One-loop master integral for order-v²ⁿ relativistic corrections to Γ[J/ψ → e⁺e^-]

We show that one-loop scalar integrals contributing to the order-v ²ⁿ relativistic corrections to the QCD vertex corrections to the spin-triplet S-wave heavy-quark-antiquark pair annihilation into a virtual photon can be expressed in terms of a single master integral. With an algebraic reduction combined with the tensor-integral reduction method, we first simplify the scalar integrals as linear combinations of two-point functions. The two-point functions are reduced into a single master integral by using the recurrence relations derived from integration by parts. Order-v² formulas are explicitly derived, and a strategy to calculate the higher-order relativistic corrections at order α_s, is also given. The result may be very useful in calculating the order-v²ⁿ relativistic corrections to the leptonic decay width of the spin-triplet S-wave heavy quarkonium at order α_s.