We demonstrate the cancellation of chiral anomalies in the Standard Model (SM) effective field theory (EFT), achieved through a consistent choice of loop momentum routing in triangle diagrams with dimension-6 operator insertions. By enforcing gauge invariance and Bose symmetry, we show that Goldstone-boson contributions cancel anomalies arising from massive gauge-boson vertices, thereby preserving the consistency of the SM EFT. We compute neutral triple gauge-boson vertices at one loop, revealing dominant contributions from dimension-6 operators at all energies below the EFT cutoff. A UV-complete anomaly-free model with a heavy vectorlike electron validates our approach, illustrating how heavy fermion decoupling generates SM EFT operators while maintaining anomaly cancellation. Our results highlight the phenomenological relevance of these vertices for probing new physics at colliders, particularly through dimension-6 effects that scale as the inverse of the center of mass energy squared, 1/s, offering a viable pathway for experimental detection.

While gluon fusion dominates Higgs pair production at the LHC, vector boson fusion (VBF) offers a unique window into Beyond the Standard Model (BSM) physics through its distinctive kinematic features and direct sensitivity to Higgs-vector boson interactions. We perform a comprehensive analysis of double Higgs production via VBF in the Standard Model Effective Field Theory (SMEFT), systematically investigating how dimension-6 and dimension-8 bosonic operators - particularly those involving field derivatives - can enhance the production rate. We identify the most relevant Wilson coefficients (WCs) affecting the trilinear Higgs coupling ($hhh$) and Higgs-vector boson interactions ($hVV$, $hhVV$). Using constraints from global fits and interpolating fit results for unconstrained WCs with Naive Dimensional Analysis, we assess their effects on the $VV \to hh$ ($V = W,Z$) scattering amplitudes and cross-sections. Our analysis includes a study of EFT convergence and validity in models with scalar extensions of the SM. Numerical simulations for the planned High-Luminosity LHC experiment (HL-LHC) in general reveal only a modest and challenging to detect enhancement of the VBF di-Higgs production rate over the SM prediction. However, we show that in optimistic scenarios, such a process could be observed at the HL-LHC. In certain cases, when the enhancement is dominated by the dimension-6 or dimension-8 operators containing field derivatives (and thus leading to stronger energy-dependent effects), this channel becomes competitive with di-Higgs production via gluon fusion. This work highlights the role of VBF di-Higgs production as a complementary channel for probing anomalous Higgs couplings and their impact on BSM physics.

In recent years, theoretical and phenomenological studies with effective field theories have become a trending and prolific line of research in the field of high-energy physics. In order to discuss present and future prospects concerning automated tools in this field, the SMEFT-Tools 2022 workshop was held at the University of Zurich from 14th-16th September 2022. The current document collects and summarizes the content of this workshop.

In recent years, theoretical and phenomenological studies with effective field theories have become a trending and prolific line of research in the field of high-energy physics. In order to discuss present and future prospects concerning automated tools in this field, the SMEFT-Tools 2022 workshop was held at the University of Zurich from 14th-16th September 2022. The current document collects and summarizes the content of this workshop.

In this study we present a universal effective action for one-loop matching of all scalar leptoquarks. We use both the Universal One-Loop Effective Action (UOLEA) and covariant diagrams to evaluate the Wilson coefficients directly in the Green basis for up-to dimension-6 operators. On the technical side, we use the newly developed method of evaluating supertraces, to further validate the results stemming from the use of covariant diagrams. As an application, we perform a fully functional matching onto Standard Model Effective Field Theory (SMEFT) of a model with two scalar leptoquark fields: a weak isospin singlet and a doublet. We demonstrate its use by calculating several observables, such as lepton magnetic and electric dipole moments, neutrino masses, proton decay rate, while we comment upon fine tuning issues in this model. Apart from its phenomenological interest, this model generates the majority of dimension-6 operators and provides an EFT benchmark towards future matching automation.

The assumption that the Standard Model is an effective field theory (SM EFT) of a more fundamental theory at a higher (than electroweak) energy scale implies a growth of cross sections for electroweak vector-boson scattering (VBS) processes signaling the appearance of a resonance (or resonances) near that scale. In this article, we investigate in detail SM EFT effects from dimension-six operators in VBS with like-sign-$W$ production in fully leptonic decay modes at the high-luminosity LHC (HL-LHC). We find that these effects are important for a handful of operators, most notably for the operator composed of three $SU(2)$ field-strength tensors responsible for strong transversely polarized vector-boson interactions. Current global fits on Wilson coefficients allow for an observable signal at the HL-LHC, if not accessible with the current LHC data set.

We present version 3 of SmeftFR, a Mathematica package designed to generate the Feynman rules for the Standard Model Effective Field Theory (SMEFT) including the complete set of gauge invariant operators up to dimension-6 and the complete set of bosonic operators of dimension-8. Feynman rules are generated with the use of FeynRules package, directly in the physical (mass eigenstates) basis for all fields. The complete set of interaction vertices can be derived, including all or any chosen subset of SMEFT operators. As an option, the user can also choose preferred gauge fixing, generating Feynman rules in unitary or $R_\xi$-gauges. The novel feature in version-3 of SmeftFR is its ability to calculate SMEFT interactions consistently up to dimension-8 in EFT expansion (including quadratic dimension-6 terms) and express the vertices directly in terms of user-defined set of input-parameters. The derived Lagrangian in the mass basis can be exported in various formats supported by FeynRules, such as UFO, FeynArts etc. Initialisation of numerical values of Wilson coefficients of higher dimension operators is interfaced to WCxf format. The package also includes a dedicated Latex generator allowing to print the result in clear human-readable form. The SmeftFR v3 is publicly available at www.fuw.edu.pl/smeft.

The assumption that the Standard Model is an Effective Field Theory (SM EFT) of a more fundamental theory at a higher, than electroweak, energy scale, implies a growth of cross-sections for electroweak Vector Boson Scattering (VBS) processes signalling the appearance of a resonance (or resonances) nearby that scale. In this article, we investigate in detail SM EFT effects from dimension-6 operators in VBS with like-sign-W production in fully leptonic decay modes at the High Luminosity LHC (HL-LHC). We find that these effects are important for a handful of operators, most notably for the operator composed of three SU(2) field strength tensors responsible for strong transversely polarized vector boson interactions. Current global fits on Wilson-coefficients allow for a signal immediately permissive at the HL-LHC if not accessible at the current LHC-dataset.

We propose a procedure to cross-validate Monte Carlo implementations of the standard model effective field theory. It is based on the numerical comparison of squared amplitudes computed at specific phase-space and parameter points in pairs of implementations. Interactions are fully linearised in the effective field theory expansion. The squares of linear effective field theory amplitudes and their interference with standard-model contributions are compared separately. Such pairwise comparisons are primarily performed at tree level and a possible extension to the one-loop level is also briefly considered. We list the current standard model effective field theory implementations and the comparisons performed to date.

The increasing interest in the phenomenology of the Standard Model Effective Field Theory (SMEFT), has led to the development of a wide spectrum of public codes which implement automatically different aspects of the SMEFT for phenomenological applications. In order to discuss the present and future of such efforts, the "SMEFT-Tools 2019" Workshop was held at the IPPP Durham on the 12th-14th June 2019. Here we collect and summarize the contents of this workshop.

We calculate the S-matrix element for the Higgs boson decay to a Z-boson and a photon, h → Zγ, at one-loop in the Standard-Model Effective Field Theory (SMEFT) framework and in linear Rξ -gauges. Our SMEFT expansion includes all relevant operators up to dimension-6 considered in Warsaw basis without resorting to any flavour or CP- conservation assumptions. Within this approximation there are 23 dimension-6 operators affecting the amplitude, not including flavour and hermitian conjugation. The result for the on-shell h → Zγ amplitude is gauge invariant, renormalisation-scale invariant and gauge-fixing parameter independent. The calculated ratio of the SMEFT versus the SM expectation for the h → Zγ decay width is then written in a semi-numerical form which is useful for further comparisons with related processes. For example, the h → Zγ amplitude contains 16 operators in common with the h → γγ amplitude and one can draw useful results about its feasibility at current and future LHC data.

We assume that New Physics effects are parametrized within the Standard Model Effective Field Theory (SMEFT) written in a complete basis of gauge invariant operators up to dimension 6, commonly referred to as “Warsaw basis”. We discuss all steps necessary to obtain a consistent transition to the spontaneously broken theory and several other important aspects, including the BRST-invariance of the SMEFT action for linear R ξ -gauges. The final theory is expressed in a basis characterized by SM-like propagators for all physical and unphysical fields. The effect of the non-renormalizable operators appears explicitly in triple or higher multiplicity vertices. In this mass basis we derive the complete set of Feynman rules, without resorting to any simplifying assumptions such as baryon-, lepton-number or CP conservation. As it turns out, for most SMEFT vertices the expressions are reasonably short, with a noticeable exception of those involving 4, 5 and 6 gluons. We have also supplemented our set of Feynman rules, given in an appendix here, with a publicly available Mathematica code working with the FeynRules package and producing output which can be integrated with other symbolic algebra or numerical codes for automatic SMEFT amplitude calculations.

We present a Peccei-Quinn (PQ)-symmetric two-Higgs doublet model that naturally predicts a fermionic singlet dark matter in the mass range 10 keV-1 GeV. The origin of the smallness of the mass of this light singlet fermion arises predominantly at the one-loop level, upon soft or spontaneous breakdown of the PQ symmetry via a complex scalar field in a fashion similar to the so-called Dine-Fischler-Sredniki-Zhitnitsky axion model. The mass generation of this fermionic Radiative Light Dark Matter (RLDM) requires the existence of two heavy vector-like SU(2) isodoublets, which are not charged under the PQ symmetry. We show how the RLDM can be produced via the freeze-in mechanism, thus accounting for the missing matter in the Universe. Finally, we briefly discuss possible theoretical and phenomenological implications of the RLDM model for the strong CP problem and the CERN Large Hadron Collider (LHC).

We perform a detailed study of an effective field theory which includes the Standard Model particle content extended by a pair of Weyl fermionic SU(2)-doublets with opposite hypercharges. A discrete symmetry guarantees that a linear combination of the doublet components is stable and can act as a candidate particle for Dark Matter. The dark sector fermions interact with the Higgs and gauge bosons through renormalizable $d=4$ operators, and non-renormalizable $d=5$ operators that appear after integrating out extra degrees of freedom above the TeV scale. We study collider, cosmological and astrophysical probes for this effective theory of Dark Matter. We find that a WIMP with a mass nearby to the electroweak scale, and thus observable at LHC, is consistent with collider and astrophysical data only when fairly large magnetic dipole moment transition operators with the gauge bosons exist, together with moderate Yukawa interactions.

We present and prove a theorem of matrix analysis, the Flavour Expansion Theorem (or FET), according to which, an analytic function of a Hermitian matrix can be expanded polynomially in terms of its off-diagonal elements with coefficients being the divided differences of the analytic function and arguments the diagonal elements of the Hermitian matrix. The theorem is applicable in case of flavour changing amplitudes. At one-loop level this procedure is particularly natural due to the observation that every loop function in the Passarino-Veltman basis can be recursively expressed in terms of divided differences. FET helps to algebraically translate an amplitude written in mass eigenbasis into flavour mass insertions, without performing diagrammatic calculations in flavour basis. As a non-trivial application of FET up to a third order, we demonstrate its use in calculating strong bounds on the real parts of flavour changing mass insertions in the up- squark sector of the MSSM from neutron Electric Dipole Moment (nEDM) measurements, assuming that CP-violation arises only from the CKM matrix.

We extend the Standard Model (SM) by adding a pair of fermionic $SU(2)$ doublets with opposite hypercharge and a fermionic $SU(2)$ triplet with zero hypercharge. We impose a discrete ${Z}_{2}$ symmetry that distinguishes the SM fermions from the new ones. Then, gauge invariance allows for two renormalizable Yukawa couplings between the new fermions and the SM Higgs field, as well as for direct masses for the doublet (${M}_{D}$) and the triplet (${M}_{T}$). After electroweak symmetry breaking, this model contains, in addition to SM particles, two charged Dirac fermions and a set of three neutral Majorana fermions, the lightest of which contributes to dark matter (DM). We consider a case where the lightest neutral fermion is an equal admixture of the two doublets with mass ${M}_{D}$ close to the $Z$-boson mass. This state remains stable under radiative corrections thanks to a custodial $SU(2)$ symmetry and is consistent with the experimental data from oblique electroweak corrections. Moreover, the amplitudes relevant to spin-dependent or spin-independent nucleus-DM particle scattering cross sections both vanish at tree level. They arise at one loop at a level that may be observed in near future DM direct detection experiments. For Yukawa couplings comparable to the top quark, the DM particle relic abundance is consistent with observation, not relying on coannihilation or resonant effects, and has a mass at the electroweak scale. Furthermore, the heavier fermions decay to the DM particle and to electroweak gauge bosons making this model easily testable at the LHC. In the regime of interest, the charged fermions suppress the Higgs decays to diphotons by 45%--75% relative to SM prediction.

In full one-loop generality and in next-to-leading order in QCD, we study rare top to Higgs boson flavour changing decay processes t → qh with q = u, c quarks, in the general MSSM with R-parity conservation. Our primary goal is to search for enhanced effects on \( \mathrm{\mathcal{B}}\left(t\to qh\right) \) that could be visible at current and high luminosity LHC running. To this end, we perform an analytical expansion of the amplitude in terms of flavour changing squark mass insertions that treats both cases of hierarchical and degenerate squark masses in a unified way. We identify two enhanced effects allowed by various constraints: one from holomorphic trilinear soft SUSY breaking terms and/or right handed up squark mass insertions and another from non-holomorphic trilinear soft SUSY breaking terms and light Higgs boson masses. Interestingly, even with \( \mathcal{O}(1) \) flavour violating effects in the, presently unconstrained, up-squark sector, SUSY effects on \( \mathrm{\mathcal{B}}\left(t\to qh\right) \) come out to be unobservable at LHC mainly due to leading order cancellations between penguin and self energy diagrams and the constraints from charge- and colour-breaking minima (CCB) of the MSSM vacuum. An exception to this conclusion may be effects arising from non-holomorphic soft SUSY breaking terms in the region where the CP-odd Higgs mass is smaller than the top-quark mass but this scenario is disfavoured by recent LHC searches. Our calculations for t → qh decay are made available in SUSY FLAVOUR numerical library.

We present SUSY_FLAVOR version 2 - a Fortran 77 program that calculates low-energy flavor observables in the general R-parity conserving MSSM. For a set of MSSM parameters as input, the code gives predictions for:1. Electric dipole moments of the leptons and the neutron.2. Anomalous magnetic moments (i.e. g - 2) of the leptons.3. Radiative lepton decays (mu -> e gamma and tau -> mu gamma, e gamma).4. Rare Kaon decays (K-L(0) -> pi(0)(nu) over bar nu and K+ -> pi(+)(nu) over barv).5. Leptonic B decays (B-s,B-d -> l(+)l(-), B -> D tau nu)6. Radiative B decays (B -> (X) over bar (s)gamma).7. Delta F = 2 processes ((K) over bar (0)-K-0, (D) over bar -D, (B) over bar (d)-B-d and (B) over bar (s)-B-s mixing).Comparing to SUSY_FLAVOR v1, where the matching conditions were calculated strictly at one-loop level, SUSY_FLAVOR v2 performs the resummation of all chirally enhanced corrections, i.e. takes into account the enhanced effects from tan beta and/or large trilinear soft mixing terms to all orders in perturbation theory. Also, in SUSY_FLAVOR v2 new routines calculation of B -> (D)tau nu, g - 2, radiative lepton decays and Br(l -> l'gamma) were added. All calculations are done using exact diagonalization of the sfermion mass matrices. The program can be obtained from http://www.fuw.edu.pl/susy_flavor.Program summaryProgram title: SUSY_FLAVOR v2Catalogue identifier: AEGV_v2_0Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEGV_v2_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 15683No. of bytes in distributed program, including test data, etc.: 89130Distribution format: tar.gzProgramming language: Fortran 77.Computer: Any.Operating system: Any, tested on Linux.Classification: 11.6.Does the new version supersede the previous version?: YesCatalogue identifier of previous version: AEGV_v1_0Journal reference of previous version: Comput. Phys. Comm. 181 (2010) 2180Nature of problem:Predicting CP-violating observables, meson mixing parameters and branching ratios for set of rare processes in the general R-parity conserving MSSM.Solution method:We use standard quantum theoretical methods to calculate Wilson coefficients in MSSM and at one loop including QCD corrections at higher orders when this is necessary and possible. The input parameters can be read from an external file in SLHA format.Reasons for new version:A major rewrite of the internal code structure to accommodate higher order corrections; new observables added.Summary of revisions:1. SUSY_FLAVOR v2.0 is able to perform resummation of chirally enhanced corrections to all orders of perturbation expansion (v1.0 included 1-loop terms only).2. Routines calculating new observables are added: g-2 lepton magnetic moment anomaly, mu to e gamma and tau to e gamma, mu gamma decays, B to D tau nu decays, B to mu e, tau e, tau mu decays.3. Parameter initialization in the sfermion sector is simplified and follows, by default, the SLHA2 conventions.Restrictions: The results apply only to the case of MSSM with R-parity conservation.Running time: For a single parameter set approximately 1 s in double precision on a PowerBook Mac G4. (c) 2012 Elsevier B.V. All rights reserved.

We review and clarify computational issues about the W -gauge boson one-loop contribution to the H → γ γ decay amplitude, in the unitary gauge and in the Standard Model. We find that highly divergent integrals depend upon the choice of shifting momenta with arbitrary vectors. One particular combination of these arbitrary vectors reduces the superficial divergency down to a logarithmic one. The remaining ambiguity is then fixed by exploiting gauge invariance and the Goldstone Boson Equivalence Theorem. Our method is strictly realised in four dimensions. The result for the amplitude agrees with the “famous” one obtained using dimensional regularisation (DR) in the limit d → 4 , where d is the number of spatial dimensions in Euclidean space. At the exact equality d = 4 , a three-sphere surface term appears that renders the Ward Identities and the equivalence theorem inconsistent. We also examined a recently proposed four-dimensional regularisation scheme and found agreement with the DR outcome.