  The anomalous exponent, $\eta_{p}$, for the decay of the reunion probability
of $p$ vicious walkers, each of length $N$, in $d$ $(=2-\epsilon)$ dimensions,
is shown to come from the multiplicative renormalization constant of a $p$
directed polymer partition function. Using renormalization group(RG) we
evaluate $\eta_{p}$ to $O(\epsilon^2)$. The survival probability exponent is
$\eta_{p}/2$. For $p=2$, our RG is exact and $\eta_p$ stops at $O(\epsilon)$.
For $d=2$, the log corrections are also determined. The number of walkers that
are sure to reunite is 2 and has no $\epsilon$ expansion.
