  In a recent work, Le Hur has shown that dissipative coupling to gate
electrodes may play an important role in a quantum box near its degeneracy
point [K. Le Hur, Phys. Rev. Lett. {\bf 92}, 196804 (2004)]: While quantum
fluctuations of the charge of the dot tend to round Coulomb blockade charging
steps of the box, strong enough dissipation suppresses these fluctuations and
leads to the reappearance of sharp charging steps. In the present paper we
study this quantum phase transition in detail using bosonization and numerical
renormalization group methods in the limit of vanishing level spacing.
