Lepton-Number Violation in Bs Meson Decays Induced by an On-Shell Majorana Neutrino

PHYSICAL REVIEW D 97, 075018 (2018)

Lepton-number violation in Bs meson decays induced by an on-shell Majorana neutrino

† ‡ Jhovanny Mejía-Guisao,1,* Diego Milanés,2, Néstor Quintero,3, and José D. Ruiz-Álvarez4,§ 1Departamento de Física, Centro de Investigación y de Estudios Avanzados del IPN, Apartado Postal 14-740, 07000 Ciudad de México, México 2Departamento de Física, Universidad Nacional de Colombia, Código Postal 11001, Bogotá, Colombia 3Facultad de Ciencias Básicas, Universidad Santiago de Cali, Campus Pampalinda, Calle 5 No. 62-00, Código Postal 76001, Santiago de Cali, Colombia 4Departamento de Física, Universidad de Los Andes, Código Postal 111711, Bogotá, Colombia

(Received 10 February 2018; published 16 April 2018)

Lepton-number violation can be induced by the exchange of an on-shell Majorana neutrino N in 0 − − þ þ semileptonic jΔLj¼2 decays of the Bs meson, Bs → P π μ μ with P ¼ K;Ds. We investigate the production of such a heavy sterile neutrino through these four-body μþμþ channels and explore the sensitivity that can be reached at the LHCb and CMS experiments. For heavy neutrino lifetimes of −1 τN ¼½1; 100; 1000 ps and integrated luminosities collected of 10 and 50 fb at the LHCb and 30, 300, and 3000 fb−1 at the CMS, we find a significant sensitivity on branching fractions of the orders 0 − − þ þ −9 −8 0 − − þ þ −8 −7 BRðBs → K π μ μ Þ ≲ Oð10 –10 Þ and BRðBs → Ds π μ μ Þ ≲ Oð10 –10 Þ. In the kinematically allowed mass ranges of mN ∈ ½0.25; 4.77 GeV and mN ∈ ½0.25; 3.29 GeV, respectively, we exclude 2 regions on the parameter space ðmN; jVμNj Þ associated with the heavy neutrino, which could slightly improve the limits from B− → πþμ−μ− (LHCb).

DOI: 10.1103/PhysRevD.97.075018

I. INTRODUCTION looking for processes with jΔLj¼2, in which the possible existence of Majorana neutrinos can be tested [1]. To discriminate if the light neutrinos are Majorana or The smoking-gun LNV signal is the neutrinoless double- Dirac fermions (i.e., if neutrinos are their own antiparticles β (0νββ) decay [4–6]. Searches of this rare nuclear or not) is one of the most important puzzles in the Standard transition have been pursued for several decades by differ- Model (SM) [1]. To date, it is already well confirmed by a ent experiments, and up to now no positive signal has been diversity of neutrino oscillation experiments (solar, atmos- observed [4–6]. Currently, the best limits on their half-lives pheric, reactors, and accelerators) [2] that light neutrinos have been obtained from the nuclei 76Ge [7] and 136Xe [8,9]. are massive particles; however, the responsible underlying Aside from the 0νββ decay, low-energy studies of rare mechanism remains unknown, and different new physics semileptonic processes in jΔLj¼2 decays of pseudoscalar (NP) scenarios beyond the SM predict the neutrinos to be mesons (K; D;Ds;B;Bc) and the τ lepton have been Dirac or Majorana massive fermions [3]. If neutrinos are considered as complementary and alternative evidence to Dirac massive particles, the total lepton number L is a prove the Majorana nature of neutrinos [10–37]. Since conserved quantity in the nature, while if neutrinos turn out these jΔLj¼2 decays can be produced (and enhanced) via to be Majorana massive particles, L will be not longer an intermediate on-shell Majorana neutrino N with a mass conserved and will be violated [1]. The most remarkable in the range ∼½0.1; 5.0 GeV, the phenomenology associ- searches of lepton-number violating (LNV) signals are by ated with such a heavy neutrino has been actively studied [10,13–37]. From the experimental side, upper limits on the branching fractions of various LNV processes have been *[email protected] † set by different experiments such NA48=2, BABAR, Belle, diego.milanes@cern.ch ‡ – [email protected] LHCb, and E791 [38 46]. See also the Particle Data §[email protected] Group [2]. Focusing on the b-quark sector, recent attention has been Published by the American Physical Society under the terms of paid to the four-body jΔLj¼2 decays of B and Bc mesons: the Creative Commons Attribution 4.0 International license. B¯ 0 → Dþπþμ−μ−, B− → D0πþμ−μ− [23,25,34,35], and Further distribution of this work must maintain attribution to − þ − − the author(s) and the published article’s title, journal citation, Bc → J=ψπ μ μ [20,21]. In addition, the jΔLj¼2 3 and DOI. Funded by SCOAP . decays of Λb baryon have been explored as well [47].

2470-0010=2018=97(7)=075018(11) 075018-1 Published by the American Physical Society JHOVANNY MEJÍA-GUISAO et al. PHYS. REV. D 97, 075018 (2018)

0 − − þ þ As a salient feature, these decay channels are not highly The Bs → P π μ μ decays are then split into two suppressed by Cabbibo-Kobayashi-Maskawa (CKM) fac- subprocesses, and the corresponding branching fraction tors, and their experimental search is within reach of can be written in the factorized form sensitivity of the LHCb and Belle II [20,21,35]. So far, − 0 − − þ þ 0 − þ the LHCb has reported the upper limit BRðB → BRðBs → P π μ μ Þ¼BRðBs → P μ NÞ 0 þ − − −6 D π μ μ Þ < 1.5 × 10 [41], and improvements are þ − × ΓðN → μ π ÞτN=ℏ; ð1Þ expected in Run 2 and the future upgrade Run 3. On the other hand, the same quark level LNV transition that jΔLj¼2 with τN as the lifetime of the Majorana neutrino. generates these four-body channels can also 0 − þ The branching ratio of Bs → P μ N is given by the produce jΔLj¼2 decays in the Bs meson, and their signals may be detected at the LHC, which offers an expression [34] B excellent environment for the s physics. 0 − þ In this work, we will explore the LNV decay channels of BRðBs → P μ NÞ 0 − − þ þ Z the Bs meson, Bs → P π μ μ with P ¼ K; Ds, via an d ðB0 → P−μþNÞ ¼jV j2 dt BR s ; ð Þ intermediate GeV-scale on-shell Majorana neutrino N.To μN dt 2 our knowledge, these jΔLj¼2 decays have not been investigated before from a theoretical nor from an exper- where imental point of view. We will work in a simplified approach in which one heavy neutrino N mixes with one flavor of SM 0 − þ dBRðBs → P μ NÞ lepton l and its interactions are completely determined by dt the mixing angle VlN [10]. Since 0νββ decay puts stringent 2 2 2 2 2 1=2 jV j2 ≲ 10−8 G τB ½λðmμ;mN;tÞλðmB ;mP;tÞ limits on the electron-heavy neutrino mixing eN ¼ F s jVCKMj2 s [48], we will focus our attention on the above four-body 384π3m3 ℏ qb t3 Bs μþμþ modes and explore their expected sensitivities at the ð½FBsPðtÞ2C ðtÞþ½FBsPðtÞ2C ðtÞÞ ð Þ LHCb and CMS experiments. We will show that their × þ þ 0 0 3 experimental search allows us to scan the parameter space 2 is the so-called differential canonical branching ratio [34], ðmN; jVμNj Þ of the heavy neutrino sector, therefore an G VCKM additional test of the existence of Majorana neutrinos. where F is the Fermi constant; qb denotes the CKM 1 Let us mention that the presence of a heavy neutrino with matrix element involved (with q ¼ u, c for P ¼ K; Ds) ; BsP BsP a mass of few GeV [∼Oð1Þ GeV] provides a realistic and and Fþ ðtÞ and F0 ðtÞ are the vector and scalar form falsifiable scenario for a common explanation of the baryon factors for the Bs → P transition, respectively, which are asymmetry of the Universe via leptogenesis [49–52] and evaluated at the square of the transferred momentum the generation of neutrino masses via the GeV-scale seesaw t ¼ðp − p Þ2 Bs P . The usual kinematic Källen function is model [53,54]. This gives us further motivation to study denoted by λðx; y; zÞ¼x2 þ y2 þ z2 − 2ðxy þ xz þ yzÞ. jΔLj¼2 decays of the Bs meson under consideration. The coefficients CþðtÞ and C0ðtÞ in (3) are defined as This work is organized as follows. In Sec. II, we study the jΔLj¼2 decays of the Bs meson. The expected 2 2 2 2 2 2 2 2 CþðtÞ¼λðm ;m ;tÞ½2t þ tðmμ þ m Þþðmμ − m Þ ; experimental sensitivities for these channels at the LHCb Bs P N N and CMS is discussed in Sec. III. Based on the results of ð4Þ Sec. III, we discuss the bounds on the parameter space 2 2 2 2 2 2 2 2 ðmN; jVμNj Þ of the heavy neutrino that can be achieved. C ðtÞ¼3ðm − m Þ½m ðt þ 2m − m Þþm ðt − m Þ; 0 Bs P μ N μ N N Our conclusions are given in Sec. V. ð5Þ

II. LNV DECAYS OF Bs MESON respectively. The total branching fraction is then obtained In this section, we study LNV signals in the jΔLj¼2 0 − − þ þ by integrating the differential canonical branching ratio decays of the Bs meson Bs → P π μ μ , with P ¼ K;Ds t ½ðm þ m Þ2; ðm − m Þ2 over the full region μ N Bs P . denoting a final-state pseudoscalar meson. These processes As mentioned at the Introduction, the coupling of the can occur via intermediate on-shell Majorana neutrino N N 0 − þ heavy neutrino (sterile) to the charged current of lepton through the semileptonic decay Bs → P μ N followed by flavor μ is characterized by the quantity VμN [10]. Without the subsequent decay N → μþπ−, with a kinematically referring to any NP scenario, we will treat mN and VμN as allowed mass in the ranges unknown phenomenological parameters that can be 0 − − þ þ Bs → K π μ μ ∶ mN ∈ ½0.25; 4.77 GeV; 1 jVCKMj¼4 09 10−3 0 − − þ þ We will use the central values ub . × and Bs → Ds π μ μ ∶ mN ∈ ½0.25; 3.29 GeV: CKM −3 jVcb j¼40.5 × 10 [2].

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þ þ þ 0 0 0 constrained (set) from the experimental nonobservation TABLE I. Coefficients ðb0 ;b1 ;b2 Þ and ðb0;b1;b2Þ of the z (observation) of jΔLj¼2 processes [10,15,20]. expansion in Eqs. (8) and (9), pole masses Mþð0Þ and tþð0Þ. On the other hand, the decay width of N → μþπ− is B → D given by the expression [10] Parameter s s [56] Mþ (GeV) 6.330 M ΓðN → μþπ−Þ¼jV j2Γ¯ ðN → μ−πþÞ; ð Þ 0 (GeV) 6.420 μN 6 t 2Þðm þ m Þ2 þ (GeV Bs Ds t 2 ðm − m Þ2 0 (GeV ) Bs Ds with þ b0 0.858 bþ −3 38 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 . G2 bþ Γ¯ ðN → μþπ−Þ¼ F jVCKMj2f2m λðm2 ;m2;m2Þ 2 0.6 16π ud π N N μ π b0 0 0.658 2 2 2 2 b0 −0 10 mμ mπ mμ 1 . 1 − − 1 þ ; ð Þ 0 × 2 2 2 7 b2 1.3 mN mN mN

CKM where jVud j¼0.97417 [2] and fπ ¼ 130.2ð1.7Þ MeV is þ þ þ 0 0 0 the pion decay constant [55]. TABLE II. Coefficients ða0 ;a1 ;a2 Þ and ða1;a2;a3Þ of the z The lifetime of the Majorana neutrino τN ¼ ℏ=ΓN in expansion in Eqs. (11) and (12), pole masses Mþð0Þ and tþð0Þ. Eq. (1) can be obtained by summing over all accessible Parameter Bs → K [57] final states that can be opened at the mass mN [10]. However, in further analysis (Secs. III and IV), we will Mþ (GeV) 5.3252 leave it as a phenomenological parameter accessible to the M0 (GeV) 5.6794 2 2 tþ (GeV ) ðmB þ mKÞ LHCb and CMS experiments. s pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi t 2 ðm þ m Þð m − m Þ2 0 (GeV ) Bs K Bs K þ a0 0.368 A. Form factors Bs → P (P = K;Ds) þ a1 −0.750 B → P þ For the form factors associated with the s tran- a2 2.72 0 sition, we will use the theoretical predictions provided by a1 0.315 0 the lattice QCD approach [56,57]. a2 0.945 FBsDs FBsDs 0 The form factors þ and 0 can be represented by a3 2.391 the z expansion through a modification of the Bourrely- Caprini-Lellouch (BCL) parametrization [56], 1 X2 n FBsKðtÞ¼ aþ zðtÞn − ð−1Þn−3 zðtÞ3 ; XJ−1 þ 2 n B P 1 n ð1 − t=M Þ 3 F s ðtÞ¼ bþ zðtÞn − ð−1Þn−J zðtÞJ ; þ n¼0 þ ð1 − t=M2 Þ n J þ n¼0 ð11Þ ð8Þ X3 BsK 0 n n F0 ðtÞ¼ anðzðtÞ − zð0Þ Þ 1 XJ−1 BsP 0 n n¼1 F ðtÞ¼ bnzðtÞ ; ð9Þ 0 ð1 − t=M2Þ 2 0 n¼0 X n þ aþ zð0Þn − ð−1Þn−3 zð0Þ3 ; ð Þ n 3 12 n¼0 respectively, where the zðtÞ function is defined as þ þ þ pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where the corresponding expansion coefficients ða0 ;a1 ;a2 Þ tþ − t − tþ − t0 0 0 0 zðtÞ¼pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð Þ and ða1;a2;a3Þ, pole masses Mþð0Þ and tþð0Þ, are displayed in t − t þ t − t 10 þ þ 0 Table II.

In Table I, we show the respective coefficients of the z J ¼ 3 III. EXPECTED EXPERIMENTAL expansion in Eqs. (8) and (9) for as well as additional SENSITIVITY AT THE LHC parameters: pole masses Mþð0Þ and tþð0Þ [56]. The masses of particles involved are taken from the Particle Data Now, let us provide an estimation of the expected Group [2]. number of events at the LHC, namely, LHCb and CMS In Ref. [57], the form factors for the Bs → K transition experiments, for the jΔLj¼2 channels of the Bs meson, 0 − − þ þ are parametrized in a modified BCL form, Bs → P π μ μ (with P ¼ K; Ds), discussed above.

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A. LHCb experiment our case, long-lived particles can only be produced on shell, The number of expected events in the LHCb experiment therefore with masses around few GeV. Thus, to account for has the form this effect, we will just add a 25% relative uncertainty to our efficiency prediction, obtaining finally LHCb N ¼ σðpp → HbXÞ fðb → BsÞBRðBs → ΔL ¼ 2Þ exp acc LHCb − − þ þ LHCb ϵD ðBs → K π μ μ ÞPN ≃ ð1.10 0.27Þ%; LHCb LHCb LHCb × ϵ ðBs → ΔL ¼ 2ÞP L ; ð13Þ D N int LHCb − − þ þ LHCb ϵD ðBs → Ds π μ μ ÞPN ≃ ð0.89 0.22Þ%: where σðpp → HbXÞ is the production cross section acc NLHCb of b hadrons inside the LHCb geometrical acceptance; With these values, the relative uncertainty on exp is of fðb → BsÞ is the hadronization factor of a b quark to 32% for both LNV modes. B LLHCb The LHCb experiment performance during LHC Run 1 the s meson; int is the integrated luminosity; ðB → ΔL ¼ 2Þ can be found in Ref. [62]. During LHC Run 2, the BR s corresponds to the branching fraction −1 LHCb expectation is to collect 10 fb at the LHC nominal of the given LNV process; and ϵD ðBs → ΔL ¼ 2Þ is its detection efficiency of the LHCb detector involving construction energy of a center of mass of 14 TeV. reconstruction, selection, trigger, particle misidentification, Already some work has been developed for the future LHCb upgrade, LHC Run 3, for which integrated lumi- and detection efficiencies. Most of the on-shell neutrinos −1 0 − − þ nosity of the order of 50 fb is expected. Assuming the produced in the decays Bs → ðK ;Ds Þμ N are expected to live a long enough time to travel through the detector and above assumptions on efficiency and cross section, Fig. 1 decay (N → μþπ−Þ far from the interaction region. This shows the number of expected events to be observed in the LHCb LHCb experiment as a function of branching fraction for effect is given by the PN factor (acceptance factor), jΔLj¼2 modes of Bs meson. The figure shows red and which accounts for the probability of the on-shell neutrino magenta functions, corresponding to LHC Run 2 and LHC N decay products to be inside the LHCb detector accep- Run 3, respectively. Table III shows the expected signal tance [30]. The reconstruction efficiency will depend on events at the LHCb experiment for some selected values of this acceptance factor as well. the branching ratio, given LHC Run 2 and LHC Run 3 The production cross section has been measured to be expected integrated luminosities. We can see that values of σðpp → HbXÞ ¼ð75.3 5.4 13.0Þ μb inside the acc the branching fractions of the order Oð10−9–10−8Þ for LHCb acceptance [58]. The world average for the hadro- 0 − − þ þ −8 −7 0 − − þ þ Bs → K π μ μ and Oð10 –10 Þ for Bs → Ds π μ μ nization factor is taken to be fðb → BsÞ¼0.103 0.005 [59]. The proper computation of the detection efficiency might be within the experimental sensitivity of the LHCb. requires fully simulated Monte Carlo samples of the exclusive decay, reconstructed in the same way as real B. CMS experiment LHCb data. Here, we perform a rough estimation of the We also consider the possible sensitivity of the CMS detection efficiency, based on extrapolation of detection experiment to the LNV signals from Bs meson decays. The efficiencies already reported by LHCb experiment of expected number of event for the CMS experiment is similar final states. written as The LHCb Collaboration has measured the detection 0 þ − þ − NCMS ¼ σðpp → B XÞ ðB → ΔL ¼ 2Þ efficiency of the Bs → ϕðK K Þμ μ decay mode to be exp s BR s 1.1% [60]. This measurement includes trigger, tracking, CMS CMS CMS ×ϵD ðBs → ΔL ¼ 2ÞPN L ; ð14Þ reconstruction, particle identification, and selection effi- int LCMS ciency. Given the content of final-state charged tracks, we where int is the integrated luminosity recorded by the 0 − − þ þ can consider the Bs → K π μ μ to be the same as for CMS experiment from proton-proton collisions delivered þ − þ − 0 the Bs →ϕðK K Þμ μ decay. Regarding the Bs → by the LHC; σðpp → BsXÞ is the Bs meson production − − þ þ Ds π μ μ decay, a golden mode to reconstruct the cross section in the CMS experiment acceptance; þ þ þ − þ CMS Ds meson hadronically is Ds → K K π , where ϵD ðBs → ΔL ¼ 2Þ is the efficiency to reconstruct and þ þ − þ −2 BRðDs → K K π Þ¼ð5.45 0.17Þ × 10 [2]. In this identify the signal events, which includes the trigger CMS situation, there will be two additional charged tracks in the efficiency; PN is a factor that accounts for the CMS final state; thus, we can multiply previous efficiency by 0.9 experiment acceptance to the decay of the neutrino; and for each additional charged track, the approximated BRðBs → ΔL ¼ 2Þ is the Bs meson branching fraction. single track reconstruction efficiency at LHCb. Finally, The CMS experiment acceptance to the signal depends in Ref. [61], reconstruction efficiencies for hypothetical on its tracker capabilities to reconstruct charged particles, long-lived particles inside the LHCb acceptance are given. especially pions and muons. Muons are reconstructed using Here, we can observe that a maximum variation of about the tracking system and the muon chambers, while pions 25% is measured in the efficiencies of particles living in the are reconstructed by the tracker solely. The decay products [5–50] ps range, with masses up to 200 GeV; however, in from the Bs are not very energetic. For this study, we

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TABLE III. Number of expected events at the LHCb for some 0 − − þ þ selected values of the branching ratio (BR) of Bs → K π μ μ 0 − − þ þ and Bs → Ds π μ μ .

LLHCb −1 NLHCb Mode int (fb ) BR exp 0 − − þ þ −6 Bs → K π μ μ 50 10 8522 2727 10−7 852 273 10−8 85 27 10−9 9 3 10 10−6 1583 506 10−7 158 51 10−8 16 5 10−9 2 1 0 − − þ þ −6 Bs → Ds π μ μ 50 10 376 120 10−7 37 12 10−8 4 1 10 10−6 70 22 10−7 7 2

With the CMS experiment, it is not possible to distinguish from a charged track left in the detector by a pion or a kaon. − Therefore, we consider all the possible decays of Ds into three charged tracks. Considering world averages for Kππ or KKπ decay branching fractions [2], we can derive that − the BRðDs → 3 charged tracksÞ¼13.00 1.96. Taking into account this additional branching fraction and the fact that we need to identify two additional charged tracks, we can plug an additional 90% efficiency factor for the 0 − − þ þ track to obtain the total efficiency for the Bs → Ds π μ μ CMS 0 − − þ þ channel. We obtain that ϵD ðBs → Ds π μ μ Þ¼ ð1.26 0.04Þ%. B0 → FIG. 1. Number of expected events of the process s Considering the distance the neutrino can fly in the K−π−μþμþ B0 → D−π−μþμþ (top) and s s (bottom) to be observed detector, we restrict the discussion to lifetimes between in the LHCb experiment as a function of their branching fractions τN ¼ 1 and 1000 ps, where the detector has sensitivity. The for a luminosity of 10 fb−1 (red) and 50 fb−1 (magenta). The neutrino originates from the decay of Bs. The mean lifetime solid black line shows the central value, while the filled area B shows the 1-σ uncertainty. of s meson is 1.505 ps [2]. Considering that the mean momentum of Bs is 20 GeV, from Table 1 in Ref. [65], the p Lorentz time dilation factor for Bs is M ≈ 4, implying a consider that the muons and pions from signal events have a decay length of 0.2 cm. For the neutrino, we consider that p < 20 p T GeV. The CMS experiment has shown to be 90% M ≈ 1 as it proceeds from the Bs decay. Therefore, the total efficient in reconstructing charged tracks in the mentioned decay length of the neutrino, taking into account the initial pT range [63]. However, these studies were performed for a decay length of Bs,isLN ¼ 0.2 cm (30.2 cm) for τN ¼ center-of-mass energy of 7 TeV; we consider that these 1 ps (1000 ps) lifetime. Accordingly, with the studies results also stand for 13 TeV. In addition, we also assume performed in Ref. [63], the reconstruction efficiency in that the reconstruction efficiency of muons is 90%, follow- the tracker is degraded in terms of the distance in the tracker ing the results from Ref. [64]. system from where the traces originate. From the same We use the same techniques as in Ref. [47] to make a study, the reconstruction efficiency of tracks originating at rough estimate of the CMS experiment efficiency to 30 cm from the collision point is 55%, while for just 1 cm, reconstruct the signal events. From some analyses per- it is 100%. The relative uncertainty applied on the overall formed with the CMS experiment for similar events reconstruction efficiency from CMS results is 18%. It can [65,66], we can assume that the efficiency for the events be expected that differences from these assumptions would 0 − − þ þ from Bs → K π μ μ will be approximately the same be found if a full study were done using the most recent 0 − − þ þ ð1.56 0.05Þ%. For the decay channel Bs → Ds π μ μ , energies used by the LHC. However, we expect to cover − we need to consider the further decay of the Ds meson. these differences by the uncertainty assigned.

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0 Additionally, the cross section of the Bs meson produc- TABLE IV. Expected number of events for the CMS experi- tion from proton-proton collisions in the geometrical ment with three branching fractions of 10−6, 10−7, and 10−8 for 0 − − þ þ 0 − − þ þ acceptance of the CMS experiment is obtained from Bs → K π μ μ and Bs → Ds π μ μ . σðpp → B0Þ ðB0 → J=ψϕÞ¼ Ref. [65]. The s ×BR s LCMS −1 NCMS 0 Mode int (fb ) BR exp 6.9 0.6 0.6 nb at 7 TeV, and taking BRðBs →J=ψϕÞ¼ −3 0 − − þ þ −6 ð1.070.08Þ×10 , the pure production cross section for Bs → K π μ μ 30 10 5616 825 0 −7 proton-proton collisions at 7 TeV is σðpp → BsÞ¼ 10 562 82 ð6.45 0.09Þ × 103 nb. Thus, assuming that the cross 10−8 56 8 0 section increases as the center-of-mass energy, the Bs 300 10−8 562 82 production cross section at 13 TeV proton-proton collisions 10−9 56 8 0 3 is σðpp → BsÞ¼ð11.98 0.17Þ × 10 nb. 0 − − þ þ −6 Bs → Ds π μ μ 30 10 591 87 Figure 2 shows the results for the expected number of 10−7 59 9 events in the CMS experiment, using the above estimations. 10−8 6 1 LCMS ¼ 30 Three benchmark luminosities are used: int , 300, 10−7 591 87 3000 −1 300 and fb . Table IV is used to quote explicitly some of 10−8 59 9 the results obtained. We observe that for 30 and 300 fb−1

the CMS experiment has sensitivity to branching fractions −9 −8 0 − − þ þ of the order Oð10 –10 Þ for Bs → K π μ μ and −8 −7 0 − − þ þ Oð10 –10 Þ for Bs → Ds π μ μ . Such a sensitivity is very similar to the one that can be reached by the LHCb (see Sec. III A). We will consider these values of branching fractions as the most conservative ones to derive limits over the parameters of the heavy sterile neutrino in the next section.

IV. BOUNDS ON THE PARAMETER 2 SPACE ðmN;jVμNj Þ The experimental nonobservation of jΔLj¼2 processes can be reinterpreted as bounds on the parameter space of a 2 heavy sterile neutrino ðmN; jVμNj Þ, namely, the squared 2 mixing element jVμNj as a function of the mass mN [10,15,20]. Based on the analysis presented in Sec. III, 2 here, we explore the constraints on the ðmN; jVμNj Þ plane that can be achieved from the experimental searches on 0 − − − þ þ Bs → ðK ;Ds Þπ μ μ at the LHC, namely, the LHCb and CMS experiments. From Eq. (1), it is straightforward to obtain the relation ℏ ðB0 → P−π−μþμþÞ 1=2 jV j2 ¼ BR s ; μN 0 − þ ¯ þ − BRðBs → P μ NÞ × ΓðN → μ π ÞτN ð15Þ

0 − þ ¯ þ − where BRðBs → P μ NÞ and ΓðN → μ π Þ are given by Eqs. (3) and (7), respectively. As was already discussed in 0 Sec. III and following the analysis of NA48=2 [38] and FIG. 2. Expected events in the CMS experiment for Bs → − − þ þ 0 − − þ þ LHCb [40], we will consider heavy neutrino lifetimes of K π μ μ (top) and Bs → Ds π μ μ (bottom) as a function of τN ¼½1; 100; 1000 ps as benchmark points in our analysis. the branching fraction of the final state considered and for three 2 benchmark luminosities: 30 (green), 300 (blue), and 3000 (gray) This will allow us to extract limits on jVμNj without any fb−1. The central value is shown with a solid line. The shaded area additional assumption on the relative size of the mixing represents the associated uncertainty in a 1-σ window. matrix elements.

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1000 1

10-1

100 10-2 B→πμμ 10-3 (LHCb revised) ps 2 |

10 N -4 μ

N 10 |V 10-5 K→πμμ(NA48/2) 1 -6 τ 10 N= 1 ps τ = 100 ps -7 N 10 -8 τ → πμμ N= 1000 ps 0.1 (a) BR(Bs K ) < 10 0.5 1.0 2.0 5.0 10-8 m N GeV 2×10-1 1 2 3 4 5 6 m [GeV] τ m N FIG. 3. Heavy neutrino lifetime N as a function of N. The 1 blue and red bands correspond to the allowed parameter spaces 2 −3 −2 2 -1 for jVτNj ¼ 10 and 10 , respectively, while jVeN j and 10 jV j2 ½10−7; 10−3 μN vary within the range . 10-2 B→πμμ 10-3 (LHCb revised) 2 |

From the theoretical point of view, is worth it to N -4 1 ≤ μ 10 justify heavy neutrino lifetimes within the domain ps |V -5 τN ≤ 1000 ps accessible to the LHCb and CMS experi- 10 K→πμμ(NA48/2) ments (see Secs. III A and III B). For that purpose, we will -6 τ 10 N= 1 ps use the approximate expression for the neutrino decay τ = 100 ps -7 N 10 -9 τ width → πμμ N= 1000 ps (b) BR(Bs K ) < 10 10-8 -1 5 2×10 1 2 3 4 5 6 GFmN 2 2 2 Γ ¼ ½8ðjV j þjV j Þþ3jV j ; ð Þ mN [GeV] N 96π3 eN μN τN 16 2 FIG. 4. Exclusion regions on the ðmN; jVμNj Þ plane for 0 − − þ þ −8 0 − − þ þ which has been previously considered in the literature (a) BRðBs →K π μ μ Þ<10 and (b) BRðBs →K π μ μ Þ< −9 [34,35,37] for neutrino masses relevant to the Bs meson 10 . The black, blue, and gray regions represent the bounds decays under consideration. By considering the current obtained for heavy neutrino lifetimes of τN ¼ 1, 100, 1000 ps, − þ − − − jV j2 l ¼ e μ τ respectively. Limits provided by K → π μ μ [38] and B → bounds on lN ( , , ) given in Ref. [10], we will þ − − 2 2 −7 −3 π μ μ [32] are also included for comparison. vary jVeNj and jVμNj within the range ½10 ; 10 and 2 −3 −2 jVτNj from 10 to 10 . In Fig. 3, we plot the heavy τ ¼ ℏ=Γ m neutrino lifetime N N as a function of N. The blue K− → πþμ−μ− jV j2 ∼ Oð10−5Þ and red bands correspond to the allowed parameter spaces , which can reach μN ,but jV j2 ¼ 10−3 10−2 only for a very narrow mass window of [0.25, 0.38] GeV. for τN and , respectively. According to m > 0 38 Fig. 3, it is possible to obtain masses at the GeV scale For N . GeV, the CKM-suppressed four-body B0 → K−π−μþμþ within the lifetime domains accessible to the LHCb and channel s would complement the region jV j2 B− → πþμ−μ− CMS experiments. of μN covered by the channel (also In Figs. 4(a) and 4(b), we show the exclusions regions on CKM suppressed). 2 B0 → D−π−μþμþ jVμNj as a function of mN obtained by taking an expected For searches on s s , in Figs. 5(a) and 5(b), ðm ; jV j2Þ sensitivity on the branching fractions of the orders we plot the exclusion curves on the N μN plane 0 − − þ þ −8 −9 ðB0 → BRðBs → K π μ μ Þ < 10 and < 10 , respectively. for expected sensitivities at the LHC of BR s − − þ þ −7 −8 In both scenarios, the black, blue, and gray regions Ds π μ μ Þ < 10 and < 10 , respectively. Again, the represent the bounds obtained for heavy neutrino lifetimes black, blue, and gray regions represent the constraints τ ¼ 1 of τN ¼ 1, 100, 1000 ps, respectively. We also plot the obtained for heavy neutrino lifetimes of N , 100, exclusion limits obtained from searches on jΔLj¼2 1000 ps, respectively. For Majorana neutrino masses larger − þ − − − 0 − − þ þ channels, K →π μ μ (NA48/2) [38] and B → than 0.38 GeV, the Bs → Ds π μ μ channel (CKM πþμ−μ− (LHCb) [43], for comparison. For the B− → allowed) would be able to exclude a slightly wider region þ − − 2 − þ − − π μ μ channel, we compare with the revised limit [32] of jVμNj than B → π μ μ . The reason for this is the from the LHCb analysis [43]. The limit from the K− → nonsuppression for the CKM elements involved. þ − − π μ μ channel is taken for τN ¼ 1000 ps [38]. We can Additionally, in Fig. 6, we show the exclusion bounds 2 observe that the most restrictive constraint is given by on the parameter space ðmN; jVμNj Þ coming from the

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1 1

-1 Belle 10-1 10 -2 10 NA3 10-2 -3 B→π 10 -3 μμ(LHCb revised) CHARMII 10 -4 2 2

| 10 | N N DELPHI

-4 μ μ 10 10-5 |V |V 10-5 K→πμμ(NA48/2) 10-6 -7 → πμμ 10-6 τ 10 Bs K N= 1 ps NuTeV τ = 100 ps -8 B →D πμμ -7 N 10 s s 10 -7 τ (a) BR(B →D πμμ) < 10 N= 1000 ps -9 s s 10 10-8 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1 2×10 1 2 3 4 5 6 mN [GeV] m [GeV] N ðm ; jV j2Þ 1 FIG. 6. Exclusion regions on the N μN plane coming from the Belle [67], DELPHI [68],NA3[69], CHARMII [70], 10-1 and NuTeV [71] experiments. Limits provided by the searches on B0 → ðK−;D−Þπ−μþμþ 10-2 s s are represented by the gray and black regions, respectively. See the text for details. B→πμμ 10-3 (LHCb revised) 2 |

N -4 μ 10 V. CONCLUDING REMARKS |V -5 →πμμ 10 K (NA48/2) We have studied the semileptonic jΔLj¼2 decays of 0 − − þ þ -6 τ B B → P π μ μ 10 N= 1 ps the s meson s via the intermediate GeV- τ = 100 ps 0 -7 N scale on-shell Majorana neutrino N, namely, Bs → 10 -8 τ (b) BR(B →D πμμ) < 10 N= 1000 ps − þ − þ s s P μ Nð→π μ Þ, with P ¼ K; Ds. To our knowledge, 10-8 B 2×10-1 1 2 3 4 5 6 these LNV decays of the s meson have not been

mN [GeV] investigated before from a theoretical nor from an experimental point of view. We investigated these same-sign 0 þ þ FIG. 5. The same caption as in Fig. 4 but for (a) BRðBs → μ μ channels and explored the sensitivity that can be − − þ þ −7 0 − − þ þ −8 Ds π μ μ Þ<10 and (b) BRðBs →Ds π μ μ Þ<10 . reached at the LHCb and CMS experiments. We considered heavy neutrino lifetimes in the experimental (LHCb and τ ¼½1; 100; 1000 Belle [67], DELPHI [68],NA3[69], CHARMII [70], CMS) accessible ranges of N ps, where N and NuTeV [71] experiments, in the mass range the probability for the on-shell neutrino decay products P [0.5,5.0] GeV.2 In comparison, the constraints obtained to be inside the detector (acceptance factor N) has been 0 − − − þ þ taken into account in our analysis. As an outcome, from the searches on Bs → ðK ;Ds Þπ μ μ are represented by the gray and black regions, for branching fractions it was found that for integrated luminosities collected of 50 −1 of BR < 10−9 and BR < 10−8, respectively. In both cases, a 10 and fb by the LHCb experiment and 30, 300, and 3000 fb−1 by the CMS experiment one would lifetime of τN ¼ 1000 ps has been taken as a representative value. It is observed that our jΔLj¼2 channels proposals expect sensitivities on the branching fractions of the ðB0 → K−π−μþμþÞ ≲ Oð10−9 − 10−8Þ are less restrictive than the bounds obtained from different orders BR s and ðB0 → D−π−μþμþÞ ≲ Oð10−8 − 10−7Þ search strategies, for instance, Belle [67] and DELPHI [68]. BR s s , as conservative m ∈ ½0 25; 4 77 Nevertheless, keeping in mind that we have taken the most values. For masses in the ranges N . . GeV m ∈ ½0 25; 3 29 conservative values for the branching fractions derived in and N . . GeV, respectively, we extracted 2 Secs. III A and III B, it is possible that branching fractions bounds on the parameter space ðmN; jVμNj Þ that might values of the order BR < 10−10 might be accessible to the be obtained from their experimental search. Depending on LHCb and CMS experiments (see Figs. 1 and 2); therefore, the τN value, it was found that for mN > 0.38 GeV these these jΔLj¼2 channels would eventually provide com- four-body channels may be capable of excluding a slightly 2 − þ − − plementary bounds. wider region of jVμNj than B → π μ μ (LHCb). Consequently, the LHCb and CMS experiments have a great chance to look for heavy Majorana neutrinos in the near future, via jΔLj¼2 decays of the Bs meson. In addition, in the best-case scenario, the experimental search 2For recent reviews on the theoretical and experimental of these LNV channels would complement the bounds statuses of different GeV-scale heavy neutrino search strategies, given by different search strategies (such as NA3, see Refs. [10,72–76] and references therein. CHARMII, NuTeV, Belle, and DELPHI).

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ACKNOWLEDGMENTS N. Quintero acknowledges support from Dirección General — The work of J. Mejía-Guisao has been financially de Investigaciones Universidad Santiago de Cali supported by Conacyt (M´exico) under Projects under Project No. 935-621717-016. J. D. Ruiz-Álvarez No. 254409, No. 221329, and No. 250607 (Ciencia gratefully acknowledges the support of COLCIENCIAS Básica) and No. 2015-2-1187 (Fronteras de la Ciencia). (Colombia).

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