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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
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LHC Is Coming, LHC Is Coming!
Only 600 days to go!
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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
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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!
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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
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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
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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)
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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
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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
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Generalized Cuts Isolate contributions of smaller set of integrals, at higher loops as well
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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
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Loop Calculations — Integral Basis
Boxes, triangles and bubbles are a basis at one loop
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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
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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?
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Recursion Relations Polynomial complexity per helicity
Berends & Giele (1988); DAK (1989) Polynomial complexity per helicity
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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
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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
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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)
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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
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Rational Parts of QCD Amplitudes
Start with cut-containing parts obtained from unitarity method, consider same contour integral
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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
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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
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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
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Double-counted: ‘overlap’
Recursion gives Subtract it off Double-counted: ‘overlap’ Compute explicitly from known Ĉ
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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)
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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
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Five-Point Example (cont.)
‘Overlap’ contributions
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Only rational terms missing
A 2→4 QCD Amplitude Bern, Dixon, Dunbar, & DAK (1994) Only rational terms missing
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A 2→4 QCD Amplitude Rational terms
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Unwinding Can we do all-n? Use shift
Apply technique from all-n massive scalar calculation
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All-Multiplicity Amplitude
Same technique can be applied to calculate a one-loop amplitude with arbitrary number of external legs
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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!
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