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Home , Sterile neutrino

D 101, 115031 (2020)

Origin of sterile dark via secret neutrino interactions with vector

† ‡ Kevin J. Kelly ,1,* Manibrata Sen,2,3, Walter Tangarife ,4, and Yue Zhang 5,§ 1Theoretical Physics Department, , P.O. Box 500, Batavia, Illinois 60510, USA 2Department of Physics, University of California Berkeley, Berkeley, California 94720, USA 3Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA 4Department of Physics, Loyola University Chicago, Chicago, Illinois 60660, USA 5Department of Physics, Carleton University, Ottawa, Ontario K1S 5B6, Canada

(Received 20 May 2020; accepted 8 June 2020; published 23 June 2020)

Secret neutrino interactions can play an essential role in the origin of . We present an anatomy of production mechanisms for sterile neutrino dark matter, a keV-scale gauge-singlet that mixes with active , in the presence of a new vector mediating secret interactions among active neutrinos. We identify three regimes of the ’s mass and coupling where it makes a distinct impact on dark matter production through the dispersion relations and/or scattering rates. We also analyze models with gauged Lμ − Lτ and B − L numbers which have a similar dark matter cosmology but different vector boson phenomenology. We derive the parameter space in these models where the observed relic abundance is produced for sterile neutrino dark matter. They serve as a well-motivated target for the upcoming experimental searches.

DOI: 10.1103/PhysRevD.101.115031

I. INTRODUCTION attributed to the small mass and mixing. As fermionic dark matter, a sterile neutrino cannot be arbitrarily light [19,20], The nature of dark matter is one of the biggest puzzles of which grants the opportunity to observe monochromatic our Universe. Many experimental searches for dark matter from its decay as an indirect detection signal [21,22]. are ongoing but with no success yet. New theoretical targets The attractiveness of sterile neutrino dark matter is further are needed to guide the path forward. Neutrinos, in many enhanced by a novel finding of how its relic abundance could aspects, are the least well-known within the be produced. Dodelson and Widrow [23] pointed out that the owing to their weak interacting nature. same mixing angle allowing a sterile neutrino to decay also In particular, neutrino self-interactions have never been directly measured in laboratories. Only indirect bounds enables it to be efficiently produced in the early Universe. have been inferred [1–18]. There is room for neutrinos to The corresponding parameter space has been scrutinized by various astrophysical probes and is virtually ruled out by the interact with themselves much more strongly than the – ordinary . This also accommodates a existing constraints [24 28]. A minimal solution to amelio- potential connection to dark matter. We investigate such rate this tension is to allow for a nonzero -number a possibility in the present work. asymmetry [29]. However, the required asymmetry does not A sterile neutrino, a gauge singlet fermion that mixes with have to be large and is challenging to observe elsewhere [30]. the Standard Model neutrinos, is one of the simplest dark Alternative proposals include decays [31], freeze-in matter candidates. It requires very little effort of model via additional particles [32,33], new interactions in the sterile building to realize such a candidate. Its cosmological sector [34,35], and gauge extensions of the Standard Model – longevity does not even require a symmetry but is simply [36 38]. These mechanisms often make dramatic changes to the ultraviolet side of the story, i.e., the initial conditions of the early Universe. *[email protected] † The focus of this work are theories that keep the most [email protected] ‡ [email protected] salient feature of the Dodelson-Widrow mechanism, the §[email protected] infrared dominance of dark matter production, and make testable predictions experimentally. The intuition is quite Published by the American Physical Society under the terms of simple. Dark matter relic abundance is set by the product of the Creative Commons Attribution 4.0 International license. the two ingredients in this mechanism, active-sterile neu- Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, trino mixing and the active neutrino reaction rate. While the and DOI. Funded by SCOAP3. former is strongly constrained by indirect searches, the

2470-0010=2020=101(11)=115031(17) 115031-1 Published by the American Physical Society KELLY, SEN, TANGARIFE, and ZHANG PHYS. REV. D 101, 115031 (2020) latter is allowed to be much stronger than the weak generated from lepton-number-violating sources beyond interaction, especially when neutrinos are interacting with the above Lagrangian. Assuming the mass hierarchy 2 ≪ ≪ themselves. This idea was recently entertained within a md=ms mν ms, the active-sterile neutrino mixing model where neutrinos self-interact via a lepton-number angle is approximately θ ≃ md=ms. The two mass eigen- charged scalar [39]. In the present work, we explore values are m1 ≃ mν and m4 ≃ ms, respectively. The heavier the impact of a new vector boson with coupling to neutrinos mass eigenstate and dark matter candidate is the following on the production of sterile neutrino dark matter. During the linear combination: dark matter production epoch, such a vector boson could manifest as a heavy mediator or a light and thermalized ν4 ¼ νs cos θ þ ν sin θ: ð2:3Þ degree of freedom. These possibilities help us to discover new variations beyond the original Dodelson-Widrow Alternatively, if the neutrino masses are Dirac (both mechanism. We consider three incarnations of the vector active and sterile), we must introduce partner fields for ν boson, in a neutrinophilic model, as well as in models with and νs for writing down the corresponding Dirac mass gauged Lμ − Lτ and B − L symmetries. We carry out terms, detailed calculations of the relic density of sterile neutrino dark matter in each model and derive the parameter space      mν 0 ν for the observed value to be reproduced. Furthermore, we NT νT C−1 þ H:c:; ð2:4Þ s ν0 confront our results with the existing and future exper- md ms s imental searches for each type of the new vector boson. pffiffiffi This paper is organized as follows. In the next section, where md ¼ yεv= 2, and mν and ms are Dirac masses. We we give an overview of sterile neutrino dark matter, and use set the (12) element to zero for simplicity. In this case, ν 0 the tension between Dodelson-Widrow and astrophysical mixes with the νs field, and the resulting dark matter state is 0 constraints as the motivation for introducing new neutrino ν4 ¼ νs cos θ þ ν sin θ, where θ ≃ md=ms. Its mass is forces. In Sec. III, we first analyze a model with a approximately m4 ≃ ms, assuming that md ≪ ms. neutrinophilic vector boson, deriving the parameter space In both cases, when θ ≪ 1, the mass eigenstate ν4 has a for dark matter relic abundance and comparing it with a ð0Þ large sterile neutrino component (i.e., ν4 ≃ νs ) and a very number of experimental probes. In Secs. IV and V,wedo small active component. Hereafter, we refer to ν4 as the the same but in the gauged U 1 and U 1 models, ð ÞLμ−Lτ ð ÞB−L sterile neutrino. The orthogonal linear combination, respectively. We draw conclusions in Sec. VI. denoted by ν1, is a mostly active neutrino mass eigenstate. With a mass between keV and MeV scales, ν4 is a viable II. MOTIVATION FOR SECRET dark matter candidate. The angle θ, parametrizing the NEUTRINO INTERACTIONS presence of its active component, is assumed to be small ν We start with the following Lagrangian which introduces enough so that 4 never reaches thermal equilibrium with a sterile neutrino field: the Standard Model sector in the early Universe. Assuming the Universe was born without a population of ν4, its relic T −1 T density could be produced through L ν ¼ L þ ν¯si=∂νs − ½yενs C H ðiσ2ÞL þ H:c:; SMþ s SM effects. In the early Universe, at temperatures before the ð2:1Þ decoupling of weak interaction, active neutrinos are con- stantly produced and destroyed by Standard Model weak ν where s is a left-handed fermion and a Standard Model interactions, as the flavor eigenstate ν. Since the produced ν − γ2γ0 T ν l− gauge singlet, C ¼ i , and L ¼ð ; Þ is the is a linear combination of mass eigenstates ν1 4, neutrino −1 2 ; Standard Model lepton doublet with = . oscillations will occur. In the early Universe, the active ν The parameter yε is a tiny Yukawa coupling between s and neutrino encounters a refractive potential due to interaction L that makes a negligible contribution to the observed with the thermal bath, which alters its dispersion relation neutrino mass. The main role of this term is to generate a from the zero temperature case, thereby changing the mixing between the active and the sterile neutrinos, after pffiffiffi oscillation probabilities. However, as the neutrino oscillates electroweak symmetry breaking, hHiT ¼ð0;v= 2Þ. on a timescale given by its energy and the mass-squared In the case where both active and sterile neutrinos are difference, it can undergo weak interactions that reset only Majorana particles, their mass terms take the form the active component. By this time, the original ν state has      already developed a νs component, which survives such a 1 mν m ν νT νT −1 d “measurement,” and eventually contributes to the relic 2 s C þ H:c:; ð2:2Þ md ms νs density of ν4 dark matter. The above oscillation process pffiffiffi can repeat for many times until weak interaction decouples. where md ¼ yεv= 2 is generated by the Yukawa coupling This is the crux of the Dodelson-Widrow mechanism [23]. in Eq. (2.1), and mν and ms are the Majorana masses It was demonstrated that a proper choice of the mixing

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θ Γ parameter can produce the correct dark matter relic dfν4 ðx; zÞ 2 ¼ sin 2θ fνðxÞ; ð2:8Þ density. d ln z 4H eff The probability for each active neutrino ν to oscillate into ≡ a νs (the latter mostly ends up as dark matter ν4) is dictated where H is the Hubble parameter and the parameter z by the effective mixing angle μ=T is introduced to label the cosmological time. Clearly, the choice of μ does not affect the result, and for Δ2sin22θ convenience, we choose μ ≡ 1 MeV throughout this work. sin22θ ≃ ; ð2:5Þ eff Δ2sin22θ þðΓ=2Þ2 þðΔ cos 2θ − V Þ2 fν is the Fermi-Dirac distribution for an active neutrino or T antineutrino, of the form fνðxÞ¼1=ð1 þ e Þ, where x ≡ which is obtained by taking the thermal average of E=T and it is z-independent. Unlike the proposal of [29], we will not consider the presence of a lepton asymmetry in time-dependent ν → νs oscillation probability. Here, Δ ≡ 2 2 2 the Universe; thus the chemical potential for an active ðm4 − m1Þ=ð2EÞ ≃ m4=ð2EÞ is the vacuum neutrino oscil- lation frequency, and E is the energy of the oscillating neutrino is set to zero. neutrino state. If the active neutrinos participate only in the This is an elegant mechanism, given its simplicity and Standard Model weak interaction, as assumed in the predictability. Indeed, sterile neutrino dark matter has been original work by Dodelson and Widrow, the effective explored extensively, from the model building to its various thermal potential V results from the self-energy of the aspects in detection (see, e.g., [41,42]), and strong tensions T emerge from the indirect detection. In particular, a nonzero neutrino, as depicted by the diagrams in Fig. 1. The ν − ν θ ν resultant potential is given by [22,40] s mixing makes the sterile neutrino dark matter 4 unstable. It could decay into either ν1 plus a or three   2 1 ν1. The former decay channel could lead to extra x-ray V ¼ −3.72G ET4 þ ; ð2:6Þ radiation from regions of accumulating dark matter. The T;SM F M2 M2 W Z nonobservation by x-ray telescopes disfavors the mixing angle θ needed for the Dodelson-Widrow mechanism to where G is Fermi’s constant and T is the temperature of F work in regions where the sterile neutrino is heavier than a the Universe. The label “SM” indicates that this expression few keV. Meanwhile, as a fermionic dark matter, it is found is only valid in the absence of any new neutrino interaction difficult to successfully fill lighter sterile neutrinos into the beyond the Standard Model (BSM). This is the result in the known dwarf galaxies [19,20,43]. As a result, the entire absence of any asymmetry. parameter space that produces the correct relic density for The interaction or collision rate Γ, due to the exchange of sterile neutrino dark matter via the Dodelson-Widrow , is given by [22,40] mechanism is almost closed [24–28].  1 27 2 4 ν It is also worth noting that there have been claims of an . GFET for e; ∼3 55 Γ ≃ ð2:7Þ unresolved . keV x-ray line from various nearby SM 0 92 2 4 ν ν . GFET for μ; τ: galaxy clusters [44,45], which is under thorough scruti- nization nowadays [27,46–48]. While the dust has not We will derive new expressions of these quantities in BSM settled, the relevant takeaway for our work is that if sterile frameworks in the upcoming sections. neutrino dark matter decays into this line, the correspond- Assuming the initial dark matter abundance to be ing mixing angle θ is too small to account for its relic negligible, the amount of dark matter produced is given density via the Dodelson-Widrow mechanism. by the integration of active neutrino production rate times A recent article [39] has suggested that a new, secret the oscillation probability over the cosmological time. interaction among the active neutrinos is effective for Because the oscillation probability is neutrino energy alleviating the above tensions. There, it was postulated dependent, we write down the equation that governs the that the secret neutrino interaction is mediated by a lepton- phase-space density evolution for the sterile neutrino with number charged scalar and much stronger than the ordinary fixed energy E [23,34,40] weak interaction, thus allowing the sterile neutrinos to be

FIG. 1. Self-energy diagrams contributing to VT in the Standard Model. In the absence of a lepton asymmetry, only the left diagram contributes.

115031-3 KELLY, SEN, TANGARIFE, and ZHANG PHYS. REV. D 101, 115031 (2020) X μ produced more efficiently in the early Universe. It was Lνν¯V ¼ λαβν¯αγ νβVμ; ð3:2Þ shown that with the new interactions, one can explain the α;β¼e;μ;τ relic density without violating all the existing constraints, for dark matter mass between a few keV and MeV scale. 2 2 where the couplings are λαβ ¼ v =ð2ΛαβÞ. This includes the point favored by the observation of the Such a new interaction, if strong enough, could keep the 3.55 keV x-ray line. As the most intriguing aspect of this neutrinos in thermal equilibrium with themselves longer model, it addresses the relic density of dark matter but with than the weak interaction, thus facilitating the production the new physics being introduced in the active neutrino rate of sterile neutrino dark matter in the early Universe. sector. There are predictions in low energy experiments, The task of this section is to quantify this statement and find such as a precision measurement of , decays, and the favored model parameter space for which the sterile accelerator neutrino facilities with a near detector (e.g., neutrino has a relic density that matches today’s observed DUNE). In turn, these probes of secret neutrino interactions amount. Meanwhile, the neutrinophilic vector boson could can test the fate of dark matter. also lead to observational effects in various processes in the In this work, we take this enticing idea further by analyzing models for secret neutrino interactions mediated laboratories where neutrinos interact. In the following by a new vector boson V. Such a new vector boson could subsections, we derive a list of existing and near-future experimental coverage on the model parameter space which naturally arise from Uð1Þ gauge extensions of the Standard Model. In the upcoming sections, we explore three of its has interesting interplay with the relic density favored incarnations: (i) a model with a neutrinophilic V; (ii) a region. Up to Sec. III G, we consider a representative case 1 1 where V couples only to the neutrino νμ, and the gauged Uð ÞLμ−Lτ model; and (iii) a gauged Uð ÞB−L sterile neutrino also mixes with νμ. The flavor dependence model. In each case, we confront the parameter space favored by relic density with existing and future exper- of our analysis will be commented afterwards, where it is imental constraints. pointed out that the relic density results remain similar for Strictly speaking, in the last two models, the neutrino other choices of flavors. Section III H comments on the interactions are not so secret because other Standard Model phenomenological implications of allowing V to couple to particles also see them, and naively they are already tightly different flavors of neutrinos. In Sec. III I, we present a constrained. Interestingly, we find that there still exists a possible ultraviolet (UV) completion for the effective viable parameter space to accommodate the correct dark operator introduced in Eq. (3.1). matter relic density in these models. Future experiments will fully probe the remaining parameter space, to either A. The anatomy of sterile neutrino discover or falsify our proposal. In contrast, the neutrino- dark matter production philic vector boson model with a genuine secret neutrino In the original Dodelson-Widrow mechanism, the relic self-interaction is less constrained and calls for new experi- density of sterile neutrino dark matter depends on the ments for it to be fully covered. neutrino weak interaction rate Γ, and the effective active- sterile neutrino mixing angle that is also controlled by the III. MODEL WITH A NEUTRINOPHILIC weak interaction, through the thermal potential VT. In the VECTOR BOSON presence of the new neutrino self-interaction mediated by Γ In the first model, we consider a new vector boson V V, both VT and are modified, although the relic density which couples only to the active neutrinos. Because can still be calculated using Eq. (2.8). 2 The new contribution to the active-neutrino thermal neutrinos exist in an SUð ÞL doublet, such a neutrinophilic nature of V is achieved via a higher dimensional operator, potential is shown in Fig. 2 and takes the following form added to the Lagrangian in Eq. (2.1), [49,50]: Z   1 1 2 2 2 μν 2 μ jλμμj ∞ m p 4Ep L ¼ L ν − VμνV þ m VμV V þ − SMþ s 4 2 V VT;VðE;TÞ¼ 2 2 dp L2 ðE;pÞ 8π E 0 2ω ω X ¯ T μ    ðLαiσ2H ÞγμðH iσ2LβÞV 2 1 m 1 þ 2 ; ð3:1Þ V þ −4 Λ × ω þ L1 ðE;pÞ Ep ; α;β¼e;μ;τ αβ e =T −1 2 ep=T þ1 4pEþm2 where the cutoff scale Λαβ characterizes the interaction þ V L1 ðE;pÞ¼ln4 − 2 ; strength of V with different combinations of lepton flavors. pE mV 2 2 ω 2 2 −2 ω 2 After electroweak symmetry breaking, the vacuum expect- þ ð pEþ E þmV Þð pE E þmVÞ L2 ðE;pÞ¼ln ; ation value of the projects out the neutrino ð−2pEþ2Eωþm2 Þð−2pE−2Eωþm2 Þ field from the lepton doublets. At low energies, the V V V couplings are neutrinophilic, ð3:3Þ

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FIG. 2. Loop contribution of the new interaction to the active- neutrino thermal potential VT . pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω 2 2 where ¼ p þ mV. In the very heavy or very light mediator V limit, the above potential simplifies into  −7π2 λ 2 4 45 4 ≪ j μμj ET =ð mVÞ;T mV; VT;VðE;TÞ¼ 2 2 ð3:4Þ þjλμμj T =ð8EÞ;T≫ mV:

It is worth noting that this potential changes sign as T passes mV. These asymptotic expressions are useful for us to infer the parametric dependence in the final relic density. FIG. 3. Scattering diagrams contributing to ΓV . The top panel In the numerical integration of the Boltzmann equation, we represents the contribution to the νν¯ → νν¯ channel, whereas the keep the most general form of the thermal potential, bottom panel shows the contribution to the νν → νν. For mV ≲ T, Eq. (3.3). the leading diagram is that of νμν¯μ → νμν¯μ with an on-shell V The new interaction mediated by the V boson provides exchange. new scattering channels among the neutrinos, including ν ν → ν ν ν ν¯ → ν ν¯ μ μ μ μ and μ μ μ μ, as shown by the Feynman those reported in Eq. (3.5). At high temperatures when diagrams in Fig. 3. The corresponding cross sections are helicity is approximately equivalent to , neutrinos    are left-handed and antineutrinos are right-handed. Both of λ 4 2 2 σ j μμj s mV 1 s them can contribute to the dark matter relic density via the νμνμ→νμνμ ¼ 2 2 þ 2 log þ 2 ; 4πm s þ m s þ 2m m mixing and oscillation (in a helicity preserving way). Thus, V V V V λ 4 2 4 4 − 2 the prefactor of 2 captures both helicity states of the active σ j μμj s − 4 2 ðmV s Þ νμν¯μ→νμν¯μ ¼ 2 2 2 m þ neutrinos and antineutrinos. 4πðs − m Þ m V s  V V  It is useful to show the asymptotic forms of the cross 10 1 s s sections and thermal rates in the heavy or light V limits. For × log þ 2 þ 3 ; ð3:5Þ ≫ mV mV T, we find ffiffiffi p 4 4 where s is the center-of-mass energy. If the initial-state jλμμj s jλμμj s ⃗ 0 ⃗0 σν ν¯ →ν ν¯ ¼ ; σν ν →ν ν ¼ ; ð3:7Þ four-momenta are denoted as ðE; pÞ and ðE ; p Þ,wehave μ μ μ μ 3πm4 μ μ μ μ 2πm4 s ¼ 2EE0 − 2p⃗· p⃗0 ¼ 2EE0ð1 − cos ϑÞ, where the tiny V V active neutrino masses are neglected, and ϑ is the relative and angle between p⃗and p⃗0. These cross sections are used for calculating the thermal 4 4 7πjλμμj ET averaged reaction rates for a neutrino with fixed energy E, Γlow-T E; T : : V ð Þ¼ 45 4 ð3 8Þ by averaging over the phase space of the other neutrino (or mV antineutrino) it scatters with, ≫ Z In contrast, for T mV, the typical center-of-mass energy 3 ⃗0 of the scattering is much higher than m . In this limit, the d p 0 0 V Γ ðE; TÞ¼2 fνðE ;TÞσ ðp;⃗ p⃗Þv ; ð3:6Þ V ð2πÞ3 tot rel cross sections in Eq. (3.5) take the forms

0 0 4 where fνðE ;TÞ¼1=½1 þ expðE =TÞ is the Fermi-Dirac jλμμj σν ν¯ →ν ν¯ ¼ σν ν →ν ν ¼ : ð3:9Þ distribution for an active neutrino or antineutrino, σ is the μ μ μ μ μ μ μ μ 4π 2 tot mV sum of the two cross sectionspffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in Eq. (3.5), and the relative 2 1 − ϑ velocity vrel is equal to ð cos Þ for ultrarelativistic These cross section blow up in the mV → 0 limit, similar to neutrinos. the case of Bhabha scattering in QED. Here the scattering Here, ΓV also includes the reaction rate of the antineu- range is regularized by a nonzero V mass. Adding the two, trino, which occurs via the charge-conjugate channels of the corresponding thermal rate is

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4 3 ζð3Þjλμμj T numerically solve the Boltzmann equation for the phase Γhigh-TðE; TÞ¼ : ð3:10Þ V π3 2 space distribution fν , Eq. (2.8), up to a sufficiently low mV 4 temperature Tf such that the value for fν4 saturates. This In addition, for T ≳ mV, the process νμν¯μ → νμν¯μ could temperature should be much lower compared to the also occur with an on-shell V exchange. In the early mediator mass mV but still higher than the dark matter Universe, this can also be interpreted as the decay and mass m4. In practice, we take Tf ¼ 100 keV. The dark inverse decay of fully thermalized V particles. In the narrow matter relic density today corresponds to width approximation, Z 1 3 ⃗ Yν4 s0m4 d p λ 4 Ω ≡ ; with Yν ≡ fν ðE; T Þ; j μμj mV 2 ρ 4 2π 3 4 f σν ν¯ →ν ν¯ ≃ δðs − m Þ; ð3:11Þ 0 sðTfÞ ð Þ μ μ μ μ 12γ V V ð3:14Þ 2 where γV is the decay rate of V, γV ¼jλμμj mV=ð12πÞ. The −5 −2 3 corresponding thermal reaction rate is approximately (here where ρ0 ¼ 1.05 × 10 h GeV=cm is the critical den- we use the Boltzmann distribution instead of Fermi-Dirac sity, s is the entropy density as a function of temperature of −3 in order to obtain an analytic expression) the Universe, and s0 ¼ 2891.2 cm is the entropy density rffiffiffiffiffiffi today. 2 2   jλμμj m T π pffiffiffiffi The result of the numerical calculation described above Γon-shell ≃ V − V 2 expð wÞþ Erfcð wÞ ; ð3:12Þ λ 8πE 4w is shown in Fig. 4, in the parameter space of μμ versus mV, where the orange contours stand for constant values of the ≡ 2 4 ∼ Ω 2 where w mV=ð ETÞ. Assuming E T, compared to the sterile neutrino dark matter relic density h . We choose a high-T 2 Γ in Eq. (3.10), the on-shell rate has an extra factor set of input parameters, m4 ¼ 7.1 keV and sin ð2θÞ¼ 4 7 10−11 ðmV =TÞ , which is more suppressed at high T. This × . This point corresponds to the potential dark corresponds to a phase space (collinear) suppression for matter explanation of the 3.55 keV x-ray line. Interestingly, two energetic neutrinos to scatter at center-of-mass energy all the curves exhibit an “S shape,” with three regimes equal to mV. However, this factor is not important when the having distinct slopes of the curve. This diverse parametric λ temperature drops to T ∼ mV. In addition, the on-shell rate dependence on μμ and mV suggests different detailed involves only two powers of the jλμμj coupling, in contrast production processes on which we shall elaborate below. to the other rates, making it easier to stand out in the small coupling regime. Numerically, we find that at T ≥ mV, ΓV is approx- Γon-shell Γhigh-T ≤ imately given by V þ , whereas at T mV Γ ≃ Γon-shell Γlow-T V V þ . These asymptotic rates in the two regimes capture the dominant contributions, and they Γon-shell match each other well at T ¼ mV because the V term dominates as long as jλμμj < 1. To summarize, in the presence of the V boson mediated neutrino self-interaction, the total thermal potential and reaction rate are given by

Γ Γ Γ VT ¼ VT;SM þ VT;V; and ¼ SM þ V; ð3:13Þ Γ where VT;SM and SM are the Standard Model contribu- tions, given in Eqs. (2.6) and (2.7), respectively. The new physics contribution VT;V is given by Eq. (3.3), and ΓV is given by Eq. (3.6). Note that the thermal potential is calculated at the amplitude level, so there will not be 2 2 any interference terms while estimating the potential. FIG. 4. Contours of Ωh for m4 ¼ 7.1 keV, sin 2θ ¼ Interference is, however, possible between the SM and 7 × 10−11, showing regions of over- and underabundance. The the V-induced contributions to the thermal rates. However, values of the relic density are labeled on the contours. Three since the new interaction is considered to be much stronger benchmark points on the contour corresponding to the observed than the ordinary weak interactions, the contribution of the relic density are chosen: A (λμμ ¼ 0.11, mV ¼ 2.77 GeV), −6 interference terms in the thermal rates are subdominant, and B(λμμ ¼ 0.003, mV ¼ 0.88 GeV), and C (λμμ ¼ 4.8 × 10 , hence neglected. With the above potential and rate, we mV ¼ 0.03 GeV).

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Along the curve where the correct relic density is repro- each active neutrino in the thermal plasma to oscillation 2 θ θ duced (Ωh ¼ 0.12), we mark three points A, B, and C, one into a sterile neutrino. At very early time, eff= is highly from each regime. suppressed because the oscillation frequency Δ is much The different parametric dependence mentioned above smaller compared to Γ and VT. The production of the sterile θ can be understood by examining Eq. (2.8), which in the neutrino state become more efficient at later time when eff very small θ limit takes the form catches up with θ. The above temperature dependence leads to interesting Γ Γ Δ2θ2 dfν4 ðx; zÞ 2 interplay in the third row of Fig. 5, which depicts the actual ≃ θ fν x ≃ fν x : eff ð Þ Γ2 4 Δ − 2 ð Þ d ln z H H = þð VTÞ dark matter production rate dfν4 ðx; zÞ=d ln z and is propor- Γ=H θ2 ð3:15Þ tional to the product of and eff . Here we focus on the energy E ¼ T (or x ¼ 1) which has the highest population We plot different factors on the right-hand side of the above of active neutrinos as the source term. We introduce three Γ 1 ≫ equation in Fig. 5. In the figure, each column corresponds useful timescales: (i) z0: when =H ¼ .Atz z0, the to case A, B, or C, respectively. The first row depicts Γ=H active neutrino self-interaction falls out of thermal equi- as a function of time (labeled by z ≡ 1 MeV=T) in each librium and dark matter production ceases; (ii) z1: when Δ ≃ Γ ≪ θ ≪ θ case. This ratio dictates the epoch when the active neutrino MaxfjVTj; ag.Atz z1, eff implies less ≳ θ ≃ θ self-interaction drops out of thermal equilibrium (when efficient production. At z z1, eff implies efficient ≡ μ ≫ Γ=H < 1). In all cases the rate Γ features a bump which production; and (iii) z2 =mV .Atz z2, the vector corresponds to the exchange of an on-shell V in the boson V is too heavy to be produced in the Universe. neutrino self-interaction. This contribution is most impor- First of all, if z0 is smaller than both z1 and z2, the new neutrino interaction via V decouples too early and is tant when T ∼ mV but becomes Boltzmann suppressed at irrelevant for dark matter production. We do not consider T ≪ mV. At low temperatures, the off-shell V exchange dominates. this case. The three cases A, B, and C mentioned above The second row of Fig. 5 depicts the effective mixing correspond to the following different hierarchies of θ angle eff [defined in Eq. (2.5)] divided by the vacuum z0;z1;z2 and thus different sterile neutrino dark matter θ θ2 production scenarios: mixing angle . The value of eff dictates the probability for

FIG. 5. The variation of the new thermal scattering rate as compared to the Hubble parameter (top), the ratio of the effective mixing angle to the vacuum angle (middle), and the differential sterile neutrino production rate (bottom) as a function of z for the three benchmark points A, B, and C, as chosen in Fig. 4.

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(i) Case A corresponds to heavy V and large coupling Like case B, this is another new production regime λμμ. In this case, the hierarchy of timescales satisfies beyond Dodelson-Widrow. z2

2 5 θ m4 Ων ∝ : ð3:17Þ 4 λ 2 5 j μμj mV

This is a new production regime beyond the Dodelson-Widrow mechanism. (iii) Case C corresponds to a very light V and tiny coupling λμμ. In this case the timescales satisfy z1

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Γ Γ For heavier sterile neutrino dark matter (black curve), we model is calculated as h→νμν¯μV= total, where the Higgs total Γ choose a correspondingly smaller mixing angle in order to width total is the sum of the Standard Model Higgs width satisfy the x-ray constraints. We find that heavier dark matter (∼4 MeV) and Γ →ν ν¯ . The present LHC upper bound on λ h μ μV generically requires larger j μμj and/or smaller mV for the Higgs invisible branching ratio, BrðH → invisibleÞ < 24% relic density, making it more constrained experimentally. On [51], leads to a constraint in the λμμ versus mV parameter the other hand, due to the small-scale structure bounds space, corresponding to the purple shaded region in Fig. 6. arising out of free-streaming considerations of the sterile The future running of high-luminosity (HL) LHC is neutrinos as well as dwarf galaxy and Lyman-α constraints, expected to further improve the above limit to BrðH → there is limited room to make the dark matter ν4 much lighter invisibleÞ < 2.5% [52], which corresponds to the purple than 7.1 keV. We have explicitly checked this for the three dashed curve in Fig. 6. We find this is the leading constraint benchmark points considered above. As a result, the gray on the model parameter space for mV > 5 MeV. In particu- curve in Fig. 6 roughly indicates the lower margin of 2 −15 lar, for the case m4 ¼ 50 keV and sin ð2θÞ¼10 , the parameter space where ν4 comprises the total dark matter present Higgs invisible limit still allows a region with relic density in the Universe. That is, all the target parameter mϕ ∼ 100 MeV, but that will be covered by the HL-LHC. space lies above the gray curve. It is also worth pointing out that the Higgs invisible decay The existence of V also mediates decays of the type ν4 → ννν¯ via an off-shell V. This decay width is proportional to constraint equally applies to V coupling to other neutrino λ4 2 2θ 5 4 ν flavors. μμ sin ð Þm4=mV. If we want 4 to be cosmologically long-lived and its lifetime to be longer than the age of the Universe, this provides an indirect constraint on the model C. Constraint from W boson decay width − − parameters. We find that, for the masses m4 and mixings The next channel we examine is the W → μ ν¯μV where 2 2θ sin ð Þ that we consider, this constraint is weaker than the V is radiated from the final state ν¯μ via its neutrinophilic other bounds to be discussed in the following subsections. coupling. Ignoring the muon mass, this decay rate is

B. Constraint from the Higgs boson invisible decay λ 2 2 5 − − j μμj Gffiffiffi FMW 2 ΓðW → μ ν¯μVÞ¼ p ð1 − wV − 12w logðwVÞ In this and the next few subsections, we present 512 2π3m2 V experimental constraints on the neutrinophilic V boson. V 8 3 − 4 To obtain the correct dark matter relic density, the mass of þ wV wVÞ; ð3:20Þ V is required to be smaller than the weak scale. It mainly ≡ 2 2 decays into neutrinos and thus appears invisible after where wV mV=MW. As with the Higgs decay, we see that production in laboratories. This leads to a number of in the small mV limit, this decay width is enhanced by 1 2 constraints by making precision measurements of processes =mV. To derive a conservative constraint, we simply that involve an active neutrino. require this decay rate to be smaller than the uncertainties The first process we consider is the Higgs boson decay. in the W total width measurement, ∼42 MeV [51]. The The operator introduced in Eq. (3.1) opens up a new Higgs resulting limit is found to be much weaker than the one invisible decay channel, h → νμν¯μV. The corresponding from Higgs invisible decay, and we do not show it in Fig. 6. partial decay rate is Including the muon mass in this calculation would drive the constraint to be slightly weaker than this calculation. λ 2 2 5 We note that the W decay limit might be further j μμj GffiffiffiFMH 2 Γðh → νμν¯μVÞ¼ p ð1 þ 12h ð6 þð3 þ 4hVÞ 3072 2π3m2 V improved by studying the kinematic edge observables of V W → μ þ MET decay at the LHC. In the Standard Model, − 64 3 − 9 4 × logðhVÞÞ hV hVÞ; ð3:19Þ when the W boson is singly produced, the final charged lepton transverse distribution features a ≡ 2 2 where hV mV=mh. We see that in the small mV limit, this Jacobian peak [53]. This feature is absent when V is 1 2 partial width is enhanced by a factor of =mV, correspond- present in the decay product. We leave a careful study of ing to the Higgs decaying into the longitudinal component this for a future work. of the V boson. The operator responsible for Higgs decay, μ 0 λðh=vÞνγ¯ νVμ, does not admit any conserved Uð1Þ gauge D. Constraint from Z boson invisible width symmetry for Vμ to be promoted as the corresponding . For example, the Uð1Þ of the neutrino We also examine the Z → νμν¯μV decay where V is number is explicitly broken by the presence of the Higgs radiated from either νμ or ν¯μ in the final state. Because V field. decays into neutrino and appears invisible at colliders, this To set a limit using this channel, we note that the decay channel contributes to additional invisible Z-boson invisible decay branching ratio of the Higgs boson in this decay width. This decay rate is

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Z 2 3 1− 2 λμμ G M ð mV =MZÞ Γ → ν ν¯ j j ffiffiffiF Z 2 2 ðZ μ μVÞ¼ p 3 gZðmV=MZ;wÞdw; 48 2π 0 " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 1 2 2 − 1 2 − 2 1 − 1 2 ðy þ Þ þ w y w þ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw wðy þ Þþðy Þ gZðy; wÞ¼ log y − w þ 1 y − w þ 1 − w2 − 2wðy þ 1Þþðy − 1Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 2 y2 − 2yðw þ 1Þþðw − 1Þ2: ð3:21Þ

Unlike the Higgs and W cases, the new Z boson decay invisible decay derived above. It is shown as the red shaded 1 2 width does not feature the =mV enhancement at small mV. region in Fig. 6. This could be understood as follows. In this decay process, ν¯ γμν the vector boson V couples to μ μ which can be defined E. Constraints from decays 1 0 ν as the neutrino current. The Uð Þ symmetry for the μ μ neutrino number associated with this current is preserved For the vector boson V lighter than the GeV scale, the νν¯ by the Standard Model Zνμν¯μ coupling. This symmetry is V coupling could also lead to new decay channels of þ þ only broken by the other Standard Model couplings (e.g., charged that will exist, m → μ νμV, where the Wμν¯ coupling considered earlier) which do not take part m ¼ π, K, B, etc. Because V will invisibly decay back in this decay rate at leading order. Due to the lack of the to neutrinos, such processes would contribute to the 1 2 mþ → μþν νν¯ =mV factor, numerically, we also find the constraint from branching ratios of the μ and get constrained. Z invisible decay is much weaker than that from Higgs The decay width mþ → lþνV is

Z 2 2 2 5 1− 2 jλμμV 0 j G Fmmm ð ml=mV Þ Γ mþ → μþν qq F 2 2 2 2 ð μVÞ¼ 3 2 fmðml=mm;mV=mm;wÞdw; 64π 2 2 m m =mm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV V 1 2 2 2 2 fmðl; v; wÞ¼ l − 2lðw þ 1Þþð1 − wÞ ½−l þ lð2w þ 1Þþwð1 − wÞðv − wÞ ð2v þ wÞ; ð3:22Þ w3 where GF is the Fermi constant, Fm is the decay constant of m, and jVqq0 j is the relevant Cabibbo-Kobayashi-Maskawa matrix element for this decay. In the small mV limit, this integral may be performed analytically,

  2 2 2 5 2 4 6 λμμV 0 G FmMm þ þ j qq j F mV mV mV Γðm → μ νμVÞ ⟶ 1 þ þ 73 þ 9 : ð3:23Þ →0 768π3 2 2 4 6 ml mV Mm Mm Mm

In practice, we have considered the charged kaon and suppression. This constraint has been calculated in the case pion decays and apply the experimental constraints, of a new-physics scalar instead of a vector in Ref. [17], þ þ −6 þ þ BrðK → μ νμνν¯Þ < 2.4 × 10 and Brðπ → μ νμνν¯Þ < where it was found to be much weaker than other 5 × 10−6 [51]. The resulting limit is shown by the blue laboratory based probes. We expect the same to be true shaded region in Fig. 6. It is weaker than that from Higgs for a vector V being emitted in this decay. invisible decay. Searches for exotic lepton decays can provide similar F. Constraints from BBN and Supernova 1987A constraints; however, they are suppressed by a larger final- If the new vector boson V is thermalized in the early − − state phase space, e.g., the decay μ → e νν¯V. Because the Universe and light enough to remain relativistic by the time kaon decay constraints are strong in this region of param- of (BBN), it will contribute to Δ eter space, we expect that this four-body phase space would Neff and affect the primordial element abundances. A lead to relatively weaker constraints than those of the conservative constraint rules out mV ≲ 5 MeV [13]. For a meson decays, which are, in turn, weaker than the Higgs detailed computation of the effects of nonstandard neutrino boson decay constraints (the Higgs decay constraint also self-interactions on BBN, see [54]. benefits from the longitudinal enhancement, with a partial If the vector V is lighter than ∼100 MeV, it can be 1 2 width that scales as =mV). For heavier V, we could use the produced from neutrino scatterings in the explosion of decay τ− → lνν¯V, which suffers from the same four-body Supernova 1987A. It could carry away a significant fraction

115031-10 ORIGIN OF STERILE NEUTRINO DARK MATTER VIA SECRET … PHYS. REV. D 101, 115031 (2020) of energy and modify the observed timescale of neutrino discussed above and would rule out the desired parameter emission. Reference [55] has estimated this effect in the space for relic sterile neutrino dark matter. 1 gauged Uð ÞLμ−Lτ model (see also Sec. IV). We rescale and apply their bound by restricting V to couple to only one I. A possible UV completion neutrino flavor. In this subsection, we present a simple UV completion for the operator introduced in Eq. (3.1). It serves as a proof G. Mono-neutrino probes at DUNE of existence of renormalizable theories that could lead to Neutrinophilic particles such as V may be emitted in the low energy effective theory considered in this work. beam neutrino experiments, if kinematically allowed. The We extend the Standard Model with a pair of chiral ν → μ− þ processes μn V p occur via initial state radiation. In NL, NR and a complex scalar ϕ. All are Standard contrast to standard neutrino quasielastic scattering, Model gauge singlets. The NL and ϕ fields are oppositely ν → μ− þ 0 μn p , the radiation of V carries away both energy charged under a new Uð1Þ gauge symmetry, whereas NR is and transverse momentum. As a result, the final state μ− neutral. With such a particle content the Uð1Þ0 still possess carries lower energy than expected in the quasielastic a gauge anomaly that could be canceled by resorting to scattering case. The resulting μ−pþ system appears to including additional heavy fermions without direct cou- have a nonzero pT with respect to the beam direction. plings to the lighter fields. We will keep those heavy These are the mono-neutrino signals introduced in [9,11] in particles implicit. a different context. In contrast to the lepton-number With the NL, NR, and ϕ fields, we could write down charged scalar radiation studied carefully in [11], here the following renormalizable Lagrangian governing their the radiation of V does not carry away a lepton number. The interactions: final state muon in the above signal process has the same L ϕ † ϕ ¯ ¯ ∂ as the quasielastic background; thus it will UV ¼ðDμ Þ ðDμ ÞþNLi=DNL þ NRi=NR not benefit from the muon charge identification capability ¯ λ ¯ ϕ ϕ of the neutrino detector. In contrast, the emission cross þ½yNRLH þ NRNL þ H:c:þVð Þ; ð3:24Þ 2 2 section for a light V features a E =mV longitudinal enhancement, making the signal stronger at low m .We where L, H are the Standard Model lepton and Higgs V ∂ 0 derive an expected 95% CL constraint assuming five years doublets, and Dμ ¼ μ ig Vμ. ϕ ϕ ϕ of DUNE [56] data collection in the neutrino mode, as We assume has a potential Vð Þ such that gets a 1 shown by the dashed green line in Fig. 6. An additional five vacuum expectation value vϕ and breaks the Uð Þ sym- years of DUNE data collection in antineutrino mode does metry. This gives a mass to the new gauge boson V, not improve this limit significantly, as the background pffiffiffi 0 processes are more difficult to resolve when an antineutrino MV ¼ 2g vϕ; ð3:25Þ scatters. Our obtained sensitivity is comparable to the current Higgs invisible limit but will be surpassed by the and also gives a Dirac mass for the NL, NR fermions high-luminosity LHC search. (preferably above the electroweak scale for the discussions of constraints in previous subsections to be valid), H. On V coupling to other neutrino flavors λ Throughout Sec. III we have focused on λμμ, the V MN ¼ vϕ: ð3:26Þ coupling to muon-flavored neutrinos. In principle, the couplings λαβ comprise a 3 × 3 matrix including couplings Next, we go to energy scales below MN and integrate out NL;NR. From the square bracket in Eq. (3.24) the equation to other flavors and even flavor-violating couplings. Since ¯ all of the early Universe reactions that lead to the of motion of NR yields (neglecting its kinetic term) production of ν4 are driven by the self-interaction of 1 neutrinos, the relic density results shown in Figs. 4 and NL ¼ − yLH: ð3:27Þ 6 are independent of the choice of flavor, as long as only MN one element of λ is turned on each time. Experimentally, the strongest constraint on λμμ comes from the Higgs boson As a result, the gauge interaction of NL with the V in the invisible decay, as shown in Fig. 6. It equally applies to all first line of Eq. (3.24) leads to the effective interaction other λαβ couplings. 0 2 On the other hand, we note that if the V coupling is flavor L g y ¯ † γμ eff ¼ 2 VμðLH Þ ðLHÞ: ð3:28Þ off diagonal, there will be strong constraints from loop MN induced μ → eV, τ → eV, and τ → μV decays. Given order-of-magnitude estimates of this loop process, we find This is same as the higher dimensional operator introduced Λ2 2 0 2 that these constraints would be stronger than all of those in Eq. (3.1), with ¼ MN=ðg y Þ. The presence of the

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new N in this UV completion could be νμ, ν¯μ, ντ, ν¯τ, μ , τ particles via the new Lμ − Lτ gauge subject to further experimental constraints (see, e.g., [57]). interaction. L As the final remark, it is worth mentioning that in UV To simplify the thermal averaging in the numerical ϕ we suppress the renormalizable couplings of NL , NR with calculation, we make the instantaneous decoupling ν the sterile neutrino field s in order to maintain the lightness approximation. For temperatures above the mass of a of dark matter candidate. We also neglect the participating in a scattering process, we calculate mass term for NR, which if it exists, will contribute to the the thermal rates assuming the fermion to be massless. In active neutrino mass via the inverted . contrast, when the temperature drops below any of the L Again, we emphasize that UV serves the role as proof of fermion mass, we assume the process vanishes immedi- existence that the effective operator Eq. (3.1) can be derived ately. Because sterile neutrino dark matter is dominantly from a renormalizable theory at higher scales. We leave the produced at temperatures below a few hundred MeV (see, detailed study of its phenomenology for a future work. e.g., Fig. 5 in Sec. III A), the τ are already decoupled from the Universe, and the presence of is marginal. This is a good approximation to simplify the IV. GAUGED Uð1ÞL − L MODEL μ τ numerical workload. The second model we explore is an extension of the Under this approximation, we report the cross sections 1 Standard Model with gauged Uð ÞLμ−Lτ symmetry. It is one and thermal rates relevant to the relic density calculation in of the simplest anomaly-free Uð1Þ models known to this model: provide an explanation to the discrepancy between the    4 2 2 Standard Model prediction of muon g − 2 and the exper- gμτ s mV s σν ν →ν ν ¼ þ log 1 þ ; imental measurement [58]. The phenomenology of this μ μ μ μ 4πm2 s þ m2 s þ 2m2 m2 V V V V model has been explored extensively [59–65]. Here, we 4 2 4 4 − 2 gμτ s 2 ðmV s Þ point out that the model also contains a target parameter σν ν¯ →ν ν¯ ¼ − 4m þ μ μ μ μ 4πðs − m2 Þ2 m2 V s space for sterile neutrino dark matter relic density.  V V  The couplings of the U 1 gauge boson V with s 10s ð ÞLμ−Lτ 1 × log þ 2 þ 3 ; Standard Model fermions are mV 4 α α α α gμτs μγ¯ μ − τγ¯ τ ν¯ γ ν − ν¯ γ ν σν ν¯ →ν ν¯ ¼ ; gμτVαð þ μ PL μ τ PL τÞ; ð4:1Þ μ μ τ τ 12π − 2 2 ðs mVÞ 4 where gμτ is the gauge coupling. gμτs σν ν →ν ν ¼ σν ν¯ →ν ν¯ ¼ ; μ τ μ τ μ τ μ τ 4π 2 2 As in the previous section, we assume that the sterile mVðs þ mVÞ neutrino dark matter ν4 has a small νμ component, 4 gμτs characterized by a mixing angle θ. To generate such a σν ν¯ →μþμ− ¼ ; μ μ 6πðs − m2 Þ2 mixing, one cannot rely on the Yukawa coupling in V  4 2 Eq. (2.1). Instead, the following higher-dimensional oper- gμτ 2 mV σν μ−→ν μ− ¼ σν μþ→ν μþ ¼ ðs þ m Þ log ator must be introduced: μ μ μ μ 4πs2 V s þ m2  V   2 2 3 2 2 4 ϕ sð s þ smV þ mV Þ T T þ 2 2 : ð4:3Þ yενs CH ðiσ2ÞL þ H:c:; ð4:2Þ 2 Λ mVðs þ mVÞ From the experience gained in the previous section (see where ϕ is a complex scalar field that Higgses the 1 Fig. 5), the most relevant neutrino reactions for dark matter Uð ÞLμ−Lτ symmetry and carries an opposite charge to production occur at T ≲ mV. We present the thermal rate ν μ T ν ≃ν that of L ¼ð μ; Þ . This is because the s field ð 4Þ must results for the heavy V and on-shell V cases. In the heavy-V 1 ≫ be a singlet under Uð ÞLμ−Lτ in order to prevent dark matter limit (mV T), from being fully thermalized in the early Universe and 8 > 37;T≥ 2mτ violating the initial condition for the production mechanism > > discussed in this work. > 33;mτ ≤ T<2mτ 7π 4 < The relic density calculation in this model proceeds in gμτ 4 Γ ≈ ET × 27; 2mμ ≤ T mV > > 25;mμ ≤ T<2mμ contribution to the thermal potential VT is the same as > ν : Fig. 2 with the gauge boson V and running in the loop. It 17;T

115031-12 ORIGIN OF STERILE NEUTRINO DARK MATTER VIA SECRET … PHYS. REV. D 101, 115031 (2020) rffiffiffiffiffiffi 4 2   2 −11 gμτm T π m4 7.1 keV, sin 2θ 7 × 10 and m4 50 keV, Γ ≃ V − ¼ ð Þ¼ ¼ V 2 2 expð wÞþ 4 ErfcðwÞ 2 2θ 10−15 48π γVE w sin ð Þ¼ , respectively. The relic density curves 8 exhibit similar S shapes, corresponding to the three > 3 ≥ 2 <> ;T mτ production regimes discussed in Sec. III A. 2 2 ≤ 2 Sterile neutrino dark matter production in this model has × > ; mμ T< mτ ð4:5Þ :> been explored in [66] in the light V limit, which corre- 1;T<2mμ; sponds to the case C defined in Sec. III A. In this regime, our result is consistent with theirs. ≡ 2 4 γ − where w mV=ð ETÞ. V is the decay width of the Lμ Figure 7 also shows the experimental constraints, mostly Lτ gauge boson, adapted from Ref. [67], where the colored shaded regions are already excluded and the regions enclosed by colored   2 Xτ gμτm dashed curves will be covered by future experiments, γ V 1 1 2 1 − 4 1=2Θ 1 − 4 V ¼ 12π þ ð þ rαÞð rαÞ ð rαÞ ; correspondingly labeled. We also include the cosmologi- α¼μ cal/astrophysical constraints from BBN and supernova ð4:6Þ 1987A [55], which mainly cover the region with mV below a few MeV scale. Compared to the neutrinophilic case, the Uð1Þ − model has a stronger interplay between the dark 2 2 Θ Lμ Lτ where rα ¼ mα=mV and is the Heaviside theta function. The above thermal potential and reaction rates are added matter relic density favored regions and experimental reaches. The present limits almost rule out the entire to their counterparts in the Standard Model and then 50 inserted into the Boltzmann equation (2.8). The dark matter parameter space for the m4 ¼ keV case. The reach of future experiments including SHiP [68,69], a dedicated relic density favored parameter space is shown in the gμτ vs NA62 analysis [70], and NA64μ [71,72], if they occur, will mV plane in Fig. 7. Similar to Fig. 6, the gray and be able to cover nearly the entire region above the m4 ¼ black curves correspond to two choices of parameters: 7.1 keV curve, that is, roughly the whole viable parameter space for dark matter relic density. The DUNE experiment, with a liquid argon near detector, can also probe the parameter space overlapping with the ðg − 2Þμ favored region, by measuring the neutrino trident production [73,74]. DUNE, if equipped with a gaseous multipurpose near detector, is also able to probe the displaced decay of V into eþe− via a loop-generated kinetic mixing [75]. Finally, a proposed muon-missing-momentum experiment [76] can probe a parameter space similar to NA64μ, for mV < 2mμ.

V. GAUGED Uð1ÞB − L MODEL The third model we consider is an extension of the 1 Standard Model with Uð ÞB−L gauge symmetry [77,78]. This is another popular framework where many phenom- enological works have been done. The interactions of the B − L gauge boson V take the form   1 ¯γα − l¯γαl − νγ¯ α ν gBLVα 3 q q PL ; ð5:1Þ

FIG. 7. Curves on the parameter space that yield the observed universal for all and lepton flavors. The new gauge relic density of dark matter today for the Lμ − Lτ vector boson interaction, under which active neutrinos are charged, could 2 −11 model and m4 ¼ 7.1 keV, sin ð2θÞ¼7 × 10 (gray) and 2 −15 potentially facilitate the production of sterile neutrino dark m4 50 keV, sin 2θ 10 (black). Existing limits from a ¼ ð Þ¼ matter. As the Uð1Þ − model, the active-sterile neutrino variety of probes are shown as shaded regions: neutrino trident Lμ Lτ scattering with CCFR (red), BABAR (blue), BBN (orange), and mixing has to be generated by a higher dimensional Supernova 1987A (purple). The green filled region corresponds operator similar to Eq. (4.2). to the preferred region for the ðg − 2Þμ anomaly. Future con- The relic density calculation proceeds the same as before. 3 λ straints from NA62 (red), NA64μ (blue), SHiP (purple), M (red), The thermal potential is calculated using Eq. (3.3) with μμ and DUNE (green for MPD decays and blue for Trident replaced by the gauge coupling gBL. To calculate the thermal scattering) are shown as dashed lines. reaction rates of neutrinos, we make the same instantaneous

115031-13 KELLY, SEN, TANGARIFE, and ZHANG PHYS. REV. D 101, 115031 (2020) decoupling approximation as described in the previous section. The thermal reaction rate in the heavy mediator V limit takes the form

8 ≳ 1 > irrelevant;T GeV > > 22; 2mμ ≤ T ≲ 1 GeV > 4 < 7πg 21;mμ ≤ T<2mμ Γ ≈ BL ET4 × ð5:2Þ V 225 4 > 17 2 ≤ mV > ; me T > 16 ≤ 2 :> ;me T< me 12;T

For on-shell V contribution, the rate is  rffiffiffiffiffiffi  g4 m2 T π Γ ≃ BL V exp ð−wÞþ ErfcðwÞ V 96π2γ E2 4w 8 V ≳ 1 > irrelevant;T GeV <> 7; 2mμ ≤ T ≲ 1 GeV FIG. 8. Curves on the parameter space that yield the observed × ð5:3Þ > 5 2 ≤ 2 relic density of dark matter today for the B − L vector boson model > ; me T< mμ 2 −11 : and m4 ¼ 7.1 keV, sin ð2θÞ¼7 × 10 (gray) and m4 ¼ 3 2 2 −15 ;T

115031-14 ORIGIN OF STERILE NEUTRINO DARK MATTER VIA SECRET … PHYS. REV. D 101, 115031 (2020) between MeV and GeV and the corresponding neutrino On the theory side, the same dark matter production 10−6 10−2 mechanisms could also work in, e.g., gauged U 1 or coupling ranges from to . For comparison, we ð ÞLe−Lμ also carried out similar calculations in the gauged 1 Uð ÞL −Lτ models, or even a gauged lepton number that has U 1 and U 1 models. The main results are e ð ÞLμ−Lτ ð ÞB−L other motivations [83,84]. The presence of electron cou- presented in Figs. 6, 7, and 8. pling for the vector boson V in these models implies strong 1 The parameter space for viable dark matter relic density experimental probes similar to Uð ÞB−L. found in this work is a well motivated target for the next experimental probes. Among the three models studied, the 1 ACKNOWLEDGMENTS Uð ÞB−L model is the most strongly constrained. Its parameter space for sterile neutrino dark matter is already We thank Andr´e de Gouvêa for helpful discussions. narrow and will be fully covered by future high intensity K. J. K. is supported by Fermi Research Alliance, LLC experiments, including BELLE-II, FASER, and LDMX. under Contract No. DE-AC02-07CH11359 with the U.S. 1 The Uð ÞLμ−Lτ case is relatively less constrained but could Department of Energy, Office of Science, Office of High also be covered by the proposed experiments, including Energy Physics. M. S. acknowledges support from the SHiP, NA62, NA64-μ, M3, and DUNE. In contrast, the National Science Foundation, Grant No. PHY-1630782, model with a neutrinophilic vector boson is the least and from the Heising-Simons Foundation, Grant No. 2017- constrained. The strongest present limit comes from the 228. The research of W. T. is supported by the College of Higgs boson invisible decay search, which will be Arts and Sciences of Loyola University Chicago. The work improved by HL-LHC. A portion of its parameter space of Y. Z. is supported by the Arthur B. McDonald Canadian has not been targeted by any experiment, to our knowledge. Astroparticle Physics Research Institute.

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