Overview of the QCD phase diagram -- Recent progress from the lattice
OOverview of the QCD phase diagram -Recent progress from the lattice
Jana N. Guenther ∗ Aix Marseille Univ., Universit´e de Toulon, CNRS, CPT, Marseille,FranceOctober 30, 2020
In recent years there has been much progress on the investigation of theQCD phase diagram with lattice QCD simulations. In this review I focus onthe developments in the last two years. Especially the addition of externalinfluences or new parameter ranges yield an increasing number of interestingresults. I discuss the progress for small, finite densities from both analyti-cal continuation and Complex Langevin simulations, for heavy quark boundstates (quarkonium), the dependence on the quark masses (Columbia plot)and the influence of a magnetic field. Many of these conditions are relevantfor the understanding of both the QCD transition in the early universe andheavy ion collision experiments which are conducted for example at the LHCand RHIC.
The strong interaction between quarks and gluons in the standard model is described byQuantum Chromo Dynamics (QCD). The investiagation of the phase diagram of QCDhas been an active subject for many years. A special focus of this research is on thetransition between hadrons at low temperature and the quark gluon plasma at hightemperature. The nature of the transition as an analytic crossover (Ref. [1–6]) has beenknown since 2006 (Ref. [1, 2]). However, different external influences may change thenature of this transition. One famous example is the addition of a chemical potential.The crossover has been established for vanishing baryon chemical potential only, whichmeans for a setting with the same number of quarks and anti-quarks. This setting ∗ [email protected] a r X i v : . [ h e p - l a t ] O c t s a good approximation for the QCD transition in the early universe. However, theexperimental investigations of the QCD transition rely on heavy ion collision experimentswhich produce the quark gluon plasma with a finite density. In addition, the collidedmatter goes through several stages and is not permanently in an equilibrium. Since, atthe moment, full QCD cannot be solved under that conditions one has to take otherpaths. Certain parameter ranges can be described efficiently by effective theories. Forother situations we depend on phenomenological models, which inturn require inputinformation that can be gained from lattice simulations.The progress on the phase diagram form lattice QCD corresponds well with, and is inpart triggered by, an increasing number of available experimental results. At the momentmost erxperimental results for heavy ion collision experiments are generated either bythe Large Hadron Collider (LHC) at Cern in Genf, Switzerland or the Relativistic HeavyIon Collider (RHIC) at the Brookhaven national laboratory in New York, USA. Whilethe results form the LHC are at low densities, RHIC accesses information for lagerchemical potential. However, there are several upcoming facilities that will allow insideto even higher densities. Some of them are for example the Nuclotron-based Ion ColliderfAcility (NICA) at the Joint Institute for Nuclear Research (JINR) in Dubna, Russia,the Compressed Baryonic Matter Experiment (CBM) at the Facility for Antiproton andIon Research in Europe (FAIR) at Darmstadt, Germany and the J-PARC heavy ionproject (J-PARC-HI) at the Japan Proton Accelerator Research Complex (J-PARC) inTokai, Japan.After this introduction, I will briefly describe the stages of a heavy ion collision ex-periment in section 2. Linking lattice QCD to heavy ion collision experiments requiresresults at low finite density which I discuss in section 3. The main challenge in thisparameter range is the infamous sign problem which is discussed in section 3.1. Forcontinuum extrapolated, physical results, one therefore has to rely on analytical con-tinuation (section 3.2) form vanishing or imaginary chemical potential. One methodthat is getting relatively close to direct lattice simulations at finite density are ComplexLangevin simulations, which are therefore reviewed in section 3.3. Other methods thatare not discussed in this review are for example reweighting techniques [7–10], densityof state methods [11, 12], using the canonical ensemble [13–15], formulations with dualvariables [16] or Lefschetz thimbles [17, 18].Another way to gain information on finite density QCD is discussed in section 4, thelattice simulations of effective field theories. Here, I focus, with section 4.1 on the resultsfor heavy quark bound states the so called heavy quarkonium.Despite the importance at finite density, also for zero density there are interestinginfluences to consider. One of them is the dependence of the transition type on the quarkmasses. It is often summarized in the Columbia plot, which is discussed in section 5.The most interesting areas of the Columbia plot are the upper right (section 5.5) andlower left corner (section 5.1). Due to the low quark masses, and the resulting expensivecomputations, different tactics are employed in the study of the lower left corner ofthe Columbia plot. In this review I discuss the use of imaginary chemical potential insection 5.2 and the variation of the number of quark flavors in section 5.3. Due to the lackof continuum extrapolated results in this parameter range, I provide an overview over2 el. hydrodynamicskinetic theory CGC / class. Yang Mills open heavy flavor glasma, pre-thermalizationt 〜 hydroexpansion QGP t 〜 freeze-out, hadronization T C =155MeV production /formation? medium interaction partial equilibration? freezeout QQ b u l k QQ Figure 1: (Ref. [19]) Overview of of the different stages of a heavy ion collision. Ingrey the different effective theories are shown which are used to describe thedifferent stages (see section 4.1).the numbers obtained from different lattice actions and lattice spacings in section 5.4.There have been many results for yet another region. The addition of a magnetic fieldto finite temperature QCD does not suffer from a sign problem and is also relevant forthe understanding of heavy ion collision experiments. The progress recently made on ourunderstanding of the phenomena triggered by a magnetic field is discussed in section 6.Finally, as common practice, this review closes with a conclusion. Here I hope toconvince you that finite temperature lattice QCD and especially the investigation of theQCD phase diagram is a fascinating, thriving topic, of which I can only cover parts inthis review.
As the experimental realization of QCD thermodynamics is strongly linked with heavyion collision experiments, I will take a small detour to explain the different stages oc-curring during such a collision which are depicted in figure 1. In a heavy ion collisionvarious nuclei can be used. Prominent examples are gold and lead which have been usedat RHIC and LHC. Two beams of nuclei are accelerated to relativisic velocities.When the two beams collide, they form a non thermaliesed state with strongly inter-acting fields, which is called a glasma and commonly treated in the color glass condensateframework (for details on this framework see Ref. [20, 21]). Here the glasma is seen inthe infinite momentum framework where there are partons of the nuclei, valence quarksand pairs of sea quarks. A main challenge is the description of different momentumscales. Due to the un-thermaliesed nature of this state, it is not accessible for latticeQCD simulations.In the next stage, the further fragmentation of the partons into quarks and gluons leads3igure 2: A schematic view on the T - µ B -plane of the QCD phase diagram.to the quark gluon plasma. This is a strongly interacting state of deconfined quarks andgluons. While first it was expected to be similar to an electromagnetic plasma, by nowit is well established that is more similar to a strongly interacting fluid. Therefore, it isoften described by relativistic hydrodynamics.The quark gluon plasma expands and cools down at the same time. When it reachestemperatures close to the QCD transition temperature, the deconfined quarks and gluonshave to recombine to colour-neutral hadrons. At this time, the chemical aboundanceof the hadrons is determined. The corresponding temperature is called the chemicalfreezeout temperature. However, the hadrons can still exchange energy and momentumuntil the point where the kinetic freezeout takes place.An important question is, whether at the stage of the Quark-Gluon-plasma and thefollowing transition to hadrons, the states thermalize. Only if this is the case, resultsfrom lattice QCD, which simulates thermal equilibrium, are directly applicable. One representation of the QCD phase diagram is in the T - µ B -plane (figure 2). The µ B =0 axis is well investigated by lattice QCD. For low chemical potential and temperature,there is a hadronic phase of colour neutral bound states. At high temperatures theeffective degrees of freedom are the quarks and gluons. This phase is called the quarkgluon plasma. These two phases are separated by a crossover at low and zero chemicalpotential. The transition is expected to change into a first order transition for higher µ B with a critical second order point in between. The position of the critical pointis under active investigation both by heavy ion collision experiments and by theoreticcalculations. For very large chemical potentials futher fascinating phenomena, like acolour super conducting phase [22], are expected.To analyse the quark gluon plasma that is created in heavy ion collision experimentsat the LHC or RHIC, a theoretical understanding of the quark gluon plasma in QCD4s needed. In the region of the deconfinement transition, QCD can not be studied withperturbative methods. To get non-perturbative results with a controlled error, one has toturn to lattice QCD. At the moment, direct simulations that are continuum extrapolatedand at physical quark masses are restricted to vanishing or imaginary chemical potentialdue to the infamous sign problem. On the other hand, the collisions especially at RHICand upcoming heavy ion collision facilities like Fair and NICA take place away from theaxis of zero µ B . Therefore, information in that region are needed. For simulation without chemical potential there are on average the same number of par-ticles and antiparticles. Therefore, the expectation value of the overall baryon numberdensity (cid:104) n B (cid:105) is zero. To describe a system with finite baryon density, we need to intro-duce a finite quark chemical potential µ q to the Lagrangian. In the continuum this isrelatively simple to achieve by adding a term of the form ψγ ψ . However, on the lattice,it is not that simple. Adding a similar term to the Dirac operator leads to a divergentenergy density in the continuum limit, which is clearly unphysical. Instead one followsthe idea of Hasenfratz and Karsch in [23] where the chemical potential is understoodas the temporal component of a vector field. The temporal hopping term has then theform − a (cid:88) n ∈ Λ (cid:16) e µT ( η ) αβ U ˆ4 ( n ) ab δ n +ˆ4 ,m + e − µT ( η ) αβ U † ˆ4 ( n − ˆ4) ab δ n − ˆ4 ,m (cid:17) . (1)It recovers the original action for µ = 0 and with µT = aµN t it reproduces the correctdensity term at linear order in aµ . However, this term breaks the γ -hermiticity of theDirac operator γ Dγ = D † (2)and leads to a complex fermion determinant for real µ . The fermion determinant enters inthe Boltzmann weight factor which has to be positive to allow a Monte Carlo simulation.There have been several ideas on how to obtain results at real finite chemical potentiallike reweighting techniques [7–10], Taylor expansion [24–28], density of state methods[11, 12], using the canonical ensemble [13–15], formulations with dual variables [16],Lefschetz thimbles [17, 18] or complex Langevin [29, 30]. Instead of conducting direct simulations, for small densities it is possible to obtain resultswith physical quark masses by analytical continuation. Since at µ B = 0 the transitionis a crossover (Ref. [1–6]), certain observables can be parameterized by an analyticfunction in the vicinity of zero as well. This fact can be exploited, by using results atzero (Ref. [24–28, 31–39]) or imaginary chemical potential (Ref. [38, 40–53]), to find afunction that can be extrapolated to real, positive chemical potential. If one aims at5 T µ B d ( p / T ) d ( µ B / T ) ( µ B /T ) = − ˆ µ Analyti al ontinuation on N t = 12 raw dataT=145MeVT=170MeV00.050.10.15 -8 -6 -4 -2 0 2 4 6 8 T µ B d ( p / T ) d ( µ B / T ) ( µ B /T ) = − ˆ µ Analyti al ontinuation on N t = 12 raw dataT=145MeVT=170MeV00.050.10.15 -8 -6 -4 -2 0 2 4 6 8 T µ B d ( p / T ) d ( µ B / T ) ( µ B /T ) = − ˆ µ Analyti al ontinuation on N t = 12 raw dataT=145MeVT=170MeV a + b ˆ µ + c ˆ µ ( a + b ˆ µ ) / (1 + c ˆ µ ) a + b ˆ µ + c sin(ˆ µ ) / ˆ µa + b ˆ µ + c ˆ µ ( a + b ˆ µ ) / (1 + c ˆ µ ) a + b ˆ µ + c sin(ˆ µ ) / ˆ µ Figure 3: (Ref. [54]) Illustration of the analytic continuation from imaginary chemicalpotential. Data points generated with purley imagniary µ B can be fitted as afunction in µ B and then extrapolated from µ B ≤ µ B > µ B . This method is therefore often called Taylor method.The other possibility to determine a function that can be continued to finite chemicalpotential is the use of simulations at purely imaginary chemical potential, where thereis no sign problem. As illustrated in figure 3, the data points can be described by afunction of µ B . For imaginary chemical potential µ B < µ B > f ( µ, T ) = a + b (cid:16) µT (cid:17) + c (cid:16) µT (cid:17) , (3) f ( µ, T ) = a + b (cid:0) µT (cid:1) c (cid:0) µT (cid:1) , (4) f ( µ, T ) = a + b (cid:16) µT (cid:17) + c sin (cid:0) µT (cid:1) µT , (5)which each having three fit parameter a , b , c . All three functions describes the availabledata well. However their continuation to µ B > T = 170 MeV. This variation has, therefore, to be taken into account as a systematicerror. 6hen utilising simulations at imaginary chemical potential, one exploits the fact thatthe γ -hermicity of the Dirac operator with a chemical potential reads γ D ( µ ) γ = D † ( − µ ) . (6)This means that in the action, the term adding the chemical potential e − µT = f isreplaced by ( e µT ) ∗ = f ∗ when switching to purely imaginary µ . For the determinant thisyields det ( D ( f )) = det (cid:18) D (cid:18) f ∗ (cid:19)(cid:19) , (7)meaning that the determinant is only real if f = 1 f ∗ . (8)For real f this is only fulfilled if f = e − µT = 1, and, therefore, µ = 0. However, if µ = i µ I is chosen to be purely imaginary, this yields f = e − i µIT = (cid:18) e i µIT (cid:19) ∗ = 1 f ∗ . (9)Thus the determinant is real for a purely imaginary chemical potential. When usingsimulations at imaginary chemical potential, less derivatives are needed than for theTaylor method. Instead, one uses different fit functions to describe a data set for severalimaginary chemical potentials. This method is sometimes called analytical continuation,but, since also the Taylor method relies on an analytical continuation, I will refer to itas imaginary chemical potential method. The different fit functions can lead to differentresults which leads to the requirement of a careful systematic analysis. The behaiviourof one type of fit functions, the Pad´e approximation, is discussed in Ref. [55]. The transition temperature:
A common observable to extrapolate is the transitiontemperature. Because of the crossover nature of the transition, its definition is am-biguous. Typical definitions are, for example, the peak of the chiral susceptibility as afunction of temperature or the inflection point of the chiral condensate. Often differentdefinitions yield consistent results within the available precision. However for an analytictransition this is not guaranteed in contrast to the situation for phase transition.A high precision determination of the transition temperature at µ B = 0 from fivedifferent observables was done in Ref. [56]. As can be seen in figure 4, all five definitionshave the same continuum limit within the available precision. This yields a combinedvalue of T c = (156 ± .
5) MeV.A new definition as the peak of the chiral susceptibility as a function of the chiralcondensate (instead of the more common definition as function of the temperature) wasintroduced this year in Ref. [53]. It allows for a precise extraction of the transitiontemperature and, therefore, also for an improvement in the extrapolation. The resultsobtained from simulations at imaginary chemical potential are shown in figure 5. They7 c o n t i nuu m N τ = N τ = N τ = N τ = HotQCD preliminary T c ( µ B = 0) [MeV] 1 /N τ χ disc χ sub Σ sub ∂ µ B Σ sub ∂ µ B χ disc (156 . ± .
5) MeV
Figure 4: (Ref. [56]) The continuum extrapolation of five different definitions of the tran-sition temperature. Towards the continuum limit the different definitions con-verge, yielding a combined result of T c = (156 ± .
5) MeV.are continuum extrapolated from the three lattices with sizes 40 ×
10, 48 ×
12 and64 ×
16. The extrapolation was done with two different functions, T c = 1 + ˆ µ B (cid:18) a + dN t (cid:19) + ˆ µ B (cid:18) b + eN t (cid:19) + ˆ µ B (cid:18) c + fN t (cid:19) (10)and T c = 11 + ˆ µ B (cid:16) a + dN t (cid:17) + ˆ µ B (cid:16) b + eN t (cid:17) + ˆ µ B (cid:16) c + fN t (cid:17) . (11)The transition temperature from imaginary chemical potential ( Analytical continua-tion in the top of figure 5) is compared to the Taylor expansion at leading and next toleading order, as well as to results from truncated Dyson-Schwinger equations (Ref. [57])and various determination of the chemical freezeout temperature in heavy ion collisionexperiments(Ref. [58–62]).The transition temperature at finite chemical potential T c ( µ B ) normalized by thetransition temperature at zero chemical potential T c (0) is often parameterized by: T c ( µ B ) T c (0) = 1 − κ (cid:18) µ B T c (cid:19) − κ (cid:18) µ B T c (cid:19) + (cid:79) ( µ B ) . (12)While there is a long history of the determination of κ [28, 38–40, 46, 47], results for κ [36, 53] only recently became available, both from the Taylor and imaginary potentialmethod. It emerges that κ is significantly smaller than κ . Within its error it is stillcompatible with zero. A comparison of the recent determinations of both κ and κ isshown in figure 6.Instead of an expansion in the baryon chemical potential, an expansion in isospin,strangeness or charge chemical potential is also an option. They were studied in detail in8 T [ M e V ] Analytical continuationNLO Taylor-expansionLO Taylor-expansion125130135140145150155160165170 0 50 100 150 200 250 300Chemical freezeout: T [ M e V ] µ B [MeV]Dyson-Schwinger: hep-ph/1906.11644nucl-th/0511071v3nucl-th/1212.2431hep-ph/1403.4903nucl-th/1512.08025nucl-ex/1701.07065 Figure 5: (Ref. [53]) Top: A comparison between the extrapolated transition tempera-ture obtained from analytical continuation from imaginary chemical potentialwith different functions, compared to the result from extrapolating with theleading order (LO) order next to leading order (NLO) Taylor coefficients. Bot-tom: A comparison of the extrapolated transition temperature to recent resultsfrom calculations with Dyson-Schwinger-Equations [57] and the freezeout tem-perature from heavy ion collision experiments [58–62].9 . - . . .
005 0 .
008 0 .
011 0 .
014 0 .
017 0 . Bonati et al, 2015, ψψ , χ ψψ Bellwied et al, 2015, ψψ , χ ψψ , χ SS Bonati et al, 2018, ψψ , dψψdµ B T c Bonati et al, 2018, ψψ , dψψdµ B T c Bazavov et al, 2018, χ , Σ This work ψψ Figure 6: (Ref. [53]) A comparison of recent results for κ and κ as defined in equa-tion (12). Ref. [36] and the lower point of Ref. [38] use the Taylor method andare depicted in blue. The green points, which are the result from Ref. [53], theupper point of Ref. [38], Ref. [47] and Ref. [46] used lattice data at imaginarychemical potential.Ref. [36]. There are two very similar possibilities to chose µ S and µ Q for the expansion in µ B that are commonly used. The first is the choice of a purely baryon chemical potential( µ S = µ Q = 0). This conditions have been used in Ref. [46]. The second possible choiceis the set up at the strangeness neutral point. In this case µ S and µ Q are chosen in away that (cid:104) n S (cid:105) = 0 and 0 . (cid:104) n B (cid:105) = (cid:104) n Q (cid:105) to match the conditions in heavy ion collisionexperiments. This was done in Ref. [38, 47]. In all cases the two values agree within theerror. Fluctuations:
Higher order fluctuations of conserved charges are calculated as the par-tial derivatives of the pressure (or the QCD partition function) with respect to thechemical potentials. Here, χ B,Q,Si,j,k = ∂ i + j + k ( p/T )( ∂ ˆ µ B ) i ( ∂ ˆ µ Q ) j ( ∂ ˆ µ S ) k (13)with ˆ µ = µT . These fluctuations are of great interest in the search for the criticalendpoint, both for heavy ion experiments and theoretical calculations (Ref. [63–65]).They are proportional to powers of the correlation length and, therefore, expected todiverge in the vicinity of a critical end point (Ref. [26, 66, 67]). Up to the 8th orderfluctuations have been calculated on finite lattices in Ref. [31, 48, 52, 68]. The baryonfluctuations up to 8th order from Ref. [52] are shown in figure 7.In principal, fluctuations can also be measured both on the lattice and experiments.A comparison between theoretical and experimental results can be used to extract thechemical freeze-out temperature T f and the corresponding chemical potential µ Bf asfunctions of the collision energy (Ref. [69–73]). To compare to experiments, one uses ra-tios of fluctuations to cancel out the explicit volume dependence. These can be matched10
35 160 185 210 T/ MeV χ B
135 160 185 210 T/ MeV χ B
135 160 185 210 T/ MeV χ B
135 160 185 210 T/ MeV χ B Figure 7: (Ref. [52]) Results for χ B , χ B , χ B and an estimate for χ B on a N t = 12 latticeas functions of the temperature, obtained from the single-temperature analysis(see text). We plot χ B in green to point out that its determination is guidedby a prior, which is linked to χ B . The red curve in each panel corresponds tothe results from the Hadron Resonance Gas model.11o the ratios of cumulants of particle number distributions in experiments. However,not all baryons can be measured in experiments. Therefore, the cumulants of the protonnumber distribution are considered a usefull proxy to compare to the baryon numbercumulants calculated on the lattice. Some combinations are: R B ( T, µ B ) = M B σ B = χ B ( T, µ B ) χ B ( T, µ B ) (14) R B ( T, µ B ) = S B σ B M B = χ B ( T, µ B ) χ B ( T, µ B ) (15) R B ( T, µ B ) = κ B σ B = χ B ( T, µ B ) χ B ( T, µ B ) (16) R B ( T, µ B ) = S HB σ B M B = χ B ( T, µ B ) χ B ( T, µ B ) (17) R B ( T, µ B ) = κ HB σ B = χ B ( T, µ B ) χ B ( T, µ B ) , (18)where the mean M B , the variance σ B , the skewness S B , the kurtosis κ B , the hyper-skewness S HB and the hyper-kurtosis κ HB are the 1st to 6th order moments of the baryonnumber distribution.New results on continuum estimates for R B , R B ( T, µ B ) and R B ( T, µ B ) from latticesizes 32 × ×
12 have been recently presented in Ref. [37]. For the comparisonwith experimental data along the transition line, R B ( T, µ B ) and R B ( T, µ B ) are shown(see top of figure 8) as a function of R B . Therefore, it is not necessary to determine thetemperature and chemical potential in a heavy ion collision separately. This allows fora comparison with data from the STAR collaboration (Ref. [74, 75]). The experimentaldatapoints fit the lattice results well.First calculations of the 5th and 6th order cumulants from a 32 × R B ( T, µ B ). While errors are still large,both on the lattice and the experimental side, there seems to be a tension between bothdata sets. Changing the lattice results such that they can describe the two availableexperimental points would require unexpectedly large higher order corrections.In general, a detailed comparison between lattice data and experiment requires theincorporation of many different aspects (Ref. [60, 77–79]). While on the lattice site,clearly, a continuum extrapolation is still needed, on the experimental side it is notalways clear which particle species allow for good comparisons to the conserved chargesmeasured in lattice QCD. While for the cumulants for the baryon number distribution theproton number distribution has been established as reasonable proxy, for the strangnessnumber distribution the situation is less clear. A recent study in the hadron resonancegas (Ref. [76]) investigated different possibilities to find comparable observables. Thecomparison for different experimental proxies to the total, which would be the latticeresult for − χ BS χ S , is shown in Fig. 9. The lattice prediction for − χ BS χ S is calculated on an48 ×
12 lattice with two different methods. One that is based on the Taylor expansion12 R Bnm (T pc ) R B (T pc ) dashed lines:joint fi t toSTAR data forR P31 , R
P42 s NN [GeV]: STAR preliminary: open symbols
NNLO, R
B31 (T pc )R B42 (T pc )STAR 2020: R p R p -3-2-1 0 1 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 200 62.4 54.4 39 27 R B =M B / σ B s NN [GeV]: NLO, R B (T pc )R B (T pc )STAR preliminary: R p Figure 8: (Ref. [37]) Top: The ratios R B ( T, µ B ) = S B σ B /M B and R B ( T, µ B ) ≡ κ B σ B as a function of R B ( T, µ B ) = M B /σ B evaluated along the transition line incomparison to the data from the STAR collaboration (Ref. [74, 75]). Thelattice calculation is an continuum estimate from N t = 8 and N t = 12 lattices.Bottom: The ratios R B ( T, µ B ) and R B ( T, µ B ) as a function of R B ( T, µ B )evaluated along the pseudo-critical line in comparison to the data from theSTAR collaboration (Ref. [75]). The lattice determination was done on an N t = 8 lattice. 13 - χ BS / χ S T [MeV]
Totalproxy: σ Λ / σ proxy: σ Λ /( σ Λ + σ )proxy: ( σ Λ + σ Ξ + σ Ω )/( σ Λ + 4 σ Ξ + 9 σ Ω + σ )proxy: σ pK /( σ Λ + σ ) Figure 9: (Ref. [76]) Comparison between different possible combinations of particlenumber cumulants that could be measured experimentally (proxies) and thetotal of − χ BS χ S that could be determined in lattice QCD calculations. Bothcalculated in the Hadron Rasonance Gas model.and one that is based on the sector expansion (see Ref. [50]), where the pressure isparameterized as P (ˆ µ B , ˆ µ S ) = P BS + P BS cosh(ˆ µ B ) + P BS cosh(ˆ µ S )+ P BS cosh(ˆ µ B − ˆ µ S )+ P BS cosh(ˆ µ B − µ S )+ P BS cosh(ˆ µ B − µ S ) . (19)As can bee seen in figure 10, both expansions agree well for small chemical potentialsand temperatures around 140 MeV. For higher temperatures, a deviation between thetwo methods becomes visible before the errorbar grows due to the increased chemicalpotential. Also the influence of different experimental cuts has been investigated. The critical endpoint:
To look for a bound on the critical endpoint in the QCD phasediagram, one can try to calculate the radius of convergence of an expansion in µ B around µ B = 0. When one wants to estimate the radius of convergence naively form thefluctuations, one defines r χ n = (cid:114) χ n χ n +2 . (20)If r χ n converges, in the limit of n → ∞ , it is guaranteed that there is no criticality withinthis radius. This has been done in Ref. [35] for the fluctuations up to 6th order.However, it has been shown in Ref. [80],that the ratio estimator as given in equa-tion (20) is never convergent in a finite volume. This is consistent with the fact that14 µ B /T
135 MeV 145 MeV 155 MeV 165 MeVTaylorSectorsHRG x12 lattice Figure 10: (Ref. [76]) Comparison between the Taylor and the sector method for − χ BS χ S on an 48 ×
12 lattice, as well as results from the hadron resonance gas model(HRG).there is never a true phase transition in a finite volume. However, the convergence of theratio estimator is problematic even when using the p n extrapolated to infinite volume. Itwill work if one uses infinite-volume Taylor coefficients and the singularity determiningthe radius of convergence corresponds to a real phase transition in the infinite-volumelimit. In the same work, alternative estimators have been discussed. Fig. 11 showsresults on an N t = 4 lattice for the standard estimator defined in equation (20), themodified Mercer-Roberts estimator r ( MMR ) k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( k + 1)( k − c k +1 c k − − k c k ( k + 2) kc k +2 c k − ( k + 1) c k +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (21)and the doubled index estimator r (2 i ) k = (cid:12)(cid:12)(cid:12)(cid:12) kc k + k c k (cid:12)(cid:12)(cid:12)(cid:12) k , (22)for a Taylor series with coefficients c k .Another way to look for a critical endpoint and find the radius of convergence is theinvestigation of the Lee-Yang zeros (Ref. [81]). These are the zeros of the partitionfunction in the plane of complex chemical potential. The radius of convergence is thedistance to the closest (leading) Lee-Yang zero in the infinite Volume limit. If this hap-pens to be on the real axis, it is a signal for the critical endpoint. While the higher orderfluctuations needed for a reliable radius of convergence estimate suffer from increasinglylarger errors and are difficult to determine, Ref. [80] discusses a cancellation that allowsfor a calculation of the leading Lee-Yang zero. However, this benefit only holds only if it15 c on ve r g e n ce r a d i u s es t i m a t o r s ( β ) Figure 11: (Ref. [80]) The radius of convergence from three different estimators deter-mined on an N t = 4 lattice. The green arrow indicates the result fromreweighting.is possible to reweight by using a reduced matrix formulation (Ref. [9, 82–85]). This isnot possible for the case of rooted staggered fermions. For this case, Ref. [86] introducesa new definition of the rooted staggered determinant which allows for a numerical studyof the Lee-Yang zeros. It is than tested on N t = 4 lattice with stout smeard staggeredfermions and a Symanzik improved gauge action. Spatial lattice extends of N s = 8 , D ( µ ) from the configuration generation which is done with aweight of |(cid:60) (det D ( µ )) | . The sign is handled separately by a discrete reweighting. One way to conduct lattice simulations, despite the sign problem that made significantprogress are complex Langevin simulations. I will only give a very brief overview over thenew developments. A more comprehensive recent review can be found in Ref. [88]. Thesesimulations are based on the Langevin process, an evolution in a fictitious Langevin time,to generate configurations with a complex measure. This involves a complexification ofall fields and, therefore, extending the SU (3) gauge group to SL (3 , C ), which is a noncompact group. This can lead to so called runaway configurations which can cause thetrajectory to converge to a wrong result (Ref. [89–92]). To keep the evolution close tothe unitary manifold and therefore to the correct result, gauge cooling (Ref. [29, 93]) was16eveloped. A recent overview on this subject can be found in Ref. [94]. Even if the gaugecooling as well as an adaptive step size in the numerical integration (Ref. [95]) or theaddition of force to the evolution [96] increase the stability of the Langevin simulations,it is still important to verify the correctness of the result. This can be done by differentcorrectness criteria, which are related to the fall-off behavior of specific observables(Ref. [97–102]).By now, results obtained with Complex Langevin simulations start to attacking theQCD phase diagram. In Ref. [103], Complex Langevin was combined with stout smear-ing and the result was compared with results from the Taylor expansion method (seesection 3.2). The simulations were performed with four flavours of staggered quarks on16 × B ( (cid:79) ) = (cid:10) (cid:79) (cid:11) (cid:104) (cid:79) (cid:105) / (23)with two different Polyakov loop related observables (cid:79) . The data is than fitted with aquadratic function to determine the curvature of the transition line κ (see equation 12).The transition line on an N s = 12 lattice from both observables is shown in Fig. 13.In addition to finite temperature studies, Complex Langevin simulations are also usedto study low or zero temperature QCD. In Ref. [102, 105], the average quark number (cid:104) N (cid:105) = N f N c N s is studied on lattices with sizes 8 ×
16 and 16 ×
32. They found aplateau for (cid:104) N (cid:105) = 24 as a function of the chemical potential. It is interpreted as themaximum number of zero momentum quarks that can exist at zero temperature. Whenthe chemical potential is increased enough, it excites the lowest non-zero momentumstates.In Ref. [106], the zero temperature transition between hadronic an nuclear matter ata chemical potential of a third of the nucleon mass µ = m N is searched for but notfound. This contradiction to the physical expectation is related to possible issues withthe Complex Langevin simulations, which faces difficulties at low or zero temperature.17 ∆ ( p / T ) µ /TTaylor exp. 6th orderTaylor exp. 4th orderTaylor exp. 2nd orderCLE4th order fit6th order fit16 *8, β =5.3, N F =4, m=0.01 T ≈
300 MeVm π ≈
655 MeV *8, Symanzik, β =3.72-stout. N F =4, m=0.02 T ≈
260 MeVm π ≈
525 MeV ∆ ( p / T ) µ /TTaylor exp. 6th orderTaylor exp. 4th orderTaylor exp. 2nd orderCLE6th order fit Figure 12: (Ref. [103]) Top: Comparison between an analytic continuation by Taylorexpansion (see sec. 3.2) and simulations with the complex Langevin equationsusing a naive action. Bottom: Comparison between an analytic continuationby Taylor expansion and simulations with the complex Langevin equationsusing a Symanzik improved action.18 s =12, β =5.9 κ =0.15, N F =2 T c / T c ( ) mu/T c (0)quadratic fi t to data from B (P-
)From B (P-
)quadratic fi t to data from B (P) (shift)From B (P) (shift) Figure 13: (Ref. [104]) The transition temperature from Complex Langevin Simulationsat heavy pion masses ( m π ≈ . N s = 12 lattice. The transition temperature is determined from the third or-der Binder cumulant B with two different Polyakov loop related observables.The data is than fitted with a quadratic function.19 Effective lattice theories
Full lattice QCD simulations for large baryon chemical potentials are out of reach for themoment. However, some insight can be gained from simulations of effective theories onthe lattice. Here usually either the fermions and the spacial gauge links are integratedout, yielding a theory only depending on Polyakov loops, which contains the temporallink variables. Or, as the second common option, spatial and temporal gauge linksare integrated out resulting in a theory with hadronic degrees of freedom. The use ofeffective theories is limited to a specific parameter range, where the reduction of thedegrees of freedom holds true. If it is possible to find overlapping parameter rangesbetween different theories, one can gain inside in a wide range of phenomena.
Heavy quarkonium refers to states of matter including heavy quarks, most commonlythe charm and bottom quark. In heavy ion collisions, cc and bb pairs can usually onlybe created in early stages of the collision due to the required large energy amount. Thisheavy bound states are then surrounded by the quark gluon plasma and can either melddue to the high temperatures or survive until freezout. In the later case, a signature forthese particles should be visible in the detectors. Since the melting of different particlesdepends on the state of the quark gluon plasma, the observation of quarkonium is auseful tool to enhance our knowledge on QCD at very high temperatures. On the otherhand it becomes more and more clear that not only the melting but also the possibilityof recombination of quarkonium states has to be taken into account. The interactionbetween the heavy bound states and the surrounding medium is one of the main questionfueling the work on effective theories. A sketch of the different stages a heavy quark pairgoes through in a heavy ion collision is shown at the bottom for figure 1 form Ref. [19]where detailed introduction and review of recent developments for heavy quarkoniumcan be found.The input from the lattice to the investigation on quarkonium can be twofold. Onone hand, the calculation of quarkonium correlators and their spectral functions as forexample done in Ref. [107, 108]. Since these studies require zero temperature simulations,they are not part of this review.On the other hand, the lattice simulations of effective field theories are a very helpfultool to investigate the behavior of quarkonium in a quark gluon plasma. To arrive at aneffective field theory, the difference in scales within the quarkonium is exploited. Themass of a heavy quark m Q is much larger than its velocity v within a bound state and,therefore, m Q (cid:29) vm Q (cid:29) v m Q (24)as well as m Q (cid:29) Λ QCD . (25)The reduction of the degrees of freedom by the integration over different scales is sketchedin figure 14. While full continuum QCD is valid at all scales, the finite volume of the20 Q vm Q Λ IR m Q v Λ QCD
T latticeQCD not dynamic not dynamic d.o.f.s |p| integrated outultrasoft gluons+wavefunctions a s -1 not dynamic not dynamic QQintegrated out d.o.f.s pintegratedout not dynamic not dynamic QCD d.o.f.s integratedout wavefunctionsQQintegratedout pNRQCD
Wilson coeff latticepNRQCD gluons soft gluons and
Paulispinors
Pauli spin ors
NRQCD latticeNRQCD
Figure 14: (Ref. [19]) Overview over the different scales treated in effective field theorieson the lattice. In the first step the lattice introduces UV cut off by the finitelattice volume and an infrared cut off by the finite lattice spacing. In the nextstep gluons and heavy quarks are integrated out yielding a non relativisticversion on QCD called NRQCD. The lattice version of NRQCD again hasthe respective cut offs. Instead of on the lattices NRQCD can also be treatedperturbatively to integrated out further degrees of freedom down to an energyscale m Q v . This leads to potential NRQCD (pNRQCD) with ultrasoft gluonsand wavefunctions as remaining degrees of freedom. pNRQCD can then bestudied on the lattice by the investigation of non-local Wilson coefficients.21attice provides an infrared cut off, while the finite lattice spacing cuts off the ultravioletdivergences. To arrive at a non relativisic version of QCD called NRQCD (Ref. [109]) thehard gluons and heavy quarks are integrated out. The relevant degrees of freedom arethe Pauli spinors and soft gluons. This effective theory it self can be treated by latticesimulation. On the other hand, a perturbative treatment of NRQCD can be applied tointegrate out further degrees of freedom down to an energy scale m Q v . This leads topotential NRQCD (pNRQCD, Ref. [110]), with ultrasoft gluons and wavefunctions asremaining degrees of freedom. In this framework the in-medium real-time potential canbe investigated. Again, it is possible to study pNRQCD on the lattice. To constructthe respective Lagrange functions for the various effective field theories, one identifiesthe relevant degrees of freedom at each energy scale and constructs a general Lagrangefunction from symmetry considerations. The relevant prefactors of each term, the socalled Wilson coefficients, are determined by matching. The strength of effective fieldtheories lies in the reduced number of degrees of freedom, that makes computationseasier. The trade off is the reduced validity range.An investigation of the validity of the perturbative treatment of effective field theorywas done in Ref. [111]. There, lattice QCD calculations of a wide temperature rangefrom 140 MeV up to temperatures of 5814 MeV with (2+1)-flavours of highly improvedstaggered fermions were performed. The evaluation of the static quark-antiquark poten-tial led to the conclusion that effective field theories can be used between 0 . (cid:46) rT (cid:46) . r beginning the separation between quark and antiquark) to describe color screen-ing reliably. NRQCD:
A large effort to use lattice NRQCD to study the behaiviour of bottemo-nium in a quark gluon plasma has been undertaken by the FASTSTUM collaborationin Ref. [112–116]. They calculate the spectrum of bottemonium around the crossovertemperature, using (2+1)-flavours of Wilson clover fermions on anisotropic lattices. Theanistropy of the lattices improves the NRQCD expansion. On the other hand it alsoleads to heavy pion masses of m π ≈
400 MeV, if computation are to be kept afordable.Another recent investigation of both bottemonium and charmonium using NRQCDhas been done in Ref. [117, 118]. Here, lattice configuration of the HOTQCD collabo-ration with 2+1 highly imporved staggered (HISQ) quarks were used. These ensembleshave a realistic pion mass of m π ≈
161 MeV. Ref. [118] was able to sort out previoustensions between Ref. [112] and Ref. [117] on the melting temperatures of quarkoniumby relating them to different uncertainties in the spectral reconstruction. It was shown,that the mass of the ground state for heavy quarkonium reduces when the temperatureis increased. pNRQCD:
To investigate not only the ground state, but also the in-medium behaiviourof exited quarkonium states, one has to turn to pNRQCD. Here the spectral functions arecomputed by solving the Schr¨odinger equation with a static potential. The results showa mass reduction for rising temperatures and agree well with lattice NRQCD results,while contradicting expectation gained from perturbative pNRQCD.22igure 15: (Ref. [19, 119]) The ratio between ψ (cid:48) and J/ψ from various heavy ion collisionexperiments: the NA50 (Ref. [120]), ALICE (Ref. [121]) and CMS (Ref. [122,123]). The orange points show the pp baseline form Ref. [124, 125]. Thepurple line comes from the statistical model of hadronization from Ref. [124].The green line is the result from a computation based on pNRQCD spectralfunctions combined with an instantaneous freezeout scenario.Recent progress with computations in lattice pNRQCD include the determination ofthe ratio between the ψ (cid:48) and J/ψ in Ref. [119]. Here a non perturbative treatmendof pNRQCD is empolyed, based on the derivation of the generalized Gauss law. Thederived potential is matched to lattice QCD results from Ref. [126–128]. The calculationof inmedium spectral functions allows for a prediction on the ψ (cid:48) and J/ψ ratio, whichis shown in figure 15. It is also compared to various results from heavy ion collisionexperiments as well as results from the statistical model of hadronization (Ref. [124]).23
Columbia plot
As discussed before, for zero chemical potential and physical quark masses, the QCDtransition is a crossover. However this changes when the quark masses are varied. Thetype of the transition between hadronic matter and the quark gluon plasma, called theQCD transition, depends on the quark masses. This is illustrated in the Columbia plotfor 2+1 quark flavours (figure 16). The upper right corner, where quark masses areinfinite, is the pure SU (3) gauge limit with static quarks and exhibits a first order phasetransition. Finite quark masses break the Z (3) center symmetry of the zero mass limitexplicitly, which weakens the phase transition until it becomes second order in the Z (2)line bordering the upper right corner of the Columbia plot.In the limit of vanishing quark masses, corresponding to the lower left corner of theColumbia plot, again a first order transition is expected for three flavours in the chirallimit, where m = 0. Finite quark masses break the chiral symmetry and weaken thetransition until it becomes second order. This yields another Z (2) line delimiting thiscorner. However, the case on the left border, for vanishing light quark masses, is stillunder investigation. Another possible scenario is that the first order region could extentall the way up to the N f = 2 limit or hit the m ud = 0 line at a finite strange quark mass,as shown in Fig. 16. The two scenarios are connected to the possible U (1) A symmetryreforestation at the transition temperature (Ref. [129, 130]) and are shown in figure 16.The computation of the Z (2) lines, especially in the lower left corner, are numericallyvery challenging. Smaller quark mass, as well as critical slowing down near a phase tran-sition increase the computational cost. At the same time, small quark masses increasethe taste breaking artifacts for the computationally relatively cheap staggered quarks,making a continuum extrapolation even more difficult. Also, scans in temperature, vol-ume and quark masses are required to determine type and position of the transition.Due to this challenges, the only continuum extrapolated point in the Columbia plot upto now is the physical point (Ref. [1–6]).To overcome the computational challenges faced when directly investigating the pa-rameter space of the Columbia plot, additional extrapolation direction are used. Sim-ulations at imaginary chemical potential can enlarge the first order region and help todetermine it at µ B = 0 by limiting the search region. Similarly, simulations at variable N f , even non integer ones, have been applied. One common direction of investigation is along the N f = 3 diagonal of the Columbia plot.However, the first order region has only be found on N t = 4 lattices with unimprovedstaggered (Ref. [131, 132]) or O ( a )-improved Wilson [133] fermions. For finer lattices orwith further improved actions, the phase transition remains elusive (Ref. [134]).A direct investigation along the N f = 2 border only yielded results for N t = 4 latticeswith unimporved staggered or Wilson quarks (Ref. [44, 135, 136]).A recent investigation of 2 + 1-flavor QCD with small quark masses is reported inRef. [137]. It studies the the Polyakov loop expectation value (cid:104) P (cid:105) and the heavy quark24 st order n d o r d e r - Z ( ) crossover n d o r d e r - Z ( ) physical point n d o r d e r - Z ( ) crossover n d o r d e r - Z ( ) physical point Figure 16: Schematic of two different, possible scenarios for the Columbia plot. Itshows the dependence of the transition between hadronic matter and thequark gluon plasma on the quark masses.25 .000.020.040.060.080.100.120.140.160.18 110 120 130 140 150 160 170 180 〈 P 〉 T [MeV] param : w 1/27 N τ =8O(2) H=0 (1- β )/ βδ χ mP z param : w 1/27 N τ =8O(2) H=1/271/401/801/1600
Figure 17: (Ref. [137]) Top: The temperature dependence of the Polyakov loop (cid:104) P (cid:105) .Bottom: The derivative of the Polyakov loop as a function of the scalingvaribale z = z tH − /βδ with t = ( T − T c ) /T c and z , T c being non-universalconstants.free energy F q ( T, H ) = − T ln (cid:104) P (cid:105) = − T | (cid:126)x − (cid:126)y |→∞ ln (cid:104) P (cid:126)x P † (cid:126)y (cid:105) (26)with H = m l /m s on N t = 8 lattices. The analysis is done for several volumes toreach the limit H −→
0. Result for the Polyakov loop and its derivative are shown inFig. 17. It is found that the scaling behavior is consistent with the scaling behaviorfor energy-like observables in the 3-d, O ( N ) universality classes. The authors observe asingular behavior of the quark mass derivatives of (cid:104) P (cid:105) and F q /T . In the chiral limit thedivergence is consistent with the 3-d, O (2) universality class.Previously, scaling studies have been carried out in Ref. [138, 139]. Here the transitiontemperature in the chiral limit T c and the magnetization M = 2 m s (cid:104) ¯ ψψ (cid:105) l − m l (cid:104) ¯ ψψ (cid:105) s f K (27)26 χ M T [MeV] m π [MeV]m s /m l N τ =8 20 16027 14040 11080 80160 55
150 200 250 300 350 400 450 135 140 145 150 155 160 165 170 χ M T [MeV] N τ =8m s /m l =80 N σ =32N σ =40N σ =56 Figure 18: (Ref. [138]) Top: The quark mass dependence of the chiral susceptibilityas defined in equation (28). Bottom: The volume dependence of the chiralsusceptibility as defined in equation (28).27s well as its derivative, the chiral susceptibility χ M = m s ( ∂ m u + ∂ m d ) M | m u = m d = m s m s χ l − (cid:104) ¯ ψψ (cid:105) s − m l χ su f K (28)were determined from lattices with N t = 6 , m π ≈
55 MeV and an O (4) scaling ansatz was used toextrapolate to the chiral limit. The final result for the transition temperature in thechiral limit determined from different analyses is given as T c = 132 +3 − MeV . (29)The dependence of the chiral susceptibility on the quark masses is shown in the topFig. 18. It was determined on an N t = 8 lattice. The spacial extent of the lattice isincreased with decreasing light quark mass m l . It is N s = 32 for H = m s m l = 20 or 27, N s = 40 for H = 40 and N s = 56 for H = 80 or 160. The dependence on the volume forfixed H = 80 is shown on the bottom for Fig. 18.A further study on the chiral limit is done in Ref. [140]. Here again pion massesdown to µ π = 55 MeV are used. Lattice spacing of a = 0 .
12 fm 0.08 fm and 0.06 fmcorresponding to N t = 8 ,
10 and 16. The authors invesgate the dependence of theDirac eigenvalue spectrum and the axial anomaly on the quark masses. For the chiralcondensate they find that while there is a clear dependence on the quark masses andlattice spacing, the volume dependence is small. This is illustated in Fig. 19.Ref. [141] investigates the Columbia plot for the N f = 3 case with improved Wilsonquarks and the Iwasaki gaugue action up to N t = 12. It uses multiensemble reweightingto determine the minimum of the kurtosis. The change in this minimum as a functionof β is then studied for different volumes. The point, where it agrees for several volumesdefines the critical β value. A sketch of this so called kurtosis intersection analysis isshown in figure 20. The results for this analysis for five different values of N t as well asthe scaling of the maximal chiral sucsptibilty χ max ∝ ( N s ) b is shown in figure 21. Theauthors conclude that the Z (2) line in the columbia plot for the N f = 3 case (along thediagonal) is located at a pion mass m π (cid:46)
110 MeV.
One way to enlarge the first order region of the Columbia plot, at least on corse lattices,is the introduction of an imaginary chemical potential. As discussed in section 3.2 thereis no sign problem in QCD the case of an purley imaginary chemical potential. A sketchof this 3d-Columbia plot is shown in Fig. 22. When adding the chemical potential axisto the Columbia plot, its curvature at µ = 0 is negative. This increases, therefore,the first order region for imaginary chemical potential and decreases it for real µ . Thisinvestigations have been done both with unimporved Wilson (Ref. [142]) and staggered(Ref. [143, 144]) fermions on N t = 4 lattices.28 .
01 0 .
02 0 .
03 0 .
04 0 .
55 80 110 140 160 m s h ¯ ψψ i / T m π [MeV] .
01 0 .
02 0 .
03 0 .
04 0 .
55 80 110 140 160 m s h ¯ ψψ i / T m π [MeV] m l / m s N τ =8 N τ =12 N τ =16 m l / m s . . . . .
002 0 .
004 0 .
006 0 .
008 0 .
01 0 .
012 0 .
014 0 . m s h ¯ ψψ i / T . . . . .
002 0 .
004 0 .
006 0 .
008 0 .
01 0 .
012 0 .
014 0 . m s h ¯ ψψ i / T (N τ / N σ ) N τ = 8 N τ = 12 N τ = 16(N τ / N σ ) Figure 19: (Ref.[140]) Top: The quark mass dependence of the chiral condensate fordifferent lattice spacing. Bottom: The volume dependence of the chiral con-densate for different lattice spacings at m π = 80 MeV. E1st order crossover K t light heavy V>V>V suscept. kurt.-2 suscept.kurt. Figure 20: (Ref. [141]) Scatch of the kurtosis instersection analysis. The minimum of thekurtosis is determined from multiensemble reweighting and the then fittet fordifferent volumes. The intersection is taken to be the value for the criticalendpoint, as here there should be no volume scaling.29 .00.51.01.52.02.53.0 1.60 1.62 1.64 1.66 1.68 1.70 1.72 1.74 1.76 1.78 1.80 1.82 1.84 b β N t =4N t =6N t =8N t =10N t =123D Z Figure 21: (Ref. [141]) The scaling exponent of the maximum sucsptibilty with the vol-umen: χ max ∝ ( N s ) b . The results are shown for N t = 4 , , ,
10 and 12 withconnecting lines. The green horizontal line shows the expectation for Z (2)-scaling. The shaded areas show the results for the critical β from the kurtosisintersection analysis. The thermal phase transition at imaginary µ
Phys. point “deeper” in crossover region than for zero density
First-order region in RW plane shrinks towards continuum µ
3d Ising m u,d m s st tr. 1 st tr. N f = N f = ✓ ⇡ ◆ ✓ µT ◆ t r i c tr i c st Z (2) Z (2) 1 st phys. point N f = 3 crossover µ
310 330 350 370 390 4100 . . . . . ⌫ c o ll a p s e crit. exponent ⌫
Figure 23: (Ref. [145]) The value of the critical pion mass for four or three staggeredquarks, for different values of N t . For four flavours the critical mass is larger,allowing to to to finer lattices. N f Another way to increase the strength of the phase transition, and therefor helping withthe investigation of the Columbia plot, is to increase the number of falvours N f . Ref. [145]studies the Columbia plot with four staggered quarks and the Wilson plaquette action.The choice of N f = 4 has the added benefit that no rooting is required. Simulationswere performed for N t = 4, 6, 8 and 10 for several spacial lattice extends. Figure 23shows the difference between the critical pion mass for four or three staggered quarks.The increased number of flavours increases the pion mass, so that simulations on latticeswith larger temporal extend are possible.To get a smooth transition between the integer number of flavours, it is even possibleto consider non-integer powers of the quark determinant (Ref. [135]). This allows for asmooth interpolation in N f . Since in the ciral limit the transition is second order for N f = 2 and first order for N f = 3 there has to be a tricritical scaling area, with a criticalpoint at N tric f , in between. Ref. [135] presents results on N t = 4 lattices for N f = 2 . .
6, 2.4, 2.2 and 2.1. The dependence of the critical mass m Z for a Z (2) scaling isexpected to follow m / Z ( N f ) = C (cid:0) N f − N tric f (cid:1) . (30)The result are shown in the top of figure 24. The rescaled mass is shown in the bottomof figure 24. A linear function was fitted to the rescaled mass to determine C and N tric f . The investigation of the Columbia plot remains a challenge for lattice QCD. For thedetermination of the second order line in the lower left corner (see figure 16) several31 m Z N f m { Z N f Figure 24: (Ref. [135]) Top: The critical mass m Z as a function of the number of flavour.The black line is the rescaled version of the fit done to the resacled massbelow. Bottom: The rescaled critical mass m Z as a function of the numberof flavour. As well as a linear fit to the data.results are available on finite lattices however there is no clear solution for the continuumlimit. Figure 25 shows an overview over the results for different fermion types. For thisplot, some conversions were necessary. Some values were converted to MeV using thevalue for the critical temperature from Ref. [146] T c = (156 ±
8) MeV for coarse latticesand staggerd quarks. Upper bounds are denoted by zero with the errorbar showing thebound. Values given without any error, are assumed to have 50% relative error. Thecolor of the points denotes the fermion type used in the simulations. Red stands forunimproved staggered, orange for improved staggered, green for unimproved Wilson andblue for improved Wilson quarks. The largest number of results is available for the threeflavour case. Here both Wilson and staggered quarks show a shrinking value when thelattices become finer. The same trend can be observed for staggerd quarks with N f = 4.For both two and three flavors, staggerd quarks show smaller values than Wilson quarks.Also for staggerd quarks an improvement of the quark formulation reduces the value forthe critical pion mass. Assuming that all quark formulations agree in the continuum,which they should, it seems that still a large amount of computational power is neededto settle the shape of the lower left corner of the Columbia plot. Investigating this regionbecomes especially challenging since both staggerd and Wilson quarks suffer from largelattice artifacts when the quark masses become small (Ref. [147]). Recent progress on the determination of the upper right corner of the Columbia plot(see figure 16) was reported in Ref. [153]. The authors determine the critical pion massfor the N f = 2 flavour case. This corresponds to the upper border of the Columbia plot.The lattice setup consisted of two dynamical, degenerate Wilson fermions on latticeswith N t = 6 , N t values correspond to lattice spacings between 0.07 fmand 0.12 fm. The results for the critical pion masses m cπ for all three lattices are shown in32 .03.54.04.55.0 N t N f = 2 [Bonati:2014kpa][Philipsen:2016hkv] N t N f = 3 [Kuramashi:2020meg][Karsch:2001nf][deForcrand:2007rq][Karsch:2003va][Endrodi:2007gc][Ding:2011du][Varnhorst:2015lea] m cπ / MeV N t N f = 4 [deForcrand:2017cgb] Figure 25: Overview over different available results for the critical pion mass in theColumbia plot for N f = 2, 3 and 4. Values are from Ref. [131–133, 141,145, 146, 148–152]. The color of the points denotes the fermion type used inthe simulations. Red stands for unimproved staggered, orange for imporvedstaggered, green for unimporved Wilson and blue for imporved Wilson quarks.Upper bounds are denoted by zero with the errorbar showning the bound.Values given without any error, are denoted with 50% relative error. Somevalues were converted to MeV using the value for the critical temperaturefrom Ref. [146] T c = (156 ±
8) MeV. Similar values for the same N t have beenshifted lightly for more clarity. 33 κ Z N τ = 6 N τ = 8 N τ = 10 .
07 0 .
08 0 .
09 0 . .
11 0 . a { fm } m π { M e V } Figure 26: (Ref. [153]) Top: The critical value for the hopping parameter κ (see equations(32) and (31)) as a function of the lattice spacing. Bottom: Value of thecritical pion mass as a function of the lattice spacing.34he bottom of figure 26. While a precise continuum limit is not possible from these threelattices alone the authors estimate a conutinuum value of m cπ ≈ κ Z has been determined. It appears in fermion matrix for the Wilson quark action as D ( n | n ) = δ n ,n − κ ± (cid:88) µ = ± [( (cid:49) − γ µ ) U µ ( n ) δ j +ˆ µ,n ] (31)and controlles the bare quark mass m b as κ = 12( am b + 4) . (32)These results do not agree with those previously computed in Ref. [154]. Here reweight-ing from quenched QCD has been used to investigate the end-line in 2+1 flavor QCD.For the N f = 2 and N t = 6 case, the results differ by about 50%. Since the same latticeaction has been used in both cases, the discrepancy is most likely related to the differentmethods and not to the cut off effects. Ref. [154] describes finite volume effects as wellas effects related to the hopping parameter expansion which seems to shift the resultsin the direction of Ref. [153]. 35 Magnetic fields
When dealing with heavy ion collision experiments, in addition to a finite chemicalpotential, also the effects of magnetic fields have to be considered (Ref. [155–157]).Form zero temperature lattice QCD studies with staggerd fermions (Ref. [158–160]) theso-called magnetic catalysis is found. It describes that the chiral condensate, which isan order parameter of the QCD chiral transition, increases with the strength of themagnetic field. From this it was derived that also the transition temperature increaseswith the magnetic field strength.A recent study (Ref.[161]) investigated the chiral phase structure of three flavourQCD with a magnetic field on N t = 4 lattices with four different volumes between N s = 8 and N s = 24. It uses a Wilson plaquette action with staggerd fermions andpion masses of m π ≈
280 MeV and two values of finite magnetic field strength. Theyfind a strengthening in the transition, which turns to first order, when a magnetic fieldis added. The results for the chiral condensate for different magnetic field strengthsand lattice volumes are shown in Fig. 27. The first order nature is derived both fromthe volume scaling as well as from meta stable states of the chiral condensate in thesimulation stream. They also see an increase in the transition temperature with themagnetic field strength.On the other hand, Ref. [162–167] find a decreasing transition temperature with grow-ing magnetic fields. Here continuum extrapolations or improved actions were employed.It is assumed that the discrepancy is related to discretization artifacts (Ref. [161]). Thedecreasing of the transition temperature as well as the chiral condensate is called inversemagnetic catalysis. Most recently, Ref. [167] uses the HISQ quark action to study theeffects of magnetic fields on a 16 × N t = 6 lattices.They use pion masses up to m π ≈
600 MeV and investigate the transition temperatureas a function of the magnetic field strength. The transition temperature is determinedboth from the chiral condensate and the Polyakov loop leading to similar results. Thechiral condensate for different strengths of the magnetic field and different pion massesis shown in figure 28 and figure 29. They find that, while the decrease of the transitiontemperature is present for all pion masses, the decrease of the chiral condensate, how-ever, is not observable for the highest pion masses. This strengthens the idea that theinverse magnetic catalysis, the decrease of the chiral condensate, is not directly relatedto the transition temperature. The authors suggest that the decrease of the transitiontemperature might be a deconfinement catalysis (Ref. [182]).Ref. [181] determines the transition point in the quark mass between magnetic catalysisand inverse catalysis further. They use the three stout smeared staggerd fermions with36 . . . . . . . . . . .
28 5 .
13 5 .
14 5 .
15 5 .
16 5 .
17 5 .
18 5 .
19 5 . .
21 5 . h ψ ¯ ψ i β ˆ b =0ˆ b =1.5ˆ b =2.0 . . . . . . . . . .
155 5 .
16 5 .
165 5 .
17 5 .
175 5 .
18 5 . h ψ ¯ ψ i β N σ =8N σ =16N σ =20N σ =24 Figure 27: (Ref. [161]) Top: The chiral condensate on an 16 × b = a √ eB . Bottom: The chiral condensate fordifferent volumes on lattices with temporal extend N t = 4 at a magnetic fieldstrength of ˆ b = a √ eB = 1 .
5. 37 m π = M e V eB=0 GeV eB=0.425 GeV eB=0.85 GeV m π = M e V
100 125 150 175 200 225 250 275
T [MeV] m π = M e V Figure 28: (Ref. [180]) The chiral susceptibility for different strengths of the magneticfield and different pion masses. Simulations were done with N f = 2 + 1 stoutsmeard staggered fermions on N t = 6 lattices. One can observe a decreasingpeak position and therefore transition temperature for all pion masses.38 m π = M e V eB= 0 GeV eB= 0.425 GeV eB= 0.85 GeV -0.05-0.02500.025 m π = M e V
100 125 150 175 200 225 250 275
T [MeV] -0.02-0.0100.01 m π = M e V Figure 29: (Ref. [180]) The chiral condensate for different strengths of the magnetic fieldand different pion masses. Simulations were done with N f = 2 + 1 stoutsmeard staggered fermions on N t = 6 lattices. One can observe that whilefor the lightest pion mass the condensate decreases for a stronger magneticfield, it increases for the highest pion mass.39 ~ IMC MC ∆ Σ M π [MeV]m/m phys L R eB [GeV ]m=8 m phys m=12 m phys m=14 m phys m=16 m phys m=18 m phys Figure 30: (Ref. [181]) Top: The dependence of the chiral condensate ∆ ˜Σ defined inequation (35) as a function of the pion mass in a magnetic field of eB =0 . × m on the border between magnetic catalysis and inverse magnetic catalysisthey look at the difference between the chiral condensate introduced by a magnetic fielddefined as ∆Σ( B, T, m ) = Σ(
B, T, m ) − Σ(0 , T, m ) (33)= 2 m phys M π F (cid:2) (cid:104) ¯ ψψ (cid:105) B, T, m − (cid:104) ¯ ψψ (cid:105) ,T,m (cid:3) , (34)for eB = 0 . at the pseudo critical transition temperature T c :∆ ˜Σ( m ) = ∆Σ( B , T c ( m, B = 0) , m ) . (35)A sign change in ∆ ˜Σ than determines the value of ˜ m since for ∆ ˜Σ > <
0. ∆ ˜Σ is shown in the top of Fig. 30, indicating ˜ mm phys = 14 . ± .
55. Thiscorresponds to a pion mass of ˜ m π = (497 ±
4) MeV. In addition also the Polyakov loop P = 1 V (cid:42)(cid:88) x (cid:60) Tr N t − (cid:89) t =0 U ( x , t ) (cid:43) (36)from which the ratio L R = P ( B, T, m ) (cid:14) P (0 , T, m ) (37)is defined, is investigated. Its behavior around ˜ m is shown in the top of Fig. 30.The connections between the transition temperature, the quark masses and the chiralcondensate in a magnetic field are further investigated in Ref. [160] for zero temperature.They connect the chiral condensate and the pion mass by the Gell-Mann-Oakes-Rennerrelation (Ref. [183]) for two flavours( m u + m d ) (cid:0) (cid:104) ¯ ψψ (cid:105) u + (cid:104) ¯ ψψ (cid:105) d (cid:1) = 2 f π M π (1 − δ π ) , (38)and its extension to the 3-flavor case (Ref. [184]) with an additional strange quark( m s + m d ) (cid:0) (cid:104) ¯ ψψ (cid:105) s + (cid:104) ¯ ψψ (cid:105) d (cid:1) = 2 f K M K (1 − δ K ) . (39)They use a 2 + 1-flavour HISQ quark action on a 32 ×
96 lattice. The results for thelight quark condensate defined asΣ l ( B ) = 2 m l M π f π (cid:0) (cid:104) ¯ ψψ (cid:105) l ( B (cid:54) = 0) − (cid:104) ¯ ψψ (cid:105) l ( B = 0) (cid:1) + 1 , (40)are shown in Fig. 31. They find that the Gell-Mann-Oakes-Renner relation for twoflavours (equation (38)) has only corrections of about 6% when a magnetic field up to eB = 3 .
35 GeV is introduced. This illuminates the connection between the reduction of41 Σ u (eB, λ UVcut ) / Σ d (eB, λ UVcut ) eB/M π eB [GeV ] λ UVcut =0 λ UVcut =0.12 λ UVcut =0.36 λ UVcut = ∞ Σ u (|q u B u |)/ Σ d (|q d B d |) |qB| [GeV ] λ UVcut =0 λ UVcut =0.12 λ UVcut =0.36 λ UVcut = ∞ Figure 31: (Ref. [160]) Top: Ratio between the renormalized up quark and the downquark condensate Σ u ( eB ) / Σ d ( eB ) as a function of eB . Bottom: Ra-tio between the renormalized up quark and the down quark condensateΣ u ( eB ) / Σ d ( eB ) as a function of | qB | = | q u B u | = | q d B d | .42he transition temperature and the pion mass. For the three flavour case the correctionsto the Gell-Mann-Oakes-Renner relation (equation (39)) are much larger namely up to56%.Another recent investigation of the magnetic catalysis and the inverse magnetic catal-ysis can be found in Ref.[185]. That study uses Dyson-Schwinger-equations instead oflattice QCD. 43 Conclusion
The investigation of the QCD phase diagram, especially in relation to heavy ion collisionexperiments, remains a lively topic for lattice QCD calculations. With newly publishedresults from LHC and RHIC as well as upcoming data from facilities like NICA, CBMand PARC-HI, a better understanding of the QCD phase diagram including variousdifferent influences that duplicate the situation in the colliders as well as possible, isneeded.After the introduction (section 1) and the brief description of the stages of a heavyion collision experiments (section 2), this review started with the discussion of latticeQCD results at low, finite density (section 3). Accessing the phase diagram with afinite chemical potential is hindered by the infamous sign problem (section 3.1). It isthe reason why, for now, continuum extrapolated, physical results, one therefore hasto rely on analytical continuation (section 3.2). Because the QCD phase transition isan analytic crossover at µ = 0, one can describe observables with an analytic functionthat is extrapolated to µ >
0. There are two techniques for analytical continuationwhich are commonly used. One can gain information on the µ dependence of the systemeither form calculating expansion coefficients from lattice simulations at zero chemicalpotential (Taylor method) or by describing simulation results at imaginary chemicalpotential with various functions. Both methods are by now agreeing well on variousresults. New publication include higher order coefficients on the Taylor expansion ofthe transition temperature, higher order cumulants of the baryon number distribution,which is used to compare to heavy ion collision experiments, as well as higher ordercumulants of the baryon number distribution, which are used to compare to heavy ioncollision experiments. In addition the possibility to find the QCD critical endpoint areunder further investigation. There the development of new observables and techniquesis necessary.One method that is getting relatively close to direct lattice simulations at finite den-sity are Complex Langevin simulations (section 3.3). These simulations are based on anevolution in a fictitious Langevin time to generate configurations with a complex mea-sure. Here results are now available for different actions (both fermion and gauge). Firstcomparisons to results from the Taylor expansion method are promising. However, fornow results are only available with heavier than physical quark masses and on relativelysmall lattices.Another way to accesses finite density QCD are effective field theories (section 4),which also can be simulated on the lattice. This review focused on the results for heavyquarkonium (section 4.1). There the separation of scales in heavy quark bound statesis used to integrate out various degrees of freedom. Lattice simulation can than employNRQCD or pNRQCD to gain information of heavy quarkonium immersed in the quarkgluon plasma.Leaving the topic of finite density also zero density simulations show fascinatingprogress. By now there are a lot of results on the investigation of the Columbia plot(section 5). The Columbia plot summarizes the dependence of the type of the QCDtransition (crossover, first or second order phase transition) on the lighter than physical44uark masses. The lower left corner (section 5.1) poses a special computational chal-lenge due to the light quark masses. Therefore, various techniques are employed to gainaccesses to the first order region located there and its critical boundary (section 5.2 andsection 5.3). For the case of three degenerate light quarks results are now available on upto N t = 12 lattices with improved Wilson quarks. However, the continuum extrapolationcannot be carried out yet. This review is therefore restricted to an overview (section 5.4)of the available results for different discretizations. Also the upper right corner of theColumbia plot has been investigated recently (section 5.5). Here the current estimatefor the critical pion mass in the continuum is m cπ ≈ Acknowledgments
The author thanks Lukas Varnhorst for proofreading and discussion. The project leadingto this publication has received funding from Excellence Initiative of Aix-Marseille Uni-versity - A*MIDEX, a French “Investissements d’Avenir” programme, AMX-18-ACE-005. 45 eferences [1] Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz, and K. K. Szabo. The Order of thequantum chromodynamics transition predicted by the standard model of particlephysics.
Nature , 443:675–678, 2006.[2] Y. Aoki, Z. Fodor, S.D. Katz, and K.K. Szabo. The QCD transition temperature:Results with physical masses in the continuum limit.
Phys. Lett. B , 643:46–54,2006.[3] Y. Aoki, Szabolcs Borsanyi, Stephan Durr, Zoltan Fodor, Sandor D. Katz, StefanKrieg, and Kalman K. Szabo. The QCD transition temperature: results withphysical masses in the continuum limit II.
JHEP , 06:088, 2009.[4] Szabolcs Borsanyi, Zoltan Fodor, Christian Hoelbling, Sandor D Katz, StefanKrieg, Claudia Ratti, and Kalman K. Szabo. Is there still any T c mystery inlattice QCD? Results with physical masses in the continuum limit III. JHEP ,09:073, 2010.[5] Tanmoy Bhattacharya et al. QCD Phase Transition with Chiral Quarks and Phys-ical Quark Masses.
Phys. Rev. Lett. , 113(8):082001, 2014.[6] A. Bazavov et al. The chiral and deconfinement aspects of the QCD transition.
Phys. Rev. , D85:054503, 2012.[7] Ian M. Barbour, Susan E. Morrison, Elyakum G. Klepfish, John B. Kogut, andMaria-Paola Lombardo. Results on finite density QCD.
Nucl. Phys. B Proc. Suppl. ,60:220–234, 1998.[8] Z. Fodor and S. D. Katz. A New method to study lattice QCD at finite temperatureand chemical potential.
Phys. Lett. , B534:87–92, 2002.[9] Z. Fodor and S. D. Katz. Lattice determination of the critical point of QCD atfinite T and mu.
JHEP , 03:014, 2002.[10] F. Csikor, G.I. Egri, Z. Fodor, S.D. Katz, K.K. Szabo, and A.I. Toth. The QCDequation of state at finite T and mu.
Nucl. Phys. B Proc. Suppl. , 119:547–549,2003.[11] Zoltan Fodor, Sandor D. Katz, and Christian Schmidt. The Density of statesmethod at non-zero chemical potential.
JHEP , 03:121, 2007.[12] Andrei Alexandru, C. Gattringer, H. P. Schadler, K. Splittorff, and J.J.M. Ver-baarschot. Distribution of Canonical Determinants in QCD.
Phys. Rev. D ,91(7):074501, 2015.[13] Andrei Alexandru, Manfried Faber, Ivan Horvath, and Keh-Fei Liu. Lattice QCDat finite density via a new canonical approach.
Phys. Rev. D , 72:114513, 2005.4614] Slavo Kratochvila and Philippe de Forcrand. The Canonical approach to finitedensity QCD.
PoS , LAT2005:167, 2006.[15] Shinji Ejiri. Canonical partition function and finite density phase transition inlattice QCD.
Phys. Rev. D , 78:074507, 2008.[16] Christof Gattringer. New developments for dual methods in lattice field theory atnon-zero density.
PoS , LATTICE2013:002, 2014.[17] Luigi Scorzato. The Lefschetz thimble and the sign problem.
PoS , LAT-TICE2015:016, 2016.[18] Andrei Alexandru, G¨ok¸ce Basar, and Paulo Bedaque. Monte Carlo algorithm forsimulating fermions on Lefschetz thimbles.
Phys. Rev. D , 93(1):014504, 2016.[19] Alexander Rothkopf. Heavy Quarkonium in Extreme Conditions.
Phys. Rept. ,858:1–117, 2020.[20] Francois Gelis, Edmond Iancu, Jamal Jalilian-Marian, and Raju Venugopalan. TheColor Glass Condensate.
Ann. Rev. Nucl. Part. Sci. , 60:463–489, 2010.[21] F. Gelis. Color Glass Condensate and Glasma.
Int. J. Mod. Phys. A , 28:1330001,2013.[22] Dirk H. Rischke. The Quark gluon plasma in equilibrium.
Prog. Part. Nucl. Phys. ,52:197–296, 2004.[23] P. Hasenfratz and F. Karsch. Chemical Potential on the Lattice.
Phys. Lett. B ,125:308–310, 1983.[24] C. R. Allton, S. Ejiri, S. J. Hands, O. Kaczmarek, F. Karsch, E. Laermann,C. Schmidt, and L. Scorzato. The QCD thermal phase transition in the pres-ence of a small chemical potential.
Phys. Rev. , D66:074507, 2002.[25] C. R. Allton, M. Doring, S. Ejiri, S. J. Hands, O. Kaczmarek, F. Karsch, E. Laer-mann, and K. Redlich. Thermodynamics of two flavor QCD to sixth order in quarkchemical potential.
Phys. Rev. , D71:054508, 2005.[26] R. V. Gavai and Sourendu Gupta. QCD at finite chemical potential with six timeslices.
Phys. Rev. , D78:114503, 2008.[27] S. Basak et al. QCD equation of state at non-zero chemical potential.
PoS ,LATTICE2008:171, 2008.[28] O. Kaczmarek, F. Karsch, E. Laermann, C. Miao, S. Mukherjee, P. Petreczky,C. Schmidt, W. Soeldner, and W. Unger. Phase boundary for the chiral transi-tion in (2+1) -flavor QCD at small values of the chemical potential.
Phys. Rev. ,D83:014504, 2011. 4729] Erhard Seiler, Denes Sexty, and Ion-Olimpiu Stamatescu. Gauge cooling in com-plex Langevin for QCD with heavy quarks.
Phys. Lett. B , 723:213–216, 2013.[30] D´enes Sexty. Simulating full QCD at nonzero density using the complex Langevinequation.
Phys. Lett. B , 729:108–111, 2014.[31] Szabolcs Borsanyi, Zoltan Fodor, Sandor D. Katz, Stefan Krieg, Claudia Ratti,and Kalman Szabo. Fluctuations of conserved charges at finite temperature fromlattice QCD.
JHEP , 01:138, 2012.[32] Sz. Borsanyi, G. Endrodi, Z. Fodor, S.D. Katz, S. Krieg, C. Ratti, and K.K.Szabo. QCD equation of state at nonzero chemical potential: continuum resultswith physical quark masses at order mu . JHEP , 08:053, 2012.[33] R. Bellwied, S. Borsanyi, Z. Fodor, S. D. Katz, A. Pasztor, C. Ratti, and K. K.Szabo. Fluctuations and correlations in high temperature QCD.
Phys. Rev. ,D92(11):114505, 2015.[34] H. T. Ding, Swagato Mukherjee, H. Ohno, P. Petreczky, and H. P. Schadler. Di-agonal and off-diagonal quark number susceptibilities at high temperatures.
Phys.Rev. D , 92(7):074043, 2015.[35] A. Bazavov et al. The QCD Equation of State to (cid:79) ( µ B ) from Lattice QCD. Phys.Rev. , D95(5):054504, 2017.[36] A. Bazavov et al. Chiral crossover in QCD at zero and non-zero chemical potentials.
Phys. Lett. , B795:15–21, 2019.[37] A. Bazavov et al. Skewness, kurtosis, and the fifth and sixth order cumulants ofnet baryon-number distributions from lattice QCD confront high-statistics STARdata.
Phys. Rev. D , 101(7):074502, 2020.[38] Claudio Bonati, Massimo D’Elia, Francesco Negro, Francesco Sanfilippo, andKevin Zambello. Curvature of the pseudocritical line in QCD: Taylor expansionmatches analytic continuation.
Phys. Rev. D , 98(5):054510, 2018.[39] G. Endrodi, Z. Fodor, S. D. Katz, and K. K. Szabo. The QCD phase diagram atnonzero quark density.
JHEP , 04:001, 2011.[40] Philippe de Forcrand and Owe Philipsen. The QCD phase diagram for smalldensities from imaginary chemical potential.
Nucl. Phys. , B642:290–306, 2002.[41] Massimo D’Elia and Maria-Paola Lombardo. Finite density QCD via imaginarychemical potential.
Phys. Rev. , D67:014505, 2003.[42] Massimo D’Elia and Francesco Sanfilippo. Thermodynamics of two flavor QCDfrom imaginary chemical potentials.
Phys. Rev. D , 80:014502, 2009.4843] Paolo Cea, Leonardo Cosmai, and Alessandro Papa. Critical line of 2+1 flavorQCD.
Phys. Rev. D , 89(7):074512, 2014.[44] Claudio Bonati, Philippe de Forcrand, Massimo D’Elia, Owe Philipsen, andFrancesco Sanfilippo. Chiral phase transition in two-flavor QCD from an imaginarychemical potential.
Phys. Rev. D , 90(7):074030, 2014.[45] Paolo Cea, Leonardo Cosmai, and Alessandro Papa. Critical line of 2+1 flavorQCD: Toward the continuum limit.
Phys. Rev. , D93(1):014507, 2016.[46] Claudio Bonati, Massimo D’Elia, Marco Mariti, Michele Mesiti, Francesco Negro,and Francesco Sanfilippo. Curvature of the chiral pseudocritical line in QCD:Continuum extrapolated results.
Phys. Rev. , D92(5):054503, 2015.[47] R. Bellwied, S. Borsanyi, Z. Fodor, J. Guenther, S. D. Katz, C. Ratti, and K. K.Szabo. The QCD phase diagram from analytic continuation.
Phys. Lett. , B751:559–564, 2015.[48] Massimo D’Elia, Giuseppe Gagliardi, and Francesco Sanfilippo. Higher order quarknumber fluctuations via imaginary chemical potentials in N f = 2 + 1 QCD. Phys.Rev. , D95(9):094503, 2017.[49] J. N. Guenther, R. Bellwied, S. Borsanyi, Z. Fodor, S. D. Katz, A. Pasztor,C. Ratti, and K. K. Szab´o. The QCD equation of state at finite density fromanalytical continuation.
Nucl. Phys. , A967:720–723, 2017.[50] Paolo Alba et al. Constraining the hadronic spectrum through QCD thermody-namics on the lattice.
Phys. Rev. D , 96(3):034517, 2017.[51] Volodymyr Vovchenko, Attila Pasztor, Zoltan Fodor, Sandor D. Katz, and HorstStoecker. Repulsive baryonic interactions and lattice QCD observables at imagi-nary chemical potential.
Phys. Lett. B , 775:71–78, 2017.[52] Szabolcs Borsanyi, Zoltan Fodor, Jana N. Guenther, Sandor K. Katz, Kalman K.Szabo, Attila Pasztor, Israel Portillo, and Claudia Ratti. Higher order fluctuationsand correlations of conserved charges from lattice QCD. 2018.[53] Szabolcs Borsanyi, Zoltan Fodor, Jana N. Guenther, Ruben Kara, Sandor D. Katz,Paolo Parotto, Attila Pasztor, Claudia Ratti, and Kalman K. Szabo. The QCDcrossover at finite chemical potential from lattice simulations. 2020.[54] Ren´e Bellwied, Szabolcs Bors´anyi, Zolt´an Fodor, Jana G¨unther, S´andor D. Katz,K´alm´an K. Szab´o, Claudia Ratti, and Attila Pasztor. Fluctuations and correlationsin finite temperature QCD.
PoS , ICHEP2016:369, 2016.[55] Attila P´asztor, Zsolt Sz´ep, and Gergely Mark´o. Apparent convergence of Pad \ ’eapproximants for the crossover line in finite density QCD. 10 2020.4956] Patrick Steinbrecher. The QCD crossover at zero and non-zero baryon densitiesfrom Lattice QCD. Nucl. Phys. A , 982:847–850, 2019.[57] Philipp Isserstedt, Michael Buballa, Christian S. Fischer, and Pascal J. Gunkel.Baryon number fluctuations in the QCD phase diagram from Dyson-Schwingerequations.
Phys. Rev. , D100(7):074011, 2019.[58] A. Andronic, P. Braun-Munzinger, and J. Stachel. Hadron production in centralnucleus-nucleus collisions at chemical freeze-out.
Nucl. Phys. , A772:167–199, 2006.[59] Francesco Becattini, Marcus Bleicher, Thorsten Kollegger, Tim Schuster, JanSteinheimer, and Reinhard Stock. Hadron Formation in Relativistic Nuclear Col-lisions and the QCD Phase Diagram.
Phys. Rev. Lett. , 111:082302, 2013.[60] Paolo Alba, Wanda Alberico, Rene Bellwied, Marcus Bluhm, Valentina Manto-vani Sarti, Marlene Nahrgang, and Claudia Ratti. Freeze-out conditions fromnet-proton and net-charge fluctuations at RHIC.
Phys. Lett. , B738:305–310, 2014.[61] V. Vovchenko, V. V. Begun, and M. I. Gorenstein. Hadron multiplicities and chem-ical freeze-out conditions in proton-proton and nucleus-nucleus collisions.
Phys.Rev. , C93(6):064906, 2016.[62] L. Adamczyk et al. Bulk Properties of the Medium Produced in Relativistic Heavy-Ion Collisions from the Beam Energy Scan Program.
Phys. Rev. , C96(4):044904,2017.[63] Y. Hatta and M.A. Stephanov. Proton number fluctuation as a signal of the QCDcritical endpoint.
Phys. Rev. Lett. , 91:102003, 2003. [Erratum: Phys.Rev.Lett. 91,129901 (2003)].[64] M.A. Stephanov. Non-Gaussian fluctuations near the QCD critical point.
Phys.Rev. Lett. , 102:032301, 2009.[65] B. Friman, F. Karsch, K. Redlich, and V. Skokov. Fluctuations as probe of theQCD phase transition and freeze-out in heavy ion collisions at LHC and RHIC.
Eur. Phys. J. C , 71:1694, 2011.[66] Misha A. Stephanov, K. Rajagopal, and Edward V. Shuryak. Event-by-event fluc-tuations in heavy ion collisions and the QCD critical point.
Phys. Rev. , D60:114028,1999.[67] M. Cheng et al. The QCD equation of state with almost physical quark masses.
Phys. Rev. , D77:014511, 2008.[68] A. Bazavov et al. Skewness and kurtosis of net baryon-number distributions atsmall values of the baryon chemical potential. 2017.[69] Frithjof Karsch. Determination of Freeze-out Conditions from Lattice QCD Cal-culations.
Central Eur. J. Phys. , 10:1234–1237, 2012.5070] A. Bazavov et al. Freeze-out Conditions in Heavy Ion Collisions from QCD Ther-modynamics.
Phys. Rev. Lett. , 109:192302, 2012.[71] S. Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti, and K. K. Szabo. Freeze-outparameters: lattice meets experiment.
Phys. Rev. Lett. , 111:062005, 2013.[72] S. Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti, and K. K. Szabo. Freeze-out parameters from electric charge and baryon number fluctuations: is thereconsistency?
Phys. Rev. Lett. , 113:052301, 2014.[73] Claudia Ratti. Lattice QCD and heavy ion collisions: a review of recent progress.
Rept. Prog. Phys. , 81(8):084301, 2018.[74] J. Adam et al. Net-proton number fluctuations and the Quantum Chromodynamicscritical point. 1 2020.[75] Toshihiro Nonaka. Measurement of the Sixth-Order Cumulant of Net-Proton Dis-tributions in Au+Au Collisions from the STAR Experiment. In , 2 2020.[76] Rene Bellwied, Szabolcs Borsanyi, Zoltan Fodor, Jana N. Guenther, JacquelynNoronha-Hostler, Paolo Parotto, Attila Pasztor, Claudia Ratti, and Jamie M.Stafford. Off-diagonal correlators of conserved charges from lattice QCD and howto relate them to experiment.
Phys. Rev. D , 101(3):034506, 2020.[77] V.V. Begun, Mark I. Gorenstein, M. Hauer, V.P. Konchakovski, and O.S. Zozulya.Multiplicity Fluctuations in Hadron-Resonance Gas.
Phys. Rev. C , 74:044903,2006.[78] Masakiyo Kitazawa and Masayuki Asakawa. Revealing baryon number fluctuationsfrom proton number fluctuations in relativistic heavy ion collisions.
Phys. Rev. C ,85:021901, 2012.[79] Masakiyo Kitazawa and Masayuki Asakawa. Relation between baryon numberfluctuations and experimentally observed proton number fluctuations in relativisticheavy ion collisions.
Phys. Rev. C , 86:024904, 2012. [Erratum: Phys.Rev.C 86,069902 (2012)].[80] Matteo Giordano and Attila P´asztor. Reliable estimation of the radius of conver-gence in finite density QCD.
Phys. Rev. D , 99(11):114510, 2019.[81] T.D. Lee and Chen-Ning Yang. Statistical theory of equations of state and phasetransitions. 2. Lattice gas and Ising model.
Phys. Rev. , 87:410–419, 1952.[82] A. Hasenfratz and D. Toussaint. Canonical ensembles and nonzero density quan-tum chromodynamics.
Nucl. Phys. B , 371:539–549, 1992.[83] Z. Fodor and S. D. Katz. Critical point of QCD at finite T and mu, lattice resultsfor physical quark masses.
JHEP , 04:050, 2004.5184] Julia Danzer and Christof Gattringer. Winding expansion techniques for latticeQCD with chemical potential.
Phys. Rev. D , 78:114506, 2008.[85] Andrei Alexandru and Urs Wenger. QCD at non-zero density and canonical par-tition functions with Wilson fermions.
Phys. Rev. D , 83:034502, 2011.[86] Matteo Giordano, Kornel Kapas, Sandor D. Katz, Daniel Nogradi, and AttilaPasztor. Radius of convergence in lattice QCD at finite µ B with rooted staggeredfermions. Phys. Rev. D , 101(7):074511, 2020.[87] Matteo Giordano, Kornel Kapas, Sandor D. Katz, Daniel Nogradi, and AttilaPasztor. New approach to lattice QCD at finite density; results for the critical endpoint on coarse lattices.
JHEP , 05:088, 2020.[88] Felipe Attanasio, Benjamin J¨ager, and Felix P.G. Ziegler. Complex Langevin andthe QCD phase diagram: Recent developments. 5 2020.[89] Jan Ambjorn and S.K. Yang. Numerical Problems in Applying the LangevinEquation to Complex Effective Actions.
Phys. Lett. B , 165:140, 1985.[90] John R. Klauder and Wesley P. Petersen. SPECTRUM OF CERTAIN NON-SELFADJOINT OPERATORS AND SOLUTIONS OF LANGEVIN EQUA-TIONS WITH COMPLEX DRIFT.
J. Stat. Phys. , 39:53–72, 1985.[91] H. Q. Lin and J. E. Hirsch. Monte carlo versus langevin methods for nonpositivedefinite weights.
Phys. Rev. B , 34:1964–1967, Aug 1986.[92] Jan Ambjorn, M. Flensburg, and C. Peterson. The Complex Langevin Equationand Monte Carlo Simulations of Actions With Static Charges.
Nucl. Phys. B ,275:375–397, 1986.[93] Gert Aarts, Lorenzo Bongiovanni, Erhard Seiler, Denes Sexty, and Ion-OlimpiuStamatescu. Controlling complex Langevin dynamics at finite density.
Eur. Phys.J. A , 49:89, 2013.[94] Juergen Berges and Denes Sexty. Real-time gauge theory simulations from stochas-tic quantization with optimized updating.
Nucl. Phys. B , 799:306–329, 2008.[95] Gert Aarts, Frank A. James, Erhard Seiler, and Ion-Olimpiu Stamatescu. Adaptivestepsize and instabilities in complex Langevin dynamics.
Phys. Lett. B , 687:154–159, 2010.[96] Felipe Attanasio and Benjamin J¨ager. Dynamical stabilisation of complex Langevinsimulations of QCD.
Eur. Phys. J. C , 79(1):16, 2019.[97] Jun Nishimura and Shinji Shimasaki. New Insights into the Problem with a Sin-gular Drift Term in the Complex Langevin Method.
Phys. Rev. D , 92(1):011501,2015. 5298] Gert Aarts, Erhard Seiler, Denes Sexty, and Ion-Olimpiu Stamatescu. ComplexLangevin dynamics and zeroes of the fermion determinant.
JHEP , 05:044, 2017.[Erratum: JHEP 01, 128 (2018)].[99] Keitaro Nagata, Jun Nishimura, and Shinji Shimasaki. Argument for justificationof the complex Langevin method and the condition for correct convergence.
Phys.Rev. D , 94(11):114515, 2016.[100] Keitaro Nagata, Jun Nishimura, and Shinji Shimasaki. Testing the criterion forcorrect convergence in the complex Langevin method.
JHEP , 05:004, 2018.[101] Manuel Scherzer, Erhard Seiler, D´enes Sexty, and Ion-Olimpiu Stamatescu. Com-plex Langevin and boundary terms.
Phys. Rev. D , 99(1):014512, 2019.[102] Shoichiro Tsutsui, Yuta Ito, Hideo Matsufuru, Jun Nishimura, Shinji Shimasaki,and Asato Tsuchiya. Exploring the QCD phase diagram at finite density by thecomplex Langevin method on a 16 ×
32 lattice.
PoS , LATTICE2019:151, 2019.[103] D´enes Sexty. Calculating the equation of state of dense quark-gluon plasma usingthe complex Langevin equation.
Phys. Rev. D , 100(7):074503, 2019.[104] M. Scherzer, D. Sexty, and I.-O. Stamatescu. Deconfinement transition line withthe complex Langevin equation up to µ/T ∼ Phys. Rev. D , 102(1):014515, 2020.[105] Yuta Ito, Hideo Matsufuru, Jun Nishimura, Shinji Shimasaki, Asato Tsuchiya,and Shoichiro Tsutsui. Exploring the phase diagram of finite density QCD at lowtemperature by the complex Langevin method.
PoS , LATTICE2018:146, 2018.[106] J.B. Kogut and D.K. Sinclair. Applying Complex Langevin Simulations to LatticeQCD at Finite Density.
Phys. Rev. D , 100(5):054512, 2019.[107] F. Karsch, E. Laermann, P. Petreczky, and S. Stickan. Infinite temperature limitof meson spectral functions calculated on the lattice.
Phys. Rev. D , 68:014504,2003.[108] Gert Aarts and Jose M. Martinez Resco. Continuum and lattice meson spectralfunctions at nonzero momentum and high temperature.
Nucl. Phys. B , 726:93–108,2005.[109] W.E. Caswell and G.P. Lepage. Effective Lagrangians for Bound State Problemsin QED, QCD, and Other Field Theories.
Phys. Lett. B , 167:437–442, 1986.[110] Nora Brambilla, Antonio Pineda, Joan Soto, and Antonio Vairo. PotentialNRQCD: An Effective theory for heavy quarkonium.
Nucl. Phys. B , 566:275,2000.[111] Alexei Bazavov, Nora Brambilla, Peter Petreczky, Antonio Vairo, and Jo-hannes Heinrich Weber. Color screening in (2+1)-flavor QCD.
Phys. Rev. D ,98(5):054511, 2018. 53112] Gert Aarts, Chris Allton, Tim Harris, Seyong Kim, Maria Paola Lombardo,Sin´ead M. Ryan, and Jon-Ivar Skullerud. The bottomonium spectrum at finitetemperature from N f = 2 + 1 lattice QCD. JHEP , 07:097, 2014.[113] G. Aarts, C. Allton, S. Kim, M.P. Lombardo, S.M. Ryan, and J.-I. Skullerud.Melting of P wave bottomonium states in the quark-gluon plasma from latticeNRQCD.
JHEP , 12:064, 2013.[114] Gert Aarts, Chris Allton, Seyong Kim, Maria Paola Lombardo, Mehmet B. Oktay,Sinead M. Ryan, D.K. Sinclair, and Jon-Ivar Skullerud. S wave bottomoniumstates moving in a quark-gluon plasma from lattice NRQCD.
JHEP , 03:084, 2013.[115] G. Aarts, C. Allton, S. Kim, M.P. Lombardo, M.B. Oktay, S.M. Ryan, D.K. Sin-clair, and J.I. Skullerud. What happens to the Upsilon and eta b in the quark-gluonplasma? Bottomonium spectral functions from lattice QCD.
JHEP , 11:103, 2011.[116] G. Aarts, S. Kim, M.P. Lombardo, M.B. Oktay, S.M. Ryan, D.K. Sinclair, andJ.-I. Skullerud. Bottomonium above deconfinement in lattice nonrelativistic QCD.
Phys. Rev. Lett. , 106:061602, 2011.[117] Seyong Kim, Peter Petreczky, and Alexander Rothkopf. Lattice NRQCD study ofS- and P-wave bottomonium states in a thermal medium with N f = 2 + 1 lightflavors. Phys. Rev. D , 91:054511, 2015.[118] Seyong Kim, Peter Petreczky, and Alexander Rothkopf. Quarkonium in-mediumproperties from realistic lattice NRQCD.
JHEP , 11:088, 2018.[119] David Lafferty and Alexander Rothkopf. Improved Gauss law model andin-medium heavy quarkonium at finite density and velocity.
Phys. Rev. D ,101(5):056010, 2020.[120] B. Alessandro et al. psi-prime production in Pb-Pb collisions at 158-GeV/nucleon.
Eur. Phys. J. C , 49:559–567, 2007.[121] Jaroslav Adam et al. Differential studies of inclusive J/ ψ and ψ (2S) production atforward rapidity in Pb-Pb collisions at √ s NN = 2 .
76 TeV.
JHEP , 05:179, 2016.[122] Vardan Khachatryan et al. Measurement of Prompt ψ (2 S ) → J/ψ
Yield Ratios inPb-Pb and p − p Collisions at √ s NN = 2.76 TeV. Phys. Rev. Lett. , 113(26):262301,2014.[123] Albert M Sirunyan et al. Relative Modification of Prompt ψ (2S) and J/ ψ Yieldsfrom pp to PbPb Collisions at √ s NN = 5 .
02 TeV.
Phys. Rev. Lett. , 118(16):162301,2017.[124] Anton Andronic, Peter Braun-Munzinger, Krzysztof Redlich, and Johanna Stachel.Decoding the phase structure of QCD via particle production at high energy.
Nature , 561(7723):321–330, 2018. 54125] Axel Drees. Relative Yields and Nuclear Modification of ψ ’ to J / ψ mesons inp+p, p( He)+A Collisions at √ s NN = 200 GeV , measured in PHENIX. Nucl.Part. Phys. Proc. , 289-290:417–420, 2017.[126] Yannis Burnier, Olaf Kaczmarek, and Alexander Rothkopf. Quarkonium at finitetemperature: Towards realistic phenomenology from first principles.
JHEP , 12:101,2015.[127] Yannis Burnier, Olaf Kaczmarek, and Alexander Rothkopf. Static quark-antiquarkpotential in the quark-gluon plasma from lattice QCD.
Phys. Rev. Lett. ,114(8):082001, 2015.[128] A. Bazavov et al. Nonperturbative QCD Simulations with 2+1 Flavors of ImprovedStaggered Quarks.
Rev. Mod. Phys. , 82:1349–1417, 2010.[129] Owe Philipsen. Constraining the QCD phase diagram at finite temperature anddensity.
PoS , LATTICE2019:273, 2019.[130] Andrea Pelissetto and Ettore Vicari. Relevance of the axial anomaly at the finite-temperature chiral transition in QCD.
Phys. Rev. D , 88(10):105018, 2013.[131] F Karsch, E Laermann, and Ch Schmidt. The Chiral critical point in three-flavorQCD.
Phys. Lett. B , 520:41–49, 2001.[132] Philippe de Forcrand and Owe Philipsen. The QCD phase diagram for threedegenerate flavors and small baryon density.
Nucl. Phys. B , 673:170–186, 2003.[133] Xiao-Yong Jin, Yoshinobu Kuramashi, Yoshifumi Nakamura, Shinji Takeda, andAkira Ukawa. Critical endpoint of the finite temperature phase transition for threeflavor QCD.
Phys. Rev. D , 91(1):014508, 2015.[134] A. Bazavov, H. T. Ding, P. Hegde, F. Karsch, E. Laermann, Swagato Mukherjee,P. Petreczky, and C. Schmidt. Chiral phase structure of three flavor QCD atvanishing baryon number density.
Phys. Rev. D , 95(7):074505, 2017.[135] Francesca Cuteri, Owe Philipsen, and Alessandro Sciarra. QCD chiral phase tran-sition from noninteger numbers of flavors.
Phys. Rev. D , 97(11):114511, 2018.[136] Owe Philipsen and Christopher Pinke. The N f = 2 QCD chiral phase transitionwith Wilson fermions at zero and imaginary chemical potential. Phys. Rev. D ,93(11):114507, 2016.[137] David Anthony Clarke, Olaf Kaczmarek, Frithjof Karsch, Anirban Lahiri, andMugdha Sarkar. Sensitivity of the Polyakov loop and related observables to chiralsymmetry restoration. 8 2020.[138] H.T. Ding et al. Chiral Phase Transition Temperature in ( 2+1 )-Flavor QCD.
Phys. Rev. Lett. , 123(6):062002, 2019.55139] Heng-Tong Ding, Prasad Hegde, Olaf Kaczmarek, Frithjof Karsch, Anirban Lahiri,Sheng-Tai Li, Swagato Mukherjee, and Peter Petreczky. Chiral phase transitionin (2 + 1)-flavor QCD.
PoS , LATTICE2018:171, 2019.[140] H.-T. Ding, S.-T. Li, Swagato Mukherjee, A. Tomiya, X.-D. Wang, and Y. Zhang.Correlated Dirac eigenvalues and axial anomaly in chiral symmetric QCD. 10 2020.[141] Yoshinobu Kuramashi, Yoshifumi Nakamura, Hiroshi Ohno, and Shinji Takeda.Nature of the phase transition for finite temperature N f = 3 QCD with nonpertur-batively O( a ) improved Wilson fermions at N t = 12. Phys. Rev. D , 101(5):054509,2020.[142] Owe Philipsen and Christopher Pinke. Nature of the Roberge-Weiss transition in N f = 2 QCD with Wilson fermions. Phys. Rev. D , 89(9):094504, 2014.[143] Philippe de Forcrand and Owe Philipsen. Constraining the QCD phase diagramby tricritical lines at imaginary chemical potential.
Phys. Rev. Lett. , 105:152001,2010.[144] Claudio Bonati, Guido Cossu, Massimo D’Elia, and Francesco Sanfilippo. TheRoberge-Weiss endpoint in N f = 2 QCD. Phys. Rev. D , 83:054505, 2011.[145] Philippe de Forcrand and Massimo D’Elia. Continuum limit and universality ofthe Columbia plot.
PoS , LATTICE2016:081, 2017.[146] F. Karsch, E. Laermann, and A. Peikert. Quark mass and flavor dependence ofthe QCD phase transition.
Nucl. Phys. B , 605:579–599, 2001.[147] Y. Iwasaki, K. Kanaya, S. Kaya, S. Sakai, and T. Yoshie. Finite temperaturetransitions in lattice QCD with Wilson quarks: Chiral transitions and the influenceof the strange quark.
Phys. Rev. D , 54:7010–7031, 1996.[148] Lukas Varnhorst. The N f =3 critical endpoint with smeared staggered quarks. PoS , LATTICE2014:193, 2015.[149] Philippe de Forcrand, Seyong Kim, and Owe Philipsen. A QCD chiral criticalpoint at small chemical potential: Is it there or not?
PoS , LATTICE2007:178,2007.[150] F. Karsch, C.R. Allton, S. Ejiri, S.J. Hands, O. Kaczmarek, E. Laermann, andC. Schmidt. Where is the chiral critical point in three flavor QCD?
Nucl. Phys. BProc. Suppl. , 129:614–616, 2004.[151] G. Endrodi, Z. Fodor, S.D. Katz, and K.K. Szabo. The Nature of the finite temper-ature QCD transition as a function of the quark masses.
PoS , LATTICE2007:182,2007. 56152] H.-T. Ding, A. Bazavov, P. Hegde, F. Karsch, S. Mukherjee, and P. Petreczky.Exploring phase diagram of N f = 3 QCD at µ = 0 with HISQ fermions. PoS ,LATTICE2011:191, 2011.[153] Francesca Cuteri, Owe Philipsen, Alena Sch¨on, and Alessandro Sciarra. The de-confinement critical point of lattice QCD with N f = 2 Wilson fermions. 9 2020.[154] Shinji Ejiri, Shota Itagaki, Ryo Iwami, Kazuyuki Kanaya, Masakiyo Kitazawa,Atsushi Kiyohara, Mizuki Shirogane, and Takashi Umeda. End point of the first-order phase transition of QCD in the heavy quark region by reweighting fromquenched QCD. Phys. Rev. D , 101(5):054505, 2020.[155] Dmitri E. Kharzeev, Larry D. McLerran, and Harmen J. Warringa. The Effectsof topological charge change in heavy ion collisions: ’Event by event P and CPviolation’.
Nucl. Phys. A , 803:227–253, 2008.[156] V. Skokov, A.Yu. Illarionov, and V. Toneev. Estimate of the magnetic field strengthin heavy-ion collisions.
Int. J. Mod. Phys. A , 24:5925–5932, 2009.[157] Wei-Tian Deng and Xu-Guang Huang. Event-by-event generation of electromag-netic fields in heavy-ion collisions.
Phys. Rev. C , 85:044907, 2012.[158] Massimo D’Elia, Swagato Mukherjee, and Francesco Sanfilippo. QCD Phase Tran-sition in a Strong Magnetic Background.
Phys. Rev. D , 82:051501, 2010.[159] Igor A. Shovkovy.
Magnetic Catalysis: A Review , volume 871, pages 13–49. 2013.[160] H.-T. Ding, S.-T. Li, A. Tomiya, X.-D. Wang, and Y. Zhang. Chiral properties of(2+1)-flavor QCD in strong magnetic fields at zero temperature. 8 2020.[161] Heng-Tong Ding, Christian Schmidt, Akio Tomiya, and Xiao-Dan Wang. Chiralphase structure of three flavor QCD in a background magnetic field. 6 2020.[162] G.S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S.D. Katz, S. Krieg, A. Schafer,and K.K. Szabo. The QCD phase diagram for external magnetic fields.
JHEP ,02:044, 2012.[163] G.S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S.D. Katz, and A. Schafer. QCDquark condensate in external magnetic fields.
Phys. Rev. D , 86:071502, 2012.[164] E. M. Ilgenfritz, M. Muller-Preussker, B. Petersson, and A. Schreiber. Magneticcatalysis (and inverse catalysis) at finite temperature in two-color lattice QCD.
Phys. Rev. D , 89(5):054512, 2014.[165] V.G. Bornyakov, P.V. Buividovich, N. Cundy, O.A. Kochetkov, and A. Sch¨afer.Deconfinement transition in two-flavor lattice QCD with dynamical overlapfermions in an external magnetic field.
Phys. Rev. D , 90(3):034501, 2014.57166] G.S. Bali, F. Bruckmann, G. Endr¨odi, S.D. Katz, and A. Sch¨afer. The QCDequation of state in background magnetic fields.
JHEP , 08:177, 2014.[167] Akio Tomiya, Heng-Tong Ding, Xiao-Dan Wang, Yu Zhang, Swagato Mukherjee,and Christian Schmidt. Phase structure of three flavor QCD in external magneticfields using HISQ fermions.
PoS , LATTICE2018:163, 2019.[168] Massimo D’Elia and Francesco Negro. Chiral Properties of Strong Interactions ina Magnetic Background.
Phys. Rev. D , 83:114028, 2011.[169] Jens O. Andersen, William R. Naylor, and Anders Tranberg. Phase diagram ofQCD in a magnetic field: A review.
Rev. Mod. Phys. , 88:025001, 2016.[170] Toru Kojo and Nan Su. The quark mass gap in a magnetic field.
Phys. Lett. B ,720:192–197, 2013.[171] Falk Bruckmann, Gergely Endrodi, and Tamas G. Kovacs. Inverse magnetic catal-ysis and the Polyakov loop.
JHEP , 04:112, 2013.[172] Kenji Fukushima and Yoshimasa Hidaka. Magnetic Catalysis Versus MagneticInhibition.
Phys. Rev. Lett. , 110(3):031601, 2013.[173] M. Ferreira, P. Costa, O. Louren¸co, T. Frederico, and C. Providˆencia. Inversemagnetic catalysis in the (2+1)-flavor Nambu-Jona-Lasinio and Polyakov-Nambu-Jona-Lasinio models.
Phys. Rev. D , 89(11):116011, 2014.[174] Lang Yu, Hao Liu, and Mei Huang. Spontaneous generation of local CP violationand inverse magnetic catalysis.
Phys. Rev. D , 90(7):074009, 2014.[175] Bo Feng, Defu Hou, Hai-cang Ren, and Ping-ping Wu. Bose-Einstein Condensa-tion of Bound Pairs of Relativistic Fermions in a Magnetic Field.
Phys. Rev. D ,93(8):085019, 2016.[176] Xiang Li, Wei-Jie Fu, and Yu-Xin Liu. Thermodynamics of 2+1 Flavor Polyakov-Loop Quark-Meson Model under External Magnetic Field.
Phys. Rev. D ,99(7):074029, 2019.[177] Shijun Mao. From inverse to delayed magnetic catalysis in a strong magnetic field.
Phys. Rev. D , 94(3):036007, 2016.[178] Umut G¨ursoy, Ioannis Iatrakis, Matti J¨arvinen, and Govert Nijs. Inverse MagneticCatalysis from improved Holographic QCD in the Veneziano limit.
JHEP , 03:053,2017.[179] Kun Xu, Jingyi Chao, and Mei Huang. Spin polarization inducing diamagnetism,inverse magnetic catalysis and saturation behavior of charged pion spectra. 7 2020.[180] Massimo D’Elia, Floriano Manigrasso, Francesco Negro, and Francesco Sanfilippo.QCD phase diagram in a magnetic background for different values of the pionmass.
Phys. Rev. D , 98(5):054509, 2018.58181] Gergely Endrodi, Matteo Giordano, Sandor D. Katz, T.G. Kov´acs, and FerencPittler. Magnetic catalysis and inverse catalysis for heavy pions.
JHEP , 07:007,2019.[182] Claudio Bonati, Massimo D’Elia, Marco Mariti, Michele Mesiti, Francesco Negro,Andrea Rucci, and Francesco Sanfilippo. Magnetic field effects on the static quarkpotential at zero and finite temperature.
Phys. Rev. D , 94(9):094007, 2016.[183] Murray Gell-Mann, R.J. Oakes, and B. Renner. Behavior of current divergencesunder SU(3) x SU(3).
Phys. Rev. , 175:2195–2199, 1968.[184] J. Gasser and H. Leutwyler. Chiral Perturbation Theory: Expansions in the Massof the Strange Quark.