In this paper we present the complete one-loop matching conditions, up to dimension-six operators of the Standard Model effective field theory, resulting by integrating out the two scalar leptoquarks S 1 ∼ (3, 1) 1 3 and S 3 ∼ (3, 3) 1 3 . This allows a phenomenological study of low-energy constraints on this model at one-loop accuracy, which will be the focus of a subsequent work. Furthermore, it provides a rich comparison for functional and computational methods for one-loop matching, that are being developed. As a corollary result, we derive a complete set of dimension-six operators independent under integration by parts, but not under equations of motions, called Green's basis, as well as the complete reduction formulae from this set to the Warsaw basis.

We perform a complete study of the low-energy phenomenology of S1 and S3 leptoquarks, aimed at addressing the observed deviations in B-meson decays and the muon magnetic dipole moment. Leptoquark contributions to observables are computed at one-loop accuracy in an effective field theory approach, using the recently published complete one-loop matching of these leptoquarks to the Standard Model effective field theory. We present several scenarios, discussing in each case the preferred parameter space and the most relevant observables.

We discuss fermion mass hierarchies within modular invariant flavour models. We analyse the neighbourhood of the self-dual point τ = i, where modular invariant theories possess a residual Z4 invariance. In this region the breaking of Z4 can be fully described by the spurion ϵ ≈ τ − i, that flips its sign under Z4. Degeneracies or vanishing eigenvalues of fermion mass matrices, forced by the Z4 symmetry at τ = i, are removed by slightly deviating from the self-dual point. Relevant mass ratios are controlled by powers of |ϵ|. We present examples where this mechanism is a key ingredient to successfully implement an hierarchical spectrum in the lepton sector, even in the presence of a non-minimal Kähler potential.

We assume that the quark-flavor coefficients matrix of the semileptonic operators addressing the neutral-current B-meson anomalies has rank-one, i.e. it can be described by a single vector in quark-flavor space. By correlating the observed anomalies to other flavor and high-p T observables, we constrain its possible directions and we show that a large region of the parameter space of this framework will be explored by flavor data from the NA62, KOTO, LHCb and Belle II experiments.

The correct equations have been substituted in the e-print version of ref [1]. We thank Konstantinos Mantzaropoulos for pointing out these corrections to us, after a comparison with results obtained via functional methods.

The ladder Bethe-Salpeter Equation of a bound (1/2) + system, composed by a fermion and a scalar boson, is solved in Minkowski space, for the first time. The formal tools are the same already successfully adopted for two-scalar and two-fermion systems, namely the Nakanishi integral representation of the Bethe-Salpeter amplitude and the light-front projection of the fulfilled equation. Numerical results are presented and discussed for two interaction kernels: i) a massive scalar exchange and ii) a massive vector exchange, illustrating both the correlation between binding energies and the interaction coupling constants, as well as the valence content of the interacting state, through the valence probabilities and the light-front momentum distributions. In the case of the scalar exchange, an interesting side effect, to be ascribed to the repulsion generated by the small components of the Dirac spinor, is pointed out, while for the vector exchange the manifestation of the helicity conservation opens new interesting questions to be addressed within a fully non-perturbative framework, as well as the onset of a scale-invariant regime.

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