The experimental CP-violation (CPV) asymmetry in long-baseline neutrino oscillations, AμeCP=P(νμ→νe)−P(νˉμ→νˉe)A^\mathrm{CP}_{\mu e}=P(\nu_\mu\to\nu_e)-P(\bar\nu_\mu\to\bar\nu_e), mixes genuine Dirac-phase CP violation with a fake, matter-induced component. Bernabéu and Segarra proved a Disentanglement Theorem splitting it into a genuine T-odd component and a matter-induced CPT-odd component, and showed that in constant-density matter the fake component has a δ\delta- and mass-ordering-independent first-rank zero at a magic energy E≃0.92 GeV (L/1300 km)E\simeq0.92\,\mathrm{GeV}\,(L/1300\,\mathrm{km}) near the second oscillation maximum, where the genuine component is maximal and proportional to sin⁡δ\sin\delta — a proposed route to direct evidence of genuine CP violation at DUNE and Hyper-Kamiokande. The open problem addressed here arises from the idealizations under which that result was derived: whether the separation survives realistic non-constant Earth matter profiles, realistic beam fluxes, detector response, finite statistics and systematic uncertainties.

Background

Neutrinos oscillate between three flavor states because the flavor eigenstates are quantum mixtures of three mass eigenstates, with mixing described by the PMNS matrix UU, which contains one Dirac CP-violating phase δ\delta. When neutrinos propagate through Earth, coherent forward scattering of electron neutrinos on ambient electrons (the MSW effect) adds a potential VV acting only on the electron-flavor component, so the effective Hamiltonian in flavor space is H=(1/2E)UM2U†+VH=(1/2E)UM^2U^\dagger+V; the standard notation sets a=2EVa=2EV for neutrinos, with a→−aa\to-a for antineutrinos. The experimentally accessible CP asymmetry AαβCP(L,E)=P(να→νβ)−P(νˉα→νˉβ)A^\mathrm{CP}_{\alpha\beta}(L,E)=P(\nu_\alpha\to\nu_\beta)-P(\bar\nu_\alpha\to\bar\nu_\beta) therefore has two physically distinct sources: the genuine phase δ\delta, and the CP asymmetry of the matter background itself, which fakes CP violation even when δ=0\delta=0. This contamination has been a long-standing obstacle to claiming direct evidence of CP violation in the lepton sector, relevant among other things to leptogenesis scenarios for the cosmic matter-antimatter asymmetry.

Bernabéu and Segarra resolved this problem theoretically through the Disentanglement Theorem: AαβCP=AαβCP,T+AαβCP,CPTA^\mathrm{CP}_{\alpha\beta}=A^{\mathrm{CP,T}}_{\alpha\beta}+A^{\mathrm{CP,CPT}}_{\alpha\beta}, where the first component is odd under time reversal T (and CPT-invariant, hence genuinely CP-violating: it vanishes unless δ≠0,π\delta\neq0,\pi) and the second is odd under CPT (T-invariant, vanishing in vacuum, hence purely matter-induced). Here T compares a transition with its reverse, P(να→νβ)P(\nu_\alpha\to\nu_\beta) versus P(νβ→να)P(\nu_\beta\to\nu_\alpha), and CPT compares neutrinos with antineutrinos; since the matter background is T-symmetric but not CPT-symmetric, the two components separate cleanly. The components have definite parities: AαβCP,TA^{\mathrm{CP,T}}_{\alpha\beta} is odd in the baseline LL and in sin⁡δ\sin\delta, even in aa and in the mass ordering h=±1h=\pm1 (the sign of Δm312\Delta m^2_{31}, i.e. normal versus inverted ordering of the neutrino masses); AαβCP,CPTA^{\mathrm{CP,CPT}}_{\alpha\beta} is even in LL and sin⁡δ\sin\delta, odd in aa, and almost odd in hh, the qualifier referring to higher-order matter corrections beyond the leading hierarchy-odd term.

For the golden appearance channel νμ→νe\nu_\mu\to\nu_e, in the regime Δm212,∣a∣≪∣Δm312∣\Delta m^2_{21},|a|\ll|\Delta m^2_{31}| with oscillation phases Δij=Δmij2L/(4E)\Delta_{ij}=\Delta m^2_{ij}L/(4E), the two components are approximately

AμeCPT≃16A(sin⁡Δ31Δ31−cos⁡Δ31)(Ssin⁡Δ31+Jrcos⁡δ Δ21cos⁡Δ31)+O(A3),AμeT≃16Jrsin⁡δ Δ21sin⁡2Δ31+O(A2),A^{\mathrm{CPT}}_{\mu e}\simeq16A\left(\tfrac{\sin\Delta_{31}}{\Delta_{31}}-\cos\Delta_{31}\right)\left(S\sin\Delta_{31}+J_r\cos\delta\,\Delta_{21}\cos\Delta_{31}\right)+\mathcal{O}(A^3),\qquad A^{\mathrm{T}}_{\mu e}\simeq16J_r\sin\delta\,\Delta_{21}\sin2\Delta_{31}+\mathcal{O}(A^2),

where AA is the matter-expansion parameter proportional to LL, S=c132s132s232S=c^2_{13}s^2_{13}s^2_{23}, and Jr=c12c132c23s12s13s23J_r=c_{12}c^2_{13}c_{23}s_{12}s_{13}s_{23} is the reduced Jarlskog invariant controlling genuine CPV. Two experimental strategies follow for the planned long-baseline programs: baseline dependence, using a second Hyper-Kamiokande water-Cherenkov detector in Korea (T2HKK, L≃1000L\simeq1000–13001300 km) in addition to the T2HK detector at 295295 km, and energy dependence, using the DUNE liquid-argon experiment at L=1300L=1300 km with a wide-band beam and reconstructed-energy bins. At energies above the first oscillation node the fake component dominates and its sign determines the mass ordering. Near the second oscillation maximum (the energy where the leading phase Δ31≃3π/2\Delta_{31}\simeq3\pi/2, roughly 0.80.8 GeV at L=1300L=1300 km) there is a magic energy E≃0.92 GeV (L/1300 km)E\simeq0.92\,\mathrm{GeV}\,(L/1300\,\mathrm{km}), fixed by tan⁡Δ31=Δ31\tan\Delta_{31}=\Delta_{31} and depending only on ∣Δm312∣|\Delta m^2_{31}|, at which the fake component has a first-rank zero — a zero with nonvanishing slope, so that averaging over an energy bin still leaves an effective zero — independent of δ\delta, hh and aa, while the genuine component is maximal and proportional to sin⁡δ\sin\delta. The zero survives a modest energy resolution ΔE∼200\Delta E\sim200 MeV.

All of these results were derived for constant matter density (the constant matter density of the Earth mantle), the νμ→νe\nu_\mu\to\nu_e appearance channel, and an idealized analytic treatment with perfect energy reconstruction and no systematic errors. The Disentanglement Theorem itself is framework-independent and its master formula is stated to remain valid for extended neutrino models with more mass eigenstates and a rectangular mixing matrix, but no explicit phenomenology in such scenarios was developed. Whether the separation strategies, and in particular the magic-energy zero, survive realistic conditions is therefore a robustness question arising from the idealizations under which the results were derived.

Problem Statement

Determine whether the separation of the experimental CPV appearance asymmetry into its genuine T-odd and matter-induced CPT-odd components survives beyond the idealized conditions in which it was derived — constant Earth matter density, the golden νμ→νe\nu_\mu\to\nu_e channel, and an analytic treatment with perfect energy reconstruction and no systematic errors — and quantify under what conditions it fails. Concretely: (1) determine the displacement and residual value of the δ\delta- and mass-ordering-independent magic-energy zero of the CPT-odd component for realistic, non-constant Earth matter density profiles along the DUNE and T2HKK baselines; and (2) determine whether, with realistic beam fluxes, interaction cross sections, detector response, energy resolution, finite statistics and systematic uncertainties, a measurement of the CP asymmetry in energy bins around the (possibly shifted) magic energy isolates the genuine component proportional to sin⁡δ\sin\delta at DUNE and Hyper-K/T2HKK, and whether the sign of the asymmetry above the first oscillation node determines the neutrino mass ordering. The objective is a quantitative characterization of the conditions under which the Disentanglement Theorem's strategy delivers direct evidence of genuine leptonic CP violation, or a demonstration that it fails under specified realistic conditions.

References

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