  It is well known that one can use B -> pi pi decays to probe the CP-violating
phase \alpha. In this paper we show that these same decays can be used to
search for new physics. This is done by comparing two weak phases which are
equal in the standard model: the phase of the t-quark contribution to the b ->
d penguin amplitude, and the phase of Bd-Bd(bar) mixing. In order to make such
a comparison, we require one piece of theoretical input, which we take to be a
prediction for |P/T|, the relative size of the penguin and tree contributions
to Bd -> pi^+ pi^-. If independent knowledge of \alpha is available, the decay
Bd(t) -> pi^+ pi^- alone can be used to search for new physics. If a full
isospin analysis can be done, then new physics can be found solely through
measurements of B -> pi pi decays. The most promising scenario occurs when the
isospin analysis can be combined with independent knowledge of \alpha. In all
cases, the prospects for detecting new physics in B -> pi pi decays can be
greatly improved with the help of additional measurements which will remove
discrete ambiguities.
