text:
\section{Introduction}

The planar 1-loop MHV amplitude of $\mathcal{N}=4$ super Yang-Mills (sYM) has proved to be one of the most important multipoint amplitudes.  It was first computed in \cite{Bern:1994zx} via unitarity methods as a sum of box functions of kinematic invariants, expressible as a sum of dilogs and infrared divergent terms.  It now provides a reference point for subsequent methods and techniques. Following Witten's discovery of twistor-string theory \cite{Witten:2003nn}, it was found that sYM amplitudes could be computed using the MHV diagram formalism (where Maximal Helicity Violating, or MHV, tree amplitudes are used as the Feynman vertices for constructing all other amplitudes \cite{Cachazo:2004kj,Risager:2005vk}).  The calculation of the planar 1-loop MHV amplitude based on the MHV diagram formalism in \cite{Brandhuber:2004yw} was the first clear indication that  this would work beyond tree level.  Subsequently, planar amplitudes and correlation functions of $\mathcal{N}=4$ sYM were found to have many more remarkable properties. In addition to superconformal symmetry, they also have dual superconformal symmetry \cite{Dualc,Drummond:2007aua,Brandhuber:2008pf,Drummond:2008vq} giving rise to Yangian symmetry \cite{Dolan:2004ps,Drummond:2009fd}. Dual superconformal symmetry stems from a duality between scattering amplitudes and null polygonal Wilson loops \cite{Alday:2007hr,Drummond:2007cf, Mason:2010yk,CaronHuot:2010ek,Bullimore:2011ni}; the dual superconformal symmetry for the amplitude is the ordinary superconformal symmetry for the Wilson-loop\footnote{There is furthermore a Wilson loop/correlator duality \cite{Alday:2010zy,Eden:2010zz,Eden:2010ce,Eden:2011yp,Eden:2011ku,Adamo:2011dq}.}.  A key early example of this duality was the computation of the 1-loop contribution to the null-polygonal Wilson loop \cite{Brandhuber:2007yx}, which gave the 1-loop MHV amplitude stripped of its  Parke-Taylor tree prefactor precisely.

Further insights emerge when gauge theories are formulated in twistor space. In an axial gauge, the Feynman vertices of the twistor action correspond to MHV tree amplitudes and can be used to generate other amplitudes and correlation functions, providing the origin for the MHV diagram formalism   \cite{Mason:2005zm,Boels:2006ir,Boels:2007qn,Bullimore:2010pj,Mason:2010yk,Adamo:2011cb}. Furthermore, the momentum twistor space that makes dual conformal invariance manifest \cite{Hodges:2009hk} is the twistor space for the null polygonal Wilson loop dual to the scattering amplitudes. This Wilson loop can be re-expressed in momentum twistor space as a holomorphic Wilson-loop \cite{Mason:2010yk} that can also be computed by MHV diagrams.  The amplitude/Wilson loop duality is then simply realized as planar duality for the corresponding MHV-diagrams, which are the the twistor space Feynman diagrams in an axial gauge.  In approaches based on MHV diagrams, the axial gauge breaks dual conformal symmetry by virtue of a choice of reference twistor $Z_*$ that determines the direction in which the twistor space gauge field is trivialized. The loop integrands constructed from MHV diagrams are otherwise dual conformal invariant and manifest the appropriate permutation symmetries.  See \cite{Adamo:2011pv} for a review of these twistor related developments.

Recently it was discovered that for certain gauge theory scattering amplitudes and correlation functions, the loop integrands can be expressed in $d\,$log form.  More precisely, for MHV amplitudes, the loop integrand is a product of exterior derivatives of logarithms of rational functions and for higher MHV degree, the loop integrand consists of $d\, \log$ s multiplied by delta functions. Indeed, the Feynman rules for the twistor holomorphic Wilson loop can be rewritten in $d\, \log$  form \cite{Lipstein:2012vs}. These Feynman rules are very similar to those for the amplitude in twistor space as described in \cite{Adamo:2011cb} as they have the same origin in the twistor action. Furthermore, many of these ideas are are not restricted to planarity or maximal supersymmetry, so the $d\, $log form is likely to apply more generally.

From the point of view of the twistor Wilson loop, the $d\,$log form of loop integrands of planar $\mathcal{N}=4$ sYM amplitudes  has a simple geometric interpretation:  the rational functions that are arguments of the $d\, $logs correspond to insertion points of propagators on MHV vertices or edges of the Wilson loop.  The external data of the scattering amplitude is encoded in the integration contours, which are given by reality conditions on the line determining the loop momentum in twistor space. An alternative $d\, \log$  form was obtained in \cite{ArkaniHamed:2012nw} from BCFW recursion using on-shell diagrams and the Grassmannan integral formula\footnote{ BCFW recursion relations (which relate higher point on-shell amplitudes to lower-point on-shell amplitudes \cite{Britto:2004ap,Britto:2005fq,ArkaniHamed:2010kv})  can also be simply realised in twistor space \cite{Mason:2009sa,ArkaniHamed:2009si}, expressing full superconformal invariance.  This  led to a formula that generates the amplitudes and leading singularities of $\mathcal{N}=4$ sYM via a contour integral over a Grassmannian \cite{ArkaniHamed:2009dn}.  There is a parallel Grassmannian dual conformal invariant formula for the Wilson loop \cite{Mason:2009qx}, and the translation between the two expresses the Yangian symmetry  in this framework \cite{ArkaniHamed:2009vw}.  BCFW recursion was extended to generate the loop integrand in \cite{ArkaniHamed:2010kv,Boels:2010nw}.    }. In the alternative approach, the rational functions in the $d\,$log form correspond to BCFW shifts which arise from an on-shell diagram formulation of  the all-loop recursion relations.  In both approaches, it is clear that the $d\,$log form of the loop integrand is not really tied to planarity, nor to maximal supersymmetry, but in this paper we nevertheless focus on these most basic and simplest examples.

The loop integrand is not however the main objective; one is really after fully integrated amplitudes and correlation functions.  The attraction of the $d\, $log form of the integrand is that it is very suggestive for direct integration without Feynman parameters. In this form, the integrand is locally exact so it should be possible to iteratively use Stokes theorem to reduce integration to combinatorics. The integrals are still nontrivial to perform, however, since one must take into account the contour of integration. Indeed, this is where the kinematic data is encoded. Furthermore, the singularities made explicit in the $d\,$log form mean that the answer very much depends on the contour. In our formulation, the integrand of an $L$-loop MHV amplitude consists of $d\,$log's of $4L$ variables, and half of these are constrained to be real by the reality constraints in momentum twistor space.  For a complex variable $s$, $d\, \log s=d s/s$ is integrable and well-defined, but for $s$ real, the integral of $d \log s$  is not well-defined unless regulated by some $i\epsilon$ prescription. We will see that these real integration variables are directly related to physical propagators in space-time and so we must use the Feynman $i\epsilon$ prescription to make these real integrals well defined.  Another puzzle is that the degree of transcendentality for an $L$-loop amplitude is expected to be $2L$, whereas naively the integral of an expression in $4L$ $d\, $logs would usually define a polylogarithm expression of transcendentality degree $4L$.  The resolution of this is that the usual definition of $Li_k$, which has of transcentality degree $k$, involves $k$ iterated {\em indefinite} integrals, but our integral is compact with no boundary.  We will see that in the integration procedure, half the $d\,$logs are used to restrict the contour to one that is a product of $2L$ intervals with the remaining integrand in $d\,$log form as required for transcendality degree $2L$.

Here we develop a systematic method for evaluating the integrals of loop amplitudes from the $d\,$log form of their integrands.  We focus on the 1-loop MHV amplitude, for which we give a complete treatment. In this case there is only one contributing MHV-diagram $K_{ij}$, where $i,j=1,\ldots,n$ index the external legs of the amplitude,  that has become known as Kermit\footnote{because of the diagram's resemblance to a puppet frog of the same name.}.  The integrand for $K_{ij}$ is the same as the integrand for the 1-loop MHV amplitude obtained by \cite{ArkaniHamed:2010kv} via BCFW recursion, although this equivalence between MHV-diagrams and BCFW expressions is a coincidence that does not persist to higher loop orders or MHV degree.
The full planar 1-loop MHV amplitude is given by
\begin{equation}\label{MHV0}
\cA(1,\ldots,n)=\frac12 \sum_{i,j}K_{ij}.
\end{equation}
  In the generic case  $ |i-j|\geq 2 $,
our new formulae for $K_{ij}$ are
\begin{equation}
K_{ij}=\Li_{2}\left(\frac{a_{i\, j}}{v_{*}}\right)+\Li_{2}\left(\frac{a_{i-1\, j-1}}{v_{*}}\right)-\Li_{2}\left(\frac{a_{i-1\, j}}{v_{*}}\right)-\Li_{2}\left(\frac{a_{i\, j-1}}{v_{*}}\right)
 + c.c.\
\label{kermitgeneric}
\end{equation}
where $a_{ij}=\la i|x_{ij}\left|j\right]/(\left\langle i \bar{\eta}\right\rangle \left[\eta j\right])=i Z_i\cdot\bar Z_j $, $v_*= x_{ij}^2/2\la \bar{\eta}|x_{ij}|\eta]$, and $\eta$ is the reference spinor. We have normalized the external twistors so that $\la i \bar{\eta}\ra=Z_i\cdot \bar Z_*=1$. If we use \eqref{eq:vstar} for $v_*$, we see that this has manifest (dual) conformal symmetry up to the choice of a reference twistor $Z_*$ (which encodes the reference spinor). Dual conformal invariance is broken only in the divergent diagrams when $|i-j| = 1$.   We regulate this case by taking $i \epsilon \rightarrow i \epsilon - m^2$ when implementing the $i \epsilon$ prescription, where $m^2$ is the mass parameter  for a mass regulation. We find that
\begin{equation}
K_{ii+1}=-\frac{1}{4}\left(\ln^{2}\left(\frac{m^{2}}{x_{ii+2}^{2}}\right)+\ln\left(\frac{x_{i-1i+1}^{2}}{x_{ii+2}^{2}}\right)\ln\left(x_{ii+2}^{2}\right)\right)-\frac{2\pi^2}{3}+\mathcal{O}(m).
\label{kermitdivergent}
\end{equation}
Finally, when $i=j$, Kermit vanishes $K_{ii}=0$. This result has the standard divergent behaviour.

Our results for the planar 1-loop MHV amplitude of $\mathcal{N}=4$
 sYM are nontrivially related to previous formulae. The formulae
 obtained in \cite{Bern:1994zx} expressed the answer in terms
of a sum of two mass easy box functions, each a sum of 5 dilogs. In \cite{Brandhuber:2004yw,Brandhuber:2007yx}  the two-mass easy box function was reduced to a simpler form containing just four dilogarithms via a 9 term dilog identity.  Although the analysis in \cite{Brandhuber:2004yw} verified the applicability of the MHV-diagram formalism for these loop amplitudes, it was not possible in that analysis to derive an expression for a single Kermit diagram since different choices of reference spinor were made for different cuts contributing to a single Kermit. Indeed, the results in \cite{Brandhuber:2004yw} do not have manifest dual conformal invariance and non-trivial dilog identities must be used to relate our formulae to theirs as we see in \ref{BSTa}. In appendix \ref{massdetails}, we verify that \eqref{kermitdivergent} reproduces the form of the 1-loop 4-point MHV amplitude obtained in \cite{Alday:2009zm}.

The method for obtaining the 1-loop MHV amplitude described here is substantially simpler than previous methods. We expect that the techniques developed in this paper will extend to higher-loop scattering amplitudes and correlation functions in $\mathcal{N}=4$ sYM as well as to other field theories.  As far as the higher-loop MHV amplitude is concerned, our argument in \cite{Lipstein:2012vs} for reduction of an $L$-loop MHV integrand into $d\, $log form first showed that it could be built essentially from $L$ Kermits, albeit with some as arguments of others.  We illustrate this  in section \ref{higherloops}, where we sketch the evaluation of a nontrivial 2-loop diagram.

This paper is organized as follows. In section \ref{background}, we review momentum twistors and the $d\, \log$  form of the planar 1-loop MHV amplitude of $\mathcal{N}=4$ sYM. In section \ref{generic1}, we we explain how to implement the Feynman $i \epsilon$ prescription and present the result for generic generic Wilson loop diagrams contributing to the 1-loop MHV amplitude. In section \ref{divergent}, we describe how to regulate divergent Wilson loop diagrams using mass-regularization. In section \ref{higherloops}, we explain how our methods can be applied to higher loop planar amplitudes, and illustrate this by giving a preliminary discussion of a nontrivial two-loop example. In section \ref{conclusion}, we present some conclusions and describe some future directions. In appendix \ref{BSTa}, we show that our result for the generic Wilson-loop diagram is equivalent to the result for the 1-loop MHV amplitude previously obtained in \cite{Brandhuber:2004yw} using unitarity methods applied to the MHV diagram formalism. In appendix \ref{massdetails} we provide more details about mass-regularization of divergent Wilson loop diagrams. In appendix \ref{symbol}, we reduce the symbol of the 1-loop MHV amplitude to a sum over terms that consist of the ingredients in Kermit as used in \S \ref{generic1} to show that our generic term gives rise to the correct symbol up to terms that cancel telescopically.

\section{Background} \label{background}

In this section, we review the basic definitions and set up the notation for the rest of the paper. First we review variables which are useful for computing planar on-shell scattering amplitudes in $\mathcal{N}=4$ sYM, notably region momenta and momentum twistors. Then we review the 1-loop MHV amplitude in $\mathcal{N}=4$ sYM, in particular the $d\, \log$  form of its loop integrand.

\subsection{Momentum Twistors}

We first write on-shell momenta in two-component spinor form as
follows: \[
p^{\alpha\dot{\alpha}}=\lambda^{\alpha}\tilde{\lambda}^{\dot{\alpha}}\]
where $\alpha=0,1$ and $\dot{\alpha}=\dot{0},\dot{1}$ are chiral
and antichiral spinor indices. The particles
also have fermionic supermomentum $$q^{\alpha a}=\lambda^\alpha \rho^a,$$ where $a$ is an $SU(4)$ R-symmetry
index and $\rho$ is fermionic.
The $n$-point superamplitudes are then functions of $n$ such $(\lambda,\rho)$ variables, one for each external particle. For example, a tree-level $n$-point MHV superamplitude has the following simple form
\begin{equation}
A_{n}^{MHV}=\frac{\delta^{4}\left(P\right)\delta^{8}\left(Q\right)}{\left\langle 12\right\rangle \left\langle 23\right\rangle ...\left\langle n1\right\rangle }
\label{MHV}
\end{equation}
where $P=\sum_{i=1}^{n}p_{i}$, $Q=\sum_{i=1}^{n}q_{i}$, and $\left\langle ij\right\rangle =\epsilon_{\alpha\beta}\lambda_{i}^{\alpha}\lambda_{j}^{\beta}$.

Dual superconformal symmetry for planar $\mathcal{N}=4$ sYM amplitudes can be seen by
arranging the external supermomenta of a colour ordered amplitude into a polygon
and writing the amplitude as a function of the vertices of this polygon, which lives in `region momentum space'. Dual superconformal symmetry then corresponds to ordinary conformal symmetry
in region momentum space. The region momentum space coordinates
are defined by
\begin{equation}
\left(x_{i}-x_{i+1}\right)^{\dot{\alpha}\alpha}=\lambda_{i}^{\alpha}\tilde{\lambda}_{i}^{\dot{\alpha}},\,\,\,\left(\theta_{i}-\theta_{i+1}\right)^{a\alpha}=\lambda_{i}^{\alpha}\rho_{i}^{a}.\label{eq:dual}\end{equation}
This representation automatically incorporates momentum conservation. The duality between amplitudes and Wilson-loops is the equivalence of the planar scattering amplitude with the planar Wilson-loop around this polygon.

The dual superconformal symmetry of the amplitudes can be made more manifest
by writing the polygon in terms of (momentum) supertwistors:
\[
\left(Z_{i}^{A},\chi_{i}^{a}\right)=\left(\lambda_{i\alpha},\mu_{i}^{\dot{\alpha}},\chi_{i}^{a}\right).\]
They transform in the fundamental representation of the dual superconformal group $SU(2,2|4)$ and relate to the region supermomenta by the
`incidence relations' \begin{equation}
\mu_{i}^{\dot{\alpha}}=-ix_{i}^{\dot{\alpha}\alpha}\lambda_{i\alpha},\,\,\,\chi_{i}^{a}=-i\theta_{i}^{a\alpha}\lambda_{i\alpha}.\label{eq:incidence}\end{equation}
These express a point $(x,\theta)$ in space-time as a complex projective line $X$ in twistor space, i.e. the point  $x_i$ corresponds to the line $X_i$ passing through both $Z_i$ and $Z_{i-1}$.
Such a projective line in
twistor space can be represented as a skew twistor
\begin{equation}
X_{i}^{AB}=\frac{Z_{i}^{[A}Z_{i-1}^{B]}}{\left\langle ii-1\right\rangle }
\label{bigX1}
\end{equation}
where we have normalized using the spinor brackets in the denominator.  Although the skew twistor is conformally invariant up to scale, its normalisation is not and requires the knowledge of the `infinity twistor' $I_{AB}$ defined by
\[
\left\langle ij\right\rangle =I_{AB}Z_{i}^{A}Z_{j}^{B},\,\,\, I_{AB}=\left(\begin{array}{cc}
\epsilon^{\alpha\beta} & 0\\
0 & 0\end{array}\right).\]
Given this, setting $x_{ij}=x_i-x_j$, the distance between two points in the dual space can be written in
terms of momentum twistors as follows:
\[
x_{ij}^2:=\left(x_{i}-x_{j}\right)^{2}=\frac{\left( ii-1jj-1\right) }{\left\langle ii-1\right\rangle \left\langle jj-1\right\rangle },\,\,\,\left( ijkl\right) =\epsilon_{ABCD}Z_i^{A}Z_j^{B}Z_k^{C}Z_l^{D}.\]

If the spacetime has Lorentzian signature, the complex conjugate of
a twistor $Z^{A}$ is a dual twistor given by $\bar{Z}_{A}=\left(\bar{\mu}^{\alpha},\bar{\lambda}_{\dot{\alpha}}\right)$.  The reality condition that the polygon lies in real Minkowski space is that
$$
\bar Z_{iA}=\alpha\varepsilon_{ABCD}Z_{i-1}^BZ_i^CZ_{i+1}^D\, ,
$$
for some $\alpha\neq 0$.

In the MHV diagram formalism, we also have a reference spinor $\eta^{\dot \alpha}$ which, in momentum twistors we incorporate as a reference twistor
$$
Z_*=(0,\eta^{\dot\alpha})\, .
$$
Ordinarily the twistors $Z_i$ do not have a natural choice of scaling (just as $\lambda_i$ does not).  However, we will find it convenient in what follows to normalize all the $\lambda_i$ and hence $Z_i$ so that
$$
\la \lambda_i\,\bar \eta\ra=Z_i\cdot \bar Z_*=1\, .
$$
With this, we can now form the invariants
\begin{equation}
a_{ij}:=iZ_i\cdot\bar Z_j =\frac{\left\langle i\right|x_{ij}\left|j\right ] }{\left\langle i \bar{\eta}\right\rangle \left[\eta j\right]},
\label{aijs}
\end{equation}
where we have included the factors in the denominator in order to indicate our normalization of the external twistors. For simplicity, we will set these normalization factors to 1 in the remainder of the paper. These invariants are, up to normalisation, the same as the invariants $(i\, j-1\, j\, j+1)$ used elsewhere, e.g.\ \cite{CaronHuot:2011ky,Golden:2013lha}. Note that they differ by a factor of $i$ from those used in \cite{Lipstein:2012vs}. These depend also on the reference twistor, and only those combinations that are independent of the choice of the scalings of the $Z_i$ are fully (dual-) conformal invariant. However, we cannot expect to obtain quantities that are independent of the  reference twistor from MHV diagrams in the first instance, so the $a_{ij}$ are a natural set of kinematic variables for us.

\subsection{The 1-loop MHV Amplitude}

In the MHV diagram formalism, loop amplitudes in $\mathcal{N}=4$ sYM can be computed using the tree-level MHV amplitudes \eqref{MHV} as the Feynman vertices with scalar propagators. The diagrams for the planar 1-loop MHV amplitude are obtained by connecting two MHV vertices by two propagators, and a generic example is depicted in Figure \ref{1lmhv}. The external region momenta  as defined by \eqref{eq:dual} are supplemented by $x_0$ for the loop momentum and the regions $(x_i,x_j)$, which are adjacent to the two propagators. The full planar 1-loop MHV amplitude is given by summing over all $(i,j)$ with $i\neq j$. As for an amplitude, a vertex must have at least three legs.  The MHV vertices  \eqref{MHV} are extended off-shell so that the off-shell momentum $x_{0i}$ corresponds to the spinor $[\eta |x_{0i}$ and so on.

\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.18]{mhv1loop.jpg}
\caption{1-loop MHV diagram in the CSW formalism.}
\label{1lmhv}
\end{center}
\end{figure}

After stripping off the MHV tree amplitude, the diagram in Figure \ref{1lmhv} is given by \cite{Brandhuber:2004yw}
\begin{equation}
K_{ij}=\int  \frac {\la i-1\, i\ra \la j-1\, j\ra \rd^4 x_0}{x_{0i}^2 x_{0j}^2 \la i-1|x_{0i}|\eta] \la i|x_{0i}|\eta] \la j-1|x_{0j}|\eta]\la j|x_{0j}|\eta]}
\label{kermitr}
\end{equation}
where $\eta$ is the reference spinor and $x_{ij}=x_i - x_j$. The object $K_{ij}$ is sometimes referred to as Kermit. In terms of momentum twistors, Kermit is given by \cite{Bullimore:2010pj}
\begin{equation}
\int \frac{\rd^{4|4} Z_A\rd^{4|4}Z_B  \left(\left(*i-1i\left[A\right.\right)\left(\left.B\right]j-1j*\right)\right)^{2}}{(\mbox{Vol GL}_2 )(A\, B \, i-1\, i)(A\, B\, j-1\, j)(A\, B\, *\, i-1)(A\, B\, *\, i)(A\, B\, *\, j-1)(A\, B\, *\, j)}
\label{kermittwist}
\end{equation}
where the reference spinor $\eta$ has been embeded in the reference twistor
$
Z_*=(0,\eta^{\dot \alpha})\, ,
$
 the loop region momentum $x_0$ corresponds to the line spanned by the twistors $(Z_A,Z_B)$, and the $GL_2$ is associated with the choice of $(Z_A,Z_B)$ from within their span and is understood to be fixed by a standard Fadeev-Popov procedure.  The singularity structure is given by the intersection of the line $X_0$ with the solid lines in twistor space depicted in figure \ref{1lmhv-kermit}  (the wiggly lines correspond to the numerator factors) which has a superficial similarity to Kermit the frog. The integrand in \eqref{kermittwist} is the same as the integrand for the 1-loop MHV amplitude obtained using all-loop BCFW recursion in \cite{ArkaniHamed:2010kv}. In that reference, the role of the reference twistor is played by $Z_1$.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.8]{kermit-pic.pdf}
\caption{Kermit diagram for 1-loop MHV diagram.}
\label{1lmhv-kermit}
\end{center}
\end{figure}

The $d\, $log form of the integrand for Kermit as given in \cite{Lipstein:2012vs} is most easily expressed in terms of the holomorphic Wilson loop in twistor space that is dual to the planar S-matrix of $\mathcal{N}=4$ sYM. In particular, the arguments of the $d\, $logs are the coordinates corresponding to insertion points of  propagators onto MHV vertices or edges of the Wilson loop. We now briefly review how this arises for Kermit, see \cite{Lipstein:2012vs} for further details.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.14]{1loop.jpg}
\caption{A generic Wilson loop diagram contributing to the 1-loop MHV amplitude.}
\label{fig1loop}
\end{center}
\end{figure}
The twistor Wilson loop Feynman diagram that corresponds to Kermit is given in Figure \ref{fig1loop}. It is related to Figure \ref{1lmhv} by planar duality, which exchanges loops in the amplitude diagram with MHV vertices in the Wilson loop diagram, as depicted in Figure \ref{duality}. Hence the loop region momentum $x_0$ in Figure \ref{1lmhv} corresponds to the line MHV vertex in Figure \ref{fig1loop} supported on the line in twistor space spanned by the twistors $(Z_A,Z_B)$.  The wavy lines in Figure \ref{fig1loop} correspond to propagators in twistor space\footnote{More generally, a twistor Wilson loop diagram with $L$ MHV vertices will correspond to an $L$-loop amplitude. In general, a twistor Wilson loop diagram with $P$ propagators and $L$ MHV vertices, corresponds to an $L$-loop $N^k$MHV amplitude, where $k=P-2L$.}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.8]{1loopMHVST.pdf}
\caption{Amplitude/Wilson-loop duality for 1-loop MHV diagram.}
\label{duality}
\end{center}
\end{figure}

Propagators in the twistor Wilson loop diagram correspond to delta functions
$$
\Delta(Z,Z'):=\frac1{2\pi i}\bar\delta^{2|4}(Z,Z_*,Z'):=\frac1{2\pi i}\int_{\C^2} \frac{\rd u}{u} \frac{\rd v}{v}\bar\delta^{4|4}(Z+u Z_* + vZ')
$$
in twistor space which is essentially a delta function restricting $Z$ and $Z'$ to lie along a line through $Z_*$ and a Cauchy pole along that line when $Z$ meets $Z'$. As explained in \cite{Lipstein:2012vs}, it is possible to define the parameters in the propagators and fix the $GL_2$ freedom so that on integrating out the $Z_A$ and $Z_B$, the delta functions enforce  $Z_A$ and $Z_B$ to be
\begin{equation}
Z_A=is_0Z_* +  \frac {Z_{i-1} +s Z_{i}}{1+s} \, , \qquad Z_B=it_0 Z_* +  \frac{ Z_{j-1} +t Z_{j}}{1+t}\,
\label{zazb}
\end{equation}
where $(s_0,t_0,s,t)$ are rational functions of the loop momentum that can be used as alternative integration variables for the twistor Wilson loop diagram. In terms of $(s_0,t_0,s,t)$, the Kermit integral reduces to
\begin{equation}{}
K_{ij}=-\frac{1}{4\pi ^2}\int
 \rd \ln{s_0}\,\rd \ln{t_0} \, \rd \ln{s} \,\rd \ln{t}\, .
\label{dlog}
\end{equation}
Now the information of the external twistors is encoded into the
contour, i.e., the condition that the line joining $Z_A$ to $Z_B$ is real.
We  normalize the external twistors so that $$Z_i\cdot \bar Z_*=1\, .$$
With this, the reality conditions are
\begin{equation}\label{real0}
0= Z_A\cdot \bar Z_A= i(s_0-\bar s_0)\, ,  \qquad 0=Z_B\cdot \bar Z_B=i(t_0-\bar t_0)\, ,
\end{equation}
and
\begin{equation}\label{real1}
 0=Z_A\cdot \bar Z_B= i(s_0 - \bar t_0) -i \frac{  a_{i-1\, j-1} + s
   a_{i\, j-1} +\bar t a_{i-1\, j} + s\bar t a_{ij}}{(1+s)(1+\bar t)}\,
\end{equation}
where we have set $a_{ij}=iZ_i\cdot\bar Z_j$. A more explicit formula for these invariants is given in \eqref{aijs}.
Thus the reality conditions imply that $s_0$ and $t_0$ are real and express $s$ as a Mobius transform of $\bar t$, which depends on $v= s_0-t_0$:
\begin{equation}
s=-\frac{\bar{t}\left(a_{i-1j}-v\right)+a_{i-1j-1}-v}{\bar{t}\left(a_{i\, j}-v\right)+a_{ij-1}-v}\, .
\label{eq:real}\end{equation}

The $(s_0,t_0,s,t)$ can be expressed as functions of $(x_0, x_i, x_j)$ as follows:
\begin{equation}\label{coord-def}
s_0 =\frac{- x_{0i}^2}{2[\eta|x_{0i}|\bar\eta\ra}\, , \quad t_0= \frac{- x_{0j}^2}{2[\eta|x_{0j}|\bar\eta\ra}\, , \quad s= \frac {\la i-1|x_{0i}|\eta]}{\la i|x_{0i}|\eta]}\, ,\quad t= \frac {\la j-1|x_{0j}|\eta]}{\la j|x_{0j}|\eta]}\, ,
\end{equation}
and these can be expressed dual conformal invariantly as
\begin{equation}
s_0 =\frac{(Z_AZ_B  Z_{i-1}Z_i)}{(Z_* Z_A Z_B  X_i \cdot \bar Z_*)}\, , \quad s= \frac {(Z_{i-1} Z_AZ_B  Z_*)}{(Z_ i Z_AZ_BZ_*)}\, , \quad \mbox{ and } \quad i, s \leftrightarrow j, t \, .
\end{equation}
These follow from the incidence relations, which correspond to
\begin{eqnarray}\nonumber
Z_A&=&(\lambda_A,-i x_0|\lambda_A\ra), \qquad Z_B=(\lambda_B,-ix_0 |\lambda_B\ra)\, ,
\\ Z_{i-1}&=&(\lambda_{i-1},-ix_i|i-1\ra)\, , \quad Z_i=(\lambda_i, -ix_i |i\ra)\, ,
\end{eqnarray}
and $i\leftrightarrow j$, together with $Z_*=(0,\eta)$ and $\la \lambda_i\, \bar\eta\ra=1$.  Writing \eqref{zazb} in  spinor parts gives
\begin{equation}\label{incidence}
\lambda_A= \frac{\lambda_{i-1}+s\lambda_i}{1+s} \, \quad \lambda_B=\frac{\lambda_{j-1}+t\lambda_j}{1+t} \, , \quad x_{0i}|\lambda_A\ra =-s_0 \eta\, , \quad x_{0j}|\lambda_B\ra=-t_0\eta.
\end{equation}
Multiplying the last two equations by $x_{0i}$ and $x_{0j}$ respectively and taking various components etc., leads to \eqref{coord-def}.

The $d\, \log$  form of the loop integrand in \eqref{dlog} is  a coordinate transformation of the original Kermit integrand in \eqref{kermitr}.
However, clearly $(s_0,t_0,s,t)$ are redundant coordinates for the real slice, so we will eliminate $s$ in favour of $t$ using the reality condition in \eqref{eq:real}.  We can then express $x_{0j}$ in terms of $(s_0,t_0,t)$ directly   by
\begin{equation}\label{t-coords}
x_{0j}= \frac { x_{ij}^2/2-v x_{ij}\cdot n }{[\bar \lambda_B|x_{ij}|\lambda_B\ra -v} \bar \lambda_B\lambda_B + t_0 n\, , \quad \mbox{ where } \quad n=\bar \eta \eta\, , \quad v=s_0-t_0.
\end{equation}
where $\lambda_B$ is given in terms of $t$, $\lambda_{j-1}$ and $\lambda_j$ in \eqref{incidence} above.  This is obtained by solving  the second and fourth equation in \eqref{incidence}, and the difference of the first two equations of \eqref{coord-def} for $x_{0j}$. Clearly a similar formula can be obtained in terms of $s$.

\section{Evaluating Kermit} \label{generic1}

The $d\, $log form  of the Kermit integrand is given in \eqref{dlog} with  contour \eqref{real0} and \eqref{eq:real}. The variables $s$ and $t$ are complex and so $\rd s/s$ and $\rd t/t$ are integrable and unambiguous.  However, $(s_0,t_0)$ are real so that the poles at $s_0,t_0=0,\infty$  must be regulated. From \eqref{coord-def}, we see that the $(s_0,t_0)$ poles are related to those of physical propagators in space-time. Hence, the Feynman $i\epsilon$ prescription for these poles will shift $(s_0,t_0)$ into the complex plane. After doing so, these real integrals will become well-defined.

\subsection{On the $i\epsilon$ prescription and the real integrals}

In terms of region momenta, the Feynman $i\epsilon$ prescription is simply
\begin{equation}{}
\frac 1{x_{0i}^2}\rightarrow \frac 1{x_{0i}^2+ i \epsilon }\, , \qquad \frac 1{x_{0j}^2}\rightarrow \frac 1{x_{0j}^2+ i \epsilon }\, .
\end{equation}
This will follow from \eqref{coord-def} if we take
\begin{equation}{}
s_0 \rightarrow s_0 + i\epsilon f_i , \quad t_0\rightarrow t_0 + i \epsilon  f_j
\qquad \mbox{ where } \qquad
f_i = \frac{-1}{2x_{0i}\cdot n} \, , \quad f_j=\frac {-1}{2x_{0j}\cdot n }\, .
\label{iepshift}
\end{equation}
In terms of the $(v,t)$ coordinates we obtain
\begin{equation} \label{fifj}
f_i=\frac {v-[\bar \lambda_B|x_{ij}|\lambda_B\ra}{x_{ij}^2-2x_{ij}\cdot n [\bar\lambda_B|x_{ij}|\lambda_B\ra } \, \qquad f_j= \frac {v-[\bar \lambda_B|x_{ij}|\lambda_B\ra}{x_{ij}^2-2v x_{ij}\cdot n }.
\end{equation}

A key point is that there is some decoupling in the integral because, from \eqref{t-coords} and \eqref{iepshift}, $f_i$ and $f_j$ depend only on $v=s_0-t_0$ and not $s_0+t_0$, so that  we can perform the $t_0$ integral while holding $v$ and $t$ constant.  Setting $s_0=v+t_0$ we are therefore left with an integral of the form
\begin{equation}
\int \frac {\rd t_0 \rd v}{(t_0+i\epsilon f_j) (v+t_0+ i\epsilon f_i)} F(v,t,\bar t)\rd t \rd \bar t\, ,
\end{equation}
where $F$, $f_i$ and $f_j$, do not depend on $t_0$.
If $f_i$ and $f_j$ have the same sign, then the contour can be contracted in the upper or lower half plane and the answer vanishes.  If they have opposite signs the contour integral picks up the residue at $- v$ or $0$ accordingly. The answer therefore reduces to
\begin{equation}
2\pi i \int_{-\infty}^\infty \frac{\sgn f_j \, \theta(-f_if_j) \, \rd v}{v + i\epsilon(f_i-f_j)} F(v,\ldots)\, ,
\label{uvint0}
\end{equation}
where $\theta$ is the Heavyside step function.

In coordinates in which $s$ is eliminated in favour of $t$,  $f_i$ and $f_j$ depend on $v$ and the step function  provide limits for the $v$ integration as follows.  Firstly, one can see immediately from their definitions that $f_i$ and $f_j$ are positive to the future of the hypersurfaces at $x_{0i}\cdot n=0$ and $x_{0j}\cdot n=0$ respectively and negative to their past.  Thus $\theta(-f_if_j)$ is supported in-between the two hypersurfaces and does not include infinity.  Thus, $f_j$ has a constant sign on the support of $\theta(-f_i f_j)$ and indeed its sign is that of $x_{ji}\cdot n$. This sign, being constant, can be taken out of the integrand.
It follows from \eqref{t-coords} that the hypersurfaces $x_{0i}\cdot n=0$ and $x_{0j}\cdot n=0$ give the following  limits on the $v$-integral
\begin{equation}
x_{0j}\cdot n= \frac {-v x_{ij}\cdot n + x_{ij}^2/2}{[\bar\lambda_B|x_{ij}|\lambda_B\ra-v}=0  \quad \mbox{ when } \quad v=v_*:= \frac{x_{ij}^2}{2x_{ij}\cdot n}
\end{equation}
and
 \begin{equation}
 x_{0i}\cdot n= x_{0j}\cdot n -x_{ij}\cdot n=0 \quad \mbox{ when } \quad v=\pm \infty\, .
 \end{equation}
 There is the possibility of a contribution from the pole at $v=0$.  However at $v=0$
 \begin{equation}
 x_{0i}\cdot n\, x_{0j}\cdot n= \frac{x_{ij}^2(x_{ij}^2-2x_{ij}\cdot n [\bar\lambda_B|x_{ij}|\lambda_B\ra)}{4 [\bar\lambda_B|x_{ij}|\lambda_B\ra^2} =-x_{ij}^2  \frac{|[\eta|x_{ij}|\lambda_B\ra|^2 }{2 [\bar\lambda_B|x_{ij}|\lambda_B\ra^2}
\end{equation}
 and this is positive in the Euclidean kinematic region usually considered, i.e., $x_{ij}^2<0$.  Thus $v=0$ lies outside the integration region and so does not contribute.

 We will see later that the integrand vanishes as $v\rightarrow \infty$, so no more prescriptions to regulate the real integral need to be made and we can now set $\epsilon=0$ (except for the divergent diagrams that will be considered separately).  Finally, noting that $\sgn \, v_*= \sgn \, x_{ij}\cdot n$ in our kinematic region, we see that   the
 real integral reduces to
\begin{equation}
 2\pi i \int_{v_*}^{ \infty }\frac{\, \rd v}{v} F(v,\ldots)\, ,\quad v_*>0 \qquad \mbox{ or } \quad    -2\pi i\int_{-\infty}^{ v_* }\frac{\, \rd v}{v} F(v,\ldots)\, , \quad v_*<0\, .
\label{uvint}
\end{equation}
We will take the former case for definiteness in what follows.

As a final remark in this subsection we note that critical value $v_*$ of $v$ can be expressed dual conformal invariantly, up to the choice of the reference twistor as
\begin{equation}
v_{*}=\frac{X_{i}\cdot X_{j}}{\bar{Z}_{*}\cdot X_{i}\cdot X_{j}\cdot Z_{*}}=\frac{a_{ij}a_{i-1j-1}-a_{ij-1}a_{i-1j}}{a_{ij} + a_{i-1j-1}-a_{i-1j}-a_{ij-1}}\, .\label{eq:vstar}\end{equation}
Thus we have implemented the standard Feynman $i\epsilon$ prescription without having to break dual conformal invariance.

\subsection{The complex integral} \label{toy}
Since the limits of the $v$ integral are independent of $t$, we can perform the $t$ integrals first and then finish with the $v$ integral in \eqref{uvint}.

The complex $(s,t)$ integrals are relatively straightforward  and we first give a toy example, first shown to us by Nima Arkani-Hamed, which we will use later:
\begin{lemma}
\begin{equation}{}
\int_\Gamma  \frac{\rd s}s\,  \frac{\rd t}t =4\pi i \log \left|\frac ba\right|\quad \mbox{where the contour $\Gamma$ is } \quad s=c\frac{\bar t -a}{\bar t -b}
\label{lem}
\end{equation}
\end{lemma}

This can be seen by expressing the integrand on the contour as the exterior derivative of
\begin{equation}{}
\log t \,\rd  \log \left( \frac{\bar t -a}{\bar t-b}  \right) =\log t\left( \frac 1{\bar t -a} -\frac 1{\bar t -b}\right )\, \rd \bar t\, \, .
\label{integrand}
\end{equation}
However, this form must have a cut from $t=0$ to $t=\infty$ and furthermore has poles at $\bar t=a$  and $b$.  So in order to use Stokes theorem, we must cut out an $\epsilon$-neighbourhood of the cut and the poles.  We can then use Stokes to reduce the integral to a contour integral around each pole and the cut.  The contour integral around the cut can then be reduced to a line integral along the cut as the contribution from the logarithmic singularities at the end vanishes as $\epsilon\rightarrow 0$, whereas the jump across the cut is $2\pi i$.  The contributions from the poles as $\epsilon\rightarrow 0$ is similarly given as $\mp 2\pi i\log t$ evaluated at the poles (noting the anti-holomorphic dependence on $t$) yielding
\begin{equation}
2\pi i\left(\int _0^\infty \rd \log \left( \frac{\bar t -a}{\bar t-b}  \right) - \log {\bar a} +\log {\bar b}\right)= -2\pi i\left(\log \frac{\bar a }{\bar b}+\log \frac {a}{b}\right)  =4\pi i\log \frac{|b|}{|a|} \, . \Box
\end{equation}

In our context, the complex integral is precisely of this form, but the contour is given by \eqref{eq:real}. With this choice of contour, the lemma for the complex integral gives
\begin{equation}
F(v)=\int_\Gamma  \frac{\rd t}t \, \frac{\rd s}s =4\pi i\ln\left|\frac{\left(a_{i-1j}-v\right)\left(a_{ij-1}-v\right)}{\left(a_{i-1j-1}-v\right)\left(a_{ij}-v\right)}\right|.
\label{Fv}
\end{equation}
As promised, $F(v)\rightarrow 0$ as $v\rightarrow \infty$ and the $v$-integral that we are left with is
\begin{equation}\label{v-int}
\int_{v_*}^\infty F(v)\rd \ln v=\int_{v_*}^\infty  4\pi i\ln\left|\frac{\left(a_{i-1j}-v\right)\left(a_{ij-1}-v\right)}{\left(a_{i-1j-1}-v\right)\left(a_{ij}-v\right)}\right|\, \rd \ln v.
\end{equation}

\subsection{The answer for generic diagrams}

From the standard definition of the dilog function  we obtain the alternative formula
\begin{equation}
\Li_2 x=-\int_0^x \log( 1-x')\frac{\rd x'}{x'}=-\int_{1/x}^\infty \log \left(1-\frac 1{x''}\right)\frac{\rd x''}{x''}\, ,
\end{equation}
where we have simply set $x''=1/x'$.  Our integral \eqref{v-int} breaks up into an alternating sum of four such terms
\begin{equation}
\int_{v_*}^{\infty}  \log\left(1-\frac {a}{v}\right)  \frac{\rd v}v=-\Li_2 \frac {a}{v_*}
\, .
\label{li2}
\end{equation}
Hence, the integral in  \eqref{v-int} reduces to
\begin{equation}
K_{ij}= \Li_{2}\left(\frac{a_{ij}}{v_{*}}\right)+\Li_{2}\left(\frac{a_{i-1j-1}}{v_{*}}\right)-\Li_{2}\left(\frac{a_{i-1j}}{v_{*}}\right)-\Li_{2}\left(\frac{a_{ij-1}}{v_{*}}\right)+c.c.
\label{generic}
\end{equation}

In \cite{Lipstein:2012vs} we verified that \eqref{generic} leads to the correct symbol for the 1-loop MHV amplitude. In particular, we showed that standard expressions for the symbol of the 1-loop MHV amplitude could be expressed as a sum over $i,j$ of
\[
\left(a_{ij-1}a_{i-1j}-a_{ij}a_{i-1j-1}\right)\otimes\frac{a_{ij-1}a_{i-1j}}{a_{ij}a_{i-1j-1}}+\mbox{c.c.}
\]
which isolates those invariants appearing in $K_{ij}$.  This discussion is repeated in appendix \ref{symbol}.  The formula \eqref{generic} above reduces to  this together with the terms
\begin{equation}\label{boundary}
\Delta_j \left( (a_{ij}-a_{i-1\, j})\otimes \frac{a_{ij}}{a_{i-1\, j}}\right)+\Delta_i\left( (a_{ij}-a_{i\, j-1})\otimes \frac{a_{ij}}{a_{i\, j-1}} \right) +\mbox{c.c.}
\end{equation}
where $\Delta_i f_{ij}=f_{ij}-f_{i-1\, j}$ and $\Delta_j f_{ij}=f_{ij}-f_{ij-1}$ are difference operators, so that the sum of such terms collapses telescopically.

Although the appearance of four dilogs in \eqref{generic} is reminiscent of the four dilogs that were obtained in \cite{Brandhuber:2004yw,Brandhuber:2007yx} for the 1-loop MHV amplitude using the CSW formalism and null polygonal spacetime Wilson loops, respectively, the formulae in these references are quite non-trivially different than ours.  In particular, their formulae are not dual conformal invariant up to the choice of the reference twistor and the reference twistor used in those references is chosen so that its underlying spinor is one of the spinors in a 2-mass-easy box that is related to the Kermit.  As such, in appendix \ref{BST}, where we describe the detailed relations of our formula to those of \cite{Brandhuber:2004yw}, we must work with one of their integral formulae in order to verify our formula against theirs.

A final remark is that the structure that is at the root of the analysis of \cite{Brandhuber:2004yw}, and the proof in \cite{Bullimore:2010pj} of the equivalence of Kermit integrand to the standard 1-loop MHV integrand, is that Kermit can be expressed as a sum of four one mass boxes (the numerator factor in the original momentum twistor representation of Kermit can be expressed as a sum of 4 terms that each cancels two of the poles leaving boxes \cite{Bullimore:2010pj}). This can be seen directly from the $d\,$log form of the integrand \eqref{dlog} by observing that $s$ and $t$ are natural ratios $s={s_{i-1}}/{s_i}$ and $ t= {t_{j-1}}/{t_j}$,  where
\begin{equation}\label{decomp}
 s_{i-1}= \la i-1|x_{0i}|\eta]\, , \quad s_i =\la i|x_{0i}|\eta]
\, ,\quad t_{j-1}=\la j-1|x_{0j}|\eta]\, , \quad t_j=\la j|x_{0j}|\eta]\, ,
\end{equation}
and this gives the decomposition into four terms.  Furthermore, the decomposition into the real part of four dilogs above can be seen to follow from this (although note that as ordinarily expressed, the 1-mass boxes diverge, but the divergent parts cancel between the four terms and is not seen in a generic Kermit).

\section{Divergent Diagrams} \label{divergent}
The usual infrared divergence arises from the $v$-integral, which becomes logarithmically divergent when $v_*=0$ and this is then compounded by a divergence in the $t$ integral at $v=0$ to make a $\log^2$ divergence.  If $i=j-2$, then $a_{ij-1}=0$, but $v_*\neq 0$  and the expression for Kermit in \eqref{generic} is still finite. On the other hand, if $i=j-1$, then $a_{ij}=a_{i-1j-1}=a_{ij-1}=0$ so that $v_*=0$ and \eqref{generic} diverges. This corresponds to a Wilson loop diagram in which the propagator connects two adjacent edges, as depicted in Figure \ref{div1}.  If $i=j$, the geometry in our setup of the $d$log form of the integrand breaks down so we must analyze this situation essentially from scratch. This situation corresponds to a Wilson loop diagram in which the propagator begins and ends on the same edge (as depicted in Figure \ref{div2}). Since the dual diagram contains a two-point MHV vertex at one end with no external leg, this diagram must vanish. Hence, the only case that needs to be regulated is $i=j-1$.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.11]{ij-1.jpg}
\caption{A propagator connecting two adjacent edges of the twistor Wilson loop. This diagram diverges.}
\label{div1}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.12]{ij.jpg}
\caption{A propagator beginning and ending on the same edge of the twistor Wilson loop. This diagram vanishes.}
\label{div2}
\end{center}
\end{figure}
In order to make \eqref{generic} well-defined, we will have to
introduce a regulator. In the following we use mass regulation
\cite{Henn:2011by}.

\subsection{Mass Regularization}
Various simplifications occur when $i=j-1$;  the contour $\Gamma$ simplifies to become
\begin{equation}
s=\frac{\bar{t}\left(a_{i-1i+1}/v-1\right)-1}{\bar{t}+1}.
\label{ijcont}
\end{equation}
Mass regularization requires that we replace $\left(x_0-x_{i}\right)^{2}\rightarrow\left(x_0-x_{i}\right)^{2}-m^{2}$,
$\left(x_0-x_{j}\right)^{2}\rightarrow\left(x_0-x_{j}\right)^{2}-m^{2}$.
Combining this with the Feynman $i\epsilon$ prescription, we see
that mass regularization is equivalent to taking $i\epsilon\rightarrow i\epsilon-m^{2}$.
  Thus performing this shift in
\eqref{iepshift} gives
\begin{equation}
\frac1{2\pi i}\int_0^\infty  \rd v \int_\Gamma \frac{\rd t}t \frac{\rd s}s\frac{1}{v+\left(i\epsilon-m^2\right)\left(f_{i}-f_{i+1}\right)}
\label{massdef}
 \end{equation}
where $f_{i},f_{j}$ are as defined above. Here we can no longer simply do the $t$ integral using the lemma above as $f_i$ and $f_j$ depend on $t$.  This is not however too complicated when $i=j-1$, since using $x_{i\, i+1}=\bar \lambda_i\lambda_i$ in equation \ref{fifj}, we have
$$
f_{i}-f_{i+1}=-\frac{1}{2}\frac{\left(v+x_{i\,i+2}^{2}\left|1+t^{-1}\right|^{-2}\right)^{2}}{v \, x_{i\,i+2}^{2}\left|1+t^{-1}\right|^{-2}}.
$$

In appendix \ref{massdetails}, we compute the integral in \eqref{massdef} by expanding the integrand in $m$ and neglecting terms of $\mathcal{O}(m)$. In the end, we obtain
\begin{equation}
-\frac{1}{4}\left(\ln^{2}\left(\frac{m^{2}}{x_{ii+2}^{2}}\right)+\ln\left(\frac{x_{i-1i+1}^{2}}{x_{ii+2}^{2}}\right)\ln\left(x_{ii+2}^{2}\right)\right)-\frac{2\pi^2}{3}+\mathcal{O}(m).
\label{massresult1}
\end{equation}
In appendix \ref{massdetails}, we also show that this expression reproduces the form of the 4-point 1-loop MHV amplitude obtained in \cite{Alday:2009zm}.

\section{Higher Loops} \label{higherloops}

Our calculation of the 1-loop MHV amplitude from Kermit is in fact already a significant first step in the computation of  higher loop amplitudes. As a start, at two loops, every diagram has at least one Kermit as a subdiagram and one can perform at least half the integrals directly and this will be true of the majority of diagrams at higher loop order.   In general, it follows from the construction of the $d\,$log form given in \cite{Lipstein:2012vs} that it will always be possible to associate two propagators to each vertex and obtain two real and two complex $d\, $log  integrals as for Kermit.  The two real integrals will then need to be regulated via an $i\epsilon$ prescription that will follow analogously to our arguments given here.

The general story at two-loops is still relatively simple and we sketch the calculation of a non-trivial two-loop MHV-diagram using the techniques described in this paper. As described in \cite{Lipstein:2012vs}, there are only two topologies of nonzero twistor Wilson loop diagrams corresponding to 2-loop MHV amplitudes. These are illustrated in Figure \ref{fig2loop}. There are also boundary versions of these diagrams, where one or two pairs of propagators end on the same edge of the Wilson loop. From these, we can see that Kermit is in some sense a basic building block for higher-loop loop amplitudes. For example, the diagram on the left in Figure \ref{fig2loop} is simply given by the product of two Kermits and therfore has the form $(\Li_2)^2$.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.15]{2loop.jpg}
\caption{The two topologies of nonzero Wilson loop diagrams contributing to the 2-loop MHV amplitude.}
\label{fig2loop}
\end{center}
\end{figure}

A less trivial case is illustrated in Figure \ref{fig1}. In this case, the amplitude does not factorize into a product of two Kermits, and we will see that one obtains also polylogs of higher degree such as $\Li_4$.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.2]{2loopdlog.jpg}
\caption{Boundary version of a Wilson loop diagram contributing to the 2-loop MHV amplitude.}
\label{fig1}
\end{center}
\end{figure}
The integral for the diagram in Figure \ref{fig1} is
\[
\int\frac{ds_{1}}{s_{1}}\frac{dt_{1}}{t_{1}}\frac{ds_{2}}{s_{2}}\frac{dt_{2}}{t_{2}}\frac{ds_{3}}{s_{3}-s_{2}}\frac{dt_{3}}{t_{3}}\frac{ds_{4}}{s_{4}}\frac{dt_{4}}{t_{4}}\]
where the momentum twistors are parametrized as follows:
\[
Z_{A}=Z_{*}+t_{1}\left(Z_{i-1}+s_{1}Z_{i}\right),\,\,\, Z_{B}=Z_{*}+t_{2}\left(Z_{j-1}+s_{2}Z_{j}\right)\]
\[
Z_{C}=Z_{*}+t_{3}\left(Z_{j-1}+s_{3}Z_{j}\right),\,\,\, Z_{D}=Z_{*}+t_{4}\left(Z_{k-1}+s_{4}Z_{k}\right).\]
In this case, we can treat the right-hand Kermit in Figure \ref{fig1} as a standard Kermit, but one in which the fixed twistors are $(Z_{j-1}+s_2Z_j,Z_j)$, and $(Z_{k-1},Z_k)$.  This factor can be done first leading to dilogs depending on $s_2$.  To implement this, we perform the following change of variables (which incorporates those we used before to make the real poles apparent):
\[
\left(s_{1},t_{1}\right)=\left(-s,-\frac{i}{s_{0}\left(1+s\right)}\right),\,\,\,\left(s_{2},t_{2}\right)=\left(t,-\frac{i}{t_{0}\left(1+t\right)}\right)\]
\[
\left(s_{3},t_{3}\right)=\left(u+t,-\frac{i}{u_{0}\left(1+u+t\right)}\right),\,\,\,\left(s_{4},t_{4}\right)=\left(-w,-\frac{i}{w_{0}\left(1+w\right)}\right).\]
In terms of the new variables, the integral becomes
\begin{equation}
\int
 \rd \ln s_{0} \rd \ln t_{0}  \rd \ln t \rd \ln s \int \rd \ln u_{0} \rd \ln w_{0}  \rd \ln u \rd \ln w
\label{eq:2loop}\end{equation}
where we have split the 2-loop integral into two factors, each of which is now very similar to Kermit. In particular, the contour is determined by:
\[
s_{0}=\bar{s}_{0},\,\,\, t_{0}=\bar{t}_{0},\,\,\, w_{0}=\bar{w}_{0},\,\,\, u_{0}=\bar{u}_{0}\]
\begin{equation}
s=\frac{\bar{t}\left(a_{i-1j}+v_{-}\right)+a_{i-1j-1}+v_{-}}{\bar{t}\left(a_{ij}+v_{-}\right)+a_{ij-1}+v_{-}}
\label{scons}
\end{equation}
\begin{equation}
w=\frac{\left(\bar{u}+\bar{t}\right)\left(a_{k-1j}+y_{-}\right)+a_{k-1j-1}+y_{-}}{\left(\bar{u}+\bar{t}\right)\left(a_{kj}+y_{-}\right)+a_{kj-1}+y_{-}}
\label{wcons}
\end{equation}
where \begin{equation}
v_{\pm}=s_{0}\pm t_{0},\,\,\, y_{\pm}=w_{0}\pm u_{0}.\label{eq:vpmypm}\end{equation}

The $u_0,w_0,u,w$ integrals in \eqref{eq:2loop} can now be evaluated exactly as in the evaluation of Kermit. This will yield a sum of dilogs which now depends on the variables $(t,\bar{t})$, since $\bar{t}$ appears on the definition of the contour of the the $u,w$ integral in \eqref{wcons}. Hence, the $s_0,t_0,s,t$ integrals in \eqref{eq:2loop} will be like those in Kermit, but with dilogs in the integrand. In more detail, the integral in \eqref{eq:2loop} has real poles in the variables $s_0,t_0,w_{0},u_{0}$ and we must make these well-defined using the Feynman $i\epsilon$ prescription.  In terms of the the $v_{\pm},y_{\pm}$ variables defined in \eqref{eq:vpmypm} we have:
\begin{equation}
\int\frac{2\rd v_{+}\rd v_{-}}{\left(v_{+}+v_{-}\right)\left(v_{+}-v_{-}\right)}\int_{\Gamma_s} \rd \ln t \rd \ln s
\int\frac{2\rd y_{+}\rd y_{-}}{\left(y_{+}+y_{-}\right)\left(y_{+}-y_{-}\right)}\int_{\Gamma_w} \rd \ln u \rd \ln w\label{eq:2loop2}\end{equation}
where $\Gamma_s$ refers to the contour defined by \eqref{scons} and $\Gamma_w$ by \eqref{wcons}.
The real variables $\left(s_{0},t_{0},w_{0},u_{0}\right)$ are related to the region momenta by
\[
s_{0}=\frac{x_{0i}^{2}}{2\left[\eta\right|x_{0i}\left|\bar{\eta}\right\rangle },\,\,\, t_{0}=\frac{x_{0j}^{2}}{2\left[\eta\right|x_{0j}\left|\bar{\eta}\right\rangle },\,\,\, w_{0}=\frac{x_{\tilde{0}k}^{2}}{2\left[\eta\right|x_{\tilde{0}k}\left|\bar{\eta}\right\rangle },\,\,\, u_{0}=\frac{x_{\tilde{0}j}^{2}}{2\left[\eta\right|x_{\tilde{0}j}\left|\bar{\eta}\right\rangle }\]
where the region momenta are depicted in Figure \ref{fig2}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.2]{2loopregion.jpg}
\caption{The CSW diagram corresponding to the Wilson loop diagram in Figure \ref{fig1}.}
\label{fig2}
\end{center}
\end{figure}

The $i\epsilon$
prescription then amounts to
\[
v_{\pm}\rightarrow v_{\pm}+i\epsilon\left(f_{i}\pm f_{j}\right),\,\,\, y_{\pm}\rightarrow y_{\pm}+i\epsilon\left(\tilde{f}_{k}\pm\tilde{f}_{j}\right)\]
where
\[
f_{i}=\frac{1}{2x_{0i}\cdot\eta},\,\,\, f_{j}=\frac{1}{2x_{0j}\cdot\eta},\,\,\,\tilde{f}_{k}=\frac{1}{2x_{\tilde{0}k}\cdot\eta},\,\,\,\tilde{f}_{j}=\frac{1}{2x_{\tilde{0}j}\cdot\eta}.\]
We can now  evaluate the $\left(v_{+},y_{+}\right)$ integrals in \eqref{eq:2loop2} as before to obtain
\begin{equation}
-4\pi^{2}\int_{v_{*}}^{\infty} \rd \ln v_{-}\int_{\Gamma_s} \rd \ln t\rd \ln s\int_{y_{*}}^{\infty}\rd \ln y_{-}\int_{\Gamma_w} \rd \ln u\rd\ln w
\label{2loop3}
\end{equation}
where
\[
v_{*}=-\frac{x_{ij}^{2}}{2x_{ij}\cdot\eta},\,\,\, y_{*}=-\frac{x_{jk}^{2}}{2x_{kj}\cdot\eta}.
\]
We now evaluate the $(u,w)$ integral using \eqref{lem} and the $y_{-}$ integral using \eqref{li2} to obtain
\begin{multline}
\int_{y_{*}}^{\infty}\rd\ln y_{-}\int_{\Gamma_w} \rd\ln u\rd\ln w=
-2\pi i\left[\Li_{2}\left(\frac{a_{k-1j}}{y_{*}}\right)-\Li_{2}\left(\frac{a_{kj}}{y_{*}}\right)\right. \\ \left. +\Li_{2}\left(\frac{\left(\bar{t}a_{kj}+a_{kj-1}\right)}{\left(\bar{t}+1\right)y_{*}}\right)-\Li_{2}\left(\frac{\left(\bar{t}a_{k-1j}+a_{k-1j-1}\right)}{\left(\bar{t}+1\right)y_{*}}\right)+c.c.\right].
\label{uwy}
\end{multline}
Since the first two dilogs do not depend on $t$ or $\bar{t}$, when we perform the $(v_{-},t,s)$ integrals in \eqref{2loop3} against these terms, the computation will be identical to that of Kermit, and we will be left with terms of form $\left(\Li_{2}\right)^{2}$. On the other hand, integrating $(v_{-},t,s)$ against the second two terms in \eqref{uwy} and their complex conjugates is nontrivial and will yield $\Li_4$'s. For example, we have integrals of the form
\begin{equation}
\int_{v_{*}}^{\infty}\rd\ln v_{-}\int_{\Gamma} \rd \ln s\rd \ln t\, \Li_{2}\left(\frac{at+b}{t+1}\right)
\label{li4}
\end{equation}
where $(a,b)$ are constants and the contour $\Gamma$ is defined by the reality constraint
\[
 s=\frac{\bar{t}-c(v_{-})}{\bar{t}-d(v_{-})},
\]
where $(c(v_{-}),d(v_{-}))$  are Mobius transformations of $v_-$ following from \eqref{scons}. Writing
\[
\int_{\Gamma} \rd \ln s\rd\ln t \,\Li_{2}\left(\frac{at+b}{t+1}\right)=\int_{\Gamma}\rd\left( \ln s  \,\Li_{2}\left(\frac{at+b}{t+1}\right) \rd \ln t\right),
\]
we see that the integral can be evaluated using Stokes theorem taking into account the singularities. We find
\begin{multline}
\int_{\Gamma} \rd \ln s \rd \ln t \,\Li_{2}\left(\frac{at+b}{t+1}\right)=2\pi i\left[\ln\left(\frac{c(v_{-})}{d(v_{-})}\right)\Li_{2}(b)+\int_{\bar{c}(v_{-})}^{\bar{d}(v_{-})}\rd\ln t \;\Li_{2}\left(\frac{at+b}{t+1}\right)\right.
\\
\left.+(b-a)\int_{1}^{\infty}\frac{\rd z}{\left(a-z\right)\left(b-z\right)}\ln\left(\frac{\left(c(v_{-})+1\right)z-c(v_{-})a-b}{\left(d(v_{-})+1\right)z-d(v_{-})a-b}\right)\ln z\right]\, .
\label{li42}
\end{multline}
The first term on the right comes from the pole at $t=0$ (the residue of the pole at $t=\infty$ vanishes), the second  from the branch cut of $\ln s$ (which runs from $s=0$ to $s=\infty$), and the third from the branch cut of the dilog in \eqref{li4} ($\Li_2(z)$ has a branch cut along the positive $z$ axis running from $z=1$ to $z=\infty$ with discontinuity $2\pi i \ln|z|$).

When the integral over $v_-$ in \eqref{li4} is performed against the first term on the right hand side of \eqref{li42}, one obtains functions of the form $(\ln)^2 \times Li_{2}$ and $\left(Li_{2}\right)^2$. Furthermore, performing the $t$ and $z$ integrals in the second and third terms on the right hand side of \eqref{li42} will yield terms of transcendentality 3, notably terms of the form $(\ln )^3$, $\ln \times  \Li_2 $, and $\Li_3$. Peforming the $v_{-}$ integral against these terms then yields functions of transcendentality 4 including $Li_4$.

\section{Conclusion} \label{conclusion}

We have seen how to compute certain scattering amplitudes and
correlation functions in $\mathcal{N}=4$ sYM without Feynman
parameters. The starting point is to express the loop integrands in
$d\, \log$  form. In our approach this is automatic for the MHV part
of the holomorphic Wilson loop in momentum twistor space to all loop orders, whose Feynman rules
can be written $d\, \log$  form \cite{Lipstein:2012vs}. An identical
approach will work for the twistor space formulation of amplitudes as
given in \cite{Adamo:2011cb} which gives formulae for amplitudes that
are remarkably similar to the formulae for the Wilson loop given
here. The integration variables in the $d\, \log$  form have a very
simple geometric interpretation: they correspond to insertion points
of propagators onto MHV vertices or edges of the Wilson loop (and for
the amplitude in ordinary twistor space, they will be insertion points
on just the MHV vertices).   An alternative, and generically
inequivalent  $d\, \log$  form for the planar amplitudes of
$\mathcal{N}=4$ sYM was given using on-shell diagrams and the
Grassmannian integral formula  in \cite{ArkaniHamed:2012nw} (although
this is essentially equivalent at 1-loop MHV).  Our methods should
apply there also if one can identify the real poles corresponding to
physical propagators from amongst the unphysical complex poles.

The external data on which the answers depend are encoded in the
integration contours that arise from reality constraints in momentum
twistor space. The main subtlety that arises is the appropriate
regularisation for the real poles that arise. Since these are
associated with physical propagators,  the standard Feynman
$i\epsilon$ prescription must be used. This can be done explicitly and
then the integrals become well-defined, and can be evaluated using
Stokes theorem.
In the non-generic divergent case, mass regularization can be
implemented by including a mass together with the Feynman $i
\epsilon$.

This procedure leads to a remarkably simple expression for the generic
`Kermit' diagram which remarkably turns out to be dual conformal invariant. Given that the Feynman $i\epsilon$ prescription breaks dual conformal invariance, this did not need to be the case.  This form of the 1-loop MHV amplitude is nontrivially related to standard ones in the literature, as we demonstrate in Appendix \ref{BSTa}.  Our method for computing the planar 1-loop MHV amplitude appears to be both direct and simpler than previous methods.  Furthermore, we provide evidence in section \ref{higherloops} that this method will scale  to higher loop amplitudes because Kermit is a  basic building block for many higher loop diagrams for the twistor Wilson loop. It seems potentially tractable  to use the techniques described here to carry out a complete evaluation of the 2-loop MHV amplitudes and compare this with the results with \cite{CaronHuot:2011ky,Golden:2013lha}, which obtained the differential of all planar 2-loop MHV amplitudes.  The Regge limit of higher loop MHV amplitudes might also provide a useful testing ground that would allow easier evaluation to compare with the results of \cite{Lipatov:2010qg,Bartels:2011ge,Dixon:2012yy}.

We believe that these methods can be used to compute loop amplitudes or correlation functions more generally.  In order to obtain the $d\,$log form we did not need to assume that the diagrams were planar when computing the holomorphic Wilson loop.  For the twistor space computation of amplitudes in \cite{Adamo:2011cb}, planarity was again not assumed, and indeed the basic framework extends to different amounts of supersymmetry.  The straightforward  $d\,$log form did seem however to require the assumption that the amplitude be MHV, and so extending the ideas beyond MHV remain a challenge. A different extension given in \cite{ArkaniHamed:2012nw} proposed an on-shell diagram formalism for the ABJM theory (a $\mathcal{N}=6$ superconformal Chern-Simons theory in three dimensions \cite{Aharony:2008ug}), and suggested that the loop integrands of the amplitudes of this theory should also have a $d\, \log$  form. In \cite{ArkaniHamed:2012nw}, they also suggest that the on-shell diagram formalism can be extended to 4d super Yang-Mills theories with $1\leq \mathcal{N}\leq4$, so it is conceivable that the integrands of loop amplitudes of these theories can also be expressed in $d\, \log$  form and so these are directions for further investigation.

\section*{Acknowledgments}
We would like to thank Nima Arkani-Hamed, Zvi Bern, Simon Caron-Huot, Lance Dixon, Gregory Korchemsky, and David Skinner for helpful discussions.  AL is supported by a Simons Postdoctoral Fellowship; LM is supported by a Leverhulme Fellowship and EPSRC grant number EP/J019518/1.

redpajama_set_name: RedPajamaArXiv


text:
Q: Oracle - query by two column combination I have two tables:
    product_table:
    ------------------------------------
     product | market | category | other
    ------------------------------------
             |        |          |
    ------------------------------------

    availability_table:
    -------------------------------------
     product | num_available | some_other
    -------------------------------------
             |               |
    -------------------------------------

I have a list of 7 markets and a list of 41 categories. For each market/category combination, I need to get several products (3 is a good number) that are currently available. I came up with a query that delivers products for one market/category combination:
SELECT product_table.product, product_table.category
FROM product_table, availability_table
WHERE   product_table.market = 1 // market is changing from query to query
    AND product_table.product = availability_table.product
    AND availability_table.num_available > 0
    AND product_table.category = 'category_1' // category is changing from query to query
    AND ROWNUM <= 3

This works well for one market/category combination, but I have to manually submit 287 queries to get all data I need. Is there any way to build a query that will do it for me with one submission?

A: Since you don't care which products are returned
SELECT product, category, market
  FROM (SELECT product,
               category,
               market,
               row_number() over (partition by market, category
                                      order by rownum) rnk
          FROM product_table
               JOIN availability_table USING (product)
         WHERE num_available > 0)
 WHERE rnk <= 3;

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text:
HomeStateCongress To Hold Protest At 300 Locations In Bengaluru Tomorrow
StateTop
Congress To Hold Protest At 300 Locations In Bengaluru Tomorrow
Bengaluru (Karnataka): Karnataka Pradesh Congress Committee (KPCC) will hold a protest against the state government at over 300 places, party leaders said on Sunday.
Congress MLA NA Harris while addressing a press conference here at KPCC Office on Sunday said that senior party leaders including former Chief Minister Siddaramaiah, State Congress president DK Shivakumar and Congress MLAs would hold the protest at Trinity Circle on January 23. This protest is against the CM Basavaraj Bommai-led Bharatiya Janta Party (BJP) government in the state on corruption and other issues.
"The state government is not ready to respond to the corruption charges. We have now decided to launch a protest and fight against it. Tomorrow our protest will be held at more than 300 locations in Bengaluru," MLA NA Harris told reporters.
All India Congress Committee (AICC) Secretary Abhishek Dutt was also present in the KPCC conference along with MLA NA Harris.
The protest will be held simultaneously at 200 signals, 26 flyovers and 25 metro stations in Bengaluru.
Earlier, Congress had alleged that the BJP charges a 40 per cent commission from the government contractors.
The party also alleged that the state government is involved in the corruption of Police, engineers, teachers and many other recruitments.
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Yerba Mate (Loose Leaf) is an antioxidant-rich, South American herb with high energizing properties, comparable to coffee without the buzz, jitters or crash.
Davidson's Yerba Mate (Loose Leaf) offers a smooth, earthy flavor profile.
Yerba mate is a super-leaf found in the Amazon Rainforest. Yerba mate is related to guayusa, as they are both sourced from trees found in the same geographic area. Davidson's Yerba Mate has been dried in a smoke-free drying process.


text:
<?xml version="1.0" encoding="utf-8"?>
<resources>
    <style name="AppTheme" parent="Theme.AppCompat.NoActionBar">
        <!--<style name="AppTheme" parent="@android:style/Theme.Holo">-->
        <!-- Customize your theme here. -->
        <!--<item name="android:windowActionBarOverlay">true</item>-->
        <item name="colorPrimary">#673AB7</item>
        <item name="colorPrimaryDark">#512DA8</item>
        <item name="colorAccent">#FF4081</item>
        <item name="android:windowTranslucentStatus">true</item>
    </style>
</resources>

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28 Nouveau Lave Glace Voiture Leclerc is free HD wallpaper. This wallpaper was upload at May 12, 2017 upload by ananda in voiture.
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I am Andrea Purtell and It is a privilege to share this Joyous Occasion with you!
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Whether you are looking for an intimate elopement, seaside ceremony, backyard celebration or large elegant affair at one of the many beautiful venues that NJ has to offer, I can provide you with the ceremony you desire.
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Not only is it a free-to-use platform, but it is also a popular search engine with over 1 billion monthly users. To help you take advantage of YouTube's marketing potential, we have prepared an interactive step-by-step guide with links to useful articles, tools and… YouTube videos!
According to Alexa.com, YouTube is the third most popular website in the world behind Google and Facebook. As the biggest video hosting platform, YouTube contains millions of videos published by large and small companies as well as individuals. It is free to watch videos on YouTube, but what is even more important, it is free to publish your own videos on the platform.
As a small business you can use YouTube to develop useful content for your customers such as visual guides, presentations or movies; advertise your products or brand using video ads and communicate with your customers via video blogging or hangouts.
To get started follow our interactive guide with questions and answers on how to use the popular video hosting platform. Our microsite contains links to useful YouTube resources from around the web. Click on the image below to go directly to the Small Business Guide to YouTube and start building your video presence online right away!


text:
The California Bureau of Investigation ("CBI" or "BI") is California's statewide criminal investigative bureau under the California Department of Justice (CA DOJ), in the Division of Law Enforcement (DLE), administered by the Office of the State Attorney General that provides expert investigative services to assist local, state, tribal, and federal agencies in major criminal investigations ranging across the state.

History

The California Office of the Attorney General (AG) was created in 1850; however, permanent law enforcement and criminal investigative elements of the CA DOJ were not established until the early 20th century. Despite this, CA DOJ's Law Enforcement personnel can trace part of their roots back to that of the original California Rangers; the first statewide criminal investigative and law enforcement agency created on May 17, 1853. Although the California Rangers were quickly disbanded (following their success in bringing the violent Five Joaquins gang to justice), their special law enforcement capabilities and statewide investigative mission now reside primarily with CA DOJ Special Agents, such as those assigned to CBI.

The CBI is the oldest, continuously operating, statewide criminal investigative agency in the CA DOJ, and in the state of California. Listed below are some key dates in the history of the CBI clarifying this lineage:

In 1905, the California Bureau of Criminal Identification (CBCI) was created for classification and identification of incarcerated criminals via various means such as fingerprints. However, the original CBCI was disbanded in 1909 and had no formal criminal investigative mission.
On January 1, 1918 (following approved legislation in 1917), CBCI was reinstated as the California Bureau of Criminal Identification and Investigation (CBCII), following recognition by state officials that unification of state law enforcement efforts was needed in collecting/maintaining a repository of criminal identification information accessible to law enforcement agencies throughout the state, in addition to investigating and apprehending criminals. This is also recognized as the founding date of the CBI, as it was when the bureau adopted its first criminal investigative authorities, assisting other law enforcement agencies throughout the state as needed in complex criminal investigations. The investigators within this bureau were recruited as experts in a variety of criminal investigative and scientific fields. CBCII, along with the later (but now-defunct) California Bureau of Narcotic Enforcement (BNE) created in 1927, provided the state with its initial state-wide criminal investigative capabilities it still has today via CBI.
On May 1, 1944, the CA DOJ was created, and the California Office of the Attorney General (OAG) was transferred to it (which also administers it), along with the BNE.
In 1945, the CBCII was also transferred under CA DOJ, and shortly after, was split up into the Bureau of Investigation (BI) and Bureau of Criminal Statistics (later to be renamed to the Bureau of Criminal Identification and Information). This is where BI would remain until present day (as the modern CBI).
In 1962, the Division of Law Enforcement (DLE) was created under the CA DOJ. The BNE and CBI were transferred under the new DLE, with a single division director (later renamed to division chief) overseeing the bureaus, and reporting directly to the CA AG.
In 1971, the 26th CA AG (a former FBI agent and U.S. Army intelligence officer), drastically reorganized the DLE in order to improve efficiency and effectiveness.
In 1998, despite many throughout the state already referring to it as such, the BI officially re-brands itself as "CBI" (which it is still widely referred to as today) with the approval of the 30th Attorney General of California, Bill Lockyer, and becomes an extremely well-known and respected, state investigative law enforcement agency. At this point in time, CBI includes field operations, an organized crime unit, a witness protection program, a major crimes program, a gang/criminal extremist unit, and an intelligence unit. The DOJ/CBI Special Agents of this time are assigned to multiple task forces (state and federal), and assist other departments throughout the state with investigations, operations, support, and advanced law enforcement training, while also still conducting investigations related to any of the California Attorney General's work/cases/matters of interest.
In 2008, the acclaim and prestige of the CBI reaches its height, when the CBS television network creates a popular new television series based on a team of CBI Special Agents. The show lasts seven seasons.
In 2009, although still essentially the same organization, the official name of CBI was changed to the Bureau of Investigation and Intelligence (BII) after it merged with the Bureau of Intelligence following a massive reorganization and consolidation of the CA DOJ's Division of Law Enforcement.
In 2012, DLE restructuring culminates with it being significantly cut, and numerous CA DOJ Special Agents (assigned to multiple bureaus) are laid off. The law enforcement capabilities of the DLE are also drastically reduced or completely eliminated. As part of this reduction, the BNE is disbanded, with a few of its remaining functions/personnel being transferred to the BII. After this, the BII was renamed back to "BI" on February 17, 2012. Since this, the DLE and CBI have slowly attempted to rebuild capabilities it once had, but is still not near the level it was prior to 2012.

Current functions

Today the CBI can be assigned at the discretion of the State Attorney General, performs investigations on public land for incidents that occur outside of local jurisdictions, or can be requested by local authority for various reasons (e.g., areas that are too small to have local detectives or cross-jurisdiction coordination of major crimes). The CBI also operates several programs including a Special Investigations Team (SIT) handling high-profile investigations and special investigative requests from the Attorney General's Office, a Special Operations Unit (SOU) targeting murder suspects and violent criminal groups with special equipment/resources, a Human Trafficking Team, a Sexual Predator and Assault Team (SPAT), an Electronic Crimes Unit (ECU), a Tax Recovery and Crime Enforcement (TRACE) Task Force, a Recycling Fraud Team (RFT), an intelligence operations clearing house/de-confliction center for law enforcement agencies (LA CLEAR), and the state anti-terrorism program. The CBI also runs and/or participates in several other highly successful state and federal task forces.

In fiction
A fictional version of CBI was featured in the television series The Mentalist. It featured several units, with the one featuring the main characters focusing mostly on the investigation of homicides, and occasionally kidnappings or missing persons. The Region 1 DVD release of the fourth season of The Mentalist includes the special CBI: Behind the Badge, featuring interaction with actual law enforcement officers in California, specifically focusing on a homicide task force with several agents and officers of different agencies (including those from CBI), who tutored those who worked on the program in providing a realistic portrayal.

A CBI Special Agent was also featured in a third-season episode of the crime television series Numb3rs, where the CBI agent worked together with the FBI to catch a murderous cult leader.

Beginning in 2007, New York Times best-selling author Jeffery Deaver wrote a series of crime drama novels starring the fictional Special Agent Kathryn Dance of the CA DOJ/CBI. The most recent novel in the series was published in 2015.

References

External links
 Bureau of Investigation (BI) – description on the California Office of the Attorney General's website

Attorney General
State Bureaus of Investigation

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text:
Read the passage and the question carefully and choose the correct answers.
Canadian Ban on 'Harmful' Single-Use Plastics
Canada is taking steps to ban "harmful" single-use plastics by 2021, Prime Minister Justin Trudeau announced Monday, making the country the latest to join a growing movement around the world to halt the use of materials deemed damaging to the planet.
Speaking at a nature reserve in Mont Saint-Hilaire, Quebec, Trudeau said that while an exact list of items that will be covered by the ban has not been determined, it will be "grounded in science." A government statement said that it could include such items as bags, straws, plates, cutlery and stir sticks. "You've all heard the stories and seen the photos, and to be honest, as a dad, it's tough trying to explain this to my kids," Trudeau said. "How do you explain dead whales washing up on beaches around the world, their stomachs jam-packed with plastic bags?"
In Toronto, Environment Minister Catherine McKenna announced the ban surrounded by people holding photos of garbage-strewn beaches and oceans, and turtles and birds entangled in plastic waste and fishing nets.
The measure also will make plastic manufacturers and companies that use plastic packaging responsible for the collection and recycling of the materials. Roughly 13 million tons of plastic end up in the ocean, where it kills nearly 100,000 marine animals each year, according to the United Nations. Some plastics can take centuries to degrade.
More than 3 million tons of plastic waste — or $6 billion worth — is chucked out by Canadians each year, according to government statistics. Without steps to crack down on the problem, that number could increase to roughly $8 billion a year by 2030.
In Canada, which has the world's longest coastline, less than 10 percent of all plastic used is recycled each year, according to government statistics. Canada is not alone in trying to clamp down on plastic pollution. Its move follows those of the European Parliament, which voted this year to ban several single-use plastic products. Countries such as Taiwan and India have pledged to implement similar bans, and several jurisdictions around the world have announced plans to ban some single-use plastic items, including bags and drinking straws.
Last year, Canada used its turn as president of the Group of Seven nations to put finding ways to deal with ocean pollution near the top of its agenda. At the Group of Seven summit in June 2018, Britain, Canada, the European Union, France, Germany and Italy endorsed a nonbinding agreement that included a pledge that all plastics in those countries be recycled, burned for energy or reused by 2040.
The announcement comes on the heels of Canada's protracted diplomatic dispute with the Philippines over more than 100 containers of Canadian waste that were shipped to the country in 2013 and 2014. The containers were labelled as recyclable plastic scraps, but an inspection by customs officials found that many of them were filled to the brim with used adult diapers and wires.
Trudeau promised on multiple occasions that a "Canadian solution" was in the works, but the waste continued to sit in ports near Manila. Canada agreed to take back the waste last month, but the Philippines recalled its ambassador and envoys to Canada after the country missed a May 15 deadline to reclaim the garbage. Trudeau's announcement also comes just months before a federal election in October, in which climate change and the environment are expected to be front and centre. The prime minister has made protecting the environment a key tenet of his government's mandate, but it hasn't been easy.
This year, the federal government imposed a carbon tax on provinces that had not yet introduced their own equivalent carbon plan. The "tax on pollution," as the Trudeau government calls it, had been scaled-down to address concerns from the business community about competition and attracting overseas investment. Trudeau has also frustrated environmental campaigners, lawmakers and indigenous groups because of his government's efforts to shore up the country's energy sector by constructing pipelines.
Last year, a federal court of appeals ruled that the government had failed to adequately consult with indigenous groups and halted the expansion of the Trans Mountain oil pipeline, which would nearly triple the amount of western Canadian crude moved from Alberta to the coast of British Columbia. The government spent $4.5 billion to buy the existing pipeline from Kinder Morgan last year and said it would continue to consult with indigenous groups on the expansion. A decision on the fate of the pipeline is expected before June 18.
Andrew Scheer, leader of the Conservative Party, told reporters in Ottawa that the ban was an attempt by Trudeau, who faced a leadership crisis this year, "to change the channel." Scheer, who has not yet introduced a climate change plan of his own and has promised to repeal the carbon tax as prime minister, said Trudeau's plan lacked details about how it would affect the economy or consumers.
"This is clearly a government clutching at straws," he said.
Source text from and copyrighted to The Washington Post (https://www.washingtonpost.com/world/trudeau-announces-canadian-ban-on-harmful-single-use-plastics/2019/06/10/54000420-8bb6-11e9-adf3-f70f78c156e8_story.html?utm_term=.ccefbd340878).
The text used here is for educational purposes. The contents and the copyright belong to the aforementioned source.
Which items could the ban of single-use plastics include?
A) Bags
B) Hats
C) Straws
D) Plates
E) Cutlery
Read explanation


Taking your dog to the beach, lake or river is always a fun outing for you and your best friend, but remember that dogs often forget their manners when out near the water. Use these tips to keep your dog safe and others at the beach happy too.
It is not unusual for people to take their dogs to pools, beaches or out on boats and never think twice about their dogs safety. It is vital that your dog be protected near water. Even if he knows how to swim there are several steps you can take to make sure his time, and your time, at the beach or pool is enjoyable.
Blue-green algae puts your dog at risk during the summer and fall. Does your local pond have this common, deadly algae?
A close look at the dog paddle sparks a scientific fire.
Like children, dogs need supervision around the pool. More safety tips await you here!
While "fun in the sun" sounds harmless, warm weather can be dangerous, even fatal.


You're used to the convenience of banking, shopping and taking care of personal business online when you're pressed for time, or when it's convenient for you. Medical care doesn't have to be any different.
Sometimes a personal or professional issue can get in the way of maintaining a healthy, productive life. The Employee Assistance Program may be able to help.
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A couple of minutes. A few questions. A lifetime of great oral health. Wouldn't it be nice to know if you are at risk for oral disease before it becomes a problem? By taking just a few minutes to answer some simple questions about yourself and your oral health, this tool from Delta Dental will assess your level of oral disease risk and your oral health needs.
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Located approximately half way between Lansing and Grand Rapids, downtown Portland is your perfect "day trip" getaway!
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©2019 City of Portland. All rights reserved. Powered by Touchpoints.


Swim group swimming at Ellerton Lake, Bishopton Lake, River Tees and North Sea. Further info see Facebook group.
It really helps other swimmers to have a brief description of a swim spot. Please could you help them by telling us a little about North East Open Water Swimmers (NEOWS).


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Home | All news | Politics | The Armenian hero whom Turkey would prefer to forget
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The Armenian hero whom Turkey would prefer to forget
Siranush Ghazanchyan Send an email May 13, 2013, 10:39
Sarkis Torossian, an Armenian-Turkish officer, was awarded medals for his courage by Mustafa Kemal
"Confronted by the chilling 100th anniversary of the genocide of 1.5 million Armenians at the hands of the Ottoman Turks in 1915, Turkey's government is planning to swamp memories of the massacres with ceremonies commemorating the Turkish victory over the Allies at the battle of Gallipoli in the same year. Already, loyalist academics have done their best to ignore the presence of thousands of Arab troops among the Turkish armies at Gallipoli – and are even branding an Armenian Turkish artillery officer who was decorated for his bravery at Gallipoli as a liar who fabricated his own biography," Robert Fisk writes in an article published by The Independent.
In fact, Captain Sarkis Torossian was personally awarded medals for his courage by Mustafa Kemal, one of the Turkish heroes of Gallipoli who later, as Ataturk, founded the modern Turkish state. But in view of the desire of some of Turkey's most prominent historians to brand Torossian a fraud, the word "modern" should perhaps be used in inverted commas.
Now these academics are even claiming that the Armenian army captain invented his two medals from the future Ataturk. Yet one of the most the outspoken Turkish historians to have fully acknowledged the 1915 genocide, Taner Akcam, has tracked down Torossian's family in America and inspected the two Ottoman medal records; one of them bears Ataturk's original signature.
"Turkey, as we all know, wants to join the EU. I also, by chance, happen to think it should. How can we Europeans claim that the Muslim world wishes to stay "apart" from our "values" when an entire Muslim country wants to share our European society? We are hypocrites indeed. Yet how can Turkey still hope to join when it still refuses to acknowledge the truth of the Armenian genocide – and symbolises this denial by a scandalous attack on a long-dead Ottoman officer?" the author rites.
Captain Torossian's memoirs, From Dardanelles to Palestine, were first published in Boston in 1947. Ayhan Aktar, professor of social sciences at Istanbul Bilgi University, first came across a copy of the book 20 years ago and was amazed to learn that there were officers of Armenian descent fighting for the Ottomans.
The eight-month battle for Gallipoli – an Allied landing dreamt up by Churchill in the hope of capturing Constantinople and breaking the deadlock on the Western Front – was a disaster for the British and French, and the mass of Australian and New Zealand troops fighting with them. They abandoned the beach-heads in January of 1916.
In his book, Torossian recounts the fighting at Gallipoli and other battles in which he participated – until, towards the end of the Great War, he found his sister among the Armenian refugees on the death convoys to Syria and Palestine. He then turned himself over to the Allies, meeting (but not liking) T E Lawrence and re-entering Turkey with French forces. He eventually travelled to the US where he died.
The gutsy Professor Aktar, however – noticing his colleagues' unwillingness to acknowledge that Arabs and Armenians fought in the Ottoman Army – decided to publish Torossian's book in the Turkish language. Initial reviews were favourable until two historians from Sabanci University took exception. Dr Halil Berktay, for example, wrote 13 newspaper columns in Taraf calling the entire book a fiction and Torossian a liar.
Taner Akcam, the Turkish historian who discovered Torossian's family, was stunned by the reaction to the Turkish edition of the book; one critic, he says, even claimed Torossian did not exist. The Turkish Foreign Minister, Ahmet Davutoglu, spoke at Gallipoli two years ago and gave a perfectly frank account of how Turkey planned to define the Armenian genocide on its hundredth anniversary. "We are going to make the year of 1915 known the whole world over," he said, "not as an anniversary of a genocide as some people claimed and slandered (sic), but we shall make it known as a glorious resistance of a nation – in other wour defence of Gallipoli."
"So Turkish nationalism is supposed to win out over history. Descendants of those who died with the Anzac troops at Gallipoli, however, might ask their Turkish hosts in 2015 why they do not honour those brave Arabs and Armenians – including Captain Torossian – who fought alongside the Ottoman Empire" Robert Fisk concludes.


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Q: Find a text using XLOOKUP I am looking for a way to find a text in a a excel using XLOOKUP (RECHERCHEX in french) and return the value of the next column.
My problem is that my text is defined using a '

For example if I want to search for C1 in the text, using =RECHERCHEX("C1";C:C;D:D;"NOT FOUND";1) it returns "B" where C11, C101, C107, C109, C112, C119, C121 but it's wrong, I would like to get the value A where C1 is.
I tried everything
=RECHERCHEX("C1";C:C;D:D;"NOT FOUND";-1)
=RECHERCHEX("C1";C:C;D:D;"NOT FOUND";0)
=RECHERCHEX("C1";C:C;D:D;"NOT FOUND";1)
=RECHERCHEX("C1";C:C;D:D;"NOT FOUND";2)

Does not work.
Do you have any solution?
Thank you

A: Use:
=XLOOKUP("*, "&"C1" & ", *",", "&C:C&", ",D:D,"NOT FOUND",2)

A: You're searching for a substring. It's not clear what you're looking for. If you want to find the first instance of C1 it would be in C5, because C1 is a substring of C143. Are you trying to find "C1," which is C1 with a comma after it? To search for C1 as a substring with no numbers following you'd need to do something more complex.
This will do what you asked for specifically, but will miss "C1" without a comma after:
=XLOOKUP("*c1,*", D3:D15, E3:E15, "not found", 2)


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Inn & Spa at Cedar Falls continues to impress 25 years later
The Wisconsin cheese ball and tapenade basket at Kindred Spirits at the Inn & Spa at Cedar Falls. CONTRIBUTED PHOTO BY ALEXIS LARSEN
Lifestyles | May 28, 2018
By Alexis Larsen
I was lucky enough to work and learn alongside former Dayton Daily News food critic Ann Heller over the course of many years together at the paper.
She helped inform my writing, my editing, my eye, my thought process, my palate and so much more and I am fortunate to still be able to call her my friend.
So when her name came up in conversation during a recent trip to dine at the Inn at Cedar Falls near Logan — a small town in Ohio's Hocking Hills region — it captured my attention completely.
I was speaking to the Inn's owners Terry Lingo and Ellen Grinsfelder after a truly excellent meal and Grinsfelder mentioned that Ann had reviewed it some time ago.
In fact, it was an article from 1992.
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Although it had published more than two decades ago, Grinsfelder was raving about the job that Ann had done capturing the spirit of the restaurant and the food and the positive impact her words had carried to her business.
When I arrived home I decided to find the review and read the insights and opinions that meant so much so many years later to the owners of the Inn and to compare her experience with my own.
Ann wrote: "Halfway between Old Man's Cave and Ash Cave, the Inn is surrounded on three sides by the Hocking Hills state park and is privately operated by Ellen Grinsfelder, daughter of the late Anne Castle. Castle was a woman who dreamed of a place where stressed urban dwellers could buy a slice of tranquility.
The kitchen at the Inn is the middle room in two 1840s log cabins that have been restored and turned into a rustic dining room and a seasonal sitting room that becomes another dining room when the temperature drops."
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The restaurant, now called Kindred Spirits, has continued to expand and update and, although the menu is not exactly the same, the warm charming setting and the delicious menu preparations that Ann raved about that greet guests at the Inn & Spa at Cedar Falls restaurant still rings true today.
Kindred Spirits focuses on seasonal farm to table recipes that bring out the flavor of the ingredients in an honest, straightforward way that has a simple elegance to it.
The restaurant's two intimate log cabins seat about 45 people and a large outdoor dining area is the place to be when the weather is cooperating.
Abby Cole was recently named executive chef of the restaurant after serving several years as sous chef, impressing guests and her bosses with her techniques and creativity. She will be responsible for designing the Inn's dining and special event menus and helping provide direction of its culinary experiences.
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ExplorePan roasted filet mignon at Kindred Spirits at the Inn & Spa at Cedar Falls. CONTRIBUTED PHOTO BY ALEXIS LARSEN Contributing Writer
The menu changes seasonally and the dishes that will be featured on the fall menu include a rack of lamb ($35) served with couscous, spiced carrots and pistachio cream; an Asian marinated portabella ($23) served with rice noodles, sumac broth, plum vinegar, spinach and salt cured yolks; pistachio crusted scallops ($29) with chorizo and apple risotto, spinach and lemon thyme sauce; cornmeal crusted trout ($25) served with artichokes, collard greens, roasted potatoes and topped with a brown butter caper sauce. The chicken paprikash ($27) features a seared chicken thigh, Ohio City Pasta's red pepper fagiolini rigati and a sour cream-paprika sauce. And then there's the pumpkin ravioli ($19). It's all as delicious as it sounds.
The Inn's best-selling pan roasted filet mignon ($41) will never be knocked down from its No. 1 spot. The preparations, sauce and side servings may change, but it is a crowd-pleaser and continues to be a favorite of visitors to the restaurant.
The Wisconsin cheese ball and tapenade basket ($7) served with flatbread, crackers and crostini and the rosemary and parmesan crème brûlée ($9) — a rich, savory version of the sweet dish you see on dessert menus — are classic appetizers that stay untouched through menu iterations.
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The triple berry cobbler with vanilla bean ice cream at Kindred Spirits at the Inn & Spa at Cedar Falls. CONTRIBUTED PHOTO BY ALEXIS LARSEN
For dessert, the restaurant is known for its triple berry cobbler with vanilla bean ice cream. And for guests lucky enough to stay at the one of the cabins or new yurts, breakfast features many delicious options including homemade granola which is sold at the Inn's gift shop.
In addition to a stellar meal in a remarkable setting, eating at the Inn also presents the opportunity to sample some of the region's handiwork.
Ann's review written 25 years ago ended, "The Inn at Cedar Falls is a place to feed both body and soul. Go for both."
I would make the same recommendation today with the addendum to look into booking a room while you are at it. The relaxing setting and charming cabins can only enhance the meal.
Dayton Eats looks at the regional food stories and restaurant news that make mouths water. Share info about your menu updates, special dinners and events, new chefs, interesting new dishes and culinary adventures. Do you know of new exciting format changes, specials, happy hours, restaurant updates or any other tasty news you think is worth a closer look at? E-mail Alexis Larsen at alexis.e.larsen@hotmail.com with the information and we will work to include it in future coverage.
What: Kindred Spirits at the Inn & Spa at Cedar Falls
Where: 21190 St. Rt. 374, Logan. Take Ohio 374, 2 1/2 miles north of Ohio 56, in Hocking County. Take U.S. 35 west to Washington Court House, then Ohio 22 east to Circleville and Ohio 56 east through Laurelville and South Bloomingville. It's roughly a 90 minute to two hour drive depending on where you live locally.
The menu: Each season the Inn presents a new menu. Dinner guests may arrive 30-minutes prior to dinner to enjoy a cocktail.
Reservations: Are required and are available starting at 5:30 p.m. Make your reservation online at https://innatcedarfalls.com or by calling (740) 569-5015.
The rooms: The Inn at Cedar Falls is open every day except Christmas. Dinner is available to the public as well as guests of the Inn, but space is limited. Breakfast is served only to overnight guests. Accommodations include log cabins, cottages, rooms and new yurts that have just been added.


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Everyone's favorite place to get memes and GIFs, Imgur, is getting its own iOS app. If your first reaction is, "wait, didn't they already have an app?" You would be correct, but Imgur says this is way better than that old app which they have pulled from the App Store.
The new app is part of the company's push to be a place for visual stories and give users a streamlined way to explore and discover those stories. It lets you browse by swiping left and right; up and down vote on images; and and participate in threaded discussions about the pictures. You can also sort the images by popularity and time.
Users can sign into their Imgur accounts and favorite images they find compelling.
Unfortunately, the ability to upload images isn't available. But Imgur tells TNW that feature is coming "very soon to mobile." Also it plans to release the app for other platforms in the near future.
But if you're a fan of the image sharing site, an on-the-go first-party browsing solution is nice.