Sensitivity of future lepton colliders and low-energy experiments to charged lepton flavor violation from bileptons

TL;DR

This paper evaluates how future lepton colliders and low-energy experiments can detect charged lepton flavor violation caused by bileptons, highlighting their complementary sensitivities and potential to explore new physics beyond the Standard Model.

Contribution

It introduces a comparative analysis of future collider and low-energy experiment sensitivities to bilepton-induced charged lepton flavor violation, emphasizing their complementary roles in probing new physics.

Findings

Future lepton colliders will complement low-energy experiments in detecting bilepton couplings.

Upcoming experiments on muonium-antimuonium conversion and tau decays will explore similar parameter spaces.

Electroweak precision tests and magnetic moment measurements will also probe bilepton effects.

Abstract

The observation of charged lepton flavor violation is a clear sign of physics beyond the Standard Model (SM). In this work, we investigate the sensitivity of future lepton colliders to charged lepton flavor violation via on-shell production of bileptons, and compare their sensitivity with current constraints and future sensitivities of low-energy experiments. Bileptons couple to two charged leptons with possibly different flavors and are obtained by expanding the general SM gauge invariant Lagrangians with or without lepton number conservation. We find that future lepton colliders will provide complementary sensitivity to the charged-lepton-flavor-violating couplings of bileptons compared with low-energy experiments. The future improvements of muonium-antimuonium conversion, lepton flavor non-universality in leptonic {\tau} decays, electroweak precision observables and the anomalous magnetic moments of charged leptons will also be able to probe similar parameter space.

Tables

Table 1. Table 1: Constraints from AMM on the CLFV couplings in units of GeV − 2 superscript GeV 2 {\rm GeV}^{-2} . Here we assume all the CLFV couplings are real and symmetric, and m h 2 = m a 2 subscript 𝑚 subscript ℎ 2 subscript 𝑚 subscript 𝑎 2 m_{h_{2}}=m_{a_{2}} for H 2 subscript 𝐻 2 H_{2} . For H 2 subscript 𝐻 2 H_{2} , H 3 subscript 𝐻 3 H_{3} , Δ 2 subscript Δ 2 \Delta_{2} and Δ 3 subscript Δ 3 \Delta_{3} , the values outside the brackets correspond to the assumption that the singly charged boson is decoupled, while values in the brackets are under the assumption that all components of the multiplet are degenerate.

	$a_e\left[3\sigma \right]$	$a_{\mu }\left[4\sigma \right]$
$H_1^{(\prime )0}$
$H_2$
$H_3$
${\Delta }_1$
${\Delta }_2$
${\Delta }_3$

Table 2. Table 2: Constraints from muonium-antimuonium conversion on the CLFV couplings in units of GeV − 2 superscript GeV 2 {\rm GeV}^{-2} . Here we assume all the CLFV couplings are real and symmetric. For H 2 subscript 𝐻 2 H_{2} , the value outside (inside) the brackets is for the extreme case m h 2 = m a 2 subscript 𝑚 subscript ℎ 2 subscript 𝑚 subscript 𝑎 2 m_{h_{2}}=m_{a_{2}} ( m h 2 ≪ m a 2 much-less-than subscript 𝑚 subscript ℎ 2 subscript 𝑚 subscript 𝑎 2 m_{h_{2}}\ll m_{a_{2}} ).

	${\mu }^{+}{e}^{-}\to {\mu }^{-}{e}^{+}$
$H_1^{(\prime )0}$
$H_2$
$H_3$
${\Delta }_1$
${\Delta }_2$
${\Delta }_3$

Table 3. Table 3: Relevant constraints from lepton flavor universality of leptonic τ 𝜏 \tau decays on the CLFV couplings in units of GeV -2 . The constraints on H 1 0 superscript subscript 𝐻 1 0 H_{1}^{0} and Δ 3 subscript Δ 3 \Delta_{3} are due to the interference with the SM and thus stronger.

	$R_{\mu e}$	$R_{\tau \mu }$
$H_1^0$
$H_2$
$H_3$
${\Delta }_2$
${\Delta }_3$

Table 4. Table 4: Input parameters for electroweak observables.

input	value
$m_Z$ [GeV]	$91.1876\pm 0.0021$Tanabashi et al. (2018)
$G_{F,\mu }$ [GeV-2]	$1.1663787\left(6\right)\times {10}^{-5}$Tanabashi et al. (2018)
${\alpha }^{-1}$	$137.035999046\left(27\right)$Parker et al. (2018)

Table 5. Table 5: Constraints from the Fermi constant and neutrino physics on the CLFV couplings in units of GeV -2 .

	Fermi constant	neutrino $\mu -e$	neutrino $\tau -e$
$H_1^0$
$H_2$
$H_3$
${\Delta }_2$
${\Delta }_3$

Table 6. Table 6: LEP limits on couplings for masses well above the center of mass energy s ∼ 130 − 207 similar-to 𝑠 130 207 \sqrt{s}\sim 130-207 GeV. The limits for H 2 subscript 𝐻 2 H_{2} from e + ​ e − → μ + ​ μ − , τ + ​ τ − → superscript 𝑒 superscript 𝑒 superscript 𝜇 superscript 𝜇 superscript 𝜏 superscript 𝜏 e^{+}e^{-}\to\mu^{+}\mu^{-},\tau^{+}\tau^{-} , i.e. e ≠ ℓ 𝑒 ℓ e\neq\ell , are obtained under the assumption that y 2 e ​ e superscript subscript 𝑦 2 𝑒 𝑒 y_{2}^{ee} or y 2 ℓ ​ ℓ superscript subscript 𝑦 2 ℓ ℓ y_{2}^{\ell\ell} vanishes.

	$e^{+}{e}^{-}\to e^{+}{e}^{-}$	$e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}$	$e^{+}{e}^{-}\to {\tau }^{+}{\tau }^{-}$
$H_{1,3}$	$\left|y_{1,3}^{ee}\right|\le 6.7\times {10}^{-4}{m}_{H_{1,3}^0}$	$\sqrt{\left|y_{1,3}^{ee}{y}_{1,3}^{\mu \mu }+y_{1,3}^{e\mu }{y}_{1,3}^{\mu e}\right|}\le 4.9\times {10}^{-4}{m}_{H_{1,3}^0}$	$\sqrt{\left|y_{1,3}^{ee}{y}_{1,3}^{\tau \tau }+y_{1,3}^{e\tau }{y}_{1,3}^{\tau e}\right|}\le 4.5\times {10}^{-4}{m}_{H_{1,3}^0}$
$H_1^{\prime 0}$	$\left|y_1^{\prime ee}\right|\le 6.8\times {10}^{-4}{m}_{H_1^{\prime 0}}$	$\sqrt{\left|y_1^{\prime ee}{y}_1^{\prime \mu \mu }+y_1^{\prime e\mu }{y}_1^{\prime \mu e}\right|}\le 5.1\times {10}^{-4}{m}_{H_1^{\prime 0}}$	$\sqrt{\left|y_1^{\prime ee}{y}_1^{\prime \tau \tau }+y_1^{\prime e\tau }{y}_1^{\prime \tau e}\right|}\le 4.7\times {10}^{-4}{m}_{H_1^{\prime 0}}$
$H_2$	$\left|y_2^{ee}\right|\le 5.3\times {10}^{-4}{m}_{h_2}$	$\left|y_2^{e\mu }\right|\le 2.5\times {10}^{-3}{m}_{h_2}$	$\left|y_2^{e\tau }\right|\le 2.4\times {10}^{-3}{m}_{h_2}$
${\Delta }_1^{++}$	$\left|{\lambda }_1^{ee}\right|\le 6.8\times {10}^{-4}{m}_{{\Delta }_1^{++}}$	$\left|{\lambda }_1^{e\mu }\right|\le 3.6\times {10}^{-4}{m}_{{\Delta }_1^{++}}$	$\left|{\lambda }_1^{e\tau }\right|\le 3.3\times {10}^{-4}{m}_{{\Delta }_1^{++}}$
${\Delta }_3$	$\left|{\lambda }_3^{ee}\right|\le 6.7\times {10}^{-4}{m}_{{\Delta }_3^{++}}$	$\left|{\lambda }_3^{e\mu }\right|\le 3.4\times {10}^{-4}{m}_{{\Delta }_3^{++}}$	$\left|{\lambda }_3^{e\tau }\right|\le 3.2\times {10}^{-4}{m}_{{\Delta }_3^{++}}$

Table 7. Table 7: Sensitivity reach from neutrino trident production in units of GeV -2 assuming 25% precision for the measurement of the cross section. In the absence of any deviation from the SM neutrino trident cross section, the Yukawa couplings of the bileptons have to satisfy the above-listed constraints.

	${\nu }_{\mu }N\to {\nu }_{\mu }{e}^{+}{e}^{-}N$	${\nu }_{\mu }N\to {\nu }_{\mu }{\mu }^{+}{\mu }^{-}N$
$H_1^0$	$-2.0\times {10}^{-5}{m}_{H_1^0}^2\le y_1^{\mu \mu }{y}_1^{ee}\le -1.6\times {10}^{-5}{m}_{H_1^0}^2$	${\left(y_1^{\mu \mu }\right)}^2\le 3.4\times {10}^{-6}{m}_{H_1^0}^2$
	$-2.2\times {10}^{-6}{m}_{H_1^0}^2\le y_1^{\mu \mu }{y}_1^{ee}\le 1.8\times {10}^{-6}{m}_{H_1^0}^2$
$H_2$	${\left|y_2^{e\mu }\right|}^2\le 3.9\times {10}^{-6}{m}_{H_2^{+}}^2$	${\left|y_2^{\mu \mu }\right|}^2\le 2.0\times {10}^{-5}{m}_{H_2^{+}}^2$
$H_3$	${\left(y_3^{e\mu }\right)}^2\le 8.8\times {10}^{-7}{m}_{H_3^{+}}^2$	${\left(y_3^{\mu \mu }\right)}^2\le 1.7\times {10}^{-6}{m}_{H_3^{+}}^2$
${\Delta }_2$	${\left|{\lambda }_2^{e\mu }\right|}^2\le 2.7\times {10}^{-6}{m}_{{\Delta }_2^{+}}^2$	${\left|{\lambda }_2^{\mu \mu }\right|}^2\le 1.4\times {10}^{-5}{m}_{{\Delta }_2^{+}}^2$
	$1.3\times {10}^{-5}{m}_{{\Delta }_2^{+}}^2\le {\left|{\lambda }_2^{e\mu }\right|}^2\le 1.7\times {10}^{-5}{m}_{{\Delta }_2^{+}}^2$
${\Delta }_3$	${\left|{\lambda }_3^{e\mu }\right|}^2\le 1.8\times {10}^{-6}{m}_{{\Delta }_3^{+}}^2$	${\left|{\lambda }_3^{\mu \mu }\right|}^2\le 3.4\times {10}^{-6}{m}_{{\Delta }_3^{+}}^2$

Table 8. Table 8: CLFV channels via Δ ​ L = 0 Δ 𝐿 0 \Delta L=0 or Δ ​ L = 2 Δ 𝐿 2 \Delta L=2 interaction at e + ​ e − superscript 𝑒 superscript 𝑒 e^{+}e^{-} collider, for probing coupling y i ​ j , λ i ​ j superscript 𝑦 𝑖 𝑗 superscript 𝜆 𝑖 𝑗 y^{ij},\lambda^{ij} .

flavor $ij$	$\Delta L=0$ CLFV channel	$\Delta L=2$ CLFV channel
$e\mu$	$e^{+}{e}^{-}\to e^{\pm }{\mu }^{\mp }{H}^0$ (s+t)	$e^{+}{e}^{-}\to e^{\pm }{\mu }^{\pm }{\Delta }^{\mp \mp }$ (s+t)
$e\tau$	$e^{+}{e}^{-}\to e^{\pm }{\tau }^{\mp }{H}^0$ (s+t)	$e^{+}{e}^{-}\to e^{\pm }{\tau }^{\pm }{\Delta }^{\mp \mp }$ (s+t)
$\mu \tau$	$e^{+}{e}^{-}\to {\mu }^{\pm }{\tau }^{\mp }{H}^0$ (s)	$e^{+}{e}^{-}\to {\mu }^{\pm }{\tau }^{\pm }{\Delta }^{\mp \mp }$ (s)

Equations

$${\mathcal{L}}_{\Delta L=0}$$

$$=\left(y_1^{ij}{H}_{1\mu }^0{\bar{\ell }}_i{\gamma }^{\mu }{P}_L{\ell }_j+y_1^{ij}{H}_{1\mu }^0{\bar{\nu }}_i{\gamma }^{\mu }{P}_L{\nu }_j\right)+y_1^{\prime ij}{H}_{1\mu }^{\prime 0}{\bar{\ell }}_i{\gamma }^{\mu }{P}_R{\ell }_j$$

$$+\left(y_2^{ij}{H}_2^{+}{\bar{\nu }}_i{P}_R{\ell }_j+y_2^{ij}{H}_2^0{\bar{\ell }}_i{P}_R{\ell }_j+h.c.\right)$$

$$+\left(y_3^{ij}\sqrt{2}{H}_{3\mu }^{-}{\bar{\ell }}_i{\gamma }^{\mu }{P}_L{\nu }_j+y_3^{ij}\sqrt{2}{H}_{3\mu }^{+}{\bar{\nu }}_i{\gamma }^{\mu }{P}_L{\ell }_j-y_3^{ij}{H}_{3\mu }^0{\bar{\ell }}_i{\gamma }^{\mu }{P}_L{\ell }_j\right),$$

$${\mathcal{L}}_{\Delta L=2}$$

$$=\left({\lambda }_1^{ij}{\Delta }_1^{++}{\ell }_i^{T}C{P}_R{\ell }_j+h.c.\right)+\left({\lambda }_2^{ij}{\Delta }_{2\mu }^{++}{\ell }_i^{T}C{\gamma }^{\mu }{P}_R{\ell }_j-{\lambda }_2^{ij}{\Delta }_{2\mu }^{+}{\nu }_i^{T}C{\gamma }^{\mu }{P}_R{\ell }_j+h.c.\right)$$

$$-\left(-{\lambda }_3^{ij}\sqrt{2}{\Delta }_3^{+}{\nu }_i^{T}C{P}_L{\ell }_j-{\lambda }_3^{ij}{\Delta }_3^{++}{\ell }_i^{T}C{P}_L{\ell }_j+{\lambda }_3^{ij}{\Delta }_3^0{\nu }_i^{T}C{P}_L{\nu }_j+h.c.\right).$$

$$\Delta a_{\mu }\equiv a_{\mu }^{\exp }-a_{\mu }^{\mathrm{SM}}=(2.74\pm 0.73)\times 10^{-9}.$$

$$\Delta a_e\equiv a_e^{\exp }-a_e^{\mathrm{SM}}=(-0.88\pm 0.36)\times 10^{-12}.$$

$$\Delta a_{\ell }(H_1^{(\prime )0})$$

$$\Delta a_{\ell }(H_3)$$

$$\Delta a_{\ell }({\Delta }_2)$$

$$\Delta a_{\ell }({\Delta }_1)=$$

$$\Delta a_{\ell }({\Delta }_3)=$$

$$\Delta a_{\ell }(H_2)=$$

$$+\sum _k\mathrm{Re}[y_2^{k\ell }{y}_2^{\ell k}]\frac{m_k{m}_{\ell }}{16{\pi }^2}\left(\frac{\ln \left(\frac{m_k^2}{m_{h_2}^2}\right)+\frac{3}{2}}{m_{h_2}^2}-\frac{\ln \left(\frac{m_k^2}{m_{a_2}^2}\right)+\frac{3}{2}}{m_{a_2}^2}\right),$$

$$\Delta a_{\ell }(H_2)$$

$$H=H_0+H_{hf}+H_Z,$$

$$H=E_{0}1+\frac{b}{2}\left(\begin{array}{cccc}\frac{1}{2}+Y& & & \\ & -\frac{1}{2}+X& 1& \\ & 1& -\frac{1}{2}-X& \\ & & & \frac{1}{2}-Y\end{array}\right),$$

$$X$$

$$|{\lambda }_1^{(M)}⟩$$

$$|{\lambda }_2^{(M)}⟩$$

$$|{\lambda }_3^{(M)}⟩$$

$$|{\lambda }_4^{(M)}⟩$$

$$s$$

$$(X,Y)$$

$$H_{LL(RR)}\equiv [\bar{\mu }{\gamma }_{\rho }{P}_{L(R)}e][\bar{\mu }{\gamma }^{\rho }{P}_{L(R)}e]$$

$$H_{LR}\equiv [\bar{\mu }{\gamma }_{\rho }{P}_{L}e][\bar{\mu }{\gamma }^{\rho }{P}_{R}e]$$

$$H_{SLL(SRR)}\equiv [\bar{\mu }{P}_{L(R)}e][\bar{\mu }{P}_{L(R)}e]$$

$$H_{SLR}\equiv [\bar{\mu }{P}_{L}e][\bar{\mu }{P}_{R}e]$$

$$\mathcal{L}(H_1^{(\prime )0})$$

$$\mathcal{L}({\Delta }_{1,3}^{++})$$

$$\mathcal{L}(H_2)$$

$$H_{M\bar{M}}(H_1^{(\prime )0})$$

$$H_{M\bar{M}}({\Delta }_{1,3}^{++})$$

$$H_{M\bar{M}}(H_2)=C{H}_{LR}+\frac{A}{2}{H}_{LL},$$

$$C$$

$$\mathcal{L}(H_2)$$

$$P(B)$$

Full text

I Introduction

The observation of neutrino oscillations and thus non-zero neutrino masses clearly established the existence of lepton flavor violation in the neutrino sector. We also expect the existence of charged lepton flavor violation (CLFV) which occurs in short-distance processes without neutrinos in the initial or final state. In the Standard Model (SM) with three massive neutrinos, the rates of CLFV processes are suppressed by $G_{F}^{2}m_{\nu}^{4}\lesssim 10^{-50}$ due to unitarity of the leptonic mixing matrix Cheng and Li (1980) and thus beyond the sensitivity of any current or planned experiments. Hence, the observation of any CLFV process implies the existence of new physics beyond the SM with three massive neutrinos. The CLFV is predicted by many different new physics models (see Refs. Lindner et al. (2018); Calibbi and Signorelli (2018) for recent reviews), including neutrino mass models such as the inverse seesaw model Tommasini et al. (1995) and radiative neutrino mass models Cai et al. (2017). It may also arise in other extensions of the SM such as the multi-Higgs doublet models Branco et al. (2012) or the minimal supersymmetric SM via gaugino-slepton loops with off-diagonal terms in the slepton soft mass matrix Raidal et al. (2008); Calibbi and Signorelli (2018).

As CLFV induces rare processes, they are generally searched at low-energy experiments with high intensity. See Refs. Carpentier and Davidson (2010); Petrov and Zhuridov (2014) for a list of constraints on the effective CLFV operators obtained from several low-energy precision measurements. The CLFV processes may also be searched for at high-energy colliders. The Large Electron-Positron Collider (LEP) sets upper limits on the branching ratio of $Z$ boson rare decays Tanabashi et al. (2018), i.e. $Z\to\ell\ell^{\prime}$ induced by loop diagrams, and still provides the most stringent constraint on the branching ratios of $Z\to\tau e(\mu)$ as $0.98(1.2)\times 10^{-5}$ up to now. The ATLAS experiment currently sets the most stringent limit of $7.3\times 10^{-7}$ on BR($Z\to e\mu$) Aad et al. (2014) and comparable limit of $5.8(2.4)\times 10^{-5}$ on BR($Z\to\tau e(\mu)$) Aaboud et al. (2018a). The future $Z$ factories could improve the sensitivity by about four orders of magnitude Dong et al. (2018). The CLFV can also occur in Higgs boson decay through the dimension-6 operator $H^{\dagger}H\bar{L}e_{R}H$ in SM effective field theory Harnik et al. (2013). The Large Hadron Collider (LHC) recently improved its limit on the effective $\tau\ell$ couplings to the level of $(1-2)\times 10^{-3}$ Sirunyan et al. (2018); The ATLAS collaboration (2019) and the proposed Higgs boson factories are expected to be sensitive to CLFV couplings down to the order of $10^{-4}$  Dong et al. (2018); Qin et al. (2018).

Besides these rare decays, CLFV can also be probed through scattering processes at colliders. A hadron collider is sensitive to effective operators with two colored particles and two leptons in processes such as $q\bar{q}\to\ell\ell^{\prime}$ Cai and Schmidt (2016) and $gg\to\ell\ell^{\prime}$ Bhattacharya et al. (2018); Cai et al. (2018). The hadron colliders are also sensitive to a number of higher dimension operators contributing to anomalous triple or quartic gauge couplings through $WW$ scattering process $pp\to 2\ {\rm jets}+W^{\ast}W^{\ast}\to 2\ {\rm jets}+\ell\nu_{\ell}\ell^{\prime}\nu_{\ell^{\prime}}$ Kalinowski et al. (2018); Chaudhary et al. (2019). A lepton collider may probe effective operators with four charged leptons via $e^{+}e^{-}\to\ell\ell^{\prime}$. These searches can also be interpreted in terms of simplified models. CLFV processes at the lepton collider can be described by seven bileptons Cuypers and Davidson (1998), which are scalar or vector bosons coupled to two leptons via a renormalizable coupling. In particular, off-shell bileptons can mediate the processes $e^{+}e^{-}\to\ell\ell^{\prime}$ whose potential observation at future lepton colliders has recently been studied by us Li and Schmidt (2019). See also Ref. Dev et al. (2018) and Refs. Sui and Zhang (2018); Bhupal Dev et al. (2018) for related studies of electroweak doublet and triplet scalar bileptons, respectively. Another promising probe for CLFV is through the on-shell production of a bilepton $X$ together with two charged leptons with different flavors, i.e. $e^{+}e^{-}\to X\ell\ell^{\prime}$. This production scenario only depends on a single CLFV coupling in each production channel and thus can be directly compared with other constraints. On-shell production has been studied in Ref. Dev et al. (2018) for an electroweak doublet scalar and Refs. Sui and Zhang (2018); Bhupal Dev et al. (2018) for an electroweak triplet scalar.

The main aim of this work is to explore the sensitivity reach to the CLFV couplings for all seven bileptons through the on-shell production of a bilepton $X$ in association with two charged leptons at proposed future lepton colliders. We compare the sensitivities of future lepton colliders with the existing constraints and future sensitivities of other experiments. Currently, the most relevant constraints are from the anomalous magnetic moments (AMMs) of electrons and muons, muonium-antimuonium conversion, lepton flavor universality (LFU) in leptonic $\tau$ decays, electroweak precision observables, and previous collider searches at the LEP and the LHC experiments. Our analysis here goes beyond the previous work by extending the study to all possible bileptons and including additional constraint from the violation of lepton flavor universality in leptonic $\tau$ decays and electroweak precision observables as well as a discussion of neutrino trident productions. We also improve the calculation of muonium-antimuonium conversion for the bileptons.

The paper is outlined as follows. In Sec. II we describe the general SM extensions with CLFV couplings. Then we discuss the relevant existing constraints on the CLFV couplings in Sec. III. In Sec. IV we present the sensitivity of neutrino trident production, future lepton colliders, and a new state-of-the-art muonium-antimuonium conversion experiment to the CLFV couplings of bileptons and compare it with the existing low-energy constraints. Our conclusions are drawn in Sec. V.

II General Lagrangian for charged lepton flavor violation

In this work we consider all possible111In principle one could extend the discussion to spin-2 fields. scalar and vector bileptons with possible CLFV couplings Cuypers and Davidson (1998). They are obtained by expanding the most general SM gauge invariant Lagrangian in terms of explicit leptonic fields. The bileptons fall in two categories depending whether they carry lepton number $L$ or not. The most general SM invariant Lagrangian of $\Delta L=0$ bileptons has four terms

[TABLE]

where $L_{i}=(\nu_{i},\ell_{i})$ denotes the left-handed SM lepton doublet with a flavor index $i$. The subscript of the new bosonic fields, i.e. 1, 2 or 3, manifests their SU$(2)_{L}$ nature as singlet, doublet or triplet, respectively. The couplings $y_{1}^{(\prime)}$ and $y_{3}$ may arise from new gauge interactions with a LFV $Z^{\prime}$ or a SU(2)L triplet gauge boson and $y_{2}$ naturally appears in two Higgs doublet models with a complex neutral scalar $H_{2}^{0}=(h_{2}+ia_{2})/\sqrt{2}$. The couplings $y_{1}$, $y_{1}^{\prime}$, and $y_{3}$ are hermitian, while $y_{2}$ may take any values. Similarly, there are three different $\Delta L=2$ lepton bilinears

[TABLE]

The neutral component of $\Delta_{3}$ only couples to the neutrino sector and thus it is irrelevant for our study of CLFV below. The couplings $\lambda_{1}$ and $\lambda_{3}$ are symmetric, while $\lambda_{2}$ may take arbitrary values. The $\Delta L=2$ coupling $\lambda_{1}$ naturally emerges in the Zee-Babu model which only couples to right-handed charged leptons Zee (1986); Babu (1988), while $\lambda_{3}$ may come from the SU$(2)_{L}$ triplet field in the Type II Seesaw model which only interacts with left-handed charged leptons Magg and Wetterich (1980); Schechter and Valle (1980); Lazarides et al. (1981); Wetterich (1981); Mohapatra and Senjanovic (1981). The coupling $\lambda_{2}$ can arise after the breaking of a unified gauge model where the lepton doublet and the charge-conjugate of charged lepton singlet $\ell^{c}$ reside in the same multiplet. One example is an SU$(3)_{c}\times$SU$(3)_{L}\times$U$(1)_{Y}$ model Pisano and Pleitez (1992). See Ref. Li and Schmidt (2019) for further details. The non-zero elements of the above couplings, i.e. $y^{(\prime)ij}_{1,2,3}$, $\lambda^{ij}_{1,2,3}$, can lead to the presence of CLFV processes. Below we focus on the off-diagonal elements of the couplings which induce CLFV on-shell production of a bilepton $X$ with two different flavor charged leptons, although we present the general results for all possible bilepton interactions.

Models with new massive vector bosons generally require the introduction of a new Higgs boson with the exception of an Abelian vectorial symmetry where the mass of the gauge boson can be generated via the Stückelberg mechanism Stueckelberg (1938). This may lead to new contributions mediated by the components of the new Higgs boson. However, the processes which we are considering do not suffer from any theoretical problems like the violation of perturbative unitarity. Thus, to remain as model-independent as possible, we will assume that the contribution of the Higgs bosons to lepton flavor violating processes is negligible. This can be realized either by making them sufficiently heavy or by suppressing the off-diagonal couplings to leptons. We restrict ourselves to the Lagrangians in Eqs. (1) and (2) for the rest of the paper. We do not take into account renormalization group corrections.

III Constraints

In this section we summarize relevant constraints on the CLFV couplings from anomalous magnetic moments of leptons, muonium-antimuonium conversion, constraints from lepton flavor universality in leptonic $\tau$ decays, electroweak precision observables, and new leptonic non-standard neutrino interactions, and the existing collider searches. We only consider constraints which are relevant for the sensitivity study in Sec. IV, e.g. we do not consider $\mu\to e\gamma$, because it depends on a product of two independent couplings. See Ref. Li and Schmidt (2019) for a study of other LFV processes mediated by bileptons. Note that, although we give the analytical results for general coupling matrices, in this and the following sections we assume that there is no additional CP violation. Thus the couplings of $H_{1\mu}^{(\prime)}$, $H_{3\mu}$, and $\Delta_{1,3}$ are real and symmetric. For simplicity we further restrict the couplings for the electroweak doublets $H_{2}$ and $\Delta_{2\mu}$ to be symmetric in the numerical analysis.

III.1 Anomalous magnetic moments

The muon magnetic dipole moment has $\sim 3.7\sigma$ discrepancy between the SM prediction Blum et al. (2018); Keshavarzi et al. (2018) and experimental measurements Bennett et al. (2006); Tanabashi et al. (2018)

[TABLE]

For the electron $g-$2, Refs. Parker et al. (2018); Davoudiasl and Marciano (2018) recently presented a precise measurement with a $2.4\sigma$ discrepancy

[TABLE]

Apparently, the muon (electron) AMM requires a positive (negative) new physics contribution to explain the discrepancy between the theoretical SM prediction and the experimental value. The one-loop diagrams contributing to the AMM by the Lagrangians in Eqs. (1) and (2) are shown in Fig. 1.

Using the general formulas provided by Lavoura in Ref. Lavoura (2003), we find that the leading contributions of the vector bosons $H_{1}^{(\prime)0}$, $H_{3}$ and $\Delta_{2}$ to the anomalous magnetic moment of the lepton $\ell$ are respectively

[TABLE]

to leading order in the charged lepton mass. For the scalars $\Delta_{1,3}$ and $H_{2}$, the new AMMs are given by

[TABLE]

to leading order in the charged lepton masses. One can see that, apart from $H_{2}$ and $H_{3}$, each new contribution to the anomalous magnetic moment has a definite sign. For $H_{2}$ in the limit of degenerate scalar masses $m_{h_{2}}=m_{a_{2}}$, the anomalous magnetic moment is not enhanced proportional to the mass of the $\tau$ lepton in the loop and the contribution obtains a definite negative sign

[TABLE]

Similarly, if all scalars apart from the CP-even neutral scalar $h_{2}$ are decoupled, the contribution is negative. In other extreme limits, such as $m_{a_{2}}\ll m_{h_{2}},m_{H_{2}^{+}}$ or $m_{H_{2}^{+}}\ll m_{h_{2}},m_{a_{2}}$, the anomalous magnetic moment may be positive. For the $H_{3}$ case, the contribution to the anomalous magnetic moment becomes negative in the limit of degenerate scalars $m_{H_{3}^{0}}=m_{H_{3}^{+}}$ and positive for $m_{H_{3}^{0}}\ll m_{H_{3}^{+}}$.

As a result, $H_{1}^{(\prime)0}$ and $\Delta_{1,3}$ can only explain the deviation in $a_{\mu}$, while $\Delta_{2}$ can explain $a_{e}$. The contributions from $H_{2,3}$ can have either sign and thus in principle address both anomalies. In this work we do not attempt to explain the deviations from the SM but rather derive a constraint on the LFV couplings described in Eqs. (1) and (2). In order to derive a constraint, we demand that the new physics contribution deviates from the experimental observation by at most $3\sigma$ for the electron and $4\sigma$ for the muon in order to account for the discrepancies in both measurements. The constraints from the AMMs are summarized in Table 1.

III.2 Muonium-antimuonium conversion

Muonium is the bound state of $\mu^{+}$ and $e^{-}$ and antimuonium is that of $\mu^{-}$ and $e^{+}$. If there is a mixing of muonium ($M=(\mu^{+}e^{-})$) and antimuonium ($\bar{M}=(\mu^{-}e^{+})$), the lepton flavor conservation of electron and muon must be violated and thus it is a sensitive probe for CLFV.

The probability of muonium-antimuonium conversion has been firstly calculated in Refs. Feinberg and Weinberg (1961a, b). Following the discussions in Refs. Feinberg and Weinberg (1961a, b); Matthias (1991); Horikawa and Sasaki (1996), we use the density matrix formalism to calculate the probability of muonium to antimuonium conversion. In contrast to previous calculations Horikawa and Sasaki (1996), we include off-diagonal elements in the Hamiltonian $H_{M\bar{M}}$ which mediates muonium-antimuonium conversion, and expand to the first order in the interaction Hamiltonian $H_{M\bar{M}}$. This is generally a good approximation for $B\gtrsim 0.1\ \mu$T assuming at most weak-scale interaction strength for $H_{M\bar{M}}$.

Muonium is described by the Hamiltonian

[TABLE]

where $H_{0}$ denotes the non-relativistic Hamiltonian for a hydrogen-like system, i.e. a bound state of two particles via a Coulomb interaction. The hyperfine splitting of the $1s$ state is described by $H_{hf}=b\,\vec{S}_{\mu}\cdot\vec{S}_{e}$ with $b\simeq 1.85\times 10^{-5}$ eV Mariam et al. (1982); Klempt et al. (1982), where $\vec{S}_{e,\mu}$ are the spins of the electron and muon, respectively. Finally, $H_{Z}=-(\vec{\mu}_{e}+\vec{\mu}_{\mu})\cdot\vec{B}$ describes the Zeeman effect with external magnetic field $\vec{B}$. The magnetic moments for electron and muon are defined as $\vec{\mu}_{e}=-g_{e}\mu_{B}\vec{S}_{e}$ and $\vec{\mu}_{\mu}=g_{\mu}\mu_{B}\vec{S}_{\mu}m_{e}/m_{\mu}$ with two $g$-factors $g_{e,\mu}\simeq 2\,(1+\alpha/2\pi)$ and the Bohr magneton $\mu_{B}=e/2m_{e}$.

In the uncoupled basis $\left|\uparrow\uparrow\right\rangle$, $\left|\uparrow\downarrow\right\rangle$, $\left|\downarrow\uparrow\right\rangle$, $\left|\downarrow\downarrow\right\rangle$, the Hamiltonian can be expressed as

[TABLE]

in terms of the ground state energy $E_{0}=-\alpha^{2}m_{red}/2$, the fine structure constant $\alpha$, the reduced mass of the two-body system $m_{red}=m_{e}m_{\mu}/(m_{e}+m_{\mu})\simeq m_{e}$ and two functions

[TABLE]

which parameterize the Zeeman effect. The energy eigenstates with their eigenenergies are thus given by

[TABLE]

and the mixing is described by

[TABLE]

For a vanishing magnetic field $s=c=1/\sqrt{2}$ and thus $\left|\lambda_{3}^{(M)}\right\rangle$ becomes the singlet state, while $\left|\lambda_{1,2,4}^{(M)}\right\rangle$ form the triplet state. The corresponding expressions for antimuonium are obtained with the replacements

[TABLE]

The interaction Hamiltonian $H_{M\bar{M}}$ inducing the muonium-antimuonium conversion may have different forms. We are particularly interested in the following vector and scalar interactions with equal and opposite chirality leptons

[TABLE]

[TABLE]

where the matrix representation on the right-hand side is in the basis $\left\langle\lambda_{i}^{(\bar{M})}|\hat{H}|\lambda_{j}^{(M)}\right\rangle$ and $a$ denotes the Bohr radius $a=(1/\alpha)(m_{e}+m_{\mu})/(m_{e}m_{\mu})\simeq 1/\alpha m_{e}$. Note that the scalar interaction leads to the same Hamiltonians in matrix form as the vector interactions, with only a different overall factor. The relevant contributions to muonium-antimuonium conversion from our Lagrangians are shown in Fig. 2 and result in

[TABLE]

The corresponding interaction Hamiltonians in the $\left|\lambda_{i}\right\rangle$ basis are

[TABLE]

[TABLE]

with

[TABLE]

Note the non-trivial dependence on the magnetic field in case of the electroweak doublet scalar $H_{2}$. In the limit of degenerate scalar masses $m_{h_{2}}=m_{a_{2}}$, the effective Lagrangian and the interaction Hamiltonian induced by $H_{2}$ simplify to

[TABLE]

which are consistent with our results in Ref. Li and Schmidt (2019).

The probability to observe a $\mu^{+}$ decay instead of a $\mu^{-}$ decay starting from an unpolarized muonium is

[TABLE]

where $\gamma=G_{F}^{2}m_{\mu}^{5}/192\pi^{3}$ is the muon decay rate, $\rho_{0}=\frac{1}{4}\sum_{i}\left|\lambda_{i}^{(M)}\right\rangle\left\langle\lambda_{i}^{(M)}\right|$ is the density matrix of the initial state muonium and $\mathcal{P}_{\bar{M}}=\sum_{i}\left|\lambda_{i}^{(\bar{M})}\right\rangle\left\langle\lambda_{i}^{(\bar{M})}\right|$ is the projection operator onto the final state antimuonium. For the interaction Hamiltonians of interest, there is only mixing between the second and third state as seen from Eqs. (17) and (18), and thus the probability can be explicitly written as

[TABLE]

and in particular for vanishing magnetic field, $B=0$, we find

[TABLE]

We compared our result with the analytic expression in Ref. Horikawa and Sasaki (1996) and the numerical values in Table II of Ref. Willmann et al. (1999) and found good agreement numerically, although the contributions from the off-diagonal entries in the interaction Hamiltonians $H_{M\bar{M}}$ were not included in Ref. Horikawa and Sasaki (1996). These additional contributions vanish given no external magnetic field $B$ and are generally subdominant at finite external magnetic field. They are suppressed by the factor of $\gamma^{2}/(\gamma^{2}+b^{2}(1+X^{2}))$ compared with the dominant contribution, because the weak decay rate $\gamma$ is much smaller than the hyperfine splitting and the Zeeman effect, i.e. $\gamma\ll b,bX,bY$.

Typically there are magnetic fields in the experimental setup. They suppress the conversion probability, because the degeneracy of the energy levels in $M$ and $\bar{M}$ is lifted. For the Hamiltonians with same chirality vector currents $H_{LL(RR)}$ and opposite chirality vector currents $H_{LR}$, the suppression factors of the probability at a finite magnetic field $B$ are

[TABLE]

respectively. In particular, we obtain the numerical values $S_{XX}(0.1\ {\rm T})=0.36$ and $S_{LR}(0.1\ {\rm T})=0.79$ for a magnetic field $B=0.1$ T. Our values are $O(1-2)\%$ larger than the results in Refs. Horikawa and Sasaki (1996); Willmann et al. (1999) due to the inclusion of additional off-diagonal entries in the Hamiltonian $\left\langle\lambda_{i}^{(\bar{M})}|H_{M\bar{M}}|\lambda_{j}^{(M)}\right\rangle$ with $(i,j)=(2,3)$ or $(3,2)$, but we agree with the overall magnitude of the suppression factor.

For the Lagrangians described in Eqs. (1) and (2), we obtain their probabilities as follows

[TABLE]

[TABLE]

where $A$ and $C$ are defined in Eq. (27). Note the non-trivial dependence of $P(H_{2})$ on the magnetic field. For real symmetric Yukawa couplings, the probability for $H_{2}$ can be written as

[TABLE]

which simplifies to

[TABLE]

The search for muonium-antimuonium conversion at the Paul Scherrer Institut (PSI) placed a constraint on the probability to observe the decay of the muon in antimuonium decay instead of the decay of the antimuon in muonium with a magnetic field of $B=0.1$ T, that is $P(B=0.1\ {\rm T})\leq 8.3\times 10^{-11}$ Willmann et al. (1999). This bound can be used to obtain the constraints on the CLFV couplings of the bileptons which we summarize in Table 2.

III.3 Lepton flavor universality

The interactions of leptons with neutrinos lead to new contributions to effective operators with two leptons and two neutrinos. In the absence of light right-handed neutrinos, there are only two types of effective operators

[TABLE]

Both Wilson coefficients in Eq. (40) are generated in the SM from the exchange of $W$ and $Z$ bosons and can be expressed in terms of the weak mixing angle $\theta_{W}$ as

[TABLE]

The second term in the expression for $g_{LL,SM}$ originates from $W$ boson exchange, while the other ones are due to $Z$ boson exchange. The new physics contributions to the two different sets of Wilson coefficients are given by

[TABLE]

and thus lepton flavor universality in lepton decays provides an interesting probe to the CLFV interactions for the bileptons. The relevant decay width for $\ell_{1}\to\ell_{2}\nu_{1}\bar{\nu}_{2}$ is Kinoshita and Sirlin (1959); Marciano and Sirlin (1988); Ferroglia et al. (2013); Fael et al. (2013); Pich (2014)

[TABLE]

in terms of the function $F(x)=1-8x+8x^{3}-x^{4}-12x^{2}\ln x$. The corrections due to the $W$ boson propagator and radiative corrections are respectively

[TABLE]

where $m_{1,2}$ is the mass of the charged lepton $\ell_{1,2}$ and $\alpha(\mu)$ is the running fine structure constant at scale $\mu$ with the result at one-loop order as

[TABLE]

The most sensitive probes of LFU are the ratios

[TABLE]

which we expanded to leading order in the new physics contribution which interferes with the SM. Decays to other neutrino flavors which do not interfere with the SM lead to

[TABLE]

where the prime indicates that we only sum over contributions which do not interfere. Taking into account both the experimental errors and the uncertainties in the SM prediction, the current experimental values and errors222We use the PDG Tanabashi et al. (2018) values and uncertainties in addition to the parameters given in Table 4.

are

[TABLE]

Thus, at the $2\sigma$ level, the relevant constraints are

[TABLE]

for the contributions interfering with the SM. The constraints on the non-interfering contributions are

[TABLE]

and thus generally weaker. In the collider analysis we always only consider a single coupling. Thus a single operator dominates and these bounds can be translated to constraints on the CLFV couplings. In Table 3 we collect the relevant constraints for the sensitivity study in Sec. IV. The constraints on $H_{1}^{0}$ and the charged component of $\Delta_{3}$ are due to the interference with the SM contribution, while others are derived from the non-interfering part.

III.4 New contribution to muon decay

As the Fermi constant is extracted from muon decay, a new contribution to muon decay via the operators in Eqs. (40) leads to an effective shift of the Fermi constant. We denote the Fermi constant extracted from muon decay by $G_{F,\mu}$ and the SM Fermi constant as $G_{F}$. The rate of muon decay to an electron and two neutrinos is given by Eq. (44). As the final neutrino flavors in muon decay are not measured there may be new contributions which do not interfere with the SM. Thus we find for the Fermi constant extracted in muon decay

[TABLE]

where the prime on the summation sign indicates that we are not summing over the interfering component with $(\alpha,\beta)=(\mu,e)$. Taking $G_{F,\mu}$ as input, we find to leading order the modification of the Fermi constant in terms of different Wilson coefficients

[TABLE]

This change of the Fermi constant leads to the modifications of other observables, in particular the weak mixing angle, the $W$ boson mass and the unitarity of the CKM matrix.

Weak mixing angle. In the SM the weak mixing angle is given by Brivio and Trott (2019); Tanabashi et al. (2018)

[TABLE]

where $\Delta r$ parameterizes the loop corrections in the SM. It depends both on the top quark and Higgs masses and is currently given by $\Delta r=0.03672\mp 0.00017\pm 0.00008$ Tanabashi et al. (2018), where the first uncertainty is from the top quark mass and the second from $\alpha(m_{Z})$. The shift in the Fermi constant leads to a shift in the weak mixing angle

[TABLE]

Using the Fermi constant extracted in muon decay $G_{F,\mu}$ together with $m_{Z}$ and $\alpha$ as input parameters, which are given in Table 4, we find numerically $s_{W}^{2}=0.22344\pm 0.00006$. A comparison with the on-shell value $s_{W,exp}^{2}=0.22343\pm 0.00007$ in PDG Tanabashi et al. (2018) leads to a constraint on the shift in the Fermi constant

[TABLE]

at the $2\sigma$ level.

$W$** boson mass.** The $W$ boson mass has been measured very precisely to be $m_{W,\rm exp}=(80.379\pm 0.012)$ GeV Tanabashi et al. (2018) and the current SM prediction is $m_{W,\rm SM}=(80.363\pm 0.020)$ GeV Tanabashi et al. (2018). Adding the errors in quadrature, we obtain

[TABLE]

i.e. the experimental measurement of the $W$ boson mass is consistent with the SM prediction at $1\sigma$. The SM prediction of the $W$ boson mass depends on the value of the Fermi constant. In the on-shell scheme, it is given by Tanabashi et al. (2018)

[TABLE]

Thus a new contribution to muon decay leads to an effective change in the SM prediction of the $W$ boson mass, even though there are no direct contributions to the $W$ boson mass. In the $(G_{F},m_{Z},\alpha)$ scheme we find to leading order for the shift in the $W$ boson mass

[TABLE]

Thus, the experimental result (61) translates into a constraint on the Fermi constant at $2\sigma$

[TABLE]

Unitarity of the CKM matrix. The unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix has been measured very precisely. In particularly the relation for the first row reads Tanabashi et al. (2018)

[TABLE]

As the CKM matrix elements $V_{uq}$ are measured in leptonic meson decays $M^{+}\to\ell^{+}\nu$, $V_{ud}$ also in beta decay, a modification of the Fermi constant extracted from muon decay leads to a violation of unitarity for the measured CKM matrix elements $V^{M}_{uq}$

[TABLE]

This can be translated in a constraint on the Fermi constant at $2\sigma$

[TABLE]

Combined constraint on the Fermi constant. Taking the most stringent constraints on the Fermi constant from Eq. (60), Eq. (64), and Eq. (67) we obtain

[TABLE]

where the lower bound comes from the weak mixing angle and the upper bound from CKM unitarity. It translates into a constraint on the Wilson coefficients

[TABLE]

The constraints for the different bileptons are collected in the second column of Table 5.

III.5 Non-standard neutrino interactions

Several constraints have been derived for the Wilson coefficients in Eq. (40) from neutrino-electron interactions

[TABLE]

The search for $\bar{\nu}_{\mu}\to\bar{\nu}_{e}$ oscillations at zero distance in the KARMEN experiment Eitel (2001) can be recast in a constraint on Biggio et al. (2009)

[TABLE]

Numerically, the constraints are weak compared with constraints from lepton flavor universality and electroweak precision observables discussed above, but complementary. Requiring the Wilson coefficients to stay within $2\sigma$ of the experimental errors, we translate the bounds in Eqs. (70), (71) and (72) to the constraints relevant for the comparison with the collider study of the bileptons in the third and fourth columns of Table 5. These constraints are not shown in the final Figs. 3 and 4, since they are weaker compared to constraints from the Fermi constant and lepton flavor universality.

III.6 Existing collider constraints

The DELPHI collaboration interpreted their searches for $e^{+}e^{-}\to\ell^{+}\ell^{-}$ in terms of 4-lepton operators Abdallah et al. (2006) which are defined by the effective Lagrangian

[TABLE]

where $\Lambda$ denotes the scale of the effective operator, $g$ is the coupling and $\eta_{ij}$ parameterizes which operators are considered at a given time and the relative sign of the operators in order to distinguish constructive (destructive) interference with the SM contribution. Conservative limits on the new physics scalars are obtained by setting the coupling to $g^{2}=4\pi$ and are summarized in the Table 30 of Ref. Abdallah et al. (2006).

These constraints can be directly applied to bileptons by comparing the effective Lagrangians. The relevant 4-lepton operators are given by

[TABLE]

As we have demonstrated in Ref. Li and Schmidt (2019), the analysis of contact interactions in Ref. Abdallah et al. (2006) does not directly apply to $\Delta_{2\mu}^{++}$, because the induced effective interactions do not fall into any of the types of effective interactions considered in Ref. Abdallah et al. (2006). Similarly for $H_{2}$, the analysis only applies in the limit of degenerate neutral (pseudo)scalar masses ($m_{h_{2}}=m_{a_{2}}$) and in the absence of one of the diagonal entries $y_{2}^{ee,\ell\ell}$, such that the scalar operator in the first term of Eq. (75) is not induced.

For the other operators we list the translated limits for masses well above the center-of-mass energy of LEP, $\sqrt{s}\sim 130-207$ GeV, in Table 6. Note that these limits are only valid when the new particle mass is much greater than $\sqrt{s}$. To make it valid for any masses, we should replace the mass in Table 6 by $(s\cos\theta/2+m^{2})^{1/2}$ after averaging over the scattering angle $\langle\cos\theta\rangle\simeq 1/2$.

Most available searches for a singly-charged scalar as well as a second neutral heavy Higgs at the LHC generally do not apply here, because they rely on couplings to quarks. If the singly-charged scalar is part of a scalar multiplet where the neutral component obtains a vacuum expectation value, the analyses in Refs. Aad et al. (2015); Sirunyan et al. (2017) place a constraint on the (electroweak) vector boson fusion production cross section and the subsequent decay to electroweak gauge bosons for singly-charged scalars with masses in the range $200-2000$ GeV. Both neutral and singly-charged scalars may also be produced in pairs via electroweak processes, but there are no applicable general searches to our knowledge. They may also be produced via $s$-channel $W$ boson exchange together with another component in the electroweak multiplet.

There are searches for doubly-charged scalars produced via electroweak pair production in both ATLAS and CMS experiments. The most stringent limits for decays to $e^{\pm}e^{\pm}$, $\mu^{\pm}\mu^{\pm}$, $e^{\pm}\mu^{\pm}$ pairs are set by the ATLAS experiment Aaboud et al. (2018b). It excludes masses $m_{\Delta^{++}_{1(3)}}\leq 320(450)$ GeV assuming BR$(\Delta^{++}_{1,3}\to\ell^{+}\ell^{+})\geq 10\%$. Assuming 100% branching ratio for a given channel, the constraints range from $650$ GeV for $\Delta_{1}^{++}\to e^{+}e^{+}$ to $850$ GeV for $\Delta_{3}^{++}\to\mu^{+}\mu^{+}$. Although the constraints set by the CMS experiment are slightly lower, it sets the most stringent lower limits on the final states with $\tau$ leptons CMS Collaboration (2017). Assuming 100% branching ratio in each channel, they range from $535$ GeV for $\Delta_{3}^{++}\to\tau^{+}\tau^{+}$ to $714$ GeV for $\Delta_{3}^{++}\to\tau^{+}e^{+}$.

IV Sensitivity of future experiments to CLFV

IV.1 Sensitivity from neutrino trident production

Neutrino trident production, the production of a charged lepton pair from a neutrino scattering off the Coulomb field of a nucleus, provides an interesting signature to search for new physics beyond the SM Mishra et al. (1991); Gaidaenko et al. (2001); Altmannshofer et al. (2014). So far, only the muonic trident has been measured with the results of $\sigma_{\rm exp}/\sigma_{\rm SM}=1.58\pm 0.64$ at CHARM-II Geiregat et al. (1990), $\sigma_{\rm exp}/\sigma_{\rm SM}=0.82\pm 0.28$ at CCFR Mishra et al. (1991) and $\sigma_{\rm exp}/\sigma_{\rm SM}=0.72^{+1.73}_{-0.72}$ at NuTeV Adams et al. (2000). While CHARM-II and CCFR achieved an accuracy of the level of 35% Altmannshofer et al. (2019), their measurements agree with the SM prediction and no signal has been established at NuTeV.

This will be improved by a measurement at the near detector of the Deep Underground Neutrino Experiment (DUNE), which can reach an accuracy of 25% Altmannshofer et al. (2019). See also Ref. Ballett et al. (2019) for a related study. The DUNE near detector is expected to measure three neutrino trident channels: $\nu_{\mu}N\to\nu_{\mu}e^{+}e^{-}N$, $\nu_{\mu}N\to\nu_{\mu}\mu^{+}\mu^{-}N$ and $\nu_{\mu}N\to\nu_{e}e^{+}\mu^{-}N$. The third one is not sensitive to new physics in a scheme where the Fermi constant $G_{F}$ is determined by muon decay, as it is directly related to muon decay by crossing symmetry. We calculate the cross sections of the former two channels in presence of the new contributions to the effective operators in Eq. (40), using the code provided by Ref. Altmannshofer et al. (2019). Assuming a precision of 25% for the cross section measurements, one can translate the expectations of the Wilson coefficients in Eqs. (42) and (43) into the sensitivities to the CLFV couplings quoted in Table 7. Note that all new physics contributions to the trident process $\nu_{\mu}N\to\nu_{\mu}e^{+}e^{-}N$ in principle result in two disconnected allowed regions of parameter space if no signal is observed at DUNE. However, some of them are not accessible by interactions of the bileptons and only one of the two regions is theoretically reasonable. We find the two reasonable regions for $H_{1}^{0}$ and $\Delta_{2}$ as shown in the left column of Table 7.

IV.2 Sensitivity of future lepton colliders to the CLFV

Apart from studying rare decays, the CLFV can be probed through scattering processes at lepton colliders. The new particles beyond the SM either mediate the scattering in off-shell channels or can be produced on-shell. In this work we focus on on-shell production of the bileptons together with a pair of different flavor leptons. The benefit of this on-shell scenario is that it only depends on one single CLFV coupling in each production channel and can be directly compared with the constraints placed by the low-energy experiments.

The proposed lepton colliders, in terms of the center of mass (c.m.) energy and the integrated luminosity used in our analysis, are

• •

Circular Electron Positron Collider (CEPC): 5 ab*-1* at 240 GeV Dong et al. (2018),

• •

Future Circular Collider (FCC)-ee: 16 ab*-1* at 240 GeV the FCC-ee design study (2017),

• •

International Linear Collider (ILC): 4 ab*-1* at 500 GeV Barklow et al. (2015), 1 ab*-1* at 1 TeV Baer et al. (2013),

• •

Compact Linear Collider (CLIC): 5 ab*-1* at 3 TeV Charles et al. (2018).

The CLFV processes can happen through the scattering of $e^{+}e^{-}$ Dev et al. (2018); Sui and Zhang (2018); Bhupal Dev et al. (2018) with on-shell new particles in final states, i.e. $e^{+}e^{-}\to\ell^{\pm}_{i}\ell^{\mp}_{j}H^{0},\ell^{\pm}_{i}\ell^{\pm}_{j}\Delta^{\mp\mp}$ with $\ell_{i},\ell_{j}=e,\mu,\tau$. The CLFV channels via $\Delta L=0$ or $\Delta L=2$ interaction at an $e^{+}e^{-}$ collider for probing the couplings $y^{ij},\lambda^{ij}$ are given in Table 8. The processes with one electron or position in final states occur through both s and t channels mediated by $Z/\gamma^{\ast}$. The processes without $e^{\pm}$ in final states only occur in s channel.

In order to estimate the lepton collider sensitivity to the CLFV couplings, we create UFO model files using FeynRules Alloul et al. (2014) and interface them with MadGraph5_aMC@NLO Alwall et al. (2014) to generate signal events. We apply basic cuts $p_{T}>10$ GeV and $|\eta|<2.5$ on the leptons in final states and assume a tau efficiency of $60\%$ Baer et al. (2013). Thus, the sensitivity reach is weakened by a factor 1.3 for the channels with one tau lepton in the final state compared with the reach for the $e\mu$ channel. The CLFV processes are not triggered by the initial state radiation (ISR) or final state radiation (FSR), and ISR/FSR barely introduces significant systematic uncertainties for our LFV signal. Moreover, the observation of CLFV does not rely on the exhibition of high energy tail induced by the radiation effects (ISR, FSR, bremsstrahlung, beamstrahlung, etc.). We thus neglect ISR/FSR effects in our analysis.

Besides the charged lepton pairs, the new bosons can decay into other SM particles, which makes the reconstruction rather model-dependent. To give a model-independent prediction, we assume 10% efficiency for the reconstruction of the new bosons. This takes into account the effects of cuts on decay products of bileptons, their decay branching fraction, and possibly missing bileptons in the detector. The dominant SM background is the Higgsstrahlung process $e^{+}e^{-}\to Zh$ followed by the mis-identification of one charged lepton from $Z$ boson decay Dev et al. (2018). The invariant mass of the two charged leptons in our signal can be easily distinguished from the background with a $Z$ resonance peak. Thus, after vetoing the $Z$ mass window, our CLFV signal is almost background free. We take the significance of $S/\sqrt{S+B}\approx\sqrt{S}$ as 3 for the observation of CLFV.

We show the sensitivity to $\Delta L=0$ and $\Delta L=2$ couplings in Figs. 3 and 4, respectively. Each of the CLFV processes only depends on one LFV coupling as shown in Table 8. Thus, the plots Figs. 3 and 4 do not rely on the values of other couplings. Note that in this work we do not expect to distinguish the chiral nature of the couplings of the mediating particles. Thus, the following results for vector $H_{1,3}^{0}$ and scalar $\Delta_{3}^{++}$ which only couple to left-handed leptons are the same as those for $H_{1}^{\prime 0}$ and $\Delta_{1}^{++}$ with only couplings to right-handed leptons, respectively. One can see that the interference between both s and t channels makes it more sensitive to probe couplings with $e\mu$ and $e\tau$ flavors, as shown in the left and middle panels of the figures. Smaller couplings can be reached for vector particles, such as $H_{1,3}^{(\prime)0}$ and $\Delta_{2}^{++}$, compared with new scalar bosons in both $\Delta L=0$ and $\Delta L=2$ interactions. The FCC-ee with the highest integrated luminosity provides the most sensitive environment and CEPC is the second most sensitive one in the low mass region. The CLIC and ILC with larger c.m. energy can probe the high mass region of the new particles.

The relevant upper bounds or projected sensitivities from low-energy experiments are also displayed for the corresponding couplings. We do not show the constraints from LFU in $\tau$ decays and electroweak precision observables, if they are weaker than the ones from the AMMs. Unless stated otherwise we assume that the components of the bilepton multiplets are degenerate in mass. It is straightforward to generalize the constraints from the effective operators of two charged leptons and two neutrinos in Eq. (40) by shifting the contours horizontally depending on the mass splitting between the different components of the bilepton multiplets. The constraints from the anomalous magnetic moments require a reevaluation. Lepton flavor universality sets the most stringent bound for $H_{1}^{0}$ and $\Delta_{3}$. The LFU bound excludes a majority of parameter space that the future lepton colliders can reach for $y_{1}^{\mu\tau}$ and $\lambda_{3}^{\mu\tau}$ couplings. Electroweak precision observables provide an even better constraint on the $e-\mu$ couplings of $H_{1}^{0}$ and $\Delta_{3}$. Muonium-antimuonium conversion also provides strong constraint on the $y^{e\mu}$ couplings. The constraints from the lepton AMMs vary with different mass spectra of the new particles and is relatively weak unless there is only one visible neutral scalar in the $H_{2}$ case. Finally, the LEP constraints from $e^{+}e^{-}\to\ell^{+}\ell^{-}$ scattering are generally weaker than the constraints from low-energy precision experiments. The neutrino trident cross section measurement at the DUNE near detector is not expected to be able to probe new parameter space.

IV.3 Sensitivity of an improved muonium-antimuonium conversion experiment

A future dedicated muonium-antimuonium conversion experiment may be able to improve the sensitivity to the Wilson coefficient of the effective operator by one order of magnitude Bernstein and Schöning (2019). This directly translates to an improvement in sensitivity by one order of magnitude compared with the constraints listed in Table 2 or about a factor of $3$ in terms of the CLFV couplings as shown in Fig. 3. Note, although muonium-antimuonium conversion can not probe the CLFV couplings of $\Delta L=2$ bileptons, it is sensitive to combinations of flavor-diagonal couplings.

V Conclusion

We studied the sensitivity of on-shell production of a bilepton with charged-lepton-flavor-violating couplings at future lepton colliders and compared it with the current constraints and future sensitivities of other experiments. We consider all possible scalar and vector bileptons with non-zero off-diagonal CLFV couplings. The bileptons are categorized into lepton number conserving ($\Delta L=0$) bileptons with $y^{ij}$ couplings and $\Delta L=2$ bileptons with $\lambda^{ij}$ couplings.

Depending on the nature of different bileptons, the most stringent constraints on the flavor off-diagonal couplings are from different measurements: Muonium-antimuonium conversions are currently most sensitive to the $e-\mu$ coupling of $\Delta L=0$ bileptons with the exception of $H_{1}^{0}$. Electroweak precision observables provide the most stringent constraint on the $e-\mu$ couplings of $H_{1}^{0}$ and $\Delta_{3}^{+}$. The $\tau-e(\mu)$ couplings of $\Delta L=0$ vector $H_{1}^{0}$ and $\Delta L=2$ scalar $\Delta_{3}$ with left-handed chirality are constrained by the absence of lepton-flavor-universality violation in $\tau$ decays, while the anomalous magnetic moment is most sensitive to an electroweak doublet scalar $H_{2}$ when only the neutral CP-even component is light. The LEP measurement of $ee\to\mu\mu(\tau\tau)$ provides a complementary constraint on the $e-\mu(\tau)$ coupling. The $\Delta L=2$ vector boson $\Delta_{2}$ is currently best constrained by the anomalous magnetic moment of the muon.

Future experiments will improve the sensitivity to several of these observables. In particular, we expect that a future muonium-antimuonium conversion experiments will lead to a factor of $3$ improvement for the $y^{e\mu}$ coupling. Furthermore, the measurement of neutrino trident scattering at the DUNE near detector (and other neutrino detectors) will provide independent probes.

Despite of the expected success and the increase in sensitivity of low-energy precision experiments, the search for on-shell production of a bilepton at future lepton colliders will provide a complementary probe of CLFV couplings. Although low-energy precision constraints provide the strongest constraints for $\mu\tau$ final state for all bileptons apart from $\Delta_{2}$, future colliders can probe new parameter space for $e\ell$ ($\ell=\mu,\tau$) final states. The FCC-ee with the highest integrated luminosity is the most sensitive machine and CEPC is the second most sensitive one in the low mass region. The CLIC and ILC with larger c.m. energy can probe the high mass region for the new bileptons.

In summary, future lepton colliders provide complementary sensitivity to the CLFV couplings of bileptons compared with low-energy experiments. The future improvements of muonium-antimuonium conversion, lepton flavor universality in leptonic $\tau$ decays, electroweak precision observables and the anomalous magnetic moments of charged leptons will probe similar parameter space.

Acknowledgements.

T.L. is supported by “the Fundamental Research Funds for the Central Universities”, Nankai University (Grant Number 63191522, 63196013).