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 $$\nu $$ SMEFT. We present the first code for solving the renormalization group equations (RGEs) of the $$\nu $$ SMEFT in an automated way. For this purpose, we have implemented the $$\nu $$ SMEFT as a new effective field theory (EFT) in the Wilson coefficient exchange format . Furthermore, we included anomalous dimensions depending on the gauge couplings and Yukawas in the python package . This novel version of allows a consistent inclusion of $$\nu $$ 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.

Switzerland

Jason Aebischer

France

Tejhas Kapoor

United States

Jacky Kumar

A package for renormalization group running in the SMEFT with sterile neutrinos

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)   <math> <mover> <mi>b</mi> <mo>¯</mo> </mover> <mo>→</mo> <mover> <mi>s</mi> <mo>¯</mo> </mover> <msup> <mi>ℓ</mi> <mo>+</mo> </msup> <msup> <mi>ℓ</mi> <mo>−</mo> </msup> </math>  $$ \overline{b}\to \overline{s}{\ell}^{+}{\ell}^{-} $$  anomalies and the (hadronic) B → πK puzzle.

Alakabha Datta

Canada

Suman Kumbhakar

David London

Uniting low-energy semileptonic and hadronic anomalies within SMEFT

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.

Kaori Fuyuto

Emanuele Mereghetti

Stefan Sandner

Chen Sun

Sterile neutrino dark matter within the νSMEFT

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.

Electric dipole moments in 5+3 flavor weak effective theory

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 <math><mi>K</mi></math> decays and nuclear beta decay through renormalization group (RG) running, and hence can influence the measurements of <math><msub><mi>V</mi><mrow><mi>u</mi><mi>s</mi></mrow></msub></math>. Interestingly, through this mechanism, a single six-dimensional effective operator <math><msub><mi>O</mi><mrow><mo>ℓ</mo><mo>ℓ</mo></mrow></msub></math> 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 <math><msup><mi>Z</mi><mo>′</mo></msup></math> model, the nonuniversal leptonic couplings of this operator can also contribute to the lepton flavor universality violating ratios such as <math><msub><mi>R</mi><msup><mi>K</mi><mrow><mo>(</mo><mo>*</mo><mo>)</mo></mrow></msup></msub></math>, which would act as stringent constraints on such scenarios. By performing a global fit of the <math><msup><mi>Z</mi><mo>′</mo></msup></math> model, we find that it is essential to have nonuniversal couplings of such a <math><msup><mi>Z</mi><mo>′</mo></msup></math> boson to all three generations of leptons.

India

Ashutosh Kumar Alok

Amol Dighe

Shireen Gangal

Germany

Leptonic operators for the Cabbibo angle anomaly with SMEFT RG evolution

Despite the observation of significant suppressions of  $$b\rightarrow s\mu ^+\mu ^-$$  <math> <mrow> <mi>b</mi> <mo>→</mo> <mi>s</mi> <msup> <mi>μ</mi> <mo>+</mo> </msup> <msup> <mi>μ</mi> <mo>-</mo> </msup> </mrow> </math>   branching ratios no clear sign of New Physics (NP) has been identified in  $$\Delta F=2$$  <math> <mrow> <mi>Δ</mi> <mi>F</mi> <mo>=</mo> <mn>2</mn> </mrow> </math>   observables  $$\Delta M_{d,s}$$  <math> <mrow> <mi>Δ</mi> <msub> <mi>M</mi> <mrow> <mi>d</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </mrow> </math>  ,  $$\varepsilon _K$$  <math> <msub> <mi>ε</mi> <mi>K</mi> </msub> </math>   and the mixing induced CP asymmetries  $$S_{\psi K_S}$$  <math> <msub> <mi>S</mi> <mrow> <mi>ψ</mi> <msub> <mi>K</mi> <mi>S</mi> </msub> </mrow> </msub> </math>   and  $$S_{\psi \phi }$$  <math> <msub> <mi>S</mi> <mrow> <mi>ψ</mi> <mi>ϕ</mi> </mrow> </msub> </math>  . Assuming negligible NP contributions to these observables allows to determine CKM parameters without being involved in the tensions between inclusive and exclusive determinations of  $$|V_{cb}|$$  <math> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>V</mi> <mrow> <mi>cb</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </math>   and  $$|V_{ub}|$$  <math> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>V</mi> <mrow> <mi>ub</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </math>  . 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  $$Z^\prime $$  <math> <msup> <mi>Z</mi> <mo>′</mo> </msup> </math>   models we point out, following the 2009 observations of Monika Blanke and ours of 2020, that despite the absence of NP contributions to  $$\varepsilon _K$$  <math> <msub> <mi>ε</mi> <mi>K</mi> </msub> </math>  , significant NP contributions to  $$K^+\rightarrow \pi ^+\nu {\bar{\nu }}$$  <math> <mrow> <msup> <mi>K</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>π</mi> <mo>+</mo> </msup> <mi>ν</mi> <mover> <mrow> <mi>ν</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>  ,  $$K_{L}\rightarrow \pi ^0\nu {\bar{\nu }}$$  <math> <mrow> <msub> <mi>K</mi> <mi>L</mi> </msub> <mo>→</mo> <msup> <mi>π</mi> <mn>0</mn> </msup> <mi>ν</mi> <mover> <mrow> <mi>ν</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>  ,  $$K_S\rightarrow \mu ^+\mu ^-$$  <math> <mrow> <msub> <mi>K</mi> <mi>S</mi> </msub> <mo>→</mo> <msup> <mi>μ</mi> <mo>+</mo> </msup> <msup> <mi>μ</mi> <mo>-</mo> </msup> </mrow> </math>  ,  $$K_L\rightarrow \pi ^0\ell ^+\ell ^-$$  <math> <mrow> <msub> <mi>K</mi> <mi>L</mi> </msub> <mo>→</mo> <msup> <mi>π</mi> <mn>0</mn> </msup> <msup> <mi>ℓ</mi> <mo>+</mo> </msup> <msup> <mi>ℓ</mi> <mo>-</mo> </msup> </mrow> </math>  ,  $$\varepsilon '/\varepsilon $$  <math> <mrow> <msup> <mi>ε</mi> <mo>′</mo> </msup> <mo>/</mo> <mi>ε</mi> </mrow> </math>   and  $$\Delta M_K$$  <math> <mrow> <mi>Δ</mi> <msub> <mi>M</mi> <mi>K</mi> </msub> </mrow> </math>   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  $$Z^\prime $$  <math> <msup> <mi>Z</mi> <mo>′</mo> </msup> </math>   coupling  $$g^L_{sd}$$  <math> <msubsup> <mi>g</mi> <mrow> <mi>sd</mi> </mrow> <mi>L</mi> </msubsup> </math>  . 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  $$Z^\prime $$  <math> <msup> <mi>Z</mi> <mo>′</mo> </msup> </math>   couplings to  $$q{\bar{q}}$$  <math> <mrow> <mi>q</mi> <mover> <mrow> <mi>q</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>  ,  $$\nu {\bar{\nu }}$$  <math> <mrow> <mi>ν</mi> <mover> <mrow> <mi>ν</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>  , and  $$\mu ^+\mu ^-$$  <math> <mrow> <msup> <mi>μ</mi> <mo>+</mo> </msup> <msup> <mi>μ</mi> <mo>-</mo> </msup> </mrow> </math>  . A characteristic feature of this scenario is a strict correlation between  $$K^+\rightarrow \pi ^+\nu {\bar{\nu }}$$  <math> <mrow> <msup> <mi>K</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>π</mi> <mo>+</mo> </msup> <mi>ν</mi> <mover> <mrow> <mi>ν</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   and  $$K_{L}\rightarrow \pi ^0\nu {\bar{\nu }}$$  <math> <mrow> <msub> <mi>K</mi> <mi>L</mi> </msub> <mo>→</mo> <msup> <mi>π</mi> <mn>0</mn> </msup> <mi>ν</mi> <mover> <mrow> <mi>ν</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   branching ratios on a branch parallel to the Grossman-Nir bound. Moreover,  $$\Delta M_K$$  <math> <mrow> <mi>Δ</mi> <msub> <mi>M</mi> <mi>K</mi> </msub> </mrow> </math>   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  $$K_L\rightarrow \mu ^+\mu ^-$$  <math> <mrow> <msub> <mi>K</mi> <mi>L</mi> </msub> <mo>→</mo> <msup> <mi>μ</mi> <mo>+</mo> </msup> <msup> <mi>μ</mi> <mo>-</mo> </msup> </mrow> </math>   which otherwise would bound NP effects in  $$K^+\rightarrow \pi ^+\nu {\bar{\nu }}$$  <math> <mrow> <msup> <mi>K</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>π</mi> <mo>+</mo> </msup> <mi>ν</mi> <mover> <mrow> <mi>ν</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>  . Of particular interest are the correlations of  $$K^+\rightarrow \pi ^+\nu {\bar{\nu }}$$  <math> <mrow> <msup> <mi>K</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>π</mi> <mo>+</mo> </msup> <mi>ν</mi> <mover> <mrow> <mi>ν</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   and  $$K_{L}\rightarrow \pi ^0\nu {\bar{\nu }}$$  <math> <mrow> <msub> <mi>K</mi> <mi>L</mi> </msub> <mo>→</mo> <msup> <mi>π</mi> <mn>0</mn> </msup> <mi>ν</mi> <mover> <mrow> <mi>ν</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   branching ratios and of  $$\Delta M_K$$  <math> <mrow> <mi>Δ</mi> <msub> <mi>M</mi> <mi>K</mi> </msub> </mrow> </math>   with the ratio  $$\varepsilon '/\varepsilon $$  <math> <mrow> <msup> <mi>ε</mi> <mo>′</mo> </msup> <mo>/</mo> <mi>ε</mi> </mrow> </math>  . We investigate the impact of renormalization group effects in the context of the SMEFT on this simple scenario.

Andrzej Buras

Kaon physics without new physics in  $$ \varepsilon _K$$  <math> <msub> <mi>ε</mi> <mi>K</mi> </msub> </math>

Basis transformations often involve Fierz and other relations that are only valid in <math><mi>D</mi><mo>=</mo><mn>4</mn></math> dimensions. In general <math><mi>D</mi></math> spacetime dimensions, however, evanescent operators have to be introduced in order to preserve such identities. Such evanescent operators contribute to one-loop basis transformations as well as to two-loop renormalization group running. We present a simple procedure on how to systematically change basis at the one-loop level by obtaining shifts due to evanescent operators. As an example we apply this method to derive the one-loop basis transformation from the Buras, Misiak and Urban basis useful for next-to-leading order QCD calculations, to the Jenkins, Manohar and Stoffer basis used in the matching to the standard model effective theory.

Andrzej J. Buras

Simple rules for evanescent operators in one-loop basis transformations

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.

Serge Hamoudou

Dimension-8 SMEFT matching conditions for the low-energy effective field theory

The present-day B-anomalies involving b → sμ + μ  −  or b → cτ  −    <math> <mover> <mi>ν</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\nu} $$  transitions can all be explained with the addition of a vector U 1 leptoquark with a mass of   <math> <msub> <mi>M</mi> <msub> <mi>U</mi> <mn>1</mn> </msub> </msub> </math>  $$ {M}_{U_1} $$  ≳ 1.8 TeV. In the scalar singlet dark matter model (SSDMM), the DM is a scalar S that couples to the Higgs via λ  hS  S 2 |H| 2. We update the fit to the data and find that the SSDMM is now viable only for M  S  ≳ 1.6 TeV. In this paper, we assume that the DM also couples to the U 1 via   <math> <msub> <mi>λ</mi> <mrow> <msub> <mi>U</mi> <mn>1</mn> </msub> <mi>S</mi> </mrow> </msub> <msup> <mi>S</mi> <mn>2</mn> </msup> <msubsup> <mi>U</mi> <mrow> <mn>1</mn> <mi>μ</mi> </mrow> <mo>†</mo> </msubsup> <msubsup> <mi>U</mi> <mn>1</mn> <mi>μ</mi> </msubsup> </math>  $$ {\lambda}_{U_1S}{S}^2{U}_{1\mu}^{\dagger }{U}_1^{\mu } $$ . In addition to leading to DM annihilation via SS →   <math> <msub> <mi>U</mi> <mn>1</mn> </msub> <msub> <mover> <mi>U</mi> <mo>¯</mo> </mover> <mn>1</mn> </msub> </math>  $$ {U}_1{\overline{U}}_1 $$ , this coupling generates SSgg and SSγγ couplings at one loop. Although naively divergent, these loop diagrams can be calculated under the assumption that the U 1 is a gauge boson of a group broken at the TeV scale. With this DM-U 1 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 (M  S  ≳   <math> <msub> <mi>M</mi> <msub> <mi>U</mi> <mn>1</mn> </msub> </msub> </math>  $$ {M}_{U_1} $$ ) and light DM (M  S  &lt;   <math> <msub> <mi>M</mi> <msub> <mi>U</mi> <mn>1</mn> </msub> </msub> </math>  $$ {M}_{U_1} $$ ).

Geneviève Bélanger

Russia

Alexander Pukhov

The B anomalies, the U 1 leptoquark and dark matter

We present for the first time Next-to-Leading (NLO) QCD renormalization group (RG) evolution matrices for nonleptonic <math><mi>Δ</mi><mi>F</mi><mo>=</mo><mn>2</mn></math> 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 (Beyond the SM) operators in the so-called Buras Misiak Urban basis into the ones in the common weak effective theory (WET) basis (the so-called Jenkins Manohar Stoffer basis) for which tree-level and one-loop matching to the SMEFT are already known. This subsequently allows us to find the two-loop QCD ADMs for the SMEFT nonleptonic <math><mi>Δ</mi><mi>F</mi><mo>=</mo><mn>2</mn></math> 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 nonleptonic <math><mi>Δ</mi><mi>F</mi><mo>=</mo><mn>2</mn></math> transitions from the new physics scale <math><mi>Λ</mi></math> down to the electroweak scale <math><msub><mi>μ</mi><mi>ew</mi></msub></math>. The main benefit of these new contributions is that they allow one to remove renormalization scheme dependences present in the one-loop matchings both between the WET and SMEFT and also between SMEFT and a chosen UV completion. But the Next-to-Leading (NLO) QCD effects, calculated here in the Naive dimensional regularisation minimal subtraction scheme, turn out to be small, in the ballpark of a few percent but larger than one-loop Yukawa top effects when only the <math><mi>Δ</mi><mi>F</mi><mo>=</mo><mn>2</mn></math> operators are considered. The more complicated class of nonleptonic <math><mi>Δ</mi><mi>F</mi><mo>=</mo><mn>1</mn></math> decays will be presented soon in another publication.