Reinhard Alkofer

University Graz, Austria

Title: Strong and electromagnetic isospin breaking in mesons

Abstract: Besides leading to interesting insights into the structure of hadrons, the effects of isospin breaking are important when judging the validity of the recently found first-row-CKM unitarity violation. In this respect, the isospin breaking effects for kaons deserve further studies. In a recent work [A. Miramontes et al., 2202.04618] electromagnetic as well as strong isospin breaking effects in the isospin mass splittings of light pseudoscalar and vector mesons are studied by employing a coupled system of quark Dyson-Schwinger and meson Bethe-Salpeter equations whose interaction kernels contain gluon, pion and photon exchange interactions. In bound states, QCD-induced isospin breaking is manifest on different levels. On the one hand, a different explicit up- and down-quark mass directly affects the propagators of the constituent quarks. On the other hand, it leads to different interaction kernels within the isospin multiplets. In addition, electromagnetic isospin breaking is induced via a photon exchange diagram. Using the kaon iso-doublet and the charged pion masses as input to determine the up, down and strange quark masses different patterns for the pion, kaon and rho meson mass splittings different each are found. Effects from two sources of mass splittings, the different quark masses and the different quark charges, do not add up linearly.

Jens Oluf Andersen

Norwegian University of Science and Technology, Norway

Title: Phase diagram and condensates at finite isospin and strangeness density

Abstract: In this talk I will map out the phase diagram at T=0 and finite isospin and strangeness chemical potentials using three-flavor Chpt. I compute the quark and pion condensates in the pion-condensed phase and compare the results with recent lattice simulations. I briefly discuss the inclusion of electromagnetic interactions and the resulting Higgs phases.

Aritra Bandyopadhyay

University of Heidelberg, Germany

Title: Lepton pair production from a hot dense magnetized QCD medium within NJL model

Abstract: We have explored the very important quantity of lepton pair production from a hot dense magnetized QCD medium within the Nambu-Jona-Lasinio model for simultaneous nonzero values of both the parallel and perpendicular components of momentum. We have also tuned our effective model to incorporate both the Magnetic Catalysis (MC) and Inverse Magnetic Catalysis (IMC) effects in our results. As opposed to the zero magnetic field case (the so-called Born rate) or the lowest Landau level approximated rate, where only the annihilation process contributes, here we observe contributions also arising out of the quark and antiquark decay processes. We found the encouraging result of considerable enhancement of lepton pair production in presence of a magnetic field. We further decompose the total rate into different physical processes and make interesting observations for both zero and nonzero baryon density. For annihilation processes we also capture the effect of IMC near the critical temperature.

Adnan Bashir

University of Michoacan, Mexico

Title: Revisiting Hlbl scattering contributions to the muon g-2 in continuum QCD

Abstract: The Standard Model has resisted more than half a century of experimental barrage of challenges successfully. The precision tests are now reaching unprecedented levels. The g-2 of the muon in particular is capable of zooming into the very limits of the Standard Model. I will review progress in this direction and revisit some hadronic light by light scattering contributions to it within continuum QCD.

Victor Braguta

JINR, Russia

Title: Influence of relativistic rotation on QCD properties within lattice simulation

Abstract: This report is devoted to our study of QCD properties effected by relativistic rotation. The study is carried out within lattice simulations which are performed in the reference frame rotating with the system under investigation. First we focus on rotating gluodynamics. We found that the critical temperature of the confinement/deconfinement transition in gluodynamics grows quadratically with increasing angular velocity. In addition, we present our first results on rotating QCD matter with dynamical fermions. The results indicate, that the effect of the rotation on fermions is opposite to gluons: it leads to the decrease of the critical temperature.

Marco Catillo

ETH Zurich, Switzerland

Title: Chiralspin symmetry group and its presence in QCD

Abstract: In Lattice QCD it is becoming more and more evident the existence of a range of temperatures (between Tc and 3Tc), in which matter becomes chiralspin symmetric (at least effectively) and the chromo-electric interaction between quarks becomes more relevant. The role of chiralspin symmetry in this regime is interesting to investigate. In this talk, I consider another possible redefinition of the chiralspin group compatible with deconfinement at high temperature QCD, describe some main features and explore the consequences.

Yong-rui Chen

Dalian University of Technology, China

Title: Critical behaviors of the O(4) and Z(2) symmetries in the QCD phase diagram

In this work we have studied the QCD phase structure and critical dynamics related to the 3-d O(4) and Z(2) symmetry universality classes in the two-flavor quark-meson low energy effective theory within the functional renormalization group approach. We have employed the expansion of Chebyshev polynomials to solve the flow equation for the order-parameter potential. The chiral phase transition line of O(4) symmetry in the chiral limit, and the Z(2) line of critical end points related to the explicit chiral symmetry breaking are depicted in the phase diagram. Various critical exponents related to the order parameter, chiral susceptibilities and correlation lengths have been calculated for the 3-d O(4) and Z(2) universality classes in the phase diagram, respectively. Moreover, the size of the critical regime in the QCD phase diagram is found to be very small.

Reference:

Yong-rui Chen,Rui Wen,and Wei-jie Fu, Phys.Rev.D 104(2021)5,054009. arXiv:2101.08484

Pietro Dall'Olio

Universidad de Huelva, Spain

Title: Gauge dependence of QCD correlation functions in linear covariant gauges

Abstract: Gauge dependence of QCD correlation functions, for linear covariant gauges, is studied within a novel formalism that introduces a gauge-invariant operator composed by the gauge field and an auxiliary Stueckelberg scalar field. This framework, recently formulated with the aim of extending the Gribov-Zwanziger scenario to linear covariant gauges in a local and BRST invariant fashion, allows to systematically separate the gauge parameter-dependent part in the calculation of any correlation function (i.e. to derive their Landau-Khalatnikov-Fradkin transformations).

Moreover, the formalism may generalize the analysis pursued by M. Tissier and N. Wschebor in the Landau gauge, by introducing a dimension 2 gauge-invariant composite operator, which reduces to the usual massive gluon term in the Landau gauge and searching for possible mass-dependent infrared safe renormalization schemes. Preliminary results are presented hereof.

Arpan Das

IFJ PAN, Poland

Title: Canonical and phenomenological formulations of spin hydrodynamics, and equivalence between different frameworks

Abstract: Recent observations of the spin polarization of weakly decaying Lambda hyperons have opened up a new direction to explore non-trivial vortical structures of strongly interacting matter produced in the heavy-ion experiments. A consistent framework of relativistic hydrodynamics with spin degrees of freedom (spin hydrodynamics) is under construction now to allow for future dynamical simulations of spin polarization. This type of hydrodynamic description is based on the conservation of the total energy and linear momentum as well as the total angular momentum which includes both the orbital and spin parts. The phenomenological approach used to construct the framework of spin hydrodynamics commonly uses a simplified form of the spin tensor. This form does not possess an expected symmetry, namely, it is not totally antisymmetric (which is a direct consequence of Noether’s Theorem applied to the Dirac Lagrangian). Consequently, in this approach, the connection between the spin hydrodynamics and the underlying field-theoretic arguments is obscured. We discuss two formulations of relativistic hydrodynamics of particles with spin 1/2 are compared. The first one uses expressions for the energy-momentum and spin tensors that have properties that follow a direct application of Noether's theorem, including a totally antisymmetric spin tensor. The other one is based on a simplified form of the spin tensor that is commonly used in the current literature under the name of phenomenological pseudo-gauge. We argue that the naive use of canonical energy-momentum tensor and the spin tensor is in contradiction with the condition that in a dissipative system entropy should be produced. Moreover, we argue that canonical spin hydrodynamics and the phenomenological formulation of spin hydrodynamics can be connected by a suitably defined pseudo-gauge transformation, provided the canonical energy-momentum tensor is properly improved. Our analysis uses arguments related to the positivity of entropy production. Our results help us to find and clarify connections between different formulations of spin hydrodynamics, which is important for the construction of a fully consistent formalism.

This talk will be based on our recent work:

"Equivalence of canonical and phenomenological formulations of spin hydrodynamics",

Asaad Daher, Arpan Das, Wojciech Florkowski, Radoslaw Ryblewski; arXiv:2202.12609.

Duive Maria van Egmond

CPHT, Ecole Polytechnique, France

Title: Finite-temperature longitudinal gluon propagator in the center-symmetric Landau gauge

Abstract: We evaluate the longitudinal or "(chromo-)electric" gluon propagator in the recently proposed center-symmetric Landau gauge at finite temperature. 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, the so-obtained longitudinal gluon propagator displays a singular behavior in the SU(2) case, characteristic of a continuous phase transition. In the SU(3) case, although there is no such singular behavior, we also observe a rather peaked variation of the longitudinal propagator around the transition. These results are in sharp contrast with corresponding ones obtained within the standard Landau gauge. We also argue why this is so.

Gernot Eichmann

LIP & IST Lisboa, Portugal

Title: Going to the light front with contour deformations

Abstract: I will talk about a new method to compute the light-front wave functions for a system of two interacting scalar particles using contour deformations. After solving the two-body Bethe-Salpeter equation, the projection onto the light front is done through a combination of contour deformations and analytic continuation methods. The resulting light-front wave functions and distribution amplitudes are in agreement with the Nakanishi method frequently used in the literature. Several extensions of the model towards unequal-mass systems and complex conjugate propagator poles are studied to make contact with QCD.

Antonio Mauricio Soares Narciso Ferreira

University of Campinas, Brazil

Title: Evidence of the Schwinger-Mechanism in QCD

Abstract: In QCD, the Schwinger mechanism endows the gluons with an effective mass through the dynamical formation of longitudinally coupled massless bound-state poles. The presence of these poles affects profoundly the infrared properties of the interaction vertices, inducing crucial modifications to their fundamental Ward identities. Within this general framework, we derive the non-Abelian Ward identity obeyed by the pole-free part of the three-gluon vertex in the soft-gluon limit, and determine the smoking-gun displacement that the onset of the Schwinger mechanism produces to the standard result. Quite importantly, this displacement coincides precisely with the Bethe-Salpeter amplitude of the massless bound state. Consequently, this signal may be computed in two independent ways: by solving an approximate version of the pertinent Bethe-Salpeter equation, or by appropriately combining the elements that enter in the aforementioned Ward identity. For the implementation of both methods, we employ two- and three-point correlation functions obtained from recent lattice simulations, and a partial derivative of the ghost-gluon kernel, which is computed from the corresponding Schwinger-Dyson equation. Our analysis reveals a striking agreement between the results obtained through either method, providing a highly nontrivial self-consistency check for the entire approach.

Christian Fischer

University of Giessen, Germany

Title: Mesons, baryons and the phase diagram of QCD

Abstract: We summarise recent theoretical results on the QCD phase diagram and the properties of hadrons at finite temperature and chemical potential based on a combination of lattice QCD and Dyson-Schwinger equations. We discuss the silver blaze property of mesons with different quantum numbers along the zero-T-finite-mu-axis and assess the influence of meson and baryon fluctuations on the location of the critical end point.

Stefan Floerchinger

Friedrich-Schiller-Universität Jena, Germany

Title: Using non-Riemannian geometry to study relativistic fluids

Abstract: I will discuss what can possibly be learned from coupling quantum fields to gauge fields appearing in non-Riemannian geometry such as the Weyl gauge field, spin connection and proper non-metricity. This will lead to equations of motion for the dilation current, spin current and shear current which also couple to the energy-momentum tensor and could be useful for a macroscopic description in the spirit of fluid dynamics.

Adrien Florio

Stony Brook, USA

Title: Dynamics of the O(4) critical point in QCD

Abstract: There are indications that the deconfinement phase transition of QCD with two-massless flavors is a second-order phase transition, likely to be in the O(4) universality class. The real-world QCD deconfinement transition is a crossover, but the smallness of the up and down quark masses makes it plausible to be in the critical region of the O(4) critical point. This hypothesis is supported by recent lattice QCD simulations, which were able to model static QCD properties using the O(4) scaling functions.

In this talk, I will review the notion of dynamical universality class. I will then introduce a hydrodynamic theory with the order parameter of the QCD phase transition as an additional hydrodynamical variable. I will recall why it shares the same universality class of QCD ("Model G") and as a result why it can be used to gain insight into the real-time dynamics of QCD. The main focus will be on our recent results obtained from classical real-time simulations (2111.03640) of this hydrodynamic theory. I will discuss the emergence of pion pseudo-particles in the broken phase and their impact on the dynamics. I will also discuss our extraction of the dynamical critical exponent 𝜁=1.47±0.01(𝑠𝑡𝑎𝑡.), which is compatible with the theoretical expectation 𝜁=3/2.

Fei Gao

Peking University

Title: Running masses in QCD and the QCD phase diagram

Abstract: The resolution of the QCD phase structure at finite temperature and density is essential for understanding the formation of matter and the evolution of the universe. It is also pivotal for interpreting data collected at running and planned heavy-ion experiments as well as further predictions. In the present talk, I discuss results obtained within the quantitative Functional QCD approach to the QCD phase structure. Within this advanced functional approach lattice benchmarks at small and vanishing density are met quantitatively. In particular, the curvature of the phase transition line coincides with the results from lattice QCD. This allows for an estimate of the critical endpoint (CEP) at around (T,\mu_B)\sim(110,600)MeV. I close with a discussion of further improvements of the approach at large densities and the prospects of the quantitative determination of the CEP or the onset of new physics.

Andreas Geissel

TU - Darmstadt, Germany

Title: Speed of sound of strong-interaction matter at supranuclear densities

Abstract: We present results for the zero-temperature thermodynamics of strong-interaction matter at high densities which have been obtained based on first-principles functional renormalization group studies. In particular, we discuss gluon screening effects on the equation of state and the speed of sound in a semi-perturbative manner. Additionally, taking into account results from studies based on the existence of a (color-)superconducting gap, we present consistent constraints for the speed of sound at supranuclear densities.

Jan Horak

Heidelberg University, Germany

Title: Spectral structure of QCD

Abstract: We investigate timelike correlation functions in QCD via spectral Dyson-Schwinger equations. Within the spectral formulation, direct access to the full complex structure of the theory is obtained. Studying the propagation of non-holomorphicities through the coupled system of DSEs in Yang-Mills theory, we obtain consistency conditions which constrain the analytic structure of possible solutions. This suggests that solutions including complex singularities are highly unlikely to exist in Yang-Mills and QCD. Making use of spectral reconstruction results for gluon and ghost propagators, we calculate the strong coupling constant in the full complex plane. In particular, we find that the propagator coupling itself obeys a spectral representation. This opens the door for calculation of timelike scattering processes of the fundamental QCD degrees of freedom. Finally, we present direct results for the quark spectral function in QCD via the spectral Dyson-Schwinger equation in the chiral limit and for light quark flavors.

Ivan Horvath

Czech Academy of Science & Univ of Kentucky, USA

Title: IR Phase of Thermal QCD

Abstract: I will discuss the rationale and evidence for the recently proposed phase of thermal QCD above the crossover temperature regime, characterized by restored scale invariance of glue in the infrared.

Chuang Huang

Dalian University of Technology, China

Title: Four-quark scatterings in QCD

Abstract: We investigate dynamical chiral symmetry breaking and the emergence of mesonic bound state from the infrared dynamics of four-quark scatterings. Both phenomena originate from the resonant scalar-pseudoscalar interaction channel, and we compute the functional renormalisation group (fRG) flows of the Fierz-complete four-quark interactions of up and down quarks with its $t$ channel momentum dependence. This is done in the isospin symmetric case, also including the flow of the quark two-point function. This system can be understood as the fRG analogues of the complete Bethe-Salpeter equations and quark gap equation. The pole mass of pion is determined from both direct calculations of the four-quark flows in the Minkowski regime of momenta and the analytic continuation based on results in the Euclidean regime, which are consistent with each other[1]. Furthermore, we compute the momentum dependence quark mass and extract the Bethe-Salpeter amplitude by computing the four-quark interactions with full Mandelstam variable dependence[2].

[1] Wei-jie Fu, Chuang Huang, Jan M. Pawlowski, Yang-yang Tan, in preparation.

[2] Wei-jie Fu, Chuang Huang, Jan M. Pawlowski, Yang-yang Tan, in preparation.

Markus Huber

University of Giessen, Germany

Title: On the glueball spectrum of Yang-Mills theory

Abstract: Results for the ground states and excited states of glueballs in Yang-Mills theory from Bethe-Salpeter equations are presented. The employed input comes from parameter-free Dyson-Schwinger calculations of the propagators and vertices.

Jamal Jalilian-Marian

Baruch College, CUNY, USA

Title: QCD at small Bjorken x

Abstract: A hadron/nucleus probed at small Bjorken x is thought to be a Color Glass Condensate; a dense system of gluons for which the standard leading twist pQCD techniques are not applicable. We describe a new Wilsonian renormalization group approach to this high-density system and investigate its properties. We outline how it may be probed in high-energy hadron/nucleus collisions and explore its experimental signatures.

Ruben Kara

University of Wuppertal, Germany

Title: The upper right corner of the Columbia plot with staggered fermions

Abstract: QCD with infinite heavy quark masses exhibits a first-order thermal transition which is driven by the spontaneous breaking of the global $\mathcal{Z}_3$ center symmetry. We analyze the corresponding order parameter, namely the Polyakov loop and its moments, in a finite volume scaling study, as well as the topological charge.

Decreasing the quark masses weakens the transition until the latent heat vanishes at the critical mass. We give an update on our exploration of the heavy mass region with three flavors of staggered quarks.

Maria Paola Lombardo

INFN, Italy

Title: The strongly interacting Quark Gluon Plasma meets the conformal window

Abstract: The chirally symmetric region of QCD is continuously connected with the conformal window, as shown by early FRG results by Braun and Gies, and further explored on the lattice. This led to the speculation that some properties of the strongly interacting Quark-Gluon Plasma, including the applicability of the AdS/CFT correspondence, may be understood by exploiting this continuity. Recent work by Alexandru and Horvath takes further steps in this direction. In this talk I will contrast and compare the behaviour of the order parameter in the QGP with that of the conformal window, monitoring the crossover from the strongly to the weakly coupled regime in the two systems.

Peter Lowdon

Goethe University Frankfurt, Germany

Title: Non-perturbative insights into the spectral properties of finite-temperature correlation functions

Abstract: Local quantum field theory (QFT) provides a framework for establishing the non-perturbative constraints imposed on finite-temperature correlation functions. In this talk I will discuss how the locality of fields has significant implications for the spectral properties of finite-temperature QFTs, in particular that the peak-broadening effects experienced by particle states can be directly extracted from imaginary-time correlation functions. As an application, I discuss the calculation of the pion spectral function peak from Euclidean data.

Cédric Mezrag

CEA - Université Paris-Saclay, France

Title: Nucleon Distribution Amplitude

Abstract: Baryon internal structure is nowadays a core research field of both theoretical and experimental QCD. If Parton Distribution Functions (PDFs) have been deeply investigated on the experimental and phenomenological side, other matrix elements encompassing information on the nucleon structure remain poorly known. Among them, Distribution Amplitudes are of specific interest, as they select a given Fock-state fluctuation, and since they appear in the description of several exclusive processes. In this talk, we will present on-going work to build a model for the leading-twist hadron distribution amplitude inspired by new results obtained regarding the diquark correlations within the nucleon. These diquark correlations are used to build a model of the Faddeev wave function, which is itself projected so that we extract the nucleon distribution amplitude. We finally compare our results with computation of moments obtained by lattice-QCD simulations.

Jose Manuel Morgado Chávez

University of Huelva, Spain

Title: Predictions from continuum Schwinger studies of pion’s structure at future electron ion colliders

Abstract: The formalism of Dyson-Schwinger equations is a powerful tool for the study of correlation functions in quantum field theory, but has also proved to yield an outstanding framework for the evaluation of hadron properties. That is precisely the baseline for this work. Starting from state-of-the-art continuum Schwinger calculations for the pion’s parton distribution function, we describe its extension to off-forward hadron kinematics, yielding the pion’s generalized parton distribution. From that point on, we evaluate the amplitude for deeply virtual Compton scattering (DVCS) in the kinematic regime covered at the foreseen Electron-Ion Collider. Pure Dyson-Schwinger predictions for the event rates and beam spin asymmetries to be observed are presented, revealing the dominance of gluon content within the pion in driving its response to DVCS at future electron-ion colliders.

Swagato Mukherjee

Brookhaven National Laboratory, USA

Title: Partonic pictures of Goldstone mesons from lattice QCD

I will present some recent state-of-the-art lattice QCD results revealing partonic structures of pion and kaon. These results will include valance parton distribution function of pion, pion distribution amplitude as well as electromagnetic form factor of kaon at large momentum transfers.

Bianca M. S. de Oliveira

University of Campinas, Brazil

Title: Four-point functions in the Background field method: Designing a Schwinger-Dyson truncation

We present an exploratory study of the four-point function composed of two background gluons, a ghost, and an antighost field, in the so-called ``all-soft'' limit, where all incoming momenta are set to zero. In this limit, the most general tensorial structure of this vertex reduces to a unique contribution, proportional to the metric. We determine the value of the associated form factor in two ways. First, we derive an exact result from the Ward-Takahashi identity that this vertex satisfies, expressing the form factor in terms of the well-known ghost dressing function. The second method is based on the Schwinger-Dyson equation satisfied by four-point vertex; the previous result emerges provided that all dressed vertices involved satisfied their own Ward-Takahashi identities. Finally, we explore different truncation scenarios for the Schwinger-Dyson equation, in order to check how the exact relation may serve as a guideline for developing reliable truncation schemes in general kinematics.

Laurin Pannullo

Goethe University Frankfurt, Germany

Title: Inhomogeneous phases in the 3+1-dimensional mean-field Nambu-Jona-Lasinio model on the lattice

Abstract: At low temperature and large chemical potential QCD might exhibit a chiral inhomogeneous phase, as indicated by various simple low-energy models. One of these models is the 3+1-dimensional Nambu-Jona-Lasinio model, which is non-renormalizable -- rendering the results possibly dependent on the employed regularization scheme. While most previously published results regarding the inhomogeneous phase in this model were obtained with the Pauli-Villars or similar regularizations, this talk explores the dependence of this phase on different lattice and related continuum regularizations.

Peter Petreczky

Brookhaven National Laboratory, USA

Title: The complex potential at non-zero temperature

Abstract: In this talk I will discuss the calculation of the static quark anti-quark potential in 2+1 flavor QCD. The potential at non-zero temperature can be defined in terms of the spectral decomposition of the Wilson line correlators in Coulomb gauge or Wilson loops. If the spectral function has a dominant peak, then the position and the width of this peak define the real and imaginary part of the potential, respectively. We identify the dominant peak of the spectral function in the lattice QCD calculations, and using a simple parametrization calculate the peak position and its width. The calculations are performed using HISQ action on lattices with temporal extent Nt=12 and varying the temperature by changing the lattices spacing [1] as well as in the fixed scale approach using lattice spacing a=0.028 fm and temporal extent Nt=20-36 [2]. Interestingly, we find that the real part of the potential is not screened, while the imaginary part of the potential is proportional to the temperatures. I will also point out how these findings corroborate recent lattice QCD results on bottomonium properties at non-zero temperature.

[1] D. Balla et al (HotQCD), Phys. Rev. D 105 (2022) 054513.

[2] HotQCD, work in progress.

Owe Philipsen

Goethe University Frankfurt, Germany

Title: Chiral spin symmetry and the QCD phase diagram

Abstract: Recently an approximate SU(4) chiral spin-flavour symmetry was discovered in multiplets of QCD meson correlation functions, in a temperature range above the chiral crossover. This symmetry is larger than the full chiral symmetry of QCD with massless u,d-quarks. It can only arise when color-electric interactions dominate the effective Dirac action of QCD, which suggests that mesons remain bound in this regime. At temperatures about two to three times the crossover temperature, this pattern disappears again, and the usual chiral symmetry is recovered. After presenting independent evidence for this phenomenon based on meson screening masses, I discuss how this chiral spin symmetric band continues into the QCD phase diagram.

Khépani Raya Montaño

University of Huelva, Spain

Title: Theoretical and Phenomenological constraints on the pion GPD

Abstract: We discuss how the valence-quark distribution function can be extended to off-forward kinematics to construct the corresponding generalized parton distribution (GPD), by employing the experimental data on its electromagnetic form factor. The discussion is heavily based upon the hypothesis of the existence of an energy scale at which the hadron can be completely understood in terms of fully dressed valence degrees of freedom, the so called hadronic scale, and an all orders evolution scheme. In addition to being compatible with empirical and lattice results, the obtained GPDs fulfill all prescriptions from QCD and exhibit agreement with contemporary theoretical predictions.

Urko Reinosa

CPHT - Ecole Polytechnique, France

Title: Physical Symmetries and Gauge Fixing

Abstract: In the context of gauge theories, depending on the gauge, it might be more or less easy to devise approximation schemes that preserve the underlying physical symmetries, such as center symmetry. The same is true when looking for models of the gauge fixing procedure in the infrared, beyond the (ultraviolet) Faddeev-Popov procedure. We critically review how background field gauges provide a favourable set-up for building symmetry-preserving approximations/models. After recalling the standard method based on the use of self-consistent backgrounds, we present a novel proposal that does not suffer from the limitations of the latter and which can be implemented both in the continuum and on the lattice. We compare the two approaches within the framework of the Curci-Ferrari model, a model for infrared Landau gauge that has been well tested in the vacuum. Within our new proposal, we obtain improved values for the transition temperatures and a better comparison of the order parameter with the lattice data.

Michael Riberdy

CEA - Université Paris-Saclay, France

Title: Nucleon Light Front Wave Functions and GPDs from Fully Projected Fadeev Wave Functions

Abstract: While hadronic structure is experimentally probed by processes such as DIS and DVCS, both lattice and continuum techniques are employed to corroborate such results from a theoretical perspective. On the continuum side, the Fadeev equations provide a fully covariant approach to three body interactions convenient for describing three quark hadronic states. However, the four-spacetime-dimensional nature of the corresponding Fadeev wave functions precludes a probabilistic interpretation, encouraging our use of their 3-spacetime-dimensional light cone projections, Light Front Wave Functions (LFWFs). An intuitional advantage of LFWFs is their role as coefficients in Fock expansions of hadronic states. We first define nucleon Fadeev wave functions in terms of off-diagonal nucleon matrix elements, and subsequently express the corresponding definite orbital angular momentum (OAM) nucleon LFWFs. With these definite quark helicity LFWFs in hand we calculate GPDs as linear combinations of their overlaps, and isolate definite OAM contributions to nucleon GPDs, PDFs, Form Factors (FFs) and the electric nucleon radius. Looking forward, this work will allow us to map dynamical effects underlying the computation of Fadeev wave functions to the multidimensional structure of the nucleon.

Craig Roberts

Nanjing University, China

Title: Parton distribution functions at a crossroad

Abstract: Parton distribution functions (DFs) are a preeminent source of hadron structure information; and experiments interpretable in terms of hadron DFs have long been a priority. For much of this time, DFs were inferred from global fits to data and viewed as benchmarks. Such fitting remains crucial, providing input for the conduct of numerous experiments worldwide. But the past decade has seen the dawn of a new theory era, with continuum and lattice studies of quantum chromodynamics beginning to yield robust DF predictions. These developments are exposing conflicts with the fitting results; and such disagreements encourage one of the following conclusions: the global fit outcomes are misconstrued; not all considered data are a true expression of intrinsic hadron qualities; or QCD, as currently understood, is not the theory of strong interactions. This presentation will sketch the background to these issues and ideas for resolving the attendant controversies.

José Rodríguez Quintero

University of Huelva, Spain

Title: Pion and proton parton distributions approached with an effective charge

Abstract: We will report on recent works featuring the pion and proton parton distribution functions at experimental scales, following an approach based on the assumption that there is an effective charge which defines an evolution scheme that is all-orders exact.

Leonardo Rodrigues dos Santos

University of Campinas, Brazil

Title: The pole structure of the ghost-gluon vertex

The activation of the Schwinger mechanism in QCD proceeds through the dynamical formation of longitudinally coupled massless poles in the fundamental vertices of the theory. In this talk we study the emergence of such a pole in the ghost-gluon vertex, through the inhomogeneous integral equation that governs the relevant form factor in general kinematics. The solution obtained reproduces the known zero-momentum patterns, and is free of the scale-fixing ambiguities typical of the homogeneous integral equations employed in previous studies.

Saga Säppi

ECT*, Trento, Italy

Title: NNNLO cold dense perturbation theory using hard thermal loops

Abstract: Understanding QCD at finite baryon density is a notoriously difficult problem, with a perturbative diagrammatic expansion in a low-coupling limit being one of the few viable first-principles options. pQCD, too, runs into trouble at higher orders, due to infrared-sensitive low-momentum soft scales. However, the divergences associated with them can be treated via effective field theory methods, and the zero-temperature theory exhibits no Linde problem, making its expansion well-defined to, in principle, all orders. Hard Thermal Loops can be used to render the zero-temperature perturbative expansion finite at NNNLO: I will explain a clean organisation of the contributions to the cold dense pressure into different kinematical regions, and go over contributions from the purely soft region, evaluated last year.

Afterwards, I will discuss a very recent computation of the "mixed" sector in the simplified setting of QED. The mixed sector consists of combined contributions combining both HTL and naïve perturbation theory. In practice, these terms are obtained by first computing higher-order corrections to the photon self-energy, done at finite temperature and density. In the computation of the pressure diagrams an explicit cancellation of the IR divergences of the naïve perturbation theory and the UV divergences of HTL is also seen.

Relevant papers:

PRD 104 (2021) 7, 074015 (hep-ph/2103.074279)

PRL 127 (2021) 16, 162003 (hep-ph/2103.05658)

arXiv hep-ph/2204.11893

arXiv hep-ph/2204.11279

Collaborators:

Tyler Gorda (TU Darmstadt), Aleksi Kurkela (U. of Stavanger), Juuso Österman, Risto Paatelainen, Philip Schicho, Kaapo Seppänen, Aleksi Vuorinen (all U. of Helsinki)

Benedikt Paul Schallmo

TU Darmstadt, Germany

Title: Equation of state of asymmetric strong-interaction matter at supranuclear densities

Abstract: We present results for the zero-temperature thermodynamics of strong-interaction matter at high densities which have been obtained based on first-principles functional renormalization group studies for isospin-asymmetric matter with two quark flavors. For trajectories relevant for astrophysical applications, we find indications of a first-order phase transition from a color-superconducting phase to an ungapped quark-matter phase. Moreover, we provide an estimate for the speed of sound in neutron star matter. Taking into account results from studies based on chiral effective field theory at low densities, our findings suggest the existence of a maximum in the speed of sound at supranuclear densities.

Sayantan Sharma

IMSc, Chennai, India

Title: The fate of axial U(1) symmetry in the chiral limit of QCD

Abstract: We study the fate of axial U(1) near the chiral crossover transition in 2+1 flavor QCD, along the line of physical strange quark mass and approaching the chiral limit for the light quarks. From a detailed analysis of the dependence of the renormalized eigenvalue spectrum of the QCD Dirac operator and related observables on the light quark mass, our study suggests that axial U(1) remains broken at the crossover region even when the chiral limit is approached. We end by discussing the implications of our work.

Ref: O. Kaczmarek, L. Mazur and S. Sharma, Phys.Rev.D 104 (2021) 9, 094518.

Igor Shovkovy

Arizona State University, USA

Title: Dilepton Emission from Magnetized Quark-Gluon Plasma

Abstract: We study the dilepton production rate from a hot quark-gluon plasma in a quantizing background magnetic field. We find that the magnetic field leads to a strong enhancement of the dilepton rate at small values of the invariant mass. In the same kinematic region, the dilepton production is characterized by a sizable ellipticity. At large values of the dilepton invariant mass, the role of the magnetic fields decreases and the result approaches the isotropic zero-field Born rate. By investigating the dependence of ellipticity on the transverse momentum, we argue that the future measurements of dilepton rate in the region of small invariant masses can constrain the magnetic field produced in heavy-ion collisions.

Paulo Silva

University of Coimbra, Portugal

Title: The ghost propagator and the transition to deconfinement in pure gauge SU(3) Yang-Mills

Abstract: The ghost propagator is studied at finite temperature below and above Tc using lattice QCD simiulations. We find that the ghost propagator is enhanced for T > Tc compared to the confined phase. Our results suggest that, similarly to the gluon propagator case, the ghost propagator can also be used to identify the phase transition.

Feliciano de Soto

Universidad Pablo de Olavide, Spain

Title: The full three-gluon vertex from quenched-lattice QCD

Abstract: We report novel lattice QCD results for the three-gluon vertex $\overline\Gamma_{\alpha\mu\nu}(q,r,p)$ from quenched lattice-QCD simulations. Using standard Wilson action, we have computed the three-gluon vertex beyond the usual kinematic restriction to the symmetric ($q^2=r^2=p^2$) and soft-gluon ($p=0$) cases where it depends on a single momentum scale. We will present a detailed analysis of the asymmetric case ($r^2=q^2\neq p^2$) where the transversely projected vertex can be cast in terms of three independent tensors. The lattice data show a clear dominance of the form-factor corresponding to the tree-level tensor. For the general kinematical configuration ($q^2\ne r^2 \ne p^2$), we have computed the projection of the three-gluon vertex providing the relevant information on the ghost-gluon kernel related function $\mathcal{W}(q^2)$ that appears in the recently discussed smoking-gun signals of the Schwinger mechanism in QCD. This projection exhibit a striking scaling in terms of $(q^2+r^2+p^2)/2$.

Jonas Stoll and Niklas Zorbach

TU - Darmstadt, Germany

Title: The (1+1)-dimensional Gross-Neveu model at non-zero µ, T and finite N

Abstract: We investigate the Gross-Neveu model for a finite number of fermions N. The solution of the Gross-Neveu model is well known in the large-N limit (N → ∞) but unknown for finite N. We approach the finite-N case with a FRG method, more precisely the Wetterich equation. By using the local potential approximation the resulting flow equation for the scale dependent effective potential can be transformed into a non-linear diffusion equation. This equation is solved numerically by applying a finite volume method. We find no discrete chiral symmetry breaking for any finite number of fermions and arbitrary chemical potentials as long as the temperature is non-zero.

Yang-yang Tan

Dalian University of Technology, China

Title: Real-time dynamics of the $O(N)$ scalar theory within the fRG approach

Abstract: In this talk, I will present our recent study [1,2] on the real-time dynamics of the $O(N)$ scalar theory within the functional renormalization group formulated on the Schwinger-Keldysh closed time path with different truncation schemes. We derived the flow equations for the $n$-point correlation functions in terms of the ``classical'' and ``quantum'' fields and its symmetry relations. An analytic expression for the flow of the four-point vertex for both real and imaginary part is obtained within different truncation schemes. We analysed the complex structure of two- and four-point functions. We explored the the spectral functions and its scaling behaviours near the critical temperature. Moreover, we calculated the dynamical critical exponent for the phase transition near the critical temperature in the $O(N)$ scalar theory in $3+1$ dimensions with different truncation schemes.

[1] Yang-yang Tan, Yong-rui Chen, and Wei-jie Fu, SciPost Phys. 12, 26 (2022).

[2] Yang-yang Tan, Huang Chuang, Yong-rui Chen, and Wei-jie Fu, in preparation.

Jonas Turnwald,

TU - Darmstadt, Germany

Title: From the Gluon Spectral Function to Glueballs: Reconstruction with Gaussian Process Regression

Abstract: We present a novel method of reconstructing spectral functions by employing Gaussian Process Regression. This is used to investigate the bound states in YM theory and the glueball masses are successfully extracted. A new method of implementing the asymptotic behavior in the kernel of the GP is presented and applied to the reconstruction of the gluon and coupling spectral functions from QCD lattice data. The knowledge of these spectral functions is crucial in order to access real-time properties of QCD.

Xiangpeng Wang

TUM, Germany

Title: Production and polarization of S-wave quarkonium in potential non-relativistic QCD

Abstract: Based on the potential nonrelativistic QCD formalism, we compute the nonrelativistic QCD long-distance matrix elements (LDMEs) for inclusive production of S-wave heavy quarkonia. This greatly reduces the number of nonperturbative unknowns and brings in a substantial enhancement in the predictive power of the NRQCD factorization formalism. We obtain improved determinations of the LDMEs and find cross-sections and polarizations of J/ψ, ψ(2S), and excited Υ states that agree well with LHC data. Our results may have important implications in pinning down the heavy quarkonium production mechanism.

Rui Wen

Dalian University of Technology, China

Title: Equation of state of the 2+1 flavor QCD within the functional renormalization group

Abstract: The fRG approach is applied to the 2+1 flavor first-principle QCD at finite temperature and densities. With the technique of the dynamical hadronization, degrees of freedom are transferred from the quarks and gluons at the high energy scale to composite hadrons, e.g., the mesons at the low energy scale. Various flow equations of propagators and vertices are derived, which allow us to calculate observables at finite temperature and densities, e.g., the quark chiral condensate. It is found that our calculated results within the 2+1 flavor fRG-QCD approach are in excellent agreement with the lattice QCD results at vanishing baryon chemical potential. The QCD phase diagram for the 2+1 flavor QCD within the fRG approach is obtained, and the location for the CEP is estimated at $(T_{\text{CEP}}=108\text{MeV}, \mu_{B,\text{CEP}}=650\text{MeV})$. Moreover, we also calculate the QCD equations of state, i.e. the pressure, the entropy density, trace anomaly and the speed of sound, at finite temperature and baryon chemical potentials. The relevant results are compared with the lattice QCD results at vanishing and small baryon chemical potentials with $\mu_B/T \lesssim 2.5$. Within the errors our results are in good agreement with the lattice results, and the equation of state at large baryon chemical potentials are also obtained.

Jonas Wessely

ITP Heidelberg, Germany

Title: Spectral functions from spectral flows

Abstract: We develop a spectral framework for the functional Renormalisation group and employ a spectral Callan-Symanzik cutoff, leading to a (renormalised) Callan-Symanzik equation with a flowing renormalisation condition. The spectral formulation gives us direct access to the full complex frequency plane for a given diagrammatic expression. This allows us to choose an on-shell renormalisation condition to control the flow in theory space, where every point on the solution trajectory corresponds to a physical theory. We show results for the propagator spectral function in a scalar theory in 2+1 dimensions and discuss applications to QCD and in particular to heavy quarks.

Nicolas Wink

TU - Darmstadt, Germany

Title: Four-ghost mechanism

Abstract: We explore a potential mechanism in Landau gauge YM theory, which would explain the emergence of dynamical mass gap in Yang-Mills theory. The four-ghost interaction is investigated in analogy to the mechanism behind chiral symmetry breaking, which has a particularly simple picture in the fRG. First results look promising that such a mechanism could indeed be realized in Yang-Mills.

Shi Yin

Dalian University of Technology, China

Title: Real-time meson two-point functions under fRG-QCD

Abstract: We investigate the scalar and pseudo-scalar meson two-point functions under Euclidean and Minkowski spacetime in the first-principle 2+1-flavour QCD under the functional renormalization group approach. The external momentum-dependent meson wave function renormalizations, the dispersion relation are computed and a moat behavior of the meson energy is observed. The analytic continuation procedure is performed to obtain the real-time meson spectral functions. The temperature, chemical potential, and momentum dependence of the spectral functions are studied.

Reference:

[1] Wei-jie Fu, Jan M. Pawlowski, Fabian Rennecke, Rui Wen, Shi Yin, in preparation.

Savvas Zafeiropoulos

CNRS & Aix-Marseille U., France

Title: Parton distribution functions from lattice QCD

Abstract: The light-cone definition of Parton Distribution Functions (PDFs) does not allow for a direct ab initio determination employing methods of Lattice QCD simulations that naturally take place in Euclidean spacetime. In this presentation, we focus on pseudo-PDFs where the starting point is the equal time hadronic matrix element with the quark and anti-quark fields separated by a finite distance. We focus on Ioffe-time distributions, which are functions of the Ioffe-time ν, and can be understood as the Fourier transforms of parton distribution functions with respect to the momentum fraction variable 𝑥. We present lattice results for the case of the nucleon and the pion addressing among others the physical point and continuum extrapolations. We also incorporate our lattice data in the NNPDF framework treating them on the same footing as experimental data and discuss in detail the different sources of systematics in the determination of the non-singlet PDFs. Finally, we will present the latest results of the HadStruc collaboration on the gluon and transversity PDF of the nucleon.

Maria Paola Lombardo

Istituto Nazionale di Fisica Nucleare, Sezione di Firenze

Title: How computers imitate humans: Machine Learning

Abstract: The talk gives an introduction into Machine Learning (ML) using the physics of the early Universe as a use case. After a short description of the history of the Universe, from the Big Bang till today, we will discuss the current knowledge of the different phase transitions encountered in this evolution, and the role of ML in sharpening and advancing this knowledge. We will see how computers can learn from data, identify patterns and make decisions with little human intervention. In a similar way, Machine Learning can have a strong impact on the insights a business gets from the available market data. We will then conclude with an attempt at collecting specific examples of improvement of business procedures by harnessing the power of ML, perhaps using the expertise of the audience for their critical assessment.

Rob Pisarski

Brookhaven National Laboratory, USA

Title: A pedestrian guide to quantum computing

Abstract: I give an introduction into quantum mechanics, and quantum computing, suitable for those with no background in physics. I first outline various aspects of quantum mechanics, including it's necessity for the stability of matter and the uncertainty principle. I then discuss how quantum bits (qubits) will enable us, eventually, to construct computers which are exponentially more powerful than those we have at present.