Analytic derivation of the three-loop basketball coefficient

Derive analytically the conjectured closed-form expression for the finite three-loop basketball integral coefficient β in the bosonic membrane energy on \(\mathbb{R}^2\times S^1\), namely \(\beta=[385\,\zeta(9)-100\,\zeta(3)^3]/8192\), rather than relying on its numerical identification.

Background

The three-loop correction to the bosonic membrane energy contains a basketball-diagram contribution whose finite part is encoded in a coefficient β\beta. The authors evaluate this coefficient numerically and, by matching it to a weight-nine transcendental basis, obtain a candidate expression involving ζ(9)\zeta(9) and ζ(3)3\zeta(3)^3.

Because the expression is inferred numerically rather than derived from the defining three-loop integral, the paper explicitly treats it as a plausible conjecture. An analytic derivation would establish the exact finite contribution and underpin the resulting three-loop membrane-energy formula.

References

We did not manage to derive this expression analytically so it may be treated as (a very plausible) conjecture. An analytic derivation may be possible using methods developed for loop diagram computations in finite temperature field theory .

— Energy of toroidal M2 brane in flat 11d background  (2608.23517 - Tseytlin et al., 24 Aug 2026) in Section 5, subsection “Membrane on \(\mathbb R^2\times\tilde S^1\),” following Eq. (Cbb) and preceding Eq. (clos)