Numerical Evaluation of One-Loop Scattering Amplitudes for

Numerical Evaluation of One-Loop

Scattering Amplitudes for e−e+ → e−e+γ

Giovanni Ossola

New York City College of TechnologyCity University of New York (CUNY)

USTRON’09 – MATTER TO THE DEEPEST

Recent Developments in Physics of Fundamental Interactions

Ustron, Poland – September 11-16, 2009

Giovanni Ossola (City Tech) OPP Reduction September 2009 1 / 30

Outline

1 Motivation & Introduction

2 A few comments on the OPP method

3 Work in Progress and New Results

4 NLO QED corrections to e−e+ → e−e+γ, e−e+ → µ

−µ

+γ


Summary and Conclusions

from USTRON 2007LHC requires NLO calculations!

There is a variety of interesting options available, but no universalmethod for one-loop multi-leg calculations

OPP is a “young” method, but it seems promising

More results soon!

G. O., C. G. Papadopoulos and R. Pittau

Nucl. Phys. B 763, 147 (2007) – arXiv:hep-ph/0609007JHEP 0707 (2007) 085 – arXiv:0704.1271 [hep-ph]


Progress in 2005-2009

Some recent calculations → Cross Sections available

pp → Z Z Z and pp → ttZ [Lazopoulos, Melnikov, Petriello]

pp → bbZ [Febres Cordero, Reina, Wackeroth]

pp → H + 2 jets, pp → WW+ jet [Campbell, Ellis, Giele, Zanderighi]

pp → VV + 2 jets via VBF [Bozzi, Jager, Oleari, Zeppenfeld]

pp → H H H [Binoth, Karg, Kauer, Ruckl]

pp → tt+jet [Ciccolini, Denner and Dittmaier]

pp → VVV [Binoth, G.O., Papadopoulos, Pittau]

pp → VVV with leptonic decays [Campanario, Hankele, Oleari et al]

pp → W+ 3 jets [Berger et al, Ellis et al]

pp → ttbb [Bredenstein et al, Bevilacqua et al]

A lot of progress on 2 → 4


Recent Progress on 2 → 4

pp → W + 3 jets

Berger et al

Blackhat + Sherpa

Ellis, Melnikov, Zanderighi

Rocket

pp → ttbb

Bredenstein, Denner, Dittmaier, Pozzorini

“traditional” approach, tensorial reduction

Bevilacqua, Czakon, Papadopoulos, Pittau, Worek

CutTools + Helac1loop + Dipoles(talk by Malgorzata Worek)


OPP Method

Three years ago (Sept.2006), we proposed a new method for the numericalevaluation of scattering amplitudes, based on a decomposition at theintegrand level.

Some of the advantages:

Universal - applicable to any process

Simple - based on basic algebraic properties

Automatizable - easy to implement in a computer code

Final Task

Produce a MULTI-PROCESS fully automatized NLO generator


“Standing on the shoulders of giants”

1 Passarino-Veltman Reduction to Scalar Integrals

M =∑

i

di Boxi +∑

i

ci Trianglei

+∑

i

bi Bubblei +∑

i

ai Tadpolei + R ,

Set the basis for our NLO calculationsExploits the Lorentz structure

2 Pittau/del Aguila Recursive Tensorial ReductionExpress qµ =

∑

i Gi ℓiµ , ℓi

2 = 0The generated terms might reconstruct denominators Di

or vanish upon integration

3 “Cut-based” Techniques (Bern, Dixon, Dunbar, Kosower in ’94)direct extraction of the coefficients of the scalar integral

Pigmaei gigantum humeris impositi plusquam ipsi gigantes vident


One-loop – Definitions

Any m-point one-loop amplitude can be written, before integration, as

A(q) =N(q)

D0D1 · · · Dm−1

where

Di = (q + pi )2 − m2

i , q2 = q2 + q2 , Di = Di + q2

Our task is to calculate, for each phase space point:

M =

∫

dnq A(q) =

∫

dnqN(q)

D0D1 . . . Dm−1


The traditional “master” formula

∫

A =

m−1∑

i0<i1<i2<i3

d(i0i1i2i3)

∫

1

Di0Di1Di2Di3

+

m−1∑

i0<i1<i2

c(i0i1i2)

∫

1

Di0Di1Di2

+m−1∑

i0<i1

b(i0i1)

∫

1

Di0Di1

+m−1∑

i0

a(i0)

∫

1

Di0

+ rational terms


OPP “master” formula - I

General expression for the 4-dim N(q) at the integrand level in terms of Di

N(q) =

m−1∑

i0<i1<i2<i3

[

d(i0i1i2i3) + d(q; i0i1i2i3)]

m−1∏

i 6=i0,i1,i2,i3

Di

+

m−1∑

i0<i1<i2

[c(i0i1i2) + c(q; i0i1i2)]

m−1∏

i 6=i0,i1,i2

Di

+

m−1∑

i0<i1

[

b(i0i1) + b(q; i0i1)]

m−1∏

i 6=i0,i1

Di

+

m−1∑

i0

[a(i0) + a(q; i0)]

m−1∏

i 6=i0

Di

This is 4-dimensional Identity


Overview Rational Terms

R = R1 + R2

R1 – The OPP expansion is written in terms of 4-dim Di , while n-dim Di

appear in scalar integrals.

A(q) =N(q)

D0D1 · · · Dm−1

R1 can be calculated in two different ways, both fully automatized.

R2 – The numerator N(q) can be also split into a 4-dim plus a ǫ-dim part

N(q) = N(q) + N(q2, q, ǫ) .

Compute R2 using tree-level like Feynman Rules.


One-Loop as a 3 step process

1) Compute the numerator N(q) numerically at given q

2) Extract coefficients/rats with OPP reduction

3) Combine with scalar integrals

M =∑

i

di Boxi +∑

i

ci Trianglei

+∑

i

bi Bubblei +∑

i

ai Tadpolei + R ,



1) Compute the numerator N(q) numerically at given q

2) Extract coefficients/rats with OPP reduction [CutTools]

3) Combine with scalar integrals [OneLOop/QCDloop]

M =∑

i

di Boxi +∑

i

ci Trianglei

+∑

i

bi Bubblei +∑

i

ai Tadpolei + R ,



1) Compute the numerator N(q) numerically at given q [???]

2) Extract coefficients/rats with OPP reduction [CutTools]

3) Combine with scalar integrals [OneLOop/QCDloop]

M =∑

i

di Boxi +∑

i

ci Trianglei

+∑

i

bi Bubblei +∑

i

ai Tadpolei + R ,

What about the numerator N(q) ?


A real proof of concept

van Hameren, Papadopoulos, Pittau – arXiv:0903.4665 [hep-ph]

1) numerator N(q) numerically with HELAC-1loop

2) coefficients via OPP reduction with CutTools

3) scalar integrals with OneLOop/QCDloop

Fully Automated numerical evaluation of ANY one-loop amplitude

All 6-particle processes in the Les Houches 2007 “Wish List’

uu → ttbb gg → ttbb uu → W +W−bb gg → W +W−bb

uu → bbbb gg → bbbb ud → W +ggg uu → Zggg

uu → ttgg gg → ttgg

Czakon, Papadopoulos, Worek – arXiv:0905.0883 [hep-ph]

Dipoles: Automated Dipole Subtraction within HELACGiovanni Ossola (City Tech) OPP Reduction September 2009 13 / 30

What’s next?

(what is still left to do??)

New Codes

OptimizationExample: Improve the system-solving algorithm in OPP-equationsDo we gain by using DFT? (work with P. Mastrolia)

Phenomenology - New processesExample: NLO QED corrections to e−e+ → e−e+

γ

and e−e+ → µ−µ

+γ

(in collaboration with S. Actis, A. Ferroglia, P. Mastrolia)


NLO QED corrections to e−e+ → e−e+γ

During the last few years, there has been a significant progress in reducing

the theoretical uncertainty in Bhabha generators used at presently running

e+e− colliders down to 0.1%

NNLO QED calculations are essential to establish the theoretical accuracy

of existing generators and, if necessary, to improve it below 0.1%

In particular, the one-loop corrections to single hard bremsstrahlung should

be calculated for full Bhabha scattering, to get a better control of the

theoretical precision

G. Balossini, C. Bignamini, C. M. Carloni Calame, G. Montagna,O. Nicrosini and F. Piccinini

“Mini-review on Monte Carlo programs for Bhabha scattering,”

talk presented at Loops and Legs in Quantum Field Theory 2008


Tree-level diagrams

QED tree-level diagrams for e−e+ → e−e+γ (full set)and e−e+ → µ−µ+γ (first line only)


One-loop diagrams

Representative one-loop diagrams for e−e+ → e−e+γ:

2d2c2b2a

2e 2f 2g

QGRAF generates 38 one-loop diagrams for the process e−e+ → µ−µ+γ

and 76 diagrams for the process e−e+ → e−e+γ


Calculation of CC4 and R1

We evaluated the one-loop corrections by means of the OPP reduction

to e−e+ → µ−µ+γ → more diagrams

to e−e+ → e−e+γ → more scales (masses)

We perform the reduction numerically for each phase space point

The interference between one-loop diagrams and the tree-level isgenerated by QGRAF, processed with FORM to produce aFORTRAN 95 output.

The latter is the input for the reduction program to get CC4 and R1

Scalar integrals evaluated with OneLOop and QCDLoop

We obtain:

1

4

∑

spins

2 Re (M1−loopM⋆

tree) = CC4 + R1 + R2 + UVct


Full mass dependence – R2 and UVct

We carried out the UV renormalization in the on-mass-shellscheme

We have retained the full dependence on the fermion masses:R2 and UVct need additional mass-counterterm diagrams


Checks that we performed

1 Two independent calculations

A) CutTools for the reduction + QCDLoop for scalar integralsB) independent code for the reduction + OneLOop for scalar integrals

2 “N = N” test (done by CutTools)

3 Double precision vs Multiple precision

4 Complete cancellation of UV and IR poles

5 Stability test on quasi-collinear configuration


The N ≡ N test

Our “master” formula again!

N(q) =

m−1∑

i0<i1<i2<i3

[

d(i0i1i2i3) + d(q; i0i1i2i3)]

m−1∏

i 6=i0,i1,i2,i3

Di

+

m−1∑

i0<i1<i2

[c(i0i1i2) + c(q; i0i1i2)]

m−1∏

i 6=i0,i1,i2

Di

+

m−1∑

i0<i1

[

b(i0i1) + b(q; i0i1)]

m−1∏

i 6=i0,i1

Di

+m−1∑

i0

[a(i0) + a(q; i0)]m−1∏

i 6=i0

Di

After determining all coefficients → this should hold for any q


Example for e−e+ → e−e+γ

Output of our FORTRAN code at a given phase space pointThe CPU time required is O(10−1) seconds

ILO = +0.75861014681036187

IV

NLO(CC4 + R) = +1

ǫ4.74506427003504525 · 10−2 + 0.50058282682639688

IV

NLO(UVct) = −1

ǫ5.28634805094575690 · 10−3 − 8.71804407858063207 · 10

IV

NLO = +1

ǫ4.21642946494046947 · 10−2 + 0.41340238604059049

Results are expressed in GeV−2

All numbers have been obtained working in double precision


Example for e−e+ → e−e+γ

Output of our FORTRAN code at a given phase space pointThe CPU time required is O(10−1) seconds

Test on the UV and IR poles!

ILO = +0.75861014681036187

IV

NLO(CC4 + R) = +1

ǫ4.74506427003504525 · 10−2 + 0.50058282682639688

IV

NLO(UVct) = −1

ǫ5.28634805094575690 · 10−3 − 8.71804407858063207 · 10

IV

NLO = +1

ǫ4.21642946494046947 · 10−2 + 0.41340238604059049

IR

NLO = −1

ǫ4.21642946495862717 · 10−2

Results are expressed in GeV−2

All numbers have been obtained working in double precision


Example for e−e+ → µ−µ

+γ

Virtual part IVNLO as a function of the energy E−of the outgoing muon:

the muon is (almost) parallel or antiparallel to the photon momentum

0.01

0.012

0.014

0.016

0.018

0.02

0.022I

V NLO

[GeV

−2]

0.1 0.2 0.3 0.4 0.5

E− [GeV]

There are no istabilities(work done in double precision)


Conclusions

LHC requires NLO calculations!

One-loop calculations are in fast evolutionOPP is a now a solid methodFull automatization is under way (fast!!)


Conclusions

LHC requires NLO calculations!

One-loop calculations are in fast evolutionOPP is a now a solid methodFull automatization is under way (fast!!)

The methods developed are general and can be used for many applications

such as NLO QED corrections to e−e+ → e−e+γ and e−e+ → µ−µ+γ

S. Actis, P. Mastrolia, and G. O.

arXiv:0909.1750 [hep-ph]

Our result is in the form of a FORTRAN code based on CutTools

full mass-dependence (no approximations)numerical stability: 9 digits for NLOcomputational speed: O(10−1) seconds/phase space point

We are now ready for phenomenological applications!


Sideboard

EXTRA SLIDES


Intermezzo: CutTools v1.1

Initialization- Choose a phase-space point (“extended” kinematics)- Define denominators Di : momenta and masses

Calculation Scalar Integrals for all combination of Di .- I used OneLOop by A. van Hameren- Store scalar integrals

Calculation of All Coefficients- Write a routine that numerically evaluates N(q) at any given q

- Use CutTools to get all coefficients- Store coefficients

Get results- Multiply scalar integral and coefficients- Add rational parts

Repeat for a new PS point

CutTools v1.1 is available!www.ugr.es/ pittau/CutTools/


Spurious Terms - I

– the recepy is not unique –

Following F. del Aguila and R. Pittau, arXiv:hep-ph/0404120

Express any q in N(q) as

qµ = −pµ

0 +∑4

i=1 Gi ℓµ

i , ℓi2 = 0

k1 = ℓ1 + α1ℓ2 , k2 = ℓ2 + α2ℓ1 , ki = pi − p0

ℓ3µ =< ℓ1|γ

µ|ℓ2] , ℓ4µ =< ℓ2|γ

µ|ℓ1]

The resulting terms Gi either reconstruct denominators Di

or vanish upon integration

→ They give rise to d , c , b, a coefficients→ They form the spurious d , c , b, a coefficients


Spurious Terms - II

d(q) term (only 1)d(q) = d T (q) ,

where d is a constant (does not depend on q)

T (q) ≡ Tr [(/q + /p0)/ℓ1/ℓ2/k3γ5]

c(q) terms (they are 6)

c(q) =

jmax∑

j=1

{

c1j [(q + p0) · ℓ3]j + c2j [(q + p0) · ℓ4]

j}

In the renormalizable gauge, jmax = 3

b(q) and a(q) give rise to 8 and 4 terms, respectively


OPP “master” formula - II

N(q) =

m−1X

i0<i1<i2<i3

h

d(i0i1 i2 i3) + d(q; i0 i1 i2 i3)i

m−1Y

i 6=i0,i1,i2,i3

Di +

m−1X

i0<i1<i2

[c(i0i1 i2) + c(q; i0 i1 i2)]

m−1Y

i 6=i0,i1,i2

Di

+

m−1X

i0<i1

h

b(i0i1) + b(q; i0 i1)i

m−1Y

i 6=i0,i1

Di +

m−1X

i0

[a(i0) + a(q; i0)]

m−1Y

i 6=i0

Di

The quantities d , c , b, a are the coefficients of all possible scalar functions

The quantities d , c , b, a are the “spurious” terms → vanish upon integration

It is now an algebraic problem:

Any N(q) just depends on a set of coefficients, to be determined!


OPP “master” formula - II

N(q) =

m−1X

i0<i1<i2<i3

h

d(i0i1 i2 i3) + d(q; i0 i1 i2 i3)i

m−1Y

i 6=i0,i1,i2,i3

Di +

m−1X

i0<i1<i2

[c(i0i1 i2) + c(q; i0 i1 i2)]

m−1Y

i 6=i0,i1,i2

Di

+

m−1X

i0<i1

h

b(i0i1) + b(q; i0 i1)i

m−1Y

i 6=i0,i1

Di +

m−1X

i0

[a(i0) + a(q; i0)]

m−1Y

i 6=i0

Di

The quantities d , c , b, a are the coefficients of all possible scalar functions

The quantities d , c , b, a are the “spurious” terms → vanish upon integration

It is now an algebraic problem:

Any N(q) just depends on a set of coefficients, to be determined!

Choose {qi} wisely

by evaluating N(q) for a set of values of the integration momentum {qi}such that some denominators Di vanish (“cuts”)


Example: 4-particles process

N(q) = d + d(q) +

3∑

i=0

[c(i) + c(q; i)]Di +

3∑

i0<i1

[

b(i0i1) + b(q; i0i1)]

Di0Di1

+

3∑

i0=0

[a(i0) + a(q; i0)] Di 6=i0Dj 6=i0Dk 6=i0

We look for a q such that

D0 = D1 = D2 = D3 = 0

→ there are two solutions q±0



N(q) = d + d(q)

Our “master formula” for q = q±0 is:

N(q±0 ) = [d + d T (q±

0 )]

→ solve to extract the coefficients d and d



N(q) − d − d(q) =

3∑

i=0

[c(i) + c(q; i)]Di +

3∑

i0<i1

[

b(i0i1) + b(q; i0i1)]

Di0Di1

+

3∑

i0=0

[a(i0) + a(q; i0)] Di 6=i0Dj 6=i0Dk 6=i0

Then we can move to the extraction of c coefficients using

N ′(q) = N(q) − d − dT (q)

and setting to zero three denominators (ex: D1 = 0, D2 = 0, D3 = 0)



N(q) − d − d(q) = [c(0) + c(q; 0)]D0

We have infinite values of q for which

D1 = D2 = D3 = 0 and D0 6= 0

→ Here we need 7 of them to determine c(0) and c(q; 0)


Numerical Evaluation of One-Loop Scattering Amplitudes for

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