Table Of ContentContinuum Results for Light Hadronic Quantities
using Domain Wall Fermions with the Iwasaki and
2
1
0 DSDR Gauge Actions
2
n
a
J
3
]
t ChristopherKelly∗
a
l ColumbiaUniversity,
-
p 905PupinHall,
e
116thSt&Broadway,
h
NewYork,NY10027,USA.
[
E-mail: [email protected]
1
v Forthe RBCandUKQCDcollaborations
6
0
7 We present preliminary continuum results for light hadronic quantities obtained by the
0
RBC/UKQCD collaboration using domain wall fermions with both the Iwasaki and the novel
.
1
DislocationSuppressingDeterminantRatio(DSDR)gaugeactions. TheDSDRactionallowsus
0
2 tosimulateatnearphysicalquarkmassesonalarger,coarserlattice(a−1=1.4GeV,L=4.6fm)
1 whileretaininggoodchiralsymmetryproperties. We discussourongoingcombinedanalysisof
:
v thethreeensemblesetsandgiveearlyresultsforthepionandkaondecayconstants,quarkmasses
Xi andBK.
r
a
TheXXIXInternationalSymposiumonLatticeFieldTheory-Lattice2011
July10-16,2011
SquawValley,LakeTahoe,California
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
ContinuumResultsforLightHadronicQuantitiesusingDomainWallFermionswiththeIwasakiandDSDRGaugeActions
ChristopherKelly
Label Size SG b Nens mupni.(MeV) #configs. mPp Q≥(MeV)
32ID 323×64×32 Iwasaki+DSDR 1.75 2 170,250 181,148 140
24I 243×64×16 Iwasaki 2.13 2 330,420 202,178 240
32I 323×64×16 Iwasaki 2.25 3 290,350,400 300,312,252 220
Table1:Asummaryofthepropertiesofthethreeensemblesetsusedinthisanalysis.Here‘S ‘denotesthe
G
gaugeaction,‘Nens’thenumberofensembles,‘mupni.’theunitarypionmassoneachofthoseensembles,‘#
configs’thenumberofgaugeconfigurationsusedinthisanalysis,andmPp Q≥thelightestavailablepartially-
quenchedpionmassontheensembleset.
1. Introduction
In these proceedings we present continuum limit predictions for the neutral kaon mixing pa-
rameter BK, the strange and average up/down quark masses and the pion decay constant fp , ob-
tainedviasimultaneouschiral/continuum fitstothethreedomainwallfermionensemblesetsgiven
in table 1. Ourmost recently generated ensemble set, referred to bythe label ‘32ID’, uses amod-
ified Iwasaki gauge action that includes the Dislocation Supressing Determinant Ratio (DSDR)
term, which allows us to simulate with near-physical pion masses on a coarser b =1.75 lattice,
whileretaining goodchiralsymmetryandtopology tunneling.
Combinedanalysesofthe24Iand32Ilattices,whichwerefertocollectivelyasthe‘2010anal-
ysis’,haverecentlybeenpublished[5,6]. Inthesepaperswedevelopedastrategyforthecombined
analysis ofmultiple ensemble sets that maximises the use ofthe available data inconstraining the
fits. Inthese proceedings wedevelop thisstrategy further toinclude 32ID ensemble set. Notethat
thenumberofconfigurations available onthelightest32IDensemblesethasalmostdoubledsince
theconference, andtheheavierensemblehasalsoincreased insizeby35%,hencetheconclusions
presented heredifferslightly fromthosegivenattheconference.
The layout of these proceedings is as follows: We provide a short discussion on the DSDR
term,followedbyasummaryofourcombinedfittingstrategy. Wethenpresentresultsforthepion
decay constant before brieflytouching onthe non-perturbative renormalisation techniques used in
the calculations of B and the physical quark masses. Results for these quantities follow. We
K
conclude withabriefsummaryandoutlook forthisanalysis.
2. The DSDR Term
The2010analysiswasperformedtodataovertherangemp =290−420MeV.Heretheextrap-
olations downtothephysical pion massof∼135 MeVprovided thedominant contribution tothe
systematic errors on the continuum predictions. For example, the chiral extrapolation systematic
on fp was∼4%. Thisprovided astrongmotivationforreaching downtolighterquarkmasses.
In order to avoid finite-volume effects, simulations with lighter quark masses require lattices
with a larger physical volume. However, at our typical couplings the required increase in the
number of lattice sites is beyond the reach of our current computing resources; we are therefore
forcedtosimulatewithcoarserlattices. Thishastheunfortunateside-effectofincreasingthesizeof
thechiralsymmetrybreaking effectsinthedomainwallfermionformulation, duetothefollowing
mechanism.
The size of the chiral symmetry breaking is parameterised by an additive mass-shift known
as the ‘residual mass’ m . This quantity is governed by the eigenvalue density r of the four-
res
dimensional Hamiltonian H =2tanh−1 HW describing quark propagation through the fifth
T 2+DW
dimension. Here D is the Wilson Dirac(cid:16)operato(cid:17)r and H =g 5D isthe hermitian Wilson Dirac
W W W
operator. The equation above implies a relationship between the eigenmodes of H and H . The
T W
modesofthelatteraredividedintotworegionsbyamobilityedgel : thoseabovel areextended
c c
2
ContinuumResultsforLightHadronicQuantitiesusingDomainWallFermionswiththeIwasakiandDSDRGaugeActions
ChristopherKelly
Figure1: AnillustrationofthemoleculardynamicsforceimpartedbytheDSDRtermatvariousvaluesof
e ande ,reproducedfromref.[3]
f b
over the lattice whereas those below are localized - in fact this property is essential in ensuring
that the four-dimensional effective theory of the fields on the boundary is local. The structure of
eigenmodes ofH impliesthatm hasthefollowingdependence onL [7]:
W res s
e−lcLs 1
m ∼R4r (l ) +R4r (0) , (2.1)
res e c L l L
s s
wherethefirsttermcontains contributions fromtheextended modesandthesecondtermfromthe
localized near-zero modes.
In modern simulations, L is typically large enough that m is dominated by the near-zero
s res
mode contribution. These modes are associated with small tears or ‘dislocations’ in the gauge
field, which occur more often as we approach the disordered strong-coupling region. In order to
retain goodchiralsymmetryatstronger coupling wemusttherefore seektosuppress thesemodes.
However we must retain enough of the very-near-zero modes that allow topological tunneling to
occur. This can be achieved by introducing a weighting term, the DSDR term, into the gauge
action, givenby[1,2,3]
W(M;e ;e )= det DW(−M+iebg 5)†DW(−M+iebg 5) =(cid:213) l i2+e 2f , (2.2)
f b det[D (−M+ie g 5)†D (−M+ie g 5)] l 2+e 2
(cid:2) W f W f (cid:3) i i b
where l are eigenvalues of H and e and e are tunable parameters. This introduces a force in
i W f b
themolecular dynamicsevolution oftheform
d l 2+e 2
F(e ,e )= −log i f , (2.3)
i f b dl l 2+e 2
i i b!
which can betuned topeak in thenear-zero region without further suppressing thevery-near-zero
modes. In figure 1 wereproduce aplot from ref. [3] which shows the force as a function of l for
severalcombinations ofe ande onatestsimulation, demonstrating thissuppression.
f b
3. Simultaneous FittingProcedure
Weobtainourfitformsviaadualexpansiontonext-to-leading order(NLO)ina2andthelight
andheavyquarkmassesm andm ,adoptingapowercountingthatdiscardstermslikeO(a2m)and
l h
O(m2)andhigheratthisorder. Expandingaroundanon-zeromasspoint(m ,m )andabsorbing
l0 h0
constant termscontaining m intotheleadingcoefficient, weobtaintheanalyticfitfunction
l0
(m˜ +m˜ )
f =Cfp 1+C a2 +Cfp x y +Cfp m˜ +Cfp (m˜ −m ) (3.1)
ll 0 f 1 2 2 l 3 h h0
(cid:0) (cid:1)
3
ContinuumResultsforLightHadronicQuantitiesusingDomainWallFermionswiththeIwasakiandDSDRGaugeActions
ChristopherKelly
2010analysis Thisanalysis(alldata) Thisanalysis(mp ≤350MeV)
fp (MeV) 124(2)(5)(2) 125(2)(2)(2) 127(3)(0.5)(3)
mMS(2GeV)(MeV) 3.59(13)(12)(6)(8) 3.48(6)(7)(3)(8) 3.39(9)(4)(2)(7)
u/d
mMS(2GeV)(MeV) 96.2(1.5)(0.2)(0.1)(2.1) 94.9(1.2)(1.4)(0.2)(2.1) 94.2(1.9)(1.0)(0.4)(2.1)
s
Bˆ 0.749(7)(21)(3)(15) 0.748(6)(15)(4)(15) 0.751(11)(8)(4)(14)
K
Table2: Resultsfor fp ,theaverageup/downquarkmass,thestrangequarkmassandBK. Thefirstcolumn
containsthe 2010analysisresult[5, 6], thesecondtheresultobtainedin thisanalysisbyfitting tothe full
rangeofavailabledata,andthethirdbyfittingonlytodatawithmp ≤350MeV.Theerrorsarestatistical,
chiral,finite-volumeandNPRerrors(whereappropriate)respectively.
for the pion decay constant and similar forms for other quantities. Here m˜ =m+m . We also
res
consider expanding about the SU(2) chiral limit (with and without finite-volume corrections), in
which case we obtain the usual NLO ChPT fit forms with an extra a2 and (m −m ) term. We
h h0
refertotheseastheChPTandChPTFVfitforms. Usingthesethreefitansätze, wesimultaneously
fit the 32I, 24I and 32ID ensemble sets, with the coefficients shared between all three data sets.
Forensemble sets i other than the primary set, chosen as the 32I set, wemustinclude appropriate
factors of the ratio of lattice spacings R =a32I/ai and quark masses R =m˜32I/m˜i in the fit
a l/h l/h l/h
forms such that the coefficients C can be assumed equal. In this analysis we adopt the so-called
‘generic scaling’ approach[5]inwhichtheseratiosaredeterminedasfreeparametersinthefit.
Before taking the continuum limit, wemust fixthe primary lattice spacing a32I and thequark
masses m32I andm32I in32I normalisation. Thisisachieved byvarying these parameters until the
u/d s
continuum values of the pion, kaon and Omega baryon masses agree with their physical values.
These quantities define the ‘scaling trajectory’ along which the continuum limit is defined. Note
that this makes the a2 coefficients of these quantities zero by definition. As the above conditions
areapplied inthecontinuum limit,theprocedure isnecessarily iterative.
The inclusion of the Iwasaki+DSDR data complicates the situation slightly over the 2010
analysis, in that the coefficients of the a2 terms must now be allowed to differ between the two
differentgaugeactions. Althoughthisintroduces oneextradegreeoffreedomperquantityintothe
fits,anyalgorithmicinstabilitythatthismaycauseisoffsetbytheincreasednumberofdatapoints.
Recall that eqn. 3.1 contains a term in the heavy sea-quark mass m . Wevary this parameter
h
using reweighting [4,5], whereby the weight of agiven gauge configuration isre-evaluated inthe
path integral atseveral strange quark masses shifted by up to20% from the simulated value. This
allows us to explore the strange sea quark dependence with minimal cost and to ultimately quote
predictions atthephysical strangequarkmassatthepenalty ofanincreaseinstatistical error.
4. Preliminary Results For fp
fp is obtained from the h0|A0|p i matrix elements in the usual way, the only difference being
that for domain wall fermions one must correctly renormalise the 4d axial current to match the
continuum current. Following ref. [5], the renormalisation factor is obtained from the ratio of the
5dDWFconservedvector currenttothe4dvectorcurrent: thiswasshowntobemoreprecisethan
the ratio of axial currents due to the unknown renormalisation coefficient between the 5d DWF
PCACcurrentandthecontinuum current.
Figure2showsthechiralextrapolation of fp downtothephysical up/downquarkmassusing
theanalytic andChPTFVansätze. Followingthe2010analysis weestimate theerroronthechiral
extrapolationthroughthedifferenceoftheChPTFVandanalyticpredictions,andthefinite-volume
error from the difference of the ChPTFV and ChPT predictions. Weobtain the value given in the
second column of table 2. Thisresult issome 4% (1.5s ) below the physical value of 130.7 MeV.
Asimilardiscrepancy wasnotedinthe2010analysis(firstcolumnofthetable),andwasattributed
4
ContinuumResultsforLightHadronicQuantitiesusingDomainWallFermionswiththeIwasakiandDSDRGaugeActions
ChristopherKelly
Figure2:Comparisonsoftheunitary fp datacorrectedtothecontinuumlimitandfitusingtheChPTFVand
analyticansätze,wherethefitsareperformedtothefulldataset(left)andonlytothedatawithmp ≤350
MeV(right). Thedataincludedinthefitaremarkedwithcircles,andthoseexcludedwithdiamonds. The
continuumlimitsaremarkedwithsquares.Pointsinpastelcoloursarecorrectedtothecontinuumusingthe
ChPTFVansatz,andinboldcoloursbytheanalyticansatz.
to the systematic error on the chiral extrapolation. There, the strategy of estimating the chiral
errorfromthedifferenceoftheChPTFVandanalyticresultsproducedasystematicerrorsufficient
to explain the discrepancy, but for this analysis it appears to be an underestimate. However, the
introduction of the light DSDR data allows us to perform stable fits even after removing some
of the heavier data from the 32I and 24I ensemble sets. We can therefore restrict our fits to a
region of lighter mass in which we would expect the fit ansätze to perform better. Cutting the
heaviest twoensembles (cf. table 1)suchthattheheaviest pionhasamassof350MeV,weobtain
the result given in the third column of table 2. Here, as a result of the cut, the central value has
increased suchthattheresultisnowconsistent withthephysicalvalue,evenwithoutincorporating
thechiralsystematic. Noticealsothatouradhocchiralsystematichasdecreasedsubstantially, and
the c 2/d.o.f. alsodecreased, whichsuggests thatthe fitsdoindeed perform better whenrestricted
tothislightermassrange. Although thisanalysisisstillpreliminary, theseresultsarepromising.
5. RenormalisationforB andthe quark masses
K
Before giving results for the quark masses and B , we present a short summary of the non-
K
perturbative renormalisation techniques thatareusedtoconvert thesequantities intothecanonical
MSschemewithunprecendented precision.
Direct conversion into the MS scheme on the lattice is not possible as this scheme is regu-
larised in non-integer dimensions. Instead we first convert to a convenient intermediate scheme,
which is then run to high energy and matched to MS using perturbation theory. We use variants
of the Rome-Southampton Regularisation-Invariant Momentum (RI-MOM) scheme, in which the
renormalisation coefficientsaredefinedfromtheratiooftheamputatedGreen’sfunction ofanop-
erator toitstree-level value intheLandau gaugeataparticular momentum scale. IntheRI-MOM
scheme, the Green’s functions are formed using massless propagators of equal momentum p, and
thescaleisdefinedasm 2= p2. Howeverthisisaso-called‘exceptional’ momentumconfiguration
in whichthe hardexternal momenta canbe routed insuch awaythat parts ofthediagram contain
only soft momenta -this enhances the effect of the spontaneous chiral symmetry breaking at high
momenta. In order to avoid this problem we follow the 2010 analysis in adopting ‘symmetric’
momentum conditions (SMOM), for which the external momenta are not equal but rather obey
m 2 = p2 = p2 =(p −p )2. Note that this applies both for the mass renormalisation, which we
1 1 2
obtain from the scalar bilinear vertex, and the four-pointVV +AAweak operator contained inthe
definition of B ; for the latter the Wick contractions comprise traces of two bilinear vertices -the
K
momenta p and p areassigned tothetwopropagators formingeachofthosebilinears.
1 2
5
ContinuumResultsforLightHadronicQuantitiesusingDomainWallFermionswiththeIwasakiandDSDRGaugeActions
ChristopherKelly
Asinthe 2010 analysis, weuse volume source propagators inthese calculations, which offer
a significant improvement over the traditional point sources, giving statistical errors of the order
0.1%evenwithonlyO(20)gaugeconfigurations.
One of the dominant systematic errors on the NPR in our earlier analyses arose through the
use ofmomenta whose unit vectors were not equivalent under hypercubic rotations, and hence do
not have the same discretisation errors; this induces a scatter in the RI-SMOM renormalisation
coefficients as a function of momentum. In ref. [6] we described how this can be corrected using
twistedboundary conditions inthevalence sector, whichallowustosmoothly varythemagnitude
ofthemomentumwhilekeepingthedirectionfixed. Wealsodiscussedhowanotherofthedominant
systematic errors, that associated with the truncation of the perturbative series, can be reduced by
afactoroftwobyperformingthematchingtotheMSschemeat3GeVratherthanthecanonical 2
GeV. For this analysis these improvements have been applied in the case of B but have yet to be
K
applied tothequarkmassrenormalisation.
6. Results fortheQuark Masses
The continuum physical quark masses are obtained in the normalisation of the 32I ensemble
set, and hence must be renormalised into the MS scheme to remove any cutoff dependence. The
renormalisation coefficientsaredeterminedbyperformingthecontinuumextrapolation ofZ /R
m l/h
overthetwoIwasakilattices,whereR arethequarkmassratiosdefinedinsection3andR32I≡1
l/h l/h
by definition. Notice that the coefficient on the 32ID lattice is not needed for this procedure. As
mentioned above, the MS(3 GeV) lattice coefficients with twisted boundary conditions have yet
to be determined, hence for this analysis we reuse those given in ref. [5], although the continuum
extrapolation is performed anew with the lattice spacings and R obtained here. Following the
l/h
2010analysis procedure, wechooseourbestNPRschemeforthecentralvaluetakethetruncation
errorontherenormalisation fromthesizeofthetwo-loopcontribution totheMSmatching.
From a fit to the full data set, we obtain the values given in the second column of table 2.
Forcomparison, wegivethe 2010 analysis result inthe firstcolumn. Wefindresults that are very
consistent, and observe a factor of two reduction in the statistical and chiral systematic errors on
the up/down quark mass over the 2010 analysis as a result of including the lighter data. We also
see a reduction in the statistical error on the strange quark mass, but also a large increase in the
chiral error. This is likely a result of allowing the mass ratios Z and Z to differ between the fit
l h
ansätze, wherebefore theywerefixedtovalues obtained bymatching thelattices atanunphysical
massscale(thefixedtrajectory method). Weintendtoinvestigate thisfurther.
Wealso investigate the effect of cutting out the heaviest twoensembles on the quark masses.
We obtain the result given in the third column of the table. Here as with fp , we see significant
improvements intheestimatedchiralerror,attheexpense ofanincreaseinstatistical error.
7. Results forB
K
We obtain results for B through an independent simultaneous fit over our three ensemble
K
sets, using the lattice spacings, quark masses, etc., obtained from the fits above. In particular, we
constrain the ChPT fits by including the lowest-order ChPT parameters f and B from the main
analysis. The fits are performed to MS-renormalised data, where we use the coefficients obtained
bymatchingtoourlatticeschemeat3GeVasdiscussedabove,afterwhichweconverttotheRGI-
scheme for the convenience of the reader. The NPR error is obtained by taking the difference of
ourtwobestschemes,asdiscussedinref.[6]. Fromthefitstothefulldataset,weobtaintheresult
giveninthesecondcolumnoftable2. Thisresultisveryconsistentwiththe2010result,andshows
a30%improvementinthechiralerror. Fittingonlytothedatawithmp ≤350MeV,weobtainthe
result given in the third column of the table. As before wesee a substantial decrease in the chiral
6
ContinuumResultsforLightHadronicQuantitiesusingDomainWallFermionswiththeIwasakiandDSDRGaugeActions
ChristopherKelly
Figure3: ComparisonsoftheunitaryBK dataintheSMOM(/q,/q)schemecorrectedtothecontinuumlimit
and fit using the ChPTFV and analytic ansätze, where the fits are performedto the full data set (left) and
onlytothedatawithmp ≤350MeV(right). Thesymbolsandtheircoloursaredescribedinthecaptionof
figure2.
error,butheretheincreaseinthestatisticalerrorislargerthanbefore. Thisislikelyduetothelack
ofstatistical resolution onthelightest32IDdatapoints.
8. Conclusions andOutlook
Using the Iwasaki+DSDR gauge action, we have been able to simulate with near physical
pions while retaining good topological tunneling properties and small finite-volume corrections.
Including these data in simultaneous fits with our Iwasaki lattices, we have been able to substan-
tially improve our continuum predictions over the 2010 analysis, especially after cutting out the
heaviest twoensembles suchthattheheaviest pionnowhasamassofonly350MeV-performing
this cut we observed factor of two reductions in our estimated chiral systematic. In addition, our
preliminary prediction for fp ,127(4) MeV,isnowconsistent withthephysical value. Weobserve
that NLOchiral perturbation theory extrapolations overthe mass range of140−350 MeV should
be expected to fail at the 5% level, hence weexpect this value to rise further towards the physical
valueaswefurtherrestrictthefitrangeinfutureanalyses.
Itisourintentiontopublishananalysisofthesedatashortly,afterwhich,alongsidecontinuing
togenerate moredataonthe32ID ensemble set, weintend tocommencethegeneration offurther
domainwallfermionensembleswithnearphysicalpions,takingadvantageofthelargeincreasein
computing power provided by our upcoming IBM Blue Gene/Q resources. This will allow us to
further refineourcontinuum predictions.
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