Light Sterile Neutrinos, Lepton Number Violating Interactions and the LSND Anomaly
aa r X i v : . [ h e p - ph ] J u l Light Sterile Neutrinos, Lepton Number ViolatingInteractions and the LSND Neutrino Anomaly
K. S. Babu
Department of Physics, Oklahoma State University, Stillwater, OK 70478
Douglas W. McKay
Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045
Irina Mocioiu
Department of Physics, The Pennsylvania State University, University Park, PA 16802
Sandip Pakvasa
Department of Physics and Astronomy, University of Hawaii, Honolulu, HI 96822
July 5, 2016
Abstract
We develop the consequences of introducing a purely leptonic, lepton number violating non-standard interaction (NSI) and standard model neutrino mixing with a fourth, sterile neutrinoin the analysis of short-baseline, neutrino experiments. We focus on the muon decay at rest(DAR) results from the Liquid Scintillation Neutrino Experiment (LSND) and the Karlsruheand Rutherford Medium Energy Neutrino Experiment (KARMEN). We make a comprehensiveanalysis of lepton number violating, NSI effective operators and find nine that affect muondecay relevant to LSND results. Two of these preserve the standard model (SM) value 3/4for the Michel ρ and δ parameters and, overall, show favorable agreement with precision dataand the ¯ ν e signal from LSND data. We display theoretical models that lead to these twoeffective operators. In the model we choose to apply to DAR data, both ν e appearance from ν µ oscillation and ν e survival after production from NSI decay of the µ + contribute to theexpected signal. This is a unique feature of our scheme. We find a range of parameters whereboth experiments can be accommodated consistently with recent global, sterile neutrino fits toshort baseline data. We comment on implications of the models for new physics searches atcolliders and comment on further implications of the lepton number violating interactions plussterile neutrino-standard neutrino mixing. Introduction
The early positive results reported by the 30 m baseline LSND ν e appearance search [1] were hardto reconcile with then-current results from ”long baseline” solar neutrino and atmospheric neutrinolimits on mixing and oscillation parameters [2]. This naturally led to conjectures about new physicsthat incorporates both short and long baseline data, invoking, for example, new interactions [3, 4],sterile neutrinos [5], [6], extra dimensions [7] and quantum decoherence [8].Among the approaches, two that have gained considerable interest are new interactions and newoscillations induced by one or more additional, super-weakly interacting, or sterile, neutrinos, withrecent global fits to short-baseline appearance and disappearance data provided in [9], [10], [11],[12] and to disappearance data alone [13]. There continues to be a great deal of interest in thepossibility of sterile neutrinos [14], [15], [16], including applications in astrophysics and cosmology[17], with some indications that, under certain assumptions, cosmology constraints may cause aproblem [18], [19]. Theoretical consistency requirements plus tight limits on µ and τ branchingfractions made lepton flavor-violating NSI models unworkable [20]. . However, as pointed out in[20], ∆L = 0 interactions can evade the charged lepton flavor constraints. Even so, high precisionmeasurements of the Michel parameters in µ decay still provide strong constraints on the form ofany proposed ∆L = 0 effective Lagrangian [4]. In this paper, we undertake a comprehensive analysis of lepton number violating muon decay,and find that among possible effective operators there are two that retain the SM prediction of ρ = δ = , We develop theoretical models that lead to these two self-consistent effective operators.We need the oscillations provided by a sterile neutrino in addition, since a model with only µ + → e + + ν µ + ν e and no oscillations as the DAR explanation of the LSND signal would directly conflictwith the absence of a signal in KARMEN data. With oscillations, the difference in baselines, 30 mfor LSND and 17 m for KARMEN, allows some leeway in fitting both signals. Moreover, MiniBoone[24] would have seen no indication of an appearance signal, since their source is the semileptonicdecay π − → µ − + ν µ , which requires a subsequent oscillation to ν e to produce an appearance signal. Only the muon DAR experiments are sensitive to the ∆L = 0 new interactions we consider here.Nonetheless, we find that our fitted oscillation parameters are compatible with those of global fitsthat include the MiniBoone and other data which rely on neutrinos from semileptonic decays.With these considerations and LSND [25] and KARMEN [26] in mind as the two experimentswith muon DAR as their ¯ ν source, we classify a set of nine possible models with ∆L = 0, retainingthose that are consistent with current experimental constraints on muon decay parameters andlepton decay branching ratios. We then add an additional, sterile neutrino that mixes with thethree standard model neutrinos. The model we choose to study in detail has the striking featurethat both antineutrinos in the flux from the source can produce the ”appearance” signal: the ν e sthat come from oscillations of both standard model (SM) and NSI decays of the µ + and the ν e s thatdon’t disappear after being directly produced in the NSI decay mode of the µ + . We develop themodels, summarize their properties and pursue the consequences of the most promising examplefor LSND and KARMEN in the following four sections. We outline further research directions, A model-independent lepton-flavor violating NSI setup with one or more sterile neutrinos can give a good globaldescription of short baseline data [21], but it considers neither lepton number violation nor consistency with chargedlepton flavor constraints. In Ref.[4], two specific models were presented (denoted ( B ) and ( B ) in our Table 2 in Sec.(2.2)). Both ofthese models have the feature that the anomalous muon decay that explains the LSND signal would also lead toshifts in all of the Michel parameters in µ decay by an amount characterized by the LSND signal strength. Resultsfrom the TWIST collaboration [22, 23] have since excluded these options. µ -decay models The effective Lagrangian we want must produce a muon decay relevant to LSND that results in afinal state with a net lepton number different from that of the decaying muon, is consistent withthe SU(2) L × U(1) symmetry of the SM, and satisfies the experimental constraints on the Micheldecay parameters of the muon [28], which have been measured to high precision [22, 23, 29]. TheSM shows overall agreement with the measured values.At the level of the muon decay process, we consider models with ν eL , µ L or µ R , e L or e R , andany flavor of active neutrino with ν aL or it’s conjugate and models with a singlet, two-componentsterile neutrino, ν R . We define three categories determined by the choice of neutrino: A : ν = ν aL , active left − handed neutrino ,B : ν = ( ν Ca ) R , active antineutrino ,C : ν = ν R , singlet right − handed sterile neutrino . Within each category, one can choose right or left helicity projections or charge conjugation pro-jections of µ , e or ν , and within these projections there are generally several ways to choose theSU(2) L multiplet contraction of indices to form a scalar and consequent effective µ -decay 4-Fermionoperator. To illustrate category A, we review the lepton-flavor violating but lepton-number conserving casesconsidered by Bergmann and Grossman [20]. Effective 4-Fermion operators responsible for theflavor-violating µ -decay can be written in category A, with choice of e and µ as left-handed SU(2) L doublet members, in Lorentz vector and scalar forms, or similarly with e and µ as right-handedSU(2) L singlets. In the doublet forms, we have( L A. ) = (¯ µ L γ λ e L )(¯ ν aL γ λ ν eL ) + H.c. (1)( L A. ) = ( ν TeL Ce L )( µ TL Cν aL ) ∗ + H.c., (2)where index a can be e , µ , or τ and C is the charge conjugation operator. Both forms lead tothe decay µ + → ¯ ν e + e + + ν a , with lepton number preserved but flavor violated, directly pro-viding a source of ¯ ν e s to produce an e + signal by inverse β -decay in the detector. SU(2) L × U(1)invariant effective Lagrangians that contain Eq.(1) and Eq.(2) follow by replacing each field by itscorresponding SU(2) L doublet, ψ Ti = ( ν i , ℓ i ):( L A. ) → ( ¯ ψ µL γ λ ψ eL )( ¯ ψ aL γ λ ψ eL ) + H.c. (3)( L A. ) → ( ψ TeL Cψ eL )( ψ TµL Cψ aL ) ∗ + H.c., (4)and then contracting the doublet indices to make an overall SU(2) L × U(1) singlet L eff . Re-gardless of the choice of SU(2) L × U(1) construction, for the V-A form one ends up with a term(¯ µ L γ λ e L )(¯ ℓ aL γ λ e L ) + ..., which gives a contribution to µ + → e + + e + + e − when a = e , a con-tribution to e − + µ + → e + + µ − when a = µ and a contribution to τ + → e + + e + + µ − , when3 = τ . All of these processes are so tightly constrained by experiment [29], that the bound on thenew physics coefficient rules out the chance that it could explain the LSND anomaly. The samesituation results from the S form, where the term ( e TL Ce L )( µ TL Cℓ aL ) ∗ + ...contributes to the sameprocesses as the V-A form for the corresponding flavor choice for the index a . Under the A-category1, one can replace the left-handed doublet members e L and µ L by right-handed singlet fields andperform the same analysis as above and find the same severe constraints. SU(2) L × U(1) Y breakingeffects, introduced through the vacuum expectation value of the Higgs field in four-fermion oper-ators with higher dimensions, cannot be significant enough to change this conclusion. The leptonnumber conserving but flavor violating new physics contributions to muon decay cannot account forthe LSND anomaly [20].
In category B, the outgoing antineutrinos and positron in µ + decay carry lepton number -3 andchange the lepton number by 2 units. A purely charged leptonic decay of µ or τ consistent withSU(2) L × U(1) cannot be constructed without violating electric charge conservation, so the limitson ”conventional” rare decays discussed above don’t apply. The two lepton helicity combinationsin this category are µ R and e L (B.1) and µ L and e R (B.2). In these first cases, the four-Fermioneffective interactions we want for the ∆L=2 µ -decay modes µ + → e + + ¯ ν e + ¯ ν a are L B. = (¯ µ R e L )(( ν Ca ) R ν eL ) and (¯ µ R ν eL )(( ν Ca ) R e L ) , while (5) L B. = (¯ µ L γ λ ν eL )(( ν Ca ) R γ λ e R ) and (¯ µ L e R )(( ν Ca ) R ν eL ) , (6)where the Hermitian conjugate terms will be understood from here on. With a scalar Higgs doubletof the standard model, H T = ( H + , H ), we can write an SU(2) L × U(1) singlet effective Lagrangianfor the first case in three ways, namely( L B. ) , , = (¯ µ R ψ ieL )( ψ T jaL Cψ keL ) H l { (1) ε ij ε kl , (2) ε ik ε jl and (3) ε il ε jk } , (7)where i, j, k, and l are indices of the components of the SU(2) L doublets and the epsilon symbolsare the constant SU(2) L antisymmetric two-index tensors.To display the effective 4-Fermion operators relevant to µ -decay, we break out the terms pro-portional to the vacuum expectation value of the neutral component of the Higgs field:( L B. ) → [(¯ µ R ν eL )( ℓ TaL Cν eL ) − (¯ µ R e L )( ν TaL Cν eL )] h| H |i , (8)( L B. ) → [(¯ µ R ν eL )( ν TaL Ce L ) − (¯ µ R e L )( ν TaL Cν eL )] h| H |i and (9)( L B. ) → [(¯ µ R ν eL )( ν TaL Ce L ) − (¯ µ R ν eL )( ℓ TaL Cν eL )] h| H |i . (10)The second terms in ( L B. ) and ( L B. ) are the same, while ( L B. ) and ( L B. ) share their firstterms. Both terms of each case must be included to ensure SU(2) L × U(1) invariance, of course. Theother member of this ∆L=2 category follows from exchanging the helicity labels on µ and e. Thebasic 4-Fermion SU(2) L × U(1) invariant forms are then( L B. ) = ( ¯ ψ iµL γ λ ψ eLj )( ψ TaLk Cγ λ e R ) H i H l H m ǫ jl ǫ km → [(¯ µ L γ λ ν eL )( ν TaL Cγ λ e R )] h| H |i , (11)( L B. ) = ( ¯ ψ iµL e R )( ψ TaLj Cψ eLk ) H i H l H m ǫ jl ǫ km → [(¯ µ L e R )( ν TaL Cν eL )] h| H |i , (12)where the 4-Fermion piece of each invariant form that is proportional to h| H |i is indicated by thearrows. 4ategory C introduces a new, right-handed, SU(2) L singlet neutrino that does not mix with thestandard model neutrinos and interacts only through its NSI . It contributes a super-weak termto the effective Lagrangian that produces a direct µ -decay mode µ + → e + + ¯ ν e + ν R . The µ -decayforms read L C. = (¯ µ L e R )(¯ ν R ν eL ) and (¯ µ L γ λ ν eL )(¯ ν R γ λ e R ) , while (13) L C. = (¯ µ R ν eL )(¯ ν R e L ) , (¯ µ R e L )(¯ ν R ν eL ) and ( e TL Cν eL )( µ TR Cν R ) ∗ . (14)The SU(2) L × U(1) forms from which these effective 4-Fermion Lagrangians arise through couplingto the Higgs doublet are( L C. ) = ( ¯ ψ iµL e R )(¯ ν R ψ eLj ) H i H k ε jk → (¯ µ L e R )(¯ ν R ν eL ) h| H |i , (15)for the first term of Eq. (13), and( L C. ) = ( ¯ ψ iµL γ λ ψ eLj )(¯ ν R γ λ e R ) H i H k ε jk → (¯ µ L γ λ ν eL )(¯ ν R γ λ e R ) h| H |i , (16)for the second. Again the arrows indicate the result of extracting the effective 4-Fermion piecesof the invariant interactions. The second version of this singlet, right-handed neutrino categoryexchanges the R and L labels on µ L and e R , and the SU(2) L × U(1) invariant forms are( L C. ) = (¯ µ R ψ ieL )(¯ ν R ψ jeL ) ε ij → (¯ µ R ν eL )(¯ ν R e L ) − (¯ µ R e L )(¯ ν R ν eL ) (17)which covers the first two terms in Eq.(14), and( L C. ) = ( ψ T ieL Cψ jeL )( µ TR Cν R ) ∗ H k H ∗ i ε jk → ( e TL Cν eL )( µ TR Cν R ) ∗ h| H |i , (18)which covers the third.This completes our summary of the effective Lagrangian models with SU(2) L × U(1) symmetrywith relevance to µ -decay. One can cast the operators in different forms by making Fierz transfor-mations on the operators we have presented, but the physics content remains the same. We devotethe remainder of this section to testing each of the lepton number violating cases for consistencywith precision Michel parameter data [22, 23, 29]. µ -decay Michel-parameters As presented in the preceding discussion, there are nine distinct variants of the ∆L = 0 NSIscheme we are studying. If no explicit breaking of the SU(2) L × U(1) symmetry at the effectiveLagrangian level is included, only the generic parameter ǫ that governs the strength of the NSIcontribution to the LSND signal needs to be considered. We assume this to be the case for thepresent discussion. The definition of ǫ in the case (L B. ) that we study in Sec. 3, for example, isgiven by L eff = G F √ (cid:2) (¯ e L γ λ ν eL )(¯ ν µL γ λ µ L ) + 2 ǫ (¯ µ R ν eL )( ν TµL Ce L ) (cid:3) . The factor 2 in the definitionof ǫ compensates for the factor 1/4 in the decay rate with the S,P structures compared to theV-A structure. When ǫ =1, the full contribution to the rate results as required for consistentnormalization. The results for the general cases are illustrated in the Appendix. In Sec. 3.2 we introduce a left-handed, sterile neutrino that mixes with the three neutrinos of the standard modeland implements new oscillation effects driven by its O(1 eV ) mass. e + in the µ -decay final state compared tothat predicted by the SM. Because the linear terms in ǫ are proportional to the neutrino masses andare completely negligible, the deviations of Michel parameters from their SM values are at order | ǫ | . Directly computing the complete spin-averaged decay distribution in the final state electronenergy for each model would give the answer for each case, or adapting the exhaustive studies of µ decay parameterizations in the literature to the particular cases of interest here can yield theresults we want [27, 28]. We rely primarily on the latter approach.The χ that tests the simultaneous fit of the Michel parameters and the LSND ¯ ν e rate for eachmodel as a function of ǫ can be written χ ( ǫ ) = ( ρ twist − ρ ( ǫ ) σ ( ρ ) twist ) + ( δ twist − δ ( ǫ ) σ ( δ ) twist ) + ( ξ twist − ξ ( ǫ ) σ ( ξ ) twist ) + ( P lsnd − | ǫ | σ ( P ) lsnd ) , (19)where ρ twist is the numerical value of the TWIST global fit for ρ and σ ( ρ ) is the 90% C.L. uncertaintyin the fit value and similarly for the δ and ξ values and their uncertainties and for those of the LSNDoscillation probability. The expressions for ρ ( ǫ ), δ ( ǫ ) and ξ ( ǫ ), the TWIST values from their globalfit given in their Table VII [23], and the best fit and its corresponding χ per degree of freedom,are listed below the respective models at the head of each column in Table 1. The standard modelvalues for ρ , δ and ξ are 3/4, 3/4 and 1, as shown in the column headed ”SM”. The χ per degreeof freedom for the SM refers only to the comparison to the Michel parameter values in the TWISTglobal fit. It is not the ǫ =0 fit to the Michel plus LSND data. If one adds the last (LSND) term tothe SM evaluation of Eq. (19) and divides by 4, the SM χ per degree of freedom is 3.34, a full unithigher than our best fit candidate models ( B ) and ( C ) . Surprisingly, the average contributionof the Michel parameters and that of the LSND signal to the SM χ are comparable.To illustrate the application of the set-up of Sec. II.C.1 in Kuno and Okada [28] to our problem,we derive the entry for the model ( L B ) ; for example ρ SM = → (1 − | ǫ | ) in Table 1. ThoughKuno and Okada consider a general flavor violating but lepton number conserving case, we canreadily adapt it to our models by casting the Lagrangian terms in the same lepton/neutrino pairingform for each Lorentz and helicity structure to read off the appropriate coefficients from theirEq.(43), which is reproduced in our Appendix as Eq.(30). The effective Michel- ρ parameter is thenidentified from Eq.(32). The second term of our Eq.(8) is the relevant piece for this discussion,and we use a Fierz rewriting to put it in the generic form: (¯ µ R e L )(¯ ν CaR ν eL )= (¯ µ R ν eL )(¯ ν CaR e L ) + (¯ µ R σ µν ν eL )(¯ ν CaR σ µν e L ). The effective Lagrangian reads L eff = − G F √ h (¯ e L γ µ ν eL )(¯ ν µL γ µ µ L ) + ǫ (¯ µ R ν eL )(¯ ν CaR e L ) + ǫ µ R σ µν ν eL )(¯ ν CaR σ µν e L ) i . (20)In the notation of Kuno and Okada, Eq.(30) in our Appendix, g SLR = ǫ ∗ and g TLR = ǫ ∗ , so the ρ parameter, Eq.(32), reads ρ = (1 − | g TLR | − Re ( g SLR g T ∗ LR )) = (1 − | ǫ | ), as shown in Table 1.The remaining cases can be evaluated similarly, and we give further discussion in the Appendix.Among the nine models, ( B ) and ( C ) show the best agreement between the limits on | ǫ | imposed by the precision measurements of the Michel parameters and the value needed to accom-modate the LSND ¯ ν e signal. A substantial amount of the χ per degree of freedom comes from The 90%C.L. value for σ ( P ) lsnd is taken to be √ . × σ expt =0.0013, which includes both statistical andsystematic errors. The η parameter is zero in the SM and in all of our nine ∆L = 0 models. The measurement of η is consistentwith zero within 1 σ , but it is an order of magnitude less precise than the others and is not included. = 0 model variants we consider is given in the column below the corresponding model’s name.The third column under ”TWIST” lists the 90%/C.L. global fit value for each parameter, as givenin Table VII of Ref.[23]. The best fit value of ǫ and the χ squared per degree of freedom at 90%C.L. for the fit are give in the last two items in each column. In the column labeled ”SM”, the χ /3value comes from the first three terms in Eq. 19, with the SM values for the Michel parameters intheir corresponding terms. See text for the discussion of the full SM comparison.Michel SM TWIST ( B ) ( B ) ( B ) ( B ) , ( C ) , ( C ) ( C ) ρ ( ǫ ) 0.75 0.74960 1 − | ǫ | − | ǫ | − | ǫ | − | ǫ | − | ǫ | ± δ ( ǫ ) 0.75 0.74997 1 − | ǫ | − | ǫ | | ǫ | | ǫ | − | ǫ | ± ξ ( ǫ ) 1.0 0.99897 1 + 2 | ǫ | | ǫ | − | ǫ | − | ǫ | − | ǫ | | ǫ | − | ǫ | ± ǫ fit χ /3 3.15 6.61 6.29 2.30 3.0 3.0 6.29 2.30the tension between the tree-level values of ρ and ξ and the global fit to data, which amounts toabout two ”90%C.L deviations” in each of these cases. For illustration, we will use models ( B ) and ( C ) , whose best fit values of | ǫ | are the largest and most promising for our LSND study,to illustrate the construction of renormalizable ∆L = 0 models and their implications in the nextsection. Notice that their fits improve the overall χ compared to the SM fit, despite the additionof the term representing the LSND appearance signal. We may interpret this as a sign that thereis room for new physics in the TWIST global fit to the Michel parameters. In this section we present briefly two explicit models that generate the anomalous lepton numberviolating decays of the muon. At low energies, these models reduce to effective interactions ofthe muon that leave both ρ and δ parameters in muon decay at their SM value of 3 /
4. Thus weprovide the basic mechanism for generating model ( B ) and ( C ) of Table 1, with the effectiveLagrangians given in Eqs. (10) and (18) respectively. We shall focus on extensions of the SM withthe introduction of scalar fields. Model ( B ) : The effective Lagrangian of Eq. (10) can be obtained by the addition of two scalarfields to the SM spectrum denoted by φ (1 , , /
2) and η + (1 , ,
1) where the quantum numbers under SU (3) C × SU (2) L × U (1) Y gauge symmetry are indicated. The φ = ( φ + , φ ) T field is a secondHiggs doublet field, but the vacuum expectation value of φ is assumed to be negligible comparedto that of the SM field H . The new couplings of these fields contain the terms L ( B ) = y ( µ R ψ ieL ) φ j ε ij + y ( ψ T ieL Cψ jaL ) η + ε ij + µ H i φ j η − ε ij + H.c., (21)where a = µ or τ . The Feynman diagram shown in Fig. 1 will induce the desired ∆ L = 2 muondecay µ + → e + ν e ν a . The strength of the interaction relative to the standard decay obtained from7 R ν eL φ + η + H e L ν aL Figure 1: Diagram inducing anomalous muon decay in model ( B ) .this diagrams is ǫ = 14 √ G F y y µvm φ m η ! (22)where we have assumed that the φ and η fields are nearly mass eigenstates. As an example,consider the choice y = 0 . , y = 1 . , m φ = 400 GeV, m η = 200 GeV and µ = 400 GeV. In thiscase, ǫ = 0 . ǫ = 3 . × − . When our ∆L = 0models are combined with the effects of a sterile neutrino that mixes with the three SM neutrinos,the requirements on ǫ weaken considerably, as we show in following sections.This explicit realization can be tested in collider experiments. First, the exchange of the neutral φ scalar would contribute in the t -channel to the cross section for the process e + e − → µ + µ − . Thiscontribution will modify the SM prediction at the level of the LSND signal, which is well withincurrent experimental limits.The model can be tested more directly at the LHC through the pair production of φ + and φ scalars. The relevant processes are pp → φ + φ via W exchange and pp → φ φ via Z exchange.The dominant decay of φ is into e + µ − and for φ + is into µ + ν . This would lead to trileptonsignature with missing energy as well as to four-lepton signals. In each case, the invariant mass of e + and µ − would show a resonant structure, corresponding to the mass of φ . The production crosssection for these processes at the LHC at 13 TeV is in the few fb range (depending on the mass of φ ). With more accumulated data, there is a chance of seeing this resonance in the future. Observethat the masses of the φ + and η + fields cannot be taken to large values, since that would diminishthe anomalous muon decay amplitude. To be relevant for the LSND anomaly, these scalars shouldhave masses below about 500 GeV, though again our comment on the effect of including a sterileneutrino appliesNote that the φ field couples primarily to e and µ , and thus carries approximate electronnumber and muon number. As a result, it does not lead to any lepton flavor violating decays.Furthermore, since the charged leptons in the muon decay diagram are distinct ( e and µ ), thisdiagram cannot be closed to generate neutrino masses. Thus, this model is quite different from theZee model [30] of neutrino masses, where m ν is induced by closing diagrams similar to Fig. 1. Weare assuming, as usual, that small neutrino masses arise via the seesaw mechanism involving heavyright-handed neutrino fields.Since there is a new decay channel for the muon, the Fermi coupling determined in nuclear betadecay will differ slightly from the coupling determined from µ decay. Such a small difference canbe interpreted as a shift in the value of | V ud | . Unitarity of the first row of the Cabibbo-Kobayashi-Maskawa (CKM) matrix would set a limit on any such differences, however, this limit at the level8f 3 × − allows some room for consistency with the LSND signal. Model ( C ) : This model includes a sterile neutrino ν R . The relevant interaction of Eq. (18) isobtained by introducing two scalar fields ∆(1 , ,
1) and η + (1 , , L ( C ) = y ( ψ TeL
C iτ τ a ψ eL )∆ a + y ( ψ T ieL Cψ jaL ) η + ε ij + α η − H † ∆ H + H.c. (23) µ R ν R η + ∆ + H e L ν eL H Figure 2: Diagram generating anomalous muon decay in model ( C ) .As in Eq. (22), we obtain ǫ ≈ × − for mass parameters of order 500 GeV and couplings y , = (0 . − field is sufficiently small as to be consistent withneutrino masses. The contribution to m ν from heavy right-handed neutrinos is assumed to bedominant. The model can be tested at the LHC in the pair production of doubly charged scalar∆ ++ , which has a unique signal of decaying dominantly into e + e + . There are limits on such scalarsfrom ATLAS [31] and CMS collaborations [32], which set the mass of ∆ ++ ≥
500 GeV if this particledecays 100% of the time time into e + e + . We note that within the model the decay ∆ ++ → ∆ + W + may also occur, which could weaken this limit. µ + ∆ L=2 decay, ν µ and ν e fluxes, and ν s -mixing-inducedoscillations For the remainder of the paper, we pursue the phenomenological consequences of applying our∆L = 0 NSI, combined with a sterile neutrino, to LSND and KARMEN data. As argued in thepreceding section, a model that is consistent with SM gauge symmetry, is safe from rare charged-lepton process like µ → eee , is consistent with constraints imposed by precision measurements ofMichel parameters and allows the most leeway for values of | ǫ | is exemplified by the ∆L=2 case( L B ) , Eq. (10). We reproduce its form here in order to make the discussion as self-contained aspossible. (¯ µ R ( ψ ke ) L )(( ψ la ) TL C ( ψ me ) L ) H n ε kn ε lm → (¯ µ R ν eL )( ν TaL Ce L ) h| H |i + ..., (24)where the ψ s represent lepton SU(2) doublets, the H is a doublet Higgs field with vacuum expec-tation value h| H |i , and a large scaling factor for h| H |i is understood. The epsilon symbols are theconstant SU(2) L antisymmetric two-index tensors and the SU(2) L doublet component indices areindicated as k, n, l and m. The lepton flavor label a on the left-handed neutrino doublet in Eq.(24),which can be µ or τ , will be taken to be µ for definiteness. The working effective Lagrangian, whichincludes the SM part and the ∆L=2 NSI part, reads9 eff = − G F √ (cid:2) (¯ e L γ λ ν eL )(¯ ν µL γ λ µ L ) + 2 ǫ (¯ µ R ν eL )( ν TµL Ce L ) (cid:3) (25)The first term in Eq.(25) is the SM, charged current effective Lagrangian, while the second termis the NSI, lepton-number violating term. As remarked earlier, the factor 2 in the definition of ǫ compensates for the factor 1/4 in the decay rate with the S,P structure compared to the V-Astructure. When ǫ =1, the rates of the two types of interaction are the same with this definition.The parameter ǫ in Eq.(25) is given in terms of the underlying model’s parameters as defined inEq.(21) by the expression given in Eq.(22). The neutrino fluxes from DAR are isotropic, with energy distributions that are determined by thedistributions of the individual muon decays. Given the same coupling coefficient in the effectiveLagrangian, the total rates of the Lorentz scalar decays are 1/4 those of the vector decays, but theshape of the energy distribution of the ν e in the SM decays is the same as that in the NSI decayand the same is true of the distribution of ν µ in the decays. Normalized to the LSND total fluxaveraged over the volume of their detector, the flux rates in cm − M eV − units as a function of theneutrino energy are dφ ( E ) dE = 5 . × × ( Em µ ) (1 − E m µ )Θ(1 − Em µ ) (26)for the ¯ ν µ flux, and dφ ( E ) dE = 5 . × × ( Em µ ) (2 − Em µ )Θ(1 − Em µ ) (27)for the ¯ ν e flux. The corresponding KARMEN fluxes are a factor 0.55 smaller than LSND, so theprefactor 5.7 in the flux expressions above is 3.1 in the KARMEN experiment. When LSND energyresolution is included, these distributions extend up to 60 MeV, as will be clear later in the eventrate plots. The KARMEN resolution is much sharper and has little effect on the flux distributionsor event rate distributions. ¯ ν µ and ¯ ν e Oscillation Probabilities
As we stressed in the Introduction, both appearance and disappearance of ¯ ν e play a role in generat-ing the inverse β -decay production of an e + signal in the detectors. We include the NSI parameter ǫ in the oscillation probabilities, though they could just as well be put in the flux factors. Denotingthe probability of appearance of ¯ ν e from oscillation of ¯ ν µ as P µe and the survival of ¯ ν e after itsdirect production in µ -decay as P ee , we write P µe ( E νµ ) = 4 | U e | | U µ | sin (∆( E νµ ))(1 + | ǫ | ) P ee ( E νe ) = | ǫ | (1 − | U e | (1 − | U e | )) sin (∆( E νe ))) , (28)where we define ∆( E νi ) = 1 . M [ eV ] L [ m ] / (4 E νi [ M eV ]), i = e, µ or τ , and M stands for thedominant difference in the square of the sterile neutrino mass and that of a standard 3-neutrinomixing mass eigenstate value. The value of L is 30m ±
4m (LSND) or 17m ±
1m (KARMEN). Inmodeling the event rate, we have two contributions. There is one originating from the ¯ ν µ component10f the flux, with a probability of P µe ( E νµ ) of oscillating to ¯ ν e plus a contribution from the NSIgenerated ¯ ν e component of the flux that survives to reach the detector with a probability P ee ( E νe ).In the absence of NSI, the ¯ ν e appearance is driven solely by the light, sterile-neutrino oscillationterm. In this case there are only two effective parameters, 4 | U e | | U µ | ≡ sin (2 θ µe ) and M . Only the product of the two mixing parameters is determined by a fit to appearance data.
This willbe important in our study of the simultaneous fits to the combined LSND and KARMEN data.
The event rates expected depend upon the flux of incoming neutrinos, the number of proton targets,the inverse beta decay cross section, σ IBD ( E ¯ νe ) [35] and the probabilities that the source neutrinosprovide ¯ ν e s after oscillation. Putting it all together, the energy distribution of events for the wholedata set as a function of energy can be written dNdE ¯ ν = ˜ ε × N p × X i = e,µ d Φ dE νi × σ IBD ( E ¯ νe ) × P ie ( E ¯ νi ) , (29)where ˜ ε is the average efficiency for detection, N p is the number of proton targets and the sumindicates that there are two sources of ¯ ν e s to trigger the signal events. If all of the ¯ ν µ were tooscillate to ¯ ν e and the probability that the ¯ ν e would be produced at the source and survive to reachthe detector were 1, the distribution of events with respect to neutrino energy that would result isshown in Fig. 3. The LSND spectra are illustrated by the solid curves and the KARMEN spectraare illustrated by the dashed curves. In each case, the top curve shows the sum of the two belowit, where the middle (above 40 MeV) curve, shows the contribution of ¯ ν µ from the source and thelower curve shows the contribution of ¯ ν e from the source.
10 20 30 40 50 6005001000150020002500 E ν ( MeV ) ν e E v en t R a t e ( c m - M e V - ) Figure 3: The solid curves are LSND events: total (top, blue), ¯ ν µ at µ -decay source (middle, black)and ¯ ν e at µ -decay source (bottom,red). The dashed curves are KARMEN events: total (top, green),¯ ν µ at µ -decay source (middle, light blue) and and ¯ ν e at µ -decay source (bottom, orange).11he curves show the hypothetical energy distribution of events that results when P µe = P ee = 1in Eq.(29). To illustrate the meaning of the curves in Fig. 3, we note that the integral of the solidblack curve (middle) from 20 MeV to 60 MeV represents the total possible number of observableevents in the LSND experiment [1], which they estimate as 33,300 ± ± P µe = 0.00264 ± ν µ to ¯ ν e is 34,300, well within their uncertainties. The size of the expectedsample from KARMEN is smaller than LSND’s because their total flux, their number of targetnuclei and their reconstruction efficiency are all smaller.With pure oscillation, no NSI, the middle curves are the relevant ones when computing the totaloscillation probability based on an observed signal of ¯ ν e + p → e + + n events at the detector. Notethat the extension of the LSND curves up to 60 MeV results from their broad energy resolution. In this section we present χ fits to the LSND and KARMEN data in our models with four parame-ters | U e | , | U µ | , M and ǫ . For LSND [25] we consider the total number of events above backgroundand the cleaner subsample of the data used in Fig. 24 and characterized in Table X of Ref.[25].This subsample contains 32.2 ± R γ >
10. For KARMEN we use the distributionof events vs. energy in Fig. 11b of Ref.[26]. The LSND oscillations subsample and the KARMENdata that we use in our combined fits are shown in Fig. 4.
Minimizing a joint LSND and KARMEN χ based on the 11 data bins in LSND’s ”excess events vsL/E” plot, Fig. 24 in Ref.[25] and the 9 data bins in KARMEN’s ”events/4 MeV vs. energy promptevent”, Fig. 11b in Ref.[26], we find a ”best fit” shown in Fig. 4. This fit’s parameter values, χ per degree of freedom (d.o.f.) and the expected number of events given by the unconstrained fitare summarized in Table 2 .Table 2: Unconstrained best fit parameter values, χ per d.o.f. and fit value of excess LSND events.The best fit χ = 17.0 value is quite democratically split between KARMEN, 8.6, and LSND, 8.4. | U e | | U µ | M ( eV ) ǫ χ /d.o.f. excess LSND events0.144 0.101 4.64 0.0 17.0/(20-4) = 1.06 21.5 (32 in data)The degeneracy we pointed out above in Sec. 2.2 comes into play here. With a best fit thatyields ǫ = 0, only the value of | U e | × | U µ | is determined by the (local) minimization procedure.The results shown for the mixing parameters are quite compatible with those reported in [13] and The measure R γ is defined in Sec. VII C of Ref. [25] as the likelihood that the γ is correlated with the promptgammas from positron annihilation divided by the likelihood that it is accidental. We should mention that eliminating the ¯ ν µ oscillations by setting | U µ | =0 and seeking the minimum of χ yieldsa solution at χ =17.2, with | U e | =0.707, M =11.2 eV and ǫ =0.025. Though the minimum is nearly degeneratewith that of the unconstrained minimum, the mixing parameter values are far outside the bounds set by many otheranalyses [10, 13, 17]. M is high, a point we return to in the following section. Thespecific values of the mixing parameters are somewhat dependent on the choice of starting valueschosen in the search for the minimum of χ . Any pair of values that yields | U e | × | U µ | = 0.0145will find | χ | = 17.0 with the accompanying values M =4.64 eV and ǫ =0. Of course the amountof χ deviation from the best fit value that develops as one allows excursions in M and/or ǫ with | U e | and | U µ | fixed, at their best fit values does depend on what pair of best fit values one chooses(consistent with the product value 0.0145). We explore excursions of parameters from best fit valuesnext. ∆ χ deviations from best fits There are 32 LSND events represented in the left panel of Fig. 4, while the fit represents 21.5 events.Even with an uncertainty of 9 in the number of events, without a constraint on the normalization ofthe fit there is evidently tension between L/E dependence of the data, the oscillation L/E modelingof the data and the overall normalization of the fit. Pursuing this question, we keep the values of | U e | , | U µ | and M fixed at the best fit values shown in Table 2 and determine the effect on χ ofletting ǫ , our lepton number violation parameter, deviate from it’s best fit value of 0. The result isshown in Fig. 5. At ǫ = 0.015, ∆ χ = 1, 68% C.L., and at ǫ = 0.02, ∆ χ = 2.71 , 90% C.L.. Thecorresponding values of the number of expected events are 27 and 31, well within the experimentalvalue 32 ±
9. The dashed curve shows the result of choosing best fit values that have a larger | U e | and correspondingly smaller | U µ | . The result shows very little sensitivity to individual values ofthe mixing parameters | U e | and | U µ | .Pursuing this point further, we first allow both ǫ and M to take on a range of values away fromthose of the best fit to see if reasonable fits with smaller M values can be found that are more With the best fit parameters shown in Table 2, we note that the values of | U e | and M and their 68% and 90%C.L. extensions that we explore later allow for the possibility that the effective mass in neutrinoless double- β decaymay vanish [33, 34]. % C.L.90 % C.L.95 % C.L. - - ε Δ χ Figure 5: The solid curve shows the deviation ∆ χ from best fit as a function of ǫ with otherparameters fixed at their values shown in Table 2. Taking the equivalent best fit values | U e | =0.20and | U µ | = 0.0725, we find the deviation shown by the dashed curve. The horizontal lines indicatethe 1 σ , 90%C.L. and 2 σ values for 1 degree of freedom.compatible with the global fit, which is in the 1.5 - 2 e V range [10]. We fix the best fit values ofthe mixing parameters | U µ | and | U e | at the values given in Table 2, and let M and ǫ vary tofind the 68%, 90% etc. allowed regions in their 2-degrees of freedom space. The result is shown inthe left-hand panel of Fig. 6. Again, to check the sensitivity of the results to the choice of mixingparameters, we set | U µ | equal to 0.0145/ | U e | to satisfy the minimization condition. With | U e | =0.20 and | U µ | = 0.0725, for example, we find that the 68% C.L. and 90% C.L. contours are withinthe width of the contour lines shown in the left panel of Fig. 6. Adding detail to the left-hand plotof Fig. 6, we plot ∆ χ vs. ǫ profiles at constant M for 1, 2 and 3 eV in Fig. 7. Both plots showthat for M above 1 eV , there is substantial parameter space consistent with global fits where the∆L-breaking effects plus sterile neutrino oscillations accommodate the LSND and KARMEN data,even at 1 σ .Similarly, the right-hand plot in Fig. 6 shows that values compatible with an interesting regionof | U µ | - ǫ values opens up within the 68% and 90% C.L. boundaries. In particular, values of ǫ in the range 0 . ≤ ǫ ≤ .
03 indicated in Fig.6, based on the LSND R γ data, agree with theallowed values we found in our study of the ∆L = 0 effects when comparing the SM predictionsand precision measurements of Michel-parameters ρ , δ and ξ in Section 2.3. Fitting the LSND data set alone, one finds that unconstrained, the 4-parameter fit produces ǫ = 0.As in the combined fit, the value of the sterile neutrino mass scale are typically about 4 - 5 eV ,larger by factors of 3 - 4 compared to the values of about 1-2 eV found in global fits [10, 9]. TheLSND total and R γ ≥
10 excess events are somewhat poorly represented by the best fit parameterset. As in the joint fit, the individual fits have χ of 8 to 9, ∼ ǫ that produces LSND’s central valueof 88 events, namely ǫ = 0.056, the resulting χ for the L/E distribution, Fig. 24 in Ref. [25], is10.7 for 11 d.o.f., which is about the same as the LSND contribution to the best fit to the joint14 % C.L.90 % C.L.0 M ε % C.L.90 % C.L.0.005 | U μ ε Figure 6: The left-hand panel shows the 68% and 90% C.L., ∆ χ = 2.3 and 4.61 contours in the ǫ - M ( eV ) plane with mixing parameters chosen at best fit values | U e | = 0.144 and | U µ | = 0.101.The right-hand panel shows the 68% and 90% C.L., ∆ χ = 2.3 and 4.61, contours in the ǫ - | U µ | plane with mixing parameters chosen at best fit values | U e | = 0.144 and M =4.64 eV .Figure 7: The ∆ χ vs. ǫ projections of the left panel of Fig. 6 at M = 1, 2 and 3 eV , wheremixing parameters are chosen at best fit values | U e | = 0.144 and | U µ | = 0.101. The constant∆ χ lines are at 2.3, 4.61 and 5.99, corresponding to 68%, 90% and 95% C.L. boundaries for 2degrees of freedom. Choosing other values for | U e | and | U µ | consistent with the product 0.0145gives essentially the same results. 15istribution. This constrained parameter set gives a very poor description of the KARMEN data,as one expects. However, if one allows a 90% C.L. excursion from the 88 ±
22 excess LSND events,one finds the value ǫ = 0.041, which is consistent with the 90% C.L. bound of signal events foundwith KARMEN’s 15 events with a background of 15.8. (Since KARMEN observed no events in the3 highest energy bins, at 42 MeV, 46 MeV and 50 MeV, it reports zero √ N errors in those bins.To estimate a statistically expected range for these points, we adopt the Poisson 68% C.L. signalmean for no events observed with the KARMEN background numbers given for those bins [36].)Comparing our results directly with the DAR results for LSND as presented in the highlydetailed statistical analysis of Ref.[37], we do not expect close agreement because their data sampleis much larger, cut at R γ ≥ − rather than at 10, and the analysis in [37] works directly withthe data, while we use a subsample that has been refined and binned and presented graphically inthe final LSND paper [25]. Our version of their data, Fig. 4, is obtained from reading off the binsand error bars in Fig. 24 of Ref.[25] and checking that the number of events and the correspondingstatistical uncertainty agrees with the R γ ≥
10 values listed there in Table X. Nonetheless, it isuseful to compare Fig. 6 in Ref.[37] with a similar version generated from our fit. In Fig. 8, weshow the 68% and 90% C.L contours in the variables M vs. sin (2 θ µe ) with ǫ = 0, it’s value atbest fit point where the M and sin (2 θ µe ) values are 5.1 eV and 0.00084. The region of overlaplies below M = 2 eV , where our 90% C.L. region covers their 95% region. At higher values of M , our 90% region lies at smaller sin (2 θ µe ) values than theirs. Our rough analysis, designed toexplore a combined ∆L = 0 and sterile neutrino mixing modeling of DAR appearance data doesnot capture the high-mass region details presented in [37], but it does overlap with the region where0 . ≤ ǫ ≤ .
03, is allowed by LSND, KARMEN and precision µ -decay data.Figure 8: Our boundaries at 68%C.L. and 90% C.L in sin (2 θ µe ) - M space with ǫ = 0, its bestfit point value, where sin (2 θ µe ) = 0.00084 and M = 5.1 eV .16 .2.2 Comments on Future Possibilities Looking ahead to (far) future possibilities where leptonic, ∆L = 2 interactions may reveal them-selves, we suggest that a neutrino factory would be ideal. The short baseline scheme proposed byGiunti, Laveder and Winter [38], for example, emphasizes the clean environment for studying newphysics in both charge modes of the muon at a ν -factory. Their focus is on oscillation studies, butthe short baselines proposed, 50 m and 200 m, lend themselves to study of excess events originatingfrom both of the exotic decays µ − → e − + ν e + ν µ and µ + → e + + ¯ ν e + ¯ ν µ , where both ∆L=2 andoscillation effects can also be looked for.In the shorter term, planned high precision DAR experiments that include µ + decays at restwould provide welcome tests for our scheme. The JSNS experiment [39, 40] with a 24m baselinesetup similar to LSND plans to look for ¯ ν e appearance from π + , K + and µ + decay at rest. All threeare sensitive to a sterile neutrino effects, but only the µ + decay provides a test of our ∆L = 2 NSI.Therefore any difference from sterile-only effects that shows up in the µ + case as compared to thehadronic decay sources would be evidence for purely leptonic new physics. Likewise, to the extentthat µ decays contributing to the atmospheric neutrino signals [41] can be distinguished from π and K decays, there is a possibility that predictions of our model can be tested.In addition, within specific renormalizable models that realize the anomalous µ -decay via ∆L =0 four-Fermion interactions, there are striking signatures that would support the whole picture.Examples were given in Sec.(2.4) of effects on the e + e − → µ + µ − cross section and of directproduction and decay of new neutral and charged scalars at the LHC. To apply the direct ∆L = 2, lepton number violating interaction Eq. (25) in µ -decay toward ex-plaining LSND [25] anomaly [4] without contradicting the absence of oscillation signal in KARMENdata [26], we added mixing between the known neutrino flavors and a sterile neutrino [11]. Unlikethe new interaction, this affects the signals in the two experiments differently because of their dif-ference in baselines. We first reviewed the lepton flavor violating, but lepton number conservingcases that, when generalized to SU(2) symmetric models, are too strictly constrained by limits oncharged lepton flavor violation to admit the LSND signal. We then constructed a set of nine can-didate ∆L = 0, consistent effective interactions that are not constrained by charged lepton data.Seven of the nine were shown to be disfavored by the precision measurements of the µ -decay Michelparameters. Two cases make no change to the SM tree-level values ρ = δ =3/4, consistent with thecurrent data and produce better overall fits with an ǫ parameter value that is consistent with therange allowed by our fitting of the combined LSND and KARMEN DAR data sets represented bythe binned distributions in Fig. 6. We then constructed renormalizable models for both of theMichel parameter preferred effective Lagrangians, connecting the underlying coupling constantsand scalar masses to the ǫ parameter. One of these was chosen for a detailed phenomenologicalstudy. The new physics interaction, whose strength is parameterized by ǫ , produces the ∆L= 2decay mode µ + → e + + ¯ ν µ + ¯ ν e . This example is compatible with experimental limits on devia-tions from the standard model description of µ decay [28] and has the special feature that thereare two sources of ¯ ν e , those from the appearance oscillation of ¯ ν µ to ¯ ν e and the survival of the Apart from Ref.[37], all fits that we’re aware of contain data beyond the DAR data, which are the only datasensitive to our new physics, ∆L = 0 interactions. ν e . We studied the interplay between the two contributions and the relative rolesthat the sterile-neutrino-driven oscillations and the ∆L=2 new physics played.We found a good, unconstrained fit to the combined DAR data sets with a fourth, sterileneutrino oscillating with parameter set | U e | = 0.144, | U µ | = 0.101, and M = 4.64 eV andthe NSI parameter ǫ = 0, see Table 2. The mixing parameter values are quite compatible withfits to appearance data [13] and global fits to both appearance and disappearance data [10], and,though the mass scale is higher than best fits reported for the combined, global fits [10], we showour 90%C.L. contour in Fig. 8, which overlaps that of [37] for masses below 2 eV and is thusconsistent with global fits. Calculating the number of excess events with our best fit parametersproduces an estimate of 21, whereas the data subsample contains 32 ± χ vs. ǫ shows that for 2 eV ≥ M ≥ eV there are points in the ranges 0 . ≤ ǫ ≤ . ≤ ǫ ≤ .
032 allowed at 90% C.L.. Many of thepoints in these regions bring the modeled numbers of events for LSND into much better alignmentwith the data, while staying within the constraints from KARMEN. A similar improvement is foundwith R γ ≥ M found in our fit, though larger than the global-fit splitting parameter, is compatible at 68% to90% C.L with the 1-2 eV region of the detailed LSND fits. Moreover, our constrained fit inves-tigations indicate that LSND fits with non-zero ǫ values and smaller values of M give acceptablefits, suggesting that the ∆L = 2 NSI may still be playing a role with values up to 0.025-0.03, inaccord with the constraints imposed by Michel parameter measurements. We conclude that thisclass of NSIs brings in new features of the sterile neutrino idea and it is well worth pursuing inanalyzing current and future neutrino data and, at the underlying model level, current and futurecollider data. Acknowledgments:
We thank KITP Santa Barbara for hospitality and support during the”Present and Future Neutrino Physics” workshop, where this work was started. D. McKay and K.Babu thank the lively CETUP 2015 workshop for providing the opportunity to present and discussthis work, and especially Barbara Szczerbinska, who kept it all together. We appreciate helpfulcommunications with Bill Louis, Geoff Mills and Thomas Schwetz. This work is supported in partby the U.S. Department of Energy Grant No. de-sc0010108 (KSB) and by de-sc0013699 (IM).
In this appendix we collect the formulas and definitions from Ref. [28] that we use in the develop-ments of Sec.2.3. The basic idea is simply that models that produce a final state positron spectrumwith the same shape as that of the V-A form of the SM (see Eq.(26)) will be compatible with thehigh-precision data, which is compatible with the SM values of Michel parameters [29, 23]. Thosemodels that give the other e + spectral behavior (see Eq.(27)) will add terms that disagree with theprediction of the V-A interaction of the SM. With appropriate modification of the leptonic labels,the Michel parameters for any model can be expressed in terms of the coupling coefficients of the18ollowing generic 4-Fermi effective Lagrangian: L µ → eνν = − G F √ g SRR (¯ e R ν eL )(¯ ν µL µ R ) + R ↔ L + g SRL (¯ e R ν eL )(¯ ν µR µ L ) + R ↔ L + g VRR (¯ e R γ µ ν eR )(¯ ν eR γ µ µ R ) + R → L + g VRL (¯ e R γ µ ν eR )(¯ ν µL γ µ µ L ) + R ↔ L ++ g TRL e R σ µν ν eL )(¯ ν µR σ µν µ L ) + R ↔ L ] + H.c.. (30)Calculation of the decay rate imposes the normalization condition1 = 14 ( | g SRR | + | g SLL | + | g SRL | + | g SLR | ) + ( | g VRR | + | g VLL | + | g VRL | + | g VLR | ) + 3( | g TRL | + | g TLR | ) . (31)Using Eq( 31) to express | g VLL | in terms of the other, presumed small, coupling constants,substituting it into the appropriate equations for the Michel parameters as summarized in thereview ”Muon Decay Parameters” in Ref.[29] and then keeping leading order terms in modulussquared couplings, one finds the following expressions for the ρ , δ and ξ -parameters [28]: ρ = 34 (cid:2) − (cid:0) | g VRL | + | g VLR | + 2 | g TRL | + 2 | g TLR | + Re ( g SRL g T ∗ RL + g SLR g T ∗ LR ) (cid:1)(cid:3) . (32) δ = 34 (cid:2) (cid:0) | g VRL | − | g VLR | + 2( | g TRL | − | g TLR | ) + Re ( g SRL g T ∗ RL − g SLR g T ∗ LR (cid:1)(cid:1) ] . (33) ξ = 1 + ( − | g SRR | − | g SLR | − | g VRR | − | g VRL | + 2 | g VLR | + 2 | g TLR | − | g TRL | − Re ( g SRL g T ∗ RL − g SLR g T ∗ LR )) . (34)The model ( B ) , where the first term in Eq.(10) with a= µ or τ is the relevant one for ourstudy, corresponds to the term with coupling g SLR in Eq.(30) and in Eq.(32), where this couplingappears only in the term multiplied by g TLR . The latter coupling is not present in this model, sothe ρ and δ parameters retain the SM value of 3/4.To illustrate the role that possible SU(2) L breaking might play, we introduce a symmetry break-ing parameter c by rewriting Eq.(8) as ( L B. ) → (¯ µ R ν eL )( ν TaL Ce L ) − c × (¯ µ R e L )( ν TaL Cν eL ), wherec is complex in general. After a Fierz reordering to match the convention of Eq.(30) and collectingterms, we find ( L B. ) → (1 + c µ R ν eL )( ν TaL Ce L ) + c µ R σ µν ν eL )( ν TaL Cσ µν e L ) . (35)Making the identifications g SLR = N (1 + c ∗ ) ǫ ∗ and g TLR = N c ∗ ǫ ∗ , we choose the normalization factorN so that it insures the ǫ → L B. ) to describe µ -decay . This requires that N = √ | ǫ | + Re ( c ) , and using Eq.(32), we find ρ = (1 − | ǫ | | c | + Re c √ | ǫ | + Re ( c ) ). For c = 1, there is nosymmetry breaking and the coefficient of | ǫ | is , as listed in Table (1).19 eferences [1] C. Athanassopoulos et al. , Phys. Rev. Lett. , 2650 (1995); C. Athanassopoulos it et al.,Phys. Rev. Lett. , 1774 (1998).[2] J. N. Bahcall, P. I. Krastev, and A. Yu. Smirnov, Phys. Rev. D, , 096016 (1998); S. M.Bilenky and C.Giunti, Phys. Lett B , 379 (1998).[3] L.M. Johnson and D.W. McKay, Phys. Lett B , 355 (1998).[4] K.S. Babu and Sandip Pakvasa, ”Lepton Number Violating Muon Decay and the LSNDAnomaly”, arXiv: hep-ph/020423v1 (2002).[5] V. Barger et al. , Phys. Lett. B , 345 (2000).[6] M. Sorel, J. M. Conrad, and M. Shaevitz, Phys. Rev. D , 073004 (2004); A. J. Anderson,J. M. Conrad, E. Figuaro-Feliciano, C. Ignarra, G. Karagiorgi, K. Scholberg, M. H. Shaevitzand J. Spitz, Phys. Rev. D , 013004 (2012).[7] H. Davoudiasl, P. Langacker and M. Perelstein, Phys. Rev. D , 105015 (2002) .[8] G. Barenboim and N.E. Mavromatos, JHEP , 034 (2005).[9] J. Kopp, P. A. N. Machado, M. Maltoni and T. Schwetz, JHEP , 050 (2013).[10] C. Giunti, M. Laveder, Y.F.Li and W.W. Long, Phys. Rev. D. , 073008 (2013).[11] M. C. Gonzalez-Garcia, M. Maltoni and T. Schwetz, arXiv:1512.06856 [hep-ph]. This referenceprovides an up-to-date review of the sterile neutrino application to short baseline anomalies.[12] S. Gariazzo, C. Giunti, M. Laveder, Y. F. Li and E. M. Zavanin, J. Phys. G , 033001 (2016).[13] C. Giunti, M. Laveder, Y. F. Lie and W. W. Long, Phys. Rev. D , 113014 (2012).[14] K. Abe et al. (The Super Kamiokande Collaboration), Phys. Rev. D , 052019 (2015) providesa bound | U µ | ≤ .
18 for M ≥ . eV , consistent with the fits of [10] and the ranges of interestto us.[15] K. Abe et al. (The T2K Collaboration), Phys. Rev. D , 051102 (2015) uses its near detectorto study ν e disappearance. The bound sin (2 θ ee ) ≤ . M ≥ eV is found, equivalentto | U e | ≤ .
27, again consistent with global fits and those we find.[16] Recent limits on | U e | (see Eq. 28) from analysis of IceCube data are quite stringent: ”Ster-ile Neutrinos in Cold Climates”, Benjamin J. P. Jones (MIT), FERMILAB-THESIS-2-15-17(Experiment:FNAL-0974); M. Lindner, W. Rodejohann and X.-J. Xu, JHEP , 124 (2016).Because both analyses make reference to the MiniBoone results as well as to LSND, it is notclear how they relate to our fits, which refer only to LSND and KARMEN DAR data. Nomi-nally, our fit gives sin (2 θ ) = 0.00084 and M = 4.6 eV .[17] ”Light Sterile Neutrinos: a Whitepaper”, K.N. Abazajian et al. , arXiv:1204.5379v1 [hep-ph](2012).[18] J. Bergstrom, M. C. Gonzalez-Garcia, V. Niro and J. Salvado, JHEP , 104 (2014).2019] B. Leistedt, H. V. Peiris and L.Verde, Phys. Rev. Lett. , 041301 (2014).[20] S. Bergmann and Y. Grossman, Phys. Rev. D , 093005 (1999).[21] E. Akhmedov and T. Schwetz, JHEP , 115 (2010).[22] R. Bayes et al. (The TWIST Collaboration), Phys. Rev. Lett. , 041804 (2011).[23] A. Hillairest et al. (The TWIST Collaboration), Phys. Rev. D , 092013 (2012).[24] A. Aguilar-Arevalo et al. , (MiniBoone Collaboration), Phys. Rev. Lett. , 161801 (2013).[25] A. Aguilar et al. (LSND Collaboration), Phys. Rev. D , 112007 (2001).[26] B. Armbruster et al. (KARMEN Collaboration), Phys. Rev. D , 112001 (2002).[27] W. Fetscher, H. J. Gerber and K.F Johnson, Phys. Lett. B , 102 (1986).[28] Y. Kuno and Y. Okada, Rev. Mod. Phys. , 151 (2001).[29] K. Olive et al. (Particle Data Group), Chin. Phys. C. , 389 (1980); Phys. Lett. B , 461 (1980).[31] G. Aad et al. [Atlas Collaboration], JHEP , 041 (2015).[32] S. Chatrchyan et al. [CMS Collaboration], Eur. Phys. J. C , 2189 (2012).[33] J. Barry, W. Rodejohann and H. Zhang, JHEP , 091 (2011).[34] C. Giunti and E. M. Zavanin, JHEP , 171 (2015).[35] P. Vogel and J.F. Beacom, Phys. Rev. D , 053003 (1999). We use the version of the crosssections that are corrected to order 1/M, as summarized by Eq. (18). This reference is quotedas the cross section source for both LSND and KARMEN.[36] G. J. Feldman and R. D. Cousins, Phys. Rev. D , 3873 (1998).[37] E. D. Church, K. Eidel, G. B. Mills and M. Steidl, Phys. Rev. D , 013001 (2002).[38] C. Giunti, M. Laveder and W. Winter, Phys. Rev. D , 073005 (2009).[39] M. Harada et al. (JSNS Collaboration), arXiv:1601.01046.[40] Fumihiko Suekane, arXiv:1604.06190v2. This reference reviews plans for the JSNS , OscSNSand KPipe experiments.[41] M. Aartsen et al.et al.