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Wednesday 25 December 2013

Penta-denominator Integrals

\[\int\frac{d^dk}{(2\pi)^d}\frac{d^dl}{(2\pi)^d} \frac{1}{k^2 (k-p)^2 l^2 (l-p)^2 (k-l)^2}={{_{0,0}F_0}}(d)\, (p^2)^{d-5}\]\[\int\frac{d^dk}{(2\pi)^d}\frac{d^dl}{(2\pi)^d} \frac{k_\mu}{k^2 (k-p)^2 l^2 (l-p)^2 (k-l)^2}={{_{1,0}F_0}}(d)\,p_\mu (p^2)^{d-5}\]\[\int\frac{d^dk}{(2\pi)^d}\frac{d^dl}{(2\pi)^d} \frac{k_\mu k_\nu}{k^2 (k-p)^2 l^2 (l-p)^2 (k-l)^2}={{_{2,0}F_0}}(d)\,p_\mu p_\nu (p^2)^{d-5}+{{_{2,0}F_1}}(d)\,\hat{\delta}_{\mu\nu}(p^2)^{d-4}\]