Lepton Number Research Articles

Search for the baryon number and lepton number violating decays τ−→Λπ− and τ−→Λ¯π− at Belle II

We present a search for the baryon number B and lepton number L violating decays τ−→Λπ− and τ−→Λ¯π− produced from the e+e−→τ+τ− process, using a 364 fb−1 data sample collected by the Belle II experiment at the SuperKEKB collider. No evidence of signal is found in either decay mode, which have |Δ(B−L)| equal to 2 and 0, respectively. Upper limits at 90% credibility level on the branching fractions of τ−→Λπ− and τ−→Λ¯π− are determined to be 4.7×10−8 and 4.3×10−8, respectively. Published by the American Physical Society 2024

First search for K+ → π0πμe decays

The first search for the lepton number violating decay K+→π0π−μ+e+ and lepton flavour violating decays K+→π0π+μ−e+, K+→π0π+μ+e− has been performed using a dataset collected by the NA62 experiment at CERN in 2016–2018. Upper limits of 2.9×10−10, 3.1×10−10 and 5.0×10−10, respectively, are obtained at 90% CL for the branching ratios of the three decays on the assumption of uniform phase-space distributions.

Spontaneous leptogenesis with sub-GeV axionlike particles

A derivative coupling of an axionlike particle (ALP) with a B−L current may lead to the baryon asymmetry of the Universe via spontaneous leptogenesis provided a lepton number breaking interaction prevails in thermal equilibrium. Conventionally, such scenario works only for heavy ALPs and high reheating temperature due to the fact that the same lepton number breaking contribution is tied up with neutrino mass generation also. In this work, we propose inert Higgs doublet assisted lepton number violating operator to relieve such tension so as to generate lepton asymmetry (of freeze-in/out type) with a much lower reheating temperature that can accommodate light (sub-GeV) ALPs sensitive to current and future ALP searches. Published by the American Physical Society 2024

Final multiplicity of a QED cascade in generalized Heitler model

We consider a generalized Heitler model for a QED cascade. An exact formula for the final number of leptons is obtained by solving the kinetic equations. We demonstrate that in such a model the final number of leptons does not depend on photon and lepton mean free paths. We derive approximate formulas for the main characteristics of cascades at high energy, including the final number of leptons and the cascade depth. We show that in general the final number of leptons is asymptotically proportional to the energy of a seed particle. It is also demonstrated how the original Heitler model is reproduced as a special case. Published by the American Physical Society 2024

Discovering heavy neutrino-antineutrino oscillations at the Z-pole

Collider-testable type I seesaw extensions of the Standard Model are generally protected by an approximate lepton number (LN) symmetry. Consequently, they predict pseudo-Dirac heavy neutral leptons (HNLs) composed of two nearly degenerate Majorana fields. The interference between the two mass eigenstates can induce heavy neutrino-antineutrino oscillations (NN¯Os\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ N\\overline{N}\ extrm{Os} $$\\end{document}) leading to observable lepton number violation (LNV), even though the LN symmetry is approximately conserved. These NN¯Os\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ N\\overline{N}\ extrm{Os} $$\\end{document} could be resolved in long-lived HNL searches at collider experiments, such as the proposed Future Circular e+e− Collider (FCC-ee) or Circular Electron Positron Collider (CEPC). However, during their Z-pole runs, the LN carried away by the light (anti)neutrinos produced alongside the HNLs prevents LNV from being observed directly. Nevertheless, NN¯Os\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ N\\overline{N}\ extrm{Os} $$\\end{document} materialise as oscillating signatures in final state distributions. We discuss and compare a selection of such oscillating observables, and perform a Monte Carlo simulation to assess the parameter space in which NN¯Os\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ N\\overline{N}\ extrm{Os} $$\\end{document} could be resolved.

Probing the neutrino seesaw scale with gravitational waves

Neutrinos are the most elusive particles of the Standard Model. The physics behind their masses remains unknown and requires introducing new particles and interactions. An elegant solution to this problem is provided by the seesaw mechanism. Typically considered at a high scale, it is potentially testable in gravitational wave experiments by searching for a spectrum from cosmic strings, which offers a rather generic signature across many high-scale seesaw models. Here we consider the possibility of a low-scale seesaw mechanism at the PeV scale, generating neutrino masses within the framework of a model with gauged U(1) lepton number. In this case, the gravitational wave signal at high frequencies arises from a first order phase transition in the early Universe, whereas at low frequencies it is generated by domain wall annihilation, leading to a double-peaked structure in the gravitational wave spectrum. The signals discussed here can be searched for in upcoming experiments, including gravitational wave interferometers, pulsar timing arrays, and astrometry observations. Published by the American Physical Society 2024

Neutrinoless double beta decay without vacuum Majorana neutrino mass

We present a proof-of-concept extension to the Standard Model that can generate a non-vanishing neutrinoless double beta decay (0νββ) signal without the existence of Majorana neutrinos or lepton number violation in the zero-density vacuum-ground-state Lagrangian. We propose that the 0νββ can be induced by the capture of an ultralight scalar field, a potential dark matter candidate, that carries two units of lepton number. This makes the observable 0νββ spectrum indistinguishable from the usual 0νββ mechanisms by any practical means. We find that a non-zero 0νββ rate does not require neutrinos to be fundamentally Majorana particles. However, for sizeable decay rates within the range of next-generation experiments, the neutrino will, generally, acquire an (effective) Majorana mass as the scalar field undergoes a transition to the Bose-Einstein condensate phase. We also discuss the distinction between the aforementioned scenario and the case in which the emission of a lepton-number-two scalar leads to 0νββ, exhibiting discernible qualitative features that make it experimentally distinguishable from the conventional scenario.

Observable gravitational waves and ΔNeff with global lepton number symmetry and dark matter

We study the possibility of testing a dark matter (DM) scenario embedded in a global lepton number symmetry U(1)L via gravitational waves (GWs) and cosmic microwave background (CMB) observations. The spontaneous breaking of U(1)L symmetry generates the seesaw scale as well as DM mass dynamically. The (pseudo-)Nambu-Goldstone boson, known as a Majoron, acquires nonzero mass due to soft symmetry-breaking terms of quadratic type in the scalar potential, which eventually breaks U(1)L to its Z2 subgroup. The spontaneous symmetry breaking, which effectively breaks Z2, leads to the formation of domain walls (DWs), posing a threat to successful cosmology, if allowed to dominate. As gravity does not respect any global symmetries, we consider higher-dimensional operators suppressed by the scale of quantum gravity (QG), namely, ΛQG, which introduces the required bias leading to DW annihilation and emission of stochastic GWs observable at near future experiments. The same operators also lead to decay of DM bringing interesting indirect detection aspects. While DM is produced nonthermally via scalar portal interactions, light Majorons can give rise to additional ΔNeff within reach of future CMB experiments. Published by the American Physical Society 2024

Rare B and K decays in a scotogenic model

A scotogenic model can radiatively generate the observed neutrino mass, provide a dark matter candidate, and lead to rare lepton flavor-violating processes. We aim to extend the model to establish a potential connection to the quark flavor-related processes within the framework of scotogenesis, enhancing the unexpectedly large branching ratio (BR) of B+→K+νν¯ observed by the Belle II Collaboration. Meanwhile, the model can address tensions between some experimental measurements and standard model (SM) predictions in flavor physics, such as the muon g−2 excess and the higher BR of Bs→μ−μ+. In the model, we introduce the following dark particles: a neutral singlet Dirac-type lepton (N); two inert Higgs doublets (η1,2), with one carrying a lepton number; a charged singlet dark scalar (χ+); and a singlet vectorlike up-type dark quark (T). The first two entities are responsible for the radiative neutrino mass, and χ+ couples to right-handed quarks and leptons and can resolve the tensions existing in muon g−2 and Bs→μ−μ+. Furthermore, the BR of B+→K+νν¯ can be enhanced up to a factor of 2 compared to the SM prediction through the mediations of the dark T and the charged scalars. In addition, we also study the impacts on the K→πνν¯ decays. Published by the American Physical Society 2024

Phenomenology of 3-3-1 models with a radiative inverse seesaw mechanism

We propose two models based on the SU(3)C×SU(3)L×U(1)X gauge symmetry, each incorporating distinct inverse seesaw mechanisms for generating neutrino masses at the radiative level. Therefore, neutrino masses are suppressed by the radiative nature of the mass generation mechanism, which occurs after the spontaneous breaking of the global lepton number symmetry. Both scenarios discussed here are characterized by the presence of vectorlike charged leptons, which are involved in generating the masses of the Standard Model charged leptons. These additional vectorlike fermions also contribute to the anomalous magnetic moments of the muon. We perform a detailed analysis of the scalar sectors, show that these models can successfully accommodate the observed baryon asymmetry through resonant leptogenesis, and compute the rates of charged lepton flavor-violating decays, such as μ→eγ. We discuss the constraints of the model arising from these processes and those associated with the nonunitarity of the lepton mixing matrix. Published by the American Physical Society 2024

Majorana phases beyond neutrinoless double beta decay

The νM SM is defined as the SM extended to include dimension-5 operators. In this model neutrino masses violate lepton number, and two parameters of the lepton mixing matrix, the Majorana phases, are yet to be constrained. One combination of these phases and the neutrino masses, often denoted by mee, is probed by neutrinoless double beta decays (0νββ). We explore what information may be obtained beyond 0νββ, and how it depends on the lightest neutrino mass. We point out that with current central values of the mixing parameters, ∆Le = 2 and ∆Le = ∆Lμ = 1 (or ∆Le = 2 and ∆Lμ = 2) processes cannot simultaneously vanish, providing a no-lose theorem, in principle, for excluding the νM SM, even in the case of normal mass ordering.

Disentangling new physics in K→πνν¯ and B→KK∗νν¯ observables

We investigate the possibility of disentangling different new physics contributions to the rare meson decays and through kinematic distributions in the missing energy . We employ dimension-6 operators within the Low-Energy Effective Field Theory (LEFT), identifying the invisible part of the final state as either active or sterile neutrinos. Special emphasis is given to lepton-number violating (LNV) operators with scalar and tensor currents. We show analytically that contributions from vector, scalar, and tensor quark currents can be uniquely determined from experimental data of kinematic distributions. In addition, we present new correlations of branching ratios for K and B-decays involving scalar and tensor currents. As there could a priori also be new invisible particles in the final states, we include dark-sector operators giving rise to two dark scalars, fermions, or vectors in the final state. In this context, we present new calculations of the inclusive decay rate for dark operators. We show that careful measurements of kinematic distributions make it theoretically possible to disentangle the contribution from LEFT operators from most of the dark-sector operators, even when multiple operators are contributing. We revisit sum rules for vector currents in LEFT and show that the latter are also satisfied in some new dark-physics scenarios that could mimic LEFT. Finally, we point out that an excess in rare meson decays consistent with a LNV hypothesis would point towards highly flavor non-democratic physics in the UV, and could put high-scale leptogenesis under tension.

Hunting for Bileptons at Hadron Colliders.

I review possible signals at hadron colliders of bileptons, namely doubly charged vectors or scalars with lepton number L=±2, as predicted by a 331 model, based on a SU(3)c×SU(3)L×U(1)X symmetry. In particular, I account for a version of the 331 model wherein the embedding of the hypercharge is obtained with the addition of three exotic quarks and vector bileptons. Furthermore, a sextet of SU(3)L, necessary to provide masses to leptons, yields an extra scalar sector, including a doubly charged Higgs, i.e., scalar bileptons. As bileptons are mostly produced in pairs at hadron colliders, their main signal is provided by two same-sign lepton pairs at high invariant mass. Nevertheless, they can also decay according to non-leptonic modes, such as a TeV-scale heavy quark, charged 4/3 or 5/3, plus a Standard Model quark. I explore both leptonic and non-leptonic decays and the sensitivity to the processes of the present and future hadron colliders.

The Time Evolution of Fast Flavor Crossings in Postmerger Disks around a Black Hole Remnant

We postprocess a three-dimensional, general relativistic, full transport neutrino radiation magnetohydrodynamics simulation of the black-hole-accretion disk-wind system thought to be a potential outcome of the GW170817 merger to investigate the presence of electron lepton number (ELN-XLN) crossings in the neutrino angular distribution. Neutrinos are evolved with an explicit Monte Carlo method and can interact with matter via emission, absorption, or scattering. Within the postprocessing framework, we find ubiquitous occurrence of ELN-XLN crossings at early times (∼11 ms), but this does not hold for later times in the simulation. At postmerger times of ∼60 ms and beyond, ELN-XLN crossings are only present near the equator. We provide a detailed analysis of the neutrino radiation field to investigate the origin and time evolution of these crossings. Previous reports have suggested ubiquitous flavor crossings persisting throughout the simulation lifetime, albeit for different sets of conditions for the merger remnant, the treatment of hydrodynamics, and neutrino transport. Even though we do not perform a direct comparison with other published works, we qualitatively assess the reasons for the difference with our results. The geometric structure and evolution of the ELN-XLN crossings found in our analysis, and by extension, fast flavor instabilities, have important implications for heavy element nucleosynthesis in neutron star mergers.

Models of accidental dark matter with a fundamental scalar

We consider models of accidental dark matter, namely models in which the dark matter is a composite state that is stable thanks to an accidental symmetry of the theory. The fundamental constituents are vectorlike fermions, taken to be fragments of representations of the grand unifying gauge group SU(5), as well as a scalar singlet. All the new fields are charged under a new confining gauge group, which we take to be SU(N), leading to models with complex dark matter. We analyse the models in the context of SU(5) grand unification with a non-standard approach recently proposed in the literature. The advantage of including the scalar mainly resides in the fact that it allows several undesired accidental symmetries to be broken, leading to a larger set of viable models with respect to previous literature, in which only fermions (or only scalars) were considered. Moreover these models present distinct novelties, namely dark states with non-zero baryon and lepton number and the existence of composite hybrid states of fermions and scalars. We identify phenomena that are specific to the inclusion of the scalar and discuss possibilities to test this setup.

Neutrino quantum kinetics in a core-collapse supernova

Our understanding of neutrino flavor conversion in the supernova core is still preliminary, despite its likely relevance to the neutrino-driven supernova mechanism. We present multi-angle and multi-energy numerical simulations of neutrino quantum kinetics within a spherically symmetric shell in the proximity of the region of neutrino decoupling. We rely on inputs from a one-dimensional core-collapse supernova model with a mass of 18.6 M ⊙ and find that, at early post-bounce times (t pb ≲ 0.5 s), no crossing is present in the angular distribution of the electron neutrino lepton number and flavor conversion is triggered by slow collective instabilities. Angular crossings appear for t pb ≳ 0.5 s and fast flavor conversion leads to flavor equipartition, with the spectral energy distribution of ν e (ν̅e) and ν x (ν̅ x ) becoming comparable. Notably, flavor equipartition is not a generic outcome of fast flavor conversion, rather it is a consequence of the relatively similar properties of neutrinos of different flavors characterizing the late accretion phase. Artificially tweaking the collision term to introduce an electron lepton number angular crossing for t pb ≲ 0.5 s, we observe that flavor equipartition is not achieved. While our findings are restricted to a specific supernova model, and they only take into account the feedback of the neutrino background on the flavor conversion, they suggest a rich phenomenology in the supernova core as a function of the post-bounce time which needs to be further explored to assess its impact on the explosion mechanism.

Could SBND-PRISM probe lepton flavor violation?

We investigate the possibility of using the Short-Baseline Near Detector (SBND) at Fermilab to constrain lepton flavor violating decays of pions and kaons. We study how to leverage SBND-PRISM, the use of the neutrino beam angular spread to mitigate systematic uncertainties, to enhance this analysis. We show that SBND-PRISM can put stringent limits on the flavor violating branching ratios BR(π+→μ+νe)=8.9×10−4, BR(K+→μ+νe)=3.2×10−3, improving previous constraints by factors 9 and 1.25, respectively. We also estimate the SBND-PRISM sensitivity to lepton number violating decays, BR(π+→μ+ν¯e)=2.1×10−3 and BR(K+→μ+ν¯e)=7.4×10−3, though not reaching previous Big European Bubble Chamber limits. Last, we identify several ways how the SBND collaboration could improve this analysis. Published by the American Physical Society 2024

Neutrino Masses from Generalized Symmetry Breaking

We explore generalized global symmetries in theories of physics beyond the standard model. Theories of Z′ bosons generically contain “noninvertible” chiral symmetries, whose presence indicates a natural paradigm to break this symmetry by an exponentially small amount in an ultraviolet completion. For example, in models of gauged lepton family difference such as the phenomenologically well motivated U(1)Lμ−Lτ, there is a noninvertible lepton number symmetry which protects neutrino masses. We embed these theories in gauged non-Abelian horizontal lepton symmetries, e.g., U(1)Lμ−Lτ⊂SU(3)H, where the generalized symmetries are broken nonperturbatively by the existence of lepton family magnetic monopoles. In such theories, either Majorana or Dirac neutrino masses may be generated through quantum gauge theory effects from the charged lepton Yukawas, e.g., yν∼yτexp(−Sinst). These theories require no bevy of new fields nor additional global symmetries but are instead simple, natural, and predictive: The discovery of a lepton family Z′ at low energies will reveal the scale at which Lμ−Lτ emerges from a larger gauge symmetry. Published by the American Physical Society 2024

Electric dipole moments of charged leptons in models with pseudo-Dirac sterile fermions

In this work, we address the impact of a small lepton number violation on charged lepton electric dipole moments — EDMs. Low-scale seesaw models protected by lepton number symmetry and leading to pseudo-Dirac pairs in the neutrino heavy spectrum provide a natural explanation for the smallness of neutrino masses with potentially testable consequences. Among which, it was thought that the small mass gap in each pair of pseudo-Dirac neutrinos may induce important contribution to the charged lepton EDMs. Recently, it has been shown that the contribution from some of the Feynman diagrams to charged lepton EDMs exactly cancel by virtue of the Ward-Takahashi identity in quantum electrodynamics. We thus consider here the Standard Model minimally extended with pairs of pseudo-Dirac sterile fermions and derive the complete analytical formula at two loops for the charged lepton EDMs. In addition, we numerically evaluate the order of the predicted EDMs consistent with the experimental bounds and constraints such as neutrino oscillation data, charged lepton flavour violating processes, sterile neutrino direct searches, meson decays, sterile neutrino decays, and cosmological and astrophysical observations. We find that, in the minimal setup accommodating neutrino data (masses and mixings) with only two pseudo-Dirac pairs, the predicted electron EDM is O10−36\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}\\left({10}^{-36}\\right) $$\\end{document}e cm, at most, which is much smaller than the current experimental bound and even future sensitivity. Hopefully, the electron EDM might reach future sensitivity, once extra pseudo-Dirac neutrinos are taken into account. The analytical formulae we derive are generic to any model involving pseudo-Dirac pairs in the heavy neutrino spectrum.

Search for leptoquark pair production decaying into te-t¯e+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$te^- \\bar{t}e^+$$\\end{document} or tμ-t¯μ+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\mu ^- \\bar{t}\\mu ^+$$\\end{document} in multi-lepton final states in pp collisions at s=13TeV\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sqrt{s} = 13\\,\ extrm{TeV}$$\\end{document} with the ATLAS detector

A search for leptoquark pair production decaying into te-t¯e+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$te^- \\bar{t}e^+$$\\end{document} or tμ-t¯μ+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\mu ^- \\bar{t}\\mu ^+$$\\end{document} in final states with multiple leptons is presented. The search is based on a dataset of pp collisions at s=13TeV\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sqrt{s}=13~\ ext {TeV} $$\\end{document} recorded with the ATLAS detector during Run 2 of the Large Hadron Collider, corresponding to an integrated luminosity of 139 fb-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{-1}$$\\end{document}. Four signal regions, with the requirement of at least three light leptons (electron or muon) and at least two jets out of which at least one jet is identified as coming from a b-hadron, are considered based on the number of leptons of a given flavour. The main background processes are estimated using dedicated control regions in a simultaneous fit with the signal regions to data. No excess above the Standard Model background prediction is observed and 95% confidence level limits on the production cross section times branching ratio are derived as a function of the leptoquark mass. Under the assumption of exclusive decays into te-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$te^{-}$$\\end{document} (tμ-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\mu ^{-}$$\\end{document}), the corresponding lower limit on the scalar mixed-generation leptoquark mass mLQmixd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_{\ extrm{LQ}_{\ extrm{mix}}^{\ extrm{d}}}$$\\end{document} is at 1.58 (1.59) TeV and on the vector leptoquark mass mU~1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_{{\ ilde{U}}_1}$$\\end{document} at 1.67 (1.67) TeV in the minimal coupling scenario and at 1.95 (1.95) TeV in the Yang–Mills scenario.