On-Shell Recursion Relations for Multi-Parton One-Loop Amplitudes

for Multi-Parton One-Loop Amplitudes Carola F. Berger Stanford Linear Accelerator Center with Zvi Bern, Lance Dixon, Darren Forde, David Kosower SUSY06 June 13th, 2006

[1] Z. Bern, L. J. Dixon, D. A. Kosower, Phys. Rev. D 71, 105013 (2005) [hep-th/0501240]; Phys. Rev. D 72, 125003 (2005) [hep-ph/0505055]; Phys. Rev. D 73, 065013 (2006) [hep-ph/0507005]. [2] CFB, Z. Bern, L. J. Dixon, D. Forde, D. A. Kosower, hep-ph/0604195. [3] CFB, Z. Bern, L. J. Dixon, D. Forde, D. A. Kosower, to appear. [4] CFB, V. Del Duca, L. J. Dixon, to appear. Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 2/18

SUSY Search Missing Energy + Jets SUSY Search Missing Energy + Jets The (In)Famous Experimenters Wishlists Feynman Graphs Color Ordering, Spinors and Twistors SM background from Z( ν ν) + jets Mangano et al. Red dots: ALPGEN, shaded histogram: PYTHIA, black dots: SUSY signal ALPGEN based on exact LO multi-parton amplitudes, PYTHIA uses shower-algorithm discrepancy for events with many hard jets pp Z + 4 jets needed at NLO! Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 3/18

The (In)Famous Experimenters Wishlists SUSY Search Missing Energy + Jets The (In)Famous Experimenters Wishlists Feynman Graphs Color Ordering, Spinors and Twistors Run II Monte Carlo Workshop 2001 Single boson Diboson Triboson Heavy flavor W + 5j W W + 5j W W W + 3j t t+ 3j W + b b+ 3j W W + b b+ 3j W W W + b b+ 3j t t + γ+ 2j W + c c+ 3j W W + c c+ 3j W W W + γγ+ 3j t t + W + 2j Z+ 5j ZZ+ 5j Zγγ+ 3j t t + Z+ 2j Z + b b+ 3j ZZ + b b+ 3j W ZZ+ 3j t t + H+ 2j Z + c c+ 3j ZZ + c c+ 3j ZZZ+ 3j t b+ 2j γ+ 5j γγ+ 5j tb b+ 3j γ + b b+ 3j γ + c c+ 3j γγ + b b+ 3j γγ + c c+ 3j W Z+ 5j W Z + b b+ 3j W Z + c c+ 3j W γ+ 3j Zγ+ 3j Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 4/18

The (In)Famous Experimenters Wishlists SUSY Search Missing Energy + Jets The (In)Famous Experimenters Wishlists Feynman Graphs Color Ordering, Spinors and Twistors process wanted at NLO (V {Z, W, γ}) background to Les Houches 2005 1. pp V V + jet t th, new physics 2. pp H + 2 jets H production by vector boson fusion (VBF) 3. pp t tb b t th 4. pp t t + 2 jets t th 5. pp V V b b VBF H V V, t th, new physics 6. pp V V + 2 jets VBF H V V 7. pp V + 3 jets new physics 8. pp V V V SUSY trilepton Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 4/18

The (In)Famous Experimenters Wishlists SUSY Search Missing Energy + Jets The (In)Famous Experimenters Wishlists Feynman Graphs Color Ordering, Spinors and Twistors process wanted at NLO (V {Z, W, γ}) background to Les Houches 2005 1. pp V V + jet t th, new physics 2. pp H + 2 jets H production by vector boson fusion (VBF) 3. pp t tb b t th 4. pp t t + 2 jets t th 5. pp V V b b VBF H V V, t th, new physics 6. pp V V + 2 jets VBF H V V 7. pp V + 3 jets new physics 8. pp V V V SUSY trilepton Large number of high-multiplicity processes that need to be computed! The LHC turns on in 2007! Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 4/18

Feynman Graphs Feynman rules are too general, not optimized, do not take into SUSY Search Missing Energy + Jets The (In)Famous Experimenters Wishlists Feynman Graphs Color Ordering, Spinors and Twistors account all symmetries of the theory Vertices and propagators involve gauge-dependent off-shell states Explosive growth of number of diagrams/terms gluon legs tree level one loop 6 220 1,034 8 34,300 3,017,490 worse than even worse than Time to panic?? No! (Semi)Numerical approaches and automatization MadEvent, ALPGEN, CompHEP, GRACE, HELAC/PHEGAS,... Kramer, Soper, Nagy; Ellis, Giele, Glover, Zanderighi; Binoth, Ciccolini, Guillet, Heinrich, Kauer, Pilon, Schubert; Czakon; Anastasiou, Daleo;... Recursion relations Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 5/18

Color Ordering, Spinors and Twistors SUSY Search Missing Energy + Jets The (In)Famous Experimenters Wishlists Feynman Graphs Color Ordering, Spinors and Twistors Strip color information, only calculate diagrams with cyclic color ordering 36 diagrams (not 220) for 6 gluons at tree level Decompose one-loop QCD amplitudes (1) A QCD = A N =4 4 A N =1 + A scalar Use the right variables to expose more symmetries - spinor helicity formalism Transformation to Penrose s twistor space = half Fourier transform of spinors (only left-handed spinors transformed) amazingly simple structure of scattering amplitudes Witten; Nair; Roiban, Spradlin, Volovich Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 6/18

Color Ordering, Spinors and Twistors SUSY Search Missing Energy + Jets The (In)Famous Experimenters Wishlists Feynman Graphs Color Ordering, Spinors and Twistors Strip color information, only calculate diagrams with cyclic color ordering 36 diagrams (not 220) for 6 gluons at tree level Decompose one-loop QCD amplitudes (1) A QCD = A N =4 4 A N =1 + A scalar Use the right variables to expose more symmetries - spinor helicity formalism Transformation to Penrose s twistor space = half Fourier transform of spinors (only left-handed spinors transformed) amazingly simple structure of scattering amplitudes Witten; Nair; Roiban, Spradlin, Volovich Recycle known amplitudes via recursion relations Berends, Giele; Mahlon; Cachazo, Svrcek, Witten; Britto, Cachazo, Feng, Witten Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 6/18

Recursion at Tree Level Recursion at Tree Level s QCD at - A Disaster? + 219 more 2 p = 0 Complex continue spinors and momenta Amplitude function of complex parameter A(z) = A(p 1,..., p j (z), p j+1,..., p l (z),..., p n ) If A(z ) 0 Cauchy s theorem (2) 1 2πi C dz z A(z) = 0 A(0) is the physical amplitude Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 7/18

Recursion at Tree Level Recursion at Tree Level s QCD at - A Disaster? (3) A(0) = = poles α configs Res z=z α A(z) z 1 A L Pl...m 2 A R Poles in z correspond to physical factorizations 1 ˆP 2 l...m = 1 Pl...m 2 z j /Pl...m k C z n 1 n 1 A n = 2 k 2 p = 0 n 1 A n k n^ ^1 A k k+1 k k 1 2 Britto, Cachazo, Feng, Witten Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 8/18

s Recursion at Tree Level s QCD at - A Disaster? Proof at tree level only relies on Cauchy s theorem and basic factorization properties. See also: Draggiotis, Kleiss, Lazopoulos, Papadopoulos; Vaman, Yao Many applications at tree level SUSY - processes with massless fermions QCD - QCD is supersymmetric at tree level Massive scalars and fermions Luo, Wen Badger, Glover, Khoze, Svrcek; Forde, Kosower; Schwinn, Weinzierl; Ferrario, Rodrigo, Talavera Higgs (top loop integrated out) Gravity Badger, Dixon, Glover, Khoze Bedford, Brandhuber, Spence, Travaglini; Cachazo, Svrcek; Bjerrum-Bohr, Dunbar, Ita, Perkins, Risager Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 9/18

QCD at - A Disaster? z Recursion at Tree Level s QCD at - A Disaster? C?? Branch cuts Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 10/18

QCD at - A Disaster? z Recursion at Tree Level s QCD at - A Disaster? C?? Branch cuts Double poles, unreal poles and nonstandard factorizations Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 10/18

QCD at - A Disaster? z Recursion at Tree Level s QCD at - A Disaster? C?? large z Branch cuts Double poles, unreal poles and nonstandard factorizations A(z ) 0 Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 10/18

On-Shell Bootstrap Method On-Shell Bootstrap Method Cut Parts for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism Here: very general consistency criteria Cuts (unitarity) Poles (factorization) (4) A(z) = C(z) + R(z) Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 11/18

Cut Parts z C?? On-Shell Bootstrap Method Cut Parts for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 12/18

Cut Parts z On-Shell Bootstrap Method Cut Parts for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism C?? C(0) contains only Li, ln, π 2 cut-constructible! via (generalized) unitarity Z dlips( l 1, l 2 )A tree ( l 1, m 1,.., m 2, l 2 )A tree ( l 2, m 2 +1,.., m 1 1, l 1 ) Trees recycled into loops Bern, Dixon, Dunbar, Kosower; Bedford, Brandhuber, McNamara, Spence, Travaglini; Quigley, Rozali; Britto, Buchbinder, Cachazo, Feng, Mastrolia; Bern, Bidder, Bjerrum-Bohr, Dixon, Dunbar, Ita, Perkins Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 12/18

for Rational Parts On-Shell Bootstrap Method Cut Parts for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism A(z) = C(z) + R(z) A(0) = C(0) + Inf R poles α = C(0) + Inf R + configs 1 2πi C Res z=z α = + + R R R dz z R(z) z 1 A L Pl...m 2 A R configs Loops recycled into loops (ignoring slight subtleties with spurious singularities) Bern, Dixon, Kosower Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 13/18

Non-Standard Factorizations z C?? A(0) = C(0)+ Inf R Res z=z α poles α R(z) z On-Shell Bootstrap Method Cut Parts for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 14/18

Non-Standard Factorizations z C?? A(0) = C(0)+ Inf R Res z=z α poles α R(z) z On-Shell Bootstrap Method Cut Parts for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism = + + R R R configs Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 14/18

Non-Standard Factorizations large z z C?? A(0) = C(0)+ Inf R Res z=z α poles α R(z) z On-Shell Bootstrap Method Cut Parts for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism _ = + + R R R _ configs = 0 double pole + ratl ( +)tree _ + _ + rational ( +)tree Factorization properties unclear at one loop. Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 14/18

Large-z Contributions On-Shell Bootstrap Method Cut Parts for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism Can pick continuations to avoid either non-standard factorizations or z contributions, but in general not both! Continuation [j, l avoids non-standard factorizations (5) A(0) = C(0) + Inf R + [j,l R[j,l recurs Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 15/18

Large-z Contributions On-Shell Bootstrap Method Cut Parts for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism Can pick continuations to avoid either non-standard factorizations or z contributions, but in general not both! Continuation [j, l avoids non-standard factorizations (5) A(0) = C(0) + Inf R + [j,l R[j,l recurs Continuation [a, b has no large-parameter contributions A(0) = C(0)+R [a,b recurs (6) + non-standard channels[a,b Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 15/18

The Bootstrap Formalism On-Shell Bootstrap Method Cut Parts for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism Solution use two continuations! Extract large-parameter contributions of primary continuation from auxiliary relation (6) (7) A(0) = C(0) + R [a,b + non-standard [a,b Inf recurs Inf R = Inf [j,l [j,l R[a,b recurs (8) if Inf [non-standard [j,l channels[a,b ] = 0 [j,l Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 16/18

The Bootstrap Formalism On-Shell Bootstrap Method Cut Parts for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism Solution use two continuations! Extract large-parameter contributions of primary continuation from auxiliary relation (6) (7) A(0) = C(0) + R [a,b + non-standard [a,b Inf recurs Inf R = Inf [j,l [j,l R[a,b recurs (8) if Inf [non-standard [j,l channels[a,b ] = 0 The complete bootstrap [j,l A(0) = C(0) + R [j,l recurs Passes all nontrivial checks! + Inf [j,l R[a,b recurs CFB, Bern, Dixon, Forde, Kosower Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 16/18

Results Results To-Do List All-multiplicity formulae for (+ +... +), ( +... +) one-loop gluon amplitudes (also with a fermion pair) Bern, Dixon, Kosower All-multiplicity formulae for (+... + +... + +... +) one-loop gluon amplitudes Forde, Kosower; CFB, Bern, Dixon, Forde, Kosower All-multiplicity formulae for ( +... +) one-loop gluon amplitudes CFB, Bern, Dixon, Forde, Kosower Some all-multiplicity results for parts of Higgs plus gluons (and fermion pair) at NNLO (effective theory - top loop integrated out) CFB, Del Duca, Dixon All of the above pages Working algorithm for all other configurations of one-loop gluon amplitudes! CFB, Bern, Dixon, Forde, Kosower Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 17/18

To-Do List Results To-Do List Understand complex factorization at one loop and beyond + connection to Lagrangian? Higher loops? Massive partons (external fermions, scalars,...) Automatization Attack the wishlists... Carola F. Berger SUSY06, June 13th, 2006 for Multi-Parton One-Loop Amplitudes - 18/18