Ee ts of expli it zero modes subtra tion

W dokumencie Uniwersytet im. Adama Mickiewicza Adam Mickiewicz University (Stron 113-122)

4.8 Expli it subtra tion of zero modes

4.8.2 Ee ts of expli it zero modes subtra tion

An ee t of the subtra tion pro edureon the pionmass and de ay onstant

(with respe t to the PP ase) an be observed in Figs. 4.27 and 4.28. For

omparison, alsothe urves orresponding tothe PP-SS ase are plotted.

Thepionmassextra tedfromthePP-SS orrelatorandthePP orrelator

with subtra ted zero modes (PPsubtr.) agree forsmallquark masses,while

the behaviour of the pion de ay onstant is very dierent. At the level of

orrelationfun tions,this resultsfromthefa tthat thePP subtr. orrelator

has a very similar slope to the one of the PP-SS orrelator, but its matrix

element is signi antlylower.

For larger quark masses (larger than

r 0 m q ≈ 0.08

), the pion masses

ex-tra tedfromthePP-SSandthePPsubtr. orrelatorsarenot onsistentwith

ea h other  the PP subtr. urve hanges slope and deviates the morefrom

thePP-SS urvethelargerthequarkmass. Thisisinapparent ontradi tion

withtheexpe tationthatexpli itsubtra tionofzeromodesremovesthe

on-tributionof thesemodes,sin e atlarger quarkmass values this ontribution

tends to zero and the PP subtr. urve should onverge to the PP (and

PP-SS) urve. Su h behaviour of the pion mass from the PP subtr. orrelator

provides a warning about the expli it subtra tion method. It was observed

beforein quen hed studieswith the xed point Dira operator, whi h is

an-other variant of a hirally improved latti e Dira operator. The studies by

Hauswirth [123℄ and Gattringer etal. [124℄ obtained asimilar pi ture  the

pion mass at small quark mass is approximately the same from the PP-SS

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

(r 0 m π ) 2

r 0 m q β=3.9 L/a=16 aµ=0.004 MTM

Overlap PP subtr.

Overlap PP-SS Overlap PP

Figure4.27: The omparisonofthequarkmass dependen e ofthe pionmass

extra ted fromPP,PP-SSandPPwithexpli itlysubtra tedzeromodes(PP

subtr.) orrelators for

β = 3.9

ensemble.

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 0.16 0.18

r 0 f π

r 0 m q β=3.9 L/a=16 aµ=0.004 MTM

Overlap PP Overlap PP-SS Overlap PP subtr.

Figure4.28: The omparisonofthequarkmassdependen eofthepionde ay

onstant extra ted from PP, PP-SS and PP with expli itlysubtra ted zero

modes (PPsubtr.) orrelators for

β = 3.9

ensemble.

0.14 β=3.9, L/a=16, aµ=0.004, am q =0.011 PP subtr. correlator

Figure 4.29: Ee tive mass plateaus for PP subtr. orrelation fun tions.

Parameters:

β = 3.9

,

L/a = 16

,

aµ = 0.004

. Upper plot:

am q = 0.011

.

Lowerplot:

am q = 0.04

.

and the PP subtr. orrelator, while at larger quark masses the PP subtr.

orrelation fun tion leads to mu h smaller pion masses than ones obtained

fromthe PPandPP-SS orrelators (whi htendtoagreeatquarkmassesfor

whi hthe ee ts of zero modes are negligible).

This on lusion isfurther onrmed by Fig. 4.29, whi hshows the

ee -tivepion mass plateausfromthe PPsubtr. orrelator,for two quarkmasses

 the mat hing mass (upper plot) and a signi antly heavier mass (lower

plot). The plateau observed for the mat hing mass looks rather normal,

fromthespe tralde omposition(1.83). However, fortheheavierquarkmass,

there is no plateau(this plot an be ompared to Fig. 4.5, whi h shows the

samequarkmass,butthe ee tivemassisextra ted fromthe(unsubtra ted)

PP orrelator). This implies that the PP subtr. orrelation fun tion might

not be a sum of exponential fun tions, but rather a sum of power

fun -tions. This results from the fa t that expli it subtra tionof zero modes isa

non-lo al pro edure, i.e. it an modify the simulated theory in a non-lo al

way, thus leading to unphysi al ee ts in the orrelation fun tions, whi h

auses that the spe tral de omposition (1.83) is not valid. It also implies

that the pion mass values obtained from the PP subtr. orrelator are not

meaningful at high values of the quark mass. However, sin e there is no

fundamental reason why dierent quark masses should lead to qualitatively

dierentbehaviourof the orrelationfun tions,we an notbesure thateven

atarelativelysmallquarkmass(su hasthemat hingmass)thesubtra tion

pro edure isvalid.

The above dis ussion leads to a on lusion that expli it subtra tion of

zero modes is adangerous hand-made pro edure, whi h may lead to

un on-trollable unphysi al ee ts in the extra ted observables. However, sin e we

have observed the onsisten y between the pion masses fromthe PP-SS and

PP subtr. orrelators (for relatively small quark masses), we may assume

here as a working hypothesis that at the mat hing mass the expli it

sub-tra tionpro edure isvalid,i.e. that the unphysi alee ts ofsubtra tion are

small. This is justied by the fa t that the ee tive mass plateau for the

mat hing mass does not show the pathology observed at the larger quark

mass.

To on lude this subse tion, we show in Fig. 4.30 the s alar orrelation

fun tion with expli itly subtra ted zero modes (SS subtr.) for two values

of the valen e quark mass  the lightest onsidered mass and the mat hing

mass. Theplotshowsthatthedominant ontributiontothefullSS orrelator

omes from the zero modes (the SS subtr. urve for

am q = 0.004

should be

omparedtothefullSS urveatthesamequarkmassFig. 4.13). Moreover,

after the zero modes are subtra ted, the s alar orrelator is negative, whi h

may be attributed to the unitarity violation ee t dis ussed in Se tion 4.1.

We have alsohypothesized in Se tion4.4that this ee t inuen es the pion

de ay onstantextra tedfromthePP-SS orrelator. Thiswouldalsoexplain

thedieren ein

f π

extra tedfromthePP-SSandPPsubtr. orrelatorsthe latter does not have the enhan ed unitarity violation ee t from the s alar

orrelator. This ee t willbe investigated further inthe next hapter.

In the next subse tion we will use the PP subtr. orrelation fun tion to

extra tthe pionde ay onstantandperformits ontinuumlimits alingtest.

-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05

0 2 4 6 8 10 12 14 16

C(t)

t

β=3.9, L/a=16, aµ=0.004 SS subtr. am q =0.004

am q =0.011

Figure4.30: TheSSsubtr. orrelationfun tion(SSwithexpli itlysubtra ted

zero modes). Parameters:

β = 3.9

,

L/a = 16

,

aµ = 0.004

, 2 valen e quark

masses:

am q = 0.004

,

am q = 0.011

(mat hing mass).

The results will not be ontaminated by the zero modes ontribution, but

we again emphasize that they have to be interpreted with aution, due to

the fa tthatthesubtra tionpro edureisnot leanfromtheeld-theoreti al

pointof view.

4.8.3 Pion de ay onstant  s aling test  PP subtr.

orrelator

Weagainbeginbyndingthe mat hingmass forea h ensemble. Theresults

ofthe mat hingpro edureareshown inFig. 4.31 andthebareoverlapquark

masses that lead to the same pion mass as inthe unitary setup are:

• β = 3.9



a ˆ m = 0.0115(15)

,

• β = 4.05



a ˆ m = 0.0065(15)

,

• β = 4.2



a ˆ m = 0.0055(15)

.

These mat hing masses are onsistent with the ones obtained from the

PP-SS orrelationfun tion, onrming againthe on lusionthat these methods

give onsistent results for smallquark masses.

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 4.31: Mat hing the pion mass (extra ted from the PP subtr.

orre-lator) for three values of the latti e spa ing, orresponding to

β = 3.9

, 4.05

and 4.2.

The urves that showthe quark mass dependen e of the pionde ay

on-stantextra tedfromthePP orrelatorwithexpli itlysubtra tedzeromodes

(PP subtr.) liewell belowthe orresponding urves for the PP and the

PP-SS ase (Fig. 4.32). This ee t has already been dis ussed in the previous

subse tion. In omparison with the PP ase, the values of the pion de ay

onstantatthemat hingmass areverymu hredu ed,whi hisshown inthe

plots by verti al lines  the dashed ones orrespond to the dieren e in the

pion de ay onstantinthe PPsubtr. ase and thesolid onestothe PP ase.

0.15 0.2 0.25 0.3 0.35 0.4

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

Figure4.32: 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

(fromPPsubtr. orrelator). Thesolidverti allines(leftofthedashedlines)

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.

Again, the inuen e of zero modes an ellation is relatively the largest for

β = 4.2

.

Fig. 4.33showsthe ontinuumlimits alingofthepionde ay onstantfor

threereferen e valuesof

r 0 m π ≈ 1.3

(herewetakethe highestvalueavailable

for the

β = 3.9

ensemble),

r 0 m π ≈ 1.0

and the one that orresponds to the mat hing riterion

r 0 m π ≈ 0.85

. As in the previous ases of the PP and

the PP-SS orrelation fun tions, we observe good s aling behaviour for all

analyzed values of

r 0 m π

, with

O(a 2 )

leading ut-o ee ts.

Moreover, the ontinuum limitofthe pionde ay onstantextra tedfrom

the PP subtr. orrelation fun tion agrees with the MTM ontinuum limit,

whi h is shown in Fig. 4.34. Therefore, both methods of subtra ting the

zero modes lead to a onsistent ontinuum limit value, whi h is the one of

the unitary approa h. This is a strong hint that the zero modes are indeed

responsible for the observed behaviourof the pion de ay onstant extra ted

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 = 0.223(14)

cont. limit = 0.311(12)

cont. limit = 0.173(13)

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 ≈ 1.3

Figure 4.33: Continuum limit s aling of the overlap pion de ay onstant

(extra ted from the PP subtr. orrelator) at the mat hing mass and two

other referen e values of

r 0 m π

.

from the PP orrelator, i.e. for the wrong ontinuum limit value of this

observable.

However, wehavetoemphasize againthat the resultsof theexpli it zero

modes subtra tion pro edure have to be treated with aution. The

onsis-ten y between both methods is a hint that the pathologi al ee ts of the

hand-made subtra tion pro edure are not very large at the mat hing mass,

but this method of an elling the zero modes ontribution is not

re om-mended,sin ethereisnosystemati wayto ontrolthepotentialnon-physi al

ee ts.

A learly safer pro edure to remove the ontribution of the zero modes

is the one with the PP-SS orrelationfun tion. Thismethoddoes not suer

from the aforementioned ee ts, sin e no modi ation atthe levelof

propa-gators is made. In this way, both the PP and SS orrelation fun tions have

theproperspe tralde ompositionand,aswehaveshown,the ontributionof

the zero modes is exa tly an elled inthe orrelator dieren e. The pri e to

pay, however, isthat the s alar orrelatormay introdu e enhan ed unitarity

violations oming from the double pole ontributionspe i to non-unitary

approa hes (quen hed, partially quen hed and mixed a tion theories). On

the other hand, su h ee ts are

O(a 2 )

latti e artefa ts and hen e should

-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 4.34: Continuum limit s aling of the dieren e of the overlap (from

the PP subtr. orrelator) and MTM pion de ay onstant at the mat hing

mass.

not ae t extrapolations to the ontinuum limit. This ee t will be further

analyzed in the next hapter.

Various further results

In the previous hapter we have performed an analysis of the ontinuum

limit s aling of the pion de ay onstant. We have dis ussed the role of the

zero modesinamixed a tionsetupof hirally-symmetri valen equarksand

non- hirally-symmetri sea quarks. To an el the non-physi al ontribution

of the zero modes we haveused the PP-SS orrelationfun tion and we have

hypothesized that while this orrelator orre tly removes the zero modes

ontribution,italsointrodu esenhan edunitarityviolations. Inthis hapter

we will analyze this ee t. We will also present some additional results

regarding the ontinuumlimits aling ofbaryon (nu leonand delta)masses,

as well assome topologi alaspe ts.

W dokumencie Uniwersytet im. Adama Mickiewicza Adam Mickiewicz University (Stron 113-122)