Abstract:We compute all the tree-level contributions to the Wilson coefficients of the dimension-six Standard-Model effective theory in ultraviolet completions with general scalar, spinor and vector field content and arbitrary interactions. No assumption about the renormalizability of the high-energy theory is made. This provides a complete ultraviolet/infrared dictionary at the classical level, which can be used to study the low-energy implications of any model of interest, and also to look for explicit completions consistent with low-energy data.

We analyze a modular invariant model of lepton masses, with neutrino masses originating either from the Weinberg operator or from the seesaw. The constraint provided by modular invariance is so strong that neutrino mass ratios, lepton mixing angles and Dirac/Majorana phases do not depend on any Lagrangian parameter. They only depend on the vacuum of the theory, parametrized in terms of a complex modulus and a real field. Thus eight measurable quantities are described by the three vacuum parameters, whose optimization provides an excellent fit to data for the Weinberg operator and a good fit for the seesaw case. Neutrino masses from the Weinberg operator (seesaw) have inverted (normal) ordering. Several sources of potential corrections, such as higher dimensional operators, renormalization group evolution and supersymmetry breaking effects, are carefully discussed and shown not to affect the predictions under reasonable conditions.

We explore alternative descriptions of the charged lepton sector in modular invariant models of lepton masses and mixing angles. In addition to the modulus, the symmetry breaking sector of our models includes ordinary flavons. Neutrino mass terms depend only on the modulus and are tailored to minimize the number of free parameters. The charged lepton Yukawa couplings rely upon the flavons alone. We build modular invariant models at levels 4 and 5, where neutrino masses are described both in terms of the Weinberg operator or through a type I seesaw mechanism. At level 4, our models reproduce the hierarchy among electron, muon and tau masses by letting the weights play the role of Froggatt-Nielsen charges. At level 5, our setup allows the treatment of left and right handed charged leptons on the same footing. We have optimized the free parameters of our models in order to match the experimental data, obtaining a good degree of compatibility and predictions for the absolute neutrino masses and the CP violating phases. At a more fundamental level, the whole lepton sector could be correctly described by the simultaneous presence of several moduli. Our examples are meant to make a first step in this direction. arXiv:1908.11867v1 [hep-ph] 30 Aug 2019 Recently, modular invariance has been invoked as candidate flavour symmetry [10]. In its simplest implementation a unique complex field, the modulus, acts as symmetry breaking parameter, thus simplifying the vacuum alignment problem. Modular invariance, in the limit of exact supersymmetry, completely determines the Yukawa couplings, to any order of the expansion in powers of the modulus. Moreover, neutrino masses, mixing angles and phases are all related to each other and, in minimal models, depend only on a few parameters. The formalism has been extended to consistently include CP transformations [11] 1 and it can involve several moduli [16,17]. The idea that Yukawa couplings are determined by a set of moduli is clearly not new, and has been naturally realized in the context of string theory [18][19][20][21][22], in D-brane compactification [23][24][25][26][27][28][29], in magnetized extra dimensions [30][31][32], and in orbifold compactification [33][34][35][36]. Modular invariance has also been incorporated in early flavour models [37][38][39][40][41]. However, the main advantage of the recent approach is that it can be implemented in a bottom-up perspective, relying on the group transformation properties of modular forms of given weight and level. Several models of lepton masses and mixing angles have been built at level 2 [42, 43], 3 [10, 44-47], 4 [48-50] and 5 [51,52]. Extensions to quarks [53,54] and to grand unified theories [55,56] have also been proposed. In most of the existing constructions, there is a unique symmetry breaking parameter: the modulus itself. While this scenario is certainly appealing since it minimizes the symmetry breaking sector, it does not yet provide a convincing explanation of the charged lepton masses. The mass hierarchy is achieved by...

MatchingTools is a Python library for doing symbolic calculations in effective field theory. It provides the tools to construct general models by defining their field content and their interaction Lagrangian. Once a model is given, the heavy particles can be integrated out at the tree level to obtain an effective Lagrangian in which only the light particles appear. After integration, some of the terms of the resulting Lagrangian might not be independent. MatchingTools contains functions for transforming these terms to rewrite them in terms of any chosen set of operators. PROGRAM SUMMARY Program Title: MatchingTools Licensing provisions: MIT Programming language: Python (compatible with versions 2 and 3) Nature of problem:The program does two kinds of calculations: computing an effective Lagrangian for the light fields of a field theory by integrating out at the tree level the heavy fields and performing algebraic manipulations with tensors in the (effective) Lagrangian. Solution method:The tree level integration of heavy fields is done by substituting them inside the Lagrangian by a covariant derivative expansion of the solution to their equations of motion. The transformation of Lagrangians is implemented as an algorithm for finding patterns of tensor products and replacing them by sums of other products.

2 More generally, any change of renormalization scheme can compensated by a change in the action. 3 Actually, the renormalization group invariance can be understood as the invariance under a particular type of field redefinition [3]. 4 In fact, this is more than a mere analogy: any quantum field theory has a BRST symmetry associated