1. Complete QCD Amplitudes: Part II of QCD On-Shell Recursion Relations
with C. Berger, Z. Bern, L. Dixon, D. Forde
From Twistors to Amplitudes, Queen Mary, November 3–5, 2005

2. LHC Is Coming, LHC Is Coming!
Only 600 days to go!

3. Precision Perturbative QCD
 Del Duca’s talk
Predictions of signals, signals+jets
Predictions of backgrounds
Measurement of luminosity
Measurement of fundamental parameters (s, mt)
Measurement of electroweak parameters
Extraction of parton distributions — ingredients in any theoretical prediction
Everything at a hadron collider involves QCD

5. NLO Jet Physics Ingredients for n-jet computations
2 → (n+1) tree-level amplitudes
2 → n one-loop amplitudes
Bern, Dixon, DAK, Weinzierl (1993–8); Kunszt, Signer, Trocsanyi (1994); Campbell, Glover, Miller (1997)
Singular (soft & collinear) behavior of tree-level amplitudes
& their integrals over phase space
Initial-state singular behavior
Incorporated into general numerical programs
Mature technology
Giele, Glover, DAK (1993); Frixione, Kunszt, Signer (1995); Catani, Seymour (1996)
 known since the ’80s
only n=3 or W+2 (known for 10 years)
 known for ~ 10 years
 known for ~ 10 years
Barrier broken!

6. on-shell recursion relations
N =4 = pure QCD + 4 fermions + 3 complex scalars
QCD = N = δN = δN = 0
chiral multiplet
scalar
cuts + rational
cuts + rational
cuts
+ rational
Brandhuber, McNamara, Spence, Travaglini (6/2005) is recent D=4-2e ref
BDDK (1997) is earlier ref
D=4−2ε unitarity
D=4 unitarity
D=4 unitarity
D=4 unitarity
bootstrap or
on-shell recursion relations

7. On-shell Recursion Relations Iteration Relation
New Structures in Gauge Theory
Complete Loop Calculations in QCD
Twistor-String Duality
Unitarity-Based Method
Higher-Loop Integral Technology

8. Three-Vertices On-shell with real momenta this vanishes
For real momenta, , so all spinor products vanish too
For complex momenta
 or
but not necessarily both!
Witten (2003)

9. Unitarity-Based Method
Use a general property of amplitudes as a practical tool for computing them
Sew loop amplitudes out of on-shell tree amplitudes: summation of Cutkosky relation
Use knowledge of possible Feynman integrals (field theory origin) & all modern techniques: identities, modern reduction techniques, differential equations, reduction to master integrals, etc.
Can sew more than two tree amplitudes: generalized unitarity

10. Unitarity-Based Calculations
Bern, Dixon, Dunbar, & DAK (1994)
At one loop in D=4 for SUSY  full answer
For non-SUSY theory, must work in D=4-2Є  full answer
van Neerven (1986): dispersion relations converge
Merge channels rather than blindly summing: find function w/given cuts in all channels

11. Generalized Cuts
Isolate contributions of smaller set of integrals, at higher loops as well

12. 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

13. Loop Calculations — Integral Basis
Boxes, triangles and bubbles are a basis at one loop

14. Rational Terms Implicitly, we’re performing a dispersion integral
in which so long as when
If this condition isn’t satisfied, there are ‘subtraction’ ambiguities corresponding to terms in the full amplitude which have no discontinuities: rational terms
In SUSY theories, or in D=4-2ε, UV convergence is improved so there are no ambiguities

15. Factorization Also a general property of field-theory amplitudes
Constrain terms to have correct collinear limit: factorization as a calculational tool
Used in to obtain simple form for rational terms
Bern, Dixon, DAK (1997)
Can this be made systematic?

16. Recursion Relations  Polynomial complexity per helicity
Berends & Giele (1988); DAK (1989)
 Polynomial complexity per helicity

17. New Representations of Tree Amplitudes
Divergent terms in one-loop NMHV amplitude must be proportional to tree — but the representation found in explicit calculations is simpler than previously-known ones
Bern, Del Duca, Dixon, & DAK (2004)
Presence of spurious singularities in individual terms seems indispensable
Suggested new kind of recursion relation
Roiban, Spradlin, & Volovich (2004)
On-shell recursion relation
Britto, Cachazo, & Feng (2004)
Simple and general proof
Britto, Cachazo, Feng, & Witten (1/2005)
‘Connected-instanton’ representation

18. On-Shell Recursion Relations
Britto, Cachazo, Feng (2004); & Witten (1/2005)
Amplitudes written as sum over ‘factorizations’ into on-shell amplitudes — but evaluated for complex momenta

19. Very general: relies only on complex analysis + factorization
Applied to gravity
Bedford, Brandhuber, Spence, & Travaglini (2/2005)
Cachazo & Svrček (2/2005)
Massive amplitudes  Glover’s talk
Badger, Glover, Khoze, Svrček (4/2005, 7/2005)
Forde & DAK (7/2005)
Integral coefficients  Bjerrum-Bohr’s talk
Bern, Bjerrum-Bohr, Dunbar, & Ita (7/2005)
Connection to Cachazo–Svrček–Witten construction
Risager (8/2005)
CSW construction for gravity  Twistor string for N =8?
Bjerrum-Bohr, Dunbar, Ita, Perkins, & Risager (9/2005)

20. On-Shell Recursion at Loop Level
Less is more. My architecture is almost nothing — Mies van der Rohe
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, there are subtleties  Dixon’s talk
Obtain simpler forms for known finite amplitudes, involving spurious singularities

21. Rational Parts of QCD Amplitudes
Start with cut-containing parts obtained from unitarity method, consider same contour integral

22. Derivation Consider the contour integral
Determine A(0) in terms of other poles and branch cuts
Cut terms have spurious singularities  rational terms do too
Rational terms
Cut terms

23. Absorb spurious singularities in rational terms into ‘completed-cut’  no sum over residues of spurious poles
entirely known from four-dimensional unitarity method
Assume as
Modified separation
Perform integral & residue sum for

24. Loop Factorization
Proven structure for real momenta, still experimental for complex momenta
Cut terms → cut terms
Rational terms → rational terms
Build up the latter using recursion, analogous to tree level

25. Double-counted: ‘overlap’
Recursion gives
Subtract it off
Double-counted: ‘overlap’
Compute explicitly from known Ĉ

26. Sometimes, as
If , can either subtract from to define a new or simply subtract it from result
Needed for computation of and when starting from the cuts computed by Bedford, Brandhuber, Spence, & Travaglini (2004)

27. Five-Point Example Look at Recursive diagrams shift
(a) Tree vertex vanishes
(b) Loop vertex vanishes
(c) Loop vertex vanishes
(d) Tree vertex vanishes
(e) Loop vertex vanishes

28. Five-Point Example (cont.)
‘Overlap’ contributions

29. Only rational terms missing
A 2→4 QCD Amplitude
Bern, Dixon, Dunbar, & DAK (1994)
Only rational terms missing

30. A 2→4 QCD Amplitude
Rational terms

31. Unwinding Can we do all-n? Use shift
Apply technique from all-n massive scalar calculation

32. All-Multiplicity Amplitude
Same technique can be applied to calculate a one-loop amplitude with arbitrary number of external legs

33. Summary
Still many branches to explore in representations of amplitudes
The combination of the unitarity-based method and on-shell recursion relations gives a powerful and practical method for a wide variety of QCD calculations needed for LHC physics
Lots of important calculations are now feasible, and are awaiting physicists eager to do them!