PoS(CORFU2018)161 https://pos.sissa.it/ , and contain a critical point in the 3D Ising model universality class. We place the ) ∗ 4 B µ ( O [email protected] Speaker. critical point in the chemical potentialat the range Relativistic covered Heavy by Ion the Collider secondthermodynamic (RHIC). Beam observables We and show Energy discuss the Scan possible effect (BESII) constraints of on themapping the critical the parameters, point Ising which model on arise phase several from diagram onto the QCD one. We build a family ofup Equations to of State (EoS) for QCD, which match the lattice QCD results ∗ Copyright owned by the author(s) under the terms of the Creative Commons c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). Corfu Summer Institute 2018 "School and(CORFU2018) Workshops on Elementary Particle Physics and31 Gravity" August - 28 September, 2018 Corfu, Greece Claudia Ratti Department of Physics, University of Houston E-mail: Lattice-based Equation of State of QCDa matter critical with point PoS(CORFU2018)161 c 2 0 T c ]. / 7 = ) (1.1) (1.2) ] and c B T µ 10 ], a first − , 9 T 29 , Claudia Ratti 8 = ( around r T ]. In [ / B 28 ]. The advantage of µ 30 12 [ = . t ) N T , 8 was presented. More recently, ( n n 2 ], and later in [ = χ  ! t 1 27 n B N T µ and reduced temperature =  h 0 ) = T ] or an analytic continuation from imaginary B ( µ n

16 2 n , c ], the first continuum extrapolated results for was presented at ) 4 1 n 8 T T 15 was shown, but only at finite lattice spacing. The 13 ∑ c / / [ , 4 4 B P 6 c T n µ 14 c ( ] , ∂ ∂ 26 ) = 13 ! and B , 1 n µ 4 , 12 c [ T , ]. The Taylor expansion of the pressure in ( ) = 2 T P T c / 24 ( . The following form for the parametrization meets the requirements B , n θ µ c ]; in Ref. [ 23 and contain a critical point in the 3D Ising model universality class [ , as a function of magnetic field 25 ], the equation of state which serves as an input to these simulations must and ) was published for the first time in [ 6 22 4 M B R , , 6 µ 5 c ( , at two values of the temperature and , 21 ]). O 8 4 , c , 31 3 20 , 0, the EoS of QCD is known with high precision, in the case of 2+1 [ 19 = , ] quark flavors. Lattice QCD simulations allow us to reconstruct the equation of state as ]). The location of the critical point depends on the model used to predict it, which makes B 2 18 11 µ , ] from a postulated first order phase transition at larger density/chemical potential. A first The parameterization of the scaling Equation of state for the Ising model is usually given in Hydrodynamical simulations have proven to be extremely successful in describing the matter The main purpose of the second Beam Energy Scan at RHIC is to discover the critical point on At After the early results for 1 17 [ [ B B where the coefficients of the expansion are the susceptibilities of the baryon number: µ µ an estimate of the temperature dependence of continuum limit for 2. Ising model Equation of State terms of magnetization can be written as: determination of the Taylor expansion method is thatpotential, all where lattice the QCD quantities simulations are do calculatedreview, not see at suffer e.g. vanishing from [ baryon the fermion chemical sign problem (for a recent (or auxiliary variables 2+1+1 [ a Taylor series in powers of were published in Ref. [ created in heavy-ion collisions. Even if hydrodynamicsa itself critical needs point to [ be modified in the vicinity of principle prediction of the existencefermionic and sign problem location that of hinders theever, Monte the critical existence Carlo of point simulations the is critical at point still finitesee has e.g. missing, chemical been predicted [ potentials. due by several to How- QCD-like models the its (for experimental a search review very challenging.the In behavior view of of experimental the observables BESII, in it the is vicinity therefore of important the to critical predict point. contain a critical point with the correctfrom singular behavior, lattice besides QCD. reproducing We all have known recently constraints constructedQCD a results family up of to equations of state, which match lattice the QCD phase diagram, which separates the known crossover transition at low chemical potential QCD EoS with a critical point 1. Introduction PoS(CORFU2018)161 = + δ b ( (2.9) (2.8) (2.1) (2.2) (2.3) (2.6) (2.4) (2.5) (2.7) : this (2.11) (2.10) (2.12) δ β / 1 = | h | α 76201, ) Claudia Ratti . − h 0 ( − sgn = . The values of the ∝ ) a ) θ ) h ( , , θ , ˜ , hence: h ( 3 0 } 0 P with ) and solving the following ) g 2 b ) = − ) h 2 θ 2 . 4 ) r θ ( = θ  − + M ) b M − 1 G a ∂ θ ( ( / 1 ( + 3 β g c 2 F , 2 ∂ θ , , ) − + 1 and a ) + ( − ) θ 2 } ) ( θ ) ) θ = ( , + ( 2 Mh b g ( b . ) = g ) ˜ 1 h α h 2 θ ) − ( + + θ θ − θ , 2 ) ) ( 2 0 + − θ a  r ˜ θ h θ R α a 1 , α + = 0 ( β . − − )( βδ 2 − M 1 h 2 R ) = M 2 c ) 1 ( R , β 0 2 0 R ( )( θ 0 ( 1 2 β F h 0 ( M h R , β 2 2 ˜ h ) + − − 2 M ) 2 0 − b = = = 1 ) = ) = = θ − h ( 1 r ) = h r h r + ( , 1 2 being the first non-trivial zero of , − M ( − ( b h a 0 1 ( { ) ) = b M M ( βθ θ ( 1 + G 1 θ 1 β 2 , c F 1 2 + 1 ( − 1 R { + 80 are 3D Ising critical exponents, and the parameters take on 0 corresponds to the one of QCD, and that the lines of first order + ( . α 2 α α 154, 0 4 α ( . 1 = θ is fixed by noticing that c β 1 2 2 2 1 − Ising h ' ) − 2 − − − P ' θ 394. 1 δ ) = = ( . 0 = = = = g 0 θ r )( θ ( θ 2 1 3 0 g ' ( c c c c ≤ ˜ h 0 | h θ | 326 and . 0, 0 is the free energy density, defined as: are normalization constants, 605, ]: . ) 11 is another critical exponent of the 3D Ising model (also, the relation 2 0 ≥ ' 0 r . 35 h , 0 R , β ' , M 0 ' 0 ( 34 , M F α M We now map the phase diagram of the 3D Ising model onto the one of QCD, so that the critical The Gibbs free energy density then follows from this parametrization: 33 , holds). The function 00804. . ) 32 To proceed with the mappingGibbs of free the energy Ising density model equals onto the pressure the up QCD to phase a diagram minus we sign: notice that the phase transition and crossover in the Ising model are mapped onto those of QCD. point of the Ising model QCD EoS with a critical point [ where 0 the values differential equation: which results in: normalization constants are such that yields where where 1 with: PoS(CORFU2018)161 h 0, re- and (2.15) (2.16) (2.14) (2.13) r . While = h B ) µ Claudia Ratti B and µ , r also being fixed T ( 1 α ]: , , 36 ) 2 ) 2 ) α 4 B α to QCD coordinates µ ) . cos ( h ]. The shape of such transition line can sin , h O r  h ( 39 − + , BC + 1 2 µ 1 38 α 0  α κ , T B 0 2 T µ cos 37 sin are scale factors for the variables 3  , thus roughly determining the shape of it.  ρ 1 ) h ρ r 0 r − ρ T ( , − ( κ w w and tan ( w r + = = 0 = 1 T α C = BC T C µ T T C − . This map makes use of six parameters, two of which correspond T 1 − lines, and T scaling of B . µ const scaling for the Ising variables, namely determining the size of the critical relative = Non-universal map from Ising variables T are the transition temperature and curvature of the transition line at global κ Figure 1: and represents a ), into the QCD phase diagram, given a choice of parameters for the map. 0 ) ρ T θ 0 has been estimated in lattice simulations [ , In the following, remembering that the aim of the EoS is to be employed in hydrodynamic sim- The simplest way to do so is through a linear map as follows [ At this point it is possible to transport the thermodynamics of the IsingIt model is (written in possible terms to impose some constraint on the parameter choice by making use of additional R = ( represents a B µ ulations for heavy-ion collisions in the BES-II program, we will consider a choice of the baryonic where spectively. The number ofby: the parameters is thus reduced to four, the angle which can be visualized in Fig. axes form with the w region, be approximated with a parabola: to the location of the critical point on the QCD phase diagram, two are the angles that the of arguments for the location of the critical point. For example, the curvature of the transition line at QCD EoS with a critical point PoS(CORFU2018)161 . ◦ 85 (3.4) (3.3) (3.2) (3.5) (3.1) . , (2.17) (2.18)

93 )) ' B 2 Claudia Ratti . To achieve µ α − BC , µ T ( − θ , ) . and . B ) ) ) is to be understood as µ . B BC T − µ ( µ , )) 3.1 , ) with the multiplication by B T T ( µ . ( Ising n , R ◦ c ( 2.12 T ) ( 0 QCD 85 crit . θ P Ising , 3 = ) , P + . B B ' ) n µ µ 2 B 2 1 , , µ  α )) + = T T , )) = B ( ( B . Note that Eq ( T T f ρ B µ 4 R µ ( C ( µ , T  , , trying to reduce their acceptable range on the T Z ) ) + ( T ρ ( T Ising θ T 4 ( ) = , θ ( P ) = B ) , ) ) , B B and B µ B Ising , µ µ µ Ising , w µ , − , T − − , ( T 1 T , n f ( T ( Non 2 T ( f Non n R = c ( 2MeV c ( . R 4 ( n w Z ∑ T ) = 350 MeV, resulting in: 4 143 Ising B T P Ising symm = µ ' ) =  , P ) ) C coefficients, which we obtain as a difference between the other two T T BC ) = B B ) below. This form does not modify the singular critical behavior at the ( ( T µ B µ µ , , µ 3.5 crit LAT n Ising , T T QCD c ( ( − T P 4 f f ( T P ), the critical pressure is obtained from Eq. ( 1 2 2 1 Non n ): c 3.2 = 3.1 is a regular function of the temperature and chemical potential, with dimension of ) = B ) µ B , to the Taylor expanded “Non-Ising” one: µ T , ) ( in Eq. ( T B ( µ ) f , B crit vanish, as they should. QCD T µ Because of the charge conjugation symmetry, in QCD the partition function needs to be an In the following, we assume that the lattice QCD expansion coefficients can be written as a sum P ( B , µ T ( f as well as the pressure. Thusthis we QCD need must to possess write a Eq. critical ( point at both even function of the baryon chemical potential: Note that in Eq. ( critical point(s) and automatically ensures thatin the odd-power coefficients in the Taylor expansion a definition for the contributions. The full pressure isany then reconstructed simply by adding the critical contribution at QCD EoS with a critical point chemical potential which is In addition, the axes are chosen to be orthogonal, as we already mentioned, so that Later we will explore different choices for Finally, the scaling parameters are initially chosen as: of an “Ising” contribution coming fromwhich the would critical contain point the of regular QCD, partof and interest: as a well “Non-Ising” as contribution, any other possible criticality present in the region basis of physical conditions for the thermodynamic quantities. 3. Thermodynamics where energy to the fourth power. We choose PoS(CORFU2018)161 (4.1) Claudia Ratti ,  ) + B µ 0 ( ] and HRG model calculations. 0  T ) T 42 ∆ B , µ − 9 0 ( 0 T T is roughly the size of the “overlap T 0 ∆  T − ∆ T tanh  − 1 tanh  1 2 + 5 ) ]). 1 B  µ 41 , 1 2 4 T ) ( T B µ 4 , HRG T T P ( P + = ) B µ , 4 T ] (list PDG2016+ in [ ( T 40 Final P works as the “switching temperature”, and we can see the comparison between the lattice data (and the extension with the HRG ) B 2 shows the comparison of the “Ising” and “Non-Ising” contributions to the parametrized µ Parametrization of baryon susceptibilities from Lattice QCD [ 3 ( 0 T To cure some pathological behavior of our EoS at smallThe temperatures, smooth we merging can perform be a obtained smooth through a hyperbolic tangent as: Fig. In Fig. where region” where both pressurespotential contribute of to the “switching the temperature” sum. is chosen to The be dependence parabolic: on this the way, the baryon “switching chemical line” Figure 2: lattice/HRG model results. 4. Results merging with the HRG model EoS. model) and the resulting parametrization. Thenot HRG model contain employed any to calculate interaction, the and pressureParticle does makes Data Group use [ of the most up to date particle list available from the QCD EoS with a critical point which will have the effect of slightlythat changing now the the form pressure of the at criticalward the pressure definition) critical (the but point main not is one its being non-zero, singular whereas behavior, leaving it all would the be even zero order derivatives in unchanged. the straightfor- PoS(CORFU2018)161 . ρ , (4.3) (4.2) w Claudia Ratti , T  B . The effect of the P 9 ∂ ∂ µ . QCD map. By keeping  B n 3 / 7→ 1 S T  = P ) ∂ε ∂ B  µ 3 , 143MeV), as well as the orientation T T ( ) = ' B B n C µ T , , T B ( µ 2 s 1, over the whole phase diagram, it is possible  c P T < 6 ∂ ∂ 2 s c 350MeV,  , 3 3 = B 1 T n T BC B ), we investigated the role of the scaling parameters = T µ ◦ µ ) B 90 + µ 3 4 , = T P T T 1 ( α in the QCD phase diagram) is shown in Fig. S − . The effect of the critical point on the thermodynamic isentropes 8 3 B − S - n 2 T with the parametrized lattice data (black, solid). / 4 α S ) , = , 4 B ◦ ) ) µ ( B B 85 µ . µ O 4 4 , , 3 T T T T ( ( ' P ε 1 α ), we can compute various thermodynamic observables of interest. In addition to the , we can see in red the points corresponding to pathological parameter choices, while the Comparison of critical (blue, dot-dashed) and “Non-Ising” (red, dashed) contributions to baryon 4.1 10 In order to complete the thermodynamic description of the finalized equation of state obtained They are shown in Figs. the location of the critical point fixed ( blue dots correspond to acceptable ones. We notice that, while most commonly specific parameter pressure, we compute the entropy density,malized baryon by density, the energy correct density power and of speed the of temperature: sound nor- (trajectories at constant to reduce the range of acceptable parameters in the non-universal Ising between the pressure from our proceduretransition and curve. the one from the HRG model is parallel to thein chiral Eq. ( Figure 3: susceptibilities up to critical point is clearly visible inquiring the distortion thermodynamic of stability, i.e. the isentropes positivity at ofdensity large pressure, chemical entropy and potentials. density, speed baryon By of density, re- energy sound, and causality, i.e of the axes ( In Fig. QCD EoS with a critical point PoS(CORFU2018)161 Claudia Ratti 25) we observe . 0 = w ( w , for very low B n 7 Pressure after merging with HRG. 1). Entropy density after merging with HRG. > 2 s c Figure 4: Figure 5: and contain a critical point in the 3D Ising model universality class. These EoSs are meant ) We presented a family of equations of state, which match available lattice QCD results up to 4 B µ ( choices are unacceptable because of the negativity of QCD EoS with a critical point violation of causality as well ( to be used as an input intoA hydrodynamic systematic simulations scan of of the the system parameter created space,BESII in and at heavy-ion relative RHIC, comparison collisions. will with hopefully experimental data allowof from us the the to critical constrain point. the size of the critical region and the location O 5. Conclusions PoS(CORFU2018)161 Claudia Ratti Preprint 675–678 ( 443 Nature 8 Energy density after merging with HRG. Baryon density after merging with HRG. Figure 7: Figure 6: ) hep-lat/0611014 This material is based upon work supported by the National Science Foundation under grant [1] Aoki Y, Endrodi G, Fodor Z, Katz S D and Szabo K K 2006 no. PHY-1654219 and byPhysics the within U.S. the Department framework of ofThe Energy, author the Office also Beam of acknowledges Energy Science, the ScanCenter use Office of of Theory of Advanced the (BEST) Computing Nuclear Maxwell and Topical Cluster Data Collaboration. and Systems at the the advanced University support of from Houston. References the Acknowledgements QCD EoS with a critical point PoS(CORFU2018)161 JHEP Claudia Ratti 16) in the QCD , 18 , Preprint ) 21 ) , 23 , ) 024912 ( 25 , ) 28 , C84 35 1804.05728 , 1712.10305 42 , 50 , Phys. Rev. 0911.1772 Preprint Preprint 68 = hep-lat/0701002 B n ) / 9 S Preprint 036006 ( Preprint D98 054012 ( 024 ( 1805.05249 Speed of sound after merging with HRG. D81 ) Phys. Rev. (from left to right B LAT2006 n Preprint / S Figure 8: Phys. Rev. PoS 1007.2580 ) Preprint Lines of constant 077 ( 11 and Stephanov M 2018 ( 1105.0622 [7] Parotto P, Bluhm M, Mroczek D, Nahrgang M, Noronha-Hostler J, Rajagopal K, Ratti[8] C, SchäferBorsanyi T S, Endrodi G, Fodor Z, Jakovac A, Katz S D, Krieg S, Ratti C and Szabo K K 2010 [6] Nahrgang M, Bluhm M, Schäfer T and Bass S A 2018 ( [5] Stephanov M and Yin Y 2018 [4] Nahrgang M, Leupold S, Herold C and Bleicher M 2011 [3] Stephanov M A 2010 [2] Stephanov M A 2006 phase diagram. The blue dot indicates the location of the critical point. Figure 9: QCD EoS with a critical point PoS(CORFU2018)161 181 ) ) ) 99–104 ) 053 ) Claudia Ratti 08 B730 EPJ Web Conf. 0705.3814 JHEP ) ) ) ) 1408.6305 ) Phys. Lett. ) hep-lat/0205016 Preprint hep-lat/0611035 Comput. Phys. Commun. hep-lat/0209146 Preprint 1407.6387 ) 0904.1400 ) Preprint 0707.1987 114509 ( 0910.0276 Preprint 0806.2233 0808.1096 Preprint D76 Preprint 851–855 ( Preprint ) ) Preprint 290–306 ( Preprint 034505 ( Preprint 1011.3130 A931 Preprint 014505 ( 1606.07494 094503 ( D76 171 ( B642 014502 ( Phys. Rev. 10 D67 012 ( 074501 ( 114503 ( D90 Preprint D80 11 Preprint D76 D78 Nucl. Phys. Phys. Rev. JHEP Nucl. Phys. ) Phys. Rev. 014504 ( ) 69–71 ( Phys. Rev. hep-lat/0204010 hep-lat/0501030 LATTICE2008 Phys. Rev. Phys. Rev. D83 539 Phys. Rev. PoS Preprint Preprint ) ) Nature 0911.5682 1607.02493 Phys. Rev. (HotQCD) 2014 2016 074507 ( 054508 ( (MILC) 2008 Preprint Preprint et al. et al. D66 D71 1204.6710 1309.5258 et al. In red (squares) the points corresponding to pathological choices of parameters, in blue (dots) 07008 ( Preprint Preprint 1715–1726 ( 137 ( and Unger W 2011 Phys. Rev. Phys. Rev. ( [9] Borsanyi S, Fodor Z, Hoelbling C, Katz S D, Krieg S and Szabo K K 2014 [24] Moscicki J T, Wos M, Lamanna M, de Forcrand P and Philipsen[25] O 2010 Borsanyi S, Endrodi G, Fodor Z, Katz S D, Krieg S, Ratti C and Szabo K K 2012 [27] Gunther J, Bellwied R, Borsanyi S, Fodor Z, Katz S D, Pasztor A and Ratti C 2017 [26] Hegde P (BNL-Bielefeld-CCNU) 2014 [23] D’Elia M and Sanfilippo F 2009 [22] de Forcrand P and Philipsen O 2008 [19] Wu L K, Luo X Q[20] and ChenD’Elia H M, S Di 2007 Renzo F and[21] Lombardo MConradi P S 2007 and D’Elia M 2007 [16] Kaczmarek O, Karsch F, Laermann E, Miao C, Mukherjee S, Petreczky[17] P, Schmidt C,de Soeldner Forcrand W P and Philipsen O[18] 2002 D’Elia M and Lombardo M P 2003 [15] Basak S [12] Allton C R, Ejiri S, Hands S J, Kaczmarek O, Karsch[13] F, Laermann E,Allton Schmidt C C R, and Doring Scorzato M, L Ejiri 2002 S, Hands S J, Kaczmarek[14] O, KarschGavai F, R Laermann V E and and Gupta Redlich S K 2008 2005 [11] Borsanyi S [10] Bazavov A the acceptable ones. Figure 10: QCD EoS with a critical point PoS(CORFU2018)161 B751 D92 ) ) ) 054503 Claudia Ratti Phys. Lett. Phys. Rev. ) D92 ) 1611.08285 Phys. Rev. ) Preprint hep-th/9610223 1508.07599 ) ) nucl-th/0410078 100001 hep-ph/0611083 094503 ( Preprint 1098–1102 Preprint C40 Preprint 23 D95 1701.04325 Preprint 1702.01113 1804.07810 014507 ( 626–652 ( 004 ( 044904 ( Preprint 11 Chin. Phys. 472–480 Phys. Rev. D93 B489 C71 Preprint Preprint A8 Phys. Rev. Lett. 054504 ( CPOD2006 Phys. Rev. 034517 ( 084301 ( ) D95 Phys. Rev. ) Nucl. Phys. Phys. Rev. PoS 81 D96 ) ) Phys. Rev. 1507.07510 (Particle Data Group) 2016 Phys. Rev. 1507.04627 2017 Rept. Prog. Phys. et al. 2017 et al. Preprint 1507.03571 1805.04445 Preprint et al. Preprint Preprint 559–564 ( 114505 ( ( ( [40] Patrignani C [41] Alba P [42] Bellwied R, Borsanyi S, Fodor Z, Katz S D, Pasztor A, Ratti C and Szabo K K 2015 [37] Cea P, Cosmai L and Papa A 2016 [39] Bellwied R, Borsanyi S, Fodor Z, Günther J, Katz S D, Ratti C and Szabo K K 2015 [36] Rehr J J and Mermin N D 1973 [38] Bonati C, D’Elia M, Mariti M, Mesiti M, Negro F and Sanfilippo F 2015 [34] Schofield P, Litster J D and[35] Ho J TBluhm 1969 M and Kampfer B 2006 QCD EoS with a critical point [28] Bazavov A [29] D’Elia M, Gagliardi G and Sanfilippo[30] F 2017 Borsanyi S, Fodor Z, Guenther J N, Katz S K, Szabó[31] K K, PasztorRatti A, C Portillo 2018 I and Ratti C 2018 [33] Guida R and Zinn-Justin J 1997 [32] Nonaka C and Asakawa M 2005