Flavor-changing Majoron interactions with leptons

When the Standard Model Higgs sector is extended with a complex singlet that breaks global lepton number symmetry spontaneously, a massless Goldstone boson called the Majoron J arises. In addition to increasing Higgs invisible decay through mixing, the Majoron can generally have flavor-changing interactions with fermions. We find that type-III seesaw model poses such interesting properties with both charged leptons and neutrinos. This opens up new channels to search for the Majoron. We use the experimental data such as muonium-anti-muonium oscillation and flavor-changing neutrino and charged lepton decays to put constraints on the couplings. As a novel way to reveal the chiral properties of these interactions, we propose an experimentally measurable polarization asymmetry of flavor-changing l to l' J decays.


I. INTRODUCTION
Since the discovery of the 125-GeV Higgs boson, it is an intriguing question whether there exists other elementary scalar bosons in nature. A class of models with an economical extension of the Higgs sector in the Standard Model (SM) involve the introduction of a SU (2) L ×U (1) Y complex singlet S that induces the breakdown of global lepton number symmetry U (1) L . Such models are of primary interest because they can generate neutrino mass [1] and house a Goldstone boson, generically called the Majoron J, from the spontaneous symmetry breaking triggered by the vacuum expectation value (VEV) of S. Due to the singlet nature of S, the Majoron has no or very weak couplings with most SM particles, and can readily evade stringent constraints on massless boson searches. The Majoron also has a lot of implications in astrophysics and cosmology [2][3][4][5]. While most existing phenomenological studies rest on its flavor-conserving interactions with fermions, we focus in this work on its flavor-changing couplings as constrained by laboratory experiments, and propose a new experimentally measurable polarization asymmetry to study the chiral structure of the couplings.
In general, the Majoron can induce flavor-changing interactions even in the simplest Type-I seesaw model if the heavy neutrino mass is generated by the VEV of S [1]. In this scenario, the Majoron only interacts with the neutrinos and the Higgs boson at tree level. In particular, the Majoron couples to the Higgs boson through mixing, inducing the Higgs decay to two Majorons manifested as invisible decay. There are also simple models where the Majoron can interact with other fermions. In this work, we use Type-III seesaw model [6], with the VEV of a singlet to provide the heavy seesaw mass, as a simple explicit example to demonstrate the possibility of having flavor-changing interactions between the Majoron and the charged leptons. Such interactions alone will induce, for example, the µ → eJ decay, as considered a long time ago [7]. Such interactions open up new channels to search for the effects of Majoron, and may have novel implications on flavor-changing lepton decays. In particular, if one can measure the helicity of charged lepton in the final state, the proposed polarization asymmetry for the flavor-changing ℓ → ℓ ′ J decays can reveal the chiral properties of the interactions without the need to know the initial-state lepton polarization. With the same Majoron couplings to neutrinos as in Type-I seesaw model, Type-III seesaw model serves as a more general framework, which we will examine in this paper.
The structure of this paper is organized as follows. In Section II, we review the Type-III seesaw model and use it as an example to motivate the Majoron with flavorchanging couplings with fermions. Section III shows major experimental constraints on the Majoron and thus the bounds on the ratio of couplings and the Majoron decay constant. In Section IV, we propose a new observable, the polarization asymmetry, in the ℓ i → ℓ j J decays to probe the chiral nature of the Majoron interactions with the fermions, and study the experimental feasibility. Section V summarizes our findings.

II. MAJORON IN TYPE-III SEESAW MODEL
In Type-III seesaw model, besides the three generations of left-handed lepton doublets L iL : (1, 2, −1/2, 1) and the right-handed charged leptons E iR : (1, 1, −1, 1) in the SM, there are also three generations of righthanded lepton triplets Σ iR : (1, 3, 0, 1), where the numbers in the parentheses indicate their SU (3) C , SU (2) L , U (1) Y and U (1) L quantum numbers, respectively. The component fields of Σ R and its charge conjugated fields Σ L are as follows: We will rename them as ν R = Σ 0 c L , ψ L = Σ − L and ψ R = Σ + c L . The Higgs sector contains the usual Higgs doublet (1, 2, 1/2, 0) with h + and I to be "eaten" by the W + and Z bosons, and an additional . The Yukawa interactions involving the leptons conserve the global lepton number and are given by The Lagrangian terms relevant to charged lepton and neutrino masses and the interactions of J to fermions are given by with where f J = v s is the Majoron decay constant that sets the seesaw scale. The Majoron interaction terms in the form of derivative couplings are The mass matrix M ν and M c can be diagonalized in Here V ν , V e L(R) are 6 × 6 unitary matrices. Breaking V ν into blocks of 3 × 3 matrices, we have Note thatM ν is diagonalized in the basis (ν L , ν c R ), therefore ν L γ µ ν L +ν R γ µ ν R = ν L γ µ ν L −ν c R γ µ ν c R after rotation will not be diagonal and lead to flavor-changing J interactions with neutrinos. Also, as V LL,LR,RL,RR are not separately unitary, there are in general flavor-changing interactions induced in the charged lepton sector. One can reduce to Type-I seesaw model by dropping the Majoron interactions with charged leptons.
Working in the basis where M e and M R are diagonal, one can approximate [8,9] LL † , with X ν R being a diagonal matrix of generally different entries [11]. Individual off-diagonal couplings can now be much larger than 10 −3 and should therefore be constrained by data. There are also constraints from mixing between heavy and light neutrinos, which can be independent of light neutrino mixing [12].
It is worth emphasizing that the Majoron generally also has flavor-changing interactions with charged leptons. The sizes of the couplings are model-dependent and are a priori unknown. We will treat them as theory parameters and constrain them using experimental data. For this purpose, we generically write the Majoron couplings to the light charged leptons and neutrinos as where i and j are flavor indices for the initial and final states, respectively. In Type-III seesaw model, on-shell fermions, we get where ν denote the light neutrinos, and g ji e1/e2 = −(m j c e ji V /A ∓ c e ji V /A m i )/f J , and g ji ν1/ν2 = (c ν ji L m νi ∓ m νj c ν ji L )/2f J with m, m ν being the eigen-mass matrices of the charged leptons and light neutrinos, respectively.
The scalar potential in this model is given by Therefore, the Higgs boson naturally mixes with the real part of S and couples with the Majoron through with tan 2θ = λ hs vv s /(λ s v 2 s − λv 2 ), leading to the Higgs decay to Majorons which increases Higgs invisible width [13]. Here we assume h 1 ≡ h is the observed 125-GeV Higgs boson.

III. CONSTRAINTS
Because of the flavor-changing Majoron interactions with neutrinos, decays of the type ν i → ν j J can occur, making neutrinos unstable. The decay width is given by Γ . At present, without information of individual neutrino masses, the measured mass differences imply two mass orderings: normal hierarchy (m 1 < m 2 < m 3 ) and inverted hierarchy (m 3 < m 1 < m 2 ). In principle, data on the lifetime-mass ratio, τ i /m i , for neutrinos can constrain the parameters, yet current data [14] do not give useful constraints. For example, taking m 1 = 0 and m 3 = 0 respectively for the normal and inverted hierarchy cases, we have in the latter case. These numbers are orders of magnitude above the current data [14] if one demands f J to be as low as the weak scale and c ν ji V,A not to exceed order O(1). For nonzero m 1 and m 3 cases, the situation gets worse. It is thus clear that currently no constraints can be placed on neutrino-Majoron flavor-changing interactions.
As the Majoron can also have flavor-changing interactions with the charged leptons, much more severe con-straints can be obtained from related processes. Concentrating on charged lepton interactions, we will drop the superscript e in c e ji V,A in the following discussions for notation simplicity. We will consider three classes of constraints: (a) muonium-anti-muonium (M -M ) oscillation, Case (a) is induced first by exchanging J to produce the (μ(c V + c A γ 5 )e) 2 operator which causes muonium and anti-muonium to oscillate. Including both s-and uchannel contributions and averaging over the spin-0 and -1 contributions, we have the oscillation probability  Table I.
The calculations for case (b) ℓ i → ℓ j J and case (c) ℓ i → ℓ j ℓ klℓ are straightforward. For case (b), we will use the strongest experimental bounds available [14] to constrain the parameters |c ji V /A |. For case (c), we will only consider the flavor-changing couplings of the Majoron and neglect the flavor-conserving ones. Such processes constrain the products |c ji V /A ||c kℓ V /A |. The upper bounds from cases (b) and (c) are given respectively in blocks II and III of Table I. From the above, we see that the muonium-anti-muonium oscillation constrains |c V /A | to be less than around 0.4 if the Majoron scale f J is 1 TeV, similar in magnitude to the constraints from the ℓ i → ℓ j ℓ klℓ decays. The most stringent constraint on the couplings comes from µ → eJ with |c V /A | 3.6 × 10 −7 for f J = 1 TeV. If one takes |c V /A | ≃ 10 −3 instead, the best constraint for f J is 3000 TeV. With improved sensitivity in branching ratio determination, the bounds can be pushed further.
At one loop level, exchanges of Majoron can contribute to g −2 of charged leptons and ℓ i → ℓ j γ decays. The Ma- joron contribution to (g − 2) e,µ is generally small, giving relatively weak bounds on the couplings, as given in block IV of Table I. Note that because of the opposite deviations, (g − 2) e ((g − 2) µ ) constrains the axial (vector) couplings. Among the ℓ i → ℓ j γ constraints, given in block V, the strongest comes from the µ → eγ decay: |c eτ V /A ||c τ µ V /A | < 0.011 f J /TeV, assuming that flavorconserving couplings are negligible. We note in passing that for these loop processes, we have explicitly checked that the same results are obtained by using on-shell current interactions, Eq. (8), and the derivative Majoron couplings, Eq. (7).
We now work out the constraint on the coupling between Higgs and Majoron, λ hs , using the invisible Higgs decay branching ratio bound, Br(h → invisible) < 19%, from the LHC [23]. This is because λ hs can mediate the h → JJ decay with Γ(h → JJ) = λ 2 hs v 2 cos 2 θ/32πm h . Hence, this process contributes to the invisible width of Higgs. Due to Higgs mixing, the width of the usual SM decay modes will be modified to Γ SM cos 2 θ. Using Γ SM = 4.07 MeV [24] and the modified invisible branching ratio, we obtain a strong constraint of λ hs < 0.014.

IV. POLARIZATION ASYMMETRY
To determine the chiral nature of the Majoron interactions with charged leptons, we propose a novel measurement using the polarizations of the final-state leptons in the ℓ i → ℓ j J decays. The polarization 4-vector spinor is , where n i is the polarization of lepton in its rest frame. For high energy leptons, i.e., E i,j ≫ m i,j , an initially left-handed or right-handed lepton ℓ iL,iR can lead to a daughter lepton that is left-handed ℓ jL with n· p = −p or right-handed ℓ jR with n · p = p. Therefore, there are all four combinations of LL, RR, LR and RL for initial and final lepton polarizations. The helicity-conserving and -flipping decay rates are given respectively by and where x ij ≡ m i /m j > 1. Our result is more general than that given in Ref. [25], in which terms proportional to Re(g ji e1 g ji * e2 ) vanish under their coupling assumption. More detailed information about deriving the above results is given in the Appendix.
In practice, the polarization of initial-state lepton, presumably produced through collisions, is not easy to determine. We thus need to average over them. But the polarization of final-state lepton can be measured. Taking τ → µJ as an example, we define the polarization asymmetry where the second expression neglects terms of order m µ /m τ . This quantity probes the Majoron interaction in more detail.
As an explicit example, consider the τ + → µ + J decay 1 and we measure the longitudinal polarization of µ + , denoted by P L . Since A = 2P L − 1, the precision on the asymmetry measurement depends on how accurately P L can be determined. Assuming c V = −c A , the final-state anti-muon is dominantly right-handed and A ≈ 1, regardless of how τ is polarized. This value would be reduced by about 3% if corrections from (m 2 µ /m 2 τ ) ln(m µ /m τ ) and m 2 µ /m 2 τ are taken into account. Currently, BR(τ → µJ) is constrained to be less than 5.7 × 10 −3 . To have an estimate about the precision one can reach for A, let's take the branching ratio to be 10 −4 , which is well below the current bound, as an example. Given the fact that BELLE-II will produce in total about 45 billion τ + τ − pairs [26], one expects to observe O(10 6 ) τ + → µ + J decays. The polarization of µ can be obtained from the µ + → e + ν e ν µ decay by measuring the energy spectrum of 1 In order to determine the muon polarization, as required in our polarization asymmetry, it is preferred to study µ + because µ − may have reactions with surrounding matter via µ − p → νµn.
positrons. Without considering the detection efficiency, the statistical error on the muon polarization determination is seen to be at the per mille level. If the decay branching ratio is different, the statistical error is then scaled by a factor of 10 −4 /Br(τ → µJ).
Another significant background source is the Michel decay τ + → µ + ν µ ν τ with a branching ratio of 17.39%. As this is a three-body decay while our signal process is a two-body decay, one can impose a cut on the kinematic variable x ≡ 2E µ /m τ in the rest frame of τ to remove most of the background. Assuming that experiment can impose the cut 0.99 ≤ x ≤ 1, then the branching ratio of the background is reduced to ≃ 3.5 × 10 −3 . Assuming again that Br(τ → µJ) = 10 −4 , we will expect a statistical error of about 0.3% from BELLE-II data. Finally, the systematic error on the muon P L measurement at Spin Muon Collaboration had been estimated to be ∼ 3% [28], making the total error at a few percent level. We therefore encourage our experimental colleagues to carry out such an analysis.
We note in passing that, in fact, the polarization asymmetry can also be obtained from τ decays at rest, in . Therefore, one can determine the chirality of Majoron interaction from τ decays at low speeds, such as those produced at threshold by BES-III, where 600 million τ pairs have been obtained [27], and the future Super Tau-Charm Factory, where a few billion τ pairs per year are expected.
It may be tempting to use µ → eJ to determine the corresponding A by measuring muon or electron polarization since a high-luminosity muon beam will be available in µ-e conversion experiment at COMET and Mu2e.
This turns out to be rather difficult for several reasons. The branching ratio of µ → eJ is bounded to be smaller than that of τ → µJ (by a factor of ∼ 10 −3 ) and, hence, can offset the gain from high-luminosity muon beam for the µ-e conversion experiment. Secondly, the energy of the electron will be half of the µ-e conversion experiment, and only a very small fraction of µ → eJ decays resides in the signal region on target. It is therefore very difficult to measure such a process at COMET and Mu2e [3]. If the polarization information of the electron is further required, it would pose more difficulty as it does not decay.
Alternatively, one may consider using polarized initialstate muons to construct an analogous asymmetry by summing over the final-state electron helicities. However, the corresponding asymmetry is identically zero.

V. CONCLUSION
We have used the type-III seesaw model as an explicit example to motivate a Majoron with flavor-changing interactions with SM fermions, though our study is largely model-independent. We have examined existing major experimental constraints on the Majoron, including the ratios of the couplings, c ℓ ′ ℓ V /A , the Majoron decay constant, f J , and the coupling between the Higgs boson and the Majoron, λ hs . Finally, we propose an experimental observable, the polarization asymmetry, in the ℓ i → ℓ j J decays. Using the τ → µJ decay as an example, we conclude that through the measurement of A, it is a promising channel to probe the chiral nature of Majoron couplings with the charged leptons. As a final remark, if the Majoron is replaced by a Majoron-like particle with a finite mass, one can carry out a similar analysis so long as the decays are kinematically allowed.
The absolute-squared matrix element is given by Using the polarization 4-vector spinor s µ i defined in the main text, we get for different helicity combinations that and where and E i,j and p i,j are the energy and 3-momentum associated with ℓ i and ℓ j , respectively. The differential decay rate of the lepton flavor-changing process is given by To get the total decay rate of the ℓ i → ℓ j J process, we must integrate over the allowed energy range for the final state lepton ℓ j , where x ij ≡ m i /m j and in the lab frame with E i ≫ m i , the energy range becomes E i /x 2 ij ≤ E j ≤ E i . After integrating over the energy range, we obtain the decay rates for the helicity-conserving and -flipping processes given in Eqs. (12) and (13), respectively.