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

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

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

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!

on-shell recursion relations N =4 = pure QCD + 4 fermions + 3 complex scalars QCD = N =4 + δN = 1 + 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

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

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)

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

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

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

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

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

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

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?

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

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

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

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)

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

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

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

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

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

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

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)

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

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

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

A 2→4 QCD Amplitude Rational terms

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

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

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!