On the exact formula for neutrino oscillation probability by Kimura,
  Takamura and Yokomakura
On the exact formula for neutrino oscillation probability by
Kimura, Takamura and Yokomakura
Osamu Yasuda∗
Department of Physics, Tokyo Metropolitan University,
Minami-Osawa, Hachioji, Tokyo 192-0397, Japan
Abstract
The exact formula for the neutrino oscillation probability in matter with constant density, which
was discovered by Kimura, Takamura and Yokomakura, has been applied mostly to the standard
case with three flavor neutrino so far. In this paper applications of their formula to more general
cases are discussed. It is shown that this formalism can be generalized to various cases where the
matter potential have off-diagonal components, and the two non-trivial examples are given: the
case with magnetic moments and a magnetic field and the case with non-standard interactions. It
is pointed out that their formalism can be applied also to the case in the long baseline limit with
matter whose density varies adiabatically as in the case of solar neutrino.
PACS numbers: 14.60.Pq, 14.60.St
∗Electronic address: yasuda˙at˙phys.metro-u.ac.jp
Typeset by REVTEX 1
http://arxiv.org/abs/0704.1531v2
mailto:yasuda_at_phys.metro-u.ac.jp
I. INTRODUCTION
Neutrino oscillations in matter (See, e.g., Ref. [1] for review.) have been discussed by
many people in the past because the oscillation probability has non-trivial behaviors in
matter and due to the matter effect it may exhibit non-trivial enhancement which could
be physically important. Unfortunately, it is not easy to get an analytical formula for the
oscillation probability in the three flavor neutrino scheme in matter, and investigation of
its behaviors has been a difficult but important problem in the phenomenology of neutrino
oscillations. In 2002 Kimura, Takamura and Yokomakura derived a nice compact formula [2,
3] for the neutrino oscillation probability in matter with constant density. Basically what
they showed is that the quantity Ũ∗αjŨβj , which is a factor crucial to express the oscillation
probability analytically, can be expressed as a linear combination of U∗αjUβj, where Ũαj and
Uαj stand for the matrix element of the MNS matrix in matter and in vacuum, respectively.
However, their formula is only applicable to the standard three flavor case. In this pa-
per we show that their result can be generalized to various cases. We also show that their
formalism can be applied also to the case with slowly varying matter density in the limit
of the long neutrino path. In Sect. II, we review briefly some aspects of the oscillation
probabilities, including a simple derivation for the formula by Kimura, Takamura and Yoko-
makura which was given in Ref. [4], because these are used in the following sections. Their
formalism is generalized to the various cases where the matter potential has off-diagonal
components, and we will discuss the case with large magnetic moments and a magnetic field
(Sect. III) and the case with non-standard interactions (Sect. IV). In Sect. V we summarize
our conclusions.
II. GENERALITIES ABOUT OSCILLATION PROBABILITIES
A. The case of constant density
It has been known [5] (See also earlier works [6, 7, 8].) that after eliminating the nega-
tive energy states by a Tani-Foldy-Wouthusen-type transformation, the Dirac equation for
neutrinos propagating in matter is reduced to the familiar form:
UEU−1 +A(t)
Ψ, (1)
where
E ≡ diag (E1, E2, E3) ,
A(t) ≡
2GFdiag (Ne(t)−Nn(t)/2,−Nn(t)/2,−Nn(t)/2) ,
ΨT ≡ (νe, νµ, ντ ) is the flavor eigenstate, U is the Maki-Nakagawa-Sakata (MNS) matrix,
m2j + ~p
(j = 1, 2, 3) is the energy eigenvalue of each mass eigenstate, and the matter
effect A(t) at time (or position ) t is characterized by the density Ne(t) of electrons and the
one Nn(t) of neutrons, respectively. Throughout this paper we assume for simplicity that
the density of matter is either constant or slowly varying so that its derivative is negligible.
The 3× 3 matrix on the right hand side of Eq. (1) can be formally diagonalized as:
UEU−1 +A(t) = Ũ(t)Ẽ(t)Ũ−1(t), (2)
where
Ẽ(t) ≡ diag
Ẽ1(t), Ẽ2(t), Ẽ3(t)
is a diagonal matrix with the energy eigenvalues Ẽj(t) in the presence of the matter effect.
First of all, let us assume that the matter density A(t) is constant. Then all the t
dependence disappears and Eq. (1) can be easily solved, resulting the flavor eigenstate at
the distance L:
Ψ(L) = Ũ exp
−iẼL
Ũ−1Ψ(0). (3)
Thus the oscillation probability P (να → νβ) is given by
P (να → νβ) =
Ũ exp (−iEL) Ũ−1
= δαβ − 4
∆ẼjkL
∆ẼjkL
, (4)
where we have defined
j ≡ ŨαjŨ∗βj ,
∆Ẽjk ≡ Ẽj − Ẽk,
and throughout this paper the indices α, β = (e, µ, τ) and j, k = (1, 2, 3) stand for those
of the flavor and mass eigenstates, respectively. Once we know the eigenvalues Ẽj and the
quantity X̃
j , the oscillation probability can be expressed analytically.
B. The case of adiabatically varying density
Secondly, let us consider the case where the density of the matter varies adiabatically as
in the case of the solar neutrino deficit phenomena. In this case, instead of Eq. (3), we get
Ψ(L) = Ũ(L) exp
Ẽ(t) dt
Ũ(0)−1Ψ(0),
where Ũ(0) and Ũ(L) stand for the effective mixing matrices at the origin t = 0 and at the
end point t = L. The oscillation probability is given by
P (να → νβ) =
Ũ(L) exp
Ẽ(t) dt
Ũ(0)−1
Ũ(L)βjŨ(L)
βkŨ(0)
αjŨ(0)αk exp
∆Ẽ(t)jk dt
. (5)
1 In the standard case with three flavors of neutrinos in matter, the energy eigenvalues Ẽj can be analytically
obtained by the root formula for a cubic equation [9]. So the only non-trivial problem in the standard
case is to obtain the expression for X̃
j , and this was done by Kimura, Takamura and Yokomakura [2, 3].
In general cases, however, the analytic expression for Ẽj is very difficult or impossible to obtain, and we
will discuss below only examples in which the analytic expression for Ẽj is known.
Eq. (5) requires in general the quantity like Ũ(t)βjŨ
∗(t)βk which has the same flavor index
β but different mass eigenstate indices j, k, and it turns out that the analytical expression
for Ũ(t)βjŨ
∗(t)βk is hard to obtain. However, if the length L of the neutrino path is very
large and if |
0 ∆Ẽ(t)jk dt| ≫ 1 is satisfied for j 6= k, as in the case of the solar neutrino
deficit phenomena, after averaging over rapid oscillations Eq. (5) is reduced to
P (να → νβ) =
j (L)X̃
j (0),
where we have defined
X̃ααj (t) ≡
∣Ũ(t)αj
In the case of the solar neutrinos deficit process νe → νe during the daylight, X̃ββj (L) at the
end point t = L and X̃ααj (0) at the origin t = 0 correspond to X
j in vacuum and [X̃
at the center of the Sun, respectively, where
j ≡ UαjU∗βj
ŨαjŨ
are bilinear products of the elements of the mixing matrices in vacuum and at the center of
the Sun, respectively. Thus we obtain
P (νe → νe) =
X̃eej
Hence we see that evaluation of the quantity X̃ααj in the presence of the matter effect is
important not only in the case of constant matter density but also in the case of adiabatically
varying density.
C. Another derivation of the formula by Kimura, Takamura and Yokomakura
In this subsection a systematic derivation of their formula is given because such a deriva-
tion will be crucial for the generalizations in the following sections.2 The arguments are
based on the trivial identities. From the unitarity condition of the matrix Ũ , we have
δαβ =
Ũ Ũ−1
ŨαjŨ
j . (6)
Next we take the (α, β) component of the both hand sides in Eq. (2):
UEU−1 +A
Ũ ẼŨ−1
ŨαjẼjŨ
ẼjX̃
j (7)
2 The argument here is the same as that in Ref. [4]. Since this derivation does not seem to be widely
known, it is reviewed here.
Furthermore, we take the (α, β) component of the square of Eq. (2):
UEU−1 +A
Ũ Ẽ2Ũ−1
ŨαjẼ
Ẽ2j X̃
j (8)
Putting Eqs. (6)–(8) together, we have
1 1 1
Ẽ1 Ẽ2 Ẽ3
Ẽ21 Ẽ
[UEU−1 +A]αβ
(UEU−1 +A)2
which can be easily solved by inverting the Vandermonde matrix:
∆Ẽ21∆Ẽ31
(Ẽ2Ẽ3, −(Ẽ2 + Ẽ3), 1)
∆Ẽ21∆Ẽ32
(Ẽ3Ẽ1, −(Ẽ3 + Ẽ1), 1)
∆Ẽ31∆Ẽ32
(Ẽ1Ẽ2, −(Ẽ1 + Ẽ2), 1)
[UEU−1 +A]αβ
(UEU−1 +A)2
. (9)
[(UEU−1 +A)j]αβ (j = 1, 2) on the right hand side are given by the known quantities:
UEU−1 +A
j + Aδαeδβe
UEU−1 +A
j + A
j + δβeX
+ A2 δαeδβe.
It can be shown that Eq. (9) coincides with the original results by Kimura, Takamura and
Yokomakura [2, 3].
A remark is in order on Eq. (9). Addition of a matrix c1 to Eq. (2) where c is a constant
and 1 is the identity matrix, or in other words, the shift
Ej → Ej + c (j = 1, 2, 3), (10)
should give the same result for X̃
j (j = 1, 2, 3), since Eq. (10) only affects the overall
phase of the oscillation amplitude and the phase has to disappear in the probability. It is
easy to show that the shift (10) indeed gives the same result as Eq. (9). The proof is given
in Appendix A. In practical calculations below, we will always put c = −E1, i.e., we will
consider the mass matrix U(E −E11)U−1 +A instead of the original one UEU−1 +A, since
all the diagonal elements (E − E11)jj = ∆Ej1 = ∆m2j1/2E are expressed in terms of the
relevant variables ∆m2j1, and therefore calculations become simpler. To save space, however,
we will use the matrix UEU−1 +A in most of the following discussions.
D. The case with arbitrary number of neutrinos
It is straightforward to generalize the discussions in sect. II C to the case with arbitrary
number of neutrinos where the matter potential is diagonal in the flavor eigenstate. The
scheme with number of sterile neutrinos is one of the example of these cases [4, 10]. The
time evolution of such a scheme with N neutrino flavors is described by
UNENU−1N +AN
where ΨTN ≡ (να1 , να2 , · · · , ναN ) is the flavor eigenstate,
EN ≡ diag (E1, E2, · · · , EN) (11)
is the energy matrix of the mass eigenstate,
AN ≡ diag (A1, A2, · · · , AN) ,
is the potential matrix for the flavor eigenstate, and UN is the N × N MNS matrix. As in
the previous sect., by taking the α, β components, we get
Ẽmj X̃
UNENU−1N +AN
for m = 0, · · · , N − 1,
which leads to the simultaneous equation
1 1 · · · 1
Ẽ1 Ẽ2 · · · ẼN
ẼN−11 Ẽ
2 · · · ẼN−1N
UNEU−1N +AN
UNEU−1N +AN
. (12)
Eq. (12) can be solved by inverting the N ×N Vandermonde matrix VN :
= V −1N
UNENU−1N +AN
UNENU−1N +AN
. (13)
The determinant of VN is the Vandermonde determinant
j<k ∆Ẽjk, and therefore V
can be analytically obtained as long as we know the value of Ẽj. The factors [(UNENU−1N +
AN)j]αβ on the right hand side of Eq. (13) can be expressed as functions of the energy Ej , the
quantity X
j in vacuum and the matter potential Aγ , since the matrix (UNENU−1N +AN)j is a
sum of products of the matrices [(UNENU−1N )ℓ]γδ =
k (0 ≤ ℓ ≤ j) and [(AN)m]ǫη =
mδǫη (0 ≤ m ≤ j). From Eq. (13) it is clear that enhancement of the oscillation
probability due to the matter effect occurs only when some of ∆Ẽjk becomes small.
III. THE CASE WITH LARGE MAGNETIC MOMENTS AND A MAGNETIC
FIELD
So far we have assumed that the potential term is diagonal in the flavor basis. We can
generalize the present result to the cases where we have off-diagonal potential terms. One of
such examples is the case where there are only three active neutrinos with magnetic moments
and the magnetic field (See, e.g., Ref. [1] for review.). The hermitian matrix3
UEU−1 B
B† U∗E(U∗)−1
B ≡ B µαβ
is the mass matrix for neutrinos and anti-neutrinos without the matter effect where neutrinos
have the magnetic moments µαβ in the magnetic field B. Here we assume the magnetic
interaction of Majorana type
µαβ ν̄α Fλκσ
λκ νcβ + h.c., (15)
and in this case the magnetic moments µαβ are real and anti-symmetric in flavor indices:
µαβ = −µβα.
If the magnetic field is constant, then the oscillation probability can be written as
P (νA → νB) = δAB − 4
X̃ABJ X̃
∆ẼJKL
X̃ABJ X̃
∆ẼJKL
, (16)
where A,B run e, µ, τ, ē, µ̄, τ̄ , and J,K run 1, · · ·, 6, respectively, and X̃ABJ ≡ UAJU∗BJ . ẼJ
(J = 1, · · · , 6) are the eigenvalues of the 6×6 matrix M. On the other hand, if the magnetic
field varies very slowly and if the length L of the baseline is so long that |∆ẼJKL| ≫ 1 is
satisfied for J 6= K, then the oscillation probability is given by
P (νA → νB) =
X̃BBJ (L)X̃
J (0). (17)
Following the same arguments as before, the quantity X̃ABJ is given by inverting the 6 × 6
Vandermonde matrix V6:
X̃AB1
X̃AB2
X̃AB6
= V −16
[M]AB
. (18)
3 See [5] for derivation of Eq. (14) from the Dirac Eq.
As in the previous sections, [(M)J ]AB (J = 0, · · · , 5) on the right hand side of Eq. (18) can
be expressed in terms of the known quantities XABK and BCD, and Eqs. (16) and (18) are
useful only when we know the eigenvalues ẼJ .
To demonstrate the usefulness of these formulae, let us consider the case where the
magnetic field is large at origin but is zero at the end point and the magnetic field varies
adiabatically. For simplicity we assume that θ13 and all the CP phases vanish.
4 In this
case the 6 × 6 matrix M in Eq. (14) becomes real, and we obtain the following oscillation
probabilities:
P (να → νβ) = P (ν̄α → ν̄β) =
(Uβj)
2[Re Ũ(0)αj]
P (να → ν̄β) = P (ν̄α → νβ) =
(Uβj)
2[Im Ũ(0)αj]
2, (19)
where Ũ(0) the 3× 3 unitary matrix which diagonalizes the 3× 3 matrix UEU−1 + iB(0) at
the origin:
UEU−1 + iB(0) = Ũ(0)Ẽ(0)Ũ−1(0).
In this example the energy eigenvalues are degenerate, i.e., the 6 × 6 energy matrix be-
comes diag(Ẽ , Ẽ), and the oscillation probability differs from Eq. (17) because the condition
|∆ẼJKL| ≫ 1(J 6= K) is not satisfied (e.g., ∆ẼJK = 0 not only for J = K = 1 but also for
J = 1, K = 4). Each probability in Eqs. (19) itself is not expressed in terms of X̃ααj (0), but
we find that the following relation holds:
P (να → νβ) + P (ν̄α → νβ) =
(Uβj)
2|Ũ(0)αj|2 =
j (0). (20)
Eq. (20) is a new result and without the present formalism it would be hard to derive it.
The details of derivation of Eq. (19) and explicit forms of X̃ααj (0) are given in Appendix B.
Eq. (20) may be applicable to the case where high energy astrophysical neutrinos, which are
produced in a relatively large magnetic field, are observed on the Earth, on the assumption
that the fluxes of neutrinos and anti-neutrinos are almost equal.
IV. THE CASE WITH NON-STANDARD INTERACTIONS
Another interesting application is the oscillation probability in the presence of new physics
in propagation [11, 12]. In this case the mass matrix is given by
UEU−1 +ANP (21)
4 In the presence of the magnetic interaction (15) of Majorana type, the two CP phases, which are absorbed
by redefinition of the charged lepton fields in the standard case, cannot be absorbed and therefore become
physical. Here, however, we will assume for simplicity that these CP phases vanish.
where
ANP ≡
2GFNe
1 + ǫee ǫeµ ǫeτ
ǫ∗eµ ǫµµ ǫµτ
ǫ∗eτ ǫ
µτ ǫττ
The dimensionless quantities ǫαβ stand for possible deviation from the standard matter effect.
Also in this case the oscillation probability is given by Eqs. (4) and (9), where the standard
potential matrix A has to be replaced by ANP . The extra complication compared to the
standard case is calculations of the eigenvalues Ẽj and the elements [(UEU−1 + ANP )m]αβ
(m = 1, 2).
Again to demonstrate the usefulness of the formalism, here we will discuss for simplicity
the case in which the eigenvalues are the roots of a quadratic equation. It is known [13] that
the constraints on the three parameters ǫee, ǫeτ , ǫττ from various experimental data are weak
and they could be as large as O(1). In Ref. [14] it was found that large values (∼ O(1)) of
the parameters ǫee, ǫeτ , ǫττ are consistent with all the experimental data including those of
the atmospheric neutrino data, provided that one of the eigenvalues of the matrix (21) at
high energy limit, i.e., ANP , becomes zero. Simplifying even further, here we will neglect the
parameters ǫeµ, ǫµµ, ǫµτ which are smaller than O(10−2) and we will consider the potential
matrix
ANP = A
1 + ǫee 0 ǫeτ
0 0 0
ǫ∗eτ 0 ǫττ
, (22)
where A ≡
2GFNe, the three parameters ǫee, ǫeτ , ǫττ are constrained in such a way that
two of the three eigenvalues become zero. We will assume that Ne is constant, and we
will take the limit ∆m221 → 0. The oscillation probability P (νµ → νe) in this case can be
analytically expressed and is given by
P (νµ → νe) = −4Re
− 4Re
(Λ+ − Λ−L)L
8A(∆E31)
Λ+Λ−(Λ+ − Λ−)
|ǫeτXeµ3 X
3 | sin(arg(ǫeµ) + δ)
× sin
(Λ+ − Λ−)L
. (23)
Eq. (23) is another new result and it would be difficult to obtain it without using the
present formalism. The details of derivation of Eq. (23), explanation of the notations and
the explicit forms of all the variables in Eq. (23) are described in Appendix C.
V. CONCLUSIONS
The essence of the exact formula for the neutrino oscillation probability in constant
matter which was discovered by Kimura, Takamura and Yokomakura lies in the fact that
the combination X̃
j ≡ ŨαjŨβj∗ of the mixing matrix elements in matter can be expressed
as polynomials in the same quantity X
j ≡ UαjUβj∗ in vacuum. In this paper we have
discussed applications of their formalism to more general cases. We have pointed out that
their formalism can be useful for the cases in matter not only with constant density but
also with density which varies adiabatically as in the case of the solar neutrino problem,
after taking the limit of the long neutrino path. We have shown that their formalism can be
generalized to the cases where the matter potential has off-diagonal components. As concrete
non-trivial examples, we discussed the case with magnetic moments and a magnetic field,
and the case with non-standard interactions. The application of the present formalism to the
case with unitarity violation has been discussed elsewhere [15]. The formalism by Kimura,
Takamura and Yokomakura is quite general and can be applicable to many problems in
neutrino oscillation phenomenology.
APPENDIX A: PROOF THAT EQ. (10) GIVES THE SAME (9)
In this appendix we show that Eq. (10) gives the same result for X̃
j (j = 1, 2, 3). The
value of X̃
j (j = 1, 2, 3) for
Ẽ + c1
Ũ−1 = UEU−1 +A+ c1
becomes at most quadratic5 in c, and all one has to do is to show that the coefficients of the
terms linear and quadratic in c vanish. Let us introduce the notation
1 1 1
Ẽ1 + c Ẽ2 + c Ẽ3 + c
(Ẽ1 + c)
2 (Ẽ2 + c)
2 (Ẽ3 + c)
≡ (V −1)(0) + c(V −1)(1) + c2(V −1)(2)
[UEU−1 +A+ c1]αβ
(UEU−1 +A+ c1)2
≡ ~B(0) + c ~B(1) + c2 ~B(2),
where V (k) is the coefficient of the inverted Vandermonde matrix which is k-th order in c,
and B
j is the coefficient of the vector (UEU−1 +A+ c1)
which is k-th order in c. Then
the terms linear in c are given by
(V −1)(1) ~B(0) + (V −1)(0) ~B(1)
∆Ẽ21∆Ẽ31
(Ẽ2 + Ẽ3, −2, 0)
∆Ẽ21∆Ẽ32
(−(Ẽ3 + Ẽ1), +2, 0)
∆Ẽ31∆Ẽ32
(Ẽ1 + Ẽ2, −2, 0)
[UEU−1 +A]αβ
(UEU−1 +A)2
5 Notice that all the factors ∆Ẽjk are invariant under the shift (10), and the only change by this shift
comes either from the terms ẼjẼk or from Ẽj + Ẽk in the inverse of the Vandermonde matrix (cf. Eq.
(9)). Hence the difference by Eq. (10) is at most quadratic in c.
∆Ẽ21∆Ẽ31
(+Ẽ2Ẽ3, −(Ẽ2 + Ẽ3), +1)
∆Ẽ21∆Ẽ32
(−Ẽ3Ẽ1, +(Ẽ3 + Ẽ1), −1)
∆Ẽ31∆Ẽ32
(+Ẽ1Ẽ2, −(Ẽ1 + Ẽ2), +1)
2 [UEU−1 +A]αβ
 = 0,
and the terms quadratic in c are given by
(V −1)(2) ~B(0) + (V −1)(1) ~B(1) + (V −1)(0) ~B(2)
∆Ẽ21∆Ẽ31
(+1, 0, 0)
∆Ẽ21∆Ẽ32
(−1, 0, 0)
∆Ẽ31∆Ẽ32
(+1, 0, 0)
[UEU−1 +A]αβ
(UEU−1 +A)2
∆Ẽ21∆Ẽ31
(Ẽ2 + Ẽ3, −2, 0)
∆Ẽ21∆Ẽ32
(−(Ẽ3 + Ẽ1), +2, 0)
∆Ẽ31∆Ẽ32
(Ẽ1 + Ẽ2, −2, 0)
2 [UEU−1 +A]αβ
∆Ẽ21∆Ẽ31
(+Ẽ2Ẽ3, −(Ẽ2 + Ẽ3), +1)
∆Ẽ21∆Ẽ32
(−Ẽ3Ẽ1, +(Ẽ3 + Ẽ1), −1)
∆Ẽ31∆Ẽ32
(+Ẽ1Ẽ2, −(Ẽ1 + Ẽ2), +1)
 = 0.
Thus X̃
j (j = 1, 2, 3) is independent of c, as is claimed.
APPENDIX B: DERIVATION OF EQ. (19)
The matrix (14) can be rewritten as
M = 1
UEU−1 + iB 0
0 UEU−1 − iB
1 −i1
−i1 1
so the problem of diagonalizing the 6 × 6 matrix (14) is reduced to diagonalizing the 3× 3
matrices UEU−1 ± iB. Since we are assuming that θ13 and all the CP phases vanish, all
the matrix elements Uαj and Bαβ = −Bβα are real, UEU−1 ± iB can be diagonalized by a
unitary matrix and its complex conjugate:
UEU−1 + iB = Ũ ẼŨ−1
UEU−1 − iB = Ũ∗Ẽ(Ũ∗)−1.
Therefore, we can diagonalize M by a 6× 6 unitary matrix Ũ as
M = Ũ
Ũ−1,
where
Ũ = 1√
1 −i1
−i1 1
0 Ũ∗
Ũ − iŨ∗
−iŨ Ũ∗
We note in passing that the reason why diagonalization of the 6 × 6 matrix is reduced to
that of the 3× 3 matrix is because the two matrices UEU−1 and B are real.
On the other hand, without a magnetic field the 6× 6 unitary matrix U is given by
where the CP phase δ has dropped out because θ13 = 0. From these we can integrate the
equation of motion and we get the fields at the end point:
Ψc(L)
= Ũ(L)
e−iΦ 0
0 e−iΦ
Ũ(0)−1
Ψc(0)
Ue−iΦŨ−1 + U∗e−iΦ(Ũ∗)−1 −i(Ue−iΦŨ−1 − U∗e−iΦ(Ũ∗)−1)
i(Ue−iΦŨ−1 − U∗e−iΦ(Ũ∗)−1) Ue−iΦŨ−1 + U∗e−iΦ(Ũ∗)−1
Ψc(0)
where
Ẽ(t) dt,
and we have assumed that a large magnetic field exists at the origin whereas there is no
magnetic field at the end point. Thus the oscillation probabilities for the adiabatic transition
are give by:
P (να → νβ) = P (ν̄α → ν̄β) = lim
Ue−iΦŨ−1 + U∗e−iΦ(Ũ∗)−1
|Uβj|2
Re(Ũαj)
P (ν̄α → νβ) = P (να → ν̄β) = lim
Ue−iΦŨ−1 − U∗e−iΦ(Ũ∗)−1
|Uβj|2
Im(Ũαj)
Hence we obtain the following relation:
P (να → νβ) + P (ν̄α → νβ) = P (να → νβ) + P (να → ν̄β) =
|Uβj |2|Ũαj |2.
To get |Ũαj |2, we need the explicit expression for the eigenvalues and the quantity X̃ααj in the
presence of a magnetic field. In the following we will subtract E11 from the energy matrix E
because it will only change the phase of the oscillation amplitude. For simplicity we will put
θ13 = 0, θ23 = π/4, and we will consider the limit ∆m
21 → 0. Defining ∆Ejk ≡ ∆m2jk/2E
Bαβ = Bµαβ ≡
0 −p −q
p 0 −r
q r 0
we have the eigenvalue equation
0 = |λ1− U(E − E11)U−1 − iB|
= λ3 −∆E31λ2 − (p2 + q2 + r2)λ+
(p− q)2. (B1)
The three roots of the cubic equation (B1) are given by
λ1 = 2R cosϕ+
, λ2 = 2R cos(ϕ+
, λ3 = 2R cos(ϕ−
where
R ≡ [(∆E31/3)2 + (p2 + q2 + r2)/3]3/2,
ϕ ≡ (1/3) cos−1
{(∆E31/3)3 +∆E31(p2 + q2 + r2)/6−∆E31(p− q)2/4}/R
The quantity X̃ααj in the presence of a magnetic field is given by
X̃αα1
X̃αα2
X̃αα3
∆λ21∆λ31
(λ2λ3, −(λ2 + λ3), 1)
∆λ21∆λ32
(λ3λ1, −(λ3 + λ1), 1)
∆λ31∆λ32
(λ1λ2, −(λ1 + λ2), 1)
Y αα2
Y αα3
 , (B2)
where
Y αα2 =
U(E −E11)U−1 + iB
= ∆E31X
0 (α = e)
∆E31/2 (α = µ, τ)
Y αα3 =
U(E −E11)U−1 + iB
= (∆E31)
2Xαα3 − (B2)αα
q2 + r2 (α = e)
r2 + p2 + (∆E31)
2/2 (α = µ)
p2 + q2 + (∆E31)
2/2 (α = τ)
. (B4)
In evaluating Y ααj , we have used the facts θ13 = 0, θ23 = π/4, ∆E21 = 0, Bαβ = −Bβα, and
that U(E −E11)U−1 is a symmetric matrix. Using all these results, it is straightforward to
obtain the explicit form for P (να → νβ) +P (ν̄α → νβ) by plugging the results of Eqs. (B2),
(B3), (B4) into the following (although calculations are tedious):
P (να → νe) + P (ν̄α → νe) = c212X̃αα1 + s212X̃αα2
P (να → νβ) + P (ν̄α → νβ) =
X̃αα1 +
X̃αα2 +
X̃αα3 (β = µ, τ),
where s12 ≡ sin θ12, c12 ≡ cos θ12.
APPENDIX C: DERIVATION OF EQ. (23)
The oscillation probability (23) is obtained in two steps. First we will obtain the eigenval-
ues of the matrix (21) with Eq. (22) and then we will plug the expressions for the eigenvalues
into Eq. (9) with A replaced by ANP given in Eq. (22).
Let us introduce notations for 3× 3 hermitian matrices:
0 −i 0
i 0 0
0 0 0
 , λ5 ≡
0 0 −i
0 0 0
i 0 0
 , λ7 ≡
0 0 0
0 0 −i
0 i 0
1 0 0
0 0 0
0 0 1
 , λ9 ≡
1 0 0
0 0 0
0 0 −1
where λ2, λ5 and λ7 are the standard Gell-Mann matrices whereas λ0 and λ9 are the notations
which are defined only in this paper. Simple calculations show that the matrix ANP in Eq.
(22) can be rewritten as
ANP = Aeiγλ9e−iβλ5
1 + ǫee + ǫττ
1 + ǫee − ǫττ
+ |ǫµτ |2
 eiβλ5e−iγλ9 , (C1)
where
β ≡ 1
tan−1
2|ǫeτ |2
1 + ǫee − ǫττ
γ ≡ 1
arg (ǫeµ).
From Eq. (C1) we see that the two potentially non-zero eigenvalues λe′ and λτ ′ of the matrix
(22) are given by
1 + ǫee + ǫττ
1 + ǫee − ǫττ
+ |ǫµτ |2
In order for this scheme to be consistent with the atmospheric neutrino data particularly at
high energy, which are perfectly described by vacuum oscillations, λτ ′ has to vanish [14]. In
this case, we have
tanβ =
|ǫeτ |
1 + ǫee
ǫττ =
|ǫeτ |2
1 + ǫee
λe′ = A(1 + ǫee)
|ǫeτ |2
(1 + ǫee)2
A(1 + ǫee)
cos2 β
Thus we have
ANP = Aeiγλ9e−iβλ5diag (λe′ , 0, 0) eiβλ5e−iγλ9 . (C2)
If we did not have β and γ, Eq. (C2) would be the same as the standard three flavor scheme
in matter, which was analytically worked out in Ref. [16] in the limit of ∆m221 → 0. It turns
out that, by redefining the parametrization of the MNS matrix Eq. (C2) can be also treated
analytically in the limit of ∆m221 → 0 as was done in Ref. [16]. The mass matrix can be
written as
UEU−1 +ANP = eiγλ9e−iβλ5
eiβλ5e−iγλ9UEU−1eiγλ9e−iβλ5 + diag (λe′, 0, 0)
eiβλ5e−iγλ9 .
Here we introduce the following two unitary matrices:
U ′ ≡ eiβλ5e−iγλ9 U
≡ diag(1, 1, eiargU ′τ3)U ′′ diag(eiargU ′e1 , eiargU ′e2 , 1),
where U is the 3× 3 MNS matrix in the standard parametrization [17] and U ′′ was defined
in the second line in such a way that the elements U ′′e1, U
e2, U
τ3 be real to be consistent with
the standard parametrization in Ref. [17] 6. Then we have
UEU−1 +ANP = eiγλ9e−iβλ5diag(1, 1, eiargU
U ′′EU ′′−1 + diag (λe′ , 0, 0)
×diag(1, 1, e−iargU ′τ3) eiβλ5e−iγλ9 . (C3)
Before proceeding further, let us obtain the expression for the three mixing angles θ′′jk and
the Dirac phase δ′′ in U ′′. Since
U ′ =
−iγUe1 + sβe
iγUτ1 cβe
−iγUe2 + sβe
iγUτ2 cβe
−iγUe3 + sβe
iγUτ3
Uµ1 Uµ2 Uµ3
−iγUτ1 − sβeiγUe1 cβe−iγUτ2 − sβeiγUe2 cβe−iγUτ3 − sβeiγUe3
where cβ ≡ cos β, sβ ≡ sin β, we get
θ′′13 = sin
−1 |U ′′e3| = sin−1 |cβe−iγUe3 + sβeiγUτ3|
θ′′12 = tan
−1(U ′′e2/U
e1) = tan
|cβe−iγUe2 + sβeiγUτ2|/|cβe−iγUe1 + sβeiγUτ1|
θ′′23 = tan
−1(U ′′µ3/U
τ3) = tan
Uµ3/|cβe−iγUτ3 − sβeiγUe3|
δ′′ = −argU ′′e3 = −arg (cβe−iγUe3 + sβeiγUτ3).
As was shown in Ref. [16], in the limit ∆m221 → 0, the matrix on the right hand side of Eq.
(C3) can be diagonalized as follows:
U ′′EU ′′−1 + diag (λe′ , 0, 0)−E11
= eiθ
λ7Γδ′′e
λ5Γ−1δ′′ e
λ2diag (0, 0,∆E31) e
−iθ′′
λ2Γδ′′e
−iθ′′
λ5Γ−1δ′′ e
−iθ′′
λ7 + diag (λe′, 0, 0)
= eiθ
λ7Γδ′′
λ5diag (0, 0,∆E31) + diag (λe′ , 0, 0)
Γ−1δ′′ e
−iθ′′
= eiθ
λ7Γδ′′e
iθ̃′′
λ5diag (Λ−, 0,Λ+) e
−iθ̃′′
λ5Γ−1δ′′ e
where Γδ′′ ≡ diag(1, 1, e−iδ
), ∆E31 ≡ ∆m231/2E, we have used the standard parametriza-
tion [17] U ′′ ≡ eiθ′′23λ7Γδ′′eiθ
λ5Γ−1δ′′ e
λ2 , and the eigenvalues Λ± are defined by
(∆E31 + λe′)±
(∆E31 cos 2θ
13 − λe′)
+ (∆E31 sin 2θ
6 The element U ′′τ2 has to be also real, but it is already satisfied because U
τ2 = Uτ2.
Having obtained the eigenvalues, by plugging these into Eq. (9) with A → ANP , Ẽ1 →
Λ−, Ẽ2 → 0, Ẽ3 → Λ+, we obtain X̃µe:
Λ−(Λ+ − Λ−)
(0, −Λ+, 1)
(−Λ+Λ−, −(Λ+ + Λ−), 1)
Λ+(Λ+ − Λ−)
(0, −Λ−, 1)
−Y µe3 + Λ+Y
Λ−(Λ+ − Λ−)
3 − (Λ+ + Λ−)Y
3 − Λ−Y
Λ+(Λ+ − Λ−)
where Y
j are defined by
UEU−1 +ANP
and are given by
2 = ∆E31 X
3 = [(∆E31)
2 + A(1 + ǫee)∆E31]X
3 + A∆E31ǫ
Furthermore, by introducing the notations
ξ ≡ [(∆E31)2 + A(1 + ǫee)∆E31]Uµ3|Ue3|
η ≡ A∆E31|ǫeτ |Uµ3Uτ3
ζ ≡ ∆E31Uµ3|Ue3|,
we can rewrite Y
2 = ζe
iδ and Y
3 = ξe
iδ + ηe−2iγ , where δ is the Dirac CP phase of the
MNS matrix U , so we have
Λ−(Λ+ − Λ−)
[ξ + ηe−i(2γ+δ) − Λ+ζ ]
[ξ + ηe−i(2γ+δ) − (Λ+ + Λ−)ζ ]
Λ+(Λ+ − Λ−)
[ξ + ηe−i(2γ+δ) − Λ−ζ ].
Notice that the phase factor eiδ in front of each X̃
j drops out in the oscillation probability
P (νµ → νe) because P (νµ → νe) is expressed in terms of X̃µej X̃
k , and the oscillation
probability (23) depends only on the combination 2γ + δ = arg (ǫeµ) + δ.
In the present case, the matrix Ũ is unitary and because of this three flavor unitarity all
the T violating terms are proportional to one factor:
∆ẼjkL
= 2 Im
∆Ẽ12L
− sin
∆Ẽ13L
+ sin
∆Ẽ23L
= −8 Im
∆Ẽ21L
∆Ẽ31L
∆Ẽ32L
This modified Jarlskog factor Im(X̃
2 ) in matter can be rewritten as
Im(X̃
2 ) =
Λ+Λ−(Λ+ − Λ−)
2 ) = −
ηζ sin(2γ + δ)
Λ+Λ−(Λ+ − Λ−)
= − A(∆E31)
Λ+Λ−(Λ+ − Λ−)
|ǫeτXeµ3 X
3 | sin(arg(ǫeµ) + δ).
This completes derivation of Eq. (23).
ACKNOWLEDGMENTS
The author would like to thank Alexei Smirnov for bringing my attention to Refs. [5,
6, 7, 8]. He would also like to thank He Zhang for calling my attention to Refs. [4, 10]
which were missed in the first version of this paper. This research was supported in part by
a Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture,
#19340062.
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http://arxiv.org/abs/hep-ph/0203099
http://arxiv.org/abs/hep-ph/0205295
http://arxiv.org/abs/hep-ph/0503118
http://arxiv.org/abs/hep-ph/0606040
http://arxiv.org/abs/hep-ph/0302093
http://arxiv.org/abs/hep-ph/0506143
http://arxiv.org/abs/hep-ph/0703098
http://arxiv.org/abs/hep-ph/9809205
	introduction
	generalities about oscillation probabilities 
	The case of constant density
	The case of adiabatically varying density
	Another derivation of the formula by Kimura, Takamura and Yokomakura 
	The case with arbitrary number of neutrinos 
	the case with large magnetic moments and a magnetic field 
	the case with non-standard interactions 
	conclusions 
	proof that Eq. (??) gives the same (??)
	Derivation of Eq. (??)
	Derivation of Eq. (??)
	Acknowledgments
	References
