The K^- p --> pi^0 pi^0 Sigma^0 reaction is studied within a chiral unitary
model. The distribution of pi^0 Sigma^0 states forming the Lambda(1405) shows,
in agreement with a recent experiment, a peak at 1420 MeV and a relatively
narrow width of Gamma = 38 MeV. The mechanism for the reaction is largely
dominated by the emission of a pi^0 prior to the K^- p interaction leading to
the Lambda(1405). This ensures the coupling of the Lambda(1405) to the K^- p
channel, thus maximizing the contribution of the second state found in chiral
unitary theories, which is narrow and of higher energy than the nominal
Lambda(1405). This is unlike the pi^- p --> K^0 \pi Sigma reaction, which gives
more weight to the pole at lower energy and with a larger width. The data of
these two experiments, together with the present theoretical analysis, provides
a firm evidence of the two pole structure of the Lambda(1405).Comment: 4 pages, 6 figure