Twistor Spinoffs for Collider Physics

1. Twistor Spinoffs for Collider Physics Lance Dixon, SLAC Fermilab Colloquium June 7, 2006

2. Electromagnetism (QED) • + weak interactions • electroweak theory • Verified to 0.1% • Hints of more: • grand unification • with supersymmetry Physics at very short distances • Unification: particle interactions simpler at short distances L. Dixon Twistor Spinoffs for Collider Physics

3. Physics at very short distances • Supersymmetry predicts a host ofnew massive particles • including a dark matter candidate • Typical masses ~ 100 GeV/c2 – 1 TeV/c2 (mproton = 1 GeV/c2) • Many other theories of electroweak scale mW,Z = 100 GeV/c2 • make similar predictions: • new dimensions of space-time • new forces • etc. How to sort them all out? • Einstein (E = mc2): heavy particles require high energies • Heisenberg (Dx Dp > h): short distances require high energies (and large momentum transfers) L. Dixon Twistor Spinoffs for Collider Physics

4. CDF D0 Geneva The Tevatron • The present energy frontier – right here! • Proton-antiproton collisions at 2 TeV center-of-mass energy L. Dixon Twistor Spinoffs for Collider Physics

5. Geneva CMS ATLAS The Large Hadron Collider • Proton-proton collisions at 14 TeV center-of-mass energy, • 7 times greater than Tevatron • Luminosity (collision rate) 10—100 times greater • New window into physics at the shortest distances – opening 2007 L. Dixon Twistor Spinoffs for Collider Physics

6. LHC Detectors ATLAS CMS L. Dixon Twistor Spinoffs for Collider Physics

7. ne m- m+ jets --- lots of jets lots of W’s ne m- m+ and Z’s top quarks with jets What will the LHC see? L. Dixon Twistor Spinoffs for Collider Physics

8. n g g ~ ~ ~ c0 c0 c+ What might the LHC see? L. Dixon Twistor Spinoffs for Collider Physics

9. Feynman told us how to do this – in principle • Feynman rules, while very general, are not optimized for these processes • Important to find more efficient methods, making use of hidden symmetries of QCD A better way to compute? • Backgrounds (and many signals) require detailed understanding of scattering amplitudes for many ultra-relativistic (“massless”) particles – especially quarks and gluons of QCD L. Dixon Twistor Spinoffs for Collider Physics

10. tree 1 loop 2 loop The loop expansion • Amplitudes can be expanded in a “small” parameter, as = g2/4p • At each successive order in g2, draw Feynman diagrams with one more loop – the number grows very rapidly! • For example, gluon-gluon scattering L. Dixon Twistor Spinoffs for Collider Physics

11. LO = |tree|2 n=8 NNLO = 2-loop x tree* + … n=2 NLO = loop x tree* + … n=3 Why do we need to do better? • Leading-order (LO), tree-level predictions are only qualitative, due to poor convergence of expansion in strong couplingas(m) ~ 0.1 • NLO corrections can be 30% - 80% of LO state of the art: L. Dixon Twistor Spinoffs for Collider Physics

12. Df NLO n=3 Tevatron Run II example Azimuthal decorrelation of di-jets at D0 due to additional radiation Z. Nagy (2003) L. Dixon Twistor Spinoffs for Collider Physics

13. LHC Example: SUSY Search Gianotti & Mangano, hep-ph/0504221 Mangano et al. (2002) • Search for missing energy + jets. • SM background from Z + jets. Early ATLAS TDR studies using PYTHIA overly optimistic • ALPGEN based on LO amplitudes, • much better than PYTHIA at • modeling hard jets • What will disagreement between • ALPGEN and data mean? • Hard to tell because of potentially • large NLO corrections L. Dixon Twistor Spinoffs for Collider Physics

14. Experimenters to theorists: “Please calculate the following at NLO” Theorists to experimenters: “Get real” Dialogue between theorists & experimenters L. Dixon Twistor Spinoffs for Collider Physics

15. Theorists to experimenters: “OK, we’ll get to work” The dialogue continues Experimenters to theorists: “OK, we’d really like these at NLO, by the time LHC starts” Les Houches 2005 L. Dixon Twistor Spinoffs for Collider Physics

16. How do we know there’s a better way? Because Feynman diagrams for QCD are “too complicated” An An from only 10 diagrams! L. Dixon Twistor Spinoffs for Collider Physics

17. Parke-Taylor formula (1986) How do we know there’s a better way? Because many answers are much simpler than expected! For example, special helicity amplitudes vanish or are very short L. Dixon Twistor Spinoffs for Collider Physics

18. (Egyptians, …, Hamilton, 1843) (Schrodinger, 1926) (200 B.C.?) (Pauli, 1925) (Dirac, 1925) (Fourier, 1807) (Penrose, 1967) (Witten, 2003) + … Mathematical Tools for Physics L. Dixon Twistor Spinoffs for Collider Physics

19. w lines appear Simplicity in Fourier space Example of atomic spectroscopy t L. Dixon Twistor Spinoffs for Collider Physics

20. Natural to use Lorentz-invariant products (invariant masses): But for particles with spin there is a better way massless q,g,g all have 2 helicities Take “square root” of 4-vectorskim (spin 1) use 2-component Dirac (Weyl) spinors ua(ki) (spin ½) The right variables Scattering amplitudes for massless plane waves of definite 4-momentum: Lorentz vectors kim ki2=0 L. Dixon Twistor Spinoffs for Collider Physics

21. Similarly, reconstructrelativisticspin-1 momenta kim from spinors: (projector onto positive-energy solutions of Dirac equation) Adding spins From two non-identical non-relativisticspin ½ objects, make spin 1 L. Dixon Twistor Spinoffs for Collider Physics

22. Instead of Lorentz products: Use spinor products: These are complex square roots of Lorentz products: Spinor products Antisymmetric product of two spin ½ is spin 0 (rotationally invariant) L. Dixon Twistor Spinoffs for Collider Physics

23. scalars gauge theory angular momentum mismatch explains denominators Spinor Magic Spinor products precisely capture square-root + phase behavior in collinear limit. Excellent variables for helicity amplitudes L. Dixon Twistor Spinoffs for Collider Physics

24. Twistor transform = “half Fourier transform”: Fourier transform , but not , for each leg Conjugate variables: Like time and frequency: Twistor space has coordinates Twistor Space Start in spinor space: L. Dixon Twistor Spinoffs for Collider Physics

25. Fourier transform of plane-wave is d-function: “Maximally Helicity Violating” amplitudes, , are plane-wave in lines appear! Twistor Transform in QCD Witten (2003) L. Dixon Twistor Spinoffs for Collider Physics

26. more lines More Twistor Magic Berends, Giele; Mangano, Parke, Xu (1988) A6 = L. Dixon Twistor Spinoffs for Collider Physics

27. Berends, Giele, Kuijf (1990) Even More Twistor Magic Now it is clear how to generalize L. Dixon Twistor Spinoffs for Collider Physics

28. off-shell MHV (Parke-Taylor) amplitudes scalar propagator, 1/p2 MHV rules Cachazo, Svrcek, Witten (2004) Twistor space picture: Led to MHV rules: More efficient alternative to Feynman rules for QCD trees L. Dixon Twistor Spinoffs for Collider Physics

29. Related approach to QCD + massive quarks • more directly from field theory Schwinn, Weinzierl, hep-th/0503015 MHV rules for trees Rules quite efficient, extended to many collider applications Georgiou, Khoze, hep-th/0404072; Wu, Zhu, hep-th/0406146; Georgiou, Glover, Khoze, hep-th/0407027 • massless quarks LD, Glover, Khoze, hep-th/0411092; Badger, Glover, Khoze, hep-th/0412275 • Higgs bosons (Hgg coupling) • vector bosons (W,Z,g*) Bern, Forde, Kosower, Mastrolia, hep-th/0412167 L. Dixon Twistor Spinoffs for Collider Physics

30. Twistor structure of loops • Simplest for coefficients of box integrals in a “toy model”, • N=4 supersymmetric Yang-Mills theory Again support is on lines, but joined into rings, to match topology of the loop amplitudes Cachazo, Svrcek, Witten; Brandhuber, Spence, Travaligni (2004) Bern, Del Duca, LD, Kosower; Britto, Cachazo, Feng (2004) L. Dixon Twistor Spinoffs for Collider Physics

31. Mass (GeV/c2) 1019 0 • A topological string has almost all of its excitations stripped • Having it move in twistor space lets the remaining ones yield QCD, plus superpartners (more or less) QCD + lots QCD + little 1019 0 What’s a (topological) twistor string? • What’s a normal string? Abstracting the lessons often the best! E.g., Bern, Kosower (1991) L. Dixon Twistor Spinoffs for Collider Physics

32. Even better than MHV rules On-shell recursion relations Britto, Cachazo, Feng, hep-th/0412308 [Off-shell antecedent: Berends, Giele (1988)] An-k+1 An Ak+1 Ak+1 and An-k+1 are on-shell tree amplitudes with fewer legs, evaluated with momenta shifted by a complex amount Trees are recycled into trees! L. Dixon Twistor Spinoffs for Collider Physics

33. 3 recursive diagrams related by symmetry A 6-gluon example 220 Feynman diagrams for gggggg Helicity + color + MHV results + symmetries L. Dixon Twistor Spinoffs for Collider Physics

34. Simpler than form found in 1980s Mangano, Parke, Xu (1988) Simple final form L. Dixon Twistor Spinoffs for Collider Physics

35. Berends, Giele, Kuijf (1990) Bern, Del Duca, LD, Kosower (2004) Relative simplicity grows with n L. Dixon Twistor Spinoffs for Collider Physics

36. Simple, general: Residue theorem+factorization how amplitudes “fall apart” in degenerate kinematic limits Inject complex momentum at leg 1, remove it at leg n. degenerate limits poles in z Cauchy: residue at zk = [kth term in relation] Proof of on-shell recursion relations Britto, Cachazo, Feng, Witten, hep-th/0501052 L. Dixon Twistor Spinoffs for Collider Physics

37. How do the new methods compare to older ones at tree level? Dinsdale, Ternick, Weinzierl, hep-ph/0602204 off-shell recursive (1988) Schwinn, Weinzierl (2005) Cachazo, Svrcek, Witten (2004) + … on-shell recursive (2005) even more gluons: Speed is of the Essence • For collider phenomenology, in the end all one needs are numbers • But one needs them many times to do integrals over phase space • For LHC, n ~ 6 – 9, they do pretty well L. Dixon Twistor Spinoffs for Collider Physics

38. 1. different collinear behavior of loop amplitudes double poles in z Bern, LD, Kosower, hep-ph/9403226, hep-ph/9708239; Britto, Cachazo, Feng, hep-th/0412103; BBCF, hep-ph/0503132; Britto, Feng, Mastrolia, hep-ph/0602178 2. branch cuts – but these can be determined efficiently using (generalized) unitarity but Trees recycled into loops! On-shell recursion at one loop Bern, LD, Kosower, hep-th/0501240, hep-ph/0505055, hep-ph/0507005 • Same techniques work for one-loopQCD amplitudes • – much harder to obtain by other methods than are trees. • New features arise compared with tree case L. Dixon Twistor Spinoffs for Collider Physics

39. Competition from semi-numerical methods – done this way Ellis, Giele, Zanderighi, hep-ph/0602185 Rational functions in loop amplitudes • After computing cuts using unitarity, there remains an additive rational-function ambiguity • Determined using - tree-likerecursive diagrams, plus - simple “overlap diagrams” • No loop integrals required in this step • Bootstrap rational functions from cuts and lower-point amplitudes • Method tested on 5-point amplitudes, used to get newQCD results: • Now working to generalize method to all helicity configurations, and to processes on the “realistic NLO wishlist”. Forde, Kosower, hep-ph/0509358 Berger, Bern, LD, Forde, Kosower, hep-ph/0604195, hep-ph/0606nnn, … L. Dixon Twistor Spinoffs for Collider Physics

40. For rational part of Example of new diagrams recursive: overlap: 7 in all Compared with 1034 1-loop Feynman diagrams (color-ordered) L. Dixon Twistor Spinoffs for Collider Physics

41. Branch cuts • Poles Revenge of the Analytic S-matrix Reconstruct scattering amplitudes directly from analytic properties Chew, Mandelstam; Eden, Landshoff, Olive, Polkinghorne; Veneziano; Virasoro, Shapiro; …(1960s) Analyticity fell somewhat out of favor in 1970s with rise of QCD; to resurrect it for computing perturbativeQCD amplitudes seems deliciously ironic! L. Dixon Twistor Spinoffs for Collider Physics

42. Conclusions • Exciting new computational approaches to gauge theories due (directly or indirectly) to development of twistor string theory • So far, practical spinoffs mostly for trees, and loops in supersymmetric theories • But now, new loop amplitudesin full, non-supersymmetricQCD – needed for collider applications – are beginning to fall to twistor-inspiredrecursive approaches • Expect therapid progressto continue L. Dixon Twistor Spinoffs for Collider Physics

43. Extra slides L. Dixon Twistor Spinoffs for Collider Physics

44. Parke-Taylor formula Initial data L. Dixon Twistor Spinoffs for Collider Physics

45. Supersymmetric decomposition for QCD loop amplitudes gluon loop N=4 SYM N=1 multiplet scalar loop --- no SUSY, but also no spin tangles N=4 SYM and N=1 multiplets are simplest pieces to compute because they are cut-constructible – determined by their unitarity cuts, evaluated using D=4 intermediate helicities L. Dixon Twistor Spinoffs for Collider Physics

46. But if we know the cuts (via unitarity inD=4), we can subtract them: rational part full amplitude cut-containing part Shifted rational function has no cuts, but has spurious poles in z because of Cn: Loop amplitudes with cuts Generic analytic properties of shifted 1-loop amplitude, Cuts andpoles in z-plane: L. Dixon Twistor Spinoffs for Collider Physics

47. L. Dixon Twistor Spinoffs for Collider Physics

48. arbitary spinor entering MHV vertices off-shell MHV (Parke-Taylor) amplitudes scalar propagator, 1/p2 Direct proof of MHV rules via OSRR K. Risager, hep-th/0508206 MHV rules: There is a differentcomplex momentum shift for which the on-shell recursion relations (OSRR) for NMHV are identical, graph by graph, to the MHV rules. Proof is inductive in L. Dixon Twistor Spinoffs for Collider Physics

49. Why does it all work? In mathematics you don't understand things. You just get used to them. L. Dixon Twistor Spinoffs for Collider Physics