Two-flavor Case Research Articles

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

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 μB predict a sharp rise in the baryon density at low temperatures and μB≳3mπ/2, where mπ 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 μB=0. 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. Published by the American Physical Society 2024

Two-loop corrections to the QCD propagators within the Curci-Ferrari model

We evaluate all two-point correlation functions of the Curci-Ferrari (CF) model in four dimensions and in the presence of mass-degenerate fundamental quark flavors, as a natural extension of an earlier investigation in the quenched approximation. In principle, the proper account of chiral symmetry breaking ($\ensuremath{\chi}\mathrm{SB}$) and the corresponding dynamical generation of a quark-mass function within the CF model requires one to go beyond perturbation theory [M. Pel\'aez, U. Reinosa, J. Serreau, M. Tissier, and N. Wschebor, Phys. Rev. D 103, 094035 (2021)]. However, it is interesting to assess whether a perturbative description applies to correlation functions that are not directly sensitive to $\ensuremath{\chi}\mathrm{SB}$, such as the gluon, ghost and quark dressing functions. We compare our two-loop results for these form factors to QCD lattice data in the two-flavor case for two different values of the pion mass, one that is relatively far from the chiral limit, and one that is closer to the physical value. Our results confirm that the QCD gluon and ghost dressing functions are well described by a perturbative approach within the CF model, as already observed at one-loop order [M. Pel\'aez, M. Tissier, and N. Wschebor, Phys. Rev. D 90, 065031 (2014)]. Our new main result is that the quark dressing function is also well captured by the perturbative approach, but only starting at two-loop order, as also anticipated [M. Pel\'aez, M. Tissier, and N. Wschebor, Phys. Rev. D 90, 065031 (2014)]. The quark-mass function predicted by the CF model at two-loop order is in good agreement with the data if the quarks are not too light but it shows some clear tension with respect to the two-loop CF dressing functions in the close to physical case, as expected. Interestingly, however, we find that there is much less tension between the nonperturbative quark-mass function, as it can be obtained from lattice simulations or from M. Pel\'aez, U. Reinosa, J. Serreau, M. Tissier, and N. Wschebor, Phys. Rev. D 103, 094035 (2021), and the two-loop CF dressing functions, which confirms the perturbative nature of the latter.

Gluodynamics and deconfinement phase transition under rotation from holography

We investigate rotating effect on deconfinement phase transition in an Einstein-Maxwell-Dilaton (EMD) model in bottom-up holographic QCD approach. By constructing a rotating black hole, which is supposed to be dual to rotating strongly coupled nuclear matter, we investigate the thermodynamic quantities, including entropy density, pressure, energy density, trace anomaly, sound speed and specific heat for both pure gluon system and two-flavor system under rotation. It is shown that those thermodynamic quantities would be enhanced by large angular velocity. Also, we extract the information of phase transition from those thermodynamic quantities, as well as the order parameter of deconfinement phase transition, i.e. the loop operators. It is shown that, in the T − ω plane, for two-flavor case with small chemical potential, the phase transition is always crossover. The transition temperature decreases slowly with angular velocity and chemical potential. For pure gluon system with zero chemical potential, the phase transition is always first order, while at finite chemical potential a critical end point (CEP) will present in the T − ω plane.

Chiral phase transition with 2+1 quark flavors in an improved soft-wall AdS/QCD model

We study the chiral phase transition with 2 + 1 quark flavors in an improved soft-wall AdS/QCD model, which can produce the light meson spectrum and many other low-energy quantities consistent with experiments in the two-flavor case. The chiral transition behaviors of the quark condensates at different quark masses are analysed in detail, and the (m_{u,d}, m_s) phase diagram for the quark sector has been obtained from the improved soft-wall model. We find that the features of the calculated phase diagram are completely consistent with the standard scenario, which is supported by lattice simulations and theoretical arguments. The evidence of a tricritical point on the m_{u,d} = 0 boundary of the (m_{u,d}, m_s) phase diagram is first clearly presented in the bottom-up AdS/QCD.

Tachyonic instability of the scalar mode prior to the QCD critical point based on the functional renormalization-group method in the two-flavor case

We establish and elucidate the physical meaning of the appearance of an acausal mode in the sigma mesonic channel, found in the previous work by the present authors, when the system approaches the $\mathrm{Z}_{2}$ critical point. The functional renormalization group method is applied to the two--flavor quark--meson model with varying current quark mass $m_q$ even away from the physical value at which the pion mass is reproduced. We first determine the whole phase structure in the three-dimensional space $(T, \mu, m_q)$ consisting of temperature $T$, quark chemical potential $\mu$ and $m_q$, with the tricritical point, $\mathrm{O}(4)$ and $\mathrm{Z}_{2}$ critical lines being located; they altogether make a wing-like shape quite reminiscent of those known in the condensed matters with a tricritical point. We then calculate the spectral functions $\rho_{\sigma, \pi}(\omega, p)$ in the scalar and pseudoscalar channel around the critical points. We find that the sigma mesonic mode becomes tachyonic with a superluminal velocity at finite momenta before the system reaches the $\mathrm{Z}_{2}$ point from the lower density, even for $m_q$ smaller than the physical value. One of the possible implications of the appearance of such a tachyonic mode at finite momenta is that the assumed equilibrium state with a uniform chiral condensate is unstable toward a state with an inhomogeneous $\sigma$ condensate. No such an anomalous behavior is found in the pseudoscalar channel. We find that the $\sigma$-to-$2\sigma$ coupling due to finite $m_q$ play an essential role for the drastic modification of the spectral function.

Prospects for neutrino spin coherence in supernovae

We present neutrino bulb model simulations of Majorana neutrino coherent spin transformation (i.e., neutrino-antineutrino transformation), coupled to neutrino flavor evolution, for conditions corresponding to the neutronization burst epoch of an Oxygen-Neon-Magnesium (O-Ne-Mg) core collapse supernova. Significant neutrino spin transformation in, for example, the neutronization burst, could alter the fluences of neutrinos and antineutrinos in a way which is potentially detectable for a Galactic core collapse supernova. Our calculations for the first time incorporate geometric dilution in the spin evolution of the neutrinos and combine two-flavor and three-flavor evolution with spin mixing physics. We find that significant spin transformations can occur, but only with a large neutrino luminosity and an electron fraction ($Y_e$) profile which facilitates adiabatic conditions for the spin-channel resonance. Using our adopted parameters of neutrino energy spectra, luminosity, density and $Y_e$ profiles, our calculations require an unrealistically large neutrino rest mass to sustain the spin transformation. It is an open question whether examining different density profiles or incorporating other sources of nonlinear feedback, such as $Y_e$ feedback, could mitigate this need. We find that spin transformations are not sensitive to the flavor structure of neutrinos, i.e., the spin transformations occur regardless of whether we simulate two- or three-flavor transformations. In the two-flavor case, spin transformations were insensitive to the choice of solar or atmospheric mass-squared splitting as well as the choice of the Majorana phase. Importantly, our three-flavor simulations, as well as our two-flavor simulations done with the atmospheric mass-squared splitting, show that the inclusion of spin degrees of freedom can significantly and qualitatively alter neutrino flavor evolution.

Pseudoscalar condensation induced by chiral anomaly and vorticity for massive fermions

We derive the pseudoscalar condensate induced by anomaly and vorticity from the Wigner function for massive fermions in homogeneous electromagnetic fields. It has an anomaly term and a force-vorticity coupling term. As a mass effect, the pseudoscalar condensate is linearly proportional to the fermion mass in small mass expansion. By a generalization to two-flavor and three-flavor cases, the neutral pion and eta meson condensates are calculated from the Wigner function and have anomaly parts as well as force-vorticity parts, in which the anomaly part of the neutral pion condensate is consistent to the previous result. We also discuss about possible observables of the condensates in heavy ion collisions such as collective flows of neutral pions and eta mesons which may be influenced by the electromagnetic field and vorticity profiles.

Three-flavor chiral effective model with four baryonic multiplets within the mirror assignment

We study three-flavor octet baryons by using the so-called extended Linear Sigma Model (eLSM). Within a quark-diquark picture, the requirement of a mirror assignment naturally leads to the consideration of four spin- ½ baryon multiplets. A reduction of the Lagrangian to the two-flavor case leaves four doublets of nucleonic states which mix to form the experimentally observed states N(939), N(1440), N(1535) and N(1650). We determine the parameters of the nucleonic part of the Lagrangian from a fit to masses and decay properties of the aforementioned states. By tracing their masses when chiral symmetry is restored, we conclude that the pairs N(939), N(1535) and N(1440), N(1650) form chiral partners.

Consistency between SU(3) and SU(2) covariant baryon chiral perturbation theory for the nucleon mass

Treating the strange quark mass as a heavy scale compared to the light quark mass, we perform a matching of the nucleon mass in the SU(3) sector to the two-flavor case in covariant baryon chiral perturbation theory. The validity of the $19$ low-energy constants appearing in the octet baryon masses up to next-to-next-to-next-to-leading order~\cite{Ren:2014vea} is supported by comparing the effective parameters (the combinations of the $19$ couplings) with the corresponding low-energy constants in the SU(2) sector~\cite{Alvarez-Ruso:2013fza}. In addition, it is shown that the dependence of the effective parameters and the pion-nucleon sigma term on the strange quark mass is relatively weak around its physical value, thus providing support to the assumption made in Ref.~\cite{Alvarez-Ruso:2013fza}.

Realization of chiral symmetry breaking and restoration in holographic QCD

With proper profiles of the scalar potential and the dilaton field, for the first time, the spontaneous chiral symmetry breaking in the vacuum and its restoration at finite temperature are correctly realized in the holographic QCD framework. In the chiral limit, a nonzero chiral condensate develops in the vacuum and decreases with temperature, and the phase transition is of 2nd order for two-flavor case and of 1st order for three-flavor case. In the case of explicit chiral symmetry breaking, in two-flavor case, the 2nd order phase transition turns to crossover with any nonzero current quark mass, and in three-flavor case, the 1st order phase transition turns to crossover at a finite current quark mass. The correct description of chiral symmetry breaking and restoration makes the holographic QCD models more powerful in dealing with non-perturbative QCD phenomena. This framework can be regarded as a general set up in application of AdS/CFT to describe conventional Ginzburg-Landau-Wilson type phase transitions, e.g. in condensed matter and cosmology systems.

Chiral phase transition in the soft-wall model of AdS/QCD

We investigate the chiral phase transition in the soft-wall model of AdS/QCD at zero chemical potential for two-flavor and three-flavor cases, respectively. We show that there is no spontaneous chiral symmetry breaking in the original soft-wall model. After detailed analysis, we find that in order to realize chiral symmetry breaking and restoration, both profiles for the scalar potential and the dilaton field are essential. The scalar potential determines the possible solution structure of the chiral condensate, except the mass term, it takes another quartic term for the two-flavor case, and for the three-flavor case, one has to take into account an extra cubic term due to the t'Hooft determinant interaction. The profile of the dilaton field reflects the gluodynamics, which is negative at a certain ultraviolet scale and approaches positive quadratic behavior at far infrared region. With this set-up, the spontaneous chiral symmetry breaking in the vacuum and its restoration at finite temperature can be realized perfectly. In the two-flavor case, it gives a second order chiral phase transition in the chiral limit, while the transition turns to be a crossover for any finite quark mass. In the case of three-flavor, the phase transition becomes a first order one in the chiral limit, while above sufficient large quark mass it turns to be a crossover again. This scenario agrees exactly with the current understanding on chiral phase transition from lattice QCD and other effective model studies.

Three-flavor chiral effective model with four baryonic multiplets within the mirror assignment

In the case of three quark flavors, (pseudo)scalar diquarks transform as antiquarks under chiral transformations. We construct four spin-1/2 baryonic multiplets from left- and right-handed quarks as well as left- and right-handed diquarks. The fact that two of these multiplets transform in a "mirror" way allows for chirally invariant mass terms. We then embed these baryonic multiplets into the Lagrangian of the so-called extended Linear Sigma Model, which features (pseudo)scalar and (axial-)vector mesons, as well as glueballs. Reducing the Lagrangian to the two-flavor case, we obtain four doublets of nucleonic states. These mix to produce four experimentally observed states with definite parity: the positive-parity nucleon $N(939)$ and Roper resonance $N(1440)$, as well as the negative-parity resonances $N(1535)$ and $N(1650)$. We determine the parameters of the nucleonic part of the Lagrangian from a fit to masses and decay properties of the aforementioned states. Studying the limit of vanishing quark condensate, we conclude that $N(939)$ and $N(1535)$, as well as $N(1440)$ and $N(1650)$ form pairs of chiral partners.

On the minimality of the order p6 chiral Lagrangian

A method to find relations between the operators in the mesonic Lagrangian of Chiral Perturbation Theory at order p6 is presented. The procedure can be used to establish if the basis of operators in the Lagrangian is minimal. As an example, we apply the method to the two-flavor case in the absence of scalar and pseudo-scalar sources (s=p=0), and conclude that the minimal Lagrangian contains 27 independent operators.

Full pseudoscalar mesonic chiral Lagrangian atp6order under the unitary group

We construct the full p6 order chiral Lagrangians for the unitary group and special unitary groups, including nf-, three- and two-flavor cases, all bilinear currents (scalar, pseudoscalar, vector, axial-vector and tensor currents) and theta parameter. The number of independent operators are 1391, 1326 and 969 for each of the flavor unitary groups. From these results, we find one extra linear relation among the traditional p4 order low-energy constants under the U(3) group, and some more linear relations with tensor sources for the p6 order low-energy constants in the special unitary groups. We develop a scheme to obtain the relations for the dependent operators in terms of independent operators.

Computation of thep6order low-energy constants with tensor sources

We present the results of calculations of the $p^4$ and $p^6$ order low-energy constants for the chiral Lagrangian with tensor sources for both two and three flavors of pseudoscalar mesons. This is a generalization of our previous work on similar calculations without tensor sources in terms of the quark self-energy $\Sigma(p^2)$, based on the first principle derivation of the low-energy effective Lagrangian and computation of the low-energy constants with some rough approximations. With the help of partial integration and some epsilon relations, we find that some $p^6$ order operators with tensor sources appearing in the literature are related to each other. That leaves 98 independent terms for $n$-flavor, 92 terms for three-flavor, and 65 terms for two-flavor cases. We also find that the odd-intrinsic-parity chiral Lagrangian with tensor sources cannot independently exist in any order of low-energy expansion.

Staggered chiral perturbation theory in the two-flavor case

I study two-flavor staggered chiral perturbation theory in the light pseudoscalar sector. The pion mass and decay constant are calculated through NLO in the partially-quenched case. In the limit where the strange quark mass is large compared to the light quark masses and the taste splittings, I show that the SU(2) staggered chiral theory emerges from the SU(3) staggered chiral theory, as expected. Explicit relations between SU(2) and SU(3) low energy constants and taste-violating parameters are given. The results are useful for SU(2) chiral fits to asqtad data and allow one to incorporate effects from varying strange quark masses.

Thermodynamics and quark susceptibilities: A Monte Carlo approach to the Polyakov–Nambu–Jona-Lasinio model

The Monte-Carlo method is applied to the Polyakov-loop extended Nambu--Jona-Lasinio (PNJL) model. This leads beyond the saddle-point approximation in a mean-field calculation and introduces fluctuations around the mean fields. We study the impact of fluctuations on the thermodynamics of the model, both in the case of pure gauge theory and including two quark flavors. In the two-flavor case, we calculate the second-order Taylor expansion coefficients of the thermodynamic grand canonical partition function with respect to the quark chemical potential and present a comparison with extrapolations from lattice QCD. We show that the introduction of fluctuations produces only small changes in the behavior of the order parameters for chiral symmetry restoration and the deconfinement transition. On the other hand, we find that fluctuations are necessary in order to reproduce lattice data for the flavor non-diagonal quark susceptibilities. Of particular importance are pion fields, the contribution of which is strictly zero in the saddle point approximation.

Unquenched Effects and Quark Mass Dependence of Lattice Gluon Propagator in Infrared Region

In this paper, the gluon propagator in Landau gauge has been studied on a lattice, including the quenched and the unquenched one. The small geometry size of lattice we use is 163 × 32, and the big one is 203 × 64. For the quenched approximation, we fit the numerical results and give a little different fitting values. We also obtain unquenched effects by comparing the gluon propagator resulting from the quenched and unquenched configurations, for both the two-flavor and three-flavor cases. For the unquenched configurations, an obvious quark mass dependence has not been found in the small quark mass case, but is found in the three-flavor case when the quark mass is big.

Crystallography of three-flavor quark matter

We analyze and compare candidate crystal structures for the crystalline color superconducting phase that may arise in cold, dense but not asymptotically dense, three-flavor quark matter. We determine the gap parameter $\ensuremath{\Delta}$ and free energy $\ensuremath{\Omega}(\ensuremath{\Delta})$ for many possible crystal structures within a Ginzburg-Landau approximation, evaluating $\ensuremath{\Omega}(\ensuremath{\Delta})$ to order ${\ensuremath{\Delta}}^{6}$. In contrast to the two-flavor case, we find a positive ${\ensuremath{\Delta}}^{6}$ term and hence an $\ensuremath{\Omega}(\ensuremath{\Delta})$ that is bounded from below for all the structures that we analyze. This means that we are able to evaluate $\ensuremath{\Delta}$ and $\ensuremath{\Omega}$ as a function of the splitting between Fermi surfaces for all the structures we consider. We find two structures with particularly robust values of $\ensuremath{\Delta}$ and the condensation energy, within a factor of 2 of those for the CFL phase which is known to characterize QCD at asymptotically large densities. The robustness of these phases results in their being favored over wide ranges of density. However, it also implies that the Ginzburg-Landau approximation is not quantitatively reliable. We develop qualitative insights into what makes a crystal structure favorable, and use these to winnow the possibilities. The two structures that we find to be most favorable are both built from condensates with face-centered cubic symmetry: in one case, the $⟨ud⟩$ and $⟨us⟩$ condensates are separately face-centered cubic; in the other case $⟨ud⟩$ and $⟨us⟩$ combined make up a face-centered cube.

Chiral extrapolations of baryon masses for unquenched three-flavor lattice simulations

We have analyzed the chiral expansion of the baryon masses to second order in the quark masses, including strong isospin violation. The calculations are performed in Lorentz-invariant baryon chiral perturbation theory. Chiral extrapolation functions for three-flavor unquenched lattice simulations are derived. The matching to the two-flavor case and the sigma terms are also considered.