CP violation in $B_{s}$ decays mediated by two nearly-degenerate Majorana neutrinos¹¹1Phys.Rev.D 105 (2022) 073001, https://link.aps.org/doi/10.1103/PhysRevD.105.073001

The neutrino oscillation experiments have confirmed that at least two of the three generations of neutrinos have nonzero masses Super-Kamiokande:1998kpq ; Super-Kamiokande:2010orq . The relation between the flavor eigenstates and mass eigenstates of three generations of neutrinos is described by the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. Thus, two questions about the nature of the neutrino arise. The first is the origin of the neutrino mass. As a well-known approach for the generation of neutrino mass, the seesaw mechanism offers a natural explanation for the tininess of the neutrino mass, for which the introduction of one or more generations of heavy Majorana neutrinos is necessary Minkowski:1977sc ; GellMann:1980vs ; Mohapatra:1979ia ; Yanagida:1980xy . Studies on the cosmological effect of the heavy neutrinos have shown that these heavy neutrinos can generate the observed baryon asymmetry of the Universe through leptogenesis Canetti:2012kh , and such heavy neutrinos can also serve as natural candidates for dark matter Dodelson:1993je ; Shi:1998km ; Asaka:2005pn ; Canetti:2012vf .

Another question is whether a neutrino is a Majorana particle, i.e., its antiparticle is identical to itself, or not. If the neutrino is a Dirac particle, then the lepton number is conserved ($\Delta L=0$). While if it is a Majorana particle, then the conservation of the lepton number can be violated by 2 units ($\Delta L=2$). Thus, the most important approach to establish the Majorana nature of the neutrino is to search for the lepton-number-violating (LNV) processes. Up to now, the neutrinoless double beta ($0\nu\beta\beta$) decay is the most explored LNV channel DellOro:2016tmg ; Dolinski:2019nrj ; Engel:2016xgb . Another way is to look for LNV processes that are induced by Majorana neutrino in the hardron or $\tau$ lepton decays. In literature, the LNV processes in the decays of mesons ($K,D,D_{s},B,B_{c}$) Abad:1984gh ; Littenberg:1991ek ; Littenberg:2000fg ; Ali:2001gsa ; Ivanov:2004ch ; Dib:2000wm ; Atre:2005eb ; Atre:2009rg ; Helo:2010cw ; Cvetic:2010rw ; Cvetic:2016fbv ; Cvetic:2017vwl ; Cvetic:2019shl ; Milanes:2016rzr ; Mejia-Guisao:2017gqp ; Milanes:2018aku ; Asaka:2016rwd ; Chun:2019nwi ; Quintero:2011yh , baryons ($\Sigma^{-},\Xi^{-},\Lambda_{b}$) Barbero:2002wm ; Barbero:2007zm ; Barbero:2013fc ; Mejia-Guisao:2017nzx ; Zhang:2021wjj , and $\tau$ lepton Castro:2012gi ; Gribanov:2001vv ; Cvetic:2002jy ; Atre:2005eb were extensively investigated. If the mass of the hypothetical Majorana neutrino lies in the range from hundreds of keV to several GeV, the decay widths of the corresponding LNV hadron/$\tau$ decays can be resonantly enhanced due to the on shellness of the intermediate Majorana neutrino. Thus, it is possible for these LNV decays to be observed by current or future hardron collider experiments. On the other hand, the nonobservation of these LNV hadron or $\tau$ decays can set strong constraints on the heavy-light neutrino mixing parameters in the resonant range.

The neutrino oscillation experiments also showed that the $\theta_{13}$ angle in the PMNS matrix has a nonzero value T2K:2011ypd ; DayaBay:2012fng ; RENO:2012mkc ; thus, the CP violation in the neutrino sector is still possible. Moreover, the introduction of one or more generations of heavy neutrinos brings in more CP-violating phases in the extended PMNS matrix, which describes the mixing between the flavor eigenstates and mass eigenstates of three normal neutrinos as well as the hypothetical sterile ones. Suppose there only exists one generation of sterile neutrino, the CP-violating phases cause no observable effect in the sterile-neutrino-induced LNV processes. However, if there exist two generations of GeV-scale sterile neutrinos that have nearly-degenerate masses, the CP-violating phases in the extended-PMNS matrix could cause observable CP violation in the sterile-neutrino-induced LNV meson decay. The idea was first pointed out in Ref. Cvetic:2013eza . The CP violation in such Majorana-neutrino-induced LNV processes arises through two different mechanisms. The first is the interference between the amplitudes contributed by the two generations of sterile neutrinos. And the second is the neutrino oscillation during the propagation of the two on shell sterile neutrinos, which was first investigated in Ref. Cvetic:2015ura in LNV $B$ meson decays. CP-violating LNV processes induced by similar mechanisms in the decays of other mesons Cvetic:2014nla ; Dib:2014pga ; Cvetic:2015naa ; Cvetic:2015ura ; Cvetic:2020lyh ; Zhang:2020hwj and $\tau$ leptons Zamora-Saa:2016ito ; Tapia:2019coy as well as $W$ boson Najafi:2020dkp were also studied in detail in the literature. Besides, the existence of two nearly-degenerate Majorana neutrinos together with the CP violating phases can lead to a significant difference between the decay widths of the LNV meson decays and those of the lepton-number-conserving ones Abada:2019bac ; Das:2017hmg , which is contrary to the common hypothesis. A remarkable conclusion about the CP violation in LNV meson decays due to intermediate sterile neutrino interference is that the relative size of the CP violation is independent of the neutrino mass while the parameter $\Delta m_{N}/\Gamma_{N}$ remains unchanged, where $\Delta m_{N}$ is the masses difference and $\Gamma_{N}$ the decay width of the neutrino. It is still unclear whether the conclusion holds true for four-body decays. Thus in this paper, we would apply the mechanism to four-body decays of $B_{s}$ mesons, of which the Feynman diagram is shown in Fig. 1. To our knowledge, no previous studies have explored the Majorana-neutrino-induced LNV decays of $B_{s}$ meson except Ref. Mejia-Guisao:2017gqp . Neither has any experimental research about the process been done. The mass difference between $B_{s}$ and $D_{s}$ reaches 3.4 GeV, which may extend the constraining mass region provided by previous channels such as three-body $D$ and $B$ meson decays. On the other hand, the branching fraction of $B_{s}\rightarrow D_{s}l\nu$ in $B_{s}$ decay reaches $8.1\%$, which means the branching fraction of the related LNV processes may be considerable since the latter is proportional to the former in case that $m_{N}$ equals zero.

The motivation for the existence of two heavy neutrinos with nearly-degenerate masses comes from the $\nu$MSM model Asaka:2005pn ; Gorbunov:2007ak , which proposes the existence of two generations of Majorana neutrinos with almost degenerate masses between 100 MeV and a few GeV as well as a light Majorana neutrino of mass $\sim$ 10 keV. The $\nu$MSM allows one to explain simultaneously neutrino oscillations, dark matter and baryon asymmetry of the Universe Gorbunov:2007ak .

In this work we study the CP violation between the Majorana-neutrino-mediated LNV process $B_{s}^{0}\rightarrow D_{s}^{-}\mu^{+}\mu^{+}\pi^{-}$ and its CP-conjugate process $\bar{B_{s}^{0}}\rightarrow D_{s}^{+}\mu^{-}\mu^{-}\pi^{+}$. These processes are induced by two Majorana neutrinos that have nearly degenerate but not equal masses. We focus on the neutrino mass region between 0.5 and 3.5 GeV where the resonant enhancement appears. We deal with the CP violation that arises from the interference between the amplitudes contributed by the two Majorana neutrinos. Moreover, we investigate the possibility of the detection of CP asymmetries in such decays during the LHCb upgrade II LHCb:2018roe .

The structure of this paper is as follows. In Sec. II, we introduce the formalism we used for calculation and give the expression for the size of CP violation. In Sec. III, we perform an experimental analysis on the processes under the experimental ability of LHCb during its upgrade II and give the upper bounds on $|U_{\mu N}|^{2}$ under the assumption that such modes were not observed in experiments. Sec. IV gives the summary and the conclusion of our work.

In this part, we evaluate the possibility that such CP violation is observed by LHCb in its upgrade II LHCb:2018roe . In the experiment, the most important quantity is the absolute size of the averaged branching ratio $\mathcal{B}r_{avr}$. In this section, we do not distinguish the two sterile neutrinos due to the degeneracy and use $N$ to represent both of them.

The expected number of $B_{s}^{0}$ mesons produced on LHCb can be estimated as

where $\mathcal{L}\approx 200~{}\mathrm{fb^{-1}}$ is the expected integrated luminosity of LHCb until 2035 LHCb:2018roe , $\sigma_{b\bar{b}}\simeq 144~{}\mathrm{\mu b}$ the $b\bar{b}$ cross section within the LHCb covered $\eta$ range ($2<\eta<5$) Aaij:2016avz , and $f(b\rightarrow B_{s}^{0})\simeq 4.4\%$ the hardronization factor of b quark to $B_{s}$ which is estimated from Ref. Aaij:2019pqz ³³3Ref. Aaij:2019pqz shows that the ratio between the production fraction of $\Lambda_{b}$ hardrons and the sum of the fraction of $B^{-}$ and $\bar{B}^{0}$ is around 0.259 (averaged between $4~{}\mathrm{GeV}<p_{T}<25~{}\mathrm{GeV}$ and $2<\eta<5$), while that between $\bar{B}^{0}_{s}$ and the sum of $B^{-}$ and $\bar{B}^{0}$ is 0.122. Since $B^{\pm}$, $B^{0}/\bar{B}^{0}$, $\bar{B}^{0}_{s}/B^{0}_{s}$ and $\Lambda_{b}^{0}/\bar{\Lambda}^{0}_{b}$ make the majority of $b\bar{b}$ products, the fraction $f(b\bar{b}\rightarrow\Lambda_{b})$ is estimated as $0.122/(1+0.259+0.122)\times 0.5\sim 0.044$.. The final result is $N_{B_{s}}\approx 1.9\times 10^{12}$.

In order to get the expected sensitivity of LHCb on the processes in consideration, we also need to know the detection efficiency of the LHCb detector $\epsilon~{}(B_{s}\rightarrow D_{s}\mu\mu\pi)$, which contains the contribution from geometrical acceptance, trigger and selection requirements and particle identification LHCb:2013svv . A systematic evaluation of the detection efficiency requires a Monte Carlo simulation under a LHCb configuration as well as considering final state radiation generation and interaction of the produced particles with the detector and its response LHCb:2013svv . Here, however, we use an indirect approach to give an approximation to the detection efficiency, whose accuracy is enough for our calculation. Reference LHCb:2015wdu shows that the simulated detection efficiency of the process $B_{s}^{0}\rightarrow\phi(K^{+}K^{-})\mu^{+}\mu^{-}$ is $1.1\%$. The main difference between $B_{s}^{0}\rightarrow\phi(K^{+}K^{-})\mu^{+}\mu^{-}$ and $B_{s}^{0}\rightarrow D_{s}^{-}\pi^{-}\mu^{+}\mu^{+}$ comes from the replacement of $K^{-}$ with $D_{s}^{-}$ ⁴⁴4The difference between the detection of $K$ and $\pi$ as well as between $\mu^{+}$ and $\mu^{-}$ is very small and can be overlooked here.. $D_{s}$ is reconstructed by the golden mode $D_{s}\rightarrow KK\pi$, which requires two additional charged tracks and thus would reduce the detection efficiency. From Ref. LHCb:2020ist , we know that the ratio between the detection efficiency of $B_{s}^{0}\rightarrow K^{-}\mu^{+}\nu$ and $B_{s}^{0}\rightarrow D_{s}^{-}\mu^{+}\nu$ is around 0.733 (averaged over the full $q^{2}$ range). Thus, we estimate the detection efficiency of $B_{s}\rightarrow D_{s}\pi\mu\mu$ to be $1.1\%\times 0.733\approx 0.81\%$. We note that the accuracy of this approximation is enough for magnitude estimation.

Another factor we need to consider is the efficiency loss due to the flight of the long-lived particle, i.e., the intermediate sterile neutrinos $N$, in the detector. The sterile neutrinos are produced nearly on shell and would travel for certain distance before decaying into its aftermath Dib:2014iga . We include this effect by adding another factor $\mathcal{P}_{N}$ to the total detection efficiency, which relies on the lifetime of the sterile neutrino $\tau_{N}$ as

where $L_{D}\sim$ 1 m is the length of the detector and

is the decay length of the sterile neutrino Dib:2014iga . Here $c$ represents the light speed and $\gamma_{N}\beta_{N}$ is the Lorentz time dilation factor of $N$. To our knowledge, the choice of $\gamma_{N}\beta_{N}$ between 1 and 10 is common in the literature such as Refs. Cvetic:2016fbv ; Dib:2014iga based on the realistic condition of collider experiments. In this study, we take that $\gamma_{N}\beta_{N}$ equals 4 instead of doing detailed calculation of $\gamma_{N}\beta_{N}$ since the former is enough for magnitude estimation. For $\tau_{N}=1000$ ps , the factor $\mathcal{P}_{N}$ is about $56\%$, while for $\tau_{N}\leq 100$ ps $\mathcal{P}_{N}$ is very close to 1 and the effect is almost negligible.

Following the practice in experimental research of sterile neutrinos such as Refs. LHCb:2014osd ; LHCb:2016inz , we take $\tau_{N}$ as a free parameter that can be measured by LHCb experiment to avoid more complication, instead of calculating $\tau_{N}$ through $\tau_{N}=\hbar/\Gamma_{N}$. We assume that the sterile neutrino lifetime $\tau_{N}=[100,1000]$ ps, which is within the acceptance of LHCb. Here, we justify that this assumption is consistent with Equation (63) under a certain choice of the size of $|U_{lN}|^{2}$. It can be read from Ref. Deppisch:2015qwa that around 1 GeV the currently known upper bound on $|U_{eN}|^{2}$ is between $10^{-7}$ and $10^{-8}$ , and the upper bound on $|U_{\mu N}|^{2}$ is between $10^{-4}$ and $10^{-5}$, while that on $|U_{\tau N}|^{2}$ is between $10^{-3}$ and $10^{-2}$. The possible region that the corresponding lifetime of the sterile neutrino can lie in is drawn in Fig. 4. The plot shows that within the mass range $[1~{}\mathrm{GeV},~{}4~{}\mathrm{GeV}]$, the choice that $\tau_{N}=[100,1000]$ ps is acceptable.

In Table. 1, we present the expected number of events for certain choices of related parameters, based on the experimental ability discussed above. We use $\mathcal{N}_{+}$/$\mathcal{N}_{-}$ to represent the event numbers of $B_{s}^{0}\rightarrow D_{s}^{-}\mu^{+}\mu^{+}\pi^{-}$/$\bar{B_{s}^{0}}\rightarrow D_{s}^{+}\mu^{-}\mu^{-}\pi^{+}$. The expected number of events at LHCb upgrade II is estimated as

For $|U_{\mu N}|^{2}=10^{-4}$, several hundreds of events can be expected, and there is significant difference between $\mathcal{N}_{+}$ and $\mathcal{N}_{-}$. For $|U_{\mu N}|^{2}=10^{-5}$, event numbers decrease to a few, and the observable CP violation is not that significant. However, it should be noted that our result is based on a relatively conservative estimation on the experimental ability of LHCb since we use the detection efficiency of previous LHCb experiments. During the upgrade II, certain detection ability of LHCb detectors will be improved LHCb:2018roe and it is possible that more events can be observed.

On the other hand, suppose that such modes were not observed on the LHCb upgrade II, we can constrain the upper bound on the heavy-light mixing parameter $|U_{\mu N}|^{2}$ by requiring the total number of events to be lower than some threshold. Reference Feldman:1997qc shows that the experimental sensitivity on $|U_{\mu N}|^{2}$ at $95\%$ confidence level (C.L.) under a background-free environment is obtained for $\mathcal{N}_{\text{events}}=3.09$. Thus, the upper bound on $|U_{\mu N}|^{2}$ at $95\%$ C.L. is obtained by requiring that

The numerical result of Equation (39) is shown in Fig. 5. For comparison, we also include currently known upper bounds on $|U_{\mu N}|^{2}$ given by other experiments including NuTeV Vaitaitis:1999wq , BEBC CooperSarkar:1985nh , Belle Liventsev:2013zz , and Delphi Abreu:1996pa in the plot. The plot shows that during the mass region $[1~{}\mathrm{GeV},3~{}\mathrm{GeV}]$, the channel gives a comparable or even stronger constraint on the size of $|U_{\mu N}|^{2}$. Specifically, for $m_{N}$ equals 2 GeV, the constraint on $|U_{\mu N}|^{2}$ goes as low as $6\times 10^{-6}$.

In this paper, we study the lepton-number-violating (LNV) process $B_{s}^{0}\rightarrow D_{s}^{-}\mu^{+}\mu^{+}\pi^{-}$ and its CP-conjugate process $\bar{B_{s}^{0}}\rightarrow D_{s}^{+}\mu^{-}\mu^{-}\pi^{+}$ that are induced by two nearly-degenerate Majorana neutrinos and explore the possibility for searching for the CP violation in such processes. We point out that the physics origin of the CP violation is $\vartheta_{12}$, which is defined as the difference between the CP-odd phases of mixing parameters between two generations of heavy neutrinos with the normal ones. The CP violation becomes considerable when the masses of the two generations of heavy neutrinos are nearly degenerate but have a nonzero difference $\Delta m_{N}$. The numerical result draws the following conclusion. First, the relative size of the CP violation $\mathcal{A}_{CP}$ is only the function of the mass difference $\Delta m_{N}$ and $\vartheta_{12}$ and $\mathcal{A}_{CP}$ is nearly independent of the absolute mass of the lighter heavy neutrino in the mass region we considered. Second, $\mathcal{A}_{CP}$ reaches its maximum when $\Delta m_{N}$ is around the size of the decay width of the intermediate sterile neutrino and the maximum value depends on the CP-odd phase difference $\vartheta_{12}$. It should be noted that the above conclusion is drawn under the assumption that the two Majorana neutrinos have approximately the same mixing parameters with the three normal neutrinos.

We also analyze the possibility that such CP violation is observed by LHCb during its upgrade II. It is shown that under a current constraint on the heavy-light neutrino mixing parameter, namely, $10^{-4}<|U_{\mu N}|^{2}<10^{-5}$, it is possible that such a CP violation can be observed with the LHCb experimental ability. We also give the upper bound on the heavy-light mixing parameter under the assumption that no positive signal of the processes is observed. The result shows that such modes can give a complementary constraint on the heavy-light mixing parameter compared with previous experiments including NuTeV, BEBC, Belle, and Delphi in the mass region 1 GeV $<m_{N}<$ 3 GeV. Thus, we note that it is worthwhile searching for such modes on LHCb due to the possibility of both observing new types of CP violation in $B_{s}$ meson decays and setting complementary constraints on $|U_{\mu N}|^{2}$.

This work is supported by National Natural Science Foundation of China (Grant No. 12075003).