Modelling the quark propagator

ANNALS

OF PHYSICS

210, 464485 (1991)

Modelling

the Quark Propagator

A. G. WILLIAMS AND G. KREIN* Department

of Physics and rhe Supercomputer Computations, Research Institute, Florida State University, Tallahassee. Florida 32306-3016

AND

C. D. ROBERTS Theory

Group,

BIdg 203, Argonne National Argonne, Illinois 60439

Laboratory,

Received January 18, 1991 We are interested in developing covariant, confining, and asymptotically free models of hadrons. With this goal in mind we have carried out a study of dynamical chiral symmetry breaking without imposing the frequently used approximation a,( - (p - k)2) N a,( - p’, ), where p: = max(p’, /c*) for the running coupling constant in the quark Schwinger-Dyson equation. We present numerical results in Landau gauge and compare these with earlier results obtained when using this approximation. We see in this context that a gluon propagator which has the form l/q4 in the infrared is too singular and must be regulated. We derive a suitably generalized expression for the pion decay constant f,. With essentially one free parameter we are able to reproduce reasonable results for various physical quantities of interest includingf,, (qq), and Aoco. c) 1991 Academic Press, Inc.

I. INTRODUCTION While there is considerable optimism that lattice gauge theory will continue to improve as a means of studying nonperturbative QCD, it is also extremely important to pursue covariant, nonperturbative approximation methods. In particular, there is a continuing need for the development of approximation techniques and models which bridge the gap between perturbative [ 1] QCD and the large amount of low- and intermediate-energy phenomenology in a single covariant framework. One such approach is to find an ansatz for the renormalized propagators, vertices, etc., of QCD which satisfy the Schwinger-Dyson equations (SDE) and respect the Slavnov-Taylor identities (STI) of the theory [2]. No fully satisfactory ansatz is known, since this would constitute a complete solution of QCD. However, some progress can be made by satisfying a subset of the SDE and ST1 and then supplementing these with information gleaned from lattice gauge theories, * Current and permanent address: Instituto de Fisica Teorica, Rua Pamplona Paula - SP, Brasil.

464 OOO3-4916/91 $7.50 Copyright 6 1991 by Academic Press, Inc. All rights of reproduction in any iorm reserved.

145, 01405 Sao

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phenomenology, etc. The test of such a scheme is to enlarge the set of SDE considered and look for a decreased dependence on the phenomenological input, although this is beyond the scope of the present study. This type of approach is well suited to model building. Since the most significant difficulties in QCD arise in the gluon sector this is the natural place to appeal to phenomenology while attempting to construct explicit solutions and maintain important symmetries in the quark sector. Two symmetries of particular importance are gauge invariance and chiral symmetry. The difficult task is to obtain solutions of the quark SDE in order to study, for example, dynamical chiral symmetry breaking (DxSB) and quark conlinement. One important feature of the present work is that the standard approximation of replacing the running coupling constant CI,(-(p - k)2) by c(,~(-p’, ), where p’, 3 max(p’, k2), is avoided. We will refer to this as the angle approximation, since its use allows the integration over the relative angle between p and k to be performed analytically in the SDE. In Ref. [3] it was shown that this appproximation may be useful for a qualitative analysis of the quark SDE when the infrared behavior is expected to be smooth, but that it was entirely inadequate for developing a detailed model of the relationship between the nonperturbative gluon and quark sectors. This is expected to be particularly true when the issue of conlinement is addressed. Avoiding this angle approximation means for the first time that detailed numerical modelling which includes the momentum-dependence of the wavefunction renormalization (i.e., A(p2) # 1) in the quark propagator can be carried out. Of course, the extent to which these and all other analyses of SDE yield results that are actually representative of QCD is not known and we proceed with this caveat clearly in mind. There have been a number of previous discussions of the quark SDE, both recent and otherwise [4-S]. There is also current interest in studies of DxSB in technicolor theories [9], where the coupling constant “walks” rather than “runs” (like c(,(Q’)) in QCD. There are, of course, similar studies in the gluon sector and of the quark-gluon interaction, (see, e.g., Baker, Ball, and Zacharaisen [lo]). The quark SDE has also been investigated in the context of bilocal meson fields obtained from functional integral studies [ 11, 121 of QCD. Where the quark SDE is solved in the above references it is either done noncovariantly (i.e., in instantaneous gluon exchange approximation), with a constant c(,~or using the angle approximation for cr,(Q2). To the best of our knowledge this is also true for the literature at large (with the exceptions of Refs. [3, 121). Our primary concern here is to extend earlier studies [S] of modelling of the quark propagator without imposing the angle approximation. However, it is clear that the results will be of interest in a much broader context, including, for example, technicolor theories [9 1. The ansatz that we consider is not the usual naive rainbow (or ladder) treatment although, with the choice of Landau gauge and with some of the approximations made here, the final form of the quark SDE is similar. The central features of this approach are attempts to ensure that important symmetries are maintained and to reproduce leading-log perturbative QCD results in the ultraviolet, through the

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inclusion of the running coupling constant CI~(Q’). We treat the infrared in a phenomenological way since it is here that our ignorance is greatest. In recent years some progress has been achieved in understanding the confining properties of the pure gauge sector, with much attention being focused on a l/q4 infrared behavior for the gluon propagator [13-151. This form for the gluon propagator has been used [14] in studies of dual QCD. It has also been obtained in a self-consistent solution of a truncated (i.e., one-loop, pure gauge) gluon SDE in the Landau gauge [15, 161. With these studies as motivation we include a discussion of this type of infrared gluon propagator. It is hoped that the correct form can eventually be derived from QCD. It is straightforward to generalize to other types of nonperturbative quark-quark interaction. In Section II we briefly review necessary QCD formalism and the asymptotic behavior of the quantities of interest. This is done both to facilitate the discussion in Section III and to make very clear the relationship of this work to perturbative QCD. In Section III we discuss the ansatz in detail. We calculate the pion decay constant .f, in a generalized form suitable when the angle approximation is not invoked. We also discuss the quark condensate (tjq) and difficulties associated with the nonperturbative definition of this when explicit chiral symmetry breaking (ExSB) quark masses are included. In Section IV we give our numerical results and show that with essentially one free parameter good values off,, (qq), and the QCD scale parameter AqcD can be obtained. We also discuss a l/q4 form for the gluon propagator in the infrared and why it must be regulated. We give our conclusions in Section V and suggest ways of obtaining more information about the infrared behavior of the quark-quark interaction.

II. QCD

FORMALISM

In order to introduce necessary formalism and to clarify subsequent discussions we summarize in this section some important properties of QCD. The fully-dressed, renormalized quark and gluon propagators (in momentum space) are denoted by S(p) and &l(p) = 6,$‘“‘(p), respectively (a, b = 1, .... 8 are SU(3) color indices). For a covariant gauge the full gluon propagator has the form [2] Y(p)

=

-f” i PUP” 1 I( P2 > 1 +zI(p2)

-aPz p2

1 1 p”

(2.1)

where n(p*) is the gluon vacuum polarization and a is the renormalized gauge parameter [ 163 at the arbitrary renormalization point p (we are suppressing a label p on D, l7, and a). The Landau and Feynman gauges are given by a = 0 and 1, respectively. The general form of the inverse of the full quark propagator in a covariant gauge is F’(p)=*-m-z(p)=Z-‘(p’)[#-M(p’)] =4P2)pl-B(P2h

G-9)

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PROPAGATOR

where m is the renormalized explicit chiral symmetry breaking (ExSB) quark mass, C(p) is the quark self-energy, M(p*) = B(p2)/A(p2) is the running quark mass, and 2 -‘(p*) = ,4(p*) is the momentum-dependence of the quark wavefunction renormalization. Note that a subscript p (to indicate renormalization-point dependence) is to be understood on all renormalized quantities. It is suppressed for clarity of presentation. In addition renormalized quantities are gauge-dependent in general, but this again is not explicitly indicated. The renormalized coupling g is related to the running coupling constant cr,(Q’) by g’~4m,(Q2)1Q’=~2, where q2 E L Q2 > 0. For Q’$ A&b leading-log (i.e., one-loop) result

(2.3)

the running coupling constant is given by the

12ll c(s(Q2)= (33 - h,) ln(Q’/n&,)

dx

(2.4)

= ln(Q*//l&.n)’

where nf is the number of quark flavors and /locn is the scale parameter of QCD. We will see later that d= 12/(33 - 2~,,) is the anomalous dimension of the mass. Note that Eq. (2.4) is renormalization-scheme and gauge independent, as is its second-order (i.e., two loop) extension [ 11. The proper (i.e., oneparticle irreducible) renormalized quark-gluon vertex is written P’(p’, p) = (i”/2) P(p’, p), where 1” are the usual SU(3) color matrices (again suppressing the renormalization point dependence). At the renormalization point p we effectively have

(2Sb)

S-‘(p)~p~=~I,2=~-m,

which implies, for example, that C(p’ = --p’) = 0, fl(p’ = -11’) = 0, Z(p’= -p*) = 1, and M(p* = -p’) =m. Gauge invariance gives rise to a set of Slavnov-Taylor identities (STI), which imply relations between the renormalized propagators and proper vertices of the theory. For example some of these ST1 imply relationships between gluon and ghost renormalization constants (recall that the renormalization constants relate bare cutoff dependent propagators and vertices to the renormalized p-dependent ones). Another of these gives p,p,D”“(p) = --a, which implies that the gluon selfenergy contributes only to the transverse part of the gluon propagator [l, 21 as indicated in Eq. (2.1). The ST1 for the quark-gluon vertex is [2]

k,~“(p’,p)Cl +Hk2)1 = Cl -fW,p)l

S-‘W-S-‘(p)Cl

-Wkp)l,

(2.6)

where again p’ = p + k. The ghost self-energy and ghost-quark scattering kernel are denoted b(k’) and B(k, p), respectively. Note that in Landau gauge B(k, p) =0

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when p’2 = p2 = k2 = - p2 to lowest order in perturbation theory. Clearly we have P(p’, p) = P(p) p’). Without ghosts, Eq. (2.6) is identical to the corresponding

Ward-Takahashi identity (WTI) of QED. Other symmetries will also give rise to STI, in general. In the limit that there are no ExSB quark masses, QCD is a chirally symmetric theory and has a ST1 for the coupling of an isovector axial-vector current (e.g., a IV-boson) to a quark, (2.7a) Since ghost terms do not contribute this actually has the form of a WTI. In the presence of ExSB quark masses the chiral symmetry WTI is replaced by

k,Wp’, p) =; CS-‘(P’) ys+ Y,S -‘CPU + 2imr,(p’, PI.

(2.7b)

The additional term contains the ExSB quark mass m and the quark pseudoscalar isovector vertex Ts. QCD is well known to be an asymptotically free theory, which means that if the characteristic momenta in some physical process are sufficiently large and space-like then QCD behaves as a free theory with logarithmic corrections. Let fi(p, g) be some renormalized proper vertex with p2 = -p2 as the characteristic momentum and g the renormalized coupling. Then if the momentum is scaled by some 2 - (Q’/p*)‘!’ B 1 we have T,(Ap, g) = (A)nj fi(p,

g)[g2/g21dJ 2: (L)nl Ti(p, g)($ln A2)“,

(2.8)

where g- g(Q’) and with the obvious definition g2(Q2) = 47ccr,(Q2).

(2.9)

The exponent ni is the naive (canonical) dimension of Ti and the exponent di is its anomalous dimension. Note that g = g(Q’ = p’) is just a restatement of Eq. (2.3). For example, for Q’ z - p2 % A&, we have the leading-log result for the running mass

where rG d= 12/(33 the scale of renormalized

is the renormalization group invariant mass parameter and 2~~) is the anomalous dimension of the mass. The parameter ti sets the ExSB in QCD and is analogous to A,,,. To lowest order the EX SB mass m is related to fi using Eq. (2.10) by mEM(-~2)=fi/[&1n(~2/A&D)]d.

(2.11)

MODELLING

The inverse quark propagator is [l, 21

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for Q2 = - p2 and Q2, pz % A&,

to lowest order (2.12)

Spl(p) = [Z p’(~2)fl = f[$ln(Q’/p’)l%

which gives Z(-Q*)= [iln(Q2/p’)lPd5, where d,v= -2a/(33 - 2n,) is the (gaugedependent) anomalous quark dimension. Note that Z( - Q2) = 1 to leading order in Landau gauge (a = 0). Similarly, the asymptotic behavior of the transverse part of the gluon propagator to leading order is D dtr,(p)

= _

g”” [

p”p’ L p2 lp’[2

&(Q’,/&2)

where d, = (39 - 9a - 4n,)/[4(33 - 2nf)] is the gluon anomalous dimension. The quark-gluon vertex for Q’ = -p2 and Q2, /A’ % A&, is given by

where d,= - (27 + 25a)/[8(33 - 2nJ]. It is straightforward to verify that 2ds - 2d, + d, = i which implies, for example, that the p-function is independent of the choice of gauge to leading order. It is also possible to define [ 1, 21 a running gauge parameter a( Q’), where a = a( Q’ = ,u’). If we consider the electromagnetic couplings of the quarks, then it follows from electromagnetic gauge invariance that the photon-quark vertex satisfies (2.15)

This is just the usual WTI of QED and has no ghost complications. The renormalized SDE for the quark propagator can be written as [2,6] s-‘(p)

= Z,[#

- WPyA)]

4 - i-Z,g’ 3

A d4k s 4(2x1 y,S(k)

D”‘(k

- PI T,(k

P),

(2.16)

where mbare(A) is the bare mass, Z, and Z, are the quark propagator and quark-gluon vertex renormalization constants, respectively, and where we have used C, 1”A”/4 = $. We have also introduced an ultraviolet (UV) cutoff A,, (often written as A for notational brevity) to regulate the integral with the understanding that we are to take the ,4”v + cc limit at the end of any calculations. The renormalization constants are functions of A,, and the renormalization point p [Z,(A, p) and Z,(A, p)]. The SDE for the (unrenormalized) quark self-energy is represented in Fig. la. The SDE for the quark self-energy can be written

L-(p)=i;X’jn$$y ,S(k) D”“(k

- p) f,(k, p),

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(b) r,”

FIG. 1. (a) Illustrates the Schwinger-Dyson equation for the quark self-energy. Open circles represent proper (i.e., one-particle irreducible) vertices and tilled circles indicate fully dressed propagators. (b) The proper quark-axial-vector vertex; (c) the proper quarkkpseudoscalar vertex coupling to a pion; and (d) the pion decay contribution to the quark-axial-vector vertex.

provided all quantities are understood as being bare or unrenormalized in this equation. A more detailed discussion of renormalization for nonperturbative quantities can be found, for example, in Baker et al. of Ref. [lo]. DxSB occurs when C develops a nonzero Lorentz scalar piece and is clearly a nonperturbative effect. In general, /iQcD and fi are renormalization-scheme dependent and so take different values, for example, in the modified minimal subtraction (MS) scheme, the minimal subtraction (MS) scheme, and the momentum subtraction scheme. However, the difference does not appear in leading order [ 1, 171, but only at second-order and higher in CI,. We retain only leading-order terms in the asymptotic region and so we do not need to concern ourselves with this.

III.

ANSATZ

We now wish to construct an approximation scheme which reproduces as far as possible the symmetries and leading-log perturbative QCD results discussed in the previous section. Of course we would also like to demonstrate how confinement can arise in such a scheme, but we do not address that issue here. The basic approximation in this ansatz approach is to neglect the effect of ghosts in the quark sector and assume that this can be compensated for by line-tuning the phenomenology in the gluon sector. This of course raises the question of how important this violation of QCD gauge invariance is. The neglect of ghosts in the quark sector implies that we have essentially abelianized the theory for the quarks, and so QCD gauge invariance becomes the requirement that color-current be conserved at the quark-gluon vertex. For this purpose it is sufficient to require that

MODELLING

Eq. (2.6) (without (p’-p+k)

fYpw=;

THE

QUARK

ghosts) is satisfied. From

Eqs. (2.2) and (2.6) we can write

c‘4(P”)+‘4(P2)l v+ (p’ + p)”

(PI2 - P’)

@’ + #I x -2

[B(p”)

471

PROPAGATOR

- B(p’)]

[4p’2)

- A(p’)]

i

+ transverse parts, 1

(3.1)

where we have indicated that only the longitudinal behavior is specified by Eq. (2.6). The unspecified transverse pieces are proportional to [g”” - (k”k”/k*)]. Since Eq. (2Sb) implies that A( -p’) = 1, then in the limit that p’ + p, p’= -p’, the first term on the RHS of Eq. (3.1) becomes yb. The corresponding contributions from the second and third terms are proportional to dA/dp* and dB/dp’, respectively, and for large 1~’ will be very small, as will the unknown transverse pieces. Thus in this limit we have f u 2: yb which is the usual perturbative vertex. The only way of determining the nonperturbative transverse pieces in Eq. (3.1) is to attempt to simultaneously solve the SDE for the quark-gluon vertex. We do not pursue this possibility here. This quark-gluon vertex has been considered earlier by Pagels and Stokar [4] and has been discussed in detail by Ball and Chiu [lo]. The specified part of Eq. (3.1) is free of any kinematic singularities as can be readily verified by considering the limit p” + p’. Even though we have argued for the use of Eq. (3.1) for the quark-gluon vertex, for this investigation we will restrict ourselves to the simpler form F’(p’, p) = i [A(p”) - A( yO. We make this choice in part because it is simpler, but most importantly because it conforms to previous numerical studies of the subject [7, 81, where the angle approximation was used, since only then can we meaningfully compare the current and earlier work. An extension of this work to include Eq. (3.1) and various types of transverse pieces is being pursued and it will be interesting to compare the results with those to be reported here. It is unfortunate that there are difficulties in using ghost-free gauges, such as axial gauges or the Coulomb gauge. Any approximation made in a noncovariant gauge tends to destroy the covariance of physical observables [ 181, In the axial gauge additional singularities of the form PI.p enter into propagators. In one study [ 191 the “gauge condition” n .p = 0 was adopted, which considerably simplified the analysis, while in another work [20] this choice of the axial gauge variable (n .p) was not enforced. The difficulty remains, however, that since one now has two fourvectors (p’ and nl’) the most general Lorentz structure of the inverse quark propagator is [ 191 S -’ = (A# - B) + (C$ - D)b, where the functions A, B, C, and D depend on each of the Lorentz scalars p’, p. n, and nr = f 1. The fact that the scalar functions A, B, C, and D can depend explicitly on n p and n2 significantly complicates the SDE analysis. For these reasons we have chosen to work in a covariant gauge in this study. In QED in Landau gauge 2, = Z, = 1 to leading order in perturbation theory. Since we are interested in constructing an abelianized model of QCD it is also

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appropriate to use this in our study. With the assumption that this is true to all orders for these momentum-independent renormalization constants we find from the SDE for the renormalized quark propagator (Eq. (2.16)) that the renormalized self-energy also has the form of Eq. (2.17) where now all quantities are understood as being renormalized. This is the usual starting point for discussions of DxSB in the literature [4-9, 111. It is well known [S-S] that approximating g’D”“(p)

Q* E -p2,

‘v g’(Q’) D(O)yp),

(3.2)

for large space-like momenta (in Landau gauge) in Eq. (2.17), for the quark selfenergy leads to the correct perturbative QCD leading-log behavior. Here D(O) is the perturbative gluon propagator, which is just D of Eq. (2.1) with IZ= 0. Thus we replace D in Eq. (2.17) by D(O) and will determine 01,(Q*) phenomenologically provided that c(,(Q*) reduces to Eq. (2.4) for Q’ $ A&,. The motivation for Eq. (3.2) arises from the asymptotic form of the quark-quark scattering kernel [21]. The behavior of the kernel (K) is known from renormalization group analysis and to leading order (in Landau gauge) is [Q’= -(q’q)*]

L,~,~,Jq’, q; f’) = - ig’(Q’) yZ,Jb%q’ - 4) YZ,,+

(3.3)

where to this order K is independent [22] of the center-of-mass momentum (P) and the renormalization point (p) and where n, m, n’, and m’ are spinor indices. In the asymptotic region K is dominated by one-gluon exchange (K 2: -ig’fDr) which leads to Eq. (3.3). Since in this region r N y in Landau gauge we can use Eq. (3.3) for g*yDr in Eq. (2.17) for Z(p). This is the meaning of Eq. (3.2). We now have in Landau gauge the renormalized SDE for the quark self-energy Z(p)=ii4zj

A d4k (2n)4r”S(k)r,(-(k-p)*)D’“‘“‘(k-p)T,(k,p).

(3.4)

As discussed previously, here we use

r”(k, p) = $C4k2) + M2)1 y”,

(3.5)

in order to compare with previous studies [8] which used the angle approximation. Using Eq. (2.2) we can write Eq. (3.4) as n d4k 3A(k2)(1/2)[,4(k2) + A(p*)] (2n)4 A*(k*) k2 - B’(k*)

A(p2)=l-ii4nJ

;+2 CAk-p)l* 3

3 p2(k-p)2

1

(3.6a)

A d4k 3B(k2)( 1/2)[A(k*) + A(p*)] A2(k2) k* - B2(k2)

d-(k-p)‘) x

(k-p)’

’

(3.6b)

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Note that we choose the renormalization point here such that p= ,4”v, which means that mbare(A) = m. We will see that our results are independent of the ultraviolet cutoff (A,,) provided it is chosen sufliciently large, since the integrals in Eq. (3.6) are convergent. Note that this would not be the case were cr,(Q’) constant. If we were to make the angle approximation [559, 111 and replace a,( - (p -k)‘) in Eq. (3.6) by a,( -PC), where p: = max(p’, k2), then we would find after an angle integration that A(p2)= 1. It is the fact that A@*)# 1 in Eqs. (3.5)-(3.6) which differentiates this ansatz from a rainbow approximation in the Landau gauge (where only perturbative quark-gluon vertices are used). When there is no ExSB renormalized quark mass (m = 0), we have exact chiral symmetry and it is well known [4-61 that conservation of the axial-vector current (in Landau gauge) leads to (cf., Eq. (2.11))

M(-Q2),,~_4Q Cln(Q2/&,)lJ-'~ where c is some constant independent of ,u. The asymptotic form for the running mass (which is renormalization point independent) can be written in convenient shorthand as

~(-Q21Q2~ oc$

[ln(Q2//i&,]“-

ln(fi2/&,)

1+ m

WQ*l&,)

1 d ’

(3.8)

where we have used Eqs. (2.10)-(2.11). Note that Eq. (3.8) is to be understood in the sense that, for exact chiral symmetry, the second term on the RHS is zero and the first term is the dominant one, while in the presence of ExSB the second term is the dominant asymptotic behavior. With ExSB there will be many terms which are suppressed by powers of In(Q*) with respect to the second term but which are larger than the first as Q’ + co. The (renormalized) quark condensate (49) is a measure of the degree of DxSB and can be defined as (44) = (vacl :4(O) q(0): jvac), where Ivac) refers to the nonperturbative vacuum and the normal ordering of the (renormalized) operators is with respect to the perturbative vacuum. Denoting S”‘(p) = (# - nz))’ as the perturbative quark propagator we have (again for the choice p = A,,) (44) = --J~III+ tr{S(x, 0) - S’O’(.u, 0)) = -12i

j

,1 d4p Z(p’) M(p*) @i$ [ p2-M2(p2)

-___ m p2-m2 1 ’

(3.9)

where S(.u, y) is the coordinate space quark propagator and where the trace over spinor (4) and color (3) indices gives the factor 12. In Appendix A we show that

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c as defined in Eqs. (3.7)-(3.8) in the limit of exact chiral symmetry (m = 0) in the Landau gauge satisfies 4n2d ’ N -

3

(44)

(3.10)

[ln(~2//i&.D)]d

which, since c is a constant independent of the renormalization point p, also implies that (44 > - CW~214&, )Id. Equation (3.7) is a result also obtained in discussions of the operator product expansion [23] (OPE) and QCD sum rules [24]. In the OPE the quark condensate is taken to be a coefficient with the above p-dependence irrespective of whether or not ExSB is present. From the discussions of Appendix A it is clear that the OPE quark condensate only corresponds to the “natural” definition of Eq. (3.9) in the limit of exact chiral symmetry. In the limit of exact chiral symmetry we have the WTI of Eq. (2.7) which can be shown to be satisfied by the equation (p’ = p + k)

(f’ + f) rgP’,P) = ; Cef2) + 4P2)l v+ $ CM”) - A(P2)1-j-( 1 - [B( jY2) + B( p’)]

I

+ transverse parts

>

+.

(3.11)

This result for the proper axial-vector vertex is similar to the expression for the proper quark-gluon vertex in Eq. (3.1) and again only the longitudinal behavior is completely specified. This can be compared with the usual perturbative axial-vector vertex (r/2) y”y5. It also follows from Eq. (3.11) that

rgp’, P) -+ -W/k’)

WP’) v,

as k -+ 0.

(3.12)

This is equivalent to the statement that as k + 0 the axial-vector vertex (Fig. lb) becomes completely dominated by the pseudoscalar coupling of a massless pion to the quark (F, in Fig. lc) and the subsequent weak decay of the pion into an axialvector current (Fig. Id). The pion decay constant f, is defined by (m, n are isospin indices) (0 lAy(

which gives from Fig. Id as k + 0

n”(k))

= if,k”cY”“,

(3.13)

MODELLING

THE

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PROPAGATOR

Then from Eq. (3.12) we find the Goldberger-Trieman vertex (as k -+ 0) rs(P,P) = -w@(P2)/f,

The simplest generalization

= -%,Z

415

relation for the pseudoscalar -‘(P*) ~(P2)/f,.

(3.15)

of Eq. (3.15) to k # 0 is

I-S(P’>P) = -N,

i CNP’*) + B(p2)l/.fn.

(3.15a)

To completely determine the pseudoscalar vertex requires solving the pseudoscalar vertex SDE (or, equivalently, the pion BetheeSalpeter equation). Using the above it is possible to obtain an expression for f i from the integral equation for pion decay illustrated in Fig. 2. Our discussion here is a relatively straightforward extension of the arguments presented in Refs. [4,25] to the case where A(p*) # 1. There are other methods of calculatingf,, for example, in the context of auxiliary meson-like fields and effective meson Lagrangians [ 11, 121. Note that to avoid double-counting lY$reg) contains only the regular piece of the axialvector vertex (i.e., the pion-pole piece of Eq. (3.15a) is excluded). Then Eytreg’ is given by Eq. (3.11) with the B terms absent. From Fig. 2 we find

xtr

[iS(p+k)][iT~(p+k,p)][iS(p)][iT;”””g’(p,p+k)]

(3.16)

As for the quark-gluon vertex we will assume that the unspecified transverse parts in Eq. (3.11) can be neglected. This is an approximation which respects chiral symmetry. To do better would require solving another SDE for the quark-axial-vector vertex. If we take k + 0 in Eq. (3.16) then we can make use of the GoldbergerTrieman relation Eq. (3.15) for r5. In the limit of exact chiral symmetry we have (see Appendix B for details)

(3.17)

FIG. 2. The integral equation for the pion decay constant (f,). The propagators and vertices are the same as those in Fig. 1 except that r;lreg’ does not contain the pion pole piece of r:. This is necessary to avoid the double counting of Feynman diagrams.

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WILLIAMS,

KREIN, AND ROBERTS

which as a physical observable must be independent of the choice of renormalization point (this has been verified numerically for our solutions). We now summarize the principal ingredients of this Landau gauge ansatz. It is assumed that: (i) ghost effects can be neglected in the quark sector with small adjustments to the nonperturbative phenomenology in the gluon sector, (ii) this then allows the quark-gluon vertex to be determined up to unknown transverse pieces by Eq. (3.1) (recall that here we use the simpler form of Eq. (3.5)) and (iii) the unspecified (nonperturbative) transverse parts of the quark-axial-vector vertex in Eq. (3.11) can be neglected. IV. NUMERICAL

SOLUTIONS

It is convenient to take the equations of the previous sections and write them in Euclidean space by performing a Wick rotation. To avoid potential confusion we use upper-case letters for Minkowski-space quantities and lower-case for their Euclidean counterparts. We then define x = Q’ = -p2 > 0 and m(x) = AI( z(x) = Z(p2), b(x) = B(p’), etc. and so, for example, m(x)= @x)/a(x). The Euclidean expression for the quark condensate is already given in Eq. (A2). The pion decay constant becomes

(4.1)

and the quark SDE can be written [cf. Eq. (3.6)] a(x)= 1+ j”‘&f(x,

(4.2a)

Y) 4~) 4x, Y),

A2 b(x) = m, +

s

dyf(x> Y) W)

(4.2b)

4x, VI,

where m, denotes the ExSB quark mass when p = Auv (as in Appendix A) and where we have defined u(x,y)=~j~dOsin20~

&cos8-f~sin2B

1

(4.3a)

u(x, y) G j” de sin’ 0 T 0

(4.3d)

MODELLING

THE

QUARK

477

PROPAGATOR

Comparison with Eq. (3.6) shows that y z -k*, s z -(k - p)*, and 0 is the angle between p and k (in Euclidean space). The quark SDE is solved by iterating the coupled integral equations in Eq. (4.2) to convergence. The angle integrations in Eqs. (4.3a), (4.3b) need only be evaluated once (fortunately), since they are independent of a(x) and h(x). It remains to choose a particular form for the infrared and intermediate Q’ behavior of cr,(Q*). This will eventually be constrained by the requirements of good phenomenology and confinement. We can of course solve for any choice. Following the discussion in Section I we would like to study a l/Q” infrared behavior of the full gluon propagator D. This implies x,~(Q”) - l/Q’ in the infrared, since in our ansatz $ D”‘(p) = cr,(Q*) D’“‘u”(p),

Q*=

-p*

(see Eq. (3.2) and associated discussion). Hence we might choose

QQ2) =

C’

dn

(Q21&m)‘-e +lnCs + (Q*/~&,)l’

(4.5)

where C’, r > 1, and E > 0 are dimensionless parameters. Both r and E regulate the Q2 --t 0 behavior. Clearly for large Q’ we recover Eq. (2.4) and asymptotic freedom. The results will be shown to be insensitive to r and so we typically choose In r = 1. It is straightforward to verify that a pure l/Q” leads to a divergent integral in the self-energy which is why the regulation parameter E > 0 is necessary. Other reguiarisations of l/Q” are possible and each will involve the introduction of some unspecified regulating parameter E. Because of this and the fact that the arguments [13-151 for l/Q” infrared behaviour are rather tenuous in any case, we study a simpler form here. We consider an effective gluon propagator whose dominant infrared behaviour is obtained through the replacement (in Euclidean space) C’

(Q’P&,,)*--

repl. CS’“‘((Q/A,,,)),

(4.6)

in Eq. (4.5) and so we use d2*)

=

C&,QWQ)

+

dn 1nCz +

(Q*/~&,)l

The d-function on the RHS has been studied earlier in related discussions [ 11,20,26,27] of the SDE as an alternative to the l/Q” behavior. The point of view adopted here is that this integrable singularity ( -S4(Q)) in the gluon propagator is not an unreasonable representation of the infrared behaviour and has the virtue that the singular structure of the kernel in the integral equations can be

478

WILLIAMS,

KREIN,

AND

ROBERTS

handled analytically. We reserve judgement on whether it is a reasonable ansatz from the point of view of QCD phenomenology. Pagels [2] has considered the alternative of keeping a divergent self-energy from l/(Q’)‘-’ with E -+O which it is argued leads to vanishing quark propagator (S- E). While this might appear to imply quark confinement it destroys any connection with asymptotic freedom and perturbative QCD. It has recently been suggested [27] that with an appropriate ansatz for the ghost self-energy (b- l/~) a finite quark propagator can result from Pagels’ arguments, however the relation to the perturbative regime remains unclear. In the approach considered here C and the quark propagator remain finite. To demonstrate quark confinement it would, for example, be sufficient to show that the quark propagator has no poles in the timelike region [ 19,28,29]. Even though propagators are gauge-dependent quantities, in general, the position of poles (or their absence), which determine particle production thresholds (or their absence), are physical observables and therefore must be gauge-invariant [ 19, 28, 293. An alternative point of view is that confinement may not manifest itself through the self-energies of the single quark or gluon, but instead might be a property of color-singlet bound states in QCD. We have also studied an alternative regularization, ol,(Q2) - l/(Q2 + E’), for small but finite E and found results consistent with those reported here. In earlier studies [S] the angle approximation was used, which itself served to “regulate” a, in the infrared. However, this approximation also distorted the infrared behavior of the quark propagator as comparison with the following results shows. Using Eq. (4.7) we find from Eq. (3.6) that the quark SDE can now be written as (D = C/i&., /e3 1) a(x) = 1 + D

a’(x)sAZ

a’(x)x + b2(x)

b(x) = m, + D

+

2a(x) b(x) a2(x)x + b*(x)

+ I

4V”(x3 14 4.4 46 Y)

(4.8a)

A2 &f(x,

(4.8b)

Y) 0)

4x> VI,

with the understanding that only the regular piece of a,(Q2) appears in Eq. (4.3) for u and u (i.e., all of CY,but the b-function). We have solved Eqs. (4.8a), (4.8b) by iterating initial guesses for a(x) and b(x) to convergence. Values for these two functions are obtained on a logarithmic grid and then IMSL double-precision subroutines ICSCCU and DCADRE are used to interpolate and integrate these functions, respectively. The results were confirmed to be completely independent of variations in initial guess for A(p2) and B(p2), the number of grid points and in the infrared and ultraviolet limits of integration. Furthermore, when a,(s) in Eq. (4.3) is replaced by CI,(X,), where x, = max(x, y) we have reproduced our earlier angle approximation results [8] to an accuracy better than a few parts in 10e4, although this agreement can obviously be improved with additional CPU time and memory requirements. Note that these earlier results were obtained by solving a second-order differential equation, which is only

MODELLING

THE

QUARK

PROPAGATOR

479

possible when using the angle approximation [7-S]. Analytic methods exist for establishing the existence of unique solutions of certain types of non-linear integral equations [30], this approach is currently being pursued in the context of these SDE studies. In Fig. 3 we show a typical solution for the case of exact chiral symmetry. Since u(Q’) 2: 2 for Q’ N 0 we see that in the infrared the delta-function piece of cr,,(Q2) appears to dominate the behavior of the u(Q’) function [26]. In the ultraviolet we find the correct perturbative QCD result a(Q*) + 1. We have verified that in the ultraviolet (i.e., the deep spacelike regime) the quark mass function M( - Q’) = m(Q’) has the correct asymptotic behavior appropriate for exact chiral symmetry (see Eqs. (3.7)-(3.8)). In Table I we present results for a variety of values of C for exact chiral symmetry (m,j = 0). Also shown for comparison is a dimensionful equivalent which we can define, for example, as M, = no,-n fi/rc. This cannot be directly related to the quantity M, in earlier work [S] which used the angle approximation, since the infrared singular behaviour is different. The results in Table I are obtained by requiring that f, = 93 MeV and then extracting the appropriate nQcD. We see that by varying C it is straightforward to obtain any value of /iQcD in the usually quoted range of 2OC~400 MeV. Two results are given for the quark condensate. One of these is obtained from the asymptotic form of the mass function (see Eqs. (3.7) and (3.10)) and corresponds with the usual definition of the quark condensate in the operator-product expansion (OPE). It is this result which should be compared with the typically extracted experimental value of the quark condensate of 225 f 25 MeV. We see that our results are in good agreement with this value. For comparison we also show the quark condensate obtained from the “natural” delinition of Eq. (3.9). These two are identical if there is no ExSB and if A(p2) = 1 as is the case with the angle approximation [8]. We also see that the change in M, is

FIG. 3. As a function loglo[m(Q’)/noco](-). b(Q*) = B($), and m(Q’) of exact chiral symmetry

595/210,'2-16

of log,,(Q*/~&.o), we plot 4QZ)(- .Note that for spacelike momenta z M(p*). The results shown are for C= (m,, = 0 in Eq. (4.8b)) (see Table I).

.-A tog10CbtQ2)/iiocDl(- - -1, and Q’= -p’>O and a(Q*)=A($), 500, In r = 1, and n, = 4 in the limit

480

WILLIAMS,

KREIN, TABLE

The Variation of Some Physical Part of the Gluon Propagator C 100 200 400 500 600 800 Experiment:

AND

ROBERTS

I

Quantities with the Dimensionless Strength C of the Regulated l/q4 (i.e., the Delta Function) in the Limit of Exact Chiral Symmetry

MC

km

f,

1.52 1.57 1.61 1.62 1.63 1.64

418 349 253 228 209 182 2WOO

93 93 93 93 93 93 93

(-(sq),)"30PE

(- <@J),P3

215 212 208 201 206 204 225 * 25

205 201 196 195 194 192

Note. Also shown is M, = Apco 8/n in GeV. The remaining quantities are all given in MeV. The results shown are for In T = 1 and nr= 4. The QCD scale parameter (Ao,,) is determined by requiring that fn = 93 MeV. The two results for the quark condensate correspond to slightly different definitions (see text).

TABLE The Variation

of the Results

Ins AQcD f, 0.1 0.5 1 2 3

220 225 228 232 234

II

for Different

((-@),A"

Values

OPE

208 208 207 204 200

93 93 93 93 93

of the Parameter

r

(- (94),)"3 196 196 195 192 188

Nole. For these cases C = 500, n, = 4, and as in Table I there is no explicit chiral symmetry breaking quark mass. All units are as for Table I.

TABLE The

Variation n/ 2 3 4 5 6

&m 231 229 228 226 223

Nore. For explicit chiral

of the Results

III

for Different

Numbers

of Quark

Flavors

fr

(- (4q>,)"' OPE

(- (&>,P3

93 93 93 93 93

203 205 207 209 212

190 192 195 198 201

these cases C = 500, In r = 1, and again there is no symmetry breaking. All units are as for Table 1.

MODELLING

THEQUARK

PROPAGATOR

481

FIG. 4. (a) The effect of including an explicit chiral symmetry breaking quark mass m,, = 10 MeV at AI = 1 GeV on the solution shown in Fig. 3. (b) The variation of the quark mass function m(Q’) with increasing explicit chiral symmetry breaking quark masses. Shown are rnP = 0, 10, 100, and 1000 MeV at p = 1 GeV. Other parameters are the same as those of Fig. 3.

is quite small even though C has been varied through almost an order of magnitude. In Tables II and III we have studied the effects of varying z which regulates the logarithmic piece of a,(Q*) and of varying the number of flavors (nf), respectively. Our results are seen to be relatively insensitive to these parameters and so we typically choose In r = 1 and n,.= 4. In a more sophisticated treatment the effects of q@pair production thresholds could be taken into account by allowing the number of flavors to be a function of Q*, but this would have little effect on our results. We see then that we essentially have a single parameter, C (or equivalently M,), and that this does little more than determine the needed value of Ao,-b. In Fig. 4 we show solutions for the quark propagator when ExSB quark masses are included. For small values of these the infrared behavior of the propagator is not changed by very much. The ultraviolet form is of course quite different. It has been verified that these solutions also have the correct leading-log behavior (see Eqs. (2.10) and (3.8)). V. CONCLUSIONS We have shown how to construct a simple model for QCD in the quark sector which is covariant, satisfies essential gauge and chiral symmetry requirements, and reproduces leading log perturbative QCD results in the deep spacelike regime. The usual angle approximation has not been invoked. We have seen for the case studied here that while this means A(p*) # 1, this momentum-dependent wavefunction renormalization did not deviate greatly from one even in the deep infrared. This suggests that the angle approximation may be qualitatively reasonable when all momentum dependence is expected to be smooth and A(p*) is not expected to deviate greatly from unity, although it is clearly unreliable for quantitative studies of particular models of the non-perturbative regime of QCD (see also [3]).

482

WILLIAMS,

KREIN,

AND

ROBERTS

Any detailed examination of the question of confinement must await an extension of the solutions into the timelike region. Such an extension raises several difficult issues as has been discussed elsewhere [31]. The inclusion of alternative quark-quark interactions in the infrared is straightforward, and there is a need to eventually derive the appropriate form of this from QCD. One particularly promising approach, which is currently being pursued [32], is to use [14] dual QCD to determine this infrared behavior and then to smoothly connect this on to the known perturbative QCD regime. Another possibility is that lattice gauge theory studies will soon be able to provide sufficiently detailed information about the quark-quark interaction in the infrared that it could be used as input into this approach. An extension of this work to include the full Ball-Chiu vertex of Eq. (3.1) is in progress [33]. Areas of current investigation include: the pion electromagnetic form factor, solving the Bethe-Salpeter equation for mesons, and the inclusion of finite temperature and density effects. APPENDIX

A

We discuss here the meaning of the quark condensate (44). It is convenient to perform a Wick rotation (x0 + -ix,, k” + ik4) and evaluate (ijq ) in Euclidean space. Define x-Q2=

-p2a0,

m(x) = WP2),

z(x) E Z( p2),

(AlI

where we use lower case to indicate Euclidean space quantities. Then Eq. (3.9) can be written

where p = Au,,, then (writing an explicit subscript n where appropriate)

‘;fz,f;’

-+ A

A2

N

Auv-m

--

3 47c2s0

x+m,

1

dx Cm/Ax)- m,l.

(A3)

The second result follows since the UV dominates for large /i,v and since in Landau gauge zA(x) 1: 1, for large x. In the case of exact chiral symmetry (i.e., m, = 0) Eq. (3.7) implies that mA(x) = (c/x) ln(x//l&.,)d‘, and from Eq. (A3) (A4) Since c is independent

of the choice of p, Eq. (3.10) imediately follows.

MODELLING

THE

QUARK

PROPAGATOR

483

When ExSB is present (i.e., m,, #O) from Eq. (2.10) m,(x)=~/[~ln(x///i&,)]d. It is clear that neither Eq. (A4) nor Eq. (3.10) will any longer result from Eq. (A3). Then the quark condensate as defined in Eq. (3.9) no longer varies with the renormalization point like (cjq) - [ln(p2/n&,)]d.

APPENDIX

B

We derive the expression [4, 251 for j’i given in Eq. (3.17). First note that only terms up to O(k) need be kept, since we will be taking k -+ 0. Since

(Bl)

A((p+k)‘)=A(p2)+2k.p$(p2)+(“(k’)

then, for example, A(p’)+k.pe

dp’]y” ;ys+O(k’),

032)

where the unknown transverse parts in Eq. (3.11) have been neglected and the pion pole term identified from Eq. (3.15a) has been removed. In Eq. (3.16) we have a trace over isospin, spinor, and color labels. The spinor trace simplifies things considerably since (in an obvious shorthand notation)

~~~C~~~~+~~~~~P~~C~;‘~~~P~+~~~~~~,~C~~P~~C~;”””~‘~P,P)+~~(~”~~~,~) = tr{k .&S(p) T;‘(p, p) S(p) f ytreg)(p, p)} + 0(k2),

033)

where we have used tr(S(p) ysS(p) yI’ys} = 0 and r,( p’, p) from the generalization of the Goldberger-Trieman (Eq. (3.15a)). The piece of Eq. (B2) which is zeroth order in k is (B4)

Note that in both Eqs. (3.1) and (3.11) we can only set k identically to zero in expressions for physical observables; however, this is a common feature in theories with massless particles. We can multiply both sides of Eq. (3.16) by (kc/k’) and sum and use tr(Pt”) = 26”” to give

(B5)

484

WILLIAMS,

KREIN,

AND

ROBERTS

where the trace is now over spinor labels only. After evaluating taking spinor traces we find the result quoted in Eq. (3.17).

(a/+“)

S(p) and

ACKNOWLEDGMENTS We thank S. Ellis, S. Sharpe, A. Szczepaniak, and J. F. Owens for a number of helpful discussions. AGW and GK acknowledge partial support from the U.S. Department of Energy and C.D.R. acknowedges support from the Australian Research Grants Scheme and the U.S. Department of Energy under Grant W-31-109-ENG-38. Part of this work was carried out while two of us (AGW and GK) were at the University of Washington.

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MODELLING

20.

THE

QUARK

H. J. MUNCZEK, Phys. Left. B 175 (1986). 215. et al., Rev. Nuooo Cimento 6 (1983).

21. P. 1. FOMIN 22. For 23. 24.

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J. S. 25. J. 26. H. 27. V.

the

Phys.

Bethe-Salpeter

Leff.

equation

B 116 (1982).

485

PROPAGATOR

1. see

171: NW/.

N. NAKANISHI,

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B 117 (1976),

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30. H. MCDANIEL, 31. D. ATKINSON

Left

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467. 32. G. KREIN AND A. G. WILLIAMS, Phys. Ret). D 43 (1991), 33. F. T. HAWES AND A. G. WILLIAMS, in preparation.

3541.

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A 146 (1990),

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