String Theory and Q C D

1. Twistor String Theory & QCD String Theory and QCD Lance Dixon, SLAC HEP2005 Europhysics Conference Lisbon, 26 July 2005

2. CMS ATLAS The Large Hadron Collider • Proton-proton collisions at 14 TeV center-of-mass energy, • 7 times greater than previous (Tevatron) • Luminosity (collision rate) 10—100 times greater • New window into physics at the shortest distances – opening 2007 String Theory and QCD

3. Physics at very short distances Lots of ideas for physics beyond the Standard Model at electroweak scale • Supersymmetry predicts a host ofnew massive particles • including a dark matter candidate • Typical masses ~ 100 GeV/c2 – 1 TeV/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? String Theory and QCD

4. n c c Signals and backgrounds • Newparticlestypically decay into old particles: quarks, gluons, charged leptons and neutrinos, photons, Ws & Zs (which in turn decay to leptons) • Kinematic signatures are not always clean (e.g. mass bumps) if neutrinos, or other escaping particles (e.g. dark matter) are present gluino cascade • Need to quantify the Standard Model backgrounds for a variety of multi-particle processes, to maximize potential for new physics discoveries String Theory and QCD

5. 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 String Theory and QCD

6. 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, tree-level predictions are only qualitative, due to poor convergence of perturbative expansion in strong couplingas(m) state of the art: String Theory and QCD

7. Parke-Taylor formula (1986) How do we know there’s a better way? Because many answers are much simpler than expected! String Theory and QCD

8. w lines appear Simplicity in Fourier space Example of atomic spectroscopy t String Theory and QCD

9. But for elementary particles with spin (e.g. all observed ones!) there is a better way: Take “square root” of 4-vectorskim (spin 1) use Dirac (Weyl) spinors ua(ki) (spin ½) q,g,g, all have 2 helicity states, The right variables Scattering amplitudes for massless plane waves of definite momentum: Lorentz 4-vectors kim ki2=0 Natural to use Lorentz-invariant products (invariant masses): String Theory and QCD

10. Singular 2 x 2 matrix: also shows even for complex momenta The right variables (cont.) Reconstruct momenta kim from spinors using projector onto positive-energy solutions of Dirac eq.: String Theory and QCD

11. These are complex square roots of Lorentz products: Spinor products Instead of Lorentz products: Use spinor products: String Theory and QCD

12. scalars gauge theory angular momentum mismatch Spinor Magic Spinor products precisely capture square-root + phase behavior in collinear limit. Excellent variables for helicity amplitudes String Theory and QCD

13. Twistor transform = “half Fourier transform”: Fourier transform , but not , for each leg Twistor space coordinates: Twistor Space Start in spinor space: String Theory and QCD

14. lines appear! Twistor Transform in QCD Witten (2003) Parke-Taylor (1986) String Theory and QCD

15. more lines More Twistor Magic Mangano, Parke, Xu (1988) = String Theory and QCD

16. Berends, Giele (1990) Even More Twistor Magic Now it is clear how to generalize String Theory and QCD

17. 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 String Theory and QCD

18. Related approach to QCD + massive quarks • but 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 String Theory and QCD

19. Bern, LD, Del Duca, Kosower; Britto, Cachazo, Feng (2004) Twistor structure of loops • Simplest for coefficients of box integrals in a “toy model”, • N=4 supersymmetric Yang-Mills theory Cachazo, Svrcek, Witten; Brandhuber, Spence, Travaligni (2004) String Theory and QCD

20. Twistor structure of loops (cont.) Bern, LD, Kosower (2004) Again support is on lines, but joined into rings, to match topology of the loop amplitudes String Theory and QCD

21. Mass (GeV/c2) 1019 0 • A topological string has almost all of its excitations stripped QCD + lots QCD + little • Having it move in twistor space lets the remaining ones yield QCD, plus superpartners (more or less) 1019 0 What’s a (topological) twistor string? • What’s a normal string? Abstracting the lessons often the best! String Theory and QCD

22. AdS/QCD QCD Another connection between string theory and QCD • AdS/CFT correspondence • Relates strongly coupled gauge theory to weakly coupled gravity in 5 dimensions • First for N=4 super-Yang-Mills theory • More recently, confining gauge theories • But coupling should still be strong in UV for gravity side to be tractable • Interesting insights into hadron masses, quark-gluon plasma, deep inelastic scattering, exclusive processes at high energy Maldacena; Gubser, Klebanov, Polyakov; Witten (1996), + … Igor Klebanov, talk at Lepton-Photon 2005 String Theory and QCD

23. Even better than MHV rules On-shell recursion relations Britto, Cachazo, Feng, hep-th/0412308 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! String Theory and QCD

24. 3 BCF diagrams related by symmetry A 6-gluon example 220 Feynman diagrams for gggggg Helicity + color + MHV results + symmetries String Theory and QCD

25. Simpler than form found in 1980s despite (because of?) spurious singularities Mangano, Parke, Xu (1988) Bern, Del Duca, LD, Kosower (2004) Relative simplicity even more striking for n>6 Simple final form String Theory and QCD

26. Cauchy: residue at zk = [kth term in relation] Proof of on-shell recursion relations Britto, Cachazo, Feng, Witten, hep-th/0501052 Very simple, general – Cauchy’s theorem + factorization Let complex momentum shift depend on z. Describe using spinors. String Theory and QCD

27. but On-shell recursion at one loop Bern, LD, Kosower, hep-th/0501240, hep-th/0505055, hep-ph/0507005 • Same techniques work for one-loop amplitudes • -- much harder to obtain by other methods than are trees. • Warm up with special tree-like one-loop amplitudes • with no cuts, only poles: • New features arise compared with tree case due to • different collinear behavior of loop amplitudes: • With a little guesswork, can still find, and solve in closed form, • recursion relations for these two infinite sequences of amplitudes String Theory and QCD

28. Loop amplitudes with cuts • Also can do loop amplitudes with cuts(hep-ph/0507005) • First compute cuts using unitarity. • Remaining rational-function terms contain “spurious singularities”, e.g. • Accounting for them properly yields simple “overlap diagrams” in addition torecursive diagrams • No loop integrals required to bootstrap rational functions from cuts and lower-point amplitudes • Method tested on 5-point amplitudes, used to compute new QCD results: String Theory and QCD

29. Branch cuts • Poles Revenge of the Analytic S-matrix? Reconstruct scattering amplitudes directly from analytic properties Chew, Mandelstam; Eden, Landshoff, Olive, Polkinghorne; …(1960s) Analyticity, tied closely to string theory, fell out of favor in 1970s with rise of QCD; to resurrect it for computing perturbativeQCD amplitudes seems deliciously ironic! String Theory and QCD

30. Conclusions • Exciting new computational approaches to gauge theories due (directly or indirectly) to development of twistor string theory, appreciation of analyticity • Most practical spinoffs to date for tree amplitudes, and loops in supersymmetric theories • But now, new loop amplitudesin full QCD are beginning to fall to this approach • Expect therapid progressto continue! String Theory and QCD

31. Why does it all work? In mathematics you don't understand things. You just get used to them. String Theory and QCD

32. Extra slides String Theory and QCD

33. New form for Bern, Del Duca, LD, Kosower (2004) Simple tree found by computing one-loop amplitude first! String Theory and QCD

34. March of the n-gluon helicity amplitudes String Theory and QCD

35. March of the tree amplitudes String Theory and QCD

36. March of the 1-loop amplitudes String Theory and QCD

37. March of the 2-loop amplitudes String Theory and QCD

38. March of the 3-loop amplitudes String Theory and QCD

39. Parke-Taylor formula Initial data String Theory and QCD