Twistors and Gauge Theory

1. Twistors and Gauge Theory DESY Theory Workshop September 30, 2005

2. 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

3. D0 event

4. Guenther Dissertori (Jan ’04)

5. 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

6. Campbell (Jan ‘04)

7. 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 amplitudes Cachazo, Svrcek, & Witten (2004) • Indirectly, new recursion relations Britto, Cachazo, Feng, & Witten (1/2005)

8. Supersymmetry Most often pursued in broken form as low-energy phenomenology "One day, all of these will be supersymmetric phenomenology papers."

9. 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

10. Color decomposition & stripping Gauge-theory amplitude  Color-ordered amplitude: function of kiand i  Helicity amplitude: function of spinor products and helicities ±1  Function of spinor variables and helicities ±1  Support on simple curves in twistor space Spinor-helicity basis Half-Fourier transform

11. 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

12. Complex Invariants These are not just formal objects, we have the explicit formulæ otherwise so that the identity always holds for real momenta

13. Complex Momenta • For complex momenta  or but not necessarily both!

14. 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  CP3orRP3(projective) twistor space Back to momentum space by Fourier-transforming 

15. 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

16. 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;

17. What does this mean in field theory?

18. Cachazo–Svrček–Witten Construction

20. Recursion Relations Berends & Giele (1988); DAK (1989)  Polynomial complexity per helicity

21. On-Shell Recurrence Relations Britto, Cachazo, Feng (2004) • Amplitudes written as sum over ‘factorizations’ into on-shell amplitudes — but evaluated for complex momenta

22. Massless momenta:

23. 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

24. C

25. Proof • Consider the contour integral • Determine A(0) in terms of other poles • Poles determined by knowledge of factorization in invariants • At tree level

26. Very general: relies only on complex analysis + factorization • Applied to gravity Bedford, Brandhuber, Spence, & Travaglini (2/2005) Cachazo & Svrček (2/2005) • Massive amplitudes Badger, Glover, Khoze, Svrček (4/2005, 7/2005) Forde & DAK (7/2005) • Integral coefficients Bern, Bjerrum-Bohr, Dunbar, & Ita(7/2005)

27. 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

28. 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 answer van 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

29. 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

30. 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

31. 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

32. Eikonal Function

33. Rational Parts of QCD Amplitudes • Start with cut-containing parts obtained from unitarity method, consider same contour integral

34. 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 Rational terms Cut terms Cut terms

35. A 2→4 QCD Amplitude Bern, Dixon, Dunbar, & DAK (1994) Only rational terms missing

36. A 2→4 QCD Amplitude Rational terms …and check this morning’s mailing for more!

37. 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!