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QCD at the LHC: What needs to be done?. Part 2: Higher Order QCD. West Coast LHC Meeting Zvi Bern, UCLA. Outline. General overview Examples of importance of higher order QCD Experimenters’ wish lists What is the problem? Evils of unphysical formalisms.

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Qcd at the lhc what needs to be done

QCD at the LHC:What needs to be done?

Part 2: Higher Order QCD

West Coast LHC Meeting

Zvi Bern, UCLA


  • General overview

  • Examples of importance of higher order QCD

  • Experimenters’ wish lists

  • What is the problem? Evils of unphysical formalisms.

  • The S-matrix reloaded: unitarity, twistors, and recursion.

  • Recent calculations and promise for the future

  • What needs to be done.


QCD at hadron collider involves a number of complex


  • Parton distribution functions

  • Parton Showers

  • Monte Carlos

  • Underlying Events

  • Hadronization

  • Resummation

  • Higher order QCD

  • very definite calculations need to be done.

Steve Ellis’ talk

Example higgs 2 jets from weak boson fusion
Example:Higgs + 2 jets from Weak Boson Fusion

Purpose: After discovery of Higgs Boson measure HWW coupling

Background uncertainty can be reduced with an NLO calculation.

Example susy search
Example: Susy Search

  • Early studies using PYTHIA

  • over optimistic.


  • PYTHIA does not properly

  • model hard jets.

  • ALPGEN is based on LO

  • matrix elements and is better

  • at modeling hard jets.

  • What will disagreement between

  • ALPGEN and data mean? Hard

  • to tell. Need NLO.

Merging nlo with parton showers
Merging NLO with Parton Showers

It is important to merge NLO with parton showering.

  • Soft and collinear emission properly treated with parton showers. Standard tool for experimenters.

  • Hard emission treated properly by NLO. Standard tool for theorists


First example of

merging NLO with shower Monte Carlo

See Dave Soper

The gold standard nnlo drell yan rapidity distributions
The Gold Standard: NNLO Drell-Yan Rapidity Distributions

  • Amazingly good stabilty

  • Theoretical uncertainties less than 1%

What needs to be done at nlo

Experimenters to theorists:

“Please calculate the following at NLO”

What needs to be done at NLO?

Theorists to experimenters:

“In your dreams”

More realistic experimenter s wish list
More Realistic Experimenter’s Wish List

Les Houches 2005

Bold action is required even for this

State of the art nlo qcd
State- of-the-Art NLO QCD

Five point is still state-of-the art in QCD:


Brute force calculations give GB expressions – numerical stability?

Amusing numbers: 6g: 10,860 diagrams, 7g: 168,925 diagrams

Much worse difficulty: integral reduction generates nasty determinants

It is time to dream
It Is Time to Dream

To attack the wish list need new ideas:

  • Numerical approaches.

    Promising recent progress.

  • Analytic on-shell methods: unitarity method, on-shell recursion, bootstrap approach

Binoth and Heinrich Kaur; Giele, Glover, Zanderighi

Binoth, Guillet, Heinrich, Pilon, Schubert;

Soper and Nagy; Ellis, Giele and Zanderighi;

Anastasiou and Daleo; Czakon;

Binoth, Heinrich and Ciccolini

Bern, Dixon, Dunbar, Kosower; Bern and Morgan; Cachazo, Svrcek and Witten;

Bern, Dixon, Kosower;

Bedford, Brandhuber, Spence, Travaglini;

Bern, Dixon, Del Duca and Kosower;

Britto, Cachazo, Feng and Witten;

Berger, Bern, Dixon, Kosower, Forde

Why are feynman diagrams clumsy for multi parton processes
Why are Feynman diagrams clumsy for multi-parton processes?

  • The vertices and propagators involve

    gauge-dependent off-shell states.

    This is the origin of the complexity.

  • To solve the problem we should rewrite perturbative quantum field theory.

  • All steps should be in term of gauge invariant

  • on-shell states.

  • Radical rewriting of perturbative expansion needed.

On shell formalisms
On-shell Formalisms

  • Curiously, an on-shell formalism was constructed at loop level prior to trees: unitarity method. (1994)

  • Solution at tree-level had to await Witten’s twistor inspiration. (2004)

    -- MHV vertices

    -- on-shell recursion

  • Combining both give one-loop on-shell bootstrap


Bern, Dixon, Dunbar, Kosower

Bern and Morgan

Cachazo, Svrcek Witten

Britto, Cachazo, Feng, Witten

Bern, Dixon, Kosower

Forde and Kosower;

Berger, Bern, Dixon, Forde amd Kosower

Spinors and twistors
Spinors and Twistors

Spinor helicity for gluon polarizations in QCD:

Penrose Twistor Transform:

Early work from Nair

Witten’s remarkable twistor-space link:

QCD scattering amplitudes Topological String Theory

Witten; Roiban, Spradlin and Volovich

Key implication: There are simple structure in gauge theory amplitudes

Amazing simplicity
Amazing Simplicity

Witten Conjectured that in twistor –space gauge theory

amplitudes should be supported on curves of degree:

Connected picture

Disconnected picture

These structures imply

an amazing simplicity

in the scattering amplitudes.

MHV vertices for

building amplitudes

Cachazo, Svrcek and Witten

Loop amplitudes

Bern, Dixon, Dunbar and Kosower (1994)

Bern and Morgran (1995)

Loop Amplitudes

  • Summary of results from early papers:

  • Key result: Any massless loop amplitude in any theory is fully determined from D-dimensional tree amplitudes and unitarity to all loop orders. Off-shell formulations are unnecessary.

  • Four-dimensional cut constructibility: At one-loop, any amplitude in a massless susy gauge theory is fully constructible from four-dimensional tree amplitudes (even in presence of IR and UV divergences). Use helicity.

  • One-loop QCD: If we use spinor helicity for the tree amplitudes we drop rational functions in loop amplitudes, but logs and polylogs all constructed correctly.

Unitarity method
Unitarity Method

Two-particle cut:

Three- particle cut:

Generalized triple cut:

Should be interpreted as demanding that cut propagators do not cancel.

Unitarity method combines very effectively with twistor-inspired ideas.

On shell bootstrap
On-Shell Bootstrap

Bern, Dixon, Kosower


Early Approach:

  • Use Unitarity Method with D = 4 helicity states. Efficient means

  • for obtaining logs and polylogs. Build from on-shell tree amplitudes.

  • Use factorization properties to find rational function part.

  • Check numerically against Feynman diagrams

Key problems preventing widespread applications:

  • Difficult to find rational functions with desired factorization properties.

  • Unclear how to automate.

Qcd at the lhc what needs to be done

Tree-Level On-Shell Recursion




New representations of tree amplitudes from IR consistency of one-loop amplitudes in N = 4 super-Yang-Mills theory.

Bern, Del Duca, Dixon, Kosower;

Roiban, Spradlin, Volovich

With intution from twistors and generalized unitarity:

Britto, Cachazo, Feng

on-shell recursion

  • On rhs only on-shell tree amplitudes with fewer legs appear.

  • Evaluate with momenta shifted by a complex amount

Simple proof of on shell recursion
Simple Proof of On-Shell Recursion

Britto, Cachazo, Feng and Witten

Consider shifted amplitude :

At tree level we know all the residues:

  • Proof relies on so little:

  • Cauchy’s theorem

  • Basic field theory factorization properties

Merging unitarity with loop level recursion
Merging Unitarity With Loop-Level Recursion

Bern, Dixon Kosower;

Forde and Kosower;

Berger, Bern, Dixon, Forde, Kosower,

  • New Features:

  • Presence of branch cuts.

  • unreal poles – poles which appear only for complex momenta.

  • double poles – S-matrices in general have double poles

  • Spurious singularities that cancel only against polylogs. Add

  • rational functions to remove these.

  • Double counts between cuts and recursion. These result in

  • overlap diagrams.

Pure phase for real momenta

Five point example
Five-point example

Assume we already have log terms computed from D = 4 cuts.

The most challenging part was rational function terms.

Only one non-vanishing recursive diagram:

Only two overlap diagrams:

The rational function terms are as easy to get as at tree level!

Six point example

What needs to be done
What needs to be done?

  • Firmer theoretical basis for formalism is needed:

    Large z behavior of loop amplitudes.

    General understanding of unreal poles.

    Complex factorization of amplitudes.

  • Attack experimenters’ wishlist.

  • Massive loops -- tree recursion understood.

  • Connection to Lagrangian – Space-cone gauge.

  • Improved evaluation of triangles or bubble integrals would be helpful.

  • Assembly of full cross-sections. Catani-Seymour dipole method.

  • Automation for general processes.

Carola Berger’s presentation

Badger, Glover, Khoze, Svrcek

Chalmers and Siegel; Vaman and Yao

Many theoretical and practical aspects

Other applications

  • Resummation of MHV N = 4 super-Yang-Mills amplitudes to all loop

  • orders.

  • UV Finiteness properties of N = 8 supergravity

  • — Definitely less divergent than people had thought.

  • —Is it finite, contrary the accepted wisdom?

  • We have the technology to find out!

Bern, Rozowsky, Yan; Anastasiou,Bern, Dixon, Kosower;

Bern, Dixon, Smirnov; Buchbinder and Cacazho; Cachazo, Spradlin,Volovich

Bern, Dixon,Dunbar,Perelstein,Rozowsky;

Howe and Stelle; Bern, Bjerrum-Bohr, Dunbar

Other Applications

A method is even more important than a discovery, since

the right method will lead to new and even more important


— L.D. Landau

On-shell method have been applied to a variety of problems.


Qcd at the lhc what needs to be done


  • Analysis of LHC experiments involve complex issues in QCD.

  • Higher order QCD is a key issue facing us.

  • Conventional approaches have failed to provide full range of

  • desired calculations – time for bold action at NLO.

  • New methods

  • (a) numerical approaches

  • (b) on-shell methods – reformulation of quantum field theory

  • New results for six partons and n-partons

  • Much more needs to be done to set up formalism, automation, and

  • construction of physical cross-sections for comparison to data.

Experimenters’ wish list awaits us.