Pseudoritial temperature and equation of statefrom Nf = 2 twisted mass lattie QCDFlorian BurgerHumboldt-Universität zu BerlinSt. Goar � Marh, 2013

2 / 25

tmfT-Collaboration:E. M. Ilgenfritz (Joint Institute for Nulear Researh, Dubna)M. Kirhner, M. Müller-Preuÿker (HU Berlin),M. P. Lombardo (INFN Frasati),C. Urbah (Uni Bonn),O. Philipsen, L. Zeidlewiz, C. Pinke (Uni Frankfurt)

OutlineMotivation and SetupT and Chiral SenariosThermodynami Equation of StateConlusions

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OutlineMotivation and SetupT and Chiral SenariosThermodynami Equation of StateConlusions

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Motivation◮ Explore �nite T phase transition/rossover for Nf = 2 QCD atvanishing hemial potential◮ Order of transition in the hiral limit not known yet for Nf = 2◮ Argued to be in universality lass of O(4) 3-dim spin modell[R. D. Pisarski, F. Wilzek, 84℄However 1st order not ruled out(e. g. C.Bonati et al. 2009, C. Bonati et al. 2011)◮ Nf = 2 equation of state → useful to study onset of massthresholds in thermodynamis◮ Alternative disretization worthwhile to study systematis◮ Considered observables (so far):Suseptibilities, renorm. L and 〈ψ̄ψ〉, equation of state,gluon and ghost propagators (see A. Sternbek's talk) 5 / 25

Twisted Mass Lattie Regularization◮ Nf = 2 Wilson twisted mass ation at maximal twist:Sf [U, ψ, ψ] =∑x χ(x) (1− κH[U] + 2iκaµγ5τ3)χ(x)

ψ =1√2 (1 + iγ5τ3)χ and ψ = χ 1√2(1 + iγ5τ3)

◮ Advantage: when κ tuned to ritial value κ :Automati O(a) improvement◮ Disadvantage: expliit �avor symmetry breaking (mostly small, buthas to be heked)◮ Tree level improved gauge ation:Sg [U] = β(0∑P [1− 13ReTr (UP)] + 1∑R [1− 13ReTr (UR)]) 6 / 25

[R. Frezzotti, G.C. Rossi, 2004℄

Simulation Points

320 400 480 700

mπ[MeV]

A B C D

Nτ = 6

8

10

12

7 / 25

OutlineMotivation and SetupT and Chiral SenariosThermodynami Equation of StateConlusions

8 / 25

Signals for the Crossover: Polyakov Loop◮ Polyakov loop suseptibility (no onvining signal)

χRe(L)χRe(L)

β

4.204.154.104.054.003.953.903.85

0.09

0.085

0.08

0.075

0.07

0.065

0.06

0.055 9 / 25

mπ ≈ 400 MeV:

Disonneted Chiral Suseptibility

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σ〈ψ̄ψ〉 = V /T (〈ψ̄ψ2〉− 〈ψ̄ψ〉2) mπ ≈ 320 MeV:Gaussian fitA12σ2ψ̄ψβ

4.003.943.883.82

0.8

0.7

0.6

0.5

0.4

0.3

0.2mπ ≈ 400 MeV:Gaussian fit

B12σ2ψ̄ψ

β

4.023.963.903.84

0.5

0.4

0.3

0.2

0.1

mπ ≈ 480 MeV:Gaussian fit

C12σ2ψ̄ψ

β

4.064.003.943.88

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

Renormalized Re(L), 〈ψ̄ψ〉

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R〈ψψ〉 = 〈ψψ〉(T , µ) − 〈ψψ〉(0, µ) + 〈ψψ〉(0, 0)〈ψψ〉(0, 0) 0

0.2

0.4

0.6

0.8

1

180 200 220 240 260 280

R<

ψψ

>

T [MeV]

Nτ = 12Nτ = 10

〈Re(L)〉R = Re(L) exp (V (r0)/2T ) 0

0.5

1

1.5

2

180 220 260 300 340 380 420

<R

e(L)

>R

T [MeV]

Nτ = 12Nτ = 10

Renormalized Re(L), 〈ψ̄ψ〉

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R〈ψψ〉 = 〈ψψ〉(T , µ) − 〈ψψ〉(0, µ) + 〈ψψ〉(0, 0)〈ψψ〉(0, 0) 0

0.2

0.4

0.6

0.8

1

180 200 220 240 260 280

R<

ψψ

>

T [MeV]

Nτ = 12Nτ = 10

〈Re(L)〉R = Re(L) exp (V (r0)/2T )B12Fit

〈Re(L)〉R

T [MeV]

360320280240200

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

Results

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Ensemble B12 C12 D8T [MeV], σ2〈ψ̄ψ〉: 217(5) 229(5) 251(10)T [MeV], 〈Re(L)〉R : 249(5) 258(5) 266(11)Signals for deon�nement and restoration of hiral symmetry atdi�erent loations in rossover region

O(4) Magneti EoS◮ Near ritial point of a 2nd order transition: universality◮ Expet data to ollaps on universal urve [J. Engels & T. Mendes, 99℄◮ 〈ψ̄ψ〉 = h1/δ f (d z) + aττh + b1h + . . .

︸ ︷︷ ︸saling violating termsτ = β − βhiral, h = 2 a µ, z = τ/h1/(β̃δ)

◮ Connetion to spin model:〈ψ̄ψ〉 ∼ M (magnetization)µ ∼ H (external �eld)β ∼ T (temperature) 14 / 25

f(z)

O(4)

z

43210-1-2-3

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

[S. Ejiri et al., 2009℄

O(4) Magneti EoS

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Pion Mass:C12: 480 MeVB12: 400 MeVA12: 320 MeVC12B12A12z

〈ψ̄ψ〉/h

1/δ

43.532.521.510.50

0.13

0.12

0.11

0.1

0.09

0.08

0.07

0.06

0.05

0.04 C12 corr.B12 corr.A12 corr.

z

〈ψ̄ψ〉/h

1/δ

-(s

caling

corr

.)

3.532.521.510.5

0.035

0.03

0.025

0.02

0.015

0.01Result: βhiral ≈ 3.76(2) →T(mπ = 0) ≈ 166 MeV

Chiral Limit Senarios from Spin Models

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◮ From universal funtion h1/δf (z) one an show:T(mπ) = T(0) + Am2/(β̃δ)π1st : T(0) = 182(14) MeVO(4): T(0) = 152(26) MeV 140 160 180

200

220

240

260

0 100 200 300 400 500

Tc

(MeV

)

mπ (MeV)

1st order

O(4)

mπ,c = 0 MeV

mπ,c = 200 MeVZ(2)

OutlineMotivation and SetupT and Chiral SenariosThermodynami Equation of StateConlusions

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Integral Method◮

ǫ =TV ∂ lnZ∂ lnT ∣∣∣∣V p = T ∂ lnZ∂V ∣∣∣∣T

◮ Trae anomaly (interation measure):I = ǫ− 3p = −TV d lnZd ln a◮ Starting point for p(T ) and ǫ(T ) viaIT 4 = T ∂∂T ( pT 4) pT 4 − p0T 40 = ∫ TT0 dτ ǫ− 3pτ5 ∣∣∣∣LCPon lines of onstant physis (LCP) 18 / 25

But: V = N3σ a3T = 1Nτ a

Trae Anomaly◮ IT 4 = ǫ− 3pT 4 = − TT 4 V d lnZd ln a

= N4τ

(adβda )(03 〈ReTrUP〉sub + 13 〈ReTrUR〉sub+∂κ∂β

〈χ̄H[U]χ〉sub − (2aµ∂κ∂β + 2κ ∂(aµ)∂β )〈χ̄iγ5τ3χ〉sub)◮ Starting point for p(T ) and ǫ(T ) by integral method◮ Subtrated expetation values: 〈. . .〉sub ≡ 〈. . .〉T>0 − 〈. . .〉T=0

→ interpolations for T = 0 data◮ Preliminary results for mπ ≈ 400 MeV and mπ ≈ 700 MeV 19 / 25

Trae anomaly - Lattie Artifats

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mπ ≈ 700 MeV:Nτ = 4Nτ = 6Nτ = 8Nτ = 10ǫ−3p

T 4

T [MeV]

800700600500400300200

12

10

8

6

4

2

0

-2

mπ ≈ 400 MeV:Nτ = 4Nτ = 6Nτ = 8Nτ = 10Nτ = 12ǫ−3p

T 4

T [MeV]

800700600500400300200

12

10

8

6

4

2

0

-2

Trae Anomaly, Tree Level Corretions◮ Observe large lattie artifats in IT 4◮ Lattie pressure in the free limit pLSB known: [P. Hegde et al., 2008℄◮ Twisted mass ation: [O. Philipsen & L. Zeidlewiz, 2010℄◮ Correted by division by pLSB/pCSB [S. Borsanyi et al., 2010℄

21 / 25mπ ≈ 700 MeV: mπ ≈ 400 MeV:

correcteduncorrectedǫ−3p

T 4

T = 318 MeV

1/N2τ

0.030.0250.020.0150.010.0050

10

9

8

7

6

5

4

3

2

correcteduncorrectedǫ−3p

T 4

T = 285 MeV

1/N2τ

0.030.0250.020.0150.010.0050

14

12

10

8

6

4

2

correcteduncorrectedǫ−3p

T 4

T = 391 MeV

1/N2τ

0.030.0250.020.0150.010.0050

6

5

4

3

2

1

0

Correted Trae Anomaly & Interpolation

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mπ ≈ 700 MeV:Nτ = 6Nτ = 8Nτ = 10

Interpolationǫ−3pT 4

T [MeV]

900700500300100

10

8

6

4

2

0

-2

mπ ≈ 400 MeV:Nτ = 4Nτ = 6Nτ = 8Nτ = 10Nτ = 12

Interpolationǫ−3pT 4

T [MeV]

900700500300100

10

8

6

4

2

0

-2Interpolation for orreted I/T 4 [S. Borsanyi et al., 2010℄ :IT 4 = exp `−h1t̄ − h2t̄2´ · „h0 + f0 {tanh f1t̄ + f2}1 + g1 t̄ + g2 t̄2 «

Pressure and Energy Density

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pT 4 − p0T 40 = ∫ TT0 dτ ǫ− 3pτ5 ∣∣∣∣LCPmπ ≈ 700 MeV:0

4

8

12

16

200 300 400 500 600 700 800

1 1.5 2 2.5 3

T [MeV]

× TTc

3pSB

D mass (ǫ− 3p)/T4

ǫ/T 4

3p/T 4

mπ ≈ 400 MeV:0

4

8

12

16

200 300 400 500 600

1 1.5 2 2.5

T [MeV]

× TTc

3pSB

B mass (ǫ− 3p)/T4

ǫ/T 4

3p/T 4

OutlineMotivation and SetupT and Chiral SenariosThermodynami Equation of StateConlusions

24 / 25

Conlusions & Outlook◮ Conlusions:- T for pion masses in the range 300 - 700 MeV- Chiral limit so far inonlusive- O(4) saling ompatible up to saling violation terms- Thermodynami equation of state for two pion masses- Have to further improve preision at T > 0 and T = 0◮ Outlook:- Nf = 2 + 1 + 1 25 / 25

Comparison with DIK-Collaboration

26 / 25

(G. Shierholz et. al) from σ〈ψ̄ψ〉:our data (tmfT)

DIK: arXiv:1102.4461DIK: arXiv:0910.2392

r0Tc

r0mπ

1.81.61.41.210.80.60.40.2

0.75

0.7

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.3

Renormalization of Re(L)〈TrL̂†(~x)TrL̂(~0)〉 = e−Fq̄q (~x,T )−F0(T )T →T→0 e−V (|~x|)T

27 / 25

Renormalization of 〈ψ̄ψ〉〈ψψ〉ren = ZP(〈χ̄iγ5τ3χ〉bare + µ P(β)a2 )+ . . .

28 / 25

T = 0 Subtrations & Interpolations

29 / 25

example: plaquette:ReTr〈UP 〉0.64

0.6

0.56

Residuals of fit

β

4.34.13.9

30

-3

Residuals of fit

β

4.34.13.9

30

-3

β = 3.70, aµ = 0.009, mπ ≈ 400 MeV 0.55

0.552

0.554

0.556

0.558

0.56

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

plaq

uette

No traj.

beta=3.70, 16^3x24

β-Funtion

30 / 25

[M. Cheng et al.: Phys.Rev. D77:014511, 2008℄“a dβda ” = − ` rχa ´ „ d( rχa )dβ «−1` rχa ´ (β) = 1+n0R(β)2d0(a2L(β)+d1R(β)2) R(β) = a2L(β)a2L(3.9) r0 = 0.420(15) fm

Interpolationr0/arχ/a

β

4.64.354.24.053.93.83.65

14

12

10

8

6

4

2-loopadβda

β

4.64.354.24.053.93.83.65

0

-0.1

-0.2

-0.3

-0.4

-0.5

-0.6

-0.7

-0.8

-0.9

Tree Level Corretions (loser look), SB (Free) LimitLattie:pLF (µ)T 4 = 3∫[0,2π)3 d3k(2π)3 1Nt Nt−1∑n=0 lnDet (|G (k)|2 + (aµ)2)G (k) = (am) + 2r∑µ

sin2(akµ2 )+ i∑µ

γµ sin(akµ)Continuum: pCF (µ/T )T 4 = 278 π290g(µ/T )g(µ/T ) = 3607π4 ∫ ∞µ/T dx x√x2 − (µ/T )2 ln(1 + e−x)Corretion: Nτ 4 6 8 10 12pLSB/pCSB 2.586 1.634 1.265 1.134 1.084 . 31 / 25

Lines of Constant Physis, onstant mπ

32 / 25

mπ ≈ 700 MeV: presently ful�lled up to 10 %D8

mPS [MeV]

β

4.54.44.34.24.143.93.83.73.6

900

850

800

750

700

650

600

550

500

mπ ≈ 400 MeV:340

360

380

400

420

440

460

480

3.70 3.8 3.9 4.05 4.2

β

mPS [MeV]

B mass

ETMCtmfT

Motivation and SetupTc and Chiral ScenariosThermodynamic Equation of StateConclusionsAppendix