Double-real contribution to the quark beam function at N$^{3}$LO QCD

Understand

We compute the master integrals required for the calculation of the double-real emission contributions to the matching coefficients of 0-jettiness beam functions at next-to-next-to-next-to-leading order in perturbative QCD.

• As an application, we combine these integrals and derive the double-real gluon emission contribution to the matching coefficient $I_{qq}(t,z)$ of the quark beam function.