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Particle Physics Model for the 17 MeV Anomaly inBeryllium Nuclear DecaysJonathan L. FengUniversity of California - Irvine

Bartosz FornalUniversity of California - Irvine

Iftah GalonUniversity of California - Irvine

Susan GardnerUniversity of Kentucky, gardner@pa.uky.edu

Jordan SmolinskyUniversity of California - Irvine

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Repository CitationFeng, Jonathan L.; Fornal, Bartosz; Galon, Iftah; Gardner, Susan; Smolinsky, Jordan; Tait, Tim M. P.; and Tanedo, Philip, "ParticlePhysics Model for the 17 MeV Anomaly in Beryllium Nuclear Decays" (2017). Physics and Astronomy Faculty Publications. 523.https://uknowledge.uky.edu/physastron_facpub/523

AuthorsJonathan L. Feng, Bartosz Fornal, Iftah Galon, Susan Gardner, Jordan Smolinsky, Tim M. P. Tait, and PhilipTanedo

Particle Physics Model for the 17 MeV Anomaly in Beryllium Nuclear Decays

Notes/Citation InformationPublished in Physical Review D, v. 95, issue 3, 035017, p. 1-25.

© 2017 American Physical Society

The copyright holder has granted permission for posting the article here.

Digital Object Identifier (DOI)https://doi.org/10.1103/PhysRevD.95.035017

This article is available at UKnowledge: https://uknowledge.uky.edu/physastron_facpub/523

Particle physics models for the 17 MeV anomaly in beryllium nuclear decays

Jonathan L. Feng,1,* Bartosz Fornal,1 Iftah Galon,1 Susan Gardner,1,2

Jordan Smolinsky,1 Tim M. P. Tait,1 and Philip Tanedo1,31Department of Physics and Astronomy, University of California, Irvine, California 92697-4575, USA2Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506-0055, USA3Department of Physics and Astronomy, University of California, Riverside, California 92521, USA

(Received 26 August 2016; published 16 February 2017)

The 6.8σ anomaly in excited 8Be nuclear decays via internal pair creation is fit well by a new particleinterpretation. In a previous analysis, we showed that a 17 MeV protophobic gauge boson provides aparticle physics explanation of the anomaly consistent with all existing constraints. Here we begin with areview of the physics of internal pair creation in 8Be decays and the characteristics of the observed anomaly.To develop its particle interpretation, we provide an effective operator analysis for excited 8Be decays toparticles with a variety of spins and parities and show that these considerations exclude simple models withscalar particles. We discuss the required couplings for a gauge boson to give the observed signal,highlighting the significant dependence on the precise mass of the boson and isospin mixing and breakingeffects. We present anomaly-free extensions of the Standard Model that contain protophobic gauge bosonswith the desired couplings to explain the 8Be anomaly. In the first model, the new force carrier is a Uð1ÞBgauge boson that kinetically mixes with the photon; in the second model, it is a Uð1ÞB−L gauge boson witha similar kinetic mixing. In both cases, the models predict relatively large charged lepton couplings ∼0.001that can resolve the discrepancy in the muon anomalous magnetic moment and are amenable to manyexperimental probes. The models also contain vectorlike leptons at the weak scale that may be accessible tonear future LHC searches.

DOI: 10.1103/PhysRevD.95.035017

I. INTRODUCTION

The existence of light, weakly coupled new particles hasbeen a well-motivated theoretical possibility for decades.The need for dark matter has motivated these particles,either to provide the dark matter itself—for example, in theform of axions—or, more recently, to mediate interactionsbetween the visible and dark sectors. Grand unificationprovides another compelling motivation for new particlesand forces. Although these particles and forces are typicallyexpected to be heavy and short range, respectively, it ispossible that a remnant of grand unification might survivedown to low energies. Another independent, but related,possibility is that some linear combination of the “acci-dental” B and Li global symmetries of the Standard Model(SM) might be gauged. If these symmetries are sponta-neously broken at low energies, they must also be weaklycoupled. All of these provide ample motivation for adiverse program of high-statistics searches for new particlesfar from the energy frontier.Nuclear transitions provide a means to probe light,

weakly coupled new physics. Indeed, in 1978, Treimanand Wilczek [1], as well as Donnelly et al. [2], proposedthat axions could be discovered through the study ofnuclear decays. Such searches are now established as part

of the corpus of constraints on axions and axionlikeparticles [3,4], as well as on light scalar particles withHiggs-like couplings [5]. The possible new particlesinclude not only scalars and pseudoscalars, but also thosewith other spin-parity assignments, which may manifestthemselves in different nuclear transitions. There are manypossible nuclear transitions to study, but particularlypromising are those that can be studied through excitednuclear states that are resonantly produced in extraordinarynumbers, providing a high-statistics laboratory to search forMeV-scale new physics.Krasznahorkay et al. have recently observed unexpected

bumps in both the distributions of opening angles andinvariant masses of electron-positron pairs produced in thedecays of an excited 8Be nucleus [6]. The bump in theangular distribution appears against monotonically decreas-ing backgrounds from SM internal pair creation (IPC), andthe anomaly has a high statistical significance of 6.8σ. Theshape of the excess is remarkably consistent with thatexpected if a new particle is being produced in thesedecays, with the best fit to the new particle interpretationhaving a χ2 per degree of freedom of 1.07.In previous work [7], we examined possible particle

physics interpretations of the 8Be signal. We showed thatscalar and pseudoscalar explanations are strongly disfa-vored, given mild assumptions. Dark photons A0, massivegauge bosons with couplings to SM particles that areproportional to their electric charge [8–11], also cannot

*Corresponding author.jlf@uci.edu

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account for the 8Be anomaly, given constraints from otherexperiments. The most stringent of these is null resultsfrom searches for π0 → A0γ. This can be circumvented ifthe new spin-1 state couples to quarks vectorially withsuppressed couplings to the proton. We concluded that anew “protophobic” spin-1 boson X, with mass around17 MeV and mediating a weak force with range 12 fm,provides an explanation of the 8Be anomaly consistent withall existing experimental constraints. cc boson have beenfurther studied in Refs. [12–14].Protophobic gauge bosons are not particularly unusual.

The Z boson is protophobic at low energies, as is any newboson that couples to B −Q, the difference of the baryon-number and electric currents. As we show below, itis extremely easy to extend the SM to accommodate alight gauge boson with protophobic quark couplings.Simultaneously satisfying the requirements on leptoncouplings requires more care. To produce the observedeþe− events, the coupling to electrons must be nonzero.This coupling is bounded from above by the shift the newboson would induce on the electron magnetic dipolemoment and from below by searches for dark photons atbeam dump experiments. The neutrino coupling, in turn, isbounded by ν–e scattering experiments, as well as by thenonobservation of coherent neutrino-nucleus scattering.Any model that consistently explains the 8Be signal mustsatisfy all of these constraints.In Sec. II we review the 8Be system and the observed

anomaly. In Sec. III, we present an effective operatoranalysis of the 8Be nuclear transitions and consider a varietyof spin-parity assignments for the new boson. We show thatmany simple candidates, including scalars and pseudosca-lars, are excluded,while a protophobic spin-1 gauge boson isa viable candidate. In the next three sections, we consider indetail the couplings such a gauge bosonmust have to explainthe 8Be anomaly: in Sec. IVwe discuss the impact of isospin

mixing and breaking in the 8Be system; in Sec. Vwe discussthe required gauge boson couplings to explain the signal,noting the sensitivity to the gauge boson’s precise mass; andin Sec. VI we evaluate the constraints imposed by all otherexperiments, refining the discussion in Ref. [7], especiallyfor the neutrino constraints. With this background, inSecs. VII and VIII, we construct simple, anomaly-freeextensions of the SM that contain protophobic gauge bosonswith the desired couplings to explain the 8Be anomaly. InSec. IX we discuss current and near-term experiments thatmay test this new particle explanation, and we conclude inSec. X by summarizing our results and noting someinteresting future directions.

II. THE 8Be ANOMALY

A. 8Be spectrum and electromagnetic decays

We review relevant properties of the 8Be system. Someof the energy levels of 8Be are shown in Fig. 1. The groundstate of the 8Be nucleus is only 0.1 MeV above thethreshold for αα breakup, and α clustering is thought toinform its structure and excitations [15,16]. The groundstate is a spin-parity JP ¼ 0þ state with isospin T ¼ 0, andits lowest-lying excitations are 2þ and 4þ rotational states,nominally of its αα dumbbell-shape, with T ¼ 0, excitationenergies 3.03 MeV and 11.35 MeV, and decay widths1.5 MeV and 3.5 MeV, respectively.1

Going up in excitation energy, the next lowest lyingstates are isospin doublets of T ¼ 0, 1 states with spin-parity assignments of 2þ, 1þ, and 3þ, respectively. The 2þ

FIG. 1. The most relevant 8Be states, our naming conventions for them, and their spin-parities JP, isospins T, excitation energies E,and decay widths Γ from Ref. [19]. Asterisks on isospin assignments indicate states with significant isospin mixing. Decays of the 8Be�

(18.15) state to the ground state 8Be exhibit anomalous internal pair creation; decays of the 8Be�0 (17.64) state do not [6].

1The 2þ excited state is notable in that it is produced throughthe β decay of 8B [17]. It is pertinent to the solar neutrino problembecause it appears with a neutrino of up to 14 MeV in energy, andit also enters in precision tests of the symmetries of the charged,weak current in the mass eight system [18].

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states are of such mass that the αα final state is the onlyparticle-decay channel open to them. Since both states areobserved to decay to the αα final state [19], which hasT ¼ 0, the 2þ states are each regarded as mixtures of theT ¼ 0 and T ¼ 1 states. The qualitative evidence forisospin-mixing in the 1þ and 3þ states is less conclusive,but each doublet state is regarded as a mixed T ¼ 0 andT ¼ 1 state [20].In this paper, our focus is on the transitions of the 1þ

isospin doublet to the ground state, as illustrated in Fig. 1.Werefer to the ground state as simply 8Be and to the 1þ excitedstates with excitation energies 18.15 MeV and 17.64 MeVas 8Be� and 8Be�0, respectively. As noted above, these lattertwo statesmix, but 8Be� is predominantlyT ¼ 0, and 8Be�0 ispredominantlyT ¼ 1. The properties of these states and theirelectromagnetic transitions have been analyzed using quan-tum Monte Carlo (QMC) techniques using realistic, micro-scopic Hamiltonians [21–24]. We discuss the current statusof this work and its implications for the properties of newparticles that may be produced in these decays in Sec. IV.The particular transitions relevant for the observed anomaly

are IPC decays. This is a process in which an excited nucleusdecays into a lower-energy state through the emission ofan electron-positron pair [25–27]. Like γ-decays—whichsatisfy selection rules based on angular momentum andparity—these decays can be classified by their parity (electric,E, or magnetic, M) and partial wave l. A p-wave magnetictransition, for example, is labeledM1. The spectra of electron-positron invariant masses and opening angles in these decaysare known to be monotonically decreasing for each partialwave in the SM [28]. It is customary to normalize the IPC ratewith respect to that of γ emission for the same nucleartransition, when the latter exists. This is because the nuclearmatrix elements, up to Coulomb corrections, as well as someexperimental systematic errors, cancel in this ratio. 8Be,moreover, is of sufficiently low-Z that the effects of itsCoulomb field on IPC are negligible [26]. 8Be� decays to7Li p most of the time, but its electromagnetic transitionshave branching fractions Brð8Be� → 8BeγÞ ≈ 1.4 × 10−5

[29] and Brð8Be�→ 8Beeþe−Þ≈3.9×10−3Brð8Be�→8BeγÞ[26,28].

B. The Atomki result

The Atomki pair spectrometer has observed the IPCdecays of 8Be� with high statistics [6,30]. A sketch of theexperiment and the new physics process being probed isshown in Fig. 2. A beam of protons with kinetic energiestuned to the resonance energy of 1.03 MeV collide with Linuclei to form the resonant state 8Be�, and a small fractionof these decay via 8Be� → 8Beeþe−. The spectrometer isinstrumented with plastic scintillators and multiwire pro-portional chambers in the plane perpendicular to the protonbeam. These measure the electron and positron energies, aswell as the opening angle of the eþe− pairs that traverse thedetector plane, to determine the distributions of openingangle θ and invariant mass mee.The experiment does not observe the SM behavior where

the θ and mee distributions fall monotonically. Instead, theθ distribution exhibits a high-statistics bump that peaks atθ ≈ 140° before returning to near the SM prediction atθ ≈ 170° [6]. To fit this distribution, Krasznahorkay et al.consider many possible sources, including the M1 compo-nent from IPC, but also others, such as an E1 componentfrom nonresonant direct proton capture [31]. They observethat the best fit comes from a 23% admixture of this E1component. Nevertheless, they are unable to explain thebump by experimental or nuclear physics effects, andinstead find that the excess in the θ distribution has astatistical significance of 6.8σ [6]. A corresponding bumpis seen in the mee distribution.If a massive particle is produced with low velocity in the

8Be� decay and then decays to eþe− pairs, it will produce abump at large opening angles. It is therefore natural toconsider a new particle X and the two-step decay 8Be� →8BeX followed by X → eþe−. With fixed background,Krasznahorkay et al. find that the best fit mass andbranching fraction are [6]

FIG. 2. Schematic depiction of the Atomki pair spectrometer experiment [6,30], interpreted as evidence for the production of a newboson X. The proton beam’s energy is tuned to excite lithium nuclei into the 8Be� state, which subsequently decays into the 8Be groundstate and X. The latter decays into an electron-positron pair whose opening angle and invariant mass are measured.

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mX ¼ 16.7� 0.35ðstatÞ � 0.5ðsysÞ MeV ð1Þ

Γð8Be� → 8BeXÞΓð8Be� → 8BeγÞ BrðX → eþe−Þ ¼ 5.8 × 10−6: ð2Þ

For the best fit parameters, the fit to this new particleinterpretation is excellent, with a χ2=dof ¼ 1.07.The new particle interpretation passes a number of

simple consistency checks. The electron-positron invariantmass and opening angle are related by

m2ee ¼ 2EeþEe− − 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2eþ −m2

e

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2e− −m2

e

qcos θ þ 2m2

e

¼ ð1 − y2ÞE2sin2θ

2þ 2m2

e

�1þ 1þ y2

1 − y2cos θ

�

þOðm4eÞ; ð3Þ

where

E≡ Eeþ þ Ee− and y≡ Eeþ − Ee−

Eeþ þ Ee−ð4Þ

are the total energy and energy asymmetry, respectively.The second term in the last line of Eq. (3) is much smallerthan the first and may be neglected. At the Atomki pairspectrometer, the 8Be� nuclei are produced highly non-relativistically, with velocity of 0.017c and, givenmX ≈ 17 MeV, the X particles are also not very relativistic.As a result, the eþ and e− are producedwith similar energies,and so one expects small jyj and mee ≈ E sinðθ=2Þ. Theexcesses in the θ and mee distributions satisfy this relation.The Atomki Collaboration verified that the excess

exclusively populates the subset of events with jyj ≤ 0.5and is absent in the complementary jyj > 0.5 domain, thatthe excess appears and then disappears as one scansthrough the proton beam resonance kinetic energy of1.03 MeV, and that the excess becomes more pronouncedwhen restricting to the subset of events with E > 18 MeVand is absent for lower energy events [6]. The latter twoobservations strongly suggest that the observed IPC eventsare indeed from 8Be� decays rather than from interferenceeffects and that the decays go to the ground state 8Be, asopposed to, for example, the broad 3 MeV JP ¼ 2þ state.Decays to the 3 MeV state would have a maximum totalenergy of 15 MeV and do not pass the E > 18 MeV cuteven when including effects of the energy resolution, whichhas a long low-energy tail, but not a high-energy one (seeFig. 2 of Ref. [30]).2

Finally, we note that IPC decays of the 17.64 MeV,isotriplet 8Be�0 state have also been investigated at the

Atomki pair spectrometer. An anomaly had previously beenreported in 8Be�0 decays [32]. This anomaly was featurelessand far easier to fit to background than the bumps discussedhere, and it has now been excluded by the present Atomkicollaboration [30]. If the observed anomaly in 8Be� decaysoriginates from a new particle, then the absence of newparticle creation in the 8Be�0 decay combined with theisospin mixing discussed in Sec. IV strongly suggestthat such decays are kinematically—not dynamically—suppressed and that the new particle mass is in the upperpart of the range given in Eq. (1). It also suggests that withmore data, a similar, but more phase space-suppressed,excess may appear in the IPC decays of the 17.64 state.

III. NUCLEAR EFFECTIVE THEORYAND PARTICLE CANDIDATES

The transition 8Be� → 8BeX followed by X → eþe−implies that X is a boson. We consider the cases in whichit is a scalar or vector particle with positive or negativeparity. In this reaction, its de Broglie wavelength isλ ∼ ð6 MeVÞ−1, much longer than the characteristic sizeof the 8Be nucleus, r ∼ ð100 MeVÞ−1. In this regime, thenucleus looks effectively pointlike, and one can organizethe corrections from the nuclear structure as a series in r=λ.This approach has a long and fruitful history in the analysisof radiative corrections in weak nuclear decays [33,34].We perform such an analysis for the case of 8Be� decaying

to a newbosonX.Many theories predict the existence of new,weakly coupled, light degrees of freedom that, prima facie,may play the role of the X boson. We show that somecommon candidates for X are excluded. We note that for thecase where X has spin-parity JP ¼ 1− and isospin mixingbetween 8Be� and 8Be�0 is neglected, nuclearmatrix elementsand their uncertainties cancel in the ratio of partial widths,Γð8Be� → 8BeXÞ=Γð8Be� → 8BeγÞ.

A. Effective operators for 8Be → 8BeX

The candidate spin/parity choices for X are as follows: a0− pseudoscalar a, a 1þ axial vectorA, and a 1− vectorV.Weargue below that there is no scalar operator in the parity-conserving limit. The leading Lorentz- and parity-invariantoperators mediating the transition 8Be� → 8BeX are

LP ¼ gP8Beð∂μaÞ8Be�μ; ð5Þ

LA ¼ gAΛA

8BeGμνFðAÞμν þ g0A

ΛAm2

A8BeAμ

8Be�μ; ð6Þ

LV ¼ gVΛV

8BeGμνFðVÞρσ ϵμνρσ; ð7Þ

where Gμν ≡ ∂μ8Be�ν − ∂ν

8Be�μ is the field strength for the

excited 8Be� state, FðVÞμν and FðAÞ

μν are the field strengths forthe new vector and axial vector bosons, respectively,

2The widths of the mee and θ distributions are determined bythe OðMeVÞ energy resolution for the electrons and positrons[30], which should not be confused with the 10 keV energyresolution for γ-rays used in testing the target thickness [6].

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and the dimensionful parameters Λi encode the dominantnuclear matrix elements relevant for the transition ineach case.In the vector case, Lorentz and parity invariance requires

that all operators containing the fields 8Be, 8Be�μ, and Vμ

must also contain two derivatives and ϵμνρσ. Any operatorsin which the two derivatives act upon the same field vanishunder antisymmetrization of the Lorentz indices, so that theonly other possible operators are

ð∂μ8BeÞ8Be�νFðVÞ

ρσ ϵμνρσ and ð∂μ8BeÞGνρVσϵ

μνρσ: ð8Þ

However, these operators can each be integrated by partsto produce a term that vanishes by antisymmetrization, andthe unique operator in LV . This is in contrast to the axialvector case, where the gauge-breaking part cannot berelated by operator identities to the gauge-invariant pieceand is thus a separate term with a separate effectivecoupling g0A.

B. Scalar candidates

A popular example of a JP ¼ 0þ scalar candidate for theX boson is a dark Higgs boson [35]. However, a scalarcannot mediate the observed 8Be� decay in the limit ofconserved parity. The initial 8Be� state has unit angularmomentum and is parity even, JP ¼ 1þ. Angular momen-tum conservation requires the final state 8BeX, whichconsists of two 0þ states, to have orbital angular momen-tum L ¼ 1. This, however, makes the final state parity oddwhile the initial state is parity even. This implies that thereare no Lagrangian terms in a parity-conserving effectivefield theory that couple a scalar to the 8Be� and 8Be�0. Thiscan also be seen at an operator level; for example, theoperator ð∂μSÞð∂ν

8BeÞGρσϵμνρσ vanishes upon integrating

by parts and using the Bianchi identity.

C. Pseudoscalar candidates

A JP ¼ 0− pseudoscalar or axionlike particle, a, generi-cally has a coupling to photons of the form gaγγaFμν ~Fμν

that is generated by loops of charged particles [36–38].For a mass ma ≈ 17 MeV, all values of this coupling in therange ð1018 GeVÞ−1 < gaγγ < ð10 GeVÞ−1 are experimen-tally excluded [39,40]. These bounds may be significantlyrevised in the presence of nonphotonic couplings, however.

D. Axial vector candidates

Axial vector candidates have several virtues. First, as weshow in Sec. VI, one of the most restrictive constraints onthe X particle comes from the decay of neutral pions,π0 → Xγ. If X is an axial vector, this decay receives nocontribution from the axial anomaly and nonanomalouscontributions to pion decay vanish in the chiral limit by theSutherland–Veltman theorem [41,42].

Second, unlike the other spin-parity combinations, theaxial candidate has two leading-order effective operatorswith different scaling with respect to the X three-momentum. The g0A term in Eq. (6) yields a 8Be� decayrate that scales as ΓX ∼ jkXj, whereas the gA;V terms inducerates that scale as ΓX ∼ jkXj3. Thus, the axial particle mayproduce the observed anomalous IPC events with smallercouplings than the vector, g0A ≪ gV . This may help avoidsome of the other experimental bounds discussed inSec. VI. At the same time, there may be more severeconstraints, as we discuss in Sec. X.Unfortunately, large uncertainties in the nuclear matrix

element for axial vectors make it difficult to extract therequired couplings for this scenario. To the best of ourknowledge, there is no reliable ab initio calculation ormeasurement of the matrix element we would need in the8Be system.

E. Vector candidates

The primary candidate for the X boson and the focus forthe remainder of this study is a JP ¼ 1− vector. A newvector couples to a current JμX that is a linear combination ofthe SM fermion currents,

L ⊃ iXμJμX ¼ iXμ

Xi¼u;d;l;ν

εieJμi ; Jμi ¼ fiγμfi: ð9Þ

Here we have introduced separate couplings to up-typequarks, down-type quarks, charged leptons, and neutrinosand assigned them charges εi in units of e. Family-universalcouplings of this type naturally avoid the introduction oftree-level flavor-changing neutral currents, which arehighly constrained. Conservation of X charge implies thatthe couplings to the proton and neutron currents, Jμp and J

μn,

are determined by εp ¼ 2εu þ εd and εn ¼ εu þ 2εd, sothat

JμX ¼X

i¼u;d;l;ν

εieJμi ¼ ð2εu þ εdÞeJμp þ ðεu þ 2εdÞJμn

þ εleJμl þ ενeJ

μν : ð10Þ

For the low energies at which we work, it is important tomap this quark-level expression to one in terms of hadrons.

Denoting the current in the previous equation by JμðquarkÞX ,we effect this by matching the requisite matrix element toits equivalent in hadronic degrees of freedom. That is, forthe proton,

Jμp ≡ hpðp0ÞjJμðquarkÞX jpðpÞi ¼ eupðp0ÞfFX1;pðq2Þγμ

þ FX2;pðq2Þσμνqν=2MpgupðpÞ; ð11Þ

where jpðpÞi denotes a proton state composed of quarksand upðpÞ is the Dirac spinor of a free proton. Note that

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QCD generates all the possible currents compatible withLorentz invariance and electromagnetic current conserva-tion. We choose FX

1;pðq2Þ and FX2;pðq2Þ to denote the

X-analogues of the familiar Dirac and Pauli form factors.Finally we form the analogue of the Sachs magnetic formfactor by introducing GX

M;pðq2Þ ¼ FX1;pðq2Þ þ FX

2;pðq2Þ,recalling that GM;pð0Þ is given by the total magneticmoment of the proton—we refer to Ref. [43] for a review.The M1 transition of interest here is determined by the totalmagnetic moment operator.The nucleon currents, written in either the quark or

hadron basis, can, in turn, be combined to form isospincurrents

Jμ0 ¼ Jμp þ Jμn; Jμ1 ¼ Jμp − Jμn: ð12Þ

Assuming isospin is conserved and the 8Be states areisospin eigenstates, h8BejJμ1j8Be�i ¼ 0, since both 8Be�

and the ground state 8Be are isosinglets. In this case,

h8BejJμXj8Be�i ¼e2ðεp þ εnÞh8BejJμ0j8Be�i; ð13Þ

h8BejJμEMj8Be�i ¼e2h8BejJμ0j8Be�i: ð14Þ

The J0 nuclear matrix elements therefore cancel in the ratioΓð8Be� → 8BeXÞ=Γð8Be� → 8BeγÞ. This observation maybe modified significantly when isospin violation isincluded, as we discuss in Sec. IV.If one sets gV ¼ e and identifies FðVÞ

ρσ with the electro-magnetic field strength in Eq. (7), then the leading operatorin LV describes the ordinary electromagnetic transition viaγ emission. Indeed, in this SM case, Lorentz- and parity-invariance require the characteristic Γð8Be� → 8BeγÞ ∝jkγj3 momentum dependence of an M1 transition. Thematrix elements in Eqs. (13) and (14) thus imply that ΛV inEq. (7) is universal for spin-1 particles. Combining all ofthese pieces, we find that

Γð8Be� → 8BeXÞΓð8Be� → 8BeγÞ ¼ ðεp þ εnÞ2

jkXj3jkγj3

¼ ðεp þ εnÞ2�1 −

�mX

18.15 MeV

�2�3=2

;

ð15Þ

when isospin is conserved. This is a convenient expression,as the experimental best fit for the anomalous decay rate toa new vector X is presented in terms of this ratio of decaywidths, as seen in Eq. (2).A simple, well-known vector boson candidate is the dark

photon A0 [8–11]. The dark photon is a light particle thatcan have small, but technically natural, couplings to theSM. For a given mass, the dark photon interactions are

controlled by a single kinetic mixing parameter, ε. This isrelated to the effective coupling in Eq. (7) by gV ¼ εe.Substituting this into Eq. (15) and comparing to theexperimental result in Eq. (2), one finds that ε2 ≈ 10−4,which is experimentally excluded by, for example, π0 → A0γsearches at NA48/2 [44].3

A generalization of the dark photon idea is to consideralso mixing between the new boson and the SM Z. Such aparticle is spin-1 with no definite parity. Unfortunately,bounds from atomic parity violation are extremely stringent[45] and constrain the dark Z couplings to be too small toexplain the 8Be anomaly.Another type of spin-1 particle is a light baryon-minus-

lepton number (B − L) boson [46–48]. This scenario isconstrained by neutrino scattering off electrons and,assuming no kinetic mixing, provides the upper limitgB−L ≲ 2 × 10−5 [49], which is again too small to accountfor the excess.As we discuss in detail in Sec. IV, Eq. (15) may receive

significant corrections in the presence of isospin mixingand breaking. We will also see, however, that in theexperimentally viable limit of εp ≪ εn, these correctionsare small. For the cases of the dark photon, dark Z, andB − L gauge boson discussed above, the size of the 8Besignal and the strength of the constraints on π0 → Xγessentially enforce protophobia, and so the argumentsagainst these candidates remain.

IV. SIGNAL DEPENDENCE ON ISOSPINMIXING AND BREAKING

The discussion of Sec. III E assumed that isospin isconserved and that the 8Be states are states of well-definedisospin. As noted in Sec. II A, however, there is substantialevidence that the 8Be states are isospin mixed, and, as wenote below, there may also be isospin breaking in theelectromagnetic transition operators stemming from theneutron-proton mass difference. In this section, we deter-mine the impact of isospin mixing and breaking on the ratefor 8Be� → 8BeX, which, of course, has implications forthe parton-level couplings required to explain the 8Besignal.The ground-state structure and excitation spectrum of

8Be, as well as its electromagnetic transitions, have beenstudied with ab initio QMC techniques, based on non-relativistic Hamiltonians with phenomenological nucleon-nucleon and three-nucleon potentials [21–24]. The latestwork, Ref. [24], uses the newer AV18þ IL7 potential.Isospin mixing is addressed in the manner of Ref. [20]:

the empirical total (hadronic) widths are used to fix theisospin mixing of the states within a particular doublet.

3Reference [6] quotes a fit of ε2 ∼ 10−7. The discrepancyappears to come from the use of expressions for axions [2] ratherthan dark photons.

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That is, for a doublet of spin J, the physical states (withlabels a and b) are given by [24]

ΨaJ ¼ αJΨJ;T¼0 þ βJΨJ;T¼1;

ΨbJ ¼ βJΨJ;T¼0 − αJΨJ;T¼1; ð16Þ

where a denotes the lower energy state. Note that αJ and βJare real and satisfy α2J þ β2J ¼ 1. The widths of the isospin-pure states are computed using the QMC approach, permit-ting the extraction of the mixing parameters in Eq. (16) fromthe measured widths, yielding, for example [24],

α1 ¼ 0.21ð3Þ and β1 ¼ 0.98ð1Þ: ð17Þ

The empirical excitation energies, which are unfolded fromthe experimental data using these mixing coefficients, agreewith the QMC energies of the states of all three mixeddoublets, to within the expected theoretical error—that is, towithin 1% uncertainty.Given this success, this procedure may be applied to the

electromagnetic transitions of these isospin-mixed states aswell, so that the M1 transitions to the ground state are of theform

hΨ0;0jjM1jjΨaJi ¼ αJM1J;T¼0 þ βJM1J;T¼1; ð18Þ

hΨ0;0jjM1jjΨbJi ¼ βJM1J;T¼0 − αJM1J;T¼1; ð19Þ

where M1J;T is the reduced matrix element of the M1operator with the isospin pure J, T states. For reference wenote that this matrix element is related to the partial widthΓM1 for the transition via

ΓM1 ¼16π

9αℏc

�ΔEℏc

�3

BðM1Þ�

ℏc2Mp½MeV�

�2

; ð20Þ

where BðM1Þ ¼ jhΨJf jjM1jjΨJiij2=ð2Ji þ 1Þ is in units of

ðμNÞ2, the squared nuclear magneton. We emphasize thatthe M1 operator can mediate both isoscalar (ΔT ¼ 0) andisovector (jΔTj ¼ 1) transitions. The JjΔTj isospin currentsare given in Eq. (12).Unfortunately, the leading one-body (impulse approxi-

mation) results compare poorly to experiment. The inclu-sion of meson-exchange currents in the M1J;T matrixelement improves matters considerably, yielding finallyΓM1 ¼ 12.0ð3Þ eV for the 17.64 MeV transition, to becompared with Γexpt

M1 ¼ 15.0ð1.8Þ eV [29], and ΓM1 ¼0.50ð2Þ eV for the 18.15 MeV transition, to be comparedwith Γexpt

M1 ¼ 1.9ð4Þ eV [29]. Nevertheless, the discrepan-cies are still significant, and it would seem that something ismissing. It is possible that the treatment of wave functionmixing is somehow inadequate. Table Vof Ref. [24] showsthat increasing the value of α1 to 0.31 makes the M1

transition rate of the 18.15 MeV state double, whiledecreasing the 17.64 MeV transition by only 5% [50].The deficiency can be redressed in a distinct way that has

not previously been considered in this context. Isospinbreaking can appear in the hadronic form of the electro-magnetic transition operators themselves [51,52] to the endthat changes in the relative strength of the isoscalar andisovector transition operators appear as a result of isospinbreaking in the masses of isospin multiplet states, such asthe nonzero neutron-proton mass difference. This is perti-nent because electromagnetic transition operators involveboth one- and two-body contributions. The nuclear struc-ture calculations of Ref. [24] employ electromagnetictransition operators from chiral effective theory in theisospin limit [53,54]. The empirical magnetic momentsof the neutron and proton are employed in the leading one-body terms in these analyses, albeit they are normalized bythe average nucleon mass, rather than the proton mass thatappears in the definition of the nuclear magneton.Consequently the isospin-breaking effects that shift therelative strength of the isoscalar and isovector transitionoperators appear in higher-order terms, namely in therelativistic corrections to leading one-body operators, aswell as in the two-body operators. These effects are likelynumerically important for the dominantly isoscalar electro-magnetic transitions because the relativistic one-bodycorrections and two-body contributions are predominantlyisovector in the isospin limit [24,55], though technicallythese corrections to a given contribution appear in higherorder in the chiral expansion.We choose to include these isospin-breaking effects

through the use of a spurion formalism [56]. That is, weinclude isospin-breaking contributions through the intro-duction of a fictitious particle, the spurion, whose purposeis to allow the inclusion of isospin-breaking effects withinan isospin-invariant framework. Since the largest effectsshould stem from the neutron-proton mass difference, thespurion acts like a new ΔT ¼ 1 operator because its size iscontrolled by ðMn −MpÞ=MN, where MN is the nucleonmass. Since the isoscalar transition operators are extremelysmall we include the “leakage” of the dominant isovectoroperators into the isoscalar channel only. This is justified bynoting that Ref. [24] used states of pure isospin andincluded meson exchange currents, to determine the iso-vector and isoscalar M1 transition strengths to be

M11;T¼1 ¼ 0.767ð9ÞμN and M11;T¼0 ¼ 0.014ð1ÞμN;ð21Þ

where the numerical dominance of the isovector M1transition strength arises from that of the empirical iso-vector anomalous magnetic moment and the charged-pion,meson-exchange contribution, which is isovector.Characterizing the strength of the ΔT ¼ 1 spurion by κ,

the matrix elements of Eqs. (18) and (19) are thus amendedby the addition of

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δhΨ0;0∥M1∥Ψa1i ¼ α1κM11;T¼1; ð22Þ

δhΨ0;0∥M1∥Ψb1i ¼ β1κM11;T¼1: ð23Þ

The size of κ is controlled by nonperturbative effects.To illustrate its role, we assume that it can be determinedby demanding that the resulting M1 transition rate ofthe 17.64 MeV decay reproduces its experimental value.The final M1 transition matrix elements thus read

hΨ0;0∥M1∥Ψa1i ¼ α1M11;T¼0 þ β1M11;T¼1 þ α1κM11;T¼1;

ð24Þ

hΨ0;0∥M1∥Ψb1i ¼ β1M11;T¼0 − α1M11;T¼1 þ β1κM11;T¼1:

ð25Þ

The needed shift in the M1 partial width of the 17.64 MeVtransition is 3.0� 2.1 eV. Employing the matrix elements

of Ref. [24], we find the central value of κ ¼ 0.549to yield hΨ0;0∥M1∥Ψb

1i ¼ 0.265μN and a M1 partialwidth of 1.62 eV, which is within 1σ of the experimentalresult.With the above discussion of both isospin mixing

and isospin breaking in hand, we now turn to theirimplications for an M1 transition mediated by an X bosonwith vector couplings εne and εpe to the neutronand proton, respectively. The M1 transition mediated byX is

hΨ0;0∥M1X∥Ψb1i ¼ ðεn þ εpÞβ1M11;T¼0 þ ðεp − εnÞ

× ð−α1M11;T¼1 þ β1κM11;T¼1Þ; ð26Þ

where the neutron and proton X couplings appear becausethe 8Be system contains equal numbers of neutrons andprotons. The resulting ratio of partial widths is, then,

ΓX

Γγ¼ jðεp þ εnÞβ1M11;T¼0 þ ðεp − εnÞð−α1M11;T¼1 þ β1κM11;T¼1Þj2

jβ1M11;T¼0 − α1M11;T¼1 þ β1κM11;T¼1j2jkXj3jkγj3

: ð27Þ

In the limit of no isospin mixing (α1 ¼ 0, β1 ¼ 1) and noisospin breaking (κ ¼ 0), Eq. (27) reproduces Eq. (15).However, substituting the isospin mixing parameters ofEq. (17) and the M1 transition strengths of Eq. (21), we find

ΓX

Γγ¼ j − 0.09ðεp þ εnÞ þ 1.09ðεp − εnÞj2

jkXj3jkγj3

κ ¼ 0;

ð28Þ

ΓX

Γγ¼ j0.05ðεp þ εnÞ þ 0.95ðεp − εnÞj2

jkXj3jkγj3

κ ¼ 0.549:

ð29ÞThe isoscalar contribution is only a small fraction of theisovector one, and so, in general, large modifications fromisospin violation are possible.In Fig. 3, we plot the ratio ΓX=Γγ in the (εp, εn) plane. In

the case of perfect isospin, the transition is isoscalar and the

FIG. 3. The ratio ΓX=Γγ in the case of perfect isospin (α1 ¼ κ ¼ 0) (left) and isospin violation (α1 ¼ 0.21, κ ¼ 0.549) (right) in the(εp, εn) plane for mX ¼ 16.7 MeV. The effects of isospin violation may be significant in general, but for the viable protophobic regionsof parameter space consistent with NA48/2 constraints (shaded), their effects are small. The best fit value of ΓX=Γγ ¼ 5.8 × 10−6 ishighlighted. The dark photon scenario corresponds to εn ¼ 0.

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ratio depends on εp þ εn, but in the case of isospinviolation, the isovector transition dominates, and the ratiodepends effectively on εp − εn. The effects of includingisospin violation are, therefore, generally significant.Interestingly, however, in the protophobic limit withεp ¼ 0, isospin violation only modifies ΓX=Γγ by a factorof about 20%. However, for larger values of jεpj, forexample, jεpj ∼ jεnj=2, isospin-breaking effects can besignificant, leading to factors of 10 changes in the branch-ing ratios, or factors of 3 modifications to the best fitcouplings. Such large excursions from protophobia areexcluded by the NA48/2 limits for the best fit values of thecouplings corresponding to mX ¼ 16.7 MeV, but may bepossible for larger values ofmX within its allowed range, aswe discuss below.

V. SIGNAL REQUIREMENTS FOR GAUGEBOSON COUPLINGS

In this section, we discuss what a gauge boson’scouplings must be to explain the 8Be signal. We beginwith the leptonic couplings, where the requirements arestraightforward to determine. To produce the IPC signal,the X boson must decay to eþe−. The Atomki pairspectrometer has a distance of OðfewÞ cm between thetarget, where the 8Be excited state is formed, and thedetectors that observe the charged particles [30]. The Xboson decay width to electrons is

ΓðX → eþe−Þ ¼ ε2eαm2

X þ 2m2e

3mX

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − 4m2

e=m2X

q; ð30Þ

with similar formulas for other fermion final states [57].Requiring that the new boson propagates no more than1 cm from its production point implies a lower bound

jεejffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBrðX → eþe−Þp ≳ 1.3 × 10−5: ð31Þ

If the X boson couples only to the charged SMfermions required to explain the 8Be anomaly, one hasBrðX → eþe−Þ ¼ 1. Note, however, that if εν ≠ 0 or ifthere exist light hidden-sector states with X charge, thenthere are generically other decay channels for X.The required quark couplings are determined by the

signal event rate, that is, the best fit ΓX=Γγ. In the Atomkiexperimental paper, the best fit branching fraction is thatgiven in Eq. (2). Combining this result with the isospin-conserving expression for the branching ratio of Eq. (15),we find

jεp þ εnj ≈1.0 × 10−2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBrðX → eþe−Þp or

jεu þ εdj ≈3.3 × 10−3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBrðX → eþe−Þp ; ð32Þ

where we have taken mX ¼ 16.7 MeV. These results,shifted slightly to mX ¼ 17 MeV, were presented previ-ously in Ref. [7].Given the discussion above, however, several refine-

ments are in order. First, one can include the isospin-violating effects discussed in Sec. IV. These modify thebranching ratio expression from Eq. (15) to Eq. (29), withthe effects shown in Fig. 3.Second, as discussed above, the presence of significant

isospin mixing strongly suggests that the absence ofanomalous IPC decays in the 8Be�0 state originates fromkinematic suppression, rather than from isospin symmetryor some other dynamical effect. This, then, argues formasses in the upper region of the allowed range of Eq. (2).Larger masses imply larger phase-space suppression,and these may significantly shift the contours of ΓX=Γγ

in the (εp, εn) plane, as can be seen by comparing the �1σvalues of mX in Fig. 4.Last, and most importantly, to determine the favored

couplings, one must know how the best fit ΓX=Γγ dependson mX. In the original experimental paper, the best fitbranching ratio ΓX=Γγ ¼ 5.8 × 10−6 was presented withoutuncertainties and only for the best fit mass of 16.7 MeV. Ina subsequent analysis, however, the experimental collabo-ration explored the implications of other masses [58]. Inpreliminary results from this analysis, the M1 and E1background normalizations were fit to the angular spectrumin the range 40° ≤ θ ≤ 120°, and confidence regions in theðmX;ΓX=ΓγÞ plane were determined with only statisticaluncertainties included. For masses larger than 16.7 MeV,the best fit branching ratio was found to be significantlysmaller. For example, formX ¼ 17.3 MeV (17.6 MeV), thebest fit was for ΓX=Γγ ≈ 2.3 × 10−6 (0.5 × 10−6) [58]. Forsuch large masses, the best fit with fixed backgrounds is notvery good, and the implications for nucleon-level couplingsare partially offset by the reduced phase space factorjkXj3=jkγj3. In a full analysis, one should also includesystematic errors which are clearly a significant source ofuncertainty in the mX determination, and also let thebackground levels float in the fit. We expect that includingthese effects will significantly improve the fit for largermasses and favor even smaller couplings. Specifically,since the anomalous events at angles between 120° and135° cannot come from signal when the X mass is heavier,larger M1 and E1 backgrounds will improve the fit and thusrequire a smaller signal to achieve the best fit to the angularspectrum.Clearly a complete understanding of the experimental

uncertainties requires a detailed analysis that incorporatesan accurate estimate of nuclear isospin violation, simulationof the experiment, systematic uncertainties, varying back-grounds, and the null 8Be�0 result. Such an analysis isbeyond the scope of this study. As a rough estimate of thehadronic couplings required to explain the 8Be signal, wetake

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jεnj ¼ ð2 − 10Þ × 10−3; ð33Þ

jεpj ≤ 1.2 × 10−3; ð34Þ

where the upper part of the εn range includes the couplingfor the best fit branching ratio formX ¼ 16.7 MeV, and thelower part presumably includes the best fit value for thelarger mX that simultaneously explain the 8Be� signal andthe 8Be�0 null results. The proton coupling constraintfollows from the NA48/2 constraints to be discussed inSec. VI A 1. In presenting our models in Secs. VII A andVIII A, we leave the dependence on εn explicit so that theimpact of various values of εn can be easily evaluated. Notethat the lower values of ΓX=Γγ are still too large toaccommodate a dark photon explanation.

VI. CONSTRAINTS FROM OTHEREXPERIMENTS

We now discuss the constraints on the gauge boson’scouplings from all other experiments, considering quark,electron, and neutrino couplings in turn, with a summary ofall constraints at the end of the section. Many of theseconstraints were previously listed in Ref. [7]. We discussthem here in more detail, update some—particularly theneutrino constraints—to include new cases and revisedestimates from other works, and include other constraints.

A. Quark coupling constraints

The production of the X boson in 8Be� decays iscompletely governed by its couplings to hadronic matter.The most stringent bound on these couplings in the mX ≈17 MeV mass range is the decay of neutral pions into Xγ.For completeness, we also list the leading subdominantconstraints on εq, for q ¼ u, d.

1. Neutral pion decay, π0 → Xγ

The primary constraint on new gauge boson couplings toquarks comes from the NA48/2 experiment, which per-forms a search for rare pion decays π0 → γðX → eþe−Þ[59]. The bound scales like the anomaly trace factorNπ ≡ ðεuqu − εdqdÞ2. Translating the dark photon boundNπ < ε2max=9 to limits on the new gauge boson couplingsgives

j2εu þ εdj ¼ jεpj≲ ð0.8–1.2Þ × 10−3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBrðX → eþe−Þp ; ð35Þ

where the range comes from the rapid fluctuations in theNA48/2 limit for masses near 17 MeV. In Ref. [7], weobserved that the left-hand side becomes small when the Xboson is protophobic—that is, when its couplings toprotons are suppressed relative to neutrons.

2. Neutron–lead scattering

A subdominant bound is set from measurements ofneutron-nucleus scattering. The Yukawa potential actingon the neutron is VðrÞ ¼ −ðεneÞ2Ae−mXr=ð4πrÞ, where A isthe atomic mass number. Observations of the angulardependence of neutron-lead scattering constrain new,weakly coupled forces [60], leading to the constraint

ðεneÞ24π

< 3.4 × 10−11�

mX

MeV

�4

: ð36Þ

3. Proton fixed target experiments

The ν-Cal I experiment at the U70 accelerator at IHEPsets bounds from X-bremsstrahlung off the initial protonbeam [61] and π0 → Xγ decays [62]. Both of these

FIG. 4. Contours of ΓX=Γγ in the (εp, εn) plane for the parametrization of isospin violation in Eq. (29). Also shown are the dark photonaxis ðεn ¼ 0Þ and the protophobic region with jεpj ≤ 1.2 × 10−3 allowed by NA48/2 constraints on π0 → Xγ. ThemX values are fixed tomX ¼ 16.1 MeV (left) and 17.3 MeV (right), corresponding to the �1σ (statistical) range of mX.

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processes are suppressed in the protophobic scenario sothat these bounds are automatically satisfied when Eq. (35)is satisfied.

4. Charged kaon and ϕ decays

There are also bounds on secondgeneration couplings. TheNA48/2 experiment places limits on Kþ→πþðX→eþe−Þ[44]. For mX ≈ 17 MeV, the bound on εn is much weakerthan the one from π0 decays in Eq. (35) [57,63]. TheKLOE-2 experiment searches for ϕ → ηðX → eþe−Þ andrestricts [64]

jεsj≲ 1.0 × 10−2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBrðX → eþe−Þp : ð37Þ

In principle εs is independent and need not be related to the8Be� coupling. However, in the limit of minimal flavorviolation, one assumes εd ¼ εs.

5. Other meson and baryon decays

The WASA-at-COSY experiment also sets limits onquark couplings based on neutral pion decays. It is bothweaker than the NA48/2 bound and only applicable formasses heavier than 20 MeV [65]. The HADES experimentsearches for dark photons in π0, η, and Δ decays andrestricts the kinetic mixing parameter to ε≲ 3 × 10−3 butonly for masses heavier than 20 MeV [66]. HADES is ableto set bounds on gauge bosons around 17 MeV in theπ0 → XX → eþe−eþe− decay channel. This, however, issuppressed by ε4n and is thus insensitive to jεnj≲ 10−2.Similar considerations suppress X contributions to otherdecays, such as πþ → μþνμeþe−, to undetectable levels.

B. Electron coupling constraints

The X boson is required to couple to electrons tocontribute to IPC events. In Eq. (31) we gave a lowerlimit on εe in order for X to decay within 1 cm of itsproduction in the Atomki apparatus. In this section wereview other bounds on this coupling.

1. Beam dump experiments

Electron beam dump experiments, such SLAC E141[67,68], search for dark photons bremsstrahlung fromelectrons that scatter off target nuclei. For mX ¼ 17 MeV,these experiments restrict jεej to live in one of two regimes:either it is small enough to avoid production, or large enoughthat the X decay products are caught in the dump [69],leading to

jεej < 10−8 orjεejffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

BrðX → eþe−Þp ≳ 2 × 10−4: ð38Þ

The region jεej < 10−8 is excluded since the new bosonwould not decay inside the Atomki apparatus. This leads to

the conclusion thatXmust decay inside the beamdump. Lessstringent bounds come from Orsay [70] and the SLAC E137[71] experiment. The E774 experiment at Fermilab is onlysensitive to mX < 10 MeV [72].

2. Magnetic moment of the electron

The upper limit on jεej can be mapped from dark photonsearches that depend only on leptonic couplings. Thestrongest bound for mX ¼ 17 MeV is set by the anomalousmagnetic moment of the electron, ðg − 2Þe, which con-strains the coupling of the new boson to be [63]

jεej < 1.4 × 10−3: ð39Þ

3. Electron–positron annihilation into Xand a photon, eþe− → Xγ

A similar bound arises from the KLOE-2 experiment,which looks for eþe− → Xγ followed by X → eþe−, andfinds jεej

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBrðX → eþe−Þp

< 2 × 10−3 [73]. An analogoussearch at BABAR is limited to mX > 20 MeV [74].

4. Proton fixed target experiments

The CHARM experiment at CERN also bounds Xcouplings through its searches for η; η0 → γðX → eþe−Þ[75]. The production of the X boson in the CHARMexperiment is governed by its hadronic couplings. Thecouplings required by the anomalous IPC events, Eq. (32),are large enough that the X boson would necessarily beproduced in CHARM. Given the lower bound from decayin the Atomki spectrometer, Eq. (31), the only way to avoidthe CHARM constraint for mX ¼ 17 MeV is if the decaylength is short enough that the X decay products do notreach the CHARM detector. The dark photon limit on εapplies to εe and yields

jεejffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBrðX → eþe−Þp > 2 × 10−5: ð40Þ

This is weaker than the analogous lower bound on jεej frombeam dump experiments. LSND data imposes an evenweaker constraint [76–78].

5. Charged kaon and ϕ decays

In charged kaon decay to leptons, the X vector bosonmay be emitted from a charged lepton line. Since the newvector interaction does not respect the precise gaugeinvariance of the SM, the interaction of the longitudinalcomponent of X is not constrained by a correspondingconserved current and thus can be significantly enhancedwith energy [79–82]. However, the most severe existinglimit comes from the nonobservation of an excess inΓðK → μþ invÞ with respect to ΓðK → μνÞ [79–81],

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which is not pertinent here as we require an appreciableΓðX → eþe−Þ in order to explain the 8Be anomaly.

6. W and Z decays

The X boson can be produced as final state radiation inW and Z decays into SM fermions. When the X thendecays into an electron-positron pair, this gives a con-tribution to ΓðZ → 4eÞ that is suppressed by Oðε2eÞ. Forthe electron couplings εe ≲ 10−3 required here, the impacton the inclusive widths is negligible compared to theorder per mille experimental uncertainties on their meas-urement [83]. The specific decay Z → 4l has beenmeasured to lie within 10% of the SM expectation byATLAS and CMS [84,85] and is consistent with thecouplings of interest here.A more severe constraint arises, however, from

the experimental value of the W width, because theenhancement mentioned in leptonic K decay appears inW → μνX as well [82]. Limiting the contribution ofW → lνX to twice the error in the W width, afterRef. [82], yields

jεlj < 4.2 × 10−3�

mX

10 MeV

�; ð41Þ

to leading order in mX=mW and ml=MW , where we haveassumed lepton universality and that l ∈ e; μ; τ can con-tribute to the W width. The resulting constraint on εe isweaker than that from the magnetic moment of the electron.

C. Neutrino coupling constraints

The interaction of a light gauge boson with neutrinos isconstrained in multiple ways, depending on the SM currentsto which the boson couples; see Refs. [81,86–88]. Theneutrino coupling is relevant for the 8Be anomaly becauseSUð2ÞL gauge invariance relates the electron and neutrinocouplings. Because neutrinos are lighter than electrons, thisgenerically opens additional X decay channels and reducesBrðX → eþe−Þ. This, in turn, reduces the lower bound on εein Eq. (31) and alleviates many of the experimental con-straints above at the cost of introducing new constraints fromX-neutrino interactions.

1. Neutrino–electron scattering

Neutrino–electron scattering stringently constrains theX boson’s leptonic couplings [49,89]. In the mass rangemX ≈ 17 MeV, the most stringent constraints are from theTEXONO experiment, where νe reactor neutrinos withaverage energy hEνi ¼ 1–2 MeV travel 28 meters andscatter off electrons. The resulting electron recoil spectrumis measured. The path length is short, so the neutrinosremain in nearly pure νe flavor eigenstates. In the SM,νee → νee scattering is mediated by both s- and t-channeldiagrams. A new neutral gauge boson that couples to both

neutrinos and electrons induces an additional t-channelcontribution.Because constraints from νee scattering are sensitive

to the interference of SM and new physics, they dependon the signs of the new gauge couplings, unlike all ofthe other constraints discussed above. The importance ofthe interference term has been highlighted in Ref. [49]in the context of a B − L gauge boson model. In thatmodel, the neutrino and electron couplings have thesame sign, and the interference was found to be alwaysconstructive.Assuming that the experimental bound is determined by

the total cross section and not the shape of the recoilspectrum, one may use the results of Ref. [49] to determinethe bounds in our more general case, where the couplingscan be of opposite sign and the interference may beeither constructive or destructive. Define the quantityg≡ jεeενj1=2. Let Δσ be the maximal allowed deviationfrom the SM cross section and g� (g0) be the values of gthat realize Δσ in the case of constructive/destructive(negligible) interference,

Δσ ¼ g40σX; ð42Þ

Δσ ¼ g2þσint þ g4þσX; ð43Þ

Δσ ¼ −g2−σint þ g4−σX; ð44Þ

where g4σX is the purely X-mediated contribution to thecross section and g2σint is the absolute value of theinterference term. Solving these equations for the g’s yieldsthe simple relation

g−gþ ¼ g20: ð45Þ

The authors of Ref. [49] found that for mX ¼ 17 MeV, themaximal allowed B − L gauge boson coupling, gB−L, is2 × 10−5 and 4 × 10−5 in the cases of constructive inter-ference and no interference, respectively. From this, includ-ing the factor of e difference between the definitions ofgB−L and our ε’s, we find

ffiffiffiffiffiffiffiffiffiffiffijεeενj

p< 7 × 10−5 for εeεν > 0

ðconstructive interferenceÞ ð46Þffiffiffiffiffiffiffiffiffiffiffijεeενj

p< 3 × 10−4 for εeεν < 0

ðdestructive interferenceÞ: ð47Þ

The relative sign of the couplings thus has a significanteffect. For a fixed value of εe, the bound on jενj is 16times weaker for the sign that produces destructiveinterference than for the sign that produces constructiveinterference.

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2. Neutrino–nucleus scatteringIn addition to its well-known motivations of providing

interesting measurements of sin θW and bounds on heavyZ0 boson [90,91], coherent neutrino-nucleus scatteringmay also provide leading constraints on light, weaklycoupled particles [92,93]. Although ν–N scattering hasnot yet been observed, it is the target of a number ofupcoming experiments that use reactors as sources. Inaddition, the process can also be probed using current andnext-generation dark matter direct detection experimentsby searching for solar neutrino scattering events [94]. Fora B − L gauge boson, this sensitivity has been estimatedin Ref. [95] for SuperCDMS, CDMSlite, and LUX,with the latter providing the most stringent constraintof gB−L ≲ 1.5 × 10−4. Rescaling this result to the case of aboson with couplings ενe and εp;ne to nucleons yields

ενεn

�ðA − ZÞ þ Z

εpεn

�<

A4πα

ð1.5 × 10−4Þ2; ð48Þ

where we approximate the LUX detector volume to becomposed of a single xenon isotope. Since the NA48/2bounds on π0 → Xγ imply the protophobic limit whereεp ≪ εn, the second term on the left-hand side may beignored. Taking A ¼ 131 and Z ¼ 54 then yieldsjενεnj1=2 < 6 × 10−4 or

εν < 2 × 10−4�0.002εn

�: ð49Þ

This bound is weaker than the ν–e scattering bound withconstructive interference and comparable to the ν–e boundwith destructive interference. As the ν–N bounds areestimated sensitivities, we use the ν–e bounds in thediscussion below.

D. Summary of constraints

Combining the required ranges of the couplings toexplain the 8Be signal from Sec. V with the strongestbounds from other experiments derived above, we nowhave the acceptable ranges of couplings for a viableprotophobic gauge boson to explain the 8Be signal.Assuming BrðX → eþe−Þ ¼ 1, the requirements are

jεnj ¼ ð2–10Þ × 10−3; ð50Þjεpj≲ 1.2 × 10−3; ð51Þ

jεej ¼ ð0.2–1.4Þ × 10−3; ð52Þffiffiffiffiffiffiffiffiffiffiffijεeενj

p ≲ 3 × 10−4: ð53ÞThe nucleon couplings are fixed to reproduce the 8Be signalrate while avoiding the π0 → Xγ decays, and the quarkcouplings are related by εu þ 2εd ¼ εn and 2εu þ εd ¼ εp.The electron coupling is bounded from above by ðg − 2Þeand KLOE-2 and from below by beam dump searches,

and the neutrino coupling is bounded by ν–e scattering.The allowed lepton coupling regions are shown in Fig. 5.

VII. Uð1ÞB MODEL FOR THE PROTOPHOBICGAUGE BOSON

In this section, we present anomaly-free extensions of theSM where the protophobic gauge boson is a light Uð1ÞBgauge boson that kinetically mixes with the photon. Thesemodels have significant virtues, which we identify inSec. VII A. One immediate advantage is that it does notdifferentiate between left- and right-handed SM fermions,and so naturally has nonchiral couplings. Depending on thebest fit couplings discussed in Sec. V, the resulting modelsmay be extremely simple, requiring only the addition of extraparticles to cancel the anomalies, as discussed in Sec. VII B.

A. Uð1ÞB gauge boson with kinetic mixing

The promotion ofUð1ÞB baryon number from a global to alocal symmetry has recently attracted attention [96–104].Gauged Uð1ÞB is not anomaly-free, but these studies haveconstructed a number of models in which the gaugeanomalies are canceled with rather minimal new mattercontent.Here we assume that the Uð1ÞB symmetry is broken

through a Higgs mechanism, as discussed below, generat-ing a mass for the B gauge boson. As with all Abeliansymmetries, the B gauge boson will generically mixkinetically with the other neutral gauge bosons of theSM. At energies well below the weak scale, this mixing isdominantly with the photon. The resulting Lagrangian is

FIG. 5. Summary of constraints and target regions for theleptonic couplings of a hypothetical X gauge boson withmX ≈ 17 MeV. Updated from Ref. [7].

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L ¼ −1

4~Fμν

~Fμν −1

4~Xμν

~Xμν þ ϵ

2~Fμν

~Xμν

þ 1

2m2

~X~Xμ

~Xμ þXf

f iDf; ð54Þ

where ~Fμν and ~Xμν are the field strengths of the photon andB gauge boson, the sum runs over all fermions f, and thecovariant derivative is

Dμ ¼ ∂μ þ ieQf~Aμ þ ieϵBBf

~Xμ: ð55Þ

Here Qf and Bf are the electric charge and baryon numberof fermion f, and ϵB is the B gauge coupling in units of e.The tildes indicate gauge-basis fields and quantities.In the mass basis, the Lagrangian is

L ¼ −1

4FμνFμν −

1

4XμνXμν þ 1

2m2

XXμXμ þXf

f iDμf;

ð56Þ

where

mX ≡ 1ffiffiffiffiffiffiffiffiffiffiffiffi1 − ϵ2

p m ~X ð57Þ

is the physical X boson mass, and

~Aμ ≡ Aμ þϵffiffiffiffiffiffiffiffiffiffiffiffi

1 − ϵ2p Xμ

~Xμ ≡ 1ffiffiffiffiffiffiffiffiffiffiffiffi1 − ϵ2

p Xμ ð58Þ

define the physical massless photon A and massive gaugeboson X. The fermions couple to photons with the usualcharge eQf, but they couple to the X boson with chargeeεf, where

εf ¼ εBBf þ εQf; ð59Þ

and the script quantities are defined by

εB ¼ ϵBffiffiffiffiffiffiffiffiffiffiffiffi1 − ϵ2

p ε ¼ ϵffiffiffiffiffiffiffiffiffiffiffiffi1 − ϵ2

p : ð60Þ

The X charges for the SM fermions, using firstgeneration notation, are

εu ¼1

3εB þ 2

3ε; ð61Þ

εd ¼1

3εB −

1

3ε; ð62Þ

εν ¼ 0; ð63Þ

εe ¼ −ε: ð64Þ

The π0 constraints we have discussed above require ε and−εB to be approximately equal to within 10% to 50%. It istherefore convenient to define ε≡ −εB þ δ, so

εu ¼ −1

3εB þ 2

3δ; ð65Þ

εd ¼2

3εB −

1

3δ; ð66Þ

εν ¼ 0; ð67Þ

εe ¼ εB − δ; ð68Þ

with corresponding nucleon charges εn ¼ εB and εp ¼ δ.This model has some nice features. For small δ, the

charges are Q − B, which satisfies the protophobic con-dition. For the same reason, the neutrino’s charge isidentically zero. As discussed in Sec. VI C, the constraintson neutrino charge are among the most stringent, bothgiven ν–e and ν–N constraints, and the 8Be signal require-ment that X decays not be dominated by the invisible decayX → νν. The model is highly constrained, and we see thatthe electron coupling is not suppressed relative to the quarkcouplings. However, for εB ≈ 0.002 and δ ≈ 0.001, thismodel provides an extremely simple and minimal explan-ation of the 8Be signal (provided gauge anomalies arecanceled, as discussed below). Note that it predicts valuesof εe ≈ 0.001, that is, in the upper part of the allowed rangeof Eq. (52). Assuming εμ ≈ εe, such couplings remove [57]at least part of the longstanding discrepancy in ðg − 2Þμbetween measurements [105] and the SM prediction [106],with important implications for the upcomingMuon (g − 2)Experiment at Fermilab [107]. They also imply promisingprospects for future searches for the protophobic X boson atlow-energy colliders, as discussed in Sec. X.We treat the kinetic mixing ε as a free parameter. In a

more fundamental theory, however, ε may be related to εB.For example, if Uð1ÞB is embedded in non-Abelian gaugegroup, ε vanishes above the symmetry-breaking scale, butwhen the non-Abelian symmetry breaks, it is generated byvacuum polarization diagrams with particles with electriccharge and B quantum numbers in the loop. Parametrically,ε ∼ ðe2=6π2ÞεB

PfQfBf ln rf [10], where the sum is over

pairs of particles in the loop, and the rf are ratios of massesof these particles. Given ∼100 particles, one would there-fore expect ε ∼ εB in general, and the particular relationε ≈ −εB, which is not renormalization group invariant, maybe viewed as providing information at low-energy scalesabout the GUT-scale particle spectrum.

B. Anomaly cancellation andexperimental implications

Models with gauged baryon number require additionalparticle content to cancel anomalies. The simplest

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experimentally viable extension of the SM with gaugedUð1ÞB requires adding three vectorlike pairs of color-singlet fields [100,103].4 These fields and their quantumnumbers are listed in Table I. The new fields carry baryoncharges that satisfy the anomaly cancellation conditionB2 − B1 ¼ 3. The χ field is naturally a dark mattercandidate [103,109], and it has to be the lightest of thenew fields to avoid stable charged matter.The Uð1ÞB symmetry is broken by the vacuum expect-

ation value (vev) hSBi ¼ vX=ffiffiffi2

pof a new SM-singlet

Higgs field carrying baryon number B ¼ 3 to allow forvectorlike mass terms and to make the χ field the lightestone. The new Yukawa terms in the Lagrangian are

LY ¼ −y1ΨLhSMηR − y2ΨL~hSMχR − y3ΨRhSMηL

− y4ΨR~hSMχL − λΨSBΨLΨR − ληSBηRηL

− λχSBχRχL þ H:c: ð69ÞIn Refs. [100,103] Uð1ÞB is assumed to be broken at theTeV scale. However, to have a light Uð1ÞB gauge boson anda gauge coupling consistent with the 8Be signal, the vev ofthe new Higgs boson cannot be so large. Defining itsvacuum expectation value by hSBi ¼ vX=

ffiffiffi2

p, the mass of

the new X gauge boson corresponding to the broken Uð1ÞBis given by

mX ¼ 3ejεBjvX; ð70Þ

implying

vX ≈ 10 GeV0.002jεBj

: ð71Þ

As a result, the new particles cannot have large vectorlikemasses from the λi couplings in Eq. (69), but must ratherhave large chiral couplings from the yi terms of Eq. (69).

The experimental constraints on the extra matter contentof this model come from several sources:

(i) First, the new particles may be produced throughDrell-Yan production at the LHC. However, forYukawa couplings yi ∼ 3, close to the perturbativelimit, the masses of the new states are ∼500 GeVand beyond current LHC sensitivity.

(ii) Second, electroweak precision measurements con-strain the properties of the new particles. The twoelectroweak doublets give an irreducible contribu-tion to the S parameter of ΔS ≈ 2=ð6πÞ≃ 0.11[110]. In the degenerate mass limit, they do notcontribute to the T and U parameters. However, thefit to electroweak precision data may be improvedwith a slight splitting of Δm ∼ 50 GeV, which givesΔT ≈ 2=ð3π sin2 2θWÞðΔm=mZÞ2 ≈ 0.09. This com-bination of ΔS and ΔT fits well within the 90% C.L.region (see, for example, Fig. 10.6 of Ref. [111]).

(iii) Third, the new particles may affect the hSM → γγdecay rate. Since these particles essentially form twoadditional families of leptons, the rate for Higgsboson decay to two photons decreases by ∼20%compared to the SM prediction [112], but this is stillwithin the experimentally allowed region [113].

In summary, a simple model with a Uð1ÞB gauge bosonthat kinetically mixes with the photon is a viable candidatefor the protophobic gauge boson. The gauge anomaliesmust be canceled by introducing additional particles, andwe have discussed the simplest realization of this fieldcontent that simultaneously explains the 8Be anomaly.

VIII. Uð1ÞB−L MODEL FOR THEPROTOPHOBIC GAUGE BOSON

In this section, we present another anomaly-free exten-sion of the SM where the protophobic gauge boson is alight Uð1ÞB−L gauge boson that kinetically mixes with thephoton. These models have significant virtues, which weidentify in Sec. VIII A. They also generically have neutrinocouplings that are too large, and we explore a mechanismfor suppressing the neutrino couplings in Sec. VIII B. Theresulting models may be extremely simple, requiring onlythe addition of one generation of vectorlike leptons whichis light and may already be probed at the LHC. Theimplications for colliders and cosmology are discussed inSec. VIII C.

A. Uð1ÞB−L gauge boson with kinetic mixing

The possibility of gauged Uð1ÞB−L has been studied formany decades [46–48,114]. The promotion of Uð1ÞB−Lfrom a global to a local symmetry is well motivated amongAbelian symmetries by its appearance in grand unifiedtheories, and the fact that it is anomaly-free once one addsto the SM three right-handed (sterile) neutrinos, which arealready strongly motivated by the existence of neutrinomasses.

TABLE I. New particle content of the simplest anomaly-freeUð1ÞB model.

Field Isospin I Hypercharge Y B

SB 0 0 3

ΨL12

− 12

B1

ΨR12

− 12

B2

ηR 0 −1 B1

ηL 0 −1 B2

χR 0 0 B1

χL 0 0 B2

4A model unifying gauged baryon number and color into anon-Abelian SU(4) has been constructed and, after symmetrybreaking, yields the same new particle content as the Uð1ÞBmodel discussed here [108].

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As in the Uð1ÞB case, we assume that the B − Lsymmetry is broken through a Higgs mechanism, generat-ing a mass for the B − L gauge boson, and that it kineticallymixes with the photon. The resulting X-charges for the SMfermions, using first generation notation, are

εu ¼1

3εB−L þ 2

3ε; ð72Þ

εd ¼1

3εB−L −

1

3ε; ð73Þ

εν ¼ −εB−L; ð74Þ

εe ¼ −εB−L − ε; ð75Þ

or, defining ε≡ −εB−L þ δ as above,

εu ¼ −1

3εB−L þ 2

3δ; ð76Þ

εd ¼2

3εB−L −

1

3δ; ð77Þ

εν ¼ −εB−L; ð78Þ

εe ¼ −δ: ð79Þ

The corresponding nucleon charges are εn ¼ εB−Land εp ¼ δ.The charges of the kinetically mixed B − L gauge boson

have nice features for explaining the 8Be anomaly. Forδ ≈ 0, the charges are Q − ðB − LÞ, which satisfies thebasic requirements of a protophobic solution to the 8Beanomaly: namely, the X boson couples to neutrons, but itscouplings to both protons and electrons are suppressed.More quantitatively, by choosing the two parametersjεB−Lj ≈ 0.002–0.008 and jδj ≲ 0.001, the up and downquark couplings give the 8Be signal and are sufficientlyprotophobic to satisfy the π0 constraints. This is no greatachievement: by picking two free parameters, two con-ditions can be satisfied. But what is nontrivial is thatwith this choice, the electron coupling satisfies the upperbound jεej ≲ 1.4 × 10−3, which is required by the com-pletely independent set of experiments that constrain leptoncouplings.Unfortunately, in contrast to the Uð1ÞB case, the neutrino

coupling does not vanish. In these models, we see thatεν ¼ −εn while the constraints discussed above require theneutrino coupling to be significantly below the neutroncoupling. In the next section, we present a mechanism toneutralize the X-charge of SM active neutrinos to satisfythese bounds.

B. Neutrino neutralization with vectorlike leptons

The B − L gauge boson with kinetic mixing predictsjενj ¼ jεnj ∼ 0.002–0.008. However, for the allowedrange of εe, the bounds from ν − e scattering requirejενj to be reduced by a factor of ∼4 or more. In thissection, we neutralize the X-charge of the active neu-trinos by supplementing the SM with vectorlike leptonswith opposite B − L quantum numbers. The B − Lsymmetry is broken by a Higgs mechanism, generatinga vacuum expectation value for the new SM-singletHiggs field hX. This symmetry breaking simultaneously(1) generates the 17 MeV mass for the X boson,(2) generates a Majorana mass for the SM sterileneutrinos, which would otherwise be forbidden byB − L symmetry, and (3) mixes the SM active neutrinoswith the new lepton states such that the resulting masseigenstates have suppressed X-charge.The fields of these models include the SM Higgs boson

hSM, and the SM lepton fields lL, eR, and νR, where the lastis the sterile neutrino required by B − L anomaly cancel-lation. To these, we add the Higgs field hX with B − L ¼ 2,and N vectorlike lepton isodoublets LiL;R and chargedisosinglets EiL;R , with B − L ¼ 1. The addition of vector-like pairs preserves anomaly cancellation. These fields andtheir quantum numbers are shown in Table II. We focushere on the first generation leptons; the mechanismmay be straightforwardly extended to the second and thirdgenerations.With these fields, the full set of gauge-invariant, renor-

malizable Lagrangian terms that determine the leptonmasses are

L ¼ LSM þ Lmix þ Lnew; ð80Þ

TABLE II. Fields and their quantum numbers in the B − Lmodel with kinetic mixing and neutrinos neutralized by mixingwith vectorlike leptons. The SM fields, including the sterileneutrino, are listed above the line. The new fields, including Ngenerations of vectorlike fields, with i ¼ 4;…; N þ 3, are listedbelow the line.

Field Isospin I Hypercharge Y B − L

hSM 12

12

0

lL ¼� νLeL

� 12

− 12

−1

eR 0 −1 −1νR 0 0 −1hX 0 0 2

LiL ¼� νiLeiL

� 12

− 12

1

LiR ¼� νiReiR

� 12

− 12

1

EiL 0 −1 1EiR 0 −1 1

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LSM ¼ð−yehSMlLeR þ yν ~hSMlLνR þ H:c:Þ − yNhXνcRνR;

ð81Þ

Lmix ¼ −λiLhXlLLiR − λiEhXEiLeR þ H:c:; ð82Þ

Lnew ¼ −MijLLiLLjR −Mij

EEiLEjR − hijhSMLiLEjR

þ kij ~hSMEiLLjR þ H:c:; ð83Þ

where i; j ¼ 4;…; N þ 3. LSM generates the Dirac andMajorana SM neutrino masses, Lmix includes the terms thatmix the SM and vectorlike fields, and Lnew contains thevectorlike masses and Yukawa couplings for the newvectorlike leptons. For simplicity, we will assume universalmasses and Yukawa couplings, so λiL ¼ λL, λiE ¼ λE,Mij

L ¼ MLδij, Mij

E ¼ MEδij, hij ¼ hδij, kij ¼ kδij.

When electroweak symmetry and B − L symmetry arebroken, the Higgs fields get vevs hhSMi ¼ v=

ffiffiffi2

p, where

v≃ 246 GeV, and hhXi ¼ vX=ffiffiffi2

p. This gives the X boson

a mass

mX ¼ 2ejεB−LjvX; ð84Þ

which constrains vX to be

vX ¼ 14 GeV0.002jεB−Lj

: ð85Þ

It also generates Dirac and Majorana masses for the SMneutrinos,mD¼ yνv=

ffiffiffi2

pandmM ¼ yNvX=

ffiffiffi2

p, and masses

MLI ¼ λLvX=

ffiffiffi2

pandME

I ¼ λEvX=ffiffiffi2

pthat mix the SM and

vectorlike leptons. The resulting neutrino masses for thefirst SM generation and the vectorlike generations areψνMM

ν ψν, where

MMν ¼

0BBBBBBBBBBBB@

0 mD 0 MLI � � � 0 ML

I

mD mM 0 0 � � � 0 0

0 0 0 ML � � � 0 0

MLI 0 ML 0 � � � 0 0

..

. ... ..

. ... . .

. ... ..

.

0 0 0 0 � � � 0 ML

MLI 0 0 0 � � � ML 0

1CCCCCCCCCCCCA

;

ð86Þ

and ψν ¼ ðνL; νR; ν4L ; ν4R ;…; νNþ3L; νNþ3R

Þ, or alterna-tively, neglecting the small SM Dirac and Majoranamasses, the remaining neutrino masses may be writtenψνLMνψ

νR þ H:c:, where

Mν ¼

0BBBBB@

0 MLI � � � ML

I

0 ML � � � 0

..

. ... . .

. ...

0 0 � � � ML

1CCCCCA; ð87Þ

and ψνL;R ¼ ðνL;R; ν4L;R ;…; νNþ3L;R

Þ. Similarly, the chargedlepton masses are ψe

LMeψeR þ H:c:, where

Me ¼

0BBBBBBBBBBBB@

0 MLI 0 � � � ML

I 0

0 MLhvffiffi2

p � � � 0 0

MEI

kvffiffi2

p ME � � � 0 0

..

. ... ..

. . .. ..

. ...

0 0 0 � � � MLhvffiffi2

p

MEI 0 0 � � � kvffiffi

2p ME

1CCCCCCCCCCCCA

; ð88Þ

and ψeL;R ¼ ðeL;R; e4L;R ;…; eNþ3L;R

Þ.Diagonalizing the neutrino mass matrix of Eq. (86)

yields N Dirac neutrino states with mass ∼ML, and twolight states: the SM sterile neutrino and the SM activeneutrino, which is the eigenstate

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2

L þ NML2I

p ð−ML; 0;MLI ; 0;M

LI ;…; 0;ML

I ; 0Þ: ð89Þ

The active neutrino’s X-charge is therefore modified by themixing with the vectorlike lepton states, with similar effectsfor the charged leptons. In the end, we find that the leptonX-charges are modified to

ενL ¼ −εB−L cos 2θνL ; ð90Þ

εeL ¼ −εB−L cos 2θeL − ε ¼ εB−Lð1 − cos 2θeLÞ − δ; ð91Þ

εeR ¼ −εB−L cos 2θeR − ε ¼ εB−Lð1 − cos 2θeRÞ − δ; ð92Þ

where

tan θνL ¼ NML2I

M2L

; ð93Þ

and θeL and θeR are determined by similar, but morecomplicated, relations derived by diagonalizing Me. Toneutralize the neutrino charge, we need

tan θνL ¼ NML2I

M2L

¼ Nλ2Lm2X

8M2Le

2ε2B−L

≈�130 GeV

ML

�2�0.002εB−L

�2� ffiffiffiffi

Np

λL4π

�2

≈ 1; ð94Þ

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where we have normalized the effective couplingffiffiffiffiN

pλL to

its ultimate perturbative limit. We see that the neutrinoX-charge may be neutralized with as few as N ¼ 1 vector-like lepton generation with mass at the weak scale. A largernumber of heavier vectorlike leptons may also neutralizethe neutrino X-charge. In addition, to preserve nonchiralelectron couplings, we require θeL ≈ θeR .

C. Implications for colliders and cosmology

Here we consider the implications of these models forcolliders and cosmology, beginning with the extremelysimple case of N ¼ 1 generation of vectorlike leptons andvanishing Yukawa couplings h ¼ k ¼ 0. In this case, themass matrices are easily diagonalized. The heavy statesinclude three “fourth generation” Dirac fermions: theisodoublet neutrino and electron with masses mν4 ≃me4 ≃ffiffiffi2

pML and the isosinglet electron with massmE4

≃ ffiffiffi2

pME.

The states ν4 and e4 have vectorlike masses and are nearlydegenerate, and so do not contribute to the S and Tparameters [110]. The light states are the usual masslessSM leptons, but mixed with opposite X-charged states,with mixing angles tan θνL ¼ tan θeL ¼ ðML

I =MLÞ2 andtan θeR ¼ ðME

I =MEÞ2. These SM fields each mix only withnew leptons with the same SM quantum numbers, and sothese mixing angles are not constrained by precisionmeasurements. ChoosingML

I =ML ¼ MEI =ME ¼ 1, we find

ενL ¼ 0 and εeL ¼ εeR ¼ εB−L − δ. For εB−L ≈ 0.002 andδ ≈ 0.001, this extension of the SM contains a protophobicgauge boson that explains the 8Be signal consistent with allcurrent constraints. As in the Uð1ÞB case, assuming εμ ≈ εeremoves at least part of the ðg − 2Þμ puzzle and impliespromising prospects for future searches at low-energycolliders, as discussed in Sec. X.The new vectorlike leptons can be produced through

Drell-Yan production at hadron and eþe− colliders, and sothis model may be explored at the LHC and future colliders.The prospects for vectorlike lepton searches at the LHChave been studied in detail in the case that they decay toWνl, Zl, and hl [115–117]. In the present case, however,the vectorlike lepton masses and decays are constrained bythe neutrino neutralization mechanism. In particular, themixing terms of Eq. (82) that neutralize the neutrinos implythat the decays ν4 → νehX, e4 → ehX, and E4 → ehX arealmost certainly dominant.The B − L Higgs boson has a variety of possible decays,

but for a moderately large Majorana Yukawa coupling yN ,the invisible decay hX → νRνR dominates. The resultingprocesses are therefore

pp → Eþ4 E

−4 → eþe−hXhX → eþe−νRνRνRνR; ð95Þ

pp → eþ4 e−4 → eþe−hXhX → eþe−νRνRνRνR; ð96Þ

pp → ν4ν4 → νLνLhXhX → νLνLνRνRνRνR; ð97Þ

pp → ν4e4 → νehXhX → eνLνRνRνRνR: ð98Þ

These signals are therefore very similar to those of selectronpair production and selectron–sneutrino pair production,leading to signatures with missing transverse energy ET,eþe− þ ET and e� þ ET . The amount of missing energy iscontrolled by

mhX ¼ffiffiffiffiffiffiλH

pvX ¼ 70 GeV

ffiffiffiffiffiffiλH4π

r0.002jεB−Lj

; ð99Þ

where λH is the Higgs boson quartic coupling appearing inthe Lagrangian term λHðhXh�XÞ2, and we have used Eq. (85).Current bounds from the combination of LEP2 and

8 TeV LHC data on the combined production of right-and left-handed selectron and smuons with mass 100 GeVallow neutralino masses of around 50 GeV [118,119].The vectorlike lepton cross section is bigger by roughly afactor of 4, but 100 GeV vectorlike leptons decaying to50–70 GeV B − L Higgs bosons may still be allowed.Existing monolepton searches based on 8 TeV LHC dataare not optimized for lepton masses as low as 100 GeVandare unlikely to have sensitivity [120,121]. Nonetheless, itmay be that future searches based on 13 TeV data willbecome sensitive, particularly if they can be optimized forlower mass vectorlike leptons. It would be interesting toinvestigate this scenario in more detail, as well as scenarioswhere other hX decays are comparable or dominant to theinvisible decay assumed above. It is also worth noting thatthe appearance of relatively strong couplings (λH, λL) in thehX sector may be an indication of compositeness, whichcould result in a richer and more complicated set of finalstates accessible to LHC energies.We now turn to the SM neutrino sector and potential

cosmological signatures. As noted above, when the hX fieldwith B − L charge 2 gets a vev, it also generates a Majoranamass for the SM singlet neutrinos. This is an importantfeature. Without a charge 2 Higgs boson, the SM neutrinosare Dirac particles. Light Dirac neutrinos are not typicallyproblematic, as the νR component does not thermalize anddoes not contribute to the number of relativistic degrees offreedom neff . In the current model, however, the processff ↔ X ↔ νRνR effectively thermalizes the νR at temper-atures T ∼mX, where the process is on-resonance. To avoidthermalization, one needs the X-charge of νR to be less than10−9 [114] or very low reheat temperatures in the windowbetween 1 MeV and mX ≈ 17 MeV. The generation of aMajorana mass avoids these problems.The Majorana mass is

mM ¼ yNvX=ffiffiffi2

p¼ yN

mX

2ffiffiffi2

pejεB−Lj

≲ 30 GeV0.002jεB−Lj

;

ð100Þ

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where the upper bound assumes yN ∼ 3. The physicalmasses of the SM active neutrinos are then determinedby the see-saw mechanism, with Dirac masses chosenappropriately. Of course, the sterile neutrino masses neednot be near their upper limit, and it is tempting to postulatethat they may be in the keV range as required for warm darkmatter. To prevent the decays X → νRνR from significantlydiluting the 8Be signal in this case, the νR X-charges mustalso be neutralized, for example, through mixing withvectorlike isosinglet neutrinos. Alternatively, the sterileneutrino masses may be in the 10–100 MeV range, as maybe helpful for reducing the standard BBN predictions forthe 7Li abundance to the observed levels [122]. We leavethese astrophysical and cosmological implications forfuture work.One might worry that having a model with an exact

Uð1ÞB−L or Uð1ÞB gauge symmetry down to the GeV orMeV energy scale would prevent any baryon numberasymmetry from being generated. This, however, is notthe case, as was discussed, for example, in Ref. [103]for a model with gauged Uð1ÞB. A lepton numberasymmetry can still be produced at a high scale and thenbe partially converted into baryon number through theelectroweak sphalerons. For the case of gauged Uð1ÞB−Lone could also invoke a Dirac leptogenesis scenariowhich relies on the fact that the right-handed neutrinosdecouple early on during the evolution of the Universe,trapping some amount of lepton number [123,124]. Theresulting lepton number deficit in the visible sector isthen again transferred to baryon number through thesphalerons.We have introduced additional fermionic matter to

render the models compatible with experimental con-straints. The step of adding extra matter may not benecessary, and it may be possible to satisfy all the existingexperimental constraints by considering a combination ofgauged U(1) quantum numbers. The possibility of multiple,new U(1) gauge bosons has been explored previously, inthe two dark-photon (“paraphoton”) case [125] and forthree Abelian groups [126]. Here we note that if one were tocombine a Uð1ÞB−L model with kinetic mixing with asecond, unbroken (or softly broken) gauge symmetry, e.g.,Le − Lτ, it is possible to bring the first-generation fermioncouplings of the B − L gauge boson to the form of theUð1ÞB model. Such relationships are completely compat-ible with the couplings needed to describe the 8Be anomalyand satisfy other constraints. However, equivalence prin-ciple constraints on new, massless gauge bosons that cancouple to the constituents of ordinary matter are severe[114,127,128]. We note that we can address this problemby making the massless gauge boson’s couplings toelectrons vanish at tree level. Further investigation isrequired to check that this suffices to render the modelcompatible with experimental constraints on new, (nearly)massless gauge bosons.

IX. FUTURE EXPERIMENTS

Current and near future experiments will probe theparameter space of interest for the protophobic gaugeboson X. The projected sensitivities of various experimentsare shown in Fig. 6 and we briefly discuss them below.

A. Other large-energy nuclear transitions

The 8Be� and 8Be�0 states are quite special in that theydecay electromagnetically to discrete final states with anenergy release in excess of 17 MeV. Other large-energygamma transitions have been observed [129], such as the19.3 MeV transition in 10B to its ground state [130] and the17.79 MeV transition in 10Be to its ground state [131]. Ofcourse, what is required is large production cross sectionsand branching fractions so that many IPC events can beobserved. It would certainly be interesting to identify otherlarge-energy nuclear transitions with these properties to testthe new particle interpretation of the 8Be anomaly.

B. LHCb

A search for dark photons A0 at LHCb experiment duringRun 3 (scheduled for the years 2021–2023) has beenproposed [132] using the charmmeson decayD�ð2007Þ0 →D0A0 with subsequent A0 → eþe−. It takes advantage of theLHCb excellent vertex and invariant mass resolution. Fordark photon masses below about 100 MeV, the experimentcan explore nearly all of the remaining parameter space in εebetween the existing prompt-A0 and beam-dump limits. Inparticular, it can probe the entire region relevant for the Xgauge boson explaining the 8Be anomaly.

FIG. 6. The 8Be signal region, along with current constraints(gray) and projected sensitivities of future experiments in theðmX; εeÞ plane. Updated from Ref. [7]. Note BrðX → eþe−Þ ¼ 1is assumed.

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C. Mu3e

The Mu3e experiment will look at the muon decaychannel μþ → eþνeνμðA0 → eþe−Þ and will be sensitive todark photon masses in the range 10 MeV≲mA0 ≲ 80 MeV[133]. The first phase (2015–2016) will probe the regionεe ≳ 4 × 10−3, while phase II (2018 and beyond) willextend this reach almost down to εe ∼ 10−4, which willinclude the whole region of interest for the protophobicgauge boson X.

D. VEPP-3

A proposal for a new gauge boson search at the VEPP-3facility was made [134]. The experiment will consist of apositron beam incident on a gas hydrogen target and willlook for missing mass spectra in eþe− → A0γ. The searchwill be independent of the A0 decay modes and lifetime. Itsregion of sensitivity in εe extends down into the beam dumpbounds, i.e., below εe ∼ 2 × 10−4, and includes the entireregion relevant for X. Once accepted, the experiment willtake 3–4 years.

E. KLOE-2

As mentioned above, the KLOE-2 experiment, lookingfor eþe− → γðX → eþe−Þ, is running and improving itscurrent bound of jεej < 2 × 10−3 [73] for mX ≈ 17 MeV.With the increased DAϕNE-2 delivered luminosity and thenew detectors, KLOE-2 is expected to improve this limit bya factor of 2 within 2 years [135].

F. MESA

The MESA experiment will use an electron beamincident on a gaseous target to produce dark photons ofmasses between ∼10 − 40 MeV with electron coupling aslow as εe ∼ 3 × 10−4, which would probe most of theavailable X boson parameter space [136]. The commission-ing is scheduled for 2020.

G. DarkLight

The DarkLight experiment, similarly to VEPP-3 andMESA, will use electrons scattering off a gas hydrogentarget to produce on-shell dark photons, which later decayto eþe− pairs [137]. It is sensitive to masses in the range10–100 MeV and εe down to 4 × 10−4, covering themajority of the allowed protophobic X parameter space.Phase I of the experiment is expected to take data in thenext 18 months, whereas phase II could run within twoyears after phase I.

H. HPS

The Heavy Photon Search experiment is using a high-luminosity electron beam incident on a tungsten target toproduce dark photons and search for both A0 → eþe− andA0 → μþμ− decays [138]. Its region of sensitivity is split

into two disconnected pieces (see Fig. 6) based on theanalyses used: the upper region is probed solely by a bumphunt search, whereas the lower region also includes adisplaced vertex search. HPS is expected to complete itsdata set by 2020.

I. PADME

The PADME experiment will look for new light gaugebosons resonantly produced in collisions of a positronbeam with a diamond target, mainly through the processeþe− → Xγ [139]. The collaboration aims to complete thedetector assembly by the end of 2017 and accumulate 1013

positrons on target by the end of 2018. The expectedsensitivity after one year of running is εe ∼ 10−3, with plansto get as low as 10−4 [140,141].

J. BES III

Current and future eþe− colliders, may also search foreþe− → Xγ. A recent study has explored the possibility ofusing BES III and BABAR to probe the 17 MeV proto-phobic gauge boson [13].

K. E36 at J-PARC (TREK)

The TREK experiment has the capacity to study K→μνeþe− decays [142]; the enhancement associated withthe interaction of the longitudinal component of X withcharged fermions should make for sensitive tests of εe inthe mass range of interest to the 8Be anomaly [143].

X. CONCLUSIONS

The 6.8σ anomaly in 8Be cannot be plausibly explainedas a statistical fluctuation, and the fit to a new particleinterpretation has a χ2=dof of 1.07. If the observed bumphas a nuclear physics or experimental explanation, thenear-perfect fit of the θ and mee distributions to the newparticle interpretation is a remarkable coincidence. Clearlyall possible explanations should be pursued. Building onour previous work [7], in this study, we presented particlephysics models that extend the SM to include a protophobicgauge boson that explains the 8Be observations and isconsistent with all other experimental constraints.To understand what particle properties are required to

explain the 8Be anomaly, we first presented effectiveoperators for various spin-parity assignments. Manycommon examples of light, weakly coupled particles,including dark photons, dark Higgs bosons, axions, andB − L gauge bosons (without kinetic mixing) are disfa-vored or excluded on general grounds. In contrast, generalgauge bosons emerge as viable candidates.In Ref. [7] we determined the required couplings of a

vector gauge boson to explain the 8Be anomaly assumingisospin conservation, and found that the particle must beprotophobic. In this work, we refined this analysis to

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include the possibility of isospin mixing in the 8Be� and8Be�0 states. Although isospin mixing and violation canyield drastically different results, these effects are relativelymild once one focuses on protophobic gauge bosons. Itwould be helpful to have a better understanding of the roleof isospin breaking in these systems and a quantitativeestimate of their uncertainties. The presence of isospinmixing also implies that the absence of an anomaly in 8Be�0decays must almost certainly be due to kinematic suppres-sion and that the X particle’s mass is above 16.7 MeV.Combining all of these observations with constraints fromother experiments, we then determined the favored cou-plings for any viable vector boson explanation.We have presented two anomaly-free extensions of the

SM that resolve the 8Be anomaly. In the first, the proto-phobic gauge boson is a Uð1ÞB gauge boson that kineticallymixes with the photon. For gauge couplings and kineticmixing parameters that are comparable in size and oppositein sign, the gauge boson couples to SM fermions withapproximate charge Q − B, satisfying the protophobicrequirement. Additional matter content is required to cancelgauge anomalies, and we presented a minimal set of fieldsthat satisfy this requirement. In the second model, thegauge boson is a Uð1ÞB−L gauge boson with kinetic mixing,and the SM fermion charges are Q − ðB − LÞ. Additionalvectorlike leptons are needed to neutralize the neutrino ifwe consider only a single U(1) gauge group. Both modelscan simultaneously resolve the ðg − 2Þμ anomaly, havelarge electron couplings that can be probed at many nearfuture experiments, and include new vectorlike leptonstates at the weak scale that can be discovered by theLHC.One may speculate that the protophobic gauge boson

may simultaneously resolve not only the 8Be and ðg − 2Þμanomalies, but also others. Possibilities include the NuTeVanomaly [14] and the cosmological lithium problemmentioned in Sec. VIII C. Another possibility is the π0 →eþe− KTeV anomaly, which may be explained by a spin-1particle with axial couplings that satisfy

ðguA − gdAÞgeA�20 MeV

mX

�2

≈ 1.6 × 10−7; ð101Þ

which is roughly consistent with the vector couplings wefound for a protophobic gauge boson [144]. Independent of

experimental anomalies, a spin-1 boson with purely axialcouplings is a promising candidate for future study [145].Such bosons need not be protophobic, because their sup-pressed contributions to neutral pion decays relax manyconstraints that existed for vector bosons. We note, how-ever, that some bounds become stronger for the axial case.For example, the decay ϕ → ηðX → eþe−Þ used in derivingthe KLOE constraints [64] is an s-wave in the axial case,implying a stronger bound than the p-wave–suppressedone in the vector case. Another example is ðg − 2Þe [146],for which an axial vector makes larger contributions than avector, for couplings of the same magnitude. In addition,there are very stringent bounds, for example, from atomicparity violation, on gauge bosons with mixed vector andaxial vector couplings [147].Finally, if the 8Be anomaly is pointing toward a new

gauge boson and force, it is natural to consider whether thisforce may be unified with the others, with or withoutsupersymmetry. In the case of Uð1ÞB−L, which is a factor ofmany well-motivated grand unified groups, it is tempting tosee whether the immediately obvious problems—for exam-ple, the hierarchy between the required Uð1ÞB−L gaugecoupling and those of the SM—can be overcome, andwhether MeV-scale data may be telling us somethinginteresting about energy scales near the Planck scale.

ACKNOWLEDGMENTS

We thank Phil Barbeau, John Beacom, Roy Holt, YoniKahn, Attila J. Krasznahorkay, Saori Pastore, TilmanPlehn, Mauro Raggi, Alan Robinson, Martin Savage,Paolo Valente, and Robert Wiringa for helpful correspon-dence. J. L. F., B. F., I. G., J. S., T. M. P. T., and P. T. aresupported in part by NSF Grants No. PHY-1316792 andNo. PHY-1620638. The work of S. G. is supported in partby the DOE Office of Nuclear Physics under ContractNo. DE-FG02-96ER40989. The work of J. L. F. is sup-ported in part by a Guggenheim Foundation grant and inpart by Simons Investigator Award No. 376204 and wasperformed in part at the Aspen Center for Physics, which issupported by NSF Grant No. PHY-1066293.

Note added.—Following this paper, the pseudoscalar andaxial vector cases have been further studied in Refs. [148]and [149], respectively.

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