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The LPC EFT workshop was held April 25-26, 2024 at the University of Notre Dame. The workshop was organized into five thematic sessions: "how far beyond linear" discusses issues of truncation and validity in interpretation of results with an eye towards practicality; "reconstruction-level results" visits the question of how best to design analyses directly targeting inference of EFT parameters; "logistics of combining likelihoods" addresses the challenges of bringing a diverse array of measurements into a cohesive whole; "unfolded results" tackles the question of designing fiducial measurements for later use in EFT interpretations, and the benefits and limitations of unfolding; and "building a sample library" addresses how best to generate simulation samples for use in data analysis. This document serves as a summary of presentations, subsequent discussions, and actionable items identified over the course of the workshop.

The complex two-Higgs doublet model (C2HDM) is one of the simplest extensions of the Standard Model with a source of CP-violation in the scalar sector. It has a $\mathbb{Z}_2$ symmetry, softly broken by a complex coefficient. There are four ways to implement this symmetry in the fermion sector, leading to models known as Type-I, Type-II, Lepton-Specific and Flipped. In the latter three models, there is a priori the surprising possibility that the 125 GeV Higgs boson couples mostly as a scalar to top quarks, while it couples mostly as a pseudoscalar to bottom quarks. This ``maximal'' scenario was still possible with the data available in 2017. Since then, there have been more data on the 125 GeV Higgs boson, direct searches for CP-violation in angular correlations of $\tau$-leptons produced in Higgs boson decays, new results on the electron electric dipole moment, new constraints from LHC searches for additional Higgs bosons and new results on $b\to s \gamma$ transitions. Highlighting the crucial importance of the physics results of LHC's Run 2, we combine all these experiments and show that the ``maximal'' scenario is now excluded in all models. Still, one can have a pseudoscalar component in $h \tau\bar{\tau}$ couplings in the Lepton-Specific case as large as $87\%$ of the scalar component for all mass orderings of the neutral scalar bosons., Comment: 24 pages, 7 figures, v2: minor changes, matches published version

The Two-Higgs Doublet Model (2HDM) is a well understood alternative to the Standard Model of particle physics. If the new particles included in the 2HDM are at an energy scale much greater than the weak scale, the theory can be matched to the Standard Model Effective Field Theory (SMEFT). We compute for the first time the complete one-loop matching at dimension-6. We compare its numerical impact with that of tree-level matching at dimension-8 by performing a global fit to single Higgs and precision electroweak measurements, and we emphasize the importance of comparing one-loop SMEFT results with corresponding one-loop results in the full 2HDM model. In the SMEFT, we consider the relative importance of both one-loop matching and the inclusion of renormalization group evolution. Our results demonstrate the necessity of studying the impact of various expansions to quantify the uncertainties of the SMEFT matching., Comment: 28 pages, 4 figures. Updated to published version. Auxiliary files at https://github.com/BDFH-2024/BDFH

Physics beyond the Standard Model (BSM) can be described in a consistent and general way through the Higgs Effective Field Theory (HEFT). Measurements of model-independent HEFT coefficients allow one to constrain the parameter space of BSM models via a matching procedure. In this work, we show that this procedure is not unique and depends on the scalings of the parameters of the Lagrangian. As examples, we consider three BSM models: the real singlet extension of the SM with a $Z_2$ symmetry, the complex singlet extension (CSE) of the SM and the 2 Higgs Doublet Model. We discuss several physical observables, and show that different scalings of the model parameters with the UV scale in the matching to the HEFT can yield quite different results. This complicates the interpretation of HEFT measurements in terms of parameters of BSM models. Additionally, as a by-product, we report the first matching of the CSE to the HEFT., Comment: Matches published version

We consider the 2 Higgs Doublet Model (2HDM) and compare two effective field theory (EFT) approaches to it, according to whether the heavy degrees of freedom are integrated out before (SMEFT) or after (HEFT) spontaneous symmetry breaking. %We show that, in the HEFT, an inconsistent EFT is obtained if one considers an expansion simply in inverse powers of the heavy masses. %We consider a HEFT expansion consistent with perturbativity %We show that, if the HEFT approach is required to comply with perturbativity, it ends up obeying the same power counting as the SMEFT one. % By requiring decoupling and perturbativity in the 2HDM, we define a consistent EFT expansion in inverse powers of the heavy masses which is applied to both the SMEFT and the HEFT matchings to the 2HDM. We organize this expansion with a dimensionless parameter $\xi$, and investigate the tree-level scatterings $hh\rightarrow hh$ and $WW\rightarrow hh$ up to $\mathcal{O}(\xi^2)$. We find no differences between the HEFT and the SMEFT approaches at this order. We show scenarios where even including dimension-8 operators of the SMEFT is insufficient to obtain an accurate matching to the 2HDM., Comment: The HEFT matching to the 2HDM is contained in the supplemental material. Version 2 is version to be published in Phys. Rev. D

We consider the one-loop corrections to the $Z b \bar b$ vertex in a $CP$-conserving left--right model (LRM), $viz$. a model with gauge group $SU(2)_L \times SU(2)_R \times U(1)$. We allow the gauge coupling constants of $SU(2)_L$ and $SU(2)_R$ to be different. The spontaneous symmetry breaking is accomplished only by doublets and/or singlets of $SU(2)_L$ and $SU(2)_R$. The lightest massive neutral gauge boson of our LRM is assumed to have the same Yukawa couplings to bottom-quark pairs as the $Z$ of the Standard Model (SM); this assumption has the advantage that, then, the infrared divergences automatically cancel down in the subtraction of the $Z b \bar b$ vertex in the SM from the same vertex in the LRM. We effect a proper renormalization of the $Z b \bar b$ vertex and check explicitly both its gauge invariance and the cancellation of all the ultraviolet divergences. We find out that a LRM with the above assumptions cannot achieve a better fit to the $Z b \bar b$ vertex than a multi-Higgs extension of the SM, $viz$. both models can only achieve a decent fit when one admits scalar particles with very low masses $\lesssim 50$ GeV. This is true even when we allow for markedly different gauge coupling constants of $SU(2)_L$ and $SU(2)_R$., Comment: 57 pages, 8 figures; v2: added section 3, figure 2 and three references; v3: matches published version

Deviations from the Standard Model (SM) can be parameterized in terms of the SM effective field theory (SMEFT), which is typically truncated at dimension-6. Including higher dimension operators -- as well as considering simultaneous insertions of multiple dimension-6 operators -- may be necessary in some processes, in order to correctly capture the properties of the underlying UV theory. As a step towards clarifying this in the Higgs boson production in gluon fusion process, we study double insertions of dimension-6 operators in the 1-loop virtual amplitude. We present needed Feynman rules up to $\mathcal{O}(1/\Lambda^4)$ and we numerically study the impact of various approximations to the $\mathcal{O}(1/\Lambda^4)$ expansion., Comment: Auxiliary file replaced

The Standard Model effective field theory (SMEFT) is the tool of choice for studying deviations of Higgs couplings from the Standard Model predictions. The SMEFT is an expansion in an infinite tower of higher dimension operators, which is typically truncated at dimension-6. We consider the effective theory including dimension-8 operators and examine the matching to the 2 Higgs Doublet Model (2HDM) that is assumed to be valid at some high scale. Both the limits from the direct production of single Higgs bosons and the indirect limits on the Higgs tri-linear coupling are considered in the context of the SMEFT matched to the 2HDM, and the importance of the assumptions about the expansions in powers of the high scale are examined numerically., Comment: Eq. 48 corrected

In the complex 2-Higgs-Doublet Model (C2HDM), the mass $m_3$ of the heaviest neutral scalar $h_3$ is usually chosen as a derived parameter. We investigate one-loop corrections to $m_3$ and their impact on decays of $h_3$. Very fine-tuned regions of the parameter space can be found where such corrections are large, not due to subtraction schemes, but rather due to the particular dependence of $m_3$ on the independent parameters. We show that even moderate corrections can have a significant impact on decays of $h_3$, as they may be several times enhanced by leading-order factors., Comment: Matches published version

In this thesis, we investigate several aspects of models with a multiplicity of Higgs bosons. We start by focusing on models with two Higgs doublets. We claim that one of the most used models in the literature -- the so-called real 2-Higgs-Doublet Model (2HDM) -- is inconsistent. We give special emphasis to the model that corrects it in the most simple way, the complex 2HDM; we study its phenomenology in the lowest order of perturbation theory and describe its renormalization in the following order. Motivated by this renormalization, we analyze in detail tadpole schemes that allow the selection of the true vacuum expectation value in models with spontaneous symmetry breaking, and we propose a theorem regarding the finitude of Green's functions with corrections to the external legs. Given the complexity of the calculations involved, it is crucial to dispose of an appropriate software. We therefore present FeynMaster, which calculates, among other things, Feynman rules, radiative corrections and counterterms in a simultaneously automatic and flexible way. We use the potential of FeynMaster to investigate two new projects related to multiple scalar bosons: first, we consider the class of models with a generic number of scalar doublets and singlets, and investigate radiative corrections to the $Z \to b \bar{b}$ process in this context; then, we explore the phenomenology of a model based on the linear seesaw hypothesis, which, in addition to two scalar doublets, includes a complex singlet., Comment: PhD thesis. Small bugs corrected

We present the one-loop electroweak renormalization of the CP-violating 2-Higgs-Doublet Model with softly broken $\mathbb{Z}_2$ symmetry (C2HDM). The existence of CP violation in the scalar sector of the model leads to a quite unique process of renormalization, since it requires the introduction of several non-physical parameters. The C2HDM will thus have more independent counterterms than independent renormalized parameters. As a consequence, different combinations of counterterms can be taken as independent for the same set of independent renormalized parameters. We compare the behaviour of selected combinations in specific NLO processes, which are assured to be gauge independent via a simple prescription. FeynMaster 2 is used to derive the Feynman rules, counterterms and one-loop processes in a simultaneously automatic and flexible way. This illustrates its use as an ideal tool to renormalize models such as the C2HDM and investigate them at NLO., Comment: One equation and one figure were slightly modified. Text and conclusions were not modified

We discuss the $Z_2$ symmetric two Higgs doublet model with a real soft breaking term (real 2HDM). We explain in detail why it is not tenable to assume CP conservation in the scalar sector to keep the dimension two term real, while CP is violated by the dimension four Yukawa couplings. We propose the calculation of the infinite tadpole of the (would-be) pseudoscalar neutral scalar. We construct a simple toy model with the same flaws, where the unrenormalizable infinity is easier to calculate. We then turn our attention to the same tadpole in the real 2HDM. We spearhead this effort focusing on diagrams involving solely bare quantities. This involves hundreds of Feynman three-loop diagrams that could feed the CP violation from the quark into the scalar sector, and is only possible with state of the art automatic computation tools. Remarkably, some intermediate results agree when using three independent derivations, including the peculiar cancellation of the leading pole divergence due to a subtle interplay between masses and the Jarlskog invariant, which we calculate analytically. The calculation is not complete however, since the full two-loop renormalization of the real 2HDM is not yet available in the literature. Still, we argue convincingly that there is an irremovable infinity., Comment: 12 pages, matches published version

We study the one-loop corrections to the $Zb\bar{b}$ vertex in extensions of the Standard Model with arbitrary numbers of scalar doublets, neutral scalar singlets, and charged scalar singlets. Starting with a general parameterization of theories with neutral and singly-charged scalar particles, we derive the conditions that, in a renormalizable model, must be obeyed by the couplings in order for the divergent contributions to cancel. Then, we show that those conditions are indeed obeyed by the models that we are interested in, and we write down the full finite expression for the vertex in those models. We apply our results to some particular cases, highlighting the importance of the diagrams with neutral scalars., Comment: 32 pages, 25 Figures in PDF. v2: version matching published version on NPB. References added

FeynMaster is a multi-tasking software for particle physics studies. By making use of already existing programs (FeynRules, QGRAF, FeynCalc), FeynMaster automatically generates Feynman rules, generates and draws Feynman diagrams, generates amplitudes, performs both loop and algebraic calculations, and fully renormalizes models. In parallel with this automatic character, FeynMaster allows the user to manipulate the generated results in Mathematica notebooks in a flexible and consistent way. It can be downloaded in https://porthos.tecnico.ulisboa.pt/FeynMaster/., Comment: 32 pages, matches published version

We examine the simplest realization of the linear seesaw mechanism within the Standard Model gauge structure. Besides the standard scalar doublet, there are two lepton-number-carrying scalars, a nearly inert SU2 doublet and a singlet. Neutrino masses result from the spontaneous violation of lepton number, implying the existence of a Nambu-Goldstone boson. Such "majoron" would be copiously produced in stars, leading to stringent astrophysical constraints. We study the profile of the Higgs bosons in this model, including their effective couplings to the vector bosons and their invisible decay branching ratios. A consistent electroweak symmetry breaking pattern emerges with a compressed spectrum of scalars in which the "Standard Model" Higgs boson can have a sizeable invisible decay into the invisible majorons., Comment: 28 pages, 15 Figures in PDF. v2: version matching published version on JHEP. References added

The complex two-Higgs doublet model is one of the simplest ways to extend the scalar sector of the Standard Model to include a new source of CP-violation. The model has been used as a benchmark model to search for CP-violation at the LHC and as a possible explanation for the matter-antimatter asymmetry of the Universe. In this work, we re-analyse in full detail the softly broken $\mathbb{Z}_2$ symmetric complex two-Higgs doublet model (C2HDM). We provide the code C2HDM_HDECAY implementing the C2HDM in the well-known HDECAY program which calculates the decay widths including the state-of-the-art higher order QCD corrections and the relevant off-shell decays. Using C2HDM_HDECAY together with the most relevant theoretical and experimental constraints, including electric dipole moments (EDMs), we review the parameter space of the model and discuss its phenomenology. In particular, we find cases where large CP-odd couplings to fermions are still allowed and provide benchmark points for these scenarios. We examine the prospects of discovering CP-violation at the LHC and show how theoretically motivated measures of CP-violation correlate with observables., Comment: 40 pages, 19 figures

We study the CP violating ZZZ vertex in the two-Higgs doublet model, which is a probe of a Jarlskog-type invariant in the extended Higgs sector. The form factor $f_4^Z$ is evaluated at one loop in a general $R_\xi$ gauge and its magnitude is estimated in the realistic parameter space. Then we turn to the decoupling limit of the two-Higgs doublet model, where the extra scalars are heavy and the physics can be described by the Standard Model supplemented by higher-dimensional operators. The leading operator contributing to $f_4^Z$ at one loop is identified. The CP violating ZZZ vertex is not generated in the effective theory by dimension-8 operators, but instead arises only at the dimension-12 level, which implies an additional suppression by powers of the heavy Higgs mass scale., Comment: 21 pages; v2: added references and comments, appendix A on method of regions, and appendix B on derivation of CP-violating effective Lagrangian. Corrected discussion of dimension-12 operators contributing to ZZZ vertex. Final JHEP version

We discuss the validity of the Standard Model Effective Field Theory (SM EFT) as the low-energy effective theory for the two-Higgs-doublet Model (2HDM). Using the up-to-date Higgs signal strength measurements at the LHC, one can obtain a likelihood function for the Wilson coefficients of dimension-6 operators in the EFT Lagrangian. Given the matching between the 2HDM and the EFT, the constraints on the Wilson coefficients can be translated into constraints on the parameters of the 2HDM Lagrangian. We discuss under which conditions such a procedure correctly reproduces the true limits on the 2HDM. Finally, we employ the SM EFT to identify the pattern of the Higgs boson couplings that are needed to improve the fit to the current Higgs data. To this end, one needs, simultaneously, to increase the top Yukawa coupling, decrease the bottom Yukawa coupling, and induce a new contact interaction of the Higgs boson with gluons. We comment on how these modifications can be realized in the 2HDM extended by new colored particles., Comment: 24 pages

With the discovery of the Higgs boson at the Large Hadron Collider the high energy physics community's attention has now turned to understanding the properties of the Higgs boson, together with the hope of finding more scalars during run 2. In this work we discuss scenarios where using a combination of three decays, involving the 125 GeV Higgs boson, the Z boson and at least one more scalar, an indisputable signal of CP-violation arises. We use a complex two-Higgs doublet model as a reference model and present some benchmark points that have passed all current experimental and theoretical constraints, and that have cross sections large enough to be probed during run 2., Comment: 16 pages

We discuss the CP nature of the Yukawa couplings of the Higgs boson in the framework of a complex two Higgs doublet model (C2HDM). After analysing all data gathered during the Large Hadron Collider run 1, the measurement of the Higgs couplings to the remaining SM particles already restricts the parameter space of many extensions of the SM. However, there is still room for very large CP-odd Yukawa couplings to light quarks and leptons while the top-quark Yukawa coupling is already very constrained by current data. Although indirect measurements of electric dipole moments play a very important role in constraining the pseudo-scalar components of the Yukawa couplings, we argue that a direct measurement of the ratio of pseudoscalar to scalar couplings should be one of the top priorities for the LHC run 2., Comment: 9 pages, 5 figures. Toyama International Workshop on Higgs as a Probe of New Physics 2015, 11-15, February, 2015

We start by presenting the current status of a complex flavour conserving two-Higgs doublet model. We will focus on some very interesting scenarios where unexpectedly the light Higgs couplings to leptons and to b-quarks can have a large pseudoscalar component with a vanishing scalar component. Predictions for the allowed parameter space at end of the next run with a total collected luminosity of $300 \, fb^{-1}$ and $3000 \, fb^{-1}$ are also discussed. These scenarios are not excluded by present data and most probably will survive the next LHC run. However, a measurement of the mixing angle $\phi_\tau$, between the scalar and pseudoscalar component of the 125 GeV Higgs, in the decay $h \to \tau^+ \tau^-$ will be able to probe many of these scenarios, even with low luminosity. Similarly, a measurement of $\phi_t$ in the vertex $\bar t t h$ could help to constrain the low $\tan \beta$ region in the Type I model., Comment: 21 pages, 10 figures

The latest LHC data confirmed the existence of a Higgs-like particle and made interesting measurements on its decays into $\gamma \gamma$, $Z Z^\ast$, $W W^\ast$, $\tau^+ \tau^-$, and $b \bar{b}$. It is expected that a decay into $Z \gamma$ might be measured at the next LHC round, for which there already exists an upper bound. The Higgs-like particle could be a mixture of scalar with a relatively large component of pseudoscalar. We compute the decay of such a mixed state into $Z \gamma$, and we study its properties in the context of the complex two Higgs doublet model, analysing the effect of the current measurements on the four versions of this model. We show that a measurement of the $h \rightarrow Z \gamma$ rate at a level consistent with the SM can be used to place interesting constraints on the pseudoscalar component. We also comment on the issue of a wrong sign Yukawa coupling for the bottom in Type II models., Comment: 31 pages, 15 figures

It has been pointed out recently that current experiments still allow for a two Higgs doublet model where the $h b \bar{b}$ coupling ($k_D m_b/v$) is negative; a sign opposite to that of the Standard Model. Due to the importance of delayed decoupling in the $h H^+ H^-$ coupling, $h \rightarrow \gamma \gamma$ improved measurements will have a strong impact on this issue. For the same reason, measurements or even bounds on $h \rightarrow Z \gamma$ are potentially interesting. In this article, we revisit this problem, highlighting the crucial importance of $h \rightarrow VV$, which can be understood with simple arguments. We show that the impacts on $k_D<0$ models of both $h \rightarrow b \bar{b}$ and $h \rightarrow \tau^+ \tau^-$ are very sensitive to input values for the gluon fusion production mechanism; in contrast, $h \rightarrow \gamma \gamma$ and $h \rightarrow Z \gamma$ are not. We also inquire if the search for $h \rightarrow Z \gamma$ and its interplay with $h \rightarrow \gamma \gamma$ will impact the sign of the $h b \bar{b}$ coupling. Finally, we study these issues in the context of the Flipped two Higgs doublet model., Comment: 13 pages, pdf figures