Particle Physics Models for the 17 MeV Anomaly in Beryllium Nuclear Decays
Jonathan L. Feng, Bartosz Fornal, Iftah Galon, Susan Gardner, Jordan Smolinsky, Tim M. P. Tait, Philip Tanedo
UUCI-TR-2016-12
Particle Physics Models for the 17 MeV Anomalyin Beryllium Nuclear Decays
Jonathan L. Feng, Bartosz Fornal, Iftah Galon, Susan Gardner,
1, 2
Jordan Smolinsky, Tim M.P. Tait, and Philip Tanedo
1, 31
Department of Physics and Astronomy,University of California, Irvine, California 92697-4575, USA Department of Physics and Astronomy,University of Kentucky, Lexington, Kentucky 40506-0055, USA Department of Physics and Astronomy,University of California, Riverside, California 92521, USA
Abstract
The 6.8 σ anomaly in excited Be nuclear decays via internal pair creation is fit well by a newparticle interpretation. In a previous analysis, we showed that a 17 MeV protophobic gauge bosonprovides a particle physics explanation of the anomaly consistent with all existing constraints. Herewe begin with a review of the physics of internal pair creation in Be decays and the characteristicsof the observed anomaly. To develop its particle interpretation, we provide an effective operatoranalysis for excited Be decays to particles with a variety of spins and parities and show that theseconsiderations exclude simple models with scalar particles. We discuss the required couplings fora gauge boson to give the observed signal, highlighting the significant dependence on the precisemass of the boson and isospin mixing and breaking effects. We present anomaly-free extensions ofthe Standard Model that contain protophobic gauge bosons with the desired couplings to explainthe Be anomaly. In the first model, the new force carrier is a U(1) B gauge boson that kineticallymixes with the photon; in the second model, it is a U(1) B − L gauge boson with a similar kineticmixing. In both cases, the models predict relatively large charged lepton couplings ∼ .
001 thatcan 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 beaccessible to near future LHC searches.
PACS numbers: 14.70.Pw, 27.20.+n, 21.30.-x, 12.60.Cn, 13.60.-r a r X i v : . [ h e p - ph ] J a n ONTENTS
I. Introduction 2II. The Be Anomaly 4A. Be Spectrum and Electromagnetic Decays 4B. The Atomki Result 5III. Nuclear Effective Theory and Particle Candidates 7A. Effective Operators for Be ∗ → Be X B Model for the Protophobic Gauge Boson 22A. U(1) B Gauge Boson with Kinetic Mixing 22B. Anomaly Cancellation and Experimental Implications 24VIII. U(1) B − L Model for the Protophobic Gauge Boson 26A. U(1) B − L Gauge Boson with Kinetic Mixing 26B. Neutrino Neutralization with Vectorlike Leptons 27C. Implications for Colliders and Cosmology 29IX. Future Experiments 32X. Conclusions 33Acknowledgments 35References 35
I. INTRODUCTION
The existence of light, weakly-coupled new particles has been a well-motivated theoreticalpossibility for decades. The need for dark matter has motivated these particles, either toprovide the dark matter itself—for example, in the form of axions—or, more recently, to2ediate interactions between the visible and dark sectors. Grand unification provides anothercompelling motivation for new particles and forces. Although these particles and forces aretypically expected to be heavy and short-range, respectively, it is possible that a remnantof grand unification might survive down to low energies. Another independent, but related,possibility is that some linear combination of the “accidental” B and L i global symmetriesof the Standard Model (SM) might be gauged. If these symmetries are spontaneously brokenat low energies, they must also be weakly-coupled. All of these provide ample motivation fora diverse program of high-statistics searches for new particles far from the energy frontier.Nuclear transitions provide a means to probe light, weakly-coupled new physics. Indeed,in 1978, Treiman and Wilczek [1], as well as Donnelly et al. [2], proposed that axions couldbe discovered through the study of nuclear decays. Such searches are now established as partof the corpus of constraints on axions and axion-like particles [3, 4], as well as on light scalarparticles with Higgs-like couplings [5]. The possible new particles include not only scalarsand pseudoscalars, but also those with other spin-parity assignments, which may manifestthemselves in different nuclear transitions. There are many possible nuclear transitions tostudy, but particularly promising are those that can be studied through excited nuclearstates that are resonantly produced in extraordinary numbers, providing a high-statisticslaboratory to search for MeV-scale new physics.Krasznahorkay et al. have recently observed unexpected bumps in both the distributions ofopening angles and invariant masses of electron–positron pairs produced in the decays of anexcited Be nucleus [6]. The bump in the angular distribution appears against monotonicallydecreasing backgrounds from SM internal pair creation (IPC), and the anomaly has a highstatistical significance of 6.8 σ . The shape of the excess is remarkably consistent with thatexpected if a new particle is being produced in these decays, with the best fit to the newparticle interpretation having a χ -per-degree-of-freedom of 1.07.In previous work [7], we examined possible particle physics interpretations of the Be signal.We showed that scalar and pseudoscalar explanations are strongly disfavored, given mildassumptions. Dark photons A (cid:48) , massive gauge bosons with couplings to SM particles that areproportional to their electric charge [8–11], also cannot account for the Be anomaly, givenconstraints from other experiments. The most stringent of these is null results from searchesfor π → A (cid:48) γ . This can be circumvented if the new spin-1 state couples to quarks vectoriallywith suppressed couplings to the proton. We concluded that a new “protophobic” spin-1boson X , with mass around 17 MeV and mediating a weak force with range 12 fm, providesan explanation of the Be anomaly consistent with all existing experimental constraints.The implications and origins of a protophobic gauge boson have been further studied inRefs. [12–14].Protophobic gauge bosons are not particularly unusual. The Z boson is protophobicat low energies, as is any new boson that couples to B − Q , the difference of the baryon-number and electric currents. As we show below, it is extremely easy to extend the SMto accommodate a light gauge boson with protophobic quark couplings. Simultaneouslysatisfying the requirements on lepton couplings requires more care. To produce the observed e + e − events, the coupling to electrons must be non-zero. This coupling is bounded fromabove by the shift the new boson would induce on the electron magnetic dipole moment andfrom below by searches for dark photons at beam dump experiments. The neutrino coupling,in turn, is bounded by ν – e scattering experiments, as well as by the non-observation ofcoherent neutrino–nucleus scattering. Any model that consistently explains the Be signalmust satisfy all of these constraints. 3n Sec. II we review the Be system and the observed anomaly. In Sec. III, we present aneffective operator analysis of the Be nuclear transitions and consider a variety of spin-parityassignments for the new boson. We show that many simple candidates, including scalars andpseudoscalars, are excluded, while a protophobic spin-1 gauge boson is a viable candidate.In the next three sections, we consider in detail the couplings such a gauge boson must haveto explain the Be anomaly: in Sec. IV we discuss the impact of isospin mixing and breakingin the Be system; in Sec. V we discuss the required gauge boson couplings to explain thesignal, noting the sensitivity to the gauge boson’s precise mass; and in Sec. VI we evaluatethe constraints imposed by all other experiments, refining the discussion in Ref. [7], especiallyfor the neutrino constraints. With this background, in Secs. VII and VIII, we constructsimple, anomaly-free extensions of the SM that contain protophobic gauge bosons with thedesired couplings to explain the Be anomaly. In Sec. IX we discuss current and near-termexperiments that may test this new particle explanation, and we conclude in Sec. X bysummarizing our results and noting some interesting future directions.
II. THE BE ANOMALYA. Be Spectrum and Electromagnetic Decays
We review relevant properties of the Be system. Some of the energy levels of Be areshown in Fig. 1. The ground state of the Be nucleus is only 0.1 MeV above the thresholdfor αα breakup, and α clustering is thought to inform its structure and excitations [15, 16].The ground state is a spin-parity J P = 0 + state with isospin T = 0, and its lowest-lyingexcitations are 2 + and 4 + rotational states, nominally of its αα dumbbell-shape, with T = 0,excitation energies 3.03 MeV and 11.35 MeV, and decay widths 1.5 MeV and 3.5 MeV,respectively. Going up in excitation energy, the next lowest lying states are isospin doublets of T = 0 , + , 1 + , and 3 + , respectively. The 2 + states are ofsuch mass that the αα final state is the only particle-decay channel open to them. Sinceboth states are observed to decay to the αα final state [19], which has T = 0, the 2 + statesare each regarded as mixtures of the T = 0 and T = 1 states. The qualitative evidence forisospin-mixing in the 1 + and 3 + states is less conclusive, but each doublet state is regardedas a mixed T = 0 and T = 1 state [20].In this paper, our focus is on the transitions of the 1 + isospin doublet to the ground state,as illustrated in Fig. 1. We refer to the ground state as simply Be and to the 1 + excitedstates with excitation energies 18.15 MeV and 17.64 MeV as Be ∗ and Be ∗(cid:48) , respectively .As noted above, these latter two states mix, but Be ∗ is predominantly T = 0, and Be ∗(cid:48) ispredominantly T = 1. The properties of these states and their electromagnetic transitionshave been analyzed using quantum Monte Carlo (QMC) techniques using realistic, microscopicHamiltonians [21–24]. We discuss the current status of this work and its implications for theproperties of new particles that may be produced in these decays in Sec. IV.The particular transitions relevant for the observed anomaly are internal pair conversion(IPC) decays. This is a process in which an excited nucleus decays into a lower-energy statethrough the emission of an electron–positron pair [25–27]. Like γ -decays—which satisfy The 2 + excited state is notable in that it is produced through the β decay of B [17]. It is pertinent to thesolar neutrino problem because it appears with a neutrino of up to 14 MeV in energy, and it also enters inprecision tests of the symmetries of the charged, weak current in the mass eight system [18]. IG. 1. The most relevant Be states, our naming conventions for them, and their spin-parities J P , isospins T , excitation energies E , and decay widths Γ from Ref. [19]. Asterisks on isospinassignments indicate states with significant isospin mixing. Decays of the Be ∗ (18.15) state to theground state Be exhibit anomalous internal pair creation; decays of the Be ∗(cid:48) (17.64) state donot [6]. selection rules based on angular momentum and parity—these decays can be classifiedby their parity (electric, E, or magnetic, M) and partial wave (cid:96) . A p -wave magnetictransition, for example, is labeled M1. The spectra of electron–positron invariant masses andopening angles in these decays are known to be monotonically decreasing for each partialwave in the SM [28]. It is customary to normalize the IPC rate with respect to that of γ emission for the same nuclear transition, when the latter exists. This is because thenuclear matrix elements, up to Coulomb corrections, as well as some experimental systematicerrors, cancel in this ratio. Be, moreover, is of sufficiently low- Z that the effects of itsCoulomb field on IPC are negligible [26]. Be ∗ decays to Li p most of the time, but itselectromagnetic transitions have branching fractions Br( Be ∗ → Be γ ) ≈ . × − [29]and Br( Be ∗ → Be e + e − ) ≈ . × − Br( Be ∗ → Be γ ) [26, 28]. B. The Atomki Result
The Atomki pair spectrometer has observed the IPC decays of Be ∗ with high statistics [6,30]. A sketch of the experiment and the new physics process being probed is shown inFig. 2. A beam of protons with kinetic energies tuned to the resonance energy of 1.03 MeVcollide with Li nuclei to form the resonant state Be ∗ , and a small fraction of these decay via Be ∗ → Be e + e − . The spectrometer is instrumented with plastic scintillators and multi-wireproportional chambers in the plane perpendicular to the proton beam. These measure theelectron and positron energies, as well as the opening angle of the e + e − pairs that traversethe detector plane, to determine the distributions of opening angle θ and invariant mass m ee .The experiment does not observe the SM behavior where the θ and m ee distributionsfall monotonically. Instead, the θ distribution exhibits a high-statistics bump that peaks at θ ≈ ◦ before returning to near the SM prediction at θ ≈ ◦ [6]. To fit this distribution,Krasznahorkay et al. consider many possible sources, including the M1 component from IPC,5 IG. 2. Schematic depiction of the Atomki pair spectrometer experiment [6, 30], interpreted asevidence for the production of a new boson X . The proton beam’s energy is tuned to excite lithiumnuclei into the Be ∗ state, which subsequently decays into the Be ground state and X . The latterdecays into an electron–positron pair whose opening angle and invariant mass are measured. but also others, such as an E1 component from non-resonant direct proton capture [31]. Theyobserve that the best fit comes from a 23% admixture of this E1 component. Nevertheless, theyare unable to explain the bump by experimental or nuclear physics effects, and instead findthat the excess in the θ distribution has a statistical significance of 6.8 σ [6]. A correspondingbump is seen in the m ee distribution.If a massive particle is produced with low velocity in the Be ∗ decay and then decays to e + e − pairs, it will produce a bump at large opening angles. It is therefore natural to considera new particle X and the two-step decay Be ∗ → Be X followed by X → e + e − . With fixedbackground, Krasznahorkay et al. find that the best fit mass and branching fraction are [6] m X = 16 . ± .
35 (stat) ± . Be ∗ → Be X )Γ( Be ∗ → Be γ ) Br( X → e + e − ) = 5 . × − . (2)For the best fit parameters, the fit to this new particle interpretation is excellent, with a χ / dof = 1 . m ee = 2 E e + E e − − (cid:113) E e + − m e (cid:113) E e − − m e cos θ + 2 m e = (1 − y ) E sin θ m e (cid:18) y − y cos θ (cid:19) + O ( m e ) , (3)where E ≡ E e + + E e − and y ≡ E e + − E e − E e + + E e − (4)are the total energy and energy asymmetry, respectively. The second term in the last line ofEq. (3) is much smaller than the first and may be neglected. At the Atomki pair spectrometer,the Be ∗ nuclei are produced highly non-relativistically, with velocity of 0 . c and, given m X ≈
17 MeV, the X particles are also not very relativistic. As a result, the e + and e − are produced with similar energies, and so one expects small | y | and m ee ≈ E sin( θ/ θ and m ee distributions satisfy this relation.The Atomki collaboration verified that the excess exclusively populates the subset ofevents with | y | ≤ . | y | > . E >
18 MeV and is absent for lower energy events [6]. The latter two observationsstrongly suggest that the observed IPC events are indeed from Be ∗ decays rather thanfrom interference effects and that the decays go to the ground state Be, as opposed to,for example, the broad 3 MeV J P = 2 + state. Decays to the 3 MeV state would have amaximum total energy of 15 MeV and do not pass the E >
18 MeV cut even when includingeffects of the energy resolution, which has a long low-energy tail, but not a high-energy one(see Fig. 2 of Ref. [30]). Finally, we note that IPC decays of the 17.64 MeV, isotriplet Be ∗(cid:48) state have also beeninvestigated at the Atomki pair spectrometer. An anomaly had previously been reported in Be ∗(cid:48) decays [32]. This anomaly was featureless and far easier to fit to background than thebumps discussed here, and it has now been excluded by the present Atomki collaboration [30].If the observed anomaly in Be ∗ decays originates from a new particle, then the absenceof new particle creation in the Be ∗(cid:48) decay combined with the isospin mixing discussed inSec. IV strongly suggest that such decays are kinematically—not dynamically—suppressedand that the new particle mass is in the upper part of the range given in Eq. (1). It alsosuggests that with more data, a similar, but more phase space-suppressed, excess may appearin the IPC decays of the 17.64 state. III. NUCLEAR EFFECTIVE THEORY AND PARTICLE CANDIDATES
The transition Be ∗ → Be X followed by X → e + e − implies that X is a boson. Weconsider the cases in which it is a scalar or vector particle with positive or negative parity. Inthis reaction, its de Broglie wavelength is λ ∼ (6 MeV) − , much longer than the characteristicsize of the Be nucleus, r ∼ (100 MeV) − . In this regime, the nucleus looks effectively point-like, and one can organize the corrections from the nuclear structure as a series in r/λ . Thisapproach has a long and fruitful history in the analysis of radiative corrections in weaknuclear decays [33, 34].We perform such an analysis for the case of Be ∗ decaying to a new boson X . Manytheories predict the existence of new, weakly-coupled, light degrees of freedom that, primafacie, may play the role of the X boson. We show that some common candidates for X areexcluded. We note that for the case where X has spin-parity J P = 1 − and isospin mixingbetween Be ∗ and Be ∗(cid:48) is neglected, nuclear matrix elements and their uncertainties cancelin the ratio of partial widths, Γ( Be ∗ → Be X ) / Γ( Be ∗ → Be γ ). A. Effective Operators for Be ∗ → Be X The candidate spin/parity choices for X are: a 0 − pseudoscalar a , a 1 + axial vector A , anda 1 − vector V . We argue below that there is no scalar operator in the parity-conserving limit. The widths of the m ee and θ distributions are determined by the O (MeV) energy resolution for theelectrons and positrons [30], which should not be confused with the 10 keV energy resolution for γ -raysused in testing the target thickness [6]. Be ∗ → Be X are: L P = g P Be ( ∂ µ a ) Be ∗ µ (5) L A = g A Λ A Be G µν F ( A ) µν + g (cid:48) A Λ A m A Be A µ Be ∗ µ (6) L V = g V Λ V Be G µν F ( V ) ρσ (cid:15) µνρσ , (7)where G µν ≡ ∂ µ Be ∗ ν − ∂ ν Be ∗ µ is the field strength for the excited Be ∗ state, F ( V ) µν and F ( A ) µν are the field strengths for the new vector and axial vector bosons, respectively, and thedimensionful parameters Λ i encode the dominant nuclear matrix elements relevant for thetransition in each case.In the vector case, Lorentz and parity invariance requires that all operators containingthe fields Be, Be ∗ µ , and V µ must also contain two derivatives and (cid:15) µνρσ . Any operators inwhich the two derivatives act upon the same field vanish under antisymmetrization of theLorentz indices, so that the only other possible operators are( ∂ µ Be) Be ∗ ν F ( V ) ρσ (cid:15) µνρσ and ( ∂ µ Be) G νρ V σ (cid:15) µνρσ . (8)However, these operators can each be integrated by parts to produce a term that vanishesby antisymmetrization, and the unique operator in L V . This is in contrast to the axialvector case, where the gauge-breaking part cannot be related by operator identities to thegauge-invariant piece and is thus a separate term with a separate effective coupling g (cid:48) A . B. Scalar Candidates
A popular example of a J P = 0 + scalar candidate for the X boson is a dark Higgs [35].However, a scalar cannot mediate the observed Be ∗ decay in the limit of conserved parity.The initial Be ∗ state has unit angular momentum and is parity-even, J P = 1 + . Angularmomentum conservation requires the final state Be X , which consists of two 0 + states, tohave orbital angular momentum L = 1. This, however, makes the final state parity-oddwhile the initial state is parity-even. This implies that there are no Lagrangian terms in aparity-conserving effective field theory that couple a scalar to the Be ∗ and Be ∗(cid:48) . This canalso be seen at an operator level; for example, the operator ( ∂ µ S )( ∂ ν Be) G ρσ (cid:15) µνρσ vanishesupon integrating by parts and using the Bianchi identity. C. Pseudoscalar Candidates A J P = 0 − pseudoscalar or axion-like particle, a , generically has a coupling to photons ofthe form g aγγ aF µν (cid:101) F µν that is generated by loops of charged particles [36–38]. For a mass m a ≈
17 MeV, all values of this coupling in the range (10 GeV) − < g aγγ < (10 GeV) − areexperimentally excluded [39, 40]. These bounds may be significantly revised in the presenceof non-photonic couplings, however. 8 . Axial Vector Candidates Axial vector candidates have several virtues. First, as we show in Sec. VI, one of the mostrestrictive constraints on the X particle comes from the decay of neutral pions, π → Xγ . If X is an axial vector, this decay receives no contribution from the axial anomaly and non-anomalous contributions to pion decay vanish in the chiral limit by the Sutherland–Veltmantheorem [41, 42].Second, unlike the other spin–parity combinations, the axial candidate has two leading-order effective operators with different scaling with respect to the X three-momentum. The g (cid:48) A term in Eq. (6) yields a Be ∗ decay rate that scales as Γ X ∼ | k X | , whereas the g A,V termsinduce rates that scale as Γ X ∼ | k X | . Thus, the axial particle may produce the observedanomalous IPC events with smaller couplings than the vector, g (cid:48) A (cid:28) g V . This may helpavoid some of the other experimental bounds discussed in Sec. VI. At the same time, theremay be more severe constraints, as we discuss in Sec. X.Unfortunately, large uncertainties in the nuclear matrix element for axial vectors make itdifficult to extract the required couplings for this scenario. To the best of our knowledge,there is no reliable ab initio calculation or measurement of the matrix element we wouldneed in the Be system.
E. Vector Candidates
The primary candidate for the X boson and the focus for the remainder of this study is a J P = 1 − vector. A new vector couples to a current J µX that is a linear combination of theSM fermion currents, L ⊃ iX µ J µX = iX µ (cid:88) i = u,d,(cid:96),ν ε i eJ µi , J µi = ¯ f i γ µ f i . (9)Here we have introduced separate couplings to up-type quarks, down-type quarks, chargedleptons, and neutrinos and assigned them charges ε i in units of e . Family-universal couplingsof this type naturally avoid the introduction of tree-level flavor-changing neutral currents,which are highly constrained. Conservation of X charge implies that the couplings to theproton and neutron currents, J µp and J µn , are determined by ε p = 2 ε u + ε d and ε n = ε u + 2 ε d ,so that J µX = (cid:88) i = u,d,(cid:96),ν ε i eJ µi = (2 ε u + ε d ) eJ µp + ( ε u + 2 ε d ) J µn + ε (cid:96) eJ µ(cid:96) + ε ν eJ µν . (10)For the low energies at which we work, it is important to map this quark-level expressionto one in terms of hadrons. Denoting the current in the previous equation by J µ (quark) X , weeffect this by matching the requisite matrix element to its equivalent in hadronic degrees offreedom. That is, for the proton, J µp ≡ (cid:104) p ( p (cid:48) ) | J µ (quark) X | p ( p ) (cid:105) = eu p ( p (cid:48) ) (cid:8) F X ,p ( q ) γ µ + F X ,p ( q ) σ µν q ν / M p (cid:9) u p ( p ) , (11)where | p ( p ) (cid:105) denotes a proton state composed of quarks and u p ( p ) is the Dirac spinor of afree proton. Note that QCD generates all the possible currents compatible with Lorentz9nvariance and electromagnetic current conservation. We choose F X ,p ( q ) and F X ,p ( q ) todenote the X-analogues of the familiar Dirac and Pauli form factors. Finally we form theanalogue of the Sachs magnetic form factor by introducing G XM,p ( q ) = F X ,p ( q ) + F X ,p ( q ),recalling that G M,p (0) is given by the total magnetic moment of the proton — we refer toRef. [43] for a review. The M1 transition of interest here is determined by the total magneticmoment operator.The nucleon currents, written in either the quark or hadron basis, can, in turn, becombined to form isospin currents J µ = J µp + J µn J µ = J µp − J µn . (12)Assuming isospin is conserved and the Be states are isospin eigenstates, (cid:104) Be | J µ | Be ∗ (cid:105) = 0,since both Be ∗ and the ground state Be are isosinglets. In this case, (cid:104) Be | J µX | Be ∗ (cid:105) = e ε p + ε n ) (cid:104) Be | J µ | Be ∗ (cid:105) (13) (cid:104) Be | J µ EM | Be ∗ (cid:105) = e (cid:104) Be | J µ | Be ∗ (cid:105) . (14)The J nuclear matrix elements therefore cancel in the ratio Γ( Be ∗ → Be X ) / Γ( Be ∗ → Be γ ). This observation may be modified significantly when isospin violation is included, aswe discuss in Sec. IV.If one sets g V = e and identifies F ( V ) ρσ with the electromagnetic field strength in Eq. (7),then the leading operator in L V describes the ordinary electromagnetic transition via γ emission. Indeed, in this SM case, Lorentz- and parity-invariance require the characteristicΓ( Be ∗ → Be γ ) ∝ | k γ | momentum dependence of an M1 transition. The matrix elementsin Eqs. (13) and (14) thus imply that Λ V in Eq. (7) is universal for spin-1 particles. Combiningall of these pieces, we find thatΓ( Be ∗ → Be X )Γ( Be ∗ → Be γ ) = ( ε p + ε n ) | k X | | k γ | = ( ε p + ε n ) (cid:20) − (cid:16) m X .
15 MeV (cid:17) (cid:21) / , (15)when isospin is conserved. This is a convenient expression, as the experimental best fit forthe anomalous decay rate to a 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 A (cid:48) [8–11]. The darkphoton is a light particle that can have small, but technically natural, couplings to the SM.For a given mass, the dark photon interactions are controlled by a single kinetic mixingparameter, ε . This is related to the effective coupling in Eq. (7) by g V = εe . Substituting thisinto Eq. (15) and comparing to the experimental result in Eq. (2), one finds that ε ≈ − ,which is experimentally excluded by, for example, π → A (cid:48) γ searches at NA48/2 [44]. A generalization of the dark photon idea is to consider also mixing between the new bosonand the SM Z . Such a particle is spin-1 with no definite parity. Unfortunately, bounds fromatomic parity violation are extremely stringent [45] and constrain the dark Z couplings to betoo small to explain the Be anomaly.Another type of spin-1 particle is a light baryon-minus-lepton number ( B − L ) boson [46–48]. This scenario is constrained by neutrino scattering off electrons and, assuming no kinetic Ref. [6] quotes a fit of ε ∼ − . The discrepancy appears to come from the use of expressions foraxions [2] rather than dark photons. g B − L (cid:46) × − [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 thepresence of isospin mixing and breaking. We will also see, however, that in the experimentallyviable limit of ε p (cid:28) ε n , these corrections are small. For the cases of the dark photon, dark Z , and B − L gauge boson discussed above, the size of the Be signal and the strength ofthe constraints on π → Xγ essentially enforce protophobia, and so the arguments againstthese candidates remain. IV. SIGNAL DEPENDENCE ON ISOSPIN MIXING AND BREAKING
The discussion of Sec. III E assumed that isospin is conserved and that the Be states arestates of well-defined isospin. As noted in Sec. II A, however, there is substantial evidencethat the Be states are isospin-mixed, and, as we note below, there may also be isospinbreaking in the electromagnetic transition operators stemming from the neutron–protonmass difference. In this section, we determine the impact of isospin mixing and breaking onthe rate for Be ∗ → Be X , which, of course, has implications for the parton-level couplingsrequired to explain the Be signal.The ground-state structure and excitation spectrum of Be, as well as its electromagnetictransitions, have been studied with ab initio
QMC techniques, based on non-relativisticHamiltonians with phenomenological nucleon-nucleon and three-nucleon potentials [21–24].The latest work, Ref. [24], uses the newer AV18+IL7 potential.Isospin mixing is addressed in the manner of Ref. [20]: the empirical total (hadronic)widths are used to fix the isospin-mixing of the states within a particular doublet. That is,for a doublet of spin J , the physical states (with labels a and b ) are given by [24]Ψ aJ = α J Ψ J,T =0 + β J Ψ J,T =1 Ψ bJ = β J Ψ J,T =0 − α J Ψ J,T =1 , (16)where a denotes the lower energy state. Note that α J and β J are real and satisfy α J + β J = 1.The widths of the isospin-pure states are computed using the QMC approach, permittingthe extraction of the mixing parameters in Eq. (16) from the measured widths, yielding, forexample [24], α = 0 . β = 0 . . (17)The empirical excitation energies, which are unfolded from the experimental data using thesemixing coefficients, agree with the QMC energies of the states of all three mixed doublets, towithin the expected theoretical error—that is, to within 1% uncertainty.Given this success, this procedure may be applied to the electromagnetic transitions ofthese isospin-mixed states as well, so that the M1 transitions to the ground state are of theform (cid:104) Ψ , || M || Ψ aJ (cid:105) = α J M J,T =0 + β J M J,T =1 (18) (cid:104) Ψ , || M || Ψ bJ (cid:105) = β J M J,T =0 − α J M J,T =1 , (19)where M J,T is the reduced matrix element of the M1 operator with the isospin pure
J, T states. For reference we note that this matrix element is related to the partial width Γ M for11he transition via Γ M = 16 π α (cid:126) c (cid:18) ∆ E (cid:126) c (cid:19) B ( M (cid:18) (cid:126) c M p [MeV] (cid:19) , (20)where B ( M
1) = |(cid:104) Ψ J f || M || Ψ J i (cid:105)| / (2 J i +1) is in units of ( µ N ) , the squared nuclear magneton.We emphasize that the M1 operator can mediate both isoscalar (∆ T = 0) and isovector( | ∆ T | = 1) transitions. The J | ∆ T | isospin currents are given in Eq. (12).Unfortunately, the leading one-body (impulse approximation) results compare poorly toexperiment. The inclusion of meson-exchange currents in the M J,T matrix element improvesmatters considerably, yielding finally Γ M = 12 . expt M = 15 . .
8) eV [29], and Γ M = 0 . expt M = 1 . α 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 consideredin this context. Isospin breaking can appear in the hadronic form of the electromagnetictransition operators themselves [51, 52] to the end that changes in the relative strength ofthe isoscalar and isovector transition operators appear as a result of isospin-breaking inthe masses of isospin multiplet states, such as the nonzero neutron-proton mass difference.This is pertinent because electromagnetic transition operators involve both one and two-body contributions. The nuclear structure calculations of Ref. [24] employ electromagnetictransition operators from chiral effective theory in the isospin limit [53, 54]. The empiricalmagnetic moments of the neutron and proton are employed in the leading one-body termsin these analyses, albeit they are normalized by the average nucleon mass, rather thanthe proton mass that appears in the definition of the nuclear magneton. Consequently theisospin-breaking effects that shift the relative strength of the isoscalar and isovector transitionoperators appear in higher-order terms, namely in the relativistic corrections to leadingone-body operators, as well as in the two-body operators. These effects are likely numericallyimportant for the dominantly isoscalar electromagnetic transitions because the relativisticone-body corrections and two-body contributions are predominantly isovector in the isospinlimit [24, 55], though technically these 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 formal-ism [56]. That is, we include isospin-breaking contributions through the introduction of afictitious particle, the spurion, whose purpose is to allow the inclusion of isospin-breakingeffects within an isospin-invariant framework. Since the largest effects should stem from theneutron-proton mass difference, the spurion acts like a new ∆ T = 1 operator because itssize is controlled by ( M n − M p ) /M N , where M N is the nucleon mass. Since the isoscalartransition operators are extremely small we include the “leakage” of the dominant isovectoroperators into the isoscalar channel only. This is justified by noting that Ref. [24] usedstates of pure isospin and included meson exchange currents, to determine the isovector andisoscalar M1 transition strengths to be M ,T =1 = 0 . µ N and M ,T =0 = 0 . µ N , (21)12here the numerical dominance of the isovector M1 transition strength arises from that ofthe empirical isovector anomalous magnetic moment and the charged-pion, meson-exchangecontribution, which is isovector.Characterizing the strength of the ∆ T = 1 spurion by κ , the matrix elements of Eqs. (18)and (19) are thus amended by the addition of δ (cid:104) Ψ , || M || Ψ a (cid:105) = α κM ,T =1 (22) δ (cid:104) Ψ , || M || Ψ b (cid:105) = β κM ,T =1 . (23)The size of κ is controlled by non-perturbative effects. To illustrate its role, we assume thatit can be determined by demanding that the resulting M1 transition rate of the 17.64 MeVdecay reproduces its experimental value. The final M1 transition matrix elements thus read (cid:104) Ψ , || M || Ψ a (cid:105) = α M ,T =0 + β M ,T =1 + α κM ,T =1 , (24) (cid:104) Ψ , || M || Ψ b (cid:105) = β M ,T =0 − α M ,T =1 + β κM ,T =1 . (25)The needed shift in the M1 partial width of the 17.64 MeV transition is 3 . ± . κ = 0 . (cid:104) Ψ , || M || Ψ b (cid:105) = 0 . µ N and a M1 partial width of 1 .
62 eV, which is within 1 σ of theexperimental result.With the above discussion of both isospin mixing and isospin breaking in hand, we nowturn to their implications for an M1 transition mediated by an X boson with vector couplings ε n e and ε p e to the neutron and proton, respectively. The M1 transition mediated by X is (cid:104) Ψ , || M X || Ψ b (cid:105) = ( ε n + ε p ) β M ,T =0 + ( ε p − ε n )( − α M ,T =1 + β κM ,T =1 ) , (26)where the neutron and proton X couplings appear because the Be system contains equalnumbers of neutrons and protons. The resulting ratio of partial widths is, then,Γ X Γ γ = | ( ε p + ε n ) β M ,T =0 + ( ε p − ε n )( − α M ,T =1 + β κM ,T =1 ) | | β M ,T =0 − α M ,T =1 + β κM ,T =1 | | k X | | k γ | . (27)In the limit of no isospin mixing ( α = 0, β = 1) and no isospin breaking ( κ = 0), Eq. (27)reproduces Eq. (15). However, substituting the isospin mixing parameters of Eq. (17) andthe M1 transition strengths of Eq. (21), we findΓ X Γ γ = | − .
09 ( ε p + ε n ) + 1 .
09 ( ε p − ε n ) | | k X | | k γ | κ = 0 (28)Γ X Γ γ = | .
05 ( ε p + ε n ) + 0 .
95 ( ε p − ε n ) | | k X | | k γ | κ = 0 . . (29)The isoscalar contribution is only a small fraction of the isovector one, and so, in general,large modifications from isospin violation are possible.In Fig. 3, we plot the ratio Γ X / Γ γ in the ( ε p , ε n ) plane. In the case of perfect isospin, thetransition is isoscalar and the ratio depends on ε p + ε n , but in the case of isospin violation,the isovector transition dominates, and the ratio depends effectively on ε p − ε n . The effectsof including isospin violation are, therefore, generally significant. Interestingly, however,in the protophobic limit with ε p = 0, isospin violation only modifies Γ X / Γ γ by a factor of13 IG. 3. The ratio Γ X / Γ γ in the case of perfect isospin ( α = κ = 0) (left) and isospin violation( α = 0 . κ = 0 . ε p , ε n ) plane for m X = 16 . X / Γ γ = 5 . × − is highlighted. The dark photon scenario corresponds to ε n = 0. about 20%. However, for larger values of | ε p | , for example, | ε p | ∼ | ε n | /
2, isospin-breakingeffects can be significant, leading to factors of 10 changes in the branching ratios, or factorsof 3 modifications to the best fit couplings. Such large excursions from protophobia areexcluded by the NA48/2 limits for the best fit values of the couplings corresponding to m X = 16 . m X within its allowed range, as wediscuss below. V. SIGNAL REQUIREMENTS FOR GAUGE BOSON COUPLINGS
In this section, we discuss what a gauge boson’s couplings must be to explain the Besignal. We begin with the leptonic couplings, where the requirements are straightforward todetermine. To produce the IPC signal, the X boson must decay to e + e − . The Atomki pairspectrometer has a distance of O (few) cm between the target, where the Be excited state isformed, and the detectors that observe the charged particles [30]. The X boson decay widthto electrons is Γ( X → e + e − ) = ε e α m X + 2 m e m X (cid:113) − m e /m X , (30)with similar formulae for other fermion final states [57]. Requiring that the new bosonpropagates no more than 1 cm from its production point implies a lower bound | ε e | (cid:112) Br( X → e + e − ) (cid:38) . × − . (31)14 IG. 4. Contours of Γ X / Γ γ in the ( ε p , ε n ) plane for the parameterization of isospin violationin Eq. (29). Also shown are the dark photon axis ( ε n = 0) and the protophobic region with | ε p | ≤ . × − allowed by NA48/2 constraints on π → Xγ . The m X values are fixed to m X = 16 . ± σ (statistical) range of m X . If the X boson couples only to the charged SM fermions required to explain the Be anomaly,one has Br( X → e + e − ) = 1. Note, however, that if ε ν (cid:54) = 0 or if there exist light hidden-sectorstates with X charge, then there are generically other decay channels for X .The required quark couplings are determined by the signal event rate, that is, the best fitΓ X / Γ γ . In the Atomki experimental paper, the best fit branching fraction is that given inEq. (2). Combining this result with the isospin-conserving expression for the branching ratioof Eq. (15), we find | ε p + ε n | ≈ . × − (cid:112) Br( X → e + e − ) or | ε u + ε d | ≈ . × − (cid:112) Br( X → e + e − ) , (32)where we have taken m X = 16 . m X = 17 MeV, werepresented previously in Ref. [7].Given the discussion above, however, several refinements are in order. First, one caninclude the isospin-violating effects discussed in Sec. IV. These modify the branching ratioexpression from Eq. (15) to Eq. (29), with the effects shown in Fig. 3.Second, as discussed above, the presence of significant isospin mixing strongly suggeststhat the absence of anomalous IPC decays in the Be ∗(cid:48) state originates from kinematicsuppression, rather than from isospin symmetry or some other dynamical effect. This, then,argues for masses in the upper region of the allowed range of Eq. (2). Larger masses implylarger phase-space suppression, and these may significantly shift the contours of Γ X / Γ γ inthe ( ε p , ε n ) plane, as can be seen by comparing the ± σ values of m X in Fig. 4.Last, and most importantly, to determine the favored couplings, one must know how thebest fit Γ X / Γ γ depends on m X . In the original experimental paper, the best fit branchingratio Γ X / Γ γ = 5 . × − was presented without uncertainties and only for the best fit massof 16.7 MeV. In a subsequent analysis, however, the experimental collaboration explored the15mplications of other masses [58]. In preliminary results from this analysis, the M1 and E1background normalizations were fit to the angular spectrum in the range 40 ◦ ≤ θ ≤ ◦ ,and confidence regions in the ( m X , Γ X / Γ γ ) plane were determined with only statisticaluncertainties included. For masses larger than 16.7 MeV, the best fit branching ratio wasfound to be significantly smaller. For example, for m X = 17 . X / Γ γ ≈ . × − (0 . × − ) [58]. For such large masses, the best fit withfixed backgrounds is not very good, and the implications for nucleon-level couplings arepartially offset by the reduced phase space factor | k X | / | k γ | . In a full analysis, one shouldalso include systematic errors which are clearly a significant source of uncertainty in the m X determination, and also let the background levels float in the fit. We expect thatincluding these effects will significantly improve the fit for larger masses and favor evensmaller couplings. Specifically, since the anomalous events at angles between 120 ◦ and 135 ◦ cannot come from signal when the X mass is heavier, larger M1 and E1 backgrounds willimprove the fit and thus require smaller signal to achieve the best fit to the angular spectrum.Clearly a complete understanding of the experimental uncertainties requires a detailedanalysis that incorporates an accurate estimate of nuclear isospin violation, simulation of theexperiment, systematic uncertainties, varying backgrounds, and the null Be ∗(cid:48) result. Suchan analysis is beyond the scope of this study. As a rough estimate of the hadronic couplingsrequired to explain the Be signal, we take | ε n | = (2 − × − (33) | ε p | ≤ . × − , (34)where the upper part of the ε n range includes the coupling for the best fit branching ratio for m X = 16 . m X that simultaneously explain the Be ∗ signal and the Be ∗(cid:48) null results. The proton couplingconstraint follows from the NA48/2 constraints to be discussed in Sec. VI A 1. In presentingour models in Secs. VII A and VIII A, we leave the dependence on ε n explicit so that theimpact of various values of ε n can be easily evaluated. Note that the lower values of Γ X / Γ γ are still too large to accommodate a dark photon explanation. VI. CONSTRAINTS FROM OTHER EXPERIMENTS
We now discuss the constraints on the gauge boson’s couplings from all other experiments,considering quark, electron, and neutrino couplings in turn, with a summary of all constraintsat the end of the section. Many of these constraints were previously listed in Ref. [7]. Wediscuss them here in more detail, update some—particularly the neutrino constraints—toinclude new cases and revised estimates from other works, and include other constraints.
A. Quark Coupling Constraints
The production of the X boson in Be ∗ decays is completely governed by its couplingsto hadronic matter. The most stringent bound on these couplings in the m X ≈
17 MeVmass range is the decay of neutral pions into Xγ . For completeness, we also list the leadingsubdominant constraints on ε q , for q = u, d .16 . Neutral pion decay, π → Xγ The primary constraint on new gauge boson couplings to quarks comes from the NA48/2experiment, which performs a search for rare pion decays π → γ ( X → e + e − ) [59]. Thebound scales like the anomaly trace factor N π ≡ ( ε u q u − ε d q d ) . Translating the dark photonbound N π < ε / | ε u + ε d | = | ε p | (cid:46) (0 . − . × − (cid:112) Br( X → e + e − ) , (35)where the range comes from the rapid fluctuations in the NA48/2 limit for masses near17 MeV. In Ref. [7], we observed that the left-hand side becomes small when the X boson isprotophobic—that is, when its couplings to protons are suppressed relative to neutrons.
2. Neutron–lead scattering
A subdominant bound is set from measurements of neutron-nucleus scattering. TheYukawa potential acting on the neutron is V ( r ) = − ( ε n e ) Ae − m X r / (4 πr ), where A is theatomic mass number. Observations of the angular dependence of neutron–lead scatteringconstrain new, weakly-coupled forces [60], leading to the constraint( ε n e ) π < . × − (cid:16) m X MeV (cid:17) . (36)
3. Proton fixed target experiments
The ν -Cal I experiment at the U70 accelerator at IHEP sets bounds from X -bremsstrahlungoff the initial proton beam [61] and π → Xγ decays [62]. Both of these processes aresuppressed in the protophobic scenario so that these bounds are automatically satisfied whenEq. (35) is satisfied.
4. Charged kaon and φ decays There are also bounds on second generation couplings. The NA48/2 experiment placeslimits on K + → π + ( X → e + e − ) [44]. For m X ≈
17 MeV, the bound on ε n is much weakerthan the one from π decays in Eq. (35) [57, 63]. The KLOE-2 experiment searches for φ → η ( X → e + e − ) and restricts [64] | ε s | (cid:46) . × − (cid:112) Br( X → e + e − ) . (37)In principle ε s is independent and need not be related to the Be ∗ coupling. However, in thelimit of minimal flavor violation, one assumes ε d = ε s .17 . Other meson and baryon decays The WASA-at-COSY experiment also sets limits on quark couplings based on neutralpion decays. It is both weaker than the NA48/2 bound and only applicable for massesheavier than 20 MeV [65]. The HADES experiment searches for dark photons in π , η , and∆ decays and restricts the kinetic mixing parameter to ε (cid:46) × − but only for massesheavier than 20 MeV [66]. HADES is able to set bounds on gauge bosons around 17 MeV inthe π → XX → e + e − e + e − decay channel. This, however, is suppressed by ε n and is thusinsensitive to | ε n | (cid:46) − . Similar considerations suppress X contributions to other decays,such as π + → µ + ν µ e + e − , to undetectable levels. B. Electron Coupling Constraints
The X boson is required to couple to electrons to contribute to IPC events. In Eq. (31)we gave a lower limit on ε e in order for X to decay within 1 cm of its production in theAtomki apparatus. In this section we review other bounds on this coupling.
1. Beam dump experiments
Electron beam dump experiments, such SLAC E141 [67, 68], search for dark photonsbremsstrahlung from electrons that scatter off target nuclei. For m X = 17 MeV, theseexperiments restrict | ε e | to live in one of two regimes: either it is small enough to avoidproduction, or large enough that the X decay products are caught in the dump [69], leadingto | ε e | < − or | ε e | (cid:112) Br( X → e + e − ) (cid:38) × − . (38)The region | ε e | < − is excluded since the new boson would not decay inside the Atomkiapparatus. This leads to the conclusion that X must decay inside the beam dump. Lessstringent bounds come from Orsay [70] and the SLAC E137 [71] experiment. The E774experiment at Fermilab is only sensitive to m X <
10 MeV [72].
2. Magnetic moment of the electron
The upper limit on | ε e | can be mapped from dark photon searches that depend only onleptonic couplings. The strongest bound for m X = 17 MeV is set by the anomalous magneticmoment of the electron, ( g − e , which constrains the coupling of the new boson to be [63] | ε e | < . × − . (39)
3. Electron–positron annihilation into X and a photon, e + e − → Xγ A similar bound arises from the KLOE-2 experiment, which looks for e + e − → Xγ followedby X → e + e − , and finds | ε e | (cid:112) Br( X → e + e − ) < × − [73]. An analogous search at BaBar18s limited to m X >
20 MeV [74].
4. Proton fixed target experiments
The CHARM experiment at CERN also bounds X couplings through its searches for η, η (cid:48) → γ ( X → e + e − ) [75]. The production of the X boson in the CHARM experiment isgoverned by its hadronic couplings. The couplings required by the anomalous IPC events,Eq. (32), are large enough that the X boson would necessarily be produced in CHARM.Given the lower bound from decay in the Atomki spectrometer, Eq. (31), the only way toavoid the CHARM constraint for m X = 17 MeV is if the decay length is short enough thatthe X decay products do not reach the CHARM detector. The dark photon limit on ε appliesto ε e and yields | ε e | (cid:112) Br( X → e + e − ) > × − . (40)This is weaker than the analogous lower bound on | ε e | from beam dump experiments. LSNDdata imposes an even weaker constraint [76–78].
5. Charged kaon and φ decays In charged kaon decay to leptons, the X vector boson may be emitted from a chargedlepton line. Since the new vector interaction does not respect the precise gauge invarianceof the SM, the interaction of the longitudinal component of X is not constrained by acorresponding conserved current and thus can be significantly enhanced with energy [79–82].However, the most severe existing limit comes from the nonobservation of an excess inΓ( K → µ + inv . ) with respect to Γ( K → µν ) [79–81], which is not pertinent here as werequire an appreciable Γ( X → e + e − ) in order to explain the Be anomaly. W and Z decays The X boson can be produced as final state-radiation in W and Z decays into SMfermions. When the X then decays into an electron–positron pair, this gives a contribution toΓ( Z → e ) that is suppressed by O ( ε e ). For the electron couplings ε e (cid:46) − required here,the impact on the inclusive widths is negligible compared to the order per mille experimentaluncertainties on their measurement [83]. The specific decay Z → (cid:96) has been measured tolie within 10% of the SM expectation by ATLAS and CMS [84, 85] and is consistent withthe couplings of interest here.A more severe constraint arises, however, from the experimental value of the W width,because the enhancement mentioned in leptonic K decay appears in W → µνX as well [82].Limiting the contribution of W → (cid:96)νX to twice the error in the W width, after Ref. [82],yields | ε (cid:96) | < . × − (cid:16) m X
10 MeV (cid:17) , (41)to leading order in m X /m W and m (cid:96) /M W , where we have assumed lepton universality andthat (cid:96) ∈ e, µ, τ can contribute to the W width. The resulting constraint on ε e is weaker than19hat from the magnetic moment of the electron. C. Neutrino Coupling Constraints
The interaction of a light gauge boson with neutrinos is constrained in multiple ways,depending on the SM currents to which the boson couples; see Refs. [81, 86–88]. The neutrinocoupling is relevant for the Be anomaly because SU(2) L gauge invariance relates the electronand neutrino couplings. Because neutrinos are lighter than electrons, this generically opensadditional X decay channels and reduces Br( X → e + e − ). This, in turn, reduces the lowerbound on ε e in Eq. (31) and alleviates many of the experimental constraints above at thecost of introducing new constraints from X –neutrino interactions.
1. Neutrino–electron scattering
Neutrino–electron scattering stringently constrains the X boson’s leptonic couplings [49,89]. In the mass range m X ≈
17 MeV, the most stringent constraints are from the TEXONOexperiment, where ¯ ν e reactor neutrinos with average energy (cid:104) E ν (cid:105) = 1 − ν e flavor eigenstates. In theSM, ¯ ν e e → ¯ ν e e scattering is mediated by both s - and t -channel diagrams. A new neutralgauge boson that couples to both neutrinos and electrons induces an additional t -channelcontribution.Because constraints from ¯ ν e e scattering are sensitive to the interference of SM and newphysics, they depend on the signs of the new gauge couplings, unlike all of the other constraintsdiscussed above. The importance of the interference term has been highlighted in Ref. [49]in the context of a B − L gauge boson model. In that model, the neutrino and electroncouplings have the same sign, and the interference was found to be always constructive.Assuming that the experimental bound is determined by the total cross section and notthe shape of the recoil spectrum, one may use the results of Ref. [49] to determine the boundsin our more general case, where the couplings can be of opposite sign and the interferencemay be either constructive or destructive. Define the quantity g ≡ | ε e ε ν | / . Let ∆ σ be themaximal allowed deviation from the SM cross section and g ± ( g ) be the values of g thatrealize ∆ σ in the case of constructive/destructive (negligible) interference,∆ σ = g σ X (42)∆ σ = g σ int + g σ X (43)∆ σ = − g − σ int + g − σ X , (44)where g σ X is the purely X -mediated contribution to the cross section and g σ int is theabsolute value of the interference term. Solving these equations for the g ’s yields the simplerelation g − g + = g . (45)The authors of Ref. [49] found that for m X = 17 MeV, the maximal allowed B − L gaugeboson coupling, g B − L , is 2 × − and 4 × − in the cases of constructive interference andno interference, respectively. From this, including the factor of e difference between the20efinitions of g B − L and our ε ’s, we find (cid:112) | ε e ε ν | < × − for ε e ε ν > (cid:112) | ε e ε ν | < × − for ε e ε ν < . (47)The relative sign of the couplings thus has a significant effect. For a fixed value of ε e , thebound on | ε ν | is 16 times weaker for the sign that produces destructive interference than forthe sign that produces constructive interference.
2. Neutrino–nucleus scattering
In addition to its well-known motivations of providing interesting measurements of sin θ W and bounds on heavy Z (cid:48) boson [90, 91], coherent neutrino–nucleus scattering, may also provideleading constraints on light, weakly-coupled particles [92, 93]. Although ν – N scatteringhas not yet been observed, it is the target of a number of upcoming experiments that usereactors as sources. In addition, the process can also be probed using current and next-generation dark matter direct detection experiments by searching for solar neutrino scatteringevents [94]. For a B − L gauge boson, this sensitivity has been estimated in Ref. [95] forSuperCDMS, CDMSlite, and LUX, with the latter providing the most stringent constraint of g B − L (cid:46) . × − . Rescaling this result to the case of a boson with couplings ε ν e and ε p,n e to nucleons yields ε ν ε n (cid:20) ( A − Z ) + Z ε p ε n (cid:21) < A πα (cid:0) . × − (cid:1) , (48)where we approximate the LUX detector volume to be composed of a single xenon isotope.Since the NA48/2 bounds on π → Xγ imply the protophobic limit where ε p (cid:28) ε n , thesecond term on the left-hand side may be ignored. Taking A = 131 and Z = 54 then yields | ε ν ε n | / < × − or ε ν < × − (cid:18) . ε n (cid:19) . (49)This bound is weaker than the ν – e scattering bound with constructive interference andcomparable to the ν – e bound with destructive interference. As the ν – N bounds are estimatedsensitivities, we use the ν – e bounds in the discussion below. D. Summary of Constraints
Combining the required ranges of the couplings to explain the Be signal from Sec. V withthe strongest bounds from other experiments derived above, we now have the acceptableranges of couplings for a viable protophobic gauge boson to explain the Be signal. Assuming21
IG. 5. Summary of constraints and target regions for the leptonic couplings of a hypothetical X gauge boson with m X ≈
17 MeV. Updated from Ref. [7].
Br( X → e + e − ) = 1, the requirements are | ε n | = (2 − × − (50) | ε p | (cid:46) . × − (51) | ε e | = (0 . − . × − (52) (cid:112) | ε e ε ν | (cid:46) × − . (53)The nucleon couplings are fixed to reproduce the Be signal rate while avoiding the π → Xγ decays, and the quark couplings are related by ε u + 2 ε d = ε n and 2 ε u + ε d = ε p . Theelectron coupling is bounded from above by ( g − e and KLOE-2 and from below by beamdump searches, and the neutrino coupling is bounded by ν – e scattering. The allowed leptoncoupling regions are shown in Fig. 5. VII. U(1) B MODEL FOR THE PROTOPHOBIC GAUGE BOSON
In this section, we present anomaly-free extensions of the SM where the protophobic gaugeboson is a light U(1) B gauge boson that kinetically mixes with the photon. These modelshave significant virtues, which we identify in Sec. VII A. One immediate advantage is thatit does not differentiate between left- and right-handed SM fermions, and so naturally hasnon-chiral couplings. Depending on the best fit couplings discussed in Sec. V, the resultingmodels may be extremely simple, requiring only the addition of extra particles to cancel theanomalies, as discussed in Sec. VII B. A. U(1) B Gauge Boson with Kinetic Mixing
The promotion of U(1) B baryon number from a global to a local symmetry has recentlyattracted attention [96–104]. Gauged U(1) B is not anomaly-free, but these studies have22onstructed a number of models in which the gauge anomalies are cancelled with ratherminimal new matter content.Here we assume that the U(1) B symmetry is broken through a Higgs mechanism, asdiscussed below, generating a mass for the B gauge boson. As with all Abelian symmetries,the B gauge boson will generically mix kinetically with the other neutral gauge bosons ofthe SM. At energies well below the weak scale, this mixing is dominantly with the photon.The resulting Lagrangian is L = − (cid:101) F µν (cid:101) F µν − (cid:101) X µν (cid:101) X µν + (cid:15) (cid:101) F µν (cid:101) X µν + 12 m (cid:101) X (cid:101) X µ (cid:101) X µ + (cid:88) f ¯ f i /Df , (54)where (cid:101) F µν and (cid:101) X µν are the field strengths of the photon and B gauge boson, the sum runsover all fermions f , and the covariant derivative is D µ = ∂ µ + ieQ f (cid:101) A µ + ie(cid:15) B B f (cid:101) X µ . (55)Here Q f and B f are the electric charge and baryon number of fermion f , and (cid:15) B 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 = − F µν F µν − X µν X µν + 12 m X X µ X µ + (cid:88) f ¯ f i /D µ f , (56)where m X ≡ √ − (cid:15) m (cid:101) X (57)is the physical X boson mass, and (cid:101) A µ ≡ A µ + (cid:15) √ − (cid:15) X µ (cid:101) X µ ≡ √ − (cid:15) X µ (58)define the physical massless photon A and massive gauge boson X . The fermions couple tophotons with the usual charge eQ f , but they couple to the X boson with charge eε f , where ε f = ε B B f + εQ f , (59)and the script quantities are defined by ε B = (cid:15) B √ − (cid:15) ε = (cid:15) √ − (cid:15) . (60)The X charges for the SM fermions, using 1st generation notation, are ε u = 13 ε B + 23 ε (61) ε d = 13 ε B − ε (62) ε ν = 0 (63) ε e = − ε . (64)23he π constraints we have discussed above require ε and − ε B to be approximately equal towithin 10% to 50%. It is therefore convenient to define ε ≡ − ε B + δ , so ε u = − ε B + 23 δ (65) ε d = 23 ε B − δ (66) ε ν = 0 (67) ε e = ε B − δ , (68)with corresponding nucleon charges ε n = ε B and ε p = δ .This model has some nice features. For small δ , the charges are Q − B , which satisfiesthe protophobic condition. For the same reason, the neutrino’s charge is identically zero.As discussed in Sec. VI C, the constraints on neutrino charge are among the most stringent,both given ν – e and ν – N constraints, and the Be signal requirement that X decays not bedominated by the invisible decay X → ν ¯ ν . The model is highly constrained, and we seethat the electron coupling is not suppressed relative to the quark couplings. However, for ε B ≈ .
002 and δ ≈ . Be signal (provided gauge anomalies are cancelled, as discussed below). Note that itpredicts values of ε e ≈ . ε µ ≈ ε e , such couplings remove [57] at least part of the longstanding discrepancy in ( g − µ between measurements [105] and the SM prediction [106], with important implications for theupcoming Muon ( g −
2) Experiment at Fermilab [107]. They also imply promising prospectsfor future searches for the protophobic X boson at low-energy colliders, as discussed inSec. X.We treat the kinetic mixing ε as a free parameter. In a more fundamental theory, however, ε may be related to ε B . For example, if U(1) B is embedded in non-Abelian gauge group, ε vanishes above the symmetry-breaking scale, but when the non-Abelian symmetry breaks,it is generated by vacuum polarization diagrams with particles with electric charge and B quantum numbers in the loop. Parametrically, ε ∼ ( e / π ) ε B (cid:80) f Q f B f ln r f [10], where thesum is over pairs of particles in the loop, and the r f are ratios of masses of these particles.Given ∼
100 particles, one would therefore expect ε ∼ ε B in general, and the particularrelation ε ≈ − ε B , which is not renormalization group-invariant, may be viewed as providinginformation at low-energy scales about the GUT-scale particle spectrum. B. Anomaly Cancellation and Experimental Implications
Models with gauged baryon number require additional particle content to cancel anomalies.The simplest experimentally viable extension of the SM with gauged U(1) B requires addingthree vectorlike pairs of color-singlet fields [100, 103]. These fields and their quantumnumbers are listed in Table I. The new fields carry baryon charges that satisfy the anomalycancellation condition B − B = 3. The χ field is naturally a dark matter candidate [103, 109],and it has to be the lightest of the new fields to avoid stable charged matter.The U(1) B symmetry is broken by the vacuum expectation value (vev) (cid:104) S B (cid:105) = v X / √ B = 3 to allow for vectorlike mass A model unifying gauged baryon number and color into a non-Abelian SU(4) has been constructed and,after symmetry breaking, yields the same new particle content as the U(1) B model discussed here [108]. χ field the lightest one. The new Yukawa terms in the Lagrangian are L Y = − y Ψ L h SM η R − y Ψ L (cid:101) h SM χ R − y Ψ R h SM η L − y Ψ R (cid:101) h SM χ L − λ Ψ S B Ψ L Ψ R − λ η S B η R η L − λ χ S B χ R χ L + h . c . (69)In Refs. [100, 103] U(1) B is assumed to be broken at the TeV scale. However, to have a lightU(1) B gauge boson and a gauge coupling consistent with the Be signal, the vev of the newHiggs boson cannot be so large. Defining its vacuum expectation value by (cid:104) S B (cid:105) = v X / √ X gauge boson corresponding to the broken U(1) B is given by m X = 3 e | ε B | v X , (70)implying v X ≈
10 GeV 0 . | ε B | . (71)As a result, the new particles cannot have large vectorlike masses from the λ i couplings inEq. (69), but must rather have large chiral couplings from the y i terms of Eq. (69).The experimental constraints on the extra matter content of this model come from severalsources: • First, the new particles may be produced through Drell-Yan production at the LHC.However, for Yukawa couplings y i ∼
3, close to the perturbative limit, the masses ofthe new states are ∼
500 GeV and beyond current LHC sensitivity. • Second, electroweak precision measurements constrain the properties of the newparticles. The two electroweak doublets give an irreducible contribution to the S parameter of ∆ S ≈ / (6 π ) (cid:39) .
11 [110]. In the degenerate mass limit, they donot contribute to the T and U parameters. However, the fit to electroweak preci-sion data may be improved with a slight splitting of ∆ m ∼
50 GeV, which gives∆ T ≈ / (3 π sin θ W )(∆ m/m Z ) ≈ .
09. This combination of ∆ S and ∆ T fits wellwithin the 90% CL region (see, for example, Fig. 10.6 of Ref. [111]). • Third, the new particles may affect the h SM → γγ decay rate. Since these particlesessentially form two additional families of leptons, the rate for Higgs decaying to twophotons decreases by ∼
20% compared to the SM prediction [112], but this is stillwithin the experimentally-allowed region [113].
TABLE I. New particle content of the simplest anomaly-free U(1) B model. Field Isospin I Hypercharge
Y BS B L − B Ψ R − B η R − B η L − B χ R B χ L B
25n summary, a simple model with a U(1) B gauge boson that kinetically mixes with thephoton is a viable candidate for the protophobic gauge boson. The gauge anomalies must becancelled by introducing additional particles, and we have discussed the simplest realizationof this field content that simultaneously explains the Be anomaly.
VIII. U(1) B − L MODEL FOR THE PROTOPHOBIC GAUGE BOSON
In this section, we present another anomaly-free extension of the SM where the protophobicgauge boson is a light U(1) B − L gauge boson that kinetically mixes with the photon. Thesemodels have significant virtues, which we identify in Sec. VIII A. They also generically haveneutrino couplings that are too large, and we explore a mechanism for suppressing theneutrino couplings in Sec. VIII B. The resulting models may be extremely simple, requiringonly the addition of one generation of vectorlike leptons which is light and may already beprobed at the LHC. The implications for colliders and cosmology are discussed in Sec. VIII C. A. U(1) B − L Gauge Boson with Kinetic Mixing
The possibility of gauged U(1) B − L has been studied for many decades [46–48, 114]. Thepromotion of U(1) B − L from a global to a local symmetry is well-motivated among Abeliansymmetries by its appearance in grand unified theories, and the fact that it is anomaly-freeonce one adds to the SM three right-handed (sterile) neutrinos, which are already stronglymotivated by the existence of neutrino masses.As in the U(1) B case, we assume that the B − L symmetry is broken through a Higgsmechanism, generating a mass for the B − L gauge boson, and that it kinetically mixes withthe photon. The resulting X -charges for the SM fermions, using 1st generation notation, are ε u = 13 ε B − L + 23 ε (72) ε d = 13 ε B − L − ε (73) ε ν = − ε B − L (74) ε e = − ε B − L − ε , (75)or, defining ε ≡ − ε B − L + δ as above, ε u = − ε B − L + 23 δ (76) ε d = 23 ε B − L − δ (77) ε ν = − ε B − L (78) ε e = − δ . (79)The corresponding nucleon charges are ε n = ε B − L and ε p = δ .The charges of the kinetically mixed B − L gauge boson have nice features for explainingthe Be anomaly. For δ ≈
0, the charges are Q − ( B − L ), which satisfies the basic requirementsof a protophobic solution to the Be anomaly: namely, the X boson couples to neutrons, butits couplings to both protons and electrons are suppressed. More quantitatively, by choosing26he two parameters | ε B − L | ≈ . − .
008 and | δ | (cid:46) . Be signal and are sufficiently protophobic to satisfy the π constraints. This isno great achievement: by picking two free parameters, two conditions can be satisfied. Butwhat is non-trivial is that with this choice, the electron coupling satisfies the upper bound | ε e | (cid:46) . × − , which is required by the completely independent set of experiments thatconstrain lepton couplings.Unfortunately, in contrast to the U(1) B case, the neutrino coupling does not vanish. Inthese models, we see that ε ν = − ε n while the constraints discussed above require the neutrinocoupling to be significantly below the neutron coupling. In the next section, we present amechanism to neutralize the X -charge of SM active neutrinos to satisfy these bounds. B. Neutrino Neutralization with Vectorlike Leptons
The B − L gauge boson with kinetic mixing predicts | ε ν | = | ε n | ∼ . − . ε e , the bounds from ν − e scattering require | ε ν | to be reduced by afactor of ∼ X -charge of the active neutrinosby supplementing the SM with vectorlike leptons with opposite B − L quantum numbers.The B − L symmetry is broken by a Higgs mechanism, generating a vacuum expectationvalue for the new SM-singlet Higgs field h X . This symmetry breaking simultaneously (1)generates the 17 MeV mass for the X boson, (2) generates a Majorana mass for the SMsterile neutrinos, which would otherwise be forbidden by B − L symmetry, and (3) mixes theSM active neutrinos with the new lepton states such that the resulting mass eigenstates havesuppressed X -charge.The fields of these models include the SM Higgs boson h SM , and the SM lepton fields (cid:96) L , e R , and ν R , where the last is the sterile neutrino required by B − L anomaly cancellation. Tothese, we add the Higgs field h X with B − L = 2, and N vectorlike lepton isodoublets L i L,R and charged isosinglets E i L,R , with B − L = 1. The addition of vectorlike pairs preservesanomaly cancellation. These fields and their quantum numbers are shown in Table II. Wefocus here on the first generation leptons; the mechanism may be straightforwardly extendedto the second and third generations.With these fields, the full set of gauge-invariant, renormalizable Lagrangian terms thatdetermine the lepton masses are L = L SM + L mix + L new (80) L SM = ( − y e h SM ¯ (cid:96) L e R + y ν (cid:101) h SM ¯ (cid:96) L ν R + h.c.) − y N h X ¯ ν cR ν R (81) L mix = − λ iL h X ¯ (cid:96) L L i R − λ iE h X ¯ E i L e R + h.c. (82) L new = − M ijL ¯ L i L L j R − M ijE ¯ E i L E j R − h ij h SM ¯ L i L E j R + k ij (cid:101) h SM ¯ E i L L j R + h.c. , (83)where i, j = 4 , . . . , N + 3. L SM generates the Dirac and Majorana SM neutrino masses, L mix includes the terms that mix the SM and vectorlike fields, and L new contains the vectorlikemasses and Yukawa couplings for the new vectorlike leptons. For simplicity, we will assumeuniversal masses and Yukawa couplings, so λ iL = λ L , λ iE = λ E , M ijL = M L δ ij , M ijE = M E δ ij , h ij = hδ ij , k ij = kδ ij .When electroweak symmetry and B − L symmetry are broken, the Higgs fields get vevs (cid:104) h SM (cid:105) = v/ √
2, where v (cid:39)
246 GeV, and (cid:104) h X (cid:105) = v X / √
2. This gives the X boson a mass m X = 2 e | ε B − L | v X , (84)27 ABLE II. Fields and their quantum numbers in the B − L model with kinetic mixing and neutrinosneutralized by mixing with vectorlike leptons. The SM fields, including the sterile neutrino, are listedabove the line. The new fields, including N generations of vectorlike fields, with i = 4 , . . . , N + 3,are listed below the line. Field Isospin I Hypercharge
Y B − Lh SM 12 12 (cid:96) L = (cid:18) ν L e L (cid:19) − − e R − − ν R − h X L i L = (cid:18) ν i L e i L (cid:19) − L i R = (cid:18) ν i R e i R (cid:19) − E i L − E i R − v X to be v X = 14 GeV 0 . | ε B − L | . (85)It also generates Dirac and Majorana masses for the SM neutrinos, m D = y ν v/ √ m M = y N v X / √
2, and masses M LI = λ L v X / √ M EI = λ E v X / √ ψ ν M Mν ψ ν , where M Mν = m D M LI · · · M LI m D m M · · · M L · · · M LI M L · · · · · · M L M LI · · · M L , (86)and ψ ν = ( ν L , ν R , ν L , ν R , . . . , ν N +3 L , ν N +3 R ), or alternatively, neglecting the small SM Diracand Majorana masses, the remaining neutrino masses may be written ¯ ψ νL M ν ψ νR + h.c., where M ν = M LI · · · M LI M L · · · · · · M L , (87)28nd ψ νL,R = ( ν L,R , ν L,R , . . . , ν N +3 L,R ). Similarly, the charged lepton masses are ¯ ψ eL M e ψ eR +h.c.,where M e = M LI · · · M LI M L hv √ · · · M EI kv √ M E · · · · · · M L hv √ M EI · · · kv √ M E , (88)and ψ eL,R = ( e L,R , e L,R , . . . , e N +3 L,R ).Diagonalizing the neutrino mass matrix of Eq. (86) yields N Dirac neutrino states withmass ∼ M L , and two light states: the SM sterile neutrino and the SM active neutrino, whichis the eigenstate 1 (cid:112) M L + N M L I ( − M L , , M LI , , M LI , . . . , , M LI , . (89)The active neutrino’s X -charge is therefore modified by the mixing with the vectorlike leptonstates, with similar effects for the charged leptons. In the end, we find that the lepton X -charges are modified to ε ν L = − ε B − L cos 2 θ ν L (90) ε e L = − ε B − L cos 2 θ e L − ε = ε B − L (1 − cos 2 θ e L ) − δ (91) ε e R = − ε B − L cos 2 θ e R − ε = ε B − L (1 − cos 2 θ e R ) − δ , (92)where tan θ ν L = N M L I M L , (93)and θ e L and θ e R are determined by similar, but more complicated, relations derived bydiagonalizing M e . To neutralize the neutrino charge, we needtan θ ν L = N M L I M L = N λ L m X M L e ε B − L ≈ (cid:20)
130 GeV M L (cid:21) (cid:20) . ε B − L (cid:21) (cid:20) √ N λ L π (cid:21) ≈ , (94)where we have normalized the effective coupling √ N λ L to its ultimate perturbative limit.We see that the neutrino X -charge may be neutralized with as few as N = 1 vectorlike leptongeneration with mass at the weak scale. A larger number of heavier vectorlike leptons mayalso neutralize the neutrino X -charge. In addition, to preserve non-chiral electron couplings,we require θ e L ≈ θ e R . C. Implications for Colliders and Cosmology
Here we consider the implications of these models for colliders and cosmology, beginningwith the extremely simple case of N = 1 generation of vectorlike leptons and vanishing Yukawacouplings h = k = 0. In this case, the mass matrices are easily diagonalized. The heavystates include three “4th generation” Dirac fermions: the isodoublet neutrino and electron29ith masses m ν (cid:39) m e (cid:39) √ M L and the isosinglet electron with mass m E (cid:39) √ M E . Thestates ν and e have vectorlike masses and are nearly degenerate, and so do not contributeto the S and T parameters [110]. The light states are the usual massless SM leptons, butmixed with opposite X -charged states, with mixing angles tan θ ν L = tan θ e L = ( M LI /M L ) and tan θ e R = ( M EI /M E ) . These SM fields each mix only with new leptons with thesame SM quantum numbers, and so these mixing angles are not constrained by precisionmeasurements. Choosing M LI /M L = M EI /M E = 1, we find ε ν L = 0 and ε e L = ε e R = ε B − L − δ .For ε B − L ≈ .
002 and δ ≈ . Be signal consistent with all current constraints. As in the U(1) B case, assuming ε µ ≈ ε e removes at least part of the ( g − µ puzzle and implies promisingprospects for future searches at low-energy colliders, as discussed in Sec. X.The new vectorlike leptons can be produced through Drell-Yan production at hadron and e + e − colliders, and so this model may be explored at the LHC and future colliders. Theprospects for vectorlike lepton searches at the LHC have been studied in detail in the casethat they decay to W ν (cid:96) , Z(cid:96) , and h(cid:96) [115–117]. In the present case, however, the vectorlikelepton masses and decays are constrained by the neutrino neutralization mechanism. Inparticular, the mixing terms of Eq. (82) that neutralize the neutrinos imply that the decays ν → ν e h X , e → eh X , and E → eh X are almost certainly dominant.The B − L Higgs boson has a variety of possible decays, but for a moderately largeMajorana Yukawa coupling y N , the invisible decay h X → ν R ¯ ν R dominates. The resultingprocesses are therefore pp → E +4 E − → e + e − h X h X → e + e − ν R ¯ ν R ν R ¯ ν R (95) pp → e +4 e − → e + e − h X h X → e + e − ν R ¯ ν R ν R ¯ ν R (96) pp → ν ¯ ν → ν L ¯ ν L h X h X → ν L ¯ ν L ν R ¯ ν R ν R ¯ ν R (97) pp → ν e → νeh X h X → eν L ν R ¯ ν R ν R ¯ ν R . (98)These signals are therefore very similar to those of selectron pair production and selectron–sneutrino pair production, leading to signatures with missing transverse energy /E T , e + e − + /E T and e ± + /E T . The amount of missing energy is controlled by m h X = (cid:112) λ H v X = 70 GeV (cid:114) λ H π . | ε B − L | , (99)where λ H is the Higgs boson quartic coupling appearing in the Lagrangian term λ H ( h X h ∗ X ) ,and we have used Eq. (85).Current bounds from the combination of LEP2 and 8 TeV LHC data on the combinedproduction of right- and left-handed selectron and smuons with mass 100 GeV allow neutralinomasses of around 50 GeV [118, 119]. The vectorlike lepton cross section is bigger by roughlya factor of 4, but 100 GeV vectorlike leptons decaying to 50 −
70 GeV B − L Higgs bosonsmay still be allowed. Existing mono-lepton searches based on 8 TeV LHC data are notoptimized for lepton masses as low as 100 GeV and are unlikely to have sensitivity [120, 121].Nonetheless, it may be that future searches based on 13 TeV data will become sensitive,particularly if they can be optimized for lower mass vectorlike leptons. It would be interestingto investigate this scenario in more detail, as well as scenarios where other h X decays arecomparable or dominant to the invisible decay assumed above. It is also worth noting thatthe appearance of relatively strong couplings ( λ H , λ L ) in the h X sector may be an indication30f compositeness, which could result in a richer and more complicated set of final statesaccessible to LHC energies.We now turn to the SM neutrino sector and potential cosmological signatures. As notedabove, when the h X field with B − L charge 2 gets a vev, it also generates a Majorana massfor the SM singlet neutrinos. This is an important feature. Without a charge 2 Higgs boson,the SM neutrinos are Dirac particles. Light Dirac neutrinos are not typically problematic, asthe ν R component does not thermalize and does not contribute to the number of relativisticdegrees of freedom n eff . In the current model, however, the process f ¯ f ↔ X ↔ ¯ ν R ν R effectively thermalizes the ν R at temperatures T ∼ m X , where the process is on-resonance.To avoid thermalization, one needs the X -charge of ν R to be less than 10 − [114] or very lowreheat temperatures in the window between 1 MeV and m X ≈
17 MeV. The generation of aMajorana mass avoids these problems.The Majorana mass is m M = y N v X / √ y N m X √ e | ε B − L | (cid:46)
30 GeV 0 . | ε B − L | , (100)where the upper bound assumes y N ∼
3. The physical masses of the SM active neutrinosare then determined by the see-saw mechanism, with Dirac masses chosen appropriately. Ofcourse, the sterile neutrino masses need not be near their upper limit, and it is tempting topostulate that they may be in the keV range as required for warm dark matter. To preventthe decays X → ν R ν R from significantly diluting the Be signal in this case, the ν R X -chargesmust also be neutralized, for example, through mixing with vectorlike isosinglet neutrinos.Alternatively, the sterile neutrino masses may be in the 10 – 100 MeV range, as may behelpful for reducing the standard BBN predictions for the Li abundance to the observedlevels [122]. We leave these astrophysical and cosmological implications for future work.One might worry that having a model with an exact U(1) B − L or U(1) B gauge symmetrydown to the GeV or MeV energy scale would prevent any baryon number asymmetry frombeing generated. This, however, is not the case, as was discussed, for example, in Ref. [103]for a model with gauged U(1) B . A lepton number asymmetry can still be produced at a highscale and then be partially converted into baryon number through the electroweak sphalerons.For the case of gauged U(1) B − L one could also invoke a Dirac leptogenesis scenario whichrelies on the fact that the right-handed neutrinos decouple early on during the evolutionof the Universe, trapping some amount of lepton number [123, 124]. The resulting leptonnumber deficit in the visible sector is then again transferred to baryon number through thesphalerons.We have introduced additional fermionic matter to render the models compatible withexperimental constraints. The step of adding extra matter may not be necessary, and it maybe possible to satisfy all the existing experimental constraints by considering a combinationof gauged U(1) quantum numbers. The possibility of multiple, new U(1) gauge bosons hasbeen explored previously, in the two dark-photon (“paraphoton”) case [125] and for threeAbelian groups [126]. Here we note that if one were to combine a U(1) B − L model withkinetic mixing with a second, unbroken (or softly broken) gauge symmetry, e.g., L e − L τ , itis possible to bring the first-generation fermion couplings of the B − L gauge boson to theform of the U(1) B model. Such relationships are completely compatible with the couplingsneeded to describe the Be anomaly and satisfy other constraints. However, equivalenceprinciple constraints on new, massless gauge bosons that can couple to the constitutentsof ordinary matter are severe [114, 127, 128]. We note that we can address this problem31
IG. 6. The Be signal region, along with current constraints (gray) and projected sensitivitiesof future experiments in the ( m X , ε e ) plane. Updated from Ref. [7]. Note Br( X → e + e − ) = 1 isassumed. by making the massless gauge boson’s couplings to electrons vanish at tree level. Furtherinvestigation is required to check that this suffices to render the model compatible withexperimental constraints on new, (nearly) massless gauge bosons. IX. FUTURE EXPERIMENTS
Current and near future experiments will probe the parameter space of interest for theprotophobic gauge boson X . The projected sensitivities of various experiments are shown inFig. 6 and we briefly discuss them below. Other Large Energy Nuclear Transitions.
The Be ∗ and Be ∗(cid:48) states are quite special inthat they decay electromagnetically to discrete final states with an energy release in excessof 17 MeV. Other large-energy gamma transitions have been observed [129], such as the19.3 MeV transition in B to its ground state [130] and the 17.79 MeV transition in Beto its ground state [131]. Of course, what is required is large production cross sectionsand branching fractions so that many IPC events can be observed. It would certainly beinteresting to identify other large energy nuclear transitions with these properties to test thenew particle interpretation of the Be anomaly.
LHCb.
A search for dark photons A (cid:48) at LHCb experiment during Run 3 (scheduled for theyears 2021 – 2023) has been proposed [132] using the charm meson decay D ∗ (2007) → D A (cid:48) with subsequent A (cid:48) → e + e − . It takes advantage of the LHCb excellent vertex and invariantmass resolution. For dark photon masses below about 100 MeV, the experiment can explorenearly all of the remaining parameter space in ε e between the existing prompt- A (cid:48) and beam-dump limits. In particular, it can probe the entire region relevant for the X gauge bosonexplaining the Be anomaly.
Mu3e.
The Mu3e experiment will look at the muon decay channel µ + → e + ν e ¯ ν µ ( A (cid:48) → e + e − ) and will be sensitive to dark photon masses in the range 10 MeV (cid:46) m A (cid:48) (cid:46)
80 MeV [133].32he first phase (2015 – 2016) will probe the region ε e (cid:38) × − , while phase II (2018 andbeyond) will extend this reach almost down to ε e ∼ − , which will include the whole regionof interest for the protophobic gauge boson X . VEPP-3.
A proposal for a new gauge boson search at the VEPP-3 facility was made [134].The experiment will consist of a positron beam incident on a gas hydrogen target and willlook for missing mass spectra in e + e − → A (cid:48) γ . The search will be independent of the A (cid:48) decaymodes and lifetime. Its region of sensitivity in ε e extends down into the beam dump bounds,i.e., below ε e ∼ × − , and includes the entire region relevant for X . Once accepted, theexperiment will take 3 – 4 years. KLOE-2.
As mentioned above, the KLOE-2 experiment, looking for e + e − → γ ( X → e + e − ),is running and improving its current bound of | ε e | < × − [73] for m X ≈
17 MeV. Withthe increased DA φ NE-2 delivered luminosity and the new detectors, KLOE-2 is expected toimprove this limit by a factor of two within two years [135].
MESA.
The MESA experiment will use an electron beam incident on a gaseous target toproduce dark photons of masses between ∼ −
40 MeV with electron coupling as low as ε e ∼ × − , which would probe most of the available X boson parameter space [136]. Thecommissioning is scheduled for 2020. DarkLight.
The DarkLight experiment, similarly to VEPP-3 and MESA, will use electronsscattering off a gas hydrogen target to produce on-shell dark photons, which later decayto e + e − pairs [137]. It is sensitive to masses in the range 10 −
100 MeV and ε e down to4 × − , covering the majority of the allowed protophobic X parameter space. Phase I ofthe experiment is expected to take data in the next 18 months, whereas phase II could runwithin two years after phase I. HPS.
The Heavy Photon Search experiment is using a high-luminosity electron beamincident on a tungsten target to produce dark photons and search for both A (cid:48) → e + e − and A (cid:48) → µ + µ − decays [138]. Its region of sensitivity is split into two disconnected pieces (seeFig. 6) based on the analyses used: the upper region is probed solely by a bump hunt search,whereas the lower region also includes a displaced vertex search. HPS is expected to completeits dataset by 2020. PADME.
The PADME experiment will look for new light gauge bosons resonantly producedin collisions of a positron beam with a diamond target, mainly through the process e + e − → Xγ [139]. The collaboration aims to complete the detector assembly by the end of 2017 andaccumulate 10 positrons on target by the end of 2018. The expected sensitivity after oneyear of running is ε e ∼ − , with plans to get as low as 10 − [140, 141]. BES III.
Current and future e + e − colliders, may also search for e + e − → Xγ . A recentstudy has explored the possibility of using BES III and BaBar to probe the 17 MeVprotophobic gauge boson [13]. E36 at J-PARC (TREK).
The TREK experiment has the capacity to study K → µνe + e − decays [142]; the enhancement associated with the interaction of the longitudinal componentof X with charged fermions should make for sensitive tests of ε e in the mass range of interestto the Be anomaly [143].
X. CONCLUSIONS
The 6.8 σ anomaly in Be cannot be plausibly explained as a statistical fluctuation, and thefit to a new particle interpretation has a χ /dof of 1.07. If the observed bump has a nuclearphysics or experimental explanation, the near-perfect fit of the θ and m ee distributions to33he new particle interpretation is a remarkable coincidence. Clearly all possible explanationsshould be pursued. Building on our previous work [7], in this study, we presented particlephysics models that extend the SM to include a protophobic gauge boson that explains the Be observations and is consistent with all other experimental constraints.To understand what particle properties are required to explain the Be anomaly, we firstpresented effective operators for various spin-parity assignments. Many common examplesof light, weakly coupled particles, including dark photons, dark Higgs bosons, axions, and B − L gauge bosons (without kinetic mixing) are disfavored or excluded on general grounds.In contrast, general gauge bosons emerge as viable candidates.In Ref. [7] we determined the required couplings of a vector gauge boson to explain the Be anomaly assuming isospin conservation, and found that the particle must be protophobic.In this work, we refined this analysis to include the possibility of isospin mixing in the Be ∗ and Be ∗(cid:48) states. Although isospin mixing and violation can yield drastically differentresults, these effects are relatively mild once one focuses on protophobic gauge bosons. Itwould be helpful to have a better understanding of the role of isospin breaking in thesesystems and a quantitative estimate of their uncertainties. The presence of isospin mixingalso implies that the absence of an anomaly in Be ∗(cid:48) decays must almost certainly be due tokinematic suppression and that the X particle’s mass is above 16.7 MeV. Combining all ofthese observations with constraints from other experiments, we then determined the favoredcouplings for any viable vector boson explanation.We have presented two anomaly-free extensions of the SM that resolve the Be anomaly.In the first, the protophobic gauge boson is a U(1) B gauge boson that kinetically mixeswith the photon. For gauge couplings and kinetic mixing parameters that are comparable insize and opposite in sign, the gauge boson couples to SM fermions with approximate charge Q − B , satisfying the protophobic requirement. Additional matter content is required tocancel gauge anomalies, and we presented a minimal set of fields that satisfy this requirement.In the second model, the gauge boson is a U(1) B − L gauge boson with kinetic mixing, and theSM fermion charges are Q − ( B − L ). Additional vectorlike leptons are needed to neutralizethe neutrino if we consider only a single U(1) gauge group. Both models can simultaneouslyresolve the ( g − µ anomaly, have large electron couplings that can be probed at many nearfuture experiments, and include new vectorlike lepton states at the weak scale that can bediscovered by the LHC.One may speculate that the protophobic gauge boson may simultaneously resolve not onlythe Be and ( g − µ anomalies, but also others. Possibilities include the NuTeV anomaly [14]and the cosmological lithium problem mentioned in Sec. VIII C. Another possibility is the π → e + e − KTeV anomaly, which may be explained by a spin-1 particle with axial couplingsthat satisfy (cid:0) g uA − g dA (cid:1) g eA (cid:18)
20 MeV m X (cid:19) ≈ . × − , (101)which is roughly consistent with the vector couplings we found for a protophobic gaugeboson [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 suppressed contributions to neutral pion decays relax many constraints thatexisted for vector bosons. We note, however, that some bounds become stronger for the axialcase. For example, the decay φ → η ( X → e + e − ) used in deriving the KLOE constraints [64]is an s -wave in the axial case, implying a stronger bound than the p -wave–suppressed one in34he vector case. Another example is ( g − e [146], for which an axial vector makes largercontributions than a vector, for couplings of the same magnitude. In addition, there are verystringent bounds, for example, from atomic parity violation, on gauge bosons with mixedvector and axial vector couplings [147].Finally, if the Be anomaly is pointing toward a new gauge boson and force, it is natural toconsider whether this force may be unified with the others, with or without supersymmetry.In the case of U(1) B − L , which is a factor of many well-motivated grand unified groups, itis tempting to see whether the immediately obvious problems—for example, the hierarchybetween the required U(1) B − L gauge coupling and those of the SM—can be overcome, andwhether MeV-scale data may be telling us something interesting about energy scales nearthe Planck scale. ACKNOWLEDGMENTS
We thank Phil Barbeau, John Beacom, Roy Holt, Yoni Kahn, Attila J. Krasznahorkay,Saori Pastore, Tilman Plehn, Mauro Raggi, Alan Robinson, Martin Savage, Paolo Valente,and Robert Wiringa for helpful correspondence. J.L.F., B.F., I.G., J.S., T.M.P.T., and P.T.are supported in part by NSF Grants PHY-1316792 and PHY-1620638. The work of S.G. issupported in part by the DOE Office of Nuclear Physics under contract DE-FG02-96ER40989.The work of J.L.F. is supported in part by a Guggenheim Foundation grant and in part bySimons Investigator Award [1] S. B. Treiman and F. Wilczek, “Axion Emission in Decay of Excited Nuclear States,”
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