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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. Weinberg,65.
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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 Weinberg,65 • soft factor is universal • receives no loop corrections • sub-leading terms are finite: for real momenta
s Sub-leading terms are singularities in complex momenta :engineering a cubic singularity
Soft Theorems Cachazo and Strominger White (subleading) Bern, Davies, Nohle
-Beyond the trees? =0 +other • Soft theorem consequence (Ward identity) of BMS symmetry • leading term protected Bondi, van der Burg Metzner Sachs
dependance upon Hodge, 2011 Berends Giele, Kuijf, 87; Mason Skinner, 2009 Gravity MHV amplitude
n=5 a b MHV “Twistor-link”-representation Nguyen, Spradlin,Volovich, Wen, 2010 connected tree diagrams involving positive helicity legs only n=6 n=7
a b = Alternate Formulation From a Seed
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
{ } B B B C A A A A C C C B -diagrams with trivalent vertex for soft leg are pure linear divergent -this matches
Collinear limit : ansatz satisfies leading soft behaviour but fails collinear limit -need to add extra term -trivial when looked at the right way
a b = N=4 One-loop, MHV n-point 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
Finite Loop Amplitudes Bern Dixon Perelstein Rosowsky,98
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 http://pyweb.swan.ac.uk/~dunbar/graviton.html
Soft Theorems??? • all-plus satisfies theorem • single minus satisfies theorem when negative leg • single minus fails sub-sub-leading result Bern, Davies, Nohle He, Huang, and Wen
Soft-Limit is a coupled BCFW shift t • sub-sub-leading directly related to double poles
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