Integrand Reduction for Multi-Loop Scattering Amplitudes Pierpaolo - PowerPoint PPT Presentation

Sofja Kowaleskaja Award Integrand Reduction for Multi-Loop Scattering Amplitudes Pierpaolo Mastrolia Max Planck Institute for Theoretical Physics, Munich Physics and Astronomy Dept., University & INFN, Padova arXiv:1107.6041 [hep-ph] , JHEP 1111 (2011) 014, with Ossola arXiv:1205.7087 [hep-ph] , with Ossola, Mirabella & Peraro HP2, MPI Munich, 7.9.12

Motivation QFT and Scattering Amplitudes from a new perspective The singularity structures from complex deformation of the kinematics Amplitudes decomposition from factorization The central role of Cauchy’s Residue Theorem (and its multivariate generalization) Reduction to Master Integrals by Integrand Decomposition Identify a unique Mathematical framework for any Multi-Loop Amplitude Based on one property of Scattering Amplitudes: the quadratic Feynman denominator Outline (recall) Integrand Reduction @ 1-Loop (recall) first steps toward Integrand-Reduction @ 2-Loop Multi-Loop Amplitude decomposition (from Partial Fractioning) Multivariate Polynomial Division

Unitarity-based Methods After Integration One-Loop Integral basis :: MI’s :: Li2(x), log(x)^2, log(x), O(x) log(x) ~ 1 ; Li2(x) ~ log(x) ; Li2(x) ~ 1 Amplitude decomposition from matching cuts Integrand-Reduction Methods Before Integration Residues are polynomials in irreducible scalar products (ISP’s) ISP’s generate MI’s Amplitude decomposition from polynomial fitting on the cuts

One-Loop Integrand Decomposition Ossola, Papadopoulos, Pittau Ellis, Giele, Kunszt, Melnikov · · · N n ( q, µ 2 ) Z Z q + p i ) 2 − m 2 i = ( q + p i ) 2 − m 2 ¯ A one − loop d − 2 ✏ µ d 4 q A n ( q, µ 2 ) , A n ( q, µ 2 ) ⌘ i − µ 2 , = D i = (¯ D 0 ¯ ¯ D 1 · · · ¯ n D n − 1 q 2 = q 2 − µ 2 . / q = / ¯ q + / with ¯ µ , We use a bar to denote objects living in d = 4 − 2 � dimensions, A one − loop = c 5 , 0 + c 4 , 0 + c 4 , 4 + c 3 , 0 + c 3 , 7 + c 2 , 0 + c 2 , 9 + c 1 , 0 d+4 d+2 d+2 n Passarino, Veltman; Tarasov

One-Loop Integrand Decomposition Ossola, Papadopoulos, Pittau Ellis, Giele, Kunszt, Melnikov · · · N n ( q, µ 2 ) Z Z q + p i ) 2 − m 2 i = ( q + p i ) 2 − m 2 ¯ A one − loop d − 2 ✏ µ d 4 q A n ( q, µ 2 ) , A n ( q, µ 2 ) ⌘ i − µ 2 , = D i = (¯ D 0 ¯ ¯ D 1 · · · ¯ n D n − 1 q 2 = q 2 − µ 2 . / q = / ¯ q + / with ¯ µ , We use a bar to denote objects living in d = 4 − 2 � dimensions, A one − loop = c 5 , 0 + c 4 , 0 + c 4 , 4 + c 3 , 0 + c 3 , 7 + c 2 , 0 + c 2 , 9 + c 1 , 0 d+4 d+2 d+2 n Passarino, Veltman; Tarasov @ the integrand-level + c 4 , 0 + c 4 , 4 µ 4 + c 3 , 0 + c 3 , 7 µ 2 + c 2 , 0 + c 2 , 9 µ 2 c 5 , 0 + c 1 , 0 A n ( q, µ 2 ) 6 = D 0 ¯ ¯ D 1 ¯ D 2 ¯ D 3 ¯ D 0 ¯ ¯ D 1 ¯ D 2 ¯ D 0 ¯ ¯ D 1 ¯ D 0 ¯ ¯ ¯ D 4 D 3 D 2 D 1 D 0

One-Loop Integrand Decomposition Ossola, Papadopoulos, Pittau Ellis, Giele, Kunszt, Melnikov · · · N n ( q, µ 2 ) Z Z q + p i ) 2 − m 2 i = ( q + p i ) 2 − m 2 ¯ A one − loop d − 2 ✏ µ d 4 q A n ( q, µ 2 ) , A n ( q, µ 2 ) ⌘ i − µ 2 , = D i = (¯ D 0 ¯ ¯ D 1 · · · ¯ n D n − 1 q 2 = q 2 − µ 2 . q = / / ¯ q + / with ¯ µ , We use a bar to denote objects living in d = 4 − 2 � dimensions, A one − loop = c 5 , 0 + c 4 , 0 + c 4 , 4 + c 3 , 0 + c 3 , 7 + c 2 , 0 + c 2 , 9 + c 1 , 0 d+4 d+2 d+2 n Passarino, Veltman; Tarasov @ the integrand-level + c 4 , 0 + c 4 , 4 µ 4 + c 3 , 0 + c 3 , 7 µ 2 + c 2 , 0 + c 2 , 9 µ 2 c 5 , 0 + c 1 , 0 A n ( q, µ 2 ) 6 = D 0 ¯ ¯ D 1 ¯ D 2 ¯ D 3 ¯ D 0 ¯ ¯ D 1 ¯ D 2 ¯ D 0 ¯ ¯ D 1 ¯ D 0 ¯ ¯ ¯ D 4 D 3 D 2 D 1 D 0 + c 4 , 0 + c 4 , 4 µ 4 + f 0123 ( q, µ 2 ) + c 3 , 0 + c 3 , 7 µ 2 + f 012 ( q, µ 2 ) + c 2 , 0 + c 2 , 9 µ 2 + f 01 ( q, µ 2 ) + c 1 , 0 + f 0 ( q, µ 2 ) = c 5 , 0 + f 01234 ( q, µ 2 ) D 0 ¯ ¯ ¯ D 0 ¯ ¯ D 1 ¯ D 2 ¯ D 3 ¯ D 0 ¯ ¯ D 1 ¯ D 2 ¯ D 0 ¯ ¯ D 1 ¯ D 1 D 0 D 4 D 3 D 2 d 4 q f i 1 i 2 ··· i n ( q, µ 2 ) Z Z Spurious Terms d − 2 ✏ µ = 0 . Ossola, Papadopoulos, Pittau D i 1 ¯ ¯ D i 2 · · · ¯ D i n

Parametric form of the residues: known � � Multi-(particle)-pole decomposition n − 1 n − 1 n − 1 n − 1 n − 1 ∆ ij (¯ q ) ∆ i (¯ q ) ∆ ijk � m (¯ q ) ∆ ijk � (¯ q ) ∆ ijk (¯ q ) � � � � � + + , A (¯ q ) = + + + D i ¯ ¯ ¯ D i ¯ ¯ D j ¯ D k ¯ D � ¯ D i ¯ ¯ D j ¯ D k ¯ D i ¯ ¯ D j ¯ D j D i D m D � D k i< <m i<j i i< < � i< <k Integrand Reduction Formula � � � � n − 1 n − 1 n − 1 n − 1 n − 1 n − 1 n − 1 n − 1 n − 1 n − 1 ¯ ¯ ¯ ¯ ¯ � � � � � � � � � � N (¯ q ) = ∆ ijk � m (¯ q ) D h + ∆ ijk � (¯ q ) D h + ∆ ijk (¯ q ) D h + ∆ ij (¯ q ) D h + ∆ i (¯ q ) D h , i< <m h � = i,j,k, � ,m i< < � h � = i,j,k, � i<j i i< <k h � = i,j,k h � = i,j h � = i Use your favourite generator, (for Feynman Diagrams, or for products of tree-amplitudes), and sample N(q) as many time as the number of unknown coefficients

Cuts and Residues X X cut-associated basis · · · For each cut ( ijk · · · ), D i = D j = D k = · · · = 0, a basis of four massless vectors n o e ( ijk ··· ) , e ( ijk ··· ) , e ( ijk ··· ) , e ( ijk ··· ) n o = 1 2 2 4 ⌘ 2 ⇣ e ( ijk ··· ) e ( ijk ··· ) · e ( ijk ··· ) = e ( ijk ··· ) · e ( ijk ··· ) = 0 , = 0 , 1 3 1 4 i e ( ijk ··· ) · e ( ijk ··· ) = e ( ijk ··· ) · e ( ijk ··· ) e ( ijk ··· ) · e ( ijk ··· ) = − e ( ijk ··· ) · e ( ijk ··· ) = 0 , = 1 2 3 2 4 1 2 3 4 use independent external momenta + auxiliary orthogonal complement: 3 2 2 3 4 2 3 1 -1 , 0 , 0 0 , 0 5 1 4 1 1 4-vectors vs components • Loop momentum decomposition 4 X x α e ( ijk ··· ) q + p i = α α =1

The Shape of Residues legs basis external ( p i ) auxiliary ( v i ) 5 4 0 4 3 1 3 2 2 2 1 3 1 0 4 ∆ -variables • ISP’s = Irreducible Scalar Products: – q -components which can variate under cut-conditions – spurious: vanishing upon integration – non-spurious: non-vanishing upon integration ⇒ MI’s ⇒ • @ 1-Loop – ( q · p i ) are ALL reducible Pittau, de l’Aguila – ISP’s could be chosen to be ALL spurious – n -ple cut identifies an n -point diagram

Integrand-Reduction beyond One-Loop Ossola & P .M. (2011) Badger, Frellesvig, Zhang (2011) Zhang (2012) Mirabella, Ossola, Peraro, & P .M (2012) Kleiss, Malamos, Papadopoulos, Verheynen (2012) >>> see also Simon and Costas’ talks

Two-Loop Integrand Reduction Ossola & P .M. (2011) Four-Dimensional Algorithm � � D 1 D 1 · · · D n N ( q, k ) � � A ( q, k ) = , d 4 − 2 � q d 4 − 2 � k A ( q, k ) , D i = ( α i q + β i k + p i ) 2 − m 2 A n = i , α i , β i ∈ { 0 , 1 } D 1 D 1 · · · D n 2 educated guess: Master-Decomposition Formula (4-dim) n n n ∆ i 1 ,...,i 8 ( q, k ) ∆ i 1 ,...,i 7 ( q, k ) ∆ i 1 ,i 2 ( q, k ) � � � A ( q, k ) = + + . . . + . D i 1 D i 2 . . . D i 8 D i 1 D i 2 . . . D i 7 D i 1 D i 2 � � � i 1 < <i 8 i 1 < <i 7 i 1 < <i 2 n n n n n n � � � � � � + . . . + ∆ i 1 ,i 2 ( q, k ) D h , N ( q, k ) = ∆ i 1 ,...,i 8 ( q, k ) D h + ∆ i 1 ,...,i 7 ( q, k ) D h + i 1 < <i 2 h � = i 1 ,i 2 i 1 < <i 8 h � = i 1 ,...,i 8 i 1 < <i 7 h � = i 1 ,...,i 7 (2.5) In Dim-Reg higher-point higher-dim MI’s can appear

Problem: what is the form of the residues? “find the right variables encoding the cut-structure”

The Shape of Residues m-particle cut the vanishing of m denominators present in that diagram. • Loop momentum decomposition 4 4 X x α e ( ijk ··· ) X y α e ( ijk ··· ) q + p i = , x k + p i = , y α α α =1 α =1 m-particle residue : of ∆ i 1 ,...,i m ISP’s variables in of ∆ i 1 ,...,i m legs basis • ISP’s = Irreducible Scalar Products: external ( p i ) auxiliary ( v i ) – spurious: vanishing upon integration 5 4 0 – non-spourios: non-vanishing upon integration ⇒ MI’s 4 3 1 ) 3 2 2 • @ 2-Loop 2 1 3 – ( q · p i ) and ( k · p i ) can be ISP’s ( 6 = 1-Loop) 1 0 4 – some ISP’s could be chosen to be spurious – ISP’s from: ⇤ direct inspection of the cut-solutions ⇤ relations among scalar products via Gram’s Id’y Badger, Frellesvig, Zhang

Multi-Loop Scattering Amp’s from Multivariate Polynomial Division

Algebraic Geometry deals with multivariate polynomials in z = ( z 1 , z 2 , . . . ) . Ideal J ≡ � ω 1 ( z ) · · · ω s ( z ) � generated by ω i � � � J = i h i ( z ) ω i ( z ) polynomial coefficients h i ( z ) Multivariate polynomial division of f ( z ) modulo ω 1 ( z ) , . . . , ω s ( z ) ? > z 2 needs an order, i.e. z 1 z 2 1 � f ( z ) = � i h i ( z ) ω i ( z ) + R ( z ) h i ( z ) & R ( z ) not unique Gröbner basis { g 1 ( z ) , . . . , g r ( z ) } exists (Buchberger’s algorithm) & generates J � unique R ( z ) Hilbert’s Nullstellensatz V ( J ) = set of common zeros of J ( f = 0 in V ( J ) ) ⇒ ( f r ∈ J for some r ) Weak Nullstellensatz: ( V ( J ) = ∅ ) ⇔ ( 1 ∈ J )