1. One-Loop Multi-Parton Amplitudes for The LHC.
Harald Ita, UCLA
Based on:
arXiv: ; arXiv: ; arXiv:
In collaboration with:
Carola Berger, Zvi Bern, Fernando Febres Cordero, Lance Dixon, Darren Forde, Daniel Maitre and David Kosower. Recently joined by Tanju Gleisberg

2. Content: Motivation: On-Shell Methods. Amplitudes from BlackHat.
NLO S-Matrix Elements.
On-Shell Approach.
On-Shell Methods.
Amplitudes from BlackHat.
Progress Report.
Conclusions.

3. Scattering processes at hadron colliders: A multi-layered problem
Understanding the underlying event
Control over parton distribution functions
The hard scattering LO
NLO (Virtual <- BlackHat, Real)
Beyond
Showering (matching)
Hadronization
taken from Rick Field

4. Used set of Tools… Parton-level LO matrix element generators:
MADGRAPH; Maltoni, Stelzer …
LO ME + shower MCs + …:
ALPGEN; Mangano, Moretti, Piccinini, Pittau, Polosa
HERWIG; Marchesini, Webber, Abbiendi, Corcella, Knowles, Moretti, Odagiri, Richardson, Seymour, Stanco
PYTHIA; Sjostrand, Mrenna, Skands
SHERPA; Gleisberg, Hoeche, Krauss, Schoenherr, Schumann, Siegert, Winter
NLO PL matrix element generators:
MCFM; Campbell, Ellis; (max 6 partons)
(BlackHat …,6,7,..(?) partons)
Frixione, Webber; +Nason, Ridolfi, Oleari
high-multiplicities + automation

5. Tevatron: Single Top Production
T. Aaltonen et al. [CDF Collaboration], arXiv:
single top production
Matrix element method uses full information of LO matrix
elements to pull the signal out of background.
It should be possible to do better by using NLO matrix elements.
A goal is to provide experimenters with necessary theoretical tools
for a wide variety of processes.

6. the LHC: an example of discovery
M L Mangano [arXiv: ]
Producing heavy colored particles
Backgrounds:
Irreducible:
Z(->neutrinos)+4 jets
Reducible:
W(->tau+neutrino)+3 jets
W(->undetected leptons)
+4 jets
top pairs
Instrumental:
Multijets
[ATLAS Collaboration]
S. Vahsen

7. How good are our tools? Wanted: LHC studies with extra jets: LO, NLO.
T. Aaltonen et al. [CDF Collaboration], arXiv:
Wanted: LHC studies with extra jets:
SMPR-model: Mrena, Richardson, 2004.
MLM-model: Alwall et al. arXiv:
LO,
MCFM; parton level; including Bern,
Dixon, Kosower, Weinzierl
matrix elements.
NLO.

8. Les Houches 2007: The Wish List
And beyond…!

9. What Has Been Done?
Most physics results done from Feynman diagram approach:
QCD corrections to vector boson pair production (W+W-, W±Z & ZZ) via vector boson fusion (VBF). (Jager, Oleari, Zeppenfeld)+(Bozzi)
QCD and EW corrections to Higgs production via VBF. (Ciccolini, Denner, Dittmaier)
pp → Higgs+2 jets. (via gluon fusion Campbell, Ellis, Zanderighi), (via weak interactions Ciccolini, Denner, Dittmaier). pp → Higgs+3 jets (leading contribution) (Figy, Hankele, Zeppenfeld).
pp→ (Beenakker, Dittmaier, Krämer, Plümper, Spira, Zerwas), (Dawson, Jackson, Reina, Wackeroth)
pp → ZZZ, (Lazopoulos, Petriello, Melnikov) pp → (McElmurry)
pp → WWZ, WWW (Hankele, Zeppenfeld, Campanario, Oleari, Prestel)
pp→WW+j+X. (Campbell, Ellis, Zanderighi). (Dittmaier, Kallweit, Uwer)
pp →W/Z (Febres Cordero, Reina, Wackeroth),
pp → +jet (Dittmaier,Uwer,Weinzierl),
(Bredenstein,Denner,Dittmaier,Pozzorini),

10. Economic techniques? Calculating using Feynman diagrams is Hard!
A Factorial growth in the number of terms, particularly bad for large number of gluons.
Calculated amplitudes simpler than expected.
For example, tree level all-multiplicity gluon amplitudes
Gauge dependant
quantities,
large cancellations
between terms.
6 gluons
~10,000 diagrams.
7 gluons
~150,000 diagrams.
Want to use on-shell
quantities only.
Park, Taylor 10

11. Think off-shell, work on-shell!
Vertices and propagators involve
unphysical gauge-dependent off-shell states.
Feynman diagram loops have to be off-shell
because they encode the uncertainty principle.
Keep particles on-shell in intermediate
steps of calculation, not in final results.
Want to reconstruct amplitude using only
on-shell information.
Fact: Off-shellness is essential for getting the correct answer.
Bern, Dixon, Dunbar, Kosower

12. On-shell methods: opening a gate to new computational possibilities...
Exploit universal physical properties for decomposition in terms of on-shell objects:
Unitarity relation.
Universal Factorization.
Complex momenta through spinor variables.

13. 2. On-Shell Methods.

14. What Has Been Done?
Past year progress using unitarity and related techniques,
gg → gggg amplitude. (Bern,Dixon,Kosower), (Britto,Feng,Mastrolia), (Bern,Bjerrum-Bohr,Dunbar,H.I.), (Berger,Bern,Dixon,Forde,Kosower), (Bedford,Brandhuber,Spence,Travaglini) (Xiao,Yang,Zhu) ,(Berger,Bern,Dixon,Forde,Kosower), (Giele,Kunszt,Melnikov)
Lots of gluons (Giele,Zanderighi), (Berger, Bern, Dixon, Febres Cordero, Forde,H.I., Kosower, Maître)
6 photons (Nagy, Soper), (Ossola, Papadopoulos, Pittau), (Binoth, Heinrich, Gehrmann, Mastrolia)
pp → ZZZ, WZZ, WWZ, ZZZ (Binoth, Ossola, Papadopoulos, Pittau),
gg → using D-Dimensional Unitarity (Ellis,Giele,Kunszt,Melnikov)
Numerical packages under construction:
BlackHat Berger, Bern, Dixon, Febres Cordero, Forde, H.I., Kosower, Maître
CutTools Ossola, Papadopoulos, Pittau
Rocket Ellis, Giele, Kunszt, Melnikov, Zanderighi

15. Reminder: one-loop basis.
See Bern, Dixon, Dunbar, Kosower, hep-ph/
All external momenta in D=4, loop momenta in D=4-2ε (dimensional regularization).
Cut Part from unitarity cuts in 4 dimensions.
Rational part from on-shell recurrence relations.
Rational part
Cut part
Process dependent D=4 rational integral coefficients

16. Unitarity Method. Unitarity Approach:
Bern, Dixon, Dunbar, Kosower, hep-ph/ , hep-th/
Recent Advances using spinorial integration techniques:
Cachazo, Svrcek, Witten; Britto, Feng, Cachazo; Britto, Feng, Mastrolia
Generalized Unitarity:
Bern, Dixon, Kosower, hep-ph/ , hep-ph/
Britto, Cachazo, Feng, hep-th/
Recent Advances: classification of surface terms.
del Aguila and Pittau, hep-ph/
Ossola, Papadopoulos and Pittau, hep-ph/
Forde, ; Badger, ,
Ellis, Giele, Kunszt, ; Giele, Kunszt and Melnikov, ; Ellis, Giele, Kunszt, Melnikov, ;

17. Unitarity: an on-shell method of calculation.
Bern, Dixon, Kosower
Cutting loops = sewing trees:
Sewing:
Cutting: 2x

18. Generalized Unitarity: isolate the leading discontinuity.
Cutting: n x
More cuts, more trees, less algebra:
Two-particle cut: product of trees contains subset of box-, triangle- and bubble-integrals. (Bern, Dixon, Kosower, Dunbar)
Triple-cut: product of three trees contains triangle- and box-integrals. (Bern, Dixon, Kosower)
Quadruple-cut: read out single box coefficient. (Britto, Cachazo, Feng)

19. Boxes: the simplest cuts.
Berger, Bern, Dixon, Febres Cordero, Forde,
H.I., Kosower, Maitre ; Risager
Un-physical (=spurious) singularities from parameterization.
Have to cancel eventually: role of rational term R.

20. Triangles: box-subtraction.
following Forde;
similar to: Ossala, Pittau, Papadopoulos;
Ellis, Giele, Kunszt
Un-physical (=spurious) singularities from parameterization.
Have to cancel eventually: role of rational term R.

21. Discrete Fourier Analysis.
See also Mastrolia, Ossola, Papadopoulos, Pittau.
Poles from factorization on box-propagators:
Subtract box contributions:
Expressions in terms of spherical harmonics for
triangle integrand:
Triangle coefficient from projection:

22. Rational Terms. Loop-level on-shell recursion. D-dim unitarity.
Used here!
Loop-level on-shell recursion.
Bern, Dixon, Kosower, hep-th/ ,hep-ph/ ,
Berger, Del Duca, Dixon, hep-ph/ ,
Forde, Kosower, hep-ph/ , Berger, Bern, Dixon, Forde, Kosower, hep-ph/
Bern, Bjerrum-Bohr, Dunbar, H.I., hep-ph/
D-dim unitarity.
Bern, Morgan, hep-ph/ ;
Bern, Dixon, Kosower, hep-ph/ ;
Anastasiou et al., hep-th/ , hep-th/ , Britto, Feng, hep-ph/ , ; Britto, Feng, Mastrolia, ; Britto, Feng, Yang, ;
Ossola, Papadopolous, Pittau, ; Mastrolia, Ossola, Papadopolous, Pittau, ;
Giele, Kunszt, Melnikov, ; Ellis, Giele, Kunszt, Melnikov, ;
Badger, ;

23. Trees in Complex Plane.
Britto, Cachazo, Feng, Witten
Translate rational amplitude into a function in the complex plane:
Shift the momentum of two external legs by a complex variable z,
Keeps both ka and kb on-shell.
Conserves momentum in the amplitude.
For example:
Only possible with
complex momenta. z
Pole at -<23>/<13> 23

24. Residues from Factorization.
Britto, Cachazo, Feng, Witten
Function of a complex variable containing only simple poles.
Position of all poles and residues from complex factorisation properties of the amplitude.
pole z
An(0), the amplitude with real momentum
Poles from here An
A<n 24

25. Rational terms in Complex Plane.
Role in QCD amplitudes:
Cancel spurious singularities of cut part in a way consistent with universal factorization.
R(z) rational in z with singularities from:
Factorization
Spurious poles
Challenges:
Factorization on unreal poles.
Large-z properties, “Inf”-terms.

26. Loop On-Shell Recursions.
Bern, Dixon, Kosower, Forde, Berger;
Bern, Bjerrum-Bohr, Dunbar, H.I.
At one-loop recursion using on-shell tree amplitudes, T, and rational pieces of one-loop amplitudes, R,
Sum over all factorisations.
“Inf” term from auxiliary recursion.
Not the complete rational result, missing “spurious” residues.
Can be done for integral coefficients, auxiliary recursions… Rn T R
+ spurious residues 26

27. Spurious box-poles: Residues of spurious poles of cuts equal and
opposite to residues of rational term R.
e.g.: A(1−, 2+, 3−, 4+, 5−, 6+)
Britto, Feng, Mastrolia

28. Numerical Spurious Pole Extraction.
Numerically extract spurious poles, use known pole locations.
Expand Integral functions at the location of the poles, Δi(z)=0, e.g.
The coefficient multiplying this is also a series in Δi(z)=0, e.g.
Numerically extract the 1/Δi poles via
BCFW-shift and discretized contour integrals.
Box or triangle
Gram determinant

29. Amplitudes From BlackHat.

30. BlackHat: A C++ implementation of on-shell techniques for 1-loop amplitudes
Portability (standard libraries for unix systems)
Modularity (object oriented)
Malleability (to accept several routines – numerics and analytics)
Numerical precision and efficiency
Ready to use with existing Monte Carlo programs
Work in progress with automated real dipole subtraction from Sherpa (with T. Gleisberg)

31. Gluon amplitudes: The Tails.
Double-precision numerical computation.
Dynamic multi-precision computation.
Reference: analytic targets from Bern, Dixon, Dunbar, Kosower, hep-ph/ , hep-ph/ , hep-ph/
Natural tail
Natural tail
Multi-
precision
(III)
Multi-
precision
PS points, ET>0.01 s,
pseudo rapidity<3, separation cut >0.4

32. Watch Instabilities.
Monitor using known IR/UV pole structure of amplitudes.
Generalization for rational part. (A consistency condition of spurious residues.)
Avoid instabilities with analytic tricks:
Use good loop momentum parametrizations & spinor variables and cancel Gram determinants.
Invert equations via discrete Fourier analysis.

33. BlackHat: a snapshot… Cross Sections Trees Trees Trees
One Loop Helicity Amplitude
Rational Part
Cut Part
boxes
triangles
bubbles
Trees
Recursive diagrams
Spurious poles
One Loop Helicity Amplitude
Rational Part
Cut Part
boxes
triangles
bubbles
Trees
Recursive diagrams
Spurious poles
One Loop Helicity Amplitude
Rational Part
Cut Part
boxes
triangles
bubbles
Trees
Recursive diagrams
Spurious poles
double
double double
quadruple double
Multiprecision arithmetic gives excellent control over numerical stability…

34. Scaling with number of legs
Berger, ZB, Dixon, Febres Cordero, Forde, Ita, Kosower, Maitre
2.33 GHz Xeon
6 gluons
7 gluons
8 gluons
8.3 ms/pt
14 ms/pt
43 ms/pt
amusing count
for 8 gluons
+ 3,017,489 Feynman diagrams

35. Progress Report.

36. Progress Report.
Berger, Bern, Dixon, Febres Cordero, Forde, HI,
Kosower, Maitre, arXiv:
With BlackHat: first leading color NLO matrix elements for:
Rely on color suppression.
Integration of virtual piece should be organized in color.
Ellis, Giele Kunszt, Melnikov, Zanderighi:
confirmed leading color and completed subleading color.
off-shell

37. Color Structure. Color dressed gggqqV amplitude. Suppress fermion loop
see also: Bern, Dixon, Kosower:
hep-ph/ , hep-ph/
Color dressed gggqqV amplitude.
Suppress fermion loop
contributions with axial/vect. coupling.
V-boson coupling: +…

38. Z+3jets: Stability Study
Berger, Bern, Dixon, FFC, Forde, Ita, Kosower,
Maître, arXiv: [hep-ph]
PS points, ET>0.01 s,
pseudo rapidity<3, separation cut >0.4
October 08 38

39. Timing: Z+3 jets. Double precision Dynamic multi precision
Helicity:
+++: 4.01ms/4.06ms ms/10.7ms (now: ~4ms)
-++: 6.41ms/6.64ms ms/42.9ms
++-: 6.47ms/6.67ms ms/28.4ms (now~6.5 ms)
-+-: 7.70ms/7.92ms ms/40.5ms
Double precision
Dynamic multi precision
Full amplitude
4-D cut-part
Same order as 6pt gluon amplitudes ~50ms: Quite an improvement compared to other numerical methods! 39

40. Conclusions
NLO QCD corrections to hard cross sections will be an important tool for LHC analyses.
On-shell methods have opened a new gate to computational power in QFTs.
BlackHat has proven good precision and scaling properties.
Hopefully soon, we expect first important phenomenological results, for example in V+3jets processes and beyond!

41. EXTRA SLIDES.

42. EXTRA SLIDES.