F ree Sreening Masses - Strength of the Anomaly

5. Thermal Transition for Two Quark Flavours 41

5.5. Strength of the Anomaly

5.5.2. F ree Sreening Masses

Sreeningmasses from thefree theory areexpetedto be realised for asymptoti

tem-peratures

T → ∞

^. ^Besides^providing^a^limiting^value^they^an^also^be^looked^at^to^learn

about volume and utodependene.

Asforinnite temperature therearenointerations, sreeningmassesfrom all

han-nels are expeted to be degenerate. This is illustrated by gure 5.13, where free

or-relators for the pseudo-salar and salaravour non-singlethannels are shown. They

havebeenalulatedbyanumerial inversionofthepropagator withrespettothefree

gaugeeld,

U = 1

1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 0.0001 0.01 1

0 5 10 15 20 25 30 35 z

ps, N _τ =2 sc, N _τ =2 ps, N _τ =12 sc, N _τ =12

Figure5.13.:Free pseudo-salarandsalarorrelatorsfor

N σ = 32

N τ

^=2,12.

The value for the free ontinuum meson sreening mass was rst onjetured by

Eletsky and Ioe [183 ℄. A omplete ontinuum alulation was then given in [184 ℄.

Aordingly, the freesreeningmassis dened by

D(p ₁ = p ₂ = 0, p ₃ =

ⁱ

M

/2, p ₄ = πT ) = 0 ,

^(5.35)

where

D(p)

^is^thepropagator'sdenominatorfortherespetivefermionformulationunder onsideration. Thefator2intheaboverelationanbeinterpretedphysiallyfromthe

fatthatthesreenedmesonidegreesoffreedomareonstitutedbytwovalenequarks.

We give a alulation for the lattie in appendix C.2 whih is based on mapping the

integrationvariablesontoaunitirle. Byinspetion ofthepolestruture,weanthen

relatetheorrelatortoanexponentialdeaygovernedbyasreeningmassinaordane

withequation(5.35).

For free staggered fermions, the leading order uto eets must be of

O (1/N _τ ² )

Aordingly, thesreening masshastheform

(M

^sr

)

^stagg

T = 2

s

π ² + m ² _q T ² − 1

3 2π ⁴ + ^m _T ₄ ⁴ ^q + 2π ² ^m _T ² ^q ₂ q

π ² + ^m _T ² ^q 2

1 N _τ ² + . . . ,

^(5.36)

where

m _q

^is ^the^free^quark^mass. ^Wilson ^fermions,^on^the^other^hand,^suer^from

O (a)

eetsthatareintroduedbyanitemass. Takingintoaountapossiblytwistedmass

with

m ² _q = µ ² ₀ + m ² ₀

^,^the^sreening ^mass^is^given^by

(M

^sr

)

^tm

T = 2 s

π ² + m ² _q

T ² − π ² + m ² _q T ²

! −1/2

m ₀ m ² _q T ³

1 N _τ

−



 1

4 π ² + m ² _q T ²

! −3/2

m ² ₀ m ⁴ _q

T ⁶

^(5.37)

+ π ² + m ² _q T ²

! −1

7 12 π ⁴ + π ²

48 + m ² _q

48T ² + m ² _q π ²

2T ² − m ² ₀ m ² _q T ⁴

! 

 1

N _τ ² + . . . .

Thus at maximaltwist, i.e.

m ₀ = 0

^,^the^quantity^is

O (a)

^-improved,

(M

)

^mtm

T = 2

s

π ² + m ² _q T ²

− π ² + m ² _q T ²

! −1

7 12 π ⁴ + π ²

48 + m ² _q

48T ² + m ² _q π ² 2T ²

! 1

N _τ ² + . . . .

^(5.38)

Fromtheaboveformulae itan beseenthattheleading disretisationartefatsofall

onsidered lattie formulations are negative, i.e. at least inleading order and

non-interatinglimitthelattiesreeningmassunderestimatesitsontinuumounterpart.

Contrary to that, perturbative ontinuum alulations [185 , 186 ℄ indiate a positive

ontribution introdued byinterations.

Anothersoureofdeviationfromtheontinuuminteratingvaluestemsfromthenite

volume usedinlattie simulations. To getanestimatefor thiseet, we have obtained

valuesforthefreesreeningmassonlattiesofdierentextentwithamaximallytwisted

mass propagator using

µ ₀ /T = 0.006

^, ^see ^gure ^5.14. ^In ^this ^ase ^we ^have ^used ^the

eetive mass as dened in equation (5.29) at

z = L/4

^to ^determine ^the ^sreening

mass. Whereas we do notnd promiment plateau behaviour for smalllatties, we nd

agreement between the eetive masses from equation (5.29) and equation (5.31) at

the given position. From Florkowski and Friman [184℄ we know that the asymptoti

behaviourofthe massless meson orrelator at hightemperaturesisgiven by

C(z) ≃ 1

z ² e ^{−2πT z} (1 + 2πT z) .

^(5.39)

This ensures the orret innite distane behaviour, i.e.

m

(z → ∞ ) = 2πT

How-ever, for atual simulations, one hasto deal with a nite extent in

z

^-diretion. ^If ^the

5 5.5 6 6.5 7 7.5 8 8.5 9

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

16 8 6 4

M scr /T

1/N ² _τ N _τ

continuum

N _σ /N _τ =2 N _σ /N _τ =4 N _σ /N _τ =6 N _σ /N _τ =8 infinite volume

Figure5.14.:Freesreeningmassasafuntionofthelattiespaingsquared,

∼ 1/N _τ ²

^,^for^four

valuesoftheaspetratio. Thedatahavebeenobtainedbynumerialinversionof

thefreepropagator,thesolidlinesonsiderallutoeetswhereasthedashed

linesonlyonsidertheleading

1/N _τ ²

ontributions.

sreeningmassisobtainedfromtheeetivemassatagivendistane,

z = L/n

^,^the

dis-repanyanbealulated,f.[177 ℄. For thispurpose,onehastoexpandthedierene

m

(L/n) − 2πT

^for ^a^small ^inverse ^spatial^extent ^nding

M

T = M

T V →∞

+ n N _τ

N _σ

^(5.40)

formassless quarks.

Ingure5.14 and 5.15we show orrespondingtsto the freesreening masses.

Fig-ure5.14shows the

N _τ

^-dependene^at ^xed ^aspet^ratio. ^F^or ^the^lattie ^spaing

depen-denewe use the omplete funtional form as impliitly dened by equation (5.35) in

orderto takepointsatalllattie spaingsinto aount. Thethinnerdashedlines

orre-spondtotheformulaupto

O (1/N _τ ² )

^. ^They^seem^to^be^anappropriatedesriptionofthe datapointsforlattieswith

N _τ & 6

^. ^This^situation^is^somewhat^better^than ^for^the^free

pressure, disussed insetion 4.2, where ner lattiesareneeded to ahieve

a ²

^-saling.

Theonlyfreet parameter leftistheoeient of theontribution linearin1/

N _σ

^,^see

equation(5.40). Aording to theaboveonsiderations,this parametershouldbeequal

to 4, we atually t 4.362(22). For this t we have inluded all points but theone at

N _τ = 4

^with^aspet^ratio ² ^that^is^learly ^out ^of ^the^range^of^anyapproximation.

The disrepany of approximately 9% between the expeted value and the tted

onemight beexplainedbynitemasseetswhiharenotinluded inequation(5.40).

However,modiationsduetoanitemass

µ ₀ /T ≪ π

^should^be^rather^small. ^Therefore,

itismorelikelythatweseehigherordersofthedoubleexpansionin

1/N _τ

^and^the^aspet

ratio.

Figure5.15showsthevolume dependene atxedlattie spaing

N _τ = 4

^. ^As^before,

thepointwithanaspetratio of2hasbeenomittedfromthet. Theremainingpoints

5 5.5 6 6.5 7

0 0.02 0.04 0.06 0.08 0.1 0.12 M scr /T

1/N _σ N _τ =4

numerical values analytical formula linear fit to [0.0.08]

Figure 5.15.:Free sreeningmassasa funtion oftheinverse spatialextent for

N τ = 4

^and

linearttoallpointsexepttheoneforthesmallestvolume.

give a linear behaviour with tted ordinate 5.28812(21) and slope

4.30(11) · N _τ

^. ^The

slopeisothenaivepreditionof4butisinneagreementwiththepreviouslyobtained

valuefor thedataat xed aspetratio.

In onlusion the leading order uto and nite volume eets for free sreening

masses an be modelled,

M

1 N τ

, 1 N σ

= M

(0, 0) + c _(L) N _τ

N σ − c _(a) 1

N _τ ² ,

^(5.41)

withpositiveoeients

c _(L,a)

^that^depend^solely^on^the^fermions'disretisationsheme thatneeds to be

O (a)

^-improved, ^of^ourse.

Notethatontraryto the above ndings,oneexpetsexponentiallysuppressednite

volume eets for the zero temperature interating spetrum[133 ℄. However, the

or-respondingproofisbasedonthealulation ofzerotemperaturesattering amplitudes.

The lineardependeneof freesreening masseshas alsobeen foundin[177 ℄ where also

arestorationofthe exponentialbehaviourisobservedwhenapproahing

T _c

^from^above.

Therefore we do not expet nite volume eets as severe as in the free ase for our

dynamial sreeningmasses at

T ∼ T _c

Im Dokument OPUS 4 | The thermal transition of quantum chromodynamics with Twisted Mass Fermions (Seite 69-72)

F ree Sreening Masses

5. Thermal Transition for Two Quark Flavours 41

5.5. Strength of the Anomaly

5.5.2. F ree Sreening Masses

T → ∞

U = 1

1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 0.0001 0.01 1

0 5 10 15 20 25 30 35 z

ps, N τ =2 sc, N τ =2 ps, N τ =12 sc, N τ =12

N σ = 32

N τ

D(p 1 = p 2 = 0, p 3 =

M

/2, p 4 = πT ) = 0 ,

D(p)

O (1/N τ 2 )

(M

)

T = 2

s

π 2 + m 2 q T 2 − 1

3

2π 4 + m T 4 4 q + 2π 2 m T 2 q 2 q

π 2 + m T 2 q 2

1

N τ 2 + . . . ,

m q

O (a)

m 2 q = µ 2 0 + m 2 0

(M

)

T = 2 s

π 2 + m 2 q

T 2 − π 2 + m 2 q T 2

! −1/2

m 0 m 2 q T 3

1 N τ

−



 1

4 π 2 + m 2 q T 2

! −3/2

m 2 0 m 4 q

T 6

+ π 2 + m 2 q T 2

! −1

7

12 π 4 + π 2

48 + m 2 q

48T 2 + m 2 q π 2

2T 2 − m 2 0 m 2 q T 4

! 

 1

N τ 2 + . . . .

m 0 = 0

O (a)

(M

)

T = 2

s

π 2 + m 2 q T 2

− π 2 + m 2 q T 2

! −1

7

12 π 4 + π 2

48 + m 2 q

48T 2 + m 2 q π 2 2T 2

! 1

N τ 2 + . . . .

µ 0 /T = 0.006

z = L/4

C(z) ≃ 1

z 2 e −2πT z (1 + 2πT z) .

m

(z → ∞ ) = 2πT

z

5 5.5 6 6.5 7 7.5 8 8.5 9

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

16 8 6 4

M scr /T

ps, N _τ =2 sc, N _τ =2 ps, N _τ =12 sc, N _τ =12

D(p ₁ = p ₂ = 0, p ₃ =

/2, p ₄ = πT ) = 0 ,

O (1/N _τ ² )

π ² + m ² _q T ² − 1

2π ⁴ + ^m _T ₄ ⁴ ^q + 2π ² ^m _T ² ^q ₂ q

π ² + ^m _T ² ^q 2

N _τ ² + . . . ,

m _q

m ² _q = µ ² ₀ + m ² ₀

π ² + m ² _q

T ² − π ² + m ² _q T ²

m ₀ m ² _q T ³

1 N _τ

4 π ² + m ² _q T ²

m ² ₀ m ⁴ _q

T ⁶

+ π ² + m ² _q T ²

12 π ⁴ + π ²

48 + m ² _q

48T ² + m ² _q π ²

2T ² − m ² ₀ m ² _q T ⁴

N _τ ² + . . . .

m ₀ = 0

π ² + m ² _q T ²

− π ² + m ² _q T ²

12 π ⁴ + π ²

48 + m ² _q

48T ² + m ² _q π ² 2T ²

N _τ ² + . . . .

µ ₀ /T = 0.006

z ² e ^{−2πT z} (1 + 2πT z) .

1/N ² _τ N _τ

N _σ /N _τ =2 N _σ /N _τ =4 N _σ /N _τ =6 N _σ /N _τ =8 infinite volume

∼ 1/N _τ ²

1/N _τ ²

+ n N _τ

N _σ

N _τ

O (1/N _τ ² )

N _τ & 6

a ²

N _σ

N _τ = 4

µ ₀ /T ≪ π

1/N _τ

N _τ = 4

1/N _σ N _τ =4

4.30(11) · N _τ

(0, 0) + c _(L) N _τ

N σ − c _(a) 1

N _τ ² ,

c _(L,a)

T _c

T ∼ T _c