Lepton-Number-Charged Scalars and Neutrino Beamstrahlung

PHYSICAL REVIEW D 97, 075030 (2018)

Lepton-number-charged scalars and neutrino beamstrahlung

Jeffrey M. Berryman,1 Andr´e de Gouvêa,2 Kevin J. Kelly,2 and Yue Zhang2,3 1Center for Neutrino Physics, Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA 2Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA 3Theoretical Physics Department, Fermilab, P.O. Box 500, Batavia, Illinois 60510, USA

(Received 26 February 2018; published 23 April 2018)

Experimentally, baryon number minus lepton number, B − L, appears to be a good global symmetry of nature. We explore the consequences of the existence of gauge-singlet scalar fields charged under B − L– dubbed lepton-number-charged scalars (LeNCSs)—and postulate that these couple to the standard model degrees of freedom in such a way that B − L is conserved even at the nonrenormalizable level. In this framework, neutrinos are Dirac fermions. Including only the lowest mass-dimension effective operators, some of the LeNCSs couple predominantly to neutrinos and may be produced in terrestrial neutrino experiments. We examine several existing constraints from particle physics, astrophysics, and cosmology to the existence of a LeNCS carrying B − L charge equal to two, and discuss the emission of LeNCSs via “neutrino beamstrahlung,” which occurs every once in a while when neutrinos scatter off of ordinary matter. We identify regions of the parameter space where existing and future neutrino experiments, including the Deep Underground Neutrino Experiment, are at the frontier of searches for such new phenomena.

DOI: 10.1103/PhysRevD.97.075030

I. INTRODUCTION B − L implies that neutrinos are Dirac fermions and nonzero masses are a consequence of neutrino Yukawa The gauge symmetry and particle content of the standard interactions, model (SM) are such that, at the renormalizable level, both 1 1 Uð ÞB (baryon number) and Uð ÞL (lepton number) are c L ⊃ yνLHν þ H:c:; ð1:1Þ exact classical global symmetries of the Lagrangian. Both Yuk turn out to be anomalous and hence violated at the quantum where H is the Higgs doublet (hypercharge þ1=2), L are the level. The combined Uð1Þ − (baryon number minus B L lepton doublets, and yν are the neutrino Yukawa couplings. B − L lepton number, ) however, turns out to be anomaly Flavor indices are suppressed. After electroweak symmetry free and is an excellent candidate for a fundamental pffiffiffi breaking, the neutrino Dirac mass matrix is mν yνv= 2, symmetry of nature, black-hole arguments notwithstanding pffiffiffi ¼ 246 2 (see, e.g., Refs. [1–3] and many references therein). where v ¼ GeV and v= is the vacuum expectation Experimentally, there is no evidence for the nonconserva- value (VEV) of the neutral component of the Higgs field. 10−12 tion of either baryon number or lepton number, in spite of Experimental constraints require yν to be of order or ambitious, ultra-sensitive, decades-long experimental smaller [4,6]. enterprises [4,5]. Here we are interested in the consequences of allowing If Uð1Þ − is a fundamental symmetry of nature, non- for the existence of new degrees of freedom charged under B L 1 − zero neutrino masses require the existence of new fermions Uð ÞB−L, assuming B L is conserved even if one allows charged under Uð1Þ − . We choose these to be left-handed for higher-dimensional operators. More specifically, we B L − antineutrinos νc (the conjugated states to the right-handed explore the physics of new scalar fields with nonzero B L neutrinos, to use a more familiar name), with lepton charge but which are otherwise singlets of the SM gauge number −1 and B − L charge þ1. Experiments require interactions. We will refer to these as lepton-number- the existence of at least two flavors of νc fields and, unless charged scalars, or LeNCSs. Different combinations of − otherwise noted, we assume there are three. Conserved LeNCS fields have different nontrivial B L charge and their couplings to the SM will be guided by B − L conservation. Since we treat B − L as an exact symmetry, we assume the scalar potential is such that none of the Published by the American Physical Society under the terms of LeNCS fields acquire VEVs. the Creative Commons Attribution 4.0 International license. νc Further distribution of this work must maintain attribution to The field content of the SM, augmented to include the the author(s) and the published article’s title, journal citation, fields, is such that all gauge-invariant, Lorentz-invariant and DOI. Funded by SCOAP3. operators have even B − L charges. Furthermore, it has

2470-0010=2018=97(7)=075030(16) 075030-1 Published by the American Physical Society BERRYMAN, DE GOUVÊA, KELLY, and ZHANG PHYS. REV. D 97, 075030 (2018)

ij – λ λαβ λαβ been shown [7 9] that, for any operator of mass dimension L c νcνcϕ ν ν ϕ ν ν ϕ O 2 int ¼ i j þ α β þ α β hþ H:c: þ ðh Þ; d and B − L charge qB−L, given the SM field content plus 2 2 v νc the fields, ð1:5Þ

d q =2 λ λ 2 Λ2 ð−1Þ ¼ð−1Þ B−L : ð1:2Þ where αβ ¼ βα ¼ v = αβ are the elements of a symmetric matrix of dimensionless couplings. We expand the As advertised, since d is an integer, the B − L charge of any physical Higgs boson field h around its vacuum expect- operator is even. Odd-dimensional operators have B − L ation value v up to linear order. In this simple setup, the ϕ charge 4n þ 2 while even-dimensional ones have B − L new scalar couples predominantly to SM neutrinos, and charge 4n, where n is an integer. This automatically implies we will demonstrate that it could lead to interesting, that all LeNCS species with odd B − L charge can only observable effects in neutrino experiments. For example, ϕ couple to the SM fields in pairs, while it is possible for could be radiated when neutrinos scatter off of regular ϕ LeNCS species with even B − L charge to couple indi- matter. Because carries away lepton number, the neutrino vidually to the SM. charged-current interaction will lead to wrong-sign ϕ In this paper, we will concentrate on a LeNCS with B − L charged leptons. This radiation would not only change 2 ϕ ϕ the charged-lepton energy spectrum but also lead to charge equal to þ , denoted hereafter by . A single 1 can couple to B − L-charge-two gauge-invariant SM oper- significant missing transverse energy in the event. We ators. As discussed above, these are odd dimensional, and explore these effects in a few accelerator neutrino experi- the lowest-order ones are νcνc (dimension three), and ments, including NOMAD, MiniBooNE, MINOS, the ν ν ðLHÞðLHÞ (dimension five). Hence, up to dimension six, NuMI Off-Axis e Appearance (NO A) experiment, and the most general Lagrangian that describes the SM aug- the Deep Underground Neutrino Experiment (DUNE). For ϕ 1 ϕ to be efficiently radiated in these experiments, we are mented by the field, assuming Uð ÞB−L is a good symmetry of nature, includes interested in mϕ values around or below the GeV scale and λ ∼ Oð1Þ. In contrast, the λc couplings between ϕ and the ij right-handed neutrinos are mostly inconsequential to λ ðLαHÞðLβHÞ L ⊃ c νcνcϕ ϕ ϕ i j þ 2 þ H:c: ð1:3Þ laboratory experiments. On the other hand, in conjunction 2 Λαβ with the λ or the cϕH couplings, λc effects can leave an imprint in cosmological observables, including the number ij Here, λc , i, j ¼ 1, 2, 3 (neutrino mass-eigenstate labels) of relativistic degrees of freedom at the time of big bang 2 are dimensionless couplings while Λαβ are dimensionful nucleosynthesis [10,11] and the formation of the cosmic couplings, α, β ¼ e, μ, τ (lepton flavor labels). We have microwave background (CMB) [12]. ignored operators that contain derivatives. In the Appendix, One last point is in order before proceeding. The ϕ we list the operators with mass-dimension eight that contain effective coupling of to neutrinos has a similar form a single ϕ field. Up to dimension five, there is only one to that of a Majoron [13,14], the (pseudo-)Goldstone boson interaction term that contains two ϕ fields, jHj2jϕj2. The in models where lepton number is broken in a controlled λ scalar potential for ϕ, therefore, contains way. The equivalent coupling for a Majoron is related to the observed neutrino masses, λ ∼ mν=f, where f is the 2 2 4 2 2 spontaneous lepton number breaking scale. The mass of the L ϕ 2 ¼ −μ jϕj − cϕjϕj − cϕ jϕj jHj ; ð1:4Þ j j ϕ H Majoron is generated by explicit lepton-number-violating interactions and, in most cases, is smaller than f. The μ ϕ where ϕ is the mass-squared parameter, while cϕ, cϕH are theory has an approximate lepton-number symmetry at dimensionless quartic couplings. Excluding the effects of energy scales above f. The new phenomena related to the ϕ nonrenormalizable operators, the mass squared of the field LeNCS ϕ to be discussed in the rest of this work cannot be 2 μ2 2 2 is mϕ ¼ ϕ þ cϕHv = . mimicked by a Majoron. The neutrino couplings to ϕ here We will concentrate on the consequences of Eq. (1.3) are in no way related to or constrained by the observed assuming that the ϕ mass is smaller than the electroweak neutrino masses. Moreover, we will be mostly interested in symmetry breaking scale. We will be especially interested in sizable couplings, λ ∼ Oð1Þ. If this were the coupling of a 2 the Λαβ couplings but will also discuss consequences of λc. Majoron, then the lepton-number-breaking scale would Throughout, we will assume that the cϕ and cϕH couplings are small enough that they do not lead to any observable 1As discussed above, LeNCS fields with odd B − L charge consequences in the laboratory or in the early Universe. We have to be emitted in pairs from neutrino beams. The corre- will briefly return to potential apparent baryon-number- sponding rate is more phase-space suppressed compared to single-ϕ radiation and, as a result, their impact on neutrino violating effects of higher-dimensional operators later. experiments is less significant. On the other hand, LeNCS fields After electroweak symmetry breaking, Eq. (1.3) with odd B − L charge serve as dark matter candidates. We will becomes elaborate on this to some extent in Sec. V.

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FIG. 1. Experimental constraints on the couplings between a LeNCS ϕ and neutrinos, λeα (left plot), λμα (right plot), with α ¼ e, μ, τ, as a function of its mass mϕ. The colorful regions in the plots are ruled out by precision measurements of the decays of pions (red), K mesons (pink), D mesons (green), Z bosons (orange), and Higgs bosons (blue). In the case of Z-boson decays, the lower limit is weaker for flavor-diagonal λ couplings (solid orange curve) than flavor-off-diagonal couplings (dashed orange curve). For the λee coupling, there is an additional constraint from measurements of double-beta nuclear decay rates. The yellow stars indicate points in the parameter spaces where one obtains a reasonably good fit to the MiniBooNE low-energy excess, as discussed in Sec. III C.

have to be very low, f ∼ OðeVÞ; under these circumstances, Γðh → νανβϕÞþΓðh → ν¯αν¯βϕ Þ Brðh Þ¼ ; ð2:2Þ it would be inconsistent to talk about Majorons with masses inv Γ → ν ν ϕ Γ → ν¯ ν¯ ϕ Γh ðh α β Þþ ðh α β Þþ SM above a MeV. is less than 34% [17], and translates into II. EXISTING CONSTRAINTS While the new scalar ϕ mainly interacts with neutrinos, jλαβj ≲ 0.7: ð2:3Þ there are several indirect constraints on the couplings λ, defined above in Eq. (1.5). In this section, we derive and This constraint is depicted in blue in Fig. 1 in the mϕ × λβα discuss these constraints, which will set the stage for the plane, for β ¼ e, μ. Identical constraints apply for λτα. discussions in the next section on the effects of ϕ in neutrino It is worth emphasizing that we consider values of λ less experiments. Many of the low-energy constraints discussed than or equal to one. These, in turn, imply Λ values [see here also apply to other models with new neutrino inter- Eq. (1.3)] of order the weak scale or higher. When using the actions mediated by new light bosons; see for example effective theory to describe processes at the electroweak Refs. [15,16]. scale (e.g., Z-boson and Higgs boson decays), the effective theory approach may still qualify as a faithful description of A. Invisible Higgs decay the system, especially if one allows the physics responsible for the dimension-six operator in Eq. (1.3) to be somewhat A light scalar could have an impact on the decay of the strongly coupled. We return to this issue in Sec. IV. 125 GeV Higgs boson. If mϕ

075030-3 BERRYMAN, DE GOUVÊA, KELLY, and ZHANG PHYS. REV. D 97, 075030 (2018) where the factor of δαβ arises for identical final-state neutrinos. The measured Z-boson invisible decay branch- ing ratio is ð20 0.06Þ% and the Z-boson total width is 2.495 GeV [18]. This translates into

jλαβj < 0.5ð1 þ δαβÞ; ð2:5Þ FIG. 2. A representative Feynman diagram for μ → 3e, which for mϕ ¼ 1 GeV. This constraint is depicted in orange in occurs at two-loop order. The time direction is from left to right. Fig. 1 in the mϕ × λβα plane, for β ¼ e, μ. The solid (dashed) line applies for α ¼ β (α ≠ β). Identical constraints apply → 2 − −ϕ for λτα. ðZ; AÞ ðZ þ ;AÞe e : ð2:7Þ

C. Charged meson decays This is identical to Majoron emission. Recent measurements of double-beta decay rates [20] translate into an If mϕ is smaller than one GeV there are strong constraints upper bound on from the decays of charged pseudoscalar mesons M ¼ π, λ ≲ 10−4 K, D [19]. In particular, the decay rate associated with ϕ j eej : ð2:8Þ ϕ emission (the is radiated from the final-state neutrino) is This constraint is depicted by a grey band in Fig. 1 in the not proportional to the well-known helicity-suppression mϕ × λ α plane, keeping in mind it only applies for α ¼ e. factor associated with two-body leptonic decays of charged e pseudoscalars. − − E. Charged-lepton flavor violation The decay width of M → lα νβϕ is At loop level, ϕ-exchange mediates charged-lepton − − ΓðM → lα νβϕÞ flavor-violating processes. At two loops, for example, 2 2 2 the exchange of ϕ and two W bosons in a double-box jλαβj G f ¼ F M ðm2 − m2 Þðm4 þ 10m2 m2 þ m4 Þ diagram leads to the rare muon decay process μ → 3e.A 768π3m3 M ϕ M M ϕ ϕ M representative Feynman diagram is depicted in Fig. 2. μ → 3 − 12 2 2 2 2 mM We estimate that the current constraint Brð eÞ < mMmϕðmM þ mϕÞ ln : ð2:6Þ −12 mϕ 10 [21] translates into

−2 We translate experimental measurements of, or constraints jλeeλeμj ≲ 10 : ð2:9Þ − − − on, M → lα ν¯β or lα ν¯βνν¯ decays as bounds on the above decay rate. These are summarized in the table below [4]. This and other similar constraints are not depicted in Fig. 1 These constraints are depicted in red/pink/green for π=K=D and will be ignored henceforth because they involve λ mesons in Fig. 1 in the mϕ × λβα plane, for β ¼ e, μ. More products of two different αβ couplings and can be avoided λ detailed estimates, consistent with the ones listed here, can simply by assuming that some of the αβ are much smaller be found in Ref. [19]. than the others.

Decay channels Decay channels Upper bound F. Cosmological constraint from PDG in our model on Br The virtual exchange of ϕ will lead to neutrino self- π → ν¯ ννπ¯ → ν ϕ 5 10−6 e e e α × interactions that can be parametrized by a four-fermion → ν¯ νν¯ → ν ϕ 6 10−5 K e e K e α × ≃ λ 2 2 −6 operator with an effective coupling constant Geff j j =mϕ, K → μν¯μνν¯ K → μναϕ 2.4 × 10 −6 for large enough values of mϕ. The strength of this effective D → eν¯e D → eναϕ 8.8 × 10 −5 D → μν¯μ D → μναϕ 3.4 × 10 interaction is constrained by cosmological observations sensitive to the neutrino free-streaming length in the early We also examined constraints from τ decays and final- Universe. A recent analysis of the CMB power spectrum ϕ 108 state radiation, and found limits that are weaker than data derives an upper bound Geff < GF [22]. This, in those estimated above using meson decays. τ decays, turn, translates into however, also provide information concerning λττ. mϕ λ ≲ 30 : ð2:10Þ D. Double-beta decays MeV

If mϕ were smaller than a few MeV, then ϕ could provide For a GeV-scale ϕ mass, this constraint is weaker than those new double-beta decay channels for certain nuclei. In discussed above; see Fig. 1. particular light ϕ particles can be produced via virtual The future sensitivity of the IceCube experiment to new neutrino annihilation, neutrino interactions was studied in Refs. [15,23]. While

075030-4 LEPTON-NUMBER-CHARGED SCALARS AND NEUTRINO … PHYS. REV. D 97, 075030 (2018) the scenario considered in those works is qualitatively contribute to Neff, though we will not consider this scenario different from the one considered here, we expect that the in detail. sensitivity of IceCube should be similar for the two models. It is important to keep in mind that λc and λ are qualitatively different couplings; λc are marginal Yukawa G. Supernova 1987A couplings while λ parametrize the consequences of higher- dimensional operators below the scale of electroweak The ϕ particle, if light enough, could be radiated through symmetry breaking. Understanding whether it is reasonable its interaction with neutrinos and accelerate the cooling of to assume that λc are tiny is akin to asking why the neutrino core-collapse supernovae. The impact of a light scalar on O 10−12 λ the observations of SN1987A is similar to that of a Yukawa couplings yν are ð Þ. Indeed, since c are Majoron, studied in Ref. [24], and revisited more recently also Yukawa couplings, the two types of interactions may in Refs. [25,26]. The analysis of Ref. [26] found that for be related in some mysterious way. Another concern is to −12 ask whether we should have also included in the effective 1 MeV ≲ mϕ ≲ 100 MeV, values of λαβ between 10 to Lagrangian the dimension-six operators 10−6 are excluded, for α, β ¼ e, μ. For larger couplings, the ϕ particles are reabsorbed by the supernova and no longer νcνc 2ϕ i j jHj affect its cooling. As a result, this bound corresponds to a Lϕ ⊃ : ð2:12Þ Λc 2 rather weak coupling, well below the region of the ð Þij parameter space depicted in Fig. 1. After electroweak symmetry breaking, these modify λij →λij 2 Λc 2 Λc ≫Λ H. Neff constraint on λc couplings c c þv =ð Þij. The constraint above requires if one is interested in λ values of order one [see Eq. (1.3)]. The couplings λ , defined in the first line of Eq. (1.3), are c Whether or not this is plausible depends on the ultraviolet mostly unconstrained as long as mϕ is heavier than the physics that leads to the effective operators in question. We neutrinos. This is easy to understand: λ mediates inter- c return to concrete examples in Sec. IV. actions between right-handed neutrinos and the ϕ field and, as long as the neutrinos are ultrarelativistic, right-handed neutrino properties are virtually unconstrained. For large III. IMPACT ON NEUTRINO-BEAM enough values of λc and light enough values of mϕ, EXPERIMENTS λ however, we anticipate c effects in neutrino decay and Since ϕ couples predominantly to neutrinos via the in the dynamics of relic neutrinos. We do not consider these coupling λ, its existence would impact neutrino scattering constraints here but hope to return to them in future work. experiments. Furthermore, since it carries lepton number, ϕ ϕ O 1 Since we are interested in GeV-scale with ð Þ emission leads to apparent lepton-number-violating effects λ ϕ coupling between the left-handed neutrinos and , in neutrino scattering. We discuss in some detail the physics however, we need to appreciate the fact that, in the early ϕ ϕ of emission in neutrino-matter scattering, followed by Universe, exchange between the right-handed and left- current constraints and the sensitivity of next-generation handed neutrino degrees of freedom may lead to a thermal neutrino scattering experiments, especially the LBNF- right-handed neutrino population. Within this context, the Δ −0 01 0 18 DUNE proposal. cosmological observation of Neff ¼ . . [12] ϕ “ λ We will be most interested in from neutrino beam- constrains the couplings c. A safe way to evade this strahlung” and accelerator-based neutrino beams. We constraint is to require the right-handed neutrinos to assume that constraints from atmospheric neutrinos are decouple from the SM plasma at temperatures above the 2 not competitive given the existing uncertainties on the QCD phase transition [27]. This translates into atmospheric neutrino flux and the fact that the incoming 2 neutrino direction is, a priori, unknown. We also assume 1 mϕ λ 10−9 that even in long-baseline, Earth-bound beam experiments, c < λ 1 : ð2:11Þ GeV neutrino decays induced by ϕ exchange are negligible. This is a safe assumption; for mϕ ≫ 1 eV, we can estimate the We note that if ϕ were lighter than a MeV, then it would be lifetime for ν → νν¯ ν¯ (mass-eigenstate indices implicit) as relativistic during big bang nucleosynthesis and thus 5 4 mμ mϕ τ ∼ τ 2There is currently a 3.4σ discrepancy between the value of the ν μ mν vλ Hubble constant indicated by local observations of the expansion 5 4 4 1 mϕ 1 of the Universe and the value determined by the Planck ∼ 2 1013 eV Collaboration. It has been argued that analyzing Planck data × years; ð3:1Þ mν 100 MeV λ jointly with the local measurements translates into ΔNeff ≲ 1 – [28 30]. Since neutrino masses require the introduction of at least τ two new fermionic degrees of freedom, this alternative constraint where μ, mμ are the muon lifetime and mass, respectively. would not significantly loosen the bound quoted here. Oscillation experiments with Earth-born neutrinos are

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Lab frame

Ebeam 2.5 GeV 1.5 m 0.6 GeV, 1.0

1500

1.0 1000 FIG. 3. The Feynman diagram of interest for ϕ emission in GeV neutrino scattering. The time direction is from left to right. A E neutrino with flavor α emits a ϕ, converting into an antineutrino 500 with flavor β before scattering off a nucleon and creating a 0.5 þ positively charged lepton lβ . In the models we consider, ϕ decays invisibly into neutrinos. 0 6 0.0 sensitive to lifetime values shorter than nanoseconds [31]. 0.5 1.0 1.5 2.0 2.5 Even solar neutrino experiments are only sensitive to life- E GeV times shorter than a tenth of a millisecond [32,33]. The right- handed neutrino couplings λ also mediate neutrino decay. FIG. 4. Correlated final-state energy distribution of the charged c lþ ϕ Since we already assume that these are very small (see lepton β and the LeNCS , for a 2.5 GeV neutrino beam mϕ 600 Sec. II H) we ignore all their effects henceforth. striking a proton at rest, for ¼ MeV. The colorful legend bar shows the number of events falling in each 50 MeV × 50 MeV bin, out of half a million simulated events. A. General discussion The Feynman diagram associated to a neutrino interact- 600 ing with a nucleon target accompanied by ϕ radiation is striking a proton target, assuming mϕ ¼ MeV. Clearly, þ for most of the events, the ϕ particle takes away most of depicted in Fig. 3. The amplitude for να þ p → l þ n þ β the beam energy. Nonetheless, the charged lepton also ϕ takes the general form typically carries enough energy to be measured in a i neutrino detector. A ¼ ACC ðiλαβÞuνðpÞ p= − =k − mν Throughout, we use MADGRAPH5 [34] (with the model 1 file implemented using FEYNRULES [35]) to simulate the ≃ λαβA =kuν p O mν ; : new physics as well as the SM processes, with the CC ð Þ 2 − 2 þ ð Þ ð3 2Þ p · k mϕ characteristic incoming neutrino beam energy spectrum for each experiment. We also calculate the differential cross where p is the four-momentum of the initial-state neutrino ν → lþ ϕ section for α þ p β þ n þ analytically (based on να, k is the four-momentum of the outgoing ϕ , and A is CC the 2 → 3 phase space integral given in Ref. [36]) and find the amplitude for the antineutrino charged-current inter- þ good agreement with the Monte Carlo simulations. In our action pν¯β → nl with the antineutrino leg amputated. β calculations, for the neutrino energies of interest, we only Superficially, the most striking signal here is the production consider the nucleon recoil, treating them as elementary of a wrong-sign charged lepton which could be identified in particles. We find that this approximation is reasonable magnetized detectors, capable of distinguishing the electric given our aspirations and will comment on it further in charge of the final-state charged lepton. Sec. III C. On the kinematics side, when mϕ → 0, the amplitude above is singular in the limit where the relative angle θ B. MINOS between p⃗and k⃗is zero. This corresponds to a collinear divergence of ϕ radiation. There is no infrared divergence The MINOS detector is magnetized, allowing charge ⃗ identification between μþ and μ−, and is hence sensitive to in the limit jkj → 0. When mϕ is comparable to the neutrino ϕ apparent changes in lepton number. The MINOS neutrino beam energy, it is possible for to be radiated at large ν ν¯ angles. As we will see, this can lead to sizable missing beam consists of 91.7% μ and 7% μ [37,38]. With it, the transverse momentum in a charged-current event and may collaboration measured the charged-current interaction rate ν¯ → μþ 3 84 provide a useful handle to identify the new physics for muon antineutrinos, μ þ p þ n,tobe . 15 contribution relative to SM backgrounds (see, e.g., 0.05 events=10 protons-on-target [39]. A nonzero λμμ Fig. 6 for more details). coupling leads to additional events with a μþ in the final Figure 4 depicts the final-state energy distribution of the state associated to the muon neutrino flux—roughly 13 þ charged lepton lβ and the LeNCS ϕ for 2.5 GeV neutrinos times greater than the antineutrino flux—by radiating a ϕ

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FIG. 5. Simulated event yields at MiniBooNE as a function of the reconstructed electron/positron energy Ee. The left plot corresponds to running the experiment with the neutrino beam and the right plot corresponds to the antineutrino beam. We calculate the þ − contributions to the signal of νμ þ p → e þ n þ ϕ or ν¯μ þ n → e þ p þ ϕ . The purple, green, and blue curves correspond to mϕ 2 equal to 10, 50, 500 MeV. All the curves assume λμe ¼ 1, and the number of signal events simply scales as jλμej . For comparison, the published data points are included with statistical error bars. The backgrounds are taken from Refs. [40,41], and the signals are stacked on top of the background. For simplicity, we assume the proton and neutron are elementary particles and ignore nuclear effects. Corroborating this approximation, we are able to reproduce the reported eþ=e− energy spectra (light grey background histograms) using the published beam νe and ν¯e energy spectra. particle, as depicted in Fig. 3. Defining the ratio of cross charge identification capabilities, such a signal is indistin- sections guishable from e− appearance. We simulate the expected spectrum of events as a þ σ νμ p → μ ϕ n R ð þ þ þ Þ function of the measured electron/positron energy for three ¼ þ ; ð3:3Þ σðν¯μ þ p → μ þ nÞ values of mϕ (10, 50, and 500 MeV) and compare with the results from the MiniBooNE experiment. The simulation and requiring that the additional contribution from Majoron uses the reported MiniBooNE muon-neutrino flux as a emission does not modify the observed rate by more than function of neutrino energy. We also simulate the expected 2σ, we arrive at R ≲ 0.002. This implies that jλμμj ≲ 1 for background using the MiniBooNE electron-neutrino flux to validate our approximation of treating the nucleons as MeV

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FIG. 6. Left: Region of the p=T × Eν plane containing most of the simulated signal and background events. The solid lines encompass − 90% of the events for the signal with mϕ ¼ 200 MeV (purple) and 1 GeV (green), and for the SM background νμ þ n → μ þ p (black). Corresponding markers for each of these show the peak of the eventpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi distribution. Right: The number of events per yearpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in a given bin of p=T , after marginalizing over Eν, for the background (dark: 20%= E½GeV hadronic energy resolution; light: 40%= E½GeV hadronic energy resolution) and the signal (purple: mϕ ¼ 200 MeV; green: mϕ ¼ 1 GeV; blue mϕ ¼ 2 GeV, all with λμμ ¼ 1) at the DUNE near detector, assuming 105 background events per year. We see that, for the more optimistic hadronic energy resolution, there are no background events with p=T > 0.5 GeV and for the less optimistic case, there are no background events with p=T > 1 GeV. have energies below 30 MeV. The LSND excess extends to E. DUNE reconstructed neutrino energies that exceed this bound. The upcoming DUNE will consist of both a near detector 5 and a far detector. The former is expected to collect Oð10 Þνμ D. NOMAD CC events per year [44]. Similar to the MINOS discussion, þ Between 1995 and 1998, the NOMAD experiment at we could use the channel νμ þ p → μ þ ϕ þ n to search λ CERN searched for oscillations of νμ (and νe)intoντ [42]. for nonzero μμ. At the DUNE near detector, however, it is The neutrino energies and baseline were such that anticipated that the charge of the muon will not be identified NOMAD was sensitive to small mixing angles but and hence one needs to account for several wrong-sign − mass-squared differences that are much larger than those background processes, including νμ þ n → μ þ p. that were ultimately revealed by neutrino oscillation In order to address this issue, we explore the kinematics experiments. NOMAD did not observe any ντ CC events of the final state, especially the presence of missing and placed an upper limit on the oscillation probability transverse momentum p=T when ϕ is radiated at large ν → ν 2 2 10−4 − Pð μ τÞ < . × . Using the published NOMAD angles. The SM background νμ þ n → μ þ p has no p=T neutrino flux and simulated scattering cross section, we assuming all final-state particles can be reconstructed with can translate this limit into one on the coupling jλμτj as a good precision. To estimate this effect, we simulate the function of mϕ (similar to the procedure discussed above background channel assumingpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi energy reconstruction 20–40% with MINOS). Since the NOMAD beam consisted of high- uncertainties betweenpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = E½GeV for the outgoing energy neutrinos, this limit extends to large values of proton and 3%= E½GeV for the outgoing muon.3 In the mϕ ≫ 1 GeV, despite the necessary energy budget left panel of Fig. 6, we display the region of the p=T × Eν ϕ τþ 4 required to produce both a and a . The resulting plane that contains most of the signal and background limit is depicted by the black curve in Fig. 7 (right). events. The solid lines encompass 90% of the events for the Similar bounds can also be extracted from the CHORUS experiment. Their results [43], however, as far Pðνμ → ντÞ 3The same background channel associated with the antineu- is concerned, are slightly less sensitive than those from þ trino beam (ν¯μ þ p → μ þ n) is expected to have much larger NOMAD. energy reconstruction uncertainties because of the neutron in the A bound on jλeτj can be similarly extracted from the final state. For this reason we concentrate on the neutrino-beam −2 NOMAD bound P ν → ντ ≲ 10 [42]. We expect the configuration as opposed to the antineutrino one. ð e Þ 4 2 → 2 extracted bound to be at least 1 order of magnitude weaker The inferred neutrino energy assumes a scattering λ process of a neutrino off an at-rest nucleon. The inferred signal than the one on j μτj discussed above and hence outside the neutrino energy, given this incorrect assumption, does not match region of the parameter space depicted in Fig. 7 (left). the real incoming neutrino energy.

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FIG. 7. Sensitivity of existing and future oscillation experiments to ϕ beamstrahlung in the λ × mϕ plane. The light blue shaded region is the union of all the constraints summarized in Fig. 1. Existing lower limits from MINOS and NOMAD are shown by the black curves. The future DUNE experiment could further probe λ values as low as indicated by the red curves. The red solid, dot-dashed, and dotted curves corresponds to a missing transverse momentum cut p=T greater than 0,0.5,1 GeV, respectively, for the signal event selection. See text for details.

signal with mϕ ¼ 200 MeV (purple) and 1 GeV (green), due to this background would be roughly a factor of 20 − and for the SM background νμ þ n → μ þ p (black). lower than the light grey curve shown in the right panel of Corresponding markers for each of these show the peak Fig. 6: a factor of 10 due to the 10% of the beam, and a of the event distribution. As expected, signal events tend to factor of 2 because the antineutrino-nucleon cross section have higher p=T and lower Eν, particularly for large values of is smaller than the neutrino-nucleon one. We expect, in mϕ. In this two-dimensional space, one can define a cut to the absence of a signal, that the experimental collaboration optimize the sensitivity to the signal. For simplicity, will be able to set a limit somewhere between the p=T > 0 5 1 however, we will consider the projection of this distribution . GeV and the p=T > GeV curves. A more thorough down to the p=T axis, as depicted in Fig. 6 (right). Here, we analysis using both p=T and Eν, of course, could improve assume 105 background events per year. this sensitivity. We perform three different analyses: one assuming zero In the near detector, a nonzero λμτ would lead to wrong- þ þ background (to establishpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the most optimistic result), one sign τ appearance, νμ þ p → τ þ n þ ϕ . Because the τ assuming 20%= E½GeV resolution for the final-state is difficult to identify in the detector and requires high- 0 5 protons, for whichp weffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi impose a p=T > . GeV cut, and energy neutrinos in order to be produced, we expect the one assuming 40%= E½GeV energy resolution, for which capability of DUNE to detect this to be significantly worse. To determine the range of λμτ to which DUNE will be we impose a p=T > 1 GeV cut. We calculate the value of sensitive, we simulate data and estimate the values for jλμμj for which DUNE can detect, above the given p=T cut, τþ one signal event per year, or 10 signal events over the which there would be one event in the near detector each experimental run of 10 years. Assuming no background, we year, whether or not that event is properly identified. The postulate this amounts to a discovery. The result is depicted resulting curve, as a function of mϕ, is depicted as a red line by the red lines in Fig. 7 (left). The most optimistic line is in Fig. 7 (right). Even in a perfect world, DUNE will not be Z solid, whereas the dot-dashed line corresponds to the p=T > able to improve on current bounds from NOMAD, -boson 0.5 GeV cut, and the dashed line corresponds to the p= > decays, and Higgs decays. T ϕ’ 1 GeV cut. The sensitivity of the DUNE experiment DUNE will also be capable of producing s with λ λ surpasses the currently disfavored region (light blue area) nonzero μτ and ττ at the far detector because, given the ν → ν for mϕ values between roughly 300 MeV and 2 GeV. 1300 km baseline, the oscillation probability Pð μ τÞ is When running in neutrino mode, DUNE still has a large for the energies of interest. The ντ can then interact via ν → μþ ϕ λ ν → τþ nonzero ν¯μ component to its flux, on the order of 10% of τ þ p þ n þ (with μτ)or τ þ p þ n þ the total beam makeup. These ν¯μ would contribute to the ϕ (with λττ). Both of these would result in large values of background; however the final state would include a p=T, similar to the near-detector discussion above, particu- (harder-to-reconstruct) neutron as well as a μþ. This larly in the latter case if the τþ subsequently decays to a μþ irreducible background will impact the sensitivity depicted and neutrinos. However, at the far detector, the dominant − − in Fig. 7 (left). We expect that the number of events per year background for this is ντ þ n → τ þ p, where, if the τ

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− decays to a μ , this background will have a signature smaller than twice the error barpffiffiffiffiffiffiffiffi in the measured value, similar to the signal in terms of its missing transverse we derive MT > ð350 GeVÞ × jλTj. As a consequence, momentum distribution. We find that this background in order for the cutoff scale Λαβ to be near the electroweak completely dominates the proposed signal for values of scale, we need a moderately large y˜αβ ∼ 2. λ λ mϕ and μτ or ττ of interest. In this scenario, the effective operator allowing inter- ν The currently running NO A experiment, as far as this actions of right-handed neutrinos with the LeNCS, dis- analysis is concerned, is similar to DUNE, particularly in its cussed in Eq. (2.12), is not generated at the tree level and ν near detector. The neutrino source for NO A generates one expects Λ ≪ Λc. It is, therefore, technically natural to neutrinos with energies near 2 GeV,and the experiment has a have the LeNCS more strongly coupled to left-handed O 104–105 near detector that collects ð Þ events per year. It can neutrinos than to right-handed ones, as discussed in also search for high-p=T events in its near detector, as Sec. II H. proposed here. We expect the corresponding line in Fig. 7 Further constraints can be derived from leptonic observ- ν for NO A to be less competitive due to the lower event rate ables. The presence of doubly and singly charged scalars and slightly worse energy reconstruction resolution com- induces a loop-level contribution to the anomalous muon pared to those expected for DUNE. We do not attempt a magnetic moment [45]. Curiously, the contribution to the detailed estimate of the sensitivity here but strongly encour- muon g − 2 is of the wrong sign to explain the current ν age searches for high-p=T events at the NO A near detector. disagreement between theory and experiment [4]; additional new physics would be needed to provide a satisfying IV. POSSIBLE ULTRAVIOLET COMPLETIONS explanation for this anomaly. While all flavors of charged In this section, we discuss possible ultraviolet-complete leptons contribute in the loop, the strongest bounds come models that, after integrating out heavy degrees of freedom, from an internal muon due to symmetry factors. Using the ≳ 500 ˜ lead to the dimension-six operator in Eq. (1.3). All models results of Ref. [45], we estimate MT ð GeVÞ × jyμμj to discussed are inspired by the tree-level realizations of the be safe with respect to this bound. Moreover, the doubly seesaw mechanism; the main difference here is that all new charged scalar permits the decay μ → 3e to occur at tree μ→3 ≤10−12 ≳ particle properties as well as their interactions are such that level. The limitpffiffiffiffiffiffiffiffiffiffiffiffi Brð eÞ [46] implies MT − the B L symmetry is preserved. ð150 TeVÞ × y˜μey˜ee, meaning some of the y˜αβ must be One option is to introduce a scalar T, a triplet under small for new physics to reside not too far above the weak 2 1 − 2 SUð ÞL with hypercharge þ and B L charge þ .We scale. We note that baryogenesis can occur in models such as will call it the type II model, because it has a structure the one considered here even though neutrinos are Dirac similar to the type II seesaw. As already highlighted, particles. This mechanism, called neutrinogenesis, has been however, unlike the seesaw mechanism, there are no studied in, for instance, Ref. [47]. − B L-violating effects here. The most general renormaliz- Another option, which we call the type I model, is to able Lagrangian in this case contains c introduce pairs of vector-like fermions Ni and Ni (i ¼ 1; 2; …;n is the number of vector-like fermions) that L ⊃ ˜ λ † ϕ − 2 † UV yαβLαTLβ þ THT H MTTrðT TÞþH:c:; are SM singlets carrying B − L charges ∓1, respectively. ð4:1Þ The most general renormalizable Lagrangian includes where y˜αβ are Yukawa couplings between the triplet T and L ⊃ ˜ c c λ ϕ UV yαiLαiHNi þ MN;iNiNi þ N;ij NiNj leptons of flavor α and β, λT are scalar couplings between ϕ λc ϕ c c λ˜c ϕ cνc the triplet, the Higgs field and the LeNCS , and MT is the þ N;ij Ni Nj þ Nν;ij Ni j þ H:c:; ð4:3Þ triplet scalar mass. When the T field is integrated out, the low-energy effective theory matches that in Eq. (1.3) with where y˜ are the strengths of the new Yukawa interactions ˜ λ 1 yαβ T and λN characterizes the strength of the interaction between Λ2 ¼ 2 : ð4:2Þ Nc and the LeNCS field ϕ.5 The constraint that the right- αβ MT ij handed neutrino couplings λc to ϕ are very small (see λ λc λ˜c At the same time, the Higgs VEV through the T term Sec. II H) implies that N;ij and Nν;ij are also small and generates a mixing between ϕ and the neutral component henceforth neglected. When all heavy fermion fields are 0 T in the triplet. In the large-MT limit, the mixing angle is integrated out, we obtain the effective operator in Eq. (1.3), 2 2 θ 0 ≃ λ 2 ϕ 2 ϕT Tv =ð MTÞ. For the range of mass of interest to ðLαHÞðLβHÞϕ=Λαβ, with this work, this mixing will lead to the decay of the Z boson ϕ’ into two s which eventually decay into neutrinos. The 5 νc We omit terms proportional to Ni j þ H:c: This can be done new contribution to the invisible Z decay width is without loss of generality and consists of a “basis choice” for the Γ 2θ4 24π 2 2θ c νc Z→2ϕ ¼ e ϕT0 MZ=ð sin WÞ. Requiring it to be N , fields.

075030-10 LEPTON-NUMBER-CHARGED SCALARS AND NEUTRINO … PHYS. REV. D 97, 075030 (2018) X 1 1 1 the scalar sector of the SM by introducing LeNCS fields ¼ y˜α λ y˜β : ð4:4Þ Λ2 i M N;ij M j (scalar fields with nonzero B − L charge), the Lagrangian αβ i;j Ni Nj of the SM plus the LeNCS fields could respect some Here, after electroweak symmetry breaking, the SM neu- accidental discrete symmetry; the LeNCS could be stable trinos mix with the new vector-like fermions. The mass and hence be an interesting dark matter candidate. matrix in the fν;Ng × fνc;Ncg basis takes the form If one were to extend the SM with any number of LeNCS species with integer B − L charges q − , then the ffiffiffi ffiffiffi ! B L p p c Lagrangian would be invariant under a Z2 symmetry where yνv= 2 yv=˜ 2 ν ð ν N Þ : ð4:5Þ all SM fields are invariant and each LeNCS species has Z2 0 c MN N charge ð−1ÞqB−L . In this scenario, the lightest odd-charged LeNCS is stable.6 Concretely, if there is only one LeNCS After diagonalization, the active-sterile mixing angles field χ with B − L charge þ1, then all of its couplings to ν θ ∼ ˜ 2 − between the and the N fields are of order as yv=MN SM fields involve operators that contain χmðχÞ n m (plus Λ λ and if is to be of order the weak scale, even for large N;ij, their Hermitian conjugates), where n, m are integers θ we expect as values to be of order one, which is clearly ruled (m≤2n). All of these operators are invariant under χ ↔−χ – out (see, for example, Refs. [48 53]). It may, however, be and this Z2 symmetry implies that χ is stable and a simple, possible to finely tune the model by taking advantage of its elegant dark matter candidate. The model here could also be flavor structure, a possibility we do not explore further here. viewed as an example of the longstanding wisdom that It also behooves us to highlight that while large y˜ Yukawa accidental symmetries can emerge once a model is endowed couplings are required, the neutrino Yukawa couplings yν, with higher symmetries [56–59]. In this section, we will which are very similar as far as the symmetry structure of the briefly discuss a few consequences of a LeNCS dark matter −12 theory is concerned, are much smaller, of order 10 . candidate, including a dark sector interacting via a LeNCS c Alternatively, the vector-like fermions Nia and Nia intro- portal, and comment on some generic features. A detailed 2 duced above can be replaced by SUð ÞL triplets, where a is discussion of a subset of these possibilities was recently 2 the SUð ÞL index in the adjoint representation: the type III presented in Refs. [60,61] (see also references therein). ˜ σa model. Their Yukawa couplings take the form yαLαi HNia, Our discussions here are based on a simple model. In σa where are the Pauli matrices. In this case, when the Nia and addition to the scalar ϕ with B − L charge þ2 discussed in c Nia fields are integrated out, the low-energy effective the preceding sections, we introduce another SM singlet a a 2 2 operator has the form ðLασ HÞðLβσ HÞϕ=Λαβ and 1=Λαβ scalar field χ carrying B − L charge −1, which serves as the is related to the ultraviolet parameters in the same way as dark matter candidate. The scalar potential of χ, ϕ and the λ˜c Eq. (4.4). In this scenario, there are no equivalent Nν;ij Higgs boson contains the following interaction terms: couplings but the concerns raised above remain. 2 2 2 2 2 All in all, both the type I and type III ultraviolet L ⊃ ðμϕχϕχ þ H:c:Þþcϕχjϕj jχj þ cHχjHj jχj completions appear to be unsuccessful, while the type II 2 ˆ þðχ O − 2 þ H:c:Þþ: ð5:1Þ model works very well. B L¼ As discussed earlier, we have been interested, for the ˆ 2 2 O − 2 most part, in effective couplings λαβ ¼ v =Λαβ [defined in B L¼ are gauge-invariant operators constructed out of the Eq. (1.5)] of order Oð0.1–1Þ. This implies that the mass SM particle content plus the right-handed neutrinos that scales of the new particles T (or N, Nc) should be around carry B − L charge equal to two. The ellipsis represents 2 the electroweak scale, unless one resorts to large (poten- higher-dimensional operators which couple χ n to SM − 2 1 tially nonperturbative) y˜ and λT;N couplings. The Large operators carrying B L ¼ n, for n> . The interactions Hadron Collider and future collider experiments have, in Eq. (5.1) will determine the thermal relic abundance of χ therefore, the opportunity to probe specific ultraviolet and the relative importance of the different interactions completions of the scenario discussed here [54,55].On govern how χ will manifest itself today. We have identified the other hand, as long as none of the new particles are a few distinct scenarios. lighter than the Higgs boson or the Z boson, our dis- (1) Neutrinophilic dark matter and a ϕ portal to the cussions in Secs. II and III, based on the effective field dark sector: If the μϕχ and cϕχ couplings defined in theory described by Eq. (1.3), remain valid. Eq. (5.1) mediate the most significant interactions controlling the physics of χ the ϕ particle plays the role of mediator between the SM sector and the V. DARK MATTER CONNECTION dark matter χ. Following the assumptions behind As discussed in the Introduction, all the gauge-invariant, 6 Lorentz-invariant effective operators constructed out of the Amusingly, it is easy to check that this Z2 charge is a 3 2 SM particle content plus any number of right-handed nonsupersymmetry version of the R-parity charge, ð−1Þ qB−Lþ s, neutrinos carry even B − L. This implies that if one extends where s is the spin quantum number.

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Eq. (1.5), where ϕ interacts with the SM sector boson and to the right-handed neutrinos. At the non- mainly through the ðLHÞ2ϕ dimension-six operator, renormalizable level, ϕ also couples to left-handed neu- χ is “neutrinophilic.” For example, two χ particles trinos via the Lagrangian spelled out in Eq. (1.3), at the will annihilate into two SM neutrinos via tree-level ϕ dimension-six level. exchange. B − L conservation implies one cannot We found that, for masses below a GeV, a LeNCS could close a neutrino loop and use Z-boson exchange to first manifest itself at intense neutrino beam experiments þ 0 0 couple χ to nucleons or charged leptons in the SM via ϕ radiation: να þ N → l þ ϕ þ N , where N, N are sector at the one-loop level. The lowest-order con- different nuclei and α, l ¼ e, μ, τ. We compiled the tributions to such interactions occur at the two-loop relevant constraints from a variety of laboratory processes, level or higher. Under these circumstances, one along with future sensitivities, on the ϕ–left-handed neu- expects a large hierarchy between the dark matter trino effective couplings λ [see Eq. (1.5)] in Figs. 1 and 7. interaction strength to neutrinos relative to the While the ϕ–right-handed neutrino couplings are only very other SM particles. This is different from other poorly constrained in the lab, bounds on the number of “leptophilic” dark matter candidates, proposed in relativistic species constrain it to be very small if the ϕ–left- Ref. [62]. There are cosmological constraints on handed neutrino effective couplings λ are accessible to dark matter–neutrino interactions from the CMB next-generation neutrino beam experiments. Note that [63,64]. On the other hand, a sizable dark matter– while we focused our discussion on a scenario where B − L neutrino interaction cross section may prove useful conservation provides guidance concerning the structure of for understanding small-scale structure formation in the new-physics Lagrangian, any scalar field that interacts the Universe [65,66]. predominantly via the operators contained in Eq. (1.3) will (2) Higgs portal dark matter: If, on the other hand, the be subject to the same constraints. cHχ coupling defined in Eq. (5.1) mediates the most We concentrated on effective operators with mass dimen- significant interaction controlling the physics of χ, χ sion less than or equal to six. At dimension eight, there are is indistinguishable from a vanilla Higgs portal many more operators, including those involving quarks. All scalar dark matter candidate. In this case, imposing are listed in the Appendix. The effective scales of many the correct thermal relic abundance for χ and dimension-eight operators are constrained to be very large satisfying the latest direct-detection constraints leads for low-mass LeNCS fields because these mediate the to χ masses larger than a few hundred GeV [67–69]. apparent violation of baryon number, including nucleon Measurements of the Higgs portal interaction are decays. For example, if ϕ is much lighter than a GeV, the insensitive to—and hence cannot reveal—the B − L operator number 10 in Table I in the Appendix—the operator charge of χ. ucdcdcðLHÞϕ—leads to a nucleon lifetime of order χ (3) Dark matter triggers nucleon decay:If interacts with the SM mainly via higher-dimensional oper- Λ 8 τ ∼ 1029 8 B L n years 8 ; ð6:1Þ ators where the SM fields carry both and 10 GeV number [contained in the last term in Eq. (5.1)] then it is possible for the dark matter to catalyze the where Λ8 is the effective scale of the operator. For heavier ϕ decay of a nuclear neutron into a neutrino, masses, one expects less severe but still relevant bounds that χ → χ − 1 ν þðZ; AÞ þðZ; A Þþ . Even though may depend on other ϕ couplings. For example, if the λ the final-state neutrino is invisible, the resulting couplings discussed here are significant then operator hadronic activity due to the removal of a neutron number 10 in Table I leads to the neutron decay n → ννν, from the target nucleus could lead to visible sig- mediated by off-shell ϕ decay. Whether it is consistent to natures if such a process occurs in a dark matter or a have λ of order one while Λ8 is much larger than a TeV neutrino detector. For a recent phenomenological depends on the ultraviolet physics responsible for the study, see Ref. [70]. effective theory. The type II example discussed in Sec. IV only generates operators in Table I that mediate apparent VI. CONCLUDING REMARKS lepton-number violation, not baryon-number violation. The We explored the hypothesis that B − L is a conserved reason is that, at the renormalizable level, there are no global symmetry of nature and that there are new SM couplings between the heavy T [colorless, SUð2Þ triplets] gauge-singlet scalar fields with integer nonzero B − L and SUð3Þ colored degrees of freedom. charge. These lepton-number-charged scalars were dubbed If B − L is an exact symmetry of nature, odd-charged LeNCSs, for short. Under this hypothesis, neutrinos are LeNCS species are interesting dark matter candidates and massive Dirac fermions and (at least two) right-handed only couple to SM degrees of freedom in pairs. The neutrino fields also exist. We concentrated our discussion stability of odd-charged LeNCS fields relies strongly on on a LeNCS field ϕ with B − L charge equal to two. At the B − L conservation. If, for example, B − L is broken by renormalizable level, ϕ only couples to the SM Higgs quantum gravity effects, we would naively expect effective

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χ νc χ − operators like LH =MP, where has B L charge one where the SM Hilbert series has been extended such that χ 1 and MP is the Planck scale, which mediate decay. In this Uð ÞB−L is an exact symmetry of the Lagrangian. χ → νν¯ τ ∼ 2 2 1 specific case, and χ→νν¯ MP=ðmχv Þ¼ year × Models of new physics generally require some ð1 GeV=mχÞ. Thus, for χ to be cosmologically long lived finagling to produce effective operators that contain and qualify as a dark matter candidate, the coefficient of this field-strength tensors; since including these in the dimension-five operator must be engineered to be small Hilbert series significantly lengthens computation time, enough, unless χ were as light as the neutrinos. There are, of we have ignored them altogether. While the inclusion of the course, ways to circumvent these constraints. One may, for (covariant) derivative operator is a necessary ingredient in 1 example, consider scenarios where Uð ÞB−L is gauged and the evaluation of the Hilbert series, we do not report spontaneously broken to a discrete subgroup which allows χ operators containing these in our list. Operators that are to be stable. This would be the case if the symmetry- products of lower-dimension operators—having the form breaking VEVs only violated B − L by two units [57,71,72]. jHj2 ×ðdimension-six operatorÞ or ν¯cν¯cϕ×ðdimension- On the other hand, our results concerning ϕ—aLeNCS four operatorÞ—are uninteresting for our purposes here, with charge two—are mostly insensitive to B − L quantum and so will be ignored. gravity effects, i.e., to Planck-suppressed higher-dimensional operators that violate B − L. For example, the Planck- 2 TABLE I. List of dimension-eight operators with exactly one suppressed Weinberg operator ðLHÞ =MP would give the LeNCS field and any number of right-handed neutrinos. Oper- left-handed neutrinos a very small Majorana mass and ators 17–23 only exist for multiple generations of fermions. render the neutrinos pseudo-Dirac fermions. It would Asterisks denote operators with multiple possible Lorentz con- not, however, modify any of the results discussed in tractions of the same fields. Note that one of the Lorentz-index Secs. II and III.7 In this case, the main structure of the configurations of operator 1 vanishes for one generation of framework discussed throughout this work (B − L as now fermions. The third column tabulates physical phenomena that an approximate global symmetry) can be maintained. can arise from each operator. Number Operator Associated phenomena ACKNOWLEDGMENTS 1* ecðLLÞðLHÞϕ ν¯e → νeϕ; l → l0ννϕ We would like to acknowledge helpful discussions with 2* dcðQLÞðLHÞϕ νp→lþnϕ; quark/meson decays Prateek Agrawal, Zackaria Chacko, Pilar Coloma, and Mark ¯ c ¯ ϕ ν →lþ ϕ Wise. J. M. B. is supported by Department of Energy Grant 3 u ðLQÞðLHÞ p n , quark/meson decays ¯ 0 No. de-sc0018327 and acknowledges the support of the 4 ν¯cðLLÞðLHÞϕ l → l ννϕ; νν → ννϕ¯ ;CνB Colegio de Física Fundamental e Interdiciplinaria de las 5a ν¯cðQQ¯ ÞðLHÞϕ ν¯N →νNð0Þϕ; quark/meson decays Am´ericas (COFI) Fellowship Program. The work of A. d. G., 5b ν¯cðLQ¯ ÞðQHÞϕ ν¯N → νNð0Þϕ; l → Mννϕ; K. J. K., and Y. Z. is supported in part by Department of quark/meson decays − − Energy Grant No. de-sc0010143. K. J. K. thanks the Fermilab 6 dcðLQ¯ ÞðQH¯ Þϕ n→νϕ; p → νπþϕ; τ → nπ ϕ Neutrino Physics Center for support during the completion of 7 ν¯cðQ¯ Q¯ ÞðQH¯ Þϕ n → νϕ; p → νπþϕ this work. This manuscript has been authored by Fermi 8 ν¯cðQ¯ Q¯ ÞðQH¯ Þϕ n → νϕ; p → νπþϕ Research Alliance, LLC under Contract No. DE-AC02- 9* ¯ c ¯cν¯c ¯ ϕ νp → lþnϕ; l → Mννϕ; 07CH11359 with the U.S. Department of Energy, Office of u e ðQHÞ quark/meson decays Science, Office of High Energy Physics. The United States 10 ucdcdc LH ϕ n → νϕ; p → νπþϕ Government retains and the publisher, by accepting the article ð Þ 11 u¯ cdce¯cðLHÞϕ νp→lþnϕ; quark/meson decays for publication, acknowledges that the United States ¯ Government retains a nonexclusive, paid-up, irrevocable, 12 dcdcν¯cðLHÞϕ ν¯N →νNð0Þϕ; b, s, meson decays worldwide license to publish or reproduce the published 13 ucu¯ cν¯cðLHÞϕ ν¯N →νNð0Þϕ; t, c, meson decays form of this manuscript, or allow others to do so, for United 14 ece¯cν¯cðLHÞϕ ν¯e → νeϕ; l → l0ννϕ States Government purposes. 15 dce¯cν¯cðQHÞϕ νp → lþnϕ; l → Mννϕ; quark/meson decays APPENDIX: DIMENSION-EIGHT OPERATORS 16 ucdcν¯cðQH¯ Þϕ n → νϕ; p → νπþϕ B − L † 0 WITH LeNCS OF CHARGE TWO 17 dcdcν¯cðQH¯ Þϕ n → νK ϕ; p → νKþϕ ¯ − þ − − In this appendix, we present a list of dimension-eight 18 dcdce¯cðQHÞϕ n → e K ϕ; τ → nK ϕ − − − operators that contain exactly one LeNCS field as well as 19 dcdcdcðLH†Þϕ n → e Kþϕ; τ → nK ϕ any number of right-handed neutrinos. To generate this list, 20 ν¯cν¯ce¯cðLH¯ Þϕ ν¯e → νeϕ; l → l0ννϕ we used the Hilbert series method described in Refs. [74,75], 21 ν¯cν¯cd¯ cðQH¯ Þϕ ν¯N →νNð0Þϕ; b, s, meson decays 22 ν¯cν¯cu¯ cðQH¯ †Þϕ ν¯N →νNð0Þϕ; t, c, meson decays 7 Things could be even safer if the model were further super- 23 ν¯cν¯cν¯cðLH¯ †Þϕ νν → ννϕ¯ ;CνB symmetrized [73].

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We are interested in the weak index structure of the (3) Apparent lepton-number-violating decays of heavy relevant operators, as this determines whether left-handed quarks and hadrons. neutrinos will experience the interaction, but we do not report (4) Apparent baryon-number-violating decays (n → νϕ, − − their Lorentz and color structures (except for one operator as p → νπþϕ, τ → nπ ϕ, etc.). an example, below). We do not concern ourselves with the (5) Neutrino-antineutrino conversion in scattering number of independent flavor components an operator (ν¯e → νeϕ and ν¯N → νNð0Þϕ). possesses; we will, however, flag operators that vanish for (6) Neutrino self-interactions (νν → ννϕ¯ ) that may one generation of fermion, as these are necessarily antisym- modify, e.g., the cosmic neutrino background (CνB). metric in the flavor indices of at least one pair of fermions, a It is beyond the scope of this work to analyze the fact that may be relevant for phenomenology. contributions of the operators in Table I to these phenom- We find that there are 23 unique field content arrangements ena; we merely intend to highlight the connection between that yield 24 different weak-index structures, which we these abstract-looking operators and processes that may tabulate in Table I. Parentheses denote pairs of weak-doublet occur in the natural world. ε δj fields whose indices are to be contracted with either ij or i, As previously stated, we will not delve into the Lorentz i j † as appropriate [e.g., ðLHÞ→εijL H , while ðLH Þ → and color structures of each operator; to do so would be δjLiðH†Þ ], where i, j (¼1, 2) are weak indices. Operators cumbersome and unilluminating. However, to provide more i j insight into the complete form of the operators, we present, 17–23 only exist for multiple generations of fermions. as an example, the full index structure(s) of operator 1: Asterisks denote operators with more than one possible contraction of their Lorentz indices; we note that one of c iα jβ kγ l c δ e ðLLÞðLHÞϕ → εαγεβδðε L L Þðε L H Þðe Þ ϕ; the Lorentz-index configurations of operator 1 vanishes for ij kl one generation of fermions. The last four operators are ðA1Þ ostensibly of the form ν¯cν¯cϕ × ðdimension-four operatorÞ but, for multiple generations of fermions, it is possible to or contract Lorentz indices such that these operators cannot iα jβ kγ l c δ factorize in this way. The operators in Table I can all be written → εαβεγδðεijL L ÞðεklL H Þðe Þ ϕ: ðA2Þ ϕ 1 as times a dimension-seven operator that violates Uð ÞB−L by two units, consistent with the findings of Refs. [7–9]. These operators have been written using two-component The third column of Table I tabulates physical phenom- spinor notation; the left-handed lepton doublet and left- ena that may arise from each operator. These include the handed positron fields (L and ec, respectively) have spinor following: indices α, β, γ, δ (¼ 1, 2). Flavor indices have been (1) Wrong-sign lepton production in neutrino-nucleon suppressed for clarity. In this form, it is clear why the scattering (νp → lþnϕ), as discussed in the main second of these vanishes for one generation of fermions: body of this work. because the first two lepton doublets are antisymmetric in (2) Apparent lepton-number-violating (semi)leptonic their Lorentz and weak indices, and because fermion fields decays of μ and τ (l → l0ννϕ and l → Mννϕ, where anticommute, the operator vanishes unless these fields are lð0Þ is a charged lepton and M is a charged meson). antisymmetric in their flavor indices.

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