Scalar Leptoquarks in Leptonic Processes

Leptoquarks are hypothetical new particles, which couple quarks directly to leptons. They experienced a renaissance in recent years as they are prime candidates to explain the so-called \textit{flavor anomalies}, i.e. the deviations between the Standard Model predictions and measurements in $b\to s\ell^{+}\ell^{-}$ and $b\to c\tau\nu$ processes and in the anomalous magnetic moment of the muon. At the one-loop level these particles unavoidably generate effects in the purely leptonic processes like $Z\to\ell^{+}\ell^{-}$, $Z\to\nu\bar\nu$, $W\to\ell\nu$ and $h\to\ell^{+}\ell^{-}$ and can even generate non-zero rates for lepton flavor violating processes such as $\ell\to \ell^{\prime}\gamma$, $Z\to\ell^+\ell^{\prime -}$, $h\to\ell^+\ell^{\prime -}$ and $\ell\to 3\ell^\prime$. In this article we calculate these processes for all five representations of scalar Leptoquarks. We include their most general interaction terms with the Standard Model Higgs boson, which leads to Leptoquark mixing after the former acquires a vacuum expectation value. In our phenomenological analysis we investigate the effects in modified lepton couplings to electroweak gauge bosons, we study the correlations of the anomalous magnetic moment of the muon with $h\to\mu^{+}\mu^{-}$ and $Z\to\mu^{+}\mu^{-}$ as well as the interplay between different lepton flavor violating decays.


Contents
1 Introduction Leptoquarks (LQs) are particles with an interaction vertex connecting leptons with quarks. These particles are predicted by Grand Unified Theories [1][2][3][4] and were systematically classified for the first time in Ref. [5] into ten possible representations under the Standard Model (SM) gauge group (five representations of scalar particles and five representations of vector particles). Their tree-level effects in low energy precision and flavor observables were studied comprehensively in Ref. [6]. After the disappearance of the HERA excess [7,8], which could have been interpreted as a LQ, the interest in LQs decreased until in recent years they experienced a renaissance due to the emergence of the flavor anomalies.
In the next section we define our conventions before we discuss the self-energies, masses and the renormalization in Sec. 3. We then present the analytic results of LQ-induced effects in leptonic amplitudes in Sec. 4. In Sec. 5 we perform our phenomenological analysis, followed by the conclusions. The Appendix contains further helpful results, in particular the generic expressions with exact diagonalization of the LQ mixing matrices.

Setup and Conventions
As outlined in the introduction, LQs are prime candidates to explain the accumulated anomalies in semi-leptonic B meson decays. Since vector LQs, as any massive vector particle, are not renormalizable without a Higgs mechanism, and since we are interested in loop processes, we will study only scalar LQs in the following. Table 1: The five different possible scalar representations of LQs under the SM gauge group and their couplings to quarks and leptons. Note that in our conventions all LQs are SU (3) c triplets. The superscript T refers to transposition in SU (2) L space, c to charge conjugation and τ to the Pauli matrices. We did not include LQ couplings to two quarks, which are possible for some representations and which would lead to proton decays. Note that such couplings can always be avoided by assigning quark or lepton number to the SM fermions and to the LQs.
The five different representations of scalar LQs transform under the SM gauge group as given in Tab. 1. Note that we have two singlets under SU (2) L (Φ 1 andΦ 1 ), two doublets (Φ 2 andΦ 2 ) and one triplet Φ 3 . The fermion fields Q (c) and L are (charge-conjugated) quark and lepton SU (2) L doublets, while u (c) , d (c) and are the corresponding SU (2) L singlets of up-quarks, down-quarks and charged leptons, respectively. The indices f and j refer to flavor and τ are the Pauli matrices, for which we use the convention We defined the hypercharge Y such that the electromagnetic charge is given by with T 3 representing the third component of the weak isospin (±1/2 for SU (2) L doublets and 1, 0, −1 for the SU (2) L triplet). According to this relation, LQs can be decomposed into the electromagnetic charge eigenstates as where the superscripts refer to the electric charge.
The LQs couple according to their representation under the SM gauge group to gauge bosons, introduced for the first time in Ref. [141], where we use the following definition for the covariant derivative Here, B µ is the U (1) Y gauge boson, W µ the one of SU (2) L and G µ of SU (3) c with the couplings g 1 , g 2 and g s , respectively. The index k runs from 1 to 3, a from 1 to 8. T k are the generators of SU (2) and λ a are the well-known Gell-Mann matrices. For SU (2) L singlets we have T k = 0, for doublets we have T k = τ k /2 with the Pauli matrices from Eq. (2.2) while the SU (2) L triplet Φ 3 is in the adjoint representation of SU (2). We use where Φ 3 is defined according to Eq. (2.4e).

Leptoquark-Higgs Interactions and Electroweak Symmetry Breaking
In addition to their couplings to fermions and the gauge interactions, LQs can couple to the SM-like Higgs doublet H (with hypercharge +1) via the Lagrangian [142] Here m 2 k andm 2 k represent the SU (2) L invariant mass terms of the LQs before EW symmetry breaking and ε IJK is the three-dimensional Levi-Civita tensor with ε 123 = 1. For simplicity, we omitted the color indices, which are always contracted among the LQs. Note that A2 1 and A2 3 have mass dimension one, while the Y couplings are dimensionless 2 . The LQ-Higgs interactions depicted in Fig. 1 lead to mixing among the LQ representations after EW symmetry breaking.
Once the Higgs acquires its vacuum expectation value (vev) v ≈ 174 GeV, this generates the mass matrices in the weak basis, with Q = {−1/3, 2/3, −4/3, 5/3} and where the eigenstates of the electric charge are assembled from the LQ field components of Eq. (2.4).
To work in the physical basis with mass eigenstates, in which the amplitudes are calculated, we need to diagonalize the mass matrices in Eq. (2.9). This can be achieved by a unitary transformationM such thatM Q is diagonal. This means that the interaction eigenstates in (2.10) are written as whereΦ Q are the mass eigenstates. The analytic expressions for the diagonalization matrices W −1/3 and W 2/3 are very lengthy or must be computed numerically. Therefore, we diagonalize the mass matrices perturbatively up to O(v 2 /m 2 LQ ), where m are the SU (2) L invariant mass terms of the LQs. The analytic expressions for the perturbative W Q read Note that the index a runs from 1 to 3 for Q = −1/3 and Q = 2/3, while for Q = −4/3 only from 1 to 2. Due to our choice of basis, the CKM matrix appears in all couplings involving left-handed up-type quarks. Similarly, also the PMNS matrix would enter in all couplings involving neutrinos in case they were taken to be massive. However, all processes that we are interested in can be calculated for massless neutrinos such that the PMNS matrix drops out. Nonetheless, we will return to the PMNS matrix in the next section when we discuss possible contributions to Majorana mass terms and the renormalization of the W ν vertex.

Leptoquark-Higgs Couplings
Let us finally consider the couplings of the SM Higgs to LQs. The interaction terms are also affected by the LQ rotations induced by EW symmetry breaking. Again, we express Eq. (2.7) in terms of mass eigenstates as with h as the physical Higgs field, a, b = {1, 2, 3} and c, d = {1, 2}. The couplings are defined asΓ where the Γ Q and Λ Q matrices read The expanded expressions forΓ Q andΛ Q are given in the Appendix A.5.

Self-Energies, Masses and Renormalization
Self-energies of SM fermions after SU (2) L breaking are directly related to their masses and enter the calculations of effective fermion-fermion-gauge-boson and fermion-fermion-Higgs couplings. In this section, we will first calculate the self-energies, then discuss the issue of renormalization and how the self-energies are included in the calculation of modified gaugeboson and Higgs couplings.
First, let us define the mass and kinetic terms of the charged lepton and neutrino Lagrangian in momentum space We allowed for the possibility of Majorana mass terms for the neutrinos, which can be generated via LQs. We then moved to the physical basis in which all mass matrices are diagonal, such that the CKM matrix V (the PMNS matrixV ) appears in the W ud (W ν) vertex. Considering only the leptonic part, we have explicitly We define the self-energies of charged leptons as follows and decompose Σ f i (p 2 ) as and similarly for neutrinos, where only the LL self-energy exists, but a possible contribution to the neutrino mass term arises.
We now expand Σ AB f i (p 2 ) with A, B = {L, R} in terms of p 2 /m 2 LQ , where m represents the LQ mass. Only the leading terms in this expansion (i.e. the ones independent of p 2 ) are UV divergent and non-decoupling. Furthermore, they are the only relevant ones in the calculation of Z , Zνν, W ν and h vertices to be discussed later. The terms linear in p 2 /m 2 are only necessary to calculate → γ. However, as they are finite and do not affect the renormalization of any parameter, they can be included in the calculation of → γ in a straightforward way and we do not give the explicit results here. The ones for Σ ,νAB

Neutrino Masses
The contribution to the Majorana mass term of the neutrinos can be calculated by considering theν c f ν i two-point function. We have generically Figure 3: The vertex diagrams which contribute to i → f γ. Depending on the electric charge of the LQ, we have (charge-conjugated) up-or down-type quarks in the loop.
with m i and m ν i being the physical masses. The unitary matrix U L is given by We used the lepton mass hierarchy to simplify U L and the fact that the self-energies are just corrections to a diagonal matrix to get an explicit expression. U R is simply obtained by exchanging L and R.
These unitary rotations (or at leading order the unit matrix plus anti-hermitian corrections) do not have a physical effect in the sense that they cannot be measured in observables. In fact, they correspond to unphysical rotations, in case of U R , or they can be absorbed by a renormalization of the PMNS matrix, in case of U L and U ν . This can also be seen by applying these rotations to gauge bosons vertices, where they drop out for the Z interaction terms and only enter the W ν vertex in the combination whereV on the left-hand side of the equation is identified with the PMNS matrix, see Eq. 3.2.
Finally, let us consider the h vertex. Here we have where Y represents the genuine vertex correction. Therefore, the effective Yukawa coupling measured in h → + − decays can be expressed in terms of the physical lepton mass and Σ LR f i as follows

Calculation of the One-Loop Effects
In this section, we compute the amplitudes governing the various purely leptonic observables. For this we take into account the Higgs-induced mixing among the different LQ representations. We will consider amplitudes involving the following fields:  [145,146] as studied in Ref. [137].

γ
In case of an on-shell photon, we define the effective Hamiltonian as Note that we have C R f i = C L * i f due to the hermiticity of the Hamiltonian. The coefficients are induced by the diagrams in Fig. 3 and for a single LQ representation only are given by where the quark index j runs from 1 to 3. We expanded the results up to the first nonvanishing order in external momenta and masses. Note that the Wilson coefficients are composed by two parts: a contribution which is proportional to m f,i and a contribution proportional to the quark mass, originating from a chirality flip on the internal quark line.
The latter term appears only if a LQ couples simultaneously to left-and right-handed upor down-type quarks. E.g. for the AMM of the muon this effect dominates in cases where we couple to third generation quarks, i.e. generates a relative enhancement by a factor m t /m µ ∼ 1600 or m b /m µ ∼ 40, respectively. Therefore, these terms are the most important ones from the phenomenological point of view. And for our results with m t and m b we also include the O(v 2 /m 2 LQ ) terms, originating from the Higgs-LQ interaction, while we only present the leading order effects for the m f,i terms.
Turning to the contributions with multiple LQ representations, i.e. the terms involving LQ mixing, we also focus on the terms proportional to m b,t and we find We first give the separate contributions of each LQ representation If we include LQ Higgs interactions, we find a new structure originating from Φ 1 -Φ 3 mixing at O(v 2 /m 2 LQ ). The quark index j runs from 1 to 3 and the loop functions are given in the Appendix A.2. Note that we again assumed that the quarks can be integrated out at the same scale as the LQs. This means that the expressions should be understood to be at the low scale and include the mixing of two-quark-two-lepton operators into four-fermion ones. Therefore, in case the quark is lighter than the corresponding leptonic process, one has to insert the scale of that process (rather than the quark mass) into the logarithms of the loop functions in Appendix A.2.

Z and Zνν
We now compute the LQ effects on the Z → − f + i and Z → ν fνi amplitudes Note that in case of mixing among LQs, the Z coupling, unlike the photon, can connect different representations with each other.
with ε µ (q) as the polarization vector of the Z boson and q 2 = (p f + p i ) 2 . In addition, there is an magnetic form factor for Z → + − . However, we do not give the form factor of this amplitude explicitly, since it does not interfere with the SM for m = 0. We perform this calculation for vanishing lepton masses and decompose the form factors as ,f i andΘ f i contain the part induced by LQ mixing. In our conventions, the tree-level SM couplings read with s w (c w ) being the sine (cosine) of the Weinberg angle. Beyond tree-level, also the SM couplings receive momentum dependent corrections, which are included in the predictions for EW observables that we study later in the phenomenological analysis.
In our calculation we only include contributions of O(m 2 EW /m 2 LQ ), i.e. effects from the top quark, the Z mass as well as the ones induced by LQ mixing, while setting all other masses to zero 3 . In case where the Z boson has a squared momentum q 2 we find 3 Similar results for the diquark contribution to Z → + − have been obtained in Ref. [147].
where the H-functions are given in Appendix A.2.
Now we turn to the Z → ν fνi amplitudes, where we show the contributions again separated by each representation Figure 5: Vertex diagrams contributing to W − → − fν i . In the case of massless down-type quarks the diagram on the left-hand side is only present with charge-conjugated quarks, since the W boson couples to purely to left-handed quarks.
Finally, we again have the contributions from LQ mixing (4.14) In case of zero momentum transfer, i.e. q 2 = 0, the form factors correspond to effective Z and Zνν couplings. We define them for later purposes in an effective Lagrangian where only the∆,Θ and the top contributions remain.

W ν
We define the amplitude of this process, also considered for generic new scalars and fermions in Ref. [140], as follows with The form factors Λ Φ f i q 2 again contain the parts with no LQ mixing, grouped by representation with Φ = {Φ 1 ,Φ 1 , Φ 2 ,Φ 2 , Φ 3 }, whileΛ f i contains the part with LQ mixing. In the SM we have at tree-level The single LQ contributions read Additionally, we have the O(v 2 /m 2 LQ ) effects from LQ mixing At the level of effective couplings, we have to evaluate the contributions at q 2 = 0, which can be treated in the context of an effective Lagrangian The effective coupling Λ W f ν i (0) then only receives LQ effects from loop-induced top quarks and from LQ mixing.

h
Let us turn next to the Higgs decays h → − f + i . We define the amplitude analogously to the leptonic W and Z decays as with The sum over Φ refers to the LQ representations Φ = {Φ 1 ,Φ 1 , Φ 2 ,Φ 2 , Φ 3 },Υ L(R),f i contain the terms which are only generated by LQ mixing and (4.24) Note that due to hermicity If f = i we can safely neglect the lighter lepton mass. The corresponding Feynman diagrams are shown in Figure 6.
We expand again in v 2 /m 2 LQ and set the lepton masses to zero. In the phenomenologically most relevant case of an internal top quark, we additionally use the fact that for Higgs decays  (4.27) The loop functions that we used in this section can be found in the Appendix A.2. In Appendix A.5 we additionally present the generic results for light quarks, i.e. for the case where m 2

4
To describe processes involving four charged leptons, we define the effective Hamiltonian as with the effective operators (4.29) Note that we sum over all flavor indices. Therefore, all other operators can be reduced to the ones in (4.28), using Fierz identities. As an advantage, we do not need to distinguish between decays involving the same or different flavors.
There are two types of diagrams which give a contribution to these operators: penguins and boxes, see Fig 7. Starting with the photon penguin, we have The Z boson gives an analogous contribution (4.32) The coefficients C V RL f iab and C V RR f iab are obtained in a straightforward way by simply exchanging L ↔ R.
The box diagrams generate the following contributions where the loop function H 1 is again given in Appendix A.2. The indices j and k run from 1 to 3. Note that we only consider the leading effects in v/m. In scenarios where the λcouplings are smaller than the gauge couplings (e ≈ 0.3 and g 2 ≈ 0.6), the box contributions are typically less important than the gauge boson penguins.

2 2ν
For these fields we use the effective Hamiltonian There are three types of contributions: Z penguins, W penguins and boxes. The Z boson yields while we have for the W boson The box diagrams yield (4.38b) Again the indices j and k run from 1 to 3 and we only considered the leading order LQ effects in v/m LQ .

Phenomenology
Let us now study the phenomenology of scalar LQs in leptonic processes. Due to the large number of observables and the many free parameters, we will choose some exemplary processes of special interest and use simplifying assumptions for the couplings in order to show the effects and the possible correlations between observables. In particular, we will consider: • EW gauge-boson couplings to leptons: the effects of scalar LQs in (effective) off-shell Z , Zνν and W ν couplings and the associated gauge-boson decays.

Electroweak Gauge-Boson Couplings to Leptons: Z , Zνν and W ν
We start our phenomenological analysis by considering the effects of scalar LQs in Z , Zνν and W ν effective couplings (at q 2 = 0) and the associated gauge boson decays (at q 2 = m 2 Z , m 2 W ), calculated in Sec. 4.2. While among Z → + − decays NP effects are strongly bounded by LEP [148] measurements, the effective W ν couplings are best constrained by low-energy observables, testing LFU of the charged current (see Ref. [149] for an overview).
We first focus on the LQ representations which generate an m 2 t /m 2 LQ effect in EW gaugeboson couplings to leptons, i.e. Φ 1 , Φ 2 and Φ 3 . In the absence of LQ mixing, we can expect this effect to be dominant and couplings to third generation quarks are well motivated by the flavor anomalies. Note that we nonetheless included the q 2 = {m 2 Z , m 2 W } terms which,   Figure 8: LQ effects in Z , Zνν and W ν couplings for the scalar LQ representations which give rise to m 2 t effects (Φ 1 , Φ 2 and Φ 3 ) as a function of the LQ mass. We neglected LQ mixing and considered only the couplings of third generation quarks to a single lepton flavor with unit strength, i.e. λ 3 = 1. Here, ∆ L,R , Θ and Λ stand for the corrections in Z , Zνν and W ν couplings, respectively (see Sec. 4.2). The solid (dashed) lines refer to the couplings entering on-shell decays (effective couplings at q 2 = 0). The green region is excluded by LEP data [148] from Z → νν decays. The blue region is excluded by Z → τ + τ − . Note that in case of Z → e + e − the limits are more constraining as constructive interference is preferred and the experimental uncertainties are smaller. However, the constraints from Z → µ + µ − are weaker. due to SU (2) L invariance, can also arise from bottom loops for some of the representations shown in Fig. 8. In order to keep the number of free parameters small, we did not include mixing among the LQs and assumed that only couplings to one lepton flavor = e, µ, τ at a time exist. This avoids limits from charged lepton flavor violating observables, which we consider later in this article. Furthermore, we normalized the LQ effect to the respective SM coupling and the LQ-quark-lepton coupling to one ( i.e. λ 3 = 1) while all other couplings are zero. Note that the effect in Fig. 8, given for couplings of unit strength, are consistent with Z → + − bounds for masses around 1.5 TeV or more. Furthermore, Z → νν is constrained by the number of neutrino families where the experimental value lies at [148] N ν = 2.9840 ± 0.0082 , while the LQ effect is predicted to be constructive. Future colliders are expected to reach a 20 times better precision [150].
Let us now turn to the case of non-vanishing LQ couplings to the SM Higgs. We study as an example the scalar doubletΦ 2 which couples only down-type quarks to leptons such that the v 2 /m 2 2 effects from the mixing with Φ 1 (generated by A2 1 ) and/or Φ 3 (generated Even though the current ATLAS and CMS results are not yet constraining this model, sizeable effects are predicted, which can be tested at future colliders. Furthermore, Φ 1 yields a constructive effect in h → µ + µ − while the one of Φ 2 is destructive such that they can be clearly distinguished with increasing experimental precision. by A2 3 ) are expected to be dominant compared to the m 2 Z /m 2 2 effects. In Fig. 9 we present the impact of LQs on on-shell Z and W couplings. Again, we setλ 2 3 = 1 and we assumẽ m 2 = m 1 = m 3 = 1 TeV, which is compatible with current LHC limits [151][152][153]. Note that a non-zero A2 1 yields a destructive effect in Z and W ν couplings while the terms with A2 3 are constructive.

5.2
Correlating the AMM of the Muon with Z → + − and h → µ + µ − In this sub-section, we focus on possible LQ explanations of the long-standing anomaly in the AMM of the muon. The discrepancy between its measurement [21] and the SM prediction [35] 4 amounts to This result is based on Refs. [154][155][156][157][158][159][160][161][162][163][164][165][166][167][168][169][170][171][172][173]. The recent lattice result of the Budapest-Marseille-Wuppertal collaboration (BMWc) for the hadronic vacuum polarization (HVP) [174] on the other hand is not included. This result would render the SM prediction of aµ compatible with experiment. However, the BMWc results are in tension with the HVP determined from e + e − → hadrons data [158][159][160][161][162][163]. Furthermore, the HVP also enters the global EW fit [175], whose (indirect) determination is below the BMWc result [176]. Therefore, the BMWc determination of the HVP would increase the tension in EW fits [177,178] and we opted for using the community consensus of Ref. [35]. First of all, we can expect a direct correlation with h → µ + µ − [179] since both processes are chirality changing and therefore involve the same couplings of LQs to fermions 5 . We can express the NP effect in terms of Υ Φ L andΥ L , defined in (4.22), as The resulting correlations are shown in Fig. 10 for Φ 1 and Φ 2 . Note that even though the current CMS and ATLAS measurements [181,182] are not able to constrain these models yet, a FCC-hh [183] can test them.
The LQ interactions with top quarks and muons also generate effects in Zµµ couplings. Therefore, let us as a next step consider the correlations of a µ with Z → + − where we refine the analysis of Ref. [107] by including the indirect effect, originating from the finite g A /g ASM LEP [148] FCC-ee [185] ILC [186] CEPC [187] CLIC [188] . We further show various expected sensitivities for future colliders (second to fifth row) under the assumption that the measurements of g A are improved by the same factor as s 2 w .
renormalization of the very precisely measured Fermi constant [184] G F = 1.166 378 7(6) × 10 −5 GeV −2 , (5.6) which can be expressed in terms of the SM parameters Since m W itself is measured in W decays, g 2 can be determined once G F is measured via the muon lifetime. However, also NP contributions enter such that resulting in a redefinition of g 2 . The SU (2) L singlet Φ 1 with non-zero real couplings λ 1L 32 and λ 1R 32 affects Zµµ as well as W µν µ , while the effect on Zνν is very small, see Fig. 11. The modified W coupling by λ 1L 32 then yields a finite, LFU renormalization of g 2 . This has been included in our analysis depicted in Fig. 11, leading to the allowed, green region deviating slightly from a circled shape.
The SU (2) L doublet Φ 2 with non-zero couplings λ 2LR 32 and λ 2RL 32 only yields a negligible contribution to W µν µ . However, there is an m t effect in Z → νν, affecting N ν , which has been precisely measured, see Eq. (5.2). This then constrains λ 2RL 32 as we show in the plot in in the right-hand side of Fig. 11. We additionally show in Fig. 11 the expected sensitivities of future experiments for Zµµ, which are summarized in Tab. 2.

Charged Lepton Flavor Violation
Let us now correlate different charged lepton flavor violating observables, i.e. → γ, Z → and → 3 . We do not study µ → e conversion in nuclei, which could be dominant in case of couplings to first generation quarks, but rather again assume only couplings to third generation quarks.
For the three body decays we have with the Wilson coefficients defined in Eq. (4.28). The analogous expression for µ → 3e can be obtained by obvious replacements. These rates have to be compared to the experimental limits given in Tab. 3 where we also quote the expected future sensitivities. We do not consider decays like τ ∓ → µ ∓ e ± e ∓ as the experimental constraints are slightly worse.
In our numerical analysis, we again assume that the LQs only couple to third generation quarks but now allow for the possibility that they couple to more than one lepton flavor at the same time. Let us start by examining the correlations between τ → µγ and Z → τ µ in  Figure 12: Correlations between τ → µγ and Z → τ µ for the three LQ representations which generate an m 2 t /m 2 LQ effect in Z couplings. We assume that Φ 1 and Φ 2 couple either to left or to right-handed leptons only such that chirally enhanced effects (which would result in dominant effects in τ → µγ) are absent. Fig. 12. One can see that this correlation is very direct under the assumption that only one representation contributes and that for Φ 1 and Φ 2 only either the left-or the right-handed couplings to leptons are non-zero at the same time such that chirality enhanced effects in τ → µγ are absent. Although currently τ → µγ is more constraining, even in the absence of chirality enhanced contributions, in the future Z → τ µ can provide competitive or even superior bounds. The situation for τ → e transitions is very similar and therefore not shown explicitly.
In Fig. 13 we show the correlations between τ → µγ and τ → 3µ. These correlations are not as clear as in the case of Z → τ µ due to the additional box contributions to τ → 3µ. Therefore, one obtains a cone instead of a straight line. Interestingly, for Φ 1 the effect in τ → µγ is smallest among the LQ representations due to the electric charge of the LQ. Hence, even though phase space suppressed, τ → 3µ is more sensitive to this particular LQ than τ → µγ. Again, the situation in τ → e transitions is very similar and therefore not shown explicitly. However, we show our analysis for µ → e transitions in Fig. 14. For the µ → eγ scenario we do not show Z → µe since the low energy bounds are so stringent that the former cannot compete, even when taking into account future prospects.  Figure 14: The analogue to the plots above for the µ → e transition. The dashed lines depict the expected sensitivity from MEG II [197] and the solid line the one of Mu3e [198].

Conclusions
Leptoquarks are prime candidates to explain the flavor anomalies, i.e. the discrepancies between measurements and the SM predictions in b → s + − , b → cτ ν and the AMM of the muon. With this motivation in mind, we calculated the one-loop amplitudes generated by scalar LQs for the purely leptonic transitions, involving: • γ • Z and Zνν Taking into account the most general set of interactions of the LQs with the SM Higgs doublet, we obtained relatively simple analytic expressions for the amplitudes by expanding the LQ mixing matrices in v/m LQ , corresponding to a mass insertion approximation.
In our phenomenological analysis, we illustrated the results of our calculation by studying: • LQ effects in effective Z , Zνν and W ν couplings and the associated gauge boson decays. Here we found for the three representations which generate m 2 t /m 2 LQ enhanced effects (Φ 1 , Φ 2 and Φ 2 ) that Z → + − is smaller than within the SM while Z → νν is enhanced. For order one couplings, the effect is at the percent level for TeV scale LQs.
• Correlations between the AMM of the muon, Z → + − , effective W µν couplings and h → µ + µ − . Here we found that, since an explanation of the (g − 2) µ anomaly requires a m t /m µ enhanced effect, also the contribution in h → µ + µ − is pronounced by the same factor. Furthermore, effects scaling like m 2 t /m 2 LQ in Z → µ + µ − are generated which are most relevant in case where the left-handed couplings are much larger than the right handed ones and vice versa.
• Correlations between τ → µγ, Z → τ µ and τ → 3µ, as well as the analogues in µ → e transitions. Here we observed that τ → µγ and Z → τ µ can be directly correlated under the assumption the LQs couple only to left-handed or to right-handed leptons (but not to both of the same time). Furthermore, in this setup τ → µγ and µ → eγ do not receive chirally enhanced effects such that τ → 3µ and µ → 3e can give competitive bounds, which is in particular the case for Φ 1 .
These interesting correlations can be tested at future precision experiments and high-energy colliders. and the one of C.G. and F.S. by the Swiss National Science Foundation grant 200020 175449/1.

A Appendix
A.1 Self-Energies Focusing on the non-decoupling, momentum-independent parts of the self-energies, we have generically with Σ RR f i and Σ RL f i obtained by interchanging chiralities and Σ νLL f i by replacing with ν. We set all quark masses within the loop equal to zero, except for the top mass. Additionally, one has to sum over all internal quarks u j , d j , u c j and d c j , as well as over their flavors j = {1, 2, 3}. The loop functions take the simple form For neutrinos we have (A.5) -36 -

A.2 Loop Functions
The loop-functions for γ with on-shell photons read see Eqs. .
contains both the hard matching part, the mixing within the effective theory and the soft contribution. For this reason, care is required if the internal quarks are lighter than the incoming lepton (e.g. the charm contribution to τ → µγ) since the RGE only contributes from the LQ scale down to the scale of the process and not to the scale of the internal quark. Therefore, we defined µ in Eq. (A.10) as follows Next we give the exact results for off-shell photons, whereof the expanded expressions are given in Eqs. (4.6) and (4.7). They read Again, j runs from 1 to 3.

A.4 Exact Results for Z , Zνν, W ν and h
In this section we give the exact expressions for the Z and W decays. TheT Q andB W i matrices, used in this section, are given in Appendix A.5. In this whole section, the M i stand for the diagonal bilinear mass terms in the charge eigenstates, given in Eq. (2.14). It is implied by the corresponding coupling matrix Γ i with same index i which of the eigenstates is concerned, e.g. Γ u c corresponds to M −1/3 . For the Z decays, we use the conventions defined in Eq. (4.8), this time with where contrary to Eq. (4.9) we show the results sorted by the charges of the LQs, since we do not distinguish between the cases with and without LQ mixing. Hence, the results cannot be grouped by representation. For Q = −1/3 we have for the m t -enhanced contributions again with a, b = {1, 2, 3} and the mixing terms read For the LQs with electric charge Q = −4/3 we have In case of Q = 2/3, we have the diagrams which include a heavy top quark For W → − fν i decays, our definition of the amplitude is given in Eq. (4.16) and contrary to Eq. (4.17), we use where we choose to group the results by the fact whether a quark (q) or a charge-conjugated quark (q c ) runs in the loop. Because of obvious reasons, a grouping by representation is again  Note that the LQ field redefinition has no impact the electromagnetic interaction, since the coupling matrix is proportional to the unit matrix and the W Q then cancel due to unitarity.
If we use the perturbative diagonalization ansatz, we obtaiñ