Future electron-position colliders, such as CEPC and FCC-ee, have the capability to dramatically improve the experimental precision for W and Z-boson masses and couplings. This would enable indirect probes of physics beyond the Standard Model at multi-TeV scales. For this purpose, one must complement the experimental measurements with equally precise calculations for the theoretical predictions of these quantities within the Standard Model, including three-loop electroweak corrections. This article reports on the calculation of a subset of these corrections, stemming from diagrams with three closed fermion loops to the following quantities: the prediction of the W-boson mass from the Fermi constant, the effective weak mixing angle, and partial and total widths of the Z boson. The numerical size of these corrections is relatively modest, but non-negligible compared to the precision targets of future colliders. In passing, an error is identified in previous results for the two-loop corrections to the Z width, with a small yet non-zero numerical impact.
In this proceeding, we highlight the computation of leading fermionic three-loop corrections to electroweak precision observables (EWPOs) accomplished recently. We summarize the numerical analysis and provide an outlook.
We present the results from a calculation of the full electroweak two-loop fermionic contributions to the effective leptonic mixing angle of the $Z$ boson, $\sin^2θ_{eff}$, in the Standard Model. On-shell renormalization and analytic calculations are performed for the three-point vertex functions at zero external momenta, whereas irreducible three-point integrals for non-vanishing external momenta are evaluated semi-analytically applying two different methods. Comparisons with a previous calculation show complete agreement.
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…
We give some details of the recently completed calculation of the full two-loop fermionic corrections to the effective leptonic weak mixing angle. Among others, we describe the C++ library DiaGen/IdSolver, which was used to reduce the two-loop light fermion vertex diagrams to linear combinations of master integrals with rational function coefficients.
We show that, unlike mass matrices, the fermionic gauge currents of the Standard Model exhibit, at the quantum level, remarkable SU(2)_f flavor properties at the observed values of the mixing angles. They accommodate all measured mixing for three families of quarks, and, for neutrinos, maximal theta_{23}, quark-lepton complementarity tan(2 theta_c)=1/2 <--> \tan (2 theta_{12})=2, and a not so small sin^2(2 theta_{13}) = .267 within the present 90% c.l. interval of the T2K experiment.
We discuss the next to the leading order (NLO) electroweak radiative corrections to the $e^- e^+ \rightarrow f^- f^+ (γ)$ cross section asymmetry, for polarized and unpolarized beam scenario. The left-right and forward-backward amplitudes, with and without radiative corrections, are evaluated and compared for various kinematics. The hard bremsstrahlung is included for arbitrary energy cuts. The radiative corrections are shown to be significant and having a non-trivial dependency on the kinematic conditions. The calculations are relevant for the ultra-precise low-energy experiment Belle~II plan…
Improved predictions for the Z-boson decay width and the hadronic Z-peak cross-section within the Standard Model are presented, based on a complete calculation of electroweak two-loop corrections with closed fermion loops. Compared to previous partial results, the predictions for the Z width and hadronic cross-section shift by about 0.6 MeV and 0.004 nb, respectively. Compact parametrization formulae are provided, which approximate the full results to better than 4 ppm.
We investigate catalysis of electroweak baryogenesis by fermionic Higgs portal dark matter using a two Higgs doublet model augmented by vector-like fermions. The lightest neutral fermion mass eigenstate provides a viable dark matter candidate in the presence of a stabilizing symmetry Z_2 or gauged U(1)_D symmetry. Allowing for a non-vanishing CP-violating phase in the lowest-dimension Higgs portal dark matter interactions allows generation of the observed dark matter relic density while evading direct detection bounds. The same phase provides a source for electroweak baryogenesis. We show that…
We present the first calculation of the two-loop electroweak fermionic correction to the flavour-dependent effective weak-mixing angle for bottom quarks, sin^2 theta_{eff}^{b anti-b}. For the evaluation of the missing two-loop vertex diagrams, two methods are employed, one based on a semi-numerical Bernstein-Tkachov algorithm and the second on asymptotic expansions in the large top-quark mass. A third method based on dispersion relations is used for checking the basic loop integrals. We find that for small Higgs-boson mass values, M_H ~ 100 GeV, the correction is sizable, of order O(10^{-4}).
One-loop radiative corrections to the charged Higgs coupling to two fermions and to the charged Higgs fermionic decay are studied. A renormalization scheme is proposed including the definition of the parameter $\tanβ$ through the coupling of the CP-odd Higgs boson to a charged lepton pair. Loops including top and bottom quarks and squarks are considered using the unitary gauge. Large corrections to the coupling are found when there is a substantial mixing in the squark mass matrix, but these corrections are suprress in the decay rate. Interesting applications to rare processes are discussed.
We present a systematic determination of the order hbar fermionic energy shift when an electroweak string is perturbed. We show that the combined effect of zero modes, bound states and continuum states is to lower the total fermionic ground state energy of the string when the Higgs instability of the string is excited. The effect of the Dirac sea is thus to destabilise the string. However, this effect can be offset by populating positive energy states. Fermions enhance the stability of an electroweak string with sufficiently populated fermionic bound states.
The result for the Higgs-dependent electroweak two-loop bosonic contributions to the effective leptonic mixing angle of the $Z$-boson in the Standard Model is presented. Together with the previously calculated fermionic contributions it yields the complete dependence of $\sin^2θ_{\rm eff}$ on the Higgs-boson mass $M_H$. Compared to the fermionic contributions, the bosonic contributions are found to be smaller and have the opposite sign, compensating part of the fermionic contributions.
Extensions of the Standard Model often come with additional, possibly electroweakly charged Higgs states, the prototypal example being the Two-Higgs-Doublet Model. While collider phenomenology does not exclude the possibility for some of these new scalar fields to be light, it is relatively natural to consider masses in the multi-TeV range, in which case the only remaining light Higgs boson automatically receives SM-like properties. The appearance of a hierarchy between the new-physics states and the electroweak scale then leads to sizable electroweak corrections, e. g. in the decays of the he…
We consider two-loop leading and next-to-leading logarithmic virtual corrections to arbitrary processes with external massless fermions in the electroweak Standard Model at energies well above the electroweak scale. Using the sector-decomposition method and alternatively the strategy of regions we calculate the mass singularities that arise as logarithms of Q^2/MW^2, where Q is the energy scale of the considered process, and 1/εpoles in D=4-2εdimensions, to one- and two-loop next-to-leading logarithmic accuracy. The derivations are performed within the complete electroweak theory with spontane…
Measurements of electroweak precision observables at future electron-positron colliders, such as the CEPC, FCC-ee, and ILC, will be sensitive to physics at multi-TeV scales. To achieve this sensitivity, precise predictions for the Standard Model expectations of these observables are needed, including corrections at the three- and four-loop level. In this article, results are presented for the calculation of a subset of three-loop mixed electroweak-QCD corrections, stemming from diagrams with a gluon exchange and two closed fermion loops. The numerical impact of these corrections is illustrated…
We consider a model with a gauge singlet Dirac fermion as a cold dark matter candidate. The dark matter particle communicates with the Standard Model via a gauge singlet scalar mediator that couples to the Higgs. The scalar mediator also serves to create a tree-level barrier in the scalar potential which leads to a strongly first order electroweak phase transition as required for Electroweak Baryogenesis. We find a large number of models that can account for all the dark matter and provide a strong phase transition while avoiding constraints from dark matter direct detection, electroweak preci…
We present the minimal model of electroweak baryogenesis induced by fermions. The model consists of an extension of the Standard Model with one electroweak singlet fermion and one pair of vector like doublet fermions with renormalizable couplings to the Higgs. A strong first order phase transition is radiatively induced by the singlet-doublet fermions, while the origin of the baryon asymmetry is due to asymmetric reflection of the same set of fermions on the expanding electroweak bubble wall. The singlet-doublet fermions are stabilized at the electroweak scale by chiral symmetries and the Higg…
Future electroweak precision measurements in the Circular Electron Positron Collider (CEPC) project would significantly improve the precision of electroweak oblique parameters. We evaluate the expected precision through global fits, and study the corresponding sensitivity to weakly interacting fermionic dark matter. Three models with electroweak multiplets in the dark sector are investigated as illuminating examples. We find that the CEPC data can probe up to TeV scales and explore some regions where direct detection cannot reach, especially when the models respect the custodial symmetry.
We present a complete calculation of the contributions to the effective leptonic weak mixing angle, sin^2theta_eff, generated by closed fermion loops at the two-loop level of the electroweak interactions. This quantity is the source of the most stringent bound on the mass, M_H, of the only undiscovered particle of the Standard Model, the Higgs boson. The size of the corrections with respect to known partial results varies between -4*10^-5 and -8*10^-5 for a realistic range of M_H from 100 to 300 GeV. This translates into a shift of the predicted (from sin^2theta_eff alone) central value of M_H…
The two-loop electroweak corrections to the anomalous magnetic moment of the muon, generated by fermionic loops, are calculated. An interesting role of the top quark in the anomaly cancellation is observed. New corrections, including terms of order $G_μαm_t^2$, are computed and a class of diagrams previously thought to vanish are found to be important. The total fermionic correction is $-(23\pm 3) \times 10^{-11}$ which decreases the electroweak effects on $g-2$, predicted from one-loop calculations, by 12\%. We give an updated theoretical prediction for $g-2$ of the muon.
Recently, the calculation of fermionic electroweak two-loop corrections to the total width of the Z boson and hadronic Z-peak cross-section in the Standard Model has been presented in arXiv:1310.2256, where "fermionic" refers to diagrams with closed fermion loops. Here, these results are complemented by presenting contributions of the same order for the Z-boson partial widths, which are the last missing pieces for a complete description of Z-pole physics at the fermionic two-loop order. The definition of the relevant observables and the calculational techniques are described in detail. Numeric…
We present the constraints on the parameter space of the Lee-Wick Standard Model coming from electroweak precision observables. The model predicts a large positive S and a negative T. We show that it is possible to find some regions in parameter space with a fermionic state as light as 2.4-3.5 TeV. We also propose a simple extension of the model including a fourth generation. In this case it is possible to pass the electroweak constraints with Lee-Wick fermionic masses of order 0.4-1.5 TeV and Lee-Wick gauge masses of order 3 TeV.
The role of the top quark in the Standard Model predictions for electroweak precision observables is reviewed and the implications of the experimental data for the indirect determination of the top mass range is discussed.
Phenomenologically relevant electroweak precision pseudo-observables related to Z-boson physics are discussed in the context of the strong experimental demands of future $e^+e^-$ colliders. The recent completion of two-loop Z-boson results is summarized and a prospect for the 3-loop Standard Model calculation of the Z-boson decay pseudo-observable is given.
One of the most precise measurements of the strong coupling constant alpha_s(MZ) is obtained in the context of global analyses of precision electroweak data. This article reviews the sensitivity of different electroweak observables to alpha_s and describes the perturbative uncertainties related to missing higher orders. The complete renormalisation scale dependence for the relevant observables is calculated at next-to-next-to-leading order and a new method is presented to determine the corresponding perturbative uncertainty for measurements of alpha_s based on these observables.
The Large N expansion in gauge models with an NJL mechanism is discussed in view of the resulting fermionic mass matrix. A hidden symmetry model is introduced.
A critical assessment is given of the comparison between the new versions of the programs TOPAZ0 40i and ZFITTER 510. The relevance for precision calculations around the $\zb$ resonance is briefly discussed. << more references added >>
Precise measurements of the interactions of electroweak vector bosons at the ILC yield information about the physics of electroweak symmetry breaking. In order to combine this with the possible observation of new resonances or the effects of new strong interactions at the LHC, we need to relate resonance parameters with low-energy observables. We derive these relations for a generic setup and draw conclusions about the new-physics reach of the ILC in the electroweak sector.
I summarise the virtual effects of the new particles predicted by supersymmetric extensions of the Standard Model on the high-precision electroweak observables measured at LEP/SLC, the Tevatron and CLEO. I will then discuss in some details the two-loop SUSY-QCD corrections to the $ρ$ parameter.
The calculation of electroweak precision observables in the MSSM is reviewed. The description of the 1997 data and the results of updated global fits are discussed in comparison with the Standard Model.
Naturalness demands that the quadratic divergence of the one-loop top contribution to the Higgs mass be cancelled at a scale below 1 TeV. This can be achieved by introducing a fermionic (spin-1/2) top partner, as in, for example, Little Higgs models. In this paper, we study the phenomenology of a simple model realizing this mechanism. We present the current bounds on the model from precision electroweak fits, flavor physics, and direct searches at the LHC. The lower bound on the top partner mass from precision electroweak data is approximately 500 GeV, while the LHC bound with 5/fb of data at …
The processes of the single W-production in hadron-hadron collisions are suggested for investigation of the nucleon spin. An approach is proposed for the determination of quark spin densities at low x. The lowest order electroweak radiative corrections to the observable quantities are calculated. The numerical calculations of the cross sections and the single spin asymmetries taking into consideration the electroweak corrections at RHIC energies have been made.
The predictions for electroweak observables following from Higgs-Free Model for Fundamental Interactions are derived. It is shown that these predictions are close to the Standard Model predictions and they are in a surprising agreement with the experimental data. The analysis of electroweak observables at low and high energy suggests that the Higgs mass $m_H$ is just the ultraviolet cutoff which increases if the process energy scale increases. We propose several experimental tests which can discriminate between the Standard and the Higgs-Free Models.
Warped extra dimensions provide a new playground to study electroweak brea-king and the nature of the higgs field. In this paper we reanalyze the electroweak observables in theories with one extra dimension and a completely general warp factor. We demonstrate that, regardless of the 5D background, SU(2)_R is needed in order to avoid excessive contributions to the T parameter. For higgsless theories cancelations between different contributions to the S parameter are needed.
Electroweak precision observables (EWPO) measured at the W and Z poles provide stringent limits on possible beyond the Standard Model physics scenarios. In an effective field theory (EFT) framework, the next-to-leading order QCD and electroweak results for EWPO yield indirect limits on possible 4-fermion operators that do not contribute to the observables at tree level. Here we calculate the next-to-leading corrections to EWPO induced by flavor non-universal 4-fermion interactions and find that the extracted limits on EFT coefficients have a strong dependence on the flavor structure of the 4-f…
We study vertex corrections to the leptonic electroweak observables in the general MSSM at tan(beta) < 35. In particular, we address the question of whether supersymmetry can be responsible for the observed 2 sigma deviation from the Standard Model prediction in the invisible width of the Z. We find that the presence of a light (around 100 GeV) chargino and sleptons hinted by the muon g-2 measurements makes the agreement with experiment slightly better and improves the electroweak fit.
We analyze the bounds on the Higgs pseudo-observables following from electroweak constraints, under the assumption that the Higgs particle is the massive excitation of an $SU(2)_L$ doublet. Using such bounds, detailed predictions for $h\to 4\ell$ decay rates, dilepton spectra, and lepton-universality ratios are presented.
We present the dominant two-loop ${\rm O}\left(G_F^2M_H^4\right)$ electroweak corrections to the fermi\-onic decay widths of a high-mass Higgs boson in the Standard Model. The corrections are negative and quite significant, and are larger in magnitude than the one-loop electroweak corrections for $M_H\gsim 400\ {\rm GeV}$. This indicates the onset of a breakdown of perturbation theory in the Higgs sector of the Standard Model at this surprisingly low value of the Higgs-boson mass.
We compute the next-to-leading order QCD and electroweak corrections to $Z$ and $W$ pole observables using the dimension-6 Standard Model effective field theory and present numerical results that can easily be included in global fitting programs. Limits on SMEFT coefficient functions are presented at leading order and at next-to-leading order under several assumptions.
The presence of an extra dimension of size R \approx TeV^{-1} introduces a tower of Kaluza-Klein gauge boson excitations that affects the standard model (SM) relations between electroweak observables. The mixing of the W and Z bosons with their excitations changes their masses and couplings to fermions. This effect depends on the Higgs field, which may live in the bulk of the extra dimension, on its boundary, or may be a combination of both types of fields. We use high-precision electroweak data to constrain 1/R. We find limits from 1 to 3 TeV from different observables, with a model independe…
The leading electroweak MSSM two-loop corrections to the electroweak precision observables are calculated. They are obtained by evaluating the two-loop O(α_t^2), O(α_t α_b), O(α_b^2) contributions to the quantity Δρin the limit of heavy scalar quarks, i.e. we consider the contributions of a Two-Higgs-Doublet model with MSSM restrictions. The full analytic result for arbitrary values of the lightest CP-even Higgs boson mass is presented. The numerical effects of the leading electroweak MSSM two-loop corrections on the precision observables M_W and sin^2(theta_eff) are analyzed. The electroweak …
We briefly review recent progress in the analysis of the Higgs sector of the Minimal R-symmetric Supersymmetric Standard Model. Importance of the interplay between W and Higgs boson masses in constraining the parameter space of the model is shown.
We perform the fit of electroweak precision observables within the Standard Model with a 126 GeV Higgs boson, compare the results with the theoretical predictions and discuss the impact of recent experimental and theoretical improvements. We introduce New Physics contributions in a model-independent way and fit for the S, T and U parameters, for the $ε_{1,2,3,b}$ ones, for modified $Zb\bar{b}$ couplings and for a modified Higgs coupling to vector bosons. We point out that composite Higgs models are very strongly constrained. Finally, we compute the bounds on dimension-six operators relevant fo…
Is the weak scale natural? This ever pending question makes the search for new particle production a highly motivated primary goal of the next LHC phase. These searches may or may not be successful. While waiting for a needed higher energy collider to extend the direct exploration, the search for signs of new physics might be confined to indirect tests for quite some time. In a few fully calculable models, weakly or semi-strongly interacting, we compare the significance to measure the Higgs couplings versus the electroweak observables.
We have calculated the complete electroweak O(alpha) radiative corrections to the single Higgs-boson production processes e+ e- --> nu_l anti-nu_l H (l=e,mu,tau) in the electroweak Standard Model. Initial-state radiation beyond O(alpha) is included in the structure-function approach. The calculation of the corrections is briefly described, and numerical results are presented for the total cross section. In the G_mu scheme, the bulk of the corrections is due to initial-state radiation, which affects the cross section at the level of -7% at high energies and even more in the ZH threshold region.…
The complete two-loop electroweak fermionic corrections to the effective leptonic weak mixing angle, sin^2(theta_eff), are now available. Here we shortly present the methods applied and illustrate the implications on indirect prediction for the Higgs boson mass within the standard model.
In these proceedings we briefly review the basic concepts underlying indirect bounds on the Wilson coefficients of some Standard Model dimension-6 operators, relevant to electroweak and Higgs observables, obtained via renormalization group (RG) mixing to strongly constrained observables. With the present data we derive RG-induced bounds, stronger than the direct constraints, on some Higgs couplings and anomalous triple gauge couplings. Any deviation from these bounds would suggest a particular pattern of correlations among the Wilson coefficients, thus offering a new window on the new physics …
Indirect information about the possible scale of supersymmetry (SUSY) breaking is provided by B physics observables (BPO) as well as electroweak precision observables (EWPO). We review the combination of the constraints imposed by recent measurements of the BPO BR(b -> s gamma), BR(B_s -> mu+ mu-), BR(B_u -> tau nu_tau) and Delta M_{B_s} with those obtained from the experimental measurements of the EWPO M_W, sin^2 theta_eff, Gamma_Z, (g-2)_mu and M_h. We perform a chi^2 fit to the parameters of the constrained minimal supersymmetric extension of the Standard Model (CMSSM), in which the SUSY-br…
High energy observables are characterized by large electroweak radiative corrections of infrared origin; double logarithms are present even for inclusive cross sections, thus violating the Bloch Nordsieck theorem. This effect, related to the initial states carrying nonabelian isospin charges, is here investigated for {\sl any} inclusive cross section in the SU(2)$\otimes$U(1) symmetric limit, that is appropriate for energies much higher than the weak scale. We develop a general formalism allowing to calculate the all order double log resummed cross sections once the hard (tree level) ones are …
Little Higgs theories, in which the Higgs particle is realized as the pseudo-Goldstone boson of an approximate global chiral symmetry have generated much interest as possible alternatives to weak scale supersymmetry. In this paper we analyze precision electroweak observables in the Minimal Moose model and find that in order to be consistent with current experimental bounds, the gauge structure of this theory needs to be modified. We then look for viable regions of parameter space in the modified theory by calculating the various contributions to the S and T parameters.
We calculate the contributions to electroweak precision observables (EWPOs) due to a fourth generation of fermions with the most general (quark-)flavour structure (but assuming Dirac neutrinos and a trivial flavour structure in the lepton sector). The new-physics contributions to the EWPOs are calculated at one-loop order using automated tools (FeynArts/FormCalc). No further approximations are made in our calculation. We discuss the size of non-oblique contributions arising from Z--quark--anti-quark vertex corrections and the dependence of the EWPOs on all CKM mixing angles involving the fourt…
The progress in the evaluation of the muon decay lifetime is reviewed. The electroweak bosonic corrections are given together with the respective shift of the W boson mass. After inclusion of the recently corrected fermionic contributions, this requires an updated fitting formula for the prediction of MW.
We extend the Standard Model (SM) by adding a pair of fermionic SU(2)-doublets with opposite hypercharge and a fermionic SU(2)-triplet with zero hypercharge. We impose a discrete Z_2-symmetry that distinguishes the SM fermions from the new ones. Then, gauge invariance allows for two renormalizable Yukawa couplings between the new fermions and the SM Higgs field, as well as for direct masses for the doublet (M_D) and the triplet (M_T). After electroweak symmetry breaking, this model contains, in addition to SM particles, two charged Dirac fermions and a set of three neutral Majorana fermions, t…
Higher-order results for electroweak precision observables in the Standard Model are analyzed in view of the experimental accuracies achievable at the present and the next generation of colliders, and the indirect prediction of the Higgs-boson mass from the precision data is discussed. Within the MSSM, two-loop results for the lightest CP-even Higgs-boson mass are confronted with the exclusion limits from LEP. Possible precision tests of the MSSM at a future linear collider are furthermore investigated.
We study the signatures of the triplet-quadruplet dark matter model at the LHC and future colliders, including the 100 TeV Super Proton-Proton Collider and the 240 GeV Circular Electron Positron Collider. The dark sector in this model contains one fermionic electroweak triplet and two fermionic quadruplets, which have two kinds of Yukawa couplings to the Higgs doublet. Electroweak production signals of the dark sector fermions in the $\text{monojet}+/\!\!\!\!E_\mathrm{T}$, disappearing track, and $\text{multilepton}+/\!\!\!\!E_\mathrm{T}$ channels at the LHC and the Super Proton-Proton Collide…
