Soft and Collinear Behaviour of Graviton Scattering Amplitudes David Dunbar, Swansea University

Soft theorems Part of General exploration of singularities of scattering amplitude as route to computation and comprehension singularity as a leg (n) becomes soft soft factor is universal receives no loop corrections sub-leading terms are finite: for real momenta Weinberg,65

Sub-leading terms are singularities in complex momenta :engineering a cubic singularity s

Soft Theorems Cachazo and Strominger Bern, Davies, Nohle White (subleading)

-Beyond the trees? Soft theorem consequence (Ward identity) of BMS symmetry leading term protected Bondi, van der Burg Metzner Sachs =0+other

Gravity MHV amplitude dependance upon Hodge, 2011 Berends Giele, Kuijf, 87; Mason Skinner, 2009

MHV “Twistor-link”-representation a b Nguyen Nguyen, Spradlin,Volovich, Wen, 2010n,Volovi2010 connected tree diagrams involving positive helicity legs only n=5 n=7 n=6

Alternate Formulation From a Seed a b =

-soft lifting from three and four point tree

Alternate Formulation:2 From Seeds

Soft-Terms from diagrams n-1 -point diagram t-dependance lies purely on green line

-diagram with soft leg attached to outside -summing contributions gives leading soft factor

diagrams with soft leg between two legs are pure quadratic

A B C A B C A B CA B C {} -diagrams with trivalent vertex for soft leg are pure linear divergent -this matches

N=4 One-loop, MHV n-point

-softlifting rational term? a b =

Collinear limit : ansatz satisfies leading soft behaviour but fails collinear limit -need to add extra term -trivial when looked at the right way

N=4 One-loop, MHV n-point a b = Rn is obtained by summing all link diagrams with a single loop

-sub-leading soft gives “anomaly” sub-leading soft can replace role of collinear limit in determining structure

Soft-Theorems for One-loop amplitudes not many amplitudes available! N=8 : all available M( ) N=6,4 MHV pure gravity 4pt+5pt completely..use what we have

Passarino-Veltman reduction Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collapsing a propagator

Bern Dixon Perelstein Rosowsky,98 Finite Loop Amplitudes

Single Minus, double poles

double Poles for real momenta amplitudes have single poles double poles arise when we use complex momenta + + +a b

-double poles not intrinsically a problem but we need a formula for sub-leading singularities

Augmented Recursion need formalism to work a off-shell (partially) but still use helicity information: -light -cone gauge methods -carry out a BCFW shift

relies upon working off-shell, (a little as possible) uses off-shell currents from Yang-Mills assumes KLT, close to off-shell produces very cumbersome but, usable, result please, please trivialise Berends-Giele, Kosower, Mahlon Alston, Dunbar and Perkins

Soft Theorems??? all-plus satisfies theorem single minus satisfies theorem when negative leg single minus fails sub-sub-leading result Bern, Davies, NohleHe, Huang, and Wen

Soft-Limit is a coupled BCFW shift sub-sub-leading directly related to double poles t

Conclusions soft theorems seem good at sub-leading fail at sub-sub-leading sub-leading constraints equivalent to collinear non-supersymmetric a long way from maximally

N=4