Volume 198, number 1 PHYSICS LETTERS B 12 November 1987

N O N P E R T U R B A T I V E SCALE A N O M A L Y AND D I L A T O N S IN GAUGE F I E L D T H E O R I E S

V.P. G U S Y N I N and V.A. M I R A N S K Y Institute for Theoretical Physics, 252130 Kiev 130, USSR

Received 10 June 1987

The scale anomaly caused by the dynamics of spontaneous chiral symmetry breaking is studied in vector-like gauge theories. The contribution of this anomaly to the gluon condensate and dilaton mass is determined. The criterion for the realization of the dynamics of a light dilaton is obtained. The applications of these results to quantum chromodynamics and technicolour models are considerd.

In the present paper we study the scale anomaly caused by the nonperturbative dynamics o f sponta- neous chiral symmetry breaking in vector-like gauge theories. In the two-loop approximation, we find the contribution o f this anomaly to the gluon condensate and to the mass o f the dilaton - a scalar meson cou- pled with the dilatation current. We consider both asymptotically free (AF) and nonasymptotically free (NAF) gauge theories (the latter can be either abe- lian theories or nonabelian theories with a suffi- ciently large number of fermions). In particular, we shall apply these results to QCD and technicolour models.

As is well known, in gauge theories the following relation for the divergence of the dilatation current takes place in all orders of perturbation theory [ 1 ]:

0 u ~ ' = 0~

=[13( a ) / 4 a ] ( F~,F ~') ....

+ [1 + r m ( a ) ] m c ( ~ , ) . . . . . (1)

where OU, is the energy-momentum tensor, (F~ x F u°) .... and (t#~) .... are properly defined com- posite operators, ~)m (O/) is the anomalous dimension of the operator ( ~ , ) .... and mc is a current mass of fermions. I f one assumes the conventional point o f view that there is only one phase in coupling con- stant with the fixed point o~ = 0 in AF theories, then the relation (1) with the ordinary fl(ot) and 7m(C~) functions should be exact there. Another situation could take place in NAF theories. As was pointed out

in refs. [2,3] (see also the reviews in ref. [4]) , the existence o f a nontrivial S-matrix in the cont inuum limit should signify the presence o f a critical cou- pling Ore> 0 (nontrivial ultraviolet stable fixed point) separating two phases with different structures of re- normalizations, in particular, with different func- tions fl(o~) and ~'m(Ot) (such a situation takes place in the ladder approximation [ 2,3]; besides, this pos- sibility is supported by computer simulations in lat- tice QED [5] ~l. We will show that in this case the scale anomaly in the supercritical (ot > otc) phase es- sentially differs f rom that in the perturbative phase with o~ < etc. We shall consider consequences o f this result and, in particular, we shall discuss the dynam- ics o f the dilaton formed in the critical regime with o~ = o~c [6,8,12].

It is essential that under certain conditions these results can be applied also in asymptotically free the- odes. In particular, we shall discuss the dynamics of scalar mesons in QCD and also we shall consider the AF theories with slowly varying coupling constant, which have recently been used in the technicolour scheme [ 10,11 ].

Let us quote the main relations derived in the present paper.

~J At present such theories are being intensively studied [ 6-9 ]. In particular, the notorious flavour-changing neutral-current problem in technicolour models can be solved along these lines [ 7,8 ] (see also a similar scenario of the technicolour dynamics in refs. [10,11]).

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Volume 198, number 1 PHYSICS LETTERS B 12 November 1987

The vacuum average of the dilatation current anomaly connected with the dynamics of sponta- neous chiral symmetry breaking in the critical re- gime with a=ac [2,3] is

(0104,10) = - (4ng/lg 4) m 4 , (2)

where n is a dimension of the fermion representation of the gauge group G, K is a number of fermion fla- vours, mo is a dynamical mass of fermions. Also, the following relation is valid [compare with eq. (I) at me=0]:

(0 [0~ ,10)= lim [//(a)/4a](OlFu~P'~lO), (ot~Otc)

(3)

where //(a) is a //-function of the supercritical ( a > a c ) phase [see eq. (14) below].

The square of the mass of the dilaton x is

2 M,~ = (16nK/zc4)F~2m 4 , (4)

where the parameter F~ is determined from the relation

- - 1 ¢ X P i.4-2 (010¢ Ix, q) - 3, . . . . . . -q,'q,)F,¢ . (5)

Moreover, assuming the validity of some relation of the linear a-model [see eq. (24) below ], we find that

M 2 =2nma/n4F~ , (6)

where F~ determines the coupling of pseudoscalar mesons to axial currents: i6abF,~P,,= (0 IJ~5 I P; b) .

Together with the recently derived relation for the chiral condensate [ 13 ] ~2

( 0 I (~¢ ~//) .... I 0 ) = -- (8nK/n4) m 3, (7)

these relations express the main characteristics of the dynamics of spontaneous chiral symmetry breaking in the critical regime in terms of the dynamical fer- mion mass.

All these relations are derived in the two-loop ap- proximation for the effective potential for composite operators [ 14 ], i.e., in the approximation which has been used in refs. [2-11 ].

In the two-loop approximation (with the bare

,2 The renormalized operator (~qu),on =limA_~Z~.(A) (~'~/)A, where in the considered approximation the renormalization constant Zm = 2ma/nA in the critical regime (limA~a(A) =ac) [3].

vertex and the bare propagator of gauge fields) the effective potential V takes the following form in the Landau gauge [ 14]:

V(B)= n g { ! 2du I ( B2 ) I n 1 }. 8 7 [ 2 U "31- U u )

2BZ(u) ]

u+B2(u)J A 2 A 2

+2 u+B2(u ) v+B2(v-------- ~ 0 0

where the fermion propagator G(p) = [/~A(p 2) - B(p 2) ] - , (in this approximation A (p 2) = 1 [ 4,14 ] ), K(u, v)=O(u-v)/u+O(v-u)/v and 2=(3a /4r t ) ×C(n ) , where C(n) is the value of the quadratic Casimir operator in the fermion representation {n} of the gauge group G. The ultraviolet cut-off A will be removed (A--.~) in all final expressions.

The extremum condition ~ V/SB=O leads to the following nonlinear equation:

A 2 vB(v)

B(u) =2 f dvK(u, v) v+B2(v ) . (9) 0

In this approximation the perturbative//-function equals zero. So it seems that in the continuum limit, A--, oo, the dilatation current has to be conserved and therefore

v(B)=(OlO°olO)=i(OlOGlO)=O, (10)

where /~ is a solution of eq. (9). At 2<2c=~ [ a < ac = n/3 C(n) ] this equation has only the trivial solution/~= 0 [2-4,15] and hence in this case the quantity V(B) is indeed zero. However, in the su- percritical (2>2c= ~) phase the situation changes. Here eq. (9) has a nontrivial solution [ 2-4,15 ] pos- sessing the following properties [ 2,3 ]:

B(pZ ) m ~ ( c thnv ),/2

× sin(2v ln(p/md) +27(v) -a rc tg 2v) , (1 1 )

where v= ½ ~ 1 ,

X(v)=arc [F(1 + 2iv)/l'2(½+ie)] ~ 4vln2.

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Volume 198, number 1 PHYSICS LETTERS B 12 November 1987

The dynamical mass rnd is determined from the relation

md-~Aexp 2v ]~¢1-- 4Aexp - ~ v (12)

and in the continuum limit it remains finite if [ 2,3 ]

7~ 7t 2

a(A) - 3C(n) ( l + ln2(4A/md))

' a c - • ( 1 3 ) A~o 3C(n)

This relation implies that in the supercritical phase the fl-function

Oa(A) 2 (3C(n)a )3/2 fl(a)=- OlnA 3C(n) 1

(14)

is nontrivial and it has a zero at a = otc = rc/3C(n). Let us show that there is a scale anomaly for which

the relations (3) - (4) are valid in the continuum limit when a ~ ac. Note that the existence of the anomaly already follows from the observation in the rather old paper ref. [ 16 ]: the value of the potential V for the solutions of eq. (9) is given by

V(/~) = - ~-~ z d u u ~ , (15) 0

where the function 7J(z) =ln(1 +z) -z/(1 +z) > 0 at z > 0. Therefore, for the nontrivial solution the quan- tity <010Uu 10> #0.

To derive eqs. ( 2 ) - (4 ) we need two relations,

(udB+ B(u) ) _ , (16)

A 2

A2B(A2)= 2 f dvvB(v) v+B2(v ) , (17) 0

following from eq. (9), and also the relation

1 0 V ( B ) ~ B ( 1 8 ) V(/~) = 2 0 lnA /

The latter follows from the extremum condition at B=B with respect to the scale transformations B~Bk=kB(PZ/k z) for the effective potential (8) [ V(Bk) =k4V(B, A2/kE)]:

dV(Bk) 5 dBk k=l= ] g-~B R= - ~ k= du=O (19)

[note that this transformation corresponds to the conventional scale transformation in coordinate space: ~(x)--*gf(x)=k3/e~(kx)]. From eqs. (8), (11 ) and eqs. ( 16)- (18) we find the relation (2):

<010~10> = lim 4V(B)= lim 20V(B)

nK 4 ( ./~E(A2)'~ = - l i m ~ - ~ 2 A l n _ l t ~ )

4nK 4 (20) " ~ - - 7~ 4 r o d .

Eq. (3) follows from eqs. (12), (14), (20) and from the well-known relation

(0 IFm, FU"lO> = -4a OV(a, A)/Oa (21)

[it can be obtained, for example, using the func- tional integral representation for the vacuum energy density V(/~)].

The relation (4) follows from eq. (20) and from the Dashen-like formula of the dynamics of the par- tial conservation of the dilatation current (PCDC) [8,17] ~3:

ME = - 4 F ~ -2<010~10) • (22)

The relation (4) leads to the following general conclusion: a light (M~ << md) dilaton can exist only if the following condition holds:

6 ~ F ~ >> md • (23)

Due to eq. (5), this means strong coupling of the di- laton with the energy-momentum tensor, i.e., with the gravitational field ~4. The nontriviality of this cri- terion is illustrated by the following point. If one as- sumes the validity of the relation of the linear a- model [ 17 ],

~3 The fact that in the critical regime with a = ctc the scalar meson is massive was shown in refs. [4,18]. Recently the same con- clusion has been drawn from the numerical computer analysis ofeq. (9).

t4 Of course, there exist other contributions to the scale anomaly

and dilaton mass in gauge theories. Therefore, this criterion should be considered only as a necessary condition for the ex- istence of such a particle.

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Volume 198, number 1 PHYSICS LETTERS B 12 November 1987

<01xl0> = - ~ / ~ F ~ , (24)

then in the PCDC situation, when [ 17 ]

O~ =F,~M~x , (25)

the relation (2) implies

M ~ = 4 n x ~ m4 /z~4 F, c F,~ , (26)

Comparing this relation with eq. (4) we find

F~ = 4 v / ~ F ~ , M2~=2nrn4/zt4~, (27)

and so in this case the criterion (23) takes the form

(10/,v/n)F,~ >> rod. (28)

However, this condition contradicts the conven- tional viewpoint that the quantity F,~/x/~ ~ I 0 - ~ ma (recall, for example, the relation

F~ --- (x/~/2n) md (29)

discussed in refs. [ 19,20]). Thus, the realization of the hypothesis of the light

dilaton [6,8,12] apparently requires a dynamics which is essentially different from that of the linear a-model.

The relations (2) and (4) have been derived in the continuum limit A ~ . However, it is clear that when the condition md/A << 1 (i.e., a - - a¢ << 1 ) holds these relations can be applied with a good accuracy in theories with finite cut-off too. In particular, if one assumes that the regime with a near-critical "frozen" running coupling [ a ( q 2) - a t < < 1] is realized in a region M 2 << q2 <<A 2 of an asymptotically free the- ory (M is a confinement scale), then the relations (2) and (4) can be used there. In this case they re- produce the effects of the part of the scale anomaly caused by the presence of the scale A at which the change of dynamic regime takes place.

Recently such theories have been considered in the technicolour scheme [ 10,11 ]. As follows from the beforesaid, the criterion (23) of realizing the light dilation dynamics remains valid there too.

Let us assume the dominating role of the super- critical [ a ( q 2 ) > a c ] regime for the dynamics of spontaneous chiral symmetry breaking in QCD [ 4 ] and consider the relations (2), (7) and (27) there (n = K = 3, a~= ~/4). In QCD the dynamical mass of quarks mo-~320 MeV and 0.6 GeV < A < 1 GeV so that m J A > 0.3 in this case [4]. It is clear that in such

a situation these relations can be regarded only as a qualitative estimate of the corresponding parameters:

(010~ 10) ~ - 4 ) < 10 -3 ( G e V ) 4 ,

( ( ~ / ' / ) . . . . ) = (l /K) <01 (~/) . . . . 10) ~

- (200 MeV) 3 , (30)

M,~~me . (31)

While the obtained value for the chiral condensate is satisfactory, that of the gluon condensate is smaller than the standard value (010~ I 0 ) = - (1.2-4) )< 10- 2 (GeV) 4 [ 21 ]. Since the value (010Uu I O) (30) reproduces only the gluon condensate part con- nected with the dynamics of spontaneous chiral sym- metry breaking, this fact agrees with the conventional viewpoint that the dominating contribution to the condensate ( 010~10) is given by the pure glu- odynamics. The small value of M~ (31 ) also agrees with the conventional scenario where the dilaton mass is basically determined by the gluodynamics and its value is greater than 1 GeV [22]. On the other hand, this estimate qualitatively agrees with the re- lation M, = 2mo for scalar mesons in QCD, which has been discussed in ref. [20], if we mean the value M~ as the part of the scalar meson mass connected with the dynamics of spontaneous chiral symmetry breaking.

In more detail these phenomenological questions will be considered elsewhere.

Thus, in vector-like gauge theories the dynamics of spontaneous chiral symmetry breaking in the crit- ical regime with a = a¢ leads to explicit scale sym- metry breaking. Is it possible to remove this nonperturbative anomaly by modifying such theo- ries? The complexity of this problem is demon- strated by the model of ref. [6 ] in which QED is supplemented with the chiral invariant four-fermion interaction. In this model the dilaton is also massive in the critical regime [6]. The regularity of this re- sult is illustrated by the fact that the relation of the type (15) for the effective potential takes place in this model too [6].

Thus, the phenomenon considered in the present paper can apparently take place for a wide class of field theories.

A preliminary version of this work has been pre-

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Volume 198, number 1 PHYSICS LETTERS B 12 November 1987

p a r e d d u r i n g t h e s tay o f o n e o f t he a u t h o r s ( V . A . M . )

at t he U n i v e r s i t y o f W e s t e r n O n t a r i o ( U W O ) . H e

t h a n k s t he m e m b e r s o f t h e D e p a r t m e n t o f A p p l i e d

M a t h e m a t i c s o f U W O , especial ly V. El ias a n d D .G .C .

M c K e o n , for t h e i r w a r m hosp i t a l i t y . H e h a s b e n e -

f i ted great ly f r o m c o n v e r s a t i o n s w i th V. Elias, D .G .C .

M c K e o n , A.A. M i g d a l a n d M . D . S c a d r o n . W e are

b o t h t h a n k f u l to P.I. F o m i n a n d Yu.A. S i t e n k o fo r

usefu l d i s cus s ions .

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