Some remarks on Coulombic effects in $pp$ and $\bar pp$ scattering and the determination of $\rho$

We point out a very simple method for calculating the mixed Coulomb-nuclear corrections to the $pp$ and $\bar pp$ scattering amplitudes that has been missed in the extensive past work on this problem. The method expresses the correction in terms of a rapidly convergent integral involving the inverse Fourier-Bessel transform of the nuclear amplitude and a known factor containing the Coulomb phase shift with form-factor corrections. The transform can be calculated analytically for the exponential-type model nuclear amplitudes commonly used in fits to the high-energy data at small momentum transfers, and gives very accurate results for the corrections. We examine the possible effects of the Martin zero in the real part of the nuclear amplitude, and the accuracy of the Bethe-West-Yennie phase approximation for the Coulomb-nuclear corrections. We then apply the method to a redetermination of the ratio $\rho$ of the real to the imaginary parts of the forward scattering amplitude in fits to high-energy ISR data previously analyzed using an approximate version of the correction. The only significant changes relative the accuracy of those fits are at 52.8 GeV. Our method is applicable more generally, and can be used also at lower energies and for proton-nucleus scattering.