Dynamical chiral-symmetry breaking in dual QCD

PHYSICAL REVIEW D VOLUME 43, NUMBER 10

Dynamical chiral-symmetry breaking in dual QCD

G . Krein* and A.G. Williams Department of Physics and the Supercomputer Computations Research Insti tute,

Florida State University, Tallahassee, Florida 32306-4052

(Received 12 October 1990)

We have extended recent studies by Baker, Ball, and Zachariasen (BBZ) of dynamical chiral- symmetry breaking in dual QCD. Specifically, we have taken dual QCD to specify the nonperturbative infrared nature of the quark-quark interaction and then we have smoothly connected onto this the known leading-log perturbative QCD interaction in the ultraviolet region. In addition, we have solved for a momentum-dependent self-energy and have used the complete lowest-order dual QCD quark-quark interaction. We calculate the quark condensate ( g q ) and the pion decay constant f, within this model. We find that the dual QCD parameters needed to give acceptable results are reasonably consistent with those extracted from independent physical considerations by BBZ.

I. INTRODUCTION

Quantum c l ~ r o m o d ~ n a i n i c s ~ ~ ~ (QCD) has given a correct theoretical description of a large class of strong- interaction phenomena a t higll energies and large momentum transfer. I ts property of asynzptotic freedom allows the use of perturbation theory for many such phenomena and hence calculations can be carried out in a systematic way. O n the other hand, low-energy nonpel,- turbative phenomena such as color confillemelit ant1 cly- ilamical chiral-symmetry breaking (DCSB) have yet t o be derived in a definitive way from the theory. This difficulty arises from the lack of a suitable systematic approximation scheme. Lattice-gauge-theory calculations hold significant promise and are proceeding as rapidly as possible within the constraints of existing computers. Lattice- gauge-theory results t o da te are qualitatively consistent with pl~enomenology, but small lattice sizes limit the ac- curacy and the range of problems that can be attacked within this framework.

From a more analytic point of view, some progress has been made in recent years by considering the QCD vacuum as a relativistic dielectric medium characterized by a momentum-dependent dielectric "constant" e(q2). A number of a u t l ~ o r s ~ - ~ have calculated E ( ~ ~ ) in the low-momentum regime by means of a self-consistent, gauge-invariant truncation of the Schwinger-Dyson equations (SDE's) of pure-gauge QCD. These calculations have been performed in different ways and in different gauges, but typically conclude that-^(^^) behaves as c(q2) -q2 /M2 as q2 -+ 0, where A4 is an undeter- mined renormalization-group-invariant mass scale. The gluon propagator in the medium, D(q2) = l/[cl%(q')], then behaves as 1/q4 as q 2 - 0 , which can he associated with a linearly rising potential in coordinate space. However, the difficulty with such an approach is tha t it is not possible t o calculate systematic corrections to t,lie propagator coming from the terms neglected in tlle trun-

15 MAY 1991

cation; the gluon SDE appears to contain uncontrollable infrared divergences beyond first order.

One way to proceed in the study of the dielectric properties of the Q C D vacuum, avoiding the above problem, was proposed by Baker, Ball, and Z a c l ~ a r i a s e n ~ - ~ (BBZ). Tliese authors suggested tha t tlle problem is related to the choice of the Yang-Mills vector potentials Ap as the ilatural variables t o study low eilergy QCD. The reason for which is based on a suggestion by 't Hooftg and R/I andelst am1' some time ago tha t many features of color confinement could be understood if the QCD vacuum possessed properties similar t o a magnetic superconductor , i.e., a superconductor with electricity and magnetisin interchanged. Therefore, BBZ argue, if this picture is re- ally relevant t o the color-confinement pl~enomenon, then one should expect tha t dual vector potentials Cp (electric vector potentials), instead of the usual A'', are the natural variables t o use. Pursuing this line of thought, BBZ were able t o express the long-range linlit of the Q C D La- grangian in ternis of the dual potentials Cp. T h e result- ing theory describes a non-Abelian dual superconductor: t l ~ e color electric flux is confined into ZN flux tubes [for an S U ( N ) gauge theory, N = 3 for QCD], which results from the spontaneous symmetry breaking associated with a magnetic condensate.

In recent work7 BBZ introduced quarks into their approach and studied the interplay between coilfine- ment and DCSB in the context of dual QCD. They used tlle SDE for the quark propagator in the so- called "rainbow" (or "ladder") approximation with a truncated quark-quark interaction and then IooBed for a constant, momentum-independent chiral-symmetry- breaking quark mass. Within their approximation scheme, they found tha t chiral symmetry is spontaneously broken only in the presence of confinement; w11en the maglletic condensate is absent, there is no clliral breakdown. In addition they concluded tha t the dyilam- ically generated mass is of the order of 100 hie\'.

3541 @ 1991 The American Physical Society

3542 G. KREIN AND A. G. WILLIAMS 43

In this paper we perform a more detailed s tudy of I)CSR in dual Q C D as forrnulat,ed by B B Z . \Ve yetmain all of t h e contributions t o the dual quark-quark interaction ( t o lowest order) a n d have not restricted our- selves t o t h e simple four-quark interaction as was clone in Ref. 7. We obtain numerical solutions t o the quark S D E \vitll momentum-dependent quark masses. We then examine means of c o n ~ l e c t i ~ l g the dual (long-distance) regime t o t h e known perturbat ive Q C D (short-distance) regime. T h e appropriate h igh-mo~nentum behavior consistent with t h e operator-product e x p a n ~ i o n ~ ~ ~ ~ ~ ~ is ob- ta ined. Finally, we calculate the pion decay c o n s t a ~ l t and t h e quark condensate as functions of the fundamental parameters of dual Q C D . As in Ref. 7 we find t h a t DCSB only occurs in the presence of the magnetic coi ldei~sate which gives rise t o the confinement. T h i s supports t l ~ e conte~l t ion of BBZ t h a t bo th confinement and DCSB are related t o the existence of a dual superconductor. While we have a t tempted t o improve on t,he ana.lysis of BBZ in several respects, we have riot here gone beyond t h e rainbow approximation since it is not clear how t o d o this a t t,he present s tage of development of dua l Q C D .

In Sec. I1 we review some of the relevant features of the dual Q C D treatment of BBZ, a n d in Sec. I11 we discuss DCSB. 111 Sec. IV we present our numerical results and our conclusions are given in Sec. V.

11. R.EVIEW OF BBZ DUAL QCD

T h e principal reason for t h e interest in dual Q C D is t h a t , as shown by 't Hooftg and Mandelstam,lo a t large distances dual gauge fields may be weakly coupled. T h i s follows from t h e fact t h a t the coupling constant of the dual fields C'' (denot,ed g by BBZ) is ii~versely proportional t o the coupling cor~s tan t of t h e usual AIL fields (denoted e by BBZ). While such general s t a t e ~ l l e ~ l t s can be made about the interactions of dual pot,entia.ls, the exact, transformation between t h e usual Yang-Mills potentials A P and t h e dual potentials Cp is unknown. Hence the exI1licit form of t h e dual l'ang-Mills Lagrangian is un- available. However, BBZ have shown how t o co~ls t ruc t a dual Lagrangian and have given plausible a rg l~ment~s t h a t it follows from t h e conveiltional Yang-hfills theory. T h i s dua l long-distance Q C D Lagrangian has many of t h e properties of a magnetic superconductor.

Let .CYM(A) denote t h e usual Yang-Mills Lagrailgian expressed in terms of t h e Ap potentials (i.e., pure gauge QCD), and note t h a t this Lagrangian appears t o be appropriate for per turbat ion theory in t h e short-distance regime. 'I'he Lagrangian of pure-gauge Q C D expressed in terms of the dual fields Cp is unknown and hopelessly complicated in any case. Let us denote t h e par t of the Cp Lagrangian which is appropriate for long distance by C ( C ) . If t h e Cp fields interact weakly a t large distances, t h e dual propagator then has a mass and the quadrat ic par t of &(C) , denoted by L(O)(C), can be determined. T h e propagator AP)(*) corresponding t o C(')(C) will then have the s ~ n a l l - ~ ~ behavior

~ v h e r e A? is some n o n v a ~ ~ i s l ~ i n g Inass (usually writtell as M by BBZ). Because of t,he weak cou~-iling a t long tlis-

tances, the A$) propagator generated by c(')(c) has , except for mass and wave-fi~nct~ion renormalizat ioi~, the same s m a l l q 2 behavior a s t h e complete Ac generated by C ( C ) . T h e Yang-Mills Lagrangian is the minimal extension of the q ~ i a d r a t ~ i c Abelian form necessary t o ensure invariance under the usual non-Abelian SlJ(3)-color- electric gauge transformations. Similarly tllen, the con- struction of C ( C ) frorn c(')(c) proceeds by adding terms t h a t ensure invariance under t,he non-Abelian rnagnet,ic gauge t ral~sformations appropriate for the dual fields. Finding the long-distance approsirnation C ( C ) t o the dual 1,agrangian has thus been reduced t o finding a \ray t o const,ruct c(')(c) in terms of t h e usual Yang-hiills theory with potentials Ap.

T h e theory represented by the quadrat ic form of C(')(C) is an Abel iar~ gauge theory \vl~ich describes a iuagnetic medium characterized by a momentum- dependent magnetic permeabil i tyg~10 p ( q 2 ) . T h i s is just the inverse of the dielectric constant t(*'):

ivllere bo th of these are familiar long-distance properties of media in Abelian gauge theories. For an Abelian theory the equations of motion are just Rlaxwell's equations

where E and D a n d B and H are related by the consti- tutive equations

T h e potentials Ap are relat,ed t,o E and B by

whereas the dual potent,ials Cp are related t o D and H by

T h e equations of motion, Eq. (3 ) , can be obtained from the action s(')(c), where

where

43 DYNAMICAL CHIRAL-SYMMETRY BREAKING IN DUAL QCD

and where

T h e Cfi propagator generated by the action in Eq. (7) in the Landau gauge is

Thus in order t o obtain a mass M in this propagator as in Eq. ( I ) , /* (q2) must have the form

which means tha t c(')(c) is given by

I t is now straightforward t o obtain from C(O)(C) the minimal extension invariant under magnetic gauge transformations:

where

T h e CM fields are now non-Abelian dual fields, where CP(x) ZE Cap(2 )Ta (summation over a = l , . . . ,8 implied), and where Ta are the generators of SU(3) color with the ilorlnalizatioil 2 t r T a T b = hab. BBZ propose the La- grangian of Eq. (13) as an appropriate starting point for studies of long-distance Yang-Mills theory.

An at tempt t o relate c(')(c) t o the long-distance behavior of the nonperturbative Ap propagator was also made by BBZ. A connection was made with earlier worli of theirs3 where they at tempted to calculate E ( ~ ' ) by means of self-consistent truncations of the Yang-Rifills SDE. As discussed in the Introduction these calculations predict

from which it follows tha t

which is just Eq. (11) if the constant is taken to be unity. T h e Lagrangian in Eq. (13) is nonlocal, lneaning that

there are extra degrees of freedom in the medium. In order t o render C(C) local, extra tensor and ghost fields, piu = - F,",, and $8, $,at, respectively, are introduced. (Note tha t $ and $t are not quark fields.) The complete

BBZ dual Lagrangian (without quarks) is

where

with

Bf = -fig and E4 = -iciJkFyk . (I91

T h e te rm proportional to is a gauge-fixilig term alid ,y and Xt are the usual Faddeev-Popov ghost fields. The "potential" W(F, $, $7) is a fourth-order polyllornial in F, $, and $1, which is required for the renormalizal~ility of the theory.

Using the Lagrangian in Eq. (17), BBZ studied the flux-tube solutions t o the classical equations of motion. They find flux-tube solutions when there is spontaneo2s symmetry breaking associated with a-non_zero value F , for the vacuum expectation value of F2. F: is negative, indicating tha t the QCD vacuum is magnetic [see Eq. (18)]. Hence BBZ argue tha t confinement in QCD is produced by the existence of a rnagnetic conde~lsa.te, i .e., dual superconductivity.

In order t o discuss chiral symmetry we need to consider quarks in dual QCD. It is not an easy task tjo couple quarks t o the dual gluons. 111 fact, BBZ were able t o do this to lowest order in g only, i.e., in the Abelian approximation. At this level of approximation the problem is equivalent t o the coupling of magnetic monopoles to pho- tons in quantulm electrodynamics. We do uot at tempt t o repeat the detailed arguments of BBZ, but instead quote the relevant results.

A new ingredient enters the problem with the quarks, namely the Dirac string. This is a way of coupling to the quarks while mai~ltaining coinpatibility with the usual field and potential relations. Of course, the string is a.n intermediary construct only and all physical quantities should be independent of it . Dual Q C D is a theory tha t necessarily involves both e and g since it is a theory in which both color-electric and -magnetic charges exist, and , as such, requires the Dirac quantization co~lditioll

where we recall tha t e is the coupling strength of the Ap fields and g is the coupling strength of the dual fields Cp.

The key ingredient for studying DCSB is the quark- quark scattering kernel, which is the colltractioll of the quark vertices with the other field propagators. Fro111 the Lagrangian of Eq. (17) we call obtain these propagators. Below we give7,'' the dual field propagator (for CpC'"), the mixed field propagator (for G'pFaP), and the tensor field propagator (for F a P F y 6 ) , respectively. The forms given for these are appropriate for the nonperturbative vacuum, where the magnetic and gluon condensates 11a.ve formed:

3544 G. KREIN A N D A . G. WILLIAMS

where M 2 and M S 2 are given below in Eqs. (30) and (31) and are zero in the perturbative vacuum.

The graphs tha t contribute to the quark-quark scattering kernel are shown in Fig. 4 of Ref. 7. Froin the contraction of the vertices and the propagators of Eqs. (21)-(23), it follows tha t the scattering kernel ( to lowest order in g) is given by

Qa r la YO q7 (y6p6 - -hO7 4 - 7 6 a 6 + w6a7)] P ,

where DP" is an "effective dual gluon propagator" that mediates the quark-quark iilteraction a t loilg distailces given by

In Eq. (24) n , rn, n', and nz' are spinor indices and in Eq. (25) the unit vector n Q i v e s the direction of the Dirac string. The functions F(ii2) and G(q2) characterize the vacuum of the theory. For the perturbative vacuum, E' and G are given by

finement), G 2 , the gluon condensate, and 9', tlie dual coupling constant:

Note tha t in the perturbative vacuum, where lrlagnetic and gluon condensates are absent, we have M = M* = 0.

It is interesting to note tha t although the propagators in Eqs. (21)-(23) are in Landau gauge, the first term of the effective gluon propagator, Eq. (25), has the ap- pearance of a spin-1 field in the axial gauge, where the Dirac string direction nfi plays the role of the axial gauge choice. However, it should be kept in mind that the dependence of tlle effective propagator on nfi arises fro111 the quark vertices through the Dirac string and is quite unrelated to the choice of ga.uge.

In a series of BBZ studied several aspects related to low-energy QCD where they fixed the independent parameters of dual QCD. T h e rclevant parameters for Eqs. (28) and (29) are

For the nonperturbative vacuum these are given by J-i',2 2 500 MeV , (32)

where M , M , and M* are mass scales which depend on the magnetic properties of t_l~le vacuum. .@ and Ad* call be expre~sed in terms of Fo2, the vacuum expectation value of F2 (whose nonzero value is responsible for con-

G2 is taken froin the QCD sum-rule approach to the char- lnoniuin spectrum. BBZ also get an approximate relation between k, p:, and g 2 , namely M 2: g 2 ~ 2 / ( - @ ) . Hence the parameters y2, G 2 , and (-G) can be used to specify the long-distance behavior of the quark-quark interaction through the quantities M , M , and Ad'.

From Eqs. (26)-(29) we see tha t the nonperturbative, confining inforination is contained in the function G(q2). As we will see in the next section, i t is also the function G(q2) which gives rise t o DCSB.

DYNAMICAL CHIRAL-SYMMETRY BREAKING IN DUAL QCD 3545

111. DYNAMICAL CHIRAL SYMMETRY BREAKING

Dynainical chiral symmetry breaking (DCSB) in QCD occurs when the nonperturbative solution to the Schwinger-Dyson equation (SDE) for the quark self- energy spontaneously generates a scalar piece and is clearly a nonperturbative effect. The general foriri of the inverse of the renorrnalized quark propagator in an arbi- trary covariant gauge is

where i n is the renormalized explicit chiral syin~netry breaking (ECSB) quark mass, C(p) is the renor~nalized quark self-energy, M(p2) z R(p2)/A(p2) is the running quark mass, and Z- l (p2) 3 A(p2) is the mornentuin- dependence of the quark wave-function renormalization. A subscript p ( to indicate renormalization-point dependence) is to be understood on all renormalized quantities, but is suppressed for clarity of presentation. Renormal- ized quantities are also gauge dependent i11 general, but this is also not explicitly indicated. The renormalized SDE for the quark propagator can be written as"

where m is the ECSB quark mass (approxirnat,ely 5- 10 MeV for the u and d quarks at p2=1 GeV2), Z s and Zr are the quark propagator and quark-gluon vertex renormalization constants respectively, and where we have used C, XaXa /4 = 4 . We have also introduced an ultraviolet (UV) cutoff Auv (written as A in equations for notational brevity) with the understanding that we are to take the Auv + ca limit at the end of any calculations. The renormalization constants are functions of Auv and the renormalization point p , [Zs(A,p) and Zr(A,p)] . The unrenormalized SDE for the quark self- energy can be written

provided all quantities are understood as being bare or unrenorlnalized in this equation. I11 an Abeliaa theory such as QED the renormalization constants satisfy Zs = Zr = 1 (to leading order) in Landau ga.uge. Since we are working in BBZ's Abelian approximation for the quark coupling to the dual fields it seems reasonable to use this here in Landau gauge. We further assume that this is appropriate to all orders for the momentum-independei~t part of the renormalization constants (at p2 = A&, say). With these assumptions we find that Ecl. (35) can 1101~

also be used for the renorrnalized quark SDE. This is the usual starting point for studies of DCSB.

As stated in the Introduction we will rest,rict our- selves to the "rainbow" approxiinatioil for the cluarl; SDE, which amounts to replacing the proper (l-particle- irreducible) quark-gluon vertex r" by its perturbative value y".

I t is a well-known fact t,hat it is necessarv to have a reasonably strong quark-gluon interaction in order to get sufficie~lt chiral breaking, and consequeiitly reasonal~le results, for example, for f, and ( q q ) . BBZ extracted6 the value of g2 = 6.3 from the Cornell potential at a momentum scale of cx1 GeV, where the potential is essen- tially Coulombic, i.e., in the perturbative vacuum region. From Eq. (20) we have a, r e2 /4 r E 0.5, which we will see is too small for getting reasonable values for the chiral parameters. In order to account for Abelianization of the quark interaction with the dual Yang-Mills fields and the neglected dressing of the quark-gluon interaction, we define a new effective coupling constant for the quarks by e' ce, a:ff = c2a, , with c a free parameter. Provided that c 21 1 we argue that this is reasonably consistent with the general approxirnatio~ls and truncatioils in the B13Z treatment of dual QCD. In Sec. IV we tliscuss the dependence of our results on the parameter c.

The expression for the quark propagator in ICq. (33) is not of t l ~ e appropriate form when there is a depenilel~cc on the Dirac string direction in the effective gluon propagator as we have here. In this case S would depeiltl 011

two extra n-dependent functions in addition to A and B, i.e., S-'(p) = (A$ - B) + (C$ + D)fi, where A, B, C , and D will all depend on p2 and p . i z in general. Solvii~g the quark SDE then requires si~nultaneously solving four coupled integral equations ill two variables and is clearly a formidable task. In order to simplify the task before us we make an approximation here of averaging out the explicit np dependence in the effective gluon propagator before attempting to solve the SDE. This involves inte- grating the effective gluon propagator in Eq. (25) over all possible string directions. We perform this integration in Euclidean space using the principal-value prescription, just as is usually done when dealing with the unphysical poles q .12 and ( q . n)' in axial gauge propagators. While we know that the string direction should not affect a.ny physical observable this suppression of the p . n dependence in the quark self-energy is an approximation. We note that had we attempted to solve for a constant self- energy, a s is often done in studies of DCSB, the angular part of the loop ~nornentum integration would exact,ly correspond to this n averaging. This n averaging is per- haps not as strong an approximation as it might a t first appear. Clearly the hope is that this approxiinatio~~ will not greatly affect the results of the calculation. To improve on this treatment will prove a significant challenge.

\?'e make one further simplificatio~~ here, namely, we solve the quark SDE for Z = 1 (i.e., A = 1). In principle, we should solve a system of two coupled integral equations for A and B, but since the nlost important nonperturbative information is contained in the equation for B, the case with A # 1 is not expected to alter the situation

3546 G . KREIN AND A. G. WILLIAMS 43

very i n u c l ~ . ~ ~ - ~ ~ To meaningfully go beyond this A = 1 approximation, as well as t o have a proper treatment of the n dependence, would require the rainbow approximation itself t o be abandoned as well as a detailed study of the quark and effective gluon proper vertex. However we d o not pursue this further here.

With these simplifications, we have tha t the SDE is given (in Euclidean space) by

where GE(q2) is just the Euclidean version of G(q') in Eq. (29) and is written

Equation (36) is solved numerically by i t e r a t i ~ n . ' ~ ~ ' ~ Once we have M ( p 2 ) , we can at tempt t o calculate tShe pion decay constant f, and the quark coildellsate ( Q q ) . These are physical quantities tha t indicate the strength of the chiral breakdown. Their expressions in terms of Ad(p2) are given by13'15-18

where again the integrals are expressed in Euclidean space.

An interesting question is how to form the connection between the dual Q C D quark-quark interaction and the known perturbative Q C D interaction. It is known tha t a t short distailces the quark-quark scattering kernel (in the Landau gauge) has the form

where n , m, n', and rn' are spinor indices, D(') is the perturbative gluon propagator (in the Landau gauge), and a , ( Q 2 ) with Q 2 f -(kl - k)' is the usual QCD running coupling constant [compare Eqs. (24) and (40) for the I R and UV scattering kernels respectively]. Specifically we have, for Q2 >> A$,--,

where X G 12/(33 - 2nf ), n f is the number of quark fla- vors, and AqcD is the usual Q C D scale parameter. To a very good approximation14,15~18 = 1 for large p2 in the deep spacelike regime and the proper quark-gluon vertex rJ' 21 7". This then gives the usua113,17'18 integral equation for M ( p 2 ) appropriate in the deep spacelike

regime (expressed in Euclidean space):

Now a comparison with our dual result in Ecl (36) sholvs tha t the equatioils are identical except tha t the dual QCD form has [ C Y : ~ ~ G ~ ( ( ~ - p)')] in place of tlie running coupling a,( - (k - p)'). As we shall show in the next section, the two effective interactions can be matched together relatively smoothly at inoinenta of the order of 2 GeV, with almost no change on the low-energy quantities such as f,, (qq), and M ( 0 ) = M ( P ' ) ~ ~ ~ = ~ .

IV. NUMERICAL RESULTS

We have solved the integral Eq. (36) by iterating an initial guess t o convergence. T h e techniques used have been discussed more fully e l ~ e w l i e r e . ' ~ ~ ~ ~ In Table I Joe present our results for pure dual QCD, i.e., u~lien no matching to the UV runiling coupling is done [Eqs. (36)- (37)l. In obtaining these results we have treated (-@) and a:' = eI2/4r c2a , = c2e2 /4 r (where e = 2x/g) as _adjustable parameters. However, we require that (-F:) N (500 MeV)' as established by BBZ and tha t c -- 1. Note tha t we have used g2 = 6.30 which implies tha t a, = 0.50. In the first column we vary (-@) from (500 MeV)' to (600 MeV)'. In the second column for each value of (-$:) we vary a:* from 1.00 to 2.00. Our results for the pion decay constant f,, the quark condensate (-(qq))lI3, and the quark mass a t zero momentum M(0) = M(p2) with p2 = 0 are shown in the remain-

TABLE I. Results obtained in our treatment when no perturbative quark-quark interaction is added in the ultraviolet region. The results shown are for the parameters g2 = 6.30 (i.e., g2/4?r = 0.50) and G2 = (330MeV)'. The parameter (-@) is in (M~v' ) and the results for f , , (-(gq))'I3, and M(0) are in MeV.

- F; m z f f f ir (-(m)I1l3 M (0)

Experiment 93 225&25 300-500

43 - DYNAMICAL CHIRAL-SYMMETRY BREAKING IN DUAL QCD 3547

ing columns. Also shown a t the bottom of the table are - the experimental valueslg for f, and (gq), and we give for comparison the typical range of quark masses in constituent quark models. The latter number is intended as a guide only since the connection between M(p2) and constituent quark models is not well understood at this time.

In Table I we see that re_asonable results can be obtained for each value of (-F:) when azff is taken in the range 1.50 to 1.75 (i.e., c rz 1.7 - 1.9). Another fea- ture of these results is the fact that tlle cliiral parameters f,, (qq), and M(0) increase as (-F:) increases. This behavior is expected, since the existence of the magnetic condensate (-@) is responsible for chiral- symmetry breaking and confinement. I t is rewarding that the results obtained for f,, (qq), and M(0) are close to their experimental (or typically quoted) values, when we use dual parameters extracted independently by BBZ in the gluon sector [ g 2 = 6.30,G2 = (330 ant1 (-@) rz (500 MeV)']. The only adjustment that was made was to allow the effective quark coupling e' ce to increase somewhat from tlie naive value e = 2 n / g , (c rz 1.7 - 1.9). While this is arguably the least satisfac- tory aspect of this study it is not unreasonable in view of the approximate, Abelian coupling of the qua~l is l o tlie dual QCD Lagrangian. I t is also worthwliile to recall that this analysis made the further approximations of A(p2) r 1 and of n averaging over the quark-quarlt interaction. With these caveats in mind it is clear that while the results are encouraging for dual QCD, caution in interpreting the results is prudent.

In Table I1 we give our results for tlie quark-quark interaction defined by the dual QCD interaction of Ecl. ( 3 6 ) in the infrared matclied on to the perturbative QCD interaction of Eq. (42) in the ultraviolet [note that both arise from scattering kernels in Landau gauge, Eqs. (24) and (40)]. Following the discussion at the end of the last section it is clear that we have the matclied equation

where we define

The matching point q; is defined in the most natural way as that intermediate momentum a t which the two effective interactions are equal, i.e., where qo is defined by the equality = [ ~ u ~ ~ 2 G ~ ( q ; ) ] . The matched effective interaction cyyatched (q2) is then continuous, and as is apparent from Fig. 1 the discontinuity in the first derivative is also acceptably small. Since as is apparent from Fig. 1 the matching point is qo 2: 2 GeV and since the perturbative QCD piece of the effective gluon propagator then only contributes for momenta 2 2 GeV, it seems appropriate to choose nj = 4. In any case we have

TABLE 11. Results obtained when the perturbative quark-quark interaction is added in the ultraviolet region with AQCD = 200 MeV and n j = 4. The dual parameters are the same as for Table I_, i.e., g2 = 6.30 and Gz = (330MeV)~. The parameter (-F:) is in (MeV2) and the results for f,, (-(9q))ll3, and M(0) are in MeV. Here the quark condensate is quoted at a momentum scale of 1 GeV. This is not the case for the condensate values in Table I since there is no connection to perturbative QCD for the pure dual case.

Experiment 93 2253~25 300-500

verified that the results are very insensitive to the exact choice of n j .

I t is clear from Fig. 1 that tlie result of tlle matching is to increase the strength of the effective interaction above q;. For this reason we naturally expect the cliiral parameters M(O), f,, and (qq) to increase, since they are a measure of the degree of DCSB and since the effective coupling strength is, as we noted, increased. We leave for a moment the discussion of the quark condensate, since

dual

FIG. 1. We plot the effective interaction i n the quark Sch\vinger-Dyson equation as a function of rnoment~~ln. Both the pure dual (-) and running coupling (- - -) interactions are shown. The matching point is qo N 2 GeV. The dual parameters are g2 = 6.30, Gz = (330MeV)~, (-F:) = ( ~ O O M ~ V ) ~ , and a", = 1.75. The running coupling has nf = 4 and AQCD = 200 MeV.

G. KREIN AND A. G. WILLIAMS

this is corriplicated by t h e issue of reiiormalizntion theory in t h e mat,rhed case. MTe see from Tables I and I1 t,liat, there is illdeed a n increase in A4(0) and fr as expected, h u t note t h a t this is extremely small (51%). This is en- tirely consistent with the usual view t h a t DCSB is an infrared and int,errnediate distance-scale plreiromena.

However, i t is certainly not t r u e t h a t tlle atlclition of the rnnning coupling ill the ultraviolet is all csscn- tially unimportant correction. I t is well frorn pert,urbative Q C D t h a t t,he quark colidelrsate increases with the re~lorirlalizatioll scale p . In fact, m,,(qq), is renon-llalizatioli-group invariant, where 772, is the usual ECSH running quark Inass. ils is well known fro111 perturbat ive QCD the quark condensale scales wit11 the renonnalization point p as

and clearly the running mass m, scales in t h e inverse way. I t is also known",18 t.llat the asymptot ic hellnvior of M ( p 2 ) in the deep spacelike regime is (ill a coiivenieirt~ shor thand)

where u is a constant given by

Tlie right-hand side (RHS) of Eq. (46) is t o be understood in t h e sense tlrat in t8he lirnit of exact chiral symmetry where the secolid t e r m is zero the leading U\I hellavior is given by tlre first t e rm, whereas if nli, # 0 then t,lle first tern1 is i g ~ ~ o r e d wit11 tlle seco l~d as the leading-log U V behavior. Naturally if m, # 0 there will be lliglier- power logaritllmic corrections t o tlie second te rm ~vlriclr will tlleinselves conrpletely donliliate the first asymptotically.

Wi th this brief summary we can now return t o consider t h e quark condensate, which is give11 in Eq. (39). Firs t , notc t h a t this expression is only related to the usual concept of the quark condensate in QCI) s u m r ~ l e s ~ ' ' ' ~ " in tlie liniit of exact chiral syrrlnletry ( 1 7 1 , = 0). Tire solutions considered here all have ~n, = 0 , and it is relatively stra.ightforward15 t,o include ECSB quark masses, so we d o not include this. If we consider pure dual QCD then M ( p 2 ) falls much faster asymptotically t h a n the first t e rm on t h e RHS of Eq. (46), arid then ( i q ) is just a nurlibcr independent of any consideration of QCD renor- rnalization or UV cutoff. Th is gives the reslilts for ( q q ) shown in Table I .

In Fig. 2 we show the solutioli M ( p 2 ) for both pure dual and ~ l l a t c l ~ e d interactions. A s is apparent the matched solution falls off niucli slower asymptotically, a n d , in fact .

--- matched

-6 - 2 4 0 2 6 8 1 0 -

loglo( P * ) -

FIG. 3. The quark mass function hl(p2) is plott,ed as a function of mornenturn squared on both linear and logarithmic scales [the units of M ( p 2 ) and in the logarithmic plot are hIeV and h l e ~ ' , respectively]. The solutions for botlt the pure dual (-) and the rnatclled ( - - -) effective interactions are shown. Not,e that on the linear scale the two results are alrnost indistinguishable and so or~ly the pure dual result is shown there. The matched resiilt has the correct leading-log ~~ltraviolet behavior as discussed in the text,.

it l ~ a s bee11 co~lfirrned n~ilnerically t h a t tlre hel~avior is just a s described in Eqs. (46)-(47). Since r11) t o a log- a r i t l ~ m M(p" -+ l / p 2 for t h e matched case, it is clear t h a t Eq. (39) diverges logarit~hmically wit11 tlie UV cutoff. I11 fact,13115-18 t h e UV cutoff is the renormalization point a t which t h e condensate is evaluated i l l this eqlla- tion. T h e results quoted in Table 11 arc the values for the condellsate a t the conventional m o r n e ~ ~ t u n i scale of' 1 GeV after being scaled using Eq . (45). For example, with a n UV cutoff of l o5 McV the unscaled condensate [for (-@) = (500 and = 1.751 is 314 MeV, which, when scaled from 10"eV t,o 1 GeV, becomes 241 MeV. I t has of course been verified numerically t h a t all solutions and results are independent of any UV cutoff provided t h a t i t is chosen sufficiently large.

I t is now apparent why the r n a g ~ ~ i t , u d e of tlle quark condensate is snraller in Table 11 than in I'ahle I even tliougll t h e extent of the DCSB has been slightly illcreased by the addition of t h e appropriate Q C D asymptotic hchavior. T h e effective mo~l len turn scale a t which tlre dual cluarl< condensate is evaluated is sonrewhat greater t11a11 1 Gel[ , since tlie integrand of E q . (39) is not negligible beyond 1 GcV for the pure dual case. There is no s t~raigl~tfor~oart l way t o scale the pure dual conclensat~e so t h a t it can be directly compared with the r r~a tc l~ed condensate a t soli-ii: arbi t rary lnoinentum scale.

We have also studied t,he effect of set t ing A? = 0 in the dual QCD effective quark interaction. To d o this we give u p t h e approximate relation for if discussed follo~ving Eqs. (30)-(32) b u t all other things remain unchanged. For pure dua l QCD with (-@) = (500 MeV)', mZff = 1.75, a n d all o ther parameters t h e same a s for Table I, we

43 DYNAMICAL CHIRAL-SYMMETRY BREAKING IN DUAL QCD 3549

find f, = 84 MeV, (-(qq))'I3 = 250 MeV, and M(0) = 428 MeV. A comparison with the corresponding entry in Table I shows that this has caused only a relatively small decrease in the DCSB quantities f,,_(-(qq))1/3, and M(0) . I t would thus seem that while M # 0 is the seed that gives rise to the complicated structure of the dual QCD Lagrangian of Eqs. (17)-(19), the end product is insensitive to its exact value (see also Ref. 8). We note that this is consistent with the suggestion by BBZ that the behavior and properties of the dual superconductor are insensitive to M.

V. CONCLUSIONS

We have studied dynainical chiral symmetry breaking in dual QCD and have attempted to improve on the earlier treatment of this by BBZ. We have used the full quark-quark dual interaction and have covariantly solved for a momentum-dependent quark inass function M ( p 2 ) by iterating the appropriate integral equation to convergence. In order to simplify our study the string direction n p , which was introduced along with quarks into dual QCD, was averaged over. While the effects of tliis are not well understood, it does not seem an unreasollahle approximation since of course no physical observahles can depend on n p .

We showed that the usual perturbative QCD behavior could be relatively smoothly matched onto the dual

QCD interaction a t an intermediate momentum scale of ~2 GeV. We verified that our solutions the11 had the appropriate ultraviolet asymptotic behavior.

We adopted the point of view that the simple tibelian- ized coupling for quarks to the dual QCD fields was an approximation and so relaxed the requirement that e = 2 ~ / 9 . We allowed an increase of the quark coupling to the dual fields by a factor c , where e' c e , hut oth- erwise used typical dual QCD parameters extracted I,y H H Z from quite different physical phenomena. It was found that with c 2: 1.7 - 1.9 good results could he found for the pion decay constant, the quark condensate, and the infrared values of the inass function.

In coi~clusion, we have found that this new study of BRZ's dual QCD has produced results tha t , within the limitations of the approxiinations made, are encouraging. This worlc has lent further support t o the idea that there may be a strong connection between DCSH, confineinent, and dual superconductivity. Extensioils of this work to studies of chiral restoration and deconfine~nent in dual QCD are underway.

ACKNOWLEDGMENTS

We would like to thank the Department of Physics a t the University of Washington, where part of this work was carried out, for its hospitality. This work was sup- ported in part by the U.S. Departine~lt of Energy.

*Permanent address: Departamento de Fisica d a UFSM, 971 19 Santa Maria/RS, Brazil.

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Dynamical chiral-symmetry breaking in dual QCD