Determination of SU(2) ChPT LECs from 2+1 flavor staggered lattice simulations

0.18 MB·English

by  Enno E. Scholz

PoS(Confinement X)111
Determination of SU(2) ChPT LECs
3
1 from 2+1 flavor staggered lattice simulations
0
2

n
a
J

1
3 EnnoE.Scholz∗a,SzabolcsBorsányib, StephanDürrb,c,ZoltánFodorb,c,d,

] Stefan Kriegb,c, AndreasSchäfera,andKalmanK.Szabob
t
a aUniversitätRegensburg,Universitätsstr.31,D-93053Regensburg,Germany
l
- bBergischeUniversitätWuppertal,Gaußstr.20,D-42119Wuppertal,Germany
p
e cJülichSupercomputingCentre,ForschungszentrumJülich,D-52425Jülich,Germany
h dInstituteforTheoreticalPhysics,EötvösUniversity,H-1117Budapest,Hungary
[
E-mail:enno.scholz(AT)physik.uni-regensburg.de,

1 durr(AT)itp.unibe.ch
v
7
5 By fitting pion masses anddecay constantsfrom2+1 flavorstaggeredlattice simulationsto the
5
7 predictions of NLO and NNLO SU(2) chiral perturbation theory we determine the low-energy
1. constants ℓ¯3 and ℓ¯4. The lattice ensembles were generated by the Wuppertal-Budapest collab-
0 oration and cover pion masses in the range of 135 to 435 MeV and lattice scales between 0.7
3
1 and2.0GeV.Bychoosingasuitablescalingtrajectory,wewereabletodemonstratethatprecise
: andstableresultsfortheLECscanbeobtainedfromcontinuumChPTtoNLO.Thepionmasses
v
i availableinthisworkalsoallowustostudytheapplicabilityofusingChPTtoextrapolatefrom
X
highermassestothephysicalpionmass.
r
a
XthQuarkConfinementandtheHadronSpectrum,
October8-12,2012
TUMCampusGarching,Munich,Germany
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
DeterminationofSU(2)ChPTLECsfrom2+1flavorstaggeredlatticesimulations EnnoE.Scholz
1. Introduction
Chiral perturbation theory (ChPT) [1, 2] is a widely used tool in many phenomenological
applications and also helpful to guide an extrapolation to lighter quark masses in lattice-QCD
simulations. Here we will report on a determination of the NLO low-energy constants (LECs) ℓ¯
3
and ℓ¯ which appear in the light quark mass dependence of the pseudo-scalar meson masses and
4
decayconstants inSU(2)ChPT.
WeanalyzeconfigurationsgeneratedbytheWuppertal-BudapestCollaboration[3,4,5,6,7,8]
usingtheSymanzikglueand2-foldstout-smearedstaggered fermionactionfora2+1flavorQCD-
simulation. The mass of the single flavor has been kept at the value of the physical strange quark
mass, whereas the two degenerate lighter quark masses have been varied such that light meson
masses in the range of 135 to 435 MeV were simulated. The simulations were performed at six
different gauge couplings b , resulting in lattice scales between 0.7 and 2.0 GeV (see next section
fordetailsonhowthescalehasbeendetermined). Figure1showsalandscapeplotofoursimulated
pionmassessquaredversusthelatticespacing. Detailsaboutthesimulatedgaugecouplings,lattice
volumes, andtuningoftheinputquarkmassvaluesarereported inourpublication [9].
The2-fold stout-smeared version of the staggered quark action has been proven to be advan-
tageous[7]inreducingtheinevitabletaste-breaking ofstaggeredfermionformulations. Therefore,
in this workweonly consider thepseudo-scalar mesons withtaste matrix g when measuring me-
5
son masses or decay constants. Again, for details of the computation of these quantities we refer
thereaderto[9].
2. Scalesetting andphysicalquark masses
To set the scale at each simulated gauge coupling b and identify the physical point, i.e. the
average up/down quark mass mphys = (m +m )/2 corresponding to a pion in the isospin limit
u d
phys
with an estimated mass of Mp =134.8MeV [10], we use a two-step procedure. First, we ex-
trapolate the ratio (aMp )2/(afp )2 of the measured squared meson masses and decay constants
to its physical value (134.8MeV/130.41MeV)2 = 1.06846, where we also used the PDG-value
b =3.85
b =3.792
b =3.75
b =3.67
b =3.55
4352 b =3.45
Figure 1: Simulated pion masses squared
3902 Mp2 vs. lattice spacing a at six different
2eV] gauge couplings b . Horizontal dashed
M
2M [p3402 linesindicatecutsonthemassrangeinour
ChPT fits. The lower left corner marked
2752
2502 bybluesolidlinesis ourfinalpreferredfit
2302
range.Seetextfordetails.
1952
11365022
0
0 0.05 0.1 0.15 0.2 0.25 0.3
a [fm]
2
DeterminationofSU(2)ChPTLECsfrom2+1flavorstaggeredlatticesimulations EnnoE.Scholz
8.0
b = 3.85 b = 3.85
quadratic fit 0.080 quadratic fit
7.0 rational fit rational fit
6.0
0.075
2) 5.0
afp
2) / ( 4.0 afp 0.070
Mp
a 3.0
(
2.0
0.065
(Mpphys/fpphys)2
1.0 amphys
0.0 0.060
0.000 0.005 0.010 0.015 0.000 0.005 0.010 0.015
am am
Figure2:Leftpanel:ratio(aMp )2/(afp )2extrapolatedto(Mpphys/fpphys)2=1.06846toobtainamphys,right
panel: afp extrapolated to amphys to obtain 1/a; both at b =3.85. Points marked by black symbols are
includedinthefit,whilethosemarkedbygreensymbolsareexcluded.
fpphys =130.41MeV [11]. In that way amphys is obtained. In the second step, we extrapolate afp
phys
to this quark massvalue and obtain thelattice scale withthe help ofthe PDG-value fp . Forthe
extrapolation we used two different ansätze: a quadratic and a rational (linear in numerator and
denominator) fitform. Anexampleoftheseextrapolations is shownfortheensemblesatb =3.85
in Fig. 2. There, only the five lightest points were used in the fits, which is a typical choice for
all other ensembles. We stress that here, like in the chiral fits to be discussed below, the data has
beencorrected forfinitevolumeeffectsbeforehand, bymeansofusingthetwo-andthree-loop re-
summedformulaeof[12]forthepiondecayconstantsandmasses,respectively. Ourspatiallattice
volumesL3 areintherange (4.3fm)3 –(6.8fm)3 withaminimalMp L≈3.3. Thisensuresthatwe
only observe small finite volume corrections. In case of the pion mass the correction factors vary
between0.1and2.7per-milleandincaseofthedecayconstantbetween0.2and7.5per-mille.
Byfixing1/a andamphys in the waydescribed above, the meson masses and decay constants
show no discretization effects at all directly at the physical point and wecan assume those effects
to be small(since of higher orderin the quark masses and/or lattice spacing) inthe vicinity ofthe
physical point,i.e.inthemassrangecoveredbyourfits. Suchdiscretization effects, ofcourse, are
present inotherobservables, whicharenotconsidered inthiswork.
3. Fits toNLO SU(2) ChPT
The quark mass dependence of the finite-volume corrected data for the meson masses and
decayconstantsisfittedsimultaneouslyatdifferentb -valuesusingtheNLO-SU(2)ChPTformulae
1 2 c c
Mp2 = (cid:18)a(cid:19) (aMp )2 = c (cid:20)1+ 16p 2f2logL 2(cid:21),
3
1 c c am
fp = (cid:18)a(cid:19)(afp ) = f(cid:20)1− 8p 2f2logL 2(cid:21), c = 2Bm = (2Bmphys)amphys ,
4
where we made use of the already determined 1/a and amphys to scale the quark masses and
the meson masses and decay constants measured in lattice units. This fit has four free param-
3
DeterminationofSU(2)ChPTLECsfrom2+1flavorstaggeredlatticesimulations EnnoE.Scholz
165 186
b =3.45 182
b =3.55
160 b =3.67 184
b =3.75
155 bb ==33..78952 2V] 182
e
150 M 180
20 180
MeV] 145 hys) [1 178 0.8 1.0 1.2
[ 140 132 pm 176
fp a
m/
135 131 a 174
130 130 2 / (Mp 172 bb ==33..4555
b =3.67
125 129 170 bb ==33..77592
0.8 1.0 1.2 b =3.85
120 168
0 2 4 6 8 10 12 0 2 4 6 8 10 12
am/amphys am/amphys
Figure 3: CombinedglobalunconstrainedNLO fit of the decay constants (leftpanel)andsquaredmeson
massesdividedbythequarkmassratio(rightpanel)toensembleswith1/a≥1.6GeVand135MeV≤Mp ≤
240MeV. Pointsincludedinthefitrangearemarkedbyblacksymbols,pointsexcludedbygreensymbols.
eters: two NLO low-energy scales L 3, L 4 (related to the LECs ℓ¯i = log[L 2i/(Mpphys)2]), the de-
cay constant intheSU(2)chiral limit f and therenormalization scheme-independent combination
c phys = (2Bmphys)oftheLOlow-energy constant Bandthe physical quark mass mphys. Sincewe
used the physical pion mass and decay constant to set the scale at each set of ensembles with a
common b ,and furthermore each set contains atleast onedata point in close vicinity tothe phys-
ical point, our ansatz should reproduce the physical point. Therefore, we also used a parameter-
reduced chiral fit, where this constraint has been implemented and only two free fit parameters
remain, whichwechose tobe c phys and f. Ourexactimplementation oftheparameter-reduced fit
formulaeisreportedin[9].
Wewouldliketopointoutthatthechiralfitformulaedonotincludeanytastebreakingeffects,
i.e., we did not use staggered ChPT. This seems justified to us, since we are only considering g -
5
taste mesons as mentioned above and use these to define our scaling trajectory at the physical
point. Inotherwords, sincethemesonmassanddecay constant atthephysical point wereusedto
setthequarkmassesandlatticescales,nodiscretization ortastebreakingeffectsarepresent inthe
ChPTformulaeforMp2 and fp asdiscussed above. Furthermore, tastebreaking effectsarereduced
anywaybythechoiceofthefermionactionasmentioned above.
Forthechiralfits,wefirstappliedseveraldifferentcutsfortheheaviestmassincludedinthefit
range. Thesecutsareindicatedbythehorizontaldashedlinesinthelandscapeplotofoursimulated
points, Fig.1. Inanextstep,westudiedtheeffectsofexcluding oneormorelatticespacingsfrom
the fitted data. In the end, judging by the resulting (uncorrelated) c 2/d.o.f of the fitand reaching
a plateau for the fitted parameters, we chose 135MeV ≤ Mp ≤ 240MeV and 1/a ≥ 1.6GeV to
be our preferred fit range. This choice is indicated by the solid blue lines in Fig. 1. The resulting
combined global fit to the unconstrained fitformulae is shown in Fig. 3. In Figure 4 we show the
impactofthedifferentmasscutsonthefittedparametersandtheresulting(uncorrelated) c 2/d.o.f.
using only ensembles with 1/a ≥ 1.6GeV. Forthe systematic uncertainty wedecided to take the
4
DeterminationofSU(2)ChPTLECsfrom2+1flavorstaggeredlatticesimulations EnnoE.Scholz
Mp range [MeV] c phys [GeV2] 1/a > 1.6 GeV Mp range [MeV] f [MeV] 1/a > 1.6 GeV
[ 135 : 160 ] [ 135 : 160 ]
[ 135 : 195 ] [ 135 : 195 ]
[ 135 : 240 ] [ 135 : 240 ]
[ 135 : 275 ] [ 135 : 275 ]
[ 135 : 340 ] [ 135 : 340 ]
[ 135 : 390 ] [ 135 : 390 ]
[ 135 : 435 ] [ 135 : 435 ]
0.0182 0.0184 0.0186 0.0188 120 122 124
Mp range [MeV] L 3 [MeV] 1/a > 1.6 GeV Mp range [MeV] L 4 [MeV] 1/a > 1.6 GeV
[ 135 : 160 ] [ 135 : 160 ]
[ 135 : 195 ] [ 135 : 195 ]
[ 135 : 240 ] [ 135 : 240 ]
[ 135 : 275 ] [ 135 : 275 ]
[ 135 : 340 ] [ 135 : 340 ]
[ 135 : 390 ] [ 135 : 390 ]
[ 135 : 435 ] [ 135 : 435 ]
400450500550600650700750800 900 1000 1100 1200
Mp range [MeV] c 2/d.o.f. 1/a > 1.6 GeV Figure4: Dependenceofthefittedparametersand(uncor-
[ 135 : 160 ] related) c 2/d.o.f.on the mass rangeof fits onlyincluding
[ 135 : 195 ]
[ 135 : 240 ] ensembles with 1/a≥ 1.6GeV. The solid, dashed, and
[ 135 : 275 ]
[ 135 : 340 ] dashed-dottedbluelinesshowourcentralvalue,statistical
[ 135 : 390 ]
[ 135 : 435 ] and statistical plussystematic errorbands, respectively, as
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 determinedfromthesefits. Seetextfordetails.
variation of all these points with respect to the central value from our preferred fit. The central
value and statistical uncertainty from our preferred fit is indicated in each panel by a solid and
dashed line, respectively, whereas the total (statistical plus systematic) uncertainty is indicated by
a dashed-dotted line. Details on the impact of removing one or more lattice spacing from the fits
canbefoundin[9].
Werepeatedthesameanalysiswiththeparameter-reduced SU(2)ChPT-formulae,whichhave
beenconstrainedtohitthephysicalpoint. InTable1wesummarizetheobtainedfitparametersfrom
each ansatz. Note that for the parameter-reduced fit only the first two lines are fitted parameters
while the two low-energy scales L , L are derived from these. The table also contains our final
3 4
estimate fortheNLOSU(2)LECs,whichwereobtained bycombining theresults fromthetwofit
ansätze in the following way: we averaged the central values and the statistical uncertainties. For
the square ofthe systematic uncertainty wesum the squares ofthe average systematic uncertainty
andthespreadofthecentralvalues.
unconstrained parameter-reduced final
c phys/(10−2GeV2) 1.8578(17)(39) 1.8639(18)(44) 1.8609(18)(74)
f/MeV 122.70(08)(41) 122.73(06)(28) 122.72(07)(35)
L /MeV 628(23)(57) 678(40)(119) 653(32)(101)
3
L /MeV 1,012(16)(83) 1,006(15)(71) 1,009(16)(77)
4
Table1: Resultswithstatisticalandsystematicuncertaintiesforfittedparametersfromunconstrained(first
column)andparameter-reduced(secondcolumn)NLOSU(2)ChPTfits. Notethatintheparameter-reduced
caseonlythefirsttwoparametersareactualfreefitparameters,whiletheothertwoarederivedtherefrom.
Thethirdcolumnshowsourcombinedfinalresults,seetextfordetails.
5
DeterminationofSU(2)ChPTLECsfrom2+1flavorstaggeredlatticesimulations EnnoE.Scholz
Mp r[a 1n3g5e :[ M16e0V ]] -l3 1/a > 1.6 GeV Mp r[a 1n3g5e :[ M16e0V ]] -l4 1/a > 1.6 GeV
[ 135 : 195 ] [ 135 : 195 ]
[ 135 : 240 ] [ 135 : 240 ]
[ 135 : 275 ] [ 135 : 275 ]
[ 135 : 340 ] [ 135 : 340 ]
[ 135 : 390 ] [ 135 : 390 ]
[ 135 : 435 ] [ 135 : 435 ]
[ 160 : 195 ] [ 160 : 195 ]
[ 160 : 240 ] [ 160 : 240 ]
[ 160 : 275 ] [ 160 : 275 ]
[ 160 : 340 ] [ 160 : 340 ]
[ 160 : 390 ] [ 160 : 390 ]
[ 160 : 435 ] [ 160 : 435 ]
[ 195 : 240 ] [ 195 : 240 ]
[ 195 : 275 ] [ 195 : 275 ]
[ 195 : 340 ] [ 195 : 340 ]
[ 195 : 390 ] [ 195 : 390 ]
[ 195 : 435 ] [ 195 : 435 ]
[ 230 : 275 ] [ 230 : 275 ]
[ 230 : 340 ] [ 230 : 340 ]
[ 230 : 390 ] [ 230 : 390 ]
[ 230 : 435 ] [ 230 : 435 ]
[ 250 : 340 ] [ 250 : 340 ]
[ 250 : 390 ] [ 250 : 390 ]
[ 250 : 435 ] [ 250 : 435 ]
2.50 2.75 3.00 3.25 3.50 3.50 3.75 4.00 4.25 4.50
Figure5: Results fortheLECs ℓ¯ (leftpanel)andℓ¯ (rightpanel)fromunconstrainedNLOSU(2)ChPT
3 4
fitsfordifferentmassrangesincluding(abovetop-mostdashedhorizontalline)andexcludingnear-physical
masses. Onlyensembleswith1/a≥1.6GeVwereincludedinthefits. Bluelinesindicateourcentralvalue
anderrorbandsinthisset-up.
Mp range [MeV] (Mpphys)2 1/a > 1.6 GeV Mp range [MeV] fpphys [MeV] 1/a > 1.6 GeV
[ 135 : 160 ] [ 135 : 160 ]
[ 135 : 195 ] [102 MeV2] [ 135 : 195 ]
[ 135 : 240 ] [ 135 : 240 ]
[ 135 : 275 ] [ 135 : 275 ]
[ 135 : 340 ] [ 135 : 340 ]
[ 135 : 390 ] [ 135 : 390 ]
[ 135 : 435 ] [ 135 : 435 ]
[ 160 : 195 ] [ 160 : 195 ]
[ 160 : 240 ] [ 160 : 240 ]
[ 160 : 275 ] [ 160 : 275 ]
[ 160 : 340 ] [ 160 : 340 ]
[ 160 : 390 ] [ 160 : 390 ]
[ 160 : 435 ] [ 160 : 435 ]
[ 195 : 240 ] [ 195 : 240 ]
[ 195 : 275 ] [ 195 : 275 ]
[ 195 : 340 ] [ 195 : 340 ]
[ 195 : 390 ] [ 195 : 390 ]
[ 195 : 435 ] [ 195 : 435 ]
[ 230 : 275 ] [ 230 : 275 ]
[ 230 : 340 ] [ 230 : 340 ]
[ 230 : 390 ] [ 230 : 390 ]
[ 230 : 435 ] [ 230 : 435 ]
[ 250 : 340 ] [ 250 : 340 ]
[ 250 : 390 ] [ 250 : 390 ]
[ 250 : 435 ] [ 250 : 435 ]
176 178 180 182 184 128 129 130 131 132 133
Figure6:Extrapolatedphysicalpionmasssquared(leftpanel)anddecayconstant(rightpanel)fromuncon-
strainedNLOSU(2)ChPTfordifferentmassrangesincluding(abovetop-mostdashedhorizontalline)and
excludingnear-physicalmasses. Bluelinesindicatetheexperimentalvaluesanderrorbandsfrom[10,11].
InadditiontoourmainNLOSU(2)ChPTfits,wealsovariedthelowercutonthepionmasses
in theunconstrained fits. Especially, this isofinterest because nowadays manylattice simulations
still do not include the physical point and use ChPT to extrapolate towards it. In Figure 5 we
show how, e.g., the low-energy constants ℓ¯ and ℓ¯ (related to the scales L and L , respectively)
3 4 3 4
change due to the various lowerand higher mass cuts (in the upper part of each plot (grey shaded
area) theresults offitsincluding thephysical point areshown, whichhavebeenused toobtain the
systematic uncertainty, seeabove). Onecanseethat ℓ¯ ,whichpredominantly influences thequark
3
mass dependence of M2, is still in agreement with results from fits including the near physical
p
points,whereasℓ¯4,whichpredominantly influencesthequarkmassdependenceof fp ,showssome
deviations when the physical point is excluded. In Figure 6 wecompare the pion mass and decay
6
DeterminationofSU(2)ChPTLECsfrom2+1flavorstaggeredlatticesimulations EnnoE.Scholz
160
LO+NLO+NNLO 186
only LO+NLO
155 only LO 184
b =3.77
150 bb ==33..78952 2eV] 118802
M
145 20 178
1
[MeV] 140 132 physm) [ 117746
fp a 172
135 131 am/ 170 182
130 130 2 / (Mp 168 LoOnl+y NLLOO++NNLNOLO 181
166 only LO
125 b =3.77
1290.8 1.0 1.2 164 bb ==33..78952 1800.8 1.0 1.2
120 162
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
am/amphys am/amphys
Figure 7: Combined global fit to NNLO SU(2) ChPT using priors for L , k and k . Only ensembles
12 M2 f
with1/a≥1.6GeVand135MeV≤Mp ≤340MeVareincludedinthefit(onlydatapointsincludedinthe
fitrangeareshownintheseplots).
L 3 [MeV] priors: L 12, kM2, kf L 4 [MeV] priors: L 12, kM2, kf
Mp range [MeV] 1/a > 1.6 GeV Mp range [MeV] 1/a > 1.6 GeV
[ 135 : 195 ] [ 135 : 195 ]
[ 135 : 240 ] [ 135 : 240 ]
[ 135 : 275 ] [ 135 : 275 ]
[ 135 : 340 ] [ 135 : 340 ]
[ 135 : 390 ] [ 135 : 390 ]
[ 135 : 435 ] [ 135 : 435 ]
[ 160 : 240 ] [ 160 : 240 ]
[ 160 : 275 ] [ 160 : 275 ]
[ 160 : 340 ] [ 160 : 340 ]
[ 160 : 390 ] [ 160 : 390 ]
[ 160 : 435 ] [ 160 : 435 ]
400 600 800 1000 1200 600 800 1000 1200 1400
Figure 8: Results forL (leftpanel)and L (rightpanel)fromNNLO SU(2) ChPT fits to differentmass
3 4
rangesusingpriorsforL ,k ,andk . Onlyensembleswith1/a≥1.6GeVwereincludedinthefits. Blue
12 M2 f
linesindicateourfinalcentralvalueanderrorbands.
constant extrapolated from our unconstrained NLO SU(2) ChPTfits (including and excluding the
nearphysicalpoints)totheexperimental values[10,11]. Alsoherethepiondecayconstant shows
more deviations from the expected result, once more and more lighter masses are excluded from
the fits. In our opinion, these observations illustrate the danger inherent in applying NLO SU(2)
ChPT-formulaetolatticedatalackingdatapointswithlightenough pionmasses.
4. Fits toNNLO SU(2) ChPT
To check the influence of higher orders in SU(2) ChPT on our results for the LECs, we also
extended ourfitstoinclude thenext-to-next-to-leading order(NNLO)contributions. Inourset-up
for the fits of the meson masses and decay constants, we had to add three new fitparameters: the
low-energyscaleL ,whichwedefinedasacombinationoftheNLOlow-energyscalesL andL :
12 1 2
logL 2 =(7logL 2+8logL 2)/15 and two parameters for the NNLO-contributions in the meson
12 1 2
massanddecayconstant dependence, k andk ,respectively (see[9]fordetails). Unfortunately,
M2 f
our amount of data at fine enough lattice spacings and light quark masses did not turn out to lead
tostable fits. Therefore, intheendweopted forusing priors onthethree additional fitparameters
7
DeterminationofSU(2)ChPTLECsfrom2+1flavorstaggeredlatticesimulations EnnoE.Scholz
in the NNLO-fits. Details on the choice for the priors are reported in [9]. In Figure 7 we show a
typical fit of our data to the NNLO formulae using priors. Since we were not able to resolve the
newparametersinasatisfactorywaywithoutaprioriinputwerefrainfromquotingresultsforthese
parameters. ButwewouldliketostressthatthechangeinthedeterminedvaluesfortheNLO-LECs
isonlyminorascanbeseenfromFig.8whereweshowtheresultsforL andL fromaNNLO-fit
3 4
(using priors for k ,k ,andL )atdifferent massranges. Therethevertical lines showour final
M2 f 12
results anderrorbandsfromourNLO-fits.
5. Conclusions
We determined the following SU(2) LECs from our NLO ChPT fits to meson masses and
decay constants measured on staggered 2+1 flavor lattice simulations of QCD, cf. also the right-
mostcolumnofTab.1.
2Bmphys = 1.8609(18) (74) ·10−2GeV2, f = 122.72(07) (35) MeV,
stat syst stat syst
ℓ¯ = 3.16(10) (29) , ℓ¯ = 4.03(03) (16) .
3 stat syst 4 stat syst
(Here we quoted the LECs ℓ¯ and ℓ¯ , which are related to the scales L and L , respectively.) In
3 4 3 4
addition, we also provide the ratio of the extrapolated physical decay constant to its value in the
phys
chiral limit fp /f = 1.0627(06)stat(27)syst. Using the value determined for the average light
quarkmassin[13,14],weobtainfromourfittedvaluefor c phys thecondensate parameter
BMS,2GeV = 2.682(36) (39) GeV, S MS,2GeV = 272.3(1.2) (1.4) MeV 3,
stat syst stat syst
(cid:0) (cid:1)
wherethecondensate wasobtained bymultiplying Bwithourresultfor f2/2.
ThespeakeracknowledgessupportfromtheDFGSFB/TR55andtheEUgrantsPIRG07-GA-
2010-268367 andPITN-GA-2009-238353 (ITNSTRONGnet).
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