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% for BibTeX - sorted numerical labels by order of
% first citation.
% A useful Journal macro
\newcommand{\simlt} {\raisebox{-.6ex}{$\stackrel{\textstyle <}{\sim}$}}
\newcommand{\simgt} {\raisebox{-.6ex}{$\stackrel{\textstyle >}{\sim}$}}
\newcommand{\jcp} {\ensuremath{J}}
\newcommand{\nua} {\nu_1}
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\newcommand{\nuc} {\nu_3}
\newcommand{\nue} {\nu_e}
\newcommand{\num} {\nu_{\mu}}
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\newcommand{\Uajst} {U^*_{\alpha j}}
\newcommand{\Ubist} {U^*_{\beta i}}
\newcommand{\Ubj} {U_{\beta j}}
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\begin{document}
\vspace*{4cm}
\title{Flavour Permutation Symmetry and Fermion Mixing \footnote{Talk given at the 43rd
Rencontres de Moriond, La Thuile, Italy, March 2008.}
}
\author{P.F.~Harrison \footnote{Speaker.} and D.R.J. Roythorne}
\addressTwo{Department of Physics, University of Warwick, Coventry,\\
CV4 7AL, England.}
\author{W.G.~Scott}
\address{Rutherford Appleton Laboratory, Chilton, Didcot,\\
OXON, OX11 0QX, England.}
\maketitle\abstracts{
We discuss our recently proposed $\Sthl\times\Sthn$
flavour-permutation-symmetric mixing observables, giving expressions for
them in terms of (moduli-squared) of the mixing matrix elements. We outline their
successful use in providing flavour-symmetric descriptions of (non-flavour-symmetric)
lepton mixing schemes. We develop our partially unified flavour-symmetric
description of both quark and lepton mixings, providing testable predictions
for $CP$-violating phases in both $B$ decays and neutrino oscillations.
}
\section{Introduction}
Flavour observables, namely quark and lepton masses and mixings are neither predicted
nor predictable in the Standard Model. Neither are they correlated with each
other in any way. However, their experimentally determined values display striking
structure: viewed on a logarithmic scale, the fermion masses of any given non-zero charge
are approximately equi-spaced; the spectrum of quark mixing angles is described by
the Wolfenstein form,~\cite{wolfenstein} suggestive of correlations between mixing
angles and quark masses, and the lepton mixing matrix is well-approximated by the
tri-bimaximal form.~\cite{TBM:1} These striking patterns are the modern-day equivalents
of the regularities observed around a century ago in hydrogen emission spectra, which
were mathematically well-described by the Rydberg formula, but nevertheless had no
theoretical basis before the advent of quantum mechanics. While consistent with the
Standard Model, they lie completely outside its predictive scope, and are surely
evidence for some new physics beyond it.
In this talk, we report on our recent attempts~\cite{HRS07} to find a new description of
fermion mixing which builds on the Standard Model and allows constraints on the mixing
observables which make no reference to individual flavours,
while describing mixing structures which are manifestly not flavour-symmetric,
as observed experimentally. This approach does not in itself constitute
a complete theory of flavour mixing beyond the Standard Model, but we hope that
it might help stimulate new developments in that direction.
\section{The Jarlskogian and Plaquette Invariance}
Jarlskog's celebrated $CP$-violating invariant,~\cite{JCP:1} $\jcp$,
is important in the phenomenology of both quarks and leptons. As well as
parameterising the violation of a specific symmetry, it has two other properties
which set it apart from most other mixing observables.
First, its value (up to its sign) is independent of any flavour labels.\footnote{We
focus first on the leptons, although many of our considerations may be
applied equally well to the quarks. In the leptonic case, neutrino mass eigenstate
labels $i=1...3$ take the analogous role to the charge $-\frac{1}{3}$ quark flavour
labels in the quark case. In this sense, we will often use the term ``flavour''
to include neutrino mass eigenstate labels, as well as charged lepton flavour labels.}
Mixing observables are in general dependent on flavour labels, eg.~the
moduli-squared of mixing matrix elements, $|\Uai|^2$, certainly depend on
%their flavour labels,
$\alpha$ and $i$. Indeed, $\jcp$ itself is often
calculated in terms of a subset of four mixing matrix elements,
namely those forming a given plaquette~\cite{BJ} (whose elements are defined by
deleting the $\gamma$-row and the $k$-column \footnote{We use a cyclic
labelling convention such that $\beta=\alpha+1$, $\gamma=\beta+1$, $j=i+1$, $k=j+1$,
all indices evaluated mod 3.} to leave a rectangle of four elements):
%
\begin{eqnarray}
\jcp={\rm Im}(\Pi_{\gamma k})={\rm Im}(U_{\alpha i}U^*_{\alpha j}U^*_{\beta i}U_{\beta j}).
\label{jarlskogian}
\end{eqnarray}
%
However, it is well-known~\cite{JCP:1} that the value of $\jcp$
does not depend on the choice of plaquette (ie.~on its flavour labels, $\gamma$ and
$k$ above) - it is
``plaquette-invariant''. This special feature originates in the fact that \jcp\ is
{\it flavour-symmetric}, carrying information sampled evenly across the whole
mixing matrix. We recently pointed-out~\cite{HRS07} that in fact, {\it any} observable function
of the mixing matrix elements, flavour-symmetrised (eg.~by summing over both
rows and columns), and written in terms of the elements of a single plaquette
(eg.~using unitarity constraints), will be similarly plaquette-invariant. Both its expression
in terms of mixing matrix elements, as well as its value, will be independent of
the particular choice of plaquette.
The second exceptional property of \jcp\ is that it may be particularly
simply related to the fermion mass (or Yukawa) matrices:
%
\begin{equation}
\jcp=-i\,\frac{{\rm Det}[L,N]}{2\LD \ND}
\label{jarlskogComm}
\end{equation}
%
where for leptons, $L$ and $N$ are the charged-lepton and neutrino mass
matrices respectively \footnote{Throughout this paper, $L$ and $N$ are taken to be
Hermitian, either by appropriate choice of the flavour basis for the right-handed fields,
or as the Hermitian squares, $MM^{\dag}$, of the relevant mass or Yukawa coupling
matrices. The symbols $m_{\alpha}$, $m_i$ generically refer to their eigenvalues in
either case.} (in an arbitrary weak basis) and
$\LD=(m_e-m_{\mu})(m_{\mu}-m_{\tau})(m_{\tau}-m_e)$ (with an analogous
definition for $\ND$ in terms of neutrino masses and likewise for the
quarks). This is useful, as, despite $\jcp$ being defined purely in terms of
mixing observables via Eq.~(\ref{jarlskogian}), by contrast,
Eq.~(\ref{jarlskogComm}) relates it to the mass matrices,
which appear in the Standard Model Lagrangian.
We will discuss our recently proposed \cite{HRS07} plaquette-invariant
(ie.~flavour-symmetric mixing) observables, which, in common with $\jcp$,
are independent of flavour labels and can be simply related to the
mass matrices. Again like $\jcp$, we find that our observables
parameterise the violation of certain phenomenological symmetries which
have already been considered
significant~\cite{SYMMSGENS}~\cite{MUTAUSYMM}~\cite{DEMOCRACY}~\cite{BHS05} in leptonic
mixing. In the next section, we define more precisely what we mean by flavour symmetry.
\section{The {\boldmath $\Sthl\times S3_{\uparrow}\!\!$}~Flavour Permutation Group}
The $\Sthl$ group is the group of the six possible permutations of the charged lepton
flavours and/or of the charge $-\frac{1}{3}$ quark flavours, while the $\Sthn$ group
is the group of the six possible permutations of the neutrino flavours (ie.~mass
eigenstates) or of the charge $\frac{2}{3}$ quark flavours (the arrow subscript
corresponds to the direction of the z-component of weak isospin of the corresponding
left-handed fields). We consider all possible such permutations,
which together constitute the direct product $\Sthl\times\Sthn$ flavour permutation group
(FPG)~\cite{HRS07} with 36 elements.
We next consider the $P$ matrix (for ``probability'')~\cite{HSW06} of moduli-squared
of the mixing matrix elements, eg.~for leptons:
%
\begin{eqnarray}
P
=\mat{|U_{e1}|^2}{|U_{e2}|^2}{|U_{e3}|^2}
{{|U_{\mu 1}|^2}}{{|U_{\mu 2}|^2}}{{|U_{\mu 3}|^2}}
{{|U_{\tau 1}|^2}}{{|U_{\tau 2}|^2}}{{|U_{\tau 3}|^2}}.
\label{Pmatrix}
\end{eqnarray}
%
It should be familiar: for quarks, semileptonic weak decay rates of hadrons are
proportional to its elements, while for leptons, the magnitudes of neutrino oscillation
probabilities may be written in terms of its elements.~\cite{HSW06} Moreover, the $P$
matrix may easily be related to the fermion mass matrices, as we will see in Section 5
below. The $P$ matrix manifestly transforms as the natural representation of
$\Sthl\times\Sthn$, the transformations being effected by pre- and/or post-multiplying
by $3\times 3$ real permutation matrices.\footnote{Less obviously, any given plaquette
of $P$ transforms as a 2-dimensional (real) irreducible representation of
$\Sthl\times\Sthn$.}
Jarlskog's invariant $J$ is a pseudoscalar under the FPG: under even permutations, it
is invariant, while under odd permutations (eg.~single swaps of rows or columns of the
mixing matrix, or odd numbers of them), it simply changes sign. This is our prototype
Flavour Symmetric Mixing Observable (FSMO). As we commented in the previous section,
it is easy to find other similar such quantities, which, surprisingly had not
appeared in the literature until recently.~\cite{HRS07} There are two types of singlets
under the S3 group:
even (\bo) which remain invariant under all permutations, and odd (\bob) which flip sign
under odd permutations. So, under the FPG, there are four types of singlet:
{$\bf 1$}$\times${$\bf 1$}, {$\bf\overline{1}$}$\times${$\bf\overline{1}$} (like $\jcp$),
{$\bf 1$}$\times${$\bf\overline{1}$} and {$\bf\overline{1}$}$\times${$\bf 1$}. By Flavour
Symmetric Observables (FSOs), we mean observables with any of these transformation
properties under the FPG. They may be functions of mixing matrix elements alone (FSMOs),
or functions of mass eigenvalues alone, or functions of both.
Starting with elements of $P$ and combining and (anti-)symmetrising them over flavour
labels in various ways,
we find that, apart from their (trivial) overall normalisation, and possibly scalar
offsets, there are a finite number of independent FSMOs at any given order in $P$.
Enumerating them, we found that there are no non-trivial ones linear
in $P$, while at 2nd order in $P$, there is only one each of
{$\bf 1$}$\times${$\bf 1$}, {$\bf\overline{1}$}$\times${$\bf\overline{1}$}.
At third order, there is exactly one each of the four types of singlet, while at
higher orders in $P$, there are multiple instances of each. Recognising that we need
only four independent variables to specify the mixing, it is clearly enough to stop
at third order, up to which, the singlets are essentially uniquely defined by their
order in $P$ and their transformation property under the FPG.
\section{Flavour-Symmetric Mixing Observables}
We introduce four FSMOs,~\cite{HRS07} uniquely defined as outlined above:
%
\begin{equation}
\renewcommand{\arraystretch}{1.50} % enlarge line spacing
\begin{array}{lcc}
& \underline{\bo\times\bo} & \underline{\bob\times\bob} \\
\underline{{\rm 2nd~Order~in}~P:} & \GG=\frac{1}{2}\,[\,\sum_{\alpha i}(P_{\alpha i})^2-1\,] & \FF=\Det\PP \\
\underline{{\rm 3rd~Order~in}~P:} & \CC=\frac{3}{2}\,\sum_{\alpha i}[\,(P_{\alpha i})^3-(P_{\alpha i})^2\,]+1
& \quad\AC=\frac{1}{18}\sum_{\gamma k}(L_{\gamma k})^3
\end{array}\label{invariants}
\end{equation}
%
where $L_{\gamma k}=(P_{\alpha i}+P_{\beta j}-P_{\beta i}-P_{\alpha j})$. Alternative,
but equivalent definitions in terms of the elements of a single plaquette are given
elsewhere.~\cite{HRS07} Note that $\FF$ is only quadratic in $P$, because of the constraints
of unitarity. We comment briefly on the normalisations and offsets we have given them.
$\FF$ and $\AC$, being anti-symmetric,
need no offset, as they are already centred on zero, which they reach for threefold
maximal mixing~\cite{TRIMAX} (uniquely defined by all 9 elements of the mixing
matrix having magnitude $\frac{1}{\sqrt{3}}$).
$\GG$ and $\CC$ are defined with offsets such that they likewise vanish
for threefold maximal mixing. All four variables are normalised so that their maximum
value is unity, which they attain for no mixing. In Ref.~\cite{HRS07},
we also give the $\bob\times\bo$ and the $\bo\times\bob\,$ FSMOs at 3rd order
(called \BB\ and \DD\ respectively), but they will not concern us here.
The four FSMOs introduced in Eq.~\ref{invariants} are the simplest ones \footnote{They also
treat the two weak-isospin sectors symmetrically, though this is not an essential feature.}
in terms of $P$ and are sufficient to completely specify the mixing, up to a number of
discrete ambiguities associated with the built-in flavour symmetry. $\jcp$ is of course
not independent, and is given by $18\JJ^2=1/6 - \GG + (4/3)\,\CC - (1/2)\,\FF^2$.
In Table \ref{table:values}, we summarise their properties and values (estimated at
90\% CL from compilations of current experimental results) for both quarks~\cite{CKMUTFIT}
and leptons.~\cite{FITS:1}
%
\begin{table*}[h]
%\renewcommand{\tabcolsep}{1pc} % enlarge column spacing
\renewcommand{\arraystretch}{1.25} % enlarge line spacing
\caption{Properties and values of flavour-symmetric mixing observables for quarks and
leptons. The experimentally allowed ranges are estimated (90\% CL) from compilations
of current experimental results, neglecting any correlations between the input
quantities.\label{table:values}}
%
\vspace{0.4cm}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Observable & Order & Symmetry: &Theoretical& Experimental Range & Experimental Range \\
Name & in \PP\ & $\Sthl\times\Sthn$ & Range & for Leptons & for Quarks \\
\hline
$\FF$ & 2 & {$\bf\overline{1}$}$\times${$\bf\overline{1}$} & $(-1, 1)$ & $(-0.14, 0.12)$ & $(0.893, 0.896)$ \\
$\GG$ & 2 & {$\bf 1$}$\times${$\bf 1$} & $(0, 1)$ & $(0.15, 0.23)$ & $(0.898, 0.901)$ \\
$\AC$ & 3 & {$\bf\overline{1}$}$\times${$\bf\overline{1}$}& $(-1, 1)$ & $(- 0.065, 0.052)$ & $(0.848, 0.852)$ \\
$\CC$ & 3 & {$\bf 1$}$\times${$\bf 1$}& $(-\frac{1}{27}, 1)$ & $(-0.005, 0.057)$ & $(0.848, 0.852)$ \\
\hline
\end{tabular}
%
\end{center}
\end{table*}
\section{Flavour-Symmetric Mixing Observables in Terms of Mass Matrices}
Equation (\ref{jarlskogComm}) gives $\jcp$, our prototype FSMO, in terms of the
fermion mass matrices, which in turn are proportional to the matrices of Yukawa
couplings which appear in the Standard Model Lagrangian. In this section, we show
how to write the FSMOs of Section 4 above also in terms of the mass matrices.
It is useful to define a reduced $P$ matrix:
%
\beq
\PT=P-D
\label{PT}
\eeq
%
where $D$ is the $3\times 3$ democratic matrix with all 9 elements equal to $\frac{1}{3}$.
We also define the reduced (ie. traceless) powers of the fermion mass matrices:
$\Lmt:=L^m-\frac{1}{3}\Tr(L^m)$ (similarly for $\Nmt$), in terms of which, we can define
the $2\times 2$ matrix of weak basis-invariants:
%
\beq
\TT_{mn}:={\rm Tr}(\widetilde{L^m}\widetilde{N^n}),\quad m,n=1,2.
\eeq
%
For known lepton masses, \TT\ is completely equivalent to \PP. In fact, it is
straightforward to show that $\PT$ is a mass-moment transform of $\TT$:
%
\beq
\PT=\widetilde{M_{\ell}}^T\cdot \TT\cdot \widetilde{M_{\nu}}
\label{PTexpan}
\eeq
%
where
%
\bea
\widetilde{M_{\ell}}=\frac{1}{\LD}
\matTwoThr{m_{\mu}^2-m_{\tau}^2}{m_{\tau}^2-m_e^2}{m_{e}^2-m_{\mu}^2}
{m_{\mu}-m_{\tau}}{m_{\tau}-m_e}{m_{e}-m_{\mu}},
\label{PtildeFromTtilde}
\eea
%
with an analogous definition for $\widetilde{M_{\nu}}$ (the inverse transform is
easily obtained).
Starting from Eq.~(\ref{invariants}) and substituting for $P$ from Eqs.~(\ref{PT})
and (\ref{PTexpan}), we find that:
%
\beq
\FF\equiv{\rm Det}\,P=3\frac{{\rm Det}\,\TT}{\LD\ND};
\qquad\quad \left[{\rm cf.~Eq.~(\ref{jarlskogComm}):}~\jcp=-i\,\frac{{\rm Det}[L,N]}{2\LD \ND}\right]
\eeq
%
\beq
\GG=\frac{\TT_{mn}\,\TT_{pq}\,\LL^{mp}\,\NN^{nq}}{(\LD\ND)^2};
\qquad\CC,\AC=\frac{\TT_{mn}\,\TT_{pq}\,\TT_{rs}\,\LL_{\CC,\AC}^{(mpr)}\,\NN_{\CC,\AC}^{(nqs)}}{(\LD\ND)^{n_{\CC,\AC}}},
\eeq
%
where the $\LL$ ($\NN$) are simple functions of traces of $\Lmt$ ($\Nmt$), given
in Ref.~\cite{HRS07}, and $n_{\CC}~(n_{\AC})=2(3)$.
\section{Application 1: Flavour-Symmetric Descriptions of Leptonic Mixing}
The tribimaximal mixing~\cite{TBM:1} ansatz for the MNS lepton mixing matrix:
%
\bea
U\simeq \left( \matrix{
-2/\sqrt{6} & 1/\sqrt{3} & \,\,\,0 \cr
\,\,\,\, 1/\sqrt{6} & 1/\sqrt{3} & \,\,\,\,\, 1/\sqrt{2} \cr
\,\,\,\, 1/\sqrt{6} & 1/\sqrt{3} & -1/\sqrt{2} \cr
} \right)
\eea
%
is compatible with all confirmed leptonic mixing measurements
from neutrino oscillation experiments, and may be considered a useful
leading-order approximation to the data. It is defined by three phenomenological
symmetries:~\cite{SYMMSGENS} $CP$ symmetry, \mt-reflection symmetry and Democracy,
which may each be expressed (flavour-symmetrically) in terms of our FSMOs. For example,
as is well known, the zero in the $U_{e3}$ position, if exact, ensures that no
$CP$ violation can arise from the mixing matrix. $CP$ symmetry is thus represented
simply by $\jcp=0$ (which is a necessary, but not sufficient condition for a single
zero in the mixing matrix, see Section 7 below). \mt-reflection symmetry~\cite{MUTAUSYMM}
means that corresponding elements in the $\mu$ and $\tau$ rows have equal moduli:
$|U_{\mu i}|=|U_{\tau i}|,~\forall i$, and this implies the two flavour-symmetric
constraints:
%
\beq
\FF=\AC=0
\label{mtSymm}
\eeq
%
(flavour symmetry means that although these two constraints imply just such a set of
equalities, they do not define {\it which} pair of rows or columns are constrained).
Democracy~\cite{DEMOCRACY}~\cite{BHS05} ensures that one row or column is trimaximally
mixed, ie.~has the form $\frac{1}{\sqrt{3}}(1,1,1)^{(T)}$, as is the case for the
$\nu_2$ column in tribimaximal mixing. Democracy is ensured flavour-symmetrically
by the two constraints:
%
\beq
\FF=\CC=0.
\label{democracy}
\eeq
%
Taking all three symmetries, tribimaximal mixing (or one of its trivial
permutations) is ensured by the complete set of constraints $\FF=\CC=\AC=\jcp=0$, which
may be written as the single flavour-symmetric condition:
%
\beq
\FF^2+\CC^2+\AC^2+\jcp^2=0.
\label{tbmConstraint}
\eeq
%
Tribimaximal mixing is manifestly not flavour symmetric. The flavour-symmetry of our
constraint, Eq.~(\ref{tbmConstraint}), is spontaneously broken by its tribimaximal
solutions. The symmetry is manifested by the existence of a complete set of solutions
of the generalised tribimaximal form, each related to the other by a member of the
flavour permutation group.
Of course, generalisations of the tribimaximal form~\cite{SYMMSGENS} possessing
subsets of its three symmetries may be similarly defined, and their corresponding
flavour-symmetric constraints may be obtained by analogy to the above. These, and
those of other special mixing forms~\cite{AF}~\cite{BIMAX} are tabulated in
Ref.~\cite{HRS07}.
\section{Application 2: A Partially Unified, Flavour-Symmetric Description of Quark
and Lepton Mixings}
A unified understanding of quark and lepton mixings is highly desirable. This is
difficult because their mixing matrices have starkly different forms:
the quark mixing matrix is characterised by small mixing angles,~\cite{CKMUTFIT}
while the lepton mixing matrix is characterised mostly by large ones.~\cite{FITS:1}
Many authors have ascribed this difference to the effect of the heavy majorana mass
matrix in the leptonic case, via the see-saw mechanism.~\cite{seesaw} Notwithstanding
the attractiveness of this explanation, it is clearly still worthwhile to ask if there
are any features of the respective mixings which the quark and lepton sectors have in
common.
Neutrino oscillation data~\cite{FITS:1} require that $|U_{e3}|^2\,\simlt\,0.05$,
significantly less than the other MNS matrix elements-squared. At least one {\em small}
mixing element is hence a common feature of both quark and lepton mixing matrices.
We are thus led first to ask the question:~``what is the flavour-symmetric condition
for at least one zero element in the mixing matrix?'' We should perhaps anticipate
two constraints, as the condition implies that both real and imaginary parts
vanish. A zero mixing element implies $CP$ conservation, so that $\jcp=0$.
A clue to the second constraint is that with \mt-reflection symmetry,
$\jcp=0$ ensures a zero somewhere in the $\nu_e$ row of the MNS matrix. However,
\mt-reflection symmetry implies two more constraints, Eq.~(\ref{mtSymm}).
In order to find a single additional constraint we consider the
$K$ matrix~\cite{Kmatrix}~\cite{HSW06} with elements:
%
\beq
K_{\gamma k}={\rm Re}(\Uai\Uajst\Ubist\Ubj),
\eeq
%
which is the $CP$-conserving analogue of $\jcp$ (cf.~the definition of \jcp,
Eq.~(\ref{jarlskogian})).
$K$ should be familiar: in the leptonic case, its elements are often
used to write the magnitudes of the oscilliatory terms in neutrino appearance
probabilities;~\cite{HSW06} in the quark case, its elements are just the CKM factors
of the $CP$-conserving parts of the interference terms in penguin-dominated decay rates.
A single zero in the mixing matrix leads to four zeroes in a plaquette of $K$ and
this clearly implies:
%
\beq
\Det\,K=0,
\label{constraint2}
\eeq
%
which is our sufficient second condition, along with $\jcp=0$.~\footnote{The two
conditions may even be expressed as one, noting that the product of all nine
elements of $P$ is given by
$\frac{1}{144}\prod_{\alpha i} P_{\alpha i}=(\Det\,K)^2+\jcp^2(2\jcp^2+{\cal{R}})^2$,
which is zero iff $\Det\, K=0$ and $\jcp=0$ (as ${\cal{R}}>0$, as long as $\jcp\neq 0$).}
We note that Eq.~(\ref{constraint2}) can easily be cast in terms of our complete
set of FSMOs, since $54\,\Det\,K\equiv 2\AC + \FF(\FF^2 - 2\CC - 1)$. Hence,
\mt-reflection symmetry, Eq.~(\ref{mtSymm}), is a special case of Eq.~(\ref{constraint2}).
Experimentally, there is no exactly zero element in the CKM matrix, so that $\Det\,K=0$
{\em and} $\jcp=0$ cannot {\em both} be exact for quarks. Moreover, for leptons,
despite there being no experimental lower limit for $|U_{e3}|$, there is no reason to
suppose that the MNS matrix has an exact zero either. In order to ensure a
small, but non-zero element in the mixing matrices, we need to consider a
modest relaxation of either condition, or of both.
For quarks, we know from experiment that $CP$ is slightly violated, with~\cite{CKMUTFIT}
$|\jcp_q/\jcp_{max}|\simeq 3\times 10^{-4}$, while~\footnote{We note that
$\jcp_{max}=\frac{1}{6\sqrt{3}}\simeq 0.1$ and $(\Det\,K)_{max}=\frac{2^6}{3^9}\simeq 0.0033$.}
for leptons, fits to oscillation
data~\cite{FITS:1} imply a fairly loose upper bound on their $CP$ violation:
$|\jcp_{\ell}/\jcp_{max}|\,\simlt\,0.33$. Turning to $\Det\,K$, we find that for quarks,
$|\Det\,K_q/(\Det\,K)_{max}|\,\simlt\, 3\times 10^{-7}$, while for leptons,
$|\Det\,K_{\ell}/(\Det\,K)_{max}|\,\simlt\, 0.6$ (the precision of lepton mixing data
does not yet allow a strong constraint). However, there is no experimental lower limit for
$|\Det\,K|$ for quarks or for leptons, each being compatible with zero, so that it is
sufficient to relax only the condition on $\jcp$.
We are thus led to conjecture that for both quarks and leptons:
%
\beq
\Det\,K=0;\quad |\jcp/\jcp_{max}|={\rm small}
\label{conjecture}
\eeq
%
(it is not implied that the small quantity necessarily has the same value in both
sectors). Equation (\ref{conjecture}) is a unified and flavour-symmetric, partial
description of both lepton and quark mixing matrices, being associated with
the existence of at least one small element in each mixing matrix, $U_{e3}$ and $V_{ub}$
respectively (it is partial in the sense that only two degrees of freedom are constrained
for each matrix). However, in the case that $\jcp$ is not exactly zero, the condition
$\Det\,K=0$ also implies that in the limit, as $\jcp\rightarrow 0$, there is at
least one unitarity triangle angle which
$\rightarrow 90^{\circ}$. This is rather obvious in the \mt-symmetry case, but is
less obvious more generally. While the flavour symmetry prevents an a priori prediction
of {\it which} angle is $\simeq 90^{\circ}$, we know from experiment~\cite{CKMUTFIT} that
for quarks, $\alpha\simeq 90^{\circ}$. A detailed calculation shows that our conjecture,
Eq.~(\ref{conjecture}), predicts, in terms of Wolfenstein parameters:~\cite{wolfenstein}
%
\beq
(90^{\circ}-\alpha)=\overline\eta\lambda^2=1^{\circ}\pm0.2^{\circ}
\label{qPrediction}
\eeq
%
at leading order in small quantities, to be compared with its current experimental
determination:~\cite{CKMUTFIT}
%
\beq
(90^{\circ}-\alpha)=0^{\circ+3^{\circ}}_{\,\,\,-7^{\circ}}.
\eeq
%
It will be interesting to test Eq.~(\ref{qPrediction}) more precisely in future
experiments with $B$ mesons, in particular, at LHCb and at a possible future Super
Flavour Factory. For leptons, experiment tells us not only that it is the $U_{e3}$ MNS
matrix element which is small but also that only the unitarity triangle
angles~\footnote{We use the
nomenclature of unitarity triangle angles we defined in reference [46] of Ref.~\cite{BHS05}.}
$\phi_{\mu 1}$ or $\phi_{\tau 1}$ can be close to $90^{\circ}$.
Then Eq.~(\ref{conjecture}) implies that:
%
\beq
|90^{\circ}-\delta|=2\sqrt{2}\,\sin{\theta_{13}}\,\sin{(\theta_{23}-\frac{\pi}{4})}\,\simlt\, 4^{\circ}
\eeq
%
at leading order in small quantities (we use the PDG convention here). It thus requires
a large $CP$-violating phase in the MNS matrix, which is promising for the discovery of
leptonic $CP$ violation at eg.~a future Neutrino Factory.
\section{Discussion and Conclusions}
Given that our flavour-symmetric variables are defined (essentially) uniquely by their
flavour symmetry properties and by their order in $P$, it is remarkable that the
leptonic data may be described simply by the constraints $\FF=\AC=\CC=\jcp=0$.
This is suggestive that these variables may be fundamental in some way.
It is furthermore tantalising that the smallness of one element in each mixing matrix,
the approximate \mt-symmetry in lepton mixing and the existence of a right
unitarity triangle may all be related to each other, through our simple partially-unified
constraint, Eq.~(\ref{conjecture}). The precision of the resulting
prediction, Eq.~(\ref{qPrediction}), motivates more sensitive tests at future
$B$ physics facilities, while the synergy with tests at a neutrino factory is manifest.
All elements of the Standard Model, apart from the Yukawa couplings of the fermions
to the Higgs, treat each fermion of any given charge on an equal footing - they are
already flavour-symmetric. The Yukawa couplings, on the other
hand, depend on flavour in such a way that each flavour has unique mass and mixing matrix
elements. Using our flavour-symmetric observables, or combinations of them appropriately
chosen, we have shown how it is also possible to specify the flavour-dependent mixings
in a flavour-independent way.~\footnote{We illustrated another variant of this in
Ref.~\cite{EXTREMISATION}.} This recovers flavour symmetry at the level of
the mixing description, the symmetry being broken only spontaneously by its solutions,
which define and differentiate the flavours in terms of their mixings.
\section*{Acknowledgments}
PFH thanks the organisers of the 43rd Rencontres de Moriond for organising a very
stimulating conference. PFH also acknowledges the hospitality of the Centre for
Fundamental Physics (CfFP) at the STFC Rutherford Appleton Laboratory. This work
was supported by the UK Science and Technology Facilities Council (STFC).
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\end{document}
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