In this section the CKM matrix is introduced, whose element $V_{td}$ we are going to investigate by measuring the branching ratio of the $K^+ \rightarrow \pi^+ \nu \bar{\nu}$ process.\\

The CKM framework provides an extension to the Cabibbo 2x2 matrix, that encodes how flavour-changing charged currents ($W^{\pm}$) couple $u, c$ and $d, s$ quark states \cite{PhysRevLett.10.531}. The coupling is described by means of the intermediate states $d'$ and $s'$ obtained from the mass eigenstates $d$ and $s$ through a rotation by an angle $\theta_C$:
\begin{align}
&\left(\begin{array}{c}
d'\\
s'
\end{array}\right) = \left(\begin{array}{cc}
\cos \theta_C & \sin \theta_C\\
- \sin \theta_C & \cos \theta_C
\end{array}\right) \left( \begin{array}{c}
d\\
s
\end{array}\right)
\end{align}
This way, Cabibbo described the quark mixing using a single real parameter $\theta_C$, named the Cabibbo angle. Information from the experiments that probed quark flavour transitions yielded the result $\theta_C \simeq 13\degree$.\\

The Cabibbo-Kobayashi-Maskawa (CKM) theory generalizes the Cabibbo matrix including also the quark states $b, t$ from the third generation \cite{Kobayashi:1973fv}:
\begin{align}
&\left(\begin{array}{c}
d'\\
s'\\
b'
\end{array}\right) = \left(\begin{array}{ccc}
V_{ud} & V_{us} & V_{ub}\\
V_{cd} & V_{cs} & V_{cb}\\
V_{td} & V_{ts} & V_{tb}
\end{array}\right) \left( \begin{array}{c}
d\\
s\\
b
\end{array}\right)
\end{align}

The current status of experimental results about quark-mixing processes yields the following evaluations \cite{PDBook}:\footnotesize
\begin{align}
&\left(\begin{array}{ccc}
|V_{ud}| = 0.97425 \pm 0.00022 & |V_{us}| = 0.2252 \pm 0.0009 & |V_{ub}| = (4.15 \pm 0.49) \,10^{-3}\\
|V_{cd}| = 0.230 \pm 0.011 & |V_{cs}| = 1.006 \pm 0.023 & |V_{cb}| = (40.9 \pm 1.1) \,10^{-3}\\
|V_{td}| = (8.4 \pm 0.6)\, 10^{-3} & |V_{ts}| = (42.9 \pm 2.6)\, 10^{-3} & |V_{tb}| = 0.89 \pm 0.07
\end{array}\right)
\end{align}\normalsize

From the above values, which were obtained by averaging various measurements, it can be inferred that the diagonal elements are clearly do\-mi\-nant. The most favoured transitions are indeed those between quarks belonging to the same family, i.e. $u \leftrightarrow d$, $c \leftrightarrow s$ and $t \leftrightarrow b$. Transitions between quarks from different families are instead suppressed at various levels.\\

Unlike the Cabibbo matrix, the CKM matrix does not represent a pure rotation, as it also includes complex parameters. It can be indeed rewritten in the following form, making use of the Wolfenstein parametrisation \cite{PhysRevLett.51.1945}:
\begin{align}
&V_{CKM} = \left(\begin{array}{ccc}
1 - \lambda^2/2 & \lambda & A\lambda^3(\rho - i \eta)\\
-\lambda & 1 - \lambda^2/2 & A\lambda^2\\
A\lambda^3(1- \rho - i \eta) & -A\lambda^2 & 1
\end{array}\right) + \mathcal{O}(\lambda^4)
\end{align}
where
\begin{align}
&A, \lambda > 0\\
&\lambda = \sin \theta_{12} = \sin \theta_C\\
&A\lambda^2 = \sin \theta_{23}\\
&A\lambda^3(\rho - i\eta) = \sin \theta_{13} e^{-i\varphi}
\end{align}

In the Wolfenstein parametrisation, $\theta_{ij}$ are three real Cabibbo-like angles, while $e^{-i\varphi}$ is a complex phase term encoding CP violation.\\

Nine conditions arise from the unitarity of the CKM matrix:
\begin{align}
&\sum_{i=u,c,t} V_{ik}^* V_{ij} = \delta_{jk} \qquad j,k = d,s,b\\
&\sum_{j=d,s,b} V_{jk}^* V_{ij} = \delta_{ik} \qquad i = u,c,t;\;\; k = d,s,b
\end{align}
The six vanishing combinations can be represented as triangles in the complex plane (as in \figurename~\ref{img:theo-triangle}).\\

\begin{figure}
\centering
\def\svgwidth{0.8\textwidth}
\input{Images/pdf/na62/triangle.pdf_tex}
\caption{Unitarity triangle defined by the relation $V_{ud}V_{ub}^*+V_{cd}V_{cb}^*+V_{td}V_{tb}^*=0$ in the complex plane.}\label{img:theo-triangle}
\end{figure}

According to the Standard Model, the $K^+ \rightarrow \pi^+ \nu \bar{\nu}$ transition is forbidden at tree-level, and thus it arises from the one-loop contributions shown in \figurename~\ref{img:feyn}.\\

Separating the contributions of the quarks $u$, $c$ and $t$, intervening as internal lines, at the leading non-trivial order the amplitude of the $s \rightarrow d \nu \bar{\nu}$ process may be expressed as
\begin{align}
&A \left(s \rightarrow d \nu \bar{\nu}\right) = \sum_{q=u,c,t} V^*_{qs} V_{qd}\, A_q \quad\qquad \text{ with }\\
&A_q \sim {\left( \frac{m_q^2}{m_W^2}\right)}^{\delta} \qquad (\delta > 0 ) \qquad q=u,c,t
\end{align}

\begin{figure}
	\centering
\begin{fmffile}{w-box}
	\begin{fmfgraph*}(110,90)
	  \fmfbottom{i1,d1,o1}%dummy vertex
	  \fmftop{i2,d2,o2}
	    \fmf{fermion,label=$\nu$,l.side=right}{i1,v1}
	    \fmf{fermion,label=$e,,\mu,,\tau$,tension=.5}{v1,v2}
	    \fmf{fermion,label=$\nu$,l.side=right}{v2,o1}
	    \fmf{fermion,label=$s$,l.side=right}{v3,i2}
	    \fmf{fermion,label=$u,,c,,t$,tension=0.5}{v4,v3}
	    \fmf{fermion,label=$d$,l.side=right}{o2,v4}
	    \fmf{photon,label=$W$,tension=0.2,l.side=left}{v1,v3}
	    \fmf{photon,label=$W$,tension=0.2,l.side=right}{v2,v4}
	\end{fmfgraph*}
\end{fmffile}
\begin{fmffile}{z-p1}
	\begin{fmfgraph*}(110,90)
		\fmfleft{i1,i2}
		\fmfright{o1,o2}
		\fmf{fermion,label=$\nu$,l.side=right}{i1,v1}
		\fmf{fermion,label=$\nu$,l.side=right}{v1,o1}
%		\fmf{fermion}{i2,v2,v3,v4,o2}
		\fmf{fermion,label=$s$,l.side=left}{i2,v2}
		\fmf{fermion,label=$u,,c,,t$,l.side=right}{v2,v3}
		\fmf{fermion,label=$u,,c,,t$,l.side=right}{v3,v4}
		\fmf{fermion,label=$d$,l.side=left}{v4,o2}
		\fmf{photon,label=$Z$}{v1,v3}
		\fmf{photon,label=$W$,left=0.5,tension=0.2}{v2,v4} % W line
	\end{fmfgraph*}
\end{fmffile}
\begin{fmffile}{z-p2}
	\begin{fmfgraph*}(110,90)
		\fmfleft{i1,i2}
		\fmfright{o1,o2}
		\fmf{fermion,label=$\nu$,l.side=right}{i1,v1}
		\fmf{fermion,label=$\nu$,l.side=right}{v1,o1}
%		\fmf{fermion}{i2,v2,v3,v4,o2}
		\fmf{fermion,label=$s$,l.side=left}{i2,v2}
		\fmf{fermion,label=$d$,l.side=left}{v4,o2}
		\fmf{photon,label=$Z$}{v1,v3}
		\fmf{fermion,label=$u,,c,,t$,l.side=left}{v2,v4} % W line
%		\fmffreeze
		\fmf{photon,label=$W$,l.side=right,right=0.5,tension=0.6}{v2,v3}
		\fmf{photon,label=$W$,l.side=right,right=0.5,tension=0.6}{v3,v4}
	\end{fmfgraph*}
\end{fmffile}
	\caption{A $W-$box and two $Z-$penguin diagrams. These are the one-loop Feynman diagrams contributing to the $s\rightarrow d \nu \bar{\nu}$ process.}\label{img:feyn}
\end{figure}

The top quark term dominates, due to its higher mass. As a consequence this process can be well described by short-distance dynamics, with the effective Hamiltonian \cite{Misiak:1999yg}
\begin{align}
H_{\text{eff}} = \frac{\alpha G_F}{2 \sqrt{2} \pi \sin^2 \theta_W} \sum_{l=e,\mu,\tau} \big( V^*_{cs} V_{cd} X_l + V^*_{ts} V_{td} \Upsilon_t  \big) \left( \bar{s}d \right) \left( \bar{\nu}_l \nu_l \right)
\end{align}
where $G_F$, $\alpha$ and $\theta_W$ are the Fermi and fine-structure constants and the Weinberg angle, respectively. $\Upsilon_t$ is a function representing the dominant top quark contribution, whose associated uncertainty is very small and mainly due to the experimental error on the top quark mass: \cite{ar}
\begin{align}
&\Upsilon_t = 1.469 \pm 0.017 \pm 0.002
\end{align}
where the two uncertainties correspond to QCD next-to-leading order and two-loops EW corrections respectively. % The ${\left\{X_l\right\}}_{l=e,\mu,\tau}$ functions encode instead the charm quark contributions and can be computed at the next-to-next-to-leading order with an error lower than 4\% \cite{Isidori:2005xm}.
The $X_l$ functions (with $ l=e,\mu,\tau $) encode instead the charm quark contributions and can be computed at the next-to-next-to-leading order with an error lower than 4\% \cite{Isidori:2005xm}. Finally, the terms $\left( \bar{s}d \right)$ and $\left( \bar{\nu}_l \nu_l \right)$ represent $V-A$ neutral weak currents.\\

Since the coupling amplitude depends on the semi-leptonic operator $\left( \bar{s}d \right) \left( \bar{\nu}_l \nu_l \right)$, the hadronic part of the amplitude of the studied process can be determined from that of the decay $K^+ \rightarrow \pi^0 e^+ \nu_e$ by means of isospin symmetry, leading to \cite{Buchalla1994106,Buchalla:1992zm}
\begin{align}
\frac{BR(K^+\rightarrow\pi^+\nu\bar{\nu})}{BR(K^+\rightarrow\pi^0 e^+ \nu_e)} = \frac{r_K}{\lambda^2} &\left\{ {\big[ \Im(V^*_{ts}V_{td}) \big]}^2 \Upsilon^2_t \right.\\
&\left.+ {\big[ \lambda^4 \Re(V^*_{cs}V_{cd}) P_0 + \Re(V^*_{ts}V_{td}) \Upsilon_t \big]}^2\right\}\notag
\end{align}
In this equation, $P_0$ describes the total charm quark contribution
\begin{align}
&P_0 = \frac{1}{\lambda^4} \left( \frac{2}{3} X_e + \frac{1}{3} X_{\tau} \right) = 0.42 \pm 0.06
\end{align}
under the assumption $X_{\mu} = X_e$ \cite{Buchalla1994106}, and $r_K = 0.901$ provides the necessary isospin-breaking corrections to be applied in order to relate $BR(K^+\rightarrow\pi^+\nu\bar{\nu})$ to $BR(K^+\rightarrow\pi^0 e^+ \nu_e)$.\\

The theoretical expectation is then
\begin{align}
BR(K^+ \rightarrow \pi^+ \nu \bar{\nu}) = \left( 7.81^{+0.80}_{-0.71} \pm 0.29 \right) \cdot 10^{-11}
\end{align}
where the uncertainties were separated in order to highlight the first contribution, which is due to the input CKM parameters \cite{PhysRevD.83.034030}.\\

In the Wolfenstein parametrisation introduced above, $V_{ts} \simeq -V_{cb}$ at the leading order, and $|V_{cb}|$, $|V_{cs}|$ and $|V_{cd}|$ are currently well known. Therefore we are left with the only free parameter $V_{td}$, to be experimentally determined with an uncertainty theoretically as low as 5 -- 7\% \cite{PhysRevD.83.034030}.\\

Our current knowledge of $|V_{td}|$ mainly derives from the analysis of the neutral strange B meson system. The most accurate measurement of the mass difference $\Delta m_s = m(B^0_s) - m(\bar{B^0_s})$ was obtained averaging CDF \cite{Abulencia:2006ze} and LHCb \cite{Aaij:2011qx} results, yielding
\begin{align}
&\Delta m_s = (17.719 \pm 0.043)~\si{\pico\second^{-1}}
\end{align}
that, through QCD calculations \cite{Laiho:2009eu}, yields to a combined $|V_{td}|$ / $|V_{ts}|$ measurement:
\begin{align}
&|V_{td}| = (8.4 \pm 0.6) \cdot 10^{-3}\\
&|V_{ts}| = (42.9 \pm 2.6) \cdot 10^{-3}\\
&|V_{td} / V_{ts}| = 0.211 \pm 0.001 \pm 0.006
\end{align}

As discussed above, an alternative determination of the $|V_{td}|$ parameter is possible through the branching ratio of $K \rightarrow \pi \nu \bar{\nu}$ decays. This kind of processes arising from loop contributions is very sensitive to new Physics, and can be used to over-constrain the CKM matrix elements and check for deviations from the Standard Model.
%only if Standard Model predictions are verified.