Volume 202, number 2 PHYSICS LETTERS B 3 March 1988

PRELUDES TO CONFINEMENT: INFRARED PROPERTIES OF T H E GLUON PROPAGATOR IN THE LANDAU GAUGE ~r

N. B R O W N ~ a n d M . R . PENNINGTON a,b a Department of Physics, University of Durham, Durham DHI 3LE, UK ~' Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA

Received 25 November 1987

We obtain a solution for the gluon propagator in the Landau gauge that self-consistently satisfies a truncation of the Schwin- ger-Dyson equations at essentially all relevant momenta. Our treatment goes beyond that of earlier work in covariant gauges and avoids a "no-go" theorem applicable to axial gauges. We deduce the propagator is as singular as l/q 4 for q<<A, indicative of confinement.

Asymptotic freedom and confinement are the fundamental features of the underlying strong interaction. While the asymptotically free nature of QCD is the key to its ability to describe the whole range of hard scattering processes, an understanding of the confinement mechanism remains an outstanding problem. In the continuum, the Schwinger-Dyson equations provide the appropriate framework in which to study the behavior of Green functions over kinematic regimes from the perturbative to the confining [ 1 ]. Extensive studies of such an equa- tion in the pure gauge sector for the gluon propagator in axial gauges [2,3 ] reveal an infrared behavior as sin- gular as 1/q4. This result implies that the Wilson loop operator obeys an area law [ 4 ], which many would regard as a signal for confinement. Unfortunately, West has shown [5], using the positivity property of the gluon spectral function in axial gauges, that it is impossible for the full (axial gauge) propagator to be more singular than 1/q 2. This result does not, however, directly contradict the conclusions of Baker et al. [6], for instance, since they consider only one of the two independent scalar functions in the axial gauge propagator. It may well be that this is indeed as singular as 1/q 4, although the full propagator is not. However, since the relation [4] of the infrared behavior of the gluon propagator to the Wilson loop involves the full propagator, the implication is that it is not possible to prove confinement via the Wilson loop in axial gauges.

It is natural then to investigate the behavior of the gluon propagator in covariant gauges, which, because of the necessary presence of negative-norm (ghost) states, circumvent the result of West [ 5 ]. Such an investigation was in fact started long ago by Mandelstam [ 7 ]. Making the particularly simple approximation that the full triple gluon vertex has the tensor structure of the bare vertex, he too obtained a 1/q 4 infrared singularity (see also ref. [ 8 ]). This has been exhaustively studied analytically in ref. [ 9 ] and numerically by us [ 10,11 ]. In this paper we go beyond this approximation by "solving" the Slavnov-Taylor identity for the triple gluon vertex [ 12 ], following more closely the treatment [ 6 ] of the axial gauge.

The Schwinger-Dyson equation for the gluon propagator is shown diagrammatically in fig. 1. With the prop- agator in the Landau gauge given by

zlw,= [ ~(q2)/q2] (6~,,,_qUq,,/q2) , (1)

the aim is to determine the gluon renormalization function f¢ (q2) at essentially all scales at which QCD should be relevant (in distance, from 0.01 fm to a few fm). The solution of the Schwinger-Dyson equations is tractable

This article has been authored under contract number DE-AC02-76CH00016 with the US Department of Energy.

0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )

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Volume 202, number 2 PHYSICS LETTERS B 3 March 1988

- i ,.'~m.e-oi.mnre,

-I +~-

/ \

! \ /

Fig. 1. Schwinger-Dyson equation for the inverse gluon propagator without fermions. Gluons are represented by solid lines, ghosts by dashed ones. The dots denote full (as opposed to bare) propagators and vertices.

only, of course, under greatly simplifying assumptions [ 1-3,6-11 ]. We first ignore the contribution of terms involving the quartic gluon vertex. This can be seen as the start of a systematic truncation of the Schwinger -Dyson equations [ 10,6 ], where the next step would be to consider the equations for both the two- and three- point functions and ignore five-gluon couplings. Here we next ignore the ghost term. Although the presence of this term is essential for ensuring the transversality of the two-point function, eq. (1), it is believed [ 7,10 ] that it will not affect the overall behavior of f# (q 2). This is supported by the one-loop perturbative calculation where the ghost loop makes less than a 10% contribution to N in the Landau gauge.

The equation still involves the unknown full triple gluon vertex (fig. 1 ), but in the absence of ghost contribu- tions we can "solve" the Slavnov-Taylor identity for the longitudinal part of the triple gluon vertex in terms of the gluon propagator renormalization ~(q2) [ 12,11 ]. As one of the external momenta goes to zero, the trans- verse part of this vertex vanishes, so the hope is that at least the low-momentum behavior of the propagator will be given by considering only the longitudinal contribution [ 2,6,12 ]. We now have a closed expression for the gluon propagator. Projecting with the tensor

Q~V = _~ (~u~ _4qUq./q2)

the tadpole term, fig. 1, gives no contribution and we have a non-linear integral equation for f¢ (q):

_ _ ~ [ ~¢(k,q) f#(k)~C(k')~(k,q) 1 _ 1 + CA~ d4k f#(k) 4q------~ + f#(q) 96n4q 2 k4k ' ~(q) k4k '4

(~U(k ' ) -~(q) '~ ~(k,q) (C~(k')-C~(k)'~ ~ ( k , q ) 1 + \ k,-TT~_q2 j -~5- -~T+ \ k,2 k2 j k2k,2q2 j , (2/

where k' = q - k , go is presently the bare coupling, CA the color Casimir = Nc and

d ( k,q) = - 12k4q 2 + 48k2( k.q) 2 -32k2q2k .q - 16(k.q) 3 +6k2q 4 + 6q2( k'q) 2 ,

:~( k,q) = - 13k4q 2 - 2k2( k.q) 2 + 38k2 q2 k.q+ 4( k.q) 3 + 12q4 k.q - 14keq 4 - 25qZ( k'q) 2 ,

~( k,q) = - 6 k 2 k.q+ 14k2 q 2 + 16(k.q) e -48q2 k.q+ 2 4q 4 , (3)

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~( k,q) = 1 2 k 4 q 2 - 4 8 k 2 ( k.q) 2 + 23k2 q2 k'q+ 48( k q ) 3 + 9q4 k q - 5k2 q 4 -40qZ( k'q) 2 • (3 cont 'd)

As in refs. [ 2,6-8,10 ] it can be argued that the only possible consistent behaviour for (c(q) as q2 ~ 0 is (C (q) ~ A/q 2. Unlike ref. [6 ] however, the loop integrals are infrared divergent if (c(q) is so singular. Such divergences are only avoided in the treatment of Baker et al. [6] by assuming that (i) the full axial gauge propagator has the tensor structure of the bare propagator and (ii) the coefficient of this singular term is independent of the gauge (but see ref. [ 13]). Nevertheless, even if the propagator is so singular, these infrared divergences should not exist. They would be artifacts of our approximations. In other words, terms we have ignored would serve to make the integrals finite. We restore this property by assuming (c(q) to be a distribution defined by a "plus prescription". Note that this does not prejudice the result, since for non-singular terms this has no effect and we allow the possibility that the coefficient of such a term is zero (i.e. A = 0). If (c(q) does behave like l lq 2 we still get finite contributions to the inverse propagator which violate the masslessness of the gluon by not vanishing as q2--*O. To preserve this is most easily done by writing (c(q2) = A / q 2 ~l (q2), where q2 (CI~0 as q2--*0. The entire contribution to the "mass" term comes from A/q 2, so we subtract the right-hand side of eq.(2) with (C equal to (A/q 2) +. Of course, ifA = 0 this has no effect.

We are now left with the usual ultraviolet divergences - logarithmic in momentum space. We deal with these by defining a running coupling a~(q) by

1 (cR(It) 1 CA J fd4k[ j { . (k ,q )_J{ (k , t~ ) ] (4)

where the gluon function, eq. (3), renormalized at q2__/~2, is then (following the method of ref. [6]) given by

(CR(#) - - 1 + CA0q(#) r d 4 k [ ~ ( k , q ) ( - ~ ( k , # ) ] + CAa~(#) 1 f d4k[~(k ,q)-£f (k ,#)](cR(k) (5) 247r 3 o 24n 3 (Ca(q) o ' %(q)

where

~ ( k ) d ( k , q ) ~R(k) Cd(k,q) (.(CR(k')-__~R(k)'~ ~(k,q) 3¢(k,q)- k4k,2q 4 (k,2+it2)kek,2q2 + \ k ,2_k2 j k2k,2q4,

(cR(k')2(k,q) ( (k '2 +#2)(cR(k ')-(q2 +#2)(cR(q)] C#(k,q) £f(k,q) = k4k,4q 2 + k,2 _q2 ] k2k,2q 2 . (6)

The truncation used here for the two-point function involves just a one-loop renormalization. Consequently, we can write c~ s in terms of a single scale parameter A, as usual. With/t conveniently chosen in the perturbative region,

oz~(It) = 4rg/flo In (/x2/A 2) , ( 7 )

where from eq. (4) flo=53CA/12. It has not proved possible to find an analytic solution to eq. (5). Instead we represent (CR (q) by a simple form

which reproduces the asymptotic behavior of the customary renormalization-group-improved perturbation the- ory, viz. for q>>A,l~

[ ( 2)1-o (ca(q2)--*(c~(q2)= 1+ fl00~(/t) In +1 (8) 4~z ~-5 ,

where Y0 = - 7CA/3. We use as an input form:

tl 2 { q2 x nb (c=( q2) =A y + (c.( +2) a.t ) , (9)

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50 L-. 1 i i inpult __

I 0 -

0 DI .I I I0 I00 q2 GeV 2

Fig. 2. Self-consistent solution for the gluon propagator renor- malization function f¢R(q) for two values of the coupling at/~2 = 10 GeV 2.

with the parameters A, qo, an, b to be determined by self-consistency. Setting fiR(#) = 1 and substituting eq. (9) into eq. ( 5 ) the four-dimensional integrals are performed numerically. With ~t ~ = 10 GeV 2, a scale we know from deep inelastic scattering to be in the perturbative regime, we impose consistency of the input and output ofeq. (5) for 0.01 < q2< 40 GeV 2. With N = 4 this matching is achieved to impressive numerical accuracy over the whole range from the deep infrared to the deep ultraviolet - a much bigger range than previously considered [ 6 ]. The results for A = 200, 500 MeV are shown in fig. 2 and the parameters detailed in table 1.

We see that the solution to the truncated Schwinger-Dyson equation, eq. (5), for the gluon propagator is indeed strongly enhanced at low momenta behaving in fact like 1/q 4 for q2~0 in the Landau gauge. Though such behavior has been previously studied in axial gauges [ 6 ] it is in covariant gauges, however, that this prop- erty generates a linearly confining potential for one-gluon exchange between static color charges and, more im- portantly, reproduces an area law for the Wilson loop operator [4,5]. Though this is the behavior for q--.0, perhaps more crucially for confinement in practice is at what scale the enhancement of the propagator sets in - this is seen to occur for q ~ A (fig. 2). The solutions of fig. 2 then amply illustrate how the logarithmic depen- dence on q2/A2 in the perturbative regime becomes, through the non-perturbative effects of the Schwin- ger-Dyson equation, essentially an inverse power behavior at low momenta exactly as confinement would require.

N.B. would like to thank the U K SERC for the award of a research studentship and W.J. Marciano and the theory group at BNL for their hospitality. M.R.P. acknowledges the receipt of a Fulbright travel grant for 1987/88.

Table l Parameters for if(q), eq, (9), for solutions shown in fig. 2.

A a s ( # ) A b q~ a~ a2 a3 a4 (MeV) ( GeV z)

200 0.172 0.01934 0.3 0.3 2.637 0.009379 -0 .4407 - 1.180 500 0.257 0.07363 0.3 0.3 1.469 -0 .09048 0.2547 - 0 . 6 2 7 4

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