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
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²
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·¸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
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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