We describe the on-shell method to deriving the Renormalization Group (RG) evolution of Wilson coefficients of high dimensional operators at one loop, which is a necessary part in the on-shell construction of the Standard Model Effective Field Theory (SMEFT), and exceptionally efficient based on the amplitude basis in hand. The UV divergence is obtained by firstly calculating the coefficients of scalar bubble integrals by unitary cuts, then subtracting the IR divergence in the massless bubbles, which can be easily read from the collinear factors we obtained for the Standard Model fields. Examples of deriving the anomalous dimensions at dimension six are presented in a pedagogical manner. We also give the results of contributions from the dimension-8 $H^4D^4$ operators to the running of $V^+V^-H^2$ operators, as well as the running of $B^+B^-H^2D^{2n}$ from $H^4D^{2n+4}$ for general $n$.
We describe on-shell methods for computing one- and two-loop anomalous dimensions in the context of effective field theories containing higher-dimension operators. We also summarize methods for computing one-loop amplitudes, which are used as inputs to the computation of two-loop anomalous dimensions, and we explain how the structure of rational terms and judicious renormalization scheme choices can lead to additional vanishing terms in the anomalous dimension matrix at two loops. We describe the two-loop implications for the Standard Model Effective Field Theory (SMEFT). As a by-product of th…
We present a map between the tree-level Standard Model Effective Theory (SMEFT) in the Warsaw basis and massive on-shell amplitudes. As a first step, we focus on the electroweak sector without fermions. We describe the Feynman rules for a particular choice of input scheme and compare them with the 3-point massive amplitudes in the broken phase. Thereby we fix an on-shell basis which allows us to study scattering amplitudes with recursion relations. We hope to open up new avenues of exploration to a complete formulation of massive EFTs in the on-shell language.
Using background-field method and super-heat-kernel expansion, we derive a master formula for the one-loop UV divergences of the bosonic dimension-6 operators in Standard Model Effective Field Theory (SMEFT). This approach reduces the calculation of all the UV divergences to algebraic manipulations. Using this formula we corroborate results in the literature for the one-loop anomalous dimension matrix of SMEFT obtained via diagrammatic methods, considering contributions from the operators $X^3, ϕ^6, ϕ^4 D^2, X^2 ϕ^2$ of the Warsaw basis. The formula is derived in a general way and can be appli…
We present for the first time NLO QCD Renormalization Group (RG) evolution matrices for non-leptonic $ΔF=2$ 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 $ΔF=2$ operators in the Warsaw basis. Ha…
This report presents the results of the Models and Effective Field Theories Subgroup of the Off-Shell Interpretations Task Force in the LHC Higgs Working Group. The main goal of the subgroup was to discuss and advance the potential impact of off-shell Higgs measurements on searches for BSM physics carried out in the EFT framework or as benchmark model studies. In the first contribution, the off-shell potential to resolve flat directions in parameter space for on-shell measurements is studied. Furthermore, the sensitivity of off-shell measurements to SMEFT dimension-6 operators for the gg $\to$…
We present the Standard Model Effective Field Theories (SMEFT) from purely on-shell arguments. Starting from a few basic assumptions such as Poincaré invariance and locality, we classify all the renormalisable and non-renormalisable interactions at the lowest order in the couplings. From these building blocks, we review how locality and unitarity enforce Lie algebra structures to appear in the S-matrix elements together with relations among couplings (and hypercharges). Furthermore, we give a fully on-shell algorithm to compute any higher-point tree-level amplitude (or form factor) in generic …
We revisit electroweak radiative corrections to Standard Model Effective Field Theory (SMEFT) operators which are relevant for the $B$-meson semileptonic decays. The one-loop matching formulae onto the low-energy effective field theory are provided without imposing any flavor symmetry. The on-shell conditions are applied especially in dealing with quark-flavor mixings. Also, the gauge independence is shown explicitly in the $R_ξ$ gauge.
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 \emph{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 violat…
The conventional oblique parameters analyses of precision electroweak data can be consistently cast in the modern framework of the Standard Model effective field theory (SMEFT) when restrictions are imposed on the SMEFT parameter space so that it describes universal theories. However, the usefulness of such analyses is challenged by the fact that universal theories at the scale of new physics, where they are matched onto the SMEFT, can flow to nonuniversal theories with renormalization group (RG) evolution down to the electroweak scale, where precision observables are measured. The departure f…
We calculate the full set of next-to-leading order (NLO) corrections to $h\to b\bar{b}$ decay in the dimension-6 Standard Model Effective Field Theory (SMEFT). Our calculation forms the basis for precision studies of this decay mode in effective field theory, providing analytic and numerical results for contributions of the 45 dimension-6 operators appearing at NLO. On the technical side, we discuss several complications in NLO SMEFT computations which have not yet been addressed in the literature. These include subtleties in Higgs-$Z$ mixing, electric charge renormalization, and especially th…
We discuss interference in the limit $\hat{m}_{W}^2/s \rightarrow 0$ in the Standard Model Effective Field Theory (SMEFT). Dimension six operators that contribute to $\barψ ψ\rightarrow \bar{ψ'}_1 ψ_2' \bar{ψ'}_3 ψ'_4$ scattering events can experience a suppression of interference effects with the Standard Model in this limit. This occurs for subsets of phase space in some helicity configurations. We show that approximating these scattering events by $2\rightarrow 2$ on-shell scattering results for intermediate unstable gauge bosons, and using the narrow width approximation, can miss interfere…
We study nuclear electric dipole moments induced by $ΔF=1$ effective operators in the Standard Model Effective Field Theory. Such contributions arise through renormalization group evolutions and matching conditions at the electroweak symmetry breaking scale. We provide one-loop formulae for the matching conditions. We also discuss correlations of these effects with $ΔF=2$ observables such as $ε_K$ and $ΔM_{B_d}$.
The Standard Model Effective Field Theory (SMEFT) provides a consistent framework for comparing precision measurements at the LHC to the Standard Model. The observation of statistically significant non-zero SMEFT coefficients would correspond to physics beyond the Standard Model (BSM) of some sort. A more difficult question to answer is what, if any, detailed information about the nature of the underlying high scale model can be obtained from these measurements. In this work, we consider the patterns of SMEFT operators present in five example models and discuss the assumptions inherent in usin…
We present a fully differential and spin-dependent $t$-channel single-top-quark calculation at next-to-leading order (NLO) in QCD including off-shell effects by using the complex mass scheme in the Standard Model (SM) and in the Standard Model Effective Field Theory (SMEFT). We include all relevant SMEFT operators at $1/Λ^2$ that contribute at NLO in QCD for a fully consistent comparison to the SM at NLO. In addition, we include chirality flipping operators that do not interfere with the SM amplitude and contribute only at $1/Λ^4$ with a massless $b$-quark. Such higher order effects are usuall…
Anomalous Dimension of a General Effective Gauge Theory I: Bosonic Sector
CERN（欧洲核子研究中心）；Dipartimento di Fisica e Astronomia “G. Galilei”, Università degli Studi di Padova（帕多瓦大学物理与天文学系）；Istituto Nazionale di Fisica Nucleare, Sezione di Padova（意大利国家核物理研究所帕多瓦分部）
In this contribution to the Snowmass 2021 process we review theoretical developments in the Standard Model Effective Field Theory (SMEFT) with a focus on effects at the dimension-8 level and beyond. We review the theoretical advances that led to the complete construction of the operator bases for the dimension-8 and dimension-9 SMEFT Lagrangians. We discuss the possibility of obtaining all-orders results in the $1/Λ$ expansion for certain SMEFT observables as well as the current status of renormalization group running and implications for positivity, and briefly present the on-shell approach t…
A model-independent anatomy of $ΔF= 2$ transitions in the context of the Weak Effective Theory (WET) below the electroweak scale (EW) and the Standard Model Effective Field Theory (SMEFT) above the EW scale is discussed. Two master formulae for the BSM contribution of the mixing amplitude $M_{12}$, in terms of Wilson coefficients are presented. The coefficients entering these formulae contain all the information below the EW scale and the NP scale $Λ$, respectively. The renormalization group evolution from the top-quark Yukawa coupling has the largest impact on the result. The obtained express…
We discuss the off-shell renormalization properties of the abelian Higgs model in the unitary gauge. The model is not renormalizable according to the usual power counting rules. In this paper, however, we show that with a proper choice of interpolating fields for the massive photon and the Higgs particle, their off-shell Green functions can be renormalized. An analysis of the nature of the extra singularities in the unitary gauge is given, and a recipe for the off-shell renormalization is provided.
We calculate one loop $y_t$ and $λ$ dependent corrections to $\barΓ_Z,\bar{R}_f^0$ and the partial $Z$ widths due to dimension six operators in the Standard Model Effective Field Theory (SMEFT), including finite terms. We assume $\rm CP$ symmetry and a $\rm U(3)^5$ symmetry in the UV matching onto the dimension six operators, dominantly broken by the Standard Model Yukawa matrices. Corrections to these observables are predicted using the input parameters $\{\hatα_{ew}, \hat{M}_Z, \hat{G}_F, \hat{m}_t, \hat{m}_h\}$ extracted with one loop corrections in the same limit. We show that at one loop …
We explore the interplay of New Physics (NP) effects in $(g-2)_\ell$ and $h \to \ell^+ \ell^-$ within the Standard Model Effective Field Theory (SMEFT) framework, including one-loop Renormalization Group (RG) evolution of the Wilson coefficients as well as matching to the observables below the electroweak symmetry breaking scale. We include both the leading dimension six chirality flipping operators including a Higgs and $SU(2)_L$ gauge bosons as well as four-fermion scalar and tensor operators, forming a closed operator set under the SMEFT RG equations. We compare present and future experimen…
Standard Model Neutrino Effective Field Theory (SMNEFT) is an effective theory with Standard Model (SM) gauge-invariant operators constructed only from SM and right-handed neutrino fields. For the full set of dimension-six SMNEFT operators, we present the gauge coupling terms of the one-loop anomalous dimension matrix for renormalization group evolution (RGE) of the Wilson coefficients between a new physics scale and the electroweak scale. We find that the SMNEFT operators can be divided into five subsets which are closed under RGE. Our results apply for both Dirac and Majorana neutrinos. We a…
Symmetry-restoring finite counterterms of SMEFT four-fermion operator insertions at one loop
Universitat de València – Consejo Superior de Investigaciones Científicas（瓦伦西亚大学 - 西班牙国家研究委员会）；University of Sussex（萨塞克斯大学）；Universidad Cardenal Herrera-CEU（卡迪纳尔埃雷罗大学-CEU）
Neural simulation-based inference of the Higgs trilinear self-coupling via off-shell Higgs production
University of California, Irvine（加州大学尔湾分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；Georgia Institute of Technology（佐治亚理工学院）；Max Planck Institute for Physics（马克斯·普朗克物理研究所）
In the context of the Standard Model effective field theory (SMEFT) we examine the constraints on the trilinear Higgs coupling that originate from off-shell Higgs production in proton-proton collisions. Our calculation of the $gg \to h^\ast \to Z Z \to 4 \ell$ process includes two-loop corrections to gluon-gluon-fusion Higgs production and one-loop corrections to the Higgs propagator and its decay. Employing a matrix-element based kinematic discriminant we determine the reach of LHC Run 3 and the high-luminosity option of the LHC in constraining the relevant SMEFT Wilson coefficients. We prese…
The Standard Model Effective Field Theory (SMEFT) provides a systematic and model-independent framework to study neutrino non-standard interactions (NSIs). We study the constraining power of the on-going neutrino oscillation experiments T2K, NO$ν$A, Daya Bay, Double Chooz and RENO in the SMEFT framework. A full consideration of matching is provided between different effective field theories and the renormalization group running at different scales, filling the gap between the low-energy neutrino oscillation experiments and SMEFT at the UV scale. We first illustrate our method with a top-down a…
