# Computation of Contour Integrals on {\cal M}_{0,n}

###### Abstract

Contour integrals of rational functions over {\cal M}_{0,n}, the moduli space of n-punctured spheres, have recently appeared at the core of the tree-level S-matrix of massless particles in arbitrary dimensions. The contour is determined by the critical points of a certain Morse function on {\cal M}_{0,n}. The integrand is a general rational function of the puncture locations with poles of arbitrary order as two punctures coincide. In this note we provide an algorithm for the analytic computation of any such integral. The algorithm uses three ingredients: an operation we call general KLT, Petersen’s theorem applied to the existence of a 2-factor in any 4-regular graph and Hamiltonian decompositions of certain 4-regular graphs. The procedure is iterative and reduces the computation of a general integral to that of simple building blocks. These are integrals which compute double-color-ordered partial amplitudes in a bi-adjoint cubic scalar theory.

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## 1 Introduction

The complete tree-level S-matrix of a large variety of field theories of massless particles are now known (or conjectured) to have a description in terms of contour integrals over {\cal M}_{0,n}, the moduli space of n-punctured Riemann sphere Cachazo:2013hca; Cachazo:2013iea; Mason:2013sva; Dolan:2013isa; Berkovits:2013xba; Adamo:2013tsa; Gomez:2013wza; Kalousios:2013eca; Dolan:2014ega; Geyer:2014fka; Cachazo:2014nsa; Cachazo:2014xea; Ohmori:2015. Some of these theories are Yang-Mills, Einstein gravity, Dirac-Born-Infeld, and the U(N) non-linear sigma model Cachazo:2014xea; Ohmori:2015. The new formulas for the scattering of n particles are given as a sum over multidimensional residues harris on {\cal M}_{0,n}.

The position of n punctures on a sphere can be given using inhomogenous coordinates as \{\sigma_{1},\sigma_{2},\ldots,\sigma_{n}\}. Three of them can be fixed using PSL(2,\mathbb{C}) transformations, say \sigma_{1},\sigma_{2},\sigma_{3}. Therefore the space is n-3 dimensional and we are working locally on a patch isomorphic to \mathbb{C}^{n-3}. The next step in the construction is a rational map from {\mathbb{C}}^{n-3}\to{\mathbb{C}}^{n-3} which is a function of the entries of a symmetric n\times n matrix, s_{ab}, with vanishing diagonal, i.e., s_{aa}=0, and all rows adding up to zero. These are the coordinates of the space of kinematic invariant for the scattering of n massless particles. The explicit form of the map is \{\sigma_{4},\sigma_{5},\ldots,\sigma_{n}\}\to\{E_{4},E_{5},\ldots,E_{n}\} with

E_{a}(\sigma)=\sum_{b=1,b\neq a}^{n}\frac{s_{ab}}{\sigma_{a}-\sigma_{b}}\quad{% \rm for}\quad a\in\{1,2,\ldots,n\}. | (1) |

Using this map, scattering amplitudes, denoted as M_{n}, are defined as the sum over the residues of

\int\prod_{a=4}^{n}d\sigma_{a}|123|^{2}\frac{H(\sigma,k,\epsilon)}{E_{4}(% \sigma)E_{5}(\sigma)\cdots E_{n}(\sigma)} | (2) |

over all the zeroes of the map \{E_{4},E_{5},\ldots,E_{n}\}. Here |123|\equiv(\sigma_{1}-\sigma_{2})(\sigma_{2}-\sigma_{3})(\sigma_{3}-\sigma_{1}) and H(\sigma,k,\epsilon) is a rational function that depends on the theory under consideration and contains all information regarding wave functions of the particles such as polarization vectors \epsilon_{a}^{\mu} and momenta k_{a}^{\mu}. The equations defining the zeroes, E_{4}=E_{5}=\cdots E_{n}=0, are known as the scattering equations Fairlie:1972; Roberts:1972; Fairlie:2008dg; Gross:1987ar; Witten:2004cp; Caputa:2011zk; Caputa:2012pi; Makeenko:2011dm; Cachazo:2012da More explicitly,

M_{n}=\sum_{\sigma^{*}\in Z(E)}\frac{|123|^{2}H(\sigma^{*},k,\epsilon)}{{\rm det% }\left.\left(\frac{\partial(E_{4},\ldots E_{n})}{\partial(\sigma_{4}\ldots% \sigma_{n})}\right)\right|_{\sigma^{*}}} | (3) |

where Z(E) is the set of all zeroes of the map. This representation of scattering amplitudes is known as the Cachazo-He-Yuan (CHY) approach Cachazo:2013hca; Cachazo:2013iea; Cachazo:2014nsa; Cachazo:2014xea

The zeroes are generically isolated and are the values of \sigma^{\prime}s for which the Morse function on {\cal M}_{0,n}

\phi(\sigma,\bar{\sigma})=\frac{1}{2}\,\sum_{a\,<\,b}\,s_{ab}\,\ln|\sigma_{a}-% \sigma_{b}|^{2} | (4) |

has local extremes^{1}^{1}1A Morse function is a real function with non-degenerate critical points morse. Gross:1987ar; Ohmori:2015.

In this paper we are not concerned with particular theories. Instead, our aim is to provide an algorithm for the analytic computation of any integral of the form

\int_{\Gamma}\prod_{a=4}^{n}d\sigma_{a}\frac{|123|^{2}}{E_{4}(\sigma)E_{5}(% \sigma)\cdots E_{n}(\sigma)}F(\sigma), | (5) |

where \Gamma is the same contour as above, i.e., a sum over all residues at Z(E). Here F(\sigma) is any rational function of only the puncture coordinates \sigma’s which transforms as

F(\sigma)\to\prod_{a=1}^{n}(\textsf{c}\,\sigma_{a}+\textsf{d})^{4}F(\sigma),% \quad{\rm under}\quad\sigma_{a}\to\frac{\textsf{a}\,\sigma_{a}+\textsf{b}}{% \textsf{c}\,\sigma_{a}+\textsf{d}}, | (6) |

with \textsf{a}\textsf{d}-\textsf{b}\textsf{c}=1, i.e., under an PSL(2,\mathbb{C}) transformation.

The transformation of F(\sigma) ensures that the integral (5) is independent of both the choice of which puncture coordinates to fix and their values. The transformation also implies that F(\sigma) is only a function of differences \sigma_{a}-\sigma_{b} which we denote as \sigma_{ab}. Clearly \sigma_{ab}=-\sigma_{ba}. The only other condition we impose on F(\sigma_{ab}) is that all its poles are of the form \sigma_{ab}^{m} for some integer m\geq 0.

The simplest kind of integrals are defined in terms of the so-called Parke-Taylor factors Parke:1986gb defined for a particular ordering of n labels (\alpha(1)\alpha(2)\cdots\alpha(n)) with \alpha\in S_{n} as

\frac{1}{(\alpha(1)\alpha(2)\cdots\alpha(n))}\equiv\frac{1}{\sigma_{\alpha(1)% \alpha(2)}\,\sigma_{\alpha(2)\alpha(3)}\cdots\sigma_{\alpha(n-1)\alpha(n)}\,% \sigma_{\alpha(n)\alpha(1)}}. | (7) |

Clearly, any Parke-Taylor factor has half the PSL(2,\mathbb{C}) weight needed to construct a valid F(\sigma_{ab}) Cachazo:2013hca; Cachazo:2014xea. One can define integrals labeled by a pair a permutations \alpha,\beta\in S_{n} using

F^{\alpha,\beta}(\sigma_{ab})=\frac{1}{(\alpha(1)\alpha(2)\cdots\alpha(n))}% \times\frac{1}{(\beta(1)\beta(2)\cdots\beta(n))}, | (8) |

or more explicitly Cachazo:2013iea

m(\alpha|\beta)\equiv\int_{\Gamma}d\mu_{n}\frac{1}{(\alpha(1)\alpha(2)\cdots% \alpha(n))}\frac{1}{(\beta(1)\beta(2)\cdots\beta(n))}, | (9) |

where we have introduced a shorthand notation for the measure

d\mu_{n}\equiv\prod_{a=4}^{n}d\sigma_{a}\frac{|123|^{2}}{E_{4}(\sigma)E_{5}(% \sigma)\cdots E_{n}(\sigma)}. | (10) |

Integrals of the form m(\alpha|\beta) have been studied in the literature and are known to evaluate to a sum over connected tree Feynman graphs with only cubic (trivalent) interactions which are compatible with the two planar orderings defined by \alpha and \beta Cachazo:2013iea. We review this result in detail in section 2 and explain how to explicitly evaluate them as a rational function of the variables s_{ab}. Here it suffices to say that these known integrals form the basic building blocks of our construction and the main result of this work is an algorithm for writing

\int_{\Gamma}d\mu_{n}F(\sigma_{ab})=R(m(\alpha|\beta)), | (11) |

where R is a rational function of its variables with only numerical coefficients.

The reason general integrals are of interest can be seen, for example, in the evaluation of an n graviton amplitude which contains a term of the form Cachazo:2013hca; Cachazo:2013iea; Cachazo:2014nsa

\int_{\Gamma}d\mu_{n}\,\frac{(\epsilon_{1}\cdot\epsilon_{2})^{2}}{\sigma_{12}^% {4}}\prod_{a=3}^{n}\left(\sum_{b=2,b\neq a}^{n}\epsilon_{a}\cdot k_{b}\frac{% \sigma_{1b}}{\sigma_{ab}\sigma_{1a}}\right)^{2}. | (12) |

In this formula \epsilon_{c},k_{c} are fixed data and after fully expanding (12) they can be factored out leaving arbitrarily complicated integrals of the form (11) to be evaluated. Also motivated by the same physical problem, Kalousios developed a technique, different from the one presented here, for the computation of general five-point integrals in Kalousios:2015fya.

The algorithm we develop is based on three key constructions. The first is a generalization of the Kawai-Lewellel-Tye (KLT) relation Kawai:1985xq; Berends:1988; Bern:1998sv. The KLT relation was originally discovered as a relation among closed and open string theory amplitudes but since then it has inspired similar relations in field theory and more recently it found a natural set up which allows vast generalizations in the CHY representation of amplitudes. We present the general KLT construction in section LABEL:kltgen.

The second result is a classic one from graph theory graph1. Consider an integrand F(\sigma) such that it does not have any zeroes. This means that it is only the product of 2n factors \sigma_{ab} in the denominator with a trivial numerator that can be set to unity. Representing each puncture by a vertex and each \sigma_{ab} by an undirected edge connecting vertices a and b one finds that each F(\sigma) leads to a unique 4-regular graph G_{F} (not necessarily simple). A classic result of Petersen guarantees that any 4-regular graph with n vertices is 2-factorable. This means that G_{F} it is always the union of two 2-regular graphs with n vertices. Petersen’s result is reviewed in section LABEL:peter.

The third and final ingredient is an observation regarding the existence of a Hamiltonian decomposition of graphs graph1; graph2. In order to state the observation let us choose any 2-regular multigraph^{2}^{2}2In this work we use the terminology graph and multigraph interchangeably. In fact, the restriction to simple graphs is never necessary. G with n-vertices and no loops. We say that a connected 2-regular graph with n-vertices, H^{\rm conn}, is compatible with G if the 4-regular graph obtained from the union of G and H^{\rm conn} contains two edge-disjoint Hamilton cycles. The observation is that out of the (n-1)! possible connected graphs the number of compatible graphs with G is always larger than (n-3)!. This is explained in section LABEL:HD.

In section LABEL:mainalgo all ingredients are combined to produce the final algorithm for computing the rational function R in (11). The algorithm is general but in particular cases it can be modified to make it much more efficient.

Section LABEL:allsix is devote to examples that not only illustrate the use of the algorithm but also give the explicit Hamiltonian decompositions needed for the computation of the most general six-point integral.

In section LABEL:disc we end with discussions including future directions and physical applications in the form of novel relations among amplitudes. The appendix has a detailed explanation of how to implement Petersen’s theorem. The implementation is not far from being the actual proof so it a good way to gain intuition on why the theorem holds.

## 2 Definition of Building Blocks

The aim of this work is to provide an algorithm for the reduction of contour integrals on the moduli space of an n-punctured sphere of the form

\int d\mu_{n}F(\sigma) | (13) |

in terms of a basis of known integrals. Ensuring that the integrand is PSL(2,\mathbb{C}) invariant implies that F(\sigma) has the form

F(\sigma)=\frac{1}{(12\cdots n)(\gamma(1)\gamma(2)\cdots\gamma(n))}f(r_{ijkl}), | (14) |

where (12\cdots n) is the canonical Parke-Taylor and (\gamma(1)\gamma(2)\cdots\gamma(n)) is a Parke-Taylor factor with a \gamma\in S_{n} ordering (see (7) for the Parke-Taylor factor definition). f is a rational function of r_{ijkl} which are general cross ratios, i.e.,

r_{ijkl}\equiv\frac{\sigma_{ij}\sigma_{kl}}{\sigma_{il}\sigma_{jk}}. | (15) |

Of course, the choice of Parte-Taylor factor is completely arbitrary and can be conveniently made depending on the case. The measure d\mu_{n} was defined in (10) and is reviewed below.

In this section we discuss the basic building blocks which are special contour integrals with f(r_{ijkl})=1 and whose values are explicitly known Cachazo:2013iea; Dolan:2013isa; Dolan:2014ega. The building blocks are labeled by a pair of permutations \alpha,\beta\in S_{n}/\mathbb{Z}_{n}. The reason one has to mod out by cyclic permutations \mathbb{Z}_{n} is obvious from the definition

m(\alpha|\beta)\equiv\int_{\Gamma}d\mu_{n}\frac{1}{(\alpha(1)\alpha(2)\cdots% \alpha(n))}\,\frac{1}{(\beta(1)\beta(2)\cdots\beta(n))}. | (16) |

Recall that (\alpha(1)\alpha(2)\cdots\alpha(n))\equiv\sigma_{\alpha(1)\alpha(2)}\sigma_{% \alpha(2)\alpha(3)}\cdots\sigma_{\alpha(n-1)\alpha(n)}\sigma_{\alpha(n)\alpha(% 1)} and the measure is

d\mu_{n}\equiv\prod_{a=4}^{n}d\sigma_{a}\frac{|123|^{2}}{E_{4}E_{5}\cdots E_{n% }}. | (17) |

An explicit evaluation of the integral m(\alpha|\beta) would involve solving the equations Cachazo:2013gna; Dolan:2013isa; Kalousios:2013eca; Dolan:2014ega; Weinzierl:2014vwa; Kalousios:2015fya

E_{a}(\sigma)=\sum_{b=1,b\neq a}^{n}\frac{s_{ab}}{\sigma_{a}-\sigma_{b}}=0% \quad{\rm for}\quad a\in\{4,5,\ldots,n\}. | (18) |

These equations have (n-3)! solutions as proven in Cachazo:2013hca; Cachazo:2013gna and the data s_{ab} can be taken to be the components of a symmetric n\times n matrix of complex entries such that s_{11}=s_{22}=\cdots s_{nn}=0 and

\sum_{b=1,b\neq a}^{n}s_{ab}=0~{}~{}{\rm for}~{}~{}a\in\{1,2,\ldots,n\}. | (19) |

Once the solutions are found one computes the Jacobian matrix

\Phi_{ab}=\left\{\begin{array}[]{cc}\frac{s_{ab}}{\sigma_{ab}^{2}}&a\neq b,\\ -\sum_{c=1,c\neq a}^{n}\frac{s_{ac}}{\sigma_{ac}^{2}}&a=b.\\ \cr\omit\span\@@LTX@noalign{ }\omit\\ \end{array} |