Table Of Content1
On the reliable measurement of the gluon condensate in lattice QCD
a
Alexander Bochkarev
Talkgiven at Lattice '94,Bielefeld, hep-lat/9411081
a
Intitute of Theoretical Physics, University of Minnesota,
Minneapolis,MN 55455, USA
5
We propose to calculate the gluon condensate in lattice QCD in an indirect way by extracting it from the
9
correlator of hadronic currents of heavy quarks. Moments (derivatives with respect to momentum at vanishing
n momentum)ofthevectorandpseudoscalar correlatorsareevaluated. Thecontribution ofthecontinuumspectrum
a in addition to the low-lying resonance is observed in accordance with qualitative expectations. Finite size e(cid:11)ects
J
are observed and shown to be insigni(cid:12)cant for heavy quarks and moderate lattices. A practical de(cid:12)nition of the
5 nonperturbatively renormalized heavy quark mass is given.
1
8 Introduction where j(cid:22) = c(cid:22)(cid:13)(cid:22)c. The function (cid:5)(q2) is com-
0 2
putableinperturbative QCDattheoriginq =0
1 Extensive analysis of the two-point correlators
since that point is far away from the leading
1
of hadronic currents by means of the QCD sum
4 threshold due to a pair of charmed quark and
rules indicates that vacuum expectation values 2
9 antiquark. (cid:5)(q = 0) is a function of the renor-
t/ of the quark and gluon (cid:12)elds receive large non- malizedheavyquarkmassmc((cid:22))and(cid:11)s((cid:22)). The
a perturbativecontributions(condensates) [1]. The relevant renormalization point here (cid:22) = 2mc is
l existenceofcondensatesimpliesaparticularform
- high enough to ensure the validity of the QCD
p of the crossover between the domain of asymp-
perturbationtheory: (cid:11)s(2mc) ' 0:2. De(cid:12)nemo-
e
totic freedom and the domainof large distances,
h ments of the polarizationoperator (1) as:
dominated by the low-lying resonance. There is (cid:26)(cid:18) (cid:19)n (cid:27)
certain range of intermediate virtualities where 1 d 2
the perturbative series inpowers of(cid:11)s is stillun- Mn = n! dq2 (cid:5)(q ) q2=0 (2)
dercontrol,whereasnonperturbativee(cid:11)ects show
Startingwithn=2themomentsMn'sarephys-
up, represented, in some channels, by the (cid:12)rst
ical quantities: all the ultraviolet singularities
few terms of the operator product expansion. It
areadsorbed bythe bareLagrangianparameters.
wasrecentlysuggested[2]thatlatticesimulations
Higher moments n (cid:29) 1 correspond to smaller
must be useful to study those intermediate dis-
virtualities, which leads to deviations from the
tances in the physics of light quarks. We (cid:12)nd it
perturbative series before that perturbative se-
very advantageous to study deviations from the
ries grows out of control [1]. For n = 4 (cid:4) 7
asymptotic freedom in the case of correlators of
those deviations are well described by the few
hadronic currents of heavy quarks. It is in this
terms of the operator product expansion, start-
case that the gluon condensate was (cid:12)rst intro- (2)
ing with the quadratic gluon condensate G (cid:17)
duced [1]. Such a way is an alternative to the a a
< 0j((cid:11)s=(cid:25))G(cid:22)(cid:23)G(cid:22)(cid:23)j0>. For the ratios of neigh-
direct evaluation of the gluon condensate as an
boring moments rn = Mn=Mn(cid:0)1 one has on
average of the local operator, initiated in [3].
the two-loop level :
Consider the polarization operator of the elec- !
(2)
tromagneticcurrents of charmed quarks: 1 2 G
Z iqx rn = 4m2c an + bn(cid:11)s(4mc) (cid:0) cn (4m2c)2 (3)
(cid:5)(cid:22)(cid:23) = i dxe <0jT (j(cid:22)(x)j(cid:23)(0))j0>
(cid:0) (cid:1) wherefan;bn;cngareknownnumbers. Thepoint
2 2
(cid:17) q(cid:22)q(cid:23) (cid:0) q g(cid:22)(cid:23) (cid:5)(q ); (1) we are making is that the l.h.s. of (3) is di-
2
rectly accessible to the Monte-Carlo lattice sim- seem to be feasible. For this reason we evaluate
2 2
ulations, and since the perturbative series in the not the Fourier transform(cid:5)(cid:22)(cid:23)(q ) at nonzero q ,
r.h.s. of (3) is reliable the nonperturbative term but rather the moments themself directly in the
2
represented by the gluon condensate is to be ex- x-space. At q =0 one can get rather simple ex-
tracted in a reliable way. Lower moments (r3;4) pressions forthemomentsintermsoftheoriginal
correspond to short distances and are dominated correlator (cid:5)(cid:22)(cid:23)(x):
by the (cid:12)rst two terms of (3), where an corre- Z
sponds to the one-loop contribution (asymptotic M~n = 22nn!(1n+1)! dx4x2n(cid:5)(cid:22)(cid:22)(x) (4)
freedom term) and bn represents a small two-
loop correction. Since the one-loop diagram is
Monte-Carlo data
a function of only one parameter - the renormal-
izedheavyquarkmass,thatparametercanbeex-
We used the clover- and tadpole-improved
tractedfromthe(cid:12)rstmoments. Inthisway,using
quenched Wilson fermionic propagators of [5],
Monte-Carlodataona(cid:12)nitelatticeonecandeter- 3
obtained on the 8 (cid:2) 16 lattice blocked from
mine the nonperturbatively renormalized heavy 3
32 (cid:2) 64 SU(3) con(cid:12)gurations at (cid:12) = 6. We
quark mass in terms of the gauge invariant cor-
have calculated the correlator (1) and the ra-
relator without any reference to the pole mass.
tios of the moments (3) for vector (Fig.1) and
This mass is to be used later to extract the last
pseudoscalar (Fig.2) currents (j = ic(cid:22)(cid:13)5c in (1))
terminther.h.s. of(3)fromthehighermoments.
for three values of the hopping parameter (cid:20) =
ForanygivenMonte-Carloevaluationofthel.h.s.
(0:9830;0:1031;0:1111). Statistical errors are no-
of (3) the perturbative part of the r.h.s. of that
ticeable but smallfor already6 propagators((cid:20)=
equation is to be built in terms of renormalised
0:1111 - the upper curves) and negligible for 10
parameters of the improved lattice perturbation
propagators ((cid:20)=0:0983- the lowercurves). The
theory [4].
givenvaluesof(cid:20)correspond tonotheavyquarks,
Thecontributionofthegluoncondensatein(3)
soourdataforrnshouldbecomparedtotheQCD
is about 10% relative to the leading (cid:12)rst term
perturbation theory as in (3). We rather analyse
(cid:24) an in the charmonium case. However, we are the formofthe spectral density (cid:26)(s) ofthe corre-
notforcedtoconsiderthephenomenologicalvalue
lator (1) by lookingat the functions r(n).
of mc. Adjusting the hopping parameter (cid:20) so The simple and phenomenologically very suc-
as to decrease mc from 1:3GeV to, say 0:9GeV cessful ansatz for (cid:26)(s):
where perturbative series in (cid:11)s(0:9GeV) is still
undercontrol[1],onecaneasilyincreasethethird 2 2 1
termin(3)upto20%. Thee(cid:14)ciencyofthistrick (cid:26)(s) = fmres(cid:14)(s(cid:0)mres) + 4(cid:25)2(cid:18)(s(cid:0)so) (5)
is due to the fact that the perturbative terms of
with a single low-lying resonance of mass mres,
the r.h.s. of (3) depend on mc logarithmically, residue f and perturbative continuumwith some
whilethegluoncondensatecontributionisrapidly 2
(cid:0)4 e(cid:11)ective threshold so >mres gives the ratios
varying power correction mc . (cid:18) 2 (cid:19)n
On the lattice one has to deal with the orig- 1 mres
inal correlator (cid:5)(cid:22)(cid:23)(q2) rather than (cid:5)(q2). We (cid:17)n (cid:17) 1 + 4(cid:25)2nf so ; (6)
therefore introduce moments M~n as derivatives 1 (cid:17)n
of (cid:5)(cid:22)(cid:22)(q2) rather than (cid:5)(q2) in (2). The new rn = m2res (cid:17)n(cid:0)1
moments are related to the initial ones (2) in a
simple way M~n = (cid:0)3Mn(cid:0)1, so the ratio r~n for At large n the ratiosrn reach a constant 1=m2res,
the new moments will be r~n+1 = rn. To know which is the resonance contribution. That con-
the Fourier transform of the correlator (1) from stant is approached from below due to the con-
the latticesimulationstothe accuracy needed for tinuum contribution, which is just what one can
the detection of the power corrections does not see on Fig.1. As n grows the ratios try to reach
the asymptotic constant value from below, but
3
at some point fail. That "crash"-point is a (cid:12)- continuum threshold so is signi(cid:12)cantly greater
nite size e(cid:11)ect. One cannot put arbitrarily high than resonance mass in that channel. For heavy
moments on a (cid:12)nite lattice because higher mo- quarks the continuum contribution will be more
mentscorrespond tolowervirtualities,largerdis- visible for small n and allow for the comparison
tances. At some point they become too sensitive with the predictions of perturbative QCD. The
to the boundaries of the box. This interpreta- Monte-Caro data with smaller lattice-spacings
tionis supported by the fact the "crash"-point is and smaller (cid:20) are necessary to see the slope of
clearly (cid:20)-dependent. At greater (cid:20), smaller mres the continuumcontribution with good resolution
(upper curves) the "crash" happens earlier. Figs. at smalln.
1,2demonstratethatonecaneasilyaccommodate
heavyfermions((cid:20)<:0983)ontypicallysimulated
lattices as far as the ratios of interest r2(cid:4)r7 [1]
are concerned.
Figure 2. Same as in Fig. 1 in the pseudoscalar
channel.
Figure 1. Ratios rn for three di(cid:11)erent hopping IamgratefultoPh. deForcrand,L.McLerran,
parameters in the vector channel. J.Smit,A.Vainshtein,M.Voloshin,R.Willeyfor
useful discussions. This work was supported by
the Pittsburgh Supercomupter Center.
The observed "asymptotic"valuesofthe ratios
2 REFERENCES
dofollowthe law (cid:24) 1=mres. Forinstance, inthe
vector channel the height of the upper curve is
1. M. Shifman, A. Vainshtein, V. Zakharov
1:69 greater than the height of the lower curve.
Nucl. Phys. B147, 385, 448 (1979).
Whereasforthecorrespondingmassesonehas[5]:
2. E. Shuryak, Rev. Mod. Phys. 65 (1993) 1.
m2res((cid:20)=0:0983) 3. A. Di Giacomo, G. Rossi Phys. Lett. B100,
= 1:68 (7)
m2res((cid:20)=0:1111) 481 (1981).
4. P. Lepage, P. Mackenzie Phys. Rev. D48,
In the pseudoscalar channel (Fig.2), where the 2250 (1993).
resonance(pion)massissmaller,theratiosrnare 5. A.Borici and Ph.de Forcrand,This Proceed-
(cid:13)atnearthe origin,whichmeansthatthe contin- ings.
uum is not seen. That is because the e(cid:11)ective
