Hadron thermodynamics from imaginary chemical potentials

We use 4stout improved staggered lattice data at imaginary chemical potentials to calculate fugacity expansion coefficients in finite temperature QCD. We discuss the phenomenological interpretation of our results within the hadron resonance gas (HRG) model, and the hints they give us about the hadron spectrum. We also discuss features of the higher order coefficients that are not captured by the HRG. This conference contribution is based on our recent papers [1, 2].,


Introduction
The vast majority of lattice results for QCD thermodynamics can be described, in the hadronic phase, by a non-interacting gas of hadrons and resonances which includes the measured hadronic spectrum up to a certain mass cut-off. This is known as the Hadron Resonance Gas (HRG) model [3,4]. The main ingredient of such a model being the hadron spectrum. It has been proposed recently to use the precise lattice QCD results on specific observables, and their possible discrepancy with the HRG model predictions, to infer the existence of higher mass states [5]. We will start this conference contribution by describing how we went further along this direction in our recent paper [1], where we separate thermodynamic contributions from different strageness and baryon number sectors, and pinpoint which chanels have a significant number of states missing.
It is also known, that near the cross-over temperature, the ideal HRG description no longer describes the thermodynamics correctly. It was pointed out recently however, that the behavior of lattice observables in the crossover region, particularly of correlations and fluctuations of conserved charges, are very sensitive to the modeling of baryonic interactions [6,7]. In many cases including interaction between hadrons changes even the qualitative features of some observables. This is of great interest, since HRG type model calculations of conserved charge fluctuations are often used to extract freeze-out parameters of heavy ion collisions [8]. Since the freeze-out temperature is close to the cross-over temperature such a sensitivity means that the extracted freeze-out parameters from such a model will depend on the modeling of the hadronic interactions. My second topic in this conference contribution will be about using lattice calculations to constrain models with repulsive interactions between baryons. This is based on our recent paper [2].
For both topics, hadron spectrum and hadronic repulsion, our discussion will be based on the very simple fact that fugacity expansion parameters at a finite real chemical potential become Fourier coefficients at imaginary chemical potential.

Lattice Action
We use the 4stout improved staggered action, introduced in [9], for both calculations. We carry out simulations at several different values of purely imaginary chemical potentials [10,11]. For the study of the hadron spectrum and the strangeness sectors, we use configurations both at µ B = µ S = 0 and Im µ S > 0. In this case we use N t = 10, 12, 16 lattices for continuum extrapolation. For studying the repulsive hadronic interactions, we use lattices with Im µ B > 0. For this project we did not carry out a continuum extrapolation, we only use N t = 12 lattices.

Ideal HRG
In the ideal HRG approach, the pressure is given by: where the sum runs over all the hadrons and resonances included in the model. Here the single particle chemical potential is defined with respect to the global conserved charges (baryonic B, electric Q and strangeness S ) as µ k = B k µ B + Q k µ Q + S k µ S . For this section of the paper, we will work in the confines of an ideal HRG model, and will try to implement every thermodynamic result in terms this model.
The HRG model provides a good description of several thermodynamic quantities below the phase transition temperature. However, discrepancies are starting to appear when more differential observables are being calculated, such as the fluctuations of conserved charges, defined as: For some observables, including new hadron states, predicted by the quark model helps in making the the discrepancy disappear. An important example of such an observable is shown in Figure 1. However, for other observables the agreement between HRG model and lattice gets worse when the QM states are included. For example the QM result overestimates both χ S 4 /χ S 2 and χ us 11 . What does this mean? χ S 4 /χ S 2 is proportional to the average strangeness squared in the system: the fact that the Here we see this combination of conserveced charges from the lattice and from the ideal HRG model. The HRG results are shown for different hadronic spectra, namely by using the PDG2012 (black solid line) and the QM (dashed red line).
QM overestimates it means that it contains either too many multi-strange states or not enough |S | = 1 states. As for χ us 11 , there the contribution is positive for baryons and negative for mesons, so the QM overestimating it means that either it contains too many strange baryons or not enough strange mesons. The bottom line is that the interpretation of these comparisons with the HRG is ambiguous. In order to more effectively settle this issue alternative observables are needed.   [12]. Right: comparison of up-strange correlator χ us 11 simulated on the lattice [9] and calculated in the HRG model using the PDG2012 (solid black line) and the QM (dashed red line) spectra.

Imaginary chemical potential and strangeness sectors
In order to solve this ambiguity, we separate the pressure of QCD as a function of the temperature into contributions coming from hadrons grouped according to their quantum numbers: whereμ i = µ i /T . This particular truncation of the fugacity expansion of the pressure is motivated by our assumption that in the confined phase, a HRG with some hadron spectra describes the thermodynamics of the QCD system. Assuming this ansatz for the pressure, the partial pressures P |B|=i,|S |= j can be expressed as linear combinations of the susceptibilities This means that in principle we can determine the strangeness sectors just by using the usual fluctuation observables. This is not the approach we pursue here. Instead, we simulate with an imaginary strangeness chemical potential µ S = iµ I . In this case by simple differentiation one gets: and similarly for χ S 2 , χ S 3 and χ S 4 . The advantage of the imaginary chemical potential approach is that, even though it is based on the exact same assumption as the direct evaluation of linear combinations, it reduces the errors considerably, since the lower derivatives already contain the information on the higher strangeness sectors. Direct method at µ=0, N t =16 Imaginary µ S method, N t =16 Figure 3. Left: Examples of continuum limit extrapolation from our N t = 10, 12, 16 lattices. Right: Comparison of our method to Taylor expansion from µ = 0 data for P 01 . The statistics would explain only a factor of 2 difference in the errorbars, but the improvement is much more drastic than that.
To obtain the Fourier cofficients (P B=0,|S |=1 + P |B|=1,|S |=1 ),P |B|=1,|S |=2 and P |B|=1,|S |=3 we perform a correlated fit with the previous ansatz for the observables χ S 1 , χ S 2 , χ S 3 and χ S 4 at every temperature. To obtain P |B|=1,S =0 we fit χ B 2 , to obtain P |B|=1,|S |=1 we fit χ B 1 . To get P |B|=0,|S |=1 we just take the difference (P B=0,|S |=1 + P |B|=1,|S |=1| ) − P |B|=1,|S |=1 . For statistical errors, we use the jacknife method. For the continuum limit we use N t = 10, 12 and 16 lattices. For systematic errors on the continuum extrapolation we use two different scale settings: w 0 and f π . More details on the scale setting can be found in [9]. We also include two different interpolating functions for temperature dependence of the P BS : a third order polynomial a + bT + cT 2 + dT 3 and a rational function (a + bT + cT 2 )/(1 + dT ). Both of these describe the data well. We also do the continuum limit with and without tree level improvement factors for the pressure. The continuum limit is done with a linear a + b/N 2 t and a 1/(a + b/N 2 t ) ansatz. These 8 different results are then weighted with the AKAIKE information criterion using the histogram method [13]. Examples of linear continuum limit extrapolations are included in Fig. 3(left). For the B = 0, |S | = 1 sector, which includes a large contribution from kaons, the continuum extrapolation could not be carried out using these lattices. Therefore we only include results for the mesonic |S | = 1 partial pressure P B=0,|S |=1 and the total S = 1 partial pressure (P B=0,|S |=1 + P |B|=1,|S |=1 ) for finite lattices in this work. The results of our lattice study are summarized in Figure 4. Our analysis clearly shows that the well-established states listed in the most updated version of the PDG are not enough to correctly reproduce the lattice results. The minimal choice which reproduces most of the data is the PDG2016 with the inclusion of one-star states which are not well-established. In some cases, such as the strange mesons and |S | = 3 baryons, there is room to add even more states. These results will potentially affect all phenomenological applications of the HRG model, including thermal fits and other calculations relevant to heavy-ion physics.   [14,15]. In the lower center panel, the dark red short-dashed curve corresponds to the PDG2016+ with inclusion of the Ω resonances predicted by the QM [16,17].

Repulsive baryonic interactions
For this section we only consider the µ S = µ Q = 0 case, and keep only µ B to be nonzero. The pressure can be written as: While the baryon number for µ B = iμ B is: where b k = kP |B|=k . The coefficients can easily be calculated by a Fourier transformation. The main difference between this and the previous section (the strangeness sectors) is that in this case we do not want to commit to a particular truncation of the series expansion (5), instead, we want the Fourier analysis tell us, when a component appears. In the ideal HRG model with no light nuclei included b 1 > 0 and b 2 = b 3 = b 3 = · · · = 0. The inclusion of nuclei would induce nonzero b 2 , b 3 , . . . , but always with a positive sign. In contrast, at very large temperatures, where we have free quarks, we expect: Since we only consider the case µ S = µ Q = 0, one has µ f = µ B /3. The net baryon susceptibility at imaginary µ B reads The coefficients b SB k are calculated by a Fourier integral to be: Since this sign alternating behavior is strikingly different from the HRG expectation of b 2 = b 3 = b 4 = · · · = 0, one might interpret the appearance of an alternating sign structure as a signal for deconfinement.
Such a conclusion might be hasty however. As we point out in [2], hadronic models can also produce such a sign structure. The simple model we use to illustrate this is the HRG with an excluded volume repulsive interaction between baryons and baryons, anti-baryon and anti-baryons, and no interactions between other hadron pairs. We call this model EV-HRG. The model has one free parameter, an average excluded volume parameter b. Model calculations of the b k can be found in [2]. The first Fourier coefficient b 1 does not depend on this b parameter, the higher coefficients do, but once b is adjusted such that b 2 is reproduced, near T c , it appears that also b 3 and b 4 are described well by the model. This can be seen in Figure 5.
The success of the EV-HRG model in describing these four Fourier coefficients does not automatically mean that such a model describes all other QCD observables, for instance the correlations and fluctuations involving the electric charge and strangeness, in the same temperature range. These observables are sensitive to the baryon-baryon interactions as well. At the same time, they are also sensitive to interactions involving mesons, as these carry both electric charge and strangeness, and to the strangeness-dependent baryonic interactions. These extensions are beyond the scope of the present work. The main point we would like to make is that these fugacity expansion parameters are quite sensitive to the modeling of hadronic interactions, and therefore can be quite useful when trying to adjust hadronic modelling.
One might wonder how the fugacity expansion parameters relate to the more conventional observables, the fluctuations of conserved charges. This is illustrated in Figure 6. When the effects of baryon-baryon interactions are small, the third and higher order coefficients can be neglected, and χ B 2 − χ B 4 is just proportional to b 2 . This fact was exploited in [18].  . The temperature dependence of χ B 2 − χ B 4 , calculated within the EV-HRG model with baryonic eigenvolume parameter b = 1fm 3 (solid line). Other lines depict the EV-HRG model calculations using truncations of the fugacity expansion, at the second (dashed blue line), third (dash-dotted red line), and fourth (dotted black line) orders. Lattice QCD data from Refs. [9] and [19] are depicted, respectively, by open and full symbols.

Summary
Lattice simulations at imaginary chemical potentials allow one to separate thermodynamic contributions from sectors with different conserved charges. In this conference contribution we briefly explained two phenomelogical applications of this fact. First, we with in an ideal HRG model, and looked for missing strange resonances by separating sectors in baryon number and strangeness, with simulations in the confined phase and with Im µ s > 0. We clearly saw, that the well-established states listed in PDG 2016 are not enough to correctly reproduce the lattice results. Next, we looked at extending the HRG model with repulsive baryonic interactions. Here the higher order fugacity expansions in the baryon number proved useful, as we demonstrated these to be sensitive to repulsion between baryons. We saw that repulsion between hadrons can reproduce the sign structure seen in the higher order fugacity expansion coefficients, and that the lattice data constrain the excluded volume parameter to be around 1fm 3 .