##### 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: **

- definitions.txt All required definitions for masters and kinematic invariants.
- B1_masters_105-113.tar (file size 89MB) All masters in the highest sector in topology B1.
- B2_masters_73-75.tar (file size 8MB) All masters in the highest sector in topology B2.
- C1_masters_1-6.tar (file size 7GB) All 3-propagator masters in topology C1.
- C1_masters_7-16.tar (file size 6.2GB) All 4-propagator masters in topology C1.
- C1_masters_17-36.tar (file size 4.2GB) All 5-propagator masters in topology C1.
- C1_masters_37-62.tar (file size 4.5GB) All 5-, 6-, 7- and 8-propagator masters in topology C1.

**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.