## Periodic Reporting for period 1 - Tips in SCQFT (Lattice gauge theories studies of timely theoretical and phenomenological questions in strongly coupled quantum field theories.)

Reporting period: 2019-04-01 to 2021-03-31

The Standard Model (SM) of particle physics is the best present description of the fundamental laws of Nature: The model has been spectacularly verified, the discovery of the Higgs boson being its crowning completion. However, in spite of this extraordinary success, there are a number of questions about the nature of fundamental interactions which remain unanswered, among which we mention the strong CP problem (related to the symmetry under the combined action of particle-antiparticle exchange and spatial reflections), the matter-antimatter asymmetry, the origin of dark matter. Some of these problems are also interrelated, think for instance of axions, which are candidate new particles which could solve the CP problem and the dark matter problem at the same time.

A full completion of our knowledge about the world around us has been a primary goal for humankind since ever, disseminating history with a series of pure knowledge breakthroughs, inducing also technological progress afterwards. In this perspective, pure research efforts trying to solve the open questions above are of fundamental importance for society as a whole.

Part of the open problems above emerge in regimes where some of the interaction couplings get large and perturbation theory is not applicable, thus demanding for a non-perturbative (NP) approach. A notable example is represented by Quantum Chromodynamics (QCD), the theory describing the interactions among quarks and gluons and how they bind together to form the hadrons we observe: the study of many aspects of QCD needs a NP approach. Many theories which are being considered as possible extensions of the SM are also characterized by NP regimes. The method of choice, in this case, are numerical lattice simulations, which are based only on first principles and are systematically improvable.

Within this context, we addressed a number of specific timely theoretical and phenomenological questions, mostly related to the spectrum of strongly coupled quantum gauge theories (within the SM or beyond) and to the dependence of QCD on the topological angle theta, whose small phenomenological value represents in fact the strong CP problem:

a) we made substantial progress towards a comprehensive determination of the spectrum of QCD and of QCD-like theories, in particular glueballs and flux tubes, which could be observed by future experiments, or be used as an input for theories beyond the SM;

b) we made progress towards setting more stringent values on the theta angle through a NP determination, by lattice QCD simulations, of the neutron electric dipole moment (nEDM). At the same time, we have addressed the problem of determining the dependence on the theta angle of many observables in QCD and QCD-like theories, with a particular focus on quantities related to axion phenomenology.

In order to reach the objectives above, we have also developed a number of side products which will seed future progress in various directions, in particular regarding numerical codes for High Performance Computing and machine learning techniques.

A full completion of our knowledge about the world around us has been a primary goal for humankind since ever, disseminating history with a series of pure knowledge breakthroughs, inducing also technological progress afterwards. In this perspective, pure research efforts trying to solve the open questions above are of fundamental importance for society as a whole.

Part of the open problems above emerge in regimes where some of the interaction couplings get large and perturbation theory is not applicable, thus demanding for a non-perturbative (NP) approach. A notable example is represented by Quantum Chromodynamics (QCD), the theory describing the interactions among quarks and gluons and how they bind together to form the hadrons we observe: the study of many aspects of QCD needs a NP approach. Many theories which are being considered as possible extensions of the SM are also characterized by NP regimes. The method of choice, in this case, are numerical lattice simulations, which are based only on first principles and are systematically improvable.

Within this context, we addressed a number of specific timely theoretical and phenomenological questions, mostly related to the spectrum of strongly coupled quantum gauge theories (within the SM or beyond) and to the dependence of QCD on the topological angle theta, whose small phenomenological value represents in fact the strong CP problem:

a) we made substantial progress towards a comprehensive determination of the spectrum of QCD and of QCD-like theories, in particular glueballs and flux tubes, which could be observed by future experiments, or be used as an input for theories beyond the SM;

b) we made progress towards setting more stringent values on the theta angle through a NP determination, by lattice QCD simulations, of the neutron electric dipole moment (nEDM). At the same time, we have addressed the problem of determining the dependence on the theta angle of many observables in QCD and QCD-like theories, with a particular focus on quantities related to axion phenomenology.

In order to reach the objectives above, we have also developed a number of side products which will seed future progress in various directions, in particular regarding numerical codes for High Performance Computing and machine learning techniques.

We extracted the spectrum of the closed flux-tube in SU(N) Gauge Theory (GT) in 4D. We demonstrate the existence of an axion on the world-sheet of the QCD string as well as a bound state of two axions. This work has been presented in two workshops with a manuscript under preparation. Additionally, the calculation of the spectrum in the presence of a θ-term and that of k=2 flux-tubes is under progress; first results for k=2 demonstrate the existence of an axion.

We obtained the spectrum of glueballs in SU(3) GT, setting up the ground for the extraction of the glueball spectrum in QCD. Our results appear in JHEP 11 (2020) 172. Additionally, we extracted the glueball spectrum in SU(N=∞); the manuscript is under preparation.

Moreover, we investigated the scalar glueball ground state mass for SU(3) and SU(4) trace deformed GT defined on R3×S1 where center symmetry is recovered even at small compactification radii. We observed that the glueball mass in the deformed theory, when center symmetry is recovered, is compatible with its value at zero temperature; this is reported in arXiv:2010.03618.

We also extracted the nEDM in lattice QCD for quarks with physical masses. Such investigation is of great importance because a finite value of the nEDM would direct towards BSM physics. This was based on an expansion of the CP-breaking topological term in the action in powers of the θ-angle and the re-weighting of 2- and 3-point functions with the topological charge. Our result appears to be statistically consistent with zero (Phys.Rev.D 103 (2021) 5, 054501). A possibility to increase its statistical precision is by using configurations generated with an imaginary θ-term. First, we investigated the topological susceptibility for full QCD with an imaginary θ-term. Continuum extrapolations demonstrated good agreement with Chiral Perturbation Theory. Next, we initiated the extraction of nEDM in QCD with an imaginary θ-term included in the action. This is a long term investigation with the first goal to study whether this method can significantly reduce the statistical uncertainty on the nEDM.

Finally, we investigated the conformal properties of the SU(2) GT with 1 adjoint fermion. Namely, we presented a major update by considering a number of larger volumes and four different values of the gauge coupling. We extracted results for the mass spectrum, Polyakov loops, and mass anomalous dimension. Our calculations hint towards the IR (near-)conformality of the theory. However, further studies at large values of coupling and smaller fermion masses are needed to fully understand the phase realized by the theory. The results have been published in arXiv:2103.10485.

We obtained the spectrum of glueballs in SU(3) GT, setting up the ground for the extraction of the glueball spectrum in QCD. Our results appear in JHEP 11 (2020) 172. Additionally, we extracted the glueball spectrum in SU(N=∞); the manuscript is under preparation.

Moreover, we investigated the scalar glueball ground state mass for SU(3) and SU(4) trace deformed GT defined on R3×S1 where center symmetry is recovered even at small compactification radii. We observed that the glueball mass in the deformed theory, when center symmetry is recovered, is compatible with its value at zero temperature; this is reported in arXiv:2010.03618.

We also extracted the nEDM in lattice QCD for quarks with physical masses. Such investigation is of great importance because a finite value of the nEDM would direct towards BSM physics. This was based on an expansion of the CP-breaking topological term in the action in powers of the θ-angle and the re-weighting of 2- and 3-point functions with the topological charge. Our result appears to be statistically consistent with zero (Phys.Rev.D 103 (2021) 5, 054501). A possibility to increase its statistical precision is by using configurations generated with an imaginary θ-term. First, we investigated the topological susceptibility for full QCD with an imaginary θ-term. Continuum extrapolations demonstrated good agreement with Chiral Perturbation Theory. Next, we initiated the extraction of nEDM in QCD with an imaginary θ-term included in the action. This is a long term investigation with the first goal to study whether this method can significantly reduce the statistical uncertainty on the nEDM.

Finally, we investigated the conformal properties of the SU(2) GT with 1 adjoint fermion. Namely, we presented a major update by considering a number of larger volumes and four different values of the gauge coupling. We extracted results for the mass spectrum, Polyakov loops, and mass anomalous dimension. Our calculations hint towards the IR (near-)conformality of the theory. However, further studies at large values of coupling and smaller fermion masses are needed to fully understand the phase realized by the theory. The results have been published in arXiv:2103.10485.

The project facilitates for first time the production of configurations in full QCD including a θ-topological term. This opens new ways for nEDM investigations and demonstrates that the conventional method of extracting the nEDM is statistically inevitable as well as the superiority of spectral projectors as a topological charge schema. This sends a message to research groups studying nEDM in which direction they should move in order to extract statistically significant results.

Our work demonstrates that traced deformed GTs materializing simplified GTs with small compactification radii, can reproduce the right properties of the theory in the infinite volume. This opens new ways of reconsidering possible extensions of such theories to describe the physical world.

We elaborate on the existence of axion modes indicating the inclusion of an extrinsic curvature term in the string theory description of the flux-tube. This provides important knowledge for those developing string theoretical models for the confining flux-tube and will potentially attract the attention of young researchers to perform analytical as well as computational investigations on this topic.

Moreover, this project brought back SU(2) GT with 1 adjoint fermion, which has recently applied on topological phases in condensed matter systems.

The fellow presented his research in two science communication events and attracted the attention of more than 700 people. Hence, he spread the words for strong force and computational physics with main goal to influence young students to follow a career path in this direction and strengthen the involvement of young researchers especially women towards this route.

Our work demonstrates that traced deformed GTs materializing simplified GTs with small compactification radii, can reproduce the right properties of the theory in the infinite volume. This opens new ways of reconsidering possible extensions of such theories to describe the physical world.

We elaborate on the existence of axion modes indicating the inclusion of an extrinsic curvature term in the string theory description of the flux-tube. This provides important knowledge for those developing string theoretical models for the confining flux-tube and will potentially attract the attention of young researchers to perform analytical as well as computational investigations on this topic.

Moreover, this project brought back SU(2) GT with 1 adjoint fermion, which has recently applied on topological phases in condensed matter systems.

The fellow presented his research in two science communication events and attracted the attention of more than 700 people. Hence, he spread the words for strong force and computational physics with main goal to influence young students to follow a career path in this direction and strengthen the involvement of young researchers especially women towards this route.