We apply a bootstrap procedure to two-loop MHV amplitudes in planar

Article funded by SCOAP3

7$ one can mutate indefinitely to produce an infinite number of $\mathcal{A}$- and $\mathcal{X}$-coordinates, but
this poses no conceptual obstacle to the bootstrap program since only finitely many can appear
in any individual generalized polylogarithm function (i.e., at any finite loop order).
For example, the symbol of $R_n^{(2)}$ contains $\frac{n}{2}(3 n^2-30n + 77)$ $\mathcal{A}$-coordinates. These can easily be enumerated by inspecting the all-$n$ result of~\cite{CaronHuot-ml-2011ky}: in the notation of that paper, there are $n(n-6)$ symbol letters of the form $\langle 1(23)(n{-}1\,n)(i\,i{+}1)\rangle$ (plus all cyclic partners), $\frac{n}{2}(n-6)(n-7)$ of the form $\langle 12\,\overline{i} \cap \overline{j} \rangle$ (plus cyclic), and $\frac{n}{2}(n-5)(n-6)$ of the form $\langle 1(n2)(i\,i{+}1)(j\,j{+}1)\rangle$ (plus cyclic). Finally, there are of course the simple Pl\"ucker coordinates $\langle ijkl \rangle$, which number $\binom{n}{4}$; however it is evident from~\cite{CaronHuot-ml-2011ky} that the only ones which appear in the two-loop MHV amplitudes are those in which at least one pair among $ij$, $jk$, $kl$ or $li$ are cyclically adjacent (for example, $\ket{1357}$ does not appear for $n>7$), so we must subtract $\frac{n}{24}(n-5)(n-6)(n-7)$ from $\binom{n}{4}$. Adding up all of these types we find a total of $\frac{n}{2}(3 n^2-30n + 77)$ symbol letters.
To determine whether a dual conformal cross-ratio $R$ formed from these letters is an $\mathcal{X}$-coordinate, we apply a simple heuristic, originally described in~\cite{Golden-ml-2013xva,Golden-ml-2013lha}:
$R$ is an $\mathcal{X}$-coordinate
if $1+R$ can also be expressed as a ratio of products of letters and
if $R$ is positive everywhere inside the positive domain (i.e., the domain in which $\ket{ijkl}> 0$ for all
$i

7$ it is obviously not feasible to enumerate all clusters. An alternative approach would be to express $x$ and $y$ as algebraic functions of the $\mathcal{X}$-coordinates $(u_1,u_2,\ldots)$ in the initial cluster and then to compute \begin{equation} \{ \log x, \log y \} = \sum_{i,j} \frac{\partial \log x}{\partial \log u_i} \frac{\partial \log y}{\partial \log u_j} \{ \log u_i, \log u_j \}. \end{equation} If for a given pair $x,y$ the right-hand side comes out to be $0$ or $\pm 1$, then it is guaranteed that there exists a cluster containing both $x$ and $y$, even if it would be computationally infeasible to find a specific path of mutations connecting that cluster to the initial cluster. However, we have found that the simplest way to compute $\{\log x, \log y\}$ for general $x,y$ is to use the fact that the Poisson bracket on $\Gr(k,n)$ is induced from the easily computible Sklyanin bracket on SL${}_n$, as described for example in~\cite{bib1057.53064}.\footnote{We are very grateful to C.~Vergu for pointing out this method to us.} The collection of all $\frac{1}{2} n(n-5)^2$ of the $v$'s and $n(n-5)^2$ of the $z$'s constitutes what we call the $\{v,z\}$ basis. Noting that $\{1+x\}_2 = - \{x\}_2$, the discussion in the previous two paragraphs suggests that it is natural to seek a representation for $\delta(R_n^{(2)})\rvert_{\Lambda^2 \B_2}$ as a linear combination of objects of the form $\{v\}_2 \wedge \{z\}_2$ which capture, at the level of the coproduct, the spirit of both the first- and last-entry constraints satisfied by the symbol. Of course, the $\wedge$-product obscures any precise notion of first or last entries for the coproduct, so our argument for restricting to $\{v\}_2\wedge \{z\}_2$ is meant to be suggestive rather than rigorous. The suitability of this ansatz is justified \emph{a posteriori} because it leads to a successful bootstrap. Based on these considerations, as well as explicit calculations at small $n$, we are motivated to hypothesize that properties 1--3 listed above are true for general $n$, so we adopt these as core elements of the cluster bootstrap for $R_n^{(2)}$. In addition we impose that $R_n^{(2)}$ should be \begin{enumerate} \item[4.]{invariant under the dihedral group acting on the $n$ particle labels, as well as under parity, and} \item[5.]{well-defined under collinear limits.} \end{enumerate} We have found that these five simple physical and mathematical conditions uniquely fix $\delta(R^{(2)}_n)\rvert_{\Lambda^2 \B_2}$ (up to a single overall multiplicative constant common to all $n$) to take the value shown explicitly in eq.~(\ref{eq:npt}). Let us emphasize that in step 5 it is not actually necessary to know the $n{-}1$-particle result in order to construct the answer for $n$ particles; it is sufficient merely to make an appropriate ansatz for the latter and impose only that the $n \parallel n{-}1$ collinear limit is well-defined. This determines both the $n$- and $n{-}1$-particle results at the same time, and in particular ties together their overall normalizations. ]]>