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We suggest a program to establish theoretical performance limits of srfcavities using modern theories of nonequilibrium superconductivity under astrong electromagnetic field. These theories will be used to calculate the mainparameter of merit of srf cavities: the quality factor Q and its dependencieson the field amplitude, temperature and frequency, which would allow us tounderstand how far the srf cavity performance could be pushed from the currentstate of the art. Given that the quality factor is determined by multiplemechanisms operating on very different length scales, we will address theinterconnected problems of a nonlinear surface resistance, rf losses ofvortices trapped in the cavity, the effect of materials defects and surfacetopography, and the opportunities to boost the srf performance by surfacenano-structuring, impurity management and multilayers. We suggest the followingdirections of theoretical srf research to address the goals of boosting theperformance of the next generation particle accelerators: 1. Establishing the Qlimit, mechanisms of nonlinear surface resistance and the residual resistancein a nonequilibrium superconductor under a strong RF field. 2. Establishing thesrf breakdown field limit, dynamic superheating field and its dependencies onfrequency, temperature and concentration of impurities. 3. Losses due totrapped vortices and extreme dynamics of ultrafast vortices driven by strong rfMeissner currents in srf cavities. 4. Optimization of srf performance due tosurface nanostructuring of the cavity surface, multilayers and impuritymanagement.

Challenges and opportunities of srf theory for next generation particle accelerators