P H Y S I C A L R E V I E W D V O L U M E 1 2 , N U M B E R 4 1 5 A U G U S T 1975

Generalized Callan-Syrnanzik equations and the renormalization group

Samuel Wallace MacDowell Yale University, *

and Pontificia Universidade Carolica do Rio de Janeiro, Brazil (Received 16 April 1975)

A set of generalized Callan-Symanzik equations derived by Symanzik, relating Green's functions with an arbitrary number of mass insertions, is shown to be equivalent to the new renormalization-group equation proposed by S. Welnberg.

INTRODUCTION

One of the important developments in high-ener- gy physics has been the formulation and interpre- tation of scaling laws within the framework of quantum field theory. The Callan-Symanzik equa- tion'" together with Weinberg's asymptotic theo- r e m has proved to be a useful tool in the analysis of the asymptotic behavior of Green ' s functions and the understanding of the scaling laws. How- ever , the method suffers f rom the limitations on the validity of Weinberg's theorem, which applies only to the so-called Euclidian region in momen- tum space.

La te r Weinberg3 proposed a homogeneous dif- ferent ial equation which looked s i m i l a r to the Callan-Symanzik equation but with the inhomoge- neous t e r m replaced by a derivative with respec t to a new m a s s parameter . This equation has sub- sequently been derived and investigated by s e v e r a l

It was apparent that i t had a wider range of applicability than the original Callan- Symanzik equation. In part icular it was shown that the new equation i s equivalent t o the differ- ential fo rm of Kadanoff's scaling lawss fo r c r i t i ca l phen~mena" '"~ '" o r the renormalization-group equations established by W i l s ~ n . ' ~

In this paper we show that the homogeneous dif- ferent ial equation of the renormalization group is equivalent to a s e t of generalized Callan-Symanzik relations13 among Green 's functions with an a rb i - t r a r y number of m a s s inser t ions (or 9' insertions).

The f i r s t two sect ions a r e devoted to the math- ematical framework. In Sec. I we show that an inhomogeneous f i r s t -o rder part ia l differential equation such a s the Callan-Symanzik equation can always be t ransformed, by what amounts to a renormalization, into a homogeneous differential equation. The inhomogeneous t e r m of the original equation is replaced, in the new equation, by a derivative with respec t to a new parameter . Therefore this homogeneous equation does not contain any m o r e information than the original one. We find that such a n equation does not coincide

with the homogeneous equation proposed by Wein- berg.

In Sec. I1 we consider a generalization of the Callan-Symanzik equation consisting of a s e t of inhomogeneous equations relating functions r,,, to r,, where in the physical problem Y i s the num- b e r of m a s s inser t ions. I t i s shown that the solu- tions of this s e t of equations can be related, by a n appropriate renormalization procedure, to deriv- atives of a function which sat isf ies a homogeneous differential equation.

In Sec. 111 we consider the generalized Callan- Symanzik equationsL3 f o r the G4 theory, and using the resul ts of the previous sect ion we der ive f rom them the homogeneous Weinberg equation. Con- versely, the s e t of generalized Callan-Symanzik relations can be obtained f rom this equation. Thus the equivalence between that s e t of relations and the homogeneous Weinberg equation i s established.

It is worth noticing that although in the s tandard derivation of the homogeneous Weinberg equation the renormalization has been done by making sub- tractions a t pi = 0, this i s by no means necessary. Actually, s o m e simplification i s achieved by p e r - forming the renormalization with on-shell sub- traction.

I . TRANSFORMATION OF AN INHOMOGENEOUS LINEAR DIFFERENTIAL EQUATION

Let us consider a f i r s t -o rder l inear differential equation of the general fo rm

We look for a transformation,

which i s such a s to turn the inhomogeneous equa- tion (1.1) fo r l?, into a homogeneous equation for r by taking

S A M U E L W A L L A C E M A C D O W E L L

Then we have Equation (1.1) could be considered a s the f i r s t of this s e t of equations.

Following the procedure of the preceding section, we look f o r a solution of this s e t of equations in the fo rm

which compared to (1.1) gives

=fi (YO, y).

Then we have

which i s a s e t of ordinary differential equations fo r the j 's and Z of the transformations (1.2), (1.3). They shal l b e determined in t e r m s of the boundary values a t s o m e a r b i t r a r y value of yo, say yo = 1. We take

(2.4)

which compared to (2.1) gives

If we consider the inverse functions yi =fi-'(yo, x i ) , the t ra jec tor ies y , b 0 ) with constant x's satisfy the differential equations

Let us introduce t = lnZ a s an independent variable instead of 31,. Then (2.5)-(2.7) may b e wri t ten a s Similarly,

Now, along these t ra jec tor ies we have 6rO(x) =0, which gives

The solutions of this s e t of equations with boundary values a t t = O given by

f t = y , , y o = l , Z 0 = 1 (2.11)

have the property that along the t ra jec tor ies y , ( t ) , which leave a l l the x's constant, the following relations hold:

which is the des i red differential equation for r. This equation i s of the s a m e f o r m a s the homo- geneous Weinberg equation but f o r a n essent ial difference in the coefficient of the t e r m ar/ay,, which, in (1.13), i s a nontrivial function a(yo, y ) ( l inear in yo). Then, a s expected, the two equa- tions a r e not equivalent.

11. A SET OF COUPLED LINEAR DIFFERENTIAL EQUATIONS

The f i r s t two a r e just reciprocity relat ions anal- ogous to (1.10) and (1.11), while the l a s t one i s obtained in the following manner: Integrating

We now consider a more general problem, name- ly that of a s e t of coupled linear differential equa- tions fo r functions r, of the following form:

12 - G E N E R A L I Z E D C A L L A N - S Y M A N Z I K E Q U A T I O N S AND T H E . . .

(2.9) we have Ill . THE GENERALIZED CALLAN-SYMANZIK EQUATIONS'~

We shal l investigate here a s e t of equations f o r a c l a s s of Green ' s functions in the q4 theory. L e t

Z-' d z -1 r: (pi. . . p,) denote the renormalized Green 's func-

(2.15) tion with n external par t ic les with momenta PI, . . . , p, and Y inser t ions of the composite op-

Hence e r a t o r q2 , with null external momentum each (mass insertions). We use an on-shell renormal- ization defined by

rD2 =p ) =o, (3.1)

(2.16)

But

d x . =dt' . Therefore

Now, along the lines x = const we have dro(x) = 0, which gives

(2.19)

taking into account (2.12)-(2.14) one obtains

with

Po =y(j)(Y0- 1 ) - 1.

Equation (2.20) i s the differential equation for r. One can now verify explicitly, by taking the r th- o rder derivative of (2.20) and set t ing yo = 1, that the ansatz (2.2) actually gives a solution to the s e t of equations (2.1).

r ; (p2=p2)= - 1, (3.3)

r ; (p i .p j ~ 5 $ ( 4 6 ~ , - I ) ) = g . (3.4)

A s shown by symanzik,13 one can establ ish a s e t of equations fo r the functions (Pi; P , g) which a r e straightforward generalizations of the Callan- Symanzik equation. They a r e

where the coefficients 0, ynz, and y , a r e functions of g alone and can be readily obtained f rom the renormalization conditions (3.2)- (3.4).

The resul t of the preceding section suggests that the function r: can be wri t ten in the fo rm

(3.6)

where Z and Z' a r e functions of g and m 2 / P and p2, g a r e related to m2, 7,g by renormalization- group equations which in differential fo rm a r e a s given by (2.5). In fact one finds

with boundary values a t m 2 = p given by

Also, following the s a m e procedure a s in the p r e - vious section one finds that rn sat isf ies the dif- ferent ial equation

1092 S A M U E L W A L L A C E M A C D O W E L L 12 -

r a a a n In concluding one could say that Weinberg's ho- LY ~ i p 2 am - ac2 iBG)7 a g + Y Q ) ] ~ ~ = o , (3'12) mogeneous differential equations fo r the Green ' s

where -

a =Y mz(ij)(m2 - p2).

functions in field theory may b e regarded a s a convenient way of expressing the generalized

(3.13) Callan-Symanzik equations (3.5) by making use of the renormalization-group t ransformations (3.6)-

This equation is of precisely the s a m e f o r m a s the homogeneous equation obtained using the meth-

(3.10).

od of soft quantization. In part icular , the coef- ficient (Y is a l inear function of the m a s s para - ACKNOWLEDGMENTS meters . It i s m o r e genera1 than the original Callan-Symanzik equation but equivalent to the s e t I would like to thank Dr. C. Sommerfield and of generalized relations (3.5). By changing the Dr. J. A. Swieca for many illuminating discussions, renormalization conditions one can modify the and members of the Institute of Physics of the coefficients of the equation, but i ts physical con- Pontificia Universidade Catolica d o Rio de Janeiro, tent remains the same. where this work was completed.

*Research supported in par t by the U. S. Atomic Energy Commission under Grant No. AT(11-1)3075.

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