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

Pair production processes for electroweak vector bosons
provide a rich spectrum of observables, which are crucial
to test in depth the $SU(2)_L\times U(1)_Y$ gauge sector
of the Standard Model.
In particular, the production of pairs of resonant vector bosons
allows for precise studies of the electroweak triple gauge couplings, while
considering off-shell vector boson pairs
is required for precision Higgs phenomenology.
Furthermore, diboson production processes are important backgrounds
in direct new physics searches.
The main production channel for pairs of vector bosons at hadron colliders is
quark-antiquark annihilation and great progress has been achieved in the last years
with the computation of the next-to-next-to-leading order (NNLO) QCD corrections to
$q \bar{q} \to \gamma \gamma$~\cite{Catani:2011qz}, $q \bar{q} \to Z \gamma$~\cite{Grazzini:2013bna},
$q \bar{q} \to ZZ$~\cite{Cascioli:2014yka} and $ q \bar{q} \to W^+W^-$~\cite{Gehrmann:2014fva} production at the LHC.
Furthermore, the fermionic NNLO corrections to $q \bar{q} \to \gamma^* \gamma^*$ were derived in~\cite{Anastasiou:2014nha}.

The gluon fusion channel contributes to
$\gamma \gamma$, $ZZ$, $Z \gamma$ and $W^+W^-$ production.
As a quark-loop induced process, its leading order (LO) cross section
is suppressed by two powers of $\alpha_s$
with respect to that of the quark channel.
This implies that it formally contributes only at NNLO in the perturbative
expansion of the hadronic process,
but numerical enhancements may be expected due to
the large gluon luminosities at typical energies
for diboson production at the LHC.
For the gluon-induced processes, the one-loop amplitudes and the corresponding
one-loop squared interference terms have been computed long ago~\cite{Costantini:1971cj,vanderBij:1988fb,Glover:1988rg,Glover:1988fe,Adamson:2002jb,Adamson:2002rm,Binoth:2006mf}.
Their impact on the total cross section was found
to range approximately from $5\%$ to more than $10\%$ for different final states at the LHC,
and to rise with increasing collider energy~\cite{Catani:2011qz,Grazzini:2013bna,Cascioli:2014yka,Gehrmann:2014fva}.
These values can substantially increase up to about $30\%$ when particular sets of
cuts, relevant for example for Higgs boson searches, are applied~\cite{Binoth:2005ua, Cascioli:2013gfa}.
It is therefore clear that the inclusion of gluon channel contributions can be important in order
to achieve a description of the full process which matches the experimental precision.
Beyond the actual size of the known leading order corrections in the gluon channel,
it is unclear how large the associated theory uncertainty actually is.
By comparison with Higgs production in gluon fusion~\cite{Dawson:1990zj,Djouadi:1991tka,Spira:1995rr},
the conventional LO scale variation is not expected to allow for a reliable estimate
of the size of neglected higher order corrections.
The recent NNLO predictions for the total $ZZ$ and $W^+W^-$ production
cross sections take into account the quark channel at NNLO and the gluon channel
at LO, resulting in a scale uncertainty of about $3 \%$~\cite{Cascioli:2014yka,Gehrmann:2014fva}.
In order to thoroughly control the theory uncertainty to this level of precision,
it is therefore very desirable to compute
the next-to-leading order (NLO) contributions for the gluon induced subprocess.
Currently this has been done only for $gg \to \gamma \gamma$~\cite{Bern:2001df,Bern:2002jx},
and the NLO corrections have been found to be not only sizeable but also
important for stabilising the theoretical predictions~\cite{Bern:2002jx}.
Finally, precise theoretical predictions for $gg \to ZZ$ can be useful
for constraining the total Higgs boson decay
width at the LHC~\cite{Campbell:2011cu,Kauer:2012hd,Caola:2013yja,Campbell:2013una}.

Technically, the computation of the NLO corrections to $gg \to V_1 V_2$ requires two ingredients, the two-loop
virtual corrections to $gg \to V_1 V_2$ and the one-loop real-virtual corrections to the corresponding
radiative processes with one more parton in the final state.
By now the computation of the one-loop amplitudes with an extra gluon does not constitute any conceptual
difficulty and can be pursued with standard techniques for one-loop
multi-legs processes~\cite{Berger:2008sj,Hirschi:2011pa,Ellis:2011cr,Bevilacqua:2011xh,Cullen:2011ac,Cascioli:2011va,Cullen:2014yla}.
The two-loop amplitudes, on the other hand, are known only for
$gg \to \gamma \gamma$~\cite{Bern:2001df} and for $gg \to Z \gamma$~\cite{Gehrmann:2013vga},
in both cases for on-shell final state photons.
In order to obtain physical predictions, both contributions need to be combined
using a subtraction scheme to isolate and cancel unphysical
IR divergences. In this case, a NLO scheme~\cite{Frixione:1995ms,Catani:1996vz} would be sufficient.

In this paper we calculate the missing two-loop massless QCD corrections to
 $gg \to V_1 V_2 $, with an off-shell vector boson pair $V_1 V_2 = \gamma^* \gamma^*, ZZ, Z\gamma^*, W^+W^-$.
The calculation builds upon the master integrals for four-point
functions with massless propagators and two massive external legs, which were computed recently
in the case of equal masses in~\cite{Gehrmann:2013cxs,Gehrmann:2014bfa}, and in
the case of different masses
in~\cite{Henn:2014lfa,Caola:2014lpa,Papadopoulos:2014hla,Gehrmann:2015ora}.
The former
were used for the first NNLO fully-inclusive calculations of $ZZ$~\cite{Cascioli:2014yka}
and $W^+W^-$~\cite{Gehrmann:2014fva} production
at the LHC, while the latter allowed
the computation of the two-loop corrections to
$q \bar{q}' \to V_1 V_2$~\cite{Caola:2014iua,Gehrmann:2015ora}. A subset of these
master integrals was also computed independently in~\cite{Chavez:2012kn,Anastasiou:2014nha}.
While the inclusion of massive top-loop mediated subprocesses would be of interest for
some phenomenological applications~\cite{Campbell:2011cu,Melnikov:2015laa}, the
computation of the two-loop amplitudes requires knowledge of challenging new
master integrals, which should be addressed in the future.

The paper is structured as follows. In Section~\ref{sec:tens} we describe
the tensor decomposition of the partonic current for the process $gg \to V_1 V_2$
and consider the possible electroweak coupling structures.
We include the vector boson decays and describe
the helicity amplitudes for the process $gg \to V_1 V_2\to 4~\mathrm{leptons}$
in terms of scalar form factors in Section~\ref{sec:helamp}.
The actual calculation of the loop contributions to these form factors is
described in Section~\ref{sec:calc}, which includes a dicussion of UV renormalisation,
IR subtraction and various checks we performed on our results.
In Section~\ref{sec:num} we present numerical results obtained with our {\tt C++} implementation.
Finally, we conclude in Section~\ref{sec:conc}.
In Appendix~\ref{sec:form} we give explicit formulae for
obtaining the physical form factors appearing in the helicity amplitudes from the
original tensor coefficients computed in this paper.
We provide computer readable files for our analytical results
and our {\tt C++} code for the numerical evaluation of the amplitudes
on our {\tt VVamp} project page on {\tt HepForge}\ at \url{http://vvamp.hepforge.org}.

\section{\texorpdfstring
{Partonic current for $gg\to V_1 V_2$}
{Partonic current for gg -> V1 V2}
}
\label{sec:tens}

We consider the production of two massive off-shell vector bosons, $V_1V_2$, in the gluon fusion channel,
\begin{equation}\label{VVprocess}
 g(p_1) + g(p_2) \longrightarrow V_1(p_3) + V_2(p_4),
\end{equation}
where $V_1V_2$ = $\gamma^* \gamma^*$,
$ZZ$, $Z\gamma^*$,  $W^+W^-$.
The final states $W^\pm \gamma^*$ and $W^\pm Z$ instead are forbidden by charge conservation.
Since the two vector bosons are off-shell we have in the general case
\begin{equation}
 p_1^2=p_2^2 =0\,, \quad p_3^2>0,\quad p_4^2>0,\quad p_3^2 \neq p_4^2,
\end{equation}
with the usual Mandelstam invariants defined as
\begin{equation}
s=(p_1+p_2)^2\,, \qquad t=(p_1-p_3)^2\,,\qquad u=(p_2-p_3)^2\,,
\end{equation}
and the relation
\begin{equation}
s+t+u = p_3^2 + p_4^2\,.
\end{equation}
The physical region for the scattering kinematics has the boundary $t\,u = p_3^2\,p_4^2$
and fulfils
\begin{equation}
s \geq \Big(\sqrt{p_3^2} + \sqrt{p_4^2}\Big)^2,\qquad
\frac{1}{2}\big(p_3^2+p_4^2-s -\kappa\big) \leq t \leq  \frac{1}{2}\big(p_3^2+p_4^2-s +\kappa\big)
\end{equation}
where $\kappa$ is the K\"all\'en function
\begin{equation}\label{kaellen}
\kappa\left(s,p_3^2,p_4^2\right) \equiv
  \sqrt{s^2 + p_3^4 + p_4^4 - 2 (s\,p_3^2 + p_3^2\,p_4^2 + p_4^2\,s)}\,.
\end{equation}
We denote the scattering amplitude for the process~\eqref{VVprocess} by
\[
S(p_1,p_2,p_3) = S_{\mu\nu\rho\sigma}(p_1,p_2,p_3)\,
 \epsilon_1^{\rho}(p_1)\,\epsilon_2^{\sigma}(p_2)\,
 \epsilon_3^{\ast\,\mu}(p_3)\,\epsilon_4^{\ast\,\nu}(p_4)
\]
where $\epsilon_1$, $\epsilon_2$ are the polarisation vectors of the incoming gluons,
$\epsilon_3$, $\epsilon_4$ are the polarisation vectors of the outgoing massive vector bosons,
$p_4=p_1+p_2-p_3$ and an overall factor $e^2$ is kept implicit
with $e$ being the positron charge.
Since we will consider leptonic decays of the massive vector bosons we will be
able to construct the full amplitude including the decays from
the partonic current
\[
S_{\mu\nu}(p_1,p_2,p_3) = S_{\mu \nu\rho\sigma}(p_1,p_2,p_3)
 \,\epsilon_1^\rho(p_1) \,\epsilon_2^\sigma(p_2)
\]
for the $2\to 2$ process.
In particular, it is only the latter which receives (pure) QCD corrections
at any order in perturbation theory.

In order to compute the partonic current
it is useful to consider its tensor decomposition.
Based on Lorentz invariance only, there are 138 independent tensor structures which can contribute
\begin{align}
S^{\mu \nu\rho\sigma}(p_1,p_2,p_3) &=
        a_1 g^{\mu\nu} g^{\rho\sigma}
       +a_2 g^{\mu\rho} g^{\nu\sigma}
       +a_3 g^{\mu\sigma} g^{\nu\rho} \nonumber \\
&+\sum\limits_{j_1,j_2=1}^{3}
\Bigl(
   b^{(1)}_{j_1 j_2}\,g^{\mu\nu}\,p_{j_1}^{\rho}\,p_{j_2}^{\sigma}
  +b^{(2)}_{j_1 j_2}\,g^{\mu\rho}\,p_{j_1}^{\nu}\,p_{j_2}^{\sigma}
  +b^{(3)}_{j_1 j_2}\,g^{\mu\sigma}\,p_{j_1}^{\nu}\,p_{j_2}^{\rho}\nonumber \\
&\phantom{\sum\limits_{j_1,j_2=1}^{3}}
+b^{(4)}_{j_1 j_2}\,g^{\nu\rho}\,p_{j_1}^{\mu}\,p_{j_2}^{\sigma}
  +b^{(5)}_{j_1 j_2}\,g^{\nu\sigma}\,p_{j_1}^{\mu}\,p_{j_2}^{\rho}
  +b^{(6)}_{j_1 j_2}\,g^{\rho\sigma}\,p_{j_1}^{\mu}\,p_{j_2}^{\nu}
\Bigr)
\nonumber \\
 &+\sum\limits_{j_1,j_2,j_3,j_4=1}^{3} c_{j_1 j_2 j_3 j_4}
p_{j_1}^{\mu}\,p_{j_2}^{\nu} p_{j_3}^{\rho}\,p_{j_4}^{\sigma}, \label{tensordec1}
\end{align}
where the coefficients $a_j$, $b_{ij}^k$ and $c_{ijkl}$ are scalar functions of the
kinematic invariants $s$, $t$, $p_3^2$, $p_4^2$ and of the space-time dimension $d$.
Not all structures are relevant for our calculation.
Many of them simply drop
due to the transversality of the gluons' polarisation vectors
\begin{equation}
\epsilon_1 \cdot p_1 = \epsilon_2 \cdot p_2 = 0\,.\label{transverseg}
\end{equation}
Moreover the tensor structure can be further simplified by fixing explicitly
the gauge for the incoming gluons. A particularly simple choice is given by the symmetrical
condition
\begin{align}
 \epsilon_1 \cdot p_2 &= \epsilon_2 \cdot p_1 = 0\,, \label{gaugeg}
\end{align}
which corresponds to the following rules for the polarisation sums
\begin{align}
 \sum_{\lambda_1} \epsilon_{1\,\lambda_1}^{\mu *}(p_1) \epsilon_{1\,\lambda_1}^{\nu}(p_1)
 &= -g^{\mu \nu} + \frac{p_1^\mu p_2^\nu + p_1^\nu p_2^\mu }{p_1 \cdot p_2}\,,\nonumber \\
  \sum_{\lambda_2} \epsilon_{2\,\lambda_2}^{\mu *}(p_2) \epsilon_{2\,\lambda_2}^{\nu}(p_2)
 &= -g^{\mu \nu} + \frac{p_1^\mu p_2^\nu + p_1^\nu p_2^\mu }{p_1 \cdot p_2}\,. \label{polsum12}
\end{align}
Further conditions can be applied on the polarisation vectors of the massive vector bosons $V_1$, $V_2$.
We employ for their polarisation vectors
\begin{align}
 \epsilon_3 \cdot p_3 = \epsilon_4 \cdot p_4 = 0\,, \label{gaugeV}
\end{align}
and for the polarisation sums
\begin{align}
 \sum_{\lambda_3} \epsilon_{3\,\lambda_3}^{\mu *}(p_3) \epsilon_{3\,\lambda_3}^{\nu}(p_3)
 &= -g^{\mu \nu} + \frac{p_3^\mu p_3^\nu }{p_3^2 }\,,\nonumber \\
  \sum_{\lambda_4} \epsilon_{4\,\lambda_4}^{\mu *}(p_4) \epsilon_{4\,\lambda_4}^{\nu}(p_4)
 &= -g^{\mu \nu} + \frac{p_4^\mu p_4^\nu }{p_4^2 }\,. \label{polsum34}
\end{align}

Imposing the constraints~\eqref{transverseg}, \eqref{gaugeg} and \eqref{gaugeV}
one is left with only $20$ independent tensor structures and
we can write the partonic current according to
\begin{align}
S^{\mu \nu}(p_1,p_2,p_3)
&=
\sum_{j=1}^{20}\, A_j(s,t,p_3^2,p_4^2)\, T_j^{\mu \nu}\,,\label{tensordec}
\end{align}
where the $A_j$ are scalar functions of $s$, $t$, $p_3^2$, $p_4^2$ and $d$.
The tensors $T^{\mu \nu}_j$ are defined as
\begin{gather}\label{tensors}
\begin{aligned}
 T_1^{\mu \nu} &= \epsilon_1 \cdot \epsilon_2\, g^{\mu \nu}\,, &
 T_2^{\mu \nu}  &= \epsilon_1^{\mu}\, \epsilon_2^{\nu}\,, &
 T_3^{\mu \nu}  &= \epsilon_1^{\nu}\, \epsilon_2^{\mu}\,, &
 T_4^{\mu \nu} &= \epsilon_1 \cdot \epsilon_2\,p_1^{\mu}\, p_1^{\nu}\,,\\
 T_5^{\mu \nu}  &= \epsilon_1 \cdot \epsilon_2\,p_1^{\mu}\, p_2^{\nu}\,,&
 T_6^{\mu \nu}  &= \epsilon_1 \cdot \epsilon_2\,p_2^{\mu}\, p_1^{\nu}\,,&
 T_7^{\mu \nu}  &= \epsilon_1 \cdot \epsilon_2\,p_2^{\mu}\, p_2^{\nu}\,,&
 T_8^{\mu \nu}   &= \epsilon_2 \cdot p_3\,\epsilon_1^\mu \,p_1^{\nu}\,,\\
 T_9^{\mu \nu}    &= \epsilon_2 \cdot p_3\,\epsilon_1^\mu \,p_2^{\nu}\,,&
 T_{10}^{\mu \nu} &= \epsilon_2 \cdot p_3\,\epsilon_1^\nu \,p_1^{\mu}\,,&
 T_{11}^{\mu \nu} &= \epsilon_2 \cdot p_3\,\epsilon_1^\nu \,p_2^{\mu}\,,&
 T_{12}^{\mu \nu} &= \epsilon_1 \cdot p_3\,\epsilon_2^\mu \,p_1^{\nu}\,,\\
 T_{13}^{\mu \nu}  &= \epsilon_1 \cdot p_3\,\epsilon_2^\mu \,p_2^{\nu}\,,&
 T_{14}^{\mu \nu}  &= \epsilon_1 \cdot p_3\,\epsilon_2^\nu \,p_1^{\mu}\,,&
 T_{15}^{\mu \nu}  &= \epsilon_1 \cdot p_3\,\epsilon_2^\nu \,p_2^{\mu}\,,&
 T_{16}^{\mu \nu} &= \epsilon_1 \cdot p_3\,\epsilon_2 \cdot p_3\, g^{\mu \nu}\,,
\end{aligned}
\nonumber\\
\begin{aligned}
 T_{17}^{\mu \nu} &= \epsilon_1 \cdot p_3\,\epsilon_2 \cdot p_3\,p_1^\mu \,p_1^{\nu}\,, &
 T_{18}^{\mu \nu}  &= \epsilon_1 \cdot p_3\,\epsilon_2 \cdot p_3\,p_1^\mu \,p_2^{\nu}\,,
\end{aligned}
\nonumber\\
\begin{aligned}
 T_{19}^{\mu \nu}  &= \epsilon_1 \cdot p_3\,\epsilon_2 \cdot p_3\,p_2^\mu \,p_1^{\nu}\,, &
 T_{20}^{\mu \nu}  &= \epsilon_1 \cdot p_3\,\epsilon_2 \cdot p_3\,p_2^\mu \,p_2^{\nu}\,.
\end{aligned}
\end{gather}
We stress that the tensor decomposition~\eqref{tensordec} is based only
on Lorentz symmetry, gauge invariance and the properties of the boson decays
and holds therefore at every order in perturbative QCD.
Moreover, no assumption has been made on the dimensionality of space-time and the
result is valid for any values of the parameter $d$.

The scalar form factors $A_j$ can be extracted from the amplitude~\eqref{tensordec}
by applying suitable projecting operators. The projectors themselves can be decomposed in the same
$20$ tensors as
\begin{align}
 P_j^{\mu \nu} = \sum_{i=1}^{20} B_{ji} \left( T_i^{\mu \nu}\right)^\dagger \qquad \text{for~}j=1,\ldots,20, \label{proj}
\end{align}
where also $B_{ji}$ are functions of the external invariants and $d$.
Their explicit form can be determined imposing
\begin{align}
 \sum_{pol} P_j^{\mu' \nu'} \left[ \epsilon_{3 \mu'} \epsilon_{4 \nu'}
 \epsilon_{3 \mu}^* \epsilon_{4 \nu}^* \right]
 S^{\mu \nu}
 &= A_j \quad \text{for~}j=1,...,20,
\end{align}
where the polarisation sums are evaluated in $d$ dimensions according to \eqref{polsum12} and \eqref{polsum34}.
The explicit results for the coefficients $B_{ji}$ are rather lengthy and we prefer not to write them here
explicitly. Computer readable files for the latter are given on our project page at {\tt HepForge}.

The partonic current is the only one which receives contributions from
QCD radiative corrections and, for two gluons of helicities $\lambda_1$ and $\lambda_2$,
can be written as
\begin{equation}
 S_{\mu \nu}(p_1^{\lambda_1},p_2^{\lambda_2},p_3) = \delta^{a_1 a_2}\,\sum_{j}
  \mathcal{C} _{V_1 V_2}^{[j]}\,S^{[j]}_{\mu \nu \rho \sigma}(p_1,p_2,p_3)
 \epsilon_{1 \lambda_1 }^{\rho}(p_1)\epsilon_{2 \lambda_2}^{\sigma }(p_2)\,,
 \label{ParCurr1}
\end{equation}
where
$\delta^{a_1 a_2}$ is the overall colour structure and
the index $j$ runs over different possible classes of diagrams
discussed below, see also Fig.~\ref{fig:diagclass}, which are characterised by
different electroweak couplings $ \mathcal{C} ^{[j]}_{V_1 V_2}$.

Before proceeding, it is convenient to introduce some notations needed in the
following.
As long as we work in QCD, we only need to consider
the coupling of electroweak vector bosons $V$ to fermions.
We follow~\cite{Denner:1991kt} and parametrise the couplings as
\begin{equation}
\mathcal{V}_\mu^{V f_1 f_2} = i\, e\, \Gamma_\mu^{V f_1 f_2}\,, \qquad
\mbox{where} \quad e = \sqrt{4\,\pi\,\alpha} \quad \mbox{is the positron charge}\,,
\end{equation}
such that all fermion charges are expressed in units of $e$ and
\begin{equation}
\Gamma_\mu^{V f_1 f_2} = L_{f_1 f_2}^V \, \gamma_\mu \left( \frac{1-\gamma_5}{2}\right)
+ R_{f_1 f_2}^V \, \gamma_\mu \left( \frac{1+\gamma_5}{2}\right)\,,
\end{equation}
with
\begin{alignat}{2}
L_{f_1f_2}^\gamma &= -e_{f_1} \,\delta_{f_1f_2} &
R_{f_1f_2}^\gamma &= -e_{f_1} \,\delta_{f_1f_2}\,,
\label{gLBcoupl}
\\
L_{f_1f_2}^Z &= \frac{I_3^{f_1} - \sin^2 {\theta_w} e_{f_1}}{\sin{\theta_w} \cos{\theta_w}} \,\delta_{f_1f_2}\,, &\qquad
R_{f_1f_2}^Z &= -\frac{\sin{\theta_w} e_{f_1}}{\cos{\theta_w}} \,\delta_{f_1f_2}\,,
\label{ZLRcoupl}
\\
L_{f_1f_2}^W &= \frac{1 }{\sqrt{2}\, \sin{\theta_w}} \,\epsilon_{f_1f_2} \,,&
R_{f_1f_2}^W &= 0\,, \label{WLRcoupl}
\end{alignat}
where $\epsilon_{f_1f_2}$ is unity for $f_1\neq f_2$, but belonging to the same isospin doublet,
and zero otherwise.

\begin{figure}
\centerline{\includegraphics[width=0.95\linewidth]{diagclass}}
\caption{\label{fig:diagclass}
Example Feynman diagrams for the process $gg \to V_1 V_2$ at the two-loop
level, where the vector bosons couple to the same fermion loop, $[A]$, to
different fermion loops, $[B]$, or to an intermediate vector boson, $[F_V]$.
The sum of all type $[B]$ contribution and the sum of all type $[F_V]$
contributions vanish, respectively.
}
\end{figure}

Let us consider the different electroweak coupling structures in detail.
It is clear that, since we do not take any electroweak radiative corrections into account,
at least one of the two vector bosons must be
coupled to an internal fermion loop.
In order to compute the one- and two-loop massless QCD corrections
we need to consider the following three possibilities, see Fig.~\ref{fig:diagclass}.
\begin{description}
 \item[Class $\mathbf{A}$:] Both vector bosons $V_1 V_2$ are attached to the same fermion loop.
 In this case the diagrams
 are proportional to the charge weighted sum of the quark flavours, which we denote as
$ \mathcal{C} _{V_1 V_2}^{[A]} = N_{V_1 V_2}$.
These diagrams could in principle yield two different contributions.
One, proportional to the sum of the vector-vector and the axial-axial couplings,
in which all dependence on $\gamma_5$ cancels out. The second, instead, contains
the vector-axial coupling and is linear in $\gamma_5$.
Due to charge parity conservation this last contribution is expected to always vanish
identically for massless
quarks running in the loops,
for any choice of $V_1$ and $V_2$~\cite{Glover:1988rg,Glover:1988fe,Melnikov:2015laa}.
One then easily finds that
\begin{align}\label{coupl1}
N_{\gamma \gamma} &= \frac{1}{2}\sum_{i} \left[ \left( L_{q_i q_i}^\gamma \right)^2 + \left( R_{q_i q_i}^\gamma \right)^2 \right], &
N_{Z \gamma} &=  \frac{1}{2} \sum_{i} \left( L_{q_i q_i}^Z L_{q_i q_i}^\gamma + R_{q_i q_i}^Z R_{q_i q_i}^\gamma \right), \nonumber\\
N_{ZZ} &= \frac{1}{2} \sum_{i} \left[  \left( L_{q_i q_i}^Z\right)^2 + \left(R_{q_i q_i}^Z \right)^2 \right], &
N_{WW} &= \frac{1}{2} \sum_{i,\,j} \left( L_{q_i q_j}^W  L_{q_j q_i}^W \right),
\end{align}
where the indices $i,j$ run over the flavours of the quarks in the loop and
$L_{q_i q_i}^\gamma = R_{q_i q_i}^\gamma$ such that $N_{\gamma\gamma}=\sum_{i} e_{q_i}^2$.

\item[Class $\mathbf{B}$:] The two vector bosons are attached to two different fermion loops.
This configuration is of course possible only starting from two loops on.
Each fermion loop contains both a vector and an axial piece.
For the case of two-loop massless QCD corrections relevant here,
both contributions can be shown to vanish.
The axial contribution cancels out for degenerate isospin doublets, while the vector
piece must sum up to zero due to Furry's theorem.

\item[Classes $\mathbf{F_V}$:] Only for the case of $V_1 V_2 = W^+W^-$, one should also take into
account the $s$-channel production diagrams, where the incoming gluons produce an intermediate electroweak
gauge boson $V=\gamma^*/Z^*$, which then decays into the outgoing $W$-pair, see Fig.~\ref{fig:diagclass}.
Charge-parity invariance ensures that the vector part of these diagrams must sum up to zero.
Again, the axial part cancels out for degenerate isospin doublets, and therefore also in
the case of massless quarks running in the loops.
\end{description}
For the case of the one- and two-loop contributions considered here, we can therefore
simplify~\eqref{ParCurr1} to
\begin{align}
 S_{\mu \nu}(p_1^{\lambda_1},p_2^{\lambda_2},p_3) &=
 \delta^{a_1 a_2}\,N_{V_1 V_2}\,S^{[A]}_{\mu \nu \rho \sigma}(p_1,p_2,p_3)
  \epsilon_{1 \lambda_1 }^{\rho}(p_1)\epsilon_{2 \lambda_2}^{\sigma }(p_2)
\,,
 \label{ParCurr2}
\end{align}
with $N_{V_1 V_2}$ given in~\eqref{coupl1}
and consider the coefficients $A_j^{[A]}$ defined by
\begin{align}
A_j(s,t,p_3^2,p_4^2) &= \delta_{a_1 a_2} N_{V_1V_2} A_j^{[A]}(s,t,p_3^2,p_4^2).
\end{align}

It is instructive to study the transformations of the partonic current~\eqref{ParCurr2}
under permutations of the external legs. We define the following two permutations
\begin{align}
&\pi_{12} := p_1 \leftrightarrow p_2 \Rightarrow \{\,t \leftrightarrow u \,\}, \nonumber \\
&\pi_{34} := p_3 \leftrightarrow p_4 \Rightarrow \{\, t \leftrightarrow u\,, \;\; p_3^2 \leftrightarrow p_4^2 \,\} \,.
\label{perm}
\end{align}
Because of Bose symmetry these two permutations must leave the partonic amplitude unchanged.
This enforces a well defined behaviour of the coefficients $A_j(s,t,p_3^2,p_4^2)$ under the action
of $\pi_{12}$ and $\pi_{34}$. From direct inspection of~\eqref{tensordec}  one finds that the following
relations must be fulfilled:
\begin{alignat}{3}
\pi_{12}:\quad
A^{[A]}_1(s,u,p_3^2,p_4^2) &= A^{[A]}_1(s,t,p_3^2,p_4^2)\,,
\qquad  &
A^{[A]}_2(s,u,p_3^2,p_4^2) &= A^{[A]}_3(s,t,p_3^2,p_4^2)\,,
\nonumber \\
A^{[A]}_4(s,u,p_3^2,p_4^2) &= A^{[A]}_7(s,t,p_3^2,p_4^2)\,,
&
A^{[A]}_5(s,u,p_3^2,p_4^2) &= A^{[A]}_6(s,t,p_3^2,p_4^2)\,,
\nonumber \\
A^{[A]}_8(s,u,p_3^2,p_4^2) &= A^{[A]}_{13}(s,t,p_3^2,p_4^2)\,,
&
A^{[A]}_{9}(s,u,p_3^2,p_4^2) &= A^{[A]}_{12}(s,t,p_3^2,p_4^2)\,,
\nonumber \\
A^{[A]}_{10}(s,u,p_3^2,p_4^2) &= A^{[A]}_{15}(s,t,p_3^2,p_4^2)\,,
&
A^{[A]}_{11}(s,u,p_3^2,p_4^2) &= A^{[A]}_{14}(s,t,p_3^2,p_4^2)\,,
\nonumber \\
A^{[A]}_{16}(s,u,p_3^2,p_4^2) &= A^{[A]}_{16}(s,t,p_3^2,p_4^2)\,,
&
A^{[A]}_{17}(s,u,p_3^2,p_4^2) &= A^{[A]}_{20}(s,t,p_3^2,p_4^2)\,,
\nonumber \\
A^{[A]}_{18}(s,u,p_3^2,p_4^2) &= A^{[A]}_{19}(s,t,p_3^2,p_4^2)\,,
&&\label{Ajx12}
\\[1ex]
\pi_{34}:\quad
A^{[A]}_1(s,u,p_4^2,p_3^2) &= A^{[A]}_1(s,t,p_3^2,p_4^2)\,,
&
A^{[A]}_2(s,u,p_4^2,p_3^2) &= A^{[A]}_3(s,t,p_3^2,p_4^2)\,,
\nonumber \\
A^{[A]}_4(s,u,p_4^2,p_3^2) &= A^{[A]}_4(s,t,p_3^2,p_4^2)\,,
&
A^{[A]}_5(s,u,p_4^2,p_3^2) &= A^{[A]}_6(s,t,p_3^2,p_4^2)\,,
\nonumber \\
A^{[A]}_7(s,u,p_4^2,p_3^2) &= A^{[A]}_7(s,t,p_3^2,p_4^2)\,,
&
A^{[A]}_{8}(s,u,p_4^2,p_3^2) &= - A^{[A]}_{10}(s,t,p_3^2,p_4^2)\,,
\nonumber \\
A^{[A]}_{9}(s,u,p_4^2,p_3^2) &= - A^{[A]}_{11}(s,t,p_3^2,p_4^2)\,,
&
A^{[A]}_{12}(s,u,p_4^2,p_3^2) &= - A^{[A]}_{14}(s,t,p_3^2,p_4^2)\,,
\nonumber \\
A^{[A]}_{13}(s,u,p_4^2,p_3^2) &= - A^{[A]}_{15}(s,t,p_3^2,p_4^2)\,,
&
A^{[A]}_{16}(s,u,p_4^2,p_3^2) &= A^{[A]}_{16}(s,t,p_3^2,p_4^2)\,,
\nonumber \\
A^{[A]}_{17}(s,u,p_4^2,p_3^2) &= A^{[A]}_{17}(s,t,p_3^2,p_4^2)\,,
&
A^{[A]}_{18}(s,u,p_4^2,p_3^2) &= A^{[A]}_{19}(s,t,p_3^2,p_4^2)\,,
\nonumber \\
A^{[A]}_{20}(s,u,p_4^2,p_3^2) &= A^{[A]}_{20}(s,t,p_3^2,p_4^2)\,,
&& \label{Ajx34}
\end{alignat}
It is interesting to notice that, upon exploiting all of these crossing relations,
only $9$ out of the $20$ coefficients $A_j^{[A]}$ turn out to be effectively
independent, while the other $11$ coefficients can be obtained by crossing of
the external legs.

\section{\texorpdfstring
{Helicity amplitudes for $gg \to V_1 V_2 \to 4~\mathrm{leptons}$}
{Helicity amplitudes for gg -> V1 V2 -> 4 leptons}
}
\label{sec:helamp}

We consider physical processes, where the two off-shell vector bosons
decay into lepton pairs
\begin{equation}
 g(p_1) + g(p_2) \rightarrow V_1(p_3) + V_2(p_4) \rightarrow l_5(p_5) + \bar{l}_6(p_6) + l_7(p_7) + \bar{l}_8(p_8)
 \label{ggVVllll}
\end{equation}
such that
$p_3= p_5+p_6\,$, $\,p_4=p_7+p_8$ and $p_5^2=p_6^2=p_7^2=p_8^2 = 0$.
As long as we consider QCD radiative corrections
the amplitudes $\mathcal{M}_{\lambda_1 \lambda_2 \lambda_3 \lambda_4}^{V_1 V_2}$ can be written, at any
order in perturbation theory, as the product of the partonic current for
$gg \to V_1 V_2$ with the two leptonic currents for the decay products,
$V_1 \to l_5 \bar{l}_6$ and $V_2 \to l_7 \bar{l}_8$,
mediated by the propagators of the two off-shell vector bosons $V_1$ and $V_2$.
We write the propagator for an off-shell vector boson in the $R_\xi$ gauge as
\begin{equation}
P_{\mu \nu}^V(q) = \frac{i\, \Delta^V_{\mu \nu}(q,\xi)}{D_V(q)}\,, \label{prop}
\end{equation}
with
\begin{equation}
\Delta^V_{\mu \nu}(q,\xi) = \left( - g_{\mu \nu} + (1 - \xi) \frac{q_\mu q_\nu}{q^2 - \xi m_V^2}\right)\,,
\label{numprop}
\end{equation}
\begin{align}
&D_{\gamma^*}(q) = q^2\,,\qquad D_{Z,W}(q) = (q^2 - m_V^2 + i\,\Gamma_V m_V)\,,\label{denprop}
\end{align}
where $m_V$ is its mass and $\Gamma_V$ is its decay width.
In our case the massive vector bosons couple to massless fermion lines such that the
term proportional to $(1-\xi)$ can be dropped.

In the following we consider fixed helicities of the external particles
and compute the amplitudes for the different helicity configurations.
While the quantities and formulae presented up to this point were treated in $d$ dimensions
throughout, we now consider 4-dimensional external states in order to compute
the amplitudes for specific helicities.
Since the decay leptons are massless, helicity is conserved along the leptonic decay currents
and the amplitude can be written as
\begin{equation}
\mathcal{M}_{\lambda_1 \lambda_2 \lambda_3 \lambda_4}^{V_1 V_2}(p_1, p_2;p_5,p_6,p_7,p_8),
\end{equation}
where $\lambda_1$ and $\lambda_2$ are the helicities of the incoming gluons, while
$\lambda_3$ and $\lambda_4$ are the helicities of the two
leptonic currents. It is clear that there are 16 different helicity configurations,
depending on the different possibilities for the initial and final states.
Each gluon has two possible helicity states, which we denote
by $L$ (-) and $R$ (+), and similarly each leptonic current occurs in either left- or
right-handed configuration, again denoted by $L$ and $R$, respectively, such that
$\lambda_j = L,R$, for $j=1,...,4$. As we will show explicitly later on,
all 16 helicity configurations can be obtained from only two independent ones, by simple
permutations of the external legs and complex conjugation.
We choose as independent configurations the following two
\begin{equation}
\mathcal{M}_{LLLL}^{V_1 V_2}(p_1, p_2;p_5,p_6,p_7,p_8)\,, \qquad
\mathcal{M}_{LRLL}^{V_1 V_2}(p_1, p_2;p_5,p_6,p_7,p_8)\,. \label{helconf}
\end{equation}

With the notations introduced above we write the two independent helicity amplitudes~\eqref{helconf},
up to two loops, as:
\begin{align}
 \mathcal{M}_{\lambda_1 \lambda_2 LL}^{V_1 V_2}(p_1, p_2;p_5,p_6,p_7,p_8) &=
 (4 \pi \alpha)^2\,
 \frac{L_{f_5f_6}^{V_1}\,L_{f_7f_8}^{V_2}}{D_{V_1}(p_3)D_{V_2}(p_4)}\,
  {\rm M} _{\lambda_1 \lambda_2 LL}(p_1, p_2;p_5,p_6,p_7,p_8)\,,\label{helampl}
\end{align}
where
the \textsl{basic amplitudes} $ {\rm M} _{\lambda_1 \lambda_2 LL}(p_1, p_2;p_5,p_6,p_7,p_8)$
are constructed from the partonic current~\eqref{ParCurr1} and
the leptonic currents~\eqref{LepCurrL} according to
\begin{align}
  {\rm M} _{\lambda_1 \lambda_2 LL}(p_1, p_2;p_5,p_6,p_7,p_8) =
 \epsilon_{1 \lambda_1 }^{\rho}(p_1)\epsilon_{2 \lambda_2}^{\sigma }(p_2)
 \,S_{\mu \nu\rho \sigma}(p_1,p_2,p_3)\,L_{L}^{\mu}(p_5^-,p_6^+)\,L_{L}^{\nu}(p_7^-,p_8^+)\,. \label{Bhelampl}
\end{align}
The leptonic decay currents do not receive
any QCD corrections and are simple tree-level objects. They can be easily
expressed in the usual spinor-helicity notation~\cite{Dixon:1996wi,Dixon:1998py} as
\begin{align}
&L^\mu_L(p_5^-,p_6^+) = \bar{u}_-(p_5) \,\gamma^{\mu} \, v_+(p_6) = [ 6\, | \gamma^{\mu } | \,5\, \rangle
= \langle 5\, | \gamma^{\mu } | \,6\, ]  \,,\label{LepCurrL}\\
&L^\mu_R(p_5^+,p_6^-) = \bar{u}_+(p_5) \,\gamma^{\mu}\, v_-(p_6)=  [ 5\, | \gamma^{\mu } | \,6\, \rangle
= \left( L^\mu_L(p_5^-,p_6^+) \right)^* = L_L^\mu(p_6^-,p_5^+)\,. \label{LepCurrR}
\end{align}
Note that, in this case, a permutation of the external
momenta is equivalent to a complex conjugation of the current and
it corresponds to a flip of the helicity $L \leftrightarrow R$.

Once the tensor decomposition of the partonic current is fixed, it is straight-forward
to express the two basic helicity amplitudes $ {\rm M} _{LLLL}$ and $ {\rm M} _{LRLL}$ in~\eqref{Bhelampl}
in the usual spinor-helicity notation~\cite{Dixon:1996wi,Dixon:1998py}.
We replace the gluon polarisation vectors according to
\begin{align}
 \epsilon_{1L}^\mu(p_1) = \frac{[ 2 | \gamma^\mu | 1 \rangle }{\sqrt{2} [12]}\,, \quad
 \epsilon_{1R}^\mu(p_1) =  \frac{\langle 2 | \gamma^\mu | 1 ] }{\sqrt{2} \langle 21 \rangle }\,, \quad
 \epsilon_{2L}^\mu(p_2) = \frac{[ 1 | \gamma^\mu | 2 \rangle }{\sqrt{2} [21]}\,, \quad
 \epsilon_{2R}^\mu(p_2) =  \frac{\langle 1 | \gamma^\mu | 2 ] }{\sqrt{2} \langle 12 \rangle }\,,
 \label{polvec}
\end{align}
which is of course compatible with the polarisation sums~\eqref{polsum12} and \eqref{polsum34}.
Note again that here we are assuming $4$-dimensional external states. This allows to
reduce considerably the number of independent structures that are required for
parametrising a specific helicity configuration.
Using~\eqref{polvec} we find that both basic amplitudes can be written in terms of $9$ independent
spinor structures as
\begin{align}
 {\rm M} _{\lambda_1 \lambda_2 LL}(p_1,p_2;p_5,p_6,p_7,p_8) &=
 C_{\lambda_1 \lambda_2} \Bigg\{ [ 2\, p \hspace{-1.0ex} /_3\, 1 \rangle\, \Big(
   E_1^{\lambda_1 \lambda_2} \langle 5 7 \rangle [ 68 ] \nonumber \\
 &+ E_2^{\lambda_1 \lambda_2}\, \langle 15 \rangle \langle 17 \rangle [16][18]
+ E_3^{\lambda_1 \lambda_2}\, \langle 15 \rangle \langle 27 \rangle [16][28] \nonumber \\
&+ E_4^{\lambda_1 \lambda_2}\, \langle 25 \rangle \langle 17 \rangle [26][18]
 +
 E_5^{\lambda_1 \lambda_2}\, \langle 25 \rangle \langle 27 \rangle [26][28] \,
  \Big) \nonumber \\
&+ E_6^{\lambda_1 \lambda_2}\, \langle 15 \rangle \langle 17 \rangle [16][28]
+   E_7^{\lambda_1 \lambda_2}\, \langle 15 \rangle \langle 17 \rangle [26][18] \nonumber \\
&+ E_8^{\lambda_1 \lambda_2}\, \langle 15 \rangle \langle 27 \rangle [26][28]
+ E_9^{\lambda_1 \lambda_2}\, \langle 25 \rangle \langle 17 \rangle [26][28] \Bigg\} \,, \label{MllLL}
\end{align}
where the $18$ newly introduced form factors $E_j^{\lambda_1 \lambda_2}$
are simple linear combinations of the scalar coefficients $A_j$.
The spinor structure of the amplitudes for the configurations $LLLL$ and $LRLL$
differs only by an overall factor which reads in the two cases
\begin{align}\label{cll}
 C_{LL} = [1\,p \hspace{-1.0ex} /_3\,2\rangle \frac{\langle 1 2 \rangle }{[12]}\,,\qquad
 C_{LR} = [2\,p \hspace{-1.0ex} /_3\,1\rangle\,,
\end{align}
but the form factors $E_j^{LL}$ and $E_j^{LR}$ are different.
We also note, in passing, that the spinor structure of~\eqref{MllLL} exhibits also
a \emph{formal} similarity to that of the $RLL$ amplitude for
$q \bar{q}' \to V_1 V_2 \to l_5\bar{l}_6 l_7 \bar{l}_8$~\cite{Caola:2014iua,Gehrmann:2015ora},
again up to an overall factor and with, of course, completely unrelated form factors.
Similar as before, we also define the functions $E_j^{\lambda_1\lambda_2\,[A]}$
\begin{equation}
E_j^{\lambda_1\lambda_2}(s,t,p_3^2,p_4^2) = \delta_{a_1 a_2} N_{V_1V_2} E_j^{\lambda_1\lambda_2\,[A]}(s,t,p_3^2,p_4^2).
\end{equation}
The explicit expressions for the form factors $E_j^{\lambda_1\lambda_2}$ in terms
of the coefficients $A_j$ are given in Appendix~\ref{sec:form}.

In order to obtain all $16$ helicity amplitudes from~\eqref{MllLL}, one should recall that
complex conjugation has the effect of reversing the helicity of the external
gluons,
\begin{align}
  \left( \epsilon_{1L}^\mu(p_1) \right)^*  = \epsilon_{1R}^\mu(p_1)\,,& \qquad
  \left( \epsilon_{2L}^\mu(p_2) \right)^*  = \epsilon_{2R}^\mu(p_2)\,,
 \end{align}
and similarly for the leptonic currents, see~\eqref{LepCurrR}~\eqref{LepCurrL}.
We define with the symbol $[...]^C$ a complex-conjugation operation which, when applied on the amplitudes
$ {\rm M} _{\lambda_1 \lambda_2 LL}$, acts \textsl{only on the spinor structures}, i.e. leaves
invariant the form factors $E_{j}^{\lambda_1 \lambda_2}$. Given the explicit
form of~\eqref{MllLL}, it is easy to see that this corresponds to simply exchanging
angle brackets with squared bracket and vice versa
\begin{equation}
 \left[  {\rm M} _{\lambda_1 \lambda_2 LL} \right]^C
 \equiv  {\rm M} _{\lambda_1 \lambda_2 LL} \left( \langle ij \rangle \leftrightarrow [ij] \right)\,.
\end{equation}
Hence, we can derive the missing helicity amplitudes for
left-handed leptonic currents from the two basic amplitudes as
\begin{align}
  {\rm M} _{RLLL}(p_1,p_2;p_5,p_6,p_7,p_8) &= \left[  {\rm M} _{LRLL}(p_1,p_2;p_6,p_5,p_8,p_7) \right]^C\,,\nonumber \\
  {\rm M} _{RRLL}(p_1,p_2;p_5,p_6,p_7,p_8) &= \left[  {\rm M} _{LLLL}(p_1,p_2;p_6,p_5,p_8,p_7) \right]^C\,,
\end{align}
where one should note that the lepton and anti-lepton momenta are exchanged in the r.h.s.
in order to have a left-handed leptonic currents on the l.h.s.
The corresponding formulae for the basic amplitudes for right-handed leptonic currents
can be obtained from the ones above by simple permutations of the lepton and anti-lepton momenta
\begin{align}
  {\rm M} _{\lambda_1 \lambda_2 RL}(p_1,p_2;p_5,p_6,p_7,p_8) &=
  {\rm M} _{\lambda_1 \lambda_2LL}(p_1,p_2;p_6,p_5,p_7,p_8)  \,,\nonumber \\
  {\rm M} _{\lambda_1 \lambda_2LR}(p_1,p_2;p_5,p_6,p_7,p_8) &=
  {\rm M} _{\lambda_1 \lambda_2LL}(p_1,p_2;p_5,p_6,p_8,p_7)  \,,\nonumber \\
  {\rm M} _{\lambda_1 \lambda_2RR} (p_1,p_2;p_5,p_6,p_7,p_8)  &=
 {\rm M} _{\lambda_1 \lambda_2LL}(p_1,p_2;p_6,p_5,p_8,p_7)  \,. \label{allBhelamp}
\end{align}
With these formulae also all the 16 physical amplitudes
$\mathcal{M}^{V_1 V_2}_{\lambda_1 \lambda_2 \lambda_3 \lambda_4}$ in~\eqref{helampl}
can be easily obtained, recalling that in the case of right-handed leptonic
currents one should, of course, exchange the corresponding
couplings $L_{f_i f_j}^{V} \leftrightarrow R_{f_i f_j}^V$.

As we already stated above, the partonic current receives contributions
form QCD radiative corrections and it can be expanded as
\begin{align}
 S_{\mu \nu \rho\sigma}(p_1,p_2,p_3) =
 \left(\frac{\alpha_s}{2 \pi} \right) S_{\mu \nu \rho\sigma}^{(1)}(p_1,p_2,p_3)
 + \left(\frac{\alpha_s}{2 \pi} \right)^2 S_{\mu \nu \rho\sigma}^{(2)}(p_1,p_2,p_3)
 + \mathcal{O}(\alpha_s^3)\,,
\end{align}
where obviously the perturbative expansion starts only at one-loop order.
Of course also the coefficients $A_j$, and equivalently the form factors
$E_j^{\lambda_1 \lambda_2}$, have the same expansion
\begin{align}
 A_j &=  \left(\frac{\alpha_s}{2 \pi} \right) A_j^{(1)}
 + \left(\frac{\alpha_s}{2 \pi} \right)^2 A_j^{(2)}
 + \mathcal{O}(\alpha_s^3)\,, \nonumber \\
 E_j^{\lambda_1 \lambda_2} &=
 \left(\frac{\alpha_s}{2 \pi} \right) E_j^{(1),\lambda_1 \lambda_2}
 + \left(\frac{\alpha_s}{2 \pi} \right)^2 E_j^{(2),\lambda_1 \lambda_2}\,
 + \mathcal{O}(\alpha_s^3)\,.
\end{align}

\section{Calculation of the form factors}
\label{sec:calc}

The calculation of the coefficients $E_j^{\lambda_1 \lambda_2}$ proceeds as follows.
We produce all one- and two-loop Feynman diagrams relevant for $gg \to V_1 V_2$
using {\tt Qgraf}~\cite{Nogueira:1991ex}. In particular we focus only on diagrams in classes
$A$ and $B$ with massless quarks, for which we find $8$ diagrams at one loop and
$138$ diagrams at two loops. Diagrams in class $F_V$, in fact, are simple three-point functions,
which sum up to zero due to charge-parity invariance.
The coefficients $A_j$ are then
calculated by applying the projectors defined in~\eqref{proj} on the different
Feynman diagrams.
We insert the Feynman rules in our diagrams, where we employ the
Feynman-'t\;Hooft gauge ($\xi = 1$) for internal gluons.
After evaluation of Dirac traces and contraction of Lorentz indices
every Feynman diagram is expressed as linear combination of a large
number of scalar integrals. The latter belong to the family of the massless
four-point functions with two off-shell legs of different virtualities and
can be reduced to a small set of master integrals using
integration-by-parts identities~\cite{Tkachov:1981wb,Chetyrkin:1981qh,Gehrmann:1999as,Laporta:2001dd}.
We employ {\tt Reduze}~2~\cite{vonManteuffel:2012np,Studerus:2009ye,Bauer:2000cp,fermat} to map
all scalar integrals to the three integral families given in~\cite{Gehrmann:2014bfa}
and their crossed versions,
and subsequently to reduce them to master integrals.
In this way, we obtain analytical expressions for the coefficients $A_j$ as linear combinations of
the latter. For the master integrals we employ the solutions presented in \cite{Gehrmann:2015ora}.
With the explicit expressions for the coefficients $A_j$ at the
different perturbative orders, it is easy to obtain the corresponding
results for the $E_j^{\lambda_1 \lambda_2}$ using the formulae given
in Appendix~\ref{sec:form}. {\tt Form}~\cite{Vermaseren:2000nd} was used
extensively for all intermediate algebraic manipulations.\newline

Because of the lack of any tree-level contribution to the process $gg \to V_1V_2$,
the UV and IR pole-structure of the one- and two-loop amplitudes is very simple.
Clearly, the one-loop amplitude must be both UV- and IR-finite, and therefore
the pole structure of the two-loop amplitude will be, \textsl{effectively},
what one usually encounters for a one-loop QCD amplitude.
As discussed above, QCD radiative corrections affect only the partonic amplitude
and can be taken into account via the $20$ independent
scalar coefficients $A_j$, see Eq.~\eqref{tensordec}.
Working in conventional dimensional regularisation, we may alternatively
consider the $18$ physically relevant form factors $E_i^{\lambda_1 \lambda_2}$ defined
as $d$ dimensional linear combinations of the $A_j$, see Appendix~\ref{sec:form}.
In what follows, all considerations regarding UV-renormalisation and the structure of
the IR poles of the partonic amplitude hold identically for any $A_j$ and $E^{\lambda_1 \lambda_2}_i$.
We will therefore focus on the scalar coefficients rather than on the full partonic amplitude,
and use the symbol $\Omega$ to refer to any of the latter,
$$\Omega \in \left\{ A_j\,, E^{\lambda_1 \lambda_2}_i \right\}\,,\qquad \mbox{for any}
\quad j=1,...,20, \quad i=1,...,9, \quad \lambda_1\lambda_2 = LL,LR\,.$$

We start by performing UV-renormalisation in the $\overline{ \rm MS}$ scheme.
In massless QCD this amounts to replacing the bare coupling, $\alpha_0$,
with the renormalised one,
$\alpha_s = \alpha_s(\mu^2)$, where $\mu$ is the renormalisation scale.
Here we only need the one-loop relation
\begin{equation}
\alpha_0\, \mu_0^{2 \epsilon}\, S_\epsilon = \alpha_s\, \mu^{2 \epsilon}
\left [ 1 - \frac{\beta_0}{\epsilon} \left( \frac{\alpha_s}{2 \pi} \right)+
\mathcal{O}(\alpha_s^2)\right]\,,
\end{equation}
where
\begin{equation}
S_\epsilon = (4 \pi)^\epsilon \,  {\rm e} ^{-\epsilon \gamma}\,,
\qquad \mbox{with the Euler-Mascheroni constant} \quad \gamma = 0.5772...\,,
\end{equation}
 $\epsilon = (4-d)/2$, $\mu_0$ is the mass-parameter introduced in dimensional regularisation to maintain
a dimensionless coupling in the bare QCD Lagrangian density, and finally $\beta_0$
is the first order of the QCD $\beta$-function
\begin{equation}
\label{beta0}
\beta_0 = \frac{11 \,C_A - 4 \,T_F\,N_f}{6}\,, \quad \mbox{with}\quad C_A = N\,,
\quad C_F = \frac{N^2-1}{2\,N}\,, \quad
T_F = \frac{1}{2}\,.
\end{equation}
The renormalisation is performed at $\mu^2 = s$, the invariant mass squared of the vector-boson
pair. The renormalised form factors read then, in terms of the un-renormalised ones,
\begin{eqnarray}
\Omega^{(1)}  &=&
S_\epsilon^{-1} \Omega^{(1),{\rm un}} ,  \nonumber \\
\Omega^{(2)} &=&
S_\epsilon^{-2} \Omega^{(2),{\rm un}}
-\frac{\beta_0 }{\epsilon} S_\epsilon^{-1}
\Omega^{(1),{\rm un}}\;.
\end{eqnarray}

After UV renormalisation, the two-loop  coefficients $\Omega^{(2)}$
contain still residual IR singularities.
In any IR-safe observable these divergences are cancelled
by the corresponding ones produced in one-loop radiative processes with one more
external parton.
In the present case of $gg \to V_1 V_2$, as discussed already above,
the IR-poles at two loops are of NLO type and their structure has been know for a long time.
Here, we choose to follow the conventions used for the NNLO corrections
to $q\bar{q}\to V_1 V_2$ in~\cite{Gehrmann:2015ora}, which required a NNLO subtraction scheme.
The exact structure of the IR poles up to NNLO in QCD was predicted
first by Catani~\cite{Catani:1998bh}.
We present our results in a slightly modified scheme described in~\cite{Catani:2013tia},
which is well suited for the $q_T$-subtraction formalism.

We define the IR finite amplitudes at renormalisation scale $\mu$
in terms of the UV renormalised ones as follows
\begin{align}
\Omega^{(1), {\rm finite} }_{q_T} &= \Omega^{(1)} \,,\nonumber \\
\Omega^{(2), {\rm finite} }_{q_T} &= \Omega^{(2)} - I_1(\epsilon)\, \Omega^{(1)} \,, \label{IRqT}
\\
\intertext{where for the gluon-fusion channel we have}
I_1(\epsilon) &= I_1^{soft}(\epsilon) + I_1^{coll}(\epsilon) \,,
\\[0.5ex]
I_1^{soft}(\epsilon) &= -\frac{ {\rm e} ^{\epsilon \gamma}}{\Gamma(1-\epsilon)} \left( \frac{\mu^2}{s}\right)^{\epsilon}\,
\left( \frac{1}{\epsilon^2} + \frac{i \pi}{\epsilon} + \delta_{q_T}^{(0)} \right)\,C_A
\,,\\
I_1^{coll}(\epsilon) &=- \frac{1}{\epsilon} \beta_0\, \left( \frac{\mu^2}{s}\right)^{\epsilon}\,.
\end{align}
Following~\cite{Catani:2013tia} we then put
$\delta_{q_T}^{(0)} = 0$.
We provide the explicit analytical results for the finite remainders of the coefficients
$A_j$ in this scheme, obtained for $\mu^2 = s$, on our project page at {\tt HepForge}.

Finally, it is straight-forward to convert these finite remainders
into the Catani's original subtraction scheme~\cite{Catani:1998bh}, as extensively described in~\cite{Gehrmann:2015ora}.
For the present case we obtain the conversion formulae
\begin{align}\label{qt2catani}
\Omega^{(1), {\rm finite} }_{\text{Catani}} &= \Omega^{(1), {\rm finite} }_{q_T} ,\nonumber\\
\Omega^{(2), {\rm finite} }_{\text{Catani}} &= \Omega^{(2), {\rm finite} }_{q_T} + \Delta I_1 \, \Omega^{(1), {\rm finite} }_{q_T},
\end{align}
with $\Delta I_1$, in the case of a $gg$ initial state, is given by
\begin{align}\label{qt2catanicoeff}
 \Delta I_1 &=   -\frac{1}{2} \pi^2 C_A + i \pi \beta_0 \,.
\end{align}

In order to test the correctness of our results we have performed a number of checks,
which we list in the following.
\begin{enumerate}
\item First of all, we computed explicitly all one- and two-loop diagrams relevant for $gg \to V_1 V_2$,
including those diagrams in class $B$ which are expected not to give any contribution due to Furry's theorem,
see Section~\ref{sec:calc}. We have verified that, after reduction to master integrals, all diagrams in
class $B$ sum up to zero.

\item We have verified explicitly that the coefficients $A_j$ respect the expected symmetry relations
derived in~\eqref{Ajx12} and \eqref{Ajx34}.

\item We have verified explicitly that the IR poles of the two-loop amplitude have the structure
predicted by Catani's formula, see Section~\ref{sec:calc}. This provides a strong
check of the correctness of the result.

\item We have performed a thorough comparison of our results with
an independent calculation of the same process~\cite{Caola:2015ila}.
Specifically, we compared our results prior to UV renormalisation and
IR subtraction. While the representation of the amplitudes in terms of
spinor structures in~\cite{Caola:2015ila} has a different form than
our decomposition~\eqref{MllLL},
we found that both are equivalent.
For the full helicity amplitudes we have found perfect numerical agreement
at one- and two-loop order.
Moreover, expressing the form factors defined in~\cite{Caola:2015ila} as linear
combinations of our form factors $E_j^{\lambda_1 \lambda_2}$, we have verified
that for each of them independently we have perfect numerical agreement
at one- and two-loop order.
\end{enumerate}

\section{Numerical {\tt C++} implementation and results}
\label{sec:num}

\begin{figure}
\includegraphics[width=\textwidth]{plot3dejll}
\caption{\label{fig:plot3dejll}
Real parts of the two loop form factors $E_j^{(2),LL\,[A]}$
for the process $gg\to V_1V_2$.
The plots illustrate their dependence on the velocity, $\beta_3$,
and the cosine of the scattering angle, $\cos\theta_3$,
of the vector boson $V_1$,
where $p_4^2 = 2 p_3^2$ is chosen for the vector boson virtualities.
}
\end{figure}

\begin{figure}
\includegraphics[width=\textwidth]{plot3dejlr}
\caption{\label{fig:plot3dejlr}
Real parts of the two loop form factors $E_j^{(2),LR\,[A]}$
for the process $gg\to V_1V_2$.
The plots illustrate their dependence on the velocity, $\beta_3$,
and the cosine of the scattering angle, $\cos\theta_3$,
of the vector boson $V_1$,
where $p_4^2 = 2 p_3^2$ is chosen for the vector boson virtualities.
}
\end{figure}

For the numerical evaluation of the helicity amplitudes for $gg \to V_1 V_2\to 4~\mathrm{leptons}$,
we implemented our results for the form factors $E_j^{\lambda_1 \lambda_2\,[A]}$
and $A_j^{[A]}$ at one- and two-loop order in a dedicated {\tt C++} code.
The implementation is based on the solutions for the master integrals presented
in~\cite{Gehrmann:2015ora}, which were specifically constructed for fast and reliable numerical
evaluations.
We organised our form factor implementation in form of a library, which is supplemented by a
simple command line interface.
We provide the software package for public download on {\tt HepForge}\ at
\url{http://vvamp.hepforge.org}.

For the numerical evaluation of the multiple polylogarithms encountered in the solutions for
the master integrals, we employ their implementation~\cite{Vollinga:2004sn} in the {\tt GiNaC}~\cite{Bauer:2000cp} library.
To identify and account for possible numerical instabilities of the form factors in collinear
or other potentially problematic regions of phase space, the code compares numerical evaluations,
which are obtained using different floating point data types, similar to the setup used in~\cite{Gehrmann:2015ora}.
If the results obtained with different precision settings differ beyond a user-defined
tolerance, the code successively increases the precision until the target precision is met.

For the rather central benchmark point of~\cite{Caola:2014iua}, the double precision mode of our code takes
roughly $600ms$ on a single computer core and results in at least $11$ significant digits for all of the
$E_j^{\lambda_1 \lambda_2\,[A]}$.
In order to estimate the actual precision, the default behaviour of our code
is to reevaluate the algebraic expressions in quad precision, which results in a total
run-time of roughly $3 s$ for this phase space point.
The run-time can increase further for regions close to the phase space
boundaries, where the multiple polylogarithms take more time to evaluate
and the precision control of our code may switch to higher precision
computations in order to return reliable numbers.
Depending on the precision setting and on the region of phase space, the
evaluations of the multiple polylogarithms and of the algebraic coefficients
may require comparable portions of the run-time.
However, in double precision mode or close to the phase space boundaries,
the run-time is dominated by multiple polylogarithm evaluations.
The described version of our code implements a minimal set of $9$ coefficients
$A_j$ and employs four evaluations of them with different kinematics
in order to derive the remaining form factors using crossing relations.
If required, it is straight-forward to further improve the evaluation
speed, either by proper caching of multiple polylogarithms or, at the
price of an increased code size, by an explicit implementation of all
form factors, as we did for the process $q \bar{q}' \to V_1 V_2$ in~\cite{Gehrmann:2015ora}.

In order to illustrate the form factors and the reliability of the code,
we used the latter to plot the real part of the two-loop form factors
for the case $p_4^2 = 2 p_3^2$ in Figures~\ref{fig:plot3dejll} and \ref{fig:plot3dejlr}.
In the plots, we vary the relativistic velocity $\beta_3$ and the cosine of
the scattering angle $\cos\theta_3$ of the vector boson $V_1$,
where $\beta_3= \kappa/(s+p_3^2-p_4^2)$ and $\cos\theta_3= (2t+s-p_3^2-p_4^2)/\kappa$.
Compared to the results for the form factors $E_j$ in the process
$q\bar{q}' \to V_1V_2$ in~\cite{Gehrmann:2015ora}, we observe
strong enhancements for the forward, backward and production threshold
regions for the form factors in the present case.
However, for the physical helicity amplitudes~\eqref{MllLL} we wish to point
out that an additional dampening (very) close to the aforementioned
phase space boundaries should be taken into account due to the additional overall
factors $C_{LL}$ and $C_{LR}$~\eqref{cll}.

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

In this paper we computed the two-loop massless QCD corrections
to the helicity amplitudes for the production of pairs of off-shell
electroweak gauge bosons, $V_1 V_2$, in the gluon fusion channel.
For the calculation we employed the solutions for the master integrals
presented in~\cite{Gehrmann:2015ora}.
Contracting the diboson amplitude with the leptonic
decay currents we have constructed the helicity amplitudes for
$gg \to V_1 V_2 \to 4~\mathrm{leptons}$.
We have compared our results to an independent calculation~\cite{Caola:2015ila} and find
perfect agreement.
Our results for these amplitudes provide the fundamental ingredient
required to compute the NLO corrections to diboson production processes in gluon fusion.
These corrections would contribute formally at N$^3$LO to the processes
$pp \to V_1V_2 + X$, but their inclusion may be important to match the
expected experimental accuracy due to the large gluon luminosity at the LHC.
In particular studying their impact is required to obtain a more reliable estimate
of the theory uncertainty and to establish more precise constraints on the total Higgs decay
width~\cite{Campbell:2011cu,Caola:2013yja,Campbell:2013una}.
We provide both analytical results and a {\tt C++} code for the numerical evaluation
of the amplitudes on {\tt HepForge}\ at \url{http://vvamp.hepforge.org}.

\section*{Acknowledgements}
We are grateful to K.~Melnikov and F.~Caola for the comparison of our results with
the results of their independent calculation prior to
publication~\cite{Caola:2015ila}. We wish to thank K.~Melnikov and T.~Gehrmann
for clarifying discussions on different aspects of the calculation,
for interesting comments on the manuscript,
and for the encouragement to carry out this calculation to the end.
We thank the {\tt HepForge}\ team for providing web space for our project.
The Feynman graphs in this article have been drawn with
{\tt JaxoDraw}~\cite{Binosi:2003yf,Vermaseren:1994je}.


China is working on new rules that could enable it to clamp down even more heavily on the Internet.
"As with all Chinese rules, they are written vaguely ... which allows the authorities the room to apply the rules selectively and flexibly," said Lento Yip, the chairman of the Hong Kong Internet Service Providers Association.
China's Ministry of Industry and Information Technology has posted the draft regulations for public feedback until April 25. Violations of the rules would bring a maximum penalty of 30,000 yuan ($4,600).
The new measures fit with China's push under President Xi Jinping for "cyber sovereignty," which has been criticized by rights activists as an attempt to upend how the global Internet is governed.
-- Shen Lu contributed to this report.


text:
These DFSK Vehicles Are Expected to Appear in Pakistan Auto Show 2021
By Waleed Shah | Published Oct 7, 2021 | 6:30 pm
Ever since its arrival in Pakistan, DFSK Glory 580 has become one of the most popular SUVs in the market. While Kia Sportage and MG HS are decidedly the top two crossover SUVs of Pakistan, Glory 580 – especially after the launch of the 'Pro' trim level – has allowed Regal Motors to establish its footing in the Pakistani market as a fierce contender, so much so, that the company is reportedly planning to introduce a few new SUV in the country.
With the Pakistan Auto Show (PAS) 2021 right around the corner, it is reckoned that the company will be putting these vehicles up for display, to catch the attention of the general public.
The following new DFSK vehicles are likely to make an appearance in the PAS 2021:
DFSK Glory 500
MG ZS is starting to become a fairly common sight on the roads, while Kia Stonic is expected to be launched in Pakistan soon. It is only fair that DFSK is also planning to introduce a subcompact SUV of its own in Pakistan, the Glory 500.
It is a B-Segment subcompact crossover SUV that will rival MG ZS, Kia Stonic, Proton X50, and other subcompact crossover SUVs.
Here's How Much Motorcycle Prices Have Risen Since January 2021
Glory 500 is expected to arrive in Pakistan with a 1.5-liter naturally aspirated 4-cylinder petrol engine that makes 118 hp and 147 Nm of torque. The power is sent to the front wheels via a CVT automatic gearbox.
In the international market, the Glory 500 is fitted with features like voice command, automatic headlights, automatic wipers, electronic parking, automatic brake hold assist, a panoramic sunroof, a floating 7-inch touchscreen, a multi-function steering wheel, two airbags, an Anti-lock Braking System (ABS) with Electronic Brakeforce Distribution (EBD) technology, and vehicle reverse assist.
Due to its array of advanced features and a somewhat appealing design, the Glory 500 is placed among the fierce contenders in the subcompact SUV segment by car enthusiasts around the world. Based on that, the Glory 500 will draw a lot of attention at the PAS 2021.
DFSK Glory 580 Star Edition
Earlier this year, reports on the internet surfaced that DFSK would introduce yet another variant of the already existing Glory 580 SUV, called Glory 580 Star Edition, however, not much came of it after a few sporadic rumors.
Glory 580 Star is almost the same compact crossover SUV that competes with the likes of Kia Sportage, Hyundai Tucson, MG HS, etc. However, unlike its competitors, it is the only SUV in its segment that offers a seating capacity for seven passengers.
These Peugeot Cars Will Appear in Pakistan Auto Show 2021
In the international market, the Glory 580 Star is offered with the same 1.5-liter turbocharged 4-cylinder petrol engine that puts out makes 150 horsepower at 2000 RPM and 220 Newton-Meters of torque at 4000 RPM, and sends the power to the front wheels, via a 6-speed CVT automatic gearbox.
The vehicle is likely to offer the same features, with the addition of lane-keeping assist, blind-spot detection, forward collision warning, and adaptive cruise control. There will be a few cosmetic differences on the outside, however, the interior will be majorly revised.
As shown in the photo, the Glory 580 Star has a more modern interior, with a huge infotainment screen plastered front and center, a gearshift dial instead of a lever, and a beautifully sculpted dash design and door panels that make the interior look more stylish.
As there is a lack of further news about the Glory 580 Star's launch in Pakistan, it is uncertain if the vehicle will appear in the PAS 2021, however it will be one of the show's highlights if it does make an appearance.
Seres 3
Along with the reports of Glory 580 Star's arrival in Pakistan, it was also stated that DFSK might be offering the Seres 3 in Pakistan as well. However, like the Glory 580 Star, the reports of the Seres 3 also gradually fizzled out.
To recap, Seres 3 is an all-electric subcompact crossover SUV, whose direct competitor in Pakistan is the MG ZS EV. It is based on the DFSK Glory 500 platform, which is why both SUVs have almost the same dimensions.
It has a single electric motor mount on the front axle and a 52 kWh battery pack in the back. The electric motor makes 160 horsepower, 300 Newton-meters of torque, and offers a range between 230 and 405 kilometers, depending on how it is driven.
Seres 3 has all same features as its internal combustion engine-powered sibling, the DFSK Glory 500. There are some improvements in the vehicle, however, such as various driver-assistance features, lane-keeping assist, blind-spot monitoring, forward collision warning, radar cruise control, autonomous emergency braking, among a few other features.
Changan Cars We Expect to See at Pakistan Auto Show 2021
Although the appearance of Series 3 is also uncertain in the Pakistani market, it is believed that due to the government's inclination toward normalization of EVs, numerous automakers will display EVs in the PAS 2021 and that DFSK is among them.
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Waleed Shah
An automotive enthusiast with a knack for written expression.

redpajama_set_name: RedPajamaCommonCrawl


text:
Canada to send more military trainers to Iraq in battle against IS: Trudeau
by Bill Curry and Steven Chase (feat. David Perry)
Justin Trudeau says he will deploy more soldiers to Iraq as part of the Liberal government's refashioning of Canada's role in the battle against Islamic State.
The Prime Minister, who remains determined to withdraw from a combat role in the region and bring home CF-18 fighters, even after IS claimed responsibility for last week's Paris attacks, said he will instead expand the military training Canadian soldiers are providing in Iraq.
He said the number of Canadian Armed Forces trainers on the ground will surpass the current contingent of 69 soldiers in northern Iraq.
Officials with the new Liberal government are in discussions with their U.S. counterparts to work out the exact details of Canada's new role. Ottawa's military contribution to the coalition against the Islamic State is under close scrutiny in the wake of the attacks.
"Obviously, we committed throughout the campaign and I've committed repeatedly to my allies that we were going to do more on the training front and that means obviously more than just 69 trainers," Mr. Trudeau told reporters on a flight between the G20 summit that wrapped up Monday in Antalya, Turkey, and the Asia Pacific Economic Co-operation summit that starts Wednesday in Manila.
"How many that will be, what form that will take, what kind of engagement we're going to have, those are things that we're going to work out, but I have reassured my allies and Canadians that, yes, we will be doing more."
The French government responded to the terrorist attacks by escalating its bombing campaign in Syria and urging a grand coalition with the United States and Russia to eradicate the Islamic State.
The Prime Minister's comments come one day after he confirmed Canada's bombing mission in the region will come to an end before March, 2016, when the current parliamentary mandate expires.
This shifting role raises a number of questions the Prime Minister's Office wasn't prepared to answer Tuesday, including whether Canadian soldiers in Iraq will continue to be authorized to identify targets for air strikes as they were under the training and mentoring mission the former Conservative government launched in September, 2014.
"Details of Canada's contribution will be determined as discussions continue within the government and with our allies on the role for Canada going forward," PMO spokesman Olivier Duchesneau said.
Mr. Trudeau, whose decision to withdraw the CF-18s was challenged by Saskatchewan Premier Brad Wall this week, continues to face pressure to change his mind.
Ujjal Dosanjh, a former federal Liberal cabinet minister in Paul Martin's government, said he thinks the Prime Minister should reverse himself and continue the airstrikes.
"I know he made a promise but the circumstances do change – or they crystallize more so – and they have with the Paris attacks," Mr. Dosanjh said.
Mr. Trudeau campaigned on a pledge to end a combat role in Iraq, but Mr. Dosanjh argues this wasn't a central theme in the 2015 vote.
"From my perspective, the election wasn't about bombing mission or no bombing mission. The election was about a change of government," he said.
Canada has about 600 Canadian Armed Forces personnel stationed in Kuwait, where they support the air-combat mission that includes surveillance aircraft and tankers.
Mr. Trudeau must come up with a revised contribution to the war against the Islamic State that doesn't leave the impression Canada is quitting the fight, said David Perry, a senior analyst with the Canadian Global Affairs Institute.
To bring back all 600 of the Kuwaiti-based airmen and air women and leave less than 70 soldiers in Iraq "would be a remarkable reduction" as other countries ramp up the battle against the Islamic State, he said.
Mr. Perry said Canada could deploy soldiers to southern Iraq to help Iraqi government forces get into better shape.
Even a year ago, Western allies were talking about the need for a new long-term training program for Iraqi government forces. The semi-autonomous Kurds in northern Iraq have proven capable of turning back Islamic State troops, but Bagdhad's forces, which received years of training from the United States, nevertheless fell apart when faced with the advancing jihadis in 2014.
Retired colonel George Petrolekas of the Conference of Defence Associations Institute said Canada could better help local forces if Canadian soldiers continue to be allowed to identify targets for air strikes and accompany Iraqi troops into the field.
Lloyd Axworthy, a former Chrétien-era cabinet minister, said he would like Canada to focus on tracking the flow of funds by militants of the Islamic State, also known as ISIS.
"I'd like to see Canada become a very major player in not just tracking individual ISIS cells, but tracking where the money goes so we can begin to provide tough containment," Mr. Axworthy said.
He said consideration should be given to "opening up havens and corridors where aid can be delivered to people in Syria."


Geography lessons at Queen Katharine Academy provide students with the wide range of skills required to make sustainable decisions as responsible global citizens and enhance their future employability. This includes a particular focus on analytical, evaluative, graphical, statistical and cartographic skills. We are continuously looking forward and determined to improve. The department meets regularly to discuss pupil progress for all Year Groups and on the personalised support that may be required by students.
The Geography department works with a shared vision for teaching and learning and supporting pupils to achieve their best. The department is very proud of every pupil's achievement and seeing some of our pupils go on to study geographical courses at university, including at PhD level.
In Key Stage Three, the department follows a curriculum that aims to introduce students to a wide variety of topics that encompass both traditional and contemporary aspects of the subject, including those that appear on the GCSE specification. The department encourages out of classroom learning through a range of independent learning projects. Students are given the opportunity to participate in a range of field trips and visits to enhance their learning in the classroom.
In Key Stage Four, the department follows the new 9-1 AQA course. The course combines a mixture of Physical and Human Geography. Pupils participate in fieldwork that is based in Hunstanton and London.
In Key Stage Five, the department follows the new Edexcel Specification course. The course combines a mixture of traditional exams with a synoptic based examination and coursework. Pupils participate in fieldwork in North Yorkshire as well as independent fieldwork based upon a unique investigation of their own.


Unfortunately, our bodies change a lot when we hit the middle-age range. Our metabolism slows down, hormones change, and muscle mass and density become different. But based on research, there are some things we do that make these changes even more drastic. If we manage to avoid these habits, it may be possible to postpone or slow down these unpleasant effects of becoming middle-aged.
At FoxSumo, we've selected a few of these bad habits and want to share them with you. Be sure to read our inspiring bonus tip at the end of the article!
Although dining out can be really fun, going out too often isn't a good idea. You've probably realized that once you became middle-aged, you started going out more often than before. The kids have grown up and you're able to have more time for yourself. Plus, you're more comfortable financially and eating out is simply convenient. But if you think about it, you most likely eat way more when you're out than you do at home.
Water is essential for your well-being. Sometimes when our bodies get dehydrated, we feel like having a snack while in reality, all we need is more water. Try to drink a glass of water before a meal or a snack and you may find that you start eating less. Remember that soda drinks are not a replacement for water.
It's always nice to have some wine with dinner. Keep in mind that wine is very high in calories and has a lot of sugar. When you add wine to your meal, you tend to eat more. The key word here is "moderation".
Instead of having a glass of wine for dinner every day, have it occasionally. Decreasing the glass size is another great solution.
Let's admit that we all like having snacks. But be careful about what you eat, especially if you are a night-snacker. Keep in mind that many snacks that claim to be "healthy" are full of sugar and extremely high in calories. Healthy snacks should consist of protein, fiber, and be low in sugar.
A recent study proved that fiber is essential for good health. It's a natural appetite suppressant and unlike carbs, fat or protein, fiber doesn't get digested by the body — it just passes through the system. Main sources of fiber are veggies, fruits, and whole grains — one of which is buckwheat. Buckwheat is gluten-free, has a very low GI and is full of different minerals and vitamins.
No matter if you've always been an active person or fitness has never been your favorite thing to do, being active is highly important when you become middle-aged. To prevent losing muscle mass, you have to combine cardio and strength exercises, including a good abdominal workout. Remember, if you are a newbie at the gym, it might be a good idea to start working with a personal trainer until you're more comfortable with the exercises.
Age is one of those things that nobody can avoid, so all we can do is learn how to get the most of what we have! Be happy, stay healthy and enjoy every day. No one can fool nature but by following our advice you can make the most of your life at this age.
Do you have your own tips on how to trick your body's clock? Share them with all of us in the comments.


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text:
Anat Saragusti Archives - What Happened to the Israeli Peace Camp?
What Happened to the Israeli Peace Camp? >
Authors >
Anat Saragusti
Film-maker, book editor and a freelance journalist and writer
Anat Saragusti is a film-maker, book editor and a freelance journalist and writer. She writes op-eds in major newspapers and websites in Israel and invited to panels, mainly on issues of state security, women participation in decision making processes and other current affairs. Saragusti is a laureate of the 2016 "Unsung Hero" Drum Major for Justice Award for "exceptional global leadership, commitment to the elevation of humankind through advocacy and awareness and for personifying the spirit of Dr. King's core philosophies."
She was a senior staff member at the weekly news magazine Ha'olam Hazeh (Lit. This World), where she was prominent in covering major events in Israel and mainly the Occupied Palestinian Territories . With the closing of Ha'olam Hazeh (1993), Saragusti joined the group that established TV Channel 2 News Company and was appointed as its reporter in Gaza. She later became the chief editor of the evening news bulletin and the weekly Friday magazine. Concurrently, she studied Law and attained a master's degree from Tel Aviv University.
She was the executive director of Agenda, a non-profit organization working to re-prioritize and reframe social change issues within the Israeli public debate and media. Later she served as the director of B'Tselem USA, and lived in DC for two years. Saragusti is a long time women activist, leading adoption of 1325 in Israel. She is considered an expert on UNSCR 1325 in the Israeli context. Anat Saragusti is a long-time member of Women Wage Peace, a member of the 100 member Central Committee and an active and leading participant in some of the movement's key professional committees. Among other documentaries, Saragusti directed the documentary "Citizen Aloni" which tells the life story of Shulamit Aloni.
Women Waging Peace


The Trump administration is urging the U.S. Supreme Court not to review the prison sentences given to Quality Egg LLC executives for their role in failing to prevent a national salmonella outbreak. This move indicates that the Trump administration will support continuance of the responsible corporate officer doctrine, despite a congressional push last year to eliminate it. This follows the Department of Justice's (DOJ) trend toward aggressive prosecution following foodborne illness outbreaks.
After reaching a plea agreement with the government, the company paid a $6.78 million fine and Quality Egg LLC owner Austin "Jack" DeCoster and his son, Chief Operating Officer Peter DeCoster, each paid a $100,000 fine. However, the District Court also gave them each a prison sentence of three months based on their pleas of guilty to strict liability misdemeanor charges of introducing adulterated food into interstate commerce, in violation of the Food, Drug, and Cosmetic Act. The United States Court of Appeals for the Eighth Circuit upheld those sentences in July.
The DeCosters are asking the U.S. Supreme Court to review this case based on an argument that giving prison time to someone based on their supervisory responsibility, as opposed to their individual conduct, breaks with a long tradition in American law. This doctrine under which the DeCosters were prosecuted is known as the responsible corporate officer doctrine, or Park doctrine. The government is opposing the DeCosters' bid for review, arguing that the sentence was actually based on the DeCosters' individual conduct found at their sentencing hearing. In other words, the government is saying an individual corporate officer can be held strictly liable and convicted based solely on his or her supervisory responsibilities, and then face prison time where the District Court determines at a sentencing hearing — where the rules of evidence do not apply and the burden of proof is lower — that the individual officer was at least negligent.
Last year, there was a bipartisan push for sentencing reform, which would have included the elimination of the responsible corporate officer doctrine. However, that proposed legislation was ultimately defeated by opposition from some, including then-Senator (now Attorney General) Jeff Sessions.
Both the lead prosecutor on the DeCoster case and the head of the DOJ Consumer Protection Branch, which oversees these matters, appeared at FaegreBD's 2016 Food & Agriculture Conference. One key takeaway from that panel was that the government will continue to rely on the responsible corporate officer doctrine if the evidence suggests that individual corporate officials were in a position to correct food safety problems and prevent substantial harm to consumers.


Rizzle Kicks believe Jamie Cullum is "the most talented person" they've ever seen play an instrument and admit they were blown away by his musical ability.
Rizzle Kicks believe Jamie Cullum is "the most talented person" they've ever seen.
The hip-hop duo – Jordan Stephens and Harley Alexander-Sule – collaborated with the 33-year-old singer-songwriter on their upcoming tracks 'The Reason I Live' and 'I Love You More Than You Think', and the pair admit they were blown away by his phenomenal ability to play an instrument.
Speaking to The Sun newspaper, Jordan said: "He's the most talented person I've ever witnessed play an instrument.
"He's got an amazing voice and his piano playing is out of this world.
ALSO READ Jamie Cullum performs live in the KGSR Music Lounge Tuesday, July 13th at Noon! Would you like to win tickets?
The pair have also teamed up with Dominic West for their new songs 'Put Your Two's Up' and 'That's Classic' and they previously admitted joining forces with the 'Wire' actor was like working with a "close friend" because they are obsessed with the US drama series.


text:
Review - Hyundai Elite i20 Diesel
30/08/2014 - 15:58 | Hyundai i20, Hyundai, Car Reviews | Anjan Ravi
"If at first you don't succeed, try again", would be an apt way of summarizing Hyundai India's tryst with the B-hatchback segment. The Korean car maker first launched the Getz in 2004, but found little success in a country which was just warming up to the idea of a premium hatchback. The Getz's successor though, the i20, fared much better, both in terms of sales and a fan following. In fact, after the Maruti Swift, the i20 is the next big seller in its segment.
The i20 is an established name in India.
After having the i20 in the market for six years, Hyundai has launched the next generation model and its global premiere was in fact held in India. The Elite i20 as it is called, promises to build on the success of its predecessor by offering more of everything for relatively the same price.
Things have changed from the time Hyundai launched its first B-segment hatchback. Now, the car maker has a design studio in Europe (Rüsselsheim, Germany, to be specific) and that is where the Elite was styled.
The Elite was styled in Germany with the Indian model getting localized touches.
The front sees an iteration of Hyundai's hexagonal grille flanked by flared headlight clusters. The sides of the car are especially characterized by black C-Pillars, which to some extent try to create a wraparound like windscreen. Also seen on this top-end model are diamond-cut 16-inch alloy wheels. The taillights of the Elite extend well into the bootlid and have an interesting pattern for their brake lights.
The light for reverse illumination has been moved down to the bumper, and sadly, there is only a single light present. While providing a single reverse light is not uncommon, it is strangely positioned on the driver's side, whilst in other cars like the SX4, Polo and Vento (to name some with this arrangement) it is positioned on the passenger's side to give the driver a better illuminated view in the wing mirror.
The Elite has good design proportions.
In terms of dimensions, the Elite is shorter than the model it replaces by a good 10 mm at 3,985 mm. The wheelbase though has increased to 2,570 mm. In terms of width and height, the Elite measures 1,734 mm and 1,505 mm. The 170 mm ground clearance is more than adequate for our road conditions.
Interior -
The dashboard is well laid out and easy to use.
The interior is an adaptation of Hyundai's dual-tone black-and-beige theme. The steering wheel, which is a 3-spoke unit, seems inspired by cars like the Xcent and Grand i10, as do a few switches here and there, the power window buttons especially.
At the center is a music system which is capable of playing CD/MP3/Radio/FM/AM/AUX-in/Bluetooth and USB. An 8-speaker audio system (4 speakers and 4 tweeters) comes on the top-end Asta trim, and the sound quality is indeed commendable.
Getting comfortable in the driver's seat takes hardly any time, for the controls for seat height, seat back recline and length adjustment are ergonomically located. The steering wheel comes with tilt and telescopic capability, and adding to the comfort factor is the center armrest which will be a boon on long drives.
There is a noticeable improvement in rear legroom and kneeroom.
Overall build quality and fit-and-finish are top notch, as it has become a key part of the car maker's DNA these days. However, the plastic quality on the Polo has a premium edge to it which is not seen on the Elite.
Comfort levels at the rear are equally good. The seat itself is designed to hold three occupants (in reality three will be a bit of a squeeze) and comes without a center armrest. The seat back recline angle is well chosen, as is the almost flat center tunnel. Compared to the older model, leg and kneeroom have improved a notch and seating two 6-footers back-to-back shouldn't be a problem on the car. On all variants save for the base trim, the Elite also offers rear AC vents.
The 285 liter boot is well shaped.
And finally, the Elite i20 comes with a 285 liter boot which is well shaped and illuminated. Additional space can be had by pushing down the rear seat (60:40 split).
To summarize, compared to its rivals, the Elite easily offers one of the most spacious cabins coupled with several comfort features that should appeal to the family car buyer.
The steering position indicator is something new in this segment.
The top of the line Asta trim seen in these images comes equipped with dual front airbags, ABS, EBD, steering position indicator and a smart pedal system. Other features include diamond-cut 16-inch alloy wheels, keyless entry with engine starter button, electrically adjustable and folding wing mirrors, multifunction steering wheel, a music system, automatic climate control, reverse camera with an electro-chromatic rear view mirror and a cooled glovebox.
The Elite i20 continues to be a feature packed car with all the bells and whistles expected from it.
Engine and Gearbox -
The diesel engine is carried over from the previous model.
Hyundai decided to carry over the engines from the older generation model. The engine powering this i20 is the 1.4-liter four-cylinder CRDI motor belting out 90 PS (88.74 bhp) and 220 Nm of torque. The gearbox is a 6-speed manual unit, and the Elite i20 is in fact the only car in its class to get a 6-speed unit, like its predecessor.
As always, this engine scores well on the refinement front and NVH levels are one of the best in segment as well. Even at higher rpms, while the sound may indicate the fuel type of the car, it doesn't get very obtrusive.
Coming to performance, the 1.4-liter engine clearly works well post 1,750 rpm. There is a noticeable surge in power beyond this point which extends till about 3,500 rpm. Keep the Elite within this powerband, and progress is bound to be quick.
It is the only vehicle to offer a 6-speed gearbox in this segment.
Below the 1,750 rpm point, turbo lag is definitely present. So how is it different compared to the older model? Well, where the older model would sometimes struggle a bit to get a move on, the Elite seems a bit more eager even when off boost. This certainly aids in city driving conditions as it minimizes gear changes.
In terms of outright power, the 1.4-liter motor feels a tad slow in comparison to the new 1.5-liter VW TDI (seen on the Polo), and lacks the sprightly nature of the 1.3-liter DDiS in the Swift. The Elite is by no means a slow car though, just that the Polo and Swift will feel noticeably quicker and more performance oriented when all their horses are called for.
The CRDI engine picks up well from 1,750 rpm.
Coming to the gearbox, the clutch pedal feels a lot more lighter than the outgoing model, which is bound to appeal to the driver's left foot in urban traffic conditions. This is further complemented by the short, yet precise gear lever which slots into gates smoothly and accurately. Thanks to the 6th gear, you can comfortably cruise from speeds as low as 80 km/h without bothering the clutch or the gear lever any further.
Overall, the performance package of the Elite is more than adequate for city and highway driving conditions. While a family car buyer will certainly appreciate the Elite's powertrain, enthusiasts may still favor the Polo or Swift.
Ride and Handling -
The new hatch feels a lot more stiffer than its predecessor.
The ride quality of the Elite is one area where Hyundai have made good improvements. Even on roads which had rough craters and potholes, the hatchback did a remarkable job of ironing them out, and seldom letting its occupants know of the terrain. Mind you, our test car came with the larger 16-inch wheels, and other variants which get 14-inch units, could feel more comfier.
Hyundai have also gotten rid of the slow suspension system seen on the earlier car, which was known to have a bouncy ride. The new Elite has almost no bouncyness to its ride, and is nearly flat.
So, how does the Elite fair dynamically? The sore points first. The steering feel is still not up there with a Maruti Swift for example. It is evident that Hyundai have tried to improve the feedback, and have done a decent job, but its still not worthy of being a driver's delight. The steering feels artificially heavy in places, and suddenly light in other places.
The steering feel could have been better in feedback and consistency.
What comes as a marked improvement is the road holding capability of the car. Taking twisty bends, it is evident that the Elite is a lot more stiffer and a lot more tauter than its predecessor. It is quick to react to direction changes and feels far more agile than the model it replaces.
Overall, the Elite i20 has a far more plusher ride than its predecessor, and yet manages to be dynamically more stable. While steering feel could have been improved, the car is dynamically stronger than the older model, and that seems to have been the focus point for this generation of the hatchback.
Brakes and Safety -
Even with only front discs, the Elite comes to a stop confidently.
The Asta (O) trim with 6 airbags is no longer in the offing. Buyers need to be content with dual airbags, one for the driver and one for the co-passenger. Other safety features come in the form of ABS, EBD, a smart pedal system (should the driver press the accelerator and brake pedal simultaneously, the brake overrides the accelerator), and a steering position indicator (warns the driver about the steering not being in the dead-ahead position upon starting).
Even with front discs only, the braking performance of the car was nothing short of impeccable. Where the old i20 could get a bit squirmy under hard braking, the Elite comes to a stop with poise. Grip levels from the 195/55 Bridgestone are equally good.
Fuel Efficiency -
The Elite i20 diesel is claimed by the ARAI to achieve 22.54 km/l for the Era and Magna variants, and 21.76 km/l for the Sportz and Asta variants.
Prices -
The Elite i20 diesel range starts from INR 6.09 lakhs and extends up to INR 7.66 lakhs, ex-Showroom, New Delhi.
Verdict -
The Elite comes off as a good Indian family car.
No doubt, the new Elite i20 is indeed a step up from the previous generation model in many aspects. The exterior and interior design, overall build quality, performance and ride and handling have certainly improved.
Loose ends? The steering of the Elite still doesn't feel as crisp as say that of the Swift's, and the engine could do with a bit more punch. Also, with manufacturers like Volkswagen providing airbags as a part of the standard kit, Hyundai needs to be wary that it may need to offer more features to stay ahead of the game.
As an overall package, the Elite i20 is hard to fault with. However, enthusiasts in the segment may find the Swift or the new Polo 1.5 TDI a bit more appealing in the driving and handling department.
It also needs to be mentioned that Hyundai Motor India Ltd. (HMIL) now has a problem many manufacturers would wish for. And that is, offering a lot of choice to the compact premium car buyer. Between the more practical Xcent and the stylish Elite, buyers may have a hard time choosing inside the brand!
Hyundai is keen to establish the i20, and generally its future products, as premium products. Here are a few things I noticed that suggest so.
1. The door lock/unlock button for the Elite is positioned right in the center of the dashboard, above the hazard light switch. All BMWs have their lock/unlock button (with a similar font) in the same area.
2. When you're locking/unlocking with the remote, the car makes a sound much like its honking. This is again seen in German cars of BMW and Mercedes.
3. The printed brochure of the Elite is much like a coffee table book, and the quality of the brochure itself (the pages, and the print) is likely to make a good and lasting impression on prospective buyers.
Hyundai Elite i20 Diesel Review
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Hyundai Alcazar Launched In India; Price Starts From INR 16.30 Lakh
Hyundai Alcazar Detailed in Yet Another Walkaround Video
Hyundai IONIQ 6 - Second IONIQ Model - Spied Testing For The First Time


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How Ranveer played Cupid for Sara and Kartik
Sanskriti Media
Published Thursday, July 04, 2019
Sara Ali Khan and Kartik Aaryan are indeed the new couple in the film industry. The actors who stopped meeting for a while in between as the filmmakers did not want too much exposure and Sara's mother Amrita Singh wanted to check out the boy who was dating his daughter, are now back officially together.
Sources inform that they share a superb bond and Kartik also calls her 'Princess' due to her royal lineage as her father Saif Ali Khan is the Nawab of Pataudi.
As the shoot of the untitled Imtiaz Ali film starring Kartik and Sara came to an end, Kartik shared on his Instagram, ""A film I never wanted to end. Thank you to my dream director. And couldn't have asked for a better saathi in this journey than Princess Sara Ali Khan. Want to work with you again and again and again."
Not one to be left behind, Sara also posted, "Thank you @kartikaaryan for instantly making me comfortable with you, for selflessly giving and for consistently looking out for me. From coffees about you to chai's with you, I wish we could do it all over again. I'm going to miss you more than you know and more than I can admit."
Sara had in fact told on Koffee With Karan that she would like to date Kartik and that's where the story started. Ranveer Singh who got them formally introduced at a party and played Cupid while telling them to go out on a date is now staking claim for their relationship.
"So cute! Bhoolna nahi sabse pehle kissne milwaya thha (Don't forget who got you to meet each other)," he posted in reply.
Well, Ranveer is now doing what his mentor Sanjay Leela Bhansali did to him and Deepika Padukone and got them together. It's time for Ranveer to give back to the world.


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Camera IconJamie Oliver favours simple meals and good company.Picture: Channel 10
Jamie Oliver on Paleo Pete, fad diets and smashed avo
RHIAN DEUTROMNews Corp Australia
December 22, 2017 7:00PM
ChristmasLifestyle
CELEBRITY chef Jamie Oliver says he's torn when it comes to fad diets but there's one simple way to ensure you are eating healthy.
Camera IconJamie Oliver advocates cooking from fresh.Picture: News Corp Australia, Paul Stuart
Talking to News Corp Australia from his base in the backstreets of London's trendy Islington, the father-of-five was asked about Australia's most controversial paleo chef, Pete Evans, who came under fire for promoting the benefits of bone broth for babies.
Sneak peek: Inside Jamie Oliver's amazing home
"I don't know Pete or what his values are but I'm caught between two opinions," Oliver said.
"In one sense I hate anything that's faddy, we get loads of points for just cooking fresh and using a rainbow of items in our meals.
"On the other hand, this kind of uprising of faddy and annoying diets is our fault for demanding these things in the first place."
According to Jamie, the public appetite for "new and exciting" content has sparked a potentially dangerous trend of radical diets.
"Paleo is not really my thing. I get that we haven't changed since caveman times and I get that we were hunter gatherers but to be honest I think that we can get so caught up in diets and fads that we become very confused," he said.
Simple meals, accompanied by good company will always be his preferred method of enjoying food.
"Cook with your mum, your aunty, your neighbour, talk about food, experiment. If you're cooking from fresh you generally tend to edit out 95 per cent of the issues and problems that everyone is yapping on about".
ADD AN AVO
Camera IconJamie Oliver keeps a keen eye on the Aussie food scene.Picture: News Corp Australia, Paul Stuart
One fad the British star is happy to embrace is Australia's smashed avocado on toast, and he is full of praise for the Australian food scene.
"There is now a rigour and dedication to learning the best of science, travelling and bringing knowledge back home that is propelling the Aussie food scene forward," he said.
"Whether it's wine or olive oil or cheese making or dudes in a garage making craft beer."
As for our latest national dish?
"The smashed avo on toast? Oh I love it," he said.
"The coffee in Australia is incredible too. If you ever get a good coffee in London it's highly likely that the person who made it was an Aussie".
Having travelled to Australia annually over the last 16 years, Jamie says he has watched the Australian food scene mature to a world leader in everything from sustainable farming to craft beer and wine.
"The old adage of 'put another prawn on the barbie' just doesn't apply any more, it's so much more than that now," Jamie said.
"The fact that you can be a surfer, go out to the clubs, but also be a total geek about varieties of coffee beans is incredible, I love that.
"You guys are completely bucking the trend of the Aussie cliche".
According to Jamie, Australia's burgeoning trend of paddock to plate experiences in rural areas such as The Farm in Byron Bay is a clear testament to how much the food industry has grown.
"The idea of having a farm that can have families on the weekend, have activities and events for kids is fantastic," he said.
"It's an area that I'm seeing more focus on where it's not just about rearing the pigs but curing them, making pies, you can just smell this sort of activity on Instagram".
Camera IconA young Jamie Oliver during filming for The Naked Chef in New York in 2000.Picture: AP
Growing up in his parent's pub, The Cricketers, in northwest Essex, a young Jamie would earn his weekly pocket money by working in the pub kitchen and learnt from a young age the importance of knowing how to cook a nutritious meal.
From there, he studied catering at the Westminster Catering College then worked in France and London before his work was picked up by the BBC and The Naked Chef was born in 1999.
"The naked chef was just this young whippersnapper, getting among it and having a laugh," he said.
"When I wrote my first book it just went immediately nuts, like One Direction-style nuts and I couldn't really believe what had happened."
His philosophy was to strip food down to its most simple components, creating simple meals for people from all walks of life.
Jamie maintains his laid-back attitude hasn't changed and is one that has resonated with his Australian fans over his two decades in the food industry.
"If you're a foodie in Britain, everyone thinks you must be rich and posh," he said.
"But the only reason I keep doing what I do is that I know the best food has always come from the most economically challenged areas.
"Aussies don't sit there going nuts about foie gras and fillet steak every day, that's not what good food is about."
Instead, the Australian food scene has wound back the clock in its approach to food, one he deeply appreciates.
"In Australia it's now about using the whole animal, finding native ingredients, going back to the art of fermenting, salting, curing and jamming," he said.
FIVE TOP TIPS FOR COOKING THIS CHRISTMAS
Aussies love fresh food and hate wasting it but unfortunately throw out $8-10 billion of food every year.
Christmas is a time of year the potential for food waste increases, so Woolworths have partnered with Jamie Oliver to introduce Food Savers, a series of clever tips designed to help you waste less and save more.
Camera IconHam is a Christmas holiday favourite.Picture: istock
1. Ham it up!
Store it correctly. Get the most out of your ham by storing it correctly. Soak a clean tea towel, pillowcase or Woolworths Ham Bag in a solution of 2 cups water and 1 tbs. vinegar. Wring out the excess liquid, then wrap any leftover meat on the bone and store in the fridge for up to two weeks, leaving plenty of room for air to circulate. Every day, remove the wrapping or bag and refresh in water and vinegar solution before rewrapping. Leftover ham can also be frozen in an airtight container for up to one month. Defrost for use in an omelet, quiche and soup.
2. Rule the roost
Before buying your turkey, plan ahead for the mouths you need to feed and be sure to measure your oven — you don't want to discover on Christmas morning that your turkey doesn't fit! Stop leftover cooked turkey from drying out by separating slices of the meat with baking paper. Keep in an airtight container in the fridge for 3-4 days so it's on hand for use in pasta dishes, toasted sandwiches or wraps.
Camera IconLeftover roast pork can be turned into tasty meals.Picture: News Limited
3. Pork out
To avoid leftovers drying out, wrap in baking paper or aluminium foil, then store in an airtight container in the fridge. Try to avoid slicing the whole roast when serving as it will store better as a larger portion. If stored correctly, cooked pork roast will last for 3-4 days in the fridge and up to four months in the freezer, but remember to remove any bones first. Transform your leftovers into smoky pulled-pork burgers, Chinese-style fried rice or golden pot pies that everyone will love for lunch or dinner.
4. Sensational Seafood
When purchasing seafood, it should be as fresh as possible and kept chilled until it can be placed in the refrigerator. If buying in advance, store your seafood on the bottom shelf of the fridge and consume within two days.
Camera IconDon't let delicious mangoes go to waste.Picture: Getty Images
5. Magnificent Mangoes
Get the most out of your mango by scraping any leftovers from the skin and serving over ice-cream, or add to salad dressing for a fresh summer taste. If you have lots of ripe mangoes, don't just leave them sitting there to go past their prime, slice the flesh into pieces and freeze in layers, separated by baking paper, in an airtight container.
Originally published as Oliver torn on food fads