Compact perturbative expressions for neutrino oscillations in matter

Journal of High Energy Physics, Jun 2016

We further develop and extend a recent perturbative framework for neutrino oscillations in uniform matter density so that the resulting oscillation probabilities are accurate for the complete matter potential versus baseline divided by neutrino energy plane. This extension also gives the exact oscillation probabilities in vacuum for all values of baseline divided by neutrino energy. The expansion parameter used is related to the ratio of the solar to the atmospheric ∆m 2 scales but with a unique choice of the atmospheric ∆m 2 such that certain first-order effects are taken into account in the zeroth-order Hamiltonian. Using a mixing matrix formulation, this framework has the exceptional feature that the neutrino oscillation probability in matter has the same structure as in vacuum, to all orders in the expansion parameter. It also contains all orders in the matter potential and sin θ 13. It facilitates immediate physical interpretation of the analytic results, and makes the expressions for the neutrino oscillation probabilities extremely compact and very accurate even at zeroth order in our perturbative expansion. The first and second order results are also given which improve the precision by approximately two or more orders of magnitude per perturbative order.

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Compact perturbative expressions for neutrino oscillations in matter

JHE Compact perturbative expressions for neutrino oscillations in matter Peter B. Denton 1 3 Hisakazu Minakata 0 2 Stephen J. Parke 3 0 Department of Physics, Yachay Tech University , San Miguel de Urcuqu ́ı 100119 , Ecuador 1 Physics & Astronomy Department, Vanderbilt University , PMB 401807, 2301 Vanderbilt Place, Nashville, TN 37235 , U.S.A 2 Instituto de F ́ısica, Universidade de Sa ̃o Paulo , C.P. 66.318, 05315-970 Sa ̃o Paulo , Brazil 3 Theoretical Physics Department, Fermi National Accelerator Laboratory , P.O. Box 500, Batavia, IL 60510 , U.S.A We further develop and extend a recent perturbative framework for neutrino oscillations in uniform matter density so that the resulting oscillation probabilities are accurate for the complete matter potential versus baseline divided by neutrino energy plane. This extension also gives the exact oscillation probabilities in vacuum for all values of baseline divided by neutrino energy. The expansion parameter used is related to the ratio of the solar to the atmospheric ∆ m2 scales but with a unique choice of the atmospheric ∆ m2 such that certain first-order effects are taken into account in the zeroth-order Hamiltonian. Using a mixing matrix formulation, this framework has the exceptional feature that the neutrino oscillation probability in matter has the same structure as in vacuum, to all orders in the expansion parameter. It also contains all orders in the matter potential and sin θ13. It facilitates immediate physical interpretation of the analytic results, and makes the expressions for the neutrino oscillation probabilities extremely compact and very accurate even at zeroth order in our perturbative expansion. The first and second order results are also given which improve the precision by approximately two or more orders of magnitude per perturbative order. CP violation; Neutrino Physics - 1 Rotations of the neutrino basis and the Hamiltonian 2.1 Overview 2.2 U23(θ23, δ) rotation 2.3 U13(φ) rotation 2.4 U12(ψ) rotation 2.5 Remarks 1 Introduction 2 3 4 Perturbation expansion 3.1 Corrections to the eigenvalues 3.2 Corrections to the eigenvectors Oscillation probabilities 4.1 The zeroth order probabilities 4.2 The first order probabilities 4.3 The second order probabilities 4.4 Precision of the perturbation expansion 5 Conclusions A Technical details A.1 Generalized approach to diagonalization A.2 Useful identities A.3 Limits A.4 Characteristic equation A.5 Unitarity of the W matrix A.6 V -matrix, S-matrix comparison A.7 CP violating term Introduction Neutrino oscillation based on the standard three flavor scheme provides the best possible theoretical paradigm which can describe most of the experimental results obtained in the atmospheric, solar, reactor, and the accelerator neutrino experiments. In matter, the propogation of neutrinos is significantly modified by the Wolfenstein matter effect [ 1 ]. The theoretical derivation and understanding of the neutrino oscillation probabilities in matter have been pursued by various means. The exact expressions of the eigenvalues, mixing angles, and the oscillation probabilities have been obtained [ 2–4 ], albeit under the assumption of uniform matter density. But the resulting expressions of the oscillation – 1 – probabilities are way too complex to facilitate understanding of the structure of the three flavor neutrino oscillations. For this reason, analytic approaches to the phenomena are mostly based on variety of perturbative frameworks. For a comprehensive treatment of neutrino oscillation in the matter, see ref. [ 5 ]. Analytic expressions for neutrino oscillations in arbitrary matter densities has also been considered, but even more simplifying arguments must be made [ 6 ]. What is the appropriate expansion parameter in such a perturbative framework? We now know that sin θ13, once used as the expansion parameter (there are an enormous number of references, see e.g., [ 7 ]), is not so small, sin θ13 ≃ 0.15. Moreover, expansion around sin θ13 = 0 misses the physics of the resonance which exists at an energy around E ∼ 10 GeV for earth densities. Therefore, in the environments in which the matter effect is comparable to the vacuum mixing effect, the only available small expansion parameter known to us is the ratio of the solar-scale ∆ m2⊙ to the atmospheric-scale ∆ m2⊕, ∆ m2⊙/∆ m2⊕ ≃ 0.03. This framework was examined in the past, to our knowledge in refs. [ 7–10 ]. Recently, two of us, see [ 11 ], presented a new perturbative framework for neutrino oscillation in matter using a modified ∆ m2⊙/∆ m2⊕ expansion. We identified a unique ∆ m2⊕ that absorb certain “first-order” terms into the “zeroth-order” Hamiltonian. The resulting expansion parameter, ǫ ≡ ∆ m221/∆ me2e where ∆ me2e ≡ ∆ m321 − sin2 θ12∆ m221 , multiplies a particularly simple perturbing Hamiltonian with zero diagonal entries. This re-organization of the perturbation expansion lead to simple and compact oscillation probabilities in all channels. The νe disappearance channel is particularly simple, being of a pure two flavor form. As was noted in [ 11 ], this new perturbation expansion, while valid in most of the baseline, L, divided by neutrino energy, E, versus matter potential plane, has issues around vacuum values for the matter potential at large values of L/E. These issues are caused by the crossing of two of the eigenvalues of the new zeroth order Hamiltonian at the solar resonance. In this paper, we solve these issues by performing an additional rotation of the neutrino basis in matter by introducing an additional matter mixing angle which is identical to θ12 in vacuum. With this extra rotation, the new eigenvalues of the unperturbed Hamiltonian do not cross and the perturbing Hamiltonian remains non-diagonal and is multiplied by an additional factor which is always less than unity and is zero in vacuum. With this additional rotation our perturbative expansion is valid in the full L/E versus matter potential plane and the zeroth order gives the exact result in vacuum. The sectional plan of this paper is as follows: in section 2 we describe in detail the sequence of rotations of the neutrino basis that leads us to the simple Hamiltonian that will be used in the perturbative expansion. The zeroth order eigenvalues and mixing matrix are given in this section. Then, in section, 3 we explicitly calculate the first and second order corrections for both the eigenvalues and the mixing matrix. In section 4, we give compact analytic expressions for νe and νμ disappearance channels as well as νμ → νe appearance channel at both zeroth and first order in our perturbative expansion. All other channels can by obtained by unitarity. Here we discuss the precision of the perturbative – 2 – H E P 0 6 ( 2 0 1 6 ) 0 5 1 treatment. Finally, in section 5 there is a conclusion. A number of technical details are contained in the appendices, see A. We have also published the new Nu-Pert code used in this paper online.1 2 Rotations of the neutrino basis and the Hamiltonian In this section we perform a sequence of rotations on the neutrino basis and the corresponding Hamiltonian such that the following conditions are satisfied: • The diagonal elements of the rotated Hamiltonian are excellent approximations to the eigenvalues of the exact Hamiltonian and do not cross for any values of the matter potential. These diagonal elements will form our H0. • The size of non-diagonal elements are controlled by our small parameter, ǫ′, which vanishes in vacuum. The non-diagonal elements will form our perturbing Hamiltonian, H1. The first two of these rotations are identical to the rotations performed in [ 11 ], while the last rotation is needed to deal with the remaining eigenvalue crossing at the solar resonance. With these three rotations the resulting Hamiltonian satisfies the conditions above and leads us to a rapidly converging perturbative expansion for the oscillation probabilities that covers all of the L/E versus matter potential plane. 2.1 Overview Neutrino evolution in matter is governed by a Schro¨dinger like equation i ∂∂x |νi = H|νi , where in the flavor basis νe  |νi = νμ , ντ  H = 21E hUMNS diag(0, ∆ m221, ∆ m321)U M†NS + diag(a(x), 0, 0)i . (2.3) UMNS is the lepton mixing matrix in vacuum, given by UMNS ≡ U23(θ23, δ)U13(θ13)U12(θ12) with2  cψ sψ U12(ψ) ≡ −sψ cψ   , 1 1 U13(φ) ≡    cφ −sφ 1 sφ  , cφ U23(θ23, δ) ≡  −s2c32e3−iδ s2c32e3iδ , 1See https://github.com/PeterDenton/Nu-Pert. 2The PDG form of UMNS is obtained from our UMNS by multiplying the 3rd row by eiδ and the 3rd column by e−iδ i.e. by rephasing ντ and ν3. The shorthand notation cθ = cos θ and sθ = sin θ is used throughout this paper. – 3 – (2.1) (2.2) (2.4) J H E P 0 6 ( 2 0 1 6 ) 0 5 1 and the matter potential, assumed to be constant, is given by a ≡ 2√2GF NeE ≈ 1.52 × 10−4 Yeρ g · cm−3 E GeV eV2 . We will perform a sequence of rotations on the flavor basis by multiplying the left and right hand side of eq. (2.1) by an appropriate unitary matrix, U † and inserting unity (U U †) between H and |νi. These rotations are chosen such that the final resulting Hamiltonian satisfies the following properties: the diagonal elements are an excellent approximations to the exact eigenvalues and the size of off-diagonal elements are controlled by a small parameter (ratio of the ∆ m2’s) and are identically zero in vacuum. The sequence of rotations applied to the eigenstates is performed in the following order |νi → |ν˜i = U2†3(θ23, δ)|νi → |νˆi = U1†3(φ)U2†3(θ23, δ)|νi → |νˇi = U1†2(ψ)U1†3(φ)U2†3(θ23, δ)|νi , with the corresponding Hamiltonians H → H˜ = U2†3(θ23, δ) H U23(θ23, δ) → Hˆ = U1†3(φ)U2†3(θ23, δ) H U23(θ23, δ)U13(φ) → Hˇ = U1†2(ψ)U1†3(φ)U2†3(θ23, δ) H U23(θ23, δ)U13(φ)U12(ψ) . The first rotation undoes the θ23 − δ rotation, whereas the φ followed by ψ rotations are matter analogues to the vacuum θ13 and θ12 rotations, respectively. In vacuum, the final Schro¨dinger equation is just the trivial mass eigenstate evolution equation. 2.2 U23(θ23, δ) rotation After the U23(θ23, δ) rotation, the neutrino basis is |ν˜i = U2†3(θ23, δ)|νi , and the Hamiltonian is given by H˜ = U2†3(θ23, δ) H U23(θ23, δ) = 1 hU13(θ13)U12(θ12) diag(0, ∆ m221, ∆ m321)U1†2(θ12)U1†3(θ13) 2E + diag(a, 0, 0)i . As was shown in [ 11 ], the Hamiltonian, H˜ , is most simple written in terms of a renormalized atmospheric ∆ m2, ∆ me2e ≡ ∆ m321 − s122∆ m221 , as defined in [ 12, 13 ], and the ratio of the ∆ m2’s ǫ ≡ ∆ m221/∆ me2e . – 4 – (2.5) (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) J H E P 0 6 ( 2 0 1 6 ) 0 5 1 In terms of the |a| → ∞ eigenvalues the exact Hamiltonian is simple given by3 H˜ = 21E   λa Note that H˜ is real and does not depend on θ23 or δ. 2.3 U13(φ) rotation Since s13 ∼ O(√ǫ), it is natural to diagonalize the (1-3) sector next, using U13(φ), again see [ 11 ]. After this rotation the neutrino basis is |νˆi = U1†3(φ)|ν˜i = U1†3(φ)U2†3(θ23, δ)|νi , (2.12) (2.13) (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) J H E P 0 6 ( 2 0 1 6 ) 0 5 1 and the Hamiltonian is given by Hˆ = U1†3(φ) H˜ U13(φ)  λ− 1 = 2E  λ0 λ+ ∆ me2e c(φ−θ13)  + ǫc12s12 2E   c(φ−θ13) s(φ−θ13)  s(φ−θ13) , – 5 – where λ∓ = 12 h(λa + λc) ∓ sign(∆ me2e)p(λc − λa)2 + 4(s13c13∆ me2e)2i , λ0 = λb = ǫc122∆ me2e , which is identical to eq. 3.1 of [ 11 ]. Also c(φ−θ13) ≡ cos(φ − θ13) and s(φ−θ13) ≡ sin(φ − θ13). The angle, φ, that achieves this diagonalization of the (1-3) sub-matrix (see appendix A.1), satisfies λa = c2φλ− + s2φλ+ , λc = s2φλ− + c2φλ+ , and sφcφ = s13c13∆ me2e , λ+ − λ− from which it is easy to derive cφ − s2φ = 2 sφ = λc − λa , λ+ − λ− s λ+ − λc , λ+ − λ− cφ = s λc − λ− . λ+ − λ− 3One can use H˜ to do a perturbative expansion, such that it is simple to recover the νμ → νe appearance probability of Cervera et al., [ 7 ] at first order. The Hamiltonian given in eq. (2.15) was used to derive simple, compact and accurate oscillation probabilities for a wide range of the L/E versus ρE plane, see [ 11 ]. However, as was noted in that paper, there is a region of this plane for which a perturbation theory based on Hˆ is insufficient to describe the physics accurately. This region is small ρE and large L/E given by |a| < 31 ∆ me2e and L/E > ∆ 4mπe2e . (2.20) To address this region of the L/E versus ρE plane, we perform one further rotation on the Hamiltonian. This rotation removes the degeneracy of the zeroth order eigenvalues at the solar resonance when λ− = λ0. This is performed in the next subsection. 2.4 U12(ψ) rotation Since λ− and λ0 cross at the solar resonance, a ≈ ǫ∆ me2e cos 2θ12/ cos2 θ13, to describe the physics near this degeneracy we need to diagonalize the (1-2) submatrix of Hˆ , using U12(ψ). The new neutrino basis is |νˇi = U1†2(ψ)|νˆi = U1†2(ψ)U1†3(φ)U2†3(θ23, δ)|νi . The resulting Hamiltonian, split into a zeroth order Hamiltonian and a perturbing Hamiltonian, is given by Hˇ = U1†2(ψ) Hˆ U12(ψ) = Hˇ0 + Hˇ1 , where 1 Hˇ0 = 2E  λ1 λ2   , λ3 The diagonal elements of the zeroth order Hamiltonian are λ1,2 = 21 h(λ0 + λ−) ∓ q(λ0 − λ−)2 + 4(ǫc(φ−θ13)c12s12∆ me2e)2i , λ3 = λ+ . The angle, ψ, that achieves this diagonalization of the (1-2) sub-matrix of Hˆ (see appendix A.1), satisfies λ− = c2ψλ1 + s2ψλ2 , λ0 = s2ψλ1 + c2ψλ2 , sψcψ = ǫc(φ−θ13)s12c12∆ me2e , ∆ λ21 where we introduce the useful shorthand notation, ∆ λij ≡ λi − λj . – 6 – (2.21) (2.22) (2.23) (2.24) (2.25) (2.26) (2.27) (2.28) J H E P 0 6 ( 2 0 1 6 ) 0 5 1 It is easy to derive that4 with the diagonal elements the zeroth order Hamiltonian and the off-diagonal elements the perturbing Hamiltonian. While the λa,b,c eigenvalues cross twice and the λ−,0,+ eigenvalues cross once, the new λ1,2,3 eigenvalues do not cross, see figure 1, which allows for the perturbation theory to be well defined everywhere. • The size of the perturbing Hamiltonian, Hˇ1, is controlled by the parameter ǫ′ ≡ ǫ s(φ−θ13) s12c12 = s(φ−θ13)s12c12 ∆∆ mm22e2e1 , where which is never larger than 1.4%. • In vacuum, s(φ−θ13) = 0 , (2.35) so that the zeroth order Hamiltonian gives the exact result. Also, in the limit where a → −∞ for NO or a → +∞ for IO s(φ−θ13) → −s13 which is of O(√ǫ). Whereas for a → +∞ for NO or a → −∞ for IO s(φ−θ13) → c13 ∼ 1, see figure 2. • Since perturbing Hamiltonian, Hˇ1, has only non-diagonal entries the first order correction to the eigenvalues are zero. The diagonal elements multiplied by 2E are, to an excellent approximation, the mass squares of the neutrinos in matter. 4Given the definition of λ1,2 in eq. (2.25), the sign term in from of cψ is not necessary, but will become necessary when we discuss the λ1 ↔ λ2 interchange symmetry. – 7 – π/2 ,ψπ/4 φ • There is a very useful interchange symmetry involving λ1,2 and ψ. The Hamiltonian is invariant under the pair of transformations λ1 ↔ λ2 and ψ → ψ ± π/2. Our expressions for sψ and cψ, see eq. (2.30), satisfy this interchange symmetry with the + in front of the π/2. Since the transition probabilities always have an even number of ψ trig functions, this interchange symmetry can be simply expressed as λ1 ↔ λ2 , c2ψ ↔ s2ψ , and cψsψ ↔ −cψsψ . (2.36) In the rest of this paper we call this the λ1,2 − ψ interchange symmetry. • An antineutrino with energy E is equivalent to a neutrino with energy −E. • The values of all of the eigenvalues in vacuum and for a → ±∞ are shown in appendix A.3. – 8 – 8 6 4 2 ) 2 V −3e0 0 1 λ−2 ( −4 −6 φ ψ NO IO 30 λ1 J H E P 0 6 ( 2 0 1 6 ) 0 5 1 10−1 10−2 10−3 To calculate the neutrino oscillation probabilities at zeroth order, all that is needed is eigenvalues and mixing matrix, given by eq. (2.25) and eq. (2.32) respectively. For higher order calculations we need not only the corrections to the eigenvalues but also the corrections to the mixing matrix. In this section we first given the corrections to the eigenvalues at both first and second order in our expansion parameter, ǫ′. This is followed by the corrections to the same order for the mixing matrix. Note that all corrections to both the eigenvalues and the mixing matrix vanish in vacuum as our expansion parameter is zero in vacuum, i.e. the zero order oscillation probabilities are exact in vacuum. 3.1 Corrections to the eigenvalues Since the diagonal terms of Hˇ1 = 0 by construction, the first order corrections to the eigenvalues are exactly zero, since J H E P 0 6 ( 2 0 1 6 ) 0 5 1 The second order corrections to the eigenvalues are given by5 5Eq. (3.2) explicitly shows why the level crossing of two of the eigenvalues (λ−, λ0) causes problems for higher orders in the perturbation theory. λ(1) = 2E(Hˇ1)ii = 0 . i k6=i λi(2) = X [2E∆ (Hˇλ1ik)ik]2 . – 9 – (3.1) (3.2) Using Hˇ1 from eq. (2.24), we see that the corrections are . We verified that the eigenvalues satisfy the characteristic equation to second order, see appendix A.4. The eigenvalues are correct at zeroth order to a fractional precision of about 10−4 or better, and through second order to a precision of 10−8 or better. In fact, the precision of λ1 + λ(1) + λ(2) for sign(∆ me2e)YeρE < 0 is completely saturated by the 1 1 limits of double precision computer calculations. 3.2 Corrections to the eigenvectors Here we present the corrections to the eigenvectors which allows us to calculate the transition probabilities to second order. Higher orders can be easily calculated by continuing this approach in a straightforward fashion. This was called the V -matrix approach in [ 14 ]. First, we relate the flavor eigenvectors to the zeroth order eigenvectors (no subscript) using UMmNS, as in eq. (2.4), Next, the exact eigenvectors of Hˇ , labeled with subscript (ex), are related to the eigenvectors of Hˇ0 (the zeroth order eigenvectors) by a unitary matrix, which we call W †, νe  νμ = UMmNS νˇ2 . ντ  νˇ3 νˇ1 νˇ1 νˇ2  νˇ3(ex) νˇ1 = W † νˇ2 . νˇ3 Combining the above gives, νe  νˇ1 νμ = V νˇ2 ντ   νˇ3(ex) where V ≡ UMmNSW . The exact V matrix transforms the exact eigenvectors of Hˇ to the flavor basis. In vacuum (a = 0), UMmNS = UMNS and W = ✶ , so V = UMNS as expected. Standard perturbation theory in Hˇ1, which contains the small parameter ǫ′, can be used to calculate W †. Here we use a slightly modified perturbation theory to calculate W directly. Expanding W as a power series in ǫ′, we define W ≡ W0 + W1 + W2 + O(ǫ′3) . It is clear from eq. (3.5) that W0 = ✶ . (3.3) (3.4) (3.5) (3.6) (3.7) J H E P 0 6 ( 2 0 1 6 ) 0 5 1 The first order correction to the W matrix is given by This series can be continued to reach arbitrary precision. However, we have found that second order provides more than sufficient precision. In summary the matrix relating the zeroth order eigenvalues of Hˇ0 to the flavor basis is given by V = UMmNSW = U23(θ23, δ)U13(φ)U12(ψ)(✶ + W1 + W2) , (3.10) to second order in ǫ′. Demonstration of the unitary nature of V , to the appropriate order, is given in appendix A.5. With the eigenvalues and eigenvectors determined to second order we can now calculate the neutrino oscillation probabilities. 4 Oscillation probabilities In vacuum and in matter with constant density, it is well known that the neutrino oscillation probabilities for να → νβ for three-flavor mixing (i, j = 1, 2, 3) can be written in the following form6 P (να → νβ) = X Vα∗iVβie−i λi(2eEx)L 2 3 i=1 = δαβ + 4C2α1β sin2 ∆ 21 + 4C3α1β sin2 ∆ 31 + 4C3α2β sin2 ∆ 32 (4.1) + 8Dαβ sin ∆ 21 sin ∆ 31 sin ∆ 32 , 6The equivalence of the V-matrix method and the S-matrix method for calculating the oscillation probabilities is addressed in appendix A.6. (3.8) (3.9) J H E P 0 6 ( 2 0 1 6 ) 0 5 1 using the exact mixing matrix, Vαi, and difference of the exact eigenvalues λi(ex). Both V and λ(ex)s depend on the energy of the neutrino E, and the matter density ρ but the i baseline L, dependence only appears in ∆ ij . By unitarity and using the fact that the sin2 functions and the triple sine function are linearly independent functions of L, as determined by their non-zero Wronskian, we have the following powerful statements, X P (να → νβ) = 1 , β X Ciαjβ = 0 , β X Dαβ = 0 . β Since Dαα = 0, we also note that Dαβ = −Dαγ for α, β, γ all different. So, up to one overall sign, there is only one D term for all channels. To determine the oscillation probability to n-th order in our perturbative expansion we (ex) to the n-th order. We denote this perturbative expansion must evaluate C, D, and ∆ λij as follows (ex) = ∆ λij + ∆ λi(j1) + ∆ λi(j2) + . . . ∆ λij Ciαjβ = (Ciαjβ)(0) + (Ciαjβ)(1) + (Ciαjβ)(2) + . . . Dαβ = (Dαβ)(0) + (Dαβ)(1) + (Dαβ)(2) + . . . . 4.1 The zeroth order probabilities At zeroth order the ∆ λ’s are given by eq. (2.25) and the C, D coefficients are the same as in vacuum with θ13, θ12 replaced with φ, ψ respectively, see eq. (3.10). Therefore where Ciαjβ = −ℜ[VαiVβ∗iVα∗j Vβj ] , Dαβ = ℑ[Vα1Vβ∗1Vα∗2Vβ2] , (Ciαjβ)(0) = −ℜ[UαiUβ∗iUα∗j Uβj ] , (Dαβ)(0) = ℑ[Uα1Uβ∗1Uα∗2Uβ2] , where here the Uαi are elements of UMmNS = U23(θ23, δ)U13(φ)U12(ψ). In table 1 we give the zeroth order coefficients for P (νe → νe), P (νμ → νe), and P (νμ → νμ), from which all remaining transitions can be easily determined by unitarity.7 4.2 The first order probabilities At first order the ∆ λ’s are again given by eq. (2.25), since λi(1) = 0, see eq. (2.24), because the diagonal elements of Hˇ1 are zero. The first order corrections to C, D only have terms proportional to ∆ λ3−11, ∆ λ3−21. This comes from the form of W1, eq. (3.8), which follows 7The ντ channels can also be obtained from the corresponding νμ channel by the following replacements c23 → −s23 and s23 → c23. (4.2) (4.3) (4.4) (4.5) (4.6) J H E P 0 6 ( 2 0 1 6 ) 0 5 1 from the position of the non-zero elements in Hˇ1. In fact, all of the coefficients can be written in the following general form, , , , , where the F1,2, G1,2 and K1,2 are related by λ1,2, ψ interchange previously discussed. Thus only three modest expressions are required to describe the C’s and D coefficients to first order for each channel. The F, G, K terms can be calculated from UMmNS by F αβ = −sψℜ (Uα1Uβ∗3 + Uα3Uβ∗1)Uα∗2Uβ2 , 1 Gαβ = −sψℜ 1 K1αβ = −sψI (Uα1Uβ∗3 + Uα3Uβ∗1)Uα∗2Uβ2 . Uα1Uβ∗3 + Uα3Uβ∗1 (2Uα∗3Uβ3 − δαβ) , F and G are even under the interchange of α and β whereas K is odd. Their explicit values are given in table 2. In the appearance channels the CP violating term must be of the following form D = ±s12c12s13c123s23c23 sin δ Qi>j ∆ m(ei2xj) , Qi>j ∆ λij where in the denominator one needs the exact eigenvalues in matter. This is the NaumovHarrison-Scott identity, see refs. [ 15, 16 ]. We have checked this identity to the appropriate order, see appendix A.7. The P (να → β) and P (ν¯α → ν¯β) probabilities are related by δ → −δ and the P (να → νβ) and P (νβ → να) transition probabilities are related by L → −L. From eq. (4.1), we see that the D term is the only term odd in L. From tables 1 and 2, we see that the D term is also the only one odd in δ, confirming the CPT invariance of these equations. Moreover, all of the Dαβ terms are the same order by order up to a coefficient of −1, 0, 1. 4.3 The second order probabilities Although we have not expanded the second order oscillation probabilities analytically, the second order corrections to the eigenvalues, λi(2), as well as the second order corrections to the mixing matrix, W2, have been used to calculate the oscillation probabilities to second order. The resulting oscillation probabilities are more than two orders of magnitude closer to the exact values than the first order probabilities. (4.7) (4.8) (4.9) J H E P 0 6 ( 2 0 1 6 ) 0 5 1 – 1 4 – να → νβ νe → νe νμ → νe νμ → νμ (C3α1β)(0) −c2φs2φc2ψ (C2α1β)(0) −c4φs2ψc2ψ s2φc2φc2ψs223 + Jrm cos δ −c2φs223(c223s2ψ + s223s2φc2ψ) −2s223Jrm cos δ c2φs2ψc2ψ(c223 − s2φs223) + c2ψJrm cos δ −(c223c2ψ + s223s2φs2ψ)(c223s2ψ + s223s2φc2ψ) −2(c223 − s2φs223)c2ψJrmr cos δ + (2Jrmr cos δ)2 (Dαβ)(0) 0 −Jrm sin δ −2sφcφsψ(s223c2φcψ − s23c23sφsψ cos δ) −s23c23cφs2ψ(c2φc2ψ − s2φ) sin δ να → νβ νe → νe νμ → νe νμ → νμ F αβ 1 −2c3φsφs3ψcψ cφs2ψ[sφsψcψ(c223 + c2φs223) −s23c23(s2φs2ψ + c2φc2ψ) cos δ] 2cφsψ(s223sφcψ + s23c23sψ cos δ)× (c223c2ψ − 2s23c23sφsψcψ cos δ + s223s2φs2ψ) −2cφsψ(s223sφcψ + s23c23sψ cos δ) ×(1 − 2c2φs223) K1αβ 0 0 10−10 The oscillation probabilities that were perturbatively calculated in this section are only useful if they are more precise than the experimental uncertainties. In figure 3, we have plotted the fractional uncertainties8 at each order of our perturbative expansion for the νμ → νe channel at the DUNE [ 17 ], baseline of 1300 km. The precision at the first oscillation maximum and minimum for DUNE are shown in table 3. We note that the precision improves at lower energies, such as for NOνA [ 18 ] and T2K/T2HK [ 19, 20 ]. The results are comparable for different values of δ, for the inverted ordering, for other channels, and for antineutrino mode. Therefore, even at zeroth order, the precision exceeds the precision of the expected experimental results. The oscillation probabilities of [ 11 ] started to become less accurate when as note therein. This restriction is removed in this paper as the eigenvalues no longer cross at the solar resonance. This improves the accuracy of the oscillation probabilities for T2K/T2HK, NOνA and DUNE. Also, for example, one could use the ν¯e → ν¯e disappearance probabilities of this paper to quantify the size of the matter effect for the medium baseline 8The exact oscillation probability were calculated using [ 3, 4 ]. 0 6 ( 2 0 1 6 ) 0 5 1 DUNE: NO, δ = 3π/2 First min First max P (νμ → νe) experiments JUNO, [ 21 ], and RENO-50, [ 22, 23 ], a setup where the oscillation probabilities of [11] miss significant physics, since the L/E varies from 6 to 25 km/MeV. 5 In this paper we have further developed and expanded upon the recent perturbative framework for neutrino oscillations in uniform matter, introduced in [ 11 ]. The new oscillation probabilities are of the same simple, compact functional form with slightly more complicated coefficients, yet, the range of applicability now includes the whole L/E versus matter potential, a, plane, i.e. the restriction that L/E be small, (L/E ≪ 1/∆ m221) around the vacuum values of the matter potential has been completely removed. In fact, with these new improvements, the oscillation probabilities in vacuum are exact at zeroth order in our perturbative expansion. This occurs because the expansion parameter s12c12∆ m221/∆ me2e = 0.014 is further multiplied by s(φ−θ13), where φ is the mixing angle θ13 in matter. In vacuum, φ = θ13 and therefore all corrections to zeroth order vanish. To achieve this extended range of applicability, an additional rotation of the Hamiltonian is performed over that in [ 11 ]. The third angle ψ is the mixing angle θ12 in matter. In the resulting Hamiltonian, the diagonal elements are the eigenvalues of the zeroth order Hamiltonian and do not cross for any values of the matter potential, especially near the solar resonance (this occurred in [ 11 ]). The non-diagonal elements of the new Hamiltonian are the perturbing Hamiltonian for our perturbative expansion and their size is controlled by the small parameter s(φ−θ13)s12c12∆ m221/∆ me2e, mentioned in the previous paragraph. The new perturbative expansion is now well defined for all values of the matter potential and gives very accurate oscillation probabilities. We have performed many cross checks on the perturbative expansion, e.g. we have checked the CP violating term recovers, order by order, the known form. We have calculated the oscillation probabilities for zeroth, first, and second order in our expansion parameter. For most practical applications related to experiments, the zeroth order oscillation probabilities are sufficiently accurate with a typical fractional uncertainty of better than 10−3. Including the first and second order corrections the accuracy improves that to better than 10−6 and 10−9, respectively. J H P 0 6 ( 2 0 1 6 ) 0 5 1 Acknowledgments P.B.D. acknowledges support from the Fermilab Graduate Student Research Program in Theoretical Physics operated by Fermi Research Alliance, LLC. This work is also supported in part by DOE grant DE-SC0011981. H.M. thanks Instituto de F´ısica, Universidade de S˜ao Paulo for the great opportunity of stay under support by Funda¸c˜ao de Amparo `a Pesquisa do Estado de S˜ao Paulo (FAPESP) with grant number 2015/05208-4. He thanks Fermilab Theory Group for warm hospitality in his visits. S.P. acknowledges partial support from the European Union FP7 ITN INVISIBLES (Marie Curie Actions, PITN-GA-2011-289442). This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 690575-InvisiblesPlus RISE. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 674896-Elusives ITN. Fermilab is operated by the Fermi Research Alliance, LLC under contract no. DE AC02-07CH11359 with the U.S. Department of Energy. A Technical details A.1 Generalized approach to diagonalization We describe the diagonalization of a particular 2 × 2 submatrix and the angle and eigenvalues. This is the approach used twice in subsections 2.3 and 2.4 to diagonalize the 1-3 and then the 1-2 submatrices. Given a general symmetric 2 × 2 matrix we wish to diagonalize with angle φ, we write J H E P 0 6 ( 2 0 1 6 ) 0 5 1 (A.1) (A.2) (A.3) (A.4) (A.5) where thus Since trace and determinant are unchanged by the U sandwich, By squaring the trace equation and subtracting 4 times the determinant equation we have λσ λρ ! = U (φ)† λa λx! U (φ) , λx λc U (φ) ≡ cφ sφ! . −sφ cφ λσ + λρ = λa + λc and λρλσ = λaλc − λ2x . (λρ − λσ)2 = (λa − λc)2 + 4λ2x , λρ,σ = 21 (λa + λc) ± q(λa − λc)2 + 4λ2x . Next, we rewrite eq. (A.1) by left (right) multiplying by U (φ) (U †(φ)), then U (φ) λσ λρ ! U (φ)† = c2φλσ + s2φλρ sφcφ(λρ − λσ)! = sφcφ(λρ − λσ) s2φλσ + c2φλρ λa λx! . λx λc This gives us three equations, The last equation is the standard equation for s2φ. Subtracting (adding) the first two gives the standard equation for c2φ (the trace). Thus the rotation angle is defined by the following λx = (λρ − λσ)sφcφ and (λc − λa) = (λρ − λσ)(c2φ − s2φ) . In addition, using only c2φ + s2φ = 1 we can write down the following useful identities λa = c2φλσ + s2φλρ , λc = s2φλσ + c2φλρ , λx = (λρ − λσ)sφcφ . c2φ = λρ − λa = λc − λσ , λρ − λσ λρ − λσ s2φ = λρ − λc = λa − λσ , λρ − λσ λρ − λσ which are used extensively throughout this paper. This set of operations will be used both for φ and ψ rotations. A.2 Useful identities From the trace and determinant identities, see eq. (A.3), λ− + λ+ = λa + λc , λ1 + λ2 = λ− + λ0 , λ+λ− = λaλc − ∆ me2ec13s13 2 , 2 λ1λ2 = λ0λ− − ǫ∆ me2ec12s12c(φ−θ13) , c(φ−θ13)s(φ−θ13) = s13c13 ∆ λa+− , a s(φ−θ13) ≈ s13c13 ∆ me2e . where we recall that the λa,b,c in the tilde basis are defined in eq. (2.12). Another useful relation is then for a ≪ ∆ me2e, A.3 Limits We list the values of the angles and the eigenvalues in vacuum and for a → ±∞ in table 4. (A.6) (A.7) (A.8) (A.9) (A.10) (A.11) (A.12) (A.13) (A.14) (A.15) J H E P 0 6 ( 2 0 1 6 ) 0 5 1 λa (λc) λb λb λc (λa) λa (λc) λc (λa) +∞ π/2 (0) π/2 λc (λa) λb λb λa (λc) λc (λa) λa (λc) The characteristic equation for neutrino oscillation in matter is λ3 − ∆ m221 + ∆ m321 + a λ2 + ∆ m221∆ m321 + a (c122 + s122s213)∆ m221 + c123∆ m321 − ac122c123∆ m221∆ m321 = 0 . The coefficient of the λ2 term is the sum of the eigenvalues, the coefficient of the λ term is the sum of pairs of the eigenvalues, and the coefficient of the λ0 term is the triple product of eigenvalues. We now verify that our matter mass eigenvalues satisfy these expressions to second order. First, the λ−,0,+ eigenvalues satisfy the first requirement exactly as was discussed in [ 11 ]. Since Pi=1,2,3 λi = Pi=−,0,+ λi, so the λ1,2,3 eigenvalues also satisfy the first requirement. Also, from eq. (3.3), Pi=1,2,3 λi(2) = 0, so the λ1,2,3 eigenvalues also satisfy the first requirement exactly through second order. We have also verified that each of the other two conditions are satisfied to second order. A.5 Unitarity of the W matrix We verify that the V matrix satisfies the unitarity requirements, V V † = ✶ . UMmNS is unitary by definition. Then we just need that the W matrix is unitary. The zeroth order requirement is W0W0† = ✶ which is immediately satisfied since W0 = ✶ . At first order the requirement is W1 + W1† = 0. This is equivalent, to the requirement that W1 is antiHermitian, or that Hˇ1 is Hermitian, which they are, respectively, see eq. (3.8). To second order, the unitarity requirement becomes, W2 + W2† = −W12. That is, that the Hermitian part of W2 must be −W12/2, which it is. An additional anti-Hermitian part is unconstrained and is calculated through perturbation theory. H P 0 6 ( 2 0 1 6 ) 0 5 1 V -matrix, S-matrix comparison In the S-matrix method, the oscillation probabilities are given by, see for example [ 11 ], SS(L) = UMmNS e−iH0LΩ( L) (UMmNS)† Ω( L) = 1 + (−i) dx eiH0xH1e−iH0x Z L 0 + (−i)2 Z L dx eiH0xH1e−iH0x Z x dx′ eiH0x′ H1e−iH0x′ + · · · , 0 0 where H0 and H1 are given by eqs. (2.23) and (2.24). (We drop the “check” in this appendix.) In the V-matrix method, used in this paper, the oscillation probabilities are given by, SV (L) = UMmNS W e−iΛL/2E W † (UMmNS)† (Λ)ij = δij (λi + λi(1) + λi(2) + · · · ) W = 1 + W1 + W2 + · · · , where the λi/2E are the eigenvalues of H0. λi(n) and Wn are given by n-th order perturbation theory. Specializing to the case when the perturbing Hamiltonian has no diagonal elements, (H1)ij = (1 − δij )hij /2E , which is relevant for the perturbation discussed in this paper, W can be calculated from eq. (3.8) for first order and eq. (3.9) for second order. Then it is trivial to show that to first order, [(UMmNS)†SS(L)UMmNS]ij = [(UMmNS)†SV (L)UMmNS]ij hij = δij e−iλiL/2E + (1 − δij ) ∆ λij e−iλiL/2E − e−iλjL/2E . (A.20) We have also checked that they are equal at second order. As this is just a consistency check of perturbation theory, we postulate that it is true to all orders, without presenting an all orders proof. A.7 CP violating term It is useful to rewrite the numerator of eq. (4.9) as ǫ(∆ me2e)3(1 − ǫ cos 2θ12 − ǫ2c212s122). We evaluate Deμ through first order, keeping terms that are explicitly second order in ǫ, noting that dividing by ∆ λ21 introduces an additional factor of ǫ in vacuum. 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Peter B. Denton, Hisakazu Minakata, Stephen J. Parke. Compact perturbative expressions for neutrino oscillations in matter, Journal of High Energy Physics, 2016, 51, DOI: 10.1007/JHEP06(2016)051