looking for the qcd critical point on the...

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Intro Tc CEP Concl.

Looking for the QCD critical point on the lattice

Philippe de ForcrandETH Zürich and CERN

Ph. de Forcrand EMMI 2009, St. Goar Crit. Pt.

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Intro Tc CEP Concl.

The minimum phase diagram of QCD

T

µ

confined

QGP

Color superconductor

Tc

• Dictated by perturbation theory at large T or large µ• Phase transitions or crossovers ?


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Intro Tc CEP Concl.

The minimum phase diagram of QCD

T

µ

confined

QGP


Tc ♥QCD critical point

• Dictated by perturbation theory at large T or large µ• Phase transitions or crossovers ?• Crossover at (µ= 0,Tc), first-order at small T → QCD critical point


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Intro Tc CEP Concl.

The sign problem

• Integrate over fermions: det(D/ +m +µγ0) complex unless µ= 0 or µ= iµi

→ standard importance sampling⇔ 〈Re(baryon density)〉= 0


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Intro Tc CEP Concl.

The sign problem



• Reweighting: - simulate theory with no sign pb., eg. |det(µ)|- reweight each measurement with ρ(U) = det(U,µ)

|det(U,µ)| complex phase

- av. “sign” 〈ρ(U)〉= Z(µ,det)Z(µ,|det |) ∼ exp(−V ∆f (µ)

T )→ large V ?, large µ ?

1. maintain statistical accuracy on 〈ρ〉: sign pb.2. ensure that Z (µ,det) is properly sampled: overlap pb.

1 and 2 require statistics ∝ exp(+V )


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Intro Tc CEP Concl.

The sign problem



• Reweighting: - simulate theory with no sign pb., eg. |det(µ)|- reweight each measurement with ρ(U) = det(U,µ)

|det(U,µ)| complex phase

- av. “sign” 〈ρ(U)〉= Z(µ,det)Z(µ,|det |) ∼ exp(−V ∆f (µ)

T )→ large V ?, large µ ?

1. maintain statistical accuracy on 〈ρ〉: sign pb.2. ensure that Z (µ,det) is properly sampled: overlap pb.

1 and 2 require statistics ∝ exp(+V )

• Measure derivatives w.r.t. µ at µ= 0: 〈W (µ)〉= 〈W (µ= 0)〉+∑k ck( µ

πT

)k

- directly at µ= 0 MILC, TARO, Bielefeld-Swansea, Gavai-Gupta,..

- by fitting polynomial to µ= iµi results D’Elia-Lombardo, PdF-Philipsen,..

Controlled thermodynamics and continuum limits⇒ derivatives only


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Intro Tc CEP Concl.

The good news: curvature of the pseudo-critical line

All with Nf = 4 staggered fermions, amq = 0.05,Nt = 4 (a∼ 0.3 fm)PdF & Kratochvila LAT05

4.8

4.82

4.84

4.86

4.88

4.9

4.92

4.94

4.96

4.98

5

5.02

5.04

5.06

0 0.5 1 1.5 2

1.0

0.95

0.90

0.85

0.80

0.75

0.70

0 0.1 0.2 0.3 0.4 0.5β

T/T

c

µ/T

a µ

confined

QGP<sign> ~ 0.85(1)

<sign> ~ 0.45(5)

<sign> ~ 0.1(1)

D’Elia, Lombardo 163

Azcoiti et al., 83

Fodor, Katz, 63

Our reweighting, 63

deForcrand, Kratochvila, 63

imaginary µ2 param. imag. µdble reweighting, LY zerosSame, susceptibilities

canonical

Agreement for µ/T . 1


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Intro Tc CEP Concl.

The good news: curvature of the pseudo-critical line

• Tc(µ) simpler/more precise by analytic continuation of Tc(iµi):determine Tc(µ= iµi) and fit results with polynomial

Tc(µ)Tc(µ=0) = 1−∑k=1 t2k

( µπT

)2k

For Nt = 4: curvature t2(Nf ,mq) varies from ∼ 0.3 to ∼ 1 Schmidt 06

• Indications (PdF& OP Nt = 6; Fodor et al. LAT08): t2 decreases as a→ 0

• Extrapolation mq →mphys, a→ 0 feasible G. Endrodi LAT09


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Intro Tc CEP Concl.

Why is curvature of pseudo-critical line important?

T

µ

confined

QGP


Tc

♥freeze-out

crit. pt.

IF Tc(µ) really flatter than freeze-out curve (a→ 0: factor ∼ 2−6 in d2Tcdµ2

∣∣∣µ=0

?)

=⇒ exp. signal from critical pt. modified/washed out by evolution until freeze-out


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Intro Tc CEP Concl. Reweighting Taylor Crit. surf.

The bad news: locating the critical point

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

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CO94

NJL01NJL89b

CJT02

NJL89a

LR04

RM98

LSM01

HB02 LTE03LTE04

3NJL05

INJL98

LR01

PNJL06

130

9

5

2

17

50

0

100

150

200

0 400 800 1000 1200 1400 1600600200

T ,MeV

µB, MeV

M. Stephanov, hep-lat/0701002

• Challenging task:detect divergent correlation length (2nd order)vs finite but large (crossover, 1rst order)on small lattice

Mission impossible?


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0. The ultimate reweightingFodor & Katz: hep-lat/0402006 (∼ physical quark masses)

•

−→−→−→Glasgow

(µqE ,TE) = (120(13),162(2)) MeV

Strategy: reweight from (µ= 0,Tc) along pseudo-critical line

Legitimate concerns:• Discretization error? Nt = 4 =⇒ a∼ 0.3 fm• Abrupt qualitative change near µE :

abrupt change of physics or breakdown of algorithm (Splittorff)?

→ repeat with conservative approach (derivative)Ph. de Forcrand EMMI 2009, St. Goar Crit. Pt.

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1. Taylor expansion

• Reweighting gives exact µ dependence, BUT limited to small V ,µError non-Gaussian, analysis subtle→breakdown may go unnoticed(Glasgow meth)

• Instead, obtain reliable V → ∞ behaviour of Taylor coefficients:

P(T ,µ) = P(T ,µ= 0)︸︷︷︸

indep. calc.

+∆P(T ,µ), ∆P(T ,µ)T 4 = ∑k=1 c2k(T )

( µT

)2k

c2k = 〈Tr( degree 2k polynomial in D/−1, ∂D/∂µ )〉µ=0→ vanilla HMC

• From {c2k}, obtain all thermodynamic info: EOS and Tc(µ) and crit. pt. and ...

• As µT increases, need higher-order c2k ’s to control truncation error


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1. Taylor expansion



P(T ,µ) = P(T ,µ= 0)︸︷︷︸

indep. calc.

+∆P(T ,µ), ∆P(T ,µ)T 4 = ∑k=1 c2k(T )

( µT

)2k


0.00

0.20

0.40

0.60

0.80

1.00

0.8 1.0 1.2 1.4 1.6 1.8 2.0

c2SB limit

SB (Nt=4)

T/T0

0.00

0.05

0.10

0.15

0.20

0.25

0.8 1.0 1.2 1.4 1.6 1.8 2.0

c4

SB limit

SB (Nt=4)T/T0

-0.10

-0.05

0.00

0.05

0.10

0.8 1.0 1.2 1.4 1.6 1.8 2.0

c6

T/T0

C. Schmidt, hep-lat/0610116


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1. Taylor expansion



P(T ,µ) = P(T ,µ= 0)︸︷︷︸

indep. calc.

+∆P(T ,µ), ∆P(T ,µ)T 4 = ∑k=1 c2k(T )

( µT

)2k


• From {c2k}, obtain all thermodynamic info: EOS and Tc(µ) and crit. pt. and ...

• As µT increases, need higher-order c2k ’s to control truncation error

• Higher order k ⇒ Tr(D/−2k , ..), ie. more noise vectors, more cancellations..

• ..and also larger volumes → work ∼ 36k (Karsch et al.) at least

• Current best: Nt =8,6th order HOTQCD, Nt =6,8th order Gavai&Gupta, 0806.2233


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Ratio method: ”effective radius of convergence”

PT 4 =

∞

∑n=0

c2n(T )( µ

T

)2n

Singularity (µE ,TE)⇒ µETE

= limn→∞

√∣∣∣∣

c2n

c2n+2

∣∣∣∣(TE) Karsch et al.

Similar, forχq

T 2 : Gavai & Gupta“µE/TE < 0.6”


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PT 4 =

∞

∑n=0

c2n(T )( µ

T

)2n


= limn→∞

√∣∣∣∣

c2n

c2n+2


Similar, forχq


Critique: • Need n→ ∞, not n = 1,2,3;

√∣∣∣

c2c4

∣∣∣ is not a lower or upper bound

• Robust criterion to choose TE?• Smallest convergence radius is NOT CEP Stephanov hep-lat/0603014

T

µ

Tc ♥(TE,µE)

0 TE

TT TE

>TE<

Complex mu plane


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PT 4 =

∞

∑n=0

c2n(T )( µ

T

)2n


= limn→∞

√∣∣∣∣

c2n

c2n+2


Similar, forχq



√∣∣∣

c2c4



Remember Fodor & Katz:


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PT 4 =

∞

∑n=0

c2n(T )( µ

T

)2n


= limn→∞

√∣∣∣∣

c2n

c2n+2


Similar, forχq



√∣∣∣

c2c4



Remember Fodor & Katz:need high-order derivatives

-0.001

0

0.001

0.002

0.003

0 0.05 0.1 0.15 0.2

Im β

0

µ

courtesy Z. Fodorthe whole story


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PT 4 =

∞

∑n=0

c2n(T )( µ

T

)2n


= limn→∞

√∣∣∣∣

c2n

c2n+2


Similar, forχq



√∣∣∣

c2c4



Can systematic errorbe controlled ?

First goal: discriminatebetween CEP and no CEP

-0.001

0

0.001

0.002

0.003

0 0.05 0.1 0.15 0.2

Im β

0

µ

courtesy Z. Fodorthe whole story


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Case study on toy model, with Roger Herrigel

• Idea: study Taylor coeffs of ∆PT 4

( µT

)≡∆P̂(µ̂)

and “effective radius of convergence” in controlled situation

• Ansatz: ∆P̂(µ̂) = ∆P̂SB2 + log(cosh(λ(T̂−1)+ ∆P̂SB

2 ))− log(cosh(λ(T̂−1)))

Why ? - correct limits high-T , low-T- c2 ≈ (1+ tanh(λ(T̂ −1)), sigmoid, no phase trans., no CEP- single parameter λ controls width of crossover (rescaling of T̂ )

• no singularity on real µ̂ axis→ test for spurious signals

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.8 1 1.2 1.4 1.6 1.8 2

P(T

,µ=

0) /

PS

B

T/Tc

Pressure


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Toy ansatz: compare with Bielefeld et al.

0.00

0.20

0.40

0.60

0.80

1.00

0.8 1.0 1.2 1.4 1.6 1.8 2.0

c2SB limit

SB (Nt=4)

T/T0

0.00

0.05

0.10

0.15

0.20

0.25

0.8 1.0 1.2 1.4 1.6 1.8 2.0

c4

SB limit

SB (Nt=4)T/T0

-0.10

-0.05

0.00

0.05

0.10

0.8 1.0 1.2 1.4 1.6 1.8 2.0

c6

T/T0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.8 1 1.2 1.4 1.6 1.8 2

c2

T/Tc

c2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.8 1 1.2 1.4 1.6 1.8 2

c4

T/Tc

c4

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.8 1 1.2 1.4 1.6 1.8 2c

6T/Tc

c6


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-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

0.8 1 1.2 1.4 1.6 1.8 2

c8

T/Tc

c8

-0.002

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.8 1 1.2 1.4 1.6 1.8 2

c1

0

T/Tc

c10

Oscillatory pattern→ ck = 0 at lower T̂ as k increases


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0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0.8 1.0 1.2 1.4 1.6 1.8 2.0

T/T0

χq

µq/T=0.0µq/T=0.5µq/T=1.0

0

0.5

1

1.5

2

2.5

3

3.5

4

0.8 1 1.2 1.4 1.6 1.8 2ch

i qT/Tc

µ/T=0.0µ/T=0.5µ/T=1.0

• Quark susceptibility rises, even without phase transition


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0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0.8 1.0 1.2 1.4 1.6 1.8 2.0

T/T0

χq

µq/T=0.0µq/T=0.5µq/T=1.0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.8 1 1.2 1.4 1.6 1.8 2ch

i qT/Tc

µ/T=1.5, exact4th order6th order8th order

• Quark susceptibility rises, even without phase transition• Truncation may increase susceptibility


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0.8 1 1.2 1.4T/T

c

-0.5

0

0.5

1 c4/c

2c

6/c

4

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.8 1 1.2 1.4 1.6 1.8 2T/Tc

c4/c2c6/c4

• Qualitative agreement below Tc although HRG not built-in


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0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

ρ0

ρ2

ρ4

Tc(µ

q)

T/T0

µq/T

0

SB(ρ2)SB(ρ

0)

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12

T/T

c

µ/Tc

rho2-4rho4-6

• Qualitative agreement

• IF we know that there is a crit. point: how to choose TE?? BJS et al.

• Can we predict whether or not there is a critical point ??


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A safer strategy ?

phys.point

00

N = 2

N = 3

N = 1

f

f

f

m s

sm

Gauge

m , mu

1st

2nd orderO(4) ?

2nd orderZ(2)

2nd orderZ(2)

crossover

1st

d

tric

∞

∞Pure

−→︸︷︷︸

if chiral CEP

* QCD critical point

crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ

•


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A safer strategy ?

phys.point

00

N = 2

N = 3

N = 1

f

f

f

m s

sm

Gauge

m , mu

1st

2nd orderO(4) ?

2nd orderZ(2)

2nd orderZ(2)

crossover

1st

d

tric

∞

∞Pure

−→︸︷︷︸

if chiral CEP

Two strategies:1 follow vertical line: m = mphys, turn on µ→ radius of convergence?2 follow critical surface: m = mcrit(µ)


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Strategy 2: follow critical surface PdF & O. Philipsen

(a) Identify µ= 0 critical line:

phys.point

00

N = 2

N = 3

N = 1

f

f

f

m s

sm

Gauge

m , mu

1st

2nd orderO(4) ?

2nd orderZ(2)

2nd orderZ(2)

crossover

1st

d

tric

∞

∞Pure

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.01 0.02 0.03 0.04

ams

amu,d

Nf=2+1

physical point

mstric - C mud

2/5

Nf = 3


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(b) Turn on imaginary µ and measure mc(µ)mc(0) = 1+∑k=1 ck

( µπT

)2k

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.01 0.02 0.03 0.04

ams

amu,d

Nf=2+1

physical point

mstric - C mud

2/5

Nf = 3

Nf = 3 Nf = 2+1, ms = mphyss

consistent 83×4 and 123×4, ∼ 5×106 traj. 163×4, Grid computing, ∼ 106 traj.mc(µ)mc(0) = 1−3.3(3)

( µπT

)2−47(20)( µ

πT

)4

︸︷︷︸

8thderivative of P

− . . . mu,dc (µ)

mu,dc (0)

= 1−39(8)( µ

πT

)2− . . .

• Higher order terms? Convergence?


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QCD critical point DISAPPEARED

crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ

No chiral crit. pt. at small chem. pot., µT . O (1), for Nt =4 (a∼0.3 fm)

cf. Ejiri 0812.1534→( µ

T

)CEP∼ 2.4


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Consistency with effective models?

• Restoration of UA(1) anomaly favors first-order finite T transitionChandrasekharan & Mehta

• Progressive restoration of UA(1) symmetry at finite density?Chen et al. 0901.2407NJL +[detq̄i(1− γ5)qj +h.c.]×exp(−µ2/µ2

0)

- backbending of crit. surface- crit. surf. ends when Tc = 0→ still no chiral critical point!

µ

crossover

mu,d

ms

X 1rst0

∞

Real world

Nf=3

• Same pattern with linear sigma model + fluctuationsBowman & Kapusta 0810.0042


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Towards the continuum limit• Critical line recedes away from physical point towards origin

first-ordera = 0

Endrodi, Fodor et al., arXiv:0710.0998

PdF & O. Philipsen, arXiv:0711.0262

Karsch et al., hep-lat/0309116

Nt =4 → Nt =∞ ⇒ change ∼ O (10) !!

IF curvature of critical surface unchanged...

* QCD critical point

crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ

•↑

•←−

µcrit increases QCD critical point DISAPPEARED

crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ

•←−


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Towards the continuum limit• Critical line recedes away from physical point towards origin

first-ordera = 0

Endrodi, Fodor et al., arXiv:0710.0998

PdF & O. Philipsen, arXiv:0711.0262

Karsch et al., hep-lat/0309116

Nt =4 → Nt =∞ ⇒ change ∼ O (10) !!

-150

-100

-50

0

50

100

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012

(aµi)2

∆B4/∆(aµi)2

B4(pbp)fit µ2

fit (µ2+µ4)

current status Nf = 3,Nt = 6

mc(µ)mc(0) = 1

{+12(11) µ2 fit−12(15) (µ2 +µ4) fit

( µπT

)2− . . .

|c1| not large: mc(µ)mc(0) ∼ 10 → µ∼ πT

Crit. surface starts almost “vertical”→need higher order to decide on crit. pt.


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Resulting phase diagram (simplest possibility)

Standard scenario Exotic scenario

T

µ

confined

QGP


m > mc(0)

Tc ♥ T

µ

confined

QGP


m > mc(0)

Tc


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Arguments for standard wisdom?

• O(4) transition for 2 massless flavors Pisarski & Wilczek⇒ tricritical points (mu,d = 0,ms = ∞,µ= µ∗) and (mu,d = 0,ms = m∗s ,µ= 0)

1rst ordercrossover

mu,d

ms

µ

X 1rst0

∞

1rst order

Real worldHeavy quarks


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• O(4) transition for 2 massless flavors Pisarski & Wilczek⇒ tricritical points (mu,d = 0,ms = ∞,µ= µ∗) and (mu,d = 0,ms = m∗s ,µ= 0)• Nf = 2 and Nf = 2+1 analytically connected

1rst ordercrossover

mu,d

ms

µ

X 1rst0

∞

1rst order



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1rst ordercrossover

mu,d

ms

µ

X 1rst0

∞

1rst order


Critique:• O(4) if strong enough UA(1) anomaly, otherwise first-order

Chandrasekharan & Mehta


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1rst ordercrossover

mu,d

ms

µ

X 1rst0

∞

1rst order


Critique:• O(4) if strong enough UA(1) anomaly, otherwise first-order

Chandrasekharan & Mehta

• Nf = 2 and Nf = 2+1 need not be connected→ study Nf = 2 crit. surf.Ph. de Forcrand EMMI 2009, St. Goar Crit. Pt.

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Intro Tc CEP Concl.

Conclusions

• Confucius: Real knowledge is to know the extent of one’s ignorance

• mc(µ)mc(0) = 1+c1

( µπT

)2+ . . . : can control systematics

Non-standard scenario c1 < 0 for Nt = 4

• a→ 0: critical surface far from physical point=⇒ need c1 > 0 and large for µE

TE<∼1, disfavored by data

• QCD critical point?

T

µ

confined

QGP


m > mc(0)

Tc

µBE

<∼ 500 MeV unlikely,or non-chiral


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Intro Tc CEP Concl.

Backup: Gavai & Gupta’s critical point

“We find the radius of convergence of the series at varioustemperatures, and bound the location of the QCD critical point to beTE/Tc ≈ 0.94 and µE/T < 0.6”

• Arbitrariness in definition of “effective radius of convergence”

µETE

= limn→∞

√∣∣∣∣

c2n

c2n+2

∣∣∣∣(TE) versus

χq

T 2 =∞

∑n=1

2n(2n−1) c2n

( µT

)2n−2

→ µETE

= limn→∞

√∣∣∣∣

2n(2n−1)c2n

(2n +2)(2n +1)c2n+2

∣∣∣∣

n = 1 → factor 1/√

6

• Criterion for TE

• Consistency of results with toy model having no critical point

• Endrodi, Fodor & Katz:peaks of all susceptibilities decrease as µ increases


looking for the qcd critical point on the...

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