First order transitions in finite temperature and density QCD with two and many flavors

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1 First order transitions in inite temperature and density QCD wit two and many lavors Sinji Ejiri (iigata University Collaboration wit Ryo Iwami (iigata, orikazu Yamada (KEK & Hirosi Yoneyama (Saga YITP Worksop HHIQCD 015 YITP, Kyoto Univ., Mar. 17, 015 1

2 Pase structure o QCD at ig temperature and density Lattice QCD Simulations Pase transition lines Equation o state Direct simulation: Impossible at µ 0. T RHIC LHC SPS quark-gluon plasma pase RHIC BES FAIR 1 st order? critical point? AGS deconinement? adron pase ciral SB? nuclear matter quarkyonic? color super conductor? µq

3 QCD pase transition at inite T and µ Expansion o te parameter space and Extrapolation Large cemical potential µ (at pysical mass Dierent quark mass at low density Large number o lavor Ciral limit o -lavor QCD (+1-lavor or (+many-lavor at inite mass Large volume limit Complex parameter: Lee-Yang zero 3

4 Quark Mass dependence o QCD pase trantion tricritical point = nd order Quenced 1 st order Pysical point ms Crossover 1 st order 0 0 mud µ = 0 µ First order crossover T Critical point QGP adron µ P. de Forcrand & O.Pilipsen, 03, 07 Bieleeld-Swansea Collab., 0, 03 On te line o pysical mass, te crossover at low density 1 st order transition at ig density. However, te 1 st order region is very small, and simulations wit very small quark mass are required. Diicult to study. 4

5 Critical surace in te eavy quark region o (+1-lavor QCD WHOT-QCD Collab., Pys.Rev.D89, (014 nd order 1 st order Polyakov loop distribution κ=0: irst order (Z(3 symmetric Im Ω Re Ω κ>κ c : crossover Critical surace at inite density µ T ms First order crossover Crossover 1 st order 0 0 mud Re Ω Im Ω κ s ~1/(strange mass κ ud 5 ~1/(up-down mass

6 Finite T and µ pase transition in (+many-lavor QCD Many-lavor QCD to construct Tecnicolor models Ciral pase transition o QCD Electroweak pase transition at inite temperature ambu-goldstone bosons 3 bosons are absorbed into gauge bosons. (3 massless bosons Te oter bosons ave not observed yet. (Te oter bosons: eavy tecni-elmions are massless, and te oters are eavy. Electro-weak baryogenesis Strong irst order transition: required. (SM: ot strong 1 st order. From te analogy o +1-lavor QCD, 1st order at small mass; nd order or crossover at large mass. It is important to determine te endpoint o te irst order region in (+many-lavor QCD. 6

7 ature o pase transition o + -lavor QCD Heavy quark region nd order tricritical point me large m eavy quark mass 1 st order -lavor Crossover Quenced 1 st order large µ = mud=ml S. E. &. Yamada, Pys. Rev. Lett. 110, (013 Assumption: -lavors are eavy. Hopping parameter κ expansion Parameter: κ t 1 m As increasing, critical mass becomes larger. Easy to investigate. Tricritical scaling: te same as (+1-lavor QCD m, ct ~ κct 1 ( m m 5 c Tricritical point ud ~ E m E c 5 m ud ~ µ 1 t At inite density?? Good test ground -lavor limit is te same as +1-lavor. 7

8 ature o -lavor QCD in te ciral limit nd order or 1 st order? Long standing problem Ligt quark mass (m l dependence o te critical line Trictitical scaling beavior? Is tere a irst order transition region in -lavor QCD? 1 m Tricritical point First order Critical points Crossover or 1 First order m o tricritical point Critical points Crossover m=, -lavor Tricritical scaling m l 5 -lavor First order transition region m l 5 Similar study in QCD wit an imaginary cemical potential: Bonati, D Elia, de Forcrand, Pilipsen, Sanilippo, arxiv: ;

9 Singularities o QCD in te complex µq plane Lee-Yang zero/ Fiser zero: partition unction Z=0 Prediction near te ciral limit, assuming O(4 universality M. Stepanov Pys. Rev. D73, (006 m = 0 m 0 nd order crossover µ q plane Z=0: distribute on tis line. Im ( µ q Z=0 Low density Re ( µ q ϕ 77 ψ = π βδ 48 Te distribution o Z=0 by Mote-Carlo simulations ature o pase transition (order & universality class

10 Pase transitions in many-lavor QCD We investigate te critical surace in -lavor QCD and QCD wit -ligt lavors + - massive lavors. (+-lavor QCD Electro-weak baryogenesis - Tecnicolor model Good testing ground or (+1-lavor QCD Plan o tis talk Histogram metod to study nature o pase transitions -dependence o te critical eavy quark mass. Ligt quark mass-dependence o te critical curve Te ciral limit o -lavor QCD: nd order or 1 st order? µ-dependence o te critical curve. Singularities in te complex µ plane, Lee-Yang zeros 10

11 Probability distribution unction Distribution unction (Histogram X: order parameters, total quark number, average plaquette etc. Z ( m, T, µ = dx W ( X, m, T, µ istogram In te Matsubara ormalism, W Z Sg ( m T µ,, DU ( detm ( m, µ e Sg ( X m T µ,,, DUδ( X-X ( detm ( m, µ e were detm: quark determinant, Sg: gauge action. Useul to identiy te nature o pase transitions e.g. At a irst order transition, two peaks are expected in W(X.

12 Z µ-dependence o te eective potential ( T, µ = dx W ( X, T, µ, e ( X = lnw ( X X: order parameters, total quark number, average plaquette, quark determinant etc. V V e Crossover ( X, T,µ Correlation lengt: sort V(X: Quadratic unction Critical point Correlation lengt: long Curvature: Zero T QGP 1 st order pase transition adron CSC? µ Two pases coexist Double well potential

13 Plaquette and Polyakov loop Dynamical variables Gauge ield: Uµ SU(3, on a link site Quark ield: ψ, ψ, Grassmann, on a site Standard gauge action S g Polyakov loop Ω = = β 1 n, µ ν 1 3 tr 1 tr 3 [ + µ + ] U ( n U ( n ˆ U ( n ˆ U ( n µ ν µ ν =P: plaquette [ U ( n,1 U ( n, U ( n, ] t s n Complex: Ω = ΩR + iω I ( β = 6 g ν link 1/T periodic boundary

14 Reweigting metod or plaquette distribution unction R W R ( P ( ( ( Pˆ P m DU Pˆ 6 β, β,, µ δ P detm, e = δ m µ = 1 ( ˆ ( ˆ 6 det (, site β β0 P M m µ P P e Eective potential: V P β, m, µ = ln W e δ ( Pˆ P plaquette P (1x1 Wilson loop or te standard action ( P β, β m, m, µ W ( P, β, m, µ W ( P, β,,0, m0 detm ( β 0, µ= 0 ( m,0 0 ( β 0, µ= 0 e 6 site site (Reweigt actor ( ˆ det (, β β0 P M m µ detm ( m0,0 P: ixed ( [ ( P, β, m, µ ] = V ( P, β, m,0 ln R( P, β, β m, m, µ, e ln R ( P = 6 ( β β site 0 P + ln detm detm ( m, µ S g ( m0,0 P: ixed = 6 Pˆ site β ( β = 6 g 14

15 First order transition point: two pases coexist Plaquette distribution unction V Perorming simulations o -lavor QCD, Dynamical eect o -lavors are included by te reweigting. We assume -lavors are eavy. Hopping parameter (κ expansion(wilson quark Eective potential e ( κ, ( 0,0 det M µ ln det M = [ R( P, κ W ( P, β, ]= ( P, β, κ = ln 0 4 t 3 t ( 88 κ P + 1 κ ( cos( µ T Ω + i sin( µ T Ω + site plaquette: P -lavor crossover V e s ( P, β,0?? + = ln R Polyakov loop: [ R( P, K ] Ω R + iω +-lavor 1st order transition Double-well I I 15

16 Rewigting o te eective potential ( ( ˆ det, site β β P M κ µ ( P ln e 0 ln R = 6 ˆ detm ( κ,0 0 P:ixed ln exp 6 3 ( s ΩR P: ixed (degenerate mass case at µ=0 +(linear term o P V e ( P β,, µ = V ( P, β,0,0 ln R( P,, µ +, e 0 (linear term o P R ( ( 3 P exp 6 s ΩR P: ixed = (or te case o µ=0 Wilson quark ( t = κ Staggered quark = ( 4( m t β-dependence is only in te linear term. 16

17 Pase structure o (+many-lavor QCD P4-imprived staggered Simulations = p4-staggered, mπ/mρ 0.7 data: Beileeld-Swansea Collab., PRD71,054508( x4 lattice. Improved-Wilson Simulations Iwasaki gauge action + = clover -Wilson ermion action, κ=0.145, 0.475, 0.150, , m π /m ρ = , , , , 16 3 x4 lattice. Dynamical eavy quark eect is added by te reweigting metod. detm: Hopping parameter expansion 17

18 Curvature o te eective potential V e ( P β,, µ = V ( P, β,0,0 ln R( P,, µ +, e 0 (linear term o P R ( ( 3 P = exp 6 s ΩR P: ixed (or te case o µ=0 Wilson quark Linear term o P is irrelevant to te curvature β-dependence is only in te linear term. Te curvature is independent o β. d V dp e ( t = κ d V dp d ln R dp ( P, µ = e ( P,0,0 ( P,, µ, -lavor Staggered quark = ( 4( m t I tere exists te negative curvature region, First order transition (double-well potential d V dp e χ P : plaquette susceptibility ( 0 6 χ site P 18

19 Eective potential at 0 p4-staggered ( P β, = V ( P, β,0 ln R( P V, e, e = p4-staggered, mπ/mρ 0.7 data: Beileeld-Swansea Collab., PRD71,054508(005 detm: opping parameter expansion. lnr increases as increasing. Te slope increases wit. V e ln R (0 19

20 Curvature o te eective potential d ln W dp at =0 = p4-staggared, mπ/mρ =? First order transition: d V e ( P, β, d V ( P, β,0 d ln R( P, dp = e dp dp < 0 ( t = κ First order transition or > 0.6 (Wilson quarks Critical value: c = (69 0

21 Slope o te eective potential V e ( P β,, µ = V ( P, β,0,0 ln R( P,, µ +, e 0 dv dp e dv dp d ln R dp e ( P,, µ = ( P,0,0 ( P,, µ + (linear term o P (constant term Te sape o dv e /dp is independent o β. I dv e /dp is an S-saped unction, First order pase transition (double-well potential. V e ( P dv e dp S-saped unction at large ( t = κ or Wilson quark

22 dependence o te critical mass c = (69 (p4-staggared, m π /m ρ 0.7 Critical mass increases as increases. Wen is large, κ is small. Ten, te opping parameter (κ expansion is good. On te and, wen is small, te κ-expansion is bad. In a quenced simulation wit t =4, te irst and second terms becomes comparable around κ=0.18. For =10, t =4, ( t = κ It may be applicable or ~10. κ c = 1 c = ( c κ c 1 t

23 Pase structure o (+many-lavor QCD using Wilson quark action -lavor QCD simulations + reweigting Ligt quark mass dependence o te critical line Is tere a irst order transition region in -lavor QCD? 1 m Tricritical point Critical points or 1 m o tricritical point Critical points m=, -lavor Tricritical scaling m l 5 -lavor First order transition region m l 5 3

24 Ligt quark mass dependence dv e dp m π /m ρ m π /m ρ large m π /m ρ m π /m ρ Color: Dierent it unction Te derivative o V e becomes an S-saped unction at large. 4 Critical point: ligt quark mass dependence is small in tis region.

25 Ligt quark mass dependence m p /m r ratio dependence ( t = κ or Wilson quarks PCAC quark mass dependence Critical point: ligt quark mass dependence is small in te region we investigated. Te red & green lines are te critical point at m l = ( =0+16. Te irst order transition in te massless -lavor QCD is not suggested. 5

26 Te eective potential at inite µ Reweigting actor ln R ( P = ln ( m, ( m,0 detm µ detm (, ( 0,0 detm µ detm ligt quarks eavy quarks P:ixed Ligt quark determinant: Taylor expansion up to O(µ 6 or staggered O(µ or Wilson ln det M ( µ det M ln det M ( κ, µ ( 0,0 = = n n 1 µ d ln det M = ( n 0 n! T d µ T n Heavy quark determinant: Hopping parameter expansion 88 site 4 κ P + 6 ( t = κ 3 s : complex pase Imln det M µ µ cos Ω R + i tan ΩI + T T control parameters 6

27 Cumulant expansion metod (SE,PRD77,014508(008, WHOT-QCD,PRD8,014508(010 Odd terms vanis rom a symmetry under µ µ ( Source o te complex pase I te distribution o is Gaussian, term dominates. Assuming te Gaussian distribution, we approximate + + = C C C C i i i e 4 3 4! 1 3! 1 exp Avoiding te sign problem at inite µ = + = = = C C C C , 3,, cumulants 0 0 C C i e 1 exp 7

28 Critical line at inite density (staggered ( t = κ or Wilson quarks = ( 4( m t or staggered quarks S. E. &. Yamada, Pys. Rev. Lett. 110, (013 First order Calculations o detm: Taylor expansion up to O(µ6 Distribution unction o te complex pase o detm: approximated by a Gaussian unction crossover Te irst order region becomes wider as increasing µ. 8

29 µ-dependence o critical C cos µ ( T tan( µ T ( t = κ Taylor expansion up to O(µ l critical surace in (µ l, µ κ t C T (ligt quark Te eect rom te pase o eavy-lavor is small ( < 30%. Te Critical κ decreases exponentially as µ. Hopping parameter expansion is good or large µ. µ µ l T 9

30 Singularities o QCD in te complex µq plane M. Stepanov Pys. Rev. D73, (006 Lee-Yang zero/ Fiser zero: partition unction Z=0 Prediction near te ciral limit, assuming O(4 universality m = 0 m 0 nd order crossover µ q plane Z=0: distribute on tis line. Im ( µ q Z=0 Low density Re ( µ q ϕ 77 ψ = π βδ 48 Singularities exist at large Im(µq even or crossover. Application range o Taylor expansion o Te distribution o Z=0 by Mote-Carlo simulations p T 4 = 1 VT ln Z ature o pase transition (order & universality class 3

31 Singularities o pure SU(3 gauge teory in te complex β plane (SE, Pys.Rev.D73,05450(006 Relation between distribution o Z=0 and plaquette distribution unction Contour plot o Z norm Z = Z ( βre, βim ( β,0 Re Plaquette data by QCDPAX, Pys.Rev.D46, 4657,(199 #con.~o(1m = 4 36 site 4 Lee-Yang zero Z=0 Plaquette distribution (istogram A 3π 1 site A π 1 site A First order transition:β Im ~ 1 V

32 Lee-Yang zeros in te complex β or pure SU(3 ormalized partition unction (reweigting or Imaginary β Z Z ( β = DU exp[ 6( β + iβ P] norm ( βre, βim ( β,0 Z = = exp P Re Z Re Re Im site Plaquette distribution unction (istogram w 6 ResiteP ( P = DUδ( P P e Z ( i6βimsitep ( = exp( i6βimsite β,0 ( β,0 1 β ( β = exp( i6β P w( PdP Z norm Im site ( i6β P 1, exp Im = site ( = 3 = V site Re ( P = P P w( P s t t Fourier transormation ( P βim site P

33 Fourier transormation ( P βim site w( P Gauss Z norm Gauss ( site = Vt on-singular: o Lee-Yang zero ormal P ( β Im site w( P A Double peak A First order transition P Znorm Distribution unction o te complex pase = 6β Im P site Z Lee-Yang zero ( β Im site β ( n +1 A Im = π 1 β Im site (n:integer ~ 1 V ( P( d ( β i norm = e w 6 Imsite

34 Singularities o ull QCD wit complex µ µ q : complex Z ( β µ R( P, µ W ( P, β, = q q dp (Reweigt actor S (Weigt actor at µq=0 g = 6 site βp * * ( det M ( µ = det M ( µ q q ( ( * P, µ = R P, µ R( P, * R µ q q q R ( iφ ( P, µ P e R( P q, µ =, µ q q Reweigt actor: complex Partition unction written by a distribution unction o P. Z Wen V e is a double-well potential And (, iφ ( P β µ = (, q e R P µ q W ( P, β dp = exp( V ( P, β, µ φ ( P φ( P π + πn, + Z e Lee-Yang zeros appear. ( ( iφ ( P (, ~ + iφ P β µ C e + e q q P- P+ Z=0? Z 0

35 umerical calculation o te reweigting actor approximations (SE, Pys.Rev.D77,014508(008 Estimation o detm by a Taylor expansion up to Sign problem: I canges its sign, Gaussian approximation Distribution unction o : Gaussian. ( = µ µ = µ n n n T M T n M 0 n d ln det d! 1 ( ln det error (statistical (0 det ( det ixed << µ P i P F e M M P C C C C P i i i F F e ! 1 3! 1 exp = + = = = C P F P F P F P F C P F P F C P F C 4 3,,,, 3 3,,,, 3,, ( 6 O µ q cumulant expansion 0 0 <..>F,P: expectation values ixed F and P. istogram o at β=3.65 i e

36 Derivative o te eective potential Assumption: Gaussian distribution o te pase. dv/dp becomes an s-saped unction. Im ( T µ q Re ( T µ q β 0 = 3.65 dv dp dv dp e ( P, β = ( P, β ( β β e 0 6 site = p4-staggared, mπ/mρ 0.7 data: Beileeld-Swansea Collab., PRD71,054508(005 0 dv e dp ( P P V e ( P P double-well potential

37 Singularities in te complex µq plane (SE & Yoneyama, in progress Position o Lee-Yang zeros or eac β. double-well Lee-Yang zeros single-well Critical line on singular Probability distribution unction becomes a double-peaked unction at large µim as well as large µre.

38 Summary We studied te pase structure o (+-lavor QCD. Tis model is interesting or te easibility study o te electroweak baryogenesis in te tecnicolor scenario. Applying te reweigting metod, we determine te critical mass o eavy lavors terminating te irst order region. Te critical mass becomes larger wit. Te irst order region becomes wider as increasing µ. Te ligt quark mass dependence o te critical eavy quark mass is small in te region we investigated. Te irst order transition in -lavor QCD is not suggested. Tis may be a good approac or te determination o boundary o te irst order region in (+1-lavor QCD at inite density. In te complex µ plane, te probability distribution unction becomes a double-peaked unction at large µim 38