NNLO subtraction for numerical integration of virtual amplitudes - - PowerPoint PPT Presentation

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NNLO subtraction for numerical integration of virtual amplitudes

Mao Zeng, ETH Zürich

arXiv:2008.12293, with Charalampos Anastasiou, Rayan Haindl, George Sterman, Zhou Yang

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Outline

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LHC challenges precision theory

• 20 fold increase in integrated luminosity in next 2 decades

Peak luminosity Integrated luminosity Luminosity [cm-2 s-1] Integrated luminosity [-1] Year

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LHC challenges precision theory

• Many more NNLO and N3LO calculaons needed

– 2, 3-loop amplitudes are a key boleneck

• Aack on all fronts:

– integraon-by-parts reducon – polylogs and iterated integrals – differenal equaons (analyc & numerical) – sector decomposion – Mellin Barnes representaon – direct parametric integraon ...

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Next-generation QCD predictions

– Convenonal methods severely challenged. To explore:

• – momentum-space numerical integraon

First NNLO result: triphoton producon [Chawdhry, Czakon, Mitov, Poncelet, '19]

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Momentum space for virtual part?

• Since long ago - real phase space integraon (Catani-Seymour, FKS,

Antenna, qT, N-jeness, CoLorFul, Stripper, nested so-collinear, geometric ...)

Numerical integraon in 4d Analyc integraon in dim. reg., universal so / collinear factorizaon

• Can we do the same for (virtual) loop integraon? So far at only

1 loop [Nagy, Soper, '06; Soper, '99; Gong, Nagy, Soper, '08; Becker, Reuschle, Weinzierl, '10; Assadsolimani, Becker, Weizierl, '10, Becker, Reuschle, Weinzierl, '12; Becker, Goetz, Reuschle, Schwan, '11; Becker, Weinzierl, '12]

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Relation with loop-tree duality

• We do not aempt to combine virtual and real integrands to cancel IR divergences.
• But LTD offers a promising opon for numerical integraon of our subtracted virtual

integrand, turning 4D integrals to 3D, with contour deformaons. Figures from [Capa, Hirschi, Kermanschah, Pelloni, Ruijl, '19]: 3D singularity surface and contour deformaon vector field, for 1-loop box integral.

LTD: Catani, Gleisberg, Krauss, Rodrigo, Winter, Bierenbaum, Draggios, Hernandez-Pinto, Sborlini, Buchta, Chachamis, Malamos, Driencourt-Mangin, Bobadilla, Baumeister, Mediger, Pecovnik, Weinzierl, Runkel, Szor, Vesga, Aguilera-Verdugo, Plenter, Ramirez-Uribe, Tracz, Capa, Hirschi, Kermanschah, Ruijl...

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First result for 2-loop subtraction

[arXiv:2008.12293, C. Anastasiou, R. Haindl, G. Sterman, Z. Yang, MZ] (1) photonic (2) fermion bubble (3) fermion box, hexagon...

• Each class of diagrams combined into one integrand

Only defined for sum of diagrams. Universal factorizaon.

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Exploiting universal IR properties

Form factor Amputated amplitude as Dirac matrix Dirac projector selecng large components for lightlike fermions

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One-loop: "global" IR subtraction

is IR finite locally, i.e. point by point. UV subtracon straighorward (but more subtle at 2 loops).

• nly diagram with

so divergence

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1-loop ward identities

by repeatedly applying

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1-loop ward identities

by repeatedly applying Two loops: vertex correcons breaks point-by-point

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2-loop case: momentum routing

Different collinear limits demand different momentum roungs for factorizaon.

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Order of subtraction

Subtract smaller regions, then larger regions in nested manner.

[Zimmermann, '69; Collins, '11, Erdogan, Sterman, '15, Ma, '19]

Implemented for individual integrals in

[Anastasiou, Sterman, '18]

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Order of "global" subtraction

• One-loop "global" IR counterterm simultaneously

cancels so and collinear divergences. Then UV subtracons.

• Two-loop "global" IR regions:

– Small region: both loop momenta are so or

• collinear. "Double-IR". Subtracted first.

– Large region: only one loop momentum is so or

• collinear. "Single-IR". Subtracted next.

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Transient singularities

• Well-known proofs of universal IR factorizaon only work for

integrated quanes (amplitudes, cross secons).

• Spurious factorizaon-breaking effects exist before loop

integraon.

• Self-energy: spurious power divergence before integraon.
• Vertex: collinear gluon / photon has "non-collinear"
• polarizaons. Non-factorized logarithmic divergence.

See also work in LTD context: [Baumeister, Mediger, Pecovnik, Weinzierl, '19]

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Achieving fully local factorization

• Vertex and self energy diagram adjacent to external legs (named

Type V and Type S diagrams) presents difficules. Type V diagrams + Type S diagrams + Regular diagrams.

• Modify Feynman diagram expressions to achieve factorizaon in

simultaneously.

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Factorization Requirements (1)

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Factorization Requirements (2)

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Factorization Requirements (3)

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Integrand modification - vertex

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Integrand modification - self energy

Re-labelling and symmetrizaon Repeated propagator causes power divergences locally Repeated propagator removed Detailed form preserves Ward identy when combined with modified vertex when

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Form factor subtraction @ 2 loops

Made possible by modified integrand exhibing fully local factorizaon.

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Form factor subtraction @ 2 loops

Removed all IR singularies in just two steps. Now need to subtract UV singularies without destroying the delicate Ward idenes responsible for IR factorizaon.

First subtract double-IR singularies, Next subtract single-IR singularies,

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Ward identity-preserving UV c.t.

i.e. vertex with scalar-polarized photon = difference between two self energy graphs.

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Ward identity-preserving UV c.t.

UV UV

Self energy c.t. is different from 1-loop work of Nagy, Soper, hep-ph/0308127, to preseve Ward identy at 2 loops.

✓

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Fermion loop contributions (1)

• Fermion loop contribuon to internal photon self energy, handled by sub-

loop tensor reducon.

tensor reducon

integrates to 0

• Scalar bubble mes 1-loop amplitude: just make each finite by subtraon.

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Fermion loop contributions (2)

• Remaining fermion loop contribuons are "loop-induced": tree-like IR

IR structure, but there is transient singularies before loop integraon.

Approximates as Gives c.t. which integrates to 0 by Ward i.d., but removes divergences locally.

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Numerical checks of finiteness

• Numerical checks at a random phase space point with raonal

components of momenta, olarizaons, and spinors.

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Numerical checks of finiteness

• Numerical checks at a random phase space point with raonal

components of momenta, polarizaons, and spinors.

• Tune the exponents to approach IR / UV limits, e.g.

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Amplitude convergence results

Similar convergence seen in fermion loop contribuons. Some IR limits show "super-convergence", to be invesgated further.

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Conclusions & Outlook