light sea quark mass
Wewouldnowliketoperforma ontinuumlimits alingtestofthepionde ay
onstant extra ted from the PP-SS orrelator
C P P −SS (t)
. We will pro eedin the same manner as before, i.e. we start by nding the mat hing mass
for ea h ensemble. The results of the mat hing pro edure are shown inFig.
4.15 and the bare overlap quark masses that lead to the same pion mass as
in the unitary setup are the following:
• β = 3.9
a ˆ m = 0.011(1)
,0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
(r 0 m π ) 2
r 0 m q β=4.2 L/a=24 aµ=0.002 MTM
Overlap
Figure 4.15: Mat hing the pion mass (extra ted from the PP-SS orrelator)
for three values of the latti e spa ing, orresponding to
β = 3.9
, 4.05 and4.2.
• β = 4.05
a ˆ m = 0.006(1)
,• β = 4.2
a ˆ m = 0.004(1)
.In omparison with the PP ase, the mat hing masses are shifted towards
larger values. This is a result of the fa t that they were arti ially lowered
due tothe zero mode ontribution.
As already dis ussed, the pion de ay onstant urve extra ted from the
PP-SS orrelationfun tionliesbelowtheoneextra tedinthePP ase. Atthe
0.2 0.25 0.3 0.35 0.4 0.45
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
r 0 f π
r 0 m q β=3.9 L/a=16 Overlap
β=4.05 L/a=20 Overlap β=4.2 L/a=24 Overlap
MTM aµ=0.004 MTM aµ=0.003 MTM aµ=0.002
Figure4.16: The dependen e of the pion de ay onstantonthe bare overlap
quark mass. The dashedlines orrespond to the mat hing quark masses
a ˆ m
(from PP-SS orrelator). The solid verti al lines (left of the dashed lines)
show the dieren e of
f π overlap
andf π M T M
(at the mat hing mass) extra tedfrom the PP orrelator.
same time, however, the mat hing masses are shifted towards larger values,
whi h orresponds to an in rease inthe pion de ay onstant. The interplay
of these two ee ts determines the dieren e between the overlap
f π overlap
and the MTM pion de ay onstant
f π M T M
at the mat hing mass. One anexpli itly omparethesedieren esfordierentensemblesbylookingatFig.
4.16, whi h shows the quark mass dependen e of the pion de ay onstant
extra ted from the PP-SS orrelator. The dashed verti al lines show the
analyzed dieren e in the PP-SS ase, while the solid verti al lines (left of
the dashed lines) show the orresponding mat hing point dieren e in the
PP ase(the lengthofthe solid linesisexa tlythe sameasthe lengthof the
verti al lines in Fig. 4.7). Cru ially, this dieren e is the most signi ant
for the
β = 4.2
ensemble and hen e it implies a large shift in the dieren ef π overlap − f π M T M
extrapolated tothe ontinuum limit.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
(a/r 0 ) 2 cont. limit = 0.196(13)
cont. limit = 0.245(10) cont. limit = 0.374(6)
L ≈ 1.3 fm
β=4.2 β=4.05 β=3.9
matching mass r 0 m π ≈ 1.0 r 0 m π ≈ 1.5
Figure 4.17: Continuum limit s aling of the overlap pion de ay onstant
(extra ted from the PP-SS orrelator) at the mat hing mass and two other
referen e values of
r 0 m π
.Fig. 4.17 shows the ontinuum limits aling of the pion de ay onstant.
We again take three referen e values of
r 0 m π ≈ 1.5
,r 0 m π ≈ 1.0
and theone that orresponds to the mat hing riterion
r 0 m π ≈ 0.85
. As before,for all analyzed values of
r 0 m π
, we observe good s alingwithO(a 2 )
leadingdis retization ee ts. A omparison tothe PP ase (Fig. 4.8) indi atesthat
the extrapolated ontinuumlimitvalueismostlyae tedforsmall
r 0 m π
andthereisalmostnoee tforthelargest
r 0 m π
. Also,inall asesthepointthatis most ae ted is the one that orresponds to the smallest latti e spa ing
and the one at
β = 3.9
pra ti allydoes not move.Clearly,su h behaviourresults from the interplay of various ee ts
in-trinsi
O(a 2 )
, unitarity violatingO(a 2 )
and the zero mode ee ts. We annot disentangleallof these ee ts,but a possiblequalitativeexplanation for
the observed behaviour an be provided by a working hypothesis that the
methodof extra ting the pion observables fromthe PP-SS orrelation
fun -tion exa tly an elsthe ontributionofthezero modes,butatthesametime
introdu esthe
O(a 2 )
unitarityviolation relatedtothe doublepoleontribu-tion to the s alar orrelator. In this way, the ee t of the zero modes may
be basi ally equal for all latti e spa ings (as expe ted for a nite volume
ee t), but the unitarityviolationee t auses thatthe pion de ay onstant
-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12
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
matching mass r 0 m π overlap =r 0 m π MTM
Figure 4.18: Continuum limit s aling of the dieren e of the overlap (from
the PP-SS orrelator) andMTM pion de ay onstant atthe mat hing mass.
in reases by an
O(a 2 )
term. Hen e, one might expe t that at even largerlatti e spa ing, the pion de ay onstant extra ted from the PP-SS
orrela-tor at the mat hing mass would be even larger than the one from the PP
orrelator, sin e then the unitarity violation ee t ould be larger than the
nite-volume ontribution of the zero modes.
The essential question is whether the ontinuum limit of the pion de ay
onstantextra tedfromthePP-SS orrelationfun tionagreeswiththeMTM
ontinuum limit. Fig. 4.18 shows the dieren es
r 0 (f π overlap − f π M T M )
for theinvestigatedlatti espa ingsandthevalueextrapolatedto
a = 0
is onsistentwith zero.
Hen e, we an on lude that the ontinuum limitof the pion de ay
on-stant omputedfortwodierentdis retizationsofvalen equarksisthesame,
provided that one takes intoa ount the role ofthe hiral zero modes of the
overlap operator, i.e. they have to be subtra ted from the overlap data in
order to ompare the ontinuum limits.
ee ts analysis
4.5.1 Simulation parameters
In order to he k the nite volume ee ts in the urrent setup, we have
investigated two additional ensembles at the oarsest latti e spa ing
a ≈ 0.079
fm, orrespondingtoβ = 3.9
and withthe same sea quark mass valueaµ = 0.004
. The parameters are (in luding the ensemble at the smallestvolume):
• 16 3 × 32
,L ≈ 1.3
fm, 544 ongurations,• 20 3 × 40
,L ≈ 1.7
fm, 239 ongurations,• 24 3 × 48
,L ≈ 2.0
fm, 435 ongurations.In order to minimize the ee t of auto orrelations, for propagator
ompu-tations we have hosen every 10th Monte Carlo traje tory (for
L/a = 16
,24) or every 20th traje tory (for
L/a = 20
). In addition, for theL/a = 24
ensemble, we have used the fully linked sour es, des ribed in Se tion 3.4.1.
Thus, only 1 inversion per gauge eld ongurationis required to onstru t
the pseudos alar orrelationfun tion. However, inthiswayitisnot possible
to al ulatethe s alar orrelator and hen e extra t the pion de ay onstant
from the PP-SS orrelator.
4.5.2 Mat hing the pion mass PP orrelator
Finite volume ee ts inthe quark mass dependen e of the pion mass an be
seen inFig. 4.19. The nite-volume ee t orresponding to a hange in the
linearextentofthelatti efrom1.3to1.7fmissigni antandapproximately
equalfortheoverlapandthe MTM ase, whereasthe ee tofgoingfrom1.7
to 2.0fm isvery smallinboth ases. The onlyex eption tothis observation
an be dis erned for the lightest valen e quark masses, where the de rease
related to the hange in volume is noti eable. This is espe ially meaningful
if the extrapolationto the hiral limitis performed. For
L/a = 20
it learlygives a non-zero value, whi h means that the ee ts of the zero modes are
stillveryimportant. Inturn,for
L/a = 24
,the hirallyextrapolatedvalueofm π
is mu h loser to zero, signallingthat the importan eof the zero modesee t de reases. However,
m π
atm q = 0
is still non-zero and therefore oneshould expe t that the ontributionof the zero modes is stillnon-negligible.
Fig. 4.19 alsoshows the mat hing mass values for ea h volume:
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
(am π ) 2
am q β =3.9 a µ =0.004 L/a=16 Overlap
MTM L/a=20 Overlap MTM L/a=24 Overlap MTM
Figure4.19: Mat hingthe pionmassfor3dierentvolumesataxedlatti e
spa ing
a ≈ 0.079
fm.• L/a = 16
a ˆ m = 0.007(1)
,• L/a = 20
a ˆ m = 0.007(1)
,• L/a = 24
a ˆ m = 0.008(1)
.For allvolumes, the mat hing mass isapproximatelythe same,whi his due
to the fa t that the nite volume ee ts (in the pion mass) of overlap and
MTM fermions are very similar.
4.5.3 Pion de ay onstant PP orrelator
Fig. 4.20 shows the quark mass dependen e of the pion de ay onstant for
three investigated volumes, together with the values in the unitary MTM
setup. The dieren e in
f π
at the mat hing point de reases as the volumeis in reased, whi h is in a ordan e with the expe tation based on the fa t
that the zero modes ontribution is a nite volume ee t. However, the
dis repan y between theoverlapandMTMvalues for
L ≈ 2
fmisstillrather0.05 0.06 0.07 0.08 0.09 0.1
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
af π
am q β=3.9 aµ=0.004 L/a=16 Overlap
MTM L/a=20 Overlap MTM L/a=24 Overlap MTM
Figure 4.20: The quark mass dependen e of the pion de ay onstant for 3
dierent volumes ata xed latti espa ing
a ≈ 0.079
fm.large (of order
15 ± 5
%) and one an suspe t that the zero modes still playa non-negligible(although mu h redu ed) role.
This is further illustrated in Fig. 4.21, whi h shows the dieren e in
f π
as a fun tion of the latti e sizeL/a
. One an estimate from this plotthat at
L/a = 32
it would be of the order of a few per ent, thus signallingthat the ontribution of the zero modes is negligible for pra ti al reasons.
This analysis isperformedatanon-zero latti espa ingand hen eit an not
be expe ted that the dieren e in
f π
goes to zero even in innite volumeat the mat hing point one expe ts an
O(a 2 )
dieren e due to dierentdis retizationee tsfromdierentsea andvalen equarksa tions. However,
a test at
L/a = 32
,whi h orresponds toL ≈ 2.6
fm is beyond the s ope ofthiswork,sin eitwouldrequireavery omputer-timeintensive omputation.
0 0.1 0.2 0.3 0.4 0.5
0 5 10 15 20 25 30 35 40
(af π ov -af π tm )/af π tm
L/a
β =3.9 a µ =0.004 PP
Figure 4.21: The relative dieren e between the overlap and MTM pion
de ay onstant atthe mat hing point.