Chromatic Effects on the PSF and Shear Measurement for the Roman Space Telescope High-Latitude Wide Area Survey

Weak gravitational lensing (WL) is a key cosmological probe that requires precise measurement of galaxy images to infer shape distortions, or shear, and constrain cosmology. Accurate estimation of the Point Spread Function (PSF) is crucial for shear measurement, but the wavelength dependence of the PSF introduces chromatic biases that can systematically impact shear inference. We focus on biases arising from spectral energy distribution (SED) differences between stars, used for PSF modeling, and galaxies, used for shear measurement. We investigate these effects in $\textit{Roman's}$ four design reference mission WL bands (Y106, J129, H158, F184) and wide filter (W146). Using $\textit{Roman}$-like image simulations, we quantify the induced shear biases and compare them to requirements on those biases. Multiplicative biases over all galaxies hover around $\sim$0.2% in the WL bands and 2% in the wide filter, exceeding the mission requirement of $|m| < 0.032\%$ and relaxed requirement of $|m| < 0.1\%$. In individual redshift bins, biases can reach 0.4$\unicode{x2013}$0.9% for the WL bands and 3$\unicode{x2013}$6% for the wide filter. Additive biases remain acceptable in the WL bands but exceed systematic limits in the wide filter. We develop and test PSF-level corrections, showing that a first-order correction reduces biases within survey requirements for the WL bands; however, higher-order terms are necessary for the wide filter. Our results highlight the necessity of chromatic corrections for precision WL with $\textit{Roman}$ and provide a framework for mitigating these biases. Finally, we compare analytical color-based corrections to self-organizing maps (SOMs) and find that both methods effectively reduce biases.