[PPT] - OpenLoops 2 M. F. Zoller in collaboration with F. Buccioni, J.-N. PowerPoint Presentation

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OpenLoops 2

M. F. Zoller

in collaboration with

F. Buccioni, J.-N. Lang, J. Lindert, P. Maierhöfer, S. Pozzorini and H. Zhang

[arxiv:1907.13071]

LoopFest XVIII – Fermilab – 08/14/2019

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OpenLoops

[Höche]

OpenLoops is a fully automated numerical tool for the tree and one-loop computation of hard scattering amplitudes required in Monte-Carlo simulations of scattering events

Full NLO QCD and NLO EW corrections available
Strong CPU performance and excellent numerical stability

Scattering probability densities in perturbation theory W00 =

hel
col

|M0|2, W01 =

hel
col

2 Re

 M∗

0M1

 ,

W11 =

hel
col

|M1|2 computed from sums of l-loop Feynman diagrams: M0 = + + . . . M1 = + + . . .

1

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Applications of OpenLoops

Interfaces to many Monte Carlo programs (unchanged from OpenLoops 1) Sherpa [Höche, Krauss, Schönherr, Siegert et al.] → NLO matching and merging , Munich/Matrix [Grazzini, Kallweit, Rathlev, Wiesemann] → (N)NLO parton level MC, Powheg [Nason, Oleari et al.], Herwig [Gieseke, Plätzer et al.] Geneva [Alioli, Bauer, Tackmann et al.], Whizard [Kilian, Ohl, Reuter et al.] Many OpenLoops applications:

NLO QCD and NLO+PS for any 2 → 2, 3, 4 SM process at LHC and future colliders
NLO EW for any 2 → 2, 3, 4 SM process [Kallweit, Lindert, Maierhöfer, Pozzorini, Schönherr]
NNLO QCD for pp → V, V V with Matrix [Grazzini, Kallweit, Rathlev, Wiesemann]

First OpenLoops 2 applications (2019):

NNLO QCD Spin correlations in t¯

t production [Behring, Czakon, Mitov, Papanastasiou, Poncelet]

NLO QCD t¯

tb¯ b+jet production [Buccioni, Kallweit, Pozzorini, M.Z.]

2

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Outline

I. Program structure of OpenLoops 2
II. Full NLO QCD and EW corrections in the SM

– Power counting in αS and α – Input schemes and parameters – Evaluation of amplitudes → Automated scale variations – Colour- and Spin-correlators

III. The on-the-fly algorithm

– Recursive construction of tree and loop diagrams – On-the-fly reduction, merging and helicity summation – On-the-fly stability system → Numerical stability and performance

IV. Summary and Outlook

3

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I. The structure of OpenLoops 2
OpenLoops program (public): User interfaces and process-independent OpenLoops routines.

Available as tar from https://openloops.hepforge.org or from git repository: git clone https://gitlab.com/openloops/OpenLoops.git

Process generator (not public): Perform analytical steps (e.g. colour factors) and generate

process-dependent code for numerical calculation → stored in process libraries

Process libraries (public) automatically downloaded by the user.

A process a library contains all partonic channels for a process class. Example: ppjj contains d ¯ d → d ¯ d, u¯ u → d ¯ d, d ¯ d → gg, gg → gg, etc and real corrections d ¯ d → d ¯ dg, u¯ u → d ¯ dg, etc Also: particle permutations + channel maps, e.g. b¯ b → gg mapped to d ¯ d → gg for MB = 0. More than 200 process libraries available for all relevant SM processes (+ HEFT) → see https://openloops.hepforge.org Additional libraries provided upon user request

Third party tools for integral evaluation (included): Collier 1.2.2 [Denner, Dittmaier, Hofer ’16],

OneLoop 3.6.1 [van Hameren ’10]

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II. Full NLO QCD and EW corrections in the SM

EW corrections enhanced by soft/collinear logarithms from virtual EW bosons:

Order

α πs2

w ln2(Q2/M2

W) ∼ 25% > αS in observables at the TeV scale

⇒ EW corrections crucial for SM tests and BSM searches at the LHC But also more challenging than NLO QCD!

Virtual corrections involve more particles and masses (γ, Z, W, H, b, t)

qi g qi g γ, Z, W γ, Z, W ℓ+ ℓ−

V → lepton decays: Effective particle multiplicity increased due to final-state interactions and

non-factorisation, e.g. pp → ZZ is 2 → 2 in QCD and 2 → 4 with EW (×400 more diagrams)

Z/γ∗

p p

ℓ− ℓ+ Z/γ∗

p p

ℓ− ℓ+ Z/γ∗

p p

ℓ− ℓ+

5

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Power counting: Nontrivial QCD-EW interplay

Simple example: q¯ q → q¯ q at Born level: M0 ∼ O(e2) + O(g2

S)

⇒ σq¯

q→q¯ q ∼ W00 ∼ O(α2 S)

QCD

+ O(α1

Sα1)

EW−QCD interf.

+ O(α2)

EW

NLO EW corrections of O(α2

Sα1) for q¯

q → q¯ q:

EW corrections to QCD Born

γ γ γ, Z

QCD corrections to EW–QCD interference

γ, Z γ, Z γ, Z

→ only full O(α2

Sα1) IR finite

→ O(α) corrections can involve emissions of γ and g, q, ¯ q In general (e.g. pp → X+jets): M0 =

˜ nq¯

q

k=0 gn−2k

S

em+2kM(k) where gn

SemM(0)

= M0

LO QCD

˜ nq¯

q = nq¯ q − 1, if nq¯ q ≥ 1 (number of external q¯

q pairs), else ˜ nq¯

q = 0

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NLO EW corrections in OpenLoops 2

M0 =

˜ nq¯

q

k=0

gn−2k

S

em+2kM(k) ⇒ W00 = M0|M0 ∼ O(αn

Sαm) + O(αn−1 S

αm+1) + . . . + O(αn−k

S

αm+k) alternating series of dominant contributions involving |M(k)

0 |2 and suppressed pure interference terms

involving M(k)

0 |M(k′)

with k = k′.

αS

α αn

Sαm

αn−1

S

αm+1 αn−2

S

αm+2 αn+1

S

αm

αn

Sαm+1

LO NLO

⇒ Mixed α αS power counting with non-trivial interference contributions ⇒ OpenLoops provides any desired order O(αn

Sαm) in a fully automated way

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Input schemes and parameters

Three EW schemes implemented:

scheme input parameters value of 1/α α(0) α(0), MW, MZ, MH + fermion masses ≈ 137 Gµ (default) Gµ, MW, MZ, MH + fermion masses ≈ 132 α(MZ) α(MZ), MW, MZ, MH + fermion masses ≈ 128 derived parameters: cos2(θw) = µ2

W

µ2

Z , . . .

⊲ α(0)-scheme: pure QED interactions at scales Q2 ≪ M2

W, production of on-shell photons

⊲ Gµ-scheme: optimal description of W-interactions at EW scale ⊲ α(MZ)-scheme: hard EW interactions at EW scale (optimal for QED, decent for SU(2))

External photons in process A → B + n

γ

n-shell

+n∗ γ∗

ff-shell

(+ γ

real emission

) ⇒ rescale with ratios of input α and αon = α(0), αoff =

          

α|Gµ if α = α(0), α if α = α|Gµ or α = α(MZ) ⇒ W →

 αon

α

 n  αoff

α

 n∗ W

(No rescaling for real emission) Optimal scale choice for external on-shell, off-shell and real-emission photons

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Complex masses, Scale variations and Renormalisation

Consistent treatment of resonances with complex mass scheme at 1-loop [Denner, Dittmaier]

→ complex mass µ2

p = M2 p − i MpΓp from real physical mass Mp and width Γp as input

⇒ implemented in a flexible way, i.e. mix between on-shell and off-shell massive particles allowed ⇒ Consistent calculation of e.g. pp → t¯ tZ → t¯ tl+l− with off-shell Z at NLO EW

Different Renormalisation schemes implemented, e.g. on-shell or MS for quark masses;

different flavour schemes for αS

Efficient QCD scale variations:

If scattering amplitudes are re-evaluated multiple times with different values of µr and αS (all other input and kinematic parameters fixed) → For each new phase-space point, matrix elements are computed and stored in a cache. → For (µr, αS) variations, only µr-dependent QCD counterterms are explicitly re-computed and the bare amplitude from the cache is re-scaled according to its αS-dependence. ⇒ Highly efficient algorithm for scale variations fully automated

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More OpenLoops 2 features

Colour and charge correlators,

→ IR subtraction methods e.g. ML|T a

j T a k |ML

αp

Sαq

and ML|QjQk|ML

αp

Sαq for L = 0, 1 (exchange of soft gluon/photon between external legs j, k)

Spin and Spin-colour correlators,

→ IR subtraction methods e.g. Bµν

j

= M|µ, j ν, j|M and B(p,q|jk|µν)

LL,LO

= ML|T a

j T a k |µ, j

ν, j|ML
αp

Sαq (soft-collinear radiation of external gluons/photons)

for L=0,1 (all L=0 correlators already available in OpenLoops 1)

Catani-Seymour I-operator → subtraction of IR poles
Selection of helicity states → polarised initial or final states

⇒ Ingredients for a wide range of applications available

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III. The OpenLoops algorithm: Tree level

Tree-level amplitudes constructed recursively from sub-trees For example M0 = + . . . → split into sub-trees Numerical recursion step: wα

a =

=

×

sub-tree wb sub-tree wc

= Xα

βγ(kb, kc)

k2

a − m2 a

universal building block

from Feynman rules wβ

b wγ c

Generic depiction:

α

wa

ka =

α

wb wc

kb kc (ki external momenta)

Highly efficient: Sub-trees constructed only once for multiple tree and loop diagrams

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III. The OpenLoops algorithm: One-loop amplitudes

High complexity in loop diagrams due to analytical structure in loop momentum q Mdiag

1

=

wN−1 wN w1 w2 D0 D1 D2 DN−1

q

= Cdiag

dDq S1(q)· · ·SN(q)

D0· · ·DN−1

Scalar propagators Di(q) = (q + pi)2 − m2

i

Factorisation into colour factor Cdiag and loop segments Si(q) =

βi−1

wi

ki Di

βi

= Xα

i (ki, pi, q)wα i

Universal building block × sub-tree(s) Open loop diagram at D0 → Dress open loop recursively: Nk(q) =

k

i=1 Si(q) = Nk−1(q)Sk(q) =

β0

w1

D1

w2

D2

wk

Dk

βk

wk+1

Dk+1

wN−1

DN−1

wN

D0

βN

dressed segments
undressed segments

=

k

r=0 N (r)

µ1...µrqµ1 . . . qµr

Completely generic and highly efficient algorithm Remaining tasks: Reduction of tensor in qµ, evaluation of q-integrals

12

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Tensor integral reduction

Challenge: High complexity in loop diagram ∼

3

µ1=0 . . .

3

µN=0 Nµ1...µN qµ1 . . . qµN

dressing steps 1 2 3 4 5 6 7 independent coefficients rank 1 2 3 4 5 6 7 5 15 35 70 126 210 330

OpenLoops 1

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Tensor integral reduction

Challenge: High complexity in loop diagram ∼

3

µ1=0 . . .

3

µN=0 Nµ1...µN qµ1 . . . qµN

OpenLoops 1: A-posteriori reduction External tools used for reduction

Collier [Denner, Dittmaier, Hofer ’16]
CutTools [Ossola, Papadopoulos, Pittau ’08]

⇒ High complexity in intermediate results Avoided entirely in OpenLoops 2 for Born-loop interference amplitudes

dressing steps 1 2 3 4 5 6 7 independent coefficients rank 1 2 3 4 5 6 7 5 15 35 70 126 210 330

OpenLoops 1

Tensor reduction

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The On-the-fly method

[Buccioni, Pozzorini, M.Z., 2018]

On-the-fly reduction via integrand-level identities [del Aguila, Pittau ’05]: qµqν = Aµν +

3

λ=0

Bµν

λ qλ

with Aµν = Aµν

−1 + Aµν 0 D0(q),

Bµν

λ

= Bµν

−1,λ + 4

i=0 Bµν

i,λDi(q)

Reconstructed Di cancel in full integrand

S1(q)S2(q) D0··· DN−1 N

i=3 Si = N µνqµqν+N µqµ+N

D0···DN−1 N

i=3 Si

dressing steps 1 2 3 4 5 6 7 independent coefficients rank 1 2 3 4 5 6 7 5 15 35 70 126 210 330

OpenLoops 1

14

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The On-the-fly method

[Buccioni, Pozzorini, M.Z., 2018]

On-the-fly reduction via integrand-level identities [del Aguila, Pittau ’05]: qµqν = Aµν +

3

λ=0

Bµν

λ qλ

with Aµν = Aµν

−1 + Aµν 0 D0(q),

Bµν

λ

= Bµν

−1,λ + 4

i=0 Bµν

i,λDi(q)

Reconstructed Di cancel in full integrand

S1(q)S1(q) D0··· DN−1 N

i=3 Si = N µνqµqν+N µqµ+N

D0···DN−1 N

i=3 Si

dressing steps 1 2 3 4 5 6 7 independent coefficients rank 1 2 3 4 5 6 7 5 15 35 70 126 210 330

OpenLoops 1

14

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The On-the-fly method

[Buccioni, Pozzorini, M.Z., 2018]

On-the-fly reduction via integrand-level identities [del Aguila, Pittau ’05]: qµqν = Aµν +

3

λ=0

Bµν

λ qλ

with Aµν = Aµν

−1 + Aµν 0 D0(q),

Bµν

λ

= Bµν

−1,λ + 4

i=0 Bµν

i,λDi(q)

Reconstructed Di cancel in full integrand

S1(q)S1(q) D0··· DN−1 N

i=3 Si = N µνqµqν+N µqµ+N

D0···DN−1 N

i=3 Si

Dressing and reduction of amplitude in a single recursion Huge reduction in complexity (rank≤ 2 at all stages)

dressing steps 1 2 3 4 5 6 7 independent coefficients rank 1 2 3 4 5 6 7 5 15 35 70 126 210 330

OpenLoops 1 OpenLoops 2

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On-the-fly reduction, merging and helicity summation

New topologies with pinched propagators

in every reduction step:

N µν qµqν D0···DN−1 = 3

i=−1

N µ

i qµ + Ni

D0 · · · / Di · · · DN−1

Merging of open loops with the same topology

and same undressed segments exploiting factorisation of dressed part N (α) and undressed segments

α N (α) Sn+2 · · · SN = N Sn+2 · · · SN

⇒ No extra cost for pinched topologies

On-the-fly helicity summation of dressed

segments interfered with Born ⇒ Factor 2 − 5 gain in CPU efficiency

w1 w2 ∼ N µνqµqν w3 wN

=                                                                                                 

w1 w2 w3 wN N µ

−1qµ

+

w1 w2 w3 wN N µ

1 qµ

+

w1 w2 w3 wN N µ

2 qµ

+

w1 w2 w3 wN N µ

3 qµ

+

w1 w2 w3 wN N µ

0 qµ

N (1)

wn

Dn

wn+1

Dn+1 Dn+1

wn+2 wN +

N (2)

wn wn+1

Dn+1

wn+2 wN + . . .

                                                 =

N

wn wn+1

Dn+1

wn+2 wN

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Numerical stability in OpenLoops

Spurious singularities due to inverse Gram determinants in the reduction can lead to large uncer- tainties in a small fraction of phase space points potentially spoiling the precision of the full MC run.

Stability system in OpenLoops 1:

Numerical accuracy of double precision (dp) result determined from rescaling test
Re-evaluation of critical phase-space points in quad precision (qp)

→ O(100) CPU cost wrt dp

Stability system in OpenLoops 2:

Ensure minimal need for quad precision

→ Exploit analytical properties in on-the-fly reduction formulas and use targeted expansions (to any order) ⇒ Instabilities postponed to the last steps of the algorithm or avoided entirely

Ensure minimal use of quad precision

→ Hybrid precision mode: Targeted use of qp only in a critical steps

16

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Hybrid precision mode in OpenLoops 2

Upgrade of dp objects to qp only triggered in a few final steps, while the bulk of the calculation is in dp

dressing reduction double precision quadruple precision

CPU cost O(1%) of full qp calculation
Excellent numerical stability at only O(10%) additional cost wrt pure dp

17

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Numerical stability improvements for hard kinematics (NLO QCD) Probability to encounter an event with accuracy Amin or less for a 2 → 4 process (

√ ˆ s = 1 TeV, 106 events)

−32 −28 −24 −20 −16 −12 −8 −4

accuracy Amin

10−6 10−5 10−4 10−3 10−2 10−1 100

fraction of events

gg → t¯ tgg at O(α5

s)

OL1+CutTools dp

OL1+CutTools: 1% of points highly unstable

18

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Numerical stability improvements for hard kinematics (NLO QCD) Probability to encounter an event with accuracy Amin or less for a 2 → 4 process (

√ ˆ s = 1 TeV, 106 events)

−32 −28 −24 −20 −16 −12 −8 −4

accuracy Amin

10−6 10−5 10−4 10−3 10−2 10−1 100

fraction of events

gg → t¯ tgg at O(α5

s)

OL1+CutTools dp OL1+Collier dp

OL1+CutTools: 1% of points highly unstable → OL1+Collier: O(103) improvement

18

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Numerical stability improvements for hard kinematics (NLO QCD) Probability to encounter an event with accuracy Amin or less for a 2 → 4 process (

√ ˆ s = 1 TeV, 106 events)

−32 −28 −24 −20 −16 −12 −8 −4

accuracy Amin

10−6 10−5 10−4 10−3 10−2 10−1 100

fraction of events

gg → t¯ tgg at O(α5

s)

OL1+CutTools dp OL1+Collier dp OL2 dp

OL1+CutTools: 1% of points highly unstable → OL1+Collier: O(103) improvement OL2 dp: extra O(10) improvement and 2–3 times faster

18

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Numerical stability improvements for hard kinematics (NLO QCD) Probability to encounter an event with accuracy Amin or less for a 2 → 4 process (

√ ˆ s = 1 TeV, 106 events)

−32 −28 −24 −20 −16 −12 −8 −4

accuracy Amin

10−6 10−5 10−4 10−3 10−2 10−1 100

fraction of events

gg → t¯ tgg at O(α5

s)

OL1+CutTools dp OL1+Collier dp OL2 dp OL2 hp 8 digits OL2 hp 11 digits

OL1+CutTools: 1% of points highly unstable → OL1+Collier: O(103) improvement OL2 dp: extra O(10) improvement and 2–3 times faster OL2 hp: extra O(100) improvement (always ≥ 7 digits) with +8% CPU time → hp target precision can be tuned (trigger for qp upgrade)

18

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Numerical stability improvements for hard kinematics (NLO QCD) Probability to encounter an event with accuracy Amin or less for a 2 → 4 process (

√ ˆ s = 1 TeV, 106 events)

−32 −28 −24 −20 −16 −12 −8 −4

accuracy Amin

10−6 10−5 10−4 10−3 10−2 10−1 100

fraction of events

gg → t¯ tgg at O(α5

s)

OL1+CutTools dp OL1+Collier dp OL2 dp OL2 hp 8 digits OL2 hp 11 digits OL2 qp

OL1+CutTools: 1% of points highly unstable → OL1+Collier: O(103) improvement OL2 dp: extra O(10) improvement and 2–3 times faster OL2 hp: extra O(100) improvement w.r.t. dp (always ≥ 7 digits) with +8% CPU time OL2 qp: always 17–32 digits with 80 times more CPU time than in dp

18

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Numerical stability improvements for hard kinematics (NLO EW) Probability to encounter an event with accuracy Amin or less for a 2 → 4 process (

√ ˆ s = 1 TeV, 106 events)

−32 −28 −24 −20 −16 −12 −8 −4

accuracy Amin

10−6 10−5 10−4 10−3 10−2 10−1 100

fraction of events

¯ uu → e+e−µ+µ− at O(α5)

OL1+CutTools dp OL1+Collier dp OL2 dp OL2 hp 8 digits OL2 hp 11 digits OL2 qp

Similar improvements for a wide range of tested processes with NLO QCD and NLO EW corrections as well as in hard, soft and collinear phase-space regions

19

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V. Summary and Outlook

OpenLoops 2: Fully automated numerical tool for tree and one-loop scattering amplitudes

Highly flexible and simple to interface to any MC generator (same interfaces as OL1)
Full NLO QCD and NLO EW corrections in the SM available (+ HEFT)
Excellent performance and numerical precision due to

– On-the-fly reduction, merging and helicity summation – On-the-fly stability system with hybrid precision Short-term and mid-term projects:

On-the-fly generation of process-dependent code
Numerical stability improvements in deep IR regions
Full NNLO automation

20

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Backup

21

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Installation of OpenLoops 2

Checkout from git repository: → takes ∼ 3 sec git clone https://gitlab.com/openloops/OpenLoops.git

r download as tar from https://openloops.hepforge.org/downloads

Compilation after checkout or download: cd OpenLoops ./scons → takes 1 − 2 min, ∼ 60 MB disk space Default requirements: Python ≥ 2.4, gfortran ≥ 4.6; Alternative compiler: ifort Change default options in file OpenLoops/openloops.cfg: [OpenLoops] fortran compiler = ifort . . . Update program and (if necessary) all process libraries with ./openloops update

22

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Process libraries

Download and compile process libraries or collections of process libraries: ./openloops libinstall <processes> ./openloops libinstall <collection>.coll for example ./openloops libinstall pptt ppttj ppttjj Predefined collections: <collection> Description bornloop all public NLO QCD born-loop libraries loop2 all public NLO QCD loop-induced libraries lhc essential LHC processes all every lib from the public process repository Time and disk space required for download+compilation of process libraries: library/collection pptt ppttj ppttjj lhc.coll (27 libs) all.coll (156 libs) time 4 sec 10 sec 2 min 24 min 105 min disk space [MB] 3.6 19.7 288 3500 13900

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Example for QCD and EW power counting

Channels and orders, e.g. W + multi-jet [Kallweit,Lindert,Maierhöfer,Pozzorini,Schönherr ’14] pp → W + n jets @LO pp → W + n jets @NLO

αn

sα

αn−1

s

α2 αn−2

s

α3 αn−3

s

α4 αn+1

s

α αn

sα2

αn−1

s

α3 αn−2

s

α4 αn−3

s

α5 ui ¯ di → W + ng

×

×

×

ui ¯

di → W + q¯ q + (n − 2)g

× × ×

×

× × ×

γui → diW + (n − 1)g
×
γui → diW + q¯

q + (n − 3)g

×

× ×

γγ → ¯

uidiW + (n − 2)g

×
ui ¯

di → W + q¯ qq′¯ q′ + (n − 3)g

×

× × × ×

. . . 24

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Computing details: Performance and memory consumption Timing ratios OL1/OL2 and timings normalised to the 2 → 2 process of the same class Process Correction OL1/OL2 (2 → n)/(2 → 2) gg → t¯ t NLO QCD 2.0 1 gg → t¯ t + g NLO QCD 2.2 23 gg → t¯ t + gg NLO QCD 2.8 700 q¯ q → t¯ t NLO QCD 1.5 1 q¯ q → t¯ t + g NLO QCD 2.4 12 q¯ q → t¯ t + gg NLO QCD 2.9 300 u¯ u → e− ¯ νe¯ µνµ NLO EW 4.3

uu → W +W +dd

NLO EW 2.3

(sample of 105 random psp)

Complexity and CPU time grow exponentially with number of external particles Performance gain of factor 2-4 w.r.t OpenLoops 1 for non-trivial processes Maximum memory allocated in RAM during calculation (RSS, Peak) Process q¯ q → t¯ tg gg → t¯ tg gg → t¯ t + gg RSS Peak [MB] 54 68 420 ⇒ 70 MB for 2 → 3 420 MB for 2 → 4 Peak RAM usage up to factor 2 lower than in OpenLoops 1

25

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Stability of OpenLoops: 2 → 3 process with single soft gluon with ξ = 10−7, where ξ ∼

Q2

soft

ˆ s

∼ Esoft

√ ˆ s ( √ ˆ s = 1 TeV, 105 events)

−32 −28 −24 −20 −16 −12 −8 −4 4

instability Amin

10−5 10−4 10−3 10−2 10−1 100

fraction of events gg → t¯ tg

* wrt OL2 qp benchmark

OL1+Collier dp * OL1+CutTools dp * OL2 dp * OL2 hybrid * OL2 qp (rescaling)

Stability: OL1 + Cuttools ∼ 5% points with zero digits OL2 → no points with less than 11 (8) digits in hp (dp) Performance: OL2 (hp) 5 times slower than OL2 (dp) → Major speed-up possible

26

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Stability of OpenLoops: 2 → 3 process with single soft gluon Dependence of stability on ξ ∼

Q2

soft

ˆ s

∼ Esoft

√ ˆ s ( √ ˆ s = 1 TeV, hard kinematics fixed)

4 −4 −8 −12 −16 −20 −24 −28 −32 −36

accuracy A

10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8 10−9

ξ gg → t¯ tg

* wrt OL2 qp benchmark

OL1+CutTools dp * OL1+Collier dp * OL2 dp * OL2 hybrid * OL2 qp (rescaling)

Stability: at least 10 digits from OL2 (hp) in ultra-soft region at least 24 digits from OL2 (qp) → excellent benchmark

27

SLIDE 36

Stability of OpenLoops: 2 → 3 process with collinear gluon pair with ξ = 10−7, where ξ ∼ Q2

coll

ˆ s

∼ k2

T

ˆ s ( √ ˆ s = 1 TeV, 105 events)

−32 −28 −24 −20 −16 −12 −8 −4 4

instability Amin

10−5 10−4 10−3 10−2 10−1 100

fraction of events gg → t¯ tg

* wrt OL2 qp benchmark

OL1+Collier dp * OL1+CutTools dp * OL2 dp * OL2 hybrid * OL2 qp (rescaling)

Stability: OL1 + Cuttools ∼ 100% points with zero digits OL2 → no points with less than 10 (7) digits in hp (dp) Performance: OL2 (hp) 8 times slower than OL2 (dp) → Major speed-up possible

28

SLIDE 37

Stability of OpenLoops: 2 → 3 process with collinear gluon pair Dependence of stability on ξ ∼ Q2

coll

ˆ s

∼ k2

T

ˆ s ( √ ˆ s = 1 TeV, hard kinematics fixed)

4 −4 −8 −12 −16 −20 −24 −28 −32

accuracy A

10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8 10−9

ξ gg → t¯ tg

* wrt OL2 qp benchmark

OL1+CutTools dp * OL1+Collier dp * OL2 dp * OL2 hybrid * OL2 qp (rescaling)

Stability: at least 9 digits from OL2 (hp) in extremely collinear region at least 23 digits from OL2 (qp) → excellent benchmark

29

SLIDE 38

Gram determinant instabilities

qµqν =

Aµν

−1 + Aµν 0 D0

+

  Bµν

−1,λ + 3

i=0 Bµν

i,λDi

   qλ,

Di = (q + pi)2 − m2

i,

p0 = 0 Aµν

i , Bµν i,λ constructed from three external momenta p1, p2, p3.

Strong dependence on inverse rank-2 Gram determinant ∆12 = (p1 · p2)2 − p2

1p2 2

Moderate dependence on inverse rank-3 Gram determinant √∆123 ∼ p3 · l(p1, p2)

→ Can become dominant in soft and collinear regions. Aµν

i

= 1 ∆12 aµν

i ,

Bµν

i,λ

= 1 ∆2

12

1 √∆123

 b(1)

i,λ

 µν

+ 1 ∆12

 b(2)

i,λ

 µν

Severe numerical instabilities for

∆12 → 0

Numerical instabilities for

∆123 → 0

30

SLIDE 39

Numerical instabilities due to small rank-2 Gram determinants

Aµν

i

= 1 ∆12 aµν

i ,

Bµν

i,λ =

1 ∆2

12

 ˜

b(1)

i,λ

 µν

+ 1 ∆12

 b(2)

i,λ

 µν

Severe numerical instabilities for ∆12 → 0

for N ≥ 4: Exploit freedom to choose p1, p2 by re-ordering at runtime:

{D1, D2, D3} − → {Di1, Di2, Di3} such that parameter ∼ |∆i1i2| is maximal ⇒ avoid small rank-2 Gram determinants until triangle reduction!

for N = 3: Identify problematic kinematic configurations → for hard kinematics only one case:

q p1 q + p1 p2 − p1 q + p2 − p2

p2

1

= −p2 < 0, p2

2

= −p2(1 + δ), 0 ≤ δ ≪ 1, (p2 − p1)2 = 0, ⇒ ∆ = −p2δ2

Master integrals: C0(p2

1, p2 2) ∼

dDq

1 D0D1D2 and B0(p2

1) ∼

dDq

1 D0D1

Reduction formulas exhibit poles in δ, e.g. for massless rank-1 topology: Cµ = 1 δ2B0(−p2) [. . .] + 1 δ2B0

−p2(1 + δ)
[. . .] + 1

δC0

−p2, −p2(1 + δ)
[. . .]

31

SLIDE 40

Any-order expansions

Expand master integrals B0, C0 in δ ∼ ∆12 ⇒ 1

δ-poles cancel (also for higher rank):

Cµ = pµ

1 + pµ 2

2p2

−B0(−p2) + 1
+ δ pµ

1 + 2pµ 2

6p2

B0(−p2)
+ . . .
General and compact formulas derived for

∂

∂δ

mB0 and ∂

∂δ

mC0 (all QCD topologies)

→ extremely fast implementation ⇒ Expansion of B0, C0 to any order M in order to reach 16, 32 or more digits ⇒ Uncertainty due to truncation of series avoided entirely.

On-the-fly reduction, no stability improvements

−15 −10 −5 5 10

accuracy A

105 1010 1015 1020 1025 1030

Q4

12/∆

2

min

100 101 102 103

(D1, D2, D3)-permutation, no expansion

−15 −10 −5 5 10

accuracy A

105 1010 1015 1020 1025 1030

Q4

12/∆

2

min

100 101 102 103

(D1, D2, D3)-permutation + any-order expansions

−15 −10 −5 5 10

accuracy A

105 1010 1015 1020 1025 1030

Q4

12/∆

2

min

100 101 102 103

32

SLIDE 41

Treating numerical instabilities in soft and collinear regions

for N ≥ 5: Exploit freedom to choose the 3 propagators for the reduction from D1, D2, D3, D4

such that parameter ∼ |∆i1i2| is maximal (as before) and parameter ∼ |∆i1i2i3| is maximal. ⇒ Avoid small rank-3 (rank-2) Gram determinants until box (triangle) reduction!

Analytical treatment of large cancellations between quasi on-shell self-energy insertions and

counterterms in unresolved regions

Numerically stable evaluation of kinematics (momenta, invariants, propagators,...)
Numerically stable construction of reduction basis → coefficients in reduction formulas
Improvements in Collier 1.2.2 (scalar boxes, triangles) [Denner, Dittmaier]

→ requires analytical understanding of the origin of instabilities and their cancellation

33

SLIDE 42

Hybrid precision

Idea: Promote only open loops enhanced by ∆−n terms and their subsequent dressing, merging and reduction to quad precision (qp), → bulk of the calculation still in double precision (dp) Example:

w1 w2 w3 wN

dp

w1 w2 w3 wN

qp

w1 w2 w3 wN

dp

w1 w2 w3 wN

dp

w1 w2 w3 wN

dp

w1 w2 w3 wN

qp

reduce 34

SLIDE 43

Hybrid precision

Idea: Promote only open loops enhanced by ∆−n terms and their subsequent dressing, merging and reduction to quad precision (qp), → bulk of the calculation still in double precision (dp) Example:

w1 w2 w3 wN

dp

w1 w2 w3 wN

qp

w1 w2 w3 wN

dp

w1 w2 w3 wN

dp

w1 w2 w3 wN

dp

w1 w2 w3 wN

qp

reduce

w1 w2 w3 wN

dp

merge

w1 w2 w3 wN

qp

merge 34

SLIDE 44

Hybrid precision

Idea: Promote only open loops enhanced by ∆−n terms and their subsequent dressing, merging and reduction to quad precision (qp), → bulk of the calculation still in double precision (dp) Example:

w1 w2 w3 wN

dp

w1 w2 w3 wN

qp

w1 w2 w3 wN

dp

w1 w2 w3 wN

dp

w1 w2 w3 wN

dp

w1 w2 w3 wN

qp

reduce

w1 w2 w3 wN

dp

merge

w1 w2 w3 wN

qp

merge

w1 w2 w3 wN

qp

dress

. . . . . .

w1 w2 w3 wN

dp

dress dress

w1 w2 w3 wN

qp

34

SLIDE 45

Hybrid precision

Idea: Promote only open loops enhanced by ∆−n terms and their subsequent dressing, merging and reduction to quad precision (qp), → bulk of the calculation still in double precision (dp) Example:

w1 w2 w3 wN

dp

w1 w2 w3 wN

qp

w1 w2 w3 wN

dp

w1 w2 w3 wN

dp

w1 w2 w3 wN

dp

w1 w2 w3 wN

qp

reduce

w1 w2 w3 wN

dp

merge

w1 w2 w3 wN

qp

merge

w1 w2 w3 wN

qp

dress

. . . . . .

w1 w2 w3 wN

dp

dress dress

w1 w2 w3 wN

qp . . . . . .

Large cancellation 34