The probability that a Gaussian tetrahedron is acute

We prove that four independent standard normal points of ℝ3 span an acute tetrahedron, that is, a tetrahedron whose six dihedral angles are all smaller than π/2, with probability (2/π) arctan(1/3) − 1/6 + (40√3 − 72)/(45π) = 0.0189404094… . For points of ℝn, n ≥ 3, let pn be this probability. For odd n, pn is a rational combination of arctan(1/3)/π − 1/12, 1/π and √3/π; for even n, π2pn is a rational combination of 1, ln(4/3) and √5 Cl2(arccos(1/9)), where Cl2 is the Clausen function. The coefficients are given for n ≤ 8; those for 4 ≤ n ≤ 8 are computer-assisted, obtained in exact rational arithmetic. The probability pn is written as a moment of the weight exp(−Φ) over the positive orthant of ℝ6, where Φ is a quadratic form of signature (1, 5) with zero diagonal. The evaluation of such moments is a translation of known results on one-loop Feynman integrals with massless propagators and on volumes of ideal hyperbolic simplices. MSC 2020: Primary 60D05; Secondary 52A22, 51M10, 33B30, 81Q30. The record contains the paper, its LaTeX source, and the programs and recorded outputs of the exact rational computations and of the Monte Carlo runs reported in the paper.