the harmonic oscillator of one-loop calculations

The Harmonic Oscillator of One-loop Calculations

Peter Uwer

SFB meeting, 09.12.2010 – 10.12.2010, Karlsruhe

Work done in collaboration with Simon Badger and Benedikt Biedermann

B5

arXiv 1011.2900, http://www.physik.hu-berlin.de/pep/tools

Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 2

Motivation

Why should we study the Harmonic Oscillator ?

Simple system which shares many properties with more complicated systems

Allows to focus on the interesting physics avoiding the complexity of more complicated systems

very well understood ideal laboratory to apply and test new methods

no complicated field content, only gauge fields in particular no fermions

general structure of one-loop corrections

well known IR structure, UV structure, color decomposition…

Despite the simplifying aspects, n-gluon amplitudes are still not trivial

Harmonic oscillator of perturbative QCD: n-gluon amplitudes in pure gauge theory


Motivation

Number of pure gluon born Feynman diagrams:

n Number of diagrams

4 4

5 25

6 220

7 2485

8 34300

9 559405

10 10525900[QGRAF]


Tree level pure gluon amplitudes

Sum over non-cyclicpermutations

Generators of SU(N)with Tr[TaTb] = ab

For large N, the color structures are orthogonal:

Color-ordered amplitudes are gauge independent quantities!

color-ordered sub-amplitudes

[?]

notation:


Tree level pure gluon amplitudes

n Number of diagramsNumber of color

ordered diag.

4 4 3

5 25 10

6 220 36

7 2485 133

8 34300 501

9 559405 1991

10 10525900 7335

Important reduction in complexity


Evaluation of color ordered amplitudes

Use color-ordered Feynman rules:

Calculate only Feynman diagrams for fixed order of external legs (“= color-ordered”)

Example:A5=

Reduction: 25 10 diagrams

+

1,2,3,4,5 1,2,3,4,5


Nicer than Feynman diagrams: Recursion

= +

External wave functions, Polarization vectors

[Berends, Giele 89]colour ordered vertices

off shell leg


Born amplitudes via recursions

Remark:

Berends-Giele works with off-shell currents BCF, CSW “on-shell” recursions use on-shell amplitudes

on-shell recursions useful in analytic approaches, in numerical approaches less useful since caching is less

efficient

Berends-Giele:

caching is trivial:


Born amplitudes via recursionscalculation

i

j


Tree amplitudes from Berends-Giele recursion

[Biedermann, Bratanov, PU]

not yet fully optimized

checked with analytically known MHV amplitudes


Color-ordered sub-amplitudes (NLO)

Leading-color amplitudes are sufficient to reconstruct the full amplitude

Color structures:

Leading-color structure:

[?]


The unitarity method I

Basic idea:

Cut reconstruction of amplitudes:

=

=

[Bern, Dixon, Dunbar, Kosower 94]

color-ordered on-shell amplitudes!

l1

l2 [Cutkosky]

Tree Tree

Tree

Tree Tree

Tree

=

×

×


The unitarity method II

[Badger, Bern, Britto, Dixon, Ellis, Forde, Kosower, Kunszt, Melnikov, Mastrolia, Ossala, Pittau, Papadopoulos,…]

After 30 years of Passarino-Veltman reduction:

Reformulation of the “one-loop” problem:

How to calculate the integral coefficients in the mosteffective way

[Passarino, Veltman ’78]


Reduction at the integrand level: OPP

Study decomposition of the integrand [Ossola,Papadopoulos,Pittau ‘08]

put internal legs on-shell products of on-shell amplitudes


Reduction at the integrand level: OPP

coefficients of the scalar integrals are computed from products of on-shell amplitudes


Rational parts

Doing the cuts in 4 dimension does not produce the rational parts

Different methods to obtain rational parts:

Recursion working in two different integer dimensions specific Feynman rules SUSY + massive complex scalar

[Bern, Dixon, Dunbar, Kosower]No rational parts in N=4 SUSY:


Codes

Rocket [Giele, Zanderighi]

Blackhat / Whitehat? [Berger et al]

Helac-1Loop Cuttools Samurai

private codes

publicly available,additional input requiredto calculate scatteing amplitudes

[Bevilaqua et al]

[Ossola, Papadopoulos, Pittau]

[Mastrolia et al]


NGluon 1.0

Publicly available code to calculate one-loop amplitudes inpure gauge theory without further input for the amplitudes

Available from: http://www.physik.hu-berlin.de/pep/tools

Required user input:

number of gluons momenta helicities

External libraries: QD [Bailey et al], FF/QCDLoop [Oldenborgh, Ellis,Zanderighi]

[Badger, Biedermann,PU ’10]


Some technical remarks

Written in C++, however only very limited use of object oriented

Operator overloading is used to allow extended floating point arithmetic i.e. double-double (real*16), quad-double (real*32) using qd

Extended precision via preprocessor macros instead of templates

Scalar one-loop integrals from FF [Oldenborgh] and QCDLoop [Ellis,Zanderighi]

Entire code encapsulated in class NGluon NGluon itself thread save, however QCDLoop, FF

most likely not


Checks

Comparison with known IR structure

Comparison with known UV structure

Analytic formulae for specific cases

Collinear and Soft limits

test of linear combination of some triangle and box integrals

test of linear combination of bubble integrals

test of entire result

powerful test, however only applicable in soft and collinear regions of the phase space


Scaling test

IR and UV check always possible, however no direct test of the finite part

Comparison with analytic results of limited use

Independent method to assess the numerical uncertainty:

Scaling testBasic idea:

in massive theories masses needs to be rescaled as well, renormalization scale needs also to be rescaled

higher contributions in DFT not easy to interpret


Scaling test

Scaling can be checked numerically i.e. we calculate the same phase space point twice

How can we learn something from this test ?For

the mantissa of all rescaled floating point numbers will become different different arithmetics at the hardware level different rounding errors

results will differ in digits which are numerically out of control


Scaling test

Remark:

test is not cheap: doubles runtime, however it gives reliable estimate of the numerical uncertainty,

for cases where no analytic results are available

In practical applications test should be used if: high reliability is requested (“luxury level”) previous (cheaper tests) indicate problems

may help saving runtime


Scaling test


Results: Numerical stability / accuracy

~ number of valid digits


Average accuracy


Bad point: bad points

rule of thumb: adding one gluon doubles the fraction of bad points


Comparison with Giele, Kunszt, Melnikov

./NGluon-demo --GKMcheck


Comparison with Giele & Zanderighi

./NGluon-demo --GZcheck


Results: Runtime measurements

no ‘tuned’ comparison done so far with competitors


Improved scaling

[Giele, Zanderighi][Badger, Biedermann, PU]


Comparison with proposal Ax2 as of 11/2009

What happened to the “Helac-1Loop” version announced for spring 2010?

achieved for“limited field” content


Summary

NGluon allows the numerical evaluation of one-loop pure gluon amplitudes without additional input

Publicly available www.physik.hu-berlin.de/pep/tools

Improved scaling behavior Fast and stable (12-14 gluons) can compete with other private codes

Can be used as framework for further developments

Outlook: add massless quarks (internal/external) add massive quarks

the harmonic oscillator of one-loop calculations

Documents

ngluon amplitudes

reconstruction of amplitudes

color decompositiondespite

eva of color

amplitudesuse color

shell products

leadingcolor structure

shell amplitudesreduction