1 NNLO subtraction for numerical integration of virtual amplitudes Mao Zeng, ETH Zürich arXiv:2008.12293, with Charalampos Anastasiou, Rayan Haindl, George Sterman, Zhou Yang
2 Outline
3 LHC challenges precision theory • 20 fold increase in integrated luminosity in next 2 decades Peak luminosity Integrated luminosity Integrated luminosity [ � -1 ] Luminosity [cm -2 s -1 ] Year
4 LHC challenges precision theory • Many more NNLO and N 3 LO calcula � ons needed – 2, 3-loop amplitudes are a key bo � leneck • A � ack on all fronts: – integra � on-by-parts reduc � on – polylogs and iterated integrals – di ff eren � al equa � ons (analy � c & numerical) – sector decomposi � on – Mellin Barnes representa � on – direct parametric integra � on ...
5 Next-generation QCD predictions First NNLO result: triphoton produc � on [Chawdhry, Czakon, Mitov, Poncelet, '19] – Conven � onal methods severely challenged. To explore: ● – momentum-space numerical integra � on
6 Momentum space for virtual part? • Since long ago - real phase space integra � on (Catani-Seymour, FKS, Antenna, qT, N-je � ness, CoLorFul, Stripper, nested so � -collinear, geometric ...) Analy � c integra � on in dim. reg., Numerical integra � on in 4d universal so � / collinear factoriza � on • Can we do the same for (virtual) loop integra � on? 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]
7 Relation with loop-tree duality LTD: Catani, Gleisberg, Krauss, Rodrigo, Winter, Bierenbaum, Draggio � s, 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... • We do not a � empt to combine virtual and real integrands to cancel IR divergences. • But LTD o ff ers a promising op � on for numerical integra � on of our subtracted virtual integrand, turning 4D integrals to 3D, with contour deforma � ons. Figures from [Capa � , Hirschi, Kermanschah, Pelloni, Ruijl, '19]: 3D singularity surface and contour deforma � on vector fi eld, for 1-loop box integral.
8 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 de fi ned for sum of diagrams. Universal factoriza � on .
9 Exploiting universal IR properties Form factor Amputated Dirac projector selec � ng amplitude large components for as Dirac matrix lightlike fermions
10 One-loop: "global" IR subtraction only diagram with so � divergence is IR fi nite locally, i.e. point by point. UV subtrac � on straigh � orward (but more subtle at 2 loops).
11 1-loop ward identities by repeatedly applying
12 1-loop ward identities Two loops: vertex correc � ons breaks point-by-point by repeatedly applying
13 2-loop case: momentum routing Di ff erent collinear limits demand di ff erent momentum rou � ngs for factoriza � on.
14 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]
15 Order of "global" subtraction • One-loop "global" IR counterterm simultaneously cancels so � and collinear divergences. Then UV subtrac � ons. • Two-loop "global" IR regions: – Small region: both loop momenta are so � or collinear. "Double-IR". Subtracted fi rst. – Large region: only one loop momentum is so � or collinear. "Single-IR". Subtracted next.
16 Transient singularities • Well-known proofs of universal IR factoriza � on only work for integrated quan �� es (amplitudes, cross sec � ons). • Spurious factoriza � on-breaking e ff ects exist before loop integra � on. • Self-energy: spurious power divergence before integra � on. See also work in LTD context: [Baumeister, Mediger, Pecovnik, Weinzierl, '19] • Vertex: collinear gluon / photon has "non-collinear" polariza � ons. Non-factorized logarithmic divergence.
17 Achieving fully local factorization • Vertex and self energy diagram adjacent to external legs (named Type V and Type S diagrams) presents di ffi cul � es. Type V diagrams + Type S diagrams + Regular diagrams. • Modify Feynman diagram expressions to achieve factoriza � on in simultaneously.
18 Factorization Requirements (1)
19 Factorization Requirements (2)
20 Factorization Requirements (3)
21 Integrand modi fi cation - vertex
22 Integrand modi fi cation - self energy Repeated propagator causes power divergences locally Re-labelling and symmetriza � on Repeated propagator removed Detailed form preserves Ward iden � ty when combined with modi fi ed vertex when
23 Form factor subtraction @ 2 loops Made possible by modi fi ed integrand exhibi � ng fully local factoriza � on.
24 Form factor subtraction @ 2 loops First subtract double-IR singulari � es, Next subtract single-IR singulari � es, Removed all IR singulari � es in just two steps. Now need to subtract UV singulari � es without destroying the delicate Ward iden �� es responsible for IR factoriza � on.
25 Ward identity-preserving UV c.t. i.e. vertex with scalar-polarized photon = di ff erence between two self energy graphs.
26 Ward identity-preserving UV c.t. ✓ UV UV Self energy c.t. is di ff erent from 1-loop work of Nagy, Soper, hep-ph/0308127, to preseve Ward iden � ty at 2 loops.
27 Fermion loop contributions (1) • Fermion loop contribu � on to internal photon self energy, handled by sub- loop tensor reduc � on. integrates to 0 tensor reduc � on • Scalar bubble � mes 1-loop amplitude: just make each fi nite by subtra � on.
28 Fermion loop contributions (2) • Remaining fermion loop contribu � ons are "loop-induced": tree-like IR IR structure, but there is transient singulari � es before loop integra � on. Approximates as Gives c.t. which integrates to 0 by Ward i.d., but removes divergences locally.
29 Numerical checks of fi niteness • Numerical checks at a random phase space point with ra � onal components of momenta, olariza � ons, and spinors.
30 Numerical checks of fi niteness • Numerical checks at a random phase space point with ra � onal components of momenta, polariza � ons, and spinors. • Tune the exponents to approach IR / UV limits, e.g.
31 Amplitude convergence results Similar convergence seen in fermion loop contribu � ons. Some IR limits show "super-convergence", to be inves � gated further.
32 Conclusions & Outlook
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