1. Bootstrapping One-loop QCD Scattering Amplitudes Lance Dixon, SLAC Fermilab Theory Seminar June 8, 2006 Z. Bern, LD, D. Kosower, hep-th/0501240, hep-ph/0505055, hep-ph/0507005; C. Berger, Z. Bern, LD, D. Forde, D. Kosower, hep-ph/0604195

2. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 2 What’s a bootstrap? Perturbation theory makes a bootstrap practical in four dimensions, because it imposes a hierarchy on S-matrix elements. Build more complicated amplitudes (more loops, more legs) directly from simpler ones, without directly using Feynman diagrams. Very general consistency criteria: Cuts (unitarity) Poles (factorization)

3. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 3 Motivation: One-loop multi-leg amplitudes for Tevatron/LHC Leading-order (LO), tree-level predictions are only qualitative, due to poor convergence of expansion in strong coupling  s (  ) ~ 0.1 NLO corrections can be 30% - 80% of LO state of the art: LO = |tree| 2 n=8 NLO = loop x tree* + … n=3 NNLO = 2-loop x tree* + … n=2

4. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 4 LHC Example: SUSY Search 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 Gianotti & Mangano, hep-ph/0504221 Mangano et al. (2002) Search for missing energy + jets. SM background from Z + jets.

5. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 5 Need a flexible, efficient method to extend the range of known tree, and particularly 1-loop QCD amplitudes, for use in NLO corrections to LHC processes, etc. Semi-numerical methods have led to some progress recently, e.g. –Higgs + 4 parton amplitudes Ellis, Giele, Zanderighi, hep-ph/0506196, 0508308 – 6-gluon amplitudes Ellis, Giele, Zanderighi, hep-ph/0602185 Here discuss a more analytical approach. Anticipate faster evaluations this way, for processes amenable to this method. Motivation (cont.)

6. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 6 Unitarity efficient for determining imaginary parts of loop amplitudes: Efficient because it recycles simple trees into loops Generalized unitarity (more propagators open) for coefficients of box and triangle integrals Cut evaluation via residue extraction (algebraic) Bootstrapping with cuts Bern, LD, Kosower, hep-ph/9403226, hep-ph/9708239; Britto, Cachazo, Feng, hep-th/0412103; BCF + Buchbinder, hep-ph/0503132; Britto, Feng, Mastrolia, hep-ph/0602178

7. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 7 Unitarity can miss rational functions that have no cut. However, n-point loop amplitudes also have poles where they factorize onto lower-point amplitudes. At tree-level these data have been systematized into on-shell recursion relations Britto, Cachazo, Feng, hep-th/0412308; Britto, Cachazo, Feng, Witten, hep-th/0501052 Efficient – recycles trees into trees Can also do the same for loops Bootstrapping with poles

8. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 8 The right variables Scattering amplitudes for massless plane waves of definite 4-momentum: Lorentz vectors k i  k i 2 =0 Natural to use Lorentz-invariant products (invariant masses): But for particles with spin there is a better way massless q,g,  all have 2 helicities Take “square root” of 4-vectors k i  (spin 1) use 2-component Dirac (Weyl) spinors u  (k i ) (spin ½)

9. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 9 The right variables (cont.) Reconstruct momenta k i  from spinors using projector onto positive-energy solutions of Dirac eq.: Singular 2 x 2 matrix: also shows even for complex momenta

10. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 10 Spinor products Use antisymmetric spinor products: Instead of Lorentz products: These are complex square roots of Lorentz products:

11. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 11 Spinor Magic Spinor products precisely capture square-root + phase behavior in collinear limit. Excellent variables for helicity amplitudes scalars gauge theory angular momentum mismatch Accounts for denominators in helicity amplitudes, e.g. Parke-Taylor (MHV):

12. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 12 On-shell tree recursion BCFW consider a family of on-shell amplitudes A n (z) depending on a complex parameter z which shifts the momenta, described using spinor variables. For example, the shift: Maintains on-shell condition, and momentum conservation,

13. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 13 Apply this shift to the Parke-Taylor (MHV) amplitudes: Under the shift: So Consider: 2 poles, opposite residues MHV example

14. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 14 MHV amplitude obeys: Compute residue using factorization At kinematics are complex collinear: so MHV example (cont.)

15. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 15 The general case A k+1 and A n-k+1 are on-shell tree amplitudes with fewer legs, evaluated with 2 momenta shifted by a complex amount. Britto, Cachazo, Feng, hep-th/0412308 In k th term: which solves

16. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 16 Proof of on-shell recursion relations Same analysis as above – Cauchy’s theorem + amplitude factorization Britto, Cachazo, Feng, Witten, hep-th/0501052 Let complex momentum shift depend on z. Use analyticity in z. Cauchy: poles in z: physical factorizations residue at = [k th term]

17. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 17 To show: Propagators: Britto, Cachazo, Feng, Witten, hep-th/0501052 3-point vertices: Polarization vectors: Total:

18. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 18 Initial data Parke-Taylor formula

19. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 19 A 6-gluon example 220 Feynman diagrams for gggggg Helicity + color + MHV results + symmetries 3 recursive diagrams related by symmetry

20. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 20 Simpler than form found in 1980s Mangano, Parke, Xu (1988) Simple final form

21. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 21 Berends, Giele, Kuijf (1990) Relative simplicity grows with n Bern, Del Duca, LD, Kosower (2004)

22. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 22 On-shell recursion at one loop Bern, LD, Kosower, hep-th/0501240, hep-ph/0505055, hep-ph/0507005; C. Berger, Z. Bern, LD, D. Forde, D. Kosower, hep-ph/0604195 Similar techniques can be used to compute one-loop amplitudes – much harder to obtain by traditional methods than are trees. However, 3 new features arise, compared with tree case: but 2) different collinear behavior of loop amplitudes leads to double poles in z, uncertainty about residues in some cases: 1) A(z) typically has cuts as well as poles 3) behavior of A(z) at large z more difficult to determine

23. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 23 Generic analytic behavior of shifted 1-loop amplitude, Loop amplitudes with cuts Cuts and poles in z-plane: But if we know the cuts (via unitarity in D=4), we can subtract them: full amplitudecut-containing partrational part Shifted rational function has no cuts, but has spurious poles in z because of how logs, etc., appear in C n :

24. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 24 However, we know how to “complete the cuts” at z=0 to cancel the spurious pole terms, using L i (r) functions: Cancelling spurious poles So we do a modified subtraction: full amplitudecompleted-cut partmodified rational part New shifted rational function has no cuts, and no spurious poles. But residues of physical poles are not given by naïve factorization onto rational parts of lower-point amplitudes, due to the rational parts of the completed-cut terms, called

25. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 25 Loop amplitudes with cuts (cont.) We need a correction term from the residue of at each physical pole z  – which we call an overlap diagram full amplitude completed-cut part recursive diagramsoverlap diagrams O n The final result: Tested method on known 5-point amplitudes, used it to compute then all adjacent-MHV: Forde, Kosower, hep-ph/0509358

26. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 26 It is possible that does not vanish at. It could even blow up there. Even if it is well-behaved, might not be. In that case, also won’t be. Subtleties at Infinity As long as we know (or suspect) the behavior of we can account for it, by performing the same contour analysis on is defined such that its large z behavior matches Leads to modified recursive + overlap formula:

27. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 27 Example: NMHV Loop Amplitudes We have determined recursively all the “split-helicity” next-to- maximally-helicity-violating (NMHV) QCD loop amplitudes, i.e. those with three adjacent negative helicities: C. Berger, Z. Bern, LD, D. Forde, D. Kosower, hep-ph/0604195 As input to the recursion relation, use (rational parts of) MHV amplitudes Forde, Kosower, hep-ph/0509358 Key issue is to determine the behavior of

28. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 28 The cut-containing terms for the general “split-helicity” case have been computed recently. Bern, Bjerrum-Bohr, Dunbar, Ita hep-ph/0507019 NMHV QCD Loop Amplitude (cont.) and are cut-constructible, and the scalar loop contribution is: generate (most of) to be determined

29. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 29 What shift to use? Problem: we don’t yet understand this loop 3-vertex But we know that it vanishes for the complex-conjugate kinematics So the shift will avoid these vertices

30. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 30 Inspecting Behavior at Infinity for n=5 Bern, LD, Kosower (1993) we find that the rational terms diverge but in a very simple way: Behavior mimicked by: Total reproduces We compute recursive + overlap diagrams + corrections from,

31. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 31 For general n Behavior at Infinity (cont.) Can use a second recursion relation, obtained by shifting to determine how the rational terms behave at large z. Find: which can be mimicked by: Compute recursive + overlap diagrams + corrections from, Obtain consistent amplitudes.

32. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 32 4 nonvanishing recursive diagrams R n n=6 where flip1 permutes: 2 nonvanishing overlap diagrams O n Compared with 1034 1-loop Feynman diagrams (color-ordered)

33. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 33 Extra rational terms, beyond L 2 terms from Result for n=6 where flip1 permutes: Bern, Bjerrum-Bohr, Dunbar, Ita hep-ph/0507019 Result also manifestly symmetric under Plus correct factorization limits

34. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 34 Conclusions On-shell recursion relations can be extended fruitfully to determine rational parts of loop amplitudes – with a bit of guesswork, but there are lots of consistency checks. Method still very efficient; compact solutions found for all finite, cut-free loop amplitudes in QCD Same technique (combined with D=4 unitarity) gives more general loop amplitudes with cuts, MHV and NMHV, which are needed for NLO corrections to LHC processes. Prospects look very good for attacking a wide range of multi-parton processes in this way

35. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 35 Extra Slides

36. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 36 Example of new diagrams recursive: overlap: For rational part of Compared with 1034 1-loop Feynman diagrams (color-ordered) 7 in all

37. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 37 Using one confirms MHV check

38. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 38 A one-loop pole analysis Bern, LD, Kosower (1993) under shift plus partial fraction ??? double pole

39. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 39 The double pole diagram To account for double pole in z, we use a doubled propagator factor (s 23 ). For the “all-plus” loop 3-vertex, we use the symmetric function, In the limit of real collinear momenta, this vertex corresponds to the 1-loop splitting amplitude, BDDK (1994) Want to produce:

40. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 40 “Unreal” pole underneath the double pole Missing term should be related to double-pole diagram, but suppressed by factor S which includes s 23 Want to produce: Don’t know collinear behavior at this level, must guess the correct suppression factor: in terms of universal eikonal factors for soft gluon emission Here, multiplying double-pole diagram by gives correct missing term! Universality?? nonsingular in real Minkowski kinematics

41. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 41 A one-loop all-n recursion relation Same suppression factor works in the case of n external legs! Know it works because results agree with Mahlon, hep-ph/9312276, though much shorter formulae are obtained from this relation shiftleads to

42. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 42 Solution to recursion relation

43. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 43 External fermions too Can similarly write down recursion relations for the finite, cut-free amplitudes with 2 external fermions: and the solutions are just as compact Gives the complete set of finite, cut-free, QCD loop amplitudes (at 2 loops or more, all helicity amplitudes have cuts, diverge)

44. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 44 Fermionic solutions and

45. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 45 March of the n-gluon helicity amplitudes

46. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 46 March of the tree amplitudes

47. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 47 March of the 1-loop amplitudes

48. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 48 Revenge of the Analytic S-matrix? Branch cuts Poles Reconstruct scattering amplitudes directly from analytic properties Chew, Mandelstam; Eden, Landshoff, Olive, Polkinghorne; … (1960s) Analyticity fell out of favor in 1970s with rise of QCD; to resurrect it for computing perturbative QCD amplitudes seems deliciously ironic!

49. June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 49 Why does it all work? In mathematics you don't understand things. You just get used to them.