1. Computational Methods in Particle Physics: On-Shell Methods in Field Theory David A. Kosower University of Zurich, January 31–February 14, 2007 Lecture III

2. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Review of Lecture II All- n forms for tree-level amplitudes: Parke-Taylor equations Mangano, Xu, Parke (1986) Maximally helicity-violating or ‘MHV’

3. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 MHV Rules

4. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Factorization Properties of Amplitudes As sums of external momenta approach poles, amplitudes factorize More generally as

5. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Factorization in Gauge Theories Tree level As but Sum over helicities of intermediate leg In massless theories beyond tree level, the situation is more complicated but at tree level it works in a standard way

6. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 What Happens in the Two-Particle Case? We would get a three-gluon amplitude on the left-hand side But so all invariants vanish, hence all spinor products vanish hence the three-point amplitude vanishes

7. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 In gauge theories, it holds (at tree level) for n  3 but breaks down for n =2: A 3 = 0 so we get 0/0 However A 3 only vanishes linearly, so the amplitude is not finite in this limit, but should ~ 1/ k, that is This is a collinear limit Combine amplitude with propagator to get a non-vanishing object

8. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Two-Particle Case Collinear limit: splitting amplitude

9. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Universal Factorization Amplitudes have a universal behavior in this limit Depend on a collinear momentum fraction z

10. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 In this form, a powerful tool for checking calculations As expressed in on-shell recursion relations, a powerful tool for computing amplitudes

11. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Example: Three-Particle Factorization Consider

12. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 As, it’s finite: expected because As, pick up the first term; with

13. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Splitting Amplitudes Compute it from the three-point vertex

14. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Explicit Values

15. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Collinear Factorization at One Loop

16. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Soft Factorization Universal factorization limits correspond to infrared-singular limits Recall that in addition to collinear singularities, amplitudes are also singular when a gluon momentum becomes soft The full amplitude does not have a simple factorization property in the soft limit, but color-ordered amplitudes do

17. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 The Soft factors are eikonal factors, and depend only on the helicity of the soft gluon, There are two singular invariants: this characterizes soft singularities and distinguishes them from collinear ones

18. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Combining Factorization Limits In performing real integrations over singular regions, we have two different kinds of singular regions — and their overlap — and different functions to integrate in each Gets worse at higher loops Desirable to combine the two limits Original method: Catani & Seymour combine squared splitting and eikonal functions to give factorization of squared amplitude Also possible at the amplitude level

19. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Dipole Factorization Catani & Seymour (1996) Unify soft & collinear limits of squared matrix element Add soft limits to collinear factorization

20. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Dipole Factorization Most widely-used subtraction method in numerical calculations at NLO

21. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Antenna Factorization DAK (1998) Combine soft & collinear limits at amplitude level Add collinear “wings” to soft “core” Use tree-level current J, computed recursively Berends & Giele (1988), Dixon (1995) Remap three momenta to two massless momenta

22. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Collinear Wings Merge a  1, 1 soft, and 1  b

23. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Antenna Factorization In any singular limit (  (a,1,b)/s ab 3  0 ),

24. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Reconstruction Functions Map three momenta to two massless momenta Conserve momentum Ensure no subleading terms contribute to singular limits Generalizes to m singular emissions: m  2

25. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Limiting Values

26. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Derivation Extract all poles in s a1 and s 1b Insert complete set of states Put reconstructed momenta on shell, momentum flows between currents

27. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Gluon Examples A particular helicity Helicity-summed square (factorization of squared matrix element)

28. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Multicollinear Limits Using CSW Consider limits where so that Different helicity configurations: Birthwright, Glover, Khoze, Marquard (2005)

29. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Power-counting and phase weight  any term in the splitting amplitude must be of the form Imagine factorizing an amplitude to extract the splitting amplitude: [ ] cannot come from vertices ( q -dependent ones cancel), must come from propagators  need j nearly on-shell propagators

30. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 In the required diagrams, use the substitutions ( c collinear leg, x non-collinear, r off-shell sum of some collinear momenta, z momentum fractions, P sum of all collinear momenta)

31. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Anomalous Dimensions & Amplitudes In QCD, one-loop anomalous dimensions of twist-2 operators in the OPE are related to the tree-level Altarelli-Parisi function, Relation understood between two-loop anomalous dimensions & one-loop splitting amplitudes DAK & Uwer (2003) Twist-2 Anomalous Dimension Altarelli- Parisi function Helicity- summed splitting amplitude =  Mellin Transform

32. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Recursion Relations Considered color-ordered amplitude with one leg off-shell, amputate its polarization vector This is the Berends–Giele current Given by the sum of all ( n +1)-point color-ordered diagrams with legs 1… n on shell Follow the off-shell line into the sum of diagrams. It is attached to either a three- or four-point vertex. Other lines attaching to that vertex are also sums of diagrams with one leg off-shell and other on shell, that is currents

33. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Berends–Giele Recursion Relations Berends & Giele (1988); DAK (1989)  Polynomial complexity per helicity

34. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007

35. Properties of the Current Decoupling identity Reflection identity Conservation

36. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Explicit Solutions Strategy: solve by induction Compute explicitly shorthand Compute five-point explicitly  ansatz

37. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Complex Momenta For real momenta, but we can choose these two spinors independently and still have k 2 = 0 Recall the polarization vector: but Now when two momenta are collinear only one of the spinors has to be collinear but not necessarily both