Darren Forde (SLAC & UCLA) arXiv: [hep-ph], hep-ph/ , hep-ph/ In collaboration with Carola Berger, Zvi Bern, Lance Dixon & David Kosower.

Motivations for precision calculations, NLO and one-loop amplitudes already discussed. Unitarity bootstrap technique combines, Unitarity cuts in D=4 for cut-constructible pieces. On-shell recurrence relations for rational pieces. Focus on the cut-constructible terms here, Directly extract the scalar bubble and triangle coefficients. Coefficients from the behaviour of the free integral parameters at infinity.

 On-shell recursion relation of rational loop pieces [Berger, Bern, Dixon, DF, Kosower] [Britto, Cachazo, Feng] +[Witten]  Only uses on-shell quantities.  Additional rational terms, requires knowledge of the (Poly)Log pieces.  Using cuts in D=4 gives “Cut-construcible” pieces. Focus on these terms Unitarity techniques Combination is the Unitarity bootstrap

 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 vertex techniques, [Bedford, Brandhuber, Spence, Traviglini], [Quigley, Rozali]  Unitarity cuts & integration of spinors, [Britto, Cachazo, Feng] + [Mastrolia] + [Anastasiou, Kunszt],  Recursion relations, [Bern, Bjerrum-Bohr, Dunbar, Ita]  Solving for coefficients, [Ossola, Papadopoulos, Pittau], [Ellis, Giele, Kunszt]  Large numbers of processes required for the LHC,  Automatable and efficient techniques desirable.

 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 around t at infinity, take only non-negative powers  =3 in renormalisable theories

 3-mass triangle of A 6 (-+-+-+)  the triple cut integrand  The complete coefficient. Extra propagator  Box terms 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] Propagator ↔ pole in t  a box. No propagator  Triangle

 In general series expansion of A 1 A 2 A 3 around t = ∞ gives,  Integrals over t vanish for chosen parameterisation, e.g. (Similar argument to [Ossola, Papadopoulos, Pittau])  In general whole coefficient given by From series expanding the box poles

 3-mass triangle coefficient of in the 14:23:56 channel. [Bern, Dixon, Kosower] Independent of t Series expand in t around infinity

 Can we do something similar?  Two delta function constraints  two free parameters y and t,  Depends upon an arbitrary massless four vector χ.  Naive generalisation, two particle cut  bubble coefficient b j of the scalar bubble integral B 0 (K j )?  Does not give the complete result.

 Series expanding around ∞ in y and then t gives  Integrals over t vanish  Integrals over y do not vanish, can show and Additional contributions?

 Integrals over t can be related to bubble contributions.  Schematically after expanding around y=∞,  Want to associate pole terms with triangles (and boxes) but unlike for previous triangle coefficients, Terms with poles in y with y fixed at pole y i Integrals over t do not vanish in this expansion  can contain bubbles ~“Inf” terms

 Extract bubble of three-mass linear triangle,  Cut l 2 and (l-K 1 ) 2 propagators, gives integrand  Depends upon χ and is not the complete coefficient. Series expand y and then t around ∞, set

 Consider all triangles sitting “above” the bubble.  Then extract bubble term from the integrals over t,  i.e. using  Integrals over t known, ( C ij a constant, e.g. C 11 =1/2)  Renormalisable theories, max power t 3.  Combining both pieces gives the coefficient,

 Using  Triangles  Bubbles  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]  Analyses of the behaviour of one-loop gravity amplitudes, including N=8 Supergravity. [Bern, Carrasco, DF, Ita, Johansson]

Direct extraction of coefficients in 2 simple steps, Specific momentum parameterisation. Series expansion in free parameters at infinity. Automatable. Use in unitarity bootstrap for complete amplitude.