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, 05315970 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 firstorder effects are taken into account in the zerothorder 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, Smatrix 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 solarscale ∆ m2⊙ to the atmosphericscale ∆ 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 “firstorder” terms into the “zerothorder” 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
reorganization 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 nondiagonal 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 –
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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 NuPert 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 nondiagonal elements are controlled by our small parameter, ǫ′, which
vanishes in vacuum. The nondiagonal 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 = Hνi ,
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/NuPert.
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)
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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 offdiagonal 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)
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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 (13) 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)
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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 (13) submatrix (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 (12) 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 (12) submatrix 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)
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It is easy to derive that4
with the diagonal elements the zeroth order Hamiltonian and the offdiagonal
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 nondiagonal 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 –
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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
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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)
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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 threeflavor 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 Vmatrix method and the Smatrix method for calculating the oscillation
probabilities is addressed in appendix A.6.
(3.8)
(3.9)
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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 nonzero 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 nth order in our perturbative expansion we
(ex) to the nth 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)
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from the position of the nonzero 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
NaumovHarrisonScott 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)
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1
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–
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
].
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DUNE: NO, δ = 3π/2
First min
First max
P (νμ → νe)
experiments JUNO, [
21
], and RENO50, [
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 nondiagonal 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.
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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 DESC0011981.
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 2015/052084. 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, PITNGA2011289442). This project has received funding from
the European Union’s Horizon 2020 research and innovation programme under the Marie
SklodowskaCurie grant agreement No 690575InvisiblesPlus RISE. This project has
received funding from the European Union’s Horizon 2020 research and innovation
programme under the Marie SklodowskaCurie grant agreement No 674896Elusives ITN.
Fermilab is operated by the Fermi Research Alliance, LLC under contract no.
DE
AC0207CH11359 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 13
and then the 12 submatrices.
Given a general symmetric 2 × 2 matrix we wish to diagonalize with angle φ, we write
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(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)
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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 antiHermitian part
is unconstrained and is calculated through perturbation theory.
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V matrix, Smatrix comparison
In the Smatrix 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 Vmatrix 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 nth 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.
(Deμ)(0) + (Deμ)(1) = sδJr ǫ(∆ me2e)3(1 − ǫ cos 2θ12) ,
∆ λ21∆ λ31∆ λ32
where Jr is the reduced Jarlskog factor, see ref. [
24
],
Jr ≡ c12s12c123s13c23s23 .
(A.17)
(A.18)
(A.19)
(A.21)
(A.22)
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The dropped higher order contribution to the numerator is
which is −ǫ2c212s122 in vacuum as desired since ∆ λ+− is ∆ me2e in vacuum.
Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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