Recurrence, Unitarity and Twistors

including work withincluding work with I. Bena, Z. Bern, V. Del Duca, D. I. Bena, Z. Bern, V. Del Duca, D. Dunbar, L. Dixon,Dunbar, L. Dixon,

D. Forde, P. Mastrolia, R. RoibanD. Forde, P. Mastrolia, R. Roiban

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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

• How many operations (multiplication, addition, etc.) does it take to evaluate an amplitude?

• Textbook Feynman diagram approach: factorial complexity

• Color ordering

exponential complexity

• O(2n) different helicities: at least exponential complexity

• 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

• Same j-point current appears in calculation of Jn as in calculation of Jm<n

• Only a polnoymial number of different currents needed

• O(n4) operations for generic helicity

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Twistors: New Representations for Trees

• Cachazo-Svrček-Witten construction Svrček’s talk

simple vertices & rules• Roiban-Spradlin-Volovich representation

Spradlin’s talk

compact representation derived from loops

inspired by trees obtained from infrared consistency equations Bern, Dixon, DAK

• Britto-Cachazo-Feng-Witten recurrence Britto’s talk

representation in terms of lower-n on-shell amplitudes

• Nice analytic forms

• In special cases, better than O(n4) operations

• Probably not the last word

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MHV Amplitudes• Pure gluon amplitudes

• All gluon helicities + amplitude = 0

• Gluon helicities +–+…+ amplitude = 0

• Gluon helicities +–+…+–+ MHV amplitude

Parke & Taylor (1986)

• Holomorphic in spinor variables

• Proved via recurrence relationsBerends & Giele (1988)

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Cachazo–Svrcek–Witten Construction

• Vertices are off-shell continuations of MHV amplitudes

• Connect them by propagators i/K2

• 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?

Can’t follow a single leg

Divide into two sets by cutting propagator

Treat as new higher-degree vertices

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• Higher degree vertices expressed in terms of lower-degree ones

• Compact formula when dressed with external legs

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Beyond Pure QCD

• Add Higgs Dixon’s talk

• Add Ws and ZsBern, Forde, Mastrolia, DAK (2004)

• Hybrid formalism: build up recursive currents using CSW construction

• W current • (W electroweak process)

• along with CSW construction

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Loop Calculations: Textbook Approach

• Sew together vertices and propagators into loop diagrams

• Obtain a sum over [2…n]-point [0…n]-tensor integrals, multiplied by coefficients which are functions of k and

• Reduce tensor integrals using Brown-Feynman & Passarino-Veltman brute-force reduction, or perhaps Vermaseren-van Neerven method

• 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

• Can use spinor-helicity method at the end to obtain helicity amplitudes

BUT

• 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

• 227,585 diagrams

• @ 1 in2/diagram: three bound volumes of Phys. Rev. D just to draw them

• @ 1 min/diagram: 22 months full-time just to draw them

• So of course one doesn’t do it that way

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Can We Take Advantage…

• Of tree-level recurrence relations?

• Of new twistor-based ideas for reducing computational effort for analytic forms?

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Unitarity

• Basic property of any quantum field theory: conservation of probability. In terms of the scattering matrix,

• In terms of the transition matrix we get,

• or

• with the Feynman i

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• This has a direct translation into Feynman diagrams, using the Cutkosky rules. If we have a Feynman integral,

• 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

• To recover the full integral, we could perform a dispersion integral

• in which so long as when

• If this condition isn’t satisfied, there are ‘subtraction’ ambiguities corresponding to terms in the full amplitude which have no discontinuities

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• But it’s better to obtain the full integral by identifying which Feynman integral(s) the cut came from.

• 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

• The cutting relation can also be applied to sums of diagrams, in addition to single diagrams

• 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.

• 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)

• Proven utility as a tool for explicit one- and two-loop calculations

– Fixed number of external legs

– All-n equations

• Tool for formal proofs: all-orders collinear factorization

• Yields explicit formulae for factorization functions: two-loop splitting amplitude

• 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

• Compute cuts in a set of channels

• Compute required tree amplitudes

• Form the phase-space integrals

• Reconstruct corresponding Feynman integrals

• Perform integral reductions to a set of master integrals

• Assemble the answer

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Unitarity-Based Calculations

Bern, Dixon, Dunbar, & DAK (1994)

• In general, work in D=4-2Є full answervan Neerven (1986): dispersion relations converge

• At one loop in D=4 for SUSY full answer

• Merge channels rather than blindly summing: find function w/given cuts in all channels

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The Three Roles of Dimensional Regularization

• Ultraviolet regulator;

• Infrared regulator;

• Handle on rational terms.

• Dimensional regularization effectively removes the ultraviolet divergence, rendering integrals convergent, and so removing the need for a subtraction in the dispersion relation

• 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 easy’ two-mass boxbox

‘‘hard’ two-mass hard’ two-mass boxbox

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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

• Need to compute only the coefficients

• Algebraic approach by Cachazo based on holomorphic ‘anomaly’ Britto’s talk Britto’s talk

Britto, Cachazo, Feng (2004)

• Knowledge of basis not required for the unitarity-based method

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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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• Cuts require two propagators to be present corresponding to a ‘massive’ channel

• Can require more than two propagators to be present: ‘generalized’ cuts

• Break up amplitude into yet smaller and simpler pieces; more effective ‘recycling’ of tree amplitudes

• Triple cuts: Bern, Dixon, DAK (1996)

all-n next-to-MHV amplitude Bern, Dixon, DAK (2004)

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Generalized Cuts

• Isolate different contributions at higher loops as well

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An Amazing Result: Planar Iteration Relation

Bern, Rozowsky, Yan (1997)

Anastasiou, Bern, Dixon, DAK (2003)

• This should generalize

Ratio to treeRatio to tree

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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)

• Use complex loop momenta to obtain solution even with three-point vertices (which vanish on-shell for real momenta)

Britto, Cachazo, Feng (2004)

Britto’s talkBritto’s talk

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Seven-Point Coefficients

collinearcollinear

multiparticlmultiparticlee

‘‘planar’planar’

‘‘cubic’cubic’3-mass3-mass

Easy 2-Easy 2-massmass

Hard 2-Hard 2-massmass

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The All-n NMHV Amplitude in N=4

• Quadruple cuts show that four-mass boxes are absent

• Triple cuts lead to simple expression for three-mass box coefficient

• Triple cuts or soft limits lead to expression for hard two-mass box coefficient as a sum of three-mass box coefficients

• 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

• Write down the three vertices, pull out cut-independent factor

• Use Schouten identity to partial fraction second factor

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• Use another partial fractioning identity with cubic denominators

• 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

• N=4 SUSY amplitudes are UV-finite, but still have infrared divergences due to soft gluons

• Leading divergences are universal to a gauge theory independent of matter content: same as QCD

• Would cancel in a physical cross-section

• General structure of one-loop infrared divergences

Giele & Glover (1992); Kunszt, Signer, Trocsanyi (1994)

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• Examine coefficients of

• Gives linear relations between coefficients of different boxes

• n (n–3)/2 equations: enough to solve for easy two-mass and one-mass coefficients in terms of three-mass and hard two-mass coefficients [odd n]

• Alternatively, gives new representation of trees

also Roiban, Spradlin, Volovich (2004)

Britto, Cachazo, Feng, Witten (2004/5)

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Infrared DivergencesBena, Bern, Roiban, DAK (2004)

• BST calculation: MHV diagrams map to cuts

• Generic diagrams have no infrared divergences

• Only diagrams with a four-point vertex have infrared divergences

• One can define a twistor-space regulator for those

• Separates the issue of infrared divergences from the formulation of the string theory

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Summary

• Unitarity is the natural tool for loop calculations with twistor methods

• Large body of explicit results useful for both phenomenology and twistor investigations