Tuning of the strange quark mass with optimal reweighting

0.25 MB·English

by  Björn Leder

Tuning of the strange quark mass with optimal
reweighting
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2 Björn Leder∗, Jacob Finkenrath
n DepartmentofPhysics,BergischeUniversitätWuppertal
a Gaussstr.20,42119Wuppertal,Germany
J
E-mail: [email protected]
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Quark mass reweighting can be used to tune the mass of dynamical quarks. The basic idea is
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t tousegaugefieldensemblesgeneratedatsomebaremassparameterstoevaluateobservablesat
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l different bare sea quark masses. This involves the computation of so called reweighing factors
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p whicharegivenasratiosoffermiondeterminants.Inthecaseofsimulationsincludingthestrange
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quark,reweightingcanbeusedtoimprovetheapproachtowardsphysicalquarkmasses. Optimal
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[ reweightingstrategiescombineachangeofthestrangequarkmasswithachangeofthelightquark

massesinordertominimizethefluctuationsofthereweightingfactor. Wepresentnumericaltest
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v ofsuchstrategiesforrecentCLS2simulationsandasoftwarepackageformassreweightingbased
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The32ndInternationalSymposiumonLatticeFieldTheory,
23-28June,2014
ColumbiaUniversityNewYork,NY
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Tuningofthestrangequarkmasswithoptimalreweighting BjörnLeder
1. Whymassreweighting?
Lattice QCD simulations proceed to generate ensembles at small lattice spacings and light
quark masses close to or at their physical values. This is possible because of the algorithmic im-
provementsofthelastdecade. Nevertheless,thecosttogenerateindependentgaugeconfigurations
grows,particularlysobecauseofthegrowingautocorrelationtimeswhenthelatticespacingislow-
ered[1]. Theproblemisfurtheraggravatedbytheinclusionofaseastrangequark,whichleadsto
anenlargedmasstuningproblem(seeforexample[2]).
In such a situation it might be more cost efficient to change the masses of a given ensemble
using reweighting instead of generating a new one. This is because although reweighting is ex-
pensiveitonlyhastobedoneforindependentgaugeconfigurationsandthereforeitscostdoesnot
increase with the autocorrelation times. Also the naively expected linear scaling with the volume
mightbemuchmilderinreality[3,4].
ReweightingsimplymeanschangingtheprobabilityP (U)(“weight”)ofeachgaugeconfigu-
a
rationU intheexpectationvalue
1 (cid:90) nf
(cid:104)O(cid:105) = D[U]P (U)O(U), P (U)=e−Sg(β,U)∏det(D(U)+m +iγ µ). (1.1)
a Z a a i 5 i
a i=1
Here we assume n quark flavors with (possibly degenerate) bare masses m and twisted masses
f i
(cid:82)
µ, the gauge action S only depends on β and normalization D[U]P(U)/Z ≡1. The subscript
i g
on the expectation value and the probability distribution specifies the bare parameter set a =
{β,m ,m ,...,m ,µ ,...,µ }. Nowanobservableatbareparametersetb={β(cid:48),m(cid:48),m(cid:48),...,m(cid:48) ,
1 2 nf 1 nf 1 2 nf
µ(cid:48),...,µ(cid:48) }isobtainedfrom
1 nf
(cid:104)OW(cid:105) P
(cid:104)O(cid:105) = a, W = b. (1.2)
b (cid:104)W(cid:105) P
a a
If one restricts oneself to changes in the masses (hence β(cid:48) =β), as we do here, the reweighting
factorW canbewrittenas
1 nf D+m +iγ µ
W = , A=∏ i 5 i . (1.3)
det(A) D+m(cid:48)+iγ µ(cid:48)
i=1 i 5 i
Thenextsectionspecifiesthestochasticestimatorusedforthedeterminantin(1.3). Insection
3 follows a summary of the properties of three important cases of mass reweighting. From these
properties an optimal strategy for the tuning of the strange quark mass is derived in section 4.
Section 5 briefly describes a publicly available implementation of this strategy and in section 6
numericaltestsandcostestimatesarepresented.
2. Fluctuations
Thenumericalestimationofreweightingfactorsoftheform(1.3)usestheMonte-Carloeval-
uationofanintegralrepresentationofthedeterminantofacomplexmatrix[5,6]
N
W (A)= 1 ∑η e−η(k)†(A−I)η(k), (2.1)
N
η N
η k=1
2
Tuningofthestrangequarkmasswithoptimalreweighting BjörnLeder
with N Gaussian distributed random vectors η(i) and W →W for N →∞ if λ(A+A†)>0
η Nη η
[5,6]. LetusassumeAcanbewrittenasA=I+εBwithε||B||(cid:28)1. Thenthestochasticerrorof
thisestimatormaybeexpanded
1
δ2(A)=var (W )/(N |W|2)= [ε2Tr(BB†)+O(ε3)]. (2.2)
η η Nη η N
η
A similar expansion is obtained for the fluctuations of the reweighting factor in the expectation
value(1.1)
σ2 =var (W)/(cid:104)W(cid:105)2=ε2var (Tr(B))+O(ε3). (2.3)
U U U
It has been noticed that the stochastic estimator (2.1) may be plagued by large fluctuations
and/or long tails in the distribution. The reason is that its variance may not be defined even if the
meanis,i.e.,thevarianceisonlydefinedforλ(A+A†)>1[7]. Therehavebeenseveralattempts
tocontrolthevariance[8,9,10]. HereweuseafactorizationofthedeterminantinN factorswith
ε(cid:48) ∝ε/N [5, 6]. Since the factorization has been described thoroughly elsewhere and for better
readability we skip the details here and stick to the formulas with N =1. The generalization is
straightforward.
3. Massreweightingfactors
3.1 Oneflavor
The simplest case of mass reweighting is the change of the mass of one quark flavor of mass
m to mass m +∆m. Using the shorthand D =D+m, the corresponding reweighting factor is
s s m
(µ =0)
s
det(D ) 1
W (m ,∆m)= ms+∆m = , A =I−∆mD−1 , (3.1)
1f s det(Dms) det(A1f) 1f ms+∆m
andthestochasticerrorandthefluctuationsscale(asymptotically)with∆m2.
3.2 Twoflavors
Now we consider the simultaneous change of the mass of two quark flavors s and l. Without
loss of generality we assume m ≤m and a change m →m −γ∆m while m →m +∆m. The
l s l l s s
correspondingreweightingfactoris(µ =µ =0)
s l
det(D D ) 1 γ∆m+γD −D
W(γ)(m ,m ,∆m)= ml−γ∆m ms+∆m = , A =I+∆m ms ml , (3.2)
2f l s det(D D ) det(A ) 2f D D
ml ms 2f ms+∆m ml−γ∆m
and the stochastic error and the fluctuations scale (asymptotically) with ∆m2. However, if m =
l
m =m and γ =1 one finds A =I+∆m2(D2 −∆m2)−1 and a scaling ∝ ∆m4 (see the isospin
s 2f m
reweighting in [6, 4]). In this case the noise and the fluctuations of decreasing the mass of one
quarkarecompensatedbyincreasingthemassofthesecondone. Forγ =0thereweightingfactor
(0)
reduces to the one flavor case: W (m ,m ,∆m) =W (m ,∆m). Therefore, in the general case
2f l s 1f s
m <m , an optimal 0≤γ∗ ≤1 may be found that minimizes the fluctuations ofW(γ). With light
l s 2f
quarksaroundthestrangequarkmassanoptimalvalueofγ∗≈0.82wasfound[5].
Ageneralizationofeq.(3.2)forfiniteµ and/orµ isstraightforward. Notethat,ingeneral,in
s l
thiscasethereweighingfactoriscomplex.
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Tuningofthestrangequarkmasswithoptimalreweighting BjörnLeder
(m,µ) (i) (ii) (iii) (iv)
up (m ,0) →(m ,µ) →(m −γ∆m,µ) →(m −γ∆m,0)
l l l l
down (m ,0) →(m ,−µ) →(m −γ∆m,−µ) →(m −γ∆m,0)
l l l l
strange (m ,0) →(m +∆m,0) →(m +2∆m,0)
s s s
type tm(sec. 3.3) 2f(sec. 3.2) 2f(sec. 3.2) tm(sec. 3.3)
Table1: Optimalreweightingstrategyforstrangequarkmassreweighting.
3.3 Twistedmassreweighting
Twisted mass reweighting was introduced [3] and implemented [11] in order to stabilize the
HMC at small quark masses. In the context of mass reweighting it can be used to ensure the
convergence of the stochastic estimator (2.1) [5]. For a doublet of quarks of mass m =m =m
1 2
and µ =µ =−µ thecorrespondingreweightingfactoris
1 2
det(D D† +µ(cid:48)2) 1 µ2−µ(cid:48)2
W (m,µ,µ(cid:48))= m m = , A =I+ , (3.3)
tm det(DmD†m+µ2) det(Atm) tm DmD†m+µ(cid:48)2
where we used (D +iγ µ)† =γ (D −iγ µ)γ . The stochastic error and the fluctuations scale
m 5 5 m 5 5
(asymptotically)with(µ2−µ(cid:48)2)2.
4. Optimalstrangequarkmassreweighting
As pointed out in the introduction, in a n =2+1 simulation with up, down, strange quark
f
masses{m ,m ,m }itmightbenecessarytotunethestrangequarkmassofanensembleinorder
l l s
toimprovetheapproachtothephysicalpoint. Wehaveseeninsection3thatitisadvantageousto
combinethechangeofthemassofonequarkwithanoppositechangeofanotherone. Furthermore,
iflightquarksarereweighted,afinitetwistedmassservesasasafeguardagainstzerocrossingsof
small eigenvalues. Taken together, these lessons lead us to propose the following reweighting
strategy:
(i) thelightquarksarereweightedtoafiniteµ,
(ii) theupandthestrangequarkarereweightedtogetherinoppositedirections,
(iii) thedownandthestrangequarkarereweightedtogetherbythesameamount,
(iv) thelightquarksarereweightedtozerotwistedmass.
ThefourstepsaresummarizedinTable1. Inthestochasticestimationthetwistedmassreweighting
in step (i) and (iv) can be combined so that the total reweighting factor W can be written as a
s
productofthreefactors
W =W W W ← (W ) (W ) (W ) , (4.1)
s (ii) (iii) (i+iv) (ii) Nη (iii) Nη (i+iv) Nη
andeachofthesefactorsisestimatedaccordingto(2.1). Thetotalchangeinthemassesis
{m ,m ,m }→{m −γ∆m,m −γ∆m,m +2∆m}.
l l s l l s
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Tuningofthestrangequarkmasswithoptimalreweighting BjörnLeder
Iftheensembleisgeneratedatfinite µ inthelightquarksector,asforexamplein[2],step(i)
canbeomitted. Dependingonwhichkindoftwistedmassreweightingisusedintheproductionof
theensemble,thelastfactoristhengivenby

Wtm(ml,0,µ)Wtm(ml−γ∆m,µ,0) none

W(i+iv)= Wtm(ml−γ∆m,µ,0) typeIin[11]. (4.2)
 √

W (m ,µ, 2µ)W (m −γ∆m,µ,0) typeIIin[11]
tm l tm l
Thereweightingfactorsforstep(ii)and(iii)areexplicitlygivenby
(γ)
W =W (m ,m ,∆m), withµ =µ (4.3)
(ii) 2f l s l
(γ)
W =W (m ,m +∆m,∆m), withµ =−µ, (4.4)
(iii) 2f l s l
andγ ≈γ∗. Asmentionedbeforefor µ (cid:54)=0thereweightingfactorW(γ) iscomplex. Butsincethe
l 2f
productW W ismanifestlyreal,thephasesofthetwofactorscancel.
(ii) (iii)
5. Publiccodeformassreweighting
Thenumericalresultspresentedinthenextsectionhavebeenobtainedwiththemrw-package
[12], an extended version of the openQCD package [13]. Keeping the structure of openQCD the
extension is implemented as a module (mrw). It provides a main program that reads in open-
QCD style input files, documentation and sample input files. The mrw-package is publicly avail-
able at https://github.com/bjoern-leder/mrw. In detail it adds the following features to
openQCD:
• oneflavormassandtwisted-massreweighting[5,6]
• interpolation(factorization)fortwistedmassreweightingtypeIandII
• factorizationwithnon-equidistantinterpolations[5,6]
• isospinmassreweighting[5,6]
• strangequarkmassreweightingofsection4
• severalcheckroutinesforallpartsofthenewmodule
6. Numericaltestandcost
The optimal strange quark mass reweighting proposed in section 4 has been tested on two
n =2+1 ensembles from the effort described in [2], see Table 2. While the first one is off the
f
quarkmasstrajectorydescribedinsection2.3of[2],thesecondonesitsonit.1 Thelatticespacing
isapproximatedtoa=0.086fm[2]andthelatticesizeis64×323.
1Thistrajectoryisdefinedby∑imi=const. Notethatforγ (cid:54)=1theoptimalstrangequarkreweightingtakesthe
ensembleoff,whereasforγ=1itwouldmovetheensemblealongthistrajectory.
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Tuningofthestrangequarkmasswithoptimalreweighting BjörnLeder
IDin[2] m [MeV] m [MeV] ∆m [MeV] γ µ
π K s
– 330(380) 450(430) -12 0.80 0.0
B105 280(200) 460(480) 12 0.80 0.001
Table 2: Ensembles used in the numerical test. In parentheses are the meson masses after reweighting.
∆m isthedifferenceoftherenormalizedstrangequarkmassesbeforeandafterthereweighting. Thefirst
s
ensemblewaspartofanexploratorystudyandisnotlistedin[2].
(cid:27)W2 =0:708(95)
(cid:30)=(cid:25)=0:0050(67)
5
4
(cid:11) 3
W
(cid:10)
=
W
2
1
0
0 50 100 150 200 250 300 350
MDU
Figure1: StrangequarkreweightingfactorW forensembleB105.
s
InFigure1thereweightingfactorW (eq. (4.1))isplottedfortheensembleB105. Thephase
s
is also plotted and is compatible with zero as expected. The fluctuations of the reweighting are
sizable, but the change in the meson masses is large: 5% (30%) for the kaon (pion). By using a
smallervalueforγ thischangecanbemademorebalanced. Assumingascalingofthefluctuations
∝∆m2Vq/m the feasibility of such a reweighting at different parameters can be projected using
l
thefollowingformula
(cid:18) ∆m (cid:19)2(cid:18)240MeV(cid:19)2 Vq
σ2 =0.71 s , q=1/4...3/4.
Ws 12MeV m (3.3fm)4q
π
where ∆m is the difference of the renormalized strange quark masses and m is the mean of the
s π
pionmassesbeforeandafterthereweighting,andV isthephysicalvolume[4]. Theuncertaintyin
thevolumescalingiscurrentlyunderinvestigation.
The cost for estimating the reweighting factors W , W , W is roughly independent of
(ii) (iii) (i+iv)
the ensemble parameters. It is fixed by demanding the stochastic noise to be much smaller than
thefluctuations[5]. TheerrorbarsinFigure1indicatethestochasticerrorforeachconfiguration.
The total number of Gaussian noise vectors was 48 for W and W , and 24 for W . Since
(ii) (iii) (i+iv)
thestochasticerrorislargeinFigure1thesenumbershavetobeincreasedinanapplicationofthe
method.
Thenumericalcostperconfiguration(corehours)is∼30%ofonetrajectoryoflength2MDU.
ButsincetheautocorrelationtimesofobservablesτO aretypicallymuchlargerthanthis,reweight-
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Tuningofthestrangequarkmasswithoptimalreweighting BjörnLeder
ingisonlyneededeveryn=τO/2(cid:29)1trajectories,effectivelycuttingthecostto∼(30/n)%.
7. Outlook
Inthefirstnumericaltestpresentedinthelastsectionavalueforγ∗isusedthatwasdetermined
in another setup and at different quark masses. A dedicated tuning needs only a small number of
configurations and has the potential to lower the fluctuations significantly. Furthermore the cost
perconfigurationcanbereducedbycombiningthereweightingfactorsW ,W intooneestima-
(ii) (iii)
tor. Finally, the mrw-package will be upgraded to openQCD-1.4, including support for periodic
boundaryconditions.
Acknowledgements: ThisworkwasfundedbytheDeutscheForschungsgemeinschaft(DFG)in
formofTransregionalCollaborativeResearchCentre55(SFB/TRR55).
References
[1] ALPHACollaboration,S.Schaefer,R.Sommer,andF.Virotta,Criticalslowingdownanderror
analysisinlatticeQCDsimulations,Nucl.Phys.B845(2011)93–119,[arXiv:1009.5228].
[2] M.Bruno,D.Djukanovic,G.P.Engel,A.Francis,G.Herdoiza,etal.,SimulationofQCDwith
N =2+1flavorsofnon-perturbativelyimprovedWilsonfermions,arXiv:1411.3982.
f
[3] M.LüscherandF.Palombi,Fluctuationsandreweightingofthequarkdeterminantonlargelattices,
PoSLATTICE2008(2008)049,[arXiv:0810.0946].
[4] J.Finkenrath,B.Leder,andF.Knechtli,IsospinEffectsbyMassReweighting,PoSLATTICE2014
(2014)297.
[5] J.Finkenrath,F.Knechtli,andB.Leder,OneflavormassreweightinginlatticeQCD,Nucl.Phys.
B877(2013)441–456,[arXiv:1306.3962].Erratum: ibid.B887(2013)574.
[6] B.Leder,J.Finkenrath,andF.Knechtli,Oneflavormassreweighting: foundations,PoS
LATTICE2013(2014)035,[arXiv:1401.1079].
[7] A.HasenfratzandA.Alexandru,Improvingthepartialglobalstochasticmetropolisupdatefor
dynamicalsmearedlinkfermions,Nucl.Phys.Proc.Suppl.119(2003)994–996,
[hep-lat/0209071].
[8] M.Hasenbusch,SpeedingupthehybridMonteCarloalgorithmfordynamicalfermions,Phys.Lett.
B519(2001)177–182,[hep-lat/0107019].
[9] A.HasenfratzandA.Alexandru,Evaluatingthefermionicdeterminantofdynamicalconfigurations,
Phys.Rev.D65(2002)114506,[hep-lat/0203026].
[10] A.Hasenfratz,R.Hoffmann,andS.Schaefer,Reweightingtowardsthechirallimit,Phys.Rev.D78
(2008)014515,[arXiv:0805.2369].
[11] M.LüscherandS.Schaefer,LatticeQCDwithopenboundaryconditionsandtwisted-mass
reweighting,Comput.Phys.Commun.184(2013)519–528,[arXiv:1206.2809].
[12] B.LederandJ.Finkenrath,mrw-Massandtwisted-massreweightingbasedonopenQCD,
https://github.com/bjoern-leder/mrw.
[13] M.LüscherandS.Schaefer,openQCD-SimulationprogramforlatticeQCD,
http://luscher.web.cern.ch/luscher/openQCD/.
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