
\section{Appendix: Why $\Delta m^2_{ee}$ is used by Daya Bay}

This section describes the advantages of reporting the Daya Bay
  measurement of electron antineutrino disappearance in terms of an
  effective mass-squared difference ${\Delta}m^2_{ee}$, which is
  independent of the unknown ordering of neutrino masses and future
  improvements in our knowledge of the solar oscillation parameters.


\subsection{Introduction}

In the three-flavor framework, the survival probability of electron antineutrino
is given by
\begin{align}
  \label{eq:exact}
 & P(\overline{\nu}_e \to \overline{\nu}_e ) \nonumber \\
 & ~~=  1   - \cos^4\theta_{13}\sin^22\theta_{12}\sin^2\Delta_{21}  \nonumber \\
 &~~~~~ - \sin^22\theta_{13}
  (\cos^2\theta_{12}\sin^2 \Delta_{31} + \sin^2\theta_{12}\sin^2\Delta_{32}),
\end{align}
where $\Delta_x = \Delta m^2_x \frac{L}{4E}$. The three
mass-squared differences are subject to the constraint $|\Delta
m^2_{31}| = |\Delta m^2_{32}| \pm |\Delta m^2_{21}|$ where
``$+$''(``$-$'') is for the normal(inverted) mass ordering (or
  hierarchy).  Therefore, determination of $\Delta m^2_{32}$
(or $\Delta m^2_{31}$) depends on knowledge of the mass
ordering and solar oscillation parameters.

The Daya Bay experiment reports a precise measurement of the
  effective mass splitting $\Delta m^2_{ee}$, which is independent of
  our knowledge of the ordering and solar parameters.  In this
  approach, we approximate the survival probability using
\begin{align}
  \label{eq:approx}
  P(\overline{\nu}_e \to \overline{\nu}_e )
  \simeq&  1
  - \cos^4\theta_{13}\sin^22\theta_{12}\sin^2\Delta_{21} \nonumber \\
&- \sin^22\theta_{13}   \sin^2 \Delta_{ee}.
\end{align}
Despite the advantage of using $\Delta m^2_{ee}$ for the
measurement, it has the disadvantage of not being a fundamental
parameter. Therefore, we must determine a relation between
  $\Delta m^2_{ee}$ and $\Delta m^2_{32}$ given knowledge of the mass
  ordering and solar oscillation parameters.

In the following sections, we are going to address the following two questions:
\begin{itemize}
\item Is Eq.~\ref{eq:approx} good enough at the current experimental precision?
\item How can we estimate the value of $\Delta m^2_{32}$ once the
  value of $\Delta m^2_{ee}$ is obtained?
\end{itemize}

\subsection{Mathematical derivation}
Using the relation  $|\Delta m^2_{31}| = |\Delta m^2_{32}| \pm
|\Delta m^2_{21}|$, Eq.~\ref{eq:exact} can be written as,
\begin{align}
  \label{eq:1}
  P(\overline{\nu}_e \to \overline{\nu}_e )
  
   
   
= &~1 - 2s^2_{13}c^2_{13}  \nonumber\\
&+ 2s^2_{13}c^2_{13}\sqrt{1-4s^2_{12}c^2_{12}\sin^2\Delta_{21}}
\cos (2\Delta_{32} \pm \phi) \nonumber\\
& - 4 c^4_{13}s^2_{12}c^2_{12}\sin^2\Delta_{21},
\end{align}
where $s_{x}  = \sin \theta_{x}$, $c_{x}  = \cos \theta_{x}$,
 and $\phi = \arctan\left( \frac{\sin 2\Delta_{21}}
{\cos 2\Delta_{21} + \tan^2 \theta_{12}}\right)$.
The last term of the above formula is the so-called ``solar term''
that governs the reactor antineutrino oscillation at O(100)~km.
For the L/E range covered by Daya Bay, $4s^2_{12}c^2_{12}\sin^2\Delta_{21} \ll 1$.
Thus, Eq.~\ref{eq:1} can be approximated as,
\begin{align}
  \label{eq:3}
&P(\overline{\nu}_e \to \overline{\nu}_e ) \nonumber \\
& \simeq 1 - 4 s^2_{13}c^2_{13} \left[\frac{1-\cos(2\Delta_{32} \pm \phi)}{2}\right]
- (solar~term)\nonumber \\
& = 1 - \sin^22\theta_{13} \sin^2(\Delta_{32}\pm \phi/2) - (solar~term).
\end{align}

By comparing Eq.~\ref{eq:3} with Eq.~\ref{eq:approx}, we obtain
the expression relating $\Delta m^2_{ee}$ to $\Delta m^2_{32}$ (or $\Delta m^2_{31}$)
\begin{align}
  \label{eq:4}
  |\Delta m^2_{ee}| &= |\Delta m^2_{32}| \pm \Delta m^2_\phi/2\\
  &= |\Delta  m^2_{31}| \mp (|\Delta m^2_{21}| - \Delta m^2_\phi/2),
  \label{eq:4a}
\end{align}
where $\Delta m^2_\phi = \phi \times \frac{4E}{L}$.

\subsection{Numerical evaluation}

By definition, $\Delta m^2_\phi$ is a function of
L/E. Using the current values of
$\Delta m^2_{21} = 7.50 \times 10^{-5}~\textrm{eV}^2$
and $\sin^22\theta_{12} = 0.857$~\cite{PDG},
Fig.~\ref{fig:dm2_phi} shows the value of
$\Delta m^2_\phi / 2$ as a function of energy for L~=~1.6~km.
We find that $\Delta m^2_\phi / 2 \simeq 5.17\times10^{-5}~\textrm{eV}^2$
is essentially a constant in our L/E region, and numerically identical to
cos$^2\theta_{12}\Delta m^2_{21}$.
Thus, this definition of $\Delta m^2_{ee}$ is similar to the definition
introduced in Ref.~\cite{Nunokawa:2005nx}:
\begin{align}
  \label{eq:6}
  \Delta m^2_{\textrm{eff}}|_e &= \cos^2 \theta_{12}|\Delta m^2_{31}| +
                                 \sin^2 \theta_{12}|\Delta m^2_{32}| \\
  &= |\Delta m^2_{32}| \pm \cos^2 \theta_{12} \Delta m^2_{21}.
\end{align}


Figure~\ref{fig:osc_prob} is a comparison of the approximated formula with
$\Delta m^2_\phi / 2 = 5.17\times10^{-5}~\textrm{eV}^2$,
\begin{align}
  \label{eq:5}
  P_{ee} \simeq&~1 - \sin^22\theta_{13} \sin^2\left[(\Delta m^2_{32} +
  5.17\times10^{-5}~\textrm{eV}^2)
  \frac{L}{4E}\right]\nonumber \\
&- (solar~term),
\end{align}
to the three-flavor formula, Eq.~\ref{eq:exact}.
In this comparison, L~=~1.6~km, $\sin^2 2\theta_{13} = 0.09$,
$\Delta m^2_{32} = 2.44 \times 10^{-3}~\textrm{eV}^2$, and
normal mass hierarchy are the inputs.
The agreement between the two, better than $10^{-4}$, is excellent and exceeds the
achievable experimental precision.

\begin{figure}[hbtp]
  \centering
  \includegraphics[width=0.45\textwidth]{dm2_phi}
  \caption{Values of $\Delta m^2_\phi/2 = |\Delta m^2_{ee} -
    \Delta m^2_{32}|$ (black solid line) at L~=~1.6~km as a function of the neutrino
  energy, with $\Delta m^2_{21} = 7.50 \times 10^{-5}~\textrm{eV}^2$
  and $\sin^22\theta_{12} = 0.857$~\cite{PDG}.
  For comparison, calculations based on other definitions of
      $\Delta m^2_{ee}$,
      $\Delta m^2_{ee} = \cos^2 \theta_{12}\Delta m^2_{31} + \sin^2
      \theta_{12}\Delta m^2_{32}$ (red dashed line) and
      $\Delta m^2_{ee} = \frac{4E}{L}\arcsin \left[\sqrt{\cos^2\theta_{12}\sin^2 \Delta_{31}
          + \sin^2\theta_{12}\sin^2 \Delta_{32} }\right]$ (blue dotted line)
      are also shown.
  }
\label{fig:dm2_phi}
\end{figure}

\begin{figure}[hbtp]
  \centering
  \includegraphics[width=0.45\textwidth]{osc_prob}
  \includegraphics[width=0.45\textwidth]{osc_prob_ratio}
  \caption{Comparison of the survival probability
    at L~=~1.6~km between the
    approximated formula with
    $\Delta m^2_{ee} = \Delta m^2_{32} +
    5.17\times10^{-5}~\textrm{eV}^2$ and the exact
    three-flavor formula (Eq.~\ref{eq:exact}).
    The oscillation parameters used in this comparison are $\sin^2 2\theta_{13} = 0.09$
    and $\Delta m^2_{32} = 2.44 \times 10^{-3}~\textrm{eV}^2$ under the
    normal mass hierarchy assumption.
    The top panel shows the survival probabilities calculated with the two formulae, and
    the bottom panel shows the ratio of the two.}
  \label{fig:osc_prob}
\end{figure}


This study demonstrates that, once we obtain the value of $|\Delta
m^2_{ee}|$ using Eq.~\ref{eq:approx},
we can reliably deduce the values of $|\Delta m^2_{32}|$ and $|\Delta
m^2_{31}|$ using Eqs.~\ref{eq:4} and~\ref{eq:4a} with
\begin{equation}
 \label{eq:7}
\Delta m^2_\phi /2
\simeq \cos^2\theta_{12}\Delta m^2_{21}.
\end{equation}
Using the current values of $\theta_{12}$ and ${\Delta}m^2_{21}$,
$\Delta m^2_\phi /2 \simeq 5.17 \times 10^{-5}{\rm eV}^2$, and
($|\Delta m^2_{21}| - \Delta m^2_\phi /2) \simeq 2.33 \times 10^{-5}\,{\rm eV}^2$.

It is important to point out that the exact solution of
$\sin^2(\Delta m^2_{ee}
  \frac{L}{4E})
= \cos^2\theta_{12}\sin^2(\Delta m^2_{31}  \frac{L}{4E})
  + \sin^2\theta_{12}\sin^2(\Delta m^2_{32} \frac{L}{4E})$
was never used to extract
the value of $\Delta m^2_{32}$ or $\Delta m^2_{31}$ from the measured
$\Delta m^2_{ee}$ in Daya Bay.


\bibliographystyle{apsrev4-1}
