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

2. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Unitarity-Based Method for Loops Start with the unitarity of the S -matrix And obtain a method of calculation relying on sewing two trees with two propagators

3. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Bern, Dixon, Dunbar, & DAK, ph/9403226, ph/9409265

4. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Example: MHV at One Loop c c c c

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

6. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Have We Seen This Denominator Before? Consider

7. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Another View of the Unitarity Method Every color-ordered amplitude can be written as Master integrals known for general process at one loop Technology exists (Laporta algorithm + differential equations or Mellin-Barnes) to find a set for a given process at higher loops Coefficients are rational functions of the spinors

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

9. Generalized Unitarity Can sew together more than two tree amplitudes Corresponds to ‘leading singularities’ Isolates contributions of a smaller set of integrals: only integrals with propagators corresponding to cuts will show up Bern, Dixon, DAK (1997) Example: in triple cut, only boxes and triangles will contribute

10. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Can we isolate a single integral? Quadruple cuts would isolate a single box Can’t do this for one-mass, two-mass, or three-mass boxes because that would isolate a three-point amplitude Unless…

11. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Cuts in Massless Channels With complex momenta, can form cuts using three-vertices too Britto, Cachazo, & Feng, th/0412103  all box coefficients can be computed directly and algebraically, with no reduction or integration N =1 and non-supersymmetric theories need triangles and bubbles, for which integration is still needed

12. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Quadruple Cuts Work in D=4 for the algebra Four degrees of freedom & four delta functions  no integrals left, only algebra 2

13. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 MHV Coefficient of a specific easy two-mass box: only one solution will contribute + + + + + + i−i− j−j− − − + − + + −+

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

15. Using momentum conservation our expression becomes

16. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Remaining Integrals Boxes done via quadruple cuts Techniques for remaining integrals still developing – Triple cuts + ordinary cuts – Double cuts with twistor-inspired extraction of spinor poles Britto, Cachazo, Feng; Britto, Feng, Mastrolia; Anastasiou, Britto, Feng, Kunszt, Mastrolia (2005–6) Goal: reduce computation of coefficients to simple algebraic computation in terms of tree amplitudes, ideally polynomial in complexity

17. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Computing QCD Amplitudes N =4 = pure QCD + 4 fermions + 3 complex scalars QCD = N =4 + δ N = 1 + δ N = 0 chiral multiplet scalar cuts + rational D=4 unitarity cuts + rational D=4 unitarity cuts + rational D=4 unitarity D=4−2ε unitarity bootstrapor on-shell recursion relations

18. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 At tree level, True in supersymmetric theories at all loop orders Non-vanishing at one loop in QCD but finite: no possible UV or IR singularities Separate V and F terms

19. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Rational Terms At tree level, we used on-shell recursion relations We want to do the same thing here Need to confront – Presence of branch cuts – Structure of factorization

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

21. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Collinear Factorization at One Loop Most general form we can get is antisymmetric + nonsingular: two independent tensors for splitting amplitude Second tensor arises only beyond tree level, and only for like helicities

22. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Explicit form of +++ splitting amplitude

23. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 No general theorems about factorization in complex momenta Just proceed Look at −+…++

24. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Amplitudes contain factors like known from collinear limits Expect also as ‘subleading’ contributions, seen in explicit results Double poles with vertex Non-conventional single pole: one finds the double-pole, multiplied by ‘unreal’ poles

25. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 On-Shell Recursion at Loop Level Bern, Dixon, DAK (1–7/2005) Finite amplitudes are purely rational We can obtain simpler forms for known finite amplitudes ( Chalmers, Bern, Dixon, DAK; Mahlon ) These again involve spurious singularities Obtained last of the finite amplitudes: f − f + g + … g +

26. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 On-Shell Recursion at Loop Level Bern, Dixon, DAK (1–7/2005) Complex shift of momenta Behavior as z   : require A (z)  0 Basic complex analysis: treat branch cuts Knowledge of complex factorization: – at tree level, tracks known factorization for real momenta – at loop level, same for multiparticle channels; and − →−+ – Avoid ± →++

27. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Rational Parts of QCD Amplitudes Start with cut-containing parts obtained from unitarity method, consider same contour integral

28. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Rational termsCut terms Consider the contour integral Determine A(0) in terms of other poles and branch cuts Derivation

29. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Cut terms have spurious singularities  rational terms do too  the sum over residues includes spurious singularities, for which there is no factorization theorem at all

30. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Completing the Cut To solve this problem, define a modified ‘completed’ cut, adding in rational functions to cancel spurious singularities We know these have to be there, because they are generated together by integral reductions Spurious singularity is unique Rational term is not, but difference is free of spurious singularities

31. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 This eliminates residues of spurious poles entirely known from four-dimensional unitarity method Assume as Modified separation so

32. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Perform integral & residue sum for so Unitarity Method???

33. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 A Closer Look at Loop Factorization Only single poles in splitting amplitudes with cuts (like tree) Cut terms → cut terms Rational terms → rational terms Build up the latter using recursion, analogous to tree level

34. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Recursion on rational pieces would build up rational terms, not Recursion gives Double-counted: ‘overlap’

35. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Subtract off overlap terms Compute explicitly from known Ĉ: also have a diagrammatic expression

36. Computational Methods in Particle Physics: On-Shell Methods in Field Theory, Zurich, Jan 31–Feb 14, 2007 Tree-level On-Shell Recursion Relations Partition P : two or more cyclicly-consecutive momenta containing j, such that complementary set contains l, The recursion relations are On shell

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