Lattice QCD at non zero chemial
potential - status and challenges
Kim Splittorff
Niels Bohr Institute / DFF - Sapere Aude
Heavy ions: Experiments Confront Theory, the Discovery Center, November 8, 2011
Lattice QCD at non zero chemial potential - status and challenges – p. 1/23
Alternative motivation !
Lattice QCD at non zero chemial potential - status and challenges – p. 2/23
Pressure (Atm)
Snowballs
Solid
1000
100
Liquid
D B
10
1
C A
0.1
Gas
0.01
0.001
0
Temperature (C)
100
374
A → B → A = Snowball
C → D → C = Fluff
Scott A. Sandford NASA Ames Research Center
http://www.astrosociety.org/pubs/mercury/9801/snowball.html
Lattice QCD at non zero chemial potential - status and challenges – p. 3/23
Fireballs
QCD
T
Chirally symmetric
Quarks Gluons
B
Tc
D
C
A
Chiral symmetry broken
Pions Protons Neutrons ...
µ
A → B → A = Jets gets eaten
C → D → C = Fluff
Lattice QCD at non zero chemial potential - status and challenges – p. 4/23
Can lattice QCD help understand the phase diagram ?
Success story µ = 0
How to include µ and the sign problem
µ 6= 0 status, challenges and progress
Lattice QCD at non zero chemial potential - status and challenges – p. 5/23
The baryon/anti-baryon symmetric world
(µ = 0)
T
Chirally symmetric
Quarks Gluons
Crossover ?
Tc
2nd order ?
1st order ?
Chiral symmetry broken
Pions Protons Neutrons ...
Landau theory: Nf ≥ 2 1st order (mf = 0)
Answer depends on values of mf and UA (1) anomaly
Lattice: @ physical quark masses crossover
Pisarski, Wilczek, PRD29 (1984) 338
Aoki, Endrodi, Fodor, Katz, Szabo, Nature 443 (2006) 675
de Forcrand, Philipsen JHEP 0701 (2007) 077
Lattice QCD at non zero chemial potential - status and challenges – p. 6/23
(µ 6= 0)
Matter antimatter asymmetry
N >0
Here: Fact which we adopt into QCD
Grand canonical approach: Fix µ determine N
1
N = ∂µ log Z(µ)
V
Lattice QCD at non zero chemial potential - status and challenges – p. 7/23
How to include µ in Z
µ is conjugate variable to N
µN = µhq † qi = µhq̄γ0 qi
LQCD = q̄(Dη γη + µγ0 + m)q + Gluons
Lattice QCD at non zero chemial potential - status and challenges – p. 8/23
How to include µ in Z
µ is conjugate variable to N
µN = µhq † qi = µhq̄γ0 qi
LQCD = q̄(Dη γη + µγ0 + m)q + Gluons
µ enters as the 0th component of the gauge field
†
q + ...
LLattice QCD = ... + eaµ q̄x γ0 Ux,x+0̂ qx+0̂ + e−aµ q̄x+0̂ γ0 Ux,x+
0̂ x
Lattice QCD at non zero chemial potential - status and challenges – p. 8/23
How to include µ in Z
µ is conjugate variable to N
µN = µhq † qi = µhq̄γ0 qi
LQCD = q̄(Dη γη + µγ0 + m)q + Gluons
µ enters as the 0th component of the gauge field
†
q + ...
LLattice QCD = ... + eaµ q̄x γ0 Ux,x+0̂ qx+0̂ + e−aµ q̄x+0̂ γ0 Ux,x+
0̂ x
Works fine for free quarks
Hasenfratz, Karsch, PLB 125 (1983) 308
Lattice QCD at non zero chemial potential - status and challenges – p. 8/23
The sign problem
ZNf =2 =
Z
dA det2 (D + µγ0 + m) e−SYM
Anti Hermitian
Hermitian
det2 (D + µγ0 + m) = |det(D + µγ0 + m)|2 e2iθ
The measure is not real and positive
Lattice QCD at non zero chemial potential - status and challenges – p. 9/23
The sign problem
ZNf =2 =
Z
dA det2 (D + µγ0 + m) e−SYM
Anti Hermitian
Hermitian
det2 (D + µγ0 + m) = |det(D + µγ0 + m)|2 e2iθ
The measure is not real and positive
No Monte Carlo sampling of Aη at µ 6= 0
Lattice QCD at non zero chemial potential - status and challenges – p. 9/23
ZNf =2 =
Z
dA det2 (D + µγ0 + m) e−SYM
Anti Hermitian
Hermitian
det2 (D + µγ0 + m) = |det(D + µγ0 + m)|2 e2iθ
What if we simply ignore e2iθ ?
Lattice QCD at non zero chemial potential - status and challenges – p. 10/23
If we ignore e2iθ : phase quenched QCD
T
Expected
Found
m π/2
mN /3
µ
Kogut, Sinclair Phys.Rev.D77:114503,2008
Lattice QCD at non zero chemial potential - status and challenges – p. 11/23
ZNf =2 =
Z
dA det2 (D + µγ0 + m) e−SYM
Anti Hermitian
Hermitian
det2 (D + µγ0 + m) = |det(D + µγ0 + m)|2 e2iθ
What if we simply ignore e2iθ ?
Lattice QCD at non zero chemial potential - status and challenges – p. 12/23
ZNf =2 =
Z
dA det2 (D + µγ0 + m) e−SYM
Anti Hermitian
Hermitian
det2 (D + µγ0 + m) = |det(D + µγ0 + m)|2 e2iθ
What if we simply ignore e2iθ ?
|det(D + µγ0 + m)|2 = det(D + µγ0 + m)det(D − µγ0 + m)
ISOSPIN CHEMICAL POTENTIAL
Alford Kapustin Wilczek PRD 59 (1999) 054502
Lattice QCD at non zero chemial potential - status and challenges – p. 12/23
The QCD Phase Diagram at nonzero isospin chemical
potential
The pions have nonzero isospin ⇒ BEC of pions
Lattice QCD at non zero chemial potential - status and challenges – p. 13/23
The QCD Phase Diagram at nonzero isospin chemical
potential
The pions have nonzero isospin ⇒ BEC of pions
T
Expected
Found
m π/2
mN /3
µ
Lattice QCD at non zero chemial potential - status and challenges – p. 13/23
The QCD Phase Diagram at nonzero isospin chemical
potential
The pions have nonzero isospin ⇒ BEC of pions
T
Expected
Found
m π/2
Conclude:
mN /3
µ
The phase e2iθ is highly relevant for the phase diagram
Son, Stephanov Phys.Rev.Lett. 86 (2001) 592
Kogut, Sinclair Phys.Rev.D77:114503,2008
Lattice QCD at non zero chemial potential - status and challenges – p. 13/23
So include the phase factor in Lattice QCD
Lattice QCD at non zero chemial potential - status and challenges – p. 14/23
Method
Idea
Challenge
Reweighting
Absorb the sign in the observable
Exponential cancellations
Taylor expansion
Expand at µ = 0
Higher order terms
Imaginary µ
Determine the analytic function
Control the extrapolation
Density of states
Use the distribution of the phase
Determine the distribution
Canonical ensemble
Work at fixed baryon number
Fix the baryon number
review Lombardo, Splittorff, Verbaarschot arXiv:0912.4410
Lattice QCD at non zero chemial potential - status and challenges – p. 15/23
The curvature and the average sign
aµ
5.06
5.04
5.02
5
4.98
4.96
4.94
4.92
4.9
4.88
4.86
4.84
4.82
4.8
0.1
0.2
0.3
0.4
0.5
1.0
QGP
<sign> ~ 0.85(1)
<sign> ~ 0.45(5)
0.95
<sign> ~ 0.1(1)
0.90
0.85
confined
T/Tc
β
0
0.80
D’Elia, Lombardo 1633
Azcoiti et al., 83
Fodor, Katz, 6
Our reweighting, 633
This work, 6
0
0.5
1
0.75
0.70
1.5
2
µ/T
de Forcrand Philipsen JHEP PoS LAT2005 (2005) 016
Progress: Karsch B-J Schaefer Wagner Wambach arXiv:1110.6038
Lattice QCD at non zero chemial potential - status and challenges – p. 16/23
Particular challenging for µ > mπ /2
2iθ
−V (1−m2π /4µ2 )2 µ2 Fπ2
1)
he
i∼e
2)
New Banks Casher relation (essential for SBχS)
3)
ψ̄ψ and nB fluctuate wildly (Lorentzian dist)
reviews
Splittorff Verbaarschot arXiv:0809.4503
Lombardo, Splittorff, Verbaarschot arXiv:0912.4410
Lattice QCD at non zero chemial potential - status and challenges – p. 17/23
Lattice measurement of endpoint (Reweighting)
Fodor Katz JHEP 0203:014,2002; JHEP 0404:050,2004
Lattice QCD at non zero chemial potential - status and challenges – p. 18/23
Lattice measurement of endpoint (Rewieghting)
- Scale the axis by TC and mπ /2
1.2
1
T/T0
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2µ/mπ
Fodor Katz JHEP 0203:014,2002; JHEP 0404:050,2004
Splittorff, hep-lat/0505001, PoS LAT2006:023,2006
Philipsen 0710.1217
Lattice QCD at non zero chemial potential - status and challenges – p. 19/23
Progress: Subset method
Lattice QCD at non zero chemial potential - status and challenges – p. 20/23
Progress: Subset method
Method
Idea
Challenge
Subset method
Sum over subset of configs
Apply it work for QCD
Bloch, PRL.107:132002,2011
Lattice QCD at non zero chemial potential - status and challenges – p. 20/23
Progress: Complex Langevin
Lattice QCD at non zero chemial potential - status and challenges – p. 21/23
Progress: Complex Langevin
Method
Idea
Challenge
Complex Langevin
Stochastic flow in complex plane
Optimization
Problems for CL are ’orthogonal’ to he2iθ i ≪ 1
Parisi, Phys. Lett. 131 B (1983) 393
Aarts, Splittorff, JHEP 1008:017,2010
Aarts, Seiler, Stamatescu Phys.Rev.D81:054508,2010
Lattice QCD at non zero chemial potential - status and challenges – p. 21/23
Progress: Complex Langevin
Illustration by Philippe de Forcrand
Lattice QCD at non zero chemial potential - status and challenges – p. 22/23
Progress: Complex Langevin
Illustration by Philippe de Forcrand
1)
shift by O(V ) in imaginary direction
2)
Amplitude: real axis O(exp(V )); complex plane O(1)
Aarts personal correspondence (2009)
de Forcrand PoS (LAT2009), 10
Lombardo Splittorff Verbaarschot arXiv:0910.5482
Lattice QCD at non zero chemial potential - status and challenges – p. 22/23
Conclusions
Interplay between lattice QCD and analytic QCD has
1) allowed us to understand the sign problem
2) understand lattice results
3) solid progress towards a solution
Lattice QCD at non zero chemial potential - status and challenges – p. 23/23
Conclusions
Interplay between lattice QCD and analytic QCD has
1) allowed us to understand the sign problem
2) understand lattice results
3) solid progress towards a solution
Lattice QCD at non zero chemial potential - status and challenges – p. 23/23
Conclusions
Interplay between lattice QCD and analytic QCD has
1) allowed us to understand the sign problem
2) understand lattice results
3) solid progress towards a solution
Lattice QCD at non zero chemial potential - status and challenges – p. 23/23