Color-dressed string disk amplitudes and the descent algebra

Inspired by the definition of color-dressed amplitudes in string theory, we define analogous color-dressed permutations replacing the color-ordered string amplitudes by their corresponding permutations. Decomposing the color traces into symmetrized traces and structure constants, the color-dressed permutations define BRST-invariant permutations, which we show are elements of the inverse Solomon descent algebra. Comparing both definitions suggests a duality between permutations in the inverse descent algebra and kinematics from the higher $alpha$ sector of string disk amplitudes. We analyze the symmetries of the $alpha$ disk corrections and obtain a new decomposition for them, leading to their dimensions given by sums of Stirling cycle numbers. The descent algebra also leads to the interpretation that the ${alpha}^2zeta_2$ correction is orthogonal to the field-theory amplitudes as well as their respective tails of BCJ-preserving interactions. In addition, we show how the superfield expansion of BRST invariants of the pure spinor formalism corresponding to ${alpha}^2$ corrections are encoded in the descent algebra.

Planar binary trees in scattering amplitudes

These notes are a written version of my talk given at the CARMA workshop in June 2017, with some additional material. I presented a few concepts that have recently been used in the computation of tree-level scattering amplitudes (mostly using pure spinor methods but not restricted to it) in a context that could be of interest to the combinatorics community. In particular, I focused on the appearance of {it planar binary trees} in scattering amplitudes and presented some curious identities obeyed by related objects, some of which are known to be true only via explicit examples.

One-loop open-string integrals from differential equations: all-order alpha-expansions at n points

We study generating functions of moduli-space integrals at genus one that are expected to form a basis for massless $n$-point one-loop amplitudes of open superstrings and open bosonic strings. These integrals are shown to satisfy the same type of linear and homogeneous first-order differential equation w.r.t. the modular parameter $tau$ which is known from the A-elliptic Knizhnik--Zamolodchikov--Bernard associator. The expressions for their $tau$-derivatives take a universal form for the integration cycles in planar and non-planar one-loop open-string amplitudes. These differential equations manifest the uniformly transcendental appearance of iterated integrals over holomorphic Eisenstein series in the low-energy expansion w.r.t. the inverse string tension $alpha$. In fact, we are led to matrix representations of certain derivations dual to Eisenstein series. Like this, also the $alpha$-expansion of non-planar integrals is manifestly expressible in terms of iterated Eisenstein integrals without referring to twisted elliptic multiple zeta values. The degeneration of the moduli-space integrals at $tau rightarrow iinfty$ is expressed in terms of their genus-zero analogues -- $(n{+}2)$-point Parke--Taylor integrals over disk boundaries. Our results yield a compact formula for $alpha$-expansions of $n$-point integrals over boundaries of cylinder- or Moebius-strip worldsheets, where any desired order is accessible from elementary operations.

Towards the n-point one-loop superstring amplitude III: One-loop correlators and their double-copy structure

In this final part of a series of three papers, we will assemble supersymmetric expressions for one-loop correlators in pure-spinor superspace that are BRST invariant, local, and single valued. A key driving force in this construction is the generalization of a so far unnoticed property at tree level; the correlators have the symmetry structure akin to Lie polynomials. One-loop correlators up to seven points are presented in a variety of representations manifesting different subsets of their defining properties. These expressions are related via identities obeyed by the kinematic superfields and worldsheet functions spelled out in the first two parts of this series and reflecting a duality between the two kinds of ingredients. Interestingly, the expression for the eight-point correlator following from our method seems to capture correctly all the dependence on the worldsheet punctures but leaves undetermined the coefficient of the holomorphic Eisenstein series ${rm G}_4$. By virtue of chiral splitting, closed-string correlators follow from the double copy of the open-string results.

Non-abelian $Z$-theory: Berends-Giele recursion for the $alpha$-expansion of disk integrals

We present a recursive method to calculate the $alpha$-expansion of disk integrals arising in tree-level scattering of open strings which resembles the approach of Berends and Giele to gluon amplitudes. Following an earlier interpretation of disk integrals as doubly partial amplitudes of an effective theory of scalars dubbed as $Z$-theory, we pinpoint the equation of motion of $Z$-theory from the Berends-Giele recursion for its tree amplitudes. A computer implementation of this method including explicit results for the recursion up to order $alpha^7$ is made available on the website http://repo.or.cz/BGap.git

The One-loop Open Superstring Massless Five-point Amplitude with the Non-Minimal Pure Spinor Formalism

We compute the massless five-point amplitude of open superstrings using the non-minimal pure spinor formalism and obtain a simple kinematic factor in pure spinor superspace, which can be viewed as the natural extension of the kinematic factor of the massless four-point amplitude. It encodes bosonic and fermionic external states in supersymmetric form and reduces to existing bosonic amplitudes when expanded in components, therefore proving their equivalence. We also show how to compute the kinematic structures involving fermionic states.