The role of the zero modes  small volume, light sea quark mass 96

W dokumencie Uniwersytet im. Adama Mickiewicza Adam Mickiewicz University (Stron 97-105)

light sea quark mass

Wewouldnowliketoperforma ontinuumlimits alingtestofthepionde ay

onstant extra ted from the PP-SS orrelator

C P P −SS (t)

. We will pro eed

in 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 and

4.2.

• 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

and

f π M T M

(at the mat hing mass) extra ted

from 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 an

expli 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 e

f 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 the

one 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 alingwith

O(a 2 )

leading

dis retization ee ts. A omparison tothe PP ase (Fig. 4.8) indi atesthat

the extrapolated ontinuumlimitvalueismostlyae tedforsmall

r 0 m π

and

thereisalmostnoee tforthelargest

r 0 m π

. Also,inall asesthepointthat

is 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 violating

O(a 2 )

and the zero mode ee ts. We an

not 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 doublepole

ontribu-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 larger

latti 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 the

investigatedlatti espa ingsandthevalueextrapolatedto

a = 0

is onsistent

with 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, orrespondingto

β = 3.9

and withthe same sea quark mass value

aµ = 0.004

. The parameters are (in luding the ensemble at the smallest

volume):

• 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 the

L/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 learly

gives a non-zero value, whi h means that the ee ts of the zero modes are

stillveryimportant. Inturn,for

L/a = 24

,the hirallyextrapolatedvalueof

m π

is mu h loser to zero, signallingthat the importan eof the zero modes

ee t de reases. However,

m π

at

m q = 0

is still non-zero and therefore one

should 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 volume

is 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

fmisstillrather

0.05 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 play

a 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 size

L/a

. One an estimate from this plot

that at

L/a = 32

it would be of the order of a few per ent, thus signalling

that 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 volume

 at the mat hing point one expe ts an

O(a 2 )

dieren e due to dierent

dis retizationee tsfromdierentsea andvalen equarksa tions. However,

a test at

L/a = 32

,whi h orresponds to

L ≈ 2.6

fm is beyond the s ope of

thiswork,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.

W dokumencie Uniwersytet im. Adama Mickiewicza Adam Mickiewicz University (Stron 97-105)