Direct determination of strange and light quark condensates from full lattice QCD

0.2 MB·English

by  C. T. H. Davies

Direct determination of strange and light quark
condensates from full lattice QCD
C. T. H. Davies∗
UniversityofGlasgow
E-mail: [email protected]
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1 C. McNeile
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2 UniversitätWuppertal

n
a A. Bazavov
J
BrookhavenNationalLaboratory
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R. J. Dowdall

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t UniversityofGlasgow
a
l
- K. Hornbostel
p
e
SouthernMethodistUniversity
h
[
G. P. Lepage

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v CornellUniversity
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0 H. Trottier
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7 SimonFraserUniversity
.
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0 WedeterminethestrangeandlightquarkcondensatesinfulllatticeQCDforthefirsttime.Thisis
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donebydirectcalculationoftheexpectationvalueofthetraceofthequarkpropagatorfollowed
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: bysubtractionoftheappropriateperturbativecontributiontoconverttoavalueforthecondensate
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i intheMSschemeat2GeV.WeuselatticeQCDconfigurationsincludingu,d,sandcquarksin
X
theseawithu/d quarkmassesgoingdowntothephysicalvalue. Wefindtheratioofthestrange
r
a
tothelightquarkcondensatetobe1.08(16).
XthQuarkConfinementandtheHadronSpectrum,
October8-12,2012
TUMCampusGarching,Munich,Germany
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
StrangeandlightquarkcondensatesfromlatticeQCD C.T.H.Davies
5 5
R R, m = physical
s l l
4 4
3 3
2 2
1 1
0 0
0 1 2 3 4 5 6 0 1 2 3 4 5 6
a-2 (GeV2) a-2 (GeV2)
Figure1:ResultsforR ,definedinEq.2.1,asafunctionofthesquareoftheinverselatticespacing.Theplot
q
ontheleftisforsquarksandontherightforlightquarksatthephysicallightquarkmass(m =(m +m )/2).
l u d
Opensquaresgiverawresultsandpluseshavetree-levelperturbativecorrections. Crosseshaveinaddition
one-loopperturbativecorrections,usingα(nf=4)(2/a).Dashedlinesgiveasimplelinearfittotherawresults.
V
1. Introduction
The condensation of quark-antiquark pairs in the vacuum signalling the breakdown of chiral
symmetryisanimportantfeatureoflowenergyQCD.Thevalueofthechiralcondensate(thequark
condensateforzeroquarkmass)isgivenbytheGell-Mann,Oakes,Rennerrelation[1]:
f2M2 m +m (cid:104)0|uu+dd|0(cid:105)
π π =− u d . (1.1)
4 2 2
A value for the chiral condensate can be extracted accurately but indirectly from chiral extrapola-
tion of lattice QCD results for meson masses and decay constants. A harder question to answer
is that of the value of the light and strange quark condensates at their physical non-zero quark
masses. This is because the condensate must be carefully defined, taking into account the quark
mass dependent mixing of mψψ (not normal-ordered) with the unit operator. On the lattice, in a
direct calculation of the condensate, such mixing gives rise to terms which diverge as m/a2. We
have calculated these terms, through O(α ) in lattice QCD perturbation theory and used these to
s
convert the lattice QCD results to a well-defined condensate in the MS scheme at a fixed scale (2
GeV).Sincewehaveresultsatmultiplevaluesofthelatticespacingwecanfitforremainingm/a2
pieces at higher order in α . We can also extrapolate our results to the continuum limit at a=0.
s
Since the conference we have finalised a paper on this work [2] and refer the reader there for all
details.
2. Results
WeusegluonfieldconfigurationsgeneratedbytheMILCcollaborationincludingu, d (m =
u
m here), s and c quarks in the sea at 3 different values of the lattice spacing and having u/d
d
quark masses down to the physical value [3]. The quark formalism used is the Highly Improved
StaggeredQuark(HISQ)action[4]thathasgoodchiralpropertiesandsmalldiscretisationerrors.
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StrangeandlightquarkcondensatesfromlatticeQCD C.T.H.Davies
Figure1showsourrawandcorrectedlatticeresultsforsquarksandforlightquarksofphysical
mass. ThequantityplottedistheratioR givenby:
q
4m (cid:104)ψψ (cid:105)
q q
R =− . (2.1)
q f2 M2
ηq ηq
Here η is the pseudoscalar meson made from quarks of mass m . This is the π for light quarks
q q
andafictitiousparticlecalledtheη forsquarks. Wedonotincludedisconnecteddiagramsinthe
s
η correlator and so it is unable to decay to gluons or to mix with the physical η and η(cid:48) mesons.
s
Because it has no valence light quarks, yet is not a heavy meson, it is a very useful particle in the
determination of the lattice spacing [5]. Its mass and decay constant in the continuum and chiral
limits can be determined accurately in terms of those of the K and the π [6]. Interestingly, values
very close to those expected in leading order chiral perturbation theory are found. R is a good
q
quantity to use in this analysis because it has reduced finite volume and quark mass dependence
overthecondensatealoneandis1forquarksofzeromass,asgivenbytheGMORrelation.
InFigure1theopensquaresgivetherawresultsfromthelattice,determiningthecondensate
from the trace of the quark propagator. The x-axis is the inverse square of the lattice spacing
and it is quite clear from these plots that there is a quark-mass dependent 1/a2 divergence. The
pluses give the results after subtraction of the difference of the tree-level lattice QCD (for HISQ
quarks) and MS perturbative expressions. The divergence is largely, but not entirely, removed.
The crosses continue this process by removing the perturbative contribution through O(α ). This
s
makeslittledifferencebecausethecoefficientoftheO(α )divergenceisverysmall. Itisthenclear
s
that the remaining divergence evident in the lattice calculations must come from higher orders in
perturbation theory, and we take account of this in our fits. We also use the fact that the form of
thedivergenceisstronglyconstrainedandfitsimultaneouslyresultsatmultiplevaluesofthequark
massandlatticespacing.
Figure 2 on the left shows the one-loop subtracted results from Figure 1 along with the final
resultsfromourfit. Thex-axisisnowthesquareofthelatticespacingandthegreyandgreenbands
showthecontinuumandchirallimitvaluesofthestrangeandlightquarkcondensatesrespectively,
intheMSschemeat2GeV.Ourfitformallowsfordiscretisationerrorsandquarkmassdependence
of the physical condensate as well as the divergent pieces discussed above. Fitting the remaining
divergenceisthekeyissueindeterminingthefinalerror,however,andthereasonwhytheerroron
the strange quark condensate is 15% and on the light quark condensate 0.5%. Full details, along
withanerrorbudget,aregivenin[2].
Ourresultsare:
(cid:104)ss(cid:105)MS(2GeV) = −0.0245(37)(3)GeV3=−(290(15)MeV)3
(cid:104)ll(cid:105)MS(2GeV) = −0.0227(1)(4)GeV3=−(283(2)MeV)3
(cid:104)ss(cid:105)MS(2GeV)
= 1.08(16)(1), (2.2)
(cid:104)ll(cid:105)MS(2GeV)
where the first error comes from our fitted result for R and the second error in each case comes
q
fromtheerrorinthequarkmasses,takenfromlatticeQCDresults[7].
Figure 2 shows on the right a comparison of our result for the ratio of the strange to light
quarkcondensatetopreviousvaluesfromQCDsumrulecalculations[8,9,10]. Ourresulthasthe
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StrangeandlightquarkcondensatesfromlatticeQCD C.T.H.Davies
Lattice QCD
this work
1.8
R
s,phys
R
1.6 l,phys
1.4
Maltmann et al
1.2
0811.1590
1
R q Dominguez et al
0.8
0711.3962
0.6
Narison et al
0.4
hep-ph/0202200
0.2
0 0 0.5 1 1.5 2
0 0.005 0.01 0.015 0.02 0.025 0.03
- -
a2 (fm2) <ss(2 GeV)>/<ll(2 GeV)>
Figure 2: The figure on the left shows the results from fitting the ratio R given by the crosses after sub-
q
tracting the perturbative contribution through O(α ). 4 different quark masses are used: m (black), m /5
s s s
(blue),m /10(red)andthephysicallightquarkmass(green). Thegreenandblackbandsshowthephysical
s
resultfromthefitwith±1σ errors. Blackisforthestrangequarkcondensateandgreenforthelightquark
condensate. Thefigureontherightshowsthecomparisonofourresultfortheratioofstrangetolightquark
condensatetothoseofprevioussumruleanalyses.
advantageofbeingadirectdeterminationwithafullerrorbudget. Ourvalueforthestrangequark
condensateagreeswellwithindependentresultsongluonfieldconfigurationsincludingu,d ands
quarksintheseaandcoveringawiderrangeofvaluesofa[2].
References
[1] M.Gell-Mann,R.Oakes,B.Renner,Phys.Rev.175:2195(1985).
[2] C.McNeileetal,arXiv:1211.6577
[3] A.Bazavovetal(MILCCollaboration),arXiv:1212.4768.
[4] E.Follanaetal(HPQCDCollaboration),Phys.Rev.D75:054502(2007),hep-lat/0610092.
[5] C.T.H.Daviesetal(HPQCDCollaboration),Phys.Rev.D81:034506(2010),arXiv:0910.1229.
[6] R.J.Dowdalletal(HPQCDCollaboration),Phys.Rev.D85:054509(2012),arXiv:1110.6887.
[7] C.McNeileetal(HPQCDCollaboration),Phys.Rev.D82:034512(2010),arXiv:1004.4285.
[8] K.Maltman,Phys.Lett.B672:257(2009),arXiv:0811.1590.
[9] C.A.Dominguezetal,JHEP0805:020(2008),arXiv: 0712.0768.
[10] S.Narison,hep-lat/0202200.
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