heavier sea quark mass
4.6.1 Motivation and simulation setup
We now investigate the ee ts of the zero modes for a heavier sea quark
mass. The motivationfor this test isprovided by Fig. 4.22. The solid urve
shows the pion mass dependen e of the pion de ay onstant for the mixed
a tion setup of overlap valen e quarks and MTM sea quarks (the
β = 4.05
,L/a = 20
ensemble). The orresponding unitary point (aµ = 0.003
) issituated below the urve and the verti al distan e from this point to the
overlap urve measures thedis repan y between the overlap and MTMpion
de ay onstants atthe mat hing mass.
The other unitary point orresponds to a heavier sea quark mass (
aµ = 0.006
) and this point lies very lose to the overlap urve. Sin e thedepen-den e of the valen e-valen e pionmass and de ay onstant onthe sea quark
mass is mu h smaller than the dependen e on the valen e quark mass 4
, we
4
This an beestimated from theformulasof PartiallyQuen hed ChiralPerturbation
Theory[120,121℄.
0.15 0.2 0.25 0.3 0.35 0.4 0.45
0 0.5 1 1.5 2 2.5 3
r 0 f π
(r 0 m π ) 2 Unitary MTM
aµ=0.003 Unitary MTM
aµ=0.006
β=4.05 L/a=20 Overlap valence on MTM sea with aµ=0.003
MTM valence on MTM sea
Figure 4.22: The pion mass dependen e of the pion de ay onstant for
over-lap valen e quarks on MTMsea. Also shown are two unitary points(MTM
valen equarks on MTM sea), diering onlyby the sea quark mass.
Param-eters:
β = 4.05
,L/a = 20
.an expe t that the overlap urve for a heavier quark mass will not move
substantially from its position for
aµ = 0.003
, thus implying that thedif-feren e between the pion de ay onstant atthe mat hing mass willbemu h
smaller than the one observed for
aµ = 0.003
.An expli it omputation of the overlap dependen e for
aµ = 0.006
willalso provide a further he k of the hypothesis that the zero modes are
re-sponsiblefor the mismat hinthe ontinuum limitvalues between the mixed
and the unitaryapproa h. Sin ethe mat hingmass willbeheavier,the
on-tribution of the zero modes will be mu h smaller. Thus, we an expe t a
smaller mismat h in the ontinuum limit. Furthermore, we an again he k
whether the pro edure of an elling the zero modes ontribution by taking
the PP-SS orrelation fun tionwilllead to a onsisten y between the mixed
and unitary ontinuum limitvalues of the pion de ay onstant. Forthis, we
will also use ensembles at
β = 3.9
andβ = 4.2
with a heavier quark masswhi h leads to approximately the same sea-sea pion mass as
aµ = 0.006
inthe ase of the
β = 4.05
ensemble.Simulationparameters are:
• 16 3 × 32
,a ≈ 0.079
fm (β = 3.9
,r 0 /a = 5.25(2)
),aµ = 0.0074
, 260on gurations,
• 20 3 × 48
,a ≈ 0.063
fm (β = 4.05
,r 0 /a = 6.61(2)
),aµ = 0.006
, 299on gurations,
• 24 3 × 48
,a ≈ 0.051
fm (β = 4.2
,r 0 /a = 8.33(5)
),aµ = 0.005
, 137on gurations.
In order to minimize the ee t of auto orrelations, we have hosen every
16th Monte Carlo traje tory (at
β = 3.9
, 4.2) or every 20th traje tory (atβ = 4.05
) for inversions with the overlap Dira operator.4.6.2 Pion de ay onstant s aling test
To perform the pion de ay onstant ontinuum limit s aling test, we rst
have to nd the mat hing quark masses for ea h ensemble. For this, we
havefound the quarkmassdependen e ofthe pionmass. The pionmass has
been extra ted from the PP orrelator hen e we expe t that it might be
ontaminated by zero modes ee ts. It is interesting to ompare the quark
mass dependen e of the pion mass for the ases of the light sea quark mass
and the heavier one. Superimposing the heavier sea quark mass urves on
the orresponding ones for light sea quark mass, one nds that they are
onsistent within statisti al error (hen e, we don't show this plot for the
heavier sea quark mass ase), i.e. that at most a mild dependen e of the
valen e-valen e pion mass on the sea quark mass an be observed. This is
inagreementwith thepredi tionsofPartiallyQuen hedChiralPerturbation
Theory, i.e. this dependen e should be very small.
However, sin e the sea-sea pion mass hangessubstantially whenthe sea
quark mass value is in reased (it is
r 0 m π ≈ 1
for all ases), there is asub-stantial hange of the mat hing mass values:
• β = 3.9
a ˆ m = 0.015(1)
,• β = 4.05
a ˆ m = 0.011(1)
,• β = 4.2
a ˆ m = 0.009(1)
.Wenowpro eedtoanalyzethequarkmass dependen eofthe pionde ay
onstant. Again, the urves orresponding to both values of the sea quark
mass for ea hensembleare very loseto ea h other. The sea-seapion de ay
onstant values are for pra ti al reasons equalfor allensembles and
onsid-erably higherthan inthe ase of lightsea quarkmass. This impliesthat the
dieren es atthe mat hing point are mu h smallerthan in the latter ase.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 r 0 (f π overlap -f π MTM )
(a/r 0 ) 2 cont. limit = 0.330(17)
L ≈ 1.3 fm cont. limit = 0.236(9)
β=4.2 β=4.05 β=3.9
matching mass heavier matching mass light
Figure 4.23: Continuum limits aling of the overlap pion de ay onstant at
the mat hing mass lightand heavier sea quark mass.
In Fig. 4.23, we show the results of the ontinuum limits aling test for
the mat hing mass, ontrastingthe ut-o ee ts in the ase of the two sea
quark masses. Also in the ase of the heavier sea quark mass, the leading
dis retization ee ts are
O(a 2 )
5. Moreover, they are smaller in the ase ofthe heavier sea quark mass, i.e. the slope of the latti e spa ing dependen e
is smaller inthis ase.
The orrespondings alingplot forthe MTM ase(onlyheavierseaquark
mass) is shown in Fig. 4.24. Sin e we work at maximal twist, the leading
ut-o ee tsare also
O(a 2 )
. However, the slopeofthe ttedlineisnegativein this ase, as opposed to apositiveslope inthe ase of the lightsea quark
mass.
Toassess therole of the zeromodes forheaviersea quark mass,asimilar
analysis has also been performed using the PP-SS orrelator to extra t the
pionmassandde ay onstant. Here wejustquotethevaluesofthemat hing
mass for this ase:
• β = 3.9
a ˆ m = 0.0165(15)
,• β = 4.05
a ˆ m = 0.012(1)
,5
Wehavealso he kedthatforotherreferen evaluesof
r 0 m π
theleading ut-oee tsarealso
O(a 2 )
.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 r 0 f π
(a/r 0 ) 2 cont. limit = 0.359(11)
L ≈ 1.3 fm
β=4.2 β=4.05 β=3.9
matching mass MTM
Figure 4.24: Continuum limit s aling of the MTM pion de ay onstant at
the mat hing mass. The ase of the heavier sea quark mass.
-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 r 0 (f π overlap -f π MTM )
(a/r 0 ) 2 L ≈ 1.3 fm β=4.2
β=4.05
β=3.9
r 0 m π overlap =r 0 m π MTM matching mass heavier PP matching mass heavier PP-SS
Figure 4.25: Continuum limit s aling of the dieren e of the overlap and
MTM pion de ay onstant at the mat hing mass. The ase of the heavier
sea quarkmass, PP and PP-SS orrelators.
• β = 4.2
a ˆ m = 0.0095(15)
.At these values of the mat hing masses, the pion de ay onstant has
been al ulated. Again, thes alingtestshows thattheleadingdis retization
ee ts are
O(a 2 )
.Theessentialissue isnowto ompare the ontinuumlimitsofthe overlap
andMTMdis retization,withtheoverlapdatafromboththePPandthe
PP-SS orrelator. InFig. 4.25weplotthedieren e
r 0 (f π overlap −f π M T M )
betweenoverlap(PP)vs. MTMandoverlap(PP-SS)vs. MTM(slightlyshiftedtothe
rightforbetterpresentation). Thedieren e
r 0 (f π overlap −f π M T M )
is onsistentwithzeroinboth ases. However,thesubtra tionofthes alar orrelatorstill
ae ts the pionobservables,whi hisespe iallyvisibleatthe oarsest latti e
spa ing. This allows us to on lude that the role of the zero modes at the
mat hing mass orresponding to the heavier sea quark mass is very mu h
redu ed with respe t to the light quark mass. It is, nevertheless, still
non-negligibleat this sea quark mass and this volume.