Sigma Meson Research Articles

DMRG study of the theta-dependent mass spectrum in the 2-flavor Schwinger model

We study the θ-dependent mass spectrum of the massive 2-flavor Schwinger model in the Hamiltonian formalism using the density-matrix renormalization group (DMRG). The masses of the composite particles, the pion and sigma meson, are computed by two independent methods. One is the improved one-point-function scheme, where we measure the local meson operator coupled to the boundary state and extract the mass from its exponential decay. Since the θ term causes a nontrivial operator mixing, we unravel it by diagonalizing the correlation matrix to define the meson operator. The other is the dispersion-relation scheme, a heuristic approach specific to Hamiltonian formalism. We obtain the dispersion relation directly by measuring the energy and momentum of the excited states. The sign problem is circumvented in these methods, and their results agree with each other even for large θ. We reveal that the θ-dependence of the pion mass at m/g = 0.1 is consistent with the prediction by the bosonized model. We also find that the mass of the sigma meson satisfies the semi-classical formula, Mσ/Mπ = 3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{3} $$\\end{document}, for almost all region of θ. While the sigma meson is a stable particle thanks to this relation, the eta meson is no longer protected by the G-parity and becomes unstable for θ ≠ 0.

Calculating composite-particle spectra in Hamiltonian formalism and demonstration in 2-flavor QED1+1d

We consider three distinct methods to compute the mass spectrum of gauge theories in the Hamiltonian formalism: (1) correlation-function scheme, (2) one-point-function scheme, and (3) dispersion-relation scheme. The first one examines spatial correlation functions as we do in the conventional Euclidean Monte Carlo simulations. The second one uses the boundary effect to efficiently compute the mass spectrum. The third one constructs the excited states and fits their energy using the dispersion relation with selecting quantum numbers. Each method has its pros and cons, and we clarify such properties in their applications to the mass spectrum for the 2-flavor massive Schwinger model at m/g = 0.1 and θ = 0 using the density-matrix renormalization group (DMRG). We note that the multi-flavor Schwinger model at small mass m is a strongly coupled field theory even after the bosonizations, and thus it deserves to perform the first-principles numerical calculations. All these methods mostly agree and identify the stable particles, pions πa (JPG = 1−+), sigma meson σ (JPG = 0++), and eta meson η (JPG = 0−−). In particular, we find that the mass of σ meson is lighter than twice the pion mass, and thus σ is stable against the decay process, σ → ππ. This is consistent with the analytic prediction using the WKB approximation, and, remarkably, our numerical results are so close to the WKB-based formula between the pion and sigma-meson masses, Mσ/Mπ = 3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{3} $$\\end{document}.

Hadronic structure on the light front. VII. Pions and kaons and their partonic distributions

This work is a continuation in our series of papers, that addresses quark models of hadronic structure on the light front, motivated by the QCD vacuum structure and lattice results. The spontaneous breaking of chiral symmetry on the light front, is shown to parallel that in the rest frame, where the non-local instanton induced $^\prime$t Hooft interaction plays a central role. By rewriting this interaction solely in terms of the good component of the fermionic field, a scalar chiral condensate emerges in the mean-field approximation, which is identical to the one obtained in the rest frame. The pions and kaons emerge as deeply bound Goldstone modes in the chiral limit, with the scalar-isoscalar sigma meson mode as a threshold state with zero binding. We explicitly derive the light front distribution amplitudes (DAs) and partonic functions (PDFs) for these mesons. The DAs and PDFs are in good agreement with those extracted from the QCD instanton vacuum in the rest frame, using the large momentum effective theory (LaMET). The QCD evolved DAs and PDFs compare well with available measurements, as well as recent lattice results.

Regression analysis of the nuclear symmetry energy for relativistic mean-field models

Regression analysis for the symmetry energy within a sample of relativistic mean-field models of nuclear matter is performed. The selected models consistently meet the experimentally obtained limitations. A proposed measure of the importance of adding the fourth-order term to the symmetry energy is analyzed. As a result of the research, it became possible to arrange the models and perform two-dimensional and one-dimensional linear regression analyses. The one-dimensional regression analysis between the input parameters confirms the appropriateness of introducing the division of the sample of considered models into four separate classes representing four statistically different fits. An additional result is the formulation of constraints on the function, which describes the energy density of the system. The performed two-dimensional regression analysis refers to an alternative method of parametrizing the symmetry energy using the power law. It was found that the Akaike information criterion of model selection involving the $\ensuremath{\gamma}$ exponent is sensitive to splitting the symmetry energy into the kinetic and potential parts. The analysis indicates that the more appropriate way of splitting is the one that does not consider the interaction with the sigma meson in the kinetic part.

Neutron star properties with careful parametrization in the vector and axial-vector meson extended linear sigma model

The existence of quark matter inside the cores of heavy neutron stars is a possibility which can be probed with modern astrophysical observations. We use a vector and axial-vector meson extended quark-meson model to describe quark matter in the core of neutron stars. We discover that an additional parameter constraint is necessary in the quark model to ensure chiral restoration at high densities. By investigating hybrid star sequences with various parameter sets, we show that low sigma meson masses are needed to fulfill the upper radius constraints and that the maximum mass of stable hybrid stars is only slightly dependent on the parameters of the crossover-type phase transition. Using this observation and results from recent astrophysical measurements, a constraint of $2.5<{g}_{V}<4.3$ is set for the constituent quark--vector meson coupling. The effect of a nonzero bag constant is also investigated, and we observe that its effect is small for values adopted in previous works.

Finite size effect on dissociation and diffusion of chiral partners in Nambu-Jona-Lasinio model * *PD thanks to Women Scientist Scheme A (WOS-A) of the Department of Science and Technology (DST) funding with (SR/WOS-A/PM-10/2019)

The masses of pion and sigma meson modes, along with their dissociation in the quark medium, provide detailed spectral structures of the chiral partners. Collectivity has been observed in pA and pp systems both at LHC and RHIC. In this research, we studied the restoration of chiral symmetry by investigating the finite size effect on the detailed structure of chiral partners in the framework of the Nambu-Jona-Lasinio model. Their diffusion and conduction have been studied using this dissociation mechanism. It is determined that the masses, widths, diffusion coefficients, and conductivities of chiral partners merge at different temperatures in the restoration phase of chiral symmetry. However, merging points are shifted to lower temperatures when finite size effect is introduced into the picture. The strengths of diffusions and conductions are also reduced once the finite size is introduced in the calculations.

Impacts of inverse magnetic catalysis on screening masses of neutral pions and sigma mesons in hot and magnetized quark matter

We investigate the screening masses of neutral pions and sigma mesons in hot and magnetized quark matter in the framework of a two-flavor lattice-improved Nambu--Jona-Lasinio (NJL) model with a magnetic-field-dependent coupling constant, which is determined by utilizing the results from lattice QCD simulations. Since such a model can well reproduce inverse magnetic catalysis (IMC), by comparing with the standard NJL model, we systemically analyze the impacts of IMC on the temperature and magnetic field dependences of the longitudinal and transverse screening masses of the chiral partners, i.e., ${\ensuremath{\pi}}^{0}$ and $\ensuremath{\sigma}$ mesons, as well as the screening mass differences between them. Particularly, it is found that the $eB$ dependences of two alternative (pseudo)critical temperatures for the chiral transition defined by $\ensuremath{\sigma}\ensuremath{-}{\ensuremath{\pi}}^{0}$ meson screening mass differences are consistent with that defined by the quark condensate.

Self-consistent O(4) model spectral functions from analytically continued functional renormalization group flows

In this paper we explore practicable ways for self-consistent calculations of spectral functions from analytically continued functional renormalization group (aFRG) flow equations. As a particularly straightforward one we propose to include parametrizations of self-energies based on explicit analytic one-loop expressions. To exemplify this scheme we calculate the spectral functions of pion and sigma meson of the $O(4)$ model at vanishing temperature in the broken phase. Comparing the results with those from previous aFRG calculations, we explicitly demonstrate how self-consistency at all momenta fixes the tight relation between particle masses and decay thresholds. In addition, the two-point functions from our new semianalytic FRG scheme have the desired domain of holomorphy built in and can readily be studied in the entire cut-complex frequency plane, on physical as well as other Riemann sheets. This is very illustrative and allows, for example, to trace the flow of the resonance pole of the sigma meson across an unphysical sheet. In order to assess the limitations due to the underlying one-loop structure, we also introduce a fully self-consistent numerical scheme based on spectral representations with scale-dependent spectral functions. The most notable improvement of this numerically involved calculation is that it describes the three-particle resonance decay of an off-shell pion, ${\ensuremath{\pi}}^{*}\ensuremath{\rightarrow}\ensuremath{\sigma}\ensuremath{\pi}\ensuremath{\rightarrow}3\ensuremath{\pi}$. Apart from this further conceptual improvement, overall agreement with the results from the considerably simpler semianalytic one-loop scheme is very encouraging, however. The latter can therefore provide a sound and practicable basis for self-consistent calculations of spectral functions in more realistic effective theories for warm and dense matter.

Locating the critical endpoint of QCD: Mesonic backcoupling effects

We study the effects of pion and sigma meson backcoupling on the chiral order parameters and the QCD phase diagram and determine their effect on the location of the chiral critical endpoint. To this end, we solve a coupled set of truncated Dyson-Schwinger equations for Landau gauge quark and gluon propagators with ${N}_{\mathrm{f}}=2+1$ dynamical quark flavors and explicitly backcoupled mesons. The corresponding meson bound-state properties and the quark-meson Bethe-Salpeter vertices are obtained from their homogeneous Bethe-Salpeter equation. We find chiral-restoration effects of the pion and/or sigma meson backcoupling and observe a (small) shift of the critical endpoint towards smaller chemical potentials. The curvature of the chiral crossover line decreases. Our results indicate that the location of the critical endpoint in the phase diagram is mainly determined by the microscopic degrees of freedom of QCD (in contrast to its critical properties).

The role of the UA(1) breaking term in dynamical chiral symmetry breaking of chiral effective theories

Abstract The spontaneous breaking of chiral symmetry is examined by chiral effective theories, such as the linear $\sigma$ model and the Nambu–Jona-Lasinio (NJL) model. We discuss the properties of the sigma meson regarded as the quantum fluctuation of the chiral condensate when the chiral symmetry is spontaneously broken, mainly by the U$_{A}$(1) anomaly. We derive a mass relation among the SU(3) flavor singlet mesons, $\eta_{0}$ and $\sigma_{0}$, in the linear $\sigma$ model to be satisfied for the anomaly-driven symmetry breaking in the chiral limit, and find that it is also supported in the NJL model. With the explicit breaking of chiral symmetry, we show that both of the chiral effective models reproducing the observed physical quantities suggest that the $\sigma_{0}$ meson should have a mass smaller than $\sim$800 MeV when anomaly-driven symmetry breaking takes place.

Low energy pion\u2013Lambda _b interaction

The low energy pion–Lambda _b interaction is study in this work, considering effective chiral Lagrangians that include pions, baryons and the corresponding resonances. Interactions mediated by a sigma meson exchange are also considered. The scattering amplitudes are calculated and then we determine the angular distributions and polarizations.

Implications of the fermion vacuum term in the extended SU(3) quark meson model on compact star properties

We study the impact of the fermion vacuum term in the SU(3) quark meson model on the equation of state and determine the vacuum parameters for various sigma meson masses. We examine its influence on the equation of state and on the resulting mass radius relations for compact stars. The tidal deformability $\Lambda$ of the stars is studied and compared to the results of the mean field approximation. Parameter sets which fulfill the tidal deformability bounds of GW170817 together with the observed two solar mass limit turn out to be restricted to a quite small parameter range in the mean field approximation. The extended version of the model does not yield solutions fulfilling both constraints. Furthermore, no first order chiral phase transition is found in the extended version of the model, not allowing for the twin star solutions found in the mean field approximation.

Gravitational form factors of light mesons

We calculate the gravitational form factors of the pion, sigma meson, and rho meson in the Nambu-Jona-Lasinio (NJL) model of quantum chromodynamics. The canonical energy-momentum tensor (EMT) is used in their derivation, allowing the possibility of an antisymmetric contribution when the hadron has intrinsic spin. We show that the asymmetric graviton vertex arising from the canonical EMT satisfies a simpler Ward-Takahashi identity (WTI) than the symmetric graviton vertex of the Belinfante EMT. The necessity of fully dressing the graviton vertex through the relevant Bethe-Salpeter equation is demonstrated for observing both the WTI and a low-energy pion theorem. Lastly, we calculate static moments of the meson EMT decompositions, obtaining predictions for the meson mass radii. We find light cone mass radii of 0.27 fm for the pion, 0.32 fm for the sigma, and 0.25 fm for the rho. For the pion and rho, these are smaller than the light cone charge radii, respectively 0.51 fm and 0.45 fm, while we have a sigma charge radius of zero. Our light cone pion mass radius agrees with a phenomenological extraction from KEKB data.

Thermodynamic properties of light mesons and phase transition in an extended SU(2) NJL model

In this work, we study the temperature and chemical potential dependence of the masses of sigma and pion mesons as well as the quark condensate by using a SU(2) flavor version of the Nambu–Jona–Lassino model, introducing a prescription that mimics confinement. We have found that as the temperature increases, the mass of sigma shifts down, while the pion mass remains almost constant. On the other hand, the quark condensate decreases as the temperature and chemical potential increases. We have also analyzed the temperature and chemical potential dependence of the spectral function of the sigma meson, from which we observe at low values of T and [Formula: see text] an absence of a peak. Furthermore, as the Mott temperature is reached, its value increases abruptly and a distinct peak emerges, which is related with the dissociation of the sigma. For the case of [Formula: see text], the Mott dissociation is exhibited about the temperature of 189 MeV. We have also obtained the chiral phase diagram and the meson dissociation for different values of [Formula: see text]. From these results, we can state a relation between chiral symmetry restoration and Mott dissociation.

The sigma-meson exchange contribution to the muonic hydrogen Lamb shift

The sigma(ξ)meson exchange contribution to the potential of the muon-proton interactionin muonichydrogen inducedbythe ξ-meson coupling to two photons is estimated. The transition form factor ξ → γγ is deduced from the quark model and experimental data on the decay widths Γσγγ. It is shown that scalar meson exchange contribution to the Lamb shift in muonic hydrogen, △ELs(2P−2S ),is rather large and relevant for a comparison with coming precise experimental data.

Finite temperature spectral functions in the O(N) model

We directly calculate spectral functions in the O(N)-model at finite temperature within the framework of the Functional Renormalization group. Special emphasis is put on a fully numerical framework involving four-dimensional regulators preserving Euclidean O(4) and Minkowski Lorentz invariance, an important prerequisite for future applications. Pion and sigma meson spectral functions are calculated for a wide range of temperatures across the phase transition illustrating the applicability of the general framework for finite temperature applications. In addition, various aspects concerning the interplay between the Euclidean and real time two-point function are discussed.

The Large-$N$ Limit with Vanishing Leading Order Condensate for Zero Pion Mass

It is conventionally assumed that the negative mass squared term in the linear sigma model version of the pion Lagrangian is $M^2 \sim \Lambda_{\rm QCD}^2$ in powers of $N_c$. We consider the case where $M^2 \sim \Lambda^2_{\rm QCD}/N_c$ so that to leading order in $N_c$ this symmetry breaking term vanishes. We present some arguments why this might be plausible. One might think that such a radical assumption would contradict lattice Monte Carlo data on QCD as function of $N_c$. We show that the linear sigma model gives a fair description of the data of DeGrand and Liu both for $N_c = 3$, and for variable $N_c$. The values of quark masses considered by DeGrand and Liu, and by Bali et. al. turn out to be too large to resolve the case we consider from that of the conventional large $N_c$ limit. We argue that for quark masses $m_{q} \ll \Lambda_{\rm QCD}/N_c^{3/2}$, both the baryon mass and nucleon size scale as $\sqrt{N_c}$. For $m_{q} \gg \Lambda_{\rm QCD}/N_c^{3/2}$ the conventional large-$N_c$ counting holds. The physical values of quark masses for QCD correspond to the small quark mass limit. We find pion-nucleon coupling strengths are reduced to order ${\cal O}(1)$ rather than ${\cal O}(N_c)$. Under the assumption that in the large $N_c$ limit the sigma meson mass is larger than that of the omega, and that the omega-nucleon coupling constant is larger than that of the sigma, we argue that the nucleon-nucleon large range potential is weakly attractive and admits an interaction energy of order $\Lambda_{\rm QCD}/N_c^{5/2} \sim 10$ MeV. With these assumptions on coupling and masses, there is no strong long range attractive channel for nucleon-nucleon interactions, so that nuclear matter at densities much smaller than that where nucleons strongly interact is a weakly interacting configuration of nucleons with strongly interacting localized cores.

Meson exchange in lepton-nucleon scattering and the proton radius puzzle

We study two-photon contribution to the elastic lepton-proton scattering, associated with sigma meson exchange, with special attention paid to the low-Q2 region. We show that the corresponding amplitude grows sharply but remains finite at Q2->0. The analytical formula for the amplitude at Q2=0 is obtained. We also estimate the shift of the muonic hydrogen energy levels, induced by sigma meson exchange. For the 2S level the shift is approximately 30 times smaller than needed to resolve the proton radius puzzle, but still exceeds the precision of the muonic hydrogen measurements.

Relation between the mass modification of heavy–light mesons and the chiral symmetry structure in dense matter

We point out that the study of the density dependences of the masses of heavy-light mesons give some clues to the chiral symmetry structure in nuclear matter. We include the omega meson effect as well as the sigma meson effect at mean field level on the density dependence of the masses of heavy-light mesons with chiral partner structure. It is found that the omega meson affects the masses of the heavy-light mesons and their antiparticles in the opposite way, while it affects the masses of chiral partners in the same way. This is because the omega meson is sensitive to the baryon number of the light degrees included in the heavy-light mesons. We also show that the mass difference between chiral partners is proportional to the mean field of sigma, reflecting the partial restoration of chiral symmetry in the nuclear matter. In addition to the general illustration of the density dependence of the heavy-light meson masses, we consider two concrete models for nuclear matter, the parity doublet model and skyrmion crystal model in the sense of mean field approximation.

The sigma meson from lattice QCD with two-pion interpolating operators

In this paper, we describe our studies of the sigma meson, [Formula: see text], using two-pion correlation functions. We use lattice quantum chromodynamics in the quenched approximation with so-called clover fermions. By working at unphysical pion masses we are able to identify a would-be resonance with mass less than [Formula: see text], and then extrapolate to the physical point. We include the most important annihilation diagram, which is “partially disconnected” or “single annihilation”. Because this diagram is quite expensive to compute, we introduce a somewhat novel technique for the computation of all-to-all diagrams, based on momentum sources and a truncation in momentum space. In practice, we use only [Formula: see text] modes, so the method reduces to wall sources. At the point where the mass of the pion takes its physical value, we find a resonance in the [Formula: see text] two-pion channel with a mass of approximately [Formula: see text] MeV, consistent with the expected properties of the sigma meson, given the approximations we are making.