Charged lepton flavor violation and the origin of neutrino masses
UULB-TH/13-16
Charged lepton flavor violation and the origin of neutrino masses
Thomas Hambye ∗ Service de Physique Th´eoriqueUniversit´e Libre de BruxellesBoulevard du Triomphe, CP225, 1050 Brussels, Belgium
The neutrino oscillations imply that charged lepton flavor violation (CLFV) processes do exist. Even if theassociated rates are in general expected very suppressed, it turns out that this is not always necessarily the case.In the framework of the three basic seesaw models, we review the possibilities of having observable rates andthus, in this way, of distinguishing these possible neutrino mass origins.
I. INTRODUCTION
As abundantly discussed during this workshop, the searchfor charged lepton flavor violation is expected to know in thenext few years a real step forward. This concerns a long seriesof processes with a µ − e , τ − µ or τ − e transition. For µ − e processes an improvement of up to 4 to 6 orders of magnitudecould be expected, in particular for the µ → eee [1] decay and µ to e conversion in atomic nuclei [2]-[6]. Important improve-ments are also expected for the µ → e γ decay [7]. On the the-ory side such transitions could be induced by a large variety ofbeyond the SM physics models. This new physics could leadto observable rates even if the associated energy scale is insome cases as large as few thousands of TeV. At the momentit is not clear which type of new physics will manifest itself atsuch energies. In this talk we will review the possibility to getobservable rates from the physics associated to the neutrinomasses, the only beyond the Standard Model (SM) physicswhich has been established so far at the laboratory level (grav-itation put apart). It is well known that neutrino oscillationsguarantee non-vanishing CLFV but at a very suppressed level.Beyond this experimentally unreachable contribution we willreview the possibilities that the seesaw states, that are gener-ally expected to be at the origin of the neutrino masses, couldinduce rates that are not suppressed by the smallness of thesemasses. This offers an opportunity to probe better the yet un-known neutrino mass origin. II. THE 3 SEESAW FRAMEWORKS
If neutrinos oscillate, CLFV processes can be inducedthrough the following lepton chain: l i W −→ ν Li → ν L j W −→ l j , withthe W index referring to a vertex involving a W boson. Nomatter the neutrinos are of Dirac or Majorana type, the ν Li → ν L j transition requires two neutrino mass insertions. As a re-sult the CLFV rates are suppressed by 4-power of the neutrinomasses. For example for µ → e γ one gets Br ( µ → e γ ) = απ ∑ i = e , µ , τ | M ν ei M † ν iµ | m W ∼ − (1) ∗ [email protected]; Review talk given at the ”1 st Conference on ChargedLepton Flavor Violation”, May 6-8 2013, Lecce, Italy where the last equality results from assuming typically M ν ij ∼ . M ν the neutrino mass matrix. The correspond-ing rate is extremely small and hopeless for experimentalists.For the Majorana case, and unlike the Dirac case, one expectsnevertheless other contributions to the CLFV rates. Majoranamasses, as in the favorite seesaw way of generating neutrinomasses, require the existence of new states with their ownmasses, that can induce CLFV transitions in ways not sup-pressed by the neutrino masses.There are 3 ways of inducing ∆ L = N i (seesaw of type-I),of a scalar triplet ∆ L (type-II) and of fermion triplets Σ i (type-III). This is illustrated in Fig. 1. In the first and third caselepton number is broken from the coexistence of Majoranamasses for the heavy states and Yukawa interactions. The lat-ter couple the heavy states to the Standard Model scalar dou-blet H = ( H + , H ) T and to a lepton doublet L = ( ν α , l − α ) T L (cid:51) − m N i N i N ci − Y N ij ˜ φ † N i L j + h . c . (2) L (cid:51) − m Σ i Tr [ ΣΣ c ] − √ Y Σ ij ˜ φ † Σ i L j + h . c . (3)with Σ = (cid:18) Σ / √ Σ + Σ − − Σ / √ (cid:19) , (4)and ˜ H = i τ H ∗ . The neutrino mass matrix one gets in thesecases is M N ν = − v Y TN m N Y N and M Σν = − v Y T Σ m Σ Y Σ , with v =
246 GeV. As for the type-II case, lepton number is brokenfrom the fact that the scalar triplet couples to both a leptondoublet pair and a scalar doublet pair L (cid:51) − m ∆ Tr [ ∆ † L ∆ L ] − L T Y ∆ Ci τ ∆ L L + µ ∆ ˜ H T i τ ∆ L ˜ H (5)with ∆ L = (cid:18) δ + / √ δ ++ δ − δ + / √ (cid:19) , (6)giving M ∆ν = Y ∆ µ ∗ ∆ v m ∆ .Neutrino masses require electroweak symmetry breaking.In all seesaw models they involve two powers of the elec-troweak symmetry breaking scale, v = √ (cid:104) H (cid:105) (cid:39)
246 GeV.As can be seen in Fig. 1, in all cases the neutrino mass gen-eration proceeds in the same way. An effective interaction a r X i v : . [ h e p - ph ] D ec FIG. 1. The 3 basic seesaw diagrams which can induce naturallysmall neutrino masses. between 2 L and to 2 H is induced, which, once we replaceboth neutral scalar fields by v , leads to the neutrino massesabove. The LLHH effective interaction induced has a sin-gle possible form, the dimension 5 Weinberg operator form, L d = e f f = c d = αβ Λ ( L c α ˜ H ∗ )( ˜ H † L β ) , with M ν αβ = − v c d = αβ / Λ . The Λ scale can be identified as the overall scale where the see-saw states lie, m N , m ∆ or m Σ . Clearly, since the effective in-teraction induced is the same in all cases, and since any ofthe 3 seesaw models above could give any possible neutrinomass matrix (i.e. any c d = αβ coefficients for this interaction),the knowledge of the neutrino mass matrix is not sufficientto be able to distinguish between the 3 seesaw models. Onewould need additional information from other kinds of exper-iments. CLFV could provide this information, provided a se-ries of conditions is satisfied. III. LARGE CLFV RATES IN SEESAW MODELS
To illustrate the fact that seesaw states can induce CLFVrates that are not suppressed by the smallness of the neutrinomasses, the type-II seesaw case is particularly clear. The see-saw scalar triplet does not induce only the lepton number vi-olating dimension 5 effective interaction. It also induces alepton number conserving dimension 6 effective interactionwhich induces CLFV. From the exchange of a scalar tripletand 2 Y ∆ Yukawa couplings, a four lepton effective interactionis induced, L d = e f f (cid:51) c d = αβδγ ( L β γ µ L δ )( L α γ µ L γ ) (7)with c d = αβδγ = ( / m ∆ ) Y † ∆ αβ Y ∆ δγ . Clearly such a process can in-duce a µ → eee or τ → l decay at tree level and l → l (cid:48) γ or µ → e conversion at one loop. In this way we get in particular[8] Br ( µ → eee ) = | c d = µeee | / G F = | Y ∆ eµ | | Y ∆ ee | / ( m ∆ G F ) . (8)The CLFV rates in this case are quadratic in the dim-6 coeffi-cient rather than quartic in the dim-5 one (as in Eq. (1), i.e. areproportional to ( Y ∆ / m ∆ ) rather than to ( Y ∆ / m ∆ ) · ( µ ∆ / m ∆ ) ).The dim-6 contribution could be many orders of magnitudelarger than the dim-5 one, especially if m ∆ is low. For ex-ample with m ∆ = µ ∆ / m ∆ ∼ Y ∆ ∼ − (so that m ν ∼ . Br ( µ → eee ) ∼ − . But µ ∆ / m ∆ hasno reason to be similar to Y ∆ . If furthermore Y ∆ >> µ ∆ / m ∆ ,one gets further enhanced rates. For example, still with m ∆ = Y ∆ ∼ − . (and µ ∆ / m ∆ ∼ − so that m ν ∼ . µ ∆ / m ∆ parameter, one canget large rates. Of course to have a so low seesaw scale goesagainst the usual seesaw expectation that the smallness of theneutrino masses is due mostly to a very large seesaw scale,but nothing forbids such a possibility. To consider a small µ ∆ parameter is technically natural because it is protected by asymmetry. In the µ ∆ → L conserving setup doesn’t show up in a so straightforwardway. From the exchange of a right-handed neutrino or fermiontriplet and 2 Yukawa interactions, and without N or Σ Ma-jorana mass insertion in their propagator, the lepton numberconserving dimension 6 effective interactions that are inducedare [9, 10] L d = e f f = c d = αβ L α ˜ φ i ∂ / ( ˜ φ † L β ) (9) L d = e f f = c d = αβ L α τ a ˜ φ iD / ( ˜ φ † τ a L β ) (10)with c d = αβ = ( Y † N M N Y N ) αβ and c d = αβ = ( Y † Σ M Σ Y Σ ) αβ respec-tively. Clearly such effective interactions also induce CLFVrates that have no reason to be suppressed in the same waythan the dimension 5 contribution. The former contribution isquadratic in the L conserving dim-6 coefficients, whereas thelatter one is quartic in the L violating dim-5 one (i.e. involve8 rather than 4 Yukawa couplings). As a result, for examplein the type-I scenario, we get Br ( µ → e γ ) = α em π ∑ i | Y N ie Y † N iµ | m N i v (11)For m N i close to the GUT scale this still results in very sup-pressed rates. However for low values of m N i the rates weget are much larger. Using the typical seesaw expectation Y N ∼ m ν m N / v , with m N the typical seesaw scale, for m N ∼
100 GeV one gets Br ( µ → e γ ) ∼ − . Therefore, as for thetype-II case above, low scale seesaw generically gives muchlarger rates than in Eq. (11), but still too small to be reachedexperimentally by the next generation of experiments. How-ever the typical seesaw expectation used here to relate Yukawacouplings to neutrino masses has no reason to be necessarilyvalid. The c d = αβ coefficients involve the Yukawa couplings in a YY † L conserving combination, whereas the c d = coefficientsinvolve them in a Y T Y L violating one. Since both combina-tions differ on the basis of a symmetry they have no reason tobe related in such a simple way. In particular lepton number isnot necessarily broken as soon as a fermion seesaw states havemasses and Yukawa interactions. It turns out that the fermionseesaw states can have masses and Yukawa interactions with-out breaking lepton number, see e.g. Refs. [10]-[22]. The mostsimple example consists in considering only 2 right-handedneutrinos (or fermion triplets), and to assign a U ( ) L leptonnumber symmetry under which N has L = N has L = − U ( ) L symmetry allows onlya Majorana mass term coupling both N (cid:48) s , ∝ m N N N c + h . c . ,and Yukawa couplings only for N , ∝ − Y N j ˜ φ † N L j . As a re-sult N and N mixes maximally to form two degenerate mass m N (cid:72) GeV (cid:76) Br (cid:72) Μ (cid:174) e Γ (cid:76) (cid:144) Br (cid:72) Μ (cid:174) eee (cid:76) Br (cid:72) Τ (cid:174) e Γ (cid:76) (cid:144) Br (cid:72) Τ (cid:174) eee (cid:76) Br (cid:72) Τ (cid:174) ΜΓ (cid:76) (cid:144) Br (cid:72) Τ (cid:174) ΜΜΜ (cid:76)
FIG. 2. R l → l (cid:48) γ µ → l (cid:48) l (cid:48) l (cid:48) = Br ( l → l (cid:48) γ ) / Br ( µ → l (cid:48) l (cid:48) l (cid:48) ) as a function of m N ,from Refs. [28, 29]. eigenstates with mass m N , that both have Yukawa couplings.In this symmetry limit, since lepton number is conserved, neu-trinos are massless, no matter how large the Y N j couplings are,so that dim-6 coefficients, and hence CLFV rates, can be largeindependently of the size of the neutrino masses. This simplyrequires large Yukawa couplings for N and a relatively low m N scale. In order that neutrino masses are induced, the U ( ) L symmetry must be broken by small parameters, Yukawa cou-plings for N and diagonal right-handed mass terms (in N N c and N N c , which gives a N mass splitting), leading to neu-trino masses proportional to these L breaking parameters. Ofcourse such a scenario is not anymore the minimal seesaw inthe sense that it requires to assume an additional approximate U ( ) L symmetry setup but it doesn’t require any additionalstates or interactions beyond the ones of the minimal seesawmodel. We will now explain how such a possibility could besingled out from CLFV processes. Note that these setups pre-dict a quasi-degenerate spectrum of right-handed neutrinos,since the mass splitting breaks L . This will play an importantrole below. IV. TYPE-I SEESAW CLFV PREDICTIONS
In the type-I seesaw model all CLFV processes are nec-essarily induced at the loop level because flavor violation in Note that these scenarios in some cases can be minimal in the sense of min-imal flavor violation, i.e. from the knowledge of the full flavor structure ofthe neutrino mass matrix, they allow to know the full flavor structure of thedim-6 coefficients [21]. This means that they predict the full flavor struc-ture of the CLFV rates. Together with the determination of the seesaw scale m N , which can be done as explained below, this allows a full reconstructionof the seesaw lagrangian. this model occurs only at the level of the neutral leptons, re-quiring at least one internal W boson inside the loop diagram.The µ → e γ rate has been calculated already very long ago,see Refs. [23]-[27]. For the µ → eee rate, see Ref. [15]. The µ → e conversion rate has been calculated by a series of ref-erences with different results. This recently motivated us toredo carefully this calculation, see Ref. [29] and Refs. therein.As pointed out in Refs. [28] and [29], to test the possible see-saw origin of one or several CLFV processes, that could beobserved in the future, the most promising way is to considerratios of two CLFV processes involving a same l → l (cid:48) transi-tion. This statement is supported by the following considera-tions. First, all process rates involving a same flavor transitionhave the same general form T l → l (cid:48) = ∑ N i | Y N il (cid:48) Y † N il | m N i · [ c + c (cid:48) log ( m N i / m W )] (12)where c and c (cid:48) are dimension four numerical factors that de-pend on m l , m W , Z , h and on the process considered (neglectingthe mass of l (cid:48) ). T l → l (cid:48) holds here for any CLFV processes, forexample Br ( µ → e γ ) , Br ( µ → eee ) or µ → e conversion rate inatomic nuclei, R Nµ → e (defined in Ref. [29] for example). Sec-ond, as we have seen, the setups that can lead in a naturalway to observable CLFV rates are based on an approximate L symmetry setup that implies a quasi-degenerate spectrumof right-handed neutrinos. As a result, in Eq. (12), the right-handed neutrino mass splittings can be neglected and the de-pendence of the rates in the Yukawa couplings factorizes out.Therefore, in the ratio of two same flavor transition rates, theYukawa coupling dependence cancels out and we are left witha function which depends only on a single unknown parame-ter, the overall m N scale T ( ) l → l (cid:48) T ( ) l → l (cid:48) = ( c + c (cid:48) log ( m N i / m W )) ( c + c (cid:48) log ( m N i / m W )) = f ct ( m N ) (13)This allows several possibilities of tests we will now discuss.In Figs. 2 and 3 are plotted the various ratios one gets forthe various CLFV processes. First of all, for the Br ( µ → e γ ) to Br ( µ → eee ) ratio, one observes [28, 29] a monotonous func-tion that is always larger than ∼ . m N that dogive an observable rate), which means that the measurementof this ratio would basically allow for a determination of m N ,or for an exclusion of the scenario if a ratio smaller than ∼ . R Nµ → e conversion rates (which depend crucially onthe nuclei considered) turn out to vanish for a particular valueof m N . Consequently the ratios are not monotonous functionsof m N . The value of m N which gives a vanishing R Nµ → e de-pends on the nuclei considered. This illustrates how impor-tant it would be to search for µ → e conversion, not only withone nuclei, but with several of them. The degeneracy in m N that a single ratio can display could be lifted up by the mea-surement of another ratio. The reason why these rates vanish (cid:45) m N (cid:64) GeV (cid:68) R (cid:208) (cid:174) eee (cid:208) (cid:174) e AuPbTiAl
FIG. 3. R µ → eµ → eee = R Nµ → e / Br ( µ → eee ) as a function of the right-handed neutrino mass scale m N , for µ → e conversion in various nu-clei, from Ref. [29]. (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) m N (cid:72) GeV (cid:76) (cid:200) (cid:83) N i k Y N i Μ Y N i e (cid:42) (cid:200) Μ (cid:174) e ΓΜ (cid:174) e e e
Μ (cid:174) e (cid:72) Al (cid:76) Μ (cid:174) e (cid:72) Ti (cid:76) FIG. 4. From Ref. [29], present bounds and future sensitivity on | ∑ ki Y Σ iµ Y Σ ∗ ie | for scenarios characterized by one right-handed neutrinomass scale. The solid lines are obtained from present experimentalupper bounds: from R Tiµ → e < . · − [30] and Br ( µ → e γ ) < . · − [31], Br ( µ → eee ) < − [32]. The dashed lines are obtainedfrom the expected experimental sensitivities: from R Tiµ → e < ∼ − [2, 3], R Alµ → e < ∼ − [2]-[6] and Br ( µ → e γ ) < − [7], Br ( µ → eee ) < − [1]. for a particular value of m N can be traced back to the factthat the up quark and down quark contributions have differentsigns in the amplitude, depending on their different chargesand weak isospins (and display different m N dependences),see Ref. [29].Fig. 4 shows the lower bounds resulting for the Yukawa couplings, if the various rates are required to be large enoughto be observed in planned experiments. It also shows the upperbounds which hold today on these quantities from the non-observation of these processes. This figure illustrates wellthe impact of future µ → e conversion measurements/bounds,as they will become increasingly dominant in exploring fla-vor physics in the µ − e charged lepton sector. Values of theYukawa couplings as low as 10 − , 10 − and 10 − could beprobed, for m N =
100 TeV, 1 TeV and 100 GeV, respectively,with Titanium experiments being the most sensitive. If theYukawa couplings are for example of order unity, the boundsof Fig. 4 can be rephrased as upper bounds on the m N scale: m N < ∼ · (cid:16) − R Tiµ → e (cid:17) · a / , m N < ∼
300 TeV · (cid:16) − R Alµ → e (cid:17) · a / , m N < ∼
100 TeV · (cid:16) − Br ( µ → e γ ) (cid:17) · a / , m N < ∼
300 TeV · (cid:16) − Br ( µ → eee ) (cid:17) · a / . with a = | ∑ N i Y N ie Y † N iµ | . Overall, this exercise shows that futureexperiments may in principle probe the type-I seesaw modelbeyond the ∼ µ → e conversion fu-ture experiments could become with time the most sensitiveones.Finally in Fig. 2 can also be found the predictions the type-Iseesaw scenario give for the ratio of 2 same flavor transitionprocesses involving the τ lepton, τ → µ and τ → e transitions.The current and future expected limits on such processes donot allow to access m N scale as large as from the µ → e pro-cesses, but neutrino data still allows that the τ processes wouldbe boosted with respect to the µ → e ones (depending on themass hierarchy considered). For specific Yukawa configura-tions of this kind, compare for example the minimal modelsconsidered in Ref. [21, 28]. V. TYPE-III SEESAW CLFV PREDICTIONS
There is a crucial difference between the type-I and type-III seesaw models for what concerns CLFV processes. Whilein the type-I case there is flavor mixing only at the level ofthe neutral leptons, for the type-III case there is flavor mix-ing directly at the level of the charged leptons. For instancea µ − - e − transition can proceed directly through charged lep-ton/charged Σ mixing, i.e. through the µ − Y −→ Σ − Y −→ e − chain,with the Y index referring to a vertex involving a Yukawa cou-pling (i.e. the insertion of a SM scalar boson or its vev). Asa result, if for the type-I case all processes necessarily occursat the loop level, for the type-III case the l → l and µ → e conversion process in atomic nuclei proceed at tree level (justattach for example a Z boson on the fermionic chain aboveand couple it on the other side to 2 leptons or 2 quarks). Only µ → e γ still has to proceed at loop level because the QEDcoupling remains flavor diagonal in the charged fermion masseigenstate basis (unlike Z couplings, see Refs. [10, 33]). Thecalculation of all corresponding rates has recently been per-formed in Refs. [33]. Obviously the tree level processes do notinduce logarithmic terms and have the simple general form T l → l (cid:48) = ∑ Σ i | Y Σ il (cid:48) Y † Σ il | m Σ i · c (14)with c a constant which depends only on m l and m W , Z (ne-glecting m l (cid:48) ). As a result the ratio of a same flavor transitionprocess are predicted to a fixed value! The measurement ofsuch a ratio could then easily rule-out the type-III scenario aspossible explanation of these processes if another value is ob-tained. Alternatively it would provide a strong indication forit if the right ratio was observed. As for the µ → e γ rate, itturns out that it also doesn’t display any logarithmic term andhas the general form of Eq. (14), leading to fixed ratios too,when compared with the other rates. Alltogether we get theneat predictions [33] Br ( µ → e γ ) = . · − · Br ( µ → eee )= . · − · R Tiµ → e Larger µ → eee and µ → e conversion rates, as compared to the µ → e γ rate, are characteristic of this model. In the majorityof other beyond the standard model scenarios that can lead toobservable CLFV rates, one finds Br ( µ → eee ) < Br ( µ → e γ ) .As for the sensitivity to the overall seesaw scale m Σ , it iseven more impressive than for the type-I seesaw case, in par-ticular for the processes which proceed at tree level, m Σ < ∼ · (cid:16) − R Tiµ → e (cid:17) · a / , m Σ < ∼ · (cid:16) − R Alµ → e (cid:17) · a / , m Σ < ∼
100 TeV · (cid:16) − Br ( µ → e γ ) (cid:17) · a / , m Σ < ∼ · (cid:16) − Br ( µ → eee ) (cid:17) · a / . with a = ∑ Σ i | Y Σ ie Y † Σ iµ | .Finally for the τ decay processes one gets the predictions Br ( τ → µ γ ) = . · − · Br ( τ → µµµ )= . · − · Br ( τ → µ − e − e + ) Br ( τ → e γ ) = . · − · Br ( τ → eee )= . · − · Br ( τ → e − µ + µ − ) VI. TYPE-II SEESAW CLFV PREDICTIONS
Each seesaw model presents a different pattern for whatconcerns the various CLFV process. As we saw above, for thetype-I model all processes are loop processes, and for the type-III case they are tree level processes, except l → l (cid:48) γ . The type-II model presents an intermediate situation. On the one hand TiAuPbAl m (cid:68) (cid:64) GeV (cid:68) R Μ (cid:174) e Γ Μ (cid:174) e (cid:72) Z (cid:76) FIG. 5. R Nµ → e / Br ( µ → e γ ) for various nuclei, as a function of m ∆ . from the exchange of a scalar triplet and 2 Y ∆ interactions onegets the 4 lepton interactions of Eq. (7), hence l → l at treelevel, Eq. (8). On the other hand the other processes can onlybe induced at one loop from the exchange of a scalar tripletbetween 4 leptons too, but necessarily contracting 2 of the lep-tons in a loop. As a result, the l → l rates are expected to becomparatively the largest one. Compared to the fermion see-saw cases, another important difference is that, by taking theratio of two same flavor transition rates, one does not alwaysget a function which depends only on the seesaw state mass.For instance the l → l rates do not involve the same Yukawacouplings than the other processes. For example Br ( µ → eee ) involves the product of Y ∆ eµ and Y † ∆ ee , Eq. (8), whereas µ → e γ and µ → e conversion processes involve a sum over l = e , µ , τ of Y ∆ lµ · Y † ∆ le . This means that a ratio involving µ → eee andanother processes depends on the Yukawa coupling configu-ration considered, but still, we would like to point out herethat any ratio involving any processes but µ → eee dependsonly on m ∆ (for m ∆ >>> m e , µ τ ). Taking the type-II µ → e γ rate in e.g. Refs. [10, 34, 35] and µ → e conversion rate ine.g. Ref. [35], in Fig. 5 we give the values of these ratios.These ratios are monotonous functions of m ∆ . In the sameway as for the type-I case, the measurement of any of theseratios would therefore gives us the value of m ∆ if these pro-cesses are due to the type-II seesaw interactions (or excludethis model as their origins). The observation of µ → e γ shouldnevertheless comes first. This can be seen from the sensitivity We thank M. Dhen for having produced this plot and for discussions aboutit. on m ∆ we get for the various processes m ∆ < · (cid:113) | Y ∆ µe || Y ∆ ee | · (cid:16) − Br ( µ → eee ) (cid:17) m ∆ <
15 TeV · (cid:113) | Y ∆ τ µ || Y ∆ µµ | · (cid:16) − Br ( τ → µµµ ) (cid:17) m ∆ < · (cid:113) | Y ∆ τ e || Y ∆ ee | · (cid:16) − Br ( τ → eee ) (cid:17) and m ∆ <
70 TeV · | ∑ l Y ∆ µl Y † ∆ le | · (cid:16) − Br ( µ → e γ ) (cid:17) m ∆ <
600 TeV · | ∑ l Y ∆ µl Y † ∆ le | · (cid:16) − R Tiµ → e (cid:17) . VII. SUMMARY
In conclusion neutrino oscillations guarantee that CLFVprocesses do exist. If the neutrino masses are of the Dirac typethe effect is nevertheless extremely suppressed ( Br ( µ → e γ ) ∼ − ). If they are of the Majorana type, and if the seesawstates at their origin lies close to the GUT scale, they are alsovery suppressed. If instead the seesaw states lie around the100 GeV-1 TeV scale they are expected hugely larger but stillgenerically too small for the next generation of CLFV exper-iments ( Br ( µ → e γ ) ∼ − / − ). However without adding any new seesaw states or interactions, it turns out that thereexist special configurations of the couplings that can lead torates saturating the present CLFV upper bounds. These aretechnically natural because can be justified on the basis ofa symmetry (approximate lepton number conservation). Forsuch cases each seesaw setup comes with a quite differentCLFV pattern. In particular for the type-III case the ratio oftwo same flavor transition processes is totally predicted (nomatter the values of the fermion triplet masses) and offers avery clear possibility of test. Similarly in the type-I case, forthe case where the right-handed neutrino would have a quasi-degenerate mass spectrum (which is precisely what happensin the setups that lead to observable rates), these ratios dependonly on the common seesaw mass scale. This also offers clearpossibility of tests. For the type-II case, the µ → eee processshould clearly be the first observed rate in the future, whereasfor the other ones µ → e conversion will become gradually themost sensitive probe. ACKNOWLEDGEMENT
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