Phenomenology and Chiral properties of Strong interactions: the $\ell_7$ low energy constant via lattice QCD simulations

Quantum Chromodynamics, or QCD, is the quantum field theory that describes the strong interaction between quarks and gluons, which is one of the three fundamental forces in nature outlined in the Standard Model of particles. This theory is a non-Abelian Gauge theory whose gauge group is a local $SU(3)$ which makes the study of its low-energy regime not feasible, since the coupling becomes $\mathcal{O}(1)$ for an energy scale on the order of $\Lambda_{\text{QCD}}\simeq 200$ MeV, where Perturbation Theory breaks down. Moreover, QCD exhibits several Non-Perturbative phenomena, such as color confinement, which cannot be studied without an alternative approach. The most successful one is Lattice QCD, which can be used to perform numerical simulations from first principles, and study much of the low-energy physics to an ideally arbitrary precision by means of Monte Carlo methods. Other powerful tools for studying the low-energy dynamics of QCD are effective field theories, such as Chiral Perturbation Theory. An Effective Field Theory (EFT) is a type of approximation, whose aims is to select the relevant degrees of freedom at the desired energy scale while neglecting the others. Chiral Perturbation theory (ChPT) is a particular EFT which focuses on the low-energy behavior of QCD exploiting the (approximate) chiral symmetry of the original theory. The basis of ChPt is the global $\mathrm{SU}(3)_L \times \mathrm{SU}(3)_R \times U(1)_V$ symmetry of the QCD Lagrangian in the massless case, which is assumed to be spontaneously broken to $\mathrm{SU}(3)_V$ giving rise to 8 massless Goldstone bosons. In order to write down the ChPT Lagrangian one writes down all the possible and most general terms not excluded by the underlying symmetries of the theory, ordering them based on the number of momenta and mass powers ($p$-expansion). The expansion will contain free parameters which are called Low Energy Constants (LECs). These parameters may be fixed phenomenologically by comparison to experimental data. At Leading-Order (LO), considering only $\mathcal{O}(p^2)$ terms, one will have only two LECs related to the pion decay constant and the quark condensate while the most general Lagrangian at $\mathcal{O}(p^4)$, i.e. at Next-Leading Order (NLO), contains ten LECs. The aim of this work takes place in this context. We want to perform the first direct measure of one of the LECs at NLO that is $\ell_7$ by means of lattice numerical simulations using staggered fermions. Among all of the LECs, $\ell_7$ is the one known less precisely, with an uncertainty larger than 50\% and its last estimate has been provided by means of a global fit involving other LECs, resulting in a strongly correlated estimate to the other LECs values. An high precision evaluation of $\ell_7$ is important for several phenomenological aspects such as for the axion phenomenology. The axion is an hypotetical particle postulated to resolve the strong CP problem in QCD and it is also known to behave as a cold dark matter candidate in all cosmologically relevant scales. Increased precision in the determination of $\ell_7$ would affect the value and uncertainty of the axion mass and its quartic self-coupling, entering also in the axion-pion scattering amplitude. Our work must be considered as a validation and extension of the RM123 framework introduced by the RM123 collaboration with rotated twisted mass fermions. The latter involves expanding the path integral around the iso-symmetric point in powers of the up and down quark mass difference. Then, one can take advantage of the RM123 approach by considering certain quantities such as the charged and neutral pion mass difference and extract an estimate for $\ell_7$, evaluating specific Euclidean correlation functions. In addition to extend the method to our staggered fermion discretization, which is know to have a lower computational cost and better numerical stability in certain situations compared to other fermion formulations, we will perform both chiral and continuum extrapolations.