Constraints on Sterile Neutrinos in the MeV to GeV Mass Range

A detailed discussion is given of the analysis of recent data to obtain improved upper bounds on the couplings $|U_{e4}|^2$ and $|U_{\mu 4}|^2$ for a mainly sterile neutrino mass eigenstate $\nu_4$. Using the excellent agreement among ${\cal F}t$ values for superallowed nuclear beta decay, an improved upper limit is derived for emission of a $\nu_4$. The agreement of the ratios of branching ratios $R^{(\pi)}_{e/\mu}=BR(\pi^+ \to e^+ \nu_e)/BR(\pi^+ \to \mu^+ \nu_\mu)$, $R^{(K)}_{e/\mu}$, $R^{(D_s)}_{e/\tau}$, $R^{(D_s)}_{\mu/\tau}$, and $R^{(D)}_{e/\tau}$, and the branching ratios $BR(B^+\rightarrow e^+\nu_e)$ and $BR(B^+\rightarrow \mu^+\nu_\mu)$ decays with predictions of the Standard Model, is utilized to derive new constraints on $\nu_4$ emission covering the $\nu_4$ mass range from MeV to GeV. We also discuss constraints from peak search experiments probing for emission of a $\nu_4$ via lepton mixing, as well as constraints from pion beta decay, CKM unitarity, $\mu$ decay, leptonic $\tau$ decay, and other experimental inputs.

A detailed discussion is given of the analysis of recent data to obtain improved upper bounds on the couplings |Ue4| 2 and |Uµ4| 2 for a mainly sterile neutrino mass eigenstate ν4. Using the excellent agreement among Ft values for superallowed nuclear beta decay, an improved upper limit is derived for emission of a ν4. The agreement of the ratios of branching ratios R (π) e/µ = BR(π + → e + νe)/BR(π + → µ + νµ), R e/τ , and the branching ratios BR(B + → e + νe) and BR(B + → µ + νµ) decays with predictions of the Standard Model, is utilized to derive new constraints on ν4 emission covering the ν4 mass range from MeV to GeV. We also discuss constraints from peak search experiments probing for emission of a ν4 via lepton mixing, as well as constraints from pion beta decay, CKM unitarity, µ decay, leptonic τ decay, and other experimental inputs.

I. INTRODUCTION
In a recent paper [1], we presented improved upper bounds on the coupling |U e4 | 2 of an electron to a sterile neutrino ν 4 from analyses of data on nuclear and particle decays, for ν 4 masses in the MeV to GeV range, and pointed out new experiments that could improve these constraints. Here we give the details of our analysis that yielded these constraints and also present a number of additional bounds on sterile neutrino mixings, in particular, on the coupling |U µ4 | 2 .
Neutrino oscillations and hence neutrino masses and lepton mixing have been established and are of great importance as physics beyond the original Standard Model (SM) [2]- [11]. Most of the data from experiments with solar, atmospheric, accelerator, and reactor (anti)neutrinos can be explained within the minimal framework of three neutrino mass eigenstates with values of ∆m 2 ij = m 2 νi − m 2 νj given approximately by ∆m 2 21 = 0.74 × 10 −4 eV 2 and |∆m 2 32 | = 2.5 × 10 −3 eV 2 , with normal mass ordering m ν3 > m ν2 favored; furthermore, the lepton mixing angles θ 23 , θ 12 , and θ 13 have been measured, with a tentative indication of a nonzero value of the CP-violating quantity sin(δ CP ) (for compilations and fits, see [12]- [18]).
The possible existence of light sterile neutrinos, in addition to the three known neutrino mass eigenstates, is a fundamental question in particle physics. These would have to be primarily electroweak-singlets (sterile), since the invisible width of the Z boson is consistent with being due to decays toν ℓ ν ℓ , where ν ℓ = ν e , ν µ , and ν τ , corresponding to the known three SM fermion families [19]. In the presence of sterile neutrinos, the neutrino interaction eigenstates ν e , ν µ , and ν τ are linear combinations that include these additional mass eigenstates. In a basis in which the charged-leptons are simultaneously flavor and mass eigenstates, the charged weak current has the form J λ =lγ λ ν ℓ , where ℓ = e, µ, τ and where n s denotes the number of additional mass eigenstates. The near-sterility of the ν i with 4 ≤ i ≤ n s is reflected in small upper bounds on the corresponding |U ℓi |. We will use the term "sterile neutrino" both in its precise sense as an electroweak-singlet interaction eigenstate and in a commonly used approximate sense as the corresponding, mainly sterile, mass eigenstate(s) in this neutrino interaction eigenstate. For technical simplicity, we will assume one heavy neutrino, n s = 1, with i = 4; it is straightforward to generalize to n s ≥ 2. Since a ν 4 in the mass range of interest here decays on a time scale much shorter than the age of the universe, it is not excluded by the cosmological upper limit on the sum of stable neutrinos, i m νi < ∼ 0.12 eV [20]. Possible sterile neutrinos are subject to many constraints from neutrino oscillation experiments using solar and atmospheric neutrinos, accelerator and reactor (anti)neutrinos, and kinematic effects in particle and nuclear decays, as well as cosmological constraints. Bounds from the non-observation of neutrinoless double beta decay are satisfied by assuming that ν 4 is a Dirac, rather than Majorana, neutrino. Although Majorana neutrino masses have often been regarded as more generic, many ultraviolet extensions of the SM contain additional gauge symmetries that forbid Majorana mass terms, so that in these models, neutrinos are Dirac fermions [21]. Much attention has been focused on possible sterile neutrinos with masses in the eV region because of results from the LSND [22] and Miniboone [23] experiments and possible anomalies in reactor antineutrino experiments (recent reviews and discussions include [24][25][26]). In addition to eV-scale sterile neutrinos, there has also been interest in possible keV-scale sterile neutrinos as warm dark matter, and in even heavier sterile neutrinos with masses extending to the GeV range, and cosmological constraints on these have been discussed [27]- [34]. These cosmological constraints involve assumptions about properties of the early universe. One valuable aspect of laboratory bounds on heavy neutrinos is that they are free of such assumptions about the early universe.
Since sterile neutrinos violate the conditions for the diagonality of the weak neutral current [35,36], ν 4 has invisible tree-level decays of the form ν 4 → ν jνi ν i where 1 ≤ i, j ≤ 3 with model-dependent 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.
This paper is organized as follows. In Sect. II we derive upper bounds on |U e4 | 2 from nuclear beta decay data. Sect. III discusses pion beta decay. Sect. IV considers connections of nuclear decay data with the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix. In Sect. V we discuss peak search experiments. In Sects. VI and VII we derive upper bounds on lepton mixing matrix coefficients from two-body leptonic decays of π + , K + , D + , D s , and B + mesons. Sects. VIII and IX are devoted to constraints from µ decay and leptonic τ decays. In Sect. X we briefly discuss other constraints on sterile neutrinos. Sect. XI contains our conclusions.

II. LIMIT ON EMISSION OF MASSIVE NEUTRINOS IN NUCLEAR BETA DECAY
The emission of a heavy neutrino ν j via lepton mixing and the associated nonzero |U ej | 2 , with a mass in the keV-MeV region can be searched for in several ways using nuclear beta decays. If the ν j mass is less than the energy release Q in a given beta decay, its emission produces a kink in the Kurie plot. Ref. [37] suggested a search for such kinks and used a retroactive data analysis to set upper bounds on this type of emission via lepton mixing of neutrinos with kinematically non-negligible masses in nuclear beta decays. In standard notation, (Z, A) denotes a nucleus with Z protons and A nucleons. For a nuclear beta decay (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 with negligibly small masses relative to the energy release in the decay plus a mass eigenstate ν 4 in ν e with non-negligible mass, the differential decay rate is where p ≡ |p| and E denote the 3-momentum and (total) energy of the outgoing e ± in the parent nucleus rest frame, 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, and C = G 2 F |V ud | 2 F F |M| 2 /(2π 3 ), 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, which takes account of the Coulomb interactions of the outgoing e ± . In general, there is also a shape correction factor, but this is not important for the superallowed decays considered here. It is understood that if the decay is to an excited state of the daughter nucleus rather than to its ground state, then there is a corresponding reduction in the maximal value of E 0 relative to its value for the decay to the ground state. The kink in the Kurie plot arises as E reaches the endpoint for the decay yielding a ν 4 and the second term in Eq. (2.1) vanishes.
Early bounds on |U e4 | 2 were set from searches for kinks in Kurie plots in [37] and analyses of particle decays [38]- [40]. Subsequently, dedicated experiments were conducted to search for kinks in the Kurie plots due to possible emission of a massive neutrino via lepton mixing for a number of nuclear beta decays over a wide range of neutrino masses from O(10) eV to the MeV range. For example, a search for kinks in the Kurie plot in 20 F beta decay reported in Ref. [41] 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. (These and other upper bounds discussed in this paper are at the 90 % confidence level unless otherwise stated.) Some recent reviews of searches for sterile neutrinos in various mass ranges include [24,25], and [42]- [50].
A general effect of the emission of a heavy neutrino ν 4 in a nuclear beta decay is to reduce the rate in a manner dependent on its mass, due to phase space suppression of the decay, and, if it is too massive to be emitted, to reduce the rate of the given decay by the factor (1−|U e4 | 2 ). Hence, in addition to examination of Kurie plots for possible kinks, a powerful method to constrain heavy neutrino emission, via lepton mixing, in nuclear beta decays is to analyze the overall rates. The apparent (app) rate, assuming no emission of a heavy neutrino, can be succinctly expressed as where F app = pE(E 0 − E) 2 is the SM kinematic function assuming no heavy neutrino emission. 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 [38][39][40], G F , given at tree level by where g and g ′ are the weak SU(2) and U(1) Y gauge couplings, and m W and m Z are the masses of the W and Z bosons. The apparent kinematic function F app is larger than the true kinematic function indicated in the square brackets in Eq. (2.1), which depends on m ν4 and |U e4 | 2 .
Since G F,app would be smaller than the true value of G F , while F app would be larger than the true F , the apparent value, |V ud,app | 2 , extracted from a particular nuclear beta decay in the context of the SM could be larger or smaller than the true value. To avoid this complication, we compare ratios of rates of different nuclear beta decays. In these ratios, the factor G 2 F,app cancels, so one can gain information about the kinematic factor and hence about |U e4 | 2 as a function of m ν4 .
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 most precisely measured superallowed 0 + → 0 + nuclear beta decays have been used for many years to infer a value of the weak mixing matrix element |V ud | [51,52]. (In our discussion of these fits, we will follow conventional notation and denote the CKM mixing matrix factor as V ud , with the implicit understanding that in our present context with possible emission of a heavy neutrino ν 4 , this is really V ud,app .) In turn, these values of |V ud | extracted from superallowed nuclear beta decays were used in early Cabibbo fits, e.g., [53], which were subsequently extended to the full CKM matrix [54][55][56]. The analyses of nuclear beta decay data have continued up to the present with significant recent progress in precision [57]- [66].
A first step in these analyses has been to establish the mutual consistency of the F t values for these superallowed 0 + → 0 + decays. The emission of a ν 4 with a mass m ν4 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 and would therefore upset this mutual consistency. Therefore, from this mutual agreement of F t values, an upper limit on |U e4 | 2 can be derived for values of m ν4 in the MeV range, such that a ν 4 could be emitted in some of these superallowed decays. F t is conventionally written as [58-61, 63, 65, 66] where K = 2π 3 ln 2/m 5 e = 0.81202776(9)×10 −6 GeV −4 − sec, G V = G F |V ud | and the radiative correction factor ∆ V R is transition-independent. Ref. [63] obtains the average F t = 3072.27 ± 0.72 sec.
The excellent mutual agreement between the F t values obtained from a set of the most precisely measured superallowed 0 + → 0 + nuclear beta decays, which involve only the vector part of the charged weak current, in comparison with the value of G F obtained from muon decay, allows one to extract, in a self-consistent manner, a value of |V ud |. In the 1990 study [58], this yielded the result |V ud | = 0.9740 ± 0.001. At present, using a set of the fourteen most precisely measured superallowed 0 + → 0 + nuclear beta decays, Hardy and Towner have obtained the considerably more precise value [64,67] (denoted HT) HT : |V ud | = 0.97420 (21) . (2.5) Another recent estimate, in agreement with these, is |V ud | = 0.97425(13) [68] (see also [69] where the superscript m refers to a metastable excited state. The maximal Q value in this set is Q = 9.4 MeV ( 74 Rb) [63,70]. The emission of a neutrino with a mass of order MeV in superallowed nuclear beta decays would cause kinematic suppression depending on the energy release Q and the neutrino mass m ν4 , which would vary from nucleus to nucleus owing to the different values of the phase space factor in the third term, proportional to |U e4 | 2 , in Eq. (2.1). Ref. [57] set upper limits on |U e4 | 2 ranging from 3 × 10 −2 to 4 × 10 −3 for m ν4 from 0.5 MeV to 4.5 MeV, while Ref. [58] obtained an upper bound on |U e4 | 2 ranging from 10 −2 down to 2 × 10 −3 for m ν4 from 0.5 to 2 MeV. Ref. [41] incorporated the phase space integration for the massive-neutrino term proportional to |U e4 | 2 in Eq. (2.1) for eight available superallowed beta decays and then derived upper bounds on |U e4 | 2 from the consistency of corrected F t values, depending nonmonotonically on ν 4 masses from 1 to 7 MeV, with the results |U e4 | 2 < 1 × 10 −3 to |U e4 | 2 < 2 × 10 −3 , shown as BD1 in Fig. 1.
A measure of the mutual agreement among F t values of the superallowed beta decays is the precision with which |V ud | 2 is determined, so a reduction in the fractional uncertainty of the value of |V ud | 2 results in an improved L. upper limits on |Ue4| 2 vs. mν 4 from various souces: PIBETA, pion beta decay (this work); BD1, previous limits from nuclear beta decay [41]; BD2, nuclear beta decay, based on our analysis using [64] and [65]; PIENU and PIENU-H, the ratio BR(π + →e + νe) BR(π + →µ + νµ) in the kinematically allowed and forbidden regions for ν4 emission [90]; πe2 PIENU, π + → e + ν4 peak searches (upper and lower curves from [84] and [91], respectively); KENU and KENU-H, the ratio BR(K + →e + νe) BR(K + →µ + νµ) in the kinematically allowed and forbidden regions for ν4 emission; Ke2 KEK, K + → e + ν4 peak search [82]; Ke2 NA62, K + → e + ν4 peak search [94]; and Ke2 NA62*, the preliminary upper limit from a K + → e + ν4 peak search [95]. Other bounds are denoted Dse2, from our analysis of , and Be2, from our analysis of peak search data in B + → e + ν4 [125]. Our new bounds are colored blue (online), while previous bounds are colored black. See text for older bounds and further discussion.
upper limit on |U e4 | 2 . Let us denote this fractional uncertainty from the i'th data analysis, as [δ (i) |V ud,i | 2 ]/|V ud,i | 2 . Then it follows that The fractional uncertainties of [δ (2) |V ud |]/|V ud | = 2 × 10 −4 and 1.4 × 10 −4 in Refs. [63,64] and [65] are improvements by the respective factors of 5 and 7.5 relative to the inputs used in the 1990 studies [41,58]. We use these improvements to infer respective improved upper bounds on |U e4 | 2 , following from the mutual agreement of the F t values among the fourteen superallowed beta decays [63][64][65]. Using the HT value in Eq. (2.5), we find the upper bound for ν 4 masses in the range from m ν4 ≃ 1 MeV to m ν4 ≃ 9.4 MeV, as indicated in Fig. 1 (BD2, upper line). Using the SGPRM value in Eq. (2.6), we find |U e4 | 2 < ∼ 2.7 × 10 −4 , (2.10) also shown in Fig. 1 (BD2, lower line). Of course, the flat line segments shown are approximations; the actual upper limits on |U e4 | 2 from the nuclear beta decay data are not precisely constant as a function of m ν4 over the range shown. If the uncertainties in the F t values for each of the superallowed nuclear beta decays used for the overall fit in [63][64][65] were equal, then one could extend this analysis to derive an upper bound on |U e4 | 2 as a function of m ν4 in this range of 1 to 9.4 MeV. However, this condition, of equal precision for the measurement of the F t value of each individual nuclear beta decay in this set, has not yet been achieved. For this reason, we have conservatively presented our upper bounds (2.9) and (2.10) as applying uniformly throughout the specified range 1 MeV < m ν4 < 9.4 MeV, i.e., as flat line segments in Fig. 1.
Since our bounds (2.9) and (2.10) above do not involve |U µ4 | 2 , they complement the upper limits on |U e4 | 2 derived from the measurement of the ratio of decay rates R (π) e/µ = Γ(π + → e + ν e )/Γ(π + → µ + ν µ ) discussed in Sect. VI A in the subset of the range of ν 4 mass values where they overlap, namely 1 < ∼ m ν4 < ∼ 10 MeV. Other methods of determining |V ud | include pion beta decay (discussed in Sec. III) and the neutron lifetime (which also has the complication of involving the axialvector part of the weak charged current), but these are not as accurate as the determination from the superallowed 0 + → 0 + beta decays.
III. LIMITS FROM π + → π 0 e + νe DECAY In this section we analyze limits on sterile neutrinos obtainable from pion beta decay, π + → π 0 e + ν e . The mass difference between the charged and neutral pions is ∆ π = m π + − m π 0 = 4.5936 ± 0.0005 MeV [13]. It will be convenient to define If ν e consists only of neutrino mass eigenstates with negligibly small masses, then the Standard-Model expression for the decay rate, denoted Γ πβ,SM , is [71] and δ incorporates radiative corrections, calculated to be δ = 0.033 [72,73]. Note that the last term in the square brackets in Eq. (3.3) is −1.20 × 10 −4 and thus is much smaller than the leading-x terms. Neglecting this last term, the function f (x) has the expansion If ν e contains the known three neutrinos with masses that are negligibly small for the kinematics here, together with an O(1) MeV ν 4 , then the rate for pion beta decay has the form 5) where Γ πβ,ν4 ≡ |U e4 | 2Γ πβ,ν4 denotes the rate for the decay π + → π 0 e + ν 4 . As in the case of nuclear beta decay, the emission of the ν 4 would produce a kink in the differential decay distribution dΓ πβ /dE e , where E e is the electron energy. In particular, while the maximum electron energy in the case of emission of neutrinos with negligibly small masses is this is reduced to in the π + → π 0 e + ν 4 decay. However, in contrast to nuclear beta decay, events ascribed to the decay π + → π 0 e + ν e were identified by the diphoton decay of the π 0 , and the e + energy was not systematically measured, e.g., in the PIBETA experiment at PSI [74,75]. Hence, one could not do a kink search for this decay, which would be quite difficult anyway because of the very small branching ratio of 10 −8 for pion beta decay. However, one can use the comparison of the measured decay rate, or equivalently, branching ratio for pion beta decay with the SM prediction to obtain a limit on possible emission of a ν 4 . We have where r πβ,ν4 denotes the ratio of the kinematic factor for the π + → π 0 e + ν 4 decay divided by that for the decay into neutrinos of negligibly small mass, and, including radiative corrections [74,75], BR(π + → π 0 e + ν e ) SM = (1.039 ± 0.001) × 10 −8 . (3.9) Defining ǫ ν4 = m 2 ν4 /∆ 2 π , the function r πβ,ν4 can be approximated to leading order in ǫ e and ǫ ν4 as The current value listed by the Particle Data Group, dominated by the PIBETA measurement [74,75], is [13] BR(π + → π 0 e + ν e ) = (1.036 ± 0.006) × 10 −8 . (3.11) This is in good agreement with the SM prediction (3.9), yielding From this we obtain the upper limit on |U e4 | 2 shown in Fig. 1 as PIBETA. As m ν4 increases, and finally exceeds the value m π + − m π 0 − m e = 4.08 MeV, the decay π + → π 0 e + ν 4 is kinematically forbidden, and hence the observed rate divided by the rate predicted in the SM with the usual mass eigenstates in ν e of negligibly small masses is reduced to the first term in Eq. (3.8), namely 1 − |U e4 | 2 . The upper bounds on |U e4 | 2 from pion beta decay are less stringent than the bounds in Eqs. (2.9) and (2.10).

IV. CONSTRAINT FROM CKM UNITARITY
If the mass of ν 4 were sufficiently large so that it could not be emitted in any superallowed nuclear beta decays used in the determination of |V ud |, then, although there would still be mutual consistency in this determination between the different superallowed nuclear decays, the result would be a spurious apparent value of |V ud | 2 , namely |V ud,app | 2 = |V ud | 2 (1 − |U e4 | 2 ) (where we again assume just one heavy neutrino). In turn, this would reduce the apparent value of |V ud | 2 + |V us | 2 + |V ub | 2 used to check the first-row unitarity of the CKM matrix. If one uses the value of |V ud | in Eq. (2.5), then the sum |V ud | 2 + |V us | 2 + |V ub | 2 is equal to unity to within the stated theoretical and experimental uncertainties. Thus, this provides another constraint on possible massive neutrino emission in the decays involved. Numerically, using the value of |V ud | in Eq. (2.5), together with the values |V us | = 0.2243(5) and |V ub | 2 = (1.55 ± 0.28) × 10 −5 from [13], Ref. [64] obtains Σ ≡ |V ud | 2 + |V us | 2 + |V ub | 2 = 0.99939 (64) . (4.1) The |V ud | 2 term dominates both the sum and the uncertainty in (4.1). Thus, with the assumption of first-row CKM unitarity, this also yields an upper limit on |U e4 | 2 , depending on m ν4 and estimates of uncertainty in |V us | 2 . If, on the other hand, one uses the lower value of |V ud | in Eq. (2.6), then, as was observed in [65], there is tension with first-row CKM unitarity. However, since the difference between the analyses in [63,64] and [65] is in the value for the transition-independent correction term ∆ V R , this does not upset the mutual agreement between the F t values, which was the key input for the bound (2.9).

V. CONSTRAINTS FROM PEAK SEARCH EXPERIMENTS
It is also of considerable interest to discuss correlated limits on sterile neutrinos from two-body leptonic decays of pseudoscalar mesons. Searches for subdominant peaks in charged lepton momenta in two-body leptonic decays of pseudoscalar mesons were suggested as a way to search for emission, via lepton mixing, of a possible heavy neutrino ν h , and to set upper limits on the associated couplings |U ℓh | 2 , also including effects on ratios of branching ratios, in [37,38]. These observations were applied retroactively to existing data to derive such limits in [37][38][39]. In particular, the upper limit |U e4 | 2 < ∼ 10 −5 was obtained from retroactive analysis of data on K + → e + ν e decays for 82 < m ν4 < 163 MeV, and upper limits on |U µ4 | 2 in the range 10 −4 − 10 −5 were obtained from data on π + → µ + ν µ ( π µ2 ) decay (Figs. 17, 22 in [38]). An analogous discussion of the emission of massive neutrino(s) in muon decay was given in [39,76], and an analysis of µ decay data was used in [39] to set upper limits on |U e4 | 2 and on |U µ4 | 2 (see Sect. VIII).
Dedicated experiments have been carried out from 1981 to the present to search for the emission, via lepton mixing, of a heavy neutrino in two-body leptonic decays of M + = π + , K + mesons and to search for effects of possible heavy neutrinos on the ratio BR(M + → e + ν e )/BR(M + → µ + ν µ ) [77]- [96]. These have set very stringent bounds. Data from the corresponding experiments with heavy-quark pseudoscalar mesons will be used below to derive new limits on sterile neutrinos. Some relevant properties of these experiments with twobody leptonic decays of charged pseudoscalar mesons will be discussed next. The peak search experiments are quite sensitive to massive neutrino emission because one is looking for a monochromatic signal and, furthermore, for a considerable range of m ν4 masses, there is a kinematic enhancement of the decays M + → e + ν 4 and M + → µ + ν 4 relative to the decays into neutrinos with negligibly small masses.
In the SM, the rate for the decay M + → ℓ + ν ℓ of a charged pseudoscalar M + , where M + = π + , K + , etc., and ℓ is a charged lepton, is, to leading order, where V ij is the relevant CKM mixing matrix element, f M is the corresponding pseudoscalar decay constant (normalized such that f π = 130 MeV), and we have used the fact that the three known neutrino mass eigenstates ν i , i = 1, 2, 3 in ν ℓ are negligibly small compared with m M for all pseudoscalar mesons M . However, because of lepton mixing, other decay modes may also occur into some number of neutrinos with nonnegligible masses. Focusing, as above, on the case of a single heavy neutrino ν 4 , the SM rate is reduced by the factor (1 − |U ℓ4 | 2 ) and there is another decay yielding the heavy neutrino with rate, 2) in which the notation is as follows [37,38]: where the factor f M arises from the square of the matrix element M, and In Eq. (5.4), λ(1, x, y) arises from the final-state twobody phase space, with λ(z, x, y) = x 2 + y 2 + z 2 − 2(xy + yz + zx) .
Note that ρ(x, y) has the symmetry property In the SM case with zero or negligibly small neutrino masses, ρ(x, 0) = x(1 − x) 2 . Here and below, it is implicitly understood that ρ(δ There is a clear signature for the decay M + → ℓ + ν 4 into a heavy neutrino, namely the appearance of a monochromatic peak in the energy or momentum distribution of the charged lepton below the dominant peak associated with the emission of neutrino mass eigenstates of negligibly small mass. The energy and momentum of this additional peak, in the rest frame of the parent meson M , are and An experiment on the two-body leptonic decay of a pseudoscalar meson M + → ℓ + ν ℓ , searching for a subdominant peak in the charged lepton momentum or energy distribution due to the decay M + → ℓν 4 , is limited to the mass range m ν4 < m M − m ℓ for which the decay is kinematically allowed. It is also limited (i) to sufficiently small m ν4 such that the momentum or energy of the outgoing ℓ + is large enough so that the event will not be rejected by the lower cut used in the event reconstruction and (ii) to sufficiently large m ν4 so that the subdominant peak can be resolved from the dominant peak.
ν4 ) increases from a minimum at δ ν4 = 0 to a maximum at δ where it has the value 2δ For decays in which m ℓ << m M and hence δ (M) ℓ << 1, this produces a large enhancement, since For example, for π e2 , K e2 , D e2 , (D s ) e2 , and B e2 decays, this ratio (5.10) has the very large values 1.87×10 4 , 2.33× 10 5 , 3.35 × 10 6 , 3.71 × 10 6 , and 2.67 × 10 7 , respectively. Physically, these large enhancement factors are due to the removal of the helicity suppression of the decay of the M + into a light ℓ + and neutrinos ν i with negligibly small masses. It is convenient to define the ratiō Thus, Note that the dominant radiative corrections divide out between the numerator and denominator of Eq. (5.13). Since a given value of lepton momentum p ℓ is uniquely determined by m ν4 for a given pseudoscalar meson M , a null observation of an additional peak in an experiment and hence an upper limit on the ratio Γ(M + → ℓ + ν 4 )/Γ(M + → ℓ + ν ℓ ) SM at a particular p ℓ yields an upper limit on |U ℓ4 | 2 for the corresponding value of m ν4 . Solving Eq. (5.13) for |U ℓ4 | 2 gives . (5.14) Hence, denoting Γ(M + → ℓ + ν 4 ) ul as the upper limit on Γ(M + → ℓ + ν 4 ), one has the resultant upper limit on |U ℓ4 | 2 : .
( 5.15) Provided that |U ℓ4 | 2 << 1, the right-hand side of Eq. (5.13) is, to a good approximation, equal to ν4 ), so that the upper limit (5.15) simplifies to . (5.16) The large values ofρ(δ ν4 ) decays over a substantial part of the kinematically allowed range of m ν4 mean that M + → e + ν 4 decays are quite sensitive to the possible emission of a heavy ν 4 . With fixed x, the functionρ(x, y) has the following Taylor series expansion in y for small y: The derivative ofρ(x, y) with respect to y is Hence, In our application, . (5.20) ν4 ) increases very rapidly from unity to values >> 1.
For a given x, the maximal value ofρ(x, y), as a function of y occurs where dρ(x, y)/dy = 0, or equivalently, dρ(x, y)/dy = 0 in the physical region. The value of y at this maximum is given by the solution for y, of the equation << 1, so that, to a very good approximation, Eq. (5.22) reduces to the equation (3y − 1)(y − 1) 2 = 0. The relevant solution of this equation, giving the value of y at which ρ(x, y) and ρ(x, y) reach their respective maxima if x << 1, is Then (with x << 1), which is >> 1. In Table I we list the maximal values ν4 ) for the various pseudoscalar mesons M + considered here, and for ℓ = e, µ, together with the respective values of m ν4 where these maxima occur. Particularly large maximal values of theρ function occur for heavy-quark pseudoscalar mesons, including 1.98 × 10 6 , 2.20 × 10 6 , and 1.58 × 10 7 for the D + → e + ν 4 , D + s → e + ν 4 , and B + → e + ν 4 decays, respectively. As is evident from this table, these maximal values are only slightly less than the maximal val- ν4 ) mentioned above. This is due to the slow falloff of the two-body phase space factor To see this, let us define, as in [38], the ratio of the phase space factor divided by its value for zero neutrino mass, This has the Taylor series expansion for small y. Hence, the phase space function normalized to its value for zero neutrino mass, i.e., ν4 )] 1/2 , decreases from unity rather slowly for small δ approaches y max from below, the phase space factor [λ(1, x, y)] 1/2 → 0, and hence so do ρ(x, y) andρ(x, y). From the factorized expression From Eq. (5.30), it follows that for any physical value of x, as y → y max from below, ρ(x, y) andρ(x, y) approach 0 with a negatively infinite slope.
For fixed x, over almost all of the kinematically allowed region in y, the reduced functionρ(x, y) is larger than 1. The fact thatρ(δ ν4 ) > 1 up to values of m ν extremely close to its upper endpoint is understandable in view of the property embodied in Eq. (5.30), that this function approaches zero with a slope that approaches −∞, i.e., nearly vertically, as δ MeV. At these respective values of m ν4 , the momentum of the e + is very small, namely 0.125 MeV, which would be below the lower cutoff for such an event to be accepted as a π e2 or K e2 event. Similar comments apply for the leptonic decays of heavy-quark pseudoscalar mesons. We will use this property in the limits that we derive below on |U e4 | 2 .

A. General Formalism
In addition to producing a subdominant peak in the charged lepton momentum p ℓ at the value (5.9), the emission of a massive neutrino in the two-body leptonic decay of a pseudoscalar meson M + would cause an apparent deviation from the SM prediction for the ratio of decay rates or branching ratios, The experimental measurements of M + → ℓ + ν ℓ include events with very soft photons; this is to be understood implicitly below. By convention, we take m ℓ ′ > m ℓ . This deviation would constitute an apparent violation of e − µ universality for the case ℓ = e, ℓ ′ = µ. In contrast to a peak search experiment with the decay M + → ℓ + ν ℓ , which places an upper bound on |U ℓ4 | 2 as a function of m ν4 , a deviation in R (M) ℓ/ℓ ′ depends, in general, on both |U ℓ4 | 2 and |U ℓ ′ 4 | 2 , as well as m ν4 (see Eqs. (6.11) and (6.12 below). The non-observation of any additional peak in the dN/dp ℓ spectrum in two-body leptonic decays of π + and K + was used via a retroactive data analysis in [37,38] and in a series of dedicated peak-search experiments to set stringent upper limits on |U e4 | 2 and |U µ4 | 2 (individually) as functions of m ν4 . Furthermore, the nonobservation of any deviation from e − µ universality in the ratio R (M) e/µ was used in [38,80,84] to obtain upper limits on lepton mixing, as will be discussed further below. As was the case with peak search experiments, in deriving a constraint from a comparison of a measured value of R (M) ℓ/ℓ ′ with the SM prediction for this ratio, one must take into account that even if m ν4 is small enough to be kinematically allowed to occur in either or both of these decays, an experiment might reject events involving emission of a ν 4 if the momentum or energy of the outgoing ℓ + or ℓ ′+ were below the cuts used in the event reconstruction and data analysis. We comment further on this below.
In Section V we reviewed the general formalism describing effects of possible massive neutrino emission in M ℓ2 decays, i.e., the decays M + → ℓ + ν ℓ , where ℓ = e, µ, and, where allowed kinematically, also ℓ = τ [37,38]. Although the actual decays and branching ratios depend on the pseudoscalar decay constants f M and the CKM mixing matrix elements, these cancel in ratios of branching ratios, which can thus be calculated to high precision and compared with experimental measurements. Let us, then, consider the ratio of branching ratios (6.1). In the Standard Model, since the neutrino mass eigenstates ν i , i = 1, 2, 3 have negligible masses, this ratio is where δ RC is the radiative correction [98]- [104], which takes into account soft photon emission, matching experimental conditions. We define the ratio of the measured ratio of branching fractions, R ℓ/ℓ ′ to the SM prediction for this ratio, R (M) ℓ/ℓ ′ ,SM , as Including the radiative correction δ RC , one has the following SM prediction for R   ℓ/ℓ ′ ,SM changes to the following [37,38]: were defined in Eq. (5.3). In Eq. (6.11) we have used the fact that the leading order radiative correction is independent of m ν4 [104]. As noted above, in analyzing experimental data, one must take account of the fact that unless an experiment is specifically searching for effects of possible heavy neutrino emission, it would normally set cuts on the energy and/or momentum of the outgoing charged lepton near to the value for the SM decay. It would thus reject events due to a sufficiently massive ν 4 and would thus measure an apparent total rate that would be reduced from the actual rate by the factor (1 − |U e4 | 2 ).
Combining Eqs. (6.2) and (6.11), we have, for the ratio of branching ratios divided by the SM prediction,R . (6.12) With a given M , one can distinguish three different intervals for m ν4 : , then both of the decays M + → ℓ + ν 4 and M + → ℓ ′+ ν 4 are kinematically forbidden. We recall the values of these intervals for the comparison of the branching ratios for M e2 and M µ2 decays with M = π + and M = K + (where we use the standard notation M ℓ2 for the decay M + → ℓ + ν ℓ ). Here, the mass intervals are The general forms of Eq. (6.12) are (6.14) Consequently, for m ν4 ∈ I (M) 2 , from the upper limit on the deviation of BR(M + → ℓ + ν ℓ )/BR(M + → ℓ ′+ ν ℓ ′ ) from its SM value, i.e., the upper limit on the deviation of R (M) ℓ/ℓ ′ from 1, an upper bound on |U ℓ4 | 2 can be obtained. Then, , then Eq. (6.12) takes the still simpler formR .

VII. BOUNDS FROM LEPTONIC DECAYS OF HEAVY-QUARK MESONS
Two-body leptonic decays of heavy-quark pseudoscalar mesons [37,38] are also valuable sources of information on sterile neutrinos. We discuss the available bounds in this section.
Searches for D + s → e + ν e have been carried out by CLEO [107], BABAR [108], and Belle [109], giving the current upper bound Hence, for the ratio of the e and τ branching ratios, one has the resultant upper limit For this ratio, from [100,104] we calculate the radiative correction 1+δ RC = 0.948. Substituting this in Eq. (6.2) with M = D s , ℓ = e, and ℓ ′ = τ , we find that in the SM, this ratio of branching ratios is is 191 MeV < m ν4 < 1.457 GeV. We restrict m ν4 to a lower-mass subset of this full interval, for the following reason. In the D + s → e + ν 4 decay, as m ν4 increases from small values to its kinematic limit, the momentum of the outgoing e + in the rest frame of the parent D s decreases from its SM value, p e ≡ |p e | = 0.984 GeV. In order for the event reconstruction procedure in a given experiment to count such a decay as a D + s → e + ν e decay, it is necessary that p e > p e,cut , where p e,cut denotes a lower experimental cut on p e . A representative value of this cut is the value p e,cut = 0.8 GeV used in the BES III experiment [112]. The e + momentum decreases to p e = 0.8 GeV as m ν4 reaches the value m ν4 = 0.85 GeV. Thus, we consider the interval 0.191 GeV < m ν4 < 0.85 GeV. For m ν4 in this interval, the ratio of branching ratios of the observed D + s → e + ν e and D + s → τ + ν τ decays is given by Eq. (6.12) with M = D + s , ℓ = e, and ℓ ′ = τ . Hence, from Eq. (6.14), this ratio of branching ratios, divided by the value in the SM, is Requiring that the emission of the ν 4 should not alter the experimentally observed upper limit onR (Ds) e/τ given above, we obtain the following upper bound on |U e4 | 2 for m ν4 in this mass range, which is the special case of (6.15) with M = D s , ℓ = e, and ℓ ′ = τ : (7.7) This limit is largely independent of the |U τ 4 | 2 term, since |U τ 4 | 2 is constrained to be less than upper bounds ranging from ∼ 0.1 to ∼ 0.01 for m ν4 in this mass range [42,44,48]. For the minimal value of m ν4 taken here, namely m ν4 = 0.191 GeV, theρ function in Eq. (7.7) is already quite large, having the value 1.37 × 10 5 . As m ν4 increases to 0.85 GeV, thisρ function increases to 1.83 × 10 6 . Thus, over this range of m ν4 , the upper limit on |U e4 | 2 in (7.7) decreases from |U e4 | 2 < 5.1 × 10 −3 to |U e4 | 2 < 3.8 × 10 −4 . We thus obtain the upper bound on |U e4 | 2 labeled D se2 in Fig. 1. In the interval 450 MeV < m ν4 < 850 MeV, these upper bounds on |U e4 | 2 (denoted as D se2 in Fig. 1) are the best available. As was pointed out in [1], dedicated peaksearch experiments to search for the heavy-neutrino decays D + s → e + ν 4 and D + → e + ν 4 would be worthwhile and could improve our upper bound on |U e4 | 2 .
In addition to the comparison of the branching ratios BR(D + s → e + ν e ) and BR(D + s → τ + ν τ ), it is also useful to comment on the comparison of BR(D + s → µ + ν µ ) and BR(D + s → τ + ν τ ), both of which have been measured. From the experimental results (7.1) and (7.2), the resultant measured ratio of branching ratios is for this decay is 191 MeV < m ν4 < 1.863 GeV, and for m ν4 in this interval, Eq. (6.12) reduces to the expression in (6.14) with M = D s , ℓ = µ, and ℓ ′ = τ . The maximum value of m ν4 to enable a large enough p µ to satisfy an experimental lower momentum cut of 0.8 GeV is m ν4 = 0.84 GeV, which is almost the same as for the D + s → e + ν 4 decay. We thus obtain an upper limit on |U µ4 | 2 which is the special case of (6.15) with M = D s , ℓ = µ, and ℓ ′ = τ , namely (7.12) Given that |U τ 4 | 2 << 1, this reduces to the special case of Eq. In the case of the D + meson, the cd annihilation amplitude is suppressed by the CKM factor |V cd | 2 relative to semileptonic and hadronic decay channels, which can proceed by c → s charged-current vertices and hence involve the much larger |V cs | 2 factor in the rates. There is significant phase-space suppression of the D + → τ + ν τ channel, since m D + − m τ is only 92.8 MeV. For one of these leptonic D decays, one has an upper limit, namely BR(D + → e + ν e ) < 0.88 × 10 −5 . (7.13) The branching ratio for D + → µ + ν µ has been measured by CLEO and BES III [13,113,115] as Recently, BES III has measured the branching ratio for D + → τ + ν τ [114] as (7.15) With the radiative correction 1 + δ RC = 0.963, the SM prediction for the ratio of these branching ratios is, from Eq. (6.1),R With ν 4 emission, this ratio would be changed tō Here we analyze constraints from two-body leptonic B + decays. These decays involve ub annihilation and hence are suppressed by the small CKM factor |V ub | 2 relative to semileptonic and hadronic B + decays involving the larger CKM factor |V cb | 2 . Currently, there is an upper limit on one leptonic B + decay, from Belle [116] and BABAR [117], and measurements of the other two, namely from Belle [118], from a Belle update [119,120], and BR(B + → τ + ν τ ) = (1.09 ± 0.24) × 10 −4 (7.22) from BABAR [121] and Belle [122,123]. Both the published and preliminary updated values of the BR(B + → µ + ν µ ) are in agreement with the SM prediction [118] BR(B + → µ + ν µ ) SM = (3.80 ± 0.31) × 10 −7 . (7.23) The measured value of BR(B + → τ + ν τ ) in (7.22) is also in agreement with the SM prediction [123,124] BR(B + → τ + ν τ ) SM = (0.75 +0.10 −0.05 ) × 10 −4 . A recent experiment to search for B + → e + X 0 and B + → µ + X 0 was carried out by Belle [125], where X 0 is a weakly interacting particle that does not decay in the detector. Assuming that X 0 = ν 4 , one can use the results of this experiment to set upper limits on |U e4 | 2 and |U µ4 | 2 . For m ν4 in the range from 0.1 GeV to 1.4 GeV, this experiment obtained an upper limit on BR(B + → e + ν 4 ) of 2.5 × 10 −6 , while in the interval of m ν4 from 1.4 GeV to 1.8 GeV, this upper limit increased to 7 × 10 −6 . In the range of m ν4 from 0.1 to 1.3 GeV, the experiment obtained (non-monotonic) upper limits on BR(B + → µ + ν 4 ) of approximately 2×10 −6 to 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 . These limits are less restrictive than the bounds (7.19) and (7.20), but have the advantage of being reported for specific values of m ν4 .
Substituting the experimental upper limit on BR(B + → e + ν 4 ) as a function of m ν4 from [125] into the relevant special case of (5.15) with M = B + and ℓ = e, we obtain the upper bound on |U e4 | 2 as a function of m ν4 shown in Fig. 1. This upper bound decreases from 0.83 to 3.4 × 10 −2 as m ν4 increases from 0.1 GeV to 1.2 GeV. Since the experimental upper limit on BR(B + → e + ν 4 ) is less stringent as m ν4 increases from 1.4 to 1.8 GeV, the same is true of the resultant upper limit on |U e4 | 2 ; for example, if m ν4 = 1.6 GeV, we get |U e4 | 2 < 5.4 × 10 −2 .
Carrying out the analogous procedure with the upper bound on BR(B + → µ + ν 4 ) from [125], we obtain an upper limit on |U µ4 | 2 that decreases from 0.83 to 3.4 × 10 −2 as m ν increases from 0.1 GeV to 1.2 GeV. As m ν4 increases from 1.2 to 1.5 GeV and then to 1.8 GeV, the upper limit on BR(B + → µ + ν 4 ) from [125] rises from approximately 3 × 10 −6 to 1.1 × 10 −5 and then decreases again to 3×10 −6 . In this interval of m ν4 masses, using the appropriate special case of (5.15), we obtain upper limits ranging from |U µ4 | 2 of 0.12 for m ν4 = 1.5 GeV to |U µ4 | 2 of 2.7 × 10 −2 at m ν4 = 1.8 GeV. See also [48]. Further peak searches for B + → e + ν 4 and B + → µ + ν 4 with Belle II would be valuable and could improve the limits from Ref. [125]. Moreover, when measurements of two-body leptonic decays of B + c mesons become available, it would also be of interest to use them to constrain lepton mixing matrix coefficients.
As was true for the other decays, in obtaining these limits from leptonic B decays, it is assumed that the only new physics is the emission of the massive ν 4 . However, in the B system there are currently several quantities whose experimental measurements are in possible tension with SM predictions, including, for example, the ratios of branching ratios R(D ( * ) ) = BR(B → D ( * ) τν τ )/BR(B → D ( * ) ℓν ℓ ), where ℓ = e, µ, and the ratio R(K ( * ) ) = BR(B → K ( * ) e + e − )/BR(B → K ( * ) µ + µ − ) (see, e.g., [126,127]).

A. General Analysis with Massive Neutrino Emission
In this section we discuss constraints from µ decays. The lifetime of the µ + was measured to 0.5 ppm accuracy by the MuLan experiment at PSI [128], yielding the value G F = 1.1663787(6)×10 −5 GeV −2 with the implicit assumption of decays only into the three known neutrino mass eigenstates. With this assumption, the uncertainty in this determination of G F is mainly from the experimental measurement; it is estimated that the uncertainty due to radiative corrections [129][130][131] is approximately 0.14 ppm and the uncertainty from the measured value of m µ is 0.08 ppm [128].
However, as was pointed out and analyzed in [37,39], in the presence of neutrino masses and lepton mixing, the decay µ → ν µ eν e would actually consist of the decays µ → ν i eν j into the individual mass eigenstates ν i andν j in the interaction eigenstates ν µ andν e , where 1 ≤ i, j ≤ 3 + n s , as allowed by phase space. The emission of massive neutrino(s) with non-negligible mass(es) in muon decay would produce several changes relative to the Standard Model. These include (i) kink(s) in the observed electron energy spectrum associated with the fact that the maximum electron energy in the rest frame of the parent µ is reduced from its SM value with neutrinos of negligibly small masses, (ii) reduction of the differential and total decay rate; (iii) a reduction in the apparent value of the Fermi coupling G F , relative to its value in the Standard Model with neutrinos of negligibly small masses; and (iv) changes in the spectral parameters ρ and η, and, for a polarized muon, ξ, and δ, that have been used to fit the differential decay spectrum of the muon. Ref. [39] calculated the changes in these spectral parameters that would be caused by emission of a massive (anti)neutrino in µ decay and used existing data to set upper limits on lepton mixing coefficients as functions of neutrino mass. From data on the ρ parameter describing the e + momentum distribution in unpolarized µ + decay, Ref. [39] derived an upper limit on |U r4 | 2 , where r = e, µ in the interval m ν4 up to 70 MeV, extending down to a few times 10 −3 at m ν4 = 30 MeV. This constraint applies to both |U e4 | 2 and |U µ4 | 2 since the ν 4 orν 4 can be emitted at either the charged-current vertex with the µ or with the e. The upper bound on |U e4 | 2 from µ decay is not as restrictive as upper bounds from π e2 or K e2 decay. However, the upper bound on |U µ4 | 2 from µ decay is valuable for an interval of m ν4 that is not covered by peak search experiments, namely the interval above the kinematic endpoint for π µ2 decay at m ν4 = 33.9 MeV and below the value of m ν4 ≃ 40 MeV, which was the lowest value at which a K µ2 peak search experiment (at KEK [82]) obtained an upper limit on |U µ4 | 2 . In [40,132] it was pointed out that because, in the presence of massive neutrino emission in µ decay, the value of G F,app extracted in the framework of the SM is smaller than the true value of G F , this would lead to predictions of the masses of the W and Z, that would be larger than the true values, and these effects were calculated. Subsequent discussions of massive neutrino effects in µ decay include [32], [133], [134], [135], and [48]. In particular, the TWIST experiment at TRIUMF measured ρ with greater accuracy [49]. Using an analysis similar to that in [39] applied to the TWIST data, one obtains upper limits on |U µ4 | 2 extending down to 2×10 −3 at m ν4 = 30 MeV (e.g., [48]). Let us consider the change in the total rate as a consequence of muon decays to a neutrino mass eigenstate ν 4 with a non-negligible mass. In the SM with neutrinos of negligibly small mass, the rate for µ decay has the form Here we have separated off a rate factor where f is a dimensionless kinematic function resulting from the integration over the three-body final-state phase space, which depends on three arguments, namely the (squares of the) ratios of each of the final-state particle masses to the muon mass. Finally, in Eq. e . For our case, from the general formulas in [39], the µ decay rate is given by wheref (x, y, z)) is the ratio of the kinematic phase space integral for each of the decays divided by the kinematic integral for the SM decay (8.5): Here and below, the kinematic function f (x, y, z) = 0 if the decay is kinematically forbidden, i.e., if The four terms in Eq. (8.8) arise from the decays (a) µ → ν i eν j ; (b) µ → ν 4 eν i ; (c) µ → ν i eν 4 ; and (d) µ → ν 4 eν 4 , where here ν i and ν j denote the known three neutrino mass eigenstates, whose masses are negligibly small in µ decay. Note that the second and third terms are present only if m µ > m e + ν 4 , and the fourth term is present only if m µ > m e +2m ν4 . Furthermore, the fourth term is strongly suppressed because it involves the product of the squares of two small leptonic mixing matrix coefficients, |U e4 | 2 |U µ4 | 2 , and because of the smaller phase space if m ν4 /m µ is substantial. Hence, to evaluate Eq. (8.8) forΓ µ , to a very good approximation, we may drop the last term, and hence we need only the kinematic function f (x, y, 0), which was calculated in Ref. [39]. A basic symmetry property of the kinematic function is that [39] f (x, y, z) = f (x, z, y) , (8.11) so the second and third terms in Eq. (8.8) have the same kinematic factor,f (a ν4 , 0). The apparent value of the Fermi coupling, G F,app , obtained from the measurement of the µ decay rate is given by 8.12) and is less than unity if (anti)neutrinos with nonnegligible masses are emitted in µ decay [39]. Explicitly, In the SM, the predicted mass of the Z is determined in terms of α = e 2 /(4π), the weak mixing angle θ W = arctan(g ′ /g), and G F,app by m Z,pred = πα 2 1/2 G F,app 14) and m W,pred = m Z,pred cos θ W , where δ Z,RC is the radiative correction [136]. As pointed out in [40,132], in the presence of massive neutrino emission in µ decay, these predicted values of m Z and m W would be larger than the true values, since G F,app < G F : m Z,true = κ 1/2 m Z,pred < m Z,pred (8.15) and m W,true = κ 1/2 m W,pred < m W,pred . (8.16) The effects on the W and Z widths were also discussed in [40,132]. The agreement between the predicted and observed masses and widths of the W and Z thus yield constraints on leptonic mixing angles as functions of m ν4 .
As mentioned above, the test of relative agreement of F t values obtained from the set of 14 superallowed nuclear beta decays in [63,64] is independent of G F,app since this divides out in the ratios of the F t values. However, depending on m ν4 , |U e4 | 2 , and |U µ4 | 2 , the result would generically be that the value of |V ud | obtained from these nuclear beta decays would not be equal to the true value, because of both the reduction of the rates for the various nuclear beta decays and the fact that the value of G F,app used in Eq. (2.1) would be different from the true value. In turn, this would generically lead to a spurious apparent violation of the first-row CKM unitarity test. Whether the apparent value of |V ud | would be larger or smaller than the true value would depend on the values of m ν4 , |U e4 | 2 , and |U µ4 | 2 and thus on the relative effects of massive neutrino emission in muon decay and in the nuclear beta decays used to obtain |V ud |.
Since the determination of |V ud | from the superallowed nuclear beta decays depends on the input value of G F,app from muon decay, an apparent violation of the first-row CKM unitarity relation Σ = 1 could indicate the presence of effects of new physics beyond the Standard Model (BSM) in muon decay. Although our discussion above has focused on the effect of the possible emission of neutrino(s) of non-neglible masses and couplings in muon decay, we note that there could also be exotic muon decays in BSM scenarios that would appear observationally to be the same as µ + →ν µ e + ν e , i.e., µ + → e + + missing neutrals, where the additional neutral particles are weakly interacting. An explicit example studied in the context of supersymmetric extensions of the SM was the decay µ + → e +γγ , whereγ denotes the photino [137]. An analogous decay involving hadrons was K + → π +γγ [138], which would appear observationally as K + → π + + missing neutrals and hence would be experimentally indistinguishable from the SM decay K + → π + νν [139]. (In modern notation, these decays would be denoted as µ + → e +χ0χ0 and K + → π +χ0χ0 , whereχ 0 is a neutralino.) As was noted in [137], the existence of the decay µ + → e +γγ by itself would lead to an apparent value of G F,app larger than the true value, opposite to the effect of massive neutrino emission. Another possibility for an exotic µ decay is µ → e + x, where x is a neutral, light, weakly interacting boson; upper limits on this were given in [140] [141]. Another example of this type of additional exotic µ decay was studied in a model with dynamical electroweak symmetry breaking [142], in which the µ + → e + transition would be mediated by a neutral virtual massive generation-changing vector boson, which then would produce a final-stateν µ ν e pair (see also [143]).

B. Limit on Exotic µ Decay Modes
If there are no light sterile neutrinos relevant for µ decay, but there are additional exotic muon decays such as in the examples above, then, since the experimentally extracted value of G F,app would be larger than the true G F , the resultant apparent value of |V ud | obtained from the superallowed nuclear beta decays, denoted |V ′ ud |, would be smaller than the true value. In turn, this would yield an apparent spurious violation of CKM unitarity in which the apparent value of Σ would be less than unity. Since an exotic BSM decay channel would increase Γ µ relative to the SM value Γ µ,SM , while emission of heavy neutrino(s) would decrease Γ µ relative to Γ µ,SM , it is possible, in principle, for both of these non-SM effects to be present and to tend to cancel each other, yielding a resultant Γ µ close to Γ µ,SM . However, in the absence of any symmetry reason, such a cancellation may be regarded as unlikely. Accordingly, in our analyses, we will treat each of these two cases individually.
If one considers the possibility that no heavy sterile (anti)neutrinos are emitted in µ decay but instead, there is an exotic extra decay channel (indicated with subscript ext) with rate Γ µ,ext , then the total decay rate would be Γ µ = Γ µ,SM + Γ µ,ext . Let us denote Γ µ,SM ≡ G 2 FΓ µ,SM and the branching ratio of the exotic decay mode as BR µ,ext = Γ µ,ext /Γ µ . Experimentalists would then extract the apparent value G F,app as Assuming that the BSM physics responsible for the additional contribution, Γ µ,ext , to µ decay does not affect nuclear beta decays, then the resultant apparent value of |V ′ ud | 2 obtained from the superallowed nuclear beta decays would be given by For our present analysis, let us further assume that the BSM physics leading to this value would not affect the decays used to determine |V us | and |V ub |. The apparent value of Σ, denoted Σ app , would then be Assuming CKM unitarity, i.e., Σ = 1, we then have Presuming that this is responsible for Σ app being less than unity and using the experimentally determined value and uncertainty in Eq. (4.1), IX. CONSTRAINTS FROM LEPTONIC τ DECAYS As with nuclear beta decay and the two-body leptonic decays of pseudoscalar mesons, semihadronic τ decays have the simplifying property of only involving a single leptonic charged-current vertex in their amplitudes, so one may define an effective mass m τ,ef f = [ i |U τ,i | 2 m 2 νi ] 1/2 . The best upper limit m ντ ,ef f < 18.2 MeV (95% C.L.) [147] comes from semihadronic τ decays.
As in the case of µ decay, one can analyze leptonic τ decays in the presence of possible sterile neutral emission; see Table II in [39] and also Ref. [135]. We denote the τ → ν τ eν e mode as τ → e and the τ → ν τ µν µ as τ → µ for short and define a reduced, dimensionless decay ratē Γ τ →ℓ via the equation where we have used the fact that the leading-order correction, δ α , is mass-independent. In the Standard Model with neutrinos of negligible masses, Using a Just as in Eq. (8.8) for µ, the term involving emission of ν 4ν4 is negligibly small relative to the other terms because it involves the product of two small leptonic mixing matrix elements squared, |U ℓ4 | 2 |U τ 4 | 2 , and because of the greater kinematic suppression of the decay into ν 4ν4 for substantial m ν4 ; one can therefore drop the final term in Eq. (9.3). The kinematic function f (x, y, 0) was calculated in [39]. It is worthwhile to inquire what can be learned from a purely leptonic observable which can be calculated and measured to high precision, namely (9.4) and the resultant ratiō We comment below on studies that also include semihadronic τ decays. Measurements of the individual branching ratios for τ → ν τ eν e and τ → ν τ µν µ have been carried out, with the results [13] BR τ →ντ eνe = 0.1782 ± 0.0004 (9.6) and BR τ →ντ µνµ = 0.1739 ± 0.0004 .  The uncertainty in the theoretical prediction (9.9) is small compared with the uncertainty in the experimental measurement (9.8). Note that the leading-order radiative correction term (1+δ α ) divides out in the ratio (9.9) since it is mass-independent. Thus, The simplest situation applies if m ν4 is sufficiently large that all of the decays τ → ν 4 eν j and τ → ν 4 µν j , where 1 ≤ j ≤ 4, are kinematically forbidden. In this case, Let us investigate a hierarchical lepton mixing situation in which |U ℓ4 | 2 << |U τ 4 | 2 for ℓ = e, µ. This is effectively equivalent to using the upper limits m νe,ef f < 2 eV and m νµ,ef f < 0.19 MeV [13] to infer that these have a negligible effect on the ratio R (τ ) e/µ . Then , (9.12) where we have used the symmetry (8.11). Solving Eq. (9.12) for |U τ 4 | 2 , we get (9.13) With (9.10), we obtain a 95 % CL upper bound on |U τ 4 | 2 that extends down to below 10 −2 as m ν4 increases to 1 GeV. More stringent constraints have been obtained from semihadronic decays [42][43][44]145]. One can also use the measured branching ratios (9.6) and (9.7) and the τ lifetime τ τ = (2.903 ± 0.005) × 10 −13 s [13] in comparison with the decay rates calculated using the MuLan value for G F to obtain limits on m ντ ,ef f . The definition of m ντ ,ef f is more complicated here than in nuclear beta decays and two-body leptonic decays of pseudoscalar mesons, where only a single charged-current vertex is involved, so m νe,ef f = [ j |U ei | 2 m 2 νj ] 1/2 and m νµ,ef f = [ j |U µi | 2 m 2 νj ] 1/2 , where the sums include all neutrino mass eigenstates that lead to the respective outgoing charged lepton with an energy or momentum such that it is included in the cuts used by a given experiment in its event reconstruction. In contrast, for leptonic τ decays, the amplitudes involve two charged-current vertices and hence products of lepton mixing matrices. If one assumes that the ν 4 is emitted via the τ − ν τ chargedcurrent coupling, then only the U τ j lepton mixing matrix element is relevant in the amplitude, and one can express m ντ ,ef f in an analogous manner, as m ντ ,ef f = [ j |U τ i | 2 m 2 νj ] 1/2 . Then, using the formulation in [146], one finds calculated values for the branching ratios (denoted by superscript (c)) of R τ →µ = 0.17293 ± 0.00030. Then, the ratios of experimental to calculated branching ratios are S τ →e = Γ τ →e /Γ τ →e,SM = 1.022 ± 0.0028 (9.14) and S τ →µ = Γ τ →µ /Γ τ →µ,SM = 1.0056 ± 0.0029. (9.15) Since the measured values exceed the calculated ones, we find the following 95% C.L. interval for the physical regions for massive neutrino emission i.e. S τ →e < 1 and S τ →µ < 1: 0.9964 < S τ →e < 1 (9.16) and 0.9982 < S τ →µ < 1 .

X. REMARKS ON SOME OTHER PARTICLE AND NUCLEAR PHYSICS CONSTRAINTS
Sterile neutrinos with masses in the range considered here are subject to a number of other constraints. We begin with a remark on K ℓ3 decays as potential sources of constraints on sterile neutrinos. These decays include K + → π 0 ℓ + ν ℓ , K 0 L → π + ℓ −ν ℓ , and K 0 L → π − ℓ + ν ℓ , where ℓ = e, µ. Since these K e3 decays are not helicitysuppressed, in contrast to M e2 decays, where M = π + , K + , etc., there is no associated enhancement of K ℓ3 decays into a massive ν 4 resulting from removal of helicity suppression, as is the case in M e2 decays. These K ℓ3 decays into a massive (anti)neutrino are subject to the usual three-body phase space suppression. The maximum ν 4 masses in the K e3 , (K 0 L ) e3 , K µ3 , and (K 0 L ) µ3 decays are 358, 362, 253, and 252 MeV, respectively. This mass range is already covered by the limits from peak search and branching ratio constraints from π ℓ2 and K ℓ2 experiments. Furthermore, the calculations of the rates for K ℓ3 and (K 0 L ) ℓ3 decays involve more uncertainty than for π ℓ2 and K ℓ2 because the hadronic amplitudes contain form factors whose dependence on q 2 (where q λ = p λ −p λ π is the four-momentum imparted to the outgoing ℓ − ν i orl + ν i pair) cannot be calculated from first principles. (For a recent discussion of parametrizations of these form factors, see [13].) The resultant uncertainty is only partially cancelled in ratios such as BR((K 0 L ) e3 )/BR((K 0 L ) µ3 ), since the (K 0 L ) e3 and (K 0 L ) µ3 involve different momenta transfers to the outgoing lepton pairs.
Next, it may be recalled that quite restrictive upper limits on mixings of mainly sterile heavy neutrinos have also been obtained from time-of-flight searches [148,149] and for neutrino decays [13,38,[150][151][152]. A recent search of this type is [153]. In the mass range of a few MeV, experiments have been performed to search for the decaȳ ν 4 → e + e − ν e usingν e beams from nuclear reactors [154][155][156]. These eventually obtained upper limits on |U e4 | 2 of 0.5 × 10 −2 at m ν4 = 1 MeV down to 3 × 10 −4 for m ν4 = 4 MeV, and then increasing to 0.6 × 10 −2 for m ν4 = 9.5 MeV [156]. From observations of the solar neutrino flux, the Borexino experiment has set upper bounds |U e4 | 2 of 10 −3 to 0.4×10 −5 for m ν4 from 1.5 MeV to 14 MeV [157]. However, since the conditions for the diagonality of the neutral weak current are violated in the presence of sterile neutrinos [35,36], a sterile neutrino may decay invisibly, as ν 4 → ν jνℓ ν ℓ . Other invisible neutrino decay modes occur in models in which neutrinos couple to a light scalar or pseudoscalar (for recent discussions and limits, see, e.g., [158]- [160] and references therein). Consequently, because of their model-dependence, we do not use limits on lepton mixing from neutrino decays here.
The situation is different for a heavy ν 4 in the mass range considered here. When considering how these limits might apply to the ν 4 , however, one must take into account the fact that there would be strong kinematic and mixing-angle suppression or exclusion of the initial emission of the heavyν 4 in the beta decays that yield theν e flux from a reactor, and a ν 4 with an MeV-scale mass would be kinematically forbidden from being emitted in the pp → D + e + + ν e reaction and the electroncapture transition e + 7 Be → 7 Li + ν e in the sun, since these have maximum energy releases of only 0.42 MeV and 0.86 MeV, respectively. Hence, one cannot necessarily apply the constraints on neutrino magnetic moments from reactor antineutrino and solar neutrino experiments to a heavy neutrino. Similarly, the constraint from stellar cooling is not directly relevant here because it only applies to neutrino mass eigenstates ν j with masses < ∼ 5 keV so that a plasmon in the star would be kinematically able to produce theν j ν j pair [13].
Finally, we comment on how a heavy neutrino could affect Higgs decays. Ref. [168] pointed out that the Higgs boson could have decays to invisible final states, and calculated rates for several of these, including decays to neutrinos. Currently, all of the decay branching ratios of the Higgs are in agreement with SM predictions, but these allow for a substantial branching ratio into invisible modes, BR(H → invisible) < ∼ 20 % [13,169,170]. The way in which the diagonal and nondiagonal couplings of neutrinos to the Higgs boson are related to the couplings U ℓ4 that enters in the weak charged current depends, in general, on details of a given model.

XI. CONCLUSIONS
One of the most important outstanding questions in nuclear and particle physics at present concerns whether light sterile neutrinos exist. In this paper we have presented a detailed analysis yielding new upper bounds on the squared lepton mixing matrix elements |U e4 | 2 and |U µ4 | 2 involved in the possible emission of a mostly sterile neutrino mass eigenstate, ν 4 , from analyses of a number of nuclear and particle decays. A brief report on the upper bounds on |U e4 | 2 was given in [1]. We have used recent advances in the precision of measured F t values for a set of superallowed nuclear beta decays to improve the upper limits on |U e4 | 2 obtained from these beta decays for a ν 4 with a mass in the range of a few MeV. From analyses of the ratios of branching ratios R (π) e/µ = BR(π + → e + ν e )/BR(π + → µ + ν µ ), R e/τ , and from B e2 and B µ2 decays, we have derived upper limits on couplings |U e4 | 2 and |U µ4 | 2 . Our bounds on |U e4 | 2 cover most of the ν 4 mass range from approximately 1 MeV to 1 GeV, and in several parts of this range they are the best bounds for a Dirac neutrino that do not make use of model-dependent assumptions on visible neutrino decays. We have also obtained a new upper bound on |U µ4 | 2 from a π µ2 peak search experiment searching for ν 4 emission via lepton mixing and have updated existing upper bounds on |U µ4 | 2 in the MeV to GeV mass range. New experiments to search for D + s → e + ν 4 and D + → e + ν 4 are suggested. These, as well as a continued search for B + → e + ν 4 and B + → µ + ν 4 decays, would be valuable and could further improve the bounds. In addition, we examined limits on |U e4 | 2 obtained from examining pion beta decay and showed that they are less stringent than those from superallowed beta decay in the same ν 4 mass range. As part of the analysis, we updated constraints from CKM unitarity on sterile neutrinos. In addition, we examined correlated constraints on lepton mixing matrix coefficients |U e4 | 2 , |U µ4 | 2 and |U τ 4 | 2 from analyses of leptonic decays of heavy-quark pseudoscalar mesons, from µ decay, and from leptonic τ decays.