Dynamical symmetry breakdown in pure Yang-Mills field theory

Nuclear Physics B95 (1975) 269-300 © North-Holland Publishing Company

DYNAMICAL SYMMETRY BREAKDOWN IN PURE YANG-MILLS FIELD THEORY F. ENGLERT * CERN, Geneva

J.M. FRERE ** Facult~ des Sciences, Universitg Libre de Bruxelles, Bruxelles

P, NICOLETOPOULOS Facultd des Sciences, Universit~ Libre de Bruxelles, Bruxelles

Received 18 November 1974 (Revised 19 March 1975)

It is shown, in a simple approximation consistent with gauge invariance, that YangMills fields associated with a compact simple Lie group, may acquire mass through their self-interactions; this phenomenon leads to a breakdown of the original group symmetry and to an eigenvalue condition for the coupling constant of the self-interaction. It is conjectured, from an analysis of Feynman graphs, that these facts are still true in the full theory. The relevance of these conclusions to the problem of fermions interacting with Yang-Millsmesons is pointed out.

1. Introduction It is well known that Yang-Mills vector mesons [1] may acquire mass when coupled to scalar fields in a gauge-invariant Lagrangian; this happens when the original symmetry is spontaneously broken through non-vanishing vacuum expectation values of some scalar fields [2, 3]; The inclusion of such scalar mesons is, however, quite arbitrary and presumably of phenomenological character; one is tempted, from a more fundamental point of view, to search for a dynamical mechanism which leads to mass generation without recourse to elementary scalar mesons [2, 4]. Along this line of thought, one is naturally led to inquire if a system of pure Yang-Mills fields associated to a compact simple Lie group is stable against mass generation. Such a phenomenon which could arise through the Yang-Mills self-interaction would provide the "purest" example of dynamical symmetry breakdown. * Permanent address: Facult6 des Sciences, Universit~ Libre de Bruxelles, Bruxelles. ** Stagiaire F.N.R.S,

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F. Englert et al. / Yang-Mills field theory

If mass generation does in fact occur, its origin is presumably to be found in the infra-red instabilities of the massless theory as suggested by the work of Coleman and Weinberg [5]. Proving the necessity of dynamical symmetry breakdown is however a difficult problem, and we have attempted to solve a simpler one, namely: is mass generation consistent with pure Yang-Mills field theory? Most previous investigations [6, 7, 15, 16] of dynamical mass generation are concerned with models in which the relevant integral equations are governed by their high-momentum regime. We shall refer to such models as type A. Theories whose solutions originate entirely from the low-momentum region will be called type B. Such a model was discussed in ref. [9]. In contrast to type A theories, the model of ref. [9] provides enough constraints to fix the low-energy coupling constant and the induced mass ratios. The main point of the present paper is that pure YangMills theory is probably of type B. Our results are contingent upon certain assumptions which we can only justify by a plausibility argument. Nevertheless we feel that our conclusions are of sufficient interest to motivate their presentation, as we are led to believe that spontaneous mass-generation may indeed occur, but only for specific values o f the coupling constant. This means that mass generation fixes the scale of the Yang-Mills fields with respect to the group parameters, a perhaps unexpected result. The self-consistency of mass generation is suggested from a simple qualitative argument. Let us consider the matrix elements of the conserved current operator J~(O) between two vector-meson states le(x)b , p) and le(X')c, p'). The indices b and c label here the conserved current, s j b and JT~ coupled to the mesons b and c. De, fining Pu = Pu + P u ' Ou = P u - P u ' one has, in absence of symmetry breaking, (e(X)b,PnlJau(O)ie(h')C,p m )

~ N e (x)b " e(X')c f a b e p u , Q--*O

(1.1)

where N is a normalization factor; indeed all other invariant form factors are multiplied by vanishing kinematical factors in this limit. The divergence of the current in (1.1) vanishes since Q u P u is equal to zero on the mass shell of both bosons. If we now tentatively assume that mesons b and c acquire unequal masses m b and m e, then Q u P u = m 2 - m 2 --/=0. A s J a is still conserved, we must add to the righthand side of (1.1) new form factors to ensure zero divergence. The simplest solution is (e(X)b , p J~(O) e (x)C,p')

=

Ne

(k)b • e (h')e [fabcPl~

Q2--*0 Qu + regular terms as Q2 ~ 0 ] . _ f a b c [(mb)2 _ (mC)2] ~_~

(1.2)

This means that the current J~ is coupled to a Nambu-Goldstone (NG) singularity which in turn is expected to generate in the usual way a mass m a 4= O. Of course this reasoning does not prove that such mass generation is consistent with the dynamics.

F. Englert et al. / Yang.Mills field theory

271

Note that the conserved current used above is not the observable one because the singularity in (1.2) is only a gauge effect; neither is it the one we shall use in the Feynman rules. Therefore the above argument should serve only as a guide to pave the way towards the subsequent formalism of Yang-Mills field theory. Our study is presented as follows. In sect. 2 we review the known Ward identities for Yang-Mills fields [8]. We emphasize the usefulness of the Landau gauge where the NG singularities are less affected by the intrinsic complications of non-Abelian gauge field quantization. In sect. 3 we construct a simple approximation to the Schwinger-Dyson equations for self-energies at zero external momentum. In this approximation no ultra-violet (u.v.) divergences are present in the symmetric theory and therefore no renormalization program is used. A simple system of algebraic equations determining induced mass ratios is obtained and it is shown that the coupling constant must obey a consistency equation. Our scheme can be interpreted diagrammatically as a type of ladder approximation and shown to be consistent with a class of Ward identities implied by gauge invariance. The proof requires tedious calculations; these are relegated to sect. 5. In sect. 4, we attempt to extend the qualitative conclusions of sect. 3 to the full theory. Under certain assumptions made plausible by the previous analysis, we show that symmetry breaking does not introduce new divergences in the renormalized theory. Moreover, the occurrence of an eigenvalue equation for the renormalized coupling constant appears as a property of the full theory and not only of the approximation used in sect. 3. We conclude the section by pointing out the relevance of these conclusions to the case where coupling to fermions is included in a gauge invariant way; the results of a previous investigation on low-energy eigenvalue equations for such systems [9] may be extended to take into account non-perturbative effects of the non-linear Yang-Mills meson self interactions.

2. Ward identities

The Ward identities pertaining to a Lagrangian exhibiting gauge invariance of the second kind are well known [8]. We shall briefly review them and discuss the simple relations they imply in presence of spontaneous symmetry breaking when the YangMills fields are described in the Landau gauge.

2.1. Feynman rules; introducing a tensor propagator Let us consider a system of pure Yang-Mills fields associated with a compact simple Lie group G. They are characterized, in the set of covariant gauges, by the Lagrangian [10] L =

1

2n

OuA a. OuA a - 8. ~*a Outpa_gfabc~ ~p*a~oeAbu ,

(2.1)

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where l ~ v = a~Aau - ~vAau + g f a b c A ubA v c .

(2.2)

Here, as throughout the text, latin letters refer to group indices and greek letters to Lorentz indices. The A~ are the Yang-Mills fields and ¢a(~0*) the Faddeev-Popov ghost [11 ] fields; ~ is a gauge parameter, g the bare coupling constant and fabe the completely antisymmetric structure constants. The Feynman graphs are drawn in terms of bare propagators corresponding to the unperturbed Lagrangian [8] LO = - 41 [ a u A ,a - a~Aau] [aUA~ - a"AUa] - ~1 buAaUa~,4~

(2.3)

This introduces not only three-point but also four-point vertices. We shall find it convenient to factorize the latter using a tensor propagator uu, az = ~ [guogur - gu~gua ]

6 ab

.

(2.4)

This propagator couples to Yang-Mills propagators through the bare interaction vertex (fig. 1) V ~ : eor = X/~ g [ g , a g,T - g ucguo ] fabe "

(2.5)

It is readily verified that the exchange of (2.4) in the three channels restores to order g2 the four-point vertex (fig. 1). In higher orders, as (2.4) is dressed both by longitudinal and transverse self-energy insertions, it will be useful to separate the bare propagator (2.4) into a transverse and a longitudinal part. Thus we write A a,b uv, o r = T ~a,b , or + L a~v,b or ,

(2.6)

b _ a i 8a b ~;;,or(q) - ~ [Gua(q) G~r(q) - G u r ( q ) G v o ( q ) ] (-~n)4 ,

(2.7)

with

Guy(q) = guy

(2.8)

q uq2q v

In this formalism the bare Yang-Mills propagator /~ub(q)= --i

F G u , ( q 2)

quq"q

8 ab

(2.9)

couples to A ~ o r as examplified in fig. 2. However, from Lorentz invariance the coupling has the form

273

F. Englert et al. / Yang-Mills fieM theory bc C~;oh(q) - [qogux - q~,gvo] ~be(q2) .

(2.10)

Therefore only the coupling of the transverse part of (2.9) to the longitudinal part of (2.6) survives. 2.2. Transversality relations The Ward identities give constraints on longitudinal parts of Green functions, that is on the divergences of the tensors

(a,a)

(d,6)

(b, ta)

X

(c,5')

(a,=)

(b,a)

X .X (d,~) (o,=)

~c,v) (b,13)

(c,v)

(ci,6) (a,c~)

(b,[3)

(d 6)

(c v) a

Fig. 1. Factorization o f the four-point bare vertex (a, ~)

(b, v) = vector bare propagator Lfl~',b

(a, ;av)

(b, ar)

(a, ta) ~

(b, v)

~- (c, ar)

/

(a, a)

= bare vertex (3.5)

(b,/3) .~

= four-point bare vertex

(d, 8)

(c,7) X (ga/3g~3 ' -

(a,IJ.) (b,v)

Fig. 2. A contribution to (b, v)

A a~'vbar

= tensor bare propagator

~ q

==.g2{fhaefhbd(gc~{3g3,8 _ gasg3,[3) + fhadfhbc

gct3,gS[3) +fhabfhcd (ga,~g38 - gasg3.t) }

(c,pX) (d,o'Q

(a,~)

(I°'V~,o'~

b,cph(q) Cv,

(c, ph) = vector-tensor transition matrix

b,e Cv,ph

F. Englert et al. / Yang-MillsfieM theory

274 G°lla2 "''an ~ 1 'U'2 ' " D n

(x 1 X 2 ..... '

Xll)=(T:~Aaull(X1)

art ... Aun(Xn))

(2.1 1)

with respect to one or more space-time variables. In particular for n > 2 one has, contracting all indices

OulSuz ... 8un G al'''an = 0

(n>2)

,

(2.12)

U 1 .-.U n

where G-Sq •"an is the connected part o f Ga.~ "an . The corresponding statement for 121 ...Prl U 1 ...Un n = 2 is the transversality o f the self-energy matrix (270 4 i Iluv. To extract the dynamical content of this property, we write IIuv = fl(1)+ II(2) II(2) contains all UV UP " contributions to Iluv which are one particle reducible with respect to the tensor propagator (2.6)• It follows kinematically from (2.10) that II(~(q) = Guy(q) q2 ~ ( q 2 ) .

(2.13)

Thus, the dynamical content of the Ward identity for n = 2 is embodied in the transversality of fl(1)(q); we write therefore II(1)(q) = Guv(q ) q2 i](q2)

(2.14)

and the dressed propagator matrix Duv has the form

Duv(q)

=

-i [ Guy(q) quqv 7 +n ( ~ ) 4 q2 _ q2 ii(q2) _ q 2 ~ ( q 2 ) q2 ]

(2.15)

We may use (2.15) if ~ 4= 0 to amputate all external vector propagators in (2.12). We then obtain for the proper vertex functions

""an = 0 , aul au~ ... aun r ~ I ...un

(2.16)

where Fu u is defined to be II u u • As was done for fl u u ' one could again 1 2 . 2 . 1 separate out kinematical effects ~ue to tensor propagation gut we shall not need to do so. We shall refer to (2.16) as the transversality relations. It is important to note that because (2.12) and (2.15) are valid for all values o f r/, (2.16) is also true b y continuity for ~ = 0 despite the fact that in this case (2.15) admits no inverse and that (2.12) is trivially satisfied. Thus the Landau gauge ~ = 0 does not play a special role in the transversality relations. Its usefulness will now appear in the remaining Ward identities where not all the Lorentz indices are contracted; we shall, however, restrict ourselves to the three-point vertex function because the relevant features o f the NG bosons may be obtained from it.

2.3. Three-point function • bc where only the vector Consider the three-point improper vertex function 7au',vo propagator attached to the/2 index has been amputated. This function satisfies the following Ward identity in momentum space

F. Englert et al. / Yang-Millsfield theory

275

(1/g)q u T~,;beu;vo(P + ~q;P - i q =iq2"Ga(q2) c p +½q) _ ~ b11, UO (p _ ~q) fuxc )",'Cb~v~bC(p,q) X ( f bxv~ Dov,~(

(2.17)

The function ~ contains graphs where at least one external Yang-Mills propagator is connected to an external ghost propagator "Gax(q2) (fig. 3). Thus', from the Feynman rules, ,I, will contain a m o m e n t u m factor q in the Landau gauge. Under the assumption (see sect. 3) that no pole is induced in qb as q2 _+ 0, we have lira ~ ( p , q) = 0 (Landau gauge). q~0

(2.18)

Thus, in the Landau gauge, as q ~ 0 (2.17) reduces to ( i / g ) qu'Tau.vo '

~. i[q2Gax(q 2) F x, Dva(P)] ,

(2.19)

q~0

where we have used matrix notation with (FX)b c =fbXe. From now on, we shall restrict ourselves to the Landau gauge. Eq. (2.19) shows that if [F x, Dvo(p)] :~ 0 a singularity develops in 7 /a. t ; v o as q -+ 0 ).

( I / g ) 7~;vo q~O tqu ['Gax(q2)FX' Dvo(P)] + regular terms. •

(2.20)

Observe that the singularity o f this "NG b o s o n " is identical to that o f a FaddeevPopov ghost. If one neglected the radiative corrections in the ghost propagator [ ~ ( q 2 ) = 6a(_i)/(2n)4q2 ], ( 2 . 1 9 ) w o u l d resemble the more conventional type Ward identities and would lead to the (incorrect) result ( I / g ) 7 a. u,uo ~q~O (qu/q 2) ( - ~1

[Fa' Dva(P)] + regular t e r m s .

(2.21)

Comparing (2.21) to (2.20) we see that for non-Abelian gauge fields, the NG boson is affected by the same radiative corrections as the Faddeev-Popov ghost; this is

~xP*q/2 (u,v) .,.Y"~f~,of q~/J

(CjO)

~p

-ql2

7X

Fig. 3. A diagram contributing to (1/iq2) ~v~ c (p, q);the explicit qh factor is exhibited on the diagram, a- - ~ - x = ghost bare propagator Gax(q2) = (-i/(2n)4)(1/q 2) 6ax.

F. Englert et al. / Yang-Mills fieM theory

276

presumably a consequence of unitarity *. Note that (2.20) is valid only for the improper vertex function because the proper vertex may develop singular form factors proportional to Pv (or Pa) which are projected out in the improper vertex due to the transversality of the external propagators in the Landau gauge. Such form factors do indeed appear, as will be seen in sect. 5, and are mandatory in order to satisfy the transversality relations (2.16) for the three-point function. In other words, while the transversality relations imply an equation analogous to (2.19) for the proper vertex function [7], namely, in matrix notation (I/g) qU p a

, _(27r)4 [q2~a(q2) F x, iivo(p)]

/~,vo q ~ 0

(2.22)

one should bear in mind that the analog of (2.20) for proper quantities is more complicated because the transversality of (2.22) in p does not originate in a simple kinematical factor, as it does in (2.19). The Ward identities (2.16), (2.19) and (2.22) will provide a convenient tool for studying spontaneous symmetry breaking solutions in pure Yang-Mills field theory through the occurrence of N-G singularities.

2.4. Renormalization We now show how u.v. divergences may be removed from (2.19) by a renormalization program so that the NG singularity persists in the renormalized theory. We assume that the only counterterms required are those of the symmetric theory; thus no mass counter terms are used for Yang-Mills fields. This convergence condition must ultimately be verified, as the symmetry of the unrenormalized Lagrangian must be identical to the symmetry underlying the finite theory. A more detailed discussion o f the re.normalization program is given in subsect. 4.1 below. If Z1, Z 3; Z 1, Z3 are respectively the vertex and wave function renormalization constants for the Yang-Mills and the ghosts fields, we may write (2.19) as

(1/gR) qu'y'a - Z3 iq 2 [ ~ a ( q 2 ) F x , Dvo(p)] ~,va q~O~ -Zl Z3 ~' ,

(2.23)

3

where gR = gZ23 Z11 is the renormalized coupling constant and the prime denotes renormalized quantities. Now, if one chooses the vertex renormalization constants in such a way as to ensure that the renormalized coupling constant o f the ghost is equal to gR, one has the usual relation [12, 13] Z1 Z1 Z 3 = Z~"

(2.24 )

Thus the renormalization constant in (2.23) is in fact Z I as seen from (2.24). If the * We are indebted to J. Zinn-Justin for pointing out to us the relation implied by unitarity between the Fadeev-Popov ghost and the NG boson.

F. Englert et al. / Yang-MillsfieM theory

277

subtractions are made at zero external momentum we will have Z'I = 1 in the Landau gauge; this follows from the same argument that led to (2,18) as applied to the ghost-ghost-Yang-Mills vertex. If one chooses another subtraction p o ~ t in order to avoid spurious infra-red divergences in the renormalization program, Z 1 remains ultra-violet finite. We therefore see that all u.v. infinities cancelin (2.19) in the renormalized theory. The preceding discussion has been limited to pure Yang-Mills fields associated to a simple group. The extension to more general gauge invariant Lagrangians, including coupling to fermions, is, however, straightforward. The Ward identities for all vertices coupled to a Yang-Mills propagator take a simple form in the Landau gauge, in analogy to (2.19); as in the present case, all u.v. divergences should be removed from Ward identities by counter terms of the symmetric theory.

3.

Spontaneoussymmetrybreaking in pure Yang-Mills field theory: a simple approximation

One may look for symmetry breaking solutions in pure Yang-Mills field theory by assuming that the Yang-Mills self-energy matrix [Iu~(p ) acquires a symmetry violating part, namely

[ F a, Iluv(p)] ~ 0 for some a.

(3.1)

It follows that as q2 _~ 0, II(q 2) as defined in (2.14) becomes a singular matrix with a 1/q 2 singularity (up to log terms). Indeed, (3.1) implies, using (2.19), (2.20) or (2.22) that such a singularity will develop in --/20~" pabc' the proper vertex function. Since this vertex contributes to the Schwinger-Dyson equation for II/(1) (fig. 4), its ,w singularity will generate, as q + 0, a non-vanishing contribution to the coefficient of q u q J q 2 in II(lv). From gauge invariance this is a contribution to q2 l-l(q2) in (2.14) which does not vanish as q2 ~ 0. As a non-zero IIu~(0 ) matrix implies that at least some Yang-Mills fields acquire mass, we see that (3.1) is in fact equivalent to the assumption that some Yang-Mills fields acquire spontaneously a mass as a consequence of NG bound state formation. As already mentioned in sect. 2, it is important to verify that assumption (3.1) does not lead to infinite symmetry violating counter terms, which would destroy the renormalizability of the theory. This question will be discussed further in sect. 4. In the present section we wish to present a simple approximation embodying many qualitative aspects o f symmetry breaking. To ensure that the dynamical character of self-generated masses is consistent with gauge invariance, our simplified scheme must satisfy the Ward identities o f sect. 2. Our approximations will be elaborated in terms of graphs whose analogs in the symmetric theory require no renormalization at zero external momenta. Therefore, any divergence encountered in the course of our development will be fatal to our program.

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F. Englert et al. / Yang-Mills field theory

(a'v) Fig. 4. A contribution to ilaa'b{~ #u /~x (a'

(b, o) //

#)

(c, r)

= rabc --pat

= dressed Yang-MiUspropagator,

In the symmetric theory, to order g2, the only diagrams contributing to gluv are shown in fig. 5. When regularized in a gauge invariant way, for instance through dimensional regularization [14], they yield Iluv -~ 0 as q2 ~ 0. To obtain a self-consistent non-vanishing lluv(0 ) one may insert a non-vanishing mass term M 2 in the Yang-Mills propagator of figs. 5a and b. Here M 2 is to be interpreted as M 2 = lim p 2 I I ( p 2 ) , p2~0

(3.2)

and we shall restrict ourselves, for simplicity, to cases where M 2 is diagonalizable in the space of the regular representation. Thus (3.3)

(M2)ab = m 2 6ab .

The consistency of approximating self-energies in the virtual loops, 5a and b by their zero momentum value (3.2) will eventually be justified by the fast convergence of the integrals at zero external momentum q.

(o)

(a,~)~"~(a~v) "-4"

( a, V.)/f "" -'-,,\(a.,v )

Fig. 5. 1-1#v aa (q) to order g2.

(b)

(c)

F. Englert et al. / Yang-Mills field theory

279

With this starting point we shall search for the minimal modifications to be made in the diagrams of fig. 5 in order to ensure that M 2 is of dynamical origin. The ~uiding principle to this effect will be the Ward identities of sect. 2. Afterwards, we will show that the approximation so obtained can be interpreted as a type of ladder approximation, whose role with respect to the full theory will be discussed in sect. 4. With (3.2) the Yang-Mills propagators become

-i guu - PuPu/P 2 L)uv(P) = (2n) 4 p2 _ M 2 "'

(3.4)

As q -+ 0, figs. 5a and 5b now yield a non-vanishing result while fig. 5c remains equal to zero in this limit (see sect. 5); it is understood that all divergent integrals are evaluated with the dimensional regularization technique. The resulting IIu,(0), however, not only diverges as the spacetime dimension n goes to 4, but is.proportional to guy and therefore inconsistent with gauge invariance. This last feature is not surprising since a contribution to IIuv(0 ) proportional to ququ/q 2 as q ~ 0 can only arise if the Yang-Mills field couple to a NG bound state at q2 = 0. From (2.20), we see that (3.4) indeed implies that such a bound state appears when M 2 is not proportional to the unit matrix. To evaluate the contribution of this bound state to IIu~, we approximate (2.20) by (2.21); this means that we shall be neglecting radiative corrections to the ghost self-energy. From (2.21) and (3.4) we find the simple result (1.2) in the form

(1/g) p a b c ( p - ~ q ; p + -pot

~q) ' , igor qu q -~0

fabc [(mb) 2 - ( m c ) 2 ] +

....

Here, as already discussed in sect. 2, the missing form factors are not only the regular ones as q -+ 0, but also those singular ones which contain longitudinal kinematical factors Po (or Pr) that were projected out in the improper vertex function ~;;bc. At any rate they do not contribute to IIu~ (0) in the Landau gauge. Inherent in the use of (2.21) to obtain (3.5) is that the function ,1~ which appears in (2.17) develops no pole when q -~ 0. Using the methods of sect. 5 it is straightforward to verify that the ansatz (3.4) and its consequence (3.5) do not lead to such a pole. The contribution o f the singular form factor (3.5) to fig. 4 is given by fig. 6c. As will be shown below by direct evaluation, the integral is convergent: its leading (logarithmic) divergence vanishes as a consequence of Lorentz and group symmetries. Since the guy term is divergent at n = 4, the inclusion of diagram 6c, while required by the Ward identity, is sufficient to ensure the transversality of the approximate IIuv in the q -+ 0 limit. To proceed with our game we now turn to the 4-point function. Fortunately it appears possible to restore transversality, and thus convergence, by introducing a suitably chosen self-energy into the tensor propagator. As will be shown in sect. 5, it suffices to use the modified tensor propagator

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F. Englert et al. / Yang-Mills field theory

where the limit n = 4 will be taken at the end. Thus we finally retain for lily(0 ) the sum of the three contributions in fig. 6. Clearly, the approximation summarized in fig. 6 cannot be taken seriously before making contact with perturbation theory. This is studied in detail in sect. 5 for a particular group (SU3), and a particular selfconsistent breaking mechanism (M 2 ~ y2). We shall summarize here the main results of sect. 5, retaining our present more general notation, since we tentatively surmise their general validity. If the diagram of fig. 6c really has a dynamical origin, it may be rewritten in the form shown in fig. 7 where the factorization of the residue at the NG pole is made explicit. The resulting vertices, approximated at zero external momenta may be determined from the Ward identity (3.5) as shown in fig. 8. On the other hand, they should also be derivable from a homogeneous Bethe-Salpeter equation; they indeed satisfy the equation shown in fig. 9. Thus our approximation may be identified with a "ladder approximation" for Green functions at zero external momenta, using the propagators (3.4) and (3.6). This fact not only enables us to establish the transversality and the convergence of II,,u as q -+ 0, but also permits us to verify diagrammatically the Ward identities (2.16) for three- and four-point vertices in our "ladder approximation". As we shall see, the use of (3.6), though crucial, is sufficient for the entire scheme to go through.

4

~,)

(a',v)

(a)

b

~(a',v) q

(b)

b

Fig. 6. Contribution to I Iaa' ~ (a) as q ~ 0 in our "ladder approximation" (a, ~ _ t ~

(b, o)

q '+

(c, r) = iggor (q~/q2) fabc (m~ - m~)

/

= vector propagator (3.4).

~,

= tensor propagator (3.6).

Auv,or = T~*v, or + Luv, or '

(3.6)

Tt*v, or - n - 1 n - 2 Tu~, o r ,

(3.7)

F. Englert et al. / Yang-Mills field theory

281

Fig. 7. Diagrammatic expression of (6c) in terms of a NG boson propagator . . . . . .

NG boson propagator - i / 2 n ) 4 • 1/q 2

(b, a)

~

--

Coupling of the NG boson to Yang-Mills vector mesons.

(e, r) We now evaluate IIu~(0 ) from the diagrams of fig. 6. It suffices to compute the quqv/q 2 term which comes from diagram 6c. Retaining only terms linear in q in the integrand, one gets

[quqv/q2lq=o m2a 6aa, = lira 3g 2 b~efabcfa,bc . q~O (m2b -- m2) (qu/q 2

- i f Pv 1 1 d4p ) ~-~J (p + ½q)2 _ m2 (p _ ~q)2 _ m 2

2 2 fa'bc] = [quqjq2]q=o ~g2 b~e , [fabc(m2 - me)

X

i f p2 d4p . (2~r~ d ( p 2 _ rob2)2 (p2 _ m2)2

(3.8)

The integral is easily evaluated upon performing a Wick rotation. We obtain 2

~aa, m 2

3 g2 b~.cX"~fabcfa,bcm2 -- mc 647r 2 , m~ + m 2 J(Xbc)'

Xbc = m2/m b / c2 '

x+i

_

Xlo x]

q

(b,o)

._~q~

°'~'~

Cc,~)

Ca,r,)~.~j~

Fig. 8. Expansion of the singular ~

(3.9)

JOb,o) -- ~NC e,~)

vertex in terms of the NG boson propagator.

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F. Englert et al. / Yang-Mills field theory IX

(b,p)

i x

(c;y) (b,p)

(c,y)

(b.13)

(c,¥)

Fig. 9. "Ladder approximation" for the coupling of the NG boson to Yang-Mills vector mesons. All external lines have zero momentum. One may check that J ( x ) = J ( 1 / x ) and that J ( x ) is a smoothly varying function in the relevant interval 0 ~
[c~i(3j] ~ (3k,

[(3i~3j1 ~ ctk .

(3.10)

For instance, in SU3, one may choose the ai's as the non-strange generators F1, F2, F 3, F 8 and the/3j's as the strange ones F4, FS,/76, F 7 . Eq. (3.5) (or (1.2)) suggests a possibility of spontaneous breakdown of symmetry where all mesons transforming like the/3j's ("strange mesons") acquire a common mass m, while all the others ("nonstrange mesons") remain massless. In this case it follows from (3.5) (or (1.2)), using (3.10), that the only mesons which can couple to NG singularities are the "strange" ones. Let us verify that this solution does satisfy the dynamical constraint (3.9). We see that (3.9) is non-trivial only if "a" is a "strange" index, in which case 6aa, m 2 = 647r 23 g2b~cfabcfa,bcm2=

6

2 g2 C m 2 6aa, '

(3.11)

where C is a characteristic of the group (C is the value of the Casimir operator ~ a ( F a ) 2 in the regular representation). The proposed solution does therefore satisfy the dynamical equation for spontaneous breakdown in our approximation, provided g2 takes the value 64zr2/3 C. This number should of course not be taken seriously, what will be relevant for further discussion, is the c o n v e r g e n c e of the loop diagrams of fig. 6 which led to the eigenvalue condition. For diagram 6c this arises because the singular vertex is symmetric in the indices b and c, while the bare vertex is antisymmetric in the same indices [see (3.8)]. A non-vanishing result is then obtained by expanding the internal propagators to first order in p • q, thereby reducing the apparently logarithmically divergent integral to a convergent one. The convergence of

F. Englert et al./ Yang-MillsfieM theory

283

the sum of diagrams 6a and 6b then follows by gauge invariance; the same is true for the diagrams of fig. 9 which by power counting alone appear logarithmically divergent. The generalization of these features to all orders in perturbation theory, and their relevance to the possible persistence of our eigenvalue condition in the full theory, will be discussed in sect. 4. 4. Low-energy eigenvalue conditions in gauge field theories

4.1. General considerations Before attempting to generalize the results of sect. 3, it might be worth while to review how eigenvalue conditions of a similar type may appear in the presence dynamically broken symmetry [9]. Consider a scale invariant Lagrangian admitting gauge invariance of the second kind, and suppose that a broken symmetry solution exists and can be characterized by scalar functions ¢i(p2) of non-zero physical dimension. In the present case, such qSi(p2) may be identified with the matrix elements of [T a, p2 l-l(p2)]; here we have included a factor o f p 2 to avoid a massless pole in the definition of ~i" In the wellknown model of mass generation of Baker, Johnson and Willey [15], and in its generalization to include axial vector gauge fields [16], the scalar part Z2(p2 ) of the fermion self-energy 22(p) =,p2;l(p2 ) + Z2(p2 ) is such a ¢(p2). The self-consistent ¢i(p2) may be determined in the following way. Expand all Green functions of the theory in Feynman graphs, in terms of bare vertices and dressed propagators. If, as we suppose, the ¢i appear in the propagators, they will obey coupled non-linear integral equations either from Schwinger-Dyson equations for self energies, or equivalently, from the vertex equations at zero-momentum transfer used in conjunction with Ward identities of the type (2.22). A simple example is riven in ref. [9] wherein eq. (3) determines E2(p2 ) when chirality is dynamically broken by vector and axial vector Abelian gauge field interactions [16, 17]. In general, we shall suppose that ¢i(p2) obeys an equation of the type ,

P

; {¢k}) G/k (p'2; {¢k}) Ck(P '2) d4q ' '

(4.1)

This is represented schematically in fig. 10; Gjk stands for the product of the two dressed propagators, and Ki/is a Bethe-Salpeter kernel. All the cohtractions of Lorentz indices have been performed so that (4.1) involves only Lorentz scalars. Before discussing the implications of eq. (4.1) we must express this equation in terms of renormalized quantities in order to remove u.v. divergences. Unfortunately, since dynamical breakdown rests on non-perturbative solutions for Green functions even the definition of renormalization poses a problem. We describe here a possible way to carry out the renormalization program, at least in principle, for the hybrid skeleton expansion introduced above.

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Fig. 10. Symbolicrepresentation of eq. (4.1) p -~/ll-x--, p

/~p

=qsi(p2)

= Kij(p2,p'2; {~k}) p = dressed propagator

A conventional way to renormalize the unbroken theory consists in using renormalized perturbation theory. By this we mean that the usual Feynman graphs are written in terms of the renormalized coupling constant gx and made convergent by performing subtractions at some external momentum p2 = _X2. In this approach the bare coupling constant go never enters, but may be evaluated in terms o f g x and the renormalization constants Z ( g x , X2, A2); the latter are calculable in terms of the diagrams involved in the subtraction procedure, using an ultraviolet cutoff. One may adapt this method to the case where Green functions are expanded in terms of bare vertices and dressed propagators. Simply replace dressed propagators by their renormalized counterparts, bare vertices by gx and adjoin the appropriate class of subtraction diagrams at p2 = _)t2. There are no overlapping divergences between terms which have been resummed into the dressed renormalized propagators, and the remaining vertices. Thus all graphs are counted correctly. To define the method completely, it remains tospecify how the subtraction terms are evaluated in our broken theory. We choose to evaluate them by the corresponding graphs of the symmetric theory (at p2 = _~k2). It follows that our renormalized coupling constant gx is related to the bare coupling constant go in the usual way. For instance, in pure Yang-Mills theorv

In this scheme the renormalized coupling constants are always those of the symmetric theory. We may now identify the hitherto arbitrary parameter X with one of the generated masses (say m) to avoid an external mass scale;g m can differ from the physical coupling constant only by an additional finite renormalization factor. This formal discussion of the renormalization procedure justifies our previous derivation of the renormalized Ward identity, despite the fact that the latter will

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285

not be verified when summing only a finite number of graphs. Also the non-perturbative eq. (4.1) may now be seen to hold for renormalized quantities as well. This is because eq. (4.1) arises from a homogeneous vertex equation obtained by projecting onto the singular part at q2 = 0. Therefore this equation does not require the overall subtraction term usually expressed in terms of a renormalization constant multiplying the inhomogeneity. Hence, from now on (4.1) and its consequences will refer to renormalized quantities. While the above renormalization procedure leads to finite results in the unbroken theory, this is not so a priori in the present case. It remains to be shown that no new divergences are introduced by the symmetry breaking terms, a question which relies on the existence of solutions to eq. (4.1). This statement merits special emphasis: a theory exhibiting dynamical symmetry breaking is renormalizable if and only if the integral equations (4.1) possess a finite non-trivial solution. We proceed to a discussion of this equation. From the Feynman rules, one has generally K..q A(p'2, p2) = K..(p2, p'2) in II Euclidean Wick ratated space; as (pZ) -+ 0% Gjk(pZ; (~bk}) tends to a symmetric limit which we shall denote 6jk G(p2). A straightforward generalization o f the argument of ref. [9] leads to a relation between the dimensionless renormalized coupling constants of the theory ifKi/is a bounded kernel, that is, if .. p '2; {~bk})G(p2) f ~]K~.(p2,

G(p'2) d4p d4p ' < oo,

q

(4.2)

in Euclidean space. This is so because the boundedness of the kernel permits the mapping of (4.1) onto a finite system of algebraic equations analogous to those determining masses in the preceding section. As in sect. 3, it follows from scale invariance that only dimensionless ratios can be determined and that there remains one consistency equation for coupling constants. Thus if (4.2) were verified in general for a pure Yang-Mills field theory, the "dimensional transmutation" [5] found in sect. 3 would persist in the full theory. We now investigate this problem.

4.2. Pure Yang-Mills theory: "ladder approximation" To appreciate the relevance of the results of sect. 3 for the full problem, we shall first rephrase our "ladder approximation" in the form of eq. (4.1) and show how the boundedness condition (4.2) arises in this case; remember that no renormalization is required in the approximation of sect. 3. From the equation in fig. 9 and the factorization of the Goldstone pole exhibited in fig. 8, one may formulate eq. (3.9) as shown in fig. 11 where we have used the Ward identity (3.5). We multiply both sides of the equation byg uu- pUpV/p2 and define

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286

= t) oqt(p2) = [ T a , q 2 I I ( q 2 ) l s t , O/(P2) (t~-s.

[,][,;

(]=~,m) p ' 2 - M2 (k ~l',m')

//, p ' 2 - M 2

K(p2,p'2)=Kstlm~< ( p 2 , p ' 2 ) =

or

ram''

~-~d

,,,

p2 J

stO, ke,, pl) , (4.3)

btldOT

•

,

~

where Kstlm is the irreducible Bether-Salpeter kernel in our "ladder approximation . With these definitions the self-consistent equation for masses takes the form

Oi(O ) = f ~ K i ~ ( O ' p'2; {0k(0)}) G] k ( p ' 2 ; {0k(0))) 0k(0 ) d4p ' " 1

(4.4)

By the Ward identity (2.22), eq. (4.4) must be equivalent to the convergent integral (3.8). This implies that the (logarithmic) divergence expected by power counting must vanish as a consequence of gauge invariance. It now follows by power counting that the integral

I =

fKiL(p 2, p'2;

{0k (0)}) G/k ( p ' 2 ; {0k(0))) d4p '

(4.5)

Ct;o')~'~') Q

P/m/I~p (s,~)

.,oy

(t,v)

p//~.~~t,~\ (

-=

)

/

(

+

p

~-..... .I-_ _,_.~<

Fig. 11. Reformulation of the eigenvalue equation in the "ladder approximation" a

(s,,)

(t,v)

(l, ~)

p'

p'(m, r)

(s, u)

p

~ (t, v) P

= [Ta' tI, uls, t

--stlm ~v, P')

287

F. Englert et al. / Yang-Mills fieM theory (1

(b,v)

(c,o)

~/%/~', . . . . . .

•

(b)

/---% •

a

, ~

....

(c)

÷.-,

,'¢

\

f

~

Fig. 12. Coupled integral equations for q# r.a.bc. The dressed (renormalized) propagators are doubly slashed; the irreducible Bethe-Salpete~"kUernelsare represented by rectangular boxes and the divergences of vertex functions by circles. The letters A, B, C, ... refer to the various vertex functions, in particular p //~kx. P (b,v) (c,o)

. __. 0 =qla l,abe l~uo(P),q

is convergent, even for non-vanishing p2, so that KiL is a convergent kernel in the sense o f (4.2). Thus one could generalize the algebraic equation (4.4) to an integral equation for fbi(p ) using K/Land still end up with an eigenvalue condition. Such a generalization would not be o f interest in itself, as there is no reason to believe that it is consistent with gauge invariance. It is however interesting to note that the reason why our "ladder approximation" leads to an eigenvalue condition is traceable to the boundedness o f the " l a d d e r " kernel; it is not simply due to the fact that the approximation ofq211(q 2) by a constant M 2 in (3.4) has reduced at the outset the self-consistent problem to a system o f algebraic equations.

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288

o

o

(b,v)

(c,o)

((:,

P

Fig. 13. Iterative solution of the system of equations represented in fig. 12

p~

l

p' :~:bc(g(p p,)

P ~'/(b,v) (c,o~''~ P

va^p

"

4.3. Pure Yang-Mills theory: the general case

In general, the quantity qU pabc as q ~ 0 does not satisfy a single integral equ'a#po tion, but a system of coupled integral equations. This is presented in fig. 12. Formally, one may solve this system by an iterative procedure where the "unperturbed" value of q u rabc is obtained by taking into account in fig. 12 only the diagrams 12a. This is then corrected by substituting for the other vertices the solutions obtained by solving their respective equations iteratively, considering the terms involving qU FabC /~Vt7 as an inhomogeneity. The result is depicted in fig. 13 where the sum of irreducible Bethe-Salpeter kernels of the diagrams 12a has been replaced by a more general kernel/~-berg ~a~o (p, p,). The method is valid only i f / ~ does not itself exhibit a NG singularity at q = 0; the latter should therefore already appear in the "unperturbed" problem defined by the irreducible kernels of the diagrams 12a or by some approximation for them. Such a "bare NG boson" appearing in the ve~ex abc "- of the kernel K would Fur o would then drive the mass instability and the remainder provide its "dressing". This situation would be in close analogy to what happens when the symmetry is broken through non-vanishing expextation values of scalar fields. In this case the bare NG boson appears already at the stage of tree diagrams, which are not present here, and is coupled to lowest order in g only t o I-~abc HigherI~Ua" order radiative corrections dress this bare propagator --i/(2~T)4 (1/q2) to all orders in g. We cannot prove that the mechanism described above is the correct one. Nevertheless it appears plausible in view of the fact that in our "ladder approximation"

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289

(fig. 12a) we have indeed obtained a "bare" NG boson. We shall therefore assume that the procedure leading to the equation shown in fig. 13 is in fact convergent. With this assumption, we shall now show why we expect the existence of an eigenvalue condition for the low-energy renormalized coupling constant. Since we have departed from our "ladder approximation", u.v. divergences will occur, at least in individual graphs. As discussed in subsect. 4.,1, these may be handled by introducing counter terms of the unbroken (symmetric) theory so that the equation shown in fig. 13 remains valid for renormalized quantities. Using the Ward identity (2.23) and factoring out the NG (or ghost) singularity as well as the finite renormalization constant, we may set q = 0. After contracting both sides of the equation with guy_ pUpV/p2, we obtain eq. (4.1). The Kernel Ki] is now

[g P~--Pv7 "'stlm~UVozf"v' P') , p2 A

Ki/(p2,p'2; {~)k))(l=st [gar P~P~7 p'2 ] uv

(4.6)

(]=-lm)

and we wish to show that (4.2) is satisfied for this kernel. In what follows, we assume the validity of the power counting procedure. However, our argument cannot rest solely on power-counting, which would simply indicate that the maximal divergence in (4.6) is logarithmic. As in sect. 4.2 we shall resort to a comparison with the Schwinger-Dyson equation for the vector self-energy. The presence of overlapping divergences slightly complicates matters. When the P graph in fig. 4 is renormalized one obtains for IIuv (= Z 3 Huv) a result proportional t to Z 1 . This means that the integrand in IIu, would lead to divergences when expressed in terms of renormalized quantities, even when the external momentum goes to zero. To cancel these divergences against those appearing in Z 1 , one should retain the cut-off in the renormalized propagators and vertices, thereby losing all benefit of the renormalization program. An alternative way is to use the following identity (which follows from the transversality of IIuv) II..(p)

= -po --

(4.7)

nOv(p).

r_..

Fig. 14. Graphical expression of (0/0 p~)Ha(p): contribution arising from the diagram of fig. 4 and the truncated vertex equation of fig. 15. The cross stands for the operation a/ap ~.

Fig. 15. Truncated vertex equation used in the example of fig. 14.

290

F. Englert et al. / Yang-MillsfieM theory

When a/O pU Ha u(p) is expanded in Feynman graphs [ 18], the contribution o f fig. 4 becomes converted into that of fig. 14. Here we have, for simplicity, kept only those contributions to pabc which arise from the truncated vertex equation of fig. 15. To ,tzpo obtain the renormalized counterpart of fig. 14, it now suffices to replace, therein, all unrenormalized quantities by renormalized ones; no explicit renormalization t constant will appear in the resulting expression for IIu~(p ). We now perform the contraction with Pa in (4.7) and use the renormalized version o f the Ward identity (2.23). It should be noted that this procedure is applicable only to the guy part o f Huv(0 ), where just the regular part of the right-hand vertex contributes*. Thus as p -+ 0, the value o f p 2 H 'aa' ( p 2 ) obtained from fig. 14 and the Ward . identity has the form •Ump_+O jr p '2 11'~~p '2") F ( p '2, p ) d4p ' in matrix notation. Here p ' 2 [ I ' ( p ' 2 ) arises from the Ward identity applied to the left-hand vertices and the kernel contains that part of the right-hand vertex which is regular as p -+ 0, except for logarithmic singularities [19]. Hence, as p'2 _+ oo, the kernel F should go to the symmetric limit: This means that the asymptotic lip 4 decrease predicted b y power counting is enhanced through the group symmetry by at least two powers of p, in the manner o f sect. 3. A similar analysis may be carried out for the remaining vertices if we retain the above assumption that the symmetry breaking is driven by the induced singularity in pabc. This simply amounts to replacing F by a more complicated kernel having, b y power counting and group symmetry, the same asymptotic behaviour. We may then relate this general F ( p '2, 0) to Kij(O, p '2 )Gj(p '2) in (4.4). This leads again, as in subsect. 4.2, to the boundedness condition (4.2). In conclusion we see that the mechanism which led to an eigenvalue condition in our "ladder approximation" may survive in the full theory. Spontaneous mass generation in pure Yang-Mills theory would then occur only for specific values o f the (renormalized) coupling constant.

4.4. Coupling to fermions For the sake o f completeness, we briefly recall our previous conclusions [9] concerning "minimal" Lagrangians. These Lagrangians only contain fermions coupled minimally to the Yang-Mills fields associated to some subgroup o f the fermion kinetic energy. It was shown in ref. [9] that when each separate Yang-Mills field couples in the Lagrangian to massless fermions o f definite chirality, a relation arises between the dimensionless coupling constants of the theory. Such a relation, * More precisely, since infra-red logarithmically divergent terms may be present, we should not really go to the limit p = 0; but we may neglect all terms linear in p which were omitted from (2.22). Had we attempted to calculate the p~pu/p2 term (from fig. 4), as was done in sect. 3 for the "ladder approximation", we would not have been allowed to use this procedure. In fact when ultraviolet divergences occur, precisely those linear terms should be responsible, in the renormalized theory, for cancelling the divergences contained in Z 1.

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291

or eigenvalue condition, is unrelated to the Gell-Mann-Low equation [19], but arises for much the same reason as for pure Yang-Mills fields. Namely, the self consistent equation for £ 2 ( p 2) has a bounded kernel. This stems from the fact that for such Lagrangians, there is no asymptotic communication between left- and right-handed fermions in the graphs, therefore nothing can drive a mass instability in the asymptotic region. In tile low-energy region, however, left-right communication in the kernel is possible due to non-linear effects of the mass generation itself; the fast decrease of the kernel in the asymptotic region is responsible in that case,just as in the present work, for the low-energy eigenvalue condition. The argument of ref. [9] is quite general, but when the gauge groups were nonAbelian, we had to limit ourselves to cases where the meson symmetry breaking terms were expressible through Feynman graphs in terms of Z 2 ( p 2) alone. Thus we assumed that the instability was driven by the fermion Yang-Mills vertex P,, in the same sense as here it is driven by pabc. We see now that this result is more general ~vO because if we use (4.1) where ~bi(q2) refers both to fermion and meson symmetry breaking terms, the boundedness condition (4.2) still follows. This, of course, remains contingent upon the validity of the mechanism described in subsect. 4.3 to calculate the pure mesonic couplings.

5. Consistency of the "ladder approximation"

5.1. Preliminaries We shall verify that the approximation presented in sect. 3 can be diagrammaticalD interpreted as a type of ladder approximation, and that it is consistent with the gauge invariance of the theory in the sense that the transversality relations (2.16) are satisfied. It will be convenient from now on to associate a specific group to out YangMills theory. We choose SU 3 and a spontaneous breakdown M 2 ~ y 2 . The strange components 4, 5, 6, 7 will thus acquire a common mass m while 8 and 1,2, 3 remain massless. The [qvqv/q 2 ]q~0 contribution to Iluv(0), which we shall call 11~tv L has been evaluated in (3.8); in the present case the resulting eigenvalue equation reduces to (3.11) with C = 3. We may rewrite (3.8), which corresponds to fig. 6c, in the form

llLaa' = t t t , [q~qJq2]q-'O ~n -- I g 2 ~ fabc fa'bc m4 9

=0 with

otherwise ,

'

i (2704

I,

(a = a' = 4,5,6,7),

(5.1)

F. Englert et al. / Yang-Millsfield theory

292

1

(5.2)

I = f d 4 p p2(p2 - m2)2 We do not set n = 4 at this time. The reason for this will soon become clear. i

5.2. Transversality of IIaua In evaluating the guy contribution to I]a'a (0) we shall be confronted with divergent O,v integrals whose divergences will eventually cancel out. To handle these divergences correctly, we will use the dimensional regularization procedure [14]. A useful property of this tool is that all linearly, quadratically, and so on, divergent integrals having no dependence on masses and external momenta may be set equal to zero. This is why the diagram of fig. 5c has not been retained in fig. 6. The diagram of fig. 6a gives, ifa is 4, 5, 6, 7 ( 6 a ) - Efabcfa'bc 2(n - 1)g 2

b,c

~

i

fdnpPuP, p2

-p2 - _ m 2

(5.3)

By the proporty stated above, we may symbolically pose

fd

np _

- 0

(n > 2 ) .

(5.4)

Hence (5.3) may be written as

(6a)=--~fabcfa'bc2(n--b,c

1)g2

~ f

Pu Pv

dnp p4

1

p2_m 2

(5.s)

which can easily be shown to be equivalent to ( 6 a ) - = - ~fabcfa'bc ( n - 1)g 2

b,c

i 4m4 F,n Pu pv 1 (5.6) (2~-) 4 4 - n j ° p p 4 (p2 _m2)2 "

To motivate the introduction of the propagator (3.6) in fig. 6b, we first evaluate this diagram with the original tensor propagator (2.6). Calling this contribution (6b) 0, we have for a = 4, 5,6, 7 (6b) 0 = ~ fabc fa'bc g2 b, c

i

2 • (2704

dnp I

($7,

lq Englert et al. / Yang-Mills field theorl,

293

Note that all diagrams where the internal vector propagator is non-strange vanish in view of (5.4); this explains the factor of ~1 in (5.7). The first term in (5.7) may, in the same manner as above, be transformed to

(6b)O, 1 = ~ fabc fa'bc b.c

g2m4 2

2

i

(

guvdnP

4 - - n (2~.)4 ,,' p 2 ( 7

(5.8)

_~m2)2

Combining (5.6) and (5.8) we get

, m4(n - 1)g 2.. i n ~

dnp

(6a) + (68) 0,1 - b, ~c (--fabc la'bc)

guy __fp2(p2 _~ rn-2)2 " (5.9)

Comparing with (5.1), we see that (5.9) is precisely the term we need to obtain a transverse expression for limq__,0 Ilaa'(q) ifa = a' = 4, 5, 6, 7. However, the last btv term in (5.7) destroys the transversality and the convergence of the resulting Ilu,. This annoying term can be eliminated by using (3.6) instead of (2.6). The selfenergy of any strange vector meson is then transverse when computed from the diagrams of fig. 6. It remains to be verified that for a = 1,2, 3 or 8 where the diagram of fig. 6c does not contribute, we indeed find llu,(0 ) = 0. We have

(6a)-

G fabc fa'bc g2 2(n-- 1) i b, c strange

(6b)--

~

b,e strange

fabcfa, bcg2(n_l) (--i) ~

(p2

dnp

pup,,

_- m~2)2'

(5.1o)

(5.11)

( 2 - 2 ~ % 2 ~ m 2"

The reason why non-strange b and c do not contribute to (5.10) and (5.11) is again (5.4). Evaluation of the integrals yields (6a) + (6b) = 0. Note that this result again depends critically on the replacement of (2.6) by (3.6). Therefore, we tentavely adopt the modified tensor propagator (3.6) and investigate its implications on the Ward Identities.

5.3. Identification with a "ladder approximation" We first evaluate the Goldstone-vector-vector coupling introduced in fig, 7 by solving the equation represented in fig. 8. This coupling must be of the form

orc = K Dbc gor Axb

(5.12)

x

where K has the dimension of a mass and the constants Dbc are dimensionless, lnx

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294

setting (5.12) in the diagrams of fig. 8, we obtain for strange index "a" 3

i

g~abe i(qU/q2)m2g°r =g e,~1;x ~rael'Dxef Dxbe K2m2i(qu/q2)gor 4 (21r)4 I , (5.13) where I is the integral (5.2), and we have introduced

5~h~,: Lbc (xb - x c ) , c/

)kb = 1 ,

b strange ,

Xb = 0 ,

b non-strange.

Hence the constant

(5.14)

Dbc must be chosen to satisfy

e, f, x

(5.15)

xeJ a 'J

One easily shows that this is solved by Dbc -- (xD)bc= ~([Fs'

Fx]+)bc,

= 0,

x strange, x non-strange.

(5.16)

Using (5.15), eq. (5.13) reduces to l=~K2_

i 1. (2n) 4

(5.17)

On the other hand, the eigenvalue condition (3.1 1) may be rewritten in the form 1 =3g2m 2

1 ,

(C=3),

(5.18)

so that

K = v~grn,

(5.19)

A solution of the equation shown in fig. 8 can therefore found: it is (5.12) with

Dbc and K given, respectively, by (5.16) and (5.19). This means that the factorization exhibited in fig. 7 is justified. To prove that our approximation is indeed the "ladder approximation" defined in sect. 3, it remains to be shown that the equation of fig. 9 is satisfied with Aaxrbcgiven by (5.12). The sum of the diagrams of fig. 9 gives (9)~(n-1)g 3

m ~ U @Deffbgefcg/]

i

{,--4p¢Py+go.r(p2-m2) d~ , (5.20)

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295

w h i c h m a y b e w r i t t e n as (9) -= (n - 1 ) g 3 m G 1

4p~p. p2 r )

g~y

if(

+ (n - 1) g3m G2 -~frc~

d-npp2(p2 = m 2)

4p~p.), I dnp g~ -p2 _ m2 / p2(p2 _ m

(5.21)

2) '

where

(71 =

(72 =

~

g non-strange

~

g strange

e~f %/'3Def fbge fc~ ", ,

~ ~ Dxeffbge fcgf" e, f

(5.22)

T h e integrals in (5.21 ) are c o n v e r g e n t w h e n n -+ 4. This can b e seen explicitly b y using (5.4). One t h u s o b t a i n s

(cl,¢x~b,[3) . (a,a. ', ~.)~~Cb,[~)• (a,ct

(a,

13)

b~8) t

(¢, Fig. 16. Graphs contributing to the three-point function in our "ladder approximation"; only terms linear in external momenta are retained. q

•~

. . . .

(Cz)a~,, x

(,7, t~)

-q -r (a, u ~

b ..... ~ r = (C2)au, be C

F. Englertet al. / Yang-MillsfieM theory

296

( 9 ) = ( G l _ G 2 ) m3g 3 2 ( n - 1) i I. n g ~ (2~r)--"-~

(5.23)

Furthermore, it turns out that (5.24)

G 1 -G2=~x/~Dbe,

so that using the eigenvalue condition (5.18), one indeed sees that (5.23) is equal to (5.12). Note the importance of the propagator (3.6) in ensuring the validity of the dynamical interpretation (fig. 9) of the approximation of sect. 3.

5.4. Transversality of the three-point function The graphs contributing to the three-point function in our approximation are represented in fig. 16. This introduces two new couplings C 1 and C2 , which in our "ladder approximation" obey the equations represented in figs. 17 and 18. The calculation of C 1 is a by-product of eq. (5.13) (see fig. 8). Using the eigenvalue equation (5.18) one has (C1)au, x = Onqu eax '

(5.2 5)

eax =X/~ b~,c~abcDbc .

(5.26)

where

Note that eax takes only the values -+1, or 0. To evaluate C2, we expand the integrand in fig. 18 to first order in r and q to obtain the kinematic factor (r - q)u. One has

( 1 8 ) - - 3 i g m 2 x,y, ~z f X qp2

[3~yxDcXZfazYl(~n)

{ ( r - q)u

d4p - £2 f" 2p" r m2)(p2 m2)(p2_m2 ) j PuLp2_m2

X

d4p

=

3 (5.27)

(p2_m2y)(p2 _m 2) ( p 2 _ mz2) J "

(

2p'q ]

~. . . .

Fig. 17. Equation determining (C1)a~,x"

F. Englert et al. / Yang-MillsfieM theory

297

-q-r

-q-~ct,l~) =

/q/~

b

"N~,,~r

c

,yq

"~r

Fig. 18. Equation determining(C2)a~,bc. Explicitly, this may be written as

i r drip (gou 4popul --(n- l)igm3 G3(r --q)o (~)4 Up4(p2- 2 m 2) - p-2 l 4popu 1 ,(5.28) -~ _ ~ 2 ] '~ (p2 -m2)2

i dnp (gou -(n - 1) igm 3 G4(r -q)o ~n-~;-'p-f where G3=

2

zy

3~yxDcxzfazy '

G4=

3~yxDxzfazy.

~

zy

x strange

(5.29)

x non strange

Both integrals in (5.28) turn out to be equal to [i/(27r)4 ] ~ Iguo; furthermore, it can be shown that (5.30)

G3 + G4 = - I fabc , so that

(C2)a#,b c = 3ig3m 2 (r - q)~

n-1

i

(2rr)4

n

I Lbc •

(5.31)

Using again the eigenvalue equation (5.18), one finds (C2)au, bc = ig(r - q)u fabc "

(5.32)

With (5.12), (5.25) and (5.32), the proof of transversality for the sum of the diagrams of fig. 16 is straightforward. The bare Yang-Mills vertex is transverse by itself. So is the sum of the remaining three graphs; this is readily checked by multi-

(a,a

Fig. ] 9 , C o n t r i b u t i o n to the

four.-poJnt

,131

(a.=

,p)

f u n c t i o n s when a]] e×terna] mesons are non-strange.

298

F. Englert et al. / Yang-Mills fieM theory

,J

J

/ps Fig. 20. Coupling of the two NG bosons to two Yang-Mills fields: all external moments are zero. plying each graph by A a Bt3 C'r and using the following identity for the group indices eax e ey fXYc = f abc "

(5.33)

5.5. Transversality o f the four-point f u n c t i o n s

If all external mesons are non-strange, then the only graphs to be considered are those of fig. 19 (and the four similar ones in the crossed channels). The transversality relation is verified channel by channel: this is because on the one hand the intermediate propagator (3.4) coincides in this case with the bare propagator, on the other hand the transverse part o f the tensor propagator does not contribute to the four-point transversality relation. Therefore, the graphs o f fig. 19 satisfy the transversality relations in the same manner as the graphs of ordinary perturbation theory to order g2. In other cases the investigation o f the transversality relation is considerably more tedious because one should exchange not only vector and tensor propagators, but also NG bosons. In the case where two external mesons are strange and two nonstrange, one obtains in this way 23 one-particle reducible graphs to be contracted with A a B ~ C ~ D ~ . The hard labouring reader will find that the result is not zero. In the spirit o f subsect. 5.4 he will be led to determine the form o f a "contact term" between two NG and two Yang-Mills vectors, in order to cancel the effect o f the previous 23 graphs. It turns out that this contact is given by the sum of the diagrams o f fig. 20 at zero external momentum. The desired result is easily obtained upon using again the eigenvalue equation (5.18). When all external mesons are strange, the transversality relations are saturated

F. Englert et al. / Yank-MillsJield theory

299

by the reducible graphs alone; we have not studied in this case the numerous possible contact terms.

5. 6. Remarks While the diagrammatic interpretation of the approximation o f sect. 3 has been clarified, the consistency achieved in this way rests strongly on the use of the tensor propagator (3.6). As the difference between (3.6) and (2.6) is mass independent, it cannot be understood directly as a symmetry breaking effect. Rather the propagator (3.6) seems to be a more convenient unperturbed propagator to study symmetry breakdown than (2.6); this suggests that one should look for a formalism where the separation o f the tensor propagator A into T and L is present ab initio in the Lagrangian. Such is the case in the so-called linear formalism o f Faddeev and Popov [21 ] which we suspect might provide a better understanding of this problem. The reason why the Faddeev-Popov ghost plays no role at the level of our ladder approximation follows from two facts: (a) there is no ghost-ghost-NG coupling and therefore no ghost contribution to the quqv/q 2 term in Iluv(0); (b) there are no radiative corrections to the ghost propagator and consequently no ghost contributions to the guy term in IIuv (0), as follows from dimensional regularization. The second requirement is consistent with the first because it is precisely the absense of radiative corrections to the ghost that justified the use o f the simple Ward identity (2.21) which results in a convergent quqv/q 2 term. In fact a ghost contribution would inevitably lead to a logarithmically divergent contribution to the self energy thereby ruining our whole scheme. Therefore we see that the very fact that the ghost plays no role as a driving term for the mass instability is crucial for obtaining a type B theory. Of course at this stage this cannot be taken as justification for our approach: it is merely a consistency assumption * Finally, we wish to point out that even within this simple approximation, tile phenomenon described in this paper admits no classical analogue; this is because the ladder summation which leads to our bare Goldstone singularity involves an infinite number of loops. This means that the mass generated thereby cannot be expanded in a power series in h [5] and therefore our dynamical symmetry breaking mechanism may be interpreted as arising from quantum mechanical fluctuations alone. This situation is distinct from the case where masses of Yang-Mills mesons are generated through expectation values o f scalar mesons wherein the classical limit is recovered in tree approximation. Also, our mechanism is unrelated to a possible mass instability in the classical theory of massless Yang-Mills fields [21 ]. We are greatly indebted to R. Brout, W. Fischler, E. Gunzig, J. Nuyts and J. ZinnJustin for useful discussions. Two of us (F.E. and P.N.) would like to thank the Theory Division at CERN for its warm hospitality. * Note in this respect that dynamical symmetry breaking in pure Yang-Mills theory from the the viewpoint of a type A theory has been recently investigated in ref. [ 7 ].

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b: Englert et al. / Yang-Mills fieM theory

References [1 ] C.N. Yang and R.L. Mills, Phys. Rev. 96 (1959) 191. [2] F. Englert and R. Brout, Phys. Rev. Letters 13 (1964) 321. [3] P.W. Higgs, Phys. Letters 12 (1964) 132; Phys. Rev. Letters 13 (1974) 508; Phys. Rev. 145 (1966) 1156. [4] F. Englert, R. Brout and M.F. Thiry, Nuovo Cimento 43 (1966) 244. [5] S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888. [6] E.C. Poggio, E. Tomboulis and S.-H.H. Tye, MIT preprint CTP 404 (1974); J.M. Cornwall, Phys. Rev. D10 (1974) 500. [7] E.J. Eichten and F.L. Feinberg, Phys. Rev. D10 (1974) 3254. [8] B.W. Lee and J. Zinn-Justin, Phys. Rev. D5 (1972) 3121;D7 (1973) 1049. [9] F. Englert, J.M. Frere and P. Nicoletopoulos, Phys. Letters 52B (1974) 433. [10] G. 't Hooft, Nucl. Phys. B33 (1971) 173. [11] L.D. Faddeev and V.N. Popov, Phys. Letters 25B (1967) 29. [12] J.C. Taylor, Nucl. Phys. B33 (1971) 436. [ 13 ] A. Slavnov, Kiev report No. ITP-71-83 E, unpublished. [14] G. 't Hooft and M.T. Veltman, Nucl. Phys. B44 (1972) 189. [15] M. Baker, K. Johnson and R. Willey, Phys. Rev. 136B (1964) 1111. [16] R. Jackiw and K. Johnson, Phys. Rev. D8 (1973) 2386. [17] J.M. Cornwall and R.E. Norton, Phys. Rev. D8 (1973) 3338.See also ref. [2]. [18] G. Mack and K. Symanzik, DESY 72/19 Hamburg (1972). [19] M. Gell-Mann and F.E. Low, Phys. Rev. 95 (1954) 1300. [20] L.D. Faddeev and V.N. Popov, NAL report THY-57, unpublished. [21] M. Veltman, Seminar given at CERN in September 1974.