twistors and gauge theory desy theory workshop september 30 september 30, 2005
TRANSCRIPT
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The Storyline
• An exciting time in gauge-theory amplitude calculations
• Motivation for hard calculations
• Twistor-space ideas originating with Nair and Witten
• Explicit calculations led to seeing simple twistor-space structure
• Explicit calculations led to new on-shell recursion relations for trees
• Combined with another class of nonconventional techniques, the unitarity-based method for loop calculations, we are at the threshold of a revolution in loop calculations
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Precision Perturbative QCD
• Predictions of signals, signals+jets
• Predictions of backgrounds
• Measurement of luminosity
• Measurement of fundamental parameters (s, mt)
• Measurement of electroweak parameters
• Extraction of parton distributions — ingredients in any theoretical prediction
Everything at a hadron collider involves QCD
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A New Duality
Topological B-model string theory (twistor space)
N =4 supersymmetric gauge theory
Weak–weak duality
• Computation of scattering amplitudes• Novel differential equations
Nair (1988); Witten (2003)Roiban, Spradlin, & Volovich; Berkovits & Motl; Vafa & Neitzke; Siegel
(2004)
• Novel factorizations of amplitudesCachazo, Svrcek, & Witten (2004)
• Indirectly, new recursion relationsBritto, Cachazo, Feng, & Witten (1/2005)
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Supersymmetry
Most often pursued in broken form as low-energy phenomenology
"One day, all of these will be supersymmetric phenomenology papers."
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Exact Supersymmetry As a Computational Tool
• All-gluon amplitudes are the same at tree level in N =4 and QCD
• Fermion amplitudes obtained through Supersymmetry Ward Identities Grisaru, Pendleton, van Nieuwenhuizen (1977); Kunszt, Mangano, Parke, Taylor (1980s)
• At loop level, N =4 amplitudes are one contribution to QCD amplitudes; N =1 multiplets give another
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Gauge-theory amplitude
Color-ordered amplitude: function of ki and i
Helicity amplitude: function of spinor products and helicities ±1
Function of spinor variables and helicities ±1
Support on simple curves in twistor space
Color decomposition & Color decomposition & strippingstripping
Spinor-helicity basisSpinor-helicity basis
Half-Fourier transformHalf-Fourier transform
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Spinors
• Want square root of Lorentz vector need spin ½
• Spinors , conjugate spinors
• Spinor product
• (½,0) (0, ½) = vector
• Helicity 1: Amplitudes as pure functions of spinor
variables
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Complex Invariants
These are not just formal objects, we have the explicit formulæ
otherwise
so that the identity always holds
for real momenta
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Let’s Travel to Twistor Space!
It turns out that the natural setting for amplitudes is not exactly spinor space, but something similar. The motivation comes from studying the representation of the conformal algebra.
Half-Fourier transform of spinors: transform , leave alone Penrose’s original twistor space, real or complex
Study amplitudes of definite helicity: introduce homogeneous coordinates
CP3 or RP3 (projective) twistor space
Back to momentum space by Fourier-transforming
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Differential Operators
Equation for a line (CP1):
gives us a differential (‘line’) operator in terms of momentum-space spinors
Equation for a plane (CP2):
also gives us a differential (‘plane’) operator
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Even String Theorists Can Do Experiments
• Apply F operators to NMHV (3 – ) amplitudes:products annihilate them! K annihilates them;
• Apply F operators to N2MHV (4 – ) amplitudes:longer products annihilate them! Products of K annihilate them;
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On-Shell Recurrence Relations
Britto, Cachazo, Feng (2004)
• Amplitudes written as sum over ‘factorizations’ into on-shell amplitudes — but evaluated for complex momenta
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Proof Ingredients
Less is more. My architecture is almost nothing — Mies van der Rohe
Britto, Cachazo, Feng, Witten (2004)
• Complex shift of momenta
• Behavior as z : need A(z) 0
• Basic complex analysis
• Knowledge of factorization: at tree level, tracks known multiparticle-pole and collinear factorization
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Proof
• Consider the contour integral
• Determine A(0) in terms of other poles
• Poles determined by knowledge of factorization in invariants
• At tree level
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• Very general: relies only on complex analysis + factorization
• Applied to gravityBedford, Brandhuber, Spence, & Travaglini (2/2005)
Cachazo & Svrček (2/2005)• Massive amplitudes
Badger, Glover, Khoze, Svrček (4/2005, 7/2005)Forde & DAK (7/2005)
• Integral coefficientsBern, Bjerrum-Bohr, Dunbar, & Ita (7/2005)
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Unitarity Method for Higher-Order Calculations
Bern, Dixon, Dunbar, & DAK (1994)
• Proven utility as a tool for explicit calculations– Fixed number of external legs– All-n equations
• Tool for formal proofs
• Yields explicit formulae for factorization functions
• Color ordering
• Key idea: sew amplitudes not diagrams
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Unitarity-Based Calculations
Bern, Dixon, Dunbar, & DAK (1994)
• At one loop in D=4 for SUSY full answer(also for N =4 two-particle cuts at two loops)
• In general, work in D=4-2Є full answervan Neerven (1986): dispersion relations converge
• Merge channels: find function w/given cuts in all channels• ‘Generalized cuts’: require more than two propagators to
be present
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Unitarity-Based Method at Higher Loops
• Loop amplitudes on either side of the cut
• Multi-particle cuts in addition to two-particle cuts
• Find integrand/integral with given cuts in all channels• In practice, replace loop amplitudes by their cuts too
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On-Shell Recursion at Loop Level
Bern, Dixon, DAK (2005)
• Subtleties in factorization: factorization in complex momenta is not exactly the same as for real momenta
• For finite amplitudes, obtain recurrence relations which agree with known results (Chalmers, Bern, Dixon, DAK; Mahlon)
• and yield simpler forms
• Simpler forms involve spurious singularities
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• Amplitudes contain factors like known from collinear limits
• Expect also as ‘subleading’ contributions, seen in explicit results
• Double poles with vertex
• Non-conventional single pole: one finds the double-pole, multiplied by
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Rational Parts of QCD Amplitudes
• Start with cut-containing parts obtained from unitarity method, consider same contour integral
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• Start with same contour integral
• Cut terms have spurious singularities, absorb them into ; but that means there is a double-counting: subtract off those residues
Cut termsRational terms
Cut terms
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Road Ahead
• Opens door to many new calculations: time to do them!
• Approach already includes external massive particles (H, W, Z)
• Reduce one-loop calculations to purely algebraic ones in an analytic context, with polynomial complexity
• Massive internal particles
• Lots of excitement to come!