Effective Gluon Propagator Research Articles

The complex heavy quarkonium potential with the Gribov–Zwanziger action

Gribov–Zwanziger prescription in Yang–Mills theory improves the infrared dynamics. In this work, we study the static potential of a heavy quark-antiquark pair with the HTL resummed perturbation method within the Gribov–Zwanziger approach at finite temperature. The real and imaginary parts of the heavy quarkonium complex potential are obtained from the one-loop effective static gluon propagator. The one-loop effective gluon propagator is obtained by calculating the one-loop gluon self-energies containing the quark, gluon, and ghost loop. The gluon and ghost loops are modified in the presence of the Gribov parameter. We also calculate the decay width from the imaginary part of the potential. We also discuss the medium effect of heavy quarkonium potential with the localized action via auxiliary fields.

Heavy quark diffusion coefficients in magnetized quark-gluon plasma

We evaluate the heavy quark momentum diffusion coefficients in a hot magnetized medium for the most general scenario of any arbitrary values of the external magnetic field. We choose to work with the systematic way of incorporating the effect of the magnetic field, by using the effective gluon and quark propagators, generalized for a hot and magnetized medium. To get gauge independent analytic form factors valid through all Landau levels, we apply the hard thermal loop technique for the resummed effective gluon propagator. The derived effective hard thermal loop gluon propagator and the generalized version of Schwinger quark propagator subsequently allow us to analytically evaluate the longitudinal and transverse momentum diffusion coefficients for charm and bottom quarks beyond the static limit. Within the static limit we also explore another way of incorporating the effect of the magnetic field, i.e. through the magnetized medium modified Debye mass and compare the results to justify the need for structural changes. Published by the American Physical Society 2024

Yukawa potential under weak magnetic field

Weak magnetic field induced corrections for the Yukawa potential due to one pion exchange between two constituent quarks (nucleons) are presented. For that, the constant magnetic field effect on the pion propagator and on the pion form factor are taken into account. An effective gluon propagator parametrized with an effective gluon mass (Mg∼0.5 GeV) is considered. In the limit of magnetic field that is weak with respect to the constituent quark mass and pion mass, analytical and semianalytical expressions can be obtained. Different types of contributions are found, isotropic or anisotropic, dependent on the pion mass and also on the constituent quark and effective gluon masses. Overall the corrections are of the order of 2% to 5% of the Yukawa potential at distances close to 2 fm, and they decrease slower than the Yukawa potential. The anisotropic corrections are considerably smaller than the isotropic components. A sizable splitting between results due to the magnetic field dependent neutral or charged pion mass is found. Published by the American Physical Society 2024

Scalar propagator in a background gluon field beyond the eikonal approximation

We investigate the path integral representation of the scalar propagator in a background gluon field, extending beyond the eikonal approximation by considering all gauge field components and incorporating its x− dependence. Utilizing the worldline formalism, we integrate the Schwinger proper time to express the scalar propagator in light-cone coordinates, facilitating a direct comparison with known results in the literature. The derived propagator captures the change of longitudinal momentum of the projectile within the medium. In the high-energy limit, our result simplifies to the effective gluon propagator employed in the BDMPS-Z formalism. Hence, we propose that our outcome serves as a foundational point for investigating corrections to the BDMPS-Z spectrum arising from the longitudinal momentum transfer of the radiated gluon with the medium, as well as for studying collisional energy loss phenomena. Lastly, by employing an expansion around the classical saddle point solution, we systematically derive an eikonal expansion in inverse powers of the boost parameter, encompassing corrections related to longitudinal momentum transfer and interactions of the projectile with the transverse component of the field.

Heavy quark dynamics in a strongly magnetized quark-gluon plasma

We present a calculation of the heavy quark momentum diffusion coefficients in a quark-gluon plasma under the presence of a strong external magnetic field, within the Lowest Landau Level (LLL) approximation. In particular, we apply the Hard Thermal Loop (HTL) technique for the resummed effective gluon propagator, generalized for a hot and magnetized medium. Using the derived effective HTL gluon propagator and the LLL quark propagator we analytically derive the full results for the longitudinal and transverse momentum diffusion coefficients as well as the energy losses for charm and bottom quarks beyond the static limit. We also show numerical results for these coefficients in two special cases where the heavy quark is moving either parallel or perpendicular to the external magnetic field.

Heavy quark dynamics in a strongly magnetized medium

We present a calculation of the heavy quark momentum diffusion coefficients in a strongly magnetized medium, within the Lowest Landau Level (LLL) approximation. In particular, we use the Hard Thermal Loop (HTL) resummed effective gluon propagator, generalized for a hot and magnetized medium. Using this effective HTL gluon propagator along with the LLL quark propagator we analytically derive the full results for the longitudinal and transverse momentum diffusion coefficients for charm and bottom quarks beyond the static limit. Going beyond the static limit of the heavy quark, we also show numerical results for these coefficients in two special cases where the heavy quark is moving either parallel or perpendicular to the external magnetic field.

Pion superfluid phase transition at finite isospin chemical potential

The pion superfluidity phase transition at $T-\mu_I$ and $\mu_q-\mu_I$ planes are studied in the framework of Dyson-Schwinger equations. The rainbow truncation and Gaussian effective gluon propagator are employed to calculate pion condensate, $\langle\pi\rangle$, as a function of $\mu_I$ at finite $T$ and $\mu_q$. At $T=(0, 80, 113, 120)$~MeV, the $\langle\pi\rangle$ keeps zero when $\mu_I$ is less than a critical value $\mu_{Ic}(T)$. The $\langle\pi\rangle$ at $T=113$~MeV agrees with the lattice QCD result. At the $T-\mu_I$ plane, the pion superfluid phase appears at low temperature and high isospin chemical potential region. At finite $\mu_q$, $\langle\pi\rangle$ varying with $\mu_I$ almost coincide for $\mu_q<400$~MeV. At $\mu_q-\mu_I$ plane, the pion superfluid phase appears at $\mu_I\gtrsim m_\pi$ when $\mu_q<400$~MeV.

Flavor-dependent U(3) Nambu–Jona-Lasinio coupling constant

A non-perturbative one gluon exchange quark-antiquark interaction is considered to compute flavor dependent U(3) Nambu-Jona-Lasinio (NJL)-type interaction of the form $G_{ij, \Gamma} (\bar{\psi} \lambda_i \Gamma \psi ) ( \bar{\psi} \lambda_j \Gamma \psi)$ for $i,j=0...8$ and $\Gamma=I, i \gamma_5$ from one loop polarization process with non degenerate u-d-s quark effective masses. The resulting NJL-type coupling constants in all channels are resolved in the long-wavelength limit and numerical results are presented for different choices of an effective gluon propagator. Leading deviations with respect to a flavor symmetric coupling constant are found to be of the order of $(M_{f_2}^*-M_{f_1}^*)^n/(M_{f_2}^*+M_{f_1}^*)^n$, for $n=1,2$, where $M_{f_i}^*$ are the effective masses of quarks $f_1,f_2=u, d$ and $s$. The scalar channel coupling constants $G_{ij, s}$ can be considerably smaller than pseudoscalar ones. The effect of the flavor-dependence of coupling constants for the masses of pions and kaons may be nearly of the same order of magnitude as the effect of the u,d and s quark mass non-degeneracy. The effect of these coupling constants is also verified for some of the light scalar mesons masses, usually described by quark-antiquark states, and for some observables of the pseudoscalar mesons.

Heavy quark potential in a static and strong homogeneous magnetic field

We have investigated the properties of quarkonia in a thermal QCD medium in the background of strong magnetic field. For that purpose, we employ the Schwinger proper-time quark propagator in the lowest Landau level to calculate the one-loop gluon self-energy, which in the sequel gives the effective gluon propagator. As an artifact of strong magnetic field approximation (eB>>T^2 and eB>>m^2), the Debye mass for massless flavors is found to depend only on the magnetic field which is the dominant scale in comparison to the scales prevalent in the thermal medium. However, for physical quark masses, it depends on both magnetic field and temperature in a low temperature and high magnetic field but the temperature dependence is very meager and becomes independent of the temperature beyond a certain temperature and magnetic field. With the above mentioned ingredients, the potential between heavy quark (Q) and anti-quark (bar{Q}) is obtained in a hot QCD medium in the presence of a strong magnetic field by correcting both short- and long-range components of the potential in the real-time formalism. It is found that the long-range part of the quarkonium potential is affected much more by magnetic field as compared to the short-range part. This observation facilitates us to estimate the magnetic field beyond which the potential will be too weak to bind Qbar{Q} together. For example, the J/psi is dissociated at eB sim 10 m_pi ^2 and Upsilon is dissociated at eB sim 100 m_pi ^2 whereas its excited states, psi ^prime and Upsilon ^prime are dissociated at smaller magnetic field eB= m_pi ^2, 13 m_pi ^2, respectively.

Chiral phase transition of QCD at finite chemical potential

In the framework of Dyson-Schwinger equation (DSE) approach, we propose a chemical potential dependent effective gluon propagator and the quark gap equation is solved with such a gluon propagator. It is found for the first time in DSE approach that with light current quark mass m not equal 0, the Wigner solution will appear at some chemical potential mu(W). Based on this, the chiral phase transition at finite chemical potential is studied and the result shows that it is a first-order phase transition. The results are compared with the known results in the literature.

Theoretical prediction of cosmological constant Λ in Veneziano ghost theory of QCD

Based on the Veneziano ghost theory of QCD, we predict the cosmological constant Л, which is related to energy density of cosmological vacuum by \( \Lambda = \frac{{8\pi G}} {3}\rho _\Lambda \). In the Veneziano ghost theory, the vacuum energy density ρЛ is expressed by absolute value of the product of quark vacuum condensate and quark current mass: \( \rho _\Lambda = \frac{{2N_f H}} {{m_{\eta '} }}c|m_q | \). We calculate the quark local vacuum condensates 〈0|: \( \bar q \)q: |0〉 by solving Dyson-Schwinger Equations for a fully dressed confining quark propagator Sf (p) with an effective gluon propagator Gµνab (q). The quark current mass mq is predicted by use of chiral perturbation theory. Our theoretical result of Л, with the resulting 〈0|: 471-4q: |0〉 = −(235 MeV)3 and light quark current mass mq ≃ 3.29–6.15 MeV, is in a good agreement with the observable of the Л ≈ 10−52 m−2 used widely in a great amount of literatures.

Low-energy effective Yang-Mills theory for quark and gluon confinement

We derive a gauge-invariant low-energy effective model of the Yang-Mills theory. We find that the effective gluon propagator belongs to the Gribov-Stingl type and agrees with it when a mass term which breaks nilpotency of the BRST symmetry is included. We show that the effective model with gluon propagator of the Gribov-Stingl type exhibits both quark and gluon confinement: the Wilson loop average has the area law and the Schwinger function violates reflection positivity. However, we argue that both quark and gluon confinement can be obtained even in the absence of such a mass term.

Width of the QCD transition in a Polyakov-loop Dyson-Schwinger equation model

We consider the pseudocritical temperatures for the chiral and deconfinement transitions within a Polyakov-loop Dyson-Schwinger equation approach which employs a nonlocal rank-2 separable model for the effective gluon propagator. These pseudocritical temperatures differ by a factor of 2 when the quark and gluon sectors are considered separately, but get synchronized and become coincident when their coupling is switched on. The coupling of the Polyakov loop to the chiral quark dynamics narrows the temperature region of the QCD transition in which chiral symmetry and deconfinement is established. We investigate the effect of rescaling the parameter ${T}_{0}$ in the Polyakov-loop potential on the QCD transition for both the logarithmic and polynomial forms of the potential. While the critical temperatures vary in a similar way, the width of the transition is more strongly affected for the logarithmic potential. For this potential, the character of the transition changes from crossover to a first-order one when ${T}_{0}&lt;210\text{ }\text{ }\mathrm{MeV}$, but it remains crossover in the whole range of relevant ${T}_{0}$ values for the polynomial form.

Nonzero mean squared momentum of quarks in the nonperturbative QCD vacuum

The non-local vacuum condensates of QCD describe the distributions of quarks and gluons in the non-perturbative QCD vacuum. Physically, this means that vacuum quarks and gluons have nonzero mean-squared momentum, called virtuality. In this paper we study the quark virtuality which is given by the ratio of the local quark-gluon mixed vacuum condensate to the quark local vacuum condensate. The two vacuum condensates are obtained by solving Dyson-Schwinger Equations of a fully dressed quark propagator with an effective gluon propagator. Using our calculated condensates, we obtain the virtuality of quarks in the QCD vacuum state. Our numerical predictions differ from the other theoretical model calculations such as QCD sum rules, Lattice QCD and instanton models.

Temperature dependence of the effective bag constant and the radius of a nucleon in the global color symmetry model of QCD

We study the temperature dependence of the effective bag constant, the mass, and the radius of a nucleon in the formalism of the simple global color symmetry model in the Dyson-Schwinger equation approach of QCD with a Gaussian-type effective gluon propagator. We obtain that, as the temperature is lower than a critical value, the effective bag constant and the mass decrease and the radius increases with the temperature increasing. As the critical temperature is reached, the effective bag constant and the mass vanish and the radius tends to infinity. At the same time, the chiral quark condensate disappears. These phenomena indicate that the deconfinement and the chiral symmetry restoration phase transitions can take place at high temperature. The dependence of the critical temperature on the interaction strength parameter in the effective gluon propagator of the approach is given.

Conformal symmetry and light flavor baryon spectra

The degeneracy among parity pairs systematically observed in the N and Delta spectra is interpreted to hint on a possible conformal symmetry realization in the light flavor baryon sector in line with AdS_5/CFT_4. The case is made by showing that all the observed N and Delta resonances with masses below 2500 MeV distribute fairly well each over the first levels of a unitary representation of the conformal group, a representation that covers the spectrum of a quark-diquark system, placed directly on the AdS_5 cone, conformally compactified to R^1*S^3. The free geodesic motion on the S^3 manifold is described by means of the scalar conformal equation there, which is of the Klein-Gordon type. The equation is then gauged by the "curved" Coulomb potential that has the form of a cotangent function. Conformal symmetry is not exact, this because the gauge potential slightly modifies the conformal centrifugal barrier of the free geodesic motion. Thanks to this, the degeneracy between P11-S11 pairs from same level is relaxed, while the remaining states belonging to same level remain practically degenerate. The model describes the correct mass ordering in the P11-S11 pairs through the nucleon spectrum as a combined effect of the above conformal symmetry breaking, on the one side, and a parity change of the diquark from a scalar at low masses, to a pseudoscalar at higher masses, on the other. The quality of the wave functions is illustrated by calculations of realistic mean-square charge radii and electric charge form-factors on the examples of the proton, and the protonic P11(1440), and S11(1535) resonances. The scheme also allows for a prediction of the dressing function of an effective instantaneous gluon propagator from the Fourier transform of the gauge potential. We find a dressing function that is finite in the infrared and tends to zero at infinity.

Gasser–Leutwyler Coefficients of Chiral Lagrangian for Pseudoscalar Goldstone Bosons in Dyson–Schwinger Equations

Based on the Dyson–Schwinger equations of quark propagator in rainbow truncation with an effective gluon propagator, the ten unknown Gasser–Leutwyler coefficients of the chiral Lagrangian for pseudoscalar Goldstone bosons are predicted. The predicted values of Li with i = 1, 2, …, 10 are in a reasonable agreement with empirical values used widely in literature, and the values predicted by many other theoretical models with QCD characteristics.

Vacuum condensates of QCD**Supported by National Natural Science Foundation of China (10647002,10565001) and Natural Science Foundation of Guangxi (0575020, 0542042, 0481030)

We study the properties of QCD vacuum state in this paper. The values of various local quark vacuum condensates, quark-gluon mixed vacuum condensates, and the structure of non-local quark vacuum condensate are predicted by the solution of Dyson-Schwinger Equations in ``rainbow'' approximation with three sets of different parameters for effective gluon propagator. The light quark virtuality is also obtained in a consistent way. Our all theoretical results here are in good agreement with the empirical values used widely in literature and many other theoretical calculations.

Chiral susceptibility in an effective interaction model

A closed integral expression for the chiral susceptibility at finite temperature is derived. The corresponding disconnected part, which proves to be of major physical relevance and free from the additive quadratic ultraviolet divergence, is identified. Then a calculation based on an effective model gluon propagator is conducted within the framework of the Dyson-Schwinger equations for two flavors in the chiral limit. A narrow, divergent peak is observed as temperature varies, and its implications are discussed.

Consequences of fully dressing quark–gluon vertex function with two-point gluon lines

We extend recent studies of the effects of quark–gluon vertex dressing upon the solutions of the Dyson–Schwinger equation for the quark propagator. A momentum delta function is used to represent the dominant infrared strength of the effective gluon propagator so that the resulting integral equations become algebraic. The quark–gluon vertex is constructed from the complete set of diagrams involving only two-point gluon lines. The additional diagrams, including those with crossed gluon lines, are shown to make an important contribution to the DSE solutions for the quark propagator, because of their large color factors and the rapid growth in their number.