DRAFT

November 10, 1999

Physics Capabilities of a Neutrino Factory

Lots of people

Nice abstract

14.60.Pq, 13.15.+g, 13.35.Bv, 07.77Ka

I. INTRODUCTION

Nice introduction

II. NEUTRINO BEAM PROPERTIES

Positive and negative muons decay via the weak interaction:-

μ^{+} → e^{+}\overline{ν}_{μ}ν_{e} ,  (1)

and

μ^{−} → e^{−}\overline{ν}_{e}ν_{μ} .  (2)

Thus, muons decaying within the straight sections of a muon storage ring will produce neutrino beams consisting of 50% ν_{μ} and 50% \overline{ν}_{e} if negative muons are stored, and 50% ν_{e} and 50% \overline{ν}_{μ} if positive muons are stored. In practice, if the detector only samples part of the beam within a small solid angle, the effective fluxes may differ from the 50% neutrino and 50% antineutrino mixtures suggested by Eqns. 1 and 2. However, the differential lepton and antilepton spectra from muon decay are well known, and the appropriate fluxes can be precisely calculated.

A. Differential spectra

The neutrino fluxes and interaction rates at a given location downstream of a muon storage ring neutrino source depend upon the number of stored muons, the beam divergence within the neutrino–beam forming straight section, and the energies and polarization of the decaying muons.

In the muon rest-frame the distribution of muon antineutrinos (neutrinos) from the decay μ^{±} → e^{±} + ν_{e} (\overline{ν}_{e}) + \overline{ν}_{μ} (ν_{μ}) is given by [41]:

\frac{d^{2}N_{ν_{μ}}}{dxdΩ} = \frac{2x^{2}}{4π} [(3 − 2x) ∓ (1 − 2x) cos θ] ,  (3)

where x ≡ 2E_{ν}/m_{μ}, θ is the angle between the neutrino momentum vector and the muon spin direction, m_{μ} is the muon rest mass, and E_{ν} the antineutrino (neutrino) energy. The corresponding expression describing the distribution of electron neutrinos (antineutrinos) is

\frac{d^{2}N_{ν_{e}}}{dxdΩ} = \frac{12x^{2}}{4π} [(1 − x) ∓ (1 − x) cos θ] .  (4)

Thus, the neutrino and antineutrino energy– and angular– distributions depend upon the parent muon energy, the decay angle, and the direction of the muon spin vector. For an ensemble of muons we must average over the polarization of the initial state muons, and the distributions become:

\frac{d^{2}N_{ν_{μ}}}{dxdΩ} ∝ \frac{2x^{2}}{4π} [(3 − 2x) ∓ (1 − 2x)P_{μ} cos θ] ,  (5)

and

\frac{d^{2}N_{ν_{e}}}{dxdΩ} ∝ \frac{12x^{2}}{4π} [(1 − x) ∓ (1 − x)P_{μ} cos θ] ,  (6)

where P_{μ} is the average muon polarization along the chosen quantization axis, which in this case is the beam direction.

FIG. 1. Calculated neutrino and antineutrino fluxes at a far site located 10000 km from a neutrino factory in which 1 × 10^{20} muons have decayed in the beam–forming straight section. The fluxes are shown as a function of the energy of the stored muons for negative muons (top two plots) and positive muons (bottom two plots), and for three muon polarizations as indicated. FIGURES NEEDS UPDATING

FIG. 2. Total neutrino flux of a given flavor versus angle at the source from the beamline at L = 732 km for E_{μ} = 10 GeV (solid line), 20 GeV (dashed), 50 GeV (dotted), and 250 GeV (dot–dashed), assuming 1.6 × 10^{20} neutrinos in the beam per operational year. A Gaussian muon beam divergence of 1 mr has been folded in. The total antineutrino flux is the same. FIGURE NEEDS UPDATING

FIG. 3. Neutrino and antineutrino spectra for muon–type neutrinos (top) and electron–type neutrinos (bottom) in the beam downstream of a muon storage ring neutrino source containing unpolarized muons.

B. Fluxes

Using Eqs. 5 and 6 the ν_{μ}, ν_{e}, \overline{ν}_{μ}, and \overline{ν}_{e} fluxes downstream of a muon storage ring neutrino source have been calculated [19] as a function of P_{μ}, the baseline length (L), and the energy of the stored muons (E_{μ}). The resulting fluxes are shown for L = 10000 km in Fig. 1. As an example, if 1×10^{20} unpolarized 20 GeV muons decay in the beam forming straight section, the resulting neutrino and antineutrino fluxes on the other side of the Earth are O(10^{10}) per m^{2}. Note that at fixed E_{μ}, a change of the muon polarization from +1 to −1 results in a change of the ν_{μ} (\overline{ν}_{μ}) flux at the distant site by a factor of approximately two. In contrast, the ν_{e} (\overline{ν}_{e}) flux is suppressed by roughly two orders of magnitude if P_{μ} = +1 (-1). Therefore, if the muons in the storage ring are polarized, changing P_{μ} can be used to statistically separate processes arising from an initial electron–type neutrino from processes originating from an initial muon–type neutrino.

For unpolarized muons, to a good approximation the calculated fluxes are reproduced by the formula:

Φ ≡ \frac{d^{2}N_{ν}}{dAdt} = \frac{n_{0}}{4πL^{2}γ^{2}(1 − β cos α)^{2}} ,  (7)

where n_{0} is the number of muon decays per unit time in the beam–forming straight–section, γ = E_{μ}/m_{μ}, β = p_{μ}/E_{μ}, α is the angle between the beam axis and the direction of interest, and dA is the differential area at the detector. In the extremely forward direction cos α ≃ 1, and using the approximation β ≃ 1 − 1/(2γ^{2}), the neutrino flux is given by:

Φ ≃ \frac{n_{0}γ^{2}}{πL^{2}} .  (8)

Note that the neutrino and antineutrino fluxes are proportional to (E_{μ}/L)^{2}.

The neutrino flux decreases if the far site is away from the beam axis (α > 0). The flux is shown as a function of α in Fig. 2 for several values of E_{μ} and L. Note that for E_{μ} < 50 GeV the flux is insensitive to beam mis-alignments (Δα) provided Δα < 1 milliradian.

FIG. 4. Distribution of energies for interaction muon–type neutrinos 730 km downstream of the neutrino source for the MINOS PH2 WBB (open histogram) compared with the corresponding distribution downstream of a neutrino factory (shaded histogram).

FIG. 5. Energy spectra far cf near from Heidi

Equations 5 and 6 can also be used to calculate the distribution of ν_{μ} and ν_{e} energies within the beam (dN_{ν}/dE_{ν}) using the approximation that, in the laboratory frame, x → E_{ν}/E_{μ} for forward neutrinos from high energy muon decays (E_{μ} >> m_{μ}). The calculated spectra are shown in Fig. 3. Note that the ν_{μ} spectrum peaks at the stored muon beam energy, whereas the ν_{e} spectrum peaks at two–thirds of the stored muon beam energy. The spectrum of interacting ν_{μ} (given by the energy weighted dN_{ν}/dE_{ν} distribution) is compared in Fig. 4 with the corressponding distribution for the MINOS WWB generated by the PH2 horn configuration. It can be seen that the neutrino factory beam can be considered to be a narrow band beam.

C. Interaction rates

Consider a neutrino with energy E_{ν} interacting with a proton of mass M_{p}. It is useful to define the energy transfer to the nucleon in the laboratory frame (ν). The fractional energy transfer y = ν/E_{ν} is related to the the center–of-mass scattering angle: (1 − y) = (1 + cos θ_{CM})/2. Within the framework of the naive quark parton model the differential cross sections for charged current (CC) neutrino and anti-neutrino scattering from a proton are given by:

\frac{dσ_{CC}^{νp}}{dxdy} = \frac{4M_{W}^{4} G_{F}^{2} M_{p}E_{ν}x}{π(Q^{2} + M_{W}^{2} )^{2}}[[d_{L}(x) + s_{L}(x)] + [\overline{u}_{R}(x) + \overline{c}_{R}(x)](1 − y)^{2}]  (9)

and

\frac{dσ_{CC}^{\overline{ν}p}}{dxdy} = \frac{4M_{W}^{4} G_{F}^{2} M_{p}E_{ν}x}{π(Q^{2} + M_{W}^{2} )^{2}}[[\overline{d}_{R}(x) + \overline{s}_{R}(x)] + [u_{L}(x) + c_{L}(x)](1 − y)^{2}]  (10)

where u_{L}, d_{L}, s_{L}, (\overline{u}_{R}, \overline{d}_{R}, \overline{s}_{R}) are the probabilities for finding a quark (antiquark) with the correct helicity carrying momentum fraction x inside the nucleon, x = Q^{2}/2Mν, and Q^{2} is the four–momentum–transfer squared.

Most heavy targets have approximately equal numbers of protons and neutrons. Isospin invariance implies that:

u(x)_{p} = d(x)_{n}, \overline{u}(x)_{p} = \overline{d}(x)_{n}  (11)
d(x)_{p} = u(x)_{n}, \overline{d}(x)_{p} = \overline{u}(x)_{n}  (12)
s(x)_{p} = s(x)_{n}, \overline{s}(x)_{p} = \overline{s}(x)_{n}  (13)
c(x)_{p} = c(x)_{n}, \overline{c}(x)_{p} = \overline{c}(x)_{n}  (14)

and provided M_{W} >> Q^{2}, the differential CC cross sections are given by:

\frac{dσ_{CC}^{νN}}{dxdy} = \frac{G_{F}^{2} M_{p}E_{ν}x}{π}[[u(x) + d(x) + 2s(x)] + [\overline{u}(x) + \overline{d}(x) + 2\overline{c}(x)](1 − y)^{2}]  (15)

and

\frac{dσ_{CC}^{\overline{ν}N}}{dxdy} = \frac{G_{F}^{2} M_{p}E_{ν}x}{π}[[u + d(x) + 2c(x)](1 − y)^{2} + [\overline{u}(x) + \overline{d}(x) + 2\overline{s}(x)]] .  (16)

Data taken on iron [?,?,?] indicate that the momentum–weighted integrals of the parton distributions xq(x) are approximately (0.26, 0.15, 0.015, < 0.01) for (xu, xd, xs, xc), and the corresponding integrals for (x\overline{u}, x\overline{d}, x\overline{s}, x\overline{c}) are (0.030, 0.035,0.015,< 0.01). This yields the approximate expressions:

\frac{dσ_{CC}^{νN}}{dy} = \frac{G_{F}^{2} M_{p}E_{ν}}{π} [0.44 + 0.065(1 − y)^{2}]  (17)

\frac{dσ_{CC}^{\overline{ν}N}}{dy} = \frac{G_{F}^{2} M_{p}E_{ν}}{π} [0.41(1 − y)^{2} + 0.095]  (18)

FIG. 6. Lepton energy spectra at a neutrino factory for CC \overline{ν}_{μ} (top left), ν_{μ} (top right), ν_{e} (bottom left), and \overline{ν}_{e} (bottom right) interactions.

The neutrino cross section is thus approximately flat in the inelasticity variable y while the antineutrino cross section favors low values of y and hence high values for the scattered lepton energy. It is convenient to define the normalized charged lepton energy

z ≡ E_{l}/E_{μ} .  (19)

At high energies (1 − y) ≃ z/x, and it can be shown that if the stored muons are unpolarized the lepton spectra are described by:

\frac{dN_{μ^{+}}}{dz} ∝ 1 + 20z^{2} − 32z^{3} + 11z^{4} ,  (20)

\frac{dN_{e^{−}}}{dz} ∝ 5 + 6z^{2} − 32z^{3} + 21z^{4} ,  (21)

\frac{dN_{μ^{−}}}{dz} ∝ 5 + 4z^{2} − 16z^{3} + 7z^{4} ,  (22)

\frac{dN_{e^{+}}}{dz} ∝ 1 + 30z^{2} − 64z^{3} + 33z^{4} .  (23)

The charged lepton spectra computed from Eqs. 20–23 are shown in Fig. 6.

Eqns 17–?? are modified by higher-twist processes (quasi-elastic scattering, non-conserved axial vector currents, resonances), final state mass effects (if a charm particle or τ lepton appears in the final state), and perturbative Quantum Chromodynamic corrections (which introduce a log Q^{2} dependence). However, if E_{ν} is greater than ∼ 20 GeV, the corrections to Eqns 17–?? are small, and the total cross section is given approximately by:

σ(ν + N → μ^{−} + X) ≈ 0.68 × 10^{−38}cm^{2} × (E_{ν}, GeV) ,  (24)

σ(\overline{ν} + N → μ^{+} + X) ≈ 0.34 × 10^{−38}cm^{2} × (E\overline{ν}, GeV) .  (25)

The number of ν_{e} and ν_{μ} CC events observed per incident neutrino on an isoscalar target of mass density ρ and length l is:

N(ν + N → μ^{−} + X) = 0.67 × 10^{−38}cm^{2} × (E_{ν}, GeV)ρlN_{A}  (26)
= 1.13 × 10^{−14}(E_{ν}, GeV)events per gr/cm^{2} ,  (27)

and

N(\overline{ν} + N → μ^{+} + X) = 0.34 × 10^{−38}cm^{2} × (E_{ν} , GeV)ρlN_{A} =  (28)
= 0.56 × 10^{−14}(E\overline{ν}, GeV) events per gr/cm^{2} .  (29)

Using Eqs. 27 and 29, the calculated ν_{e} and ν_{μ} CC interaction rates resulting from 2×10^{20} muon decays in the beam– forming straight–section of a neutrino factory are compared in Table I with expectations for the corresponding rates at the next generation of accelerator–based neutrino experiments. Note that event rates at a neutrino factory increase as E_{μ}^{3}, and are significantly larger than expected for the next generation of approved experiments if E_{μ} > 20 GeV.

FIG. 7. Tau–neutrino CC cross-section compared with the corresponding muon–neutrino cross-section.

Tau neutrino interaction rates are substantially different than those for lighter quarks, especially near the production threshold at xxx.gev. The presence of the tau mass in the final state shifts the definitions of the kinematic variables. Furthermore, the mass terms in the leptonic current can no longer be ignored and allow the axial vector structure functions W4 and W5 to play a more important role. Figure 7 shows a calculation [43] of the interaction rate ratios for tau and muon charged current events. At near threshold, contributions from quasi-elastic and single pion production dominate. If the ν_{τ} cross sections from Ref. [44] are used, the predicted event rates are 20–30% higher.

In addition to CC interactions, neutrinos also interact via neutral current (NC) scattering. For an isoscalar target the differential cross sections are described by:

\frac{dσ_{NC}^{νN}}{dxdy} = \frac{M_{W}^{4} G_{F}^{2} M_{p}E_{ν}}{π(Q^{2} + M_{W}^{2} )^{2}} x[[L_{d}^{2} + R_{d}^{2}(1 − y)^{2}](d(x) + u(x) + 2s(x)] + [L_{u}^{2}(1 − y)^{2} + R_{u}^{2}][\overline{u}(x) + \overline{d}(x) + 2\overline{c}(x)]] , (30)

\frac{dσ_{NC}^{\overline{ν}p}}{dxdy} = \frac{2M_{W}^{4} G_{F}^{2} M_{p}E_{ν}}{π(Q^{2} + M_{W}^{2} )^{2}} x[[L_{u}^{2}(1 − y)^{2} + R_{u}^{2}](u(x) + d(x) + 2c(x)] + [L_{d}^{2} + R_{d}^{2}(1 − y)^{2}][\overline{u} + \overline{d}(x) + 2\overline{s}(x)]] (31)

where the L_{q} and R_{q} are the left and right handed weak couplings for u-type and d-type quarks, with L_{u,c} = 0.5−0.67 sin^{2} θ_{W}, L_{d,s} = −0.5+0.33 sin^{2} θ_{W}, R_{u,c} = −0.67 sin^{2} θ_{W}, and R_{d,s} = 0.33 sin^{2} θ_{W}. To a first approximation the total NC cross sections for high energy neutrinos:

σ(ν + N → ν + X) ∼ 0.5 × σ(ν + N → l^{−} + X) ,  (32)

σ(\overline{ν} + N → ν + X) ∼ 0.5 × σ(\overline{ν} + N → l^{+} + X) .  (33)

D. Beam divergence, momentum spread, and straight-section length

In practice the detailed properties of the muon beam within the straight section of the storage ring will affect the neutrino fluxes at near and far detector sites. In particular the beam divergence (σ_{θ}) and the beam momementum spread σ_{p} must be considered, and in addition, for near experiments, the finite length of the straight section.

A finite σ_{θ} will result in a decrease of the neutrino flux at a distant site. Current design studies have taken as a reasonable design criteria σ_{θ} ≤ 0.1/γ. If the beam divergence σ_{θ} = 0.1/γ in both transverse directions the flux at the far site will be decreased by ∼ 8%. A finite σ_{p} tends to increase event rates at a distant site. For example, with σ_{p}/p = 0.02 and p = 50 GeV/c, the event rates are increased by ∼ 1%.

This is where Heidi adds a few lines plus a plot showing effect of finite straight section length on short baseline rates.

TABLE I. Muon neutrino and electron antineutrino CC interaction rates in the absence of oscillations, calculated for baseline length L = 732 km (FNAL → Soudan), for MINOS using the wide band beam and a muon storage ring with E_{μ} = 10, 20, 50 and 250 GeV.

⟨E_{ν_{μ}}⟩  ⟨Eν̄_{e}⟩  N(ν_{μ} CC)  N(ν̄_{e} CC)
Experiment  (GeV)  (GeV)  (per kt–yr)  (per kt–yr)
MINOS  Beam
Low energy  3  –  458  1.3
Medium energy  6  –  1439  0.9
High energy  12  –  3207  0.9
Muon ring  E_{μ} (GeV)
10  7  6  2200  1300
20  14  12  18000  11000
50  35  30  2.9×10^{5}  1.8×10^{5}
250  175  150  3.6×10^{7}  2.3×10^{7}

[1] See for example B. Kayser, “Neutrino mass: Where do we stand, and where are we going ?”, hep-ph/9810513, 28 Oct. 1998; J. Conrad, “Recent results on neutrino oscillations”, hep-ex/9811009, 5 Nov. 1998.

[2] B.T. Cleveland et al., Nucl. Phys. B (Proc. Suppl.) 38, 47 (1995); GALLEX collaboration, W. Hampel et al., Phys. Lett. B388, 384 (1996); SAGE collaboration, J.N. Abdurashitov et al., Phys. Rev. Lett. 77, 4708 (1996); Kamiokande collaboration, Y. Fukuda et al., Phys. Rev. Lett, 77, 1683 (1996); Super–Kamiokande collaboration, Y. Fukuda et al., Phys. Rev. Lett. 82, 2430 (1999); Phys. Rev. Lett. 82, 1810 (1999).

[3] See, e.g., J.N. Bahcall, S. Basu, and M.H. Pinsonneault, Phys. Lett. B 433, 1 (1998), and references therein.

[4] Super–Kamiokande Collaboration, Y. Fukuda et al., Phys. Lett. B433, 9 (1998); Phys. Lett. B436, 33 (1998); Phys. Rev. Lett. 81, 1562 (1998); Phys. Rev. Lett. 82, 2644 (1999).

[5] Kamiokande collaboration, K.S. Hirata et al., Phys. Lett. B280, 146 (1992); Y. Fukuda et al., Phys. Lett. B335, 237 (1994); IMB collaboration, R. Becker-Szendy et al., Nucl. Phys. Proc. Suppl. 38B, 331 (1995); Soudan-2 collaboration, W.W.M. Allison et al., Phys. Lett. B391, 491 (1997); MACRO collaboration, M. Ambrosio et al., Phys. Lett. B434, 451 (1998).

[6] G.L. Fogli, E. Lisi, A. Marrone, and G. Scioscia, Phys. Rev. D59, 033001 (1999).

[7] V. Barger, T.J. Weiler, and K. Whisnant, Phys. Lett. B440, 1 (1998).

[8] V. Barger and K. Whisnant, Phys. Rev. D59, 093007 (1999).

[9] R. Barbieri, L.J. Hall, D. Smith, A. Strumia, and N. Weiner, JHEP 9812, 017 (1998).

[10] C. Athanassopoulos et al. (LSND Collab.), Phys. Rev. Lett. 77, 3082 (1996).

[11] C. Athanassopoulos et al. (LSND Collab.), Phys. Rev. Lett. 81, 1774 (1998).

[12] S.M. Bilenky, C. Giunti, and W. Grimus, Eur. Phys. J. C1, 247 (1998); V. Barger, S. Pakvasa, T.J. Weiler, and K. Whisnant, Phys. Rev. D58, 093016 (1998).

[13] MINOS Collaboration, “Neutrino Oscillation Physics at Fermilab: The NuMI-MINOS Project,” NuMI-L-375, May 1998.

[14] K.Nishikawa et al. (KEK-PS E362 Collab.), “Proposal for a Long Baseline Neutrino Oscillation Experiment, using KEK-PS and Super-Kamiokande”, 1995, unpublished; INS-924, April 1992, submitted to J.Phys.Soc.Jap.; Y. Oyama, Proc. of the YITP Workshop on Flavor Physics, Kyoto, Japan, 1998, hep-ex/9803014.

[15] CHOOZ Collaboration, M. Apollonio et al., Phys. Lett. B420, 320 (1998).

[16] Y. Declais et al., Nucl. Phys. B434, 503 (1995).

[17] P. Alivisatos et al., “KamLAND Proposal”, Preprint Stanford-HEP-98-03; G. Gratta, hep-ex/9905011, “Neutrino oscillation experiments at nuclear reactors,” Proc. of the 17th International workshop on Weak Interactions and Neutrinos (WIN99), Cape Town, South Africa, Jan. 1999.

[18] Palo Verde Collaboration, F. Boehm et al., Prog. Art. Nucl. Phys. 40, 253 (1998).

[19] S. Geer, Phys. Rev. D57, 6989 (1998).

[20] A. De Rujula, M.B. Gavela, and P. Hernandez, Nucl. Phys. B547, 21 (1999).

[21] M. Campanelli, A. Bueno, and A. Rubbia, hep-ph/9905240.

[22] P. Lipari and M. Lusignoli, Phys. Rev. D58, 073005 (1998); Q.Y. Liu and A. Yu. Smirnov, Nucl. Phys. B524, 505 (1998); R. Foot, R.R. Volkas, and O. Yasuda, Phys. Rev. D58, 013006 (1998); M.C. Gonzalez–Garcia, H. Nunokawa, O. Peres, and J.W.F. Valle, Nucl. Phys. B543, 2 (1999).

[23] T. Blazek, S. Raby, and K. Tobe, hep-ph/9903340.

[24] L. Wolfenstein, Phys. Rev. D17, 2369 (1978).

[25] V. Barger, K. Whisnant, S. Pakvasa, and R.J.N. Phillips, Phys. Rev. D22, 2718 (1980).

[26] P. Langacker, J.P. Leveille, and J. Sheiman, Phys. Rev. D 27, 1228 (1983).

[27] P. Lipari, hep-ph/9903481.

[28] N. Cabbibo, Phys. Lett. B72, 333 (1978).

[29] V. Barger, K. Whisnant, R.J.N. Phillips, Phys. Rev. Lett. 45, 2084 (1980)

[30] S. Pakvasa, in Proc. of the XXth International Conference on High Energy Physics, ed. by L. Durand and L.G. Pondrum, AIP Conf. Proc. No. 68 (AIP, New York, 1981), Vol. 2, pp.1164.

[31] For recent discussions of CP violation effects in neutrino oscillations, see V. Barger, Y.-B. Dai, K. Whisnant, and B.- L. Young, Phys. Rev. D59, 113010 (1999); D.J. Wagner and T.J. Weiler, Phys. Rev. D59, 113007 (1999); A.M. Gago, V. Pleitez, R.Z. Funchal, hep-ph/9810505; K.R. Schubert, hep-ph/9902215; K. Dick, M. Freund, M. Lindner and A. Romanino, hep-ph/9903308; J. Bernabeu, hep-ph/9904474; S.M. Bilenky, C. Giunti, and W. Grimus, Phys. Rev. D58, 033001 (1998); M. Tanimoto, Prog. Theor. Phys. 97, 901 (1997); J. Arafune, J. Sato, Phys. Rev. D55, 322 (1997); T. Fukuyama, K. Matasuda, H. Nishiura, Phys. Rev. D57, 5844 (1998); M. Koike and J. Sato, hep-ph/9707203, Proc. of the KEK Meetings on CP Violation and its Origin; H. Minakata and H. Nunokawa, Phys. Lett. B413, 369 (1997); H. Minakata and H. Nunokawa, Phys. Rev. D57, 4403 (1998); J. Arafune, M. Koike and J. Sato, Phys. Rev. D56, 3093 (1997); M. Tanimoto, Phys. Lett. B435, 373 (1998).

[32] “Status of Muon Collider Research and Development and Future Plans”, C. Ankenbrandt et al. (The Muon Collider Collaboration), Fermilab-Pub-98/179, Physics Review Special Topics: Accelerators and Beams, in press.

[33] Workshop on the Potential for Neutrino Physics at Future Muon Colliders, Brookhaven National Laboratory, 13-14 August, 1998, see http://pubweb.bnl.gov/people/bking/nushop/workshop.html

[34] “Muon Storage Rings at CERN”; http://muonstoragerings.cern.ch/Welcome.html/

[35] V. Barger, T.J. Weiler, and K. Whisnant, Phys. Lett. B427, 97 (1998).

[36] A. Bueno, M. Campanelli, and A. Rubbia, hep–ph/9808485; hep–ph/9809252; B. Autin et al., CERN–SPSC/98-30; M. Doucet, J. Panman, and P. Zucchelli, hep-ex/9905029; S. Dutta, R. Gandhi, and B. Mukhopadhyaya, hep-ph/9905475.

[37] S. Geer, C. Johnstone, and D. Neuffer, “Muon Storage Ring Neutrino Source: The Path to a Muon Collider ?”, FERMILAB-TM-2073, 1999; “Design concepts for a muon storage ring neutrino source,” FERMILAB-PUB-99/121.

[38] S. Holmes et al., “A Development Plan for the Fermilab Proton Source,” FERMILAB-TM-2021, September 1997, unpublished.

[39] D. Finley, S. Geer, and J. Sims, “Muon Colliders: A vision for the future of Fermilab,” FERMILAB-TM-2072, 1999.

[40] B. Autin, talk at Muon Collider Collaboration Workshop, St. Croix, May 1999.

[41] See for example: T. K. Gaisser, “Cosmic Rays and Particle Physics,” Cambridge University Press 1990.

[42] CCFR?

[43] H.M. Gallagher and M. Goodman, “Neutrino Cross Sections”, NuMI-112, November 1995.

[44] D. Casper, private communication.

[45] y dependence (?)

[46] Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theor. Phys. 28, 870 (1962).

[47] L. Wolfenstein, Phys. Lett. B 107, 77 (1981); P.B. Pal and L. Wolfenstein, Phys. Rev. D 25, 766 (1982); F. del Aguila and M. Zralek, Nucl. Phys. B447, 211 (1995); Acta Phys. Pol. B27, 971 (1996).

[48] V. Barger, K. Whisnant, R.J.N. Phillips, Phys. Rev. D22, 1636 (1980).

[49] T.K. Kuo and J. Pantaleone, Rev. Mod. Phys. 61, 937 (1989).

[50] Y. Totsuka, talk at The Innerspace-Outerspace Conference, Fermilab, May 1999.

[51] Parameters of the Preliminary Reference Earth Model are given by A. Dziewonski, Earth Structure, Global, in: “The Encyclopedia of Solid Earth Geophysics”, D. E. James (Ed), (Van Nostrand Reinhold, New York, 1989) p. 331; also see R. Gandhi, C. Quigg, M. Hall Reno, and I. Sarcevic, Astroparticle Physics 5, 81 (1996).

[52] R. Bernstein and S. Parke, Phys. Rev. D44, 2069 (1991).

[53] E. Akhmedov, P. Lipari, and M. Lusignoli, Phys. Lett. B300, 128 (1993); P. Lipari and M. Lusignoli, Phys. Rev. D58, 073005 (1998); E. Akhmedov, A. Dighe, P. Lipari, and A.Yu. Smirnov, Nucl. Phys. B542, 3 (1999).

[54] S.T. Petcov, Phys. Lett. B434, 321 (1998); M.V. Chizhov, M. Maris, and S.T. Petcov, hep-ph/9810501; M.V. Chizhov and S.T. Petcov, hep-ph/9903424.

[55] H.W. Zaglauer and K.H. Schwarzer, Z. Phys. C40, 273 (1988); Q. Liu, S. Mikheyev, and A.Yu. Smirnov, Phys. Lett. B440, 319 (1998); E.K. Akhmedov, Nucl. Phys. B538, 25 (1998).

[56] J. Pruet and G.M. Fuller, astro-ph/9904023.

[57] J. Pantaleone, Phys. Rev. D49, 2152 (1994); Phys. Rev. Lett. 81, 5060 (1998).

[58] V. Barger, N. Deshpande, P.B. Pal, R.J.N. Phillips, and K. Whisnant, Phys. Rev. D43, 1759 (1991).

[59] L.J. Hall and H. Murayama, Phys. Lett. B436, 323 (1998); hep-ph/9810468.

[60] F. Vissani and A.Yu. Smirnov, Phys. Lett. B432, 376 (1998).

[61] KARMEN collaboration, K. Eitel and B. Zeitnitz, at Neutrino–98, Takayama, Japan, June 1998, hep-ex/9809007; newer reference with no constraint!

[62] J.N. Bahcall, P. Krastev, and A.Yu. Smirnov, Phys. Rev. D58, 096016 (1998).

[63] J.N. Bahcall, P. Krastev, and A.Yu. Smirnov, hep-ph/9905220.

[64] V. Barger, K. Whisnant, and R.J.N. Phillips, Phys. Rev. D 24, 538 (1981); S.L. Glashow and L.M. Krauss, Phys. Lett. B190, 199 (1987).

[65] E. Church et al. (BooNE Collab.), “A letter of intent for an experiment to measure ν_{μ} → ν_{e} oscillations and ν_{μ} disappearance at the Fermilab Booster”, May 16, 1997, unpublished.