General formulae for f1 --> f2 gamma

L. Lavoura

REVIEW

General formulae for f1 --> f2 gamma

L. Lavoura

High Energy Physics - Phenomenology · Submitted 25 Feb 2003 · 7 references

Pith's verdict: A solid, usable formula sheet for one-loop f1→f2γ; the scalar half is clean, the vector half is right but under-validated, and one numerical cross-check would have settled it.

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arXiv IDhep-ph/0302221 (version 1)

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CitationL. Lavoura. "General formulae for f1 --> f2 gamma." arXiv:hep-ph/0302221 (2003). https://pith.science/paper/OT4PQCZ4

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Editor's note

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Lavoura's paper is a formula sheet, and a useful one: complete one-loop σ_L and σ_R for f1→f2γ mediated by a scalar or a vector boson plus an internal fermion, with arbitrary charges, arbitrary external fermion masses, and both chiralities in the gauge coupling. The scalar case, Eqs. (32)–(39), is a clean transcription of standard integrals and can be used as it stands. The vector case, Eqs. (51)–(60), is the part people actually need, since V+A couplings are exactly what left-right and 3-3-1 models bring and the Cheng–Li result covers only left-handed couplings. The massless-limit functions in Eqs. (61), (63), (64) are stated to be new, and I see no reason to doubt it. The off-shell-photon generality is a nice touch for anyone embedding this in a three-body decay like f1→f2 e+e−, and the δ_L, δ_R operators are given for that purpose even though the paper only tabulates σ_L, σ_R. That is honest scoping rather than an omission. Two things to flag. The reader's worry that the Goldstone couplings in Eq. (49) are an unproved ansatz is misplaced: those are fixed by gauge invariance of the fermion mass term, not chosen. The triple vertex in Eq. (50) is the standard Yang–Mills form and would fail only for a boson with anomalous magnetic or quadrupole moments, which is outside what the paper claims. The real soft spot is the other one: the three new vector loop functions are not checked against anything. Eq. (62) is identified with Cheng–Li; the other three stand alone. For a paper whose entire value is that people will substitute these expressions into their own models, one numerical comparison against an independent evaluation would have cost little and bought a lot. The cancellation of the divergent x against two-point integrals is asserted, not displayed — standard, but in a formula paper you would like to see the step. Minor: the Conclusions cite undefined 'b1, b2' integrals; a typo. Who should read it: anyone computing radiative lepton-flavour-violating decays in a model with right-handed or vector couplings. It saves a week of algebra. Worth a referee, and worth citing if you work in that corner of the field.

Referee report

Major objections (2)

[§5, Eqs. (49)–(50)] The vector-boson result rests on the assumed triple-gauge vertex of Eq. (50) and the Goldstone couplings of Eq. (49). The paper states this assumption explicitly in §5 (''I assume that, just as in the Standard Model''), but the abstract and the §6 conclusions describe the results as ''completely general'' without qualification. The scope of ''general'' should be stated precisely: general in masses, charges, and chiral couplings, but restricted to minimal Yang–Mills-like triple and Goldstone vertices. This is adequate for the elementary gauge bosons of the left–right-symmetric and 3-3-1 models cited in §1, but it would not cover a spin-1 boson with anomalous magnetic or quadrupole moments. The Goldstone couplings of Eq. (49) are less of an independent ansatz than they may appear, since they are fixed by gauge invariance of the fermion mass term; the triple vertex is the genuine assumption and should be flagged as such in the abstract and conclusions.
[§5, Eqs. (61), (63), (64)] Three of the four massless-limit vector functions are new and are not validated against any independent computation; only Eq. (62) is cross-identified with ref. [3]. The paper does give the explicit combinations of a, c, d that enter the left-hand sides of Eqs. (61)–(64), so a reader can in principle verify them by substituting Eqs. (20)–(25), but no such check is shown. For a formula paper whose value is substitution-ready expressions, an explicit algebraic or numerical verification of the new functions would materially increase confidence. A short appendix showing the substitution, or a numerical comparison of the exact Eqs. (51)–(60) against the massless-limit Eqs. (61)–(64), would close this gap.

Minor objections (6)

[§2, Eq. (7)] The partial-width formula should state the convention for σ_L and σ_R explicitly, in particular that the electric charge e is factored out of the amplitude as stated below Eq. (1). The dimension and massless limit of Eq. (7) are consistent with the standard dipole formula, but the convention is worth spelling out for users.
[§4 and §5] The claim that the results are ''fully general — they hold even when the photon is off-shell, q² ≠ 0'' applies only to σ_L and σ_R. The full off-shell amplitude also requires δ_L and δ_R, which the paper explicitly does not give (as acknowledged in §2). The wording in §4 and §5 should be qualified accordingly.
[§3, Eqs. (11)–(19)] The divergent coefficient x in Eq. (13) is not given explicitly; the paper states only that it cancels against two-point integrals. It would help to state the cancellation explicitly or to give x, so that a reader reconstructing the two-point contributions can check the finite result.
[§4, Eqs. (40)–(43)] The cross-identification of these functions with H(r), G(r), K(r), and I(r) of ref. [4] (with r = t − 1 and apart from a common factor 2) is a useful check and should be highlighted as such. The endpoint behaviour quoted below Eq. (43) is consistent with the explicit expressions.
[§5, Eqs. (61)–(64)] The statement that the function in Eq. (62) has been given in ref. [3] while those in Eqs. (61), (63), and (64) are new is clear and appropriately specific. The sign and monotonicity checks quoted below Eq. (64) are consistent with the explicit expressions and are a helpful sanity check.
[Title and abstract] The title and abstract use ''f1 --> f2 gamma'' while the body uses ''f1 → f2γ''. This is a cosmetic inconsistency but worth unifying for the final version.

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The paper

The paper claims

Lavoura writes down the complete one-loop amplitude for the radiative decay $f_1$ to $f_2$ plus a photon, covering both a general scalar (Yukawa) mediator and a general vector mediator with mixed left- and right-handed couplings, for any in

The reading

Radiative decays such as $mu$ to $e$ plus a photon are standard probes of physics beyond the Standard Model, but each new model normally forces its authors through a fresh one-loop calculation. This paper's aim is to make that calculation unnecessary: it presents closed, general expressions for the two coefficients $sigma_L$ and $sigma_R$ that carry the whole on-shell $f_1$ to $f_2 gamma$ amplitude, for two broad classes of interaction. The first class is a spin-0 boson $B$ coupled to fermions through the identity and $gamma_5$ (a general Yukawa coupling); the second is a spin-1 boson coupled through arbitrary mixtures of $V-A$ and $V+A$. In both cases the external fermion masses, the internal fermion and boson masses, and all electric charges are left arbitrary, and the intermediate steps keep the photon momentum $q^2$ general. If the formulae hold, a model builder identifies the internal fermion $F$ and boson $B$, reads off the coupling products $lambda$, $rho$, $zeta$, $upsilon$ (or their primed counterparts), substitutes into the listed equations, and obtains the decay amplitude without redoing the loop integrals. One internal wrinkle: the closing section refers to loop integrals $b_1$, $b_2$ that are never defined anywhere earlier in the paper.

What carries the argument

The carrying device is a reduction of the one-loop vertex onto a basis of two gauge-invariant Dirac structures, $Sigma^L_mu = (p_1+p_2)_mu gamma_L - gamma_mu(m_2 gamma_L + m_1 gamma_R)$ and its right-handed partner, so that the whole calculation is compressed into two numbers $sigma_L$ and $sigma_R$ with dimensions of inverse mass. The loop content is packaged into a small set of finite three-point integrals: $a$ (scalar), $c_1, c_2$ (one power of loop momentum), and $d_1, d_2, f$ (two powers), defined with the boson or the fermion running in the loop, together with their barred partners obtained by swapping which propagator carries the external momenta. The divergent piece $x$ of the two-power integral cancels against two-point integrals once the photon attaches to the external legs, which is why only the finite coefficients appear in the final answer.

Extended reading notes

Core claim

The central claim is that the one-loop vertex $f_1(p_1)$ to $f_2(p_2) gamma(q)$ can be written, after the two-point self-energy diagrams are folded into three-point integrals, as $e epsilon^{*mu}(q) M_mu$ with $M_mu = bar u_2 (sigma_L Sigma^L_mu + sigma_R Sigma^R_mu + delta_L Delta^L_mu + delta_R Delta^R_mu) u_1$, where only $sigma_L$ and $sigma_R$ survive for an on-shell photon. For the Yukawa case the author obtains $sigma_L = Q_F(rho k_1 + lambda k_2 + upsilon k_3) + Q_B(rho bar k_1 + lambda bar k_2 + upsilon bar k_3)$ and $sigma_R = Q_F(lambda k_1 + rho k_2 + zeta k_3) + Q_B(lambda bar k_1 + rho bar k_2 + zeta bar k_3)$, with $k_i$ and $bar k_i$ built from the finite integrals $a, c_1, c_2, d_1, d_2, f$ and their barred versions. For the gauge case the analogous $sigma_L$ and $sigma_R$ are given in terms of eight functions $y_1$ to $y_4$ and $bar y_1$ to $bar y_4$, assembled from the same integrals plus mass ratios $m_F^2/m_B^2$ and $m_i^2/m_B^2$. The author stresses that these expressions involve no approximation and remain valid when $q^2$ is not zero, and that they collapse to compact functions of $t = m_F^2/m_B^2$ in the limit $m_1 = m_2 = 0$, $q^2 = 0$; those limit functions reproduce the known result of Cheng and Li and the functions of He, Cheng and Li, while three of them are stated to be new. The partial width follows as $Gamma = (m_1^2 - m_2^2)^3 (|sigma_L|^2 + |sigma_R|^2) / (16 pi m_1^3)$. The author notes explicitly that only $sigma_L$ and $sigma_R$ are given, so the $delta_L$ and $delta_R$ coefficients needed when the photon is a sub-process of a larger decay such as $f_1$ to $f_2 e^+ e^-$ are left for elsewhere.

Load-bearing premise

For the vector-boson case the author assumes that the photon coupling to two vector bosons and the couplings of the would-be Goldstone bosons take the Standard-Model forms of Eqs. (50) and (49), computed in Feynman-'t Hooft gauge, and a genuinely different gauge sector need not respect that ansatz.

Editorial extensions

If this is right

Any model whose new physics enters through a single fermion $F$ and a single boson $B$ of the stated types can obtain its $f_1$ to $f_2 gamma$ amplitude by substitution into Eqs. (32)-(39) or (51)-(60), without a dedicated loop computation.
The results apply directly to $mu$ to $e gamma$, $tau$ to $e gamma$, $tau$ to $mu gamma$ and to quark-level flavour-changing decays, at the electroweak level where QCD corrections are absent.
Because $V+A$ couplings are retained, the formulae cover left-right-symmetric models, models with vector-like fermions, and the 3-3-1 model with its singly and doubly charged vector bosons.
The mass-dependent expressions allow the external fermion masses to be kept, so decays between fermions of comparable mass are not forced into the massless-external-fermion approximation.
The compact functions of $t = m_F^2/m_B^2$ in Eqs. (40)-(43) and (61)-(64) give ready-made decoupling behaviour as the internal fermion becomes heavy.

Reading between the lines

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

The paper's real economy is reusability: a single table of integrals plus two algebraic assembly rules replaces an entire genre of repeated one-loop computations, which suggests the natural next step is to publish the analogous assembly rules for the remaining tensor structures, starting with the $delta_L$ and $delta_R$ coefficients the author sets aside.
Because the results hold off-shell in $q^2$, they can be inserted as sub-diagrams into three-body decays such as $f_1$ to $f_2 e^+ e^-$, but only once the omitted $delta_L$, $delta_R$ terms are supplied; without them the off-shell claim is not yet usable in practice.
The restriction to a single internal fermion and a single internal boson means models with several contributing generations or with mixed neutral-current states must sum the formulae coherently, including the relative phases of the coupling products $lambda$, $rho$, $zeta$, $upsilon$; interference between the $V-A$ and $V+A$ pieces is where a genuine model dependence re-enters.

Original abstract

At one-loop level the decay f1 --> f2 gamma, where f1 and f2 are two spin-1/2 particles with the same electric charge, is mediated by a boson B and a spin-1/2 fermion F. The boson B may have either spin 0 - interacting with the fermions through Dirac matrices 1 and gamma5 - or spin 1 - with V+A and V-A couplings to the fermions. I give general formulae for the one-loop electroweak amplitude of f1 --> f2 gamma in all these cases.

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At one-loop level the decay f1 --> f2 gamma, where f1 and f2 are two spin-1/2 particles with the same electric charge, is mediated by a boson B and a spin-1/2 fermion F. The boson B may have either spin 0 - interacting with the fermions through Dirac matrices 1 and gamma5 - or spin 1 - with V+A and V-A couplings to the fermions. I give general formulae for the one-loop electroweak amplitude of f1 --> f2 gamma in all these cases.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	At one-loop level the decay f1 --> f2 gamma, where f1 and f2 are two spin-1/2 particles with the same electric charge, is mediated by a boson B and a spin-1/2 fermion F.	background 0.96	0.15	0.08	asserted 1.00
s1	The boson B may have either spin 0 - interacting with the fermions through Dirac matrices 1 and gamma5 - or spin 1 - with V+A and V-A couplings to the fermions.	background 0.55	0.14	0.07	hedged possibility 0.98
s2	I give general formulae for the one-loop electroweak amplitude of f1 --> f2 gamma in all these cases.	contribution 0.98	0.95	0.72	asserted 1.00

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References

Works cited (7)

[4]

B. He, T.-P. Cheng, and L.-F. Li, Phys. Lett. B 553, 277 (2003)

View on Pith 2003
[3]

T.-P. Cheng and L.-F. Li, Gauge theory of elementary particle physics — problems and solutions (Oxford University Press, 2000), pp. 244–250

View on Pith 2000
[1]

For τ → eγ: CLEO Collaboration, K. W. Edwards et al., Phys. Rev. D 55, R3919 (1997); for µ → eγ: MEGA Collaboration, M. L. Brooks et al., Phys. Rev. Lett. 83, 1521 (1999); for τ → µγ: CLEO Collaboration, S. Ahmed et al., Phys. Rev. D 61 071101(R) (2000)

View on Pith 1997
[2]

See for instance L. M. Barkov et al., Search for µ+ → e+γ down to 10−14 branching ratio (research proposal to PSI, 1999)

View on Pith 1999
[5]

R. N. Mohapatra and J. C. Pati, Phys. Rev. D 11, 566 (1975); R. N. Mohapatra and J. C. Pati, Phys. Rev. D 11, 2558 (1975); G. Senjanovi´ c and R. N. Mohapatra, Phys. Rev. D 12, 1502 (1975). See also G. Sen- janovi´ c, Nucl. Phys. B153, 334 (1979)

View on Pith 1975
[6]

For an introduction to those models together with many re ferences, see G. C. Branco, L. Lavoura, and J. P. Silva, CP violation (Oxford Uni- versity Press, 1999), chapter 24

View on Pith 1999
[7]

F. Pisano and V. Pleitez, Phys. Rev. D 46, 410 (1992); P. H. Frampton, Phys. Rev. Lett. 69, 2889 (1992). See also R. Foot, O. F. Hern´ andez, F. Pisano, and V. Pleitez, Phys. Rev. D 47, 4158 (1993); T. V. Duong and E. Ma, Phys. Lett. B 316, 307 (1993). 11

View on Pith 1992

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A gauged L_mu - L_tau symmetry with a Z' boson at 200-300 GeV per unit coupling fits electroweak data and can resolve the muon g-2 anomaly while generating mu-tau symmetric neutrino mixing.
keV sterile neutrino dark matter in gauge extensions of the standard model
hep-ph 2009-12 accept novelty 5.0 of 10

Out-of-equilibrium decay of a heavier right-handed neutrino can dilute a thermally overproduced keV sterile neutrino to the observed dark matter density, but only if active neutrino masses arise from a type II rather ...
$\mu \to e \gamma$ and $\tau \to l \gamma$ decays in the fermion triplet seesaw model
hep-ph 2008-03 conditional novelty 5.0 of 10

In the type-III seesaw, the ratio of radiative to three-body lepton-flavor-violating decays is fixed near 10^-3, so a positive mu-to-e-gamma or tau-to-l-gamma signal would contradict existing mu-to-eee and tau-to-3l b...
A Call for New Physics : The Muon Anomalous Magnetic Moment and Lepton Flavor Violation
hep-ph 2016-10 accept novelty 4.5 of 10

The paper derives general one-loop master integrals linking the muon g-2 and mu to e gamma, then applies them across simplified and UV-complete models to map where signals in one or both observables can survive.
Freeze-in at all couplings
hep-ph 2026-06 unverdicted novelty 4.0 of 10

In low-reheating-temperature charged-parent freeze-in dark matter models, stronger couplings are viable if both dark matter and mediator number densities are tracked, with updated LHC and lepton-flavor-violation constraints.
Revisiting $\mu$-$e$ conversion in $R$-parity violating SUSY
hep-ph 2026-01 unverdicted novelty 4.0 of 10

RG running changes limits on certain RPV SUSY λ and λ' couplings by up to 80 percent, with upcoming μ-e conversion experiments expected to set stronger bounds than μ→eγ or μ→eee decays.
Les Houches 2019 Physics at TeV Colliders: New Physics Working Group Report
hep-ph 2020-02 conditional novelty 4.0 of 10

The report compiles new-physics analyses from the 2019 Les Houches workshop, releasing public SMEFT likelihoods from ATLAS 8 TeV dilepton data and HL-LHC projections for leptoquark and dark-sector searches.
Lepton flavor violation and non-unitary lepton mixing in low-scale type-I seesaw
hep-ph 2011-07 conditional novelty 4.0 of 10

In inverse and linear TeV-scale seesaw models, non-unitarity in lepton mixing up to the percent level survives current lepton-flavor-violation bounds, including the MEG mu to e gamma limit.
Weak Triplet Models of Neutrino Magnetic Moments
hep-ph 2026-05 unverdicted novelty 3.0 of 10

Weak triplet models permit decoupling of neutrino magnetic moment from mass in minimal realizations but require delicate parameter adjustments for observable effects, and lose the decoupling in extended scenarios.
The Triadic Texture: Neutrino Predictions, Viable Vacuum, and Phenomenological Constraints
hep-ph 2026-05 unverdicted novelty 3.0 of 10

A new neutrino mass matrix texture under A4 x multiple Z symmetries predicts normal hierarchy, theta23 octant, and bounds on phases and masses in a seesaw model.
Physics of leptoquarks in precision experiments and at particle colliders
hep-ph 2016-03 accept novelty 3.0 of 10

The paper is a comprehensive review of light leptoquark phenomenology, collecting constraints from precision flavor, electroweak, Higgs, and collider observables for proton-stable scenarios.
Flavour physics of leptons and dipole moments
hep-ph 2008-01 accept novelty 3.0 of 10

The report synthesizes current bounds and future prospects for charged-lepton flavor violation and electric dipole moments as probes of TeV-scale physics.
keV Neutrino Model Building
hep-ph 2013-02 accept novelty 2.0 of 10

The review organizes the model-building literature for keV sterile neutrino dark matter, showing which mechanisms can produce the keV scale and which models survive the X-ray, seesaw, and neutrino-oscillation constraints.
Phenomenological Aspects of Models with Low Scale Seesaw
hep-ph 2024-12

How later papers used it

146 later papers cite it

Evidence: substantial. 145 of the citing papers share no author with it, and 77 of those carry a confident label from Pith's citation classifier.

Pith read 20 of the 219 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

7 later papers by other authors are listed below: 7 used its method or tool.

Uses its method or tool 7 of 67 papers with this label

Wrote the branching fractions for these processes according to this paper and calculated muon and tau decay branching ratios as a function of the heaviest fermion mass.

Radiative accidental matter (2016) · 0.92 · paraphrased: the licence does not allow quoting
Wrote the amplitude for the radiative decay using this paper's expression.

Prospects for Finding Sterile Neutrino Dark Matter at KATRIN (2019) · 0.99 · paraphrased: the licence does not allow quoting
Used the general results given in this paper to find the leading order contributions to the partial width.

A "Vector-like chiral" fourth family to explain muon anomalies (2017) · 0.97 · paraphrased: the licence does not allow quoting
Similarly, to compute the branching ratios for the radiative decays, $l_i →l_j γ$, we use [ref] : Γ[l_i →l_j γ] = αg_X^4/4 π (1-m_l_j^2/m_l_i^2)^3 m_l_i^4/M_X^4 m_l_i ( |c_L^γ|^2 + |c_R^γ|^2 ) .Here, $α$ is the fine-structure constant.

Loop-induced masses for the first two generations with optimum flavour violation (2024) · $l_i →3 l_j$ and $l_i →l_j γ$ · 0.99
The partial width for $ℓ_1→ℓ_2γ$ is [ref] [equation] Using the Eqs. [ref] and [ref], the coefficients $σ_L, R$ for $μ→e γ$ mediated by the VL-lepton $F$ are the following, where the subscript $i=1,2$ corresponds to $Z_1,2$: [equation] where, [equation] The factors $y_i$, with $c= i96m_Z_i^2 (m_Z_i^2-m_F^2)^4 π^2$, are [equation] The dominant contribution to radiative decays arise from the chirally enhanced VL-lepton mediated diagrams.

Flavor Physics Constraints on Left-Right Symmetric Models with Universal Seesaw (2023) · Radiative Decays of Charged Leptons · 0.95
Extracted the relevant expressions for flavor violating radiative decays from this paper.

Graph Reinforcement Learning for Exploring BSM Model Spaces (2024) · 0.88 · paraphrased: the licence does not allow quoting
The decay width of $l_i →l_j γ$ can be described by [equation] At the same time, we can also extract the lepton anomalous magnetic moment and EDM from the vertex as it is generated by similar diagrams in Figure [ref] [equation] [equation] From Eq. ([ref]), we obtain [ref] [equation] [equation] where $m_k$ is the mass of fermion in the middle of the loop diagram and the assumption that new vector mass is much bigger than the SM lepton mass is used.

Constraints on $\Delta L=2$ Vector Bosons with Tree Couplings to SM Particles (2025) · $l_i →l_j γ$, magnetic dipole moment and electric dipole moment of charged leptons · 0.84

How the literature uses it

141 arXiv papers cite this in 214 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 107 sentences without a confident label

Contests it 1 paper

This agrees with Eq. (7) of Ref. [ref] , that has a factor $e^2$ missing, though.

Two-body lepton-flavour-violating decays in a 2HDM with soft family-lepton-number breaking · $ℓ_1^±→ℓ_2^±γ$ · contest 0.83

Uses its method or tool 65 papers

The analytic formulas of one-loop form factors for leptoquark exchange relating to $Δa_e_a$ and cLFV decay amplitudes are [ref] :

LFV decays in a 3-3-1 model with singlet leptoquarks · Couplings and analytic formulas for $(g-2)_e_a$ anomalies and LFV decay rates · 0.87
The admixture of the heavy sterile neutrinos in the left-handed charged current $SU(2)_L ⊗U(1)_Y$ weak interaction gives rise to the decay $l_i →l_j γ$ at one-loop level, with a branching ratio given by [ref] [equation] where $Γ_μ ≃3×10^-19$ GeV is the total muon decay width and $R$ is given in Eq. [ref].

Three-Loop Inverse Scotogenic Seesaw Models · Charged lepton flavor violation · 0.90
We use the analytic formulas for computing one-loop contributions to cLFV decay amplitudes and $(g-2)_e_a$ given in Ref. [ref] , which consistent with previous results [ref] .

$(g-2)_{e,\mu}$ and Lepton flavor violating decays in a left-right model · Decays $e_b→e_a γ$ and $(g-2)_e_a$ · 0.96

Compares with it 12 papers

These Lagrangians [ref] and [ref] are consistent with those in Ref. [ref] .

One-loop contributions to decays $e_b\to e_a \gamma$ and $(g-2)_{e_a}$ anomalies, and Ward identity · General amplitudes and notations · 0.89
This coincides — in the limit of massless leptons — with the results from Ref. [ref] , once the logarithms (stemming from the low-energy behaviour and, hence, not contributing to the matching) are removed.

Constraints on leptoquarks from lepton-flavour-violating tau-lepton processes · Leading contribution to Cgamma from leptoquarks · 1.00
As for $μ→eγ$ a similar treatment can be applied to find, [equation] where in agreement with [ref] , and $I_f, 21 ≡4 (2 I_f, 2 + I_f, 1)$.

A Call for New Physics : The Muon Anomalous Magnetic Moment and Lepton Flavor Violation · Doubly Charged Scalar · 0.84

Cites it as background 8 papers

Expressions for LFV rates within the Zee model have long been derived in the literature [ref] .

Zee-model predictions for lepton flavor violation · LFV and other observables · 0.83
The portal couplings induce lepton flavor violation [ The lepton flavor violation in the LPDM model is studied in Refs. [ref] . ], especially $ℓ_i →ℓ_jγ$, whose branching fraction is given by [ref] , [equation] where the functions $F$ and $G$ are defined in Eq. [ref].

Taming lepton portal dark matter by a non-invertible selection rule · Flavor problem · 0.84
For further details on the form factors $D_L,R$, the reader is referred to Refs. [ref] .

Fermion Mass and Mixing in a Low-Scale Seesaw Model based on the S4 Flavor Symmetry · Lepton masses and mixings · 0.84

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [4]

B. He, T.-P. Cheng, and L.-F. Li, Phys. Lett. B 553, 277 (2003)

View on Pith 2003

[2] [3]

T.-P. Cheng and L.-F. Li, Gauge theory of elementary particle physics — problems and solutions (Oxford University Press, 2000), pp. 244–250

View on Pith 2000

[3] [1]

For τ → eγ: CLEO Collaboration, K. W. Edwards et al., Phys. Rev. D 55, R3919 (1997); for µ → eγ: MEGA Collaboration, M. L. Brooks et al., Phys. Rev. Lett. 83, 1521 (1999); for τ → µγ: CLEO Collaboration, S. Ahmed et al., Phys. Rev. D 61 071101(R) (2000)

View on Pith 1997

[4] [2]

See for instance L. M. Barkov et al., Search for µ+ → e+γ down to 10−14 branching ratio (research proposal to PSI, 1999)

View on Pith 1999

[5] [5]

R. N. Mohapatra and J. C. Pati, Phys. Rev. D 11, 566 (1975); R. N. Mohapatra and J. C. Pati, Phys. Rev. D 11, 2558 (1975); G. Senjanovi´ c and R. N. Mohapatra, Phys. Rev. D 12, 1502 (1975). See also G. Sen- janovi´ c, Nucl. Phys. B153, 334 (1979)

View on Pith 1975

[6] [6]

For an introduction to those models together with many re ferences, see G. C. Branco, L. Lavoura, and J. P. Silva, CP violation (Oxford Uni- versity Press, 1999), chapter 24

View on Pith 1999

[7] [7]

F. Pisano and V. Pleitez, Phys. Rev. D 46, 410 (1992); P. H. Frampton, Phys. Rev. Lett. 69, 2889 (1992). See also R. Foot, O. F. Hern´ andez, F. Pisano, and V. Pleitez, Phys. Rev. D 47, 4158 (1993); T. V. Duong and E. Ma, Phys. Lett. B 316, 307 (1993). 11

View on Pith 1992