Annihilation signatures of neutron dark decay models in neutron oscillation and proton decay searches

We point out that two models that reconcile the neutron lifetime anomaly via dark decays of the neutron, also predict dark matter-neutron ($\bar{\chi}-n$) annihilation that may be observable in neutron-antineutron oscillation and proton decay searches at Super-Kamiokande, Hyper-Kamiokande and DUNE. We study signatures of $\bar{\chi}n\to \gamma\pi^0$ (or multi-pions) and $\bar{\chi}n\to \phi\gamma\pi^0$ (or $\phi+$multi-$\pi^0$), where $\phi$ is an almost massless boson in one of the two models.


I. INTRODUCTION
In the standard model (SM), the neutron almost exclusively decays through beta decay, n → p + e − +ν e , with Br(n → p + anything) = 1. The neutron lifetime is measured in bottle experiments and beam experiments which use different methodologies. In bottle experiments, the total neutron lifetime τ bottle n is measured by counting the number of neutrons trapped in a container as function of time. On the other hand, beam experiments count the number of protons resulting from neutron decay in a neutron beam. In this case, the neutron lifetime is given by where N n is the number of neutrons in the beam. In the SM, the two methods should give the same neutron lifetime. However, there is tension between the bottle [1][2][3] and beam [4,5] measurements of the neutron lifetime at about the 4σ level [6]: To explain the discrepancy, the neutron decay width into channels without protons must be ∆Γ(n → no proton) 7.1 × 10 −30 GeV , which is about 1% of the total neutron decay width.
Two models, dubbed Model I and Model II, invoking dark decays of the neutron were proposed in Refs. [7,8]. The basic idea is to introduce a tiny mixing between the neutron and a new particle in the dark sector, which allows the neutron to decay into dark sector particles χ (a Dirac fermion) and φ (a scalar) through n → χ + γ and n → χ + φ. The χ couples very weakly with the SM sector so as to not trigger a detectable signal in the beam experiments. Meanwhile, the mass of χ is restricted in a narrow window, 937.900 MeV < m χ < 938.783 MeV , to simultaneously satisfy the requirement of 9 Be stability and prevent the decay, χ → p + e − +ν e [7]. These criteria make χ a good dark matter (DM) candidate if other decay modes in the dark sector are forbidden. Based on these assumptions,χ − n annihilation provides detectable signals with energy of O(GeV) in Super-Kamiokande (Super-K) or in the future experiments, Hyper-Kamiokande (Hyper-K) and DUNE. The signals are similar to that for proton decay and neutron-antineutron oscillations. For instance in Model I, prompt photon and multi-pion signals are obtained fromχn annihilation. Dark matter-nucleon annihilation has been studied in a broader context in Ref. [9].
Note that these models have trouble with neutron star stability [10][11][12]. The conversion of neutrons to dark matter in the neutron star softens the nuclear equation of state to the point that neutron stars above two solar masses are not possible, which is in contradiction with observations. In a recent paper [13] it was shown that extending Model II with strong repulsive DM-baryon interactions solves the problem. This extension does not contribute additional diagrams to the annihilation signatures we discuss. We view the models of Refs. [7,8,13] as examples that produce neutron dark decays that may be extended with complex dark sectors to address various experimental and astrophysical constraints.
The paper is organized as follows. In section II, we present the relevant effective interactions and parameterize the required form factor. We introduce Model I and -II for the dark decays of the neutron in section III and IV, respectively. Signatures of DM-neutron annihilation are discussed in section V, and the signal event numbers at underground experiments are estimated in section VI. We summarize our results in section VII.

II. EFFECTIVE INTERACTIONS AND FORM FACTOR
Since the neutron has no electric charge, it couples to the photon via the magnetic dipole interaction, where g n −3.826 is the neutron g-factor and Fn γn (Q 2 ) is the corresponding form factor.
On the other hand, the pion-neutron and pion-proton effective interactions satisfy the isospin symmetry and are given by 13.54 [14], and Fn πn (Q 2 ) is the form factor. For thenπn vertex with onlyn off-shell at momentum squared −Q 2 , we parameterize the form factor as where y ⊂ [0, 2] is an unknown exponent. y = 0 represents the case of no form factor suppression, while y = 2 gives a good fit to the electromagnetic form factor of the proton.
We determine the value of y by comparing with the experimentally measurednp annihilation cross section in the nonrelativistic limit. Here, Λ n characterizes the size of the nucleon, and is typically 4πf π ≈ 1.2 GeV. The form factor is normalized to unity for −Q 2 = m 2 n . Furthermore, we assume the same behavior for the magnetic form factor of the neutron.
The complete amplitude also involves the transitional vertices betweenχ − n. We simply set where n and χ are almost degenerate in mass.

III. MODEL I
Model I allows the dark decay n → χ + γ by introducing two dark sector particles, a Dirac fermion χ (whose antiparticleχ we identify as the DM candidate) and a heavy scalar mediator Φ (color triplet, weak singlet, hypercharge Y 2 = − 1 3 ). The new interaction Lagrangian terms that contribute to χ − n mixing are [7] As stated earlier, 937.900 MeV < m χ < 938.783 MeV .
The colored Φ must be much heavier than 1 TeV to be compatible with LHC data. It is therefore reasonable to work in the effective theory framework to describe processes at the GeV scale. The DM-triquark operator is derived by integrating out the heavy scalar Φ: where form factor β = 0|u R d R d R |n 0.0144 GeV 3 [15]. This operator is effectively an off-diagonal mass term between the DM and neutron, leading to a m 2 Φ suppressed mixing angle, where ε ≡ βλ 1 λ 1 /m 2 Φ . The DM-neutron mixing gives rise to a DM-neutron-photon coupling, which is responsible not only for neutron dark decay n → χγ, but also predicts the DM-neutron annihilation channel,χ + n → γ + π 0 , as shown in Fig. 1. A nonrelativistic DM particle in the halo interacts with a static neutron target and produces a photon and pion back-to-back, each with an energy of about a GeV.
The topology of the signal is similar to that of the proton decay channel p → e + π 0 at experiments like Super-K. Both produce 3 electron-like Cherenkov rings in Super-K [16], and the reconstructed total momentum P tot tends to be small. However, the reconstructed invariant mass M tot 2 GeV fromχ + n → γ + π 0 , is higher than M tot 1 GeV from A multi-pion final state can also result from DM-neutron annihilation. It is analogous tō n-nucleon annihilation in the SM which predominantly yields a multi-pion final state. The DM-nucleon annihilation cross section is related to then-nucleon cross section via [9] σ(χN → multi-pions) = θ 2 σ(nN → multi-pions) .
This multi-pion signal can be detected in Super-K as well, and the signal kinematics is similar to that for n−n oscillations [17]. However, because of the compositeness of the hadron and a lack of experimental measurements of thenπn form factor, a perturbative calculation using Eqs. (1), (2) cannot give a precise estimate of the annihilation cross section. To overcome this, we match our calculation to the experimentally measurednp annihilation cross section by varying the exponent y of the form factor.

IV. MODEL II
Model II has a richer structure than Model I with two additional dark sector particles: a Dirac fermionχ and a complex scalar φ [7]. After the heavy scalar Φ is integrated out,χ mixes with the neutron through the mixing angle which is the same as Eq. (3) with m χ replaced by mχ. Then χ couples to φ andχ via the new interaction, So, in addition to n → γχ, a new neutron dark decay channel n → φχ is allowed. Hence, the sum of the decay widths, ∆Γ n→γχ + ∆Γ n→φχ 7.1 × 10 −30 GeV , to reconcile the tension between beam and bottle experiments.
For m χ > m φ , the the annihilation channelχχ → φφ via t-channelχ exchange can provide the correct DM relic density if λ φ 0.04. The three masses m χ , m φ , and mχ should satisfy the relations, to prevent 9 Be decays to 8 Be + χ + φ and 8 Be +χ, and to prohibit χ → p + e − +ν e , respectively [8]. We choose three benchmark points with λ φ 0.04: where m χ m φ and mχ are in MeV. All three points explain the neutron lifetime anomaly with the corresponding values of θ listed in Table I. P1 and P2, respectively, are the points from Refs. [7,8], with n →χγ kinematically allowed for P1 but not for P2. Sinceχ plays the role of a propagator in the DM-neutron annihilation process, the signal event distributions are different for P1 and P2. For P3, the DM-neutron annihilation cross section is maximized, as we will see in the next section.
Feynman diagrams for the DM-neutron annihilation process, are shown in Fig. 2. The event distributions will be different from Model I, due to the additional dark sector particle φ in the final state. Since φ can escape the detector, the reconstructed invariant mass M tot and total momentum P tot from γ and π 0 have different distributions from Model I.

V. DARK MATTER-NUCLEON ANNIHILATION CROSS SECTION
We now calculate the DM-neutron annihilation cross section for several possible signals in underground experiments. First, we determine the value of the mixing angle θ by requiring the neutron dark decay widths to be ∆Γ n→χγ 7.1 × 10 −30 GeV and ∆Γ n→χγ + ∆Γ n→χφ 7.1 × 10 −30 GeV for Models I and II, respectively. For Model II, we also fix λ φ = 0.04, to obtain the correct DM relic density.
For the benchmark points in Table I, the typical values of the mixing angles are respectively, θ 10 −10 and 10 −11 , in Model I and Model II. In general, θ in Model II is one order magnitude smaller than that in Model I, because 3 with x 1 ≡ m χ /m n and x 2 ≡ m φ /m n . In the static limit, the diagrams in Fig. 1 forχn → γπ 0 in Model I yield the spin averaged amplitude squared, The cross section features a 1/v behavior in the nonrelativistic limit (applicable for an average DM velocity v 10 −3 c). We present values of v c σ in Table I, which are independent of the DM velocity. This process produces γ and π 0 at about a GeV, which makes Super-Kamiokande well suited to detect this signal.
We analytically estimate the form factor suppression forχn → γπ 0 as follows. While the virtual momentum flows in the two diagrams in Fig. 1 may be different, in the static limit they are the same: −Q 2 = ( P 2 − k 1 ) 2 . Therefore, the cross section is suppressed by the common factor, 1 − m 2 n /Λ 2 The maximum suppression (y = 2) for a typical value of Q 2 is O(10 −5 ). Note that the form factor suppression forχn → φγπ 0 in Model II depends on the kinematics and a full numerical integration is required. The values of v c σ including form factor suppression with y = 2 are provided in Table I We now compute the DM-neutron annihilation cross section to multi-pions. For Model I, the process isχn → pions, and for Model II, φ is associated produced with pions,χn → φ + pions. Since the multi-pion channel is the dominant mode for antinucleus-nucleus annihilation, we expect the same for DM-neutron annihilation.
Since a perturbation calculation is not valid for a largenπn coupling, we use the experimentally measured value of thenp annihilation cross section in the lown velocity limit [18][19][20][21]: v c σ(np → multi-pions) exp = 44 ± 3.5 mb , which is s-wave dominant and independent of then velocity. We assume that σ(nn) σ(np).
Then for Model I, the DM-neutron annihilation cross section is given by Numerical values are provided in Table I. Forχn → φ+pions in Model II, there is no experimental dataset that can be used directly.
With water as the target for Super-K and Hyper-K, and m χ = 938.783 MeV, the interaction rate per second per gram of water is where v DM = 10 −3 c is the thermal average DM velocity, the DM number density is n χ = ρ χ /m χ per cm 3 in terms of the local DM density ρ χ = 0.3 GeV/cm 3 , N A = 6.022 × 10 23 is the Avogadro number, and N A /18 is the total number of H 2 O molecular per gram of water.
For the liquid Argon target at DUNE, the interaction rate per second per gram of target is where ρ Ar = 1.3954 g/cm 3 is the density of liquid Argon. In the nonrelativistic limit, the interaction rate is independent of the v DM , and therefore independent of the velocitydistribution of the DM in the galactic halo.
The signal events are obtained by multiplying the above interaction rates with the total exposure. For Super-K [16], the current total exposure is 306.3 kiloton-years. For Hyper-K [23], we use a fiducial mass of 372 kiloton with 20 years of data-taking. For DUNE [24], we take a 40 kiloton fiducial mass with 20 years of data-taking. The events numbers for the different signal channels in the three experiments are displayed in Table I. The kinematic cuts applied in Super-K's searches for proton decay and n −n oscillations are summarized in Table II. We adopt the same cuts (cut-1, cut-2, cut-3) and definitions of total visible momentum, P tot ≡ | all−rings vector of the i th ring, the invariant mass, M tot ≡ E 2 tot − P 2 tot , and the total visible energy, where m i is the mass of the i th ring assuming that showering and nonshowering rings are from γ and π ± , respectively [17]. For our case, m i = 0. Kinematic  allowed number of signal events for cut-2 and cut-3 are as in Table II. To evaluate the expected number of signal events at Hyper-K and DUNE, we assume that the observed event rate is compatible with the expected background rate, and scale Super-K's exposure. The 3σ ranges are provided in Table.II.
To calculate the number of events that satisfy the kinematic cuts, we perform a Monte Carlo simulation by assuming 10% momentum uncertainty for each ring in Super-K [26]. 1 We take the momentum resolution at Hyper-K and DUNE to be 10%. The event distributions region, and only a tiny fraction of events are inside the cut-2 and cut-3 regions. Therefore, these signals do not contaminate the proton decay search. In Table III, we tabulate the percentage of events for each channel that pass the three kinematic cuts.

VII. RESULTS AND SUMMARY
Model I is comfortably ruled out by the current n−n oscillation search at Super-K because it predicts O(10 6 )χn → pions events in the cut-1 region, while Super-K has observed 24 events with an expected background of 24.1 events. Note that theoretical uncertainties do not affect this exclusion because the calculation of theχn → pions cross section is driven by experimental data, and so is not impacted by the hadron form factor uncertainty.
It is difficult to completely explore the parameter space of Model II because of its many degrees of freedom, and hence difficult to rule it out. We therefore focused on specific benchmark points. The expected numbers of φ3π 0 + φ5π 0 signal events for P1, P2, and P3 at Super-K after applying cut-1 are 17.1, 0.038, and 545, respectively, where we used  χn → φγπ 0χ n → φ3π 0χ n → φ5π 0χ n → φγπ 0χ n → φ3π 0χ n → φ5π 0  is excluded by Super-K at more than 3σ for the above values of y. P1 does not contribute a significant event excess at Super-K. P2 is three orders of magnitude beyond the reach of Super-K because of the heavierχ.
If cut-1 is extended to M tot = 2 GeV, the signal events increase to 23.0, 0.039, and 680 for P1, P2, and P3, respectively, and more than 95% of the signal events fall inside the extended kinematic region for P1 and P3.
We show the sensitivities of Super-K and the future experiments Hyper-K and DUNE in terms of y in Table IV, where we applied kinematic cut-1. The table gives the minimum value of y that ensures that the number of signal events lies within the 3σ range in Table II.
Negative values of y mean that although there is no form factor suppression, the experiment cannot probe the parameter point. Clearly, DUNE will have better sensitivity than Super-K, and Hyper-K will have the best sensitivity as evidenced by the higher minimum values of y.