arXiv subjects

J. J. Toscano

Publications and source records attributed to J. J. Toscano.

At least 19 recordsLinked to original sources

arXiv2026

Batalin-Vilkovisky description of Yang-Mills theories with universal extra dimensions

The Batalin-Vilkovisky (BV) framework is a powerful technique for quantizing a wide range of gauge systems. This formalism employs fields and antifields--introducing a symplectic structure through the antibracket--to implement BRST symmetry, which captures the essence of gauge invariance. Within this context, we study the symmetry structure of a pure Yang-Mills theory with universal extra dimensions. We construct a BRST-invariant extended action for the $(4+n)$-dimensional theory depending on fields and antifields, which constitutes a proper solution to the master equation. The higher-dimensional spacetime and gauge symmetries are then hidden within their four-dimensional counterparts via canonical transformations. Gauge invariances are fixed by combining gauge-fixing procedures for both the standard and Kaluza-Klein (KK) fields. This procedure is implemented covariantly in the adjoint representation of the gauge group. While the standard gauge fields are fixed using the Background Field Method (BFM), the KK gauge excitations are fixed covariantly, as they transform as matter fields. Finally, the manifest $SU(N)$ gauge symmetry of the quantized theory is emphasized.

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arXiv2025

Effects of CPT violation on the neutrino charge radius in the Standard Model Extension

CPT-odd effects on the neutrino charge radius are studied within the Standard Model Extension. We consider CPT violation from the electroweak Yang-Mills sector, characterized by Lorentz-violating coefficients $(k_1)_μ$ and $(k_2)_μ$, which have positive mass units. The $\barννγ$ vertex arises at tree level via exchange of two $Z$ bosons. Although suppressed by $\frac{1}{m_Z^4}$, this process is notable. The vertex function $Γ_μ$ includes three independent gauge structures satisfying the Ward identity $q^μΓ_μ= 0$, characteristic of neutral particles, and induces gauge-independent electromagnetic form factors. Besides charge and anapole, two novel form factors appear. The charge form factor $f_Q(q^2)$ contains an energy-dependent term causing $f_Q(0)\neq 0$, so electromagnetic properties are undefined for real photons with CPT violation. Instead, $f_Q$ and the charge radius are defined in the static limit: $q^0=0$ and $\mathbf{q}\to 0$. Here, $f_Q(0, \mathbf{0})=0$ and a correction to the SM neutrino charge radius is found: \[ \langle r^2_ν\rangle_{CPTV} = \frac{3c_{2W}}{2c_W^4} \left[\frac{k_2^2}{m_Z^2} + \frac{\mathbf{k}_2^2}{m_Z^2}\cos^2θ_γ\right] \frac{1}{m_Z^2}, \] with $θ_γ$ the angle between $\mathbf{q}$ and $\mathbf{k}_2$. Using recent bounds on $(k_i)_μ$ and reasonable assumptions, we obtain a small correction $\langle r^2_ν\rangle_{CPTV} \leq 0.83 \times 10^{-51}\ \mathrm{cm}^2$.

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arXiv2025

Gauge structure of Yang-Mills theories with extra dimensions

An effective Lagrangian for Yang-Mills theories with $n$ extra dimensions is constructed. We start from a field theory governed by the extra-dimensional Poincaré group $ISO(1,3+n)$ and the extended gauge group $SU(N,M^{4+n})$, characterized by an energy scale $Λ$ and assumed to be valid at energies far below this scale. Assuming that the size of the extra dimensions is much larger than the distance scale at which this theory is valid, an effective theory with symmetry groups $ISO(1,3)$ and $SU(N,M^{4})$ is constructed. Such theories are connected by a canonical transformation that hides $ISO(1,3+n)\otimes SU(N,M^{4+n})$ into $ISO(1,3)\otimes SU(N,M^{4})$, and endows the KK gauge fields with mass. Using a set of orthogonal functions $\{f^{(\underline{0})},f^{(\underline{m})}(\bar x)\}$, generated by the Casimir invariant $\bar{P}^2$ associated with the translations subgroup $T(n)\subset ISO(n)$, the degrees of freedom of $ISO(1,3+n)\otimes SU(N,M^{4+n})$ are expanded via a general Fourier series, whose coefficients are the degrees of freedom of $ISO(1,3)\otimes SU(N,M^{4})$. These functions, corresponding to the projection on the coordinates basis $\{|\bar{x} \big >\}$ of the discrete basis $\{|0\big >,|p^{(\underline{m})}\big >\}$ generated by $\bar {P}^2$, are central in defining the effective theory. Components along the base state $f^{(\underline{0})}=\big <\bar x|0\big>$, identified as the standard Yang-Mills fields, do not receive mass at the compactification scale; components along excited states $f^{(\underline{m})}=\big <\bar x|p^{(\underline{m})}\big>$, corresponding to KK excitations, receive mass at this scale. Associated with any direction $|p^{(\underline{m})}\neq0\big >$ there are a massive gauge field and a pseudo-Goldstone boson. Resemblances of this mass-generating mechanism with the Brout-Englert-Higgs mechanism are stressed.

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arXiv2024

CPT-Odd effects on the electromagnetic properties of charged leptons in the Standard Model Extension

The impact of the CPT-Odd electroweak gauge sector of the Standard Model Extension on the electromagnetic properties of charged leptons is studied. This gauge sector is characterized by the $(k_1)_μ$ and $(k_2)_μ$ Lorentz violation (LV) coefficients, which have positive mass dimension because they are associated with a $U_Y(1)$-invariant and with an $SU_L(2)$-invariant dimension-three operators, respectively. We present a comprehensive study on the impact of this sector on the magnetic dipole moment (MDM) and the electric dipole moment (EDM) of charged leptons, up to second order in these LV coefficients, both at the tree and one-loop levels.The contributions of $O(k_i)$ to the MDM are found to be suppressed relative to the corresponding contributions to the EDM by approximately three orders of magnitude. Using a recent experimental limit on the electron EDM the $|(k_2)_0-|\mathbf{k_2}|\cosθ_γ|<0.86\, m_e$ bound was obtained. As far as the contributions of $O(k^2_i)$ are concerned, we find that the tree-level contributions are suppressed with respect to the one-loop ones by at least a factor of $\left(m^2_l/m^2_Z\right)$. We find that the contribution to the electron MDM is by far the dominant one, as it can be up to four and seven orders of magnitude greater than those of the muon and tau, respectively. The Lorentz coefficient $(k_{AF})_μ$ of the Carroll-Field-Jackiw's QED is given by a linear combination of $(k_1)_μ$ and $(k_2)_μ$. Assuming that $|k^2_1|, |k^2_2|\gg |k^2_{AF}|$ and taking $(k_{AF})_μ=0$, which implies that $(k_1)_μ$ and $(k_2)_μ$ are collinear, we obtain an upper bound of $\left|\frac{ k^2_2}{m^2_e} \right|<4.36\times 10^{-10}$. The fact that $k^2_2$ is an observer Lorentz invariant allows us to introduce a new-physics scale through $\sqrt{k^2_2}=Λ_{CPT}$, for which we obtain the upper limit $Λ_{CPT}< 2.08 \times 10^{-5}\, m_e$.

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arXiv2023

Vacuum polarization in Yang-Mills theories with Lorentz violation

The renormalizable extension of a pure Yang-Mills theory with Lorentz violation is characterized by the CPT-Even $(k_F)_{μνλρ}$ and the CPT-Odd $(k_{AF})_μ$ constant Lorentz coefficients. In this paper, the one-loop structure of the theory up to second order in these Lorentz violating coefficients is studied using the BFM-gauge. Results for the diverse beta functions are derived and contrasted with those given in the literature at first order in these parameters. Special emphasis is putted on the beta function $β(g)$, which is studied in both mass-independent and mass-dependent renormalization schemes. It is found that in a mass-independent scheme the $(k_{AF})_μ$ Lorentz coefficient does not contribute to the $β(g)$ function, but it does in a mass-dependent scheme with contributions that are gauge-dependent and IR divergent.

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arXiv2022

Effects of Lorentz violation in the Higgs sector of the Minimal Standard Model Extension

A bound on the CPT-odd four vector coefficient $k^μ_ϕ$ that appears in Higgs sector of the Minimal Standard Model Extended (MSME) is presented. The analysis is based on the contributions arising from the sector in question to the anomalous dipole moment for leptons calculated at the one loop level, for which an analytical expression is obtained. The largest contribution of this Lorentz violating coefficient is on the lightest lepton, which results as a consequence of a strong non-decoupling effect. By using the experimental uncertainty of the electron anomalous dipole moment we predict that $|k^2_{ ϕ\,R}|<3.29\times 10^{-29}$ GeV$^2$.

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arXiv2020

Gauge-invariant approach to the beta function in Yang-Mills theories with universal extra dimensions

The radiative correction to beta function is comprehensively studied at 1 loop in the context of universal extra dimensions. Instead of using cutoffs to regularize 1-loop divergences, the dimensional regularization scheme is used. Large momenta effects are removed from physical amplitudes by adjusting the parameters of the appropriate counterterms. The use of a SU(N)-covariant gauge-fixing procedure is stressed. 1-loop contributions of KK excitations are characterized by discrete KK sums and continuous momenta sums, which can diverge. Two types of UVs are identified, one arising from poles of the gamma function and associated with short-distance effects in the usual 4-dimensional spacetime manifold, and the other emerging either from poles of the 1-dimensional Epstein function or from the gamma function, and corresponding to short-distance effects in the compact manifold. We address the cases of 5 and 4+n dimensions (n>1) separately. In 5 dimensions the 1-dimensional Epstein function is convergent, so the usual counterterm renormalizes the vacuum polarization function. For 4+n dimensions, the 1-dimensional Epstein function is divergent, so renormalization is implemented by interactions of canonical dimension higher than 4, already present in the effective theory. The polarization function is renormalized using both mass-dependent and mass-independent schemes, with extra-dimensions effects decoupling in the former case but not in the latter. The beta function is calculated for an arbitrary number of extra dimensions. Our main result is that Yang-Mills theory remains perturbative at 1 loop, which is in disagreement with the results obtained in the literature by using a cutoff regulator, which suggest that Yang-Mills theory in more than 4 dimensions ceases to be perturbative. We emphasize the advantages of a mass-dependent scheme in this type of theories, in which decoupling is manifest.

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arXiv2020

Diphoton Higgs signal strength in universal extra dimensions

The signal strength of the $gg \to H \to γγ$ reaction in $pp$ collisions at the LHC is studied within the context of the SM with UED. The impact of an arbitrary number $n$ of UED on both the $gg\to H$ and $H\to γγ$ subprocesses is studied. The 1-loop contributions of Kaluza-Klein excitations to these subprocesses are proportional to discrete and continuous sums, which can diverge. By implementing dimensional regularization, it is shown that discrete regularized sums can naturally be expressed as multidimensional Epstein functions, and that divergences, if exist, emerge through the poles of these functions. It is found that continuous sums converge, but the discrete ones diverge, with the exception of the $n=1$ case, in which the 1-dimensional Epstein function converges. It is argued that divergences that arise from discrete sums for $n\geq 2$ are genuine UV divergences, since they correspond to short-distance effects in the compact manifold. Then, the amplitudes are renormalized in a modern sense by incorporating interactions of canonical dimension higher than four that allow us to generate the required counterterms, which are determined using a $\overline{\rm MS}$-like renormalization scheme. We find that the $gg\to H$ subprocess is quite sensitive to both the size and the dimension of the compact manifold, but the SM prediction for $H\to γγ$ subprocess is practically unchanged. In the $n=1$ case, it is found that the experimental constraint on the compactification scale $R^{-1}\geq 1.5$ TeV allow us to reproduce the experimental limit on the signal strength $1.01\leq μ^{(1)}_{γγ}\leq 1.2$. In the $n\geq 2$ cases, it is found that the experimental limit on $μ^{(n)}_{γγ}$ leads to stronger lower bounds for the compactification scale given by $R^{-1}\geq 1.55, 2.45, 3.57, 5.10, 7.25$ TeVs for $n=2, 4, 6, 8, 10$, respectively.

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arXiv2020

One-loop order effects from one universal extra dimension on $λϕ^{4}$ theory

The self-interacting $λϕ^{4}$ scalar field theory is a warhorse in quantum field theory. Here we explore the one-loop order impact from one universal extra dimension, $S^{1}/\mathbb{Z}_{2}$, to the self-energy and four point vertex functions associated to this theory. Such effects come as an infinite number of UV divergences corresponding to an infinite superposition of excited KK particles around the loop. We show that dimensional regularisation is adequate enough to control them in terms of the product of the one dimensional inhomogenous Epstein zeta function times the gamma function. From the analytical properties of these functions, the UV divergences are extracted and the counterterms defined; the latter turn out to be of canonical dimension four at the Lagrangian level. We use both, the MS-scheme and a mass-dependent subtraction scheme to remove divergences. Only the latter manifestly satisfy the decoupling theorem.

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arXiv2020

Implications of extra dimensions on the effective charge and the beta function in quantum electrodynamics

A comprehensive analysis on the photon self-energy, the fermion self-energy, and the fermion vertex function is presented at one loop in the context of quantum electrodynamics (QED) with 1 extra dimension. In 5-dimensional theories, characterized by an infinite number of Kaluza-Klein fields, one-loop amplitudes involve discrete as well as continuous sums, $\sum^\infty_{n=1}\int d^4k$, that could diverge. Using dimensional regularization, we express such sums as products of gamma and Epstein functions, both defined on the complex plane, with divergences arising from poles of these functions in the limit as $ D \to 4$. Using the analytical properties of the Epstein function, we show that the ultraviolet divergences generated by the Kaluza-Klein sums can be consistently renormalized, which means that the corresponding renormalized quantities reduce to the usual ones of QED at the limit of a very large compactification scale $R^{-1}$. The main features of QED at the one-loop level were studied. We use the mass-dependent $μ$-scheme to calculate, in QED with an arbitrary number $n$ of extra dimensions, a beta function fulfilling all desirable physical requirements. We argue that in this type of theories, with a large mass spectrum covering a wide energy range, beta functions should not be calculated by using mass-independent renormalization schemes. We show that the beta function is finite for any energy $μ$. In particular, it reduces to the usual QED result $e^3/12π^2$ for $m\ll μ\ll R^{-1}$ and vanishes for $m\gg μ$, with $m$ the usual fermion mass. Throughout the work, the decoupling nature of all our results obtained from the analytical properties of the Epstein function is stressed.

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arXiv2018

One-loop structure of the photon propagator in the Standard Model Extension

We study radiative corrections on the photon propagator from the electroweak sector of the minimal Lorentz- and $CPT$-violating Standard Model Extension. We derive the most general Lorentz-violating ghost sector from BRST symmetry and renormalization theory. We introduce a Lorentz-violating nonlinear gauge that simplifies both the Higgs and gauge-sector extensions, which can be helpful in radiative corrections. At one loop, these sectors contribute to the $CPT$-even part of the photon propagator, characterized by the Riemann-type tensor $(k_F)_{αβμν}$. We give exact results for the contributions to the SO(1,3) irreducible parts of $(k_F)_{αβμν}$, namely, the Weyl-type tensor $(\hat{k}_F)_{αβμν}$, the Ricci-type tensor $(k_F)_{αβ}$, and the curvature-type scalar $k_F$. In the Yukawa sector, one-loop contributions are ultraviolet finite, but most of them are unobservable due to finite renormalization. The only observable effect is a contribution proportional to $(k_F)_{αβ}$ that emerges via a dimension-6 term that is observer and gauge invariant. In the Higgs and gauge sectors, all the irreducible parts of the corresponding Riemann-type tensors receive divergent contributions, so they are observable. The only finite contribution corresponds to the dimension-6 term. We think of these contributions as radiative corrections to the renormalized tensors and assume that both effects are of the same order of magnitude to find bounds from vacuum birefringence and compare with the literature. Bounds on $(k_F)_{αβ}$ contributions, innocuous to birefringence, are also derived using limits on the renormalized tensor from Laser-Interferometer-Gravitational-Wave-Observatory data. We compare these bounds with the literature. Beta functions associated with $(\hat{k}_F)_{αβμν}$ and $(k_F)_{αβ}$ are derived.

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arXiv2018

The Feynman kernel of a dimensionally reduced scalar field theory

We construct a consistent quantum field theory of a dimensionally reduced self-interacting scalar field. The Kaluza-Klein dimensional reduction on the well-known $Φ^{4}$ scalar theory, on a certain $(4+n)$ spacetime with an arbitrary number of extra dimensions, induces a four dimensional reduced theory with scalar fields: the zeroth mode (`light' field) and an infinite number of KK-excited modes (`heavy' fields). This theory is quantized by Hamiltonian path integral methods. It is shown, from first principles, that non-trivial measure factors at the level of the functional measure are absent even if the whole set of heavy fields is taken into account. Hints on the regularization and renormalization process are briefly discussed.

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arXiv2018

About heavy neutrinos: Lepton-flavor violation in decays of charged leptons

The fundamental description of nature, beyond the Standard Model (SM), may include heavy neutrinos that mix and thus allow processes in which lepton flavor is not preserved. We investigate the impact of charged currents that couple heavy gauge bosons to heavy neutrinos and SM leptons on lepton-flavor-violating decays of SM leptons into three charged leptons, with no final-state neutrinos. We implement our expressions for the leading contributions to ${\rm Br}(l_α\to l_β\,l_σ\,l_σ)$, which hold for either Dirac or Majorana neutrinos, to the trilepton decay $μ\to3e$, of the muon, and so determine sets of masses of heavy neutrinos and the heavy gauge boson, within GeVs to few TeVs, that are consistent with the upper bounds provided by the SINDRUM Collaboration. We find, however, that constraints dictated by the upper bound on ${\rm Br}(μ\to eγ)$, from the MEG Collaboration, are more stringent. We utilize such parameters to find that the contributions to tau decays are $\sim10^{-15}-10^{-13}$, well below bounds from $B$ factories. The mixing of heavy and SM charged bosons is also investigated. We find that current experimental data from MEG and SINDRUM would allow mixing angles as large as $\sim10^{-2}$, for a relatively light new charged boson, but the expected sensitivity of the Mu3e experiment would be capable of setting an upper bound on this angle as small as $\sim10^{-4}$ if the mass of this boson is within the range of few TeVs.

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arXiv2018

The Standard Model in extra dimensions and its Kaluza-Klein effective Lagrangian

We construct an effective theory for the SM with extra dimensions. We start from a theory governed by the extra-dimensional groups ISO$(1,3+n)$ and $G_{\rm SM}({\cal M}^{4+n})=SU_C(3,{\cal M}^{4+n})\times SU_L(2,{\cal M}^{4+n})\times U_Y(1,{\cal M}^{4+n})$, characterized by an unknown energy scale $Λ$ and valid at energies far below this scale. Assuming that the extra dimensions are much larger than the distance scale of this theory, we construct an effective theory with symmetry groups ISO(1,3), $G_{\rm SM}({\cal M}^{4})$. The theories are connected by a canonical transformation that hides ISO$(1,3+n)$, $G_{\rm SM}({\cal M}^{4+n})$ into ISO(1,3), $G_{\rm SM}({\cal M}^{4})$; KK fields receive mass. Using a set of orthogonal functions $\{f^{(\underline{0})},f^{(\underline{m})}\}$, generated by the Casimir invariant $\bar{P}^2$ associated with the translations subgroup $T(n)\subset$ISO$(n)$, we expand the degrees of freedom of ISO$(1,3+n)$, $G_{\rm SM}({\cal M}^{4+n})$ in general Fourier series, whose coefficients are the degrees of freedom of ISO(1,3), $G({\cal M}^{4})$. These functions, which correspond to the projection on $\{|\bar{x}\big>\}$ of the discrete basis $\{|0\big>,|p^{(\underline{m})}\big>\}$ of $\bar{P}^2$, are central to define the effective theory. Components along the ground state $f^{(\underline{0})}=\big <\bar x|0\big>$ do not receive mass at the compactification scale, so they are the SM fields; components along excited states $f^{(\underline{m})}=\big<\bar x|p^{(\underline{m})}\big>$ get mass at this scale, so they are KK excitations. For any direction $|p^{(\underline{m})}\neq0\big>$ there are a massive gauge field and a pseudo-Goldstone boson. We stress resemblances of this mass-generating mechanism with the Englert-Higgs mechanism and discuss physical implications. We include a full catalog of Lagrangian terms that can be used to calculate Feynman rules.

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arXiv2018

The role of hidden symmetries and Kaluza-Klein mass generation in extra-dimensional gauge theories

The transition from formulations with extra dimensions to Kaluza-Klein theories, aimed at extending the Standard Model, bears the ingredients of hidden symmetries and the Kaluza-Klein mechanism for mass generation. We explore these essential aspects in detail, and find that much can be said about them with no reference to the specific geometry of compact extra dimensions: the low-energy theory is determined, included dynamic variables and symmetries; mass terms arise; eigenfunctions that define Kaluza-Klein fields are fixed by the appropriate choice a Casimir invariant; there is a set of Kaluza-Klein pseudo-Goldstone bosons. Throughout our presentation, similarities and differences among spontaneous symmetry breaking, commonly present in conventional Standard Model extensions, and what happens in Kaluza-Klein theories are signaled and discussed.

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arXiv2017

Electric dipole moments of charged leptons at one loop in the presence of massive neutrinos

Violation of $CP$ invariance is a quite relevant phenomenon that is found in the Standard Model (SM), though in small amounts. This has been an incentive to look for high-energy descriptions in which $CP$ violation is increased, thus enhancing effects that are suppressed in the SM, such as the electric dipole moments (EDMs) of elementary particles. In the present investigation, we point out that charged currents in which axial couplings are different from vector couplings are able to produce one-loop contributions to EDMs of charged leptons if neutrinos are massive and if these currents violate $CP$. We develop our discussion around charged currents involving heavy neutrinos and a $W'$ gauge boson coupling to SM charged leptons. Using the most stringent bound on the electron EDM, provided by the ACME Collaboration, we determine that the upper bound on the difference between axial and vector currents lies within $\sim10^{-10}$ and $\sim10^{-7}$ for heavy-neutrino masses between $0.5\,{\rm TeV}$ and $6\,{\rm TeV}$ and if the $W'$ mass is within $0.45\,{\rm TeV}-7\,{\rm TeV}$. This possibility is analyzed altogether with the anomalous magnetic moments of charged leptons, among which we estimate, for the $τ$ lepton, an anomalous magnetic moment contribution between $\sim10^{-8}$ and $\sim10^{-10}$ for neutrino masses ranging from $0.5\,{\rm TeV}$ to $6\,{\rm TeV}$ and a $W'$ mass between $0.45\,{\rm TeV}$ and $7\,{\rm TeV}$. The general charged currents are also used to calculate the branching ratio for $μ\to eγ$, which gets suppressed if the set of masses of heavy neutrinos is quasidegenerate. In a scenario of nondegenerate neutrino masses, we find that regions of neutrino and $W'$ masses in which the contributions to this flavor changing branching ratio are lower than the current upper bound exist. We show that such regions can be widened if the $W'$ gauge boson mass is larger.

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arXiv2016

Distinctive ultraviolet structure of extra-dimensional Yang-Mills theories by integration of heavy Kaluza-Klein modes

One-loop Standard Model observables produced by virtual heavy Kaluza-Klein fields play a prominent role in the minimal model of universal extra dimensions. Motivated by this aspect, we integrate out all the Kaluza-Klein heavy modes coming from the Yang-Mills theory set on a spacetime with an arbitrary number, $n$, of compact extra dimensions. After fixing the gauge with respect to the Kaluza-Klein heavy gauge modes in a covariant manner, we calculate a gauge independent effective Lagrangian expansion containing multiple Kaluza-Klein sums that entail a bad divergent behavior. We use the Epstein-zeta function to regularize and characterize discrete divergences within such multiple sums, and then we discuss the interplay between the number of extra dimensions and the degree of accuracy of effective Lagrangians to generate or not divergent terms of discrete origin. We find that nonrenormalizable terms with mass dimension $k$ are finite as long as $k>4+n$. Multiple Kaluza-Klein sums of nondecoupling logarithmic terms, not treatable by Epstein-zeta regularization, are produced by four-dimensional momentum integration. On the grounds of standard renormalization, we argue that such effects are unobservable.

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arXiv2016

Implications of Lorentz violation on Higgs-mediated lepton flavor violation

The lepton flavor violating decay of the Higgs boson $H\to l_Al_B$ is studied within two qualitatively different extensions of the Yukawa sector: one renormalizable and the other nonrenormalizable; both incorporating Lorentz violation in a model-independent fashion. These extensions are characterized by Yukawa-like matrices, the former by a constant Lorentz 2-tensor $Y^{AB}_{μν}$, whereas the latter by a constant Lorentz vector $Y^{AB}_μ$. It is found that the experimental constraints on the decays $l_A\to l_Bγ$ severely restrict lepton flavor violating Higgs signals in the renormalizable scenario. In this context, it is found that $BR(H\to μ^\pm e^\mp)$ and $BR(H\to τ^\pm μ^\mp)$ cannot be larger than $10^{-18}$ and $10^{-11}$, respectively. In the nonrenormalizable scenario, transitions mediated by the Higgs or the $Z$ gauge boson are induced at tree level, and we find mild restrictions on lepton flavor violation. Using the experimental limits on the three-body decays $l_A \to l_B \bar{l}_Cl_C$ to constraint the vector $Y^{AB}_μ$, it is found that the branching ratio for the decays $H\to μ^\pm e^\mp$ is of about $4\times 10^{-9}$, more important, a branching ratio of $7\times 10^{-4}$ is found for the $τ^\pm μ^\mp$ mode. Accordingly, the decay $H \to τ^\pm μ^\mp$ could be at the reach of future measurements. The lepton flavor violating decays of the $Z$ gauge boson were also studied. In the renormalizable scenario, it was found the undetectable branching ratios $BR(Z\to μ^\pm e^\mp)<5.7\times 10^{-21}$ and $BR(Z\to τ^\pm μ^\mp)<2.0\times 10^{-12}$. In the nonrenormalizable scenario, it was found that $BR(Z\to μ^\pm e^\mp)<0.67\times 10^{-12}$ and $BR(Z\to τ^\pm μ^\mp)<1.12\times 10^{-7}$. Although the latter branching ratio is relatively large, it still could not be within the range of future measurements.

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