Finding new physics (NP) is the most important problem in particle physics today. Studying “anomalies”, i.e., measurements of low-energy observables whose values disagree with the predictions of the Standard Model (SM), is a powerful search strategy. The SM Effective Field Theory (SMEFT) provides a general model-independent framework for parameterizing NP; it is natural to try to find the SMEFT operator(s) that can explain such anomalies. This is a challenging task because (i) the number of SMEFT operators is enormous, and (ii) at loop level there are very complicated correlations among the operators. Analyses by humans typically rely on phenomenological intuition to decide which operators are relevant. This is often biased and does not explore the complete SMEFT operator space. Interestingly, reinforcement learning (RL) techniques excel at tasks that require decision making to achieve their goals. In this paper, we introduce an RL method that can be used to find the SMEFT operators that explain any anomalies. We test it on the CDF W-mass anomaly, and show that it reproduces (and improves upon) known results. We then consider a far more complicated situation with multiple anomalies and show that, even here, this method is able to find the SMEFT operators that explain the data. Our RL method can therefore be used to efficiently search for NP at the level of SMEFT.

The Standard Model Effective Field Theory (SMEFT) based on the unbroken gauge group SU(3)_C⊗SU(2)_L⊗U(1)_Y and containing only particles of the Standard Model (SM) has developed in the last decade to a mature field. It is the framework to be used in the energy gap from scales sufficiently higher than the electroweak scale up to the lowest energy scale at which new particles show up. We summarize the present status of this theory with a particular emphasize on its role in the indirect search for new physics (NP). While flavour physics of both quarks and leptons is the main topic of our review, we also discuss electric dipole moments, anomalous magnetic moments (g-2)_μ,e, Z-pole observables, Higgs observables and high-p_T scattering processes within the SMEFT. We group the observables into ten classes and list for each class the most relevant operators and the corresponding renormalization group equations (RGEs). We exhibit the correlations between different classes implied both by the operator mixing and the SU(2)_L gauge symmetry. Our main goal is to provide an insight into the complicated operator structure of this framework which hopefully will facilitate the identification of valid ultraviolet completions behind possible anomalies observed in future data. Numerous colourful charts, and 85 tables, while representing rather complicated RG evolution from the NP scale down to the electroweak scale, beautify the involved SMEFT landscape. Over 950 references to the literature underline the importance and the popularity of this field. We discuss both top-down and bottom-up approaches as well as their interplay. This allows us eventually to present an atlas of different landscapes beyond the SM that includes heavy gauge bosons and scalars, vector-like quarks and leptons and leptoquarks.

Sterile neutrinos are well-motivated beyond the Standard Model (BSM) particles. The Standard Model Effective Field Theory (SMEFT) augmented with these new fields is known as the ν SMEFT. We present the first code for solving the renormalization group equations (RGEs) of the ν SMEFT in an automated way. For this purpose, we have implemented the ν SMEFT as a new effective field theory (EFT) in the Wilson coefficient exchange format WCxf. Furthermore, we included anomalous dimensions depending on the gauge couplings and Yukawas in the python package wilson. This novel version of wilson allows a consistent inclusion of ν SMEFT renormalization group (RG) running effects above the electroweak (EW) scale in phenomenological studies involving sterile neutrinos. Moreover, this new release allows us to study EW, strong, and Yukawa running effects separately within the SMEFT.

Whenever an anomaly in the flavour sector appears, analyses are performed examining whether it can be explained by adding a small number of carefully-chosen flavour non-universal four-fermion SMEFT operators. These analyses are typically carried out in the down or the up basis, i.e., it is assumed that the weak and mass eigenstates are the same for the left-handed down-type or up-type quarks. In these bases, there is no dependence on the matrices that transform from the weak to the mass basis, and which are unmeasurable in the Standard Model. In this paper, we argue that it is better to use a generic weak basis, in which no assumptions about the alignment of weak and mass eigenstates are made. The analysis now directly includes elements of the transformation matrices. By doing a fit to the data, it is possible to both determine if the flavour anomaly can be explained and extract the transformation matrices. In principle, this can be extended to a complete reconstruction of the Yukawa matrices.

Abstract Sterile neutrinos with masses at the keV scale and mixing to the active neutrinos offer an elegant explanation of the observed dark matter (DM) density. However, the very same mixing inevitably leads to radiative photon emission and the non-observation of such peaked X-ray lines rules out this minimal sterile neutrino DM hypothesis. We show that in the context of the Standard Model effective field theory with sterile neutrinos (νSMEFT), higher dimensional operators can produce sterile neutrino DM in a broad range of parameter space. In particular, νSMEFT interactions can open the large mixing parameter space due to their destructive interference, through operator mixing or matching, in the X-ray emission. We also find that, even in the zero mixing limit, the DM density can always be explained by νSMEFT operators. The testability of the studied νSMEFT operators in searches for electric dipole moments, neutrinoless double beta decay, and pion decay measurements is discussed.

Two categories of four-fermion SMEFT operators are semileptonic (two quarks and two leptons) and hadronic (four quarks). At tree level, an operator of a given category contributes only to processes of the same category. However, when the SMEFT Hamiltonian is evolved down from the new-physics scale to low energies using the renormalization-group equations (RGEs), due to operator mixing this same SMEFT operator can generate operators of the other category at one loop. Thus, to search for a SMEFT explanation of a low-energy anomaly, or combination of anomalies, one must: (i) identify the candidate semileptonic and hadronic SMEFT operators, (ii) run them down to low energy with the RGEs, (iii) generate the required low-energy operators with the correct Wilson coefficients, and (iv) check that all other constraints are satisfied. In this paper, we illustrate this method by finding all SMEFT operators that, by themselves, provide a combined explanation of the (semileptonic) b→sℓ^+ℓ^- anomalies and the (hadronic) B → πK puzzle.

Abstract A fully generic treatment of electric dipole moments (EDMs) is presented in the CP-violating and flavor-conserving weak effective field theory (WET) with five flavors of quarks and three flavors of leptons. We systematically analyze leading contributions to EDMs originating from QCD and QED renormalization group running between the electroweak scale and low energy scales of about 2 GeV. We include the full one-loop anomalous dimension and a subset of two-loop corrections, as well as threshold corrections at the bottom, charm and τ masses. This allows us to derive master formulae in the space of generic WET for the neutron and proton EDMs, for EDMs of diamagnetic atoms, and for the precession frequencies constrained in molecular EDM experiments, from which bounds on the electron EDM are extracted. In particular, our master formulae capture the contributions of WET CP-violating operators with heavy quark and lepton flavors. As an application, we study EDM constraints on the Yukawa couplings of the Higgs boson, in both the linear and non-linear realizations of electroweak symmetry breaking.

The measurements of the Cabibbo-Kobayashi-Maskawa (CKM) elements can be contaminated by newphysics effects. We point out that purely leptonic operators at the high scale can influence semileptonic K decays and nuclear beta decay through renormalization group (RG) running, and hence can influence the measurements of Vus. Interestingly, through this mechanism, a single six-dimensional effective operator Oll at the high scale can alleviate the tension due to the Cabibbo angle anomaly, by generating the desired operators at the low scale through RG running. When generated as a result of a Z' model, the nonuniversal leptonic couplings of this operator can also contribute to the lepton flavor universality violating ratios such as RK(*), which would act as stringent constraints on such scenarios. By performing a global fit of the Z' model, we find that it is essential to have nonuniversal couplings of such a Z' boson to all three generations of leptons.

Abstract The present-day B-anomalies involving b → sμ+μ− or b → cτ−$$ \overline{\nu} $$ ν ¯ transitions can all be explained with the addition of a vector U1 leptoquark with a mass of $$ {M}_{U_1} $$ M U 1 ≳ 1.8 TeV. In the scalar singlet dark matter model (SSDMM), the DM is a scalar S that couples to the Higgs via λhSS2|H|2. We update the fit to the data and find that the SSDMM is now viable only for MS ≳ 1.6 TeV. In this paper, we assume that the DM also couples to the U1 via $$ {\lambda}_{U_1S}{S}^2{U}_{1\mu}^{\dagger }{U}_1^{\mu } $$ λ U 1 S S 2 U 1 μ † U 1 μ . In addition to leading to DM annihilation via SS →$$ {U}_1{\overline{U}}_1 $$ U 1 U ¯ 1 , this coupling generates SSgg and SSγγ couplings at one loop. Although naively divergent, these loop diagrams can be calculated under the assumption that the U1 is a gauge boson of a group broken at the TeV scale. With this DM-U1 coupling term, there are additional contributions to the various DM observables (relic density, direct and indirect detection). We find that the constraints on the SSDMM are relaxed for both heavy DM (MS ≳ $$ {M}_{U_1} $$ M U 1 ) and light DM (MS<$$ {M}_{U_1} $$ M U 1 ).

Abstract In particle physics, the modern view is to categorize things in terms of effective field theories (EFTs). Above the weak scale, we have the SMEFT, formed when the heavy new physics (NP) is integrated out, and for which the Standard Model (SM) is the leading part. Below M W , we have the LEFT (low-energy EFT), formed when the heavy SM particles (W ± , Z 0, H, t) are also integrated out. In order to determine how low-energy measurements depend on the underlying NP, it is necessary to compute the matching conditions of LEFT operators to SMEFT operators. These matching conditions have been worked out for all LEFT operators up to dimension 6 in terms of SMEFT operators up to dimension 6 at the one-loop level. However, this is not sufficient for all low-energy observables. In this paper we present the momentum-independent matching conditions of all such LEFT operators to SMEFT operators up to dimension 8 at tree level.

The basis transformations of the effective operators often involve Fierz and other relations which are only valid in $D=4$ space-time dimensions. In general, in $D$ space-time dimensions, however, the evanescent operators have to be introduced to preserve such identities. Such operators contribute to one-loop basis transformations as well as to two-loop renormalization group running. In this talk, I discussed a simple procedure for systematically changing of basis at 1-loop level including shifts due to evanescent operators. As an example, we apply this method to derive the 1-loop basis transformation from the BMU basis useful for NLO QCD calculations, to the JMS basis used in the matching to the SMEFT.

Despite the observation of significant suppressions of b→ sμ ^+μ ^- branching ratios no clear sign of New Physics (NP) has been identified in Δ F=2 observables Δ M_d,s , ε _K and the mixing induced CP asymmetries S_ψ K_S and S_ψϕ . Assuming negligible NP contributions to these observables allows to determine CKM parameters without being involved in the tensions between inclusive and exclusive determinations of |V_cb| and |V_ub| . Furthermore this method avoids the impact of NP on the determination of these parameters present likely in global fits. Simultaneously it provides SM predictions for numerous rare K and B branching ratios that are most accurate to date. Analyzing this scenario within Z^' models we point out, following the 2009 observations of Monika Blanke and ours of 2020, that despite the absence of NP contributions to ε _K , significant NP contributions to K^+→π ^+νν̅ , K_L→π ^0νν̅ , K_S→μ ^+μ ^- , K_L→π ^0ℓ ^+ℓ ^- , ε '/ε and Δ M_K can be present. In the simplest scenario, this is guaranteed, as far as flavour changes are concerned, by a single non-vanishing imaginary left-handed Z^' coupling g^L_sd . This scenario implies very stringent correlations between the Kaon observables considered by us. In particular, the identification of NP in any of these observables implies automatically NP contributions to the remaining ones under the assumption of non-vanishing flavour conserving Z^' couplings to qq̅ , νν̅ , and μ ^+μ ^- . A characteristic feature of this scenario is a strict correlation between K^+→π ^+νν̅ and K_L→π ^0νν̅ branching ratios on a branch parallel to the Grossman-Nir bound. Moreover, Δ M_K is automatically suppressed as seems to be required by the results of the RBC-UKQCD lattice QCD collaboration. Furthermore, there is no NP contribution to K_L→μ ^+μ ^- which otherwise would bound NP effects in K^+→π ^+νν̅ . Of particular interest are the correlations of K^+→π ^+νν̅ and K_L→π ^0νν̅ branching ratios and of Δ M_K with the ratio ε '/ε . We investigate the impact of renormalization group effects in the context of the SMEFT on this simple scenario.

The measurements of the Cabibbo--Kobayashi--Maskawa (CKM) elements can be contaminated by new-physics effects. We point out that purely leptonic operators at the high scale can influence semileptonic $K$ decays and nuclear beta decay through renormalization group (RG) running, and hence can influence the measurements of $V_{us}$. Interestingly, through this mechanism, a single six-dimensional effective operator $O_{\ell\ell}$ at the high scale can alleviate the tension due to the Cabibbo angle anomaly, by generating the desired operators at the low scale through RG running. When generated as a result of a $Z'$ model, the non-universal leptonic couplings of this operator can also contribute to the lepton flavor universality violating ratios such as $R_{K^{(*)}}$, which would act as stringent constraints on such scenarios. By performing a global fit of the $Z'$ model, we find that it is essential to have non-universal couplings of such a $Z'$ boson to all three generations of leptons.

Electroweak interactions assign a central role to the gauge group SU(2)L ?? U(1)Y, which is either realized linearly (SMEFT) or nonlinearly (e.g., HEFT) in the effective theory obtained when new physics above the electroweak scale is integrated out. Although the discovery of the Higgs boson has made SMEFT the default assumption, nonlinear realization remains possible. The two can be distinguished through their predictions for the size of certain low-energy dimension-6 four-fermion operators: for these, HEFT predicts O(1) couplings, while in SMEFT they are suppressed by a factor v2=??2NP, where v is the Higgs vev. One permit its non-SMEFT coefficient to have a HEFTy size. We also note that the angular distribution in B?? ??? D*(??? D??0)?????(??? ????? ????)?????? contains enough information to extract the coefficients of all new-physics operators. Future measurements of this angular distribution can therefore tell us if non-SMEFT new physics is really necessary.

A bstract We study implications of the four-fermion semileptonic operators at the low-energy and at electroweak (EW) scale in the framework of Standard Model Effective Field Theory (SMEFT). We show how the renormalization group (RG) running effects can play an important role in probing the generic flavour structure of such operators. It is shown that at the 1-loop level, through RG running, depending upon the flavour structure, these operators can give rise to sizable effects at low energy in the electroweak precision (EWP) observables, the leptonic, quark, as well as the Z boson flavour violating decays. To this end, we isolate the phenomenologically relevant terms in the full anomalous dimension matrices (ADMs) and discuss the impact of the QED+QCD running in the Weak effective field theory (WET) and the SMEFT running due to gauge and Yukawa interactions on the dim-4 and dim-6 operators at the low energy. Considering all the relevant processes, we derive lower bounds on new physics (NP) scale Λ for each semileptonic operator, keeping a generic flavour structure. In addition, we also report the allowed ranges for the Wilson coefficients at a fixed value of Λ = 3 TeV.

We stress the importance of precise measurements of rare decays K^+→π^+νν̅, K_L→π^0νν̅, K_L,S→μ^+μ^- and K_L,S→π^0ℓ^+ℓ^- for the search of new physics (NP). This includes both branching ratios and the distributions in q^2, the invariant mass-squared of the neutrino system in the case of K^+→π^+νν̅ and K_L→π^0νν̅ and of the ℓ^+ℓ^- system in the case of the remaining decays. In particular the correlations between these observables and their correlations with the ratio ε'/ε in K_L→ππ decays, the CP-violating parameter ε_K and the K^0-K̅^0 mass difference Δ M_K, should help to disentangle the nature of possible NP. We stress the strong sensitivity of all observables with the exception of Δ M_K to the CKM parameter |V_cb| and list a number of |V_cb|-independent ratios within the SM which exhibit rather different dependences on the angles β and γ of the unitarity triangle. The particular role of these decays in probing very short distance scales far beyond the ones explored at the LHC is emphasized. In this context the role of the Standard Model Effective Field Theory (SMEFT) is very important. We also address briefly the issue of the footprints of Majorana neutrinos in K^+→π^+νν̅ and K_L→π^0νν̅.

Electroweak interactions assign a central role to the gauge group $SU(2{)}_{L}\ifmmode\times\else\texttimes\fi{}U(1{)}_{Y}$, which is either realized linearly (SMEFT) or nonlinearly (e.g., HEFT) in the effective theory obtained when new physics above the electroweak scale is integrated out. Although the discovery of the Higgs boson has made SMEFT the default assumption, nonlinear realization remains possible. The two can be distinguished through their predictions for the size of certain low-energy dimension-6 four-fermion operators: for these, HEFT predicts $O(1)$ couplings, while in SMEFT they are suppressed by a factor ${v}^{2}/{\mathrm{\ensuremath{\Lambda}}}_{\mathrm{NP}}^{2}$, where $v$ is the Higgs vev. One such operator, ${O}_{V}^{LR}\ensuremath{\equiv}(\overline{\ensuremath{\tau}}{\ensuremath{\gamma}}^{\ensuremath{\mu}}{P}_{L}\ensuremath{\nu})(\overline{c}{\ensuremath{\gamma}}_{\ensuremath{\mu}}{P}_{R}b)$, contributes to $b\ensuremath{\rightarrow}c{\ensuremath{\tau}}^{\ensuremath{-}}\overline{\ensuremath{\nu}}$. We show that present constraints permit its non-SMEFT coefficient to have a HEFTy size. We also note that the angular distribution in $\overline{B}\ensuremath{\rightarrow}{D}^{*}(\ensuremath{\rightarrow}D{\ensuremath{\pi}}^{\ensuremath{'}}){\ensuremath{\tau}}^{\ensuremath{-}}(\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\nu}}_{\ensuremath{\tau}}){\overline{\ensuremath{\nu}}}_{\ensuremath{\tau}}$ contains enough information to extract the coefficients of all new-physics operators. Future measurements of this angular distribution can therefore tell us if non-SMEFT new physics is really necessary.

We present for the first time NLO QCD Renormalization Group (RG) evolution matrices for non-leptonic $\Delta F=2$ transitions in the Standard Model Effective Field Theory (SMEFT). To this end we transform first the known two-loop QCD anomalous dimension matrices (ADMs) of the BSM operators in the so-called BMU basis into the ones in the common Weak Effective Theory (WET) basis (the so-called JMS basis) for which tree-level and one-loop matching to the SMEFT are already known. This allows us subsequently to find the two-loop QCD ADMs for the SMEFT non-leptonic $\Delta F=2$ operators in the Warsaw basis. Having all these ingredients we investigate the impact of these NLO QCD effects on the QCD RG evolution of SMEFT Wilson coefficients for non-leptonic $\Delta F=2$ transitions from the new physics scale $\Lambda$ down to the electroweak scale $\mu_\text{ew}$. The main benefit of these new contributions is that they allow to remove renormalization scheme dependences present both in the one-loop matchings between the WET and SMEFT and also between SMEFT and a chosen UV completion. But the NLO QCD effects, calculated here in the NDR scheme, turn out to be small, in the ballpark of a few percent but larger than one-loop Yukawa top effects when only the $\Delta F=2$ operators are considered. The technology developed in our paper allows to obtain the ADMs in the SMEFT from the ones of the BMU basis also for non-leptonic $\Delta F=1$ decays and the results of this more involved analysis will be presented soon in another publication.