text:
\section{Introduction}

The study of gauge-theory scattering amplitudes at the multi-loop level is central to precision collider physics, as well as to the fundamental understanding of these theories.
A salient property of gauge-theory scattering amplitudes is the presence of infrared singularities. Understanding the origin and structure of these singularities is essential for cross-section calculations, where one of the biggest challenges is the intricate cancellation of the singularities between virtual and real corrections. For sufficiently inclusive quantities, this knowledge can be translated into all-order resummation of various towers of logarithms.
From a more theoretical perspective, understanding the singularity structure of amplitudes is interesting since it provides deep insight into all-order properties of amplitudes and their iterative structure.
Indeed, one of the main features of infrared singularities is that they exponentiate, and can thus be encoded into an anomalous dimension. The latter can be computed from the ultraviolet singularities of correlators of semi-infinite Wilson line
operators~\cite{Polyakov:1980ca,Arefeva:1980zd,Dotsenko:1979wb,Brandt:1981kf,Korchemsky:1985xj,Korchemsky:1985xu,Korchemsky:1987wg}.
Infrared singularities are therefore simpler than finite corrections, and they feature a transparent iterative structure which is similar across different gauge theories.

Further insight into the all-order structure of scattering amplitudes may be gained by considering special kinematic limits, where new factorization properties arise.
The simplification owing to special kinematic limits may render both infrared-divergent and finite terms computable to high loop orders, and potentially even to all orders.
For example, the high-energy (Regge) limit of two-parton scattering (forward scattering with $s\gg -t$) lends itself to an effective description in terms infinite Wilson lines~\cite{Korchemskaya:1996je,Korchemskaya:1994qp}, or alternatively in terms of Reggeized gluons propagating in transverse space, i.e. in $2-2\epsilon$ dimensions. The wavefunction describing Reggeised gluons admits a rapidity evolution equation, the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation and its generalizations~\cite{Kuraev:1977fs,Balitsky:1978ic,Lipatov:1985uk,Mueller:1993rr,Mueller:1994jq,Balitsky:1995ub,Balitsky:1998kc,Kovchegov:1999yj,JalilianMarian:1996xn,JalilianMarian:1997gr,Iancu:2001ad},  making it possible to sum up high-energy logarithms to all orders~\cite{Caron-Huot:2013fea,Caron-Huot:2017fxr,Caron-Huot:2017zfo}.
Another example where major simplifications occur are collinear limits of multileg amplitudes. Specifically, one may identify a relation between an $n$-parton amplitude in which two partons are collinear and an $(n-1)$-parton amplitude, where the collinear divergences in the former are factorized into a \emph{splitting amplitude}. The universality of the splitting amplitude translates into constraints on both the finite and infrared-divergent parts of amplitudes.

Infrared singularities have been studied in QCD for many decades for processes
involving two partons and any number of colour singlet
particles~\cite{Mueller:1979ih,Collins:1980ih,Sen:1981sd,Sen:1982bt,Gatheral:1983cz,Frenkel:1984pz,Sterman:1981jc,Magnea:1990zb}. The
singularity structure of amplitudes involving several partons has been
examined more recently, in both the
massless~\cite{Korchemsky:1993hr,Korchemskaya:1996je,Korchemskaya:1994qp,Catani:1996vz,Catani:1998bh,Sterman:2002qn,Dixon:2008gr,Kidonakis:1998nf,Bonciani:2003nt,Dokshitzer:2005ig,Aybat:2006mz,Gardi:2009qi,Becher:2009cu,Becher:2009qa,Gardi:2009zv,
  Dixon:2009gx,Dixon:2009ur,Bret:2011xm,DelDuca:2011ae,Caron-Huot:2013fea,Ahrens:2012qz,Naculich:2013xa,Erdogan:2014gha,Gehrmann:2010ue}
and
massive~\cite{Kidonakis:2009ev,Mitov:2009sv,Becher:2009kw,Beneke:2009rj,Czakon:2009zw,Ferroglia:2009ii,Chiu:2009mg,Mitov:2010rp,Mitov:2010xw,Gardi:2013saa,Falcioni:2014pka,Henn:2013fah,Dukes:2013gea,Dukes:2013wa,Gardi:2010rn,Gardi:2011wa,Gardi:2011yz,Gardi:2013ita,Laenen:2008gt,Vladimirov:2015fea}
cases.
In ref.~\cite{Almelid:2015jia} the multileg soft anomalous dimension for massless particles has been determined at three-loop order. This diagrammatic
calculation was reviewed  in ref.~\cite{Gardi:2016ttq}.
The present talk concerns primarily the properties of three-loop soft anomalous dimension following ref.~\cite{Almelid:2017qju}.
We begin by briefly reviewing the factorization and exponentiation properties of soft singularities, followed by a description of the general structure of the soft anomalous dimension.  Next, rather than analysing the three-loop result directly, we will present
general considerations allowing one to identify the relevant kinematic variables and the class of iterated integrals in terms of which the result can be written. Subsequently, we will examine the two kinematic limits mentioned above, where information regarding soft singularities can be deduced using other methods. Considering the high-energy limit we will summarise recent conclusions  regarding soft singularities based on rapidity evolution equations according to~Refs.~\cite{Caron-Huot:2017fxr,Caron-Huot:2017zfo}.
Finally, we will discuss the bootstrap procedure of ref.~\cite{Almelid:2017qju}, which  showed that, quite remarkably, it is possible to start with a completely general
ansatz of basis functions, and recover the three-loop result of~\cite{Almelid:2015jia}, up to an overall normalization factor, by imposing symmetry considerations along with properties of the singularities in the collinear and high-energy limits.

\section{Factorization and exponentiation of soft singularities}

The most important fact about infrared singularities is that they can be factorized:
the singularities emerge from soft or collinear modes, which do not resolve many details of the process associated with highly virtual particles, and therefore factorize.
Our interest in this talk is in scattering of \emph{massless} quarks and gluons, which means that there are collinear singularities, in addition to soft ones.
In \emph{fixed-angle factorization} we consider all the hard scales in the problem, $s_{ij}$, to be much larger than the QCD scale $\Lambda^2$:
\[
s_{ij}=2p_i\cdot p_j=2\beta_i\cdot \beta_j \,Q^2\gg \Lambda^2\,,
\]
where the 4-velocity vector $\beta_i$ is proportional to the momentum $p_i$, having extracted an overall hard scale factor $Q$.
Infrared singularities emerge from soft and collinear regions, which are captured by separate soft and jet functions. The amplitude for $n$ parton scattering in $d=4-2\epsilon$
spacetime dimensions  assumes the factorised
form~\cite{Sen:1982bt,Kidonakis:1998nf,Sterman:2002qn,Dixon:2008gr,Aybat:2006mz,Gardi:2009qi,Gardi:2009zv}
\begin{equation}
{\cal A}_n(\{p_i\},\epsilon,\alpha_s)=
{\cal S}(\{\beta_i\},\{{\bf T}_i \},\epsilon,\alpha_s)\,
{\cal H}_n(\{p_i\},\{n_i\},\epsilon,
\alpha_s)\,
\prod_{i=1}^n \frac{J(p_i,n_i,\epsilon,\alpha_s)}
{{\cal J}(\beta_i,n_i,\epsilon,\alpha_s)},
\label{softcolfac}
\end{equation}
where dependence on colour degrees of freedom is shown in terms of colour operators ${\bf T}_i$ for each particle $i$, $\alpha_s$ is the $d$-dimensional running coupling, ${\cal H}_n$ is a process-dependent {\it hard function} that is finite as
$\epsilon\rightarrow 0$, and ${\cal S}(\{\beta_i\},\{{\bf T}_i \},\epsilon,\alpha_s)$ and $J(p_i,n_i,\epsilon,\alpha_s)$ are the {\it soft} and {\it
  jet functions} that collect infrared singularities originating from
emissions that are soft and collinear to particle $i$
respectively. The function ${\cal J}(\beta_i,n_i,\epsilon,\alpha_s)$ in (\ref{softcolfac}) removes the double counting of soft-collinear divergences accounted for in both ${\cal S}$ and $J$.

For our discussion the soft function ${\cal S}$ will be most important, since only this function involves singularities depending on the colour and kinematic degrees of freedom of several partons, and as such becomes significantly more complicated in multileg scattering beyond the planar limit as compared to two-parton scattering, or indeed the planar limit in multileg scattering.
The soft function can be defined and computed using a correlator of products of semi-infinite Wilson lines, emanating from the point of the hard interaction:
\begin{equation}
{\cal S}(\{\beta_i\},\{{\bf T}_i\} , \epsilon,\alpha_s)=\left\langle 0\left|
{\rm T}\left[\Phi_{\beta_1}\,\Phi_{\beta_2}\ldots\Phi_{\beta_n}\right]
\right|0\right\rangle,
\label{Sdef}
\end{equation}
where ${\rm T}[\cdots]$ represents a time-ordered product and
\begin{equation}
\Phi_{\beta_i}={\cal P}\exp\left[i\mu^\epsilon g_s {\bf
    T}_i^a\int_0^\infty ds\,\beta_i\cdot A^a(s\beta_i)\right]
\label{Phidef}
\end{equation}
is a Wilson-line operator along a trajectory of the $i^{\rm th}$
parton.
The soft function therefore depends on the angles between the lines. It also has a non-trivial colour structure which acts as a matrix in colour space on the hard interaction.  The jets, in turn, are  colour singlet and hence simple: collinear singularities do not depend on the colour flow of the other partons and they depend on the momentum of a single parton.

The next essential property of infrared singularities is that they exponentiate. This can be shown most easily by establishing that the soft function (and similarly the jet function) admits a renormalization-group equation, whose solution is a path-ordered exponential. Upon combining the singular factors in eq.~(\ref{softcolfac}), the factorized amplitude may be expressed as~\cite{Becher:2009qa,Gardi:2009zv,Gardi:2016ttq}
\begin{equation}
{\cal
  A}_n\left(\{p_i\},\epsilon,\alpha_s(\mu^2)
\right)=
Z_n\left(\{p_i\},\{{\bf T}_i\} ,\epsilon,\alpha_s(\mu_f^2)
\right)
{\cal H}_n\left(\{p_i\},\frac{\mu_f}{\mu}
,\epsilon,\alpha_s(\mu^2)\right)\,,
\label{Andef}
\end{equation}
where the factor $Z_n$, which captures all the singularities, is a matrix in colour-flow space, and can be written as
\begin{equation}
Z_n\left(\{p_i\},\{{\bf T}_i\} ,\epsilon,\alpha_s(\mu_f^2)
\right)={\cal
  P}\exp\left\{-\frac12\int_0^{\mu_f^2}\frac{d\lambda^2}{\lambda^2}
\Gamma_n\left(\{p_i\},\{{\bf T}_i \},\lambda,
\alpha_s(\lambda^2)\right)\right\},
\label{ZnRG}
\end{equation}
where $\mu_f^2$ is a factorization scale, ${\cal P}$ denotes path-ordering and $\Gamma_n$ is the so-called \emph{soft anomalous dimension}. The latter encodes all infrared singularities, order by order in the perturbative expansion. It is itself a finite quantity, and the singularities in eq.~(\ref{ZnRG}) are generated by integrating over the $d$-dimensional coupling from zero momentum. In the rest of the talk I will focus on the kinematics and colour dependence of the soft anomalous dimension $\Gamma_n$.

\section{The structure of the soft anomalous dimension}

At one loop a single gluon may be exchanged between any of the Wilson lines in (\ref{Sdef}), leading to a very simple structure in  $\Gamma_n$: it is just a sum over colour dipoles, ${\bf T}_i\cdot {\bf T}_j$ in colour space, multiplied by the logarithm of the corresponding kinematic invariant $s_{ij}$. Remarkably, it turns out that this dipole sum formula generalises to two-loops. This was first observed in an explicit computation~\cite{Aybat:2006mz} demonstrating
that potential three-particle correlations were absent at this
order; subsequently it was explained~\cite{Gardi:2009qi,Becher:2009cu,Becher:2009qa} to be a direct consequence of the factorisation formula,
eq.~(\ref{softcolfac}), together with invariance under rescaling of
the four-velocities $\{\beta_i\}$.
Invariance of the soft function alone
is broken for lightlike Wilson lines due to the appearance of
collinear singularities. It is restored however,
upon dividing by the eikonal jets, thus linking the breakdown of scale
invariance in the soft function to the jet function, and hence to the
cusp anomalous dimension (see ref.~\cite{Gardi:2009qi}). These
considerations lead to a differential equation for the soft anomalous
dimension, whose \emph{minimal solution} up to three-loop order is the
so-called {\it dipole
  formula}~\cite{Gardi:2009qi,Becher:2009cu,Becher:2009qa},
\begin{equation}
\Gamma_n^{\rm dip.}\left(\{p_i\},\{ {\bf T}_i\},\mu,\alpha_s\right)
=-\frac12\widehat{\gamma}_K(\alpha_s)\sum_{i<j}\log\left(\frac{-s_{ij}-i0}
{\mu^2}\right){\bf T}_i\cdot {\bf T}_j
+\sum_{i=1}^n\gamma_{J_i}(\alpha_s).
\label{gamdip}
\end{equation}
Here $\widehat{\gamma}_K$ is the cusp anomalous dimension~\cite{Korchemsky:1985xj,Korchemsky:1987wg,Moch:2004pa}, with the quadratic Casimir of the representation of the Wilson lines scaled out (see discussion below regarding the violation of Casimir scaling beyond three loops);
$\gamma_{J_i}$ is an anomalous dimension associated with hard collinear
singularities, and is also known to three-loop order for both quark
and gluon jets~\cite{Moch:2005tm,Gehrmann:2010ue}.

Refs.~\cite{Gardi:2009qi,Becher:2009cu,Becher:2009qa}  identified two classes of potential corrections to the soft anomalous dimension going beyond the dipole formula of eq.~(\ref{gamdip}).
The first is related to the fact that
eq.~(\ref{gamdip}) contains the cusp anomalous dimension with all
colour dependence scaled out, thus assuming that {\it Casimir scaling}
holds to all orders. In fact, this breaks for the first time at
four-loop order due to the appearance of new colour structures, quartic Casimirs, as has very recently been shown explicitly in
refs.~\cite{Boels:2017skl,Boels:2017ftb,Moch:2017uml,Grozin:2017css}. This implies that the form of
eq.~(\ref{gamdip}) will have to be modified beyond three loops~\cite{Gardi:2009qi,Becher:2009cu,Becher:2009qa}. The
second source of corrections starts already at three loops, and
constitutes a homogeneous solution to the differential equation for
the soft anomalous dimension derived in ref.~\cite{Gardi:2009qi}. This
implies dependence on kinematics only through conformally invariant
cross ratios
\begin{equation}
\rho_{ijkl}\equiv\frac{(-s_{ij})(-s_{kl})}{(-s_{ik})(-s_{jl})}
=\frac{(\beta_i\cdot \beta_j)\,(\beta_k\cdot \beta_l)}
{(\beta_i\cdot \beta_k)\,(\beta_j\cdot\beta_l)},
\label{rhodef}
\end{equation}
such that the complete soft anomalous
dimension through three-loop order assumes the form
\begin{equation}
\Gamma_n(\{p_i\},\{ {\bf T}_i\},\mu,\alpha_s)=
\Gamma_n^{\rm dip.}(\{p_i\},\{ {\bf T}_i\},
\mu,\alpha_s)+\Delta_n(\{\rho_{ijkl}\},\{ {\bf T}_i\},\alpha_s)\,,
\label{gamnfull}
\end{equation}
where the correction
\begin{equation}
\Delta_n(\{\rho_{ijkl}\},\{ {\bf T}_i\},\alpha_s) =\left(\frac{\alpha_s}{4\pi}\right)^3 \Delta_n^{(3)}(\{\rho_{ijkl}\},\{ {\bf T}_i\})+{\cal O}(\alpha_s^4)
\end{equation}
begins at three-loop order. Whether or not it is nonzero at this order
remained conjectural for a number of
years~\cite{Dixon:2009ur,Bret:2011xm,DelDuca:2011ae,Ahrens:2012qz,Naculich:2013xa,Caron-Huot:2013fea}.
Recently, however, it was calculated in
ref.~\cite{Almelid:2015jia}. Before we quote its form, let us note that for
any given four particles $\{i,j,k,l\}$, there are potentially 24 cross
ratios. However, eq.~(\ref{rhodef}) implies
\begin{equation}
\rho_{ijkl}=\rho_{jilk}=\rho_{klij}=\rho_{lkji},
\label{rhotrans1}
\end{equation}
which reduces the number of cross ratios to 6. In fact, further relations such as
\begin{equation}
\rho_{ijkl}=\frac{1}{\rho_{ikjl}},\quad
  \rho_{ijlk}\rho_{ilkj}=\rho_{ijkl}
\label{rhotrans2}
\end{equation}
can be used to write all the cross ratios in terms of just 2
independent cross ratios, which can be taken to be $\{\rho_{ijkl},
\rho_{ilkj}\}$. The explicit form of the three-loop
correction to the dipole formula can then be written
\begin{align}
\label{eq:expected_Delta}
& \Delta_n^{(3)}\left(\left\{\rho_{ijkl}\right\},\{{\bf T}_i\}\right) = 16\,f_{abe}f_{cde}
\Big\{
-C\, \sum_{i=1}^n \ \sum_{\substack{{1\leq j<k\leq n}\\ j,k\neq i}}\left\{{\rm \bf T}_i^a,  {\rm \bf T}_i^d\right\}   {\rm \bf T}_j^b {\rm \bf T}_k^c  \\
&+\!\!\!{\sum_{1\leq i<j<k<l\leq n}}
\Big[
 {\rm \bf T}_i^a  {\rm \bf T}_j^b   {\rm \bf T}_k^c {\rm \bf T}_l^d   \, {\cal F}(\rho_{ikjl},\rho_{iljk})
+{\rm \bf T}_i^a  {\rm \bf T}_k^b {\rm \bf T}_j^c   {\rm \bf T}_l^d
\, {\cal F}(\rho_{ijkl},\rho_{ilkj})
+ {\rm \bf T}_i^a   {\rm \bf T}_l^b  {\rm \bf T}_j^c    {\rm \bf T}_k^d
\, {\cal F}(\rho_{ijlk},\rho_{iklj}) \Big]        \nonumber
\Big\}\,,
\end{align}
where
\begin{equation}
C=\zeta_5+2\zeta_2\zeta_3,
\label{Cdef}
\end{equation}
and the explicit form of the function ${\cal F}$ can be most elegantly written by
introducing variables $\{z_{ijkl},\bar{z}_{ijkl}\}$ satisfying
\begin{equation}
z_{ijkl}\bar{z}_{ijkl}=\rho_{ijkl},\quad
(1-z_{ijkl})(1-\bar{z}_{ijkl})=\rho_{ilkj}.
\label{zdef}
\end{equation}
With these definitions, one has
\begin{equation}
{\cal F}(\rho_{ijkl},\rho_{ilkj})=F(1-z_{ijkl})
-F(z_{ijkl}),
\label{calFdef}
\end{equation}
where
\begin{equation}
F(z)={\cal L}_{10101}(z)+2\zeta_2[{\cal L}_{001}(z)
+{\cal L}_{100}(z)].
\label{Fdef}
\end{equation}
The function ${\cal L}_w(z)$ (where $w$ is a word
composed of zeroes and ones) is a single-valued harmonic polylogarithm (SVHPL)~\cite{Brown:2004}.

\section{Single-valued harmonic polylogarithms and the Riemann sphere}
\label{sec:SVHPL}

The single-valued harmonic polylogarithms (SVHPLs)~\cite{Brown:2004,Remiddi:1999ew,Dixon:2012yy}  used to express the result in eq.~(\ref{Fdef}) are special combinations of harmonic polylogarithms~\cite{Remiddi:1999ew}
that are free of discontinuities, and thus single-valued, in the
kinematic region where $\bar{z}$ is equal to the complex conjugate of
$z$. This region is a subset of the so-called Euclidean region, where
all Mandelstam invariants are spacelike with $s_{ij}<0$. Unitarity of
massless scattering amplitudes dictates that they can only have
singularities due to the vanishing of Mandelstam invariants, which do
not occur in the Euclidean region of fixed angle scattering. Thus,
single-valuedness of the function $F$ reflects directly the analytic
structure of the underlying amplitude.

It is remarkable that such a fundamental physics principle dictates the specific class of functions in terms of which the result for the soft anomalous dimension is expressed. This also provides the starting point for the bootstrap approach~\cite{Almelid:2017qju}, namely determining the result by
constraining an ansatz, as an alternative to computing it directly. Ref.~\cite{Almelid:2017qju} started by explaining the geometrical origin of the kinematic dependence through the variables in eq.~(\ref{zdef}) and then how the class of weight-five SVHPLs of these variables can be identified as the relevant functions at three loops. Let us briefly review these considerations.

The correlator of Wilson lines in eq.~(\ref{Sdef}) defining the soft function is invariant under rescaling of individual Wilson-line velocity vectors, $\beta_i\to \kappa_i\beta_i$, and one can therefore fix the value of $\beta_i^2$ at will, without affecting the correlator. Upon considering all velocity vectors timelike and future pointing, and choosing a common value for $\beta_i^2$ for all $i$, one concludes that the kinematics is completely determined by a
set of $n$ points in three-dimensional hyperbolic space ${\mathbb H}^3$, namely the locus of all coordinates
\begin{equation}
\beta^\mu:\quad
 (\beta^0)^2-(\beta^1)^2-(\beta^2)^2 -(\beta^3)^2=R^2,\qquad \beta^0>0,
\end{equation}
for some constant $R$, as illustrated in fig.~\ref{fig:Riemann_Sphere} (left).
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.5]{Riemann_Sphere_param.pdf}\qquad\qquad\qquad
\includegraphics[scale=0.4]{Riemann_Sphere_param_sphere.pdf}
\caption{\label{fig:Riemann_Sphere}Left -- the non-lightlike case: an illustration of the hyperbolic space (in one dimension less) with $\beta^0$ as the vertical axis, pierced by the $n$ velocity vectors; Right -- the lightlike case: the kinematics is described by $n$ points on a Riemann sphere, which is the boundary of the aforementioned hyperbolic space.}
\end{center}
\end{figure}
Feynman integrals are known to evaluate to iterated integrals, but the class of iterated integrals depending on $n$ points on ${\mathbb H}^3$ is yet to be fully understood. The situation is simpler however upon considering the lightlike limit relevant to massless scattering.
Parametrizing the velocity vectors by
\begin{equation}
\label{upper_half}
\beta_i=
\left(1 + \frac{r_i^2 + z_i \bar{z}_i}{4} , \,\,
\frac{ z_i + \bar{z}_i}2 , \,\,
\frac{ z_i - \bar{z}_i}{2i},\,\,
 1 - \frac{r_i^2 + z_i \bar{z}_i}{4} \right)
\end{equation}
where $\beta_i^2=r_i^2$ and  $z_i$ and $\bar{z}_i$ are complex conjugates,
the hyperbolic 3-space is represented by  the so-called upper half space model, ${\mathbb H}^3\simeq {\mathbb R}^2\times {\mathbb R}_+$. Now we can consider the lightlike limit, $r_i\to 0$, corresponding to the boundary of the aforementioned hyperbolic space. The velocity vector is then represented by a point on a Riemann sphere parametrized by the complex variable $z_i$. We thus see that while for massive scattering the kinematics is described by a configuration of $n$ points on ${\mathbb H}^3$, in massless scattering it is described by a configuration of $n$ points on a Riemann sphere, illustrated in fig.~\ref{fig:Riemann_Sphere} (right). The class of iterated integrals on the latter space,  ${\cal M}_{0,n}$, is well understood~\cite{Brown:2009qja}, as we now briefly discuss.

Let us first examine the relevant kinematic variables. In the coordinate system of (\ref{upper_half}) the product of two velocity vectors is given by
\begin{equation}
2\beta_i\cdot \beta_j=
|z_i-z_j|^2+r_i^2+r_j^2\,\,\,\xrightarrow{r_i,r_j\rightarrow \,0}\,\,\, |z_i-z_j|^2\,,
\label{betaibetaj}
\end{equation}
so in the lightlike limit, conformally invariant cross ratios (\ref{rhodef}) are given by
\begin{equation}
\rho_{ijkl}=\frac{(\beta_i\cdot \beta_j)(\beta_k\cdot \beta_l)}
{(\beta_i\cdot \beta_k)(\beta_j\cdot \beta_l)}\,\,\,
\xrightarrow{r_i,r_j,r_k,r_l\rightarrow \,0}\,\,\,
|{z}_{ijkl}|^2\,=\, {z}_{ijkl} {\bar{z}}_{ijkl}\,,\quad \text{with}\quad {z}_{ijkl}\equiv\frac{(z_i-z_j)\,(z_k-z_l) }{(z_i-z_k)\,(z_j-z_l)}.
\label{CICRz}
\end{equation}
Furthermore, we can use the symmetries of the sphere, namely an ${\rm
  SL}(2,{\mathbb C})$ invariance, to
fix the positions of three of the $n$ points. Specifically, considering a subset of
four Wilson lines $\{i,j,k,l\}$,  one may choose
\begin{equation}
z_i=z_{ijkl},\quad z_j=0,\quad z_k=\infty,\quad z_l=1,
\label{zchoice}
\end{equation}
so that the only nontrivial position $z_i$ on the Riemann sphere may
be identified with the variable of eqs.~(\ref{zdef}). Specifically, three-loops webs span up to four Wilson lines (see~\cite{Gardi:2013ita,Almelid:2015jia,Gardi:2016ttq}), so it is clear at the outset that the function ${\cal F}$ can only depend on \emph{one} complex variable, $z_{ijkl}$, independently of the number of scattered particles $n$.

Having identified the kinematic variables, we may use the fact that the nature of iterated integrals on the space of configurations of $n$ points on the sphere, ${\cal M}_{0,n}$ is completely known: they can always be expressed as linear combinations of products of multiple polylogarithms, and the coefficients of this linear combination are rational functions~\cite{Brown:2009qja}.
At this point we can use further considerations to restrict the class of functions (see ref.~\cite{Almelid:2017qju} for further details):
\begin{itemize}
\item{} The physical restriction of having strictly no branch points in the Euclidean region, where all $s_{ij}<0$, selects the class of \emph{single valued} polylogarithms.
\item{} Given that three-loops webs span up to four Wilson lines the relevant functions depend on
a single (complex) variable $z_{ijkl}$. This uniquely identifies them as single-valued \emph{harmonic} polylogarithms. This is because singularities appear only when two of the points coincide, e.g. $z_i\to z_j$, corresponding to collinear limits; this dictates the symbol alphabet:
$\left\{z_{ijkl},\bar{z}_{ijkl}, 1-z_{ijkl}, 1-\bar{z}_{ijkl}\right\}$.
\item{} While a priori both the polylogarithms and their rational coefficients depend on the kinematic variables $z_{ijkl}$, the fact the three-loop $\Delta_n^{(3)}$ is the same in all gauge theories, along with the conjectural property that ${\cal N}=4$ amplitudes (in the maximally-helicity-violating sector) are pure functions of uniform (maximal) weight, leads to the expectation that ${\cal F}$ should be pure and of uniform weight $5$.
\end{itemize}
Having identified the relevant functions it is possible to set up a bootstrap procedure, as was done in ref.~\cite{Almelid:2017qju}. We will describe this after reviewing the available information from the high-energy and collinear limits.

\section{Soft singularities in the high-energy limit}

One of the most important and theoretically interesting limits to consider is the high-energy limit  of parton scattering where the centre-of-mass energy $s$ is much larger than the momentum transfer $-t$, see refs.~\cite{Sotiropoulos:1993rd,Korchemsky:1993hr,Korchemskaya:1996je,Korchemskaya:1994qp,DelDuca:2001gu,DelDuca:2013ara,DelDuca:2014cya,Bret:2011xm,DelDuca:2011ae,Caron-Huot:2013fea,Caron-Huot:2017fxr,Caron-Huot:2017zfo}.
Specifically, let us focus here on nearly forward  $2 \to 2$ parton scattering, which can be scattering of two gluons, two (anti-) quarks, or a gluon and an (anti-) quark. At tree-level this limit is dominated by the $t$-channel gluon-exchange diagram. At higher orders in perturbation theory it is corrected by large logarithms that, to leading logarithmic accuracy, simply exponentiate by modifying the $t$-channel gluon propagator according to
\begin{equation}
\label{LL_Regge}
\frac{1}{t}\to \frac{1}{t}\, \left(\frac{s}{-t}\right)^{\alpha(t)}\,,
\end{equation}
dubbed ``Gluon Reggeization'', where $\alpha(t)$ is the gluon Regge trajectory, which is infrared-singular.
Gluon Reggeization can be extended to next-to-leading logarithms (NLL) for the real part of the amplitude by including ${\cal O}(\alpha_s^2)$ corrections in the trajectory and in addition
introducing impact factors that describe the interaction of the Reggeized gluon
with the target and the projectile. These impact factors depend on the scattered particle and on the momentum transfer $-t$, but are independent of the energy. This factorization is represented by diagram (a) in fig.~\ref{Regge_exchanges}, and is referred to as Regge-pole high-energy factorization.

The infrared singularity structure implied by gluon Reggeization according to eqs.~(\ref{LL_Regge}) and its extension to NLL for the real part of the amplitude,  is fully consistent with soft-gluon exponentiation where the soft anomalous dimension is taken to be the dipole formula of eq.~(\ref{gamdip}).
However, this simple picture does not extend beyond NLL due to \emph{multiple
Reggeized gluon exchange}, which forms Regge cuts.

The possibility to compute directly logarithmically-enhanced terms associated with multiple Reggeized gluons exchange in partonic scattering is an important theoretical development of the recent years, starting with the seminal paper of Simon Caron-Huot~\cite{Caron-Huot:2013fea}. In this framework Reggeized gluons are sourced by the logarithm of an infinite Wilson-line operator along the trajectory of the target or projectile, defined at a given point in the  $(2-2\epsilon)$ dimensional transverse space. These effective degrees of freedom are governed by rapidity evolution equations, namely BFKL and its non-linear generalizations by Balitsky-JIMWLK~\cite{Kuraev:1977fs,Balitsky:1978ic,Lipatov:1985uk,Mueller:1993rr,Mueller:1994jq,Balitsky:1995ub,Balitsky:1998kc,Kovchegov:1999yj,JalilianMarian:1996xn,JalilianMarian:1997gr,Iancu:2001ad}, which can be solved order-by-order in perturbation theory~\cite{Caron-Huot:2017fxr,Caron-Huot:2017zfo}.

Considering the real part of the amplitude at NNLL accuracy, Regge-cut contributions were computed explicitly in ref.~\cite{Caron-Huot:2017fxr} through three loops (see also~\cite{Fadin:2016wso,Fadin:2017nka}). At this order one encounters a couple of new effects including the evolution of a state consisting of three Reggeized gluons and the mixing between the latter and
a single Reggeized gluon, as shown in fig.~\ref{Odd_three_loop_Regge_exchanges}.
Once this mixing is taken into account one can also define the three-loop gluon Regge trajectory~\cite{Caron-Huot:2017fxr} (the latter is known in ${\cal N}=4$ super Yang-Mills thanks to the computation of the amplitude in ref.~\cite{Henn:2016jdu}, but not in QCD).

The imaginary part of the amplitude is affected already at NLL accuracy by Regge cuts due to the exchange of two Reggeized gluons~\cite{Caron-Huot:2013fea} (see also~\cite{DelDuca:2001gu,Bret:2011xm,DelDuca:2011ae,DelDuca:2013ara,DelDuca:2013dsa,DelDuca:2014cya}) which form the famous BFKL ladder graphs shown in diagram (b) in fig.~\ref{Regge_exchanges}. The corresponding infrared singularities were computed very recently to all orders in perturbation theory~\cite{Caron-Huot:2017zfo}.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.8]{Reggeons123.pdf}
\caption{\label{Regge_exchanges}From left to right, exchange of one, two and three Reggeized gluons, respectively. We draw the Reggeized gluons as double wavy lines. Single Reggeon exchange in the first diagram contributes at LL accuracy to the real part of the amplitude, while two-Reggeon exchange in the second diagram contributes at NLL accuracy to the imaginary part. Last, three Reggeons exchange starts contributing to the real part at NNLL accuracy. The shaded blobs in the first and second diagram account for single- and two-Reggeon impact factors, which give additional contributions at subleading logarithmic accuracy to these diagrams.}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.7]{1to3-3to1.pdf}\qquad\quad
\includegraphics[scale=0.7]{3-Reggeons-1-Rung.pdf}
\caption{\label{Odd_three_loop_Regge_exchanges}Three loop contributions associated with $1\to 3$, $3\to 1$ and $3\to 3$ Reggeized gluon exchanges. These contributions were evaluated in Ref.~\cite{Caron-Huot:2017fxr}.}
\end{center}
\end{figure}

Let us now summarise the state-of-the-art knowledge of infrared singularities in the high-energy limit of
$2\to 2$ QCD scattering according to refs.~\cite{Caron-Huot:2017fxr,Caron-Huot:2017zfo}. To this end we write the soft anomalous dimension in this limit by collecting towers of increasing logarithmic accuracy as follows:
\begin{equation}
{\Gamma} \left(\alpha_s\right) =
{\Gamma}_{\rm LL}\left(\alpha_s,L\right)
+ {\Gamma}_{\rm NLL}\left(\alpha_s,L\right)
+ {\Gamma}_{\rm NNLL} \left(\alpha_s,L\right)
+ \ldots\,
\end{equation}
where  ${\Gamma}_{{\rm N}^k{\rm LL}}$ collects all terms with $\alpha_s^n L^{n-k}$. We choose the high-energy logarithm, $L$, to be of even signature (i.e. symmetric under $u$ to $s$ exchange):
\begin{equation}
L=\frac12\left(\ln \frac{-s-i0}{-t}+\ln \frac{-u-i0}{-t}\right)=\ln \left|\frac{s}{t}\right| -i\frac{\pi}{2}\,,
\end{equation}
so that terms with odd (even) signature in the amplitude are real (imaginary) (see section 2 in ref.~\cite{Caron-Huot:2017fxr}). Given that the tree-level hard amplitude is itself odd, real (imaginary) terms in the soft anomalous dimension are even (odd).
The leading-logarithmic anomalous dimension is one-loop exact,
\begin{equation}
{\Gamma}_{\rm LL}\left(\alpha_s,L\right) =\frac{\alpha_s}{\pi}L {\bf T}_t^2\,,
\end{equation}
and the real part of the NLL one is two-loop exact:
\begin{equation}
{\Gamma}_{\rm NLL}^{(+)} =  \frac{\alpha_s}{\pi} \sum_{i=1}^{2} \left( \frac{\widehat{\gamma}^{(1)}_{K}}{2} C_i \log \frac{-t}{\lambda^2}
+ 2{\gamma}^{(1)}_{J_i} \right)
\,\,+\,\,
\left(\frac{\alpha_s}{\pi}\right)^2 \frac{\widehat{\gamma}^{(2)}_{K}}{2} L \, {\bf T}_t^2\,,
\end{equation}
as can be seen from eq.~(\ref{gamdip}) (the coefficients of the cusp and collinear anomalous dimensions are summarised in Appendix A of ref.~\cite{Caron-Huot:2017fxr}).
The full soft anomalous dimension at NLL in the high-energy limit, ${\Gamma}_{\rm NLL}$, also has a signature-odd imaginary part, ${\Gamma}_{\rm NLL}^{(-)}$.
At one loop it can be deduced from eq.~(\ref{gamdip}) (see refs.~\cite{Bret:2011xm,DelDuca:2011ae,DelDuca:2013ara,DelDuca:2013dsa,DelDuca:2014cya}),
\begin{equation}
{\Gamma}_{\rm NLL}^{(-)} = i\pi \frac{\alpha_s}{\pi} \,\frac{{\bf T}_{s}^2-{\bf T}_{u}^2}{2}\,+
{\cal O} (\alpha_s^4)\,,
\end{equation}
but it also receives corrections starting at four loops~\cite{Caron-Huot:2013fea}. These have recently been computed to all orders in ref.~\cite{Caron-Huot:2017zfo}:
\begin{equation}
{\Gamma}_{\rm NLL}^{(-)} = i\pi \frac{\alpha_s}{\pi} \,G\left(\frac{\alpha_s}{\pi}L\right)\frac{{\bf T}_{s}^2-{\bf T}_{u}^2}{2}\,,
\end{equation}
where
\begin{equation}
G(x) \equiv \sum_{l=1}^{\infty} G^{(l)} x^{l-1}\quad\text{with}\quad
G^{(l)} \equiv
\frac{1}{(l-1)!}\left[ \frac{(C_A-{\bf T}_t^2)}{2}\right]^{l-1} \left.\left( 1 - \frac{C_A}{C_A -{\bf T}_t^2}R(\epsilon)  \right)^{-1}\right\vert_{\epsilon^{l-1}}
\label{G}
\end{equation}
and where
\begin{equation}
R(\epsilon) =
\frac{\Gamma^{3}(1-\epsilon)\Gamma(1+\epsilon)}{\Gamma(1-2\epsilon)} -1
= -2\zeta_3 \, \epsilon^3 -3\zeta_4 \, \epsilon^4 -6\zeta_5 \epsilon^5
-\left(10 \zeta_6-2\zeta^2_3 \right) \epsilon^6 + {\cal O}(\epsilon^7)\,.
\end{equation}
We can see that new colour structures emerge at higher orders, depending on ${\bf T}_t^2$ and $C_A$. It is not surprising that the result has maximal weight at any order, as it applies to
${\cal N}=4$ super Yang-Mills.
What is interesting, and unusual, is that this anomalous dimension sums up to an entire function of the coupling times the logarithm, so it has an infinite radius of convergence -- this is due to the factorial suppression of the coefficients in eq.~(\ref{G}). We refer the interested reader to ref.~\cite{Caron-Huot:2017zfo} for further details.

At the next logarithmic order, NNLL, three-loop results are known explicitly~\cite{Caron-Huot:2017fxr}:
\begin{equation}
\label{NNLL_results}
{\Gamma}_{\rm NNLL}^{(+)} ={\cal O} (\alpha_s^4)\,;\qquad
{\Gamma}_{\rm NNLL}^{(-)} = i\pi\left[ \frac{\zeta_3}{4}(C_A-{\bf T}_t^2)^2 \,\left(\frac{\alpha_s}{\pi}\right)^3 L+
{\cal O} (\alpha_s^4)
\right]\frac{{\bf T}_{s}^2-{\bf T}_{u}^2}{2}\,,
\end{equation}
but higher orders remain so far unknown. The fact that the real part, ${\Gamma}_{\rm NNLL}^{(+)}$, vanishes at three loops was deduced from the computation~\cite{Caron-Huot:2017fxr} based on Balitsky-JIMWLK equation. In other words, at this logarithmic order the rapidity evolution equation reproduces precisely the poles that are generated by the dipole formula of eq.~(\ref{gamdip}).
This was shown to be consistent with the result for the three-loop soft anomalous dimension~\cite{Almelid:2015jia}, which does not produce any new contributions proportional to $L^1$ upon taking the high-energy limit.
The non-vanishing imaginary value ${\Gamma}_{\rm NNLL}^{(-)}$ quoted in eq.~(\ref{NNLL_results}) was deduced by taking the high-energy limit of the soft anomalous dimension of~ref.~\cite{Almelid:2015jia}; a corresponding direct computation in the high-energy limit at this order would require accounting for  the exchange of four Reggeised gluons.

To summarise: we have seen that significant progress was achieved recently in our ability to compute logarithmic corrections in the high-energy limit of $2\to 2$ scattering associated with the exchange of multiple Reggeized gluons by iteratively solving the BFKL and JIMWLK rapidity evolution equations.
The interplay between the exponentiation of soft singularities owing to infrared factorization and the exponentiation of high-energy logarithms by means of rapidity evolution equations has proven to be highly insightful in both directions.
Specifically, at three loops, the absence of logarithmically enhanced contributions with $\alpha_s^3 L^k$ for any $k\geq 1$  for the real part, and any $k\geq 2$ in the imaginary part, provides a very strong constraint on the structure of the soft anomalous dimension.

\section{Collinear limits}

Next, let us consider the collinear limit of an $n$-parton amplitude following refs.~\cite{Becher:2009qa,Dixon:2009ur,Almelid:2017qju}.
This is the limit where two of the external hard partons become collinear. In general, this limit gives rise to another factorization property we have not discussed so far: this is the so-called collinear or splitting-amplitude factorization.

Working in the kinematic region where all $n$ coloured particles carrying momenta $\{p_k\}$ are outgoing, we consider the limit in which two of these partons, $i$ and $j$, become collinear.  In this limit $s_{ij}=2p_i\cdot p_j\rightarrow
0$, resulting in kinematic divergences proportional to $1/s_{ij}$. It is well known that for final-state collinear partons\footnote{In
  the case of a space-like splitting, factorisation is
  violated~\cite{Catani:2011st,Forshaw:2012bi}.} these divergences factorise~\cite{Bern:1999ry,Kosower:1999xi,Feige:2014wja,Catani:2003vu}, such that one may write
\begin{equation}
{\cal A}_n(p_1,p_2,\{p_j\};\mu,\epsilon,\alpha_s)
\xrightarrow{1\parallel 2}{\bf Sp}(p_1,p_2,{\bf T}_1,{\bf T}_2;\mu,\epsilon,\alpha_s)\,
{\cal A}_{n-1}(P,\{p_j\},\mu,\epsilon,\alpha_s).
\label{colfac}
\end{equation}
Without loss of generality, we have taken particles 1 and 2 collinear,
where $\{p_j\}$, $j=3\ldots n$ denotes the set of remaining
momenta. The right-hand side contains the $(n-1)$-particle amplitude in
which the momenta $p_1$ and $p_2$ are replaced by the sum $P=p_1+p_2$,
multiplied by a universal {\it splitting function} ${\bf Sp}$, which collects
all singular contributions to the amplitude due to particles
1 and 2 becoming collinear.
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=0.7]{Collinear_limit_wo_gluons.pdf}
\caption{\label{Collinear_limit}Left: An $n$ parton amplitude (in this case $n=4$) with $p_1$ nearly collinear to $p_2$; Right: The same, factorized into a corresponding $(n-1)$ parton amplitude where partons $1$ and $2$ have been replaced by a single particle carrying momentum $P=p_1+p_2$ and colour charge ${\bf T}={\bf T}_1+{\bf T}_2$, times a splitting amplitude describing the collinear pair.}
\end{center}
\end{figure}
A central property of the splitting amplitude ${\bf Sp}$ is that it depends only on the degrees of freedom (momenta, colour and helicities) of the collinear particles, and is entirely independent of the rest of the process.

It can be shown that the singularities of the splitting amplitude are controlled by an anomalous dimension which is the difference between the soft anomalous dimensions of the $n$ parton amplitude and the $(n-1)$ one~\cite{Becher:2009qa,Dixon:2009ur,Almelid:2017qju}:
\begin{equation}
\begin{split}
\Gamma_{\bf Sp}&(p_1,p_2,{\bf T}_1,{\bf T}_2,\mu_f,\alpha_s(\mu_f^2))\\
&\,\equiv \Gamma_n(p_1,p_2,\{p_j\},{\bf T}_1,{\bf T}_2,\{{\bf T}_j \},\mu_f,\alpha_s(\mu_f^2))
-\Gamma_{n-1}(P,\{p_j\},{\bf T},\{ {\bf T}_j \},\mu_f,\alpha_s(\mu_f^2))\,.
\label{GamSpdef}
\end{split}
\end{equation}
Given that the quantity on the left-hand side of this equation depends
manifestly on the quantum numbers of partons 1 and 2 only, this must
also be true for the right-hand side.
At the level of the dipole formula $\Gamma_{\bf Sp}$ can be computed and upon using colour conservation it immediately satisfies the requirement that it only involves the degrees of freedom of the collinear pair.
At three loops there is potential for this to break down, since there are diagrams that explicitly correlate the collinear pair and the rest of the process, as illustrated in fig.~\ref{Collinear_limit_gluons}.
This leads to an important constraint on the $\Delta_n$ in eq.~(\ref{gamnfull}), namely that the difference
\begin{equation}
\Delta^{(3)}_{\bf Sp}({\bf T}_1,{\bf T}_2)=\left[
\Delta^{(3)}_n(\{\rho_{ijkl}\},{\bf T}_1,{\bf T}_2,\{{\bf T}_j\})
-\Delta^{(3)}_{n-1}(\{\rho_{ijkl}\},{\bf T},\{{\bf T}_j\})
\right]_{p_1\parallel p_2}
\label{delcoll}
\end{equation}
can only depend on the quantum numbers of the two particles
that are becoming collinear. Note that the right-hand side of eq.~\eqref{delcoll} is
evaluated in the limit where $p_1$ and $p_2$ have become collinear.
Since $\Delta^{(3)}_{\bf Sp}({\bf T}_1,{\bf T}_2)$ is expected to be universal, it can be extracted from any $n$-point amplitude. Special simplification occurs for $n=3$, since in this case $\Delta^{(3)}_{(n-1)}$ vanishes and it follows that
\begin{equation}\begin{split}
\label{DelSp3}
\Delta^{(3)}_{\bf Sp}({\bf T}_1,{\bf T}_2)&\,=\left[\Delta^{(3)}_3(-{\bf T}_1-{\bf T}_2,{\bf T}_1,{\bf T}_2)-\Delta_2^{(3)}({\bf T}_1,{\bf T}_2)
\right]_{p_1\parallel p_2}\\
&\,=\left.\Delta^{(3)}_3(-{\bf T}_1-{\bf T}_2,{\bf T}_1,{\bf T}_2)\right|_{p_1\parallel p_2}\,,
\end{split}
\end{equation}
but since there are no conformally invariant cross ratios that one can form
from three particle momenta, the right-hand side evaluates to a
constant.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.7]{Collinear_limit_3-loops.pdf}
\caption{\label{Collinear_limit_gluons}As in fig.~\ref{Collinear_limit}, but where the the collinear pair interacts with the rest of the process via a three-loop web, that on the face of it, could violate collinear factorization. In effect, such a violation does not occur (when all partons are ourgoing) owing to intricate cancellations involving colour conservation.}
\end{center}
\end{figure}

Indeed, using the result of eqs.~(\ref{eq:expected_Delta}) through (\ref{Fdef}) we obtain:
\begin{equation}
\label{Split_delta}
\Delta_{\rm \bf Sp}^{(3)}=\left.(\Delta_n^{(3)}-\Delta_{n-1}^{(3)})\right\vert_{1\parallel 2}=-24 \left(\frac{\alpha_s}{4\pi}\right)^3\, (\zeta_5+2\zeta_2\zeta_3)\,\left[f^{abe}f^{cde}\left\{{\rm \bf T}_1^a,  {\rm \bf T}_1^c\right\}\left\{{\rm \bf T}_2^b,  {\rm \bf T}_2^d\right\} +\frac12C_A^2  {\rm \bf T}_1\cdot {\rm \bf T}_2\right]\,.
\end{equation}
Note that $\Delta_n^{(3)}$ and $\Delta_{n-1}^{(3)}$ are separately dependent on both colour and kinematics of the rest of the process. Their difference reduces to (\ref{Split_delta}) through intricate cancellations involving colour conservation and an interplay between the constant $C$ and kinematic-dependent function ${\cal F}$.
This can be regarded as a highly non-trivial confirmation of the properties and universality of splitting amplitudes, as well as a check of the calculation itself. Conversely, the requirement that the right hand side of eq.~(\ref{delcoll}) only depends on the degrees of freedom of the collinear pair can be used as a constraint on $\Delta_n^{(3)}$ as was done in ref.~\cite{Almelid:2017qju}, feeding into the determination of this quantity by bootstrap.

\section{Bootstrapping the three-loop soft anomalous dimension}

The bootstrap procedure of ref.~\cite{Almelid:2017qju} to determine the three-loop soft anomalous dimension starts off with two ingredients. The first is the  colour structure, which is known to be the set of fully-connected colour factors based on the non-Abelian exponentiation theorem for multiple Wilson lines~\cite{Gardi:2013ita}. The second is the identification of the relevant class of functions as weight-five SVMPLs as established in~ref.~\cite{Almelid:2017qju} and briefly reviewed in section~\ref{sec:SVHPL} above.

Considering the full set of connected colour factors,
the most general form that $\Delta_n^{(3)}$ can take is:
\begin{align}
\begin{split}
\label{eq:Delta_general_color}
\Delta_n^{(3)}&= \sum_{\{ i,j,k,l \}} f_{abe}f_{cde}\, {\bf T}_i^a {\bf T}_j^b {\bf T}_k^c {\bf T}_l^d \,A_{ijkl}
+\sum_{\{ i,j,k \}}  f_{abe} f_{cde} \left\{{\bf T}_i^a,{\bf T}_i^d\right\} {\bf T}_j^b {\bf T}_k^c \,B_{ijk}\\
&+\sum_{\{ i,j,k \}}  f_{abc}{\bf T}_i^a{\bf T}_j^b{\bf T}_k^c \,C_{ijk}
+\sum_{\{ i,j \}}   f_{abe} f_{cde} \left\{{\bf T}_i^a,{\bf T}_i^d\right\} \left\{{\bf T}_j^b,{\bf T}_j^c\right\} \,D_{ij}
+\sum_{\{ i,j \}}  {\bf T}_i\cdot {\bf T}_j \,E_{ij}\ ,
\end{split}
\end{align}
where the sums run over all sets $\{ i,j,\ldots \}$ of distinct Wilson
lines, and the coefficients $A_{ijkl}$, $B_{ijk}$ etc. are functions
of the four-velocities of the Wilson lines entering each colour factor.
Equation~\eqref{eq:Delta_general_color}, however, is still largely
over-complete. First, it was shown in
ref.~\cite{Gardi:2009qi,Becher:2009cu,Becher:2009qa} that colour
tripoles of the form $f_{abc}{\bf T}_i^a{\bf T}_j^b{\bf T}_k^c$ are
absent at any loop order (a corresponding kinematic function would violate rescaling invariance) and so we must have $C_{ijk}=0$ in
eq.~\eqref{eq:Delta_general_color}. Second, the soft anomalous dimension, and hence
$\Delta_n^{(3)}$, depends on the colour generators subject to colour conservation, namely
\begin{equation}
\label{eq:colour_conservation}
\left(\sum_{i=1}^n{\bf
  T}_i^a\right){\cal H}_n = 0\,.
\end{equation}
In refs.~\cite{AlmelidPhD,Gardi:2016ttq,longinprep} the role of colour
conservation in the context of $\Delta_n^{(3)}$ was analysed in
detail, and a basis of colour structures that are independent after
imposing eq.~\eqref{eq:colour_conservation} was worked out. In this basis the general ansatz takes the much simpler form:
\begin{align}
\begin{split}
\label{eq:ansatz_colour}
\Delta_n^{(3)}&= \sum_{\{i,j,k,l\}} f_{abe}f_{cde}\, {\bf T}_i^a {\bf T}_j^b {\bf T}_k^c {\bf T}_l^d \,A_{ijkl}(\rho_{ikjl},\rho_{iljk})
\,-16\,C\,\sum_{i=1}^n \ \sum_{\substack{{1\leq j<k\leq n}\\ j,k\neq i}}  f_{abe} f_{cde} \left\{{\bf T}_i^a,{\bf T}_i^d\right\} {\bf T}_j^b {\bf T}_k^c\,\,.
\end{split}
\end{align}
Finally, using Bose symmetry one can show that eq.~(\ref{eq:ansatz_colour}) takes the form of  eq.~(\ref{eq:expected_Delta}) with ${\cal F}$ given by eq.~(\ref{calFdef}), where we now regard $F(z)$ as an unknown function and $C$ as an unknown constant.

The next step is to write a general ansatz for these in terms of weight-five SVHPLs.
A general ansatz for $F(z)$ consists of 32 weight-five SVHPLs, 8
weight-three SVHPLs multiplied by $\zeta_2$, 4 weight-two SVHPLs
multiplied by $\zeta_3$, 2 weight-one SVHPLs multiplied by $\zeta_4$,
and finally a general linear combination of the 2 weight-five
constants $\zeta_5$ and $\zeta_{2}\zeta_3$, all with rational
prefactors. This gives 48 distinct terms. Similarly, a general ansatz
for the constant $C$ consists of a general linear combination of the
2 independent weight five multiple zeta values, with rational prefactors. However, this
na\"ive ansatz is overly large for two reasons. First, because of the
invariance under the interchange $z_{ijkl}\leftrightarrow\bar{z}_{ijkl}$ and the way
this symmetry acts on SVHPLs via reversal of words ${\cal L}_w(\bar{z})={\cal L}_{\tilde{w}}(z)+\ldots$, only palindromic combinations of weight five
SVHPLs need to be considered.
Second, the function $F(z_{ijkl})$ only appears in our ansatz, eq.~(\ref{eq:expected_Delta}), via
the differences:
\begin{align}
F(1-1/z)-F(1/z),\quad F(1-z)-F(z),\quad F(1/(1-z))-F(z/(z-1)).
\label{Fdiffs}
\end{align}
Taking these considerations into account results in an ansatz for $F(z)$ involving only 11 parameters:
\begin{align}\label{ansatz1}
F(z)&=a_1{\cal L}_{00000}+a_2{\cal L}_{00100}+a_3{\cal L}_{10001}
+a_4{\cal L}_{10101}+a_5\left({\cal L}_{01001} + {\cal L}_{10010}\right)\nonumber \\
&\quad+a_6\left[{\cal L}_{00101} + {\cal L}_{10100} + 2 ( {\cal L}_{00011}
+ {\cal L}_{11000})  \right]
+a_7\left[{\cal L}_{11010} + {\cal L}_{01011} + 3 ( {\cal L}_{00011}
+ {\cal L}_{11000}) \right]\nonumber \\
&\quad+a_8\,\zeta_2{\cal L}_{000}
+a_9\,\zeta_2\left({\cal L}_{001}+{\cal L}_{100}\right)
+a_{10}\,\zeta_3\,{\cal L}_{00}+a_{11}\,\zeta_2^2\,{\cal L}_0,
\end{align}
where each $a_i\in{\mathbb Q}$ is an undetermined rational numerical
coefficient, and we have suppressed the dependence of the SVHPLs on their
argument, ${\cal L}_w \equiv {\cal L}_w(z)$. Writing our ansatz for the
constant~$C$ explicitly as
\begin{equation}
C=a_{12}\,\zeta_5+a_{13}\,\zeta_2\,\zeta_3\,,
\label{ansatz2}
\end{equation}

The final step in the bootstrap calculation is to constrain the parameters $a_1$ though $a_{13}$ using the information from the high-energy and the collinear limits reviewed in the last two sections.

To use the high-energy limit one needs to consider the analytic continuation of the ansatz for $F(z)$ given by eq.~(\ref{ansatz1}) to the physical region of $2\to 2$ scattering, before taking the high-energy limit, $s\gg -t$. This process requires familiarity with the analytic properties of SVHPLs and is described in detail in section 5 of ref.~\cite{Almelid:2017qju}. It turns out that the  general ansatz of eq.~(\ref{ansatz1}) gives rise to multiple logarithmically-enhanced terms, $c_k L^k$ ranging all the way to $k=5$, which is the maximal power for weight-five polylogarithms. Setting to zero the real part of $c_k$ for any $k\geq 1$, and similarly the imaginary parts for any $k\geq 2$, leads to six constraints on the $a_i$ coefficients. This is then repeated for the three different forward limits, yielding a total of 8 independent linear relations between $a_i$:
\begin{align}\label{aconditions_regge}
\begin{split}
\!\!\!\!(a_1,\ldots,a_8) &= \left(0,\frac{a_{10}}{10},-\frac{a_{10}}{10}-\frac{a_{11}}{48},\frac{a_9}{2}-\frac{3a_{10}}{20}-\frac{a_{11}}{12},\frac{a_{10}}{10}+\frac{a_{11}}{48},\frac{7a_{10}}{20},-\frac{a_{10}}{4},-\frac{3a_{10}}{5}\right).
\end{split}
\end{align}

Finally, as shown in section 6 of ref.~\cite{Almelid:2017qju}, the collinear-limit constraint, namely the fact that the right hand side of eq.~(\ref{delcoll}) is a constant, independent of the colour degrees of freedom of the non-collinear partons, translates into the requirement that
\begin{align}
\Big[ \mathcal{F}(\rho_{1234}, \rho_{1432} ) - \mathcal{F}(\rho_{1243}, \rho_{1342} ) +  2 \mathcal{F}(\rho_{1324}, \rho_{1423} ) \Big]_{p_1 \parallel p_2} &= 0,
\notag \\
\Big[ \mathcal{F}(\rho_{1234}, \rho_{1432} ) + \mathcal{F}(\rho_{1243}, \rho_{1342} ) \Big]_{p_1 \parallel p_2} &= 8 C.
\label{Fcalcoll}
\end{align}
This, in turn,  implies the following relations between the coefficients $a_i$:
\begin{align}
(a_1,a_7,a_8,a_{11},a_{12},a_{13})=
\left(0,-\frac{a_{10}}{4},0,0,
3a_2-\frac{3a_3}{2}+a_4+\frac{9a_5}{2}-\frac{3a_6}{2}
+\frac{a_7}{2}, a_9\right).
\label{aconditions_coll2}
\end{align}
Upon implementing the full set of constraints, the
ans\"atze of eqs.~(\ref{ansatz1}) and~(\ref{ansatz2}) reduce to
\begin{equation}
F(z)=a_4\left({\cal L}_{10101}+2\zeta_2({\cal L}_{100}+{\cal L}_{001})
\right) {\rm~~~and~~~~}
C = a_4(\zeta_5+2\zeta_2\zeta_3)\,.
\label{ansatz4}
\end{equation}
Thus, both $F$ and $C$ have been uniquely determined by the bootstrap procedure,
up to an overall rational number $a_4$. Since $\Delta_n^{(3)}$ depends linearly on $F$ and $C$,
the form of the three-loop correction to the dipole formula is fixed completely by symmetries and physical constraints.
Comparing the expressions in eq.~\eqref{ansatz4} with the result of ref.~\cite{Almelid:2015jia} (quoted here in
eqs.~(\ref{Fdef}) and~(\ref{Cdef})), we see that the latter can be
reproduced by setting $a_4 = 1$.

\section{Conclusions}

Infrared singularities of massless scattering amplitudes in gauge theories are known in full to three-loop accuracy~\cite{Almelid:2015jia}. This is the first order in which corrections to the dipole sum occur. For three coloured partons this is a constant contribution, while for four or more it involves non-trivial kinematic dependence.

Analysing special kinematic limits has proven very insightful, both as a way to learn about the QCD dynamics in these limits per se, and as a way to provide input for a bootstrap procedure. As done in ref.~\cite{Almelid:2017qju} and reviewed here,  at three loops the available information was sufficient to reproduce the result of the diagrammatic computation of the soft anomalous dimension up to a single overall numerical factor.

In the high-energy limit there has been major progress in computing logarithmically-enhanced terms in $2\to 2$ amplitudes based on an order-by-order solution of the corresponding  rapidity evolution equations~\cite{Caron-Huot:2013fea,Caron-Huot:2017fxr,Caron-Huot:2017zfo}. In particular, at NLL we now know the soft anomalous dimension to all loop orders, and at NNLL we have an independent computation of both infrared-singular and finite terms at three loops for the real part of the amplitude.  The same methods can be applied to obtain explicit results at higher orders and infer the iterated structure of amplitudes in this limit.

We saw that understanding the space of functions in which the answer for the soft anomalous dimension should be written opens the way to use the general constraints from symmetries and the information from special kinematic limits to construct the answer. With these methods, the four-loop soft anomalous dimension may be within reach. Furthermore, other physical quantities, such as the single-emission radiative soft function may be amenable to similar methods. Finally, it is conceivable that bootstrap techniques could eventually be applied to multi-loop QCD amplitudes themselves. Of course, this still requires a significant advancement of our understanding of their properties. Their infrared singularities and special kinematic limits is a good place to start.

\begin{acknowledgments}
I would like to thank {\O}yvind Almelid, Claude Duhr, Simon Caron-Huot, Andrew McLeod,  Joscha Reichel, Leonardo Vernazza and Chris White for very stimulating and enjoyable collaboration on the recent work reported here. This work is supported by the STFC
Consolidated Grant ``Particle Physics at the Higgs Centre''.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


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text:
Local cases of flu are few so far
by Madison White Franks
KILMARNOCK—Although cases of flu have been confirmed locally, health officials here do not consider it an epidemic.
"Epidemic is a term used frequently to describe flu season this year, but it's better to describe the flu as widespread this season," said Three Rivers Health District director Dr. Richard Williams.
Northern Neck Free Health Clinic nurse practitioner Tamara Hall said there has been one positive flu test at the clinic in the past two weeks.
"We are not seeing it as much as everyone else," she said.
She said some patients are coming in with flu-like symptoms, but they are not testing positive for the flu.
According to Northumberland Public Schools director of instruction Amy Lamb there have been five confirmed cases of the flu at the elementary school and one confirmed case between the middle and high school.
"There have been no spikes in absences compared to last year," reported Lancaster assistant school superintendent Dan Russell.
Dr. Williams said the predominant strain of flu in the Northern Neck is Influenza A, H1 type.
"This [type] is proving to cause a less serious illness than the H3 type flu that is affecting much of the U.S., including central and western areas of Virginia. This may change and we may see the H3 type flu increase in our area as the flu season progresses," he said.
"It is difficult to know how many people have influenza in a community because many people recover from influenza at home without visiting a healthcare provider, many people who go to a healthcare provider do not have laboratory testing conducted and not all cases of influenza are reported to public health authorities," said Dr. Williams.
"Flu can cause serious illness requiring hospitalization and can even be fatal in the worst cases," he continued. "If anyone suspects they have the flu, they should contact their physician quickly. Antiviral agents can be effective in shortening the course of the flu and in making the disease less serious."
To prevent the flu, people should get their flu shot, wash their hands frequently, and limit exposure to other people who might be sick, said Dr. Williams.
"We do believe you should get the flu shot and we provide them at the clinic," said Northern Neck Free Health Clinic executive director Jeannie Nelson.
"Contact your doctor early if you think you might be coming down with the flu," said Dr. Williams. "It can be difficult to tell if you have the flu or a common cold. Flu symptoms include rapid onset of dry cough, fever, aches and pains in your muscles and joints, headache, nasal congestion, sore throat and severe fatigue. Nausea and vomiting can also occur."
He said a doctor can do a rapid flu test by swabbing your throat or nose to determine whether or not you have the flu.

redpajama_set_name: RedPajamaCommonCrawl


text:
Carolyn L. Westhoff, MD
Department of Obstetrics & Gynecology, Division of Family Planning and Preventive Services, Department of Epidemiology, Department of Population and Family Health
Board Certifications:
Sarah Billinghurst Solomon Professor of Reproductive Health in the Department of Obstetrics and Gynecology and Professor of Population and Family Health and Epidemiology at the Columbia University Medical Center
S. Zev Williams, MD, PhD
Department of Obstetrics & Gynecology, Division of Reproductive Endocrinology & Infertility
Reproductive Endocrinology/Infertility, Obstetrics and Gynecology
Associate Professor of Obstetrics & Gynecology at CUMC
New and Existing Patients:
Jason D. Wright, MD
Department of Obstetrics & Gynecology, Division of Gynecologic Oncology
Gynecologic Cancers, Cancer Care
Gynecologic Oncology, Obstetrics and Gynecology
Sol Goldman Associate Professor of Gynecologic Oncology (in Obstetrics and Gynecology)
Ladin Yurteri-Kaplan, MD
Department of Obstetrics & Gynecology, Division of Gynecologic Specialty Surgery
Urogynecology, Female Pelvic Medicine and Reconstructive Surgery
Assistant Professor of Obstetrics & Gynecology at CUMC
Noelia M. Zork, MD
Department of Obstetrics & Gynecology, Division of Maternal Fetal Medicine
Maternal and Fetal Medicine
CUMC/NewYork-Presbyterian Morgan Stanley Children's Hospital


We are currently looking for someone to join us in the tasting room/art gallery. Saturdays are a must.
Have you got a great personality?
Are you upbeat and hard working?
We're looking for someone who can work alone and with others.
Responsibilities would be to maintain the look and feel of our tasting room/art gallery, assist customers, replenish stock, dust, refill growlers, offer tastings, chat about beer and more. Do you also take great pride in providing an incredible customer experience to everyone who walks through our door? The right person must be hard working, dependable, outgoing and have a good sense of humor.
*Knowledge of craft beer and the desire to learn more is needed.
We also have a part time position available on Tuesdays in packaging.
If this is you and you are interested please email me a cover letter and resume.


Rahul leads the software development and computer systems teams at the CSF Center. He is in charge of the entire software development process and his responsibilities include project planning, requirements management, quality assurance and final deployment. Rahul provides architectural and technical guidance during the design, development and delivery of all projects.
Before joining the CSF Center Rahul was the Principal Software Engineer at STMicroelectronics and led many software development projects. He earned his Bachelors of Engineering in Electronics and Communication Engineering from the University College of Engineering, Osmania University, India and Masters of Science in Computer Engineering from NJIT. His master's thesis work offered pioneering research in Powerline communications which contributed to the creation of Arkados, a co-founding member of the Homeplug Alliance. He was part of the small core team that developed a re-configurable hardware and software platform for high-speed transmission of converged multimedia, voice, and data traffic over AC electrical wires with applications in Smartgrid.


Emerson Street Automotive Inc at 1406 Emerson Street was recently discovered under Leesburg Oldsmobile catalytic converter repair.
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The Environmental Science and Design Research Initiative (ESDRI) was excited to be able to offer competitively awarded seed grant funding to research teams in Spring Semester 2018. ESDRI was established in 2017 to support faculty efforts to achieve continued scholarly success in areas of research concerning both natural and built systems.
The projects funded demonstrate the broad nature of scholarly activity addressing the environment at Kent State University, including projects related to climate change, urban health and forest restoration. The teams were chosen based on their creative and cutting-edge nature, breadth of the interdisciplinary team and future funding potential.
See the ESDRI website for more details.


text:
You Can Two
From mental and emotional prep to feeding and everything in between, this action-oriented guide covers the broad spectrum of how to tackle the first few months after the babies arrive.
Author: Jennifer Bonicelli
Publisher: Rockridge Press
The founders of the blog Two Came True deliver essential information and practical prep strategies to get expecting parents ready to discover the joys of twin parenting. From mental and emotional prep to feeding and everything in between, this action-oriented guide covers the broad spectrum of how to tackle the first few months after the babies arrive.
Tale Of Two Twins A The Langevin Experiment Of A Traveler To A Star
A Trip to the End of the Universe and the Twin Paradox, Amer. J. Phys. ... Twin Paradox: A Complete Treatment from the Point of View of Each Twin, Amer. ... Index 162 A Tale of Two Twins: The Langevin Experiment of a Traveler to a Star.
Author: Lucien Gilles Benguigui
Publisher: World Scientific
The thought experiment proposed by Langevin in 1911, known under the popular names, 'Clock Paradox' or 'Twin Paradox', is the most surprising result of the theory of Relativity: A twin who travels to a star at nearly the velocity of light comes back to Earth and finds his twin brother much older. In over a century, several thousands of published articles debated both in favor of and against this result. Unique to the physics of Relativity, this baffling phenomenon is analyzed as a main goal of this book.Among an incredible number of solutions, is there one of simplicity and clarity which may be accepted unanimously by all of the physics community? The answer is yes and this solution, which has its origin in Einstein himself, is developed in the framework of the Special Theory of Relativity. In detailing this solution, it is shown that the essential ingredient to understand the theory is the acceleration of the twins. All the models which do not include acceleration are incompatible with the original idea of Langevin. If one considers this phenomenon, several questions come to mind. Why did physicists debate excessively on the paradox and struggle to reach an agreement? Why was there resistance to integrate acceleration into their studies? Why is the solution developed in this book known only by a minority of scientists?Written for physicists, historians and philosophers of science, this book seeks to answer these questions based on (1) the psychological difficulty to accept the theoretical results, and (2) the fact that scientific knowledge is not uniformly distributed among scientists.
She had two heads, two sets of arms but only a single body and a single pair of legs. She was what scientists call a coalescent twin. This type of twin results from the incomplete separation of a single egg into two parts.
Our Twins Journal
"A good neighbor will baby-sit, a great neighbor will baby-sit twins" "Multiple Scraps Fill the Books of Our Lives! " "One plus one is two-and that's the both of you. " "Twice as much to love,two blessings from above.
Publisher: Lulu.com
Her Twin Baby Secret
She was the one carrying the twins. 'Two? As in twins?' She merely raised her brows in answer. 'Of course it's twins. Two are twins.' He stood, began to pace. 'You're sure?' Though she hadn't quite anticipated this reaction, she nodded, ...
Author: Therese Beharrie
Publisher: Harlequin
Could her biggest competition……become her babies' father? Alexa Moore is the definition of independent. She runs her own successful business and she's decided to start a longed-for family—by herself! But after business rival Benjamin Foster comes to her rescue and pretends to be her boyfriend for a night, their earth-shattering kiss sparks Alexa's desire to share much more with him… But having Benjamin in her life means she must confide in him—she's pregnant with twins! "...this is my favourite book by this author. It made me laugh out loud several times...this was well-written and introspective with convincing, 3 dimensional characters. Highly recommended and chuckle-worthy." —Goodreads on Island Fling with the Tycoon "All in all, From Heiress to Mom is a delightful, funny, sweet, excellently paced, and very real book. You can tell that a lot of love and care has been put into this book, and if I'm honest, that's really refreshing for a romance novel." —Goodreads
Two are Twins
Examines the phenomenon of twins and the effects of new twins on existing children in the family.
Author: June Albertsen
Publisher: Double Talk
Category: Brothers and sisters
The Spectacle of Twins in American Literature and Popular Culture
We focus on who we've both become as people" (57). Twin Kurt Froman appreciates others' attempts to distinguish him from his twin, commenting, "I like when people make an effort to recognize 'these are your traits and these are his'" ...
Author: Karen Dillon
Publisher: McFarland
Category: Social Science
The cultural fantasy of twins imagines them as physically and behaviorally identical. Media portrayals consistently offer the spectacle of twins who share an insular closeness and perform a supposed alikeness—standing side by side, speaking and acting in unison. Treating twinship as a cultural phenomenon, this first comprehensive study of twins in American literature and popular culture examines the historical narrative—within the discourses of experimentation, aberrance and eugenics—and how it has shaped their representations in the 20th and 21st centuries.
The Twins of Fairland
Then she hugged both twins tightly and said, "I could not be more proud of the both of you. I am so happy that I lived to see this day and hold my children in my arms." Tears ran down the queen's face. Mallrok led the three of them to ...
Author: sb white
Twins separated at birth are reunited. As they get to know each other, they discover an ancient magic hidden for years. They team up to protect the hidden kingdom of Fairland alongside their unique animal guardians. When they are challenged by a forgotten enemy with dark powers, they must trust and depend on each other. Tre and Skylin are guided on their journey into unknown territory by Mallrok, a mystic shaman. The twins may look alike, but are very different in other ways. Together they complete each other as the Twins of Fairland. Join the Twins of Fairland on their journey of discovery filled with rescues, hidden kingdoms, magical powers, and forgotten enemies. The Twins of Fairland is a well crafted, easy to read story for the preteen/teen reader.
The Twins Two by Two
Inspired by a bedtime story about Noah's ark, twins minnie and maz become pairs of crocodiles, elephants, bats and other animals as they prepare for bed.
Author: Catherine Anholt
Publisher: Candlewick Press (MA)
New Understandings of Twin Relationships
This is closely intertwined with the third singleton discriminatory perspective of twins, that both twins within the set are two identical people devoid of any differentiation in physiology, personality, and behavior; such supposed ...
Author: Barbara Klein
Category: Psychology
New Understandings of Twin Relationships takes an experience-based approach to exploring how twin attachment and estrangement are critical to understanding the push and pull of closely entwined personal relationships. Based on the research expertise of each of the authors (all identical twins in their own right), and vignettes from twins across the globe, this book describes the inner workings of the twin-world, showing how the twin-world creates experiences that are often more intense and intricately textured than those in the singleton-world. Chapters debunk myths surrounding twinship and analyze the developmental stages of the twin relationship as well as the effect of being a twin on one's mental health from different perspectives. The authors articulate how attachment, separation anxiety, loneliness, estrangement, and the subjective experience of the twin and non-twin "other" impact behavior, thinking, and feeling. Through its careful study of the many psychological challenges that twins face throughout their lifetime, this text will help psychologists, scholars, clinicians, and twins themselves attain a deeper understanding of all interpersonal relationships.
Circus Bee
The Timeless Racer
PROCEEDINGS OF THE 18TH INTERNATIONAL CONFERENCE ON FLUIDIZED BED COMBUSTION: PRINT (H01310)
A Cowboy Carol
The Art of Dancing Explained by Reading and Figures and Six Dances
Failing Students or Failing Schools?
JORDAN HARPER UNTITLED 2 PA
Flash Cards: Numbers 0 - 100
Out of the House of Bondage: The Transformation of the Plantation Household
The Passion Test
The Official Handbook of Hitting and Bunting
40 Great Glass Fusing Projects
Flame of Coven
Medinet Habu IX
Harold Garfinkel: The Creation and Development of Ethnomethodology
Sterilization of Medical Devices
Identity Marks in Deir el-Medina During the Ramesside Period
The Collected Works of Count Rumford, Volume II: Practical Applications of Heat
The Numerati
Modern Bargello
The Roots of American Exceptionalism: Institutions Culture and Policies
William Walton Edition Complete Set
Scissors Make Cents
City of Caesar, City of God
The Days of Wine and Roses Are Over
Raven Winter
My Blood Approves
Transitions to School - International Research, Policy and Practice


I once was very active with http://musped.org. It is now DEAD but I moved and updated the links to journals section – and it is now here. Please let me know of the ones I've missed!
Finally Olle and me have submitted Internet as Teacher to the Journal of Music, Technology and Education Now we cross our fingers!


Description Literacy Lessons: Designed for Individuals is a two-volume resource. Part One: Why? When? and How? focuses on the theory surrounding successful Reading Recovery programs, and how individual lessons for young children at risk can be highly successful.
Description Literacy Lessons: Designed for Individuals is a two-volume resource. Part Two: Teaching Procedures is a training manual for practising teachers.
Description One of the Concepts About Print test readers supporting the No Shoes and Stone titles. This title has been developed in direct response to teachers' requests for more test readers. The book is in full-colour, and provides teachers with a wide choice of test materials.
Description Innovative teacher resources by Marie M. Clay. This title has been developed in direct response to teachers' requests for more test readers. The book is in full-colour, and provides teachers with a wide choice of test materials.
Description This collection of readings is about all children and the early years of schooling. It brings together for educators and classroom teachers significant new, previously unpublished articles (Chapters 4, 10, and 14) as well as several of Dr. Clay's seminal papers. Here she reiterates for new teachers many of the concerns that lie at the heart of her work with young children - the sensitive observation of the constructive child, the challenges of early encounters with how we put language down in print (Concepts About Print), introducing storybooks to children, and how we can better adapt to diversity in our classrooms. Other emphases that emerge in her new articles call for changes in how teachers think about three things: literacy awareness as it develops before and after the transition to school, the power of writing for all children in early literacy encounters, and conversation as a tool for vastly improving teaching interactions. Marie Clay focuses on many reciprocal connections - how one kind of learning supports another - between oral language and reading, between writing and reading, between theory and practice. She believes that practice informs theory and theory informs practice in a circular and continuing set of relationships. Teachers who want to go beyond their present practice must get into the writings of those who challenge their expectations on how best to meet the needs of their very diverse children. Each chapter in this collection - from the glimpses of some fascinating children to some strong challenges to basic assumptions about literacy teaching - could provide a centerpiece for a workshop, the background reading for several group discussions, or an opportunity for practicing teachers to bring their experience face to face with a text that challenges.
Description Only when Running Records are used in standard ways with standard recording and scoring procedures will they provide a reliable and valid assessment of text reading. Classroom teachers use Running Records from time to time: to record what progress children are making to compare and group children to check how well texts are matched to the child's need for challenge.
Description These have been specifically developed as literacy lesson plan sheets, and are available in pads of 200 double-sided sheets.
Description Development psychologist, Dr Marie Clay, shares her non-traditional methods of dealing with literacy learning difficulties in Change Over Time in Children's Literacy Development. It has been written for early intervention researchers, graduate students and academics, tutors or teachers who are preparing early intervention teaching programs. Change Over Time in Children's Literacy Development discusses the theories, questions, and developments in early literacy intervention, that have made Marie Clay a leader in this area.
Innovative teacher resources by Marie M. Clay.


text:
Home Insight Wales provides fertile ground for independent presses both old and new
Wales provides fertile ground for independent presses both old and new
Published May 10, 2019 by Natasha Onwuemezi
James Griffiths of up and coming indie Panther Publishing and Ashley Drake, founder of industry stalwart Welsh Academic Press, discuss how their businesses have evolved....
The Welsh independent publishing industry is a thriving ecosystem consisting of established players and younger presses, with new entrants hitting the market with regularity. This year, while new indie Panther Publishing is still in its first 12 months, Welsh Academic Press is celebrating 25 years in the business.
Ashley Drake (pictured below right), founder of the Cardiff-based Welsh Academic Press, had early exposure to the business: his father was a director of Blackwell's. "I remember a copy of The Bookseller being delivered to our home every week. I also remember, as a boy, shaking hands with Sir Basil Blackwell when he welcomed his staff's children to the annual Christmas party at Oxford Town Hall. I was there so often that the Norrington Room in Broad Street felt like a second home and, as a teenager, I worked for a summer at Blackwell's distribution arm, Marston on the Cowley Road [in Oxford]." He is quick to thank his parents, and says that "with that background, working in publishing seemed a natural thing to do."
In 1994, after two years covering the north of England and Scotland for a range of scholarly and academic publishers as a sales rep, Drake noted that the genre of books he wanted to read were not being published, and so decided to start his own company while continuing to work as a rep. The press initially focused on Drake's personal interests and knowledge of Welsh, Scottish and Irish history and politics, as well as of the stateless nations of Europe.
Drake also established two Welsh-language imprints: trade list Y Ddraig Fach (meaning "The Little Dragon"), and educational list Gwasg Addysgol Cymru ("Wales Educational Press") to enable him to publish children's books in Welsh. "As someone who learnt Welsh at university, I was passionate to see children in Wales have access to bestsellers in their own language, so sought out and acquired the rights to publish the classic Ladybird/Disney mini-hardbacks in Welsh with Y Ddraig Fach, and The Diary of Anne Frank for Gwasg Addysgol Cymru."
Horror show
New press on the scene Panther had a different route in. With a background in photography and admin, founder and "big fiction reader" James Griffiths made the decision to launch his press in 2018. "I enjoy novels, I love authors, and it made sense to me to put both of my interests together and form Panther Publishing," says Griffiths.
"I knew of authors who were struggling to find publishers for all sorts of reasons. There are so many writers, and so many publishers not willing, or not able, to take a punt on new talent, so a lot of voices go unheard, which is a shame, both for the writers and the publishers. We also found that there didn't seem to be a Welsh publisher specialising in crime, mystery, thriller, horror and paranormal fiction, which is what we publish. S
o our decision to launch Panther was one of wanting to find new talent, wanting to nurture Welsh writing and Welsh settings in novels, and also wanting to publish in those genres."
Panther has signed two authors: J S Strange and W D Jackson-Smart. Strange's first novel, Murder on the Rocks, is based in Cardiff, and features a gay male detective. The story centres around a murdered author in a prestigious writing group, and how the detective's mother's death is related to the killing. Panther is publishing two of Jackson-Smart's DI Graves series, The Demons Beneath and From Inside the House. The first book follows Graves tracking a killer, and From Inside the House, features brutal murders, a murderer on the loose, and body parts being taken from each scene.
For Drake, the 25 years his press has been operating for have "really flown by". He says: "The Welsh Disney books were hugely popular; we sold over 60,000 units in just two or three years, but as our sales were primarily outside the mainstream UK trade (via Welsh independent booksellers, and we were the first publisher to sell Welsh-language books direct to supermarkets such as Tesco and Safeway) those sales figures unfortunately never featured in The Bookseller's bestseller charts I had read avidly as a boy. It was all 'under the radar'."
The journey has not been without its challenges. Over the past 25 years, Drake has only published full-time for seven years, so he says finding the time to manage the business while working for another employer has always been a struggle. In 2003, Drake was offered the opportunity to become director of the University of Wales Press—the youngest in its history. "Understandably, my company took a back seat for the next five years and many projects I'd been working on were either transferred to UWP or acquired by other publishers, including Mainstream. Effectively the core publishing programme was removed from the business, so when my five-year tenure at the university came to an end in 2008, it was tough having to effectively re-start the company from scratch."
While Drake has had trouble juggling workload among other commitments, Griffiths says the biggest struggle for his business has been promotion. "Bloggers are life-savers," he says. "Those who participated on our first blog tour really made me happy, because they gave Panther and they gave J S Strange a chance. That's a big thing for us. We've also found some great bloggers for the new W D Jackson-Smart novels. But without massive budgets, I am struggling to keep pre-orders coming in and to get the books seen as much as we'd like them to be—and as much as I know they deserve. Obviously I knew this would be a struggle from the start, but it's something I'm willing to face and keep looking at different avenues."
Promotion and visibility of titles is a problem shared by many indies. Drake says: "It's never going to be easy. It's a matter of selecting the right projects and marketing them as strongly as you can. Independents are usually more passionate and dynamic than the corporates, and that can make a big difference."
Ultimately, both are positive about the future of independent publishing. Griffiths says: "I see opportunities for new talent to be seen and heard. I see great opportunities for more independent publishers to be found and to be developed. I also see great ways for independent publishers to work together, and to be recognised by the publishing industry."
This piece is part of The Bookseller's country focus on Wales. For more in the series, click here.
Wales Country Focus
Panther Publishing
Welsh Academic Press
Welsh presses debate merits of translation as number of native speakers swells
Six questions for...the National Library of Wales' Pedr ap Llwyd
Peter Conradi | 'God and the afterlife are childish bribes... you have to be good, as she put it, for nothing'
Retail Spotlight: Palas Print
Ones to Watch: Illustration trio Collins, Davies and Low make their mark alongside author duo


Only 1 week of February left and only 2 short weeks until my Spring Break! I'm in desperate need of it too because I have been plagued with Spring Fever. Last year, it was my senior year of college so my roommates and I went to Miami (the perfect Co-Ed spring break location). Technically I'm still in school so I get a week off too, although I don't have any crazy plans this year, I've made a list of essentials that will always come in handy... Miami beach trip or stay-cation!
Headphones: These Frends Headphones are super cute and are a necessity for a plane ride, chillin' on the beach or lounging in bed. They're like a pair of earrings that plays music.
BB Cream: I always love using BB cream on my days off. It gives the perfect amount of coverage without looking to done-up and this one by Maybelline has SPF 30. It's been a staple of mine for years. I even went on a whole trip to Ghana, Africa using just this on my face... I looked flawless and was sun-burn free, so it's perfect for the sun-rays at the beach.
Tinted Lip Balm: Like BB cream, this tinted lip balm can make you look like you put more effort into your look than you did. I love wearing lipstick, but tinted lip balm is so much easier to swipe on and is more moisturizing. One of my favorites is this one by Fresh, I received a sample from Sephora for my birthday one year and fell in love. I wear this around the house and out and about. It makes my lips feel super soft, while giving the perfect amount of color. The line also has many different shades of balm to choose from!


text:
Shi'ites dare police, says 'We'll continue to march on the streets of Abuja'
Friday Ajagunna
Shi'ite protesters in Abuja
Members of Islamic Movement in Nigeria (IMN) have vowed to continue their daily protest on the streets of the federal capital territory (FCT) until Ibrahim Zakzaky, their leader, is freed.
Last week, the police had restricted all protests in Abuja to the Unity Fountain following clashes between the sect and security agencies. But the Shi'ites dared the police to be ready to kill more of their members who will be out on the streets.
Ibrahim Musa, a member of IMN who spoke to journalists in Abuja accused the federal government of refusing to obey the court order which freed Zakzaky and restored his fundamental human rights.
"Why is government hunting somebody that is out to fight for his or her right? You have killed more than a thousand of his (Zakzaky's) followers and put them in a mass grave. You keep on violating the law of the land and you think nobody should come out and challenge you? That is bad. This shows that the government of is oppressing its citizens. And you do not expect me to cross my legs and allow President Muhammadu Buhari to kill my leader," he said.
"The protest will continue, and if they do not want to see us on the street, government should free our leader. If they want to kill, they should go ahead and kill. We are ready to die for our leader. Failure to free our leader, the government should get ready to kill more of us.
"The protests on the streets of Abuja are as a result of the government's disregard for the judgment of a court of competent jurisdiction, the Federal High Court Abuja, which freed the revered leader of the movement from an illegal detention described by the government as 'protective custody'."
Musa also dismissed the claim by the federal government that it is not involved in the ongoing trial of Zakzaky.
The IMN leader has been in the custody of the Department of State Services (DSS) since December 2015.
Musa said Buhari's administration cannot be exonerated from the "attacks of the armed forces on members of the group".
"It is not possible under every logical reasoning to exonerate President Buhari from the genocide perpetrated by the Nigerian Army under his command as the Commander-in-Chief, where over 1,000 unarmed members of the movement, including men, women and children, were killed from December 12 to 14, 2015," he said.
"The government has declared illegally burying 347 members of the movement in a mass grave at Mando Village in Kaduna after the mass murder, in violation of the Geneva Convention and every known law in the country.
"The movement has never been violent or taken over streets, as falsely claimed by the presidency. The Islamic Movement has been in existence for about 40 years. It is on record that the movement has been peaceful and has never resorted to violence.
"It is only when government sends its police or Army to attack the peaceful processions that fracas sets in, courtesy of the security agents planted in the movement. We once again dissociate ourselves from any act of brigandage during our protests on the streets."
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General F-16 News
F-16 Unit News
F-16 Conflicts
F-16 Industry News
F-16 Mishap News
F-16 Fighting Falcon News
Lockheed Martin sets sale of 50 F-16s to Greece
March 9, 2000 (by Lieven Dewitte) - The governments of Greece and the United States signed a Letter of Offer and Acceptance (LoA) today providing for Greece's purchase of 50 new Lockheed Martin F-16 aircraft. The agreement reaffirms Greece's selection of the upgraded F-16 block 50+ as its new fighter in spring of 1999.
The F-16 block 50+ was selected after a long and intense international competition that included the F-15E, Mirage 2000, Eurofighter and Sukhoi 27.
The total sales package including the aircraft, mission equipment and support provisions is worth about $2.1 billion to Lockheed Martin and various suppliers. The agreement includes an option for Greece to purchase 10 additional F-16 aircraft.
The current purchase by the Hellenic Air Force is the third acquisition of the F-16 aircraft by the Government of Greece. The first was in 1986 while the second buy was in 1992. The previous procurements were for 40 aircraft.
In the current order, Greece is acquiring an advanced version of the F-16 with new equipment, including conformal fuel tanks for extended aircraft range and an upgraded version of the Northrop Grumman F-16 radar, designated the APG-68(V)XM. Final engine selection between the General Electric F110-GE-129 and the Pratt & Whitney F100-PW-229 will be made within the next 45 days.
Deliveries of the new aircraft will begin approximately 30 months after LoA signature and continue about two years. The sale will extend F-16 production at Lockheed Martin's Fort Worth, Texas, facility well into the 21st century, when combined with the additional F-16s on order for Israel, Egypt and the United States Air Force and the recently signed contract with the United Arab Emirates. Other countries are also considering new orders.
F-16 Fighting Falcon news archive
Block 50/52
Egyptian Air Force
Israel Air Force


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Northowram Surgery 01422 206121 Hipperholme Surgery 01422 205154
The Northolme Practice
Right Care Consultation
Blood Tests and Test Results
Statement of Purpose Kos Clinic
Statement of Purpose Northowram Surgery
Conditions A to Z
BBC Health News
Covid vaccine: WHO warns of 'catastrophic moral failure'18 Jan 2021 23:15It accuses richer nations of a "me-first" approach to Covid vaccine, and hoarding precious supplies.
Covid-19: 'Don't end up like us, please'18 Jan 2021 18:05Martin Freeborn's wife, Helen, lost her fight against Covid at the Royal London Hospital, where staff are struggling to cope with rising admissions.
Military medics to help in Shropshire hospitals due to staff shortages19 Jan 2021 05:40Sickness levels have risen at Shropshire's hospitals, with staff also shielding or self-isolating.
Oxford research tackles threat of antibiotic resistance19 Jan 2021 01:55Oxford says the "public cannot get enough of experts" with a £100m donation for antibiotics research.
Covid-19: Vaccine rollout extended to over-70s in England18 Jan 2021 16:15More than five million more people will be offered a jab as 10 new mass vaccination hubs open in England.
Covid: Brazil approves and rolls out AstraZeneca and Sinovac vaccines18 Jan 2021 10:35Brazil has the world's second-highest Covid death toll but has seen delay and discord over vaccines.
Kos Clinic Northowram
Monday 07:30 - 18:30 07:30 - 18:00
Tuesday 07:30 - 18:30 07:30 - 18:00
Wednesday 07:30 - 18:30 07:30 - 18:00
Thursday 07:30 - 18:30 07:30 - 18:00
Friday 07:30 - 18:30 07:30 - 18:00
Appointments are available at both surgeries between 7:30am and 8:00am Monday to Friday. These extended hours appointments are for patients who cannot attend during normal surgery hours, and not for people to walk in and expect to be seen.
During the extended hours (7:30am - 8:00am) there is no telephone access to the practice and you will not be able to pick up prescriptions or letters etc.
SURGERY CLOSED ON THE FOLLOWING AFTERNOONS FOR TRAINING
Wednesday 12th February 2020 Surgery closes at 12.30pm
Wednesday 29th April 2020 Surgery closes at 12.30pm
Wednesday 9th September 2020 Surgery closes at 12.30pm
Wednesday 11th November 2020 Surgery closes at 12.30pm
SURGERY CLOSED ALL DAY - BANK HOLIDAYS
Friday 10th April 2020 Good Friday
Monday 13th April 2020 Easter Monday
Friday 8th May 2020 Early May bank holiday (VE Day)
Monday 25th May 2020 Spring bank holiday
Monday 31st August 2020 Summer bank holiday
Friday 25th December 2020 Christmas Day
Monday 28th December 2020 Boxing Day (substitute day)
Friday 1st January 2021 New Year's Day
Northowram Surgery, Northowram Green, Northowram, Halifax , HX3 7JE
Hipperholme Surgery
Roydlands Street, Hipperholme, Halifax, HX3 8AF
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Pocket doors, a curbless shower and high end finishes are the focal points of this bathroom remodel.
Kathy's master bath was outdated and lacked natural light. The swinging doors for the closet and bath took up valuable bedroom space. By installing pocket doors, a larger window in the bath and a curbless shower we made the bathroom much more functional and Kathy's choice of high end materials and fixtures added style and beauty to the room.