Spontaneous Symmetry Breaking in Qcdt

INS—918 JP9211001 INSTITUTE FOR NUCLEAR STUDY iNS-Rep.-9i8 UNIVERSITY OF TOKYO March 1992 Tanashi, Tokyo 188 Japan Spontaneous Symmetry Breaking in QCDt HlROFUMIYAMADA Institute for Nuclear Study, University of Tokyo Midoridio, Tanaslii-shi, Tokyo 188 Japa.n 'This is a rovised version of the previous work; INS-Rep.-910, January (1992). INS-Rep.-918 March 1992 Spontaneous Symmetry Breaking in QCDt HlROFUMI YAMADA Institute for Nuclear Study, University of Tokyo Midoricho, Tanashi-shi, Tokyo 188 Japan ABSTRACT We study dynamical chiral symmetry breaking in QCD by the use of the generalized Hartree-Pock method. The low energy quark mass is calculated to the second order of diagrammatic expansion around shifted perturbative vacuum where quarks are massive. We show that the low energy mass is finite and rcnormalization group invariant. We find that the finite mass gap emerges as the solutions of gap equation and stationarity condition, thereby breaking the chiral symmetry. We also discuss the possibility that the breaking solution may exist up to all orders. 'This is a revised version of the previous work; INS-Rep.-910, January (1992). 1 Introduction The spontaneous breaking of chiral symmetry in QCD in the limit of vanishing current quark mass is one of crucial issues in understanding the theory. Most of the analytic studies on dynamical symmetry breaking are based on Schwinger-Dyson equation in the framework of efFective potential [1,2]. However, so far, it is not yet settled whether chiral symmetry is dynamically broken in QCD and what is responsible for the phenomenon. The idea of mass generation as a consequence of dynamical breakdown of chiral symmetry is first explored by Nambu and Jona-Lasinio [3]. In their pioneering work they used the so-called generalized Hartree-Fock method [4,5,6,7] to obtain non- perturbative solution of the fermion mass. The point of the method is to shift the perturbative vacuum from massless to massive one. This trick enables us to perform non-trivial diagrammatic expansion which would be very difficult to obtain if we start out with massless vacuum. Then, it is natural to apply the method to QCD and see what would come out. In the present paper we investigate dynamical chiral symmetry breaking in QCD with 7!f massless quarks by the use of the generalized Hartree-Fock method. As the order parameter we choose the low energy quark mass which is defined by the zero momentum limit of the effective quark mass. To examin the symmetry realization we incorporate the color and flavor independent mass terms of quarks into the free and interaction parts with the opposite signs and then calculate the low energy mass by performing diagrammatic expansion. In contrast to the four-fermion model studied in ref.[3] QCD is a renormalizable theory so that we will obtain the finite dynamical quark mass if the theory is still renormalizable under the fluctuation around the massive vacuum. This paper is organized as follows. In the next section we make preliminary analysis on diagrammatic calculation of low energy quark mass. Some special aspects in the diagrammatic expansion in the generalized Hartree-Fock scheme are illustrated. In the third section we obtain the low energy mass up to the second order as a func- tion of mass parameter of the shifted perturbative vacuum. We show that the low energy mass is finite and renormalization group invariant. In section 4 we improve the result, obtained in the preceding section by the method of effective charges [8]. Then we evaluate the low energy mass under the optimization procedure by stationarity condition [9,6] and the self-consistent gap condition. In the last section we discuss the possibility that the breaking solution may survive to all orders. We also comment on the earlier work of Chang and Chang [10] who made analysis similar to ours. Through out this paper we work in Landau gauge and dimensional regulariza- tion scheme (dimension of space-time= 4 — e) with MS subtraction procedure[ll]. 2 Diagrammatic calculation of the low energy quark mass Let us start with the Lagrangian, •C = Cfree + £inf i n Sh Ctni =mf fcfc - g £ hA^T-fti + C£r - °", (1) i=i i=i where T denotes the fundamental representation matrix of the gauge group SU(NC). Here we have suppressed the color indices for the sake of simplicity. The quark mass terms incorporated into CfTrc and £,-„( are special ingredients in performing diagrammatic expansion. We regard the mass piece in £jn( as interaction term that generates the vertex shown in Fig.l. The effect of mass insertion in the Feynman diagrams is computed by means of the following technique due to Neveu [6]. First, let us consider how the incorporation - A - of the quark mass term affects the original massless propagator. The effect of the trick is easily seen by first rewriting the original massless propagator as # ft— m + m fr— m(l — S)' (2) and then expanding (2) into the series; #r p- m[ jr-m \fr~ m The power of S counts the number of mass vertex on quark line (see Fig.2). From (2) and (3) we find that if one wants the finite sum of quark poropagators shown in Fig.2 one can obtain it from -— by first shifting m —t m(l — 6) and then expanding in powers of 6 up to the interested degree. This technique can be used for general Feynman diagrams and reduces the calculational effort considerably. Let us illustrate how it works by taking simple diagrams as example. Consider the one-loop self- energy diagram shown in Fig.3. For zero external momentum it is calculated to be A , m2 1* 2 2 £2( a( log—r-f-l, - = 7 + log4jr, a = —- where CR - {N\ - I)/2iVc. After the shift, m -» m(l - 6), we expand it in powers of 8, The result is 2 2 4TT ^e fi 3' L 4JT ^£ /< 3' 4TT J The part proportional to 6 represents the contribution of Fig.4. Keeping terms up to the order, Sl, and setting 5=1, we have -rJ^a. (6) Thus we find that the diagram shown in Fig.4 cancels the divergence of Fig.3. This cancellation mechanism is a typical one in the generalized Hartree-Fock scheme. i <-* Also when the diagram shown in Fig.3 is involved as a sub-diagram in a given bigger diagram, the ultraviolet divergence contained is cancelled as long as the diagram shown in Fig.4 is kept in pairs. Let us turn to another aspect in the generalized Hartree-Fock scheme. Given a Feynman diagram one obtains a series of diagrams by incorporating successively the mass vertex into the diagram. And the power of gauge coupling constant is not affected by the mass insertion. For example all the diagrams in the sum shown in Fig.5 contributes with the order a. The sum of them up to the first N diagrams is computed by means of <5 expansion technique from (4). Since the part of (4) simply proportional to m, j-n(f + 5) j is present only in the orders, <5° and Sl, and they cancel with each other when 6 = 1, let us consider the term, mlog(m//<). After the shift the above term is expanded in powers of 6 as „.„. ,„ Hence we obtain 3iC m a. (8) 2TT N-l We find from (8) that the result in the dimensionally regularized perturbation theory with the massless quark propagator is recovered in the N —» 00 limit. Also to other orders of a there are contributions of infinite series of diagrams. In a rough sense the generalized Hartrec-Fock scheme extend the ordinary series in powers of coupling constant into the double scries in which the diagrams are classified according to the two labels, coupling constant and mass insertion. Hence the definition of the expansion scheme becomes a non-trivial task. In the present paper we expand M into a power series of g* in a similar but not the same manner as the ordinary perturbation expansion. When M is given •- as M = J2k=ia k{g2)ki we regard its order as n. However, since an infinite set of Feynman diagrams can contribute a*, we allow a* to vary with the order n. This n dependence of ak is determined according to the choice of subset of the infinite series of diagrams. The problem of defining systematic expansion scheme should be investigated but it is beyond the scope of this work. In this paper we suffice ourselves to define only rather heuristic expansion which is restricted to the second order. 3 Renormalization structure and low energy mass to the second order The quark propagator can be put into the form where the effective mass M{p2) is renormalization group invariant [2]. Since the quark propagator is diagonal in flavors and colors we have omitted those indices. The low energy mass M is defined by, M = lim Mtp2). (10) pi—Q Since A and B behave as (11) M begins with the order g2. Here the second term in B comes from Fig.l. We calculate M up to the second order. Since the contribution of order g* in A is absent in Landau gauge, it is sufficient to calculate only B and therefore we evaluate the quark self-energy diagrams with the external momentum set zero. Renormalizability is one of the guiding principles in defining expansion scheme. This is obvious because the introduction of mass counter terms spoils the original - (i - cliiral symmetry. We therefore investigate the renormalization structure of Feynman diagrams and then determine the second order low energy mass.

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