Lepton number violation in B_s meson decays induced by an on-shell Majorana neutrino
Jhovanny Mejia-Guisao, Diego Milanes, Nestor Quintero, Jose D. Ruiz-Alvarez
LLepton number violation in B s meson decays induced by an on-shell Majorana neutrino Jhovanny Mejía-Guisao, ∗ Diego Milanés,
2, †
Néstor Quintero,
3, ‡ and José D. Ruiz-Álvarez
4, § Departamento de Física, Centro de Investigación y de Estudios Avanzados del IPN,Apartado Postal 14-740, 07000 Ciudad de México, México Departamento de Física, Universidad Nacional de Colombia, Código Postal 11001, Bogotá, Colombia Facultad de Ciencias Básicas, Universidad Santiago de Cali, Campus Pampalinda,Calle 5 No. 62-00, Código Postal 76001, Santiago de Cali, Colombia Departamento de Física, Universidad de Los Andes, Código Postal 111711, Bogotá, Colombia
Lepton-number violation can be induced by the exchange of an on-shell Majorana neutrino N in semileptonic | ∆ L | = B s meson, B s → P − π − µ + µ + with P = K , D s . We investigate the production of such aheavy sterile neutrino through these four-body µ + µ + channels and explore the sensitivity that can be reachedat the LHCb and CMS experiments. For heavy neutrino lifetimes of τ N = [1, 100, 1000] ps and integratedluminosities collected of 10 and 50 fb − at the LHCb and 30, 300, and 3000 fb − at the CMS, we find asignificant sensitivity on branching fractions of the orders BR ( B s → K − π − µ + µ + ) (cid:46) O ( − − − ) andBR ( B s → D − s π − µ + µ + ) (cid:46) O ( − − − ) . In the kinematically allowed mass ranges of m N ∈ [ . , . ] GeV and m N ∈ [ . , . ] GeV, respectively, we exclude regions on the parameter space ( m N , | V µ N | ) associatedwith the heavy neutrino, which could slightly improve the limits from B − → π + µ − µ − (LHCb). I. INTRODUCTION
To discriminate if the light neutrinos are Majorana or Diracfermions (i.e. if neutrinos are their own antiparticles or not)is one of the most important puzzles in the Standard Model(SM) [1]. To date, it is already well confirmed by a diversityof neutrino oscillation experiments (solar, atmospheric, reac-tors and accelerators) [2] that light neutrinos are massive parti-cles; however, the responsible underlying mechanism remainsunknown, and different new physics (NP) scenarios beyondthe SM predict the neutrinos to be Dirac or Majorana mas-sive fermions [3]. If neutrinos are Dirac massive particles, thetotal lepton number L is a conserved quantity in the nature,while if neutrinos turn out to be Majorana massive particles, L will be not longer conserved and will be violated [1]. Themost remarkable searches of lepton-number violating (LNV)signals are by looking for processes with | ∆ L | =
2, in whichthe possible existence of Majorana neutrinos can be tested [1].The smoking-gun LNV signal is the neutrinoless double- β (0 νβ β ) decay [4–6]. Searches of this rare nuclear transi-tion have been pursued for several decades by different exper-iments and up to now no positive signal has been observed [4–6]. Currently, the best limits on their half-lives have beenobtained from the nuclei Ge [7] and
Xe [8, 9]. Asidefrom the 0 νβ β decay, low-energy studies of rare semilep-tonic processes in | ∆ L | = K , D , D s , B , B c ) and the τ lepton have been considered ascomplementary and alternative evidence to prove the Majo- ∗ jmejia@fis.cinvestav.mx † [email protected] ‡ [email protected] § [email protected] rana nature of neutrinos [10–37]. Since these | ∆ L | = N with a mass in the range ∼ [0.1,5.0]GeV, the phenomenology associated with such a heavy neu-trino has been actively studied [10, 13–37]. From the experi-mental side, upper limits on the branching fractions of variousLNV processes have been set by different experiments suchNA48/2, BABAR, Belle, LHCb, and E791 [38–46]. See alsothe Particle Data Group [2].Focusing on the b -quark sector, recent attention has beenpaid to the four-body | ∆ L | = B and B c mesons:¯ B → D + π + µ − µ − , B − → D π + µ − µ − [23, 25, 34, 35], and B − c → J / ψπ + µ − µ − [20, 21]. In addition, the | ∆ L | = Λ b baryon have been explored as well [47]. As a salientfeature, these decay channels are not highly suppressed byCabbibo-Kobayashi-Maskawa (CKM) factors, and their ex-perimental search is within reach of sensitivity of the LHCband Belle II [20, 21, 35]. So far, the LHCb has reported theupper limit BR ( B − → D π + µ − µ − ) < . × − [41], andimprovements are expected in Run 2 and the future upgradeRun 3. On the other hand, the same quark level LNV tran-sition that generates these four-body | ∆ L | = | ∆ L | = B s meson and their sig-nals may be detected at the LHC, which offers an excellentenvironment for the B s physics.In this work, we will explore the LNV decay channels ofthe B s meson, B s → P − π − µ + µ + with P = K , D s , via an in-termediate GeV-scale on-shell Majorana neutrino N . To ourknowledge, these | ∆ L | = N mixes with one flavor of SM lepton (cid:96) andits interactions are completely determined by the mixing an-gle V (cid:96) N [10]. Since 0 νβ β decay puts stringent limits to the a r X i v : . [ h e p - ph ] A p r electron-heavy neutrino mixing | V eN | (cid:46) − [48], we willfocus our attention on the above four-body µ + µ + modes andexplore their expected sensitivities at the LHCb and CMS ex-periments. We will show that their experimental search allowsus to scan the parameter space ( m N , | V µ N | ) of the heavy neu-trino sector, therefore, an additional test of the existence ofMajorana neutrinos.Let us mention that the presence of a heavy neutrino witha mass of few GeV [ ∼ O ( ) GeV] provides a realistic andfalsifiable scenario for a common explanation of the baryonasymmetry of the Universe via leptogenesis [49–52] andthe generation of neutrino masses via the GeV-scale seesawmodel [53, 54]. This give us further motivation to study | ∆ L | = B s meson under consideration.This work is organized as follows. In Sec. II, we studythe | ∆ L | = B s meson. The expected experi-mental sensitivities for these channels at the LHCb and CMSis discussed in Sec. III. Based on the results of the previous,in Sec. IV, we discuss the bounds on the parameter space ( m N , | V µ N | ) of the heavy neutrino that can be achieved. Ourconclusions are given in Sec. V. II. LNV DECAYS OF B s MESON
In this section, we study LNV signals in the | ∆ L | = B s meson B s → P − π − µ + µ + , with P = K , D s de-noting a final-state pseudoscalar meson. These processes canoccur via intermediate on-shell Majorana neutrino N throughthe semileptonic decay B s → P − µ + N followed by the subse-quent decay N → µ + π − , with a kinematically allowed massin the ranges B s → K − π − µ + µ + : m N ∈ [ . , . ] GeV , B s → D − s π − µ + µ + : m N ∈ [ . , . ] GeV . The B s → P − π − µ + µ + decays are then split into two subpro-cesses and the corresponding branching fraction can be writ-ten in the factorized formBR ( B s → P − π − µ + µ + ) = BR ( B s → P − µ + N ) × Γ ( N → µ + π − ) τ N / ¯ h , (1)with τ N as the lifetime of the Majorana neutrino. The branch-ing ratio of B s → P − µ + N is given by the expression [34]BR ( B s → P − µ + N ) = | V µ N | (cid:90) dt d BR ( B s → P − µ + N ) dt , (2)where d BR ( B s → P − µ + N ) dt = G F τ B s π m B s ¯ h | V CKM qb | (cid:2) λ ( m µ , m N , t ) λ ( m B s , m P , t ) (cid:3) / t × (cid:16)(cid:2) F B s P + ( t ) (cid:3) C + ( t ) + (cid:2) F B s P ( t ) (cid:3) C ( t ) (cid:17) , (3) is the so-called differential canonical branching ratio [34],where G F is the Fermi constant; V CKM qb denotes the CKMmatrix element involved (with q = u , c for P = K , D s ) ; and F B s P + ( t ) and F B s P ( t ) are the vector and scalar form factorsfor the B s → P transition, respectively, which are evaluatedat the square of the transferred momentum t = ( p B s − p P ) .The usual kinematic Källen function is denoted by λ ( x , y , z ) = x + y + z − ( xy + xz + yz ) . The coefficients C + ( t ) and C ( t ) in (3) are defined as C + ( t ) = λ ( m B s , m P , t )[ t + t ( m µ + m N ) + ( m µ − m N ) ] , (4) C ( t ) = ( m B s − m P )[ m µ ( t + m N − m µ )+ m N ( t − m N )] , (5)respectively. The total branching fraction is then obtained byintegrating the differential canonical branching ratio over thefull t region [( m µ + m N ) , ( m B s − m P ) ] .As mentioned at the Introduction, the coupling of the heavyneutrino (sterile) N to the charged current of lepton flavor µ ischaracterized by the quantity V µ N [10]. Without referring toany NP scenario, we will treat m N and V µ N as unknown phe-nomenological parameters that can be constrained (set) fromthe experimental non-observation (observation) of | ∆ L | = N → µ + π − is givenby the expression [10] Γ ( N → µ + π − ) = | V µ N | ¯ Γ ( N → µ − π + ) , (6)with¯ Γ ( N → µ + π − ) = G F π | V CKM ud | f π m N (cid:113) λ ( m N , m µ , m π ) × (cid:20)(cid:18) − m µ m N (cid:19) − m π m N (cid:18) + m µ m N (cid:19)(cid:21) , (7)where | V CKM ud | = . f π = . ( . ) MeV is thepion decay constant [60].The lifetime of the Majorana neutrino τ N = ¯ h / Γ N in Eq. (1)can be obtained by summing over all accessible final statesthat can be opened at the mass m N [10]. However, in furtheranalysis (Secs. III and IV), we will leave it as a phenomeno-logical parameter accessible to the LHCb and CMS experi-ments. We will use the central values | V CKM ub | = . × − and | V CKM cb | = . × − [2]. TABLE I. Coefficients ( b + , b + , b + ) and ( b , b , b ) of the z expan-sion in Eqs. (8) and (9), pole masses M +( ) and t +( ) .Parameter B s → D s [61] M + (GeV) 6.330 M (GeV) 6.420 t + (GeV ) ( m B s + m D s ) t (GeV ) ( m B s − m D s ) b + b + -3.38 b + b b -0.10 b A. Form factors B s → P ( P = K , D s ) For the form factors associated with the B s → P transition,we will use the theoretical predictions provided by the latticeQCD approach [61, 62].The form factors F B s D s + and F B s D s can be represented by the z expansion through a modification of the Bourrely-Caprini-Lellouch (BCL) parametrization [61], F B s P + ( t ) = ( − t / M + ) J − ∑ n = b + n (cid:104) z ( t ) n − ( − ) n − J nJ z ( t ) J (cid:105) , (8) F B s P ( t ) = ( − t / M ) J − ∑ n = b n z ( t ) n , (9)respectively, where the z ( t ) function is defined as z ( t ) = √ t + − t − √ t + − t √ t + − t + √ t + − t . (10)In Table I, we show the respective coefficients of the z ex-pansion in Eqs. (8) and (9) for J = M +( ) and t +( ) [61]. The masses ofparticles involved are taken from the Particle Data Group [2].In Ref. [62], the form factors for the B s → K transition areparametrized in a modified BCL form F B s K + ( t ) = ( − t / M + ) ∑ n = a + n (cid:104) z ( t ) n − ( − ) n − n z ( t ) (cid:105) , (11) F B s K ( t ) = ∑ n = a n (cid:0) z ( t ) n − z ( ) n (cid:1) + ∑ n = a + n (cid:104) z ( ) n − ( − ) n − n z ( ) (cid:105) , (12)where the corresponding expansion coefficients ( a + , a + , a + ) and ( a , a , a ) , pole masses M +( ) and t +( ) , are displayed inTable II. TABLE II. Coefficients ( a + , a + , a + ) and ( a , a , a ) of the z -expansion in Eqs. (11) and (12), pole masses M +( ) and t +( ) .Parameter B s → K [62] M + (GeV) 5.3252 M (GeV) 5.6794 t + (GeV ) ( m B s + m K ) t (GeV ) ( m B s + m K )( √ m B s − √ m K ) a + a + -0.750 a + a a a III. EXPECTED EXPERIMENTAL SENSITIVITY AT THELHC
Now, let us provide an estimation of the expected numberof events at the LHC, namely, LHCb and CMS experiments,for the | ∆ L | = B s meson, B s → P − π − µ + µ + (with P = K , D s ), discussed above. A. LHCb experiment
The number of expected events in the LHCb experiment hasthe form N LHCbexp = σ ( pp → H b X ) acc f ( b → B s ) BR ( B s → ∆ L = ) × ε LHCb D ( B s → ∆ L = ) P LHCb N L LHCbint , (13)where σ ( pp → H b X ) acc is the production cross section of b -hadrons inside the LHCb geometrical acceptance; f ( b → B s ) is the hadronization factor of a b -quark to the B s meson; L LHCbint is the integrated luminosity; BR ( B s → ∆ L = ) cor-responds to the branching fraction of the given LNV pro-cess and ε LHCb D ( B s → ∆ L = ) is its detection efficiency ofthe LHCb detector involving reconstruction, selection, trigger,particle misidentification, and detection efficiencies. Mostof the the on-shell neutrinos produced in the decays B s → ( K − , D − s ) µ + N are expected to live a long enough time totravel through the detector and decay ( N → µ + π − ) far fromthe interaction region. This effect is given by the P LHCb N fac-tor (acceptance factor), which accounts for the probability ofthe on-shell neutrino N decay products to be inside the LHCbdetector acceptance [30]. The reconstruction efficiency willdepend on this acceptance factor as well.The production cross section has been measured to be σ ( pp → H b X ) acc = ( . ± . ± . ) µ b inside the LHCbacceptance [63]. The world average for the hadronizationfactor is taken to be f ( b → B s ) = . ± .
005 [64]. The -10 -9 -8 -7 -6 BR ( B s K − π − µ + µ + ) -1 E x pe c t ed e v en t s a t L HC b e x pe r i m en t L = 10 fb − L = 50 fb − -10 -9 -8 -7 -6 BR ( B s D − s π − µ + µ + ) -3 -2 -1 E x pe c t ed e v en t s a t L HC b e x pe r i m en t L = 10 fb − L = 50 fb − FIG. 1. Number of expected events of the process B s → K − π − µ + µ + (top) and B s → D − s π − µ + µ + (bottom) to be observedin the LHCb experiment as a function of their branching fractionsfor a luminosity of 10 fb − (red) and 50 fb − (magenta). The solidblack line shows the central value, while the filled area shows the1- σ uncertainty. proper computation of the detection efficiency requires fullysimulated Monte Carlo samples of the exclusive decay, re-constructed in the same way as real LHCb data. Here, weperform a rough estimation of the detection efficiency, basedon extrapolation of detection efficiencies already reported byLHCb experiment of similar final states.The LHCb Collaboration has measured the detection ef-ficiency of the B s → φ ( K + K − ) µ + µ − decay mode to be1 .
1% [65]. This measurement includes trigger, tracking, re-construction, particle identification, and selection efficiency.Given the content of final-state charged tracks, we can con-sider the B s → K − π − µ + µ + to be the same as for the B s → φ ( K + K − ) µ + µ − decay. Regarding the B s → D − s π − µ + µ + decay, a golden mode to reconstruct the D + s meson hadron-ically is D + s → K + K − π + , where BR ( D + s → K + K − π + ) =( . ± . ) × − [2]. In this situation, there will be twoadditional charged tracks in the final state; thus, we can mul-tiply previous efficiency by 0.9 for each additional chargedtrack, the approximated single track reconstruction efficiencyat LHCb. Finally, in Ref. [66], reconstruction efficiencies forhypothetical long-lived particles inside the LHCb acceptanceare given. Here, we can observe that a maximum variation ofabout 25% is measured in the efficiencies of particles living inthe [5 - 50] ps range, with masses up to 200 GeV; however, inour case, long-lived particles can only be produced on-shell,therefore with masses around few GeV. Thus, to account forthis effect, we will just add a 25% relative uncertainty to ourefficiency prediction, obtaining finally ε LHCb D ( B s → K − π − µ + µ + ) P LHCb N (cid:39) ( . ± . ) % , ε LHCb D ( B s → D − s π − µ + µ + ) P LHCb N (cid:39) ( . ± . ) % . With these values, the relative uncertainty on N LHCbexp is of 32%for both LNV modes.The LHCb experiment performance during LHC-Run1 canbe found in Ref. [67]. During LHC-Run2 the expectation isto collect 10 fb − at the LHC nominal construction energy ofa center of mass of 14 TeV. Already some work has been de-veloped for the future LHCb upgrade, LHC-Run3, for whichintegrated luminosity of the order of 50 fb − is expected. As-suming the above assumptions on efficiency and cross section,Fig. 1 shows the number of expected events to be observed inthe LHCb experiment as a function of branching fraction for | ∆ L | = B s meson. The figure shows red and ma-genta functions, corresponding to LHC-Run2 and LHC-Run3,respectively. Table III shows the expected signal events at theLHCb experiment for some selected values of the branchingratio, given LHC-Run2 and LHC-Run3 expected integratedluminosities. We can see that values of the branching frac-tions of the order O ( − − − ) for B s → K − π − µ + µ + and O ( − − − ) for B s → D − s π − µ + µ + might be within theexperimental sensitivity of the LHCb. B. CMS experiment
We also consider the possible sensitivity of the CMS experi-ment to the LNV signals from B s meson decays. The expectednumber of event for the CMS experiment is written as N CMSexp = σ ( pp → B s X ) BR ( B s → ∆ L = ) × ε CMS D ( B s → ∆ L = ) P CMS N L CMSint , (14)where L CMSint is the integrated luminosity recorded by theCMS experiment from proton-proton collisions delivered bythe LHC; σ ( pp → B s X ) is the B s meson production cross sec-tion in the CMS experiment acceptance; ε CMS D ( B s → ∆ L = ) TABLE III. Number of expected events at the LHCb for some se-lected values of the branching ratio (BR) of B s → K − π − µ + µ + and B s → D − s π − µ + µ + .Mode L LHCbint (fb − ) BR N LHCbexp B s → K − π − µ + µ +
50 10 − ± − ± − ± − ±
310 10 − ± − ± − ± − ± B s → D − s π − µ + µ +
50 10 − ± − ± − ±
110 10 − ± − ± is the efficiency to reconstruct and identify the signal events,which includes the trigger efficiency; P CMS N is a factor that ac-counts for the CMS experiment acceptance to the decay of theneutrino; and BR ( B s → ∆ L = ) is the B s meson branchingfraction.The CMS experiment acceptance to the signal depends onits tracker capabilities to reconstruct charged particles, espe-cially pions and muons. Muons are reconstructed using thetracking system and the muon chambers, while pions are re-constructed by the tracker solely. The decay products fromthe B s are not very energetic. For this study, we consider thatthe muons and pions from signal events have a p T <
20 GeV.The CMS experiment has shown to be 90% efficient in recon-structing of charged tracks in the mentioned p T range [68].However, these studies were performed for a center of massenergy of 7 TeV; we consider that these results also stand for13 TeV. In addition, we also assume that the reconstruction ef-ficiency of muons is 90%, following the results from Ref. [69].We use the same techniques as in Ref. [47] to make a roughestimate of the CMS experiment efficiency to reconstruct thesignal events. From some analyses performed with the CMSexperiment for similar events [70, 71], we can assume that theefficiency for the events from B s → K − π − µ + µ + will be ap-proximately the same ( . ± . ) %. For the decay channel B s → D − s π − µ + µ + , we need to consider the further decay ofthe D − s meson. With the CMS experiment, it is not possible todistinguish from a charged track left in the detector by a pionor a kaon. Therefore, we consider all the possible decays of D − s into three charged tracks. Considering world averages for K ππ or KK π decay branching fractions [2], we can derive thatthe BR ( D − s → . ± .
96. Taking intoaccount this additional branching fraction and the fact that we need to identify two additional charged tracks, we can plugan additional 90% efficiency factor for the track to obtain thetotal efficiency for the B s → D − s π − µ + µ + channel. We obtainthat ε CMS D ( B s → D − s π − µ + µ + ) = ( . ± . ) %.Considering the distance the neutrino can fly in the detec-tor, we restrict the discussion to lifetimes between τ N = B s . The mean lifetime of B s mesonis 1.505 ps [2]. Considering that the mean momentum of B s is20 GeV, from Table 1 in Ref. [70], the Lorentz time dilationfactor for B s is pM ≈
4, implying a decay length of 0.2 cm. Forthe neutrino, we consider that pM ≈ B s decay. Therefore, the total decay length of the neutrino,taking into account the initial decay length of B s , is L N = . τ N = B s meson produc-tion from proton-proton collisions in the geometrical accep-tance of the CMS experiment is obtained from [70]. The σ ( pp → B s ) × BR ( B s → J / ψφ ) = . ± . ± . ( B s → J / ψφ ) = ( . ± . ) × − , the pureproduction cross section for proton-proton collisions at 7 TeVis σ ( pp → B s ) = ( . ± . ) × nb. Thus, assuming thatthe cross section increases as the center of mass energy, the B s production cross section at 13 TeV proton-proton collisions is σ ( pp → B s ) = ( . ± . ) × nb.Figure 2 shows the results for the expected number ofevents in the CMS experiment, using the above estimations.Three benchmark luminosities are used: L CMSint =
30, 300,and 3000 fb − . Table IV is used to quote explicitly someof the results obtained. We observe that for 30 and 300fb − the CMS experiment has sensitivity to branching frac-tions of the order O ( − − − ) for B s → K − π − µ + µ + and O ( − − − ) for B s → D − s π − µ + µ + . Such a sensitivity isvery similar to the one that can be reached by the LHCb (seeSec. III A). We will consider these values of branching frac-tions as the most conservative ones to derive limits over theparameters of the heavy sterile neutrino in the next section. -10 -9 -8 -7 -6 BR ( B s K − π − µ + µ + ) -1 E x pe c t ed e v en t s a t C M S e x pe r i m en t L = 30 fb − L = 300 fb − L = 3000 fb − -10 -9 -8 -7 -6 BR ( B s D − s π − µ + µ + ) -2 -1 E x pe c t ed e v en t s a t C M S e x pe r i m en t L = 30 fb − L = 300 fb − L = 3000 fb − FIG. 2. Expected events in the CMS experiment for B s → K − π − µ + µ + (top) and B s → D − s π − µ + µ + (bottom) as a functionof the branching fraction of the final state considered and for threebenchmark luminosities: 30 (green), 300 (blue), and 3000 (gray)fb − . The central value is shown with a solid line. The shaded arearepresents the associated uncertainty in a 1- σ window. IV. BOUNDS ON THE PARAMETER SPACE ( m N , | V µ N | ) The experimental non-observation of | ∆ L | = ( m N , | V µ N | ) , namely, the squared mix-ing element | V µ N | as a function of the mass m N [10, 15, 20].Based on the analysis presented in Sec. III, here, we explorethe constraints on the ( m N , | V µ N | ) plane that can be achievedfrom the experimental searches on B s → ( K − , D − s ) π − µ + µ + at the LHC, namely the LHCb and CMS experiments. TABLE IV. Expected number of events for the CMS experimentwith three branching fractions of 10 − , 10 − , and 10 − for B s → K − π − µ + µ + and B s → D − s π − µ + µ + .Mode L CMSint (fb − ) BR N CMSexp B s → K − π − µ + µ +
30 10 − ± − ± − ± − ± − ± B s → D − s π − µ + µ +
30 10 − ± − ± − ± − ± − ± From Eq. (1), it is straightforward to obtain the relation | V µ N | = (cid:34) ¯ h BR ( B s → P − π − µ + µ + ) BR ( B s → P − µ + N ) × Γ ( N → µ + π − ) τ N (cid:35) / , (15)where BR ( B s → P − µ + N ) and Γ ( N → µ + π − ) are given byEqs. (3) and (7), respectively. As was already discussedin Sec. III and following the analysis of NA48/2 [38] andLHCb [40], we will consider heavy neutrino lifetimes of τ N = [ , , ] ps as benchmark points in our analysis.This will allow us to extract limits on | V µ N | without any ad-ditional assumption on the relative size of the mixing matrixelements.From the theoretical point of view, is worth it to justifyheavy neutrino lifetimes within the domain 1 ps ≤ τ N ≤ m N (cid:72) GeV (cid:76) Τ N (cid:72) p s (cid:76) FIG. 3. Heavy neutrino lifetime τ N as a function of m N . The blue andred bands correspond to the allowed parameter space for | V τ N | = − and 10 − , respectively, while | V eN | and | V µ N | vary within therange [ − , − ] . [GeV] N m -1 · | N m | V -8 -7 -6 -5 -4 -3 -2 -1 (NA48/2) mmpfi K ( L H C b r e v i s e d ) mmpfi B -8 ) < 10 mmp K fi s (a) BR(B = 1 ps N t = 100 ps N t = 1000 ps N t [GeV] N m -1 · | N m | V -8 -7 -6 -5 -4 -3 -2 -1 (NA48/2) mmpfi K ( L H C b r e v i s e d ) mmpfi B -9 ) < 10 mmp K fi s (b) BR(B = 1 ps N t = 100 ps N t = 1000 ps N t FIG. 4. Exclusion regions on the ( m N , | V µ N | ) plane for (a) BR ( B s → K − π − µ + µ + ) < − and (b) BR ( B s → K − π − µ + µ + ) < − . Theblack, blue, and gray regions represent the bounds obtained for heavyneutrino lifetimes of τ N = , , K − → π + µ − µ − [38] and B − → π + µ − µ − [32] are alsoincluded for comparison. Secs. III A and III B). For that purpose, we will use the ap-proximate expression for the neutrino decay width Γ N = G F m N π (cid:2) (cid:0) | V eN | + | V µ N | (cid:1) + | V τ N | (cid:3) , (16)which has been previously considered in the literature [34, 35,37] for neutrino masses relevant to the B s meson decays underconsideration. By considering the current bounds on | V (cid:96) N | ( (cid:96) = e , µ , τ ) given in Ref. [10], we will vary | V eN | and | V µ N | within the range [ − , − ] and | V τ N | from 10 − to 10 − .In Fig. 3, we plot the heavy neutrino lifetime τ N = ¯ h / Γ N asa function of m N . The blue and red bands correspond to theallowed parameter space for | V τ N | = − and 10 − , respec-tively. According to Fig. 3, it is possible to obtain masses atthe GeV-scale within the lifetimes domains accessible to theLHCb and CMS experiments.In Figs. 4(a) and 4(b) we show the exclusions regionson | V µ N | as a function of m N obtained by taking an ex-pected sensitivity on the branching fractions of the orders [GeV] N m -1 · | N m | V -8 -7 -6 -5 -4 -3 -2 -1 (NA48/2) mmpfi K ( L H C b r e v i s e d ) mmpfi B -7 ) < 10 mmp s D fi s (a) BR(B = 1 ps N t = 100 ps N t = 1000 ps N t [GeV] N m -1 · | N m | V -8 -7 -6 -5 -4 -3 -2 -1 (NA48/2) mmpfi K ( L H C b r e v i s e d ) mmpfi B -8 ) < 10 mmp s D fi s (b) BR(B = 1 ps N t = 100 ps N t = 1000 ps N t FIG. 5. The same caption as in Fig. 4 but for (a) BR ( B s → D − s π − µ + µ + ) < − and (b) BR ( B s → D − s π − µ + µ + ) < − . BR ( B s → K − π − µ + µ + ) < − and < − , respectively.In both scenarios, the black, blue, and gray regions repre-sent the bounds obtained for heavy neutrino lifetimes of τ N = , , | ∆ L | = K − → π + µ − µ − (NA48/2) [38] and B − → π + µ − µ − (LHCb) [43], for compar-ison. For the B − → π + µ − µ − channel, we compare with therevised limit [32] from the LHCb analysis [43]. The limitfrom K − → π + µ − µ − channel is taken for τ N = K − → π + µ − µ − , which can reach | V µ N | ∼ O ( − ) , butonly for a very narrow mass window of [0.25, 0.38] GeV.For m N > .
38 GeV, the CKM-suppressed four-body channel B s → K − π − µ + µ + would complement the region of | V µ N | covered by the channel B − → π + µ − µ − (also CKM-suppre-ssed).For searches on B s → D − s π − µ + µ + , in Figs. 5(a) and5(b), we plot the exclusion curves on the ( m N , | V µ N | ) plane for an expected sensitivities at the LHC of BR ( B s → D − s π − µ + µ + ) < − and < − , respectively. Again, theblack, blue, and gray regions represent the constraints ob-tained for heavy neutrino lifetimes of τ N = , , [GeV] N m | N m | V -9 -8 -7 -6 -5 -4 -3 -2 -1 BelleNA3 CHARMIINuTeV DELPHI mmp K fi s B mmp s D fi s B FIG. 6. Exclusion regions on the ( m N , | V µ N | ) plane coming fromthe Belle [72], DELPHI [73], NA3 [74], CHARMII [75], andNuTeV [76] experiments. Limits provided by the searches on B s → ( K − , D − s ) π − µ + µ + are represented by the gray and black regions,respectively. See the text for details. respectively. For Majorana neutrino masses larger than 0.38GeV, the B s → D − s π − µ + µ + channel (CKM-allowed) wouldbe able to exclude a slightly wider region of | V µ N | than B − → π + µ − µ − . The reason for this is the non-suppressionfor the CKM elements involved.Additionally, in Fig. 6, we show the exclusion bounds onthe parameter space ( m N , | V µ N | ) coming from the Belle [72],DELPHI [73], NA3 [74], CHARMII [75], and NuTeV [76]experiments, in the mass range [0.5,5.0] GeV . In compar-ison, the constraints obtained from the searches on B s → ( K − , D − s ) π − µ + µ + are represented by the gray and black re-gions, for branching fractions of BR < − and BR < − ,respectively. In both cases, a lifetime of τ N = | ∆ L | = < − might beaccessible to the LHCb and CMS experiments (see Figs. 1and 2), therefore, these | ∆ L | = V. CONCLUDING REMARKS
We have studied the semileptonic | ∆ L | = B s meson B s → P − π − µ + µ + via the intermediate GeV-scale on-shell Majorana neutrino N , namely, B s → P − µ + N ( → π − µ + ) , with P = K , D s . To our knowledge, these LNV decaysof the B s meson have not been investigated before from atheoretical nor from an experimental point of view. We in-vestigated these same-sign µ + µ + channels and explored thesensitivity that can be reached at the LHCb and CMS ex-periments. We considered heavy neutrino lifetimes in theexperimental (LHCb and CMS) accessible ranges of τ N =[ , , ] ps, where the probability for the on-shell neu-trino N decay products to be inside the detector (accep-tance factor P N ) has been taken into account in our analy-sis. As an outcome, it was found that for integrated lumi-nosities collected of 10 and 50 fb − by the LHCb experi-ment and 30, 300, and 3000 fb − by the CMS experimentone would expect sensitivities on the branching fractions ofthe orders BR ( B s → K − π − µ + µ + ) (cid:46) O ( − − − ) andBR ( B s → D − s π − µ + µ + ) (cid:46) O ( − − − ) , as conservativevalues. For masses in the ranges m N ∈ [ . , . ] GeV and m N ∈ [ . , . ] GeV, respectively, we extracted bounds onthe parameter space ( m N , | V µ N | ) that might be obtained fromtheir experimental search. Depending on the τ N value, it wasfound that for m N > .
38 GeV these four-body channels maybe capable of excluding a slightly wider region of | V µ N | than B − → π + µ − µ − (LHCb).Consequently, the LHCb and CMS experiments have agreat chance to look for heavy Majorana neutrinos in the nearfuture, via | ∆ L | = B s meson. In addition, inthe best-case scenario, the experimental search of these LNVchannels would complement the bounds given by differentsearch strategies (such as NA3, CHARMII, NuTeV, Belle, andDELPHI). ACKNOWLEDGMENTS
The work of J. Mejía-Guisao has been financially supportedby Conacyt (México) under Projects No. 254409, No. 221329and No. 250607 (Ciencia Básica), and No. 2015-2-1187(Fronteras de la Ciencia). N. Quintero acknowledges supportfrom Dirección General de Investigaciones - Universidad San-tiago de Cali under Project No. 935-621717-016. J. D. Ruiz-Álvarez gratefully acknowledges the support of COLCIEN-CIAS (Colombia). For recent reviews on the theoretical and experimental status of different GeV-scale heavy neutrino search strategies see Refs. [10, 55–59] and ref-erences therein. [1] A. de Gouvêa and P. Vogel, Lepton flavor and number conserva-tion, and physics beyond the standard model, Prog. Part. Nucl.Phys. , 75 (2013) [arXiv:1303.4097 [hep-ph]] .[2] C. Patrignani et al. (Particle Data Group Collaboration), Chin.Phys. C , 100001 (2016) [http://pdg.lbl.gov] .[3] See, for instance: R. N. Mohapatra et al. , Theory of neu-trinos: A White paper, Rep. Prog. 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