P H Y S I C A L R E V I E W D V O L U M E 8 . N U M B E R 8 1 5 O C T O B E R 1 9 7 3

Short-Distance Behavior of a g+4 Theory, and Conformal-Invariant Four-Point Functions*

Enrico C. Poggio Department of Physics, Columbia University, New York, New York 10027

(Received 7 March 1973)

The "zero-mass" limiting short-distance theory for the g +4 interaction is explicitly constructed. The propagator is computed to third order and the vertex to fourth order in g , the zero-mass coupling constant. The Gell-Mann and Low short-distance equations which result as exact consequences can be explicitly determined to all orders in perturbation theory. If the eigenvalue condition is imposed, one can construct a theory in terms of the massless free propagator and a conformal-invariant four-point vertex function only. The energy-momentum tensor of this theory is the same as that proposed by Callan, Coleman, and Jackiw.

INTRODUCTION

In their pioneering work, Gell-Mann and Low1 (GML) studied the short-distance behavior of quan- tum electrodynamics (QED) by what is now known a s the renormalization group method. Their r e - sults not only included some facts about the as- ymptotic behavior of the photon and electron Green's functions, but also put forth some very striking conditions that the bare, unrenormalized electron-photon coupling constant should obey. These were, in the short-distance limit, that it be either infiilite, a s perturbation theory seems to predict, o r that i t be a finite number independent of the physical coupling constant, and that it must be a small positive root x of some equation (GML equation)

now known a s the eigenvalue equation. Unfortu- nately from the formalism leading to these results one i s not capable of telling which of the two al- ternatives, if either, represent physical reality.

Even though much work and soul searching i s still to be done, some recent theoretical studies seem to indicate that the interesting eigenvalue situation might be of some significance to the understanding, in t e rms of field theory, of the basic dynamics underlying the high-energy be- havior of the fundamental interactions and other physical phenomena. The f i r s t hint came from Johnson, Baker, and collaborator^,^-^ who in their model theory of QED, where photon propagators do not contain photon self-energy insertions, r e - alized that the theory could be made finite if an eigenvalue equation i s satisfied. In fact, they found that the photon renormalization constant Z3 takes the form (2, = Z, can be made separately finite order by order by choosing an appropriate gauge)

Z3 = F1(a,) l n ( ~ ~ / g ' ) +finite te rms, (1.2) !I 2 >>gz

where a, is the bare coupling (fine-structure) con- stant. The function Fl(x) has the property that if for some x i t has a zero, then the same x is also a zero at .k(x), the GML function. Later on, Wil- son6 suggested that eigenvalue-type equations might be of a wider dynamical interest, especially that they might be relevant in understanding strong interactions and phase transition phenomena.' More recently, Adler,8 using the very elegant techniques developed by Symanzikg and Callan" was able to reproduce the J - B (Johnson-Baker) results and went further on to state that1' if a different approach is taken in summing up graphs contributing to Z3, then the J -B function Fl(x) should also satisfy

where a! is now the physical coupling constant. Furthermore, he points out that the solution to (1.3) must be an essential zero.

In this work,'' we will use the physically simple, but nontrivial, g@4 field theory to car ry out two studies. First , we will show that one can derive, in a simple and compact manner, the GML equa- tions, exactly', once an asymptotic zero-mass theory has been properly defined. Secondly, we will t ry to add a modest contribution toward the understanding of the significance of eigenvalue- type equations in field theories a t short distances by analyzing their effects on our zero-mass theory.

The existence of a zero-mass limiting theory i s the crucial stepping stone, in the GML analysis, to a r r ive a t the short-distance equations. Most renormalization group approaches take this step for granted.13 An explicit construction of the limiting theory will not only put GML results on solid theoretic ground, but will also allow for a well-defined theoretical formalism which one can use to analyze i t s possible consequences. A s i t

2432 E N R I C O C . P O G G I O 8 -

tu rns out, our approach i s compact and s traight- forward and allows u s to compute al l of the short- dis tance functions in a s imple way. The simplicity is an inheritance f r o m the method we devised to c a r r y out the renormalization of the theory in question.14 Basically, we feel that the determina- tion of a proper renormalization program i s c ru- cial if one wants to construct, simply, z e r o - m a s s (short-distance) theories explicitly. We reca l l that our renormalization procedure w a s charac- t e r ized by avoiding the introduction of renormal - ization constants and a canonical Lagrangian for - malism, which, we explained, "non-commute" with the short-distance analysis of relat ivis t ic field theories.

The zero-mass formalism, in effect, should allow u s to calculate the f o r m of the eigenvalue equation in a very d i rec t way and so determine the eigenvalue, if it exists. Here, unfortunately, one is faced with the stumbling block which has, thus f a r , prevented any well-defined conclusion to the problem; namely one i s in the position of t rying to make s tatements which a r e beyond perturbation theory and yet one s t i l l has perturbation theory as the only calculational tool available. Only the full determination of the eigenvalue function and i t s eigenvalues would represen t the f i r s t r e a l break- through in understanding the short-distance be- havior along these theoretical lines. We a r e able to tell , however, what happens to the short-dis- tance theory if an eigenvalue condition is obeyed. F o r the g44 interaction we observe that the theory becomes what we shal l cal l a "classical-like a s - ymptotic" theory. By "classical-like" we mean that the theory at the eigenvalue resembles , o r can be abstracted to resemble, a c lassical theory. F o r instance, the m a s s l e s s g d 4 interaction i s well known to b e both canonically scale- and conformal- invariant. The renormalizat ion procedure, by necessity, introduces l a r g e cutoff masses , thus destroying those invariances. The canonical field dimension d, defined f o r a boson field by

where D i s the generator of dilatations, will lose meaning through the perturbative quantum c o r r e c - tions procedure. Similarly, the vertex function will be neither scale- nor conformal-invariant, f o r the s a m e reasons. The Callan-Coleman- Jackiw15 (CCJ) energy-momentum tensor , which is seen to be the relevant one in the study of those invariances of the theory, becomes ill-defined (i.e., it will not behave a s a CCJ tensor) through the quantum correct ions. If the eigenvalue equa- tion i s valid, one observes that, fo r the shor t - distance theory, the propagator and vertex func-

tion become scale- and conformal-invariant, and the concept of dimensionality is reestablished, the dimension being, of course, anomalous:

d = l +F(go) . (I. 5)

F(g,) is a constant grea te r than zero, i t s value being determined by go, the eigenvalue. Fur ther - more, f r o m the theory a t the eigenvalue we can construct a theory which is in t e r m s of the mass - l e s s f r e e propagator and an explicitly conformal- invariant four -point ver tex function. The con- struction of our theory allows for the determina- tion of such a function (within a perturbative ex- pansion). The energy-momentum tensor of such a theory i s the s a m e a s the CCJ tensor .

This paper is divided into th ree parts : In the f i r s t the ze ro-mass theory is constructed; in the second the short-distance GML equations a r e derived; in the third the eigenvalue condition and i t s validity a r e discussed, and then, assuming the existence of an eigenvalue, the short-distance theory is analyzed accordingly and the conformal- invariant theory is constructed. Appendix A will spel l out some of the techniques used to calculate the ze ro-mass and GML functions. Appendix B contains various tables with the resu l t s of these calculations.

I. THE "ZERO-MASS" THEORY

A. The Zero-Mass Limit

If we take the l imit of the renormalized theory as the physical m a s s approaches zero, we s e e that we obtain infrared logarithmic divergences coming f r o m subtraction constants defined a t on- shell points. Effectively these divergences a re , then, of the s a m e nature a s the original ultraviolet ones of the unrenormalized theory. This procedure, therefore, will not present u s with a well-defined asymptotic theory, and s o we must spel l out what it i s we mean exactly by a "zero-mass" theory. Our object is to find a "zero-mass" propagator function 8, and a "zero-mass" ver tex function T defined by

fP2[1 +5(P2, ~ ~ ~ ; m ~ ; g ) ] (1.1)

and

such that

(A) (;:)z and

l im T =fini te functions to a l l o r d e r s in g , (m-o)

8 - S H O R T - D I S T A N C E B E H A V I O R O F A g@' T H E O R Y , A N D ... 2433

By g, we mean the "zero-mass7' coupling constant which i s defined by evaluating (lim,,,) T at the symmetry point, off the mass-shell, defined by

By (lim,,,) we understand that all the momenta of the theory, including the off-shell momenta P;, must be taken to be much la rger than m2. This is, essentially, GML's program. Here the off-shell momentum P: will se t the scale and it will r e - place nz2 in the logarithmic expressions. The success of the theory will then depend on showing that remaining rn2 t e rms will vanish a s m becomes small compared to all the momenta. Because of the simple analytic s tructure of the equations de- fining T, in our interactive theory, the validity of condition(B), Eq. (1.3), i s the crucial one for the completion of our program.

Since the theory is renormalizable, we expect that the off-shell theory must be related to the on-shell theory. So there must be a function S(P:, m2) such that

h- ' (p2 , P t ; m 2 ; g ) =S(P t ,m2 , g) ~ - ' ( P t r n ~ ; g),

where A and g a r e the renormalized propagator and coupling constant, respectively. If (lim,,,) Zi-l i s to be finite, then the expansion of S(P,2,m2;g) will contain logarithmic t e rms depending on the mass which will cancel those in A-'when S and A-I

a r e multiplied. The residual t e rms must be of O(rn/P ', m 2 / P t , P '/P:)

S ( P , ~ ; ~ ~ ; g) ~ - l ( ~ ' ; m ~ ; g)

=finite t e rm + O ($; $) . (1.6)

A simple and explicit determination of S ( P t , m2; g) i s by no means obvious. But next we will show that our particular construction of the renormal- ized theory allows u s to specify it in a natural and simple way.

B. The "Zero-Mass" Functions and Their Equations

Assuming that we have found a connecting func- tion S(P;, m2; g) such that a, the zero-mass prop- agator defined by (1.5), will satisfy the condition (1.3), the zero-mass vertex function, and i t s dif- ferential equations, can be readily determined. Let u s recall from Paper 114 that if the propagator A is known, then the vertex function T obeys the

formal graphical equation in Fig. 1, which we r e - write, for completeness, in symbolic form a s

T=I I ,+$K, (AA) ,T ,

K , = v + ~ K , ( A A ) , T ++K, (AA) ,T , (1.7)

K, = v + ~ K , ( A A ) , T +$K,(AA),T,

V and K,, K t , K, a r e the two-particle irreducible (T.P.I.) and reducible functions in the s, t , u chan- nel, respectively.14 It was then also noticed that if we redefine a new vertex and propagator by

where R i s some arbitrary constant, then TR and ZiR obey (1.7) except for an undefined constant term. Furthermore, T, and ZR obey, uniquely, the differentiated form of (1.7), which we will also rewrite for completeness

T'=K:+T:, , K A = V ' + T ~ + T L ,

Ki = V' + T: + T: , K L = V ' + T ~ + T ~ ,

where

FIG. 1. Graphical "Bethe-Salpeter-Dyson-Schwinger" equation for the zero-mass vertex function. Single lines represent the zero-mass propagator A.

2434 E N R I C O C . P O G G I O 8 -

Equation (1.9b) i s shown graphically in Fig. 2. All primed quantities a r e to be understood a s being differentiated by an appropriate differential oper - ator.

Specializing these results to our case, if H i s related to A via S(P$, m2) a s given by (1.5), then we can define

such that (lim,,,) T i s finite, if (1.3) i s satisfied, and

Then T and h satisfy the differentiated equation (1.9). In order to have an explicit se t of equations, we must define the appropriate differential oper- ator that will make (1.9) a(over1apping) divergence- f ree equation. Because the mass t e rm i s not pres- ent in the propagator, the scaling derivative ion- shell)

will not give well-defined results upon integration. As suggested by Ward, one has to introduce the off -shell scaling derivative16

where the momenta P: obey condition (1.4a). If A(Pi) i s a function of the external momenta P , , then

Therefore, using (1.13) and (1.14) and the boundary conditions (1.4b), if the zero-mass propagator ii i s known, one can determine the zero-mass vertex from (1.9) a s follows

(1.15)

So, once S(P2, m2) and &(P ') a r e found, the f i r s t part of our project will be concluded. Their de- termination will be made in analogy to the method

FIG. 2 . Graphical representation of the differentiated T.P.I. s-channel contribution to T . These a r e analyti- cally finite.

introduced by Baker and Johnson4 in order to find the asymptotic form of the electron propagator in QED.

Let u s recall, f rom Paper I, the equations that a r e obeyed by the renormalized propagator

where

M(P', nz2) turned out to be the t race of the forward field energy-momentum tensor. It can be explicitly computed by evaluating the integral-differentia1 equation

a = n ' / !2~ )~ (1.18)

subject to the boundary condition

M ( P ' ) = ~ ~ at P 2 = - m 2 . (1.19)

K , i s the same kernel defined by (1.7) and c2(>.i) i s the D-wave Gegenbauer polynomial defined by

If we integrate (1.17) with respect to the off-shell point P t , we will get instead of (1.16)

so that

So, if we define

S H O R T - D I S T A N C E B E H A V I O R O F A g 9 4 T H E O R Y , A N D ... P 2 B ( P 2 ; p;,m2) = P 2 4 ( P 2 ; m 2 ) S-1(P;,m2) , where

(1.23) a = 7~'/(27r)~ ,

where and so the propagator is given by

then (lim,,,) P 2 & ( P 2 , P,",m2) will be a finite, well- defined quantity if

(1.25) and

The connecting function will then be P o 2 ~ ( ~ 0 2 , m2) . To investigate the possibilities (1.25) and (1.26), l e t u s define the function

F(P2/P:; g) = l im [M(P2) A (P;)] . (1.27) r n + O 1

If F(P '/Po2, g) exists a s a finite, well-defined func- tion, then for (1.26) to be t r u e one must impose on it the condition

otherwise P2 a) ( P 2 , Po2) is either a constant o r diverges a s P 2 - 0 . Assuming that (1.27) is true, then one can calculate F ( P 2 ) &om a n equation anal- ogous to (1.18) wri t ten in t e r m s of the ze ro-mass function

It can be shown by induction that (1.9), (1.15), (1.27), (1.28), (1.29), and (1.30) constitute, o r d e r by o r d e r in perturbation theory, a completely con- sistent, unique, finite and exact representat ion of the short-dis tance limit: ( q 2 , q;)>>m2, f o r the two- and four-point functions of our theory. The proof is straightforward and will not be given h e r e with full details [it is given in Ref. 121. Let u s just r e m a r k that, because of the s t ruc ture of the ver - tex function ( s e e Appendix A) the P 2 - 0 behavior of the kernel K , is such that Eq. (1.29) will always be finite and well-defined.14 We should point out how cr i t i ca l the procedure of the renormalizat ion program is to carrying out the z e r o - m a s s limiting procedure. Had we used Eq. (3.35) of Paper I in- s tead of (3.43), which gives (1.16), we would have run into complications, fo r the s imple constraint given by (1.28) would have to have been replaced by a sequence of constraints on Ql(P2) and a l l i t s derivatives, for the p rogram to work.

The simplicity and compactness of the renormal - ized equation has been t ransmit ted, then, to the equations fo r & and F. Fur thermore , s ince now the propagators have a s impler s t ruc ture with r e - spect to the renormalized ones, eas ie r explicit analytic i terat ive solutions can be c a r r i e d out, if enough patience and interest a r e a t hand. In Ap- pendix A we d i scuss our i terat ive method for solv- ing Eqs. (1.9), and we repor t the resu l t s fo r to third o r d e r in 2 and f o r T t o fourth o r d e r in g in Appendix B.

XI. THE SHORT-DISTANCE BEHAVIOR

Using only the fact that the theory in question is renormalizable, a f te r a sequence of clever manipulations of equations expressing such a fact, a s for example (1.8), one can derive the following relat ionship between propagators and ver tex functions defined a t various points1

where

t m2 Q," ' Qo

2436 E N R i C O C . P O G G I O

The functions T, T I , T,, etc. a r e the vertex functions defined at the various symmetry points, $m2, $Pt, $ Q t , etc. The functions d, 4 , d2, etc. represent the propagators defined at those points

and obey functional equations of the type

The imposition of the boundary conditions defining the T's and d's implies that function I must satisfy:

Equations (2.1) and (2.5) imply that function I i s independent of the different subtraction points. It i s called the invariant charge. It should be stressed, again, that Eq. (2.1) a s it stands, beyond saying that the the- ory i s a renormalizable one, cannot be used to extract any physical prediction.'" As Gell-Mann and Low pointed out, only if [(lim,,,)I] exists and gives a finite result, then one might draw some conclusions about the asymptotic behavior of the theory.

In the previous chapter we have shown that the (lirn,,,) does exist. In fact we have exact, explicit ex- pressions for the (lirn,,,) a and T functions and, therefore, a s a consequence of (1-10) and (1.24) we can construct the exact (lim,,,) invariant function I,

If we now se t @(Io(P2)) = 2IO(P2) F ( 1 , Io (P2) )

the invariant charge i s seen to obey the functional (2.11)

relationship Because of the connection between & to the renor- malized propagator via the (lirn,,,) process (2.10)

Io(P2) = [P2&(p2/P,2; IO(P:))l2 determines, exactly, i ts asymptotic behavior

where we have used the boundary condition (2.5) to express g in t e rms of I. This relationship can be made more explicit by differentiating (2.8) :

Furthermore, because of Eqs. (1.17) and (1.27), we have that

therefore

Coupled Eqs. (2.9), (2.11), and (2.10) a r e exact statements of the GML equations (also known by the awkward name of renormalization group equa- tions) which specify the short-distance behavior of the theory.g110 As they stand, their solution appears to involve quite a formidable task. One can calculate + ( I ) and F(1, I ) in perturbation theory a s a power ser ies in I using the methods of Sec. I [the lowest-order results a r e compiled in Appen-

a [A-'(Pz/P:; I,(P:)) ] dix B (sec. 211. Conversely, the knowledge of @(I) X~ P , allows for the determination of all the log t e rms

of T [ Q ~ / Q , ~ , z ( Q , ~ ) ] . Whether these t e rms can be (2'10) summed up into a well-defined analytic function

is not at al l clear yet. As we mentioned in the

8 - S H O R T - D I S T A N C E B E H A V I O R O F A g @ 4 T H E O R Y , A N D .. Introduction, Eq. (2.9) could allow for the solution 1 p2 F(go) ( ) 9

Io(P =go , if (2.13) and thus expresses the exact asymptotic behavior

of the propagator function. Some interesting con- @(go)=O , sequences a r e attached to this resul t . F i r s t , we

that is, go is a root of the function a(%). This notice that

"bare" asymptotic coupling, which will obviously F(&) > 0 (3.3) be independent of 2 and g, the ze ro-mass and physical coupling constant, respectively, is called Since Otherwise the theory have unwanted

the eigenvalue and (2.13) the eigenvalue equation. ghosts a s P 2 - 0 . This is an additional constraint

The nature of the eigenvalue and i t s consequences that go, the eigenvalue, must obey. The third

with respec t to our short-distance theory will be o r d e r in the go resul t f o r ~ ( g , ) is given in Appen-

explored in Sec. 111. dix B. The second consequence of th i s resu l t is that the renormalization constant 2, defined by

111. THE THEORY OF THE "EIGENVALUE"

A. Existence of the Eigenvalue

In what follows, we will a ssume that (2.13) i s a valid s tatement and i t s impact on the theory will thus be analyzed. Whether this assumption i s justified o r not, we a r e not able to answer. An intensive, semiquantitative study of (2.13) has been c a r r i e d out within our only known and avail- able means, perturbation theory.12 Basically i t w a s concluded that, because of the nature of re le - vant integrals leading to i t s computation to some high o r d e r of perturbation (say iV= 5 to N- lo ) , @(N'(go) i s a polynomial with coefficients of a l t e r - nating signs, the magnitude of the coefficients appearing to grow a s N increased. Whether this behavior p e r s i s t s a t higher and higher o r d e r s , we cannot te l l with certainty. One is, unfortunately, faced with the problem of making predictions which a r e beyond perturbation theory and yet one has, seemingly, just perturbation theory a s a mecha- nism to work with. It is not a t a l l improbable that by studying the consequences of t h e eigenvalue condition on the theory, one could get, a t least , a n inkling of the general nature of these roots." We will t r y to r e p o r t fu r ther on th i s mat te r in a future paper.

B. Green's Functions at the Eigenvalue and the Conformal Group

If (2,13) is sat isf ied by s o m e g=go, then the o ther GML equation becomes

which can be readily solved to give [in the (lim,,,) region]

(3.2)

L l i m h U ( P 2 ) g-

-p2,,2 P 2 + m 2

where A U is the unrenormalized propagator and m 2 is the physical m a s s , will be in the (lim,,,)

The theory at the eigenvalue is, thus, noncanonical. This behavior can also be understood if we look a t the situation f r o m a different angle, namely by introducing the 15-parameter space-t ime t rans - formation group, the conformal group,'' given by the Poincar6 group and

dilatations: x,- hx, , (3.6)

inversions: xP-xp /x2 , (3.7)

and the special conformal t ransformation:

The genera tors of dilatations, D, obey the canonical field commutation ru le

where d is the dimension of the field. F o r the f r e e field c a s e the canonical dimension is equal to the physical dimension of the field. Once interactions a r e introduced one rea l izes that, in perturbation theory, the concept of dimensionality s t a r t s losing meaning, a s can be observed by analyzing the s t ruc ture of the propagator.'* This is, of course, due to the s t rong dilatation-breaking effects intro- duced by the regular izat ions which a r e necessar i ly brought in to make the theory finite, At the eigen- value, we note f r o m (3.2), a l l the breaking t e r m s , coming in this theory a s a sequence of logari thmic t e r m s , s u m u p to a s imple power behavior. The concept of dimensionality i s thus again recovered, but now the dimension of the field has the noncan- onical value

2438 E N R I C O C . P O G G I O 8 -

The eigenvalue propagator is thus a conformal co- variant object like the f r e e propagator. Some in- terest ing conclusions, along these lines, can also b e made for the eigenvalue ver tex function. Let u s reca l l Eq. (2.8) defining the invariant charge. At the eigenvalue i t will be now wri t ten a s

and, therefore, the vertex function evaluated a t the invariant symmetry point defined by (2.7) is

In t e r m s of the ze ro-mass coupling g, T can be written, in perturbation theory

so, when @(go) = 0 the coefficients +, must be such that the power behavior (3.12) is obtained. The condition is that (at the eigenvalue only)

Equation (3-12) i s a c lea r indication that the ze ro- m a s s vertex, a t the eigenvalue, is not just sca le covariant but a l so that i t s coordinate space rep- resentation is conformal covariant. That is, we c a n find a function S(P12, pZ2, P3', P4'; S, t ) such that

7 (P l , p2, p 3 , P4)

has a coordinate space t ransform

where T i s re lated to the connected piece of (01 T*($(xl) @(x2) Q(x3) Q(x4)) I 0) , which i s invariant under the r u l e s (3.6)-(3.9). These constrain -r to be of the f o r m

where X and Y a r e the four-dimensional harmonic ra t ios

where xi ,=(xi - xj)' for i # j . Conformal. invariance, pegr se , cannot specify m o r e than (3.17). Since our theory must be invariant under the coordinate charges xi -xj, i ij, O i s fu r ther constrained to obey

O(X, Y) = O( Y, X)

Once the obvious choice of S(P12, PZ2, P3', Pq2; S, t) is specified we will show that, within our theoret- ical framework, r (P1, P z , P3, P 4 ) can be computed in principle.

C. Conformal Scattering Amplitude

A s we have mentioned in P a p e r I, the nature of the interaction allows u s to identify the vertex function of the theory with a two-body, off-shell, scat ter ing amplitude. In o r d e r to work with the ver tex function itself, l e t u s define

s o that, in coordinate space, C will be a conformal density, and f r o m (3.17)

The s implest choice of S, in accordance with (3.11) and (3.12) is , obviously,

such that

The "conformal vertex" C can be determined by using the eigenvalue condition and the auxiliary conditions (3.14), and by imposing the condition that the right-hand s ide of (3.23) be independent of Q;. The s t ruc ture of T f r o m the perturbation analysis ( s e e Appendix A) is given to b e

S H O R T - D I S T A N C E B E H A V I O R O F A g @ 4 T H E O R Y , AND ...

where t and the Cn's a r e functions of the ratios of the external masses and s, t only, and vanish at the sym- metry boundary condition point. The function [ t(PlP2; P3P4) +(2-3) +(2-4)] i s effectively the finite part, within a constant, of the integrals obtained by evaluating the formal "Dyson-Schwinger" Eq. (1.7) and set- ting m=O. They a r e the characteristic elements of the interaction and should contain the essential dynam- ical information. The presence of the log terms i s due only to renormalization procedure. The relevance of these characteristic te rms will become more transparent at the eigenvalue. From (3.23) and (3.24) and the eigenvalue condition, we obtain, in fact,

C(P,, P2 ,P3, p4) = E + ~ ( P , P , ; P,P,;~,;~ fi ( w r / 2 + ( 2 - 3 ; s-u) +(2-4; s-t) i = j 3

The third-order contribution to t i s given inAppen- dix B. One can further obtain a much simpler form for C if we exploit the invariant nature, Eq. (1.8), of the vertex equations. By applying (3.23) to the equation for T and defining (the notation should be obvious)

and using Eq. (3.2), the "conformal amplitude" C will satisfy the formal equation (symbolically)

A, i s the free, massless propagator. Since C obeys the boundary condition at the symmetry point

one can define differentiated equations analogous to (1.9). We should remark that the integrals in (3.27) a r e finite because of the good convergence of the functions t defined by (3.24). Furthermore, because of (3.5) there will be no ambiguous con- stant in V,. From the result (3.25) the function C, once go i s known, i s strictly dependent only on the ratio of invariants and external masses and one cannot properly talk about a "series expansion in te rms of go." Equation (3.25) i s just the inheri- tance of perturbation theory which i s the only method we know to compute F(go) and t(QlQ2Q3Q4; go). Equation (3.27) and i t s differentiated form a re

I

a more precise, though less explicit (since their solution i s quite hopeless) statement about C. With this in mind, let u s now try to solve (3.27) via its appropriate differentiated form and boundary con- ditions with respect to some coupling parameter. The result would effectively be the same a s solving the "zero-mass" theory with f ree propagatorsonly. If we now force the expansion parameter to be go, the eigenvalue, then because of (3.25) we would have a s a result

C(P1, P2, P3, P,; go) =gb + tF=O(P1, P2, P3, P4; go)

and

$;=O (go) = 0 . By the subscript, F = 0, we mean that we a r e com- puting graphs without propagator selfenergy in- sertions. Equation (3.31) i s a somewhat simpler version of the eigenvalue condition (2.13). Effec- tively, it i s a reflection of the fact that the "zero- mass7' vertex function, without propagator self- energy insertions, i s the relevant "invariant charge." The function tF=O, at the eigenvalue, i s then related to the function calculated with the full propagator by

Its conformal invariance reflects the conformal- invariant nature of the massless quartic boson selfinteraction. Because, a t the eigenvalue, the theory has some characteristics which a r e remi-

2440 E N R I C O C . P O G G I O 8 -

niscent of the unquantized, canonical theory, we could cal l it a "classical-like" asymptotic theory. This "classical-like" nature i s a lso observed if we analyze a relevant energy-momentum tensor in this theory.

D. The Energy -iblomentum Tensor

The existence of the eigenvalue condition has some interesting consequences in regard to the short-distance behavior of the energy-momentum tensor . Some y e a r s ago Callan, Coleman, and ~ a c k i w ' ~ pointed out that, canonically, the energy- momentum tensor which is relevant to studying the breaking of sca le and conformal t rans forma- tions is given by

where Q g U is the canonical tensor. @g:, has the s a m e Poincare s t ruc ture a s 9;' but it has the additional property that

where D p is the dilatation current . D p i s con- se rved for the m a s s l e s s c a s e and @;:, is t race- l ess . When quantization effects a r e taken into consideration, condition (3.34) i s never satisfied and renormalization effects leave a t e r m of the f o r m (8,au -gpU U)@ unspecified.14 At shor t dis- tances, and assuming the existence of the eigen- value, Schroerlg h a s shown that t h e t r a c e of the tensor vanishes. Within our theoret ical f ramework th i s can be easily seen, a s follows. In Paper I, we defined a forward energy-momentum tensor by exploiting the invariance of the theory with respec t to general coordinate t ransformations through the insertion of a zero-energy graviton into the propagator. T h e resul t ing Ward identity thus defined the tensor. If we now, instead, make an inser t ion into the vertex function and go to our (lim,,,) region, the resulting t r a c e identity will be proportional, a t the symmetry point, to P2 a / aP2 of the invariant charge, and thus proportional to @(g,)=O if there is an eigenvalue. More inter- esting, yet, i s the fact that within the "classical- l ike asymptoticJ' theory defined by A, and C, the relevant energy-momentum tensor i s the CCJ (Callan-Coleman-Jackiw) tensor . Le t u s reca l l f r o m Paper I that the t r a c e l e s s energy-momen- tum tensor obeys the equation

The f o r m fac tors A and B a r e connected via Ward 's identity

Q p @ $ u = - P u ~ - ' ( ~ + Q ) - ( P + Q ) u ~ - ' ( ~ ) , (3.36)

where the full tensor O '" is given by

The f o r m factor C(f, Q) specifies the t r a c e of B p U , o n c e A and B a r e known. A scheme to com- pute to al l o r d e r s finite A , B, and C has been given to within specifying C(0,O) = C, =undetermined con- stant. We can now define a "zero-mass" energy- momentum tensor by taking the (lim,,,) of

+Su ( P , P + Q; P,Z) =P,Z A(PO2/m2) Qg" ( P , P + Q; m2)

so that i t s t r a c e l e s s par t obeys the fo rmal equation

In analogy to the analysis given in Paper I, one can reduce the above equation into a se t of dif- ferential-integral equations f o r the ze ro-mass fo rm fac tors A , B, and C, analogous to Eqs. (4.51), (4.54)) and (4.59) of that paper. If we a r e a t the eigenvalue, s ince the fo rm fac tors a r e proportional to A-', their P; dependence i s - ( P : ) ~ . SO we can define a t r a c e l e s s tensor which i s scale-covariant with respec t to P and Q a s follows:

Because of condition (3.5), the homogeneous t e r m i s not present , s o that we obtain an equation

Since the f o r m fac tors a r e scale-invariant, if we take a scaling derivative of the above, we obtain, obviously

But, within the context of our theory, the Ward identity which specifies the boundary conditions on A and B is

and therefore 2, = -1, B, = -1. At this s tage noth- ing can be said about c, beyond recognizing that it must be some (a rb i t ra ry) scale-invariant func - tion of P, Q, Since the existence of the eigenvalue is consistent with t h e vanishing of the t r a c e of the asymptotic tensor e p U , if we impose th i s condition, then C, =: and therefore

8 - S H O R T - D I S T A N C E B E H A V I O 1R O F A g $ * T H E O R Y , A N D ... ACKNOWLEDGMENT

- + ( P @ Q ~ + P " Q " - + ~ ~ ~ P . Q )

- + ( Q ~ Q " - ~ ~ ~ ~ Q ~ ) , (3.44)

which is the s a m e a s the CCJ tensor . We will then cal l CpU the "classical-like" asymptotic energy- momentum tensor .

IV. CONCLUSIONS

We have shown that one can construct the short- distance Green's functions by an appropriate def- inition of the "zero-mass" l imit of the full, r e - normalized theory. The resulting equations a r e exact, in the sense that no additional limiting procedures have to be performed, and can be solved explicitly to all o r d e r s in the ze ro-mass coupling constant. A direct consequence is that we can obtain exactly the renormalization group (RG) equations. Of course these equations can be ob- tained m o r e elegantly by using the techniques of Callan and Symanzik (CS), and, in a sense, the proof of the exis tence of the ze ro-mass l imit i s equivalent to showing that the right-hand s ide of the CS equation vanishes in the region of interest . (As can be seen f r o m o u r proof, th i s in essence is just a consequence of Weinberg's theoremsz0 on the asymptotic behavior of graphs.) What our analysis loses in elegance, it gains in practicality. The full determination of the CS differential operator (and therefore the RG functions and parameters ) requ i res the knowledge of the relevant renormal - ization constants. Our computational scheme, which is a consequence of a renormalization of the theory that avoided introducing explicitly these constants, gives an exact and well defined recipe for determining these p a r a m e t e r s to a l l o r d e r s without having to deal with infinite quantities.

By imposing the interesting eigenvalue condition we observed that one could extract , a t this par - t icular point, a theory which resembles the clas- s ica l ze ro-mass one, where the propagators a r e the mass less , f r e e ones, the interaction i s con- formal-invariant, and the energy -momentum ten- s o r relevant to these is the one proposed by Callan, Coleman, and Jackiw. One has, a lso, a well-de- fined scheme f o r determining the conformal am- plitudes which, in a perturbation sense, can be determined f r o m the finite-part contributions to the ver tex function of the unrenormalized, ze ro- m a s s theory. Unfortunately (and probably unsur- prisingly) we have no way to conclude for the exis- tence of the eigenvalue, even though we could com- pute the eigenvalue function to a l l o r d e r s . Con- clusions about the possible nature of the eigen- value, a s in Adler 's QED, if i t exists, a r e at this s tage negative.

I would like to thank Professor Kenneth A. John- son for his guidance and suggestions a t the various s tages of th i s work.

APPENDIX A: CALCULATIONAL TECHNIQUES

1 . Feynman Rules

The Feynman r u l e s a r e defined, symbolically, by the graphical equation of Fig. 1 and by the Eqs. (1.7) and (1.9). We will summar ize them h e r e a lit t le more explicitly:

(1) Replace every nondifferentiated ver tex by T(Q,, Qz, QS, Q4)'

(2) Replace every nondifferentiated propagator line by

(3) Replace every differentiated kernel and prop- agator by

where A = {z,, R,, R,, K) and (Q: + Qy) =$ ( 1 + 2 6 i J Q:.

(4) F o r every integration loop w r i t e a fac tor

2. Iteration of the Vertex Function Equations

a. Iteration Sclzenze

Assuming that the propagator i s known, f r o m (1.9a) and (1.9b), we note that we can w r i t e

Tr=7: ,+7;+7: (-41)

and integrating

T = T , + T , + T , ,

where

Computation of 7: and i t s subsequent integration a r e very s t raightforward to third o r d e r s ince they involve an integration over a s ingle loop. Fur ther - more, we want now to show that, in a?z oordeor by order sense, 71 can b e written, f o r the o r d e r N

2442 E N R I C O C . P O G G I O

i ; ( ~ j = k ( ~ i - Q ; ) . & [ F $ N ~ ( T ( N ~ ) ; ~ j ~ 2 ) ; ~ ( * n ) ) ] , More explicitly, one iteration of (1.7) will yield i = ~ T , =$v+$T( iGi ) , T - a T ( F d ) , R t ( h Z l t T ; (A5)

(A41 . ,

up to second order the t e r m depending on IT,, R, , where N,, N,, and N3 a r e smal le r than N, so that R, will not be present, so TCN1j, TIN2), and &(N3) a r e explicitly known f r o m a previous iteration step; and TIN) involves only TI" ,$ ~ ( 1 ) ( L a ) , ~ ( 1 ) - (gi = ~ p ) . (A61 the evaluation of a single-loop integral. Resul t (A4) follows essentially f r o m the fact that if we By repeating the iteration N t imes, we will have

i t e ra te the formal equation repeatedly, we obtain T = T , + T , + T , ,

a s u m of t e r m s that depends only on T and ;I plus a single t e r m that depends on the undefined kernels . where

(-1)N - - - - + 2 ~ + ' T ( A A ) , T . . . T(LZ), ~ ( n a ) , ~ , ( a n ) , T , (A71

where the factor T(LL), T in the next-to-last t e r m o c c u r s N t imes. The las t t e rm, depending on Kt, will not contribute to Nth-order perturbation theory. We will define Ti.') to be given by the above formula with- out the gt t e r m . Furthermore, by using the analogous expressions fo r the lower o r d e r s we can r e e x p r e s s it, compactly, by the following one loop integral which i s to be used to evaluate (A4) o rder by o r d e r

where the coefficients A : ~ ~ ~ ~ have the values where

= 1 - '6 (-49) 1

OOn 2 nN 4, (Q,,Q,) = /da"2ln~" 0 ' 45 Q02 and

A ( N ~ - - [Qi = Q41 n ln2n - - i6n-nl-n2+ N ' (A10)

(A15)

and The coefficients bf-"l-"2 turn out to take the values

b(N-n1-n2j = (1 - 6 )gv-nl-n2) n = n2/(2n)4. (-416)

on j ( A l l ) The generalized integrals will contain log t e r m s

where that depend on the external momenta and will thus i ( N - q - n z j =

n + ' n , ~ - n l - n z - n ). (A121 involve much inore complex angular dependences.

L e t u s now analyze the basic s t ruc ture of the one loop integrals that a r e involved in the above scheme.

b . Bas ic Integvals and Gegenbnuev Polynomials I T I A typical graphical contribution to (A8) is given

i n Fig. 3. It i s easy t o rea l ize that a l l these con- tributions a r e complicated generalizations of the following integrals:

which, in Euclidean form, becomes

In (Q1,Q4) = - a Jn (Ql,Q4),

FIG. 3. Typical one-loop graphs occurring in the dete~mination of the vertex function. Convenient choices

(A14) of internal momenta are shown.

8 - S H O R T - D I S T A N C E B E H A V I O R O F A gq4 T H E O R Y , AND ... Interestingly enough, i t i s easy to verify that one can compute, to al l orders, the coefficients of the logarithmic powers by evaluating a sequence of J, integrals. In fact, for this purpose i t i s sufficient to se t either Q, o r Q, equal to zero in (A16). Furthermore, explicit evaluation of J, will determine the basic structure of the (3;-indepen- dent t e rms of the vertex function which a r e re le- vant to the discussion of conformal invariant ampli- tudes. Of course, explicit determination of these contributions becomes already quite cumbersome a t the fourth-order level. The function J , (Q,,Q,) defined by (A15) can be evaluated exactly by ex - panding the angular dependent quantities in t e rms of Gegenbauer polynomials. The properties of these have already been given, but for reasons of completeness we will summarize again. They obey:

(1) the generating formula

(2) the normalization condition

(3) the addition property

(4) the reflection property

(5) the recursion formulas

clcn = e n + I + c, -I, (A2 l a )

The integrals a r e then expressed by

where

Q 2 Q .Q n 0 .

symrn. pt.

c . Dete~minatio??, of the $tiz-Orde~ V-Graph Contribution to (g)

To this order the graph is of the skeleton form and therefore there i s only a single log t e r m . Since i t is symmetric with respect to Q,Q2Q,Q4 (see Fig. 4 ) i t must be of the form

and v, is independent of 8: and symmetric with respect to the external momenta. Explicit de ter - mination of v4(Q,Q,Q3Q4) is quite tedious and in- volved, and i t will not be given. The constant V4 can be easily determined if we go into coordi- nate space:

Because of the definition of J, (Q,,&,) we recall that

FIG. 4. The fourth-order graph contributions to the vertex function.

2444 E N R I C O C . P O G G I O

where lro (Q) = A41n (&'/A2)

xij = (xj -xi )2 , i S; j (A26) and thus A, i s the coefficient we a r e looking for .

by translational invariance, we have Equation (A29) can be evaluated in a very straight- forward manner using the Gegenbauer polynomfal

e x p E Q=,Q, exl ) i;'(Q,Q2Q3) = / d 4 r l d 4 x , d 4 x 3 ~ expansion. '1 '2 x32x12x13x23 '

Now, suppose that we se t Q, = Q, = 0, Q3 = Q; then

The x,-x, integral i s convergent and well-defined, and by dimensional arguments

Therefore

d. Stvuctzrre of the Vertex Fzd?zct~o~z 112 P e ~ t u ~ b a f ~ o n Theoql

Using the above facts, we have explicitly carried out computations to fourth order in g. This is, in fact, the f i r s t nontrivial order since the V graphs s ta r t contributing. The coefficients of the logs a r e determined, a s we explained, by computing the re l - evant basic integrals. The contribution to the sin- gle log power from the V graph has been discussed separately in sec . c . The results a r e summarized in Appendix B (secs. 1). By a straightforward, but cumbersome and tedious induction analysis, one can show that to al l orders, the vertex function takes the simple form

where

7(Q1Q,Q3Q4) = t (QlQ2;Q3Q4;Z) + (2- 3) + (2 - 4 )

(A311

i s the cutoff-independent part of T . The functions C, a r e also cutoff -independent. The polynomials 4, ( g ) behave a s

The results up to fourth order a r e included in Appendix B(subsection 1). The relevant graphs con- sidered a r e shown in Fig. 4.

APPENDIX B: TABLES OF CALCULATIONAL RESULTS

la . Zero-Mass Vertex Function to Fourth Order in g

v(") = contributions from V graphs,

Ti:),,, = contributions from T.P.R. graphs.

If

- T?) = 7'") (Q1Q2;Q3Q4;QO2),

then

Results.

(1) T(l) = 1,

-(3) - 8 2(Q1+Q2)' (3) T s - s h - - - 3 - 7 (Q3 + 8,)'

3 Qo $ Q,"

I ( 8 4 A Q212 - + ln (Q" '22 -, ln ; eo" ; Q,"

8 S H O R T - D I S T A N C E B E H A V I O R O F A g</>4 T H E O R Y , AND . . . 2445

= « ? ! - <?,), for «?,< Q2),

^ ^ i • - 16 111 4 n 2 +16 in 4 ^ 2 - ? 8 i n 4 ^ 2 48 i n 4 £) 2 3^0 3^0 3^0 3^0

+ r4 |Q 02 |Qo2

+ ^1(Q1Q2;Q3Q4)lnT7rT+ e4(Q1,Q2)+ 04(Q3Q4), 3 ^ 0

n - 6 + 1 2 ^ 3 " 4 111 y 8 ^ ( n + 1 ) 3 ' (n+ l)3

^ ( Q i f e ^ s ^ ^ o 2 ) a n d e4 (Qi^ 2 ) h a v e n o t b e e n deter­

mined explicitly.

v4 has not been determined explicitly.

lb. Zero-Mass Propagator to Third Order in g

22^a)"D„=-j -*F(x;g.),

— Q2

^ M = I3>, D2= - ^ l n Q ~ 2 ,

j6 (* )= i* [ ln*- ( | + l n | ) ] ,

A = - ^ l n 2 ^ + ^ ( | + l n | ) l n ^ .

2. Eigenvalue-Theory Functions

(A) F^o) ̂ g02[F2 + g0F3 + O(^0

2)],

**2 12 9

F 3 = - 1 ( 1 + 1 lnf),

(B) &(«,) -g-02[B? + ftBi + «»J5J + OteS)],

B1 1 - 2 .

S 2 = f + i l n | + f ln 2 |

- £ ( ^ 0 - 3 ( 6 - 5 0 , ( 1 ) ) ,

(C) * (g 0 )=2^o i r (« ' o ) + i/'i(^o)

> i " f ,

> 3 = £ , y3"~ 1 12" in3 ""T" 1 U 3

*Research supported in par t by the U. S. Atomic Energy Commission.

1M. Gell-Mann and F. Low, Phys. Rev. 95, 1300 (1954). 2K. Johnson, R. Willey, and M. Baker, Phys. Rev. 163,

1699 (1967). 3M. Baker and K. Johnson, Phys. Rev. 183, 1292 (1969). 4M. Baker and K. Johnson, Phys. Rev. D 3 , 2516 (1971). 5M. Baker and K. Johnson, Phys. Rev. D 3 , 2541 (1971). 6K. Wilson, Phys. Rev. D 3, 1818 (1971). 7K. Wilson, Phys. Rev. D 4, 3174 (1971); 4, 3184 (1971). 8S. L. Adler and W. A. Bardeen, Phys. Rev. D 4, 3045

(1971); 6, 734(E) (1972). 9K. Symanzik, Comm. Math. Phys. 1£, 227 (1970). 10C. G. Callan, J r . , Phys. Rev. D 2, 1541 (1970). U S . L. Adler, Phys. Rev. D 5, 3021 (1972). i 2 E. C. Poggio, Ph.D. thes i s , M. I. T. (unpublished). i3N. N. Bogoliubov and D. V. Shirkov, Introduction to the

Theory of Quantized Fields (Interscience, New York, 1957), Chap. VIII.

1 4E. C. Poggio, Ann. Phys. (N.Y.) (to be published), hereafter referred to as Paper I.

15C. G. Callan, J r . , S. Coleman, and R. Jaekiw, Ann.

2446 E N R I C O C . P O G G I O 8 -

Phys. (N.Y.) 59, 42 (1970). 18S. Coleman and R. Jackiw, Ann. Phys. (N.Y.) g, 552 1 6 ~ . C , Ward, Phys. Rev. 84, 897 (1951). (1971). IIK. Huang and F. E. Low, Zh. Eksp. Teor. Fiz. 46, 845 "B. Schroer, Nuovo Cimento Lett. l7, 867 (1971).

(1964)ISov. Phys.--JETP El, 579 (1964)J.

P H Y S I C A L R E V I E W D VOLTJRIE 8 . N U M B E R 8 1 5 O C T O B E R 1 9 7 3

Explicit Poinear6 Embedding of the Harmonic-Oscillator Model of Composite Elementary Particles

L. P. Staunton Depnrftnenr of Physics and Ast~onomy, Unrverslty 01 North Carolma, Chapel Wdl, North Carolzncl 27514

(Rece~ved 14 March 1973)

There are two unitarily inequivalent realizations of the Poincare Lie algebra of the recently proposed harmonic-oscillator model of elementary particles. One of these, which may be called the usual type, is trivial. The nature of the second type of realization, in which the structural information carried by continuous internal variables is not lost, suggests a very general method for avoiding the pitfalls suggested by the analyses of Wigner and O'Raifeartaigh for the construction of Poincare states with nontrivial internal structure.

Recently, Biedenharn, Han, and van Dam1 have interpreted the new positive-energy relativistic wave equation of Dirac2 a s a description of a sys- tem composed of two particles interacting via a harmonic-oscillator potential within a new quan- tum version of the front form3 of classical rela- tivistic mechanics of Dirac. Their interpretation involves the identification of the (scaled) set of eigenfunctions of a (2 + 1) Galilean Hamiltonian, H,, with certain selected solutions of generalized, higher-order relativistic wave equations of the type suggested by the spin-zero example of Dirac, the complete set of eigenfunctions describing a set of s tates having a Chew-Frautschi mass-spin spectrum (massL - spin). They emphasize that the essential content of Dirac's new equation and of their higher-order generalizations i s the existence of a covariant model of interacling particles.

It i s hardly necessary to detail the relevance of such a model, particularly in view of the exten- sive l i terature on the essentially nonrelativistic parton model, o r to point out that relativistic co- variance i s the crux of the matter. The question then naturally a r i s e s whether the somewhat c i r - cuitous route of matching the eigenfunctions of H, with solutions to higher-order wave equations may be avoided in favor of directly incorporating the differential operator H, into a complete realiza- tion of the Poincar6 algebra.

In fact, the Poincar6 embedding of this differ- ential operator can be accomplished in two ways, with very different physical content. The f i r s t realization (called A below) provides a closed Poirrcar6 algebra only on the se t of equivalence

classes of the eigenfunctions of H,, while the manifold of these eigenfunctions i s decomposed into disjoint Poincar6 Hilbert spaces, one for each spin value, which may be unified only in an infinite-direct-sum fashion. The connection be- tween the mass and spin pointed out in Ref. 3 i s seen to a r i s e from the Poincare-embedding pro- cess and to require disjoint Hilbert spaces.

On the other hand, the second type of embedding (called B below) realizes a closed Poincar6 alge- bra in a unified Hilbert space, but in a trivial direct-product fashion which implies that the spin quantum number of .the states i s zero, regardless of the excitation level of the harmonic oscillators. This second realization i s presented only for con- t r a s t with A arid to illustrate the remarks below.

In general, given the Poincare-operator algebra within a Hilbert space, a complete se t of com- muting quantum observables for a state consists of the mass, spin quanta, arid 3-momenta of the state, a s pointed out by Wigner.* It follows that any further continuous variables which a r e thought to ca r ry additional information 011 the internal s tructure of a given mass-spin-momentum state a r e redundant, the action of the Poincar6 group being effectively intransitive for the supposed ad- ditional information. Realization B i s an example of this effect.

Realization A below effects Poincare closure only when acting on the (classes of) eigenfunctions of If,, separately upon each for each spin value in a separate Bilbert space. These functions depend on two continuous oscillator liariables to which the remarks immediately above apply within each