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Integrand reduction for five-parton two-loop scattering amplitudes - - PowerPoint PPT Presentation

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Integrand reduction for five-parton two-loop scattering amplitudes in QCD High Precision for Hard Processes Christian Brnnum-Hansen in collaboration with Simon Badger, Bayu Hartanto, and Tiziano Peraro 1/15 Christian Brnnum-Hansen High

slide-1
SLIDE 1

1/15

Integrand reduction for five-parton two-loop scattering amplitudes in QCD

High Precision for Hard Processes

Christian Brønnum-Hansen

in collaboration with

Simon Badger, Bayu Hartanto, and Tiziano Peraro

Christian Brønnum-Hansen High Precision for Hard Processes

slide-2
SLIDE 2

2/15

some desired processes at NNLOQCD pp → H + 2 jets pp → γγ + jet pp → 3 jets

Les Houches 2017: Physics at TeV Colliders Standard Model Working Group Report

Christian Brønnum-Hansen High Precision for Hard Processes

slide-3
SLIDE 3

2/15

some desired processes at NNLOQCD pp → H + 2 jets pp → γγ + jet pp → 3 jets

Les Houches 2017: Physics at TeV Colliders Standard Model Working Group Report

Christian Brønnum-Hansen High Precision for Hard Processes

slide-4
SLIDE 4

2/15

some desired processes at NNLOQCD pp → H + 2 jets pp → γγ + jet pp → 3 jets

Les Houches 2017: Physics at TeV Colliders Standard Model Working Group Report

measurement of strong coupling from jet ratio pp→3j

pp→2j

αS(mZ) = 0.1148 ± 0.0014(exp.) ± 0.0018(PDF) ± 0.0050(theory)

CMS Collaboration @ 7 TeV: arXiv:1304.7498

Christian Brønnum-Hansen High Precision for Hard Processes

slide-5
SLIDE 5

3/15

all-plus sector including non-planar contribution long known

  • A two-loop five-gluon helicity amplitude in QCD

Badger, Frellesvig, and Zhang 2013

  • A complete two-loop, five-gluon helicity amplitude in Yang-Mills theory

Badger, Mogull, Ochirov, and O’Connell 2015

  • Analytic form of the two-loop planar five-gluon all-plus helicity amplitude in QCD

Gehrmann, Henn, and Lo Presti 2015

  • Local integrands for two-loop all-plus Yang-Mills amplitudes

Badger, Mogull, and Peraro 2016

Christian Brønnum-Hansen High Precision for Hard Processes

slide-6
SLIDE 6

3/15

all-plus sector including non-planar contribution long known

  • A two-loop five-gluon helicity amplitude in QCD

Badger, Frellesvig, and Zhang 2013

  • A complete two-loop, five-gluon helicity amplitude in Yang-Mills theory

Badger, Mogull, Ochirov, and O’Connell 2015

  • Analytic form of the two-loop planar five-gluon all-plus helicity amplitude in QCD

Gehrmann, Henn, and Lo Presti 2015

  • Local integrands for two-loop all-plus Yang-Mills amplitudes

Badger, Mogull, and Peraro 2016

several recent results

  • A first look at two-loop five-gluon amplitudes in QCD

Badger, CBH, Hartanto, and Peraro 2017

  • Planar two-loop five-gluon amplitudes from numerical unitarity

Abreu, Febres-Cordero, Ita, Page, and Zeng 2017

  • Planar two-loop five-parton amplitudes from numerical unitarity

Abreu, Febres-Cordero, Ita, Page, and Sotnikov 2018

  • Two-loop five-point massless QCD amplitudes within the IBP approach

Chawdhry, Lim, and Mitov 2018

Christian Brønnum-Hansen High Precision for Hard Processes

slide-7
SLIDE 7

4/15

leading colour gluon contribution A(2)(1, 2, 3, 4, 5) =

  • σ∈S5/Z5

tr (T aσ(1) · · · T aσ(5)) × A(2) (σ(1), σ(2), σ(3), σ(4), σ(5)) (1)

Christian Brønnum-Hansen High Precision for Hard Processes

slide-8
SLIDE 8

4/15

leading colour gluon contribution A(2)(1, 2, 3, 4, 5) =

  • σ∈S5/Z5

tr (T aσ(1) · · · T aσ(5)) × A(2) (σ(1), σ(2), σ(3), σ(4), σ(5)) (1) colour-ordered amplitude A(2) (1, 2, 3, 4, 5) =

T

∆T({k}, {p})

  • α∈T Dα

(2)

Christian Brønnum-Hansen High Precision for Hard Processes

slide-9
SLIDE 9

4/15

leading colour gluon contribution A(2)(1, 2, 3, 4, 5) =

  • σ∈S5/Z5

tr (T aσ(1) · · · T aσ(5)) × A(2) (σ(1), σ(2), σ(3), σ(4), σ(5)) (1) colour-ordered amplitude A(2) (1, 2, 3, 4, 5) =

T

∆T({k}, {p})

  • α∈T Dα

(2) sum over planar topologies

Christian Brønnum-Hansen High Precision for Hard Processes

slide-10
SLIDE 10

4/15

leading colour gluon contribution A(2)(1, 2, 3, 4, 5) =

  • σ∈S5/Z5

tr (T aσ(1) · · · T aσ(5)) × A(2) (σ(1), σ(2), σ(3), σ(4), σ(5)) (1) colour-ordered amplitude A(2) (1, 2, 3, 4, 5) =

T

∆T({k}, {p})

  • α∈T Dα

(2) sum over planar topologies irreducible numerator

Christian Brønnum-Hansen High Precision for Hard Processes

slide-11
SLIDE 11

4/15

leading colour gluon contribution A(2)(1, 2, 3, 4, 5) =

  • σ∈S5/Z5

tr (T aσ(1) · · · T aσ(5)) × A(2) (σ(1), σ(2), σ(3), σ(4), σ(5)) (1) colour-ordered amplitude A(2) (1, 2, 3, 4, 5) =

T

∆T({k}, {p})

  • α∈T Dα

(2) sum over planar topologies irreducible numerator integrand reduction

Ossola, Papadopoulos, Pittau, Mastrolia, Badger, Frellesvig, Zhang, Peraro, Mirabella, . . . (2005-)

Christian Brønnum-Hansen High Precision for Hard Processes

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SLIDE 12

5/15

57 topologies, 425 irreducible numerators. examples: pentabox, maximal topology bubble insertion, maximal topology divergent cut

Abreu, Febres Cordero, Ita, Jaquier, and Page 2017

(one-loop)2

Christian Brønnum-Hansen High Precision for Hard Processes

slide-13
SLIDE 13

6/15

reconstruct integrand from d-dimensional unitarity cuts ∆

  • = Cut
  • (3)

Christian Brønnum-Hansen High Precision for Hard Processes

slide-14
SLIDE 14

6/15

reconstruct integrand from d-dimensional unitarity cuts ∆

  • = Cut
  • (3)

generalised unitarity cuts

Bern, Rozowsky, Yan, Dixon, Kosower, de Freitas, Wong, . . . 1997-

Christian Brønnum-Hansen High Precision for Hard Processes

slide-15
SLIDE 15

6/15

reconstruct integrand from d-dimensional unitarity cuts ∆

  • = Cut
  • (3)

generalised unitarity cuts

Bern, Rozowsky, Yan, Dixon, Kosower, de Freitas, Wong, . . . 1997-

finite field reconstruction

Peraro 2016

Christian Brønnum-Hansen High Precision for Hard Processes

slide-16
SLIDE 16

6/15

reconstruct integrand from d-dimensional unitarity cuts ∆

  • = Cut
  • (3)

generalised unitarity cuts

Bern, Rozowsky, Yan, Dixon, Kosower, de Freitas, Wong, . . . 1997-

finite field reconstruction

Peraro 2016

Berends-Giele recursion & six-dimensional spinor-helicity

Cheung, O’Connell 2009

Christian Brønnum-Hansen High Precision for Hard Processes

slide-17
SLIDE 17

6/15

reconstruct integrand from d-dimensional unitarity cuts ∆

  • = Cut
  • (3)

generalised unitarity cuts

Bern, Rozowsky, Yan, Dixon, Kosower, de Freitas, Wong, . . . 1997-

finite field reconstruction

Peraro 2016

Berends-Giele recursion & six-dimensional spinor-helicity

Cheung, O’Connell 2009

momentum twistors

Hodges 2009

Christian Brønnum-Hansen High Precision for Hard Processes

slide-18
SLIDE 18

7/15

split loop momentum in parallel and perpendicular components ki = k,i + k⊥,i (4)

Christian Brønnum-Hansen High Precision for Hard Processes

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SLIDE 19

7/15

split loop momentum in parallel and perpendicular components ki = k,i + k⊥,i (4) k,i = aijpj, aij = aij(ki · pj)

Christian Brønnum-Hansen High Precision for Hard Processes

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SLIDE 20

7/15

split loop momentum in parallel and perpendicular components ki = k,i + k⊥,i (4) k,i = aijpj, aij = aij(ki · pj) k[4]

⊥,i + k[−2ǫ] ⊥,i

Christian Brønnum-Hansen High Precision for Hard Processes

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SLIDE 21

7/15

split loop momentum in parallel and perpendicular components ki = k,i + k⊥,i (4) k,i = aijpj, aij = aij(ki · pj) k[4]

⊥,i + k[−2ǫ] ⊥,i

k[4]

⊥,i = bijωj,

bij = bij(ki · ωj)

Christian Brønnum-Hansen High Precision for Hard Processes

slide-22
SLIDE 22

7/15

split loop momentum in parallel and perpendicular components ki = k,i + k⊥,i (4) k,i = aijpj, aij = aij(ki · pj) k[4]

⊥,i + k[−2ǫ] ⊥,i

k[4]

⊥,i = bijωj,

bij = bij(ki · ωj) relation to extra-dimensional ISPs µij = −k[−2ǫ]

⊥,i

· k[−2ǫ]

⊥,j

= ki · kj − k,i · k,j − k[4]

⊥,i · k[4] ⊥,j

(5)

Christian Brønnum-Hansen High Precision for Hard Processes

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SLIDE 23

8/15

going back to the all-plus case ∆

  • ∝ F(ds, µij)
  • tr+(1235)(k1 + p5)2 + s12s34s45
  • (6)

Christian Brønnum-Hansen High Precision for Hard Processes

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SLIDE 24

8/15

going back to the all-plus case ∆

  • ∝ F(ds, µij)
  • tr+(1235)(k1 + p5)2 + s12s34s45
  • (6)

F(ds, µij) = (ds − 2)

  • µ11µ22 + (µ11 + µ22)2 + 2µ12(µ11 + µ22)
  • + 16
  • µ2

12 − µ11µ22

  • ds is spin dimension, FDH results obtained for ds = 4

Christian Brønnum-Hansen High Precision for Hard Processes

slide-25
SLIDE 25

8/15

going back to the all-plus case ∆

  • ∝ F(ds, µij)
  • tr+(1235)(k1 + p5)2 + s12s34s45
  • (6)

F(ds, µij) = (ds − 2)

  • µ11µ22 + (µ11 + µ22)2 + 2µ12(µ11 + µ22)
  • + 16
  • µ2

12 − µ11µ22

  • ds is spin dimension, FDH results obtained for ds = 4

same pattern for the other contributing topologies

Christian Brønnum-Hansen High Precision for Hard Processes

slide-26
SLIDE 26

9/15

construction of a simple “no-µ” basis ∆T =

  • i

ci

  • mj∈S

mαij

j

(7)

Christian Brønnum-Hansen High Precision for Hard Processes

slide-27
SLIDE 27

9/15

construction of a simple “no-µ” basis ∆T =

  • i

ci

  • mj∈S

mαij

j

(7) rational coefficient in external kinematics

Christian Brønnum-Hansen High Precision for Hard Processes

slide-28
SLIDE 28

9/15

construction of a simple “no-µ” basis ∆T =

  • i

ci

  • mj∈S

mαij

j

(7) rational coefficient in external kinematics S = {k · p} ∪ {k · ω}

Christian Brønnum-Hansen High Precision for Hard Processes

slide-29
SLIDE 29

9/15

construction of a simple “no-µ” basis ∆T =

  • i

ci

  • mj∈S

mαij

j

(7) rational coefficient in external kinematics S = {k · p} ∪ {k · ω} for example the pentabox has 76 terms

Christian Brønnum-Hansen High Precision for Hard Processes

slide-30
SLIDE 30

9/15

construction of a simple “no-µ” basis ∆T =

  • i

ci

  • mj∈S

mαij

j

(7) rational coefficient in external kinematics S = {k · p} ∪ {k · ω} for example the pentabox has 76 terms construction of “µ” containing basis i) take an over-complete set of monomials in ki · pj, ki · ωj, and µij

Christian Brønnum-Hansen High Precision for Hard Processes

slide-31
SLIDE 31

9/15

construction of a simple “no-µ” basis ∆T =

  • i

ci

  • mj∈S

mαij

j

(7) rational coefficient in external kinematics S = {k · p} ∪ {k · ω} for example the pentabox has 76 terms construction of “µ” containing basis i) take an over-complete set of monomials in ki · pj, ki · ωj, and µij ii) choose a set of criteria to order the set of monomials

Christian Brønnum-Hansen High Precision for Hard Processes

slide-32
SLIDE 32

9/15

construction of a simple “no-µ” basis ∆T =

  • i

ci

  • mj∈S

mαij

j

(7) rational coefficient in external kinematics S = {k · p} ∪ {k · ω} for example the pentabox has 76 terms construction of “µ” containing basis i) take an over-complete set of monomials in ki · pj, ki · ωj, and µij ii) choose a set of criteria to order the set of monomials iii) map each monomial containing µij from the set of step i) onto a linear combination of monomials of the simple basis

Christian Brønnum-Hansen High Precision for Hard Processes

slide-33
SLIDE 33

9/15

construction of a simple “no-µ” basis ∆T =

  • i

ci

  • mj∈S

mαij

j

(7) rational coefficient in external kinematics S = {k · p} ∪ {k · ω} for example the pentabox has 76 terms construction of “µ” containing basis i) take an over-complete set of monomials in ki · pj, ki · ωj, and µij ii) choose a set of criteria to order the set of monomials iii) map each monomial containing µij from the set of step i) onto a linear combination of monomials of the simple basis iv) solve for the independent monomials

Christian Brønnum-Hansen High Precision for Hard Processes

slide-34
SLIDE 34

10/15

new integrand parametrisation procedure

  • no polynomial division

Christian Brønnum-Hansen High Precision for Hard Processes

slide-35
SLIDE 35

10/15

new integrand parametrisation procedure

  • no polynomial division
  • integrand basis in preferred variables

Christian Brønnum-Hansen High Precision for Hard Processes

slide-36
SLIDE 36

10/15

new integrand parametrisation procedure

  • no polynomial division
  • integrand basis in preferred variables
  • contains spurious monomials

Christian Brønnum-Hansen High Precision for Hard Processes

slide-37
SLIDE 37

10/15

new integrand parametrisation procedure

  • no polynomial division
  • integrand basis in preferred variables
  • contains spurious monomials

make additional monomials spurious

Christian Brønnum-Hansen High Precision for Hard Processes

slide-38
SLIDE 38

10/15

new integrand parametrisation procedure

  • no polynomial division
  • integrand basis in preferred variables
  • contains spurious monomials

make additional monomials spurious

k2 k1 + k2

  • k2 → −k1 − k2

Christian Brønnum-Hansen High Precision for Hard Processes

slide-39
SLIDE 39

10/15

new integrand parametrisation procedure

  • no polynomial division
  • integrand basis in preferred variables
  • contains spurious monomials

make additional monomials spurious

k2 k1 + k2

  • k2 → −k1 − k2
  • µ12 → −µ11 − µ12

Christian Brønnum-Hansen High Precision for Hard Processes

slide-40
SLIDE 40

10/15

new integrand parametrisation procedure

  • no polynomial division
  • integrand basis in preferred variables
  • contains spurious monomials

make additional monomials spurious

k2 k1 + k2

  • k2 → −k1 − k2
  • µ12 → −µ11 − µ12
  • µ11 + 2µ12 is spurious

Christian Brønnum-Hansen High Precision for Hard Processes

slide-41
SLIDE 41

11/15

helicity flavour non-zero coefficients non-spurious coefficients contributions @ O(ǫ0)

+++++

(ds − 2)0 50 50 (ds − 2)1 175 165 50 (ds − 2)2 320 90 60

−++++

(ds − 2)0 1153 761 405 (ds − 2)1 8745 4020 3436 (ds − 2)2 1037 100 68

−−+++

(ds − 2)0 2234 1267 976 (ds − 2)1 11844 5342 4659 (ds − 2)2 1641 71 48

−+−++

(ds − 2)0 3137 1732 1335 (ds − 2)1 15282 6654 5734 (ds − 2)2 3639 47 32

Christian Brønnum-Hansen High Precision for Hard Processes

slide-42
SLIDE 42

12/15 s12 = 113 7 , s23 = − 152679950 96934257 , s34 = 1023105842 138882415 , s45 = 10392723 3968069 , s15 = − 8362 32585 Christian Brønnum-Hansen High Precision for Hard Processes

slide-43
SLIDE 43

12/15 s12 = 113 7 , s23 = − 152679950 96934257 , s34 = 1023105842 138882415 , s45 = 10392723 3968069 , s15 = − 8362 32585 ǫ−4 ǫ−3 ǫ−2 ǫ−1 ǫ0

  • A(2),[0]

−−+++ 12.5

  • 9.17716 + 47.12389 i
  • 107.40046 - 25.96698 i

17.24014 - 221.41370 i 388.44694 - 167.45494 i P(2),[0] −−+++ 12.5

  • 9.17716 + 47.12389 i
  • 107.40046 - 25.96698 i

17.24013 - 221.41373 i —

  • A(2),[0]

−+−++ 12.5

  • 9.17716 + 47.12389 i
  • 111.02853 - 12.85282 i
  • 39.80016 - 216.36601 i

342.75366 - 309.25531 i P(2),[0] −+−++ 12.5

  • 9.17716 + 47.12389 i
  • 111.02853 - 12.85282 i
  • 39.80018 - 216.36604 i

— Christian Brønnum-Hansen High Precision for Hard Processes

slide-44
SLIDE 44

12/15 s12 = 113 7 , s23 = − 152679950 96934257 , s34 = 1023105842 138882415 , s45 = 10392723 3968069 , s15 = − 8362 32585 ǫ−4 ǫ−3 ǫ−2 ǫ−1 ǫ0

  • A(2),[0]

−−+++ 12.5

  • 9.17716 + 47.12389 i
  • 107.40046 - 25.96698 i

17.24014 - 221.41370 i 388.44694 - 167.45494 i P(2),[0] −−+++ 12.5

  • 9.17716 + 47.12389 i
  • 107.40046 - 25.96698 i

17.24013 - 221.41373 i —

  • A(2),[0]

−+−++ 12.5

  • 9.17716 + 47.12389 i
  • 111.02853 - 12.85282 i
  • 39.80016 - 216.36601 i

342.75366 - 309.25531 i P(2),[0] −+−++ 12.5

  • 9.17716 + 47.12389 i
  • 111.02853 - 12.85282 i
  • 39.80018 - 216.36604 i

—

  • A(2),[i] =

A(2),[i] ALO

Christian Brønnum-Hansen High Precision for Hard Processes

slide-45
SLIDE 45

12/15 s12 = 113 7 , s23 = − 152679950 96934257 , s34 = 1023105842 138882415 , s45 = 10392723 3968069 , s15 = − 8362 32585 ǫ−4 ǫ−3 ǫ−2 ǫ−1 ǫ0

  • A(2),[0]

−−+++ 12.5

  • 9.17716 + 47.12389 i
  • 107.40046 - 25.96698 i

17.24014 - 221.41370 i 388.44694 - 167.45494 i P(2),[0] −−+++ 12.5

  • 9.17716 + 47.12389 i
  • 107.40046 - 25.96698 i

17.24013 - 221.41373 i —

  • A(2),[0]

−+−++ 12.5

  • 9.17716 + 47.12389 i
  • 111.02853 - 12.85282 i
  • 39.80016 - 216.36601 i

342.75366 - 309.25531 i P(2),[0] −+−++ 12.5

  • 9.17716 + 47.12389 i
  • 111.02853 - 12.85282 i
  • 39.80018 - 216.36604 i

—

  • A(2),[i] =

A(2),[i] ALO

(ds − 2)i

Christian Brønnum-Hansen High Precision for Hard Processes

slide-46
SLIDE 46

12/15 s12 = 113 7 , s23 = − 152679950 96934257 , s34 = 1023105842 138882415 , s45 = 10392723 3968069 , s15 = − 8362 32585 ǫ−4 ǫ−3 ǫ−2 ǫ−1 ǫ0

  • A(2),[0]

−−+++ 12.5

  • 9.17716 + 47.12389 i
  • 107.40046 - 25.96698 i

17.24014 - 221.41370 i 388.44694 - 167.45494 i P(2),[0] −−+++ 12.5

  • 9.17716 + 47.12389 i
  • 107.40046 - 25.96698 i

17.24013 - 221.41373 i —

  • A(2),[0]

−+−++ 12.5

  • 9.17716 + 47.12389 i
  • 111.02853 - 12.85282 i
  • 39.80016 - 216.36601 i

342.75366 - 309.25531 i P(2),[0] −+−++ 12.5

  • 9.17716 + 47.12389 i
  • 111.02853 - 12.85282 i
  • 39.80018 - 216.36604 i

—

  • A(2),[i] =

A(2),[i] ALO

(ds − 2)i universal pole structure P(2) = I(1)A(1) + I(2)A(0)

Catani 1996; Becher, Neubert 2009; Gnendiger, Signer, St¨

  • ckinger 2014

Christian Brønnum-Hansen High Precision for Hard Processes

slide-47
SLIDE 47

12/15 s12 = 113 7 , s23 = − 152679950 96934257 , s34 = 1023105842 138882415 , s45 = 10392723 3968069 , s15 = − 8362 32585 ǫ−4 ǫ−3 ǫ−2 ǫ−1 ǫ0

  • A(2),[0]

−−+++ 12.5

  • 9.17716 + 47.12389 i
  • 107.40046 - 25.96698 i

17.24014 - 221.41370 i 388.44694 - 167.45494 i P(2),[0] −−+++ 12.5

  • 9.17716 + 47.12389 i
  • 107.40046 - 25.96698 i

17.24013 - 221.41373 i —

  • A(2),[0]

−+−++ 12.5

  • 9.17716 + 47.12389 i
  • 111.02853 - 12.85282 i
  • 39.80016 - 216.36601 i

342.75366 - 309.25531 i P(2),[0] −+−++ 12.5

  • 9.17716 + 47.12389 i
  • 111.02853 - 12.85282 i
  • 39.80018 - 216.36604 i

—

  • A(2),[i] =

A(2),[i] ALO

(ds − 2)i universal pole structure P(2) = I(1)A(1) + I(2)A(0)

Catani 1996; Becher, Neubert 2009; Gnendiger, Signer, St¨

  • ckinger 2014

reduced to master integrals

Gehrmann, Henn, Lo Presti, Papadopoulos, Tommasini, Wever

Christian Brønnum-Hansen High Precision for Hard Processes

slide-48
SLIDE 48

12/15 s12 = 113 7 , s23 = − 152679950 96934257 , s34 = 1023105842 138882415 , s45 = 10392723 3968069 , s15 = − 8362 32585 ǫ−4 ǫ−3 ǫ−2 ǫ−1 ǫ0

  • A(2),[0]

−−+++ 12.5

  • 9.17716 + 47.12389 i
  • 107.40046 - 25.96698 i

17.24014 - 221.41370 i 388.44694 - 167.45494 i P(2),[0] −−+++ 12.5

  • 9.17716 + 47.12389 i
  • 107.40046 - 25.96698 i

17.24013 - 221.41373 i —

  • A(2),[0]

−+−++ 12.5

  • 9.17716 + 47.12389 i
  • 111.02853 - 12.85282 i
  • 39.80016 - 216.36601 i

342.75366 - 309.25531 i P(2),[0] −+−++ 12.5

  • 9.17716 + 47.12389 i
  • 111.02853 - 12.85282 i
  • 39.80018 - 216.36604 i

— ǫ−4 ǫ−3 ǫ−2 ǫ−1 ǫ0

  • A(2),[1]

+++++

  • 2.5

0.60532 - 12.48936 i 35.03354 + 9.27449 i P(2),[1] +++++

  • 2.5

0.60532 - 12.48936 i —

  • A(2),[1]

−++++

  • 2.5
  • 7.59409 - 2.99885 i
  • 0.44360 - 20.85875 i

P(2),[1] −++++

  • 2.5
  • 7.59408 - 2.99885 i

—

  • A(2),[1]

−−+++

  • 0.625
  • 0.65676 - 0.42849 i
  • 1.02853 + 0.30760 i
  • 0.55509 - 6.22641 i

P(2),[1] −−+++

  • 0.625
  • 0.65676 - 0.42849 i
  • 1.02853 + 0.30760 i

—

  • A(2),[1]

−+−++

  • 0.625
  • 0.45984 - 0.97559 i

1.44962 + 0.53917 i

  • 0.62978 + 2.07080 i

P(2),[1] −+−++

  • 0.625
  • 0.45984 - 0.97559 i

1.44962 + 0.53917 i —

  • A(2),[2]

+++++

  • A(2),[2]

−++++

  • A(2),[2]

−−+++

  • A(2),[2]

−+−++ ǫ0 0.60217 - 0.01985 i

  • 0.10910 - 0.01807 i
  • 0.06306 - 0.01305 i
  • 0.03481 - 0.00699 i

Christian Brønnum-Hansen High Precision for Hard Processes

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SLIDE 49

13/15

process flavour numerical analytic gg → gg (ds − 2)0 (ds − 2)1 (ds − 2)2 gg → ggg (ds − 2)0 (ds − 2)1 ( ) (ds − 2)2 q¯ q → gg q¯ q → q¯ q (ds − 2)0 (ds − 2)1 (ds − 2)2 q¯ q → ggg q¯ q → q′¯ q′g (ds − 2)0 (ds − 2)1 (ds − 2)2

Christian Brønnum-Hansen High Precision for Hard Processes

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SLIDE 50

13/15

process flavour numerical analytic gg → gg (ds − 2)0 (ds − 2)1 (ds − 2)2 gg → ggg (ds − 2)0 (ds − 2)1 ( ) (ds − 2)2 q¯ q → gg q¯ q → q¯ q (ds − 2)0 (ds − 2)1 (ds − 2)2 q¯ q → ggg q¯ q → q′¯ q′g (ds − 2)0 (ds − 2)1 (ds − 2)2 Badger, CBH, Gehrmann, Hartanto, Henn, Lo Presti, Peraro arXiv:1807.09709

Christian Brønnum-Hansen High Precision for Hard Processes

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SLIDE 51

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summary

  • d-dimensional integrand reduction, generalised unitarity cuts, and

IBP reduction

  • finite field reconstruction
  • pole structure check

Christian Brønnum-Hansen High Precision for Hard Processes

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SLIDE 52

14/15

summary

  • d-dimensional integrand reduction, generalised unitarity cuts, and

IBP reduction

  • finite field reconstruction
  • pole structure check
  • utlook
  • analytic results
  • improved integrand basis
  • reduction to master integrals or directly to pentagon functions

Christian Brønnum-Hansen High Precision for Hard Processes

slide-53
SLIDE 53

14/15

summary

  • d-dimensional integrand reduction, generalised unitarity cuts, and

IBP reduction

  • finite field reconstruction
  • pole structure check
  • utlook
  • analytic results
  • improved integrand basis
  • reduction to master integrals or directly to pentagon functions

Gehrmann, Henn, Lo Presti arXiv:1807.09812

Christian Brønnum-Hansen High Precision for Hard Processes

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SLIDE 54

15/15

thank you

Christian Brønnum-Hansen High Precision for Hard Processes

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SLIDE 55

1/5 Christian Brønnum-Hansen High Precision for Hard Processes

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SLIDE 56

2/5 Christian Brønnum-Hansen High Precision for Hard Processes

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SLIDE 57

3/5 Christian Brønnum-Hansen High Precision for Hard Processes

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SLIDE 58

4/5

NNLOQCD 2 → 3 observable

Christian Brønnum-Hansen High Precision for Hard Processes

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SLIDE 59

5/5

simultaneously determine coefficients for simpler cut solutions ∆

  • +

∆

  • Doff-shell

= Cut

  • (8)

simultaneously determine coefficients to avoid divergent cuts ∆

  • +

∆

  • Doff-shell

= Cut

  • − subtractions

(9)

Christian Brønnum-Hansen High Precision for Hard Processes