1. Maximal Unitarity at Two Loops David A. Kosower Institut de Physique Théorique, CEA–Saclay work with Kasper Larsen & Henrik Johansson; & work of Simon Caron- Huot & Kasper Larsen 1108.1180, 1205.0801, 1208.1754 & in progress Amplitudes 2013 Schloss Ringberg on the Tegernsee, Germany May 2, 2013

2. Amplitudes in Gauge Theories Workshop is testimony to recent years’ remarkable progress at the confluence of string theory, perturbative N =4 SUSY gauge theory, and integrability One loop amplitudes have led to a revolution in QCD NLO calculations at the multiplicity frontier: first quantitative predictions for LHC, essential to controlling backgrounds For NNLO & precision physics: need two loops Sometimes, need two-loop amplitudes just for NLO: gg  W + W − LO for subprocess is a one-loop amplitude squared down by two powers of α s, but enhanced by gluon distribution 5% of total cross section @14 TeV 20–25% scale dependence 25% of cross section for Higgs search 25–30% scale dependence Binoth, Ciccolini, Kauer, Krämer (2005) Experiments: measured rate is 10–15% high? Need NLO to resolve

3. Two-loop amplitudes On-shell Methods Integrand Level Mastrolia & Ossola; Badger, Frellesvig, & Zhang; Zhang; Mastrolia, Mirabella, Ossola, Peraro; Kleiss, Malamos, Papadopoulos, Verheyen Generalization of Ossola–Papadopoulos–Pittau at one loop Integral Level “Minimal generalized unitarity”: just split into trees Feng & Huang; Feng, Huang, Luo, Zheng, & Zhou “Maximal generalized unitarity”: split as much as possible Generalization of Britto–Cachazo–Feng & Forde This talk

4. On-Shell Master Equation Focus on planar integrals Terms in c j leading in ε Work in D=4 for states, integrals remain in D=4 −2 ε Seek formalism which can be used either analytically or purely numerically

5. Generalized Discontinuity Operators Cut operations (or ‘projectors’) which satisfy so that applying them to the master equation yields solutions for the c j Important constraint

6. Putting Lines on Shell Cutkosky rule

7. Quadruple Cuts of the One-Loop Box Work in D=4 for the algebra Four degrees of freedom & four delta functions … but are there any solutions?

8. Do Quadruple Cuts Have Solutions? The delta functions instruct us to solve 1 quadratic, 3 linear equations  2 solutions With k 1,2,4 massless, we can write down the solutions explicitly Yes, but…

9. Solutions are complex The delta functions would actually give zero! Need to reinterpret delta functions as contour integrals around a global pole [ other contexts: Vergu; Roiban, Spradlin, Volovich; Mason & Skinner ] Reinterpret cutting as contour modification

10. Global poles: simultaneous on-shell solutions of all propagators & perhaps additional equations Multivariate complex contour integration: in general, contours are tori For one-loop box, contours are T 4 encircling global poles

11. Two Problems Too many contours (2) for one integral: how should we choose it? Changing the contour can break equations: is no longer true if we modify the real contour to circle one of the poles Remarkably, these two problems cancel each other out

12. Require vanishing Feynman integrals to continue vanishing on cuts General contour  a 1 = a 2

13. Four-Dimensional Integral Basis

14. Take a heptacut — freeze seven of eight degrees of freedom One remaining integration variable z Six solutions, for example S 2 : Performing the contour integrals enforcing the heptacut  Jacobian Localizes z  global pole  need contour for z within S i Planar Double Box

15. How Many Global Poles Do We Have? Caron-Huot & Larsen (2012) Parametrization All heptacut solutions have Here, naively two global poles each at z = 0, −χ  12 candidate poles In addition, 6 poles at z =  from irreducible-numerator ∫s 2 additional poles at z = −χ−1 in irreducible-numerator ∫s  20 candidate global poles

16. But:  Solutions intersect at 6 poles  6 other poles are redundant by Cauchy theorem (∑ residues = 0) Overall, we are left with 8 global poles (massive legs: none; 1; 1 & 3; 1 & 4)

17. Picking Contours Two master integrals A priori, we can take any linear combination of the 8 tori surrounding global poles; which one should we pick? Need to enforce vanishing of all parity-odd integrals and total derivatives: – 5 insertions of ε tensors  4 independent constraints – 20 insertions of IBP equations  2 additional independent constraints – In each projector, require that other basis integral vanish

18. Master formulæ for coefficients of basis integrals to O ( ε 0 ) where P 1,2 are linear combinations of T 8 s around global poles More explicitly,

19. More Masses Legs 1 & 2 or 1, 2, &3 massive Three master integrals: I 4 [1], I 4 [ ℓ 1 ∙ k 4 ] and I 4 [ ℓ 2 ∙ k 1 ] 16 candidate global poles …again 8 global poles 5 constraint equations (4 , 1 IBP)  3 independent projectors Projectors again unique (but different from massless or one-mass case)

20. Four Masses

21. Simpler Integrals

22. Slashed Box

23. Multivariate Contour Integration

25. Summary First steps towards a numerical unitarity formalism at two loops Knowledge of an independent integral basis Criterion for constructing explicit formulæ for coefficients of basis integrals Four-point examples: double boxes with all external mass configurations; massless slashed box