Limits on the Mixing of Tau Neutrino to Heavy Neutrinos

Limits at 90 % c.l. on the square of the mixing strength |U_tau4|^2 between nu_tau and a mostly isosinglet heavy neutrino with mass in the range 10-290 MeV/c^2 are reported. The results were derived using the negative result of a search for neutral particles decaying into two electrons conducted by the CHARM collaboration in a neutrino beam dump experiment. Upper limits ~10^-4 were obtained for neutrino masses larger than 160 MeV/c^2.

(1) Such a mixing could result in neutrino oscillations when the mass differences are small, and in neutrino decays when the mass differences are large.
In this paper we report limits on the square of the mixing strength |U τ 4 | 2 between ν τ and a heavy neutrino, ν 4 , mostly isosinglet under the Standard SU (2) L gauge group, with mass in the range 10-290 MeV/c 2 .The limits were obtained using the negative result of a search for events produced by the decay of neutral particles into two electrons performed by the CHARM Collaboration in a neutrino beam dump experiment [1][2][3][4].The decays of the neutral particles, produced in the dumping of 400 GeV protons in a Cu target, were looked for in a volume located at a distance of L = 480 m from the beam dump.
The decay detector, shown in Fig. 1, has already been described elsewhere [5].It had an empty decay region of D = 35 m length and 3 m × 3 m surface area defined by a veto scintillator plane (SC1) and a scintillator hodoscope (SC2).The volume was subdivided into three regions using two sets of four proportional tube planes (P1 and P2) [6].One module of the CHARM fine-grain calorimeter [6] was displaced to the end of the decay region.In order to improve the resolution of the shower angle measurement and to reconstruct better the decay point, three sets of four proportional tube planes (P3, P4 and P5) were installed in front of the module.Lead converters of 0.5X 0 each were placed in front of P1, P2, P4 and P5.The detector was parallel to the neutrino beam line at a mean distance of 5 m, corresponding to an angle with respect to the incident proton beam of 10 mrad, and covered a solid angle of 3.9 × 10 −5 sr.The signature of the neutral particles decaying into two electrons would be events originating in the decay region at a small angle with respect to the neutrino beam axis with one or two separate electromagnetic showers.
The detector was exposed to a neutrino flux produced by 1.7 × 10 18 protons on a solid copper target [7] and 0.7 × 10 18 protons on a copper target laminated with an effective density of one-third of that of solid copper [7].In the combined exposures, 21 000 events were collected satisfying the trigger requirements of no hit in the scintillator planes SC1 and a hit in at least four out of the six scintillator planes of the calorimeter module.The events were further selected requiring that the transverse co-ordinates of the shower vertex lie in a square of 2.5 m × 2.5 m centred on the detector axis and that the electron energy, E el , measured in the calorimeter module, be larger than 2 GeV.The events recognised as cosmic rays were also rejected.The remaining sample of 7185 events is dominated by inelastic scattering of electron-and muon-neutrinos and antineutrinos producing hadron showers.Compared with the decay of neutral particles into two electrons, such events have a broader reconstructed angular distribution because of the intrinsic resolution and leakage effects.The regularity of the development of electromagnetic showers was used to distinguish further between the signal and the background events.In particular, the distribution of the deviation of the reconstructed direction of the (two) shower(s) from that of the incoming beam and the fraction of the energy detected by the proportional drift tubes of the calorimeter module outside a narrow cone around the shower were evaluated for the decay events by a Monte Carlo method [4].No event compatible with the features of the decay of a neutral particle into two electrons was found.
g replacements In this analysis we assumed that the ν 4 couples to matter essentially via mixing with ν τ (|U e4 | 2 , |U µ4 | 2 << |U τ 4 | 2 ) and we studied the proton production of D s 's in the target and their decay into τ .The heavy neutrino is produced in the charged current decays and τ → ν 4 + .... .
The Feynman diagrams of the process (2) and of the decay τ − → ν 4 + l − + νl (l = e, µ) are shown in Fig.s 2a and 2b respectively.On the basis of the assumptions made, the isosinglet heavy neutrino decays only via neutral current interactions according to the modes ν 4 → ν τ + e + + e − (4) The Feynman diagram illustrating the signal decay channel ( 4) is shown in Fig. 2c.The channels ( 5)-( 7) contribute to the beam attenuation.The branching ratio of mode ( 6) is negligible and the decay ( 7) opens for neutrino masses larger than the π 0 mass.The total decay width is then given by For heavy neutrinos with mass larger than 290 MeV/c 2 other decay modes open.
The leptonic partial width is predicted to be [8]: where gL = g L − 1 = −1/2 + sin 2 θ w and g R = sin 2 θ w , θ w is the weak angle.In this study the neutrinos were assumed to have Dirac masses and then K = 1.For Majorana neutrinos K is equal to 2. The leptonic partial width is dominated by the mode (5): The partial width for the decay ( 7) is predicted to be [9] Γ For a given heavy neutrino mass m ν 4 the number of the decay events (4) expected in the detector is where Φ (E ν 4 ) is the differential flux of heavy neutrinos, P ν 4 →ντ e + e − (E ν 4 ) is the probability for a heavy neutrino of energy E ν 4 to decay in the decay fiducial volume according to reaction (4), and ε (m ν 4 ) is the efficiency of the selection criteria of one or two electrons in the calorimeter based on the regularity of the development of electromagnetic showers and the collinearity between the (two) shower(s) and neutrino direction.The flux Φ (E ν 4 ) is given by The number of protons on the target corrected for the detector dead time, 13.6% for the solid target and 21.7% for the laminated one, is N p = 2.0×10 18 .The fraction of proton inelastic interactions leading to a charged D s is given by [10] where the copper mass number is A Cu = 63.55.Linear A dependence is assumed for charm production.The used values of the inelastic proton cross-section, σ Cu inel [11], of the ratio of the production cross-section for D ± s over the production cross-section for D ± + D 0 , σ (D s )/σ (D) [12], and of the inclusive cross-sections for the production of D mesons, σ nucleon D [13], are reported in Table 1.The ratio σ (D s )/σ (D) was obtained by the Beatrice experiment studying charmed particles produced by π − 's of 350 GeV/c [12].It is compatible with the results obtained by e + e − experiments at center of mass energies equal to 10 GeV and at Z 0 mass [14].The value of σ nucleon D was obtained by the NA27 Collaboration studying the production of D's in the interactions of 400 GeV protons in an H 2 target [13].
In equation ( 13) one has The value of the branching ratio of the D s decay into a zero mass neutrino, BR (D s → ν τ + τ ), is reported in Table 1 [15].Its uncertainty dominates the systematic error of this study.The factor ρ Ds describes phase space and helicity effects [16].In the case of τ decay into ν 4 where BR (τ → ν τ + X i ) is the branching ratio of the τ decay into a zero mass neutrino according to the considered mode i [15], see Table 2, and the ρ i τ are factors depending on heavy neutrino mass.For i = 1-3 they take into account phase space and helicity effects [17].For the modes 4-11 the ρ i τ 's were computed using only i Mode Branching ratio Table 2: The τ decay modes and corresponding branching ratio values used in the analysis phase space [18].The numerical values of ρ 4 τ as a function of the neutrino mass are smaller than the ones of Ref. [19] by about 10%.The latter were obtained taking into account also helicity effects and the experimental width of the vector meson ρ.
For a given neutrino mass, the acceptances of the heavy neutrino flux coming from the ], and the corresponding energy spectrum normalized to one, φ Ds ν 4 (E ν 4 ) [φ τ ν 4 (E ν 4 )] were obtained using a Monte Carlo simulation.The production of strange charm by protons was parametrized using the semi-empirical expression where x F is the meson longitudinal momentum in the collision center of mass frame divided by its maximum value √ s/2, and p T is the meson transverse momentum.Since there are few experimental results available on the production of D ± s , the values of n and b were inferred from the measurements of D production.Assuming the hadronization process to be independent of the cc production mechanism, the parameters n and b are independent of the meson produced.Most measurements agree with a value of b equal to 1 (GeV/c) −2 .The used values reported in Table 1 were obtained by the NA27 Collaboration studying the production of D's in the interactions of 400 GeV protons in an H 2 target [13].Cascade production was neglected.
The energy spectrum of heavy neutrinos from τ decay is given by where φ i ν 4 (E ν 4 ) is the normalized energy distribution of neutrinos produced in the decay mode i (see Table 2).In the case of leptonic channels, i = 1 and 2, the spectra were obtained using the matrix element The quantities p τ , p ν l , p l and p ν 4 are the four-momenta of τ , light neutrino, electron or muon, and heavy neutrino, respectively.The spectrum of heavy neutrino produced by channel 11 was obtained using phase space.The multi-pion decay modes were simulated using two models.In model (a) the spectra of channels 4-10 and their relative contributions as a function of the heavy neutrino mass were computed using phase space [18].In model (b) channel 4 was assumed to be produced through the resonance ρ and channels 5-10 through the resonance a 1 .The resonances were assumed to have zero width.The ρ i τ 's in (18) are given by: where The values of Table 3 were computed using the average of the spectra obtained in the two models.The systematic error in Table 1 reflects the differences of the spectra.
The decay (4) was simulated using the matrix element [8] which neglects the electron mass.The quantities p e − , p e + and p ντ are the fourmomenta of electron, positron and tau neutrino, respectively.In the center of mass of the decaying heavy neutrino the four-vector p ν 4 is given by [20] where η is a unit vector parallel to the direction of the heavy neutrino in the rest frame of the particle decaying into neutrino, D s , or τ , and |h| is the absolute value of the neutrino (antineutrino) helicity.In the case of heavy neutrinos coming from D s decay, the values of |h| obtained in Ref. [15] were used.As the polarization of τ produced in D s decay is negligible, |h| = 0 was assumed for the heavy neutrinos produced in τ decay.The acceptances and the mean momenta of decaying heavy neutrinos expected to be detected in the detector are reported in Table 3 for different values of neutrino mass.The efficiency of the cut E el > 2 GeV is about 85% for heavy neutrinos coming from D s and larger than 95% for the ones coming from τ .In Eq. ( 12) the probability for a heavy neutrino of energy E ν 4 to decay in the decay fiducial volume is given by where λ = (γβ c)/Γ tot is the heavy neutrino mean decay path (γ = E ν 4 /m ν 4 , β = p ν 4 /E ν 4 ).According to Eqs. ( 9) and (11) The quantity ε (m ν 4 ) in Eq. ( 12) is the efficiency of the selection criteria based on the regularity of the development of electromagnetic showers and the collinearity between the (two) shower(s) and the neutrino direction.The values, obtained using a detailed Monte Carlo simulation of the detector response, decrease with increasing heavy neutrino mass.It ranges from 91% for m ν 4 = 10 MeV/c 2 to 65% for m ν 4 = 290 MeV/c 2 [4].
Since no decay event was detected, upper limits at 90% confidence level on |U τ 4 | 2 were obtained in the neutrino mass range 10-290 MeV/c of N = N l = 6.42 events was used in Eq. (12).Since the contribution to the systematic error of the uncertainty on the spectra of heavy neutrinos coming from τ decay is negligible, N l does not depend on the neutrino mass.It corresponds to an average probability of observing no events, P 0 (N l ) , equal to 10% [21]: where f (0; N ′ l ) = e −N ′ l is the Poisson probability of obtaining zero events.In the case of negative N ′ l , f (0; N ′ l ) = 1 was used.The probability density function W (N ′ l ; N l , σ) takes into accounts the systematic errors summarized in Table 1 and was assumed to be a Gaussian distribution.Combining in quadrature the uncertainties reported in the table, one gets σ/N l = 0.64.In the case of no uncertainty, W (N ′ l ; N l , σ) = δ (N ′ l − N l ) and the integral gives N l = 2.30: the upper limit at 90% confidence level for the mean value of a Poisson distribution in the case of zero observations.The chosen value N l = 6.42 is a safely conservative number: taking for instance W (N ′ l ; N l , σ) equal to a log-normal distribution to avoid negative values of N ′ l [21], one obtains N l = 3.21, a factor 2 smaller than the value used in the analysis.
In conclusion, the negative results of a search of decays of neutral particles into two electrons performed by the CHARM Collaboration in a neutrino beam dump experiment, allowed limits to be set at 90% c.l. on the square of the mixing strength, |U τ 4 | 2 , between ν τ and a mostly isosinglet fourth neutrino, ν 4 , having a mass in the range 10-290 MeV/c 2 .Values of ∼ =10 −4 were obtained for masses larger than 160 MeV/c 2 .
We thank the members of the CHARM Collaboration who allowed us to use their data.We would like to thank B. Van de Vyver for the program simulating the generation of prompt neutrinos.We thank G. Barbiellini, L. Di Lella, A.D. Dolgov, S.N.Gninenko, P. Loverre, M. Mangano, J. Swain and K. Winter for many interesting discussions and comments.We would also like to thank S.H. Hansen for advice on the astrophysical and cosmological limits.Big Bang Nucleosynthesis have also been derived for the corresponding mass range shown, respectively [9].All the limits were obtained assuming that neutrinos have Dirac masses (K = 1).

Figure 1 :
Figure 1: Layout of the decay detector.