Constraining the QCD phase diagram

. Lattice QCD and functional methods are making signiﬁcant progress in constraining the QCD phase diagram. As an important milestone, the chiral phase transition with massless u , d -quarks at zero density is now understood to be of second order for all strange quark masses, and a smooth crossover as soon as m u , d , 0. Together with information on ﬂuctuations and reﬁned reweighted simulations, this bounds a possible critical point to be at µ B / T > ⇠ 3. On the other hand, an approximately chiral-spin symmetric temperature window has been discovered above the chiral crossover, T ch < ⇠ T < ⇠ 3 T ch , with distinct correlator multiplet patterns and a pion spectral function suggesting resonance-like degrees of freedom, which dissolve graduallly with temperature.


Introduction
Knowledge of the QCD phase diagram is of great importance for the physics of heavy ion collisions and neutron stars.Theoretical predictions from lattice QCD are severely hampered by a fermion sign problem, which prohibits straightforward Monte Carlo simulations at finite baryon chemical potential.Nevertheless, methods working for µ B < ⇠ 3T as well as studies of the general parameter dependence of the chiral phase transition have reached a new level of maturity, providing phenomenologically relevant constraints on the location of a possible critical point.
The expected scenario for physical QCD is intimately connected to the situation in the chiral limit of massless u, d-quarks [1][2][3], as sketched in Fig. 1.For m u,d = 0 there is an exact chiral symmetry, whose breaking/restoration across T c (µ B ) must proceed by a non-analytic phase transition.If this transition is second-order at µ B = 0 and first-order at T = 0, as in several low-energy models, there must be a tricritical point where the order of the transition changes.On quite general grounds, a Z(2)-critical line emanates from a tricritical point in in the direction of the symmetry breaking field ( i.e. m u,d ), with a known tricritical exponent [4], and represents the critical endpoint at physical mass values.However, if nothing is known about the chiral limit, other possibilities are transition lines that are entirely first or entirely second order for m u,d = 0, and the situation at physical masses would be di↵erent.This illustrates the importance of the chiral limit to both constrain and understand the physical phase diagram.
2 The chiral transition at µ B = 0 In the chiral limit the quark determinant in the partition function diverges, prohibiting direct simulations and making its approach computationally expensive.For decades expectations have thus been based on an analysis of 3d sigma models as low energy e↵ective theories, augmented by a 't Hooft term for the U(1) A anomaly, whose renormalisation group flow based on the epsilon expansion [5] predicts the chiral phase transition to be first-order for N f 3. The case of N f = 2 was found to crucially depend on the anomalous U(1) A symmetry: If it remains broken at T c , the chiral transition should be second order in the O(4)universality class, whereas its e↵ective restoration would enlarge the symmetry and push the transition to first order.
Early QCD simulations on coarse lattices were consistent with the scenario shown in Fig. 2 (left): A firstorder region could be seen for N f = 3, whereas the smallest available masses showed a continuous crossover for N f = 2.However, the location of the Z(2)-boundary varies widely between di↵erent discretisations, indicating large cuto↵ e↵ects.The general pattern is for the first-order region to shrink when the lattice is made finer, while improved staggered actions see no trace of a first-order transition at all.For a more detailed discussion and list of references see [7].
Figure 2. The order of the QCD thermal transition as a function of the quark masses.Left: Scenario proposed in [5] and observed on coarse lattices.Right: Emerging continuum limit [6].The universality class in the three-flavour chiral limit is not yet known.From [6].
Recent investigations using the Highly Improved Staggered Quark (HISQ) action start at the N f = 2 + 1 physical point and then gradually reduce the light quark mass until m PS ⇡ 55 MeV [8].Fig. 3 (left) shows the chiral susceptibility normalised to its peak value at the physical pion mass.The peak location defines the pseudo-critical temperature T pc .However, the peak height stays finite and no sign of a first-order transition is detectable.Analytic fRG calculations in infinite volume [9] are compatible with this behaviour, Fig. 3 (left).Similar results are found using twisted mass Wilson fermions at quark masses at and above the physical point [10].Thus either a Z(2)critical point bounding a very narrow first-order region is approached, or a second-order transition in the chiral limit.
Eventually one would like to calculate the critical exponents characterising the approach to criticality, and thus determine the universality class from first principles.However, this requires exponential accuracy in the numerically expensive light mass regime and is not feasible.In practice the data are fitted to scaling formulae with exponents fixed to known values for the universality classes in question, such as the approach of the crossover temperature to the critical temperature T c in the chiral limit, The extrapolated chiral critical temperatures are 4 MeV twisted mass Wilson [10] , T c = 142 +0.5 0.5 MeV fRG methods [9] .
The variation between the possible sets of critical exponents is covered by the errors, so that an extrapolation makes sense even without definite knowledge of the true universality class.A first-order chiral transition for N f 2 [2,7] could recently be practically ruled out by investigating the e↵ects of the number of flavours, their masses and the lattice spacing [6] on the transition.The study exploits the fact that a change from a first-order to a second-order chiral transition necessarily passes through a tricritical point, such as in the scenario Fig. 2 (left).This implies tricritical scaling with known exponents for the Z(2) boundary line as it approaches the chiral limit, allowing for a controlled extrapolation.Rather than varying quark masses independently, ref. [6] continuously varies the number N f of degenerate quarks [12].A tricritical m tric s as in Fig. 2 (left) then translates into a tricritical 2 < N tric f < 3. On the lattice, such a tricritical point additionally depends on the lattice spacing, N tric f (a), which can be inverted to a tric (N f ).Fig. 3 (middle) shows the Z(2)-critical boundary line separating the light bare quark mass region featuring firstorder transitions from the crossover region as a function of lattice spacing for the unimproved staggered fermion action.In agreement with previous studies, the first-order region grows as more flavours are added, and it shrinks as the lattice is made finer.Those N f -theories with three available lattice spacings show tricritical scaling and a tricritical point in the lattice chiral limit, which is moving to the left as N f is increased.Note that the continuum limit is in the origin at (am, aT ) = (0, 0).This implies a tricritical point in the continuum limit to be beyond Conversely, the first-order regions observed in these simulations are not connected to the continuum limit and must be regarded as lattice artefacts.A powerful check on this finding is provided by N f = 3 O(a)-improved Wilson fermions [11].Plotting those data in terms of the appropriate scaling variable, perfect tricritical scaling is observed, Fig. 3 (right), so that the continuum transition is of second order as in the staggered case.Two further discretisations with N f = 3, HISQ fermions [13] and domain wall fermions [14], do not see any phase transition at the smallest available quark masses, and thus are fully consistetnt with those findings.A recent fRG study of 3d sigma models including a 6 term [15] and numerical bootstrap methods applied to U(m)⇥U(n) [16] models are also compatible with a second-order chiral transition.Thus, the Columbia plot in the continuum looks as in Fig. 2 (right), with a second-order chiral transition for m u,d = 0 and any value of m s , and crossover as soon as m u,d , 0.
From the physical point towards the chiral limit  Left: Pseudo-critical temperature of the crossover defined by the chiral susceptibility , the inflection point of the chiral condensate or an additively renormalised chiral condensate 3 , for N f " 2 `1 `1 twisted mass Wilson fermions close to the continuum.Lines represent chiral extrapolations according to the Op4q second-order or finite critical Zp2q-mass scenario.From [30].Right: Columbia plot expressed in , -masses in units of the Wilson flow parameter t 0 .Critical points have been determined using an Opaq-improved Wilson action.The first-order region includes the physical point on coarse lattices, but shrinks drastically as N is increased.From [31].
employing either Op4q exponents or Zp2q-exponents and a critical pseudo-scalar mass up to m " 100 MeV.Again, it is not possible to distunghuish between these scenarios.As in the previous case, the extrapolated value of the critical temperature in the chiral limit is therefore robust under changes of the critical exponents and quoted as in remarkable agreement with the staggered result.is the al pion mass, see Eq. ( 5).The lattice QCD data has been taken from Refs.[29,62].mass.For the his ratio in our out a factor of l) found in lowexample, 28 rk-meson (QM) stead find 10 ,  4) value.However, we observe that it is consistent within fit errors with the value for p which we obtained from our analysis of the light-quark susceptibility.Overall, we therefore cautiously conclude that QCD is not within the scaling regime for the range of pion masses considered here, providing us with m 30 MeV as a conservative estimate for the upper bound of this regime.An actual determination of the size of the scaling regime is beyond the scope of present work as it requires to study very small pion masses.
In analogy to the definition ( 8), we can also define the relative dependence D (l,s) (m ) of the pseudocritical temperature on the pion mass in case of the reduced susceptibility.For m = 140 MeV, we then find that this quantity is only slightly smaller than the corresponding quantity associated with the light-quark susceptibility.
In Fig. 3 (right panel), we finally compare our fRG results for the reduced susceptibility to very recent results from the HotQCD collaboration [29].We observe excellent agreement between the results from the two approaches for pion masses m 100 MeV.The deviations of the results from the two approaches for smaller pion masses may at least partially be attributed to cuto artefacts in the lattice data.Note that cuto e ects are expected to shift the maxima to smaller temperatures.We refer to Ref. [18] for a respective discussion.
It is also worthwhile to compare the peak positions of the reduced susceptibilities extracted from the lattice QCD data with those from our fRG study, see Tab.I and Fig. 3 (left panel).As discussed above, the peak position can be used to define a pseudocritical temperature.For the presently available pion masses on  [8] and from fRG calculations [9].From [9].The Z(2)-critical line separating first-order transitions (below) from crossover (above), for unimproved staggered fermions [6] (middle) and O(a)-improved Wilson fermions [11] (right), with tricritical scaling fits to both.From [6].

The search for a critical point
Lattice searches for a critical point are based on indirect methods to extract information about the phase structure for small baryon density, µ B /T < ⇠ 3: (i) Reweighting [17], (ii) Taylor expansion in µ/T [18] and (iii) analytic continuation from imaginary chemical potential, for which there is no sign problem [19,20].When the QCD pressure is expressed as a series in baryon chemical potential, , the Taylor coe cients are the baryon number fluctuations evaluated at zero density, which can also be computed by fitting to untruncated results at imaginary µ B .This permits full control of the systematics between (ii) and (iii).These coe cients are presently known up to 2n = 8.Quite generally power series are limited by their radius of convergence, corresponding to the nearest singularity of the full function (relative to the expansion point) in the complex variable.If such a singularity is located on the real axis, it signals a phase transition.
In [21] a cluster expansion model (CEM) was developed, with all expansion coe cients recursively related to the lower ones, and the first two are fixed by lattice data.As in the virial expansion, such a recursion is possible as long as the system is su ciently dilute and dominated by two-body interactions.The model is quantitatively consistent with all known lattice pressure coe cients, as e.g. in Fig. 4 (left), and since its coe cients are known to all orders, it allows a controlled extraction of the radius of convergence.The only phase transition predicted in this way is the known Roberge-Weiss transition in the direction of imaginary chemical potential µ B = i⇡T [22], which implies that any phase transition in the real direction is further away than that.While this is just a model, it tells us that nothing in the available lattice fluctuation data enforces singular behaviour.
Another option is to use the coe cients calculated directly from the lattice.From these one can construct Padéapproximants, which are rational functions (i.e.infinite order in the expansion variable) whose first coe cients agree with the explicitly computed ones.These approximants can be further constrained by simulation results for the full pressure at imaginary chemical potential.Their singularities are then taken as estimates for the singularities of the full function.Locations of the Lee-Yang edge singularities (indicating a branch point in the pressure) extracted in this way are shown in Fig. 4 (middle), based on HISQ fermion data [23].So far all singularities are at complex values of the chemical potential, but note the closing in toward the real axis as temprerature is decreased.This bounds a true phase transition to lower temperatures than those for which coe cients are currently available.
Taylor expansions are avoided by using reweighting techniques in Monte Carlo simulations.Recent new calculations with stout smeared staggered fermions [24], using techniques considerably refined compared to earlier ones, evaluate the renormalised chiral condensate, as shown in Fig. 4 (right).The reweighted real µ B simulations are fully compatible with the analytic continuation from imaginary µ B simulations, but with smaller errors.
Neither method shows a sign of non-analyticity so far.In summary, lattice information on physical QCD in equilibrium at µ B /T  3 is increasing and increasingly controlled.Table 1 collects the current bounds on the location of a critical endpoint resulting from these analyses.These are also consistent with the critical endpoint candidates found in the most recent truncations of Dyson-Schwinger equations [26] and their combination with functional renormalisation group methods [27], which predict (µ B /T ) cep ⇡ 5.6.8 , calculated within CEMof Wuppertal-Budapest [20] and HotQCD [18,19] collaborations are shown by the blue and green ibilities eptibilities B k = @ k 1 (⇢B/T 3 )/@(µB/T ) k 1 in the CEM read  [20] and HotQCD collaborations [18,19].The CEM al-Budapest data [11] for b1(T ) and b2(T ) as an input and are therefore labeled results are in quantitative agreement with the lattice data for B 2 and B 4 / B 2 .ith the lattice data for B 6 / B 2 and B 8 , although these data are still preliminary ne interesting qualitative feature is the dip in the temperature dependence of s negative.It was interpreted as a possible signature of chiral criticality [21].o present in CEM (see red stars in Fig. 2c), i.e. in a model which has no critical egative dip in B 6 / B 2 cannot be considered as an unambiguous signal of chiral ier coe cients b1 and b2 from susceptibilities ptibilities at a given temperature are determined in the CEM by two parameters e cients b1 and b2.One can now consider a reverse prescription -assuming atz one can extract the values of b1 and b2 at a given temperature from two f baryon number susceptibilities by reversing Eq. ( 6).We demonstrate this CD data of the HotQCD collaboration for B 2 and B 4 / B 2 .The temperature 2 coe cients, reconstructed from the HotQCD collaboration's lattice data on is shown in Fig. 3 by the green symbols.The extracted values agree rather data of the Wuppertal-Budapest collaboration, shown in Fig. 3 by the blue

Lattice results for fluctuations + resummations
Physics A 00 (2018) 1-4 3 2 , and (d) B 8 , calculated within CEM-HotQCD [18,19] collaborations are shown by the blue and green 3 )/@(µB/T ) k 1 in the CEM read , although these data are still preliminary feature is the dip in the temperature dependence of ted as a possible signature of chiral criticality [21].stars in Fig. 2c), i.e. in a model which has no critical ot be considered as an unambiguous signal of chiral rom susceptibilities rature are determined in the CEM by two parameters can now consider a reverse prescription -assuming lues of b1 and b2 at a given temperature from two bilities by reversing Eq. ( 6).We demonstrate this collaboration for B 2 and B 4 / B 2 .The temperature d from the HotQCD collaboration's lattice data on green symbols.The extracted values agree rather dapest collaboration, shown in Fig. 3 Expressing the relation given in Eq. 28 in terms of the mulants ¯ B,n 0 entering the Taylor series for the presre, Eq. 7, we have in the region of complex poles, The positions of the poles in the complex μB -plane e shown in Fig. 6.Only the two poles in the region e(μ B ) 0 are shown.With decreasing temperature the poles move closer to the real axis as c 8,2 approaches c + 8,2 , i.e. c,4 = 0 for c 8,2 = c + 8,2 .Furthermore, it is clear from Eq. 29 that c,4 and r c,4 are correlated, which leads to the orientation of the 1-error ellipse in the complex µ B,c plane arising from the errors on c 6,2 and c 8,2 , which are assumed to given by independent Gaussian distributions of the variables c 6,2 and c 8,2 .
In Fig. 7 we show as symbols and bands, respectively, the distance of poles of the [2,2] and [4,4] Padé approximants from the origin as function of temperature.The bands shown in Fig. 7 have been obtained by using the spline interpolations of ¯ B,6 0 and ¯ B,8 0 on N = 8 lattices and the continuum extrapolated results for ¯ B,2 0 and ¯ B,4 0 , shown in Fig. 1, respectively.As can be seen the two estimators yield a similar magnitude for r c,2 and r c,4 .Their location in the complex µ B -plane, however, is quite di erent.While the poles of the [2,2] Padé are always on the real axis, the poles of the [4,4] Padé are in the complex plane in the entire interval 135 MeV  T  165 MeV.
For 135 MeV  T  165 MeV we find that the poles of the [4,4] Padé appear at a distance from the origin corresponding to |μ B |> 2.5 at T 135 MeV and rises to values larger than |μ B |> 3 for T > T pc .This also are the best estimates for a temperature dependent bound on the radius of convergence of the Taylor series for the pressure, based on the Mercer-Roberts estimator.The information extracted from the [4,4] Padé approximants on the location of poles in the analytic function representing the pressure as function of a complex valued chemical potential μB thus seems to be consistent with the good convergence properties of the Taylor series itself.FIG. 7. Magnitude of poles nearest to the origin obtained from the [2,2] (squares and circles) and [4,4] (bands) Padé approximants for Taylor expansions at µQ = µS = 0 and for strangeness neutral, isospin symmetric media, respectively.

Poles of the
Expressing the relation given in Eq. 28 in terms of the cumulants ¯ B,n 0 entering the Taylor series for the pressure, Eq. 7, we have in the region of complex poles, The positions of the poles in the complex μB-plane are shown in Fig. 6.Only the two poles in the region Re(μB) 0 are shown.With decreasing temperature the poles move closer to the real axis as c8,2 approaches c + 8,2 , i.e. c,4 = 0 for c8,2 = c + 8,2 .Furthermore, it is clear from Eq. 29 that c,4 and rc,4 are correlated, which leads to the orientation of the 1-error ellipse in the complex µB,c plane arising from the errors on c6,2 and c8,2, which are assumed to given by independent Gaussian distributions of the variables c6,2 and c8,2.
In Fig. 7 we show as symbols and bands, respectively, the distance of poles of the [2,2] and [4,4] Padé approximants from the origin as function of temperature.The bands shown in Fig. 7 have been obtained by using the spline interpolations of ¯ B,6 0 and ¯ B,8 0 on N = 8 lattices and the continuum extrapolated results for ¯ B,2 0 and ¯ B,4 0 , shown in Fig. 1, respectively.As can be seen the two estimators yield a similar magnitude for rc,2 and rc,4.Their location in the complex µB-plane, however, is quite di erent.While the poles of the [2,2] Padé are always on the real axis, the poles of the [4,4] Padé are in the complex plane in the entire interval 135 MeV  T  165 MeV.
For 135 MeV  T  165 MeV we find that the poles of the [4,4] Padé appear at a distance from the origin corresponding to |μB|> 2.5 at T 135 MeV and rises to values larger than |μB|> 3 for T > Tpc.This also are the best estimates for a temperature dependent bound on the radius of convergence of the Taylor series for the pressure, based on the Mercer-Roberts estimator.The information extracted from the [4,4] Padé approximants on the location of poles in the analytic function representing the pressure as function of a complex valued chemical potential μB thus seems to be consistent with the good convergence properties of the Taylor series itself.
r coefficients are the baryon number fluctuations evaluated at zero density, which be computed by fitting to untruncated results at imaginary µ B .This permits full f the systematics between (ii) and (iii).These coefficients are presently known up to n N " 16 lattices, Figure 13   From [25].Middle: Lee-Yang zeros indicating the radius of convergence of the pressure series in complex chemical potential.From [23].Right: Renormalised chiral condensate as a function of real and imaginary chemical potential.From [24].

Emergent chiral spin symmetry
While a critical point remains elusive on the lattice so far, an emergent approximate chiral spin symmetry has been discovered above the chiral crossover at µ B = 0.A S U(2) CS chiral spin transformation acts on Dirac fields as Here k = 1, . . . 4 can be any of the euclidean gamma matrices.From the generators it is apparent that S U(2) CS U(1) A .When combined with ordinary vectorial isospin transformations, S U(2) CS ⌦S U(2) V can be embedded into the larger S U(4), which contains the usual chiral symmetry of the massless QCD Lagrangian, S U(4) S U(2) L ⇥ S U(2) R ⇥ U(1) A .The QCD Lagrangian is not invariant under chiral spin transformations.A thermal medium implies a preferred Lorentz frame, and the massless quark action can be decomposed as Explicit calculation shows the colour-electric part of the quark-gluon interaction to be CS-and S U(4)-invariant, while the colour-magnetic interaction and kinetic terms (and thus the free Dirac action) are not.Hence, chiral spin symmetry is never exact in physical QCD, but its approximate realisation is possible if the colour-electric quarkgluon interaction dominates the quantum e↵ective action in some dynamical range, which would then be strongly coupled.
On the lattice, symmetries become apparent in degeneracy patterns of euclidean correlation functions, with some appropriate Dirac matrix.Information about all excitations in each quantum number channel is carried by the spectral functions ⇢ (!, p), For an isotropic system in equilibrium, it is su cient to probe the spatial and temporal correlators averaged over the orthogonal directions, Numerical results for spatial J = 0, 1 meson correlators from N f = 2 JLQCD domain wall fermions with physical quark masses, good chiral symmetry and lattice spacings < 0.1 fm [28] are shown in Fig. 5. Three multiplets of spatial correlators, E 1,2,3 , at di↵erent temperatures are seen.Of these, E 1 is due to U(1) A restoration whereas E 3 requires the full chiral symmetry.Both multiplets are expected above the chiral crossover.Not expected is the multiplet E 2 , which does not correspond to a representation of chiral symmetry, but to one of the larger S U(4).For the representations and associated meson states, see [29,30].Appearance of E 2 demonstrates the dynamical emergence of chiral spin symmetry in this regime.As temperature is further increased, E 2 gradually disappears as a separate multiplet and only those belonging to the expected chiral symmetry survive.Similar findings are reported from temporal correlators at T = 220 MeV and the same lattice spacing [31].Recently, the same patterns were observed with the quark content increased to N f = 2 + 1 + 1 QCD, including physical strange and charm quarks [32].
A related observable which is sensitive to the entry and exit of the CS-symmetric temperature range are screening masses [33].These correspond to the exponential decay of the large-separation spatial correlators in (8), C s (z) ⇠ exp( m scr z) for z ! 1.While not directly accessible experimentally, they can be readily evaluated non-perturbatively and perturbatively.Around the chiral crossover temperature, the screening masses show the expected degeneracy due to chiral symmetry.However, the temperature dependence predicted by pertubation theory is only attained for T > ⇠ 500 MeV.This can be understood by the approximate chiral spin symmetry in between, which perturbation theory about free quarks cannot reproduce.
If there is a chiral-spin symmetric band at zero density, it must necessarily continue to finite baryon density because the µ B -term respects that symmetry.From the known behaviour of screening masses with µ B one  4) chiral spin symmetry, at temperatures above the crossover.At large temperatures, these reduce to the multiplets of the ordinary chiral symmetry.From [28].
infers that the chiral-spin symmetric band curves downwards, as in the possible phase diagram Fig. 6 (left).In the absence of lattice data the continuation to higher densities and lower temperatures is speculative, of course, but baryon parity doublets and quarkyonic matter both are CSsymmetric to populate such a band [33].
Information about the nature of the e↵ective degrees of freedom in di↵erent regimes is encoded in the spectral functions, Eq. ( 8).Unfortunately, their extraction from discrete sets of lattice correlator data represents an illposed inversion problem.Recently, a new method was attempted [34], which applies to stable scalar particles in a heat bath, i.e. to the pion in the case of QCD, which allows to circumvent the integral inversion.
The method exploits locality of quantum field theories to ensure a representation of the spectral function [35,36], In an isotropic medium the spatial correlators and the spectral density are then related by [34] C s PS (z) = For temperatures below the threshold to the scattering states the first term in Eq. ( 10) should dominate.Once the continuum part is neglected, the calculation is straightforward.First, spatial pion correlators from [28] are fitted by the sum of two exponentials representing the ⇡, ⇡ ⇤ .This provides the D m, (|x|) = ↵ ⇡,⇡ ⇤ exp( ⇡,⇡ ⇤ |x|), from which the spectral function can be reconstructed using Eqs.(9,10) and the known vacuum masses m ⇡ , m ⇡ ⇤ .The result is shown in Fig. 6 (middle) and, as a non-trivial check, correctly predicts the temporal lattice correlator [31] for ⌧ > m 1 ⇡ ⇤ , Fig. 6 (right).The spectral function shows resonance-like peaks for both the pion and its first excitation.As the temperature increases, the peaks widen and gradually disappear, consistent with sequential hadron melting, but at temperatures significantly above the chiral crossover.This suggests non-perturbative, hadron-like excitations within the approximately chiral-spin symmetric temperature range.

Conclusions
The last few years have seen remarkable progress in the determination of the QCD phase structure.A major milestone is the understanding of the chiral transition at zero density in the massless limit, which is nearly completed.The transition temperature is known fairly accurately and there is strong evidence for the transition to be of second order for all N f 2 [2,7].Together with data on baryon number fluctuations, this constrains the location of a critical point to (µ B /T ) cep > 3 and T cep < 132 MeV.
A new development is the discovery of an approximate chiral spin symmetry, which emerges dynamically in a temperature band above the chiral crossover, T pc < ⇠ T < ⇠ 3T pc .It can be identified in the multiplet structure of correlation functions and a↵ects associated observables like screening masses and spectral functions.Together these suggest a regime with chiral symmetry restored but hadron-like degrees of freedom.It would be most interesting to investigate if and how this a↵ects experimental observables in heavy ion collisions. .Left: a possible QCD phase diagram with a chiral-spin symmetric band.From [33].Middle: Pion spectral function extracted from spatial lattice correlators.Right: Temporal correlator predicted by that spectral function, compared to lattice data.From [34].

Figure 1 .
Figure 1.Connection of the putative QCD phase diagram for physical light quark masses to the chiral limit in the front plane.

Figure 3 .
Figure 3. Left: Pseudo-critical temperature of the crossover defined by the chiral susceptibility , the inflection point of the chiral condensateor an additively renormalised chiral condensate 3 , for N f " 2 `1 `1 twisted mass Wilson fermions close to the continuum.Lines represent chiral extrapolations according to the Op4q second-order or finite critical Zp2q-mass scenario.From[30].Right: Columbia plot expressed in , -masses in units of the Wilson flow parameter t 0 .Critical points have been determined using an Opaq-improved Wilson action.The first-order region includes the physical point on coarse lattices, but shrinks drastically as N is increased.From[31].

129Fig. 3 (
Fig. 3 (right) shows an investigation of sections of the chiral critical line using Opaq

8
fRG results for the pseudocritical temperature as a function of the pion mass to .The various dashed lines represent fits to the numerical data, see main text for erature Tc have been obtained from an extrapolation of the fits to m 0. The rapolated results for the chiral critical temperature obtained from a definition of not involve the peak position of the susceptibility, see main text for details.Right educed condensate as a function of the temperature.The normalisation ¯ (l,s) M . Thus, the eement with the ht-quark suscepthe exponent p, m the expected O(

= 4 201IG. 6 .
edge (LYE) singularities in the complex -plane B -a new method to detect the QCD critical point ?ielefeld-Parma Collaboration: K. Zambello, S. Singh, MeV < T < 145 MeV • Robust identification of LYE from analytic continuation via multi-point Padé approximation of the net-baryon density • Find Z(2) scaling close to the RW-transition and a candidate chiral LYE, preliminary results: 2101.02254• Radius of convergence is limited by LYE • Advantage: no regular part involved in the analysis, the determination of non-universal parameter thus more precise LYE in the complex -plane ( ) Location of poles nearest to the origin obtained from the [4,4] Padé approximants in the complex μB-plane.Only poles ith Re(µB) > 0 are shown.Shown are results the case µQ = µS = 0 (left) and the strangeness neutral, isospin symmetric se (right).IG. 7. Magnitude of poles nearest to the origin obtained om the[2,2] (squares and circles) and[4,4] (bands) Padé proximants for Taylor expansions at µQ = µS = 0 and for rangeness neutral, isospin symmetric media, respectively.

5 B 2 [
on of Taylor series using (standard) Padéts [n,4]-Pade are identical to the ing Mercer-Roberts approximation of the nvergence (if poles are complex) bound for QCD critical point: served temperature scaling of the poles does not resemble universal scaling f approximation not sufficient?y from scaling region?aper: the EoS at non-zero , well controlled ressure and number density for and , respectively -consistent with .< 125 MeV, B /T > 2.Hot-QCD], arXiv:2202.09184[hep-lat] ation which can be controlled as long as µ{T " 1: (i) Reweighting [79], (ii) Taylor n in µ{T [80] and (iii) analytic continuation from imaginary chemical potential [63,64].QCD pressure is expressed as a series in baryon chemical potential, (left), and in principle also observable experimentally.iew of the equation of state relating to heavy ion phenomenology, see [81,82].Note this low density regime appears to be accessible by complex Langevin simulations recourse to series expansions, albeit not yet for physical quark masses [83].This additional cross check between different methods.search for radius of convergence uster expansion model, infinite order, scribes all available lattice data [Vovchenko et al.PRD 18] < l a t e x i t s h a 1 _ b a s e 6 4 = " N I L / 3 h 7 5 D a / J 6 w u 9 5 y D

From 8 B 2 (Figure 4 .
Figure 4. Left: Baryon number fluctuations from the lattice in comparison with the CEM model.From[25].Middle: Lee-Yang zeros indicating the radius of convergence of the pressure series in complex chemical potential.From[23].Right: Renormalised chiral condensate as a function of real and imaginary chemical potential.From[24].

3 Figure 5 .
Figure 5. Spatial correlation functions with domain wall fermions show distinct E 1 , E 2 , E 3 multiplets of the approximate S U(4) chiral spin symmetry, at temperatures above the crossover.At large temperatures, these reduce to the multiplets of the ordinary chiral symmetry.From[28].
, s) , with = 1/T , the thermal spectral density e D (u, s), and the standard Källen-Lehmann vacuum representation arising as T !0. For stable massive particles the analytic vacuum structure of the spectral density is preserved in the absence of a true phase transition.The authors therefore propose an ansatz with separable particle and scattering contributions, e D (u, s) = e D m, (u) (s m 2 ) + e D c, (u, s) .

1 ]Figure 6
Figure 6.Left: a possible QCD phase diagram with a chiral-spin symmetric band.From[33].Middle: Pion spectral function extracted from spatial lattice correlators.Right: Temporal correlator predicted by that spectral function, compared to lattice data.From[34].
) ber susceptibilities at µB = 0 have recently been computed in lattice QCD[16,n with these lattice data can test the predictive power of the CEM.