We find explicit expressions for the invariant-amplitude contributions of the box diagrams in the interaction of pseudoscalar bosons with spin-½ fermions. The spin complications are reduced using unitarity such that we can express the $A$ and $B$ amplitudes as single-variable integrals over closed-form analytic functions. The problem is solved for the elastic-process amplitudes and is useful in calculations including fourth-order terms. Previous calculations used mainly numerical methods; we rely on exact partial-fraction expansions to obtain amplitudes from a few key analytic functions. The amplitudes are calculated independently for both the $s$ and $t$ channels and are tested by three different consistency checks. Various useful dispersion-integral forms are given, with emphasis on those suitable for partial-wave expansions. The closed forms obtained also yield "natural" subtraction points for the $s$-variable integrals. In writing the crossed box amplitude for convenient partial-wave expansion, we find a relation free from any formal dependence on a subtraction. This is due to an improved convergence of the absorptive parts in the new functional dependence which arises. The partial-wave contributions are also given explicitly.

• Received 2 March 1970

DOI:https://doi.org/10.1103/PhysRevD.2.3064