Boson Pair Research Articles

Electroweak W+W+ production in association with three jets at NLO QCD matched with parton shower

Abstract We present a next-to-leading order QCD calculation for the electroweak production of a pair of same-sign weak gauge bosons in association with three jets in the region dominated by vector-boson scattering, and its matching with parton-shower programs according to the POWHEG formalism. Our calculation allows for an accurate description of jet observables crucial for the optimization of experimental analyses of vector boson scattering processes. We find QCD corrections and parton-shower effects moderate in size for inclusive quantities and distributions of the tagging jets. A larger impact of these corrections is observed for subleading jets.

Investigating the excitation function of HBT radii for Lévy-stable sources

Abstract Contemporary heavy-ion physics research aims to explore the phase diagram of strongly interacting matter and search for signs of the possible critical endpoint on the QCD phase diagram. Femtoscopy is among the important tools used for this endeavor; there have been indications that combinations of femtoscopic radii parameters (referred to as HBT radii for identical boson pairs) can be related to the system's emission duration. An apparent non-monotonic behavior in their excitation function thus might signal the location of the critical point. In this paper, we show that conclusions drawn from the results obtained with a Gaussian approximation for the pion source shape might be altered if one utilizes a more general Lévy-stable source description. We find that the characteristic size of the pion source function is strongly connected to the shape of the source and its possible power-law behavior. Taking this into account properly changes the observed behavior of the excitation function.

Extending hh→bb̄bb̄ searches into the HL-LHC era

Searches for pairs of Higgs bosons are the best tools to precisely measure the Higgs boson self-coupling [Formula: see text] in future colliders. We study various strategies for the [Formula: see text] search in the HL-LHC era with focus on constraining [Formula: see text]. We implement a machine-learning-based approach to separate signal and background and apply recent advances in machine learning interpretability, compare the traditional 4 b-jet reconstruction to final states with 1 or 2 large-radius jets, and test scenarios with different top-quark Yukawa couplings, among other factors. This talk was presented as part of the Mini-workshop on Machine Learning at the 10th International Conference on New Frontiers in Physics (ICNFP-2021). a

Improved constraints on Higgs boson self-couplings with quartic and cubic power dependencies in the cross section

Abstract The precise determination of the Higgs boson self-couplings is essential for understanding the mechanism behind electroweak symmetry breaking. However, due to the limited number of Higgs boson pair events at the LHC, only loose constraints have been established so far. 
Current constraints are based on the assumption that the cross section is a quadratic function of the trilinear Higgs self-coupling within the $\kappa$ framework. Incorporating higher-order quantum corrections from virtual Higgs bosons would significantly alter this function form, introducing new quartic and cubic power dependencies on the trilinear Higgs self-coupling. 
To derive this new function form, we propose a specialized renormalization procedure that tracks all Higgs self-couplings at each calculation step. 
Additionally, we introduce renormalization constants for the coupling modifiers within the $\kappa$ framework to ensure the cancellation of all ultraviolet divergences.
With the new function forms of the cross sections in both the gluon-gluon fusion and vector boson fusion channels,
the upper limit of $\kappa_{\lambda_{\rm 3H}}=\lambda_{\rm 3H}/\lambda_{\rm 3H}^{\rm SM}$ by the ATLAS (CMS) collaboration is reduced from 6.6 (6.49) to 5.4 (5.37).
However, extracting a meaningful constraint on the quartic Higgs self-coupling $\lambda_{\rm 4H}$ from Higgs boson pair production data remains challenging.
We also present the invariant mass distributions of the Higgs boson pair at different values of $\kappa_{\lambda}$, which could aid in setting optimal cuts for experimental analysis.Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Science and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.

Electroweak corrections to Higgs boson pair production: the top-Yukawa and self-coupling contributions

We present results for the Yukawa-enhanced and Higgs self-coupling type electroweak corrections to di-Higgs production in gluon fusion. The calculation of the corresponding four-scale, two-loop amplitude is carried out retaining the exact symbolic dependence on all masses and scales during the reduction to master integrals. The resulting integrals are then evaluated at high precision using both the series expansion of the differential equations and sector decomposition. Differential cross sections for the di-Higgs invariant mass and the transverse momentum of a Higgs boson are shown, where we find that the corrections are most pronounced at low invariant mass and transverse momentum.

Analytic amplitudes for a pair of Higgs bosons in association with three partons

The pair production of Higgs bosons at the LHC can give information about the triple Higgs boson coupling. We perform an analytic one-loop calculation of the amplitudes for a pair of Higgs bosons in association with three partons, retaining the exact dependence on the quark mass circulating in the loop. These amplitudes constitute the real radiation corrections in the calculation of Higgs boson pair production at next-to-leading order in the strong coupling. The results of an analytic generalised-unitarity computation are simplified via analytic reconstruction in spinor variables. Compact ansätze for kinematic pole residues are iteratively fitted via p-adic evaluations near said poles and subtracted until no pole remains. A new ansatz construction is introduced to minimally parametrise coefficients of amplitudes with multiple massive external legs. The simplified expressions are faster to evaluate than automatic codes and can lead to more stable results near singular regions.

Testing Bell inequalities and probing quantum entanglement at a muon collider

A muon collider represents a promising candidate for the next generation of particle physics experiments after the expected end of LHC operations in the early 2040s. Rare or hard-to-detect processes at the LHC, such as the production of multiple gauge bosons, become accessible at a TeV muon collider. We present here the prospects of detecting quantum entanglement and the violation of Bell inequalities in H → ZZ → 4ℓ events at a potential future muon collider. We show that the spin density matrix of the Z boson pairs can be reconstructed using the kinematics of the charged leptons from the Z boson decays. Once the density matrix is determined, it is straightforward to obtain the expectation values of various Bell operators and test the quantum entanglement between the Z boson pair. Through a detailed study based on Monte-Carlo simulation, we show that the generalized CGLMP inequality can be maximally violated, and testing Bell inequalities could be established with high significance.

CP-sensitive simplified template cross-sections for tt¯H

The CP structure of the Higgs boson is a fundamental property which has not yet been constrained with high precision. CP violation in the Yukawa coupling between the Higgs boson and top quark pair can be probed directly at the Large Hadron Collider by measuring top-quark-associated Higgs production. Multivariate analysis techniques commonly developed so far by the experiments are designed for a specific signal model and, therefore, complicate reinterpretations and statistical combinations. With this motivation in mind, we propose a CP-sensitive extension of the simplified template cross-section (STXS) framework. Considering multiple Higgs decay channels, we perform an in-depth comparison of CP-sensitive observables and combinations thereof. Our resulting proposal is to extend the existing binning in the transverse momentum of the Higgs boson pT,H by either the pseudorapidity difference of the two top-quarks ∆ηtt¯, or a variable that is based on the top quark momenta, namely b2 or the Collins-Soper angle |cosθ∗|. We demonstrate that this variable selection provides close to optimal sensitivity to the CP mixture in the top Yukawa coupling for an integrated luminosity of 300 fb−1, by comparing it to the results of a multivariate analysis. Our results also suggest a benefit of the two-dimensional STXS extension at 3000 fb−1.

Effective field theory descriptions of Higgs boson pair production

Higgs boson pair production is traditionally considered to be of particular interest for a measurement of the trilinear Higgs self-coupling. Yet it can offer insights into other couplings as well, since – in an effective field theory (EFT) parameterisation of potential new physics – both the production cross section and kinematical properties of the Higgs boson pair depend on various other Wilson coefficients of EFT operators. This note summarises the ongoing efforts related to the development of EFT tools for Higgs boson pair production in gluon fusion, and provides recommendations for the use of distinct EFT parameterisations in the Higgs boson pair production process. This document also outlines where further efforts are needed and provides a detailed analysis of theoretical uncertainties. Additionally, benchmark scenarios are updated. We also re-derive a parameterisation of the next-to-leading order (NLO) QCD corrections in terms of the EFT Wilson coefficients both for the total cross section and the distribution in the invariant mass of the Higgs boson pair, providing for the first time also the covariance matrix. A reweighting procedure making use of the newly derived coefficients is validated, which can be used to significantly speed up experimental analyses.

Search for Higgs boson pair production with one associated vector boson in proton-proton collisions at s = 13 TeV

A search for Higgs boson pair (HH) production in association with a vector boson V (W or Z boson) is presented. The search is based on proton-proton collision data at a center-of-mass energy of 13 TeV, collected with the CMS detector at the LHC, corresponding to an integrated luminosity of 138 fb−1. Both hadronic and leptonic decays of V bosons are used. The leptons considered are electrons, muons, and neutrinos. The HH production is searched for in the bb¯bb¯ decay channel. An observed (expected) upper limit at 95% confidence level of VHH production cross section is set at 294 (124) times the standard model prediction. Constraints are also set on the modifiers of the Higgs boson trilinear self-coupling, kλ, assuming k2V = 1, and vice versa on the coupling of two Higgs bosons with two vector bosons, k2V. The observed (expected) 95% confidence intervals of these coupling modifiers are −37.7 < kλ < 37.2 (−30.1 < kλ < 28.9) and −12.2 < k2V < 13.5 (−7.2 < k2V < 8.9), respectively.

Double-Higgs Production

A major focus in particle physics has been on understanding the interactions of the Higgs boson. Tremendous progress has been made in determining the strength of the couplings of the Higgs boson to fermions and vector bosons, but its self-interaction has yet to be established. Understanding the Higgs self-coupling and the form of the potential function of the Higgs field will illuminate the process by which the Higgs boson acquires a vacuum expectation value and could provide insight into the early Universe and, perhaps, its eventual fate. The most natural way to probe the Higgs self-interaction is via searches for Higgs boson pair (HH) production. Since the Standard Model makes a definite prediction for the Higgs self-coupling, enhanced rates and modified kinematic properties of HH production are a smoking-gun signature for new physics. This article reviews the current experimental status of HH searches, discusses the experimental challenges and limitations, and provides an outlook for the future of the field.

Search for pair production of boosted Higgs bosons via vector-boson fusion in the [formula omitted] final state using pp collisions at [formula omitted] with the ATLAS detector

A search for Higgs boson pair production via vector-boson fusion is performed in the Lorentz-boosted regime, where a Higgs boson candidate is reconstructed as a single large-radius jet, using 140 fb−1 of proton–proton collision data at s=13 TeV recorded by the ATLAS detector at the Large Hadron Collider. Only Higgs boson decays into bottom quark pairs are considered. The search is particularly sensitive to the quartic coupling between two vector bosons and two Higgs bosons relative to its Standard Model prediction, κ2V. This study constrains κ2V to 0.55<κ2V<1.49 at the 95% confidence level. The value κ2V=0 is excluded with a significance of 3.8 standard deviations with other Higgs boson couplings fixed to their Standard Model values. A search for new heavy spin-0 resonances that would mediate Higgs boson pair production via vector-boson fusion is carried out in the mass range of 1–5 TeV for the first time under several model and decay-width assumptions. No significant deviation from the Standard Model hypothesis is observed and exclusion limits at the 95% confidence level are derived.

Gamma-ray signal from ZN≥3 dark matter-companion models

In Ref. [10], we proposed to replace the final dark matter (DM) particle in the semi-annihilation mode DM+DM→antiDM+Higgs boson with its ZN≥3 companion, thus reducing DM number density without DM-nucleon scattering. In this work, we study the indirect detection signals from DM annihilation, the Higgs boson pair with one of them from the companion decay being on- or off- shell, depending on the DM-companion mass splitting. We generate the photon spectrum by using PYTHIA8 and study the properties of the spectrum, to find that the hard part of the spectrum in our model is mainly shaped by the direct Higgs boson and thus does not differ much from that of the conventional semi-annihilation mode. Using the Fermi-LAT data of dwarf spheroidal galaxies, we derive the current limit of the DM annihilation cross section for DM+DM→companion⁎+Higgs boson, and for the relatively light DM, it reaches the typical thermal cross section. However, for the TeV scale DM, we have to rely on the Cherenkov Telescope Array, which is able to rule out the whole parameter space except for the coannihilation region.

D-wave charge-4e superconductivity from fluctuating pair density waves

We present a theory for charge-4e superconductivity as a leading low-temperature instability with a nontrivial d-wave symmetry. We show that in several microscopic models for the pair-density-wave (PDW) state, when the PDW wave vectors connect special parts of the Fermi surface, the predominant interaction is in the bosonic pairing channel mediated by exchanging low-energy fermions. This bosonic pairing interaction is repulsive in the s-wave channel but attractive in the d-wave one, leading to a d-wave charge-4e superconductor. By analyzing the Ginzburg-Landau free energy including higher-order fluctuation effects of PDW, we find that the charge-4e superconductivity emerges as a vestigial order of PDW, and sets in via a first-order transition. Both the gap amplitude and the transition temperature decay monotonically with increasing superfluid stiffness of the PDW order. Our work provides a microscopic mechanism of higher-charge condensates with unconventional ordering symmetry in strongly-correlated materials.

Search for non-resonant Higgs boson pair production in final states with leptons, taus, and photons in pp collisions at s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{s} $$\\end{document} = 13 TeV with the ATLAS detector

A search is presented for non-resonant Higgs boson pair production, targeting the bbZZ, 4V (V = W or Z), VVττ, 4τ, γγVV and γγττ decay channels. Events are categorised based on the multiplicity of light charged leptons (electrons or muons), hadronically decaying tau leptons, and photons. The search is based on a data sample of proton-proton collisions at s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{s} $$\\end{document} = 13 TeV recorded with the ATLAS detector during Run 2 of the Large Hadron Collider, corresponding to an integrated luminosity of 140 fb−1. No evidence of the signal is found and the observed (expected) upper limit on the cross-section for non-resonant Higgs boson pair production is determined to be 17 (11) times the Standard Model predicted cross-section at 95% confidence level under the background-only hypothesis. The observed (expected) constraints on the HHH coupling modifier, κλ, are determined to be −6.2 < κλ< 11.6 (−4.5 < κλ< 9.6) at 95% confidence level, assuming the Standard Model for the expected limits and that new physics would only affect κλ.

Gauge field dynamics in multilayer Kitaev spin liquids

The Kitaev spin liquid realizes an emergent static Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb{Z}}}_{2}$$\\end{document} gauge field with vison excitations coupled to Majorana fermions. We consider Kitaev models stacked on top of each other, weakly coupled by Heisenberg interaction ∝ J⊥. This inter-layer coupling breaks the integrability of the model and makes the gauge fields dynamic. Conservation laws and topology keep single visons immobile. However, an inter-layer vison pairs can hop with a hopping amplitude linear in J⊥ confined to the layer, but their motion is strongly influenced by the type of stacking. For AA stacking, an interlayer pair has a two-dimensional motion but for AB or ABC stacking, sheet conservation laws restrict its motion to a one-dimensional channel within the plane. For all stacking types, an intra-layer vison-pair is constrained to move out-of-plane only. Depending on the anisotropy of the Kitaev couplings Kx, Ky, Kz, the intra-layer vison pairs can display either coherent tunnelling or purely incoherent hopping. When a magnetic field opens a gap for Majorana fermions, there exist two types of intra-layer vison pairs - a bosonic and a fermionic one. Only the bosonic pair obtains a hopping rate linear in J⊥. We use our results to identify the leading instabilities of the spin liquid phase induced by the inter-layer coupling.

Three-loop corrections to Higgs boson pair production: reducible contribution

We compute three-loop corrections to the process gg → HH originating from one-particle reducible diagrams. This requires the computation of two-loop corrections to the gluon-gluon-Higgs vertex with an off-shell gluon. We describe in detail our approach to obtain semi-analytic results for the vertex form factors and present results for the two form factors contributing to Higgs boson pair production.

Search for the nonresonant production of Higgs boson pairs via gluon fusion and vector-boson fusion in the bb¯τ+τ− final state in proton-proton collisions at s=13 TeV with the ATLAS detector

A search for the nonresonant production of Higgs boson pairs in the HH→bb¯τ+τ− channel is performed using 140 fb−1 of proton-proton collisions at a center-of-mass energy of 13 TeV recorded by the ATLAS detector at the CERN Large Hadron Collider. The analysis strategy is optimized to probe anomalous values of the Higgs boson self-coupling modifier κλ and of the quartic HHVV (V=W,Z) coupling modifier κ2V. No significant excess above the expected background from Standard Model processes is observed. An observed (expected) upper limit μHH&lt;5.9(3.3) is set at 95% confidence-level on the Higgs boson pair production cross section normalized to its Standard Model prediction. The coupling modifiers are constrained to an observed (expected) 95% confidence interval of −3.1&lt;κλ&lt;9.0 (−2.5&lt;κλ&lt;9.3) and −0.5&lt;κ2V&lt;2.7 (−0.2&lt;κ2V&lt;2.4), assuming all other Higgs boson couplings are fixed to the Standard Model prediction. The results are also interpreted in the context of effective field theories via constraints on anomalous Higgs boson couplings and Higgs boson pair production cross sections assuming different kinematic benchmark scenarios. © 2024 CERN, for the ATLAS Collaboration 2024 CERN

Search for Higgs boson pair production in the bb¯W+W−\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extrm{b}\\overline{\ extrm{b}}{\ extrm{W}}^{+}{\ extrm{W}}^{-} $$\\end{document} decay mode in proton-proton collisions at s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{s} $$\\end{document} = 13 TeV

A search for Higgs boson pair (HH) production with one Higgs boson decaying to two bottom quarks and the other to two W bosons are presented. The search is done using proton-proton collisions data at a centre-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 138 fb−1 recorded by the CMS detector at the LHC from 2016 to 2018. The final states considered include at least one leptonically decaying W boson. No evidence for the presence of a signal is observed and corresponding upper limits on the HH production cross section are derived. The limit on the inclusive cross section of the nonresonant HH production, assuming that the distributions of kinematic observables are as expected in the standard model (SM), is observed (expected) to be 14 (18) times the value predicted by the SM, at 95% confidence level. The limits on the cross section are also presented as functions of various Higgs boson coupling modifiers, and anomalous Higgs boson coupling scenarios. In addition, limits are set on the resonant HH production via spin-0 and spin-2 resonances within the mass range 250–900 GeV.

Enhanced four-body decays of charged Higgs bosons into off-shell pseudoscalar Higgs and W±\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^\\pm $$\\end{document} boson pairs in a lepton-specific 2-Higgs doublet model

We study the time-honoured decay H±→AW±\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^\\pm \\rightarrow A W^\\pm $$\\end{document} but for the first time, we do so for the case of both A and W±\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^\\pm $$\\end{document} being off-shell, therefore computing a 1→4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\rightarrow 4$$\\end{document} body decay. We show that the corresponding decay rate not only extends the reach of H±\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^\\pm $$\\end{document} searches to small masses of the latter but also that the results of our implementation differ significantly from the yield of the 1→3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\rightarrow 3$$\\end{document} body decay over the phase space region in which the latter is normally used. We show the phenomenological relevance of this implementation in the case of the so-called lepton-specific 2-Higgs Doublet Model (2HDM) over the mass region wherein the aforementioned 1→4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\rightarrow 4$$\\end{document} body decay can dominate just beyond the top (anti)quark mass. This mass region is accessible in the lepton-specific 2HDM as the Yukawa couplings are such that limits from b→sγ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b \\rightarrow s \\gamma $$\\end{document} and τ→μντνμ¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au \\rightarrow \\mu \ u _{\ au } \\bar{\ u _\\mu }$$\\end{document} observables on MH±\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_{H^\\pm }$$\\end{document} are rather mild. However, we emphasize that similar effects may occur in other 2HDM types, as the W±H∓A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^\\pm H^\\mp A$$\\end{document} vertex is 2HDM type independent.