Bootstrapping One-Loop 2 n Amplitudes
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1 Bootstrapping One-Loop 2 n Amplitudes Carola F. Berger Stanford Linear Accelerator Center with Zvi Bern, Lance Dixon, Darren Forde, David Kosower Loopfest V June 19th, 2006
2 [1] Z. Bern, L. J. Dixon, D. A. Kosower, Phys. Rev. D 71, (2005) [hep-th/ ]; Phys. Rev. D 72, (2005) [hep-ph/ ]; Phys. Rev. D 73, (2006) [hep-ph/ ]. [2] CFB, Z. Bern, L. J. Dixon, D. Forde, D. A. Kosower, hep-ph/ [3] CFB, Z. Bern, L. J. Dixon, D. Forde, D. A. Kosower, to appear. Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 2/23
3 The (In)Famous Les Houches 2005 Wishlist The (In)Famous Les Houches 2005 Wishlist Feynman Graphs Color Ordering, Spinors and Twistors process wanted at NLO (V {Z, W, γ}) background to 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 Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 3/23
4 The (In)Famous Les Houches 2005 Wishlist The (In)Famous Les Houches 2005 Wishlist Feynman Graphs Color Ordering, Spinors and Twistors process wanted at NLO (V {Z, W, γ}) background to 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 Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 3/23
5 Feynman Graphs The (In)Famous Les Houches 2005 Wishlist Feynman Graphs Color Ordering, Spinors and Twistors Feynman rules are too general, not optimized, do not take into account all symmetries of the theory Vertices and propagators involve gauge-dependent off-shell states In real kinematics no on-shell 3-point vertex Explosive growth of number of diagrams/terms 219 more 2 p = 0 Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 4/23
6 Feynman Graphs The (In)Famous Les Houches 2005 Wishlist Feynman Graphs Color Ordering, Spinors and Twistors Feynman rules are too general, not optimized, do not take into account all symmetries of the theory Vertices and propagators involve gauge-dependent off-shell states In real kinematics no on-shell 3-point vertex Explosive growth of number of diagrams/terms Time to panic?? Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 4/23
7 Feynman Graphs The (In)Famous Les Houches 2005 Wishlist Feynman Graphs Color Ordering, Spinors and Twistors Feynman rules are too general, not optimized, do not take into account all symmetries of the theory Vertices and propagators involve gauge-dependent off-shell states In real kinematics no on-shell 3-point vertex Explosive growth of number of diagrams/terms 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 Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 4/23
8 Color Ordering, Spinors and Twistors The (In)Famous Les Houches 2005 Wishlist Feynman Graphs Color Ordering, Spinors and Twistors Strip color information, only calculate diagrams with cyclic color ordering 36 diagrams instead of 220 for n = 6 gluons Use the right variables to expose more symmetries - spinor helicity formalism λ i = u (p i ) = 1 2 (1 γ 5)u(p i ) λi = u (p i ) = 1 2 (1 γ 5)u(p i ) i j = i j = ū (p i )u (p j ) [i j] = i j = ū (p i )u (p j ) Transformation to Penrose s twistor space (Fourier transform in λ) 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 Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 5/23
9 Recursion at Tree Level Recursion at Tree Level Proof at Tree-Level s QCD at - A Disaster? Complex continue spinors and momenta (1) (2) [j, l : λ j λ j z λ l λ l λ l zλ j p µ j p µ j (z) ˆpµ j = pµ j z j γ µ l 2 p µ l p µ l (z) ˆpµ l = p µ l z j γ µ l 2 n 1 n n 1 n^ ^1 2 1 A n = 2 k A n k A k 2 p = 0 k1 k k 1 Britto, Cachazo, Feng Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 6/23
10 Proof at Tree-Level Recursion at Tree Level Proof at Tree-Level s QCD at - A Disaster? Amplitude function of complex parameter A(z) = A(p 1,..., p j (z), p j1,..., p l (z),..., p n ) If A(z ) 0 - Cauchy s theorem 1 dz (3) 2πi C z A(z) = 0 A(0) = A(z) (4) Res z=z α z poles α Poles in z correspond to physical factorizations (5) 1 ˆP 2 l...m = 1 Pl...m 2 z j /P l...m k Britto, Cachazo, Feng, Witten C z Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 7/23
11 s Recursion at Tree Level Proof 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 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 Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 8/23
12 QCD at - A Disaster? z Recursion at Tree Level Proof at Tree-Level s QCD at - A Disaster? C?? Branch cuts with spurious singularities ln(s 1/s 2 ) (s 1 s 2 ) 2 Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 9/23
13 QCD at - A Disaster? z Recursion at Tree Level Proof at Tree-Level s QCD at - A Disaster? C?? Branch cuts with spurious singularities ln(s 1/s 2 ) (s 1 s 2 ) 2 a b Double poles, unreal poles [a b] 2 and nonstandard factorizations a b [a b], Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 9/23
14 QCD at - A Disaster? z Recursion at Tree Level Proof at Tree-Level s QCD at - A Disaster? C?? large z Branch cuts with spurious singularities ln(s 1/s 2 ) (s 1 s 2 ) 2 a b Double poles, unreal poles [a b] 2 and nonstandard factorizations A(z ) 0 a b [a b], Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 9/23
15 On-Shell Bootstrap Method On-Shell Bootstrap Method Cut Parts via Unitarity Cut Parts and Spurious Singularities for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism 6-Point Example Here: very general consistency criteria Cuts (unitarity) Poles (factorization) (6) A(z) = c Γ [C(z) R(z)] Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 10/23
16 Cut Parts via Unitarity z C?? On-Shell Bootstrap Method Cut Parts via Unitarity Cut Parts and Spurious Singularities for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism 6-Point Example Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 11/23
17 Cut Parts via Unitarity z On-Shell Bootstrap Method Cut Parts via Unitarity Cut Parts and Spurious Singularities for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism 6-Point Example 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 SUSY: R = 0 fully cut-constructible 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 Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 11/23
18 Cut Parts and Spurious Singularities On-Shell Bootstrap Method Cut Parts via Unitarity Cut Parts and Spurious Singularities for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism 6-Point Example C contains spurious singularities, ln(s 1/s 2 ) (s 1 s 2. These ) 2 cancel in the full amplitude. Reshuffle (= complete cut) terms between C and R, [( ) )] A(z) = c Γ C(z) ĈR (R(z) ĈR [ ] 1 c Γ Ĉ(z) R(z) 2πi C dz z Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 12/23
19 Cut Parts and Spurious Singularities On-Shell Bootstrap Method Cut Parts via Unitarity Cut Parts and Spurious Singularities for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism 6-Point Example C contains spurious singularities, ln(s 1/s 2 ) (s 1 s 2. These ) 2 cancel in the full amplitude. Reshuffle (= complete cut) terms between C and R, [( ) )] A(z) = c Γ C(z) ĈR (R(z) ĈR [ ] 1 c Γ Ĉ(z) R(z) 2πi ) A(0) = Inf A (Ĉ(0) Inf Ĉ R(z) (7) Res z=z α z = R R R configs Loops recycled into loops poles α Bern, Dixon, Kosower C dz z Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 12/23
20 for Rational Parts = R R R configs On-Shell Bootstrap Method Cut Parts via Unitarity Cut Parts and Spurious Singularities for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism 6-Point Example Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 13/23
21 for Rational Parts On-Shell Bootstrap Method Cut Parts via Unitarity Cut Parts and Spurious Singularities for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism 6-Point Example = R R R configs Ĉ has a rational part from cut completion, ĈR. Recursion for rational parts over all poles, including those already included in Ĉ. avoid double counting by subtracting off overlap terms O (not unique) (8) Res z=z α poles α (9) R(z) z R = R ĈR = Res z=z α poles α Res z=z α poles α R(z) z R(z) z Res z=z α poles α O ĈR(z) Bern, Dixon, Kosower z Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 13/23
22 Non-Standard Factorizations On-Shell Bootstrap Method Cut Parts via Unitarity Cut Parts and Spurious Singularities for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism 6-Point Example z C?? A(0) = Inf A c Γ [Ĉ(0) Inf Ĉ Res z=z α poles α = R R R configs R(z) z ] O??? Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 14/23
23 Non-Standard Factorizations On-Shell Bootstrap Method Cut Parts via Unitarity Cut Parts and Spurious Singularities for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism 6-Point Example large z z C?? A(0) = Inf A c Γ [Ĉ(0) Inf Ĉ Res z=z α poles α = R R R _ configs = 0 1 [a b] _ R(z) z a b [a b] 2 ] O??? rational 1 [a b] Factorization properties unclear at one loop. Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 14/23
24 Large-z Contributions On-Shell Bootstrap Method Cut Parts via Unitarity Cut Parts and Spurious Singularities for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism 6-Point Example Can pick shifts to avoid either non-standard factorizations or z contributions, but in general not both! [j, l avoids non-standard factorizations (10) A(0) = Inf Ac Γ [Ĉ(0) Inf [j,l [j,l Ĉ R [j,l recurs O[j,l ] Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 15/23
25 Large-z Contributions On-Shell Bootstrap Method Cut Parts via Unitarity Cut Parts and Spurious Singularities for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism 6-Point Example Can pick shifts to avoid either non-standard factorizations or z contributions, but in general not both! [j, l avoids non-standard factorizations (10) A(0) = Inf Ac Γ [Ĉ(0) Inf [j,l [j,l Ĉ R [j,l recurs O[j,l ] [a, b has no large-parameter contributions (11) A(0) = c Γ [Ĉ(0) Inf Ĉ R [a,b recurs [a,b non-standard channels [a,b O [a,b ] Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 15/23
26 The Bootstrap Formalism On-Shell Bootstrap Method Cut Parts via Unitarity Cut Parts and Spurious Singularities for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism 6-Point Example Solution use two shifts! Extract large-parameter contributions of primary shift from auxiliary relation (11) A(0) = c Γ» b C(0) Inf [a,b» Inf A [a,b = c Γ Inf [j,l [j,l (12) if bc(0) Inf [a,b bc R [a,b recurs O [a,b non-std [a,b Inf [j,l bc R [a,b recurs O [a,b Inf [non-standard [j,l channels[a,b ] = 0 Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 16/23
27 The Bootstrap Formalism On-Shell Bootstrap Method Cut Parts via Unitarity Cut Parts and Spurious Singularities for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism 6-Point Example Solution use two shifts! Extract large-parameter contributions of primary shift from auxiliary relation (11) A(0) = c Γ» b C(0) Inf [a,b» Inf A [a,b = c Γ Inf [j,l [j,l (12) if bc(0) Inf [a,b bc R [a,b recurs O [a,b non-std [a,b Inf [j,l bc R [a,b recurs O [a,b Inf [non-standard [j,l channels[a,b ] = 0 The complete bootstrap A(0) = Inf [j,l A[a,b c Γ [Ĉ(0) Inf [j,l Ĉ R [j,l recurs O[j,l ] Passes all nontrivial checks! CFB, Bern, Dixon, Forde, Kosower Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 16/23
28 Example: A (1) (1, 2, 3, 4, 5, 6 ) (13) X(1, 2, 3, 4, 5, 6) X(3, 2, 1, 6, 5, 4) flip 1 bc 6 (1, 2, 3, 4, 5, 6 ) = 1 3c Γ A N =1 6;1 (1, 2, 3, 4, 5, 6 ) (14) 2 9 Atree 6 (1, 2, 3, 4, 5, 6 ) Ĉ a 6 Ĉ a 6 flip 1 Ĉ6 a = 2 i [2 4] 1 (34) 2 h P (3 4) 1 P i [2 3] 5 (34) [4 5][5 6] 5 (12) 6 h 3 (5 4) 6 P (45) 6 P i [1 2] [1 6] 5 (3 4) 2 L 2 ( P P 2 34 (P 2 34) 3 ) L 2 ( P P 2 34 (P 2 34) 3 ) L 2 (r) = ln(r) (r 1/r)/2 (1 r) 3 Bern, Bjerrum-Bohr, Dunbar, Ita Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 17/23
29 Example: A (1) (1, 2, 3, 4, 5, 6 ) (13) X(1, 2, 3, 4, 5, 6) X(3, 2, 1, 6, 5, 4) flip 1 bc 6 (1, 2, 3, 4, 5, 6 ) = 1 3c Γ A N =1 6;1 (1, 2, 3, 4, 5, 6 ) (14) 2 9 Atree 6 (1, 2, 3, 4, 5, 6 ) Ĉ a 6 Ĉ a 6 flip 1 Ĉ6 a = 2 i [2 4] 1 (34) 2 h P (3 4) 1 P i [2 3] 5 (34) [4 5][5 6] 5 (12) 6 h 3 (5 4) 6 P (45) 6 P i [1 2] [1 6] 5 (3 4) 2 L 2 ( P P 2 34 (P 2 34) 3 ) L 2 ( P P 2 34 (P 2 34) 3 ) L 2 (r) = ln(r) (r 1/r)/2 (1 r) 3 Shift [1, 2 Bern, Bjerrum-Bohr, Dunbar, Ita Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 17/23
30 Example: A (1) (1, 2, 3, 4, 5, 6 ) contd. On-Shell Bootstrap Method Cut Parts via Unitarity Cut Parts and Spurious Singularities for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism 6-Point Example Inf [1,2 Ĉ 6 = lim Ĉ 6 z Recursive and overlap contributions in channels P61 2 P61 2 z 1 /P61 2 (15) P23 2 P23 2 z 1 /P23 2 (16) P234 2 P234 2 z 1 (17) /P ^ L T 3 L T 3 6 ^ 2 1 ^ ^ 2 1 ^ (f) (g) T L 6 3 T T ^ 2 (h) 1 ^ 2 ^ (i) 1 ^ (a) ^ 1 ^ 2 ^ (b) 2 (c) CR 6 CR 6 CR ^ Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 18/23
31 Example: A (1) (1, 2, 3, 4, 5, 6 ) contd. On-Shell Bootstrap Method Cut Parts via Unitarity Cut Parts and Spurious Singularities for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism 6-Point Example Auxiliary recursion relation for Inf [1,2 A ^ 4^ 3 ^3 4 ^ ^3 5 2 T L T L 6 L T 1 6 (a) 5 2 (b) 1 2 (c) Inf A 6;1(1, 2, 3, 4, 5, 6 ) = [1,2 4^ ^ L T Inf A 5;1(1, 2, ˆ3, ˆK 45, [1,2 6 ) i P45 2 A tree 3 ( ˆK 45, ˆ4, 5 ) (18) (d) 4^ 5 Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 19/23
32 Example: A (1) (1, 2, 3, 4, 5, 6 ) contd. (19) R 6 = R 6 a R 6 a flip 1 On-Shell Bootstrap Method Cut Parts via Unitarity Cut Parts and Spurious Singularities for Rational Parts Non-Standard Factorizations Large-z Contributions The Bootstrap Formalism 6-Point Example br a 6 = i < [2 3] (3 4) 2 : [4 6]3 [2 5] 5 6 [1 2] [3 4] [6 1] 1 (2 3) [3 4] 6 1 P ( ) [4 6]2 ( ) = [3 4] [6 1] ; [2 3] ! [4 6] [6 1] all-n solution CFB, Bern, Dixon, Forde, Kosower Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 20/23
33 Results Results To-Do List Summary 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 Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 21/23
34 To-Do List Results To-Do List Summary 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 Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 22/23
35 Summary One of the most remarkable discoveries in elementary particle physics has been that of the existence of the complex plane. in J. Schwinger, Particles, Sources, and Fields, Vol. I. Results To-Do List Summary Carola F. Berger Loopfest V, June 19th, 2006 Bootstrapping One-Loop 2 n Amplitudes 23/23
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