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 anal- ysis 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 station- arity 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. Since the one-loop diagrams are already discussed in the preceding section we discuss here the two-loop diagrams. As a first example let us consider the ladder type diagrams shown in Fig.6. We find that, although each diagram involves the ultraviolet divergence, the sum of the two diagrams is finite:
Fig.6(a) + Fig.6(b) = -m^-a • £s£(8 - 5C(2)). (12)
Note that, among three diagrams having single mass insertion, we added only one diagram to Fig.6-(a). Next we consider the diagrams shown in Fig.7. The result of the calculation is
2 2 a2nF( 2 1/ 5\ 1 2m 5 m a -— (^-+-(2(^+(277 +-+)) + -lologg - --logl - (13-6) 2 19x 1 2rn? 97 m -)+-log_--log- (13-c) 2 2 20x 1. 2m 11 )+ll (13-d) where Ca — Nc. Hcie we comment on the appearance of logarithmic terms and the absence of (t)~'log(m//i) which are essential in the present work. At the two- loop level, the momentum integration yields the pole relevant parts for the diagrams involving no counter terms,
s ±(m>)-' = J - ;log(m ) + ilogV), (14) i(m2r = 7-log(m2), (15)
and for the diagrams each involving single counter term,
J 2 2 2 i(m )-/ = I - ilog(m ) + ilog (m'). (16)
In our case the dangerous term, (e)~'log(m2), in (14) is cancelled by the same term in (16) and Iog2(m2) survives. Since the pole terms are decoupled from logarithms it is easy to see that the divergences in (13) are cancelled by the corresponding diagrams shown in Fig.S: the poles shift as
m\- + const/)—* m(l — S)\-+ const.),
m(—- H—const. + const.)—» m(l — 6)[—j- + -const. + const.), (17) and therefore vanish when 6 = 1. Taking into account that mlog(m//j) —» m|log(m/^)+ 6(-log(m/fi)-l)] and
2 2 2 mlog (m//0 -» m [log (m/^) + 6{-log (m/fi) - 2log(m/(i)j\ (18) under m —» ni(l - 6), we obtain the the sums of Fig.7-a and Fig.8-a etc.;
Fzg.7(a) + Fig.&(a) = -m^a - £ JL, (19 - a)
Flg.Ub) + FigW) = -J-^a • ~f{^ - |), (19 -
(19 -
Note that the logarithm, mlog(m/^), in (19) comes from mlog2(m//i) in (13). Since the divergences are thus cancelled, we simply define the second order contribution as the sum of Fig.-l, Fig.C, Fig.7 and Fig.S. By adding the contributions of Fig.l and Fig.3 which may be taken as of the first order, we ootain up to the second order,
b M = JJ^a[l- ja(\og^ _^)], (20)
_ UCa-2nF _ (7fi(8 - 5C(2)) + 5/2JVc - 10rtF/9 + 295CG/36
We stress that M in (20) is renormalization group invariant up to 0{a7).
Before closing this section, we show the renonnalizability at the two-loop level. Note that all two-loop diagrams are generated from the diagrams, Fig.6-(a) and Fig.7, through the shift, m —» m(l — 6), and following expansion in powers of 6. Let us first consider the sets of diagrams generated from the diagrams shown in Fig.7. It is already found that the pole terms are present only in the contributions of orders, <5° and <5', and cancells with each other. Hence the contributioins of higher orders in S are finite. The set of diagrams generated from the diagram, Fig.6-(a), has slightly complicated renormalization structure. To show the renormalizability we need to clarify the reason that the sum, Fig.6-(a) + Fig.6-(b), becomes finite. Since the large momentum behavior of sub-diagrams is the key to the solution we examin it below.
It is well known that the diagram, Fig.3, is given in Landau gauge as (see for example, ref.[12]),
SimCn
where p represents the external momentum. For large -p2, (21) is expanded as The leading large momentum behavior tc given by mlog(— p2/fi2) and this is the origin of the divergence to be generated through p integration for outside loop. The large — p2 behavior is damped when the diagram shown in Fig.4 is added to it. Under the shift of mass parameter, the result (22) shifts to
(23)
Thus we obtain by setting 5=1,
ZimCn [2)?22. /—p*\ _/TO4\l
It should be noted that the leading mlog(— p2//'2) term is subtracted simultaneously with the pole. In this manner the ultraviolet behavior becomes soft when the diagram having single mass vertex is added. Higher order (in 8) contributions also behave softly. For instance if one further add higher order diagrams to (22) one finds that the logarithmic factor vanishes at the order 83 and the following contribution is left to all orders,
2 -ZimCR 2m 1 2 4TT p N{N -
This soft behavior would be relevant to the high energy behavior of effective mass M{p2) suggested in the context of operator product expansion in theories where chiral symmetry is spontaneously broken [13]. Now, higher order diagrams thus behave soft enough to yield finite results after the momentum integration for outside loop, we find that the set under consideration is finite as long as the subdiagrams, Fig.3 and Fig.4, are kept in pairs. This complete the proof of renormalizability at the two-loop level.
-10- 4 Chiral symmetry breaking solution
The parameter m is a free parameter because the trick of the incorporation of mass terms should m t affect the theory. However due to the approximation or truncation of the full theory the m dependence remains in M as shown in (20). Then the next task is to fix the value of m and M. Before plunging into it, however, it is convenient for the following arguments to rewrite (20) in terms of manifestly rcnormalizalion group invariant quantities. To achieve this purpose the method of effective charges due to Grunberg [S] is adequate.
To begin with we consider the general form of M expected to n-th order of ex- pansion. What should be noted first is the appearance of overall scaling factor, 1/(JV - 1), in the contribution of order a (see (S)). We show that the same factor appears in the contribution of order a2 below. The most essential point is whether the logarithm takes the form,
to the order N in 6. This is proved by applying the <5 expansion to mlog2(m//i) in (13). The result of the expansion is
m\og2{m//i) ~>
(27) where we have truncated the series at 0(5N~l). Then setting 6 = 1 we obtain
n N-2 1 9
which prove (26). Hence by choosing an appropriate expansion scheme we would have (29)
to the n-th order. We define the effective charge
which leads from (29) and (30) to
2 a(m) = afl - ^a(log^ - An) + O(a )]. (31)
The behavior of 67 with respect to m is governed by the associated beta function, ~0{a) — mda/dm. One can obtain the explicit form of ~${a) by reorganizing mda/dm obtained from (31) into a series of a. In the course of this reorganization one finds that (i disappears with a{fi) in the expression because of renormalization group invariant property of a. Thus /? is a function of only a as it should be. It is known that the first two terms agree with those of the ordinary beta function [8],
(32)
By solving the equation, fi(a) = mda/dm, we obtain
m = ±AexpU— + -plog-2— + / ^(=5) = ±Aexp / dx.. , (33)
boa' where A denotes the integration constant. The plus (minus) sign is for positive
(negative) in. From now on we confine ourselves to the case of positive m. We thus obtain from (30) and (33)
> r 3C/? A fa 1 /or"\ M = -r— a exp / dx=-—. (35) 2TT n - 1 '; d(x)
12- One can relate A to Aj^g as the following manner. By expanding Ja dxl/J3{x) in (33) in terms of a we obtain,
Since A is renormalization group invariant the R.H.S. of (36) is equal to Ajfg we obtain
A — e '\-MS- \*i)
Let us concentrate on the second order case. To this order we have from (20) and (30), [|^] (38)
and (38) leads to
0 = -boa*. (39)
We then have
By solving (39) for a we obtain
log (me-
The approximation, /" dxl/0(x) ~ l/6o»i then leads to
(43) which shows the difference between (41) and the purely second order result (20).
Since the low energy mass M is thus formulated in terms of 57 and A, let us turn to the evaluation of M. There are two ways to fix the value of M. One is the optimization procedure [9,6] which picks up the stationary value of M with respect to tn and the other is to use the gap equation following the spirit of original generalized IJartrce-Fock scheme. Let us first make the optimization by using the stationarity condition,
^ ^ (a)) = 0. (44)
From (39) the solutions of (44) are
57 = 0, i. (45)
As explicitly shown in (41) M is not analytic at the origin of a. On convergence grounds we interpret the first solution in (45) as representing the a —• 0~ limit. The first solution then leads m = 0 and consequently, M = 0. The fact that the unbroken solution is obtained as the limit that a goes to zero from the negative side may be a signature of the instability of the solution. Now the second solution gives chiral symmetry breaking solution. It yields m = A~exp(l) and
M = A^i-exp(l)w3.8Aw for nF = 2. (46)
The ratio of the first+second order contributions, which is given by m( —1 + ^jfa), to the zero-th order m is ~ —0.59. Hence our result (46) would be reliable [7] though the higher order contribution may not be negligible for attaining enough numerical accuracv. Note that, in our second order analysis, the existence of the non-trivial positive extremum point of M(a) is a direct cosequence of the asymptotic freedom [14,15] manifested in (39). Next, let us apply the gap condition,
m = m~-a. (47)
Besides the trivial solution, m = 0, (47) leads to the solution
__ 2T _ ff °~ 3CH~ 2'
Also in this case the un-broken solution corresponds to the a -+ Q~ limit. From (40), (48) gives
A/ = m = Aexpf—£)SB4.6ATTC, for nF = 2. (49)
Thus, we have demonstrated that the both procedure give the broken solution.
The status of the unbroken solution become worse at higher orders. At higher orders expja dx-~- includes the contribution of !og(a) and therefore develops the imaginary part of m when a is negative. Therefore the unbroken solution would, after all, be discarded. Thus the only consistent solution may be the symmetry breaking solution.
From the point of view of our approach, the presence of log(m/^) in (20) is neces- sary to give the chiral symmetry breaking solution. Since the logarithm comes from the set of Feynman diagrams generated by Fig.7-(b,c,d) this shows the importance of the effect of gluon polarization and gluon-quark vertex correction.
5 Disscussion and conclusion
Let us discuss the large order behavior of the low energy mass and the possibility that the breaking solution may exist up to all orders. It is convenient to back to (44); a + ]3(a} = 0. (50)
In the n -* oo limit (50) must be fulfilled for any finite m. However, this does not necessarily mean that (50) should hold for any a in the limit. This is because, as shown in the Gross-Ncvcu model [7] , the correspondence of a and m would become ill-defined in the n —> oo limit. Then, let us consider that, in which region of Q, (50) must hold at least approximately.
First we note from (7) and (28) that m in (20) is eventually replaced by mjn for large n. This means that the effect of m is suppressed by n and therefore our expan- sion around tiie massive vacuum approaches to that, of massless vacuum. However the presence of large logarithms due to small mjn^i 'invalidates the expansion ot M in powers of a as (20) or (29). Hence the behavior of M is yet non-trivial for large 7i. Note that these large logarithms can not be absorbed into the running coupling constant a because its argument, /i/Ajyj ~ m/nA-^g decreases when n increases. In fact this is another motivation to apply the method of effective charges. For a we are available to use (50) to fix the infra-red behavior as we will see in the next paragraph. Now, then, we guess to all orders that, for large n, m and n would enter a in the combined form of m/n l. For ensureing the conjecture at the order of g2, it should be shown that An behaves as log(n) for large n to give from (37),
X -* constant • n • AJ^J, (n —* oo). (51)
The above behavior is verified if the contributions of the sets generated by Fig.6-(a) and Fig.7-(a) to An converge to finite constants. It is easy to see that the set of diagrams generated by Fig.7-(a) indeed satisfy the convergence condition but it is not checked yet for the other sets due to the lack of systematic summation procedure.
It would satisfy, however, since the set is finite without any counter term. Therefore 'This is shown to be satisfied by the Gross-Neveu model in the leading order oflarge N expansion PI-
- Hi- we proceed under the assumption that, at large orders, a is a function of the single argument, m/nAjfj. Then, from (35), it is shown that /? exists in the n —> oo limit.
Now the argument of a becomes small for m which satisfy approximately m < nA^yj, most of m correspond to a valued in the infra-red region. Hence it is necessary and sufficient that, when n is very large, (50) has to be satisfied approximately in the infra-red region of a. Then, stationarity condition, (50), leads to the result that /?(a) does not have the fixed point. This is because if otherwise the behavior of J3(a) near the fixed point contradicts to (50). Hence, with m fixed, a increases unlimitedly as n increases. Thus the effective charge becomes stronger with n for most perturbative vacuum satisfying m < nAj^j. This feature is also seen in the gap condition because it gives a — 2ir(n — 1)/3CR for chiral symmetry breaking solution. Thus, we arrive at the conclusion that (50) should hold for large a. Then, from (50), fi(a) has to behave for large a as
?(5) = -5 + 0(l/5). (52)
We stress that the invariance of M with respect to m fixes the behavior of /?(
Non-zero optimal M survives in the n -» oo limit as the following manner. Let us shift in (33) the reference point of a from 0 to 1. The result is f^ [^^/>(^J (53)
Then we have
Now in the n —> oo limit, the optimal value of M is given by lim ( lim JV/(Q) )=£JATJ5 lim lim exp / dx(= + -), C = —- lim '• ws r — ooVn— oo '/ o-.»n-
where C is finite numerical constant. Recalling that 1//J behaves as -I/a + O(l/a3) (see (52)) when a is very large, we find
Jim f"dx(l + -)= finite. (57) a—x Ji \ff x'
Thus M is non-vanishing in the n —» oo limit.
Before closing this paper we comment on the work of Chang and Chang [10] who also made analysis on dynamical symmetry breaking in QCD by using the generalized Hartree-Fock method. They suggested to perform diagramatic expansion by taking the mass vertex as the same order as the gauge coupling. We think this may be the source of the problems involved in their analysis. The mass parameter, m, is determined to be const. Aj^ with the help of gap equation and accordingly m depends non-analytically on the gauge coupling constant. This exhibits the inconsistency of their order assignment. The most crucial difference lies in the point that they introduced the divergent quark mass terms as counter terms to make M finite. This causes the anomalous behavior of M with respect to the variation of (i. For example, their M is not renormalization group invariant. Further the mass correction, SM = M — m, is shown to obey the same renormalization group equation as that for the current quark mass. Therefore their gap equation, M — m = 0, does not have renormalization group invariant meaning. As we have seen in section 3, M is actually finite without introducing mass counter terms and is renormalization group invariant, as it should be.
In conclusion we have succeeded to obtain the symmetry breaking solution to the second order of generalized Hartree-Fock expansion. The use of the method of effective charges to formulate the low energy quark mass is found to be effective.
-18- We stress the close relationship between the associated beta function and the exis- tence of symmetry breaking solution. Also it should be noted that the stationarity requirement expressed by (50) is effective to disscuss the large order behavior of low energy mass. By this constraint we could extrapolate its behavior at small a to the infra-red strong coupling region.
The auther would like to thank Dr. T. Hashimoto for help in the calculation of Feynman diagrams (Fig. 7-(d)).
- 19- References
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-20- Figure Captions
Fig. 1 The vertex due to the incorporated mass term.
Fig.2 Massless quark propagator expanded in powers of mass vertex. Solid line represents the massive quark propagator. i/{fC— rn).
Fig.3 A first order self-energy diagram.
Fig.4 The diagram which cancels the ultraviolet divergence in Fig.2.
Fig.5 The sum of the one-loop diagrams. These contributions are all proportional
to Q.
Fig.6 (a) A ladder two-loop diagram which contributes to the low energy mass, (b) A two-loop diagram which cancels the divergence contained in the diagram (a).
Fig.7 Feynman diagrams at the two-loop level without mass vertex. The cross in the diagrams represents the vertex due to the ordinary counter terms.
Fig.8 Feynman diagrams at the two-loop level with single mass vertex. These dia- grams cancel the divergences in Fig.5. Fig.l
+ • +
Fig.2 Fig.3
Fig.4 H _! •_•—• L+ I « » > « I + ..
Fig.5
(a)
(b) Fig.6 (a)
(b)
(c) (a)
(b)
(C)
(d) Fig.8