Radiative Decays of Cosmic Background Neutrinos in Extensions of MSSM with a Vector Like Lepton Generation
PPreprint No: NSF-KITP-13-080.
Radiative Decays of Cosmic Background Neutrinos in Extensionsof MSSM with a Vector Like Lepton Generation
Amin Aboubrahim b , Tarek Ibrahim a,b and Pran Nath c,d a. Department of Physics, Faculty of Science, University of Alexandria,Alexandria 21511, Egypt b. Department of Physics, Faculty of Sciences, Beirut Arab University, Beirut 11 - 5020,Lebanon c. Department of Physics, Northeastern University, Boston, MA 02115-5000, USA d. KITP, University of California, Santa Barbara, CA 93106-4030 Abstract
An analysis of radiative decays of the neutrinos ν j → ν l γ is discussed in MSSM extensionswith a vector like lepton generation. Specifically we compute neutrino decays arising fromthe exchange of charginos and charged sleptons where the photon is emitted by the chargedparticle in the loop. It is shown that while the lifetime of the neutrino decay in the StandardModel is ∼ yrs for a neutrino mass of 50 meV, the current lower limit from experimentfrom the analysis of the Cosmic Infrared Background is ∼ yrs and thus beyond the reachof experiment in the foreseeable future. However, in the extensions with a vector like leptongeneration the lifetime for the decays can be as low as ∼ − yrs and thus withinreach of future improved experiments. The effect of CP phases on the neutrino lifetime isalso analyzed. It is shown that while both the magnetic and the electric transition dipolemoments contribute to the neutrino lifetime, often the electric dipole moment dominateseven for moderate size CP phases. Keywords: Cosmic background neutrinos, radiative neutrino decay, vector lepton multipletsPACS numbers: 13.40Em, 12.60.-i, 14.60.Fg Email: [email protected] Email: [email protected] Emal: [email protected] Permanent address. Current address. Permanent address a r X i v : . [ h e p - ph ] J u l Introduction
It is well known that a neutrino can decay radiatively to neutrinos with lower masses. Thusfor the neutrino mass eigenstates ν , ν , ν , with m ν > m ν > m ν one can have radiativedecays so that ν → ν γ, ν γ . In the Standard Model this process can proceed by theexchange of a charged lepton and a W boson so that ν → l − W + ( loop ) → ν , γ . However,the lifetime for the neutrino decay in the Standard Model is rather large [1], i.e., τ SMν ∼ yrs , (1)for a ν with mass 50 meV. Now the current lower limit based on data from galaxy surveyswith infrared satellites AKARI [2], Spitzer [3] and Hershel [4] as well as the high precisioncosmic microwave background (CMB) data collected by the Far Infrared Absolute Spec-trometer (FIRAS) on board the Cosmic Background explorer (COBE) [5] for the study ofradiative decays of the cosmic neutrinos[6] using the Cosmic Infrared Background (CIB)gives [6] τ expν ≥ yrs (2)This lower limit is below the Standard Model prediction of Eq.(1) by over 30 orders ofmagnitude and thus the study of cosmic neutrinos using the Cosmic Infrared Backgroundis unlikely to be fruitful in testing the radiative decays of the neutrinos in the StandardModel. However, much lower lifetimes for the neutrino decays can be achieved when onegoes beyond the Standard Model. For example, radiative decays of the neutrinos have beendiscussed in extensions of the standard model with a heavy mirror generation [7]. Usingtheir result one finds a neutrino lifetime ∼ yrs which while much smaller than the onegiven by the Standard Model is still eight orders of magnitude above the current level of sen-sitivity. Similarly in the left-right symmetric models, calculations show that one can lowerthe lifetime for the decay of the neutrino significantly so that [6] τ LRν ∼ . × yrs. Theexperimental measurement using radiative decays provides a way to measure the absolutemass of the neutrino. Thus consider the decay ν j → ν l γ . In the rest frame of the decay of ν j the photon energy is given by E γ = ( m j − m l ) / (2 m j ). Since neutrino oscillations provideus with the neutrino mass difference m j − m l , a measurement of the photon energy allowsa determination of m j . Thus the study of Cosmic Infrared Background provides us with analternative way to fix the absolute value of the neutrino mass aside from the neutrino less1ouble beta decay.In this work we will discuss a new class of models where the neutrino lifetimes as lowas close to the current experimental lower limits can be obtained which makes the study ofthe lifetimes of the cosmic neutrinos using CIB interesting. Specifically we consider neutrinodecay via a light vector like generation. Light vector like generations have been discussedin a variety of works recently. Specifically these include the neutrino magnetic moments [8],contribution to EDMs of leptons [9] and quarks EDMs [10, 11], contribution to radiativedecay of charged leptons [12] and to variety of other phenomena [13, 14, 15, 16, 17]. Likethe flavor changing radiative decay of the charged leptons (for a review see [18] ) the ra-diative decays of the neutrinos provide a window to new physics. With the inclusion ofthe vector generation we also expect the radiative decays of the neutrinos could be signifi-cantly larger than in the Standard Model. The reason for this expectation is the following:In the analysis of the decay τ → µγ it is found [12] that the decay for this process ismuch larger in models with vector like multiplets than in conventional models. We ex-pect that a similar phenomenon will occur in the analysis of the radiative decay of theneutrinos. This is so because the diagrams that enter in the neutrino radiative decayare very similar to the diagrams that enter in the analysis of the radiative decay of the τ . Thus we expect that the analysis would yield a decay lifetime which would be ordersof magnitude closer to the current experimental limits than the result from the StandardModel. In the analysis we will impose the most recent constraints from the Planck satelliteexperiment [19], i.e., that (cid:80) i m ν i < .
85 eV (95% CL ) as well as the neutrino oscilla-tion constraints [20] on the mass differences ∆ m ≡ m − m = 2 . +0 . − . × − eV , and∆ m ≡ m − m = 7 . +0 . − . × − eV .We note in passing that the radiative decays of the cosmic neutrinos in a supersymmetricframework was discussed in early work in [21]. However, in their work the radiative decayof neutrinos with testable lifetimes make flavor changing processes in the charged leptonsector exceed the experimental limits. Thus these authors had to consider broken R paritymodels to circumvent these constraints. In our work there are no problems of this sort in theanalysis presented here. Indeed the flavor changing neutral currents in the charged sector The recent data from the Planck experiment [19], gives two upper limits on the sum of the neutrinomasses, i.e., 0.66 eV and 0.85 eV (both at 95% CL), where the latter limit includes the lensing likelihood. f , f (cid:48) , f (cid:48)(cid:48) inEq.(6) enter the charged lepton sector, they do not enter the neutrino sector. Further, whilethe couplings f , f (cid:48) , f (cid:48)(cid:48) enter the neutrino sector they do not enter the charged lepton sector.This allows one to suppress the neutral current processes in the charged lepton sector with-out a problem. In a similar fashion the muon g-2 experiment does not put any constrainton the current analysis. This is so because the contribution of the vector-like multiplet to g µ − f , f (cid:48) , f (cid:48)(cid:48) which as already indicated above do not enterin the radiative decays of the neutrinos and these couplings can be adjusted so that thecontribution of the vector like multiplet to g µ − g µ − Vector like multiplets arise in a variety of unified models [22] some of which could be lowlying. Here we simply assume the existence of one low lying leptonic vector multiplet whichis anomaly free in addition to the MSSM spectrum. Before proceeding further it is useful torecord the quantum numbers of the leptonic matter content of this extended MSSM spectrumunder SU (3) C × SU (2) L × U (1) Y . Thus under SU (3) C × SU (2) L × U (1) Y the leptons of thethree generations transform as follows ψ iL ≡ (cid:18) ν iL l iL (cid:19) ∼ (1 , , −
12 ) , l ciL ∼ (1 , , , ν ciL ∼ (1 , , , i = 1 , , ∼ is the value of the hypercharge Y defined so that Q = T + Y . These leptons have V − A interactions. We can now adda vector like multiplet where we have a fourth family of leptons with V − A interactionswhose transformations can be gotten from Eq.(3) by letting i run from 1-4. A vector likelepton multiplet also has mirrors and so we consider these mirror leptons which have V + A interactions. Their quantum numbers are as follows χ c ≡ (cid:18) E cL N cL (cid:19) ∼ (1 , ,
12 ) , E L ∼ (1 , , − , N L ∼ (1 , , . (4)3he MSSM Higgs doublets as usual have the quantum numbers H ≡ (cid:18) H H (cid:19) ∼ (1 , , −
12 ) , H ≡ (cid:18) H H (cid:19) ∼ (1 , ,
12 ) . (5)As mentioned already we assume that the vector multiplet escapes acquiring mass atthe GUT scale and remains light down to the electroweak scale. As in the analysis ofRef.[9] interesting new physics arises when we consider the mixing of the second and thirdgenerations of leptons with the mirrors of the vector like multiplet. Actually we will extendour model to include the mixing of the first generation as well, for the computation of thedecay ν → ν , γ . Thus the superpotential of the model may be written in the form W = − µ(cid:15) ij ˆ H i ˆ H j + (cid:15) ij [ f ˆ H i ˆ ψ jL ˆ τ cL + f (cid:48) ˆ H j ˆ ψ iL ˆ ν cτL + f ˆ H i ˆ χ cj ˆ N L + f (cid:48) H j ˆ χ ci ˆ E L + h H i ˆ ψ jµL ˆ µ cL + h (cid:48) H j ˆ ψ iµL ˆ ν cµL + h H i ˆ ψ jeL ˆ e cL + h (cid:48) H j ˆ ψ ieL ˆ ν ceL ]+ f (cid:15) ij ˆ χ ci ˆ ψ jL + f (cid:48) (cid:15) ij ˆ χ ci ˆ ψ jµL + f ˆ τ cL ˆ E L + f ˆ ν cτL ˆ N L + f (cid:48) ˆ µ cL ˆ E L + f (cid:48) ˆ ν cµL ˆ N L + f (cid:48)(cid:48) (cid:15) ij ˆ χ ci ˆ ψ jeL + f (cid:48)(cid:48) ˆ e cL ˆ E L + f (cid:48)(cid:48) ˆ ν ceL ˆ N L (6)where ˆ ψ L stands for ˆ ψ L , ˆ ψ µL stands for ˆ ψ L and ˆ ψ eL stands for ˆ ψ L . Here we assume a mixingbetween the mirror generation and the third lepton generation through the couplings f , f and f . We also assume mixing between the mirror generation and the second lepton genera-tion through the couplings f (cid:48) , f (cid:48) and f (cid:48) . The same is true for the mixing between the mirrorgeneration and the first lepton generation through the couplings f (cid:48)(cid:48) , f (cid:48)(cid:48) and f (cid:48)(cid:48) . The abovenine mass terms are responsible for generating lepton flavor changing process.
We will focushere on the supersymmetric sector. Then through the terms f , f , f , f (cid:48) , f (cid:48) , f (cid:48) , f (cid:48)(cid:48) , f (cid:48)(cid:48) , f (cid:48)(cid:48) one can have a mixing between the third generation, the second and the first generationleptons which allows the decay of ν → ν , γ through loop corrections that include charginosand scalar lepton exchanges with the photon being emitted by the chargino or by a chargedslepton. The mass terms for the leptons and mirrors arise from the term L = − ∂ W∂A i ∂A j ψ i ψ j + H.c. (7)where ψ and A stand for generic two-component fermion and scalar fields. After spontaneousbreaking of the electroweak symmetry, ( (cid:104) H (cid:105) = v / √ (cid:104) H (cid:105) = v / √ − L m = ( ¯ ν τR ¯ N R ¯ ν µR ¯ ν eR ) f (cid:48) v / √ f − f f v / √ − f (cid:48) − f (cid:48)(cid:48) f (cid:48) h (cid:48) v / √ f (cid:48)(cid:48) h (cid:48) v / √ ν τL N L ν µL ν eL + H.c. (8)Here the mass matrices are not Hermitian and one needs to use bi-unitary transformationsto diagonalize them. Thus we write the linear transformations ν τ R N R ν µ R ν e R = D νR ψ R ψ R ψ R ψ R , ν τ L N L ν µ L ν e L = D νL ψ L ψ L ψ L ψ L , (9)such that D ν † R f (cid:48) v / √ f − f f v / √ − f (cid:48) − f (cid:48)(cid:48) f (cid:48) h (cid:48) v / √ f (cid:48)(cid:48) h (cid:48) v / √ D νL = diag ( m ψ , m ψ , m ψ , m ψ ) . (10)In Eq.(10) ψ , ψ , ψ , ψ are the mass eigenstates for the neutrinos, where in the limit of nomixing we identify ψ as the light tau neutrino, ψ as the heavier mass eigen state, ψ as themuon neutrino and ψ as the electron neutrino. To make contact with the normal neutrinohierarchy we relabel the states so that ν = ψ , ν = ψ , ν = ψ , ν = ψ (11)which we assume has the mass hierarchical pattern m ν < m ν < m ν < m ν (12)We will carry out the analytical analysis in the ψ i notation but the numerical analysis will becarried out in the ν i notation to make direct contact with data. Next we consider the mixing5f the charged sleptons and the charged mirror sleptons. The mass squared matrix of theslepton - mirror slepton comes from three sources, the F term, the D term of the potentialand the soft susy breaking terms. Using the superpotential of Eq.(6) the mass terms arisingfrom it after the breaking of the electroweak symmetry are given by the Lagrangian L = L F + L D (13)where L F and L D are given in the Appendix along with the matrix elements of the sleptonmass squared matrix. The chargino exchange contribution to the decay of the tau neutrino into a muon neutrino(electron neutrino) and a photon arises through the loop diagram in Fig.(1). The relevantpart of the Lagrangian that generates this contribution is given by − L ν − ˜ τ − χ + = (cid:88) j =1 2 (cid:88) i =1 8 (cid:88) k =1 ¯ ψ j [ C Ljik P L + C Rjik P R ] ˜ χ + i ˜ τ k + H.c. (14)where C Ljik = − f (cid:48) V ∗ i D ν ∗ R j ˜ D τ k − f (cid:48) V ∗ i D ν ∗ R j ˜ D τ k + gV ∗ i D ν ∗ R j ˜ D τ k − h (cid:48) V ∗ i D ν ∗ R j ˜ D τ k − h (cid:48) V ∗ i D ν ∗ R j ˜ D τ k ,C Rjik = − f U i D ν ∗ L j ˜ D τ k − h U i D ν ∗ L j ˜ D τ k + gU i D ν ∗ L j ˜ D τ k + gU i D ν ∗ L j ˜ D τ k − h U i D ν ∗ L j ˜ D τ k − f U i D ν ∗ L j ˜ D τ k , (15)where ˜ D τ is the diagonalizing matrix of the scalar 8 × U and V are the matrices that diagonalizethe chargino mass matrix M C so that U ∗ M C V − = diag ( m +˜ χ , m +˜ χ ) . (16) ψ j → ψ l + γ decay width The decay ψ j → ψ l + γ is induced by one-loop electric and magnetic transition dipolemoments, which arise from the diagrams of Fig.(1). In the dipole moment loop, the incoming6igure 1: The diagrams that allow decay of the ψ j into ψ l + γ via supersymmetric loopsinvolving the charginos and the staus where the photon is either emitted by the chargino(left) or by the stau (right) inside the loop. ψ j is replaced by a ψ l . For an incoming ψ j of momentum p and a resulting ψ l of momentum p (cid:48) , we define the amplitude (cid:104) ψ l ( p (cid:48) ) | J α | ψ j ( p ) (cid:105) = ¯ u ψ l ( p (cid:48) )Γ α u ψ j ( p ) (17)where Γ α ( q ) = F jl ( q ) iσ αβ q β m ψ j + m ψ l + F jl ( q ) σ αβ γ q β m ψ j + m ψ l + ..... (18)with q = p − p (cid:48) and where m f denotes the mass of the fermion f . The decay width of ψ j → ψ l + γ is given byΓ( ψ j → ψ l + γ ) = m ψ j π ( m ψ j + m ψ l ) (cid:32) − m ψ l m ψ j (cid:33) {| F jl (0) | + | F jl (0) | } (19)where the form factors F jl and F jl arise from the left and the right loops of Fig. (1) asfollows F jl (0) = F jl left + F jl right F jl (0) = F jl left + F jl right (20)The chargino contribution F jl left is given by F jl left = − (cid:88) i =1 8 (cid:88) k =1 (cid:20) ( m ψ j + m ψ l )64 π m ˜ χ i + { C Llik C R ∗ jik + C Rlik C L ∗ jik } F (cid:32) M τ k m χ i + (cid:33) + m ψ j ( m ψ j + m ψ l )192 π m χ i + { C Llik C L ∗ jik + C Rlik C R ∗ jik } F (cid:32) M τ k m χ i + (cid:33) (cid:21) (21)7here F ( x ) = 1( x − { x − x + 1 − x ln x } (22)and F ( x ) = 1( x − { x + 3 x − x + 1 − x ln x } (23)The right contribution F jl right is given by F jl right = (cid:88) i =1 8 (cid:88) k =1 (cid:20) ( m ψ j + m ψ l )64 π m ˜ χ i + { C Llik C R ∗ jik + C Rlik C L ∗ jik } F (cid:32) M τ k m χ i + (cid:33) + m ψ j ( m ψ j + m ψ l )192 π m χ i + { C Llik C L ∗ jik + C Rlik C R ∗ jik } F (cid:32) M τ k m χ i + (cid:33) (cid:21) (24)where F ( x ) = 1( x − { − x + 2 x ln x } (25)and F ( x ) = 1( x − {− x + 6 x − x − − x ln x } (26)The left contribution F jl left is given by F jl left = − (cid:88) i =1 8 (cid:88) k =1 ( m ψ j + m ψ l ) m ˜ χ i + π M τ k { C Ljik C R ∗ lik − C Rjik C L ∗ lik } F (cid:32) m χ i + M τ k (cid:33) (27)where F ( x ) = 12( x − (cid:26) − x + 3 + 2 ln x − x (cid:27) (28)The right contribution F jl right is given by F jl right = (cid:88) i =1 8 (cid:88) k =1 ( m ψ j + m ψ l ) m ˜ χ i + π M τ k { C Ljik C R ∗ lik − C Rjik C L ∗ lik } F (cid:32) m χ i + M τ k (cid:33) (29)where F ( x ) = 12( x − (cid:26) x + 2 x ln x − x (cid:27) (30)Now for the numerical analysis below we switch from the ψ i notation to the ν i notation.Here ν , ν , ν are the three neutrino mass eigenstates and we assume the mass hierarchyso that ν is heavier than ν and ν is heavier than ν . For the cosmic neutrinos we are8nterested in the decay of the ν to ν and ν . Thus the total decay width of ν is given byΓ total ( ν ) = Γ( ν → ν + γ ) + Γ( ν → ν + γ ). The lifetime of the tau neutrino is calculatedfrom the equation τ ( ν ) = (cid:126) Γ total ( ν ) (31)where the Γ total ( ν ) is in GeV and (cid:126) = 2 . × − GeV.Year. ν lifetime In this section we give a numerical estimate of the neutrino lifetime for the heaviest neutrino ν and investigate its dependence on the input parameters. In the analysis we ensure thatthe constraint of Σ i m ν i < .
85 eV from the Planck Satellite experiment [19] is satisfied andthat ∆ m and ∆ m lie in the 3 σ range of the neutrino oscillation experiment [20], i.e., inthe range of (2 . − . × − eV and (7 . − . × − eV respectively. In Table (1),we give a benchmark point where the constraints mentioned above are satisfied. The formfactors and the lifetime of the ν decay are calculated and given in Table (1).We now begin by exhibiting the dependence of the ν lifetime on the SU (2) gauginomass m . The chargino masses are sensitive to m and increasing m implies a larger av-erage chargino mass which affects the ν decay width and the lifetime. This is exhibitedin Fig. (2) for values of tan β = 30, 40, 50 while the values of the other input parametersare shown in the caption of Fig. (2). It is found that both the magnetic and the electrictransition dipole moments enter in the analysis. The magnetic transition dipole momentdepends on F jl while the electric transition dipole moment depends on F jl . Typically theelectric transition dipole moment dominates the decay even for moderate size CP phasessince F jl turns out to be much larger than F jl .In Fig. (3) we investigate the effect of the variation of m on ν lifetime, where m =˜ M τL = ˜ M E = ˜ M τ = ˜ M χ = ˜ M µL = ˜ M µ = ˜ M eL = ˜ M e (see Appendix). Three curves areshown on the figure, corresponding to tan β = 30, 40, 50, starting from the upper curve(tan β = 30) and going down. The analysis shows that the lifetime of ν increases as m increases. This is as expected since a larger m implies larger sfermion masses that enter inthe loop which gives a smaller decay width and a larger lifetime. It is seen that with valuesof the input parameters in reasonable ranges the lifetime can be as low as few times 10 yrs9eutrino Mass Eigenvalues (GeV) m ν = 5 . × − m ν = 8 . × − m ν = 1 . × − Process: F jl left (1 . × − ) exp( − . i ) F jl right (1 . × − ) exp(+0 . i ) ν → ν + γ F jl (0) (2 . × − ) exp(+0 . i ) F jl left (7 . × − ) exp(+2 . i ) F jl right (1 . × − ) exp(+2 . i ) F jl (0) (9 . × − ) exp(+2 . i )Decay Width 1 . × − GeVProcess: F jl left (2 . × − ) exp(+1 . i ) F jl right (2 . × − ) exp(+0 . i ) ν → ν + γ F jl (0) (2 . × − ) exp(+0 . i ) F jl left (1 . × − ) exp( − . i ) F jl right (2 . × − ) exp( − . i ) F jl (0) (1 . × − ) exp( − . i )Decay Width 3 . × − GeVLife time 1 . × YearsTable 1: Sample numerical values for the neutrino masses and the calculated form factorsand decay widths of the two processes ν → ν + γ and ν → ν + γ . The lifetime is also given.The analysis corresponds to the parameter set: | m | = 150, | µ | = 100, | f | = 1 . × − , | f (cid:48) | = 2 × − , | f (cid:48)(cid:48) | = 8 × − , | f | = | f (cid:48) | = | f (cid:48)(cid:48) | = 50, | f | = 8 . × − , | f (cid:48) | = 9 . × − , | f (cid:48)(cid:48) | = 4 × − , m N = 212, | A | = 600, m E = 260, m = 300, tan β = 50, χ m = 1 . χ µ = 0 . χ = 0 . χ (cid:48) = 0 . χ (cid:48)(cid:48) = 0 . χ = 1 . χ (cid:48) = 1 . χ (cid:48)(cid:48) = 1 . χ = 1 . χ (cid:48) = 0 . χ (cid:48)(cid:48) = 0 . χ A = 2 .
4. All masses are in GeV and phases in rad.just within the reach of improved CIB experiment.In Fig. (4) we investigate the effect on ν lifetime of the variation of χ which is the phaseof the coupling term f in the neutrino mass matrix. The analysis is done for two values ofits magnitude | f | (see the figure caption). The analysis shows that the ν lifetime dependssensitively on the phase χ and also on its magnitude. Fig. (4) exhibits several oscillationsin the lifetime as a function of χ .One possible origin of such oscillations could be constructive and destructive interferencebetween F jl left and F jl right , and between F jl left and F jl right . Such interference was noticed10igure 2: Variation of ν lifetime versus | m | for three values of tan β . Starting with the uppercurve, tan β = 30, 40, 50. Other parameters have the values | µ | = 100, | f | = 1 . × − , | f (cid:48) | = 2 × − , | f (cid:48)(cid:48) | = 8 × − , | f | = | f (cid:48) | = | f (cid:48)(cid:48) | = 35, | f | = 1 . × − , | f (cid:48) | = 5 . × − , | f (cid:48)(cid:48) | = 4 × − , m N = 200, | A | = 500, m E = 260, m = 300, χ m = 1 . χ µ = 0 . χ = 0 . χ (cid:48) = 0 . χ (cid:48)(cid:48) = 0 . χ = 1 . χ (cid:48) = 1 . χ (cid:48)(cid:48) = 1 . χ = 1 . χ (cid:48) = 0 . χ (cid:48)(cid:48) = 0 . χ A = 0 . F is muchlarger than F for this region of the parameter space, we focus on the F terms. Here onefinds that the F left is larger than F right and further each of the terms have phases of thesame sign. Thus this possibility does not appear to be the reason for large oscillations in ν lifetime. The above suggests that it is the interference in the F left terms themselvesthat is the origin of such rapid variation. This can come about because there are sixteendifferent contribution to F left each with their own phases and thus multiple constructiveand destructive interference can occur which is what Fig. (4) exhibits.11igure 3: Exhibition of the dependence of ν lifetime on m for three values of tan β . Startingwith the upper curve, tan β = 30, 40, 50. Other parameters have the values | µ | = 100, | f | = 1 . × − , | f (cid:48) | = 2 × − , | f (cid:48)(cid:48) | = 8 × − , | f | = | f (cid:48) | = | f (cid:48)(cid:48) | = 35, | f | = 1 . × − , | f (cid:48) | = 5 . × − , | f (cid:48)(cid:48) | = 4 × − , m N = 200, | A | = 500, m E = 260, | m | = 100, χ m = 1 . χ µ = 0 . χ = 0 . χ (cid:48) = 0 . χ (cid:48)(cid:48) = 0 . χ = 1 . χ (cid:48) = 1 . χ (cid:48)(cid:48) = 1 . χ = 1 . χ (cid:48) = 0 . χ (cid:48)(cid:48) = 0 . χ A = 0 .
4. All masses are in GeV and phases in rad. χ F jl left (1 . × − ) exp(+0 . i ) (3 . × − ) exp(+1 . i ) F jl right (5 . × − ) exp( − . i ) (1 . × − ) exp( − . i ) F jl (0) (1 . × − ) exp(+0 . i ) (2 . × − ) exp(+1 . i ) F jl left (2 . × − ) exp(+0 . i ) (1 . × − ) exp( − . i ) F jl right (2 . × − ) exp(+0 . i ) (1 . × − ) exp( − . i ) F jl (0) (2 . × − ) exp(+0 . i ) (1 . × − ) exp( − . i )Decay width 1 . × − GeV 7 . × − GeVTable 2: A list of the right and left contributions, the form factors and the decay width ofthe process ν → ν + γ for two values of χ , with | f | = 0.1 GeV.In Fig. (5) we exhibit the variation of the lifetime as a function of the trilinear coupling | A | for two values of | µ | . In the analysis we make the simple approximation A τ = A E = A µ = A e = A .Finally we discuss the effect of | f | on the tau neutrino lifetime. This analysis is exhibited12igure 4: Exhibition of the dependence of ν lifetime on the phase χ for two values of | f | . Solid curve is for | f | = 0.1 and dashed curve is for | f | = 0.05. Other parametershave the values | m | = | µ | = 100, | f | = 1 . × − , | f (cid:48) | = 2 × − , | f (cid:48)(cid:48) | = 8 × − , | f | = | f (cid:48) | = | f (cid:48)(cid:48) | = 35, | f (cid:48) | = 5 . × − , | f (cid:48)(cid:48) | = 4 × − , m N = 200, | A | = 500, m E = 260, m = 300, tan β = 40, χ m = 1 . χ µ = 0 . χ = 0 . χ (cid:48) = 0 . χ (cid:48)(cid:48) = 0 . χ = 1 . χ (cid:48) = 1 . χ (cid:48)(cid:48) = 1 . χ (cid:48) = 1 . χ (cid:48)(cid:48) = 0 . χ A = 0 .
4. All masses are in GeV and phases in rad.in Fig. (6) for two values of tan β (see figure caption). While f appears both in the sleptonand the neutrino mass matrix, the major effect of f arises via the variations in the neutrinomass matrix. In summary the analysis of Figs.(2) - (6) shows that the neutrino lifetime aslow as the current experimental lower limits can be obtained in models with a vector likegeneration. These lifetimes are over 30 orders of magnitude smaller than in the StandardModel and thus within the reach of improved experiment.13igure 5: Exhibition of the dependence of ν lifetime on | A | for two values of | µ | . Solidcurve is for | µ | = 150 and dashed curve is for | µ | = 100. Other parameters have the values | m | = 100, | f | = 1 . × − , | f (cid:48) | = 2 × − , | f (cid:48)(cid:48) | = 8 × − , | f | = | f (cid:48) | = | f (cid:48)(cid:48) | = 35, | f | = 1 . × − , | f (cid:48) | = 5 . × − , | f (cid:48)(cid:48) | = 4 × − , m N = 200, m E = 260, m = 350,tan β = 50, χ m = 1 . χ µ = 0 . χ = 0 . χ (cid:48) = 0 . χ (cid:48)(cid:48) = 0 . χ = 1 . χ (cid:48) = 1 . χ (cid:48)(cid:48) = 1 . χ = 1 . χ (cid:48) = 0 . χ (cid:48)(cid:48) = 0 . χ A = 0 .
4. All masses are in GeV and phases in rad.Figure 6: Exhibition of the dependence of the ν lifetime on | f | for two values of tan β .Solid curve is for tan β = 30 and dashed curve is for tan β = 40. Other parameters havethe values | m | = 100, | µ | = 100, | f (cid:48) | = 2 × − , | f (cid:48)(cid:48) | = 8 × − , | f | = | f (cid:48) | = | f (cid:48)(cid:48) | = 35, | f | = 1 . × − , | f (cid:48) | = 5 . × − , | f (cid:48)(cid:48) | = 4 × − , m N = 200, | A | = 500, m E = 260, m = 400, χ m = 1 . χ µ = 0 . χ = 0 . χ (cid:48) = 0 . χ (cid:48)(cid:48) = 0 . χ = 1 . χ (cid:48) = 1 . χ (cid:48)(cid:48) = 1 . χ = 1 . χ (cid:48) = 0 . χ (cid:48)(cid:48) = 0 . χ A = 0 .
4. All masses are in GeV and phases in rad.14
Conclusion
Lepton flavor changing processes provide an important window to new physics beyond theStandard Model. In this work we have analyzed the radiative decay of the neutrinos ν i → ν j γ in an extension of the MSSM with a vector like leptonic multiplet. Specifically we considermixing between the Standard Model generations of leptons with the mirror leptons in thevector multiplet. It is because of these mixing which are parametrized by f , f , f , f (cid:48) , f (cid:48) , f (cid:48) , f (cid:48)(cid:48) , f (cid:48)(cid:48) and f (cid:48)(cid:48) as defined in Eq.(6) that the neutrino can have a radiative decay. The com-putation of the neutrino decay is done in the supersymmetric sector where we compute thecontributions to the neutrino decay arising from diagrams with exchange of charginos andstaus in the loop with the chargino or the stau emitting the photon. The effects of CPviolation were also included in the analysis. In the presence of CP phases both the magneticand the electric transition dipole moments contribute to the neutrino lifetime. However, itis found that the electric transition dipole moment often dominates for moderate size CPphases in the region of the parameter space investigated. A numerical analysis shows thatthe neutrino lifetime can be smaller than the one predicted in the Standard Model by sev-eral orders of magnitude. Thus the Standard Model gives a lifetime for the decay of theheaviest neutrino ν so that τ SMν ∼ yrs for a ν with mass 50 meV. However, in theclass of models where the three generations of sleptons can mix with the vector like sleptongeneration one finds that the decay lifetime of ν can be as low as 10 years and thus muchsmaller than the Standard Model prediction. Thus improved experiments in the future givethe possibility of observation of such effects. In this Appendix we give further details of the interactions of the vector like multiplet. Thetotal lagrangian is constituted of L F and L D where L F = L L + L N . (32)Here 15 L L = (cid:18) v | f (cid:48) | | f | + | f (cid:48) | + | f (cid:48)(cid:48) | (cid:19) ˜ E R ˜ E ∗ R + (cid:18) v | f (cid:48) | | f | + | f (cid:48) | + | f (cid:48)(cid:48) | (cid:19) ˜ E L ˜ E ∗ L + (cid:18) v | f | | f | (cid:19) ˜ τ R ˜ τ ∗ R + (cid:18) v | f | | f | (cid:19) ˜ τ L ˜ τ ∗ L + (cid:18) v | h | | f (cid:48) | (cid:19) ˜ µ R ˜ µ ∗ R + (cid:18) v | h | | f (cid:48) | (cid:19) ˜ µ L ˜ µ ∗ L + (cid:18) v | h | | f (cid:48)(cid:48) | (cid:19) ˜ e R ˜ e ∗ R + (cid:18) v | h | | f (cid:48)(cid:48) | (cid:19) ˜ e L ˜ e ∗ L + (cid:26) − f µ ∗ v √ τ L ˜ τ ∗ R − h µ ∗ v √ µ L ˜ µ ∗ R − f (cid:48) µ ∗ v √ E L ˜ E ∗ R + (cid:18) f (cid:48) v f ∗ √ f v f ∗ √ (cid:19) ˜ E L ˜ τ ∗ L + (cid:18) f v f (cid:48)∗ √ f v f ∗ √ (cid:19) ˜ E R ˜ τ ∗ R + (cid:18) f (cid:48) v f (cid:48)∗ √ h v f (cid:48)∗ √ (cid:19) ˜ E L ˜ µ ∗ L + (cid:18) f (cid:48) v f (cid:48)∗ √ f (cid:48) v h ∗ √ (cid:19) ˜ E R ˜ µ ∗ R + (cid:18) f (cid:48)(cid:48)∗ v f (cid:48) √ f (cid:48)(cid:48) v h ∗ √ (cid:19) ˜ E L ˜ e ∗ L + (cid:18) f (cid:48)(cid:48) v f (cid:48)∗ √ f (cid:48)(cid:48)∗ v h ∗ √ (cid:19) ˜ E R ˜ e ∗ R + f (cid:48) f ∗ ˜ µ L ˜ τ ∗ L + f f (cid:48)∗ ˜ µ R ˜ τ ∗ R + f f (cid:48)(cid:48)∗ ˜ e R ˜ τ ∗ R + f (cid:48)(cid:48) f ∗ ˜ e L ˜ τ ∗ L + f (cid:48)(cid:48) f (cid:48)∗ ˜ e L ˜ µ ∗ L + f (cid:48) f (cid:48)(cid:48)∗ ˜ e R ˜ µ ∗ R − h µ ∗ v √ e L ˜ e ∗ R + H.c. (cid:27) (33)and − L N = (cid:18) v | f | | f | + | f (cid:48) | + | f (cid:48)(cid:48) | (cid:19) ˜ N R ˜ N ∗ R + (cid:18) v | f | | f | + | f (cid:48) | + | f (cid:48)(cid:48) | (cid:19) ˜ N L ˜ N ∗ L + (cid:18) v | f (cid:48) | | f | (cid:19) ˜ ν τR ˜ ν ∗ τR + (cid:18) v | f (cid:48) | | f | (cid:19) ˜ ν τL ˜ ν ∗ τL + (cid:18) v | h (cid:48) | | f (cid:48) | (cid:19) ˜ ν µL ˜ ν ∗ µL + (cid:18) v | h (cid:48) | | f (cid:48) | (cid:19) ˜ ν µR ˜ ν ∗ µR + (cid:18) v | h (cid:48) | | f (cid:48)(cid:48) | (cid:19) ˜ ν eL ˜ ν ∗ eL + (cid:18) v | h (cid:48) | | f (cid:48)(cid:48) | (cid:19) ˜ ν eR ˜ ν ∗ eR + (cid:26) − f µ ∗ v √ N L ˜ N ∗ R − f (cid:48) µ ∗ v √ ν τL ˜ ν ∗ τR − h (cid:48) µ ∗ v √ ν µL ˜ ν ∗ µR + (cid:18) f v f (cid:48)∗ √ − f v f ∗ √ (cid:19) ˜ N L ˜ ν ∗ τL + (cid:18) f v f ∗ √ − f (cid:48) v f ∗ √ (cid:19) ˜ N R ˜ ν ∗ τR (cid:18) h (cid:48) v f (cid:48)∗ √ − f (cid:48) v f ∗ √ (cid:19) ˜ N L ˜ ν ∗ µL + (cid:18) f (cid:48)(cid:48) v f ∗ √ − f (cid:48)(cid:48)∗ v h (cid:48) √ (cid:19) ˜ N R ˜ ν ∗ eR + (cid:18) h (cid:48)∗ v f (cid:48)(cid:48) √ − f (cid:48)(cid:48)∗ v f √ (cid:19) ˜ N L ˜ ν ∗ eL + (cid:18) f (cid:48) v f ∗ √ − h (cid:48) v f (cid:48)∗ √ (cid:19) ˜ N R ˜ ν ∗ µR + f (cid:48) f ∗ ˜ ν µL ˜ ν τ ∗ L + f f (cid:48)∗ ˜ ν µR ˜ ν ∗ τR − h (cid:48) µ ∗ v √ ν eL ˜ ν ∗ eR + f (cid:48)(cid:48) f ∗ ˜ ν eL ˜ ν ∗ τL + f f (cid:48)(cid:48)∗ ˜ ν eR ˜ ν ∗ τR + f (cid:48)(cid:48) f (cid:48)∗ ˜ ν eL ˜ ν ∗ µL + f (cid:48) f (cid:48)(cid:48)∗ ˜ ν eR ˜ ν ∗ µR + H.c. (cid:27) . (34)Similarly the mass terms arising from the D term are given by − L D = 12 m Z cos θ W cos 2 β { ˜ ν τL ˜ ν ∗ τL − ˜ τ L ˜ τ ∗ L + ˜ ν µL ˜ ν ∗ µL − ˜ µ L ˜ µ ∗ L +˜ ν eL ˜ ν ∗ eL − ˜ e L ˜ e ∗ L + ˜ E R ˜ E ∗ R − ˜ N R ˜ N ∗ R } + 12 m Z sin θ W cos 2 β { ˜ ν τL ˜ ν ∗ τL +˜ τ L ˜ τ ∗ L + ˜ ν µL ˜ ν ∗ µL + ˜ µ L ˜ µ ∗ L + ˜ ν eL ˜ ν ∗ eL + ˜ e L ˜ e ∗ L − ˜ E R ˜ E ∗ R − ˜ N R ˜ N ∗ R + 2 ˜ E L ˜ E ∗ L − τ R ˜ τ ∗ R − µ R ˜ µ ∗ R − e R ˜ e ∗ R } . (35)In addition we have the following set of soft breaking terms V soft = ˜ M τL ˜ ψ i ∗ τL ˜ ψ iτL + ˜ M χ ˜ χ ci ∗ ˜ χ ci + ˜ M µL ˜ ψ i ∗ µL ˜ ψ iµL + ˜ M eL ˜ ψ i ∗ eL ˜ ψ ieL + ˜ M ν τ ˜ ν c ∗ τL ˜ ν cτL + ˜ M ν µ ˜ ν c ∗ µL ˜ ν cµL + ˜ M ν e ˜ ν c ∗ eL ˜ ν ceL + ˜ M τ ˜ τ c ∗ L ˜ τ cL + ˜ M µ ˜ µ c ∗ L ˜ µ cL + ˜ M e ˜ e c ∗ L ˜ e cL + ˜ M E ˜ E ∗ L ˜ E L + ˜ M N ˜ N ∗ L ˜ N L + (cid:15) ij { f A τ H i ˜ ψ jτL ˜ τ cL − f (cid:48) A ν τ H i ˜ ψ jτL ˜ ν cτL + h A µ H i ˜ ψ jµL ˜ µ cL − h (cid:48) A ν µ H i ˜ ψ jµL ˜ ν cµL + h A e H i ˜ ψ jeL ˜ e cL − h (cid:48) A ν e H i ˜ ψ jeL ˜ ν ceL + f A N H i ˜ χ cj ˜ N L − f (cid:48) A E H i ˜ χ cj ˜ E L + H.c. } (36)From L F,D and by giving the neutral Higgs their vacuum expectation values in V soft we canproduce the mass squared matrix M τ in the basis (˜ τ L , ˜ E L , ˜ τ R , ˜ E R , ˜ µ L , ˜ µ R , ˜ e L , ˜ e R ). We labelthe matrix elements of these as ( M τ ) ij = M ij where M = ˜ M τL + v | f | | f | − m Z cos 2 β (cid:18) − sin θ W (cid:19) ,M = ˜ M E + v | f (cid:48) | | f | + | f (cid:48) | + | f (cid:48)(cid:48) | + m Z cos 2 β sin θ W , = ˜ M τ + v | f | | f | − m Z cos 2 β sin θ W ,M = ˜ M χ + v | f (cid:48) | | f | + | f (cid:48) | + | f (cid:48)(cid:48) | + m Z cos 2 β (cid:18) − sin θ W (cid:19) ,M = ˜ M µL + v | h | | f (cid:48) | − m Z cos 2 β (cid:18) − sin θ W (cid:19) ,M = ˜ M µ + v | h | | f (cid:48) | − m Z cos 2 β sin θ W ,M = ˜ M eL + v | h | | f (cid:48)(cid:48) | − m Z cos 2 β (cid:18) − sin θ W (cid:19) ,M = ˜ M e + v | h | | f (cid:48)(cid:48) | − m Z cos 2 β sin θ W ,M = M ∗ = v f (cid:48) f ∗ √ v f f ∗ √ ,M = M ∗ = f ∗ √ v A ∗ τ − µv ) ,M = M ∗ = 0 , M = M ∗ = f (cid:48) f ∗ ,M ∗ = M ∗ = 0 , M ∗ = M ∗ = f (cid:48)(cid:48) f ∗ , M ∗ = M ∗ = 0 , M = M ∗ = 0 ,M = M ∗ = f (cid:48)∗ √ v A ∗ E − µv ) , M = M ∗ = v f (cid:48) f (cid:48)∗ √ v h f ∗ √ ,M = M ∗ = 0 , M = M ∗ = v f (cid:48)(cid:48) f (cid:48)∗ √ v h f (cid:48)∗ √ , M = M ∗ = 0 ,M = M ∗ = v f f (cid:48)∗ √ v f f ∗ √ , M = M ∗ = 0 , M = M ∗ = f f (cid:48)∗ ,M = M ∗ = 0 , M = M ∗ = f f (cid:48)(cid:48)∗ , M = M ∗ = 0 , M = M ∗ = v f (cid:48) f (cid:48)∗ √ v f (cid:48) h ∗ √ ,M = M ∗ = 0 , M = M ∗ = v f (cid:48) f (cid:48)(cid:48)∗ √ v f (cid:48)(cid:48) h ∗ √ ,M = M ∗ = h ∗ √ v A ∗ µ − µv ) , M = M ∗ = f (cid:48)(cid:48) f (cid:48)∗ , M = M ∗ = 0 , M = M ∗ = 0 ,M = M ∗ = f (cid:48) f (cid:48)(cid:48)∗ , M = M ∗ = h ∗ √ v A ∗ e − µv ) (37)Here the terms M , M , M , M arise from soft breaking in the sector ˜ τ L , ˜ τ R , the terms M , M , M , M arise from soft breaking in the sector ˜ µ L , ˜ µ R , the terms M , M , M , M arise from soft breaking in the sector ˜ e L , ˜ e R and the terms M , M , M , M arise from softbreaking in the sector ˜ E L , ˜ E R . The other terms arise from mixing between the staus, smuons18nd the mirrors. We assume that all the masses are of the electroweak size so all the termsenter in the mass squared matrix. We diagonalize this hermitian mass squared matrix bythe unitary transformation ˜ D τ † M τ ˜ D τ = diag ( M τ , M τ , M τ , M τ , M τ , M τ , M τ , M τ ). For afurther clarification of the notation see [12]). Acknowledgments : One of the authors (PN) acknowledges the hospitality of KITP, Santa Bar-bara, where part of this work was done. This research was supported in part by the NationalScience Foundation under Grant Nos. PHY-0757959, PHY-0704067 and NSF PHY11-25915.
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