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

The growing precision of high energy collider experiments puts increasing strain on our ability to
make reliable theoretical predictions. Standard techniques for the computation of perturbative
scattering amplitudes often fail when applied to the multi-loop and multi-leg processes currently
produced in abundance at the LHC. Constantly evolving methods have led to differential predictions at
next-to-next-to-leading (NNLO) for $2\to2$ scattering and N${}^3$LO for $2\to1$
processes\footnote{For recent summaries of the state-of-the-art see
\cite{Badger:2016bpw,Bendavid:2018nar}.}.

The need to match experimental precision has led to increasing efforts from the theoretical community
to develop new techniques for $2\to3$ predictions at NNLO. The first hurdle has been to compute
unknown two-loop amplitudes in which the analytic and algebraic complexity causes conventional approaches to
integral reduction to fail. Major new advances that exploit numerical evaluations over finite
fields~\cite{vonManteuffel:2014ixa,Peraro:2016wsq,Peraro:2019svx}
have recently produced the first analytic results for five-parton amplitudes in the leading colour
approximation. Combined with the recently computed analytic master
integrals~\cite{Papadopoulos:2015jft,Gehrmann:2018yef} using the canonical
basis approach to differential equations, a form suitable for combination with the unresolved
contributions to the cross section has been
obtained~\cite{Gehrmann:2015bfy,Badger:2018enw,Abreu:2018zmy,Abreu:2019odu}.

The production of a $W$-boson together with jets at hadron colliders are important signatures that can
be used as precision probes of the Standard Model. QCD corrections to $W$+jets have been a traditional testing
ground for new technology. $pp\to W+j$ was among the first $2\to2$ process computed at NLO
\cite{Giele:1993dj}. The amplitudes for $pp\to W+2j$ were computed using the recently developed on-shell unitarity
method~\cite{Bern:1994zx,Bern:1994cg,Bern:1996ka,Bern:1997sc}
and were implemented into \textsc{MCFM} to provide differential cross-section predictions~\cite{Campbell:2002tg}.
NLO results for associated $W$-boson production with three or more jets are accessible through
automation and the use of the generalised unitarity
method~\cite{Ellis:2008qc,Ellis:2009zw,Berger:2008sj,Berger:2009ep,Berger:2010zx,Bern:2013gka}.

NNLO corrections to $pp\to W+2j$ will open up possibilities for further precision tests of the
Standard Model. The two-loop amplitudes are obvious targets for the new technology developed for massless
five-point amplitudes, yet the off-shell vector boson adds an extra scale and therefore a new layer
of complexity. The first step towards a complete analytic computation is to set up a procedure that
could evaluate the amplitudes numerically using rational kinematics. It is this benchmark evaluation
of the amplitudes that is the subject of this paper.

The computation of higher order corrections to perturbative scattering amplitudes is a well studied
problem. Amplitudes with two or more loops have relied on the technology of
integration-by-parts (IBP)~\cite{Tkachov:1981wb,Chetyrkin:1981qh} reduction, which in recent times
has involved following Laporta's algorithm~\cite{Laporta:2001dd}, together with numerical or
analytic methods for the evaluation of the resulting basis of master integrals. For these
multi-scale basis integrals with massless internal propagators the differential equation
technique~\cite{Kotikov:1990kg,Gehrmann:1999as,Henn:2013pwa}
has been employed to find analytic expressions, most recently for the complete set of
planar~\cite{Gehrmann:2015bfy,Gehrmann:2018yef} and
non-planar integrals~\cite{Abreu:2018rcw,Boehm:2018fpv,Abreu:2018aqd,Chicherin:2018mue,Chicherin:2018old}.
For the case, in which the amplitudes considered here fall, only one of the
three planar families has been
evaluated~\cite{Papadopoulos:2015jft}.
Combining the master integrals into complete amplitudes requires the solution of increasingly
complicated linear systems of IBP equations. Considerable effort has led to a variety of efficient
solutions~\cite{Gluza:2010ws,vonManteuffel:2014ixa,Larsen:2015ped,Ita:2015tya,Kosower:2018obg} and
public
implementations~\cite{Lee:2012cn,Smirnov:2014hma,Smirnov:2019qkx,vonManteuffel:2012np,Maierhoefer:2017hyi}.
Applications to five-particle problems have been possible though yielded large IBP reduction
tables~\cite{Boels:2018nrr,Chawdhry:2018awn}.
In this paper, we only perform the IBP reduction numerically over
finite fields in order obtain the coefficients of the amplitude in
terms of master integrals.  As shown e.g.\ in
refs.~\cite{Badger:2018enw,Peraro:2019svx}, when combined with
functional reconstruction techniques, this approach also allows to
directly reconstruct analytic results for amplitudes, sidestepping the need of
computing and using large analytic IBP tables, which are often
significantly more complicated.

Another important ingredient has been the development of efficient methods to construct
on-shell integrands and integral coefficients. Integrand reduction techniques~\cite{Ossola:2006us}
combined with the use of a Feynman diagram approach or generalised
unitarity have been very successful for the computation of one-loop amplitudes, in particular to construct
scalar integral coefficients numerically. These techniques have been extended to two
loops~\cite{Mastrolia:2011pr,Mastrolia:2012an,Zhang:2012ce,Badger:2012dp,Mastrolia:2012wf,Mastrolia:2013kca,Mastrolia:2016dhn}
and methods to employ unitarity cuts~\cite{Bern:1994zx,Bern:1994cg} to build amplitudes by directly
incorporating IBP decomposition have been
established~\cite{Kosower:2011ty,CaronHuot:2012ab,Ita:2015tya,Abreu:2017xsl,Abreu:2017idw}.

The first steps towards helicity amplitudes for five-point amplitudes were taken through
numerical evaluations of two-loop five point amplitudes in QCD using modular arithmetic~\cite{Badger:2017jhb,
Abreu:2017hqn,Badger:2018gip,Abreu:2018jgq}. These algorithms have been generalised to allow for a full
reconstruction of the coefficients of the pentagon functions classified in~\cite{Gehrmann:2018yef} leading to an analytic form of the
single-minus helicity amplitudes~\cite{Badger:2018enw} and the complete leading colour
five-parton helicity amplitudes within the numerical unitarity framework~\cite{Abreu:2018zmy,Abreu:2019odu}.
The success of computations in the planar sector has shifted focus to the
non-planar sector of massless two-loop five-point amplitudes with a series of new results in
super-symmetric Yang-Mills~\cite{Abreu:2018aqd,Chicherin:2018yne} and
gravity~\cite{Abreu:2019rpt,Chicherin:2019xeg}
as well as in the all-plus sector of QCD~\cite{Badger:2019djh}.

In this paper we consider the case of planar amplitudes with an off-shell external leg.
We apply the recently developed technology for the computation of
two-loop five-particle amplitudes using sampling of Feynman diagrams over finite fields.
Using a modular approach, recently presented as part of the \textsc{FiniteFlow} algorithms~\cite{Peraro:2019svx}, we are able to numerically evaluate the diagrams and perform an integrand reduction, subsequently reducing
the resulting integrals using integration-by-parts identities. Since the complete set
of analytic master integrals is not known, some of the integrals were evaluated numerically
using sector decomposition~\cite{Binoth:2000ps,Smirnov:2015mct,Borowka:2017idc}. Analytic results for the following classes of master integral are available:
one of the three families of the off-shell five-point pentagon-box~\cite{Papadopoulos:2015jft} and four-point functions with
one~\cite{Gehrmann:2000zt} and two off-shell~\cite{Gehrmann:2015ora,vonManteuffel:2015msa,Henn:2014lfa,Caola:2014lpa} legs.
For master integral topologies for which a numerical evaluation through sector decomposition is challenging,
we identified a basis of master integrals using
local numerators~\cite{ArkaniHamed:2010kv,ArkaniHamed:2010gh}
with simplified divergence structure and therefore easier numerical evaluation.
We consider both the $q\bar{Q}Q\bar{q}^\prime\bar\nu\ell$ and $qgg\bar{q}^\prime\bar\nu\ell$ sub-processes
in our computation where the decay of the $W$-boson is also incorporated.

We describe our integrand reduction setup that is subsequently interfaced to IBP reduction in
Section~\ref{sec:setup}. In Section~\ref{sec:w4pamp} we discuss the structure of the leading colour
$W$+4 parton amplitude at two loops including its singularity structure.
The identification of a master integral basis with local numerator insertions is elaborated in
Section~\ref{sec:localMIs}. Finally, we present numerical benchmark results for
both sub-processes
in Section~\ref{sec:results} and draw our conclusions in
Section~\ref{sec:conclusions}.

\section{Calculational framework \label{sec:setup}}

The framework described in this section is a modification of
the numerical algorithm for two-loop amplitudes presented in~\cite{Badger:2018enw} to allow for the use of
integrands built from Feynman diagrams as an alternative to generalised
unitarity cuts in six dimensions. While the unitarity method can be very efficient, a fully
numerical approach, with rational reconstruction, is also able to avoid the traditional problems
associated with the Feynman diagram approach.

We start by generating a set of Feynman diagrams using \textsc{Qgraf}~\cite{Nogueira:1991ex} and
performing colour decomposition to separate the colour parts of the amplitude from the kinematic
parts that depend only on external momenta $\lrbrace{p}$. We obtain
\begin{equation}
\mathcal{A}^{(2)}_n(\lrbrace{p})  = \sum_{c} \mathcal{C}_c \; A^{(2)}_{n,c} (\lrbrace{p}),
\label{eq:colourordering}
\end{equation}
where $\mathcal{A}^{(2)}_n(\lrbrace{p})$ is the two-loop colour-dressed $n$-point amplitude,
$A^{(2)}_n(\lrbrace{p})$ is the two-loop colour-stripped $n$-point amplitude and
$\mathcal{C}_c$ is the corresponding colour factor. The colour-stripped amplitude is made up of numerator functions,
$N_T(\lrbrace{k},\lrbrace{p})$, and a set of loop propagator denominators,
$D_\alpha(\lrbrace{k},\lrbrace{p})$, for each diagram topology $T$
\begin{equation}
A^{(2)}_n(\lrbrace{p})  = \int \prod_{i=1}^{2} \frac{d^d k_i}{i \pi^{d/2}e^{-\epsilon \gamma_E}}
\sum_{T}  \frac{N_T(d_s,\lrbrace{k},\lrbrace{p})}{\prod_{\alpha \in T}
D_\alpha(\lrbrace{k},\lrbrace{p})},
\label{eq:numeratorfunction}
\end{equation}
where $k_i$ is the loop momenta, $d=4-2\epsilon$ is the space-time dimension
and $d_s = g^\mu_\mu$ is the spin dimension.
The loop amplitude in t'Hooft-Veltman (HV) scheme~\cite{tHooft:1972tcz} can be obtained by
setting $d_s = d$, while the Four-Dimensional-Helicity (FDH) scheme~\cite{Bern:2002zk} can be achieved by setting
$d_s=4$. Each numerator function, $N_T(d_s,\lrbrace{k},\lrbrace{p})$, that contains numerators of Feynman
diagrams that share the same diagram topology, is processed by applying
the t'Hooft algebra. This is carried out with the help of \textsc{Form}~\cite{Kuipers:2012rf,Ruijl:2017dtg} and the \textsc{Spinney} library~\cite{Cullen:2010jv}.
In general, the explicit functional dependence of the numerator function at this point is given by
\begin{equation}
N_T(d_s,\lrbrace{k},\lrbrace{p}) =
N_T\big(d_s,k_i.k_j,\mu_{ij},k_i.q_j,q_i.q_j,\bar{u}(p_i)f(k,q)u(p_j)\big),
\label{eq:numeratordep}
\end{equation}
where $q_i = \lrbrace{p_i,\varepsilon_i}$
($\varepsilon_i$ is the polarisation vector of the external
vector boson) and $\bar{u}(p_i)f(k,q)u(p_j)$ is a spinor string made
up of slashed momenta ($\slashed{q}_i$ and $\slashed{k}_i$).
The $d$-dimensional loop momenta can be decomposed into a
four-dimensional part and an extra-dimensional part
\begin{equation}
k_i = \bar{k}_i + \tilde{k}_i\label{eq:loopmomdecomposition}.
\end{equation}
Due to rotational invariance in the extra dimensions, $\tilde{k}_i$ can only appear in the
numerator function as $\mu_{ij} = - \tilde{k}_i \cdot \tilde{k}_j$.

We obtain helicity amplitudes by specifying the helicity/polarisation of each external particles
and we further parametrise the dependence on the external
kinematics by using momentum twistor variables, $x_i$~\cite{Hodges:2009hk}.
This allows us to express the spinor
products of external momenta $(\spA ij, \spB ij)$ and Mandelstam invariants ($s_{ij}$) uniformly in terms
of momentum twistor variables, where momentum conservation and spinor product relations like
Schouten identities are already built in. At this point we are considering the two-loop $n$-point helicity amplitude
\begin{equation}
A^{(2),h}_n(\lrbrace{p})  = \int \prod_{i=1}^{2} \frac{d^d k_i}{i \pi^{d/2}e^{-\epsilon \gamma_E}}
\sum_{T}  \frac{N^h_T(d_s,\lrbrace{k},\lrbrace{p})}{\prod_{\alpha \in T}
D_\alpha(\lrbrace{k},\lrbrace{p})},
\label{eq:helicitynumerator}
\end{equation}
where the explicit functional dependence on the helicity-dependent numerator function
$N^h_T(\lrbrace{k},\lrbrace{p})$ is
\begin{equation}
N^h_T(d_s,\lrbrace{k},\lrbrace{p}) =
N^h_T\big(d_s,x_i,k_i.k_j,\bar{k}_i.p_j,\mu_{ij},
\spAB {p_a}{\bar{k}_{i}}{p_b}, \spaa {p_a}{\bar{k}_i}{\bar{k}_j}{p_b}, \spbb {p_a}{\bar{k}_i}{\bar{k}_j}{p_b}
\big).
\label{eq:helicitynumeratordep}
\end{equation}
In processing the algebraic expressions in
Eqs.~\eqref{eq:colourordering},~\eqref{eq:numeratorfunction}~and~\eqref{eq:helicitynumerator},
we have used in-house \textsc{Form}~\cite{Kuipers:2012rf,Ruijl:2017dtg} and \textsc{Mathematica}
scripts.

In this form the helicity-dependent numerator functions in Eq.~\eqref{eq:helicitynumerator} must be
re-expressed in terms of integral families that can later be reduced using IBP equations. To achieve
this we apply an integrand reduction algorithm to obtain
\begin{equation}
A^{(2),h}_n(\lrbrace{p})  = \int \prod_{i=1}^{2} \frac{d^d k_i}{i \pi^{d/2}e^{-\epsilon \gamma_E}}
\sum_{T}  \frac{\Delta^h_T(d_s,\lrbrace{k},\lrbrace{p})}{\prod_{\alpha \in T}
D_\alpha(\lrbrace{k},\lrbrace{p})},
\label{eq:reducedamp}
\end{equation}
where $\Delta$ is the irreducible numerator for an independent topology $T$.
In order to determine $\Delta$, we first need to construct a basis of irreducible
scalar products (ISPs). We opt to use a basis of ISPs in terms of auxiliary propagators
that is suitable for IBP reduction.

To build an IBP compatible integrand basis,
we define an integral family
\begin{align}
G_{a_1 a_2 a_3 a_4 a_5 a_6 a_7 a_8 a_9 a_{10} a_{11}} &=
\int \frac{d^d k_1}{i \pi^{d/2} e^{-\epsilon \gamma_E}} \frac{d^d k_2}{i \pi^{d/2} e^{-\epsilon \gamma_E}} \nonumber \\
&\times \frac{1}{k_1^{2a_1}} \frac{1}{(k_1-p_1)^{2a_2}} \frac{1}{(k_1-p_1-p_2)^{2a_3}} \frac{1}{(k_1+p_4+p_{56})^{2a_4}} \nonumber \\
&\times \frac{1}{k_2^{2a_5}} \frac{1}{(k_2-p_{56})^{2a_6}} \frac{1}{(k_2-p_4-p_{56})^{2a_7}} \frac{1}{(k_1+k_2)^{2a_8}} \nonumber \\
&\times \frac{1}{(k_1+p_{56})^{2a_9}} \frac{1}{(k_2+p_1)^{2a_{10}}} \frac{1}{(k_2+p_1+p_2)^{2a_{11}}},
\label{eq:integralfamily}
\end{align}
where $p_{ij \cdots k} = p_i + p_j + \cdots + p_k$.
Up to cyclic permutations of the external legs,
all integrals appearing in Eq.~\eqref{eq:reducedamp} can be written in the form of Eq.~\eqref{eq:integralfamily}.
We follow the conventions used in~\cite{Badger:2018enw} where
negative exponents, $a_i < 0$, correspond to the ISPs of the irreducible numerator, $\Delta_T^h$.
The irreducible numerators are the most general polynomials in the ISPs with exponents bounded by
renormalisability conditions. As an example, the parametrisation for the two-mass double-box topology is
\begin{align}
\Delta^h \bigg(\usepix{1.6cm}{figs/twomassdbox}{0}{0}{25}{0}\bigg) =
\sum c^h_{(1,a_2,1,1,1,1,1,1,a_9,a_{10},a_{11})}&(k_1 - p_1)^{-a_2} (k_1+p_{56})^{-2 a_{9}} \nonumber \\
\times &(k_2+p_1)^{-2 a_{10}} (k_2+p_1+p_2)^{-2 a_{11}},\label{eq:twomassdboxbasis}
\end{align}
where the figure represents the topology $T$. The bounds on the exponents are
\begin{alignat}{2}
-4 & \leq a_2 + a_9 &&\leq 0,  \\
-4 & \leq a_{10}+a_{11} &&\leq 0, \\
-6 & \leq a_2 + a_9+a_{10}+a_{10} &&\leq 0.
\end{alignat}
The helicity-dependent coefficients are functions of the spin dimension, $d_s$, and the external kinematics, $c^h = c^h (d_s, \{p\})$. To determine the coefficients we express the numerators in terms of the propagators and ISPs. This is achieved by expanding the loop momenta in terms of external momenta
\begin{align}
\bar{k}_i^\mu &= \sum_{j=1}^{4} a_{ij} p_j^\mu.
\label{eq:loopmomparameterisation1}
\end{align}
The coefficients of the spanning vectors, $p_j$, are functions of the inverse propagators and ISPs, $a_{ij} = a_{ij} (D_\alpha, \text{ISPs})$.
$a_{ij}$ can be determined by solving a linear system of equations constructed by contracting Eq.~\eqref{eq:loopmomparameterisation1}
with the spanning vectors, $p_j$.
All variables in the numerators Eq.~\eqref{eq:helicitynumeratordep} can then be expressed in terms of these coefficients. For example
\begin{align}
\spAB {p_a}{\bar{k}_{i}}{p_b} &= \sum_{j=1}^{4} a_{ij} \spAB {p_a}{p_{j}}{p_b},
\label{eq:loopmomparameterisation2} \\
\mu_{ij} &= - \frac{1}{2} \left( (k_i + k_j)^2 - k_i^2 - k_j^2 \right) + \sum_{m=1}^{4} \sum_{n=1}^{4} a_{im} a_{jn}\, p_m \cdot p_n.
\label{eq:loopmomparameterisation3}
\end{align}
The last relation is obtained by squaring Eq.~\eqref{eq:loopmomdecomposition}.
The variables are straightforwardly evaluated on generalised unitarity cuts by
setting all propagators to zero without relying on explicit loop momenta solutions to the cut constraints.
We observe that this form is the starting point for the derivation of the Baikov
representation~\cite{Baikov:1996iu}, which is obtained by integrating out angular dependence in the
space transverse to the external momenta. We note that the change of variables to rewrite the
numerators in terms of propagators could be performed directly. However, the choice to apply the substitution
using the integrand reduction approach breaks the problem into a series of linear systems with fewer
parameters, rather than one large system.

At this point we can solve for the coefficients of the integrand parametrisations by equating them to the diagram numerators.
Using the two-mass double-box as example again, we have the cut equation
\begin{align}
\Delta^h \bigg(\usepix{1.6cm}{figs/twomassdbox}{0}{0}{25}{0}\bigg) +
\frac{\Delta^h \bigg(\usepix{1.5cm}{figs/T431_1}{0}{0}{40}{0}\bigg)}{(k_1-p_1)^2}
 = N^h \bigg(\usepix{1.6cm}{figs/twomassdbox}{0}{0}{25}{0}\bigg)
   + \frac{N^h \bigg(\usepix{1.5cm}{figs/T431_1}{0}{0}{40}{0}\bigg) }{(k_1-p_1)^2},
\end{align}
which is valid only when both sides are evaluated on the hepta-cut
for the two-mass double box, i.e. $D_\alpha = 0$ for $\alpha = 1,3, \dots ,8$.

After setting up the integrand reduction system, we write the helicity amplitude as
a linear combination of integrals in the integral family of Eq.~\eqref{eq:integralfamily},
\begin{equation}
A^{(2),h}_n(\lrbrace{p})  = \sum_{\mathbf{a}} c^h_{\mathbf{a}}(x_i) \;
G_{\mathbf{a}},
\label{eq:ampreadyforIBP}
\end{equation}
where we sum over tuples $\mathbf{a} = (a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8,a_9,a_{10},a_{11})$.
At this stage IBP reduction can be applied to the integrals $G_{\mathbf{a}}$.
This basis choice allows for a simple interface to the IBP reduction. The
integrand is never reconstructed analytically but only sampled numerically. The final
results are independent of the particular parametrisation and therefore the exact form of the integrand is not
the main concern here. Nevertheless, alternative representations of the integrand,
e.g.~\cite{Bourjaily:2017wjl}, may lead to improved efficiency.

The integrals with non-zero coefficients after numerical sampling of the integrand are reduced to
a set of master integrals via IBP identities.
The IBP relations are generated in \textsc{Mathematica} using the Laporta
approach~\cite{Laporta:2001dd}  with the aid of \textsc{LiteRed}~\cite{Lee:2012cn},
and solved numerically over finite fields within the \textsc{FiniteFlow}
framework~\cite{Peraro:2019svx}.
We can finally write the helicity amplitudes in the
master integral basis $J_k$
\begin{equation}
A^{(2),h}_n(\lrbrace{p})  = \sum_{k} c^{\mathrm{IBP},h}_k(x_i,\epsilon)
\; J_k(\lrbrace{p},\epsilon).
\label{eq:mirepsamp}
\end{equation}

\section{Planar two-loop $W$ plus four parton scattering \label{sec:w4pamp}}

The number of Feynman diagrams contributing to $q\bar{Q}Q\bar{q}^\prime\bar\nu\ell$
and $qgg\bar{q}^\prime\bar\nu\ell$ processes at leading colour are 210 and 603, respectively,
and the leading colour partial amplitudes are extracted according to,
\begin{align}
\mathcal{A}^{(L)}(1_{q},2_{\bar Q},3_{Q},4_{\bar q ^\prime},5_{\bar\nu},6_\ell) &= n^L g_s^2 g_W^2
 \; \;
 \delta_{i_1}^{\;\;\bar i_2} \delta_{i_3}^{\;\;\bar i_4}  \;\;
 A^{(L)}(1_{q},2_{\bar{Q}},3_{Q},4_{\bar q^\prime},5_{\bar\nu},6_\ell),
 \\
\mathcal{A}^{(L)}(1_{q}, 2_{g}, 3_{g}, 4_{\bar q^\prime}, 5_{\bar\nu}, 6_\ell) &= n^L g_s^2 g_W^2
 \; \bigg[
 \left( T^{a_{2}} T^{a_{3}} \right)_{i_1}^{\;\;\bar i_4}
 A^{(L)}(1_{q},2_g, 3_g ,4_{\bar q^\prime}, 5_{\bar\nu}, 6_\ell)
 + (2 \leftrightarrow 3)
 \bigg],
\label{eq:colourdecompositionW}
\end{align}
where $n= m_\epsilon N_c \alpha_s/(4\pi),\ \alpha_s = g_s^2/(4\pi)$ and $m_\epsilon=i (4\pi)^{\epsilon} e^{-\epsilon\gamma_E}$. $g_s$ and $g_W$ are the strong and weak coupling constants respectively.
We note that the vector boson only couples to the quark line connecting $q$ and $\bar q^\prime$ and
does not couple to the equal flavour quark pair $Q,\bar Q$.

We choose a rational parametrisation of the massless $2\to4$ kinematics using
the momentum twistor parametrisation~\cite{Hodges:2009hk}
\begin{align}
  Z =
  \begin{pmatrix}
    1 & 0 & y_1 & y_2 & y_3 & y_4\\
    0 & 1 & 1   & 1   & 1   & 1 \\
    0 & 0 & 0   & \tfrac{x_5}{x_2} & x_6 & 1 \\
    0 & 0 & 1   & 1   & x_7 & 1-\tfrac{x_8}{x_5}
  \end{pmatrix},
  \label{eq:ZmatrixW}
\end{align}
where $y_i = \sum_{j=1}^{i} \prod_{k=1}^{j} \frac{1}{x_k}$ and
\begin{align}
  x_1 &= s_{12}, &
  x_2 &= -\frac{\spA23\spA41}{\spA12\spA34}, &
  x_3 &= -\frac{\spA34\spA51}{\spA13\spA45}, &
  x_4 &= -\frac{\spA45\spA61}{\spA14\spA56}, \nonumber\\
  x_5 &= \frac{s_{23}}{s_{12}}, &
  x_6 &= -\frac{\spAB{5}{3+4}{2}}{\spA51\spB12}, &
  x_7 &= \frac{\spAA{5}{(2+3+4)(2+3)}{1}}{\spA51 s_{23}}, &
  x_8 &= \frac{s_{123}}{s_{12}}. &
\end{align}
We stress that while we generate a complete parametrisation for the $2\to4$
scattering process, analytic expressions could be obtained with only six independent parameters
since the decay of the $W$ boson completely factorises. Since it is easy to generate a rational
parametrisation for $n$-particle scattering of massless particles, it is simplest to start from a
configuration including the decay of the $W$ boson.

\begin{figure}[h]
  \begin{center}
    \includegraphics[width=0.85\textwidth]{figs/w4p_maxcuts.eps}
  \end{center}
  \caption{Independent maximal cut topologies contributing to planar $W+4$ parton scattering at
  two-loops. The full set of 15 maximal cuts can be obtained by including 2 permutations of
  $A_1$, $A_3$, $B_1$, $B_2$, $C_1$ and $C_2$ topologies.}
  \label{fig:W4pmaxcuts}
\end{figure}

The leading colour partial amplitude is passed through an integrand reduction stage which projects
onto a basis of 453 topologies with irreducible numerators written into the basis of the 15 maximal cuts
shown in Figure~\ref{fig:W4pmaxcuts}. The remaining integrals
are then passed through a Laporta style IBP reduction to find a basis of 202 master integrals
(including the 5 cyclic permutations).
The distinct master integral topologies are shown in
Figures~\ref{fig:W4pMasters5pt}~and~\ref{fig:W4pMastersNon5pt}.
The most complicated integrals that need to be reduced are
rank 5 pentagon-boxes, e.g. $G_{11111111-3-1-1}$ according to the notation defined in
Eq.~\eqref{eq:integralfamily}.

\begin{figure}[h]
  \begin{center}
  \includegraphics[width=0.8\textwidth]{figs/w4p_master_5pt.eps}%
  \end{center}
  \caption{Master integrals for leading colour $W+4$ parton scattering at two loops with five external legs. $(a,b)$
  represents the number of crossing of external legs ($a$) and the number master integral for a given
  topology ($b$). A massless (massive) external leg is indicated by a single (double) line external
  leg. The $\ast$ sign identifies master integral topologies that are not known analytically.}
  \label{fig:W4pMasters5pt}
\end{figure}

\begin{figure}[h]
  \begin{center}
  \includegraphics[width=0.8\textwidth]{figs/w4p_master_non5pt.eps}%
  \end{center}
  \caption{Master integrals for leading colour $W+4$ parton scattering at two loops with four external legs
  or fewer. $(a,b)$ represents the number of crossing of external legs ($a$) and the number master integral
  for a given topology ($b$). A massless (massive) external leg is indicated by a single (double) line external leg.
  All master integral topologies shown are known analytically.}
  \label{fig:W4pMastersNon5pt}
\end{figure}

Once the amplitude is decomposed in terms of master integrals and the evaluations of master integrals
are available (either analytically or numerically),
we can perform a Laurent expansion in the dimensional regularisation parameter, $\epsilon$.
The $\epsilon$-expanded partial amplitude contains a divergent part, manifested
by the poles in $\epsilon$, and a finite part.
The infra-red (IR) divergent part of the partial amplitude,
obtained after removing the ultra-violet (UV) divergences by introducing a set of counter-terms,
is universally known~\cite{Catani:1998bh,Becher:2009qa,Becher:2009cu,Gardi:2009qi}.

The pole structure of the unrenormalised amplitude in the HV scheme at one and two loops is given by
\begin{align}
{\mathcal{P}}^{(1)} & = 2 I_1(\epsilon) + \frac{b_0}{\epsilon},  \\
{\mathcal{P}}^{(2)} & =   2 I_1(\epsilon) \bigg( \hat{A}^{(1)} - \frac{b_0}{\epsilon} \bigg) + 4 I_2(\epsilon)
                + \frac{2b_0}{\epsilon} \hat{A}^{(1)} - \frac{b_0^2}{\epsilon^2} + \frac{b_1}{2\epsilon},
\label{eq:poles}
\end{align}
where $\hat{A}^{(1)}$ is the unrenormalised one-loop amplitude normalised to the tree-level amplitude.
The $I_2(\epsilon)$ operator is defined by
\begin{equation}
I_2(\epsilon) =   - \frac{1}{2}I_1(\epsilon) \left[ I_1(\epsilon)
              + \frac{\beta_0}{\epsilon} \right]
              + \frac{N(\epsilon)}{N(2\epsilon)} \left[ \frac{\beta_0}{2\epsilon}
                        + \frac{\gamma_{1}^{{\mathrm{cusp}}}}{8} \right] I_1(2\epsilon)
              + H^{(2)}(\epsilon),
\end{equation}
while the $I_1(\epsilon)$ operators for the $W+4$ parton process at leading colour are
\begin{align}
I^{q\bar{Q}Q\bar{q}^\prime\bar\nu\ell}_1(\epsilon) &= -N_c \frac{N(\epsilon)}{2} \bigg( \frac{1}{\epsilon^2} + \frac{3}{2\epsilon} \bigg) \big[ \left(-s_{12}\right)^{-\epsilon} + \left(-s_{34}\right)^{-\epsilon} \big], \\
I^{qgg\bar{q}^\prime\bar\nu\ell}_1(\epsilon) &= -N_c \frac{N(\epsilon)}{2} \bigg\lbrace \bigg( \frac{1}{\epsilon^2} + \frac{5}{3\epsilon} \bigg) \big[ \left(-s_{12}\right)^{-\epsilon} + \left(-s_{34}\right)^{-\epsilon} \big]
                 + \bigg( \frac{1}{\epsilon^2} + \frac{11}{6\epsilon} \bigg) \left(-s_{23}\right)^{-\epsilon} \bigg\rbrace,
\end{align}
where $N(\epsilon) = {e^{\epsilon \gamma_E}}/{\Gamma(1-\epsilon)}$ and
\begin{align}
H_{q\bar{Q}Q\bar{q}^\prime\bar\nu\ell}^{(2)}(\epsilon) &= \frac{1}{16\epsilon} \bigg\lbrace 4 \gamma_1^q - \gamma_1^{\mathrm{cusp}} \gamma_0^q + \frac{\pi^2}{4} b_0 \gamma_0^{\mathrm{cusp}} C_F \bigg\rbrace, \\
H_{qgg\bar{q}^\prime\bar\nu\ell}^{(2)}(\epsilon) &=  \frac{1}{16\epsilon} \bigg\lbrace 2\left(\gamma_1^q+\gamma_1^g\right)
                                                - \frac{1}{2}\gamma_1^{\mathrm{cusp}} \left(\gamma_0^q+\gamma_0^g\right)
                                                + \frac{\pi^2}{8} b_0 \gamma_0^{\mathrm{cusp}} \left(C_F+C_A\right) \bigg\rbrace.
\end{align}
Note that the $H_{q\bar{Q}Q\bar{q}^\prime\bar\nu\ell}^{(2)}(\epsilon)$ and $H_{qgg\bar{q}^\prime\bar\nu\ell}^{(2)}(\epsilon)$ functions are given in the leading colour limit.
The $\beta$ function coefficients and anomalous dimensions without the contribution from closed fermion loops $N_f$ are
\begingroup
\allowdisplaybreaks
\begin{align}
\beta_0 = & \;\frac{11}{3}C_A, \\
\beta_1 = & \; \frac{34}{3} C_A^2, \\
\gamma_0^g = & \; -\frac{11}{3}C_A, \\
\gamma_1^g = & \;  C_A^2 \left( -\frac{692}{27} + \frac{11\pi^2}{18} + 2 \zeta_3\right), \\

\gamma_0^q = & \; -3 C_F, \\
\gamma_1^q = & \; C_F^2 \left( -\frac{3}{2} + 2 \pi^2 - 24 \zeta_3 \right)
                  + C_F C_A \left( -\frac{961}{54} -\frac{11\pi^2}{6} + 26 \zeta_3 \right), \\

\gamma_0^{\mathrm{cusp}} = & \; 4, \\
\gamma_1^{\mathrm{cusp}} = & \; \left( \frac{268}{9} - \frac{4\pi^2}{3} \right) C_A,
\end{align}
\endgroup
where $C_A = N_c$, $C_F = (N_c^2-1)/(2N_c)$.

With the pole structures available, we can check that the divergent part of the
two-loop amplitude agrees with the predicted UV and IR poles. On the other hand, we can obtain the so-called
two-loop finite remainder by subtracting the UV and IR poles from the two-loop amplitude where
 the pole structure in Eq.~\eqref{eq:poles} is
expanded to $\mathcal{O}(\epsilon^0)$. The analytic form
of the finite remainder is in general much simpler than the finite part, as demonstrated
in~\cite{Badger:2018enw,Abreu:2018zmy,Abreu:2019odu,Badger:2019djh}.

\section{Local master integrals \label{sec:localMIs}}

The master integrals appearing in the two-loop leading colour
$q\bar{Q}Q\bar{q}^\prime\bar\nu\ell$ and $qgg\bar{q}^\prime\bar\nu\ell$ amplitudes, that are not known analytically, are evaluated numerically using sector
decomposition~\cite{Binoth:2000ps,Smirnov:2015mct,Borowka:2017idc}.
In general, it is challenging to obtain results with good numerical accuracy for
complicated master integral topologies (e.g. topologies with 6, 7 or 8 propagators in
Figs.~\ref{fig:W4pMasters5pt}~and~\ref{fig:W4pMastersNon5pt}) within
a reasonable amount of time, even for an evaluation in the Euclidean region.
Having numerical accuracy under control is particularly essential when large cancellations occur between
different terms in the amplitude. One way in which this can be achieved is to use
a basis of master integral with local numerator
insertions~\cite{ArkaniHamed:2010kv,ArkaniHamed:2010gh} to regulate divergences. This is the approach
we explore in this work. Another approach well suited to numerical evaluation is to use a quasi-finite basis
of integrals~\cite{Panzer:2014gra,vonManteuffel:2014qoa}. We did not attempt to compare the two
approaches but note that at least two factors are involved: firstly, in the reorganisation of the
amplitude through the change of basis and secondly, the improved convergence of the resulting master
integrals when evaluated with sector decomposition.

As an example, we consider one of the master integral topologies with 8 propagators, $A_1$ pentagon-box topology in Figure~\ref{fig:W4pmaxcuts}.
There are three master integrals for this topology
\begin{equation}
I\bigg(\usepix{1.5cm}{figs/T431_1}{0}{0}{40}{0}\bigg)\big[ 1 \big],\quad
I\bigg(\usepix{1.5cm}{figs/T431_1}{0}{0}{40}{0}\bigg)\big[ (k_1+p_{56})^2 \big],\quad
I\bigg(\usepix{1.5cm}{figs/T431_1}{0}{0}{40}{0}\bigg)\big[ (k_2+p_1)^2 \big],
\label{eq:standardmiA1}
\end{equation}
or
\begin{equation}
G_{11111111000},\quad G_{11111111-100},\quad G_{111111110-10},
\end{equation}
labelled according to Eq.~\eqref{eq:integralfamily}.
In Eq.~\eqref{eq:standardmiA1}, we use a notation for the integral with a numerator insertion
$N(k_i,p_i,\mu_{ij})$
\begin{equation}
 I\big( T \big) \big[ N(k_i,p_i,\mu_{ij}) \big] =
\int \frac{d^d k_1}{i \pi^{d/2} e^{-\epsilon \gamma_E}} \frac{d^d k_2}{i \pi^{d/2} e^{-\epsilon \gamma_E}}
\frac{N(k_i,p_i,\mu_{ij})}{\prod_{\alpha \in T} D_\alpha(k_i,p_i)},
\label{eq:integralwnumerator}
\end{equation}
where $T$ is the diagram topology that we will specify by drawing it, and $D_\alpha$ is a set
of massless propagator denominator for a given topology $T$.
The Laurent expansion for those master integrals starts at
$\mathcal{O}(\epsilon^{-4})$ for $G_{11111111000}$ and $\mathcal{O}(\epsilon^{-3})$ for $G_{11111111-100}$ and
$G_{111111110-10}$, hence, getting accurate numerical results for the finite part
is a demanding task. Here we follow the notation used in~\cite{Badger:2016ozq} where the local numerators were applied to
six-gluon all-plus helicity amplitudes in Yang-Mills theory.
For the master integral topology under consideration, we choose the following basis of master
integral that exhibits improved IR behaviour
\begin{gather}
 I\bigg(\usepix{1.5cm}{figs/T431_1}{0}{0}{40}{0}\bigg)\big[ \spaa{4}{k_2}{p_{56}}{4} \mu_{11} \big], \nonumber \\
 I\bigg(\usepix{1.5cm}{figs/T431_1}{0}{0}{40}{0}\bigg)\big[ \spbb{4}{k_2}{p_{56}}{4} \mu_{11} \big],
 \label{eq:localmisA1} \\
 I\bigg(\usepix{1.5cm}{figs/T431_1}{0}{0}{40}{0}\bigg)\big[ \tr_-(1(k_1-p_1)(k_1-p_{12})3)
\spaa{4}{k_2}{p_{56}}{4} \big], \nonumber
\end{gather}
where $\tr_{\pm}(ijkl) = \frac{1}{2}\tr(
(1\pm\gamma_5)\slashed{p}_i\slashed{p}_j\slashed{p}_k\slashed{p}_l)$.
The first two integrals in Eq.~\eqref{eq:localmisA1} evaluate to $\mathcal{O}(\epsilon)$ and do not contribute to the two-loop
amplitude, while the last integral is finite. Note that, for this topology the master integral
coefficients do not contain any poles in $\epsilon$, therefore, the master integrals need to be expanded
to $\mathcal{O}(1)$. To evaluate the last integral in Eq.~\eqref{eq:localmisA1} using sector decomposition
method, we first need to write the numerator insertion in terms of scalar products and momentum
twistor variables ($k_i\cdot k_j$, $k_i\cdot p_j$, $x_i$) using loop momentum decomposition given in
Eqs.~\eqref{eq:loopmomparameterisation1}~-~\eqref{eq:loopmomparameterisation3}
\begin{equation}
 I\bigg(\usepix{1.5cm}{figs/T431_1}{0}{0}{40}{0}\bigg)\big[ \tr_-(1(k_1-p_1)(k_1-p_{12})3)
\spaa{4}{k_2}{p_{56}}{4} \big] = I\bigg(\usepix{1.5cm}{figs/T431_1}{0}{0}{40}{0}\bigg)\big[
f(k_i\cdot k_j,k_i\cdot p_j,x_i)  \big].
\label{eq:localmiSecDec}
\end{equation}
The integral on the RHS can be directly evaluated using \textsc{pySecDec}~\cite{Borowka:2017idc}
by passing the whole numerator into sector decomposition algorithm. For topologies with four point
kinematics an extra stage of transverse integration is necessary to convert the integrals into a
form compatible with the sector decomposition approach.

\begin{table}
\begin{center}
\begin{tabular}{ m{7em} m{8em} m{3em} m{3em} m{10em}  }
\hline \hline
$T$ & $N(k_i,p_i,\mu_{ij})$ & $\delta_1$ & $\delta_2$ & Crossing \\ \hline\hline
\multirow{3}{7em}{\usepix{1.5cm}{figs/T431_1}{0}{0}{40}{0}}
& $\Omega_{2;4|56}\; \mu_{11}$        & $\epsilon$ & 1 & \multirow{3}{10em}{ $(1\leftrightarrow
4,2\leftrightarrow 3)$ } \\
& $\Omega_{2;4|56}^*\; \mu_{11}$      & $\epsilon$ & 1 \\
& $\Psi_{1;1|2|3} \; \Omega_{2;4|56}$ & 1      & 1  \\ \hline
\multirow{3}{7em}{\usepix{1.5cm}{figs/T431_2}{0}{0}{40}{0}}
& $\Phi_{2;2|3}\; \mu_{11}$        & $\epsilon$ & 1 & \multirow{3}{10em}{ --- } \\
& $\Phi_{2;2|3}^*\; \mu_{11}$   & $\epsilon$ & 1 \\
& $\Psi_{1;4|56|1} \; \Phi_{2;2|3}$ & 1      & 1  \\ \hline
\multirow{3}{7em}{\usepix{1.5cm}{figs/T3315L}{0}{0}{30}{0}}
& $\mu_{12}$                         & $\epsilon$ & 1 & \multirow{3}{10em}{ $(1\leftrightarrow
4,2\leftrightarrow 3)$ } \\
& $ \Omega_{1;4|56}\;\Phi_{2;2|3}$   & 1 & 1 \\
& $ \Omega_{1;4|56}\;\Phi_{2;2|3}^*$ & 1 & 1 \\ \hline
\usepix{1.3cm}{figs/T4215L}{5}{0}{35}{0} & $\mu_{11}$ & $\epsilon$ & 1 & $\;(1\leftrightarrow
4,2\leftrightarrow 3)$ \\ \hline
\multirow{6}{7em}{\usepix{1.6cm}{figs/T3215L}{0}{0}{25}{0}}
& 1 & $\epsilon^{-2}$ & $\epsilon$ & \multirow{6}{9em}{ --- } \\
& $\Phi_{1;2|3}$   & 1 & $\epsilon$ \\
& $\Phi_{1;2|3}^*$ & 1 & $\epsilon$ \\
& $\Phi_{1;2|3}\;\Omega_{2;1|56}$   & 1 & $\epsilon$ \\
& $\Phi_{1;2|3}\;\Omega_{2;1|56}^*$ & 1 & $\epsilon$ \\
& $\Phi_{1;2|3}^*\;\Omega_{2;1|56}$ & 1 & $\epsilon$ \\ \hline
\multirow{2}{7em}{\usepix{1.1cm}{figs/T4115L_1}{10}{0}{35}{0}}
& $\Psi_{1;2|3|4}$   & $\epsilon^{-1}$ & $\epsilon$ & \multirow{2}{10em}{ $(1\leftrightarrow
4,2\leftrightarrow 3)$  } \\
& $\Psi_{1;2|3|4}^*$ & $\epsilon^{-1}$ & $\epsilon$ \\ \hline
\multirow{2}{7em}{\usepix{1.1cm}{figs/T4115L_2}{10}{0}{30}{0}}
& $\Psi_{1;4|56|1}$   & $\epsilon^{-1}$ & $\epsilon$ & \multirow{2}{10em}{ ---  } \\
& $\Psi_{1;4|56|1}^*$ & $\epsilon^{-1}$ & $\epsilon$ \\ \hline
\end{tabular}
\end{center}
\caption{
Master integral topologies made up of five external legs with local numerator insertions.
The topology $T$ and numerator $N(k_i,p_i,\mu_{ij})$ correspond to the integral definition in
Eq.~\eqref{eq:integralwnumerator}. Numerator building blocks $\Psi$, $\Phi$ and $\Omega$ are defined in
Eq.~\eqref{eq:localnumshorthand}. $\delta_1$ is the order at which the  expansion
of the master integral starts, while $\delta_2$ the highest order in $\epsilon$ needed from the master integral for
the amplitude evaluation.
}
\label{tab:localMI1}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{ m{7em} m{8em} m{3em} m{3em} m{10em}  }
\hline \hline
$T$ & $N(k_i,p_i,\mu_{ij})$ & $\delta_1$ & $\delta_2$ & Crossing \\ \hline\hline
\multirow{2}{7em}{\usepix{1.6cm}{figs/T3312M_1}{0}{0}{25}{0}}
& $ \Omega_{1;1|23}\;\Omega_{2;4|56} $     & 1 & 1 & \multirow{2}{10em}{ $(1\leftrightarrow
4,2\leftrightarrow 3)$ } \\
& $ \Omega_{1;1|23}\;\Omega_{2;4|56}^* $   & 1 & 1 \\ \hline
\multirow{2}{7em}{\usepix{1.6cm}{figs/T3312M_2}{0}{0}{25}{0}}
& $ \Omega_{1;4|56}\;\Omega_{2;3|12} $     & 1 & 1 & \multirow{2}{10em}{ $(1\leftrightarrow
4,2\leftrightarrow 3)$ } \\
& $ \Omega_{1;4|56}\;\Omega_{2;3|12}^* $   & 1 & 1 \\ \hline
\usepix{1.6cm}{figs/T3212M_2}{5}{0}{25}{0} & $ \Omega_{1;3|12} $     & 1 & 1 & $\;(1\leftrightarrow
4,2\leftrightarrow 3)$ \\ \hline
\usepix{1.6cm}{figs/T3212M_1}{5}{0}{25}{0} & $ \Omega_{1;1|23} $     & 1 & 1
& \makecell[l]{
  $\;(1\leftrightarrow 4)$ \\
  $\;(2\leftrightarrow 5,3\leftrightarrow 6)$ \\
  $\;(1\leftrightarrow 4,2\leftrightarrow 5,3\leftrightarrow 6)$
  } \\ \hline\hline
\end{tabular}
\end{center}
\caption{
Master integral topologies made up of four external legs with local numerator insertions.
The topology $T$ and numerator $N(k_i,p_i,\mu_{ij})$ correspond to the integral definition in
Eq.~\eqref{eq:integralwnumerator}. Numerator building blocks $\Psi$, $\Phi$ and $\Omega$ are defined in
Eq.~\eqref{eq:localnumshorthand}. $\delta_1$ is the order at which the  expansion
of the master integral starts, while $\delta_2$ the highest order in $\epsilon$ needed from the master integral for
the amplitude evaluation.
}
\label{tab:localMI2}
\end{table}

In Tables~\ref{tab:localMI1}~and~\ref{tab:localMI2}, we present a list of master integral
topologies with local numerator insertions, where we use
 the following shorthand notation for the numerator insertions
\begin{align}
\Psi_{i;a|b|c}   & = \tr_-(a (k_i - p_a) (k_i - p_{ab}) c), \nonumber \\
\Phi_{i;a|b}     & = \spAB{a}{k_i}{b}, \\
\Omega_{i;a|b} & = \spaa{a}{k_i}{p_b}{a}. \nonumber
\label{eq:localnumshorthand}
\end{align}
We also include in Tables~\ref{tab:localMI1}~and~\ref{tab:localMI2} the order at which the  expansion
of the master integral starts, $\mathcal{O}(\delta_1)$, the highest order in $\epsilon$ needed from the master integral for
the amplitude evaluation, $\mathcal{O}(\delta_2)$, as well as the list of possible permutations/crossings to obtain the
full list of master integral topologies with local numerator insertions.
$\delta_1>\delta_2$ indicates that an integral does not contribute to the finite part of the amplitude, while
$\delta_1=\delta_2$ means that the integral contributes only to the finite part of the
amplitude. In this latter case, it means only the leading order term in the $\epsilon$ expansion of the
integral is required.

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

We select a random Euclidean phase-space point by choosing rational values in the momentum twistor
parametrisation from Eq.~\eqref{eq:ZmatrixW},
\begin{gather}
  x_{1} = -1,\quad
  x_{2} = \frac{79}{270},\quad
  x_{3} = \frac{64}{61},\quad
  x_{4} = -\frac{37}{78},\nonumber \\  \nonumber \\
  x_{5} = \frac{83}{102},\quad
  x_{6} = \frac{4723}{9207},\quad
  x_{7} = -\frac{12086}{7451},\quad
  x_{8} = \frac{3226}{2287}.
\label{eq:kinematicpts}
\end{gather}

The numerical results of the leading colour partial amplitudes are obtained by evaluating the master integrals
in three different ways:
\begin{enumerate}
\item We make use of the master integral solutions  that are known analytically
and readily available for evaluation
in the Euclidean kinematics~\cite{Gehrmann:2000zt,Papadopoulos:2015jft}.
\item Master integrals defined with the local numerator insertions,
shown in Tables~\ref{tab:localMI1}~and~\ref{tab:localMI2}, are numerically evaluated using
\textsc{pySecDec}~\cite{Borowka:2017idc}.
\item The remaining master integrals, which contain either one-loop squared
topologies or 5 propagator or fewer two-loop topologies, are evaluated directly using
\textsc{Fiesta}~\cite{Smirnov:2015mct} and
\textsc{pySecDec}~\cite{Borowka:2017idc}.\footnote{The one-loop integrals, required to
$\mathcal{O}(\epsilon^2)$, are evaluated numerically. Since these terms do not cause any stability
issues there was no need to consider optimisation with analytic expressions.}
\end{enumerate}
Note that there is a class of four-point master integrals with two off shell legs, that are not
covered in~\cite{Papadopoulos:2015jft}, but available
in~\cite{Henn:2014lfa,Gehrmann:2015ora,vonManteuffel:2015msa}, where the solutions are derived
in the physical region. Instead of taking those results and perform analytic continuations
to the Euclidean region for the numerical evaluation, we choose finite local master integral bases
for the 6- and 7-propagator two-loop topologies, as shown in Table~\ref{tab:localMI2}, and
directly evaluate the rest of the integrals in this class numerically\footnote{
These integrals belong to the third type of evaluation discussed above. We see no problem in
performing the analytic continuation on the expressions for the double off-shell $2\to 2$
integrals. However, since the approach with local numerators and the sector decomposition worked with sufficient
accuracy it was unnecessary to do this for our example.}.
To assess uncertainties from the numerical evaluations via the sector decomposition method,
we perform numerical integrations with three different random number seeds.
The final results are obtained by taking the average
and the error is computed by averaging the difference among the three results.

\begin{table}[t]
\begin{center}
\begin{tabularx}{0.95\textwidth}{|C{0.9}|C{0.7}|C{1.1}|C{1.2}|C{1.1}|C{1.0}|}
  \hline
  $qgg\bar{q}^\prime\bar\nu\ell$  & $\epsilon^{-4}$ & $\epsilon^{-3}$ & $\epsilon^{-2}$ & $\epsilon^{-1}$ & $\epsilon^{0}$ \\
  \hline
  $\widehat{A}^{(2),[0]}_{-++++-}$ & 4.50000 & -2.38581(9) & 17.0943(2)   & 69.09(3)    & -137.1(3)   \\
  $\widehat{A}^{(2),[1]}_{-++++-}$ & 0.00000 & -0.62498(3) & -147.7288(3) & -347.221(9) & 863.63(8)   \\
  $\widehat{A}^{(2),[2]}_{-++++-}$ & 0.00000 & 0.00000     & -0.031254(7) & -3.72543(2) & -65.7967(5) \\
  \hline
  $qgg\bar{q}^\prime\bar\nu\ell$  & $\epsilon^{-4}$ & $\epsilon^{-3}$ & $\epsilon^{-2}$ & $\epsilon^{-1}$ & $\epsilon^{0}$ \\
  \hline
  $\widehat{A}^{(2),[0]}_{-+-++-}$ & 4.50000 & -2.3858(1)   & -16.3282(4)   & -1.397(3)    & 70.05(4) \\
  $\widehat{A}^{(2),[1]}_{-+-++-}$ & 0.00000 & -0.624996(6) & 0.7603(2)     & 4.457(1)     &  1.04(1) \\
  $\widehat{A}^{(2),[2]}_{-+-++-}$ & 0.00000 & 0.00000      & -0.0312498(2) & -0.025800(5) & 0.72620(5) \\
  \hline
  $qgg\bar{q}^\prime\bar\nu\ell$  & $\epsilon^{-4}$ & $\epsilon^{-3}$ & $\epsilon^{-2}$ & $\epsilon^{-1}$ & $\epsilon^{0}$ \\
  \hline
  $\widehat{A}^{(2),[0]}_{--+++-}$ & 4.50000 & -2.38579(5)  & -22.18117(7)  & -16.113(6)   & 90.06(4)   \\
  $\widehat{A}^{(2),[1]}_{--+++-}$ & 0.00000 & -0.625000(1) &  1.131987(9)  &  5.7364(2)   & -2.1289(6) \\
  $\widehat{A}^{(2),[2]}_{--+++-}$ & 0.00000 & 0.00000      & -0.0312502(3) &  0.005162(2) & 1.21279(4) \\
  \hline
\end{tabularx}
\end{center}
\caption{The numerical results of
$\hat{A}^{(2),[i]}\left(1_q,2_{g},3_{g},4_{\bar{q}^\prime},5_{\bar\nu},6_\ell \right)$
using kinematic point in Eq.~\eqref{eq:kinematicpts} for each $(d_s-2)$ component.
}
\label{tab:numggDs}
\end{table}

\begin{table}[t]
\begin{center}
\begin{tabularx}{0.95\textwidth}{|C{0.9}|C{0.7}|C{1.1}|C{1.2}|C{1.1}|C{1.0}|}
  \hline
  $q\bar{Q}Q\bar{q}^\prime\bar\nu\ell$  & $\epsilon^{-4}$ & $\epsilon^{-3}$ & $\epsilon^{-2}$ & $\epsilon^{-1}$ & $\epsilon^{0}$ \\
  \hline
  $\widehat{A}^{(2),[0]}_{-+-++-}$ & 2.00000 & -6.00283(9)  & -12.7724(2) & 31.869(6)     & 158.89(6)   \\
  $\widehat{A}^{(2),[1]}_{-+-++-}$ & 0.00000 & -0.583333(1) & 0.96122(6)  & 5.2453(4)     & 2.853(3)    \\
  $\widehat{A}^{(2),[2]}_{-+-++-}$ & 0.00000 & 0.00000      & -0.055555  & -0.240170(1) & -0.25365(2) \\
  \hline
  $q\bar{Q}Q\bar{q}^\prime\bar\nu\ell$  & $\epsilon^{-4}$ & $\epsilon^{-3}$ & $\epsilon^{-2}$ & $\epsilon^{-1}$ & $\epsilon^{0}$ \\
  \hline
  $\widehat{A}^{(2),[0]}_{--+++-}$ & 2.00000 & -6.00282(8)  & -18.0013(1)  & 34.592(7)    & 222.52(6)    \\
  $\widehat{A}^{(2),[1]}_{--+++-}$ & 0.00000 & -0.583334(1) &  2.059832(8) & 5.4211(2)    & -13.5049(5)  \\
  $\widehat{A}^{(2),[2]}_{--+++-}$ & 0.00000 & 0.00000      & -0.055555    & -0.081689(2) &  1.10832(2)   \\
  \hline
\end{tabularx}
\end{center}
\caption{The numerical results of
$\hat{A}^{(2),[i]}\left(1_q,2_{\bar{Q}},3_{Q},4_{\bar{q}^\prime},5_{\bar\nu},6_\ell \right)$
using kinematic point in Eq.~\eqref{eq:kinematicpts} for each $(d_s-2)$ component.
}
\label{tab:numQQDs}
\end{table}

\begin{table}[t]
\begin{center}
\begin{tabularx}{0.95\textwidth}{|C{0.9}|C{0.7}|C{1.1}|C{1.2}|C{1.1}|C{1.0}|}
  \hline
  $qgg\bar{q}^\prime\bar\nu\ell$ & $\epsilon^{-4}$ & $\epsilon^{-3}$ & $\epsilon^{-2}$ & $\epsilon^{-1}$ & $\epsilon^{0}$ \\
  \hline
  $\widehat{A}^{(2)}_{-++++-}$ & 4.50000 & -3.63577(3) & -277.2182(7) & -344.56(1)  & 2051.1(2) \\
  $          P^{(2)}_{-++++-}$ & 4.5     & -3.63576    & -277.2186    & -344.569(6) & --- \\
  \hline
  $\widehat{A}^{(2)}_{-+-++-}$ & 4.50000 & -3.63581(9) & -13.6826(2) & 6.143(5)   & 66.21(7) \\
  $          P^{(2)}_{-+-++-}$ & 4.5     & -3.63576    & -13.6824    & 6.145(1)   & --- \\
  \hline
  $\widehat{A}^{(2)}_{--+++-}$ & 4.50000 & -3.63579(5) & -18.79219(7) & -6.633(6)  & 79.02(4) \\
  $          P^{(2)}_{--+++-}$ & 4.5     & -3.63576    & -18.79212    & -6.6303(5) & --- \\
  \hline
\end{tabularx}
\end{center}
\caption{The numerical comparison of
$\hat{A}^{(2)}\left(1_q,2_{g},3_{g},4_{\bar{q}^\prime},5_{\bar\nu},6_\ell \right)$
with the universal pole structure ${\mathcal{P}}^{(2)}$ defined in Eq.~\eqref{eq:poles}, using kinematic point of
Eq.~\eqref{eq:kinematicpts}, in the HV scheme.}
\label{tab:numggHV}
\end{table}

\begin{table}[t]
\begin{center}
\begin{tabularx}{0.95\textwidth}{|C{0.9}|C{0.7}|C{1.1}|C{1.2}|C{1.1}|C{1.0}|}
  \hline
  $q\bar{Q}Q\bar{q}^\prime\bar\nu\ell$ & $\epsilon^{-4}$ & $\epsilon^{-3}$ & $\epsilon^{-2}$ & $\epsilon^{-1}$ & $\epsilon^{0}$ \\
  \hline
  $\widehat{A}^{(2)}_{-+-++-}$ & 2.00000 & -7.16949(9) &  -9.9055(2) & 39.922(6)  & 154.79(7) \\
  $          P^{(2)}_{--+++-}$ & 2       & -7.16944    &  -9.9054    & 39.9245(8) & ---       \\
  \hline
  $\widehat{A}^{(2)}_{--+++-}$ & 2.00000 & -7.16948(8) & -12.9371(1) & 41.432(8)  & 189.53(6) \\
  $          P^{(2)}_{--+++-}$ & 2       & -7.16944    & -12.9370    & 41.4353(6) & --- \\
  \hline
\end{tabularx}
\end{center}
\caption{The numerical comparison of
$\hat{A}^{(2)}\left(1_q,2_{\bar{Q}},3_{Q},4_{\bar{q}^\prime},5_{\bar\nu},6_\ell \right)$
with the universal pole structure ${\mathcal{P}}^{(2)}$ defined in Eq.~\eqref{eq:poles}, using kinematic point of
Eq.~\eqref{eq:kinematicpts}, in the HV scheme.}
\label{tab:numQQHV}
\end{table}

We present the results for unrenormalised
$q\bar{Q}Q\bar{q}^\prime\bar\nu\ell$ and $qgg\bar{q}^\prime\bar\nu\ell$
helicity amplitudes, normalised to the tree level amplitude
\begin{equation}
\hat{A}^{(2)}_{\lambda_1\lambda_2\lambda_3\lambda_4\lambda_5\lambda_6}
= \frac{A^{(2)}\left(1^{\lambda_1},2^{\lambda_2},3^{\lambda_3},4^{\lambda_4},5^{\lambda_5},6^{\lambda_6}\right)}
   {A^{(0)}\left(1^{\lambda_1},2^{\lambda_2},3^{\lambda_3},4^{\lambda_4},5^{\lambda_5},6^{\lambda_6}\right)},
\label{eq:treenormalisation}
\end{equation}
with helicities $\lambda_i$. We can further split the amplitude into components of $d_s-2$
\begin{equation}
A^{(2)}\big(1^{\lambda_1},2^{\lambda_2},3^{\lambda_3},4^{\lambda_4},5^{\lambda_5},6^{\lambda_6}\big) =
\sum_{i=0}^{2} \left(d_s-2\right)^i A^{(2),[i]}
\big(1^{\lambda_1},2^{\lambda_2},3^{\lambda_3},4^{\lambda_4},5^{\lambda_5},6^{\lambda_6}\big).
\end{equation}

In Tables~\ref{tab:numggDs}~and~\ref{tab:numQQDs} we display numerical evaluations of the helicity amplitudes for
$qgg\bar{q}^\prime\bar\nu\ell$ and $q\bar{Q}Q\bar{q}^\prime\bar\nu\ell$ channels, respectively, using
the kinematic point given in Eq.~\eqref{eq:kinematicpts} for each $(d_s-2)$ component.
In Tables~\ref{tab:numggHV}~and~\ref{tab:numQQHV},
we compare the divergent part of our numerical results against the
universal two-loop pole structure in Eq.~\eqref{eq:poles} in the HV scheme.
To obtain the two-loop pole structure of Eq.~\eqref{eq:poles} up to the single pole,
we need to compute the one-loop amplitude up to $\mathcal{O}(\epsilon)$ for both the
$qgg\bar{q}^\prime\bar\nu\ell$ and $q\bar{Q}Q\bar{q}^\prime\bar\nu\ell$ processes.
The one-loop amplitude is computed by processing the Feynman diagrams through our integrand reduction
setup, followed by IBP decomposition into the one-loop master integral basis consisting of six
bubbles,
a three-mass triangle, two one-mass boxes, a two-mass easy box, two two-mass hard boxes
and a one-mass pentagon,
in a similar fashion to the two-loop case that is discussed in Section~\ref{sec:setup}. The $\mathcal{O}(\epsilon)$ part of the
two-mass easy box and one-mass pentagon integrals are evaluated numerically using
$\textsc{Fiesta}/\textsc{pySecDec}$, while the rest are obtained from available analytic
expressions~\cite{Ellis:2007qk,Papadopoulos:2014lla}. Therefore, the numerical values quoted for the poles in
Tables~\ref{tab:numggHV}~and~\ref{tab:numQQHV} are exact up to $\mathcal{O}(\epsilon^{-2})$.
We assess the uncertainty of the $\mathcal{O}(\epsilon^{-1})$ part of the pole structure using the same
method as in the two-loop numerical evaluations.

We additionally perform another check of our results by independently
processing the two-loop Feynman diagrams
through a numerical diagram-based integrand reduction into an integrand representation consisting of
four-dimensional ISPs and the extra-dimensional part of the loop momenta $\mu_{ij}$. The integrals
containing $\mu_{ij}$ are first written as dimension-shifted integrals and all integrals appearing
in the amplitude are evaluated directly using $\textsc{Fiesta}/\textsc{pySecDec}$.
This approach is similar to the method we employed for the numerical evaluation of
the planar two-loop five gluon amplitude~\cite{Badger:2017jhb}.
The results obtained using this method are in perfect agreement
with the results reported in this paper.

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

In this article we have presented numerical results for the planar two-loop helicity amplitudes
for the scattering of a $W$-boson with four partons for the first time. This computation is the first
step towards obtaining analytic expressions using the reconstruction of rational functions
with finite field arithmetic.

A number of important steps remain to be completed in order for this
to be a feasible target. Firstly, the complete list of master integrals should be evaluated
analytically since the sector decomposition approach is still too CPU intensive for phenomenological
applications. This seems a reasonable aim owing to the recent success of the planar and non-planar pentagon
functions~\cite{Chicherin:2017dob,Abreu:2018aqd,Abreu:2019rpt,Chicherin:2018yne,Chicherin:2018mue,Chicherin:2019xeg,Chicherin:2018old}, though the efficient numerical implementation of the resulting analytic functions will require
further study. Secondly, the coefficients of the master integrals still have a high degree of
algebraic complexity. As shown in applications to five-parton scattering, direct reconstruction of
the finite remainder after subtraction of UV and IR poles leads to a substantial reduction in
complexity~\cite{Badger:2018enw,Abreu:2018zmy,Abreu:2019odu}.

We hope that the work presented here presents valuable information that can be used to achieve
these goals, as well as providing encouragement that they are realistic in the near future.

\begin{acknowledgments}
We thank  Johannes Henn, Thomas Gehrmann, Andreas von Manteuffel for helpful discussions.
We also thank Johannes Schlenk for his assistance with \textsc{pySecDec}.
SB is supported by an STFC Rutherford Fellowship ST/L004925/1 and CBH and HBH received partial
support from Rutherford Grant ST/M004104/1. This project has received funding from the European Union's Horizon
2020 research and innovation programme under grant agreement No 772099. This project
has received funding from the European Union's Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement 746223.

\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
The Origin of Emotions with Jaak Panksepp (BSP 91)
Dr. Jaak Panksepp
In his new book, The Archaeology of Mind: Neuroevolutionary Origins of Human Emotions, Jaak Panksepp set out to make his life's work more accessible to a general audience. To be honest, reading this book requires a significant commitment, but I think he does a wonderful job of updating his classic textbook, Affective Neuroscience. Anyone who is interested in this field will definitely want this book as a reference.
The other strength of Archeology of Mind is its evolutionary approach. The primary emotional processes that Panksepp has spent his career studying have their origins in the ancient parts of the brain that are shared by all mammals. This contradicts longstanding assumptions in neuroscience, but it has important implications for both humans and other animals.
In Episode 91 of the Brain Science Podcast, Dr. Panksepp and I talked about some of the new information contained in Archaeology of Mind, with a particular focus on FEAR, which, contrary to what many researchers claim, does NOT begin in the amygdala, but begins much lower. We do talk briefly about the experimental evidence, but this was covered in more detail during Dr. Panksepp's previous appearance on the Brain Science Podcast in BSP 65.
New episodes of the Brain Science Podcast are always FREE. All episodes posted after January 1, 2013 are FREE. Please see individual show notes for links to audio files and transcripts.
The Archaeology of Mind: Neuroevolutionary Origins of Human Emotions, by Jaak Panksepp and Lucy Biven.
Affective Neuroscience: The Foundations of Human and Animal Emotions, by Jaak Panksepp.
Please see the episode transcript for additional links and references.
Related Episodes:
BSP 32: An introduction to brain anatomy.
BSP 47: A review of brain evolution.
BSP 65: Previous interview with Jaak Panksepp (click here for Premium version).
BSP 89: Interview with Evan Thompson, author of Mind in Life: Biology, Phenomenology, and the Sciences of Mind.
BSP 90: Review of Self Comes to Mind: Constructing the Conscious Brain, by Antonio Damasio.
The earliest episodes of the Brain Science Podcast are now disappearing from iTunes but they remain freely available here. They are also available within the Brain Science Podcast app for mobile devices. By the way, the mobile app has been updated, and I need users to post new reviews.
Don't forget to get your copy of my eBook, Are You Sure? The Unconscious Origins of Certainty, from Amazon.com. You can also buy the PDF version HERE.
The Brain Science Podcast is supported by listener Donations. It also relies on your word of mouth, so don't forget to share it with others.
Next month's Brain Science Podcast will be our annual review episode. Meanwhile don't forget to check out my other podcast, Books and Ideas. The most recent episode is an interview with Emily Reese from Minnesota Classical Radio.
Join the Brain Science Podcast Fan Page on Facebook, Google+, and share your thoughts in our Discussion Forum on Goodreads. Of course, you can also send me email at gincampbell at mac dot com.
To get show notes automatically and never miss an episode of the Brain Science Podcast sign up for the BSP Newsletter.
November 30, 2012 / Ginger Campbell, MD/
"Limbic System", Brain Evolution, Brain Research, Emotion, Interviews, Neuroscience, Podcast Show Notes, Psychology
Archeology of Mind, Emotion, Jaak Panksepp, Neuroscience, affective neuroscience, brain, brain science podcast, fear
"The Self Illusion" with Bruce Hood (BSP 88)
Bruce Hood, PhD
The Self Illusion: How the Social Brain Creates Identity, by Bruce Hood, is a fascinating look at how our brains create both our experience of the world and our sense of being a single, coherent self. As the word "illusion" in the title indicates, neither is exactly what it seems. When I interviewed Dr. Hood (BSP 88), he explained that The Self Illusion is a broad introduction to this somewhat surprising idea. TheSelf Illusion was written with a general audience in mind. For those already familiar with the topic, he also puts a new emphasis on the role of development. All readers should come away with a new appreciation for the critical role social interactions play through out human life.
Buy Audio mp3 for $1.
Books and Ideas #34: Bruce Hood talks about his first book, SuperSense. (mp3)
BSP 29: Dr. Maryanne Wolf, author of Proust and the Squid: The Story and Science of the Reading Brain. (mp3)
BSP 57: Dr. Chris Frith, author of Making up the Mind: How the Brain Creates Our Mental World. (mp3)
BSP 67: Thomas Metzinger, PhD, author of The Ego Tunnel: The Science of the Mind and the Myth of the Self. (mp3)
Books and Ideas #43: Dr. Carol Tavris explains the basics of cognitive dissonance. (mp3)
BSP 72: The Neuroscience of Magic. (Listen Now)
BSP 75: Dr. David Eagleman, author of Incognito: The Secret Lives of the Brain. (mp3)
Books and Ideas #48: Jonathan Gottschall, author of The Storytelling Animal: How Stories Make Us Human. (mp3)
Additional References:
SuperSense: Why We Believe in the Unbelievable, by Bruce M. Hood. (Audible link)
The Self Illusion: How the Social Brain Creates Identity, by Bruce Hood (Audible link)
Willpower: Rediscovering the Greatest Human Strength, by Roy F. Baumeister & John Tierney (Audible link)
Please see the FREE episode transcript for many additional references, including references to the scientific studies discussed in this podcast.
I will be in Philadelphia, PA October 16-21 to attend the AAFP annual meeting. Listeners who live in the area and physicians who are attended this meeting are invited to drop me an email if interested in getting together in person.
The latest episode of Books and Ideas is an interview with Jonathan Gottschall, author of The Storytelling Animal: How Stories Make Us Human. This episode continues some of the same themes explored in BSP 88.
Upcoming episodes: Discussion of Self Comes to Mind, by Antonio Damasio, and interviews with Evan Thompson (Mind in Life) and Jaak Panksepp (The Archaeology of Mind).
Sign up for the Brain Science Podcast Newsletter: You will get the show notes automatically and never miss a new episode.
Don't forget to post your review of my eBook, Are Your Sure? The Unconscious Originis of Certainty, on Amazon.com, Goodreads.com or your favorite website. (Send me your Amazon receipt to get a free PDF.)
Connect with other BSP fans: BSP Facebook Fan Page, Google +, Discussion Forum on Goodreads.com.
Follow me on Twitter: @docartemis.
Send me feedback at brainsciencepodcast@gmail.com
September 07, 2012 / Ginger Campbell, MD/
Consciousness, Development, Interviews, Neuroscience, Podcast Show Notes, Psychology
Bruce Hood, Neuroscience, Podcast, The Self Illusion, brain, brain science podcast, cognition, developmental psychology, science
BSP Fans Are Enthusiastic About Newsletter
I have been getting a great response to the BSP Newsletter, but those of you who receive blog posts via email don't seem to be getting the sign-up form. If you want to sign up just Click Here to go the sign-up page or go to /brainsciencepodcast/newsletter/.
brain science podcast, newsletter
Review of Year 2 of the Brain Science Podcast (BSP 52)
Brain Science Podcast #52 is our Second Annual Review Episode. We review some of the highlights from 2008. I also discuss the various other on-line resources that I have created for listeners. Then we look ahead to what I have planned for 2009. This episode is aimed at all listeners, including those who are new to the show.
Highlights from 2008:
Early in the year (#30) I discussed language evolution. My criticism of Noam Chomsky's claim that human language results from a Universal Grammar Module generated quite a bit of discussion. My main purpose was to emphasize that current neuroscience does not support this hypothesis. I discussed Chomsky's work in follow-up interviews with Dr. Michael Arbib (BSP 39) and linguist Alice Gaby (BSP 41).
It is my impression that, at least to some extent, this debate comes back to the age-old "nature versus nurture" controversy, which I discussed more explicitly way back in Episode 4. The evidence seems to be mounting that human intelligence is a product of both processes.
There is no doubt that the capacity for language is inherited, but brain plasticity appears to be equally important. One piece of evidence for this is that the changes in the brain that occur when people learn to read are different between languages like English and German and those like Chinese and Japanese. (Episode 24 and Episode 29)
We had 17 guests on the Brain Science Podcast in 2008, so I can't mention them all here.
John Ratey, MD: In Episode 33 we talked about exercise and the brain, while in Episode 45 we talked about ADD.
Robert Burton, MD: In Episode 43 talked about the implications of the discovery that our sense of knowing (feeling certain) is generated by parts of the brain that are outside our conscious control!
John Medina, PhD: In Episode 37 we considered the practical implications of neuroscience, such as the importance of getting enough sleep and why true multi-tasking is actually impossible.
Dr. Brenda Milner: In Episode 49 this pioneering neuroscientist shared highlights from her long career.
Another highlight was our first live podcast, which was recorded at Dragon*Con in Atlanta, Georgia on August 31.
In the fall I returned to the subject of evolution with a three part series on the evolution of the brain.
Episode 47, Episode 48, and Episode 51.
Episode 51 is an outstanding interview with Dr. Seth Grant in which we discuss the surprising discovery that synapse complexity seems to have evolved BEFORE larger more complex brains.
Online Resources for Listeners:
I encouraged listeners to frequent this website and to subscribe to the RSS feed so as to receive information between posts.
I encouraged listeners to explored the sidebars and tabs on the website for links to other sites of interest.
I reminded listeners that this website includes a complete listing of previous episodes as well as a list of all the guests that have been on the show.
It is now possible to support the Brain Science Podcast via both PayPal and by direct mail.
I encouraged listeners to participate in our Discussion Forum and to post pictures to our Flickr Group.
I invited listeners to contribute content to the Brain Science Podcast Room on FriendFeed and the new Neuroscience News Network on SocialMedian.
I reminded listeners that my personal blog is now at http://gingercampbellmd.com. This site includes abridged show notes for the Brain Science Podcast as well as the complete show notes for Books and Ideas.
Listeners are encouraged to continue to post reviews on iTunes™, Podcast Pickle, Podcast Alley, Digg, and similar sites. All blog posts and tweets are greatly appreciated.
Send email feedback to Ginger Campbell, MD at brainsciencepodcast@gmail.com
December 19, 2008 / Ginger Campbell, MD/ 3 Comments
Brain Anatomy, Brain Evolution, Brain Research, Neuroscience, Podcast Show Notes
Brenda Milner, Ginger Campbell, John Ratey, Michael Arbib, Podcast, Robert Burton, brain science, brain science podcast, language evolution
Dr. Brenda Milner: Pioneer in Memory Research (BSP 49)
Brain Science Podcast #49 is an interview with pioneering neuroscientist, Brenda Milner, PhD. Dr. Milner is known for her contributions to understanding memory and her work with split-brain patients. Her work as an experimental psychologist has been fundamental to the emergence of the field of cognitive neuroscience.
This interview is a follow-up of Dr. Milner's recent interview with Dr. Marc Pelletier on Futures in Biotech. I highly recommend listening to both interviews.
Listen to Dr. Milner on Futures in Biotech (Episode33)
Brenda Milner, PhD: Montreal Neurological Institute. (Read her MNI biography)
Wilder Penfield, MD: neurosurgeon and found of the Montreal Neurological Institute.
Roger Sperry, PhD: received the Nobel Prize for his work with split-brain patients
Brain Research, Interviews, Memory, Neuroscience, Podcast Show Notes, Psychology
Brenda Milner, Ginger Campbell, Memory, Neuroscience, brain, brain science podcast, experimental psychology
Welcome to the New Brain Science Podcast Website!
As of today, June 14, 2008 the official Brain Science Podcast and Blog has moved here. The old site will remain up because it has an extensive archive of show notes and posts, but no new show notes will be posted there. Make sure you have bookmarked the new site at http://docartemis.com/brainsciencepodcast/. Within the next day or so you should also be able to get here by using http://brainsciencepodcast.com. Please explore the new site and send email to docartemis at gmail.com if you find any band links or other problems.
Ginger Campbell, brain science podcast
Mirror Neurons with Michael Arbib (BSP 39)
Episode 39 of the Brain Science Podcast is an interview with Dr. Michael Arbib from the University of Southern California. Dr. Arbib's work with functional brain imaging has established the presence of mirror neurons in the human brain. In our interview, we focused on the role of mirror neurons in imitation and language. In particular, I questioned Dr. Arbib about the Mirror System Hypothesis (MSH) of Language Evolution that he proposed in 1998 with Giacomo Rizzolatti. We also explored how this hypothesis diverges from the universal grammar proposed by Noam Chomsky. Dr. Arbib also shared his enthusiasm for future research and we talked about the special challenges caused by the interdisciplinary nature of modern neuroscience.
Brain Science Podcast #7: A discussion of bonobos with Stuart Shanker, PhD.
Brain Science Podcast #30: a discussion of language evolution including my thoughts on Chomsky.
Brain Science Podcast #35: a detailed discussion of Mirrors in the Brain, by Giacomo Rizzolatti.
Scientists mentioned in the interview:
Giacomo Rizzolatti: His team discovered mirror neurons at the University of Parma, Italy. Other team members: Vittorio Gallese, Luciano Fadiga, and Leo Fogassi.
Ursula Bellugi (Salk Institute): pioneered the neurobiology of sign language.
Richard Byrne (University of St. Andrews): studies how gorillas learn in the wild.
Michael Tomasello (Max Planck Institute for Comparative Anthropology): studies social behavior of primates, including how communicative gestures vary between groups.
Noam Chomsky (MIT): famous linguist who has proposed an inborn universal grammar.
DL Cheney and RM Seyfarth: research about primate vocal behavior, especially the use of calls in the wild.
Cybernetics, by Norbert Wiener- this classic inspired Dr. Arbib's career in computational neuroscience.
Action to Language via the Mirror Neuron System, (2006) edited by Michael Arbib.
Mirrors in the Brain by Giacomo Rizzolatti and Corrado Sinigaglia, 2008. Discussed in Episode 37.
Mirroring People by Marco Iacoboni, 2008. The author studies mirror neurons at UCLA.
Rizzolatti G & Arbib MA Language within our grasp. Trends in Neuroscience 21(5): 188-194- the first paper describing the Mirror System Hypothesis, 1998.
Iacoboni M, Molnar-Szakacs I, Gallese V, Buccino G, Mazziotta JC, et al. Grasping the intentions of others with one's own mirror neuron system. PLoS Biol 3 (3): e79, 2005.
Arbib, M.A., "From Monkey-like Action Recognition to Human Language: An Evolutionary Framework for Neurolinguistics" (with commentaries and author's response), Behavioral and Brain Sciences, 28:105-167. (Click here for supplemental commentaries and the author's "electronic response.") 2005.
Arbib MA 2006 "Broca's Area in System Perspective: Language in the Context of Action-Oriented Perception" in Broca's Region, Oxford University Press, pp 153-168.
Arbib MA 2008 "Invention and Community in the Emergence of Language: Insights from New Sign Languages," in Foundations in Evolutionary Cognitive Neuroscience, Cambridge University Press.
Click here for more papers by Dr. Arbib. He has also written over 40 books.
Michael Arbib's website.
Brain Research, Cognitive Science, Interviews, Language, Mirror Neurons, Neuroscience, Podcast Show Notes
Giacomo Rizzolatti, Ginger Campbell, Michael Arbib, Mirror Neurons, Neuroscience, Noam Chomsky, Podcast, brain, brain science podcast, computation, language evolution, mirror system hypothesis
"Brain Rules" with John Medina (BSP 37)
John Medina, PhD
Episode 37 of the Brain Science Podcast is an interview with Dr. John Medina, author of Brain Rules: 12 Principles for Surviving and Thriving at Work, Home, and School. We talk about how exercise, sleep, and stress effect our brains, with an emphasis on practical advice for healthier brain function. We also look at how research on memory, vision, and the brain's attention system suggests how we can improve our ability to learn and our ability to share ideas with others.
Dr. Medina's focus is on considering real world examples of how our schools and work environments could be reformed to utilize the growing knowledge of neuroscience. But he also stresses the importance of compiling sufficient experimental data before embarking on new programs.
Links and References:
Brain Rules: 12 Principles for Surviving and Thriving at Work, Home, and School (Book & DVD), by John Medina
The book's website contain extensive references and instructive videos.
YouTube Video of John Medina speaking at Google™.
May 16, 2008 / Ginger Campbell, MD/ 9 Comments
Books, Brain Research, Interviews, Neuroscience, Podcast Show Notes, learning
Brain Rules, Brain Rules: 12 Principles for Surviving and Thriving, Ginger Campbell, John Medina, Neuroscience, Pear Press, Podcast, brain science podcast
Embodied Cognition with Art Glenberg (BSP 36)
Art Glenberg, PhD
Episode 36 of the Brain Science Podcast is an interview with Arthur Glenberg, PhD, about embodied cognition. Dr. Glenberg recently moved to Arizona State University, after over 30 years at the University of Wisconsin's Laboratory of Embodied Cognition. His research focuses on the relationship between embodiment and language. In this interview, we explore the experimental evidence for a theory of language that embraces the concept that our language abilities are actually rooted in our perceptual and motor abilities. Dr. Glenberg also explains how his work has practical implications in helping children learn how to read.
Since Dr. Glenberg has had a long career as a working research scientist, this interview also provided an opportunity to explore how scientific hypotheses are formed and how experiments are designed to test these hypothesis. I think this interview will give you a fascinating look into the real world of cognitive psychology.
New of the Brain Science Podcast are always FREE. All episodes posted after January 1, 2013, are free. See the individual show notes for links the audio files.
Arthur Glenberg, PhD
email: arthur.glenberg@asu.edu.
Laboratory for Embodied Cognition.
Havas, D.A., Glenberg, A.M., and Rink, M. (2007) Emotion simulation during language comprehension. Psychonomic Bulletin & Review14 (3), 436-441
Numerous references available as PDF.
Thinking With the Body: blog post of March 3, 2008.
Other scientists mentioned in the Episode:
George Lakoff: pioneering linguist.
James Gibson: known for his ideas about affordances.William Epstein-emeritus professor at the University of Wisconsin.
Joseph Campos: University of California (Berkelely).
Amy Needham and Amanda Woodard-experiments with velcro mits and infant cognition.
David A Havas: graduate student and co-author with Dr. Glenberg.
Mike Kashak: Florida State University.
Mike Rinck: German co-author-see paper under Glenberg (more papers).
Vittorio Gallese, Dept of Neuroscience, University of Parma, Italy (where mirror neurons were discovered): extensive experimental with motor neurons in monkeys.
Fritz Stack (Germany): experiments showing that facial experiments affect mood and cognition.
Havas, D.A., Glenberg, A.M., and Rink, M. (2007) Emotion simulation during language comprehension. Psychonomic Bulletin & Review14 (3), 436-441.
more publications by Arthur Glenberg.
Sommerville, J.A., Woodard, A.L., and Needham, A., Action experience alters 3-month-old infants' perception of others' actions, Cognition 96 (2005) B1-B11..Strack, F., Martin, L. L., & Stepper, S. (1988). Inhibiting and facilitating conditions of the human smile: A nonobtrusive test of the facial feedback hypothesis. Journal of Personality & Social Psychology, 54, 768-777.
Women, Fire, and Dangerous Things (1987), by George Lakoff.
Recent essays by George Lakoff written for the Rockridge Institute
Brain Research, Cognitive Science, Consciousness, Embodiment, Interviews, Language, Mirror Neurons, Neuroscience, Podcast Show Notes, learning
Art Glenberg, Ginger Campbell, brain science, brain science podcast, cognitive psychology, embodied cognition, experiemental psychology
Transcript of György Buzsáki's Interview Is Now Online
Episode 31 of the Brain Science Podcast was a challenging interview with György Buzsáki about his book Rhythms of the Brain. Thanks to listener Diane Jacobs, we now have a transcript available for Episode 31.
Click here for the transcript.
Be sure to visit Diane's Blog at http://humanantigravitysuit.blogspot.com/. When it comes to reading books about neuroscience, Diane makes me feel like a slacker.
April 27, 2008 / Ginger Campbell, MD/ 1 Comment
Announcements, Other Websites, Transcripts
Diane Jabobs, Gyorgy Buzsaki, brain science podcast, podcast transcripts
A Closer Look at Mirror Neurons (BSP 35)
Brain Science Podcast #35 is a discussion of Mirrors in the Brain: How Our Minds Share Actions, Emotions, and Experience, by Giacomo Rizzolatti and Corrado Sinigaglia. Mirror neurons were discovered in Rizzolatti's lab in Parma, Italy, in the early 1990s, and his book is a detailed to discussion of the experimental evidence in both monkeys and humans. Direct single neuron recordings have been made in monkeys. The evidence in humans is indirect, since it is based on mainly on neuro-imaging studies like PET scans and fMRI scans. Even so, mirror neurons appear to be essential to our ability to understand both the actions and emotions of others.
In this episode, we also explore the evidence that there are other neurons in the motor areas of the brain that have sensory properties and that the areas of the brain traditionally thought to be devoted to sensory functions also contain neurons with motor properties. Another fascinating discovery is the fact that there are neurons that respond not only to somatosensory inputs (such as being touched) but also to visual or auditory inputs from objects within our peri-personal space. For background on these body maps, I recommend listening to Episode 21 and Episode 23. If you are new to the Brain Science Podcast, you may want to listen to those episodes first, because this week's episode is a little more technical than most.
I will be exploring the importance of these discoveries in future episodes.
Giacomo Rizzolatti: University of Parma.
Mirror neurons (Wikipedia entry).
Mirror neurons (Scholarpedia entry written by Dr. Rizzolatti),
Books, Brain Research, Cognitive Science, Mirror Neurons, Neuroscience, Podcast Show Notes
Corrado Sinigaglia, Giacomo Rizzolatti, Ginger Campbell, MD, Mirror Neurons, Mirrors in the brain, Neuroscience, body maps, brain imaging, brain science, brain science podcast, single neuron recordings
Dan Rather's Reports on Neuroplasticity
Today HDNet™ is re-showing an episode of Dan Rather Reports called "Mind Science." It is an excellent review of neuroplasticity. It includes interviews with several leading scientists in the field. I especially enjoyed seeing Nobel Laureate, Eric Kandel, talk about his work with memory. (I talked about Kandel's work on the Brain Science Podcastin Episode 3 and Episode 12.)
"Mind Science" also features the Dalai Llama and scientist, Richard Davidson, talking about the evidence that meditation can change the brain. Rather interviews Sharon Begley about her book, Train Your Mind, Change Your Brain: How a New Science Reveals Our Extraordinary Potential to Transform Ourselves (which I discussed in detail in Episode 10 of the Brain Science Podcast). Other scientists featured in the episode include Michael Merzenich on improving brain function as we age, and Dr. Edward Taub on his revolutionary approach to stroke rehabilitation. (My show notes for Episode 10 include links for all the scientists interviewed by Rather.)
It was particulary gratifying to see Dr. Kandel endorse Ed Taub's Constraint-Induced Movement Therapy. Dr. Taub was interviewed in Episode 28 of the Brain Science Podcast. If you don't get HDNet™, you can watch Dan Rather Reports on-line, via podcast or on Facebook.
Summary of relevant episodes of the Brain Science Podcast::
Eric Kandel's work on Memory: Episode 3 and Episode 12.
Neuroplasticity: Episode 10 and Episode 26.
Interview of Dr. Edward Taub: Episode 28.
Dalai Llama, Dan Rather, Dan Rather Reports, Edward Taub, Eric Kandel, HdNet, Michael Merzenich, Richard Davidson, Sharon Begley, Train your mind change your brain, brain science podcast, meditation, meditation and the brain, mind science, neuroplasticity
More on BDNF: "Miracle Grow" for the Brain
In Episode 33 of the Brain Science Podcast, Harvard's Dr. John Ratey introduced us to brain-derived neurotropic factor (BDNF), which he described as "Miracle Grow for the Brain" because it actually stimulates the grow of new neurons in the brain. The emphasis in our discussion was on the importance of exercise in stimulating the release of BDNF.
If you are interested in checking out some further references on BDNF, you may want to check out Charles Daney's Science and Reason Blog. Daney also does a good job of explaining exactly what a neurotropic factor is and does.
April 07, 2008 / Ginger Campbell, MD/ 2 Comments
Brain Chemistry, Exercise and Play, Neuroscience, Neurotransmitters, Other Websites, Topics
BDNF, Charles Daney, Ginger Campbell, John Ratey, MD, Science and Reason, brain science, brain science podcast, brain-derived neurotropic factor, exercise and the brain
Treating Vets with Mirrors
Jamie Davis of MedicCast sent me a link to an interesting article from the CNN website. It describes how mirror box therapy is being used to help veterans who have suffered amputations in Iraq and Afghanistan. Those of you who heard Sandra Blakeslee's interview about body maps back in Episode 23 will remember that she described how her son built the first mirror box for VS Ramachandran several years ago, while working as a graduate student.
It is good to see that military physicians are beginning to apply some of the recent findings of neuroscience to helping injured vets, but you may also recall that when we talked with Dr. Edward Taub about stroke rehab, he reported the difficulty of getting new methods of head injury treatment into the VA clinics.
link to article about mirrors: http://www.cnn.com/2008/HEALTH/03/19/mirror.therapy/index.html
Brain Research, Mirror Neurons, Neuroscience, Other Websites, Stroke Rehabilitation
Edward Taub, Jamie Davis, Mediccast, VS Ramachandran, brain science podcast, mirror box therapy, pain, treating vets with amputations
Rachel Herz Talks About Smell (BSP 34)
Rachel Herz, PhD
Episode 34 of the Brain Science Podcast is an interview with Rachel Herz, author of The Scent of Desire: Discovering Our Enigmatic Sense of Smell (2007). Dr. Herz teaches at Brown University, and she is a leading authority on the psychology of smell. We talk about the how smell works, its role in emotion and memory, why it is so vulnerable, and why smell is much more important than most of us realize. We also consider some of the questions that remain unanswered.
Rachel Herz:
Brown Medical School faculty page.
Wikipedia: a good source for further references.
RachelHerz.com.
thescentofdesire.com.
The 2004 Nobel Prize for Medicine and Physiology was awarded to Richard Axel and Linda B. Buck for their discoveries of "odorant receptors and the organization of the olfactory system."
BuckL, Axel R.: "A novel multigene family may encode odorant receptors: a molecular basis for odor recognition."
Cell. 1991 Apr 5;65(1):175-87.
"The (Shocked) Nose Knows" by Gisela Telis Science NOW Daily News 27 March 2008.
The Scent of Desire: Discovering Our Enigmatic Sense of Smell (2007), by Rachel Herz
Books, Interviews, Neuroscience, Podcast Show Notes, Psychology, smell
Ginger Campbell, MD, Neuroscience, Rachel Herz, The Scent of Desire: Discovering our enigmatic sense of, brain science podcast, olfaction, psychology of smell, smell
More on Brain Oscillation (from the Human Anti-Gravity Suit)
Diane Jacobs has put a great deal of effort into digesting Rhythms of the Brain, by György Buzsáki. She has just finished an extensive discussion of Buzsáki's interview in Episode 31 of the Brain Science Podcast. You can find this series on her blog, which she calls the Human Anti-Gravity Suit. She has also done a transcript of the episode that we are going to post as soon as we get some final corrections from Dr. Buzsáki.
Other Websites, Rhythms and Oscillations
Gyorgy Buzsaki, Human Anti-Gravity Suit, Rhythms of the Brain by György Buzsáki, brain oscillations, brain science podcast, rhythms of the brain
Exercise and the Brain (BSP 33)
John J. Ratey, M.D.
Episode 33 of the Brain Science Podcast is an interview with Harvard physician, Dr. John Ratey, about his new book, Spark: The Revolutionary New Science of Exercise and the Brain.
We explore the exciting evidence about how exercise helps the brain. It stimulates the release of a number of different neurotransmitters and neuropeptides, but probably more importantly, it helps keeps these compounds balanced. We consider why exercise is so important in dealing with stress, in treating a wide range of mental illnesses including depression, anxiety, and attention deficit disorder. There is also evidence that exercise improves our ability to learn and our ability to avoid the loss of mental agility associated with aging.
This episode contains information that everyone can use. I hope you will share it with your friends and family.
Spark: The Revolutionary New Science of Exercise and the Brain (2008), by John J. Ratey.
Dr. Ratey's website: Go here for more interview of Dr. Ratey and also to find links to the latest research about exercise and the brain.
March 20, 2008 / Ginger Campbell, MD/ 17 Comments
Attention Deficit Disorder (ADD), Books, Brain Chemistry, Brain Research, Exercise and Play, Interviews, Neuroscience, Podcast Show Notes
John J Ratey, John Ratey, MD, Spark: The Revolutionary New Science of Exercise and th, brain fitness, brain science, brain science podcast, exercise and the brain
Enhanced (AAC) Version of Episode 32: Now available in My Libsyn Feed
With the help of Rob Walch at Wizzard™ Media, the enhanced version of Episode 32 (Brain Anatomy) is now available in the regular feed. If you don't use iTunes™ or Quicktime™ you will probably want to stick with the illustrations posted on this website. Subscribe via iTunes™ Subscribe in a reader or podcatcher Subscribe to Brain Science Podcast with Dr. Ginger Campbell by Email
Announcements, Brain Anatomy, Neuroscience, Podcast Show Notes
Brain Anatomy, brain science podcast
Here Is a Working Link to the Enhanced Version of Episode 32
I want to apologize to anyone who got the "enhanced version" of episode 32 in their feed, but found that it had no graphics. The Brain Science Podcast is currently part of an ad campaign on Wizzard Media (Libsyn). They failed to tell us that the technology they use to insert ads will not work with enhanced files, so they just converted it back to a stripped down mp3.
Since I can't put the episode into the normal feed, but you can get it here. It should play in Quicktime™ on your computer. To save the file on to your computer, right click on the link and choose "save as." All iPods can handle this format, but many other players, such as the Zune™ will not. Sorry about the inconvenience. (Note: If you are using Windows make sure you have the latest version of Quicktime™.)
If you just want to look at the illustrations on your computer click here.
Announcements, Podcast Show Notes
Beyond the Zonules of Zinn, Brain Anatomy, David Bainbridge, Ginger Campbell, Podcast, anatomy, brain, brain science podcast, neuroanatomy, science
Brain Anatomy: Illustrations for BSP 32
I have posted the illustrations I promised last week when I released Episode 32 of the Brain Science Podcast. Except for the colored diagram of the lobes of the brain, these illustrations come from Beyond the Zonules of Zinn: A Fantastic Journey Through Your Brain, and have been used with the permission of the author, David Bainbridge.
Click here to see thumbnails of all the illustrations, along with the approximate time they were discussed during the episode. (Note: because of the automatic ad insertion process the times may be off by up to 67 seconds.) http://brainsciencpodcast.wordpress.com/episodes/figures-for-episode-32-a-whirlwind-tour-of-brain-anatomy/
Here is a link to the enhanced version of the podcast, which includes the illustrations but not the captions. Unfortunately, due to some technical issues with Libsyn's ad-server software, I am currently unable to put the enhanced version into the regular feed.
Announcements, Books, Brain Anatomy, Podcast Show Notes

redpajama_set_name: RedPajamaCommonCrawl


text:
Filip IV, Filip den sköne, (fr. Philippe IV, Philippe le Bel), född 1268 i Fontainebleau, död 29 november 1314 i Fontainebleau, kung av Frankrike, 1285-1314. Son till Filip III av Frankrike och Isabella av Aragonien, far till Ludvig X av Frankrike, Filip V av Frankrike och Karl IV av Frankrike. Ska inte förväxlas med Filip I av Kastilien, som också kallades "Filip den sköne".

Biografi

Politik
Filip IV var allierad med skottarna och åren 1293-1303 förde han krig mot England som upphörde med Parisfreden. Därvid stötte han på motstånd främst i Akvitanien, som tillhörde den engelske kungen. Vid denna tid höll den långa europeiska högkonjunkturen på att ebba ut. Kronans inkomster hotades. De ständiga krigen var också kostsamma. Filip beskattade kyrkan hårt, vilket ledde till en fejd med påven Bonifatius VIII. Bonifatius förbjöd därför sina präster att betala skatt. Efter Bonifatius död år 1303, säkrade Filip kontrollen över kyrkan genom att flytta den heliga stolen till Avignon, (Babyloniska fångenskapen).

Genom sin offensiva politik och sitt bruk av en romersk-rättslig statslära, framstår Filip IV som en av de tidigare representanterna för den starka kungamakt som under senmedeltiden växte fram runt om i Europa.

Förföljelser
För att visa sin lojalitet mot kyrkan och berika Frankrike som var fattigt på grund av krig inledde Filip IV förföljelser av judarna och tempelherreorden. 1306 lät han kasta ut alla judar ur Frankrike och beslagta deras egendom. 1307 lät han gripa alla riddare av den rika och självständiga tempelherreorden, bränna deras ledare på bål och konfiskera deras förmögenheter.

Familj
Gift 1284 med Johanna I av Navarra (1273-1305), dotter till Henrik I av Navarra och Brie, även kallad Henrik den tjocke.

Barn
 Margareta av Frankrike (1288-1312)
 Ludvig X av Frankrike, (1289-1316), gift med (i tur och ordning) Margareta av Burgund (1290-1315) och Clemence d'Anjou (1293-1328)
 Blanche (1290-1294)
 Isabella av Frankrike (1292-1358), gift med Edvard II av England (1284-1327)
 Filip V av Frankrike (1294-1322), gift med Johanna II av Burgund (1291-1330)
 Karl IV av Frankrike (1295-1328), gift med (i tur och ordning) Blanche av Burgund (1296-1326), Maria av Luxemburg, (1304-1324) och Johanna av Évreux (1310-1371)
 Robert av Frankrike (1297-1308)

Galleri

Anfäder

Källor

Personer i Frankrike under 1200-talet
Personer i Frankrike under 1300-talet
Frankrikes regenter
Huset Capet
Avlidna 1314
Män
Födda 1268

redpajama_set_name: RedPajamaWikipedia


text:
Busy Week-End In Prospect
With the weather forecast predicting improved conditions, the Grounds Sub-Committee are optimistic that Hamilton Park will be in good condition for the week-end's fixtures. The Connacht Cup U-14 semi-final between Sligo and Westport will take place in Hamilton Park on this Saturday at 3pm 12pm.
This is certain to be a very competitive game. Sligo won the U-13 Cup last season, and following an unbeaten U-14 league run this season in the northern section, they slipped up in the league semi-final against Buccaneers.
However, their cup defeat of Galwegians was very impressive, and they will be very determined to qualify for the final.
At the time of going to press, the arrangements for the refixture of the U-13 Connacht Cup semi-final between Sligo and Buccaneers had not been decided on by the Connacht Branch. Sligo are hopeful that this game will also be in Hamilton Park, but the liklihood is that it will be at a neutral venue. It is certain that the game will go ahead at the week-end, and parents and supporters should check the club's website for details of this fixture.
At Hamilton Park on Sunday next [Kick-Off 3 p.m.], Sligo's 2nd team play Tuam in the first round of the Ard na Cregg Cup. Sligo will field a strong side, and on home ground will be more than hopeful of victory. Having used a big squad of players in the league, the team management of Michael Storey, Des Butler, Declan Lynch and Finian O'Driscoll, will select from quite a large panel, with a good balance of youth and experience. As many supporters as possible are urged to get behind the team on Sunday. The Sligo 'man of the match' will be the recipient of an award after the game. The award is very generously donated by the sponsors of senior rugby in the club, Martin Reilly Motors.
← Week-End Of Inactivity As All Games Called Off Former Sligo Underage Players In Big Time →


Log in to your Accounts Greg Weis and Marty Weis are the investment professionals at Plan 1 Financial. Based on what is right for you, they will suggest a single investment or a total plan from the complete range of investment options available.
Come in and take advantage of our knowledge and experience. We will be happy to visit with you in complete confidence.
Plan 1 Financial is located at 403 West Water Street in downtown Decorah, Iowa. Office hours are Mondays through Fridays, 8:30 a.m. to 5 p.m.; and evenings and Saturdays by appointment.
Plan 1 offers a complete range of investment options, including stocks, bonds, mutual funds, annuities, individual retirement accounts, and much more.
Full service and nationally-competitive discount stock trading is available.
At Plan 1 Financial, Greg Weis and Marty Weis offer the knowledge and experience to assist you with your retirement goals.
Call 563-382-8258 to schedule a free investment consultation today.
Interested in exploring a new path for your investment future?
Greg Weis and Marty Weis can help you work toward your financial goals. At Plan 1, you can work with people you know ... people who know you!
Call 563-382-8258 to learn more.
To be well balanced these days, a typical investor needs more ways to grow and protect themselves. To learn more about the Plan 1 Financial balanced approach for investing, see Greg Weis and Marty Weis at Plan 1 Financial in downtown Decorah. They are here to help guide you in planning and protecting your estate. Call 382-8258 or stop by the Plan 1 Investment Center at 403 West Water Street in downtown Decorah to learn more today.
Do you have American Funds accounts in your investment portfolio?
Gregory T. Weis and Martin T. Weis are Registered Representatives of and securities are offered through LaSalle St. Securities LLC (LSS), Member FINRA/SIPC. Plan 1 Financial & Heartland Investment Associates Inc are not affiliated with LSS.
Securities in accounts carried by National Financial Services LLC ("NFS"), a Fidelity Investments company, are protected in accordance with the Securities Investor Protection Corporation ("SIPC") up to $500,000. The $500,000 total amount of SIPC protection is inclusive of up to $250,000 protection for claims for cash, and is subject to periodic adjustments for inflation in accordance with terms of the SIPC statute and approval by SIPC's Board of Directors. NFS also has arranged for coverage above these limits. Neither coverage protects against a decline in the market value of securities, nor does either coverage extend to certain securities that are considered ineligible for coverage. For more details on SIPC or to request an SIPC brochure, visit www.sipc.org or call 202.371.8300.
The "excess of SIPC" coverage would only be used when SIPC coverage is exhausted. Like SIPC protection, "excess of SIPC" protection does not cover investment losses in customer accounts due to market fluctuation. It also does not cover other claims for losses incurred while broker-dealers remain in business. Total aggregate "excess of SIPC" coverage available through NFS's "excess of SIPC" policy is $1 billion. Within NFS's "excess of SIPC" coverage, there is no per customer dollar limit on coverage of securities, but there is a per customer limit of $1.9 million on coverage of cash. This is the maximum "excess of SIPC" protection currently available in the brokerage industry.
1 Fidelity's "excess of SIPC" insurance is provided by Lloyd's of London together with Axis Specialty Europe SE, Markel International Insurance Company, XL Specialty Insurance Company, and Munich Reinsurance Co.


Thailand Bitumen Import January to December 2018.
Japan Bitumen Export January to December 2018.
Japan Bitumen Import January to December 2018.
India Bitumen Import January to December 2018.
Singapore Bitumen Export January to September 2018.
Thailand Bitumen Export January to September 2018.
Thailand Bitumen Import January to September 2018.
Japan Bitumen Export January to September 2018.
Japan Bitumen Import January to September 2018.
India Bitumen Import January to September 2018.
Singapore Bitumen Export January to June 2018.
Thailand Bitumen Export January to June 2018.
Thailand Bitumen Import January to June 2018.
Japan Bitumen Export January to June 2018.
Japan Bitumen Import January to June 2018.
India Bitumen Import January to June 2018.
Thailand Bitumen Export January to March 2018.
Thailand Bitumen Import January to March 2018.
Japan Bitumen Export January to March 2018.
Japan Bitumen Import January to March 2018.
Singapore Bitumen Export January to March 2018.
India Bitumen Import January to March 2018.
China Bitumen Import January to March 2018.


Forums » arachnode.net » General Questions » how to crawl a site with specific path starting from http://xyz.com/en/?
how to crawl a site with specific path starting from http://xyz.com/en/?
Is there any way to start crawling a site that starts with such path?
Do you want to crawl the de-DE directory, or no?


Manoomin, wild rice in Ojibwe, is a traditional staple in the Anishinaabe diet. Growing on the shores of lakes in Manitoba and Ontario, it is an important food for both cultural identity, and health. Manoomin processing has been experienced at many of our programs, including the recent 2016 Youth Gathering.
Harvesting manoomin is a small feat in itself but the hard work starts once you are back on dry land.
The first step in processing manoomin is to spread the manoomin on tarps outside to dry it. This also lets all the spiders that were harvested with the manoomin to escape.
The next step is to parch the manoomin which is done over a fire using poplar wood which adds to the flavouring of the rice. You will need a very large pot and a paddle or long stick to stir it. Continuous stirring of the rice is critical to prevent the manoomin from burning. The first sign that the rice is done is the smell of toasted manoomin; which is followed by a crackling of the manoomin. Take the pot of immediately and continue to stir the manoomin so it doesn't burn.
Once the manoomin has cooled off, place it in a shallow hole and dance on. The dancing movement separates the manoomin from the husk, preparing it to be winnowed. Winnowing is done by placing some of the danced on manoomin in trays and gently tossing it. A winder day is a benefit as the wind will blow away the empty husks.
The next step is to sort through the winnowed manoomin and separate the ready to be cooked manoomin and the manoomin that needs to be processed more.
Final step: cook it and eat it!