Small volume analysis

For onvenien e, werewrite here the formulafor the s alar orrelation

fun -tion at the mat hing mass:

C SS (t) −−−→ − ^t→∞ B ₀ ² 2L ³

e ^{−2M V V t}

M _{V V} ³ (γ V V + γ SS − 2γ ^{V S} ) a ² t.

^(5.1)

The low-energy onstants

γ V V = γ V S = 0

^{, due to exa t hiral symmetry}

in the valen e se tor [110℄, but

γ SS

^is non-vanishing, sin e the sea Dira operator is not hirally-symmetri .

Formula(5.1)impliesthatthes alar orrelationfun tionforthemat hing

quark mass an be ome negative at large times (provided that

γ SS > 0

^).

However, the shape of this orrelator is basi ally the one observed in Fig.

4.13, whi h shows the

β = 3.9

^{ase at a quark mass below the mat hing}

mass. Clearly, this orrelation fun tion does not be ome negative, sin e

it has a large positive ontribution from the zero modes and the unitarity

violation ee t is obs ured.

In order to analyze the ee t predi ted by eq. (5.1), we would have to

removethezeromode ontributionfromthes alar orrelatororworkatlarge

enoughvolumeandquarkmasssothatthis ontributionwouldbenegligible.

The latterrequiresavery omputer-timeintensive omputationandishen e

beyond the s ope of the urrent proje t. However, su h analysis is planned

for the future and would provide the leanest way of testing the predi tion

of eq. (5.1).

Inthe urrentproje t,wethereforehavetorestri tourselvestotheformer

method, i.e. to remove the zero mode ontribution from the SS orrelator.

This is possible by following the pro edure of expli it subtra tion of zero

modes at the level of propagators. As we have shown in the previous

hap-ter, this is a dangerous pro edure with hard to ontrol systemati ee ts.

Therefore, the results of this analysis have to be interpreted with aution

and treated as an outlook on this kind of analysis, whi h will be later

per-formed in a lean setup of large volume and large enough quark mass, so

that the zero mode ee ts willbe negligible. Analternativeapproa h ould

onsistinusingonly ongurationsinthetrivialtopologi alse tor,whi hare

ongurations in this se tor is too small to allow for meaningful ts of eq.

(5.1).

Fig. 4.30 shows that after the zero mode ontribution is removed, the

s alar orrelator at the mat hing mass be omes negative indeed. We have

also he ked that the SS orrelator on topologi ally trivial ongurations is

negative at large time (the error bands are too large to perform ts of eq.

(5.1), however, the on lusion about the sign of the orrelator is

unambigu-ous), whi h onrms that the unitarity ee t is really present in our mixed

a tion setup.

Ourstrategyisthefollowing. Weusethreesmall-volumeensembleswhose

parameters are given inSe tion4.2.1 and expli itlysubtra t the zero modes

at the level of propagators, as des ribed in Se tion 4.8.1. In this way, we

obtain for ea h ensemble the SS subtr. orrelator at the mat hing mass.

Then, we teq. (5.1) to the latti edata.

Spe i ally, we write this equationas:

C SS (t) ^t→∞ = −γ t e ^{−2M V V t} .

^(5.2)

where we havedened aparameter

γ

^:

γ ≡ B ₀ ² γ SS

2(M V V L) ³ a ² ≡ ˜γa ² .

^(5.3)

Sin e the temporal extent of the latti e is nite and equals

T

^{for ea h}

en-semble(with periodi boundary onditions intime), the ttingformulathat

we use reads:

C SS (t) ^t

^large

= −γ t e ^{−2M V V t} + (T − t) e ^{−2M V V (T −t)}
.

^(5.4)

The parameters that we t are

γ

^{and the pion mass}

M V V

^{. The denition}

of the parameter

γ

^{implies that}

γ

^{should have a quadrati dependen e on}

the latti e spa ing,sin e

B 0

^and

γ SS

^{are low-energy onstantsand}

M V V L

^is

approximatelythe same for ea hensemble.

Thetfortheensembleatthe oarsest latti espa ing(

β = 3.9

^)isshown

inFig. 5.3. Thettingintervalis

t ∈ [9, 23]

^{and inthis intervalthet}

repre-sentsaverygooddes ription oflatti edata. Qualitativelysimilarbehaviour

is observed also inthe

β = 4.05

^and

β = 4.2

^ases.

Oneof the ttingparametersisthe pion mass

M V V

^{. Its valuesextra ted}

fromthets anbe omparedwithvaluesofthe mat hingpionmass (known

pre isely from the maximallytwisted mass PP orrelator). This provides a

onsisten y he kforthets. Inall asesthettedvaluesof

M V V

^arearound

-0.1 -0.08 -0.06 -0.04 -0.02 0

0 5 10 15 20 25 30

C SS (t)

t

β=3.9, L/a=16, aµ=0.004, am _q =0.011 SS subtr.

fit t=[9,23]

Figure 5.3: The SS subtr. orrelation fun tion at the mat hing mass. The

solid line represents the t of eq. (5.4).

2 standard deviations below the mat hing pion mass. This is a reasonable

agreement,taking intoa ount the unknown systemati ee trelated tothe

subtra tion pro edure.

The ttingparameter

γ

^{has a few sour es of} un ertainties, related to:

1. statisti alerrors in

C SS (t)

^{(errorbars in Fig. 5.3),}

2. the hoi e of the tting interval  to estimate it we have performed

several ts with dierenttting intervals,

3. errors of the mat hing pro edure  to estimate it we have performed

ts not only forthe mat hing quarkmasses, but alsofor quarkmasses

diering in latti e units by

±0.001

^{(whi h} orresponds to the error in the mat hing mass),

4. the fa t that the produ t

M _{V V} L

^{is not exa tlymat hed for all}

ensem-bles,

5. unknownvaluesoftherenormalization onstant

Z S

^{ofthes alar urrent}

 we assumethat

Z _S

^{is equalfor all ensembles,}

6. anunknownsystemati errorintrodu edbythezeromodessubtra tion

pro edure.

-0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

γ

(a/r ₀ ) ² SS subtr. correlator

L ≈ 1.3 fm

Figure 5.4: Continuum limit s aling of the tting parameter

γ

^{, dened by}

eq. (5.3).

Fig. 5.4 shows the tted values of

γ

^{. The errors on ea h value in lude}

sour es 1-4 from the above list of un ertainties. The error related to the

unknown value of

Z _S

^{should be rather small ompared to the overall size of}

the error from sour es 1-4. We have not tried to estimate the error related

to the zero modes subtra tion pro edure.

We observe good s aling of the parameter

γ

^{with leading}

O(a ² )

^ut-o

dependen e. The value of this parameter extrapolated to the ontinuum is

onsistent with zero. This result is in very good agreement with the

hy-pothesis that the s alar orrelator is inuen ed by the unitarity violation

ee t predi ted and analyzed in [107, 108, 109, 110℄. It also provides an

explanation for the behaviour des ribed in the previous subse tion, i.e. the

seemingly in onsistent with the hypothesisaboutthe role of the zero modes

la k of ee t of subtra ting the SS orrelator at

β = 3.9

^{. The unitarity}

vio-lations analysis suggests that there are indeed two ompeting ee ts in the

PP-SS orrelatorthezero mode ontributionisremoved,but the orrelator

is ontaminatedbyaunitarityviolationoriginatingfromanenhan eddouble

pole ontribution. These two ee ts are roughly balan ed at

β = 3.9

^{, but at}

β = 4.05

^and

β = 4.2

^{, thenite volume ee t ofthe zero modes} an ellation dominates overthe

O(a ² )

^{unitarityviolationee t, whi hissmalleratthese}

latti e spa ings.

ay onstant extra ted from the PP-SS and the PP subtr. orrelator, sin e

the latter does not have the double pole ontribution of the SS orrelator.

However, ithastoberememberedthatthisanalysishasbeenperformedwith

the unphysi al zeromodessubtra tion pro edureand itmay suer from

un-predi table ee ts. Therefore, this analysis has to be treated with aution.

It providesaplausibleexplanationofthe observed ee ts. However, inorder

to quantitatively analyze the ee t of unitarity violations in the s alar

or-relator and reliably extra t the low-energy onstant

γ SS

^{, a simulation with}

large enough volume and quark mass would have to be performed in order

to have a negligible ontributionfrom the zero modes to the full s alar

or-relator (without expli it subtra tion pro edure). In addition, eq. (5.1) was

derived for an innite volume and hen e it would be very advantageous to

have large volume data for the SS orrelation fun tion in order to use the

tting ansatz of this formulain anappropriate way.

W dokumencie Uniwersytet im. Adama Mickiewicza Adam Mickiewicz University (Stron 125-129)