The QCD Phase Diagram fromLattice Simulations

Simon Hands University of Wales Swansea

Difficulties at µ 6= 0

Progress at small µ/T

Taylor Expansion of the Free Energy

Color Superconductivity

Superfluidity in the NJL model

Daresbury 3rd March

. – p.1/24

The QCD Phase Diagram

µ

T(MeV)

(MeV)

T

GSI ?quarkmatter?

quark−gluon plasma

nuclearmatter

c

µ

100

200

500 1500

colorsuperconductor

hadronic fluid

RHIC/A

LICE

compact stars

critical endpoint

crossover(µ Ε ,Τ )Ε

onset

crystalline

. – p.2/24

Bluffer’s Guide to Lattice QCD

Feynman Path Integral for QCD

〈O(ψ, ψ̄, Aµ)〉 =1

Z

∫

DψDψ̄DAµOei � �

x(ψ̄M [Aµ]ψ+ 1

4FµνF

µν)

with Z ≡ 〈1〉

Two technical tricks:

• Analytically continue from Minkowski to Euclidean spacet 7→ ix4 FPI has better convergence properties• Discretise Fµν and M on a 4d spacetime latticeFPI becomes an ordinary multi-dimensional integral

〈O〉 can now be estimated numerically using Monte Carloimportance sampling, in effect “simulating” quantumfluctuations of the ψ, ψ̄ and Aµ fields

. – p.3/24

States can be analysed by choosing O with appropriatequantum numbers and then measuring the energy viadecay in Euclidean time:

〈O(0)O†(x4)〉 ∝ e−Ex4

Thermal effects modelled by restricted the time extent ofthe Euclidean universe to 0 < x4 < β ⇒

Z includes all excitations with Boltzmann weight e−βE , ie.

〈O〉 =1

Z

∑

i

Oie−

EikT

with temperature T = β−1

Lattice currently the most systematic way of studying QCDwith T > 0Best estimate for deconfining transition:

Tc ' 170(5)MeV

. – p.4/24

Equation of State at µB = 0 (Lt = 4)Bielefeld group (2000)

0.0

1.0

2.0

3.0

4.0

5.0

1.0 1.5 2.0 2.5 3.0 3.5 4.0

T/Tc

p/T4

pSB/T4

3 flavour2+1 flavour

2 flavour

170

16

14

12

10

8

6

4

2

01.0 1.5 2.0 2.5 3.0 3.5 4.0

210 250 340 510

LHC RHIC

680

εSB / T4

T / Tc

T (MeV)

ε / T

4

6.34.3

2.9

1.8

0.6 GeV / fm3 = εc �

• For Nf = 2 transition is crossover

• For Nf = 3 and m < mc transition is first order

• For realistic “Nf = 2 + 1” a crossover currently favoured

NB pSB

T 4=

8π2

45+Nf

[

7π2

60+

1

2

(µq

T

)2

+1

4π2

(µq

T

)4]

. – p.5/24

The Sign Problem for µ 6= 0

In Euclidean metric the QCD Lagrangian reads

LQCD = ψ̄(M +m)ψ +1

4FµνFµν

with M(µ) = D/ [A] + µγ0

Straightforward to show γ5M(µ)γ5 ≡M †(−µ) ⇒detM(µ) = (detM(−µ))∗

ie. Path integral measure is not positive definite for µ 6= 0Fundamental reason is explicit breaking of time reversal symmetry

Monte Carlo importance sampling, the mainstay of latticeQCD, is ineffective

. – p.6/24

A formal solution to the Sign Problem is reweighting ie. toinclude the phase of the determinant in the observable:

〈O〉 ≡〈〈O arg(detM)〉〉

〈〈arg(detM)〉〉

with 〈〈. . .〉〉 defined with a positive measure |detM |e−Sboson

Unfortunately both denominator and numerator areexponentially suppressed:

〈〈arg(detM)〉〉 =〈1〉

〈〈1〉〉=ZtrueZfake

= exp(−∆F ) ∼ exp(−#V )

Expect signal to be overwhelmed by noise inthermodynamic limit V → ∞

. – p.7/24

Two Routes into the Plane

� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �

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� � � � � � � � �� � � � � � � � �� � � � � � � � �� � � � � � � � �� � � � � � � � �� � � � � � � � �� � � � � � � � �� � � � � � � � �� � � � � � � � �

� � � � � � � � �� � � � � � � � �� � � � � � � � �� � � � � � � � �� � � � � � � � �� � � � � � � � �� � � � � � � � �� � � � � � � � �� � � � � � � � �

µ

T

Analytic continuation

µ

Reweighting in , (T)µ

in /T

β

(I) Analytic continuation inµ/T by eitherTaylor expansion @ µ = 0

Gavai & Gupta; QCDTARO

Simulation with imaginaryµ̃ = iµ de Forcrand & Philipsen;

d’Elia & Lombardo

effective for µT< min

(

µE

TE, π

3

)

(II) Reweighting along transition line Tc(µ) Fodor & Katz

Overlap between (µ, T ) and (µ+ ∆µ, T + ∆T ) remainslarge, so multi-parameter reweighting unusually effective

. – p.8/24

3.64 3.65 3.66 3.67

1.2

1.4

1.6

1.8

χψψ

µ=0.00µ=0.04µ=0.08

βc(µ=0)

3.75 3.76 3.770.6

0.7

0.8

0.9

1.0

1.1

1.2

χL

µ=0.00µ=0.05µ=0.10µ=-0.05µ=-0.10

βc(µ=0)

The Bielefeld/Swansea group used a hybrid approach; ie. reweightusing a Taylor expansion of the weight:

Allton et al , PRD66(2002)074507

ln

(

detM(µ)

detM(0)

)

=∑

n

µn

n!

∂n ln detM

∂µn

∣

∣

∣

∣

µ=0

This is relatively cheap and enables the use of large spatial volumes(163 × 4 using Nf = 2 flavors of p4-improved staggered fermion).Note with Lt = 4 the lattice is coarse: a−1(Tc) ' 700MeV

. – p.9/24

The (Pseudo)-Critical Line

0 200 400 600 800µ

B/MeV

145

150

155

160

165

170

175

180

185

T/M

eV

Nf=2, [115]

Nf=2+1, [116]

Nf=2, [119]

Nf=4, [121]

[E. Laermann & O. Philipsen, Ann.Rev.Nucl.Part.Sci.53:163,2003]

Remarkable consensus on the curvature. . .

RHIC collisions operate in region µB ∼ 45MeV. – p.10/24

0

50

100

150

200

250

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Fodor, Katz

µB [GeV]

T [MeV] nuclear matter density:MC data, am=0.2MC data, am=0.1ideal quark gas

Tc

const. energy densityTf, J.Cleymans et. al.

The pseudocritical line found lies well above the (µB, T )trajectory marking chemical freezeout in RHIC collisions

⇒ is there a region of the phase diagram where hadronsinteract very strongly (ie. inelastically)? So what?

. – p.11/24

The Critical Endpoint µE/TE

Reweighting estimatevia Lee-Yang zeroesµE/TE = 2.2(2)

Z. Fodor & S.D. Katz JHEP0404(2004)050

Taylor expansion estimatefrom apparent radius ofconvergenceµE/TE >∼ |c4/c6| ∼ 3.3(6)

Allton et al PRD68(2003)014507

0 2 4 6 8 100

0.5

1

1.5

2

2.5

ρ0

ρ2

ρ4

Tc(µ

q)

T/T0

µq/T

0

SB(ρ2)SB(ρ

0)

Analytic estimate via Binder cumulant 〈(δO)4〉/〈(δO)2〉2

evaluated at imaginary µ ⇒ µE/TE ∼ O(20)!P. de Forcrand & O. Philipsen NPB673(2003)170

. – p.12/24

Taylor Expansion

0.8 1 1.2 1.4 1.6 1.8 2

T/T0

0

0.2

0.4

0.6

0.8

1 c2

c2

ISB limit

SB (Nτ=4)

0.8 1 1.2 1.4 1.6 1.8 2

T/T0

0

0.05

0.1

0.15

0.2

0.25

c4

c4

I

SB limit

SB (Nτ=4)

0.8 1 1.2 1.4 1.6 1.8 2

T/T0

-0.1

-0.05

0

0.05

0.1

c6

I

c6

I

In our most recent work we develop the Taylor expansionof the free energy to O((µq/T )6) (recall cSB6 = 0):

p

T 4=

∞∑

n=0

cn(T )(µq

T

)n

with cn(T ) =1

n!

∂n(p/T 4)

∂(µq/T )n

∣

∣

∣

∣

µq=0

Similarly we define expansion coefficients

cIn(T ) =1

n!

∂n(p/T 4)

∂(µI/T )2∂(µq/T )n−2

∣

∣

∣

∣

µq=0,µI=0

. – p.13/24

Equation of State Allton et al PRD68(2003)014507

0.8 1 1.2 1.4 1.6 1.8 2

T/Tc0

0

0.2

0.4

0.6

µq/T

c0=0.8

µq/T

c0=1.0

µq/T

c0=0.6

µq/T

c0=0.4

µq/T

c0=0.2

Pressure change ∆p/T 4

0.8 1 1.2 1.4 1.6 1.8 2

T/Tc0

0

0.5

1

1.5

2

nq/T

3

nuclear matter density

µq/T

c0=0.8

µq/T

c0=1.0

µq/T

c0=0.6

µq/T

c0=0.4

µq/T

c0=0.2

Quark density nq/T3

∆p(µ, T )

T 4=p(µ, T ) − p(0, T )

T 4=

nmax∑

n=1

cn(T )(µ

T

)n

; nq =∂p

∂µ

. – p.14/24

Growth of Baryonic Fluctuations

0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4µ

q/T=1.0

µq/T=0.8

µq/T=0.6

µq/T=0.4

µq/T=0.2

µq/T=0.0

χq/T

2

T/T0

0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

µq/T=1.0

µq/T=0.8

µq/T=0.6

µq/T=0.4

µq/T=0.2

µq/T=0.0

χI/T

2

T/T0

Allton et al PRD68(2003)014507

Quark number susceptibility χq = ∂2 lnZ∂µ2

qappears singular

near µq/T ∼ 1; isospin susceptibility χI = ∂2 lnZ∂µ2

I

does not

Massless field at critical point a combination of theGalilean scalar isoscalars ψ̄ψ and ψ̄γ0ψ?

. – p.15/24

The QCD Phase Diagram

µ

T(MeV)

(MeV)

T

GSI ?quarkmatter?

quark−gluon plasma

nuclearmatter

c

µ

100

200

500 1500

colorsuperconductor

hadronic fluid

RHIC/A

LICE

compact stars

critical endpoint

crossover(µ Ε ,Τ )Ε

onset

crystalline

. – p.16/24

χSB vs. Cooper Pairing

pairing instability

FE

k

E

2Σ

E

k

E

k

E

k

instabilitypairing

Dirac Sea

Fermi Sea

2∆

. – p.17/24

Color Superconductivity

In the asymptotic limit µ→ ∞, g(µ) → 0, the ground stateof QCD is the color-flavor locked (CFL) state characterisedby a BCS instability, [D. Bailin and A. Love, Phys.Rep. 107(1984)325]

ie. diquark pairs at the Fermi surface condense via

〈qαi (p)Cγ5qβj (−p)〉 ∼ εAαβεAij × const.

breaking SU(3)c⊗SU(3)L⊗SU(3)R ⊗U(1)B ⊗U(1)Q−→SU(3)∆⊗U(1)Q̃

The ground state is simultaneouslysuperconducting (8 gapped gluons, ie. get mass O(∆)),

superfluid (1 Goldstone),and transparent (all quasiparticles with Q̃ 6= 0 gapped).

[M.G. Alford, K. Rajagopal and F. Wilczek, Nucl.Phys.B537(1999)443]

. – p.18/24

The NJL Model

Effective description of soft pions interacting with nucleons

LNJL = ψ̄(∂/ +m+ µγ0)ψ −g2

2[(ψ̄ψ)2 − (ψ̄γ5~τψ)2]

∼ ψ̄(∂/ +m+ µγ0 + σ + iγ5~π.~τ)ψ +2

g2(σ2 + ~π.~π)

Introduce isopsin indices so full global symmetry isSU(2)L⊗SU(2)R⊗U(1)B

Dynamical χSB for g2 > g2c ⇒ isotriplet Goldstone ~π

Scalar isoscalar diquark ψtrCγ5 ⊗ τ2 ⊗Acolorψ breaks U(1)B

⇒ diquark condensation signals high density ground stateis superfluid

. – p.19/24

Model is renormalisable in 2+1d so GN analysis holds

In 3+1d, an explicit cutoff is required. We follow thelarge-Nf (Hartree) approach of Klevansky (1992) andmatch lattice parameters to low energy phenomenology:

Phenomenological Lattice ParametersObservables fitted extractedΣ0 = 400MeV ma = 0.006

fπ = 93MeV 1/g2 = 0.495

mπ = 138MeV a−1 = 720MeV

The lattice regularisation preservesSU(2)L⊗SU(2)R⊗U(1)B

. – p.20/24

Equation of State and Diquark Condensation

� �� �� �� �� � ���� � � � �� � � � � ���� ����� �

!

"$# %"&# '&# (&# )&# %&&# '

&# *&# (

&# +&# )

&# ,&# %

&# "&

. – p.21/24

Equation of State and Diquark Condensation

- ./ 0�1 2 34 56 7 5�86 7 3 9 3: 2 5 ; 1 : <�=> 3?@AB B C

D

E$F GEHF IHF JHF KHF GHHF I

HF LHF J

HF MHF K

HF NHF G

HF EH

O PRQ ST U VW SQ X Y[Z \Q VT] S^`_ _ ab$c d e

^ _ _ a

f

g$h igjh kjh ljh mjh ijjh k

jh njh l

jh ojh m

jh pjh i

jh gj

qsr rut v wxwyz {|} ~�� �|} �� �|} �� �|} �� �|} �� �

�

��� ��

~��� �� � ���� �� � ����� ��

��� ���� � �

��� ���� ~ �

��� ~��� ��

�

Add source j[ψtrψ + ψ̄ψ̄tr]

Diquark condensate esti-mated by taking j → 0

Our fits exclude j ≤ 0.2

. – p.21/24

The Superfluid Gap

Quasiparticlepropagator:

〈ψu(0)ψ̄u(t)〉 = Ae−Et +Be−E(Lt−t)

〈ψu(0)ψd(t)〉 = C(e−Et − e−E(Lt−t))

Results from 96 × 122 × Lt, µa = 0.8 extrapolated toLt → ∞ (ie. T → 0) then j → 0

� �� �� �� �� �� �

� ��� ����� ���

��� ��

� � ��

��     ¡   �¢ £¤¥ ¦ £§¨ª© �   «©¬­ ® ® £¯   ¦ £§¨ª© �   «©° ±

²

³ ´µ ¶

·¸¹ ·º·»· º¼¼© ½

¼¾¼© ½

¾¼© °¾¼© ¿

¾¼© À¾¼© Á

¾¼© ¾¼© Ã

¾¼© ľ¼© Å

The gap at the Fermi surface signals superfluiditySJH & D.N. Walters PLB548(2002)196 PRD69(2004)076011

. – p.22/24

ÆÇ�È ÉÊ ËÍÌ Î ÏÐÒÑ ÓÔ Õ Ó ÖÆ ÇÈ ÉÊ Ë Ì Î ÏÐ Õ ÓÔ×ªØ ØÙ ÚÛ ÔÒÜ ÝÔÒÜ ÞÔÒÜ ßÔÒÜ ÐÔÒÜ àÔÒÜ ÖÔÒÜ áÔÒÜ ÓÔÜ ÏÔ

ÔÜ ÓÔÒÜ Ï à

ÔÜ ÏÔÒÜ Ô à

Ôâ ÔÒÜ Ô à

• ∆/Σ0 ' 0.15 ⇒ ∆ ' 60MeVin agreement with self-consistent approaches

• Similar formalism to study non-relativistic model forEITHER nuclear matter (with or without pions)⇒ calculation of E/A D. Lee & T. Schäfer nucl-th/0412002

OR Cold atoms with tunable scattering length⇒ study of BEC/BCS crossover M. Wingate cond-mat/0502372

In either case non-perturbative due to large dimensionlessparameter kF |a| � 1, with a the s-wave scattering length.

. – p.23/24

Summary

. – p.24/24

Summary

Approaches with different systematics are yieldingencouraging agreement on the critical line Tc(µ)

. – p.24/24

Summary

Approaches with different systematics are yieldingencouraging agreement on the critical line Tc(µ)

Still no consensus on location of the critical endpoint,but NO OBVIOUS OBSTACLE to calculation of µE/TE

. – p.24/24

Summary

Approaches with different systematics are yieldingencouraging agreement on the critical line Tc(µ)

Still no consensus on location of the critical endpoint,but NO OBVIOUS OBSTACLE to calculation of µE/TE

Need better control over: statistics; approach to thechiral limit; above all over approach to continuum limit

. – p.24/24

Summary

Approaches with different systematics are yieldingencouraging agreement on the critical line Tc(µ)

Still no consensus on location of the critical endpoint,but NO OBVIOUS OBSTACLE to calculation of µE/TE

Need better control over: statistics; approach to thechiral limit; above all over approach to continuum limit

Evidence for superfluidity in NJL3+1

. – p.24/24

Summary

Approaches with different systematics are yieldingencouraging agreement on the critical line Tc(µ)

Still no consensus on location of the critical endpoint,but NO OBVIOUS OBSTACLE to calculation of µE/TE

Need better control over: statistics; approach to thechiral limit; above all over approach to continuum limit

Evidence for superfluidity in NJL3+1

For the future:is there a model with long-range interactions whichinterpolates between BEC and BCS?what is the physical origin of the sign problem?

. – p.24/24

Summary

Approaches with different systematics are yieldingencouraging agreement on the critical line Tc(µ)

Still no consensus on location of the critical endpoint,but NO OBVIOUS OBSTACLE to calculation of µE/TE

Need better control over: statistics; approach to thechiral limit; above all over approach to continuum limit

Evidence for superfluidity in NJL3+1

For the future:is there a model with long-range interactions whichinterpolates between BEC and BCS?what is the physical origin of the sign problem?

And finally. . .. – p.24/24