Gluon Propagator Research Articles

Probing the singularities of the Landau-Gauge gluon and ghost propagators with rational approximants

We employ Padé approximants in the study of the analytic structure of the four-dimensional SU(2) Landau-gauge gluon and ghost propagators in the infrared regime. The approximants, which are model independent, serve as fitting functions for the lattice data. We carefully propagate the uncertainties due to the fitting procedure, taking into account all possible correlations. For the gluon-propagator data, we confirm the presence of a pair of complex poles at {p}_{textrm{pole}}^2 = [(−0.37 ± 0.05stat± 0.08sys) ± i (0.66 ± 0.03stat± 0.02sys)] GeV2, where the first error is statistical and the second systematic. The existence of this pair of complex poles, already hinted upon in previous works, is thus put onto a firmer basis, thanks to the model independence and to the careful error propagation of our analysis. For the ghost propagator, the Padés indicate the existence of a single pole at p2 = 0, as expected. In this case, our results also show evidence of a branch cut along the negative real axis of p2. This is corroborated with another type of approximant, the D-Log Padés, which are better suited to studying functions with a branch cut and are applied here for the first time in this context. Due to particular features and limited statistics of the gluon-propagator data, our analysis is inconclusive regarding the presence of a branch cut in the gluon case.

A correspondence between the free and interacting field

We discover a correspondence between the free field and the interacting states. This correspondence is firstly given from the fact that the free propagator can be converted into a tower of propagators for massive states, when expanded with the Hermite function basis. The equivalence of propagators reveals that in this particular case the duality can naturally be regarded as the equivalence of one theory on the plane wave basis to the other on the Hermite function basis. More generally, the Hermite function basis provides an alternative quantization process with the creation/annihilation operators that correspond directly to the interacting fields. As an illustration, we apply this basis to the 3+1 dimensional Yang–Mills theory, where the three-dimensional space being reduced through the Hermite function basis, and an auxiliary parameter omega denotes for string tension. At large omega limit, with then considering only the lowest order Hermite function (Lowest Landau Level), the equivalent action becomes the Banks–Fischler–Shenker–Susskind (BFSS) matrix model. At small omega limit, the perturbative series summed over all orders of Hermite function gives a massive gluon propagator.

Estimation of the diffusion coefficient of heavy quarks in light of Gribov-Zwanziger action

The heavy quark momentum diffusion coefficient (κ) is one of the most essential ingredients for the Langevin description of heavy quark dynamics. In the temperature regime relevant to the heavy ion collision phenomenology, a substantial difference exists between the lattice estimations of κ and the corresponding leading order (LO) result from the hard thermal loop (HTL) perturbation theory. Moreover, the indication of poor convergence in the next-to-leading order (NLO) perturbative analysis has motivated the development of several approaches to incorporate the non-perturbative effects in the heavy quark phenomenology. In this work, we estimate the heavy quark diffusion coefficient based on the Gribov-Zwanziger prescription. In this framework, the gluon propagator depends on the temperature-dependent Gribov mass parameter, which has been obtained self-consistently from the one-loop gap equation. Incorporating this modified gluon propagator in the analysis, we find a reasonable agreement with the existing lattice estimations of κ within the model uncertainties.

Screened massive expansion of Schwinger-Dyson equations

A general formal derivation of the screened massive expansion is provided by Schwinger-Dyson equations. Some known issues of the expansion are clarified and a more general framework is established for a natural extension of the method to two-loop or to amplitudes which are not directly defined by a generating functional. For instance, a one-loop screened expansion is given for the effective gauge-parameter-independent gluon propagator which arises from the pinch-technique.

Spectral functions of gauge theories with Banks-Zaks fixed points

We investigate spectral functions of matter-gauge theories that are asymptotically free in the ultraviolet and display a Banks-Zaks conformal fixed point in the infrared. Using perturbation theory, Callan-Symanzik resummations, and UV-IR connecting renormalization group trajectories, we analytically determine the gluon, quark, and ghost propagators in the entire complex momentum plane. At weak coupling, we find that a K\"all\'en-Lehmann spectral representation of propagators is achieved for all fields, and determine suitable ranges for gauge-fixing parameters. At strong coupling, a proliferation of complex conjugated branch cuts renders a causal representation impossible. We also derive relations for scaling exponents that determine the presence or absence of propagator nonanalyticities. Further results include spectral functions for all fields up to five loop order, bounds on the conformal window, and an algorithm to find running gauge coupling analytically at higher loops. Implications of our findings and extensions to other theories are discussed.

Gluon Propagator in Background Field at Next-to-Eikonal Order

Gluon Propagator in Background Field at Next-to-Eikonal Order

Two-loop three-gluon vertex from the Curci-Ferrari model and its leading infrared behavior to all loop orders

We evaluate the three-gluon vertex with one vanishing external momentum within the Curci-Ferrari (CF) model at two-loop order and compare our results to Landau-gauge lattice simulations of the same vertex function for the SU(2) and SU(3) gauge groups in four dimensions. This extends previous works [J. A. Gracey et al., Phys. Rev. D 100, 034023 (2019); N. Barrios et al., Phys. Rev. D 102, 114016 (2020)] that considered similarly the two-loop ghost and gluon two-point functions as well as the two-loop ghost-antighost-gluon vertex (with vanishing gluon momentum). With the parameters of the model being adjusted by fitting the two-point functions to available lattice data, our evaluation of the three-gluon vertex arises as a pure prediction. We find that two-loop corrections systematically improve the agreement between the model and the lattice data as compared to earlier one-loop calculations, with a better agreement in the SU(3) case as already seen in previous studies. We also analyze the renormalization scheme dependence of our calculation. In all cases, this dependence diminishes when two-loop corrections are included, which is consistent with the perturbative CF paradigm. In addition, we study the low momentum regime of the three-gluon vertex in relation with the possibility of zero crossing. Within the CF model, we show that the leading infrared behavior of the exact vertex is given by the same linear logarithm that arises at one-loop order, multiplied by the all orders cubic ghost dressing function at zero momentum (we provide similar exact results for other vertex functions). We argue that this property remains true within the Faddeev-Popov framework under the assumption that the resummed gluon propagator features a decoupling behavior. This shows that the zero crossing is a property of the exact three-gluon vertex function. Within the CF model, we find however that the scale of the zero crossing is considerably reduced when going from one- to two-loop order. This seems consistent with some recent lattice simulations [G. T. R. Catumba et al., EPJ Web Conf. 258, 02008 (2022)]. Finally, our analysis also allows us to support recent claims about the dominance of the tree-level tensor component [F. Pinto-G\'omez et al., arXiv:2208.01020].

Study of nonstrange quark stars within a modified NJL model

In this work, a modified Nambu-Jona-Lasinio (NJL) model with proper-time regularization is employed to study the properties of hypothetical nonstrange quark stars. The coupling constant of the four-fermion interaction in the conventional NJL model is modified as $G={G}_{1}+{G}_{2}⟨\overline{\ensuremath{\psi}}\ensuremath{\psi}⟩$ to highlight the feedback of the quark propagator to the gluon propagator. To study the dependence of the equation of state (EOS) on this modification as well as the vacuum pressure, we choose nine representative EOSs for comparison. It is found that a smaller ${G}_{1}$ leads to a stiffer EOS, and a higher vacuum pressure (i.e., a smaller bag constant) yields a softer EOS at low energy density. It is further shown that the heaviest quark star under this modified NJL model satisfies not only the recent mass measurement of PSR $\mathrm{J}0740+6620$, but also the radius constraints from x-ray timing observations. The corresponding tidal deformability is also in agreement with the observations of GW170817.

Neural network approach to reconstructing spectral functions and complex poles of confined particles

Reconstructing spectral functions from propagator data is difficult as solving the analytic continuation problem or applying an inverse integral transformation are ill-conditioned problems. Recent work has proposed using neural networks to solve this problem and has shown promising results, either matching or improving upon the performance of other methods. We generalize this approach by not only reconstructing spectral functions, but also (possible) pairs of complex poles or an infrared (IR) cutoff. We train our network on physically motivated toy functions, examine the reconstruction accuracy and check its robustness to noise. Encouraging results are found on both toy functions and genuine lattice QCD data for the gluon propagator, suggesting that this approach may lead to significant improvements over current state-of-the-art methods.

Signatures of the Yang-Mills deconfinement transition from the gluon two-point correlator

We evaluate the longitudinal or (chromo-)electric Yang-Mills gluon propagator in the recently proposed center-symmetric Landau gauge at finite temperature [D. M. van Egmond et al., SciPost Phys. 12, 087 (2022).]. To model the effect of the Gribov copies in the infrared, we use the Curci-Ferrari model which, in turn, allows us to rely on perturbative calculations. At one-loop order in the SU(2) case, the so-obtained longitudinal gluon propagator provides a clear signature for ${Z}_{2}$ center-symmetry breaking with a singular behavior, characteristic of a continuous phase transition. This is in sharp contrast with what is found within the standard Landau gauge. We also identify various signatures for ${Z}_{3}$ center-symmetry breaking in the SU(3) case in the form of genuine order parameters. Among those, we find that the gluon propagator, although degenerate along the diagonal color directions in the confining phase, becomes nondegenerate in the deconfined phase. Our results open new ways of identifying the transition from correlation functions both within continuum approaches and on the lattice.

Restoring rotational invariance for lattice QCD propagators

This note presents a method to reduce the discretization errors appearing when solving a Quantum Field Theory in a hypercubic lattice in both position and momentum-space. The method exploits the artifacts that break rotational symmetry to recover rotationally invariant results for two-point Green functions. We show that a combination of the results obtained in position and momentum space can be useful to signal the presence of rotationally invariant artifacts making use of their approximate Fourier transforms in the continuum. The method will be introduced using a Klein-Gordon propagator, and a direct application to gluon propagator in quenched lattice QCD will be given.

Emergence of mass in the gauge sector of QCD* *Supported by the grant PID2020-113334GB-I00/AEI/10.13039/501100011033 of the Spanish AEI-MICINN, and the grant Prometeo/2019/087 of the Generalitat Valenciana

It is currently widely accepted that gluons, while massless at the level of the fundamental QCD Lagrangian, acquire an effective mass through the non-Abelian implementation of the classic Schwinger mechanism. The key dynamical ingredient that triggers the onset of this mechanism is the formation of composite massless poles inside the fundamental vertices of the theory. These poles enter the evolution equation of the gluon propagator and nontrivially affect the way the Slavnov-Taylor identities of the vertices are resolved, inducing a smoking-gun displacement in the corresponding Ward identities. In this article, we present a comprehensive review of the pivotal concepts associated with this dynamical scenario, emphasizing the synergy between functional methods and lattice simulations and highlighting recent advances that corroborate the action of the Schwinger mechanism in QCD.

Gluon condensates and effective gluon mass

Lattice simulations along with studies in continuum QCD indicate that non-perturbative quantum fluctuations lead to an infrared regularisation of the gluon propagator in covariant gauges in the form of an effective mass-like behaviour. In the present work we propose an analytic understanding of this phenomenon in terms of gluon condensation through a dynamical version of the Higgs mechanism, leading to the emergence of color condensates. Within the functional renormalisation group approach we compute the effective potential of covariantly constant field strengths, whose non-trivial minimum is related to the color condensates. In the physical case of an $SU(3)$ gauge group this is an octet condensate. The value of the gluon mass obtained through this procedure compares very well to lattice results and the mass gap arising from alternative dynamical scenarios.

Impact of dynamical fermions on the center vortex gluon propagator

The impact of $SU(3)$ center vortices on the Landau-gauge gluon propagator is calculated in the presence of dynamical fermions and compared to the pure Yang-Mills case. The presence of dynamical fermions is found to alter the behavior of the center vortex propagator when compared to the established pure-gauge result. The gluon spectral representation is also explored from the center vortex perspective, where center vortices are shown to exhibit clear signs of positivity violation, which is an indicator of confinement. Vortex removal subsequently restores positivity, demonstrating the crucial role center vortices play in the confinement of gluons.

Influence of light quark loops on the Wigner phase with Dyson–Schwinger equations approach

We study the influence of light quark loops on the Wigner phase by solving coupled Dyson-Schwinger equations for quark propagator and gluon propagator. We take the gluon propagator in the Nambu phase from $N_f$ = 2 unquenched lattice QCD and choose various phenomenological models for the quark-gluon vertex. The gluon propagator in Winger phase is assumed to be different from that in the Nambu phase only due to the vacuum polarization of the quark loop. We obtain the Wigner solution of the coupled equations, compared with that from solving only the equation of the quark propagator. We discussed the corrections by the light quark loops and the dependence on various models of the quark-gluon vertex.

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.

Ghost dynamics from Schwinger-Dyson equations

We discuss the coupled dynamics of the ghost dressing function and the ghost-gluon vertex through the Schwinger-Dyson equations that they satisfy. In order to close the system of equations, we combine recent lattice data for the gluon propagator and an approximate STI-derived Ansatz for the general kinematics three-gluon vertex. The numerical solution of the resulting coupled system exhibits excellent agreement to lattice data, for both the ghost dressing function and the ghost-gluon vertex, and allows the determination of the coupling constant. Next, in the soft gluon limit the full three-gluon vertex appearing in the ghost-gluon equation reduces to a special projection that is tightly constrained by lattice simulations. Specializing the ghost-gluon Schwinger-Dyson equation to this limit provides a nontrivial consistency check on the approximations employed for the three-gluon interaction and shows that the latter has an important quantitative effect on the ghost-gluon vertex. Finally, our results stress the importance of eliminating artifacts when confronting lattice data with continuum predictions.

Rigorous reconstruction of gluon propagator in the presence of complex singularities

It has been suggested that the Landau-gauge gluon propagator has complex singularities, which invalidates the Källén–Lehmann spectral representation. Since such singularities are beyond the standard formalism of quantum field theory, the reconstruction of Minkowski propagators from Euclidean propagators has to be carefully examined for their interpretation. In this talk, we present rigorous results on this reconstruction in the presence of complex singularities. As a result, the analytically continued Wightman function is holomorphic in the usual tube, and the Lorentz symmetry and locality are kept valid. On the other hand, the Wightman function on the Minkowski spacetime is a non-tempered distribution and violates the positivity condition. Finally, we discuss an interpretation and implications of complex singularities in quantum theories, arguing that complex singularities correspond to zero-norm confined states.

Studying mass generation for gluons

In covariant gauges, the gluonic mass gap in Yang-Mills theory manifests itself in the basic observation that the massless pole in the perturbative gluon propagator disappears in nonperturbative calculations, but the origin of this behavior is not yet fully understood. We summarize a recent study of the respective dynamics with Dyson-Schwinger equations in Landau-gauge Yang-Mills theory. We identify the parameter that distinguishes the massive Yang-Mills regime from the massless decoupling solutions, whose endpoint is the scaling solution. Similar to the PT-BFM scheme, we find evidence that mass generation in the transverse sector is triggered by longitudinal massless poles.

Emergent Hadron Mass in Strong Dynamics

Emergent Hadron Mass (EHM) is a mechanism capable of explaining both the unnaturally small (pion) and large (proton) masses of hadrons and if not the origin of confinement, then intimately connected with it. Even though the modern formulation of EHM has only recently been completed, it is rapidly becoming a fundamental focus for modern and future experimental efforts aimed at understanding the strong force within the Standard Model. Using Dyson-Schwinger equations for the gluon, ghost and quark propagators, I will introduce the main EHM concepts and illustrate how far this framework can take us in understanding strong interaction phenomenology.