CoDEx is a Mathematica ® package that calculates the Wilson Coefficients (WCs) corresponding to effective operators up to mass dimension-6. Once the part of the Lagrangian involving single as well as multiple degenerate heavy fields, belonging to some Beyond Standard Model (BSM) theory, is given, the package can then integrate out propagators from the tree as well as 1-loop diagrams of that BSM theory. It then computes the associated WCs up to 1-loop level, for two different bases: "Warsaw" and "SILH". CoDEx requires only very basic information about the heavy field(s), e.g., Colour, Isospin, Hyper-charge, Mass, and Spin. The package first calculates the WCs at the high scale (mass of the heavy field(s)). We then have an option to perform the renormalisation group evolutions (RGEs) of these operators in "Warsaw" basis, a complete one (unlike "SILH"), using the anomalous dimension matrix. Thus, one can get all effective operators at the electro-weak scale, generated from any such BSM theory, containing heavy fields of spin: 0, 1/2, and 1. We have provided many example models (both here and in the package-documentation) that more or less encompass different choices of heavy fields and interactions. Relying on the status of the present day precision data, we restrict ourselves up to dimension-6 effective operators. This will be generalised for any dimensional operators in a later version. Program SummaryProgram Title: CoDEx Version: 1.0.0 Licensing provisions: CC By 4.0 Programming language: Wolfram Language ® URL: https://effexteam.github.io/CoDEx Send BUG reports and QuestionsHere, d i is the mass dimensionality of the operator O i (starts from 5), and C i is the corresponding Wilson coefficient -function of BSM parameters. It is important to note that the choice of operator basis, i.e., explicit structure of O i 's is not unique. Among different choices we restrict us to "SILH" [1,2] and "Warsaw" [3][4][5] bases. These bases can be transformed from one to another. Λ is the cut off scale at which all WCs are computed (C i (Λ)), and usually identified as the mass of the heavy field being integrated out. This EFT approach relies on the validity of the perturbative expansion of the S-matrix in the powers of Λ −1 (UV-scale), and the resultant series is expected to pass the convergence test. As this scale is higher than the scale M Z , where the precision test is performed, dimension-6 operators are more suppressed than the dimension-5 ones and so on. Now, where to truncate the 1/Λ series? This decision is made case by case, based on the achieved(expected) precision level of the observables at present(future) experiments 1 . One can consult these lectures [7][8][9][10][11] where effective field theory has been introduced and discussed in great detail. Several other packages and libraries are available in the literature, which do various things regarding SMEFT operators and the corresponding Wilson Coefficients, from basis transformation to running of the coefficients [12][13][14][15][16]. Now the nagging questions are:...

Effective Field Theory (EFT) technique is one of the most elegant ways to capture the impact of high scale theory, if any, at some low energy by incorporating higher mass dimensional (≥ 5) effective operators (O i ). The low energy EFT is described in terms of only light degrees of freedom (DOF) which can appear on-shell. An essential task while developing the EFT framework is to compute these O i 's. Hilbert Series (HS) is a novel and mathematically robust method to construct the complete set of gauge invariant independent, effective operators. The HS requires the knowledge of the transformation properties of the light DOF and the covariant derivatives under the internal gauge symmetries and conformal groups. The Hilbert Series method, by its virtue, automatically takes care of the redundancies in the operator set due to the Equations of Motion (EOMs) of fields and Integration by Parts (IBPs) with impeccable accuracy.In this paper, we have adopted this methodology to construct the complete set of independent operators up to dimension-6 in the "Warsaw"-like basis for two different Beyond Standard Model scenarios -Two Higgs Doublet Model (2HDM) and Minimal Left-Right Symmetric Model (MLRSM). For both these cases, we have calculated the corrections to the scalar, gauge boson and fermion mass spectra due to the dimension-6 operators. The additional contributions to all the Feynman vertices are computed and their impact on different observables, namely Weak mixing angle, Fermi constant, ρ and oblique (S, T, U ) parameters. We have further discussed how the magnetic moments of charged leptons and production and decay of the massive BSM particles, e.g., charged scalar and different rare processes are affected in the presence of effective operators. We have also constructed the effective scalar four-point interactions and commented on the possible reinvestigation of the theoretical constraints, e.g., unitarity and vacuum stability within these frameworks.

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The Standard Model Effective Field Theory (SMEFT) is an established theoretical framework that parametrises the impact a UV theory has on low-energy observables. Such parametrization is achieved by studying the interactions of SM fields encapsulated within higher mass dimensional (≥ 5) operators. Through judicious employment of the tools of EFTs, SMEFT has become a source of new predictions as well as a platform for conducting a coherent comparison of new physics (beyond Standard Model) scenarios. We, for the first time, are proposing a diagrammatic approach to establish selection criteria for the allowed heavy field representations corresponding to each SMEFT operator. We have elucidated the links of a chain connecting specific CP conserving dimension-6 SMEFT operators with unique sets of heavy field representations. The contact interactions representing each effective operator have been unfolded into tree- and (or) one-loop-level diagrams to reveal unique embeddings of heavy fields within them. For each case, the renormalizable vertices of a UV model serve as the building blocks for all possible unfolded diagrams. Based on this, we have laid the groundwork to construct observable-driven new physics models. This in turn also prevents us from making redundant analyses of similar models. While we have taken a predominantly minimalistic approach, we have also highlighted the necessity for non-minimal interactions for certain operators.