I want to know the scattering amplitude involving both open and closed string, more specifically, the amplitude between two gluons and one graviton in open closed set up. Is there a reference where such an amplitude is computed explicitly? (I just want to know in the bosonic string set up to understand the physics)

If not, how should I compute it? I am confused by the $c$ ghost for the closed string vertex operator. Using $PSL(2,R)$, we can fixed two gluon vertex operator position at $x_1,x_2$, and furthermore the real coordinate of graviton at $x_3$, but we are still left with the undetermined imaginary coordinate $y_3$. We need to integrate over $y_3$, but how about the $c \tilde c$ ghost? In eq. 2.23 of https://arxiv.org/abs/0907.2211, they seem to do the computation for superstring where they only have $\tilde c$ ghost? But how about $c$ ghost? I am also confused by the statement after eq. 2.24. It says the amplitude will vanish unless the transverse momentum of graviton is not zero. Why?

I am now even more confused by one thing: naive computation involves
$$ <\dot X^m e^{i p_1\cdot X)}(x_1) \dot X^n e^{i p_2\cdot X)}(x_2) \partial X^r \bar\partial X^s e^{i k\cdot X}(z,\bar z) > $$

If we set all the momenta to zero, then we can still have non-trivial contraction $$ \frac{1}{|x_1-z|^2|x_2-z|^2 } $$ which has no momentum dependence. This suggests that the amplitude has a momentum independence pieces which only depend on the polarization. But this is very strange because for minimally coupled gluon and graviton, their 3 point amplitude has momentum dependences. What is wrong with my logic? I am really confused. Thanks!


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