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\sect{Chemical equilibrium for gamma == e and gamma == pi}

The reaction  ${\gamma+\gamma \leftrightarrow e^{+}+ e^{-}}$ 
occurs inside a star, where ${\gamma}$ is a photon and $e^{\pm}$  
are the positron and electron (mass $\mass$).
In general $e^{\pm}$ are relativistic. Assume overall charge neutrality,
and that the system is in equilibrium at temperature ${T}$.  \\

(1) Find an expression for the densities of ${e^{\pm}}$. 

(2) Find these densities in the limit ${T \ll \mass c^{2}}$. 

(3) Solve the same problem for the reaction ${\gamma +\gamma \leftrightarrow \pi^{+}+\pi^{-}}$,   
where ${\pi^{±}}$ are bosons with mass ${M}$. 

(4) Can the $\pi$ bosons become Bose-condensed if the temperature is
sufficiently lowered?  Explain the result physically.



