Chiral zero modes and their ontribution to mesoni orrelators 88

to mesoni orrelators

4.3.1 Chiral zero modes

There is a lear dieren e in the eigenvalue spe tra of hiral and non- hiral

massless latti e Dira operators. In the ase of the former, it is possible

that eigenmodes

φ(x)

^with ^zero^eigenvalue ^appear^at^any ^value ^of ^the^latti
e

spa ing

a

^. ^Moreover, ^su
h ^zero ^modes ^have ^a^denite ^hirality,^i.e. ^they ^are

eigenmodes of

γ ₅

γ 5 φ(x) = ±φ(x)

^(4.8)

with eigenvalue

±1

^. ^If ^the ^eigenvalue ^equals

+1

^, ^we ^all ^su
h ^eigenmode ^a

zero mode inthe positive hirality se tor(ora right-handed zero mode) and

for eigenvalue

−1

^we ^speak ^of ^a ^zero ^mode ⁱⁿ ^the ^negative ^hirality ^se
tor

between the zero modes and topologi alproperties of gaugeelds.

Non- hiral latti e Dira operators an alsodevelop zero modes, but this

anonlyhappenatsu ientlysmallvaluesofthelatti espa ing. Thevalues

that are presently rea hed insimulationsare fartoo largetohaveexa t zero

modesofnon- hiralDira operatorsfromthepra ti alpointofviewwe an

therefore assumethatinour mixeda tion setupwithoverlap valen equarks

and twisted mass sea quarks the valen e Dira operator admits zero modes

and the sea Dira operator doesnot. Hen e, ina nitevolume situationthe

zero modes of the valen e Dira operator lead to a ontribution that is not

ompensatedbythe fermioni determinantand an ae t ertain orrelation

fun tionsandhen esomeobservables. ItwasshownbyBlumetal. [114℄that

the ontributionofthezeromodes(e.gtomesoni orrelators)isproportional

1/ √

V

^, ^where

V

^is ^the ^latti
e ^volume, ^and ^therefore ^it ^is ^a ^nite ^volume

artefa t.

It is interesting to spe ulate about the role of zero modes in an unitary

overlap simulation 3

. In su h ase, the ontribution of the zero modes would

be suppressed by the (overlap) fermioni determinant. In other words, an

ee tofthezeromodesthatwewanttoinvestigateinthemixeda tionsetup

orthe analogous ee t inthe quen hed approximation[119℄results fromthe

fa t that the ontribution of the zero modes is not properly suppressed by

the fermioni determinant, sin e it is a determinant that originates from a

non- hirallysymmetri a tion(the MTM ase)orthereisnodeterminantat

all (i.e. it is set to a onstant in the quen hed approximation). Moreover,

it an be hypothesized that very lose to the ontinuum limit, zero modes

of the MTM Dira operator would also appear and the ontribution of the

zero modesinthevalen ese torwouldbesuppressed bytheMTMfermioni

determinant. In this way, it would lead to a lowered ontinuum limitof the

overlap pionde ay onstantinthePP asewith respe t totheoneextra ted

fromthelinearextrapolationin

a ²

^and^under^the^assumption^ofuniversalitya limitmore onsistentwiththeunitaryMTMvalue. However, su hhypothesis

isnottestableinlatti e al ulations,sin eprobablyasimulationwithavery

small latti e spa ing would have to be performed. Nevertheless, the pion

de ay onstant ontinuum limit s aling test in the unitary overlap setup

would be interesting from this point of view and should onrm that the

ontinuumlimitofboth unitaryoverlapand unitaryMTMisthe same,even

when one looks at the PP orrelator in the former ase, as we have done in

the previous se tion.

Forunitaryoverlapsimulationsappropriatealgorithmsneedtobeused,whi htakethe

zeromodesintoa ount,e.g. thePolynomialHMC(PHMC)algorithm[115,116,117,118℄.

orrelators

We now pro eed to show how the ontribution of the zero modes an be

al ulated and subtra ted fromthe observables. In this way,we willbe able

to ompute the overlap pion de ay onstant withoutthe ontribution of the

zero modes and then perform the ontinuum limit s aling test of Se tion

4.2 again and he k whether the nite volume ee t of the zero modes is

responsible for the dieren ein the ontinuum value.

Letus onsider the spe tral de omposition of the propagator

S(x, y)

S(x, y) = X

i

φ i (x)φ ^† _i (y) λ i + m q

,

^(4.9)

where

λ i

^are ^the eigenvalues of the masslessDira operator

D ˆ

^,^i.e.:

Dφ ˆ _i (x) = λ _i φ _i (x),

^(4.10)

and

m q

^is^the ^bare ^quark ^mass.

Inserting this de omposition intothe expression for the mesoni

orrela-tion fun tion (1.91), we obtain:

C(t) = X

Let usnow isolate the ontribution of the zero modes:

C(t) = C 00 (t) + 2C 0N (t) + C N N (t),

^(4.12)

Let us now onsider the ontributions of the zero modes

C 00 (t)

C 0N (t)

to the pseudos alar (

Γ ¹ = Γ ² = γ 5

⁾ ^and ^s
alar ⁽

Γ ¹ = Γ ² =

¹⁾ orrelation fun tions. In both ases we obtain the same result:

C ₀₀ ^{P P,SS} (t) = X

where in the s alar ase we have used eq. (4.8). The terms that ontain

the zero modes ontribution are proportional to

1/m ² _q

^and

1/m q

^and ^hen
e

diverge in the hiral limit

m q = 0

^. ^Sin
e ⁱⁿ ^our ^simulation ^setup ^the ^sea

quarkmass isratherlight,atthemat hingmassalsothe valen equarkmass

is lightand thereforethe zero modes ontribution an be important.

However, sin e the zero mode ontribution tothe pseudos alar(

C P P (t)

⁾

and s alar (

C SS (t)

⁾ orrelation fun tions is equal, it is possible to exa tly an el this ontribution by taking the dieren e of these two orrelators.

This was rst suggested by Blumet al. [114℄. Wedene:

C P P −SS (t) = C P P (t) − C ^SS (t).

^(4.18)

This is a valid orrelation fun tion with a proper transfer matrix

de ompo-sition. Therefore, it should be possible to extra t the pion mass and de ay

onstant from this orrelation fun tion.

C P P −SS (t)

^is ontaminated by the s alar ex itation. However, sin e the lightest s alar meson is mu h heavier

than the lightest pseudos alar meson, if we look at large enough time, the

ontributionof thes alar states shouldbe absent andwe an indeedextra t

the pion observables of interest.

In a mixed a tion setup there is an additional ompli ation. The s alar

orrelator is parti ularly vulnerable to the double pole ontribution, whi h

has already been dis ussed in Se tion 4.1. The residue from this double

pole does not vanish even in the ase of mat hed pion masses. Hen e, by

onsideringthe orrelationfun tion

C P P −SS (t)

^we^ex
hange ^the ontribution of thezero modes foraunitarity violationrelatedtothe mixeda tionsetup.

However, this is an ee t of

O(a ² )

^, ^whi
h ^an ^be ^onsidered ^to ^be ^an ^extra

dis retization ee t, in addition to the standard

O(a ² )

^s
aling ^violations

present in all observables. Therefore, su h unitarity violations vanish in the

ontinuum and they should not ae t the extrapolation of the pion de ay

onstant ( omputed from

C P P −SS (t)

⁾ ^to^the ^ontinuum.

The ee t of the zero modes on the pion mass an be observed in Fig.

4.11, whi hshows the bare overlapquark mass dependen e of the pion mass

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 a µ =0.004 MTM

Overlap PP-SS Overlap PP

Figure4.11: The omparisonofthequarkmass dependen e ofthe pionmass

extra ted fromPP and PP-SS orrelators for

β = 3.9

^ensemble.

extra ted from the pseudos alar (PP) orrelator and the PP-SS orrelator

C _{P P −SS} (t)

^. ^As^expe
ted^from onsiderationsinthis se tion, the ee t is the most pronoun ed for small quark masses, while for larger masses the pion

massextra tedfromboth orrelatorsisthesame(uptostatisti alerror). The

pionmassextrapolatedtothe hirallimit(

m q = 0

⁾^is^zero,^when^the^ee
ts^of

thezeromodeshavebeensubtra ted. Thisisina ordan ewiththe

leading-order predi tion of PartiallyQuen hed Chiral Perturbation Theory

m ² _π ∝ m q

^[120,^121℄. ^Also,^the^shape^of^the^quark^mass^dependen
e^of^the^pion^mass

agrees with this predi tion in this range of masses the urvature implied

by thenext-to-leadingorderpredi tionisonlyslightlyvisibleand,espe ially,

there is noeviden e for hiral logarithms

∝ m ^q log m q

^. ^An extrapolation to the hiral limitin the PP ase yields a non-zero value. The observed shape

ould be mistaken for a hiral logarithm relevant for small quark masses,

but itisentirelyduetothe hiral zeromodes, i.e. itisanite-volumeee t.

Theplotalsoshows thatthezeromodeshaveasigni antee twithrespe t

to the mat hing mass, whi h moves towards larger values of the bare quark

mass.

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

Figure4.12: The omparisonofthequarkmassdependen eofthepionde ay

onstant extra ted fromPP and PP-SS orrelators for

β = 3.9

^ensemble.

We also show the inuen e of the zero modes on the quark mass

depen-den eofthe pionde ay onstant(Fig. 4.12). Asforthe pionmass,the ee t

is signi ant forsmallquark masses and the PP-SS urve liesbelow the PP

one. This ee t brings the de ay onstant towards the twisted mass value.

However, sin ethemat hing massin reases, the ee tatthe mat hing mass

is rather small (for

β = 3.9

⁾ ^and ^hen
e ^an investigation of the ontinuum limit s aling is needed to he k whether the zero modes are enough to

ex-plain the dieren e between the ontinuum limit values of the pion de ay

onstant. This willbeperformed inthe following se tion.

4.3.3 Comparison of orrelation fun tions

To illustrate the ee ts of subtra ting the zero modes intwo dierent ways,

we plot in Fig. 4.13 the following orrelation fun tions: PP, SS and

PP-SS. We also plot the PP and PP-SS orrelation fun tions for one hosen

gauge eld onguration. Ensemble parameters are:

β = 3.9

L/a = 16

aµ = 0.004

am q = 0.004

^,^i.e. ^we ^hoose ^the ^lightest ^available^valen
e ^quark

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0 2 4 6 8 10 12 14 16

C(t)

t

PP

SS

PP-SS

averages PP SS PP-SS

0 0.1 0.2

0 2 4 6 8 10 12 14 16

PP PP-SS

single conf.

Figure4.13: Ensembleaveragesforthe following orrelationfun tions:

pseu-dos alar (PP), s alar (SS), the dieren e of PP and SS (PP-SS). The inset

shows the PP and PP-SS orrelation fun tions on a single onguration.

Parameters:

β = 3.9

L/a = 16

aµ = 0.004

am q = 0.004

mass to havethe biggest ontributionof the zero modes.

Letus summarize the on lusions fromthis plot.

•

^The ^PP-SS ^orrelator ^has ^a ^smaller ^slope^(with ^respe
t ^to^the ^PP

or-relator) in the plateau region thus it orresponds to a smaller pion

mass. This was already observed in Fig. 4.11 (the valen e quark mass

in Fig. 4.13 orresponds to the leftmost pair of points in Fig. 4.11).

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

10 11 12 13 14 15 16

C(t)

t

averages PP SS PP-SS

Figure4.14: Ensembleaveragesforthe following orrelationfun tions:

pseu-dos alar(PP),s alar(SS),thedieren eofPPand SS(PP-SS).Parameters:

β = 3.9

L/a = 16

aµ = 0.004

am q = 0.04

^(mu
h^larger^valen
e^quark^mass

than in Fig. 4.13).

From Fig. 4.11, one an also on lude that the ee t of the hange of

slope in the plateau region is smaller for larger valen e quark masses.

This is in a ordan e with our previous onsiderations the leading

quark-mass dependen e of the zero-mode ontribution to the PP and

SS orrelatorsis

O(1/m ² q )

•

^The ^matrix ^element ^of ^the ^PP-SS ^orrelator

|h0|P |πi| ^{P P −SS}

^is ^largely

redu ed with respe t to the PP orrelator matrix element

|h0|P |πi|

However, this leads to a relatively small de rease in the pion de ay

onstant (observed in Fig. 4.12), sin e the de rease in this matrix

element is almost ompensated for by a de rease in

m ² _π

^, ^whi
h ^omes

in the denominatorof eq. (1.95).

•

^The ^ee
t ^of ^the ^zero ^modes ^on ^a ^single onguration onsists in

pro-du ing an unphysi al peak at the timesli e (

t = 12

ⁱⁿ ^Fig. ^4.13) ^that

orresponds to the lo ation ofthe zeromode. This peakis removed in

the PP-SS orrelator.

We also onsider (Fig. 4.14) the ase of a heavier valen e quark mass

am q = 0.04

^(the ^remaining^parameters ^are ^the ^same). ^The ^plot ^shows ^only

the large-timebehaviour of the orrelation fun tions.

•

^The ^SS ^orrelator ^is ^onsistent ^with ^zero. ^F^or

t ∈ [10, 16]

^, ^there ^is ^no

ontributionfromthes alarex itation,asthes alarmesonistooheavy.

Sin e the s alar orrelator is zero, also the ontribution of the zero

modes is negligibleand hen e the PPand PP-SS orrelationfun tions

lead to the same result.

•

^F^or^this ^value^of ^quark^mass, ^one^also^expe
ts ^a^negligible ontribution from the double pole to the s alar orrelator eq. (4.2) implies that

for large

M V V

^this ontributionis very small.

•

^Hen
e, ^the ^pion ^mass ^and ^de
ay ^onstant ^extra
ted ^at ^this ^mass ^from

the PP/PP-SS orrelatordonotseemtobe ontaminatedbyeitherthe

ontributionof the zero modes orunitarity violations.

The analysis of this subse tion implies that, as expe ted, the role of the

zero modes de reases as the quark mass is in reased. In the next se tion

we will use the PP-SS orrelation fun tion to perform an analysis of the

behaviour of the pion de ay onstant with the zero modes ontribution

re-moved. Inparti ular,wewould liketo he kits ontinuumlimitifthe zero

modes are indeedresponsible forthe mismat hof ontinuumlimitsobserved

in Fig. 4.10, their removal should lead to the same ontinuum limit of the

pion de ay onstant asthe one of the unitary approa h.

W dokumencie Uniwersytet im. Adama Mickiewicza Adam Mickiewicz University (Stron 89-97)

Chiral zero modes and their ontribution to mesoni orrelators 88

φ(x)

a

γ 5

γ 5 φ(x) = ±φ(x)

±1

+1

−1

1/ √

V

V

a 2

S(x, y)

S(x, y) = X

i

φ i (x)φ † i (y) λ i + m q

,

λ i

D ˆ

Dφ ˆ i (x) = λ i φ i (x),

m q

C(t) = X

C(t) = C 00 (t) + 2C 0N (t) + C N N (t),

C 00 (t)

C 0N (t)

Γ 1 = Γ 2 = γ 5

Γ 1 = Γ 2 =

C 00 P P,SS (t) = X

1/m 2 q

1/m q

m q = 0

C P P (t)

C SS (t)

C P P −SS (t) = C P P (t) − C SS (t).

C P P −SS (t)

C P P −SS (t)

O(a 2 )

O(a 2 )

C P P −SS (t)

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 a µ =0.004 MTM

Overlap PP-SS Overlap PP

β = 3.9

C P P −SS (t)

m q = 0

m 2 π ∝ m q

∝ m q log m q

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

β = 3.9

β = 3.9

β = 3.9

L/a = 16

aµ = 0.004

am q = 0.004

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0 2 4 6 8 10 12 14 16

C(t)

t

PP

SS

PP-SS

averages PP SS PP-SS

0 0.1 0.2

0 2 4 6 8 10 12 14 16

PP PP-SS

single conf.

β = 3.9

L/a = 16

aµ = 0.004

am q = 0.004

•

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

10 11 12 13 14 15 16

C(t)

t

γ ₅

a ²

φ i (x)φ ^† _i (y) λ i + m q

Dφ ˆ _i (x) = λ _i φ _i (x),

Γ ¹ = Γ ² = γ 5

Γ ¹ = Γ ² =

C ₀₀ ^{P P,SS} (t) = X

1/m ² _q

C P P −SS (t) = C P P (t) − C ^SS (t).

O(a ² )

O(a ² )

r ₀ m _q β =3.9 L/a=16 a µ =0.004 MTM

C _{P P −SS} (t)

m ² _π ∝ m q

∝ m ^q log m q

r ₀ m _q β=3.9 L/a=16 aµ=0.004 MTM

O(1/m ² q )

|h0|P |πi| ^{P P −SS}

m ² _π