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From Twistors to Gauge-Theory Amplitudes

From Twistors to Gauge-Theory Amplitudes. WHEPP, Bhubaneswar, India January 7 , 2006. The Storyline. An exciting time in gauge-theory amplitude calculations Motivation for hard calculations Twistor-space ideas originating with Nair and Witten

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From Twistors to Gauge-Theory Amplitudes

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  1. From Twistors to Gauge-Theory Amplitudes WHEPP, Bhubaneswar, India January 7, 2006

  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, and parts of loops • 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 WHEPP, Bhubaneswar, India, January 7, 2006

  3. D0 event WHEPP, Bhubaneswar, India, January 7, 2006

  4. Guenther Dissertori (Jan ’04)  De Roeck’s talk WHEPP, Bhubaneswar, India, January 7, 2006

  5. Precision Perturbative QCD  Del Duca’s talk • 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  Harlander’s talk Everythingat a hadron collider involves QCD WHEPP, Bhubaneswar, India, January 7, 2006

  6. WHEPP, Bhubaneswar, India, January 7, 2006

  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) WHEPP, Bhubaneswar, India, January 7, 2006

  8. Supersymmetry Most often pursued in broken form as low-energy phenomenology "One day, all of these will be supersymmetric phenomenology papers." WHEPP, Bhubaneswar, India, January 7, 2006

  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 WHEPP, Bhubaneswar, India, January 7, 2006

  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 WHEPP, Bhubaneswar, India, January 7, 2006

  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 WHEPP, Bhubaneswar, India, January 7, 2006

  12. Complex Invariants These are not just formal objects, we have the explicit formulæ otherwise so that the identity always holds for real momenta WHEPP, Bhubaneswar, India, January 7, 2006

  13. Complex Momenta • For complex momenta  or but not necessarily both! WHEPP, Bhubaneswar, India, January 7, 2006

  14. WHEPP, Bhubaneswar, India, January 7, 2006

  15. 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  WHEPP, Bhubaneswar, India, January 7, 2006

  16. 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 WHEPP, Bhubaneswar, India, January 7, 2006

  17. 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; WHEPP, Bhubaneswar, India, January 7, 2006

  18. What does this mean in field theory? WHEPP, Bhubaneswar, India, January 7, 2006

  19. Cachazo–Svrček–Witten Construction WHEPP, Bhubaneswar, India, January 7, 2006

  20. Recursion Relations Berends & Giele (1988); DAK (1989)  Polynomial complexity per helicity WHEPP, Bhubaneswar, India, January 7, 2006

  21. On-Shell Recursion Relations Britto, Cachazo, Feng (2004) • Amplitudes written as sum over ‘factorizations’ into on-shell amplitudes — but evaluated for complex momenta WHEPP, Bhubaneswar, India, January 7, 2006

  22. Massless momenta: WHEPP, Bhubaneswar, India, January 7, 2006

  23. Proof Ingredients Less is more. My architecture is almost nothing — Mies van der Rohe Britto, Cachazo, Feng, Witten (2005) • 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 WHEPP, Bhubaneswar, India, January 7, 2006

  24. C WHEPP, Bhubaneswar, India, January 7, 2006

  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 WHEPP, Bhubaneswar, India, January 7, 2006

  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) • Connection to Cachazo–Svrček–Witten construction Risager (8/2005) • CSW construction for gravity  Twistor string for N =8? Bjerrum-Bohr, Dunbar, Ita, Perkins, & Risager (9/2005) WHEPP, Bhubaneswar, India, January 7, 2006

  27. Unitarity-Based Methodfor Loop Calculations Bern, Dixon, Dunbar, & DAK (1994) • Use a basic property of amplitudes as a calculational tool • Key idea: sew amplitudes not diagrams • Proven utility as a tool for explicit calculations • Fixed number of external legs • All-n equation • Formal proofs WHEPP, Bhubaneswar, India, January 7, 2006

  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 WHEPP, Bhubaneswar, India, January 7, 2006

  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 WHEPP, Bhubaneswar, India, January 7, 2006

  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 • Again, use properties of amplitude as calculational tool WHEPP, Bhubaneswar, India, January 7, 2006

  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 WHEPP, Bhubaneswar, India, January 7, 2006

  32. Eikonal Function WHEPP, Bhubaneswar, India, January 7, 2006

  33. Rational Parts of QCD Amplitudes • Start with cut-containing parts obtained from unitarity method, consider same contour integral WHEPP, Bhubaneswar, India, January 7, 2006

  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 WHEPP, Bhubaneswar, India, January 7, 2006

  35. A 2→4 QCD Amplitude Bern, Dixon, Dunbar, & DAK (1994) Only rational terms missing WHEPP, Bhubaneswar, India, January 7, 2006

  36. A 2→4 QCD Amplitude Rational terms WHEPP, Bhubaneswar, India, January 7, 2006

  37. Also computed Berger, Bern, Dixon, Forde, DAK WHEPP, Bhubaneswar, India, January 7, 2006

  38. All-Multiplicity Amplitude • Same technique can be applied to calculate a one-loop amplitude with arbitrary number of external legs WHEPP, Bhubaneswar, India, January 7, 2006

  39. 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! WHEPP, Bhubaneswar, India, January 7, 2006

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