Surprises in the Columbia plot Philippe de Forcrand ETH Zürich & CERN Fire and Ice, Saariselkä, April 7, 2018
Fire and Ice Launch Audio in a New Window BY ROBERT FROST (1874-1963) Some say the world will end in fire, Some say in ice. From what I ve tasted of desire I hold with those who favor fire. But if it had to perish twice, I think I know enough of hate To say that for destruction ice Is also great And would suffice.?
Fire and Ice?
QCD at finite temperature!
QCD at finite temperature! No: in this talk, keep µ =0 but vary quark masses
Columbia plot: expectations for QCD-like theories vs Temperature N. Christ et al, PRL 1990 Hot: deconfined + chirally symmetric Cold: confined + chirally broken } via crossover or phase transition (1rst, 2nd ) m s? O(4) N f =2 PURE 1 st GAUGE m tri s physical point N f =3? cross over N f =1 m c? 1 st Upper right: YM w/ center symmetry Upper/Lower left: chiral symmetry
Columbia plot: expectations for QCD-like theories vs Temperature Goal: determine red lines & blue line, i.e. 2nd transitions To check our understanding of phase diagram m s? O(4) N f =2 PURE 1 st GAUGE m tri s physical point N f =3? cross over N f =1 m c? 1 st µ Real world µ Real world Heavy quarks To help, e.g., with QCD critical point * QCD critical point QCD critical point DISAPPEARED 0 X crossover 1rst 0 X crossover 1rst m s m s
Numerical simulations µ =0 No sign problem! Still, computer cost at T T c at T 0 : Nb. of Monte Carlo iterations to explore both phases 2 for crossover or second- transition exp L2 s for first- transition Need box L 3 s 1/T, (L s T ) 1 cf. finite-size effects Further large factor for chiral limit m q! 0 Continuum limit: T T c, N t =4 time-slices! a 0.3fm =) Need N t & 12
Wilson versus staggered in the crossover region The good news: thermodynamics of stout-smeared staggered and Wilson agree perfectly (Fodor et al, 1205.0440) MeV 125 150 175 200 225 250 275 1 0.8 staggered continuum Wilson continuum s /T 2 0.6 0.4 0.2 0 0.08 0.1 0.12 0.14 0.16 T/m N t 2 [4, 28] Note: N f =2+1,m = 545 MeV,m K = 614 MeV,a& 0.057 fm
Warm-up: Yang-Mills SU(3), upper right corner m s O(4) N f =2 PURE 1 st GAUGE L(x) Polyakov loop (closed by finite- T b.c.) m tri s physical point? m c? cross over N f =3 N f =1 Order parameter htrli =exp( F q /T ) 1 st Global center symmetry: L(x)! exp(i 2 3 )L(x) 8x ( large gauge transformation! action invariant) Deconfinement transition: htrli = 0(low T )! 6= 0(high T ) Order of the transition: Svetitsky-Yaffe conjecture
Svetitsky-Yaffe (1982): any gauge group G,(d + 1) dimensions Suppose transition is second- (!1) Long-range physics governed by fluctuations of parameter If H e (TrL) TrL! H e is short-range, then only symm. group and dimension matter Universality class is that of ker(g) -symm. d-dim. scalar field theory Consequences: IF second- transition THEN SU(2) 3d Ising? True SU(3) 3d Z 3?? no known such univ. class! first-? Sp(2) 3d Ising? NO: first- hep-lat/0312022 Svetitsky-Yaffe does NOT predict of transition
SU(3) Yang-Mills deconfinement transition is first- Distribution of TrL in complex plane at T c Allowed domain in complex plane for TrL, L 2 SU(3)
Upper right corner? m s O(4) physical point N f =2 N f =3 PURE 1 st GAUGE Deconfinement transition First- for pure gauge m crit q O(2 3) GeV T c m tri s? cross over N f =1! need large N t m c? (and N t N s : multiscale problem) 1 st So far, N max t = 8 (am crit 2) Philipsen et al. 1609.05745
Upper and lower left corners? Pisarski & Wilczek (1984) m s O(4) N f =2 PURE 1 st GAUGE Chiral transition: all m q =0 Global symmetry SU(N f ) A m tri s physical point N f =3? cross over N f =1 spontaneously broken/restored at T c Order parameter h i m c? IF 2nd, 1 st! THEN univ. class of 3d SU(N f ) N f =2! O(4) N f 3! first- from Ginzburg-Landau analysis
Pisarski & Wilczek critique Argument does not exclude first- (for ) N f =2 Global symmetry SU(N f ) A! U(N f ) A if U(1) A restored at T c Ginzburg-Landau analysis of effective potential for h i may FAIL: Vicari et al.: 3d ACP N 1 (1706.04365) 3d ARP N 1 (1711.04567) 6-loop G-L analysis: No stable fixed point! first- Monte Carlo: solid evidence for second- transition Explanation?? - 6-loop not enough?? - gauge d.o.f. absent from Ginzburg-Landau potential (w/ T. Rindlisbacher)
Pisarski & Wilczek critique Summary: When 2nd- predicted, may still be first- When first- predicted, may still be 2nd- Zero predictive power! Renewed interest in Monte Carlo simulations...
Upper left corner m s O(4) N f =2 PURE 1 st GAUGE m tri s physical point? cross over N f =3 N f =1 Technical difficulty: chiral limit m c? 1 st Bypass difficulty with imaginary chemical potential (µ/t) 2 m s - Extended Columbia plot 0 - Red surfaces are 2nd- PT -(/3) 2 bounded by tricritical lines Magenta point separates O(4) [above] and first- [below]! track blue line
Upper left corner (mu tric (m=0)/t) 2 = 0.85(5) (µ/t) 2 m s 0-0.2 (mu tric (m=0)/t) 2 = 0.85(5) Di Giacomo 2008 mc1 mc2 0 (mu/t) 2-0.4-0.6 Crossover -(/3) 2-0.8-1 First- -(pi/3) 2 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 (a m q ) ❷ Results: staggered fermions, N t =4 1311.0473, 1408.5086 Scaling consistent with tricritical point (mean field in 3d) (µ =0,m q = 0) point is in the first- region Pisarski & Wilczek wrong? to be confirmed on finer lattices 1711.05658
Chiral limit with physical m s? Unger, Karsch et al. m s O(4) N f =2 PURE 1 st GAUGE 0909.5122 Improved staggered fermions, N t =4 m tri s physical point N f =3? cross over N f =1 + (not ), or then first- O(4) O(2) Z 2 m c? 1 st Results: Consistent with O(2) chiral transition Cannot distinguish from Z 2 at small (minimum m 75 MeV )
Lower left corner: critical pion mass N f =3 m s N f =2 2 nd O(4) physical point N f =3 m tri s? m c? 1 st cross over PURE 1 st GAUGE N f =1 Tune quark mass for 2nd thermal transition Measure T =0 pion mass or m /T c Varnhorst, LAT14: More improvement, more smearing + smaller m N t action m p,c Ref. 4 stagg.,unimproved 260 MeV [5] 6 stagg.,unimproved 150 MeV [6] 4 stagg.,p4 70 MeV [7] 6 stagg.,stout apple 50 MeV [8] 6 stagg.,hisq apple 45 MeV [9] 6 Wilson-Clover 135 MeV [4] Karsch et al, 2001 PdF & OP, 2007 Karsch et al, 2004 Endrodi et al, 2007 Karsch et al, 2011 Ukawa et al, 2014
The first- region is small m s O(4) N f =2 PURE 1 st GAUGE Endrodi et al, 0710.0998 N t =4, 6 smeared KS m tri s physical point? cross over N f =3 N f =1 m c? 1 st more smearing
Staggered vs Wilson, N f =3: agreement when a!0? 4.5 Nf=3, Nt=4,6,8 4 m /T c m p i/t c 3.5 3 2.5 2 Wilson (clover-imp) (Ukawa et al, 1411.7461) Staggered (PdF et al, 0711.0262) 1.5 1 0.5 0 Clover Wilson + Iwasaki Standard KS 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 2 1/N t continuum / a 2 Considerable discrepancy for N t =4, 6. Extrapolate?
Staggered vs Wilson, N f =3: agreement when a!0? 4.5 Nf=3, Nt=4,6,8 4 m /T c m p i/t c 3.5 3 2.5 2 2.60 Wilson (clover-imp) (Ukawa et al, 1411.7461) Staggered (PdF et al, 0711.0262) 1.5 1 0.5 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 2 1/N t continuum 0.37 Clover Wilson + Iwasaki Standard KS / a 2 Considerable discrepancy for N t =4, 6. Extrapolate? Discretization error O(100%) for staggered and Wilson!
Staggered vs Wilson, N f =3: agreement when a!0? Need finer lattices for reliable extrapolation: N t =6) a 0.2 fm Careful: (am q ) decreases much faster than a : Can rooting (det KS ) 3/4 be playing tricks on us? Famous quote: rooting is evil (Mike Creutz) Check universality for N f =4 (bonus: cheaper, because m /T c increases)
Staggered N f =4, N t =4, 6, 8, 10 w/ M.D Elia 1702.00330 Determine critical quark mass the usual way: B 4 ( ) =1.604 N f =4, N t =4 ratio of moments of distribution 3 2.5 L=8 L=12 L=16 N f =2 m s 2 nd O(4) m tri s physical point? cross over N f =3 PURE 1 st GAUGE N f =1 B 4 (pbp) 2 1.5 first crossover second (scale-invariant) m c? 1 st N f =4 1 0.02 0.04 0.06 0.08 0.1 0.12 0.14 a m q 3d-Ising finite-size scaling Here, N t =4! am crit q =0.0548(11) Then, at T =0: m (am crit q )/T c =2.39(1)
Staggered N f =4, N t =4, 6, 8, 10 w/ M.D Elia 1702.00330 m pi /T c 3 2.5 2 1.5 1 0.5 0 Complete [preliminary] results 10 8 Nf=4, Nt=4,6,8,10 6 N f =3 Standard KS, Nf=4 Standard KS, Nf=3 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 2 1/N t / a 2 Dramatic plunge as a! 0!! 4 m /T c consistent with zero (with large error) in continuum limit Caveats: spatial size too small, finite-size scaling incomplete, stats..
Why??? 3 2.5 2 Nf=4, Nt=4,6,8,10 6 4 m pi /T c 1.5 1 0.5 0 10 8 N f =3 Standard KS, Nf=4 Standard KS, Nf=3 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 2 1/N t N f =4: no rooting here Taste doublers? effective Larger N f N f increases as a! 0 makes transition stronger (ie. opposite effect) Vicari et al.: failure of Pisarski & Wilczek?
Wilson-clover, N f =3, N t =4, 6, 8, 10 4 Ukawa et al, 1706.01178, 1710.08057 6 10 8 m p t0 Note: slope increases for finer lattices (like staggered) T c p t0 m /T c Wilson qualitatively similar to staggered N f =3 N f =4
Conclusions Pisarski & Wilczek make NO trustworthy prediction The time is ripe for revisiting the Columbia plot Careful: cutoff errors are very large m s O(4) N f =2 PURE 1 st GAUGE m s O(4) N f =2 PURE 1 st GAUGE m tri s physical point? cross over N f =3 N f =1 m tri s physical point? cross over N f =3 N f =1 m c? 1 st? m c? 1 st
Thank you Aleksi!
Thank you Aleksi! Now I believe in Santa Claus!
Backup
Center symmetry Consider center transformation ( large gauge transformation in continuum): U 4 (x, 0 )! exp i 2 N k {z } z k 2Z N - space-like plaquettes unaffected U 4 (x, 0 ) 8x; 0 fixed 1/T - time-like plaquettes at = 0 multiplied by z k z k = 1 (z k commutes with all links) Action S L = X 1 N ReTr invariant But Polyakov loop rotated: L(x)! z kl(x) 8x, ie. htrli! z k htrli =) htrli =0 - Center symmetry realized, ie. confinement =) htrli 6=0 - Center symmetry spontaneously broken, ie. deconfinement Note: inverse symmetry breaking, ie. at high temperature (YM: less dis at high T )
ACP N 1 Complex generalization of non-linear -model z x 2 C N, z x 2 =1 X H = J z x z y 2 hx,yi Global symmetry: z x! G z x 8x, G 2 U(N) Local symmetry: z x! L (x) z x, L (x) 2 U(1) ACP N 1 : J<0! z x? z y in ground-state RP N 1 : same but z x 2 R N! G 2 O(N), L (x) 2 Z 2