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Prospects for the Prospects for the Future:Future:
The Message from The Message from AnalyticityAnalyticityHigh Precision for Hard Processes at the LHCHigh Precision for Hard Processes at the LHC
Zurich, 2006Zurich, 2006
Zvi Bern, UCLAZvi Bern, UCLA
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LHC PhysicsLHC Physics
The LHC will start operations in 2007!
We are all extremely excited by this.
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OutlineOutline
• A few comments on the impressive progress we have heard about in this conference.
• Mostly I wish to discuss scattering amplitudes and the hope the newly uncovered structures and simplicity gives for the future.
• The challenges awaiting us.
This will not be a traditional summary talk.
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Summary of Recent ProgressSummary of Recent Progress
• NLO -- Explicit results. Applications to QCD and Electroweak -- new theoretical understanding
• Parton showering Monte Carlos and resummation -- merging with matrix elements
• NNLO -- progress towards general differential cross sections -- splitting functions and DIS
• Theoretical issues -- Consistent treatment of decay widths -- Applications to gravity
Areas of progress we heard about at this conference:
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NLONLO• Traditional analytic with various improvements: talks from Denner, Oleari, Spira, Uwer and Zeppenfeld• Semi-numerical talks from Binoth, Dittmaier, Pittau, Weinzierl, Zanderighi• Pure numerical talk from Catani, Passarino and Soper• On-shell methods talks from Brandhuber, Britto, Dunbar, Feng, Forde, Glover, Kosower, Travaglini• Hybrid methods talk from Zhu and Papadopoulos• Construction of Cross Sections talk from Catani
New milestones: In QCD, one-loop six gluon amplitudes with all helicities and n gluon amplitudes with 2 and 3 negative helicities;In electroweak
“Best” approach probablywill combine a number of these ideas plus new ones.
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To open a bottle perform complete six and seven point calculations.
Which is the best bottle?Which is the best bottle?
PassarinoVeltman Semi-
Numerical ImprovedTraditional
Numerical On Shell
New Ideas?
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NNLONNLOImpressive progress for fully differential processes:
• Subtraction or Antenna approach. Drell-Yan and jets discussed. talks from Gehrmann, Kilgore, Del Duca, Somogyi • Sector decomposition approach. no talk but lots of progress: Binoth, Heinrich; Anastasiou, Melnikov and Petriello.
• NNLO PDF’s and DIS Coefficient Functions Talks from Marchesini, Moch, Stirling
PDF’s and DIS
Improved understanding of IR singularitiesTalks from Dixon and Mitov
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Parton Showering or ResummationParton Showering or ResummationFor realistic prediction need parton showering or resummationNeed to import NLO matrix elements into this.
W+ W-: MC@NLO vs Resummation
Grazzini’stalk
Talks from de Florian, Giele, Grazzini, Nason, Webber
W+ W-: Parton Shower vs NLO
Webber’stalk
MC@NLO: Good features of parton showers merged withgood features of NLO.
Heard new ideas formerging NLO with parton showers.
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The quest for simplicityThe quest for simplicity
We all know that loop computations are a serious and complicated business.
Might amplitudes in general be a lot simpler than we thought?
In this talk I want to argue that in general amplitudesare much simpler than we thought, though more needsto be done to fully expose the simplicity.
This seems hard to believe given the factorial explosions we normally face.
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NLO QCD: where we are nowNLO QCD: where we are now
We are only now starting to achieve six point calculations
Typical examples of current calculations:
Electroweak 4 fcross-sections
Denner, Dittmaier, Roth and Wieder
Bern, Dixon and KosowerDixon, Kunszt and SignerCampbell and Ellis
Dittmaier, Uwer, Weinzierl
Six gluon amplitudesZB, Dixon, Dunbar and KosowerBritto, Cachazo, FengBidder, Bjerrum-Bohr, Dixon, DunbarBedford, Brandhuber, Spence and Travaglini ZB, Bjerrum-Bohr, Dunbar, Ita Ellis, Giele and ZanderighiXiao, Yang and Zhu
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Les Houches NLO WishlistLes Houches NLO Wishlist
Bold action required!
Les Houches 2005
Talks fromDissertori andHuston
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Example: Susy SearchExample: Susy Search
Early ATLAS TDR studies using PYTHIA overly optimistic.
ALPGEN vs PYTHIA
• ALPGEN is based on LO matrix elements and much better at modeling hard jets.
• What will disagreement between ALPGEN and data mean? Hard to tell. Need NLO.
An important task for people in this room is to figure out how to actually compute this.
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What we needWhat we need
• Numerical stability.• Scalable to 6 and 7 external partons.• A general solution that applies to any process.• Automation.
What we’re dreaming ofWhat we’re dreaming of• Modest growth in complexity with increasing
number of legs.• “Compact” analytic expressions.
We heard a number of analytic and numerical proposalsfor trying to do so.
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Where is the simplicity?
This is from a 4-point evaluation which is very simple to do by today’s standards.
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Why are Feynman diagrams clumsy for high-multiplicity processes?
• Vertices and propagators involve
gauge-dependent off-shell states.
Origin of the complexity.
• All steps should be in terms of gauge invariant on-shell states. • To fully expose the simplicity a radical rewriting of perturbative QCD is needed.
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“One of the most remarkable discoveries
in elementary particle physics has been
that of the existence of the complex
plane.” J. Schwinger in “Particles, Sources and Fields” Vol 1
• We saw a beautiful application of this comment in Denner’s talk – complex mass scheme for unstable particles.• A second application are on-shell methods.
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Spinors and TwistorsSpinors and Twistors
Spinor helicity for gluon polarizations in QCD:
Penrose Twistor Transform:
Witten’s remarkable twistor-space link:
Scattering amplitudes Topological String Theory
Early work from Nair
Witten; Roiban, Spradlin and Volovich
Witten
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Remarkable SimplicityRemarkable Simplicity
Witten conjectured that in twistor–space gauge theoryamplitudes have delta-function support on curves of degree:
Connected picture
Disconnected picture
Structures imply an amazing simplicity in the scattering amplitudes. Massless gauge theory tree amplitudes are much simpler than anyone imagined.
WittenRoiban, Spradlin and VolovichCachazo, Svrcek and WittenGukov, Motl and NeitzkeBena Bern and Kosower
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MHV AmplitudesMHV Amplitudes
At tree level Parke and Taylor conjectured a very simple form for n-gluon scattering.
+ +
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Proven by Berends and Giele
Parke and Taylor (1984)
In twistor space this is represented by a straight line
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MHV VerticesMHV Vertices
These MHV amplitudes can be thought of as vertices for building new amplitudes.
Cachazo, Svrcek and Witten
momentum spacetwistor space
Define
EquivalentlyKosower
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Six gluon example
A “nifty” or “numerically ready” calculation
Kunszt via susy
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““Nifty” CalculationsNifty” CalculationsSuppose you know all the amplitudes up to n-1 points,if you can obtain the n-point amplitude by drawing somepictures and writing down the answer we define itto be a “nifty” calculation.
(This is not the same as the maximally efficient calculation.) Weinziel’s talk
“Nifty” calculations exhibit the newly uncovered structures and simplicity.
Very nice, but: What about higher points?What about masses?What about one loop?What about massive loops?What about higher loops?
We shall consider eachof these questions in turn.
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Britto, Cachazo, Feng
BCF + Witten
An
Ak+1
An-k+1
Tree-Level On-Shell RecursionTree-Level On-Shell Recursion
New representations of tree amplitudes from IR consistency of one-loop amplitudes in N = 4 super-Yang-Mills theory.
Using intuition from twistors and generalized unitarity:
Bern, Del Duca, Dixon, Kosower;Roiban, Spradlin, Volovich
Proof relies on so little. Power comes from generality• Cauchy’s theorem• Basic field theory factorization properties
On-shell conditions maintained by shift.
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• Simple processes with masses looked at – no problem in principle for going on.
• If you wish to go more than one level in the recursion then should clean up before inserting in the next level.
Tree-Level On-Shell RecursionTree-Level On-Shell Recursion
Helicities or statesPartitions of legs separatinglegs j and l
Frozen value of the shift
All you do write down the term corresponding to the diagramand numerically insert the shifted (complex) momenta -- “nifty” calculation.
Badger, Glover, Khoze and Svrcek; Forde and Kosower
Talk from Forde
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One LoopOne LoopAt one loop there are only a limited number of nearly “nifty” complete calculations of amplitudes.However all one-loop calculations have important piecesthat can be calculated in a “nifty” way:
Near D=4 any one-loop amplitude can be expressed as a sumover box, triangle, bubble and rational function contributions
Box coefficients can be obtained in a “nifty” way.
+ rational
I want to discuss box coefficients and then show you a “nearlynifty” calculation of a full amplitude.
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Unitarity MethodUnitarity Method
Two-particle cut:
Application of generalized unitarity:
Three- particle cut:
Generalized cut interpreted as cut propagators not canceling.
A number of recent refinements to method discussed in this conference:
Bern, Dixon, Dunbar and Kosower
Bern, Dixon and Kosower
Talks from Britto, Feng, Forde, Kosower, Pittau
“Unitarity method” turns unitarity into a practical method for obtaining complete amplitudes with an arbitrary number of kinematic variables. Completely equivalent to Feynman diagram results.
The statement that box coefficients are simple is best understoodin the context of the unitarity method.
Used to obtain (Matrix elements implemented in MCFM)
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Quadruple CutQuadruple Cut
As observed by Britto, Cachazo and Feng quadruple cut freezes integral: hep-th/0412104
What about bubbles and triangles and rational terms? Not “nifty” but singificant movement in this direction. Talks from Britto, Feng, Forde, Kosower and Pittau
Consider massless case:
4 integrals and 4 delta functions
Works very nicely even when some Ki2 vanish.
etc
Box coefficient nifty:
Solve system of equations:
}
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One Loop
The existence of such twistor structures connected with loop-level simplicity of box coefficients in massless case
At one-loop the coefficients of all box integral functions in anymassless gauge theories have beautiful twistor space interpretation
Twistor space supportBox integral
Bern, Del Duca, Dixon and KosowerBritto, Cachazo and Feng
Three negativehelicities
Four negativehelicities
• Twistor structure of complete amplitudes not mapped out.• Higher loops not mapped out at all.
Profound implicationrestricting box coefficients
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Massive LoopsMassive Loops
What about massive loops?
Much less is understood about the analytic structure of this case. Nevertheless, quadruple cut is frozen.
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3
4
Again solve the system and plug into product of trees
See hep-th/0506068 for a simple example with a uniform mass in the loop. Brandhuber, McNamara, Spence and Travaglini
Pittau’s talk discussed a reformulation of this in a more traditional framework.
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Example at One LoopExample at One Loop Can we any compute complete amplitudes at one loop in a nifty way?
Used to determine all log terms for split helicity configuration
ZB, Bjerrum-Bohr, Dunbar, Ita
If all loop momentum dependent poles are unmodified by the z shift an on-shell recursion determines the coefficients rather straightforwardly.
All coefficients of boxes, triangles and bubbles obtainable in a “nifty” way for QCD split helicity configuration.
• Consider 6 point:Scalar loop
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There is still much to do to get to generic “nearly nifty” calculations, but we heard significant progress from Binoth, Britto, Brandhuber, Dunbar, Feng, Kosower, Forde, Glover, Papadopoulos, Pittau, Travaglini and Zhu.
Berger, ZB, Dixon, Forde and Kosower
The rational function can also be obtained from on-shell recursion
Not quite “nifty”.Also overlap contributionsobtained from extractingresidues.
All terms “nearly nifty”
Talks from Kosower and Forde
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Two Loops: Where is the simplicity?Two Loops: Where is the simplicity?Typical example of 2 loop amplitude: From ZB, De Freitas and Dixon
Similar results fromAnastasiou, Glover, Oleari and Tejeda-Yeomans
+ another 15 pages
This definitely doesnot look simple.
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Where is the simplicity?Where is the simplicity?
We have a long way to go to uncover structures and“simplicity” at two loops and beyond.
1) What can twistor space tell us? I don’t have much to say here, but someone should study this.
2) Are there any hints of simplicity?
3) Trouble with the integrals.
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A Two-Loop Hint A Two-Loop Hint Consider the four gluon all-positive helicity amplitude in QCD.This is the simplest example. If we can’t find simplicity herethere is no hope for any other QCD amplitudes.
Why do the planar and non-planar double boxes look the same? I believe this is a clue.
If you expand it in polylogs it is some moderate mess.Instead let’s write it in a special basis of integrals
planar
non-planar
p q qp qp
Bern, Dixon, Kosower hep-th/0001001
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IntegrationIntegrationAt two-loops and beyond there are serious roadblocks to fullyunraveling the structure of amplitudes.
• Laporta algorithm can solve for a given set of integrals, but we don’t know ahead of time what the basis of master integrals will be. Output basis can look a very different depending on choices. (Smirnov)2 have a proposal for systematizing the basis choice – Groebner basis.
• Fantastic new tool: publicly available MB package from Czakon for (numerically) evaluating loop integrals. (Similar program from Anastasiou and Daleo, but not public.)
• Rather non-trivial issues with construction of cross-sections. talks from Del Duca, Gehrmann, Kilgore, Somogyi
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The Road AheadThe Road Ahead
• Attack NLO calculations on experimenters’ wishlist.
• Automation for general processes.
• Assembly of full cross-sections, e.g., Catani-Seymour formalism or improved versions
• Inclusion of more matrix elements into MC@NLO, Vincia, Powheg, etc.
• NNLO: Finish jets. jets.
jet. Improved PDF’s and fragmentation.
This is a difficult road
What do we need to do to make this happen?
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Galileo Galilei Institute: Physics ChallengeGalileo Galilei Institute: Physics Challenge
GGI Physics Challenges:• Jet observable at NNLO; Average thrust for jets.
• All one loop amplitudes for pp 4 jets pp W + 3 jets.
• Full one-loop top production with decays folded in. Unstable particles within loop.
• Evaluator for higher loop integrals. Program or web page where you feed kinematics, get back a number. Compilation of known results.
See the organizers: Brandhuber, Del Duca, Glover, Kosower,Passarino, Spence, Travaglini, Zeppenfeld Aug 27-Oct 26, 2007
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More GGI ChallengesMore GGI Challenges• at one loop.
• Parton showers. At LO merge with W + 3 partons At NLO merge with W + 1 and W + 2 partons • Produce automated program to construct IR subtraction counterterms for evaluating cross sections. Plug in color ordered amplitudes out comes cross sections. Plug and play with standard interface. Make it flexible to add new physics.
• Electroweak corrections to W + jet production
• Two-loop renormalization of electroweak Lagrangian in the complex pole (mass) scheme
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SummarySummary
Get the job done in time for LHC data
• We need energetic action to provide the
full range of precision calculations for the LHC.
• Lots of great progress described in this conference.
• Many concrete new results.
• Many important issues remain. GGI Challenge.
