Charged tau leptons emerging in a long baseline experiment with a muon
storage ring and a far-away detector will positively establish neutrino
oscillations. We study the conversion of $\nu_\mu$ ($\bar{\nu}_\mu$) and of
$\bar{\nu}_e$ ($\nu_e$) to $\nu_\tau$ or $\bar{\nu}_\tau$ for neutrinos from a
20 GeV muon storage ring, within the strong mixing scheme and on the basis of
the squared mass differences which are compatible with all reported neutrino
anomalies, including the LSND data. In contrast to other solutions which ignore
the Los Alamos anomaly, we find charged tau production rates which should be
measurable in a realistic set up. As a consequence, determining the complete
mass spectrum of neutrinos as well as all three mixing angles seems within
reach. Matter effects are discussed thoroughly but are found to be small in
this situation.Comment: 11 pages, 5 postscript figures (eps

Barenboim, G.

Scheck, F.

Physics Letters B

English

arXiv

Charged tau leptons emerging in a long baseline experiment with a muon storage ring and a far-away detector will positively establish neutrino oscillations. We study the conversion of v(mu) ((v) over bar(mu)) and of (v) over bar (v(e)) to v(tau) or (v) over bar(tau) for neutrinos from st 20 GeV muon storage ring, within the strong mixing scheme and on the basis of the squared mass differences which are compatible with all reported neutrino anomalies, including the LSND data. In contrast to other solutions which ignore the Los Alamos anomaly, we find charged tau production rates which should be measurable in a realistic set up. As a consequence, determining the complete mass spectrum of neutrinos as well as all three mixing angles seems within reach. Matter effects are discussed thoroughly but are found to be small in this situation

Barenboim Szuchman, Gabriela Alejandra

Scheck, Florian

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:hep-ph/0004160v1  17 Apr 2000MZ-TH/00-14April 00Tau neutrinos from muon storage ringsGabriela Barenboim∗ and Florian Scheck†Institut fu˝r Physik - Theoretische ElementarteilchenphysikJohannes Gutenberg-Universita˝t, D-55099 Mainz, GermanyAbstractCharged tau leptons emerging in a long baseline experiment with a muonstorage ring and a far-away detector will positively establish neutrino oscillations.We study the conversion of νµ (νµ) and of νe (νe) to ντ or ντ for neutrinos froma 20 GeV muon storage ring, within the strong mixing scheme and on the basisof the squared mass differences which are compatible with all reported neutrinoanomalies, including the LSND data. In contrast to other solutions which ignorethe Los Alamos anomaly, we find charged tau production rates which shouldbe measurable in a realistic set up. As a consequence, determining the completemass spectrum of neutrinos as well as all three mixing angles seems within reach.Matter effects are discussed thoroughly but are found to be small in this situation.∗gabriela@thep.physik.uni-mainz.de†Scheck@dipmza.physik.uni-mainz.de1 IntroductionNeutrino beams from a muon storage ring provide an ideal tool for the next roundof experiments, aiming at establishing quantitatively oscillations and, possibly, CP-violation in the leptonic sector [1, 2]. The main reason for this is that both compositionand spectrum of such beams are perfectly known and, in addition, the beam energymay be tuned. For instance, the neutrino beams from a storage ring such as the onesdescribed in [1, 2] would contain an equal number of muon neutrinos and electronantineutrinos, or, depending on the sign of the parent muon, an equal number of muonantineutrinos and electron neutrinos, without any contamination from other neutrinospecies.The physics potential of a storage ring and a far-away detector, with regard tooscillations, was studied, e.g., in [3, 4], and, with regard to CP-violation, in [5, 6]. Inthis paper we address the question of appearance of tau neutrinos in the processesνµ −→ ντ and νe −→ ντ ,due to oscillations. We calculate the number of produced τ+ and τ−, taking intoaccount matter effects on the neutrino beam on its way from the source to the detector,within the strong mixing scheme of three flavours only that we had proposed earlier[7]. The latter assumption is receiving growing support by the fact that new data fromKAMIOKANDE seem to disfavour the existence of a fourth, sterile, neutrino, whilestill showing the characteristic modulation of events as a function of zenith angle. As aconsequence of the strong mixing which helps to populate strongly the ντ and ντ finalchannels from both the νe (νe) and the νµ (νµ) initial states, we find τ± productionrates one to two orders of magnitude larger than within a model ignoring the LSNDdata.In the next section we summarize some relevant formulae and collect the parametersextracted earlier from the analysis of all neutrino anomalies. We then describe anddiscuss matter effects in the Earth’s crust and calculate the event rates for the τ±appearance channels. The article ends with our results as well as some conclusions.2 Formulae and parametersIn a scenario with three families of leptons, the mixing between three neutrino speciesis described by a conventional Cabibbo-Kobayashi-Maskawa matrix U relating flavourstates to mass eigenstates. If neutrinos are Dirac particles, U has the formU = c12c13 s12c13 s13e−iδ−s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ s23c13s12s23 − c12c23s13eiδ −c12s23eiδ − s12c23s13 c23c13 (1)1where cij = cos θij and sij = sin θij . In the case they are of Majorana type, extraphases appear but their effects on neutrino observables are of order mν/Eν [8] and aregenerally negligible.The oscillation probabilities in vacuum take the form [9]P (νi → νj) = PCP−even(νi → νj) + PCP−odd(νi → νj) (2)P (ν¯i → ν¯j) = PCP−even(νi → νj)− PCP−odd(νi → νj), (3)wherePCP−even(νi → νj) = δij − 4ReJ ji12 sin2∆12 − 4ReJ ji23 sin2∆23 − 4ReJ ji31 sin2∆31,(4)PCP−odd(νi → νj) = −8σijJ sin∆12 sin∆23 sin∆31,with J the Jarlskog invariant andJ ijkh ≡ UikU †kjUjhU †hi , ∆ij ≡ ∆m2ijL/4E , σij ≡∑kεijk (5)In order to account for all reported neutrino anomalies one needs only two squaredmass differences. The LSND result can be understood in terms the mass difference∆M2 := m23 −m22 ≈ 0.3 eV2 (6)The second mass difference is tuned so as explain the observed deficit of electronneutrinos coming from the sun,10−4 eV2 ≤ ∆m2 ≤ 10−3 (7)with m22−m21 ≡ ∆m2, while the atmospheric neutrino data depend on both differencesand are found in agreement with the prediction. Given the squared mass differences,the mixing angles are easily found to be [7] (solution I)θ12 ≈ 35.50 , θ23 ≈ 27.30 , θ13 ≈ 13.10 . (8)This solution is favoured by the existing oscillation data and, thus, implies simultaneousand strong mixing of all three flavours.23 Matter effectsOf all neutrino species, only electron neutrinos can scatter elastically in the forwarddirection off electrons in matter, via charged current interactions. When the electronneutrinos oscillate into either muon or tau neutrinos, this introduces an additionalterm in the diagonal element of the neutrino flavour evolution matrix correspondingto νe → νe [10]. It is useful to define an effective mass term which stems from elasticscattering due to charged weak currents,a = 2√2GFneE = 7.7 · 10−5eV2(ρgr/cm3)(EνGeV)(9)where ne is the electron number density in matter of density ρ and E is the neutrinoenergy.As is well-known matter effects become important [11] only when a is comparableto, or larger than, the quantity ∆m2ij= m2i −m2j for some mass difference and neutrinoenergy. Given our neutrino mass spectrum and taking into account that for the Earth’scrust ρ ≃ 3gr/cm3, we are far from being in a range where matter effects would bedominant and could not be neglected. This would be the case, for example, in the caseof νµ → νe where large CP asymmetries are expected but will be masked by mattereffects. In the case that we discuss here, matter effects are not expected to play asignificant role but we will include them anyway. (In the case of electron neutrinososcillating into tau neutrinos, and with our parameters, they represent at most a 2 %effect.)In order to exhibit the essential mechanisms we assume for a moment the twolightest neutrinos to be degenerate in mass. Indeed, this limiting case is close to therealistic situation where ∆M2 ≫ ∆m2, cf. eqs. (6), (7). In this case, the transitionprobabilities in vacuum for the “terrestrial” experiments depend only on three variables,i.e., θ23, θ13 and ∆M2, as follows,P (νe → ντ ) ≈ 4U213U233 sin2(∆M2L4E)= cos2(θ23) sin2(2θ13) sin2(∆M2L4E), (10)P (νµ → ντ ) ≈ 4U223U233 sin2(∆M2L4E)= cos4(θ13) sin2(2θ23) sin2(∆M2L4E). (11)Note that in this limit of setting ∆m2 equal to zero the probabilities are independentof the angle θ12.3When matter effects are included the above formulae are still valid provided onemakes the replacements,∆M2 −→ ∆M2 + (3s213 − 1) a2(12)s213 −→ s213[1− 2a (s213 − 1)∆M2], (13)with s23 unchanged. Here we have assumed that ∆M2 ≫ a,∆m2, a hierarchy whichis respected by the mass spectrum we chose.From these formulae it is clear that the probability for a muon neutrino to oscillateinto a tau neutrino in matter will not be different from its vacuum value. The reasonfor this is clear: the interpretation of the modulation of neutrino events with zenithangle reported by SuperKamiokande in terms of a muon neutrino oscillating into atau neutrino either requires a nearly maximal θ23, in two-neutrino mixing schemes,or still a sizeable one, in three-neutrino mixing schemes. (As noted above, the datastrongly disfavour schemes with three active and one sterile neutrinos.) Furthermore,reactor experiments set a strict upper bound on θ13, giving sin2 θ13 ≤ .005. Thus, thefactor cos4(θ13) in eq. (11), to a good approximation, is equal to 1. Likewise the factorsin2(2θ23) in the same equation is also close to 1. Therefore, νµ → ντ oscillations arefavoured and the influence of matter effects on them is negligible.Introducing the terms containing ∆m2 does not modify this simple picture becausethey are suppressed by the huge gap between the two squared mass differences. It isimportant to notice that although we have assumed degeneracy between the two lightestneutrinos, this degeneracy is broken by matter effects which introduce an “effective”mass difference of the order a(s213 − 1)/2.In the case of electron neutrinos oscillating into tau neutrinos, however, the storyis different. In this case, the factor sin2(2θ13) in eq. (10) is a suppressing one andcould compensate, at least partially, the gap in the squared mass differences. Withour parameters, the contributions proportional to ∆m2 do not really compete with theone proportional to ∆M2 but they account for a sizeable correction. In our resultspresented below we use the full expression for the transition probabilities.At this point it is important to stress, that unlike our scheme, in schemes whereonly two anomalies are taken into account, disregarding the Los Alamos result, thetypical mass differences are ∆M2 ≈ 10−3 eV2 and ∆m2 ≈ 10−10 eV2, or 10−6 eV2 to10−5 eV2, depending on whether in the vacuum solution the small-angle MSW solutionor the large-angle MSW solution to the solar neutrino problem is chosen. Under theseassumptions, ∆M2 is just in the appropriate range to exhibit sizeable matter effectsso that these cannot be neglected.The conclusion so far is clear: If MiniBooNe confirms the LSND evidence for os-cillations, then ∆M2 is too large to cause a significant modification of the oscillation4probabilities due to matter effects. As we will see below, the prospects of discover-ing both νµ → ντ and νe → ντ oscillations then are promising indeed. However, ifKamLAND obtains a positive result in disappearance of electron neutrinos, corrobo-rating the large-angle MSW solution for solar oscillations, or if KARMEN definitelyand conclusively excludes the LSND result, then ∆M2 will be in a range to producea significant modification of the oscillation probability due to matter. In this case,the rates for tau appearance being lower by more than an order of magnitude, an ex-periment would be more difficult and would probably require a more intense neutrinosource than that assumed here.4 Event ratesThe calculation of event rates for tau lepton (anti tau lepton) production from electronneutrinos (anti muon neutrinos) as a result of oscillations is straightforward. The totalnumber of events in the two channels is given bynτ− = Nµ+ NkT109 NA E3µm2µ π L2∫ EµEmingνe ǫ σCCντP (νe → ντ ) dE (14)nτ+ = Nµ+ NkT109 NA E3µm2µ π L2∫ EµEmingν¯µ ǫ σCCν¯τP (ν¯µ → ν¯τ ) dE (15)where Nµ+ is the number of positive muon decays, σντ is the charged current crosssection per nucleon , P (νe → ντ ) and P (ν¯µ → ν¯τ ) are the oscillation probabilities forneutrinos traveling inside the Earth taking into account matter effects. NkT is thesize of the detector in kilotons, 109 NA is the number of nucleons in a kiloton, Eµ isthe energy of the muons in the ring and Emin = 5 GeV is a lower cut on the neutrinoenergies that we assume, for the sake of an example. For our numerical calculation,the energy spectrum (normalized to 1) of the neutrinos is taken to begνe = 12x2(1− x) (16)gν¯µ = 2x2(3− 2x) (17)where x = Eν/Eµ is the fractional neutrino energy. It is straightforward to obtain thecorresponding expression in the case of either electron anti neutrinos or muon neutrinos.Regarding the cross section, our calculation is based on the renormalization groupimproved parton model, focusing on the inclusive process ντ (ν¯τ ) + N −→ τ−(τ+) +anything. Note that in this case, unlike the charged current cross section for the muon,the terms proportional to the charged lepton mass are not negligible. The differentialcross section is [12],dσdx dy=G2FMEνπ(M2WM2W +Q2)2 {F2(1− y − Mxy2E)+ F1xy2 ± xF3(y − y22)5(18)+m2τME(−F2(M4E+12x)+F1y2∓ F3y4)}where we use the Bjorken scaling variables x = Q2/2Mν and y = ν/Eν . Here −Q2is the invariant momentum transfer between the incident neutrino and outgoing tau,ν = Eν − Eτ is the energy loss in the laboratory frame while M and MW are thenucleon and W-gauge boson masses respectively.The Callan-Gross relation 2xF1 = F2 simplifies the above equation further. It isthen dependent on only two form factors, F2 and F3, which are given in terms of theparton distributions byF2 =∑ix (qi + q¯i) (19)F3 =∑i(qi − q¯i) (20)For our calculations we have used the MRST99 [13] parton distribution. Althoughthere does not seem to be a general consensus about the best way of combining thequasi-elastic resonance and the deep inelastic scattering cross sections at fixed energyto form the total cross section, a number of different approaches were proposed in theliterature or were used in practice [14]. We adopt the simplest of them, and includeonly the deep inelastic part. In any case, the error one makes in the prediction due touncertainties in the neutrino spectrum and by assuming a constant matter distributionin the Earth’s crust is larger than the uncertainties in the cross section itself.5 Results and conclusionsWe consider a 20 GeV muon storage ring at CERN with a neutrino beam from oneof its straight sections that points to the Gran Sasso underground laboratory, at adistance of 732 km. This is about the same distance as the one between FNAL andthe Soudan mine.In order to mimic a nearly realistic experimental situation we set a lower cutoff inenergy at 5 GeV and we assume a detection efficiency of 30%. Also, for two cases,production of τ+ from the process νµ → ντ and production of τ− from the processνe → ντ , we show our results in bins of 3 GeV so as to get a feeling for the number ofevents to be expected with our parameters.Fig. 1 shows the predicted spectrum for tau lepton appearance coming from νe → ντ ,while Fig. 2 shows the predicted spectrum of anti taus coming from νµ → ντ . For thesake of comparison in Figs. 3 and 4 we plot the same observables for the choice ofparameters reported in Ref [3] which disregards the LSND data (note the different6scales in the ordinates!). Results similar to these were also reported in [15]. Fig. 5,finally, shows our result for τ− and τ+ production, grouped in bins of 3 GeV each.Provided the difference ∆M2 is in the range required by the LSND data, it is clearfrom Figs. 1, 2, and 5 that an experiment using a 20 GeV muon storage ring and abaseline of about 730 km between source and detector (corresponding to the distanceCERN-Gran Sasso, or FNAL to the Soudan mine) would have a fair chance to seethe appearance of charged tau leptons. The average oscillation probability could bemeasured with a statistical precision better than 3% [16] and ∆M2 could be determinedwith a precision of about 3%. Note that since matter effects are small in this case,there is no additional uncertainty on this mass difference arising from an incompleteknowledge of the oscillation mode.All in all, with the choice of parameters that we obtained by a simultaneous ex-planation of all neutrino anomalies, a 10 kT detector some 732 km downstream wouldprobe cos2(θ23) sin2(2θ13) to very low values. When combined with measurements ofνe → νµ oscillations (see for example [17]) and with further results from atmosphericneutrinos, a precise determination of the complete mass spectrum of neutrinos and ofall mixing angles seems possible.AcknowledgementsWe are very grateful to Jose´ Bernabeu, Daniel de Florian and Karl Jakobs forenlightening discussions, comments and explanations. Financial support from the DFGis also acknowledged.References[1] S. Geer, C. Johnstone, and D. Neuffer, FERMILAB-TM-2073;D. Finley, S. Geer, and J. Sims, FERMILAB-TM-2072;Prospective Study of Muon Storage Rings at CERN,B. Autin, A. Blondel, and J. Ellis (eds.), CERN 99-02/ECFA 99-197.[2] S.H.Geer, Phys. Rev. D 57 (1998), 6989.[3] A. deRujula, M.B. Gavela and P. Hernandez, Nucl. Phys. B 547 (1999), 21.[4] A. Donini, M.B. Gavela ,P. Hernandez and S. Rigolin, hep-ph/9909254.[5] A. Romanino, hep-ph/9909425.[6] G. Barenboim and F. Scheck; Phys. Lett. B475 (2000), 95; (hep-ph/0001208).[7] G. Barenboim and F. Scheck, Phys. Lett. B 440 (1998), 332.7[8] L. Wolfenstein, Phys. Lett. B 107 (1981), 77;P.B. Pal and L. Wolfenstein, Phys. Rev. D 25 (1982), 766;F. del Aguila and M. Zralek, Nucl. Phys. B 447 (1995), 211.[9] K. Dick, M. Freund, M. Lindner and A. Romanino, hep-ph/9903308.[10] J. Arafune, M. Koike and J. Sato, Phys. Rev. D 56 (1997), 3093M. Tanimoto, Phys. Lett. B345 (1988) 373;H. Minakata and H. Nunokawa, Phys. Lett. B 413 (1997) 369.[11] L. Wolfenstein, Phys. Rev. D 17 (1978), 2369;S.P. Mikheyev and A.Y. Smirnov, Sov. J. Nucl. Phys. 42 (1986) 913;V. Barger et al., Phys. Rev. D 22 (1980) 2718.[12] H.M. Gallagher and M.C. Goodman, Neutrino Cross Sections, NuMI-112, Novem-ber 1995.[13] A.D. Martin, R.G. Roberts, W.J. Stirling and R.S. Thorne, hep-ph/9907231[14] P. Lipari, M. Lusignoli and F. Sartogo, Phys. Rev. Lett. 74 (1995), 4384 andreferences therein.[15] V. Barger, S. Geer and K. Whisnant, hep-ph/9906487.[16] A. Cervera, F. Dydak and J.J. Gomez-Cadenas, Nufact ’99 Workshop, Lyon,France.[17] M. Freund, M. Lindner, S. Petcov and A. Romanino, hep-ph/9912457.8fromνντ−number   ofτeE   (GeV)ν8 10 12 14 16 18 2010203040Figure 1: Number of tau leptons detected in one year’s time in a 732 km baselineand assuming 30% detecting efficiency. The solid line correspond to setting the CPviolating phase δ = 0 while the dashed line correspond to δ = π/2fromνντnumber   ofτµ+E   (GeV)ν8 10 12 14 16 18 20100200300400500600Figure 2: Number of antitau leptons detected in one year’s time in a 732 km baselineand assuming 30% detecting efficiency. The solid and the dashed curves (defined asin Fig.1) are indistinguishable.9fromνντ−number   ofτeνE   (GeV)8 10 12 14 16 18 200.050.10.150.20.25Figure 3: Number of tau leptons detected in one year’s time in a 732 km baselineand assuming 30% detecting efficiency for the parameters of ref. [3]. Dashed andsolid lines as before.fromνντnumber   ofτµ+E   (GeV)ν8 10 12 14 16 18 205101520253035Figure 4: Number of tau leptons detected in one year’s time in a 732 km baselineand assuming 30% detecting efficiency for the parameters of ref. [3]. The solid andthe dashed curves are indistinguishable.10fromνντnumber   ofτµ+fromνντ−number   ofτeE   (GeV)νE   (GeV)ν204060801001201402505007501000125015005-8 8-11 11-14 14-17 17-205-8 8-11 11-14 14-17 17-20Figure 5: Number of tau (above) and antitau (below) leptons detected in one year’stime in a 732 km baseline and assuming 30% detecting efficiency grouped in bins of3 GeV.11

Tau neutrinos from muon storage rings

Gabriela Barenboim

Florian Scheck

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http://dx.doi.org/10.1016/S0370-2693(00)00708-5

Tau neutrinos from muon storage rings

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Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

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