The main goal of our research program is the study of the nonperturbative features of strongly interacting gauge theories, and notably of quantum chromodynamics (QCD).  Due to the limitations of perturbative computations, nonperturbative methods are required to investigate several fundamental properties of strongly interacting particles and strongly interacting matter. Characterizing the phase diagram of QCD as a function of temperature, baryon density and other external parameters is a typical task for which nonperturbative approaches are essential. The same is true for the determination of the dependence of QCD properties on the value of the topological theta angle, not to mention the study of the confinement mechanism.

The main approach that is used to investigate from first principles the nonperturbative properties of QCD is the lattice formulation, in which the continuum euclidean theory is discretized on a finite space-time lattice, and expectation values (thermal or vacuum ones) are estimated by means of Monte-Carlo simulations. Since reliable QCD simulations require huge computational resources, significant efforts have to be devoted to the writing of efficient and portable code, in order to use in the most effective way present and future computational resources. Effective field theories can also be profitably used to supplement or support the numerical approach, for example when a direct numerical analysis is too computationally expensive.

The study of simplified models sharing some nonperturbative features with QCD (e.g.  QCD with unphysical quark masses or chemical potentials, Yang-Mills theories, spin models) can be extremely useful in several respects: to test new theoretical ideas and algorithms, to investigate the accuracy and possible limitations of some approximation schemes, and to perform precision studies to verify some fine theoretical predictions.

The main activities of the various research units follows (in square brackets the relevant references from the arxiv)

The color confinement mechanism [Bari, Cosenza, Pisa]

The long-distance linear static potential between color sources is naturally associated with the presence of a tube-like structure (“flux tube”) of the chromo-electric field in the longitudinal direction, i.e. along the line connecting the static quark and antiquark. In recent works we have obtained a comprehensive numerical description of the color field generated by static sources in pure SU(3) gauge theory, measuring all components of both chromoelectric and chromomagnetic fields. Moreover, we have given first evidence of the persistence in the continuum limit of the chromomagnetic currents around the flux tubes in the SU(3) vacuum, thus demonstrating that the confining force can be understood using an electromagnetic analogy, as suggested by the dual superconductor model of the vacuum. We plan to extend our analysis to (2+1)-flavor QCD with physical quark masses, where only preliminary results are available so far, and to the finite temperature case (both in pure SU(3) and in full QCD) across the deconfinement transition. We also plan to assess the impact of perturbing the zero-temperature vacuum with an external chromomagnetic field on the shape of the flux tube and, in general, on the distribution of all color fields and currents around the static sources. An ambitious step, strongly dependent on the development of an efficient algorithm and on the availability of computer time, could be the study in pure gauge SU(3) of the 3-quark static potential and of the corresponding flux-tube structure.

Beyond the linear leading term, the long distance behavior of the static potential in gauge theories is characterized by universal subleading terms, predicted by effective string theory. In the 3d U(1) gauge model the Polyakov solution is however not fully consistent with these expectations, a fact confirmed by numerical simulations of the model. To better understand this discrepancy we will study variations of the standard 3d U(1) model (e.g. adding a coupling to monopoles or investigating models coupled to scalars, which display strongly interacting charged fixed points). Whether effective string predictions remain valid for 4d SU(N) Yang-Mills theories in the presence of a nonvanishing theta term in 4d SU(N) Yang-Mills theories is another topic that will be scrutinized. Finally we plan to study the static potential and the color flux-tube in trace deformed Yang-Mills theories, to better understand their relevance for analytical approaches to the confinement problem.

Theta dependence and QCD axion [Pisa]

We plan to extend our numerical study of the topological susceptibility in full QCD to higher temperatures. In order to do that we will need to solve issues related to topological freezing, by extending to full QCD the algorithm developed in [2012.14000].  This study will be important in order to make prediction on the cosmological axion abundance based on the misalignment mechanism for axion production; at the same time, we will investigate the sphaleron rate in full QCD, by the analysis of the temporal correlators of the topological charge density, which is relevant to thermal axion production. The relation between chiral properties and topology at zero temperature will be studied by an investigation of the chiral condensate and of the mass dependence of the topological susceptibility through the staggered spectral projectors method developed in [1908.11832].

The topological properties of the QCD vacuum, the properties of the QCD axion, and their relationships with chiral symmetries (in particular with the U(1) axial symmetry), can also be analytically investigated, both at zero and at finite temperature, by means of chiral effective Lagrangians. In particular, we plan to investigate the properties at finite temperature (around the chiral transition) of the QCD axion also in those cases in which the quarks (and, therefore, the mesons) are charged under the U(1) Peccei-Quinn symmetry. We also plan to investigate in detail the role of the strange quark for the above-mentioned topics and for the scalar and pseudoscalar meson mass spectrum of the theory at finite temperature, around the chiral transition.

Restricting to the pure gauge case, we plan to extend the study of the theta dependence of the deconfinement temperature [1205.0538, 1306.2919] to SU(N) gauge theories with N>3. In four dimensional Yang-Mills theories we also plan to study the dependence of the spectrum (i.e. string tension and glueball masses) on the theta parameter. The use of the analytic continuation method can be instrumental in significantly improving the exploratory results already present in the literature. An analogous study can be carried out also at finite temperature, thus investigating the theta-dependence of the screening masses, for which no result at all is known. In the finite temperature setting, we plan to investigate the shape, size and interaction strengths of finite-temperature topological excitations, to shed light on the temperature range for which
semiclassical instanton computations in the dilute approximation can be reliably applied. Depending on the results of this pure gauge study, an extension of this type of analysis to the QCD case can also be considered.

QCD in extreme conditions [Bari, Cosenza, Ferrara, Pisa]

We have recently shown that the crossover transition of QCD with physical quark masses becomes a first order phase transition for large values of the external magnetic field [2111.11237]. We plan to continue our investigation on the location of the critical endpoint of this first order line in the T-eB plane, and to investigate various physical observables at or around the transition line. In particular, we plan to determine the latent heat and to study how various properties related to confinement and chiral symmetry change when crossing the first order transition, in particular the localization properties of the Dirac spectrum and the structure of the color flux tube between static sources. Another issue of interest will be the determination of the transport properties of the thermal medium, in particular the electric conductivity, in the presence of strong magnetic background fields and across the phase transition.

We plan to explore the QCD phase diagram with (2+1) HISQ flavors in the presence of a constant abelian chromomagnetic field. This study will be done on the line of constant physics with a pion mass of 140 MeV. Our objective is to examine the relationship between the deconfinement temperature and the intensity of the external field.

Dual formulations of lattice models [Cosenza, Pisa]

Dual formulation approaches to solve the sign problem, which prevents obtaining reliable results at finite density, have been applied to two-dimensional nonlinear sigma models at non-zero chemical potential [1808.07810] and to three-dimensional SU(3) Polyakov loop models at finite baryon potential [2011.08285, 2112.00043]. While in the former case the dual representation is exact, in the latter case the positive dual weight has been obtained adopting certain approximations. The most essential of these approximations are the strong coupling expansion and the static approximation for the full quark determinant. Within these approximations it is possible to derive some Polyakov loop models which thus become effective models describing high-temperature QCD. The dual theory is then constructed for such effective models.  We plan to extend this work to QCD with non-zero baryon density, improving and relaxing the approximations made so far. In particular we plan to improve the strong coupling approximation by constructing and simulating the dual model at arbitrary values of the temporal gauge coupling constant and to extend the dual formulation to the full lattice gauge action with the static fermion determinant. Corrections to the quark static determinant can be included by expanding the non-static contribution in the ratio of lattice spacings on the anisotropic lattice with staggered and/or Wilson fermions. A sign-free formulation of scalar QCD at finite density will also be investigated.

With a positive weight formulation available, we plan to study the phase diagram of the dual model in the temperature-baryon chemical potential plane, and to calculate baryon density, quark condensate and Polyakov loop expectation values. Appealing appears the possibility of using these methods to compute the screening chromo-electric and chromo-magnetic masses in the high-temperature deconfined phase.

High performance computing and quantum computing [Bari, Ferrara, Pisa]

The reference RHMC LQCD code used by our group has been developed using MPI, accelerated for NVidia GPUs through the OpenACC framework, and has been optimized targeting the Kepler and Pascal based GPU architectures (K80 and P100) [1701.00426,1801.01473]. We now plan to target the more recent NVidia GPU accelerators (V100, A100 and H100). To optimize the code for these architectures we plan to profile the performance of these accelerators, identifying possible kernels of our code for which improvements are possible, and performing the corresponding implementation changes required to overlap at best the kernel execution with the transfer time between device and hosts and also between MPI processes. This might require careful modification of the data structure, in order to exploit the largest possible fraction of computing power offered by these devices. For this activity we plan to develop specific micro-benchmarks and mini-app to measure the computing performance and assess the corresponding offload overheads.

Another activity will be focused on making our code easily portable to different computing architectures, possibly also with different accelerators, investigating the potentiality offered by OpenMP. Indeed the latest release of OpenMP now supports GPU acceleration computing, allowing to annotate the  kernels that should be offloaded and executed on GPUs through OpenMP pragma directives. This allows to keep unchanged the structure of the code that can be easily recompiled to target different architectures. For this activity we plan to initially annotate specific kernels of our LQCD code to execute on GPUs, and assess the computing performance and compare with that achieved with the OpenACC framework.

Quantum computing provides a radical solution to some algorithmic problems encountered in the standard formulation of lattice QCD when dealing with finite density or real time problems. However, quantum computing also presents several challenges, both related to the present time limited hardware resources and to algorithm development. We plan to benchmark quantum algorithms for the evaluation of thermal averages using simple discrete gauge models, and to investigate the new challenges posed by gauge theories in the application of other already quite developed quantum algorithms, like those using variational ansatzes.

Strongly coupled lattice models [Pisa]

In the context of quantum gravity with the causal dynamical triangulation approach, we plan to extend our previous work on the two dimensional interaction between gravity and gauge fields to higher dimensions. This will require developing suitable algorithms for the update of geometry and gauge fields, and an implementation of gauge related observables on triangulated geometries.

We plan to complete the study of the three dimensional strongly coupled gauge models coupled to scalar fields, then to move to the case of fermionic models. This will be done starting from the Abelian case, which is both computationally simpler and theoretically more solid (results for a large number of flavor fields are available), considering both the compact and the non-compact formulations of the U(1) gauge group. This part of the study will give us enough information to judge the feasibility of an investigation involving the nonAbelian case. Also for the fermionic case it will be possible to make contact with the problematics discussed in the above “color confinement”
subsection.