A new Wolfenstein-like expansion of lepton flavor mixing
towards understanding its fine structure

Zhi-zhong Xing1,2 ***E-mail<EMAIL_ADDRESS>
1Institute of High Energy Physics and School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China
2Center of High Energy Physics, Peking University, Beijing 100871, China

###### Abstract

Taking the tri-bimaximal flavor mixing pattern as a particular basis, we
propose a new way to expand the $3\times 3$ unitary Pontecorvo-Maki-Nakagawa-
Sakata (PMNS) lepton flavor mixing matrix $U$ in powers of the magnitude of
its smallest element $\xi\equiv\left|U_{e3}\right|\simeq 0.149$. Such a
Wolfenstein-like parametrization of $U$ allows us to easily describe the
salient features and fine structures of flavor mixing and CP violation, both
in vacuum and in matter.

##  1 Motivation

Among the three Euler-like rotation angles of the Cabibbo-Kobayashi-Maskawa
(CKM) quark flavor mixing matrix $V$ [1, 2], it is the largest one — the
Cabibbo angle $\theta_{\rm C}\simeq 13^{\circ}$ that was most accurately
measured from the very beginning [3]. That is why Wolfenstein proposed a
remarkable parametrization of the CKM matrix $V$ in 1983 by expanding its nine
matrix elements in powers of a small parameter $\lambda\equiv\sin\theta_{\rm
C}\simeq 0.225$ [4], from which the hierarchical structure of quark flavor
mixing can be well understood. For example, one may easily arrive at the four-
layered ordering

$\displaystyle\underline{|V_{tb}|>|V_{ud}|>|V_{cs}|}\gg\underline{|V_{us}|>|V_{cd}|}\gg\underline{|V_{cb}|>|V_{ts}|}\gg\underline{|V_{td}|>|V_{ub}|}$
(1)

of respective ${\cal O}(1)$, ${\cal O}(\lambda)$, ${\cal O}(\lambda^{2})$ and
${\cal O}(\lambda^{3})$ as a natural consequence of the unitarity of $V$ [5].
In particular, the CKM matrix $V$ approaches the unique identity matrix $I$ in
the $\lambda\to 0$ limit, implying that the up- and down-type quark sectors
should have an underlying parallelism between their flavor textures. Such a
conceptually interesting limit is quite suggestive, and it has widely been
considered for explicit model building [6].

In comparison, two of the three Euler-like rotation angles of the Pontecorvo-
Maki-Nakagawa-Sakata (PMNS) lepton flavor mixing matrix $U$ (i.e.,
$\theta_{12}\simeq 33.4^{\circ}$ and $\theta_{23}\sim 45^{\circ}$ [3]) are so
large that a naive Wolfenstein-like parametrization of $U$ seems quite
unlikely 111But see, e.g., Refs. [7, 8, 9], for the early attempts in this
regard.. A very real possibility is that the leading-order term of the PMNS
matrix $U$ is a constant flavor mixing pattern $U_{0}$ consisting of two large
angles and originating from a kind of discrete flavor symmetry, while the
smallest flavor mixing angle $\theta_{13}\simeq 8.6^{\circ}$ and CP-violating
effects arise after small perturbations or quantum corrections to $U_{0}$ are
taken into account [10, 11]. From the point of view of model building [6, 12,
13, 14], the most popular choice of $U_{0}$ has been the tri-bimaximal flavor
mixing pattern $U_{\rm TBM}$ [15, 16, 17] which predicts
$\theta^{(0)}_{12}=\cot\sqrt{2}\simeq 35.26^{\circ}$,
$\theta^{(0)}_{13}=0^{\circ}$ and $\theta^{(0)}_{23}=45^{\circ}$. Given the
smallness of $\theta_{13}$, several attempts have been made along the above
line of thought to expand the $U_{\rm TBM}$-based PMNS matrix $U$ in powers of
the small Wolfenstein parameter $\lambda$ that is borrowed from quark flavor
mixing (see, e.g., Refs. [18, 19, 20, 21, 22]).

Guided by the best-fit values and $1\sigma$ intervals of three lepton flavor
mixing angles extracted from a global analysis of the currently available
experimental data on neutrino oscillations [23, 24],

$\displaystyle{\rm
NMO:}\quad\theta_{12}=0.583^{+0.013}_{-0.013}\;,\quad\theta_{13}=0.150^{+0.002}_{-0.002}\;,\quad\theta_{23}=0.736^{+0.019}_{-0.015}\;$
(2)

for the normal mass ordering (NMO) of three active neutrinos, or

$\displaystyle{\rm
IMO:}\quad\theta_{12}=0.583^{+0.013}_{-0.013}\;,\quad\theta_{13}=0.150^{+0.002}_{-0.002}\;,\quad\theta_{23}=0.855^{+0.018}_{-0.020}\;$
(3)

in the inverted mass ordering (IMO) case, we find that the smallest angles
$\theta_{13}$ is most accurately determined and thus suitable for serving as
an optimal expansion parameter 222It is worth pointing out that the previous
phenomenological conjecture $|U_{e3}|=\lambda/\sqrt{2}\simeq 0.159$ [25] is no
more favored by the Daya Bay precision measurements (i.e.,
$\sin\theta_{13}\simeq 0.149$) [26].. So we are going to study an expansion of
the $U_{\rm TBM}$-based PMNS matrix $U$ in powers of the small parameter
$\xi\equiv|U_{e3}|=\sin\theta_{13}\simeq 0.149\simeq 0.662\lambda$. Moreover,
the best-fit values of $\theta_{12}$ and $\theta_{23}$ lead us to

$\displaystyle{\rm NMO:}\quad\theta_{12}-\theta^{(0)}_{12}$
$\displaystyle\simeq$ $\displaystyle-0.032\simeq-1.463\hskip
1.13791pt\xi^{2}\;,$ $\displaystyle\theta_{23}-\theta^{(0)}_{23}$
$\displaystyle\simeq$ $\displaystyle-0.049\simeq-2.225\hskip
1.13791pt\xi^{2}\;;\hskip 14.22636pt$ (4)

as well as

$\displaystyle{\rm IMO:}\quad\theta_{12}-\theta^{(0)}_{12}$
$\displaystyle\simeq$ $\displaystyle-0.032\simeq-1.463\hskip
1.13791pt\xi^{2}\;,$ $\displaystyle\theta_{23}-\theta^{(0)}_{23}$
$\displaystyle\simeq$ $\displaystyle+0.070\simeq+3.135\hskip
1.13791pt\xi^{2}\;.\hskip 14.22636pt$ (5)

Eqs. (4) and (5) imply that the observed values of $\theta_{12}$ and
$\theta_{23}$ are most likely to deviate from their respective tri-bimaximal
flavor mixing limits at the level of ${\cal O}(\xi^{2})$. This observation
provides us with a new angle of view, which is quite different from those in
the previous attempts, to expand $U$ in the basis of $U_{\rm TBM}$.

In what follows we shall propose a new expansion of the PMNS matrix $U$ by
starting from the standard Euler-like parametrization of $U$ and taking
account of

  * •

$\sin\theta_{13}\equiv\xi\simeq 0.149$ as the lepton flavor mixing expansion
parameter;

  * •

$\theta_{12}=\theta^{(0)}_{12}-A\xi^{2}$ with
$\theta^{(0)}_{12}=\cot\sqrt{2}\simeq 35.26^{\circ}$ and $A\sim{\cal O}(1)$;

  * •

$\theta_{23}=\theta^{(0)}_{23}-B\xi^{2}$ with $\theta^{(0)}_{23}=45^{\circ}$
and $|B|\sim{\cal O}(1)$.

It is obvious that $B\neq 0$ characterizes the effect of $\mu$-$\tau$
permutation symmetry breaking of $U$, and the sign of $B$ determines the
octant of $\theta_{23}$ [27, 28]. In this case the PMNS matrix will be
expressed in terms of the tri-bimaximal flavor mixing pattern $U_{\rm TBM}$,
the three real parameters $\xi$, $A$ and $B$, and the poorly known CP-
violating phase $\delta$. Here we leave aside the two possible extra CP phases
associated with the Majorana nature of massive neutrinos, as they are
completely unknown and have no effect on the fine structure of $U$.

##  2 The expansion of $U$

As advocated by the Particle Data Group (PDG), the standard Euler-like
parametrization of the unitary $3\times 3$ PMNS matrix $U$ is explicitly of
the form [3]

$\displaystyle U=\left(\begin{matrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-{\rm
i}\delta}\cr-s_{12}c_{23}-c_{12}s_{13}s_{23}e^{{\rm
i}\delta}&c_{12}c_{23}-s_{12}s_{13}s_{23}e^{{\rm i}\delta}&c_{13}s_{23}\cr
s_{12}s_{23}-c_{12}s_{13}c_{23}e^{{\rm
i}\delta}&~{}-c_{12}s_{23}-s_{12}s_{13}c_{23}e^{{\rm
i}\delta}~{}&c_{13}c_{23}\cr\end{matrix}\right)\;,$ (6)

in which $c_{ij}\equiv\cos\theta_{ij}$ and $s_{ij}\equiv\sin\theta_{ij}$ (for
$ij=12,13,23$) with $\theta_{ij}$ lying in the first quadrant, $\delta$
denotes the irreducible CP-violating phase responsible for CP violation in
neutrino oscillations, and possible additional CP phases of the Majorana
nature can always be factored out on the right-hand side of $U$ and thus have
been omitted. Following the strategy outlined above for a Wolfenstein-like
expansion of the $U_{\rm TBM}$-based PMNS matrix $U$, we have

$\displaystyle s_{13}$ $\displaystyle\equiv$ $\displaystyle\xi\;,$
$\displaystyle c_{13}$ $\displaystyle=$ $\displaystyle
1-\frac{1}{2}\xi^{2}-\frac{1}{8}\xi^{4}+{\cal O}(\xi^{6})\;,$ $\displaystyle
s_{12}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\left[1-\sqrt{2}A\xi^{2}-\frac{1}{2}A^{2}\xi^{4}\right]+{\cal
O}(\xi^{6})\;,$ $\displaystyle c_{12}$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{6}}\left[1+\frac{1}{\sqrt{2}}A\xi^{2}-\frac{1}{2}A^{2}\xi^{4}\right]+{\cal
O}(\xi^{6})\;,\hskip 14.22636pt$ $\displaystyle s_{23}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[1-B\xi^{2}-\frac{1}{2}B^{2}\xi^{4}\right]+{\cal
O}(\xi^{6})\;,$ $\displaystyle c_{23}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[1+B\xi^{2}-\frac{1}{2}B^{2}\xi^{4}\right]+{\cal
O}(\xi^{6})\;.$ (7)

Substituting Eq. (7) into Eq. (6), we immediately arrive at the nine elements
of $U$ in the PDG-advocated phase convention as follows:

$\displaystyle U_{e1}$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{6}}-\frac{1-\sqrt{2}A}{\sqrt{6}}\xi^{2}-\frac{1+2\sqrt{2}A+4A^{2}}{4\sqrt{6}}\xi^{4}\;,$
$\displaystyle U_{e2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}-\frac{1+2\sqrt{2}A}{2\sqrt{3}}\xi^{2}-\frac{1-4\sqrt{2}A+4A^{2}}{8\sqrt{3}}\xi^{4}\;,$
$\displaystyle U_{e3}$ $\displaystyle=$ $\displaystyle\xi e^{-{\rm
i}\delta}\;,$ $\displaystyle U_{\mu 1}$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{6}}-\frac{1}{\sqrt{3}}\xi e^{{\rm
i}\delta}+\frac{\sqrt{2}A-B}{\sqrt{6}}\xi^{2}-\frac{A-\sqrt{2}B}{\sqrt{6}}\xi^{3}e^{{\rm
i}\delta}+\frac{A^{2}+B^{2}+2\sqrt{2}AB}{2\sqrt{6}}\xi^{4}\hskip 14.22636pt$
$\displaystyle+\frac{A^{2}+B^{2}+\sqrt{2}AB}{2\sqrt{3}}\xi^{5}e^{{\rm
i}\delta}\;,$ $\displaystyle U_{\mu 2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{6}}\xi e^{{\rm
i}\delta}+\frac{A+\sqrt{2}B}{\sqrt{6}}\xi^{2}+\frac{\sqrt{2}A+B}{\sqrt{6}}\xi^{3}e^{{\rm
i}\delta}-\frac{A^{2}+B^{2}-\sqrt{2}AB}{2\sqrt{3}}\xi^{4}$
$\displaystyle+\frac{A^{2}+B^{2}+2\sqrt{2}AB}{2\sqrt{6}}\xi^{5}e^{{\rm
i}\delta}\;,$ $\displaystyle U_{\mu 3}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}-\frac{1+2B}{2\sqrt{2}}\xi^{2}-\frac{\left(1-2B\right)^{2}}{8\sqrt{2}}\xi^{4}\;,$
$\displaystyle U_{\tau 1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}-\frac{1}{\sqrt{3}}\xi e^{{\rm
i}\delta}-\frac{\sqrt{2}A+B}{\sqrt{6}}\xi^{2}-\frac{A+\sqrt{2}B}{\sqrt{6}}\xi^{3}e^{{\rm
i}\delta}-\frac{A^{2}+B^{2}-2\sqrt{2}AB}{2\sqrt{6}}\xi^{4}$
$\displaystyle+\frac{A^{2}+B^{2}-\sqrt{2}AB}{2\sqrt{3}}\xi^{5}e^{{\rm
i}\delta}\;,$ $\displaystyle U_{\tau 2}$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{6}}\xi e^{{\rm
i}\delta}-\frac{A-\sqrt{2}B}{\sqrt{6}}\xi^{2}+\frac{\sqrt{2}A-B}{\sqrt{6}}\xi^{3}e^{{\rm
i}\delta}+\frac{A^{2}+B^{2}+\sqrt{2}AB}{2\sqrt{3}}\xi^{4}$
$\displaystyle+\frac{A^{2}+B^{2}+2\sqrt{2}AB}{2\sqrt{6}}\xi^{5}e^{{\rm
i}\delta}\;,$ $\displaystyle U_{\tau 3}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}-\frac{1-2B}{2\sqrt{2}}\xi^{2}-\frac{\left(1+2B\right)^{2}}{8\sqrt{2}}\xi^{4}\;,$
(8)

up to ${\cal O}(\xi^{6})$ or equivalently ${\cal O}(10^{-5})$. This degree of
precision and accuracy for the elements of $U$ should be good enough to
confront the present and future precision measurements of various neutrino
oscillation channels. Some comments and discussions are in order.

  * •

The two off-diagonal asymmetries of the unitary PMNS matrix $U$, which largely
characterize its geometrical structure about the $U_{e1}$-$U_{\mu 2}$-$U_{\tau
3}$ and $U_{e3}$-$U_{\mu 2}$-$U_{\tau 1}$ axes, are given by

$\displaystyle{\cal A}_{1}$ $\displaystyle\equiv$
$\displaystyle|U_{e2}|^{2}-|U_{\mu 1}|^{2}=|U_{\mu 3}|^{2}-|U_{\tau
2}|^{2}=|U_{\tau 1}|^{2}-|U_{e3}|^{2}$ $\displaystyle\simeq$
$\displaystyle+\frac{1}{6}-\frac{\sqrt{2}}{3}\xi\cos\delta-\frac{1}{3}\left(2+2A^{2}+B^{2}+2\sqrt{2}AB\right)\xi^{2}\;,\hskip
14.22636pt$ $\displaystyle{\cal A}_{2}$ $\displaystyle\equiv$
$\displaystyle|U_{e2}|^{2}-|U_{\mu 3}|^{2}=|U_{\mu 1}|^{2}-|U_{\tau
2}|^{2}=|U_{\tau 3}|^{2}-|U_{e1}|^{2}$ (9) $\displaystyle\simeq$
$\displaystyle-\frac{1}{6}+\frac{1}{6}\left(1-4\sqrt{2}A+6B\right)\xi^{2}\;.$

In comparison, the corresponding off-diagonal asymmetries of the CKM quark
flavor mixing matrix $V$ are respectively of ${\cal O}(\lambda^{6})$ and
${\cal O}(\lambda^{2})$ [5]. So the PMNS matrix $U$ is geometrically not so
symmetrical as the CKM matrix $V$. Given the fact that either ${\cal A}_{1}=0$
or ${\cal A}_{2}=0$ would imply the congruence of three pairs of the PMNS
unitarity triangles in the complex plane [29], we find that the relatively
large off-diagonal asymmetries of $U$ means that its six unitarity triangles
are not very similar to one another in shape.

  * •

The three $\mu$-$\tau$ interchange asymmetries of the PMNS matrix $U$, which
describe small effects of the $\mu$-$\tau$ flavor symmetry breaking, are found
to be

$\displaystyle\Delta_{1}$ $\displaystyle\equiv$ $\displaystyle|U_{\tau
1}|^{2}-|U_{\mu
1}|^{2}\simeq-\frac{2}{3}B\xi^{2}\left[1-2\left(1+\sqrt{2}A\right)\xi^{2}\right]-\frac{2}{3}\xi\left(\sqrt{2}-A\xi^{2}\right)\cos\delta\;,\hskip
14.22636pt$ $\displaystyle\Delta_{2}$ $\displaystyle\equiv$
$\displaystyle|U_{\tau 2}|^{2}-|U_{\mu
2}|^{2}\simeq-\frac{2}{3}B\xi^{2}\left[2-\left(1-2\sqrt{2}A\right)\xi^{2}\right]+\frac{2}{3}\xi\left(\sqrt{2}-A\xi^{2}\right)\cos\delta\;,$
$\displaystyle\Delta_{3}$ $\displaystyle\equiv$ $\displaystyle|U_{\tau
3}|^{2}-|U_{\mu 3}|^{2}\simeq 2B\xi^{2}\left(1-\xi^{2}\right)\;.$ (10)

Of course, $\Delta_{1}+\Delta_{2}+\Delta_{3}=0$ holds, as assured by the
unitarity of $U$. One can simply see that $\Delta_{1}=\Delta_{2}=\Delta_{3}=0$
requires both $B=0$ and $\delta=\pm\pi/2$, the conditions that allow $U$ to
have the exact $\mu$-$\tau$ reflection symmetry [30]. The preliminary T2K
measurement hints at $\delta\sim-\pi/2$ [31], and thus $\Delta_{1}$,
$\Delta_{2}$ and $\Delta_{3}$ are all expected to be of ${\cal O}(10^{-2})$.
It is therefore expected that the $\mu$-$\tau$ permutation or reflection
symmetry may serve as a minimal lepton flavor symmetry which is greatly
helpful for explicit model building [27, 28].

  * •

The well-known Jarlskog invariant of leptonic CP violation [32, 33], which
measures the universal strength of CP-violating effects in neutrino
oscillations, is given as

$\displaystyle{\cal J}_{\nu}\equiv{\rm Im}\left(U_{e2}U_{\mu
3}U^{*}_{e3}U^{*}_{\mu
2}\right)\simeq\frac{1}{6}\xi\left[\sqrt{2}-\left(\sqrt{2}+A\right)\xi^{2}\right]\sin\delta\;.\hskip
8.5359pt$ (11)

So ${\cal J}_{\nu}\simeq 3.5\times 10^{-2}\sin\delta$ holds in the leading-
order approximation. Given $\delta\sim-\pi/2$, for instance, the size of the
leptonic Jarlskog invariant will be about a thousand times larger than that of
its counterpart in the quark sector [3].

Note that one may only keep the leading-order terms of ${\cal A}_{1}$, ${\cal
A}_{2}$, $\Delta_{1}$, $\Delta_{2}$, $\Delta_{3}$ and ${\cal J}_{\nu}$ in most
cases, as they are analytically simple enough and numerically accurate enough.

##  3 The ordering of $|U_{\alpha i}|$

Taking account of the $U_{\rm TBM}$-based expansion of the PMNS matrix $U$ in
powers of $\xi\simeq 0.149$ in Eq. (8), we find that the analytical
approximation

$\displaystyle U$ $\displaystyle\simeq$
$\displaystyle\left(\begin{matrix}\displaystyle\frac{2}{\sqrt{6}}-\frac{1-\sqrt{2}A}{\sqrt{6}}\xi^{2}&\displaystyle\frac{1}{\sqrt{3}}-\frac{1+2\sqrt{2}A}{2\sqrt{3}}\xi^{2}&\xi
e^{-{\rm
i}\delta}\cr\vspace{-0.45cm}\cr\displaystyle-\frac{1}{\sqrt{6}}-\frac{1}{\sqrt{3}}\xi
e^{{\rm
i}\delta}+\frac{\sqrt{2}A-B}{\sqrt{6}}\xi^{2}&\displaystyle\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{6}}\xi
e^{{\rm
i}\delta}+\frac{A+\sqrt{2}B}{\sqrt{6}}\xi^{2}&\displaystyle\frac{1}{\sqrt{2}}-\frac{1+2B}{2\sqrt{2}}\xi^{2}\cr\vspace{-0.45cm}\cr\displaystyle\frac{1}{\sqrt{6}}-\frac{1}{\sqrt{3}}\xi
e^{{\rm i}\delta}-\frac{\sqrt{2}A+B}{\sqrt{6}}\xi^{2}&\hskip
5.69046pt\displaystyle-\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{6}}\xi e^{{\rm
i}\delta}-\frac{A-\sqrt{2}B}{\sqrt{6}}\xi^{2}\hskip
5.69046pt&\displaystyle\frac{1}{\sqrt{2}}-\frac{1-2B}{2\sqrt{2}}\xi^{2}\end{matrix}\right)\hskip
14.22636pt$ (12)

is actually good enough to fit current neutrino oscillation data. Let us
examine to what extent one may identify the ordering of $|U_{\alpha i}|$ (for
$\alpha=e,\mu,\tau$ and $i=1,2,3$).

  * •

For the matrix elements in the first row of $U$, it is easy to identify
$|U_{e1}|>|U_{e2}|>|U_{e3}|$. In fact, $|U_{e1}|$ and $|U_{e3}|$ are the
largest and smallest moduli among the nine elements of $U$.

  * •

For the matrix elements in the third column of $U$, the sign of $B$ is crucial
as it determines whether $|U_{\mu 3}|$ is larger or smaller than its
counterpart $|U_{\tau 3}|$. This point is obviously supported by $|U_{\tau
3}|^{2}-|U_{\mu 3}|^{2}\simeq 2B\xi^{2}$ obtained from Eq. (10). We are
therefore left with $|U_{\tau 3}|\geq|U_{\mu 3}|>|U_{e3}|$ for $B\geq 0$, or
$|U_{\mu 3}|\geq|U_{\tau 3}|>|U_{e3}|$ for $B\leq 0$.

  * •

For the matrix elements in the second and third rows of $U$, the smallness of
$\xi$ assures that $|U_{\mu 1}|<|U_{\mu 2}|<|U_{\mu 3}|$ and $|U_{\tau
1}|<|U_{\tau 2}|<|U_{\tau 3}|$ hold. This observation is independent of the
values of $A$, $B$ and $\delta$ in the Wolfenstein-like expansion of $U$
proposed above.

  * •

To compare between the magnitudes of $U_{\tau i}$ and $U_{\mu i}$ (for
$i=1,2$), we may simplify the expressions of $\Delta_{1}$ and $\Delta_{2}$ in
Eq. (10) as follows:

$\displaystyle|U_{\tau 1}|^{2}-|U_{\mu 1}|^{2}$ $\displaystyle\simeq$
$\displaystyle-\frac{2}{3}\xi\left(\sqrt{2}\cos\delta+B\xi\right)\;,$
$\displaystyle|U_{\tau 2}|^{2}-|U_{\mu 2}|^{2}$ $\displaystyle\simeq$
$\displaystyle+\frac{2}{3}\xi\left(\sqrt{2}\cos\delta-2B\xi\right)\;.\hskip
14.22636pt$ (13)

It becomes clear that $|U_{\tau 1}|\leq|U_{\mu 1}|$ will hold if
$\sqrt{2}\cos\delta+B\xi\geq 0$ is satisfied; and $|U_{\tau 1}|\geq|U_{\mu
1}|$ will hold if $\sqrt{2}\cos\delta+B\xi$ flips its sign. On the other hand,
$|U_{\tau 2}|\geq|U_{\mu 2}|$ will hold if $\cos\delta\geq\sqrt{2}B\xi$ is
satisfied; and $|U_{\tau 2}|\leq|U_{\mu 2}|$ will hold if
$\cos\delta\leq\sqrt{2}B\xi$ is satisfied.

  * •

Whether $|U_{e2}|$ can be larger or smaller than $|U_{\mu 2}|$ or $|U_{\tau
2}|$ is another open question before the quadrant of the CP-violating phase
$\delta$ is surely determined. The reason is simply that

$\displaystyle|U_{e2}|^{2}$ $\displaystyle\simeq$ $\displaystyle|U_{\mu
2}|^{2}-\frac{1}{2}\left(1+2\sqrt{2}A\right)\xi^{2}+\frac{\sqrt{2}}{3}\xi\left(\cos\delta-\sqrt{2}B\xi\right)\;$
(14) $\displaystyle\simeq$ $\displaystyle|U_{\tau
2}|^{2}-\frac{1}{2}\left(1+2\sqrt{2}A\right)\xi^{2}-\frac{\sqrt{2}}{3}\xi\left(\cos\delta-\sqrt{2}B\xi\right)\;\hskip
14.22636pt$

holds to the accuracy of ${\cal O}(\xi^{2})$. In case of $\cos\delta=0$,
however, the sign of $B$ will play a part.

A summary of the above discussions leads us to the following most likely
ordering of the nine matrix elements of $U$ in magnitude:

$\displaystyle|U_{e1}|>\\{|U_{\mu 3}|,|U_{\tau 3}|\\}>\\{|U_{e2}|,|U_{\mu
2}|,|U_{\tau 2}|\\}>\\{|U_{\mu 1}|,|U_{\tau 1}|\\}>|U_{e3}|\;,$ (15)

where the ordering of the bracketed moduli remains unidentifiable from the
present neutrino oscillation data. That is why the next-generation long-
baseline neutrino oscillation experiments aim to pin down the octant of
$\theta_{23}$ (or equivalently, the sign of $B$) and the quadrant of $\delta$.

To be more specific, let us simply take the best-fit value of $\delta$ to
illustrate the ordering of the nine PMNS moduli $|U_{\alpha i}|$ (for
$\alpha=e,\mu,\tau$ and $i=1,2,3$). Given [23, 24]

$\displaystyle{\rm NMO:}\quad\delta\simeq-0.711\pi\;,\quad A\simeq
1.463\;,\quad B\simeq 2.225\;,$ (16)

where the values of $A$ and $B$ are directly extracted from Eq. (4), we obtain
$|U_{\tau 1}|>|U_{\mu 1}|$ from Eq. (13), $|U_{\tau 2}|<|U_{e2}|<|U_{\mu 2}|$
from Eq. (14) and $|U_{\tau 3}|>|U_{\mu 3}|$ from Eq. (10). As a result,

$\displaystyle{\rm NMO:}\quad|U_{e1}|>|U_{\tau 3}|>|U_{\mu 3}|>|U_{\mu
2}|>|U_{e2}|>|U_{\tau 2}|>|U_{\tau 1}|>|U_{\mu 1}|>|U_{e3}|\;.$ (17)

In the IMO case, we input [23, 24]

$\displaystyle{\rm IMO:}\quad\delta\simeq-0.467\pi\;,\quad A\simeq
1.463\;,\quad B\simeq-3.135\;,$ (18)

where the values of $A$ and $B$ are directly extracted from Eq. (5), and then
find $|U_{\tau 1}|>|U_{\mu 1}|$ and $|U_{e2}|<|U_{\mu 2}|<|U_{\tau 2}|$ from
Eq. (14) and $|U_{\tau 3}|<|U_{\mu 3}|$ from Eq. (10). As a consequence,

$\displaystyle{\rm IMO:}\quad|U_{e1}|>|U_{\mu 3}|>|U_{\tau 3}|>|U_{\tau
2}|>|U_{\mu 2}|>|U_{e2}|>|U_{\tau 1}|>|U_{\mu 1}|>|U_{e3}|\;.$ (19)

We see that the nine PMNS matrix elements do not have a clearly layered
hierarchy in magnitude, as compared with the four-layered ordering of the nine
CKM moduli shown in Eq. (1). The reason behind this difference should be
closely related to the underlying mechanism responsible for the origin of tiny
neutrino masses, although it remains vague and unclear at present.

##  4 The unitarity triangle

The “appearance” neutrino oscillation $\nu_{\mu}\to\nu_{e}$ and its CP-
conjugated process $\overline{\nu}_{\mu}\to\overline{\nu}_{e}$ are the only
realistic channels to measure leptonic CP violation in a long-baseline
oscillation experiment like T2K [31]. It is the so-called unitarity triangle
$\triangle_{\tau}$ [34] defined by the orthogonality relation
$U_{e1}U^{*}_{\mu 1}+U_{e2}U^{*}_{\mu 2}+U_{e3}U^{*}_{\mu 3}=0$ in the complex
plane that is directly related to $\nu_{\mu}\to\nu_{e}$ and
$\overline{\nu}_{\mu}\to\overline{\nu}_{e}$ oscillations. In view of the PDG-
advocated phase convention of $U$ taken in Eq. (6) or (8), we find that it is
more convenient to use the side $U_{e3}U^{*}_{\mu 3}$ to rescale the three
sides of $\triangle_{\tau}$ [35]. In this case, we simply arrive at

$\displaystyle\triangle^{\prime}_{\tau}:\quad\frac{U_{e1}U^{*}_{\mu
1}}{U_{e3}U^{*}_{\mu 3}}+\frac{U_{e2}U^{*}_{\mu 2}}{U_{e3}U^{*}_{\mu
3}}+1=0\;,$ (20)

where the two sloping sides are given by

$\displaystyle\frac{U_{e1}U^{*}_{\mu 1}}{U_{e3}U^{*}_{\mu 3}}$
$\displaystyle\simeq$ $\displaystyle-\frac{\sqrt{2}}{3\xi}e^{{\rm
i}\delta}-\frac{2}{3}\;,$ $\displaystyle\frac{U_{e2}U^{*}_{\mu
2}}{U_{e3}U^{*}_{\mu 3}}$ $\displaystyle\simeq$
$\displaystyle+\frac{\sqrt{2}}{3\xi}e^{{\rm i}\delta}-\frac{1}{3}\;,\hskip
14.22636pt$ (21)

to a good degree of accuracy. Just taking $\delta\simeq-\pi/2$ and $\xi\simeq
0.149$ for example, we obtain the numerical results $|U_{e1}U^{*}_{\mu
1}|/|U_{e3}U^{*}_{\mu 3}|\simeq 3.23$ and $|U_{e2}U^{*}_{\mu
2}|/|U_{e3}U^{*}_{\mu 3}|\simeq 3.18$. Namely, two of the three sides of
$\triangle^{\prime}_{\tau}$ are about three times longer than the shortest one
in magnitude, as illustrated by the solid black triangle in Fig. 1. Note that
the height of $\triangle^{\prime}_{\tau}$, denoted as ${\cal
J}^{\prime}_{\nu}$, is correlated with the Jarlskog invariant ${\cal J}_{\nu}$
as follows:

$\displaystyle{\cal J}^{\prime}_{\nu}=\frac{{\cal J}_{\nu}}{|U_{e3}U^{*}_{\mu
3}|^{2}}\simeq\frac{1}{3\xi}\left[\sqrt{2}-\left(A-2\sqrt{2}B\right)\xi^{2}\right]\sin\delta\;,$
(22)

where the expression of ${\cal J}_{\nu}$ obtained in Eq. (11) has been used.
We are then left with the result ${\cal
J}^{\prime}_{\nu}\simeq\sqrt{2}\sin\delta/\left(3\xi\right)$ in the leading-
order approximation, as clearly indicated by the imaginary parts of the two
sloping sides of $\triangle^{\prime}_{\tau}$ in Eq. (21).

Figure 1: An illustration of the geometrical shapes of the rescaled unitarity
triangle $\triangle^{\prime}_{\tau}$ (solid black) and its effective matter-
corrected counterpart $\widetilde{\triangle}^{\prime}_{\tau}$ (dashed red for
the NMO case and dashed blue for the IMO case) in the complex plane, where
$\delta\simeq-\pi/2$ has typically been input.

It is well known that the terrestrial matter effects on $\nu_{\mu}\to\nu_{e}$
and $\overline{\nu}_{\mu}\to\overline{\nu}_{e}$ oscillations are not very
significant in the T2K and Hyper-Kamiokande experiments with the baseline
length $L\simeq 295~{}{\rm km}$ and the typical beam energy $E\simeq
0.6~{}{\rm GeV}$ [3], but such effects can modify the shape of the above
unitarity triangle and thus modify the Jarlskog invariant of CP violation to
some extent. To see this point more clearly, let us consider the effective
rescaled unitarity triangle

$\displaystyle\widetilde{\triangle}^{\prime}_{\tau}:\quad\frac{\widetilde{U}_{e1}\widetilde{U}^{*}_{\mu
1}}{\widetilde{U}_{e3}\widetilde{U}^{*}_{\mu
3}}+\frac{\widetilde{U}_{e2}\widetilde{U}^{*}_{\mu
2}}{\widetilde{U}_{e3}\widetilde{U}^{*}_{\mu 3}}+1=0\;,$ (23)

where $\widetilde{U}_{ei}$ and $\widetilde{U}_{\mu i}$ (for $i=1,2,3$) denote
the effective PMNS matrix elements in matter. Following the analytical
approximations made in Ref. [36], we obtain
$\widetilde{U}_{e3}\widetilde{U}^{*}_{\mu 3}\simeq U_{e3}U^{*}_{\mu 3}$ and

$\displaystyle\frac{\widetilde{U}_{e1}\widetilde{U}^{*}_{\mu
1}}{\widetilde{U}_{e3}\widetilde{U}^{*}_{\mu 3}}$ $\displaystyle\simeq$
$\displaystyle\frac{\alpha}{\epsilon}\cdot\frac{U_{e1}U^{*}_{\mu
1}}{U_{e3}U^{*}_{\mu 3}}-\frac{\epsilon-\alpha-\beta}{2\epsilon}\;,$
$\displaystyle\frac{\widetilde{U}_{e2}\widetilde{U}^{*}_{\mu
2}}{\widetilde{U}_{e3}\widetilde{U}^{*}_{\mu 3}}$ $\displaystyle\simeq$
$\displaystyle\frac{\alpha}{\epsilon}\cdot\frac{U_{e2}U^{*}_{\mu
2}}{U_{e3}U^{*}_{\mu 3}}-\frac{\epsilon-\alpha+\beta}{2\epsilon}\;,\hskip
14.22636pt$ (24)

to a good degree of accuracy, where $\alpha\equiv\Delta m^{2}_{21}/\Delta
m^{2}_{31}$, $\beta\equiv\mathbb{A}/\Delta m^{2}_{31}$, $\mathbb{A}\equiv
2\sqrt{2}\hskip 1.42271ptG_{\rm F}N_{e}E$, and

$\displaystyle\epsilon\equiv\sqrt{\alpha^{2}-2\left(|U_{e1}|^{2}-|U_{e2}|^{2}\right)\alpha\beta+\left(1-|U_{e3}|^{2}\right)^{2}\beta^{2}}\;$
(25)

with $\Delta m^{2}_{ij}$ (for $i,j=1,2,3$) being the neutrino mass-squared
differences, $G_{\rm F}$ being the Fermi constant, $N_{e}$ being the
terrestrial background density of electrons, and $E$ being the neutrino beam
energy [37, 38]. Substituting Eq. (21) into Eq. (24), we arrive at

$\displaystyle\frac{\widetilde{U}_{e1}\widetilde{U}^{*}_{\mu
1}}{\widetilde{U}_{e3}\widetilde{U}^{*}_{\mu 3}}$ $\displaystyle\simeq$
$\displaystyle-\left(\frac{\sqrt{2}}{3\xi}e^{{\rm
i}\delta}+\frac{1}{6}\right)\frac{\alpha}{\epsilon}-\frac{1}{2}\left(1-\frac{\beta}{\epsilon}\right)\;,$
$\displaystyle\frac{\widetilde{U}_{e2}\widetilde{U}^{*}_{\mu
2}}{\widetilde{U}_{e3}\widetilde{U}^{*}_{\mu 3}}$ $\displaystyle\simeq$
$\displaystyle+\left(\frac{\sqrt{2}}{3\xi}e^{{\rm
i}\delta}+\frac{1}{6}\right)\frac{\alpha}{\epsilon}-\frac{1}{2}\left(1+\frac{\beta}{\epsilon}\right)\;.\hskip
14.22636pt$ (26)

This result shows that the terrestrial matter effects can obviously modify the
rescaled unitarity triangle $\triangle^{\prime}_{\tau}$ in vacuum. Taking
account of $\widetilde{\cal J}_{\nu}/{\cal J}_{\nu}\simeq|\alpha|/\epsilon$
obtained in Ref. [36], where $\widetilde{\cal J}_{\nu}$ denotes the effective
Jarlskog invariant in matter, we may similarly achieve the height of
$\widetilde{\triangle}^{\prime}_{\tau}$:

$\displaystyle\widetilde{\cal J}^{\prime}_{\nu}=\frac{\widetilde{\cal
J}_{\nu}}{\big{|}\widetilde{U}_{e3}\widetilde{U}^{*}_{\mu
3}\big{|}^{2}}\simeq\frac{\alpha}{3\epsilon\xi}\left[\sqrt{2}-\left(A-2\sqrt{2}B\right)\xi^{2}\right]\sin\delta\;,$
(27)

whose leading term is certainly consistent with the imaginary parts of the two
sloping sides of $\widetilde{\triangle}^{\prime}_{\tau}$ that can directly be
seen from Eq. (26).

To illustrate, one may typically take $\mathbb{A}\simeq 2.28\times
10^{-4}~{}{\rm eV}^{2}\left(E/{\rm GeV}\right)$ for a neutrino trajectory
through the Earth’s crust [39], which is suitable for the realistic ongoing
and upcoming long-baseline neutrino oscillation experiments. Of course, the
matter parameter $\mathbb{A}$ should flip its sign for an antineutrino beam,
and both $\alpha$ and $\beta$ are sensitive to the neutrino mass ordering
(i.e., the sign of $\Delta m^{2}_{31}$). Given the best-fit values $\Delta
m^{2}_{21}\simeq 7.41\times 10^{-5}~{}{\rm eV}^{2}$, $\Delta m^{2}_{31}\simeq
2.51\times 10^{-3}~{}{\rm eV}^{2}$ (NMO) or $\Delta
m^{2}_{31}\simeq-2.41\times 10^{-3}~{}{\rm eV}^{2}$ (IMO) [23, 24], we have
$\alpha\simeq 2.95\times 10^{-2}$ (NMO) or $\alpha\simeq-3.07\times 10^{-2}$
(IMO), together with $\beta\simeq 5.45\times 10^{-2}$ (NMO) or
$\beta\simeq-5.68\times 10^{-2}$ (IMO) for the T2K and Hyper-Kamiokande
neutrino oscillation experiments with $E\simeq 0.6~{}{\rm GeV}$. In this case
we obtain $\epsilon\simeq 4.97\times 10^{-2}$ (NMO) or $\epsilon\simeq
5.18\times 10^{-2}$ (IMO) after taking into account the best-fit values of
$\theta_{12}$ and $\theta_{13}$ given in Eqs. (2) and (3). Fig. 1 illustrates
how the geometrical shape of $\triangle^{\prime}_{\tau}$ changes as a
consequence of the terrestrial matter effects on neutrino oscillations in the
NMO and IMO cases. It becomes clear that the area of
$\widetilde{\triangle}^{\prime}_{\tau}$ is remarkably smaller than that of
$\triangle^{\prime}_{\tau}$, and their ratio is simply governed by
$\widetilde{\cal J}_{\nu}/{\cal J}_{\nu}\simeq|\alpha|/\epsilon$ as discussed
above.

It is finally worth pointing out that $\theta_{12}$, $\theta_{23}$ and
$\delta$ are essentially insensitive to terrestrial matter effects in the
$E\lesssim 1~{}{\rm GeV}$ region [40, 41, 42], and $\sin
2\widetilde{\theta}_{12}/\sin 2\theta_{12}\simeq\widetilde{\cal J}_{\nu}/{\cal
J}_{\nu}\simeq|\alpha|/\epsilon$ holds as a good approximation [36]. These
observations imply that $\xi$, $B$ and $\delta$ in our Wolfenstein-like
expansion of the PMNS matrix $U$ are also expected to be insensitive to
terrestrial matter effects in an accelerator-based neutrino oscillation
experiment with $E\lesssim 1~{}{\rm GeV}$, and only $A$ is an exception. To be
specific, we take
$\widetilde{\theta}_{12}\simeq\theta^{(0)}_{12}-\widetilde{A}\xi^{2}$ and then
obtain

$\displaystyle\widetilde{A}\xi^{2}\simeq
A\xi^{2}\frac{|\alpha|}{\epsilon}+\sqrt{2}\left(1-\frac{|\alpha|}{\epsilon}\right)\;.$
(28)

The second term on the right-hand side of Eq. (28) is in general unsuppressed
as it arises from $\theta^{(0)}_{12}\simeq 35.26^{\circ}$. Although
$\widetilde{A}$ itself may be far above ${\cal O}(1)$, it does not really
point to a significant matter effect simply because it is not directly related
to any observable of neutrino oscillations.

##  5 Summary

Motivated by the fact that the fine structure of quark flavor mixing can well
be understood in the Wolfenstein expansion of the CKM matrix $V$, we have
proposed a similar expansion of the PMNS matrix $U$ in the basis of the tri-
bimaximal mixing pattern $U_{\rm TMB}$ towards understanding the fine
structure of lepton flavor mixing. The corresponding expansion parameters are
$\lambda$ for $V$ and $\xi$ for $U$, and thus we can easily arrive at

$\displaystyle\lim_{\lambda\to 0}V=I\;,\quad\lim_{\xi\to 0}U=U_{\rm TBM}\;,$
(29)

in the limits of vanishing or vanishingly small $\lambda$ and $\xi$. While $I$
is unique in the quark sector, $U_{\rm TBM}$ is just our choice in the lepton
sector. However, we have argued that it is rather reasonable to choose $U_{\rm
TBM}$ as an expansion basis of $U$ since this constant pattern is particularly
favored from the point of view of model building with the help of an
underlying discrete flavor symmetry.

We have discussed the ordering of the nine PMNS moduli, the leptonic unitarity
triangle $\triangle_{\tau}$ and its effective counterpart in terrestrial
matter by using the Wolfenstein-like expansion of $U$ proposed in this paper.
Our approach proves to be very useful and instructive. More applications of
this new parametrization of $U$ in neutrino physics will be explored
elsewhere.

## Acknowledgements

This work was supported in part by the National Natural Science Foundation of
China under grant No. 12075254 and grant No. 11835013.

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