arXiv subjects

F. Karsch

Publications and source records attributed to F. Karsch.

At least 19 recordsLinked to original sources

arXiv2025

A generalized definition of the isothermal compressibility in (2+1)-flavor QCD

We introduce a generalized definition of the isothermal compressibility ($\kappa_{T,\sigma_Q^2}$) calculable by keeping net conserved charge fluctuations rather than total number densities constant. We present lattice QCD results for this isothermal compressibility, expressed in terms of fluctuations of conserved charges that are related to baryon ($B$), electric charge ($Q$) and strangeness ($S$) quantum numbers. This generalized isothermal compressibility is compared with hadron resonance gas model calculations as well as with heavy-ion collision data obtained at RHIC and the LHC. We find $\kappa_{T,\sigma_Q^2}=13.8(1.3)$~fm$^3$/GeV at $T_{pc,0}=156.5(1.5)$~MeV and $\hat{\mu}_B=0$. This finding is consistent with the rescaled result of the ALICE Collaboration, where we replaced the number of charged hadrons ($N_{\rm ch}$) by the total number of hadrons ($N_{\rm tot}$) at freeze-out. Normalizing this result with the QCD pressure ($P$) we find that the isothermal compressibility on the pseudo-critical line stays close to that of an {\it ideal gas}, {\it i.e.} $P \kappa_{T,\sigma_Q^2}\simeq 1$.

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arXiv2024

Strangeness-Correlations on the pseudo-critical line in (2+1)-flavor QCD

We present some lattice QCD results on first ($\chi_1^i$) and second ($\chi_2^i$) cumulants of and correlations ($\chi_{11}^{ij}$) among net baryon-number ($B$), strangeness ($S$) and electric charge ($Q$) along the pseudo-critical line ($T_{pc}(\mu_B)$) in the temperature ($T$)--baryon chemical potential ($\mu_B$) phase diagram of (2+1)-flavor QCD. We point out that violations of the isospin symmetric limit of vanishing electric charge chemical potential are small along the $T_{pc}(\mu_B)$ for the entire range of $\mu_B$ covered in the RHIC beam energy scan. For the strangeness neutral matter produced in heavy-ion collisions this leads to a close relation between $\chi_{11}^{BS}$ and $\chi_{11}^{QS}$. We compare lattice QCD results for $\chi_{11}^{BS}/\chi_2^S$ along the $T_{pc}(\mu_B)$ line with preliminary experimental measurements of $\chi_{11}^{BS}/\chi_2^S$ for collision energies $7.7~{\rm GeV}\le \sqrt{s_{_{NN}}}\le 62.4~{\rm GeV}$. While we find good agreements for $\sqrt{s_{_{NN}}}\ge 39$~GeV, differences are sizeable at smaller values of $\sqrt{s_{_{NN}}}$. Moreover, we compare lattice QCD results for the ratio of the strangeness ($\mu_S$) to baryon ($\mu_B$) chemical potentials, which define a strangeness neutral system with fixed electric charge to baryon number density, with experimental results obtained by the STAR collaboration for $\mu_S/\mu_B$ using strange baryon yields on the freeze-out line. Finally, we determine the baryon chemical potential at the freeze-out ($\mu_B^f$) by comparing $\chi_1^B/\chi_2^B$ along the $T_{pc}(\mu_B)$ with the experimentally measured net-proton cumulants $\chi_1^p/\chi_2^p$. We find that $\{\mu_B^f, T_{pc}(\mu_B^f) \}$ are consistent with the freeze-out parameters of the statistical-model fits to experimentally measured hadron yields for $\sqrt{s_{_{NN}}} \geq 11.5$ GeV.

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arXiv2024

Curvature of the chiral phase transition line from the magnetic equation of state of (2+1)-flavor QCD

We analyze the dependence of the chiral phase transition temperature on baryon number and strangeness chemical potentials by calculating the leading order curvature coefficients in the light and strange quark flavor basis as well as in the conserved charge ($B, S$) basis. Making use of scaling properties of the magnetic equation of state (MEoS) and including diagonal as well as off-diagonal contributions in the expansion of the energy-like scaling variable that enters the parametrization of the MEoS, allows to explore the variation of $T_c(\mu_B,\mu_S) = T_c ( 1 - (\kappa_2^B \hat{\mu}_B^2 + \kappa_2^S \hat{\mu}_S^2 + 2\kappa_{11}^{BS} \hat{\mu}_B \hat{\mu}_S))$ along different lines in the $(\mu_B,\mu_S)$ plane. On lattices with fixed cut-off in units of temperature, $aT=1/8$, we find $\kappa_2^B=0.015(1)$, $\kappa_2^S=0.0124(5)$ and $\kappa_{11}^{BS}=-0.0050(7)$. We show that the chemical potential dependence along the line of vanishing strangeness chemical potential is about 10\% larger than along the strangeness neutral line. The latter differs only by about $3\%$ from the curvature on a line of vanishing strange quark chemical potential, $\mu_s=0$. We also show that close to the chiral limit the strange quark mass contributes like an energy-like variable in scaling relations for pseudo-critical temperatures. The chiral phase transition temperature decreases with decreasing strange quark mass, $T_c(m_s)= T_c(m_s^{\rm phy}) (1 - 0.097(2) (m_s-m_s^{\rm phys})/m_s^{\rm phy}+{\cal O}((\Delta m_s)^2)$.

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arXiv2023

QCD material parameters at zero and non-zero chemical potential from the lattice

Using an eighth-order Taylor expansion in baryon chemical potential, we recently obtained the (2+1)-flavor QCD equation of state (EoS) at non-zero conserved charge chemical potentials from the lattice. We focused on strangeness-neutral, isospin-symmetric QCD matter, which closely resembles the situation encountered in heavy-ion collision experiments. Using this EoS, we present here results on various QCD material parameters; in particular we compute the specific heat, speed of sound, and compressibility along appropriate lines of constant physics. We show that in the entire range relevant for the beam energy scan at RHIC, the specific heat, speed of sound, and compressibility show no indication for an approach to critical behavior that one would expect close to a possibly existing critical endpoint.

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arXiv2023

Charm degrees of freedom in hot matter from lattice QCD

We study the nature of charm degrees of freedom in hot strong interaction matter by performing lattice QCD calculations of the second and fourth-order cumulants of charm fluctuations, and their correlations with net baryon number, electric charge and strangeness fluctuations. We show that below the chiral crossover temperature thermodynamics of charm can be very well understood in terms of charmed hadrons. Above the chiral transition charm quarks show up as new degrees of freedom contributing to the partial charm pressure. However, up to temperatures as high as 175 MeV charmed hadron-like excitations provide a significant contribution to the partial charm pressure.

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arXiv2022

Equation of state and speed of sound of (2+1)-flavor QCD in strangeness-neutral matter at non-vanishing net baryon-number density

We update results on the QCD equation of state in (2+1)-flavor QCD with non-zero conserved charge chemical potentials obtained from an eighth-order Taylor series. We present results for basic bulk thermodynamic observables of strangeness-neutral strong-interaction matter, i.e. pressure, number densities, energy and entropy density, and resum Taylor series results using Pad\'e approximants. Furthermore, we calculate the speed of sound as well as the adiabatic compression factor of strangeness-neutral matter on lines of constant entropy per net baryon number. We show that the equation of state ($P(n_B), \epsilon (n_B)$) is already well described by the $4^{\rm th}$-order Taylor series in almost the entire range of temperatures accessible with the beam energy scan in collider mode at the Relativistic Heavy Ion Collider.

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arXiv2022

Towards the chiral phase transition in the Roberge-Weiss plane

We discuss the interplay between chiral and center sector phase transitions that occur in QCD with an imaginary quark chemical potential $\mu=i(2n+1) \pi T/3$. Based on a finite size scaling analysis in (2+1)-flavor QCD using HISQ fermions with a physical strange quark mass and a range of light quark masses, we show that the endpoint of the line of first-order Roberge-Weiss (RW) transitions between center sectors is second order for light quark masses $m_l\ge m_s/320$, and that it belongs to the $3$-d, $Z(2)$ universality class. The operator for the chiral condensate behaves like an energy-like operator in an effective spin model for the RW phase transition. As a consequence, for any non-zero value of the quark mass, the chiral condensate will have an infinite slope at the RW phase transition temperature, $T_{RW}$. Its fluctuation, the disconnected chiral susceptibility, behaves like the specific heat in $Z(2)$ symmetric models and diverges in the infinite volume limit at the RW phase transition temperature $T_{RW}$ for any non-zero value of the light quark masses. Our analysis suggests the critical temperatures for the RW phase transition and the chiral phase transition coincide in the RW plane. On lattices with temporal extent $N_\tau=4$, we find in the chiral limit $T_{\chi}=T_{RW}=195(1)~$MeV.

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arXiv2022

Taylor expansions and Pad\'e approximants for cumulants of conserved charge fluctuations at non-vanishing chemical potentials

Using high statistics datasets generated in (2+1)-flavor QCD calculations at finite temperature we present results for low order cumulants of net baryon-number fluctuations at non-zero values of the baryon chemical potential. We calculate Taylor expansions for the pressure (zeroth order cumulant), net baryon-number density (first order cumulant) and the variance of the distribution on net-baryon number fluctuations (second order cumulant). We obtain series expansions from an eighth order expansion of the pressure and compare these to diagonal Pad\'e approximants. This allows us to estimate the range of values for the baryon chemical potential in which these expansions are reliable. We find $\mu_B/T\le 2.5$, $2.0$ and $1.5$ for the zeroth, first and second order cumulants, respectively. We furthermore, construct estimators for the radius of convergence of the Taylor series of the pressure. In the vicinity of the pseudo-critical temperature, $T_{pc}\simeq 156.5$ MeV, we find $\mu_B/T \gtrsim\ 2.9$ at vanishing strangeness chemical potential and somewhat larger values for strangeness neutral matter. These estimates are temperature dependent and range from $\mu_B/T \gtrsim\ 2.2$ at $T=135$ MeV to $\mu_B/T\ \gtrsim\ 3.2$ at $T=165$ MeV. The estimated radius of convergences is the same for any higher order cumulant.

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arXiv2021

Lattice QCD at Imaginary Chemical Potential in the Chiral Limit

We report on an ongoing study on the interplay between Roberge-Weiss (RW) and chiral transitions in simulations of (2+1)-flavor QCD with an imaginary chemical potential. We established that the RW endpoint belongs to the 3-$d$, $Z_2$ universality class when calculations are done with the Highly Improved Staggered Quark (HISQ) action in the RW plane with physical quark masses. We also have explored a range of quark masses corresponding to pion mass values, $m_\pi\geq40$~MeV and found that the transition is consistent with $Z_2$ universality class. We argue that observables that were usually used to determine the chiral phase transition temperature, e.g. the chiral condensate and chiral susceptibility, are sensitive to the RW transition and are energy-like observables for the $Z_2$ transition, contrary to the magnetic-like (order parameter) behavior at vanishing chemical potential. Moreover the calculations performed at $m_\pi\sim40$~MeV also put a stringent constraint for a critical pion mass at zero chemical potential for a possible first-order chiral phase transition.

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arXiv2021

Second order cumulants of conserved charge fluctuations revisited I. Vanishing chemical potentials

We update lattice QCD results for second order cumulants of conserved charge fluctuations and correlations at non-zero temperature and vanishing values of the conserved charge chemical potentials. We compare these results to hadron resonance gas calculations with and without excluded volume terms as well as S-matrix results in the hadronic phase of QCD, and comment on their current limitations. We, furthermore, use these results to characterize thermal conditions in the vicinity of the pseudo-critical line of the chiral transition in QCD. We argue that the ratio of strange to baryon chemical potentials is a robust observable that, on the one hand, deviates only little from hadron resonance gas results, but, on the other hand, is very sensitive to the spectrum of strange baryon resonances.

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arXiv2020

Higher order cumulants of electric charge and strangeness fluctuations on the crossover line

We present lattice QCD calculations of higher order cumulants of electric charge distributions for small baryon chemical potentials $μ_B$ by using up to NNNLO Taylor expansions. Ratios of these cumulants are evaluated on the pseudo-critical line, $T_{pc}(μ_B)$, of the chiral transition and compared to corresponding measurements in heavy ion collision experiments by the STAR and PHENIX Collaborations. We demonstrate that these comparisons give strong constraints on freeze-out parameters. Furthermore, we use strangeness fluctuation observables to compute the ratio $μ_S/μ_B$ on the crossover line and compare it to $μ_S/μ_B$ at freeze-out stemming from fits to strange baryon yields measured by the STAR Collaboration.

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arXiv2020

Skewness, kurtosis and the 5th and 6th order cumulants of net baryon-number distributions from lattice QCD confront high-statistics STAR data

We present new results on up to $6^{th}$ order cumulants of net baryon-number fluctuations at small values of the baryon chemical potential, $μ_B$, obtained in lattice QCD calculations with physical values of light and strange quark masses. Representation of the Taylor expansions of higher order cumulants in terms of the ratio of the two lowest order cumulants, $M_B/σ_B^2=χ_1^B(T,μ_B)/χ_2^B(T,μ_B)$, 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 beam energy $\sqrt{s_{_{NN}}}=54.4$ GeV, agree well with lattice QCD results on cumulants of net baryon-number fluctuations close to the pseudo-critical temperature, $T_{pc}(μ_B)$, for the chiral transition in QCD. We also present first results from a next-to-leading order expansion of $5^{th}$ and $6^{th}$ order cumulants on the line of pseudo-critical temperatures.

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arXiv2019

Chiral phase transition temperature in (2+1)-Flavor QCD

We present a lattice QCD based determination of the chiral phase transition temperature in QCD with two degenerate, massless quarks and a physical strange quark mass. We propose and calculate two novel estimators for the chiral transition temperature for several values of the light quark masses, corresponding to Goldstone pion masses in the range of $58~{\rm MeV}\lesssim m_π\lesssim 163~{\rm MeV}$. The chiral phase transition temperature is determined by extrapolating to vanishing pion mass using universal scaling analysis. Finite volume effects are controlled by extrapolating to the thermodynamic limit using spatial lattice extents in the range of $2.8$-$4.5$ times the inverse of the pion mass. Continuum extrapolations are carried out by using three different values of the lattice cut-off, corresponding to lattices with temporal extent $N_τ=6,\ 8$ and $12$. After thermodynamic, continuum and chiral extrapolations we find the chiral phase transition temperature $T_c^0=132^{+3}_{-6}$ MeV.

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arXiv2019

Chiral crossover in QCD at zero and non-zero chemical potentials

We present results for pseudo-critical temperatures of QCD chiral crossovers at zero and non-zero values of baryon ($B$), strangeness ($S$), electric charge ($Q$), and isospin ($I$) chemical potentials $μ_{X=B,Q,S,I}$. The results were obtained using lattice QCD calculations carried out with two degenerate up and down dynamical quarks and a dynamical strange quark, with quark masses corresponding to physical values of pion and kaon masses in the continuum limit. By parameterizing pseudo-critical temperatures as $ T_c(μ_X) = T_c(0) \left[ 1 -κ_2^{X}(μ_{X}/T_c(0))^2 -κ_4^{X}(μ_{X}/T_c(0))^4 \right] $, we determined $κ_2^X$ and $κ_4^X$ from Taylor expansions of chiral observables in $μ_X$. We obtained a precise result for $T_c(0)=(156.5\pm1.5)\;\mathrm{MeV}$. For analogous thermal conditions at the chemical freeze-out of relativistic heavy-ion collisions, i.e., $μ_{S}(T,μ_{B})$ and $μ_{Q}(T,μ_{B})$ fixed from strangeness-neutrality and isospin-imbalance, we found $κ_2^B=0.012(4)$ and $κ_4^B=0.000(4)$. For $μ_{B}\lesssim300\;\mathrm{MeV}$, the chemical freeze-out takes place in the vicinity of the QCD phase boundary, which coincides with the lines of constant energy density of $0.42(6)\;\mathrm{GeV/fm}^3$ and constant entropy density of $3.7(5)\;\mathrm{fm}^{-3}$.

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arXiv2019

Chiral phase transition of (2+1)-flavor QCD

We present here results on the determination of the critical temperature in the chiral limit for (2+1)-flavor QCD. We propose two novel estimators of the chiral critical temperature where quark mass dependence is strongly suppressed compared to the conventional estimator using pseudo-critical temperatures. We have used the HISQ/tree action for the numerical simulation with lattices with three different temporal extent $N_τ=$6, 8, 12 and varied the aspect ratio over the range $4 \leq N_σ/N_τ \leq 8$. To approach the chiral limit, the light quark mass has been decreased keeping the strange quark mass fixed at its physical value. Our simulations correspond to the range of pion masses, 55 MeV $\leq m_π \leq$ 160 MeV.

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arXiv2017

Skewness and kurtosis of net baryon-number distributions at small values of the baryon chemical potential

We present results for the ratios of mean ($M_B$), variance ($σ_B^2$), skewness ($S_B)$ and kurtosis ($κ_B$) of net baryon-number fluctuations obtained in lattice QCD calculations with a physical light to strange quark mass ratio. Using next-to-leading order Taylor expansions in baryon chemical potential we find that qualitative features of these ratios closely resemble the corresponding experimentally measured cumulants ratios of net proton-number fluctuations for beam energies down to $\sqrt{s_{_{NN}}} \ge 19.6$ GeV. We show that the difference in cumulant ratios for the mean net baryon-number, $M_B/σ_B^2=χ_1^B(T,μ_B)/χ_2^B(T,μ_B)$ and the normalized skewness, $S_Bσ_B=χ_3^B(T,μ_B)/χ_2^B(T,μ_B)$, naturally arises in QCD thermodynamics. Moreover, we establish a close relation between skewness and kurtosis ratios, $S_Bσ_B^3/M_B=χ_3^B(T,μ_B)/χ_1^B(T,μ_B)$ and $κ_Bσ_B^2=χ_4^B(T,μ_B)/χ_2^B(T,μ_B)$, valid at small values of the baryon chemical potential.

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arXiv2017

The QCD Equation of State to $\mathcal{O}(μ_B^6)$ from Lattice QCD

We calculated the QCD equation of state using Taylor expansions that include contributions from up to sixth order in the baryon, strangeness and electric charge chemical potentials. Calculations have been performed with the Highly Improved Staggered Quark action in the temperature range $T\in [135~{\rm MeV}, 330~{\rm MeV}]$ using up to four different sets of lattice cut-offs corresponding to lattices of size $N_σ^3\times N_τ$ with aspect ratio $N_σ/N_τ=4$ and $N_τ=6-16$. The strange quark mass is tuned to its physical value and we use two strange to light quark mass ratios $m_s/m_l=20$ and $27$, which in the continuum limit correspond to a pion mass of about $160$ MeV and $140$ MeV espectively. Sixth-order results for Taylor expansion coefficients are used to estimate truncation errors of the fourth-order expansion. We show that truncation errors are small for baryon chemical potentials less then twice the temperature ($μ_B\le 2T$). The fourth-order equation of state thus is suitable for the modeling of dense matter created in heavy ion collisions with center-of-mass energies down to $\sqrt{s_{NN}}\sim 12$ GeV. We provide a parametrization of basic thermodynamic quantities that can be readily used in hydrodynamic simulation codes. The results on up to sixth order expansion coefficients of bulk thermodynamics are used for the calculation of lines of constant pressure, energy and entropy densities in the $T$-$μ_B$ plane and are compared with the crossover line for the QCD chiral transition as well as with experimental results on freeze-out parameters in heavy ion collisions. These coefficients also provide estimates for the location of a possible critical point. We argue that results on sixth order expansion coefficients disfavor the existence of a critical point in the QCD phase diagram for $μ_B/T\le 2$ and $T/T_c(μ_B=0) > 0.9$.

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arXiv2017

Chiral phase structure of three flavor QCD at vanishing baryon number density

We investigate the phase structure of QCD with 3 degenerate quark flavors as function of the degenerate quark masses at vanishing baryon number density. We use the Highly Improved Staggered Quarks on lattices with temporal extent $N_{t}=6$ and perform calculations for six values of quark masses, which in the continuum limit correspond to pion masses in the range $80~{\rm MeV} \lesssim m_π \lesssim 230~$MeV. By analyzing the volume and temperature dependence of the chiral condensate and chiral susceptibility we find no direct evidence for a first order phase transition in this range of pion mass values. Relying on the universal scaling behaviors of the chiral observables near an anticipated chiral critical point, we estimate an upper bound for the critical pion mass, $m_π^c \lesssim$ 50 MeV, below which a region of first order chiral phase transition is favored.

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