Abstract
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.
Highlights
In the last few years, a variety of results for scattering amplitudes in field theory at loop level have been derived using string theoretic methods
In this paper we analyzed some aspects related to the definition of the loop momentum in string and field theory
This formalism was mostly developed to be applied to the monodromy relations, but it would be very interesting to see if the global integrand defined in this way has any nice physical properties
Summary
In the last few years, a variety of results for scattering amplitudes in field theory at loop level have been derived using string theoretic methods. The seminal papers [2,3,4,5] laid the foundations for the definition of the loop momentum in string theory amplitudes in their modern formulation as conformal field-theory correlation functions integrated over the moduli space of Riemann surfaces. More precisely it will support the conjecture that in all higher loop relations, the monodromy relations always combine the numerators appearing in the field-theory limit into groups of graphs called BCJ triplets. It should be noted that in this paper we will exclusively be concerned with the bosonic part of the string amplitudes, which is the one that carries the loop-momentum zero modes Further applications of these results are presented in the discussion in Sec. V together with open questions
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