Generalised $\mu$-$\tau$ symmetries and calculable gauge kinetic and mass mixing in $U(1)_{L_\mu-L_\tau}$ models

Anjan S. Joshipura; Ketan M. Patel; Namit Mahajan

REVIEW 3 major objections 4 minor 1 cited by

Generalised $\mu$-$\tau$ symmetries and calculable gauge kinetic and mass mixing in $U(1)_{L_\mu-L_\tau}$ models

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Generalized mu-tau symmetries turn the arbitrary kinetic and mass mixing of U(1)_{L_mu-L_tau} models into finite loop predictions, with a seesaw benchmark that has practically no kinetic mixing and a vectorlike-lepton benchmark with…

desk verdict Solid, original one-loop calculations of kinetic and mass mixing in Lμ-Lτ models, but the abstract oversells the seesaw case by ignoring its own non-decoupling mass-mixing subcase. read the letter →

arxiv 1909.02331 v2 pith:3VZX24KV submitted 2019-09-05 hep-ph

Anjan S. Joshipura , Namit Mahajan , Ketan M. Patel This is my paper · ORCID

classification hep-ph

keywords kineticmixingmassU(1)L_mu-L_taumu-tauinterchangesymmetryreflectionseesawmodelvectorlikechargedleptonsnon-decoupling

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that in gauged $U(1)_{L_\mu-L_\tau}$ extensions of the Standard Model, the otherwise arbitrary kinetic mixing $\sin\chi$ and mass mixing $\delta M^2$ between hypercharge and the new $Z'$ boson become calculable loop-level quantities once a generalized $\mu$-$\tau$ symmetry forbids them at tree level. The authors show that exact $\mu$-$\tau$ interchange or reflection symmetries are incompatible with observed lepton mixing, so the symmetry must be spontaneously or softly broken. In the seesaw model with $\mu$-$\tau$ broken only by right-handed neutrino masses, the kinetic mixing is suppressed and practically absent, while a $\mu$-$\tau$-breaking Dirac mass term produces a finite non-decoupling contribution to $\delta M^2$. In a model with vectorlike charged leptons, a finite gauge mixing survives even as the vectorlike masses go to infinity. If correct, this turns two free parameters of the theory into predictions testable through precision electroweak data and rare tau decays.

What carries the argument

The central object is a family of generalized $\mu$-$\tau$ interchange and reflection symmetries acting on leptons together with sign flips of the gauge fields ($Z'_\mu\to-Z'_\mu$ or $B_\mu\to-B_\mu$). These symmetries are engineered so that the diagonalizing matrices obey Eq. (8), $|(U_f)_{\mu i}|^2=|(U_f)_{\tau i}|^2$, which eliminates the diagonal $Z'$ couplings that would otherwise generate kinetic and mass mixing at one loop through vacuum-polarization diagrams. The actual calculations are carried by trace identities: Eqs. (57,58) express the divergent part of $\mathcal{B}_{BZ'}$ in the seesaw model as traces of $m_D^\dagger m_D X_3$ and $m_D m_D^\dagger X_3$, which vanish for the assumed $m_D$ textures, and Eq. (81) performs the analogous cancellation in the vectorlike model. These identities are what convert an a priori divergent and incalculable quantity into a finite one-loop prediction.

What would settle it

Compute the coefficient of the $1/\epsilon$ pole in $\mathcal{B}_{BZ'}$ in the seesaw model with a generic $\mu$-$\tau$-breaking Dirac mass matrix $m_D$ that still fits neutrino oscillation data; the identities (57,58) fail and the pole returns, so the finiteness claim dies. Alternatively, measure the $Z$-$Z'$ mixing angle $\xi$ in precision electroweak data at the level predicted by Eq. (87) for the vectorlike model; the non-decoupling contribution stays finite at arbitrarily large $m_{4,5}$, so a null result at that sensitivity would rule it out.

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Extended reading notes

Core claim

The central claim is that imposing a generalized $\mu$-$\tau$ symmetry in the full theory, and breaking it softly or spontaneously, makes the tree-level-forbidden kinetic and mass mixing parameters finite and calculable. In the seesaw completion, the divergent part of $\mathcal{B}_{BZ'}$ cancels through the trace identities (57) and (58) when the Dirac neutrino mass matrix $m_D$ is $\mu$-$\tau$ symmetric or diagonal with unbroken $L_\mu-L_\tau$; the resulting $\mathcal{A}_{BZ'}$ is suppressed by light-neutrino masses and vanishes as the right-handed neutrinos decouple, whereas $\mathcal{B}_{BZ'}$ retains a finite non-decoupling piece proportional to $m_2^2-m_3^2$ when $m_D$ breaks $\mu$-$\tau$. In the vectorlike charged-lepton model, both parameters receive non-decoupling contributions that survive $m_{4,5}\to\infty$, so the gauge mixing can be large. The paper also establishes that exact versions of the forbidding symmetries are excluded: $\mu$-$\tau$ interchange gives $\theta_{13}=\theta_{23}=0$ in the lepton mixing matrix, and $\mu$-$\tau$ reflection forces vanishing CP violation.

Load-bearing premise

The calculation is load-bearing on the assumption that the Dirac neutrino mass matrix $m_D$ is exactly $\mu$-$\tau$ symmetric, or else diagonal with unbroken $L_\mu-L_\tau$; for a generic $m_D$ the divergent part of $\delta M^2$ returns and the mixing parameters cease to be calculable.

Editorial extensions

If this is right

In the seesaw model with $\mu$-$\tau$ symmetric $m_D$, $\sin\chi$ and $\delta M^2$ are finite, suppressed by the right-handed neutrino masses, and vanish as $M_R\to\infty$; the model is a complete $L_\mu-L_\tau$ framework with practically no kinetic mixing.
If $m_D$ preserves $L_\mu-L_\tau$ while breaking $\mu$-$\tau$, $\delta M^2$ acquires a finite non-decoupling contribution proportional to $m_2^2-m_3^2$ that survives $M_R\to\infty$.
With vectorlike charged leptons, both mixing parameters receive non-decoupling contributions that survive $m_{4,5}\to\infty$ and can be sizable, set by $\ln(m_4^2/m_5^2)$ and the mixing angles $\varphi_{L,R}$, $\theta_{L,R}$.
The seesaw scenario sidesteps the usual bounds from atomic parity violation, Borexino, COHERENT, and beam-dump experiments because the $Z'$ has essentially no coupling to electrons or quarks; the decisive probes are muon and tau processes.
The model reconciles $(g-2)_\mu$ with the observed $1.6\sigma$ excess in $\mathrm{BR}(\tau^-\to\mu^-\nu_\mu\nu_\tau)$ for $g'$ between $0.004$ and $0.006$ and $M_{Z'}$ between $1.12$ and $1.24$ GeV, which can be tested at Belle II and a muon collider.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the non-decoupling $\delta M^2$ result is correct, then integrating out the heavy leptons leaves behind a finite local mass-mixing counterterm; precision electroweak measurements of the $Z$-$Z'$ mixing angle $\xi$ can therefore probe the heavy sector even when the vectorlike leptons are far beyond direct collider reach.
The same forbid-at-tree-level, break-softly recipe should transfer to other abelian flavour symmetries, such as $U(1)_{L_e-L_\mu}$ or $U(1)_{B-L}$, where analogous residual symmetries would make their kinetic and mass mixing parameters calculable as well.
A sharper test of the vectorlike model would be to measure $\xi$ in coherent elastic neutrino-nucleus scattering or atomic parity violation with sensitivity to Eq. (87); because the contribution does not decouple, a null result at that level would exclude the model rather than merely push the new leptons to higher mass.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Worth a real look. This paper does something the literature mostly waves at: it actually computes the loop-generated kinetic mixing (sinχ) and mass mixing (δM²) in two concrete U(1)_{Lμ−Lτ} completions, and shows the results are finite under specific symmetry assumptions. The generalized μ−τ reflection/interchange classification in Sections II–III is a systematic treatment that extends earlier work, and the non-decoupling of the mass-mixing parameter in the vectorlike model — and, notably, in the diagonal-m_D seesaw subcase — is a real qualitative point, not a contrivance. The general one-loop formulas (34,35) and the finiteness identities (57,58) and (81) are laid out clearly enough to be checked; I did the algebra on the obvious pieces and it hangs together. The limiting expressions in Sec. V follow from the stated approximations. This is honest, careful work.

The softest spot is packaging, not math. The abstract says the seesaw model has 'practically no kinetic mixing' and that mixing vanishes when the right-handed neutrinos decouple. That is true for the μ−τ symmetric m_D subcase they emphasize, but in the diagonal m_D subcase their own Eq. (68) gives a finite, non-decoupling mass mixing, BBZ′ ≈ −(3g_Y g′/32π²)(m₂²−m₃²). The body does acknowledge this ('the neutrino mass mixing parameter can be large and independent of the right-handed neutrino masses if the Dirac mass matrix also breaks μ−τ'), so it is a summary defect, not a calculation error. Still, an abstract that leads readers toward the opposite conclusion in the very model advertised as 'complete' is something to fix before publication. A second, smaller gap: the two-loop claim in Section III is asserted without derivation; it is plausible from the symmetry structure, but it deserves an appendix or a clear reference. I could not fully verify a few signs and factors in Eq. (68), but the overall structure and the vanishing-mass limits are right.

The stress-test worry about generic μ−τ-breaking m_D is correct but not fatal: for such m_D, the one-loop BBZ′ is indeed divergent and incalculable, and the finiteness claims are conditional on m_D being μ−τ symmetric or diagonal (Lμ−Lτ symmetric). That condition is stated explicitly in the text, so a careful reader is not misled; only the abstract overreaches.

Who should read this: model builders working on gauged Lμ−Lτ, Z′ phenomenology, and the interplay of discrete lepton symmetries with calculable mixing parameters. It deserves a serious referee and, after the abstract is tightened and the two-loop statement is supported, publication.

Referee Report

3 major / 4 minor

Summary. The paper studies the one-loop generation of the kinetic mixing parameter sinχ and the mass mixing parameter δM² between hypercharge and a U(1)_{L_μ−L_τ} Z′. It classifies generalized μ-τ interchange/reflection symmetries that forbid these terms at tree level, shows that exact versions are incompatible with the observed lepton mixing pattern and/or leptonic CP violation, and then presents two UV frameworks in which the symmetry breaking that makes the mixing calculable is controlled: (A) the standard seesaw model with μ-τ breaking confined to the right-handed neutrino mass matrix, and (B) a model with vectorlike charged leptons whose soft μ-τ breaking generates finite contributions. The paper derives general one-loop formulas for A_BZ′ and B_BZ′, gives finiteness identities, and presents approximate expressions in the decoupling limit. It concludes that in the seesaw case the kinetic mixing is generically small and vanishes when the right-handed neutrinos decouple, while in the vectorlike model non-decoupling contributions survive; it also discusses constraints from rare tau decays and the muon (g−2).

Significance. If the derivation is correct, the paper is a useful step toward turning a priori arbitrary kinetic and mass mixing parameters into calculable functions of the neutrino and charged-lepton sector parameters in explicit models. Its strengths are the general one-loop formulas with the full divergent structure, the finiteness identities in Eqs. (57)–(58) and (81), and the concrete limiting expressions in Eqs. (65), (68), and (87) that can be confronted with experiment. The appearance of the atmospheric angle θ as an input from neutrino phenomenology is legitimate, and I do not see a circularity problem: θ is not tuned to reproduce the mixing parameters being computed. The main caveat is that the headline “practically no kinetic mixing” conclusion for the seesaw model is not uniform across the parameter space, because the diagonal-m_D subcase has a non-decoupling mass mixing in Eq. (68).

major comments (3)

[Abstract, §V.A, Eq. (68)] The blanket statement in the abstract that in case (i) “the kinetic mixing parameters are suppressed and vanish if the right-handed neutrinos decouple from the theory” is not correct for B_BZ′. For the diagonal-m_D subcase with unbroken L_μ−L_τ, Eq. (68) gives B_BZ′ → −(3 g_Y g′/32π²)(m_2² − m_3²) as M_R → ∞, i.e., a finite non-decoupling mass mixing. The body’s Summary later acknowledges this (“the neutrino mass mixing parameter can be large and independent of the right handed neutrino masses if the Dirac mass matrix also break the µ-τ symmetry”), but the abstract and the concluding “practically no kinetic mixing” statement do not. The abstract and Section VI should distinguish sinχ, which decouples, from δM², which does not for this subcase, and should present case (i) as two separate subcases.
[§V.A, Eqs. (57)–(58)] The finiteness of B_BZ′ in the seesaw model is conditional on a special structure of m_D. The divergent E-pole in Eq. (35) is removed only when the right-hand sides of Eqs. (57) and (58) either vanish individually (μ-τ symmetric m_D of Eq. (44)) or cancel (diagonal m_D with unbroken L_μ−L_τ, as in Eq. (66)); for a generic μ-τ-breaking m_D the divergent coefficient is proportional to Tr[(m_D†m_D − m_D m_D†)X_3], and δM² is not calculable without a symmetry-protecting counterterm. This restriction is stated in passing after Eq. (65), but it is load-bearing for the claim that the seesaw model is “a complete framework with practically no kinetic mixing.” It should be promoted to a clearly stated condition in the abstract and in Section VI.
[§VI, phenomenological discussion] The statement that constraints from precision electroweak tests, atomic parity violation, Borexino, and COHERENT “do not hold in the type of seesaw model discussed in section V A due to very suppressed Z-Z′ mixing” is too broad. In the diagonal-m_D subcase of §V.A, the non-decoupling B_BZ′ of Eq. (68) contributes to the observable mixing angle ξ through Eq. (39), so the Z−Z′ mixing is not necessarily suppressed enough to evade those constraints. The phenomenological claims in Section VI should be stated per subcase rather than for the seesaw model as a whole.

minor comments (4)

[Eq. (65)] The quantity Δ_atm appears without a definition; it should be identified explicitly (presumably m_ν3² − m_ν2², or Δm_31², with a stated sign convention) so that the numerical size of the first term in A_BZ′ can be assessed.
[Eq. (78)] In Eq. (78) the right-handed hypercharge matrix is written as the scalar “−1” rather than as −1 times the 5×5 identity; displaying it as a diagonal matrix would avoid ambiguity in later traces such as Eq. (81).
[Sec. II, Eq. (8)] Equation (8) is described as “termed as the µ-τ reflection symmetry,” but Eq. (8) is a condition on the mixing matrix that follows from the reflection symmetry, not the symmetry itself. The wording should be adjusted to avoid conflating the two.
[Appendix A] There is a typographical spacing error in the heading “Form2≪q2,” which should read “For m² ≪ q²”; the same type of spacing issue appears in a few other places, for example after Eq. (87).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the one-loop kinetic- and mass-mixing calculations are self-contained loop evaluations from stated model inputs, not fits to gauge-mixing data.

full rationale

The central derivation is self-contained. In Sections IV and V, the paper starts from the general one-loop vacuum-polarization formulas, Eqs. (34)-(35), proves the finiteness identities, Eqs. (57)-(58), using the seesaw relations, Eqs. (55)-(56), and then evaluates ABZ' and BBZ' in the stated limits. The resulting expressions, Eqs. (65), (68), and (86)-(87), depend on Yukawa/mass parameters, gauge couplings, and lepton mixing angles that are model inputs or neutrino-oscillation data; none of these are fitted to the kinetic-mixing or mass-mixing quantities being 'predicted.' The atmospheric angle θ entering Eq. (65) is taken from neutrino phenomenology, not tuned to reproduce sinχ. The paper's own flagged limitations—that a generic μ-τ-breaking mD would make BBZ' divergent (Section IV: 'the resultant Z-Z′ mixing remains divergent and incalculable in the minimal set up with general mass matrices') and that the diagonal-mD subcase gives a finite non-decoupling BBZ' in Eq. (68)—are conditionality/accuracy caveats on the abstract's blanket 'practically no kinetic mixing' statement, not circular reductions. The self-citations, e.g., Refs. [46]-[48] and [61], are used for standard lepton-mixing consequences or numerical inputs; the load-bearing radiative-correction framework is followed from Refs. [58]-[60], with the identities and integrals exhibited in the present paper. No equation is defined in terms of its own output, and no fitted parameter is renamed as a prediction. Under the hard rules, the absence of such reductions gives score 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper's results depend on the assumed U(1)_{L_mu-L_tau} charge assignments, the discrete mu-tau symmetries and their spontaneous or soft breaking, and the mass hierarchies used to expand the loop integrals. These are model inputs, not fitted to the kinetic-mixing output.

free parameters (4)

theta (2-3 neutrino mixing angle in KL, Eq. 63)
Input from the neutrino sector that enters KM expressions through cos 2theta; constrained by oscillation data but not fitted to the KM parameters in this paper.
M2, M3 (right-handed neutrino masses)
Free model parameters that set the Z' mass scale and the suppression of kinetic mixing in the seesaw model.
m (common Dirac mass in the 2-3 sector, model A)
Free parameter appearing in the m^2/M^2 ratios that control the magnitude of the calculated mixing.
m4, m5, m, mtilde (vectorlike sector parameters, model B)
Mass and Yukawa parameters of the vectorlike lepton sector; they determine the size and non-decoupling behavior of the computed mixing.

assumptions (5)

domain assumption The Standard Model is extended with a gauged U(1)_{L_mu-L_tau} with the charge assignments in Eq. (3).
The entire calculation assumes this gauge extension and its fermionic charge assignments as the framework.
domain assumption The discrete mu-tau interchange or reflection symmetries are symmetries of the Yukawa interactions and are spontaneously or softly broken such that tree-level KM is forbidden.
The paper relies on these symmetries to forbid KM at tree level and to control the structure of mass matrices; the breaking pattern determines the calculability.
standard math Dimensional regularization and the standard loop functions in Appendix A are used for the one-loop computation.
The loop integrals and renormalization scheme are standard QFT tools.
domain assumption The seesaw limit mD << MR and the decoupling hierarchy m_l << M_f are assumed for the simplified KM formulas.
The explicit formulas in Sec. V are derived in these limits, which are typical for such models.
domain assumption The singlet fields that break L_mu-L_tau do not couple directly to the Z boson, so their contribution to Z-Z' mixing is suppressed.
In model A, the paper argues that singlet VEVs only induce mixing through a suppressed quartic coupling; this is needed for the 'practically no kinetic mixing' conclusion.

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Pith. "Pith review of Generalised $\mu$-$\tau$ symmetries and calculable gauge kinetic and mass mixing in $U(1)_{L_\mu-L_\tau}$ models." pith.science (2026). https://pith.science/paper/3VZX24KV

@misc{pith2026190902331,
  author       = {Pith},
  title        = {Pith review of: Generalised $\mu$-$\tau$ symmetries and calculable gauge kinetic and mass mixing in $U(1)_L_\mu-L_\tau$ models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3VZX24KV}},
  note         = {Machine review of arXiv:1909.02331}
}

abstract

Extensions of the standard model with a $U(1)$ gauge symmetry contain gauge invariant kinetic mixing, $\sin\chi$, and gauge non-invariant mass mixing, $\delta M^2$, between the hypercharge and the new gauge boson $Z'$. These represent a priori incalculable but phenomenologically important parameters of the theory. They become calculable if there exist spontaneously or softly broken symmetries which forbid them at tree level but allow their generation at the loop level. We discuss various symmetries falling in this category in the context of the gauged $L_\mu-L_\tau$ models and their interplay with lepton mixing. It is shown that one gets phenomenologically inconsistent lepton mixing parameters if these symmetries are exact. Spontaneous breaking of these symmetries can lead to consistent lepton mixing and also generates finite and calculable values of these parameters at one or two loop order depending on the underlying symmetry. We calculate these parameters in two specific cases: (i) the standard seesaw model with $\mu$-$\tau$ symmetry broken by the masses of the right-handed neutrinos and (ii) in a model containing a pair of vectorlike charged leptons which break $\mu$-$\tau$ symmetry. In case (i), the right-handed neutrinos are the only source of gauge mixing. The kinetic mixing parameters are suppressed and vanish if the right-handed neutrinos decouple from the theory. In contrast, there exists a finite gauge mixing in case (ii) which survives even when the masses of vectorlike leptons are taken to infinity, exhibiting non-decoupling behaviour. The seesaw model discussed here represents a complete framework with practically no kinetic mixing and hence can survive a large number of experimental probes used to rule out specific ranges in the coupling $g^\prime$ and mass $M_{Z^\prime}$. The model can generate non-universality in tau decays, which can be tested in future experiments.

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Works this paper leans on

64 extracted references · 22 canonical work pages · cited by 1 Pith paper

[1]

Foot, Mod

R. Foot, Mod. Phys. Lett. A6, 527 (1991)

1991
[2]

X. G. He, G. C. Joshi, H. Lew, and R. R. Volkas, Phys. Rev. D43, 22 (1991)

1991
[3]

seesaw-like

+q2 (m2 2 M 2 2 − m2 3 M 2 3 )) . (68) 16 Unlike in the previous case, there is a ﬁnite non-decoupling contribution in BBZ ′ which does not vanish when the right handed neutrino masses are taken to inﬁnity. This contribution is proportional to the amount of µ-τ breaking, m2 2−m2 3, in mD. ABZ ′ is still seen to vanish when the RH neutrino masses go to inﬁ...

work page 2015
[4]

R. Foot, X. G. He, H. Lew, and R. R. Volkas, Phys. Rev. D50, 4571 (1994), arXiv:hep- ph/9401250 [hep-ph]

arXiv 1994
[5]

X.-G. He, G. C. Joshi, H. Lew, and R. R. Volkas, Phys. Rev. D44, 2118 (1991)

1991
[6]

E. Ma, D. P. Roy, and S. Roy, Phys. Lett. B525, 101 (2002), arXiv:hep-ph/0110146 [hep-ph]

arXiv 2002
[7]

S. Baek, N. G. Deshpande, X. G. He, and P. Ko, Phys. Rev. D64, 055006 (2001), arXiv:hep- ph/0104141 [hep-ph]

arXiv 2001
[8]

Soft breaking of $L_\mu-L_\tau$ symmetry: Light neutrino spectrum and Leptogenesis

B. Adhikary, Phys. Rev. D74, 033002 (2006), arXiv:hep-ph/0604009 [hep-ph]

work page Pith review arXiv 2006

Show all 64 references

[9]

A. S. Joshipura and S. Mohanty, Phys. Lett. B584, 103 (2004), arXiv:hep-ph/0310210 [hep- ph]

2004 arXiv
[10]

Baek and P

S. Baek and P. Ko, JCAP 0910, 011 (2009), arXiv:0811.1646 [hep-ph]

2009 arXiv
[11]

E. J. Chun and K. Turzynski, Phys. Rev. D76, 053008 (2007), arXiv:hep-ph/0703070 [hep-ph]

2007 arXiv
[12]

Heeck and W

J. Heeck and W. Rodejohann, Phys. Rev. D84, 075007 (2011), arXiv:1107.5238 [hep-ph]

2011 arXiv
[13]

Heeck and W

J. Heeck and W. Rodejohann, J. Phys. G38, 085005 (2011), arXiv:1007.2655 [hep-ph]

2011 arXiv
[14]

Das and S

M. Das and S. Mohanty, Phys. Rev. D89, 025004 (2014), arXiv:1306.4505 [hep-ph]

2014 arXiv
[15]

Harigaya, T

K. Harigaya, T. Igari, M. M. Nojiri, M. Takeuchi, and K. Tobe, JHEP 03, 105 (2014), arXiv:1311.0870 [hep-ph]

2014 arXiv
[16]

Altmannshofer, S

W. Altmannshofer, S. Gori, M. Pospelov, and I. Yavin, Phys. Rev. D89, 095033 (2014), arXiv:1403.1269 [hep-ph]

2014 arXiv
[17]

S. S. Chatterjee, A. Dasgupta, and S. K. Agarwalla, JHEP 12, 167 (2015), arXiv:1509.03517 [hep-ph]. 24

2015 arXiv
[18]

Altmannshofer, S

W. Altmannshofer, S. Gori, S. Profumo, and F. S. Queiroz, JHEP 12, 106 (2016), arXiv:1609.04026 [hep-ph]

2016 arXiv
[19]

Altmannshofer, C.-Y

W. Altmannshofer, C.-Y. Chen, P. S. Bhupal Dev, and A. Soni, Phys. Lett. B762, 389 (2016), arXiv:1607.06832 [hep-ph]

2016 arXiv
[20]

M. Ibe, W. Nakano, and M. Suzuki, Phys. Rev. D95, 055022 (2017), arXiv:1611.08460 [hep- ph]

2017 arXiv
[21]

Biswas, S

A. Biswas, S. Choubey, and S. Khan, JHEP 02, 123 (2017), arXiv:1612.03067 [hep-ph]

2017 arXiv
[22]

Araki, S

T. Araki, S. Hoshino, T. Ota, J. Sato, and T. Shimomura, Phys. Rev. D95, 055006 (2017), arXiv:1702.01497 [hep-ph]

2017 arXiv
[23]

Patra, S

S. Patra, S. Rao, N. Sahoo, and N. Sahu, Nucl. Phys. B917, 317 (2017), arXiv:1607.04046 [hep-ph]

2017 arXiv
[24]

Chen and T

C.-H. Chen and T. Nomura, Phys. Lett. B777, 420 (2018), arXiv:1707.03249 [hep-ph]

2018 arXiv
[25]

Chen and T

C.-H. Chen and T. Nomura, Nucl. Phys. B940, 292 (2019), arXiv:1705.10620 [hep-ph]

2019 arXiv
[26]

Arcadi, T

G. Arcadi, T. Hugle, and F. S. Queiroz, Phys. Lett. B784, 151 (2018), arXiv:1803.05723 [hep-ph]

2018 arXiv
[27]

Nomura and H

T. Nomura and H. Okada, Phys. Rev. D97, 095023 (2018), arXiv:1803.04795 [hep-ph]

2018 arXiv
[28]

Nomura and H

T. Nomura and H. Okada, Phys. Lett. B783, 381 (2018), arXiv:1805.03942 [hep-ph]

2018 arXiv
[29]

Banerjee, P

H. Banerjee, P. Byakti, and S. Roy, Phys. Rev. D98, 075022 (2018), arXiv:1805.04415 [hep- ph]

2018 arXiv
[30]

Banerjee and S

H. Banerjee and S. Roy, Phys. Rev. D99, 035035 (2019), arXiv:1811.00407 [hep-ph]

2019 arXiv
[31]

Singirala, S

S. Singirala, S. Sahoo, and R. Mohanta, Phys. Rev. D99, 035042 (2019), arXiv:1809.03213 [hep-ph]

2019 arXiv
[32]

Escudero, D

M. Escudero, D. Hooper, G. Krnjaic, and M. Pierre, JHEP 03, 071 (2019), arXiv:1901.02010 [hep-ph]

2019 arXiv
[33]

Biswas and A

A. Biswas and A. Shaw, JHEP 05, 165 (2019), arXiv:1903.08745 [hep-ph]

2019 arXiv
[34]

Heeck, M

J. Heeck, M. Lindner, W. Rodejohann, and S. Vogl, SciPost Phys. 6, 038 (2019), arXiv:1812.04067 [hep-ph]

2019 arXiv
[35]

T. K. Poddar, S. Mohanty, and S. Jana, (2019), arXiv:1908.09732 [hep-ph]

2019 arXiv
[36]

(34) in the limit q2≪m2 µ

result ABZ ′≈− gYg′ 16π2 lnm2 µ m2 τ (40) from Eq. (34) in the limit q2≪m2 µ. If µ-τ interchange or reﬂection symmetry is imposed on Ml then the resulting condition Eq. (8) leads to vanishing diagonal elements in XL,R which givesABZ ′ = 0. Finiteness of ABZ ′ obtained at 1-loo...
[37]

S. N. Gninenko and N. V. Krasnikov, Phys. Lett. B783, 24 (2018), arXiv:1801.10448 [hep-ph]

2018 arXiv
[38]

Holdom, Phys

B. Holdom, Phys. Lett. 166B, 196 (1986)

1986
[39]

del Aguila, G

F. del Aguila, G. D. Coughlan, and M. Quiros, Nucl. Phys. B307, 633 (1988), [Erratum: Nucl. Phys.B312,751(1989)]

1988
[40]

K. S. Babu, C. F. Kolda, and J. March-Russell, Phys. Rev. D54, 4635 (1996), arXiv:hep- ph/9603212 [hep-ph]

1996
[41]

Altmannshofer, S

W. Altmannshofer, S. Gori, M. Pospelov, and I. Yavin, Phys. Rev. Lett. 113, 091801 (2014), arXiv:1406.2332 [hep-ph]

2014 arXiv
[42]

Kaneta and T

Y. Kaneta and T. Shimomura, PTEP 2017, 053B04 (2017), arXiv:1701.00156 [hep-ph]. 25

2017 arXiv
[43]

Abdullah, J

M. Abdullah, J. B. Dent, B. Dutta, G. L. Kane, S. Liao, and L. E. Strigari, Phys. Rev. D98, 015005 (2018), arXiv:1803.01224 [hep-ph]

2018 arXiv
[44]

Bauer, P

M. Bauer, P. Foldenauer, and J. Jaeckel, JHEP 07, 094 (2018), arXiv:1803.05466 [hep-ph]

2018 arXiv
[45]

Kamada and H.-B

A. Kamada and H.-B. Yu, Phys. Rev. D92, 113004 (2015), arXiv:1504.00711 [hep-ph]

2015 arXiv
[46]

P. F. Harrison and W. G. Scott, Phys. Lett.B547, 219 (2002), arXiv:hep-ph/0210197 [hep-ph]

2002 arXiv
[47]

Grimus and L

W. Grimus and L. Lavoura, Phys. Lett. B579, 113 (2004), arXiv:hep-ph/0305309 [hep-ph]

2004 arXiv
[48]

A. S. Joshipura and K. M. Patel, Phys. Lett. B749, 159 (2015), arXiv:1507.01235 [hep-ph]

2015 arXiv
[49]

Gupta, A

S. Gupta, A. S. Joshipura, and K. M. Patel, JHEP 09, 035 (2013), arXiv:1301.7130 [hep-ph]

2013 arXiv
[50]

A. S. Joshipura, Eur. Phys. J. C53, 77 (2008), arXiv:hep-ph/0512252 [hep-ph]

2008 arXiv
[51]

Grimus and L

W. Grimus and L. Lavoura, Phys. Lett. B774, 325 (2017), arXiv:1708.09809 [hep-ph]

2017 arXiv
[52]

Altarelli and F

G. Altarelli and F. Feruglio, Rev. Mod. Phys. 82, 2701 (2010), arXiv:1002.0211 [hep-ph]

2010 arXiv
[53]

A. Yu. Smirnov, Proceedings, 2nd Symposium on Prospects in the Physics of Discrete Symme- tries (DISCRETE 2010): Rome, Italy, December 6-11, 2010 , J. Phys. Conf. Ser. 335, 012006 (2011), arXiv:1103.3461 [hep-ph]

2011 arXiv
[54]

S. F. King and C. Luhn, Rept. Prog. Phys. 76, 056201 (2013), arXiv:1301.1340 [hep-ph]

2013 arXiv
[55]

Ishimori, T

H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada, and M. Tanimoto, Prog. Theor. Phys. Suppl. 183, 1 (2010), arXiv:1003.3552 [hep-th]

2010 arXiv
[56]

del Aguila, M

F. del Aguila, M. Masip, and M. Perez-Victoria, Nucl. Phys. B456, 531 (1995), arXiv:hep- ph/9507455 [hep-ph]

1995
[57]

Choubey and W

S. Choubey and W. Rodejohann, Eur. Phys. J. C40, 259 (2005), arXiv:hep-ph/0411190 [hep- ph]

2005 arXiv
[58]

Dev, (2017), arXiv:1710.02878 [hep-ph]

A. Dev, (2017), arXiv:1710.02878 [hep-ph]

2017 arXiv
[59]

K. Asai, K. Hamaguchi, N. Nagata, S.-Y. Tseng, and K. Tsumura, Phys. Rev. D99, 055029 (2019), arXiv:1811.07571 [hep-ph]

2019 arXiv
[60]

Pilaftsis, Z

A. Pilaftsis, Z. Phys. C55, 275 (1992), arXiv:hep-ph/9901206 [hep-ph]

1992 arXiv
[61]

J. G. Korner, A. Pilaftsis, and K. Schilcher, Phys. Lett. B300, 381 (1993), arXiv:hep- ph/9301290 [hep-ph]

1993
[62]

Grimus and L

W. Grimus and L. Lavoura, Phys. Lett. B546, 86 (2002), arXiv:hep-ph/0207229 [hep-ph]

2002 arXiv
[63]

Broggio, E

A. Broggio, E. J. Chun, M. Passera, K. M. Patel, and S. K. Vempati, JHEP 11, 058 (2014), arXiv:1409.3199 [hep-ph]

2014 arXiv
[64]

M. E. Peskin and D. V. Schroeder, An Introduction to quantum ﬁeld theory (Addison-Wesley, Reading, USA, 1995). 26

1995

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Foot, Mod

R. Foot, Mod. Phys. Lett. A6, 527 (1991)

1991

[2] [2]

X. G. He, G. C. Joshi, H. Lew, and R. R. Volkas, Phys. Rev. D43, 22 (1991)

1991

[3] [3]

seesaw-like

+q2 (m2 2 M 2 2 − m2 3 M 2 3 )) . (68) 16 Unlike in the previous case, there is a ﬁnite non-decoupling contribution in BBZ ′ which does not vanish when the right handed neutrino masses are taken to inﬁnity. This contribution is proportional to the amount of µ-τ breaking, m2 2−m2 3, in mD. ABZ ′ is still seen to vanish when the RH neutrino masses go to inﬁ...

work page 2015

[4] [4]

R. Foot, X. G. He, H. Lew, and R. R. Volkas, Phys. Rev. D50, 4571 (1994), arXiv:hep- ph/9401250 [hep-ph]

arXiv 1994

[5] [5]

X.-G. He, G. C. Joshi, H. Lew, and R. R. Volkas, Phys. Rev. D44, 2118 (1991)

1991

[6] [6]

E. Ma, D. P. Roy, and S. Roy, Phys. Lett. B525, 101 (2002), arXiv:hep-ph/0110146 [hep-ph]

arXiv 2002

[7] [7]

S. Baek, N. G. Deshpande, X. G. He, and P. Ko, Phys. Rev. D64, 055006 (2001), arXiv:hep- ph/0104141 [hep-ph]

arXiv 2001

[8] [8]

Soft breaking of $L_\mu-L_\tau$ symmetry: Light neutrino spectrum and Leptogenesis

B. Adhikary, Phys. Rev. D74, 033002 (2006), arXiv:hep-ph/0604009 [hep-ph]

work page Pith review arXiv 2006