Transcript From Twistors to Calculations

From Twistors
to
Gauge-Theory Amplitudes
WHEPP, Bhubaneswar, India
January 7, 2006
The Storyline
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An exciting time in gauge-theory amplitude calculations
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Motivation for hard calculations
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Twistor-space ideas originating with Nair and Witten
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Explicit calculations led to seeing simple twistor-space structure
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Explicit calculations led to new on-shell recursion relations for
trees, and parts of loops
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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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D0 event
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 De Roeck’s talk
Guenther Dissertori (Jan ’04)
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Precision Perturbative QCD
 Del Duca’s talk
 Harlander’s talk
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Predictions of signals, signals+jets
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Predictions of backgrounds
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Measurement of luminosity
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Measurement of fundamental parameters (s, mt)
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Measurement of electroweak parameters
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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
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Computation of scattering amplitudes
Novel differential equations
Nair (1988); Witten (2003)
Roiban, Spradlin, & Volovich; Berkovits & Motl; Vafa & Neitzke; Siegel (2004)
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Novel factorizations of amplitudes
Cachazo, Svrcek, & Witten (2004)
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Indirectly, new recursion relations
Britto, Cachazo, Feng, & Witten (1/2005)
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Supersymmetry
Most often pursued in broken form as low-energy
phenomenology
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"One day, all of these will be supersymmetric phenomenology papers."
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Exact Supersymmetry As a
Computational Tool
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All-gluon amplitudes are the same at tree level in N =4 and
QCD
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Fermion amplitudes obtained through Supersymmetry Ward
Identities Grisaru, Pendleton, van Nieuwenhuizen (1977); Kunszt,
Mangano, Parke, Taylor (1980s)
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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 decomposition & stripping
Color-ordered amplitude: function of ki and i
 Spinor-helicity basis
Helicity amplitude: function of spinor products and helicities ±1

Function of spinor variables and helicities ±1
 Half-Fourier transform
Support on simple curves in twistor space
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Spinors
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Want square root of Lorentz vector  need spin ½
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Spinors
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Spinor product
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(½,0)  (0, ½) = vector
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, conjugate spinors
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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Complex Momenta
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For complex momenta

or
but not necessarily both!
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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 momentumspace spinors
Equation for a plane (CP2):
also gives us a differential (‘plane’) operator
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Even String Theorists Can Do Experiments
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Apply F operators to NMHV (3 – ) amplitudes:
products annihilate them! K annihilates them;
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Apply F operators to N2MHV (4 – ) amplitudes:
longer products annihilate them! Products of K annihilate them;
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What does this mean in field theory?
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Cachazo–Svrček–Witten Construction
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Recursion Relations
Berends & Giele (1988); DAK (1989)
 Polynomial complexity per helicity
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On-Shell Recursion Relations
Britto, Cachazo, Feng (2004)
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Amplitudes written as sum over ‘factorizations’ into on-shell
amplitudes — but evaluated for complex momenta
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Massless momenta:
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Proof Ingredients
Less is more. My architecture is almost nothing — Mies van der Rohe
Britto, Cachazo, Feng, Witten (2005)
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Complex shift of momenta
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Behavior as z  : need A(z)  0
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Basic complex analysis
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Knowledge of factorization: at tree level, tracks known
multiparticle-pole and collinear factorization
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C
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Proof
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Consider the contour integral
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Determine A(0) in terms of other poles
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Poles determined by knowledge of factorization in invariants
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At tree level
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Very general: relies only on complex analysis + factorization
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Applied to gravity
Bedford, Brandhuber, Spence, & Travaglini (2/2005)
Cachazo & Svrček (2/2005)
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Massive amplitudes
Badger, Glover, Khoze, Svrček (4/2005, 7/2005)
Forde & DAK (7/2005)
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Integral coefficients
Bern, Bjerrum-Bohr, Dunbar, & Ita (7/2005)
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Connection to Cachazo–Svrček–Witten construction
Risager (8/2005)
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CSW construction for gravity  Twistor string for N =8?
Bjerrum-Bohr, Dunbar, Ita, Perkins, & Risager (9/2005)
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Unitarity-Based Method
for Loop Calculations
Bern, Dixon, Dunbar, & DAK (1994)
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Use a basic property of amplitudes as a calculational tool
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Key idea: sew amplitudes not diagrams
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Proven utility as a tool for explicit calculations
– Fixed number of external legs
– All-n equation
– Formal proofs
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Unitarity-Based Calculations
Bern, Dixon, Dunbar, & DAK (1994)
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At one loop in D=4 for SUSY  full answer
(also for N =4 two-particle cuts at two loops)
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In general, work in D=4-2Є  full answer
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Merge channels: find function w/given cuts in all channels
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‘Generalized cuts’: require more than two propagators to be
present
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van Neerven (1986): dispersion relations converge
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Unitarity-Based Method at Higher Loops
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Loop amplitudes on either side of the cut
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Multi-particle cuts in addition to two-particle cuts
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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)
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Subtleties in factorization: factorization in complex momenta is
not exactly the same as for real momenta
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For finite amplitudes, obtain recurrence relations which agree
with known results (Chalmers, Bern, Dixon, DAK; Mahlon)
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and yield simpler forms
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Simpler forms involve spurious singularities
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Again, use properties of amplitude as calculational tool
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Amplitudes contain factors like
limits
known from collinear
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Expect also
results
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Double poles with vertex
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Non-conventional single pole: one finds the double-pole,
multiplied by
as ‘subleading’ contributions, seen in explicit
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Eikonal Function
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Rational Parts of QCD Amplitudes
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Start with cut-containing parts obtained from unitarity method,
consider same contour integral
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Start with same contour integral
Rational terms
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Cut terms
Cut terms have spurious singularities, absorb them into ; but
that means there is a double-counting: subtract off those
residues
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A 2→4 QCD Amplitude
Bern, Dixon, Dunbar, & DAK (1994)
Only rational terms missing
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A 2→4 QCD Amplitude
Rational terms
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Also computed
Berger, Bern, Dixon, Forde, DAK
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All-Multiplicity Amplitude
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Same technique can be applied to calculate a one-loop
amplitude with arbitrary number of external legs
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Road Ahead
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Opens door to many new calculations: time to do them!
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Approach already includes external massive particles (H, W, Z)
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Reduce one-loop calculations to purely algebraic ones in an
analytic context, with polynomial complexity
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Massive internal particles
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Lots of excitement to come!
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