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

TL;DR

This paper provides improved upper bounds on the coupling of sterile neutrinos to electrons across the MeV to GeV mass range by analyzing various nuclear and particle decay data.

Contribution

It presents new constraints on sterile neutrino couplings using a comprehensive analysis of multiple decay processes, extending the mass range of previous bounds.

Findings

Stronger upper bounds on $|U_{e4}|^2$ across MeV to GeV range.

Constraints derived from nuclear beta decays and meson decay ratios.

Enhanced limits improve the parameter space for sterile neutrino models.

Abstract

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.

Equations

$${\nu }_{\ell }=\sum _{i=1}^{3+n_s}{U}_{\ell i}{\nu }_i,$$

$$\frac{dN}{dE}=C\bigg{[}(1-|U_{e4}|^{2})pE(E_{0}-E)^{2}+|U_{e4}|^{2}pE(E_{0}-E)\Big{[}(E_{0}-E)^{2}-m_{\nu_{4}}^{2}\Big{]}^{1/2}\,\theta(E_{0}-E-m_{\nu_{4}})\bigg{]}\ ,$$

$$|U_{e4}{|}^2<4\times 10^{-4}$$

$$|U_{e4}{|}^2<2.7\times 10^{-4}$$

$$R_{\ell /{\ell }^{\prime }}^{(M)}\equiv \frac{BR(M^{+}\to {\ell }^{+}{\nu }_{\ell })}{BR(M^{+}\to {\ell }^{\prime +}{\nu }_{{\ell }^{\prime }})}$$

$$R_{\ell /{\ell }^{\prime },SM}^{(M)}$$

$${\bar{R}}_{\ell /{\ell }^{\prime }}^{(M)}\equiv \frac{R_{\ell /{\ell }^{\prime }}^{(M)}}{R_{\ell /{\ell }^{\prime },SM}^{(M)}}.$$

$${\bar{R}}_{e/\mu }^{(\pi )}=0.9980\pm 0.0019.$$

$$R_{e/\mu }^{(K)}=(2.488\pm 0.009)\times 10^{-5}.$$

$$R_{e/\mu ,SM}^{(K)}=(2.477\pm 0.001)\times 10^{-5},$$

$${\bar{R}}_{e/\mu }^{(K)}=1.0044\pm 0.0037.$$

$$R_{\ell /{\ell }^{\prime }}^{(M)}$$

$$\times$$

$$\rho (x,y)=[x+y-(x-y{)}^2][\lambda (1,x,y){]}^{1/2},$$

$$\lambda (z,x,y)=x^2+y^2+z^2-2(xy+yz+zx).$$

$$\bar{\rho}(x,y)=\frac{\rho(x,y)}{\rho(x,0)}=\frac{\rho(x,y)}{x(1-x)^{2}}\$$

$${\bar{R}}_{\ell /{\ell }^{\prime }}^{(M)}=\frac{1-|U_{\ell 4}{|}^2+|U_{\ell 4}{|}^2\bar{\rho }({\delta }_{\ell }^{(M)},{\delta }_{{\nu }_4}^{(M)})}{1-|U_{{\ell }^{\prime }4}{|}^2+|U_{{\ell }^{\prime }4}{|}^2\bar{\rho }({\delta }_{{\ell }^{\prime }}^{(M)},{\delta }_{{\nu }_4}^{(M)})}.$$

$$|U_{\ell 4}{|}^2<\frac{{\bar{R}}_{\ell /{\ell }^{\prime }}^{(M)}-1}{\bar{\rho }({\delta }_{\ell }^{(M)},{\delta }_{{\nu }_4}^{(M)})-1}\mathrm{for}{m}_{{\nu }_4}\in I_2^{(M)}.$$

$$|U_{\ell 4}{|}^2<1-{\bar{R}}_{\ell /{\ell }^{\prime }}^{(M)}\mathrm{for}{m}_{{\nu }_4}\in I_3^{(M)}.$$

$$|U_{\ell 4}{|}^2<\frac{{\bar{R}}_{e/{\mu }^{\prime }}^{(\pi )}-1}{\bar{\rho }({\delta }_e^{(\pi )},{\delta }_{{\nu }_4}^{(\pi )})-1}<\frac{0.0014}{\bar{\rho }({\delta }_e^{(\pi )},{\delta }_{{\nu }_4}^{(\pi )})-1}.$$

$$|U_{\ell 4}{|}^2<\frac{{\bar{R}}_{e/{\mu }^{\prime }}^{(K)}-1}{\bar{\rho }({\delta }_e^{(K)},{\delta }_{{\nu }_4}^{(K)})-1}<\frac{0.010}{\bar{\rho }({\delta }_e^{(K)},{\delta }_{{\nu }_4}^{(K)})-1}.$$

$$R_{e/\tau ,SM}^{(D_s)}=2.29\times 10^{-6}.$$

$$\frac{BR(M^{+}\to {\ell }^{+}{\nu }_4)}{BR(M^{+}\to {\ell }^{+}{\nu }_{\ell }{)}_{SM}}=\frac{|U_{\ell 4}{|}^2\bar{\rho }({\delta }_{\ell }^{(M)},{\delta }_{{\nu }_4}^{(M)})}{1-|U_{\ell 4}{|}^2}.$$

Full text

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). Most oscillation experiments with solar, atmospheric, accelerator, and reactor (anti)neutrinos can be explained within the minimal framework of three neutrino mass eigenstates with values of $\Delta m^{2}_{ij}=m_{\nu_{i}}^{2}-m_{\nu_{j}}^{2}$ given approximately by $\Delta m^{2}_{21}=0.74\times 10^{-4}$ eV2 and $|\Delta m^{2}_{32}|=2.5\times 10^{-3}$ eV2, with normal mass ordering $m_{\nu_{3}}>m_{\nu_{2}}$ favored; furthermore, the lepton mixing angles $\theta_{23}$, $\theta_{12}$, and $\theta_{13}$ have been measured, with a tentative indication of a nonzero value of the CP-violating quantity $\sin(\delta_{CP})$ pdg2018 -cggnov .

In addition to the three known neutrino mass eigenstates, there could be others, which would necessarily be primarily electroweak-singlets (sterile) st . 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 $\nu_{\ell}$ would be given by

[TABLE]

where $\ell=e,\ \mu,\ \tau$; $n_{s}$ denotes the number of sterile neutrinos; and $U$ is the lepton mixing matrix anom .

Here we obtain improved upper limits on $|U_{ei}|^{2}$ for a sterile neutrino $\nu_{i}$ 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 $n_{s}\geq 2$. Since a $\nu_{4}$ in this mass range decays, it is not excluded by the cosmological upper limit on the sum of effectively stable neutrinos, \sum_{i}m_{\nu_{i}}\mathrel{\raisebox{-2.58334pt}{\stackrel{{\scriptstyle\textstyle<}}{{\sim}}}}0.12 eV planck2018 . Such a $\nu_{4}$ is subject to a number of constraints from cosmology (e.g., cosm ); 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 $\nu_{4}$ is a Dirac neutrino dirac . Since sterile neutrinos violate the conditions for the diagonality of the weak neutral current leeshrock77 ; sv80 , $\nu_{4}$ has invisible tree-level decays of the form $\nu_{4}\to\nu_{j}\bar{\nu}_{i}\nu_{i}$ where $1\leq i,j\leq 3$ 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 $\nu_{4}$ via lepton mixing in nuclear beta decay has several effects, including producing a kink in the Kurie plot and reducing the decay rate shrock80 . For the nuclear beta decays $(Z,A)\to(Z+1,A)+e^{-}+\bar{\nu}_{e}$ or $(Z,A)\to(Z-1,A)+e^{+}+\nu_{e}$ into a set of neutrino mass eigenstates $\nu_{i}\in\nu_{e}$, $i=1,2,3$ of negligibly small masses, plus a $\nu_{4}$ of of non-negligible mass, the differential decay rate is

[TABLE]

where $p\equiv|{\mathbf{p}}|$ and $E$ denote the 3-momentum and (total) energy of the outgoing $e^{\pm}$, $E_{0}$ denotes its maximum energy for the SM case, the Heaviside $\theta$ function is defined as $\theta(x)=1$ for $x>0$ and $\theta(x)=0$ for $x\leq 0$, and $C=G_{F}^{2}|V_{ud}|^{2}F_{F}|{\cal M}|^{2}/(2\pi^{3})$, where ${\cal M}$ 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 shrock80 and analyses of particle decays shrock81a -shrock_vpi , and from dedicated experiments. For example, a search for kinks in the Kurie plot in 20F beta decay reported in Ref. deutsch1990 yielded an upper bound on $|U_{e4}|^{2}$ decreasing from $5.9\times 10^{-3}$ for $m_{\nu_{4}}=0.4$ MeV to $1.8\times 10^{-3}$ for $m_{\nu_{4}}=2.8$ MeV. (Some recent reviews of searches for sterile neutrinos include kusenkorev -batell2018 .)

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 $\mu$ decay, the measurement of the $\mu$ 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 $G_{F}/\sqrt{2}=g^{2}/(8m_{W}^{2})$, where $g$ is the SU(2) gauge coupling shrock81a ; shrock81b ; shrock_vpi . 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, $t\equiv t_{1/2}$, yields the product $ft$. Incorporation of nuclear and radiative corrections yields the corrected $ft$ value for a given decay, denoted ${\cal F}t$. Conventionally, analyses of the ${\cal F}t$ values for the most precisely measured superallowed $0^{+}\to 0^{+}$ nuclear beta decays have been used, in conjunction with the value of $G_{F,app}$ from $\mu$ decay, to infer a value of the weak mixing matrix element, $|V_{ud}|$ ht75 -vudapp . A first step in these analyses has been to establish the mutual consistency of the ${\cal F}t$ values for these superallowed $0^{+}\to 0^{+}$ decays. Since the emission of a $\nu_{4}$ 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.

Hence, from this mutual agreement of ${\cal F}t$ values, an upper limit on $|U_{e4}|^{2}$ can be derived for values of $m_{\nu_{4}}$ such that a $\nu_{4}$ could be emitted in some of these superallowed decays. Ref. hardy_towner1990 obtained upper bounds on $|U_{e4}|^{2}$ ranging from $10^{-2}$ down to $2\times 10^{-3}$ for $m_{\nu_{4}}$ from 0.5 to 2 MeV, while Ref. deutsch1990 obtained the limits $|U_{e4}|^{2}<1\times 10^{-3}$ to $|U_{e4}|^{2}<2\times 10^{-3}$ for $m_{\nu_{4}}$ from 1 to 7 MeV. The maximum $Q$ value in the current set of 14 superallowed $0^{+}\to 0^{+}$ beta decays used for the ${\cal F}t$ fit in hardy_towner2015 ; hardy_towner2018 is 9.4 MeV (for 74Rb). A measure of the mutual agreement 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 upper limit on $|U_{e4}|^{2}$. Ref. hardy_towner1990 obtained $|V_{ud}|=0.9740\pm 0.001$. The recent analyses in hardy_towner2018 and ramsey_musolf obtain $|V_{ud}|=0.97420(21)$ and $|V_{ud}|=0.97370(14)$, respectively ckmu . Applying these factors of improvement from hardy_towner2018 and ramsey_musolf to the previous bounds in hardy_towner1990 , improved upper bounds are obtained as

[TABLE]

and

[TABLE]

for $\nu_{4}$ masses in the range 0.5\ {\rm MeV}\mathrel{\raisebox{-2.58334pt}{\stackrel{{\scriptstyle\textstyle<}}{{\sim}}}}m_{\nu_{4}}<9.4 MeV, indicated in Fig. 1 as BD2. (These and other limits presented are at the 90 % confidence level.)

We next discuss upper bounds from two-body leptonic decays of charged pseudoscalar mesons (generically denoted as $M^{+}$) shrock80 ; shrock81a . This method is quite powerful, because the signal is a monochromatic peak in $dN/dp_{\ell}$ 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 $\nu_{4}$ also changes the ratio $BR(M^{+}\to e^{+}\nu_{e})/BR(M^{+}\to\mu^{+}\nu_{\mu})$ from its SM value,, and this was used to set further bounds shrock80 ; shrock81a ; bryman83b . A number of dedicated experiments have been performed to search for a peak due to heavy neutrino emission and also to measure $BR(M^{+}\to e^{+}\nu_{e})/BR(M^{+}\to\mu^{+}\nu_{\mu})$ with $\pi^{+}_{\ell 2}$, $K^{+}_{\ell 2}$, and $B^{+}_{\ell 2}$, where $\ell=e,\ \mu$ abela81 -triumf_pimu2 .

In the SM with only the three known neutrinos with negligibly small masses, the ratio

[TABLE]

is given by

[TABLE]

where $\delta^{(M)}_{\ell}=m_{\ell}^{2}/m_{M}^{2}$ and $\delta_{RC}$ is the radiative correction (RC) earlyrad -sdb .

We denote the ratio of the experimental measurement of $R^{(M)}_{\ell/\ell^{\prime}}$ to the SM prediction as

[TABLE]

The most precise measurement of $R^{(\pi)}_{e/\mu}$ is from the PIENU experiment at TRIUMF, with the result $R^{(\pi)}_{e/\mu}=(1.2344\pm 0.0023_{stat}\pm 0.0019_{syst})\times 10^{-4}$ pienu2015 . The resultant PDG world average is $R^{(\pi)}_{e/\mu}=(1.2327\pm 0.0023)\times 10^{-4}$  pdg2018 , in agreement with the SM prediction with RC, $R^{(\pi)}_{e/\mu}=(1.2352\pm 0.0002)\times 10^{-4}$  ms93 ; cirigliano07 ; annrev2011 . Using the PDG value of $R^{(\pi)}_{e/\mu}$, one finds

[TABLE]

The ratio $R^{(K)}_{e/\mu}$ has recently been measured by the NA62 experiment at CERN na62_kemu , dominating the world average pdg2018

[TABLE]

The SM prediction with RC cirigliano07 ; sdb is

[TABLE]

resulting in

[TABLE]

With emission of a heavy neutrino $\nu_{4}$, the ratio $R^{(M)}_{\ell/\ell^{\prime},SM}$ for general $\ell,\ \ell^{\prime}$ changes to

[TABLE]

where $\delta^{(M)}_{\nu_{4}}=m_{\nu_{4}}^{2}/m_{M}^{2}$, and the kinematic function $\rho(x,y)$ is shrock80 ; shrock81a

[TABLE]

with

[TABLE]

Thus, in the SM case, $\rho(x,0)=x(1-x)^{2}$. Here and below, it is implicitly understood that $\rho(\delta^{(M)}_{\ell},\delta^{(M)}_{\nu_{4}})=0$ if $m_{\nu_{4}}\geq m_{M}-m_{\ell}$, where the decay $M^{+}\to\ell^{+}\nu_{4}$ is kinematically forbidden. We define

[TABLE]

so

[TABLE]

With no loss of generality, we order $\ell$ and $\ell^{\prime}$ such that $m_{\ell^{\prime}}>m_{\ell}$ and define the mass intervals (i) $I^{(M)}_{1}:\ m_{\nu_{4}}<m_{M}-m_{\ell^{\prime}}$; (ii) $I^{(M)}_{2}:\ m_{M}-m_{\ell^{\prime}}<m_{\nu_{4}}<m_{M}-m_{\ell}$; and (iii) $I^{(M)}_{3}:\ m_{\nu_{4}}>m_{M}-m_{\ell}$. Thus, a $\nu_{4}$ with $m_{\nu_{4}}\in I^{(M)}_{1}$ contributes to both $M^{+}_{\ell 2}$ and $M^{+}_{\ell^{\prime}2}$ decays, while if $m_{\nu_{4}}\in I^{(M)}_{2}$, then $\nu_{4}$ contributes to $M^{+}_{\ell 2}$, but not to $M^{+}_{\ell^{\prime}2}$ decay, and if $m_{\nu_{4}}\in I^{(M)}_{3}$, then $\nu_{4}$ cannot be emitted in either $M^{+}_{\ell 2}$ or $M^{+}_{\ell^{\prime}2}$ decay.

If for a given $m_{\nu_{4}}$, one knows, e.g., from peak-search experiments, that $|U_{\ell^{\prime}4}|^{2}$ is sufficiently small that the denominator of (18) can be approximated well by 1, then an upper bound on the deviation of $\bar{R}^{(M)}_{\ell/\ell^{\prime}}$ from 1 yields an upper bound on $|U_{\ell 4}|^{2}$. Thus, one has the bound

[TABLE]

This gives very stringent upper limits on $|U_{\ell 4}|^{2}$ because $\bar{\rho}(\delta^{(M}_{e},\delta^{(M)}_{\nu_{4}})>>1$ over much of the interval $I^{(M)}_{2}$ (see Figs. 3-5 in shrock81a ). If $m_{\nu_{4}}\in I^{(M)}_{3}$, then (18) reduces to $\bar{R}^{(M)}_{\ell/\ell^{\prime}}=(1-|U_{\ell 4}|^{2})/(1-|U_{\ell^{\prime}4}|^{2})$, so if $|U_{\ell^{\prime}4}|^{2}<<1$ in this interval, then the upper limit is

[TABLE]

We now apply this analysis to $R^{(\pi)}_{e/\mu}$, using (19) and (20) with $M^{+}=\pi^{+}$, $\ell=e$, and $\ell^{\prime}=\mu$. From previous $\pi^{+}_{\mu 2}$ peak search experiments abela81 -triumf_pimu2 and the calculation of $\bar{\rho}(\delta^{(\pi)}_{\mu},\delta^{(\pi)}_{\nu_{4}})$, it follows that $|U_{\mu 4}|^{2}$ is sufficiently small for $m_{\nu_{4}}\in I^{(\pi)}_{2}$ that we can approximate the denominator of Eq. (18) by 1. From $\bar{R}^{(\pi)}_{e/\mu}$ in Eq. (8), using the procedure from feldman-cousins , we obtain the limit $\bar{R}^{(\pi)}_{e/\mu}<1.0014$. Then, for $\nu_{4}\in I^{(\pi)}_{2}$, we find

[TABLE]

This bound is labelled as PIENU in Fig. 1. If $m_{\nu_{4}}\in I^{(\pi)}_{3}$, i.e., $m_{\nu_{4}}>139$ MeV, then, using (20), we obtain the upper bound on $|U_{e4}|^{2}$ given by the flat line labelled PIENU-H in Fig. 1.

We next obtain a bound on $|U_{e4}|^{2}$ by applying the same type of analysis to $R^{(K)}_{e/\mu}$. From $K_{\mu 2}$ peak search experiments asano81 ; bnl949 ; na62_2018 and the calculation of $\bar{\rho}(\delta^{(K)}_{\mu},\delta^{(K)}_{\nu_{4}})$, $|U_{\mu 4}|^{2}$ is sufficiently small that we can approximate the denominator of Eq. (18) well by 1. Using Eq. (11) for $\nu_{4}\in I^{(K)}_{2}$, we find

[TABLE]

This upper limit on $|U_{e4}|^{2}$ is labelled KENU in Fig. 1. For $m_{\nu_{4}}\in I^{(K)}_{3}$, i.e., $m_{\nu_{4}}>493$ MeV, using (20), we obtain the flat upper bound labelled KENU-H in Fig. 1.

One can also apply these methods to two-body leptonic decays of heavy-quark hadrons. We first consider $D_{s}^{+}\to\ell^{+}\nu_{\ell}$ decays cc , using (19) and (20) with $M^{+}=D_{s}^{+}$, $\ell=e$, and $\ell^{\prime}=\tau$. Experimental data from CLEO, BABAR, Belle, and BES have determined $BR(D_{s}^{+}\to\mu^{+}\nu_{\mu})=(5.49\pm 0.17)\times 10^{-3}$ and $BR(D_{s}^{+}\to\tau^{+}\nu_{\tau})=(5.48\pm 0.23)\times 10^{-2}$ alexander09 -ablikim19 . Furthermore, searches by CLEO alexander09 , BABAR babar10 , and Belle zupanc13 have yielded the limit $BR(D_{s}^{+}\to e^{+}\nu_{e})<0.83\times 10^{-4}$. Hence, $R^{(D_{s})}_{e/\tau}<1.6\times 10^{-3}$. For $R^{(D_{s})}_{e/\tau}$, using the results of ms93 , we calculate $1+\delta_{RC}=0.948$. Substituting this in Eq. (6) with $M=D_{s}$, $\ell=e$, $\ell^{\prime}=\tau$, we find

[TABLE]

Therefore, $\bar{R}^{(D_{s})}_{e/\tau}<7.0\times 10^{2}$. For $R^{(D_{s})}_{e/\tau}$, the interval $I^{(D_{s})}_{2}$ is $191\ {\rm MeV}<m_{\nu_{4}}<1.457$ GeV. Actually, we restrict $m_{\nu_{4}}$ to a lower-mass subset of this interval, because for sufficiently great $m_{\nu_{4}}$, even though the $D_{s}^{+}\to e^{+}\nu_{4}$ decay is kinematically allowed to occur, the momentum $p_{e}$ (in the $D_{s}$ rest frame) would be below the minimal value set by experimental cuts in the BES III event reconstruction. With $p_{e,cut}\simeq 0.8$ GeV bes3_pc , this means that $m_{\nu_{4}}$ must be less than 0.85 GeV for the event to be accepted. Thus, we consider $0.191\ {\rm GeV}\ <m_{\nu_{4}}<0.85$ GeV. Substituting the experimental limit on $\bar{R}^{(D_{s})}_{e/\tau}$ in the special case of (18) with $M=D_{s}$, $\ell=e$, $\ell^{\prime}=\tau$ and using the fact that $|U_{\tau 4}|^{2}<<1$ for this $m_{\nu_{4}}$ mass range pdg2018 , we obtain a resultant limit from (19). For $m_{\nu_{4}}=0.191$ GeV, $\bar{\rho}(\delta_{e}^{(D_{s})},\delta^{(D_{s})}_{\nu_{4}})=1.37\times 10^{5}$, increasing to $\bar{\rho}(\delta_{e}^{(D_{s})},\delta^{(D_{s})}_{\nu_{4}})=1.83\times 10^{6}$ for $m_{\nu_{4}}=0.85$ GeV. We thus obtain the upper bound on $|U_{e4}|^{2}$ labelled $D_{se2}$ in Fig. 1.

A dedicated peak-search experiment to search for the heavy-neutrino decay $D_{s}^{+}\to e^{+}\nu_{4}$ would be worthwhile and could improve the upper bound on $|U_{e4}|^{2}$. Similarly, a search for leptonic $D$ decays like $D^{+}\to e^{+}\nu_{4}$ would be valuable and will be discussed elsewhere DB_RS . The very large values of $\bar{\rho}(\delta_{e}^{(D_{s})},\delta^{(D_{s})}_{\nu_{4}})$ and $\bar{\rho}(\delta_{e}^{(D)},\delta^{(D)}_{\nu_{4}})$ over a large portion of the kinematically allowed ranges of $m_{\nu_{4}}$ in $D_{s}^{+}\to e^{+}\nu_{4}$ and $D\to e^{+}\nu_{4}$ mean that there would be quite strong kinematic enhancement of the heavy neutrino decay relative to the corresponding $(D_{s}^{+})_{e2}$ and $D^{+}_{e2}$ decays. In particular, these searches could be performed by the BES III experiment, which recently reported results from a data sample of 3.19 fb*-1* and expects to collect considerably higher statistics.

Finally, we consider $B^{+}\to\ell^{+}\nu_{\ell}$ decays. There is an upper limit $BR(B^{+}\to e^{+}\nu_{e})<0.98\times 10^{-6}$ from Belle satoyama07 and BABAR aubert09 . For the other two leptonic decay modes, $BR(B^{+}\to\mu^{+}\nu_{\mu})=(6.46\pm 2.22_{stat}\pm 1.60_{syst})\times 10^{-7}$ from Belle belle_bmunu2018 , with a recent update $BR(B^{+}\to\mu^{+}\nu_{\mu})=(5.3\pm 2.0_{stat}\pm 0.9_{syst})\times 10^{-7}$ belle_moriond2019 ; moriond2019 , and $BR(B^{+}\to\tau^{+}\nu_{\tau})=(1.09\pm 0.24)\times 10^{-4}$ from BABAR lees13 and Belle hara13 ; kronenbitter15 . The measured values of $BR(B^{+}\to\mu^{+}\nu_{\mu})$ are in agreement with the SM prediction $BR(B^{+}\to\mu^{+}\nu_{\mu})_{SM}=(3.80\pm 0.31)\times 10^{-7}$ belle_bmunu2018 . The measured value of $BR(B^{+}\to\tau^{+}\nu_{\tau})$ is also in agreement with the SM prediction $BR(B^{+}\to\tau^{+}\nu_{\tau})_{SM}=(0.75^{+0.10}_{-0.05})\times 10^{-4}$ kronenbitter15 ; ckmfitter .

We focus on data from a $B^{+}_{\ell 2}$ peak search experiment by Belle park2016 . In general shrock80 ,

[TABLE]

For $m_{\nu_{4}}$ in the range from 0.1 GeV to 1.4 GeV, the Belle experiment obtained an upper limit on $BR(B^{+}\to e^{+}\nu_{4})$ of $2.5\times 10^{-6}$, while in the interval of $m_{\nu_{4}}$ from 1.4 GeV to 1.8 GeV, this upper limit increased to $7\times 10^{-6}$. In the range of $m_{\nu_{4}}$ from 0.1 to 1.3 GeV, the Belle experiment obtained (non-monotonic) upper limits on $BR(B^{+}\to\mu^{+}\nu_{4})$ of approximately $2-4\times 10^{-6}$, and in the interval of $m_{\nu_{4}}$ from 1.3 GeV to 1.8 GeV, it obtained upper limits varying from $2\times 10^{-6}$ to $1.1\times 10^{-5}$. Substituting the $BR(B^{+}\to e^{+}\nu_{4})$ limits in Eq. (24) with $M=B$ and $\ell=e$, we obtain the upper limits on $|U_{e4}|^{2}$ shown as the curve $B_{e2}$ in Fig. 1 banomaly . From the $BR(B^{+}\to e^{+}\nu_{4})$ limits we infer upper limits on $|U_{\mu 4}|^{2}$ that decrease from 0.83 to $3.4\times 10^{-2}$ as $m_{\nu_{4}}$ increases from 0.1 GeV to 1.2 GeV. Further peak searches for $B^{+}\to\ell^{+}\nu_{4}$ with $\ell=e,\ \mu$ at Belle II would be worthwhile as a higher-statistics extension of park2016 .

We briefly remark on other constraints on a Dirac $\nu_{4}$ in the mass range considered here. From the results of p77 ; leeshrock77 , it follows that there is a negligibly small contribution to decays such as $\mu\to e\gamma$ and $\mu\to ee\bar{e}$. Similarly, there is no conflict with bounds on neutrino magnetic moments fs80 ; pdg2018 , and contributions to invisible Higgs decays invis are well below the current upper limit of $BR(H\to{\rm invis.})<19\%$ lhc_invis .

In this work, improved upper limits on $|U_{e4}|^{2}$ have been presented covering most of the range from $m_{\nu_{4}}=0.5$ MeV to $m_{\nu_{4}}\simeq 1$ GeV, representing the best available laboratory bounds for a Dirac neutrino $\nu_{4}$ 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 $\nu_{4}$ decays. For example, for $m_{\nu_{4}}=30$ MeV, our upper bound is $|U_{e4}|^{2}<0.8\times 10^{-6}$, while the best bound for this value of $m_{\nu_{4}}$ from experiments searching for neutrino decays is $|U_{e4}|^{2}<1\times 10^{-6}$ ps191 . New peak search experiments to search for $D_{s}^{+}\to e^{+}\nu_{4}$ and $D^{+}\to e^{+}\nu_{4}$ as well as a continued search for $B^{+}\to e^{+}\nu_{4}$ would be valuable; these could improve the bounds further. Other constraints on sterile neutrinos such as from $\pi^{+}\to\pi^{0}e^{+}\nu_{e}$ decay,and a detailed report of the results presented here will be published elsewhere DB_RS .

We thank J. Benitez, J. Hardy, V. Luth, W. Marciano, M. Ramsey-Musolf, C. Yuan, and G. Zhao for useful discussions. This work was supported in part by the Natural Sciences and Engineering Research Council and the National Research Council of Canada (D.B.) and by the U.S. National Science Foundation Grant NSF-PHY-16-1620628 (R.S.).