P. M. FISHBANE AND J. D. SULLIVAN

P. M. Fishbane and J. D. Sullivan, Phys. Rev. D 6,645 (1972) (hereafter known as paper II).

3P. M. Fishbane and J. D. Sullivan, Phys. Rev. D 6,3568 (1972) (hereafter known as paper III).

4S. D. Drell, D, J. Levy, and T.-M. Yan, Phys. Rev.Letters 22, 744 (1969); Phys. Rev. 187, 2159 (1969);Phys. Rev. D 1, 1035 (1970); 1, 1617 (1970); 1, 2402(1970).

5P. M. Fishbane and J. D. Sullivan, Phys. Rev. D 4,458 (1971).

6V. N. Gribov and L. N. Lipatov, Phys. Letters 37B,78 (1971); Yad. Fiz. 15, 781 (1972) @ov. J. Nucl. Phys.15, 438 (1972)]; Yad. Fiz. 15, 1218 (1972).~Such graphs have been studied in renormalizabletheories by, among others, S.-J. Chang and P. M.Fishbane, Phys. Rev. D 2, 1084 (1970); by V. N.Gribov and L. N. Lipatov, Ref. 6. For an alternateand efficient way to carry out such leading-logarithmcalculations see N. Christ, B. Hasslacher, and

A. H. Mueller, Phys. Rev. D 6, 3543 (1972).C. G. Callan and D. J. Gross, Phys. Rev. Letters

22, 156 (1969).9P. V. Landshoff, J. C. Polkinghorne, and R. D.

Short, Nucl. Phys. B28, 225 (1971).R. L. Jaffe and C. H. Llewellyn Smith (unpublished).

~~Actually the cutoff choice z, z' ~ A2x 2 for thesimple one-loop diagrams generates results whichsatisfy both the analytic continuation and the reciprocalrelation. We are unaware of any physical interpreta-tion of this cutoff or the appropriate generalization tomultiloop diagrams.

Throughout this section we will use = to denotequantities calculated according to the leading-logarith-mic approximation of Sec. III.

R. P. Feynman (unpublished); J. D. Bjorken andE. Paschos, Phys. Rev. 185, 1975 (1969).

~4S. L. Adler, Phys. Rev. 143, 1144 (1966).

PHYSICAL REVIEW D VOLUME 7, NUMBER 6 15 MARCH 1973

Radiative Corrections as the Origin of Spontaneous Symmetry Breaking*

Sidney Coleman

and

Erick WeinbergLyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138

(Received 8 November 1972)

We investigate the possibility that radiative corrections may produce spontaneous symmetrybreakdown in theories for which the semiclassical (tree) approximation does not indicatesuch breakdown. The simplest model in which this phenomenon occurs is the electrodynamicsof massless scalar mesons. We find (for small coupling constants) that this theory moreclosely resembles the theory with an imaginary mass (the Abelian Higgs model) than one witha positive mass; spontaneous symmetry breaking occurs, and the theory becomes a theory ofa massive vector meson and a massive scalar meson. The scalar-to-vector mass ratio iscomputable as a power series in e, the electromagnetic coupling constant. We find, to low-est order, m (8)/m (V) = (3/2~){e /4m). We extend our analysis to non-Abelian gauge the-ories, and find qualitatively simQar results. Our methods are also applicable to theories inwhich the tree appraximation indicates the occurrence of spontaneous symmetry breakdown,but does not give complete information about its character. (This typically occurs when thescalar-meson part of the Lagrangian admits a greater symmetry group than the total Lagran-gian. ) We indicate how to use our methods in these cases.

I. INTRODUCTION

Massless scalar electrodynamics, the theory ofthe electromagnetic interactions of a mass-zerocharged scalar field, has had a bad name for along time now; the attempt to interpret this theoryconsistently has led to endless paradoxes. ' Inthis paper we describe how nature avoids theseparadoxes: Massless scalar electrodynamicsdoes not remain massless, nor does it remainelectrodynamics; both the scalar meson and the

photon acquire a mass as a result of radiative cor-rections.

The preceding statement may appear less orac-ular if we imbed massless scalar electrodynamicsin a larger family of theories with a mass. If wewrite the complex scalar fields in terms of tworeal fields, y, and y„ the Lagrange density is'

2 = --,'(F„.)'+ —,'(s„q, —eA „y,)'+ —,'(a „q,+ eA„y,)'

RA DIA TIVE CORRE C TIONS AS TH E ORIGIN OF. . . 1889

plus renormalization counterterms. (The quarticself-coupling is required for renormalizability,to cancel the logarithmic divergence that arisesin the amplitude for scalar Coulomb scattering. )It is widely believed that if p. is positive, this isa normal field theory. ' The particle spectrumconsists of a charged scalar particle, its antipar-ticle, and a massless photon. On the other hand,if p.

' is negative, the theory is unstable; spon-taneous symmetry breaking takes place. 4 Thescalar field acquires a vacuum expectation value,the Higgs phenomenon occurs, and at the end weare left with a massive neutral scalar meson anda massive neutral vector meson. '

Our assertion is that the massless theory islike the second case rather than the first; the onlydifference is that the driving mechanism of theinstability is not a negative mass term in the La-grangian, but certain effects of higher-order pro-cesses involving virtual photons.

Other interesting things happen in this theory.The Lagrange density (1.1), with vanishing mass,involves two free parameters, e and A. . Afterspontaneous breaking, the theory is still expressedin terms of two parameters; these are most con-veniently chosen to be e and the vacuum expecta-tion of the field, (rp). The surprising thing is thatwe have traded a dimensionless parameter, A., onwhich physical quantities can depend in a com-plicated way, for a dimensional one, (cp), onwhich physical quantities must depend in a trivialway, governed by dimensional analysis. We callthis phenomenon dimensional transmutation, andargue that it is a general feature of spontaneoussymmetry breaking in fully massless field theo-ries.

One consequence of dimensional transmutationis that all dimensionless quantities must be func-tions of e alone. In particular we find that thescalar-to-vector mass ratio is given by

m'(S) 3 e2

m'(V) 2v 4n '

plus higher-order corrections. We think that thisis the most interesting result of our analysis; webelieve it is the first time a small mass ratio(rather than a small mass difference) has beenfound as a natural consequence of the presence ofa small dimensionless coupling constant.

The outline of our paper is as follows: SectionII is a brief review of the formalism upon whichour work is based, the method of generatingfunctionals and effective potentials. Section IG isa detailed sample computation in the simplestcase, massless quartically self-interacting me-son theory. Here we show how to treat in our for-malism the infrared divergences which occur in

massless field theories. Section IV contains thecomputations for massless scalar electrody-namics, and a discussion of their reliability. Sec-tion V is an explanation of how to apply the re-normalization group of Gell-Mann and Low to ourproblem; we use this method to extend the domainof reliability of our earlier results. The readerwho just wants to get a general idea of what weare up to is strongly advised to skip Sec. V, whichis independent of nearly everything else in thepaper. Section VI extends our methods to theo-ries of multiplets of massless scalar mesons in-teracting with massless non-Abelian gauge fieldsand massless fermions. We obtain closed formsfor the lowest-order effects of virtual gauge par-ticles in the general case; these forms are valideven in theories in which there is a negative-massterm driving symmetry breakdown, and thereforemay be of use to readers who do not share ourfascination with fully massless theories. SectionVII lists our conclusions and makes some specula-tions.

Two appendixes deal with matters peripheral toour main line of argument. One shows that certaininteresting properties of our explicit lowest-ordercomputations persist in higher orders; the otherdiscusses the sense in which the massless theo-ries we study are the limits of massive theoriesas the mass goes to zero.

II. FUNCTIONAL METHODS ANDTHE EFFECTIVE POTENTIAL

In this section we review the functional methodsintroduced into quantum field theory by Schwinger,and extended to the study of spontaneous symmetrybreakdown by Zona-Lasinio. ' It is common tostudy spontaneous symmetry breakdown in the so-called semiclassical approximation, that is tosay, to search for minima of the potential, thenegative sum of all the nonderivative terms in theLagrange density. The method of Zona-Lasinioenables us to define a function, called the effec-tive potential, such that the minima of the effec-tive potential give, without any approximation,the true vacuum states of the theory. Further-more, there exists a diagrammatic expansion forthe effective potential, such that the first term inthis expansion reproduces the semiclassical ap-proximation. From our viewpoint, the virtue ofthis method is that it enables us to compute higher-order corrections while still retaining a greatadvantage of the semiclassical approximation:the ability to survey all possible vacua simultan-eously. Thus we are able to investigate the char-acter of the unstable but symmetric theory de-scribed by the same Lagrangian which governs a

1890 SIDNE Y COLEMAN AND E RICK WE IN BERG

given Goldstone-type asymmetric theory, even inthe presence of renormalization effects. In otherformalisms for computing higher-order correc-tions to spontaneously broken symmetries, suchinvestigations are much more difficult. In addi-tion, we are able to investigate cases in whichthe radiative corrections qualitatively change thestructure of the theory (e.g. , by turning minimain the effective potential into maxima); in otherformalisms, it is much more difficult to detectthe occurrence of such phenomena.

A. General Formalism and Definitions

&&I((z) (p+ Ip-) (2.2)

We can expand W in a functional Taylor series

1W=g — d'x ~ ~ d'x d"'(x ~ x )d(x)" d(» )n n!

(2.3)

It is well-known that the successive coefficientsin this series are the connected Green's functions;G "' is the sum of all connected Feynman diagramswith n external lines.

The classical field, cp„ is defined by

6Wpc(x) p )

(o'I «x) I og

(O' IO )(2.4)

The effective action I'((p, ), is defined by a func-tional Legendre transformation

For notational simplicity, we shall restrict ourselves in this section to the theory of a singlescalar field, y, whose dynamics are describedby a Lagrange density, Z(y, S„(p). The generaliza-tion to more complicated cases is trivial. Let usconsider the effect of adding to the Lagrange den-sity a linear coupling of y to an external source,J(x), a c-number function of space and time:

Z(rp, S„rp) Z+ d-(x)(p(x) .The connected generating functional, W(d), is de-fined in terms of the transition amplitude from thevacuum state in the far past to the vacuum statein the far future, in the presence of the sourceZ(x),

This equation will shortly turn out to be criticalin the study of spontaneous breakdown of sym-metry. The effective action may be expanded ina manner similar to that of (2.3):

I = d'x[-V((p, )+-,'(S„(p,)'Z(rp, )+"~ ] . (2.8)

V((p, ) —an ordinary function, not a functional —iscalled the effective potential. By comparing theexpansions (2.7) and (2.8), it is easy to see thatthe noh derivative of V is the sum of all 1PI graphswith n vanishing external momenta. In tree ap-proximation (that is to say, neglecting all dia-grams with closed loops), V is just the ordinarypotential, the negative sum of all nonderivativeterms in the Lagrange density.

The usual renormalization conditions of pertur-bation theory can be expressed in terms of thefunctions that occur in (2.8). For example, if wedefine the squared mass of the meson as the valueof the inverse propagator at z ro momentum, then

d Vdoc' 0

(2.9a)

Likewise, if we define the four-point function atzero external momenta to be the coupling con-stant, A, , then

d V4

dPc 0(2.9b)

Similarly, the standard condition for the normal-ization of the field becomes

I'= — d x," d »„I'"'(x,~ ~ x„)y,(x,) ~ y,(x„) .(n)

n

(2.'I)

It is possible to show that the successive coeffi-cients in this series are the II'I Green's functions(sometimes called proper vertices); I'i"' is thesum of all 1PI Feynman diagrams with n externallines. [A IPI (one-particle-irreducible) Feynmandiagram is a connected diagram that cannot bedisconnected by cutting a single internal line. Byconvention, 1PI diagrams are evaluated with no

propagators on the external lines. ] There is analternative way to expand the effective action:Instead of expanding in powers of y„we can ex-pand in powers of momentum (about the pointwhere all external momenta vanish). In positionspace, such an expansion looks like

( ))=)()'(J) —f&'*&(*)w(*)*(2.5) z(0)=I. (2.9c)

From this definition, it follows directly that

6I( )

=-d(x) . (2.6)

[The alert reader may wonder what happens tothese conditions in massless field theories, forwhich the Green's functions have logarithmic sin-gularities (infrared divergences) when the exter-

RA DIA TIVE CORRE C TIONS AS THE ORIGIN OF. . . 1891

nal momenta vanish. The answer is that theysurvive with only minor modifications, as we willshow in Sec. III.]

We are now ready to apply this apparatus to thestudy of spontaneous symmetry breaking. Let ussuppose our Lagrange density possesses an in-ternal symmetry; for simplicity, let us imagineit to be the transformation y- -p. Then, spon-taneous symmetry breaking occurs if the quantumfield cp develops a nonzero vacuum expectationvalue, even when the source Z(x) vanishes. FromEqs. (2.4) and (2.6), this occurs if

5r0

Gap,(2.10)

for some nonzero value of y, . Further, since weare typically only interested in cases where thevacuum expectation value is translationally invar-iant (that is to say, we are not interested in thespontaneous breakdown of momentum conserva-tion), we can simplify this to

dv =0d97c

(2.11)

for some nonzero value of q, . The value of y, forwhich the minimum occurs, which we denote by

(y), is the expectation value of p in the new (asym-metric) vacuum. It is easy to see that if thissituation is to be stable under small external per-turbations, the stationary point given by Eq. (2.11)must be a minimum of the effective potential.

To explore the properties of the spontaneouslybroken theory, we define a new quantum fieldwith vanishing vacuum expectation value,

(2.12a)

This generates a corresponding redefinition ofthe classical field,

expansion: first summing all diagrams with noclosed loops (tree graphs), then those with oneclosed loop, etc.' Of course, each stage in thisexpansion also involves an infinite summation,but, as we shall show in Sec. III, this summationis trivial.

Let us introduce a parameter a into our La-grange density, by defining

(2.13)

We shall now show that the loop expansion isequivalent to a power-series expansion in a. LetP be the power of a associated with any graph.Then it is easy to see that

P=I —V, (2.14)

where I is the number of internal lines in thegraph and V is the number of vertices. This isbecause the propagator, being the inverse of thedifferential operator occuring in the quadraticterms in g, carries a factor of a, while everyvertex carries a factor of a '. (Note that it is im-portant that we are dealing with 1PI graphs, forwhich there are no propagators attached to exter-nal lines. ) On the other hand, the number ofloops, L is given by

L=I- V+1 . (2.16)

This is because the number of loops in a diagramis equal to the number of independent integrationmomenta; every internal line contributes one in-tegration momentum, but every vertex contributesa 5 function that reduces the number of indepen-dent momenta by one, except for one 5 functionthat is left over for over-all energy-momentumconservation. Combining Eqs. (2.14) and (2.15),we find that

(2.12b) P =L-1, (2.16)

from which it immediately follows that the actualmass, coupling constant, etc. , are computablefrom equations exactly like the Eqs. (2.9), exceptthat the derivatives are evaluated at (y), ratherthan at zero.

B. The Loop Expansion

Knowledge of the effective potential is knowledgeof the structure of spontaneous symmetry break-down. Unfortunate1y, except for trivial models,we do not know the effective potential; to calculateit requires an infinite summation of Feynman dia-grams, a task beyond our computational abilities.Thus, it is important to know a sensible approxi-mation method for V. We shall now attempt toshow that one such sensible method is the loop

the desired result.The point of this analysis is not that the loop

expansion is a good approximation scheme becausea is a small parameter; indeed, a is equal to one.(However, it is certainly no worse than ordinaryperturbation theory for small coupling constants,since the set of graphs with n loops or less cer-tainly includes, as a subset, all graphs of nthorder or less in the coupling constants. ) Thepoint is, rather, since the loop expansion corre-sponds to expansion in a parameter that multipliesthe total Lagrange density, it is unaffected byshifts of fields [such as in Eq. (2.12)], and by theredefinition of the division of the Lagrangian intofree and interacting parts associated with suchshifts. The n-loop approximation to the effectivepotential thus preserves what we have identified

1892 SIDNE Y COLEMAN AND ERICK STEINBERG

as the main advantage of the effective potentialmethod; it enables us to survey all possible vacu-um states at once, and to compute higher-ordercorrections before deciding which vacuum thetheory finally picks.

III. A SAMPLE COMPUTATION AND THE PROBLEMOF INFRARED DIVERGENCES

In this section we shall present a detailed corn-putation of the one-loop approximation to the ef-fective potential in the simplest possible case,the theory of a massless, quartically self-inter-acting meson field. The Lagrange density forthis theory is

& = a(spp) —( p + a&(st%) kf!9' 4 ( C9'

(3.2a)

To next order (one-loop approximation), we havethe infinite series of polygon graphs shown in Fig.2, as well as the contributions from the mass andcoupling-constant counterterms. Thus we obtain

d'k ~ 1 2A.y, '(271) „-g 2n k + ie

(3.2b)

where B and C are, as usual in renormalizationtheory, only to be evaluated to lowest order in ourexpansion parameter, in this case the loop-count-ing parameter a.

(3.1)

where A, B, and C are the usual wave-function,mass, and coupling-constant renormalizationcounterterms to be determined self-consistently,order by order in the expansion, by imposing thedefinitions of the scale of the renormalized field,the renormalized mass, and the renormalizedcoupling constant (Not.e that a mass renormal-ization term is present, even though we are study-ing the massless theory; this is because the the-ory posesses no symmetry that would guaranteevanishing bare mass in the limit of vanishing re-normalized mass. '}

To lowest order (tree approximation) only onegraph contributes, shown in Fig. 1. Thus we ob-tain

Certain numerical factors in this expressionrequire further explanation:

(1) The over-all factor of i comes from the de-finition of W, Eq. (2.2).

(2) The factor of —,' in the numerator of the frac-

tion is a Bose statistics factor; interchange ofthe two external lines at the same vertex does notlead to a new graph, and therefore the 1/4! in thedefinition of the coupling is incompletely canceled.

(3) The 1/2nis acombinatoric factor; rotation orreflection of the n-sided polygon does not lead toa new contraction in the Wick expansion, andtherefore the 1/n! in Dyson's formula is incom-pletely canceled.

At first glance, the expression (3.2) seemshideously infrared divergent; each term in thesum is worse than the one before. However, con-siderable improvement is obtained if we sum theseries

9 c -B9c~

pc

d4k

(2.) '" "2i' (3.3)

where, in this expression, we have rotated theintegral into Euclidean space and dropped the i&.As we see, the apparent infrared divergence hasbeen turned into a logarithmic singularity at qr,equals zero. This is reminiscent of the phenom-enon we would have encountered had we attemptedto compute the radiative corrections to the propa-gator in this theory. As is well known, these be-have, to lowest nontrivial order, like P'lnP', hadwe been so foolish as to attempt to calculate thisfunction by computing its power-series expansionat p' equals zero, we would have found a sequenceof increasingly infrared-divergent terms, just asin Eq. (3.2}. The two situations are preciselyparallel; in momentum space, we can avoid theinfrared divergences by staying away from vanish-ing momentum; here, even though all our mo-menta vanish, we can avoid them by staying awayfrom vanishing y, .

[Equation (3.3) also has an apparent logarithmicsingularity in the coupling constant. However, aswe shall show immediately, this singularity isillusory; it is eaten by the renormalizationcounterterms. ]

Of course, the integral in Eq. (3.3) is still ultra-

FIG. 1. The no-loop approximation for the effectivepotential.

/ X q !l-

FIG. 2. The one-loop approximation for the effectivepotential.

RADIATIVE CORRECTIONS AS THE ORIGIN OF. . . 1893

violet-divergent; to evaluate it, we cut off the in-tegral at O' =A', and obtain

avoided the previous one, by replacing Eq. (2.9c)by

V= —,y, + —Bcp, +—,Cy,

64 +~ 256 2A ~ 2(3.4)

Z(M) =1 .Imposing Eq. (3.7}, we find

3x' xM 1132 2g 3

(3.8)

(3.9)

where we have thrown away terms that vanish asA' goes to infinity. We can now determine thevalue of the renormalization counterterms by im-posing the definitions of the renormalized massand coupling constant.

We want the renormalized mass to vanish; fol-lowing Eq. (2.9a), we find

d V

dt's o

This implies thatXA'B=-32r

(3.8)

(3 8)

Unfortunately, we cannot follow Eq. (2.9b) todefine the renormalized coupling constant; thefourth derivative of V at the origin does not exist,because of the logarithmic infrared singularity.Once again, this is reminiscent of the situationthat exists in momentum space for the same the-ory. There, the coupling constant can not be de-fined at the usual mass-shell symmetry point, be-cause it is on top of the logarithmic singularity inmomentum space; the standard strategy is to de-fine the coupling constant at some off-mass-shellposition in momentum space, away from the sin-gularity. Here, we should lose much of the sim-plicity of our computation if we attempted to ex-tend it away from vanishing momentum to do ex-actly the same thing; however, it is easy to do aparallel thing, and define the coupling constant ata point away from the singularity in classical-field space. That is to say, we define the renor-malized coupling constant by

d4V=A. ,

dVc(3.'I)

where M is some number with the dimensions of amass. We emphasize that M is completely arbi-trary, just as is the corresponding quantity in themomentum-space analysis; different choices fori' will lead to different definitions of the couplingconstant, different parametrizations of the theory,but any nonzero 4I is as good as any other. Al-though we do not need to know the wave-functionrenormalization counterterm, A, for the computa-tion we are now doing, we remark that the standardcondition for defining the scale of the field, Eq.(2.9c), is afflicted by the same infrared singular-ity as Eq. (2.9b). We avoid this difficulty as we

Putting all of this together, we find

4t&~ 256„& "M& 6(3.10)

d'V 3Z' M"dkp, 32m' M

Equation (3.10) may be rewritten as

(3.11)

V=—,y, '++ ln ', , ——+ O(X') . (3.12)

This is simply a reparametrization of the samefunction, to the order in which we are working;it is a change of definitions, not a change of phys-ics.

(The learned reader may remember that, in mo-mentum space, where similar arbitrary renorm-alization masses enter for massless field theo-ries, there exists a method of "improving" per-turbation theory to obtain an expansion in whichthe independence of the renormalization massesis exact to every order. This method is the re-normalization group of Ge11-Mann and Low. Itcan be transferred bodily to the formalism we areusing, and we will do so in Sec. V.)

This is our final expression for the effective po-tential in the one-loop approximation and com-pletes our sample computation.

Several comments are in order:(1) This is a renormalizable theory; so we

should expect all dependence on the cutoff to dis-appear from our final expression for V; this isindeed the case.

(2}As we remarked earlier, the violent infraredsingularities in the individual diagrams have be-come a singularity at the origin of classical-fieldspace. We show in Appendix A that this is trueto all orders in the loop expansion.

(3) More surprisingly, but as promised, the log-arithmic dependence on the coupling constant hasalso disappeared. In Appendix A we show that thisis also true to higher orders; when all the dust ofrenormalization has settled, the n-loop contribu-tion to V is simply proportional to A."".

(4) It is easy to check explicitly that the renor-malization mass, M, is indeed an arbitrary pa-rameter, with no effect on the physics of theproblem. If we pick a different mass, M', thenwe define a new coupling constant

1894 SIDNE Y COLEMAN AND ERICK WEINBERG

(5}Since the logarithm of a small number isnegative, it appears as though the one-loop cor-rections have turned the minimum at the origininto a maximum, and caused a new minimum toappear away from the origin- that is to say, thatthe one-loop corrections have generated spontane-ous symmetry breaking. Alas, appearances aredeceptive: The apparent new minimum occurs ata value of y, determined by

Aln, = ——m'+O(A. ) .(y)' 32 (3.13)

Since we expect higher orders to bring in higherpowers of X in(y, '/M'), the new minimum liesvery far outside the expected range of validity ofthe one-loop approximation, even for an arbi-trarily small coupling constant, and must be re-jected as an artifact of our approximation. As weshall see shortly, though, there do exist physical-ly interesting theories for which the effective po-tential has a very similar form, and to which thesame criticism can not be applied. Among themis massless scalar electrodynamics.

IV. MASSLESS SCALAR ELECTRODYNAMICS

+ —,'(s„y, + eA„(p,)' ——,(y, + cp, ')'

where

+ counterterms, (4.1)

F„„=B,A„- 8„&„. (4.2}

The calculation of the effective potential can besomewhat simplified if we realize that it can onlydepend on

2 — 2 29 c =0'zc +9 2c (4.3)

Thus, we need only compute graphs with all theexternal lines ~I(),'s. Furthermore, if we work in

We shall now apply the apparatus developed inthe last two sections to massless scalar electro-dynamics, the theory of a massless charged me-son minimally coupled to the electrodynamic field.We write the charged meson field in terms of tworeal fields, cp, and y„and write the Lagrangedensity as

', (E„„)'+5-(-s„rp, —eA„qr, )'

Landau gauge, where the photon propagator is

. gp v —kgkv/kp v k2+i (4.4)

then the contribution of any graph of the type shownin Fig. 3 vanishes. This is because the externalmomentum is zero; therefore the momentum ofthe internal meson is the same as that of the in-ternal photon, and vanishes when it is contractedwith the photon propagator.

Thus there are only three classes of graphs tocompute: those of the type shown in Fig. 2, witha y, running around the polygon, those of the sametype, but with a cp, running around the polygon,and those shown in Fig. 4, with a photon runningaround the polygon. Aside from trivial numericalfactors, all are of exactly the same structure asthe graphs considered in Sec. III. After somestraightforward computation, we obtain

4~ 9 c + 1152' 64' Wc M2 6

(4.5)

This function, like the one discussed in Sec. III,has a minimum away from the origin. Here, how-ever, the minimum need not be illusory. In Sec.III, the minimum arose from balancing a term oforder Z against a term of order )Pin(rp/M); thus,for small A, , it inevitably occurred at large1n(y/M), outside the expected domain of validityof our approximation. Here, even for an arbi-trarily small coupling constant, we can obtain aminimum by balancing a term of order A. againsta term of order e ln(y/M). Even though the sec-ond term formally arises in a higher order of ourexpansion than the first, there is no reason in theworld why A. cannot be of the same order of mag-nitude as e'. Indeed, this is what we should ex-pect if we think of the quartic meson self-interac-tion as being forced on us by renormalization, tocancel the divergence in Coulomb scattering, it-self of order e4.

We think this point is so important that we willrestate it in slightly different language: At firstglance, the central idea of this paper, that higher-order effects may qualitatively change the char-

FIG. 3. Some diagrams which do not contribute to theeffective potential in scalar quantum electrodynamics.The wiggly lines represent photons, the solid lines, spin-less mesons.

+ 0 ~ ~

FIG. 4. The photon contribution to the one-loop approx-imation for the effective potential.

RADIATIVE CORREC TIONS AS THE ORIGIN OF. . . 1895

aeter of a physical theory, seems to be in flatcontradiction with the basic idea of perturbationtheory, that higher-order effects are always smallcorrections to lower-order ones. This is not so.In the case under consideration, the first effectsof the electromagnetic coupling, the radiative cor-rections (Fig. 3) are pro forma of higher orderthan the effects of the direct quartic coupling (Fig.1). However, this says nothing about their actualrelative magnitudes, which can be comparable,even for very small e and A. .

Therefore, for the time being, we will restrictourselves to the case where X is indeed of ordere'. In Sec. V, we shalt. use the renormalizationgroup to show that all of our results extend with-out alteration to a more general case, arbitrary(but still small) e and X.

Under this assumption, the term of order A.' in

Eq. (4.5} is negligible compared to the otherterms, and we can drop it. Indeed, consistencydemands we drop it, since it is of the same orderof magnitude as the two-loop electromagnetic cor-rections, which we have not computed. Also,since the renormalization mass M is arbitrary, tothe order in which we are working [see remark(4) at the end of Sec. III], we might as well sim-plify our computational task by choosing it to bethe actual location of the minimum, &y&. Thuswe obtain

realize that dimensional transmutation is an in-evitable feature of spontaneous symmetry break-down in a massless field theory; it is nothing buta reflection of the simple fact that, for a fixedtheory, a change in the arbitrary renormalizationmass leads to a change in the numerical value ofthe dimensionless coupling constants.

Thus, we obtain our final expression for the ef-fective potential

(4.9)

This is parametrized in terms of e and &rp& alone;all reference to A. has disappeared. If we hadadapted a different definition of A. , the intermediateequations (4.5)-(4.8) would have changed, but Eq.(4.9) would have remained unaltered.

From this point on the analysis is the same asfor the familiar Abelian Higgs model. ' Aftershifting the field, the mass of the scalar mesonis given by

m'(S) = V "(&q»)

(4.10)

The would-be Goldstone boson combines with thephoton to make a massive vector meson; its massis given by the conventional formula

4 ~9c +64&29 c ln( &2 6

(4.6)m'(V) =e'&y&' .

Combining these two equations, we find

(4.11)

Since &y& is defined to be the minimum of V, we

deducem'(S) 3 e'm (V) 2w 4v' (4.12)

0= V (&y&)

11e'=

6 16m' &~&'' (4.7)

or,

33er (4.8)

Note that the redefinition of the coupling con-stant associated with shifting the fields has led toa determination of X in terms of e. Phrased moredramatically, after we have done the shift, thefinal value of A. is independent of the initial valueof A. . This smells suspiciously like some sort ofbootstrap or eigenvalue condition, but this is notthe case: We have not lost any free parameters;we started out with two (e and A.}, and we haveended up with two (e and &y&). What is slightlysurprising is that we have traded a dimensionlessparameter, A. , for a dimensional one, &rp&. Wecall this phenomenon dimensional transmutation.The reader who has followed our arguments should

the result announced in the Introduction. Weemphasize that Eqs. (4.8)-(4.11) are exact only inour approximation. Their right-hand sides havecorrections of higher order in e. Like all otherradiative corrections, these are unambiguouslycalculable in principle, although, as the numberof loops increases, they grow rapidly more dif-ficult to compute in practice. Also, if we wereto go to higher orders, we would have to defineparticle masses as the locations of poles in prop-agators rather than as the values of inversepropagators at zero momentum, as we have beendoing. This is because the first of these defini-tions is gauge-invariant„while the second, in gen-eral, is not. Fortunately, to the order in whichwe are working, the two definitions coincide.

To what extent do we expect these yet higher-order corrections to affect the qualitative behaviorwe have found'? Near &rp&, in(y, /&y&) is small,so we expect the effects of higher orders on theeffective potential in this region to be small,despite the fact that graphs with more loops in-

i896 SIDNE Y COLEMAN AND ERICK WEINBERG

3e'(y)'128e ' (4.13)

plus corrections which we expect to be small.Therefore, although graphs with more loops mightturn the maximum at the origin back into a mini-mum, they cannot turn it into an absolute mini-mum, and the asymmetric vacuum we have foundremains a local minimum definitely lower thanthat at the origin.

troduce higher powers of ln(rp, /(cp)) into the po-tential along with higher powers of e. For ex-tremely large values of cp„where the logarithmbecomes large, our entire expansion schemebreaks down. There may, perhaps, be new min-ima lurking in this region, but it is beyond thecapabilities of any perturbative computationalmethod to detect them.

Of course, logarithms grow large for small val-ues of their arguments as well as large, so ourexpansion scheme is as unreliable near the originas it is for large y, . Thus, although we shouldtrust the minimum we have found at (y), we shouldnot trust the maximum we have found at zero;graphs with more loops might well turn this intoa minimum again. However, no graph can changethe value of V at zero, which stays firmly fixedat zero. On the other hand,

BM

8 8M +P—+ny r'"' x, ~ ~ x„=o .

8$(5.2)

and

8 8 8M +P—+yy V=O

BM BX 'By,

(M +P—+yy, + 2y X=0.BM 8& By

(5.3)

(5 4)

It will turn out to be convenient, instead of work-ing with V, to work with the dimensionless func-tion

84@

Bgc(5.5)

Since dimensionless quantities, such as V ' andZ, can only depend on y, and M through the ratioy, /M, it is convenient to define

t =In(y, /M) .It is also useful to define

(5 6)

These are the familiar differential equations ofthe renormalization group. This should be nosurprise, for the argument given in the precedingparagraph is just the standard argument used toderive the renormalization group. '

For our purposes, though, it is better to ex-ploit Eq. (5.1) by applying it to the expansion (2.8).In this way we obtain

V. THE RENORMALIZATION GROUPP =P/(1 y), - (5.7)

For simplicity, let us begin by restricting our-selves to the theory of a self-interacting mesonfield, discussed in Sec. III; later we will extendour analysis to massless quantum electrodynam-ics. As we have seen, the explicit expression forthe effective action in this theory involves a mass,M. But, as we have emphasized, this mass isarbitrary; its only function is to define the re-normalized coupling constant, A. , through Eq.(3.7), and the scale of the renormalized field,through Eq. (3.8). Thus, a small change in M canalways be compensated for by an appropriatesmall change in A, and an appropriate small re-scaling of the field. This statement can be ex-pressed as an equation

8 8 4+ p—+y d'xy, (x), , v=0,~QgL&)a

(5.1)

for an appropriate choice of the coefficients P andy. (By dimensional analysis, P and y can dependonly on z.)

lf we apply Eq. (5.1) to the expansion of F interms of 1PI Green's functions, Eq. (2.7), we findthat

and

y = y/(1 —y) (5.8)

In terms of these quantities, Eqs. (5.3) and (5.4)become

48 —8——+p—+4y v'"(t ~)=0Bt (5 9)

(8 —8——+p —+2y z(t, z)=0.Bt BA.

(5.10)

z(o, z) =1. (s.i2)If we combine these with Eqs. (5.9) and (5.10), wefind that

1 8y= ——z(0, z), (s.i3)

This is the form of the renormalization-groupequations that we shall use.

In terms of the quantities we have defined, therenormalization conditions, (3.7) and (3.8), become

v"'(0, ~) =~ (s.ii)and

RADIATIVE CORRE C TIONS AS THE ORIGIN OF. . . 1897

and

p =—' v'"(0, ~) —4y ~ .at

(5.14)

Thus, if we know the derivatives that occur on theright-hand sides of these equations, we know Pand y. Of course, even if we are very industrious,we cannot know these derivatives exactly; at best,we can only know the first few terms in their loopexpansions. Thus, we can only know the first fewterms in the power-series expansions of P and y.Nevertheless, as we shall see, even this is usefulinformation.

For the moment, though, let us imagine we

know P and y exactly. Then it is easy to constructthe general solution of the equation

a —a——+P—+ny F(t, k) =0,at N

(5.15)

of which Eqs. (5.9) and (5.10}are special cases.Since this is of the same structure as the much-studied Eq. (5.2), we will merely describe the con-struction, and refer the reader to the literature"for its derivation. W'e begin by finding the func-tion A. '(t, X) defined as the solution of the ordinarydifferential equation

dt=P(&'), (5.16)

with the boundary condition

z'(O, x) =z . (5.1t)

Then, the general solution of Eq. (5.15) is of theform

tF(t, X) =f{X'(t,X))exp n dt y{X'(t, a)), (5.18)

0

where f is an arbitrary function.For the special cases of Z and V ', the renor-

maiization equations, (5.11) and (5.12), fix thearbitrary function f. Thus we obtain

tZ(t, x)=exp 2 dtyg'(t, ~))

0

and

v'"(t, ~) = ~'(t, ~)[z(t, ~)]' .

(5.19}

(5.20)

This is a remarkable result, and shows the powerof the renormalization group; Z and V ' are com-pletely determined, for all t, in terms of P andy, that is to say, in terms of their first deriva-tives at t=o.

We can go further, and use Eqs. (5.19) and (5.20)to compute how the renormalized coupling con-stant changes when we change the renormalizationmass. Let us suppose we change our renormaliza-

tion point from y, =M to

c= M

If y' is the new field, then

(s„y',)' = (e„y,)'Z {ln(M'/M), z),whence

y,' = y, z(ln(M'/M), A. ) '~

(5.21)

(5.22)

(5.23)

By definition, the new coupling constant, A, ', isgiven by

a4 Vgt4Pc pc=

=Z 'V 'i(in(M'/M), X)

= A. '{ln(M'/M), A.) . (5.24)

This elucidates the meaning of the function~'(t, x).

It is easy to extend all this analysis to a generalmassless renormalizable field theory: The re-normalization group equations will have one P-like term for every coupling constant and one y-like term for every field. Equation (5.16) will be-come a system of coupled ordinary differentialequations, one for each coupling constant. Other-wise, all will be as before; those functions in theeffective action that fix the renormalization con-ditions will be determined in terms of their de-rivatives at the renormalization point, and we willbe able to trace out the changes in the renormal-ized coupling constants as the renormalizationpoint changes.

So much for the dream world in which we know

P and y exactly. What about the real world, inwhich we know them only to lowest order 7 Let ussuppose that we construct an approximation forX'(A, , t) by integrating, from some small X, an ap-proximate version of Eq. (5.16), obtained by re-placing P by its one-loop approximation. Wewould expect this approximation to be reliable forthat range of t for which A. '(t, X) remains small,for, in this domain, the terms we have neglectedby truncating the power series for P are indeedsmall compared to the terms we have retained.However, if, for some range of t, X'(t, x) becomeslarge, the approximation becomes evident non-sense, for then the terms we have neglected arelarge compared to the terms we have retained. Ifwe are lucky, the approximation for X' will staysmall for a large range of t, and we will obtainan improved approximation —"improved" in thesense that it is valid over a larger range of t.(It is important to remember that the domain ofreliability of the one-loop approximation is de-termined not only by the condition that the cou-

1898 SIDNE Y COLEMAN AND ERICK WEINBE RG

pling constant be small, but also by the conditionthat the logarithmic factor t not be too large. Itis the latter restraint that the renormalizationgroup may allow us to escape, not the former. )

Let us apply these ideas to the one-loop approx-imation in self-interacting meson theory, dis-cussed in Sec. III. Differentiating Eq. (3.10), we

find that

gin by expanding the effective action and retainingonly those terms which define the renormalizationconstants:

r= d'x —Vq, --,'H q, F„., '

+ 2~(pc)[(sp&c —e&pc p2c)

(4) 3A, tV =A, + 62) (5.25)

+ (8p9'2 + e+p 9'x ) ]+ ' ' ' } ~

(5.32)

Z=1 . (5.26)

in the one-loop approximation. In Sec. III, we didnot compute the one-loop corrections to Z; how-ever, the computation is straightforward, and theresult is that, in the one-loop approximation,they vanish:

Note that the last group of terms has a commoncoefficient function, Z; this is a consequence ofgauge invariance. We impose the same renormal-ization conditions on Z and V as before; in addi-tion, we fix the scale of the renormalized electro-magnetic field by

Thus, in this approximation, H(M) =1 . (5.33)

y=O (5.27) We have computed all these functions in the one-loop approximation. The results are

and

The approximate differential equation is

dA. ' 3A. '

dt 16'2

Its solution is

(5.28)

(5.29) and

(4) 5X2 9e4

3e2Z=1+, t,8w

(5.34)

(5.35)

(5.36)

A.

1 —3gt/16v' ' (5.30)

From this we obtain the improved approximationfor the effective potential,

where t, as before, is In(cp, /M).The renormalization-group equation for this

theory is

8 8 8M— +P +P,—+y

V(4)1-3zt/16v' ' (5.31)

Note that this agrees with the one-loop approxima-tion in its expected domain of validity ((A( «1,(Xt~«l). However, Eq. (5.31) is valid in a muchlarger range of t, including all negative t, for inthis range A,

' remains small. In Sec. III, we foundthat the one-loop corrections turned the minimumat the origin of classical-field space into a maxi-mum, but we mistrusted this, because the regionnear the origin (large negative t) was outside thedomain of validity of the approximation. We nowsee that our mistrust was justified; Eq. (5.31) isa good approximation near the origin, and it pre-dicts a maximum, not a minimum. In Sec. III wealso found a phony minimum at large t; in placeof this, Eq. (5.31) has a pole at large t, but it isequally phony —this is a region of large A. ', whereour new approximation is as untrustworthy as ourold one.

%e now turn to scalar electrodynamics. We be-

+y cp„+ (p, I =0.

(5.37)

Applying this to Eg. (5.32), we deduce that

P =-ey (5.38)

8 —8 — 8—+P—+P —+2y Z t, p, e =0, (5.40)

and

(8 —8 — 8——+ P—+ P —+ 2y H(t, A. , e) = 0 .8t 8A. ' 8e

Here the barred coefficients are defined as before,

This is the reflection in our formalism of the oldresult that Zy Z2 We also find that

8 —8 — 8+ P—+ P,—+ 4y V'"(t, a, e) = 0, (5.39)

RA D IA TI V E CORR E C T ION S AS THE ORIGIN OF. . . 1899

by dividing by (1 —y).Evaluating these equations at the renormaliza-

tion point, we obtain the one-loop approximationto the coefficient functions:

and

y = 3e'/16 m',

p = (~sr' —3e"x+ 9e')/4v',

P, = e'/48m' .

(5.42)

(5.43)

(5.44)

Thus, the approximate differential equations we

must solve are

and

de = e "/48m',dt

(5.45)

dz'=( x" 3e"x'+9e")/4w' .

dt(5.46)

The first of these can be solved trivially, byquadrature:

2e1 —e't/24m' (5.47)

The second looks more fearsome; however, if wedefine

R = A. '/e",Eq. (5.46} becomes

(5.48)

e", =5fP-19ft+54.d e")

This can also be solved by quadrature:

A.' =~o e"[v'719 tan(~4719 Ine' + 8}+19],

(5.49)

(5.50)

where 8 is an integration constant, to be chosensuch that A, '=A. when e'=e.

We can hardly praise these solutions for theirbeauty. What about their utility? As t becomeslarge and positive, e' becomes large, so our ap-proximation certainly breaks down in this region,just as in the preceding case. As t becomeslarge and negative, e' stays small, but A.

' be-comes large, since the argument of the tangentinevitably passes through a multiple of n. Thusour approximation also breaks down in this re-gion; unlike the preceding case, we cannot usethe renormalization group to obtain an approxima-tion to the effective potential that is valid nearthe origin of classical-field space as well as nearthe renormalization point.

Nevertheless, all is not lost; we can still ob-tain useful information. For we can make theargument of the tangent change by 2w by varyingt so as to move e" through a very small range.

(By the crudest estimates, the range from —,'e' to2e' is more than sufficient. ) In the course of thisvariation, X' traverses the entire real axis. Ofcourse, we cannot trust the further reaches ofthis excursion, but we can trust it for the regionof smaLl A. '.

Thus, even for very small e, we can move A.

from any small value to any other small value bya change in the renormalization mass that doesnot change the order of magnitude of e. Thismeans that the restriction we imposed in Sec.IV- that X be of the order of magnitude of e'- isin fact no restriction at all. For, if A. is not ofthe order of magnitude of e', we can always makeit such, by appropriately changing the renormal-ization mass. Thus, the effective potential formassless scalar electrodynamics develops aminimum away from the origin, and spontaneoussymmetry breakdown occurs, for arbitrarysmall e and A. .

Vl. NON-ABELIAN GAUGE THEORIES

In this section we compute in closed form theone-loop corrections to the effective potential ina general massless renormalizable gauge-fieldtheory. The expressions we obtain involve tracesof functions of certain matrices constructed fromthe coupling constants of the theory. For anygiven theory, it is simple to compute these quan-tities, and it is then a straightforward calculusexercise to find the minima of V. Unfortunately,we do not have enough skill in the manipulationof arbitrary representations of arbitrary gaugegroups to be able to give even a qualitative dis-cussion of the structure of spontaneous symmetrybreaking in the general case; therefore we re-strict our detailed discussion to three specificmodels for which we have done explicit computa-tions. One is a Yang-Mills triplet coupled to ascalar isovector; another is the same tripletcoupled to a scalar isotensor; the third is amassless version of the Weinberg-Salam theoryof leptons. These display a range of interestingphenomena, but in their gross features they areall the same as massless scalar electrodynam-ics: Radiative corrections induce spontaneoussymmetry breaking, and all mass ratios are com-putable in terms of dimensionless coupling con-stants.

We should emphasize that all of our analysis ison the level of that of Sec. IV. In particular, thismeans that we will always assume that the quarticscalar self-couplings are of the order of magni-tude of e'. In Sec. V me mere able to remove thisrestriction from our study of massless scalarelectrodynamics with the aid of the renormaliza-

i900 SIDNE Y COLEMAN AND ERICK WEINBERG

tion group, but we have not yet extended thesemethods to the non-Abelian case.

A. Computation of the Effective Potential

The class of Lagrangians we will study involvesa set of real spinless boson fields, which we de-note by qP, a set of Dirac bispinor fields, whichwe denote by 4 ', and a set of real vector fields,which we denote by A'„. The index a runs overthe appropriate range in each case. Sometimeswe will find it convenient to assemble all the spin-less fields into a vector, which we denote by y,All these fields are massless, and the interac-tions between them are of renormalizable type:quartic self-interactions of the spinless bosons,Yukawa-type boson-fermion couplings (not nec-essarily parity-conserving}, and minimal gauge-invariant couplings of the vector fields.

If we quantize the theory in Landau gauge, theonly graphs we need consider are polygon graphswith either spinless mesons (Fig. 2), gauge me-sons (Fig. 4), or fermions (Fig. 5) runningaround the loops. Note that in the fermion sum,only graphs with an even number of external linesoccur; this is because the trace of an odd numberof y matrices vanishes. These are not the onlyinternal lines in a non-Abelian gauge theory;there are also the famous ghost fields. " How-ever, in Landau gauge, these have no direct cou-pling to the spinless fields, and thus need not beconsidered for our computation. (Of course, wewould have to take account of ghost fields if wewere to compute terms in the effective actionthat depend on classical gauge fields, or if wewere to compute two-loop corrections to the ef-fective potential, or even if we were to work ina less well-chosen gauge. )

Thus, the one-loop approximation to the effec-tive potential can be written as a sum of terms:

V(rp, ) = V, + V, + Vf + V, + V, ,

where V, is the zero-loop approximation, V, , V&,and V~ are the contributions from spinless-me-son, fermion, and gauge-field loops, and V, isthe contribution from coupling-constant-renormal-ization counterterms. V, is a quartic polynomialdetermined in terms of the other four terms oncewe have stated our renormalization conditions.

To compute V, , we need to know the vertex that

occurs at the corner of a polygon graph, connect-ing an internal meson of type a with one of type b.This vertex, shown in Fig. 6, is given by

a'y,W, (i4,) = (6.2)

Vis a real symmetric matrix and a quadraticfunction of y, . To compute the n-sided polygon,we must sum over all possible internal mesons.This corresponds to multiplying the W matricesaround the loop and then taking their trace. Thus,copying directly from Sec. III, we find that

V, = &4, TrLW'(y, ) lnW(y, }j, (6.3)

plus cutoff-dependent quartic terms, which weabsorb in V, .

The contribution of the gauge-field loops maybe computed in a similar way. We define a ma-trix M'(y) in terms of the nonderivative couplingof gauge fields to spinless mesons:

2 = ~ ~ ~ + -'Q M'(q )A A" + ~ ~ ~ .ab

(5.4)

M'.n= r.zn(T;e T'~m} (6.5)

where T, is the representation of the ath infini-tesimal transformation of the gauge group, and

g, is the coupling constant of the associated gaugefield. (If the gauge group is simple, all the g'sare equal; otherwise, this is not necessarily thecase. ) Following the same reasoning as before,

Wob (Pc)

&c

mob (&P~)

0 b

M~ob (&c)

Like O', M' is a real symmetric matrix and aquadratic function of y. We call this matrix M'because, to first nonvanishing order M', (g)) isthe square of the vector-meson mass matrix. Be-cause the vector mesons are minimally coupled,we can write

+ rs 1& +

FIG. 5. The fermion contribution to the effective po-tential. The directed lines represent fermions.

FIG. 6. The vertices which are needed to compute theone-loop appraximation to V(y, ) in a general masslessgauge theory. These are to be inserted in the vertices ofthe polygon graphs of Figs. 2, 4, and 5.

RADIATIVE CORREC TIONS AS THE ORIGIN OF. . . 1901

we find that B. First Model: Yang-Mills Fields and a Scalar Isovector

V = 2Tr[M (cp, )lnM (y, )], (6.6)

maa Aaa+ iBa~ y5 . (6 6)

(We use a Hermitian y, .) A and B are Hermitianmatrices and linear functions of y. To first non-vanishing order, m((y}) is the fermion mass ma-trix, whence its name. We can exploit the factthat only graphs with an even number of internalfermions contribute to the sum by grouping termsin the matrix product pairwise:

1 1 1~ ~ ~ pg p7g ~ ~ ~ ~ ~ ~ ppg pg 0 ~ ~ (6.9)

The sum is then seen to be the same as in theother cases, except, of course, for the standardminus sign for fermion loops, "and we obtain

V~ ==,Tr[(m m (y,})'ln(m m" (y,))],

plus quartic terms, which we absorb in V, . Notethat in this equation the trace runs over Diracindices as well as internal indices.

The fermion contribution to the effective poten-tial has opposite sign to the other terms; thismeans that the effect of fermion closed loops canmake shallower, or even eliminate altogether,minima caused by the other terms. However, forthe only model with fermions we shall consider(the Weinberg-Salam theory of leptons), this ef-fect is completely negligible, because, as we seefrom Eqs. (6.6) and (6.10), Vf is smaller than V,by a factor on the order of the fourth power of thefermion-to-vector-meson mass ratio. For thistheory, this ratio is so small that the effect offermion loops is tiny even when compared withthat of two-loop electromagnetic corrections.We should bear in mind, though, that if we builda model in which the Yukawa couplings are suchthat superheavy fermions appear, with massescomparable to the vector-meson masses, thiseffect can be important.

plus quartic terms, which we absorb in V, . Theextra factor of three in this equation comes fromthe trace of the numerator of the Landau-gaugepropagator.

The contribution of the fermion loops may becomputed in a similar way. We define a matrixm(p) in terms of the Yukawa coupling:

2 = ~ ~ ~ —Q 4', m, , (rp)+, + ~ ~ ~ .ab

The matrix m is a matrix in Dirac space as wellas in internal space:

We now turn to our first model, the theory of atriplet of SU(2) gauge fields minimally coupled toa set of scalar mesons transforming accordingto the vector representation of the group. We as-semble the scalar fields into a vector in the stan-dard way. By SU(2) invariance, the effective po-tential can depend only on the length of this vec-tor; therefore it suffices to compute it in the casewhen only the third component is nonzero:

(C let 'P2cs 93c) ( 1 1 9 c) (6.11)

We begin by computing V, , since, from our ex-perience with the Abelian theory, we expect thisto be the dominant term in the effective potential.From Eq. (6.5}we find that

2 (6.12)

where e is the gauge-field coupling constant; allother entries vanish. Thus we find, from Eq.(6.6), that

3eV =32, g, in@,

w(6.13}

This is, except for the coefficient, of exactly thesame form as the corresponding term in masslessscalar electrodynamics, Eq. (4.5). Thus, thediscussion of its minimum, and of the effects ofshifting the renormalization point to the minimum,is the same as that given in Sec. III.

Hence, spontaneous symmetry breakdown oc-curs, and the field develops a vacuum expectationvalue. The vacuum expectation value is neces-sarily invariant under a U(1) subgroup of theoriginal SU(2) group, which we identify with elec-tric charge. The gauge field associated with thissubgroup (the photon) remains massless, whilethe two other (charged) gauge fields eat the corre-sponding scalar mesons and acquire a mass. Theratio of the masses of the charged vector mesonsto that of the remaining (neutral) scalar meson isgiven by

m'(S) 2 e'm'(V) v 4v ' (6.14)

and the quartic scalar coupling constant is givenby

33e'A, =

4 2 (6.i6)

Because the potential is twice as great as that ofthe Abelian theory, these formulas differ by a fac-tor of two from the corresponding ones we foundin Sec. IV. Since in this model a massless photonremains after symmetry breakdown, it is not

1902 SIDNE Y COLEMAN AND ERICK WEINBERG

C. Second Model: Yang-Mills Fields and

a Scalar Isotensor

Our second model is the same as the first, ex-cept that the scalar fields transform according tothe five-dimensional (tensor) representation ofSU(2)." We represent these in the standard wayas a traceless symmetric matrix. In computingthe effective potential, we can, with no loss ofgenerality, choose this matrix to be diagonal:

0 0

0 b 0

0 0 c)(6.16)

where

a+ b+ c=0 . (6.17)

foolish to identify e with the actual electromagnet-ic coupling constant; if we do this, Eq. (6.14) pre-dicts that the charged vector mesons are abouttwelve times more massive than the neutral scalarmeson.

(6.18)Tr y' = 2(Tr rp')' .Thus there is only one term in Vo.

[Digression: This has an amusing consequence,for even if we had destabilized the vacuum by hand,

by introducing a negative mass term in the Lagran-gian, V, would still possess a larger symmetrythan SU(2}, to wit, SO(5)." Thus, even in thiscase, if we analyzed the theory in the semiclassi-cal (no loop) approximation, we would be in troublefor two reasons: (1} We would only determine a+ b'+ c' at the minimum, and would have no ideahow to fix the other SU(2) invariant, abc; (2) evenif we picked a vacuum expectation value arbitrari-ly from this over-rich set, we would find that thedoubly charged scalars were spurious Goldstonebosons, and had zero mass. To pick the rightvacuum, and to obtain a nonzero mass for thedoubly charged scalars, it would be necessary tocompute the effects of gauge-field loops. Moral:The techniques developed in this paper may be ofuse even to someone who thinks that all this talkof massless scalars is nonsense. ]

To compute V, we begin by calculating the vec-tor-meson mass-squared matrix:

We would be especially happy if the minimum ofthe effective potential corresponded to a situationfor which

((b- c)'2 2+2I 0 (c- a)' 0

(a- b)')

(6.20)

1a=b=-&c . (6.18) Hence,

For, in this case, there would remain an unbrokenU(1) subgroup of the original SU(2) group, whichwe could identify with electric charge, and as-sociated with this subgroup, a massless vectormeson, which we could identify with the photon.The other particles remaining in the theory wouldbe a pair of charged vector mesons, a neutralscalar, and a pair of doubly charged scalars.

We emphasize that there is nothing in the sym-metry properties of this theory that guaranteesthat the minimum of V will obey Eq. (6.18). Thus,in contrast to the preceding model, if a masslessphoton emerges, it will be as a consequence ofdetailed dynamics, not just of trivial group theory.

We now turn to the actual computation of the ef-fective potential. We begin by investigating the re-strictions placed by SU(2) symmetry upon thezeroth-order potential V,. In contrast to the pre-vious case, a cubic interaction, Try', is allowedby the group. To simplify our problem, we ex-clude this by adding to our continuous symmetrygroup the discrete transformation y--p. Thereare apparently two possible quartic couplings,Try' and (Try')'; however, these are related bythe tracelessness of y'.

4

V~=6 2(a —b) ln[(a —b)'/p']

+ cyclic permutations, (6.21)

V =Vo+Vg3e', (a- b)' 1, (a-b)' ln

16@2

+ cyclic permutations

From this expression it is evident that

a=b = 3p, ,2C=-3' ~

(6.22)

(6.23)

is a minimum of V, for it is a minimum of two of

where p' is an arbitrary renormalization mass.Just as before, a change in p,

' can always be com-pensated for by a change in the quartic couplingconstant, A. . Just as in Sec. IV, we now assumethat X is of the order of magnitude of e", then V,is small compared with V, , and we can neglect it.Further, we can always adjust p.

' to give the quar-tic coupling any desired value; we choose to dothis such that

RADIATIVE CORREC TIONS AS THE ORIGIN OF. . . 1903

the three terms in Eq. (6.22) and a maximum (butwith vanishing second derivative) of the third. Itis possible to show that this is the only minimumof V (other than those obtained from it by trivialpermutations), but we will not give the proof here.Note that this is of the form (6.18); thus, electriccharge remains a manifest symmetry and there isa massless photon.

The mass of the charged vector follows directlyfrom (6.20):

sons form a complex doublet; the Lagrangian con-tains a negative mass term for this doublet, whichforces spontaneous symmetry breaking. Becauseof the symmetry, we can, with no loss of general-ity, arrange that only one component of the doublethas nonzero vacuum expectation value, and we canchoose this value, which we denote by (y), to bereal. After symmetry breaking, the vector me-sons are a massless photon, with coupling

m'(V) = 2e'p, ' . (6.24) g +g' (6.30)

Fortunately, to compute the scalar masses, weneed only consider diagonal perturbations in Eq.(6.16):

two massive charged mesons, 8", identified withthe intermediate bosons of the weak interactions,with masses

S D3

S D3 vY

(6.25)

m'(W') =-,' g'(q)',and a massive neutral meson, Z, with mass

m'@) =-'(g'+g")(c)' .

(6.31)

(6.32)

2 p. 2SC=-3 vY'

and

m (S) =9e g'/2v

m (D) =3e p. /2v

(6.26)

(6.2V)

Whence,

m'(S)m'(D)

and

(6.28)

where S is the neutral scalar and D is a Hermitiancombination of the two charged scalars. [Thesquare roots are to give these fields the right nor-malization in the kinetic energy, —,

' Tr(e„ye y).]Expanding (6.22), we find

There is a.iso a neutral scalar meson, S, the onlysurviving member of the original complex doublet;its mass is not determined in terms of the othermasses in the theory. Also, of course, there arethe leptons, but for our immediate purposes theseare of no interest, since they are just the usualelectron, muon, and neutrinos, and by the remarkat the end of Sec. VIA, they make a negligible con-tribution to the effective potential.

We want to investigate what new information weobtain if we assume that, before symmetry break-down, the scalar masses are zero, and that sym-metry breakdown is driven by radiative correc-tions. Playing the game just as before, we seethat the important thing is to compute V, . For-tunately, the matrix M'is given to us, in diagonalform, by Eqs. (6.31) and (6.32); all we have to dois to substitute y, for (y). Thus we obtain

m'(D) 3 e'm'(V) v 4~ (6.29) V, =1024, [2g'+ (g'+g")'] y, 'In@, ', (6.33)

3

[We should note that had we put a negative massterm in the theory, as discussed in the precedingdigression, Eq. (6.29) would still have survived,because the no-loop approximation would still havemade no contribution to the mass of the doublycharged scalars. ]

D. Weinberg-Salam Theory of Leptons

The model of leptons proposed by Weinberg andSalarn" has been so much discussed in the recentlite "ature that, rather than explaining it in detail,we shall just remind the reader of some of itssalient features. The gauge group is SU(2)xU(1);the coupling constant of the triplet gauge fields iscalled g, that of the singlet g'. The spinless me-

plus quartic terms, which are of no importance.Comparing this with our previous work, we im-mediately find

m'(S) =,[2g'+ (g'+g")']( y)'3

, [2g'm'(W')+(g'+g")m'(Z)] .3

(6.34)

The important observation is that this puny re-lation is al/ the new information we obtain. Thisis what we would expect from simple parametercounting. We have added one new assumption-vanishing scalar mass before symmetry break-down- and have obtained one new relation- a pre-

1904 SIDNE Y COLE MAN AND ERICK WEINBE RG

diction of the scalar mass after symmetry break-down. This is obviously what will happen if we

apply our ideas to any of the current horde ofgauge theories of the weak and electromagneticinteractions. Since these theories typically eithercontradict experiment whatever the scalar masses,or have so many free parameters that they can bemade to fit experiment whatever the scalar mass-es, we do not believe that the ideas developed inthis paper will make a significant contribution tocurrent weak-interaction phenomenology. Never-theless, some day the grail may be found; theo-rists may discover a simple and compelling modelwith only a small number of adjustable parametersthat fits experimental reality. If and when thishappens, it will be interesting to see what con-straints are put on the parameters of this modelby the condition that the theory be fully massless.

VII. CONCLUSIONS AND A SPECULATION

(1) We hope that the reader who has followed ourarguments will now agree with our basic conten-tions: that radiative corrections can be the domi-nant driving force of spontaneous symmetry break-ing; that, at least for weak coupling constants,this possibility can be investigated in a systematicway; and that, in particular, massless scalarelectrodynamics undergoes radiatively inducedspontaneous symmetry breakdown.

We are very much aware that we are exploringunconventional ideas and that there may be somebasic flaw in our whole approach which we havebeen too stupid to see. Barring this possibility,though, we believe we have done an honest com-putation by the standards of perturbation theory,being careful of our approximations and not dis-carding terms obviously large compared to thoseretained. We therefore feel that our computationshave the same a priori plausibility as any otherperturbative computation in a renormalizable fieldtheory with weak coupling constants, such as, forexample, the computation of the anomalous mag-netic moment of the electron.

(2) Since our work does lead to the determina-tion of some coupling constants in terms of others[for example, Eq. (4.8)], it is natural to ask whatconnection it has with the generalized eigenvaluecondition, "which also seeks to determine couplingconstants by demanding that, in the real world,the P coefficients in the renormalization-groupequations vanish. The answer is that the two ideascannot both be true. For, if the P's vanish, so dothe P 's, by Eq. (5.7). If the P 's vanish, the effec-tive potential is a simple power. If the effectivepotential is a simple power, then its only possibleminimum is at the origin.

(~) As we explained at the end of Sec. VI, we donot believe that our ideas will be of any immediateuse in the currently popular gauge theories of weakand electromagnetic interactions. These theoriestypically contain so many arbitrary parametersthat the additional constraints imposed by demand-ing that the scalar mesons be massless beforesymmetry breakdown offer only a slight improve-ment. Nevertheless, if a model is found that sat-isfies experiment and has only a small number offree parameters, these constraints may play animportant role. Meanwhile, as shown by the modelof VI B, some of our computational techniquesmay be of use even in theories in which symmetrybreakdown is driven by a negative mass term, tocompute the masses of spurious Goldstone bosons.

(4) The speculation: A bold way of interpretingour results would be to say that nature abhorsmassless particles with long-range interactionsbetween them, and escapes these abhorrent sys-tems by the Goldstone-Higgs mechanism, whichgives the particles a mass, or makes the interac-tions short-range, or both. If we accept thisstatement, then we should expect symmetry break-down even in theories mthout fundamental spinlessbosons, such as massless spinor electrodynamics,or the theory of non-Abelian gauge fields coupledto massless fermions.

Unfortunately, we know of no computationalscheme, analogous to the one we have used, tostudy whether this in fact happens, even for weakcoupling constants. The trouble is that the objectwhich we would expect to develop a vacuum ex-pectation value is a composite field, like 0 4. Inprinciple, there is no obstacle to extending theformalism of Sec. II to such objects'. the formalmachinery of the functional Legendre transforma-tion does not depend on whether we couple our ex-ternal sources to fundamental or composite fields.In practice, though, we have not been able to finda sensible approximation method on this idea.

Nevertheless, we are speculating, so let us as-sume spontaneous symmetry breakdown does takeplace in such theories. Then we are led to a re-markable situation: As we have seen, when sym-metry breakdown occurs in a fully massless fieldtheory, so does dimensional transmutation; onedimensionless coupling constant disappears, to bereplaced by a mass parameter. But spinor electro-dynamics (or, for that matter, any theory of gaugefields and fermions based on a simple Lie group)has only one dimensionless coupling constant tobegin with. Thus, after symmetry breakdown, weare left with no free dimensionless coupling con-stants.

Thus ue are led to a Program for comPuting thefine-structure constant. (We emphasize that this

RADIATIVE CORRECTIONS AS THE ORIGIN OF. . . 1905

is not the same as the eigenvalue condition [see(2), above]. ) To propose such a program is an

act of hubris; but we can moderate our ambitionand imagine working in a theory in which the gauge

group has two simple factors, and thus there aretwo coupling constants. In this case one would

survive, and the fine structure constant would stillbe a free parameter, but all mass ratios (such asthe muon-electron ratio) could be computed in

terms of it.Even this last speculation is certainly very am-

bitious, and therefore most likely false. Neverthe-

less, it would be very pleasant to be able to in-vestigate the question in a quantitative way. Thuswe feel that the outstanding theoretical challengeposed by our work is to extend our methods to the-ories without fundamental spinless fields.

ACKNOWLEDGMENTS

One of us (S.C.) would like to express his thanksto the Aspen Institute for Physics and to the Stan-ford Linear Accelerator Center for their hospital-ity while this work was being completed. We areindebted to James Bjorken, Curtis Callan, RogerDashen, Howard Georgi, Sheldon Glashow, DavidGross, Benjamin Lee, and Steven Weinberg forinstructive comments. We are especially gratefulto H. David Politzer, who has followed this workfrom its beginnings and who has shed light in manydark places.

APPENDIX A. MORE LOOPS

1. The Disappearance of Infrared Divergences

In Sec. III we were able to show that the infiniteset of one-loop graphs for the effective potentialcould be summed to give a single integral. Weshall now extend this result, and show that the sumof all n-loop graphs can be expressed as a finitesum of n-tuple integrals. Just as in the one-loopcase, each of these integrals will be obtained bysumming an infinite series of graphs, each ofwhich is infrared-divergent; however, the onlyrelic of the infrared divergences remaining in thesum will be a singularity at the origin of classicalfield space.

For simplicity, we begin by restricting ourselvesto the self-interacting meson theory of Sec. III.For a general graph contributing to the effectivepotential, we define a type-n vertex to be one withn internal lines (and therefore, 4n external lines)attached to it, and denote the number of type-nvertices in the graph by V„. For a 1PI graph, allvertices are either type-two, type-three, or type-four. The total number of vertices is given by

V = V2+ V3+ V4 . (A1)

Since each internal line has two ends, and eachend terminates on a vertex, the total number ofinternal lines is given by

2I =2V, + 3V, + 4V, .Thus, the number of loops in the graph is

L=I —V+1

—~V +V +1.

(A2)

(As)

(A4)

This rule may readily be extended to masslessscalar electrodynamics; the only difference is

FIG. 7. Above: The only two-loop prototype graphsin self-interacting meson theory. These are V4

——1,VS=0, and V4=0, V3=2. Below: Two of the infiniteseries of two-loop graphs obtained from these by insert-ing type-two vertices on the internal lines.

Thus, for any fixed L, there exist only a finitenumber of graphs with L loops and no type-twovertices. We shall refer to these as prototypegraphs. All other L-loop graphs are generatedfrom the prototype graphs by sticking an arbitrarynumber of type-two vertices onto the internal linesof the prototype graphs. Figure 7 shows this pro-cess for the two possible two-loop prototypegraphs.

From this viewpoint, the one-loop graphs of Sec.III are a degenerate case; for them the prototypegraph is a graph with no vertices, a simple circle.This is responsible for the peculiar cyclic sym-metry of these graphs, which led to the factor of1/n in Eq. (3.2b). For all other cases, the ends ofthe internal lines in the prototype graph are pinneddown to prototype vertices. Therefore, the inser-tion of type-two vertices does not introduce anynew symmetry into the graph, and it is trivial tosum up the result of all such insertions, since wehave an independent geometric series for each in-ternal prototype line. Thus we obtain the followingComputational Rule: To evaluate the sum of alln-loop graphs for n greater than one, compute thesum of all n-loop prototype graphs, but make thefollowing substitution for every internal propaga-tor:

i906 SIDNE Y COLEMAN AND ERICK WEINBERG

that for internal photon lines, the substitution is

. gp u—

krak u/k . gg u —krak&, /k'Z 2k'+ i~ k -ey, +jr (A5)

In both cases we see that the infrared divergencesin the individual graphs become, in the sum, asingUlarity at the origin of classical-field space.

Until now, we have only discussed graphs forwhich all external lines carry zero momentum,but we may sometimes wish to compute graphs forwhich some of the external momenta are nonzero[for example, to compute Z(cp, )]. The computa-tional rule here is the same as before, except thatthe class of prototype graphs will include somegraphs with type-two vertices, those for whichone or both of the external momenta attached tothe type-two vertices are nonzero. The reasonfor this is simple: In counting combinations toproduce a geometric series, the nonzero momen-tum serves to distinguish these type-two verticesfrom the ruck of the others.

2. The Disappearance of Logarithmsof Coupling Constants

(Ae)

If we express the propagator in terms of the newvariable, we find that

1k ——Ap

(A7)

In addition, for every integration momentum, wehave

de =g2d4y' . (AS)

Please note that this rescaling of momenta doesnot affect our renormalization procedure, sinceour prescription is to subtract graphs at fixed cp, .Had we followed the more usual renormalizationprocedure, and subtracted at some fixed Euclideanmomentum, the rescaling would change our re-normalization conditions, and would not be legiti-mate.

We can now add up all the powers of the couplingconstant associated with a given prototype graph.In addition to the powers explicitly displayed in thepreceding two equations, there is, of course, an

In Sec. III we showed that, after renormaliza-tion, the sum of all the one-loop graphs for theeffective potential was proportional to ~'. We shallnow extend this result to many-loop graphs in self-interacting meson theory.

We shall use the analysis in terms of prototypegraphs, explained above. In every prototypegraph let us define a new momentum variable foreach loop,

u =X'~20'

p ~l/2p p (A10)

Then, reasoning as before, we find a result of theform

y, 'g(P' /M, rp, /M ) +'= y, g(p/A, M, y, '/M')A

(A11)

where g is a function which depends on the proto-type graph under consideration. Here the loga-rithms of the coupling constant have not disap-peared, but are associated with the logarithmicdependence of the function on p'.

This has the interesting consequence that if wecompute Z by differentiating the above functionwith respect to p' at the origin, we find that theL-loop contribution is proportional to A~. This isa natural arrangement of powers from the view-point of the renormalization group; it implies thatin the L-loop approximation, both terms in theformula for P, Eq. (5.14), are proportional to A.~".

It is much more difficult to perform a similaranalysis for massless scalar electrodynamics, be-cause there is no rescaling of momenta that willsimultaneously eliminate the coupling constantsfrom the denominators of both the meson propaga-tor, Eq. (A4), and the photon propagator, Eq.(A5). Indeed, at this time, we do not know whetherthe n-loop contributions to the effective potentialare simple polynomials in e and A. , or whetherthey contain logarithms of the coupling constants.

Nevertheless, using an ingenious trick suggestedby Politzer, "we can investigate the dependenceon e of the L-loop contribution to the effective po-tential, after we have determined A. as a functionof e by shifting the renormalization point to theminimum of V. Politzer's suggestion is to para-

explicit factor of A. for every vertex. Thus, thecontribution to the effective potential is of the form

g. 'f(y. /M)(&)"" '=q. 'f(q, /M)(&)"',

(A9)

where M is the renormalization mass and f is afunction depending on the graph under considera-tion.

The situation is somewhat different if we con-sider graphs in which some external lines carrynonzero momenta. For example, let us considersumming a set of L-loop graphs for which one ex-ternal line carries a momentum p, another carriesa momentum -p, and all others carry momentumzero. That is to say, let us consider computingthe corrections to the two-point function in a fixedclassical field. In this case, to preserve momen-tum conservation, we must also rescale the exter-nal momentum,

RADIATIVE CORRECTIONS AS THE ORIGIN OF. . . 1907

metrize the expansion not in terms of A. , as wehave been doing, but in terms of another parame-ter, A. „defined by

6 ~V3iV 8+

(A12)

This is certainly as good (or as bad) a definitionof the renormalized coupling constant as our oldone; both parameters equal the bare values of A.

in zero-loop approximation, which is all that isneeded to define an iterative renormalization pro-cedure. In any event, X (or A, ) exists only to beeliminated in favor of e at the end of the computa-tion; which parameter we use in the intermediatestages should not be important. The advantage of

is that it is easy to write in closed form theequation that eliminates it, for, if we shift the re-normalization point to a minimum of V, Eq. (A12)becomes

z, =0. (A13)

APPENDIX B. MASSIVE SCALAR MESONS

In the body of this paper we restricted ourselves,except for brief digressions, to theories in whichthe spinless fields were massless, that is to say,in which the renormalization conditions forcedV"(0) to vanish. In this appendix we discuss theextension of our techniques to theories in whichthe scalar mesons have masses, either real orimaginary, that is to say, in which the renormal-ization conditions force V"(0) to be some fixednonzero quantity, either positive or negative.

1. Self-Interacting Mesons with Positive Mass

Thus, all graphs that involve the quartic-mesonself-coupling disappear, and it is easy to constructa rescaling argument to determine the e depen-dence of the loop expansion. We shall not give thedetails here, but merely state that the results arethe same as those in the previous case, exceptthat A, is replaced by e'. Thus, the L-loop contri-bution to the effective potential is proportional to(e') ", etc.

where, as before, B and C are renormalizationcounterterms, and the integral is over Euclideanmomentum space. Since in this region (k'+ p, ')never vanishes, we can drop the i&. We determineB and C just as in Sec. III, by imposing the re-normalization conditions

and

d V

d9c q =oC

d Vt4dy,

(B2)

The expression for V we obtain in this way issomewhat cumbersome. For simplicity, we willgive it explicity only in the regime of interest forthe study of the zero-mass limit, p, «M:

A.V=-,' 'y, '+—,p, '+, (p'+-'Py, ')'ln 1+

+ 2~0 Pc S~ gc2 2 3 2 4

2+, (p'+-,'xy, ')' In 1+7T

2I 2 2 ~ 2 4 1 2 4 ~P+2~v y, —24~ y. +4~ y. ln

phd

(B4)

As the mass goes to zero, this clearly goessmoothly into the corresponding expression forthe massless theory, Eq. (3.10).

Of course, since the massive theory is free ofinfrared divergences, we could choose M to bezero. This is obviously a stupid thing to do if oneis studying the zero-mass limit; we could hardlyexpect a smooth limit as p, goes to zero if weabruptly change the definition of X at the last mo-ment. Nevertheless, for the reader who might beinterested in the massive theory for its own sake,we give the form of V with this renormalizationconvention:

We begin with the self-interacting meson theoryof Sec. III, except that we now assume that the me-son has a positive mass, p, . The only effect ofthis is to change the meson propagator from themassless to the massive form. Thus, the one-loop approximation to the effective potential, Eq.(3.3), becomes

For the same hypothetical reader, we state, with-out proof, that, in the many-meson case„ the rele-vant formulas in Sec. VI, Eqs. (6.2) and (6.3), re-main valid in the massive theory if Vo is inter-preted as including quadratic mass terms as wellas quartic self- interactions.

2. Self-Interacting Mesons with Imaginary Mass

We now investigate the theory for which p.' is

negative. In this case, (k'+ p, ') can vanish even

1908 SIDNE Y COLEMAN AND ERICK WEINBERG

for Euclidean k; thus we cannot drop the i e in Eq.(Bl), and the effective potential has an imaginarypart. " This cannot be canceled by the I3 and C

counterterms; these must remain real, to pre-serve the reality of the Lagrangian.

It is easy to compute this imaginary part:

lmV=- [(y'+ —,'Ay, ')'8(-p, ' ——,&y, ') P-] .64m

(B6)

p'+~6A(y)' =0 . (B8}This is a point at which the 8 function in Eq. (B6)vanishes; thus all derivatives of V at this pointare real.

The second term in this equation is just an addi-tive constant; it has no effect on the physics of thesystem, and can be dropped with impunity. Thefirst term, though, cannot be swept under the rug;it represents a genuine physical effect.

We can gain some heuristic understanding of thiseffect if we return to the analogy between V and themomentum-space propagator, discussed in Sec.IO. In ordinary perturbation theory, the propaga-tor has an imaginary part at those momenta forwhich the off-mass-shell particle is kinematicallyunstable. Here, the vacuum itself is kinematicallyunstable, because of the negative mass term inour initial Lagrangian; thus V develops an imagin-ary part. We cannot claim credit for discoveringthis phenomenon; it was first observed in theclassic calculation by Euler and Heisenberg" ofthe effective Lagrangian for constant electromag-netic fields in quantum electrodynamics. TheEuler-Heisenberg function is real for constantmagnetic fields, but imaginary for constant elec-tric fields, because, in the presence of a constantelectric field, the vacuum can decay into electron-positron pairs.

Just as in the usual theory of unstable particles,the presence of an imaginary part forces us tomodify our mass renormalization condition. Sincewe can only add real counterterms to the Lagran-gian, we can only control the real part of V"(0};thus, for negative p', Eq. (B2) must be replacedby

d V'

Re (Bv)dpi( c (jc=O

The physical meaning of p,' is now even more re-

mote than before, but it is still a finite quantitywhich serves to parametrize the theory.

We should observe that the imaginary part of theeffective potential does not affect the reality ofzero-momentum Green's functions in the physical(asymmetric) theory For, to.lowest order, thevacuum expectation value of y is given by

3. Massive Scalar Electrodynamics

For scalar electrodynamics, "even if the mesonhas a mass, the photon does not. Thus, diagramsinvolving photon loops are still infrared divergent,and we must use a nonzero renormalization mass,M, to define the quartic coupling constant, A.. Fol-lowing Sec. VI, we will assume that X is of theorder of magnitude of e', so we can neglect the ef-fects of meson loops compared with those of pho-ton loops. Then, no matter what the value of A,

and M, we can find a mass m such that

2 2V=~g fj{), +„.2y, ln64K m

(B9)

Se M2ln—+-

8m m2 3 (Blo)

If we wish to make contact with the conventionaldescription of the massive theory, then we shouldchoose M to be on the order of p., the only massin the theory. It would be perverse to do other-wise; we can create spurious contradictions bydescribing a physical situation in two differentformalisms in which we assign the same symbolto two drastically different objects. (Of course,when we are studying the zero-mass limit, wemust keep M fixed. Thus M becomes many timeslarger than p- but here we are asking a differentquestion. )

Now, if V is to have a second minimum at all(let alone one that is lower than the one at theorigin), then it is easy to show that

Note that m is not a renormalization mass but agenuine parameter which, together with e and p. ,characterizes the theory; it replaces the redundantpair of variables, X and M. From Eq. (B9), it istrivial to verify the smooth approach to the zero-mass limit, Eq. (4.9).

Nevertheless, there is still a surprise lurkingin this formula: For sufficiently small positivep', V evidently has two local minima, one at theorigin and one near m, and its value at the secondminimum is evidently lower than its value at thefirst. Thus, even for positive p.', spontaneoussymmetry breaking occurs.

This seems to contradict the conventional wis-dom stated in Sec. I." Is ordinary real-mass sca-lar electrodynamics unstable' No, it is not, aswe shall now explain.

The point is that the conventional wisdom statesthat the theory does not suffer spontaneous sym-metry breaking if p' is positive and A. is positive.As we have said many times, the value of A. de-pends on our choice of M; in the case at hand, wedifferentiate Eq. (B9) four times and find

RA DIA TIVE CORRE C TION S AS THE ORIGIN OF. . . 1909

ln 4 2 &-I .16p2 p2

3e4m'

For small e (in particular, for e as small as thephysical electron charge), this implies that A. isnegative if M is on the order of p, .

Thus, properly interpreted, the theory in whichwe have found a second minimum is not conven-tional massive scalar electrodynamics at all, buta theory with negative quartic coupling constant.At first glance, we would expect such a theory tohave no stable vacuum at all, because, as we moveaway from the origin, V decreases without bound.However, it is not too surprising to be told that,

at least for small x (on the order of e'), the effectsof radiative corrections dominate those of thequartic self-coupling, and turn V upwards again.

We have argued that this is the natural way tointerpret the two minima in Eq. (88). Even if thereader does not believe this, we think he mustadmit that it is no less natural than the alternativeinterpretation (that radiative corrections causenormal massive scalar electrodynamics to sufferspontaneous symmetry breaking), and thereforeshould be preferred, if only because, when giventwo otherwise equivalent descriptions of the samephysical phenomenon, we should choose the onethat does least violence to our intuition.

*Work supported in part by the National Science Foun-dation under Grant No. 30819X.

~We refer here to such familiar problems as infinitecross sections, etc. In fact, it will turn out that thesewell-known difficulties are somewhat off the point, sincethey are evidence against the existence of an 8 matrixbut not against the existence of a theory of off-mass-shell Green's functions. We shall show that even thelatter type of theory does not exist.

We use a metric with signature (+—-).3We will return to (and verify in our formalism) this

common belief in Appendix B.4For the sophisticated reader, we emphasize that p2

is not the bare mass; it is the renormalized mass aboutthe symmetric vacuum. If p2 is positive, this is indeedthe renormalized mass of the meson. If p2 is negative,it possesses no such simple interpretation, but it isstill a completely well-defined renormalized quantitywhich can be used to parametrize the theory. For theunsophisticated reader, we emphasize that these re-marks will be explained in greater detaQ later.

SWe give here a brief list of some of the standardpapers on these phenomena. Spontaneous symmetrybreaking: J. Goldstone, Nuovo Cimento 19, 154 (1961);Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345(1961);J. Goldstone, A. Salam, and S. Weinberg, ibid.127, 965 (1962). Higgs phenomenon: F. Englert andR. Brout, Phys. Rev. Letters 13, 321 (1964); P. Higgs,Phys. Letters 12, 132 (1964); G. S. Guralnik, C. R.Hagen, and T. W. B. Kibble, Phys. Rev. Letters 13, 585(1964); P. Higgs, Phys. Rev. 145, 1156 (1966); T. W. B.Kibble, ibid. 155, 1554 (1967). Renormalization:G. 't Hooft, Nucl. Phys. B33, 173 (1971);B35, 167 (1971);B. W. Lee, Phys. Rev. D 5, 823 (1972); B. W. Lee andJ. Zinn-Justin, ibid. 5, 3121(1972);5, 3137 (1972);5, 3155 (1972).

6J. Schwinger, Proc. Natl. Acad. Sci. U. S. 37, 452(1951);37, 455 (1951);G. Jona-Lasinio, Nuovo Cimento34, 1790 (1964). The effective potential was first intro-duced by Goldstone, Salam, and Weinberg (Ref. 5), butwe follow the development of Jona-Lasinio here. A goodreview of these methods is the contribution of B. Zuminoto Lectures in Elementary Particles and Quantum FieldTheory, edited by S. Deser et al. (MIT, Cambridge,

Mass. , 1970).'Y. Nambu, Phys. Letters 26B, 626 (1968); S. Coleman,

J. Wess, and B. Zumino, Phys. Rev. 177, 2238 (1969).Except for scale invariance, and that is afflicted with

anomalies. Nevertheless, as we shall see in Sec. V, wecan define masslessness in a general abstract way, bydemanding that the theory obey the homogeneous renor-malization-group equations rather than the inhomoge-neous version of these equations, the Callan-Symanzikequations.

M. Gell-Mann and F. Low, Phys. Rev. 95, 1300 (1954);N. N. Bogoliubov and D. V. Shirkov, Introduction to theTheory of Quantized Fields (Interscience, New York,1959). Our notation follows the review of S. Coleman,in Proceedings of the 1971 International Summer School"Ettore Majorana" (Academic, New York, to be pub-lished).

For a detailed explanation, see Coleman (Ref. 9).L. D. Faddeev and V. N, Popov, Phys. Letters 25B,

29 (1967)~

Except that the 2 from the reflection symmetry of thediagram is missing; but this is replaced by a ~ that comesfrom only summing the even terms in the series.

~3This model was suggested to us by S. L. Glashow andH. Georgi, who collaborated with us on the analysis of itsstructure. They were attracted to this model becauseloop effects are important in it even if V contains a massterm, as will be explained in the text.

The discovery that, for certain theories, the scalar-meson part of the Lagrangian inevitably is invariantunder a larger symmetry group than the total Lagran-gian was made independently by S. Weinberg. More cleverthan we, he realized, as we did not, that this phenomenonmight provide an explanation for the hierarchy of (ap-parently) broken symmetries found in the real world.[S. Weinberg, Phys. Rev. Letters 29, 1698 (1972).]

S. Weinberg, Phys. Rev. Letters 19, 1264 (1967);A. Salam, in Elementary Particle Theory: RelativisticGrouPs and Analyticity (Nobel Symposium No. 8), editedby N. Svartholm (Wiley, New York, 1969).

The eigenvalue condition was first advanced as acondition for the bare charge in quantum electrodynamicsby GeQ-Mann and Low (Ref. 9), and by M. Baker,K. Johnson, and R. Winey tPhys. Rev. 136, B1111(1964)].

19j.0 SIDNEY COLEMAN AND ERICK WEINBERG

It has been advocated as a condition for the physicalcoupling constants by K. Wilson [Phys. Rev. D 3, 1818(1971)] and by S. Adler [ibid. 5, 3021 (1972)l.

'H. D. Politzer (private communication).This was pointed out to us by R. Dashen and D. Gross

(private communication).

W. Heisenberg and H. Euler, Z. Physik 98, 714 (1936);J. Schwinger, Phys. Rev. 82, 664 (1951).

The main content of this section is an answer by H. D.Politzer to a question raised by J. D. Bjorken (privatecommunication); our role is reportorial only.

Sec. I, second paragraph.

PHYSICAL REVIEW D VOLUME 7, NUMBER 6 15 MARCH 1973

Divergence Cancellations in Spontaneously Broken Gauge Theories*

Wolfgang KummergDepartment of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104

and

Kenneth LaneDepartment of Physics, University of California, Berkeley, California 94720

(Received 16 October 1972)

An extremely simple method is presented to calculate unambiguously higher-order correc-tions in the unitary (U) gauge of theories with spontaneously broken gauge symmetries.Manipulating Feynman integrals in coordinate space, the spurious nonrenormalizable in-finities of this gauge are isolated in the form of (contracted) Feynman graphs. Withoutmaking reference to any specific global regularization scheme, the complete cancellationof these graphs is demonstrated in the cases of fermion-fermion scattering to fourthorder in the Abelian model considered by Appelquist and Quinn and for the similar neutrinoscattering in Weinberg's SU(2)z x Y model. The reason for such complete cancellation isseen to be a consequence of the algebraic structure of the equal-time commutators amongcurrents, their divergences, and various fields. This structure, of course, is dictated bythe original gauge symmetry. As a check on our methods, the weak muon anomaly in Wein-berg's model is calculated, and agreement is found with the (gauge-invariant) results ofother authors.

I. INTRODUCTION

One of the most interesting recent developmentsof field theory is the discovery of models of"quasirenormalizable" type. Weinberg' presentedthe first such realistic model, in which he proposedto unify the weak and electromagnetic interactionsof leptons through the spontaneous breakdown ofsymmetry in a gauge-invariant Lagrangian via theHiggs phenomenon. By now a large number ofmodels of this type have been proposed which alsoinclude hadronic interactions. '

A common feature of all these theories is thefact that, to a given order of perturbation theory,individual Feynman diagrams contributing to aspecific process contain nonrenormalizable diver-gences. We shall call these infinities "spurious"because they happen to cancel when all individualdiagrams are added. Cancellations of this typehave been shown to occur in the Weinberg theory"and in a simplified Abelian model' by means of asimple cutoff prescription to regulate the integralsinvolved.

It is well known for renormalizable theories ofthe usual type that different arbitrary regulariza-tions yield different finite additions to the divergentparts of Feynman graphs. ' This is not seriousbecause the infinite together with such (ambiguous)finite parts are absorbed into mass and charge re-normalization anyhow. In quasirenormalizabletheories some of the infinities —the spurious ones—cancel against infinities in other graphs. Herefinite, regularization-dependent terms can be leftover. Such an ambiguity has already been en-countered in the early calculations of the anoma-lous magnetic moment of the muon in Weinberg'stheory. '

We have been referring so far to the "U-gauge"formulation of the theory only. In this gauge thefields are redefined so as to eliminate superfluousscalar bosons. Just because of the spurious-di-vergence problem mentioned above, the proof ofrenormalizability for these theories has not beengiven in the U gauge. In a series of fundamentalpapers 't Hooft' and Lee and Zinn-Justin9 havegiven the proof in the so-called "R gauge, " where