††thanks: orcid # 0000-0003-2028-6782

# Interplay between the factorization of the Jarlskog Invariant
and location of the Solar and Atmospheric Resonances
for Neutrino Oscillations in Matter

Stephen J. Parke<EMAIL_ADDRESS>Theoretical Physics Dept., Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA

(December 14, 2020)

###### Abstract

The Jarlskog invariant which controls the size of intrinsic CP violation in
neutrino oscillation appearance experiments is modified by Wolfenstein matter
effects for neutrinos propagating in matter. In this paper we give the exact
factorization of Jarlskog invariant in matter into the vacuum Jarlskog
invariant times two, two-flavor matter resonance factors that control the
matter effects for the solar and atmospheric resonances independently. We
compare the location of the minima of the factorizing resonance factors with
the location of the solar and atmospheric resonances, precisely defined. They
are not identical but the fractional differences are both found to be less
than 0.1%. In addition, we explain why symmetry polynomials of the square of
the mass of the neutrino eigenvalues in matter, such as inverse of the square
of the Jarlskog invariant in matter, can be given as polynomials in the matter
potential.

††preprint: FERMILAB-PUB-20-609-T

##  I Introduction

In the current and future long baseline experiments, T2K Itow  _et al._
(2001), NOvA Ayres  _et al._ (2004), DUNE Acciarri  _et al._ (2016) and
T2HK(K) Abe  _et al._ (2014, 2018), neutrinos propagate between 300 to 1300 km
through the earth’s crust between source and detector. Neutrino oscillations
in matter are significantly different than in vacuum because of the coherent
forward scattering of the neutrinos off the electrons in matter. This effect
is known as the Wolfenstein matter effect Wolfenstein (1978).

The neutrino mass eigenvalues and the mixing angles of the neutrinos in matter
are strongly impacted by the matter effect which depends on density of the
matter and the energy of the neutrino through the Wolfenstein matter potential
Wolfenstein (1978), $a$, given by

$a\equiv 2\sqrt{2}G_{F}N_{e}E_{\nu}\,,$ (1)

where $G_{F}$ is the Fermi constant, $N_{e}$ is the number density of
electrons, and $E_{\nu}$ is the neutrino energy in the matter rest frame.
These changes in the masses and the mixing angles have a significant effect on
the oscillation probabilities as the neutrinos transverse through matter.

One of the primary goal of the above long baseline accelerator experiments is
to measure intrinsic CP violation in the neutrinos sector. In vacuum, the CP-
violating part of the neutrino oscillation probability in the appearance
channels, e.g. $\nu_{\mu}\rightarrow\nu_{e}$, is given by Bilenky and Petcov
(1987)

$8J\sin\Delta_{31}\sin\Delta_{32}\sin\Delta_{21}\,,$ (2)

where the kinematic phases are given by $\Delta_{jk}=\Delta
m^{2}_{jk}L/4E_{\nu}$ with $\Delta m^{2}_{jk}=m^{2}_{j}-m^{2}_{k}$ for an
experiment of baseline $L$ and neutrino energy $E_{\nu}$. $J$ is the Jarlskog
invariant Jarlskog (1985), that controls the size of CP violation. Using the
standard parameterization of the Pontecorvo, Maki, Nakagawa and Sakata (PMNS)
matrix Maki  _et al._ (1962); Pontecorvo (1968), the Jarlskog invariant is
given by

$J\equiv s_{23}c_{23}s_{13}c^{2}_{13}s_{12}c_{12}\sin\delta\,,$ (3)

where $s_{ij}=\sin\theta_{ij}$, $c_{ij}=\cos\theta_{ij}$.

For neutrinos propagating in matter, the part of the appearance oscillation
probability that depends on the intrinsic CP violation is given by

$8\widehat{J}\sin\widehat{\Delta}_{31}\sin\widehat{\Delta}_{32}\sin\widehat{\Delta}_{21}\,,$
(4)

where $\widehat{x}$ is the matter value for the vacuum variable $x$. The
Jarlskog invariant in matter, $\widehat{J}$, is given by same expression as
eq. 3, but with the mixing angles and phase replaced by their matter values
Zaglauer and Schwarzer (1988); Krastev and Petcov (1988); Parke and Weiler
(2001). The variables $\widehat{\theta}_{12}$, $\widehat{\theta}_{13}$ and
$\Delta\widehat{m^{2}}_{jk}$, that appear in eq. 4, have a strong dependence
on the Wolfenstein matter potential, $a$.

In Denton and Parke (2019) by Denton and Parke (DP), a simple and precise
factorization of the Jarlskog invariant in matter was given as follows:

$\widehat{J}\approx\frac{J}{\mathcal{R}_{\odot}^{\text{appx}}\,\mathcal{R}_{\rm
atm}^{\text{appx}}}\,,$ (5)

where

$\displaystyle\mathcal{R}_{\odot}^{\text{appx}}$
$\displaystyle=\sqrt{1-2c_{13}^{2}\cos 2\theta_{12}(a/\Delta
m^{2}_{21})+c_{13}^{4}(a/\Delta m^{2}_{21})^{2}}\,,$
$\displaystyle\mathcal{R}_{\rm atm}^{\text{appx}}$
$\displaystyle=\sqrt{1-2\cos 2\theta_{13}(a/\Delta m^{2}_{ee})+(a/\Delta
m^{2}_{ee})^{2}}\,.$ (6)

The fractional precision of this factorization is better than 0.07%, for all
$a$ and both mass orderings. A factorization with a precision of a few per
cent was given in Wang and Zhou (2019) with a follow up paper, Wang and Zhou
(2020), that reproduced the precise factorization of DP. In this paper we give
an extension of the analysis of DP to derive an exact factorization of this
Jarlskog invariant in matter as well as a perturbative expansion in the two
small quantities $s^{2}_{13}$ and $\Delta m^{2}_{21}/\Delta m^{2}_{ee}$.

Then, after defining precisely and calculating the location of the solar and
atmospheric resonances to the relevant accuracy in the matter potential for
the first time, we compare the location of the resonance with the location of
the minima of the factorizing two flavor resonance factors. Although they are
identical at the order given by DP, what is show here, is they are not
identical at higher orders but the fractional differences are found to be
small, less than 0.1%. A qualitative understanding of why they differ is also
given.

As part of our analysis, we show the simplicity of variables that can be
expressed as a symmetric polynomial of the neutrino eigenvalues in matter,
$\widehat{m^{2}}_{j}$, such as the Jarlskog invariant in matter as well as
other examples, Section II. We give the exact factorization of the Jarlskog
invariant in matter as a product of two, two flavor resonance factors, Section
III as well as a perturbative expansion. In Section IV, we compare the minima
of these exact factorizing factors to the actual location of the solar and
atmospheric resonance for the matter potential, followed by a summary and
conclusion section. There are five Appendices that contain computational
details. Thus this paper provides additional understanding of the Wolfenstein
matter effect for three neutrinos propagating through matter.

##  II Simplicity of Symmetric Polynomials of the Eigenvalues

In the flavor basis, the neutrino propagation Hamiltonian in matter is given
by

$(2E)H=U\begin{pmatrix}0\\\ &\Delta m^{2}_{21}\\\ &&\Delta
m^{2}_{31}\end{pmatrix}U^{\dagger}+\begin{pmatrix}a&&\\\ &0\\\
&&0\end{pmatrix}\,.$ (7)

$U$ is the PMNS Pontecorvo (1968); Maki  _et al._ (1962) lepton mixing matrix,
parameterized by

$U=U_{23}(\theta_{23},\delta)U_{13}(\theta_{13})U_{12}(\theta_{12})\equiv\\\
\begin{pmatrix}1\\\ &c_{23}&s_{23}e^{i\delta}\\\
&-s_{23}e^{-i\delta}&c_{23}\end{pmatrix}\\!\\!\\!\begin{pmatrix}c_{13}&&s_{13}\\\
&1\\\ -s_{13}&&c_{13}\end{pmatrix}\\!\\!\\!\begin{pmatrix}c_{12}&s_{12}\\\
-s_{12}&c_{12}\\\ &&1\end{pmatrix},$ (8)

where $s_{ij}=\sin\theta_{ij}$ and $c_{ij}=\cos\theta_{ij}$ are given by

$\displaystyle\sin^{2}\theta_{13}$ $\displaystyle\equiv|U_{e3}|^{2}\approx
0.022\,,$ $\displaystyle\sin^{2}\theta_{12}$
$\displaystyle\equiv|U_{e2}|^{2}/(1-|U_{e3}|^{2})\approx 0.32\,,$ (9)
$\displaystyle\sin^{2}\theta_{23}$ $\displaystyle\equiv|U_{\mu
3}|^{2}/(1-|U_{e3}|^{2})\approx 0.55\,.$

The sine and cosine of $\delta$ are given by

$\displaystyle e^{i\delta}$
$\displaystyle\equiv\frac{(1-|U_{e3}|^{2})U_{e2}U^{*}_{\mu 2}U^{*}_{e3}U_{\mu
3}+|U_{e2}|^{2}|U_{e3}|^{2}|U_{\mu 3}|^{2}}{|U_{e1}U_{e2}U_{e3}U_{\mu
3}U_{\tau 3}|}\,.$ (10)

At this time, circa 2020, the numerical value of $\delta$ is still to be
determined. Our definition of the CP-violating phase $\delta$ is invariant
under re-phasing111The commonly used definition, $\delta=-\rm{Arg}(U_{e3})$,
is not re-phasing invariant and requires an explicit phase choice especially
in the $U_{\alpha j}$ elements, for $\alpha=\mu$ and $\tau$, j=1 and 2. of the
rows and columns of $U$ and thus shifting it from it’s PDG location, i.e. from
next to $s_{13}$ to next to $s_{23}$, does not affect any observable222The
sign of $\delta$ given by eq. 10 is consistent with the sign in the PDG
expression for PMNS matrix.. This choice is made because the
$\theta_{23},\delta$ sub-matrix of U, commutes with the matter potential and
factors these variables from many expressions. For other choices for the
ordering of the factorization of the PMNS matrix see Denton and Pestes (2020).

For the long baseline neutrino oscillation experiments, T2K, NOvA, DUNE and
T2HK(K) it is sufficient to consider the matter density along the path of the
neutrino to be a constant, $a$, as has been discussed in detail in Kelly and
Parke (2018); King  _et al._ (2020). Therefore, for the rest of this paper we
will consider the matter potential to be a constant for a neutrino of a given
energy. This simplifies the solution to the evolution of the neutrino state in
matter significantly, by allowing for exact analytical solution. However, the
resulting solution is still analytically impenetrable as we will see later in
this section.

The characteristic equation for the matrix (2E)H gives the eigenvalues of the
square of the neutrino masses in matter, $\widehat{m^{2}}_{j}$, all satisfying

$\displaystyle\left(\widehat{m^{2}}_{j}\right)^{3}-A\,\left(\widehat{m^{2}}_{j}\right)^{2}+B\,\widehat{m^{2}}_{j}-C=0\,,$
(11)

where $A$, $B$, and $C$ are the sum of the eigenvalues, sum of the products of
the eigenvalues, and the triple product of the eigenvalues:

$\displaystyle A$ $\displaystyle\equiv\sum_{j}\widehat{m^{2}}_{j}=\Delta
m^{2}_{31}+\Delta m^{2}_{21}+a\,,$ $\displaystyle B$
$\displaystyle\equiv\sum_{j>k}\widehat{m^{2}}_{j}\widehat{m^{2}}_{k}=$ (12)
$\displaystyle\Delta m^{2}_{31}\Delta m^{2}_{21}+a(\Delta
m^{2}_{31}c^{2}_{13}+\Delta m^{2}_{21}(c^{2}_{12}+s^{2}_{13}s^{2}_{12}))\,,$
$\displaystyle C$ $\displaystyle\equiv\prod_{j}\widehat{m^{2}}_{j}=a\Delta
m^{2}_{31}\Delta m^{2}_{21}c^{2}_{13}c^{2}_{12}\,.$

We use the convention that in vacuum
$(\widehat{m^{2}}_{1},\widehat{m^{2}}_{2},\widehat{m^{2}}_{3})=(0,\Delta
m^{2}_{21},\Delta m^{2}_{31})$.

From refs. Cardano (1545), Barger  _et al._ (1980), Zaglauer and Schwarzer
(1988), the exact eigenvalues in matter are,

$\displaystyle\widehat{m^{2}}_{1}$
$\displaystyle=\frac{1}{3}A-\frac{1}{3}\sqrt{A^{2}-3B}\left(\mathcal{Z}+\sqrt{3}\sqrt{1-\mathcal{Z}^{2}}\right)\,,$
$\displaystyle\widehat{m^{2}}_{2}$
$\displaystyle=\frac{1}{3}A-\frac{1}{3}\sqrt{A^{2}-3B}\left(\mathcal{Z}-\sqrt{3}\sqrt{1-\mathcal{Z}^{2}}\right)\,,$
(13) $\displaystyle\widehat{m^{2}}_{3}$
$\displaystyle=\frac{1}{3}A+\frac{2}{3}\sqrt{A^{2}-3B}\,\mathcal{Z}\,.$

Where $\mathcal{Z}$ contains the $\cos\\{\frac{1}{3}\cos^{-1}[\cdots]\\}$
terms, given by

$\displaystyle\mathcal{Z}$
$\displaystyle=\cos\left\\{\frac{1}{3}\cos^{-1}\left[\frac{2A^{3}-9AB+27C}{2(A^{2}-3B)^{3/2}}\right]+\zeta\right\\}\,,$
(14)

with $\zeta=0$ for normal ordering (NO) that gives
$\widehat{m^{2}}_{1}<\widehat{m^{2}}_{2}<\widehat{m^{2}}_{3}$ and
$\zeta=2\pi/3$ for inverted ordering (IO) that gives
$\widehat{m^{2}}_{3}<\widehat{m^{2}}_{1}<\widehat{m^{2}}_{2}$, the usual
conventions. This convoluted term, $\cos\\{\frac{1}{3}\cos^{-1}[\cdots]\\}$,
which is a generic feature of the analytic solution to cubic equations, does
not lend itself to a useful perturbative expansion for arbitrary values of the
matter potential, even though, for neutrino oscillations there are two small
parameters $\sin^{2}\theta_{13}\sim 0.02$ and $\Delta m^{2}_{21}/\Delta
m^{2}_{31}\sim 0.03$. This unfortunate fact333As an example of the analytic
impenetrability of $\mathcal{Z}$, setting $a=0$ and recovering the vacuum
values for the eigenvalues, $(0,\Delta m^{2}_{21},\Delta m^{2}_{31})$, is a
highly non-trivial exercise. In vacuum $\mathcal{Z}=\frac{1}{2}(\Delta
m^{2}_{31}+\Delta m^{2}_{32})/\sqrt{(\Delta m^{2}_{32})^{2}+\Delta
m^{2}_{21}\Delta m^{2}_{31}}$ for NO. makes the exact analytical solution, eq.
13, only useful, in general, for numerical studies, not analytic
understanding.

However, while the exact eigenvalues have a very complicated analytic form,
any symmetric polynomial in the eigenvalues can be uniquely expressed in terms
of A, B & C. This follows from the Fundamental Theorem of Symmetric
Polynomials (FTSP), Wikipedia (2020), and allows any symmetric polynomial of
the $\widehat{m^{2}}_{i}$ to be written as a polynomial in the matter
potential, substantially simplifying such expressions.

An example of this is given in Harrison and Scott (2000), Yokomakura  _et al._
(2000) & Denton and Parke (2019),

$\left(\prod_{j>k}\Delta\widehat{m^{2}}_{jk}\right)^{2}=\\\
(A^{2}-4B)(B^{2}-4AC)+(2AB-27C)C\,,$ (15)

which given the expressions for A, B & C in eq. 12 is a fourth order
polynomial in the matter potential, $a$. No impenetrable
$\cos\\{\frac{1}{3}\cos^{-1}[\cdots]\\}$ terms appear.

The Eigenvector-Eigenvalue Identity, Denton  _et al._ (2020) & Denton  _et
al._ (2019), gives us the elements of the PMNS matrix in matter as follows

$|\widehat{U}_{\alpha
i}|^{2}=\frac{(\widehat{m^{2}}_{i})^{2}-(\xi+\chi)_{\alpha}\widehat{m^{2}}_{i}+(\xi\chi)_{\alpha}}{\Delta\widehat{m^{2}}_{ij}\Delta\widehat{m^{2}}_{ik}}\,,$
(16)

where $i$, $j$ and $k$ are all different. The variables $(\xi+\chi)_{\alpha}$
and $(\xi\chi)_{\alpha}$, which are the trace and determinant, respectively,
of (2E)H with the $\alpha$-row and $\alpha$-column removed. All
$(\xi+\chi)_{\alpha}$ and $(\xi\chi)_{\alpha}$ are linear polynomials in the
matter potential, $a$, and are given in Appendix A.

Combining eq. 15 & 16, we have that

$\left(\prod_{j>k}\Delta\widehat{m^{2}}_{jk}\right)^{2}\
\left(\prod_{i}|\widehat{U}_{\alpha i}|^{2}\right)=\\\
\prod_{i}\left((\widehat{m^{2}}_{i})^{2}-(\xi+\chi)_{\alpha}\widehat{m^{2}}_{i}-(\xi\chi)_{\alpha}\right)$
(17)

is a symmetric polynomial in the eigenvalues, $\widehat{m^{2}}_{i}$, and thus
can be written as a polynomial in A, B, C, $(\xi+\chi)_{\alpha}$ and
$(\xi\chi)_{\alpha}$, as follows

$\displaystyle\prod_{i}\left((\widehat{m^{2}}_{i})^{2}-(\xi+\chi)_{\alpha}\widehat{m^{2}}_{i}+(\xi\chi)_{\alpha}\right)$
$\displaystyle=$
$\displaystyle~{}~{}C[C-B(\xi+\chi)_{\alpha}+A(\xi+\chi)_{\alpha}^{2}-(\xi+\chi)_{\alpha}^{3}]$
(18)
$\displaystyle-(\xi+\chi)_{\alpha}(\xi\chi)_{\alpha}[(AB-3C)-B(\xi+\chi)_{\alpha}+A(\xi\chi)_{\alpha}]$
$\displaystyle+(\xi\chi)_{\alpha}[(B^{2}-2AC)+(A^{2}-2B)(\xi\chi)_{\alpha}+(\xi\chi)_{\alpha}^{2}].$

This is also a polynomial in the matter potential of maximum fourth order. In
Appendix B we give this expression which is straightforward to evaluate for
$\alpha=e,~{}\mu$ and $\tau$. Again, no
$\cos\\{\frac{1}{3}\cos^{-1}[\cdots]\\}$ terms appear.

For $\alpha=e$ for eq. 17, one finds that the right hand side is independent
of the matter potential, i.e.

$\left(\prod_{j>k}\Delta\widehat{m^{2}}_{jk}\right)^{2}\left(\prod_{i}|\widehat{U}_{ei}|^{2}\right)=\\\
\left(\prod_{j>k}\Delta{m^{2}}_{jk}\right)^{2}\
\left(\prod_{i}|U_{ei}|^{2}\right),$ (19)

this is the well known Naumov-Harrison-Scott (NHS) identity Naumov (1992);
Harrison and Scott (2000), divided by the Toshev Identity Toshev (1991),
squared. In this form, this invariance is not a surprise as in the
$a\rightarrow+\infty$ limit

$\displaystyle\left(\prod_{j>k}\Delta\widehat{m^{2}}_{jk}\right)^{2}$
$\displaystyle\rightarrow a^{4}\,,$
$\displaystyle|\widehat{U}_{ei}|^{2}\rightarrow\frac{1}{a^{2}}\quad$
$\displaystyle\left\\{\begin{array}[]{l}i=1,2\quad\rm{for~{}NO}\\\
i=1,3\quad\rm{for~{}IO}\end{array}\right.$ (22)

This implies that the fourth order polynomial must just be a constant.

For $\alpha=\mu$ or $\tau$, again in the $a\rightarrow+\infty$ limit

$\displaystyle|\widehat{U}_{\mu i}|^{2}\rightarrow\frac{1}{a^{2}}\quad$
$\displaystyle\left\\{\begin{array}[]{l}i=3\quad\rm{for~{}NO}\\\
i=2\quad\rm{for~{}IO}\end{array}\right.$ (25)

This implies that RHS of eq. 17 is a quadratic polynomial in the matter
potential, $a$. This fact is confirmed by explicit calculation where the
coefficients of the powers the matter potential, $a$, depend on $\theta_{23}$
and $\cos\delta$. Further details are also given in Appendix B.

##  III Exact Jarlskog Invariant in Matter

We start from the exact Naumov-Harrison-Scott (NHS) identity Naumov (1992);
Harrison and Scott (2000), which is that the Jarlskog factor in matter times
the product of the $\Delta m^{2}$ matter is an exact invariant444The simplest
way to “derive” this expression is that in the $L/E\rightarrow 0$, eq. 2 and 4
must be equal. :

$\frac{J}{\widehat{J}}=\frac{\Delta\widehat{m^{2}}_{32}\Delta\widehat{m^{2}}_{31}\Delta\widehat{m^{2}}_{21}}{\Delta
m^{2}_{32}\Delta m^{2}_{31}\Delta m^{2}_{21}}\,.$ (26)

While the exact eigenvalues have a very complicated analytic form Zaglauer and
Schwarzer (1988) due to the presence of the
$\cos\\{\frac{1}{3}\cos^{-1}[\cdots]\\}$ terms, $J^{2}/\widehat{J}^{2}$ can be
written as a simple fourth order polynomial of the vacuum parameters and the
matter potential, because of the FTSP.

Combining eq. 12 and 15 one obtains the exact expression for
$\left(\prod_{i>j}\Delta\widehat{m^{2}}_{ij}\right)^{2}$ as a fourth order
polynomial in the matter potential, $a$. This guarantees its factorization in
two quadratics as shown by Lodovico de Ferrari in 1540. We start by writing
eq. 26 as follows:

$\displaystyle\frac{J^{2}}{\widehat{J}^{2}}$
$\displaystyle=1+\sum^{4}_{n=1}f_{n}\left(\frac{a}{\Delta
m^{2}_{21}}\right)^{n}$ (27)

where the $f_{n}$’s are dimensionless functions of the vacuum oscillation
parameters which can be easily derived from eq. 15. Explicit expressions for
the $f_{n}$’s are given in Appendix C. $f_{1}$ and $f_{2}$ are of order 1,
where as $f_{3}$ is of order $\epsilon$ and $f_{4}$ is of order
$\epsilon^{2}$, where

$\epsilon\equiv\Delta m^{2}_{21}/\Delta m^{2}_{ee}\sim 0.03\,,$ (28)

and $\Delta m^{2}_{ee}\equiv c_{12}^{2}\Delta m^{2}_{31}+s_{12}^{2}\Delta
m^{2}_{31}$, Nunokawa  _et al._ (2005). Therefore, there is a distinct
hierarchy in the $f_{n}$’s.

Eq. 27 can be exactly factorized as

$\displaystyle\left(\frac{J}{\widehat{J}}\right)^{2}$
$\displaystyle=\left(1-2S^{ex}_{sol}\left(\frac{a}{\Delta
m^{2}_{21}}\right)+T^{ex}_{sol}\left(\frac{a}{\Delta
m^{2}_{21}}\right)^{2}\right)$ $\displaystyle\times$
$\displaystyle~{}\left(1-2S^{ex}_{atm}\left(\frac{a}{\Delta
m^{2}_{ee}}\right)+T^{ex}_{atm}\left(\frac{a}{\Delta
m^{2}_{ee}}\right)^{2}\right)$ (29)

with

$\displaystyle S^{ex}_{sol}$
$\displaystyle=\frac{1}{4}\left(-f_{1}+\sqrt{f_{1}^{2}+4(y-f_{2})}\right)$
(30) $\displaystyle T^{ex}_{sol}$
$\displaystyle=\frac{1}{2}\left(y+\sqrt{y^{2}-4f_{4}}\right)$ (31)
$\displaystyle S^{ex}_{atm}$
$\displaystyle=\frac{1}{4}\left(-f_{1}-\sqrt{f_{1}^{2}+4(y-f_{2})}\right)/\epsilon$
(32) $\displaystyle T^{ex}_{atm}$
$\displaystyle=\frac{1}{2}\left(y-\sqrt{y^{2}-4f_{4}}\right)/\epsilon^{2}\,.$
(33)

The $1/\epsilon$ and $1/\epsilon^{2}$ terms for $S^{ex}_{atm}$ and
$T^{ex}_{atm}$, respectively, are need to change the $(a/\Delta m^{2}_{21})$
to $(a/\Delta m^{2}_{ee})$ in second bracket of eq. 29. Except for the $a^{3}$
term, the coefficients for all other powers of $a$ are satisfied for any value
of $y$. To get the correct coefficient for the $a^{3}$ terms, $y$ must satisfy
the following cubic equation:

$\displaystyle y^{3}+g_{2}y^{2}+g_{1}y+g_{0}=0$ (34) $\displaystyle{\rm with}$
$\displaystyle\quad g_{2}=-f_{2},\quad g_{1}=f_{1}f_{3}-4f_{4}$
$\displaystyle{\rm and}$ $\displaystyle\quad
g_{0}=(4f_{2}-f^{2}_{1})f_{4}-f^{2}_{3}.$

The relevant, exact solution to this cubic equation is

$\displaystyle y$
$\displaystyle=2\sqrt{Q}\cos\left\\{\frac{1}{3}\arccos\left[R/\sqrt{Q^{3}}\right]\right\\}-g_{2}/3$
(35) $\displaystyle Q$ $\displaystyle=(g_{2}^{2}-3g_{1})/9,\quad
R=(9g_{1}g_{2}-27g_{0}-2g^{3}_{2})/54\,,$

where $y\approx f_{2}\sim 1$. The other solutions give complex coefficients in
the factorization of eq. 29.

Unfortunately the solution to the cubic equation does not provide any analytic
insight even though it is the exact solution. However, because the $f_{n}$’s
have a distinct hierarchy in $\Delta m^{2}_{21}/\Delta m^{2}_{ee}$, so do the
$g_{n}$’s: $g_{2}$ is of order 1, whereas $g_{1}$ and $g_{0}$ are first and
second order in $\Delta m^{2}_{21}/\Delta m^{2}_{ee}$, respectively. This
allows an iterative solution to eq. 34,

$\displaystyle y^{(n)}$
$\displaystyle=-g_{2}-g_{1}/y^{(n-1)}-g_{0}/(y^{(n-2)})^{2}$ (36)

with $y^{(0)}=-g_{2}$ and $y^{(1)}=-(g^{2}_{2}-g_{1})/g_{2}$. Therefore, $y$
can easily be calculated to the required order in $\Delta m^{2}_{21}/\Delta
m^{2}_{ee}$.

Then to obtain the factorization coefficients, $S_{sol}$ thru $T_{atm}$, eqs.
30 \- 33, we use the fact that for neutrino oscillations there are two small
parameters,

$\sin^{2}\theta_{13}\sim 0.02\quad{\rm and}\quad\epsilon\equiv\Delta
m^{2}_{21}/\Delta m^{2}_{ee}\sim 0.03$ (37)

so that one can easily perform a Taylor Series expansion in these two small
qualities to obtain

$\displaystyle S_{sol}$ $\displaystyle\approx c^{2}_{13}\cos
2\theta_{12}-(s^{2}_{13}\epsilon)-\cos
2\theta_{12}(s^{2}_{13}\epsilon^{2})+\cdots$ (38) $\displaystyle T_{sol}$
$\displaystyle\approx c^{4}_{13}-2\cos 2\theta_{12}(s^{2}_{13}\epsilon)+2\cos
2\theta_{12}(s^{4}_{13}\epsilon)$
$\displaystyle\quad-2(1-6s^{2}_{12}c^{2}_{12})(s^{2}_{13}\epsilon^{2})+\cdots$
(39) $\displaystyle S_{atm}$ $\displaystyle\approx\cos
2\theta_{13}+s^{2}_{12}c^{2}_{12}(\epsilon^{2})-4s^{2}_{12}c^{2}_{12}(s^{2}_{13}\epsilon^{2})$
$\displaystyle\quad+s^{2}_{12}c^{2}_{12}\cos
2\theta_{12}(\epsilon^{3})+\cdots$ (40) $\displaystyle T_{atm}$
$\displaystyle\approx
1+2s^{2}_{12}c^{2}_{12}(\epsilon^{2})+2s^{2}_{12}c^{2}_{12}\cos
2\theta_{12}(\epsilon^{3})+\cdots$ (41)

The first term in each of these equations gives the approximation discussed in
DP and the correction to the first term are of ${\cal O}(\epsilon^{2})$ or
${\cal O}(s^{2}_{13}\epsilon)$ as expected.

Figure 1: The top panel is the ratio between the Jarlskog invariant in matter
and the vacuum Jarlskog invariant for both NO (blue) and IO (red) as function
of the energy of the neutrino. Here $Y_{e}\rho$=1.5 g.cm-3. The bottom panel
is the fractional difference between the various approximation to the
factorization of the Jarlskog invariant compared with the exact Jarlskog
invariant in matter. The dashed lines are using the DP approximation, eq. 5
Denton and Parke (2019) , whereas the solid lines are the third order
approximation given in this paper, eqs. 38 \- 41, giving more than two orders
of magnitude improvement. The results for the exact factorization, eqs. 30 \-
33, are not shown on this figure because they depend on the machine precision;
using “double precision” one obtains $10^{-14}$ fractional precision. Positive
(negative) energy is for neutrinos (anti-neutrinos) and zero is vacuum for
both neutrinos and antineutrinos and all energies.

In Table I we numerical evaluate the exact factorization along with those of
DP and the approximation given by eq. 38 to 41.

NO (IO) | $~{}~{}S_{sol}/S^{ex}_{sol}-1$ |  $T_{sol}/T^{ex}_{sol}-1$ |  $S_{atm}/S^{ex}_{atm}-1$ |  $T_{atm}/T^{ex}_{atm}-|~{}~{}$
---|---|---|---|---
DP approx |  $+1.9~{}(-1.9)\times 10^{-3}$ |  $+4.7~{}(-5.0)\times 10^{-4}$ |  $-1.9~{}(-1.9)\times 10^{-4}$ |  $-4.0~{}(-4.0)\times 10^{-4}$
eq. 38 to 41 | $-1.6~{}(+1.4)\times 10^{-7}$ | $-6.1~{}(+2.5)\times 10^{-7}$ | $+2.8~{}(+2.8)\times 10^{-7}$ | $-6.9~{}(-6.9)\times 10^{-8}$
Table 1: The fractional differences for the coefficients of the factorizing
quadratics between DP, Denton and Parke (2019), and the exact, eqs. 30 \- 33,
(middle row), as well as between the perturbative expansions given in this
paper eqs. 38 \- 41 and the exact (bottom row). For this Table the following
parameters where used: $\sin^{2}\theta_{13}=0.022$, $\sin^{2}\theta_{12}=0.32$
and $\Delta m^{2}_{21}=7.5\times 10^{-5}$ eV2. $\Delta m^{2}_{ee}=\pm
2.5\times 10^{-3}$ eV2 where the plus (minus) sign is for NO (IO). The matter
density times electron fraction used is $Y_{e}\rho=1.5$ g/cm3.

Thus the factorization of the Jarlskog invariant in matter is given by

$J\approx\mathcal{R}_{\odot}\,\mathcal{R}_{\rm atm}\,\widehat{J}\,,$ (42)

where

$\displaystyle\mathcal{R}_{\odot}$
$\displaystyle=\sqrt{1-2S_{sol}\left(\frac{a}{\Delta
m^{2}_{21}}\right)+T_{sol}\left(\frac{a}{\Delta m^{2}_{21}}\right)^{2}}\,,$
(43) $\displaystyle\mathcal{R}_{\rm atm}$
$\displaystyle=\sqrt{1-2S_{atm}\left(\frac{a}{\Delta
m^{2}_{ee}}\right)+T_{atm}\left(\frac{a}{\Delta m^{2}_{ee}}\right)^{2}}\,.$
(44)

This is the same functional form as in DP, eq. 5. Depending on which
approximation one uses for the $S$’s and $T$’s one obtains a fractional
precision of $7\times 10^{-4}$ using DP, eq. 6, $4\times 10^{-6}$ using eq.
38-41 and machine precision using the numerical solution to the cubic equation
and eqs. 30 \- 33. See Fig. 1 for both the ratio of $\widehat{J}/J$, as well
as the fractional difference between the exact and the approximation of DP,
eq. 6 and that of eq. 38-41. If by machine precision one means “double
precision,” then the fractional difference is $\sim 10^{-14}$. For
phenomenological purposes the precision using the simple result of DP Denton
and Parke (2019) is accurate enough for most applications.

##  IV Relationship with the Solar and Atmospheric Resonances

Its easily shown that $\mathcal{R}_{\odot}$ is minimum when555For the
numerical calculations in this section, we use the same parameters as in Table
1.

$\displaystyle\frac{a^{\text{sol}}_{\text{min}}}{\Delta m^{2}_{21}}$
$\displaystyle=\frac{S^{ex}_{sol}}{T^{ex}_{sol}}=\frac{\cos
2\theta_{12}}{c^{2}_{13}}+\cos 4\theta_{12}\,(s^{2}_{13}\epsilon)+\cdots$ (45)
$\displaystyle=\frac{\cos
2\theta_{12}}{c^{2}_{13}}+\left\\{\begin{array}[]{ll}-5.2\times
10^{-4}&\text{NO}\\\ ~{}~{}5.0\times 10^{-4}&\text{IO}\end{array}\right.\,,$
(48)

and that $\mathcal{R}_{\rm atm}$ is minimum when

$\displaystyle\frac{a^{\text{atm}}_{\text{min}}}{\Delta m^{2}_{ee}}$
$\displaystyle=\frac{S^{ex}_{atm}}{T^{ex}_{atm}}=\cos
2\theta_{13}-s^{2}_{12}c^{2}_{12}(\epsilon^{2})+\cdots$ (49)
$\displaystyle=\cos 2\theta_{13}-\left\\{\begin{array}[]{ll}2.0\times
10^{-4}&\text{NO}\\\ 1.9\times 10^{-4}&\text{IO}\end{array}\right.\,.$ (52)

from eq. 43 and 44 respectively. The “$\cdots$” are higher order terms in
$s^{2}_{13}$ and $\epsilon$. Clearly, these two values for the matter
potential are close to the values for the solar and atmospheric resonances.
But what is the precise relationship? It cannot be exact as both
$\mathcal{R}_{\odot}$ and $\mathcal{R}_{\rm atm}$ are symmetric functions
about the value that minimizes them, whereas for the solar and atmospheric
resonances there must be some asymmetry caused by the fact that the solar
resonance is below (above) atmospheric resonance for the NO (IO).

To answer this question we first have to define the solar and atmospheric
resonance. For the solar resonance we define it to be the value of the matter
potential that minimizes the separation between the matter mass eigenstates
$|\widehat{\nu}_{1}\rangle$ and $|\widehat{\nu}_{2}\rangle$, that is when

$\displaystyle\frac{d(\widehat{m^{2}}_{2}-\widehat{m^{2}}_{1})}{da}=0\,.$ (53)

It can be easily shown by taking the derivative of A, B & C with respect to
$a$, see Appendix D, or using the evolution equations of Xing  _et al._ (2018)
that

$\displaystyle\frac{d\widehat{m^{2}}_{i}}{da}=|\widehat{U}_{ei}|^{2}\,.$ (54)

Therefore the solar resonance condition also implies that

$\displaystyle|\widehat{U}_{e1}|^{2}=|\widehat{U}_{e2}|^{2},\quad\sin^{2}\widehat{\theta}_{12}=0.5\,.$
(55)

That is, by minimizing the separation of the matter mass eigenstates
$|\widehat{\nu}_{1}\rangle$ and $|\widehat{\nu}_{2}\rangle$, we also have
exact maximal mixing between the matter eigenstates
$|\widehat{\nu}_{1}\rangle$ and $|\widehat{\nu}_{2}\rangle$.

For the atmospheric resonance, one could consider the resonance condition for
NO to be

$\displaystyle\frac{d\,(\widehat{m^{2}}_{3}-\widehat{m^{2}}_{2})}{da}=0\,,\quad|\widehat{U}_{e3}|^{2}=|\widehat{U}_{e2}|^{2}\,,$
(56)

which implies $\sin^{2}\widehat{\theta}_{13}\approx 0.5.$ For IO

$\displaystyle\frac{d\,(\widehat{m^{2}}_{3}-\widehat{m^{2}}_{1})}{da}=0\,,\quad|\widehat{U}_{e3}|^{2}=|\widehat{U}_{e1}|^{2}\,,$
(57)

which also implies $\sin^{2}\widehat{\theta}_{13}\approx 0.5$, Xing  _et al._
(2018). That is, almost maximal mixing between $|\widehat{\nu}_{3}\rangle$ and
$|\widehat{\nu}_{2}\rangle$ ( $|\widehat{\nu}_{1}\rangle$ ) for NO (IO). For
both NO and IO, the fractional difference between
$\sin^{2}\widehat{\theta}_{13}$ and 0.5 is $\sim 2\times 10^{-4}$ with these
definitions.

However, we present here a mass ordering independent definition of the
atmospheric resonance, given by

$\displaystyle\frac{d\,(\widehat{m^{2}}_{3}-\widehat{m^{2}}_{2}-\widehat{m^{2}}_{1})}{da}=0\,.$
(58)

This gives an excellent approximation666The fractional difference between the
solution to eqs. 56 and 58 or eqs. 57 and 58 is $7\times 10^{-5}$. Using eqs.
56 and 57 as the definition of the atmospheric resonance does not change our
conclusions.  to eq. 56 for NO as the mass of the matter eigenstate
$|\widehat{\nu}_{1}\rangle$ is essentially independent of the matter potential
at the atmospheric resonance as
$d\widehat{m^{2}}_{1}/da=|\widehat{U}_{e1}|^{2}\approx 2\times 10^{-4}$ here.
Similarly for eq. 57 for IO.

This mass ordering independent definition also gives

$\displaystyle|\widehat{U}_{e3}|^{2}=|\widehat{U}_{e2}|^{2}+|\widehat{U}_{e1}|^{2},\quad\sin^{2}\widehat{\theta}_{13}=0.5\,,$
(59)

from eq. 54, independent of the mass ordering. That is, exact maximal mixing
between the matter eigenstate $|\widehat{\nu}_{3}\rangle$ and the state
$\cos\widehat{\theta}_{12}|\widehat{\nu}_{1}\rangle+\sin\widehat{\theta}_{12}|\widehat{\nu}_{2}\rangle$.
Therefore, the mass ordering independent definition, eq. 58, is what will be
used for the atmospheric resonance in the rest of this paper.

Independent of this discussion, it was argued in Denton and Parke (2018) that
combination $(\widehat{m^{2}}_{3}-\widehat{m^{2}}_{2}-\widehat{m^{2}}_{1})$ is
the effective $\Delta m^{2}_{ee}(a)$ in matter with the addition of a constant
that depends on ones conventions: $\Delta m^{2}_{ee}(a)$ gives the effective
frequency for a $\nu_{e}$ disappearance oscillations in matter at the
atmospheric minima and coincides with $\Delta m^{2}_{ee}$ in vacuum, i.e.
$\Delta m^{2}_{ee}(a=0)=\Delta m^{2}_{ee}$. In the conventions of this paper,

$\displaystyle\Delta
m^{2}_{ee}(a)\equiv\widehat{m^{2}}_{3}-\widehat{m^{2}}_{2}-\widehat{m^{2}}_{1}+\Delta
m^{2}_{21}c^{2}_{12}$ (60) $\displaystyle\approx\Delta
m^{2}_{ee}\sqrt{(1-2\cos 2\theta_{13}(a/\Delta m^{2}_{ee})+(a/\Delta
m^{2}_{ee})^{2}}\,.$

Note, this approximation is just the leading terms in $\Delta
m^{2}_{ee}\mathcal{R}_{\rm atm}$.

With the above definitions, the exact position of the solar and atmospheric
resonances, $a_{\text{xsol}}$ and $a_{\text{xatm}}$ respectively, can be
calculated perturbatively, see Appendix E, and numerical from eq. 13 and we
find that

$\displaystyle\frac{a_{\text{xsol}}}{\Delta m^{2}_{21}}$
$\displaystyle=\frac{\cos
2\theta_{12}}{c^{2}_{13}}+\cos^{2}2\theta_{12}(s^{2}_{13}\epsilon)+\cdots$
(61) $\displaystyle=\frac{\cos
2\theta_{12}}{c^{2}_{13}}+\left\\{\begin{array}[]{ll}~{}~{}9.7\times
10^{-5}&\text{NO}\\\ -8.2\times 10^{-5}&\text{IO}\end{array}\right.\,,$ (64)

and

$\displaystyle\frac{a_{\text{xatm}}}{\Delta m^{2}_{ee}}$ $\displaystyle=\cos
2\theta_{13}-s^{2}_{12}c^{2}_{12}(1-2s_{13})\,(\epsilon^{2})+\cdots\,$ (65)
$\displaystyle=\cos 2\theta_{13}-\left\\{\begin{array}[]{ll}1.52\times
10^{-4}&\text{NO}\\\ 1.49\times 10^{-4}&\text{IO}\end{array}\right.\,.$ (68)

Thus the accuracy of the leading terms is at the $10^{-4}$ level, provide one
includes the $c^{2}_{13}$ for the solar resonance and $\Delta m^{2}_{ee}$
instead of $\Delta m^{2}_{31}$ or $\Delta m^{2}_{32}$ for the atmospheric
resonance, Zaglauer and Schwarzer (1988); Denton  _et al._ (2016). The
magnitude of $\Delta m^{2}_{31}$ and $\Delta m^{2}_{32}$ differs from $\Delta
m^{2}_{ee}$ at the 1-2% level.

Comparing the eqs. 45 & 61 and eqs. 49 & 65, it clear that the minima of
$\mathcal{R}_{\odot}$ and $\mathcal{R}_{\rm atm}$ do not exactly coincide with
the exact solar and exact atmospheric resonances but the difference is at the
0.06% for solar and 0.005% for atmospheric which are negligible for
phenomenological purposes. The order of magnitude difference between these two
is probably due to the fact that the variable
$(\widehat{m^{2}}_{3}-\widehat{m^{2}}_{2}-\widehat{m^{2}}_{1})$ is more the
symmetric about the atmosphere resonance than
$(\widehat{m^{2}}_{2}-\widehat{m^{2}}_{1})$ is about the solar resonance, see
Fig. 2.

Figure 2: The solid lines are
$(\widehat{m^{2}}_{3}-\widehat{m^{2}}_{2}-\widehat{m^{2}}_{1}+\Delta
m^{2}_{21}c^{2}_{12})$ for both NO (blue) and IO (red) as a function of the
neutrino energy. Note, the approximate symmetry about the minima, $|E|\sim$
10.5 GeV, for these two functions. At $E=0$, i.e. vacuum, this combination is
exactly $\pm\Delta m^{2}_{ee}$, plus NO, minus for IO. The dashed lines are
$(\widehat{m^{2}}_{2}-\widehat{m^{2}}_{1})$ for NO (blue) and IO (red). These
two functions have a significant asymmetry about the minima, E $\sim$ 0.12
GeV, especially for $|E|>$ 10 GeV. Here, $Y_{e}\rho=1.5$ g.cm-3. Positive
(negative) energy is for neutrinos (anti-neutrinos) and zero is vacuum for
both neutrinos and antineutrinos and all energies.

It is also worth noting that by using $\Delta m^{2}_{ee}|_{IO}=-\Delta
m^{2}_{ee}|_{NO}$ rule for switching between the two mass orderings (NO
$\leftrightarrow$ IO), the fractional difference between $|a_{\text{xatm}}|$
for the two orderings is $\sim 3\times 10^{-6}$ and in fact, the fractional
difference between

$-(\widehat{m^{2}}_{3}-\widehat{m^{2}}_{2}-\widehat{m^{2}}_{1}+\Delta
m^{2}_{21}c^{2}_{12})|_{IO}(-a)\\\
\quad\text{and}\quad(\widehat{m^{2}}_{3}-\widehat{m^{2}}_{2}-\widehat{m^{2}}_{1}+\Delta
m^{2}_{21}c^{2}_{12})|_{NO}(a)$ (69)

is smaller than $10^{-5}$ for all $a$. As compared to $\sim$1%, if one uses
any of the following rules to flip the mass ordering:

$\Delta m^{2}_{31}|_{IO}=-\Delta m^{2}_{31}|_{NO}\quad\text{or}\quad\Delta
m^{2}_{32}|_{IO}=-\Delta m^{2}_{32}|_{NO}\\\ \text{or}\Delta
m^{2}_{32}|_{IO}=-\Delta m^{2}_{31}|_{NO}\,.\quad\quad$ (70)

##  V Summary and Conclusions

In this paper we have derived three new results for three flavor neutrino
oscillations in matter. First, we have shown that any quantity that can be
represented by a symmetric polynomial of the eigenvalues of the Hamiltonian,
$\widehat{m^{2}}_{j}$, can be written as a polynomial in the matter potential,
$a$. Examples of such quantities are

$\displaystyle\left(\prod_{j>k}\Delta\widehat{m^{2}}_{jk}\right)^{2},\quad\left(\frac{J}{\widehat{J}}\right)^{2}\,$
and
$\displaystyle\left(\prod_{j>k}\Delta\widehat{m^{2}}_{jk}\right)^{2}\times\left(\prod_{i}|\widehat{U}_{\alpha
i}|^{2}\right)\,.$

The first two are fourth order polynomials in the matter potential whereas the
last one is a constant for $\alpha=e$ and a quadratic polynomial in the matter
potential for $\alpha=\mu,~{}\tau.$ Clearly there are many other such
quantities that are symmetric in the eigenvalues.

Second, for the Jarlskog invariant in matter we give the exact factorization
of the fourth order polynomial, such that

$\widehat{J}=\frac{J}{\mathcal{R}_{\odot}^{\text{ex}}\,\mathcal{R}_{\rm
atm}^{\text{ex}}}\,,$

where

$\displaystyle\mathcal{R}_{\odot}^{ex}$
$\displaystyle=\sqrt{1-2S^{ex}_{sol}\left(\frac{a}{\Delta
m^{2}_{21}}\right)+T^{ex}_{sol}\left(\frac{a}{\Delta
m^{2}_{21}}\right)^{2}}\,,$ $\displaystyle\mathcal{R}_{\rm atm}^{ex}$
$\displaystyle=\sqrt{1-2S^{ex}_{atm}\left(\frac{a}{\Delta
m^{2}_{ee}}\right)+T^{ex}_{atm}\left(\frac{a}{\Delta
m^{2}_{ee}}\right)^{2}}\,.$

The exact coefficients $S^{ex}_{sol}$ to $T^{ex}_{atm}$ are given in eqs. 30
\- 33. Since the exact coefficients involve the solution to a cubic polynomial
which are notoriously challenging to penetrate analytically, a Taylor series
expansion of these coefficients to ${\cal O}(s^{4}_{13}\epsilon),~{}{\cal
O}(s^{2}_{13}\epsilon^{2}),~{}{\cal O}(\epsilon^{3}),$ is also given in eqs.
38 – 41. The fractional accuracy of these coefficients is better than
$10^{6}$, as shown in Table 1.

Third, we define the solar resonance to be when

$\displaystyle\frac{d(\widehat{m^{2}}_{2}-\widehat{m^{2}}_{1})}{da}=0\quad\Longleftrightarrow\quad\sin^{2}\widehat{\theta}_{12}=0.5\,,$

and calculate that this occurs when

$\displaystyle\frac{a}{\Delta m^{2}_{21}}$ $\displaystyle=\frac{\cos
2\theta_{12}}{c^{2}_{13}}+\cos^{2}2\theta_{12}(s^{2}_{13}\epsilon)+\cdots\,.$

where the “$\cdots$” are higher order terms in $s^{2}_{13}$ and
$\epsilon\equiv\Delta m^{2}_{21}/\Delta m^{2}_{ee}$. Independent of the mass
ordering, we define the atmospheric resonance to be when

$\displaystyle\frac{d(\widehat{m^{2}}_{3}-\widehat{m^{2}}_{2}-\widehat{m^{2}}_{1})}{da}=0\quad\Longleftrightarrow\quad\sin^{2}\widehat{\theta}_{13}=0.5\,,$

and calculate that this occurs when

$\displaystyle\frac{a}{\Delta m^{2}_{ee}}$ $\displaystyle=\cos
2\theta_{13}-s^{2}_{12}c^{2}_{12}(1-2s_{13})\,(\epsilon^{2})+\cdots\,.$

For three flavors of neutrinos, these are the most accurate values for the
locations of the solar and atmospheric resonances available in the literature.
The value of the matter potential typically quoted for the atmospheric
resonance, $a/\Delta m^{2}_{32}=\cos 2\theta_{13}$, differs from above value
by 2%, a phenomenological significant difference.

We have also made a comparison between the location of the minima in the
matter potential for the two factorizing factors, $\mathcal{R}_{\odot}^{ex}$
and $\mathcal{R}_{\rm atm}^{ex}$ and the location in matter potential of the
solar and atmospheric resonances. The difference in the value of the matter
potential between the location of the minima and the resonance values are
0.06% for solar and 0.005% for atmospheric, therefore they do not coincide.
However, these difference are negligible for phenomenological purposes.

To conclude, the understanding of the physics of neutrino propagation in
matter is of essential importance to the current NOvA and T2K, and upcoming
DUNE and T2HK(K) long baseline neutrino oscillation experiments. The
Wolfenstein matter effect significantly complicates three flavor neutrino
oscillations in matter compared to vacuum. The exact analytic expressions,
even for constant matter density, are analytically impenetrable due to the
presence of the $\cos\\{\frac{1}{3}\cos^{-1}[\cdots]\\}$ terms. In this paper
we show that variables that depend on the symmetric polynomials of the mass
squared of the neutrino eigenstates in matter are just simple polynomials of
the matter potential, a significant simplification. We give a number of
examples and in particular the Jarlskog invariant in matter which controls the
size of intrinsic CP violation in the above neutrino appearance experiments.
We give for the first time the exact factorization of the Jarlskog invariant
in matter into two quadratic resonance factors. The location of the minima of
these resonance factors is compared to the location of the solar and
atmospheric resonances to the appropriate order in the small quantities
$s^{2}_{13}$ and $\Delta m^{2}_{21}/\Delta m^{2}_{ee}$. This required, for the
first time, the calculation of the location of the solar and atmospheric
resonances beyond leading order. All of this further enhances our
understanding of neutrino propagation in matter relevant for the long baseline
neutrino oscillation experiments being currently performed and for the next
generation of experiments.

###### Acknowledgements.

I thank Peter Denton for many enlightening discussions during the course of
this work. Fermilab is operated by the Fermi Research Alliance under contract
no. DE-AC02-07CH11359 with the U.S. Department of Energy. This project has
received funding/support from the European Union’s Horizon 2020 research and
innovation programme under the Marie Sklodowska-Curie grant agreement No
860881-HIDDeN. I dedicate this paper to the memory of the outstanding neutrino
theorist and friend, Sandip Pakvasa. His insights on neutrino physics and the
neutrinos he emitted, will live on until the end of the Universe.

##  Appendix A The variables $\xi$ and $\chi$

The quantities $(\xi+\chi)_{\alpha}$ and $(\xi\chi)_{\alpha}$ are the trace
and determinant of the sub-matrix of (2E)H with the $\alpha$-row and
$\alpha$-column removed. In the flavor basis, since $H_{ee}$ is the only
element that depends on the matter potential these traces and determinants are
at most linear polynomials in the matter potential, a. In fact,
$(\xi+\chi)_{e}$ and $(\xi\chi)_{e}$ are independent of the matter potential.
Explicitly,

$\displaystyle\xi_{e}+\chi_{e}$
$\displaystyle=\left(\mathcal{H}_{\mu\mu}+\mathcal{H}_{\tau\tau}\right)=\Delta
m^{2}_{31}c^{2}_{13}+\Delta
m^{2}_{21}(c^{2}_{12}+s^{2}_{13}s^{2}_{12})=dB/da\,,$ (71)
$\displaystyle\xi_{e}\chi_{e}$
$\displaystyle=\left(\mathcal{H}_{\mu\mu}\mathcal{H}_{\tau\tau}-\mathcal{H}_{\mu\tau}^{2}\right)=\Delta
m^{2}_{31}\Delta m^{2}_{21}c^{2}_{13}c^{2}_{12}=dC/da\,,$ (72)
$\displaystyle\xi_{\mu}+\chi_{\mu}$
$\displaystyle=\left[c_{23}^{2}(\mathcal{H}_{ee}+\mathcal{H}_{\tau\tau})+s_{23}^{2}(\mathcal{H}_{ee}+\mathcal{H}_{\mu\mu})-2s_{23}c_{23}\cos\delta(\mathcal{H}_{\mu\tau})\right]\,,$
(73) $\displaystyle\xi_{\mu}\chi_{\mu}$
$\displaystyle=\left[c_{23}^{2}(\mathcal{H}_{ee}\mathcal{H}_{\tau\tau}-\mathcal{H}_{e\tau}^{2})+s_{23}^{2}(\mathcal{H}_{ee}\mathcal{H}_{\mu\mu}-\mathcal{H}_{e\mu}^{2})-2s_{23}c_{23}\cos\delta(\mathcal{H}_{ee}\mathcal{H}_{\mu\tau}-\mathcal{H}_{e\mu}\mathcal{H}_{e\tau})\right]\,.$
(74)

The $\xi_{\tau}$ and $\chi_{\tau}$ eigenvalues are the same as $\xi_{\mu}$ and
$\chi_{\mu}$ under the interchange $s_{23}^{2}\leftrightarrow c_{23}^{2}$ and
$s_{23}c_{23}\to-s_{23}c_{23}$. Where

$\mathcal{H}=\begin{pmatrix}a+\Delta m^{2}_{ee}s_{13}^{2}+\Delta
m^{2}_{21}s_{12}^{2}&\quad&c_{13}s_{12}c_{12}\Delta
m^{2}_{21}&&s_{13}c_{13}\Delta m^{2}_{ee}\\\\[5.0pt] \cdot&&\Delta
m^{2}_{21}c_{12}^{2}&\quad&-s_{13}s_{12}c_{12}\Delta m^{2}_{21}\\\\[5.0pt]
\cdot&&\cdot&&\Delta m^{2}_{ee}c_{13}^{2}+\Delta
m^{2}_{21}s_{12}^{2}\end{pmatrix}\,,$ (75)

where $\mathcal{H}_{\alpha\beta}=\mathcal{H}_{\beta\alpha}$ and $\Delta
m^{2}_{ee}\equiv c_{12}^{2}\Delta m^{2}_{31}+s_{12}^{2}\Delta m^{2}_{31}$
Nunokawa  _et al._ (2005). The relationship between $\mathcal{H}$ and H of eq.
7 is
$\mathcal{H}=(2E)~{}U^{\dagger}_{23}(\theta_{23},\delta)HU_{23}(\theta_{23},\delta)$.
See Denton  _et al._ (2020) for further details.

Defining $\xi_{\alpha}\leq\chi_{\alpha}$, then the Cauchy interlace theorem
tells us that
$\widehat{m^{2}}_{1}\leq\xi_{\alpha}\leq\widehat{m^{2}}_{2}\leq\chi_{\alpha}\leq\widehat{m^{2}}_{3}$
for NO.

##  Appendix B  Evaluation of
$\prod_{i}\left((\widehat{m^{2}}_{i})^{2}-(\xi+\chi)_{\alpha}\widehat{m^{2}}_{i}+(\xi\chi)_{\alpha}\right)$

It is straight forward to show that

$\displaystyle\prod_{i}\left((\widehat{m^{2}}_{i})^{2}-(\xi+\chi)_{\alpha}\widehat{m^{2}}_{i}+(\xi\chi)_{\alpha}\right)$
$\displaystyle=$
$\displaystyle~{}~{}C[C-B(\xi+\chi)_{\alpha}+A(\xi+\chi)_{\alpha}^{2}-(\xi+\chi)_{\alpha}^{3}]$
(76)
$\displaystyle-(\xi+\chi)_{\alpha}(\xi\chi)_{\alpha}[(AB-3C)-B(\xi+\chi)_{\alpha}+A(\xi\chi)_{\alpha}]$
$\displaystyle+(\xi\chi)_{\alpha}[(B^{2}-2AC)+(A^{2}-2B)(\xi\chi)_{\alpha}+(\xi\chi)_{\alpha}^{2}]\,.$

For $\alpha=e$, both $(\xi+\chi)_{e}$ and $(\xi\chi)_{e}$ are independent of
$a$ and this is at most a quadratic polynomial in $a$. The quadratic and
linear terms in $a$ cancel and one is left with the constant, so this
expression is independent of the matter potential.

For $\alpha=\mu$ (or $\tau$), both $(\xi+\chi)_{\mu}$ and $(\xi\chi)_{\mu}$
are linear in $a$ and this polynomial is at most fourth order in $a$. Again
the top two powers of $a$ cancel and one is left with a quadratic polynomial
in $a$. Since $(\xi+\chi)_{\mu}$ and $(\xi\chi)_{\mu}$ depend on $\theta_{23}$
and $\delta$ it is convenient to reorganize the calculation by multiplying
$(\widehat{m^{2}}_{i})^{2}$ by $(c^{2}_{23}+s^{2}_{23})$ as follows:

$\displaystyle\prod_{i}\left((\widehat{m^{2}}_{i})^{2}-(\xi+\chi)_{\mu}\widehat{m^{2}}_{i}+(\xi\chi)_{\mu}\right)=\prod_{i}\left(c^{2}_{23}X_{i}+s^{2}_{23}Y_{i}+2s_{23}c_{23}\cos\delta~{}Z_{i}\right)$
(77)

where

$\displaystyle X_{i}$
$\displaystyle=(\widehat{m^{2}}_{i})^{2}-(\mathcal{H}_{ee}+\mathcal{H}_{\tau\tau})(\widehat{m^{2}}_{i})+(\mathcal{H}_{ee}\mathcal{H}_{\tau\tau}-\mathcal{H}_{e\tau}^{2})=(\widehat{m^{2}}_{i}-\mathcal{H}_{ee})(\widehat{m^{2}}_{i}-\mathcal{H}_{\tau\tau})-\mathcal{H}^{2}_{e\tau}\,,$
$\displaystyle Y_{i}$
$\displaystyle=(\widehat{m^{2}}_{i})^{2}-(\mathcal{H}_{ee}+\mathcal{H}_{\mu\mu})(\widehat{m^{2}}_{i})+(\mathcal{H}_{ee}\mathcal{H}_{\mu\mu}-\mathcal{H}_{e\mu}^{2})=(\widehat{m^{2}}_{i}-\mathcal{H}_{ee})(\widehat{m^{2}}_{i}-\mathcal{H}_{\mu\mu})-\mathcal{H}^{2}_{e\mu}\,,$
(78) $\displaystyle Z_{i}$
$\displaystyle=\mathcal{H}_{\mu\tau}(\widehat{m^{2}}_{i})-(\mathcal{H}_{ee}\mathcal{H}_{\mu\tau}-\mathcal{H}_{e\mu}\mathcal{H}_{e\tau})=\mathcal{H}_{\mu\tau}(\widehat{m^{2}}_{i}-\mathcal{H}_{ee})-\mathcal{H}_{e\mu}\mathcal{H}_{e\tau}\,,$

which satisfy the following condition that $X_{i}Y_{i}=Z^{2}_{i}$. Then the
LHS of eq. 76 is a homogeneous third order polynomial in $c^{2}_{23}$,
$s^{2}_{23}$ and $(2s_{23}c_{23}\cos\delta)$:

$\displaystyle\prod_{i}\left(c^{2}_{23}X_{i}+s^{2}_{23}Y_{i}+2s_{23}c_{23}\cos\delta~{}Z_{i}\right)$
$\displaystyle=$
$\displaystyle~{}c_{23}^{6}~{}[X_{1}X_{2}X_{3}]+c_{23}^{4}s^{2}_{23}~{}[X_{1}X_{2}Y_{3}]+c_{23}^{2}s^{4}_{23}~{}[X_{1}Y_{2}Y_{3}]+s_{23}^{6}~{}[Y_{1}Y_{2}Y_{3}]$
$\displaystyle+\,(2s_{23}c_{23}\cos\delta)(~{}c_{23}^{4}~{}[Z_{1}X_{2}X_{3}]+c_{23}^{2}s_{23}^{2}~{}[X_{1}Y_{2}Z_{3}]+s_{23}^{4}~{}[Z_{1}Y_{2}Y_{3}]~{})$
$\displaystyle+\,(2s_{23}c_{23}\cos\delta)^{2}(c_{23}^{2}~{}[Z_{1}Z_{2}X_{3}]+s_{23}^{2}~{}[Z_{1}Z_{2}Y_{3}]~{})+\,(2s_{23}c_{23}\cos\delta)^{3}~{}[Z_{1}Z_{2}Z_{3}]$
(79)

where each $[\cdots]$ is symmeterized over the labels (1,2,3) which guarantees
that each term is a polynomial in $a$. This is easy to calculate using an
algebraic program like FORM. Some, but not all of the terms are very simple,
we give here examples of the simplest terms.

The coefficient of $s^{6}_{23}$ is given by

$Y_{1}Y_{2}Y_{3}=s^{4}_{13}c^{2}_{13}~{}s^{2}_{12}c^{2}_{12}~{}(\Delta
m^{2}_{21})^{2}\biggr{(}\Delta m^{2}_{31}\Delta m^{2}_{32}+a\,\Delta
m^{2}_{ee}\biggr{)}^{2},$

the coefficient of $c^{6}_{23}$ is

$X_{1}X_{2}X_{3}=s^{2}_{13}c^{2}_{13}~{}s^{4}_{12}c^{4}_{12}~{}(\Delta
m^{2}_{21})^{4}\,a^{2},$

and the coefficient of $(2s_{23}c_{23}\cos\delta)^{3}$

$Z_{1}Z_{2}Z_{3}=s^{3}_{13}c^{2}_{13}~{}s^{3}_{12}c^{3}_{12}~{}(\Delta
m^{2}_{21})^{3}(\Delta m^{2}_{31}\Delta m^{2}_{32}+a\Delta m^{2}_{ee})a\,.$

Note $Z_{1}Z_{2}Z_{3}=\sqrt{X_{1}X_{2}X_{3}\,Y_{1}Y_{2}Y_{3}}$.

For $\alpha=\tau$, just interchange $s^{2}_{23}\leftrightarrow c^{2}_{23}$ and
flip sign of $s_{23}c_{23}$.

##  Appendix C Derivation of the $f_{n}$’s

Writing A, B and C, defined in eq. 12 , as constant plus linear in $a$ terms,
as follows,

$\displaystyle A$ $\displaystyle=A_{0}+a,\quad\quad~{}A_{0}=\Delta
m^{2}_{21}+\Delta m^{2}_{31},$ $\displaystyle B$
$\displaystyle=B_{0}+aB_{1},\quad B_{0}=\Delta m^{2}_{31}\Delta
m^{2}_{21},\quad\quad\quad B_{1}=\Delta m^{2}_{31}c^{2}_{13}+\Delta
m^{2}_{21}(c^{2}_{12}+s^{2}_{13}s^{2}_{12}),$ $\displaystyle C$
$\displaystyle=aC_{1},\quad\quad\quad~{}~{}C_{1}=\Delta m^{2}_{31}\Delta
m^{2}_{21}c^{2}_{13}c^{2}_{12}.$ (80)

Then from eq. 15, the $f_{n}$’s of eq. 27 are given by

$\displaystyle f_{1}$ $\displaystyle=2\Delta
m^{2}_{21}[(A_{0}^{2}-4B_{0})(B_{0}B_{1}-2A_{0}C_{1})+B_{0}^{2}(A_{0}-2B_{1})+A_{0}B_{0}C_{1}\,]/f_{0}\,,$
$\displaystyle f_{2}$ $\displaystyle=(\Delta
m^{2}_{21})^{2}\,[(A_{0}^{2}-4B_{0})(B_{1}^{2}-4C_{1})+B_{0}^{2}+4(A_{0}-2B_{1})(B_{0}B_{1}-2A_{0}C_{1})+2(B_{1}A_{0}+B_{0})C_{1}-27C_{1}^{2}\,]/f_{0}\,,$
$\displaystyle f_{3}$ $\displaystyle=2(\Delta
m^{2}_{21})^{3}[(B_{0}B_{1}-2A_{0}C_{1})+(B_{1}^{2}-4C_{1})(A_{0}-2B_{1})+B_{1}C_{1}]/f_{0}\,,$
$\displaystyle f_{4}$ $\displaystyle=(\Delta
m^{2}_{21})^{4}(B_{1}^{2}-4C_{1})/f_{0}\,,$ (81)

where $f_{0}=B_{0}^{2}(A_{0}^{2}-4B_{0})=(\Delta m^{2}_{21}\Delta
m^{2}_{31}\Delta m^{2}_{32})^{2}$.

Note, $f_{1},~{}f_{2},~{}f_{3}~{}\&~{}f_{4}$ are of order 1, 1, $\epsilon$ &
$\epsilon^{2}$, respectively, where $\epsilon\equiv\Delta m^{2}_{21}/\Delta
m^{2}_{ee}$.

##  Appendix D Proof of $d\,\widehat{m^{2}}_{i}/da=|\widehat{U}_{ei}|^{2}$

Differentiate A, B & C of eq. 12 with respect to $a$, to obtain the following:

$\displaystyle 1$
$\displaystyle=\frac{d\widehat{m^{2}}_{1}}{da}+\frac{d\widehat{m^{2}}_{2}}{da}+\frac{d\widehat{m^{2}}_{2}}{da}\,,$
(82) $\displaystyle(\xi+\chi)_{e}$
$\displaystyle=(\widehat{m^{2}}_{2}+\widehat{m^{2}}_{3})\frac{d\widehat{m^{2}}_{1}}{da}+(\widehat{m^{2}}_{3}+\widehat{m^{2}}_{1})\frac{d\widehat{m^{2}}_{2}}{da}+(\widehat{m^{2}}_{1}+\widehat{m^{2}}_{2})\frac{d\widehat{m^{2}}_{3}}{da}\,,$
(83) $\displaystyle(\xi\chi)_{e}$
$\displaystyle=(\widehat{m^{2}}_{2}\widehat{m^{2}}_{3})\frac{d\widehat{m^{2}}_{1}}{da}+(\widehat{m^{2}}_{3}\widehat{m^{2}}_{1})\frac{d\widehat{m^{2}}_{2}}{da}+(\widehat{m^{2}}_{1}\widehat{m^{2}}_{2})\frac{d\widehat{m^{2}}_{3}}{da}\,.$
(84)

Multiple eq. 82 by $(\widehat{m^{2}}_{i})^{2}$, eq. 83 by
$(-\widehat{m^{2}}_{i})$ and add these to eq. 84, one obtains

$\displaystyle\frac{d\,\widehat{m^{2}}_{i}}{da}$
$\displaystyle=\frac{(\widehat{m^{2}}_{i})^{2}-(\xi+\chi)_{e}(\widehat{m^{2}}_{i})+(\xi\chi)_{e}}{\Delta\widehat{m^{2}}_{ij}\Delta\widehat{m^{2}}_{ik}}=|\widehat{U}_{ei}|^{2}\,.$
(85)

with the indicies $i$, $j$ and $k$ all different. See eq. 16.

##  Appendix E Perturbative Expansions about the Resonances

A perturbative expansion is used to obtain the values of the matter potential
for the atmospheric and solar resonances.

###  E.1 Atmospheric Resonance

In this Appendix we perform an perturbative expansion in the matter potential
about the approximate value of the matter potential that minimizes
$(\widehat{m^{2}}_{3}-\widehat{m^{2}}_{2}-\widehat{m^{2}}_{1})$, that is

$a=\Delta m^{2}_{ee}\cos 2\theta_{13}+\delta a$ (86)

Starting with the Hamiltonian from eq. 7, a (2-3) rotation using
$U_{23}(\theta_{23},\delta)$ is performed to remove the $\theta_{23}$ and
$\delta$ dependence. Then a $\pi/4$ rotation is performed in the (1-3) sector
using the value of $a$ given in eq. 86. The resulting Hamiltonian is given by

$\displaystyle
U^{\dagger}_{13}(\pi/4)U^{\dagger}_{23}(\theta_{23},\delta)~{}(2E)H~{}U_{23}(\theta_{23},\delta)U_{13}(\pi/4)=H_{a}+h$
(87)

with

$\displaystyle H_{a}=\text{\bf diag}(~{}$ $\displaystyle\Delta
m^{2}_{ee}(c^{2}_{13}-s_{13}c_{13})+\Delta m^{2}_{12}s^{2}_{12},~{}~{}\Delta
m^{2}_{21}c^{2}_{12},~{}~{}\Delta m^{2}_{ee}(c^{2}_{13}+s_{13}c_{13})+\Delta
m^{2}_{12}s^{2}_{12}~{})\,,$ (88) $\displaystyle\quad
h=\frac{1}{2}\left(\begin{array}[]{ccc}\delta a&\delta m^{2}_{+}&\delta a\\\
\cdot&0&\delta m^{2}_{-}\\\ \cdot&\cdot&\delta a\end{array}\right)$ (92)

where $\delta m^{2}_{\pm}=\sqrt{2}\Delta m^{2}_{21}s_{12}c_{12}(c_{13}\pm
s_{13})$. Note, because the solar crossing has not been resolved the mass
eigenstates $\nu_{1}$ and $\nu_{2}$ are interchanged. Since $h$ is symmetric,
the $\cdots$’s are given by the appropriate term above the diagonal.

Now perturbation theory can be used to calculate
$(\widehat{m^{2}}_{3}-\widehat{m^{2}}_{2}-\widehat{m^{2}}_{1})$, which to
third order in $h$, is given by

$\displaystyle\widehat{m^{2}}_{3}-\widehat{m^{2}}_{2}-\widehat{m^{2}}_{1}\approx~{}d_{31}+2(h^{2}_{13}/d_{31}+h^{2}_{23}/d_{32})+2h_{11}h_{23}(2h_{12}d_{32}-h_{23}d_{31})/(d_{32}^{2}d_{31})\,,$
(93)

where $h_{jk}$ are the elements of matrix $h$, eq. 92, and
$d_{jk}=H_{a}(j,j)-H_{a}(k,k)$, e.g. $d_{31}=\Delta
m^{2}_{ee}(2s_{13}c_{13})$. Eq. 93 is a quadratic in the variable $\delta a$
as both $h_{11}$ and $h_{13}$ are linear in $\delta a$. To obtain an estimate
of the shift one needs to go to third order, as first order vanishes, second
order is quadratic and third order gives the first odd term in $\delta a$. The
value of $\delta a$ that minimizes
$(\widehat{m^{2}}_{3}-\widehat{m^{2}}_{2}-\widehat{m^{2}}_{1})$ is given by

$\displaystyle\delta a$
$\displaystyle=-h_{23}(2h_{12}d_{32}-h_{23}d_{31})/d_{32}^{2}\,\approx-s^{2}_{12}c^{2}_{12}(1-2s_{13})(\Delta
m^{2}_{21})^{2}/\Delta m^{2}_{ee}\,.$ (94)

This gives eq. 65. The minimum of eq. 93 reproduces the exact minimum to
better than one part in $10^{8}$.

Using the above perturbation theory, one can also show that the value of
$\delta a$ that minimizes $(\widehat{m^{2}}_{3}-\widehat{m^{2}}_{2})$ for NO
is given by

$\displaystyle\delta a$ $\displaystyle\approx-s^{2}_{12}c^{2}_{12}(\Delta
m^{2}_{21})^{2}/\Delta m^{2}_{ee}\,.$ (95)

Thus the alternative definition of the atmospheric resonance, eq. 56, changes
the corrections at ${\cal O}(s_{13}\epsilon^{2})$ and similarly for IO.

###  E.2 Solar Resonance

In this Appendix we perform an perturbative expansion in the matter potential
about the approximate value of the matter potential that minimizes
$(\widehat{m^{2}}_{2}-\widehat{m^{2}}_{1})$, that is

$a=a_{0}+\delta a,\quad\quad a_{0}=\Delta m^{2}_{21}\cos
2\theta_{12}/\cos^{2}\theta_{13}$ (96)

Starting with the Hamiltonian from eq. 7, a (2-3) rotation using
$U_{23}(\theta_{23},\delta)$ is performed to remove the $\theta_{23}$ and
$\delta$ dependence. Then a $\theta_{13}$ rotation is performed in the (1-3)
sector followed by a $\pi/4$ rotation is performed in the (1-2) sector using
the value of $a$ given in eq. 96. The resulting Hamiltonian is given by

$\displaystyle
U^{\dagger}_{12}(\pi/4)U^{\dagger}_{13}(\theta_{13})U^{\dagger}_{23}(\theta_{23},\delta)~{}(2E)H~{}U_{23}(\theta_{23},\delta)U_{13}(\theta_{13})U_{12}(\pi/4)=H_{s}+h$
(97)

where

$\displaystyle H_{s}=\text{\bf diag}($ $\displaystyle~{}\Delta
m^{2}_{21}(c^{2}_{12}-s_{12}c_{12}),~{}~{}\Delta
m^{2}_{21}(c^{2}_{12}+s_{12}c_{12}),~{}~{}\Delta m^{2}_{31}+a_{0}~{})\,,$ (98)
$\displaystyle h=\frac{1}{2}\left(\begin{array}[]{ccc}\delta
a\,c^{2}_{13}&\delta a\,c^{2}_{13}&\sqrt{2}\,a\,s_{13}c_{13}\\\ \cdot&\delta
a\,c^{2}_{13}&\sqrt{2}\,a\,s_{13}c_{13}\\\ \cdot&\cdot&2\delta
a\,s^{2}_{13}\end{array}\right)\,.$ (102)

Since $h$ is symmetric, the $\cdots$’s are given by the appropriate term above
the diagonal. There are only three independent elements in $h$: we will use
$h_{11},~{}h_{13},~{}h_{33}$ as the independent ones.

Now perturbation theory can be used to calculate
$(\widehat{m^{2}}_{2}-\widehat{m^{2}}_{1})$, which to third order in $h$, is
given by

$\displaystyle\widehat{m^{2}}_{2}-\widehat{m^{2}}_{1}\approx$
$\displaystyle~{}d_{21}+2h^{2}_{11}/d_{21}-h^{2}_{13}\,d_{21}/(d_{32}d_{31})$
$\displaystyle-h^{2}_{13}(d_{31}+d_{32})\,[\,2h_{11}/(d_{32}d_{31}d_{21})+(h_{11}-h_{33})d_{21}/(d^{2}_{31}d^{2}_{32})\,]$
(103)

where here $h_{jk}$ are the elements of matrix $h$, eq. 102, and
$d_{jk}=H_{s}(j,j)-H_{s}(k,k)$, e.g. $d_{21}=\Delta
m^{2}_{21}(2s_{12}c_{12})$.

Using the fact that $h_{13}$ is approximately constant, $h_{13}\approx
a_{0}\,s_{13}c_{13}/\sqrt{2}$, and keeping only the terms with $d_{21}$ in
denominator, then the value of $\delta a$ that minimizes
$(\widehat{m^{2}}_{2}-\widehat{m^{2}}_{1})$ is approximately given by

$\displaystyle\delta a$ $\displaystyle\approx
h^{2}_{13}(d_{31}+d_{32})/(d_{31}d_{32})\approx\cos^{2}2\theta_{12}\,s^{2}_{13}\,(\Delta
m^{2}_{21})^{2}/\Delta m^{2}_{ee}\,.$ (104)

This gives eq. 61. The minimum of eq. 103 reproduces the exact minimum to one
part in $10^{7}$.

## References

  * Itow  _et al._ (2001) Y. Itow _et al._ (T2K), in  _Neutrino oscillations and their origin. Proceedings, 3rd International Workshop, NOON 2001, Kashiwa, Tokyo, Japan, December 508, 2001_ (2001) pp. 239–248, arXiv:hep-ex/0106019 [hep-ex] .
  * Ayres  _et al._ (2004) D. S. Ayres _et al._ (NOvA), (2004), arXiv:hep-ex/0503053 [hep-ex] .
  * Acciarri  _et al._ (2016) R. Acciarri _et al._ (DUNE), (2016), arXiv:1601.05471 [physics.ins-det] .
  * Abe  _et al._ (2014) K. Abe _et al._ (Hyper-Kamiokande Working Group) (2014) arXiv:1412.4673 [physics.ins-det] .
  * Abe  _et al._ (2018) K. Abe _et al._ (Hyper-Kamiokande), PTEP 2018, 063C01 (2018), arXiv:1611.06118 [hep-ex] .
  * Wolfenstein (1978) L. Wolfenstein, Phys. Rev. D17, 2369 (1978), [294(1977)].
  * Bilenky and Petcov (1987) S. M. Bilenky and S. T. Petcov, Rev. Mod. Phys. 59, 671 (1987), [Erratum: Rev. Mod. Phys.60,575(1988)].
  * Jarlskog (1985) C. Jarlskog, Phys. Rev. Lett. 55, 1039 (1985).
  * Maki  _et al._ (1962) Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theor. Phys. 28, 870 (1962), [,34(1962)].
  * Pontecorvo (1968) B. Pontecorvo, Sov. Phys. JETP 26, 984 (1968), [Zh. Eksp. Teor. Fiz.53,1717(1967)].
  * Zaglauer and Schwarzer (1988) H. W. Zaglauer and K. H. Schwarzer, Z. Phys. C40, 273 (1988).
  * Krastev and Petcov (1988) P. I. Krastev and S. T. Petcov, Phys. Lett. B205, 84 (1988).
  * Parke and Weiler (2001) S. J. Parke and T. J. Weiler, Phys. Lett. B501, 106 (2001), arXiv:hep-ph/0011247 [hep-ph] .
  * Denton and Parke (2019) P. B. Denton and S. J. Parke, Phys. Rev. D 100, 053004 (2019), arXiv:1902.07185 [hep-ph] .
  * Wang and Zhou (2019) X. Wang and S. Zhou, JHEP 05, 035 (2019), arXiv:1901.10882 [hep-ph] .
  * Wang and Zhou (2020) X. Wang and S. Zhou, Nucl. Phys. B 950, 114867 (2020), arXiv:1908.07304 [hep-ph] .
  * Denton and Pestes (2020) P. B. Denton and R. Pestes, (2020), arXiv:2006.09384 [hep-ph] .
  * Kelly and Parke (2018) K. J. Kelly and S. J. Parke, Phys. Rev. D 98, 015025 (2018), arXiv:1802.06784 [hep-ph] .
  * King  _et al._ (2020) S. F. King, S. Molina Sedgwick, S. J. Parke, and N. W. Prouse, Phys. Rev. D 101, 076019 (2020), arXiv:2001.05505 [hep-ph] .
  * Cardano (1545) G. Cardano,  _Ars Magna_ (1545).
  * Barger  _et al._ (1980) V. D. Barger, K. Whisnant, S. Pakvasa, and R. J. N. Phillips, Phys. Rev. D22, 2718 (1980), [300(1980)].
  * Wikipedia (2020) Wikipedia, “Elementary symmetric polynomials,”  (11/2020).
  * Harrison and Scott (2000) P. F. Harrison and W. G. Scott, Phys. Lett. B476, 349 (2000), arXiv:hep-ph/9912435 [hep-ph] .
  * Yokomakura  _et al._ (2000) H. Yokomakura, K. Kimura, and A. Takamura, Phys. Lett. B 496, 175 (2000), arXiv:hep-ph/0009141 .
  * Denton  _et al._ (2020) P. B. Denton, S. J. Parke, and X. Zhang, Phys. Rev. D 101, 093001 (2020), arXiv:1907.02534 [hep-ph] .
  * Denton  _et al._ (2019) P. B. Denton, S. J. Parke, T. Tao, and X. Zhang, (2019), arXiv:1908.03795 [math.RA] .
  * Naumov (1992) V. A. Naumov, Int. J. Mod. Phys. D1, 379 (1992).
  * Toshev (1991) S. Toshev, Mod. Phys. Lett. A6, 455 (1991).
  * Nunokawa  _et al._ (2005) H. Nunokawa, S. J. Parke, and R. Zukanovich Funchal, Phys. Rev. D72, 013009 (2005), arXiv:hep-ph/0503283 [hep-ph] .
  * Xing  _et al._ (2018) Z.-Z. Xing, S. Zhou, and Y.-L. Zhou, JHEP 05, 015 (2018), arXiv:1802.00990 [hep-ph] .
  * Denton and Parke (2018) P. B. Denton and S. J. Parke, Phys. Rev. D 98, 093001 (2018), arXiv:1808.09453 [hep-ph] .
  * Denton  _et al._ (2016) P. B. Denton, H. Minakata, and S. J. Parke, JHEP 06, 051 (2016), arXiv:1604.08167 [hep-ph] .