Antigravity Agent
1929 lines
81 KiB
TeX
1929 lines
81 KiB
TeX
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\title{\hspace{-0.0cm}{\LARGE Into the Amplituhedron}}
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\author{\vspace{-.5cm}Nima Arkani-Hamed$^{a}$ and Jaroslav Trnka$^{b}$\\
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{\footnotesize{\it $^{a}$ School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA}\\
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{\it $^{b}$ California Institute of Technology, Pasadena, CA 91125,
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USA}}\vspace{-.5cm}} \preprint{2013}
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\abstract{We initiate an exploration of the physics and geometry of
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the amplituhedron, starting with the simplest case of the integrand
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for four-particle scattering in planar ${\cal N} = 4$ SYM. We show
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how the textbook structure of the unitarity double-cut follows from
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the positive geometry. We also use the geometry to expose the
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behavior of the multicollinear limit, providing a direct motivation
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for studying the logarithm of the amplitude. In addition to
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computing the two and three-loop integrands, we explore various
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lower-dimensional faces of the amplituhedron, thereby computing
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non-trivial cuts of the integrand to all loop orders. }
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\preprint{CALT-68-2873}
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\begin{document}
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\newpage
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\section{Geometry and Physics of the Amplituhedron}
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In \cite{P1}, we introduced a new geometric object--the
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Amplituhedron--underlying the physics of scattering amplitudes for
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${\cal N}=4$ SYM in the planar limit. At tree level, the
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amplituhedron is a natural generalization of
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``the inside of a convex polygon". Loops arise by extending the
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geometry to incorporate the idea of ``hiding particles" in the only
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natural way possible.
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The amplituhedron ${\cal A}_{n,k,L}$ for $n$-particle N$^k$MHV
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amplitudes at $L$ loops, lives in $G(k,k+4;L)$, which is the space
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of $k$-planes $Y$ in $k+4$ dimensions, together with $L$ 2-planes
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${\cal L}_1, \cdots, {\cal L}_L$ in the $4$ dimensional complement
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of $Y$. The external data are given by a collection of $n$ $(k+4)$
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dimensional vectors $Z_a^I$. Here $a=1, \cdots n$, and $I = 1,
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\cdots, (k+4)$. This data is taken to be ``positive", in the sense
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that all the ordered $(k+4) \times (k+4)$ determinants $\langle
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Z_{a_1} \cdots Z_{a_{k+4}} \rangle > 0$ for $a_1 < \cdots <
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a_{k+4}$. The subspace of ${\cal A}_{n,k,L}$ of $G(k,k+4;L)$ is
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determined by a ``positive" linear combination of the (positive)
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external data. The $k$-plane is $Y_{\alpha}^I$, and the 2-planes
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are ${\cal L}_{\gamma (i)}^I$, where $\gamma = 1,2$ and $i = 1,
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\dots, L$ . The amplituhedron is the space of all $Y, {\cal
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L}_{(i)}$ of the form
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\begin{equation}
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Y_{\alpha}^I = C_{\alpha a} Z_a^I, \qquad {\cal L}_{\gamma (i)}^{I}
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= D_{\gamma a (i)} Z_a^I
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\end{equation}
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where the $C_{\alpha a}$ specifies a $k$-plane in $n$-dimensions,
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and the $D_{\gamma a (i)}$ are $L$ 2-planes living in the $(n-k)$
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dimensional complement of $C$, with the positivity property that for
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any $0 \leq l \leq L$, all the ordered maximal minors of the $(k + 2
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l) \times n$ matrix
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\begin{equation}
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\left(\begin{array}{ccc} & D_{(i_1)} & \\
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\hdashline & \vdots & \\ \hdashline & D_{(i_l)} & \\ \hdashline & C
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\end{array} \right)
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\end{equation}
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are positive.
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There is a canonical rational form $\Omega_{n,k;L}$
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associated with ${\cal A}_{n,k;L}$, with the property of having
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logarithmic singularities on all the lower-dimensional boundaries
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of ${\cal A}_{n,k;L}$. The loop integrand form for the
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super-amplitude is naturally extracted from $\Omega_{n,k;L}$
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\cite{P1}.
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The amplituhedron can be defined in a few lines, as we have just
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done. But the resulting geometry is incredibly rich and
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intricate--as it must be, to generate all the structure found in
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planar ${\cal N} = 4$ SYM scattering amplitudes to all loop orders!
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For instance, the singularity structure of the amplitude is
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reflected in the geometry of the various boundaries of the
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amplituhedron; studying this geometry in some of the simplest cases
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allows us to see the emergence of locality and unitarity from
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positive geometry.
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Even just the tree amplituhedron generalizes the positive
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Grassmannian $G_+(k,n)$ \cite{alex}. A complete understanding of $G_+(k,n)$
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revealed many surprising connections to other structures, from the
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fundamentally combinatorial backbone of affine permutations, to
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cluster algebras, to the physical connection with on-shell processes
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\cite{alex, FG, positive}. It is natural to expect the full amplituhedron
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${\cal A}_{n,k,L}$ to have a much richer structure. A complete
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understanding of the full geometry of the amplituhedron, at the same
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level as our understanding of the positive Grassmannian, will likely
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involve further physical and mathematical ideas. Our goal in this
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note is to begin laying the groundwork for this exploration, by
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looking at various simple aspects of amplituhedron geometry in the
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simplest non-trivial case of clear physical interest.
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While the tree amplituhedron generalizes the positive Grassmannian
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in a direct way, extending the notion of positivity to external
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data, the extension of positivity associated with ``hiding
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particles" which gives rise to loops is more novel and interesting.
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The very simplest case of four-particle scattering has $k=0, n=4$.
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Here, we don't have the additional structure of Grassmann components
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for the external data \cite{P1}, the external data are just the
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ordinary bosonic momentum-twistor\cite{A1} variables
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$Z^I_1,Z^I_2,Z^I_3,Z^I_4$, for $I=1, \cdots, 4$. Furthermore, the
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constraint of positivity for external data is trivial in this case;
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indeed using a $GL(4)$ transformation we can set the $4 \times 4$
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matrix $(Z_1, \cdots, Z_4)$ to identity. The loop variables are
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just lines in momentum-twistor space (or better, two-planes in
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four-dimensions), which correspond to points in the (dual)
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space-time. Having set the $Z$ matrix to the identity, each $2
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\times 4$ matrix for the lines ${\cal L}^I_{\gamma (a)}$ is simply
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identified with the $D$ matrices $D_{(i)}$.
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The amplituhedron positivity
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constraints are that the all the ordered minors of each $D_{(i)}$
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matrix are positive
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\begin{equation}
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(12)_i, (13)_i, (14)_i, (23)_i, (24)_i,(34)_i > 0
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\end{equation}
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We also have mutual positivity, that the $4 \times 4$ determinant
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$\langle D_{(i)} D_{(j)} \rangle > 0$, which tells us that
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\begin{align}
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(12)_i (34)_j + (23)_i (14)_j + (34)_i
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(12)_j + (14)_i (23)_j - (13)_i (24)_j - (24)_i (13)_j > 0
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\end{align}
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We can also express these conditions in a convenient gauge, where
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\begin{equation}
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D_{(i)} = \left(\begin{array}{cccc} 1 & x_i & 0 & -w_i \\ 0 &
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y_i & 1 & z_i
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\end{array} \right)
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\end{equation}
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Then the positivity of each $D_{(i)}$ simply tells us that
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\begin{equation}
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x_i,y_i,z_i,w_i > 0
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\end{equation}
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while the mutual positivity conditions become
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\begin{equation}
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(x_i - x_j)(z_i - z_j) + (y_i - y_j)(w_i - w_j)<0
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\end{equation}
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In this note we study various aspects of the geometry defined by these
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inequalities, as well as the corresponding canonical form $\Omega$,
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which directly gives us the loop integrand for four-particle scattering. Of course the
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four-particle amplitude has been an object of intensive study
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for many years \cite{Bern:2006ew, Bern:2005iz, Bern:2007ct,
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Bourjaily:2011hi, Eden:2012tu}, with loop integrand now available
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through seven loops. But our approach will be fundamentally
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different from previous works. We will not begin by drawing planar
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diagrams made out of ``boxes", we will make no mention of recursion
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relations, we will not make ansatze for the integrand which are
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checked against cuts, and we will make no mention of physical
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constraints from exponentiation of infrared divergences etc.
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Instead, we will discover all the known general properties of the
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loop integrand, and many other properties besides, directly by
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studying the positive geometry of the amplituhedron.
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We will start with a lightning review of the one-loop geometry,
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which is just that of $G_+(2,4)$, mostly to define some notation and
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nomenclature. We then do some warm-up exercises for associating
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canonical forms $\Omega$ with spaces specified by particularly
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simple inequalities, which will come in handy in later sections. The
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first non-trivial case with mutual positivity is obviously two
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loops, and we show how to triangulate the space and extract the
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loop integrand, matching the well-known result given as a
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sum of two double-boxes. Interestingly, while our triangulation of
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the two-loop amplituhedron is manifestly ``positive", the sum of
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double-boxes is not, with each term having singularities outside the amplituhedron
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that only cancel in the sum.
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We then make some general observation on the structure of certain
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cuts of the amplitude, which correspond to various boundaries of the
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amplituhedron. In particular, the textbook understanding of
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unitarity as following from the break-up of the loop integrand into two
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parts sewed together on the ``unitarity cut" follows in a beautiful way from
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positive geometry. These general
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results and some further explicit triangulations also allow us to
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determine the three-loop integrand. We move on to exploring another
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natural set of cuts that take the amplitude into the multi-collinear
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region. This exposes a fascinating property of cuts of the
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multi-loop integrand: the residues depend not only on
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the final cut geometry, but also on the path taken to reach that
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geometry. Studying the combinatorics of this path dependence naturally
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motivates looking at the logarithm
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of the amplitude, and explains why the log has such good IR
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behavior.
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From our new perspective, the determination of the integrand to all loop orders
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requires a complete understanding of the full
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amplituhedron geometry. We have not yet achieved this yet, but we
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believe that a systematic approach to this problem is
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possible. As a prelude, we give a survey of
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some of the lower-dimensional ``faces" of amplituhedron. We can
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explicitly triangulate these faces and find their corresponding
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canonical forms, which give us cuts of the full integrand. This
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already gives us highly non-trivial all-loop order information about
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the integrand, in many cases not readily available from any other approach.
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\section{One Loop Geometry}
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At one loop we have a single line ${\cal L}_1 {\cal L}_2$, which we
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often also called ``$(AB)$". The geometry is given by the positive
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Grassmannian $G_+(2,4)$. The external data form a polygon in
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$\mathbb{P}^3$ with vertices $Z_1$, $Z_2$, $Z_3$, $Z_4$ and edges
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$Z_1Z_2$, $Z_2Z_3$, $Z_3Z_4$, $Z_1Z_4$.
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$$
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\includegraphics[scale=.65]{pic00.pdf}
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$$
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The line $AB={\cal L}_1 {\cal L}_2$ is parametrized as
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\begin{equation}
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{\cal L}^I_\gamma = D_{\gamma a}Z_a^I
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\end{equation}
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where $\gamma=1,2$ and $a,I=1,\dots,4$. The matrix $D$ represents a
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cell of positive Grassmannian $G_+(2,4)$, in the generic case it is a
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top cell. In one particularly convenient gauge-fixing we can write
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\begin{equation}
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D = \left(
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\begin{array}{cccc}
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1 & x &0 & -w \\
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0 & y & 1 & z \\
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\end{array}
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\right)\label{Cmatrix}
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\end{equation}
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where $x,y,z,w>0$. This gauge-fixing of the $D$ matrix covers all
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boundaries by sending variables $x,y,z,w$ to zero or infinity.
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The form with logarithmic singularities on the boundaries of the
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space is trivially
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\begin{equation}
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\Omega = \frac{dx}{x}\frac{dy}{y}\frac{dw}{w}\frac{dz}{z}
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\end{equation}
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The boundaries occur when one of the variables approaches $0$ or
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$\infty$. We can easily translate this expression back to
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momentum twistor space by solving two linear equations:
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\begin{equation}
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Z_A = Z_1 + xZ_2 - w Z_4,\qquad Z_B = yZ_2 + Z_3 + zZ_4
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\end{equation}
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which gives
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\begin{equation}
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\Omega = \frac{\la AB\,d^2Z_A\ra\la AB\,d^2Z_B\ra\la1234\ra^2} {\la
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AB12\ra\la AB23\ra\la AB34\ra\la AB14\ra}
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\end{equation}
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We now describe the boundaries of this space in detail--these are
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nothing but all the cells of $G_+(2,4)$, which have also been
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described at length in e.g. \cite{positive}. We describe them in
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detail here since the same geometry will arise repeatedly in the context of cuts of
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the multiloop amplitudes. At the level of the form they correspond
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to logarithmic singularities. In giving co-ordinates for the boundaries,
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we will freely use different gauge-fixings as convenient for
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any given case, with all parameters positive. They will always be
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trivially related to boundaries of (\ref{Cmatrix}).
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The first boundaries occur when line
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$AB$ intersects one of the lines $Z_1Z_2$, $Z_2Z_3$, $Z_3Z_4$ or
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$Z_1Z_4$. In the gauge-fixing (\ref{Cmatrix}) this sets one of the
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variables to $x,y,z,w$ to $0$. In particular,
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\begin{equation}
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\la AB12\ra =
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w,\quad \la AB23\ra = z,\quad \la AB34\ra = y,\quad \la AB14\ra = x
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\end{equation}
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where we suppressed $\la1234\ra$. For cutting $Z_1Z_2$, $\la AB12\ra
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= w=0$ we get
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$$
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\begin{minipage}[c]{0.23\textwidth}
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\includegraphics[scale=.65]{pic0.pdf}
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\end{minipage}
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\begin{minipage}[c]{0.2\textwidth}
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$$\left(
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\begin{array}{cccc}
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1 & x & 0 & 0 \\
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0 & y & 1 & z \\
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\end{array}
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\right)$$
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\end{minipage}
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$$
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In all four cases the form is the dlog of remaining three variables;
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we will suppress writing it explicitly.
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The second boundaries occur when the line $AB$
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intersects two lines $Z_iZ_{i\pl1}$ and $Z_j Z_{j \pl 1}$. There are two
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distinct cases. If we cut two non-adjacent lines $Z_1Z_2$, $Z_3Z_4$
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or $Z_2Z_3$, $Z_1Z_4$ there is just one solution. For the first one
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$\la AB12\ra=\la AB34\ra=0$ we have
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$$
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\begin{minipage}[c]{0.23\textwidth}
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\includegraphics[scale=.65]{pic1.pdf}
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\end{minipage}
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\begin{minipage}[c]{0.2\textwidth}
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$$\left(
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\begin{array}{cccc}
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1 & x & 0 & 0 \\
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0 & 0 & 1 & z \\
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\end{array}
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\right)\qquad
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%\Omega = \frac{dx}{x}\frac{dy}{y}
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$$
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\end{minipage}
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$$
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In the second case we intersect two adjacent lines. Let us cut
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$Z_1Z_2$, $Z_2Z_3$ (the other three cases are cyclically related), ie.
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$\la AB12\ra=\la AB23\ra=0$. There are two different solutions --
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either the line $AB$ passes through $Z_2$ or the line $AB$ lies in
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the plane $(Z_1Z_2Z_3)$.
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$$
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\begin{minipage}[c]{0.23\textwidth}
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\includegraphics[scale=.65]{pic2.pdf}
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\end{minipage}
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\begin{minipage}[c]{0.2\textwidth}
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$$\left(
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\begin{array}{cccc}
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1 & 0 & 0 & 0 \\
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0 & y & 1 & z \\
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\end{array}
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\right)
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%\qquad \Omega = \frac{dx}{x}\frac{dy}{y}
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$$
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\end{minipage}\qquad
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\begin{minipage}[c]{0.23\textwidth}
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\includegraphics[scale=.65]{pic3.pdf}
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\end{minipage}
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\begin{minipage}[c]{0.2\textwidth}
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$$
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\left(
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\begin{array}{cccc}
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1 & x & 0 & 0 \\
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0 & y & 1 & 0 \\
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\end{array}
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\right)
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%\qquad \Omega = \frac{dx}{x}\frac{dy}{y}
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$$
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\end{minipage}
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$$
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There are two different types of third boundaries. The first type is
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a triple cut -- the line $AB$ intersects three of four lines. One
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representative is $\la AB12\ra = \la AB23\ra= \la AB34\ra = 0$.
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There are two solutions to this problem. Either $AB$ passes through
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$Z_2$ and intersects the line $Z_3Z_4$ or $AB$ passes through $Z_3$
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and intersects the line $Z_1Z_2$.
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$$
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\begin{minipage}[c]{0.23\textwidth}
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\includegraphics[scale=.65]{pic4.pdf}
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\end{minipage}
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\begin{minipage}[c]{0.2\textwidth}
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$$\left(
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\begin{array}{cccc}
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0 & 1 & 0 & 0 \\
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0 & 0 & 1 & \alpha \\
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\end{array}
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\right)
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%\qquad \Omega=\frac{dx}{x}
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$$
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\end{minipage}\qquad
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\begin{minipage}[c]{0.23\textwidth}
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\includegraphics[scale=.65]{pic4b.pdf}
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\end{minipage}
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\begin{minipage}[c]{0.2\textwidth}
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$$\left(
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\begin{array}{cccc}
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0 & 0 & 1 & 0 \\
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-\alpha & -1 & 0 & 0 \\
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\end{array}
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\right)
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%\qquad \Omega=\frac{dx}{x}
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$$
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\end{minipage}
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$$
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There is also a ``composite" cut when we cut only two lines while
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imposing three constraints. We can pass $AB$ through $Z_2$ while lying in the plane
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plane $(Z_1Z_2Z_3)$.
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$$
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\begin{minipage}[c]{0.23\textwidth}
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\includegraphics[scale=.65]{pic5.pdf}
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\end{minipage}
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\begin{minipage}[c]{0.2\textwidth}
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$$\left(
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\begin{array}{cccc}
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0 & 1 & 0 & 0 \\
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-\alpha & 0 & 1 & 0 \\
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\end{array}
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\right)
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%\qquad \Omega= \frac{dx}{x}
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$$
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\end{minipage}
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$$
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Finally, for the quadruple cuts we can either cut all four lines
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which localizes $AB$ to $AB=Z_1Z_3$ or $AB=Z_2Z_4$, or we can
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consider the "composite'' cut $AB=Z_1Z_2$ (and cyclically related)
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which cuts only three lines (not $Z_3Z_4$) while still imposing four
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|
constraints.
|
|
$$
|
|
\begin{minipage}[c]{0.23\textwidth}
|
|
\includegraphics[scale=.65]{pic6.pdf}
|
|
\end{minipage}
|
|
\begin{minipage}[c]{0.2\textwidth}
|
|
$$\left(
|
|
\begin{array}{cccc}
|
|
1 & 0 & 0 & 0 \\
|
|
0 & 0 & 1 & 0 \\
|
|
\end{array}
|
|
\right)
|
|
%\qquad \Omega=1
|
|
$$
|
|
\end{minipage}\qquad
|
|
\begin{minipage}[c]{0.23\textwidth}
|
|
\includegraphics[scale=.65]{pic7.pdf}
|
|
\end{minipage}
|
|
\begin{minipage}[c]{0.2\textwidth}
|
|
$$\left(
|
|
\begin{array}{cccc}
|
|
1 & 0 & 0 & 0 \\
|
|
0 & 1 & 0 & 0 \\
|
|
\end{array}
|
|
\right)
|
|
%\qquad \Omega=1
|
|
$$
|
|
\end{minipage}
|
|
$$
|
|
|
|
\section{Warmup Exercises}
|
|
|
|
The amplituhedron is defined by various positivity conditions. We
|
|
will shortly be ``triangulating" the spaces defined by these inequalities and finding the
|
|
canonical form $\Omega$ associated with them. But it will be helpful
|
|
to practice on some simpler cases, which will also later be useful to
|
|
determining amplitudes and cuts of amplitudes.
|
|
|
|
Let us start with a trivial example; suppose we have
|
|
\begin{equation}
|
|
a < x < b
|
|
\end{equation}
|
|
It's obvious that the form is $\frac{1}{x - a} - \frac{1}{x - b}$,
|
|
but lets reproduce this in a heavy-handed way, from the viewpoint
|
|
using ``positive co-ordinates". In this case, we can write
|
|
\begin{equation}
|
|
x = a + (b - a) \frac{\alpha}{1 + \alpha}
|
|
\end{equation}
|
|
Note that for $\infty > \alpha > 0$, we cover the entire range of $a < x < b$. The canonical form is just
|
|
$\frac{d \alpha}{\alpha}$, which can be re-written in the original co-ordinates as
|
|
\begin{equation}
|
|
\frac{d \alpha}{\alpha} = \frac{dx}{x -a} - \frac{dx}{x - b} =
|
|
\frac{(a - b) dx}{(x - a)(x - b)}
|
|
\end{equation}
|
|
|
|
Next, consider $0 < x_1 < x_2$. Once again, we can use positive
|
|
variables
|
|
\begin{equation}
|
|
x_1 = \alpha_1, \qquad x_2 = \alpha_1 +
|
|
\alpha_2
|
|
\end{equation}
|
|
and the form is quite trivially
|
|
\begin{equation}
|
|
\frac{d \alpha_1}{\alpha_1} \frac{d \alpha_2}{\alpha_2} = \frac{dx_1
|
|
dx_2}{x_1 (x_2 - x_1)}
|
|
\end{equation}
|
|
We will henceforth skip the step of parametrization with
|
|
positive variables, and also omit the measure factor in
|
|
presenting results.
|
|
|
|
Next consider $0 < x_1 < x_2 < a$, the form
|
|
is
|
|
\begin{equation}
|
|
\left(\frac{1}{x_1} - \frac{1}{x_1 - a}\right)\left(\frac{1}{x_2 -
|
|
x_1} - \frac{1}{x_2 - a}\right) = \frac{a}{x_1 (x_2 - x_1) (a -
|
|
x_1)}
|
|
\end{equation}
|
|
This extends trivially to e.g. $0 < x_1 < x_2 < a < x_3 < x_4 < b$,
|
|
for which the form is
|
|
\begin{equation}
|
|
\frac{a b}{x_1 (x_2 - x_1) (a - x_2) (x_3 - a) (x_4 - x_3)(b - x_4)}
|
|
\end{equation}
|
|
We will find it convenient to use a notation to represent these
|
|
forms. Consider a chain of inequalities of the form $0 < X_1
|
|
< X_2 \cdots < X_N$. Some of the $X$'s are the variables our form
|
|
depends on, and some are constants like $a,b$ in our previous
|
|
examples. We will represent the constants by underlining the
|
|
corresponding $X$'s. In this notation, the form accompanying our two
|
|
examples above are denoted as $[x_1, x_2, \underline{a}]$ and
|
|
$[x_1,x_2,\underline{a},x_3,x_4,\underline{b}]$. As yet another
|
|
example,
|
|
|
|
\begin{align} [x_1, \underline{a},
|
|
\underline{b},x_2,x_3,\underline{c},x_4] = \left(\frac{1}{x_1} -
|
|
\frac{1}{x_1 - a}\right)\left(\frac{1}{x_2 - b} - \frac{1}{x_2 -
|
|
c}\right)\left(\frac{1}{x_3 - x_2} - \frac{1}{x_3 - c}\right)\left(
|
|
\frac{1}{x_4 - c}\right)
|
|
\end{align}
|
|
|
|
Next, suppose we have $x_i, y$ with $y > x_i$
|
|
for all $i$. Then, if the $x's$ are ordered so that $x_1 < \cdots, <
|
|
x_n$, we have $y> x_n$, and the form is
|
|
\begin{equation}
|
|
[x_1, \cdots, x_n, y] = \frac{1}{x_1} \frac{1}{x_2 - x_1} \cdots
|
|
\frac{1}{x_n - x_{n-1}} \frac{1}{y - x_n}
|
|
\end{equation}
|
|
Then we simply sum over all the permutations
|
|
\begin{equation}
|
|
\sum_\sigma [x_{\sigma_1}, \cdots, x_{\sigma_n}, y]
|
|
\end{equation}
|
|
Note that individual terms in this sum have spurious poles $(x_i -
|
|
x_j)$, which cancel in the sum. Indeed, in this simple case, it is
|
|
trivial to do the sum explicitly, and find
|
|
\begin{equation}
|
|
\frac{y^{n-1}}{(y - x_1)(y-x_2) \cdots (y - x_n) x_1 \cdots x_n}
|
|
\end{equation}
|
|
Extremely naively, we may have expected the product in the
|
|
denominator, but why is there is a numerator factor? The reason is
|
|
that otherwise, the form would not have only logarithmic
|
|
singularities! For instance, the residues on $x_1, \cdots, x_n \to
|
|
0$ would give $1/y^n$; it is the numerator that makes this $1/y$.
|
|
We can extend this to $y_I > x_i$ for a collection of $m$ $y$'s. This
|
|
means that the smallest $y$ is larger than the largest $x$. Thus the
|
|
form is
|
|
\begin{equation}
|
|
\sum_{\sigma, p} [x_{\sigma_1}, \cdots, x_{\sigma_n},y_{p_1}
|
|
\cdots, y_{p_m}]
|
|
\end{equation}
|
|
Again the spurious poles cancel in the sum, but the forms are more
|
|
interesting. In the simplest new case where $n=3, m=2$ the form is
|
|
\begin{equation}
|
|
\frac{x_1 x_2 x_3 y_1 + x_1 x_2 x_3 y_2 - x_1 x_2 y_1 y_2 - x_1 x_3
|
|
y_1 y_2 - x_2 x_3 y_1 y_2 + y_1^2 y_2^2}{x_1 x_2 x_3 (y_1 - x_1)(y_1
|
|
- x_2)(y_1 - x_3)(y_2 - x_1)(y_2 - x_2)(y_3 - x_3)}
|
|
\end{equation}
|
|
|
|
Let us now consider the inequality $x, y > 0$ and
|
|
also $x+ y < 1$, or $x + y > 1$. The first case is just the inside
|
|
of a triangle, while the second case is a quadrilateral:
|
|
$$
|
|
\includegraphics[scale=.75]{pic8.pdf}
|
|
$$
|
|
Obviously the form in the first case $x + y < 1$ is
|
|
\begin{equation}
|
|
\frac{-1}{x y (x + y - 1)}
|
|
\end{equation}
|
|
For the second case, the region can be broken into two pieces in
|
|
obvious ways. For instance, if $x > a$, there is no further
|
|
restriction on $y$, while if $x<1$, we must have $y > 1 - x$
|
|
$$
|
|
\includegraphics[scale=.75]{pic9.pdf}
|
|
$$
|
|
The form is then
|
|
\begin{equation}
|
|
\frac{1}{x - 1} \frac{1}{y} + \frac{1}{x (1 - x)} \frac{1}{y + x
|
|
- 1} = \frac{x + y}{x y (x + y - 1)}
|
|
\end{equation}
|
|
|
|
This form could have also been derived without any triangulation.
|
|
The denominator reflects all the inequalities as it should. However,
|
|
with a random numerator, we would have non-vanishing residue at the
|
|
origin $x=y=0$, which is clearly not in the space. The numerator
|
|
kills that residue, and the resulting form has logarithmic
|
|
singularities on the boundary of our space. We could
|
|
have also arrived at this form in another way. We know the form for $x +
|
|
y < 1$. Since the form with no restriction (other than positivity)
|
|
on $x,y$ is just $1/(xy)$, we conclude that the form for $x + y > 1$
|
|
is
|
|
\begin{equation}
|
|
\frac{1}{x y } - \frac{-a}{x y (x + y - 1)} = \frac{x + y}{x y (x +
|
|
y - 1)}
|
|
\end{equation}
|
|
|
|
As a final example, let us consider $x,y,a_1,b_1,a_2,b_2 > 0$, together with the two constraints
|
|
\begin{equation}
|
|
\frac{x}{a_1} + \frac{y}{b_1} > 1, \quad \frac{x}{a_2} +
|
|
\frac{y}{b_2}
|
|
> 1
|
|
\end{equation}
|
|
We will find the form by triangulating the space in two different
|
|
ways. In the first triangulation, begin by ordering $a_1 < a_2$ without
|
|
loss of generality; the final form will be obtained by symmetrizing
|
|
$1 \leftrightarrow 2$. The shape of the allowed region $x,y$ space
|
|
depends on whether $b_1 < b_2$ or $b_1 > b_2$:
|
|
$$
|
|
\includegraphics[scale=.83]{pic9a.pdf}
|
|
$$
|
|
If $b_1 < b_2$, then the space is essentially the same as the
|
|
quadrilateral we just studied. The associated form obtained by
|
|
breaking it up into the two regions $x> a_2$, and $0<x<a_2$, is
|
|
given by
|
|
\begin{equation}
|
|
[a_1,a_2][b_1,b_2] \left([\underline{a_2},x] \frac{1}{y} + [x, \underline{a_2}] \frac{1}{y + \frac{b_2 x}{a_2} -b_2} \right)
|
|
\end{equation}
|
|
|
|
If $b_1 > b_2$, we have a pentagonal shape. We can break this up into three regions, where $x>a_2$, $a_2 > x > a_{12}$ and $a_{12}> x > 0$. Here
|
|
$a_{12} = \frac{a_1 a_2 (b_1 - b_2)}{a_2 b_1 - a_1 b_2}$. The associated form is
|
|
\begin{equation}
|
|
[a_1,a_2] [b_2,b_1] \left([\underline{a_2},x] \frac{1}{y} + [\underline{a_{12}},x,\underline{a_2}] \frac{1}{y+ \frac{b_2 x}{a_2} - b_2} +
|
|
[x, \underline{a_{12}}] \frac{1}{y + \frac{b_1 x}{a_1} - b_1} \right)
|
|
\end{equation}
|
|
|
|
Summing these forms and symmetrizing in $1 \leftrightarrow 2$, all
|
|
the spurious poles cancel and we find for the final form
|
|
\begin{equation}
|
|
\frac{(\frac{x}{a_1} + \frac{y}{b_1})(\frac{x}{a_2} +
|
|
\frac{y}{b_2})}{x y a_1 b_1 a_2 b_2 (\frac{x}{a_1} + \frac{y}{b_1} -
|
|
1)(\frac{x}{y_2} + \frac{y}{b_2} - 1)}
|
|
\end{equation}
|
|
|
|
Note that we could also have arrived at this result in another
|
|
simpler way, by thinking of the constraints in $(a_1,b_1)$ and
|
|
$(a_2,b_2)$ spaces separately. For fixed $x,y$, if we redefine $A_i
|
|
= x/a_i$ and $B_i = y/b_i$, we just have $A_1 + B_1 > 1, A_2 + B_2 >
|
|
1$. We then get for the form
|
|
\begin{equation}
|
|
\frac{1}{x y} \times \frac{A_1 + B_1}{A_1 B_1 (A_1 + B_1 - 1)}
|
|
\times\frac{(A_2 + B_2)}{A_2 B_2 (A_2 + B_2 - 1)}
|
|
\end{equation}
|
|
which, including the trivial Jacobian factors from the change of
|
|
variables, reduces immediately to our above result obtained using
|
|
triangulation.
|
|
|
|
\section{Two Loops}
|
|
|
|
We now move on to studying the inequalities defining the
|
|
amplituhedron for four-particle scattering, starting at two-loops,
|
|
where we just have a single mutual positivity condition to deal
|
|
with, simply
|
|
\begin{equation}
|
|
(x_1 - x_2)(z_1 - z_2) + (y_1 - y_2)(w_1 - w_2) < 0\label{twoloop}
|
|
\end{equation}
|
|
Without loss of generality we can take $x_1 < x_2$. Then we have
|
|
\begin{equation}
|
|
z_1 - z_2 > \frac{(y_1 - y_2)(w_1 - w_2)}{x_2 - x_1}
|
|
\end{equation}
|
|
If either $y_1> y_2, w_1 > w_2$ or$y_1 < y_2, w_1 < w_2$, we have
|
|
$(y_1 - y_2)(w_1 - w_2) > 0$; the form is then
|
|
\begin{equation}
|
|
[x_1, x_2] \frac{1}{z_2} \frac{1}{z_1 - z_2 - \frac{(y_1 - y_2)(w_1
|
|
- w_2)}{x_2 - x_1}} \left([y_1,y_2][w_1,w_2] + [y_2,y_1][w_2,w_1]
|
|
\right)
|
|
\end{equation}
|
|
But if $y_1 < y_2, w_1 > w_2$ or $y_1> y_2, w_1 < w_2$, we have
|
|
\begin{equation}
|
|
z_2 - z_1 < - \frac{(y_1 - y_2)(w_1 - w_2)}{x_2 - x_1}
|
|
\end{equation}
|
|
Then the form is
|
|
\begin{align}
|
|
\frac{1}{x_1} \frac{1}{x_2 - x_1} \frac{1}{z_1} \left(\frac{1}{z_2} - \frac{1}{z_2 - z_1 + \frac{(y_1 - y_2)(w_1 - w_2)}{x_2
|
|
- x_1}}\right) \left([y_1,y_2] [w_2,w_1] + [y_2,y_1][w_1,w_2]
|
|
\right)
|
|
\end{align}
|
|
Finally, we just have to swap $1 \leftrightarrow 2$. The sum of
|
|
these terms is then
|
|
\begin{equation}
|
|
\frac{x_1 z_2 + x_2 z_1 + y_1 w_2 + y_2 w_1}{x_1 x_2 y_1 y_2 z_1 z_2
|
|
w_1 w_2 [(x_1 - x_2)(z_1 - z_2) + (y_1 - y_2)(w_1 -
|
|
w_2)]}\label{twoloopform}
|
|
\end{equation}
|
|
|
|
We can expand it as a sum of four terms by canceling terms in
|
|
numerator and denominator,
|
|
\begin{align}
|
|
\left(\frac{1}{x_2 y_1 y_2 z_1 w_1 w_2 [(x_1 - x_2)(z_1 - z_2) +
|
|
(y_1 - y_2)(w_1 - w_2)]} + 1 \leftrightarrow 2 \right) \nonumber\\ +
|
|
\left(\frac{1}{x_1 x_2 y_2 z_1 z_2 w_1 [(x_1 - x_2)(z_1 - z_2) +
|
|
(y_1- y_2)(w_1 - w_2)]} + 1 \leftrightarrow 2
|
|
\right)\label{doublebox}
|
|
\end{align}
|
|
We can solve for all variables in terms of momentum twistors, finding
|
|
\begin{align}
|
|
\left[\frac{\la1234\ra^3}{\begin{array}{c}\la AB12\ra\la AB23\ra\la
|
|
AB34\ra\la ABCD\ra\\ \la CD34\ra\la CD14\ra\la CD12\ra\end{array}}+
|
|
\frac{\la1234\ra^3}{\begin{array}{c}\la AB23\ra\la AB34\ra\la
|
|
AB14\ra\la ABCD\ra\\ \la CD14\ra\la CD12\ra\la
|
|
CD23\ra\end{array}}\right] + {\rm symmetrization} \label{doublebox2}
|
|
\end{align}
|
|
where by symmetrization we mean adding another two terms where we swap
|
|
$(AB)\leftrightarrow(CD)$. The expression
|
|
(\ref{doublebox2}) is the integrand for two double boxes
|
|
$$
|
|
\includegraphics[scale=.98]{pic10a.pdf}
|
|
$$
|
|
which is the standard representation of the two-loop amplitude. Note
|
|
that our approach gives the fully symmetrized (in
|
|
$(AB)\leftrightarrow(CD)$) integrand, so we get four terms
|
|
instead of two.
|
|
|
|
It is natural to ask whether some other triangulation of the space
|
|
may have directly given us this local expansion, but it is easy to
|
|
see that this is impossible: each double box individually has a cut which
|
|
is not allowed by the positivity conditions and is therefore
|
|
``outside" the amplituhedron. The cut is a simple one: suppose
|
|
we double cut one loop variable so that $D_{(1)}$ passes through the
|
|
point 1, while $D_{(2)}$ passes through the point $3$. The $D$
|
|
matrices on this cut have the form
|
|
\begin{equation}
|
|
D_{(1)} = \left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & y & 1 & z
|
|
\end{array} \right),\quad D_{(2)} = \left(\begin{array}{cccc} 1 &
|
|
x & 0 & -w \\ 0 & 0 & 1 & 0 \end{array}\right)
|
|
\end{equation}
|
|
But note that the mutual positivity condition between $D_{(1)}$ and
|
|
$D_{(3)}$ is automatically satisfied,
|
|
\begin{equation}
|
|
\langle D_{(1)} D_{(2)} \rangle = x z + y w > 0
|
|
\end{equation}
|
|
Because of this, we conclude that taking a further residue where
|
|
$(AB)_1 (AB)_2$ is cut must vanish, since there is no way to set
|
|
this to zero without further setting one of $x,z$, and $y,w$, to
|
|
zero.
|
|
|
|
This is a very simple and striking prediction of positivity, which
|
|
is true at any loop order: if we single out any two loops $(AB)_1,
|
|
(AB)_2$, and consider double cutting each so one line passes through
|
|
$1$ and the other through $3$, then the residue cutting $(AB)_1
|
|
(AB)_2$ vanishes. The vanishing of this cut is not manifest from
|
|
the local expansion. Even at two loops, each double box individually
|
|
obviously has support on this 5-cut, but the residue cancels in the
|
|
sum
|
|
$$
|
|
\includegraphics[scale=.98]{pic10.pdf}
|
|
$$
|
|
By contrast, obviously each term in our triangulation is compatible
|
|
with all positivity conditions. This mirrors familiar features of
|
|
the BCFW expansion for tree amplitudes: they correspond to
|
|
triangulations that are ``inside" the amplituhedron and manifestly
|
|
consistent with positivity properties (and therefore also the
|
|
symmetries of the theory), at the expense of manifest locality.
|
|
|
|
\section{Generalities on Cuts}
|
|
|
|
Before starting our more detailed exploration of multiloop
|
|
amplitudes, let us make some general observations about cuts of the
|
|
integrand.
|
|
|
|
\subsection*{Reconstruction from Single Cuts}
|
|
|
|
We are familiar with reconstructing the integrand from BCFW shifts
|
|
of the external data \cite{BCFWloop}. For instance, if we shift $Z_1 \to \hat{Z}_1 =
|
|
Z_1 + \alpha Z_4$, the integrand at $\alpha = 0$ is (the negative
|
|
of) the residues of the single cuts where $\langle (AB)_i\,\hat{1} 2
|
|
\rangle \to 0$. This is trivially reflected from positivity. We can
|
|
divide the $w_i$ space into the pieces where $w_1$ is smallest,
|
|
$w_2$ is smallest and so on. Suppose $w_1$ is smallest; then we can
|
|
set $w_i = w_1 + \hat{w}_i$. The remaining positivity conditions are
|
|
then exactly the same as the same as the computation of the single
|
|
cut where $w_1 \to 0$. And we have to sum over the single cuts for
|
|
setting each of the $w_i \to 0$.
|
|
|
|
Obviously, we can extend this to all the variables $(x,y,z,w)$. We
|
|
can always take sum $x_{i_x}, y_{i_y}, z_{i_z},w_{i_w}$ to be
|
|
smallest, and sum over all possible $i_x,i_y,i_z,i_w$. Then, we can
|
|
compute the integrand directly by summing over all these 4-cuts.
|
|
This naturally corresponds to using a residue theorem using an
|
|
extended BCFW deformation under which $Z_1 \to \hat{Z}_1 = Z_1 + \alpha
|
|
Z_4 + \beta Z_2, \, Z_3 \to \hat{Z}_3 = Z_3 + \gamma Z_2 + \rho
|
|
Z_4$.
|
|
|
|
\subsection*{Emergent Planarity and Leading Singularities}
|
|
Let us now consider the opposite extreme, and look at the
|
|
zero-dimensional faces of the amplituhedron. Here, each $D_{(i)}$ is
|
|
taken to be one of the zero dimensional cells of $G(2,4)$, where the
|
|
columns $i,j$ can be set to the identity and the remaining entries
|
|
are zero. From the mutual positivity of equation (\ref{twoloop}), we
|
|
learn something very simple right away: the configuration will
|
|
satisfy positivity in all cases except one: we can't have the $(13)$
|
|
cells and $(24)$ cells at the same time. It is trivial to see that
|
|
this fact extends to all $n$ MHV amplitudes at all loop
|
|
orders. If all the $AB_i = (ab)_i$ are drawn as chords on a disk,
|
|
then a configuration with lines that don't cross is allowed, but
|
|
a configuration with lines that cross violates positivity and must
|
|
have vanishing residue. Examples of allowed and non-allowed
|
|
configurations are shown below:
|
|
$$
|
|
\includegraphics[scale=.63]{pic11.pdf}
|
|
$$
|
|
|
|
|
|
The fact that our form can ultimately be expressed as a sum over
|
|
planar local integrands is not obviously built into the
|
|
geometry, but of course does emerge from it. We see this planarity
|
|
very vividly in the above structure of leading
|
|
singularities--clearly planar diagrams can only give us leading
|
|
singularities of the allowed type, while all other objects can give
|
|
us the illegal ``crossing" configurations. Indeed there are many
|
|
meaningful local integrands, compatible with the cyclic structure on
|
|
external data, which can nonetheless not be considered as ``planar".
|
|
A simple example is the square of one-loop amplitude, whose
|
|
integrand can be written in momentum twistor space as
|
|
\begin{equation}
|
|
\frac{\la1234\ra^4}{\la AB12\ra\la AB23\ra\la AB34\ra\la AB14\ra\la
|
|
CD12\ra\la CD23\ra\la CD34\ra\la CD14\ra}
|
|
\end{equation}
|
|
This integrand has an obvious leading singularity where e.g. $AB =
|
|
13$ and $CD = 24$, which cross in index space. This is a ``not
|
|
allowed" leading singularity that is incompatible with positivity.
|
|
Thus, we see that the planar structure of the integrand is not a
|
|
trivial consequence of cyclically ordered external data, but
|
|
actually emerges from the positive geometry of the amplituhedron. Note that
|
|
``planarity" is not an obvious invariant
|
|
property of the full integrand, but is only a natural statement
|
|
about a particular expansion of the integrand in terms of (local)
|
|
Feynman diagrams. It is thus perhaps not surprising that planarity
|
|
should be one of many derived properties of the integrand from the
|
|
amplituhedron point of view.
|
|
|
|
\section{Unitarity from Positivity}
|
|
|
|
In much of the recent work on scattering amplitudes in planar ${\cal
|
|
N}=4$ SYM, the unitarity of loop amplitudes has been directly
|
|
associated with correctly matching the single cut of the loop
|
|
integrand \cite{BCFWloop}, determined by the the forward limit \cite{Simon} of the lower-loop
|
|
amplitude. However there is an even simpler manifestation of
|
|
unitarity, familiar from the textbooks, in the double-cut (or
|
|
``unitarity cut") of the integrand, which is given by sewing
|
|
together two lower loop integrand.
|
|
$$
|
|
\includegraphics[scale=.75]{pic12.pdf}
|
|
$$
|
|
This is easy to translate to momentum-twistor language. Starting
|
|
with the $L$-loop integrand, we take one loop variable, $(AB)_L$,
|
|
to cut $(12)$ and $(34)$. The corresponding $D$ matrix is then of the form
|
|
\begin{equation}
|
|
D_L = \left(
|
|
\begin{array}{cccc}
|
|
1 & x & 0 & 0 \\
|
|
0 & 0 & y & 1 \\
|
|
\end{array}
|
|
\right)
|
|
\end{equation}
|
|
If we compute the residue of the integrand on this configuration,
|
|
unitarity tells us that the result must be
|
|
\begin{equation}
|
|
\frac{dx}{x} \frac{dy}{y} \times \hspace{-0.2cm}\sum_{L_1+L_2=L-1}
|
|
M_4^{L_1}(Z_1-xZ_2,Z_2,Z_3,Z_4-yZ_3)\,\,
|
|
M_4^{L_2}\left(Z_1,Z_2-\frac{1}{x}Z_1,Z_3-\frac{1}{y}Z_4,Z_4\right)
|
|
\end{equation}
|
|
We will now see that this result follows in a simple and beautiful
|
|
way from the positive geometry of the amplituhedron.
|
|
|
|
On the unitarity cut, the positivity conditions are the usual ones
|
|
for the $(L-1)$ loop variables. For $D_L$ we just have that $x,y >
|
|
0$. The mutual positivity between $D_L$ and the remaining $D_i$ just
|
|
tells us that
|
|
\begin{equation}
|
|
(23)_i + xy(14)_i - x (13)_i - y(24)_i > 0\label{eq1}
|
|
\end{equation}
|
|
This condition also tells us that
|
|
\begin{align}
|
|
&[(13) - y(14)][(24) - x(14)] =(13)(24) - x(13)(14) - y(14)(24) + x y (14)^2 \\
|
|
&\hspace{3cm}= (12)(34) + (14)[(23) + x y (14) - x(13) - y(24)]
|
|
> (12)(34) > 0\nonumber
|
|
\end{align}
|
|
where in the second line we used $(13)(24) = (12)(34) + (23)(14)$.
|
|
Now, obviously we can divide the space of each $D_i$ into ones where
|
|
$(13)_i - (14)_i> 0$, and $(13)_i - y (14)_i< 0$, similarly $(24)_i
|
|
- x(14)_i > 0$ or $(24)_i - x (14)_i <0$. However, if the product of
|
|
these two factors is negative it is impossible to satisfy equation
|
|
(\ref{eq1}). Thus, for each $i$, we have {\it either} that
|
|
\begin{equation}
|
|
(13) - y (14) > 0 \quad {\rm and} \quad (24) - x (14) > 0
|
|
\end{equation}
|
|
or
|
|
\begin{equation}
|
|
(13) - y (14) > 0, \quad {\rm and} \quad (24) - x (14) > 0
|
|
\end{equation}
|
|
|
|
Let us say that $L_1$ of the lines $D_{a}$ satisfy the first
|
|
inequality and the remaining $L_2 = L - 1 - L_1$ lines $D_{A}$
|
|
satisfy the second inequality. Explicitly, in the first case the
|
|
region is represented by positivity conditions
|
|
$$
|
|
(12)_a>0,(13)_a-y(14)_a>0,(14)_a>0,(23)_a>0,(24)_a-x(14)_a>0,(34)_a>0
|
|
$$
|
|
\begin{equation}
|
|
(23)_a + xy(14)_a - x (13)_a - y(24)_a> 0
|
|
\end{equation}
|
|
Let us define shifted columns
|
|
\begin{equation}
|
|
(\hat{3})_a = (3)_a- y(4)_a,\qquad (\hat{2})_a = (2)_a-x(1)_a
|
|
\end{equation}
|
|
Thus the set of positivity conditions become
|
|
\begin{equation}
|
|
(1\hat{2})_a>0,(1\hat{3})_a>0,(14)_a>0,(\hat{2}\hat{3})_a>0,(\hat{2}4)_a>0,(\hat{3}4)_a>0
|
|
\end{equation}
|
|
In the second region we have
|
|
$$
|
|
(12)_A>0,y(14)_A-(13)_A>0,(14)_A>0,(23)_A>0,x(14)_A-(24)_A>0,(34)_A>0
|
|
$$
|
|
\begin{equation}
|
|
(23)_A + xy(14)_A - x (13)_A - y(24)_A > 0
|
|
\end{equation}
|
|
Let us define shifts
|
|
\begin{equation}
|
|
(\hat{1})_A = (1)_A - \frac{1}{x}(2)_A,\qquad (\hat{4})_A =
|
|
(4)_A-\frac{1}{y}(3)_A
|
|
\end{equation}
|
|
Then the set of positivity conditions become
|
|
\begin{equation}
|
|
(\hat{1}2)_A>0,(\hat{1}3)_A>0,(\hat{1}\hat{4})_A>0,(23)_A>0,(2\hat{4})_A>0,(3\hat{4})_A>0
|
|
\end{equation}
|
|
|
|
Now, we come to the positivity conditions internal to the $D_{a}$'s, internal to the $D_{A}$'s, and also the ones between $D_{a}$ and $D_{A}$'s. Actually quite strikingly, the $D_a$'s and $D_A$'s are
|
|
automatically mutually positive! We look at
|
|
\begin{equation}
|
|
(12)_a(34)_A+(23)_a(14)_A+(34)_a(12)_A+(14)_a(23)_A-(13)_a(24)_A-(24)_a(13)_A \label{pos4}
|
|
\end{equation}
|
|
Rewriting this in terms of the natural shifted variables
|
|
\begin{equation}
|
|
(2)_a =(\hat{2})_a + x(1)_a,\quad (3)_a = (\hat{3})_a+y(4)_a \quad
|
|
(1)_A = (\hat{1})_A + \frac{1}{x}(2)_A,\quad (4)_A=
|
|
(\hat{4})_A+\frac{1}{y}(3)_A
|
|
\end{equation}
|
|
and plugging into (\ref{pos4}) we find
|
|
\begin{align}
|
|
&(1\hat{2})_a(3\hat{4})_A+[(\hat{2}\hat{3})_a+xy(14)_a+y(\hat{2}4)_a+x(1\hat{3})_a]
|
|
\left[\frac{1}{xy}(23)_A+(\hat{1}\hat{4})_A+\frac{1}{x}(2\hat{4})_A+\frac{1}{y}(\hat{1}3)_A\right]\nonumber\\
|
|
&\hspace{-0.7cm}+(\hat{3}4)_a(\hat{1}2)_A+(14)_a(23)_A -
|
|
[(1\hat{3})_a+y(14)_a]\left[(2\hat{4})_A+\frac{1}{y}(23)_A\right]-[(\hat{2}4)_a+x(14)_a]\left[(\hat{1}3)_A+\frac{1}{x}(23)_A\right]\nonumber\\
|
|
&=(1\hat{2})_a(3\hat{4})_A+(\hat{3}4)_a(\hat{1}2)_A+\frac{1}{y}[(\hat{2}\hat{3})_a+x(1\hat{3})_a](\hat{1}3)_A
|
|
+x[(1\hat{3})_a+y(14)_a](\hat{1}\hat{4})_A\nonumber\\
|
|
&+[(\hat{2}\hat{3})_a+y(\hat{2}4)_i](14)_A+\frac{1}{x}[(\hat{2}\hat{3})_a+y(\hat{2}4)_i](2\hat{4})_A
|
|
+\frac{1}{xy}(\hat{2}\hat{3})_a(23)_A>0
|
|
\end{align}
|
|
|
|
The positivity here is quite non-trivial; the expression many terms
|
|
with plus and minus signs that cancel each other, leaving only
|
|
pluses.
|
|
|
|
The mutual positivity internally for the $D_a$'s (or the $D_A$'s) are
|
|
exactly the same for the shifted and unshifted columns, since the
|
|
$(4 \times 4)$ determinants are unchanged in shifting a column by a
|
|
multiple of another. These can be easily translated in shifts of
|
|
external twistors. Under ${\cal A}_\gamma = D_{\gamma a} \cdot Z_a$,
|
|
we have for the first shift
|
|
\begin{align}
|
|
A = D\cdot Z &= (1)Z_1+(\hat{2})Z_2+(\hat{3})Z_3
|
|
+(4)Z_4\nonumber\\&=
|
|
(1)Z_1+(2)Z_2-x(1)Z_2+(3)Z_3-y(4)Z_3+(4)Z_4\nonumber\\ &=
|
|
(1)\hat{Z_1}+(2)Z_2+(3)Z_3+(4)\hat{Z_4}
|
|
\end{align}
|
|
Thus, the form for the $L_1$ lines is
|
|
\begin{equation}
|
|
M_4^{L_1}(Z_1-xZ_2,Z_2,Z_3,Z_4-yZ_3)
|
|
\end{equation}
|
|
and analogously the form for the $L_2$ lines is
|
|
\begin{equation}
|
|
M_4^{L_2}\left(Z_1,Z_2-\frac{1}{x}Z_1,Z_3-\frac{1}{y}Z_4,Z_4\right)
|
|
\end{equation}
|
|
|
|
Thus, we conclude that the unitarity cut is
|
|
\begin{equation}
|
|
\frac{dx}{x} \frac{dy}{y} \times\hspace{-0.3cm} \sum_{L_1+L_2=L-1}
|
|
M_4^{L_1}(Z_1-xZ_2,Z_2,Z_3,Z_4-yZ_3)\,\,
|
|
M_4^{L_2}\left(Z_1,Z_2-\frac{1}{x}Z_1,Z_3-\frac{1}{y}Z_4,Z_4\right)
|
|
\end{equation}
|
|
precisely as needed to enforce unitarity.
|
|
|
|
\section{Three Loops}
|
|
Having established these general results, let us turn to the three-loop
|
|
amplitude. Recall from our general discussion that it
|
|
suffices to look at various cuts of the amplitude, coming from
|
|
taking $x_{\sigma_1}, y_{\sigma_2}, z_{\sigma_3}, w_{\sigma_4}$ to
|
|
be smallest. For the case of three loops, at least one pair of
|
|
$\sigma_1, \sigma_2, \sigma_3, \sigma_4$ will correspond to the same
|
|
loop, thus, to compute the full three-loop integrand, it suffices to
|
|
compute the cut of the integrand where one loop is double-cut. We
|
|
have already verified that the unitarity double-cut is correctly
|
|
reproduced at any loop order. It thus suffices to compute the
|
|
remaining double cuts, which we call the ``corner cuts": where the
|
|
line passes through one of the points $Z_i$, or its parity
|
|
conjugate, where the line lies in the plane $(Z_{i-1} Z_i Z_{i+1})$.
|
|
Since these are parity conjugate, it is enough to compute one of
|
|
them, which we take to be the cut where the line corresponding to
|
|
the third loop passes through point 4. It will be convenient to use
|
|
a different gauge-fixing for the third loop
|
|
\begin{equation}
|
|
D_{(3)} = \left(\begin{array}{cccc} a & 1 & b & 0 \\ 0 & 0 & 0 & 1
|
|
\end{array} \right)
|
|
\end{equation}
|
|
If we further rescale the variables for the remaining loop variables
|
|
as $w_i \to w_i/b, y_i \to b y_i; z_i \to z_i/a, x_i \to a x_i$, the
|
|
remaining positivity conditions become
|
|
\begin{equation}
|
|
x_i + y_i > 1, \quad (x_1 - x_2)(z_1 - z_2) + (y_1 - y_2)(w_1 - w_2)
|
|
< 0
|
|
\end{equation}
|
|
We can assume that $x_1 < x_2$, so that just as for two-loops we
|
|
have then sum at the end over $1 \leftrightarrow 2$. Let us also
|
|
define
|
|
\begin{equation}
|
|
Z_+ = \frac{1}{z_2} \frac{1}{z_1 - z_2 - \frac{(y_1 - y_2)(w_1 -
|
|
w_2)}{x_2 - x_1}}, \quad Z_- = \frac{1}{z_1} \left(\frac{1}{z_2} -
|
|
\frac{1}{z_2 - z_1 - \frac{(y_2 - y_1)(w_1 - w_2)}{x_2 - x_1}}
|
|
\right)
|
|
\end{equation}
|
|
Then, by dividing the space into pieces much as we did at 2-loops,
|
|
we find that the form is
|
|
\begin{align}
|
|
&[x_1, x_2, \underline{1}] ( [\underline{1 - x_1}, y_1, y_2]
|
|
([w_1,w_2] Z_+ + [w_2,w_1]
|
|
Z_-)\\
|
|
&+([\underline{1 - x_2}, y_2, \underline{1 - x_1}, y_1] + [
|
|
\underline{1 - x_1},y_2, y_1]) ([w_2,w_1] Z_+ + [w_1,w_2]
|
|
Z_-))\nonumber\\
|
|
&+ [x_1, \underline{1}, x_2]( [\underline{1 - x_1}, y_1, y_2] ([w_1,
|
|
w_2] Z_+ + [w_2,w_1] Z_-)\nonumber\\
|
|
&+ ([y_2, \underline{1 - x_1}, y_1] + [\underline{1 - x_1}, y_2,
|
|
y_1])([w_2,w_1] Z_+ + [w_1,w_2] Z_-))\nonumber\\
|
|
&+ [\underline{1}, x_1, x_2] ([y_1, y_2]([w_1,w_2] Z_+ + [w_2,w_1]
|
|
Z_-) + [y_2,y_1]([w_2,w_1] Z_+ + [w_1,w_2] Z_-)\nonumber
|
|
\end{align}
|
|
Adding $1 \leftrightarrow 2$, all spurious poles cancel and we
|
|
obtain
|
|
\begin{align} \frac{\Bigg\{
|
|
\begin{array}{c}
|
|
w_2 x_1 x_2 y_1 + w_2 x_2 y_1^2 + w_1 x_1 x_2 y_2 - w_1 y_1 y_2 -
|
|
w_2 y_1 y_2 + w_2 x_1 y_1 y_2 + w_1 x_2 y_1 y_2 \\
|
|
+ w_2 y_1^2 y_2 +w_1 x_1 y_2^2 + w_1 y_1 y_2^2 - x_1 x_2 z_1 + x_1
|
|
x_2^2 z_1 + x_2^2 y_1 z_1 + x_1 x_2 y_2 z_1 \\
|
|
+ x_2 y_1 y_2 z_1 - x_1 x_2 z_2 + x_1^2 x_2 z_2+ x_1 x_2 y_1 z_2 +
|
|
x_1^2 y_2 z_2 + x_1 y_1 y_2 z_2
|
|
\end{array}\Bigg\}}{abx_1 x_2 y_1 y_2 z_1 z_2 w_1 w_2 (x_1 + y_1 - 1) (x_2 + y_2 - 1)
|
|
((x_2 - x_1)(z_1 - z_2) + (y_2 - y_1)(w_1 - w_2))}
|
|
\end{align}
|
|
This matches what we get from the familiar local expansion, as a sum
|
|
over ladders and ``tennis court" diagrams:
|
|
$$
|
|
\includegraphics[scale=.9]{pic18.pdf}
|
|
$$
|
|
|
|
\section{Multi-Collinear Region}
|
|
|
|
We have seen that a particular double cut of a single loop--the unitarity cut-- is
|
|
simply expressed in terms of (shifted) lower-loop objects. It is thus
|
|
natural to look at the other two kinds of double cuts. Let us
|
|
consider the cut where the $L^{th}$ line passes through 2. It will
|
|
be convenient to use a different gauge-fixing for this last line
|
|
\begin{equation}
|
|
D_{(L)} = \left( \begin{array}{cccc}0 & 1 & 0 & 0 \\ -\alpha & 0 & 1
|
|
& \gamma \end{array} \right)
|
|
\end{equation}
|
|
Then the mutual positivity conditions between $D_{(L)}$ and the
|
|
other lines is simply
|
|
\begin{equation}
|
|
\alpha w_i + z_i > \gamma
|
|
\end{equation}
|
|
It is amusing that from the point of view of the
|
|
lower-loop problem, we are simply putting a simple additional
|
|
restriction on the allowed region for $w_i,z_i$.
|
|
|
|
Despite the apparent simplicity of this deformation of the $L-1$
|
|
loop problem, unlike the unitarity cut, this double cut can't be
|
|
determined in terms of shifts of lower-loop problems in a
|
|
straightforward way. However, there is a further, triple cut, which
|
|
does have a very simple interpretation. Consider the limit where
|
|
$\beta \to 0$. This is the collinear region, where the line passes
|
|
through the point 2 while lying in the plane (123) \cite{localintegrand}. Note that the
|
|
positivity condition is now automatically satisfied, and so the cut
|
|
is trivial:
|
|
\begin{equation}
|
|
A_L^{{\rm coll.}} = \frac{d \alpha}{\alpha} \times A_{L-1}
|
|
\end{equation}
|
|
|
|
In this discussion we assumed that all the lines but one are
|
|
generic. We now investigate what happens when $l$ lines are sent
|
|
into the collinear region. The most general way this can happen is
|
|
to start with $L_{12}$ lines cutting $(12)$ and $L_{23}$ lines
|
|
cutting $(23)$. Let us gauge-fix in a convenient way, and write for
|
|
the two sets of lines
|
|
\begin{equation}
|
|
D_{(i)} = \left(\begin{array}{cccc} \beta_i & 1 & 0 & 0 \\ -\alpha_i
|
|
& 0 & 1 & \gamma_i \end{array} \right) \quad D_{(I)} =
|
|
\left(\begin{array}{cccc} 0 & 1 & \rho_I & 0
|
|
\\ \alpha_I & 0 & 1 & \gamma_I \end{array} \right)
|
|
\end{equation}
|
|
|
|
In order to reach the collinear limit, we must send $\beta_i,
|
|
\gamma_i \to 0$, and $\rho_I, \delta_I \to 0$. We can send these to
|
|
zero in different ways, but let us focus on one for definiteness,
|
|
the other cases can be treated similarly. For the lines cutting
|
|
$(12)$, we first take them to pass through $2$, and then move them
|
|
into the collinear region where they lie in $(123)$; in other words,
|
|
we first send $\beta_i \to 0$, and then $\gamma_i \to 0$. Similarly
|
|
for the lines intersecting $(23)$, we first send them to pass
|
|
through $2$, then to lie in $(123)$, so that we put the $\rho_I \to
|
|
0$, then send $\gamma_I \to 0$. Now, the positivity conditions
|
|
between these lines are just
|
|
\begin{align}
|
|
(\beta_i -\beta_j)(\gamma_i - \gamma_j) > 0, \quad (\rho_I -
|
|
\rho_J)\left(\frac{\gamma_I}{\alpha_I} -
|
|
\frac{\gamma_J}{\alpha_J}\right) > 0, \quad (\beta_i - \alpha_I
|
|
\rho_I) (\gamma_i - \gamma_I) > 0
|
|
\end{align}
|
|
|
|
Collectively, these tell us something simple. Suppose we take the
|
|
lines to pass through $2$ in some particular order, say by first
|
|
taking $\beta_ 1 \to 0$, then $\beta_2 \to 0$, then $\rho_1 \to 0$,
|
|
then $\rho_2 \to 0$, then $\beta_3 \to 0$ etc. Then, the cut
|
|
vanishes unless the lines are taken into the collinear limit in
|
|
exactly the same order! In this case, the cut is just
|
|
\begin{equation}
|
|
\prod_{a=1}^{l} \frac{d \alpha_a}{\alpha_a} \times M^{L-l}
|
|
\end{equation}
|
|
|
|
\section{Log of the Amplitude}
|
|
|
|
Scattering amplitudes have well-known double-logarithmic infrared
|
|
divergences, arising precisely from loop integration in the
|
|
collinear region. At $L$ loops, we have a log$^{2L}$ divergence,
|
|
which exponentiates in a well-known way; the logarithm of the
|
|
amplitude only has a log$^2$ divergence. This is a
|
|
motivation for looking at the log of the amplitude from a
|
|
physical point of view. But as we have just seen, the loop integrand
|
|
form also has an extremely simple behavior in the multicollinear
|
|
limit. We will now see that this behavior, together with some very
|
|
simple combinatorics, already motivates looking at the logarithm of
|
|
the amplitude directly at the level of the integrand. While the
|
|
amplitude itself has a non-vanishing residue when one loop momentum
|
|
is brought into the collinear region, we will see that the log of
|
|
the amplitude vanishes in the multicollinear region, unless {\it
|
|
all} $L$ loop momenta are taken into the collinear region. The
|
|
residue depends in a non-trivial way on the specific path taken into
|
|
the collinear region. Furthermore, we will see that the log of the
|
|
amplitude naturally leads us to consider {\it all} the natural
|
|
``positive regions" we can think of related to amplituhedron
|
|
geometry.
|
|
|
|
Let us start by introducing a generating function combining together
|
|
the amplitude at all loop order, otherwise known as the amplitude
|
|
itself:
|
|
\begin{equation}
|
|
M = 1 + g M_1 + g^2 M_2 + \cdots
|
|
\end{equation}
|
|
Now, consider for any function $f$, the expansion for $f(M)$.
|
|
Suppose that
|
|
\begin{equation}
|
|
f(1 + x) = x + a_2 x^2 + a_3 x^3 + \cdots
|
|
\end{equation}
|
|
then
|
|
\begin{eqnarray}
|
|
f(M) &=& (g M_1 + g^2 M_2 + \cdots) + a_2(g^2 A_1^2 + 2 g^3 M_1 M_2
|
|
+ \cdots) + a_3 g^3 M_3 + \cdots \nonumber \\ &=& g M_1 + g^2(M_2 +
|
|
a_2 M_1^2) + g^3(M_3 + 2 a_2 M_1 M_2 + a_3M_1^3) + \cdots
|
|
\end{eqnarray}
|
|
|
|
We'd now like to extract the permutation-invariant integrand from
|
|
this expression at $L$ loops. For instance,
|
|
\begin{eqnarray}
|
|
M_3 &=& \int d^4 x_1 d^4 x_2 d^4 x_3\,\, M_3(x_1,x_2,x_3) \nonumber \\
|
|
M_1 M_2 &=& \int d^4 x_1 d^4 x_2 d^4 x_3\,\, \left[(M_1(x_1) M_2(x_2,x_3) + M_1(x_2) M_2(x_1,x_3) + M_1(x_3) M_2(x_1,x_2) \right] \nonumber \\
|
|
M_1^3 &=& \int d^4 x_1 d^4 x_2 d^4 x_3\,\, M_1(x_1)M_1(x_2)M_1(x_3)
|
|
\end{eqnarray}
|
|
Actually, for the combinatorics, the ``$\int d^4 x$" are irrelevant.
|
|
Instead, we define a generating function
|
|
\begin{equation}
|
|
M = 1 + \sum_i x_i (i) + \sum_{i<j} x_i x_j (ij) + \sum_{i<j<k} x_i
|
|
x_j x_k (ijk) + \cdots
|
|
\end{equation}
|
|
Here $``(1)"$ stands for $M_1(x_1)$, $``(134)"$ stands for
|
|
$M_3(x_1,x_3,x_4)$ and so on. Then, the integrand for $f(M)$ at $L$
|
|
loops is just the coefficient of $(x_1 \cdots x_L)$ in the expansion
|
|
of $f(A)$, or put another way
|
|
\begin{equation}
|
|
f(M)^{L-loop} = \partial_{x_1 \cdots x_L} f(M)|_{x_1 = \cdots =
|
|
x_L = 0}
|
|
\end{equation}
|
|
|
|
Obviously, if we are only interested in $L$ loops, we can truncate
|
|
the $x_i$ to just $x_1, \cdots, x_L$ if we like. Thus, explicitly,
|
|
for $L=3$, we have
|
|
\begin{align}
|
|
M = 1 + x_1 (1) + x_2 (2) + x_3(3) + x_1 x_2 (12) + x_1 x_3 (13) +
|
|
x_2 x_3 (23) + x_1 x_2 x_3 (123)
|
|
\end{align}
|
|
and foreseeing our future interest, for $f(M) =$ log$(M)$, we have
|
|
for the 3-loop log of the amplitude
|
|
\begin{equation}
|
|
({\rm log} M)^{3-loop} = (123)+ 2 (1) (2) (3)-[(1) (23) + (2)(13) +
|
|
(3)(12)] \label{log3loop}
|
|
\end{equation}
|
|
%\begin{eqnarray}
|
|
%({\rm log} M)^{3-loop} &=& M_3(x_1,x_2,x_3)+ 2 M_1(x_1) M_1(x_2)
|
|
%M_1(x_3) \nonumber \\ &-& [M_1(x_1) M_2(x_2,x_3) + M_1(x_2)
|
|
%M_2(x_1,x_3) - M_1(x_3) M_2(x_1,x_2)]\label{log3loop}
|
|
%\end{eqnarray}
|
|
|
|
We would now like to compute the cut of $f(M)$ in the
|
|
multi-collinear limit. Suppose $L$ lines are
|
|
sent to pass through $2$ in some order $(1,\cdots, L)$. We already
|
|
know that the cut of the amplitude in the multi-collinear limit
|
|
depends on the order in which the lines are then sent to the
|
|
collinear region--indeed for the amplitude the cut vanishes unless
|
|
the lines are sent to the collinear limit in the same order
|
|
$(1,\cdots, L)$. This will not in general be true for $f(M)$.
|
|
Suppose that a set of $l$ lines are moved into the collinear limit
|
|
in some order $\sigma = \{\sigma_1, \cdots, \sigma_l\}$. For
|
|
instance for $L=3$, we could take $\sigma = \{2\}$ or $\sigma=
|
|
\{13\}$ or $\sigma = \{231\}$. Then, it is easy to see that the
|
|
multi-collinear cut of $f(M)$ can be computed as follows.
|
|
|
|
We first
|
|
play the following game, to produce a new generating function
|
|
$M^{\sigma}$: (I) if an ordered subset of $\sigma$ occurs out of
|
|
order in the brackets of $M$, we drop that term. (II) We then delete
|
|
all the labels in $\sigma$. Let's illustrate this for $L=3$ with
|
|
$\sigma = \{31\}$. The terms $x_1 x_3 (13)$ and $x_1 x_2 x_3 (123)$
|
|
in $M$ are dropped, and the rest are kept. Then, we drop the
|
|
indices $3,1$, and are left with
|
|
\begin{equation}
|
|
M^{\{31\}} = 1 + x_1 + x_2 (2) + x_3 + x_1 x_2 (2) + x_2 x_3 (2)
|
|
\end{equation}
|
|
|
|
The multi-collinear cut of $f(M)$ is easily seen to be just
|
|
\begin{equation}
|
|
\prod \frac{d \alpha_{\sigma_i}}{\alpha_{\sigma_i}} \times
|
|
\partial_{x_1, \cdots x_L} f(M^\sigma)|_{x_1 = \cdots = x_L = 0}
|
|
\end{equation}
|
|
|
|
We are now ready to see why the logarithm of the amplitude is so
|
|
natural from a purely combinatorial point of view. Let us return to
|
|
looking at $M^{\{ 31 \}}$. Observe that while $M$ is itself and
|
|
irreducible polynomial, $M^{\{31\}}$ factorizes as
|
|
\begin{equation}
|
|
M^{\{31\}} = [1+ x_1 + x_3][1 + x_2 (2)]
|
|
\end{equation}
|
|
Note that the second factor is just the $M$ polynomial made of the
|
|
undeleted variables, while the first factor is the generating
|
|
function made of the variables with only terms in correct order
|
|
kept.
|
|
|
|
This is a general statement. For any $\sigma$, let
|
|
$\bar{\sigma}$ be the complementary set. Then
|
|
\begin{equation}
|
|
M^{\sigma} = \left(1 + \sum_i x_{\sigma_i} + \sum_{i<j; \sigma_i <
|
|
\sigma_j} x_{\sigma_i} x_{\sigma_j} + \cdots\right) \times
|
|
M^{\bar{\sigma}}
|
|
\end{equation}
|
|
To illustrate with a more non-trivial example, say for $L=5$ and
|
|
$\sigma = \{415\}$, we have
|
|
\begin{align}
|
|
M^{\{415\}} = (1 + x_1 + x_4 + x_5 + x_1 x_5 + x_4 x_5) \times (1 +
|
|
x_2(2) + x_3(3) + x_2 x_3 (23))
|
|
\end{align}
|
|
|
|
Note that if $\sigma$ is anything other than the empty set, the
|
|
factorization in non-trivial. Because of this factorization, it is
|
|
natural to consider the log of the object. Then, we see that for any
|
|
string $\sigma$ of length $0 < l < L$,
|
|
\begin{equation}
|
|
\partial_{x_1\cdots x_L} {\rm log} M^{\sigma} = 0
|
|
\end{equation}
|
|
|
|
We thus learned something remarkable: if we take the log of the
|
|
amplitude, then the cut taking any number $l < L$ of the loop
|
|
variables into the collinear region vanishes! Only if all $L$ lines
|
|
are taken into the collinear region together, can we get something
|
|
non-zero. This explains why the log of the amplitude only has a
|
|
log$^2$ divergence. We get a log$^2$ divergence from each loop
|
|
momentum brought into this collinear region. For the amplitude
|
|
itself, we can bring all $L$ lines into the collinear region one at
|
|
a time, and thus we get the log$^{2L}$ IR divergence. However for
|
|
the log of the amplitude, since all $L$ lines must be brought in the
|
|
collinear region together we just get a single overall log$^2$
|
|
divergence. (Note that had even this limit given us no residue, the
|
|
log would have have been completely IR divergence free!).
|
|
|
|
Very interestingly, the logarithm of the amplitude doesn't have the
|
|
property, familiar for the amplitude itself, of having ``unit
|
|
leading singularities". If all $L$ lines are taken into the
|
|
collinear region in an order $\sigma = (\sigma_1, \cdots,
|
|
\sigma_L)$, then the residue is
|
|
\begin{equation}
|
|
\partial_{x_1 \cdots x_L} {\rm log}\left(1 + \sum_i x_i + \hspace{-0.2cm}\sum_{i<j; \sigma_i < \sigma_j} x_i x_j + \cdots\right)|_{x_i=0}
|
|
\end{equation}
|
|
|
|
Note that unlike the amplitude itself, which is only non-vanishing
|
|
for $\sigma = (1,2,\cdots, n)$, the log of the amplitude vanishes in
|
|
this case, since
|
|
\begin{equation}
|
|
\left(1 + \sum_i x_i + \hspace{-0.2cm}\sum_{i<j; \sigma_i <
|
|
\sigma_j} x_i x_j + \cdots\right) = (1 +x_1) \cdots (1+x_n)
|
|
\end{equation}
|
|
maximally factorizes! In the other extreme, if $\sigma = (n,
|
|
\cdots, 1)$ is oppositely ordered to $(1, \cdots, n)$, then we have
|
|
\begin{equation}
|
|
\partial_{x_1 \cdots x_L} (1 + x_1 + \cdots x_L)|_{x_i = 0} = (L-1)!
|
|
\end{equation}
|
|
In general, we can find residues ranging from $1$ to $(L-1)!$. For instance, at 4 loops,
|
|
we have the non-vanishing residues 1,2,3, 4 and 6, coming from the following paths:
|
|
\begin{align}
|
|
&1: (2,3,4,1)(2,4,1,3)(3,1,4,2)(4,1,2,3)\quad
|
|
2:(2,4,3,1)(3,2,4,1)(4,1,3,2)(4,2,1,3) \nonumber\\
|
|
&3:(3,4,1,2) \quad4: (3,4,2,1)(4,2,3,1)(4,3,1,2) \quad 6:(4,3,2,1)
|
|
\quad 0:\mbox{other}
|
|
\end{align}
|
|
|
|
The log of the amplitude has another fascinating feature, which we
|
|
can see already starting at 2-loops, where
|
|
\begin{equation}
|
|
{\rm log} M_2 = (12) - (1)(2)
|
|
\end{equation}
|
|
|
|
Note that the 2-loop amplitude puts the positivity restriction
|
|
$\langle D_{(1)} D_{(2)} \rangle > 0$ on the lines, but
|
|
``$1-$loop$^2$" part does not put any positivity restrictions on
|
|
them. Indeed, we can think of this as the sum over two regions, with
|
|
$\langle D_{(1)} D_{(2)} \rangle > 0$ and $\langle D_{(1)} D_{(2)}
|
|
\rangle < 0$. Thus, the sum that gives the log is the form
|
|
associated with the region where $\langle D_{(1)} D_{(2)} \rangle <
|
|
0$ ! The pattern continues at all higher loops. At 3-loops we have
|
|
three positivity conditions involving
|
|
\begin{equation}
|
|
\{\la D_{(1)} D_{(2)}\ra,\la D_{(1)} D_{(3)}\ra,\la
|
|
D_{(2)}D_{(3)}\ra\}
|
|
\end{equation}
|
|
For the amplitude they are all positive, $M_3 = \{+++\}$ while for
|
|
the log of the amplitude (\ref{log3loop}) we get a sum of terms
|
|
\begin{equation}
|
|
({\rm log}\,M)^{3-loop} = \{+--\} \oplus \{-+-\} \oplus \{--+\}
|
|
\oplus 2\{---\}
|
|
\end{equation}
|
|
|
|
At 4 loops we have 6 positivity conditions,
|
|
\begin{equation}
|
|
\{\la D_{(1)}D_{(2)}\ra,\la D_{(1)} D_{(3)}\ra,\la D_{(1)}D_{(4)}
|
|
\ra, \la D_{(2)}D_{(3)}\ra, \la D_{(2)}D_{(4)}\ra, \la
|
|
D_{(3)}D_{(4)}\ra\}
|
|
\end{equation}
|
|
For the amplitude we have $M_4=\{++++++\}$. The log is
|
|
\begin{align}
|
|
&({\rm log}\,M)^{4-loop}=
|
|
(1234) -[(12)(34) + (13)(24) + (14)(23)] \nonumber\\
|
|
&\hspace{1cm}+[(12)(3)(4) + (13)(2)(4) + (14)(2)(3) + (23)(1)(4) +
|
|
(24)(1)(3)+(34)(1)(2)] \nonumber\\
|
|
&\hspace{1cm}+2\,[(123)(4) + (124)(3) + (134)(2) + (234)(1)] -
|
|
6\,(1)(2)(3)(4) \label{log4}
|
|
\end{align}
|
|
%\begin{align}
|
|
%&({\rm log}\,M)^{4-loop}=
|
|
%M_4(x_1,x_2,x_3,x_4)-6M_1(x_1)M_1(x_2)M_1(x_3)M_1(x_4)\label{log4}\\
|
|
%&+2[M_3(x_1,x_2,x_3)M_1(x_4)+M_3(x_1,x_2,x_4)M_1(x_3)+M_3(x_1,x_3,x_4)M_1(x_2)+M_3(x_2,x_3,x_4)M_1(x_1)]\nonumber\\
|
|
%&-[M_2(x_1,x_2)M_2(x_3,x_4)+M_2(x_1,x_3)M_2(x_2,x_4)+M_2(x_1,x_4)M_2(x_2,x_3)]\nonumber\\
|
|
%&+M_2(x_1,x_2)M_1(x_3)M_1(x_4)+M_2(x_1,x_3)M_1(x_2)M_1(x_4)+M_2(x_1,x_4)M_1(x_2)M_1(x_3)\nonumber\\
|
|
%&+M_2(x_2,x_3)M_1(x_1)M_1(x_4)+M_2(x_2,x_4)M_1(x_1)M_1(x_3)+M_2(x_3,x_4)M_1(x_1)M_1(x_2)\nonumber
|
|
%\end{align}
|
|
and can be decomposed into a sum of regions as
|
|
\begin{equation}
|
|
({\rm log}\,M)^{4-loop} = R_1 \oplus 2R_2 \oplus 3R_3 \oplus
|
|
4R_4\oplus 6R_6
|
|
\end{equation}
|
|
where
|
|
\begin{align*}
|
|
R_1 =&
|
|
\{---+++\}\oplus\{--++-+\}\oplus\{--+++-\}\oplus\{-+--++\}\\&\oplus\{-+-++-\}\oplus\{-++--+\}\oplus\{-++-+-\}\oplus\{-+++--\}\\
|
|
&\oplus\{+---++\}\oplus\{+--+-+\}\oplus\{+-+--+\}\oplus\{+-+-+-\}\\&\oplus\{+-++--\}\oplus\{++---+\}\oplus\{++--+-\}\oplus\{++-+--\}\\
|
|
R_2=&\{----++\}\oplus\{---+-+\}\oplus\{---++-\}\oplus\{--+--+\}\\&\oplus\{--+-+-\}\oplus\{-+---+\}\oplus\{-+-+--\}\oplus\{-++---\}\\
|
|
&\oplus\{+---+-\}\oplus\{+--+--\}\oplus\{+-+---\}\oplus\{++----\}\\
|
|
R_3=&\{--++--\}\oplus\{-+--+-\}\oplus\{+----+\}\\
|
|
R_4=&\{-----+\}\oplus\{----+-\}\oplus\{---+--\}\oplus\{--+---\}\\
|
|
&\oplus\{-+----\}\oplus\{+-----\}\\
|
|
R_6= &\{------\}
|
|
\end{align*}
|
|
While the expansion of the logarithm itself includes terms with both plus and minus signs,
|
|
remarkably, in all cases we get a sum over regions, with all
|
|
positive integer coefficients, reflecting the allowed leading
|
|
singularities for different orderings of approaching the collinear
|
|
region.
|
|
|
|
\newpage
|
|
|
|
\section{Some Faces of the Amplituhedron}
|
|
|
|
In this section, we study a few classes of lower-dimensional faces
|
|
of the amplituhedron, that are particularly easy to triangulate. The
|
|
canonical form associated with these faces computes corresponding
|
|
cuts of the full integrand.
|
|
|
|
\subsection*{Ladders and Next-to-Ladders}
|
|
Already in \cite{P1}, we discussed a set of faces that are
|
|
extremely easy to understand. Let's take all $L$ loops to cut
|
|
the line $(12)$, by sending all the $w_i \to 0$. The positivity
|
|
conditions just become $(x_i - x_j)(z_i - z_j) < 0$. In whatever
|
|
configuration of $x$'s we have, they are ordered in some way, say
|
|
$x_1 < \dots < x_L$, and this condition tells us that the $z$'s are
|
|
oppositely ordered $z_1 > \dots
|
|
> z_L$. The $y_i$ just have to be positive. The associated form is
|
|
then trivially
|
|
\begin{equation}
|
|
\frac{1}{y_1} \dots \frac{1}{y_L} \frac{1}{x_1} \frac{1}{x_2 - x_1}
|
|
\dots \frac{1}{x_L - x_{L-1}} \frac{1}{z_L} \frac{1}{z_{L-1} - z_L}
|
|
\dots \frac{1}{z_1 - z_2}
|
|
\end{equation}
|
|
which corresponds to the unique ``ladder" local diagrams that can
|
|
contribute to this cut; to see this propagator structure explicitly,
|
|
we simply regroup the terms in the product as $1/(y_1 \cdots y_L)$
|
|
multiplying
|
|
\begin{equation}
|
|
\frac{1}{x_1} \times \frac{1}{(x_2 - x_1)(z_1 - z_2)} \times
|
|
\cdots \times \frac{1}{(x_L - x_{L-1})(z_{L-1} -z_L)} \times
|
|
\frac{1}{z_L}
|
|
\end{equation}
|
|
|
|
We can move on to consider ``next-to-ladder" cuts. Suppose for
|
|
instance that $(L-1)$ of the loop variables cutting $(12)$, while
|
|
the $L$'th loop cuts $(34)$ so that $y_L \to 0$. The positivity for
|
|
the $(L-1)$ lines is simply $x_1<x_2< \cdots < x_{L-1}$ and $z_1 >
|
|
z_2 > \cdots> z_{L-1}$ as above. The mutual positivity conditions
|
|
are just that
|
|
\begin{equation}
|
|
w_L y_i > (x_i - x_L)(z_i - z_L)
|
|
\end{equation}
|
|
|
|
The canonical form is very easy to determine. We simply consider all
|
|
the for $L$ orderings of the $x$'s for which $x_1 < \cdots, <
|
|
x_{L-1}$, i.e. the orderings $[x_1, \cdots, x_{L-1}, x_L]$, $[x_1,
|
|
\cdots, x_L, x_{L-1}]$, $\cdots$, $[x_L, x_1, \cdots, x_{L-1}]$;
|
|
similarly, we consider all the analogous orderings of the $z$'s:
|
|
$[z_L, z_{L-1},\cdots, z_1]$, $[z_{L-1},z_L,\cdots,z_1]$, $\cdots$,
|
|
$[z_{L-1}, \cdots, z_1, z_L]$, Then if in the ordering, either both
|
|
$x_k > x_L$ , $z_k > z_L$ or $x_k < x_L, z_k < z_L$, we have $y_k >
|
|
(x_k - x_L)(z_k - z_L)/w_L$, otherwise we just have $y_k > 0$. The
|
|
corresponding form is
|
|
|
|
\begin{align}
|
|
&\sum_{\sigma_1 < \cdots < \sigma_{L-1}, \rho_1
|
|
> \cdots > \rho_{L-1}} [x_{\sigma^{-1}_1}, \cdots,
|
|
x_{\sigma^{-1}_L}] [z_{\rho^{-1}_1}, \cdots, z_{\rho^{-1}_L}]\\
|
|
&\hspace{0.5cm}\times \prod_{k = 1}^{L-1} \left\{
|
|
\begin{array}{c} [y_k - \frac{1}{w_L}(x_k - x_L)(z_k - z_L)]^{-1} \,
|
|
\sigma_k > \sigma_L,
|
|
\rho_k > \rho_L \quad {\rm or}\quad \sigma_k < \sigma_L, \rho_k < \rho_L\\
|
|
y_k^{-1} \,\,\, {\rm otherwise} \end{array} \right\}\nonumber
|
|
\end{align}
|
|
|
|
This expression sums the cuts for local diagrams of the form
|
|
$$
|
|
\includegraphics[scale=.85]{pic19.pdf}
|
|
$$
|
|
|
|
\subsection*{Corner Cuts}
|
|
We can systematically approach the faces of the amplituhedron where
|
|
every line is one of the double-cut configurations. We already know
|
|
what happens with the unitarity double-cut on general grounds. So we
|
|
are left with the ``corner cuts", where any line either passes
|
|
through $Z_i$, or lies in the plane $(Z_{i-1} Z_i Z_{i+1})$. We use
|
|
different convenient gauge fixings: for the case of lines passing
|
|
through $1$, and lines in the plane $(412)$, we use
|
|
\begin{equation}
|
|
D_{{\rm through \, 1}} = \left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\
|
|
0 & y & 1 & z
|
|
\end{array} \right), \quad D_{{\rm in \, (412)}} =
|
|
\left(\begin{array}{cccc} u & 1 & 0 & 0 \\ - v & 0 & 0 & 1
|
|
\end{array} \right)
|
|
\end{equation}
|
|
|
|
Note that
|
|
\begin{equation}
|
|
\langle D_{{\rm through\, 1}} D_{{\rm in \,(412)}} \rangle = - 1
|
|
\end{equation}
|
|
is negative, and so we immediately learn that it is impossible to
|
|
have lines of both types in one corner! We can either have a
|
|
collection of lines passing through 1, {\it or} a collection of
|
|
lines lying in the plane $(412)$. Suppose we approach the
|
|
configuration where all the lines path through $1$, by starting with
|
|
all the lines intersecting $(41)$, and sending the lines into the
|
|
corner in some order, first $w_1 \to 0, \cdots,$ then $w_L > 0$.
|
|
This orders $w_1 < \cdots < w_L$ and so $y_1 > \cdots > y_L$, thus
|
|
the form on this final corner cut is just $[y_L, \cdots, y_1]$. Note
|
|
that we see again something we have observed already a number of
|
|
times: the form on the cut depends not just on the geometry of the
|
|
ultimate configuration of lines, but also on the path taken to that
|
|
configuration.
|
|
|
|
We can easily determine completely general
|
|
corner cuts where all the lines are of one type or the other. For
|
|
instance, suppose we start with $L_1$ lines cutting $(14)$, and
|
|
$L_2$ lines cutting $(12)$, and that we send these lines to pass
|
|
through the corners $1$ and $2$ in some order. If we parametrize the
|
|
matrices as
|
|
\begin{equation}
|
|
\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & y_i & 1 & z_i
|
|
\end{array} \right) \, \left(\begin{array}{cccc} 0 & 1 & 0 & 0 \\ -
|
|
\alpha_I & 0 & \beta_I & 1 \end{array} \right)
|
|
\end{equation}
|
|
then the positivity conditions are just $y_{L_1} > \cdots > y_1,
|
|
\beta_{L_2}> \cdots > \beta_1$, with the mutual positivity condition
|
|
$z_i \beta_I > 1$, which just means $z_i > 1/\beta_1$ for all $i$.
|
|
Then the form is trivially
|
|
\begin{equation}
|
|
\prod_I \frac{1}{\alpha_I} \times [y_{L_1}, \cdots, y_1] [\beta_{L_2}, \cdots, \beta_1] \prod_i
|
|
\frac{1}{z_i - \beta_1^{-1}}
|
|
\end{equation}
|
|
|
|
This result generalizes trivially to the case with $L_1$ lines
|
|
cutting $(41)$, $L_2$ lines cutting $(12)$, $L_3$ lines cutting
|
|
$(23)$ and $L_4$ lines cutting $(34)$, then taken to pass through
|
|
$1,2,3,4$.
|
|
$$
|
|
\includegraphics[scale=.75]{pic20.pdf}
|
|
$$
|
|
|
|
These results are very simple and arise from a single local term.
|
|
Much more interesting are the mixed corner cuts, where we have the
|
|
two different types of lines passing through different corners. One
|
|
case is still extremely simple, where the two different lines pass
|
|
through consecutive corners. Suppose we have $L_1$ lines passing
|
|
through $1$, and $L_2$ lines lying in $(123)$. It is trivial to see
|
|
that
|
|
\begin{equation}
|
|
\langle D_{{\rm through \, 1}} D_{{\rm lying \, in \, (123)}}
|
|
\rangle = (14)_{{\rm through \, 1}} (23)_{{\rm lying \, in (123)}} >
|
|
0
|
|
\end{equation}
|
|
and so the mutual positivity between these two sets is automatically
|
|
satisfied. The form is then just the product of the form for the
|
|
$L_1$ lines and the $L_2$ lines separately.
|
|
|
|
The non-trivial case is
|
|
when the corner cuts are different lines in opposite corners.
|
|
Suppose we have $L_1$ lines cutting $(41)$ that were then sent to
|
|
pass through $1$ in order $(L_1, \cdots, 1)$, and $L_3$ lines
|
|
cutting $(23)$ that were made to pass through $(234)$ in order
|
|
$(1,\cdots,L_3)$.
|
|
$$
|
|
\includegraphics[scale=.75]{pic21.pdf}
|
|
$$
|
|
For notational convenience we'll parametrize the $D$ matrices using different
|
|
variable names in this case:
|
|
\begin{equation}
|
|
D_{{\rm through \, 1}} = \left(\begin{array}{cccc} 1 & 0 & 0 & 0
|
|
\\ 0 & x_i & 1 & y_i
|
|
\end{array} \right),\quad D_{{\rm lying \, in (234)}} =
|
|
\left(\begin{array}{cccc} 0 & 1 & a_I^{-1} & 0 \\ 0 & 0 & b_I^{-1} &
|
|
1 \end{array} \right)
|
|
\end{equation}
|
|
Then the positivity conditions are
|
|
\begin{equation}
|
|
x_1 < \cdots < x_{L_1}, \, a_1 <
|
|
\cdots < a_{L_3}, \, \, {\rm and} \, \, \frac{x_i}{a_I} +
|
|
\frac{y_i}{b_I} > 1
|
|
\end{equation}
|
|
|
|
It is quite straightforward to triangulate this space; let us work
|
|
out the case $L_3 = 2$ explicitly. Here the geometry is very similar
|
|
to the last of our warmup exercises. Suppose first that $b_1 < b_2$.
|
|
Then the inequalities are just $x_i/a_2 + y_i/b_2 > 1$ together with
|
|
the restriction $x_1 < \cdots < x_{L_1}$. We simply order the $x_i$
|
|
relative to $a_2$. If $x_i > a_2$, then we just have $y_i > 0$ and
|
|
the form is $1/y_i$, while if $x_i < a_2$, we have $Y_{i,2} > 0$ and
|
|
the form is $1/Y_{i,2}$. Here we have defined
|
|
\begin{equation}
|
|
Y_{i,1} = y_i + \frac{b_1 x_i}{a_1} - b_1, Y_{i,2} = y_i + \frac{b_2
|
|
x_i}{a_2} - b_2.
|
|
\end{equation}
|
|
Thus, for $b_1 < b_2$, the form is just
|
|
\begin{equation}
|
|
\frac{1}{a_1 (a_2 - a_1)} \frac{1}{b_1(b_2 - b_1)} \sum_m [\cdots,
|
|
x_m, \underline{a_2},x_{m+1} \cdots ] \prod_k
|
|
\left\{\begin{array}{c} Y_{k,2}^{-1} \, \, k \leq m \\ y_k^{-1} \,
|
|
\, k > m \end{array} \right\}
|
|
\end{equation}
|
|
|
|
If instead $b_2 < b_1$, then we have to break $x$ space up into the
|
|
three regions between $0,a_{12},a_2$ where $a_{12} = \frac{a_1 a_2
|
|
(b_1 - b_2)}{a_2 b_1 - a_1 b_2}$. We have to sum over all the
|
|
orderings of the $x$'s relative to $a_{12},a_2$'; for all the $x_i >
|
|
a_2$, the form in $y$ space is just $1/y_i$, for the $x_i$ in the
|
|
range $a_2 > x_i > a_{12}$ the $y$ form is just $1/Y_{i,2}$, while
|
|
for $a_{12} > x_i > 0$ the $y$ form is $1/Y_{i,1}$. Thus in this
|
|
case the form is
|
|
\begin{equation}
|
|
\frac{1}{a_1 (a_2 - a_1)} \frac{1}{b_1(b_2 - b_1)} \sum_{m \leq l}
|
|
[\cdots, x_m,\underline{a_{12}},x_{m+1}, \cdots, x_l, \underline{a_2},x_{l+1}, \cdots]
|
|
\prod_k \left\{\begin{array}{c} Y_{k,1}^{-1}, \, \, k \leq m \\
|
|
Y_{k,2}^{-1}, \, \, m < k \leq l \\ y_k^{-1} \, \, k > l \end{array}
|
|
\right\}
|
|
\end{equation}
|
|
|
|
The full form is just the sum of these two pieces. While this result
|
|
is completely straightforward from triangulation, it gives rise to
|
|
highly non-trival local expressions even at comparatively low loop
|
|
order. In the first really interesting case at 5 loops, with $L_1 =
|
|
3$ and $L_3 = 2$, 19 local terms contribute to this cut, and when they are all combined under a common
|
|
denominator, the numerator has 325 terms.
|
|
|
|
There is another interesting feature of these cuts, that is not
|
|
evident from any traditional point of view but is obvious from the
|
|
positive geometry. We have seen that fixing the order in which the
|
|
lines are brought to pass through $1$, imposes the constraint $x_1 <
|
|
\cdots < x_{L_1}$. But, if we sum over all the different orderings,
|
|
we simply remove these ordering constraints! We then expect that the
|
|
form simplifies greatly. Indeed, if we stick with the case $L_3 =
|
|
2$, then we just get several copies of the problem $x/a_1 + y/b_1 >
|
|
1, x/a_2 + y/b_2 > 1$, which we analyzed in our warmup section.
|
|
Thus, the sum over all the ways to start with $L_1$ lines on $(41)$
|
|
which are then sent through $1$ (while sending $L_3 = 2$ lines to
|
|
lie in (234) in the usual fixed order), is
|
|
\begin{align}
|
|
\frac{1}{a_1 (a_2 - a_1)}&\left[ \frac{1}{b_1 (b_2 - b_1)} \prod_i
|
|
\left([x_i, \underline{a_2}] \frac{1}{Y_{i,2}} +
|
|
[\underline{a_2},x_i] \frac{1}{y_i}\right)\right.\\
|
|
&\hspace{0.3cm}\left.+ \frac{1}{b_2 (b_1 - b_2)} \prod_i \left([x_i,
|
|
\underline{a_{12}}] \frac{1}{Y_{i,1}} + [\underline{a_{12}}, x_i,
|
|
\underline{a_2}] \frac{1}{Y_{i,2}} + [\underline{a_2},x_i]
|
|
\frac{1}{y_i}\right) \right]\nonumber
|
|
\end{align}
|
|
|
|
|
|
\subsection*{Internal Cuts}
|
|
|
|
It is interesting that up to 4 loop order, every loop in the local
|
|
expansion of the amplitude touches the external lines, but this
|
|
behavior is obviously not generic. Starting at 5 loops, we have
|
|
diagrams with purely internal loops, such as
|
|
$$
|
|
\includegraphics[scale=.8]{pic16.pdf}
|
|
$$
|
|
and it is interesting to probe these from positivity. Let us look at
|
|
a particularly simple set of cuts that exposes the structure in a
|
|
nice way. Suppose we take 4 lines $(AB)_1, \cdots, (AB)_4$, and take
|
|
them to pass through $1,2,3,4$ respectively. But additionally, we
|
|
take the cut where $\langle AB_1 AB_2 \rangle \to 0$, $\langle AB_2
|
|
AB_3 \rangle \to 0$, $\langle AB_3 AB_4 \rangle \to 0$, $\langle
|
|
AB_4 AB_1\rangle \to 0$, i.e. the lines are taken to one intersect
|
|
the next. The $D$ matrices
|
|
are simply
|
|
\begin{align}
|
|
D_{(1)} = \left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\
|
|
0 & \alpha^{-1} & 1 & \beta \end{array} \right), \quad D_{(2)} =
|
|
\left(\begin{array}{cccc} 0 & 1 & 0 & 0 \\ \gamma & 0 & \beta^{-1} &
|
|
1 \end{array} \right),\nonumber\\ D_{(3)} =
|
|
\left(\begin{array}{cccc} 0 & 0 & 1 & 0 \\ -1 & -\sigma & 0 &
|
|
\gamma^{-1} \end{array} \right),\quad D_{(4)} =
|
|
\left(\begin{array}{cccc} 0 & 0 & 0 & 1 \\ - \sigma^{-1} & -1 & -
|
|
\alpha & 0 \end{array} \right)
|
|
\end{align}
|
|
|
|
Note that the mutual positivity between $D_{(1)} D_{(3)}$ and
|
|
$D_{(2)} D_{(4)}$ is automatic. The geometry of the lines is
|
|
$$
|
|
\includegraphics[scale=.7]{pic17.pdf}
|
|
$$
|
|
and so we can think of the lines as $(AB)_1 = \hat{1} \hat{2},
|
|
(AB)_2 = \hat{2} \hat{3}, (AB)_3 = \hat{3} \hat{4}, (AB)_4 = \hat{1}
|
|
\hat{4}$, where
|
|
\begin{align}
|
|
\hat{1} = 1 + \sigma(2 + \alpha(3 + \beta 4)), \quad \hat{2} = 2 +
|
|
\alpha(3 + \beta(4 - \gamma 1)) \nonumber\\ \hat{3} = 3 + \beta(4 -
|
|
\gamma(1 + \sigma 2)),\quad \hat{4} = 4 - \gamma(1 + \sigma(2 +
|
|
\alpha 3))
|
|
\end{align}
|
|
|
|
Now, it is easy to see that the remaining mutual positivity
|
|
conditions between $D_{(1)}, \cdots, D_{(4)}$ and the other
|
|
$D_{(i)}$ are just satisfied by the lower-loop shifted amplitude;
|
|
thus we conclude that on this cut the form is
|
|
\begin{equation}
|
|
\frac{d \alpha}{\alpha} \frac{d \beta}{\beta} \frac{d
|
|
\gamma}{\gamma} \frac{d \sigma}{\sigma} \times M^{L-4}(\hat{1},
|
|
\hat{2}, \hat{3}, \hat{4})
|
|
\end{equation}
|
|
|
|
\section{Four Particle Outlook}
|
|
|
|
We have only scratched the surface of the rich amplituhedron
|
|
geometry controlling four-particle scattering in planar ${\cal N} =
|
|
4$ SYM at all loop order. There is obviously much more to be done
|
|
just along the elementary lines of this note, minimally in further continuing a
|
|
systematic exploration of other facets of the geometry,
|
|
corresponding to different classes of cuts of physical interest. But
|
|
we close with a few comments about some different avenues of
|
|
exploration.
|
|
|
|
In this note we have approached the determination of the integrand
|
|
for four-particle scattering by directly ``triangulating" the
|
|
amplituhedron geometry. The $L-$ loop geometry is defined in a
|
|
self-contained way, as a subspace living inside $L$ copies of
|
|
space-time realized as $G(2,4)$. In particular, no-where do we need
|
|
to refer to lower-loop, higher-$k$ amplitudes, as in necessary in
|
|
the BCFW recursion approach \cite{BCFW2} to loop integrands
|
|
\cite{BCFWloop}. Nonetheless, it is likely that some natural
|
|
connection exists with the full problem, and perhaps a broader view
|
|
of the bigger amplituhedron geometry in which the four-particle
|
|
problem sits will be important for systematically determining the
|
|
all-loop integrand. Certainly, experience with the positive Grassmannian
|
|
\cite{alex, positive} strongly suggests that different faces can't be properly understood in
|
|
isolation.
|
|
|
|
As we have seen, the approach to computing the integrand by
|
|
triangulating the amplituhedron does not give us the familiar
|
|
expansions that are manifestly local. This is of course not
|
|
surprising; however, what {\it is} surprising is that some special
|
|
local expansions expose yet {\it another} aspect of positivity, that
|
|
we are not making apparent in the triangulation approach. As also
|
|
mentioned in \cite{P1}, we are still clearly missing a picture of
|
|
the form $\Omega$ which is analogous to one available for convex
|
|
polygons, determined by a literal volume of the {\it dual} polygon.
|
|
We don't yet have a notion of a ``dual amplituhedron", but there is
|
|
a powerful indication that such a formulation must exist: the form
|
|
$\Omega$ is itself positive, inside the amplituhedron! More
|
|
specifically, we can write the $L-$loop integrand as
|
|
\begin{equation}
|
|
\Omega_L(AB_i) = \prod_{i=1}^L \langle AB_i d^2 A_i \rangle \langle
|
|
AB_i d^2 B_i \rangle M_L(AB_i)
|
|
\end{equation}
|
|
Then, we claim that when the $(AB)_i$ are taken to lie inside
|
|
the amplituhedron,
|
|
\begin{equation}
|
|
M_L(AB_i) > 0
|
|
\end{equation}
|
|
We will return to exploring this fact at greater in length in
|
|
\cite{withandrew}. We stress that this property is {\it not}
|
|
manifest term-by-term in the amplituhedron triangulation expansion
|
|
of the integrand. Random forms of the local expansion also don't
|
|
make this remarkable property manifest term-by-term, but there are
|
|
particularly nice forms of local expansion that do make this
|
|
manifest. As we will discuss in \cite{withandrew}, we suspect that
|
|
this surprising positivity property of the integrand is pointing the
|
|
way to a more direct and intrinsic, triangulation-independent
|
|
definition for the canonical form $\Omega$ associated with the
|
|
amplituhedron.
|
|
|
|
From a mathematical point of view, it is interesting that the study
|
|
of amplitudes leads to stratifications of various collections of
|
|
objects in projective space. If we consider a collection of $n$
|
|
vectors in $k$ dimensions, together with a cyclic structure on this
|
|
data, we are led to the beautiful stratification of the space given
|
|
by the positive Grassmannian. Even just with the the four-particle
|
|
amplituhedron, we see something new, not needing a cyclic structure
|
|
on the objects: given a collection of $L$ 2-planes in 4 dimensions,
|
|
the positivity conditions are fully permutation invariant between
|
|
the $L$ lines. Just as with the positive Grassmannian, it is natural to expect the cell structure of
|
|
the amplituhedron to be determined in a fundamentally combinatorial
|
|
way. The fascinating path-dependence of the forms associated with
|
|
the cuts, together with the combinatorics that arise just in the
|
|
simple discussion of the multi-collinear limit, are perhaps
|
|
indications of an underlying combinatorial structure.
|
|
|
|
The four-particle amplitude is a truly remarkable object. At the
|
|
level of the integrand, at multi-loop order it contains non-trivial
|
|
information about all the more complicated multi-particle amplitudes
|
|
in the theory. At the level of the final integrated expression, we
|
|
have a function that smoothly interpolates between a picture of
|
|
``interacting gluons" at weak coupling to ``minimal area surface in
|
|
AdS space"\cite{DCI2} at strong coupling. We have explored a reformulation of
|
|
this physics in terms of a simple to define, yet rich
|
|
and intricate geometry. We hope that this will lead us a more direct
|
|
understanding of how the picture of ``gluons" and ``strings" arise
|
|
as different limits of a single object. As a small step in this
|
|
direction, it is encouraging to find a natural understanding,
|
|
intrinsic to the geometry, of the behavior of the amplitude in the
|
|
multi-collinear region, and an associated intrinsic-to-the-geometry
|
|
rationale for taking the log of the amplitude. Trying to more
|
|
completely determine the IR singular behavior of the integrand of
|
|
the amplitude is an ideal laboratory to connect our approach to the
|
|
loop integrand with the final integrated expressions, and especially
|
|
to ideas related to integrability. Indeed the coefficient of the
|
|
log$^2$ infrared divergence of the log of the amplitude is given by
|
|
the cusp anomalous dimension, which was brilliantly determined using
|
|
integrability in \cite{BS, Beisert:2006ez,Eden:2006rx}. It is
|
|
notable that this approach makes crucial use of a spectral
|
|
parameter, something which is absent in our present discussion of
|
|
the amplituhedron. Given the spectral deformation of
|
|
on-shell diagrams given in \cite{Ferro:2013dga,Ferro:2012xw}, it is
|
|
natural to ask whether a similar deformation can be found directly at the
|
|
level of the amplituhedron.
|
|
|
|
\section*{Acknowledgements}
|
|
|
|
We thank Jake Bourjaily, Freddy Cachazo, Simon Caron-Huot, Johannes
|
|
Henn, Andrew Hodges, Jan Plefka, Dave Skinner and Matthias
|
|
Staudacher for stimulating discussions. N.~A.-H. is supported by the
|
|
Department of Energy under grant number DE-FG02-91ER40654. J.~T. is
|
|
supported in part by the David and Ellen Lee Postdoctoral
|
|
Scholarship and by DOE grant DE-FG03-92-ER40701 and also by NSF
|
|
grant PHY-0756966.
|
|
|
|
\begin{thebibliography}{99}
|
|
|
|
\bibitem{P1}
|
|
N.~Arkani-Hamed and J.~Trnka,
|
|
%``The Amplituhedron,''
|
|
arXiv:1312.2007 [hep-th].
|
|
|
|
\bibitem{alex}
|
|
A.~Postnikov, arXiv:math/0609764.
|
|
|
|
\bibitem{FG}
|
|
V. V. Fock and A. B. Goncharov,
|
|
Ann. Sci. L'Ecole Norm. Sup. (2009) , arXiv:math.AG/0311245.
|
|
|
|
%\bibitem{N4}
|
|
% N.~Arkani-Hamed, F.~Cachazo, C.~Cheung and J.~Kaplan,
|
|
% %``A Duality For The S Matrix,''
|
|
% JHEP {\bf 1003}, 020 (2010)
|
|
% [arXiv:0907.5418 [hep-th]].
|
|
|
|
\bibitem{positive}
|
|
|
|
N.~Arkani-Hamed, F.~Cachazo, C.~Cheung and J.~Kaplan,
|
|
%``A Duality For The S Matrix,''
|
|
JHEP {\bf 1003}, 020 (2010)
|
|
[arXiv:0907.5418 [hep-th]]; N.~Arkani-Hamed, J.~L.~Bourjaily, F.~Cachazo, A.~B.~Goncharov, A.~Postnikov and J.~Trnka,
|
|
%``Scattering Amplitudes and the Positive Grassmannian,''
|
|
arXiv:1212.5605 [hep-th].
|
|
%%CITATION = ARXIV:1212.5605;%%
|
|
|
|
\bibitem{A1}
|
|
A.~Hodges,
|
|
%``Eliminating spurious poles from gauge-theoretic amplitudes,''
|
|
JHEP {\bf 1305}, 135 (2013)
|
|
[arXiv:0905.1473 [hep-th]].
|
|
|
|
|
|
\bibitem{Bern:2006ew}
|
|
Z.~Bern, M.~Czakon, L.~J.~Dixon, D.~A.~Kosower and V.~A.~Smirnov,
|
|
%``The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory,''
|
|
Phys.\ Rev.\ D {\bf 75}, 085010 (2007)
|
|
[hep-th/0610248].
|
|
|
|
\bibitem{Bern:2005iz}
|
|
Z.~Bern, L.~J.~Dixon and V.~A.~Smirnov,
|
|
%``Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond,''
|
|
Phys.\ Rev.\ D {\bf 72}, 085001 (2005)
|
|
[hep-th/0505205].
|
|
|
|
\bibitem{Bern:2007ct}
|
|
Z.~Bern, J.~J.~M.~Carrasco, H.~Johansson and D.~A.~Kosower,
|
|
%``Maximally supersymmetric planar Yang-Mills amplitudes at five loops,''
|
|
Phys.\ Rev.\ D {\bf 76}, 125020 (2007)
|
|
[arXiv:0705.1864 [hep-th]].
|
|
|
|
\bibitem{Bourjaily:2011hi}
|
|
J.~L.~Bourjaily, A.~DiRe, A.~Shaikh, M.~Spradlin and A.~Volovich,
|
|
%``The Soft-Collinear Bootstrap: N=4 Yang-Mills Amplitudes at Six and Seven Loops,''
|
|
JHEP {\bf 1203}, 032 (2012)
|
|
[arXiv:1112.6432 [hep-th]].
|
|
|
|
\bibitem{Eden:2012tu}
|
|
B.~Eden, P.~Heslop, G.~P.~Korchemsky and E.~Sokatchev,
|
|
%``Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N=4 SYM,''
|
|
Nucl.\ Phys.\ B {\bf 862}, 450 (2012)
|
|
[arXiv:1201.5329 [hep-th]].
|
|
|
|
\bibitem{BCFWloop}
|
|
N.~Arkani-Hamed, J.~L.~Bourjaily, F.~Cachazo, S.~Caron-Huot and J.~Trnka,
|
|
%``The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM,''
|
|
JHEP {\bf 1101}, 041 (2011)
|
|
[arXiv:1008.2958 [hep-th]].
|
|
|
|
\bibitem{Simon}
|
|
S.~Caron-Huot,
|
|
%``Loops and trees,''
|
|
JHEP {\bf 1105}, 080 (2011)
|
|
[arXiv:1007.3224 [hep-ph]].
|
|
|
|
\bibitem{localintegrand}
|
|
N.~Arkani-Hamed, J.~L.~Bourjaily, F.~Cachazo and J.~Trnka,
|
|
%``Local Integrals for Planar Scattering Amplitudes,''
|
|
JHEP {\bf 1206}, 125 (2012)
|
|
[arXiv:1012.6032 [hep-th]].
|
|
|
|
\bibitem{BCFW2}
|
|
R.~Britto, F.~Cachazo, B.~Feng and E.~Witten,
|
|
%``Direct proof of tree-level recursion relation in Yang-Mills theory,''
|
|
Phys.\ Rev.\ Lett.\ {\bf 94}, 181602 (2005)
|
|
[hep-th/0501052].
|
|
|
|
|
|
\bibitem{withandrew}
|
|
N.~Arkani-Hamed, A.~Hodges and J.~Trnka, to appear.
|
|
|
|
\bibitem{DCI2}
|
|
L.~F.~Alday and J.~M.~Maldacena,
|
|
%``Gluon scattering amplitudes at strong coupling,''
|
|
JHEP {\bf 0706}, 064 (2007)
|
|
[arXiv:0705.0303 [hep-th]].
|
|
|
|
\bibitem{BS}
|
|
N.~Beisert and M.~Staudacher,
|
|
%``The N=4 SYM integrable super spin chain,''
|
|
Nucl.\ Phys.\ B {\bf 670}, 439 (2003)
|
|
[hep-th/0307042].
|
|
|
|
\bibitem{Beisert:2006ez}
|
|
N.~Beisert, B.~Eden and M.~Staudacher,
|
|
%``Transcendentality and Crossing,''
|
|
J.\ Stat.\ Mech.\ {\bf 0701}, P01021 (2007)
|
|
[hep-th/0610251].
|
|
|
|
\bibitem{Eden:2006rx}
|
|
B.~Eden and M.~Staudacher,
|
|
%``Integrability and transcendentality,''
|
|
J.\ Stat.\ Mech.\ {\bf 0611}, P11014 (2006)
|
|
[hep-th/0603157].
|
|
|
|
|
|
\bibitem{Ferro:2013dga}
|
|
L.~Ferro, T.~Lukowski, C.~Meneghelli, J.~Plefka and M.~Staudacher,
|
|
%``Spectral Parameters for Scattering Amplitudes in N=4 Super Yang-Mills Theory,''
|
|
arXiv:1308.3494 [hep-th].
|
|
|
|
\bibitem{Ferro:2012xw}
|
|
L.~Ferro, T.~Lukowski, C.~Meneghelli, J.~Plefka and M.~Staudacher,
|
|
%``Harmonic R-matrices for Scattering Amplitudes and Spectral Regularization,''
|
|
Phys.\ Rev.\ Lett.\ {\bf 110}, no. 12, 121602 (2013)
|
|
[arXiv:1212.0850 [hep-th]].
|
|
|
|
\end{thebibliography}
|
|
|
|
\end{document}
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