There are no alkene double bonds left in either the styrene or the butene once the reaction is complete. Instead, you have pendant phenyl groups and pendant methyl ketones.
Just guessing here, but it is possible that the UV light is removing a hydrogen radical (homolytic bond dissociation) at the alpha position to the benzene ring, on the polymer chain, and that the radical is stabilized by the benzene ring. This provides a site for radical oxidation by the atmosphere, and oxidative deradation of the chain. The corresponding position on the butene section of the polymer is a quaternary carbon with no benzene ring to stabilize a radical.