Improved Constraints on Sterile Neutrinos in the MeV to GeV Mass Range

Improved upper bounds are presented on the coupling $|U_{e4}|^2$ of an electron to a sterile neutrino $\nu_4$ from analyses of data on nuclear and particle decays, including superallowed nuclear beta decays, the ratios $R^{(\pi)}_{e/\mu}=BR(\pi^+ \to e^+ \nu_e)/BR(\pi^+ \to \mu^+ \nu_\mu)$, $R^{(K)}_{e/\mu}$, $R^{(D_s)}_{e/\tau}$, and $B^+_{e 2}$ decay, covering the mass range from MeV to GeV.

In addition to the three known neutrino mass eigenstates, there could be others, which would necessarily be primarily electroweak-singlets (sterile) [7]. Indeed, sterile neutrinos are present in many ultraviolet (UV) extensions of the SM. Whether sterile neutrinos exist in nature is one of the most outstanding questions in particle physics, and therefore, improved constraints on their couplings are of fundamental and far-reaching importance. Taking account of the possibility of sterile neutrinos, the neutrino interaction eigenstates ν ℓ would be given by where ℓ = e, µ, τ ; n s denotes the number of sterile neutrinos; and U is the lepton mixing matrix [8].
Here we obtain improved upper limits on |U ei | 2 for a sterile neutrino ν i in a wide range of masses from the MeV to GeV scale and point out new experiments that would be worthwhile and could yield further improvements. For simplicity, we assume one heavy neutrino, n s = 1, with i = 4; it is straightforward to generalize to n s ≥ 2. Since a ν 4 in this mass range decays, it is not excluded by the cosmological upper limit on the sum of effectively stable neutrinos, i m νi < ∼ 0.12 eV [9]. Such a ν 4 is subject to a number of constraints from cosmology (e.g., [10]); however, since these depend on assumptions about the early universe, we choose here to focus on direct laboratory bounds. Constraints from the non-observation of neutrinoless double beta decay are satisfied by assuming that ν 4 is a Dirac neutrino [11]. Since sterile neutrinos violate the conditions for the diagonality of the weak neutral current [12,13], ν 4 has invisible tree-level decays of the form ν 4 → ν jνi ν i where 1 ≤ i, j ≤ 3 with model-dependent invisible branching ratios. Because our bounds are purely kinematic, they are complementary to bounds from searches for neutrino decays, which involve model-dependent assumptions on branching ratios into visible versus invisible final states.
We first obtain improved upper bounds on |U e4 | 2 from nuclear beta decay data. The emission of a ν 4 via lepton mixing in nuclear beta decay has several effects, including producing a kink in the Kurie plot and reducing the decay rate [14]. For the nuclear beta decays (Z, A) → (Z + 1, A) + e − +ν e or (Z, A) → (Z − 1, A) + e + + ν e into a set of neutrino mass eigenstates ν i ∈ ν e , i = 1, 2, 3 of negligibly small masses, plus a ν 4 of of non-negligible mass, the differential decay rate is where p ≡ |p| and E denote the 3-momentum and (total) energy of the outgoing e ± , E 0 denotes its maximum energy for the SM case, the Heaviside θ function is defined as θ(x) = 1 for x > 0 and θ(x) = 0 for x ≤ 0, where M denotes the nuclear transition matrix element, V is the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix, and F F is the Fermi function. Early bounds on |U e4 | 2 were set from searches for kinks in Kurie plots in [14] and analyses of particle decays [15]- [17], and from dedicated experiments. For example, a search for kinks in the Kurie plot in 20 F beta decay reported in Ref. [18] yielded an upper bound on |U e4 | 2 decreasing from 5.9 × 10 −3 for m ν4 = 0.4 MeV to 1.8 × 10 −3 for m ν4 = 2.8 MeV. (Some recent reviews of searches for sterile neutrinos include [19]- [24].) In addition to kink searches, a powerful method to set constraints on massive neutrino emission, via lepton mixing, in nuclear beta decays is to analyze the decay rates. Since, in general, the heavy neutrino would also be emitted in µ decay, the measurement of the µ lifetime performed assuming the SM would yield an apparent (app) value of the Fermi constant, denoted G F,app , that would be smaller than the true value, G F , given at tree level by where g is the SU(2) gauge coupling [15][16][17]. To avoid this complication, the ratios of rates of different nuclear beta decays are compared.
The integration of dN/dE over E gives the kinematic rate factor f . The combination of this with the half-life for the nuclear beta decay, t ≡ t 1/2 , yields the product f t. Incorporation of nuclear and radiative corrections yields the corrected f t value for a given decay, denoted F t. Conventionally, analyses of the F t values for the most precisely measured superallowed 0 + → 0 + nuclear beta decays have been used, in conjunction with the value of G F,app from µ decay, to infer a value of the weak mixing matrix element, |V ud | [25]- [34]. A first step in these analyses has been to establish the mutual consistency of the F t values for these superallowed 0 + → 0 + decays. Since the emission of a ν 4 with mass of a few MeV would have a different effect on the kinematic functions and integrated rates for nuclear beta decays with different Q (energy release) values, it would upset this mutual consistency.
We next discuss upper bounds from two-body leptonic decays of charged pseudoscalar mesons (generically denoted as M + ) [14,15]. This method is quite powerful, because the signal is a monochromatic peak in dN/dp ℓ and for M + e2 decays, the strong helicity suppression in the SM case is removed when a heavy neutrino is emitted. The presence of a massive ν 4 also changes the ratio BR(M + → e + ν e )/BR(M + → µ + ν µ ) from its SM value,, and this was used to set further bounds [14,15,38]. A number of dedicated experiments have been performed to search for a peak due to heavy neutrino emission and also to measure BR(M + → e + ν e )/BR(M + → µ + ν µ ) with π + ℓ2 , K + ℓ2 , and B + ℓ2 , where ℓ = e, µ [36]- [49]. In the SM with only the three known neutrinos with negligibly small masses, the ratio is given by where δ (M) ℓ = m 2 ℓ /m 2 M and δ RC is the radiative correction (RC) [50]- [55].
We denote the ratio of the experimental measurement of R (M) ℓ/ℓ ′ to the SM prediction as The most precise measurement of R (π) e/µ is from the PIENU experiment at TRIUMF, with the result R (π) e/µ = (1.2344 ± 0.0023 stat ± 0.0019 syst ) × 10 −4 [45]. The resultant PDG world average is R The ratio R (K) e/µ has recently been measured by the NA62 experiment at CERN [43], dominating the world average [1] The SM prediction with RC [52,55] is resulting inR (K) e/µ = 1.0044 ± 0.0037 .
We next obtain a bound on |U e4 | 2 by applying the same type of analysis to R (K) e/µ . From K µ2 peak search experiments [37,44,48] and the calculation ofρ(δ ν4 ), |U µ4 | 2 is sufficiently small that we can approximate the denominator of Eq. (16) well by 1. Using Eq. (11) for This upper limit on |U e4 | 2 is labelled KENU in Fig. 1.
In the range of m ν4 from 0.1 to 1.3 GeV, the Belle experiment obtained (non-monotonic) upper limits on BR(B + → µ + ν 4 ) of approximately 2 − 4 × 10 −6 , and in the interval of m ν4 from 1.3 GeV to 1.8 GeV, it obtained upper limits varying from 2 × 10 −6 to 1.1 × 10 −5 . Substituting the BR(B + → e + ν 4 ) limits in Eq. (22) with M = B and ℓ = e, we obtain the upper limits on |U e4 | 2 shown as the curve B e2 in Fig. 1 [74]. From the BR(B + → e + ν 4 ) limits we infer upper limits on |U µ4 | 2 that decrease from 0.83 to 3.4 × 10 −2 as m ν4 increases from 0.1 GeV to 1.2 GeV. Further peak searches for B + → ℓ + ν 4 with ℓ = e, µ at Belle II would be worthwhile as a higher-statistics extension of [46]. We briefly remark on other constraints on a Dirac ν 4 in the mass range considered here. From the results of [12,75], it follows that there is a negligibly small contribution to decays such as µ → eγ and µ → eeē. Similarly, there is no conflict with bounds on neutrino magnetic moments [1,76], and contributions to invisible Higgs decays [77] are well below the current upper limit of BR(H → invis.) < 19% [78].
In this work, improved upper limits on |U e4 | 2 have been presented covering most of the range from m ν4 = 0.5 MeV to m ν4 ≃ 1 GeV, representing the best available laboratory bounds for a Dirac neutrino ν 4 that do not make model-dependent assumptions concerning visible neutrino decay modes. Over parts of this range, the bounds obtained are competitive with those that assume specific visible ν 4 decays. For example, for m ν4 = 30 MeV, our upper bound is |U e4 | 2 < 0.8 × 10 −6 , while the best bound for this value of m ν4 from experiments searching for neutrino decays is |U e4 | 2 < 1 × 10 −6 [79]. New peak search experiments to search for D + s → e + ν 4 and D + → e + ν 4 as well as a continued search for B + → e + ν 4 would be valuable; these could improve the bounds further. Other constraints on sterile neutrinos such as from