Gauge independence of subleading contributions to the operator product pole mass

Volume 173, number 2

PHYSICS LETTERS B

5 June 1986

GAUGE INDEPENDENCE OF SUBLEADING CONTRIBUTIONS T O T H E O P E R A T O R P R O D U C T POLE M A S S V. E L I A S Physics Department, University of Winnipeg, Winnipeg, Manitoba, R3B 2E9 Canada and Department of Applied Mathematics, University of Western Ontario, London, Ontario, N6A 5B9 Canada

M. S C A D R O N Physics Department, University of Arizona, Tucson, Arizona 85721, USA

and R. T A R R A C H Departament de F~tsicaTebrica, Unit,ersitat Autbnoma de Barcelona. Bellaterra (Barcehma), Spain

Received 23 October 1985

We prove that the mass-shell gauge independence of the quark condensate contribution to the quark self-energy is not accidental: the same phenomenon is shown to occur for the mixed condensate contribution. This allows one to conclude that the pole mass is gauge-independent in the operator product expansion.

QCD is expected to be a confining field theory. Consequently the true quark propagator should ultimately have no poles on the real positive p2 axis. However, such a pole does occur in perturbative QCD. This pole mass is gauge parameter independent and appears to be physically meaningful for heavy flavours [1 ]. To discuss this pole mass for light flavours, it is necessary to include nonperturbative effects in a systematic way. These effects can be incorporated into perturbative QCD through the operator product expansion (OPE). Since the OPE is a short distance expansion whose domain in p2 is compatible with perturbative QCD, it is no great surprise that OPE-augmented perturbative QCD continues to support a quark propagator mass pole. For this to be physically meaningful, however, its location must continue to be gauge parameter independent. The dominant nonperturbative contribution to the quark propagator arises from the quark condensate. This contribution was first obtained by Politzer in the Landau gauge [2]. Pascual and De Rafael subsequently derived the quark condensate contribution to the quark selfenergy for arbitrary covariant gauges. Their result may be expressed in the following form [3], ~ , ~ ) = (g2 ( ~t~)/9p2)13 + a(1 - m~/p 2) + O(m2 /p 2, m 3~b/(p2) 2 .... )1.

(1)

If higher powers o f m are disregarded, all gauge parameter dependence in (1) manifestly cancels at the/b = m pole, O(m2/p 2) and O(rn3~b/(p2) 2) contributions to the square bracketted quantity in (1) have been shown to vanish in ref. [4], suggesting that (1) is not corrected by terms higher order in mass, and leading to the gauge parameter independent dynamical mass of ref. [5]. It would buttress our confidence in effective-quark-mass phenomenology (and deepen our understanding of the applicability of the OPE within gauge theories) to know whether this gauge parameter independence is truely general, as opposed to an accident arising from the simplicity of the only graph (fig. la), contributing terms linear 184

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Volume 173, number 2

PHYSICS LETTERS B

5 June 1986

a)

× ~_. . . . . . . .

+

~

R

X

X

b)

+

e) ,__

~ ......

'

•

R

X

.....

•

f)

--

c)

d)

g)

Fig. 1. Lines entering the box regions represent fields that condense at different spacetime points. The dots within the box represent the fields that form the local condensates according to the corresponding short distance expansion.

in ( ~ ) to 2;(#). Indeed, the question we would like to address in this paper is whether or not gauge parameter independence of the pole mass is a general property of OPE-augmented QCD. As present knowledge of nonperturbarive field theory is limited, a general proof of gauge parameter independence is beyond our level of expertise. Rather, we have chosen to calculate the contribution of the mixed quark-gluon vacuum condensate [(ffGff) ( : ~(igXa/2)Ga~v oUV~ : )] to the quark selfenergy. This computation is quite cumbersome, with terms of comparable magnitude arising from higher-order OPE insertions into two-vertex Feynman graphs (figs. lb and lc) as well as from the full set of three-vertex Feynman graphs (figs. l d - l g ) . Despite such complexities, we find once again that gauge parameter dependence within the selfenergy cancels when ~ = m. Such a cancellation can hardly be accidental, thereby providing compelling evidence that the pole mass in perturbative QCD remains gauge parameter independent even when nonperturbative contributions are allowed to arise through vacuum condensates. To obtain these results, we begin by relating the propagator and selfenergy in the usual way, S ( # ) = [~ -- m - ~ ( # ) ]

- 1 __(# _ m ) - I

+ (# _ m)-I

~(#)(#

_ m)-I

+ ...,

(2)

with S expressed in terms of Heisenberg fields by, iS(#)

- f d4x eipx (OlT~b(x)~(0)[0).

(3)

In the interaction picture where such fields are free, eq. (3) becomes

iS(#):fd4xeipx(oTt~(x)~(O)(exp

i f d 4 y £ i n t @) ) 0 ) .

(4)

£int(Y) is the QCD interaction lagrangian density which, to the order we are computing, can be written as £int(Y) = £q(V) + £g(V),

(5)

where £q is the quark-gluon vertex and £g the triple-gluon vertex. The first mixed condensate contributions to the quark propagator arise from the following term in the expansion o f e q . (4),

185

Volume 173, number 2 iS2q(/~) = _1

PHYSICS LETTERSB

fd4x eip~fd4y fd4z (OIT~(x)'~(O)£q(y)gq(Z)lO).

5 June 1986 (6)

Wick's theorem permits mixed condensate contributions of two types, graphically represented by figs. lb and lc. Fig. lb corresponds to Wick contractions yielding two quark propagators, one gluon propagator, and one nonperturbative vacuum expectation value of the form (:~r(Z) ¢n(y): ). Expansion of this nonperturbafive quantity in terms of local gauge-invariant condensates is most easily performed in the Fock-Schwinger gauge [6]

x B~Z(x) = O,

-1 a a B ( x ) - ~ l:g~, Btz(X ).

(7)

In this gauge, the Taylor series expansion of if(x) about zero may be expressed as oo

~(X) = n~=O~. XOal ...xwnDoa1(O) ... Dton(0)~(O),

(8)

where the gauge-covariant derivatives are defined by Dw(x ) = 0¢o - Boo(x ).

(9)

No gauge-variant condensates appear in the Fock-Schwinger gauge expansion of (:~r(Z) ~n(y): ). Such gauge invariance is not present in other gauges, but the coefficients of gauge-invariant condensates, which are all we require, do not depend on the gauge chosen. Using the Fock-Schwinger gauge, we are able to extract the mixed condensate contribution to (:~r(z) ~n(Y): ) [7],

- 16rn ~ ( y - z)2 (2rn2(~ff) + i(~Gff)) + 9Agyoqzw2G~r,t%('~G~)l + a-~(v - z ) 2 0 - ¢ ) ( 2 i m S ( ~ ) - m(~G~)) + (fourth and higher-derivative terms; six and higher-dimensional condensate terms).

(10)

The mass in (10) enters through the equation of motion iD(x) qJ(x) = md/(x).

(11)

The contribution of fig. lb to iS2q is now straightforward to calculate, and is given by iS2(2q)= __~ g2 [(~Gff)/fp2)3] (1 - a)ll + m~/p 2 + o(me)l.

(12)

The whole O(m) term in (12) is due to the masses entering through the fermion propagator, <01T~ ~(x)~ts(0)10) = SO ~ i

f d 4 p e_iP x p2~ _+ mm 2

(13)

The gauge parameter a enters (12) through the gluon propagator

= [--i/(2rr)4](Sab/X2)[½(1 + a)gtav + (1 -- a)x u xv/x2 ] .

(14)

The contribution of fig. lc to iS2q corresponds to contractions yielding three quark propagators and one nonperturbative vacuum expectation value of the type ( : Ba.(z)Bav (y):). Obviously, this contribution cannot depend on the gauge parameter a because of the absence of a gl~on propagator. In fact, there is no mixed condensate component in the expansion of(:Ba(z)BavO'): ) analogous to (10), in which case is(2Cq ) = 0. We now examine 186

Volume 173, number 2

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5 June 1986

iS2q,g~) = - ½f d 4 x eipx f d4y f d 4 z f d4w (01Jff (x)~(0)£q (y)£q (z)£g(W)10,.

(15)

This contribution, graphically represented by fig. 1d, arises from a Wick decomposition involving two quark propagators, two ghion propagators, and one nonperturbative vacuum expectation value of the type [7]

(:~r(z)Bo(W)~n(Y):)= 1-~ (~G~b)(2wO(oOp)nr + ml(z + y)~wtff° - (z +Y)o ~b + iCw~%o - iwr~o~oPlnr) + (second and higher-derivative terms; six and higher-dimensional condensate terms).

(16)

This expression is obtained by using (8) and the Fock-Schwinger gauge Taylor series expansion for the gluon field: oo

1

Bu (x) = n=0 ~ n !(n + 2-----3x to xt° 1 ... x C°n[Dto1(0), [Dto2 (0), [... [Dton (0), Gtou (0)] ...] ] ] ,

(17)

where

1-?taGatou~.(x ). ~ G u(x ) = [Du(x ), Dto(x)] = i ~g

(18)

Using (16), we f'md that the contribution of fig. 1d is given by iS2q,g = _ ~ g 2 [(~Gff)/(p2)3 ] (3 + a)ll + m$/p 2 + O(m2)l.

(19)

Finally, we examine

iS3q(p) = -~i

fdax eiPXfd4yfd4z fdaw <01Tff(x)~(0)£qfy)£q(Z)£q(W)10>.

(20)

Wick contractions leave three quark propagators, one gluon propagator, and one nonperturbative vacuum expectation value of the type (16). These contractions can be performed in three different ways, graphically represented by figs. le, l f a n d lg. Figs. 1f and 1g, however, correspond to doing perturbation theory on zero momentum propagators. The work of Shifman, Vainshtein and Zakharov [8] has shown that such contributions are fully accounted for in the calculations corresponding to figs. lb and lc. The contribution of fig. le is given by iS~eq) = -2--~8g2 [(~G~)/(p2)3 ] 13(1 _ a) + (mtb/p 2) (1 - 3a) + O(m2)l.

(21)

Summing all nonvanishing contributions [eqs. (12), (19), (21)] and extracting the selfenergy following eq. (2), we find, iE(/b) = -2--~sg2 [(~Gff)/(p2)2] 165 + 7a - (m~b/p2) (67 + 7a) + O(m2)l.

(22)

This is the result of our computation. All gauge parameter dependence manifestly cancels at the ~b= m pole. This is precisely the same sort of on-shell gauge independence observed in (2) for the (5 if) condensate. The complexity of the calculations for (~Gff) suggests that such gauge independence could hardly be accidental. This leads us to conclude that on-shell gauge independence is a general property of all higher condensate contributions to the quark selfenergy. Two of us, V.E. and R.T., appreciate the hospitality of the University of Arizona. V.E. acknowledges support from the Natural Sciences and Engineering Research Council of Canada. M.S. acknowledges support from the U.S. Department of Energy. R.T. acknowledges support from CAICYT, Spain.

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References [1 ] [2] [3] [4] [5] [6]

R. Tarrach, Nucl. Phys. B183 (1981) 384. II.D. Politzer, Nucl. Phys. Bl17 (1976) 397. P. Pascual and E. de Rafael, Z. Phys. C12 (1984) 127. V. Elias, M.D. Scadron and R. Tarrach, Phys. Lett. 162B (1985) 176. V. Elias and M.D. Scadron, Phys. Rev. D30 (1984) 647. V.A. Fock, Sov. Phys. 12 (1937) 404; J. Schwinger, Particles and fields (Addison-Wesley, New York, 1970); M.A. Shffman, Nucl. Phys. B173 (1980) 13; C. Cronstroem, Phys. Lett. 90B (1980) 267; M. Dubikov and A. Smilga, Nucl. Phys. B185 (1981) 109. [7] P. Pascual and R. Tarrach, QCD: Renormalization for the practitioner, Lecture Notes in Physics, Vol. 194 (Springer, Berlin, 1984). [8] M. Shifman, A. Vainshtein and V. Zakharov, Nucl. Phys. B147 (1979) 385,448.

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