Two loop expansion of the Schroedinger functional coupling alpha_SF in SU(3) lattice gauge

TwoloopexpansionoftheSchr¨odingerfunctionalcouplingαSFinSU(3)latticegaugetheory

A.Bodea∗

a

arXiv:hep-lat/9710043v1 10 Oct 1997Institutf¨urPhysik,Humboldt-Universit¨atzuBerlin,Invalidenstr.110,D-10115Berlin,Germany

ThetwoloopcoefficientoftheexpansionoftheSchr¨odingerfunctionalcouplingintermsofthelatticecouplingiscalculatedfortheSU(3)Yang-Millstheory.Thiscoefficientisrequiredtorelatelatticedatatothe

2.DefinitionofαSF

2

ThecouplingαSF(q)=gSF(L)/(4π),atq=−1

LisdefinedviatheeffectiveactionΓ:

󰀃

exp(−Γ)=D[U]exp(−S[U])(1)

withperturbationtheoryispossible.The

relationcanbecalculatedbyexpandingbothα

MS

wheretheWilsonactionS[U]=

1

andαSF

wasrequiredinordertoavoidasignificantsourceoferrorintheconversiontoα

MS

isnormalisedviatheclassicalactionminimumΓ0.Thederivativeiswithrespecttoη,whichparametrisestheboundaryfieldsappliedatt=0,L.Theremainingthreespacedimensionshaveperiodicboundaryconditions.Detailsaboutthechosendiagonalandconstantboundaryfieldscanbefoundin[6].

3.PerturbativeexpansionandcalculationTheperturbativeexpansionhastobetakenaroundtheinducednonzerobackgroundfield.Sincethepropagatorsarenotknownanalytically,wecalculatethemnumericallyforeachIandsum

󰀆󰀆󰀆′Γ(L,η)󰀆η=0

(3)

∗Supported

byDeutscheForschungGemeinschaft,Grant

No.Wo389/3-2

2

themupwithineachofthe13diagrams[10].Theuseofsymmetries(forinstancetranslationinvari-anceandthecubicgroupinspace)inthiscontextreducesthenumericaleffort.ThepropagatorsarediagonalintherootbasisofSU(3)andinspa-tialmomentumspace,andhavebeencalculatedtherebyarecurrencerelationintime.Numericalefficiencyandpreservingmaximalprecisionwastakenintoaccountintheprogram.Inparticulartheηderivativesweretakenanalytically.

Theexpectedindependenceofthegaugepa-rameterλ0ofΓ′2,thevalidityofthesymmetries,acomparisonofΓ′recalculation2withanumericalderivativeofΓ2andtheoftheSU(2)resultsareappliedastestsaswellasanindependentcalcu-lationofPeterWeiszforsmallI.

In2monthsCPU-timeonaHP735with128MBmainmemoryweobtainedmX2(I)intherangeI=4...32,where:

g2L)=g20+m1(I)g40+m2(I)g6SF(0...

(4)m1(I)=

ma1(I)

+

c(1)

tmb1(I)

(5)m2(I)=ma

(1)2(I)+ctmb2(I)+(2)

(I)󰀄

c(1)t󰀅2mc2(I)

(6)

+ctmd2(I)+m12

From[9]ma1isknownandmb1,mc2,md

2arederivedanalytically.

Symanzik’sanalysissuggestsasasymptoticex-pansionofmX2

󰀁∞mXrX2(I)

=

n+sXnln(I)+tXnln2(I)

n=0

∂n

MS

ing0[4]

weget:α

MS

2

[7]weareabletoquotebSF0.4827(88)/(4π)3

,whereafittothedata2

=ofnumericalsimulationsgivesbeff=1.5(8)/(4π)3.Thereisnoreasonthatthecoefficient2

ofthefitcoincideswithbSF

2,sinceαSFistracednonpertur-bativlyonthelattice.Neverthelesstheorderisthesameandweexpectasquotedin[6]thesameerrorreductionasintheSU(2).

Theperturbativerelation(12)betweenαSFandα

MS

(2.048q)=αSF(q)+0.271(11)αSF(q)3

(15)

α

forsmallenoughcouplings.UsingthesmallestnumericallydeterminedcouplingαSFwegetfortheSU(3)gaugetheory:α

MS

(78.2GeV)=0.08407(121)(5)(18)

wherethefirsterrorarisesfromthenumericalun-certaintiesandthescaledeterminationatlowen-ergies.Theseconderrorresultsfromtheresidualevolutionandtheconversiontothe

MS

(14.5GeV)=0.1145(23)(15)(19)

α

3

MS

in

theSU(3)gaugetheory.Ithasleadtoareduc-tionofsystematicerrorscomparabletothecaseofSU(2).

7.Acknowledments

IamgratefultoPeterWeiszforprovidingchecks.IwishtothankUlliWolffforguidanceduringthecalculationandhelpfulcommentsonthemanuscript.REFERENCES

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(1995)234

2.R.Narayanan,U.Wolff,Nucl.Phys.B444

(1995)4253.M.L¨uscher,P.Weisz,Nucl.Phys.B445

(1995)4294.M.L¨uscher,P.Weisz,Phys.Lett.B349(1995)

165

5.G.deDivitiis,R.Frezzotti,M.Guag-nelli,M.L¨uscher,R.Petronzio,R.Sommer,P.Weisz,U.Wolff,Nucl.Phys.B437(1995)4476.M.L¨uscher,R.Sommer,P.Weiszand

U.Wolff,Nucl.Phys.B413(1994)4817.O.V.Tarasov,A.A.Vladimirov

A.Yu.Zharkov,Phys.Lett.B93(1980)4298.M.L¨uscherandP.Weisz,Nucl.Phys.B266

(1986)3099.M.L¨uscher,privatecommunication10.inpreparation