik) :( 1: )i-écil’mkm‘zt.‘ Vat-1A) (21+)

From the generalized unitarity condition
Me
(See Appendix), the spectral function § (x,y) is in

fourth order

 

1.1 9~
Swim) KT” 1 51
- 2 ._. “T 1 2
1W5 Al'MIYXUh/ioj h’ibV-Xll; 4m; Mgflk
ah ah
g (x,y) : 0, for x or y less than 4M2-and Y (x,y)

complex.

Using this spectral function, Iéb(x,y) has been
approximated to within 30% as follows:

1m Iab(X.y1‘-=

 

 

(flirty; sky-11111! y _ (ht-MEN .
’1 "T (CX’E-NtEl’h V31 -&CX‘&(L"K"—CDX+CEY’ ,
L
(26)
+ 9111-111?) 9,1. _ B

 

 

11111-1211": ” h-iRy—ua‘yagsyy.

47

Re Iab(x2Y) 2:"

11,, {eel-1v 1111:) =1 3
._.... 15,, 4M __ x
Q 4- $71.16 (7" 4‘“ x91’) “1* 111111 31‘s AWE-wivvt-i-w-Llh" (27)

2°43 (ax/11011111431- _ )1“ %/@R1iq.fi+me1u17+ “-11-,
3W11x-31 + ax-eim \ and AW

MM 9111:1111 i [(4- qm'fi/VJ a 11111-111 “RX 01101-3) Axe-(>141 (Milli-3))
\

at- x elude: (hex

 

 

 

 

 

 

 

 

 

 

 

(13'1“) n3) '1' sic-31101:) + 3;. 0145 £1931) “LET
0‘ L (001110131 (3 ' 31

+ 11- “111/1: «11-11'»%(

1, (“NATHAN hit—i ‘91:“ Aim s__.1_________ c1 is >{3Jéd-911M-f? )]
7 34/111- 91114) (3 ' - CK

 

 

 

 

 

(V3411 1')”-

_Ne(‘ Y______)_ (J _ i _ _ ' A aha-@1114“ +é‘gil‘
M LI'M“..__;,yf1my ‘ifl‘YYL {33%11 b) (A h)“ 13) 1433K rs-“

 

 

 

1M1

+ (Ix-KW 11-11114» 1,.Q(:m:1:b-m‘+ “m“ B)+ (h~a)m 91-11) £4,3(i1x-411x-Plhx-fl!)

 

 

+ (k'q)4(X-q)\K-G) 901-1111111041 (fl/73111913) + M 01-13 )]g

01—13

1+8

where

4:1mdmbv.
{3:1W‘A‘ Mb);

A :1X-c111X-15)

B : 4MXR+4W12M§X

c = 113-4145!)
0 = 31135131531 C+ 07mm)!
E: c1130

 

1m‘ ______, ‘fi
a=1m£+mc1+ 3......”‘6 W +1/(1mgmmwmé11‘ - 019
3

 

 

 

U.
H

ova (5m +m51+w pug?

 

C =1miwi1- 13.1““ 13me 1.3“”51-40
in“ )n‘

 

d = __ s. ). Mir/ll 3. ‘1
Fa A (M‘4M13+Teqrb)fd\\3

I
M ..
o - L d1d-c11xdtx'd1+xc1c~x)

1
ab1b-u.)+7\1o\x'\o\ + M1341

 

9(x):0 for x( O
9(x)=1for x) 0.

1+9

t

An estimate of the size of Iab(x,y) is

L
given in Table 3 by a comparison of Iaa(t,s) and +‘/1M:'t)

for ébrward scattering.

 

 

 

 

TABLE 3
T (ts) for t=0 m=lal- m
..aa 2 2 3 a - fl
2
fa :f2
2
ma-—t a
lm :[aa(t,s):.5fal+ x 10-4

 

_ ‘* -4
Re Iaa(t,s)._ 4fa x l0

 

Iab is the fourth perturbation theory con-
tribution to the scattering amplitude. However,

unlike the usual treatment, it is not evaluated in the

static approximation, and it is completely covariant.

50

We see from Table 3 that if the meson
coupling constant is not significantly larger than the

3.
o I ° 0’ .
pion-nucleon pseudoscalar coupling constant (:L leg )

of the spectral function in fourth order perturbation
theory contributes only 10%tof the pole term to the
scattering amplitude at this energy. The similaritbs
of the fourth order expressions for scalar and spinor
nucleons suggest that the fourth order contribution in
the charged spinor case will not be more than /\J 10%
of the pole term. Thus, either the spectral function
is unimportant in the NN problem, which is unlikely,
or the fourth order approximation is not very good.
Since evaluation of the spectral function to a ;attcr
approximation is very di ficult, we consider only poie
terms in what follows. These have a chance of being

‘accurate at energies which are neither too high nor too

low (150-300 Mev.).

CHAPTER V
CHARGED SPlNOR NUCLEONS

l. The Scattering Amplitude
Since we write the scattering amplitude
in terms of nucleon spin and i-spin wave functions, we
record the following quantities for future reference.

The Dirac spinor for a nucleon or antinucleon with
. . . . 1
four-momentum q and spin prOJectnon (r us:

 

( ‘11: fl '\
H' 10' 11,}- ‘5‘1'3-
l %.+i%:__
[\M /)_ lG’1fi-‘C‘V‘
UKWWA) :- J‘ ' H. fig, “‘1 1. r
I 1 3. H‘ 3033/“): 1:.) 33'1‘64'1‘1l‘ff‘5:
‘* A HF“
‘14 (1)
‘Bl’ri‘ln.
\ "(64‘1'14

 

 

 

 

0111,63A) = WWW, >11 1’,

2 2 2 2... q = M . z 1'73. corresponds to

spin projections parallel and antiparallel to the three-

4
momentum ‘1), )1 :: 1 corresponds to a nucleon, and

)1: -l corresponds to an antinucleon.
1. See D.R.Bates,“Quantum Theory", Vol

.lll,Academic Press,
New York, 1962. .

52

In this representation, the spinor oper-

ators are:

(0 o 0 .41
0 -i 0

X1.: 0 i 0 0

O

 

 

 

Ki 0 0 0,1

/0 o 01}.
0 —l 0

O

‘9'<
[l
O

 

 

000i (2)

 

 

 

 

r.
O
O
O
.L

L.

0
OOO

 

 

53

2

The nucleon and antinucleon i-spin wave functions are:

Proton X1p) =3 (A)

Neutron X114) 3 1(1)) (3)
Antineutron X171) '3 1:3)

Antiproton 71(13): 13‘)
In this representation, the i-spin matrices are:
t“ — 1°‘1
X " l 0
0-1.
TY (I. 0) (1*)
z” (I o)
2 0'1

For nucleon-nucleon and antinucleon-nucleon

elastic scattering, there are ten independent scattering
amplitudes, five for i-spin triplet scattering and five
for i-spin singlet scattering. This can be seen as
follows. Assuming charge independence, we get no singlet-
triplet i'Spin transitions. There are four states of
total angular momentum J , orbital angular momentum Q ,

¥

2. The isotopic spin of a nucleon is one-half. See
J. Hamilton, op. cit., p. 198. ‘

54

and spin S, for a particular total i-spin T. The
five allowed, independent transitions between these

four states are shown in Table 4.

TABLE #

Allowed Transitions in N-N and N-N Elastic Scattering

 

SPIN SINGLET

J=£{"‘) J =3;

 

SPIN TRIPLET
J‘lg‘léJ =£*-|

J=2i J=£+

J :Lgl—u : 1,,”

 

55

That these are the only independent
stransitions can be seen from the following arguments.
For a particular i-spin state in nucleon-nucleon scatter-
ing, the Pauli principle3 requires that the parity of
singlet and triplet spin states is opposite since the
total wave function must be antisymmetric under the
interchange of the nucleons. Thus, a singlet-triplet
spin transition violates parity conservation insofar as
T is conserved. In the spin triplet state, transitions
from Jz/ti to 13,“: l and from J=£iil to 4:1,
also violate parity conservation. The transition from
le-L-l to J=£++I is equivalent to the transition
from J=Li+ I. to‘Jaflq-I by time reversal invariance.

All of the arguments above apply to anti-
nucleon-nucleon elasticscattering except the application
of the Pauli principle. vHowever, in this case, G-parity '

Si+T (-])S++T

and since T is conserved, there are no singlet-triplet

and parity conservation imply (-l)

spin transitions (See Chapter VI).

 

3. Notice that in the case of neutral, scalar nucleons
(equations (23). Chapter IV) Bose statistics were
applied. This was done to avoid the inconsistancies invdv-
ed when Fermi statistics are applied to scalar particles.
See F. Mandl, “Introduction to Quantum Field Theory",
lnterscience, New York, I959, pp. l6 and #8.

56

‘ #
Following Goldberger, et al., we take the

ten invariant amplitudes shown below.

Channel I
ii, = 1E°I s.-’§.)+ F,°1T,+T,1’ri:1H-;-fi.1+E,°l\li'\7.1+alt-15.125213»

4- [Eh-31+ a: 1TliTi1liF§1Flrfiill FlthValiFs‘lii-illil

where 13 and 13 are singlet and triplet i-spin
projection operators.

“PO-— L“ It? "Ca

 

’(6)
13‘: 3+ ’5, ’1’],

Channel I I or

 

~

#3,” = [15,0 15,"_§“)+ ELO(-El+il)+ 153° 1 Ha .fill 11’ if; (v'fiVu) +1350 ( VII-10] R

 

(7)
+ 1 "(31' (Sn-i1 i 132 (1311+ '13; 1Ru'fi..)+ ii: unfinlfit‘ kill-ill] I3,
where ‘
i51: L::ii° ii.
° ’ 4' (8)

'l

 

.)
-— 3+TQ'TB
P. ’4 _

 

4. M. L. Goldberger, rqu,Grisaru, S.W. MacDowell,

- and D. Y. Wong, Phys. Rev. l20, 2250 (I960). A slightly
different amplitude is used by D. Amati, E. Leader,
and B. Vitale, Nuovo Cimento 11” 68 (1960).

 

57

Channel Ill
'3“; =1E° (S..- S...1+F° 1T...+T T1+E°(l:1...-i:i...1+E°lV...+i/j,.1+E°1?..;-E.1)E
(91

[E 1s...-3§..+1+,‘1E T... T..1+F 1+1.- ’13'1...1+'F 1v...+V...1+1F: it...i..1lE

where fPo= 1" (13°13 . (10)
:: 4:, 9
4.-

The operators Sl’ etc. are analogous to the

{3- decay operators.

Channel I, for 8.11:1, VI ‘1):
1111,61. 1,141. 11113693111 11.6118 .n. “(WA M1

”MN”

for 5 RIM?
1M 1.611.151. u 1136313101 worn/111 n Miami/v) (11)

AA = An :/\(.:/\D=I
where:
1 for S 8' g
. KM for v 8 V
.fl-z &%v for T 8' T
1.1:; KM for R 5 a
X; for 'P 8' ’9

for example:

H , = (“113(5) >115)ng AilM‘hBG‘gha) Q1%¢Wc&)¥1M£\A 11'an AR)

 

58

Channel II, for 511 HuTuvu Pll

 

111151111151 what 1,1 01 1°1th 1,111. uI‘lmWa An)

~~

for gll fill-Tu VH1)"
{111.111.151 11111.6, 11.1 011th 13111 mimifl 11:1 (121
)15 3 )‘L :- I

Ag=+AD=‘l

Similarly, we define the operators for Channel III.

S,T,A,V,P are related to S,T,A,V,P as follows:‘

 

 

 

1? f1 1 I- 1 11 isl
’17 4-2. 0 2-1 v
’f :-—l-- 60-2 06 - T (13)
X 4' ti 2 0-2-1 A
151 .ngl 1-1 14 11>,

 

 

 

r—

Equation (13) can be verified directly using the
definitions of the spinors and spinor operators (equations

(I) and (2) ). 'We also record the following relations

59

which result directly from equation (13).

 

 

 

 

 

 

 

 

 

 

 

 

'5" ”2110 o‘é-‘E
v 420-2 4 v+l7
T1,]. ~60 20-6 T-f"? (14)
A +- o 2 o 2 o A-fi
~PJ k0-110 2J e-a
no ,r h’
(5" (2110? 5-3
ii 420 2-4 v+'\7
~ I ,
T=-4: 60 20 6 -I-T (15)
K 0 20-20 A-‘A
A- w
P o.-110-2 P-P
LJ .\ J~ J

2. Crossing Relations
_ V We can determine the behavior of‘E} (s,t,u)
under the interChange of t and u from the Pauli pwgnciple,
From the definition of t and u (equations (7), Chapter IV),
this interchange corresponds to the exchange ofparticles
C and D. In Channel I, the.wave function must be anti-

symmetric under the exchange, and

'3] (s,t,u) '5 wk}. (s,u,t) . (16)

60

The exchange C H0 gives:

CHO

PdT’TPo

SH 15

v H V (17)
,0

T'++ T

A all
1"

P a P

Equations (16) and (l7) imply:

a} Uri Ir

9 9 = -1 , , 0 l8
Eetu) () FL (sut) ()

The crossing relation between Channels l

and II has the following form (See Reference #).
“'1

F3: FR Bit FR
Fifi: E; B“ E:

where B is the i-spin crossing matrix and F
u (R

(19)

the spin crossing matrix.

(3th]? [3:00 gazi [l l] (20)

 

 

 

rm rt; [:3 R4 as) (1' ad“) -0
(F): PM fitfl‘sfiiflc -_-_-'— ‘1 ° °‘ (21)
“‘ IE‘E‘E’CIC‘ 0” 22:3
r F l" V I‘ ~
I 6:5: (35:55) 'I L ‘I 4U

 

 

is

The crossing relation between Channels I and III can

be determined from equations (18) and (I9), but it is
not necessary to give it explicitly. Equations (18)

and (19) contain sufficient information to determine the

features of the dispersion relation.

3. Double Dispersion Relations
Assuming each of the ‘ Few. (s,t,u) has:‘
only the singularities required by causality and unitarity;

we get the Mandelstam representation (equation (l9).Chapter

A '?
IV) for each R (s,t,u).

 

 

4, I I-Ilil“? R 1'
F;(fs,t,u) = z“: Rmakl m*-I+ "T“ )+ ( iii)

0° 00

l 0° 0 )3 ' (LIA) 1:80 I). I (5”)

+——- .11: a W: 45'413‘1
1“) u (t'- t)(u'- u) + (s‘- SSW-t) (n)
4"? 4'”; 4:94:11}.

 

 

60

+ .L. )4“ )4 («W futiliu'l
1" x (5‘- s)(u‘- u)
‘I'M 4H;

 

where 5:31,‘ (tl,ul) - (I) 1' 824.1% (u',t|).

We have applied equation (18) in equation (22).
The residues Rm. _& and R'M‘l’ and the

spectral functions can be

itvfi and f;3\3

determined Using unitarity, similarly to equations (l#),

Chapter IV. However, since equation (19) has related

62

3:st to fit”; and the u pole term to the t pole
term, unitarity applied in Channels l and II gives all
of these residues and spectral functions. This is the
reason we do not need explicit crossing relations

between Channels l and III.

h. Cross Sections

We now relate the.invariant amplitude
defined in equation (5) to the cross section. First
we define the amplitude for transitions between states

of given helicities, following the formalism of Jacob

and Wick.5

(WCG‘DI‘PLIG'HQ'Q

The helicity, CV , is the projection of the spin of
a nucleon along the direction of motion of the nucleon.
(P‘ is defined such that the relation between the

helicity amplitude and a particular i-spin component of

3‘ (s,t,u)-is:

my“ «cw Ivar»: F Is- §l

 

(23)

+ FailTI*A—fl)+ F3i(Hl-fi|)+ F4: (VII‘.~\’/l)+F§i (Pf?)

where i = 0,l.
5. M. Jacob 8 G.C.Wick, Ann.Phys.‘1, 40# (I959).

63

The cross section for scattering in a given
i-spin state between states of definite helicities

is:

i¢i(c’80§" Gig-‘0) ' a_
on; "" = WWII “(WNW (2F)

 

There are no singlet-triplet i-spin transitions. The

ten independent helicity amplitudes are5
KP}: (tal WI t I.) = (4.4.) Wig-I.)
- 50:: <51“ ‘Pl VJ.) : <‘h’kl LP‘lI/x 93>

IP,‘=<t-I.I IP‘I ’.-'/..>= <—F.9.\ Ip‘I-u.)

(25)

LR}: <I/‘-}3.‘ will. 91) : <J/x'IL\ \‘pil l/x' '/1>

LP} = (I. M) U.) = -<'«.-'/.\ till '42.)
<- m I‘I-m = - (w I“ Its-’4.)

<2 '«I Fifi-mam Ill I...)
= Q 'I.’ ‘1.) W 7N.) : “(J/f“ Qty-9:9,)

where i .-. 0,l.

6h

The connection between the F3 and the
helicity amplitude, )PS' , can be determined fran
equation (23) and the definitions of the F¥* 1*

(equations (5) and (II) ).

 

. I I , , , 9 .
LP! ‘ my? [Mil F.‘+ IF£+ Félmel - I 354+ 7;} 5‘]
. %AWRL5E/+*l I’M‘I‘ffi In‘fi (mean 3-1.3]

493 {l'gmr-I (an? +sF./l+%IF,+1F fFHflme/L (26)

l. l I. L' 'I 4‘ g 1 g -.
“RI" RT: [M 3+ s Fm- I (-F. +1F.+F.)l sulfa/1

i

-_.M i I .~
ng- .Trlf.+F+l we

where s -.-t+(252+ M2)
_ +2
t--2q (l-cose)

Ila-22’;2 (II-case)

 

is. [AI—174+ [Ia-Jr'i‘IFJm a'lme (1")

 

a 3" ' "a
+ W E'- ‘W? I +-I- I1M‘E'+ SF~'/.+ I I-E‘HF'Jre‘ll’m‘llx
+Ll1M1F‘ 5F" 4" A—' " I 3‘ ' ‘I-
s ,.+. q/L-e‘g.(—I",+1F;+Fg)l$m 9/3.

all FF'llSWlel

 

+4M‘

Jfiinlp): ‘
an . 3m"

Wigwam:I-I-z—WIIUFWWH

 

{-31 {ML [E1 F.‘+ (Rot FJIFJI- F4) c0161—
(30)
IN F4°+ E; )1 mean" F;- i’Fg' Isn‘fiii‘fi |1+
' | 9 I I o o -"
3|1M°IF1°F F. )+ .3. ($331. 7,1. F.°-F. Hr. III-3w. ”all“?
+ g.l1M1(F,°+F,‘)+-‘;_(F4°+F+‘)—‘?(- F,°- fi‘I-LFfI-IR'I F;+I:5') j‘sad‘l'fi

+ II M)“ l F.°+ F.‘+ F4°+ H“ l a‘8 («4‘ 93

 

65

Neutron-neutron and proton-proton scatter-
ings are in pure i-spin triplet states. The center
of mass cross sections for these scatterings with
unpolarized incident and target nucleons is the
square of the a1? olitude obtained by averaging over
initial and sum fling over final helicity states.

I
‘la. 5" 9:.

clLInnL _cm;—LZ ZS: |<VHlehl>lL

TS)- All. 4' but 3:.“- ". km;

Applying equations (25), we get:

 

Ail“): ‘Wflz {AM IHH’ lw‘rkltpal (27)

I i I i
+1l\l’,l+i{l‘l’cl l
Neutron-proton scattering is in an

equal mixture of i-spin triplet and singlet states

and, similarly, the center of mass cross section

for unpolarized incident and target nucleons is:

Amine- I {Hipgtp‘rilwu pm“-

an. T
+1l)():+ Ifl‘l-i ilkffflfl‘l"
I xlflfifi'l‘fi

Applying equations (26) to equations (27) and

 

(28)

(28), we get:

CHAPTER VI
PIONS AND HEAVY MESONS

l. G-parity

The quantum numbers specifying a state
of strongly interacting particles with zero baryon
number,B, and strangeness, 2, , are the spin, J, the
parity, P, the i-spin,T, and the charge conjugation
parity of the neutral member of the i-spin multiplet,
C. Lee and Yang‘ introduced the operation 6, which is
the product of charge conjugation and a I80o rotation

about the Z-axis in i-spin space.

lfl’T}. (I)

All members of an i-spin multiplet with
B = 0, B = 0 are eigenstates of G with the same G-parity.
We specify the G-parity of all charge states instead of C
for the neutral state. Also, since G is conserved in
strong interactions, it can lead to selection rules.
. The i-spin wave function,)L , transforms under
tfifl; as follows:

For i-spin zero,
IwT;

e X0 7- X0 (2)

 

l. T.D. Lee and C.N. Yang, Nuovo Cimento‘i, 7R9 (l956).

68

For i-spin one, since

 

 

 

X”: X.+ ix,
X- 7- .XI ' L13;
X9: X3
and since
r (_ '1
“urT, X. X.
8? 1K; : )kt
~.X: L'XSJ
then .
iirT, )0 X-
C x“ =- x“
X° )0

2. G-parity of Baryon-Antibaryon States
For a nucleon-antinucleon state with zero
charge, charge conjugation interchanges spacial and
spin coordinates of the particle and antiparticle. The

nucleon-antinucleon wave function is

\lfméNfi): th.VI)l'va,¢,) i [px.G‘.)\Ex,Ir.)

‘

W and x are spin and space coordinates.

C if. 'INN)= lVIXIWI>Inx.W.) t. Isiah): pm.)

EQI

If al' and bl' are antinucleon and nucleon creation

operators then:

C II... Wt («Iv-Fa Saws) 1 aim I»: W I o>

69

Since the fermion operators anticommute

C ‘KM, mm = - { bum aim.) at him.) damn} 10>
C 41‘th n): - lnxtmflfi xx.) 3; lme>||3xm>

The interchange of the X; and (Ft gives a factor

(-l)L (-l)s+l. Thus, we get

9me = WWW“ m. mm (a)

L and S are the orbital angular momentum and the total
spin. Since the G-parity of all members of an i-spin
multiplet is the same, equations (2), (3), and (h)

give the G-parity of nucleon-antinucleon states as:
L+S+T
G==(-1) (5)

, Equation (4) shows that the parity of an RN state is

(-l)L*'l. Thus, N and N have opposite intrinsic parity.

70

Table 5 shows the G-parity of nucleon-antinucleon

states with J < 2.

TABLE 5
G-Parity of NH States With J4 2

 

NR State T J P L s G
'50 0 0 - 0 0 1..
‘so I o - o o -
3po 0 0+ l l +
3P0 I 0 + I I "
'PI 0 I + I o -
'P' l I + I o 4-
3Pl 0 I + I I l-
3P, I I + I I -
35! 0 I - 0 l -
35! I I - o I +
3D. o I - 2 I -

 

71

For a lambda-antilambda state, the wave

function is:

YIN/I) -.-. Mm.) mm»

C ‘I’WII = III N.) mm = I—II"*'I-II“‘ WAR)
Thus, the G-parity of a lambda-antilambda state is:
G = (ml-+5 ~ (6)

since the i-spin of the lambda is zero.
For a sigma-antisigma state with zero charge,

the wave function is:

‘i’w I 23:) = I I B°x£>lf1°ufi> + I. I n‘x.fi‘.>l mm

+ clfi'xnmli‘xsm

where a, b, and c determine the.i-spin state.

C Wntutlxi) = (K ‘ ioxuwt)l XOXLWD+ b I if XIWI> ‘ 2*)“ W1.»
+ a: l '2?me BRAD

Lil S+l ‘
= I...) M mm
Thus, the G~parity of a sigma-antisigma state is:

G‘:(-l)L+S+T (7)

72

, A complete discussion of the G-parity
of baryon-antibaryon states should include 2 R and
i A states. However, the G-parity of these states
.depends on the relative A—E parity, which has not

been determined.

3. Quantum Numbers of Mesons
A meson with strangeness zero is
spedified by the quantum numbers T, J, P and G. Table
6 shows the sixteen combinations of these quantum
numbers with J<2, T<2. Only ten of the mesons shown
in Table ‘6 have the same quantum numbers as a nucleon-
antinucleon state. Notice that equivalence to a nucleon-

antinucleon state places the following restrictions

on the G-parity.

Scalar meson G = (-|)T
Vector meson G :(-l)TH

(3)

Axial, Vector meson G = t (-l')T

Pseudoscalar meson G = (-l)T

73

TABLE 6
Mesons With J<2, T<2

 

 

 

 

 

Meson JPG T NN STATE

0H 0 3

PO

o+" O .-
SCALAR 0+* 1 -

o*’ I 390

I’* o -
VECTOR _+

l l 35' 5 30'

l'-‘ I -

1** o 3P,

1*” o 1?,
AXIAL
VECTOR 1++ I ‘ lpl

I+‘ I 39'

o‘* 0 'so

0" o -
PSEUDO 0'* I -

§CAFA§
,-'.-. 0-- l . SO

 

7A

' Table 7 shows the meson-nucleon interactions

that are linear in the meson field and display strong

interaction symmetries (conservation of J, P, G, and T).

"TABLE 2

Meson-Nucleon Interactions

 

 

 

 

 

 

 

 

MESON T INTERACTION;
5m... 0 I, II N up, B. IIIMNIJJ
‘ ”I l; RTN‘EPS [ii Wffi‘lt’N-JAIQ
0- ' o- 8 u
NV NIP” r N949 5 ‘9-3 ‘9
VECTOR OBVNNA‘NPJ 1" A V v 3- (A; A)
"ICN’EMAM IJ OVEN-VJ“ t;a£:_I,-:N.(a_¢if-IA¢J‘
L
olfimxgwfl ' Ismwn“ Imam WWII?)
cééTtR + " 4 ;
' 1 ‘ “ I— 4' u *3 A
{l“~x‘wl“M ln”“‘“t""ofl tQBXCGf’TNwEEMR
o 3; “I? m II, B: Wm???
PSEUDO ’
SCALAR

-l

 

 

_ ~9 '9 4 «a
Time-WV. panning.)

 

 

 

 

 

*We do not use the bracketed interactions for reasons
explained in the text.

75

_ For vector and axial vector mesons, Wentzel2
shows that Quipn’ must vanish identically to avoid
negative energy states. The vector and axial vector
.interactions for the scalar and pseudoscalar mesons

can be transformed as follows:

+, I”: «MI. w, -. - I, l 3,3» m.) I: <9)
I, VIIIIIII III, Ru. ii. MIMI). “0)

Application of Euler's equations:

E” I:
W ” liafll,

93:; I:
MP “ éiM’l

gives the same field equations for both interactions

in equation (9).

(KM éN'l‘ljl‘l/sz *5 X“ KdquYlJ (ll-a)

(CI-m I: IS mm, «m»

2. G. Wentzel, "Quantum Theory of Fields,“ Inter-
science, New York, 1949, Chapter 3. . .

 

\

76

The Euler equations also give the same field

equations for both interactions in equation (10).

(XM AM+M)YN={|D ngM(éA\PP)‘I/ . “2‘37
(U'Wl‘n Lpp:"i{'p AMP-l)” X‘XM‘I’N) (lZ-b)

Expanding the right hand side of equation

(9), we get:

Mammal); - filisflaxul ‘Kv flinnmm

Applying equation (ii-a) and the conjugate equation,

we get: , _
l'sq’av XI: ‘VN 9A LP: = " ls (MR/lat 4‘ I?” “fill?" Xi‘YNQN‘) Lg

: - d-‘l': ‘T’N XMYNKQMLPS) LP: (13)

Equation (i3) shows that the vector interaction for

scalar mesons is equivalent to an interaction that is

quadratic in the meson field.
Expanding the right hand side of equation

(10), we get:
'lpq’N X: X» Wu dflpb 7‘ “lb ({ JMQN x581“ YN+ 47““ «A anal/N} ) LPD

since “‘5: KM]+= o

I, <9, mm m:- - i. H AMT/M I‘m- '9. m Mimi) I?»

77

Applying equation (iZ-a) and the conjugate equation,

we get:
fill»: X5 X» llv 3MP)? l]: (3 W?" Xs‘l’N ‘ “bill 3A ll: 3.3“ ‘(b '( l ‘0

Equation (14) shows that the axial vector interaction

for pseudoscalar mesons is equivalent to the pseudo-

scalar interaction to linear terms in the meson field.
Since we calculate only the pole terms

explicitly, and since we eliminate negative energy states

we take:

0_ I- o: I: O ‘ 0
.ls'ls'll’ ll, }v=}v=%fl=?q=°
The G-parity of the mesons in the interactions

‘ without brackets in Table 7 is determined by the con-

servation of G-parity. The G-operation has the following

effect on the nucleon fields:

0 “7,9. V,G"= at 59,9. ‘l’N

GCl-llllfl’?%Gd=-w§ {ll/11%“! US)

where

Uzi-l for .n. MANNY;

(«7‘34 for .0." XMIU'é"); X‘g’“

78

The mesons have the same G-parity given in equations

' (l5), since G-parity is conserved.

Scalar Meson G =(-l)T
Vector Meson G 2: (-])TH
Pseudoscalar Meson 5::(-1)T (l6)

G_:(-l)T for 1L": (S AIM
C=(-I)T+l fern: x; 912.0

Axial Vector Meson

_ Equations (8) and (lb) are the same, as
we would expect, since both sets of equations derive
from strong interaction symmetries. Notice that the
vector meson has two linear interactions, and there are
two axial vector mesons.“

Five mesons have been observed. Table 8

shows these mesons with their masses and quantum

numbers.

6

79

TABLE 8

Observed Mesons

 

 

MESON MASS T JPG DECAY
-_. 1T°aax
’I‘r IIIO Mev I . o ”14“,”
3
LI) 780 Mev ‘0 ‘1'“ w—+ 3 17
S’ 4 750 Mev I I"" f-9 2 TT
§
'1 550 Mev 2 ? n°—> 3 rr
° +
(if)? 575 Mev I ? l"’ 211’

 

h. Mesons and the Nucleon-Nucleon Scattering
Amplitude. '

The (Q, f) ”L and if mesons have been
studied primarily by their decays.7 However, Lichtenberg8
and others have considered the effects of mesons with

various quantum numbers on the static nuclear potential.

 

3. 8.6. Maglic, et al., Phys. Rev. Letters , l78(1961);

8 M.L. Stevenson, et al.,Phys. Rev. 12§,687 1962).

. A. J.A. Anderson, et al., Phys. Rev. Letters é.365 (1960;
D. Stonehill, et al.,Phys. Rev. Letters 6, 62A (1960; and
A.R. Erwin, et al., Phys. Rev. Letters 6, 628 (1961).

5. A. Pevsner, et al., Phys. Rev. Letters 1, #21-(1962);

P.L. Bastien, et al., Phys. Rev. Letters §, 114 (1962);
8.0.0, Carmony, et al., Phys. Rev. Letters g, 117 (1962).

6. R. Barloutaud, et al., Phys. Rev. Letters §, 32 (1962);
and B. Sechi Zorn, Phys. Rev. Letters §J 282 (1962).

7- See for example 0.3. Lichtenberg E G.C. Summerfield,
Phys. Rev. (to be published).
8.‘ 0.8. Lichtenberg, Nuovo Cimento (to be published);

D.B. Lichtenber , J. Kovacs, 8 H. McManus, Bull. Am. Phys.
Soc..Z,55 (1962?; and N. Hoshizaki, 1. Lin, 6 S. Machida,

Prog. Theor. Phys. gé, 680 (1961).

80

We determine the mesons' pole terms in the nucleon-
nucleon dispersion relation.

The pole terms are given by single particle
exchange in perturbation theory. The meson exchange.
diagrams that give poles in the nucleon-nucleon amplitude

are shown in Figure 6.

 

 

 

 

qB Cl q' q
t L D B x \ C
r' f f' I'
m m
- \ >__ k v >_
q ’ q q ’ q
A C A. D
FIGURE 6

 

Meson Exchange Diagrams

The contributions of these diagrams for the ten
mesons that interact linearly with nucleons are given

in Table 9.

In equation (22), Chapter V, the pole terms for

the ten? invariant amplitudes F. j are given as:
l-
. \-I)“l’
u ' 17
F? (s,t) Zngilm-TLg Wu) ( )
Equations (14) & (15), Chapter V, give the expressions

 

in Table 9 in the representation of equation (17). The

R are given in terms of the OOUpling constants and

mij
masses in Table 10. Equation (17) and equation (29)

81

TABLE- 2

' Meson Pole Terms

 

MESON ' POLE TERM

 

 

JPG T Mass

 

0“. ' 0 M (3': )7 (TOTE) S'/ (1‘1: " “l + (.‘so )7 CPO-R) KI Ant" t)
o“ I M2 (TSTBTJ'TIH. Imi-ui + lI;1‘isP.I’I’.1'§I/mt-ti
l 0 M3 _ " (“0 V‘TbOS.)l?o+?‘l/( “3- u) " (0.“ 771+ Kg: 7 lfi'?I)/(H;'-'t)

' w 1 M, 41‘I+I‘s.IIyI,-III/m;-urloci/3+5“§'.1is?o+’P.l/Inr-ti
1” - 0 “5 (Wm?)HI/Inz-uwVsil‘ii’rliifii/mi-ti

I“ 1 M6 ' (IAI‘Is’I.-IIII:I./mz-AI*liii‘wil’Fi/Inz-I)
1+- 0 M7 {Mi (til); WW.) P, /m;-u3+ H: Harm-1’.) E/lfli‘fil
IH I M8 -{f’1;(t£.)‘(3?.'l’.il’x/inf-u) l “llfill(3fillilg/l"3"ll
o o M, II:I‘I’I.+'IIIR/m:-.I +II11‘IR~II)I3;/In;-II
o” I Mm manna-IIIRAMMwIzgi‘IsRIiIIE/Int-t)

 

a°= winner 3 4': (1%th

If: M§It31‘-4Mtiiii+mt3) i B=M$itiI‘-4Mt1u¢+mt&)

82

and (50), Chapter V give the differential cross sections.

clam C- b)
—-I—+> L—III. I11: 1I+I+IIAI+I+I+I+ Amie-Ismai-

§[%+ WHRMJ -: 3114-1; ~1§1~§lefiithf+ful+ . Us).
[ii-131;)R“Millilfi’filmg‘lllwkmfl-

1 IIIIliI-aI I-L1I11"1+ -‘-\IIII+++*RIII+‘IRIII11I:’TI m 4111+
11-11”). IIIIIIII‘IIIIM III M‘wllll 1.0149, + 3133 [WRMJI
éRrIIllift'fifu—l'i‘t ( Rmimmflmnllmi t m+u llsII‘ie +

4Htl gt. )fimu‘l' KHAN“ 11:" “I‘M“ 5"“) 9}

433%):‘5'fixlj' l2|“xix-niol#tén%u\+RMlllNILt-¥l;l:1n 1+[l3m2o*em4o)mit M’- lull'

(Rm|+gw1(r;t- M u—L11m9‘1-2,1§+1][11m.i.,at I}? I1+R»3.(.,,.t $131+
£11,1-}; limit», FLA-I, S 3min“: mu)+l([‘l'*?llmo “Rm4.1(-Il“tl
——1I1[,II+IIIIII IIIJII-Ié-I I—I-IIIII. I Isn’IIII-iIanIIi-I-
ItIIIII=III.-i*III.III-‘I:I+Ii-III11—1-12II+++1I+I+ (I;
s ill-Majln t M’- in] Tl“) Ruin :Rwil]tm+ -t “+147“?! [Bum
Wm. +R..;.]1M+ -1- ‘-m° 41111-1- [-RMIIi'JHIITRms-IJLW-r tea-«11x

- M4941- i5l211H1?m:o+s'RMIo:ll_mT.tmlqlllM‘RwI 1'
§RM4.1[-1:::-,,\-{_-u]- 7, i-KIIII + mm + Imsole—rt- in
9) Emu-1mm” +RMS‘1][M- .1 1" 1:117“ Swag/ALT“

4M lElllRMxol RW+O][M3.:T m1+u~l+l~fimilfim+fll~mlr

@111" sm 91

83

In equations (18) and (19).
sznm’ 2+ M?)
t:-2q’2 (l-cosG)
u::-232 (H-cose)

and the sum on m is taken from M1 to M10.

Equations (18) and (19) can be used to
study the properties of mesons from nucleon-nucleon
phenomena3or,when the masses and coupling constants
are known,they can be used to determine the NN cross
section. Because of the large number of parameters
in equations (l8) and (19), comparison with experiment
must await further clarification of heavy mesons.

We have considered only the pole terms in
equation (18) and (19). The effects of heavy mesons
should include an evaluation of the spectral function
in equation (22), Chapter V. We showed in Chapter IV
that fourth order perturbation theory does not give
adequate evaluation of the spectral function for scalar
particles,and we expect it is not any better fer spinor

particles. A more complete evaluation than this is

beyond the scope of this work.

84

TABLE 10
Residues, Rmij
mh m6 m7
-3fx‘ o o o
5
33

{9|
L

 

L

a° = Mums? a' = (Hum?

b°= {M§(t$>‘-4MIS WHMC’R b' = WW): +mtbk+¢+wt¢ )2

We?!

APPENDIX
SINGULARITIES IN PERTURBATION
THEORY

l. lnteqral Transform;1

Consider a function F(Z) defined in

a region, R, of the complex Z-plane by:

F(Z)== S 9(W.Z) dW (l)
C .

C is a contour in the W-plane, as shown in Figure 7.

W-Plane

 

rjwom

 

FIGURE_Z
Contour Defining F(Z)

 

1. Our discussion follows that of R.J. Eden, Univ.
of Maryland, Phys. Dept. Tech. Report #le(1961),
and Jan Tarski, J. Math. Phys. L, 1&9 (1960).

86

F(Z) is well defined and analytic in R, if there
are no Singularities of g(Z,H) in the neighborhood
of C for Z in R.

F(Z) can be analytically continued into
any region adjacent to R for which no singularities
of g(Z,W) are in the neighborhood of C. Suppose that
as a point 20 is approached from a certain direction
a singularity, wo, of g(Z,W) approaches C. F(Z) can
be analytically continued to 20 by deforming the
contour C to C' ahead of the approaching singularity,

and without crossing any other singularity.

F(zo)= 9(20.W) dw C(2)

I

C

Applying the Cauchy theorem before the

approaching singularity gets to C, we get:

1Sg(Z,W)dW ; g(z.w)dw (3)
. I

C ‘ C

F(Z) is analytic in the region away from 20 and in
the neighborhood of lo, and from equation (3)F(Z)

is continuous across the boundary of the two regions.

There are two cases when the prescription
for analytically continuing F(Z) breaks down:
I) Pinching Singularities
Two singularities of g(Z,W) approach the

same point on C from opposite sides.

r}.

87

2) End Point Singularitieg
A singularity of g(Z,W) approaches

either end point of C.

Polkinghorne and Screaton2 show that the

.singularities of continued, multiple integral transforms

are given by l) or 2) applied to each variable of

integration.

2. 'Perturbation integral;

The general perturbation theory
term for the Feynman amplitude of a graph with n

internal lines is3

BM.)
.1: SJ*K.'--§=1°‘Kz . (4)
{II (“oi-meme)
2,is less than n since the vertex delta functions.have

been integrated out. The qi are linearly related to

the Ki and the external momenta pj. BUB-A is a
polynomial for particles with spin and a constant for

scalar particles. It does not effect the singularities

in either case3 and we set it equal to one.

: T .u T n 5
F‘ P K‘ V K” .TTl‘tf-mfne) ( )

. . . 4
Usnn the Identity, "
g l éU'gfi)

l l
: ... n
2. J.C. Polkinghorne 8 G.R. Screaton, Nuovo Cimento
15, 289 and 925 (1960) .

 

 

.3. For example, See J.Hamilton, "The Theory of Elementary

Particles," Clarendon Press, London, 1959.
h. J.S.R. Chishohn, Proc. Camb. Phil.Soc.&§,300 (1952).

r$'

\‘q

88

 

We get:
I . .
r: = C A :1 4 44K éh- Ml
e 80 4"" 'E cl“ 84k." '1) [“2“ Kill (7)
where (3)

n .
‘l’élck,i<.l>)= Edd?" Mhie)

We take the limit€;-) O, remembering the convention
+ie., when defining the physical scattering amplitude.
F has possibly the following singularities. i

1) End Point Sinqularitieg

 

W210
di‘ 0,: (9)
K}=»0

2) PinchinggSinqularities

$40

and (10)

AK;
Either end point or pinching singularities

must occur in all variables of integration. The form
of ‘Y is such that:

.. . ‘l'
V“ ng—a, , (m

\

89
The end point conditions are redundant, since 0!; : 1
implies that 93-3,: 0. Now transform the K; to put

° C

‘Y in canonical form.
Ki.: Ki+£i (12)

where 1.; is a linear combination of the Kj(jj: i)

and

I ,L
Y: imam) K: + be») (13)

Then I (14)
. I
”I __ ALA“) KL
3 c
The pinching conditions are also redundant,
since KEr-O implies that Ly, :0 ‘ . Since 3.3., -.-. if
3K; 3K5 an
we get the following necessary conditions for singularit-
ies of F. '
1) 93:0 L: “...”!
aka

(15)

2) either 0‘} ‘-‘- 0

or 7);: ma}

9O

 

Consider a closed loop within a pertur-

bation graph.

Pa: , F:
R KL j

 

FIGURE 8

Closed Loop Within a Feynman Graph

The p's neednot be single lines. In the evaluation
of the Feynman amplitude we can choose K; as shown in
Figure 8.- The Ri—l four-momenta delta functions at

the vertices give R equations relating the qi, Ki' and

R+l
pi, and the condition 2 Pigo . Thus we write:
i
ql = Ki " PI
R

91

and

SLY- a , ,t . R , ,.
m - 5?; {Mn ~wn) +§=fhllfwlfl

or (17)
B itchy-=0

Where the sum is taken around a closed loop in the

graph. This gives the Landau-Bjorken5 conditions for

the singularities of a Feynman amplitude associated

with a Feynman graph.

1) quk‘ :: 0, sum taken around closed

loops in the graph.
2) Either' C(C: (J
1
or '33: “a;

Notice that C(;= O removes the line i from consid-
eration, leading to a reduced diagram in which the

line does not appear.

3. Normal Thresholds

Consider the fourth order diagram shown

 

 

 

in Figure 9.
t “3 L .
Pb q pd
1+ qz
> < >
pa q] pc
FIGURE 9

 

Fourth Order Box Graph
5. L.D. Landau, Nuc.Phys. ii, 18] (1959).

 

92

The Feynman amplitude of this graph has a singularity

for the following conditions:
a) dxqu-tt' 0

2 2
b) ql : fm

2 _ 2
q , m
3 3

c) “M‘B.*°':‘b,,* ”(3153+ d414= 0
From conditions a) and c), we get:
d‘%.+o(3cb3=0 (18)

Taking the scalar product of equation (18) with qI and
q3, we get: .

d.°o.‘+ a, (Am-(133:0

d‘cb|.1)3+ “313:: O

From condition b), we get:

‘3‘: WM: (brig = 0

:_ (l9)
‘quh'1¥'+<d5vn3‘.o
For non zero solutions of these equations,
m 2
:0 (20)
9:93 m3 .

 

 

mlz “‘32 ' (ql0q3)2:0

qi'q3 = t mums

93

The positive solution leads to negative
values of ck , which are not on the contour.6 So

there is only one singularity from equations (20).

ql a q3 ': -m|m3
Since 5: (pl-p )2— (q -q )2 this singularity is
a b — l 3 '
.. 2 _ 2 2- . _ 2
at - (qi- <13) .. ml+m3 2qiq3—(mli-m3) (21)

4. Anomalous Thresholds
Consider the singularity of the Feynman

amplitude for Figure 9 for the following conditions.

a) CNh‘: 0
b) q2= m2
l I
2__ 2
C12 — m2.
2 _ 2

c> dntfidfibfids‘bco

Condition c) gives:
Q‘MB'J- dxckg' $31} 43 abl'%'3:f (22)
“:‘H‘lrfl' cum} 4 “3 78%;-"
dI Gkidkg + «J%$‘%3+ “3 max: 0

Defining .
x; :- (6:..CE} [”4th
solutions 0 equations (22), only if

l my,3 *
XL-' )3! : O
X3'y:3l I

l

X; 7' YILYI3 i {(l‘YI:)(l'X:R/J'

5. This needs further justification by a detailed study
of the method of continuation. See References l,2,or 5.

we get non zero

(23)

 

 

94

If none of the particles in Figure 9

can energetically decay into the other two at the

same vertex, we get7

'l‘<>/I;él

-|$'Y135:l

(24)

Equations (24) and the positive solution of equation
(23) give negative values of the 0k . For positive
ok the negative solution of (23) must be taken and

the following condition must hold.
I; ' l/
Yull'l’a;}/ +Y13{l-‘/,:ix<0 (25)
9
and X3: Yu>/;3— {("X:)()'Y£)il (26)

In terms of the Y; , s is

2 2
s : ml +- m3, - 2 thmlq3°

For X3 given by-equation (26), Y“) -I and the
threshold is below 5 : (ml-Tm3)2, in contradiction

to the results of Chapter III. This is an anomalous

 

7. See Reference l.

95

threshold, and it will not appear unless equation (25)
is satisfied. Expressing equation (25) in terms of
the masses of the particles in Figure 9, we get the
following results: _

l) if m32+ (“1+2 ( Md2 and mlz+ m“2 > Maz'

there is an anomalous threshold, if

2 ’ 2
Md -m32-ml+2 ) mlz-l-m42 - Ma .

2 2 2 2 2 2
2) If m + m < Md and ml + m“ < Ma ,

3 4

there is an anomalous threshold.

3) If neither l) or 2) are satisfied there

is no anomalous threshold.

If the internal masses of Figure 9 are much

smaller than the exterior masses singularities occur for

complex values of s.

5.. Fourth Order Diagrams

Consider the singularities of the
amplitude for Figure 9, for

qizzm.2 , i: I: 2: 3: 1+9
' (27)

4
$4,190

This gives the following condition for singularities:

96

l | XLYH 714'

Yn.‘ yx'i thl
y” >133 I y“;

Y” Yul Y34 l

n
O

 

 

(28)

where

S = kpfi+ p531: ((B‘_:B‘3)‘L= meg} M31.- 1 Yr; MOMS

: (FA'PQY: (ellfcbiil‘l: M:+M;‘J~Y;4 Mfmq

FDaZ” Mt: UL," (64),} m3’+M;-)Yflm,m4(29)

P132: M: -.: [154"?an m“+m4‘-LY,4M,M4

Pcz '-'- M:: (aUt'okilz: MF+M11-lYI1mIml-

2_ L_ ;
Pd - My (in-1,): m3+m;-:y,,m,m3

F th M=M=M=M: =- =,'
or e case a b c d ml m3 M
equations (28) and (29) give singularities of the Feynman

amplitude for s and t such that:'

(s-Wlltfi :kmimg‘ lt+ lm} ngm- 4 mm,“ o

S > 4M1 (30)
t 7 (”fir/hi);

97

 

FIGURE l0 L

Singularities in Fourth Order Perturbation Theory

A
E:Equation (30)

——~d--{-—-------

 

Figure 10 shows all the singularities for Figure 9,

in the case Ma=:Mb:.Mc = Md:.ml= m3: M. The dashed

lines are normal thresholds and the solid curve is the
solution of equation (30). For this case there are no

8

anomalous thresholds or complex singularities.

6. Generalized Unitarity

Cutkosky9

has shown that the discontinuity
of the Feynman across a cut starting at one of the
singularities discussed above is given as follows: In
equation (4), for each pinching singularity l/(cs: W1; )

is replaced by 3.1Tl Sk‘k: Wm)

 

8. See Reference l.

9. R.E. Cutkosky, J. Math. Phys. 1, 429 (1960).

98

Discontinuity F :

(will lam-Um (CS—Elm)" NEW? (.31)
T>p+l "lb-h)” 4‘6“" mu) ‘
p is the number of pinching singularities. For example,
the discontinuity for a normal threshold (equation (20) )
is:
Discontinuity F- =

._qqu £44K “Ii-Mflcfl‘sI-ml’;
(%.‘-M.‘)(‘b§'-M3)

The discontinuity for an anomalous threshold
(equation (26) ) is:

(32)

Discontinuity F -'-

‘ (33)
.. 31TH SJ‘IK 3(1):- mf) Sl‘bi—mil é(%3"m3‘)/(‘L;' Will)

The discontinuity for equation (30) is:

Discontinuity F 3

‘W 4 WK Mf-mn $<‘l£-mll((°b}m§) Stu-m2) ‘3‘”

 

: 1T1 l y,, 1.3%.; 14,1
M‘m m 7'“ V»! M
T 4 Yup/,3: V34
y“) ya‘l 7’34 l

 

7. Crossinq Relation;

Consider the Feynman amplitude for the

graph in Figure 9, representing the reaction A+B —>C-\-D.

(35)

 

FI : 8% PAWS PL” Pa)

(‘4 Pa 1,- ‘m (“amt-1,)

(=l

99

Now consider the Feynman amplitude for the

graph in Figure ll, representing the reaction A+5E-;§+'D.

FIGURE ll

Crossed Box Diagram

 

 

 

 

P _J 1 L P
b “\ )\ q ,- /- d

3

qz 94

q]

Pa > < 4 P;
P Fm '

(36)

 

4 I 8 44%.", A41, (liven-3134(-rim-1.1Sql-Wt‘nllqmegg)
8 (be pew? ‘34) (LT (cfit' WI?)

By inspection of equations (35) and (36)

FIll can be obtained from F' by reversing the signs of
pb and pc. Thus, FI and FIII are the same function of
pa, pb, pc, and pd and consequently they are the same

function of s, t, and u. However, the connection between
5, t, and u and the center of mass energy and scattering

angle must be different for Fl and Fill“ s, t, and u

100

for F are:
Ill 2

s '2 (PA'PB)
t -.-. (pA-PD)2 (37)
u = (PA+P )2

The center of mass conditions for equal masses are:

(BAT: ITSBizlECl-slsol z [Ell

_ +2 2',
poA: p08: 13°C: poo—(q+M )L(38)

4 .> a 4

The equations corresponding to (37) and (38)
for F. are equations (4) and (6), Chapter IV. Equations
(37) and (38) give the results shown in Table I.
Lehmann, Symanzik and Zimmermann,lo and Goldberger,
Nambu and OehmeH derive crossing relations without

reference to perturbation theory.

 

l0. H. Lehmann, K. Symanzik and W. Zimmermann, Nuovo
Cimento I, 205 (1955), and 6, 3l9 (1957)-

ll. M2 L. Goldberger, Y. Nambu, and R. Oehme, Ann.
Phys.2 226 (1957) . H , m u

Pursue .1qu '9.

nICHIan STATE UNIV. LIBRARIES
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