Maximal Unitarity at Two Loops

David A. KosowerInstitut 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 progressAmplitudes 2013

Schloss Ringberg on the Tegernsee, GermanyMay 2, 2013

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 squareddown by two powers of αs, but enhanced by gluon distribution5% of total cross section @14 TeV 20–25% scale dependence25% 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

Two-loop amplitudes

On-shell Methods

Integrand LevelMastrolia & 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 treesFeng & Huang; Feng, Huang, Luo, Zheng, & Zhou

“Maximal generalized unitarity”: split as much as possibleGeneralization ofBritto–Cachazo–Feng & FordeThis talk

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

Generalized Discontinuity Operators

• Cut operations (or ‘projectors’) which satisfy

so that applying them to the master equation

yields solutions for the cj

Important constraint

Putting Lines on Shell

• Cutkosky rule

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?

Do Quadruple Cuts Have Solutions?

The delta functions instruct us to solve

1 quadratic, 3 linear equations 2 solutionsWith k1,2,4 massless, we can write down the solutions explicitly

Yes, but…

• 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

• 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

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

• Require vanishing Feynman integrals to continue vanishing on cuts

• General contour

a1 = a2

Four-Dimensional Integral Basis

• Contains integrals with up to 8 propagatorsGluza & 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

• 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 z within Si

Planar Double Box

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

• 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)

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

• Master formulæ for coefficients of basis integrals to O (ε0)

where P1,2 are linear combinations of T8s around global poles

More explicitly,

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) 3 independent projectors

• Projectors again unique (but different from massless or one-mass case)

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) 4 independent & unique projectors

• Equal-mass case: &• Three masters: need to impose additional

symmetry eqn• 3 independent & unique projectors

Simpler Integrals

• 7 propagators can choose these

to be absentfrom integral basis

• 6 propagators

• 5 propagators

• Along with double box , these are all “parents” of the slashed box and share its cuts

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)

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);• ε1 À 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)

• 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 (, , )

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