Extended QCD phase diagram from the lattice...Outline 1 QCD phase diagram ta real and imaginary 2...

Extended QCD phase diagramfrom the lattice

F. Cuteri, C. Czaban, O. Philipsen, C. Pinke, A. Sciarra

Goethe University - Frankfurt am Main

University of Giessen - February 1, 2017

[10.1103/PhysRevD.93.054507, 10.1103/PhysRevD.93.114507]

10.1103/PhysRevD.93.05450710.1103/PhysRevD.93.114507

Monte Carlo methods & cutoff effects

Z =∫DU

∏Nf

detD(µf ,mf ;U)e−SG (β;U), β =

2Ncg2

http://www.jicfus.jp

Importance sampling ↔ positive Boltzmann weightSign problem for real µ 6= 0↔ detD ∈ CThermodynamical studies with T = 1

a(β)Nτ

Thermodynamic and continuum limit at �xed T byNτ ,Nσ →∞, a→ 0

Slat = Scont + c1a + c2a2 + . . .

Non-unique discretization: di�erent lattice actions ↔ di�erent cuto� e�ectsWilson: LO O(a)

I local, doublers-free, correct CLI no chiral symmetry

Staggered: LO O(a2)I theoretically not fully understood (rooting)I remnants of chiral symmetry

F. Cuteri - Goethe University Extended QCD phase diagram from the lattice Feb 01, 2017 2 / 33

http://www.jicfus.jp

The sign problem in lattice QCD

-2 -1 1 2x

-1.0

-0.5

0.5

1.0

ã-x2+ä Μ x

Μ=40

Μ=0


Outline

1 QCD phase diagram at real and imaginary µ

2 Extended QCD Columbia plot (µ2 6= 0)

3 The order of the thermal phase transition

4 Extrapolation from imaginary chemicalpotentialNature of the Roberge-Weiss endpoint at Nf = 2

Z2 line in mπ −(µT

)2at Nf = 2

5 Extrapolation from non-integer Nf at µ = 0Z2 line in mZ2 − (Nf )

6 Outlook


Outline






)2at Nf = 2


6 Outlook


The QCD phase diagram at real µ

http://nica.jinr.ru/

PQCD

Models Tµ � 1Lattice QCD Tµ � 1heavy-ion collisions

I RHIC-BESI NICAI Nuclotron-MI LHCI FAIR

Location and order of thermal phase transitions as a function of µ?

Existence of a critical endpoint (CEP) at µ 6= 0?


http://nica.jinr.ru/

The QCD phase diagram at µ = iµithe Roberge-Weiss symmetry

End or meeting points

First order lines

Roberge and Weiss (1986),10.1016/0550-3213(86)90582-1

Z (µ) = Z (−µ)

Z( µT

)= Z

(µ

T+ i

2πN

Nc

)µci = (2k + 1)

πT

Nc, k ∈ Z

|L(n)|e−iϕ = 13TrC

[Nτ−1∏n0=0

U0(n0,n)

]

〈ϕ〉 = 2nπ3

with n ∈ {0, 1, 2}

0 < µi <π

3

RW endpoint: true phase transition

Meeting of RW and chiral/decon�nement phase transitions continued to µ

Rich phase structure depending on Nf and mq


10.1016/0550-3213(86)90582-1

T −(µT

)2 −mu,d phase diagram

Z(2)1st triple

1st triple

1st

1st

Crossover

Tricritical points

1st order

Roberge-Weiss plane

de Forcrand and Philipsen (2002), 10.1016/S0550-3213(02)00626-0,D'Elia, Lombardo (2003), 10.1103/PhysRevD.67.014505

Coarse lattices

All β critical at µci1st order regionsextend till µ = 0

I mu,d → 0I mu,d → ∞

2nd order criticalmasses mc1,m

c2 at

µ = 0


10.1016/S0550-3213(02)00626-0 10.1103/PhysRevD.67.014505

Outline






)2at Nf = 2


6 Outlook


2D Columbia plot ms −mu,d at µ = 0Dependence of the order of the thermal phase transition on mq

Breaking of global Z (3) for mu,d ,ms →∞Restoration of global SUL(Nf )× SUR(Nf ) for mu,d ,ms → 0

?Current knowledge (no CL)

Order parameters

Polyakov loop for mu,d ,ms →∞P = Tr e i

∫Atdt

Chiral condensate for mu,d ,ms → 0ψ̄ψ = ∂ logZ/∂mu,d

Svetitsky, Ya�e (1982), 10.1016/0550-3213(82)90172-9Pisarski, Wilczek (1984), 10.1103/PhysRevD.29.338

Aoki et al. (2006), 10.1038/nature05120de Forcrand and Philipsen (2007), 10.1088/1126-6708/2007/01/077


10.1016/0550-3213(82)90172-910.1103/PhysRevD.29.338 10.1038/nature05120 10.1088/1126-6708/2007/01/077

2D Columbia plot ms −mu,d at µ = 0At least two possible scenarios on the nature of Nf = 2 chiral transition

Second order scenario First order scenario

Direct (expensive) approach: µ = 0 and mu,d → 0Indirect (cheaper) approach: µ = iµi , bigger mu,d and scaling laws

Svetitsky, Ya�e (1982), 10.1016/0550-3213(82)90172-9Pisarski, Wilczek (1984), 10.1103/PhysRevD.29.338

Aoki et al. (2006), 10.1038/nature05120de Forcrand and Philipsen (2007), 10.1088/1126-6708/2007/01/077


10.1016/0550-3213(82)90172-910.1103/PhysRevD.29.338 10.1038/nature05120 10.1088/1126-6708/2007/01/077

2D Columbia plot ms −mu,d at µ = iπT3Assuming the ��rst order scenario� is realized

µ = 0 µ = iπT3

Tricritical

Tricritical


3D Columbia plot(µT

)2 −ms −mu,d

Tricritical

Tricritical

Tric

ritical

Tricritical��7MKR�TVSFPIQ

��23��WMKRTVSFPIQ

courtesy of A. Sciarra arXiv:1610.09979


arXiv:1610.09979

(µT

)2 −mu,d Columbia plot

Tric

ritical

Tricritical

��7MKR�TVSFPIQ

��23��WMKRTVSFPIQ

courtesy of A. Sciarra arXiv:1610.09979

From mtric1 in RW plane µ = iµi =iπT3

Send µi → 0 mapping µZ2i (mu,d) (3D Ising)Expected tricritical scaling m

2/5u,d = C

[(µT

)2 − ( µT

)2tric

]


arXiv:1610.09979

Towards real-µ Columbia plot

Analytical continuation on constraint that no non-analyticity is met

Existence of 2nd order CEP depending on the bending of the Z2 surfacescontinued from µi


Outline






)2at Nf = 2


6 Outlook


Establishing the order of the transition

Analysis of the order parameter 〈O〉 distribution and its moments

Bn(〈O〉, β,mq, µ,V ) ≡ Bn(β,mq, µ,Nσ) =〈(O − 〈O〉)n〉〈(O − 〈O〉)2〉n/2

O = LIm, ψ̄ψ

∀mq identify βc by{the condition B3(β,mq, µ,Nσ) = 0 ∀Nσ;the crossing of B4(β,mq, µ,Nσ) for di�erent Nσ.

B4(β(c),mq, µ,Nσ) gives the order of the transition as function of β,mq or µ

B4(β(c),mq, µ,Nσ) =〈(O − 〈O〉)4〉〈(O − 〈O〉)2〉2

∣∣∣∣βc

∼V→∞

1, 1storder;

1.604, 2ndorderZ2;

3, crossover.


Finite Size Scaling (FSS) analysisCritical exponent ν by either �t or quantitative collapse

V1 V2>V1 V3>V2>V1 V=¥

0

1

1.604

3

HX-XcL

B4

B4(X ,Nσ) = B4(Xc ,∞) + a1 x + a2 x2 +O(x3)

x ≡ (X − Xc)N1/νσ , X = β, m,µiT


Outline






)2at Nf = 2


6 Outlook


Roberge-Weiss transition in Nf = 2 QCD

(Unimproved) Wilson fermions on Nτ = 4, 6 [10.1103/PhysRevD.93.054507]

µiT

= π ± 2πk, k ∈ Z

T =1

a(β)Nτ→ aµi =

π

Nτ

O = LIm

κ =1

2(am + 4)


10.1103/PhysRevD.93.054507



−0.08 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 0.08Lwith zero meanIm

0

1000

2000

3000

4000

5000

6000

7000

coun

ts

Histogram of Lwith zero meanIm5.7800 5.7840 5.7880 5.7900 5.7920 5.7940

µiT

= π ± 2πk, k ∈ Z

T =1

a(β)Nτ→ aµi =

π

Nτ

O = LIm

κ =1

2(am + 4)


10.1103/PhysRevD.93.054507



5.775 5.780 5.785 5.790 5.795

β

1.0

1.5

2.0

2.5

3.0

B4(L

wit

hze

rom

ean

Im

)

Nσ=20 (raw)Nσ=24 (raw)Nσ=30 (raw) µi

T= π ± 2πk, k ∈ Z

T =1

a(β)Nτ→ aµi =

π

Nτ

O = LIm

κ =1

2(am + 4)


10.1103/PhysRevD.93.054507



1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

5.78 5.785 5.79 5.795 5.8

B4pL

Imq

β

Nσ “ 20Nσ “ 24Nσ “ 30

µiT

= π ± 2πk, k ∈ Z

T =1

a(β)Nτ→ aµi =

π

Nτ

O = LIm

κ =1

2(am + 4)

�t reweighted data by B4(β,Nσ) = B4(βc ,∞) + a1 (β − βc)N1/νσ + . . . .Ferrenberg, Swendsen (1989), 10.1103/PhysRevLett.61.2635


10.1103/PhysRevD.93.054507 10.1103/PhysRevLett.61.2635



0.3

0.4

0.5

0.6

0.7

0.08 0.10 0.12 0.14 0.16

ν

κ

2nd order

Tricritical

1st triple

Nτ = 4 Nτ = 6

Nτ=4 Nτ=60.6301

13

12

κ

νκtricr.heavy = 0.10(1), κ

tricr.light = 0.155(5)

κtricr.heavy = 0.11(1), κtricr.light = 0.1625(25)


10.1103/PhysRevD.93.054507



Pinke, Philipsen (2014), 10.1103/PhysRevD.89.094504, F.C. et al (2015), 10.1103/PhysRevD.93.054507

Bonati et al. (2011), 10.1103/PhysRevD.83.054505

0.3

0.4

0.5

0.6

0.7

500 1000 1500 2000 2500 3000 3500 4000 4500

ν

mπ {MeV}

2nd order

Tricritical

1st triple

ë staggeredNτ “ 4 Nτ “ 6 Nτ “ 8 Nτ “ 4

mtricr. heavyπ = 3659+589−619 MeV, m

tricr. lightπ = 669

+95−81 MeV

0.12 fm . a . 0.18 fm


10.1103/PhysRevD.93.05450710.1103/PhysRevD.89.09450410.1103/PhysRevD.93.05450710.1103/PhysRevD.83.054505



Pinke, Philipsen (2014), 10.1103/PhysRevD.89.094504, F.C. et al (2015), 10.1103/PhysRevD.93.054507


0.3

0.4

0.5

0.6

0.7

500 1000 1500 2000 2500 3000 3500 4000 4500

ν

mπ {MeV}

2nd order

Tricritical

1st triple

ë staggeredNτ “ 4 Nτ “ 6 Nτ “ 8 Nτ “ 4

Tension between results for di�erent discretizations (Wilson, staggered)

Need larger Nτ to attain continuum limit


10.1103/PhysRevD.93.05450710.1103/PhysRevD.89.09450410.1103/PhysRevD.93.05450710.1103/PhysRevD.83.054505


(Unimproved) Staggered fermions on Nτ = 6 [Philipsen and Sciarra arXiv:1610.09979]

0.003 0.005 0.007 0.009 0.011

0.3

0.4

0.5

0.6

0.7

Only 2 volumes

Still running

ν

0.20 0.30 0.40 0.50 0.60 0.70

Larger volumeneeded

Still running

m

Tricritical

2nd order

1st triple

Simulations ongoing

Waiting for pion mass measurementShift wrt Nτ = 4 (d'Elia, San�lippo (2009), 10.1103/PhysRevD.80.111501Bonati et al. (2011), 10.1103/PhysRevD.83.054505)


arXiv:1610.09979 10.1103/PhysRevD.80.111501 10.1103/PhysRevD.83.054505

Z2 critical line in mπ −(µT

)2in Nf = 2 QCD

(Unimproved) Wilson fermions on Nτ = 4 [10.1103/PhysRevD.93.114507]

Starting point mtric1 at Nτ = 4 (Pinke, Philipsen (2014), 10.1103/PhysRevD.89.094504)

chiraltransition

(crossover/first order)

RW transition

triple point

2.orderendpoint

simulation point

µiT /

π3

T

0 1

O = ψ̄ψ = Nf TrD−1

T =1

a(β)Nτ→ aµi =

π

3Nτ

κ =1

2(am + 4)

�xed κ←→ B4(µiT,Nσ) = B4(

µciT,∞) + a1

[(µiT

)2−(µciT

)2]N1/νσ + . . .

�xed µ = 0←→ B4(κ,Nσ) = B4(κ,∞) + b1[1

κ− 1κc

]N1/νσ + . . .


10.1103/PhysRevD.93.11450710.1103/PhysRevD.89.094504

Different discretizations compared

( µRWIT )2

-1

-0.8

-0.6

-0.4

-0.2

0

0 100 200 300 400 500 600 700 800 900 1000

(µI T)2

mπ

mZ(2),WilsonπmZ(2),Staggeredπmtric,Staggeredπ

mtric,Wilsonπ (Nτ = 4)mtric,Wilsonπ (Nτ = 6)

mCrossoverπ (Twisted Mass, Nτ = 12)mCrossoverπ (Clover, Nτ = 16)

a & 0.25 fm

κc(µ = 0) = 0.1815(1)

mcπ(µ = 0) ≈ 560 MeV

Wilson O(a) improved results from Burger et al. (2013), 10.1103/PhysRevD.87.074508(Twisted Mass) and Brandt et al. (2014), arXiv:1310.8326 (clover)

No need for extrapolation using tricritical scaling law

Only qualitative agreement between di�erent discretizations(cf. staggered Bonati et al. (2014), 10.1103/PhysRevD.90.074030)

Expected shift of µZ2i (mu,d) to smaller mu,d as a→ 0


10.1103/PhysRevD.87.074508arXiv:1310.8326 10.1103/PhysRevD.90.074030

Outline






)2at Nf = 2


6 Outlook


Nf continuous parameter in the pathintegral

Partition function describing Nf �avours of degenerate mass m

ZNf (m, µ,Nf ) =

∫DA [detM(A,m, µ)]Nf e−SG




ZNf (m, µ,Nf ) =


Investigation of the phase diagram in the Nf vs m plane




ZNf (m, µ,Nf ) =



I mapping out the mZ2(Nf ) line




ZNf (m, µ,Nf ) =




Chiral limit extrapolation constrained by universality




ZNf (m, µ,Nf ) =





I tricritical scaling




ZNf (m, µ,Nf ) =





I tricritical scaling

I Non-unique interpolation between Nf = 2 and Nf = 3;

ZNf=2.# =

∫DU [detM(U,m, µ)]2.# e−SG

ZNf=2+1 =

∫DU [detM(U,m1, µ)]2 [detM(U,m2, µ)] e−SG


Further insights into the phase diagram

We still do not know the order of chiral transition for Nf = 2

?

?Only two possible scenarios out of more are discussed ( Vicari (2007), arXiv:0709.1014,...)


arXiv:0709.1014


mZ2(Nf ) according to the second order scenario



mZ2(Nf ) according to the second order scenario

Ntricf > 2



mZ2(Nf ) according to the �rst order scenario



mZ2(Nf ) according to the �rst order scenario

Ntricf < 2


Code & investigated parameter space

All simulations employ the OpenCL-based CL2QCD codePhilipsen et al. (2014) arXiv:1411.5219 https://github.com/CL2QCD/cl2qcd

I Unimproved rooted staggered fermion discretization (RHMC algorithm)

No. of �avours −→ Nf = 2.8, 2.6, 2.4, 2.2, 2.1, 2.0Coarse lattices −→ Nτ = 4Chemical potential −→ µ = 0Scan in mass −→ m ∈ [0.0250, 0.0007]Finite size scaling −→ Nσ = 8, 12, 16, 20Scan in temperature −→ (3− 5) β valuesStatistics −→ ∼ (200k − 400k) per β


arXiv:1411.5219https://github.com/CL2QCD/cl2qcd

Checking the order of the transition...

Chiral condensate 〈ψ̄ψ〉 and moments of its distribution

Bn(〈ψ̄ψ〉, β,m,V ) ≡ Bn(β,m,Nσ) =

〈(ψ̄ψ −

〈ψ̄ψ〉)n〉〈(

ψ̄ψ −〈ψ̄ψ〉)2〉n/2





〈(ψ̄ψ −

〈ψ̄ψ〉)n〉〈(

ψ̄ψ −〈ψ̄ψ〉)2〉n/2

∀m,Nσ identify βc by the condition B3(β,m,Nσ) = 0





〈(ψ̄ψ −

〈ψ̄ψ〉)n〉〈(

ψ̄ψ −〈ψ̄ψ〉)2〉n/2


0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009

β +5.182

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

B3(〈ψ̄

ψ〉)

〈ψ̄ψ〉〈ψ̄ψ〉





〈(ψ̄ψ −

〈ψ̄ψ〉)n〉〈(

ψ̄ψ −〈ψ̄ψ〉)2〉n/2


B4(βc ,m,Nσ) to extract the order of the transition as function of m

B4(βc ,m,Nσ) =

〈(ψ̄ψ −

〈ψ̄ψ〉)4〉〈(

ψ̄ψ −〈ψ̄ψ〉)2〉2

∣∣∣∣∣∣∣βc

∼V→∞

1, 1storder;

1.604, 2ndorderZ2;

3, crossover.


...and locating mZ2(Nf )

Fitting B4(βc ,m,Nσ) at �xed Nf and various Nσ, as function of m

B4(βc ,m,Nσ) = B4(βc ,mZ2 ,∞) + a(m −mZ2)N1/νσ + . . .

1.35

1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

0.017 0.018 0.019 0.02 0.021 0.022 0.023 0.024 0.025 0.026

B4pβc,m

,Nσq

m

Nf “ 2.8

Nσ “ 8Nσ “ 12Nσ “ 16

B4(βc ,mZ2 ,∞) = 1.604

a = 0.59(5)

ν = 0.6301

mZ2 = 0.0211(3)

χ2ndf=10 = 0.27


How (wide) small is the tricritical scalingregion in mZ2(Nf )?

0

0.005

0.01

0.015

0.02

0.025

0.03

0 0.5 1 1.5 2 2.5 3 3.5 4

mZ

2

Nf

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0 0.5 1 1.5 2 2.5 3 3.5 4

m2

{5Z

2

Nf



0

0.005

0.01

0.015

0.02

0.025

0.03

1.5 2 2.5 3 3.5

mZ2

Nf

0

0.05

0.1

0.15

0.2

0.25

1.5 2 2.5 3 3.5

m2

{5Z2

Nf

de Forcrand and Philipsen (2003), 10.1016/j.nuclphysb.2003.09.005


10.1016/j.nuclphysb.2003.09.005


0

0.005

0.01

0.015

0.02

0.025

0.03

1.5 2 2.5 3 3.5

mZ2

Nf

0

0.05

0.1

0.15

0.2

0.25

1.5 2 2.5 3 3.5

m2

{5Z2

Nf

de Forcrand and Philipsen (2003), 10.1016/j.nuclphysb.2003.09.005



10.1016/j.nuclphysb.2003.09.005 10.1103/PhysRevD.90.074030


0

0.005

0.01

0.015

0.02

0.025

0.03

1.5 2 2.5 3 3.5

mZ2

Nf

0

0.05

0.1

0.15

0.2

0.25

1.5 2 2.5 3 3.5

m2

{5Z2

Nf

m2/5Z2

(Nf ) = C(Nf − Ntricf

)


Nf = 2.2 ∈ tricritical scaling region?

0

0.005

0.01

0.015

0.02

0.025

0.03

1.5 2 2.5 3 3.5

mZ2

Nf

0

0.05

0.1

0.15

0.2

0.25

1.5 2 2.5 3 3.5

m2

{5Z2

Nf

m2/5Z2

(Nf ) = C(Nf − Ntricf

)χ2ndf=1 = 0.28


Nf = 2.2 ∈ tricritical scaling region?

0

0.005

0.01

0.015

0.02

0.025

0.03

1.5 2 2.5 3 3.5

mZ2

Nf

0

0.05

0.1

0.15

0.2

0.25

1.5 2 2.5 3 3.5

m2

{5Z2

Nf

Assuming that the tricritical scaling sets in at Nf = 2.2I Same scaling window in m as found in the extrapolation from µiI N

tric

f = 1.75(32)

Waiting for ongoing simulations at Nf = 2.0...


Outline






)2at Nf = 2


6 Outlook


Outlook

First investigationI Just entered the scaling region...I Coarse Nτ = 4 lattices exploredI Rooted staggered fermion discretization: cross check needed

Theoretical caveat: at non-integer Nf our theory lacks locality and does NOTcorrespond to a well de�ned QFT in the CL

Theoretical caveat: ∞ many interpolations between Nf = 2 and Nf = 3limNf→2,3Z(Nf = 2.#) = Z(Nf = 2, 3)However

I the relative position of Ntricf w.r.t. Nf = 2 has to be uniquely determined byevery interpolation

I possible matching among interpolations

In view of a continuum limit: the width of scaling regions �xed in physicalunits will shrink in lattice units

I Which extrapolation is more expensive?F from imaginary chemical potential µi at Nf = 2F from non-integer Nf at µ = 0F from Nf = 2+ 1 while increasing ms at µ = 0


THANK YOU!


QCD phase diagram at real and imaginary Extended QCD Columbia plot (2=0)The order of the thermal phase transitionExtrapolation from imaginary chemical potentialNature of the Roberge-Weiss endpoint at Nf=2Z2 line in m- (T)2 at Nf=2

Extrapolation from non-integer Nf at =0Z2 line in mZ2-(Nf)

Outlook

Extended QCD phase diagram from the lattice...Outline 1 QCD phase diagram ta real and imaginary 2...

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Transcript of Extended QCD phase diagram from the lattice...Outline 1 QCD phase diagram ta real and imaginary 2...