The role of the zero modes  small volume, heavier sea quark

W dokumencie Uniwersytet im. Adama Mickiewicza Adam Mickiewicz University (Stron 105-110)

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

) is

situated 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 the

depen-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 the

dif-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

will

also 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

and

β = 4.2

with a heavier quark mass

whi h leads to approximately the same sea-sea pion mass as

aµ = 0.006

in

the 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

, 260

ongurations,

• 20 3 × 48

,

a ≈ 0.063

fm (

β = 4.05

,

r 0 /a = 6.61(2)

),

aµ = 0.006

, 299

ongurations,

• 24 3 × 48

,

a ≈ 0.051

fm (

β = 4.2

,

r 0 /a = 8.33(5)

),

aµ = 0.005

, 137

ongurations.

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 (at

β = 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 a

sub-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 of

the 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 slopeofthettedlineisnegative

in 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 ts

arealso

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 )

between

overlap(PP)vs. MTMandoverlap(PP-SS)vs. MTM(slightlyshiftedtothe

rightforbetterpresentation). Thedieren e

r 0 (f π overlap −f π M T M )

is onsistent

withzeroinboth 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.

W dokumencie Uniwersytet im. Adama Mickiewicza Adam Mickiewicz University (Stron 105-110)