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In string theory, it's possible to collide a number of strings only in the asymptotic past, and get a final asymptotic state which contains a D0-brane, an anti D0-brane, and a number of strings. If space is compactified, the D0-brane can be replaced by some wrapped D-brane particle, or a stable bound state of a number of them.

How is this handled using string worldsheets only? It's true if the string coupling is weak, many strings have to collide to produce a pair of D-branes, but the string coupling is always nonzero, even if tiny. Does this mean a worldsheet formalism only is incomplete? It's true this is a nonperturbative effect, but does this mean perturbation theory is incomplete?

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You must be careful about the ordering of the limits.

Perturbation theory is usually "effective and accurate enough" for all questions in which the coupling is weak. However, the actual description of perturbation theory is that it is a systematic expansion around $g=0$. For $g=0$, the number of strings and/or their energy needed to produce a Dirichlet brane-antibrane pair – whose mass goes like $1/g$ – is infinite simply because $1/g=\infty$. So the problem is ill-posed for $g=0$ and we can't expand around $g=0$ in the naive way.

In reality, the process you are referring to is omnipresent in hundreds of string theory papers, especially those written a decade ago. I am referring to Ashoke Sen's minirevolution on tachyon condensation. Take the time-reversed process to yours. You have a D0-brane and a D0-antibrane in the initial state, perhaps with some extra strings which are really not needed for the essence of the problem in this picture, and you calculate the probability amplitude that they annihilate and create certain strings in the final state.

My/Sen's chronological ordering is the more natural one because the mass concentrated in heavy D-branes clearly carries a lower entropy than the state of very many light strings. The second law of thermodynamics implies that entropy is increasing so it's much more likely that an unstable collection of D-branes will annihilate than the process in which a family of many fundamental strings will conspire and invest almost all of their combined energy into the production of a single D0-brane and a single D0-antibrane; this process is of course very unlikely.

The D-brane annihilation has been described in various effective field theory approximations. We really care what happens with the tachyon. It has to roll down to a minimum in which the unstable brane(s) disappear(s). Which strings will be exactly produced is a difficult question that can only be accessed approximately. It's because it's really hard to define the initial state. Because it's unstable, it's not unique – for the same reason why there's no unique state of a particle near the maximum of a potential (imagine the inverted harmonic oscillator) and all these details affect the precise final state.

However, in principle, it's fair to say that it's a nonperturbative process given by a D-instanton (an unstable one, like in Coleman's papers), more precisely D$(p-1)$-instanton if you have $D$p-branes in your state with D-branes.

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