Skewness, kurtosis, and the fifth and sixth order cumulants of net baryon-number distributions from lattice QCD confront high-statistics STAR data

We present new results on up to sixth-order cumulants of net baryon-number fluctuations at small values of the baryon chemical potential, ${\mu }_B$, obtained in lattice QCD calculations with physical values of light and strange quark masses. Representing the Taylor expansions of higher-order cumulants in terms of the ratio of the two lowest-order cumulants, $M_B/{\sigma }_B^2={\chi }_1^B(T,{\mu }_B)/{\chi }_2^B(T,{\mu }_B)$, allows for a parameter-free comparison with data on net proton-number cumulants obtained by the STAR Collaboration in the Beam Energy Scan at RHIC. We show that recent high-statistics data on skewness and kurtosis ratios of net proton-number distributions, obtained at a beam energy $\sqrt{s_{NN}}=54.4\mathrm{GeV}$, agree well with lattice QCD results on cumulants of net baryon-number fluctuations close to the pseudocritical temperature, $T_{pc}({\mu }_B)$, for the chiral transition in QCD. We also present first results from a next-to-leading-order expansion of fifth- and sixth-order cumulants on the line of the pseudocritical temperatures.

Figure 1

Cumulants of net baryon-number fluctuations from second to eighth order (top to bottom) evaluated at ${\mu }_B=0$ on lattices of size $N_{\sigma }^3\times N_{\tau }$ with $N_{\sigma }=4N_{\tau }$. For further details see text.

Figure 2

Expansion of $R_{12}^B\equiv M_B/{\sigma }_B^2$ at a fixed temperature close to the pseudocritical line $T_{pc}({\mu }_B)$ versus the baryon chemical potential. Shown are results from up to NNNLO expansions on lattices of size ${32}^3\times 8$ and ${48}^3\times 12$.

Figure 3

Continuum estimate for $R_{12}^B$ based on NNNLO expansion results obtained on lattices of size ${32}^3\times 8$ and ${48}^3\times 12$.

Figure 4

The cumulant ratios $R_{31}^B(T,{\mu }_B)\equiv S_B{\sigma }_B^3/M_B$ (top) and $R_{42}^B(T,{\mu }_B)\equiv {\kappa }_B{\sigma }_B^2$ (bottom) versus ${\mu }_B/T$ for four different values of the temperature calculated from LO, NLO and NNLO Taylor expansions of the cumulants ${\chi }_n^B(T,{\mu }_B)$ on lattices with temporal extent $N_{\tau }=8$.

Figure 5

The ${\mu }_B$-dependent correction to $R_{31}^B$ compared to one third of the correction for $R_{42}^B$. The inset shows a comparison of the second derivatives of $R_{31}^B$ and $R_{42}^B/3$ with respect to ${\mu }_B/T$.

Figure 6

Continuum estimates for the skewness ratio, $R_{31}^B\equiv S_B{\sigma }_B^3/M_B$ (top), and kurtosis ratio $R_{42}^B\equiv {\kappa }_B{\sigma }_B^2$ (bottom) at ${\mu }_B=0$ based on results obtained on lattices of size ${32}^3\times 8$ and ${48}^3\times 12$, respectively. The inset in the bottom figure shows the difference $R_{42}^B-R_{31}^B$ at ${\mu }_B=0$ as function of $T$.

Figure 7

Continuum estimates for the skewness (top) and kurtosis (bottom) ratios obtained from joint fits to data obtained on lattices with temporal extent $N_{\tau }=8$ and 12.

Figure 8

The cumulant ratios $R_{51}^B(T,{\mu }_B)$ and $R_{62}^B(T,{\mu }_B)$ versus ${\mu }_B/T$ from LO and NLO Taylor expansions of the cumulants calculated on lattices with temporal extent $N_{\tau }=8$.

Figure 9

The cumulant ratio $R_{12}^B(T,{\mu }_B)$ evaluated on the pseudocritical line $T_{pc}({\mu }_B)$ for the case $n_S=0$ and $n_Q/n_B=0.4$. Also shown is the corresponding result obtained in HRG model calculations. In the latter case the width of the line reflects differences resulting from using particle spectra for a noninteracting HRG listed in the Particle Data Tables as well as resulting within quark model calculations.

Figure 10

The cumulant ratios (bands) $R_{31}^B(T,{\mu }_B)\equiv S_B{\sigma }_B^3/M_B$ and $R_{42}^B(T,{\mu }_B)\equiv {\kappa }_B{\sigma }_B^2$ versus $R_{12}^B(T,{\mu }_B)\equiv M_B/{\sigma }_B^2$ on the pseudocritical line calculated from NNLO Taylor series. Data are results on cumulant ratios of net proton-number fluctuations obtained by the STAR Collaboration [18]. Also shown are preliminary results obtained at $\sqrt{s_{NN}}=54.4\mathrm{GeV}$ [20]. Dashed lines show joint fits to the data as described in the text.

Figure 11

The cumulant ratios $R_{51}^B(T,{\mu }_B)$ and $R_{62}^B(T,{\mu }_B)$ versus $R_{12}^B(T,{\mu }_B)$ evaluated on the pseudocritical line. Data are preliminary results for the cumulant ratio $R_{62}^P$ of net proton-number fluctuations obtained by the STAR Collaboration at $\sqrt{s_{NN}}=200$ and 54.4 GeV for the 0–40% centrality class [20].