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Three generations mixing and τ decay puzzle

arXiv:hep-ph/9207263v1 27 Jul 1992IFT-P.023IFUSP/P-997July 1992Three generations mixing and τ decay puzzleC. O. Escobara, O.L.G.

Peresb, V. PleitezbandR. Zukanovich Funchalaa Instituto de F´ısica da Universidade de S˜ao Paulo01498-970 C.P.

20516–S˜ao Paulo, SPBrazilbInstituto de F´ısica Te´oricaUniversidade Estadual PaulistaRua Pamplona, 14501405-900–S˜ao Paulo, SPBrazilWe consider the possibility that the τ decay puzzle is a consequence of theKobayashi-Maskawa mixing in the leptonic sector.PACS numbers:14.60-z, 12.15.Ff, 13.35+sTypeset Using REVTEX1

The physics of the τ lepton will provide in the near future evidence concerning thequestion if this lepton, with its neutrino partner, is a sequential lepton or not. So far, aswas stressed in Ref.

[1], all experiments are internally consistent with the Standard Model.Notwithstanding, it is well known that the accuracy of the τ data are still poor and it shouldbe possible that new physics will come up when the proposed τ-Charm factory gives newand more accurate data about τ decays and properties [2].For example, new data could confirm the so called “τ decay puzzle” [3], which is a dif-ference between the measured world average [4] τ lifetime, ττ and the theoretically expectedvalue within two standard deviations. Explicitly [5]τ expτ−τ thτ = (0.16 ± 0.09) × 10−13 s.(1)although this discrepancy is not yet statistically significant, it can be translated into discrep-ancies in particular branching ratios, implying that the expected leptonic branching ratiosare about 2.3σ higher than the average measurements [3].

The branching ratio conflictssuggest that a shift in ττ and/or τ mass, mτ, should occur when more precise measurementsbecome available. In fact, preliminary results from BES in Beijing point to a down shift of2σ in mτ in relation to the world average value [6].

Notwithstanding, in order to solve thediscrepancy a down shift of 6.4σ is required as pointed out in Ref. [3].Other possibility is that no such a shift on ττ or mτ is needed but the problem withthe branching ratios would still exist.

This would imply a new physics. The relationship ofEq.

(1) with a possible deviation from universality can be see as follows. It is well knownthat the decay diagram of the muon into electron being similar to that of the τ decay intoelectron or muon.

For example, implies the following relationship GτGµ!2=τµττ mµmτ5B.R. (τ →e¯νν),(2)where Gτ and Gµ are the coupling constant of the τ and µ to the charged weak currentrespectively.

Assuming eµ universality an average leptonic branching ratio can be definedas in Ref. [6]:2

⟨Bτl ⟩= Bτe +Bτµ0.9732= 17.88 ± 0.26 %. (3)where the factor 0.973 is due to the mass of the muon.Using the world average value for the branching ratio Eq.

(3) in Eq. (2) it follows thatGτ/Gµ = 0.975 ± 0.010 using non-LEP and LEP data or Gτ/Gµ = 0.985 ± 0.0009 usingthe BEPC value for the τ mass [6].

Of course in the Standard Model universality impliesGτ = Gµ.There have been some speculations about this possible deviation from the StandardModel. Some examples, usually discussed in the literature, of the new physics needed tosolve the puzzle are:1. the introduction of new gauge bosons [7],2. four-generation leptons, mixing mainly with the third generation [8,9],3. scalar particles which interfere destructively with the W-exchange amplitude in the τdecay [10].Of course, each of the above possibilities, and their variants, have their own difficulties.

Thefirst one implies a drastic modification of the Standard Model; the second implies also amodification of the quark sector, necessary in order to avoid anomalies; finally, the third oneneeds a larger Higgs sector (two triplets) in the Standard Model.On the other hand, the simplest solution has not been even mentioned in the literature.It is possible, if the neutrinos are massive, that a mixing similar to the Kobayashi-Maskawaone [11] occurs with three lepton generations. Here we will consider the analysis of theexperimental data with five free parameters, three angles and two neutrino mass differencesfrom Ref.

[12]. The effects of the Kobayashi-Maskawa mixing in the leptonic sector wereconsidered in Ref.

[13]. In particular the effects of such a mixing for the case of leptonicdecays of the τ-lepton were explicitly considered in Ref.

[14]. Here we will consider thisscenario in the context of a possible “τ decay puzzle”.3

The Kobayashi-Maskawa lepton mixing matrix isν′eν′µν′τ=Vee Veµ VeτVµe Vµµ VµτVτe Vτµ Vττνeνµντ. (4)The unprimed fields are mass eigenstates and we will not consider the hierarchical mix-ing [15].

In Ref. [12] the Maiani parameterization [16] of the mixing matrix was chosen andtwo solutions were found for the oscillation parameters which imply the following ranges forthe diagonal matrix elements (Vee, Vµµ, Vττ)solution a) 1.00—0.98, 1.00—0.99, 1.00—0.98solution b) 1.00—0.97, 1.00—0.98, 1.00—0.98In the theoretical predictions of the τ partial widths we will neglect neutrino masses.

Infact the neutrino mass is only important if mντ > 100 MeV for τ →eντ ¯νe-decay, or ifmντ > 50 MeV for τ →µντ ¯νµ decay [14] while the present current limit is mττ < 35MeV [4].We use the following notation: ¯Bτi = ¯Γτi /Γτtot, ¯Γτi being the τ partial decay width into the icharged particle (e−, µ−, π−, K−) considering the mixing and Γτtot the total width.The widths with this kind of mixing are given by¯Γτl = |Vττ|2|Vll|2|Vµµ|2|Vee|2Γτl ,(5)and¯Γτh =|Vττ|2|Vµµ|2|Vee|2Γτh,(6)where l = e, µ and h = π, K and Γτl,h are the τ partial widths without mixing which haveappeared in the literature [3]. In Eqs.

(5) and (6) the denominator comes from the definitionof the Gµ constant in the µ decay. Numerically we will consider the hadronic partial decaywidth as Γτh = Γτπ + ΓτK.Let us start by the theoretical result for the partial width from Refs.

[3,17], which includesradiative corrections, using the current data from Ref. [4] and presented with the rangeimplied by the solution a) above:4

¯Γ(τ →e−ντ ¯νe) = (3.95+0.03−0.04—4.19+0.03−0.04) × 10−13GeV,(7)¯Γ(τ →µ−ντ ¯νµ) = (3.84+0.03−0.04—4.17+0.03−0.04) × 10−13GeV,(8)¯Γ(τ →πντ) = (2.45 ± 0.06—2.71 ± 0.07) × 10−13GeV,(9)¯Γ(τ →Kντ) = (1.60 ± 0.04—1.76 ± 0.04) × 10−14GeV,(10)¯Γ(τ →hντ) = (2.61 ± 0.06—2.88 ± 0.07) × 10−13GeV. (11)With this mixing we have instead of Eq.

(2) GτGµ!2= |Vµµ|2|Vττ|2τµττ mµmτ5B.R. (τ →e¯νν),(12)and using the current τ lifetime and mass [6]mτ = 1776.9 ± 0.4 ± 0.3, MeVττ = (3.00 ± 0.05) × 10−13s.

(13) GτGµ!2= |Vµµ|2|Vττ|2 ×0.941 ± 0.024 (world average)0.967 ± 0.018 (BEPC). (14)With the values of the diagonal matrix elements given in solution a) we obtain |Vµµ|2/|Vee|2 =0.98—1.04.

We see that there is consistency with the value Gτ/Gµ = 0.975 ± 0.010 [6].Eqs. (6)-(10) are also compatible with the respective experimental branching ratios.

Similarresults arise using the matrix elements given in (b).We can also verify that ratios of partial widths are consistent with leptonic mixing¯Γτµ¯Γτe= |Vµµ|2|Vee|2ΓτµΓτe= 0.95 ± 0.01—1.01+0.01−0.02 ,(15)¯Γτh¯Γτµ=1|Vµµ|2ΓτµΓτe= 0.68 ± 0.02—0.69 ± 0.02,(16)¯Γτh¯Γτe=1|Vee|2ΓτhΓτe= 0.66 ± 0.02—0.69 ± 0.02. (17)As before, we show in Eqs.

(15)-(17) the range of the ratios of the partial widths takinginto account the range of the matrix elements. Again it is possible to verify that there isconsistency with experimental data.5

We have shown in this comment that a possible deviation from µ −τ universality ifconfirmed by future experiments is sufficiently small to be accounted by three generationsmixing in the leptonic sector.ACKNOWLEDGMENTSWe are very gratefull to Funda¸c˜ao de Amparo `a Pesquisa do Estado de S˜ao Paulo(FAPESP) (R.Z. ), Coordenadoria de Aperfei¸coamento de Pessoal N´ıvel Superior (CAPES)(O.L.G.P.) for full financial support and Conselho Nacional de Desenvolvimento Cient´ıficoe Tecnol´ogico (CNPq) (V.P.) for partial financial support.6

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