One-loop matrix elements of effective superstring interactions: α′-expanding loop integrands

Abstract

In the low-energy effective action of string theories, non-abelian gauge interactions and supergravity are augmented by infinite towers of higher-mass-dimension operators. We propose a new method to construct one-loop matrix elements with insertions of operators D2kFn and D2kRn in the tree-level effective action of type-I and type-II superstrings. Inspired by ambitwistor string theories, our method is based on forward limits of moduli-space integrals using string tree-level amplitudes with two extra points, expanded in powers of the inverse string tension α′. Similar to one-loop ambitwistor computations, intermediate steps feature non-standard linearized Feynman propagators which eventually recombine to conventional quadratic propagators. With linearized propagators the loop integrand of the matrix elements obey one-loop versions of the monodromy and KLT relations. We express a variety of four- and five-point examples in terms of quadratic propagators and formulate a criterion on the underlying genus-one correlation functions that should make this recombination possible at all orders in α′. The ultraviolet divergences of the one-loop matrix elements are crosschecked against the non-separating degeneration of genus-one integrals in string amplitudes. Conversely, our results can be used as a constructive method to determine degenerations of elliptic multiple zeta values and modular graph forms at arbitrary weight.

One-loop amplitudes in Einstein-Yang-Mills from forward limits

“…• uplift to higher-mass-dimension operators: It would be rewarding to incorporate αcorrections into our forward-limit approach to EYM one-loop amplitudes as done for loop integrands of pure SYM and supergravity in [46]. Adapting the methods of the reference to genus-one correlators mixing gauge and gravity multiplets should yield one-loop matrix elements of higher-mass-dimension operators 17 D 2k F m R n and the nonanalytic contributions to the α -expansion of the respective genus-one string amplitudes.…”

“…all admissible contributions from gauge and/or gravity multiplets in the loop. The conversion between linearized and quadratic propagators in the loop is performed via elementary partial-fraction manipulations in the examples of this work, see [46][47][48][49][50][51][52] for more general recent discussions of this conversion. However, the supersymmetryagnostic amplitude relations of this work still feature linearized propagators in intermediate steps.…”

“…The amplitude relations in this work are derived from forward limits of tree-level building blocks, where divergences in intermediate expressions are bypassed via ambitwistor-string methods [37][38][39][40]. 2 This way of implementing forward limits has been applied to explicitly construct loop integrands of gauge theories [43][44][45] and one-loop matrix elements of highermass-dimension operators in the low-energy effective action of superstrings [46]. However, the ambitwistor methods at work lead to an unconventional form of the Feynman propagators in the loop integrand: The inverse propagators of the partial integrands are linearized in the loop momentum , i.e.…”

The SAGEX Review on Scattering Amplitudes, Chapter 2: An Invitation to Color-Kinematics Duality and the Double Copy

“…Note that the (DF ) 2 theory we have introduced in the beginning of this section appears (in its non-minimal version) in several of the entries. While the discussion here focuses on tree-level amplitudes, some extensions to loop level are available in the literature [208][209][210][211][212][213][214][215][216][217][218][219][220]. See also [221,222] for a construction of string amplitudes in terms of field-theory amplitudes using the scattering-equations formalism.…”

Off-Shell Color-Kinematics Duality for Chern-Simons

“…For tree-level amplitudes in purely-adjoint gauge theories, the application of the double copy is equivalent to the field-theory limit of the Kawai-Lewellen-Tye (KLT) relations [73] between open and closed string amplitudes. The double-copy applies to a large web of gravitational theories [2,40,[74][75][76][77][78][79][80][81][82][83][84][85][86][87][88][89][90][91][92][93], including string theory [94][95][96][97][98][99][100] and ambi-twistor string constructions [101][102][103][104][105][106]. The double copy has been extended to classical solutions in general relativity, such as Schwarzschild and Kerr black holes and calculations relevant to gravitational-wave physics [132][133][134][135][136][137][138][139][140][141]…”