Anomalies in (semi)-leptonic $B$ decays $B^{\pm} \to \tau^{\pm} \nu$, $B^{\pm} \to D \tau^{\pm} \nu$ and $B^{\pm} \to D^* \tau^{\pm} \nu$, and possible resolution with sterile neutrino

TL;DR

This paper investigates anomalies in semileptonic B decays involving tau leptons, proposing sterile neutrinos as a possible explanation for deviations from Standard Model predictions and exploring future experimental tests at Belle-II.

Contribution

It introduces sterile neutrino scenarios to explain anomalies in B decay ratios and evaluates new ratios for future experimental validation.

Findings

Sterile neutrinos can account for observed anomalies in $R(D^{(*)})$.

Predicted ratios align with current experimental deviations.

Future measurements at Belle-II can test the sterile neutrino hypothesis.

Abstract

The universality of the weak interactions can be tested in semileptonic $b \to c$ transitions, and in particular in the ratios $R(D^{(*)}) \equiv \Gamma(B \rightarrow D^{(*)} \tau \nu )/\Gamma(B \rightarrow D^{(*)} \ell \nu )$ (where $\ell = \mu$ or $e$). Due to the recent differences between the experimental measurements of these observables by BaBar, Belle and LHCb on the one hand and the Standard Model predicted values on the other hand, we study the predicted ratios $R(D^{(*)}) = \Gamma(B \rightarrow D^{(*)} \tau +$ "missing" $)/\Gamma(B \rightarrow D^{(*)} \ell \nu )$ in scenarios with an additional sterile heavy neutrino of mass $\sim 1$ GeV. Further, we evaluate the newly defined ratio $R(0) \equiv \Gamma(B \to \tau +$ "missing" $)/\Gamma(B \to \mu \nu )$ in such scenarios, in view of the future possibilities of measuring the quantity at Belle-II.

Tables

Table 1. Table 1: Experimental results  Huschle:2015rga ; Aaij:2015yra ; Lees:2012xj and the SM predictions  Na:2015kha of R ​ ( D ) 𝑅 𝐷 R(D) and R ​ ( D ∗ ) 𝑅 superscript 𝐷 R(D^{*}) . The SM prediction with the scalar form factor (SFF) variations Kim:2016yth is also given. The first and second experimental errors are statistical and systematic, respectively.

	$R\left(D\right)$	$R\left(D^{\ast }\right)$
BaBar	$0.440\pm 0.058\pm 0.042$	$0.332\pm 0.024\pm 0.018$
Belle	$0.375\pm 0.064\pm 0.026$	$0.302\pm 0.030\pm 0.011$
LHCb	$-$	$0.336\pm 0.027\pm 0.030$
Experimental average HFAG:leptonic ; HFAG	$0.397\pm 0.040\pm 0.028$	$0.316\pm 0.016\pm 0.010$
SM Prediction Na:2015kha	$0.300\pm 0.008$	$0.252\pm 0.003$
SM Prediction (SFF Variation) Kim:2016yth	$0.335$	$-$

Table 2. Table 2: Branching fractions of B + → e + ​ ν , μ + ​ ν , τ + ​ ν → superscript 𝐵 superscript 𝑒 𝜈 superscript 𝜇 𝜈 superscript 𝜏 𝜈 B^{+}\to e^{+}\nu,~{}\mu^{+}\nu,~{}\tau^{+}\nu in units of 10 − 6 superscript 10 6 10^{-6}   PDG2014 ; HFAG:leptonic ; HFAG .

$B^{+}\to$	$e^{+}\nu$	${\mu }^{+}\nu$	${\tau }^{+}\nu$
BABAR			$179\pm 48$
Belle			$91\pm 19\pm 11$
Experimental average			$106\pm 19$

Table 3. Table 3: Predicted values of R 0 subscript 𝑅 0 R_{0} : The R 0 subscript 𝑅 0 R_{0} values are calculated in two ways: (i) considering the U 𝑈 U to be nonunitary; or (ii) considering U 𝑈 U to be unitary. The results are given to four digits to facilitate comparison. The values of | U τ ​ N | 2 superscript subscript 𝑈 𝜏 𝑁 2 |U_{\tau N}|^{2} for various values of M N subscript 𝑀 𝑁 M_{N} are taken equal to the known CHARM and DELPHI upper bounds CHARM ; DELPHI .

$M_N\left(\mathrm{GeV}\right)$	${\left|U_{\tau N}\right|}^2$	$R_0$  [nonunitarity]	$R_0$  [unitarity]
0	0	$2.2255\times {10}^2$ [SM]	$2.2255\times {10}^2$ [SM]
0.1	$8.0\times {10}^{-4}$	$2.2273\times {10}^2$	$2.2255\times {10}^2$
0.2	$2.0\times {10}^{-4}$	$2.2259\times {10}^2$	$2.2255\times {10}^2$
0.3	$1.5\times {10}^{-1}$	$2.5708\times {10}^2$	$2.2370\times {10}^2$
0.4	$6.0\times {10}^{-2}$	$2.3672\times {10}^2$	$2.2336\times {10}^2$
0.5	$2.5\times {10}^{-2}$	$2.2864\times {10}^2$	$2.2307\times {10}^2$
0.6	$1.4\times {10}^{-2}$	$2.2608\times {10}^2$	$2.2297\times {10}^2$
0.7	$9.0\times {10}^{-3}$	$2.2491\times {10}^2$	$2.2291\times {10}^2$
0.8	$6.0\times {10}^{-3}$	$2.2420\times {10}^2$	$2.2286\times {10}^2$
0.9	$4.0\times {10}^{-3}$	$2.2370\times {10}^2$	$2.2281\times {10}^2$
1.0	$3.0\times {10}^{-3}$	$2.2345\times {10}^2$	$2.2278\times {10}^2$
2.0	$3.0\times {10}^{-4}$	$2.2268\times {10}^2$	$2.2262\times {10}^2$
3.0	$4.5\times {10}^{-5}$	$2.2257\times {10}^2$	$2.2256\times {10}^2$

Equations

$$R(D^{(\ast )})\equiv \frac{\Gamma (B\to D^{(\ast )}\tau \nu )}{\Gamma (B\to D^{(\ast )}\ell \nu )},$$

$${\mathcal{B}}_{\mathrm{SM}}(B^{+}\to {\tau }^{+}\nu )=(0.848_{-0.055}^{+0.036})\times 10^{-4}.$$

$$R(0)\equiv \frac{\Gamma (B\to \tau \nu )}{\Gamma (B\to \mu \nu )},$$

$${\nu }_{\ell }=\sum _{k=1}^3{U}_{\ell {\nu }_k}{\nu }_k+U_{\ell N}N,$$

$$\sum _{k=1}^3|U_{\ell {\nu }_k}{|}^2=1-|U_{\ell N}{|}^2.$$

$$\Delta \mathcal{L}\sim \tilde{g}\bar{\tau }{\gamma }^{\mu }{\tilde{W}}_{\mu }^{-}N+h.c.\Rightarrow \delta \mathcal{L}=\left(-\frac{g}{\sqrt{2}}\right)U_{\tau N}\bar{\tau }{W}_{X\mu }^{-}N+h.c.,$$

$${\tilde{W}}_R^{\pm }=-e^{\mp i\lambda }\sin \xi W^{\pm }+\cos \xi W_2^{\pm },$$

$$\xi \approx \frac{2|vw|}{|v{|}^2+|w{|}^2}{\left(\frac{M_W}{M_2}\right)}^2,$$

$$-\frac{g}{\sqrt{2}}\bar{\tau }{\gamma }^{\mu }\left(\frac{1+{\gamma }_5}{2}\right)N{\tilde{W}}_{R\mu }^{-}+h.c.$$

$$=$$

$$|U_{\tau N}{|}^2\approx {\left(\frac{M_W}{M_2}\right)}^4.$$

$$\left(\frac{g_{\tau }}{g_{\mu }}\right)=1.0010\pm 0.0015,$$

$${\left(\frac{g_{\tau }}{g_{\mu }}\right)}^2={\left(\frac{M_{\mu }}{M_{\tau }}\right)}^5\frac{\Gamma ({\tau }^{-}\to {\nu }_{\tau }{e}^{-}{\bar{\nu }}_e(\gamma ))}{\Gamma ({\mu }^{-}\to {\nu }_{\mu }{e}^{-}{\bar{\nu }}_e(\gamma ))}\times \left[\frac{R_{\gamma }(\mu )}{R_{\gamma }(\tau }\frac{R_W(\mu )}{R_W(\tau )}\frac{f(M_e^2/M_{\mu }^2)}{f(M_e^2/M_{\tau }^2)}\right]=1+(2\pm 3)\times 10^{-3}.$$

$$\Gamma ({\tau }^{-}\to {\nu }_{\tau }{e}^{-}{\bar{\nu }}_e)$$

$$=$$

$$\overline{\Gamma }(\tau \to N{\ell }^{-}{\bar{\nu }}_{\ell })$$

$$\Gamma ({\tau }^{-}\to {\nu }_{\tau }{e}^{-}{\bar{\nu }}_e)=\sum _{k=1}^3|U_{\tau {\nu }_k}{|}^2\overline{\Gamma }(\tau \to {\nu }_k{e}^{-}{\bar{\nu }}_e)=\overline{\Gamma }(\tau \to \nu e^{-}{\bar{\nu }}_e){|}_{M_{\nu }=0},$$

$$G_{\ell }(M_N)=\frac{\Gamma ({\tau }^{-}\to N{\ell }^{-}{\bar{\nu }}_{\ell })}{\Gamma ({\tau }^{-}\to {\nu }_{\tau }{\ell }^{-}{\bar{\nu }}_{\ell })}=|U_{\tau N}{|}^2{\overline{G}}_{\ell }(M_N),$$

$${\overline{G}}_{\ell }(M_N)=\frac{\overline{\Gamma }({\tau }^{-}\to N{\ell }^{-}{\bar{\nu }}_{\ell })}{\overline{\Gamma }({\tau }^{-}\to \nu {\ell }^{-}{\bar{\nu }}_{\ell }){|}_{M_{\nu }=0}}$$

$$G_e(M_N)=(2\pm 3)\times 10^{-3}\Rightarrow |U_{\tau N}{|}^2=\frac{(2\pm 3)\times 10^{-3}}{{\overline{G}}_e(M_N)}.$$

$$|U_{\tau N}{|}^2<\frac{(-1)\times 10^{-3}}{({\overline{G}}_e(M_N)-1)}=\frac{(+1)\times 10^{-3}}{|{\overline{G}}_e(M_N)-1|}.$$

$$⟨D(p_D)|\overline{c}{\gamma }^{\mu }b|B^{-}(p_B)⟩=\left[(2p_D+q{)}^{\mu }-\frac{(M_B^2-M_D^2)}{q^2}{q}^{\mu }\right]F_1(q^2)+\frac{(M_B^2-M_D^2)}{q^2}{q}^{\mu }{F}_0(q^2),$$

$$\Gamma (B\to D\ell N)=|U_{\ell N}{|}^2\overline{\Gamma }(B\to D\ell N),$$

$$\overline{\Gamma }(B\to D\ell N)$$

$$w$$

$$z(w)$$

$$F_1(q^2)=F_1(w=1)\left(1-8{\rho }^{2}z(w)+(51{\rho }^2-10)z(w{)}^2-(252{\rho }^2-84)z(w{)}^3\right),$$

$${\rho }^2$$

$$|V_{cb}|F_1(w=1)$$

$$F_0(q^2)=(1+\alpha q^2+\beta q^4)F_1(q^2),$$

$$R(D)$$

$$R(D)=\frac{\overline{\Gamma }(B\to D\tau \nu )+|U_{\tau N}{|}^2\overline{\Gamma }(B\to D\tau N)}{\overline{\Gamma }(B\to D\ell \nu )}(\ell =e,\mu ).$$

$$H_{\eta }^{\mu }$$

Full text

I Introduction

The lepton universality of weak gauge theory can be tested in exclusive semileptonic $B$ decays, possibly through the existence of new charged currents, as well as the quark flavor mixing structure Cabibbo:1963yz of the Standard Model (SM). To achieve these goals, it is usually necessary to calculate accurately the corresponding hadronic matrix elements. However, in the ratios like

[TABLE]

with $\ell=e~{}\text{or}~{}\mu$, most of the hadronic uncertainties cancel, making such ratios particularly relevant for testing the universality of weak interactions. In Table 1 we show the SM predictions which we will use. The shown (hadronic) uncertainties originate, respectively, from lattice calculations Na:2015kha and from estimated higher order correction in HQET to the ratio $A_{0}/A_{1}$ of form factors Fajfer:2012vx . Recently, new improved experimental results from Belle Huschle:2015rga and LHCb Aaij:2015yra Collaborations appeared, in addition to the older BaBar results Lees:2012xj . As a result, the world average values reported by the HFAG groupHFAG:leptonic ; HFAG are larger than the SM predicted values for $R(D)$ by 1.9 $\sigma$, and for $R(D^{*})$ by 3.3 $\sigma$ (cf. Table 1).

Many theoretical explanations have been proposed to explain these indications of possible lepton universality violation, among them the charged scalar exchanges Fajfer:2012jt , vector resonances Buttazzo:2016kid or a $W^{\prime}$ boson He:2012zp ; Freytsis:2015qca ; Bhattacharya:2016mcc , leptoquarks (or, equivalently, R-parity violating supersymmetry) Freytsis:2015qca ; Tanaka:2012nw ; Calibbi:2015kma ; Bhattacharya:2016mcc . Effects of exchange of on-shell sterile neutrinos have also been evaluated, cf.  Abada:2013aba ; Cvetic:2016fbv . Explanation of the anomalies within an effective field theory approach with dimension-6 scalar, vector and tensor operators appeared in Biancofiore:2013ki ; Calibbi:2015kma . It was shown in Fan:2015kna that the perturbative QCD (pQCD) combined with lattice results reduces the difference from the experimental values. The electroweak effects in $B$-anomalies were investigated in Ref. Paradisi .

In Ref. Kim:2016yth , the authors checked how robust are the SM predictions for $R(D^{(*)})$ ratios. In contrast to the predictions of the vector form factors for $B\to D\ell\nu$ decays, which have been determined well in measurements of the branching ratios and $q^{2}$-distributions in light lepton channels, the scalar form factor (SFF) is measurable only in decays with $\tau$ leptons. Even small deviations of the SFF from the lattice values can bring the SM prediction closer to the current measured values of $R(D)$, as shown in Table 1.

As closely related to the decays, $B\rightarrow D^{(*)}\tau\nu$, the branching fraction (BF) of $B^{+}\to\tau^{+}\nu$ decay111Throughout this paper, formulas can be applied also to charge-conjugate modes. was measured by Belle and BaBar BtotaunuBFs , as shown in Table II PDG2014 ; HFAG:leptonic ; HFAG The SM prediction for the decay BF is given by CKMfitter

[TABLE]

This implies that if the measurement is improved in future $B$-factory experiments such as Belle-II BelleIIwww , the comparison can clarify whether new physics scenarios are needed.

As a probe of new physics beyond the SM, the leptonic decays $B^{+}\to\ell^{+}\nu$ are very interesting. This is so because these decay rates can be evaluated very precisely, and even at the tree-level new physics effects may appear, e.g., contributions of charged Higgs Hou in two-Higgs doublet models HHG . The leptonic decay rates of $B^{+}\to\ell^{+}\nu$ are in the SM proportional to the square of the charged lepton mass, $m_{\ell}^{2}$. Thus, the decays of $B^{\pm}$ to $e^{\pm}\nu$ and $\mu^{\pm}\nu$ are strongly suppressed in comparison with the decays to $\tau^{\pm}\nu$. Here we define new ratio $R(0)$ Hou as

[TABLE]

which is one of the most interesting to test the universality of weak interactions, since all the hadronic uncertainties cancel in the ratio, and the ratio is a function of $M_{\tau}^{2}/M_{B}^{2}$ (and $M_{\mu}^{2}/M_{B}^{2}$).

Heavy sterile neutral particles (a.k.a. “heavy neutrinos”) have suppressed mixing with SM neutrinos and appear in various new physics scenarios, among them the original seesaw seesaw with very heavy neutrinos, seesaw with neutrinos with mass $\sim 0.1$-$1$ TeV 1TeVNu , or with mass $\sim 1$ GeV 1GeVNu . For some studies of the production of very heavy neutrinos with mass $\sim 100$ GeV at the LHC we refer to DasLHC . We will include in our considerations the reactions $B^{\pm}\to\tau^{\pm}N,B^{\pm}\to D\tau^{\pm}N,B^{\pm}\to D^{*}\tau^{\pm}N$, where $N$ is any heavy sterile neutrino of the Dirac or Majorana type, and interpret the measured branching fractions in the new physics scenario. For example, even if $N$ is invisible in the detector, we can still distinguish $B^{\pm}\to\ell^{\pm}N$ signals from $B^{+}\to\ell^{+}\nu$ for $\ell=e$ or $\mu$, because these are two-body decays and therefore the momentum of the charged lepton in the $B$ meson rest frame is fixed by the mass of $N$. However, in the case of decays $B^{\pm}\to\tau^{\pm}N$, the produced $\tau^{\pm}$ particle decays fast and hence there are more than one neutrinos in the final state, and the decay signature of $B^{\pm}\to\tau^{\pm}N$ cannot be distinguished from the ordinary $B^{\pm}\to\tau^{\pm}\nu$. Therefore, the experimentally observed signal of $B^{\pm}\to\tau^{\pm}\nu$ may include contributions from $B^{\pm}\to\tau^{\pm}N$, and this signal we will denote as $B^{\pm}\to\tau^{\pm}+{\rm``missing"}$.

Massive neutrinos $N$ mix in general with the standard flavor neutrinos, $e.g.$ as in a seesaw type new physics scenario. We denote as $U_{\ell N}$ the mixing coefficient for the heavy mass eigenstate $N$ with the standard flavor neutrino $\nu_{\ell}$ ($\ell=e,\mu,\tau$).222 Other notations for $U_{\ell N}$ exist in the literature, among others $V_{\ell 4}$ in Atre ; $B_{\ell N}$ in PilZPC . The standard sub-eV neutrino $\nu_{\ell}$ ($\ell=e,\mu,\tau$) can then be represented as

[TABLE]

where $\nu_{k}$ ($k=1,2,3$) are the light mass eigenstates. The $3\times 3$ matrix $U_{\ell\nu_{k}}$ is the usual PMNS matrix Pmns . In the relations (4) we assume the existence of only one additional massive sterile neutrino $N$, however, it can be extended with any number of $N$. Then the extended (unitary) PMNS matrix $U$ would be in this case a $4\times 4$ matrix, implying the relations

[TABLE]

One of our scenarios will be with this unitarity assumption. This will modify the decay width, due to the existence of a massive neutrino $N$, by the amount $\Gamma(B^{+}\to\tau^{+}N)-\Gamma(B^{+}\to\tau^{+}N)|_{M_{N}=0}$ and $\Gamma(B^{+}\to D^{(*)}\tau^{+}N)-\Gamma(B^{+}\to D^{(*)}\tau^{+}N)|_{M_{N}=0}$, where the minus terms are due to the unitarity of $U$.

In the other scenario, the $3\times 3$ PMNS mixing matrix is unitary, and $N$ will be regarded as a neutral fermion which does not mix with the SM flavor neutrinos $\nu_{\ell}$, but couples with charged leptons such as $\tau$ in the same form as in the first scenario, for example,

[TABLE]

via mediation of a new physics charged gauge boson $\widetilde{W}^{\pm}$, and subscript $X$ denotes either $L$ or $R$ (left or right-handed projection). Here $\widetilde{W}^{\pm}$ has the light SM gauge boson $W^{\pm}$ component, and this may lead to couplings (6), where the suppression effects of such (or similar) scenarios are parameterized in the parameter $U_{\tau N}$. These couplings have the same form as in the previous scenario, but now we have no condition of unitarity (5). And such violation of the unitarity manifests unknown new physics beyond the SM.

For our analysis, we want to keep in a most generic form both the scenarios which lead to Eqs. (4)-(5) and those which lead to Eq. (6). Nonetheless, we wish to mention, as a representative example for the mechanism of Eq. (6), the LR-models LR1 ; LR2 ; LangSan with the gauge group $SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$. Such models have in the scalar sector a $2\times 2$ LR-doublet $\phi$ and triplets $\Delta_{L}$, $\Delta_{R}$. The vacuum expectation value (VEV) $u_{R}$ of $\Delta_{R}$ is much larger than the other VEVs, $|u_{R}|\gg|v|,|w|\gg|u_{L}|$ (where $v,w\sim 10^{2}$ GeV are VEVs in $\phi$), leading to a hierarchical mixing of the charged flavor bosons ${\widetilde{W}}_{L}^{\pm}$ and ${\widetilde{W}}_{R}^{\pm}$. The flavor boson ${\widetilde{W}}_{R}^{\pm}$ has, as a result, a small component ($\sim|v^{2}/u_{R}^{2}|$) of the SM mass eigenstate boson $W^{\pm}$. More specifically,

[TABLE]

where $W_{2}^{\pm}$ is the heavy mass eigenstate with mass $M_{2}^{2}\approx g^{2}|u_{R}|^{2}/4\gg M_{W}^{2}$, $\lambda$ is a real phase, and the mixing angle $\xi$ is

[TABLE]

where $(|v|^{2}+|w|^{2})=1/(\sqrt{2}G_{F})$. In such models we can have a heavy neutrino $N_{R}$ which forms with $\tau_{R}$ an $SU(2)_{R}$-doublet $(N_{R},\tau_{R})$ and $N_{R}$ is a flavor and mass eigenstate, i.e., it does not mix with other flavor neutrinos such as $(\nu_{\ell,L})^{c}$ ($\ell=e,\mu,\tau$) and $\nu_{\ell^{\prime},R}$ ($\ell^{\prime}=e,\mu$). Then the gauge boson mixing (7) gives us in the terms which couple ${\widetilde{W}}_{R}^{\pm}$ with leptons the following contributions (we take $g_{R}=g_{L}=g$):

[TABLE]

where the ellipsis stands for the couplings with the heavy $W_{2}^{\pm}$ boson. Here we see that in such a case the heavy-light mixing parameter is $|U_{\tau N}|^{2}=\sin^{2}\xi\approx\xi^{2}$, which, according to Eq. (8) is then

[TABLE]

According to the analysis of general LR-models of Ref. LangSan , there is a lot of freedom in $V_{R}$, the right-handed quark mixing matrix, resulting in a relatively generous bound $M_{2}>300$ GeV coming predominantly from the $K_{L}$-$K_{S}$ mass difference $\Delta m_{K}$ and by $B_{d}{\bar{B}}_{d}$ mixing (and assuming $g_{R}=g_{L}$). This implies the upper bound $|U_{\tau N}|^{2}<5\times 10^{-3}$ in such LR-scenarios.333If the LR-models are restricted to the minimal (symmetric) versions, where $V_{R}$ is closely related with $V_{L}$($=V_{\rm CKM}$), the resulting bound is more restrictive, $M_{2}>2.3$ TeV NemSen . The signals of $W_{R}$ were searched also at LHC (CMS Collaboration) CMSPAS , in specific minimal LR-model scenarios where $W_{R}^{\pm}$ would decay in the $\ell=e,\mu$ channels to $\ell N_{\ell,R}$; mass exclusion regions $M_{2}>3.3$ TeV were found CMSPAS in such scenarios, but only if $M_{N_{\ell}}>200$ GeV.

We further point out that the right-handedness of the coupling $\tau$-$W$-$N$, Eq. (9b), does not affect the formulas for the decay widths $\Gamma(B\to(D^{(*)})\tau\nu)$ that we use in the present paper: we checked that these formulas turn out to be the same as in the case of the left-handedness of the $\tau$-$W$-$N$ coupling; we recall that the couplings of quarks to $W$ are, of course, always left-handed.

In our numerical analysis, we will derive the results for two scenarios: either the ($4\times 4$) matrix $U$ is unitary, or $U$ without unitarity assumption: $i.e.$ analyses with and without unitarity assumption. Our formulas, to be derived in the following Sections, will be applicable also to cases with more than one additional massive neutral fermion $N$ where the second fermion has a mass $M_{N^{\prime}}\gtrsim 10$ GeV, such particles being too heavy to be produced on-shell in the considered decays.

At present, the upper bounds for the mixing parameters $|U_{\tau N}|^{2}$ are available from the measurements and analyses of the CHARM CHARM and DELPHI DELPHI Collaborations. These were dedicated direct measurements and analyses for decays producing heavy neutrinos $N$.444 CHARM limits CHARM were obtained for $M_{N}<300$ MeV, based on the absence of signals $N\to\nu_{\tau}Z^{0}\to\nu_{\tau}e^{+}e^{-}$ in a neutrino beam dump experiment, where $N$ is produced from decays of $D_{s}$ mesons. DELPHI limits DELPHI were obtained for $M_{N}>250$ MeV, based on the absence of signals $e^{+}e^{-}\to Z^{0}\to\nu_{\ell}N$ for long-lived and short-lived $N$, and they apply to all three parameters $|U_{\ell N}|^{2}$ ($\ell=e,\mu,\tau$). On the other hand, there are indirect indications that the heavy-light mixing parameters $|U_{\tau N}|^{2}$ have more restrictive upper bounds, coming from the $\tau$ lepton decays where the analyses were made under the assumption of the SM scenario of (practically) massless neutrinos and unitary $3\times 3$ PMNS matrix $U_{\rm PMNS}$, and there the lepton universality of the electroweak coupling $g$ was shown to a large precision HFAG (Sec. 9.2 there). However, in these latter analyses, unlike in Refs. CHARM ; DELPHI , it was assumed that heavy neutrinos do not exist. In view of the lack of any new updated dedicated measurements and analyses of the $\tau$ decays with heavy neutrinos, we will usually present here the upper bounds on the mixing parameters $|U_{\tau N}|^{2}$ as those from Refs. CHARM ; DELPHI . Nonetheless, in the next Section we will present an analysis of the lepton universality results HFAG in the scenario of one additional heavy neutrino $N$, and in the subsequent analyses in this work we will keep in mind the restrictions on the heavy-light mixing from such an analysis.

In this paper, we will consider the recent experimental anomalies, $R(D)$ and $R(D^{*})$, with the theoretical assumption of one not very heavy neutrino $N$: $M_{N}\sim 1$ GeV. We will also predict the newly defined $R(0)$, which can be measured at Belle-II, as a function of the unknown parameters, $M_{N}$ and $U_{\tau N}$.

II Restrictions on $|U_{\tau N}|^{2}$ from lepton universality tests in $\tau$ decays

The Heavy Flavor Averaging Group (HFAG) HFAG obtained restrictions coming from the lepton universality tests of SM. They analyzed, among other things, the measured widths $\Gamma(\tau\to e+{\rm missing})$ and $\Gamma(\mu\to e+{\rm missing})$, where “missing” stands for $\nu_{\tau}{\bar{\nu}}_{e}(\gamma)$ and $\nu_{\mu}{\bar{\nu}}_{e}(\gamma)$. They thus obtained

[TABLE]

where the above notation stands for

[TABLE]

The above ratio in their analysis includes the QED [’$(\gamma)$’] MS and other corrections [$\sim(M_{\tau}/M_{W})^{2}$, $\sim(M_{e}/M_{\mu})^{2}$]. However, the net contribution of all these effects to the above quantity, represented as the correction factor in the brackets in Eq. (12), turns out to be $\sim 10^{-4}$ and will thus be ignored in the analysis here.

If, on the other hand, we have an additional heavy neutrino $N$ which couples to $\tau$ (but not to $e$ or $\mu$), either in a scenario in which the $4\times 4$ matrix $U$ is nonunitary [e.g., scenarios as in Eq. (6)], or in a scenario where the $4\times 4$ mixing matrix $U$ is unitary [cf. Eq. (5)], the above analysis changes significantly.

Namely, in the scenario where the formal $4\times 4$ matrix $U$ is nonunitary (and the $3\times 3$ PMNS mixing matrix is unitary), the accounting for the nonzero mass $M_{N}$ changes the above analysis in the following way. The quantity $\Gamma(\tau^{-}\to\nu_{\tau}e^{-}{\bar{\nu}}_{e})$ gets replaced by

[TABLE]

where ${\overline{\Gamma}}(\tau\to N\ell^{-}{\bar{\nu}}_{\ell})$ stands for the decay width to a (massive) neutrino $N$ and the coupling parameter $|U_{\tau N}|^{2}=1$

[TABLE]

Here, $z_{N}=(M_{N}/M_{\tau})^{2}$, $z_{\ell}=(M_{\ell}/M_{\tau})^{2}$ ($\ell=e$ or $\ell=\mu$), and $z=p_{W}^{2}/M_{\tau}^{2}$ where $p_{W}^{2}$ is square of the invariant mass of $W^{*-}=(\ell{\bar{\nu}}_{\ell})$. The first term on the right-hand side of Eq. (13b) denotes the usual SM contribution

[TABLE]

where the above identities hold because of the (practical) masslessness of the SM neutrinos $\nu_{k}$ ($k=1,2,3$) and the unitarity of the $3\times 3$ PMNS matrix.

The following ratio is crucial in the present analysis:

[TABLE]

where $\ell=e$ or $\ell=\mu$, and

[TABLE]

is the corresponding canonical ratio.

The values (12), together with Eq. (13b) and the notations (16)-(17), then lead in the mentioned scenario to the following predictions for $|U_{\tau N}|^{2}$:

[TABLE]

We recall that $N$ here is massive, on-shell, couples only to $\tau$ (and not to $\mu$ or $e$), and the $3\times 3$ PMNS matrix is as in SM, i.e., unitary. The ratio function $G_{\ell}(M_{N})$ is presented in Fig. 1 for the cases $|U_{\tau N}|^{2}=1$ and $|U_{\tau N}|^{2}=10^{-3}$.

If, on the other hand, the $4\times 4$ heavy-light mixing matrix $U$ is unitary, i.e., Eq. (5), an analogous analysis leads to the conclusion that $|U_{\tau N}|^{2}$ has a very restrictive upper bound. Namely, in such a case the ratio on the left-hand side of Eq. (12) must be below the value of unity, and the most generous upper bound for $|U_{\tau N}|^{2}$ is then obtained if the right-hand side of Eq. (12) is $1-1\times 10^{-3}$

[TABLE]

The restrictions (18) and (19) are presented in Figs. 2.

From Fig. 2 (a) we can see that in the scenario of the $3\times 3$ unitary PMNS matrix (the $4\times 4$ matrix $U$ is then nonunitary), the bounds (12) imply for the heavy-light mixing parameter $|U_{\tau N}|^{2}$ stringent upper bounds $|U_{\tau N}|^{2}\lesssim 0.5\times 10^{-2}$ if the mass of $N$ is low ($M_{N}<0.4$ GeV), and for higher masses the bounds are not stringent: $|U_{\tau N}|^{2}\lesssim 10^{-1}$ for $M_{N}>1$ GeV.

On the other hand, if the $4\times 4$ matrix $U$ is considered unitary, Fig. 2 (b) implies that the upper bounds for $|U_{\tau N}|^{2}$ are quite stringent ($\lesssim 10^{-3}$) for higher masses $M_{N}>0.6$ GeV, while for light masses less stringent upper bounds apply. The results of Fig. 2 (b) are consistent with the results of the analysis Becir of the experimental constraints for the well-measured decay ratios ${\cal B}(\tau\to\ell\nu\nu^{\prime})$ where $\ell=e,\mu$, and $\nu$ and $\nu^{\prime}$ are light neutrinos or the heavy neutrino $N$, in the scenario which corresponds here to the unitary $4\times 4$ $U$ matrix. The authors of Ref. Becir obtain the upper bound $|U_{\tau N}|^{2}<0.01$ for $0.3\ {\rm GeV}<M_{N}<1\ {\rm GeV}$.

We will keep in mind these restrictions which come indirectly from the lepton universality tests Eq. (12), i.e., from the decays $\tau\to e+{\rm missing}$ and $\mu\to e+{\rm missing}$. Nonetheless, in the graphs we will account for the bounds on $|U_{\tau N}|^{2}$ coming from the dedicated direct measurements of the CHARM CHARM and DELPHI DELPHI Collaborations mentioned in the Introduction.

III Theoretical details of $R(D)$, $R(D^{*})$ and $R(0)$ with Light Sterile Neutrino

III.1 $R(D)$ and decays of $B\to D\ell N$

In the SM the amplitude for hadronic transition $B\to D$ is given in terms of vector and scalar form factors, $F_{1}(q^{2})$ and $F_{0}(q^{2})$, defined as

[TABLE]

where $q=p_{B}-p_{D}$ is the momentum of the virtual $W^{-}$. For the process $B\to D\ell N$, the decay width is given in Ref. Cvetic:2016fbv as

[TABLE]

where the kinematically allowed values of $q^{2}$ are $(M_{N}+M_{\ell})^{2}\leq q^{2}\leq(M_{B}-M_{D})^{2}$. Notice that if $N$ is replaced by $\nu_{\ell}$ (i.e., $M_{N}\approx 0$), ${\overline{\Gamma}}(B\to D\ell N)$ in Eq. (22) becomes the SM decay width of $B\to D\ell\nu_{\ell}$, i.e., $\Gamma_{\rm SM}(B\to D\ell\nu_{\ell})$.

The form factor $F_{1}(q^{2})$ is well known CLN .555 For early attempts to account for the flavor symmetry breaking in form factors of heavy pseudoscalars, cf. Ref. Khlopov . It can be expressed in terms of the variable $w$

[TABLE]

in the following approximate form CLN :

[TABLE]

where the free parameters $\rho^{2}$ and $F_{1}(w=1)$ have been recently determined with high precision by the Belle Collaboration, Ref. Belle1

[TABLE]

The value (25b) was deduced from their value of $\eta_{EW}{\cal G}(1)|V_{cb}|=\eta_{EW}F_{1}(w=1)\sqrt{4r}/(1+r)=(42.29\pm 1.37)\times 10^{-3}$, where $r=M_{D}/M_{B}$ and $\eta_{EW}=1.0066\approx 1$ Sirlin . In our numerical evaluations, we will use the central values $\rho^{2}=1.09$ and $|V_{cb}|F_{1}(w=1)=48.14\times 10^{-3}$. A recent study Kim:2016yth has shown that the mostly unknown scalar form factor $F_{0}(q^{2})$, being expressed as

[TABLE]

can enhance the value of $R(D)$ up to $0.335$ within the SM, as shown in Table I. We will use this scaling relation for $F_{0}(q^{2})$ with $\alpha=+0.16~{}{\rm GeV^{-2}}$ and $\beta=-0.003~{}{\rm GeV^{-2}}$ as used in Ref. Kim:2016yth .

Taking the contribution from $B\to D\tau N$ decays into account, and assuming the unitarity (5) of the matrix $U$, the ratio of branching fractions $R(D)$ is

[TABLE]

where ${\overline{\Gamma}}(B\to D\tau\nu)$ is the expression (22) for zero neutrino mass $M_{\nu}=0$ and $M_{\ell}=M_{\tau}$. If the unitarity is not assumed, $R(D)$ becomes

[TABLE]

III.2 $R(D^{*})$ and decays of $B\to D^{*}\ell N$

The matrix elements for $B\to D^{*}$ transition are more complicated than those for $B\to D$ transition, because of the vector character of $D^{*}$, including four form factors:

[TABLE]

where

[TABLE]

The four form factors are

[TABLE]

Here, $R_{*}=2\sqrt{M_{B}M_{D^{*}}}/(M_{B}+M_{D^{*}})$, the variables $w$ and $z(w)$ are given by Eqs. (23), and the values of the free parameters determined in Ref. Belle2 are

[TABLE]

We will use the central values of these parameters.

For the decays $B\to D^{*}\ell N$, the width is given in Ref. Cvetic:2016fbv as

[TABLE]

where the canonical decay width (i.e., without the heavy-light neutrino mixing) is

[TABLE]

where ${\overline{\lambda}}=\lambda\left(1,\frac{M_{N}^{2}}{q^{2}},\frac{M_{1}^{2}}{q^{2}}\right)$ and $|{\vec{q}}|=\frac{1}{2}M_{B}\lambda^{1/2}\left(1,\frac{q^{2}}{M_{B}^{2}},\frac{M_{D^{*}}^{2}}{M_{B}^{2}}\right)$. As in the case of $B\to DlN$, ${\overline{\Gamma}}(B\to D^{*}\ell N)$ expresses the SM decay width $\Gamma_{\rm SM}(B\to D^{*}\ell\nu_{\ell})$, when $N$ is replaced by $\nu_{\ell}$ (i.e., $M_{N}\approx 0$).

Analogously as in the case of $R(D)$, including the possible contribution from $B\to D^{*}\tau N$ decays and assuming unitarity (5) of the full $U$ matrix, the ratio of branching fractions $R(D^{*})$ can be written as

[TABLE]

where ${\overline{\Gamma}}(B\to D^{*}\tau\nu)$ is the expression (LABEL:bGBDstlN) for zero neutrino mass $M_{\nu}=0$ and $M_{\ell}=M_{\tau}$. When the matrix $U$ is not assumed to be unitary, $R(D^{*})$ becomes

[TABLE]

III.3 $R(0)$ and decays of $B\to\ell N$

In the decay $B^{+}\to\ell^{+}\nu_{\ell}~{}~{}(\ell=e,\mu,\tau)$, within the SM with $M_{\nu_{\ell}}\approx 0$, the decay width is given by

[TABLE]

where $y_{\ell}=M_{\ell}^{2}/M_{B}^{2}$. Here, $M_{B}$ and $f_{B}$ are the $B^{+}$ meson mass and the decay constant, respectively, $|V_{ub}|$ is the corresponding CKM matrix element, and $G_{F}=1.1664\times 10^{-5}\ {\rm GeV}^{-2}$ is the Fermi coupling constant.

The decay width for the process $B^{+}\to\ell^{+}N$ $(\ell=e,\mu,\tau)$ is, as shown in Cvetic:2016fbv :

[TABLE]

where the canonical width ${\overline{\Gamma}}$ (i.e., without the heavy-light mixing factor $|U_{\ell N}|^{2}$) is

[TABLE]

where $y_{N}=M_{N}^{2}/M_{B}^{2}$ and the function $\lambda^{1/2}$ is given by

[TABLE]

It is obvious that if $N$ is replaced by $\nu_{\ell}$ (i.e., $y_{N}\approx 0$), ${\overline{\Gamma}(B^{+}\to\ell^{+}N)}$ in Eq. (39) becomes $\Gamma_{\rm SM}(B^{+}\to\ell^{+}\nu_{\ell})$ in Eq. (37). And the SM expectation for the newly defined $R(0)$, which is completely independent of hadronic uncertainty, is given by

[TABLE]

where the uncertainty comes almost entirely from the uncertainty in the $\tau$ lepton mass PDG2016 .

For the decays $B^{+}\to\tau^{+}+$ ”missing momentum”, taking the contribution from the decays $B^{+}\to\tau^{+}N$ into account, the widths can be obtained with the unitarity assumption (5) by

[TABLE]

where ${\overline{\Gamma}}(B^{+}\to\tau^{+}\nu)$ is for zero neutrino mass $M_{\nu}=0$, i.e., it is equal to the SM expression (37) with $\ell=\tau$. Considering the contribution from $B^{+}\to\tau^{+}N$, the ratio $R(0)$ in the considered scenario with one heavy neutrino $N$ and the unitarity (5) of the $U$ matrix is

[TABLE]

Here, ${\overline{\Gamma}}(B^{+}\to\ell^{+}N)$ is given in Eq. (39), and ${\overline{\Gamma}}(B^{+}\to\ell^{+}\nu)$ is the same expression with zero mass of neutrino $M_{\nu}=0$, i.e., Eq. (37). On the other hand without the unitarity assumption, we have instead of the relation (42) the following relation:

[TABLE]

and the ratio $R(0)$ becomes

[TABLE]

As we shall see later, the observation of $R(0)$ can give very useful information on the values of $|U_{\tau N}|$ and the mass of a sterile neutrino $M_{N}$.

IV Numerical Analysis and Discussions

IV.1 $R(D)$, $R(D^{*})$ and $B\to D\tau N$, $D^{*}\tau N$

In this numerical analysis, first we study $R(D)$ and $R(D^{*})$ anomalies in our scenarios: in one scenario the matrix $U$ without the unitarity assumption; in the other scenario, the full $4\times 4$ matrix $U$ is considered unitary. As mentioned, the latter scenario is realized when seesaw-type mechanisms are used, and the former may appear when the heavy neutral fermion $N$ originates from a different, unknown, mechanism in a high energy framework beyond the SM. Also, we examine the possibility of finding certain “direct” bounds on magnitudes of the matrix elements $|U_{\ell N}|$ from this analysis of $R(D)$ as well as $R(D^{*})$.

We first calculate $R(D)$ including the effect of the process $B\to D\tau N$ as given in Eqs. (28) and (27). By comparing the theoretical result with the experimental data shown in Table 1, the allowed parameter space for $|U_{\tau N}|^{2}$ is found in terms of the mass of the sterile neutrino, $M_{N}$. The results are shown in Fig. 3 for the two scenarios.

In Fig. 3(a), the result is found for the scenario without the assumption of unitarity for the $4\times 4$ matrix $U$. It shows the allowed parameter space for $|U_{\tau N}|^{2}$ and $M_{N}$ obtained from the experimental average value of $R(D)$. The red (light grey) region is allowed by the experimental data at $1\sigma$ level, i.e., when the deviation does not surpass $1\sigma$. The blue (dark grey) region is allowed by the data at $2\sigma$ level, i.e., when the deviation does not surpass $2\sigma$ (and is above $1\sigma$). The white region could be regarded as excluded, the deviation there surpasses $2\sigma$. For comparison the known available upper bounds from CHARM and DELPHI experiments CHARM ; DELPHI for $|U_{\tau N}|^{2}$ are also included, as tiny black squares666 These upper bounds were obtained from various physical processes searching for heavy sterile neutrino $N$ and are given in Table 3, see footnote 4 and Refs. CHARM ; DELPHI ; Atre .. We see that certain range of values of $|U_{\tau N}|^{2}$ and $M_{N}$ can fit the experimental data of $R(D)$. At $1\sigma$ level, there is a tendency that for a smaller value of $M_{N}$, a smaller $|U_{\tau N}|^{2}$ can fit the data. For instance, for $M_{N}=0.3$ GeV, the smallest value of $|U_{\tau N}|^{2}$ allowed by the $1\sigma$ data is $2.8\times 10^{-2}$, while for $M_{N}=1.0$ GeV, the smallest allowed value of $|U_{\tau N}|^{2}$ is $7.1\times 10^{-2}$.

In particular, for $M_{N}=0.3$ GeV, the values of $2.8\times 10^{-2}\lesssim|U_{\tau N}|^{2}\lesssim 3.5\times 10^{-1}$ are allowed by the $1\sigma$ data, in comparison with the known CHARM ; DELPHI upper bound $|U_{\tau N}|^{2}=1.5\times 10^{-1}$. Similarly, for $M_{N}=0.4$ GeV, the $1\sigma$ data allows the values of $3.0\times 10^{-2}\lesssim|U_{\tau N}|^{2}\lesssim 3.7\times 10^{-1}$, compared to the DELPHI DELPHI upper bound $|U_{\tau N}|^{2}=6.0\times 10^{-2}$. We note that for $M_{N}=0.3$ GeV and $0.4$ GeV, the smallest allowed value of $|U_{\tau N}|^{2}$ is smaller than its available CHARM ; DELPHI upper bound. However, if $M_{N}\geq 0.5$ GeV, the values of $|U_{\tau N}|^{2}$ allowed by the $1\sigma$ data are larger than the known DELPHI upper bounds. However, if using the indirect upper bounds coming from the lepton universality measurements, Fig. 2(a), all the allowed points move out of the $1\sigma$ region in Fig. 3(a).

In Fig. 3 (b), the result corresponds to the case of considering unitarity of the ($4\times 4$) matrix $U$, cf. Eq. (5). In this scenario, there is no allowed parameter space for $|U_{\tau N}|^{2}$ and $M_{N}$ by the experimental data at $1\sigma$ level. At $2\sigma$ level, certain region of the parameter space is allowed. For example, for $M_{N}\lesssim 0.3$ GeV, the values of $0\lesssim|U_{\tau N}|^{2}\lesssim 1$ are allowed, while for $M_{N}=0.6$ GeV and $1.0$ GeV, $0\lesssim|U_{\tau N}|^{2}\lesssim 0.4$ and $0\lesssim|U_{\tau N}|^{2}\lesssim 0.2$ are allowed, respectively. 2. 2.

In the case of $R(D^{*})$, it is harder to fit the experimental data by including the contributions from $B\to D^{*}\tau N$ together with those from the SM processes. Similarly to the analysis of $R(D)$, we compute $R(D^{*})$ including the effect of the decay $B\to D^{*}\tau N$ as given in Eqs. (35) and (36), and again examine the two scenarios: (a) one considering the $4\times 4$ matrix $U$ to be nonunitary, Eq. (36); (b) the other considering $U$ to be unitary, Eqs. (5) and (35). It turns out that in the latter scenario (b), there are no allowed values of $|U_{\tau N}|^{2}$ and $M_{N}$. Namely, Table 1 shows that the SM value of $R(D^{*})$ is by more than $3\sigma$ below the central experimental value; on the other hand, the theoretical value in scenario (b) becomes even lower when $U_{\tau N}\not=0$, cf. Eq. (35). In contrast, in the former scenario (a), certain values of $|U_{\tau N}|^{2}$ and $M_{N}$ are allowed at $2\sigma$ level, but not at $1\sigma$ level. The allowed parameter space is shown in Fig. 4 for the scenario (a). For instance, for $M_{N}=0.3$ GeV and $0.4$ GeV, the values of $2.0\times 10^{-1}\lesssim|U_{\tau N}|^{2}\lesssim 3.6\times 10^{-1}$ and $2.1\times 10^{-1}\lesssim|U_{\tau N}|^{2}\lesssim 3.9\times 10^{-1}$, respectively, would be allowed at $1\sigma$ level. However, as the tiny black squares in Fig. 4 indicate, the present CHARM and DELPHI CHARM ; DELPHI upper bounds on $|U_{\tau N}|^{2}$ give compatibility of the experimental $R(D^{*})$ with the scenario (a) to at best $2\sigma$ level, and this only if $M_{N}\approx 0.3$ GeV. Further, if we include the indirect upper bounds coming from the lepton universality measurements, Fig. 2(a), the allowed points move out of the $2\sigma$ region in Fig. 4.

We comment on a feature of the results for $R(D^{(*)})$ when the $4\times 4$ $U$ matrix is unitary. To be specific, let us consider the canonical decay widths ${\overline{\Gamma}}$ for the decays $B\to D^{(*)}\tau N$ given in Eqs. (22) and (LABEL:bGBDstlN). We find that $\bar{\Gamma}(B\to D^{(*)}\tau N)<\bar{\Gamma}(B\to D^{(*)}\tau\nu)$, i.e., $\bar{\Gamma}(B\to D^{(*)}\tau N)$ is a monotonously decreasing function of $M_{N}$. Thus the $|U_{\tau N}|^{2}$ term in $R(D^{(*)})$, i.e., in the numerator of Eqs. (27) and (35), becomes negative and the effect of this term is to reduce $R(D^{(*)})$ with respect to its SM value. Only in the scenario where $U$ is nonunitary, cf. Eqs. (28) and (36), does the presence of a heavy neutral fermion $N$ increase the ratio $R(D^{*})$.

IV.2 $R(0)$ and $B\to\tau N$

We now analyze the decay process $B\to\tau+{\rm``missing~{}momentum"}$ by including the process $B\to\tau N$. We will first consider BF of $B\to\tau+{\rm``missing~{}momentum"}$. Similarly to the previous cases of $R(D)$ and $R(D^{*})$, the two scenarios are examined: (a) one scenario considering the matrix $U$ is nonunitary, and where $\Gamma(B^{+}\to\tau^{+}+{\rm``missing~{}momentum"})$ is consequently given by Eq. (44); (b) the other scenario where the $4\times 4$ matrix $U$ is considered to be unitary, Eq. (5), and where $\Gamma(B^{+}\to\tau^{+}+{\rm``missing~{}momentum"})$ is consequently given by Eq. (42). The results of this analysis are presented in Figs. 5 (a) and (b). The allowed values of $|U_{\tau N}|^{2}$ and $M_{N}$ are in a wide range for both scenarios.

For future experiments, we make predictions for $R(0)$ given in Eqs. (43) and (45). They are summarized in Table 3. The known upper bounds for values of $|U_{\tau N}|^{2}$, for various specific values of $M_{N}$ are taken from Refs. CHARM ; DELPHI ; Atre . Due to the smallness of these upper bounds for $|U_{\tau N}|^{2}$, the predicted values of $R(0)$ for various values of $M_{N}$ do not deviate much from the SM predicted value $R(0)_{\rm SM}=2.2255\times 10^{2}$, cf. Eq. (41). Only for $0.3~{}{\rm GeV}\leq M_{N}\leq 0.8~{}{\rm GeV}$, sizable deviations from $R(0)_{\rm SM}$ are expected in the scenario of nonunitarity of $U$. For example, for $M_{N}=0.3$ GeV and $0.4$ GeV, the predicted values are $R_{0}=2.5708\times 10^{2}$ and $2.3672\times 10^{2}$, respectively.

More interesting predictions relevant to $R(0)$ are depicted in Fig. 6 (a) and (b), corresponding to the above two scenarios, respectively. The figures show the graphs of $|U_{\tau N}|^{2}$ versus $M_{N}$ for given values of $R(0)$. Provided that the value of $R(0)$ is determined in future experiments (e.g., by measuring the BFs of $B\to\mu\nu$ and $B\to\tau+{\rm``missing"}$ precisely), one can obtain useful information on $|U_{\tau N}|^{2}$ and $M_{N}$ from the figures. For example, if $R(0)$ is determined to be $2.250\times 10^{2}$, it corresponds to the case (ii) (i.e., red thick line) in Fig. 6 (a) and (b) from which the value of $|U_{\tau N}|^{2}$ can be extracted for a given $M_{N}$. However, to discriminate experimentally between the cases (i) (SM) and (ii), the branching ratios for $B\to\tau$ + “missing” and $B\to\mu\nu$ will have to be measured with precision of $1\%$ or better; it is possible that such a precision cannot be achieved at Belle-II. Further, if $U$ is nonunitary and if, simultaneously, the $\tau$-$N$-$W$ coupling comes from LR-model scenarios, then we have the upper bound $|U_{\tau N}|^{2}<5\times 10^{-3}$, cf. discussion after Eq. (10). In such a case, we cannot expect to get values of $R(0)$ over $2.24\times 10^{2}$ in such scenarios. Furthermore, if we take into account the indirect upper bounds on $|U_{\tau N}|^{2}$ coming from the lepton universality measurements, cf. Fig. 2(a), the three black squares at $M_{N}^{2}=0.3$, $0.4$ and $0.5$ GeV in Fig. 6 (a) decrease to the values $\approx 10^{-2}$, and we cannot get values of $R(0)$ over $2.26\times 10^{2}$.

V Conclusions

In this work we studied the experimental anomalies of the ratios $R(D)$ and $R(D^{*})$ related to the semileptonic $B$ decays $B\to D\tau\nu$ and $B\to D^{*}\tau\nu$, and the newly suggested observable $R(0)$ related to the purely leptonic $B$ decays $B\to\tau\nu$ and $B\to\mu\nu$, considering possible effects from presence of a neutral fermion (sterile neutrino) $N$ with mass $\sim 1$ GeV. In theoretical estimation of $R(D)$, $R(D^{*})$, and $R(0)$, the possible effects of the processes $B\to D\tau N$, $B\to D^{*}\tau N$, and $B\to\tau N$ were included. For generality we considered two possible scenarios: with the assumption of unitarity of the full $4\times 4$ mixing matrix $U$ (extended PMNS matrix), and without the assumption of unitarity.

We analyzed each of $R(D)$, $R(D^{*})$ and $R(0)$ separately. Our findings are summarized as follows.

For the observable $R(D)$, assuming $U$ to be nonunitary, the discrepancy between the experimental data and the theoretical prediction can be resolved at $1\sigma$ level [cf. Fig. 3(a)] for $M_{N}<0.5$ GeV. The possible values of $|U_{\tau N}|^{2}$ are found for various values of the mass $M_{N}$. Especially, we have found that the values of $|U_{\tau N}|^{2}$ allowed by the $1\sigma$ data are $2.8\times 10^{-2}\lesssim|U_{\tau N}|^{2}\lesssim 3.5\times 10^{-1}$ and $3.0\times 10^{-2}\lesssim|U_{\tau N}|^{2}\lesssim 3.7\times 10^{-1}$ for $M_{N}=0.3$ GeV and $0.4$ GeV, which fall within these intervals, respectively. These values can be compared with the known CHARM and DELPHI CHARM ; DELPHI upper bounds $|U_{\tau N}|^{2}=1.5\times 10^{-1}$ and $6.0\times 10^{-2}$ for $M_{N}=0.3$ GeV and $0.4$ GeV, respectively. However, if the nonunitary $U$ has its origin in general $SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$ models LangSan , these models would imply that $|U_{\tau N}|^{2}<5\times 10^{-3}$. Further, if accounting for the indirect upper bounds on $|U_{\tau N}|^{2}$ coming from the lepton universality measurements, Fig. 2(a), the otherwise generous CHARM and DELPHI upper bounds at $M_{N}\leq 0.5$ GeV get decreased to $|U_{\tau N}|^{2}\lesssim 10^{-2}$. 2. 2.

When the full $4\times 4$ matrix $U$ is assumed to be unitary, in contrast to the above case there is no resolution for the anomaly of $R(D)$ within $1\sigma$ level, as shown in Fig. 3(b). 3. 3.

We found it to be more difficult to resolve the anomaly of $R(D^{*})$, compared with the $R(D)$ one. The discrepancy in $R(D^{*})$ can be resolved only when we assume that $U$ is nonunitary, and only at best at $2\sigma$ level and at $M_{N}\approx 0.3$ GeV, cf. Fig. 4. When taking into account the indirect upper bounds on $|U_{\tau N}|^{2}$ coming from the lepton universality measurements, then the discrepancy cannot be resolved even at $2\sigma$ level. 4. 4.

We demonstrated that certain useful information on the parameters $|U_{\tau N}|$ and $M_{N}$ can be extracted from the purely leptonic $B$ meson decays $B\to\tau\nu$, $B\to\tau N$ and $B\to\mu\nu$. If the observable $R(0)$, involving the rates of these decays, is measured in the future experiments, such as at Belle-II, the value of $|U_{\tau N}|$ could be determined without any hadronic uncertainties, depending on $M_{N}$, cf. Fig. 6.

We assumed that the sterile heavy particle $N$ is stable and invisible, hence does not decay inside the detector, and thus manifests itself as “missing momentum” in the measurements. However, depending on the values of $U_{lN}$, $M_{N}$ and the detector size, the produced sterile neutrino $N$ can decay within or beyond the actual detector. When $N$ is produced as $B^{+}\to\tau^{+}N$ or $B^{+}\to D^{(*)}\tau^{+}N$, and if $N$ also decays within the detector, the main signature of $N$ will be $N\to l^{+}\pi^{-}$ (if $N$ is Majorana) or $N\to l^{-}\pi^{+}$ (if $N$ is Dirac or Majorana), which will appear experimentally as a resonance in $M(l^{\pm}\pi)$. Then the experimental signatures would be $B\to D^{(*)}l^{\pm}l^{\pm}\pi$ and $B\to l^{\pm}l^{\pm}\pi$ (if $N$ is Majorana) or $B\to D^{(*)}l^{\pm}l^{\mp}\pi$ and $B\to l^{\pm}l^{\mp}\pi$ (if $N$ is Dirac or Majorana). The details of such decays at Belle-II and LHCb have been discussed in Ref. Cvetic:2016fbv , for sufficiently small values of $|U_{\ell N}|$, and $M_{N}\lesssim 2$ GeV.

Acknowledgements.

G.C. acknowledges the support by FONDECYT (Chile) Grant No. 1130599. The work of C.S.K. was supported in part by the NRF grant funded by the Korean government of the MEST (No. 2016R1D1A1A02936965).