JHEP11(2018)069
loop order ` 1 2 3 4 5
# of integrals 1 1 3 19 141
Table 1. The number of `-loop integrals I
(`)
(u, v) contributing to the correlators (2.13). For the
sake of simipicity we mode out: 1). integrals, which factorize in a product of lower-loop integrals;
2). permutations of external points; 3). rational factors in cross-ratios u, v accompanying conformal
integrals.
so it can be represented as
I(1, 2, 3, 4) =
1
x
2
13
x
2
24
I(u, v) (2.10)
where I(u, v) is a conformally invariant function and, consequently, it depends on conformal
cross-ratios
u =
x
2
12
x
2
34
x
2
13
x
2
24
, v =
x
2
14
x
2
23
x
2
13
x
2
24
. (2.11)
Several examples of five-loop conformally covariant integrals are given in eq. (5.2).
The number of linear independent conformal integrals is smaller than one could naively
expect on the basis of the discrete symmetries of their integrands. The conformal symmetry
implies nontrivial relations among them, e.g.
I(1, 2, 3, 4) = I(3, 4, 1, 2) (2.12)
immediately follows from (2.10). The latter relation reduces the number of independent
orientations of the given integral. Applying (2.12) to the conformal `
0
-loop subintegrals
(`
0
< `) of the `-loop integrals we generate ‘magic’ identities [36] among `-loop integrals of
the different topology. Also some of the `-loop integrals trivially factorize in a product of
several lower-loop conformal integrals, and some of the integrals differ only by a rational
factor in cross-ratios u, v. These observations enable us to reduce the number of conformal
integrals we have to deal with. The number of non-trivially distinct `-loop integrals is
given in table 1. The asymptotic expansion of the integrals at u → 0, v → 1 is discussed in
appendix A and the results are collected in an ancillary file.
In the following we denote (2.9) — the integrated contribution of the {b
ij
} harmonic
structure to the r.h.s. of eq. (2.6) — by
F
(`)
{b
ij
}
(u,v)
x
2
13
x
2
24
. As we discussed above it is given by a
linear combination of the conformal integrals
F
(`)
{b
ij
}
(u, v) =
X
m
˜c
(m)
{b
ij
}
I
(`)
m
(u, v) , (2.13)
where numerical coefficients ˜c
(m)
{b
ij
}
are linear combinations of c
(k)
{b
ij
}
originating from con-
formal polynomials P
(`)
{b
ij
}
. Let us stress that the integrated expression (2.13) contains
less coefficients than the integrand. Thus we obtain the following representation for the
– 5 –
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