Observables

5. Thermal Transition for Two Quark Flavours 41

5.2. Observables

We present the observables relevant for our study of the thermal transition beginning

with the hiral ondensate in ontinuation of thelast setion. We thenintrodue the

gaugeobservables, inpartiular thePolyakovloop thatissensitive to deonnement.

5.2.1. Chiral Condensate

The most important observable for our analysis approahing the massless limit is the

hiral ondensate,

ψψ

= T V

∂ ln Z

∂m _q ,

^(5.11)

that is the order parameter for hiral symmetry. Thus it an be assoiated with the

magnetisation

M

^of^a ^spin^model^of^the^same universalitylassapplyingall thesaling properties presented insetion5.1.

If the anomaly is suiently strong to break the

U _A (1)

^symmetry^, ^the universality lass aording to Pisarski and Wilzek is the three-dimensional

O(4)

^. ^Also ⁱⁿ^ase ^of

a rst order transition some kind of eetive saling behaviour an be expeted [166 ℄.

More importantly, the rst order senario inludes a seond order endpoint inthe

3d-Ising (

Z (2)

⁾ universality lass, f. gure 5.1. In this ase, one should thus enounter a situation that is dominated by

Z(2)

^-saling ^for ^a ^ritial ^point ^at ^nite ^quark ^mass

when approahing the hiral limit from the rossover regime. A subtlety in the latter

ase is thatfor the ritialpoint at nite massno exat hiral symmetryof theation

exists. Thus

ψψ

isnolongeran exatorderparameterandthequarkmassannotbe

identied to be the symmetry breaking eld. The indued unertainty might be small

as we ould think of it as a perturbation due the small quark mass just like in hiral

perturbation theory. Sine our dataare not aurate enough to ath suh eets, we

do not onsider them for the rest of this work. They must, however, be kept inmind

for future workwithinreased auray.

The appropriate quantity to loatethe hiral phase transitionis the hiral

susepti-bility,

χ _σ = ∂ ψψ

∂m _q .

^(5.12)

Evaluatingthederivativewithrespettothequarkmass,oneobtainstwodierent

on-tributions to

χ _σ

^,^see ^e.^g.^{[150 ℄.} ^W^e ^neglet^the^piee^that^isproportional to

M ⁻²

where

M

^is ^the ^fermion determinant, and onsider only the remaining part, whih is

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

200 210 220 230 240 250 260 270

< ψ  ψ >

T (MeV)

< ψ ^ ψ > (A12) 2µ|π| ² (A12)

<ψ ^ ψ> (B12) 2µ|π| ² (B12)

< ψ ^ ψ > (C12) 2µ|π| ² (C12)

Figure5.2.:Chekfor theonsistenybetween

ψψ

and

| π | ²

^by ^omparing^the^expetation

values

ψψ

and

2µ ⁰ | π | ²

^for^our^simulation^runs.

thevariane peronguration,

σ _ψψ ² = V /T

(ψψ) ²

− ψψ 2

.

^(5.13)

This quantity shows a peak assoiated with the hiral transition. Moreover, it is

ex-petedto dominate the signalof

χ σ

^,^see^e.^g.^{[167 ℄.}

Thepionnorm,

| π | ² = X

x

ψ(x) 1

2 γ 5 τ ⁺ ψ(x) ψ(0) 1

2 γ 5 τ ⁻ ψ(0)

,

^(5.14)

is interesting for twisted masssimulations beause its denition is independent of the

twist angle. Itis onneted tothehiral ondensatevia a Ward identity,

2m _q | π | ² = − ψψ

,

^(5.15)

whih hasbeenproven for twisted massQCDby Frezzotti etal.[85℄.

The ondensate that we have used for analysis has been alulated as

ψψ

by our

ollaborators in Berlin from stohasti noisy estimators, f. [86 ℄, on all available

on-gurations. We have performeda double hek by omparingthese results to the pion

norm. At leastfor the runs in Rome, B10 and C12 (see setion 5.3), the two

alula-tions are ompletely independent of eah other. Most importantly, the ode in Rome

performs the alulationof the pion norm with point soures. In any ase, pionnorm

and

ψψ

an be used to hek for onsisteny sine they desribe the same physial

quantity basedon dierent operators. We showorresponding heks in gure5.2and

gure5.3.

Notethat we usethe unrenormalised

ψψ

whih is relatedto the renormalised

hi-ral ondensate by both multipliative and additive renormalisation [85℄. The additive

4e-05 6e-05 8e-05 0.0001 0.00012 0.00014 0.00016 0.00018 0.0002 0.00022 0.00024

220 225 230 235 240 245 250 255 T (MeV)

σ ² (| π | ² ), rescaled σ ² (ψ ^ ψ)

Figure 5.3.:ChekwithB12datafortheonsistenybetweenthevarianesof

ψψ

and

| π | ²

wherethelatteronehasbeenresaledbyaonstantfator.

renormalisationisproportionaltothemassandaetssalingviolationsifpresent

onlyquantitatively,f.[105 ℄. Thereforeasalinganalysisbasedonour

ψψ

isperfetly

admissible ifwe keep

N _τ

^xed.

5.2.2. Gauge Observables

The gaugeobservables thatwe onsiderarethetraed plaquette

P = 1

6N _c N _τ N _σ ³

^ReTr

X

x

X

µ>ν

U _µ (x)U _ν (x + ˆ µ)U _µ ^† (x + ˆ ν)U _ν ^† (x)

^(5.16)

and thereal partofthe Polyakov loop,i.e.the real partof

L = 1 N _c

1 N _σ ³

^Tr

X

x N τ −1

Y

x 4 =0

U ₄ (x, x ₄ ) .

^(5.17)

Along withthe above observables,we lookat their suseptibilities,

χ _O = N _σ ³ O ²

− h O i ²

.

^(5.18)

ThePolyakovloopisof partiularinterest sineitistheorderparameter ofthepure

gauge deonnement transition. This transition is related to the breaking of entre

symmetry[168℄. Thetransformation for this symmetryis givenby

U _µ (x) →

^eⁱ

^2π/3n 1 _SU(N _c ₎ U _µ (x) , n ∈ N ,

^(5.19)

where thefator multiplying thelinkisfrom theentreof

SU (N _c )

^,^i.^e.^from

C(SU (N _c )) = { z ∈ SU (N _c ) | ∀ U ∈ SU (N _c ) : zU z ⁻¹ = U } .

^(5.20)

0 0.005 0.01 0.015 0.02 0.025

3.75 3.78 3.81 3.84 3.87 3.9 3.93

Re(L)

β

0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

3.75 3.78 3.81 3.84 3.87 3.9 3.93

χ (Re(L))

β

Figure5.4.:RealpartofthePolyakovloop(left)anditssuseptibility(right)at

aµ ⁰ = 0.040

for

κ c (β )

^on^a^lattie^of^extent

16 ³ × 8

Thegaugeation itselfisinvariant underentretransformationsbutthePolyakovloop

piksup a phasefator. Thus for unbroken entre symmetrythePolyakovloop

expe-tationvaluehasto vanish,whih orrespondsto onnement. Inthehightemperature

phasewherethePolyakovloopisnon-zero,entresymmetryisaordinglybroken. The

onnetion to onnement an be read o easily from the relation to the quark free

energy,

h| L |i ∼

^−β(F ^q ^−F ⁰ ⁾ .

^(5.21)

F ₀

^is^the ^neessary^vauumsubtration. In ase ofperfet onnement, asingle quark aquires an innite free energy suh that

h| L |i = 0

^. ^Note ^that ^the ^expliit ^entre

symmetrybreakingfor small quarkmasses isso severe for thevolumes relevant to our

studies that out of the three possible phases for

L

^only ^the ^one ^on ^the ^real ^axis ^as

ditatedbythequarkmasspersists. Therefore itissuient tousetherealpartofthe

Polyakov loop asthequantity signallingonnement for our simulations.

Im Dokument OPUS 4 | The thermal transition of quantum chromodynamics with Twisted Mass Fermions (Seite 54-57)

5. Thermal Transition for Two Quark Flavours 41

5.2. Observables

ψψ

= T V

∂ ln Z

∂m q ,

M

U A (1)

O(4)

Z (2)

Z(2)

ψψ

χ σ = ∂ ψψ

∂m q .

χ σ

M −2

M

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

200 210 220 230 240 250 260 270

< ψ  ψ >

T (MeV)

< ψ  ψ > (A12) 2µ|π| 2 (A12)

<ψ  ψ> (B12) 2µ|π| 2 (B12)

< ψ  ψ > (C12) 2µ|π| 2 (C12)

ψψ

| π | 2

ψψ

2µ 0 | π | 2

σ ψψ 2 = V /T

(ψψ) 2

− ψψ 2

.

χ σ

| π | 2 = X

x

ψ(x) 1

2 γ 5 τ + ψ(x) ψ(0) 1

2 γ 5 τ − ψ(0)

,

2m q | π | 2 = − ψψ

,

ψψ

ψψ

ψψ

4e-05 6e-05 8e-05 0.0001 0.00012 0.00014 0.00016 0.00018 0.0002 0.00022 0.00024

220 225 230 235 240 245 250 255 T (MeV)

σ 2 (| π | 2 ), rescaled σ 2 (ψ  ψ)

ψψ

| π | 2

ψψ

N τ

P = 1

6N c N τ N σ 3

X

x

X

µ>ν

U µ (x)U ν (x + ˆ µ)U µ † (x + ˆ ν)U ν † (x)

L = 1 N c

1 N σ 3

X

x N τ −1

Y

x 4 =0

U 4 (x, x 4 ) .

χ O = N σ 3 O 2

− h O i 2

.

U µ (x) →

2π/3n 1 SU(N c ) U µ (x) , n ∈ N ,

SU (N c )

C(SU (N c )) = { z ∈ SU (N c ) | ∀ U ∈ SU (N c ) : zU z −1 = U } .

0 0.005 0.01 0.015 0.02 0.025

3.75 3.78 3.81 3.84 3.87 3.9 3.93

Re(L)

β

0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

3.75 3.78 3.81 3.84 3.87 3.9 3.93

χ (Re(L))

∂m _q ,

U _A (1)

χ _σ = ∂ ψψ

∂m _q .

χ _σ

M ⁻²

< ψ ^ ψ > (A12) 2µ|π| ² (A12)

<ψ ^ ψ> (B12) 2µ|π| ² (B12)

< ψ ^ ψ > (C12) 2µ|π| ² (C12)

| π | ²

2µ ⁰ | π | ²

σ _ψψ ² = V /T

(ψψ) ²

| π | ² = X

2 γ 5 τ ⁺ ψ(x) ψ(0) 1

2 γ 5 τ ⁻ ψ(0)

2m _q | π | ² = − ψψ

σ ² (| π | ² ), rescaled σ ² (ψ ^ ψ)

| π | ²

N _τ

6N _c N _τ N _σ ³

U _µ (x)U _ν (x + ˆ µ)U _µ ^† (x + ˆ ν)U _ν ^† (x)

L = 1 N _c

1 N _σ ³

U ₄ (x, x ₄ ) .

χ _O = N _σ ³ O ²

− h O i ²

U _µ (x) →

^2π/3n 1 _SU(N _c ₎ U _µ (x) , n ∈ N ,

SU (N _c )

C(SU (N _c )) = { z ∈ SU (N _c ) | ∀ U ∈ SU (N _c ) : zU z ⁻¹ = U } .

aµ ⁰ = 0.040

16 ³ × 8

^−β(F ^q ^−F ⁰ ⁾ .

F ₀