We describe a general algorithm which builds on several pieces of data available in
the literature to construct explicit analytic formulas for two-loop MHV amplitudes in

Article funded by SCOAP3

7$, but we believe
that this presents
no obstruction to our algorithm since it
is evident from the result of~\cite{CaronHuot-ml-2011ky} that
only finitely many (in fact,
precisely $\frac{3}{2}n(n-5)^2$) of the
$\mathcal{A}$-coordinates actually appear in the two-loop MHV amplitude
$R_n^{(2)}$, and our experience has shown that the ``most complicated
part'' of these amplitudes (see~\cite{Golden-ml-2014xqa} for details)
can be expressed in terms of the $\mathcal{X}$-coordinates
belonging to finitely many $A_3$ subalgebras of $\Gr(4,n)$.
For the special cases $n=6,7$, we expect that the two-loop symbol alphabet
(which contains already all available $\mathcal{A}$-coordinates)
will be sufficient to express all amplitudes (whether MHV or not) to all
loop order, but for $n>7$ we know of no reason to exclude the possibility
that the symbol alphabet could grow larger at higher loops (indeed
we expect it to become infinite for ten-point
N${}^3$MHV amplitudes starting already at only two loops).
A salient feature of cluster $\mathcal{X}$-coordinates is that they are positive when evaluated inside the positive Grasmmannian, defined as the subset of the Euclidean domain where $\ket{ijkl}>0$ whenever $i

6$, by exploring for example the space of ``heptagon functions''. Our trepidation to take this route stems from the fact that the required symbol alphabet grows rapidly with $n$: as mentioned above, the symbol alphabet for $R_n^{(2)}$ has $\frac{3}{2} n (n-5)^2$ entries~\cite{CaronHuot-ml-2011ky}, so the space of weight-four symbols has dimension\footnote{This is rather too pessimistic; the analyticity condition cuts this down by one power of $n$ and the integrability condition no doubt cuts down by some more powers of $n$.} $\mathcal{O}(n^{12})$. We have pursued instead the somewhat orthogonal approach of organizing our calculations not from left-to-right in the symbol, but rather in order of decreasing mathematical complexity of the functional constituents. At weight four, this means that we first focus our attention on the ``non-classical'' part of the amplitude: the $\Lambda^2 B_2$ component of its coproduct. The remaining purely classical pieces of an amplitude can be systematically computed in order from most to least complicated by following the procedure outlined in~\cite{Goncharov-ml-2010jf}. This approach has the disadvantage of leaving the analytic properties of amplitudes obscure, while it has the advantage of making some remarkable mathematical properties --- the relation to the cluster structure on the kinematic domain --- manifest. The very first step in this approach is the one most fraught with peril, as we now explain. The $\Lambda^2 B_2$ component of the coproduct of $R_n^{(2)}$ can be expressed~\cite{Golden-ml-2013xva,futureWork} as a linear combination of various $\{x_i\}_2 \wedge \{x_j\}_2$ where the $x$'s are drawn from the $\mathcal{X}$-coordinates of the $\Gr(4,n)$ cluster algebra. Moreover, the $x$'s always appear together in pairs satisfying $\{ x_i, x_j \} = 0$ with respect to the natural Poisson structure on the kinematic domain $\Conf_n(\mathbb{P}^3)$; this implies that each pair of variables generates an $A_1 \times A_1$ subalgebra of the $\Gr(4,n)$ cluster algebra. For several years a guiding aim of this research program, strongly advocated by Goncharov, has been that it should be possible to write each amplitude under consideration as a linear combination of special functions associated with smaller building blocks (``atoms''). For example, it is well-known that the function\footnote{We caution the reader that several variants of this function exist in the literature, beginning with~\cite{G91a}, all of which differ from each other by the addition of terms proportional to $\Li_4$, or products of lower-weight $\Li_k$'s. In fact even in this short paper we will use a second variant $K_{2,2}$ momentarily. All of these variants have the same $\Lambda^2 B_2$ coproduct component. The particular $L_{2,2}(x,y)$ used here may also be expressed as $L_{2,2}(x,y) = \frac{1}{2} \Li_{2,2}(x/y,-y) - (x \leftrightarrow y)$ in terms of the $\Li_{2,2}$ function.} \begin{equation} L_{2,2}(x,y) = \frac{1}{2} \int_0^1 \frac{dt}{t} \Li_2(-t x) \Li_1(-t y) - (x \leftrightarrow y) \end{equation} has the simple $\Lambda^2 B_2$ coproduct component $\{x\}_2 \wedge \{y\}_2$. Therefore one might be tempted to construct the non-classical part of a desired $R_n^{(2)}$ by writing down an appropriate linear combination of $L_{2,2}(x_i,x_j)$ functions; the difference between this object and $R_n^{(2)}$ must then be expressible in terms of the classical functions $\Li_k$ only. The fatal flaw in this approach is that while $L_{2,2}(x_i,x_j)$ indeed has a simple coproduct, it is poorly adapted to applications where one wants to manifest cluster structure because its symbol has some entries of the form $x_i - x_j$, which is never expressible as a product of cluster $\mathcal{A}$-coordinates (and thus can never be an $\mathcal{X}$-coordinate). Therefore one would have to considerably enlarge the symbol alphabet under consideration in order to fit all of the classical pieces of the amplitude left over by subtracting a linear combination of $L_{2,2}$'s. Just as bad, one would almost inevitably generate $\Li_k$ functions whose arguments range over the entire real line, greatly complicating the problem of arranging all of the branch cuts of the individual terms to conspire to cancel out everywhere in the positive domain. So if we want to maintain a connection to the cluster structure (and, more practically, to avoid enormously complicating the calculation by being forced to clean up unwanted mess in the symbol), we should abandon the idea that each individual term $\{x_i\}_2 \wedge \{x_j\}_2$ may be thought of as an atom.\footnote{Instead they are perhaps quarks: never allowed to appear alone, but always bound safely together in $A_2$ functions or perhaps other, not yet discovered, more exotic baryons.} The problem of identifying the smallest building block manifesting all of the known cluster properties of $R_n^{(2)}$ was solved (at least, for a few of the simplest cluster algebras, and more generally conjectured) in~\cite{Golden-ml-2014xqa}. The solution is a function associated to the $A_3$ cluster algebra which we can write in the form \begin{equation} \label{eq:fA3def} f_{A_3}(x_1,x_2,x_3) = \sum_{i=1}^3 K_{2,2}(x_{i,1}, x_{i,2}), \end{equation} where \begin{alignat}{2}\label{eq:A3coords} x_{1,1} &= x_1, &\quad x_{1,2} &= 1/x_3,\nonumber \\ x_{2,1} &= \left(x_1 x_2+x_2+1\right) x_3, & x_{2,2} &=\frac{x_1 x_2+x_2+1}{x_1}, \\ x_{3,1} &= \frac{x_2 x_3+x_3+1}{x_2}, & x_{3,2} &= \frac{x_2 x_3+x_3+1}{x_1 x_2 x_3} \nonumber \end{alignat} and \begin{equation} K_{2,2}(x,y) = L_{2,2}(x,y) - \left[ \Li_4(x/y) - \frac{1}{3} \Li_3(x/y) \log(x/y) - (x \leftrightarrow y)\right] - \frac{1}{2} \Li_2(-x) \Li_2(-y). \end{equation} The expression for $K_{2,2}$ given here differs from the one presented in~\cite{Golden-ml-2014xqa} by the addition of terms proportional to products of logarithms as well as the final $\Li_2 \Li_2$ term, none of which affect the coproduct of $K_{2,2}$. As long as the three $x_i$ generate an $A_3$ algebra $x_1 \to x_2 \to x_3$ (which could be a subalgebra of a larger algebra), the $A_3$ function accomplishes a remarkable feat: \begin{itemize} \item the $\Lambda^2 B_2$ component of its coproduct, $\sum_{i=1}^3 \{x_{i,1}\}_2 \wedge \{x_{i,2}\}_2$, involves only pairs of Poisson commuting $\mathcal{X}$-coordinates; \item the $B_3 \otimes \mathbb{C}^*$ component of its coproduct can be written in terms of $\mathcal{X}$-coordinates (the $\Li_4$ term in $K_{2,2}$ is crucial here); \item its symbol can be written entirely in terms of $\mathcal{A}$-coordinates (here the $\Li_3 \log$ term is crucial); \item and it is smooth and real-valued everywhere inside the positive domain (i.e., as long as $x_1,x_2,x_3>0$), thanks to the terms which were added compared to~\cite{Golden-ml-2014xqa}. \end{itemize} The $\Li_2 \Li_2$ term in~(\ref{eq:fA3def}) is completely innocuous and was chosen for inclusion because it was observed to nicely package together most of the $\Li_2 \Li_2$ terms in the amplitude $R_7^{(2)}$. It would be very interesting to see if a more optimal packaging of subleading terms could be obtained, whether for $n=7$ or even for all $n$. Working with $A_3$ functions, rather than the underlying individual $L_{2,2}$'s, therefore allows us to avoid having to enlarge the symbol alphabet beyond the set of cluster $\mathcal{A}$-coordinates. Moreover, when expressing the classical contributions to an amplitude we are able to restrict our attention to the functions $\Li_k(-x)$, which are smooth and real-valued throughout the positive domain as long as the arguments $x$ are always taken from the set of cluster $\mathcal{X}$-coordinates. ]]>

0$ such that $(Z_1,\ldots,Z_6,Z_7(t))$ lies in the positive domain for all $0 < t < t_0$. Then the collinear limit\footnote{We caution the reader that our normalization convention for $R_7^{(2)}$ agrees with that of~\cite{CaronHuot-ml-2011ky}, which differs by a factor of four from that of~\cite{Goncharov-ml-2010jf}, so the $R_6^{(2)}$ appearing on the right-hand side of eq.~(\ref{eq:collinear}) should be four times the function $R_6^{(2)}$ given in the latter reference.} \begin{equation} \label{eq:collinear} \lim_{t \to 0^+} R^{(2)}_7(Z_1,\ldots,Z_6,Z_7(t)) = R^{(2)}_6(Z_1,\ldots,Z_6), \end{equation} together with the known formula~\cite{Goncharov-ml-2010jf} for $R_6^{(2)}$, determines the overall additive constant in $R_7^{(2)}$. Each cross-ratio appearing our formula for $R_7^{(2)}$ approaches either $0$, $\infty$, or a finite value in the limit $t \to 0^+$, so it is a simple matter to compute the limit of the formula using the asymptotic behavior of the polylogarithm functions \begin{align} \Li_2(-1/t) &\sim - \frac{1}{2} \log^2 t - \frac{\pi^2}{2}, \\ \Li_3(-1/t) &\sim + \frac{1}{6} \log^3 t + \frac{\pi^2}{6} \log t, \\ \Li_4(-1/t) &\sim - \frac{1}{24} \log^4 t - \frac{\pi^2}{12} \log^2 t - \frac{7 \pi^4}{360}\,, \end{align} together with the asymptotic expansions (when $x$, $t$ and $a$ are positive) \begin{align} L_{2,2}(x,t) &\sim 0, \\ L_{2,2}(x,1/t) &\sim \frac{1}{4} \Li_2(-x) \log^2t + \Li_3(-x) \log t + \Li_4(-x) + \frac{\pi^2}{12} \Li_2(-x), \\ L_{2,2}(1/t,a/t^2) &\sim - \frac{5}{24} \log^4 t + \frac{1}{3} \log a \log^3 t - \frac{1}{8} \log^2 a \log^2 t + \frac{\pi^4}{24} \log^2 t - \frac{\pi^2}{24} \log^2 a - \frac{\pi^4}{30}, \end{align} where $\sim$ signifies the omission of terms which vanish as powers of $t$ (or powers of $t$ times powers of $\log t$). We have taken the limit of $R_7^{(2)}$ by choosing various random initial kinematic points in the positive domain with all momentum twistors having integer entries. Then, after taking the limit $t \to 0^+$, the two sides of eq.~(\ref{eq:collinear}) can be evaluated numerically with arbitrary precision. In this manner we find that that we have to add $- \frac{13}{36} \pi^4$ to our formula for $R_7^{(2)}$ in order for eq.~(\ref{eq:collinear}) to be satisfied. ]]>