arXiv subjects

Carlos R. Mafra

Publications and source records attributed to Carlos R. Mafra.

At least 19 recordsLinked to original sources

arXiv2026

Manifest Moebius invariance of massive tree-level three-point amplitudes in pure spinor superspace

Using BRST cohomology properties in pure spinor superspace and identities for OPE brackets of non-free fields, we obtain a new compact nested-bracket representation of massive tree-level three-point open-string amplitudes in which Moebius invariance is manifest. Explicit superspace calculations for amplitudes with level-one massive states confirm this finding, and we derive new BRST recurrence relations among three-point numerators to extend the result to arbitrary mass levels. This provides a manifestly Moebius-invariant expression for massive three-point amplitudes in the pure spinor formalism.

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arXiv2025

The massive one-loop four-point string amplitude in pure spinor superspace

The open- and closed-string three- and four-point one-loop amplitudes involving massless states and one first-level massive state are computed in pure spinor superspace. For the open string, we show that their one-loop correlators can be rewritten in terms of tree-level kinematic factors. We then analyze the closed string. For three points, this is immediate. For four points, we show that it is possible to rewrite the one-loop closed-string correlator using tree-level kinematic factors, but only for certain combinations of massive and massless states (different for type IIA and IIB).

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arXiv2025

Massless representation of massive superfields and tree amplitudes with the pure spinor formalism

We construct the unintegrated vertex operator at the first mass level of the open superstring from the OPE of massless vertices. Using BRST cohomology manipulations, the tree amplitude of two massless and one massive state is rewritten in terms of the pure spinor superspace kinematic expression of the massless four-point amplitude at the ${α^\prime}^2$ level. A generalization relating the partial $n$-point tree amplitudes with one massive state and linear combinations of the ${α^\prime}^2$ corrections to n+1 massless amplitudes is found and shown to be consistent with unitarity.

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arXiv2024

Towards massive field-theory amplitudes from the cohomology of pure spinor superspace

By analogy with the formula for the massless string disk amplitudes, we define massive field-theory tree amplitudes and conjecture that the BRST cohomology structure of pure spinor superspace fixes their form. We give evidence by deriving the pure spinor superspace expression of the massive field-theory n-point tree amplitude with one first-level massive and n-1 massless states in two ways: 1) from BRST cohomology arguments in pure spinor superspace and 2) from the ${α^\prime}^2$ correction to the massless string amplitudes by inverting the unitarity constraint in superspace.

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arXiv2023

A relation between massive and massless string tree amplitudes

We uncover a relation between the scattering amplitudes of massive strings and the $α'$ expansion of the massless string amplitude at tree level. More precisely, the n-point tree amplitude of n-1 massless and one massive state is written as a linear combination of n+1 massless string amplitudes at the $α'^2$ order.

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arXiv2023

Tree-level amplitudes from the pure spinor superstring

We give a comprehensive review of recent developments on using the pure spinor formalism to compute massless superstring scattering amplitudes at tree level. The main results of the pure spinor computations are placed into the context of related topics including the color-kinematics duality in field theory and the mathematical structure of $α'$-corrections.

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arXiv2022

Pure spinor formulation of the superstring and its applications

The pure spinor formalism for the superstring has the advantage over the more conventional Ramond-Neveu-Schwarz formalism of being manifestly spacetime supersymmetric, which simplifies the computation of multiparticle and multiloop amplitudes and allows the description of Ramond-Ramond backgrounds. In addition to the worldsheet variables of the Green-Schwarz-Siegel action, the pure spinor formalism includes bosonic ghost variables which are constrained spacetime spinors and are needed for covariant quantization using a nilpotent BRST operator. In this review, several applications of the formalism are described including the explicit computation in D=10 superspace of the general disk amplitude with an arbitrary number of external massless states, genus one amplitudes with up to seven external states, genus two amplitudes with up to five external states, and the low-energy limit of the genus three amplitude with up to four external states. The pure spinor formalism has also been used to covariantly quantize the superstring in an $AdS_5\times S^5$ background and might be useful for proving the AdS-CFT correspondence in the limit of small AdS radius. This is an overview written for the "Handbook of Quantum Gravity", eds. C. Bambi, L. Modesto and I. Shapiro.

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arXiv2022

KK-like relations of $α^{\prime}$ corrections to disk amplitudes

Inspired by the definition of color-dressed amplitudes in string theory, we define analogous {\it 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 {\it BRST-invariant permutations}, which we show are elements of the inverse Solomon descent algebra and we find a closed formula for them. We then present evidence that these permutations encode KK-like relations among the different $α'$ corrections to disk amplitudes refined by their motivic MZV content. In particular, the number of linearly independent amplitudes at a given ${α^{\prime}}$ order and motivic MZV content is given by (sums of) Stirling cycle numbers. In addition, we show how the superfield expansion of BRST invariants of the pure spinor formalism corresponding to ${α^{\prime}}^2 f_2$ corrections is encoded in the descent algebra.

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arXiv2022

Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality

In an earlier paper, we constructed the genus-two amplitudes for five external massless states in Type II and Heterotic string theory, and showed that the alpha' expansion of the Type II amplitude reproduces the corresponding supergravity amplitude to leading order. In this paper, we analyze the effective interactions induced by Type IIB superstrings beyond supergravity, both for U(1)_R-preserving amplitudes such as for five gravitons, and for U(1)_R-violating amplitudes such as for one dilaton and four gravitons. At each order in alpha', the coefficients of the effective interactions are given by integrals over moduli space of genus-two modular graph functions, generalizing those already encountered for four external massless states. To leading and sub-leading orders, the coefficients of the effective interactions D^2 R^5 and D^4 R^5 are found to match those of D^4 R^4 and D^6 R^4, respectively, as required by non-linear supersymmetry. To the next order, a D^6 R^5 effective interaction arises, which is independent of the supersymmetric completion of D^8 R^4, and already arose at genus one. A novel identity on genus-two modular graph functions, which we prove, ensures that up to order D^6 R^5, the five-point amplitudes require only a single new modular graph function in addition to those needed for the four-point amplitude. We check that the supergravity limit of U(1)_R-violating amplitudes is free of UV divergences to this order, consistently with the known structure of divergences in Type IIB supergravity. Our results give strong consistency tests on the full five-point amplitude, and pave the way for understanding S-duality beyond the BPS-protected sector.

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arXiv2021

Towards closed strings as single-valued open strings at genus one

We relate the low-energy expansions of world-sheet integrals in genus-one amplitudes of open- and closed-string states. The respective expansion coefficients are elliptic multiple zeta values in the open-string case and non-holomorphic modular forms dubbed "modular graph forms" for closed strings. By inspecting the differential equations and degeneration limits of suitable generating series of genus-one integrals, we identify formal substitution rules mapping the elliptic multiple zeta values of open strings to the modular graph forms of closed strings. Based on the properties of these rules, we refer to them as an elliptic single-valued map which generalizes the genus-zero notion of a single-valued map acting on multiple zeta values seen in tree-level relations between the open and closed string.

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arXiv2021

Local BCJ numerators for ten-dimensional SYM at one loop

We obtain local numerators satisfying the BCJ color-kinematics duality at one loop for super-Yang-Mills theory in ten dimensions. This is done explicitly for six points via the field-theory limit of the genus-one open superstring correlators for different color orderings, in an analogous manner to an earlier derivation of local BCJ-satisfying numerators at tree level from disk correlators. These results solve an outstanding puzzle from a previous analysis where the six-point numerators did not satisfy the color-kinematics duality.

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arXiv2020

A Lie bracket for the momentum kernel

We develop new mathematical tools for the study of the double copy and colour-kinematics duality for tree-level scattering amplitudes using the properties of Lie polynomials. We show that the $S$-map that was defined to simplify super-Yang--Mills multiparticle superfields is in fact a new Lie bracket on the dual space of Lie polynomials. We introduce {\it Lie polynomial currents} based on Berends-Giele recursion for biadjoint scalar tree amplitudes that take values in Lie polynomials. Field theory amplitudes are obtained from the Lie polynomial amplitudes by numerators characterized as homomorphisms from the free Lie algebra to kinematic data. Examples are presented for the biadjoint scalar, Yang--Mills theory and the nonlinear sigma model. That these theories satisfy the Bern-Carrasco-Johansson amplitude relations follows from the identities we prove for the Lie polynomial amplitudes and the existence of BCJ numerators. A KLT map from Lie polynomials to their dual is obtained by nesting the S-map Lie bracket; the matrix elements of this map yield a recently proposed `generalized KLT matrix', and this reduces to the usual KLT matrix when its entries are restricted to a basis. Using this, we give an algebraic proof for the cancellation of double poles in the KLT formula for gravity amplitudes. We finish with some remarks on numerators and colour-kinematics duality from this perspective.

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arXiv2020

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.

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arXiv2020

A closed-formula solution to the color-trace decomposition problem

In these notes we present a closed-formula solution to the problem of decomposing traces of Lie algebra generators into symmetrized traces and structure constants. The solution is written in terms of Solomon idempotents and exploits a projection derived by Solomon in his work on the Poincare-Birkhoff-Witt theorem.

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arXiv2020

Two-loop superstring five-point amplitudes I: Construction via chiral splitting and pure spinors

The full two-loop amplitudes for five massless states in Type~II and Heterotic superstrings are constructed in terms of convergent integrals over the genus-two moduli space of compact Riemann surfaces and integrals of Green functions and Abelian differentials on the surface. The construction combines elements from the BRST cohomology of the pure spinor formulation and from chiral splitting with the help of loop momenta and homology invariance. The $α' \to 0$ limit of the resulting superstring amplitude is shown to be in perfect agreement with the previously known amplitude computed in Type~II supergravity. Investigations of the $α'$ expansion of the Type~II amplitude and comparisons with predictions from S-duality are relegated to a first companion paper. A construction from first principles in the RNS formulation of the genus-two amplitude with five external NS states is relegated to a second companion paper.

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arXiv2020

All-order alpha'-expansion of one-loop open-string integrals

We present a new method to evaluate the $α'$-expansion of genus-one integrals over open-string punctures and unravel the structure of the elliptic multiple zeta values in its coefficients. This is done by obtaining a simple differential equation of Knizhnik-Zamolodchikov-Bernard-type satisfied by generating functions of such integrals, and solving it via Picard iteration. The initial condition involves the generating functions at the cusp $τ\to i\infty$ and can be reduced to genus-zero integrals.

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arXiv2019

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 $τ$ which is known from the A-elliptic Knizhnik--Zamolodchikov--Bernard associator. The expressions for their $τ$-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 $α'$. In fact, we are led to matrix representations of certain derivations dual to Eisenstein series. Like this, also the $α'$-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 $τ\rightarrow i\infty$ 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 $α'$-expansions of $n$-point integrals over boundaries of cylinder- or Moebius-strip worldsheets, where any desired order is accessible from elementary operations.

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arXiv2019

Towards the n-point one-loop superstring amplitude I: Pure spinors and superfield kinematics

This is the first installment of a series of three papers in which we describe a method to determine higher-point correlation functions in one-loop open-superstring amplitudes from first principles. In this first part, we exploit the synergy between the cohomological features of pure-spinor superspace and the pure-spinor zero-mode integration rules of the one-loop amplitude prescription. This leads to the study of a rich variety of multiparticle superfields which are local, have covariant BRST variations, and are compatible with the particularities of the pure-spinor amplitude prescription. Several objects related to these superfields, such as their non-local counterparts and the so-called BRST pseudo-invariants, are thoroughly reviewed and put into new light. Their properties will turn out to be mysteriously connected to products of one-loop worldsheet functions in packages dubbed "generalized elliptic integrands", whose prominence will be seen in the later parts of this series of papers.

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