The strange contribution to $a_{\mu}$ with physical quark masses using M\"obius domain wall fermions

0.28 MB·

by  Matt Spraggs

The strange contribution to a with physical quark
µ
masses using Möbius domain wall fermions
6
1
0
2 Matt Spraggs∗a, Peter Boyleb, Luigi Del Debbiob, Andreas Jüttnera, Christoph

n Lehnerc, Kim Maltmand,e, Marina Marinkovicf, Antonin Portellia,c
a
J aSchoolofPhysicsandAstronomy,UniversityofSouthampton,SouthamptonSO171BJ,UK
4 bSUPA,SchoolofPhysics,TheUniversityofEdinburgh,EdinburghEH93JZ,UK
cPhysicsDepartment,BrookhavenNationalLaboratory,Upton,NY11973,US

t] dDepartmentofPhysicsandAstronomy,YorkUniversity,Toronto,Ontario,M3J1P3,Canada
a
eCSSM,UniversityofAdelaide,Adelaide,SA5005,Australia
l
p- fCERN,PhysicsDepartment,1211Geneva23,Switzerland
e Email: [email protected]
h
[

1 Wepresentpreliminaryresultsforthestrangeleading-orderhadroniccontributiontotheanoma-
v lous magnetic moment of the muon using RBC/UKQCD physical point domain wall fermions
7
3 ensembles. Wediscussvariousanalysisstrategiesinordertoconstrainthesystematicuncertainty
5 inthefinalresult.
0
0
.
1
0
6
1
:
v
i
X
r
a
The33rdInternationalSymposiumonLatticeFieldTheory,
14-18July,2015
KobeInternationalConferenceCenter,Kobe,Japan
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Strangecontributiontoa usingMöbiusdomainwallfermions MattSpraggs
µ
Table1: Ensemblesusedinthisstudy[7].
Parameter 48I 64I
L3×T×L 483×96×24 643×128×12
s
am 0.00078 0.000678
l
am 0.0362 0.02661
s
a−1 /GeV 1.730(4) 2.359(7)
L/fm 5.476(12) 5.354(16)
m /MeV 139.2(4) 139.2(5)
π
m /MeV 499.0(12) 507.6(16)
K
m L 3.863(6) 3.778(8)
π
1. Introduction
The anomalous magnetic moment of the muon, a , is one of the most accurately determined
µ
quantitiesinparticlephysics,withanaccuracyoftheorderofonepartpermillion[1]. Thereiscur-
rentlya3σ to4σ tensionbetweentheexperimentalandtheoreticaldeterminationsofthisquantity.
Thenewmuong−2experimentatFermilabisexpectedtoreducetheuncertaintyfromexperiment
by a factor of four, making a reduction in the theoretical error desirable. The leading-order (LO)
hadronic contribution is the main source of this uncertainty. In addition, current estimates of this
valuearecomputedusingaσ(e+e−→hadrons)data[2,3],makingafirst-principlescomputation
desirable. Herewepresentthecomputationoftheconnectedstrangecontributiontothisquantity.
We use a variety of analysis techniques in order to test both the techniques and their effect on the
finalvalueofas.
µ
TheLOstrangehadroniccontribution,as,canbecomputedasfollows[4]:
µ
as =(cid:16)α(cid:17)2(cid:90) ∞dQ2Πˆ (cid:0)Q2(cid:1)f(cid:0)Q2(cid:1), (1.1)
µ π
0
whereα istheQEDcoupling,Πˆ(Q2)=4π2(cid:0)Πs(Q2)−Πs(0)(cid:1)istheinfra-redsubtractedhadronic
vacuum polarization (HVP) scalar function and f is the integration kernel derived in perturbation
theory, with a singularity at Q2 =0. The resulting integrand is highly peaked near Q2 ≈m2/4,
µ
meaningthatthefinalvalueofa ishighlysensitivetovariationsinthevaluesofΠˆ(Q2).
µ
Our analysis can be broadly divided into two strategies. The first makes use of the hybrid
method outlined in [5]. The second uses continuous momenta in the lattice Fourier transform to
computethescalarHVPfunctiondirectlyatarbitrarymomentum[6].
2. SimulationDetails
Simulationshavebeenperformedonthetwo2+1flavourdomainwallfermion(DWF)ensem-
bleswithnear-physicalpionmassesdescribedin[7]. Forconveniencewesummarizetheproperties
oftheseensemblesinTable1.
We compute the lattice vacuum polarisation,C , using Z wall sources and Möbius domain
µν 2
wall fermions, with a local vector current at the source and the DWF conserved vector current at
thesink,i.e.:
C (x)= ZV (cid:10)V (x)V (0)(cid:11), (2.1)
µν µ ν
9
2
Strangecontributiontoa usingMöbiusdomainwallfermions MattSpraggs
µ
where Z is the vector renormalization constant, a is the lattice spacing, V is the local vector
V ν
currentandwedefinetheconservedMöbiusDWFvectorcurrentV (x)asdescribedin[7].
µ
Toaccountforasmallmistuninginthestrangequarkmassoneachensemble,weperformeda
setofpartiallyquenchedmeasurementsusingthephysicalvalueofthestrangequarkmass. These
wereperformedinadditiontotheunitarymeasurements[7].
3. Analysis
Weimplementedavarietyofanalysisstrategiesinordertoascertainthedependenceofa on
µ
theanalysistechnique.
3.1 HVPComputation
We can compute the HVP tensor in momentum space by performing a Fourier transform of
thepositionspaceHVPcorrelator,i.e.:
Π (Q)=∑e−iQ·xC (x)−∑C (x), (3.1)
µν µν µν
x x
where the second summation effectively subtracts the zero-mode [8]. In the infinite volume limit
this term is zero, and subtracting it greatly reduces the noise in the low-Q2 region. For the lowest
momentum value of Π(cid:0)Qˆ2(cid:1) the improvement in the statistical error is approximately a factor of
five.
We then perform a tensor decomposition of the HVP tensor, so that it may be related to the
scalarHVPfunctionasfollows:
Π (cid:0)Qˆ(cid:1)=(cid:0)δ Qˆ2−Qˆ Qˆ (cid:1)Π(cid:0)Qˆ2(cid:1)+···, (3.2)
µν µν µ ν
where the ellipis denotes contributions from Lorentz symmetry breaking, discretisation and finite
volume effects and Qˆ =2sin(Q/2) is the momentum of the intermediate photon. We remove a
potential source of lattice cut-off effects by considering only the diagonal component of the HVP
tensorwhereQˆ =0[9].
µ
3.2 HybridMethod
We used the hybrid method as described in [5]. This method consists of partitioning the
integrandin(1.1)intothreenon-overlappingadjacentregionsusingcutsatlow-andhigh-Q2. The
integrandisthencomputedforthethreeregionsindifferentways. Thelow-Q2 regionisintegrated
by modellingΠ(Q2)to extrapolate toΠ(0), which is subtracted to computeΠˆ(Q2). This result is
thencombinedwiththekernel f(Q2)toproducetheintegrandofinterest,whichisthenintegrated
numerically. The mid-Q2 region is integrated directly by multiplying the lattice data by f(Q2)
before using the trapezium method. Finally, the high-Q2 region is integrated by using the result
from perturbation theory [10, 5]. Restricting the use of an HVP parameterisation to the low-Q2
regionallowsustominimisesystematiceffects[5].
We use two classes of parameterisations for the low-Q2 region when performing the integral
in Equation (1.1): Padé approximants and conformal polynomials. The Padé approximants are
writtenasfollows[11]:
(cid:32) (cid:33)
R (cid:0)Qˆ2(cid:1)=Π +Qˆ2 m∑−1 a2i +δ c2 , n=m,m+1, (3.3)
mn 0 b2+Qˆ2 mn
i=0 i
3
Strangecontributiontoa usingMöbiusdomainwallfermions MattSpraggs
µ
wherea,b,Π andpossiblycareparameterstobedetermined.
i i 0
Theconformalpolynomialsarewrittenasfollows[5]:
√
PE(cid:0)Qˆ2(cid:1)=Π +∑n pwi, w= 1−√1+z, z= Qˆ2, (3.4)
n 0 i 1+ 1+z E2
i=1
where p andΠ areparameterstobedetermined. TheparameterE isthetwo-particlemassthresh-
i 0
old.
We use two techniques for constraining the low-Q2 models: χ2 minimisation and continuous
time moments [12]. The χ2 minimization involves a fit where the covariance matrix is approx-
imated by its diagonal, i.e. the fit is uncorrelated. This technique lends weight to points in the
computedHVPwithasmallerstatisticalerroratlargervaluesofQ2.
The moments method defines a relationship between the HVP scalar function and the lattice
space-averagedcurrent-currentcorrelator,C (t).
µµ
∑e−iQ0tC (t)=Qˆ2Π(cid:0)Qˆ2(cid:1) (3.5)
µµ 0 0
t
TakingthenthderivatewithrespecttoQˆ atQˆ =0allowsustowrite
0 0
(−1)n∑t2nCµµ(t)= ∂∂Q22nn(cid:0)Qˆ20Π(cid:0)Qˆ20(cid:1)(cid:1)(cid:12)(cid:12)(cid:12)(cid:12) (3.6)
t 0 Q0=0
WetheninsertoneoftheaboveanalyticalansätzefortheHVPscalarfunction,settingupasystem
ofequationsthatcanbesolvedtodeterminethemodelparameters.
The moments method uses continuous derivatives, meaning an infinite volume is assumed.
Whenperformingthemomentsmethod,weuseamodelthatisafunctionofQˆ2. However,within
themomentsmethod,derivativesaretakenwithrespecttoQ andnotQˆ . Withinthedetermination
0 0
ofthemodelparameters,thelow-Q2 cutisnotusedasaninputforthistechnique,sotheresulting
parametersdonotdependonthelowcutusedinthehybridmethod[12].
3.3 ContinuousMomenta
One alternative to the hybrid method is to compute the HVP directly at an arbitrary mo-
mentum by performing the Fourier transform at said momentum [6]. Whereas before we used
Q = 2πn withn ∈Z, −T/2≤n <T/2,wenowletn lieanywhereonthehalf-closedinterval
0 T 0 0 0 0
[−T/2,T/2). Thisallowsforthecomputationofas withoutusingaparameterisationoftheHVP.
µ
Because we are computing the HVP tensor for momenta that are non-Fourier modes on the
lattice, there may be some finite volume errors associated with this method. However, it can be
shownthattheseareexponentiallysuppressedbythelatticevolume[6]. Usingthistechnique, we
compute the HVP at arbitrary momenta up to some high cut, after which the perturbative result is
used.
4. Results
We used nine different parameterisations of the HVP when performing the hybrid method:
P0.5GeV,P0.5GeV,P0.5GeV,P0.6GeV,P0.6GeV,P0.6GeV,R ,R andR . Wescanthreelowcutsand
2 3 4 2 3 4 0,1 1,1 1,2
threehighcuts: 0.5GeV2,0.7GeV2 and0.9GeV2,and4.5GeV2,5.0GeV2 and5.5GeV2. Weused
4
Strangecontributiontoa usingMöbiusdomainwallfermions MattSpraggs
µ
the same high cuts when computing as using continuous momenta, where we used a step size of
µ
0.005forn .
t
Figure1illustratesanexampleextrapolationtothecontinuumandthephysicalstrangequark
mass. Weperformatwo-dimensionallinearfitina2 andtherelativedeviationofthestrangemass
fromthephysicalvalue. WedothisbecausedomainwallfermionsareO(a)improved,andinthe
lattercaseweassumealineardependenceofas onthestrangequarkmass. Inthiscaseweusedthe
µ
R parameterisation,whichwasconstrainedusinganuncorrelated χ2 minimisation. Thelowcut
0,1
inthiscasewas0.5GeV2 andthehighcutwas4.5GeV2. Theeffectofthestrangequarkmistuning
isclearlyvisible,withthefinalvalueofas shiftingfromapproximatley50×10−10to53.0×10−10.
µ
(a) (b)
Figure1: Continuumandstrangequarkmassextrapolations. Intheright-handsetofplotswehave
subtractedtheeffectsfromthestrangequarkmass(top)andlatticespacing(bottom).
Figure 2 illustrates the variation of as as the low cut in the hybrid method is varied. All the
µ
computedvaluesofas agreewithinstatistics,andmostofthevaluesareinstrongagreementwith
µ
one another. Furthermore, our results agree with those of HPQCD [12] and ETMC [13] to within
statistics. The models with the fewest parameters, i.e. P0.5GeV, P0.6GeV and R , deviate slightly
2 2 0,1
from 53.0×10−10. This is more apparent in the case where the models are constrained with χ2
fits. ThisislikelyaresultofthefitfavouringdataatlargerQ2,wherethestatisticalerrorissmaller,
whilstthemomentsuseanexpansionaroundQ2=0,favouringdataaroundthispoint.
Figure3demonstratesthevariousvaluesofa computedinthisanalysis. Goodagreementis
µ
found between all values of as. This suggests that the systematic error resulting from the various
µ
analysistechniquesissmall.
5. Summary
We have computed the strange contribution to the anomalous magnetic moment of the muon
usingdomainwallfermionswithphysicalquarkmasses. Weusedavarietyofanalysistechniques,
in particular the hybrid method proposed in [5] and continuous momenta [6]. Our final values of
as show good agreement with each other, suggesting that the systematic error from the choice of
µ
analysis technique is small. Furthermore, we find good agreement with the work of HPQCD [12]
andETMC[13].
We are now in the process of finalising the analysis of possible sources of systematic error,
particularly finite volume effects. We are simultaneously extending our analysis to the connected
5
Strangecontributiontoa usingMöbiusdomainwallfermions MattSpraggs
µ
(a) (b)
Figure2: Computedvaluesofas againstvariouslowcutsforfits(left)andmoments(right).
µ
Figure3: Errorbarplotillustratingthevariousvaluesofa computedinthisanalysis.
µ
lightcontributioncontributiontoa . Inthefutureweplantoaccountfortheeffectofdisconnected
µ
diagrams.
6. Acknowledgements
This work is part of a programme of research by the RBC/UKQCD collaboration. This re-
search was funded by the European Research Council under the European Community’s Seventh
Framework Programme (FP7/2007-2013) ERC grant agreement No 279757. The authors also ac-
knowledgeSTFCgrantsST/J000396/1andST/L000296/1. M.S.isfundedbyanEPSRCDoctoral
Training Centre grant (EP/G03690X/1) through the ICSS DTC. The calculations reported here
have been done on the DiRAC Bluegene/Q computer at the University of Edinburgh’s Advanced
ComputingFacility.
6
Strangecontributiontoa usingMöbiusdomainwallfermions MattSpraggs
µ
References
[1] F. Jegerlehner and A. Nyffeler. The Muon g-2. Phys. Rept. 477 pp. 1–110 (2009).
doi:10.1016/j.physrep.2009.04.003
[2] M. Davier, A. Hoecker, B. Malaescu and Z. Zhang. Reevaluation of
the Hadronic Contributions to the Muon g-2 and to alpha(MZ). Eur.
Phys. J. C71 p. 1515 (2011). [Erratum: Eur. Phys. J.C72,1874(2012)],
doi:10.1140/epjc/s10052-012-1874-8, 10.1140/epjc/s10052-010-1515-z
[3] K. Hagiwara et al. (g−2) and alpha(M2) re-evaluated using new precise data. J. Phys.
µ Z
G38p.085003(2011). doi:10.1088/0954-3899/38/8/085003
[4] T. Blum. Lattice calculation of the lowest order hadronic contribution to the
muon anomalous magnetic moment. Phys. Rev. Lett. 91 p. 052001 (2003).
doi:10.1103/PhysRevLett.91.052001
[5] M. Golterman, K. Maltman and S. Peris. New strategy for the lattice evaluation of the
leading order hadronic contribution to (g−2) . Phys.Rev. D90(7) p. 074508 (2014).
µ
doi:10.1103/PhysRevD.90.074508
[6] L.DelDebbioandA.Portelli(toappear)
[7] T. Blum et al. (RBC, UKQCD). Domain wall QCD with physical quark masses (2014).
arXiv:1411.7017
[8] D.BerneckerandH.B.Meyer.VectorCorrelatorsinLatticeQCD:Methodsandapplications.
Eur.Phys.J.A47p.148(2011). doi:10.1140/epja/i2011-11148-6
[9] P. Boyle, L. Del Debbio, E. Kerrane and J. Zanotti. Lattice Determination of the Hadronic
ContributiontotheMuong−2usingDynamicalDomainWallFermions. Phys.Rev.D85p.
074504(2012). doi:10.1103/PhysRevD.85.074504
[10] K. G. Chetyrkin, J. H. Kuhn and M. Steinhauser. Three loop polarization function and O
(alpha-s**2) corrections to the production of heavy quarks. Nucl. Phys. B482 pp. 213–240
(1996). doi:10.1016/S0550-3213(96)00534-2
[11] C. Aubin, T. Blum, M. Golterman and S. Peris. Model-independent parametrization of the
hadronicvacuumpolarizationandg-2forthemuononthelattice. Phys.Rev.D86p.054509
(2012). doi:10.1103/PhysRevD.86.054509
[12] B. Chakraborty et al. (HPQCD). Strange and charm quark contributions to the
anomalous magnetic moment of the muon. Phys. Rev. D89(11) p. 114501 (2014).
doi:10.1103/PhysRevD.89.114501
[13] X.Fengetal.Computingthehadronicvacuumpolarizationfunctionbyanalyticcontinuation.
Phys.Rev.D88p.034505(2013). doi:10.1103/PhysRevD.88.034505
7