The strange degrees of freedom in QCD at high temperature. Christian Schmidt

Transcription

1 The strange degrees of freedom in QCD at high temperature Christian chmidt Christian chmidt XQCD 213 1

2 Abstract We use up to fourth order cumulants of net strangeness fluctuations and their correlations with net baryon number fluctuations to extract information on the strange meson and baryon contribution to the low temperature hadron resonance gas, on the dissolution of strange hadronic states in the crossover region of the QCD transition, on the quasi-particle nature of strange quark contributions to the high temperature quark-gluon plasma phase. T. T c T c. T. 2T c T & 2T c T c := chiral crossover temperature Christian chmidt XQCD 213 2

3 Outline 1) Introduction Definitions of cumulants and correlations Motivations to study fluctuations of conserved charges 2) The lattice setup and results The HIQ action 4 th order fluctuations and correlations 3) A closer look to strangeness trangeness in the HRG model Disentangling different strangeness sectors 4) ummary based on BNL-BI: arxiv: BNL-Bielefeld Collaboration: A. Bazavov, H.-T. Ding, P. Hegde, O. Kaczmarek, F. Karsch, E. Laermann,. Mukherjee, P. Petreczky, C. chmidt, D. mith, W. oeldner, M. Wagner Christian chmidt XQCD 213 3

4 Generalized usceptibilites / Cumulants Expansion of the pressure: p T 4 = 1X i,j,k= 1 i!j!k! BQ ijk, µb T i µq T j µ T k X = B, Q, : conserved charges X n, = 1 VT n ln X /T ) n µ X = generalized susceptibilities only at µ X =! Experiment VT 3 X 2 = ( N X ) 2 VT 3 X 4 = ( N X ) 4 3 ( N X ) 2 2 VT 3 X 6 = ( N X ) 6 15 ( N X ) 4 ( N X ) 2 +3 ( N X ) 2 3 cumulants of net-charge fluctuations N X N X hn X i only at freeze-out ( µ f ( p s),t f ( p s) )! ratios of cumulants are volume independent! Christian chmidt XQCD 213 4

5 Motivations 1) Discover a critical point (if exists) Analyze Taylor series/pade resummations of various susceptiblities: find region of large fluctuations Analyze the radius of convergence directly 2) Analyze freeze-out conditions Expected phase diagram of QCD: 154(9) critical end-point QCD quark-gluon-plasma Match various cumulant ratios of measured electric charge fluctuations to (lattice) QCD results: determine freeze-out parameters.! see talk by M. Wagner 3) Identify the relevant degrees of freedom (this talk) Compare (lattice) QCD fluctuations to various hadronic/quasiparticle models: hadron gas nuclear matter neutron stars Christian chmidt XQCD vacuum chemical potential µ B Does deconfinement take place above the chiral crossover temperature?

6 Motivations Does deconfinement take place above the chiral crossover temperature? The chiral crossover line: T c = 154(9) MeV T c (µ B )=T c () 1.66(7)µ 2 B HotQCD, PRD 85 (212) 5453 BNL-BI, PRD 83 (211) 1454 Karsch, CPOD 213, arxiv: T f [MeV] s NN = 2 GeV s NN = 62.5 GeV LQCD: T c (µ B ) TAR + HRG: PRC 79, 3499 (29) TAR charge flucn. (prelim.) + LQCD PHENIX charge flucn. (prelim.) + LQCD TAR proton flucn. (prelim.) + LQCD baryon flucn. µ B f [MeV] freeze-out points are in agreement with the chiral crossover line apparent discrepancies among the freeze-out points that need to be resolved Christian chmidt XQCD 213 6

7 The Lattice etup Action: highly improved staggered quarks (HIQ) Lattice size: , , Mass: m q = m s /2! m 16 MeV tatistics: O(1 3 ) - O(1 4 ) Observables: traces of combinations of M and ln = 1 Z DU Tr M 1 M e Tr lnm e G Z = Tr M 1 2 ln 2 = Tr M 1 M Tr M 1 M M 1 + DTr M 1 M 2 E Method: stochastic estimators with Tr [Q] 1 NX i N Q i i=1 N = 15 with random vectors 1 NX i,x N i,y = lim N!1 i=1 x,y Christian chmidt XQCD 213 7

8 Generalized usceptibilites / Cumulants.3 2 B B T c =154(9)MeV N =6 8 BNL-Bielefeld preliminary T c =154(9)MeV N =6 8 BNL-Bielefeld preliminary structure consistent with O(4) critical behavior at µ B =, m = Christian chmidt XQCD 213 8

9 B Correlations 2 nd order 4 th order B - 11 N = B - 13 N =6 8 T c =154(9)MeV B 22 T c =154(9)MeV BNL-Bielefeld preliminary CB [Koch et al., PRL 95 (25) 18231] B 11 / 2 HRG BNL-Bielefeld preliminary Cont N =6 N =8 N =12 free B Christian chmidt XQCD 213 9

10 Figure We will 1: The consider ratiovarious of quartic combinations and quadratic of mixed net strangeness susceptibilities fluctuations χ B nm which versus are motivated temperature. by their Results interpretation are compared in terms to theof HRG a free model quark calculation. gas at highalso temperature shown is or the a result free hadron for trangeness a HRG gas at model low with temperature. within = the 1 hadrons The HRGlatter only. is Atgiven high temperatures in terms of aresult noninteracting are compared gaswith of mesons fluctuations and baryons, in an ideal i.e., the strange hadron quark resonance gas. The gas grey (HRG) bandmodel, shows HRG: the result for the pseudo-critical temperature and its error band [24]. p HRG = 1 ln Z M T 4 VT 3 m i (T,V,µ )+ 1 trangeness in Hadron Resonance ln Z B VT Gas 3 m i (T,V,µ B,µ ), (3) where the partition i function mesons for mesonic (M) or baryonic i baryons (B) particle species i with P mass m i is = M+B given by, 1 +B 2 +B 3 1 Throughout with M: partial pressure of =1 mesons this work we set the electric charge chemical potential tozero,µ Q =. χ 2 = M+B B ( B ln Zm M/B i VT3 π d mi ) 3 2 (±1) k+1 B 1 : partial pressure of =1 baryons χ B 11 = B 2 1 2B i 2 3B 3 T k=1 B 2 : partial pressure of =2 baryons χ B 31 = B 1 2B 2 3B 3 B 3 : partial here pressure we have of =3 set µ baryons Q χ B 22 = B B B 3 For baryons χ B and 13 = B strange Bparticles B we can validity of Boltzmann approx. 3 assume Boltzmann approximation: χ 4 = M+B B B 3 ln Z M/B m i / f(m i,t) cosh(b i ˆµ B + i ˆµ ) pressure Boltzmann obtains contributions approx works from very different well hadronic for strange sectors, hadrons, defined by all possible (B i, i )-combinations k 2 K 2 (km i /T )cosh(k(b iˆµ B + iˆµ )). (4) Here upper signs correspond to mesons and lower signs to baryons; d i is the degeneracy factor of particles with mass m i and K 2 (x) is a modified Bessel function. Note that in Eq. (4) the sum is taken over particle states only. Contributions from particles and the corresponding anti-particles combine to the cosh-term in Eq. (4). With these relations it is straightforward to calculate the mixed susceptibilities defined in Eq. (1). µ B,µ -dependence factorizes In the low temperature range that is relevant for a comparison with HRG model calculations, T < 17 MeV, it is a good approximation to treat the entire strange deviations < 3% hadron sector of the HRG in Boltzmann approximation, i.e., we restrict the sum in Eq. (4) to the k = 1 term only. The HRG than has a simple representation in terms 7 Christian chmidt XQCD 213 1

11 The pressure obtains contributions from different hadronic sectors: p HRG trangeness within the HRG = f T 4 (,) (T )+f (,1) (T ) cosh( ˆµ ) } mesons + f (1,) cosh(ˆµ B )+f (1,1) (T ) cosh(ˆµ B µ ) } baryons + f (1,2) (T ) cosh(ˆµ B 2ˆµ )+f (1,3) (T ) cosh(ˆµ B 3ˆµ ) χ 4 /χ2 HRG HRG =1 B n,m 6 N τ =6 8 B n B n+2 for diagonal fluctuations:, whereas (multi-strange hadrons) for correlations: B n+2,m free at T. 16 MeV we find reasonable agreement with the HRG (within 2%) at T & 16 MeV deviations from HRG become large Figure 1: The ratio of quartic and quadratic net strangeness fluctuations versus Christian temperature chmidt (left) and the ratio of mixed XQCD susceptibilites 213 of net strangeness and 11 1 B B χ 31 /χ11 n 6 n+2 HRG N τ =6.2 free

12 Disentangling different strangeness sectors The pressure obtains contributions from 4 different strangeness sectors: p HRG = f T 4 (,) (T )+f (,1) (T ) cosh( ˆµ ) } mesons + f (1,) cosh(ˆµ B )+f (1,1) (T ) cosh(ˆµ B µ ) } baryons + f (1,2) (T ) cosh(ˆµ B 2ˆµ )+f (1,3) (T ) cosh(ˆµ B 3ˆµ ) ) p HRG T 4! 6= = M 1 + B 1 + B 2 + B 3 ) diagonal fluctuations and correlations are given as linear combinations of the different strangeness sectors 2 =( 1)2 M 1 +( 1) 2 B 1 +( 2) 2 B 2 +( 3) 2 B 3 4 =( 1)4 M 1 +( 1) 4 B 1 +( 2) 4 B 2 +( 3) 4 B 3 B 11 =( 1)B 1 +( 2)B 2 +( 3)B 3 B =( 1)2 B 1 +( 2) 2 B 2 +( 3) 2 B 3 invert this relation! Christian chmidt XQCD

13 The most general combination of all diagonal and o -diagonal fluctuations operators up to the 4 th order is given by sensitive to strangeness fluctuations, we have six independent observables (χ 2, χ 4, χ B 13, χb 22, χb 31 A 1 ) Bat handb that we can use to B analyzebfluctuations and correlations 11 + A A A A 5 13 A 6 4. (1) in different strangeness sectors of the HRG. Using these six observables it is easy to It will project to a particular combination of the meson (M) construct observables that project onto different strangeness sectorsofthehrg.we Idea: and various separate strange strangeness baryon contributions sectors by making (B =1,Buse =2 of,ball =3 diagonal ), denoted strangeness as will consider here a set of observables that give, in the low temperature regime, the fluctuations and baryon-strangeness correlations up to the 4 th order contribution1 of M strange + 2 Bmesons =1 + 3 and B =2 strange + 4 Bbaryons =3. with strangeness (2) = 1, 2and B 3 to the (2+1)-flavor x QCD x 2 B 31 partition + x function. x 4 B 22 + They x 5 B 13 all + x depend 6 4 on 2 free parameters, with Disentangling different strangeness sectors Based on the strangeness content of the fluctuation operators we can find a linear mapping F : R 6! R 4 that can be expressed = y 1 Mby 1 a+ matrix y 2 B 1 F+ ij y2 3 BR (6 4) 2 + y, 4 B 3 i.e. wem(c can 1 write,c 2 ) = χ 2 χb 1 solve: A~x =ê i with 1 ( B (c 1,c 2 ) = χ 2 4 χ 2 +5χB 13 F ij A j = i, with A F = B χB C A (3) χ 4 χ χB χB B 2 (c 1,c 2 ) = ( χ 4 χ 2 +3χB 22 + c 1v 1 + c 2 v 2 (6) ) + c1 v 1 + c 2 v 2 (7) ) + c1 v 1 + c 2 v 2 (8) ) + c1 v 1 + c 2 v 2 (9) As oneb defined can easily see, F has Rank(F )=4,anditskernelK is a 2-dimensional 3 (c 1,c 2 by ) powers = of strangeness charges 13 vector space (dim(k) = 618 Rank(F)). olving +3χB 22 the homogenous Rank=4 system F ij x j =wefindtwovectorsv 1,v 2 that span the kernel K, we have dim (Kernel)=2, spanned by v 1,v 2 v1 t = (1,, 1/6, 2, 1, 1/6) (4) v 1 = χ B 31 χ B 11 (1) v t 2 = (, 1, 1/6, 2, 1, 1/6) (5) K = {x 2 R 6 : x = av 1 + bv 2 with a, b 2 R} (6) v 2 = 1 3 (χ 2 χ 4 ) 2χ B 13 4χ B 22 2χ B 31 (11) In Order to project onto a specific strangeness sector we have to solve the inhomogenousmust systemvanish F ij A j if = HRG j, with is a valid t =(1, description!,, ), (, 1,, ), (,, 1, ) and (,,, 1), respectively. In general the4 solution set of the inhomogenous Christian chmidt XQCD

14 Disentangling different strangeness sectors.35.3 v 1 free HTL.35.3 χ 2 B - χ4 B free HTL.25 v N τ =6 (open) N τ =8 (filled) N τ = strange baryons carry baryon number 1 partial pressure from strange particles is all baryons carry baryon number 1 Figure 2: The difference between the correlations of net strangeness fluctuations hadronic with different powers of net baryon number fluctuations (left). The right hand figure shows the difference of quadratic and quartic baryon number fluctuations. indicator for the validity of the HRG The particular differences shown should vanish in the low temperature phase, if this can be described at T. by16 a superposition MeV we find of reasonable uncorrelated agreement hadrons, with which the HRG only have baryon number B at =or±1. T & 16This MeVisdeviations the case in from thehrg hadron become resonance large gas model. Lines at high temperature show the ideal gas value (free) and the O(g 4 ) perturbative result using one-loop hard thermal loop resummation (HTL). Christian chmidt XQCD

15 least one derivative with respect to the strangeness chemical potential and thus are sensitive to strangeness fluctuations, we have six independent observables (χ 2, χ 4, χ B 13 Disentangling, χb 22, χb 31 ) at hand different that westrangeness can use to analyze sectors fluctuations and correlations in different strangeness sectors of the HRG. Using these six observables it is easy to construct observables that project onto different strangeness sectorsofthehrg.we willsolving consider the here 4 inhomogenous a set of observables systems A~x that=ê give, i in the low temperature regime, the contribution of strange mesons and strange baryons with strangeness = 1, 2and 3 to the (2+1)-flavor QCD partition function. They all depend on 2 free parameters, with the solutions are translations of the kernel M(c 1,c 2 ) = χ 2 χb 22 + c 1v 1 + c 2 v 2 (6) 1 ( ) B 1 (c 1,c 2 ) = χ 2 4 χ 2 +5χB 13 +7χB 22 + c1 v 1 + c 2 v 2 (7) B 2 (c 1,c 2 ) = 1 ( ) χ 4 4 χ 2 +4χB 13 +4χB 22 + c1 v 1 + c 2 v 2 (8) 1 ( ) B 3 (c 1,c 2 ) = χ 18 4 χ 2 +3χB 13 +3χB 22 + c1 v 1 + c 2 v 2 (9) v 1 = χ B 31 χ B 11 (1) v 2 = 1 3 (χ 2 χ 4 ) 2χ B 13 4χ B 22 2χ B 31 (11) 4 Christian chmidt XQCD

16 Disentangling different strangeness sectors B 1 (,) B 1 (,3/2) B 1 (3,3/2) HRG P =1,B BNL-BI: arxiv: M(,) M(,-3) M(-6,-3) HRG P =1,M B 3 (,) B 3 (,1/6) B 3 (1/3,1/6) HRG P =3,B B 2 (,) B 2 (,-3/4) B 2 (-3/2,-3/4) HRG P =2,B Christian chmidt XQCD

17 Disentangling different strangeness sectors For T. 155 MeV different strange sectors agree separately with the HRG For higher temperatures the deviations from HRG set in abruptly and rapidly become large modifications of the strange hadron contribution to bulk thermodynamics become apparent in the crossover region and follow the same pattern present also in the light quark sector. Christian chmidt XQCD

18 trangeness in the QGP We now probe the quasi-particle picture, i.e. to what extent the susceptibilities involving strangeness contributions can be understood in terms of elementary degrees of freedom that carry quantum numbers = ±1,B = ±1/3,Q = ±1/3 1.8 BQ BQ - χ 211 /χ121 free for a free (uncorrelated) strange quarks we have B 2 Q = BQ 2 =1 lim T!1 BQ 211 / BQ HRG N τ =6 8 for free (uncorrelated) strange hadrons we have contributions from two charge sectors. < 1 BQ 211 / BQ Christian chmidt XQCD

19 trangeness in the -.2 QGP HRG: =2 baryons -.4 HRG: =2 baryons B We now probe the quasi-particle picture, 2 (,) -.2 i.e. to what extent the susceptibilities B 2 (-1.5,)-.5.2 involving strangeness -.3 contributions B 2 can (-1.5,-1.5) be understood 15 2 in terms 25 of elementary 3 degrees of freedom that carry quantum numbers FIG = ±1,B = ±1/3,Q = Figure ±1/ : Combinations 15 of2 mixed25 susceptibilities 3 tha in a HRG model, where they project onto specific st Baryon-strangeness (top) and electric charge- 242 strangeness Adding correlations two (bottom), of theseproperly susceptibilities scaled by the yields observables that vanish in the high Christian strangeness chmidt fluctuations and powers of the fractional XQCD 213 bary B 2 (,) B 2 (-1.5,) HRG: =3 baryons B 2 (-1.5,-1.5) B 3 (,) B 3 (1.5,) B 3 (1.5,1.5) 3.5 Figure 3: Combinations -3χ B 11 /χ2 of mixed 9χ B 22 /χ4 susceptibilities 226 that have identical interpretations 3. B B -3χ in a HRG model, where 13 /χ4-27χ they 31 project /χ4 onto specific strangeness sectors rescaled such 227 that they all approach the same high t 228 the chiral crossover region. ( ) ( ) n 229 the χ B sdof around T c is quite 1 similar 6 to that inv nm T =( 1)m,n π 1.5 rescaled such that they all approach the same high temperature free gas limit, 1. non-int. quarks( ) ( ) n χ B 1 6 nm T =( 1)m 3.5 Q Q /χ2 /χ4 3 π χ 11 Q 3χ 13 /χ4 9χ 22 Q 27χ 31 /χ4 232 Note that Eq. (13),n+ ( also m =4,m> holds for ) non-zero. values (13) of t 234 The perturbative behavior of these mixed suscept 235 charges for T > T c provide unambiguous evidenc from the expansion ) 236 relevant of the pressure degrees of 3-flavor freedom QCD g 237 interacting quark gas only for T & 2T in one loop HTL resummed perturbation theory c. For the int (wi O(g 4 diateare temperatures, that of a Tweakly c. T. interact T c, strangeness i ) in the Debye quark andgas thermal for quark masses). Note 2.5 that Eq. (13) also holds for non-zero values of the chemical potentials The perturbative behavior of these mixed susceptibilities can be easily obtained pert. est. 1. from the expansion of the pressure of 3-flavor QCD given in [25] for massless quarks T & 2T c in one loop HTL resummed perturbation theory (with HTL terms expanded up to O(g ) in the Debye and thermal quark masses). 225 those obtained from the uncorrelated hadron gas vacuum masses of the strange hadrons. uch a ha description of the sdof breaks down immediately Moreover, the behav the light up and down quarks. Altogether, these r suggest that the deconfinement of strangeness seem takes place at the chiral crossover temperature. O other hand, our LQCD results involving correlati strangeness with higher powers of baryonic and e the sdof in the QGP are compatible with the w trivially correlated with the baryonic and electric ch indicating that the QGP in this temperature regim Adding 241 twomains of these stronglysusceptibilities interacting. yields obser temperature limit. Although We consider the results presented in particular, here were not ob in the limit of zero lattice spacing, the e ects of c 1

20 ummary We have provided evidence that in QCD the strange hadron sector gets modified strongly in the vicinity of the pseudo-critical temperature determined from the light quark chiral susceptibilities. Deviations from the HRG model start becoming large in the transition region and follow a pattern similar to that known for the light quark sector. There thus is no evidence that deconfinement and the dissolution of hadronic bound states may be shifted to higher temperatures for strange hadrons. We also showed that at temperatures larger than T>2Tc a simple quasi-particle model may be sufficient to describe properties of mixed strangeness-baryon number susceptibilities. Closer to Tc the structure of these susceptibilities becomes more complicated. A feature well-known also from bulk thermodynamic quantities like the pressure. Christian chmidt XQCD 213 2