  Pulsar timing array (PTA) is expected to detect gravitational wave background
(GWB) in the nanohertz band within the next decade. This provides an
opportunity to test the gravity theory and cosmology. A typical data analysis
method to detect GWB is cross-correlation analysis. The overlap reduction
function (ORF) plays an important role in the correlation data analysis of GWB.
The present approach to dealing with the intricate integration in ORF is to use
short-wave approximation to drop out the tricky terms. In this paper, we
provide the full analytic expression of the ORF for PTA without any
approximation for all possible polarizations allowed by modifications of
general relativity. Compared with the numerical simulation and short-wave
approximation, our results are more efficient and widely applicable. Especially
for the scalar-longitudinal mode where the short-wave approximation is not
available, our analytical expression is particularly significant.
