Probing Sterile Neutrino via Lepton Flavor Violating Decays of Mesons
aa r X i v : . [ h e p - ph ] A p r Prepared for submission to JHEP
Probing Sterile Neutrino via Lepton Flavor ViolatingDecays of Mesons
Shiyong Hu, Sam Ming-Yin Wong, Fanrong Xu Department of Physics and Siyuan Laboratory, Jinan University,Guangzhou 510632, P.R. China
E-mail: [email protected]
Abstract:
A sterile neutrino at GeV mass scale is of particular interest in this work.Though not take part in neutrino oscillation, the sterile neutrino can induce flavor violatingsemileptonic and leptonic decay of
K, D and B mesons. We calculated a box diagramcontribution in these processes. By making use of current experiment limit of lepton flavorviolating decays M + h → M + l ℓ +1 ℓ − and M → ℓ − ℓ +2 , we explore the allowed parameter spaceof U e and U µ in different mass ranges. Generally speaking, both channels give a limit tothe product of U e and U µ . When sterile neutrino mass is located in between pion andkaon mass, K + → π + e ± µ ∓ gives the strongest constraint while B + → π + e ± µ ∓ providesthe dominated constraint when its mass in between of kaon and B meson. If sterile neutrinois even heavier than B mesons, the B s → µ ± e ∓ experiment which is performed at LHCbgives the strongest constraint. Keywords:
Sterile neutrino, Lepton flavor violation, semileptonic decay, leptonic decay Corresponding author. ontents A ( x, y ) 73.3 The combining analysis 7 Now it has been well established that at least two active neutrinos are massive with tinymasses. The origin of neutrino mass is still an open question. Sorts of ideas have beenproposed to solve this fundamental question, including seesaw mechanism [1] and radiativecorrection mechanism (for example,[2], [3] and for a recent review see [4]). General speak-ing, new particles out of SM particle spectrum will appear associated with neutrino massmodels. As a hypothetic particle, though does not participate weak interaction, sterileneutrino is unavoidable in some neutrino mass models beyond SM. For example, in TypeI seesaw mechanism the heavy right-handed neutrino singlet contributing the tiny massof left-handed neutrino is absent from SU(2) interaction and hence appears as a sterileneutrino.The prediction of sterile neutrino mass in theory is model dependent thus is not unique.In the view of experiment, there are some hints to indicate the existence of sterile neutrinoas well as its mass. One type of experiment is neutrino oscillation. In 2001 the LSNDexperiment searched ¯ ν µ → ¯ ν e oscillations, suggesting that neutrino oscillations occur in the0 . < ∆ m <
10 eV range [5]. Later the MiniBooNE experiment indicated a two-neutrinooscillation, ¯ ν µ → ¯ ν e , occurred in the 0 . < ∆ m < . range [6]. An updated globalfit [7], taking into account recent progress, gives ∆ m ≈ . (best-fit), 1 . (at 2 σ ),2 . (at 3 σ ). Hence if its mass is located at eV, sterile neutrino effect can be unfoldedby oscillation experiments. On the other hand, sterile neutrino mass can be even heavier.The operation of LHC provides an opportunity to search TeV scale heavy sterile neutrino[8], [9]. The IceCube Neutrino Observatory, which locates in Antarctic, gives an uniquevision to observe PeV neutrino [11]. In between eV and TeV-PeV, the GeV sterile neutrino– 1 –ould appear in weak decays of bound states of heavy quarks. Thus the sterile neutrino,with its undermined mass varied from eV to TeV-PeV, provides a port to connect NewPhysics beyond SM.In recent years the semileptonic decays B → K ∗ ℓ + ℓ − and B → πℓ + ℓ − have beenstudied extensively both theoretically and experimentally. Though the expectation of newphysics in forward-backward asymmetry of lepton pairs in B → K ∗ ℓ + ℓ − has alreadyfaded away, NP hope still holds at the so-called observable P ′ [12, 13]. Similar situationhappened in the leptonic decays of B and B s . The SM-like B ( B s → µ + µ − ) and perhapsnew physics allowed B ( B d → µ + µ − ) give strong constraints to theories beyond SM [14, 15],but the windows to NP is not closed. It is known that both types of decays are FCNCprocess, giving a chance to put NP particles in the loop. Then it is nature to considerthe possibility of a sterile neutrino in the loop. In fact there have been continuous effortsto study GeV scale sterile neutrino indirectly via some certain semileptonic and leptonicdecays of B, D and K mesons. In the semileptonic decay processes, if its mass is in betweenthe meson masses of initial and final states, the sterile neutrino can be on-shell produced[18]. A popular consideration is to take sterile neutrino as the Majorana neutrino, thuslepton number violating decays are induced[16],[17], [19],[20],[21],[23]. The idea to makeuse of leptonic decay with neutrino final state, in which sterile neutrino is involved attree level, is also proposed[24], [25]. In above works the final state leptons, though withlepton number violation, are mostly with same flavors thus only single PMNS matrix isrelevant. The lepton flavor violating decays from mesons, on the other hand, is related totwo PMNS matrix elements, thus could give complementary information to correspondinglepton number violating decays.The early quest for lepton flavor violating processes can be traced back to the leptoniccecay K L → e ± µ ∓ in 1998 [29]. So far K L → e ± µ ∓ still gives a very strong constraint toNP models. The latest experiment for leptonic decay is carried out in LHCb by searching B → e ± µ ∓ giving the upper limit 1 . × − [37]. As for the semileptonic decays of K, D and B mesons, most of them are still results from BaBar [31] and it is hoped that LHCbcan bring new limit in the near future. A detailed summary for related experiments isgiven in Table 3. In this paper, we will analysis both leptonic and semileptonic decaysfrom K, D and B mesons induced by sterile neutrino. By combing all the currently relatedexperiments, we will give the constraints to relevant PMNS matrix elements.This paper is organized as follows. In section 2.1, we will give a brief introduction to themodel related to heavy sterile neutrino. In section 2.2 we will discuss a set of semileptonicdecay processes and derive the exact formulas with heavy sterile neutrino contribution. Asystematic formalism for leptonic decays of K, D and B mesons are given in section 2.3.In section 3, we will perform a numerical study and give the allowed coherent parameterspace. Discussion and conclusion will be made in section 4. In this section, a brief introduction of heavy sterile neutrino is given firstly. Then we derivethe required analytical formulas in semileptonic and leptonic decays separately.– 2 – .1 Model setting
As introduced in section 1, here we are only interested in the GeV scale sterile neutrino.Due to its heavy mass, in flavor space the sterile neutrino will decouple from other threeactive neutrinos in the oscillation processes. With the appearance of a sterile neutrino andwithout involving the details of a concrete model, the mass mixing can always written viaa non-unitary mixing matrix, ν e ν µ ν τ = U e U e U e U e U µ U µ U µ U µ U τ U τ U τ U τ ν ν ν ν , (2.1)which characterizes the rotation between mass eigenstate and flavor eigenstate in vacuum.A direct consequent for the non-unitary mixing is zero-distance effect [26], the oscillationcould happen even without propagate few distance. Such effect has been pointed out to bedetected in oscillation experiment by a near detector, which will be discussed in a separatework. In the following context, we will focus on the mass of ν and the mixing elements U e ,µ ,τ . And hereafter we adopt the notation N to denote sterile neutrino for the purposeof emphasize. The sterile neutrino, if its mass is in between with the initial heavy meson and final massmeson, can be produced on-shell and then decay shortly. As for the heavy meson, we areespecially interested in those charged ones. The reason for such a choice is due to the factthat tree-level annihilation diagram not only gives dominated contribution to the decay ofheavy meson, but also provides a chance to produce sterile neutrino from W boson sourcedfrom quark annihilation, see Fig. 1. M + h M + l ℓ +1 ℓ − N Figure 1 . The semileptonic decay of charged heavy mesons, in which M + h ( M + l ) means a chargedheavy (light) meson. The branching fraction for the three-body decay can be simplified to the multiplicationof two-body decays. In general we can write down B ( M + h → M + l ℓ +1 ℓ − ) = B ( M + h → ℓ +1 N ) B ( N → M + l ℓ − ) (2.2)in which M h ( M l ) denotes heavy (light) meson, and ℓ , ℓ represent charged lepton ( e, µ, τ )with different flavors. A straightforward calculation gives the decay of heavy meson B ( M + h → ℓ +1 N ) = 18 π G F f M h m M h m N τ M h | U ℓ | X ( M h ) . (2.3)– 3 –hich relies on two unknown parameters, the PMNS matrix element U ℓ and the mass of N . Especially the heavy meson dependent function X ( M h ) introduced in Eq. (2.3) reflectsthe features of M h , X ( M h ) = | ξ | λ (1 , x N , x ℓ )[(1 + x ℓ − x N ) + y ℓ (1 + x N − x ℓ )] . (2.4)in which ξ is a particular CKM matrix element corresponding to the mother particle, andthe definition of the auxiliary function is given as λ ( x, y, z ) = x + y + z − xy + xz + yz ).The two parameters correlated to initiated and final state particle are defined as x i ≡ m i m Mh , y i ≡ m i m N .For the further decay of sterile neutrino, one can calculate its decay width,Γ( N → M l ℓ − ) = 116 π G F m N | U ℓ | Y ( M l ) . (2.5)Similar to the decay of M h , besides the PMNS matrix U ℓ , the width depends on the finalstate dependent function Y , given Y ( M l , ℓ ) = | ξ | f M l λ (1 , y ℓ , y M l )[(1 + y ℓ − y M l )(1 + y ℓ ) − y ℓ ] , (2.6)with another CKM matrix element ξ which is determined by M l . The lepton final statefrom W is assumed negligible, accordingly the branching fraction of N decay is B ( N → M l ℓ − ) = | U ℓ N | Y ( M l , m ℓ ) P ℓ ; q | V ℓN | | V uq | Y ( M q , m ℓ ) (2.7)In the denominator, the summation is performed only to the first two generations for bothlepton and quark sector. As for the function Y , the value for its first parameter M q shouldbe chosen as M q = π + ( K + ). Table 1 . The detailed parameters for semileptonic decay M + h → M + l ℓ +1 ℓ − , in which M h ( M l ) meansheavy (light) meson and ℓ , = e, µ, τ . M + h ξ f M h m M h τ M h x i M + l ξ f M l B + V ub f B m B + τ B + m i m B + K + V us f K π + V ud f π D + V cd f D m D + τ D + m i m D + K + V us f K π + V ud f π K + V us f K m K + τ K + m i m K + π + V ud f π One should keep in mind that Eq. (2.2) gives a general description of this type process,which actually contains many modes when different initial and final states are chosen. InTable 1 we have summarized explicitly corresponding parameters for such modes.– 4 – .3 Leptonic decay
The leptonic decay with different final state flavors in SM is induced by active neutrinoin box diagram, however, its effect is very tiny thus can be negligible. On the other handthe smallness in SM might be made use of searching new physics. If the processes canbe observed in experiment, it will be definitely a signal for desired new physics. As anillustration, we will consider such a process induced by heavy sterile neutrino. ℓ − ℓ +1 M Figure 2 . Flavor violating leptonic decay of neutral mesons.
To calculate the amplitude of usual final states with same flavor int theory, we willinclude both penguin diagram and box diagram contribution. In this work we only focuson the final state leptons with different flavor, thus only box diagram contributes since thevector boson in penguin diagram does not change flavor, see Fig. 3. The initial neutralmeson M , could be B , D and K L and the intermediate particle N is off-shell. Genericallywe have the branching fraction for such decays B ( M → ℓ +1 ℓ − ) = G F α π sin θ W A ( z h , z N ) | U ℓ N | | U ℓ N | Z ( M , ℓ , ℓ ) (2.8)which relies on PMNS matrix U ℓ N , U ℓ N and a function correlated with initial and finalstates Z ( M , ℓ , ℓ ) ≡ | ξ | τ ( M ) m M f M λ (1 , x ℓ , x ℓ ) [ x ℓ (1 + x ℓ − x ℓ ) + x ℓ (1 + x ℓ − x ℓ )](2.9)with the product of relevant CKM elements ξ . The other relevant parameter x i is definedas z i ≡ m i m W . All the explicit parameters involved in different decay modes are summarizedin Table 2. Table 2 . The detailed parameters for semileptonic decay M → ℓ +1 ℓ − . M ξ ( M ) τ ( M ) f M m M z h B s V ∗ tb V ts τ ( B s ) f B s m B s z t B V ∗ tb V td τ ( B ) f B m B z t D V ∗ cb V ub τ ( D ) f D m D z b K L V ∗ ts V td τ ( K L ) f K m K L z t – 5 –he loop function A ( x, y ) in Eq.(2.8), which is obtained by calculating the box diagramin Fig. 3, A ( x, y ) = 14 (cid:20) x − y (1 − x )(1 − y ) + (1 − y ) x ln x − (1 − x ) y ln y (1 − x ) (1 − y ) ( x − y ) (cid:21) , (2.10)reveals the inner information of M . It can be checked that function A ( x, y ) is an extensionof standard loop function B ( x ) = h x − x + x ln x ( x − i , and can return to B case when thesecond parameter vanishes. A more qualitative analysis for A ( x, y ) will be given in nextsection. There have been about 20 years history for the search for flavor violating decays. Wesummarize all the relevant experiments in Table 3 as the input of our numerical study.
Table 3 . The status of flavor violating semileptonic or leptonic decays related to
K, D and B . Theabbreviations are as follows: LED means light eigenstate dominate and HED is heavy eigenstatedominate. channel 90% CL limits collaboration K + → π + µ + e − . × − [27] K + → π + µ − e + . × − [28] K L → e ± µ ∓ . × − BNL [29] D + → π + µ + e − . × − BaBar [30] D + → π + µ − e + . × − BaBar [30] D + → K + µ + e − . × − BaBar [30] D + → K + µ − e + . × − BaBar [30] D → eµ . × − LHCb [36] B + → π + e ± µ ∓ . × − BaBar [31] B + → π + e ± τ ∓ . × − BaBar [32] B + → π + µ ± τ ∓ . × − BaBar [32] B + → K + e ± µ ∓ . × − BaBar [33] B + → K + e ± τ ∓ . × − BaBar [32] B + → K + µ ± τ ∓ . × − BaBar [32] B + → K ∗ + e ± µ ∓ . × − BaBar [33] B → e ± µ ∓ . × − LHCb [34] B → e ± τ ∓ . × − BaBar [35] B → µ ± τ ∓ . × − BaBar [35] B → e ± µ ∓ . × − LHCb [37] B s → e ± µ ∓ . × − (LED) LHCb [37] B s → e ± µ ∓ . × − (HED) LHCb [37]– 6 – .2 Property of A ( x, y )The branching fractions of leptonic decays largely relies on the A ( x, y ), hence before nu-merical studies of phenomenology it is necessary to explore the features of this function. InFig.3 we plot the dependence its behaviors respect to sterile neutrino mass. Typic featuresof A ( x, y ) are shown below. • A singularity appears at m N = m W , and more close to W mass, more enhanced thefunction value is. • In SM such a diagram actually also gives contribution, given A ∼ − . • There is a particular choice that A = 0. Take B decay as an example, the internalheavy quark loop comes from top. And the zero point is located at top quark massregion. However, in SM such an effect is not appear as this Feynman diagram doesnot appear individually. • When sterile neutrino mass is larger than electroweak scale, the behavior is asymp-totic stable, giving a value smaller than SM. Since we are only interested in GeVscale sterile neutrino, such a range will not be involved in this paper.
Figure 3 . The behavior of function A , in which the black dot stands for SM situation. Now with the prepared necessary analytical formulas in above, we will present our numericalstudy in the following section.The semileptonic decay happens, in our working frame, is due to the on-shell produc-tion of sterile neutrino, which actually requires the sterile neutrino mass in between ofthe initial heavy meson and the final light meson. However, the effect of off-shell sterileneutrino can play a role in leptonic decays. Thus whatever mass of sterile neutrino is,the contribution from leptonic decays cannot be negligeble. In other words, if the mass is– 7 –ot located in between initial and final mesons, only the leptonic decay experiments giveconstraints to corresponding mixing matrix, we call this scenario D. In addition to scenarioD, in the mass region m π < m N < m B , we classify the mass range into three other differentscenarios, named as Scenario A, B and C. • Scenario A: m π < m N < m K If sterile neutrino mass located at this region, the semileptonic decays induced by theon-shell sterile neutrino contains B + → π + µ ± e ∓ , B + → π + τ ± µ ∓ , B + → π + τ ± e ∓ , D + → π + µ ± e ∓ and K + → π + µ ± e ∓ . In principle, all the leptonic decays from K L , D and B , B s , including K L → µ ± e ∓ , D → µ ± e ∓ and B s ) → µ ± e ∓ , should also be takeninto account. However, from the numerical analysis all the parameter spaces are fullyallowed by these leptonic decays, which is too weak to give an efficient constraint.Thus only these semileptonic ones provide some effective information.Taking µ, e final states as an example, we compare the decays from three differentparent particle and find K + decay provides the most stringent constraint shown inTable 4, while B + and D + decays give a much wide allowed region. It is easily tosee the product of U eN and U µN is strictly constrained to O (10 − ), but a furtherrestriction to U αN requires other input experiment, which will be discussed in aseparate work. Figure 4 . The allowed parameter space in scenario A, in which we have taken m N = 0 . Similar analysis can be performed for µ, τ or τ, e final states. The output of thecorrelated constraint for relevant PMNS matrix elements is somehow too weak, thuswe will not show them explicitly. • Scenario B: m K < m N < m D As pointed in above context, leptonic decays always appear. For the semileptonicdecays in this case, only B + and D + decays while K + is forbidden otherwise the– 8 –other particle is lighter than its daughter particle. The explicit modes whichare incorporated into our numerical simulations are: B + → K + ( π + ) µ ± e ∓ , B + → K + ( π + ) τ ± µ ∓ , B + → K + ( π + ) τ ± e ∓ and D + → K + ( π + ) µ ± e ∓ .As the first step, let’s focus on e, µ final states. First by comparing various B + decaymodes with different final states, one can find the constraint to PMNS matrix from B + → π + e ± µ ∓ dominates the corresponding ones from B + → K + e ± µ ∓ , as shownin Table 5. Second for the allowed region extracted from D + decays, D + → π + e + µ − is much stronger than D + → K + e ± µ ∓ . Looking at the same π + final states, thenumerical analysis tells that B + decay gives the strongest restriction,which actuallygives an upper limit for the product of U eN and U µN , O (10 − ). As for the individualmatrix elements, one has to resort to other way. Figure 5 . The allowed parameter space in scenario B, in which we have taken m N = 1 . It is noted that so far no more stringent constraint can be obtained from leptonicdecay. And the constraint of the correlation of PMNS matrix V τN and V eN , V µN isstill too weak from τ, µ or τ, e final states, which is also neglected here. • Scenario C: m D < m N < m B In addition to leptonic decays, only semileptonic decays from B + can appear in thissituation, which actually gives more stringent constraints.We still stick to the µ, e final states with the same reason as previous scenarios.Though sterile neutrino mass is taken 4 GeV, the numerical simulation leads to thesame conclusion as scenario B. Thus we will not show its corresponding plot here. • Scenario D: m N < m π or m N > m B In this scenario, semileptonic decays are forbidden and only leptonic decays happen.If the sterile neutrino mass is lighter than the lightest meson π , the mass dependentfunction A is close to the SM situation giving a small amplitude (module to PMNSmatrix element), then the further constraint to PMNS matrix from experiment mea-surement is weak. Such behavior has been checked and we will not show in graphshere. – 9 – igure 6 . The allowed parameter space in scenario D, in which we have taken m N = 70 GeV asan illustration. It is interesting to explore the mass range larger than B mesons. We take Fig. 6 asan illustration, with sterile neutrino mass m N = 70 GeV. Among all the 4 chargedneutral meson decays, the parameter space from D and B decays are still fully filledthus not marked in the figure. The experiment upper limit for K L and B s indeedtouch the restriction to U eN − U µN parameter space. As shown in Fig. 6, the recentLHCb experiment B s → e ± µ ∓ now catch up with the classical BNL experiment on K L → e ± µ ∓ .In a summary, the allowed parameter of PMNS matrix is sterile neutrino mass depen-dent. When the mass is lighter than m π , parameter space does not receive a constraint fromcurrent meson decay experiments. If the mass is located between m π and m K , the semilep-tonic decay K + → π + e + µ − provides the most stringent constraint, V µN V eN ∼ O (10 − ).When its mass is in between kaon mass and B meson, BaBar experiment B + → π + e ± µ ∓ in fact dominate the constraint, giving V µN V eN ∼ O (10 − ). If sterile neutrino is heavierthan B meson, leptonic decay B s → e ± µ ∓ provides the strongest constraint. In this work, we consider various semileptonic and leptonic decays of neutral mesons in-duced by a heavy sterile neutrino, which can in turn constrain parameter space of theunknown PMNS matrix elements. Especially we calculated the loop function of a boxdiagram contributing to leptonic decays. Making use of the two types of decays from dif-ferent parent particle, we find the allowed range of parameter space of sterile neutrino ismass dependent. If sterile neutrino is lighter than pion mass, these meson decays havenull restriction. When sterile neutrino mass is located in between pion and kaon mass, K + → π + e ± µ ∓ gives the strongest constraint while B + → π + e ± µ ∓ provides the domi-nated constraint when sterile neutrino mass in between of kaon and B meson. If sterile– 10 –eutrino is even heavier than B mesons, the measurement performed at LHCb B s → µ ± e ∓ gives the strongest constraint. It should be noted that so far we can only extract therestriction information to parameter space from the decays with e, µ final states while thedecays with a τ in final state is not incorporated. From the analysis, we provide a globalconstraint for | V µN V eN | in different m N mass region, however, the magnitude of an indi-vidual PMNS matrix element cannot determined in this work and will be discussed in aseparate work. Acknowledgments
F. Xu was supported partially by NSFC under Grant No. 11605076, as well as the Funda-mental Research Funds for the Central Universities in China under the Grant No. 21616309.
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