Transcript From Twistors to Calculations

Recurrence, Unitarity and Twistors
including work with I. Bena, Z. Bern, V. Del Duca, D. Dunbar, L. Dixon,
D. Forde, P. Mastrolia, R. Roiban
We’ve heard a lot about twistors and about amplitudes in gauge
theories, N=4 supersymmetric gauge theory in particular.
What are the motivations for studying amplitudes?
 Bern’s talk
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Computational Complexity of Tree Amplitudes
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How many operations (multiplication, addition, etc.) does it
take to evaluate an amplitude?
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Textbook Feynman diagram approach: factorial complexity
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Color ordering
 exponential complexity
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O(2n) different helicities: at least exponential complexity
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But what about the complexity of each helicity amplitude?
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Recurrence Relations
Berends & Giele (1988); DAK (1989)
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Complexity of Each Helicity Amplitude
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Same j-point current appears in calculation of Jn as in calculation
of Jm<n
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Only a polnoymial number of different currents needed
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O(n4) operations for generic helicity
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Twistors: New Representations for Trees
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Cachazo-Svrček-Witten construction
 Svrček’s talk
simple vertices & rules
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Roiban-Spradlin-Volovich representation
 Spradlin’s talk
compact representation derived from loops
inspired by trees obtained from infrared consistency equations
Bern, Dixon, DAK
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Britto-Cachazo-Feng-Witten recurrence
 Britto’s talk
representation in terms of lower-n on-shell amplitudes
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Nice analytic forms
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In special cases, better than O(n4) operations
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Probably not the last word
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MHV Amplitudes
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Pure gluon amplitudes
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All gluon helicities +  amplitude = 0
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Gluon helicities +–+…+  amplitude = 0
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Gluon helicities +–+…+–+  MHV amplitude
Parke & Taylor (1986)
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Holomorphic in spinor variables
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Proved via recurrence relations
Berends & Giele (1988)
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Cachazo–Svrcek–Witten Construction
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Vertices are off-shell continuations of MHV amplitudes
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Connect them by propagators i/K2
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Draw all diagrams
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Recursive Formulation
Bena, Bern, DAK (2004)
Recursive approaches have proven powerful in QCD; how can we
implement one in the CSW approach?
Divide into two sets by cutting propagator
Can’t follow a single leg
Treat as new higher-degree vertices
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Higher degree vertices expressed in terms of lower-degree ones
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Compact formula when dressed with external legs
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Beyond Pure QCD
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Add Higgs
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Add Ws and Zs
 Dixon’s talk
Bern, Forde, Mastrolia, DAK (2004)
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Hybrid formalism: build up recursive currents using CSW
construction
W current • (W  electroweak process)
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along with CSW construction
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Loop Calculations: Textbook Approach
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Sew together vertices and propagators into loop diagrams
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Obtain a sum over [2…n]-point [0…n]-tensor integrals,
multiplied by coefficients which are functions of k and 
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Reduce tensor integrals using Brown-Feynman & PassarinoVeltman brute-force reduction, or perhaps Vermaseren-van
Neerven method
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Reduce higher-point integrals to bubbles, triangles, and boxes
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Can apply this to color-ordered amplitudes, using color-ordered
Feynman rules
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Can use spinor-helicity method at the end to obtain helicity
amplitudes
BUT
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This fails to take advantage of gauge cancellations early in the
calculation, so a lot of calculational effort is just wasted.
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Traditional Methods in the N=4 One-Loop
Seven-Point Amplitude
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227,585 diagrams
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@ 1 in2/diagram: three bound volumes of Phys. Rev. D just to
draw them
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@ 1 min/diagram: 22 months full-time just to draw them
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So of course one doesn’t do it that way
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Can We Take Advantage…
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Of tree-level recurrence relations?
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Of new twistor-based ideas for reducing computational effort
for analytic forms?
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Unitarity
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Basic property of any quantum field theory: conservation of
probability. In terms of the scattering matrix,
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In terms of the transition matrix
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or
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with the Feynman i
we get,
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This has a direct translation into Feynman diagrams, using the
Cutkosky rules. If we have a Feynman integral,
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and we want the discontinuity in the K2 channel, we should
replace
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When we do this, we obtain a phase-space integral
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In the Bad Old Days of Dispersion
Relations
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To recover the full integral, we could perform a dispersion
integral
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in which
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If this condition isn’t satisfied, there are ‘subtraction’
ambiguities corresponding to terms in the full amplitude which
have no discontinuities
so long as
when
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But it’s better to obtain the full integral by identifying which
Feynman integral(s) the cut came from.
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Allows us to take advantage of sophisticated techniques for
evaluating Feynman integrals: identities, modern reduction
techniques, differential equations, reduction to master integrals,
etc.
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Computing Amplitudes Not Diagrams
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The cutting relation can also be applied to sums of diagrams, in
addition to single diagrams
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Looking at the cut in a given channel s of the sum of all
diagrams for a given process throws away diagrams with no cut
— that is diagrams with one or both of the required
propagators missing — and yields the sum of all diagrams on
each side of the cut.
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Each of those sums is an on-shell tree amplitude, so we can
take advantage of all the advanced techniques we’ve seen for
computing them.
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Unitarity Method for Higher-Order Calculations
Bern, Dixon, Dunbar, & DAK (1994)
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Proven utility as a tool for explicit one- and two-loop calculations
– Fixed number of external legs
– All-n equations
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Tool for formal proofs: all-orders collinear factorization
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Yields explicit formulae for factorization functions: two-loop splitting
amplitude
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Recent work also by Bedford, Brandhuber, Spence, Travaglini; Britto, Cachazo,
Feng; Bidder, Bjerrum-Bohr, Dixon, Dunbar, & Perkins;
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Unitarity-Based Method at One Loop
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Compute cuts in a set of channels
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Compute required tree amplitudes
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Form the phase-space integrals
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Reconstruct corresponding Feynman integrals
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Perform integral reductions to a set of master integrals
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Assemble the answer
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Unitarity-Based Calculations
Bern, Dixon, Dunbar, & DAK (1994)
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In general, work in D=4-2Є  full answer
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At one loop in D=4 for SUSY  full answer
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van Neerven (1986): dispersion relations converge
Merge channels rather than blindly summing: find function
w/given cuts in all channels
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The Three Roles of Dimensional Regularization
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Ultraviolet regulator;
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Infrared regulator;
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Handle on rational terms.
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Dimensional regularization effectively removes the ultraviolet
divergence, rendering integrals convergent, and so removing the
need for a subtraction in the dispersion relation
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Pedestrian viewpoint: dimensionally, there is always a factor of
(–s)–, so at higher order in , even rational terms will have a
factor of ln(–s), which has a discontinuity
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Basis in N=4 Theory
‘easy’ two-mass box
‘hard’ two-mass box
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Example: MHV at One Loop
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We obtain the result,
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Knowledge of basis opens door to new methods of computing
amplitudes
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Need to compute only the coefficients
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Algebraic approach by Cachazo based on holomorphic
‘anomaly’
 Britto’s talk
Britto, Cachazo, Feng (2004)
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Knowledge of basis not required for the unitarity-based method
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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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Cuts require two propagators to be present corresponding to a
‘massive’ channel
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Can require more than two propagators to be present:
‘generalized’ cuts
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Break up amplitude into yet smaller and simpler pieces; more
effective ‘recycling’ of tree amplitudes
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Triple cuts:
Bern, Dixon, DAK (1996)
all-n next-to-MHV amplitude Bern, Dixon, DAK (2004)
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Generalized Cuts
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Isolate different contributions at higher loops as well
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An Amazing Result:
Planar Iteration Relation
Ratio to tree
Bern, Rozowsky, Yan (1997)
Anastasiou, Bern, Dixon, DAK (2003)
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This should generalize
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With knowledge of the integral basis, quadruple cut gives
general numerical solution for N=4 one-loop coefficients (four
equations for four-vector specify it)
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Use complex loop momenta to obtain solution even with threepoint vertices (which vanish on-shell for real momenta)
 Britto’s talk
Britto, Cachazo, Feng (2004)
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Seven-Point Coefficients
3-mass
‘cubic’
collinear
Easy 2mass
‘planar’
multiparticle
Hard 2mass
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The All-n NMHV Amplitude in N=4
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Quadruple cuts show that four-mass boxes are absent
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Triple cuts lead to simple expression for three-mass box
coefficient
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Triple cuts or soft limits lead to expression for hard two-mass
box coefficient as a sum of three-mass box coefficients
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Infrared equations lead to expressions for easy two-mass (and
one-mass) box coefficients as sums of three-mass box
coefficients
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Triple Cuts
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Write down the three vertices, pull out cut-independent factor
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Use Schouten identity to partial fraction second factor
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Use another partial fractioning identity with cubic denominators
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Isolate box with three cut momenta and
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All-n Results & Structure
 Dixon’s talk
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Infrared Consistency Equations
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N=4 SUSY amplitudes are UV-finite, but still have infrared
divergences due to soft gluons
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Leading divergences are universal to a gauge theory
independent of matter content: same as QCD
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Would cancel in a physical cross-section
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General structure of one-loop infrared divergences
Giele & Glover (1992); Kunszt, Signer, Trocsanyi (1994)
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Examine coefficients of
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Gives linear relations between coefficients of different boxes
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n (n–3)/2 equations: enough to solve for easy two-mass and
one-mass coefficients in terms of three-mass and hard twomass coefficients [odd n]
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Alternatively, gives new representation of trees
also Roiban, Spradlin, Volovich (2004)
Britto, Cachazo, Feng, Witten (2004/5)
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Infrared Divergences
Bena, Bern, Roiban, DAK (2004)
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BST calculation: MHV diagrams map to cuts
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Generic diagrams have no infrared divergences
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Only diagrams with a four-point vertex have infrared
divergences
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One can define a twistor-space regulator for those
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Separates the issue of infrared divergences from the
formulation of the string theory
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Summary
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Unitarity is the natural tool for loop calculations with twistor
methods
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Large body of explicit results useful for both phenomenology
and twistor investigations
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