On cancellations of ultraviolet divergences in supergravity amplitudes

461

1,015

We propose a dual formulation for the S Matrix of N = 4 SYM. The dual provides a basis for the "leading singularities" of scattering amplitudes to all orders in perturbation theory, which are sharply defined, IR safe data that uniquely determine the full amplitudes at tree level and 1-loop, and are conjectured to do so at all loop orders. The scattering amplitude for n particles in the sector with k negative helicity gluons is associated with a simple integral over the space of k planes in n dimensions, with the action of parity and cyclic symmetries manifest. The residues of the integrand compute a basis for the leading singularities. A given leading singularity is associated with a particular choice of integration contour, which we explicitly identify at tree level and 1-loop for all NMHV amplitudes as well as the 8 particle N 2 MHV amplitude. We also identify a number of 2-loop leading singularities for up to 8 particles. There are a large number of relations among residues which follow from the multi-variable generalization of Cauchy's theorem known as the "global residue theorem". These relations imply highly non-trivial identities guaranteeing the equivalence of many different representations of the same amplitude. They also enforce the cancellation of non-local poles as well as consistent infrared structure at loop level. Our conjecture connects the physics of scattering amplitudes to a particular subvariety in a Grassmannian; space-time locality is reflected in the topological properties of this space.

Section: N = 8 Sugramentioning

confidence: 99%

A duality for the S matrix

Arkani-Hamed

Cachazo

Cheung

et al. 2010

461

1,015

“…The past two decades have seen a growing realization that scattering amplitudes in gauge theory and gravity exhibit amazing properties that are invisible in the usual local formulation of field theory [1][2][3][4], ranging from the stunning simplicity of MHV amplitudes [5,6] to the recent discovery of dual-superconformal invariance [7][8][9][10][11][12][13][14][15][16] (Dual conformal invariance also gives an ansatz for the basis of integrals at higher loops: [17][18][19]) to the surprisingly good UV behavior of (super)gravity amplitudes [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. These discoveries strongly suggest the existence of a holographically dual formulation of (Super) Yang-Mills and (Super)…”

Section: Towards a Dual Theory Of The S-matrixmentioning

confidence: 99%

The S-matrix in twistor space

Arkani-Hamed

Cachazo

Cheung

et al. 2010

137

204

Abstract: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 YangMills 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 Open Access doi:10.1007/JHEP03 (2010)110 JHEP03 (2010)110 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 N = 4 SYM and 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 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 natu...

“…Furthermore, N = 8 supergravity exhibits the same UV behavior, when continued to higher spacetime dimensions, as does N = 4 super-Yang-Mills (sYM) [3][4][5]. Surprising cancellations are also visible at lower loop orders [6][7][8][9][10][11][12][13][14][15][16][17], and even at tree level where the amplitudes are nicely behaved at large (complex) momenta [16,[18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

$ \mathcal{N} \geqslant {4} $ supergravity amplitudes from gauge theory at two loops

Boucher-Veronneau

Dixon

2011

101

148

We present the full two-loop four-graviton amplitudes in N = 4, 5, 6 supergravity. These results were obtained using the double-copy structure of gravity, which follows from the recently conjectured color-kinematics duality in gauge theory. The twoloop four-gluon scattering amplitudes in N = 0, 1, 2 supersymmetric gauge theory are a second essential ingredient. The gravity amplitudes have the expected infrared behavior: the two-loop divergences are given in terms of the squares of the corresponding one-loop amplitudes. The finite remainders are presented in a compact form. The finite remainder for N = 8 supergravity is also presented, in a form that utilizes a pure function with a very simple symbol.