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 aling}

^{with}

### 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,}

^{in}

^{all}

^{ ases}

^{the}

^{point}

^{that}

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.}

^{W}

^{e}

^{ 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} )

^{unitarity}

^{violation}

^{related}

^{to}

^{the}

^{double}

^{pole}

^{}

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}

^{with}

^{the}

^{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}

^{ hirally}extrapolatedvalueof

### m π

^{is}

^{mu h}

^{ loser}

^{to}

^{zero,}

^{signalling}

^{that}

^{the}

^{importan e}

^{of}

^{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

^{fm}

^{is}

^{still}

^{rather}

### 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.