Bootstrapping One-loop Q C D Scattering Amplitudes

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. 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) L. Dixon Bootstrapping QCD Amplitudes

3. LO = |tree|2 n=8 NNLO = 2-loop x tree* + … n=2 NLO = loop x tree* + … n=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 couplingas(m) ~ 0.1 • NLO corrections can be 30% - 80% of LO state of the art: L. Dixon Bootstrapping QCD Amplitudes

4. 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 Bootstrapping QCD Amplitudes

5. Motivation (cont.) • 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. L. Dixon Bootstrapping QCD Amplitudes

6. Bootstrapping with cuts • 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) 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 L. Dixon Bootstrapping QCD Amplitudes

7. Bootstrapping with poles • Unitarity can missrational 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 L. Dixon Bootstrapping QCD Amplitudes

8. 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 The right variables Scattering amplitudes for massless plane waves of definite 4-momentum: Lorentz vectors kim ki2=0 Take “square root” of 4-vectorskim (spin 1) use 2-component Dirac (Weyl) spinors ua(ki) (spin ½) L. Dixon Bootstrapping QCD Amplitudes

9. 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.: L. Dixon Bootstrapping QCD Amplitudes

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

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

12. On-shell tree recursion • BCFW consider a family of on-shell amplitudes An(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, L. Dixon Bootstrapping QCD Amplitudes

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

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

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

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

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

18. Parke-Taylor formula Initial data L. Dixon Bootstrapping QCD Amplitudes

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

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

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

22. 1) A(z)typically has cuts as well as poles 2) different collinear behavior of loop amplitudes leads to double poles in z, uncertainty about residues in some cases: but 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 featuresarise, compared with tree case: 3) behavior of A(z) at large zmore difficult to determine L. Dixon Bootstrapping QCD Amplitudes

23. But if weknow 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 how logs, etc., appear in Cn: Loop amplitudes with cuts Generic analytic behavior of shifted 1-loop amplitude, Cuts andpoles in z-plane: L. Dixon Bootstrapping QCD Amplitudes

24. Cancelling spurious poles However, we know how to “complete the cuts” at z=0 to cancel the spurious pole terms, using Li(r) functions: So we do a modified subtraction: modified rational part full amplitude completed-cut 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 L. Dixon Bootstrapping QCD Amplitudes

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

26. 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: Subtleties at Infinity • 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. L. Dixon Bootstrapping QCD Amplitudes

27. 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 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 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 • and are cut-constructible, • and the scalar loop contribution is: generate (most of) to be determined NMHV QCD Loop Amplitude (cont.) L. Dixon Bootstrapping QCD Amplitudes

29. 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 What shift to use? L. Dixon Bootstrapping QCD Amplitudes

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

31. which can be mimicked by: Compute recursive + overlap diagrams + corrections from , Obtain consistent amplitudes. Behavior at Infinity (cont.) • For generaln Can use a second recursion relation, obtained by shifting to determine how therational terms behave at largez. Find: L. Dixon Bootstrapping QCD Amplitudes

32. n=6 • 4 nonvanishing recursive diagramsRn where flip1 permutes: • 2 nonvanishing overlap diagramsOn Compared with 1034 1-loop Feynman diagrams (color-ordered) L. Dixon Bootstrapping QCD Amplitudes

33. Result also manifestly symmetric under Plus correct factorization limits Result for n=6 Bern, Bjerrum-Bohr, Dunbar, Ita hep-ph/0507019 • Extra rational terms, beyondL2terms from where flip1 permutes: 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 inQCD • Same technique (combined with D=4unitarity) 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 L. Dixon Bootstrapping QCD Amplitudes

35. Extra Slides L. Dixon Bootstrapping QCD Amplitudes

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

37. MHV check • Using one confirms L. Dixon Bootstrapping QCD Amplitudes

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

39. To account for double pole in z, we use a doubled propagator factor (s23). 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) The double pole diagram Want to produce: L. Dixon Bootstrapping QCD Amplitudes

40. Missing term should be related to double-pole diagram, but suppressed by factor S which includes s23 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?? “Unreal” pole underneath the double pole nonsingular in real Minkowski kinematics Want to produce: L. Dixon Bootstrapping QCD Amplitudes

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

42. Solution to recursion relation 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, QCDloop amplitudes (at 2 loops or more, all helicity amplitudes have cuts, diverge) L. Dixon Bootstrapping QCD Amplitudes

44. Fermionic solutions and L. Dixon Bootstrapping QCD Amplitudes

45. March of the n-gluon helicity amplitudes L. Dixon Bootstrapping QCD Amplitudes

46. March of the tree amplitudes L. Dixon Bootstrapping QCD Amplitudes

47. March of the 1-loop amplitudes L. Dixon Bootstrapping QCD Amplitudes

48. Branch cuts • Poles Revenge of the Analytic S-matrix? 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 perturbativeQCD amplitudes seems deliciously ironic! L. Dixon Bootstrapping QCD Amplitudes

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