New Techniques for the Reduction of One-Loop Scattering Amplitudes - PowerPoint PPT Presentation

New Techniques for the Reduction of One-Loop Scattering Amplitudes Giovanni Ossola New York City College of Technology City University of New York (CUNY) LoopFest 2009 Radiative Corrections for the LHC and ILC University of Wisconsin at Madison – May 7-9, 2009 Giovanni Ossola (City Tech) OPP Reduction LoopFest 2009 1 / 28

Outline 1 LHC needs NLO 2 A walk through the OPP method 3 Applications and Results Giovanni Ossola (City Tech) OPP Reduction LoopFest 2009 2 / 28

LHC needs NLO The experimental programs of LHC require high precision predictions for multi-particle processes The current need of precision goes beyond tree order. At LHC, most analyses require at least next-to-leading order calculations (NLO) The search and the interpretation of new physics requires a precise understanding of the Standard Model backgrounds. We need accurate predictions and reliable error estimates In summary: One-loop corrections for multi-particle processes! Giovanni Ossola (City Tech) OPP Reduction LoopFest 2009 3 / 28

Did NLO need LHC? Some recent calculations → Cross Sections available pp → Z Z Z and pp → t ¯ tZ [Lazopoulos, Melnikov, Petriello] p → b ¯ p ¯ bZ [Febres Cordero, Reina, Wackeroth] pp → H + 2 jets, pp → WW + jet [Campbell, Ellis, Giele, Zanderighi] pp → VV + 2 jets via VBF [Bozzi, J¨ ager, Oleari, Zeppenfeld] pp → H H H [Binoth, Karg, Kauer, Ruckl] pp → t ¯ t +jet [Ciccolini, Denner and Dittmaier] pp → VVV [Binoth, G.O., Papadopoulos, Pittau] pp → VVV with leptonic decays [Campanario, Hankele, Oleari et al ] pp → W + 3 jets [Berger et al , Ellis et al ] tb ¯ pp → t ¯ b [Bredenstein, Denner, Dittmaier, and Pozzorini] A lot of progress on 2 → 4 Giovanni Ossola (City Tech) OPP Reduction LoopFest 2009 4 / 28

Did NLO need LHC? Some recent calculations → Cross Sections available pp → Z Z Z and pp → t ¯ tZ [Lazopoulos, Melnikov, Petriello] p → b ¯ p ¯ bZ [Febres Cordero, Reina, Wackeroth] pp → H + 2 jets, pp → WW + jet [Campbell, Ellis, Giele, Zanderighi] pp → VV + 2 jets via VBF [Bozzi, J¨ ager, Oleari, Zeppenfeld] Also many New Techniques pp → H H H [Binoth, Karg, Kauer, Ruckl] pp → t ¯ t +jet [Ciccolini, Denner and Dittmaier] pp → VVV [Binoth, G.O., Papadopoulos, Pittau] pp → VVV with leptonic decays [Campanario, Hankele, Oleari et al ] pp → W + 3 jets [Berger et al , Ellis et al ] tb ¯ pp → t ¯ b [Bredenstein, Denner, Dittmaier, and Pozzorini] A lot of progress on 2 → 4 Giovanni Ossola (City Tech) OPP Reduction LoopFest 2009 4 / 28

OPP Method In 2007, we proposed a new method for the numerical evaluation of scattering amplitudes, based on a decomposition at the integrand level . G. O., C. G. Papadopoulos and R. Pittau Nucl. Phys. B 763 , 147 (2007) Some of the advantages: Universal - applicable to any process Simple - based on basic algebraic properties Automatizable - easy to implement in a computer code Giovanni Ossola (City Tech) OPP Reduction LoopFest 2009 5 / 28

OPP Method In 2007, we proposed a new method for the numerical evaluation of scattering amplitudes, based on a decomposition at the integrand level . G. O., C. G. Papadopoulos and R. Pittau Nucl. Phys. B 763 , 147 (2007) Some of the advantages: Universal - applicable to any process Simple - based on basic algebraic properties Automatizable - easy to implement in a computer code Final Task Produce a MULTI-PROCESS fully automatized NLO generator Giovanni Ossola (City Tech) OPP Reduction LoopFest 2009 5 / 28

“Standing on the shoulders of giants” 1 Passarino-Veltman Reduction to Scalar Integrals � � M = d i Box i + c i Triangle i i i � � + b i Bubble i + a i Tadpole i + R , i i Giovanni Ossola (City Tech) OPP Reduction LoopFest 2009 6 / 28

“Standing on the shoulders of giants” 1 Passarino-Veltman Reduction to Scalar Integrals � � M = d i Box i + c i Triangle i i i � � + b i Bubble i + a i Tadpole i + R , i i Set the basis for our NLO calculations Exploits the Lorentz structure q µ � B µ = 1 ] = p 1 µ B 1 ( p 1 , m 0 , m 1 ) d 4 q [ q 2 − m 2 0 ][( q + p 1 ) 2 − m 2 Giovanni Ossola (City Tech) OPP Reduction LoopFest 2009 6 / 28

“Standing on the shoulders of giants” 1 Passarino-Veltman Reduction to Scalar Integrals � � M = d i Box i + c i Triangle i i i � � + b i Bubble i + a i Tadpole i + R , i i 2 Pittau/del Aguila Recursive Tensorial Reduction Express q µ = � i G i ℓ i µ , ℓ i 2 = 0 The generated terms might reconstruct denominators D i or vanish upon integration Giovanni Ossola (City Tech) OPP Reduction LoopFest 2009 6 / 28

“Standing on the shoulders of giants” 1 Passarino-Veltman Reduction to Scalar Integrals � � M = d i Box i + c i Triangle i i i � � + b i Bubble i + a i Tadpole i + R , i i 2 Pittau/del Aguila Recursive Tensorial Reduction Express q µ = � i G i ℓ i µ , ℓ i 2 = 0 The generated terms might reconstruct denominators D i or vanish upon integration 3 “Cut-based” Techniques Aim at the direct extraction of the coefficients that multiply the scalar integral Giovanni Ossola (City Tech) OPP Reduction LoopFest 2009 6 / 28

“Standing on the shoulders of giants” 1 Passarino-Veltman Reduction to Scalar Integrals � � M = d i Box i + c i Triangle i i i � � + b i Bubble i + a i Tadpole i + R , i i 2 Pittau/del Aguila Recursive Tensorial Reduction Express q µ = � i G i ℓ i µ , ℓ i 2 = 0 The generated terms might reconstruct denominators D i or vanish upon integration 3 “Cut-based” Techniques Aim at the direct extraction of the coefficients that multiply the scalar integral Pigmaei gigantum humeris impositi plusquam ipsi gigantes vident Giovanni Ossola (City Tech) OPP Reduction LoopFest 2009 6 / 28

One-loop – Definitions Any m -point one-loop amplitude can be written, before integration, as N (¯ q ) A (¯ q ) = D 0 ¯ ¯ D 1 · · · ¯ D m − 1 where q + p i ) 2 − m 2 q 2 = q 2 + ˜ ¯ ¯ q 2 q 2 D i = (¯ , ¯ , D i = D i + ˜ i Our task is to calculate, for each phase space point: N (¯ q ) � � d n ¯ d n ¯ M = q A (¯ q ) = q D 0 ¯ ¯ D 1 . . . ¯ D m − 1 Giovanni Ossola (City Tech) OPP Reduction LoopFest 2009 7 / 28

The traditional “master” formula m − 1 � � A = d ( i 0 i 1 i 2 i 3 ) D 0 ( i 0 i 1 i 2 i 3 ) i 0 < i 1 < i 2 < i 3 m − 1 � + c ( i 0 i 1 i 2 ) C 0 ( i 0 i 1 i 2 ) i 0 < i 1 < i 2 m − 1 � + b ( i 0 i 1 ) B 0 ( i 0 i 1 ) i 0 < i 1 m − 1 � + a ( i 0 ) A 0 ( i 0 ) i 0 + rational terms Giovanni Ossola (City Tech) OPP Reduction LoopFest 2009 8 / 28

OPP “master” formula - I General expression for the 4-dim N ( q ) at the integrand level in terms of D i m − 1 m − 1 � � � d ( i 0 i 1 i 2 i 3 ) + ˜ � N ( q ) = d ( q ; i 0 i 1 i 2 i 3 ) D i i 0 < i 1 < i 2 < i 3 i � = i 0 , i 1 , i 2 , i 3 m − 1 m − 1 � � + [ c ( i 0 i 1 i 2 ) + ˜ c ( q ; i 0 i 1 i 2 )] D i i 0 < i 1 < i 2 i � = i 0 , i 1 , i 2 m − 1 � m − 1 � � b ( i 0 i 1 ) + ˜ � + b ( q ; i 0 i 1 ) D i i 0 < i 1 i � = i 0 , i 1 m − 1 m − 1 � � + [ a ( i 0 ) + ˜ a ( q ; i 0 )] D i i � = i 0 i 0 This is 4-dimensional Identity Giovanni Ossola (City Tech) OPP Reduction LoopFest 2009 9 / 28

OPP “master” formula - II m − 1 m − 1 m − 1 m − 1 h i X d ( i 0 i 1 i 2 i 3 ) + ˜ Y X Y N ( q ) = d ( q ; i 0 i 1 i 2 i 3 ) D i + [ c ( i 0 i 1 i 2 ) + ˜ c ( q ; i 0 i 1 i 2 )] D i i 0 < i 1 < i 2 < i 3 i � = i 0 , i 1 , i 2 , i 3 i 0 < i 1 < i 2 i � = i 0 , i 1 , i 2 m − 1 i m − 1 m − 1 m − 1 h X b ( i 0 i 1 ) + ˜ Y X Y + b ( q ; i 0 i 1 ) D i + [ a ( i 0 ) + ˜ a ( q ; i 0 )] D i i 0 < i 1 i � = i 0 , i 1 i 0 i � = i 0 The quantities d , c , b , a are the coefficients of all possible scalar functions The quantities ˜ c , ˜ d , ˜ b , ˜ a are the “spurious” terms → vanish upon integration It is now an algebraic problem: Any N(q) just depends on a set of coefficients, to be determined! Giovanni Ossola (City Tech) OPP Reduction LoopFest 2009 10 / 28

OPP “master” formula - II m − 1 m − 1 m − 1 m − 1 h i X d ( i 0 i 1 i 2 i 3 ) + ˜ Y X Y N ( q ) = d ( q ; i 0 i 1 i 2 i 3 ) D i + [ c ( i 0 i 1 i 2 ) + ˜ c ( q ; i 0 i 1 i 2 )] D i i 0 < i 1 < i 2 < i 3 i � = i 0 , i 1 , i 2 , i 3 i 0 < i 1 < i 2 i � = i 0 , i 1 , i 2 m − 1 i m − 1 m − 1 m − 1 h X b ( i 0 i 1 ) + ˜ Y X Y + b ( q ; i 0 i 1 ) D i + [ a ( i 0 ) + ˜ a ( q ; i 0 )] D i i 0 < i 1 i � = i 0 , i 1 i 0 i � = i 0 The quantities d , c , b , a are the coefficients of all possible scalar functions The quantities ˜ c , ˜ d , ˜ b , ˜ a are the “spurious” terms → vanish upon integration It is now an algebraic problem: Any N(q) just depends on a set of coefficients, to be determined! Choose { q i } wisely by evaluating N(q) for a set of values of the integration momentum { q i } such that some denominators D i vanish (“cuts”) Giovanni Ossola (City Tech) OPP Reduction LoopFest 2009 10 / 28

Example: 4-particles process 3 3 � � d + ˜ � � b ( i 0 i 1 ) + ˜ N ( q ) = d ( q ) + [ c ( i ) + ˜ c ( q ; i )] D i + b ( q ; i 0 i 1 ) D i 0 D i 1 i =0 i 0 < i 1 3 � + [ a ( i 0 ) + ˜ a ( q ; i 0 )] D i � = i 0 D j � = i 0 D k � = i 0 i 0 =0 We look for a q such that D 0 = D 1 = D 2 = D 3 = 0 → there are two solutions q ± 0 Giovanni Ossola (City Tech) OPP Reduction LoopFest 2009 11 / 28