We have performed high-statistics lattice simulations using 4HEX improved staggered fermions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>16</mn><mn>3</mn></msup><mo>×</mo><mn>8</mn></math> lattices. We calculated the Taylor expansion coefficients of the pressure with respect to the baryochemical potential to the tenth order at zero and the fourth order at purely imaginary chemical potentials. We used this data to construct rational function approximations of the free energy. We use a rational ansatz that explicitly satisfies the charge conjugation symmetry and the Roberge-Weiss periodicity, which are exact properties of the quantum chromodynamics free energy. We use this ansatz to estimate the position of Lee-Yang zeros in the complex chemical potential plane. The temperature dependence of the imaginary part of the Lee-Yang zeros is then fitted with ansätze motivated by the universal behavior of the free energy near a 3D Ising critical point. In principle, this allows one to estimate the temperature of the critical endpoint. We consider several sources of systematic errors. On this single lattice spacing, we find that with 84% probability the chiral critical endpoint is either below 103 MeV temperature or it does not exist. We also identify some caveats of the method, which do not disappear even with the extremely high statistics of this present study. We discuss to what extent these can be eliminated by future high statistics lattice analyses.

Germany

Alexander Adam

Szabolcs Borsányi

United States

Hungary

Zoltán Fodor

Jana N. Guenther

Piyush Kumar

Italy

Paolo Parotto

Attila Pásztor

Chik Him Wong

High-precision baryon number cumulants from lattice QCD in a finite box: Cumulant ratios, Lee-Yang zeros, and critical endpoint predictions

Charting the phase diagram of quantum chromodynamics (QCD) at large density is a challenging task due to the complex action problem in lattice simulations. Through simulations at imaginary baryon chemical potential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>μ</mi><mi>B</mi></msub></math> we observe that, if the strangeness neutrality condition is imposed, both the strangeness chemical potential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>μ</mi><mi>S</mi></msub><mo>/</mo><msub><mi>μ</mi><mi>B</mi></msub></math> and the strangeness susceptibility <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>χ</mi><mn>2</mn><mi>S</mi></msubsup></math> take on constant values at the chiral transition for varying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>μ</mi><mi>B</mi></msub></math>. We present new lattice data to extrapolate contours of constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>μ</mi><mi>S</mi></msub><mo>/</mo><msub><mi>μ</mi><mi>B</mi></msub></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>χ</mi><mn>2</mn><mi>S</mi></msubsup></math> to finite baryon chemical potential. We argue that they are good proxies for the QCD crossover because, as we show, they are only mildly influenced by criticality and by finite volume effects. We obtain continuum limits for these proxies up to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>μ</mi><mi>B</mi></msub><mo>=</mo><mn>400</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, through a next-to-next-to-leading order (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi mathvariant="normal">N</mi><mn>2</mn></msup><mi>LO</mi></math>) Taylor expansion based on large-statistics data on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>16</mn><mn>3</mn></msup><mo>×</mo><mn>8</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>20</mn><mn>3</mn></msup><mo>×</mo><mn>10</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>24</mn><mn>3</mn></msup><mo>×</mo><mn>12</mn></math> lattices with our 4HEX improved staggered action. We show that these are in excellent agreement with existing results for the chiral transition and, strikingly, also with analogous contours obtained with the hadron resonance gas (HRG) model. On the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>16</mn><mn>3</mn></msup><mo>×</mo><mn>8</mn></math> lattice, we carry out the expansion up to next-to-next-to-next-to-next-to-leading order (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi mathvariant="normal">N</mi><mn>4</mn></msup><mi>LO</mi></math>), and extend the extrapolation beyond <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>μ</mi><mi>B</mi></msub><mo>=</mo><mn>500</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, again finding perfect agreement with the HRG model. This suggests that the crossover line constructed from this proxy starts deviating from the chemical freeze-out line near <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>μ</mi><mi>B</mi></msub><mo>≈</mo><mn>500</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, as expected but not yet observed.

Finite density QCD phase structure from strangeness fluctuations

In this paper we employ lattice simulations to search for the critical point of QCD. We search for the onset of a first-order QCD transition on the phase diagram by following contours of constant entropy density from imaginary to real chemical potentials under conditions of strangeness neutrality. We scan the phase diagram and investigate whether these contours meet to determine the probability that the critical point is located in a certain region on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>T</mi><mo>−</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>B</mi></mrow></msub></mrow></math> plane. To achieve this we introduce a new, continuum extrapolated equation of state at zero density with improved precision using lattices with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>τ</mi></msub><mo>=</mo><mn>8</mn></math>, 10, 12, 16 time slices, and supplement it with new data at imaginary chemical potential. The current precision allows us to exclude, at the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><mi>σ</mi></math> level, the existence of a critical point at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>μ</mi><mi>B</mi></msub><mo>&lt;</mo><mn>450</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>.

Claudia Ratti

Volodymyr Vovchenko

Lattice QCD constraints on the critical point from an improved precision equation of state

We present a new equation of state for QCD in which the temperature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi></math> and the three chemical potentials for baryon number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>μ</mi><mi>B</mi></msub></math>, electric charge <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>μ</mi><mi>Q</mi></msub></math>, and strangeness <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>μ</mi><mi>S</mi></msub></math> can be varied independently. This result is based on a generalization of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>T</mi><mo>′</mo></msup></math> expansion scheme, thanks to which the diagonal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>μ</mi><mi>B</mi></msub></math> extrapolation was pushed up to a baryo-chemical potential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>μ</mi><mi>B</mi></msub><mo>/</mo><mi>T</mi><mo>∼</mo><mn>3.5</mn></math> for the first time. This considerably extended the coverage of the Taylor expansion, limited to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>/</mo><mi>T</mi><mo>&lt;</mo><mn>2.5</mn><mi>–</mi><mn>3</mn></mrow></math>. As a consequence, we are able to offer a substantially larger coverage of the four-dimensional QCD phase diagram as well, compared to previously available Taylor expansion results. Our findings are based on new continuum estimated lattice data on the full set of second- and fourth-order fluctuations.

Ahmed Abuali

Johannes Jahan

Micheal Kahangirwe

Hitansh Shah

Seth A. Trabulsi

New 4D lattice QCD equation of state: Extended density coverage from a generalized <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>T</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math> expansion

The crossover from hadronic to quark matter is understood to be both a deconfinement as well as a chiral symmetry–restoring transition. The precise definition of these aspects may be ambiguous. Here we use the static quark free energy and its temperature derivative as proxies for deconfinement and the chiral condensate and its mass derivatives for the chiral transition. At zero baryochemical potential, and infinite volume, the chiral and deconfinement crossover temperatures almost agree. However, as we vary the spatial volume of the simulation, we observe that chiral and deconfinement-related observables have a qualitatively different chemical potential and volume dependence. In general, deconfinement-related observables have a milder volume dependence. Furthermore, while the deconfinement transition appears to get broader with increasing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>μ</mi><mi>B</mi></msub></math>, the width as well as the strength of the chiral transition is approximately constant. Our results are based on simulations at zero and imaginary chemical potentials using 4-stout-improved staggered fermions with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>τ</mi></msub><mo>=</mo><mn>12</mn></math> time slices and physical quark masses.

Ruben Kara

Ludovica Pirelli

Chiral versus deconfinement properties of the QCD crossover: Differences in the volume and chemical potential dependence from the lattice

The QCD crossover line in the temperature (<math><mi>T</mi></math>)-baryochemical potential (<math><msub><mi>μ</mi><mi>B</mi></msub></math>) plane has been computed by several lattice groups by calculating the chiral order parameter and its susceptibility at finite values of <math><msub><mi>μ</mi><mi>B</mi></msub></math>. In this work we focus on the deconfinement aspect of the transition between hadronic and quark gluon plasma phases. We define the deconfinement temperature as the peak position of the static quark entropy [<math><msub><mi>S</mi><mi>Q</mi></msub><mo>(</mo><mi>T</mi><mo>,</mo><msub><mi>μ</mi><mi>B</mi></msub><mo>)</mo></math>] in <math><mi>T</mi></math>, which is based on the renormalized Polyakov loop. We extrapolate <math><msub><mi>S</mi><mi>Q</mi></msub><mo>(</mo><mi>T</mi><mo>,</mo><msub><mi>μ</mi><mi>B</mi></msub><mo>)</mo></math> based on high statistics finite temperature ensembles on a <math><msup><mn>16</mn><mn>3</mn></msup><mo>×</mo><mn>8</mn></math> lattice to finite density by means of a Taylor expansion to eighth order in <math><msub><mi>μ</mi><mi>B</mi></msub></math> (NNNLO) along the strangeness neutral line. For the simulations the 4HEX staggered action was used with <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> flavors at physical quark masses. In this setup the phase diagram can be drawn up to unprecedentedly high chemical potentials. Our results for the deconfinement temperature are in rough agreement with phenomenological estimates of the freeze-out curve in relativistic heavy ion collisions. In addition, we study the width of the deconfinement crossover. We show that up to <math><msub><mi>μ</mi><mi>B</mi></msub><mo>≈</mo><mn>400</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, the deconfinement transition gets broader at higher densities, disfavoring the existence of a deconfinement critical endpoint in this range. Finally, we examine the transition line without the strangeness neutrality condition and observe a hint for the narrowing of the crossover towards large <math><msub><mi>μ</mi><mi>B</mi></msub></math>.

Kálmán K. Szabó

QCD deconfinement transition line up to <math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>=</mo><mn>400</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math> from finite volume lattice simulations

Fluctuations play a key role in the study of QCD phases. Lattice QCD is a valuable tool to calculate them, but going to high orders is challenging. Up to the fourth order, continuum results have been available since 2015. We present the first continuum results for sixth-order baryon fluctuations for temperatures between <math><mi>T</mi><mo>=</mo><mn>130</mn></math> and 200 MeV and eighth order at <math><mi>T</mi><mo>=</mo><mn>145</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> in a fixed volume. Comparison with earlier studies with imaginary chemical potential suggests that the volume effect is under control for <math><mi>T</mi><mo>&lt;</mo><mn>145</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. Our results are in sharp contrast with well-known results in the literature obtained at finite lattice spacing.

Sándor D. Katz

Dávid Pesznyák

Continuum-extrapolated high-order baryon fluctuations

The precision with which hadronic vacuum polarization (HVP) is obtained determines how accurately important observables, such as the muon anomalous magnetic moment <math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow></msub></mrow></math> or the low-energy running of the electromagnetic coupling <math><mi>α</mi></math>, are predicted. The two most precise approaches for determining HVP are dispersive relations combined with <math><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><mtext>hadrons</mtext></mrow></math> cross section data and lattice QCD. However, the results obtained in these two approaches display significant tensions, whose origins are not understood. Here we present a framework that sheds light on this issue and—if the two approaches can be reconciled—allows them to be combined. Via this framework, we test the hypothesis that the tensions can be explained by modifying the R-ratio in different intervals of center-of-mass energy <math><msqrt><mi>s</mi></msqrt></math>. As ingredients, we consider observables that have been precisely determined in both approaches. These are the leading hadronic contributions to <math><msub><mi>a</mi><mi>μ</mi></msub></math>, to the so-called intermediate window observable, and to the running of <math><mi>α</mi></math> between spacelike virtualities 1 and <math><mn>10</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mn>2</mn></msup></math> (for which only a preliminary lattice result exists). Our tests take into account all uncertainties and correlations, as well as uncertainties on uncertainties in the lattice results. For instance, using this framework we show that results obtained in the two approaches can be made to agree, for all three observables, by modifying the <math><mi>ρ</mi></math> peak in the experimental spectrum. More specifically, we show that this requires a common <math><mo>∼</mo><mn>5</mn><mo>%</mo></math> increase in the contributions of the peak to each of the three observables. This result is robust against the presence or absence of the running of <math><mi>α</mi></math> in the comparison. However, such an increase is much larger than the uncertainties on the measured R-ratio. We also discuss a variety of generalizations of the methods used here, as well as the limits in the information that can be extracted from the R-ratio via a finite set of observables.

France

Michel Davier

Antoine Gérardin

Laurent Lellouch

Bogdan Malaescu

Finn M. Stokes

Balint C. Toth

Lukas Varnhorst

Zhiqing Zhang

Hadronic vacuum polarization: Comparing lattice QCD and data-driven results in systematically improvable ways

Research on the quantum chromodynamics (QCD) phase diagram with lattice field theory methods is dominated by the use of rooted staggered fermions, as they are the computationally cheapest discretization available. We show that rooted staggered fermions at a nonzero baryochemical potential <math><msub><mi>μ</mi><mi>B</mi></msub></math> predict a sharp rise in the baryon density at low temperatures and <math><msub><mi>μ</mi><mi>B</mi></msub><mo>≳</mo><mn>3</mn><msub><mi>m</mi><mi>π</mi></msub><mo>/</mo><mn>2</mn></math>, where <math><msub><mi>m</mi><mi>π</mi></msub></math> is the Goldstone pion mass. We elucidate the nature of the nonanalyticity behind this sharp rise in the density by a comparison of reweighting results with a Taylor expansion of high order. While at first sight this nonanalytic behavior becomes apparent at the same position where the pion condensation transition takes place in the phase-quenched theory, the nature of the nonanalyticity in the two theories appears to be quite different: While at nonzero isospin density the data are consistent with a genuine thermodynamic (branch-point) singularity, the results at nonzero baryon density point to an essential singularity at <math><msub><mi>μ</mi><mi>B</mi></msub><mo>=</mo><mn>0</mn></math>. The effect is absent for four flavors of degenerate quarks, where rooting is not used. For the two-flavor case, we show numerical evidence that the magnitude of the effect diminishes on finer lattices. We discuss the implications of this technical complication on future studies of the QCD phase diagram.

Matteo Giordano

Can rooted staggered fermions describe nonzero baryon density at low temperatures?

The equation of state of the quark gluon plasma is a key ingredient of heavy-ion phenomenology. In addition to the traditional Taylor method, several novel approximation schemes have been proposed with the aim of calculating it at finite baryon density. In order to gain a pragmatic understanding of the limits of these schemes, we compare them to direct results at <math><msub><mi>μ</mi><mi>B</mi></msub><mo>&gt;</mo><mn>0</mn></math>, using reweighting techniques free from an overlap problem. We use 2stout improved staggered fermions with eight time slices and cover the entire Relativistic Heavy Ion Collider Beam Energy Scan range in the baryochemical potential, up to <math><msub><mi>μ</mi><mi>B</mi></msub><mo>/</mo><mi>T</mi><mo>=</mo><mn>3</mn></math>.

Equation of state of a hot-and-dense quark gluon plasma: Lattice simulations at real <math><msub><mi>μ</mi><mi>B</mi></msub></math> vs extrapolations