Transition in the Chiral Limit

5. Thermal Transition for Two Quark Flavours 41

5.4. Transition in the Chiral Limit

230 235 240 245 250 255

0 0.002 0.004 0.006 0.008 0.01 T c (MeV)

1/N _τ ² Re(L)

chiral cond.

plaquette

Figure5.9.:

N τ

^-dependene ^of pseudo-ritialtemperature obtainedfromthe utuationsof thethreeobservablesforB10andB12. Thepointsforthehiralondensateand

theplaquettehavebeenshiftedby

± 0.0002

ofutoeetsinoursimulationsissmallomparedtotheombinedunertainties from

statistisandsale setting.

160 180 200 220 240 260

0 100 200 300 400 500

T c (MeV)

m _π (MeV) 1 ^st order

O(4)

Re(L) plaquette chiral cond.

Z(2) m _π,c =0 MeV

m _π,c =200 MeV

Figure 5.10.:Chiralextrapolationfor

T c (m π )

^. ^The^ts^assuming^seond^order

O(4)

^or^rst

or-derexponentsarebasedontheritialtemperaturesfromthehiralondensate.

The

Z (2)

^t^is^based^on^all^available^data^points.

mimitheorretbehaviour[79 ,171℄. Forthepossibleritial

Z(2)

^point^at^nite^quark

mass, we have to replae

m π → m ² _π − m ² _π,c

^. ^Sine^the ^value^of

m π,c

^is^not ^restrited^by

our tseither, we have looked at two extremal situations:

m _π,c = 0

^and²⁰⁰^MeV.

The resulting extrapolations for all senarios are shown in gure 5.10. Obviously

our data are not apable of a lear separation of the dierent senarios. Assuming

O(4)

universality, we obtaina hiral ritialtemperature

T _c = 154(38)

^MeV. ^The^other

possible universality lasses lead to slightly dierent values. Most importantly, the

value for the rst order t,

T c = 191(23)

^MeV, ^seems ^to ^be ^slightly ^above ^the ^values

expeted inother investigations [12 ℄. But the extrapolation relying on the rst order

exponents that we have applied for the sake of omparison sine it has also been

introdued elsewhere [79 , 171℄ should really be questioned as we have based on the

smoothness of our signals strong reason to think thatour pion masses fall into the

rossoverregime and thus, before entering the rst orderregion, the ritialend point

has to be enountered. Note that the existene of a ritial point at nite mass also

renders a hiral extrapolationimpossible.

For xed

N _τ = 12

^,^assuming^to ^be^lose ^enough ^to^the^ontinuum, ^we ^an ^also^apply

equation (5.9) withexternal eld

h = 2aµ ₀

^,^see ^gure^5.11. ^A ^t ^to ^all ^three ^masses^is

notfeasiblewith

O(4)

^oeients^indiating^large^saling^violationsⁱⁿ^the^heaviest^mass.

Restriting to the two lighter masses,there areas manydata points ast parameters.

However,wean still estimate

β

hiral

(N _τ = 12) ≈ 3.63 .

^(5.24)

Thisorrespondsto

T _c (m _π = 0) ≈ 138(54)

^Me^V ^where ^the^errors^are^due^to ^the

extrap-olation of the sale setting to very small values of

β

^. ^In ^any ^ase, ^this ^value ^of

T c

^is

onsistent withthe one obtained fromtheprevious t. Weusethat estimatefor

β

hiral toompareourdatawiththemagnetiequationofstate(5.5),wherewefollowprevious

3.85 3.9 3.95 4 4.05 4.1 4.15

0.006 0.008 0.01 0.012 0.014 0.016

β c

h=2aµ ₀ chiral cond.

Re(L) plaquette fit to chiral cond.

Figure5.11.:Critialouplings

β

^as^funtion ^of^the^external^eld

h = 2aµ ⁰

^at

N τ = 12

^. ^The

tinludestheouplingsobtainedfromthevarianeof

ψψ

forthetwolightest

masses,A12andB12.

studies [161, 151 , 105℄. Inluding possible saling violations [105 ℄ and printing all t

parameters expliitly,wehave

ψψ

= h ^1/δ cf (dτ /h ^1/(δβ) ) + a _t τ h + b ₁ h + . . . .

^(5.25)

We have tted our data by using either one or both violation terms. The dataset

C12 annot be aommodated by any of these possibilities, leading to large values of

χ ²

^. ^On^the^other^hand,^ts^to^A12+B12^are^feasibleⁱⁿ^allombinations, givinga

β

^hiral

onsistent withour previous determination. The ts work with either orretion term

alone, but when both are admitted

a _t ≈ 0

^within ^errors, ^see ^table ^5.4. ^In ^gure ^5.12,

we show a ombined t to A12 and B12 xing

β

hiral

= 3.63

^from ^our independent determination and

a _t = 0

^with

χ ² /

^dof

= 0.52

^. ^The ^fat ^that^these ^ts^are ^not ^able ^to

inlude the C12 data indiates a mass whih is outside the regime where the leading

orretions,equation(5.25),areappliable. Thisisinagreement withgure5.11,where

also the heaviest point annot be inluded in the saling desription. Furthermore,

the relation for the pseudo-ritial oupling,

β(h)

^, ^has ^been ^derived ^using ^a ^double

derivative,

∂ ² χ _σ (x(h, τ)))

∂h∂τ = 0 ,

^(5.26)

andthusfromtheleadingorretionsofequation(5.25)onlythetermproportionalto

a _t

ontributestoviolations in

β(h)

^. ^As^we^expet^a^small

a _t

^from^our^ts, ^thisobservation strengthens the ondene that the two lighter masses are properly disribed by the

salingt asshowningure5.11,andthatthepointfor theheaviestmass,C12, suers

fromhigher orderviolations.

Sinewe areinarangeof thesalingvariable

τ /h ^1/(δβ)

^where ^the^saling^funtion ^is

ratherat,judgementon whetherthereareadditionalviolations ofthe

O(4)

^behaviour

ornotisdiult. Repeatingthisexerisefortherstordersenariowithendpointdoes

0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085

3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6

< ψ  ψ >/h 1/ δ

τ h ^-1/( ^δβ ⁾

fit to A12+B12 A12

B12 C12

Figure 5.12.:Saling for the bare

ψψ

for the data at

N τ = 12

^and ^modelling ^of ^saling

violations. ThetshownisfortheombinedA12andB12data.

ID data

β

hiral

c d a _t b ₁ χ ²

^/dof

1 A12 3.53(13) 0.149(72) 0.354(45) 0 0 0.44

2 B12 3.38(19) 0.24(20) 0.38(14) 0 0 0.98

3 C12 4(18) 0.2(18.1) 0.5(12.2) 4e+8

4 A12+B12 3.29(2) 1(2) 1.0(1.3) 0 0 1.8

5 A12+B12 3.63(4) 0.37(62) 1.5(1.7) 0 1.2(2) 0.55

6 A12+B12 3.55(4) 0.8(1.6) 1.6(2.1) 1.2(3) 0 0.8

7 A12+B12 3.67(7) 0.4(1.3) 2(5) -0.79(99) 1.8(7) 0.52

8 A12+B12 3.63 0.6(1.3) 2.1(3.4) -0.3(5) 1.4(3) 0.52

9 A12+B12 3.63 0.4(4) 1.6(1.2) 0 1.19(2) 0.52

10 A12+B12 3.63 0.7(1.7) 2(3) 1.97(4) 0 1.3

11 A12+B12+C12 3.63 0 5e+7

12 A12+B12+C12 3.63 5e+7

13 A12+B12+C12 5e+7

Table5.4.:Fits forthe (violated)saling funtion

cf(dx) + a t τ h ¹⁻¹ ^/δ + b 1 h ¹⁻¹ ^/δ

^. ^Numbers

in boldfae havebeenxed before tting. TheC12data annotbebroughtinto

agreementwiththesalingfuntion. Iftheviolatingterms

b 1

^and

a t

^are^omitted,

the saling violations seem to be absorbed by

β

^hiral ^that ^beomes onsiderably smaller (seets 1,2,4). Weannotdisentangle theterms

b ¹

^and

a t

^but ^the^ts ⁷

and8wherebothparametersarefreeseemtosuggest

a t ≈ 0

not give further insight asthe ombinations of exponentsare too lose. Therefore our

dataare fullyonsistent withthe

O(4)

^senario, ^but ^do ^not ^rule ^out ^thepossibility of therst orderase. This wouldrequire drastially smallerpion masses ombined with

nitesize studies, asthehiral salingseemsto bevalid if

m _π . 300

^MeV.

Im Dokument OPUS 4 | The thermal transition of quantum chromodynamics with Twisted Mass Fermions (Seite 63-67)

5. Thermal Transition for Two Quark Flavours 41

5.4. Transition in the Chiral Limit

230 235 240 245 250 255

0 0.002 0.004 0.006 0.008 0.01 T c (MeV)

1/N τ 2 Re(L)

chiral cond.

plaquette

N τ

± 0.0002

160 180 200 220 240 260

0 100 200 300 400 500

T c (MeV)

m π (MeV) 1 st order

O(4)

Re(L) plaquette chiral cond.

Z(2) m π,c =0 MeV

m π,c =200 MeV

T c (m π )

O(4)

Z (2)

Z(2)

m π → m 2 π − m 2 π,c

m π,c

m π,c = 0

O(4)

T c = 154(38)

T c = 191(23)

N τ = 12

h = 2aµ 0

O(4)

β

(N τ = 12) ≈ 3.63 .

T c (m π = 0) ≈ 138(54)

β

T c

β

3.85 3.9 3.95 4 4.05 4.1 4.15

0.006 0.008 0.01 0.012 0.014 0.016

β c

h=2aµ 0 chiral cond.

Re(L) plaquette fit to chiral cond.

β

h = 2aµ 0

N τ = 12

ψψ

ψψ

= h 1/δ cf (dτ /h 1/(δβ) ) + a t τ h + b 1 h + . . . .

χ 2

β

a t ≈ 0

β

= 3.63

a t = 0

χ 2 /

= 0.52

β(h)

∂ 2 χ σ (x(h, τ)))

∂h∂τ = 0 ,

a t

β(h)

a t

τ /h 1/(δβ)

O(4)

0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085

3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6

< ψ  ψ >/h 1/ δ

τ h -1/( δβ )

fit to A12+B12 A12

B12 C12

ψψ

N τ = 12

β

c d a t b 1 χ 2

cf(dx) + a t τ h 1−1 /δ + b 1 h 1−1 /δ

b 1

a t

β

b 1

a t

1/N _τ ² Re(L)

m _π (MeV) 1 ^st order

Z(2) m _π,c =0 MeV

m _π,c =200 MeV

m π → m ² _π − m ² _π,c

m _π,c = 0

T _c = 154(38)

N _τ = 12

h = 2aµ ₀

(N _τ = 12) ≈ 3.63 .

T _c (m _π = 0) ≈ 138(54)

h=2aµ ₀ chiral cond.

h = 2aµ ⁰

= h ^1/δ cf (dτ /h ^1/(δβ) ) + a _t τ h + b ₁ h + . . . .

χ ²

a _t ≈ 0

a _t = 0

χ ² /

∂ ² χ _σ (x(h, τ)))

a _t

a _t

τ /h ^1/(δβ)

τ h ^-1/( ^δβ ⁾

c d a _t b ₁ χ ²

cf(dx) + a t τ h ¹⁻¹ ^/δ + b 1 h ¹⁻¹ ^/δ

b ¹

m _π . 300