While the anomalous magnetic moments of the electron and the muon have been measured with remarkable precision, the magnetic moment of the $\textit{τ}$-lepton is only known to rather limited precision. A promising approach to measure it exploits $\textit{τ}$ $\textit{τ}$ production in ultraperipheral collisions of lead ions at the LHC. In this article, a state-of-the-art theory prediction for $\textit{τ}$ $\textit{τ}$ production including leptonic $\textit{τ}$-decays is provided. The impact of spin correlations between the $\textit{τ}$-leptons, of the masses of final-state leptons, of next-to-leading-order electroweak corrections, and of the parametrization of the photon flux are discussed.

Germany

Stefan Dittmaier

Tim Engel

José Ariza

Mathieu Pellen

Electroweak corrections to $\textit{τ}$

Radiative corrections in pole approximation, which are based on the leading contribution in a systematic expansion of amplitudes about resonance poles, naturally decompose into factorizable corrections attributed to the production or decay of the resonance and non-factorizable corrections induced by soft photon (or gluon) exchange between those subprocesses. In this paper we complete an earlier calculation of mixed QCD × electroweak corrections of  $$\mathcal{O}\left({\alpha }_{{\text{s}}}\alpha \right)$$  to the neutral-current Drell-Yan cross section in pole approximation by including the previously neglected corrections that are solely related to the Z-boson production process. We present numerical results both for differential distributions and for the forward-backward asymmetry differential in the lepton-pair invariant mass, which is the key observable in the measurement of the effective weak mixing angle at the LHC. Carefully disentangling the various types of factorizable and non-factorizable corrections, we find (as expected in our earlier work) that the by far most important contribution at  $$\mathcal{O}\left({\alpha }_{{\text{s}}}\alpha \right)$$  originates from the interplay of initial-state QCD corrections and electroweak final-state corrections.

CERN

Alexander Huss

Jan Schwarz

Mixed NNLO QCD × electroweak corrections to single-Z production in pole approximation: differential distributions and forward-backward asymmetry

The described Dark Abelian Sector Model (DASM) extends the Standard Model (SM) by a “dark” sector containing a spontaneously broken U(1)d gauge group. Keeping this dark sector quite generic we only add one additional Higgs boson, one Dirac fermion, and right-handed SM-like neutrinos to the SM. Using the only two singlet operators of the SM with dimension less than 4 (the U(1)Y field-strength tensor and the SM Higgs mass operator |Φ|2) as well as the right-handed neutrino fields we open up three portals to the dark sector. Dark sectors, such as the one of the DASM, that introduce an additional Higgs boson H as well as an additional Z′ gauge boson can have a large influence on the predictions for electroweak precision observables and even accommodate possible dark matter candidates. We consider one of the two Higgs bosons to be the known 125 GeV Higgs boson and parameterize the extension of the scalar sector by the mass of the second Higgs boson, the Higgs mixing angle, and a Higgs self-coupling. We do not assume any mass hierarchy in the gauge sector and use the mass of the additional Z′ boson and a corresponding gauge-boson mixing angle to parameterize the extension of the gauge sector. The fermion sector is parameterized by the mass of the additional fermion and a fermion mixing angle. We describe an on-shell as well as an   <math> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\textrm{MS}} $$  renormalization scheme for the DASM sectors and give explicit results for the renormalization constants at the 1-loop level, and, thus, prepare the ground for full NLO predictions for collider observables in the DASM. As a first example, we provide the DASM prediction for the W-boson mass derived from muon decay.

Jonas Rehberg

Heidi Rzehak

Renormalization of a Standard Model extension with a Dark Abelian Sector and predictions for the W-boson mass

In this paper, the existence of a massive dark photon, associated with a new gauge group is considered. The dark photon can be kinetically mixed with the photon. To study some applications, the thermo field dynamics formalism is used. Exploring the topological structure of this approach, the influence of dark photons on the Stefan–Boltzmann law and the Casimir effect at zero and finite temperature is calculated.

Brazil

V. Prata

A. Santos

Canada

Faqir Khanna

Stefan–Boltzmann law and Casimir effect for dark photons

We present a new calculation of next-to-leading-order corrections of the strong and electroweak interactions to like-sign W-boson scattering at the Large Hadron Collider, implemented in the Monte Carlo integrator Bonsay. The calculation includes leptonic decays of the W bosons. It comprises the whole tower of next-to-leading-order contributions to the cross section, which scale like   <math> <msubsup> <mi>α</mi> <mi>s</mi> <mn>3</mn> </msubsup> </math>  $$ {\alpha}_{\textrm{s}}^3 $$  α 4,   <math> <msubsup> <mi>α</mi> <mi>s</mi> <mn>2</mn> </msubsup> </math>  $$ {\alpha}_{\textrm{s}}^2 $$  α 5, α s α 6, and α 7 in the strong and electroweak couplings α s and α. We present a detailed survey of numerical results confirming the occurrence of large pure electroweak corrections of the order of ~ −12% for integrated cross sections and even larger corrections in high-energy tails of distributions. The electroweak corrections account for the major part of the complete next-to-leading-order correction, which amounts to 15–20% in size, depending on the details of the event selection chosen for analysing vector-boson-scattering. Moreover, we compare the full next-to-leading-order corrections to approximate results based on the neglect of contributions that are not enhanced by the vector-boson scattering kinematics (VBS approximation) and on resonance expansions for the W-boson decays (double-pole approximation); the quality of this approximation is good within ≲ 1.5% for integrated cross sections and the dominating parts of the differential distributions. Finally, for the leading-order predictions, we construct different versions of effective vector-boson approximations, which are based on cross-section contributions that are enhanced by collinear emission of W bosons off the initial-state (anti)quarks; in line with previous findings in the literature, it turns out that the approximative quality is rather limited for applications at the LHC.

Philipp Maierhöfer

Christopher Schwan

Belgium

Ramon Winterhalder

Like-sign W-boson scattering at the LHC — approximations and full next-to-leading-order predictions

A recently proposed scheme for a gauge-invariant treatment of tadpole corrections in spontaneously broken gauge theories — called Gauge-Invariant Vacuum expectation value Scheme (GIVS) — is applied to a singlet Higgs extension of the Standard Model and to the Two-Higgs Doublet Model. In contrast to previously used tadpole schemes, the GIVS unifies the gauge-invariance property with perturbative stability. For the Standard Model this was demonstrated for the conversion between on-shell and   <math> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\mathrm{MS}} $$  renormalized masses, where the GIVS leads to very moderate, gauge-independent electroweak corrections. In models with extended scalar sectors, issues with tadpole renormalization exist if Higgs mixing angles are renormalized with   <math> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\mathrm{MS}} $$  conditions, which is the major subject of this article. In detail, we first formulate non-linear representations of the extended scalar sectors, which is an interesting subject in its own right. Then we formulate the GIVS which employs these non-linear representations in the calculation of the tadpole renormalization constants, while actual higher-order calculations in the GIVS proceed in linear representations as usual. Finally, for the considered models we discuss the next-to-leading-order (electroweak and QCD) corrections to the decay processes h/H → WW/ZZ → 4 fermions of the CP-even neutral Higgs bosons h and H using   <math> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\mathrm{MS}} $$ -renormalized Higgs mixing angles with the GIVS and previously used tadpole treatments.

Electroweak renormalization based on gauge-invariant vacuum expectation values of non-linear Higgs representations. Part II. Extended Higgs sectors

The renormalization of vacuum expectation value parameters, such as v in the Standard Model (SM), is an important ingredient in electroweak renormalization, where this issue is connected to the treatment of tadpoles. Tadpole counterterms can be generated in two different ways in the Lagrangian: in the course of parameter renormalization, or alternatively via Higgs field redefinitions. The former typically leads to small corrections originating from tadpoles, but in general suffers from gauge dependences if   <math> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\mathrm{MS}} $$  renormalization conditions are used for mass parameters. The latter is free from gauge dependences, but is prone to very large corrections in   <math> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\mathrm{MS}} $$  schemes, jeopardizing perturbative stability in predictions. In this paper we propose a new scheme for tadpole renormalization, dubbed Gauge-Invariant Vacuum expectation value Scheme (GIVS), which is a hybrid scheme of the two mentioned types, with the benefits of being gauge independent and perturbatively stable. The GIVS is based on the gauge-invariance property of Higgs fields, and the corresponding parameters like v, in non-linear representations of Higgs multiplets. We demonstrate the perturbative stability of the GIVS in the SM by discussing the conversion between on-shell and   <math> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\mathrm{MS}} $$  renormalized masses.

Electroweak renormalization based on gauge-invariant vacuum expectation values of non-linear Higgs representations. Part I. Standard Model

Building on an older method used to derive non-decoupling effects of a heavy Higgs boson in the Standard Model, we describe a general procedure to integrate out heavy fields in the path integral. The derivation of the corresponding effective Lagrangian including the one-loop contributions of the heavy particle(s) is particularly transparent, flexible, and algorithmic. The background-field formalism allows for a clear separation of tree-level and one-loop effects involving the heavy fields. Using expansion by regions the one-loop effects are further split into contributions from large and small momentum modes. The former are contained in Wilson coefficients of effective operators, the latter are reproduced by one-loop diagrams involving effective tree-level couplings. The method is illustrated by calculating potential non-decoupling effects of a heavy Higgs boson in a singlet Higgs extension of the Standard Model. In particular, we work in a field basis corresponding to mass eigenstates and properly take into account non-vanishing mixing between the two Higgs fields of the model. We also show that a proper choice of renormalization scheme for the non-standard sector of the underlying full theory is crucial for the construction of a consistent effective field theory.

Sebastian Schuhmacher

Maximilian Stahlhofen

Integrating out heavy fields in the path integral using the background-field method: general formalism

The electric charge renormalization constant, as defined in the Thomson limit, is expressed in terms of self-energies of the photon–Z-boson system in an arbitrary <math><msub><mi>R</mi><mi>ξ</mi></msub></math> gauge to all perturbative orders. The derivation as carried out in the Standard Model holds in all spontaneously broken gauge theories with the <math><mrow><mrow><msub><mrow><mi>SU</mi><mo>(</mo><mn>2</mn><mo>)</mo></mrow><mrow><mi>w</mi></mrow></msub><mo>×</mo></mrow><msub><mrow><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>Y</mi></mrow></msub></mrow></math> gauge group in the electroweak sector and is based on the application of charge universality to a fake fermion with infinitesimal weak hypercharge and vanishing weak isospin, which effectively decouples from all other particles. Charge universality, for instance, follows from the known universal form of the charge renormalization constant as derived within the background-field formalism. Finally, we have generalized the described procedure to gauge theories with gauge group <math><mrow><msub><mrow><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mi>Y</mi></msub><mo>×</mo><mi>G</mi></mrow></math> with any Lie group <math><mi>G</mi></math>, only assuming that electromagnetic gauge symmetry is unbroken and mixes with <math><mrow><msub><mrow><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mi>Y</mi></msub></mrow></math> transformations in a nontrivial way.

Electric charge renormalization to all orders

First results on the radiative corrections of order   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (N  f  α s α) are presented for the off-shell production of W or Z bosons at the LHC, where N  f  is the number of fermion flavours. These corrections comprise all diagrams at   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (α s α) with closed fermion loops, form a gauge-invariant part of the next-to-next-to-leading-order corrections of mixed QCD×electroweak type, and are the ones that concern the issue of mass renormalization of the W and Z resonances. The occurring irreducible two-loop diagrams, which involve only self-energy insertions, are calculated with current standard techniques, and explicit analytical results on the electroweak gauge-boson self-energies at   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (α s α) are given. Moreover, the generalization of the complex-mass scheme for a gauge-invariant treatment of the W/Z resonances is described for the order   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (α s α). While the corrections, which are implemented in the Monte Carlo program Rady, are negligible for observables that are dominated by resonant W/Z bosons, they affect invariant-mass distributions at the level of up to 2% for invariant masses of ≳ 500 GeV and are, thus, phenomenologically relevant. The impact on transverse-momentum distributions is similar, taking into account that leading-order predictions to those distributions underestimate the spectrum.