Revisiting the strong coupling limit of lattice QCD

Transcription

1 Revisiting the strong coupling limit of lattice QCD Philippe de Forcrand ETH Zürich and CERN with Michael Fromm (ETH)

2 Motivation Intro Algorithm Results Concl years of analytic predictions: 80 s: Kluberg-Stern et al., Kawamoto-Smit, Damgaard-Kawamoto T c (µ= 0) = 5/3, µ c (T = 0) = s: Petersson et al., 1/g 2 corrections 00 s: detailed (µ, T) phase diagram: Nishida, Kawamoto,... 08: Ohnishi, Münster & Philipsen,... How accurate is mean-field (1/d) approximation? Almost no Monte Carlo crosschecks: 89: Karsch-Mütter MDP formalism µ c (T = 0) : Karsch et al. T c (µ= 0) : Azcoiti et al., MDP ergodicity?? 06: PdF-Kim, HMC hadron spectrum 2% of mean-field Can one trust the details of analytic phase-diagram predictions?

3 Phase diagram according to Nishida (2004) Very similar to conjectured phase diagram of N f = 2 QCD

4 Intro Algorithm Results Concl. Strong coupling SU(3) with staggered quarks Z = D UD ψd ψexp( ψ(d/(u)+m)ψ), no plaquette term (β = 0) One KS fermion field (ie. 4 tastes"): 6 d.o.f. per site D/(U) = 1 2 x,ν η ν (x)(u ν (x) U ν(x ˆν)), η ν (x) = ( ) x 1+..+x ν 1 Chemical potential µ exp(±aµ)u ±4

5 Intro Algorithm Results Concl. Strong coupling SU(3) with staggered quarks Z = D UD ψd ψexp( ψ(d/(u)+m)ψ), no plaquette term (β = 0) One KS fermion field (ie. 4 tastes"): 6 d.o.f. per site D/(U) = 1 2 x,ν η ν (x)(u ν (x) U ν(x ˆν)), η ν (x) = ( ) x 1+..+x ν 1 Chemical potential µ exp(±aµ)u ±4 Alternative 1: integrate over fermions Z = D U det(d/(u)+m) HMC, etc...

6 Intro Algorithm Results Concl. Strong coupling SU(3) with staggered quarks Z = D UD ψd ψexp( ψ(d/(u)+m)ψ), no plaquette term (β = 0) One KS fermion field (ie. 4 tastes"): 6 d.o.f. per site D/(U) = 1 2 x,ν η ν (x)(u ν (x) U ν(x ˆν)), η ν (x) = ( ) x 1+..+x ν 1 Chemical potential µ exp(±aµ)u ±4 Alternative 1: integrate over fermions Z = D U det(d/(u)+m) HMC, etc... Alternative 2: integrate over links Rossi & Wolff Color singlet degrees of freedom: Monomer (meson ψψ) M(x) {0,1,2,3} Dimer (meson hopping), non-oriented n ν (x) {0,1,2,3} Baryon hopping, oriented BBν (x) {0,1} self-avoiding loops C

7 Strong coupling SU(3) with staggered quarks Z = D UD ψd ψexp( ψ(d/(u)+m)ψ), no plaquette term (β = 0) One KS fermion field (ie. 4 tastes"): 6 d.o.f. per site D/(U) = 1 2 x,ν η ν (x)(u ν (x) U ν(x ˆν)), η ν (x) = ( ) x 1+..+x ν 1 Chemical potential µ exp(±aµ)u ±4 Alternative 1: integrate over fermions Z = D U det(d/(u)+m) HMC, etc... Alternative 2: integrate over links Rossi & Wolff Color singlet degrees of freedom: Monomer (meson ψψ) M(x) {0,1,2,3} Dimer (meson hopping), non-oriented n ν (x) {0,1,2,3} Baryon hopping, oriented BBν (x) {0,1} self-avoiding loops C 3! (3 n ν (x))! Z(m,µ) = {M,n ν,c} x M(x)! mm(x) x,ν 3!n ν (x)! ρ(c) loops C with constraint (M + ±ν n ν )(x) = 3 x / {C}

8 MDP Monte Carlo Intro Algorithm Results Concl. 3! (3 n ν (x))! Z(m,µ) = {M,n ν,c} x M(x)! mm(x) x,ν 3!n ν (x)! ρ(c) loops C with constraint (M + ±ν n ν )(x) = 3 x / {C} 3 difficulties: sign of C ρ(c): associate ± baryon loops with ( & ) polymer loops weight: ±cosh µ +1 much milder sign problem T MDP ensemble Karsch & Mütter

9 MDP Monte Carlo Intro Algorithm Results Concl. 3! (3 n ν (x))! Z(m,µ) = {M,n ν,c} x M(x)! mm(x) x,ν 3!n ν (x)! ρ(c) loops C with constraint (M + ±ν n ν )(x) = 3 x / {C} 3 difficulties: sign of C ρ(c): associate ± baryon loops with ( & ) polymer loops weight: ±cosh µ +1 much milder sign problem T MDP ensemble Karsch & Mütter changing monomer number difficult: weight m x M(x) monomer-changing update (Karsch & Mütter) restricted to m O (1)

10 MDP Monte Carlo Intro Algorithm Results Concl. 3! (3 n ν (x))! Z(m,µ) = {M,n ν,c} x M(x)! mm(x) x,ν 3!n ν (x)! ρ(c) loops C with constraint (M + ±ν n ν )(x) = 3 x / {C} 3 difficulties: sign of C ρ(c): associate ± baryon loops with ( & ) polymer loops weight: ±cosh µ +1 much milder sign problem T MDP ensemble Karsch & Mütter changing monomer number difficult: weight m x M(x) monomer-changing update (Karsch & Mütter) restricted to m O (1) tight-packing constraint local update inefficient, esp. as m 0

11 MDP Monte Carlo Intro Algorithm Results Concl. 3! (3 n ν (x))! Z(m,µ) = {M,n ν,c} x M(x)! mm(x) x,ν 3!n ν (x)! ρ(c) loops C with constraint (M + ±ν n ν )(x) = 3 x / {C} 3 difficulties: sign of C ρ(c): associate ± baryon loops with ( & ) polymer loops weight: ±cosh µ +1 much milder sign problem T MDP ensemble Karsch & Mütter changing monomer number difficult: weight m x M(x) monomer-changing update (Karsch & Mütter) restricted to m O (1) tight-packing constraint local update inefficient, esp. as m 0 Solved with worm algorithm (Prokof eev & Svistunov)

12 Intro Algorithm Results Concl. Worm algorithm for MDP Here for chiral limit m = 0 (no monomers: M(x) = 0 x) Break a dimer bond and introduce a pair of adjacent monomers M(x),M(y) Choose among neighbours of y by local heatbath and move M(y) there heatbath: sampling of 2-point function 1 Z M(x)M(y)exp( S ) Keep moving head y until y x, ie. worm closes new configuration in Z

13 Intro Algorithm Results Concl. Worm algorithm for MDP Here for chiral limit m = 0 (no monomers: M(x) = 0 x) Break a dimer bond and introduce a pair of adjacent monomers M(x),M(y) Choose among neighbours of y by local heatbath and move M(y) there heatbath: sampling of 2-point function 1 Z M(x)M(y)exp( S ) Keep moving head y until y x, ie. worm closes new configuration in Z Global change obtained from sequence of local updates Each local step gives information on 2-point function Very close to Adams & Chandrasekharan for U(N)

14 Worm algorithm for MDP Here for chiral limit m = 0 (no monomers: M(x) = 0 x) Break a dimer bond and introduce a pair of adjacent monomers M(x),M(y) Choose among neighbours of y by local heatbath and move M(y) there heatbath: sampling of 2-point function 1 Z M(x)M(y)exp( S ) Keep moving head y until y x, ie. worm closes new configuration in Z 1 Local Metropolis, 4 3 x2 at µ c, m q = n B # iterations x 10 4 Worm, same parameter set 1 n B # iterations x 10 4

15 Consistency check with HMC 0.8 Worm MDP vs. HMC (Forcrand and Kim 06) β = 0, same volume (µ = T = 0) m π, HMC σ/n c σ/n c, Mean Field m π σ/n c, Worm m π m q

16 Consistency check with HMC Worm MDP vs. HMC (Forcrand and Kim 06) β = 0, same volume (µ = T = 0) m, HMC π m, Mean Field π m π, Worm m q

17 Sign problem? Intro Algorithm Results Concl. Worst case m = 0: 1 Average sign, L 3 x2, m q = σ± L = 10 L = µ Can reach µ, ie. adequate

18 Transition T = 0,µ= µ c Puzzle: Mean-field baryon mass is 3 expect µ c = 1 3 F B(T = 0) 1 Mean-field estimate µ c much smaller Baryon mass 3 checked by HMC µ c 0.63 checked by Karsch & Mütter for T = 1/4 only PdF & Kim Explanation? Problem with m 0 or T 0 extrapolation of MC data? Or nuclear attraction 1/3 baryon mass! Check with m = 0, T 0 worm simulations

19 Consistency check with Karsch & Mütter Baryon number density n B, 8 3 x4, m q = 0.1, Worm vs. Metropolis N t = 4, Worm N t = 4, Karsch n B µ Agreement except at µ= 0.68 µ c ergodicity of local update

20 Reducing the quark mass Baryon number density n, L 3 x2 (4), m = 0.1, Worm vs. Metropolis B q N = 4, L = 8 t N = 2, L = 10 t N = 4, L = 8, t Karsch n B µ As m 0, µ c decreases and transition becomes stronger

21 Reducing the quark mass Nt = 2, L = 4 L = 10 Nt = 4, L = 4 L = 10 Baryon number density n B, L 3 x2 (4), m q = n B Phase coexistence N t = µ As m 0, µ c decreases and transition becomes stronger

22 Reducing the quark mass Nt = 2, L = 4 L = 10 Nt = 4, L = 4 L = 10 Baryon number density n B, L 3 x2 (4), m q = 0 n B Phase coexistence N t = µ As m 0, µ c decreases and transition becomes stronger

23 Varying the mass at fixed T = 1/2 1 Baryon number density n B, 10 3 x2, varying m q n B x2, m = 0.1 q 0.1 m = q m = 0 q µ From first-order (m = 0) to crossover (m = 0.1) critical mass m c?

24 Critical mass m c (T = 1/2)? 10 0 Histogram n B, µ = µ c, T = 1/2, m = m = m = x x m = x x x x Critical mass m c (T = 1/2) 0.05

25 CEP: compare with Nishida (2004) Qualitative agreement, but not quantitative

26 m = 0: compare µ c (T = 1/2,T = 1/4) with Nishida (2004) Qualitative agreement, but not quantitative

27 m = 0: compare µ c (T = 1/2,T = 1/4) with Kawamoto (2005) Take T c = 5/3 (mean-field) [MC: 1.40 Karsch] qualitative agreement, but not quantitative

28 Conclusions Intro Algorithm Results Concl. Summary For m = 0, µ c (T = 1/4) 0.62 (< m B /3) and µ c (T = 1/2) 0.54 Critical end-point (not chiral) moves to larger µ as m increases Outlook Improve systematics: Multicanonical MC for first-order transition at low T Asymmetry γ in Dirac coupling to vary T continuously Check mean-field scaling T = γ 2 /N t Compare real and imaginary µ Determine phase diagram: Tricritical point for m = 0 Critical end-point as a function of m Extend to 2 KS fields: Baryon no longer self-avoiding Bπ scattering etc.. Isospin µ