On integrands and loop momentum in string and field theory

The notion of a unique integrand does not a priori makes sense in field theory: different Feynman diagrams have different loop momenta and there should be no reason to compare them. In string theory, however, a global integrand is natural and allows one, for instance, to make explicit the separation between left- and right-moving degrees of freedom. However, the significance of this integrand had not really been investigated so far. It is even more important in view of the recently discovered loop monodromies that are related to the duality between color and kinematics in gauge and gravity loop amplitudes. This paper intends to start filling this gap, by presenting a careful definition of the loop momentum in string theory and describing precisely the resulting global integrand obtained in the field-theory limit. We will then apply this technology to write down some monodromy relations at two and three loops and make contact with the color-kinematics duality.