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Like fluctuations, nondiagonal correlators of conserved charges provide a tool for the study of chemical freeze-out in heavy ion collisions. They can be calculated in thermal equilibrium using lattice simulations, and be connected to moments of event-by-event net-particle multiplicity distributions. We calculate them from continuum-extrapolated lattice simulations at

The study of the phase diagram of quantum chromodynamics (QCD) has been the object of intense effort from both theory and experiment in the last decades. Relativistic heavy ion collision experiments both at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) have been able to create the quark gluon plasma (QGP) in the laboratory, and explore the low-to-moderate baryon density region of the QCD phase diagram.

At low baryon density, the transition from a hadron gas to a deconfined QGP was shown by lattice QCD calculations to be a broad crossover

The structure of the QCD phase diagram cannot currently be theoretically calculated from first principles, as lattice calculations are hindered by the sign problem at finite density. Several methods have been utilized in order to expand the reach of lattice QCD to finite density, like full reweighting

We remark here that there are alternative approaches to lattice QCD for the thermodynamical description. Specific truncations of the Dyson-Schwinger equations allow the calculation of the crossover line and also to extract baryonic fluctuations

The confined, low-temperature regime of the theory is well described by the hadron resonance gas (HRG) model, which is able to reproduce the vast majority of lattice QCD results in this regime

Although the net number of individual particles may change after the chemical freeze-out through resonance decays, the net baryon number, strangeness and electric charge are conserved. Their event-by-event fluctuations are expected to correspond to a grand canonical ensemble. In general, when dealing with fluctuations in QCD, and in particular in relation to heavy ion collisions, it is important to relate fluctuations of such conserved charges and the event-by-event fluctuations of observed (hadronic) species. The former have been extensively studied with lattice simulations

Previous studies found that, for certain particle species, fluctuations are more sensitive to the freeze-out parameters than yields

Correlations between different conserved charges in QCD provide yet another possibility for the comparison of theory and experiment. They will likely receive further contribution from measurements in the future, with new species being analyzed and increased statistics allowing for better determination of moments of event-by-event distributions

In this manuscript, we present continuum-extrapolated lattice QCD results for all second-order nondiagonal correlators of conserved charges. We then identify the contribution of the single particle species to these correlators, distinguishing between measured and nonmeasured species. Finally, we identify a set of observables, which can serve as proxies to measure the conserved charge correlators. The manuscript is organized as follows. In Sec.

The lattice formulation of quantum chromodynamics opens a nonperturbative approach to the underlying quantum field theory in equilibrium. Its partition function belongs to a grand canonical ensemble, parametrized by the baryochemical potential

There is a conserved charge corresponding to each flavor of QCD. The grand canonical partition function can be then written in terms of quark number chemical potentials (

Susceptibilities are then defined as

It is straightforward to express the derivatives of

Such derivatives play an important role in experiment. In an ideal setup the mean of a conserved charge

The procedure to define the chemical potential on the lattice

In this work, we extend our previous results

These expectation values are naturally volume dependent. Their leading volume dependence can, however, be canceled by forming ratios. In

The gauge action is defined by the tree-level Symanzik improvement, and the fermion action is a one link staggered with four levels of stout smearing. The parameters of the discretization as well as the bare couplings and quark masses are given in

The charm quark is also included in our simulations, in order to account for its partial pressure at temperatures above 200 MeV, where it is no longer negligible

In this work we use the lattice sizes of

We show here the continuum extrapolated cross-correlators at zero chemical potential. In Fig.

Examples for the continuum extrapolation. We show the three cross-correlators on the lattices (from right to left):

The baryon-electric charge cross correlator from the lattice at finite lattice spacing and its continuum limit.

The

The baryon-strangeness cross correlator from the lattice at finite lattice spacing and its continuum limit.

The

The continuum-extrapolated results at

The electric charge-strangeness cross correlator from the lattice at finite lattice spacing and its continuum limit.

Since lattice QCD can be defined at finite values of the

There are several options to extract physics at finite densities, nevertheless. It seems natural to use algorithms that were designed to work on complex actions—both the complex Langevin equation

Instead, we use here the parameter domain that is available for mainstream lattice simulations. In fact, besides zero chemical potential, simulations at imaginary

A conceptually very similar method, the Taylor method, provides the extrapolation in terms of calculating higher derivatives with respect to

Since we relate the baryon-strangeness correlator to experimental observables later on we use

In most phenomenological lattice studies the chemical potentials are selected such that the strangeness vanishes for each set of

Comparison of two approaches to the finite density extrapolation of two observables. The Taylor result is truncated such that only the leading

The Taylor coefficients for correlators can be easily obtained by considering the higher derivatives with respect to

Whether we extract the required derivatives from a single simulation (one per temperature) at

In

In the plots we show results from a specific lattice

In Fig.

We are building on our earlier work in

The partitioning of the QCD pressure in sectors is very natural in the space of imaginary chemical potentials

Thus, for this work we considered the next-to-leading order of the sector expansion, including the

It is somewhat ambiguous how the

In Fig.

The magnitude of the various sector coefficients in the temperature region relevant for freeze-out studies. In the first panel we show the standard sectors on a logarithmic scale as published in our earlier work

The alert reader may ask why we do not include the

Now we can compare the results to the Taylor expansion. In Fig.

In conclusion, we consider only the chemical potential range where our two methods agree in the extrapolation. At present, our lattice data allow a continuum extrapolation from the sector method only, which we do using

Continuum extrapolation of the

The large error bars in comparison to the

The HRG model is based on the idea that a gas of interacting hadrons in their ground state can be well described by a gas of noninteracting hadrons and resonances. The partition function of the model can thus be written as a sum of ideal gas contributions of all known hadronic resonances

The temperature and the three chemical potentials are not independent, as the conditions in Eqs.

In this work we utilize the hadron list PDG2016+ from

In the HRG model the

The HRG model has the advantage, when comparing to experiment, of allowing for the inclusion of acceptance cuts and resonance decay feed-down, which cannot be taken into account in lattice QCD calculations.

The acceptance cuts on transverse momentum and rapidity (or pseudorapidity) can be easily taken into account in the phase space integrations via the change(s) of variables,

The rich information contained in the system created in a heavy ion collision about the correlations between conserved charges is eventually carried over to the final stages through hadronic species correlations and self-correlations. It is convenient, in the framework of the HRG model, to consider the hadronic species which are stable under strong interactions, as these are the observable states accessible to experiment. However, due to experimental limitations, charged particles and lighter particles are easier to measure, and so we cannot access every relevant hadron related to conserved charges. Thus, historically protons have served as a proxy for baryon number, kaons have served as a proxy for strangeness, and net electric charge is measured through

In our framework, we consider the following species, stable under strong interactions:

A few remarks are in order here. First of all, we refer to the listed species as commonly measured because, although some others are potentially measurable (especially the charged

It is straightforward to adapt the HRG model so that it is expressed in terms of stable hadronic states only. The sum over the whole hadronic spectrum is converted into a sum over both the whole hadronic spectrum and the list of states which are stable under strong interactions,

In light of the above considerations, it is useful to define the contribution to the conserved charges from final state stable hadrons. In the following, we adopt the convention where the net number of particles of species

With this definition, we can express conserved charges as

Using this decomposition, we can write as an example the

The result of this decomposition is that each of the correlators one can build between conserved charges is formed from the sum of many different particle-particle correlations. In particular, the sum of those correlators which entirely consist of observable species yields the measured part of a certain correlator, while its nonmeasured part consists of all other terms, which include at least one nonobservable species. In Fig.

Second-order correlators of the conserved charges

We notice that both the

The decomposition in Eq.

Breakdown of the different final state hadronic contributions to the cross-correlators of the conserved charges

Breakdown of the different final state hadronic contributions to the diagonal correlators of the conserved charges

A few features can be readily noticed. First, in all cases only a handful of the most sizable contributions account for the measured portion of the corresponding observable. As stated above, the

The case of

In the diagonal case, a similar picture appears. The

In Figs.

Breakdown of the different final state hadronic contributions to the cross-correlators of the conserved charges

Breakdown of the different final state hadronic contributions to the diagonal correlators of the conserved charges

Another important effect we have not addressed yet, which is present in experiment, is the isospin randomization

The distributions of protons and neutrons then factorize and the correlation between the two is erased. The average number of protons and neutrons, as well as antiprotons and antineutrons, and consequently the average net-proton and net-neutron number, are left unchanged by such reactions, but fluctuations are not. In particular, this results in an enhancement of both the net-proton and net-neutron variance, at the expense of the correlation between the two (note that the variance of net nucleon

Thus there is information lost through the process of randomization, and the original

The issue of having particles that cannot be detected poses the problem of a loss of conserved charges. Historically, the proxies for baryon number, electric charge, and strangeness have been the protons, the

We have seen in Figs.

Both in theory and experiment, it is customary to consider ratios of fluctuations, in order to eliminate, at least at leading order, the dependence on the system volume. For this reason, we focus on the ratios

Let us start considering the

In the upper panel of Fig.

The temperature dependence of the ratios

The case of

Now, we wish to consider the correlator

Temperature dependence of the ratio

From Fig.

Having discussed all three combinations of the off-diagonal cross-correlators we are lacking a good proxy for a correlator ratio involving only the light quarks. As a detour from the main line of the discussion we show that this is also a difficult task in the case of the diagonal correlators. Consider the ratio

Temperature dependence of the ratio

These last two proxies are also shown in Fig

We have seen in this section how to construct good proxies for ratios including both diagonal and cross-correlators of conserved charges. The proxies including strangeness make use of only a couple of hadronic observables, namely the variances

Since experimental measurements for moments of net-particle distributions are currently available both from the LHC and RHIC, it is interesting to analyze the behavior of the quantities we are studying also at finite values of the baryon chemical potential. In the left panel of Fig.

Behavior of the ratios

In Sec.

Finally, in the right panel of Fig.

The third row in Fig.

In the previous section we have considered the impact of including kinematic cuts on the proxies we have defined previously, by considering some exemplary cuts which were chosen to be the same for all particle species. However, experimental measurements exist for different species, and it is possible to test how the proxies we constructed compare to the experimental results, this time including the corresponding cuts on a species-by-species (or measurement-by-measurement) basis.

In Fig.

Behavior of the proxies

We see that the proxy (it is only one independent quantity as discussed above) works well also in comparison with available experimental data, when the considered freeze-out line is the one with a temperature

One more remark is in order: by comparing, e.g., the curves in Figs.

In this work, we first presented new continuum-extrapolated lattice QCD results for second-order nondiagonal correlators of conserved charges. While the continuum extrapolation is a straightforward task at

We performed an HRG-model-based study on the second-order correlators, both diagonal and nondiagonal. At

In order to compare either to lattice QCD results or experimental measurements, we focused on ratios of fluctuations, whose behavior can be reproduced through commonly measured hadronic observables, i.e., proxies.

In the following we summarize the findings for a ratio with each of the three possible cross-correlators of baryon (

The

Luckily, neither the isospin randomization nor the introduction of cuts on the kinematics had a significant effect on either

Thus, we have a ratio at hand that is available both from lattice simulations and for experimental measurement. The ratio

Finally, we compare our results to experiment. A direct use of lattice data in the experimental context would require the use of the same kinematic cuts for

These high temperatures for the chemical freeze-out motivates the use of lattice QCD in future studies since they fall at the limit of the validity of the HRG model.

This project was partly funded by the DFG Grant No. SFB/TR55 and also supported by the Hungarian National Research, Development and Innovation Office, NKFIH Grants No. KKP126769 and No. K113034. The project also received support from the BMBF Grant No. 05P18PXFCA. Parts of this work were supported by the National Science Foundation under Grant No. PHY-1654219 and by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, within the framework of the Beam Energy Scan Theory (BEST) Topical Collaboration. A. P. is supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and by the ÚNKP-19-4 New National Excellence Program of the Ministry of Innovation and Technology. The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. for funding this project by providing computing time on the GCS Supercomputer JUWELS and JURECA/Booster at Jülich Supercomputing Centre (JSC), and on SUPERMUC-NG at LRZ, Munich as well as on HAZELHEN at HLRS Stuttgart, Germany. C. R. also acknowledges the support from the Center of Advanced Computing and Data Systems at the University of Houston. J. N. H. acknowledges the support of the Alfred P. Sloan Foundation, support from the U.S. DOE Nuclear Science Grant No. de-sc0019175. R. B. acknowledges support from the U.S. DOE Nuclear Physics Grant No. DE-FG02-07ER41521. We acknowledge PRACE for awarding us access to Piz Daint hosted at CSCS, Switzerland.

In this appendix we give the results of the continuum extrapolations explained in Fig.

The continuum extrapolation of the cross-correlators at