Abstract
In this article we present the worldsheet integrand for one-loop amplitudes in maximally supersymmetric superstring theory involving any number n of massless open string states. The polarization dependence is organized into the same BRST invariant kinematic combinations which also govern the leading string correction to tree level amplitudes. The dimensions of the bases for both the kinematics and the associated worldsheet integrals is found to be the unsigned Stirling number S_3^{n-1} of first kind. We explain why the same combinatorial structures govern on the one hand finite one-loop amplitudes of equal helicity states in pure Yang Mills theory and on the other hand the color tensors at quadratic alpha prime order of the color dressed tree amplitude.
Highlights
The recursive BRST cohomology method obtained in [8] leads to compact and elegant supersymmetric answers and makes use of so-called BRST building blocks which can be regarded as superspace representatives of cubic diagrams
Amplitudes computed with the pure spinor formalism give rise to superspace kinematic factors in the cohomology of the BRST operator
That is why the zi integrals do not introduce any poles in kinematic invariants,9 i.e. that all massless open string propagators enter through the BRST invariants C1
Summary
In recent years the pure spinor formalism [4] allowed striking compactness in the computations of scattering amplitudes both in string theory [5, 6, 39,40,41, 48,49,50, 57, 60, 61] and directly in its field-theory limit [6, 8, 17, 18, 58, 59]. The open superstring did inspire the color organization of gauge theory amplitudes and provided an elegant proof for Bern-Carrasco-Johansson (BCJ) relations among color-ordered tree amplitudes [23, 24], based on monodromy properties on the worldsheet Another difficult field theory problem which found a string-inspired answer is the explicit construction of local kinematic numerators for gauge theory tree amplitudes which satisfy all the dual Jacobi identities, see [28]. After these tree-level examples of cross-fertilization between superstring and field theory amplitudes, we hope that this work helps to provide further guidelines to organize multileg one-loop amplitudes in maximally supersymmetric SYM in both ten and four dimensions.. According to subsection 7.3, the representation (7.21) of the colordressed O(α′2) tree manifests a duality to the one-loop kinematic factor (5.34)
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