for Scattering Amplitudes MHV @ 30, FermiLab 18.3.2016 Pierpaolo - PowerPoint PPT Presentation

Adaptive Unitarity and Magnus Exponential for Scattering Amplitudes MHV @ 30, FermiLab 18.3.2016 Pierpaolo Mastrolia Physics and Astronomy Department Galileo Galilei University of Padova - Italy

Motivation Amplitudes & Phenomenology masses do matter non-planar diagrams may contribute integrals diverge from the beauty of simple formulas (in special kinematics) to the beauty of the structures (in arbitrary kinematics) Path Multiloop Integrand Decomposition: exploiting dimensional regularisation Magnus Series for Master Integrals

High Energy Physics Goals: Loops vs Legs Legs 0 1 2 3 4 5 6 7 8 9 5 Indirect searches High precision 4 Loops 3 2 1 Direct discovery 0 High multiplicity

Complexity: Loops vs Legs Legs 0 1 2 3 4 5 6 7 8 9 5 4 Slow progress : Loops 3 one unit O(10ys) 2 1 limitations: many kinematic invariants 0 many particle masses 2006 with Parke and Taylor in good company up to

Complexity: Loops vs Legs Legs 0 1 2 3 4 5 6 7 8 9 5 4 Loops 3 2 1 0 A u t o m a t i o n One-Loop Revolution Dramatic impact on Collider Phenomenology 2006 2015 >> Kunszt, Kosower

Why is it all that difficult? Feynman Diagrams ~ The realm of Integral Calculus ~ dx dy dz ... f(x, y, z,...)

Why is it all that difficult? Feynman Diagrams ~ The realm of Integral Calculus ~ dx dy dz ... f(x, y, z,...) Turning Integral Calculus into an Algebraic Problem

Amplitudes Decomposition: the algebraic way a = a x i + a y j + a z k Basis: {i j k} Scalar product/Projection: to extract the components a x = a.i a y = a.j a z = a.k

Projections :: On-Shell Cut-Conditions vanishing denominators 1 p 2 − m 2 − i 0 → � ( p 2 − m 2 ) 0 →

Completeness Relations: cutting “1” the richness of factorization i (-i) = 1 � | ψ n � � ψ n | = 1 . n ( p 2 − m 2 ) = (/ p − m )(/ p + m ) " µ ν = " µ " ν

Completeness Relations: cutting “1” the richness of factorization =

SuperGravity @ 40 MHV @ 30 TASI lectures @ 20 Integrand-Reduction @10 unitarity at integrand level Ossola Papadopoulos Pittau (2006) Ellis Giele Kunszt Melnikov (2007) Ossola & P.M. (2011) Badger, Frellesvig, Zhang (2011) Zhang (2012) Mirabella, Ossola, Peraro, & P.M. (2012)

One-Loop Integrand Decomposition · · · 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 + / µ , with ¯ 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 d+4 + c 3 , 0 + c 3 , 7 d+2 + c 2 , 0 + c 2 , 9 + c 1 , 0 d+2 n @ the integrand-level Ossola, Papadopoulos, Pittau + 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 5 , 0 + f 01234 ( q, µ 2 ) + c 1 , 0 + f 0 ( q, µ 2 ) 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 f’s are “ spurious” ==> integrate to 0 !!!

Improved Integrand Red’n Integrand Reduction universal 5 N ( q, µ 2 ) ∆ i 1 i 2 ...i k ( q, µ 2 ) ⇢ � X X ∆ i 1 ...i m ( q, µ 2 ) = Res i 1 ...i m D i 1 ¯ ¯ D i 2 . . . ¯ D i 1 ¯ ¯ D i 2 . . . ¯ − D i n D i k polynomial k =( m +1) i 1 <i 2 <...<i k non-polynomial non-polynomial a + b x + c x^2 + ... Ossola Papadopoulos Pittau = = P + = + P + P = P + P + P + = P + P + P + P + integrand subtraction required!

Improved Integrand Red’n Integrand Reduction with Laurent series expansion Forde; Kilgore; Badger; universal ∞ ∞ 5 N ( q, µ 2 ) ∆ i 1 i 2 ...i k ( q, µ 2 ) ⇢ � X X ∆ i 1 ...i m ( q, µ 2 ) = Res i 1 ...i m D i 1 ¯ ¯ D i 2 . . . ¯ D i 1 ¯ ¯ D i 2 . . . ¯ − D i n D i k polynomial k =( m +1) i 1 <i 2 <...<i k polynomial polynomial a + b x + c x^2 + ... a’+ b’ x + c’ x^2 + ... a’’ + b’’ x + c’’ x^2 + ... coefficients of MI’s :: a = a’+ a’’ Laurent series implemented via univariate Polynomial Division Mirabella Peraro & P .M. (2012)

2.2.2 Quintuple cut � � N (¯ q ) The residue of the quintuple-cut, ¯ D i = . . . = ¯ D m = 0, defined as, q ) = c ( ijk � m ) µ 2 . ∆ ijk � m (¯ q ) = Res ijk � m ¯ D 0 · · · ¯ ¯ 5 , 0 D n − 1 2.2.3 Quadruple cut The residue of the quadruple-cut, ¯ D i = . . . = ¯ D � = 0, defined as, � �� n − 1 � � ∆ ijk � m (¯ q ) N (¯ q ) � � µ 2 � � � c ( ijk � ) + c ( ijk � ) = c ( ijk � ) + c ( ijk � ) µ 2 + c ( ijk � ) ∆ ijk � (¯ q ) = Res ijk � µ 4 − ( K 3 · e 4 ) x 4 − ( K 3 · e 3 ) x 3 ( e 1 · e 2 ) , D 0 · · · ¯ ¯ − D i ¯ ¯ D j ¯ D k ¯ D � ¯ 4 , 1 4 , 3 4 , 0 4 , 2 4 , 4 D n − 1 D m i< <m 2.2.4 Triple cut The residue of the triple-cut, ¯ D i = ¯ D j = ¯ D k = 0, defined as, � n − 1 n − 1 � � � � � N (¯ q ) ∆ ijk � m (¯ q ) ∆ ijk � (¯ q ) � � ∆ ijk (¯ q ) = Res ijk − · · D 0 · · · ¯ ¯ − D i ¯ ¯ D j ¯ D k ¯ D � ¯ − D i ¯ ¯ D j ¯ D k ¯ D n − 1 D m D � i< <m i< < � � � � � SAMURAI � � c ( ijk ) 4 + c ( ijk ) c ( ijk ) 4 + c ( ijk ) ( e 1 · e 2 ) 3 . q ) = c ( ijk ) + c ( ijk ) ( c ( ijk ) + c ( ijk ) 3 , 8 µ 2 ) x 4 + ( c ( ijk ) + c ( ijk ) 3 , 2 x 2 3 , 5 x 2 ( e 1 · e 2 ) 2 − 3 , 3 x 3 3 , 6 x 3 3 , 7 µ 2 − 3 , 9 µ 2 ) x 3 + ¯ ( e 1 · e 2 ) + 3 3 3 , 0 3 , 1 3 , 4 Ossola Reiter Tramontano P.M. (2010) 2.2.5 Double cut S cattering AM plitudes from U nitarity-based R eduction A lgorithm at The residue of the double-cut, ¯ D i = ¯ D j = 0, defined as, the I ntegrand-level � � n − 1 n − 1 n − 1 N (¯ q ) ∆ ijk � m (¯ q ) ∆ ijk � (¯ q ) ∆ ijk (¯ q ) � � � ∆ ij (¯ q ) = Res ij D 0 · · · ¯ ¯ − D i ¯ ¯ D j ¯ D k ¯ D � ¯ − D i ¯ ¯ D j ¯ D k ¯ − D i ¯ ¯ D j ¯ D n − 1 D m D � D k i< <m i< < � i< <k � � · − − · · · � � ( e 1 · e 2 ) 2 . c ( ij ) 1 + c ( ij ) 4 + c ( ij ) 3 − c ( ij ) 2 , 7 x 1 x 4 − c ( ij ) 2 , 9 µ 2 + � � 2 , 2 x 2 2 , 4 x 2 2 , 6 x 2 = c ( ij ) 2 , 0 + c ( ij ) c ( ij ) 2 , 1 x 1 − c ( ij ) 2 , 3 x 4 − c ( ij ) + 2 , 8 x 1 x 3 ( e 1 · e 2 ) + 2 , 5 x 3 2.2.6 Single cut � The residue of the single-cut, ¯ D i = 0, defined as, · · · n − 1 n − 1 � n − 1 n − 1 � ∆ ijk (¯ q ) ∆ ij (¯ q ) ∆ ijk � m (¯ q ) ∆ ijk � (¯ q ) N (¯ q ) � � � � ∆ i (¯ q ) = Res i + D i ¯ ¯ D j ¯ D i ¯ ¯ − − D 0 · · · ¯ ¯ − D i ¯ ¯ D j ¯ D k ¯ D � ¯ − D i ¯ ¯ D j ¯ D k ¯ D k D j D n − 1 D m D � i< <k i<j i< <m i< < � · · � � = c ( i ) c ( i ) 1 , 1 x 2 + c ( i ) 1 , 2 x 1 − c ( i ) 1 , 3 x 4 − c ( i ) 1 , 0 + ( e 1 · e 2 ) . 1 , 4 x 3