It has long been known that the gluon propagator in Landau-gauge QCD exhibits a mass gap; and its emergence has been ascribed to the action of the Schwinger mechanism in the gauge sector of QCD. In the present work, we relate this property to the physical mass gap of QCD by considering two observables associated with confinement and chiral symmetry breaking, namely the confinement-deconfinement transition temperature and the pion decay constant, respectively. It turns out that the first observable is linearly proportional to the gluon mass gap, a fact that allows us to assign a direct physical meaning to this scale. Moreover, we identify three distinct momentum regimes in the gluon propagator, ultraviolet, intermediate, and deep infrared, and assess their impact on the aforementioned observables. Both observables are sensitive to the first two regions of momenta, where functional approaches essentially coincide, but are insensitive to the third, deep infrared, regime. The combined information is used for a simple fit for the gluon propagator, all of whose parameters admit a clear physical interpretation. Finally, we discuss how this fit can help us access the intertwined dynamics of confinement and chiral symmetry breaking in QCD-type theories.

China

Mauricio Ferreira

Spain

Germany

Joannis Papavassiliou

Jan Pawlowski

Nicolas Wink

Physics of the gluon mass gap

<p>QCD at large density reveals a rich phase structure, ranging from a potential critical end point and inhomogeneous phases or moat regimes to color superconducting ones with competing order effects. Resolving this region in the phase diagram of QCD with functional approaches requires a great deal of quantitative reliability already for a qualitative access. In the present work, we systematically extend the functional renormalization group approach to low-energy QCD by setting up a fully self-consistent approximation scheme in a low-energy effective quark-meson theory. In this approximation, all pointlike multiscattering events of the mesonic pion and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>σ</mi></math> mode are taken into account in terms of an effective potential as well as all higher quark-antiquark-mesonic scattering orders. As a first application we compute the phase structure of QCD including its low temperature, large chemical potential part. The quantitative reliability of the approximation and systematic extensions is also discussed.</p>

Friederike Ihssen

Jan M. Pawlowski

Franz R. Sattler

Toward quantitative precision for QCD at large densities

Thermal photons produced in heavy-ion collision experiments are an important observable for understanding quark-gluon plasma (QGP). The thermal photon rate from the QGP at a given temperature can be calculated from the spectral function of the vector current correlator. Extraction of the spectral function from the lattice correlator is known to be an ill-conditioned problem, as there is no unique solution for a spectral function for a given lattice correlator with statistical errors. The vector current correlator, on the other hand, receives a large ultraviolet contribution from the vacuum, which makes the extraction of the thermal photon rate difficult from this channel. We therefore consider the difference between the transverse and longitudinal part of the spectral function, only capturing the thermal contribution to the current correlator, simplifying the reconstruction significantly. The lattice correlator is calculated for light quarks in quenched QCD at <math><mi>T</mi><mo>=</mo><mn>470</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> (<math><mo>∼</mo><mn>1.5</mn><msub><mi>T</mi><mi>c</mi></msub></math>), as well as in <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> flavor QCD at <math><mi>T</mi><mo>=</mo><mn>220</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> (<math><mo>∼</mo><mn>1.2</mn><msub><mi>T</mi><mrow><mi>p</mi><mi>c</mi></mrow></msub></math>) with <math><msub><mi>m</mi><mi>π</mi></msub><mo>=</mo><mn>320</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. In order to quantify the nonperturbative effects, the lattice correlator is compared with the corresponding <math><mi>NLO</mi><mo>+</mo><msup><mi>LPM</mi><mi>LO</mi></msup></math> estimate of correlator. The reconstruction of the spectral function is performed in several different frameworks, ranging from physics-informed models of the spectral function to more general models in the Backus-Gilbert method and Gaussian process regression. We find that the resulting photon rates agree within errors.

Pakistan

Sajid Ali

Dibyendu Bala

Taiwan

Anthony Francis

United States

France

Greg Jackson

Olaf Kaczmarek

Jonas Turnwald

Tristan Ueding

Lattice QCD estimates of thermal photon production from the QGP

We compute spectral functions in a scalar <math><msup><mi>ϕ</mi><mn>4</mn></msup></math>-theory in three spacetime dimensions via the spectral functional renormalization group. This approach allows for the direct, manifestly Lorentz covariant computation of correlation functions in Minkowski spacetime, including a physical on-shell renormalization. We present numerical results for the spectral functions of the two- and four-point correlation functions for different values of the coupling parameter. These results agree very well with those obtained from another functional real-time approach, the spectral Dyson-Schwinger equation.

Jan Horak

Jonas Wessely

Scalar spectral functions from the spectral functional renormalization group

We compute the bound state properties of three-dimensional scalar <math><msup><mi>ϕ</mi><mn>4</mn></msup></math> theory in the broken phase. To this end, we extend the recently developed technique of spectral Dyson-Schwinger equations to solve the Bethe-Salpeter equation and determine the bound state spectrum. We employ consistent truncations for the two-, three- and four-point functions of the theory that recover the scaling properties in the infinite coupling limit. Our result for the mass of the lowest-lying bound state in this limit agrees very well with lattice determinations.

Austria

Gernot Eichmann

Andrés Gómez

Bound states from the spectral Bethe-Salpeter equation

We investigate the dissipation rate of a scalar field in the vicinity of the phase transition and the ordered phase, specifically within the universality class of model A. This dissipation rate holds significant physical relevance, particularly in the context of interpreting effective potentials as inputs for dynamical transport simulations, such as hydrodynamics. To comprehensively understand the use of effective potentials and other calculation inputs, such as the functional renormalization group, we conduct a detailed analysis of field dependencies. We solve the functional renormalization group equations on the Schwinger-Keldysh contour to determine the effective potential and dissipation rate for both finite and infinite volumes. Furthermore, we conduct a finite-size scaling analysis to calculate the dynamic critical exponent <math><mi>z</mi></math>. Our extracted value closely matches existing values from the literature.

Laura Batini

Italy

Eduardo Grossi

Dissipation dynamics of a scalar field

We compute masses of the two lightest glueballs from spectral reconstructions of timelike interaction channels of the four-gluon vertex in Landau gauge Yang-Mills theory. The Euclidean spacelike dressings of the vertex are calculated with the functional renormalization group. For the spectral reconstruction of these Euclidean data, we employ Gaussian process regression. The glueball resonances can be identified straightforwardly and we obtain <math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>s</mi><mi>c</mi></mrow></msub><mo>=</mo><mn>1870</mn><mo>(</mo><mn>75</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math> as well as <math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msub><mo>=</mo><mn>2700</mn><mo>(</mo><mn>120</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, in accordance with functional bound state and lattice calculations.

Coralie S. Schneider

Julian M. Urban

Yang-Mills glueball masses from spectral reconstruction

We investigate the renormalization group (RG)-time integration of the effective potential in the functional renormalization group in the presence of spontaneous symmetry breaking and its subsequent convexity restoration on the example of a scalar theory in <math><mi>d</mi><mo>=</mo><mn>3</mn></math>. The features of this setup are common to many physical models and our results are, therefore, directly applicable to a variety of situations. We provide exhaustive work-precision benchmarks and numerical stability analyses by considering the combination of different discrete formulations of the flow equation and a large collection of different algorithms. The results are explained by using the different components entering the RG-time integration process and the eigenvalue structure of the discrete system. Particularly, the combination of Rosenbrock methods, implicit multistep methods, or certain (diagonally) implicit Runge-Kutta methods with exact or automatic differentiation Jacobians proves to be very potent. Furthermore, a reformulation in a logarithmic variable circumvents issues related to the singularity bound in the flat regime of the potential.

Numerical RG-time integration of the effective potential: Analysis and benchmark

We compute the strong coupling constant of Landau gauge QCD in the full complex momentum plane, both directly and via spectral reconstruction. In particular, we consider the Taylor coupling given by the product of ghost and gluon dressing functions. Assuming spectral representations for the latter, we first show that also the coupling obeys such a representation. The subsequent spectral reconstruction of the coupling data, obtained from <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> flavor lattice QCD results for the ghost and gluon, is based on a probabilistic inversion of this representation using Gaussian process regression with analytically enforced asymptotics. In contradistinction, our direct calculation relies on earlier reconstruction results for the ghost and gluon spectral functions themselves, as well as data obtained in functional QCD. Apart from its relevance for studies of resonances or scattering processes, the calculation also serves as a nontrivial benchmark of our reconstruction approach. The results show remarkable agreement, testifying to the reliability of the method.

Savvas Zafeiropoulos

Nonperturbative strong coupling at timelike momenta

We reconstruct ghost and gluon spectral functions in <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> flavor QCD with Gaussian process regression. This framework allows us to largely suppress spurious oscillations and other common reconstruction artifacts by specifying generic magnitude and length scale parameters in the kernel function. The Euclidean propagator data are taken from lattice simulations with domain wall fermions at the physical point. For the infrared and ultraviolet extensions of the lattice propagators as well as the low-frequency asymptotics of the ghost spectral function, we utilize results from functional computations in Yang-Mills theory and QCD. This further reduces the systematic error significantly. Our numerical results are compared against a direct real-time functional computation of the ghost and an earlier reconstruction of the gluon in Yang-Mills theory. The systematic approach presented in this work offers a promising route toward unveiling real-time properties of QCD.

José Rodríguez-Quintero

Reconstructing QCD spectral functions with Gaussian processes