Darren Forde (SLAC & UCLA) arXiv: (To appear this evening)

Motivations for precision calculations  NLO and one-loop amplitudes already given. Unitarity bootstrap technique combining, Unitarity cuts in D=4 for cut-constructible pieces. On-shell recurrence relations for rational pieces. Focus on the cut-constructable terms here, A new method for extracting scalar bubble and triangle coefficients. Coefficients from the behaviour of the free integral parameters at infinity.

 Cut-constructible from gluing together trees in D=4,  i.e. unitarity techniques in D=4.   missing rational pieces in QCD. [Bern, Dixon, Dunbar, Kosower]  Rational from one-loop on-shell recurrence relation. [Berger, Bern, Dixon, DF, Kosower]  Alternatively work in D=4-2ε, [Bern, Morgan], [Anastasiou, Britto, Feng, Kunszt, Mastrolia]  Gives both terms but requires trees in D=4-2ε. Unitarity bootstrap technique Focus on these terms

 A one-loop amplitude decomposes into  Quadruple cuts freeze the integral  boxes [Britto, Cachazo, Feng] l l3l3 l2l2 l1l1 Rational terms

 What about bubble and triangle terms?  Triple cut  Scalar triangle coefficients?  Two-particle cut  Scalar bubble coefficients?  Disentangle these coefficients. Additional coefficients Isolates a single triangle

 Approaches,  Unitarity technique, [Bern, Dixon, Dunbar, Kosower]  MHV one-loop cut-constructible by joining MHV vertices at two points, [Bedford, Brandhuber, Spence, Traviglini], [Quigley, Rozali]  Integration of spinors, [Britto,Cachazo,Feng] + [Mastrolia] + [Anastasiou, Kunst],  Solving for coefficients, [Ossola, Papadopoulos, Pittau]  Recursion relations, [Bern, Bjerrum-Bohr, Dunbar, Ita]  Large numbers of processes required for the LHC,  Automatable and efficient techniques desirable.  Can we do better?

 Coefficients, c ij, of the triangle integral, C 0 (K i,K j ), given by Single free integral parameter in l Triple cut of the triangle C 0 (K i,K j ) K3K3 K2K2 K1K1 A3A3 A2A2 A1A1 Masslessly Projected momentum Series expansion in t at infinity

 3-mass triangle of A 6 (-+-+-+)  the triple cut integrand  The complete coefficient. Propagator ↔ pole in t, Extra propagator  Box terms No propagator  Triangle 6 λ ‘s top and bottom 2 solutions to γ  divide by 2 The scalar triangle coefficient [Nagy, Soper], [Binoth, Heinrich, Gehrmann, Mastrolia], [Ossola, Papadopoulos, Pittau]

 In general higher powers of t appear in [Inf A 1 A 2 A 3 ] (t).  Integrals over t vanish for chosen parameterisation, e.g. (Similar argument to [Ossola, Papadopoulos, Pittau])  In general whole coefficient given by

 3-mass triangle coefficient of in the 14:23:56 channel. [Bern, Dixon, Kosower] 2 λ ‘s top and bottom Independent of t Series expand in t around infinity

 The bubble coefficient b j of the scalar bubble integral B 0 (K j ) Two-particle cut of the bubble B 0 (K i ) K1K1 A1A1 A2A2 Two free integral parameter in l max y≤4

 Similar to triangle coefficients, but depends upon t.  Two free parameters implies Two-particle cut contrib Box and triangle coeff’s y fixed at pole One extra Pole in y, looks like a triangle Contains bubbles

 Example: Extract bubble of three-mass linear triangle,  Cut l 2 and (l-K 1 ) 2 propagators, gives integrand  Complete coefficient. Series expand y and then t around ∞, set Single pole No “triangle” terms as

 Multiple poles  Can’t choose χ so that all integrals in t vanish.  Sum over all triangles containing the bubble,  Renormalisable theories, max of t 3.  Integrals over t known, C ij a constant, e.g. C 11 =1/2  Gives equivalent, χ independent result

 Comparisons against the literature  Two minus all gluon bubble coefficients for up to 7 legs. [Bern, Dixon, Dunbar, Kosower], [Bedford, Brandhuber, Spence, Travigini]  N=1 SUSY gluonic three-mass triangles for A 6 (+-+-+-), A 6 (+-++--). [Britto, Cachazo, Feng]  Various bubble and triangle coefficients for processes of the type. [Bern, Dixon, Kosower]  Bubble and three-mass triangle coefficients for six photon A 6 (+-+-+-) amplitude. [Nagy, Soper], [Binoth, Heinrich, Gehrmann, Mastrolia], [Ossola, Papadopoulos, Pittau]

“Compact” coefficients from 2 steps, Momentum parameterisation. Series expansion in free parameters at infinity. Efficient and easy to implement. Scalar bubble and triangle coefficients. Use with unitarity bootstrap for complete amplitude.

 Recursion using on-shell amplitudes with fewer legs, [Britto, Cachazo, Feng] + [Witten]  Final result independent of the of choice shift.  Complete amplitude at tree level.  At one loop need the cut pieces [Berger, Bern, Dixon, DF, Kosower]  Combining both involves overlap terms. Two reference legs “shifted”, Intermediate momentum leg is on-shell.