Meson Equation Research Articles

Spin Alignment of Vector Mesons in Heavy-Ion Collisions.

Polarized quarks and antiquarks in high-energy heavy-ion collisions can lead to the spin alignment of vector mesons formed by quark coalescence. Using the relativistic spin Boltzmann equation for vector mesons derived from Kadanoff-Baym equations with an effective quark-meson model for strong interaction and quark coalescence model for hadronizaton, we calculate the spin density matrix element ρ_{00} for ϕ mesons and show that anisotropies of local field correlations with respect to the spin quantization direction lead to ϕ meson's spin alignment. We propose that the local correlation or fluctuation of ϕ fields is the dominant mechanism for the observed ϕ meson's spin alignment and its strength can be extracted from experimental data as functions of collision energies. The calculated transverse momentum dependence of ρ_{00} agrees with STAR's data. We further predict the azimuthal angle dependence of ρ_{00} which can be tested in future experiments.

Supersymmetry in the double-heavy hadronic spectrum

Relativistic light-front bound-state equations for double-heavy mesons, baryons and tetraquarks are constructed in the framework of supersymmetric light front holographic QCD. Although heavy quark masses strongly break conformal symmetry, supersymmetry and the holographic embedding of semiclassical light-front dynamics still holds. The theory, derived from five-dimensional anti-de Sitter space, predicts that the form of the confining potential in the light-front Hamiltonian is harmonic even for heavy quarks. Therefore, the basic underlying supersymmetric mechanism, which transforms meson-baryon and baryon-tetraquark wave functions into each other, can also be applied to the double-heavy sector; one can then successfully relate the masses of the double-heavy mesons to double-heavy baryons and tetraquarks. The dependence of the confining potential on the hadron mass scale agrees completely with the one derived in heavy light systems from heavy quark symmetry. We also make predictions for higher excitations of the charmonium and bottomonium states. In particular, the remarkable equality of the Regge slopes in the orbital angular momentum, $L$, and the principal quantum number, $n$, is predicted to remain valid.

Light mesons in QCD and unquenching effects from the 3PI effective action

We investigate the impact of unquenching effects on QCD Green's functions, in the form of quark-loop contributions to both the gluon propagator and three-gluon vertex, in a three-loop inspired truncation of the three-particle irreducible (3PI) effective action. The fully coupled system of Dyson-Schwinger equations for the quark-gluon, ghost-gluon and three-gluon vertices, together with the quark propagator, are solved self-consistently; our only input are the ghost and gluon propagators themselves that are constrained by calculations within lattice QCD. We find that the two different unquenching effects have roughly equal, but opposite, impact on the quark-gluon vertex and quark propagator, with an overall negative impact on the latter. By taking further derivatives of the 3PI effective action, we construct the corresponding quark-antiquark kernel of the Bethe-Salpeter equation for mesons. The leading component is gluon exchange between two fully dressed quark-gluon vertices, thus introducing for the first time an obvious scalar-scalar component to the binding. We gain access to time-like properties of bound states by analytically continuing the coupled system of Dyson-Schwinger equations to the complex plane. We observe that the vector axial-vector splitting is in accord with experiment and that the lightest quark-antiquark scalar meson is above 1 GeV in mass.

Extending the Truncated Dyson-Schwinger Equation to Finite Temperatures

In view of the properties of mesons in hot strongly interacting matter the properties of the solutions of the truncated Dyson-Schwinger equation for the quark propagator at finite temperatures within the rainbow-ladder approximation are analysed in some detail. In Euclidean space within the Matsubara imaginary time formalism the quark propagator is not longer a O(4) symmetric function and possesses a discrete spectra of the fourth component of the momentum. This makes the treatment of the Dyson-Schwinger and Bethe-Salpeter equations conceptually different from the vacuum and technically much more involved. The question whether the interaction kernel known from vacuum calculations can be applied at finite temperatures remains still open. We find that, at low temperatures, the model interaction with vacuum parameters provides a reasonable description of the quark propagator, while at temperatures higher than a certain critical value $T_c$ the interaction requires stringent modifications. The general properties of the quark propagator at finite temperatures can be inferred from lattice QCD calculations. We argue that, to achieve a reasonable agreement of the model calculations with that from lattice QCD, the kernel is to be modified in such a way as to screen the infra-red part of the interaction at temperatures larger than $T_c$. For this, we analyse the solutions of the truncated Dyson-Schwinger equation with existing interaction kernels in a large temperature range with particular attention on high temperatures in order to find hints to an adequate temperature dependence of the interaction kernel to be further implemented in to the Bethe-Salpeter equation for mesons. This will allow to investigate the possible in medium modifications of the meson properties as well as the conditions of quark deconfinement in hot matter.

Supersymmetry across the light and heavy-light hadronic spectrum

Relativistic light-front bound-state equations for mesons and baryons can be constructed in the chiral limit from the supercharges of a superconformal algebra which connect baryon and meson spectra. Quark masses break the conformal invariance, but the basic underlying supersymmetric mechanism, which transforms meson and baryon wave functions into each other, still holds and gives remarkable connections across the entire spectrum of light and heavy-light hadrons. We also briefly examine the consequences of extending the supersymmetric relations to double-heavy mesons and baryons.

Outer crust of a cold non-accreting magnetar

The outer crust structure and composition of a cold, non-accreting magnetar is studied. We model the outer crust to be made of fully equilibrated matter where ionized nuclei form a Coulomb crystal embedded in an electron gas. The main effects of the strong magnetic field are those of quantizing the electron motion in Landau levels and of modifying the nuclear single particle levels producing, on average, an increased binding of nucleons in nuclei present in the Coulomb lattice. The effect of an homogeneous and constant magnetic field on nuclear masses has been predicted by using a covariant density functional, in which induced currents and axial deformation due to the presence of a magnetic field that breaks time-reversal symmetry have been included self-consistently in the nucleon and meson equations of motion. Although not yet observed, for $B\gtrsim 10^{16}$G both effects contribute to produce different compositions and to enlarge the range of pressures typically present in common neutron stars. Specifically, in such a regime, the magnetic field effects on nuclei favor the appearance of heavier nuclei at low pressures. As $B$ increases, such heavier nuclei are also preferred up to larger pressures. In the most extreme case, the whole outer crust is almost made of ${}_{40}^{92}$Zr$_{52}$.

Light-front holography and superconformal quantum mechanics: A new approach to hadron structure and color confinement

A primary question in hadron physics is how the mass scale for hadrons consisting of light quarks, such as the proton, emerges from the QCD Lagrangian even in the limit of zero quark mass. If one requires the effective action which underlies the QCD Lagrangian to remain conformally invariant and extends the formalism of de Alfaro, Fubini and Furlan to light-front Hamiltonian theory, then a unique, color-confining potential with a mass parameter [Formula: see text] emerges. The actual value of the parameter [Formula: see text] is not set by the model – only ratios of hadron masses and other hadronic mass scales are predicted. The result is a nonperturbative, relativistic light-front quantum mechanical wave equation, the Light-Front Schrödinger Equation which incorporates color confinement and other essential spectroscopic and dynamical features of hadron physics, including a massless pion for zero quark mass and linear Regge trajectories with the identical slope in the radial quantum number [Formula: see text] and orbital angular momentum [Formula: see text]. The same light-front equations for mesons with spin [Formula: see text] also can be derived from the holographic mapping to QCD (3+1) at fixed light-front time from the soft-wall model modification of AdS5 space with a specific dilaton profile. Light-front holography thus provides a precise relation between the bound-state amplitudes in the fifth dimension of AdS space and the boost-invariant light-front wavefunctions describing the internal structure of hadrons in physical space-time. One can also extend the analysis to baryons using superconformal algebra – [Formula: see text] supersymmetric representations of the conformal group. The resulting fermionic LF bound-state equations predict striking similarities between the meson and baryon spectra. In fact, the holographic QCD light-front Hamiltonians for the states on the meson and baryon trajectories are identical if one shifts the internal angular momenta of the meson ([Formula: see text]) and baryon ([Formula: see text]) by one unit: [Formula: see text]. We also show how the mass scale [Formula: see text] underlying confinement and the masses of light-quark hadrons determines the scale [Formula: see text] controlling the evolution of the perturbative QCD coupling. The relation between scales is obtained by matching the nonperturbative dynamics, as described by an effective conformal theory mapped to the light-front and its embedding in AdS space, to the perturbative QCD regime. The data for the effective coupling defined from the Bjorken sum rule [Formula: see text] are remarkably consistent with the Gaussian form predicted by LF holographic QCD. The result is an effective coupling defined at all momenta. The predicted value [Formula: see text] is in agreement with the world average [Formula: see text]. We thus can connect [Formula: see text] to hadron masses. The analysis applies to any renormalization scheme.

Matrix algorithms for solving (in)homogeneous bound state equations

In the functional approach to quantum chromodynamics, the properties of hadronic bound states are accessible via covariant integral equations, e.g. the Bethe–Salpeter equation for mesons. In particular, one has to deal with linear, homogeneous integral equations which, in sophisticated model setups, use numerical representations of the solutions of other integral equations as part of their input. Analogously, inhomogeneous equations can be constructed to obtain off-shell information in addition to bound-state masses and other properties obtained from the covariant analogue to a wave function of the bound state. These can be solved very efficiently using well-known matrix algorithms for eigenvalues (in the homogeneous case) and the solution of linear systems (in the inhomogeneous case). We demonstrate this by solving the homogeneous and inhomogeneous Bethe–Salpeter equations and find, e.g. that for the calculation of the mass spectrum it is as efficient or even advantageous to use the inhomogeneous equation as compared to the homogeneous. This is valuable insight, in particular for the study of baryons in a three-quark setup and more involved systems.

THE CJT CALCULATION IN STUDYING NUCLEAR MATTER BEYOND MEAN FIELD APPROXIMATION

We have introduced the Cornwall–Jackiw–Tomboulis (CJT) resummation scheme in studying nuclear matter. Based on the CJT formalism and using Walecka model, we have derived a set of coupled Dyson equations of nucleons and mesons. Neglecting the medium effects of the mesons, the usual mean field theory (MFT) results can be obtained. The beyond MFT calculations have been performed by thermodynamic consistently determining the meson effective masses and solving the coupled gap equations for nucleons and mesons together. The numerical results for the nucleon and meson effective masses at finite temperature and chemical potential in nuclear matter are discussed.

Interior Multiplications and Deformations with Meson Algebras

A meson algebra is involved in the Duffin wave equation for mesons in the same way as a Clifford algebra is involved in the Dirac wave equation for electrons. Therefore meson algebras too should have geometrical properties after the manner of Grassmann. Actually it is possible to define interior multiplications with similar properties, and deformations too. Every meson algebra is a deformation of a neutral meson algebra, in the same way as (almost) every Clifford algebra is a deformation of an exterior algebra. Some applications follow: the PBW-property is proved for all meson algebras, the injectiveness of Jacobson’s diagonal morphism is proved with the minimal hypothesis, and the existence of Lipschitz monoids is established at least for meson algebras over fields.

Studying unquenching effects in QCD with Dyson-Schwinger equations

We summarise recent results on the properties of gluons, quarks and light mesons from the Green's functions approach to QCD. We discuss a self-consistent, infrared power law solution for the Schwinger-Dyson equations of the 1PI-Greens functions of Yang-Mills theory. The corresponding running coupling has a universal fixed point at zero momentum. Based on these analytical results a truncation scheme for the coupled system of Schwinger-Dyson equations for the propagators of QCD and the Bethe-Salpeter equation for light mesons has been formulated. We compare numerical results for charge eigenstate vector and pseudoscalar meson observables with corresponding lattice data. The effects of unquenching the system are found to be small but not negligible.

Probing unquenching effects in the gluon polarization in light mesons

We introduce an extension to the ladder truncated Bethe-Salpeter equation for mesons and the rainbow truncated quark Schwinger-Dyson equations which includes quark-loop corrections to the gluon propagator. This truncation scheme obeys the axialvector Ward-Takahashi identity relating the quark self-energy and the Bethe-Salpeter kernel. Two different approximations to the Yang-Mills sector are used as input: the first is a sophisticated truncation of the full Yang-Mills Schwinger-Dyson equations, the second is a phenomenologically motivated form. We find that the spectra and decay constants of pseudoscalar and vector mesons are overall described well for either approach. Meson mass results for charge eigenstate vector and pseudoscalar meson masses are compared to lattice data. The effects of unquenching the system are small but not negligible.

A new approach related with group analysis and hodograph type transformation for constructing exact solutions

The method suggested in the manuscript uses the idea of the hodograph transformation method, which exchanges the independent and dependent variables. Here a change of the independent variables into dependent variables is applied to first derivatives. For the derivatives one obtains a system of differential equations. Group analysis is applied to this system. New invariant solutions, which are not invariant for the original equations, are obtained. The approach is illustrated by the semi-linear wave equation. For example, for Pion Meson equation one obtains a solution, which is reduced to quadrature.

Mott effect and $J/\psi$ dissociation at the quark-hadron phase transition

We investigate the in-medium modification of pseudoscalar and vector mesons in a QCD motivated chiral quark model by solving the Dyson-Schwinger equations for quarks and mesons at finite temperature for a wide mass range of meson masses, from light (pi, rho) to open-charm (D, D*) states. At the chiral / deconfinement phase transition, the quark-antiquark bound states enter the continuum of unbound states and become broad resonances (the hadronic Mott effect). We calculate the in-medium cross sections for charmonium dissociation due to collisions with light hadrons in a chiral Lagrangian approach, and show that the D and D* meson spectral broadening lowers the threshold for charmonium dissociation by pi and rho meson impact. This leads to a step-like enhancement in the reaction rate. We suggest that this mechanism for enhanced charmonium dissociation may be the physical mechanism underlying the anomalous J/Psi suppression observed by NA50.

Large N limit of SO(N) gauge theory of fermions and Bosons

In this paper we study the large Nc limit of SO(Nc) gauge theory coupled to a Majorana field and a real scalar field in 1+1 dimensions extending ideas of Rajeev [Int. J. Mod. Phys. A 9, 5583 (1994)]. We show that the phase space of the resulting classical theory of bilinears, which are the mesonic operators of this theory, is OSp1(H|H)/U(H+|H+), where ℋ|ℋ refers to the underlying complex graded space of combined one-particle states of fermions and bosons and H+|H+ corresponds to the positive frequency subspace. In the begining to simplify our presentation we discuss in detail the case with Majorana fermions only [the purely bosonic case is treated in Toprak and Turgut, J. Math. Phys. 43, 1340 (2002)]. In the Majorana fermion case the phase space is given by O1(H)/U(H+), where ℋ refers to the complex one-particle states and H+ to its positive frequency subspace. The meson spectrum in the linear approximation again obeys a variant of the ’t Hooft equation. The linear approximation to the boson/fermion coupled case brings an additonal bound state equation for mesons, which consists of one fermion and one boson, again of the same form as the well-known ’t Hooft equation.

Nonperturbative determination of heavy meson bound states

In this paper we obtain a heavy meson bound state equation from the heavy quark equation of motion in heavy quark effective theory (HQET) and the heavy meson effective field theory we developed very recently. The bound state equation is a covariant extention of the light-front bound state equation for heavy mesons derived from light-front QCD and HQET. We determine the covariant heavy meson wave function variationally by minimizing the binding energy �. Subsequently the other basic HQET parameters �1 and �2, and the heavy quark masses mb and mc can also be consistently determined.

New perspective on the scalar meson puzzle, from spontaneous chiral symmetry breaking beyond BCS

We introduce coupled channels of Bethe-Salpeter mesons both in the boundstate equation for mesons and in the mass gap equation for chiral symmetry. Consistency is insured by the Ward Identities for axial currents, which preserve the Goldstone boson nature of the pion and prevents a systematic shift of the hadron spectrum. We study the decay of a scalar meson coupled to a pair of pseudoscalars. We also show that coupled channels reduce the breaking of chiral symmetry, with the same Feynman diagrams that appear in the coupling of a scalar meson to a pair of pseudoscalar mesons. Exact calculations are performed in a particular confining quark model, where we find that the groundstate $I=0, ^3P_0 q \bar q$ meson is the f_0(980) with a partial decay width of 40MeV. We also find a 30% reduction of the chiral condensate due to coupled channels.

Nonperturbative determination of heavy meson bound states

In this paper we obtain a heavy meson bound state equation from the heavy quark equation of motion in heavy quark effective theory (HQET) and the heavy meson effective field theory we developed very recently. The bound state equation is a covariant extention of the light-front bound state equation for heavy mesons derived from light-front QCD and HQET. We determine the covariant heavy meson wave function variationally by minimizing the binding energy $\bar{\Lambda}$. Subsequently the other basic HQET parameters $\lambda_1$ and $\lambda_2$, and the heavy quark masses $m_b$ and $m_c$ can also be consistently determined.

Stability of the spectator, Dirac, and Salpeter equations for mesons

Mesons are made of quark-antiquark pairs held together by the strong force. The one channel spectator, Dirac, and Salpeter equations can each be used to model this pairing. We look at cases where the relativistic kernel of these equations corresponds to a time-like vector exchange, a scalar exchange, or a linear combination of the two. Since the model used in this paper describes mesons which cannot decay physically, the equations must describe stable states. We find that this requirement is not always satisfied, and give a complete discussion of the conditions under which the various equations give unphysical, unstable solutions.

Diquark confinement in an extended NJL model

In a Nambu-Jona-Lasinio model supplemented with an infrared cutoff in addition to the ultraviolet cutoff we study the issue whether diquarks are confined when the model is extended beyond the rainbow-ladder approximation. The gap equation, obtained in a truncation scheme motivated via a non-trivial quark-gluon vertex function, is solved to determine the constituent quark mass if chiral symmetry is spontaneously broken. In a second step, the Bethe-Salpeter equations for mesons and diquarks beyond the ladder approximation are derived, taking care to preserve Goldstone's theorem in the pion channel. While the obtained masses of pseudo-scalar and vector mesons are only moderately shifted compared to the values in the ladder approximation, we observe that scalar diquarks disappear from the physical spectrum and therefore are confined. For axial-vector diquarks we observe indications that the same mechanism may also work, but the NJL model allows no conclusive answer in this channel.