Effective Gluon Mass Research Articles

All-order equation of the effective gluon mass

We present the general derivation of the full nonperturbative equation that governs the momentum evolution of the dynamically generated gluon mass, in the Landau gauge. The entire construction hinges crucially on the inclusion of longitudinally coupled vertices containing massless poles of nonperturbative origin, which preserve the form of the fundamental Slavnov-Taylor identities of the theory. The mass equation is obtained from a previously unexplored version of the Schwinger-Dyson equation for the gluon propagator, particular to the pinch technique-background field method formalism, which involves a reduced number of two-loop dressed diagrams, thus simplifying the calculational task considerably. The two-loop contributions turn out to be of paramount importance, modifying the qualitative features of the full mass equation and enabling the emergence of physically meaningful solutions. Specifically, the resulting homogeneous integral equation is solved numerically, subject to certain approximations, for the entire range of physical momenta, yielding positive-definite and monotonically decreasing gluon masses.

Computation of pion and kaon heavy ion multiplicities in a gluon-meson model

In high energy Heavy Ion Collisions, the onset of the quark-gluon plasma is the colour glass condensate, dominated by gluons. The final state is hadronic, and dominated by pions and kaons. Here we investigate an effective approach of QCD with these bosonic fields and which can help to describe the transition of gluons into light mesons. Formally, our approach consists in integrating out the quark fields from the QCD path integral, utilizing auxiliary fields. In this way the fermionic fields are replaced by light mesons, such as the pions and sigma field. We apply our effective action to compute the number of pions and kaons per gluon emitted by a Boltzmann gluon gas, their multiplicities as function of a gluon mass. We conclude that an effective gluon mass remains finite at T=Tc.

Unquenching the gluon propagator with Schwinger-Dyson equations

In this article we use the Schwinger--Dyson equations to compute the nonperturbative modifications caused to the infrared finite gluon propagator (in the Landau gauge) by the inclusion of a small number of quark families. Our basic operating assumption is that the main bulk of the effect stems from the ``one-loop dressed'' quark loop contributing to the full gluon self-energy. This quark loop is then calculated, using as basic ingredients the full quark propagator and quark-gluon vertex; for the quark propagator we use the solution obtained from the quark-gap equation, while for the vertex we employ suitable Ans\"atze, which guarantee the transversality of the answer. The resulting effect is included as a correction to the quenched gluon propagator, obtained in recent lattice simulations. Our main finding is that the unquenched propagator displays a considerable suppression in the intermediate momentum region, which becomes more pronounced as we increase the number of active quark families. The influence of the quarks on the saturation point of the propagator cannot be reliably computed within the present scheme; the general tendency appears to be to decrease it, suggesting a corresponding increase in the effective gluon mass. The renormalization properties of our results, and the uncertainties induced by the unspecified transverse part of the quark-gluon vertex, are discussed. Finally, the gluon propagator is compared with the available unquenched lattice data, showing rather good agreement.

Deconfinement phase transition in the Hamiltonian approach to Yang-Mills theory in Coulomb gauge

The deconfinement phase transition of SU(2) Yang--Mills theory is investigated in the Hamiltonian approach in Coulomb gauge assuming a quasi-particle picture for the grand canonical gluon ensemble. The thermal equilibrium state is found by minimizing the free energy with respect to the quasi-gluon energy. Above the deconfinement phase transition the ghost form factor remains infrared divergent but its infrared exponent is approximately halved, while the gluon energy, being infrared divergent in the confined phase, becomes infrared finite in the deconfined phase. For the effective gluon mass we find a critical exponent of 0.37. Using the lattice results for the gluon propagator to fix the scale, the deconfinement transition temperature is obtained in the range of 275 to 290 MeV.

Effective Action and Phase Transitions in Yang-Mills Theory on Spheres

We study the covariantly constant Savvidy-type chromomagnetic vacuum in finite-temperature Yang-Mills theory on the four-dimensional curved spacetime. Motivated by the fact that a positive spatial curvature acts as an effective gluon mass we consider the compact Euclidean spacetime $S^1\times S^1\times S^2$, with the radius of the first circle determined by the temperature $a_1=(2\pi T)^{-1}$. We show that covariantly constant Yang-Mills fields on $S^2$ cannot be arbitrary but are rather a collection of monopole-antimonopole pairs. We compute the heat kernels of all relevant operators exactly and show that the gluon operator on such a background has negative modes for any compact semi-simple gauge group. We compute the infrared regularized effective action and apply the result for the computation of the entropy and the heat capacity of the quark-gluon gas. We compute the heat capacity for the gauge group SU(2N) for a field configuration of $N$ monopole-antimonopole pairs. We show that in the high-temperature limit the heat capacity is well defined in the infrared limit and exhibits a typical behavior of second-order phase transition $\sim (T-T_c)^{-3/2}$ with the critical temperature $T_c=(2\pi a)^{-1}$, where $a$ is the radius of the 2-sphere $S^2$.

Massless bound-state excitations and the Schwinger mechanism in QCD

The gauge invariant generation of an effective gluon mass proceeds through the well-known Schwinger mechanism, whose key dynamical ingredient is the nonperturbative formation of longitudinally coupled massless bound-state excitations. These excitations introduce poles in the vertices of the theory, in such a way as to maintain the Slavnov-Taylor identities intact in the presence of massive gluon propagators. In the present work we first focus on the modifications induced to the nonperturbative three-gluon vertex by the inclusion of massless two-gluon bound-states into the kernels appearing in its skeleton-expansion. Certain general relations between the basic building blocks of these bound-states and the gluon mass are then obtained from the Slavnov-Taylor identities and the Schwinger-Dyson equation governing the gluon propagator. The homogeneous Bethe-Salpeter equation determining the wave-function of the aforementioned bound state is then derived, under certain simplifying assumptions. It is then shown, through a detailed analytical and numerical study, that this equation admits non-trivial solutions, indicating that the QCD dynamics support indeed the formation of such massless bound states. These solutions are subsequently used, in conjunction with the aforementioned relations, to determine the momentum-dependence of the dynamical gluon mass. Finally, further possibilities and open questions are briefly discussed.

Gluon mass through ghost synergy

In this work we compute, at the "one-loop-dressed" level, the nonperturbative contribution of the ghost loops to the self-energy of the gluon propagator, in the Landau gauge. This is accomplished within the PT-BFM formalism, which guarantees the gauge-invariance of the emerging answer. In particular, the contribution of the ghost-loops is automatically transverse, by virtue of the QED-like Ward identities satisfied in this framework. Using as nonperturbative input the available lattice data for the ghost dressing function, we show that the ghost contributions have a rather sizable effect on the overall shape of the gluon propagator, both for $d=3,4$. Then, by exploiting a recently introduced dynamical equation for the effective gluon mass, whose solutions depend crucially on the characteristics of the gluon propagator at intermediate energies, we show that if the ghost loops are removed from the gluon propagator then the gluon mass vanishes. These findings strongly suggest that, at least at the level of the Schwinger-Dyson equations, the effects of gluons and ghosts are inextricably connected, and must be combined suitably in order to reproduce the results obtained in the recent lattice simulations.

Small-xnonlinear evolution with impact parameter and the structure function data

The nonlinear Balitsky-Kovchegov equation at small x is solved numerically, incorporating impact parameter dependence. Confinement is modeled by including effective gluon mass in the dipole evolution kernel, which regulates the splitting of dipoles with large sizes. It is shown, that the solution is sensitive to different implementations of the mass in the kernel. In addition, running coupling effects are taken into account in this analysis. Finally, a comparison of the calculations using the dipole framework with the inclusive data from HERA on the structure functions F2 and FL is performed.

Dynamical equation of the effective gluon mass

In this article we derive the integral equation that controls the momentum dependence of the effective gluon mass in the Landau gauge. This is accomplished by means of a well-defined separation of the corresponding "one-loop dressed" Schwinger-Dyson equation into two distinct contributions, one associated with the mass and one with the standard kinetic part of the gluon. The entire construction relies on the existence of a longitudinally coupled vertex of nonperturbative origin, which enforces gauge invariance in the presence of a dynamical mass. The specific structure of the resulting mass equation, supplemented by the additional requirement of a positive-definite gluon mass, imposes a rather stringent constraint on the derivative of the gluonic dressing function, which is comfortably satisfied by the large-volume lattice data for the gluon propagator, both for SU(2) and SU(3). The numerical treatment of the mass equation, under some simplifying assumptions, is presented for the aforementioned gauge groups, giving rise to a gluon mass that is a non-monotonic function of the momentum. Various theoretical improvements and possible future directions are briefly discussed.

Gluon-propagator functional form in the Landau gauge in SU(3) lattice QCD: Yukawa-type gluon propagator and anomalous gluon spectral function

We study the gluon propagator ${D}_{\ensuremath{\mu}\ensuremath{\nu}}^{ab}(x)$ in the Landau gauge in SU(3) lattice QCD at $\ensuremath{\beta}=5.7$, 5.8, and 6.0 at the quenched level. The effective gluon mass is estimated as $400--600\text{ }\text{ }\mathrm{MeV}$ for $r\ensuremath{\equiv}({x}_{\ensuremath{\alpha}}{x}_{\ensuremath{\alpha}}{)}^{1/2}=0.5--1.0\text{ }\text{ }\mathrm{fm}$. Through the functional-form analysis of ${D}_{\ensuremath{\mu}\ensuremath{\nu}}^{ab}(x)$ obtained in lattice QCD, we find that the Landau-gauge gluon propagator ${D}_{\ensuremath{\mu}\ensuremath{\mu}}^{aa}(r)$ is well described by the Yukawa-type function ${e}^{\ensuremath{-}mr}/r$ with $m\ensuremath{\simeq}600\text{ }\text{ }\mathrm{MeV}$ for $r=0.1--1.0\text{ }\text{ }\mathrm{fm}$ in the four-dimensional Euclidean space-time. In the momentum space, the gluon propagator ${\stackrel{\texttildelow{}}{D}}_{\ensuremath{\mu}\ensuremath{\mu}}^{aa}({p}^{2})$ with $({p}^{2}{)}^{1/2}=0.5--3\text{ }\text{ }\mathrm{GeV}$ is found to be well approximated with a new-type propagator of $({p}^{2}+{m}^{2}{)}^{\ensuremath{-}3/2}$, which corresponds to the four-dimensional Yukawa-type propagator. Associated with the Yukawa-type gluon propagator, we derive analytical expressions for the zero-spatial-momentum propagator ${D}_{0}(t)$, the effective mass ${M}_{\mathrm{eff}}(t)$, and the spectral function $\ensuremath{\rho}(\ensuremath{\omega})$ of the gluon field. The mass parameter $m$ turns out to be the effective gluon mass in the infrared region of $\ensuremath{\sim}1\text{ }\text{ }\mathrm{fm}$. As a remarkable fact, the obtained gluon spectral function $\ensuremath{\rho}(\ensuremath{\omega})$ is almost negative definite for $\ensuremath{\omega}gm$, except for a positive $\ensuremath{\delta}$-functional peak at $\ensuremath{\omega}=m$.

Nonlinear Iteration Solution for the Full Gluon Propagator as a Function of the Mass Gap

We have explicitly shown that QCD is the color gauge invariant theory at non-zero mass gap as well. It has been defined as the value of the regularized full gluon self-energy at some finite point. The mass gap is mainly generated by the nonlinear interaction of massless gluon modes. All this allows one to establish the structure of the full gluon propagator in the explicit presence of the mass gap. In this case, the two independent general types of formal solutions for the full gluon propagator as a function of the regularized mass gap have been found. The nonlinear iteration solution at which the gluons remain massless is explicitly present. The existence of the solution with an effective gluon mass is also demonstrated.

Screened potential and quarkonia properties at high temperatures

We perform a quark model calculation of the quarkonia $b\overline{b}$ and $c\overline{c}$ spectra using smooth and sudden string breaking potentials. The screening parameter is scale dependent and can be related to an effective running gluon mass that has a finite infrared fixed point. A temperature dependence for the screening mass is motivated by lattice QCD simulations at finite temperature. Qualitatively different results are obtained for quarkonia properties close to a critical value of the deconfining temperature when a smooth or a sudden string breaking potential is used. In particular, with a sudden string breaking potential quarkonia radii remain almost independent of the temperature up to the critical point, only well above the critical point the radii increase significantly. Such a behavior will impact the phenomenology of quarkonia interactions in medium, in particular for scattering dissociation processes.

RevisitingB→ πK,πK* and ρKdecays: CP violations and implication for new physics

Combining the up-to-date experimental information on B → πK,πK* and ρK decays, we revisit the decay rates and CP asymmetries of these decays within the framework of QCD factorization. Using an infrared finite gluon propagator of Cornwall prescription, we find that the time-like annihilation amplitude could contribute a large strong phase, while the space-like hard spectator scattering amplitude is real. Numerically, we find that all the branching ratios and most of the direct CP violations, except ACP(B± → K±π0), agree with the current experimental data with an effective gluon mass mg 0.5 GeV. Taking the unmatched difference in direct CP violations between B → π0K± and πK± decays as a hint of new physics, we perform a model-independent analysis of new physics contributions with a set of (1+γ5)b⊗(1+γ5)q (q=u,d) operators. Detail analyses of the relative impacts of the operators are presented in five cases. Fitting the twelve decay modes, parameter spaces are found generally with nontrivial weak phases. Our results may indicate that both strong phase from annihilation amplitude and new weak phase from new physics are needed to resolve the πK puzzle. To further test the new physics hypothesis, the mixing-induced CP violations in B → π0KS and ρ0KS are discussed and good agreements with the recent experimental data are found.

Holographic glueball structure

I derive and systematically analyze scalar glueball correlation functions in both the hard-wall and dilaton soft-wall approximations to holographic QCD. The dynamical content of the holographic correlators is uncovered by examining their spectral density and by relating them to the operator product expansion, a dilatational low-energy theorem and a recently suggested two-dimensional power correction associated with the short-distance behavior of the heavy-quark potential. This approach provides holographic estimates for the three lowest-dimensional gluon condensates or alternatively their Wilson coefficients, the two leading moments of the instanton size distribution in the QCD vacuum and an effective UV gluon mass. A remarkable complementarity between the nonperturbative physics of the hard- and soft-wall correlators emerges, and their ability to describe detailed QCD results can be assessed quantitatively. I further provide the first holographic estimates for the decay constants of the ${0}^{++}$ glueball and its excitations. The hard-wall background turns out to encode more of the relevant QCD physics, and its prediction ${f}_{S}\ensuremath{\simeq}0.8\char21{}0.9\text{ }\text{ }\mathrm{GeV}$ for the phenomenologically important ground-state decay constant agrees inside errors with recent QCD sum rule and lattice results.

Power law running of the effective gluon mass

The dynamically generated effective gluon mass is known to depend non-trivially on the momentum, decreasing sufficiently fast in the deep ultraviolet, in order for the renormalizability of QCD to be preserved. General arguments based on the analogy with the constituent quark masses, as well as explicit calculations using the operator-product expansion, suggest that the gluon mass falls off as the inverse square of the momentum, relating it to the gauge-invariant gluon condensate of dimension four. In this article we demonstrate that the power law running of the effective gluon mass is indeed dynamically realized at the level of the non-perturbative Schwinger-Dyson equation. We study a gauge-invariant non-linear integral equation involving the gluon self-energy, and establish the conditions necessary for the existence of infrared finite solutions, described in terms of a momentum-dependent gluon mass. Assuming a simplified form for the gluon propagator, we derive a secondary integral equation that controls the running of the mass in the deep ultraviolet. Depending on the values chosen for certain parameters entering into the Ansatz for the fully dressed three-gluon vertex, this latter equation yields either logarithmic solutions, familiar from previous linear studies, or a new type of solutions, displaying power law running. In addition, it furnishes a non-trivial integral constraint, which restricts significantly (but does not determine fully) the running of the mass in the intermediate and infrared regimes. The numerical analysis presented is in complete agreement with the analytic results obtained, showing clearly the appearance of the two types of momentum dependence, well-separated in the relevant space of parameters. Several technical improvements, various open issues, and possible future directions, are briefly discussed.

Analyzing dynamical gluon mass generation

We study the necessary conditions for obtaining infrared finite solutions from the Schwinger-Dyson equation governing the dynamics of the gluon propagator. The equation in question is set up in the Feynman gauge of the background field method, thus capturing a number of desirable features. Most notably, and in contradistinction to the standard formulation, the gluon self-energy is transverse order-by-order in the dressed loop expansion, and separately for gluonic and ghost contributions. Various subtle field-theoretic issues, such as renormalization group invariance and regularization of quadratic divergences, are briefly addressed. The infrared and ultraviolet properties of the obtained solutions are examined in detail, and the allowed range for the effective gluon mass is presented.

Gluon mass generation in the PT-BFM scheme

In this article we study the general structure and special properties of the Schwinger-Dyson equation for the gluon propagator constructed with the pinch technique, together with the question of how to obtain infrared finite solutions, associated with the generation of an effective gluon mass. Exploiting the known all-order correspondence between the pinch technique and the background field method, we demonstrate that, contrary to the standard formulation, the non-perturbative gluon self-energy is transverse order-by-order in the dressed loop expansion, and separately for gluonic and ghost contributions. We next present a comprehensive review of several subtle issues relevant to the search of infrared finite solutions, paying particular attention to the role of the seagull graph in enforcing transversality, the necessity of introducing massless poles in the three-gluon vertex, and the incorporation of the correct renormalization group properties. In addition, we present a method for regulating the seagull-type contributions based on dimensional regularization; its applicability depends crucially on the asymptotic behavior of the solutions in the deep ultraviolet, and in particular on the anomalous dimension of the dynamically generated gluon mass. A linearized version of the truncated Schwinger-Dyson equation is derived, using a vertex that satisfies the required Ward identity and contains massless poles belonging to different Lorentz structures. The resulting integral equation is then solved numerically, the infrared and ultraviolet properties of the obtained solutions are examined in detail, and the allowed range for the effective gluon mass is determined. Various open questions and possible connections with different approaches in the literature are discussed.

Nonlinear dynamics of soft boson collective excitations in hot QCD plasma III: Bremsstrahlung and energy losses

Within the framework of semiclassical approximation a general formalism for deriving an effective current generating bremsstrahlung of arbitrary number of soft gluons (longitudinal or transverse ones) in scattering of higher-energy parton off thermal parton in hot quark-gluon plasma with subsequent extension to two and more scatterers is obtained. For the case of static color centers, an expression for energy loss induced by usual bremsstrahlung of lowest-order with allowance for an effective temperature-induced gluon mass and finite mass of the projectile (heavy quark) is derived. The detailed analysis of contribution to radiation energy loss associated with existence of effective three-gluon vertex induced by hot QCD medium is performed. It is shown that in general, the bremsstrahlung associated with this vertex has no sharp direction (as in the case of usual bremsstrahlung) and therefore here, we can expect an absence of suppression effect due to multiple scattering. For the case of two-color static scattering centers it was shown that the problem of calculation of bremsstrahlung induced by four-gluon hard thermal loop (HTL) vertex correction can be reduced to the problem of the calculation of bremsstrahlung induced by three-gluon HTL correction. It was shown that for limiting value of soft gluon occupation number N k ∼ 1/ α s all higher processes of bremsstrahlung of arbitrary number of soft gluons become of the same order in coupling, and the problem of resummation of all relevant contributions to radiation energy loss of fast parton, arises. An explicit expression for matrix element of two soft gluon bremsstrahlung in small angles approximation is obtained.

Properties of the deconfining phase transition in SU(N) gauge theories

We extend our earlier investigation of the finite temperature deconfinement transition in SU(N) gauge theories, with the emphasis on what happens as N->oo. We calculate the latent heat in the continuum limit, and find the expected quadratic in N behaviour at large N. We confirm that the phase transition, which is second order for SU(2) and weakly first order for SU(3), becomes robustly first order for N>3 and strengthens as N increases. As an aside, we explain why the SU(2) specific heat shows no sign of any peak as T is varied across what is supposedly a second order phase transition. We calculate the effective string tension and electric gluon masses at T=Tc confirming the discontinuous nature of the transition for N>2. We explicitly show that the large-N `spatial' string tension does not vary with T for T Tc it increases as T-squared to a good approximation, and the k-string tension ratios closely satisfy Casimir Scaling. Within very small errors, we find a single Tc at which all the k-strings deconfine, i.e. a step-by-step breaking of the relevant centre symmetry does not occur. We calculate the interface tension but are unable to distinguish between linear or quadratic in N variations, each of which can lead to a striking but different N=oo deconfinement scenario. We remark on the location of the bulk phase transition, which bounds the range of our large-N calculations on the strong coupling side, and within whose hysteresis some of our larger-N calculations are performed.

Two loop effective potential for langleA2μrangle in the Landau gauge in quantum chromodynamics

We construct the effective potential for the dimension two composite operator 1/2 A^{a 2}_\mu in QCD with massless quarks in the Landau gauge for an arbitrary colour group at two loops. For SU(3) we show that an estimate for the effective gluon mass decreases as N_f increases.