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SFB meeting, 09.12.2010 – 10.12.2010, Karlsruhe. The Harmonic Oscillator of One-loop Calculations. B5. Peter Uwer. Work done in collaboration with Simon Badger and Benedikt Biedermann. arXiv 1011.2900, http://www.physik.hu-berlin.de/pep/tools. Motivation. - PowerPoint PPT Presentation
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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
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 3
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]
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 4
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:
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 5
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
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 6
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
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 7
Nicer than Feynman diagrams: Recursion
= +
External wave functions, Polarization vectors
[Berends, Giele 89]colour ordered vertices
off shell leg
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 8
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:
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 9
Born amplitudes via recursionscalculation
i
j
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 10
Born amplitudes via recursionscalculation
i
j
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 11
Tree amplitudes from Berends-Giele recursion
[Biedermann, Bratanov, PU]
not yet fully optimized
checked with analytically known MHV amplitudes
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 12
Color-ordered sub-amplitudes (NLO)
Leading-color amplitudes are sufficient to reconstruct the full amplitude
Color structures:
Leading-color structure:
[?]
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 13
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
=
×
×
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 14
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]
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 15
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
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 16
Reduction at the integrand level: OPP
coefficients of the scalar integrals are computed from products of on-shell amplitudes
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 17
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:
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 18
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]
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 19
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]
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 20
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
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 21
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
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 22
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
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 23
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
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 24
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
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 25
Scaling test
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 26
Results: Numerical stability / accuracy
~ number of valid digits
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 27
Results: Numerical stability / accuracy
~ number of valid digits
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 28
Average accuracy
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 29
Bad point: bad points
rule of thumb: adding one gluon doubles the fraction of bad points
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 30
Comparison with Giele, Kunszt, Melnikov
./NGluon-demo --GKMcheck
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 31
Comparison with Giele & Zanderighi
./NGluon-demo --GZcheck
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 32
Results: Runtime measurements
no ‘tuned’ comparison done so far with competitors
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 33
Improved scaling
[Giele, Zanderighi][Badger, Biedermann, PU]
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 34
Comparison with proposal Ax2 as of 11/2009
What happened to the “Helac-1Loop” version announced for spring 2010?
achieved for“limited field” content
Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 35
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