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Dark Interpretation of the Decay Anomaly

Bartosz Fornal and Benjam´ın Grinstein Department of Physics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA (Dated: May 10, 2018) There is a long-standing discrepancy between the neutron lifetime measured in beam and bottle experiments. We propose to explain this anomaly by a dark decay channel for the neutron, involving one or more dark sector particles in the final state. If any of these particles are stable, they can be the dark matter. We construct representative models consistent with all experimental constraints.

INTRODUCTION In this paper we focus on the latter possibility. We as- sume that the discrepancy between the neutron lifetime mea- The neutron is one of the fundamental building blocks of surements arises from an incomplete theoretical description matter. Along with the and it makes up most of neutron decay and we investigate how the of the visible . Without it, complex atomic nuclei sim- (SM) can be extended to account for the anomaly. ply would not have formed. Although the neutron was discov- ered over eighty years ago [1] and has been studied intensively thereafter, its precise lifetime is still an open question [2, 3]. NEUTRON DARK DECAY The dominant neutron decay mode is β decay − Since in beam experiments neutron decay is observed by n → p + e +ν ¯e , detecting decay , the lifetime they measure is related described by the matrix element to the actual neutron lifetime by 1 µ M = √ GF Vud gV [p ¯ γµn − λ p¯ γ5γµn ] [e ¯ γ (1 − γ5)ν ] . 2 beam τn τn = . (1) The theoretical estimate for the neutron lifetime is [4–7] Br(n → p + anything) 4908.7(1.9) s In the SM the branching fraction (Br), dominated by β decay, τn = 2 2 . |Vud| (1 + 3 λ ) is 100% and the two lifetimes are the same. The neutron decay rate obtained from bottle experiments is The (PDG) world average for the axial- −28 vector to vector coupling ratio is λ = −1.2723 ± 0.0023 [8]. Γn ' 7.5 × 10 GeV. Adopting the PDG average |Vud| = 0.97417 ± 0.00021 gives τn between 875.3 s and 891.2 s within 3 σ. The discrepancy ∆τn ' 8.4 s between the values measured in There are two qualitatively different types of direct neutron bottle and beam experiments corresponds to [18] lifetime measurements: bottle and beam experiments. exp bottle beam −30 In the first method, ultracold are stored in a con- ∆Γn = Γn − Γn ' 7.1 × 10 GeV . tainer for a time comparable to the neutron lifetime. The re- maining neutrons that did not decay are counted and fit to a We propose that this difference be explained by the exis- tence of a dark decay channel for the neutron, which makes decaying exponential, exp(−t/τn). The average from the five bottle experiments included in the PDG [8] world average is Br(n → p + anything) ≈ 99% . [9–13] τ bottle = 879.6 ± 0.6 s . n There are two qualitatively different scenarios for the new arXiv:1801.01124v3 [hep-ph] 9 May 2018 Recent measurements using trapping techniques [14, 15] yield dark decay channel, depending on whether the final state con- a neutron lifetime within 2.0 σ of this average. sists entirely of dark particles or contains visible ones: In the beam method, both the number of neutrons N in a beam and the protons resulting from β decays are counted, (a) n → invisible + visible , and the lifetime is obtained from the decay rate, dN/dt = (b) n → invisible . −N/τn. This yields a considerably longer neutron lifetime; the average from the two beam experiments included in the Here the label “invisible” includes dark sector particles, as PDG average [16, 17] is well as . Such decays are described by an effective beam operator O = Xn, where n is the neutron and is a spin τn = 888.0 ± 2.0 s . 1/2 operator, possibly composite, e.g. X = χ1χ2...χk, with The discrepancy between the two results is 4.0 σ. This sug- the χ’s being fermions and combining into spin 1/2. gests that either one of the measurement methods suffers from From an experimental point of view, channel (a) offers a de- an uncontrolled systematic error, or there is a theoretical rea- tection possibility, whereas channel (b) relies on higher order son why the two methods give different results. radiative processes. We provide examples of both below. 2

Proton decay constraints One can also have a scalar DM particle φ with mass mφ < The operator O generally gives rise to via 938.783 MeV and χ being a Dirac right-handed .

∗ + Trivial model-building variations are implicit. The scenarios p → n + e + νe , with a Majorana fermion χ or a real scalar φ are additionally followed by the decay of n∗ through the channel (a) or (b) constrained by neutron-antineutron oscillation and dinucleon and has to be suppressed [19]. Proton decay can be elimi- decay searches [23, 24]. nated from the theory if the sum of masses of particles in the minimal final state f of neutron decay, say Mf , is larger than MODEL-INDEPENDENT ANALYSIS mp − me. On the other hand, for the neutron to decay, Mf must be smaller than the neutron mass, therefore it is required that Based on the discussed experimental constraints, the avail- mp − me < Mf < mn . able channels for the neutron dark decay are: n → χ γ , n → χ φ , n → χ e+e− , Nuclear physics bounds In general, the decay channels (a) and (b) could trigger nu- as well as those involving additional dark particle(s) and/or clear transitions from (Z,A) to (Z,A−1). If such a transition (s). is accompanied by a prompt emission of a state f 0 with the 0 sum of masses of particles making up f equal to Mf 0 , it can Neutron → dark matter + photon be eliminated from the theory by imposing Mf 0 > ∆M = M(Z,A) − M(Z,A − 1) . Of course Mf 0 need not be the This decay is realized in the case of a two-particle interac- 0 same as Mf , since the final state f in nuclear decay may tion involving the fermion DM χ and a three-particle interac- not be available in neutron decay. For example, Mf 0 < Mf tion including χ and a photon, i.e., χ n , χ n γ. Equations (2) 0 when the state f consists of a single particle, which is not and (3) imply that the DM mass is an allowed final state of the neutron decay. If f 0 = f then f 0 must contain at least two particles. The requirement becomes, 937.900 MeV < mχ < 938.783 MeV therefore,  and the final state photon energy ∆M < min Mf 0 ≤ Mf < mn . 0.782 MeV < E < 1.664 MeV . (4) The most stringent of such nuclear decay constraints comes γ from the requirement of 9Be stability, for which ∆M = We are not aware of any experimental constraints on such 937.900 MeV, thus monochromatic . The search described in [25–27]  measured photons from radiative β decays in a neutron beam, 937.900 MeV < min M 0 ≤ M < 939.565 MeV . (2) f f however, photons were recorded only if they appeared in co- The condition in Eq. (2) circumvents all nuclear decay limits incidence with a proton and an electron, which is not the case listed in PDG [8], including the most severe ones [20–22]. in our proposal. To describe the decay n → χ γ in a quantitative way, we Dark matter consider theories with an interaction χ n, and an interaction Consider f to be a two-particle final state containing a dark χ n γ mediated by a mixing between the neutron and χ. An sector spin 1/2 particle χ. Assuming the presence of the in- example of such a theory is given by the effective Lagrangian teraction χ n, the condition in Eq. (2) implies that the other eff gne µν  L =n ¯ i∂/ − mn + σ Fµν n particle in f has to be a photon or a dark sector particle φ with 1 2mn mass mφ < 1.665 MeV (we take it to be spinless). The decay +χ ¯ i∂/ − m  χ + ε (¯nχ +χn ¯ ) , (5) − χ χ → p + e +ν ¯e is forbidden if where gn '−3.826 is the neutron g-factor and ε is the mixing mχ < mp + me = 938.783 MeV . (3) parameter with dimension of mass. The term corresponding Provided there are no other decay channels for χ, Eq. (3) en- to n → χ γ is obtained by transforming Eq. (5) to the mass sures that χ is stable, thus making it a DM candidate. On the eigenstate basis and, for ε  mn − mχ, yields − other hand, if χ → p + e +ν ¯e is allowed, although this pre- eff gne ε µν χ Ln→χγ = χ¯ σ Fµν n . (6) vents from being the DM, its lifetime is still long enough to 2mn (mn − mχ) explain the neutron decay anomaly. In both scenarios φ can be a DM particle as well. Therefore, the neutron dark decay rate is Without the interaction χ n, only the sum of final state g2 e2  m2 3 m ε2 masses is constrained by Eq. (2). Both χ and φ can be DM n χ n ∆Γn→χγ = 1 − 2 2 candidates, provided 8π mn (mn − mχ) 2 exp 1+x 3 1−x  ε [GeV]  |mχ − mφ| < mp + me . ≈ ∆Γn 2 1.8×10−3 9.3×10−14 , (7) 3 where x = mχ/mn. The rate is maximized when mχ satu- where rates the lower bound mχ = 937.9 MeV. A particle physics f(x, y) = [(1 − x)2 − y2] [(1 + x)2 − y2]3 realization of this case is provided by model 1 below. The testable prediction of this class of models is a with x = mχ/mn and y = mφ/mn. A particle physics real- monochromatic photon with an energy in the range specified ization of this scenario is provided by model 2 below. by Eq. (4) and a branching fraction For mχ˜ > mn the missing energy signature has a branching fraction ≈ 1%. There will also be a very suppressed radiative ∆Γn→χγ ≈ 1% . process involving a photon in the final state with a branching Γ n fraction 3.5 × 10−10 or smaller. + − A signature involving an e e pair with total energy As discussed earlier, in the case 937.9 MeV < mχ˜ < mn Ee+e− < 1.665 MeV is also expected, but with a suppressed both the visible and invisible neutron dark decay channels are branching fraction of at most 1.1 × 10−6. present. The ratio of their branching fractions is If χ is not a DM particle, the bound in Eq. (3) no longer ∆Γ 2g2 e2 (1 − x˜2)3 applies and the final state monochromatic photon can have an n→χγ˜ = n , 2 p (13) energy in a wider range: ∆Γn→χφ |λφ| f(x, y) 0 < E < 1.664 MeV , (8) γ where x˜ = mχ˜/mn, while their sum accounts for the neutron decay anomaly, i.e. entirely escaping detection as Eγ → 0. ∆Γ + ∆Γ n→χγ˜ n→χφ ≈ 1% . Neutron → two dark particles Γn The branching fraction for the process involving a photon Denoting the final state dark fermion and scalar by χ and in the final state ranges from ∼ 0 to 1%, depending on the φ, respectively, and an intermediate dark fermion by χ˜, con- masses and couplings. A suppressed decay channel involving sider a scenario with both a two- and three-particle interaction, e+e− is also present. χ˜ n , χ n φ. The requirement in Eq. (2) takes the form

937.900 MeV < mχ + mφ < 939.565 MeV , (9) Neutron → dark matter + e+e− and both χ, φ are stable if |m − m | < 938.783 MeV . This case is realized when the four-particle effective inter- χ φ action involving the neutron, DM and an e+e− pair is present + − Also, mχ˜ > 937.900 MeV. and Br(n → χ e e ) ≈ 1%. The requirement on the DM If mχ˜ > mn, the only neutron dark decay channels are mass from Eq. (2) is ∗ − n → χ φ and n → χ˜ → p + e +ν ¯e, with branching frac- tions governed by the strength of the χ n φ interaction. Even if 937.900 MeV < mχ < 938.543 MeV this coupling is zero, the lifetime of χ˜ is long enough for the and the allowed energy range of the e+e− pair is anomaly to be explained. In the case 937.9 MeV < mχ˜ < mn, the particle χ˜ can be produced on-shell and there are 2 me ≤ E + − < 1.665 MeV . three neutron dark decay channels: n → χ˜ γ, n → χ φ and e e ∗ − n → χ˜ → p + e +ν ¯e (when mχ˜ > 938.783 MeV), with Assuming the effective term for n → χ e+e− of the form branching fractions depending on the strength of the χ n φ ∗ − eff coupling. The rate for the decay n → χ˜ → p + e +ν ¯e Ln→χe+e− = κ χ¯ n e¯ e is negligible compared to that for n → χ˜ γ. In the limit of a vanishing χ n φ coupling this case reduces to n → χ˜ γ. and a suppressed two-particle interaction χ n, the neutron An example of such a theory is dark decay rate is

eff eff ¯ 2 5 (1−x)2 L2 = L1 (χ → χ˜) + (λφ χ˜ χ φ + h.c.) κ m Z dξ 3 n √ 2 2  2   ∗ µ 2 2 ∆Γn = 3 ξ − 4z (1 + x) − ξ +χ ¯ i∂/ − mχ χ + ∂µφ ∂ φ − mφ|φ| . (10) 128 π 4z2 ξ p The term corresponding to n → χ φ is × (1 − x2 − ξ)2 − 4 ξ x2 ,

eff λφ ε ∗ where x = mχ/mn and z = me/mn. It is maximized Ln→χφ = χ¯ n φ . (11) √ mn − mχ˜ for mχ = 937.9 MeV, in which case it requires 1/ κ ≈ 670 GeV to explain the anomaly. We will not analyze fur- This yields the neutron dark decay rate ther this possibility, but we note that a theory described by the 2 2 + − |λφ| p mn ε Lagrangian (10) with φ coupled to an e e pair could be an ∆Γn→χφ = f(x, y) 2 , (12) 16π (mn − mχ˜) example. 4

FIG. 1. Dark decay of the neutron in model 1. FIG. 2. Dark decay of the neutron in model 2.

PARTICLE PHYSICS MODELS If χ is a DM particle, without an efficient annihilation chan- nel one has to invoke non-thermal DM production to explain We now present two microscopic renormalizable models its current abundance. This can be realized via a late decay of that are representative of the cases n → χ γ and n → χ φ. a new heavy scalar, as shown in [40] for a similar model. Cur- rent DM direct detection experiments provide no constraints [41]. Model 1 The parameter choice in Eq. (15) is excluded if χ is a The minimal model for the neutron dark decay requires Majorana particle, as in the model proposed in [42], by only two particles beyond the SM: a scalar Φ = (3, 1)−1/3 the neutron-antineutron oscillation and dinucleon decay con- (color triplet, weak singlet, hypercharge −1/3), and a Dirac straints [23, 24]. Neutron decays considered in [43] are too fermion χ (SM singlet, which can be the DM). This model is a suppressed to account for the neutron decay anomaly. realization of the case n → χ γ. The neutron dark decay pro- ceeds through the process shown in Fig. 1. The Lagrangian of the model is Model 2 ijk c ∗i c ∗i A representative model for the case n → χ φ involves four L1 = λq  uLi dRjΦk + λχΦ χ¯ dRi + λl QRi lLΦ new particles: the scalar Φ = (3, 1) , two Dirac fermions +λ ijk Qc Q Φ + h.c. − M 2 |Φ|2 − m χ¯ χ , (14) −1/3 Q Ri Lj k Φ χ χ˜, χ, and a complex scalar φ, the last three being SM singlets. c where uL is the complex conjugate of uR. We assign The neutron dark decay in this model is shown in Fig. 2. The numbers Bχ = 1, BΦ = −2/3 and, to forbid proton decay Lagrangian is [28–30], assume conservation, i.e. set λl = 0 L = L (χ → χ˜) + (λ χ˜¯ χ φ + h.c.) − m2 |φ|2 − m χ¯ χ . [31]. For simplicity, we choose λQ = 0. The rate for n → χγ 2 1 φ φ χ is given by Eq. (7) with (16) β λ λ q χ B = B = 1 B = 0 ε = 2 Assigning χ˜ φ and χ , baryon number is con- MΦ served. We have also imposed an additional U(1) symmetry and β defined by under which χ and φ have opposite charges. For mχ > mφ ρ ijk c  ρ 1+γ5  σ the annihilation channel χ χ¯ → φ φ¯ via a t-channel χ˜ ex- h0| uLidRj dRk|ni = β 2 σ u . change is open. The observed DM relic density, assuming Here u is the neutron spinor, σ is the spinor index and the mχ = 937.9 MeV and mφ ≈ 0, is obtained for λφ ' 0.037. parenthesis denote spinor contraction. Lattice QCD calcu- Alternatively, the DM can be non-thermally produced. 3 lations give β = 0.0144(3)(21) GeV [32], where the er- The rate for n → χ φ is described by Eq. (12) with ε = rors are statistical and systematic, respectively. Assuming 2 β λqλχ/MΦ. For mχ˜ =mχ, the anomaly is explained with mχ = 937.9 MeV to maximize the rate, the parameter choice explaining the anomaly is |λ λ | |λ | q χ φ ≈ 4.9 × 10−7 TeV−2. M 2 0.04 |λqλχ| −6 −2 Φ 2 ≈ 6.7 × 10 TeV . (15) MΦ For λφ ≈ 0.04 this is consistent with LHC searches, pro- In addition to the monochromatic photon with energy Eγ < vided again that MΦ & 1 TeV. Direct DM detection searches 1.664 MeV and the e+e− signal, one may search directly also present no constraints. For similar reasons as before, χ and χ˜ for Φ. It can be singly produced through p p → Φ or pair cannot be Majorana particles. produced via gluon fusion g g → ΦΦ. This results in a dijet As discussed above, in this model the branching fractions or four-jet signal from Φ → dcuc, as well as a monojet plus for the visible (including a photon) and invisible final states missing energy signal from Φ → d χ. Given Eq. (15), Φ is not can be comparable, and their relative size is described by + − excluded by recent LHC analyses [33–38] provided MΦ & Eq. (13). A final state containing an e e pair is also pos- 1 TeV [39]. sible. The same LHC signatures are expected as in model 1. 5

CONCLUSIONS Note added: Inspired by the proposal in this paper, several experimental The puzzling discrepancy between the neutron lifetime and theoretical efforts have already been undertaken. measurements has persisted for over twenty years. We could In [50] the scenario n → χγ has been challenged exper- not find any theoretical model for this anomaly in the litera- imentally for 0.782 MeV < Eγ < 1.664 MeV. The case ture. In this paper we bring the neutron enigma into attention Eγ < 0.782 MeV remains unexplored. by showing that it can be explained by a neutron dark decay In [51] the case Br(n → χ e+e−) = 1% has been excluded channel with an unobservable particle in the final state. Our for Ee+e− & 2 me + 100 keV. proposal is phenomenological in its nature and the simple par- In [52–54] the implications for neutron stars have been con- ticle physics models provided serve only as an illustration of sidered. For dark matter with self-interactions, the neutron selected scenarios. dark decays in a neutron star can be sufficiently blocked for Despite most of the energy from the neutron dark decay the observed neutron star masses to be allowed. escaping into the dark sector, our proposal is experimentally In [55] it has been argued that the unexpectedly high pro- verifiable. The most striking signature is monochromatic pho- duction of 10Be in the decay of 11Be [56] can be explained by tons with energies less than 1.664 MeV. Furthermore, if the the neutron dark decay proposed in this paper. dark particle is the dark matter, the energy of the photon is bounded by 0.782 MeV from below. The simplest model pre- dicts the neutron decay into dark matter and a photon with a branching fraction of approximately 1%. Another signature consists of electron- pairs with total energy less than [1] J. Chadwick, The Existence of a Neutron, Proc. R. Soc. A 136, 1.665 MeV. 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