Constructing QCD One-loop Amplitudes

1. Darren Forde (SLAC & UCLA) Constructing QCD One-loop Amplitudes arXiv:0704.1835 (To appear this evening)

2. Overview

3. The unitarity bootstrap Focus on these terms • Cut-constructible from gluing together trees in D=4, • i.e. unitarity techniques in D=4. •  missingrational 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

4. One-loop integral basis • A one-loop amplitude decomposes into • Quadruple cuts freeze the integral  boxes [Britto, Cachazo, Feng] Rational terms l l1 l3 l2

5. Two-particle and triple cuts • 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

6. Disentangeling coefficients • 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?

7. Triangle coefficients • Coefficients, cij, of the triangle integral, C0(Ki,Kj), given by Triple cut of the triangle C0(Ki,Kj) Single free integral parameter in l K3 A3 A2 A1 K1 K2 Series expansion in t at infinity Masslessly Projected momentum

8. six photons 6 λ‘s top and bottom [Nagy, Soper], [Binoth, Heinrich, Gehrmann, Mastrolia], [Ossola, Papadopoulos, Pittau] • 3-mass triangle of A6(-+-+-+)  the triple cut integrand • The complete coefficient. Extra propagator  Box terms No propagator  Triangle Propagator ↔ pole in t, 2 solutions to γ divide by 2 The scalar triangle coefficient

9. Vanishing integrals • In general higher powers of t appear in [Inf A1A2A3](t). • Integrals over t vanish for chosen parameterisation, e.g.(Similar argument to [Ossola, Papadopoulos, Pittau]) • In general whole coefficient given by

10. Another Triangle Coefficient • 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

11. What about bubbles? • The bubble coefficient bj of the scalar bubble integral B0(Kj) Two-particle cut of the bubble B0(Ki) Two free integral parameter in l A2 max y≤4 K1 A1

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

13. Triple-cut contributions • Example: Extract bubble of three-mass linear triangle, • Cut l2 and (l-K1)2propagators, gives integrand • Complete coefficient. Single pole Series expand y and then t around ∞, No “triangle” terms as set

14. Triple cut contributions cont. • Multiple poles Can’t choose χso that all integrals in t vanish. • Sum over all triangles containing the bubble, • Renormalisable theories, max of t3. • Integrals over t known, Cij a constant, e.g. C11=1/2 • Gives equivalent, χindependent result

15. other Applications • 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 A6(+-+-+-), A6(+-++--). [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 A6(+-+-+-) amplitude. [Nagy, Soper], [Binoth, Heinrich, Gehrmann, Mastrolia], [Ossola, Papadopoulos, Pittau]

16. Conclusion

17. A<n R<n Rn On-shell recursion relations Two reference legs “shifted”, • 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. Intermediate momentum leg is on-shell.