\section{Introduction}
\emph{Path-pairability} is a graph theoretical notion that emerged from a practical networking problem introduced by Csaba, Faudree, Gy\'arf\'as, Lehel, and Schelp \cite{CS}, and further studied by Faudree, Gy\'arf\'as, and Lehel \cite{mpp,F,pp} and by Kubicka, Kubicki and Lehel \cite{grid}. Given a fixed integer $k$ and a simple undirected graph $G$ on at least $2k$ vertices, we say that $G$ is {\it $k$-path-pairable} if, for any pair of disjoint sets of distinct vertices $\{x_1,\dots,x_k\}$ and $\{y_1,\dots,y_k\}$ of $G$, there exist $k$ edge-disjoint paths $P_1,P_2,\dots,P_k$, such that $P_i$ is a path from $x_i$ to $y_i$, $1\leq i\leq k$. The path-pairability number of a graph $G$ is the largest positive integer $k$, for which $G$ is $k$-path-pairable, and it is denoted by $\pp(G)$. A $k$-path-pairable graph on $2k$ or $2k+1$ vertices is simply said to be {\it path-pairable}.

Path-pairability is related to the notion of \textit{linkedness}. A graph is $k$-\emph{linked} if for any choice of $2k$ vertices $\{s_1, \ldots ,
s_k, t_1, \ldots , t_k\}$ (not necessarily distinct), there are internally vertex disjoint paths $P_1, \ldots , P_k$ with $P_i$ joining $s_i$ to $t_i$ for $1 \le i \le k$. Bollob{\'a}s and Thomason~\cite{BollobasThomason} showed that any $2k$-connected graph with a lower bound on its edge density is $k$-linked. On the other hand, a graph being path-pairable imposes no constraint on the connectivity or edge-connectivity of the graph. The most illustrative examples of this phenomenon are the stars $K_{1, n-1}$. Indeed, it is easy to see that stars are path-pairable, while they are neither $2$-connected nor $2$-edge-connected. Note that, for any pairing of the vertices of $K_{1, n-1}$, joining two vertices in a pair is straightforward due to the presence of a vertex of high degree, and the fact that the diameter is small. This example motivates the study of two natural questions about path-pairable graphs: given a path-pairable graph $G$ on $n$ vertices, how small can its maximum degree $\Delta(G)$ be, and how large can its diameter $d(G)$ be? This note addresses some aspects of the second question. To be precise, for a family of graphs $\mathcal{G}$ let us define $d(n, \mathcal{G})$ as follows:
\[
	d(n, \mathcal{G}) = \max\{d(G): G \in \mathcal{G} \text{ and } G \text{ is path-pairable on } n \text{ vertices}\}.
\]
When $\mathcal{G}$ is the family of path-pairable graphs, we shall simply write $d(n)$ instead of $d(n, \mathcal{G})$.

\begin{comment}
It was proved by Faudree et al. that $\Delta_\text{min}(n)$ has to grow with the size of the graph; in particular, if $G$ is a path-pairable graph on $n$ vertices with maximum degree $\Delta$, then $n\leq 2\Delta^\Delta$ holds. The result places sublogarithmic lower bound on $\Delta_\text{min}(n)$, that is, $\Delta_\text{min}(n) = \Omega\left(\frac{\log n}{\log\log n}\right)$. To date the best known asymptotic upper bound is $\Delta_\text{min}(n) = O(\log n)$  due to Gy\H ori et al \cite{ntp}.
\end{comment}
The maximum diameter of arbitrary path-pairable graphs was investigated by M\'esz\'aros \cite{me_diam} who proved that $d(n) \le 6 \sqrt{2} \sqrt{n}$.
Our aim in this note is to investigate the maximum diameter of path-pairable graphs when we impose restrictions on the number of edges and on how the edges are distributed.
To state our results, let us denote by $\mathcal{G}_m$ the family of graphs with at most $m$ edges.
The following result determines $d(n, \mathcal{G}_m)$ for a certain range of $m$.

\begin{thm}\label{diam_m}
    If $2n \le m \le \frac{1}{4}n^{3/2}$ then
    \[
        \sqrt[3]{\frac{1}{2}m-n} \le d(n, \mathcal{G}_m) \le 16 \sqrt[3]{m}.
    \]
\end{thm}
We remark that the upper bound in the Theorem~\ref{diam_m} holds for $m$ in any range, but when $m \ge \frac{1}{4}n^{3/2}$ the bound obtained by M\'esz\'aros \cite{me_diam} is sharper.
Determining the behaviour of the maximum diameter among path-pairable graphs on $n$ vertices with fewer than $2n$ edges remains an open problem.
In particular, we do not know if the maximum diameter must be bounded  (see Section \ref{sec:final}).

Following this line of research, it is very natural to consider the problem of determining the maximum attainable diameter for other classes of graphs. For example, what is the behaviour of the maximum diameter of path-pairable \emph{planar} graphs? Although we could not give a satisfactory answer to this particular question, we were able to do so for graphs which are $c$-\emph{degenerate}.
As usual, we say that an $n$-vertex graph $G$ is $c$-\emph{degenerate} if there exists an ordering $v_1,\ldots,v_n$ of its vertices such that $|\{v_j: j > i, v_iv_j\in E(G) \}|\leq c$ holds for all $i=1,2,\ldots,n$. We let $\mathcal{G}_{c\text{-deg}}$ denote the family of $c$-degenerate graphs. Clearly all $c$-degenerate graphs have a linear number of edges, so Theorem~\ref{diam_m} implies that $d(n, \mathcal{G}_{c\text{-deg}}) = O(\sqrt[3]{n})$. However, as the next result shows, this bound is far from the truth.

\begin{thm}\label{diam_cdeg}
Let $c \ge 5$ be an integer. Then
\[
	(2+o(1)) \frac{\log(n)}{\log({\frac{c}{c-2}})}  \leq d(n, \mathcal{G}_{c\text{-deg}}) \leq  (12+o(1)) \frac{\log(n)}{\log(\frac{c}{c-2})}
\]
	as $n \rightarrow \infty$.
\end{thm}

We remark that we have not made an effort to optimize the constants appearing in the upper and lower bounds of Theorems~\ref{diam_m} and~\ref{diam_cdeg}.
\subsection{The Cut-Condition}

While path-pairable graphs need not be highly connected or edge-connected, they must satisfy certain `connectivity-like' conditions that we shall need in the remainder of the paper. We say a graph $G$ on $n$ vertices satisfies the \emph{cut-condition} if for every $X \subset V(G)$, $|X| \le n/2$, there are at least $|X|$ edges between $X$ and $V(G)\setminus X$. Clearly, a path-pairable graph has to satisfy the cut-condition. On the other hand, satisfying the cut-condition is not sufficient to guarantee path-pairability in a graph; see \cite{me_pp} for additional details.

\subsection{Organization and Notation}
The proofs of the lower bounds in Theorems~\ref{diam_m} and~\ref{diam_cdeg} require constructions of path-pairable graphs with large diameter. In Section~\ref{sec:blowup}, we
show how to obtain such graphs by proving that a more general class of graphs is path-pairable. In Sections~\ref{sec:proofdiam_m} and~\ref{sec:proofdiam_cdeg} we shall complete the proofs of
Theorems~\ref{diam_m} and~\ref{diam_cdeg}, respectively. Finally, we mention some open problems in Section~\ref{sec:final}.

Our notation is standard. Thus, for a (simple, undirected) graph $G$ we shall denote by $V(G)$ and $E(G)$ the vertex set and edge set of $G$, respectively. We also let $|G|$ and $d(G)$ denote the number of vertices and diameter of $G$, respectively. For a vertex $x \in V(G)$ we let $N_{G}(x)$ denote the neighbourhood of $x$ in $G$, and we shall omit the subscript `$G$' when no ambiguity arises.
\section{Path-pairable graphs from blowing up paths}\label{sec:blowup}

In this section, we will show how to construct a quite general class of graphs which have high diameter and are path-pairable.
Let $G$ be a graph with vertex set $V(G)=\{v_1,\ldots,v_k\}$, and let $G_1,\ldots, G_k$ be graphs. We define the \textit{blown-up graph} $G(G_1,\ldots, G_k)$ as follows:  replace every vertex $v_i$ in $G$ by the corresponding graph $G_i$, and for every edge $v_iv_j \in E(G)$ insert a complete bipartite graph between the vertex sets of $G_i$ and $G_j$.

Let $P_k$ denote the path on $k$ vertices. The following lemma asserts that if we blow-up a path with graphs $G_1, \ldots , G_k$, such that $G_{i}$ is path-pairable for $i \le k-1$, and certain properties inherited from the cut-condition hold, then the resulting blow-up is path-pairable.

\begin{lemma}\label{blown-up}
    Suppose that $G_1, \ldots ,G_k$ are graphs on $n_1, \ldots , n_k$ vertices, respectively, where $G_{i}$ is path-pairable for $i \le k-1$.
    Let $n = \sum_{i =1}^k n_i$ and let $u_i = \sum_{j=1}^i n_j$ for $i=1, \dots , k-1$. Then $P_k(G_1,\ldots, G_k)$ is path-pairable if and only if
\begin{equation}\label{eq_1}
    n_i\cdot n_{i+1}\geq \min(u_i,n-u_i)
\end{equation}
holds for $i=1,\ldots,k-1$.
\end{lemma}
\begin{proof}
For each $i = 1, \ldots , k$, let $U_i = \bigcup_{j=1}^i V(G_j)$ so that $u_i = |U_i|$. Now, if $P_k(G_1, \ldots , G_k)$ is path-pairable, then we may apply the cut-condition to the cut $\{U_i, V(G)\setminus U_i\}$. This implies $n_i\cdot n_{i+1}\geq \min(u_i,n-u_i)$ must hold for $i=1,\ldots , k-1$. In the remainder, we show that this simple condition is enough to yield the path-pairability of
$G := P_k(G_1, \ldots , G_k)$. Assume that a pairing $\mathcal{P}$ of the vertices of $G$ is given. If $\{u, v\} \in \mathcal{P}$ we shall say that $u$ is a \emph{sibling} of $v$ (and vice-versa).
We shall define an algorithm that sweeps through the classes $G_1,G_2,\ldots, G_k$ and joins each pair of siblings via edge-disjoint paths.

First we give an overview of the algorithm.
We proceed by first joining pairs $\{u, v\} \in \mathcal{P}$ via edge-disjoint paths such that $u$ and $v$ belong to different $G_i$'s, and then afterwards joining pairs that remain inside some $G_j$ (using the path-pairability of $G_j$).
Before round $1$ we use the path-pairability property of $G_{1}$ to join those siblings which belong to $G_{1}$.
In round $1$ we assign to every vertex $u$ of $G_1$ a vertex $v$ of $G_2$.
If $\{u, v\} \in \mathcal{P}$ are siblings, then we simply choose the edge $uv$.
Then we join the siblings which are in $G_{2}$ again using the path-pairability property of $G_{2}$.
For those paths $uv$ that have not ended (because $\{u, v\} \notin \mathcal{P}$) we shall continue by choosing a new vertex $w$ in $G_3$ and continue the path with edge $vw$, and so on.
Paths which have not finished joining a pair of siblings we shall call \emph{unfinished}; otherwise, we say the path is \emph{finished}.
The last edge which completes a finished path we shall call a \emph{path-ending edge}.
During round $i$ we shall first choose those vertices in $G_{i+1}$ which, together with some vertex of $G_i$, form path-ending edges.
At the end of round $i$, in $G_{i+1}$ we will have endpoints of unfinished paths and perhaps also some endpoints of finished paths.
Note that the vertices of $G_{i+1}$ might be endpoints of several unfinished paths.
For $x \in G_{i+1}$ let $w(x)$ denote the number of unfinished paths $P\cup \{x\}$ with $P \subset U_i$ at the end of round $i$ which are to be extended by a vertex of $G_{i+2}$ (including the single-vertex path $x$ in the case when $x$ was not joined to its sibling in the latest round).
Note that every such path corresponds to a yet not joined vertex in $U_{i+1}$ as well as to another vertex yet to be joined lying in $V(G)\setminus U_{i+1}$.
It follows that
\begin{equation}\label{eq:weights}
	\sum_{x \in G_{i+1}}w(x) \le \min(u_{i+1}, n-u_{i+1}).
\end{equation}

Let us now be more explicit in how we make choices in each round.
We shall maintain the following two simple conditions throughout our procedure (the first of which has been mentioned above):
\begin{itemize}
\item[(a)] During round $i$ ($1\le i \le k-1$), if $w \in G_i$ is the current endpoint of the path which began at some vertex $u\in U_i$ (possibly $u=w$), and $\{u, v\} \in \mathcal{P}$ for $v \in G_{i+1}$, then we join $w$ to $v$. Informally, we choose path-ending edges when we can.
\item[(b)] $w(x) \leq n_{i+1}$ for all $x \in G_i$, for $i =1, \ldots , k-1$.
\end{itemize}
The second condition above is clearly necessary in order to proceed during round $i$, as ${|N(x)\cap G_{i+1}| = n_{i+1}}$ for every $x \in G_i$, and hence we cannot continue more than $n_{i+1}$ unfinished paths through $x$.

We claim that as long as both of the above conditions are maintained, the proposed algorithm finds a collection of edge-disjoint paths joining every pair in $\mathcal{P}$.
Both conditions are clearly satisfied for $i=1$ as $w(x) \le 1\leq n_2$ for all $x\in G_1$.
Let $i \ge 2$ and suppose both conditions hold for rounds $1, \ldots , i-1$.
Our aim is show that an appropriate selection of edges between $G_i$ and $G_{i+1}$ exists in round $i$ to maintain the conditions.
We start round $i$ by choosing all path-ending edges with endpoints in $G_i$ and $G_{i+1}$; this can be done since, by induction, $w(x) \le n_{i+1}$ for every $x \in G_i$.
Observe that if $i = k-1$ then the only remaining siblings are in $G_{k}$.
Then for every $\left\{ u, v \right\} \in \mathcal{P}$ such that $u,v \in G_{k}$ we can find a vertex $w$ in $G_{k-1}$ and join $u, v$ with the path $uwv$.
When $i < k-1$ then the remaining paths can be continued by assigning arbitrary vertices from $G_{i+1}$ (without using any edge multiple times).
We choose an assignment that balances the `weights' in $G_{i+1}$.
More precisely, let us choose an assignment of the vertices that minimizes
\[
	\sum\limits_{a\in G_{i+1}}w(a)^2.
\]
If for every $x \in G_{i+1}$ we have that $w(x) \le n_{i+2}$ we are basically done.
It remains to find edge-disjoint paths inside $G_{i+1}$ for those pairs $\{x, y\} \in \mathcal{P}$ whose vertices belong to $G_{i+1}$.
But this is possible because of the assumption that $G_{i+1}$ is path-pairable.

Suppose then that in the above assignment there exists $x \in G_{i+1}$ with $w(x) \geq n_{i+2}+1$.
We first claim that, under this assignment, no other vertex of $G_{i+1}$ has small weight.
\begin{claim}\label{claim:y-big}
Every vertex $y\in G_{i+1}$ satisfies $w(y)\geq n_{i+2}-1$.
\end{claim}
\begin{proof}
Suppose there is $y \in G_{i+1}$ such that $w(y) \le n_{i+2}-2$.
Then, as $w(x)>w(y)+2$, there exist vertices $v_1,v_2\in G_i$ such that certain paths ending at $v_1$ and $v_2$ were joined in round $i$ to $x$ ($x$ was assigned as the next vertex of these paths) but no paths at $v_1$ or $v_2$ were assigned $y$ as their next vertex.
Observe that at least one of the edges $v_1x$ and $v_2x$ is not a path ending edge which could have been replaced by the appropriate $v_1y$ or $v_2y$ edge, respectively.
That operation would result in a new assignment with a smaller square sum $\sum_{a\in G_{i+1}}w(a)^2$, which is a contradiction.
\end{proof}

Therefore, we may assume $w(y)\geq n_{i+2} - 1$ for all $y\in G_{i+1}$.
In this case, partition the vertices of $G_{i+1}$ into three classes:
\begin{align*}
X &= \{v\in G_{i+1}: w(v) \geq n_{i+2} +1\}\\
Y &= \{v\in G_{i+1}: w(v) = n_{i+2} - 1\} \\
Z &= \{v\in G_{i+1}: w(v) = n_{i+2}\}.
\end{align*}

Observe first that $1 \le |X| \leq |Y|$, since otherwise using~(\ref{eq:weights}) we have \[n_{i+1}n_{i+2}+1\leq \sum\limits_{s\in G_{i+1}}w(s)\leq \min(u_{i+1},n-u_{i+1}),\] contradicting condition~(\ref{eq_1}).
Notice also that the same argument as in Claim~\ref{claim:y-big} shows that $w(v) \le n_{i+2}+1$ for every $v \in G_{i+1}$, hence we can actually write
\[
    X = \left\{ v \in G_{i+1}: w(v) = n_{i+2}+1 \right\}.
\]

We will need the following claim which asserts that if there are siblings in $G_{i+1}$ then they must belong to $Z$.
\begin{claim}\label{claim:Z_pairs}
     If $\{u,v\} \in \mathcal{P}$ and $u, v \in G_{i+1}$, then $u, v \in Z$.
\end{claim}
\begin{proof}
We first show that every $y \in Y$ is incident to a path-ending edge. Suppose, to the contrary, that there is $y \in Y$ such that there is no path-ending edge which ends at $y$.
    It follows that there are at most $w(y)$ vertices in $G_{i}$ which had been joined to $y$.
    Hence we can take any $x \in X$ and find $z \in G_{i}$ which was not joined to $y$, and such that $xz$ is not a path-ending edge.
    Replacing $zx$ by $zy$ would result in a smaller square sum $\sum_{a \in G_{i+1}}w(a)^{2}$, which gives a contradiction.

    Now, let $\{u, v\} \in \mathcal{P}$ such that $u, v \in G_{i+1}$.
    Since every $y \in Y$ is incident to a path-ending edge, we have that $u, v \not\in Y$.
    Suppose, for contradiction, that $u \in X$.
    Then $u$ was joined to $w(u) = n_{i+2}+1$ vertices in $G_{i}$, and hence for every $y \in Y$, there is $z \in G_{i}$ which was joined to $u$ but not $y$.
    Replacing $zu$ by $zy$ would result in a smaller square sum $\sum_{a \in G_{i+1}}w(a)^{2}$, which again gives a contradiction.
\end{proof}

Finally, we shall show that we can reduce the weights of the vertices in $X$ (and pair the siblings inside $G_{i+1}$) using the path-pairable property of $G_{i+1}$.
For every $x \in X$ pick a different vertex $y_{x} \in Y$ (which we can do, since $|Y| \ge |X|$) and let $\mathcal{P'} = \left\{ \{u, v\} \in \mathcal{P} : u, v \in G_{i+1}  \right\} \cup \left\{ \{x, y_{x}\} : x \in X  \right\}$.
Since $G_{i+1}$ is path-pairable, we can find edge-disjoint paths joining the siblings in $\mathcal{P'}$ (note that by Claim~\ref{claim:Z_pairs} none of the pairs $\{x, y_x\}$ interfere with any siblings $\{u, v\} \in \mathcal{P}$ with $u, v \in G_{i+1}$).
Observe now that for every $x \in X$ one path has been channeled to a vertex $y\in Y$, thus the number of unfinished path endpoints at $x$ has dropped to $n_{i+2}$ and so the condition is maintained.

\end{proof}

    We close the section by pointing out that the condition that the graphs $G_{i}$ are path-pairable is necessary.
    We do this by giving an example of a blown-up path $P_k(G_{n_1},\ldots,G_{n_k})$ that satisfies the cut-conditions of Lemma~\ref{blown-up} yet it is not path-pairable unless some of $G_i$'s are path-pairable as well.
    For the sake of simplicity we set $k = 5$ and prove that $G_3$ has to be path-pairable.
    Let $n = 2t^2 + t$ for some even $t \in \mathbb{N}$ and let $n_1 = n_5 = t^2-t$, $n_2=n_3 = n_4 = t$.
    Clearly $P_5(G_{n_1},\ldots,G_{n_5})$ satisfies the Condition~\ref{eq_1} of Lemma~\ref{blown-up}.
    Observe, that any pairing of the vertices in $G_{1} \cup G_{2}$ with the vertices in $G_{4} \cup G_{5}$ has to use all the edges between $G_{3}$ and $G_{2} \cup G_{4}$.
    Therefore if we additionally pair the vertices inside $G_{3}$, then the paths joining those vertices can only use the edges in $G_{3}$, therefore $G_{3}$ has to be path-pairable.

\section{Proof of Theorem \ref{diam_m}}\label{sec:proofdiam_m}

    Take $x,y\in V(G)$ such that $d(x,y) = d(G)$ and let $V_{i}$ be the set of vertices at distance exactly $i$ from $x$, for every $i$.
    Observe that $V_0 = \{x\}$ and $y\in V_{d(G)}$.
    For $i \in \left\{ 1, \dots, d(G) \right\}$ define $n_{i}$ to be the size of $V_{i}$ and let $u_{i} = \sum_{j=0}^{i} n_{j}$.

    We need the following claim.
    \begin{claim}
        $u_{2k+1} \geq \binom{k+2}{2}$ as long as $u_{2k+1}\leq\frac{n}{2}$.
    \end{claim}
    \begin{proof}
        We shall use induction on $k$.
        For $k = 0$ it is clear.
        Assume that $u_{2k-1} \ge \binom{k+1}{2}$.
        By the cut-condition we have that the number of edges between $V_{2k}$ and $V_{2k+1}$ is at least $u_{2k-1}$, hence $n_{2k}\cdot n_{2k+1} \ge u_{2k-1} \ge \binom{k+1}{2}$.
        By the arithmetic-geometric mean inequality, $n_{2k} + n_{2k+1} \ge 2\sqrt{\binom{k+1}{2}} \ge k+1$.
        As $u_{2k+1} = u_{2k-1} + n_{2k} + n_{2k+1}$, we have $u_{2k+1} \ge \binom{k+2}{2}$.
    \end{proof}

    Now, let $A = \bigcup_{i=0}^{\lfloor d / 3 \rfloor}V_{i}$, $B = \bigcup_{i = \lfloor d / 3 \rfloor +1}^{2d/3} V_{i}$, $C = \bigcup_{i = \lfloor 2d/3 \rfloor + 1}^{d} V_{i}$.
    Observe, that $|A|, |C| \ge \min\left\{\frac{n}{2}, \frac{d^{2}}{100}\right\}$, so joining vertices in $A$ with vertices in $C$ requires at least $ \min\left\{\frac{n}{2},\frac{d^2}{100}\right\}\cdot\frac{d}{3}$ edges.
    Hence,
    \[ \min\left\{\frac{n}{2},\frac{d^2}{100}\right\}\cdot\frac{d}{3}\leq m,\]
    which implies
    \[d\leq \max\left\{\frac{6m}{n},16\sqrt[3]{m}\right\}.\]
    Notice that whenever $m \le 4n^{3/2}$ we have $d \le 16\sqrt[3]{m}$.
    Let us remark that if $m \ge \frac{1}{4}n^{3/2}$ then the upper bound is trivially satisfied by the general upper bound obtained in \cite{me_diam}.

    For the lower bound, let $n$ and $2n \le m \le \frac{1}{4}n^{3/2}$ be given.
    For any natural number $\ell$ we shall denote by $S_\ell$ the star $K_{1, \ell-1}$ on $\ell$ vertices.
    Consider the graph $G = P_{k}(G_{1},\dots,G_{k})$ on $n$ vertices,
    where $k = \left\lfloor \sqrt[3]{\frac{m}{2}-n} \right\rfloor$ and $G_{1} = G_{2} = \dots = G_{k} = S_{k}$, $G_{k+1} = S_{k^{2}}$, $G_{k+2} = S_{2}$, and $G_{k+3}$ is an empty graph on $n-2k^{2}-2$ vertices.

    Straightforward calculation shows that $u_i = i\cdot k$, for $i\leq k, u_{k+1}= 2k^2$, and $u_{k+2} = 2k^2 + 2$.
    Also $n_1n_2=n_2n_3=\ldots=n_{k-1}n_{k}=k^2$, $n_{k}n_{k+1}=k^3$, $n_{k+1}n_{k+2}=2k^2$, and $n_{k+2}n_{k+3}=2n-4k^2-4$.
    Therefore, for $i \in \left\{ 1, \dots, k+1 \right\}$ we have $n_{i} \cdot n_{i+1} \ge u_{i} \ge \min(u_{i}, n-u_{i})$ and $n_{k+2} \cdot n_{k+3} \ge n_{k+3} \ge \min(u_{k+2}, n-u_{k+2})$.
    Hence it follows from Lemma~\ref{blown-up} that $G$ is path-pairable.

    It is easy to check that the number of edges in $G$ is at most $2n + 2k^{3} \le m$.
    On the other hand, the diameter of $G$ is $k+2 \ge \sqrt[3]{\frac{m}{2} - n}$.

\section{Proof of Theorem \ref{diam_cdeg}}\label{sec:proofdiam_cdeg}

    In this section, we investigate the maximum diameter a path-pairable $c$-degenerate graph on $n$ vertices can have.
    We shall assume that $c$ is an integer and $c\geq 5$.

    Let $G$ be a $c$-degenerate graph on $n$ vertices with diameter $d$.
    We shall show first that $d \le 4\log_{\frac{c+1}{c}}(n)+3$.
    Let $x \in G$ be such that there is $y \in G$ with $d(x,y) = d$.
    For $i \in \left\{ 0, \dots, d \right\}$, write $V_{i}$ for the set of vertices at distance $i$ from $x$.
    Let $n_{i} = |V_{i}|$ and $u_{i} = \sum_{j = 0}^{i}n_{j}$.
    Observe that $|V_{i}| \ge 1$ for every $i \in \left\{ 0, \dots, d \right\}$.
    We can assume that $u_{\lfloor \frac{d}{2} \rfloor} \le \frac{n}{2}$ (otherwise we repeat the argument below with $V'_{i} = V_{d-i})$.

    The result will easily follow from the following claim.
    \begin{claim}
        $u_{2k+1} \ge \left( \frac{c+1}{c} \right)^{k}$ as long as $u_{2k+1} \le \frac{n}{2}$.
    \end{claim}
    Let us assume the claim and prove the result.
    Letting $k = \frac{\lfloor\frac{d}{2}\rfloor -1}{2}$, we have that $n/2 \ge u_{2k+1} \ge \left( \frac{c+1}{c} \right)^{\frac{\lfloor \frac{d}{2} \rfloor-1}{2}}$.
    Hence $d \leq 4\log_{\frac{c+1}{c}}(n)+3 = 4 \frac{\log(n)}{\log(\frac{c+1}{c})} + 3 \le 4 \frac{\log(n)}{\log(\frac{c}{c-2})}\frac{\log(\frac{c}{c-2})}{\log(\frac{c+1}{c})} + 3\le 12 \frac{\log(n)}{\log(\frac{c}{c-2})}+3$, where the last inequality follows from the easy to check fact that $\frac{\log(\frac{c}{c-2})}{\log(\frac{c+1}{c})} \le 3$, for all $c \ge 5$.
    \begin{proof}[Proof of the Claim]
        We shall prove the claim by induction on $k$.
        The base case when $k=0$ is trivial as $u_{1} \ge 2$.
        Suppose the claim holds for every $l \le k-1$.
        Since $G$ is $c$-degenerate we have that $e(V_{2k}, V_{2k+1}) \le c\left( n_{2k} + n_{2k+1} \right)$.
        On the other hand, it follows from the cut-condition that $e(V_{2k}, V_{2k+1}) \ge u_{2k} = u_{2k-1}+ n_{2k}$.
        Therefore, by the induction hypothesis, we have
        $n_{2k} + n_{2k+1} \ge
        \frac{1}{c}\left( u_{2k-1}+n_{2k} \right)
        \ge \frac{1}{c} \left( \left( {\frac{c+1}{c}}\right)^{k-1} +n_{2k} \right)
        \ge \frac{1}{c}\left( \frac{c+1}{c} \right)^{k-1}$. Hence, $u_{2k+1}=u_{2k-1}+n_{2k}+n_{2k+1}\geq \left( \frac{c+1}{c} \right)^{k-1}+ \frac{1}{c}\left( \frac{c+1}{c} \right)^{k-1}= \left( 1+\frac{1}{c} \right) \left( \frac{c+1}{c} \right)^{k-1} \ge \left( \frac{c+1}{c} \right)^{k}$, which proves the claim.
    \end{proof}
    We shall prove the lower bound assuming $c$ is an odd integer; when $c$ is even we apply the same argument for $c-1$.

    To do so, consider the graph $G = P(G_{1},\ldots, G_{2m^{\prime}-1})$ for some $m^{\prime}\in \mathbb{N}$, which we specify later.
    Firstly, we shall define the sizes of $G_i$ for $i \in \{1,\ldots, 2m'-1\}$.
    To do so, let us define a sequence  $\{n_i\}_{i\in\mathbb{N}}$ where $n_{2i} =\frac{c-1}{2} $ and $n_{2i+1}$ is defined recursively in the following way:

    \begin{equation}
        n_{2i+1}= \left \lceil \frac{2}{c-1} \cdot \sum_{j=1}^{2i} n_j \right \rceil \leq \frac{2}{c-1}\sum_{j=1}^{2i} n_j +1
    \end{equation}

    Let $m$ be the largest integer such that  $\sum_{j=1}^{m} n_j \leq n/2$.
    We let $m'=m$ when $m$ is odd  and $m'=m-1$ when $m$ is even.
    Moreover, let $|G_{m'}|=n-2\sum_{j=1}^{j=m'-1} n_j$ and let $|G_i|=n_i$ for $1\leq i < m'$ and $|G_{m'+j}|= |G_{m'-j}|$ for $ j \in \{1,\ldots, m'-1\}$.

    For all $i\in \{1,\ldots 2m'-1\}$ let $G_i = S_{n_i}$ be a star on $n_i$ vertices.
    It is easy to check that the graph $P_{2m'-1}(G_1,\ldots,G_{2m'-1}) $ is path-pairable by Lemma \ref{blown-up}.
    It has diameter at least $2m-4$ and $m \geq \log_{\frac{c+1}{c-1}}(n)(1+o(1))$.
    Again an easy verification shows that the graph $G$ is $c$-degenerate.

\section{Final remarks and open problems}\label{sec:final}

    We obtained tight bounds on the parameter $d(n, \mathcal{G}_{m})$ when $(2+\epsilon)n \leq m \leq  \frac{1}{4} n^{3/2}$, for any fixed $\epsilon >0$.
    It is an interesting open problem to investigate what happens when the number of edges in a path-pairable graph on $n$ vertices is around $2n$. We ask the following:
    \begin{question}
        Is there a function $f$ such that for every $\epsilon>0$ and for every path-pairable graph $G$ on $n$ vertices with at most $(2-\epsilon)n$ edges, the diameter of $G$ is bounded by $f(\epsilon)$?
    \end{question}

    Another line of research concerns determining the behaviour of $d(n, \mathcal{P})$, where $\mathcal{P}$ is the family of planar graphs. Since planar graphs are $5$-degenerate, it follows from Theorem~\ref{diam_cdeg} that the diameter of a path-pairable planar graph on $n$ vertices cannot be larger than $c \log{n}$.
    This fact makes us wonder whether there are path-pairable planar graphs with unbounded diameter.
    \begin{question}
        Is there a family of path-pairable planar graphs with arbitrarily large diameter?
    \end{question}

    The graph constructed in the proof of the lower bound in Theorem~\ref{diam_cdeg} when $c=5$ is not planar since it contains a copy of $K_{3,3}$. Therefore, it cannot be used to show that the diameter of a path-pairable planar graph can be arbitrarily large (note, however, that this graph does not contain a $K_7$-minor nor a $K_{6,6}$-minor). We end by remarking that we were able to construct an infinite family of path-pairable planar graphs with diameter $6$, but not larger.

    \begin{comment}
    \begin{thm} \label{diameter_thm}
        For every even $n\geq 6$ there is a path-pairable planar graph on $n$ vertices and with diameter equal $6$.
    \end{thm}
    \begin{proof}
        \textbf{There are few errors here.
                First, the diameter is $5$.
                When $k_{1} = 1$ the graph seems not to be path-pairable.
                Also, when applying the induction we have to make sure that $A_{3}$ doesn't have too few vertices\dots}
        Consider the following planar graph on $n=k_1+k_2+6$ vertices, where $k_1,k_2\geq 1$ and  $k_1+k_2$ is even.
        Partition the vertex set of $G$ into $7$ non-empty subsets $A_1, A_2,A_3,A_4,A_5,A_6$ and $A_7$, namely $A_1=\{a_1\}$,$A_2=\{a_2\}$, $A_3=\{v_1,v_2,...,v_{k_1}\}$, $A_4=\{a_3,a_4\}$ and $A_5=\{w_1,w_2,...,w_{k_2}\}$, $A_6=\{a_5\}$ and $A_7=\{a_6\}$.
        Connect $a_1$ to $a_2$ and $a_2$ to every vertex in $A_3$. Let $a_3$ be joined to every vertex in $A_3\cup A_5$ and $a_4$ to the set $\{v_1,v_k,a_3,w_1,w_{k_2}\}$. Symmetrically, let $a_5$ be connected to every vertex in $A_4$ and finally let $a_6$ be connected to $a_5$. We also add the edges of respective paths inside $A_3, A_4,A_5$, that is, we join consecutive vertices of the sequences $(v_1, v_2,...,v_{k_1})$, $(a_3,a_4)$, and  $(w_1,w_2,...,w_{k_2})$.

    Clearly $d(a_1,a_6)=6$, so the diameter of $G$ is $6$. It is also easy to see that the graph is planar. We need to show it is path-pairable. The statement is fairly obvious for small values of $n$; we leave the verification of these cases to the reader. On the other hand, if $k_1+k_2>6$ (i.e. $n>12$), then every pairing $\mathcal{P}$ of the vertices contains at least one pair of terminals $(u,v)$ such that their corresponding vertices lie in $A_3\cup A_5$. We can join this pair by a path of length 2 through the vertex $a_3$ and complete the pairing process via induction that completes our proof.
    \end{proof}
\end{comment}

\begin{comment}
-------------------------------------Need to re-write this proof------------------\\

For every pair between vertices in $\{a_3\}\cup A_2\cup A_3$ use the vertex $a_2$ to join them. If $(a_3,a_2) \in \mathcal{P}$ then just join them via the edge between them. Also if whenever $a_1$ is paired with some vertex $v_t$ in $A_2$, use the edge $(a_1,v_t)$ and similarly if $a_4$ is paired with a vertex in $A_4$.

We split our analysis into three cases, regarding how $\mathcal{P}$ pairs the vertices in  ${a_1,a_2,a_3,a_4}$.
\begin{itemize}
\item[i)]$(a_1,a_4)\in \mathcal{P}$; \\
To join $(a_1,a_4)$ use the path $(a_1,v_1,a_3,w_1,a_4)$.
Then either $a_3$ is paired with $a_2$ (which we can solve) or $a_3$ is paired with something in $A_2\cup A_3$, say $v_i$, then use the path $a_3,v_k,v_{k-1},...,v_i$.

\item[ii)] $(a_1,a_2) \in \mathcal{P}$; (when $(a_4,a_2) \in \mathcal{P}$, it is symmetric)\\
Firstly assume $a_3$ and $a_4$ are paired with vertices $v_{j_1},v_{j_2} \in A_2$ with $j_1\leq j_2$ (the other case is symmetric), respectively. Then use the path $(a_1,v_j,a_2)$ to join $a_1$ to $a_2$ and the path $(a_3,v_1,v_2,...,v_{j_1})$ to join $a_3$ and the path $(a_4,w_1,a_3,v_k,v_{k-1},...,v_{j_2})$. The case when $a_3$ is paired with a vertex $w_j \in A_4$ the argument works. If $a_3$ is paired with $a_4$ then use the path $(a_1,v_1,a_3,a_2)$ to join $a_1$ to $a_2$ and the path $(a_3,w_1,a_4)$. When $a_3$ is paired with some vertex $w_l \in A_4$ and $a_4$ is paired with some vertex $v_j \in A_2$ then, as before, use the path $(a_1,v_j,a_2)$ to join $a_1$ to $a_2$, use the path $(a_3,w_1,w_2,...,w_l)$ to join $a_3$ to $w_l$ and the path $(a_4,w_k,a_3,v_1,v_2,...,v_j)$.

\item[iii)] $a_1$ and $a_4$ are paired with some vertex $w_j \in A_4$ and $ v_j \in A_2$, respectively.
Then use the paths $(a_1,v_1,a_3,w_1,a_2,w_j)$ and $(a_2,w_k,a_3,v_k,a_1,w_j)$ to join $a_1$ to $w_j$ and $a_4$ to $v_j$, respectively. Now if $a_3$ is paired $w_l$ (or $v_l$) then join them via $(a_3,a_2,w_l)$ (or $(a_3,a_2,v_l)$).

\end{itemize}

Our case analysis exhausted all possibilities,  so we proved our graph is path-pairable.
\end{comment}

\bibliographystyle{acm}


\section{Introduction}\label{sec:intro}

In two recent studies the subject of {\em semi-inclusive} charge-changing neutrino (CC$\nu$) reactions with nuclei \cite{semi} and application to the special case of deuterium \cite{deut} were presented. Analogous to the semi-inclusive reaction $(e,e'x)$ where one assumes that the scattered electron and some particle $x$ are detected in coincidence, in the weak interaction case one considers reactions of the type $(\nu_\ell, \ell^- x)$ and $(\bar{\nu}_\ell, \ell^+ x)$. These involve incident neutrinos or anti-neutrinos of specific flavor ($\ell = e$, $\mu$ or $\tau$) together with coincident detection of the corresponding charged leptons and some particle $x$. In the present work we shall focus on nucleons ejected from the nucleus, and hence $x=N$, where $N=p$ or $n$. Note that in the nuclear case the ``natural'' type of nucleon may not be the one of interest, whereas for a single-nucleon target and when no other particle is produced other that the final-state nucleon ({\it i.e.,} no pion production, kaon production, {\it etc.}) charge conservation forces the final-state nucleon to be only of one type. Namely, in this latter case one only has reactions of the type $\nu_\ell + n\rightarrow \ell^- + p$ and $\bar{\nu}_\ell + p \rightarrow  \ell^+ + n$. In the present work we shall specialize still further and consider only incident neutrinos, final-state negative leptons and emission of protons ($x=p$). For completeness in defining the terminology commonly being used, we note that reactions where only the final-state leptons are detected, such as $(e,e')$, $(\nu_\ell, \ell^-)$ or $(\bar{\nu}_\ell, \ell^+)$, are called {\em inclusive} reactions.

As has become quite clear in recent years, the typical high-energy neutrino beams used in studies of neutrino oscillations, typically at neutrino energies of around a GeV to tens of GeV, $E_\nu$, have rather broad spreads in energy, which introduces model dependence in the specification of the distance over energy ratio $L/E_\nu$ that enters in the standard oscillation expressions. However, as discussed in \cite{deut}, deuterium provides, at least in principle, an exception to the typical case of heavier nuclei. Namely, once so-called ``no-pion'' events are isolated, all that can occur for the case of incident neutrinos is the reaction $\nu_\ell + {}^2$H$ \rightarrow \ell^- + p + p$. Upon detecting two of the three particles in the final state and knowing the direction of the incident neutrino the neutrino's energy can be reconstructed using nothing beyond the kinematics of the reaction. In \cite{deut} a specific relativistic model for the deuterium ground state and final $NN$ scattering state was employed to model this reaction; in the present study we use the same model for the $A=2$ states and the required electroweak current matrix elements.

This said, there are still practical issues of which to be aware. Namely, making very large target/detectors of hydrogen or deuterium is problematical because of the safety issues involved and the difficulty of providing very large amounts of these nuclei. Using target/detectors of something involving large fractions of deuterium together with other light nuclei, such as heavy water (D$_2$O) or deuterated methane (CD$_4$), might alleviate the safety issue and could provide practical amounts of deuterium, although having other nuclei such as oxygen or carbon present potentially can bring in new considerations. In this study we have focused on a specific case to explore how such mixed nuclear cases behave; specifically, here we consider the case of ${}^2$H$_2 {}^{16}$O. The goal is to take what we have already done for deuterium, add model results for CC$\nu$ semi-inclusive reactions on ${}^{16}$O and determine the degree to which events from the two nuclear species can be separated. One expects the deuteron events to be very peaked and to occur in a different part of the kinematic space involved from the oxygen events, and, as well, the oxygen events to be much more spread in the appropriate kinematic variables so that the ratio of deuterium to oxygen becomes quite favorable. Indeed, we shall show that this is the case.

We will be drawing on our previous study of semi-inclusive CC$\nu$ reactions in \cite{semi} to highlight and quantify the differences of deuterium and a more typical nucleus such as oxygen (here the nucleus could be chosen to be carbon or any other relatively light nucleus). As a specific model for the oxygen case we employ the spectral function approach of \cite{Benhar:1994hw,Benhar:2005dj}. The goal will be to optimize the selection of semi-inclusive events for the case of deuterium and then see what emerges for the ``background'' from the oxygen events.

The paper is organized as follows. In Sect.~\ref{sec:semi} we summarize the necessary formalism for the semi-inclusive CC$\nu$ reaction, taking as a basis the previous study reported in \cite{semi}, and include some of the relevant formalism needed to inter-relate the experimental ``lab frame'' to the so-called ``$q$-frame''. In Sect.~\ref{sec:deut} we specialize the results of the previous section to the case of deuterium to make very clear the advantage provided by this particular nucleus. We do not repeat the discussion of the formalism for the dynamics and currents involved in the deuterium case, since these have been reported in \cite{deut}. For the case of oxygen we present the required formalism in the context of the spectral function in Sect.~\ref{sec:spect}, following which we employ the two models discussed above to obtain typical results for heavy water and present these in Sect.~\ref{sec:results}. In Sect.~\ref{sec:concl} we offer our conclusions, while in the Appendix we collect expressions for the off-shell single-nucleon response functions employed for the oxygen spectral function case.

\section{Semi-Inclusive Cross Section}\label{sec:semi}

Semi-inclusive CC$\nu$ scattering is represented by the Feynman diagram shown in Fig.~\ref{fig:semi}, where $Q^{\mu}$ is the four-momentum of the W-boson,
\begin{figure}
	\centerline{\includegraphics[height=2in]{semi.pdf}}
	\caption{(color online) Feynman diagram for semi-inclusive charge-changing neutrino reactions involving a target nucleus with nucleon number $A$ with emission and detection of a nucleon with four-momentum $P_N^{\mu}$ together with detection of a final-state charged lepton with four-momentum $K'^{\mu}$ }
	\label{fig:semi}
\end{figure}
\begin{equation}
K^\mu=(\varepsilon,\bm{k})
\end{equation}
is the incident lepton four-momentum and
\begin{equation}
{K'}^\mu=(\varepsilon',\bm{k}')
\end{equation}
is the four-momentum of the lepton in the final state,
where $\varepsilon=\sqrt{k^2+m^2}$ and $\varepsilon'=\sqrt{{k'}^2+{m'}^2}$ are the energies of the incident and final leptons with respective masses $m$ and $m'$. Then the four-momentum transfer is
\begin{equation}
Q^\mu=K^\mu-{K'}^\mu=(\varepsilon-\varepsilon',\bm{k}-\bm{k}')=(\omega,\bm{q})\,.
\end{equation}
The four-momentum of the target nucleus with nucleon number $A$ can be written in its rest frame as
\begin{equation}
P_A^\mu=(M_A,\bm{0})\,.
\end{equation}
The four-momentum of the detected nucleon is
\begin{equation}
P_N^\mu=(E_N,\bm{p}_N)\,,
\end{equation}
where  $m_N$ is the nucleon mass, $E_N \equiv \sqrt{\bm{p}_N^2+m_N^2}$ and
the four-momentum of the residual $A-1$ system is
\begin{equation}
P_{A-1}^\mu=(\sqrt{\bm{p}_m^2+W_{A-1}^2},\bm{p}_m)
\end{equation}
with the invariant mass $W_{A-1}$.

The energy of an incoming neutrino can be determined by measuring the three-momenta of the outgoing charged lepton, which we take to be a muon in what follows (although clearly the $e$ or $\tau$ cases can also be considered), and nucleon, corresponding to kinematics B of \cite{semi}.  In this case the four-fold differential cross section in the laboratory frame is then
\begin{align}
\frac{d\sigma}{dk'd\Omega_{k'}dp_Nd\Omega_N^L}=&\frac{G^2\cos^2\theta_cm_N{k'}^2p_N^2W_{A-1}}{2(2\pi)^5k\varepsilon'E_N}\int \frac{d^3p_m}{\sqrt{p_m^2+W_{A-1}^2}}\eta_{\mu\nu}W^{\mu\nu}\nonumber\\
&\times\delta^4(K+P_A-K'-P_N-P_{A-1})\nonumber\\
=&\frac{G^2\cos^2\theta_cm_N{k'}^2p_N^2W_{A-1}}{2(2\pi)^5k\varepsilon'E_N}\int \frac{d^3p_m}{\sqrt{p_m^2+W_{A-1}^2}}\eta_{\mu\nu}W^{\mu\nu}\nonumber\\
&\times\delta(\varepsilon+M_A-\varepsilon'-E_N-\sqrt{p_m^2+W_{A-1}^2})\delta(\bm{k}-\bm{k}'-\bm{p}_N-\bm{p}_m) \, ,\label{eq:dsig0}
\end{align}
where $G$ is the weak interaction coupling constant and $\theta_c$ is the Cabibbo mixing angle. Defining
\begin{equation}
E_B=\varepsilon'+E_N-M_A
\end{equation}
and
\begin{equation}
\bm{p}_B=\bm{k}'+\bm{p}_N \, ,
\end{equation}
the cross section becomes
\begin{align}
\frac{d\sigma}{dk'd\Omega_{k'}dp_Nd\Omega_N^L}
=&\frac{G^2\cos^2\theta_cm_N{k'}^2p_N^2W_{A-1}}{2(2\pi)^5k\varepsilon'E_N}\int \frac{d^3p_m}{\sqrt{p_m^2+W_{A-1}^2}}\eta_{\mu\nu}W^{\mu\nu}\nonumber\\
&\times\delta(\varepsilon-E_B-\sqrt{p_m^2+W_{A-1}^2})\delta(\bm{k}-\bm{p}_B+\bm{p}_m)\nonumber\\
=&\frac{G^2\cos^2\theta_cm_N{k'}^2p_N^2W_{A-1}}{2(2\pi)^5k\varepsilon'E_N} \frac{1}{\sqrt{(\bm{p}_B-\bm{k})^2+W_{A-1}^2}}\eta_{\mu\nu}W^{\mu\nu}\nonumber\\
&\times\delta(\varepsilon-E_B-\sqrt{(\bm{p}_B-\bm{k})^2+W_{A-1}^2}) \, .
\end{align}
Using the remaining $\delta$-function, the incident neutrino momentum and energy are given by
\begin{equation}
k_0=\frac{1}{a_B}\left(X_Bp_B\cos\theta_B+E_B\sqrt{X_B^2+m^2a_B}\right) \label{Xk0}
\end{equation}
and
\begin{equation}
\varepsilon_0=\frac{1}{a_B}\left(E_BX_B+p_B\cos\theta_B\sqrt{X_B^2+m^2a_B}\right)\,, \label{Xe0}
\end{equation}
where
\begin{equation}
X_B=\frac{1}{2}\left(p_B^2-E_B^2+W_{A-1}^2-m^2\right)
\end{equation}
and
\begin{equation}
a_B=p_B^2\cos^2\theta_B-E_B^2\,.
\end{equation}
The energy-conserving $\delta$-function can be rewritten as
\begin{equation}
\delta(\varepsilon-E_B-\sqrt{(\bm{p}_B-\bm{k})^2+W_{A-1}^2})=\frac{\varepsilon_0\sqrt{(\bm{p}_B-\bm{k})^2+W_{A-1}^2}}{\sqrt{X_B^2+m^2a_B}}\delta(k-k_0)\,.
\end{equation}
The cross section then becomes
\begin{equation}
\frac{d\sigma}{dk'd\Omega_{k'}dp_N^2d\Omega_N^L}
=\frac{G^2\cos^2\theta_cm_N{k'}^2\varepsilon\, p_N^2W_{A-1}v_0}{2(2\pi)^5k\varepsilon'E_N\sqrt{X_B^2+m^2a_B}} \mathcal{F}^2_\chi
\delta(k-k_0) \, ,\label{eq:sigma_lab}
\end{equation}
where $\mathcal{F}^2_\chi \equiv \eta_{\mu\nu}W^{\mu\nu}/v_0$ with $v_0 \equiv (\varepsilon + \varepsilon')^2 - q^2$. The resulting response may be written
\begin{align}
{\cal F}_\chi^2=&\hat{V}_{CC}(w_{CC}^{VV(I)}+w_{CC}^{AA(I)})
+2\hat{V}_{CL}(w_{CL}^{VV(I)}+w_{CL}^{AA(I)})
+\hat{V}_{LL}(w_{LL}^{VV(I)}+w_{LL}^{AA(I)})\nonumber\\
&+\hat{V}_T(w_{T}^{VV(I)}+w_{T}^{AA(I)})\nonumber\\
&+\hat{V}_{TT}\left[(w_{TT}^{VV(I)}+w_{TT}^{AA(I)})\cos 2\phi_N+(w_{TT}^{VV(II)}+w_{TT}^{AA(II)})\sin 2\phi_N\right]\nonumber\\
&+\hat{V}_{TC}\left[(w_{TC}^{VV(I)}+w_{TC}^{AA(I)})\cos\phi_N
+(w_{TC}^{VV(II)}+w_{TC}^{AA(II)})\sin\phi_N)\right]\nonumber\\
&+\hat{V}_{TL}\left[(w_{TL}^{VV(I)}+w_{TL}^{AA(I)})\cos\phi_N
+(w_{TL}^{VV(II)}+w_{TL}^{AA(II)})\sin\phi_N\right]
\nonumber\\
&+\chi\left[\hat{V}_{T'}w^{VA(I)}_{T'}+\hat{V}_{TC'}(w^{VA(I)}_{TC'}\sin\phi_N
+w_{TC'}^{VA(II)}\cos\phi_N)\right.\nonumber\\
&\left.+\hat{V}_{TL'}(w^{VA(I)}_{TL'}\sin\phi_N
+w^{VA(II)}_{TL'}\cos\phi_N)\right]\label{eq:cal_F2_chi}
\end{align}
with
\begin{equation}
\chi=\left\{
\begin{array}{rl}
-1 &\mathrm{for\ neutrinos}\\
1  &\mathrm{for\ antineutrinos}
\end{array}\right.\,.
\end{equation}
The kinematic functions $V_a$ and response functions $w^i_j$  are as defined in \cite{semi} with the explicit dependence on azimuthal angle $\phi_N$ defined in the $q$-fixed frame. Response functions labeled by the superscript $(II)$ vanish in the plane-wave limit.

If the neutrino momentum distribution normalized to unity is designated as $P(k)$,
the cross section weighted by this distribution is then given by
\begin{align}
\left<\frac{d\sigma}{dk'd\Omega_{k'}dp_Nd\Omega_N^L}\right>
=&\int_0^\infty dk \frac{G^2\cos^2\theta_cm_N{k'}^2\varepsilon\, p_N^2W_{A-1}}{2(2\pi)^5k\varepsilon'E_N\sqrt{X_B^2+m^2a_B}} v_0 \mathcal{F}^2_\chi
\delta(k-k_0)P(k)\nonumber\\
=&\frac{G^2\cos^2\theta_cm_N{k'}^2\varepsilon_0\, p_N^2W_{A-1}v_0}{2(2\pi)^5k_0\varepsilon'E_N\sqrt{X_B^2+m^2a_B}} \mathcal{F}^2_\chi
P(k_0)\, .\label{eq:sigma_lab}
\end{align}

Next it is useful to inter-relate the variables in the laboratory frame shown in Fig.~\ref{fig:kinelab} to those in the so-called $q$-system shown in Fig.~\ref{fig:kineq}.
\begin{figure}
	\centerline{\includegraphics[height=3in]{lab_system_color.pdf}}
	\caption{(color online) Semi-inclusive $(\nu_{\mu},\mu^-p)$ CC$\nu$ reaction in the laboratory frame. Here the incident neutrino with three-momentum ${\bm k}$ is along the $3'$ direction, the neutrino and the final-state muon with three-momentum ${\bm k}'$  lie in the $1'$--$3'$ plane and the normal to the plane defines the $2'$ direction. The outgoing nucleon (here a proton) has three-momentum ${\bm p}_N$ and is traveling in the direction characterized by polar angle $\theta_N^L$ and azimuthal angle $\phi_N^L$ in the lab system, as shown.}
	\label{fig:kinelab}
\end{figure}
\begin{figure}
	\centerline{\includegraphics[height=3in]{q_system_color.pdf}}
	\caption{(color online) Semi-inclusive $(\nu_{\mu},\mu^-p)$ CC$\nu$ reaction in the $q$-system. Here the three-momentum transfer ${\bm q}$ defines the $3$ direction, the neutrino and the final-state muon lie in the $1$--$3$ plane and the normal to the plane defines the $2$ direction. The outgoing nucleon (here a proton) has three-momentum ${\bm p}_N$ and is traveling in the direction characterized by polar angle $\theta_N$ and azimuthal angle $\phi_N$ in the $q$-system, as shown.}
	\label{fig:kineq}
\end{figure}
We have the following identities relating the angles in the two systems:
\begin{align}
\cos\theta_N=&\cos\theta_N^L \cos\theta_{q}-\cos\phi^L_N \sin\theta_N^L \sin\theta_{q}\label{eq:cos_theta_N}\\
\sin\theta_N=&\sqrt{1-\cos^2\theta_N}\label{eq:sin_theta_N}\\
\cos\phi_N=&\frac{\cos\phi^L_N\sin\theta_N^L\cos\theta_{q}+\cos\theta_N^L\sin\theta_{q}}
{\sin\theta_N}\label{eq:cos_phi_N}\\
\sin\phi_N=&\frac{\sin\phi_N^L\sin\theta_N^L}{\sin\theta_N}\label{eq:sin_phi_N}
 \end{align}
and the inverse relations are given by
\begin{align}
\cos\theta_N^L=&\cos\theta_N \cos\theta_{q}+\cos\phi_N \sin\theta_N \sin\theta_{q}\label{eq:cos_theta_NL}\\
\sin\theta_N^L=&\sqrt{1-\cos^2\theta_N^L}\label{eq:sin_theta_NL}\\
\cos\phi^L_N=&\frac{\cos\phi_N\sin\theta_N\cos\theta_{q}+\cos\theta_N\sin\theta_{q}}{\sin\theta_N^L}\label{eq:cos_phi_NL}\\
\sin\phi^L_N=&\frac{\sin\phi_N\sin\theta_N}{\sin\theta_N^L}\label{eq:sin_phi_NL}\,.
\end{align}
Note that as the neutrino energy changes, even for fixed directions for the outgoing muon and nucleon, the direction of the momentum transfer also changes, and, therefore, through these relationships, the polar and azimuthal angles in the $q$-system also change. The lab system is relevant when experimental issues are being considered; however, the $q$-system with the $3$-direction along the momentum of the exchanged boson has special symmetries that are masked in the lab system.

Here, we want to express the cross section in lab frame. This can be done by using Eqs.~(\ref{eq:cos_phi_N}) and (\ref{eq:sin_phi_N}) to replace the azimuthal angular dependence in Eq.~(\ref{eq:cal_F2_chi}) and by defining the three-momenta
\begin{equation}
\bm{k}=k\hat{\bm{u}}_{3'}\,,
\end{equation}
\begin{equation}
\bm{k}'=k'\left(\sin\theta_l\hat{\bm{u}}_{1'}+\cos\theta_l\hat{\bm{u}}_{3'}\right)
\end{equation}
and
\begin{equation}
\bm{p}_N=p_N\left(\cos\phi_N^L\sin\theta_N^L\hat{\bm{u}}_{1'}+\sin\phi_N^L\sin\theta_N^L\hat{\bm{u}}_{2'}+\cos\theta_N^L\hat{\bm{u}}_{3'}\right)\,,
\end{equation}
where $\theta_l$ is the lepton scattering angle. The unit vectors in the lab frame are $\hat{\bm{u}}_{1'}$, $\hat{\bm{u}}_{2'}$ and $\hat{\bm{u}}_{3'}$, as shown in Fig.~\ref{fig:kinelab}. The three-momentum transfer is
\begin{equation}
\bm{q}=\bm{k}-\bm{k}'
\end{equation}
and its square is
\begin{equation}
q^2=k^2+{k'}^2-2kk'\cos\theta_l.
\end{equation}
The angle between $\bm{k}$ and $\bm{q}$ can be obtained from
\begin{equation}
\bm{k}\cdot\bm{q}=kq\cos\theta_{q}=\bm{k}\cdot\bm{k}-\bm{k}\cdot\bm{k}'=k^2-kk'\cos\theta_l \, ,
\end{equation}
which can be solved to yield
\begin{equation}
\cos\theta_{q}=\frac{k-k'\cos\theta_l}{q}\,.
\end{equation}
Similarly we can use
\begin{equation}
\bm{k}\cdot\bm{p}_B=kp_B\cos\theta_B=\bm{k}\cdot(\bm{k}'+\bm{p}_N)=(kk'\cos\theta_l+kp_N\cos\theta_N^L)
\end{equation}
to obtain
\begin{equation}
p_B\cos\theta_B=k'\cos\theta_l+p_N\cos\theta_N^L\,.
\end{equation}
The remaining expressions needed to obtain the cross section in the lab frame are
\begin{equation}
p_B^2={k'}^2+p_N^2+2\bm{k}'\cdot\bm{p}_N={k'}^2+p_N^2+2k'p_N\left(\cos\phi_N^L\sin\theta_N^L\sin\theta_l+\cos\theta_N^L\cos\theta_l\right)
\end{equation}
and
\begin{equation}
p_m^2=k^2+p_B^2-2\bm{k}\cdot\bm{p}_B=k^2+p_B^2-2kp_B\cos\theta_B\,.
\end{equation}

As noted in \cite{deut}, Eq.~(\ref{eq:sigma_lab}) applies also to the case of exclusive scattering from the deuteron by making the substitutions $M_A\rightarrow M_d$ and $W_{A-1}\rightarrow m_N$.

\section{Deuterium}\label{sec:deut}

For the purpose of determining whether the deuterium cross section can be separated from that of oxygen, we wish to choose kinematics which are optimal for the deuteron and then use the values $\bm{k}'$ and $\bm{p}_N$ determined from the deuteron in calculating the semi-inclusive scattering from oxygen.

To obtain the optimal kinematics for scattering from the deuteron we start with Mandelstam $s$ for the virtual $W$ and the deuteron. This is given by
\begin{equation}
s=(P_d+Q)^2=(M_d+\omega)^2-q^2\,.
\end{equation}
The scaling variables \cite{Day:1990mf}
\begin{equation}
y=\frac{(M_d+\omega)\sqrt{s(s-4m_N^2)}}{2s}-\frac{q}{2}
\end{equation}
and
\begin{equation}
Y=y+q
\end{equation}
can be used to obtain limiting values for the magnitude of the missing momentum $p_m$ as
\begin{equation}
|y|\leq p_m\leq Y\,.
\end{equation}

Since the deuteron cross section behaves roughly as the deuteron momentum distribution $n(p_m)$, which peaks at $p_m=0$, the cross section can be optimized by choosing kinematics such that $y=0$. Solving this for the incident neutrino energy yields
\begin{align}
\varepsilon_0=&\frac{1}{2
	\left[(\varepsilon'-M_d+m_N)^2-{k'}^2 \cos^2(\theta_l)\right]}\left\{\zeta k' \cos(\theta_l)\left[ -2
{\varepsilon'}^2 \left(m^2-2
(M_d-m_N)^2\right)\right.\right.\nonumber\\
&\left.\left.-4\varepsilon' (M_d-m_N)\left(-m^2+M_d^2-2 M_d m_N+{m'}^2\right)+2
{k'}^2 m^2 \cos (2\theta_l)+m^4-2 m^2 M_d^2\right.\right.\nonumber\\
&\left.\left.+4 m^2 M_d
m_N-4 m^2
m_N^2+M_d^4-4 M_d^3
m_N+4 M_d^2 m_N^2+2
M_d^2 {m'}^2-4 M_d
m_N
{m'}^2+{m'}^4\right]^{\frac{1}{2}}\right.\nonumber\\
&\left.-2
{\varepsilon'}^2 M_d+2
{\varepsilon'}^2 m_N+\varepsilon'
m^2+3 \varepsilon' M_d^2-6
\varepsilon' M_d m_N+2
\varepsilon' m_N^2+\varepsilon'
{m'}^2-m^2 M_d+m^2
m_N\right.\nonumber\\
&\left.-M_d^3+3 M_d^2
m_N-2 M_d
m_N^2-M_d
{m'}^2+m_N {m'}^2\right\} \, ,
\end{align}
where
\begin{equation}
\zeta=\left\{
\begin{array}{rr}
-1 & \mathrm{for}\ \theta_l\leq \frac{\pi}{2}\\
1 &  \mathrm{for}\ \theta_l> \frac{\pi}{2}
\end{array}\right.
\end{equation}

Four-momentum conservation for the deuteron requires that
\begin{align}
0=&M_d+\omega-\sqrt{p_N^2+m_N^2}-\sqrt{p_m^2+m_N^2}\label{eqn:deuteron_energy}\\
\bm{0}=&\bm{q}-\bm{p}_N+\bm{p}_m \, .\label{eqn:deuteron_momentum}
\end{align}
Using Eq.~(\ref{eqn:deuteron_energy}) the square of the detected nucleon momentum is
\begin{equation}
p_N^2=\left(M_d+\omega-\sqrt{p_m^2+m_N^2}\right)^2-m_N^2\,.\label{eq:p_N_squared}
\end{equation}
Using Eq.~(\ref{eqn:deuteron_momentum}),
\begin{equation}
\bm{p}_m=\bm{q}-\bm{p}_N\,,
\end{equation}
yields
\begin{equation}
p_{m}^2=q^2+p_N^2-2p_Nq\cos\theta_N\,.
\end{equation}
Solving this for $\cos\theta_N$ gives
\begin{equation}
\cos\theta_N=\frac{q^2+p_N^2-p_m^2}{2p_Nq}\,.\label{eq:cos_theta_N_2}
\end{equation}
By specifying $k'$, $\theta_l$, $\phi_N$ and using Eqs.~(\ref{eq:p_N_squared}) and (\ref{eq:cos_theta_N_2}), the lab frame angles are then given by Eqs.~(\ref{eq:cos_theta_NL}), (\ref{eq:sin_theta_NL}), (\ref{eq:cos_phi_NL}) and (\ref{eq:sin_phi_NL}). This provides a complete set of input variables to evaluate the deuteron and oxygen cross sections. Note that Eq.~(\ref{eq:cos_theta_N_2}) results in a correlation of the values of $p_N$ and $\cos\theta^L_N$.

All of the conditions required by these constrained kinematics can only be satisfied by limiting
\begin{equation}
0\leq\theta_l\leq\left\{
\begin{array}{ll}
\cos^{-1}\left(\frac{\varepsilon'-M_d+m_N}{k'}\right)& \mathrm{for}\ -k'<\varepsilon'-M_d+m_N\leq k'\\
\pi & \mathrm{for}\ \varepsilon'-M_d+m_N\leq -k'
\end{array}
\right.\,.\label{eq:theta_l_max}
\end{equation}

The deuterium matrix elements needed to construct the cross section are described in \cite{deut}.

\section{Spectral Function}\label{sec:spect}

\begin{figure}
	\centerline{\includegraphics[height=2in]{PWIA_semi.pdf}}
	\caption{(color online) Feynman diagram for a factorized approximation to the semi-inclusive charge-changing neutrino reaction illustrated for the general case in Fig.~1. }
	\label{fig:PWIA_semi}
\end{figure}

For this work we estimate the oxygen semi-inclusive cross sections using a factorized spectral function model. The current matrix element for this model can be written as
\begin{equation}
\left<\bm{p}_N,s_N;\bm{P}_{A-1},s_{A-1}\left|J^\mu(q)\right|\bm{P}_A,s_A\right>=\bar{u}(\bm{p}_N,s_N)_a J^\mu(q)_{ab}\Psi(P_{A-1},s_{A-1};P_A,s_A)_{bc} \, ,
\end{equation}
where $s_N$, $s_A$ and $s_{A-1}$ are the spins of the ejected proton, target nucleus and residual system, respectively, and $\Psi(P_{A-1},s_{A-1};P_A,s_A)$ represents a three-point function with the $A$ line truncated. The Dirac indices are explicitly indicated. The nuclear response tensor is then given by
\begin{align}
W^{\mu\nu}=&\sum_{s_N}\sum_{s_A}\sum_{s_{A-1}}
\bar{u}(\bm{p}_N,s_N)_a J^\nu(q)_{ab}\Psi(P_{A-1},s_{A-1};P_A,s_A)_{bc}\nonumber\\
&\times\bar{\Psi}(P_{A-1},s_{A-1};P_A,s_A)_{cd} J^\mu(-q)_{de}u(\bm{p}_N,s_N)_e\nonumber\\
=&\sum_{s_N}
\bar{u}(\bm{p}_N,s_N)_a J^\nu(q)_{ab}\frac{1}{8\pi}\Lambda^+(\bm{p}_m)_{bd}S(p_m,E_m) J^\mu(-q)_{de}u(\bm{p}_N,s_N)_e\nonumber\\
=&\frac{1}{8\pi} \mathrm{Tr}\left[J^\mu(-q)\Lambda^+(\bm{p}_N)J^\nu(q)\Lambda^+(\bm{p}_m)\right]S(p_m,E_m)\nonumber\\
=&\frac{1}{8\pi} w^{\mu\nu}(P_{A}-P_{A-1},Q)S(p_m,E_m)\,,
\end{align}
where $w^{\mu\nu}(P_{A}-P_{A-1},Q)$ is an off-shell single-nucleon response tensor and $S(p_m,E_m)$ is the spectral function.
The missing energy is approximated by
\begin{equation}
E_m\cong E_s+\mathcal{E} \, ,
\end{equation}
where $E_s$ is the separation energy,
\begin{equation}
\mathcal{E}=\sqrt{p_m^2+W_{A-1}^2}-\sqrt{p_m^2+{W_{A-1}^0}^2}\,,
\end{equation}
and $W_{A-1}^0$ is the invariant mass of the lowest state of the residual system. Energy conservation requires that
\begin{align}
0=&M_A+\omega-\sqrt{p_N^2+m_N^2}-\sqrt{p_m^2+W_{A-1}^2}\nonumber\\
=&M_A+\omega-\sqrt{p_N^2+m_N^2}-\sqrt{p_m^2+W_{A-1}^2}+\sqrt{p_m^2+{W_{A-1}^0}^2}-\sqrt{p_m^2+{W_{A-1}^0}^2}\nonumber\\
=&M_A+\omega-\sqrt{p_N^2+m_N^2}-\mathcal{E}-\sqrt{p_m^2+{W_{A-1}^0}^2}\,.
\end{align}
So $\mathcal{E}$ can also be written as
\begin{equation}
\mathcal{E}=M_A+\omega-\sqrt{p_N^2+m_N^2}-\sqrt{p_m^2+{W_{A-1}^0}^2}\,.
\end{equation}
From momentum conservation $\bm{p}_N=\bm{q}-\bm{p}_m$, and therefore
\begin{equation}
\mathcal{E}=M_A+\omega-\sqrt{(\bm{q}-\bm{p}_m)^2+m_N^2}-\sqrt{p_m^2+{W_{A-1}^0}^2}\,.
\end{equation}
The range of $\mathcal{E}$ is then limited by
\begin{equation}
\mathcal{E}_+\leq\mathcal{E}\leq\mathcal{E}_-\,,
\end{equation}
where
\begin{equation}
\mathcal{E}_-=M_A+\omega-\sqrt{(p_m-q)^2+m_N^2}-\sqrt{p_m^2+{W_{A-1}^0}^2}
\end{equation}
and
\begin{equation}
\mathcal{E}_+=\max(M_A+\omega-\sqrt{(p_m+q)^2+m_N^2}-\sqrt{p_m^2+{W_{A-1}^0}^2},0)\,.
\end{equation}

The normalization of the spectral function $S(p_m,E_m)$ is defined here such that
\begin{equation}
\int_0^\infty dE_m S(p_m,E_m)=n(p_m)
\end{equation}
is the momentum distribution and
\begin{equation}
\frac{1}{(2\pi)^3}\int_0^\infty dp_m p_m^2 n(p_m)=A-Z\,.
\end{equation}

Expressing the four-momentum of the struck nucleon as
\begin{equation}
P_A^\mu-P_{A-1}^\mu=(M_A-\sqrt{p_m^2+W_{A-1}^2},-\bm{p}_m)\,,
\end{equation}
defining
\begin{equation}
\bm{p}=-\bm{p}_m
\end{equation}
and using energy conservation
\begin{equation}
M_A-\sqrt{p^2+W_{A-1}^2}=\sqrt{p_N^2+m_N^2}-\omega\,,
\end{equation}
one has
\begin{align}
P_A^\mu-P_{A-1}^\mu=&(\sqrt{p_N^2+m_N^2}-\omega,\bm{p})\nonumber\\
=&(\sqrt{p_N^2+m_N^2}-\omega-\sqrt{p^2+m_N^2}+\sqrt{p^2+m_N^2},\bm{p})\nonumber\\
=&(\sqrt{p_N^2+m_N^2}-\omega-\sqrt{p^2+m_N^2},\bm{0})+(\sqrt{p^2+m_N^2},\bm{p})\nonumber\\
=&(\delta,\bm{0})+(\sqrt{p^2+m_N^2},\bm{p})=\Delta^\mu+P^\mu \, ,
\end{align}
where
\begin{equation}
P^{\mu}=(\sqrt{p^2+m_N^2},\bm{p})
\end{equation}
is an on-shell four-vector and
\begin{equation}
\Delta^\mu=(\delta,\bm{0})
\end{equation}
is off-shell with
\begin{equation}
\delta=\sqrt{p_N^2+m_N^2}-\sqrt{p^2+m_N^2}-\omega\,.
\end{equation}

The quantity $\mathcal{F}^2_\chi$ in Eq.~(\ref{eq:sigma_lab}) is then given by
\begin{equation}
\mathcal{F}^2_\chi\cong\frac{1}{8\pi}\widetilde{\mathcal{F}}^2_\chi S(p_m,E_m) \, ,
\end{equation}
where
\begin{align}
\widetilde{\mathcal{F}}^2_\chi=&  \widehat{V}_{CC}\left(\widetilde{w}^{VV(I)}_{CC}+\widetilde{w}^{AA(I)}_{CC}\right)+2\widehat{V}_{CL}\left(\widetilde{w}^{VV(I)}_{CL}+\widetilde{w}^{AA(I)}_{CL}\right)+\widehat{V}_{LL}\left(\widetilde{w}^{VV(I)}_{LL}+\widetilde{w}^{AA(I)}_{LL}\right)\nonumber\\
&+\widehat{V}_{T}\left(\widetilde{w}^{VV(I)}_{T}+\widetilde{w}^{AA(I)}_{T}\right)
 +\widehat{V}_{TT}\left(\widetilde{w}^{VV(I)}_{TT}+\widetilde{w}^{AA(I)}_{TT}\right)\cos 2\phi_N  \nonumber\\
 &+\widehat{V}_{TC}\left(\widetilde{w}^{VV(I)}_{TC}+\widetilde{w}^{AA(I)}_{TC}\right)\cos\phi_N+\widehat{V}_{TL}\left(\widetilde{w}^{VV(I)}_{TL}+\widetilde{w}^{AA(I)}_{TL}\right)\cos\phi_N \nonumber\\
&
+\chi \left[ \widehat{V}_{T^{\prime }}\widetilde{w}^{VA(I)}_{T^{\prime }}+\widehat{V}_{TC^{\prime }}\widetilde{w}^{VA(I)}_{TC^{\prime }}\sin\phi_N+\widehat{V}_{TL^{\prime }}\widetilde{w}^{VA(I)}_{TL^{\prime }}\sin\phi_N\right] \, .
\end{align}
The off-shell single-nucleon response functions $\widetilde{w}^i_j$ are listed in the Appendix.

Since the invariant mass of the residual $A-1$ system is not measured, it is necessary that the semi-inclusive cross section be integrated over all possible values of $W_{A-1}$ to give
\begin{align}
\left<\frac{d\sigma}{dk'd\Omega_{k'}^Ldp_Nd\Omega_N^L}\right>
=&\int_{W^0_{A-1}}^{\infty}dW_{A-1}\frac{G^2\cos^2\theta_cm_N{k'}^2\varepsilon_0 p_N^2W_{A-1}v_0}{8(2\pi)^6k_0\varepsilon'E_N\sqrt{X_B^2+m^2a_B}} \widetilde{\mathcal{F}}^2_\chi S(p_m,E_m)
P(k_0) \, ,
\end{align}
where $W^0_{A-1}$ is the lowest possible mass for the residual system which in some cases may not be a bound state. For the specific case considered in the present study this corresponds to the ground-state mass of $^{15}$O.

Note that the integral over the invariant mass requires that $k_0$ and $\varepsilon_0$ in Eqs.~(\ref{Xk0}) and (\ref{Xe0}) must take on a range of values rather than being fixed as in the case of the deuteron.

\section{Results}\label{sec:results}

For the purposes of this paper, we have chosen to weight the cross sections using the flux momentum distribution for the DUNE experiment \cite{Alion:2016uaj} normalized to unit area, represented $P(k)$ as shown in Fig. \ref{fig:DUNE_flux}. The spectral function for oxygen is from \cite{Benhar:1994hw,Benhar:2005dj} renormalized according to the units and conventions used here.
\begin{figure}
	\centerline{\includegraphics[height=3in]{DUNE_probability.pdf}}	\caption{(color online) DUNE flux converted to a probability density as a function of $k$ in GeV.  }\label{fig:DUNE_flux}
\end{figure}

Figures \ref{fig:k_prime_1}, \ref{fig:k_prime_2} and \ref{fig:k_prime_3} show cross sections for $^2\mathrm{H}$ and $^{16}\mathrm{O}$ for $k'=1,2\ \mathrm{and}\ 3\,\mathrm{GeV}$ respectively, as a function of the polar angle of the detected proton $\theta_N^L$, for a variety of lepton scattering angles subject to the $y=0$ constraint (see Sect.~\ref{sec:deut}) and the restriction required by Eq.~(\ref{eq:theta_l_max}). For each scattering angle, the values of the incident neutrino energy $k$ and the momentum transfer $q$ are given for the deuteron. For oxygen these quantities cover a range of values due to their dependence on the invariant mass $W_{A-1}$ which is integrated over to the  semi-inclusive cross section. For completeness, each figure contains the momentum of the detected proton $p_N$ as a function of $\theta_N^L$ with values given by the right-hand scale. Since for $^2\mathrm{H}_2{}^{16}\mathrm{O}$ there are two deuterium nuclei for each oxygen nucleus, the cross sections for deuterium are multiplied by a factor of 2.
\begin{figure}
	\centerline{\includegraphics[height=2in]{sigma_1_5.pdf}\includegraphics[height=2in]{sigma_1_15.pdf}}
	\centerline{\includegraphics[height=2in]{sigma_1_25.pdf}\includegraphics[height=2in]{sigma_1_35.pdf}}
	\centerline{\includegraphics[height=2in]{sigma_1_45.pdf}\includegraphics[height=2in]{sigma_1_55.pdf}}
	\centerline{\includegraphics[height=2in]{sigma_1_65.pdf}\includegraphics[height=2in]{sigma_1_75.pdf}}
	\caption{(color online) Probability weighted cross sections for $k'=1$ GeV for various scattering angles $\theta_l$. The solid lines represent twice the deuteron cross section and the dashed lines are for the oxygen cross section versus $\theta^L_N$. The value of $p_N$ is represented by the dotted lines.}\label{fig:k_prime_1}
\end{figure}
\begin{figure}
	\centerline{\includegraphics[height=2in]{sigma_2_5.pdf}\includegraphics[height=2in]{sigma_2_15.pdf}}
	\centerline{\includegraphics[height=2in]{sigma_2_25.pdf}\includegraphics[height=2in]{sigma_2_35.pdf}}
	\centerline{\includegraphics[height=2in]{sigma_2_45.pdf}}
	\caption{(color online) As for Fig. \ref{fig:k_prime_1} but now for $k'=2$ GeV.  }\label{fig:k_prime_2}
\end{figure}
\begin{figure}
	\centerline{\includegraphics[height=2in]{sigma_3_5.pdf}\includegraphics[height=2in]{sigma_3_15.pdf}}
	\centerline{\includegraphics[height=2in]{sigma_3_25.pdf}}
	\caption{(color online)  As for Fig. \ref{fig:k_prime_1} but now for $k'=3$ GeV.   }\label{fig:k_prime_3}
\end{figure}
In all cases the maximum value of the oxygen cross section is at most one tenth of the deuterium cross section at its maximum value with the relative size decreasing for increased muon energy and scattering angle. It should be remembered, however, that these cross sections are evaluated and kinematics chosen to maximize the contribution of deuterium.

The size of the deuterium cross sections relative to those of oxygen may still seem rather startling. The explanation for this is straightforward.  The semi-inclusive cross sections are roughly proportional to the neutron momentum distributions for the two nuclei as shown in Fig \ref{fig:momentum_dist}. Note that the maximum value of the deuterium momentum distribution is roughly five times as large as that for oxygen.  Given that there are two deuterium nuclei for each oxygen nucleus, this difference in the peak values of the deuterium and oxygen momentum distributions explains the difference in the size of the cross section shown above.
\begin{figure}
	\centerline{\includegraphics[height=3in]{momentum_distributions.pdf}}
	\caption{(color online) Neutron momentum distributions for $^2$H (solid line) and $^{16}$O (dashed line).}\label{fig:momentum_dist}
\end{figure}
Figure \ref{fig:inclusive} shows the inclusive cross sections for deuterium and oxygen as a function of the incident neutrino momentum. This shows that integrating over all possible values of proton three-momentum results in a much larger and broader quasielastic peak for oxygen than for deuterium, as should be expected. This indicates that the unconstrained semi-inclusive cross section is distributed over a much larger region of phase space than that for deuterium.
\begin{figure}
	\centerline{\includegraphics[height=3in]{inclusive.pdf}}
	\caption{(color online) Inclusive CC$\nu$ cross sections for $^2$H (solid line) and $^{16}$O (dashed line).}\label{fig:inclusive}
\end{figure}

\section{Conclusions}\label{sec:concl}

The study presented in this paper of the semi-inclusive charge-changing neutrino reaction $(\nu_{\mu},\mu^- p)$ on a target of heavy water (D$_2$O) indicates that by careful choice of muon and proton three-momenta it is theoretically possible to separate deuterium events from those for oxygen. While naive considerations such as simply counting the number of neutrons provided by the two nuclei, namely, two for the two deuterium nuclei versus eight for the oxygen might lead one to expect that the latter will constitute a large background when the goal is to focus on events from the former, this proves not to be the case for events selected to favor the deuterium. As discussed in the previous section where results are given, this expectation is not necessarily the case:  the spectral function for deuterium is sharply peaked at small values of the missing momentum, whereas that for oxygen peaks at larger missing momenta where contributions from the 1p-shell are dominant and at low missing momenta but at higher missing energies where the 1s-shell contributions occur. Furthermore, these contributions to the oxygen spectral function are spread much more widely in missing momentum than the corresponding sharply peaked ones for deuterium, roughly by the factor of four obtained by forming the ratio of the Fermi momenta for the two nuclei, namely 55 MeV/c for deuterium and 230 MeV/c for oxygen.
In passing we note that the high missing energy/missing momentum region, while contributing perhaps 20\% to the inclusive cross section, is essentially irrelevant for the semi-inclusive cross section as the strength there is very broadly distributed and little is picked up when only limited parts of the ``nuclear landscape'' of the spectral function for a nucleus like oxygen are involved.

In summary, from this theoretical study it appears that targets such as heavy water or deuterated methane containing significant amounts of deuterium together with light nuclei such as oxygen or carbon have the potential to provide unique information for studies of charge-changing neutrino reactions. Upon isolating the deuterium events using semi-inclusive reactions the kinematics alone will yield the incident neutrino energy on an event-by-event basis. Moreover, the cross section for such reactions on deuterium are arguably the best known throughout the periodic table even at quite high energies where relativistic modeling of the type used in the present work is undertaken. This being the case, such measurements hold the promise of determining the incident neutrino flux, thereby providing a very high-quality calibration of other existing or planned near detectors for neutrino oscillation experiments. Additionally, the fact that the nuclear structure issues are so well under control for the case of deuterium means that measurements of this type could serve in determining other aspects of the reaction, for instance, yielding new insights into the nature of the isovector axial-vector form factor of the nucleon. The issue now is an experimental one: can a practical target/detector of heavy water be realized? How are the protons in the final state to be detected? Can layers of (normal, un-deuterated) scintillator be used, as some have suggested, or are there other techniques to employ? Also: what is the optimal oscillation experiment using heavy water? While a near detector of heavy water appears worth contemplating, a far detector would be more challenging. Perhaps this last issue should be viewed in reverse, starting with the largest practical heavy water detector, then using the cross section to find how far from the neutrino source it could be placed, and then, finally, determining from the ``sweet spot'' for oscillation studies what beam energy is appropriate.