BCFW Shifts Research Articles

General expressions for on-shell recursion relations for tree-level open string amplitudes

In this paper, we present a systematic derivation aimed at obtaining general expressions for on-shell recursion relations for tree-level open string amplitudes. Our approach involves applying the BCFW shift to an open string amplitude written in terms of multiple Gaussian hypergeometric functions. By employing binomial expansions, we demonstrate that the shifted amplitudes manifest simple poles, which correspond to scattering channels of intermediate states. Using the residue theorem, we thereby derive a general expression for these relations.

On-shell functions on the Coulomb branch of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 4 SYM

We study on-shell functions in the kinematic space for the Coulomb branch of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 4 SYM. We construct BCFW bridges that help us build bigger on-shell functions. As a consequence, we provide on-shell diagram formulations for BCFW shifts that correspond to various mass configurations. We will use this to calculate the quadruple cut for the one-loop amplitude on the Coulomb branch and maximal cuts for higher-loops. We make preliminary comments on finding the inequivalent set of on-shell functions for the Coulomb branch.

Celestial recursion

We examine the BCFW recursion relations for celestial amplitudes and how they inform the celestial bootstrap program. We start by recasting the celestial incarnation of the BCFW shift as a generalization of the action of familiar asymptotic symmetries on hard particles, before focusing on two limits: z → ∞ and z → 0. We then discuss how the celestial CFT data encodes the large-z behavior determining which shifts are allowed, while the infinitesimal limit is tied to the celestial bootstrap program via the BG equations that constrain the MHV sector. The extension to super-BCFW is also presented. We close by remarking on several open questions for future study.

Non-planar BCFW Grassmannian geometries

In this paper, we study non-adjacent BCFW recursion relations and their connection to positive geometry. For an adjacent BCFW shift, the n-point NkMHV tree-level amplitude in mathcal{N} = 4 SYM theory is expressed as a sum over planar on-shell diagrams, corresponding to canonical “dlog” forms on the cells in the positive Grassmannian G+(k, n). Non-adjacent BCFW shifts naturally lead to an expansion of the amplitude in terms of a different set of objects, which do not manifest the cyclic ordering and the hidden Yangian symmetry of the amplitude. We show that these terms can be interpreted as dlog forms on the non-planar Grassmannian geometries, generalizing the cells of the positive Grassmannian G+(k, n) to a larger class of objects which live in G(k, n). We focus mainly on the case of NMHV amplitudes and discuss in detail the Grassmannian geometries. We also propose an alternative way to calculate the associated on-shell functions and dlog forms using an intriguing connection between Grassmannian configurations and the geometry in the kinematical space.

Wilson lines and boundary operators of BCFW shifts

Boundary operators are gauge invariant operators whose form factors correspond to boundary contributions of BCFW shifts. In gauge theory, the boundary operators contain infinite series, which are constrained by gauge symmetry. We compute the boundary operators of all possible BCFW shifts in Yang-Mills theory and QCD, and show that the infinite series can be elegantly organized into Wilson lines, which are natural building blocks for non-local gauge invariant operators. We comment on their connection to jet functions and gauge invariant off-shell amplitudes. We also verify our results by studying various BCFW shifts of four and five-point amplitudes.

Scattering Amplitudes and BCFW in $\mathcal{N}=2^*$ Theory

We use massive spinor helicity formalism to study scattering amplitudes in \mathcal{N}=2^*𝒩=2* super-Yang-Mills theory in four dimensions. We compute the amplitudes at an arbitrary point in the Coulomb branch of this theory. We compute amplitudes using projection from \mathcal{N}=4𝒩=4 theory and write three point amplitudes in a convenient form using special kinematics. We then compute four point amplitudes by carrying out massive BCFW shift of the amplitudes. We find some of the shifted amplitudes have a pole at z=\inftyz=∞. Taking the residue at z=\inftyz=∞ into account ensures little group covariance of the final result.

Massive on-shell recursion relations for n-point amplitudes

We construct two and three-line shifts for tree-level amplitude with massless and/or massive particles, and provide a method to construct general multi-line shifts for all masses. We choose the massless-massive BCFW shift from these shifts and examine its validity in renormalizable theories. Using such a shift, we find that amplitudes with at least one massless vector boson are constructible. This reveals the importance of gauge theory in the construction of amplitudes with massive particles. We also find that this kind of amplitudes have a cancellation related to group structure among different channels, which is essential for constructibility. Furthermore, we show that in the limit of large shift parameter z, the amplitude with four massive vector bosons, which can include transverse massive vector particles, have structures proportional to the amplitude with shifted vector particles replaced by Goldstone bosons in the leading order. This is responsible for the failure of massive-massive BCFW recursion relations in the amplitudes with four massive vector bosons.

Recursion relations for scattering amplitudes with massive particles

We use the recently developed massive spinor-helicity formalism [1] of Arkani-Hamed et al. to study a new class of recursion relations for tree-level amplitudes in gauge theories. These relations are based on a combined complex deformation of massless as well as massive external momenta. We use these relations to study tree-level amplitudes in scalar QCD as well as amplitudes involving massive vector bosons in the Higgsed phase of Yang-Mills theory. We prove the validity of our proposal by showing that in the limit of infinite momenta of two of the external particles, the amplitude once again is controlled by an enhanced Spin-Lorentz symmetry paralleling the proof of BCFW shift for massless gauge theories. Simple examples illustrate that the proposed shift may lead to an efficient computation of tree-level amplitudes.

UV consistency conditions for Cachazo-He-Yuan integrands

We extend on-shell bootstrap methods from spacetime amplitudes to the worldsheet objects of the CHY formalism. We find that the integrands corresponding to tree-level non-linear sigma model, Yang-Mills and $(DF)^2$ theory are determined by demanding enhanced UV scaling under BCFW shifts. Back in spacetime, we also find that $(DF)^2$ theory is fixed by gauge invariance/UV scaling and simple locality assumptions.

UV considerations on scattering amplitudes in a web of theories

The scattering predictions of a web of theories including Yang-Mills (YM), gravity, bi-adjoint scalar, the non-linear sigma model (NLSM), Dirac-Born-Infeld-Volkov-Akulov (DBI-VA) and the special Galileon (sGal) form a class of special objects with two fascinating properties: they are related by the double-copy procedure, and they can be defined purely by on-shell constraints. We expand on both of these properties. First we show that NLSM tree-level amplitudes are fully determined by imposing color-dual structure together with cyclic invariance and locality. We then consider how hard-scaling can be used to constrain the predictions of these theories, as opposed to the usual soft-scaling. We probe the UV by generalizing the familiar BCFW shift off-shell to a novel single hard limit. We show that UV scalings are sufficient to fully constrain: 1. Bi-adjoint doubly-ordered amplitudes, assuming locality; 2. NLSM and BI, assuming locality and unitarity; 3. Special Galileon, assuming locality, unitarity, and a UV bound for the general Galileon vertex. We see how potentially distinct aspects of this UV behavior can be understood and unified via double-copy relations. Surprisingly, we find evidence that assuming unitarity for these theories may not be necessary, and can emerge via UV considerations and locality alone. These results complete the observations that, like IR considerations, UV scaling is sufficient to fully constrain a wide range of tree-level amplitudes, for both gauge, gravity, and effective field theories.

Uniqueness from locality and BCFW shifts

We introduce a BCFW shift that can be used to recursively build the full YangMills tree-level amplitude as a function of polarization vectors. Furthermore, in line with the recent results of [1], we conjecture that the Yang-Mills tree-level scattering amplitude is uniquely fixed by locality and demanding the usual asymptotic behavior under a sufficient number of shifts. Unitarity therefore emerges from locality and constructability. We prove this statement at the leading order in the soft expansion.

Recursion relations for graviton scattering amplitudes from Bose symmetry and bonus scaling laws

Modern on-shell S-matrix methods may dramatically improve our understanding of perturbative quantum gravity, but current foundations of on-shell techniques for General Relativity still rely on off-shell Feynman diagram analysis. Here, we complete the fully on-shell proof of Ref.~\cite{ST} that the recursion relations of Britto, Cachazo, Feng, and Witten (BCFW) apply to General Relativity tree amplitudes. We do so by showing that the surprising requirement of "bonus" $z^{-2}$ scaling under a BCFW shift directly follows from Bose-symmetry. Moreover, we show that amplitudes in generic theories subjected to BCFW deformations of identical particles necessarily scale as $z^{\rm even}$. When applied to the color ordered expansions of Yang-Mills, this directly implies the improved behavior under non-adjacent gluon shifts. Using the same analysis, three-dimensional gravity amplitudes scale as $z^{-4}$, compared to the $z^{-1}$ behavior for conformal Chern-Simons matter theory.

On form factors in N = 4 $$ \mathcal{N}=4 $$ SYM theory and polytopes

In this paper we discuss different recursion relations (BCFW and all-line shift) for the form factors of the operators from the $\mathcal{N}=4$ SYM stress-tensor current supermultiplet $T^{AB}$ in momentum twistor space. We show that cancelations of spurious poles and equivalence between different types of recursion relations can be naturally understood using geometrical interpretation of the form factors as special limit of the volumes of polytopes in $\mathbb{C}\mathbb{P}^4$ in close analogy with the amplitude case. We also show how different relations for the IR pole coefficients can be easily derived using momentum twistor representation. This opens an intriguing question - which of powerful on-shell methods and ideas can survive off-shell ?

Yang-Mills amplitude relations at loop level from non-adjacent BCFW shifts

This article studies methods to obtain relations for scattering amplitudes at the loop level, with concrete examples at one loop. These methods originate in the analysis of large so-called Britto-Cachazo-Feng-Witten shifts of tree level amplitudes and loop level integrands. In particular BCFW shifts for particles which are not color adjacent and some particular generalizations of this situation are analyzed in some detail in four and higher dimensions. For generic non-adjacent shifts our results are independent of loop order for integrands and hold for generic minimally coupled gauge theories with possible scalar potential and Yukawa terms. By a standard argument this result indicates a generalization of the Bern-Carrasco-Johansson relations for tree level amplitudes exists to the integrand at all loop levels. A concrete relation is presented at one loop. Furthermore, inspired by results in QED it is shown that the results on generalized BCFW shifts of tree level amplitudes imply relations for the so-called rational, bubble and triangle terms of one loop amplitudes in pure Yang-Mills theory. Bubble and triangle terms for instance are shown to obey a five photon decoupling identity, while a three photon decoupling identity is demonstrated for the rational terms. Along the same lines recently conjectured relations for helicity equal amplitudes at one loop are shown to generalize to helicity independent relations for the massive box coefficient of the rational terms.

Pomerons and BCFW recursion relations for strings on D-branes

We derive pomeron vertex operators for bosonic strings and superstrings in the presence of D-branes. We demonstrate how they can be used in order to compute the Regge behavior of string amplitudes on D-branes and the amplitude of ultrarelativistic D-brane scattering. After a lightning review of the BCFW method, we proceed in a classification of the various BCFW shifts possible in a field/string theory in the presence of defects/D-branes. The BCFW shifts present several novel features, such as the possibility of performing single particle momentum shifts, due to the breaking of momentum conservation in the directions normal to the defect. Using the pomeron vertices we show that superstring amplitudes on the disc involving both open and closed strings should obey BCFW recursion relations. As a particular example, we analyze explicitly the case of 1 → 1 scattering of level one closed string states off a D-brane. Finally, we investigate whether the eikonal Regge regime conjecture holds in the presence of D-branes.