Fluctuations of conserved charges from imaginary chemical po- tential

When comparing lattice calculation to experimental data from heavy ion collision experiments, the higher order fluctuations of conserved charges are important observables. An efficient way to study these fluctuations is to determine them from simulations at imaginary chemical potential. In this talk we present results up to the six order derivative in μB (with up to eighth order included in the fit), calculated on a 483 × 12 lattice with staggered fermions using different values of μB while μS = μQ = 0.


Introduction
To analyse the quark gluon plasma that is created in heavy ion collision experiments at the LHC or RHIC a theoretical understanding of the deconfinement region of QCD is needed. Lattice QCD is a good tool to study QCD since this area can not be accessed perturbatively. At the moment direct simulations that are continuum extrapolated and at physical quark masses are restricted to vanishing or imaginary chemical potential. On the other hand the collisions especially at RHIC take place away from the axis of zero µ B [1]. Therefore information in that region are needed. Even though it is not possible to do direct lattice simulations, it is possible to extrapolate observables from zero or imaginary chemical potential. This method is called analytical continuation. The analytical continuation from imaginary potential is by now well established (see for example [2][3][4][5]).
In this proceeding we present preliminary results on the fluctuations of conserved charges. These fluctuations can be measured on the particle distributions in heavy ion collisions (see for example [6]). A comparison between the experimental measurements and the theoretical calculations allows then for the determination of the order of the transition. A similar analysis form simulations at imaginary chemical potential was done in [5], that we will improve upon by presenting calculations to higher orders in µ B . Results on the same observables were also studied by the Taylor expansion method and published in [7]. This method relies on simulations at µ B = 0 and calculates the analytical continuation from the measured derivatives.

Lattice details
We use a tree-level Symanzik improved gauge action, with four times stout smeared (ρ = 0.125) staggered fermions. We simulate 2 + 1 + 1 dynamical quarks where the light flavors are tuned in a way to reproduce the physical pion and kaon mass and we set m c m s = 11.85 [8]. For the zero temperature runs we use large volumes which full fill Lm π > 4. The scale is determined via f π . More Details can be found in [9].
The maximal useful value of m B is m B = iπT because of the Roberge-Weiss transition [10]. We simulate at eight different values of m B given as: µ ( j) B = iT jπ 8 for j ∈ {0, 1, 2, 3, 4, 5, 6, 7}. The analysis is done purely on a 48 3 × 12 lattice without continuum extrapolation, at twenty-two temperatures in the temperature range 140. . . 250 MeV. All simulations are done at µ Q = µ S = 0. The ratios of the cumulates however are calculated at n S = 0 and n Q = 0.4 n B to match the conditions in heavy ion collisions.

Analysis
We present our analysis in three steps. We start with the analysis for χ B 2 , χ B 4 , χ B 6 and χ B 8 where we use the notation First we do the analysis for each temperature separately (section 3.1). Afterwards we use the information that the results are expected to lie on a smooth curve, by introducing a spline through the results (section 3.2). Finally we use the same techniques introduced for χ B i to calculate three different ratios of the cumulants of the baryon distribution at n S = 0 and n Q = 0.4 n B in terms of the χ B,Q,S i, j,k (section 3.3).

Single Temperature
As a first step we analyse the data for a single temperature. For each and χ B 4 , while for µ B = 0 only χ B 2 and χ B 4 are measured since χ B 1 and χ B 3 are odd functions in µ B and therefore equal to zero.
We make the ansatz for the partition sum: From this we can calculate the derivatives that we can measure on the lattice: In a first step we determine the coefficients c 2 , c 4 and c 6 from a correlated fit to the data, while c 8 is set to zero. The results are shown as blue points in figure 1. This ansatz does not account for the systematic uncertainties that arise from higher order contributions in µ B T . As a first estimate for this uncertainties we include c 8 in our fit and check for changes in c 2 to c 6 . The results are shown in figure  2. While the results for χ B 2 and χ B 4 are mostly unchanged, the changes for χ B 6 after the inclusion of χ B 8 are significant. Here it becomes obvious that a careful investigation of the influences of higher orders are necessary.

Spline Fit
We expect our results for χ B i (T ) to lie on a smooth curve. We implement this information by fitting the results with a spline. Therefore the fit parameters c 2 , c 4 , c 6 and c 8 now become functions of T themselves. For the spline fitting procedure the choice of note points is crucial. To reduce the bias that is implemented with a specific choice we have three different modes. The first and possibley simplest method is just to have all note points evenly spaced. However at higher temperatures we expect slower changes in our curves. Also for µ B = 0 our simulation points at high temperatures are more sparse. Therefore as a second choice we double the distance between the last three note points. As a last possibility we chose our note points randomly, with a uniform distribution. However to avoid the case of two note points that are very close to each other we only except a configuration if the distance between two neighbouring note points is at least 10 MeV. An illustration of this different node point configurations is shown in figure 3. Also the number of note points is varied between six and eight. The results are shown by the blue band in figure 1 and figure 2.

Cumulants
For a comparison with heavy ion collision experiments the cumulants of the net baryon distribution are a useful tool. The first four cumulants are the mean M B , the variance σ 2 B , the skewness S B and the kurtosis κ B . By forming appropriate ratios, we can cancel out explicit volume factors. However the measured distributions themselves may still depend on the volume, which one should take into account, when comparing to experiments.
Heavy ion collisions with lead or gold take place with at µ B > 0, n S = 0 and n Q = 0.4 n B . Since our simulations are done at µ S = µ Q = 0 and µ B 0 we have to do some calculations to arrive at the same observables that are measured in experiments (see for example [6]). We investigate three different rations of cumulants and write each as a Taylor expansion: The µ B dependence of the χ B i (T,μ B ) can be again written in terms of the Taylor expansion: with We can now use the constrains n S = 0 and n Q = 0.4 n B which can be rewritten as to determine r B,k i j coefficients form the equations 8, 9 and 10. However we now need to know not only the behaviour of the χ B i but also of derivatives with respect to µ S and µ Q . For now our simulations are restricted to ensembles with finite µ B . Therefore the µ S and µ Q derivatives have to be calculated directly and without the support from the fit that we used in the µ B direction. We calculate various χ B,Q,S i, j,k with the appropriate values of j and k and all possible values for i so that i + j + k ≤ 4.
For each group of fluctuations with the same j and k we perform a fit analogous to the procedure described in the sections 3.1 and 3.2. This is sufficient to determine the first to r B,k i j coefficients for all three observables. The results are shown in figure 4, 5 and 6. For higher order coefficients, higher order derivatives in µ S and µ Q are needed. The direct measurements have a rapidly increasing error with each derivative and very large statistics would be needed to improve our calculations in that manner. Another possibility would be add ensembles with finite µ S and µ Q and do a similar fit as for the µ B direction. This approach has been used in [5].

Error Analysis
For a reliable comparison between experimental measurements and theoretical calculations the error estimation is an important ingredient. As we present work in progress results the error estimation process is not yet finished. Our statistical error is estimated by the Jack-Knife-Method. For our systematic error there are several sources, which we have not yet completely covered. We determine our systematic error by the histogram method described in [11], where each analysis is weighted with the akaike information criteria. We include the influence of the number of points in the µ B direction, by either including or ignoring the data from our highest value for µ B . We also try to estimate the influence of the spline node points as described in section 3.2. However a very important source for our systematic error is the influence of the higher order contributions in µ B that are not included in our fit ansatz. A rough idea of this influences can be gained from the comparison of the results shown in figure 1 and figure 2, where one order more was included. However a more detailed analysis is necessary to obtain a reliable error on our result.

Conclusion
We presented preliminary results on fluctuations of conserved charges χ B 2 , χ B 4 and χ B 6 and on the three combinations of cumulants M B