1. On-Shell Methods in Gauge Theory David A. Kosower IPhT, CEA–Saclay Taiwan Summer Institute, Chi-Tou ( 溪頭 ) August 10–17, 2008 Lecture III

2. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 On-Shell Recursion Relation = Britto, Cachazo, Feng & Witten (2005)

3. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Unitarity-Based Calculations Bern, Dixon, Dunbar, & DAK (1994)

4. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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

5. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008

6. 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

7. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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…

8. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Cuts in Massless Channels With complex momenta, can form cuts using three-vertices too Britto, Cachazo, & Feng (2004)  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

9. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Quadruple Cuts Work in D=4 for the algebra Four degrees of freedom & four delta functions  no integrals left, only algebra 2

10. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 MHV in N = 4 SUSY Coefficient of a specific easy two-mass box: only one solution will contribute + + + + + + i−i− j−j− − − + − + + −+

11. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008

12. Using momentum conservation our expression becomes

13. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 We obtain the result,

14. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Remaining Integrals In QCD, we also need triangle and bubble integrals Triangles coefficients can be extracted from triple cuts But boxes have triple cuts too  need to isolate triangle from them – Subtract box integrands (Ossola, Papadopoulos, Pittau) – Isolate using different analytic behavior (Forde) – Spinor residue extraction (Britto, Feng, Mastrolia)

15. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Analytic Isolation in Triangles Define massless momenta where 1 2 3 ← l 1

16. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Analytic Isolation Parametrize 1 2 3 ← l 1

17. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Forde (2007) t is remaining unfrozen degree of freedom Each box corresponds to a pole in t Triangle has no poles at finite t, only higher-order poles at 0 and ∞ Integrals of powers of t vanish (‘total derivatives’) Constant term gives triangle coefficient: take large- t expansion of produce of trees, then set t = 0

18. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Similar technique can be applied to bubbles For numerical purposes, replace limit with discrete Fourier transform & also use OPP subtraction BlackHat: Berger, Bern, Febres Cordero, Dixon, Ita, DAK, Maître (2008) Reduces computation of coefficients to simple algebraic computation in terms of tree amplitudes

19. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Factorization at One Loop

21. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Explicit form of +++ splitting amplitude

23. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 No general theorems about factorization in complex momenta Just proceed Look at −+…++

24. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 No funny business except for channels with three-point amplitudes No funny business for three-point amplitudes with opposite- helicity gluons

26. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 On-Shell Recursion at Loop Level Bern, Dixon, DAK (2005) Complex shift of momenta, same as at tree level 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. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Rational Parts of QCD Amplitudes Start with cut-containing parts obtained from unitarity method, consider same contour integral

28. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Rational termsCut terms Consider the contour integral Determine A(0) in terms of other poles and branch cuts What poles do we have? Derivation

29. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 The Easy Two-Mass Box Seeming singularity when But canceled by dilogarithms & logarithm

30. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Spurious Singularities When evaluating the two-mass triangle, we will obtain functions like and There can be no physical singularity as and there isn’t but cancellation happens non-trivially

31. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008  the sum over residues includes spurious singularities, for which there is no factorization theorem at all

32. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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

33. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 This eliminates residues of spurious poles entirely known from four-dimensional unitarity method Assume as Modified separation so

34. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Perform integral & residue sum for so Unitarity Method???

35. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 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

36. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Recursion on rational pieces would build up rational terms, not Recursion gives Double-counted: ‘overlap’

37. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Subtract off overlap terms Compute explicitly from known Ĉ: also have a diagrammatic expression

38. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Recursive Diagrams

39. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 One-Loop Amplitudes From the point of view of the final user – Accurate and reliable codes – Large number of processes – Rapid codes – Several suppliers, to cross-check results – Easy integration into existing tools (MCFM, S HERPA, A LPGEN, H ELAC P YTHIA,…), want

40. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Numerical Approach To date: bespoke calculations Need industrialization  automation Easiest to do this numerically Numerical approach overall: worry about numerical stability Do analysis analytically Do algebra numerically Only the unitarity method combined with a numerically recursive approach for the cut trees can yield a polynomial-complexity calculation at loop level

41. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 BlackHat Carola Berger, Z. Bern, L. Dixon, Fernando Febres Cordero, Darren Forde, Harald Ita, DAK, Daniel Maître Written in C++ Framework for automated one-loop calculations – Organization in terms of integral basis (boxes, triangles, bubbles) – Assembly of different contributions Library of functions ( spinor products, integrals, residue extraction ) Tree amplitudes ( ingredients ) Caching Thus far: implemented gluon amplitudes

42. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Outline of a Calculation Unitarity freezes all propagators in box integrals: coefficients are given by a product of tree amplitudes with complex momenta ( Britto, Cachazo, Feng ) — compute them numerically Unitarity leaves one degree of freedom in triangle integrals: coefficients are the residues at ∞ ( Forde ): compute these numerically using discrete Fourier projection after subtracting boxes à la Ossola, Papadopoulos, Pittau Two degrees of freedom in bubbles: two-dimensional discrete Fourier projection, after subtractions Physical poles: on-shell recursion numerically (complex momenta) Spurious poles (‘cut completion’): locations numerically, functional form from each integral known analytically Integral functions known analytically

43. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Numerics and Examples Usually ordinary double precision is sufficient for stability For exceptional points (1-2%), use quad precision Check MHV amplitudes (− −++…+), against all- n analytic expressions Check all six-point amplitudes

44. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Unitarity-Based Method at Higher Loops Loop amplitudes on either side of the cut Multi-particle cuts in addition to two-particle cuts Find integrand/integral with given cuts in all channels

45. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Generalized Cuts In practice, replace loop amplitudes by their cuts too

46. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 At a Frontier Integral basis not known a priori Collinear factorization known to two loops Recursion relations for rational parts not derived — potential subtleties Work out cuts to derive possible integrals as well as their coefficients Some QCD calculations at two loops Most work in N = 4 supersymmetric theory

47. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 At one loop, Green, Schwarz, & Brink (1982) Two-particle cuts iterate to all orders Bern, Rozowsky, & Yan (1997) Three-particle cuts give no new information for the four-point amplitude N =4 Cuts at Two Loops = =  for all helicities one-loop scalar box

48. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Two-Loop Four-Point Result Integrand known Bern, Rozowsky, & Yan (1997) Integrals known only later

49. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 N =4 Integrand at Higher Loops Bern, Rozowsky, & Yan (1997)

50. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Four-Point Amplitude at Four Loops Motivation: compute anomalous dimension for comparison with Beisert–Eden–Staudacher equation in AdS/CFT connection Construct integral basis: – Impose no-triangle condition – Write down all four-loop integrals with maximal number of propagators – Construct additional integrals by omitting propagators

51. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008

52. Cuts Compute a set of six cuts, including multiple cuts to determine which integrals are actually present, and with which numerator factors

53. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 8 integrals present 6 given by ‘rung rule’; 2 are new. Coefficients are ±1 UV divergent in D = (vs 7, 6 for L = 2, 3) Integrals in the Amplitude

54. On-Shell Methods in Gauge Theory, Taiwan Summer Institute ( 溪頭 ), Aug 10–17, 2008 Summary: On-Shell Methods Use only information from physical states Use properties of amplitudes as calculational tools – Factorization → on-shell recursion relations – Unitarity → unitarity method – Underlying field theory → integral basis Formalism Known integral basis: Unitarity On-shell Recursion