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arXiv:0809.3251v1 [hep-lat] 18 Sep 2008

PoS(LATTICE2008)095

Physical results from 2+1 flavor Domain Wall QCD

RBC and UKQCD Collaborations

Enno E. Scholz∗

Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA

E-mail: scholzee@quark.phy.bnl.gov

We review recent results for the chiral behavior of meson masses and decay constants and the

determination of the light quark masses by the RBC and UKQCD collaborations. We find that

one-loop SU(2) chiral perturbation theory represents the behavior of our lattice data better than

one-loop SU(3) chiral perturbation theory in both the pion and kaon sectors.

The simulations have been performedusing the Iwasaki gauge action at two different lattice spac-

ings with the physical spatial volume held approximately fixed at (2.7fm)3. The Domain Wall

fermion formulation was used for the 2+1 dynamical quark flavors: two (mass degenerate) light

flavors with masses as light as roughly 1/5 the mass of the physical strange quark mass and one

heavier quark flavor at approximately the value of the physical strange quark mass.

On the ensembles generated with the coarser lattice spacing,

physicalaverageup-and down-quarkand

3.72(0.16)stat(0.33)ren(0.18)systMeV and mMS

spectively, while we find for the pion and kaon decay constants fπ= 124.1(3.6)stat(6.9)systMeV,

fK= 149.6(3.6)stat(6.3)systMeV. The analysis for the finer lattice spacing has not been fully

completed yet, but we already present some first (preliminary) results.

we obtain for the

masses

mMS

strange quark

ud(2GeV) =

s (2GeV) = 107.3(4.4)stat(9.7)ren(4.9)systMeV, re-

The XXVI International Symposium on Lattice Field Theory

July 14 - 19, 2008

Williamsburg, Virginia, USA

∗Speaker.

c ? Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence.

http://pos.sissa.it/

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Physical results from 2+1 flavor Domain Wall QCD

Enno E. Scholz

1. Introduction

Dueto computer and algorithmic constraints weare not able to simulate directly atthe physical

light quark mass. This necessitates performing a chiral extrapolation. There are various ways that

this extrapolation can bedone. Wefound that applying SU(2)partially quenched chiral perturbation

theory (PQChPT) is working more reliable at next-to-leading order (NLO) compared to SU(3)

PQChPT [1, 2]. The reason is that the strange quark mass is already too heavy to be described

by the NLO terms in SU(3) ChPT. To be able to also extract quantities from the kaon sector, we

introduced the SU(2) ChPT for kaon physics in [1, 2]. Recently other collaborations made similar

observations about the limitations of NLO-SU(3) ChPT and also successfully applied (kaon) SU(2)

ChPT in their analyses, e.g. [3].

We simulated QCD using Nf= 2+1 flavors of Domain Wall fermions. Currently the mass

of the heavy single flavor mhis kept fixed at a value close to the physical strange quark mass. We

generated ensembles at multiple values for the mass mlof the two (degenerate) light quark flavors.

Here we will focus on the extraction of the light quark masses, the pion and kaon decay

constants and the low energy constants (LECs) of the SU(2) chiral Lagrangian. For a discussion

of the treatment of the kaon bag parameter we refer to [2, 4] and [5] for recent developments.

The remainder is organized as follows: in Sec. 2 we briefly describe our method to extract the

physical results and estimate the systematic error and quote the results obtained at the ensembles

with a lattice cut-off 1/a = 1.73GeV. Before we conclude, we briefly present preliminary results

obtained at a finer lattice spacing in Sec. 3.

For any unexplained notation and further details, we refer to [2]; especially App. A therein

contains an overview of the conventions followed here as well.

2. Physical results at 1/a = 1.73GeV

To obtain physical results on the 243×64, Ls= 16 lattices (generated using the Iwasaki gauge

action at β =2.13), weonly used the ensembles withthe twolightest dynamical light quark masses,

ml= 0.005 and 0.01, which correspond to pion masses of 331 and 419 MeV, respectively. In

the subsequent analysis, partially quenched (valence) masses mx,y∈ {0.001,0.005,0.01,0.02,0.03,

0.04} have been used as well. The lattice scale 1/a = 1.729(28)GeV (a = 0.1141(18)fm), the

physical average light and strange quark masses are fixed by the masses of the Ω−-baryon, the pion,

and the kaon. In case of the Ω−-baryon this procedure includes an extrapolation in the dynamical

light quark mass to the physical average up- and down-quark mass and a (valence) interpolation

in the heavy dynamical mass to the point of the physical strange quark mass, cf. [2] for details.

The residual mass parameter, measuring the remaining breaking of the chiral symmetry, turned out

to be mres= 0.00315(2). In the following we will briefly describe our fit strategy and how the

extrapolations in the pion and kaon sectors were performed and how the systematic errors were

estimated.

2.1 PQChPT fits

As we already discussed extensively in [1, 2], fitting to SU(3) NLO PQChPT including the

physical strange quark mass is problematic. As shown for example in the left panel of Fig. 1, the

decay constant receives large NLO-contributions (around 60–70%) when extrapolated from pion

2

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Physical results from 2+1 flavor Domain Wall QCD

Enno E. Scholz

80

f0

100

f

120

140

160

0 1402

2502

3302

4202

fPS [MeV]

m2

PS [MeV2]

fπ

mll = 331 MeV

mll = 419 MeV

SU(2) fit

SU(3) fit

ml=

0.01

ml=

0.005

-0.01 -0.005 0 0.005 0.01

rel. error of fit [ (fit - value)/value ]

meson masses

meson decay constants

Figure 1: Right panel: Comparing the extrapolationto the SU(2) (dashed green curve) and SU(3) (dashed-

dotted blue curve) chiral limit for the degeneratepseudoscalardecay constant. Left panel: Relative deviation

of the SU(2) PQChPT fit from the data.

masses in the range of 331–419 MeV to the SU(3) chiral limit (f0). The decay constant in the

SU(2) chiral limit f (in which the strange quark mass is not sent to zero but kept fixed (close) to

its physical value) receives a much smaller (30–40%) NLO-contribution. Also we observed that

applying PQChPT to data with meson masses in the region of the physical kaon mass, does not

lead to reasonable fits if only terms up to NLO are considered. Therefore, we simultaneously fitted

our data for the meson masses and decay constants to SU(2) NLO PQChPT imposing a cut on the

average quark mass of mavg≤ 0.01 (corresponding to mPS≤ 420MeV), see Fig. 10 from [2]. From

the meson mass fit we are able to determine the value mud= (mu+md)/2 for the physical average

light quark mass. Finally, we extrapolated the meson decay constant to this point to predict fπ. We

are aware that our data is correlated within the two ensembles (correlations between the different

valence masses and between the meson masses and decay constants) but our statistics (for each

ensemble 45 jackknife blocks made from 2 measurements) was not sufficient to obtain a reliable

estimate of the (inverse) correlation matrix for the 2x6 data points per ensemble as needed in a

correlated fit. For that reason, we refrained from using a correlated fit. From the uncorrelated

(simultaneous) fit we obtained a χ2/d.o.f. of 0.3. As shown in the right panel of Fig. 1, the relative

deviation of the fit from the data is always less than 1%. Note, that we are not fitting to an exact

theory, ChPT is an expansion around zero quark masses and higher orders (which were omitted

here) are expected to account for those deviations.

The extrapolation to mudin the kaon sector was done using kaon SU(2) as presented in [1, 2]

and references therein. We did the extrapolation at two different (valence) masses for the heavy

quark, my= 0.03 and 0.04 and linearly interpolated between those. From the physical value of the

(quadratically averaged) kaon mass we obtain the strange quark mass msand then in turn the kaon

decay constant fKat that point. Example plots are shown in Figs. 11 and 12 of [2].

2.2 Systematic errors

We have to include estimates for the systematic errors due to the following four sources: finite

volume of the simulated lattice box, the absence of a continuum extrapolation, corrections from

higher orders in (PQ)ChPT, and the fact that our simulated heavy quark mass turned out to be

roughly 15% higher than the physical strange quark mass.

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Physical results from 2+1 flavor Domain Wall QCD

Enno E. Scholz

In case of the finite volume effects in our simulated (2.74fm)3box, we repeated the SU(2) fits

this time including finite volume correction terms (see App. C in [2] and references therein). We

assigned the difference between those fits and our original fits as the FV-systematic error. Plots of

the correction factor can be found in Fig. 13 of [2]. A comparison of our finite volume correction

factors for our meson masses and decay constants at the dynamical points with the resummed

method of [6] shows good agreement, see Tab. 1.

Since the analysis on the ensembles generated at a finer lattice spacing is not yet finished (for

preliminary results see Sec. 3) for the moment we estimate the effect from the missing continuum

extrapolation to be 4%, which corresponds to (aΛQCD)2.

The higher order effects in (PQ)ChPT are taken into account as the difference between our

original fits and fits using a larger cut-off in the average quark mass (mavg≤ 0.02). Here we had

to introduce analytic NNLO-terms to obtain a reasonable agreement between our data and the fits.

Also, since with only a limited set of dynamical quark masses we could not include all possible

analytic NNLO-terms, we conservatively doubled the difference to estimate the systematic error

due to higher order terms in (PQ)ChPT.

With only one value for the dynamical heavy quark mass, an exploration of the effects due

to shifting mhwas not possible. Therefore, we had to rely on the predictions of SU(3) ChPT to

estimate the size of the moderate (15%) shift from mhto ms. More details on the conversion from

SU(3) LECs to those of SU(2) and how to obtain the “ms?= mh” systematic error therefrom can be

found in [2].

The final results given in the following subsection contain the systematic errors discussed

above added in quadrature. Table XII of [2] gives a detailed breakdown of the total error into

the different sources. In case of quantities which have to be renormalized in a certain scheme, we

provide the renormalization error separately. (We usually quote results in the MS-scheme at 2 GeV,

using the Rome-Southampton RI-MOM method. See [7] and references therein.)

2.3 Final results

Including the (estimates of the) systematic errors discussed in the previous subsection, we

quote the following physical results from our SU(2) (PQ)ChPT analysis at 1/a = 1.73GeV:

fπ= 124.1(3.6)stat(6.9)systMeV,

fK= 149.6(3.6)stat(6.3)systMeV, fK/fπ= 1.205(0.018)stat(0.062)syst,

mMS

ud(2GeV) = 3.72(0.16)stat(0.33)ren(0.18)systMeV,

mMS

s (2GeV) = 107.3(4.4)stat(9.7)ren(4.9)systMeV,

˜ mud: ˜ ms= 1 : 28.8(0.4)stat(1.6)syst.

Furthermore, the SU(2) LECs were determined as

f = 114.8(4.1)stat(8.1)systMeV, BMS(2GeV) = 2.52(0.11)stat(0.23)ren(0.12)systGeV,

¯l3= 3.13(0.33)stat(0.24)syst, ¯l4= 4.43(0.14)stat(0.77)syst.

3. First results at larger cut-off

Currently, our collaborations are in the middle of finishing measurements on a second set of

ensembles, generated at a finer lattice spacing. We simulated three different light quark masses

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Physical results from 2+1 flavor Domain Wall QCD

Enno E. Scholz

Rm[%]−Rf[%]

mll[MeV]

331

419

SU(2)

0.09(.01)

0.03(.00)

CDH

0.13(.03)

0.04(.01)

SU(2)

0.36(.03)

0.10(.01)

CDH

0.32(.00)

0.09(.00)

243, V ≈ (2.74fm)3

323, V ≈ (2.60fm)3

307

364

419

0.16(.01)

0.07(.01)

0.04(.00)

0.26(.07)

0.12(.03)

0.06(.02)

0.62(.03)

0.28(.01)

0.14(.01)

0.64(.01)

0.28(.00)

0.13(.00)

Table 1: Finite volume correction factors obtained from our SU(2) PQChPT fits including FV-terms com-

pared to results interpolated from [6] (CDH).

ml= 0.004, 0.006, and 0.008 at a fixed heavy quark mass, mh= 0.03 on 323×64, Ls= 16 lat-

tices with the gauge coupling set to β = 2.25 (Iwasaki gauge action). A first estimate of the

lattice cut-off obtained from measuring the Sommer-parameter r0/a gives 1/a = 2.42(4)GeV

(a≈0.08fm), where r0=0.47fm has been assumed. The PQChPT fits will include valence masses

mx,y∈{0.002,0.004,0.006,0.008,0.025,0.03}. Using theabove lattice cut-off, our dynamical pion

masses are 307, 366, and 418 MeV, respectively, whereas the lightest valence pion mass reaches

236 MeV. The preliminary value for the residual mass parameter is 6.76(0.11)·10−4, i.e. almost

by a factor of 5 smaller than on the coarser lattices used in the previous analysis.

Since we have not reached a sufficiently high statistics on the three ensembles, we will refrain

from quoting any physical results from this analysis. The following subsections contain the pre-

liminary fits to SU(2) PQChPT and also (unquenched) ChPT, since here we have enough data to

even perform a fit just including dynamical data points.

3.1 PQChPT fits

In Fig. 2 we show simultaneous (uncorrelated) fits of the meson decay constants and masses

to NLO-PQChPT formulae, where a cut of mavg≤ 0.008 (mPS≤ 420MeV) in the average quark

mass has been applied. The obtained χ2/d.o.f. of 0.6 is reasonable, although for some points the

fit deviates as much as 1.0(0.7)% from the data. But since here the statistical uncertainty of 0.7

percent-points is rather large, we will have to wait for the higher statistics to see if these deviations

will disappear or remain.

Finite volume effects may also be of more importance in the analysis of the 323ensembles,

since (given the preliminary number for the lattice cut-off quoted above) the spatial volume V ≈

(2.6fm)3is slightly smaller compared to our 243ensembles. For the dynamical pion mass we

still have mllL ≈ 4.1–5.5, whereas for our lightest valence pion mass, we only have mxxL ≈ 3.1.

In Tab. 1 we give the finite volume correction factors for our dynamical points as obtained from

our SU(2) fits including finite volume terms and compare them to the results from the resummed

Lüscher formula of [6] and the results from the 243ensembles. The correction factors for our

lightest valence meson (mx= my= 0.002) are Rm= 0.96(.04)%, −Rf= 1.00(.04)% at ml= 0.004

and Rm= 2.00(.08)%, −Rf= 0.41(.02)% at ml= 0.008.

3.2 ChPT fit

Having three dynamical light quark masses which can be considered to be light enough to

be described by NLO SU(2) ChPT, a combined fit just including those dynamical points becomes

5