One-Loop Multi-Parton Amplitudes for The LHC. Harald Ita, UCLA Based on: arXiv:0803.4180 ; arXiv:0807.3705; arXiv:0808.0941 In collaboration with: Carola Berger, Zvi Bern, Fernando Febres Cordero, Lance Dixon, Darren Forde, Daniel Maitre and David Kosower. Recently joined by Tanju Gleisberg

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

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

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) MC@NLO,POWHEG-method; Frixione, Webber; +Nason, Ridolfi, Oleari high-multiplicities + automation

Tevatron: Single Top Production T. Aaltonen et al. [CDF Collaboration], arXiv:08092581 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.

the LHC: an example of discovery M L Mangano [arXiv:0809.1567] 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

How good are our tools? Wanted: LHC studies with extra jets: LO, NLO. T. Aaltonen et al. [CDF Collaboration], arXiv:0711.4044 Wanted: LHC studies with extra jets: + . . . SMPR-model: Mrena, Richardson, 2004. MLM-model: Alwall et al. arXiv:0706.2569. LO, MCFM; parton level; including Bern, Dixon, Kosower, Weinzierl matrix elements. NLO.

Les Houches 2007: The Wish List And beyond…!

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),

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

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

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.

2. On-Shell Methods.

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

Reminder: one-loop basis. See Bern, Dixon, Dunbar, Kosower, hep-ph/9212308. 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

Unitarity Method. Unitarity Approach: Bern, Dixon, Dunbar, Kosower, hep-ph/9403226, hep-th/9409265. Recent Advances using spinorial integration techniques: Cachazo, Svrcek, Witten; Britto, Feng, Cachazo; Britto, Feng, Mastrolia Generalized Unitarity: Bern, Dixon, Kosower, hep-ph/9708239, hep-ph/0001001. Britto, Cachazo, Feng, hep-th/0412103. Recent Advances: classification of surface terms. del Aguila and Pittau, hep-ph/0404120. Ossola, Papadopoulos and Pittau, hep-ph/0609007. Forde, 0704.1835; Badger, 0806.4600, 0807.1245 . Ellis, Giele, Kunszt, 0708.2398; Giele, Kunszt and Melnikov, 0801.2237; Ellis, Giele, Kunszt, Melnikov, 0806.3467;

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

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)

Boxes: the simplest cuts. Berger, Bern, Dixon, Febres Cordero, Forde, H.I., Kosower, Maitre 0803.4180; Risager 0804.3310. Un-physical (=spurious) singularities from parameterization. Have to cancel eventually: role of rational term R.

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.

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:

Rational Terms. Loop-level on-shell recursion. D-dim unitarity. Used here! Loop-level on-shell recursion. Bern, Dixon, Kosower, hep-th/0501240,hep-ph/0505055, Berger, Del Duca, Dixon, hep-ph/0608180, Forde, Kosower, hep-ph/0509358, Berger, Bern, Dixon, Forde, Kosower, hep-ph/0607014 Bern, Bjerrum-Bohr, Dunbar, H.I., hep-ph/0507019. D-dim unitarity. Bern, Morgan, hep-ph/9511336; Bern, Dixon, Kosower, hep-ph/9708239; Anastasiou et al., hep-th/0609191, hep-th/0612277, Britto, Feng, hep-ph/0612089, 0711.4284; Britto, Feng, Mastrolia, 0803.1989; Britto, Feng, Yang, 0803.3147; Ossola, Papadopolous, Pittau, 0802.1876; Mastrolia, Ossola, Papadopolous, Pittau, 0803.3964; Giele, Kunszt, Melnikov, 0801.2237; Ellis, Giele, Kunszt, Melnikov, 0806.3467; Badger, 0806.4600;

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

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

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.

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

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

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

Amplitudes From BlackHat.

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)

Gluon amplitudes: The Tails. Double-precision numerical computation. Dynamic multi-precision computation. Reference: analytic targets from Bern, Dixon, Dunbar, Kosower, hep-ph/9403226, hep-ph/9409265, hep-ph/0507005. Natural tail Natural tail Multi- precision (III) Multi- precision 100 000 PS points, ET>0.01 s, pseudo rapidity<3, separation cut >0.4

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.

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…

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

Progress Report.

Progress Report. Berger, Bern, Dixon, Febres Cordero, Forde, HI, Kosower, Maitre, arXiv:0808.0941. 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

Color Structure. Color dressed gggqqV amplitude. Suppress fermion loop see also: Bern, Dixon, Kosower: hep-ph/9708239, hep-ph/9409393. Color dressed gggqqV amplitude. Suppress fermion loop contributions with axial/vect. coupling. V-boson coupling: +…

Z+3jets: Stability Study Berger, Bern, Dixon, FFC, Forde, Ita, Kosower, Maître, arXiv:0808.0941[hep-ph] 100 000 PS points, ET>0.01 s, pseudo rapidity<3, separation cut >0.4 October 08 38

Timing: Z+3 jets. Double precision Dynamic multi precision Helicity: +++: 4.01ms/4.06ms 10.6ms/10.7ms (now: ~4ms) -++: 6.41ms/6.64ms 41.7ms/42.9ms ++-: 6.47ms/6.67ms 28.0ms/28.4ms (now~6.5 ms) -+-: 7.70ms/7.92ms 39.2ms/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

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!

EXTRA SLIDES.

EXTRA SLIDES.