Chiral Condensate Research Articles

Gross-Neveu model with O(2)L×O(2)R chiral symmetry: Duality with Zakharov-Mikhailov model and large N solution

The two-flavor Gross-Neveu model with ${\mathrm{U}(2)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{U}(2{)}_{R}$ chiral symmetry in $1+1$ dimensions is used to construct a novel variant of four-fermion theories with ${\mathrm{O}(2)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{O}(2{)}_{R}$ chiral symmetry. The spontaneous breaking of the group O(2), a continuous group with two connected components (rotations and reflections), gives rise to new phenomena. It is ideally suited to describe a situation where two distinct kinds of condensation compete, in particular chiral symmetry breaking (particle-hole condensation) and Cooper pairing (particle-particle condensation). After solving the O(2) chiral Gross-Neveu model in detail, we demonstrate that it is dual to another classically integrable model due to Zakharov and Mikhailov. The duality enables us to solve the quantum version of this model in the large $N$ limit with semiclassical methods, supporting its integrability at the quantum level. The resulting model is the unique four-fermion theory sharing the full Pauli-G\"ursey symmetry with free, massless fermions (``perfect Gross-Neveu model'') and provides us with a solvable model for competing chiral and Cooper pair condensates, including explicit soliton dynamics and the phase diagram.

Equation of state for strange quark matter: Linking the Nambu–Jona-Lasinio model to perturbative QCD

Neutron star constraints and {\it ab initio} pQCD evaluations require the EoS representing cold quark matter to be stiff at intermediate baryonic densities and soft at high-$n_B$. Here, I suggest that the three flavor NJL model with a density dependent repulsive coupling, $G_V(\mu)$, can generate an EoS which interpolates between these two regimes. Such an interpolation requires repulsion to start decreasing with the chemical potential just after chiral transition takes place. The conjecture behind this mechanism is that repulsion should be necessary only as long as the quark condensates, which dress the effective masses, have non-vanishing values. This assumption guarantees that an initially hard EoS suffers a conspicuous change of slope at ${\cal E} \simeq 0.7 \,{\rm GeV fm^{-3}}$ converging to the pQCD results at higher energy densities. Then, the speed of sound naturally reaches a non-conformal maximum at $n_B = 3.23 \, n_0 = 0.52 \, {\rm fm}^{-3}$ while the trace anomaly remains positive for all densities, in agreement with recent investigations. These non-trivial results {\it cannot} be simultaneously obtained when $G_V$ vanishes or has a fixed value. Therefore, the simple model proposed here is able to link the (non-perturbative) region of intermediate densities to the region where pQCD becomes reliable.

Generating the electro-weak scale by vector-like quark condensation

We show that vector-like quarks in the fundamental or higher-dimensional representations of QCD can generate the electro-weak scale in a phenomenologically viable way by chiral symmetry breaking condensates. The thereby generated scales are determined by numerically solving the Dyson-Schwinger equation and these scales are sizable, because they grow with the hard vector-like mass. Communicating such a scale to the Standard Model via a conformally invariant scalar sector can dynamically generate the electro-weak scale without a naturalness problem, because all non-dynamical mass scales are protected by chiral symmetry. We present a minimal setup which requires only a new neutral scalar with mass not too far above the electro-weak scale, as well as vector-like quarks at the (multi-)TeV scale. Both are consistent with current bounds and are attractive for future experimental searches at the LHC and future colliders. Depending on the hypercharge of the vector-like quarks, hadrons made of them are color-neutral bound states which would be interesting Dark Matter candidates.

Gauge dependence of the quark gap equation: An exploratory study

We study the gauge dependence of the quark propagator in quantum chromodynamics by solving the gap equation with a nonperturbative quark-gluon vertex which is constrained by longitudinal and transverse Slavnov-Taylor identities, the discrete charge conjugation and parity symmetries and which is free of kinematic singularities in the limit of equal incoming and outgoing quark momenta. We employ gluon propagators in renormalizable $R_\xi$ gauges obtained in lattice QCD studies. We report the dependence of the nonperturbative quark propagator on the gauge parameter, in particular we observe an increase, proportional to the gauge-fixing parameter, of the mass function in the infrared domain, whereas the wave renormalization decreases within the range $0 \leq \xi \leq 1$ considered here. The chiral quark condensate reveals a mild gauge dependence in the region of $\xi$ investigated. We comment on how to build and improve upon this exploratory study in future in conjunction with generalized gauge covariance relations for QCD.

Behavior of a chiral condensate around astrophysical-mass Schwarschild and Reissner-Nordström black holes

In this work, we develop a perturbative method to describe the behavior of a chiral condensate around a spherical black hole whose mass is astrophysically realistic. We use the inverse mass as the expansion parameter for our perturbative series. We test this perturbative method in the case of a Schwarzschild black hole, and we find that it agrees well with previous numerical results. For an astrophysical-mass Schwarzschild black hole, the leading order contribution to the condensate is much larger (in most of space) than the next-to-leading order contribution, providing further evidence for the validity of the perturbative approach. The size of the bubble of restored chiral symmetry is directly proportional to the size of the black hole.Next, we apply this perturbative method to a Reissner-Nordström (RN) black hole. We find that, as the charge-to-mass ratio increases, the bubble of restored chiral symmetry becomes larger relative to the black hole. This effect is particularly pronounced for near-extremal RN black holes. The case of an extremal RN black hole provides an interesting counterexample to the standard thermal explanation for the formation of a bubble of restored chiral symmetry around a black hole.

Dirac sea effects on heavy quarkonia decay widths in magnetized matter: A field theoretical model of composite hadrons

We study the partial decay widths of charmonium (bottomonium) states to $D\bar D (B\bar B)$ mesons in magnetized (nuclear) matter using a field theoretic model of composite hadrons with quark (and antiquark) constituents. These are computed from the mass modifications of the decaying and produced mesons, including the nucleon Dirac sea effects within a chiral effective model. The Dirac sea contributions are observed to lead to an rise (drop) in the quark condensates as the magnetic field is increased, an effect called the (inverse) magnetic catalysis. In the presence of a magnetic field, there are mixings of spin 0 (pseudoscalar) and spin 1 (vector) states (PV mixing) The nucleon Dirac sea effects are observed to be significant and the anomalous magnetic moments (AMMs) of the nucleons are observed to play an important role. For $\rho_B$=0, there is observed to be magnetic catalysis (MC) without and with AMMs, whereas, for $\rho_B=\rho_0$, the inverse magnetic catalysis (IMC) is observed when the AMMs are taken into account, contrary to MC, when the AMMs are ignored. The magnetic field effects on the heavy quarkonium decay widths should have observable consequences on the production of the hidden and open heavy flavour mesons from ultra-relativistic peripheral heavy ion collision experiments, at RHIC and LHC, where the produced magnetic fields are extremely large.

Thermalization and prethermalization in the soft-wall AdS/QCD model

The real-time dynamics of chiral phase transition is investigated in a two-flavor ($N_f=2$) soft-wall AdS/QCD model. To understand the dynamics of thermalization, we quench the system from initial states deviating from the equilibrium states. Then, we solve the nonequilibrium evolution of the order parameter (chiral condensate $\langle \sigma\equiv\bar{q}q\rangle$). It is shown that the system undergoes an exponential relaxation at temperatures away from the critical temperature $T_c$. The relaxation time diverges at $T_c$, presenting a typical behavior of critical slowing down. Numerically, we extract the dynamic critical exponent $z$, and get $z\approx 2$ by fitting the scaling behavior $\sigma\propto t^{-\beta/(\nu z)}$, where the mean-field static critical exponents (order parameter critical exponent $\beta=1/2$, correlation length critical exponent $\nu=1/2$ ) have been applied. More interestingly, it is remarked that, for a large class of initial states, the system would linger over a quasi-steady state for a certain period of time before the thermalization. It is suggested that the interesting phenomenon, known as prethermalization, has been observed in the framework of holographic models. In such prethermal stage, we verify that the system is characterized by a universal dynamical scaling law and described by the initial-slip exponent $\theta=0$.

Hadron spectra and pion form factor in dynamical holographic QCD model with anomalous 5D mass of scalar field* *Supported in part by the National Natural Science Foundation of China (NSFC) (12235016, 12221005, 11725523, 11735007, 12275108), the Strategic Priority Research Program of Chinese Academy of Sciences (XDB34030000, XDPB15), the start-up funding from University of Chinese Academy of Sciences (UCAS), and the

The simplest version of the dynamical holographic QCD model is described by adding the KKSS model action on a dilaton-graviton coupled background, in which the AdS metric is deformed by the gluon condensation and further deformed by the chiral condensation. In this framework, both the chiral symmetry breaking and linear confinement can be realized. The light-flavor hadron spectra and the pion form factor were investigated, but it was difficult to reconcile the light-flavor hadron spectra and pion form factor. By considering the anomalous 5-dimension mass correction of the scalar field from QCD running coupling, it is found that the light flavor hadron spectra and pion form factor can be described well simultaneously. In particular, the ground state and lower excitation states of the scalar, pseudo scalar, and axial vector meson spectra are improved. However, the vector meson spectra are not sensitive to the anomalous 5-dimension mass correction of the scalar field.

Compact stellar structures in Weyl geometric gravity

We consider the structure and physical properties of specific classes of neutron, quark, and Bose-Einstein condensate stars in the conformally invariant Weyl geometric gravity theory. The basic theory is derived from the simplest conformally invariant action, constructed, in Weyl geometry, from the square of the Weyl scalar, the strength of the Weyl vector, and a matter term, respectively. The action is linearized in the Weyl scalar by introducing an auxiliary scalar field. To keep the theory conformally invariant the trace condition is imposed on the matter energy-momentum tensor. The field equations are derived by varying the action with respect to the metric tensor, Weyl vector field and scalar field. By adopting a static spherically symmetric interior geometry, we obtain the field equations, describing the structure and properties of stellar objects in Weyl geometric gravity. The solutions of the field equations are obtained numerically, for different equations of state of the neutron and quark matter. More specifically, constant density stellar models, and models described by the stiff fluid, radiation fluid, quark bag model, and Bose-Einstein condensate equations of state are explicitly constructed numerically in both general relativity and Weyl geometric gravity, thus allowing an in depth comparison between the predictions of these two gravitational theories. As a general result it turns out that for all the considered equations of state, Weyl geometric gravity stars are more massive than their general relativistic counterparts. As a possible astrophysical application of the obtained results we suggest that the recently observed neutron stars, with masses in the range of $2{M}_{\ensuremath{\bigodot}}$ and $3{M}_{\ensuremath{\bigodot}}$, respectively, could be in fact conformally invariant Weyl geometric neutron or quark stars.

Restriction on the form of the quark anomalous magnetic moment from lattice QCD results* *Supported in part by the National Natural Science Foundation of China (12235016, 12221005, 11725523, 11735007), the Strategic Priority Research Program of Chinese Academy of Sciences (XDB34030000, XDPB15), start-up funding from the University of Chinese Academy of Sciences (UCAS), and the Fundamental Research Funds for the Central

The quark anomalous magnetic moment (AMM) is dynamically generated through spontaneous chiral symmetry breaking. It has been revealed that, even though its exact form is still unknown, the quark AMM is essential to exploring quark matter properties and QCD phase structure under external magnetic fields. In this study, we take three different forms of the quark AMM and investigate its influence on the chiral phase transition under a magnetic field. In general, a negative (positive) quark AMM acts as a magnetic-catalyzer (magnetic-inhibitor) for chiral symmetry breaking. It is found that a constant quark AMM drives an unexpected 1st order chiral phase transition, a quark AMM proportional to the chiral condensate flips the sign on the chiral condensate, and a quark AMM proportional to the square of the chiral condensate suppresses the magnetic enhancement in the chiral condensate at finite temperatures while retaining the chiral crossover phase transition. We also evaluate the intrinsic temperature dependence of the effective AMM form by fitting the effective model result of the chiral condensate to lattice QCD data, which may have a nontrivial correlation with the chiral phase transition.

Chiral symmetry restoration at high matter density observed in pionic atoms

According to quantum chromodynamics, vacuum is not an empty space, because it is filled with quark–antiquark pairs. The pair has the same quantum numbers as the vacuum and forms a condensate because the strong interaction of the quantum chromodynamics is too strong to leave the vacuum empty. This quark–antiquark condensation, the chiral condensate, breaks the chiral symmetry of the vacuum. The expectation value of the chiral condensate is an order parameter of the chiral symmetry, which is expected to decrease at high temperatures or high matter densities where the chiral symmetry is partially restored. Head-on collisions of nuclei at ultra-relativistic energies have explored the high-temperature regime, but experiments at high densities are rare. Here we measure the spectrum of pionic 121Sn atoms and study the interaction between the pion and the nucleus. We find that the expectation value of the chiral condensate is reduced at finite density compared to the value in vacuum. The reduction is linearly extrapolated to the nuclear saturation density and indicates that the chiral symmetry is partially restored due to the extremely high density of the nucleus. In quantum chromodynamics, the condensation of quark–antiquark pairs breaks the chiral symmetry of vacuum. Experiments with pionic tin atoms demonstrate that the symmetry is partially restored at high densities.

Construction of Chiral Cyclobutanone-Fused 4-Aminoquinolines via Sequential Chiral Phosphoric Acid and Palladium Catalysis.

A one-pot catalytic asymmetric route to novel chiral quaternary-carbon-containing cyclobutanone-fused 4-aminoquinoline derivatives in good to high yields and enantioselectivities is described. This process consists of a chiral phosphoric acid-catalyzed desymmetric carbonyl-amine condensation of prochiral cyclobutane-1,3-diones with 2-halogenated anilines and a Pd-catalyzed coupling reaction of the chiral enaminone intermediates with isocyanides.

Consistency of chiral symmetry breaking in chiral Yang-Mills theory with adiabatic continuity

We study the pattern of chiral symmetry breaking in the $\ensuremath{\psi}\ensuremath{\chi}\ensuremath{\eta}$ model [with the chiral fermion sector containing ${\ensuremath{\psi}}^{{ij}}$, ${\ensuremath{\chi}}_{[ij]}$, and ${\ensuremath{\eta}}_{i}^{A}$; see Armoni and Shifman Phys. Rev. D 85, 105003 (2012)] on ${\mathbb{R}}^{3}\ifmmode\times\else\texttimes\fi{}{S}_{L}^{1}$ and derive implications forcing ${\mathbb{R}}^{4}$ physics. Center-symmetric vacua are stabilized by a double-trace deformation. With the center symmetry maintained at small $L({S}^{1})\ensuremath{\ll}{\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{-}1}$, i.e., at weak coupling, no phase transitions are expected in passing to large $L({S}^{1})\ensuremath{\gg}{\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{-}1}$ (here, $\mathrm{\ensuremath{\Lambda}}$ is the dynamical Yang-Mills scale). Starting with the small-$L$ limit, we find the leading-order nonperturbative corrections in the given theory. The instanton-monopole operators induce the adjoint chiral condensate $⟨{\ensuremath{\psi}}^{{ij}}{\ensuremath{\chi}}_{[jk]}⟩\ensuremath{\ne}0$ at weak coupling, i.e., at $L({S}^{1})\ensuremath{\ll}{\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{-}1}$. Then adiabatic continuity tells us that $⟨{\ensuremath{\psi}}^{{ij}}{\ensuremath{\chi}}_{[jk]}⟩\ensuremath{\ne}0$ exists on ${\mathbb{R}}^{4}$, in full accord with the prediction from Bolognesi et al. [Phys. Rev. D 97, 094007 (2018)]. Simultaneously with $⟨{\ensuremath{\psi}}^{{ij}}{\ensuremath{\chi}}_{[jk]}⟩\ensuremath{\sim}{\mathrm{\ensuremath{\Lambda}}}^{3}{\ensuremath{\delta}}_{k}^{i}$, the $\mathrm{SU}({N}_{c})$ gauge symmetry is spontaneously broken at strong coupling down to its maximal Abelian subgroup.

Self-learning Monte Carlo for non-Abelian gauge theory with dynamical fermions

In this paper, we develop the self-learning Monte-Carlo (SLMC) algorithm for non-Abelian gauge theory with dynamical fermions in four dimensions to resolve the autocorrelation problem in lattice QCD. We perform simulations with the dynamical staggered fermions and plaquette gauge action by both in the hybrid Monte-Carlo (HMC) and SLMC for zero and finite temperature to examine the validity of SLMC. We confirm that SLMC can reduce autocorrelation time in non-Abelian gauge theory and reproduce results from HMC. For finite temperature runs, we confirm that SLMC reproduces correct results with HMC, including higher-order moments of the Polyakov loop and the chiral condensate. Besides, our finite temperature calculations indicate that four flavor ${\mathrm{QC}}_{2}\mathrm{D}$ with $\stackrel{^}{m}=0.5$ is likely in the crossover regime in the Colombia plot.

Renormalization group flow to effective quantum mechanics in the IR in an emergent dual holographic description for spontaneous chiral symmetry breaking

Implementing the Wilsonian renormalization group (RG) transformation in a nonperturbative way, we construct an effective holographic dual description with an emergent extradimension identified with an RG scale. Taking the large$-N$ limit, we obtain an equation of motion of an order-parameter field, here the chiral condensate for our explicit demonstration. In particular, an intertwined structure manifests between the first-order RG flow equations of renormalized coupling functions and the second-order differential equation of the order-parameter field, thus referred to as a nonperturbative RG-improved mean-field theory. Assuming translational symmetry as a vacuum state, we solve these nonlinear coupled mean-field equations based on a matching method between UV- and IR-regional solutions. As a result, we find an RG flow from a weakly-coupled chiral-symmetric UV fixed point to a strongly-correlated chiral-symmetry broken IR fixed point, where the renormalized velocity of Dirac fermions vanishes most rapidly and effective quantum mechanics appears at IR. Furthermore, we translate these RG flows of coupling functions into those of emergent metric tensors and extract out geometrical properties of the emergent holographic spacetime constructed from the UV- and IR-regional solutions. Surprisingly, we obtain the volume law of entanglement entropy in the Ryu-Takayanagi formula, which implies appearance of a black hole type solution in the limit of infinite cutoff even at zero temperature. We critically discuss our field theoretic interpretation for this solution in terms of potentially gapless multi-particle excitation spectra.

Effect of QCD Phase Transition on Axion Properties for Two Flavors of NJL Model

The effect of chiral phase transition on the properties of the axion in the Nambu-Jona-Lasinio two-flavor model was studied at finite temperature. We tend to study and analyze the changes of thermodynamic potential, and note that the changes of temperature, coupling parameters and quark chemical potential have great influence on thermodynamic potential. Then the variations of axion mass, self-coupling and quark condensation are given by numerical calculation.

Deriving the dilaton potential in improved holographic QCD from the chiral condensate

Abstract We derive an explicit form of the dilaton potential in improved holographic QCD (IHQCD) from the QCD lattice data of the chiral condensate as a function of the quark mass. This establishes a data-driven holographic modeling of QCD—machine-learning holographic QCD. The modeling consists of two steps for solving inverse problems. The first inverse problem is to find the emergent bulk geometry consistent with the lattice QCD simulation data at the boundary. We solve this problem with the refinement of the neural ordinary differential equation, a machine-learning technique. The second inverse problem is to derive a bulk gravity action with a dilaton potential such that its solution is the emergent bulk geometry. We solve this problem at non-zero temperature, and derive the explicit form of the dilaton potential. The dilaton potential determines the bulk action, the Einstein-dilaton system; thus we derive holographically the bulk system from the QCD chiral condensate data. The usefulness of the model is shown in the example of the prediction of the string breaking distance, whose value is found to be consistent with other lattice QCD data.

QCD spectral sum rules 2022

We present a compact review of the status of QCD spectral sum rules until 2022. We emphasize the recent progresses for determining the QCD input parameters (αs, running quark masses, quark and gluon condensates) where their correlations have been taken into account. Some selected phenomenological uses of the sum rules in different channels (light and heavy quarks, gluonia/glueballs, hybrids and four-quark states) are briefly reviewed and commented. The estimate of the 1−+ light hybrid mass is revised which confirms the hybrid nature of the π1(1600) but not the π1(2050).

Dynamically induced topological inflation

We propose an inflation model in which the inflationary era is driven by the strong dynamics of $Sp(2)$ gauge theory. The quark condensation in the confined phase of $Sp(2)$ gauge theory generates the inflaton potential comparable to the energy of the thermal bath at the time of phase transition. Afterwards, with a super-Planckian field value at the global minimum, the inflation commences at a false vacuum region lying between true vacuum regions and hence the name ``topological inflation.'' Featured by the huge separation between the scale of the false vacuum ($V(0{)}^{1/4}\ensuremath{\sim}{10}^{15}\text{ }\text{ }\mathrm{GeV}$) and the field value at the global minimum (${\ensuremath{\phi}}_{\mathrm{min}}\ensuremath{\sim}{M}_{P}$), the model can be consistent with cosmic microwave background (CMB) observables without suffering from the initial condition problems. Crucially, this is achieved only with some mild tuning of parameters in $V(\ensuremath{\phi})$. In addition to $Sp(2)$, this model is based on an anomaly free ${Z}_{6R}$ discrete $R$ symmetry. Remarkably, while all parameters are fixed by CMB observations, the model predicts a hierarchy of energy scales including the inflation scale, supersymmetry-breaking scale, R-symmetry breaking scale, Higgsino mass and the right-handed neutrino mass given in terms of the dynamical scale of $Sp(2)$.

Phenomenology of strong interactions: towards an effective theory for low energy QCD

In this paper, we develop models applicable to phenomenological particle physics by using the string analogy of particles. These theories can be used to investigate the phenomenology of confinement, deconfinement, chiral condensate, QGP phase transitions, and even the evolution of the early universe. Other confining properties such as scalar glueball mass, gluon mass, glueball-meson mixing states, QCD vacuum, and color superconductivity can also be investigated in these model frameworks. We use one of the models to describe the phenomenon of color confinement among glueballs at the end of the paper. The models are built based on the Dirac–Born–Infeld (DBI) action modified for open strings with their endpoints on a Dp-brane or brane–anti-brane at a tachyonic vacuum.