Recurrence, Unitarity and Twistors including work with I. Bena, Z. Bern, V. Del Duca, D. Dunbar, L. Dixon, D. Forde, P. Mastrolia, R. Roiban

2 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

3 Goals of Explicit Calculations of Amplitudes Analytic results –Understand structure of results –Insight into theory –Develop new tools –Anomalous dimensions Numerical values to be integrated over phase space –LHC Physics: background to discoveries at the energy frontier  Bern’s talk –Computational complexity of algorithm important

4 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 (2 n ) different helicities: at least exponential complexity But what about the complexity of each helicity amplitude?

5 Recurrence Relations Berends & Giele (1988); DAK (1989)

6

7 Complexity of Each Helicity Amplitude Same j-point current appears in calculation of J n as in calculation of J m<n Only a polnoymial number of different currents needed O (n 4 ) operations for generic helicity

8 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 (n 4 ) operations Probably not the last word

9 Twistors Are an Experimental Subject! Analysis of known results revealed simple structure of amplitudes at tree level and one loop Cachazo, Svrček, & Witten (2004) Simpler than anticipated in Witten’s original formulation of twistor string theory Lead to new ideas for calculational techniques  Svrček’s talk No ab initio derivation, so checks and independent calculations are essential

10 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 relations Berends & Giele (1988)

11 Cachazo–Svrcek–Witten Construction Vertices are off-shell continuations of MHV amplitudes Connect them by propagators i / K 2 Draw all diagrams

12 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

13 Higher degree vertices expressed in terms of lower-degree ones Compact formula when dressed with external legs

14 Beyond Pure QCD Add Higgs  Dixon’s talk Add Ws and Zs Bern, Forde, Mastrolia, DAK (2004) Hybrid formalism: build up recursive currents using CSW construction W current (W  electroweak process) along with CSW construction

15 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

16 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.

17 Traditional Methods in the N=4 One-Loop Seven-Point Amplitude 227,585 1 in 2 /diagram: three bound volumes of Phys. Rev. D just to draw 1 min/diagram: 22 months full-time just to draw them So of course one doesn’t do it that way

18 Can We Take Advantage… Of tree-level recurrence relations? Of new twistor-based ideas for reducing computational effort for analytic forms?

19 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 

20 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 K 2 channel, we should replace

21 When we do this, we obtain a phase-space integral

22 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

23 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.

24 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.

25 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;

26 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

27 Unitarity-Based Calculations Bern, Dixon, Dunbar, & DAK (1994) In general, work in D=4-2 Є  full answer van 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

28 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

29 Integral Reductions At one loop, all n  5-point amplitudes in a massless theory can be written in terms of nine different types of scalar integrals: boxes (one-mass, ‘easy’ two-mass, ‘hard’ two-mass, three-mass, and four-mass); triangles (one-mass, two-mass, and three-mass); bubbles In an N =4 supersymmetric theory, only boxes are needed.

30 Basis in N =4 Theory ‘easy’ two-mass box ‘hard’ two-mass box

31 Example: MHV at One Loop

32 Start with the cut Use the known expressions for the MHV amplitudes

33 Most factors are independent of the integration momentum

34 We can use the Schouten identity to rewrite the remaining parts of the integrand, Two propagators cancel, so we’re left with a box — the  5 leads to a Levi-Civita tensor which vanishes What’s left over is the same function which appears in the denominator of the box:

35

36 We obtain the result,

37 Knowledge of basis opens door to new methods of computing amplitudes Need to compute only the coefficients  Britto’s talk Algebraic approach by Cachazo based on holomorphic ‘anomaly’  Britto’s talk Britto, Cachazo, Feng (2004) Knowledge of basis not required for the unitarity-based method

38 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

39 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)

40 Generalized Cuts Isolate different contributions at higher loops as well

41 An Amazing Result: Planar Iteration Relation Bern, Rozowsky, Yan (1997) Anastasiou, Bern, Dixon, DAK (2003) This should generalize Ratio to tree

42 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 talk

43 Loops From MHV Vertices Sew together two MHV vertices Brandhuber, Spence, & Travaglini (2004)

44 Simplest off-shell continuation lacks i  prescription Use alternate form of CSW continuation  DAK (2004) to map the calculation on to the cut Brandhuber, Spence, & Travaglini (2004)

45 One-Loop Seven-Point NMHV Amplitude Bern, Del Duca, Dixon Results remarkably simple (~6 pages, each coefficient essentially a one-liner) to draw all the Feynman diagrams in 6 pages, each would have to fit in 1 mm 2 Structure 3-mass, easy & hard 2-mass, 1-mass boxes

46 Seven-Point Coefficients collinear multiparticle ‘planar’ ‘cubic’ 3-mass Easy 2- mass Hard 2- mass

47 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

48 Triple Cuts Write down the three vertices, pull out cut-independent factor Use Schouten identity to partial fraction second factor

49 Use another partial fractioning identity with cubic denominators Isolate box with three cut momenta and

50 All-n Results & Structure  Dixon’s talk

51 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)

52 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)

53 Infrared Divergences Bena, 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

54 Summary Unitarity is the natural tool for loop calculations with twistor methods Large body of explicit results useful for both phenomenology and twistor investigations