Cosmological transverse momentum fields, whose directions are perpendicular to Fourier wave vectors, induce temperature anisotropies in the cosmic microwave background via the kinetic Sunyaev-Zeldovich (kSZ) effect. The transverse momentum power spectrum contains the four-point function of density and velocity fields, $langledeltadelta v vrangle$. In the post-reionization epoch, nonlinear effects dominate in the power spectrum. We use perturbation theory and cosmological $N$-body simulations to calculate this nonlinearity. We derive the next-to-leading order expression for the power spectrum with a particular emphasis on the connected term that has been ignored in the literature. While the contribution from the connected term on small scales ($k>0.1,h,rm{Mpc}^{-1}$) is subdominant relative to the unconnected term, we find that its contribution to the kSZ power spectrum at $ell = 3000$ at $z<6$ can be as large as ten percent of the unconnected term, which would reduce the allowed contribution from the reionization epoch ($z>6$) by twenty percent. The power spectrum of transverse momentum on large scales is expected to scale as $k^2$ as a consequence of momentum conservation. We show that both the leading and the next-to-leading order terms satisfy this scaling. In particular, we find that both of the unconnected and connected terms are necessary to reproduce $k^2$.
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