Table Of ContentIsospin Effects by Mass Reweighting
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1 JacobFinkenrath ,a, FrancescoKnechtlia, BörnLedera,b
∗
a DepartmentofPhysics,BergischeUniversitätWuppertal
]
t Gaussstr. 20,D-42119Wuppertal,Germany
a
l b DepartmentofMathematics,BergischeUniversitätWuppertal
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p Gaussstr. 20,D-42119Wuppertal,Germany
e
E-mail: [email protected]
h
[
2 Mostoftoday’slatticesimulationsareperformedintheisospinsymmetriclimitofthelightquark
v
sector. Mass reweightingis a techniqueto includeeffectsofisospin breakingin the sea quarks
1
at moderate numericalcost. We will give a summary of our recentresults on fine lattices with
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4 lightquarkmassesandwillshowhowlightquarkmassescanbeextractedbyintroducingsuitable
6
tuningconditionsforthebaremassparameters.
0
. Ingeneralthereweightingfactorintroducesadditionalfluctuationsandthusincreasesthestatisti-
1
0 caluncertainties. Inthecaseofisospinreweightingthisfactorisaratiooffermiondeterminants.
5 Thestochasticevaluationofthedeterminantspotentiallyleadstostochasticnoiseinobservables.
1
: Weshowthequarkmassandthevolumedependenceofthesefluctuations.
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The32ndInternationalSymposiumonLatticeFieldTheory
23-28June,2014
ColumbiaUniversityNewYork,NY
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
IsospinEffectsbyMassReweighting JacobFinkenrath
1. Introduction
MassReweighting[1]isaninterestingandefficientmethodtocorrectandtoincludeeffectsof
quark masses. Itcan beused for tuning, e.g. thestrange quark massm in a2+1simulation orthe
s
isospinsplittingintheup-quarkmassm andthedown-quarkmassm . Moreoveritcanbeapplied
u d
to understand the mass behavior of observables. For small corrections it is applicable and more
efficient than new simulations. Mass reweighting involves the evaluation of fermion determinants
whichcanberewrittenbyanintegral representation. Thisintegralrepresentation canbeestimated
byMonteCarlointegration whichneedsaround100inversions oftheDiracoperatortocontrolthe
stochastic noiseefficiently.
Thereweighting factorentersthemeasurement ofanobservable by[2]
OW
O = h i = OW˜ (1.1)
W
h i W h i
h i
wherethemassreweightingfactorforn flavorsofquarks
f
nf detD
W =(cid:213) mnew,i (1.2)
detD
i=1 mold,i
isnormalized withW˜ =W/ W . Here,theDiracoperator isgivenbythecloverimprovedWilson
h i
Dirac operator D =D+m. The reweighting factor can be rewritten as a determinant of a ratio
m
matrixM with
1 1
W = = . (1.3)
det(cid:213) nf D 1 D detM
i=1 −mnew,i mold,i
In general lattice simulations are done in th(cid:2)e isospin sym(cid:3)metric limit in the light quark sector by
setting the light quark masses tothe average light quark massm =0.5(m +m ). Theidea isto
ud u d
use mass reweighting tointroduce isospin breaking. Thereweighting isperformed bysplitting up
thelightquarkmassesbykeepingtheaveragequarkmassconstant2m =m +m =const andit
ud u d
followswiththemassshiftD m =m m
ud d u
−
m =m 0.5 D m m m +0.5 D m =m . (1.4)
u ud ud ud ud ud d
− · ← → ·
Thisleadstotheisospinreweighting factor
1 1
W = = . (1.5)
iso det D−mu1DmudD−md1Dmud detMiso
Now,thedeterminantofthenon–herm(cid:2)itianmatrixM canb(cid:3)erewrittenbyanintegralrepresentation
givenby
1
= D[h ]exp h †Mh (1.6)
detM {− }
Z
which holds for l (M+M†) > 0 [3] and the normalized integral measure is given by D[h ] =
(cid:213) n dx dy /p withh =x +iy . Theintegraleq.(1.6)canbeestimatedstochastically
j=1 j j j j j
1 = 1 (cid:229)Nh e−h i†(M−I)h i+O(1/ Nh ) (1.7)
detM Nh
i=1
p
2
IsospinEffectsbyMassReweighting JacobFinkenrath
Table 1: The table shows the analyzed CLS - ensembles generate with n =2 - O(a) improved Wilson
f
fermionswithm =m =m [4]. Weusedensemblesofthreedifferentlatticespacingsawithpionmasses
ud u d
mp from 580 MeV down to 192 MeV and lattice volumeV/a4. The number of configurationsNcnfg are
seperatedbyMDUs/configwithR therelativenumberofactivelinks. Themaximalreweightingrangeis
act
givenbythequarkmassshiftintheMS-schemeD m . Therenormalizedmassisdefinedasin[4].
R,max
ID V/a4 a[fm] mp [MeV] Ncnfg MDUs/config Ract D mR,max [MeV]
·
A5 64 323 0.076 330 202 20 1 4.43(60)
× ·
E4 64 323 0.066 580 100 16 0.37 7.1(17)
× ·
D5 48 243 “ 440 503 8 1 5.9(10)
× ·
E5 64 323 “ 440 99 160 0.37 6.01(96)
× ·
F7 96 643 “ 270 350 16 0.37 5.01(38)
× ·
G8 128 643 “ 192 90 8 1 5.86(56)
× ·
O7 96 483 0.049 270 98 40 1 5.63(40)
× ·
bydrawingNh pseudofermion fieldsh distributed viathenormalized function (cid:181) exp h †h and
{− }
I the unit matrix with dimension of M. Note for every drawn field h n –inversions of the Wilson
f
Diracoperator D havetobeperformed.
mi
Ingeneral massreweighting introduces fluctuations whichincreasethestatistical error. These
fluctuations are the ensemble fluctuations, introduced by the ensemble average in eq. (1.1), and
the stochastic fluctuations, introduced by the stochastic estimation of the integral eq. (1.6). The
fluctuations are given by the variance averaged over the ensemble and the pseudofermions h . We
willdefinethevarianceoftheintegralrepresentation eq.(1.6)bys 2= ww† h w h w† h
hh i i−hh i ihh i i
withthestochasticestimatew(U,h )=1/Nh (cid:229) iexp{−h i†(M−I)h i}. Byperformingtheh –average
h (i.e.allh i independently) thefluctuations aregivenforfiniteNh by
hi
s 2= 1 1 +Nh −1 1 1 1 (1.8)
Nh det(M+M† I) Nh detMM† − detM detM†
(cid:28) − (cid:29) (cid:28) (cid:29) (cid:28) (cid:29)(cid:28) (cid:29)
whichholdsforl (M+M† I)>0. Thestochasticfluctuations foroneconfiguration aregivenby
−
neglectingtheensembleaverage andvanishforNh ¥ . Moreoverbyintroducing amassinter-
hi →
polation betweenthestartandthetarget massthestochastic fluctuations canbefurther controlled,
i.e.ifnoDiracoperatorhasazeroeigenvalueduringtheinterpolationtheconditionl (M+M†)>1
canbeinsured[5]. InthiscasethenumberofinversionsisNinv(cid:181) N Nh ,whereN isthenumberof
·
interpolation steps. Note for many reweighting cases it is effecient to use the even-odd precondi-
tionedWilson–Diracoperator (e.g.see[6]),howeverwedonotfindanimprovementinthecaseof
isospin reweighting. The ensemble fluctuations can be tamed by including additional quarks into
the reweighting process, e.g. in the case of isospin reweighting the fluctuations are minimized by
keeping theaveragequarkmassm =0.5(m +m )constantduring thereweighting.
ud u d
Here,wewilldiscussmassreweightingbyintroducinganisospinbreakinginthelightquarks.
We will show the scaling of the different fluctuations (ensemble and stochastic) and how the up–
anddown–quark masscanbeextracted fromtheanalyzed ensembles(seetab.1).
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IsospinEffectsbyMassReweighting JacobFinkenrath
E4
V] 104 D5
* E5
4 R
m A5
O7
D / [v F7
Nin102 G8
>
2
W
2/st
s<
100
0 0.05 0.1 0.15
m *r
R 0
Figure1: Therelativestochasticfluctuationsoftheisospinreweightingfactorisshownasafunctionofthe
renormalizedquarkmass. s s2t istheaverageofthestochasticvarianceestimatedusingNh =6.
2. IsospinReweighting
Theisospinreweighting factoreq.(1.5)canbeexpanded inD m2
ud
detD detD
W = mu md =1+D m2 Tr(D 2)+O(D m4 ) (2.1)
iso detD2 ud −mud ud
mud
by using detM = exp Tr(ln(M)) . The fluctuations s 2 in eq. (1.8) of the isospin reweighting
{ }
factor can be expanded in D m2 . It can be shown that the stochastic fluctuations s 2 decouple
ud st
fromtheensemblefluctuationss 2 withs 2=s 2+s 2 . Thestochasticfluctuationsintheisospin
ens st ens
reweighting casearegivenby
s 2(N ) D m4 1
st inv = ud Tr +O D m6 . (2.2)
W2 N 2 ud
(cid:28) (cid:29) inv * DmudD†mud + (cid:0) (cid:1)
(cid:16) (cid:17)
Theensemblefluctuations are
s 2 1
ens =D m4 var Tr +O D m6 (2.3)
W 2 ud D2 ud
h i (cid:18) (cid:20) mud(cid:21)(cid:19)
(cid:0) (cid:1)
with the variance var(O) = O2 O 2. Note this is true because the Dirac operator is g -
5
−h i
hermitian. Now, the cost can be derived by demanding that the stochastic fluctuations do not
(cid:10) (cid:11)
dominate theensemblefluctuations, e.g.thats 2(N )/s 2 ! 0.1.
st inv ens
∼
By using the analyzed ensembles, listed in table 1, it is possible to deduce numerically the
scaling behavior of the fluctuations in the quark mass and the volume. However in the case of
the stochastic fluctuations the trace of the Wilson Dirac operator is known in chiral perturbation
theory, e.g. asin [7], by Tr 1 (cid:181) S V. Thenumerical analysis isconsistent with this behavior.
h (DD†)2i m3R
Itfollowsforthestochastic fluctuations (seefig.1)
D m4V 1
s 2 k R (2.4)
st ≈ stNinvmrR′ r0r′
4
IsospinEffectsbyMassReweighting JacobFinkenrath
4
106 10
E4 E4
0.25V r ]0 DE55 0.75V r ]phys0103 DE55
4m * R104 AO57 4m * R102 AO57
F7 F7
2D 2D
W| G8 W| G8
/ [|ns102 / [|ns101
2e 2e
s s
0
0 0.05 0.1 0.15 100 0.05 0.1 0.15
m r m r
R 0 R 0
Figure2: Thescalingoftheensemblefluctuationsfordifferentvolumebehaviorsisshown. Theleftfigure
showsthequarkmassbehaviorforavolumebehaviorof√4V andtherightfigureforavolumebehaviorof
V3/4.
by using the scale r [8] to form dimensionless quantities. By fixing the volume behavior to V
0
we perform a fit (black, solid) to the quark mass behavior by using the ensembles E4 (green,
triangle), D5 (red, diamond), E5 (black, triangle), A5 (cyan, circle), O7 (magenta, diamond), F7
(blue, square) and G8(green, star). Foreach ensemble wecompute s 2 ands 2 for twovalues of
st ens
D m=D m /2,D m (seetab. 1). Thequarkmassbehavior isgivenbyr =2.63(5). Forthered
max max ′
lines the quark mass behavior and the volume behavior isfixed toV/m3 and only ensembles with
q
pion masses<340MeV areincluded. Thedatashowagoodagreement withtheexpectation from
chiralperturbation theoryforpionmasses<340MeV.
Theleadingtermoftheensemblefluctuationseq.(2.3)isproportionaltovar(TrD 2). Numer-
−
ically weobserveaweakvolumedependenceVq withq<1. Similartothestochastic fluctuations
theensemblefluctuations canbewrittenas
D m4Vq 1
s 2 k R . (2.5)
ens≈ ens mrR r0r−4−q
In general the simultaneous deduction of the volume and the quark mass behavior is difficult.
A varied volume behavior changes simultaneously the mass behavior. A good fit is given for a
volume scaling of q=0.25 (see left figure 2) which gives amassbehavior of r=3.85(13) for all
ensembles (black line) and r=3.94(31) for ensembles with pion masses <340MeV (red dashed
line). Aweakervolumebehavior isalsosupported bycomparisonofD5andE5ensembles, which
givesq 0.46. Howeverbyassumingasimilarquarkmassbehaviorasinthecaseforthestochastic
∼
fluctuations with r=3 the scaling in the volume is roughly given by q 0.75. In the right figure
∼
2 we fixed the volume behavior to q = 0.75 which gives a mass behavior of r = 2.83(13) by
including every ensemble (black line) and r=3.04(31) byincluding ensembles withpion masses
smaller than < 340MeV (red, dashed line). We conclude that the volume behavior is given by
q 0.25...0.75byasimultaneous variation ofthequarkmassbehavior fromr 4...3.
≃ ≃
Thecostofisospinreweighting canbeestimatedfromtheratio
s 2(N )/s 2 ks′t (Lmp )2L with ks′t =1e 3 (2.6)
st inv ens ∼ k N r k −
e′ns inv 0 e′ns
·
5
IsospinEffectsbyMassReweighting JacobFinkenrath
3
30
]
] V
V e
e M
M 20 [
[d 0.076 fm ud2.5 0.076 fm
R,u10 0.066 fm mR, 0.066 fm
m
0.049 fm 0.049 fm
D
0 2
0 2 4 6 8 0 5 10
R3 R3
Figure3:Thefiguresshowtheaveragequarkmassm (left)andthemasssplittingm m plottedversus
ud d u
−
R afterfixingR andR . ThemassesarerenormalizedintheMS–schemeat2GeV.
3 1 2
forq=0.25andr=4. FortheG8ensemblefollowsN 200foraratioof0.1.
inv
≈
3. QuarkMasses
Thecontinuum limitcanbeperformed onalineofconstant physics. Thisline canbedefined
bykeepingdimensionlessratiosofphysicalquantitiesconstant. Thesefixthebaremassparameters,
here,aquenchedstrangequarkwithm ,theisospinmasssplittingD m andtheaveragelightquark
s ud
massm . Wetaketheratios
ud
0.5(m2 +m2 ) m2 m2 m2
R1= (0.5(fK0+ f K±))2 , R2= (0.5(Kf0−+ fK±))2 and R3= (0.5(f p+± f ))2 (3.1)
K0 K K0 K K0 K
± ± ±
withthemesonmasses,thepionmp ,theneutralkaonmK0 andthechargedkaonmK andthekaon
± ±
decay constants f and f . The physical values of the ratios are taken from [9] and we assume
K0 K
±
0.5(f + f )=155MeV. Now, the strategy is to use R to fix m , which is done in [4] and R
K0 K 1 s 2
±
to fix the isospin splitting D m . Afterwards the ratio R is used to extrapolate the light quarks
ud 3
towardsthephysicallimit.
We measure the PCAC mass on the analyzed ensembles (see tab. 1) and convert them into
theMS-renormalization scheme. Thedimensionless ratios R andR aregiveninthelowestorder
2 3
chiral perturbation theory up to O(D m2 ,m2 )by R = B D m (1+Cm ) and R = 2Bm with
ud ud 2 F2 ud ud 3 F2 ud
constants B,C and F2. Now, itis possible toperform extrapolations towards the physical point in
thelightquarkmasses. Fortheaverage lightquarkmassthisisshownintheleftfigureoffig.3by
assuming m (R ) a R . Byusing the F7 and G8ensemble the average light quark mass at the
ud 3 1 3
≈
physical point atfinitelattice spacing ofa=0.066fmisgivenbym =3.19(11)MeV. Forthe
ud,R
masssplittinginthelightquarks(seerightplotinfigure3)weassumedD m (R ) b +b R . By
ud 3 0 1 3
≈
usingthedataoftheE5,F7andG8ensembleitfollowsforthemasssplittingD m =2.49(10)MeV
ud
atfinitelatticespacing a=0.066fm.
Theisospineffectsentertheobservablebytheisospinreweightingfactorwhichscalespropor-
tional toD m2 . InthecaseofthePCACmassthestatistical erroristoobigcompared totheeffect
ud
of the isospin reweighting correction. A determination of this effect is only possible for larger
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IsospinEffectsbyMassReweighting JacobFinkenrath
statistics. Neclecting theseaquarkeffectsproportional toD m2 ,bysettingtheisospinreweighting
ud
factor to one, the isospin quark mass splitting is given by D m = 2.52(10) MeV. However the
ud
isospinreweightingeffectsincreasesforsmallerquarkmassesandwewanttoreducethestatistical
errortofigureouttheisospineffectsforexampleinthepionmass. Inordertoperformacontinuum
limitthestatistics hastobeincreased andotherensembles havetobeincluded.
4. Conclusion
Isospinmassreweighting needsamoderatenumerical effort. Theanalysis showsthatthecost
scales with (LM )2L for a volume scaling of the ensemble fluctuations with √4V and is around
PS
200 inversions of the Dirac operator for the G8 ensemble which has a pion mass of 192 MeV at
a volume ofV/a4 =128 643. By using the introduced dimensionless ratios R , R and R it is
1 2 3
×
possible toextractthelightquark masses. Theisospin massspliting isD m =2.49(10)MeVand
ud
theaveragequarkmassism =3.19(11)MeVatfinitelatticespacingofa=0.066fm. Although
ud,R
amorecarefulanalysisisneededtoextractcompetitivenumbersitshowsthatthetuningconditions
are suitable to extract the light quark masses. In order to extract continuum physics the statistics
hastobeimprovedandQED-effectshavetobeincluded. Asoftwarepackageformassreweighting
[10](seealso[11])ispubliclyavailable intheframeworkoftheopenQCDcode[12].
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