We consider a singly electrically charged scalar h coupling to the lepton doublets, L⊃|∂μh|2+|∂μk|2-mh2|h|2+2habνaℓbh+kabℓaℓbk+c.c.(1)The doubly electrically charged scalar k, which does not contribute to NTP, has been included to make connection with UV completions. The singly charged scalar contributes to NTP via diagrams like the one shown in Fig. 1 and results in the amplitude shown in Eq. (2). In the following, we use xα and yα˙† to denote left- and right-handed initial states respectively, and yα and xα˙† to denote right- and left-handed final states, following [23]. We assign the labels {1,2,3,4,5}={ν,γ,ν′,ℓ+,ℓ-}, and we use the mostly minus metric ημν={1,-1,-1,-1}.

In the context of the equivalent photon approximation [19,24], the matrix element for γνa→ℓb+νcℓd- can be summarized succinctly as Mh=-2habhcd*e(P1-P4)2-mh2[A14qd2-md2+A35qb2-mb2]A14={(x1y4)(x3†q¯dε2x5†)+(x1y4)(x3†ε¯2y5)md}A35={(x1qbε¯2y4)(x3†x5†)+(x1ε2x4†)(x3†x5†)mb}qb≡P2-P4qd≡P2-P5ℓ=ℓ·σℓ¯=ℓ·σ¯,(2)where {a,b,c,d} label lepton generations.

The above matrix element contains contributions from four different diagrams. Two contain mass insertions appearing in the second terms of A14 and A35. The two amplitudes correspond to the photon interacting with either the negatively or positively charged lepton. The following identities, u=(x,y†)Tv=(y,x†)Tu¯=(y,x†)v¯=(x,y†),(3)can be used to rewrite the amplitudes in terms of the Dirac spinors, A14=v¯1PLv4u¯3PR(qd+md)ε2v5,A35=v¯1PL(qb+mb)ε2v4u¯3PRv5.(4)As a check of our calculations, we used the symbolic manipulation language FORM [25] and compared our results to [20]. LEP searches rule out charged Higgs for mh±≲100  GeV on general grounds based solely on its electromagnetic interactions with the photon and Z boson [26] and so we have ignored the four-momentum in the scalar’s propagator. The full cross section is obtained from Eq. (2) by σNν=Z2απ∫mjk2Sdssσγν(s)∫(s/2Eν)2∞dQ2Q2F2(Q2),(5)where F(Q2) above is the Woods-Saxon form factor [15,27].

For generic NTP final states the SM and BSM contributions can both be treated as real. The sign of the interference is dictated by the symmetry or antisymmetry of the couplings in Eq. (2), as well as the relative sign of the SM contribution. For a given NTP process, the presence of Z and/or W vector mediators induces an axial (CA) and vector (CV) coupling, upon which the matrix element depends linearly [15]. If the SM mediators are both W and Z bosons (CV,A>0), we find a positive relative sign. When the mediator is only a Z boson (CV,A<0), we get a negative sign. When the mediator is only a W boson (CV,A=1), we find a positive sign for m+>m- and a negative one when m+<m-; this effect is related to subtle helicity properties [15]. For antisymmetric couplings heμ=-hμe the new physics part of the matrix element carries an additional negative sign, while for the symmetric case (heμ=hμe), there is a positive sign. The final results for the sign of the interference terms are shown in Appendix C. For symmetric (antisymmetric) couplings, we have mostly constructive (destructive) interference.

Many flavor combinations for the incoming neutrino, outgoing neutrino, and charged leptons are possible. In deciding which reaction channel is ideally suited to one’s purposes, two strategies should be considered. First, a channel with a relatively high SM contribution could be chosen, allowing for interference effects, which will be dominant in the limit of small coupling.¹

This interference is not sensitive to the phases of the couplings, which can be expected on general grounds related to the arbitrary definitions of phases in hab [28].

A complementary approach is to consider a production channel that is closed in the SM, but open in the case of new physics. To ensure low backgrounds, one needs to be able to control the flux of incident (anti)neutrinos. To see this consider νμ→e-μ+ν which is SM forbidden, but possible in the presence of BSM scalars. If, however, the beam was contaminated with ν¯μ, then the SM allowed ν¯μ→μ+e-ν¯e would present a substantial background. DUNE has the capability to eliminate contamination with its neutrino horn. In contrast, SHiP has a much more complicated incident flux profile and cannot separate the neutrino and antineutrino fluxes.

The sensitivities we present in this paper are based on future experiments measuring rates consistent with the irreducible SM coherent NTP backgrounds. The details of the procedure are outlined in Appendix C, and numerical sensitivity equations are shown in Tables III and IV. A full simulation would have to be performed by the collaborations prior to their analysis, but we believe our analysis provides a good approximation. For simplicity, we focus the discussion on SHiP; however DUNE is also well equipped to tackle the same backgrounds.

The SHiP tau neutrino detector, modeled after the OPERA experiment [29], is based on emulsion cloud chambers (ECC) technology. The ECC is composed of a series of thin films interleaved with lead plates, followed by a muon spectrometer. A qualitatively similar setup to this was used in the CCFR experiment [14,30], which featured iron plates interleaved with liquid scintillators and drift chambers. The use of fine emulsion film layers will provide SHiP with more accurate track ID capabilities as compared to CCFR. That said, CCFR was able to observe a μ+μ--trident rate of 37 events given a theoretical SM prediction of 45 events. They isolated their signal by collecting μ- and μ+ events and imposing cuts on the energy, angles, total invariant mass, hadronic activity, and vertex resolution. SHiP can implement similar cuts; however one caveat is that CCFR was dealing with much larger incoming neutrino average energies (∼160  GeV) as compared with SHiP and DUNE providing it with an enhanced signal [15,19]. Since the bulk of these trident events are expected to come from SM processes, the kinematics of the outgoing pair of charged leptons is well captured in [20,21].

Consider a mixed-flavor ℓb+ℓd- lepton pair search with a hadronic veto. Final e+e- states can arise from resonant π0 production followed by a Dalitz decay where one of the photons is lost. For μ-μ+ final states, the dominant backgrounds are from νμA→μ-YX, where Y represents either a pion, kaon, charm- or D-meson which decays to a final state involving μ+ [31], as seen by NuTeV. Production of vector meson final states is also likely, but these can be distinguished from NTP since they deposit more hadronic energy and lead to a larger invariant mass for the lepton pair. The decay length of pions is on the order of a few meters, and therefore these backgrounds could also contribute to μ-e+ mixed-flavor final states if the meson fakes a charged lepton before decaying. The fake rate suppression at SHiP is very competitive. In particular, for electron ID efficiencies greater than 80%, the pion contamination rate is roughly ηπ→e≤0.5% [32]. In the SHiP detector at the end of the decay chamber, pion contaminations of ηπ→μ=0.1% can be achieved for muon identification efficiencies of roughly 1. For e- and μ+ final states, it is difficult to imagine how this would be produced outside of NTP. One possibility is coherent pion production from a ν¯e incoming state and a negatively charged pion. This background is expected to be small for a number of reasons, owing to the differences in the ℓ+ and ℓ- energy spectrum, the much smaller lifetime flux of ν¯e at SHiP. Combinatorial backgrounds where one observes an electron and an antimuon from two unrelated processes could be eliminated by the micron vertex resolution available at both SHiP and DUNE.

In this section, we illustrate the sensitivity of mixed-flavor NTP to charged scalars. We also highlight how certain flavor configurations precluded in the SM give superior sensitivity to existing constraints. As an illustrative example, we consider the model described by Eq. (1) and assume that haa=|h|, and hab=0 for a≠b. As is eventually discussed in Appendix B, most of the strong existing constraints commonly considered for these types of models [10,11] drop out and NTP provides the dominant constraint, outperforming the (g-2) for the muon. Our results are shown in Fig. 2.

We have forecasted the SM backgrounds at SHiP and DUNE using the rates presented in [15]. The best performing mode is the μ+e- channel at DUNE. A priori, the irreducible backgrounds to this process are ν¯μ→μ+ν¯ee- and νe→μ+νμe-. However, DUNE will have the ability to run in neutrino and antineutrino mode independently. This, coupled to the fact that the νe luminosities are low at this experiment, makes this channel a 0 irreducible background search. Hence, we can use this channel to investigate the interplay between 0 background and the lack of interference term in the cross section. We make the interesting observation that the mixed-flavor final states in both experiments provide stronger constraints than the μ+μ- states, while probing a Yukawa diagonal theory. In going from the muon final states to more general final states, the sensitivities to |h| at DUNE are improved by 50% whereas at SHiP, they can be improved up to 20%.

We now show how NTP compares to other constraints when taking into account doubly charged scalars, assuming that hab=kab in Eq. (2). This is analogous to the Higgs triplet (HT) model to be discussed later, without imposing the requirements of reproducing neutrino masses. The introduction of a doubly charged scalar k±± implies additional constraints from μ+e-→μ-e+. A natural question to ask is the following: What improvements in sensitivity are required to make trident competitive with these stronger constraints? We assume that one could measure the NTP cross section to within a given percentage of the SM cross section, for various benchmark precisions. These results are presented in Fig. 3. As a reference, the 10% curve for DUNE’s μ+e- channel corresponds roughly to the 90% C.L. bounds shown in Fig. 2. As can be seen, very high precision in the measured NTP cross section would be required to compete with the leading constraints on scalar couplings assuming k±±.

We consider the ZB model to demonstrate the effects of negative interference and the requirements of reproducing neutrino textures. Using Eqs. (A2) to (A4), we express all of the fab as a function of only feμ and Pontecorvo-Maki-Nakagawa-Sakata matrix data [34]. Note that due to the vanishing faa couplings of the ZB model, we are now probing nondiagonal couplings. We do this for both the normal and inverted hierarchies, and derive constraints on |feμ| as a function of mF using the best performing mixed-flavor trident channels. For the normal hierarchy, we set the CP-violating phase δ to its best fit value. For the inverted hierarchy, the dependence on δ factors out, and so the ZB model’s contribution to NTP is independent of δ. Our results are presented in Figs. 4(a) and 4(b).

The νμ→νiμ+μ- final state was observed at the CCFR and CHARM-II experiments, and we can calculate experimental bounds on the triplet model using their data. Singlet scalars cannot be probed using this data due to the antisymmetry of the couplings fab.

CHARM-II had a neutrino beam of ⟨Eν⟩≈20  GeV [12,13] with a glass target (Z=11) and the CCFR Collaboration had a neutrino beam of ⟨Eν⟩≈160  GeV using an iron target (Z=26) [12,14]. The two experiments measured production cross sections of [12] σCHARM-II/σSM=1.58±0.57,σCCFR/σSM=0.82±0.28.(7)Using CCFR as an example, we set bounds by demanding σSM+Triplet≤σSM(0.82+1.64×0.28),(8)where 1.64 standard deviations encompasses 90% of a Gaussian likelihood function. For mT in units of TeV, |tμμ|2[26.38mT2+1.59|teμ|2+|tμμ|2+|tτμ|2mT4]≤691.36(9)for CHARM-II, and |tμμ|2[34.87mT2+1.97|teμ|2+|tμμ|2+|tτμ|2mT4]≤168.07(10)for CCFR. Assuming |tab|=|t|, at 90% C.L. the two collaborations impose the following constraints: |t|≤3.10(mTTeV)CHARM-II,|t|≤1.77(mTTeV)CCFR.(11)The stronger bounds from CCFR are a result of the fact that this experiment saw a deficit of events in comparison to the SM prediction and so the upper bound at 90% C.L. is lower than CHARM-II.