Two-color QCD Research Articles

Chiral Effective Model of Cold and Dense Two-Color QCD: The Linear Sigma Model Approach

This review is devoted to summarizing recent developments of the linear sigma model (LSM) in cold and dense two-color QCD (QC2D), in which lattice simulations are straightforwardly applicable thanks to the disappearance of the sign problem. In QC2D, both theoretical and numerical studies derive the presence of the so-called baryon superfluid phase at a sufficiently large chemical potential (μq), where diquark condensates govern the ground state. The hadron mass spectrum simulated in this phase shows that the mass of an iso-singlet (I=0) and 0− state is remarkably reduced, but such a mode cannot be described by the chiral perturbation theory. Motivated by this fact, I have invented a LSM constructed upon the linear representation of chiral symmetry, more precisely Pauli–Gürsey symmetry. It is shown that my LSM successfully reproduces the low-lying hadron mass spectrum in a broad range of μq simulated on the lattice. As applications of the LSM, topological susceptibility and sound velocity in cold and dense QC2D are evaluated to compare with the lattice results. Additionally, the generalized Gell–Mann–Oakes–Renner relation and hardon mass spectrum in the presence of a diquark source are analyzed. I also introduce an extended version of the LSM incorporating spin-1 hadrons.

Lattice study on finite density QC2D towards zero temperature

We investigate the phase structure and the equation of state (EoS) for dense two-color QCD (QC2D) at low temperature (T = 40 MeV, 324 lattice) for the purpose of extending our previous works [1, 2] at T = 80 MeV (164 lattice). Indeed, a rich phase structure below the pseudo-critical temperature Tc as a function of quark chemical potential μ has been revealed, but finite volume effects in a high-density regime sometimes cause a wrong understanding. Therefore, it is important to investigate the temperature dependence down to zero temperature with large-volume simulations. By performing 324 simulations, we obtain essentially similar results to the previous ones, but we are now allowed to get a fine understanding of the phase structure via the temperature dependence. Most importantly, we find that the hadronic-matter phase, which is composed of thermally excited hadrons, shrinks with decreasing temperature and that the diquark condensate scales as ⟨qq⟩ ∝ μ2 in the BCS phase, a property missing at T = 80 MeV. From careful analyses, furthermore, we confirm a tentative conclusion that the topological susceptibility is independent of μ. We also show the temperature dependence of the pressure, internal energy, and sound velocity as a function of μ. The pressure increases around the hadronic-superfluid phase transition more rapidly at the lower temperature, while the temperature dependence of the sound velocity is invisible. Breaking of the conformal bound is also confirmed thanks to the smaller statistical error.

Order and chaos in the SU(2) matrix model: Ergodicity and classical phases

We study the classical nonlinear dynamics of the SU(2) Yang-Mills matrix model introduced in [] as a low-energy approximation to two-color QCD. Restricting to the spin-0 sector of the model, we unearth an unexpected tetrahedral symmetry, which endows the dynamics with an extraordinarily rich structure. Among other things, we find that the spin-0 sector contains coexisting chaotic subsectors as well as nested chaotic “basins”, and displays alternation between regular and chaotic dynamics as energy is varied. The symmetries also grant us a considerable amount of analytic control which allows us to make several quantitative observations. We see that the classical spin-0 sector has a rich phase structure, arising from ergodicity breaking. Surprisingly, we find that many of these classical phases display numerous similarities to previously discovered phases of the spin-0 sector [], and we explore these similarities in a heuristic fashion. Published by the American Physical Society 2024

Sound velocity peak induced by the chiral partner in dense two-color QCD

Recently, the peak structure of the sound velocity was observed in the lattice simulation of two-color and two-flavor QCD at the finite quark chemical potential. The comparison with the chiral perturbation theory (ChPT) result was undertaken; however, the ChPT failed in reproducing the peak structure. In this study, to extend the ChPT framework, we incorporate contributions of the σ meson, that is identified as the chiral partner of pions, on top of the low-energy pion dynamics by using the linear sigma model (LSM). Based on the LSM, we derive analytic expressions of the thermodynamic quantities as well as the sound velocity within a mean-field approximation. As a result, we find that those quantities are provided by sums of the ChPT results and corrections, where the latter is characterized by a mass difference between the chiral partners, the σ meson and pion. The chiral partner contributions are found to yield a peak in the sound velocity successfully. We furthermore show that the sound velocity peak emerges only when mσ>3mπ and μq>mπ, with mσ(π) and μq being the σ meson (pion) mass and the quark chemical potential, respectively. The correlation between the sound velocity peak and the sign of the trace anomaly is also addressed. Published by the American Physical Society 2024

Mass spectrum of spin-one hadrons in dense two-color QCD: Novel predictions by extended linear sigma model

We construct an extended version of the linear sigma model in such a way as to describe spin-1 hadrons as well as spin-0 hadrons in two-color QCD (QC2D) by respecting the Pauli-Gürsey SU(4) symmetry. Within a mean-field approximation, we therefrom examine a mass spectrum of the spin-1 hadrons at finite quark chemical potential (μq) and zero temperature. Not only mean fields of scalar mesons and scalar-diquark baryons but also of vector mesons and vector-diquark baryons are incorporated. As a result, we find that, unless all of those four types of mean fields are taken into account, neither lattice result for the critical μq that corresponds to the onset of baryon superfluidity nor for μq dependence of the pion mass can be reproduced. We also find that a slight suppression of the ρ meson mass in the superfluid phase, which was suggested by the lattice simulation, is reproduced by subtle mixing effects between spin-0 and spin-1 hadrons. Moreover, we demonstrate the emergence of an axial-vector condensed phase and possibly of a vector condensed phase by identifying the values of μq at which the corresponding hadron masses vanish. The possible presence of isotriplet 1− diquarks that may be denoted by a tensor-type quark bilinear field is also discussed. Published by the American Physical Society 2024

Fate of the topological susceptibility in two-color dense QCD

We explore the topological susceptibility at finite quark chemical potential and zero temperature in two-color QCD (QC2D) with two flavors. Through the Ward-Takahashi identities of QC2D, we find that the topological susceptibility in the vacuum solely depends on three observables: the pion decay constant, the pion mass, and the η mass in the low-energy regime of QC2D. Based on the identities, we numerically evaluate the topological susceptibility at finite quark chemical potential using the linear sigma model with the approximate Pauli-Gursey SU(4) symmetry. Our findings indicate that, in the absence of U(1)A anomaly effects represented by the Kobayashi-Maskawa-’t Hooft-type determinant interaction, the topological susceptibility vanishes in both the hadronic and baryon superfluid phases. On the other hand, when the U(1)A anomaly effects are present, the constant and nonzero topological susceptibility is induced in the hadronic phase, reflecting the mass difference between the pion and η meson. Meanwhile, in the superfluid phase it begins to decrease smoothly. The asymptotic behavior of the decrement is fitted by the continuous reduction of the chiral condensate in dense QC2D, which is similar to the behavior observed in hot three-color QCD matter. In addition, effects from the finite diquark source on the topological susceptibility are discussed. We expect that the present study provides a clue to shed light on the role of the U(1)A anomaly in cold and dense QCD matter.

Phase Diagram of Dense Two-Color QCD at Low Temperatures

This review is devoted to the modern understanding of the two-color QCD phase diagram at finite baryon density and low temperatures. First, we consider the theoretical picture of this phase diagram. It is believed that at low baryon density, two-color QCD can be described by chiral perturbation theory (ChPT), which predicts a second-order phase transition with Bose-Einstein condensation of diquarks at μ=mπ/2. At larger baryon chemical potentials, the interactions between baryons become important, and ChPT is not applicable anymore. At sufficiently large baryon chemical potential, the Fermi sphere composed of quarks is formed, and diquarks are condensed on the surface of this sphere. In this region, two-color baryon matter reveals properties similar to those of the Quarkyonic phase. Particular attention in this review is paid to lattice studies of dense two-color QCD phase diagram. In the low-density region, the results of lattice studies are in agreement with ChPT predictions. At sufficiently large baryon densities, lattice studies observe a Fermi sphere composed of quarks and condensation of diquarks on its surface. Thus, available lattice studies support most of the theoretical predictions. Finally, we discuss the status of the deconfinement in cold dense two-color matter, which was observed in lattice simulation with staggered fermions.

Probing the hadron mass spectrum in dense two-color QCD with the linear sigma model

We investigate modifications of hadron masses at finite quark chemical potential in two-flavor and two-color QCD, data of which are available from lattice simulations, within a linear sigma model based on approximate Pauli-Gursey $SU(4)$ symmetry. The model describes not only ground-state scalar diquarks and pseudoscalar mesons but also the excited pseudoscalar diquarks and scalar mesons; each ground-state diquark (meson) has the corresponding excited diquark (hadron) with opposite parity as a chiral partner. Effects of chiral symmetry breaking and diquark condensates are incorporated by a mean-field treatment. We show that various mixings among the hadrons, which are triggered by the breakdown of baryon number conservation in the superfluid phase, lead to a rich hadron mass spectrum. We discuss the influence of $U(1{)}_{A}$ anomaly on the density dependence of the mass spectrum and also manifestations of the chiral partner structures as the density increases in the superfluid phase. The predicted hadron masses are expected to provide future lattice simulations with useful information on such symmetry properties in dense two-color QCD.

The θ-angle and axion physics of two-color QCD at fixed baryon charge

We analyze the impact of the θ-angle and axion dynamics for two-color (in fact any Sp(2N )) QCD at nonzero baryon charge and as a function of the number of matter fields on the vacuum properties, the pattern of chiral symmetry breaking as well as the spectrum of the theory. We show that the vacuum acquires a rich structure when the underlying CP violating topological operator is added to the theory. We discover novel phases and analyse the order of their transitions characterizing the dynamics of the odd and even number of flavours. We further determine the critical chemical potential as function of the θ angle separating the normal from the superfluid phase of the theory. Our results will guide numerical simulations and novel tests of the model’s dynamics. The results are also expected to better inform phenomenological applications of the model ranging from composite Higgs physics to strongly interacting massive dark matter models featuring number changing interactions. In the companion work [1] we repurpose and upgrade the approach to determine the impact of the θ-angle and axion physics on non-perturbative near conformal dynamics related to the fixed baryon charge sector.

Chiral Asymmetry and Phase Diagram of the Two-Color QCD

Chiral Asymmetry and Phase Diagram of the Two-Color QCD

Topological Aspects of Dense Matter: Lattice Studies

Topological fluctuations change their nature in the different phases of strong interactions, and the interrelation of topology, chiral symmetry and confinement at high temperature has been investigated in many lattice studies. This review is devoted to the much less explored subject of topology in dense matter. After a short overview of the status at zero density, which will serve as a baseline for the discussion, we will present lattice results for baryon rich matter, which, due to technical difficulties, has been mostly studied in two-color QCD, and for matter with isospin and chiral imbalances. In some cases, a coherent pattern emerges, and in particular the topological susceptibility seems suppressed at high temperature for baryon and isospin rich matter. However, at low temperatures the topological aspects of dense matter remain not completely clear and call for further studies.

Delineating chiral separation effect in two-color dense QCD

We study the chiral separation effect (CSE) in two-color and two-flavor QCD to delineate quasiparticle pictures in dense matter from low to high temperatures. Both massless and massive quarks are discussed. We particularly focus on the high density domain where diquarks form a color-singlet condensate with the electric charge $1/3$. The condensate breaks the baryon number and $U(1{)}_{A}$ axial symmetry, and induces the electromagnetic Meissner effects. Within a quark quasiparticle picture, we compute the chiral separation conductivity at one-loop. We have checked that Nambu-Goldstone modes, which should appear in the improved vertices as required by the Ward-Takahashi identities, do not contribute to the chiral-separation conductivity due to their longitudinal natures. In the static limit, the destructive interferences in the particle-hole channel, as in the usual Meissner effects, suppress the conductivity (in the chiral limit, to $1/3$ of that of the normal phase). This locally breaks the universality of the CSE coefficients, provided quasiparticle pictures are valid in the bulk matter.

Numerical study of the chiral separation effect in two-color QCD at finite density

We study the Chiral Separation Effect (CSE) in finite-density SU(2) lattice gauge theory with dynamical quarks. We find that the CSE is well described by the free quark result in the high-temperature quark-gluon plasma phase. As one enters the confinement regime with broken chiral symmetry at chemical potential smaller than half of the pion mass, the CSE response is gradually suppressed towards low temperatures in comparison to the free quark result. This suppression can be approximately described by assuming that the CSE current is proportional to the charge density, rather than the chemical potential, as suggested in the literature (ArXiv:1712.01256, Phys.Rev.D97(2018)085020). We also provide an upper bound on the contribution of disconnected fermionic diagrams to the CSE, which is consistent with zero within our statistical errors and small compared to that of the connected diagrams. Our results are obtained mainly in the QCD-like regime of SU(2) gauge theory at low densities, and hence should be at least qualitatively applicable to QCD as well.

New universality classes of the non-Hermitian Dirac operator in QCD-like theories

In non-Hermitian random matrix theory there are three universality classes for local spectral correlations: the Ginibre class and the nonstandard classes ${\mathrm{AI}}^{\ifmmode\dagger\else\textdagger\fi{}}$ and ${\mathrm{AII}}^{\ifmmode\dagger\else\textdagger\fi{}}$. We show that the continuum Dirac operator in two-color QCD coupled to a chiral U(1) gauge field or an imaginary chiral chemical potential falls in class ${\mathrm{AI}}^{\ifmmode\dagger\else\textdagger\fi{}}$ (${\mathrm{AII}}^{\ifmmode\dagger\else\textdagger\fi{}}$) for fermions in pseudoreal (real) representations of SU(2). We introduce the corresponding chiral random matrix theories and verify our predictions in lattice simulations with staggered fermions, for which the correspondence between representation and universality class is reversed. Specifically, we compute the complex eigenvalue spacing ratios introduced recently. We also derive novel spectral sum rules.

Unitary matrix integral for two-color QCD and the GSE-GUE crossover in random matrix theory

We analytically evaluate a unitary matrix integral which appears in the low-energy limit of two-color QCD at finite chemical potential. The result is expressed as a pfaffian. We illustrate its application to the GSE-GUE crossover in random matrix theory.

Thermal quarks and gluon propagators in two-color dense QCD

We study Landau gauge gluon propagators in two-color QCD at finite quark chemical potential ($\mu_q$) and temperature ($T$). We include medium polarization effects at one-loop by quarks into massive gluon propagators, and compared the analytic results with the available lattice data. We particularly focus on the high density phase of color-singlet diquark condensates whose critical temperature is $\sim 100$ MeV with weak dependence on $\mu_q$. At zero temperature the color singlet condensates protect the IR limit of electric and magnetic gluon propagators from the medium screening effects. At finite temperature, this behavior remains true for the magnetic sector, but the electric screening mass should be generated by thermal, and hence gapless, particles which are unbound from the diquark condensates. Treating thermal excitations as quasi-quarks, we found that the electric screening develops too fast compared to the lattice results. Beyond the critical temperature for diquark condensates the analytic results are consistent with the lattice results.

Quark distribution inside a pion in many-flavor ( 2+1 )-dimensional QCD using lattice computations: UV listens to IR

We study the changes in the short-distance quark structure of the Nambu-Goldstone boson when the long-distance symmetry-breaking scales are depleted controllably. We achieve this by studying the valence Parton Distribution Function (PDF) of pion in 2+1 dimensional two-color QCD, with the number $N$ of massless quarks as the tunable parameter that slides the theory from being strongly broken for $N=0$ to the conformal window for $N>4$, where the theory is gapped by the fixed finite volume. We perform our study non-perturbatively using lattice simulations with $N=0,2,4,8$ flavors of nearly massless two-component Wilson-Dirac sea quarks and employ the leading-twist formalism (LaMET/SDF) to compute the PDF of pion at a fixed valence mass. We find that the relative variations in the first few PDF moments are only mild compared to the changes in decay constant, but the shape of the reconstructed $x$-dependent PDF itself dramatically changes from being broad in the scale-broken sector to being sharply peaked in the near-conformal region, best reflected in PDF shape observables such as the cumulants.

Gluons in two-color QCD at high baryon density

Landau gauge longitudinal and transverse gluon propagators are studied in lattice QCD with gauge group [Formula: see text] at varying temperature and quark density. In particular, it is found that the longitudinal propagator decreases with increasing quark chemical potential at all temperatures under study, whereas the transverse propagator increases with increasing quark chemical potential at [Formula: see text] MeV and does not depend on it at higher temperatures. The relative strength of chromoelectric and chromomagnetic interactions is also discussed.

Effects of dense quark matter on gluon propagators in lattice QC2D

The transverse and longitudinal gluon propagators in the Landau gauge are studied in the two-color lattice QCD at nonzero quark chemical potential $\mu_q$. Parameterization of the momentum dependence of the propagators is provided for all values of chemical potential under study. We find that the longitudinal propagator is infrared suppressed at nonzero $\mu_q$ with suppression increasing with increasing $\mu_q$. The transverse propagator dependence on $\mu_q$ was found to be opposite: it is enhanced at large $\mu_q$. It is found, respectively, that the electric screening mass is increasing while the magnetic screening mass is decreasing with increasing $\mu_q$. Nice agreement between the electric screening mass computed from the longitudinal propagator and the Debye mass computed earlier from the singlet static quark-antiquark potential was found. We discuss how the dependence of the propagators on the chemical potential correlates with the respective dependence of the string tension. Additionally, we consider the difference between two propagators as a function of the momentum and make interesting observations.

The dual properties of two-color QCD with baryon, chiral and isospin densities

The phase structure of dense quark matter in the two color case has been investigated with nonzero baryon μB , isospin μI and chiral isospin μ I5 chemical potentials. It has been demonstrated that due to the duality properties the phase diagram is extremely symmetric and the whole phase diagram in the case of two colors can be obtained just by dualities from the phase structure of three color case. This shows that the dualities are rather useful tool.