Breaking 8–Fold Neutrino Parameter Degeneracies

K. Whisnant

Dept. of Physics and Astronomy, Iowa State Univ., Ames, IA 50011, USA

Abstract.

Parameter degeneracies are a generic problem in the three–neutrino analysis of long–baseline neutrino appearance measurements. Degeneracies can lead to different inferred values for the neutrino mixing angle θ_{13}, and they often mix CP violating and CP conserving solutions. Appropriate experimental strategies can reduce or eliminate degeneracies and/or the CP V/CP C confusion.

1. Introduction

With the strong evidence for neutrino oscillations from solar [1] and atmospheric [2] neutrino experiments, the focus is now changing from establishing the existence of neutrino oscillations to obtaining a complete understanding of the parameters in the neutrino mass matrix. Solar neutrino data suggest |δm_{31}^{2}| ∼ 3 × 10^{−3} eV^{2} and sin^{2} 2θ_{23} ≃ 1.0 (in the standard three–neutrino description), while atmospheric neutrino data provide give δm_{21}^{2} ∼ 5 × 10^{−5} eV^{2} and sin^{2} 2θ_{12} ≃ 0.8. Reactor experiments [3] put an upper bound on sin^{2} 2θ_{13} of about 0.1. In the future, KamLAND [4] will likely provide a more precise measurement of the solar neutrino parameters and ν_{μ} → ν_{μ} survival in long–baseline experiments will provide a more precise measurement of the atmospheric neutrino parameters (to ≤ 10%).

The heretofore unknown neutrino parameters are the CP phase δ that can appear in three–neutrino oscillations, the mixing angle θ_{13}, and the sign of δm_{31}^{2}, which determines the ordering of the neutrino mass eigenstates. In this talk I discuss the difficulties in making a unique determination of these neutrino parameters, and how experiments might be designed to overcome them.

2. Neutrino parameter degeneracies

The standard method for determining the parameters θ_{13} and δ is to measure the rates for ν_{μ} → ν_{e} and ν̄_{μ} → ν̄_{e} in a long–baseline neutrino experiment. For our analysis we use approximate expressions for the the oscillation probabilities similar to those derived in Ref. [5]. For δm_{31}^{2} > 0

P (ν_{μ} → ν_{e}) = Ax^{2} + Bxy(cos δ cos Δ − sin δ sin Δ) + Cy^{2} ,  (1)
P̄(ν̄_{μ} → ν̄_{e}) = Āx^{2} + B̄xy(cos δ cos Δ + sin δ sin Δ) + Cy^{2} ,  (2)

where x ≡ sin θ_{23} sin 2θ_{13}, y ≡ |δm_{21}^{2}/δm_{31}^{2}| cos θ_{23} sin 2θ_{12}, and Δ = 1.27|δm_{31}^{2}/eV^{2}|(L/km)/(E_{ν}/GeV) is the oscillation argument. The coefficients A, B, C, etc., depend on |δm_{31}^{2}|, Δ, and the density of matter. Matter effects lead to a suppression of Ā (B̄) relative to A (B). The probabilities for δm_{31}^{2} < 0 are given by making the exchanges A ↔ Ā, B ↔ −B̄, and δ → −δ. More details, including a discussion of their range of validity, are given in Ref. [6]. These approximations have a similar form to those for low energy [7] and the exact expressions for constant density [8].

The standard method for determining θ_{13} and δ is to measure P and P̄ at one L and E_{ν}. If θ_{13} is held fixed and δ is varied from 0 to 2π, an ellipse is traced out in the (P, P̄) plane [9] (it is assumed that the other oscillation parameters are well–measured from other experiments, as mentioned above). The measured values of P and P̄ then in principle should determine θ_{13} and δ. However, there are three possible two–fold degeneracies that can occur:[6]

(i) The ellipses for two different values of θ_{13} can overlap, so that the same P and P̄ are described by two different sets of parameters (δ, θ_{13}) and (δ′, θ′_{13}) (the (δ, θ_{13}) ambiguity). The values of θ_{13} and θ′_{13} can be very different, and if δ is CP violating, then δ′ can be CP conserving (CP V/CP C confusion). In some cases solutions with maximal CP violation can give the same predictions for P and P̄ as a solution with CP conserved.

(ii) Ellipses for δm_{31}^{2} > 0 are shifted towards the P axis, while those for δm_{31}^{2} < 0 are shifted towards the P̄ axis, due to different matter effects. These ellipses can overlap, so that for many values of P and P̄, the parameters (δ, θ_{13}) for δm_{31}^{2} > 0 and a different set of parameters (δ′, θ′_{13}) for δm_{31}^{2} < 0 can both reproduce the data (the sgn(δm_{31}^{2}) ambiguity). Here θ_{13} and θ′_{13} are not as far apart as they can be for the (δ, θ_{13}) ambiguity, but a large CP V/CP C confusion is possible.

(iii) Since only sin^{2} 2θ_{23} is measured in atmospheric and ν_{μ} → ν_{μ} experiments, there are two possible values for θ_{23}, one with θ_{23} < π/4 and one with θ_{23} > π/4 [10]. The ellipses for these two solutions can overlap, once again with different solutions for θ_{13} and δ (the (θ_{23}, π/2 − θ_{23}) ambiguity). As with the (δ, θ_{13}) ambiguity, the uncertainties in θ_{13} can be large and a large CP V/CP C confusion is also possible. This ambiguity vanishes for θ_{23} = π/4, the value preferred by atmospheric data.

Each of these two–fold ambiguities may exist for a given data set, so that it is possible to have an eight–fold ambiguity in some cases, i.e., eight distinct sets of the parameters (δ, θ_{13}) that describe the data.

3. Strategies for breaking parameter degeneracies

Appropriate experimental strategies may be used to either reduce or eliminate the effects of parameter degeneracies. The simplest is to choose your baseline so that the cos δ terms in the oscillation probabilities vanish. For a narrow spectrum this can be achieved by choosing L/⟨E_{ν}⟩ (where ⟨E_{ν}⟩ is the average energy) such that Δ = (n − 1/2)π, at the peak of the oscillation in vacuum (which also is near the peak after matter is taken into account). Then the ellipses in (P, P̄) space collapse to lines, and the line for one value of θ_{13} no longer overlaps the lines for other θ_{13} (for the same sgn(δm_{31}^{2}) and θ_{23}), thereby removing θ_{13} from the (δ, θ_{13}) ambiguity. Since only sin δ is being measured, there is a residual (δ, π − δ) ambiguity; however, CP V and CP C solutions are no longer mixed for the same sgn(δm_{31}^{2}) and θ_{23}. There still can be large CP V/CP C confusion due to the sgn(δm_{31}^{2}) ambiguity, unless δ ≃ 3π/2 (π/2) for δm_{31}^{2} > 0 (< 0), in which case the sgn(δm_{31}^{2}) ambiguity is avoided. On a peak the CP V/CP C confusion due to the (θ_{23}, π/2 − θ_{23}) ambiguity is small, although the uncertainty in sin^{2} 2θ_{13} can be as much as a factor of two for sin^{2} 2θ_{23} = 0.90 [6].

Another tactic is to go to longer baseline L to increase the matter effect, thereby splitting apart the predictions for δm_{31}^{2} > 0 and δm_{31}^{2} < 0. We have found that predictions for the two signs of δm_{31}^{2} do not overlap for sin^{2} 2θ_{13} > 0.01(0.005)[0.002] for L = 1290(1770)[2900] km, assuming the values of the solar and atmospheric parameters given above [6]. We have also found that the first peak of the oscillation (i.e., Δ = π/2) is best for resolving the sgn(δm_{31}^{2}) ambiguity, since higher peaks tend to have more overlap between δm_{31}^{2} > 0 and δm_{31}^{2} < 0 [6].

Currently there is an experiment planned [12] to go from the Japan Hadron Facility to the Super–Kamiokande detector (L = 295 km and ⟨E_{ν}⟩ ≃ 0.7 GeV). This lies approximately on the first peak of the oscillation, but the L is not long enough to resolve the sgn(δm_{31}^{2}) ambiguity except for extreme values of δ. Therefore it is interesting to know what additional measurements are needed to resolve the remaining (δ, π − δ), sgn(δm_{31}^{2}) and (θ_{23}, π/2 − θ_{23}) ambiguities.

In general, for two measurements, ambiguities occur across wide areas of the (δ, θ_{13}) plane. We have shown [11] that, in principle, (i) a third measurement can resolve the (δ, π − δ) ambiguity, and the remaining ambiguities occur only along lines in the (δ, θ_{13}) plane, (ii) a fourth measurement may still have sgn(δm_{31}^{2}) and (θ_{23}, π/2 − θ_{23}) ambiguities, but they only occur at isolated points in the (δ, θ_{13}) plane, and (iii) a fifth measurement can remove all of the ambiguities. All of the degeneracies do not necessarily have to be completely removed in a given experiment, since they are only a problem if they occur in the region of parameter space accessible to the experiment. We have found that making the measurements at longer L pushes the degeneracies to lower values of θ_{13}; furthermore, making measurements at different L tends to push the degeneracies to lower θ_{13} than if all of the measurements are at the same L [11]. For example, ignoring experimental uncertainties, the degeneracies for two ν and two ν̄ measurements at the same L lie below sin^{2} 2θ_{13} = 0.01(0.002) for L = 300(2900) km; with only three measurements at different L, they can be pushed to sin^{2} = 0.001 or lower. For comparison, the sensitivity of the SuperJHF to Hyper–Kamiokande experiment [12] is expected to be around sin^{2} 2θ_{13} = 0.002.

In summary, sitting on an oscillation peak reduces the (δ, θ_{13}) ambiguity to a (δ, π − δ) ambiguity with no CP V/CP C confusion, which a third measurement can then resolve. Going to long enough L can resolve the sgn(δm_{31}^{2}) ambiguity, and perhaps the (θ_{23}, π/2 − θ_{23}) ambiguity, especially if more than two measurements are made. Even if the (θ_{23}, π/2 − θ_{23}) ambiguity cannot be resolved, the resulting uncertainty in δ is small, although θ_{13} will not be as well determined.

I gratefully acknowledge the collaboration of V. Barger and D. Marfatia.

References

[1] See talk by T. Kajita, these proceedings.

[2] See talk by S. Biller, these proceedings.

[3] See plenary talk by D. Markoff, these proceedings.

[4] See parallel talk by D. Markoff, these proceedings.

[5] A. Cervera et al. 2000 Nucl. Phys. B 579 17; M. Freund 2001 Phys. Rev. D 64 053003.

[6] V. Barger, D. Marfatia and K. Whisnant 2002 Phys. Rev. D 65 073023; hep-ph/0108090.

[7] J. Arafune, M. Koike and J. Sato 1997 Phys. Rev. D 56 3093.

[8] K. Kimura, A. Takamura and H. Yokomakura 2002 Phys. Lett. B 537 86; hep-ph/0205295.

[9] H. Minakata and H. Nunokawa 2001 JHEP 0110 001.

[10] G. L. Fogli and E. Lisi 1996 Phys. Rev. D 54 3667.

[11] V. Barger, D. Marfatia and K. Whisnant 2002 Phys. Rev. D 66 053007.

[12] Y. Itow et al., hep-ex/0106019.