Deforming Photons

So, a mass-less collision with a photon is possible.  During the process the photon gain mass as its $\psi$ changes to counter the impact.  At the extreme, the photon has the mass of a proton.$\require{cancel}$

$m_{\rho\,ph} c^2=m_{\rho\,po} c^2-\cancelto{0}{\int^{x_a}_{0}{\psi}dx}$

where $x_z=0$ and

$\cfrac{\partial\,F}{\partial\,x}=maximum=-\cfrac{\partial^2\psi}{\partial\,x^2}$

such that,

$\Delta F=-\cfrac{\partial^2\psi}{\partial\,x^2}\Delta x$

At $x_z=0$, a change in $\Delta x$ encounters a maximum change in force density.  $\psi$ deformed and the photon gained mass, but at which point ($x_z=0$), a change in $x$ results in the largest change in $F$ and the photon readily returns to its original form.  There is no new equilibrium/stable point.