The KLT relations (Kawai, Lewellen, Tye) relate a closed string amplitude to a product of open ones. While I get the physics behind this I don't really understand the derivation in the original paper (see https://doi.org/10.1016/0550-3213(86)90362-7 ). Especially I don't understand the contour deformations starting around eq. (3.4) and following. I'd be glad if someone could either explain that a bit more in detail.
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The specific question you are asking is unclear to me. But I would like to point out to you a useful reference. The KLT relations were originally derived in string theory, but in 2010-11, Thomas Sondergaard and colleagues at the Niels Bohr Institute found a purely field-theoretic proof based on new ideas in the theory of scattering amplitudes. This proof is algebraic and does not make use of contour deformations. It is also less specialized, in that it does not require the equipment of string theory.
Sondergaard wrote a nice review paper covering the KLT derivation and the new field-theoretic derivation. It is published in Advances in High Energy Physics, but you can find it more easily at https://arxiv.org/abs/1106.0033. You will find all of the original references there. This paper also introduces the Bern-Carrasco-Johanssen (BCJ) representation of gravity amplitudes, which is an important step beyond KLT.
Michael Peskin
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