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Maximal Unitarity at Two Loops

Maximal Unitarity at Two Loops. David A. Kosower Institut de Physique Th é orique , CEA– Saclay work with Kasper Larsen & Henrik Johansson; & work of Simon Caron- Huot & Kasper Larsen 1108.1180, 1205.0801, 1208.1754 & in progress

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Maximal Unitarity at Two Loops

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  1. Maximal Unitarityat Two Loops David A. KosowerInstitut de Physique Théorique, CEA–Saclay work with Kasper Larsen & Henrik Johansson; & work of Simon Caron-Huot & Kasper Larsen1108.1180, 1205.0801, 1208.1754 & in progress Amplitudes 2013SchlossRingberg on the Tegernsee, GermanyMay 2, 2013

  2. Amplitudes in Gauge Theories • Workshop is testimony to recent years’ remarkable progress at the confluence of string theory, perturbativeN=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 αs, 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 cj leading in ε • Work in D=4 for states, integrals remain in D=4−2ε • 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 cj 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 k1,2,4 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 T4 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  a1 = a2

  13. Four-Dimensional Integral Basis • Contains integrals with up to 8 propagators Gluza & DAK; Schabinger • Irreducible numerators: not expressible in terms of propagator denominators & invariants • External momenta in D=4, loop momenta in D=4-2ε • We’ll examine integrals with up to four external legs  up to 7 propagators • Structure of basis depends on external masses, unlike one-loop case

  14. Planar Double Box • Take a heptacut — freeze seven of eight degrees of freedom • One remaining integration variable z • Six solutions, for example S2: • Performing the contour integrals enforcing the heptacut Jacobian • Localizes z  global pole  need contour for zwithin Si

  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 (ε0) where P1,2 are linear combinations of T8s around global poles More explicitly,

  19. More Masses • Legs 1 & 2 or 1, 2, &3 massive • Three master integrals:I4[1], I4[ℓ1∙k4] and I4[ℓ2∙k1] • 16 candidate global poles…again 8 global poles • 5 constraint equations(4 , 1 IBP)  3independent projectors • Projectors again unique (but different from masslessor one-mass case)

  20. Four Masses • Four master integrals for generic masses:I4[1], I4[ℓ1∙k4], I4[ℓ2∙k1], and I4[ℓ1∙k4 ℓ2∙k1] • 12 candidate global poles…again 8 global poles • 4 constraint equations(4 , 0 IBP)  4independent & unique projectors • Equal-mass case: & • Three masters: need to impose additional symmetry eqn •  3independent & unique projectors

  21. Simpler Integrals • 7 propagators can choose theseto beabsentfrom integral basis • 6 propagators • 5 propagators • Along with double box , these are all “parents” of the slashed box and share its cuts

  22. Slashed Box • Two loops with 5 propagators — 3 additional degrees of freedom z1, z2, z3 • 4 distinct solution sets, each a C3~ CP3 • Example (S1) • Jacobian ~ • Solutions intersect pairwise in CP1 CP1 • Solutions all intersect in an S2 (z2=z3=0)

  23. Multivariate Contour Integration • One-dimensional contour integrals are independent of the contour’s shape • Not true in higher dimensions! • Size of torus: C(ε1)  C(ε2); • ε1À ε2: 0 • ε1¿ε2: 1 • Independent of size if we change variables (tilted in original variables) — but can’t do this globally • Practical solution: be careful & choose shape (do iteratively)

  24. Look at slashed box along with its parents • ~1000 candidate global poles • ~100 candidate global poles after removing duplicates • 27 independent global poles (enforce Cauchy conditions) • 16 global poles after imposing parity • 1 pole has a residue for ; the others need to be included in order to project out the other integrals (,,)

  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

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