We compute the one-loop matching between the Standard Model Effective Field Theory and the low-energy effective field theory below the electroweak scale, where the heavy gauge bosons, the Higgs particle, and the top quark are integrated out. The complete set of matching equations is derived including effects up to dimension six in the power counting of both theories. We present the results for general flavor structures and include both the $CP$-even and $CP$-odd sectors. The matching equations express the masses, gauge couplings, as well as the coefficients of dipole, three-gluon, and four-fermion operators in the low-energy theory in terms of the parameters of the Standard Model Effective Field Theory. Using momentum insertion, we also obtain the matching for the $CP$-violating theta angles. Our results provide an ingredient for a model-independent analysis of constraints on physics beyond the Standard Model. They can be used for fixed-order calculations at one-loop accuracy and represent a first step towards a systematic next-to-leading-log analysis.
We match the electroweak chiral Lagrangian with two singlet scalars to the next-to-minimal composite Higgs model with $ SO(6)/SO(5) $ coset structure and extract the scalar divergences to one loop. Assuming the additional scalar to be heavy, we integrate it out and perform a matching to the well-established electroweak chiral Lagrangian with one light Higgs.
We determine the real parts of electroweak matching conditions relevant for top quark pair production close to threshold in e+e- annihilation at next-to-next-to-leading logarithmic (NNLL) order. Numerically the corrections are comparable to the NNLL QCD corrections.
We investigate the effect of fermionic electroweak multiplet dark matter models on the stability of the electroweak vacuum using two-loop renormalization group equations (RGEs) and one-loop matching conditions. Such a treatment is crucial to obtain reliable conclusions, compared with one-loop RGEs and tree-level matching conditions. In addition, we find that the requirement of perturbativity up to the Planck scale would give strong and almost mass-independent constraints on the Yukawa couplings in the dark sector. We also evaluate these models via the idea of finite naturalness for the Higgs m…
Electroweak corrections associated with the instability of the top quark to the next-to-next-to-leading logarithmic (NNLL) total top pair threshold cross section in e+e- annihilation are determined. Our method is based on absorptive parts in electroweak matching conditions of the NRQCD operators and the optical theorem. The corrections lead to ultraviolet phase space divergences that have to be renormalized and lead to NLL mixing effects. Numerically, the corrections can amount to several percent and are comparable to the known NNLL QCD corrections.
We report on recent advances on the computation of the matrix elements of the electroweak penguins Q_7 and Q_8 which are relevant for the Delta I=3/2 contribution to epsilon'_K in the chiral limit. The matching of scale and scheme dependences between Wilson coefficients and these matrix elements is done analytically at NLO in alpha_S.
We use the POWHEG method to perform parton-shower matching for Drell-Yan production of W and Z bosons at the LHC at NLO QCD and NLO electroweak accuracy. In particular, we investigate an improved treatment of the vector-boson resonances within the POWHEG method. We employ an independent implementation of the POWHEG method and compare to earlier results within the POWHEGBOX. On the technical side, we provide the FKS formalism for photon-radiation off fermions within mass regularization.
I describe a new way for baryogenesis to proceed, which evades many of the problems of GUT and electroweak scenarios. If the reheat temperature after inflation is below the electroweak scale, neither GUT baryon production nor traditional electroweak baryogenesis can occur. However, non-thermal production of sphaleron configurations via preheating could generate the observed baryon asymmetry of the universe. Such low scale baryon production is particularly attractive since it evades a number of strong constraints on reheating from gravitino and moduli production.
Wilson is a Python library for matching and running Wilson coefficients of higher-dimensional operators beyond the Standard Model. Provided with the numerical values of the Wilson coefficients at a high new physics scale, it automatically performs the renormalization group evolution within the Standard Model effective field theory (SMEFT), matching onto the weak effective theory (WET) at the electroweak scale, and QCD/QED renormalization group evolution below the electroweak scale down to hadronic scales relevant for low-energy precision tests. The matching and running encompasses the complete…
We construct a complete basis of dimension-8 operators in the Low-Energy Effective Field Theory below the Electroweak Scale (LEFT). We find there are 35058 dimension-8 operators in the LEFT for two generations of up-type quarks and three generations of down-type quarks, charged leptons, and left-handed neutrinos. The existence of this operator basis is a necessary prerequisite for matching to the Standard Model Effective Field Theory at the dimension-8 level.
We consider a minimal extension of the Standard Model which advocates a dark neutrino sector charged under a hidden $U(1)^\prime$. We show that neutrino masses can arise radiatively in this model. The observed values are compatible with a light dark sector below the electroweak scale and would imply new heavy fermions which may be testable in the next generation of beam dump searches at DUNE, NA62 and SHIP.
If the neutrino is Majorana type and the electroweak phase transition is second or weak first order, neutrino-induced interactions together with sphaleron transitions have the potential to erase a previously generated baryon asymmetry of the universe. Taking correctly into account the evolution of the vacuum expectation of the Higgs field the effective light neutrino masses are constrained to be lighter than order 10 MeV, while the effective heavy masses are constrained to be heavier than order 10^7 GeV.
This note presents a study of matching and merging schemes and weak showering at the LHC as applied to the production of electroweak bosons in association with jets. These advanced theoretical tools are seen to provide a good description of event shapes in the central region when compared to measurements performed by the ATLAS and CMS collaborations at a centre-of-mass energy of 7 TeV. Matching and merging schemes also provide a superior description of forward $Z+$jets production. The study constitutes a test of matching and merging schemes in a novel region of phase space and can be considere…
We compute the complete two-loop O(alpha) Wilson coefficients relevant for radiative decays of the B meson in the SM. This is a necessary step in the calculation of the O(alpha alpha_s^n ln^n m_b/M_W) corrections and improves on our previous analysis of electroweak effects in B -> X_s gamma. We describe in detail several interesting technical aspects of the calculation and include all dominant QED matrix elements. In our final result, we neglect only terms originated from the unknown O(alpha alpha_s) evolution of the Wilson coefficients and some suppressed two-loop matrix elements. Due to the …
We estimate the thermal masses and damping rates of active (m < eV) and sterile (M ~ GeV) neutrinos with thermal momenta k ~ 3T at temperatures below the electroweak crossover (5 GeV < T < 160 GeV). These quantities fix the equilibration or "washout" rates of Standard Model lepton number densities. Sterile neutrinos interact via direct scatterings mediated by Yukawa couplings, and via their overlap with active neutrinos. Including all leading-order reactions we find that the washout rate generally exceeds the Hubble rate for 5 GeV < T < 30 GeV. Therefore it is challenging to generate a large l…
The low-energy effective field theory is constructed by integrating out Standard Model states with masses proximate to the electroweak scale. We report the equations of motion for this theory, including corrections due to higher dimensional operators up to mass dimension six. We construct the corresponding symmetry currents, and discuss how the $\rm SU(2)_L \times U(1)_{\mathsf{y}}$ symmetry, and global symmetries, are manifested when Standard Model states are integrated out. Including contributions from higher dimensional operators to the equations of motion modifies the interpretation of con…
The effective renormalizable theory describing electromagnetic and strong interactions of quarks of five light flavors ($n_f = 5$ QCD$\times$QED) is considered as a low-energy limit of the full Standard Model. Two-loop relation between the running strong coupling constants $α_s$ defined in either theories is found by simultaneous decoupling of electroweak gauge and Higgs bosons in addition to the top quark. The relation potentially allows one to confront "low-energy determination of $α_s$ with a high-energy one with increased accuracy. Numerical impact of new $\mathcal{O}(α_sα)$ terms is studi…
Exact analytical expressions for the $ΔS=1$ coupling $\im G_E$ in terms of observable spectral functions are given. This coupling determines the size of the $ΔI=3/2$ contribution to $ε'$. We show analytically how the scheme-dependence and scale dependences vanish to all orders in $1/N_c$ and NLO in $α_S$ explicitly both for $Q_7$ and $Q_8$. Numerical results are derived for both $Q_7$ and $Q_8$ from the $τ$-data and known results on the scalar spectral functions. In particular we study the effect of all higher dimension operators. The coefficients of the leading operators in the OPE of the nee…
I review some progress made on the problem of calculating electroweak processes of mesons at low energy with the use of an approximation to large-Nc QCD which we call the Minimal Hadronic Approximation. An update of results for the matrix elements of the electroweak penguin operators Q7 and Q8 is also given.
We derive the conditions for matching high-energy renormalizable Quantum Field Theories onto low-energy nonrenormalizable ones by means of the FDR approach described in Fortsch.Phys. 63 (2015) 132-141. Our procedure works order-by-order in the loop expansion and avoids the addition of higher dimensional interactions into the nonrenormalizable Lagrangian. To illustrate our strategy, we match the high-energy fermion-loop corrections computed in the complete electroweak theory onto the nonrenormalizable four-fermion Fermi model. As a result, the Fermi Lagrangian can be used without modifications …
General analyses of $B$-physics processes beyond the Standard Model require accounting for operator mixing in the renormalization-group evolution from the matching scale down to the typical scale of $B$ physics. For this purpose the anomalous dimensions of the full set of local dimension-six operators beyond the Standard Model are needed. We present here for the first time a complete and non-redundant set of dimension-six operators relevant for $B$-meson mixing and decay, together with the complete one-loop anomalous dimensions in QCD and QED. These results are an important step towards the au…
In the two Higgs doublet model, perturbativity breaks down below the Planck scale: For top mass values 170-180 GeV, the Yukawa coupling reaches the pole below 1000 TeV, if the ratio of vev's v_u/v_d is below 1. We discuss top condensate scenarios for 150 GeV < m_top < 200 GeV, showing that a version with two composite Higgses is a viable option. Within electroweak gauge extensions the second Higgs does not come with additional parameters, but is dictated by the symmetry.
We propose a new alternative for baryogenesis which resolves a number of the problems associated with GUT and electroweak scenarios, and which may allow baryogenesis even in modest extensions of the standard model. If the universe never reheats above the electroweak scale following inflation, GUT baryon production does not occur and at the same time thermal sphalerons, gravitinos and monopoles are not produced in abundance. Nevertheless, non-thermal production of sphaleron configurations via preheating could generate the observed baryon asymmetry of the universe.
Effective Lagrangians are a useful tool for a data-driven approach to physics beyond the Standard Model at the LHC. However, for the new physics scales accessible at the LHC, the effective operator expansion is only relatively slowly converging at best. For tree-level processes, it has been found that the agreement between the effective Lagrangian and a range of UV-complete models depends sensitively on the appropriate definition of the matching. We extend this analysis to the one-loop level, which is relevant for electroweak precision data and Higgs decay to photons. We show that near the sca…
We report on dispersive and finite energy sum rule analyses of the electroweak penguin matrix elements < (pi pi) | Q_{7,8} | K > in the chiral limit. We accomplish the correct perturbative matching (scale and scheme dependence) at NLO in alpha_s, and we describe two different strategies for numerical evaluation.
The Standard Model extended with right-handed neutrinos whose masses are below the electroweak scale provides a simultaneous solution for the origin of neutrino masses and of the baryon asymmetry of the Universe, that can be tested in current experiments. If three right-handed neutrinos participate to the processes, their parameter space of solutions extends to very large mixing angles, saturating the current experimental constraints. Solutions with right-handed neutrino masses at the GeV scale can be probed in the decay of $B$ mesons at the LHC. For this channel the collision of isotopes of i…
We present a semiclassical study of the suppression of topology changing, baryon number violating transitions induced by particle collisions in the electroweak theory. We find that below the sphaleron energy the suppression exponent is remarkably close to the analytic estimates based on a low energy expansion about the instanton. Above the sphaleron energy, the relevant semiclassical solutions have qualitatively different properties from those below the sphaleron: they correspond to jumps on top of the barrier. Yet these processes remain exponentially suppressed, and, furthermore, the tunnelin…
We study Radiative Electroweak Symmetry Breaking in the Minimal Supersymmetric Standard Model (MSSM). We employ the 2-loop Renormalization Group equations for running masses and couplings taking into account sparticle threshold effects. The decoupling of each particle below its threshold is realized by a step function in all one-loop Renormalization Group equations (RGE). This program requires the calculation of all wavefunction, vertex and mass renormalizations for all particles involved. Adapting our numerical routines to take care of the succesive decoupling of each particle below its thres…
For the minimal wino and Higgsino benchmark models we provide accurate energy spectra of high-energy photons from TeV scale dark-matter annihilation $χχ\to γ+X$ by merging electroweak Sudakov resummation near maximal energy with the electroweak parton-shower PPPC4DM, and accounting for the Sommerfeld effect. Electroweak resummation significantly changes the shape of the photon-energy spectrum in the wide range $E_γ\sim (0.6\ldots 1)\, m_χ$ and hence the form of the so-called "line-signal".
New physics realized above the electroweak scale can be encoded in a model independent way in the Wilson coefficients of higher dimensional operators which are invariant under the Standard Model gauge group. In this article, we study the matching of the $SU(3)_C \times SU(2)_L \times U(1)_Y$ gauge invariant dim-6 operators on the standard $B$ physics Hamiltonian relevant for $b \to s$ and $b\to c$ transitions. The matching is performed at the electroweak scale (after spontaneous symmetry breaking) by integrating out the top quark, $W$, $Z$ and the Higgs particle. We first carry out the matchin…
In a class of supersymmetric gauge theories with asymptotic freedom, the low energy effective theory below the confinement scale is described by the composite superfields of the fundamental representation fields. Based on the supersymmetric gauge theory with N_c=2 and N_f=3 with an additional unbroken Z_2 symmetry, we propose a new model where neutrino masses, dark matter, and baryon asymmetry of the Universe can be simultaneously explained by physics below the confinement scale. This is an example for the ultraviolet complete supersymmetric extension of so-called radiative seesaw scenarios wi…
Precision Electroweak Physics of Neutral Currents Below the Z-pole
PRISMA + Cluster of Excellence, Institute for Nuclear Physics and Helmholtz-Institut Mainz, Johannes Gutenberg-University（PRISMA卓越集群、核物理研究所与亥姆霍兹美因茨研究所，约翰内斯·古腾堡大学）；Amherst Center for Fundamental Interactions and Physics Department, University of Massachusetts（阿默斯特基础相互作用中心与物理系，马萨诸塞大学）
We study Radiative Electroweak Symmetry Breaking in the Minimal Supersymmetric Standard Model (MSSM). We employ the 2-loop Renormalization Group equations for running masses and couplings taking into account sparticle threshold effects. The decoupling of each particle below its threshold is realized by a step function in all one-loop Renormalization Group equations (RGE). This program requires the calculation of all wavefunction, vertex and mass renormalizations for all particles involved. Adapting our numerical routines to take care of the succesive decoupling of each particle below its thres…
The origin of the matter-antimatter asymmetry remains one of the most fundamental problems of cosmology. In this talk I present a novel scenario for baryogenesis at the electroweak scale, without the need for a first order phase transition. It is based on the out of equilibrium resonant production of long wavelength Higgs and gauge configurations, at the end of a period of inflation, which induces a large rate of sphaleron transitions, before thermalization at a temperature below critical.
A review of the calculation of the two loop bosonic corrections to $Δr$ is presented. Factorization and matching onto the Fermi model are discussed. An approximate formula, describing the quantity over the interesting range of Higgs boson mass values from 100 GeV to 1 TeV is given.
We present a global analysis of the Higgs and electroweak sector, in the SMEFT framework and matched to a UV-completion. As the UV-model we use the triplet extension of the electroweak gauge sector. The matching is performed at one loop, employing functional methods. In the SFitter analysis, we pay particular attention to theory uncertainties arising from the matching. Our results highlight the complementarity between SMEFT and model-specific analyses.
We consider a recent proposal by Dimopoulos and Kaplan to embed the electroweak SU(2)_L X U(1)_Y into a larger group SU(3)_W X SU(2) X U(1) at a scale above a TeV. This idea is motivated by the prediction for the weak mixing angle sin^2 theta_W = 1/4, which naturally appears in these models so long as the gauge couplings of the high energy SU(2) and U(1) groups are moderately large. The extended gauge dynamics results in new effective operators that contribute to four-fermion interactions and Z pole observables. We calculate the corrections to these electroweak precision observables and carry …
The scotogenic neutrino seesaw model is a minimal extension of the standard model with three $\mathbb{Z}_2$-odd right-handed singlet fermions $N$ and one $\mathbb{Z}_2$-odd Higgs doublet $η$ that can accommodate the tiny neutrino mass and provide a dark matter candidate in a unified picture. Due to lack of experimental signatures for electroweak scale new physics, it is appealing to assume these new particles are well above the electroweak scale and take the effective field theory approach to study their effects on low energy observables. In this work we apply the recently developed functional…
The standard electroweak theory admits a string solution, the Z string, in which only the electrically neutral Higgs fields are excited. This solution is unstable at zero temperature: Z strings decay by exciting charged Higgs modes. In the early Universe, however, there was a long period during which the Higgs particles were out of equilibrium but the photon field was in thermal equilibrium. We show that in this phase Z strings are stabilized by interactions of the charged Higgs modes with the photons. In a first temperature range immediately below the electroweak symmetry breaking scale, the …
Electroweak Sudakov logarithms at high energy, of the form alpha/(sin^2theta_W)^n log^m s/M_{Z,W}^2, are summed using effective theory methods. The corrections are computed to processes involving two external particles in the standard model. The results include non-zero particle masses, such as the t-quark mass, electroweak mixing effects which lead to unequal W and Z masses, and radiative Higgs corrections proportional to the Yukawa couplings. We show that the matching at the scale M_{W,Z} has a term at most linear in log s/mu^2 to all orders. The effective theory formalism is compared with, …
We study the properties of the electroweak theory in the vicinity of the critical point --- the endpoint of the first order phase transition line that exists in a wide class of theories, including the Standard Model and a part of the parameter space of the Minimal Supersymmetric Standard Model. We find that the critical point belongs to the 3d Ising universality class. The endpoint in the Standard Model is located at the Higgs mass below the existing experimental lower bound, implying that there is no electroweak phase transition in this theory.
We present a renormalization-group (RG) analysis of dark matter interactions with the standard model, where dark matter is allowed to be a component of an electroweak multiplet, and has a mass at or below the electroweak scale. We consider, in addition to the gauge interactions, the complete set of effective operators for dark matter interactions with the standard model above the weak scale, up to and including mass dimension six. We calculate the RG evolution of these operators from the high scale Lambda down to the weak scale, and perform the matching to the tower of effective theories below…
If there are no light Higgs bosons found below $\cal O$(800 GeV) or so, the interactions among longitudinally-polarized vector bosons ($\wl$) will become strong at the TeV region, and new physics that is responsible for the electroweak symmetry breaking must emerge at this energy scale. We discuss the phenomenological prospects of the Strongly-interacting Electro-Weak Sector (SEWS) at future TeV linear colliders and hadronic supercolliders.
In order to extend the Standard Model to TeV scale energies one must address two basic questions: (1) What is the complete description of the effective theory of fundamental particles at and below the electroweak scale? and (2) What is the dynamics responsible for electroweak symmetry breaking? The answers to these questions are crucial for addressing the third outstanding question of particle physics: What are the origins of the Standard Model parameters? We briefly summarize current theoretical approaches to answering these questions.
We present the matching of the next-to-leading order QCD calculation for W+W+jj production via vector-boson fusion in hadronic collisions to parton-shower Monte-Carlo programs according to the POWHEG method. Our implementation complements existing code for QCD-induced W+W+jj production in the POWHEGBOX package, thereby providing a platform for the complete Standard Model production of W+W+jj events via QCD and electroweak interactions. The impact of parton-shower effects is discussed for various distributions and found to be small in most cases. However, few observables, that are relevant for …
An independent result for the two loop fermionic contributions to the muon lifetime in the Standard Model is obtained. Deviations are found with respect to \cite{Freitas:2000gg}, which result in a shift of the W boson mass by approximately -1.3 MeV over the range of Higgs boson masses from 100 GeV to 1 TeV. Supplied with the bosonic contributions from \cite{Awramik:2002wn,Onishchenko:2002ve,Awramik:2002vu}, this shift, due to the complete electroweak contributions, varies from -2.4 MeV to -0.6 MeV. Additionally, a new test of the matching procedure defining the Fermi constant is presented, whi…
Physics beyond the Standard Model, realized above the electroweak scale, can be incorporated in a model independent way in the Wilson coefficients of higher dimensional gauge invariant operators. In these proceedings we review the matching of the $SU(3)_C\times SU(2)_L\times U(1)_Y$ gauge invariant dimension-six operators on the effective Hamiltonian governing $b\to s$ and $b\to c$ transitions, including the leading 1-loop effects.
By using effective field theory techniques for the standard model, we discuss the issue of what $μ$ scale is the appropriate one in the QCD corrections to the large-$\mt$ electroweak contributions to $Δr$. This needs the construction of an effective field theory below the top quark. We argue that, while matching corrections do verify the simple prescription of taking $μ\simeq \mt$ in $\a(\m)$, logarithmic (i.e. $\sim \log \mt$) corrections do not, and require the use of the running $\a(μ)$ in the corresponding renormalization group equation.
We calculate the full two-loop electroweak matching corrections to the operator governing the decay B_q --> l^+ l^- in the Standard Model. Their inclusion removes an electroweak scheme and scale uncertainty of about 7% of the branching ratio. Using different renormalization schemes of the involved electroweak parameters, we estimate residual perturbative electroweak and QED uncertainties to be less than 1% at the level of the branching ratio.
If the Standard Model (SM) is an effective theory, as currently believed, it is valid up to some energy scale $Λ$ to which the Higgs vacuum expectation value is sensitive throughout radiative quadratic terms. The latter ones destabilize the electroweak vacuum and generate the SM hierarchy problem. For a given perturbative Ultraviolet (UV) completion, the SM cutoff can be computed in terms of fundamental parameters. If the UV mass spectrum involves several scales the cutoff is not unique and each SM sector has its own UV cutoff $Λ_i$. We have performed this calculation assuming the Minimal Supe…
I describe, from the bottom up, a sequence of natural effective field theories. Below a TeV we have the minimal standard model with a light Higgs, and an extra neutral scalar. In the 1-10 TeV region these scalars are part of a multiplet of pseudo Nambu-Goldstone Bosons. Interactions with additional TeV mass scalars, gauge bosons, and vector-like charge 2/3 quarks stabilize the Higgs mass squared parameter without finetuning. Electroweak superconductivity may be determined in this effective theory as a UV insensitive vacuum alignment problem. Above the 10 TeV scale we have strongly coupled new …
In the standard model with electroweak symmetry breaking through the Higgs mechanism, electroweak gauge-boson scattering amplitudes are large if the Higgs boson is heavy, and electroweak gauge interactions become strong. In theories with electroweak symmetry breaking through alternative mechanisms, there could be a strongly interacting gauge sector, possibly with resonances in an accessible energy region. In general, the scattering of longitudinally polarized massive gauge bosons can give information on the mechanism of spontaneous symmetry breaking. At energies below the symmetry breaking sca…
If electroweak symmetry breaking is driven by a new strongly-coupled dynamical sector, one expects their bound states to appear at the TeV scale or slightly below. However, electroweak precision data imposes severe constraints on most of the existing models, putting them under strain. Conventional models require the new composite states to come in pairs of rather heavy, close to degenerate spin-1 resonances. In this paper I argue that spin-1 states can actually be lighter without clashing with experimental bounds. As an example, I consider a composite model with a light pseudovector resonance …
Electroweak radiative corrections will play a major role in the analysis of several upcoming ultra-precision experiments such as Belle-II, so is crucial to make sure that they are fully under control. The article outlines the recent developments in the theoretical and computational approaches to one-loop (NLO) electroweak radiative corrections to the parity-violating asymmetry in $e^- e^+ \to μ^- μ^+ (γ)$ process with longitudinally polarized electrons. We derive asymptotic expressions for low and high energy regions (well below or above $Z$-resonance, correspondingly) and analyze the leading …
We complete the list of one loop renormalization group equations and matching conditions relevant for the computation of the electroweak precision parameters $S$ and $T$ in the three site Higgsless model. We obtain one-loop formulas for $S$ and $T$ expressed in terms of physical observables such as the KK gauge boson mass $M_{W'}$, the KK fermion mass $M$, and the KK gauge boson ($W'$) couplings with light quarks and leptons $g_{W'ff}$. It is shown that these physical observables, $M_{W'}$, $M$ and $g_{W'ff}$ are severely constrained by the electroweak precision data. Unlike the tree level ana…
We study analytically and numerically the bounds imposed by the electroweak precision tests on a minimal composite Higgs model. The model is based on spontaneous SO(5)/SO(4) breaking, so that an approximate custodial symmetry is preserved. The Higgs arises as a pseudo-Goldstone boson at a scale below the electroweak symmetry breaking scale. We show that one can satisfy the electroweak precision constraints without much fine-tuning. This is the case if the left-handed top quark is fully composite, which gives a mass spectrum within the reach of the LHC. However a composite top quark is strongly…
