Towards the Chiral Limit

Our runs at pion masses below 500MeV (A12, B10, B12, C12) form the basisfor our

investigations of properties towards the hiral limit. Unfortunately, not always large

statistis from the existing tmfT ongurations are available for thedetermination of

sreeningmasses. Therefore,theneessarytsareaetedbyratherlargeunertainties.

Wehavefoundthatapplyingtheosh-relationgiveninequation(5.28)to ourdatawith

trangesfrom anoptimisation of

χ ² /

^{dof leadsto massvalues whihare}systematially about 10% larger than the results based on the long distane behaviour of the loal

eetive masses, equation (5.31). We attribute this to therather large error bars due

tosmallstatistis. Thisimpliesthattheosh-t appearstobereliableevenin

z

^-ranges

where the desriptionbyasingle exponential deayis no longer valid. For this reason,

wehaveresortedtodeterminethesreeningmassesfromtsofaonstanttothetailsof

theloaleetive masses. Figure5.18givesan examplefor ourrun A12. Thelefthand

panelshows the ttedonstant whereasthe right hand sidepresentstheappliation of

1 1.5 2 2.5 3 3.5 4 4.5 5

3.76 3.78 3.8 3.82 3.84 3.86 3.88 3.9 3.92 3.94 (M sc scr -M ps scr )/T

β aµ ₀ = 0.040

a µ ₀ = 0.025

Figure 5.17.:Dierene of salar and pseudo-salar avour multiplet sreening masses for

aµ ⁰ = 0.025

^and

aµ ⁰ = 0.04

^ona

16 ³ × 8

^lattie.

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

4 6 8 10 12 14

am eff (z/a)

z/a β=3.900 β=3.960 β=4.015

5 10 15 20 25 30

C(z)

z/a

β=3.900 β=3.960 β=4.015

Figure 5.18.:Comparisonofloaleetivemassesandorrelatorsforthreevaluesofthelattie

ouplingin runA12.

the t result to the osh-behaviour (withan additionally determined prefatorfor the

osh).

All ts are olleted in appendix D.4. Nevertheless, the possiblylarge intrinsi

un-ertainties from the available data samples also reeted by poor values of

χ ²

^for

some of thesreening masses mustbe onsidered before drawing strong onlusions

from those results. We give an example for salarand pseudo-salar hargedsreening

masses in gure5.19, left. Espeiallythelarge utuations ofthe salarmassindiate

the problemati situation for the tting proedure. Note that a rm determination of

thesalar masshasnot been possible inall ases.

Theright handsideofgure5.19showsthepseudo-salarsreeningmassfor thetwo

runs B10 and B12. In the (pseudo-)ritial region, these masses are expeted to show

an inrease whih we indeed observe. Fromomparison of theB10and B12 data, itis

also lear that the disretisation errors are smaller than the ombined statistial and

0 2 4 6 8 10 12 14 16 18

200 210 220 230 240 250

M scr /T

T (MeV) pseudo scalar

scalar

2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3

220 225 230 235 240 245 250 255 260 M ps scr /T

T (MeV) N _τ =10

N _τ =12

Figure5.19.:Sreening masses for dierent runs. We have negleted the unertainty in

T

from the salesetting for these plots. Left: Charged salarand pseudo-salar

massesforA12. Right: Chargedpseudo-salarmassforB10andB12.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0 2 4 6 8 10

am eff (z/a)

z/a

vector axial vector

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0 2 4 6 8 10

am eff (z/a)

z/a

vector axial vector

Figure5.20.:Comparison of loal eetivemasses for the vetor and axial vetorhannels.

Left: A12,

β = 3.960

^{. Right: B12,}

β = 4.000

^.

systemati errors. For the following we onentrate on the salar and pseudo-salar

masses whih are important to quantify the axial anomaly. For the vetor and axial

vetorhannels, we nddegeneray lose to and above

T _c

^{,seegure5.20 for examples}

at

β

^{-valueslose to}

β _c

^{for A12 and B12.}

Thesplittingofthepseudo-salarandsalarsreeningmassesfromrunsA12,B12and

C12is showningure 5.21, left. Notsurprisingly,theresults arevery noisy. However,

omparing to gure 5.17 the splitting seems to be slightly enhaned for the smaller

masseseven though the lattiespaing is dierent. Inorder to getmore stableresults,

we onsider the integrated suseptibilities for the sreening orrelators as dened in

equation (5.34). For those quantities no ts are needed. Hene the main soure of

unertainty for the sreening masses is disarded. Consequently, the splitting of the

salar and pseudo-salar suseptibilities as shown in gure 5.21, right, gives a muh

moreoherent piture than the masssplittingdoes.

For both the splitting of sreening masses and suseptibilities, we have determined

the value in the transition region. For this purpose we have used the pseudo-ritial

temperatures related to

σ ²

ψψ

^{as olleted in table 5.3. In gure 5.21, this is indiated}

by the orrespondingly shaded areas. The result is shown in gure 5.22. Whereas all

0 2 4 6 8 10 12 14 16

200 210 220 230 240 250 260 270 (M sc scr -M ps scr )/T

T (MeV) A12 B12 C12

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

200 210 220 230 240 250 260 270 χ π - χ a 0

T (MeV) A12 B12 C12

Figure 5.21.:Strengthoftheanomalyasafuntionoftemperaturedeterminedfromthe

split-tingofsreeningmasses(left)andtheirsuseptibilities(right)forrunsA12,B12,

C12.

0 2 4 6 8 10 12 14 16

300 350 400 450 500

(M sc scr -M ps scr )/T

m _π

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

300 350 400 450 500

χ π - χ a 0

m _π (MeV)

Figure 5.22.:Strengthoftheanomalyasafuntionofpionmassdeterminedfromthesplitting

ofsreeningmasses (left)and theirsuseptibilities(right).

that an be read o from the mass splittings is that they seem to be more or less

onsistent within error bars, thesplitting from the suseptibilities exhibitsan inrease

towardssmallerpionmasses. Thisisanimportantobservationindiatingthattheaxial

anomaly might not be negligibleinthehiral limit.

Conentrating on the suseptibilities, we aompany our previous plots bythe ratio

χ _π /χ _{a 0}

^{whih approahes 1for vanishing anomaly. Figure5.23, left, presents theratio}

forallofourruns. ThedataforB10aremoresatteredthanthosefortherunsat

N _τ = 12

^{. The onlusionsfor}

N _τ = 12

^{are thesame as thosefrom gure 5.21. F}urthermore, the lattiespaing dependene judged byrunsB10 and B12againappears to be small

ompared to other unertainties. Note however, that an extrapolation to vanishing

anomalyi.e.unitratiointheontinuumlimitisnot exludedasanbeestimated

from gure 5.23, right. Espeially,the B10point has too large errors to pose a strong

onstraint on theextrapolation.

The remaining question is whether our numbers for the

U _A (1)

^{splitting have to be}

onsideredlargeorsmall. Thereisnouniqueansweranddierentstudieshaveproposed

one interpretation or the other, see e.g.[150 ℄ and[187℄ respetively. Chandrasekharan

and Mehta have foundintheir modelstudy thattheanomaly hastobequitestrong in

order to hange the nature of the transition from rst to seond order [172℄, possibly

2 2.5 3 3.5 4 4.5 5

200 210 220 230 240 250 260 270 χ π / χ a 0

T (MeV) A12 B12 C12 B10

1 1.5 2 2.5 3 3.5 4

0 0.002 0.004 0.006 0.008 0.01 χ π / χ a 0

1/N _τ ²

Figure5.23.:Ratioofthepseudo-salarandsalarsuseptibilities. Left: Temperature

depen-deneforall available pionmasses. Right: Lattie spaing dependene for the

intermediatepionmass(runsB10andB12).

largerthan inQCD.However, astheyarguethemselves, thestrengthof theanomaly is

alulatedfromnon-ritialsalessothatadiretomparisontoQCDisnotadmissible.

Altogether,weinterpretourndings,inpartiular therisingbehaviourtowardssmaller

pionmasses,suhthat theanomlymight not be negligibleinthehiral limit.

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