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

\IEEEPARstart{S}{pin}-orbit interaction in a nonmagnetic heavy metal
generates pure spin current flowing along the direction normal to an electric current.
The phenomenon called spin Hall effect [1-3] has attracted much attention
as a new method to excite spin transfer torque [4,5] on a magnetization in a ferromagnet layer
adjacent to the nonmagnetic layer [6-20].
In particular, magnetization switching of a perpendicular ferromagnet is an important issue
for practical application.

Unfortunately however, it is difficult to switch the perpendicular magnetization solely by the spin Hall effect
due to the following reason.
Let us assume that an electric current flows along $x$-direction,
while a ferromagnet is set in $z$-direction.
Then, the direction of the spin polarization of the pure spin current
generated by the spin Hall effect is geometrically determined to be induced in $y$-direction [21].
The spin Hall torque exerted by this pure spin current tries to move the magnetization
from the perpendicular ($z$) direction to the $y$-direction,
and finally, the magnetization stops its dynamics
when it becomes parallel to the $y$-axis.
Since the spin Hall torque does not break the symmetry with respect to the film-plane,
the magnetization does not cross the film plane.
Therefore, we cannot switch the magnetization deterministically
from one equilibrium direction to the other.
To overcome this problem, usually an in-plane field along the $x$-direction is applied [6].
Using a ferromagnet with a tilted anisotropy or non-uniform anisotropy are other solutions [14,16,20].

\begin{figure}
  \centerline{\includegraphics[width=1.0\columnwidth]{fig1.eps}}
  \caption{
           (a) Schematic view of the system consisting of the two ferromagnets
               and three nonmagnets.
               The thicknesses of the N${}_{k}$ and F${}_{k}$ ($k=1,2$) layers are denoted as
               $d_{{\rm N}_{k}}$ and $d_{{\rm F}_{k}}$, respectively.
               The unit vector pointing in the magnetization direction of the F${}_{k}$ layer is $\mathbf{m}_{k}$.
           (b) The spin Hall torques acting on the magnetization of the F${}_{2}$ layer are schematically shown.
  \vspace{-3.0ex}}
  \label{fig:fig1}
\end{figure}

In this paper, we propose another solution to switch perpendicular ferromagnet deterministically
by using the spin Hall effect.
As schematically shown in Fig. \ref{fig:fig1}(a),
the system consists of two ferromagnets, F${}_{1}$ and F${}_{2}$,
and three nonmagnets, N, N${}_{1}$, and N${}_{2}$.
The two nonmagnets, N${}_{1}$ and N${}_{2}$, show spin-orbit interactions,
and inject pure spin current into the F${}_{1}$ and F${}_{2}$ layers via the spin Hall effect.
The N layer is just a spacer,
and does not show spin-orbit interaction.
Current is passed along the $x$-direction and flows within the N${}_{1}$ and N${}_{2}$ layers.
The F${}_{2}$ layer is a free layer,
and has a perpendicular anisotropy.
The spin Hall torque acting on the magnetization of the F${}_{2}$ layer
consists of three contributions.
The first one is the conventional spin Hall torque caused by the pure spin current injected from the N${}_{2}$ layer,
which is denoted as $\bm{\tau}^{(1)}$ in Fig. \ref{fig:fig1}(b).
The second contribution comes from the electrons
passing through from N${}_{2}$ to N${}_{1}$ layer.
Due to the diffusion between the N${}_{1}$ and N${}_{2}$ layer,
these electrons can return back to the F${}_{2}$ layer, and excites the spin Hall torque denoted as $\bm{\tau}^{(2)}$ in Fig. \ref{fig:fig1}(b).
The third contribution comes from the pure spin current originated in the N${}_{1}$ layer,
and is denoted as $\bm{\tau}^{(3)}$ in Fig. \ref{fig:fig1}(b).
While the first torque $\bm{\tau}^{(1)}$ points to the $y$-direction, as in the case of the conventional spin Hall system,
the directions of $\bm{\tau}^{(2)}$ and $\bm{\tau}^{(3)}$ depend on
the magnetization of the F${}_{1}$ layer.
This is because the longitudinal and transverse spin currents [22-28] have different relaxation length scales;
here the longitudinal and transverse correspond to parallel and perpendicular to the local magnetization, respectively.
Deterministic switching of the F${}_{2}$ layer is possible
when the F${}_{1}$ layer tilts from the $z$-axis.
In the following, we derive the explicit forms of the spin Hall torques, $\bm{\tau}^{(i)}$ ($i=1,2,3$),
based on the diffusion equation of the spin accumulation and the Landauer formula.

This paper is organized as follows.
In Sec. \ref{sec:Spin Hall Torque Formula}, we derive a theoretical formula of the spin torque
acting on the magnetization of the F${}_{2}$ layer in the geometry given by Fig. \ref{fig:fig1}(a).
In Sec. \ref{sec:LLG Equation}, we show evidence of the deterministic switching in the F${}_{2}$ layer
by solving the Landau-Lifshitz-Gilbert (LLG) equation.
The conclusion is summarized in Sec. \ref{sec:Conclusion}.

\section{Spin Hall Torque Formula}
\label{sec:Spin Hall Torque Formula}

The system we consider is schematically shown in Figs. \ref{fig:fig1}(a) and \ref{fig:fig1}(b).
We denote the thicknesses of the N${}_{k}$ and F${}_{k}$ layers as $d_{{\rm N}_{k}}$ and $d_{{\rm F}_{k}}$, respectively ($k=1,2$).
The unit vector pointing in the magnetization direction of the F${}_{k}$ layer is
denoted as $\mathbf{m}_{k}$.
The electric field $E_{x}$ is applied along the $x$-direction,
and generates an electric current density in the nonmagnet [21]
\begin{equation}
  J_{e}
  =
  \sigma_{{\rm N}_{k}}
  E_{x}
  +
  \frac{\vartheta_{{\rm N}_{k}} \sigma_{{\rm N}_{k}}}{2e}
  \mathbf{e}_{y}
  \cdot
  \partial_{z}
  \bm{\mu}_{{\rm N}_{k}},
  \label{eq:electric_current}
\end{equation}
where $\sigma_{{\rm N}_{k}}$ and $\vartheta_{{\rm N}_{k}}$ are
the conductivity and the spin Hall angle of the N${}_{k}$ layer, respectively.
The spin accumulation in the N${}_{k}$ layer is denoted as $\bm{\mu}_{{\rm N}_{k}}$.
Similarly, the spin current density in the nonmagnet flowing along the $z$-direction is [21]
\begin{equation}
  \mathbf{J}_{s}
  =
  -\frac{\hbar \sigma_{{\rm N}_{k}}}{4e^{2}}
  \partial_{z}
  \bm{\mu}_{{\rm N}_{k}}
  +
  \frac{\hbar \vartheta_{{\rm N}_{k}}}{2e}
  E_{x}
  \mathbf{e}_{y},
  \label{eq:spin_current}
\end{equation}
where the vector represents the direction of the spin polarization.
The spin accumulation in the nonmagnet obeys the diffusion equation,
and its solution can be expressed by a linear combination of $e^{\pm z/\lambda_{\rm N}}$,
where $\lambda_{\rm N}$ is the spin diffusion length.

The pure spin current injected from the nonmagnet creates a spin accumulation in the ferromagnet.
The spin accumulation in a ferromagnet, $\bm{\mu}_{\rm F}$,
can be decomposed into the longitudinal and transverse components as
$\bm{\mu}_{\rm F}^{\rm L}=(\mathbf{m}\cdot\bm{\mu}_{\rm F})\mathbf{m}$ and
$\bm{\mu}_{\rm F}^{\rm T}=\mathbf{m}\times(\bm{\mu}_{\rm F}\times\mathbf{m})=\bm{\mu}_{\rm F}-\bm{\mu}_{\rm F}^{\rm L}$, respectively.
The relaxation length scale of the longitudinal spin accumulation, called the spin diffusion length,
depends on the spin-flip scattering time, and is on the order of 1-10 nm [22].
The relaxation length of the transverse spin accumulation, called the penetration depth of the transverse spin current,
depends on both the spin-flip scattering time and precession period of spin around the local magnetization due to the exchange coupling.
Since the penetration depth of the transverse spin current is usually shorter than the spin diffusion length [23-28],
let us assume that the spin accumulation in the F layer, $\bm{\mu}_{\rm F}$, has only the longitudinal component, for simplicity.
In other words, the direction of the spin polarization of the electrons becomes
parallel to the magnetization direction when it is injected into the ferromagnet.
We note that a finite penetration depth of the transverse spin current can be taken into account in the following calculations,
as done in the case of spin pumping [26].
It does not however affect the main conclusion significantly, in spite of the fact that it makes the calculations complex.
The important point is that the longitudinal and transverse spin currents relax with different length scales,
and therefore, the direction of the spin polarization of the spin current is modified
by passing through a ferromagnet.
The idea is similar to what is used in Ref. [29].

At both ends of the nonmagnets, N${}_{1}$ and N${}_{2}$,
the spin currents are zero.
On the other hand, the spin current density at the F/N interface
flowing from F to N layer is given [23]
\begin{equation}
\begin{split}
  \mathbf{J}_{s}^{\rm F \to N}
  =&
  \frac{1}{4\pi S}
  \left[
    \frac{(1-\gamma_{\rm F/N}^{2})g_{\rm F/N}}{2}
    \mathbf{m}
    \cdot
    \left(
      \bm{\mu}_{\rm F}
      -
      \bm{\mu}_{\rm N}
    \right)
    \mathbf{m}
  \right.
\\
  &
    -
  \left.
    g_{\rm r(F/N)}
    \mathbf{m}
    \times
    \left(
      \bm{\mu}_{\rm N}
      \times
      \mathbf{m}
    \right)
  \right],
  \label{eq:spin_current_FN}
\end{split}
\end{equation}
where $S$ is the cross section area of the $xy$-plane
(Eq. (\ref{eq:spin_current_FN}) is applicable to both F${}_{k}$/N and F${}_{k}$/N${}_{k}$ interfaces).
Here $g=g^{\uparrow}+g^{\downarrow}$ and $\gamma=(g^{\uparrow}-g^{\downarrow})/(g^{\uparrow}+g^{\downarrow})$
are the total F/N interface conductance
and its spin polarization, respectively.
We neglect spin-flip scattering at the interface.
The conductance $g$ is related to the F/N interface resistance $r$ via $g/S=h/(e^{2}r)$.
The real part of the mixing conductance is denoted as $g_{\rm r}$.
For simplicity, we neglect the imaginary part of the mixing conductance,
which is usually negligible in a metallic current-perpendicular-to-plane system [23,24]
but might be large in spin Hall geometry [11-16].
The spin Hall torque acting on the magnetization of the F${}_{k}$ layer
is given by
\begin{equation}
  \frac{{\rm d}\mathbf{m}_{k}}{{\rm d}t}
  \bigg|_{\rm STT}
  =
  \frac{\gamma_{0}}{M_{k}Sd_{{\rm F}_{k}}}
  \mathbf{m}_{k}
  \times
  \left[
    \left(
      \mathbf{J}_{s}^{{\rm F}_{k} \to {\rm N}_{k}}
      +
      \mathbf{J}_{s}^{{\rm F}_{k} \to {\rm N}}
    \right)
    \times
    \mathbf{m}_{k}
  \right],
  \label{eq:STT_def}
\end{equation}
where $\gamma_{0}$ is the gyromagnetic ratio of the F${}_{k}$ layer.

We assume that the N layer sandwiched by the two ferromagnets
is sufficiently thin compared to its spin diffusion length,
and therefore, the spin current in the N layer is conserved,
i.e., $\mathbf{J}_{s}^{{\rm F}_{1} \to {\rm N}} + \mathbf{J}_{s}^{{\rm F}_{2} \to {\rm N}}=\bm{0}$.
Solving the diffusion equations of the N${}_{k}$ and F${}_{k}$ layers,
then substituting the solutions to Eq. (\ref{eq:spin_current_FN}),
and using the relation $\mathbf{J}_{s}^{{\rm F}_{1} \to {\rm N}} + \mathbf{J}_{s}^{{\rm F}_{2} \to {\rm N}}=\bm{0}$,
Eq. (\ref{eq:STT_def}) becomes
\begin{equation}
\begin{split}
  \frac{{\rm d}\mathbf{m}_{k}}{{\rm d}t}
  \bigg|_{\rm STT}
  =&
  (-1)^{k}
  \tau_{k}^{(1)}
  \mathbf{m}_{k}
  \times
  \left(
    \mathbf{e}_{y}
    \times
    \mathbf{m}_{k}
  \right)
\\
  &
  -
  (-)^{k+1}
  \left[
    \tau_{k}^{(2)}
    m_{k y}
    \mathbf{m}_{1}
    \cdot
    \mathbf{m}_{2}
    -
    \tau_{k}^{(3)}
    m_{\ell y}
  \right]
\\
  &
  \ \ \ \ \ \ \ \times
  \frac{\mathbf{m}_{k} \times (\mathbf{m}_{\ell} \times \mathbf{m}_{k})}{1-\lambda_{1}^{\prime}\lambda_{2}^{\prime}(\mathbf{m}_{1}\cdot\mathbf{m}_{2})^{2}},
  \label{eq:LLG_STT}
\end{split}
\end{equation}
where $(k,\ell)=(1,2)$ or $(2,1)$.
The following notations are introduced;
\begin{equation}
  \tau_{k}^{(1)}
  =
  \frac{\gamma_{0} \hbar \vartheta_{{\rm N}_{k}} g_{{\rm r}({\rm F}_{k}/{\rm N}_{k})}^{\prime} \sigma_{{\rm N}_{k}} E_{x}}{2eg_{{\rm sd}({\rm N}_{k})}M_{k}d_{{\rm F}_{k}}}
  \tanh
  \left(
    \frac{d_{{\rm N}_{k}}}{2 \lambda_{{\rm N}_{k}}}
  \right),
\end{equation}
\begin{equation}
  \tau_{k}^{(2)}
  =
  \lambda_{\ell}^{\prime}
  \frac{\gamma_{0} \hbar \vartheta_{{\rm N}_{k}} g_{{\rm r}({\rm F}_{k}/{\rm N})} g_{{\rm F}_{k}/{\rm N}}^{\prime} \sigma_{{\rm N}_{k}} E_{x}}
    {2e \tilde{g}_{k} [g_{{\rm r}({\rm F}_{\ell}/{\rm N})}+g_{{\rm F}_{k}/{\rm N}}^{\prime}]M_{k} d_{{\rm F}_{k}}},
\end{equation}
\begin{equation}
  \tau_{k}^{(3)}
  =
  \frac{\gamma_{0} \hbar \vartheta_{{\rm N}_{\ell}} g_{{\rm r}({\rm F}_{k}/{\rm N})} g_{{\rm F}_{\ell}/{\rm N}}^{\prime} \sigma_{{\rm N}_{\ell}} E_{x}}
    {2e \tilde{g}_{\ell} [g_{{\rm r}({\rm F}_{k}/{\rm N})}+g_{{\rm F}_{\ell}/{\rm N}}^{\prime}]M_{k} d_{{\rm F}_{k}}},
\end{equation}
\begin{equation}
  \lambda_{k}^{\prime}
  =
  \frac{g_{{\rm r}({\rm F}_{k}/{\rm N})} - g_{{\rm F}_{k}/{\rm N}}^{\prime}}
    {g_{{\rm r}({\rm F}_{\ell}/{\rm N})} + g_{{\rm F}_{k}/{\rm N}}^{\prime}},
\end{equation}
where
\begin{equation}
  \frac{1}{g_{{\rm r}({\rm F}_{k}/{\rm N}_{k})}^{\prime}}
  =
  \frac{1}{g_{{\rm r}({\rm F}_{k}/{\rm N}_{k})}}
  +
  \frac{1}{g_{{\rm sd}({\rm N}_{k})} \tanh(d_{{\rm N}_{k}}/\lambda_{{\rm N}_{k}})},
\end{equation}
\begin{equation}
\begin{split}
  \frac{1}{g_{{\rm F}_{k}/{\rm N}_{k}}^{\prime}}
  =&
  \frac{2}{(1-\gamma_{{\rm F}_{k}/{\rm N}_{k}}^{2})g_{{\rm F}_{k}/{\rm N}_{k}}}
  +
  \frac{1}{g_{{\rm sd}({\rm F}_{k})} \tanh(d_{{\rm F}_{k}}/\lambda_{{\rm F}_{k}})}
\\
  &+
  \frac{1}{g_{{\rm sd}({\rm N}_{k})} \tanh(d_{{\rm N}_{k}}/\lambda_{{\rm N}_{k}})},
\end{split}
\end{equation}
\begin{equation}
\begin{split}
  \frac{1}{g_{{\rm F}_{k}/{\rm N}}^{\prime}}
  =&
  \frac{2}{(1-\gamma_{{\rm F}_{k}/{\rm N}}^{2})g_{{\rm F}_{k}/{\rm N}}}
  +
  \frac{1}{g_{{\rm sd}({\rm F}_{k})} \tanh(d_{{\rm F}_{k}}/\lambda_{{\rm F}_{k}})}
\\
  &-
  \frac{g_{{\rm F}_{k}/{\rm N}_{k}}^{\prime}}{[ g_{{\rm sd}({\rm F}_{k})} \sinh(d_{{\rm F}_{k}}/\lambda_{{\rm F}_{k}}) ]^{2}},
\end{split}
\end{equation}
\begin{equation}
\begin{split}
  \frac{1}{\tilde{g}_{k}}
  =
  \frac{g_{{\rm F}_{k}/{\rm N}_{k}}^{\prime} \tanh[d_{{\rm N}_{k}}/(2 \lambda_{{\rm N}_{k}})]}
    {g_{{\rm sd}({\rm F}_{k})} g_{{\rm sd}({\rm N}_{k})} \sinh(d_{{\rm F}_{k}}/\lambda_{{\rm F}_{k}})},
\end{split}
\end{equation}
\begin{align}
 g_{{\rm sd}({\rm F}_{k})}
 =
 \frac{h (1-\beta_{{\rm F}_{k}}^{2}) S}{2e^{2} \rho_{{\rm F}_{k}} \lambda_{{\rm F}_{k}}},
&&
 g_{{\rm sd}({\rm N}_{k})}
 =
 \frac{h S}{2e^{2} \rho_{{\rm N}_{k}} \lambda_{{\rm N}_{k}}},
\end{align}
and $\rho=1/\sigma$.
The spin polarization of the conductivity is denoted as $\beta=(\sigma^{\uparrow}-\sigma^{\downarrow})/(\sigma^{\uparrow}+\sigma^{\downarrow})$.
A calculation of spin Hall torque in a system including two ferromagnets
with a different geometry was developed in Ref. [30].

In Eq. (\ref{eq:LLG_STT}),
the first torque proportional to $\tau_{k}^{(1)}$ originates from
the pure spin current injected from the N${}_{k}$ layer via the spin Hall effect,
and points to the $y$-direction, as in the case of the conventional spin Hall torque.
The second term proportional to $\tau_{k}^{(2)}$ originates from
the electrons passing from the N${}_{k}$ layer through the F${}_{k}$ layer
and being reflected from the N${}_{\ell}$ layer ($\ell \neq k$)
This torque becomes zero when $m_{ky}=0$
because the spin current from the N${}_{k}$ layer is
completely absorbed at the F${}_{k}$/N${}_{k}$ interface,
and therefore the electrons passing through the F${}_{k}$ layer
and diffusing between the N${}_{1}$ and N${}_{2}$ layers do not have net spin polarization.
Similarly, this torque also becomes zero when $\mathbf{m}_{1}\cdot\mathbf{m}_{2}=0$
because in this case the spin current is completely absorbed
at the F${}_{k}$/N${}_{k}$ or F${}_{\ell}$/N interface.
The third term proportional to $\tau_{k}^{(3)}$ originates from
the pure spin current injected from the N${}_{\ell}$ layer.
Due to the absorption of the transverse spin current in the N${}_{\ell}$ layer,
this torque is zero when $m_{\ell y}=0$.

The directions of the second and third torques acting on the F${}_{k}$ layer, $\mathbf{m}_{k}\times(\mathbf{m}_{\ell}\times\mathbf{m}_{k})$,
can be changed by controlling the magnetization direction of the other layer, $\mathbf{m}_{\ell}$.
This is because the direction of the spin polarization is modified by passing through the ferromagnet
due to the different relaxation lengths between the longitudinal and transverse spin currents.
Now let us assume that the magnetization in the F${}_{1}$ layer, $\mathbf{m}_{1}$, tilts from the $z$-axis.
Then the second and third torques acting on the magnetization of the F${}_{2}$ layer
remain finite even when $\mathbf{m}_{2}$ arrives at the film-plane,
contrary to the first torque which becomes zero when $\mathbf{m}_{2}$ becomes parallel to the $y$-axis,
as in the case of the conventional spin Hall torque.
Depending on the direction of the electric field $E_{x}$,
the second and third torques move the magnetization $\mathbf{m}_{2}$ to the positive or negative $z$-direction.
Therefore, a deterministic switching of $\mathbf{m}_{2}$ can be expected.
A finite tilted angle of $\mathbf{m}_{1}$ is necessary
to break the symmetry with respect to the film-plane
and induces the deterministic switching.
If $\mathbf{m}_{1}\parallel\mathbf{e}_{z}$,
and when $\mathbf{m}_{2}$ arrives at the film-plane,
the second and third torques in Eq. (\ref{eq:LLG_STT}) become zero.
Then, the magnetization will be stopped at the plane, as in the case of the conventional spin Hall system.

\section{LLG Equation}
\label{sec:LLG Equation}

Using Eq. (\ref{eq:LLG_STT}),
the LLG equation of the F${}_{k}$ layer becomes
\begin{equation}
\begin{split}
  \frac{{\rm d}\mathbf{m}_{k}}{{\rm d}t}
  =&
  -\gamma_{0}
  \mathbf{m}_{k}
  \times
  \mathbf{H}_{k}
  +
  \bm{\tau}_{k}
  +
  \alpha
  \mathbf{m}_{k}
  \times
  \frac{{\rm d}\mathbf{m}_{k}}{{\rm d}t},
  \label{eq:LLG}
\end{split}
\end{equation}
where
the spin Hall torque $\bm{\tau}_{k}$ is given by Eq. (\ref{eq:LLG_STT}).
As mentioned above, we assume that the magnetization of the F${}_{1}$ layer is fixed,
while that of the F${}_{2}$ layer changes its direction according to Eq. (\ref{eq:LLG}).
The magnetic field acting on $\mathbf{m}_{2}$, $\mathbf{H}_{2}=H_{\rm K}m_{2z} \mathbf{e}_{z}$,
consists of the perpendicular anisotropy field $H_{\rm K}$.
The Gilbert damping constant of the F${}_{2}$ layer is denoted as $\alpha$.
In the absence of the electric field $E_{x}$,
the F${}_{2}$ layer has two stable states, $\mathbf{m}_{2}=\pm\mathbf{e}_{z}$.
In this calculation, we assume that the magnetization $\mathbf{m}_{2}$ initially points to the positive $z$-direction.

We assume that the parameters of the N${}_{1}$ and N${}_{2}$ layers,
as well as those of the F${}_{1}$ and F${}_{2}$ layers, are identical.
We define the current density $j$ as $j=\sigma_{{\rm N}_{1}}E_{x}=\sigma_{{\rm N}_{2}}E_{x}$.
Linearizing Eq. (\ref{eq:LLG}) around the initial state,
we found that the magnetization of the free (F${}_{2}$) layer is destabilized
when the current magnitude is larger than a critical value,
\begin{equation}
  j_{\rm c}
  =
  \frac{2 \alpha eM_{2}d_{2}}{\hbar \vartheta \mathcal{P}}
  H_{\rm K},
  \label{eq:critical_current}
\end{equation}
where $\mathcal{P}$ is given by
\begin{equation}
\begin{split}
  \mathcal{P}
  =&
  \frac{g_{{\rm r}({\rm F}_{k}/{\rm N})} g_{{\rm F}_{k}/{\rm N}}^{\prime}}{2 \tilde{g}_{k} [g_{{\rm r}({\rm F}_{k}/{\rm N})} + g_{{\rm F}_{k}/{\rm N}}^{\prime}]}
\\
  &
  \times
  \left[
    \frac{2 (1 - \lambda_{1}^{\prime} \lambda_{2}^{\prime})}{(1-\lambda_{1}^{\prime}\lambda_{2}^{\prime} m_{1z}^{2})^{2}}
    +
    \frac{\lambda_{1}^{\prime}}{1-\lambda_{1}^{\prime} \lambda_{2}^{\prime} m_{1z}^{2}}
  \right]
  m_{1y}
  m_{1z}
\end{split}
\end{equation}

\begin{figure}
  \centerline{\includegraphics[width=1.0\columnwidth]{fig2.eps}}
  \caption{
           The time evolution of $m_{2z}$ for $j=2.0$, $2.5$, $2.9$, and $3.0 \times 10^{6}$ A/cm${}^{2}$.
           The inset shows the dependence of the balance current density $j(E)$ on
           $\theta=\cos^{-1}m_{z}=\cos^{-1}\sqrt{-2E/(MH_{\rm K})}$.
  \vspace{-3.5ex}}
  \label{fig:fig2}
\end{figure}

We confirm the deterministic switching of $\mathbf{m}_{2}$ from $\mathbf{m}_{2}=+\mathbf{e}_{z}$ to $\mathbf{m}_{2}=-\mathbf{e}_{z}$
by numerically solving Eq. (\ref{eq:LLG}).
The values of parameters are taken from typical experiments in CoFeB/TaN heterostructure [16]
or similar metallic F/N multilayer [22-26] as
$\lambda_{\rm F}=12$ nm,
$\rho_{\rm F}=1600$ $\Omega$nm,
$\beta=0.56$,
$d_{\rm F}=1$ nm,
$\lambda_{\rm N}=2.5$ nm,
$\rho_{\rm N}=3750$ $\Omega$nm,
$d_{\rm N}=4$ nm,
$\vartheta=0.1$,
$r=0.28$ k$\Omega$nm${}^{2}$,
$\gamma=0.70$,
$g_{\rm r}/S=15$ nm${}^{-2}$,
$\gamma_{0}=1.764 \times 10^{7}$ rad/(Oe s),
$\alpha=0.005$,
$M_{2}=1250$ emu/cm${}^{3}$,
and $H_{\rm K}=450$ Oe,
where, we assume that the interface conductances, $g$ and $g_{\rm r}$, at the F${}_{k}$/N${}_{k}$ and F${}_{k}$/N interfaces are the same.
We assume that the magnetization of the F${}_{1}$ layer points to the direction
$\mathbf{m}_{1}=(\sin\theta_{\rm p}\cos\varphi_{\rm p},\sin\theta_{\rm p}\sin\varphi_{\rm p},\cos\theta_{\rm p})$
where $\theta_{\rm p}=30^{\circ}$ and $\varphi_{\rm p}=90^{\circ}$.
The critical current density estimated by Eq. (\ref{eq:critical_current}) is $1.98 \times 10^{6}$ A/cm${}^{2}$.
Figure \ref{fig:fig2} shows the time evolution of $m_{2z}$
obtained from Eq. (\ref{eq:LLG}) with these parameters and several current densities.
The dynamics becomes relatively slow around $m_{2z}\sim 0.6-0.8$ where $\mathbf{m}_{2}$ becomes almost parallel to $\mathbf{m}_{1z}$,
and therefore, the second and third torques in Eq. (\ref{eq:LLG_STT}) become small.
As shown, the deterministic switching from the positive to negative $z$-direction is confirmed.

The switching occurs at the current density of $j_{\rm sw}\simeq 2.9 \times 10^{6}$ A/cm${}^{2}$,
which is larger than the critical current density.
Note that the critical current determines the instability of the initial state,
while the instability does not guarantee the switching.
Let us define the balance current density $j(E)$ as a current satisfying
$\oint {\rm d}t ({\rm d}E/{\rm d}t)=0$,
where $E=-M \int {\rm d}\mathbf{m}_{2}\cdot\mathbf{H}_{2}=-MH_{\rm K}m_{2z}^{2}/2$ is the energy density,
and the integral region is over the constant energy curve.
The constant energy curve corresponds to
a precession trajectory with a constant angle $\theta=\cos^{-1}m_{z}=\cos^{-1}\sqrt{-2E/(MH_{\rm K})}$.
As discussed in Ref. [31],
the critical and switching current densities are defined as
$j_{\rm c}=\lim_{E \to E_{\rm min}}j(E)$ and $j_{\rm sw}={\rm max}[j(E)]$, respectively,
which are not necessarily same.
The analytical solution of $j(E)$ for an arbitrary $E$ is complex.
Instead, we show the numerically calculated $j(E)$ as a function
of $\theta$ in the inset of Fig. \ref{fig:fig2}.
As shown, $j(E)$ has a maximum at $\theta \simeq 55^{\circ}$.
Therefore, $j(\theta \simeq 55^{\circ})=2.9 \times 10^{6}$ A/cm${}^{2}$ is the theoretical switching current density,
which is consistent with the numerical result in Fig. \ref{fig:fig2}.
On the other hand, $j_{\rm c}$ corresponds to $j(\theta \to 0^{\circ})$.
When the current density is larger than $j_{\rm c}$ and smaller than $j_{\rm sw}$,
the self-oscillation of the magnetization can be expected,
which is beyond the scope of this paper.

\section{Conclusion}
\label{sec:Conclusion}

In conclusion,
we propose a model of deterministic magnetization switching
in a perpendicular ferromagnet by spin Hall effect.
The system consists of two ferromagnets and three nonmagnets.
The nonmagnets generate pure spin current by the spin Hall effect,
whose spin polarization points to the in-plane ($y$) direction.
Passing through a ferromagnet however, the direction of the spin polarization is modified
due to the different relaxation length scales of the longitudinal and transverse spin currents.
Consequently, a tilt of the magnetization in one ferromagnet breaks the symmetry of the system,
and the spin Hall torque excited in such geometry
enables to switch a perpendicular magnetization in another ferromagnet deterministically.
The idea was confirmed by deriving the spin Hall torque formula
and solving the LLG equation numerically.

\section*{Acknowledgment}

T. T. expresses gratitude to
Sinji Yuasa, Kay Yakushiji, Koji Ando, Hitoshi Kubota, Akio Fukushima, Takayuki Nozaki, Makoto Konoto,
Hidekazu Saito, Satoshi Iba, Aurelie Spiesser, Yoichi Shiota, Sumito Tsunegi, Ryo Hiramatsu, Takehiko Yorozu,
Hiroki Maehara, and Ai Emura for their support and encouragement.
This work was supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) 25790044.

\ifCLASSOPTIONcaptionsoff
  \newpage
\fi


text:
\section{Basic facts about the top quark}
\label{section:1}

The top quark is the heaviest elementary particle known up to today.
It has been discovered at the Tevatron and it is currently studied at the LHC.
The top quark can be characterised by three essential numbers.
These are the top quark mass,
the top quark width
and the branching ratio for the decay into a bottom quark.
The current experimentally measured values are \cite{Nakamura:2010zzi}
\begin{eqnarray*}
 & & m_t = 173.1 \pm 0.6 \pm 1.1 \; \mbox{GeV},
 \nonumber \\
 & & \Gamma_t = 2.0^{+0.7}_{-0.6} \; \mbox{GeV},
 \nonumber \\
 & & \frac{\Gamma(W b)}{\Gamma(W q)} = 0.99^{+0.09}_{-0.08}.
\end{eqnarray*}
The first number tells us that the top quark is heavier than all other known elementary particles,
from the second one we deduce that the lifetime of the top quark is shorter than the characteristic
hadronisation time scale.
The third number indicates that the top quark decays predominately into a bottom quark and a $W$-boson.
These numbers have several implications:
The top quark mass is close to the electro-weak symmetry breaking scale $v = 246 \; \mbox{GeV}$.
If there is new physics associated with electro-weak symmetry breaking, top quark physics is a place to look
for.
Secondly, the large top mass sets a hard scale.
The short lifetime of the top quark implies that the top quark decays before it can form bound states.
Therefore, top quark physics is described by perturbative QCD.
The absence of hadronisation effects allows also that spin information of the top quark
is transferred to its decay products.
Finally, the large top mass implies also that the top quark
contributions are relevant to precision physics.
For example, the top gives a significant contribution to the Higgs self-energy
and a precise knowledge of the top mass is required for an indirect determination of the Higgs mass
from electro-weak precision fits.

This raises immediately the question how precise the top quark mass can be extracted from experiments.
There are some theoretical subtleties which should be taken into account.
The starting point for a theoretical description is the Lagrange density, the relevant part reads
\begin{eqnarray*}
 {\cal L} & = & \bar{\psi} \left( i D\!\!\!/ - m_{\mathrm{bare}} \right) \psi.
\end{eqnarray*}
Here, the bare top quark mass appears. Renormalisation relates the bare mass to a renormalised mass
\begin{eqnarray*}
 m_{\mathrm{bare}} & = & Z_m \; m_{\mathrm{renorm}}.
\end{eqnarray*}
It should be stressed that $Z_m$ and hence $m_{\mathrm{renorm}}$ depend on the renormalisation scheme.
Popular choices for a renormalisation scheme are the on-shell scheme, where the
the mass $m_{\mathrm{pole}}$ is defined as the pole of the propagator
(and $m_{\mathrm{pole}}$ is therefore called the pole mass),
or the $\overline{\mathrm{MS}}$-scheme, in which the
$\overline{\mathrm{MS}}$-mass $m_{\overline{\mathrm{MS}}}(\mu)$ is scale-dependent.
Within perturbation theory one can convert between the different schemes.
Naively, the pole mass seems to be the natural choice.
However, the exact definition of the pole mass assumes the concept of stable colour-less particle.
The top quark is neither stable nor colour-less, and this re-intro\-duces non-perturbative
effects.
As a result, the pole mass is ambiguous by an
amount ${\cal O}\left(\Lambda_{\mathrm{QCD}}\right)$ \cite{Bigi:1994em,Beneke:1994rs,Smith:1996xz,Skands:2007zg}.
This ambiguity limits the precision by which the pole mass can be extracted from experiment.
As an alternative one can use a mass definition, which is only sensitive to short distances.
The $\overline{\mathrm{MS}}$-mass $m_{\overline{\mathrm{MS}}}(\mu)$
is an example of a short-distance mass.
The direct extraction of $m_{\overline{\mathrm{MS}}}(\mu)$ from the measurements avoids the above-mentioned
ambiguities \cite{Langenfeld:2009wd}.
Furthermore it is possible to use alternative short-distance mass definitions.
For example, for the process $e^+ e^- \rightarrow t \bar{t}$ a top quark jet mass
has been defined and studied in detail \cite{Fleming:2007qr}.
It should be emphasised that the use of short-distance mass definitions will in general
lead to distortions from a perfect Breit-Wigner shape.

Recent experimental results from the Tevatron on top quark mass measurements can be found in \cite{Lancaster:2011wr}.

\section{Top pair production}
\label{section:2}

Top quark pair production proceeds at the Tevatron mainly through the quark-antiquark channel, while at the
LHC the gluon-gluon channel dominates, reflecting the different parton fluxes at the two accelerators.
\begin{figure}
\resizebox{0.9\columnwidth}{!}{\includegraphics{fig1.eps} }
\caption{The leading-order Feynman diagrams for top pair production.}
\label{fig1}
\end{figure}
The relevant leading-order Feynman diagrams are shown in fig.~\ref{fig1}.
For top pair production not only the inclusive process
\begin{eqnarray*}
 p p & \rightarrow & t \bar{t} + X,
\end{eqnarray*}
but also the exclusive processes
\begin{eqnarray*}
 p p \rightarrow t \bar{t} + 0 \; \mbox{jets},
 \;\;
 p p \rightarrow t \bar{t} + 1 \; \mbox{jet},
 \;\;
 p p \rightarrow t \bar{t} + 2 \; \mbox{jets},
 \;\;
 ...
\end{eqnarray*}
are of interest. (For the Tevatron on should replace the initial state $p p$ by $p \bar{p}$.)
In the inclusive top-quark sample a substantial number of events is accompanied by additional jets.
These processes are all known to next-to-leading order (NLO) in QCD.
The NLO corrections to $p p \rightarrow t \bar{t}$ \cite{Nason:1987xz,Nason:1989zy,Beenakker:1988bq}
have been known for a long time,
while the NLO corrections to $p p \rightarrow t \bar{t} + 1 \; \mbox{jet}$ \cite{Dittmaier:2007wz,Dittmaier:2008uj,Melnikov:2010iu,Melnikov:2011ta}
and $p p \rightarrow t \bar{t} + 2 \; \mbox{jets}$ \cite{Bevilacqua:2010ve}
are of a more recent vintage.
The NLO corrections of the important sub-process $p p \rightarrow t \bar{t} + b \bar{b}$ of
$p p \rightarrow t \bar{t} + 2 \; \mbox{jets}$ have been calculated
by two groups \cite{Bredenstein:2009aj,Bredenstein:2008zb,Bredenstein:2010rs,Bevilacqua:2009zn}.
Furthermore, the NLO QCD corrections to the processes
$p p \rightarrow t \bar{t} + H$ \cite{Beenakker:2002nc,Dawson:2003zu},
$p p \rightarrow t \bar{t} + Z$ \cite{Lazopoulos:2008de,Kardos:2011na} and
$p p \rightarrow t \bar{t} + \gamma$ \cite{PengFei:2009ph,Melnikov:2011ta}
are known.
It should be mentioned that these calculation either treat the top quarks as stable particles or include
the top decay within the narrow width approximation.
A full NLO calculation including top decays and non-factorisable corrections has been performed
for the process $p p \rightarrow b \bar{b} W^+ W^-$ \cite{Denner:2010jp,Bevilacqua:2010qb}.
The inclusion of NLO corrections reduces the scale dependence of the predictions.
\begin{figure}
\resizebox{0.75\columnwidth}{!}{\includegraphics{ttj_NLO_Tev.eps} }
\caption{Scale dependence of the NLO and LO cross section for $p \bar{p} \rightarrow t \bar{t} + 1 \; \mbox{jet}$
at the Tevatron.
The figure is taken from \cite{Dittmaier:2007wz}.}
\label{fig:ttj_NLO_Tev}
\end{figure}
\begin{figure}
\resizebox{0.75\columnwidth}{!}{\includegraphics{ttj_NLO_LHC.eps} }
\caption{Scale dependence of the NLO and LO cross section for $p p \rightarrow t \bar{t} + 1 \; \mbox{jet}$ at
the LHC.
The figure is taken from \cite{Dittmaier:2007wz}.}
\label{fig:ttj_NLO_LHC}
\end{figure}
Examples are shown in fig.~\ref{fig:ttj_NLO_Tev} and fig.~\ref{fig:ttj_NLO_LHC},
where the scale dependence of the NLO predictions and the LO
prediction for the processes $p \bar{p} \rightarrow t \bar{t} + 1 \; \mbox{jet}$ at the Tevatron
and $p p \rightarrow t \bar{t} + 1 \; \mbox{jet}$ at the LHC are shown.

In the near future one can expect results at next-to-next-to-leading order (NNLO) for the process
$p p \rightarrow t \bar{t}$.
The squared one-loop amplitudes $g g \rightarrow t \bar{t}$ and $q \bar{q} \rightarrow t \bar{t}$
are known \cite{Korner:2008bn,Kniehl:2008fd,Anastasiou:2008vd}, work on
the corresponding two-loop amplitudes \cite{Czakon:2007ej,Czakon:2007wk,Bonciani:2010mn,Bonciani:2009nb,Bonciani:2008az}
is in progress.
In addition a method to handle the infrared divergences at NNLO, in particular with respect to initial state partons
and massive partons, is needed.
Again, work on this topic is in progress \cite{Bierenbaum:2011gg,Czakon:2011ve,Bernreuther:2011jt,Gehrmann:2011wi,Glover:2010im,GehrmannDeRidder:2011aa,Boughezal:2010mc}.

In multi-scale problems there can be large logarithms of the form
$\alpha_s^n \ln^j\beta$ in the perturbative expansion.
An example is given for the process of top pair production in the threshold region, where
\begin{eqnarray*}
 \beta & = & \sqrt{ 1 - \frac{4m_t^2}{\hat{s}}}.
\end{eqnarray*}
In addition there can be Coulomb singularities of the form $1/\beta^k$.
These terms can potentially spoil the perturbative expansion.
The solution is to resum these terms.
For top pair production the resummation has been carried with
next-to-next-to-leading logarithmic (NNLL)
accuracy \cite{Moch:2008qy,Czakon:2009zw,Cacciari:2011hy,Kidonakis:2010dk,Ahrens:2010zv,Beneke:2011mq}.
The resummed results can be re-expanded and give an ``approximate NNLO'' result \cite{Kidonakis:2010dk,Aliev:2010zk,Czakon:2011xx,Ahrens:2011px,Kidonakis:2011ca}.
In comparing the results from the different groups one should keep in mind that there are some differences
in the approaches.
This concerns the exact definition of the resummation variable and the question which terms are resummed,
e.g. soft gluons only or also Coulomb terms.

An alternative approach to resummation of large logarithms is provided by parton showers. These are usually
only at the leading-log (LL) level, but offer fully exclusive final states.
The current interest lies in matching fixed-order NLO calculations with parton showers.
The frameworks of MC@NLO \cite{Frixione:2002ik} and POWHEG \cite{Nason:2004rx} offer the possibility to do this
and avoid double-counting.
A convenient tool in this respect is the POWHEG-BOX \cite{Alioli:2010xd} and several processes related
to top pair production have been implemented into the
POWHEG-BOX \cite{Kardos:2011qa,Garzelli:2011vp,Garzelli:2011is,Alioli:2011as}.

Experimental results on top pair production cross sections can be found in \cite{Aaltonen:2011tm,Abazov:2011mi,Abazov:2011cq,Aad:2012qf,Aad:2011yb,Aad:2010ey,Chatrchyan:2011yy,Chatrchyan:2011ew,Chatrchyan:2011nb}.

Within the Standard Model the top quark decays purely through left-handed weak decay,
and spin information is transferred to the decay products.
In the dilepton channel of top pair production it is therefore possible
to measure correlations between the angles of the two leptons.
One has
\begin{eqnarray*}
\lefteqn{
\frac{1}{\sigma} \frac{d^2\sigma}{d\cos \theta_l d \cos \theta_{\bar{l}}}
 = } & &
 \nonumber \\
 & &
 \frac{1}{4} \left( 1
 + B_1 \cos \theta_l
 + B_2 \cos \theta_{\bar{l}}
 - C \cos \theta_l \cos \theta_{\bar{l}},
\right)
\end{eqnarray*}
where the coefficient $C$ gives the correlation \cite{Mahlon:1997uc,Bernreuther:2000yn,Bernreuther:2001bx,Bernreuther:2001rq,Bernreuther:2004jv}.
Here, $\theta_{\bar{l}}$ is the angle between the lepton $\bar{l}^+$ in the rest frame of the top and a reference
direction $\hat{\bf a}$.
Similar, $\theta_l$ is the angle between the lepton $l^-$ in the rest frame of the anti-top and a reference direction
$\hat{\bf b}$.
Convenient choices for $\hat{\bf a}$ and $\hat{\bf b}$ are the helicity basis, the beam basis and the off-diagonal basis.
For these choices QCD yields vanishing coefficients $B_1$ and $B_2$.
Experimental results can be found in \cite{Aaltonen:2010nz,Abazov:2011gi}.

QCD predicts a charge asymmetry in top pair production starting at ${\cal O}\left(\alpha_s^3\right)$.
The charge asymmetry results from an interference of $C$-odd parts of the amplitude with $C$-even parts.
At the Tevatron, which is a proton-antiproton collider, the charge asymmetry manifests itself in a forward-backward asymmetry.
Since the asymmetry starts at ${\cal O}\left(\alpha_s^3\right)$, the existing NLO calculation of the differential cross section
for the process $p \bar{p} \rightarrow t \bar{t}$ gives only a LO prediction for the forward-backward asymmetry \cite{Kuhn:1998jr,Kuhn:1998kw}.
The situation is different for the process $p \bar{p} \rightarrow t \bar{t} + \mbox{jet}$.
Here, the NLO calculation of the differential cross section gives a true
NLO prediction for the forward-backward asymmetry \cite{Halzen:1987xd,Bowen:2005ap,Dittmaier:2007wz,Dittmaier:2008uj,Melnikov:2010iu,Melnikov:2011ta}.
The differential cross section starts at LO with ${\cal O}\left(\alpha_s^3\right)$ and includes at NLO
terms of order ${\cal O}\left(\alpha_s^4\right)$.
For the process $p \bar{p} \rightarrow t \bar{t} + \mbox{jet}$ the LO prediction for the asymmetry is about $-8\%$, however including NLO corrections
the asymmetry is almost washed out.
\begin{figure}
\resizebox{0.75\columnwidth}{!}{\includegraphics{NLOasym_Tev.eps} }
\caption{The forward-backward asymmetry at Tevatron. The figure is taken from \cite{Dittmaier:2007wz}.}
\label{fig:NLOasym_Tev}
\end{figure}
This is shown in fig.~\ref{fig:NLOasym_Tev}.
For the process $p \bar{p} \rightarrow t \bar{t}$
the LO prediction for the asymmetry in the laboratory frame is $(4.7 \pm 0.7)\%$ \cite{Kuhn:2011ri}.
CDF has measured a significant larger value of $(15.0 \pm 5.5)\%$ \cite{Aaltonen:2011kc,Abazov:2011rq}.
This has triggered a significant amount of papers on new physics contributions to the forward-backward asymmetry, see ref.~\cite{Westhoff:2011tq} and
the references therein.
One should however investigate if uncalculated higher order corrections can explain the discrepancy.
It is known that the electro-weak corrections are small \cite{Hollik:2007sw,Bernreuther:2010ny,Kuhn:2011ri}, as well as
soft gluon corrections \cite{Kidonakis:2011zn,Ahrens:2011mw,Ahrens:2011uf}.
In this context one should welcome a differential NNLO calculation of $p \bar{p} \rightarrow t \bar{t}$, which would provide
the NLO correction to the forward-backward asymmetry of this process.
It should also be noted that the CDF collaboration compares their result to a theory prediction of $(3.8 \pm 0.6)\%$, which they obtain
from MCFM \cite{Campbell:1999ah,Campbell:2010ff}. This number corresponds to a normalisation of the asymmetry to the NLO cross section.
As pointed out in ref.~\cite{Kuhn:2011ri}, the asymmetry is only known to LO and a normalisation to the LO cross section is more appropriate.
This explains the numerical difference between $(3.8 \pm 0.6)\%$ and $(4.7 \pm 0.7)\%$.

The LHC is a proton-proton collider and due to the symmetric initial state the charge asymmetry does not manifest itself in a forward-backward
asymmetry.
However, in the process $q \bar{q} \rightarrow t \bar{t}$, the top quark $t$ tends to follow the initial quark $q$, while the anti-top quark $\bar{t}$
tends to follow the initial $\bar{q}$.
Furthermore, the initial quarks tend to have a larger momentum fraction (valence-like) than the initial anti-quarks (which are always sea quarks).
Therefore the rapidity distribution of the top quarks is expected to be broader than the rapidity distribution of the anti-top quarks.
A measurement will not be simple due to the fact that at the LHC the $q \bar{q}$-initial state gives only a small fraction of the events, which
are dominated by the $g g$-initial state.
However, in view of the Tevatron results, this measurement should be worth the effort.
First experimental results are to be found in \cite{Chatrchyan:2011hk}.

\section{Single top production}
\label{section:3}

A top quark can also be produced singly by an electro-weak $W t b$-vertex.
\begin{figure}
\resizebox{0.9\columnwidth}{!}{\includegraphics{fig2.eps} }
\caption{The leading-order Feynman diagrams for single top production.}
\label{fig2}
\end{figure}
The LO diagrams of the single top production mechanisms are shown in fig.~\ref{fig2}.
The flavour excitation channel and the $W$-gluon fusion channel are often also collectively referred to as $t$-channel.
Flavour excitation corresponds to a five-flavour scheme, $W$-gluon fusion to a four-flavour scheme.
There are several motivations from physics to study single top production:
First of all, the process is sensitive to the electro-weak $W t b$-vertex and non-standard couplings can give a hint on physics beyond
the Standard Model.
Secondly, the top quark is produced left-handed.
Since no hadronisation occurs, spin correlations survive in the final decay products.
Thirdly, the flavour excitation channel can be used to extract the $b$-quark density.
Finally, single top production allows a direct measurement of the CKM matrix element $V_{tb}$ and a verification of the unitarity of the
CKM-matrix.
Here, a few remarks are in order:
$V_{tb}$ is known indirectly from unitarity
\begin{eqnarray*}
\left| V_{ub} \right|^2 + \left| V_{cb} \right|^2 + \left| V_{tb} \right|^2 & = & 1
\end{eqnarray*}
to a very high precision: $\left| V_{tb} \right| = 0.9990 - 0.9993$.
There is no way to measure $\left| V_{tb} \right|$ directly to this precision!
The indirect determination assumes three generations and unitarity.
From top pair production at the Tevatron one knows
\begin{eqnarray*}
\frac{BR(t \rightarrow Wb)}{BR(t \rightarrow Wq)}
 =
 \frac{\left| V_{tb} \right|^2}
      {\left| V_{td} \right|^2 + \left| V_{ts} \right|^2 + \left| V_{tb} \right|^2}
 =
 0.99^{+0.09}_{-0.08}.
\end{eqnarray*}
Assuming three generations and unitarity, the denominator $\left| V_{td} \right|^2 + \left| V_{ts} \right|^2 + \left| V_{tb} \right|^2$
equals one.
However, if we do not wish to make this assumption, then it follows only
$\left| V_{tb} \right| >> \left| V_{ts} \right|, \left| V_{td} \right|$.
Single top production allows a direct $\left| V_{tb} \right|$-measurement
without any assumptions on the number of generations.

For all single top production channels fixed-order NLO calculations are available
\cite{Bordes:1995ki,Stelzer:1997ns,Harris:2002md,Cao:2004ky,Cao:2004ap,Cao:2005pq,Heim:2009ku,Campbell:2009ss,Campbell:2009gj,Campbell:2004ch,Falgari:2011qa,Falgari:2010sf,Campbell:2005bb,Frixione:2008yi}.
In addition resummation at the NNLL level has been carried out \cite{Kidonakis:2011wy,Kidonakis:2010tc,Zhu:2010mr}.
Single top quark production has been implemented into the MC@NLO framework \cite{Frixione:2005vw} as well as into POWHEG framework \cite{Alioli:2009je}.

Experimental results on single top production can be found in \cite{Aaltonen:2010jr,Abazov:2011pt,Chatrchyan:2011vp}.

Of particular interest in single top production is the fact that the top quark is produced through the
electro-weak $Wtb$-vertex left-handed.
Since no hadronisation occurs, spin correlations survive in the final decay products \cite{Jezabek:1994zv,Mahlon:1999gz,Mahlon:1997pn,vanderHeide:2000fx}.
In $W$-gluon fusion or flavour excitation the top quark is highly
polarised along the direction of the $d$-quark.
In addition the $u$-quark density is the largest among the quark densities.
Therefore the cross section receives the dominant contribution
from the configuration where the $u$-quark is in the initial state and the $d$-quark in the final state, which
in turn produces the non-$b$ tagged jet $q$. A suitable observable is therefore the variable
\begin{eqnarray*}
a & = & \frac{1}{2} \left( 1 + \cos \theta_{q\bar{l}} \right)
\end{eqnarray*}
where $\theta_{q\bar{l}}$ is the angle between the light quark jet and the charged lepton in the rest frame of the top.
For the angular correlation of the decaying top quark one has
\begin{eqnarray*}
\frac{d\sigma}{da} & = & \sigma \left( 2 P a + (1-P) \right),
\end{eqnarray*}
where $P$ denotes the polarisation of the top quark along the spin axis defined by the spectator jet $q$.
Including background processes, the slope of this distribution is given by
\begin{eqnarray*}
2 P_{\mathrm{signal}} \sigma_{\mathrm{signal}} + 2 P_{\mathrm{background}} \sigma_{\mathrm{background}}.
\end{eqnarray*}
Although in principle other distributions may serve to infer the spin correlations, the $a$ distribution seems particularly
attractive due to its simple shape.

The LHC is a $p p$-collider. As a consequence the cross sections for single top production and single
anti-top production are not the same.
As far as spin correlations are concerned, one finds for single anti-top production that in the $W$-gluon fusion channel
or flavour excitation channel the dominant contribution comes now from configurations where the $d$-quark is in the initial
state.
As a consequence, a suitable observable for spin correlations in single anti-top production is the angle between
the charged lepton and one of the beam directions.

A measurement of spin correlations in single top quark production would be very interesting.

redpajama_set_name: RedPajamaArXiv


text:
\section{Stochastic calculus}\label{appendix:stochastic calculus}
Typically, a stochastic differential equation is represented by two different forms, the Ito and Stratonovich forms, which are respectively given by
\begin{align}
    f(t)dW(t)&\equiv f(t)\left(W(t+dt)-W(t)\right),\\
     f(t)\circ dW(t)&\equiv f\left(t+\frac{dt}{2}\right)\left(W(t+dt)-W(t)\right).
\end{align}
In the Stratonovich interpretation, a calculus is performed at the midpoint of the interval $[t, t+dt]$. The two forms can be converted to each other via
\begin{align}
    f(t)\circ dW(t)=f(t)dW(t)+\frac{1}{2}df(t)dW(t).
\label{eq:conversion}
\end{align}
The conversion of Eq.~\eqref{eq:conversion} is performed according to the Ito rule:
\begin{align}
    dW(t)^2&=dt,\\
    dW(t)dt&=0.
\end{align}

\section{Ito and Stratonovich conversion of stochastic Schr\"{o}dinger equation}\label{appendix:conversion}
A diffusive stochastic SSE is represented by the Ito form \cite{Gisin_1992, gardiner2004quantum}
\begin{align}
    d\ket{\psi} =& dt\left[ -iH_{\mathrm{eff}}+\sum_{k=1}^M \frac{\braket{X_k}}{2}\left(L_k - \frac{\braket{{X_k}}}{4}\right)\right]\ket{\psi}\nonumber\\
    &+\sum_{k=1}^M dW_k(t)\left(L_k - \frac{\braket{X_k}}{2}\right)\ket{\psi}.
    \label{eq:SSE_Ito}
\end{align}
According to Appendix \ref{appendix:stochastic calculus}, Eq.~\eqref{eq:SSE_Ito} can be converted to the equivalent Stratonovich form as follows:
\begin{align}
    d\ket{\psi} =& \left[ -iH_{\mathrm{eff}}+\sum_{k=1}^M +\frac{1}{2}\braket{L_k^{\d}L_k} +\braket{X_k}\left(L_k-\frac{\braket{X_k}}{2}\right)\right.\nonumber\\
    &\left.+\frac{1}{4}\left(-2L_k^2+\braket{L_k^2}+\braket{L_k^{\d 2}}\right)\right]\ket{\psi}dt\nonumber\\
    &+\sum_{k=1}^M \left(L_k - \frac{\braket{X_k}}{2}\right)\ket{\psi}\circ dW_k(t).
    \label{eq:SSE_Str}
\end{align}
The conversion from Eq.~\eqref{eq:SSE_Ito} to Eq.~\eqref{eq:SSE_Str} is performed according to the multivariate Ito rule \cite{gardiner2004handbook}:
\begin{align}
        dW_m(t)dW_n(t)&=\delta_{m,n}dt,\\
    dW_m(t)dt&=0,
\end{align}
where $\delta_{m,n}$ is Kronecker's delta.

\section{Generators of $SU(3)$}\label{appendix:Lie}
The generators of $SU(3)$ correspond to Gell-Mann matrices $\{\lambda_l\}$.
\begin{align}
\lambda_1&=    \begin{bmatrix}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0 \\
\end{bmatrix},
\lambda_2=    \begin{bmatrix}
0 & -i & 0 \\
i & 0 & 0 \\
0 & 0 & 0 \\
\end{bmatrix},
\lambda_3=    \begin{bmatrix}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 0 \\
\end{bmatrix},\nonumber\\
\lambda_4&=    \begin{bmatrix}
0 & 0 & 1 \\
0 & 0 & 0 \\
1 & 0 & 0 \\
\end{bmatrix},
\lambda_5=    \begin{bmatrix}
0 & 0 & -i \\
0 & 0 & 0 \\
i & 0 & 0 \\
\end{bmatrix},
\lambda_6=    \begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0 \\
\end{bmatrix},\nonumber\\
\lambda_7&=    \begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & -i \\
0 & i & 0 \\
\end{bmatrix},
\lambda_8=    \frac{1}{\sqrt{3}}\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & -2 \\
\end{bmatrix}.
\end{align}
Gell-Mann matrices are orthogonal with respect to the Hilbert-Schmidt inner product:
\begin{align}
    \mathrm{Tr}[\lambda_m\lambda_n]&=2\delta_{mn}.
\end{align}
The generators $E_l$ of $SU(N)$ are defined so that they are normalized with respect to the trace norm:
\begin{align}
    E_l&\equiv \frac{1}{\sqrt{2}}\lambda_l\nonumber.
\end{align}
\section{Phase transformation}\label{appendix:phase transformation}
Since $U(1)$ has no physical effect on the state $\ket{\psi}$, by phase transformation $\ket{\psi'} \equiv e^{i\phi(t)}\ket{\psi}$, we can obtain a equation physically equivalent to an SSE~\eqref{eq:SSE}:
\begin{align}
    d\ket{\psi'}=\left(id\phi - \frac{1}{2}(d\phi)^2\right)\ket{\psi'}+e^{i\phi}(1+id\phi)d\ket{\psi}.
\end{align}
In particular, Eq.~\eqref{eq:phaseSSE} is obtained by the phase transformation $\ket{\psi'} \equiv e^{i\phi(t)}\ket{\psi}$, which satisfies the differential equation as follows:
\begin{align}
    d\phi=\frac{i}{4}\sum_{k=1}^M\braket{X_k}\braket{L_k-L_k^{\dagger}}dt+\frac{i}{2}\sum_{k=1}^M\braket{L_k-L_k^{\dagger}}dW_k.
\end{align}