Simple and compact expressions for neutrino oscillation probabilities in matter
Received: May
Simple and compact expressions for neutrino oscillation probabilities in matter
Hisakazu Minakata 0 2
Stephen J. Parke 0 1
0 P. O. Box 500, Batavia, IL 60510 , U.S.A
1 Theoretical Physics Department, Fermi National Accelerator Laboratory
2 Instituto de F sica, Universidade de Sa~o Paulo
We reformulate perturbation theory for neutrino oscillations in matter with an expansion parameter related to the ratio of the solar to the atmospheric Unlike previous works, we use a renormalized basis in which certain rstorder e ects are taken into account in the zerothorder Hamiltonian. We show that the new framework has an exceptional feature that leads to the neutrino oscillation probability in matter with the same structure as in vacuum to rst order in the expansion parameter. It facilitates immediate physical interpretation of the formulas, and makes the expressions for the neutrino oscillation probabilities extremely simple and compact. We nd, for example, that the e disappearance probability at this order is of a simple two avor form with an appropriately identi ed mixing angle and m2. More generally, all the oscillation probabilities can be written in the universal form with the channeldiscrimination coe cient of 0; functions of 23. Despite their simple forms they include all order e ects of 13 and all order e ects of the matter potential, to rst order in our expansion parameter.
Neutrino Physics; Beyond Standard Model

3
Results of our perturbative expansion for all oscillation probabilities
Mass squared eigenvalues in matter
The mixing angle 13 and mixing matrix in matter
Compact formulas for the oscillation probabilities in matter
3.3.1
3.3.2
3.3.3
3.3.4
3.3.5
3.3.6
e ! e disappearance channel
e !
An example of the power of our oscillation probabilities
Comparison with the existing perturbative frameworks
oscillation probabilities
4
Formulating the renormalized helioperturbation theory
Choosing the basis for the renormalized helioperturbation theory
Hat basis S matrix and the oscillation probability Mass eigenstate in matter: V matrix method
5
More about the renormalized helioperturbation theory
Exact versus zerothorder eigenvalues in matter
Eigenvalues in vacuum and in the asymptotic regions a !
1
Neutrino mixing matrix in matter and the
1 Introduction form matter 2 Structure of the neutrino oscillation probabilities in vacuum and in uni
Simplicity and compactness Raison d'^etre and the requirements for perturbative treatment
Neutrino oscillation based on the standard three avor scheme provides the best
possible theoretical framework available to date to describe most of the experimental results
obtained in the atmospheric, solar, reactor, and the accelerator neutrino experiments.
Although numerically calculated neutrino oscillation probabilities su ce to analyze
experimental data, understanding of the framework, in particular the one in matter [1], has not
yet reached a su cient level, in our opinion. Under the assumption of uniform matter
density distribution, the exact expressions of the eigenvalues, mixing angles, and the oscillation
probabilities in matter have been obtained [2{4]. Yet, the results for these quantities are
generally too complicated to facilitate understanding of the structure of the three
avor
neutrino oscillations in matter primarily due to the complexities of solving the cubic
eigenvalue characteristic equation. For a recent comprehensive treatment of neutrino oscillation
in the matter, see ref. [5].
parameter
in refs. [7{10].
Analytic approaches to the neutrino oscillation phenomenon, so far, are mostly based
on variety of perturbative frameworks. If the matter e ect is small one can treat it as a
small perturbation [6]. In the environments in which the matter e ect is comparable to
the vacuum mixing e ect, the only available small expansion parameter known to us is the
ratio of the solarscale
m2 to the atmosphericscale
m2 ,
m2 =
m2
' 0:03. sin 13
has been often used as an expansion parameter (there are enormous number of references,
q
see e.g., [7]), but it is now known that its value is not so small, sin 13 ' 0:15, which is of
the order of
m2 = m2 . Moreover, expansion around sin 13 = 0 misses the physics of
the resonance which exists at an energy around E
10 GeV for earth densities. Therefore,
it appears that the suitable perturbative framework is the one with the unique expansion
m2 = m2 . This framework was indeed examined in the past, to our knowledge
In this paper, we present a new framework of perturbative treatment of neutrino
oscillation in matter. We follow the reasoning stated above which led to identi cation of
the unique expansion parameter
m2 =
m2 . But, unlike the preceding works, we use a
\renormalized basis" as the basis of perturbation. That is, we absorb certain terms of order
to our \zerothorder" Hamiltonian around which we perturb. Or, in other word, we take
the zerothorder eigenvalues in matter such that it matches the exact eigenvalues to order
, see section 5. We will show that use of the renormalized basis makes the structure of the
perturbation theory exceptionally transparent, as we will explain in the next section. It
allows us to obtain simple, elegant and compact expressions for the oscillation probabilities,
which have a universal form even to rst order in our expansion parameter.
For example, e survival probability takes the form to order
as
denote the eigenvalues of the states which participate
the 13 level crossing. Despite its extremely simple form, P ( e !
e) in (1.1) takes into
account all order e ects of 13 and the matter potential. Since we will only consider terms
up to order , in this paper, our results here are not applicable to the region near the solar
MSW resonance [1, 11]. Our perturbative framework will be called as the \renormalized
helioperturbation theory" in the rest of this paper.
The section plan of this paper is somewhat unusual: in the next section 2 we summarize
the key features of the perturbative framework we develop in this paper. Then, in section 3
we describe the principle results of this work including the oscillation probabilities for all
channels in matter. This section does not describe the derivations but provides a self
contained summary of the results of this paper. Following this section, see section 4, we
present a systematic exposition of our perturbative framework and how the results of the
section 3 are derived. In the appendices A, B, and C we present, respectively, calculational
details of the S matrix, the results of oscillation probabilities to order in the standardized
form, and useful relationships to verify the equivalence of various expressions.
2
Structure of the neutrino oscillation probabilities in vacuum and in
uniform matter
probabilities for
following form1
We start by giving a precise de nition of what we mean by \simplicity and compactness"
of the expressions for the neutrino oscillation probabilities. Then, we explain the need for
reformulating perturbative treatment, followed by outlining \what is new" in this paper.
2.1
Simplicity and compactness
In vacuum and in matter with constant density, it is well known that the neutrino oscillation
for three avor mixing (i; j = 1; 2; 3) can be written in the
P (
!
) =
X V iV i e i 2iEL
4 X Re[V iV iV j V j ] sin2 ( j
i)L
2
=
( 3
4E
j>i
sin
2)L
( 2
4E
1)L
sin
( 1
4E
3)L
(2.1)
4E
:
!
3
i=1
+ 8 Im[V 1V 1V 2V 2] sin
The avor mixing matrix elements, V i, relate the avor states,
, to the eigenstates of
the Hamiltonian, i, with eigenvalues i=2E by
= V i i. Equation (2.1) applies both in
vacuum and in uniform matter since the matrix V diagonalizes the full Hamiltonian, which
includes the Wolfenstein matter potential, a [1]. In vacuum, i = mi2 where mi denotes
the mass of the ith neutrino state and V = UMNS.
Notice that the identical form of the oscillation probabilities in vacuum and in matter,
eq. (2.1), apart from replacement of
of the formulas in matter is as transparent as in vacuum.
mi2j by ( i
j ), implies that physical interpretation
1We have used for the last term in (2.1) Im[V 1V 1V 2V 2] = Im[V 2V 2V 3V 3] = Im[V 1V 1V 3V 3]
which follows from unitarity, and the identity (C.2).
{ 3 {
What is less well appreciated is that the expressions of the oscillation probabilities
in (2.1) are maximally simple and compact. That is, they contain 5 (including the constant
term) functions of L=E which are linearly independent. This property can be easily veri ed
by showing the Wronskian is nonvanishing, which implies that none of the functions can be
written in terms of the other functions for all L=E. Hence, in a true mathematical sense,
eq. (2.1) and it's equivalents give the simplest and most compact form for the complete set
of three avor oscillation probabilities in vacuum and in uniform matter.2
A special feature of the oscillation probabilities which is also worth noting is that each
term in eq. (2.1) factorizes into the characteristic sin [( j
i)L=4E] factor and the
products of the V matrix elements which control the amplitude of the oscillation. Both the
eigenvalues, i and the matrix elements of V are independent of the baseline, L, but are functions
of the mixing angles, 's, the
mj2i, and the product of the energy of neutrino times the
matter density via the matter potential.3 CP and T violation is described by the last term
in (2.1), which has the universal, channelindependent form in the three neutrino mixing.
2.2
Raison d'^etre and the requirements for perturbative treatment
With the simplest form of the oscillation probability eq. (2.1) and by knowing both the
exact form of the eigenvalues, i=2E, and the elements of the V matrix, see [3], one might
expect we have all that is needed for theoretical discussions. Unfortunately, the analytic
expressions for these eigenvalues, i=2E, as well as the V matrix elements are notoriously
complex and give no analytic insight into the oscillation physics in uniform matter. This is
true even when one of
m221, sin 13, sin 12 or a is set equal to zero. In any one of these
limits, the characteristic equation for the eigenvalues factorizes. But, the form of the general
solution does not simplify trivially to yield the correct eigenvalues, even though it must.
The structure of the general solutions of the cubic characteristic equation is the root cause
of this rather unusual behaviour. Hence, there is a need for a reformulated perturbative
framework so that we can obtain approximate but much simpler expressions for the
eigenvalues, i, and the mixing matrix elements, V i, which provide the necessary physics insight.
In this paper, we formulate perturbation theory by which we can calculate the
eigenvalues i and the elements of V matrix as a simple power series of the small expansion
parameter, a renormalized
m2 =
m2 . At the same time the structurerevealing form
of the oscillation probabilities (2.1) is kept intact. While the existing frameworks do not
satisfy the latter requirement, the key to the success in our case is due to the correct
decomposition of the Hamiltonian into the unperturbed and the perturbed terms. Use of
the renormalized zerothorder basis, allows us to correctly determine the eigenvalues to the
appropriate order in our expansion parameter. The resultant renormalized eigenvalues and
3
a /
2There are equivalent ways to write this set of oscillation probabilities with 5 independent L=E functions,
e.g one could use the trigonometric identity 2 sin2 x = 1
cos 2x to replace the sin2 x. Any other way of
identity (C.2) increases the number of independent functions by 2.
the mixing parameters e ectively absorb the additional terms that arise in the conventional
perturbative frameworks, as extra functions to the minimally required 5 linearly
independent functions. This occurs automatically and we believe that such a framework has never
before been formulated.4 Because of the structural simplicity of the L=E dependence, the
expressions of the oscillation probabilities calculated by our method are extremely simple
and compact, as will be fully demonstrated in the next section.
3
Results of our perturbative expansion for all oscillation probabilities
In this section, we describe the main results of this paper without derivations and with
only minimal discussion. In later sections we provide the derivation and more in depth
discussions. We start with the approximate eigenvalues of the Hamiltonian, the approximate
neutrino mixing matrix and then give the oscillation probabilities for all channels to rst
order in the expansion parameter, , see eq. (3.3) for the precise de nition.
3.1
Mass squared eigenvalues in matter
In vacuum the three eigenvalues of the full Hamiltonian which governs the neutrino
oscillation is given in the form mi2=2E, where mi is the mass of ith mass eigenstate of neutrinos,
i = 1; 2; 3. Similarly, in matter we write the three eigenvalues as
where the state label runs over i =
; 0; + for the approximate Hamiltonian of three
avor mixing system. To treat the normal and the inverted mass orderings (NO and IO
respectively) in a uni ed way, we de ne the eigenvalues as follows5
mr2en + a
sign( mr2en) ( mr2en
mr2en + a + sign( mr2en) ( mr2en
mmr222e1n ;
In eq. (3.1), the renomalized
stein matter potential [1], a, are de ned as follows:6
m2
mr2en, the expansion parameter , and the
Wolfen4See, however, the clarifying remark at the end of subsection 3.3.6.
5We note that the eigenvalues in (3.1) above appear in ref. [5]. See section 4.1 for the derivation and a
comment on the treatment in [5].
6The following notation is used throughout:
mi2j
mi2
mj2, sij = sin ij and cij = cos ij where
ij are the standard neutrino mixing angles and GF is the Fermi constant, Ne is the number density of
electrons, E is the energy of the neutrino, Ye the electron fraction and
is the density of matter.
q
q
i
2E
;
(3.3)
of
3.2
p
and
a = 2 2GF NeE
g:cm 3
GeV
This choice of
mr2en is crucial to the compact formulas for the oscillation probabilities that
will be given in this paper. Note also that the sign of
mr2en signals the mass ordering, both
mr2en and
are positive (negative) for NO (IO). However, for both orderings
m221 > 0, as required by nature. Notice that 0 is the same for the both mass orderings,
and when we switch from NO to IO we also switch the sign in front of the square root in
eq. (3.1). The nicest feature of the sign choice is that the oscillation probability has a uni ed
expression and the solar resonance is in
 0 level crossing for the both mass orderings.
mr2en is equal to the e ective atmospheric
disappearance experiment in vacuum,
me2e
m2 measured in a electron (anti) neutrino
me2e recently measured by the reactor 13 experiments [13{15] up to e ects
m231)2. Whether the coincidence between
me2e and
mr2en re ects a deep
aspect of neutrino oscillation or not will be judged depending upon what happens at second
order in . This point as well as the relevance of the other e ective
me2e, will be discussed in depth in a forthcoming communication.
The mixing angle 13 and mixing matrix in matter
We use the angle
to represent the mixing angle 13 in matter. With the de nitions of
the eigenvalues (3.1), the following massordering independent expressions for cosine and
sine 2 (see section 4.2) are given by
(3.5)
(3.6)
(3.7)
(3.8)
and as a goes from + 1 to
1 for the IO. In vacuum (a = 0),
the atmospheric resonance, when a =
mr2en cos 2 13, for both mass orderings.
1 to + 1 for the NO
= 13 and
=
=4 at
The mixing matrix in matter, V , relates the
avor eigenstates, e
,
, to the
perturbatively de ned matter mass eigenstates,
, 0
, + as follows (see section 4.4):
where the matrix V is unitary. It is convenient to split V into a zeroth order term, V (0),
and a rst order term, V (
1
) in our expansion,
where the zeroth order matrix is given by
V (0) = 6
2
4
0
B
+
V (0) + V (
1
);
0
c23
with
being the CP violating phase, whereas the rst order correction is given by
c
by (3.7) with (3.8) and (3.9) must be unitary to order . In fact it is, since the following
two conditions are satis ed
Of course, none of what follows is self consistent without unitarity here.
With the matter eigenvalues, 's , de nite by eq. (3.1) and the matter mixing matrix, V ,
given by eq. (3.7), simple and compact expressions can be easily derived for the oscillation
probabilities in matter for all channels, to leading order in , as will be shown in the next
section.
3.3
Compact formulas for the oscillation probabilities in matter
In this section we start by presenting the shortest path to the oscillation probabilities of
the erelated channel by using the eigenvalues,
;0, and mixing matrix, V , given in the
previous section to order . Then, we derive the universal expression of the oscillation
probabilities which is applicable to all channels.
3.3.1
e !
e disappearance channel
The derivation of the e survival oscillation probability, P ( e !
e), in our renormalized
helioperturbation theory is extremely simple. We start from the general expression
4E
4E
where, L, is the baseline. Because jVe0j2 = O(
2
), we obtain to order eq. (1.1), or
where j +
{ 7 {
Notice that the formula in eq. (3.11) takes into account the matter e ect as well as the
e ect of s13 to all orders. Nonetheless, it keeps an exceptional simplicity, an e ective
twoavor form in matter which consists of single term with the unique eigenvalue di erence
+
, the feature we believe to be unique in the market. The feature stems from the fact
that there is no e component at zeroth order in in the \0" state in matter. It is expressed
in the zero in the Ve0 element of the zerothorder V matrix as in (3.8), see also section 4.4.
3.3.2
e !
and
e !
appearance channels
Now, we discuss the appearance channels e !
simplest way to derive the formulas for the oscillation probabilities starting from the V
matrix by using unitarity. The oscillation probability P ( e !
) can be computed as
appearance channel probability is quite compact, despite
that it contains allorder contributions of s13 and a. In particular, it keeps the similar
structure as the one derived by the Cervera et al. [7], which retains terms of order 2 but
is expanded by s13 only up to second order.
This method of computing P ( e !
expression of the oscillation probability which is manifestly free from the apparent
singularity as
!
0 because 1=(
0) always appears adjacent to sin [(
0)L=4E].
We will refer this method as the \shortcut method" in the rest of this paper.
) in the above o ers the shortest path to the
{ 8 {
where the common shorthand notation for the kinematic phase
ij = ( i
j )L=4E is
used. Again, since jVe0j2 = O(
2
), we have to order
(3.13)
(3.14)
(3.15)
here Jr, the reduced Jarlskog factor, is
( +
0)
( +
sin
( +
0)
4E
a)
sin
)L
(
+
4E
mr2en
0)L
cos
2
( +
), eq. (3.14), is not quite of the form of eq. (2.1) because of
( +
4E
0)L . This can be easily remedied,
2 sin
+ sin
0 cos
+0 = sin2
+0
sin2
0
sin2
+
(3.16)
which leads to exactly the form of eq. (2.1). However, if this is used then some of the terms
are singular when
= 0, yet the total expression is equivalent to eq. (3.14) and is nite.
This is the reason why we prefer the form of eq. (3.14).
The oscillation probability for e !
channel can be obtained by all of the following
three methods: (
1
) the similar calculation by the shortcut method, (
2
) using the unitarity
relation P ( e !
s23; s23 ! c23) [10], whose derivation is sketched in section 4.1. We
note that Jr changes sign by the transformation. In general the oscillation probability in
arbitrary channel can be obtained by (
1
) the shortcut method, or by (
2
) rewriting the
expressions in appendix B using the formulas in appendix C.
3.3.3
The general form of the oscillation probabilities
!
The expressions of the oscillation probabilities in the
sector can be derived by one
of the methods mentioned in the previous subsection. Then, one observes a remarkable
feature that all the oscillation probabilities P (
written in a universal form:
!
) (including the e sector) can be
sin
( +
( +
4E
{ 9 {
"
"
"
P (
!
) =
+4 fA+ g s2 c2 +
fB+ g (Jr cos )
+4 fA+0g c2 +
fB+0 g Jr cos =c123
+4 fA 0g s2 +
fB 0g Jr cos =c123
+8 Jr ( +
)( +
0)(
0)
( mr2en)2 ( +
( +
1, or the simple functions of 23.
and S
are given in table 1. Notice that they
In table 1, we observe that three relations hold between the coe cients
A 0 = A+0;
B 0 = B+0
and
B+
This invariance must be respected by the oscillation probabilities because the two cases in
eq. (3.19) are both equally valid ways of diagonalizing the zerothorder Hamiltonian. Look
at eq. (3.5) to con rm that it is invariant under (3.19). Then, the former two identities
in (3.18) trivially follow, but the last one requires use of the kinematic relationship7
sin
+ sin
+0 cos
0 = sin
+ sin
0 cos
+0 + sin2
+ :
We note that 23 dependence in the oscillation probabilities, apart from that in Jr,
is con ned into the
ve independent coe cients. All the other parameters,
m2's, 12,
13, , E and a, are contained in the remaining part of the probabilities, which takes the
universal form for all the oscillation channels.
The antineutrino oscillation probabilities P (
) can be easily obtained from the
neutrino oscillation probabilities. One can show that by taking complex conjugate of the
evolution equation for antineutrinos of energy E that it is equivalent to the evolution
equation for neutrino with energy equal to
E. Therefore,
!
: E) = P (
:
E) :
(3.20)
Here, we give some critical observations regarding the formulas (3.17) and the
associated table 1:
The oscillation probability P ( e !
e), to rst order in , is of a simple two avor
form with the mattermodi ed mixing angle
13 and the mass squared di erence
given by
of P ( e !
frameworks so far studied.
mr2en. As far as we know, having single two avor form as the expression
e) in matter at this order is the unique case among all the perturbative
7Notice that the relation B+
= C
needs to be satis ed only to order 0 because these terms are
already suppressed by .
Formulating the perturbation theory of neutrino oscillation with automatic
implementation of the oscillation probability formulas (2.1), which is to be carried out
in section 4, is the principal result of this paper. However, the formula (3.17) goes
one step further beyond eq. (2.1) by displaying the channelindependent universal
function of L=E, up to the coe cients A , B , etc. which depend only on 23.
in the limit
perfectly
nite.
!
With the universal form of the probabilities, unitarity can be trivially checked just
by adding up the appropriate columns of table 1 with proper care of using the correct
signs in the last row for the intrinsic CPviolating terms.8 One can easily con rm it
by adding, for example, the e ! e
, e !
, and e !
columns to give zero in
If one uses the identity (3.16) to transform the sin ( +
term to the righthand side of (3.16), then the transformational invariance given by
eq. (3.19) will be manifest and eq. (3.17) will become the form of eq. (2.1). However,
0 some terms will diverge even though the total expression is
In the universal form, the amplitude of a given oscillation frequency, to leading
(zeroth) order in , is controlled by . The amplitudes of the frequencies ( +
0), (
0) and ( +
) are given proportional to A+0 cos2 , A 0 sin2
A+ sin2 cos2 , respectively. For the channels involving
e, only the ( +
frequency appears whereas in the
sector all frequencies appear and are of
equal magnitude near the 13 resonance, when
=4, but every where else either
and
)
the frequency ( +
0)
mr2en or (
0)
mr2en dominates.
Finally, we give more clarifying remarks on generic features of the oscillation
probability formulas presented in this section 3.
The term proportional to sin in P (
), in eq. (3.17), is equal to
0)
!
sin
8 Jr sin
( +
sin
(
4E
0)L
sin
( +
0)L
4E
up to an overall sign. This is because the quantity in f
g is just the Jarlskog factor
in matter due to the NaumovHarrisonScott identity, [17, 18].
Due to the above theorem sin terms are always accompanied by the Jarlskog factor
Jr = c12s12c23s23c123s13. Though less well known, it can be proven that cos
must be proportional to the reduced Jarlskog factor Jr=c213 [19]. It can also be shown
terms
that both of the dependent terms must come with the suppression factor of [17{
19]. It is nice to observe that these general properties are realized in our formulas.
Association of the factor and the all angle factors to the dependent terms indicates
that they are genuine three avor e ects. In our general formula (3.17), all the terms
8Here, we assume that the oscillation probability formulas are derived without using unitarity, excluding
one of the options we mentioned earlier to derive them.
with explicit factor are of this dependent terms. Notice that other corrections of
order , though they do not show up explicitly, are implicitly contained in
mr2en,
;0 and
which do not contain any
or 23 dependence.
In vacuum, a = 0, the above oscillation probabilities reproduce the standard results,
to
rst order in . The form is somewhat unusual but we have checked that the
matter e ects are negligible, the coe cients of the (L=E)n, for n
expressions are identical. Also, we have checked that in the limit L=E ! 0, where
3, terms in eq. (3.17), are identical to those in vacuum for all oscillation channels.
Range of applicability of the formulas
To discuss the range of applicability of our expressions, it is useful to rst consider the
vacuum expressions to
rst order in the expansion parameter . For all channels, the
expansion of the vacuum oscillation probabilities to rst order in
does not include terms
proportional to sin2
21 which starts at second order in ,
sin2
=2, that is at the rst atmospheric oscillation
0:002 which is small for the channels e !
x where x = e; , or
since 1
at the second atmospheric oscillation maximum,
) are all of order sin2 2 13
31 = 3 =2 and 2 2
2
31
0:1. However,
0:02, which is
signi cant compared to the sin2 2 13 term. So in vacuum our rst order expansion is only
a good approximation for
the other channels,
!
2
,
!
or L=E . 1000 km/GeV for these e channels. For
and
!
, our rst order expansion is a good
approximation to somewhat beyond L/E = 1000 km/GeV because the leading terms are
not suppressed by the smallness of sin2 2 13.
Then, what about the validity in matter? In section 5 we will argue that our
perturbative description is valid outside the solar resonance. Notice that the region without
validity (no guarantee for approximation being good) is rather wide and includes the
vacuum because the solar resonance width j aj = p3(sin 2 12= cos2 13) m221 is larger than
the solar resonance position a = (cos 2 12= cos2 13)
m221. We expect then that our
helioperturbation theory works for the matter potential a larger than a few tenth of j
mr2enj.
To give the reader a sense of the precision of our approximation we have plotted in
gure 1, the contours of equal probability for the exact and the approximate solutions for
the channels e !
, e !
e and
!
. The right (left) half plane of each panel of
gure 1 corresponds to the neutrino (antineutrino) channel. As expected, for large values
of the matter potential, jaj > 13 j
mr2enj we
nd we have no restrictions on L/E, to have
a good approximation to the exact numerical solutions. Whereas for small values of the
matter potential, jaj < 13 j mr2enj we still need the restriction L=E . 1000 km/GeV.9
9Of course, the boundary between these two regions should be interpreted as an approximate one. In
fact, an exact boundary would sensitively depend on the de nition of the di erence allowed between the
exact and the approximate probabilities.
oscillation probabilities for upper left, e !
e, upper right, e !
and lower,
!
. The
upper (lower) half plane is for normal ordering (inverted ordering), whereas positive (negative) L/E
is for neutrinos (antineutrinos). For treatment of antineutrinos, see section 3.3.3. The order of
the contours given in the title is determined from the line L/E=0. The discontinued as one crosses
Ye jEj = 0 is because we are switching mass orderings at this point. In most of parameter space the
approximate and exact contours sit on top of one another so the lines appear to alternate bluered
dashed. Note that, for L/E >1000 km/GeV and jYe Ej < 5 g.cm 3.GeV, the di erence between
the exact and approximate contours becomes noticeable at least for e ! e and e !
.
We note that most of the settings for the ongoing and the proposed experiments,
except possibly for the one which utilizes the second oscillation maximum, fall into the
region L=E . 1000 km/GeV. To improve the accuracy to larger values of L/E, especially
for values of jaj < 13 j
mr2enj, second order perturbation theory in
be the subject of a future publication.
is needed, which will
An example of the power of our oscillation probabilities
In this section we present an example of the power of our compact expression of the
oscillation probabilities in understanding the physics of oscillations to rst order in our
expansion parameter . For simplicity we consider the e disappearance channel. In gure 2
the e disappearance probability is shown, as a function of neutrino energy E, for baselines,
L, of 3000 km (upper panel) and 5000 km (lower panel). As in gure 1, the solid blue curves
are drawn by using the exact expression of P ( e ! e), and the dashed red curves by our
compact formula in eq. (3.11). The black dotted curves are for the vacuum case.
We rst notice that our approximate formula agrees well with the exact expression in
particular at higher energy. Secondly, because of the long baselines, the matter e ect is
sizeable, producing not only a large shift in the position of the rst oscillation minimum but
also an signi cant increase in the depth of the minimum. These two e ects are correlated
by the energy dependent quantity, ( +
depth of the oscillation minimum changes from
) which can be seen graphically in gure 3: the
whereas the energy at which the the rst oscillation minimum occurs changes from
!
+
mr2en
2
Because of the simplicity of our expression for P ( e !
e), eq. (3.11), these shifts are
accurate to rst order in the expansion parameter . This simple understanding of the
features of P ( e ! e) is new to this paper.
3.3.6
Comparison with the existing perturbative frameworks
As we emphasized in section 2, our machinery has an advantage over the existing
perturbative frameworks by having the minimum number of terms composed of sin [( j
i)L=4E]
(i; j = 1; 2; 3) in the oscillation probabilities. This contrasts with the features of the
existing perturbative frameworks in which much larger number of terms than those minimally
necessary as in (2.1) are produced. They include, typically, the terms with either extra
L=E dependences or di erent frequencies in the sine functions, or often both, which easily
obscures the physical interpretation.
To give a feeling to the readers on how simple and compact our formulas are, we
compare our expressions to the ones in the existing literatures to the same order in expansion.
For de niteness, we pick the ones in ref. [10] to make the comparison, the most recent one
among the reference list given in section 1.
(upper panel) and 5000 km (lower panel). We have used the earth matter density 2.8 g/cm3.
Our expression of P ( e !
e) in (3.11) which has only a single term (ignoring unity)
may be compared with eqs. (4.6) and (4.7) which consist of total 3 terms. With regard to
), if we count numbers of terms with di erent L=E dependence we have one
independent and 2 dependent terms in (3.14), total 3. Whereas eqs. (4.8) and (4.9) in [10]
have total 7 terms, 3 independent and 4 dependent ones. Our expression of P (
in (3.17) has 3 independent and 2 dependent terms, adding up to total 5. On the other
hand, eqs. (4.10) and (4.11) in [10] contain total 8 independent and 10 dependent terms
(2 sin
and 8 cos
terms), which adds up to 18. So not only do our expressions for the
oscillation probabilities have less than half the number of L=E structures, but the form of
!
)
the L=E dependence is made manifest and identical to the vacuum form.
We note that when our formulas are expanded by
m221=
m231, they agree, of course,
with the existing ones calculated previously. Therefore, our formalism may be regarded
as a systematic way of organizing the equivalent expressions to order
neater, structurerevealing ones. The bene t for having such simple expressions is that the
physical interpretation of the formulas is transparent, as emphasized in section 2.
m221=
m231 into
4
Formulating the renormalized helioperturbation theory
In this section, we formulate the heliototerrestrial ratio perturbation theory, for short the
helioperturbation theory, which has the unique expansion parameter
(4.1)
We will show that use of its renormalized version is the key to the very simple formulas
of the oscillation probabilities exhibited in section 3.3 and appendix B. In fact, there are
mmr222e1n :
two ways of deriving the oscillation probabilities, the S matrix method and the wave
function method. Here we sketch both of them, leaving technical or computational parts
into appendices A and B. The meaning of the agreement between results obtained by both
the S matrix and the wave function methods will be discussed at the end of this section.
The S matrix describes neutrino avor changes
after traversing a distance L,
and the oscillation probability is given by
(L) = S
(0);
P (
!
When the neutrino evolution is governed by the Schrodinger equation, i ddx
= H , S
matrix is given as
S = T exp
dxH(x)
where T symbol indicates the \time ordering" (in fact \space ordering" here). In the
standard three avor neutrinos, Hamiltonian is given by
H =
8
1 <>
2E >
:
where the symbols are de ned in an earlier footnote. For the case of constant matter
density, the righthand side of (4.4) may be written as e iHx. We recapitulate here the
earlier footnote: in (4.5)
mj2i
mj2
mi2 where mi denotes the mass of ith mass eigenstate
neutrinos. Position dependent function a(x)
2 2GF Ne(x)E is a coe cient for measuring
the matter e ect on neutrinos propagating in medium of electron number density Ne(x) [1]
where GF is the Fermi constant and E is the neutrino energy.
The neutrino
avor mixing matrix U is usually taken to be the standard form UPDG
given by Particle Data Group. We, however, prefer to work in a slightly di erent basis,
for this paper, in which the avor mixing matrix has a form (with the obvious notations
sij
sin ij etc. and
being the CP violating phase)
(4.2)
(4.3)
(4.4)
(4.5)
under the understanding that the left phase matrix in the rst line in eq. (4.6) is to
be absorbed into the
neutrino wave functions. Being connected by the phase matrix
rotation, it is obvious that our U and UPDG give rise to the same neutrino oscillation
probabilities.
i
Z L
0
3
0
0
!
j2:
It is convenient to work with the tilde basis de ned as ~
Hamiltonian is related to the avor basis one as
= (U2y3)
, in which the
H~ = U2y3HU23;
where U23 is de ned in eq. (4.6). The S matrix in the avor basis is related to the S matrix
in the tilde basis S~ as
S(L) = U23S~(L)U2y3;
with (4.10) and (4.11) is identical with the tildeHamiltonian in (4.9). Note the simplicity
of the perturbing Hamiltonian, H~1, especially the zeros on the diagonal elements. The
diagonalization of H~0 leads to the eigenvalues given in section 3.1, eq. (3.1), and because
of the zeros along the diagonal of H~1, there are no corrections to these eigenvalues at rst
order in a perturbative expansion.
The authors of [5] treat the order
e ect in the Hamiltonian as a renormalization of
the matter potential, whereas we regard it as a renormalization of
m231.
mm222311 as
H~ (x) =
Notice that the matter term in the Hamiltonian (4.5) is invariant under the U23 rotation.
Hence, the dynamics of neutrino propagation in matter is governed by only the two mixing
angles 12 and 13 [22], which is also independent of
thanks to our convention of U
in (4.6). From the rst equation in (4.8), the relationships P ( e !
c23 !
s23; s23 ! c23) etc. simply follow as used in section 3.3.
The simplest formulation of helioperturbative treatment in the tildebasis includes
decomposition of H~ into the zeroth and the rst order terms in the expansion parameter
) = P ( e !
:
m231 ><
2E
6
4
>
:
To derive the compact formulas of oscillation probabilities we use slightly di erent
basis, a renormalized basis, to formulate the perturbation theory. That is, we absorb a
certain order
terms into the zeroth order Hamiltonian, H~ (x) = H~0(x) + H~1(x):
2 0
mr2en 64 c13
2E
0
c13
0
s13
m221=
3
0
0
(4.7)
(4.8)
Our treatment can be easily generalized to cases with matter density variation as far
as the adiabatic approximation holds. See e.g., [10] for such a treatment. However, we
prefer to remain, for simplicity and compactness of the expressions, into the formulas with
constant matter density approximation in the rest of this paper.
We transform the Hamiltonian H^ = H^0 + H^1, from the\tilde" basis to the \hat" basis,
H^0 = U yH~0U ;
H^1 = U yH~1U
where the unperturbed Hamiltonian H^0 is diagonal,
We take the following form of unitary matrix U to diagonalize H~0:
H^0 =
2
1
i
Z L
0
(L) = eiH^0LS^(L):
(4.12)
(4.13)
(4.14)
(4.15)
(4.16)
(4.18)
HJEP01(26)8
The expressions of the zeroth order eigenvalues
Similarly, cosine and sine are given in eq. (3.5).
Also the perturbing Hamiltonian, H^1, retains it's simple form thanks to that the U
rotation keeps \zeros" in H~1 unchanged both on the diagonal and in the topright and
i
Z L
0
To calculate S^(L) we de ne (L) as
In fact, H^1 is identical to H~1 with 13 replaced by ( 13
4.3
S matrix and the oscillation probability
The S matrix in the avor basis is related to the S matrix in the tilde and the hat bases as
S(L) = U23S~(L)U2y3 = U23U S^(L)U yU2y3
where we have used explicitly the fact that the matter density is constant:
Then, (L) obeys the evolution equation
where
Then, (x) can be computed perturbatively as
i
d
dx
H1
(x) = H1 (x)
eiH^0xH^1e iH^0x:
Collecting the formulas the S matrix can be written as
Z L
0
(L) = 1 + ( i)
dxH1(x) + O(
2
):
S(L) = U23U e iH^0L (L)U yU2y3
Thus, we are left with perturbative computation of
(L) with use of H1 in (4.20) to
calculate the S matrix. With the S matrix in hand it is straightforward to compute the
oscillation probabilities by using (4.3). We leave these tasks to appendices A and B.
Mass eigenstate in matter: V matrix method
H^0 is diagonal, we have
In this section we calculate the V matrix directly using our perturbation theory. If we
switch o the perturbation H^1, the mass eigenstates in matter, to lowest order, are given
by the hatbasis wave function ^i(0), which are the eigenstates of H^0 in (4.12), and since
^(0) = (U23U )iy
i
:
Thus, the V matrix is given to zeroth order by V (0) = U23U
whose explicit form is given
in section 5.3, and also in eq. (3.8).
In order to obtain the mass eigenstates in matter to rst order in , i = ^i(0) + ^i(
1
), let
us compute the rst order correction to the hat basis wave functions. Using the familiar
perturbative formula for the perturbed wave functions
i
^(
1
) = 2E
j6=i i
X (H^1)ji ^(0)
j
j
with H^1 in (4.12), and the i's are given by the eigenvalues of H^0, see (3.1). Then the
mass eigenstate in matter i can be written to rst order in
as:
0
1
0
13)
mr2en c12s12c(
mr2en c12s12s(
+
1
0
0
13)
13)
0
1
mr2en c12s12s(
+
0
1 0 ^(0) 1
Using (4.23), this equation is of the form i = V y
which can be inverted to easily obtained
the V matrix given in eq. (3.7),
2
4
V = U23U 61 + c12s12
mr2en
8
>< c(
>
:
0 0
13) B 1 0 0 C +
A
>
=
>
;
A
5
7 : (4.25)
(4.19)
(4.20)
(4.21)
(4.22)
(4.23)
(4.24)
HJEP01(26)8
This can be used to directly compute the oscillation probabilities as was performed in
section (3.3), or by inserting the V matrix elements into the expressions of the oscillation
probabilities in (4.6).
This V
matrix method was used to calculate the matter e ect
correction in the oscillation probabilities [20]. For a di erent approach toward simpler
expressions of the V matrix elements see [21].
In sections 4.3 and 4.4, we have sketched the two di erent methods for calculating
the oscillation probabilities, the Smatrix method (section 4.3) and the V matrix method
(section 4.4). The results obtained by the both methods agree with each other, and the
expressions of the oscillation probabilities are of the form given in eq. (2.1).
Let us make a comment on what happens if we use a di erent decomposition of the
Hamiltonian into unperturbed and perturbed parts as in (4.9). Then, the perturbed
Hamiltonian has nonzero diagonal terms, and this produces rst order corrections to the
eigenvalues. If we use these expressions for the eigenvalues and expand by the small expansion
parameter we obtain terms that do not exist in (2.1), such as the ones proportional to
sin [( j
i)L=2E]. Of course, the Smatrix method will give the same expressions.
Therefore, absence of the explicit rstorder correction to the eigenvalues is essential to keep
perturbative expressions of the oscillation probabilities of the form (2.1) to order . In our
case it is guaranteed by vanishing diagonal elements of the perturbed Hamiltonian (4.11).
5
More about the renormalized helioperturbation theory
In this section, we critically examine the framework of the renormalized helioperturbation
theory. Despite a drawback of the current framework (which is to be described below) we
argue that our perturbation theory works apart from the vicinity of the solar resonance
crossing.
5.1
Exact versus zerothorder eigenvalues in matter
The three eigenvalues of the Hamiltonian are written as 2Ei , where i runs over 1; 2; 3 for
the exact eigenvalues, and i =
; 0; + for the zerothorder eigenvalues in our perturbative
framework. In
gure 3 the
i are plotted as a function of a, the Wolfenstein matter
potential, for both the exact and the zerothorder ones given by eq. (3.1). It is clear in
gure 3 that our zerothorder eigenvalues fail to treat the
solarm2 scale level crossing
correctly. As one goes through the solar resonance in the exact solution the two eigenvalues
involved, the red and green, repel one another, whereas in our perturbative solution the
two corresponding eigenvalues cross with each other.
We point out here that the drawback is the common feature among the similar
perturbative treatments of neutrino oscillation available in the market. In this section we argue,
for the rst time, that despite the drawback, our perturbative framework successfully treat
the avor content of these two states in region reasonably far from the solar resonance.
We rst note that, despite the feature, the atmospheric and solar resonances occur at
the correct values of the matter potential with our zerothorder eigenvalues. The
atmospheric resonance occurs when
+
is a minimum, which is at
a =
panel) and in the inverted mass ordering, abbreviated as IO, (right panel). The exact eigenvalues
are depicted with colored lines, green for 1, red for 2, and
ight blue for 3
. The eigenvalues
calculated by using our renormalized helioperturbation theory are drawn by the black dashed lines
whose state labels are marked on the gures. The approximate eigenvalues for 0 and
cross at
the solar resonance, whereas the exact eigenvalues for 1 and
2 repelled at solar resonance. To
make these features visible we used a solar
m221 three times as large as the measured value.
and the minimum di erence between
+ and
resonance occurs when
0 is a minimum; this occurs when
is
mr2en sin 2 13 as expected. The solar
a =
mr2en cos 2 12= cos2 13
and the minimum di erence is zero! This is not the value determined by the full
Hamiltonian which is
mr2en sin 2 12. Therefore, while our perturbative scheme treats the
atmospheric resonance correctly to order , it misses the e ects of the solar resonance.
5.2
Eigenvalues in vacuum and in the asymptotic regions a !
1
In this and the next subsections, we will give the arguments to indicate that our
renormalized helioperturbation theory works apart from the vicinity of the solar resonance despite
the issue mentioned above.
We rst show that the zerothorder eigenvalues given in (3.1) agrees with the exact
ones to order
in the asymptotic regions a !
1. We use the characteristic equation of
the full Hamiltonian (4.5) to derive (i = 1; 2; 3)
X
i
X
i;j
i = a +
m321 +
m221 ;
i j =
m221
m321 + a (c122 + s122s213) m221 + c123
m321 ;
1 2 3 = c122c213a
m221
m321:
(5.1)
Then, by using the asymptotic expansion of 's one can obtain to leading order in the 1=a
expansion (at a ! +1 in the NO case)
1 =
with our zerothorder eigenvalues given in (3.1):
must be interchanged. They are identical to the ones computed
For the IO, the analytic expressions of i (i =
i at a !
1 for the case of NO.10
the both mass orderings by
When the matter potential vanishes, a = 0, the mass squared eigenvalues are given for
; 0; +) at a !
1 is the same as those of
0
B
0
B
2
c12
mr2en
a + (s213 + s122 ) mr2en
0 a + (s213 + s212 ) mr2en
mr2en
mr2en
1
C
A
1
C
A
0
B
0 s212
mr2en AC
m231
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
terms are di erent from the exact values, 1 = 0, 2 =
m231. To understand the meaning of the failure at order we need to discuss what happens
to the avor content of the matter mass eigenstates as the matter potential changes from
m221, and 3 =
below to above the solar resonance. This will be done in the next subsection.
Similarly, the asymptotic behaviour of the angle , i.e., 13 in matter can be easily
worked out. With the de nition of in (3.5), it is easy to show that
takes on the following
values as a is varied from
1 to + 1. In the case of normal mass ordering,
NO = <
8
>
>
>
>
>
>
>
>
>
>
:
0;
13;
4
2
;
;
2
13;
1
a =
a = 0
a =
a = 2
10In fact, the rst line in (5.1) holds exactly.
while in the case of inverted mass ordering,
IO = <>> 2
8
>
>
>
>
13;
2
4
;
;
13
0;
a =
a = 2
a =
a = 0
(5.7)
(5.8)
(5.9)
It re ects a natural view that physics at a ! +1 for the normal mass ordering corresponds
to the one at a !
1 for the inverted mass ordering at least to leading order in .
Neutrino mixing matrix in matter and the
Since the two levels cross at the solar resonance in our perturbative treatment, one may
expect that our treatment fails completely beyond the solar resonance, i.e., in the region
with matter density higher than the resonance. However, we will show in this subsection
that the
avor contents of the three eigenstates are correctly reproduced at least in the
asymptotic region. That is, the zerothorder V matrix describes correctly the asymptotic
behaviour of the exact eigenstates in matter.
Suppose we denote the exact avor mixing matrix in matter as the (almost standard)
U matrix de ned in (4.6). Let us discuss the case of NO
but V (0) in (3.8). Notice that s23 in matter is frozen to its vacuum value for
< 0:1, and
s12 ' 0 at a !
1 [3]. At a ! +1, s12 ' 1 and c12 ' 0 the matter U matrix is identical
to (3.8) if we interchange 1 and 2 apart from rephasing factor
1 for the new second
rst. At a !
1 U is nothing
mass eigenstate.
To make the meaning of this feature clearer we write down here the avor content of
the states i (i = 1; 2; 3) at a !
1. Noticing that i = (V y)i
at a !
1 is given at zeroth order by
, the avor composition
Whereas the composition at a ! +1 is given by
1 = c e
2 = fc23
s (s23e i
s23e
i
g = 0
+ c23 ) =
3 = s e + c (s23e i
+ c23 ) = +
1 =
2 = c e
fc23
s23e
i
s (s23e i
3 = s e + c (s23e i
g =
0
+ c23 ) =
+ c23 ) = +
The avor compositions given in (5.8) and (5.9) imply that the avor content of the lower
two mass eigenstate in matter is correctly described in our perturbative framework despite
the failure of describing the solar level crossing. Note, the combinations of
and
in
the (
) and f
g in (5.8) and (5.9) are orthogonal with each other.
In the case of IO, essentially the same discussion goes through. The asymptotic
behavior of 12 at a !
1 is the same as that in NO. The asymptotic behavior of 13 at
a !
1 for NO is mapped into the one at a !
1 for IO. But, this is already taken care
of by the de nition of
given in (3.5). Therefore, the same U matrix in matter as the one
in NO are obtained at a !
1. Then, the same avor compositions as in (5.8) and (5.9)
follow for IO. Again it is consistent with those we expect from the level crossing diagram.
We add that our + state always corresponds to 3 state. At a ! +1 in NO and at
a !
1 in IO the electron neutrino component is all in this + = 3 state. At a !
1
in NO and at a ! +1 in IO the electron neutrino component is all in
for IO) state. In vacuum for both NO and IO, Ve(+0) = s = s13, which is the correct value
( 1 for NO, 2
To summarize: despite that the eigenvalues calculated by our helioperturbation theory
do not show the correct behavior at around the solar level crossing, the avor composition
of the states are correctly represented by our zerothorder states. It is the very reason why
HJEP01(26)8
our perturbative formulas work at much higher and lower values of Ye E compared to the
one at the solar resonance. This point escaped detection in the previous literature to our
knowledge. We have seen that they work practically whole regions except for the vicinity
of the solar resonance.
6
Concluding remarks
We have developed a new perturbative framework of neutrino oscillations, which allows us
to derive compact expressions of the oscillation probabilities in matter, which we examined
in this paper to order
m221=
mr2en '
m221=
m231. Although vacuum like in their
forms, our compact results keep allorder e ects of s13 and the matter potential a, while
using
as the unique expansion parameter.
The characteristic features of our perturbative framework (dubbed as the renormalized
helioperturbation theory), in contrast to the ones previously studied by various authors,
are as follows:
We use the renormalized basis (4.10) de ned with the atmospheric masssquared
di erence corrected by an order
quantity,
mr2en
m231
2
s12
m221. The
decomposition of the Hamiltonian into the unperturbed (4.10) and perturbed terms (4.11),
is done in such a way that there is no diagonal entries in the perturbed one. It makes
the eigenvalues of the zeroth order Hamiltonian the correct ones to order . Usage of
this decomposition is crucial to obtain the form of the oscillation probabilities (2.1)
akin to the form in vacuum. This facilitates an immediate physical interpretation
that the frequencies of the oscillations are determined by ( +
), ( +
0),
and (
0), where the 's are the eigenvalues of the zeroth order Hamiltonian,
see eq. (3.1). They determine the oscillation pattern, the functional form of L=E
dependence, for all the oscillation probabilities as in eq. (2.1) to this order.
The amplitudes of the sin [( j
i)L=4E] terms in the oscillation probabilities, (2.1),
can be determined either by the Smatrix method, or the V matrix method. The
latter may be simpler because of no
rst order correction to the eigenvalues. In
contrast to the complexity of the exact solution, this perturbative approach ensures
calculability of the V matrix elements in the form of power series, see eq. (4.25). It
leads to the simple analytic expressions of the oscillation probabilities which have
a universal form to rst order in , as shown in eq. (3.17). The coe cients in this
universal form are 0; 1 or simple functions of 23, see table 1.
From the universal form of the oscillation probabilities, eq. (3.17), we readily observe
the followings:
(
1
) Each one of the zerothorder terms in the oscillation probabilities, in all channels,
takes the same form as the corresponding two avor oscillation probability in
vacuum but with use of the eigenvalues and 13 in matter. In the special case of
e ! e, only the term with the particular frequency /
+
is nonzero, so
the L=E dependence is of the two avor form. The dominance of this frequency
also occurs in the oscillation probability for e !
.
(
2
) All the rstorder correction terms with the explicit factor
(
m2 = m2 )
in the oscillation probabilities are proportional to either cos
or sin . They
are also suppressed by the angle factor / c12s12c23s23s13. Hence, they are the
genuine three avor e ects. The explicit expressions of the dependent terms
in our formulas are in agreement with the general theorems.
(3) Unitarity can be trivially checked.
Further comments are made in section 3.3.3.
Despite the success of our current framework in almost all regions including the one
around the rst oscillation maximum, it has a clear drawback. It fails to accommodate the
physics at around the solarscale resonance. This is related to the unphysical feature that
the two eigenvalues (
and 0) cross with each other at the solar resonance, the universal
fault in all the perturbative treatment which include the expansion parameter
in the
market. However, we have o ered, for the rst time to our knowledge, an explanation why
our framework works beyond the resonance despite the failure in treating the solar level
crossing. We hope that we can return to the problem as a whole in the future.
As we noticed at the end of section 3.1 our renormalized
mr2en is identical, to order ,
to the e ective
m2 measurable in an (anti) e disappearance experiment in vacuum. It is
a tantalizing question whether it is just a coincidence, or is an indication of something deep.
It is quite possible that the features of the current framework mentioned above
naturally generalizes to higher orders. That is, one can demand that the oscillation probabilities
of the general form in (2.1) calculated by the V matrix method, with the carefully chosen
zerothorder eigenvalues, agree with the ones from the Smatrix method and are correct to
certain order in . Our result in this paper may be regarded as an existence proof of this
concept to order . Since we know that this is true in the exact form of the oscillation
probabilities (assuming adiabaticity) [3], it is likely to be correct in each order in perturbation
theory. It may or may not require higher order renormalization in
mr2en.
Acknowledgments
We thank Nordic Institute for Theoretical Physics (NORDITA) and the Kavli Institute for
Theoretical Physics in UC Santa Barbara for their hospitalities, where part of this work
A
A.1
pute H1(x)
calculated as
Calculation of S matrix elements
Computation of S^ matrix elements
Z x
0
2
4
1(x) = ( i)
dx0H1(x0)
;
By using H^1 in (4.15) and eiH^0x = diag ei x=2E; ei 0x=2E; ei +x=2E , one can easily
comeiH^0xH^1e iH^0x. Then, using eq. (4.21), the rst order term of (x) can be
where we have introduced the simpli ed notations, c(
sin (
13), etc. The simplicity in the structure of (A.1) with many zeros is the
mathematical reason why the expressions of neutrino oscillation probabilities are so simple in
13), s(
13)
our renormalized helioperturbation theory.
The S^ matrix is given by S^ = e iH^0x
= e iH^0x [1 +
matrix. Then, the elements of S^ matrix are given by:
1(x)] where 1 denotes the unit
0
0
s
13)
13)
i( +
ei( +
0)x=2E 1
0)
0)
13)
;
;
s
(
cos (
0
0
13)
e i( +
0)x=2E 1 7
i( +
0)
was done. This research was supported in part by the National Science Foundation under
Grant No. NSF PHY1125915.
H.M. thanks Universidade de S~ao Paulo for the great opportunity of stay under
\Programa de Bolsas para Professors Visitantes Internacionais na USP". He is grateful to
Theory Group of Fermilab for supports and warm hospitalities during the several visits in
20122015 where this work was started and made corner turning progresses.
S.P. acknowledges partial support from the European Union FP7 ITN INVISIBLES
(Marie Curie Actions, PITN GA2011 289442). Fermilab is operated by the Fermi
Research Alliance under contract no. DEAC0207CH11359 with the U.S. Department of
HJEP01(26)8
^
See = e i x=2E;
Se = S^ e = 0;
^
Se = S^ e =
^
^
S
= S^
i
i
S
= e i 0x=2E;
S
= e i +x=2E;
mr2enc12s12c(
mr2enc12s12s(
13)
13)
e i 0x=2E
e i x=2E
e i 0x=2E
e i +x=2E
i(
i( +
0)
0)
;
:
A.2
The relationships between S^, S~, and S matrices
The relationships between S^, S~, and S matrices are summarized as
S~(x) = U S^(x)U y
S(x) = U23S~(x)U2y3:
(A.1)
3
5
(A.2)
(A.3)
The left equation can be written explicitly as
S
2 S~ee S~e Se
~
6 S~ e S~
4 ~
S e S~
S
S
3
5
2 c
4
0 s 3 2 S^ee S^e Se
^
3 2 c 0
s 3
7 = 6 0 1 0 7 6 S^ e S^
s 0 c
5 4 ^
S e S^
^
S
^
S
Hence, the elements of S~ can by written in terms of the elements of S^ as
See = c2 S^ee + s2 S^
~
+ c s
Se + S^ e ;
^
S
s S^ e + c S^ ;
s2 S^e + c2 S^ e
s S^e + c S^
= s2 S^ee + c2 S^
c s
= S~
c s
^
See
^
S
;
^
See
^
S
= S~e ;
Se + S^ e :
^
Similarly, the elements of S can be written in terms of the elements of S~ matrix as
See = S~ee;
S e = c23S~ e + s23ei S~ e = Se (
);
S e = c23S~ e
s23e i S~ e = Se (
);
S
S
S
S
= c223S~
= c223S~
= c223S~
= s223S~
+ s223S~
+ c23s23(e i S~
+ ei S~ );
s223e2i S~
+ c23s23ei (S~
S~ );
s223e 2i S~
+ c23s23e i (S~
S~ ) = S (
c23s23(e i S~
+ ei S~ ):
(A.4)
(A.5)
(A.6)
; L) = jS
j2, eq. (4.3).
B
Expressions of neutrino oscillation probabilities
With the expressions of S matrix elements obtained in appendix A, and using eq. (4.3)
it is straightforward to calculate the neutrino oscillation probabilities. Similarly, one can
insert the V matrix elements given in (3.7) into (2.1) to obtain the equivalent results. In
this appendix we only give the resulting expressions, but of all the oscillation channels
under the viewpoint that unitarity is not to be imposed but must be proven to show the
consistency of the calculation. To compare these with the results of section (3.3), use of
the identities given in appendix C may be of some help.
The only comment worth to give here is about the simple method for transformation
c23 !
P (
s23 and s23 ! c23 to obtain P ( e !
) from P ( e !
), or P (
!
), which is utilized in section 3. Though we work with the rephased
mixing matrix de ned in (4.6), the transformations produces Se from Se , and S
S , up to an overall phase, see eq. (A.6).
) from
avor
from
B.1
Oscillation probabilities in
e row
P ( e ! e) is extremely simple as
where L is the baseline. The reasons for the simplicity is discussed in depth in section 3.3.
4E
(B.1)
It is almost trivial to verify unitarity in the e row: P ( e ! e) + P ( e !
) + P ( e !
1
1
c c(
8
4
4E
) = 4c2 s2 c223 sin2 ( +
4E
0)L
mr2enc12s12c23s23c s cos
c c(
sin
4E
( 0
)L
c c(
13) (
1
13) (
0)
+ s s(
0)
+ s s(
+8
mr2enc12s12c23s23c s sin
sin
( +
4E
sin
4E
13) ( +
13) ( +
c c(
sin
0)L
( 0
4E
0)L
4E
0)L
(B.2)
(B.3)
1
13) (
4E
1
13) (
4E
1
0)
1
0)
0)
1
:
0)
1
:
P (
P (
)
Oscillation probabilities in
row
e : ) = P ( e !
e) is related to the Tconjugate channel probability P ( e !
) as P (
!
:
), whose latter can be obtained by replacing
by
in (B.2).
Therefore, we only give the expressions of P (
!
) and P (
!
):
s223c s c c( 13)(
c s223s c c( 13)(
4E
1
1
+4 mr2enc12s12c23s23 c223 s223 cos
0)L + s2 sin2 (
4E
4E
0)L
With the above results the unitarity in
row can also be veri ed: P( ! e) + P( !
) + P(
! ) = 1.
B.3 Oscillation probabilities in
row
P( ! e) and P(
) can be given by their Tconjugate channels: P( ! : ) =
P( ! : ). Therefore, we only give the expressions of P(
! ) below.
P( ! )
= 1 4c243c2s2 sin2 ( +
8 mr2enc12s12c23s23 cos
c223c s c c( 13)(
c c223s c c( 13)(
4E
1
1
+ s s223 c223s2 c( 13)(
)L 4c223s223 c2 sin2 ( +
0)L + s2 sin2 (
4E
4E
0)L
0) s s( 13)( +
1
0) sin2 ( + )L
4E
0) + s223 c223c2 s( 13)( +
1
0) sin2 ( + 0)L
4E
1
0)+c223c s s( 13)( + 0) sin2 (
1
4E
0)L : (B.6)
The unitarity in row can also be veri ed: P( ! e) + P( ! ) + P( ! ) = 1.
2 X Im[V iV iV j V j ] sin
( j
2E
i)L
where i = 0; ; + (or 1; 2; 3). To cast this term to the one in (2.1), one needs the following
identity
HJEP01(26)8
sin
( +
2E
sin
4E
0)L
2E
sin
+ sin
4E
0)L
2E
sin
0)L
( 0
4E
+)L
:
Here, we list some more formulas which may be useful to understand the relationship
between di erent expressions of the oscillation probabilities:
Some useful identities
A straightforward derivation of the general expressions of the oscillation probabilities
contains the following form of CP or Tviolation terms
(C.1)
(C.2)
(C.3)
(C.4)
(C.5)
cos 2(
c c(
s s(
c s(
s c(
13) =
13) =
13) =
13) =
13) =
a cos 2 13 ;
sin 2(
) + ( mr2en
( mr2en
( mr2en + a) ;
13) =
a) ;
a) :
) + ( mr2en + a) ;
2
s12
2
s12
mr2en ;
mr2en :
( +
a sin 2 13 :
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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