Dynamical symmetry breaking in QCD-like gauge theories

Volume 150B. number 4 PHYSICS LETTERS 10 January 1985

DYNAMICAL SYMMETRY BREAKING IN QCD-LIKE GAUGE THEORIES

R. CASALBUONI a,b, S. DE CURTIS a,c, D. DOMINICI a,b,d and R. GATTO b

a Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, Florence, Italy b Ddpartement de Physique Th dorique, Universitd de Gen~ve, Geneva, Switzerland e International SchoolforAdvanced Studies, Trieste, Italy d Dipartimento di Fisica dell'UniversitY, Florence, Italy

Received 6 September 1984

We present an explicit derivation of an effective action for composite operators which has the same local extrema as the effective action of CornwaU, Jackiw and Tomboulis. Our effective action, which we have already used in a particular appli- cation, can be shown to reproduce in quadrilinear field theories the effective action obtained with the method of eoUective variables at one-loop level. It has the advantage of eliminating the problem of unboundness-below of the potential in the free case, and when applied to QCD-like theories at one-loop level, gives, with the theoretically suggested ansatz, only the local minima. We discuss different ansatz with or without logarithmic corrections in the coupling constant and in the fermion self-energy, and we find that the result of spontaneous breaking of chiral symmetry for large coupling remains stable.

1. The effective action. In a previous paper [1 ] we introduced a modified version of the Cornwa l l - J ack iw-Tom. boulis (CJT) [2] effective action to deal with dynamical symmetry breaking (DSB) in gauge theories. We would like to show in this section the precise relation between the two actions and in which sense they are equivalent for computat ional purposes. To this end let us consider the CJT formalism for fermionic fields ff (x) (of course, the argument applies also to bosonic fields). One introduces bilocal sources K(x , y) , coupled to ~b (x) ~O (y), and de- fines the effective action as the double Legendre transform, F, of the generating functional of the connected Green functions. CJT have proved that F (in euclidean space) can be expanded into the following formal series:

FCJT(~. S) = I ( ~ ) - Tr log S - 1 _ T r ( D - 1 S ) - F 2 , (1)

where ~b and S are the Legendre-conjugate variables of the local and bilocal sources, and D is the free fermion propagator as given by the classical action I(~b). 1-' 2 is given by the two-particle irreducible vacuum diagrams evaluated by using S as the fermion propagator. It is easy to show that the bilocal source is the derivative o f F with respect to S, K = 6F/6S. From eq. (1) one then gets

K =S -1 - D -1 - 6U2/~S. (2)

By using this expression in eq. (1) one obtains

PCJT = I ( ~ ) -- Tr l og (D-1 + ~I,2/6S ) + Tr(~F2]6S)S - F2 - Tr log[1 + ( /9-1 + ~1-,2]~S)-1K] + Tr K S , (3)

and again using eq. (2) one gets our final expression:

I'CJT = F + Tr log(1 - KS) + Tr K S , (4)

where

F = l(~b) - Tr I o g ( / ) - I + 6F2/6S) + Tr(6F2/6S) S - F 2 (5)

is the form of the effective action we have used in our previous paper [1]. I t is clear from eqs. (4) and (2) that the following identities hold:

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Volume 150B, number 4 PHYSICS LETTERS 10 January 1985

FCJTIk= 0 = FIk= 0 , 8FcJT/aSIk= o = 8I'/SSIx= 0 = 0 . (6)

The second equation means that PCJT and P both give rise to the same Schwinger-Dyson equation:

S -1 = D - 1 + 8F2/8S, (7)

which is the extremum condition for the effective action. We see that in order to determine the extrema of the effective action the choice between FCj T and F is purely a matter of convenience. In particular, in the next section it will be shown that practical calculations are greatly simplified by the use of F, eq. (5).

We would like also to mention an attractive property of the modified action, eq. (5). Let us consider a field theory with quadrilinear interaction like the O(N) scalar model, or the non-renormalizable Nambu-Jona-l_asinio model [3], or the two-dimensional Gross-Neveu model [4]. All these theories can be reformulated in terms of collec- tive variables by introducing into the generating functional, composite fields bilinear in the elementary ones. Def'm- ing the effective action for both elementary and composite fields one can show (at least at one-loop) that it coincides with P, eq. (5), after convenient identification between the composite field and the variable S(x , x). In the case of the O(N) scalar model this has been shown in ref. [5] where, although no explicit use was made of the action (5), the way of evaluating the effective action was in fact equivalent to the use of definition (5). In the same model, in the limit N-+ 0% a similar trick was also used in ref. [2]. It is also possible to show explicitly the equiva- lence for the Nambu-Jona-Lasinio [6] and the Gross-Neveu t l models.

Finally, we want to mention a problem with the CJT approach: it is not guaranteed that the effective potential is bounded from below. This problem already arises at the level of the free theory [7,8]. This means that particular care must be taken in order to choose the correct minimum of FCJT, and that the finding of an unbounded potential is not an "a priori" index of the illness of the procedure. However, as we shall see, in the case of gauge theories at one-loop order and for any of the ansatz we shall use, F [as defined by eq. (5)] turns out to be bounded from below. Also, the effective action of eq. (5) does not have problems in the free case, simply because by turn- ing off the interaction F reduces to a constant functional.

2. Chiral symmetric breaking; ef fect o f the logarithmic corrections. We now turn to the problem of gauge theories. The calculation [ 1 ] we have done, made use of some simplifying assumptions, namely we did not take into account the fact that the coupling constant is "running", and we neglected logarithmic corrections in the leading asymptotic behaviour of the self-energy. The motivation was mainly that in this way the problem could be entire- ly treated analytically. Furthermore, a phenomenological analysis we have performed [9] in this context of the pseudoscalar masses did not show any obvious inadequacy of our treatment. In view of these facts, it is obvious that the next step one has to take is to see whether the neglected corrections, which we know must be there, can substantially alter the obtained picture of DSB. We shall see that the overall qualitative picture will remain un- alterated.

We will consider the case of an SU(N), QCD-like theory with n massless flavors. In order to evaluate F2 we make use of the Landau gauge and we take into account only the lowest order two-particle-irreducible vacuum diagrams. The result can be expressed in the following way

I" d4p d4q tr[B F 2=6NC2 J(-~n)4 (-~)4 t f P ) B ( q ) l D ( p - q ) g Z f P , q ) f d4y , (8)

where C2 is the quadratic Casimir of the fermionic representation of the gauge group, and we have put

Aa A • a^ + Bab) A , B 1 ... . . N , b 1 ... . . n . (9) [S(P)]Bb = 8 B (tA bp , = a, =

+ l For the Gross-Neveu effective potential, see eq. (4.12) in ref. 14], the identification between their composite field Oc(X) and S (x, x) is ac(X) = -Ng Tr [S (x, x ) ] .

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V o l u m e 150B, n u m b e r 4 PHYSICS L E T T E R S 10 J a n u a r y 1985

Fur thermore ,D(p) = 1/p2,g(p, q) is the running coupling constant, and the trace in eq. (8) is taken on the flavor indices. Due to the non-renormalization of the wave function in the Landau gauge at this order, there is no con- tribution to F2 from the term A of eq. (9).

The attitude we take here is that the DSB of gauge theories is dominated by short distance effects. Consequent- ly, we will make some assumptions about the behaviour o fg (p , q). First of all, in the leading log approximation, we can write [10-12]

gZ(p, q) = 0 (p2 _ q2) gZ(p2) + 0 (q2 _ pC) gZ(q2). (10)

Then we will substitute the infrared behaviour o f g ( p 2) with a constant, introducing in this way a scale/2 which characterizes the infrared region. We will thus assume the following expression for g [ 11 ]:

g2 (rl) = (b/a) [0 (-rl) + 0 (r?) (1 + rl/a)- 1] , (11)

where, for future convenience, we have introduced the variables ~ = log(p//2), a = log(/2/A), and b = 241r2/(11N-2n). A is the renormalization group invariant mass.

By putting E = -6F2/6S [we notice that 2; is the self-energy when the Schwinger-Dyson equation is satisfied, as it follows from eq. (7)] one finds the following relation:

+ o o

~(r/) = -(3C2/STr2)/2 -1 e - n f e - ( t - n ) 0(~ - 7/) + g2(rl) e - ( n - O 0 ( r / - ~)] , (12) - - o o

with q3B(q) = o(~). We can invert this relation by applying an appropriate differential operator to both sides of the expression. We get

a(~) -- - ~ / 2 e -'~ (7) e2'7 --aTd ~(n) , (13)

with

h(rl) = [1/g2(rl)] [d log g2(rl)/drl - 2 ] - i . (14)

We notice that in deriving eq. (13) it is crucial to assume eq. (10). The expression (13) is however valid for any choice of g07 ). Then, for arbitrary Z07 ) one gets the following expression for F2:

P2 = ~ 2 / 2 2 [ h ( r l ) e2n tr{[d~(r/)/drl] ~(r/)}]+~, 2N/22 f drlh(~)e2ntr([dX(rl)/d~?] 2) d4y , (15) - - o o

where we have used the property 2F2 = Tr [(5F2/6S)S] and we have performed an integration by parts. We see that, due to eqs. (12) and (13), given any ansatz for E(r/) or for B(r/), we can reduce the evaluation of

F, eq. (5), to the calculation of two one-dimensional integrals, one in eq. (15), and the other coming from Tr log(D -1 - Z).

Now we will look for a convenient ansatz for X(r/). First of all let us discuss its asymptotic behaviour for large euclidean momenta . This was evaluated by using the operator product expansion in ref. [13]. The result, con- firmed by a recent analysis [12], is

(p) y ~ [g2 (p)/p2] [log (p/A)] d , (16)

where d = 9C2/(11N - 2n) = 3C2b/81r 2. In the OPE evaluation of Z(p ) the factors g2(p) and [log~p/A)] a are due on 1 1 u . s to the renormalization group improvement of the Wils coeffic 'ent, and in particular [log(p/A)] come from

the anomalous dimension of ( ~ ) . What can we say about the behaviour for p -~ 07 We notice that at the minimum of F, the identification of Z with the self-energy leads us to the relation (remember that A = 1 at this order in the Landau gauge):

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i/3 +B = (--i/3 -- ~ ) -1 , (17)

from which

B = _~/(p2 + )-,2). (18)

This equation inserted into eq. (12) gives us the non-linear Schwinger-Dyson equation. Going back to momentum variables and in the limit of zero momentum we get

00 q2~(q) dq lim Z(p) = (3C2/87r 2) f p~0 0 ~- q2 + Z2(q) g2(q) . (19)

Due to the asymptotic behaviour (16) the integral is convergent and therefore the limit in eq. (19) is finite. Taking all this into account, we will assume the following form for E (r/):

(7"1) = UX [0 (-71) + (a/b) g2 (r/) e -zn F(r/) 0 (7"/)] , (20)

where we will consider three possibilities: (i) F(r/) = 1 and g07) = constant; this leads to the ansatz we have studied in our previous work.

(ii) F(~/) = 1 and g(r/) given by eq. (11); this corresponds to studying the effects of introducing the running coupling constant in the theory, but neglecting the effects of anomalous dimensions.

(iii) F(r/) = (1 + B/aft, and gOD from eq. (11); this choice corresponds to the asymptotic behaviour of eq. (16). We notice that with all these choices the first term in eq. (15) vanishes. Inserting now eq. (20) into eq. (15) and

into Tr log(D-I _ ~), performing a further convenient integration by parts, and diagonalizing ~ by a chiral rota- tion (see ref. [ 1 ]), we get our final expression for r of eq. (5):

n

r = (N/47r2)/a 4 ~ V l ( X a ) f d 4 y , (21) a=l

with

oo

-- f dy y3 log[y2 + (a[b)Z(x2[y4) g4(v) F2(y)] , (22) 1

where the integration variable y is p/la. With the ansatz (i) expression (22) coincides with the potential we derived in ref. [1 ] with the identification

c = 81r2a]3C2 b. We notice also that all previous results were parametrized in terms of the variable c. However, in the present context, it appears more natural to use as a parameter the ratio of the two scales ta and A. We also re- call that with the ansatz (i) the potential (22) is bounded from below with the asymptotic behaviour [1]

V(I i) > cX 2 , (23) X---~oo

where the superscript specifies the ansatz used. Furthermore, one obtains that the theory is spontaneously broken for c < 1 [1], which corresponds to/s[A < 1.560 for QCD with three flavors.

Using the expression for g(r/) [eq. (11)] in eq. (14), we see that h 01) < 0, and therefore the first integral in eq. (22) is negative definite. From this we can derive a lower bound for the potentials V(1 ii) and V(1 iii), valid for each X:

V(li)(x) ~ v(lii)(x) and v(~ii)(x ) . (24)

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Vohtmc 150B, number 4 PHYSICS LETTI:RS 10 January 1985

L • 0[58

i

\

• I } q 0 . _

B. OOE 06

/ /

/

COE 06

Fig. 1. Behaviour of V~ ii) near to the critical point. The curves shown correspond to the values t~[A = 1.331 (dash-dotted line), 1.332 (solid line), 1.333 (dashed line).

- . 0 6 0

I

1

- . 0Lt0 "~ , 0 2 0 ."- - . I

F q . OOE 08

/ / J

/

/ i

GO ~ ~ . . 0 2 0 / . OLt

= 1 , OOE - 0 6

Fig. 2. Behaviour of V~ iii) near to the critical point. The curves shown correspond to the values #/A = 1.354 (dash- dotted line), 1.355 (solid line), 1.357 (dashed line).

In particular this shows that both d ii) and V(1 iii) are bounded from below. The numerical analysis of eq. (22) in cases (i.i) and (iii) is straightforward and one sees that in both cases there

is spontaneous symmetry breaking characterized by the following critical values of ~/A: in case (ii) (g/A)erit. -- 1.333, and in case (iii) (ju/A)crit. = 1.355. The behaviour of the potential near to the criticalpoints in cases (ii) and (iii) is shown in figs. 1 and 2 respectively. Furthermore in fig. 3 the potentials 6 ii) and V(liii)- are represented at the same value o f g / A = 1.221.

L

1 .5

. 1 5

t ~ t~//J o - - ~ . ~¸-'I "

. 1 0

Fig. 3. The potentials V~U)t;; (solid line), and V~ iii) (dash-dotted line) for #/A = 1.221.

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The conclusion we can draw from this analysis is that the effect of the running coupling constant does not change the picture of dynamical symmetry breaking of QCD-like gauge theories obtained in ref. [ 1 ]. Furthermore, the quantitative differences of cases (ii) and (iii) with respect to case (i) are not dramatic (~13% for the critical value of p/A). Taking into account that the various potentials should be compared for different values o f p / A (this quantity should be recalculated case by case from the experimental data), we do expect very small changes for the observable quantities we have estimated in ref. [9].

S. De Curtis would like to thank the Department of Theoretical Physics of the University of Geneva for the kind hospitality. This work has been supported in part by the Swiss National Science Foundation.

Note added. After having completed this work we received a preprint by Castorina and Pi [14] dealing with the same problem discussed here and in our previous paper [1 ]. These authors use the original CJT variational principle. On the other hand we have used, both here and in ref. [1 ], the modified variational principle which eliminates unboundness-below in the free case. Direct comparisons even for similar ansatz are therefore not easy to make. Nev- ertheless the conclusions on the chiral symmetry breaking are the same we had found in ref. [1] with the non- logarithmic ansatz and which we confirm here with the logarithms. The only difference concerns the lack of finite symmetry breaking minimum in their calculation with non-logarithmic ansatz. Within our modified variational principle we had such a finite minimum also with the non-logarithmic ansatz(see ref. [1 ]). But this only reflects the difference in the variational principles used.

References

[1] R. Casalbuoni, S. De Curtis, D. Dominici and R. Gatto, Phys. Lett. 140B (1984) 357. [2] J. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D10 (1974) 2428. [3] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345. [4] D. Gross and A. Neveu, Phys. Rev. D10 (1974) 3235. [5 ] R. Casalbuoni, E. Castellani and S. De Curtis, Phys. Lett. 131B (1983) 95. [6] S. De Curtis, thesis (1983), unpublished. [7] T. Banks and S. Raby, Phys. Rev. D14 (1976) 2182. [8] M. Peskin, Lectures Les Houches Summer School (1982). [9] A. Barducci, R. Casalbuoni, S. De Curtis, D. Dominici and R. Gatto, Phys. Lett. 147B (1984) 460.

[10] W. Marciano and H. Pagels, Phys. Rep. 36C (1978) 137. [11] K. Higashijima, Phys. Lett. 124B (1983) 257; Phys. Rev. D29 (1984) 1228. [12] P.I. Fomin, V.P. Gusynin, V.A. Miranski and Yu. Sitenko, Riv. Nuovo Cimento 6 (1984) 1. [13] K. Lane, Phys. Rev. D10 (1974) 2605;

D. Politzer, Nucl. Phys. Bl17 (1976) 397. [14] P. Castorina and S.Y. Pi, QCD chiral symmetry breaking in a Rayleigh-Ritz variational calculation, MIT preprint.

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Dynamical symmetry breaking in QCD-like gauge theories