I'm a freshman in string theory. My question is: when we compute the full closed string propagator in a theory that contains both closed and open string, do we need to include open string as the intermediate state?
I can imagine the simplest case with only one intermediate open string. The world sheet is a cylinder with a slit cut on it. Is there any reference where people compute this diagram?
Yes, in computing the full closed string two-point amplitude one sums over all intermediate configurations, which includes summing over intermediate states with boundaries (aka open strings) if there are D-branes present.*
The cylinder with a slit that you mention can be mapped to a two-point disc amplitude (in particular a disc with two closed string vertex operator insertions). There are many references, one particularly clear one is by Becker, Guo and Robbins.
That this two-point disc amplitude should be non-vanishing also follows from the fact that D-branes are dynamical gravitating objects, because upon integrating over the moduli of this two-point disc amplitude there are regimes in the integrand which are best described as closed string (which includes graviton) exchange with the D-brane.
*An aside: There is no issue about whether a slit in the cylinder breaks conformal invariance. The intermediate states are always generically offshell (unless one focuses on the imaginary part), so naively conformal invariance would be broken. But intermediate states are not only offshell but also summed over, and this sum usually restores conformal invariance. In fact, loops and intermediate boundaries can nevertheless break conformal invariance but it is always restored when it needs to be, sometimes in a subtle way (e.g., by shifting the string background).
