The S-matrix in twistor space
By N. Arkani-Hamed, F. Cachazo, C. Cheung, Jared Kaplan. 2009
199 citations recorded by the source corpus.
The marvelous simplicity and remarkable hidden symmetries recently uncovered in (Super) Yang-Mills and (Super)Gravity scattering amplitudes strongly suggests the existence of a “weak-weak” dual formulation of these theories where these structures are made more manifest at the expense of manifest locality. In this note we suggest that in four dimensions, this dual description lives in (2, 2) signature and is naturally formulated in twistor space. We begin at tree-level, by recasting the momentum-space BCFW recursion relation in a completely on-shell form that begs to be transformed into twistor space. Our transformation is strongly inspired by Witten’s twistor string theory, but differs in treating twistor and dual twistor variables on a more equal footing; a related transcription of the BCFW formula using only twistor space variables has been carried out independently by Mason and Skinner. Using both twistor and dual twistor variables, the three and four-point amplitudes are strikingly simple–for Yang-Mills theories they are “1” or “-1”. The BCFW computation of higher-order amplitudes can be represented by a simple set of diagrammatic rules, concretely realizing Penrose’s program of relating “twistor diagrams” to scattering amplitudes. More specifically, we give a precise definition of the twistor diagram formalism developed over the past few years by Andrew Hodges. The “Hodges diagram” representation of the BCFW rules allows us to compute amplitudes and study their remarkable properties in twistor space. For instance the diagrams for Yang-Mills theory are topologically disks and not trees, and reveal striking connections between amplitudes that are not manifest in momentum space. Twistor space also suggests a new representation of the amplitudes directly in momentum space, that is naturally determined by the Hodges diagrams. The BCFW rules and Hodges diagrams also enable a systematic twistorial formulation of gravity. All tree amplitudes can be combined into an “S-Matrix” scattering functional which is the natural holographic observable in asymptotically flat space; the BCFW formula turns into a simple quadratic equation for this “S-Matrix” in twistor space, providing a holographic description of $ \mathcal{N} = 4 $ SYM and $ \mathcal{N} = 8 $ Supergravity at tree level. We move on to initiate the exploration of loop amplitudes in (2, 2) signature and twistor space, beginning with a discussion of their IR behavior. We find that the natural pole prescriptions needed for transformation to twistor space make the amplitudes perfectly well-defined objects, free of IR divergences. Indeed in momentum space, the loop amplitudes so regulated vanish for generic momenta, and transformed to twistor space, are even simpler than their tree-level counterparts: the full 4-pt one-loop amplitudes in $ \mathcal{N} = 4 $ SYM are simply equal to “1” or “0”! This further supports the idea that there exists a sharply defined object corresponding to the S-Matrix in (2, 2) signature, computed by a dual theory naturally living in twistor space.
Subfields: Nuclear and High Energy Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/dd0269e1342742a8e56cb71a4998c8036f30464d
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Citations in the corpus
- The Grassmannian origin of dual superconformal invariance (2009, methodology): “It is amusing that different “gauge-fixings” of the GL(k) symmetry are convenient for seeing different symmetries: the gauge-fixing introduced in [ 1 ] is more convenient for manifesting parity, while the one exploited in this paper is better suited to manifesting dual superconformal invariance.”
- Scattering Amplitudes and the Positive Grassmannian (2012, unknown): “If we strip-off the (unshifted) Parke-Tyalor prefactor, the integrand for the one-loop ratio function—the loop amplitude divided by the tree-amplitude—is simply [160], A 4 A 4 = dα1 α1 dα2 α2 dα3 α3 dα4 α4 〈1 2〉〈2 3〉〈3 4〉〈4 1〉 〈1̂ 2̂〉〈2̂ 3̂〉〈3̂ 4̂〉〈4̂ 1̂〉 ,”
- TASI Lectures on Scattering Amplitudes (2017, background): “On-shell tools have excavated a landscape of structures: hidden Yangian and dual conformal symmetries [80, 81, 82, 83, 84], reformulations of the S-matrix in twistor space [85, 86, 22, 87, 88, 89, 90, 91] and Grassmannian space [92, 93, 94], as well as connections to abstract volumes [12, 95] like t”
- Scattering amplitudes and Wilson loops in twistor space (2011, methodology): “This operation has become known as the inverse soft limit [54, 55].”
- Dual superconformal invariance, momentum twistors and Grassmannians (2009, background): “Most of the material here is readily available in standard twistor theory texts (see e.g. [44–47]), but since many of the following formulæ are useful in translating the dual superconformal invariants into momentum twistor space, we include their derivation to make this paper self-contained.”
- Unification of residues and Grassmannian dualities (2009, unknown): “2) into the link representation of [2].”
- New Recursion Relations and a Flat Space Limit for AdS/CFT Correlators (2012, methodology): “eq′,−kq′,...eπn,kπn(p)). (4.12) The notation is the same as that used above. 4.5 Another form of the relations Let us specialize the recursion relations to w = 0. Then we can rewrite (4.12) following [18] X ± Z∞ 0 iT 2 p2 +( Pm l o=1k πo)2 dp2 2 w∓(p) w±(p)−w∓(p) = Z∞ 0 dp2 2 Z H dw w " −iT 2δ ”
- Local integrals for planar scattering amplitudes (2010, background): “This particular space will not play a significant role in this work, so we refer the interested reader to [4,35] for more details.”
- The all-loop integrand for scattering amplitudes in planar $ \mathcal{N} = 4 $ SYM (2010, background): “As we have seen, loop amplitudes can easily hide in plain sight in subtle ways, masquerading as a formal way of representing “1” in terms of IR-divergent integrals in (3, 1)-signature! It is likely that a deeper understanding of the contours associated with the “Hodges diagrams” [70,77], already for”
- Explicit BCJ numerators from pure spinors (2011, unknown): “where C(m) is the Catalan number which counts the number of kinematic channels appearing in a color-ordered (m + 2)-point amplitude [40].”