We study contact four-point amplitudes in the spinor-helicity formalism in anti-de Sitter space. We find that these amplitudes can be brought to an especially simple form, which we call canonical. Next, we classify consistent contact amplitudes by requiring correct transformation properties with respect to the AdS isometry algebra. Finally, we establish a connection between the canonical form of AdS amplitudes and scalar multi-trace conformal primaries in flat space.
In this communication we briefly report an unexpected theoretical discovery which emerge from the mapping of Elko mass-dimension-one spinors into single helicity spinors. Such procedure unveils a class of spinor which is classified as type-4 spinor field within Lounesto classification. In this paper we explore the underlying physical and mathematical contents of the type-4 spinor.
In this report we advance into a mapping procedure transmuting a single helicity spinor to a dual helicity spinor. Such a mathematical mechanism reveal us a class of spinor which fits into fourth class within Lounesto classification. The focus of the present manuscript is to expose the algebraic construction of flag-dipole spinors and briefly explore its underlying physical and mathematical contents.
The spinor helicity formalism in four dimensions has become a very useful tool both for understanding the structure of amplitudes and also for practical numerical computation of amplitudes. Recently, there has been some discussion of an extension of this formalism to higher dimensions. We describe a particular implementation of the spinor-helicity method in ten dimensions. Using this tool, we study the tree-level S-matrix of ten dimensional super Yang-Mills theory, and prove that the theory enjoys a dual conformal symmetry. Implications for four-dimensional computations are discussed.
In this paper we use the spinor-helicity formalism to calculate 3-point functions involving scalar operators and spin-$s$ conserved currents in general 3d CFTs. In spinor-helicity variables we notice that the parity-even and the parity-odd parts of a correlator are related. Upon converting spinor-helicity answers to momentum space, we show that correlators involving spin-$s$ currents can be expressed in terms of some simple conformally invariant conserved structures. This in particular allows us to understand and separate out contact terms systematically, especially for the parity-odd case. We…
In this letter we suggest a natural spinor-helicity formalism for massless fields in AdS${}_4$. It is based on the standard realization of the AdS${}_4$ isometry algebra $so(3,2)$ in terms of differential operators acting on $sl(2,\mathbb{C})$ spinor variables. We start by deriving the AdS counterpart of plane waves in flat space and then use them to evaluate simple scattering amplitudes. Finally, based on symmetry arguments we classify all three-point amplitudes involving massless spinning fields. As in flat space, we find that the spinor-helicity formalism allows to construct additional cons…
Lecture notes on Poincaré-invariant scattering amplitudes and tree-level recursion relations in spinor-helicity formalism. We illustrate the non-perturbative constraints imposed over on-shell amplitudes by the Lorentz Little Group, and review how they completely fix the three-point amplitude involving either massless or massive particles. Then we present an introduction to tree-level BCFW recursion relations, and some applications for massless scattering, where the derived three-point amplitudes are employed.
In a recent letter we suggested a natural generalization of the flat-space spinor-helicity formalism in four dimensions to anti-de Sitter space. In the present paper we give some technical details that were left implicit previously. For lower-spin fields we also derive potentials associated with the previously found plane wave solutions for field strengths. We then employ these potentials to evaluate some three-point amplitudes. This analysis illustrates a typical computation of an amplitude without internal lines in our formalism.
It is shown that the coefficient of the cubic interaction vertex, in higher spin Lagrangians, has a very simple form when written in terms of spinor helicity products. The result for a higher-spin field, of spin $λ$, is equal to the corresponding Yang-Mills coefficient raised to the power $λ$. Among other things, this suggests perturbative ties, similar to the KLT relations, between higher spin theories and pure Yang-Mills. This result is obtained in four-dimensional flat spacetime.
We discuss the helicity spinors interpolating between the instant form dynamics (IFD) and the front form dynamics, or the light-front dynamics (LFD), and present the interpolating helicity amplitudes as well as their squares for the scattering of two fermions, and the annihilation of fermion and anti-fermion. We parametrize the interpolation between the two dynamics, IFD and LFD, by an interpolation angle and derive not only the generalized helicity spinors in the $(0,J)\oplus(J,0)$ chiral representation that links naturally the two typical IFD vs. LFD helicity spinors but also the generalized…
We study the consistency conditions for interactions of massless fields of any spin in four-dimensional flat space using the light-cone approach. We show that they can be equivalently rewritten as the Ward identities for the off-shell light-cone amplitudes built from the light-cone Hamiltonian in the standard way. Then we find a general solution of these Ward identities. The solution admits a compact representation when written in the spinor-helicity form and is given by an arbitrary function of spinor products, satisfying well-known homogeneity constraints. Thus, we show that the light-cone c…
We make an explicit link between the cubic interactions of off-shell fields and the on-shell three-point amplitudes in four dimensions. Both the cubic interactions and the on-shell three-point amplitudes had been independently classified in the literature, but their relation has not been made explicit. The aim of this note is to provide such a relation and discuss similarities and differences of their constructions. For the completeness of our analysis, we also derive the covariant form of all parity-odd massless vertices.
In this work, the library spinney is presented, which provides an implementation of helicity spinors and related algorithms for the symbolical manipulation program Form. The package is well suited for symbolic amplitude calculations both in traditional, Feynman diagram based approaches and unitarity-based techniques.
Methods to apply the spinor helicity technique and string reorganization to multiloop amplitudes using the Feynman- and Schwinger-parameter representations are reviewed and expanded.
The success of the spinor helicity technique for tree processes is reviewed. To apply it to multiloop diagrams one is naturally led to the Schwinger proper-time representation, whose properties are discussed. This representation also serves as a useful link between field theories and string theories.
The genuine Kaluza-Klein-like theories (with no fields in addition to gravity) have difficulties with the existence of massless spinors after the ompactification of some of dimensions of space\cite{witten}. We assume a $M^{(1+3)} \times$ a flat finite disk in $(1+5)$-dimensional space, with the boundary allowing spinors of only one handedness. Massless spinors then chirally couple to the corresponding background gauge gravitational field, which solves equations of motion for a free field, linear in the Riemann curvature.
Manifestly Lorentz covariant Feynman rules are given in terms of a "scalar" field for each helicity, dramatically simplifying the calculation of amplitudes with massless particles. The spinor helicity formalism is properly identified as a null complex spacelike (not lightlike) gauge, where two massless external momenta define the reference frame. Usually this gauge is applied only to external line factors; we extend this method to vertices and propagators by modifying the action itself using lightcone methods.
The technique for representing spinors and the definition of the discrete symmetries is used to illustrate on a toy model properties of massless and massive spinors states, in the first and the second quantized picture. Since in this toy model the number of the starting massless representations is well defined as well as the origin of masses and charges in $d=(3+1)$ space, this contribution might help to clarify the problem about Dirac, Weyl and Majorana kinds of representations in physically more interesting cases.
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.
The genuine Kaluza-Klein-like theories--with no fields in addition to gravity--have difficulties with the existence of massless spinors after the compactification of some space dimensions \cite{witten}. We proposed in previous paper a boundary condition for spinors in d=(1+5) compactified on a flat disk that ensures masslessness of spinors (with all positive half integer charges) in d=(1+3) as well as their chiral coupling to the corresponding background gauge gravitational field. In this paper we study the same toy model, proposing a boundary condition allowing a massless spinor of one handed…
The spinor-helicity formalism is an essential technique of the amplitudes community. We draw on this method to construct a scheme for classifying higher-dimensional spacetimes in the style of the four-dimensional Petrov classification and the Newman-Penrose formalism. We focus on the five-dimensional case for concreteness. Our spinorial scheme naturally reproduces the full structure previously seen in both the CMPP and de Smet classifications, and resolves longstanding questions concerning the relationship between the two classifications.
Five-dimensional gauge and gravity theories are known to exhibit striking properties. D=5 is the lowest dimension where massive tensor states appear naturally, providing a testing ground for perturbative insights into six-dimensional tensor theories. Five-dimensional supergravities are highly constrained and admit elegant geometric and algebraic formulations, with global symmetries manifest at the Lagrangian level. In this paper, we take a step towards the systematic investigation of amplitudes in five dimensions, and present a five-dimensional version of the spinor-helicity formalism, applica…
Using the pure spinor formalism we prove identities which relate the tree-level, one-loop and two-loop kinematic factors for massless four-point amplitudes. From these identities it follows that the complete supersymmetric one- and two-loop amplitudes are immediately known once the tree-level kinematic factor is evaluated. In particular, the two-loop equivalence with the RNS formalism (up to an overall coefficient) is obtained as a corollary.
The spinor-helicity formalism has become an invaluable tool for understanding the S-matrix of massless particles in four dimensions. In this paper we construct a spinor-helicity formalism in six dimensions, and apply it to derive compact expressions for the three, four and five point tree amplitudes of Yang-Mills theory. Using the KLT relations, it is a straightforward process to obtain amplitudes in linearized gravity from these Yang-Mills amplitudes; we demonstrate this by writing down the gravitational three and four point amplitudes. Because there is no conserved helicity in six dimensions…
We consider massless fields of arbitrary spin in de Sitter space. We introduce a spinor-helicity formalism, which encodes the field data on a cosmological horizon. These variables reduce the free S-matrix in an observer's causal patch, i.e. the evolution of free fields from one horizon to another, to a simple Fourier transform. We show how this result arises via twistor theory, by decomposing the horizon<->horizon problem into a pair of (more symmetric) horizon<->twistor problems.
The quantum field theory is expanding in de Sitter space-time. The ambient space formalism let us to develop the quantum field theory in a careful mathematical framework. Using the gauge-covariant derivative in the de Sitter ambient space, the gauge invariant Lagrangian density has been found. In this paper, the equations of the conjugate spinor for massless spin-$\frac{3}{2}$ gravitational field is obtained by Euler-Lagrange equation. Finally, these equations can be written in terms of the operator of the Casimir group de Sitter.
The genuine Kaluza-Klein-like theories (with no fields in addition to gravity with torsion) have difficulties with the existence of massless spinors after the compactification of some of dimensions of space\cite{witten}. We demonstrate in this letter on an example of a flat torus - as a compactified part of an (1+5)-dimensional space - that for constant fields and appropriate boundary conditions there exists in the 1+3 dimensional space a massless solution, which is mass protected and chirally coupled with a Kaluza-Klein charge to the corresponding gauge field.
We calculate analytically tree level amplitudes for Higgs production via Weak Bosons Fusion (WBF) and QCD corrections to the Standard Model prediction (SM) by using the helicity spinor formalism. We provide the explicit expression for the amplitude with two gluons emission in the final state together with massless final two quarks. The massive generalization of the Britto-Cachazo-Feng-Witten (BCFW) recursion scheme is applied successfully obtaining compact results which agree with, and overcome in simplicity, the conventional Feynman diagrams approach.
Spinor description for the curvatures of $D=5$ Yang-Mills, Rarita-Schwinger and gravitational fields is elaborated. Restrictions imposed on the curvature spinors by the dynamical equations and Bianchi identities are analyzed. In the absence of sources symmetric curvature spinors with $2s$ indices obey first-order equations that in the linearized limit reduce to Dirac-type equations for massless free fields. These equations allow for a higher-spin generalization similarly to $4d$ case. Their solution in the form of the integral over Lorentz-harmonic variables parametrizing coset manifold $SO(1,…
We present a systematic procedure to compute complete, analytic form factors of gauge-invariant operators at loop level in pure Yang-Mills. We consider applications to operators of the form $\mathrm{Tr}\, F^n$ where $F$ is the gluon field strength. Our approach is based on an extension to form factors of the dimensional reconstruction technique, in conjunction with the six-dimensional spinor-helicity formalism and generalised unitarity. For form factors this technique requires the introduction of additional scalar operators, for which we provide a systematic prescription. We also discuss a gen…
Same amplitudes evaluated independently using RNS and pure spinor formalism are expected to agree. While for massless states, this fact has been firmly established, for massive states such an explicit check has been lacking so far. We compute all massless-massless-massive 3-point functions in open supertrings in pure spinor formalism for the first massive states and compare them with the corresponding RNS results. We fix the normalization of the vertex operators of the massive states by comparing same set of 3-point functions for a fixed ordering in the two formalisms. Once fixed, the subseque…
We study in detail the irreducible representation of E theory that corresponds to massless particles. This has little algebra Ic(E9) and contains 128 physical states that belong to the spinor representation of SO(16). These are the degrees of freedom of maximal supergravity in eleven dimensions. This smaller number of the degrees of freedom, compared to what might be expected, is due to an infinite number of duality relations which in turn can be traced to the existence of a subaglebra of Ic(E9) which forms an ideal and annihilates the representation. We explain how these features are inherite…
The genuine Kaluza-Klein-like theories--with no fields in addition to gravity--have difficulties with the existence of massless spinors after the compactification of some space dimensions \cite{witten}. We proposed (Phys. Lett. B 633 (2006)771) such a boundary condition for spinors in 1+5 compactified on a flat disk that ensures masslessness of spinors in d=1+3 as well as their chiral coupling to the corresponding background gauge field (which solves equations of motion for a free field linear in the Riemann curvature). In this paper we study the same toy model: M^{(1+3)} x M^{(2)}, looking th…
We studied properties of spinors in a toy model in $d=(5+1)$, when ${\cal M}^{(5+1)}$ breaks to an infinite disc with a zweibein which makes a disc curved on an almost $S^2$ and with a spin connection field which allows on such a sphere only one massless spinor state, as a step towards realistic Kaluza-Klein theories in non compact spaces. Previously we allowed on $S^2$ two kinds of the spin connection fields, those which are gauge fields of spins in and those which are the gauge fields of the family quantum numbers, both as required for this toy model by the spin-charge-family theory. This ti…
The tensionless string theory with perimeter action has pure massless spectrum of higher-spin gauge fields. The multiplicity of these massless states grows linearly. It is therefore much less compared with the standard string theory and is larger compared with the field theory models of the Yang-Mills type. It is important to define nontrivial interaction between infinite amount of massless particles of the perimeter string theory. The appropriate vertex operators were defined recently and I study the lowest order vertex operators and the corresponding scattering amplitudes in tree approximati…
We consider the covariant quantization of the D=11 massless superparticle (M0-brane) in the spinor moving frame or twistor-like Lorentz harmonics formulation. The action involves the set of 16 constrained 32 component Majorana spinors, the spinor Lorentz harmonics parametrizing (as homogeneous coordinates, modulo gauge symmetries) the celestial sphere S9. There presence allows us to separate covariantly the first and the second class constraints of the model. After taking into account the second class constraints by means of Dirac brackets and after further reducing the first class constraints…
The two-by-two representation of the SL(2,c) group is for spin-1/2 particles. Starting from this two-by-two representation, it is possible to construct the four-by-four matrices for spin-1 particles. For massless particles, it is possible to construct four-potentials from two-component SL(2,c) spinors. Four potentials are subject to gauge transformations and are gauge-dependent. Then this gauge dependence necessarily comes from the two-component spinors which make up the four-potential. Then there must be a gauge-dependent spinor. This gauge-dependent spinor is discussed in detail. It is shown…
We show that 10D spinor helicity formalism can be understood as spinor moving frame approach to supersymmetric particles extended to the description of amplitudes. This allows us to develop the spinor helicity formalism for 11D supergravity and a new constrained superfield formalism for 10D SYM and 11D SUGRA amplitudes. We show how the constrained on-shell superfields, one-particle counterparts of the superamplitudes, can be obtained by quantization of massless superparticle mechanics. We make some stages towards the calculation of amplitudes of 10D SYM and 11D SUGRA in this framework. In part…
In this paper we discuss fundamental aspects related to the helicity and dynamics of the spin-$1/2$ fermions encompassed within the very well-known Lounesto's classification. More specifically, we investigate how the bi-spinorial structures behave under discrete symmetries, as well as analyse some consequences on the spinors dynamics. In addition, we find an interesting relation between the spinor helicity and the Lounesto spinor classification.
This thesis discusses how the pure spinor formalism can be used to efficiently compute superstring scattering amplitudes. We emphasize the pure spinor superspace form of the kinematic factors, where the simplifying features of this language have allowed an explicit relation among the massless four-point amplitudes at tree-level, one- and two-loops to be found. We show how these identities elegantly simplify the task of computing the amplitudes for all possible external state combination related by supersymmetry. In particular, the two-loop amplitudes involving fermionic states had never been c…
