P H Y S I C A L R E V I E W V O L U M E 1 5 3 , N U M B E R 5 25 J A N U A R Y 1 9 6 7

Bethe-Salpeter Equation and Goldstone Bosons in Quantum Electrodynamics

R . S. WlLLEY

University of Pittsburgh, Pittsburgh, Pennsylvania (Received 17 August 1966)

The relevance of the Goldstone theorem to quantum electrodynamics with zero bare mass for the electron is studied. Subject to the necessary approximations, it is concluded that the Goldstone theorem has no physical consequences for this theory.

I. INTRODUCTION

IT has been proposed1 in the context of a particular nonperturbative approach to quantum electro­

dynamics that the electron should have zero bare mass. Then, under certain conditions discussed in Ref. 1, the usually divergent self-energy integrals become conditionally convergent integrals, and the Dyson-Schwinger equation for the electron propagator has finite solutions to any order in a perturbation expansion of the Bethe-Salpeter kernel which appears in the equation for the vertex function. With no bare mass, there is no mass scale in the theory; and the self-energy equation has a solution for any value of the physical electron mass. The solution with the actual electron mass is picked out by imposing the boundary condition that the inverse propagator be zero at the physical electron mass.

With no bare mass the field theory underlying the Dyson-Schwinger Greene-function equations is y5-invariant; the existence of solutions with nonzero mass implies the "spontaneous" breakdown of this symmetry. In the field-theoretic context this can only occur through a y5 degeneracy of the vacuum. Also in the field-theoretic context, there exist general proofs2

that spontaneous breakdown of invariance under a continuous group of transformations implies the exist­ence of zero-mass states of appropriate quantum numbers. The theorem customarily goes by the name of the Goldstone theorem, and the corresponding zero-mass particles by the name of Goldstone particles. None of these proofs provide any information about the coupling of the Goldstone particles to other par­ticles in the theory. Thus, the Goldstone theorem, by itself, has no observable consequences. Such conse­quences can only follow when the coupling of the Goldstone particles to the other particles in the theory is determined.3

The original suggestion that spontaneous symmetry breakdown might play a role in field theory was by Nambu and Jona-Lasinio4 in a paper specifically de-

1 K. Johnson, M. Baker, and R. Willey, Phys. Rev. 136, B l l l l (1964).

2 J. Goldstone, Nuovo Cimento 19, 154 (1961); J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev. 127, 965 (1962).

3 N. G. Deshpande and S. Bludman, Phys. Rev. 146, 1186 (1966).

4 Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961).

153

voted to the question of spontaneous breakdown of 75 invariance in a different theory. In that paper the observability of the ensuing massless particles was demonstrated by displaying the corresponding pole in the solution of the Bethe-Salpeter equation for the scattering amplitude. The residue of this pole is identi­fied as the square of the coupling constant. The primary purpose of the present paper is to carry out the ana­logous calculation5 for the version of quantum electro­dynamics presented in Ref. 1. The result is that there is no pole in the electron-positron scattering amplitude corresponding to a massless pseudoscalar Goldstone particle.

In recent years there has also been considerable interest in the Bethe-Salpeter equation in its own right as a nonperturbative source of information on the analytic properties and high-energy behavior of the relativistic scattering amplitude. With very little additional work we are able to find the contribution of a certain infinite set of diagrams to the high-energy behavior of one of the scalar amplitudes for electron-positron forward scattering.

II. THE BETHE-SALPETER EQUATION

The Bethe-Salpeter equation for the off-mass-shell electron-positron scattering amplitude is

a bTcd (p,p',k) = a bKcd (p,p',k)

+J (dl) a*TfMk)S<0(h) XghKeK(!,p',k)Shf(L). (2.1)

a a

p-l

+ i+ X 2

9r^\"

FIG. 1. Bethe-Salpeter equation for electron-position scattering.

5 In Appendix A, we review briefly that part of the Nambu-Jona-Lasinio model which is of interest in the present context. Since the mathematics involved is much less complicated than in the analogous quantum electrodynamic calculations, it may be useful to read this Appendix before the main body of the paper.

1364

153 B E T H E - S A L P E T E R E Q U A T I O N 1365

The choice of momentum variables is illustrated in Fig. 1, where the equation is given in diagrammatic form. The spin-matrix indices are written out explicitly. (dl) means dH/(2irY and l±=l±k/2.

If there is a massless pseudoscalar particle which couples to the electron, a pole will occur in T in the form6

^ , / n K K g(P+,P-)g(p-',P+) abTcd{p,p,k) = y \d7cb

k2

+terms regular at k2=0. (2.2)

This is illustrated in Fig. 2. The matrix indices of T imply a lengthy expansion in terms of invariant func­tions and Dirac matrices. We concentrate on a single invariant function defined by the 75 projection

y5da abTcd(p,p',k)yhc= I2ie2f{p,p',k). (2.3)

A sufficient condition for no zero-mass pseudoscalar pole to appear in T is that no such pole appear in / .

The ofT-mass-shell scalar amplitude / is a function of six scalar variables:

liP,P',k) = f(k*,p • P',P\P'\P • W • *) • (2-4)

The mass-shell scalar amplitude is obtained by setting

(#±§*)»=(^±*)»=-»* , (2.5)

which requires

p-k = p'-k = 0, (2.6a)

p*=p'2=-m2-lk2. (2.6b)

We write the mass-shell invariant amplitude as

f(k% p-p>, -m*-\k\ -m*-\k\ 0, 0) = f(k\p-p'). (2.7)

Since — k2 is the square of the total center-of-mass energy, the physical region for k2 is —4w2>&2. Thus, a massless pseudoscalar particle, if coupled to the elec­tron, appears as a pole at the unphysical value k2=0. We demonstrate the nonexistence of this pole by showing that f(0,p-pf) exists. In the Bethe-Salpeter equation we cannot restrict the scattering amplitude to the mass shell, but we can obtain the function f{0,p-pf) by solving the Bethe-Salpeter equation for

mp-p',p\p'\ofi)=f{p,p') (2.8)

and at the end setting p2= p'2 — = m2. Thus, to compute the mass-shell amplitude f(k2,p-p') at the unphysical value, k2=Q; we solve the Bethe-Salpeter equation for k=0 and at the end set p2=p,2= ~tn2. In terms of the

6 If one takes Eq. (2.2) as an anzatz, it follows that the function g(p+,p~) must satisfy a homogeneous vertex equation. M. Baker, K. Johnson, and B. W. Lee [Phys. Rev. 133, B209 (1964)] have shown that this equation necessarily has a solution when the 75-invariant Dyson-Schwinger propagator equation has a 75-nonvariant solution. They further point out that this does not imply a pole in the corresponding inhomogeneous equation because of the non-Fredholm nature of the equations.

FIG. 2. Pole contribution if pseudoscalar particle is coupled to

usual Mandelstam variables,

v 1

A s=-k*, t=~(p-p')\ (2.9a)

F(s,t) = f(k\p-p'), (2.9b)

we look at the amplitude for physical values of t, t<0; and unphysical value of s, s=0.

Now we must determine the kernel to be used in the Bethe-Salpeter equation. Since the exact kernel includes an infinite sum of terms, clearly we must use an approximate kernel. The approximation should be one which, when applied to the self-energy problem, leads to a symmetry-breaking solution. In Ref. 1, a systematic approximation scheme is presented for the calculation of the electron propagator for large off-mass-shell momenta. This scheme includes the boundary condition S~1(p) = 0 when yp=—tn. The behavior of S(p) near the mass shell is to be determined by ordinary renormalized perturbation theory. Thus, according to Ref. 1, the electron propagator equation has approxi­mate solutions with the properties

1 m/m2Y S(p)->-+- - ) + •

yp pAp2/ 1

>o, s(P)

€ = — + • • 4?r

pt-*cc

yp—> —m

(2.10)

Because of the zero mass of the photon, the point yp=—m is an infinite branch point rather than a simple pole. The combination of the approximation scheme for the far-off-mass-shell behavior with re-normalized perturbation theory for the near-mass-shell behavior can be achieved as follows. The first non-trivial approximation to the Dyson-Schwinger equation used in the asymptotic approximation scheme with mo=0 is

S~1(p) = yp+ie02 J (dp')Dap(p-p')y«S(p')ye. (2.11)

This is illustrated in Fig. 3. In this approximation we may write

S-Kp)=yp+A(p*), (2.12)

with A (p2) the solution of the integral equation

A{f) = ie02 / W)Da, •e(p-p'ha-

A(p»)

p^+A'iff*) V . (2.13)

FIG. 3. Approximate Dyson-Schwinger equation for the elec­tron propagator. S(p) 7P +

1366 R. S. W I L L E Y 153

FIG. 4. The ladder-approximation Bethe-Salpeter equation.

Since the approximation (2.11) is only intended for the asymptotic region, (2.13) is used only to determine the asymptotic behavior of A(p2). Thus, in Ref. 1, the A2

term in the denominator is dropped and the solution of the resulting linear integral equation yields the asymptotic behavior of (2.10). To get an equation which also has the correct properties near the mass shell, we can replace A(p2) in the denominator by its perturbation expansion

A(p2) = m+aoA1(p2)+ao2A2(p

2)+'" (^finite). (2.14)

Thus, to order e02 in the combined perturbative and

asymptotic non-perturbative approximation scheme, we have the equation7

f A{p'2) A(p2) = iet? (dp')Da0(p-p')y« y*. (2.15)

J fi'2+ni2

This equation has as solution a hypergeometric function which has a branch cut from p2=—m2 to - co and asymptotic behavior (ni2/p2)*. With the boundary condition A (—m2) = m, this provides an explicit solution with the properties (2.10). Now we may determine the corresponding approximation to the Bethe-Salpeter equation for the electron-positron scattering amplitude. From Fig. 3 it is evident that the corresponding kernel is just the one-photon-exchange diagram.8

abKcd(p,pfJk) = -ie2Da0(p-p,)ya

abyPcd

(P-Pf)a{p~-P')f r ( = -ie2\ gap— (p-py

1

Now following the discussion of Eq. (2.8) we set £=0 and then carry out the projection indicated in in Eq. (2.3). In Appendix B we show that for k = 0, the amplitude / decouples from all other invariant ampli­tudes. Thus, we obtain an integral equation for f{p>pf):

X yW^d. (2.16) (p-py

Furthermore, we are to make the approximation of replacing S(p) by {yp-\rm)~l everywhere that the modified behavior of the second term in the large p expansion of S{p) is not required for convergence of an integral. We arrive at the ladder-approximation Bethe-Salpeter equation illustrated in Fig. 4.

7 This equation was first proposed by Th. A. J. Maris, V. E. Herscovitz^ and G. Jacob, Phys. Rev. Letters 12, 313 (1964). The extension of this combined approximation scheme to higher orders has been considered by M. Baker and F. H. Lewis (unpublished).

8 Formally, this follows from Eq. (11) and the definition K——82/dS. Furthermore, we neglect vacuum polarization so we may replace eo2 by e2.

f(P, pT) = Me2 [ (dl)

1 1 X-

(p'-l)2l2+m • /&*). (2.17)

III. SOLUTION OF THE APPROXIMATE BETHE-SALPETER EQUATION

Although we eventually desire the solution of (2.17) for p2=p/2=—m2, we will proceed9 by solving the equation for p2,p'2>0 (space-like), and then obtain the desired solution by analytic continuation. For p2,p/2

space-like, we can rotate the IQ integration contour and work in a four-dimensional Euclidean metric. The only angle in the problem is the angle between p and pf, so we write

where

and

f(p,p>)=— Z (n+ l)fn(p\p'*)Un(cosB), (3.1) 2w2 n=o

cos0=-p'pf

Un(cOSd)--

(p2p'2)112

sin (^+1)#

sin0

(3.2)

(3.3)

is a Tchebycheff polynomial of the second kind.10

Substituting (3.1) into (2.17), we can carry out the angular integrations and obtain a one-dimensional integral equation for the "partial-wave" amplitude fnM^tn'Mp2^2),

Jn\XjX ) 2TT2 1 fx<\ n/2

-,y)

where

n-\-l %>\%>J

\ cx' y [y\nl2

n+lJo x'(y+l)\x'J

x r i fx\ni2

+ dy - Mx,y), (3.4) n+lj* y+l\y/

\=3a/4-w, x=p2/ni2, xf=p,2/m2. 9 In obtaining the solution of Eq. (2.17), we make use of tech­

niques developed by a number of people for the Bethe-Salpeter equation for scalar particles, e.g., J. D. Bjorken, J. Math. Phys. 5, 192 (1964) and M. Baker and I. Muzinich, Phys. Rev. 132 2291 (1963). I am also indebted to Professor Muzinich for a copy of an unpublished manuscript by M. K. Banerjee, L. S. Kisslinger, C. A. Levinson, and I. J. Muzinich dealing with the Bethe-Salpeter equation for the <£4 theory.

10 We use the definition and properties of 27n(cos0) given in the Bateman Manuscript Project, edited by A. Erdelyi (McGraw-Hill Book Company, Inc., New York, 1953), Vol. I I .

153 B E T H E - S A L P E T E R E Q U A T I O N 1367

By differentiation we may convert Eq. (3.4) into a differential equation

-d2 2d — +

—(MJO X (J/JV

n{n+2)

Ax2

= - 2 7 T 2 -

J n\X,X )

d(x—xf) (3.5)

with the boundary conditions implied by the integral equation

aw/a+2/»(s,a?') - > 0

1 fnfax') —» 0

/*.n/2

(x->0)

(x —» oo ) .

(3.6a)

(3.6b)

The details of the reduction of (2.17) to (3.4) and to (3.5) are given in Appendix C. Here we only note that in obtaining (3.5) from (3.4) we have interchanged x and %' at the end for typographical convenience. The solution is symmetric in these variables.

The particular solution of the inhomogeneous equation (3,5) is

fn(x,x') = -27r2yi^(x<)y2M(x>), (3.7)

where yiin\ y%(n) are two independent solutions of the homogeneous equation normalized such that the Wronskian yxy^—yxyi— 1/x2.

The homogeneous differential equation is simply transformed into the hypergeometric equation; two appropriate independent solutions are

yi(n)(x) = xn/2F(a, a—v;n+2; —x),

T(a)T(a+i)

(3.8a)

y2(n)(x)= —

where

r (»+2)r (H- i )

Xxnl*-aF(a-l~n, a;v+l; -1/x), (3.8b)

FIG. 5. The Sommerfeld-Watson contour C. -0

n plane

^ z i c

to infinity and back just above it, including the poles of (sin?™)-1 at integer values of n (Fig. 5). The ampli­tude exists if the sum (integral) converges. From the properties of the hypergeometric functions, we have

2TT2 1 /x<\w 1 fn(x,x') —> ( —) ~-e~~n ln(*</*>);

n-*°° n+1 s>Vr>/ n

so the integral converges for x^x*. This demonstrates the existence of f(p,p') in the Euclidean region p2,p'2>0 and | cos^| < 1. This already indicates the noncoupling of the Goldstone boson since, if coupled, it should produce a pole in the off-mass-shell amplitude as well However, it is still worthwhile to check that our solution can be analytically continued to give the mass-shell amplitude.

According to Eqs. (1.6) and (1.9) the desired con­tinuation is to

pi=p">=~mz for k2=0,

t=-(p-p')2=2ni2(l-cos0)<0,

cos0> 1.

We start by deforming the integration contour. Since F(a,f3;y;z)/r(y) is an entire function of a, 13, and 7, the product yi(n)(x<)y2(n)(x>) is an entire function of the parameters a, n, and v. But v involves a square root (3.9), which implies branch points in the n plane at n== — 1±2\A. For \=3a/47r<| , we may restrict the branch cut to the left half-plane. Then we can open the contour to enclose the entire right half-plane (Fig. 6). Let

»=£+«?. (3.11) a=(n+l+v)/2, v=l(n+l)2-4\Jt2. (3.9) Then, asymptotically

Some useful properties of these functions are collected in Appendix D.

To construct the complete amplitude f{p,pf), we have to do the sum (3.1) which we can rewrite as a contour integral by a Sommerfeld-Watson transformation

1 f»*f(p>P') = — £ ( » + 1 ) / « ( * , * 0 Unicosd)

2w2 n

sm-irn -~e-*\v\f Un(co${Tr-6))<^e^-6)W,

Mx,x') en* !"<*</*>) # (O<0<7r) (3.12) n

For x?*x'j the integral converges everywhere on C and

1 r dn

47r2i J c sinirn > smirn

X (if + l)/»(*,*') ^n(C0S(7T~ 0)) . (3.10)

The contour C runs just below the real n axis from zero

FIG. 6. The contour a (x,=2v

/x).

1368 R. S. W I L L E Y 153

T

- I-X ;"H+X' I T

L

in fact, there is no contribution from the infinite semicircle. So we may write

1 r dn m2f(p,p') =

AirH J L sm7ra X(n+l)fn(x,xr)Un(cos(ir-0)). (3.13)

The contour L runs parallel to the imaginary n axis and is displaced by an infinitesimal amount into the left half-plane to avoid the pole of (sin7ra)_1 at n=0

All terms exist for Re0=O, Im0^O (cos> 1). Thus, there is no pole in F(s,t) for =0 and fixed negative L

There is one other point which should be considered. The general solution of the differential equation (3,5) includes an arbitrary linear combination of the inde­pendent solutions of the homogeneous equation, subject to the boundary conditions. Thus, we might add to the

(Fig. 7). At this point we may set #=#'=—-1 and the integral still converges. Evaluating the hypergeometric functions gives

UM | w _ i = H V * » * - - 1 ] , (3.14) ik

where b*=(n+l-v)/2.

It remains to continue to cos0>l (fixed negative t). To this end we first give 0 an imaginary part and then let the real part of 0 go to zero. In this limit the integral diverges. However, this just reflects a singu­larity at t=0 which is associated with the exchange of massless photons. To determine the finite amplitude for fixed negative /, it is necessary to separate and sum explicitly that part of (3.1) which contributes to the singularity at 2=0. Expanding (3.14) for large n we find

particular solution (3.7) terms of the form

[ci(n)yiM(x)+c2(n)y2(n)(%)l

X [ c i < » > y i < » ^ ) W B V n V ) ] , where C\{n) and c2

(n) are arbitrary functions of n. Each of these four extra terms satisfies the boundary con­ditions (3.6). However, since /»(£,;*/) is a function of

(3.15)

7r r HA 1 / 1 \-j (H-l)-[e-^&-l]=~27r2 1 +(A-§7r2\2)+ +01 )

i\ L n+1 (n+1)2 \(»+l)VJ

= ~2TT2 1 +(X-f7r2X2) +01-) . L n+1 (»+l)(n+2) \ »VJ

We define a new function <j>n{x,xf) by writing

r iwX 1 n 1 /%<\n

(n+l)fn(x,x') = 2ir*\ 1 +(\-§7r2A2) — ( — J +*»(*,*')• (3.16) L n+1 (n+l)(n+2)Jx>\x>J

The function <£„(#>#') has the property

<t>n{x,xf)| z-s '^-i '^l/V for n —><*>. (3.17) From (3.3), we see that for 0=0

Un(l) = n+1. Thus,

I > n ( - 1 ~l)Un(cOSd) n

converges for 0=O(/=O). The entire /=0 singularity comes from the terms separated explicitly in (3.16). The sum over n for these terms can be done using the generating function for the Tchebycheff polynomials. The sums are evaluated in Appendix E. The sum over <£„ may be replaced by the Sommerfeld-Watson integral and the contour deformed as described above. The result is

f(p,p') 1 ( 1 ixX rCcose - l^ -Ccose+l ] 1 / 2 - ! / 2s-2 \ r l r cos0 - l - | 1 / 2 f 1 «rX r l c o s 0 - - l r z - l c o s 0 + l I1'2! / 2^ x r l r c o s e - l n

In +(X X2 J -l2(l-cos0) 2[cos20-l]1 '2 LCcatf-lJ 'M-CcosH-l]1 '^ \ 3 /L2Lcos0+lJ

rrcose-ll i^+Ccose+l]1 /2-! -ii 1 r dn Xln -iln2(l-costf) + * „ ( - l , -l)t/„(cos(7r-0)). (3.18)

Lrcos0-l]1 /2-[cos0+l]1 /2J J J MirVJLsmTm

153 B E T H E - S A L P E T E R E Q U A T I O N 1369

FIG. 8. Sum of ladder diagrams for ee scattering.

two variables, the integral equation is more restrictive than the boundary conditions (3.6) which involve only dependence on one variable while the other is fixed. In particular, y<2,{n)(x)y<2,{n)(%f) is not a solution of the homogeneous integral equation since it behaves like (xr)~nl2~~l for x' —>0, while the integral obtained by substituting this proposed solution behaves like (x')~~n/2

in this same limit. Thus, C2(n) = 0. On the other hand, we have not been able to find any restrictions on ci(n). Thus, the solution (3.18) obtained by summing the "partial-wave" series with the particular solution (3.7) is not proven to be unique. However, the important point is the existence of a solution in which the Gold-stone boson is not coupled to the physical particles. We note that in the Nambu-Jona-Lasinio model in which the Goldstone boson does appear in the scattering amplitude, there is no solution to the Bethe-Salpeter equation for k = 0 because the pole appears in the particular solution (Appendix A).

IV. HIGH-ENERGY BEHAVIOR IN THE t CHANNEL

We have determined the contribution of all ladder diagrams (Fig. 8) to F(s,t) for s=0. Crossing symmetry for e+e—> e+e implies

+ L'

F(s,t) = F(t,s). (4.1)

The approximate F(s,t) computed from just the ladder diagrams does not have this property, but there is another set of "crossed" diagrams (Fig. 9) whose contribution to F(s,t) is obtained from the ladder approximation F(s,t) by simply interchanging s and /. The contribution of these diagrams to physical forward electron-positron scattering may thus be determined from the ladder approximation F(0,t) by making the analytic continuation to />4w2 rather than /<0.

We start with the integral representation (3.10) of the off-mass-shell solution f(p,p') and deform the contour as described in the previous section. The con­tinuation to t>4:m2 means cos0< — 1 which requires 0=7r+ir. The fact that the real part of 0 is to be ir rather than zero means that there will be no difficulty with the convergence of the final integral and hence no need for any subtraction. The t dependence (0 de­pendence) of the integrand is all in the Techebycheff polynomial Un(cos(ir— 0)). We write

0=7r+fr;

FIG. 9. "Crossed" diagrams.

FIG. 10. The contour L'+y.

* L

then

*7»(cosGi—0))= . (4.2) eT—e~

The behavior for large r depends on the value of £. It is minimized for £= —1. With this in mind, we start with the integral representation (3.10) for x,x'>l and real 0, and deform the contour to C (Fig. 6). We now open the contour farther to the left so that it consists of an infinite semicircle closed by a straight line with £= — 1 except for an indentation around the cut on the negative real axis (Fig. 10). Again there is no contri­bution from the integral along the infinite semicircle so we have

m2f(p. 1 r dn

, * > ' ) = — / — 4:7T2i J L'+y Sin7Ttf

X(n+l)fn(x,xr)Un(co&(ir--e)). (4.3)

We may now perform the analytic continuation by setting %=x'= — 1 and d=T+ir. According to (4.2), at each point along L' the integrand behaves like e~T

as r —>oo. Along the contour y encircling the right end of the cut, the integrand behaves like e*T, where £ ranges from - 1 to — 1+2\A. Thus, this is the dominant term for r—>oo. The integral along the contour y is equal to the line integral from —1 to — 1 + 2 \ A of the discontinuity of the integrand across the cut. Thus,

m2f(p:

1 r -1+2 V\ di T

(£+1)—e~i2*tt+v sin7r£ iX

X{exp[-~27r[4A-(£+l)2]i/2]

-exp[27r[4X-(B-l)2]i /2]}^ (4.4)

The integral is difficult to do exactly, but the asymp­totic behavior is determined by the upper limit

m2f(P,P') ~ e(~1+2V»T~ 1

tl~2^ (4.5)

ACKNOWLEDGMENTS

I have benefitted from several conversations with Professor M. Baker and Professor I. Muzinich con­cerning the solutions of Bethe-Salpeter equations.

APPENDIX A: THE NAMBU-JONA-LASINIO MODEL

We outline some of the results of the Nambu-Jona-Lasinio (NJL) paper4 in a form which makes explicit

1370 R . S . W I L L E Y 153

msrP*

ft = X FIG. 11, The approximate self-

energy and Bethe-Salpeter equa­tions in the Nambu-Jona-Lasinio model.

APPENDIX B : DECOUPLING OF THE AMPLITUDE f(p9p')

We have to take the 75, 75 projection of

/ (dl) T(p,l,k)S(l+)K(l,p',k)S(l-) (Bl)

the parallel between their calculation and ours. NJL w i t h ^ = 0 w i t h ^ i n o r i n d i c e s n d t a n d ^ ^ consider femuons mteractmg through a direct four- t u t i n g E q (2>l f i ) for K { l j ^ t h e i n t e g r a n d is fermion interaction. The approximate self-energy equa tion corresponding to our (2.11) is

—ip%-

ys(p) — yp—i\ol (dp' )TrS(p'). (Al)

1 r ( (i-P>)a(i-p>)^ te' 1 ga8

/)» J (l~p')2l (l-p>)

XSeg(l+)yagby

!ibcyPchShf(lJ)=—3iez-

[Y^da aeTfd(p,l,k)

1

NJL look for a solution of the form of Eq. (2.12) with 4 = ?«= const. Then the self-energy equation reduces to

m Xo f

= — / dp'2p'2-4w2J

m

p,2+m2 (A2)

(l-p'Y

Xyha aeT,d(p,l,k)Seg(l+)y\^hf(lJ). (B2)

Using S(p) = (yp+m)"1, the last three factors are

~m—yl+ m—yl~~\ [~ m-^-yl+ m—yl~ -76-

This equation has the trivial solution m—Q. I t also has the desired nontrivial, nonperturbative solution, « ^ 0 , provided the condition

[ m—yl+ m—yl-~l |~

Xo r p' L = — / dp'2—

4TT2 J fi'H p'2+m2 (A3)

f 2 + w 2 l-2-jrm2Jef L / +2 + m 2 L 2 + m 2 J e /

^yKhLm2+P~~lk2+myk+h(ylyk~ykyl)2hf. (B3)

For k = 0, substitution of (B3) into (B2) gives

1 1 -3ie2- y*da aeTfd(p,l)y5ef . a-^)2

/2_j_w2

is satisfied. The integral is divergent and requires a cutoff to define the model. If this model is considered as an approximation to a Lagrangian theory, the 75 invariance of the Lagrangian implies conservation of the axial-vector current. N J L show tha t the existence (2 1 7 ) m a v be written of the conserved current implies the existence of mass-less pseudoscalar particles.11 The Bethe-Salpeter equa- 1 3a f tion in the approximation corresponding to (1) is the f(p>p')=~ ~—I / l3dld£lt

analog of the ladder approximation in a Yukawa \P—P) %w J coupling theory,

f(p,p') = 1 + ^ f dPzr—J(p,l). (A4)

APPENDIX C: SEPARATION OF ANGULAR VARIABLES

After rotation of the h integration contour, Eq.

m - ( C D

4TT2 J l2+m2

Since the kernel is a constant, the solution is trivial:

(p-pf)2 4TT 3J {p'-l)2{l2+m2)

In four-dimensional hyperspherical coordinates

^0=s in 2 0i sme2deid02d<l>. (C2)

We introduce the expansions

/(A r Xo r i2 -r

,p') = \l / dl2 + c o n s t . (A5) L 4 W l2+m2J

1 • 1 iV.

= E (p'-iy ..of

1

Us{co$Bpn), (C3a)

/ & / ) = — £ (r+l)fr(p2,l2)Ur(cosdpl), (C3b)

2ir2 r=o The extra constant is an arbitrary solution of the homogeneous equation when (A3) is satisfied. However, the particular solution is seen to be infinite, indicating w h e r e l<f(f>f) i s t h e l e s s e r ( g r e a t e r ) of lp,% U s m t h e

a pole m f(p,p ,k) at * = 0, precisely when (A3) is orthogonality relation satisfied, providing for a 75-noninvariant solution of (Al). r 8rs

I dQi Ur(cos6pi) U8(cos$pf i) = 2w2—-Ur(co$6pp>), (C4) 11K. Johnson, Phys. Letters 5, 253 (1963) has pointed out that formal arguments based on conserved axial-vector currents are unreliable.

r+1 lUr(C0Sepp)=Ur(l) = r+ll,

153

we compute

B E T H E - S A L P E T E R E Q U A T I O N

r (»+2) r«

— [dnp un(cosdpp,)f(p,p')=fn(p*,p'*) n+1 J

yi(x) - >

y2(x) —>

.x(v-D/2

T(a)T(a+l)

-T(a)r(a+1) 1

1371

(D4)

2TT2 1 /p<\n 3a 1

n+1 p>

/p<y 3a 1

\£>/ 2irn+l

f I' (l'<\n

x di ( — ) UW). (cs)

T(n+l)T(v+l) x<-v+1^2

yi(x) —>xn/2,

1 1

for x-+°o

yi(x) • n+lxnl2+l

for w—»oo . (D5)

In terms of the variables x and x', and X = 3o:/4x,

2 7 r 2 r l / A w / 2 1 /x\n/2 rl/x\nl2 l/x\n/2 -i

/ » ( ^ 0 = — -( —) <%-OH—( —) 00'-*) -lLxVx/ xf\xfJ J

lJo xf{y+l)\xf)

X /*°° 1 /x\nl2

/ ^ ~ ) /»(*o0- (C6) n+ljx> y+l\y/

n+1

APPENDIX E: EVALUATION OF SUMS

We evaluate the sums

/ = E Unto, w=0

» l /= E Un{z),

K=O n-\-1 1

K= L Un(Z) ,

» - O ( M + 1 ) ( » + 2 )

( - 1 < S = C O S 0 < 1 ) .

(Ela)

(Elb)

(Elc)

This equation may be written in abbreviated form as The generating function for the Tchebycheff poly­nomials of the second kind is

1 Mx')=In(x')+ Fn(x') + X'nl*Gn(x') , 1

*"•/»« /&*) = — =£ t f . (* )*" , U < 1 - (E2)

where the dependence on * has been suppressed. Dif- T h e ieft_hand side provides the analytic continuation ferentiation of this equation twice with respect to x' of t h e right-hand side for |$| > 1 . Specializing to | = 1 leads to the differential equation (3.5).

APPENDIX D: PROPERTIES OF THE HYPERGEOMETRIC FUNCTIONS

gives

Next we write 7=1/2(1-2) . (E3)

We collect some useful properties of the functions (3.8). Using the integral representation The integral is

Xte)=° £ un(z)—= / . (E4) »-o n+1 Jc, l-2?z+1'2

F(fxfi;y;z) = T(y)

rOs)r(7-/s)

or the formulas of the Bateman Manuscript Project (Vol. I, Sec. 2.10) we obtain

h ^ (i-fe)« *' (D1) 2[22-i]i/2 n L a - 2 - &2- ln c—^— &2- n1/^J •

Setting | = 1 , we have

( 1 - f e )« * ' ( D 1 ) 2[^l] l /2nLa-^[^l] l /2 ) (^,_ [^1 ] l /2 ) -

ipt Project Setting f=l, we have [2-l]l/2-[2+l]l/2-

yi(x) —> xn/2,

yz(x) -

J--

Finally, ;'{

1

2[z2-\j'2 L o - r p / H - ^ + i ] 1/2 (E5)

y1(—i) = e-inrl2-

n+lxn'2+1

T(n+2)

K^,z) = ZUn(z)- - = / d?JWrf. (E6) »-o (»+l)(»+2) J<>

for x^O (D2) - - - 0 — ( w + 1 ) ( ;

y2(—1) = ««("-»/«•

r (o+i)r( j+i)

r(«)

Setting £=1 and carrying out the integral, we find after some combination of terms

lp-if2 ro-rp+Cz+i] (D3) K. i r » - i T ^ l n r ^ - u i " + L » + i j i " - |

2L3+1J Lcz—in1/2—Cs+ii1/^ r(i-4)r(»+2)

O- i j ^ -DH- i ] 1 / 2 . - J ln2( l -« ) . (E7)