amplitudes

Triangular IBP equations

Results of the paper Liu, Mitov arXiv2512.05923. Please refer to this article for explanations.

List of results: We provide all the triangular equations for the three topologies given in the paper:

  • 2-loop non-planar off-shell di-boson (37 master integrals). Download here (0.39 MB)
  • 2-loop planar 3-jet C_1 (62 master integrals). Download here (3.45 MB)
  • 2-loop non-planar 3-jet B_1 (113 master integrals). Download here (71.67 MB)

For each topology, we provide the triangular equations that solve the system up to −5 in the top sector, up to −4 in the next lower sectors, up to −3 in the subsequent ones, and so on.

IMPORTANT: Restoration of dimensionful parameter.

In our calculations we have set s=1 for the four-point topology and s12=1 for the five-point topology. This means that after retrieving the projection of any integral onto a master integral, one must restore an overall power of s or s12 by comparing the mass dimensions of the two sides of the equation.

Organisation of the equations. For each topology, we provide the following four Mathematica readable dat files:

  • “config.dat”: this includes the definition of internal and external momenta, propagators, kinematics variables and the physical topology. The physical topology is a list of 1 and 0, which indicates the physical propagators and the irreducible numerators respectively. For example, topology={1,1,1,1,1,1,1,0,0} means the first seven propagators are physical and the last two are irreducible numerators.
  • “allequations.dat”: this is formatted as a Mathematica list, which contains the LHS of all the triangular equations, where the RHS is zero. The list is ordered in lower-triangular form, i.e. each new equation will solve a new integral.
  • “allintegrals.dat”: This file is a Mathematica-formatted list containing all integrals that appear in the equations. The ordering is as follows: the list begins with the master integrals, followed by all remaining integrals in the order in which they are solved by the equations. For example, the first non-master integral in the list corresponds to the integral solved by the first equation in “allequations.dat”.
  • “allmasters.dat”: This file is a Mathematica-formatted list containing all the master integrals.
  • The integrals are named as B[nu1,nu2,…..], where nu1 corresponds to the power of the first propagators defined in “config.dat” , nu2 to the second, and so on.

 

IBP reductions for two-loop five-points amplitudes in QCD

Results of the paper Chawdhry, Lim and Mitov arXiv:1805.09182. Please refer to this article for explanations.

List of results: We classify integrals by “degree,” which is defined to be the sum of all numerator powers in the integrand. In all cases, we allow the integrand to have a maximum of 1 squared denominator. We provide all projections satisfying the following criteria:

  • C1 topology:
    • Masters 1-7: all integrals of degree <= 4; and also the 5 integrals of degree 5 which appear in the amplitude for qq->QQg
    • Masters 8-36: all integrals of degree <= 4; and also all 21 integrals of degree 5 from the highest sector (this includes the
      5 integrals of degree 5 which appear in the amplitude qq->QQg)
    • Masters 37-62: all integrals of degree <= 5
  • B1 topology:
    • Masters 105-113 (highest sector for this topology): all integrals of degree <= 6
  • B2 topology:
    • Masters 73-75 (highest sector for this topology): all integrals of degree <= 6

Results to download: 

 

IMPORTANT: Restoration of dimensionful parameter s12: In our calculations we have set s12=1. This means that after retrieving the  projection of any integral onto a master integral, one must restore an overall power of s12 by comparing the mass dimensions of the two sides of the equation.

Organization of results: The files are arranged by topology (B1, C1, etc.) and by master integral. They are grouped into .tar files which each contain several .bz2 files. Each .bz2 file contains the projections of all solved integrals onto a specific master. For example, C1-Master-62.bz2 contains the projections of 2878 integrals on to the 62nd master in the C1 topology. The masters for the five topologies are enumerated in the file definitions.txt. To obtain the full reduction of an integral, one must retrieve its projection onto each master in the topology and then sum.

Naming of the integrals: Inside the .bz2 files, integrals are named based on the indices of the individual propagators in the integrand. For the purposes of labelling, there is no distinction between master and non-master integrals. Positive indices denote propagators appearing in the denominator of the integrand, whilst negative indices denote propagators appearing in the numerator of the integrand. An integral is named by concatenating the indices, using the letter ‘m’ to denote negative indices (i.e. numerator powers) and the letter ‘x’ as a separator. The resulting string is then prefixed by the family of the topology (‘B’ or ‘C’), again using ‘x’ as a separator.

For example, Cx1x1xm1x1x1x1x0xm1xm1x0xm1 denotes the integral with indices {1,1,-1,1,1,1,0,-1,-1,0,-1} from the C family of topologies.

Zero-valued projections: Many integrals have a projection of 0 onto one or more masters. In many cases, these trivial projections will not appear in the results files. The user can safely assume that if a projection of a certain integral onto a certain master is not given, and yet is within the range stated above (“list of results”), then that projection is zero.