Abstract
We consider the tetrahedral three-loop diagram in Ed exceptional field theory evaluated as a scalar diagram for four external gravitons. At lowest order in momenta, this diagram contributes to the ∇6R4 term in the low-energy effective action for M-theory. We evaluate explicitly the sums over the discrete exceptional field theory loop momenta that become sums over 1/2-BPS states in the compact exceptional space. These sums can be rewritten as Eisenstein series that solve the homogeneous differential equations that supersymmetry implies for the ∇6R4 coupling. We also show how our results, even though sums over 1/2-BPS states, are consistent with expected 1/4-BPS contributions to the couplings.
Highlights
2y yyyy 1345 yy d−1 d sometimes prove uniqueness of the perturbative and non-perturbative contributions to certain higher-derivative corrections [24]
We consider the tetrahedral three-loop diagram in Ed exceptional field theory evaluated as a scalar diagram for four external gravitons
At lowest order in momenta, this diagram contributes to the ∇6R4 term in the low-energy effective action for M-theory
Summary
The constant logarithmic term in ξ(2)ξ(3) log(πμ2) exhibits the need of introducing a renormalisation scale in the non-analytic component of the amplitude, which is a consequence of the logarithmic divergence in the supergravity form factor of the E(0,0)R4 type invariant with four external gravitons at 2-loop. Exhibits the need of introducing a renormalisation scale in the non-analytic component of the amplitude, which is a consequence of the logarithmic divergence in the supergravity form factor of the E(1,0)∇4R4 type invariant with four extrenal gravitons at 1-loop in four dimensions. As in the other cases the non-degenerate orbit provides the homogeneous solution to the differential equations of [20] and it was shown in particular in [28] that the above Eisenstein series encodes all the relevant information about supergravity divergences up to three loops.
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