Dark particle interpretation of the neutron decay anomaly

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 a dark sector particle in the final state. If this particle is stable, it can be the dark matter. Its mass is close to the neutron mass, suggesting a connection between dark and baryonic matter. In the most interesting scenario a monochromatic photon with energy in the range 0.782 MeV – 1.664 MeV and branching fraction 1% is expected in the final state. We construct representative particle physics models consistent with all experimental constraints.


Introduction
In the previous talk, A. Holley presented the status of measurements of neutron lifetime. We will not repeat this here. The purpose of this talk is to contemplate the possibility that the discrepancy between neutron lifetime measurements in bottle and beam experiments is due to a new phenomenon, rather than a systematic error or a statistical fluctuation [1] Basic principles demand that the particle lifetime does not depend on the measurement technique or decay mode. For a new physics explanation of the discrepancy we posit that one of the experiments is not reporting a lifetime. The bottle experiment fits to a decaying in time exponential, measuring the lifetime by definition. An important observation is that the beam experiment, by contrast, measures the instantaneous β-decay rate: τ n Br(n → p + anything) .
This gives the decay lifetime only if the neutron β-decays 100% of the time. Therefore, the discrepancy could be explained if the neutron decays about 1% of the time into a protonless final state. A very direct test of this hypothesis could be performed, but has not been done, in both bottle and beam experiments. In a bottle experiment one could measure the number of decays that result in final state protons: the hypothesis asserts that these would account only for 99% of the neutrons lost. In a beam experiment one could fit the number of observed protons to an exponential decaying along the beam: here the fit to the exponential would result in a 1% smaller lifetime than the one determined from the instantaneous rate at one point along the beam. The question immediately arises as to what is the new decay channel of the neutron, a protonless state with 1% branching fraction. It is incumbent upon us to present a scenario that is consistent with our current understanding of elementary particle interactions.

Model independent analysis
The final state of neutron decay must have unit fermion number and be overall electrically neutral. The simplest possibility for a protonless decay channel consists of a chargeless, "dark" fermion, χ, and a neutral boson. In principle the dark fermion could be a neutrino and the neutral boson could be a photon, but soon we will see that the sum of their masses must be within 1.572 MeV of the neutron mass, so at least one of the two must be a new elementary particle. We consider two possible final states: (i) a dark fermion, χ, and a photon, and (ii) a dark fermion, χ, and a new spinless elementary dark boson, φ.
In either case, the mass M f of the final state f must be sufficiently large that proton stability is not vitiated: to avoid p → n * e + ν followed by n * → f , we must have M f > m p − m e . A slightly stronger bound follows from stability of 9 Be against decay to 2 4 He + f , yielding [2]: For the case f = χ + γ this immediately gives 937.992 MeV < m χ < 939.565 MeV. For m χ < m p + m e = 938.783 MeV, β decay of χ is not allowed giving the interesting possibility that χ is stable (or very long lived) and could be a candidate for a dark matter component of the universe. Similarly, for the case f = χ + φ both χ and φ are dark matter candidates if |m χ − m φ | < m p + m e .

Effective theory for χγ final state
To describe the decay n → χ γ in a quantitative way, we consider theories with a mass mixing term χ n, and an induced interaction χ n γ. An example of such a theory is given by the effective Lagrangian where g n −3.826 is the neutron g-factor and ε is the mixing parameter with dimension of mass.
The term corresponding to n → χ γ is obtained by transforming Eq. (3) to the mass eigenstate basis and, for ε m n − m χ , yields Therefore, the neutron dark decay rate is , where x = m χ /m n . The rate is maximized when m χ saturates the lower bound, m χ = 937.992 MeV. Below we will give a microscopic particle physics realization of this case.
The testable prediction of this class of models is a monochromatic photon with an energy in the range 0 < E γ < 1.572 MeV and a branching fraction If the dark fermion χ is to be sufficiently light that it may be stable, and hence a DM candidate, then 0.782 MeV < E γ < 1.572 MeV. A null search for monochromatic photons rules out this  [3]. A signature involving an e + e − pair with total energy E e + e − < 1.572 MeV is also expected, but with a suppressed branching fraction of at most 1.1 × 10 −6 . A null search for n → χe + e − sets a bound of ∼ 10 −4 on the branching fraction with 99.9997% significance for 937.900 MeV < m χ < 938.543 MeV [4].

Effective theory for χφ final state
Consider now a scenario in which the neutron decays to an intermediate off-shell dark fermion, χ, which subsequently decays to a final state dark fermion and a scalar, χ and φ. The masses of the final state particles are constrained by Eq. (2), with M f = m χ + m φ , and both χ and φ are stable if |m χ − m φ | < 938.783 MeV. It is also possible to have an on-shell decay toχγ if 937.992 MeV < mχ < 939.565 MeV, but we are mostly interested in the case mχ > 939.565 MeV for which the neutron decay is dark.
An example of such a theory is Going to the mass eigenstate basis, the term corresponding to n → χ φ is which yields the neutron dark decay rate where 3 with x = m χ /m n and y = m φ /m n . A microscopic, particle physics realization of this scenario is provided below. For mχ > m n the missing energy signature has a branching fraction ≈ 1%. There will also be a very suppressed radiative process involving a photon in the final state with a branching fraction 3.5 × 10 −10 or smaller.
As discussed earlier, in the case 937.992 MeV < mχ < m n both the visible and invisible neutron dark decay channels are present. The ratio of their branching fractions is wherex = mχ/m n , while their sum accounts for the neutron decay anomaly, i.e.
The branching fraction for the process involving a photon in the final state ranges from about 0 to 1%, depending on the masses and couplings. A suppressed decay channel involving e + e − is also present.

Microscopic Models
While the effective theory analysis presented above is useful in estimating the dark decay parameters that a fundamental microscopic model must produce, the existence of a viable model with renormalizable interactions that satisfies all experimental constraints is not obvious. We will The minimal model for the n → χγ requires only two particles beyond the SM: a scalar Φ = (3, 1) −1/3 (color triplet, weak singlet, hypercharge −1/3), and a Dirac fermion χ (SM singlet). The interaction Lagrangian of the model includes where u c L is the charge conjugate of u R . Assigning baryon numbers B χ = 1, B Φ = −2/3 we see that proton decay is forbidden [5][6][7]. The rate for n → χγ is given by Eq. (5) with and β defined by 0| ijk u c Li d Rj d ρ Rk |n = β 1 2 (1 + γ 5 ) ρ σ u σ , where u is the neutron spinor, σ is the spinor index and the parenthesis denote spinor contraction. Lattice QCD calculations give β = 0.0144(3)(21) GeV 3 [8], where the errors are statistical and systematic, respectively. Assuming m χ = 937.992 MeV to maximize the rate, in order to explain the anomaly the parameters must satisfy In addition to the monochromatic photon with energy E γ < 1.572 MeV and the e + e − signal, one may search directly also for Φ. It can be singly produced through p p → Φ or pair produced via gluon fusion g g → Φ Φ. This results in a dijet or four-jet signal from Φ → d c u c , as well as a monojet plus missing energy signal from Φ → d χ. Given Eq. (12), Φ is not excluded by recent LHC analyses [9][10][11][12][13][14] provided M Φ 1 TeV. 1 The parameter choice in Eq. (12) is excluded if χ is a Majorana particle, as in the model proposed in [15], by the neutron-antineutron oscillation and dinucleon decay constraints [16,17]. Neutron decays considered in [18] are too suppressed to account for the neutron decay anomaly.
A representative model for the case n → χ φ involves four new particles: the scalar Φ = (3, 1) −1/3 , two Dirac fermionsχ, χ, and a complex scalar φ, the last three being SM singlets. The interaction Lagrangian is an extension of the one in (10) We have imposed an additional U (1) symmetry under which χ and φ have opposite charges. For m χ > m φ the annihilation channel χχ → φφ via a t-channelχ exchange is open. The observed DM relic density, assuming m χ = 937.992 MeV and m φ ≈ 0, is obtained for λ φ 0.037. Alternatively, the DM can be non-thermally produced.
The rate for n → χ φ is described by Eq. (7) with ε = β λ q λ χ /M 2 Φ . For mχ = m χ , the anomaly is explained with For λ φ ≈ 0.04 this is consistent with LHC searches, provided again that M Φ 1 TeV. Direct DM detection searches present no constraints. For similar reasons as before, χ andχ cannot be Majorana particles. IOP Conf. Series: Journal of Physics: Conf. Series 1308 (2019) 012010 IOP Publishing doi:10.1088/1742-6596/1308/1/012010 5 As discussed above, in this model the branching fractions for the visible (including a photon) and invisible final states can be comparable, and their relative size is described by Eq. (8). A final state containing an e + e − pair is also possible. The same LHC signatures are expected as in model 1.

Unstable Nucleus Decay
While the lower bound in (2) with M f = m χ forbids the decay (A, Z) → (A − 1, Z) + χ for any stable nucleus, it is still permitted for some unstable nuclei. There are several unstable isotopes with a neutron binding energy S(n) < 1.572 MeV and a sufficiently long lifetime to probe the dark decay channel when the dark particle mass m χ < m n − S(n). One example presented in [1] is 11 Li, for which S(n) = 0.396 MeV. 11 Li β decays with a lifetime 8.75 ms. However, in the presence of a dark particle χ the decay chain 11 Li → 10 Li + χ → 9 Li + n + χ becomes available. Ref. [2] proposes to search for 11 Be → 10 Be + χ. This has S(n) = 501.6(3) keV and half-life 13.76(7) s. Since this is a halo neutron nucleus, the calculation of the decay rate 11 Be → 10 Be + χ + γ can be well estimated from that of n → χ + γ by accounting for the phasespace factor. This leads to a branching fraction of 10 −6 -10 −4 for 938 MeV m χ 939 MeV. However, the dark decay of 11 Be does not necessarily involve a photon in the final state. For example, if the model (3) is supplemented by a π-nucleon interaction, πnn, or a two-body nucleon interaction,nnnn, then in terms of the mass eigenstates one obtains interactions with the dark fermion, πχn andχnnn that can directly mediate (A, Z) → (A − 1, Z) + χ. Ref. [19] estimates the decay width for 11 Be → 10 Be + χ to be an order of magnitude larger than the measured total width of 11 Be for m χ 939 MeV, excluding this model. However, we note that the result of Ref. [19] exhibits unphysical behavior: the decay amplitude, expressed as a function of x = m n − m χ − S(n), has singular behavior as x → 0. No independent verification of the result is available. Yet, the computation uses the mixing termχn, rather than the induced interactions, πχn orχnnn, to mediate the transition, and it is likely this artifact is responsible for the unphysical behavior.
The calculated branching fraction for dark 11 Be decay in Ref. [2] applies just as well to the purely dark decay of the model in (5). The branching fraction for 11 Be decay into 10 Be has been measured, and the result is some 400 times larger than that expected from β-delayed proton emission [20]. This could be explained if a yet unobserved very low lying and very narrow 11 B resonance existed. The authors of Ref. [2] propose an experiment measuring protons in 11 Be decay. This may discover the putative resonance, or, in the absence of the expected proton signal, give supporting evidence for the dark decay hypothesis.

Neutron Stars
The impact of neutron dark decays on neutron stars was considered in Refs. [21][22][23]. The resulting production of dark particles changes the energy density and pressure inside a neutron star, modifying its equation of state. This in turn changes the predictions for the maximum allowed neutron star masses, since they are derived from integrating the Tolman-Oppenheimer-Volkoff equation that explicitly depends on the equation of state.
It was shown that the observed neutron star masses (2M for the heaviest neutron stars discovered) are allowed if strong repulsive self-interactions are present in the dark sector of our models. Such interactions are easily introduced in the representative Models 1 and 2 discussed in Sec. 3 by simply adding a dark vector boson coupled strongly to the dark particle χ.
Interestingly, a strongly self-interacting dark sector lies along the lines of the self-interacting dark matter paradigm, which was introduced two decades ago [24] to solve the core-cusp and missing satellite problem of the ΛCDM model. A repulsive interaction between neutrons and dark particles can also be effective in allowing observed, 2M , neutron star masses [25]. For appropriate strength and range of the interaction there is an energy cost associated with creating dark particles and therefore pure neutron matter is preferred.

Models with a self-interacting dark sector
A model of this type was constructed in Ref. [26], where a neutron dark decay involving a dark fermion and a dark photon in the final state was considered, i.e., n → χ A . The effective Lagrangian is where the covariant derivative D µ = ∂ µ − i g A µ . It was shown that the strength of the dark photon coupling to the dark particle χ, governed by the parameter g and resulting in repulsive interactions between the χ particles, can be chosen such that the neutron lifetime discrepancy is explained and, at the same time, all astrophysical bounds are satisfied, including constraints from neutron stars, galaxy clusters, cosmic microwave background, Big Bang nucleosynthesis and supernovae. If the dark particle χ in this model is stable, it can contribute to the dark matter in the universe, but cannot account for all of the dark matter. Many of the astrophysical constraints are alleviated if one assumes non-thermal dark matter production. This was shown in Ref. [27], where a model for the neutron dark decay n → χ φ was constructed, based on our Model 2, but with a dark boson introduced to mediate large self-interactions of χ. The Lagrangian for the dark sector is There exists a choice of parameters for which this model satisfies neutron star constraints, remains consistent with all other astrophysical bounds and χ makes up all of the dark matter in the universe. In addition, due to the self-interactions of χ, the model is shown to solve the small-scale structure problems of the ΛCDM model.

Hadron dark decays
The idea of dark decays can be applied also to other neutral hadrons. In Ref. [28] it was argued that the mesons K 0 L and B 0 can decay to dark sector particles at measurable rates. An explicit model was constructed with a dark sector consisting of several families of dark fermions. An analogous mechanism that prevents neutron beta decays in neutron stars, i.e., Pauli blocking, also forbids neutron dark decays inside a neutron star in this model.

Baryogenesis
It has recently been shown that the model addressing the neutron lifetime puzzle based on the Lagrangian in Eq. (15) provides a successful framework for low-scale baryogenesis [29]. In addition, a model very similar to our Model 2, with couplings ofχ to other quark flavors and a Majorana (instead of Dirac) fermion χ, has been proposed in the context of low-scale baryogenesis as well [30].

Related solutions
Taking into consideration only the experimental data for g A from experiments performed after the year 2002, the bottle neutron lifetime is favored [31]. Based on this observation, explanations of the neutron lifetime discrepancy have been put forward in which it is the bottle lifetime that is equivalent to the Standard Model prediction for τ n . The difference in outcomes of the bottle and beam measurements is explained via neutron-mirror neutron oscillations resonantly enhanced in large magnetic fields thus affecting only beam measurements [32], or by invoking a sizable Fierz interference term canceling the dark decay contribution to the neutron decay rate [33].