On-Shell Recursion Relations for Q C D Loop Amplitudes (Part I)

1. On-Shell Recursion Relationsfor QCDLoop Amplitudes(Part I) Lance Dixon, SLAC “From Twistors to Amplitudes” Workshop Queen Mary U. of London November 3, 2005 Z. Bern, LD, D. Kosower, hep-th/0501240, hep-ph/0505055, hep-ph/0507005; C. Berger, Z. Bern, LD, D. Forde, D. Kosower, to appear

2. Motivation • 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. • Unitarity is an efficient method for determining imaginary parts of loop amplitudes: • Efficient because it recycles trees into loops L. Dixon On-Shell Recursion Relations at One Loop

3. Motivation (cont.) • Unitarity can missrational functions that have no cut. • These functions can be recovered using dimensional analysis, computing cuts to higher order in e, in dim. reg. with D=4-2e: Bern, LD, Kosower, hep-ph/9602280; Brandhuber, McNamara, Spence, Travaglini, hep-th/0506068 • But tree amplitudes are more complicated in D=4-2e than in D=4. • Seems too much information is used: • n-point loop amplitudes also factorize onto lower-point amplitudes. • At tree-level these data have been systematized into on-shell recursion relationsBritto, 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 On-Shell Recursion Relations at One Loop

4. 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 (n,1) shift: • Maintains on-shell condition, and momentum conservation, L. Dixon On-Shell Recursion Relations at One Loop

5. MHV example • Apply this shift to the Parke-Taylor (MHV) amplitudes: • Under the (n,1) shift: • So • Consider: • 2 poles, opposite residues L. Dixon On-Shell Recursion Relations at One Loop

6. MHV example (cont.) • MHV amplitude obeys: • Compute residue using factorization • At kinematics are complex collinear: • so L. Dixon On-Shell Recursion Relations at One Loop

7. 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 On-Shell Recursion Relations at One Loop

8. 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 On-Shell Recursion Relations at One Loop

9. First consider special tree-like one-loop amplitudes • with no cuts, only poles: • However, new featuresstill arise, compared with tree case, due to different collinear behavior of loop amplitudes: but On-shell recursion at one loop Bern, LD, Kosower, hep-th/0501240, hep-th/0505055 • Same techniques can be used to compute one-loop amplitudes • – which are much harder to obtain by other methods than are trees. • Again we are trying to determine a rational function of z. L. Dixon On-Shell Recursion Relations at One Loop

10. under shift plus partial fraction ??? double pole A one-loop pole analysis Bern, LD, Kosower (1993) L. Dixon On-Shell Recursion Relations at One Loop

11. 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 On-Shell Recursion Relations at One Loop

12. 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 On-Shell Recursion Relations at One Loop

13. 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 On-Shell Recursion Relations at One Loop

14. Solution to recursion relation L. Dixon On-Shell Recursion Relations at One Loop

15. 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 On-Shell Recursion Relations at One Loop

16. Fermionic solutions and L. Dixon On-Shell Recursion Relations at One Loop

17. Loop amplitudes with cuts • Can also extend same recursive technique (combined with unitarity) to loop amplitudes with cuts(hep-ph/0507005) • Here 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 the rational functions from the cuts and lower-point amplitudes • Tested method on 5-point amplitudes, used it to compute and more recently Forde, Kosower, hep-ph/0509358 L. Dixon On-Shell Recursion Relations at One Loop

18. 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 (cont.) • See also Part II (David Kosower’s talk Saturday) Generic analytic properties of shifted 1-loop amplitude, Cuts andpoles in z-plane: L. Dixon On-Shell Recursion Relations at One Loop

19. Loop amplitudes with cuts (cont.) 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 different subtraction: modified rational part full amplitude completed-cut part New shifted rational function has no cuts, and no spurious poles. But the residues of the physical poles are not given by naïve factorization onto (the rational parts of) lower-point amplitudes, due to the rational parts of the completed-cut terms, L. Dixon On-Shell Recursion Relations at One Loop

20. So we need a correction term from the residue of at each physical pole – which we call an overlap diagram The final result: recursive diagrams overlap diagrams full amplitude completed-cut part Loop amplitudes with cuts (cont.) L. Dixon On-Shell Recursion Relations at One Loop

21. As long as we know (or suspect) the behavior of • we can account for it, by performing yet another • (rational) modification to , and hence to , • and then using the master formula again. • ( is required to cancel spurious singularities, • but there is freedom to use it also to account for behavior). 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 On-Shell Recursion Relations at One Loop

22. An NMHV QCD Loop Amplitude • David Kosower will discuss the application of this formalism • to the “MHV” QCD loop amplitudes, • Here I describe the application to an NMHV QCD loop amplitude, • The general “split-helicity” case • works identically to this case Berger, Bern, LD, Forde, Kosower, to appear L. Dixon On-Shell Recursion Relations at One Loop

23. 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 On-Shell Recursion Relations at One Loop

24. Problem: we don’t • yet understand • this loop 3-vertex • But we know “experimentally” • that it vanishes for the • complex-conjugate kinematics So the shift will avoid these vertices What shift to use? L. Dixon On-Shell Recursion Relations at One Loop

25. which can be mimicked by: Add this term to definition of Compute recursive + overlap diagrams + boundary correction: 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 On-Shell Recursion Relations at One Loop

26. which can be mimicked by: Add this term to definition of . Compute recursive + overlap diagrams + boundary correction. Obtain consistent amplitudes. Behavior at Infinity (cont.) • For generaln we assume that therational terms diverge inthe analogous way L. Dixon On-Shell Recursion Relations at One Loop

27. 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 On-Shell Recursion Relations at One Loop

28. 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 • Recently extended same technique (combined with D=4unitarity) to some of the 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 On-Shell Recursion Relations at One Loop

29. Why does it all work? In mathematics you don't understand things. You just get used to them. L. Dixon On-Shell Recursion Relations at One Loop

30. Extra Slides L. Dixon On-Shell Recursion Relations at One Loop

31. 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 On-Shell Recursion Relations at One Loop

32. March of the n-gluon helicity amplitudes L. Dixon On-Shell Recursion Relations at One Loop

33. March of the tree amplitudes L. Dixon On-Shell Recursion Relations at One Loop

34. March of the 1-loop amplitudes L. Dixon On-Shell Recursion Relations at One Loop

35. MHV check • Using one confirms L. Dixon On-Shell Recursion Relations at One Loop

36. To show: Britto, Cachazo, Feng, Witten, hep-th/0501052 Propagators: 3-point vertices: Polarization vectors: Total: L. Dixon On-Shell Recursion Relations at One Loop

37. Parke-Taylor formula Initial data L. Dixon On-Shell Recursion Relations at One Loop