Chiral zero modes and their ontribution to mesoni orrelators 88

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

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 zeroeigenvalue appearatany value of thelatti e

spa ing

a

. Moreover, su h zero modes have adenite 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

to

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

andundertheassumptionofuniversalitya 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.

3

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

isthe 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)

weex 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)

) tothe 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)

. Asexpe tedfrom 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

)iszero,whentheee tsof

thezeromodeshavebeensubtra ted. Thisisina ordan ewiththe

leading-order predi tion of PartiallyQuen hed Chiral Perturbation Theory 

m 2 π ∝ m q

[120,121℄. Also,theshapeofthequarkmassdependen eofthepionmass

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 availablevalen 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 tothe 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 hlargervalen equarkmass

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 remainingparameters are the same). The plot shows only

the large-timebehaviour of the orrelation fun tions.

The SS orrelator is onsistent with zero. For

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.

Forthis valueof quarkmass, onealsoexpe ts anegligible 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)