We simplify the one-loop functional matching formalism to develop a streamlined prescription. The functional approach is conceptually appealing: all calculations are performed within the UV theory at the matching scale, and no prior determination of an Effective Field Theory (EFT) operator basis is required. Our prescription accommodates any relativistic UV theory that contains generic interactions (including derivative couplings) among scalar, fermion, and vector fields. As an example application, we match the singlet scalar extended Standard Model (SM) onto SMEFT.
This paper presents STrEAM (SuperTrace Evaluation Automated for Matching), a Mathematica package that calculates all functional supertraces which arise when matching a generic UV model onto a relativistic Effective Field Theory (EFT) at one loop and to arbitrary order in the heavy mass expansion. STrEAM implements the covariant derivative expansion to automate the most tedious step of the streamlined functional matching prescription presented in arXiv:2011.02484 . The code and an example notebook are available at https://www.github.com/EFTMatching/STrEAM .
This note gives an overview of the tools for the precision matching of ultraviolet theories to the Standard Model effective field theory (SMEFT) at the tree level and one loop. Several semi- and fully automated codes are presented, as well as some supplementary codes for the basis conversion and the subsequent running and matching at low energies. A suggestion to collect information for cross-validations of current and future codes is made.
We present the results of the first complete one-loop matching calculation between the real singlet scalar extension of the Standard Model and the Standard Model effective field theory (SMEFT) at dimension six. Beyond their immediate relevance to the precision calculation of observables in singlet extensions of the Standard Model, our results illustrate a variety of general features of one-loop matching. We explore the interplay between non-supersymmetric non-renormalization theorems, the logarithmic dependence of Wilson coefficients, and the relevance of mixed diagrams in theories with large …
We study axion effective field theories (EFTs), with a focus on axion couplings to massive chiral gauge fields. We investigate the EFT interactions that participate in processes with an axion and two gauge bosons, and we show that, when massive chiral gauge fields are present, such interactions do not entirely originate from the usual anomalous EFT terms. We illustrate this both at the EFT level and by matching to UV-complete theories. In order to assess the consistency of the Peccei--Quinn (PQ) anomaly matching, it is useful to introduce an auxiliary, non-dynamical gauge field associated to t…
We present SuperTracer, a Mathematica package aimed at facilitating the functional matching procedure for generic UV models. This package automates the most tedious parts of one-loop functional matching computations. Namely, the determination and evaluation of all relevant supertraces, including loop integration and Dirac algebra manipulations. The current version of SuperTracer also contains a limited set of output simplifications. However, a further reduction of the output to a minimal basis using Fierz identities, integration by parts, simplification of Dirac structures, and/or light field …
The effective field theory (EFT) construction when the objects of interest are the In-In (or real time) correlators rather than the In-Out S-matrix elements is constructed. This is done using the formal equivalence between the In-In and In-Out correlation functions, and demanding that the EFT and the full theory provide the same answers for the In-In correlation functions. Matching coefficients for a simple example are provided including for operators with two or more space or time derivatives. It is also pointed out that the Schrodinger representation of QFT captures some of the desired featu…
IFAE and BIST, Universitat Autònoma de Barcelona（巴塞罗那自治大学）；School of Physics, Peking University（北京大学物理学院）；International Centre for Theoretical Physics Asia-Pacific (ICTP-AP), University of Chinese Academy of Sciences (UCAS)（中国科学院大学亚太理论物理中心）；School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University（北京大学物理学院）；Center for High Energy Physics, Peking University（北京大学高能物理中心）；Beijing Laser Acceleration Innovation Center（北京激光加速创新中心）；CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）
We apply effective field theory (EFT) methods to compute the renormalization group improved effective potential for theories with a large mass hierarchy. Our method allows one to compute the effective potential in a systematic expansion in powers of the mass ratio, as well as to sum large logarithms of mass ratios using renormalization group evolution. The effective potential is the sum of one-particle irreducible diagrams (1PI) but information about which diagrams are 1PI is lost after matching to the EFT, since heavy lines get shrunk to a point. We therefore introduce a tadpole condition in …
Low-energy effective field theories (EFT) encode information about the physics at high energies--i.e., the high-energy theory (HET). To extract this information the EFT and the HET have to be matched to each other. At the one-loop level, general results for the matching of renormalizable operators have already been obtained in the literature. In the present paper, we take a step towards a better understanding of renormalizable operator matching at the two-loop level: Focusing on the diagrammatic method, we discuss in detail the various contributions to two-loop matching conditions and compare …
We discuss the conditions for an effective field theory (EFT) to give an adequate low-energy description of an underlying physics beyond the Standard Model (SM). Starting from the EFT where the SM is extended by dimension-6 operators, experimental data can be used without further assumptions to measure (or set limits on) the EFT parameters. The interpretation of these results requires instead a set of broad assumptions (e.g. power counting rules) on the UV dynamics. This allows one to establish, in a bottom-up approach, the validity range of the EFT description, and to assess the error associa…
In this paper, we continue to construct the low-energy effective field theories (EFTs) of the canonical seesaw models, which are natural extensions of the Standard Model (SM) to accommodate tiny but nonzero neutrino masses. Different from three right-handed neutrino singlets in the type-I seesaw model, the Higgs triplet in the type-II seesaw model participates directly in the electroweak gauge interactions, rendering the EFT construction more challenging. By integrating out the heavy Higgs triplet in the functional-integral formalism, we carry out a complete one-loop matching of the type-II se…
The 2 Higgs Doublet Model (2HDM) of type III has renormalisable Lepton Flavour-Violating couplings, and its one and two-loop ("Barr-Zee") contributions to $μ\to e γ$ are known. In the decoupling limit, where the mass scale M of the second doublet is much greater than the electroweak scale, the model can be parametrised with an Effective Field Theory(EFT) containing dimension six operators. The $1/M^2$ terms of the exact calculation are reproduced in the EFT, provided that the four-fermion operator basis below the weak scale is enlarged with respect to the SU(2)-invariant Buchmuller-Wyler list.…
We develop tools for performing effective field theory (EFT) calculations in a manifestly gauge-covariant fashion. We clarify how functional methods account for one-loop diagrams resulting from the exchange of both heavy and light fields, as some confusion has recently arisen in the literature. To efficiently evaluate functional traces containing these "mixed" one-loop terms, we develop a new covariant derivative expansion (CDE) technique that is capable of evaluating a much wider class of traces than previous methods. The technique is detailed in an appendix, so that it can be read independen…
The computation of one-loop corrections to Reggeon-Particle-Particle effective vertices with two scales of virtuality is considered in the framework of gauge-invariant effective field theory for Multi-Regge processes in QCD. Rapidity divergences arising in loop integrals are regulated by "tilted Wilson lines" prescription. General analysis of rapidity divergences at one loop is given and necessary scalar integrals with one and two scales of virtuality are computed. Two examples of effective vertices at one loop are considered: the effective vertex of interaction of (space-like) virtual photon …
An Effective Field Theory (EFT) analysis of the magnetic moment of the muon is discussed. The expression for the dipole moment is given in terms of operator coefficients of the low-energy effective field theory (LEFT) and the Standard Model effective field theory (SMEFT), where one-loop renormalization group improved perturbation theory, the one-loop matching from SMEFT onto LEFT as well as one-loop lepton matrix elements of the effective-theory operators were taken into account. Interestingly only a very limited set of the SMEFT operators is able to explain the current deviation of the magnet…
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 2H…
SUSY meets SMEFT: Complete one-loop matching of the general MSSM
Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS/IN2P3（亚原子与宇宙物理学实验室，格勒诺布尔阿尔卑斯大学，法国国家科学研究中心/国家核物理与粒子物理研究所）；Centro de Ciências Naturais e Humanas, Universidade Federal do ABC（自然与人文科学中心，ABC联邦大学）；Departament de Física Teòrica, IFIC (Universitat de València - CSIC)（理论物理系，瓦伦西亚大学-西班牙国家科学研究委员会联合研究所）；Institute for Theoretical Particle Physics and Cosmology, RWTH Aachen University（理论粒子物理学与宇宙学研究所，亚琛工业大学）
Dismissing traditional naturalness concerns while embracing the Higgs boson mass measurement and unification motivates careful analysis of trans-TeV supersymmetric theories. We take an effective field theory (EFT) approach, matching the Minimal Supersymmetric Standard Model (MSSM) onto the Standard Model (SM) EFT by integrating out heavy superpartners, and evolving MSSM and SMEFT parameters according to renormalization group equations in each regime. Our matching calculation is facilitated by the recent covariant diagrams formulation of functional matching techniques, with the full one-loop SU…
Extensions of the Standard Model (SM) often contain new particles with masses far above the electroweak scale. Due to the presence of a mass hierarchy, effective field theory (EFT) is a suitable tool for the study of such extensions. In this thesis, we provide a solution to the matching problem at the one-loop level in the form of the Bispinor Universal One-Loop Effective Action (BSUOLEA), which is a pre-computed expression for the effective action, parameterised in terms of derivatives of the Lagrangian of the ultraviolet (UV) model. The BSUOLEA is derived using functional methods and is suff…
The shift in focus towards searches for physics beyond the Standard Model (SM) employing model-independent Effective Field Theory (EFT) methods necessitates a rigorous approach to matching to guarantee the validity of the obtained results and constraints. The limits on the leading dimension-six EFT effects can be rather inaccurate for LHC searches that suffer from large uncertainties while exploring an extensive energy range. Similarly, precise measurements can, in principle, test the subleading effects of the operator expansion. In this work, we present an algorithmic approach to automatise m…
To obtain a first order phase transition requires large new physics corrections to the Standard Model (SM) Higgs potential. This implies that the scale of new physics is relatively low, raising the question whether an effective field theory (EFT) description can be used to analyse the phase transition in a (nearly) model-independent way. We show analytically and numerically that first order phase transitions in perturbative extensions of the SM cannot be described by the SM-EFT. The exception are Higgs-singlet extension with tree-level matching; but even in this case the SM-EFT can only captur…
One of the simplest extensions of the Standard Model is the inclusion of an additional scalar multiplet, and we consider scalars in the $SU(2)_L$ singlet, triplet, and quartet representations. We examine models with heavy neutral scalars, $m_H\sim 1-2$ TeV, and the matching of the UV complete theories to the low energy effective field theory. We demonstrate the agreement of the kinematic distributions obtained in the singlet models for the gluon fusion of a Higgs pair with the predictions of the effective field theory. The restrictions on the extended scalar sectors due to unitarity and precis…
We show that the well-known obstacle for working with the zero-mass variable flavor number scheme, namely, the omission of O(1) mass power corrections close to the conventional heavy flavor matching point (HFMP) mu_b=m, can be easily overcome. For this it is sufficient to take advantage of the freedom in choosing the position of the HFMP. We demonstrate that by choosing a sufficiently large HFMP, which could be as large as 10 times the mass of the heavy quark, one can achieve the following improvements: 1) above the HFMP the size of missing power corrections O(m) is restricted by the value of …
Many studies of possible new physics employ effective field theory (EFT), whereby corrections to the Standard Model take the form of higher-dimensional operators, suppressed by a large energy scale. Fits of such a theory to data typically use parton level observables, which limits the datasets one can use. In order to theoretically model search channels involving many additional jets, it is important to include tree-level matrix elements matched to a parton shower algorithm, and a suitable matching procedure to remove the double counting of additional radiation. There are then two potential pr…
Mapping UV theories onto low energy effective descriptions is a procedure known as matching. The last decade has seen tremendous progress in the development of new tools for efficiently performing matching calculations, by relying on so-called functional methods. This white paper summarizes the status of functional matching. Specifically, matching for relativistic theories is a fully solved problem up to one-loop order in perturbation theory, and to arbitrary order in the effective field theory expansion. A streamlined prescription that has been partially automated facilitates the application …
Effective field theories are an incredibly powerful tool in order to study and understand the true nature of the symmetry breaking sector dynamics of the Standard Model. However, they can suffer from some theoretical problems such as that of unitarity violation. Nevertheless, in order to interpret experimental data correctly a fully unitary prescription is needed. To this purpose, unitarization methods are addressed, but each of them leads to a different (unitary) prediction. Because of this, there is an inherent theoretical uncertainty in the determination of the effective field theory parame…
Coupling hadronic kinetic theory models to fluid dynamics in phenomenological studies of heavy ion collisions requires a prescription for ``particlization''. Existing particlization models are based on implicit or explicit assumptions about the microscopic degrees of freedom that go beyond the information provided by the preceding fluid dynamical history. We propose an alternative prescription which uses only macroscopic information provided by the hydrodynamic output. This method follows directly from the connections between information theory and statistical mechanics.
We present the universal one-loop effective action for all operators of dimension up to six obtained by integrating out massive, non-degenerate multiplets. Our general expression may be applied to loops of heavy fermions or bosons, and has been checked against partial results available in the literature. The broad applicability of this approach simplifies one-loop matching from an ultraviolet model to a lower-energy effective field theory (EFT), a procedure which is now reduced to the evaluation of a combination of matrices in our universal expression, without any loop integrals to evaluate. W…
A near-future detection of the 21cm signal from the epoch of reionization will provide unique opportunities to probe the underlying cosmology, provided that such cosmological information can be extracted with precision. To this end, we further develop effective field theory (EFT) inspired techniques for the 21cm brightness temperature field during the epoch of reionization, incorporating renormalized bias and a treatment of redshift space distortions. Notably, we confirm that in redshift space, measures of the 21cm brightness, e.g the power spectrum, should have irreducible contributions that …
We show how a careful analysis of the behaviour of a parton shower Monte Carlo in the vicinity of the soft and collinear regions allows one to formulate a modified MC@NLO-matching prescription that reduces the number of negative-weight events with respect to that stemming from the standard MC@NLO procedure. As a first practical application of such a prescription, that we dub MC@NLO-$Δ$, we have implemented it in the MadGraph5_aMC@NLO framework, by employing the Pythia8 Monte Carlo. We present selected MC@NLO-$Δ$ results at the 13 TeV LHC, and compare them with MC@NLO ones. We find that the for…
Effective field theory (EFT) approaches are widely used at the LHC, such that it is important to study their validity, and ease of matching to specific new physics models. In this paper, we consider an extension of the SM in which a top quark couples to a new heavy scalar. We find the dimension six operators generated by this theory at low energy, and match the EFT to the full theory up to NLO precision in the simplified model coupling. We then examine the range of validity of the EFT description in top pair production, finding excellent validity even if the scalar mass is only slightly above …
We assess the status of a wide class of WIMP dark matter (DM) models in light of the latest experimental results using the global fitting framework $\textsf{GAMBIT}$. We perform a global analysis of effective field theory (EFT) operators describing the interactions between a gauge-singlet Dirac fermion and the Standard Model quarks, the gluons and the photon. In this bottom-up approach, we simultaneously vary the coefficients of 14 such operators up to dimension 7, along with the DM mass, the scale of new physics and several nuisance parameters. Our likelihood functions include the latest data…
We present a practical three-step procedure of using the Standard Model effective field theory (SM EFT) to connect ultraviolet (UV) models of new physics with weak scale precision observables. With this procedure, one can interpret precision measurements as constraints on a given UV model. We give a detailed explanation for calculating the effective action up to one-loop order in a manifestly gauge covariant fashion. This covariant derivative expansion method dramatically simplifies the process of matching a UV model with the SM EFT, and also makes available a universal formalism that is easy …
Recently, a general result for evaluating the path integral at one loop was obtained in the form of the Universal One-Loop Effective Action. It may be used to derive effective field theory operators of dimensions up to six, by evaluating the traces of matrices in this expression, with the mass-dependence encapsulated in the universal coefficients. Here we show that it can account for loops of mixed heavy-light particles in the matching procedure. Our prescription for computing these mixed contributions to the Wilson coefficients is conceptually simple. Moreover it has the advantage of maintain…
Effective field theory of coupled dark energy and dark matter
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Universidad de Salamanca（萨拉曼卡大学）；Mahidol University（玛希隆大学）；Sogang University（西江大学）；Waseda University（早稻田大学）
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…
