Structural Aspects of Numerical Loop Calculus

Transcription

1 Structural Aspects of Numerical Loop Calculus Do we need it? What is it about? Can we handle it? Giampiero Passarino Dipartimento di Fisica Teorica, Università di Torino, Italy INFN, Sezione di Torino, Italy 7th DESY Workshop on Elementary Particle Theory L & L, April G. Passarino, L & L 04

2 A Paradigm: FD & HTF When possible diagrams are written in terms of multiple Nielsen - Goncharov polylogarithms Li m1,,m n (z 1,, z n ) = >i 1 > >i n >0 n l=1 z i l l i m l l. they have two nice properties

3 A Paradigm: FD & HTF When possible diagrams are written in terms of multiple Nielsen - Goncharov polylogarithms Li m1,,m n (z 1,, z n ) = >i 1 > >i n >0 n l=1 z i l l i m l l. they have two nice properties

4 A Paradigm: FD & HTF When possible diagrams are written in terms of multiple Nielsen - Goncharov polylogarithms Li m1,,m n (z 1,, z n ) = >i 1 > >i n >0 n l=1 z i l l i m l l. they have two nice properties Induce an expansion in terms of Bernoulli numbers (miracolous acceleration of convergence), e.g. S 2,p (z) = Li 3,{1}p 1 (z, {1} p 1 ) = 1 p! l=0 B l (l + p) l! ζl+p, ζ = ln(1 z),

5 A Paradigm: FD & HTF When possible diagrams are written in terms of multiple Nielsen - Goncharov polylogarithms Li m1,,m n (z 1,, z n ) = >i 1 > >i n >0 n l=1 z i l l i m l l. they have two nice properties Induce an expansion in terms of Bernoulli numbers (miracolous acceleration of convergence), e.g. S 2,p (z) = Li 3,{1}p 1 (z, {1} p 1 ) = 1 p! l=0 B l (l + p) l! ζl+p, ζ = ln(1 z), the expansion parameter has the same cut of the function G. Passarino, L & L 04

6 Essentials of analytical approach FD are transformed into multiple sums by means of Mellin - Barnes transforms, Multiple sums are traded for integrals by means of clever tricks, from integrals algebra structures are derived Tricks in Multiple Sums

7 Essentials of analytical approach FD are transformed into multiple sums by means of Mellin - Barnes transforms, Multiple sums are traded for integrals by means of clever tricks, from integrals algebra structures are derived Tricks in Multiple Sums S 1,2 (z) = Li 2,1 (z, 1) = n=2 n 1 p=1 z n n 2 1 p,

8 Essentials of analytical approach FD are transformed into multiple sums by means of Mellin - Barnes transforms, Multiple sums are traded for integrals by means of clever tricks, from integrals algebra structures are derived Tricks in Multiple Sums S 1,2 (z) = Li 2,1 (z, 1) = n=2 n 1 p=1 z n 1 n 2 p, Use n 1 p=1 Obtain S 1,2 (z) = 1 = Li 3 (z) p = ψ(n) ψ(1) = 1 0 dx [ zn n 2 1 x dx 1 xn 1 1 x, 0 1 x n=2 dx x 1 [Li 2 (z x) Li 2 (z)]. (x z) n ] = 1 n 2 0 dx 1 x [Li 2 (z) 1 x Li 2 (z x)] G. Passarino, L & L 04

9 continued... For multiple polylogarithms we can derive similar results by using the Lerch Φ function: n 1 l=1 z l 1 = Φ(z, p, 1) z n 1 Φ(z, p, n), Φ(z, p, n) = ( 1)p 1 l p Γ (p) 0 n 1 z l giving l = z p ( 1)p 1 1 Γ (p) dx 1 (x 0 lnp 1 z)n 1 x. 1 x z l=1 1 dx xn 1 1 x z lnp 1 x, For instance we have Li n1,n 2 (z 1, z 2 ) = ( 1)n 2 (n 2 1)! {I n 1,n 2 1(z 1 z 2 ) z dx lnn2 1 x 1 x z 2 [Li n1 (z 1 ) Li n1 (x z 1 z 2 )]}, I n1,n 2 (z 1 z 2 ) = 1 0 dx x lnn 2 x Li n1 (x z 1 z 2 ), I n1,n 2 = ( 1) n 1 1 n 2! (n 1 + n 2 1)! I 1,n 1 +n 2 1(ζ) = ( 1) n 2 n 2! Li n1 +n 2 +1 (ζ). G. Passarino, L & L 04

10 But, in general, it doesn t work

11 But, in general, it doesn t work The simple, fully massive, two-loop Lauricella functions, leading to hugly multiple sums with multiple binomial coefficients and argument 1. Sunset becomes a combination of F C (a, b; c 1,, c n ; z 1,, z n ) = n i=1 m i =0 (a) M (b) M n i=1 (c i ) mi n i=1 z m i i, M = m m n, n i=1 x 1/2 i < 1 p 2 < (m 1 + m 2 + m 3 ) 2. p 2 > (m 1 + m 2 + m 3 ) 2 but around- can be analytically continued in the region threshold behavior is not available. Actually we know that the of products of Bessel functions (analytical cont. J I and K H). Sunset has an integral representation in terms

12 But, in general, it doesn t work The simple, fully massive, two-loop Lauricella functions, leading to hugly multiple sums with multiple binomial coefficients and argument 1. Sunset becomes a combination of F C (a, b; c 1,, c n ; z 1,, z n ) = n i=1 m i =0 (a) M (b) M n i=1 (c i ) mi n i=1 z m i i, M = m m n, n i=1 x 1/2 i < 1 p 2 < (m 1 + m 2 + m 3 ) 2. p 2 > (m 1 + m 2 + m 3 ) 2 but around- can be analytically continued in the region threshold behavior is not available. Actually we know that the of products of Bessel functions (analytical cont. J I and K H). Sunset has an integral representation in terms Sunset (p, m 1, m 2, m 3 ) 0 dx x 1 2 ν J ν (qx) 3 i=1 K ν (m i x), ν = n 2 1, q2 = p 2. G. Passarino, L & L 04

13 Arbitrary FD are Generalized Sunsets Combining q 1, q 2 and q 1 q 2 propagators in any two-loop diagram (left) we obtain the integral of a Sunset (right), = 1 0 dx 1 But GSunset (α 1,, α 3 ) (1 p2 M 2)2 (n+1) α, α = i α i, M 2 = i m 2 i,

14 Arbitrary FD are Generalized Sunsets Combining q 1, q 2 and q 1 q 2 propagators in any two-loop diagram (left) we obtain the integral of a Sunset (right), = 1 0 dx 1 But GSunset (α 1,, α 3 ) (1 p2 M 2)2 (n+1) α, α = i α i, M 2 = i m 2 i, So that integration over external Feynman parameters has severe stability problems since the {x}-dependent normal threshold is always integration region. G. Passarino, L & L 04

15 For all one is worth although it s fun Sunset S112 into a combinations of inte- With some effort we can cast the grals X 0 X 0 dx y {Li 1 2 ; ln(1 1 x x )}, dx (x x 0 ) y {Li 1 2 ; ln(1 1 x x )}, where y 2 = a 0 x a 1 x a 2 x a 3 x + a 4, and where X, x 0, a 0, a 4 are functions of p 2 and of internal masses. Physicist s elliptic polylogarythms? Examples:

16 For all one is worth although it s fun Sunset S112 into a combinations of inte- With some effort we can cast the grals X 0 X 0 dx y {Li 1 2 ; ln(1 1 x x )}, dx (x x 0 ) y {Li 1 2 ; ln(1 1 x x )}, where y 2 = a 0 x a 1 x a 2 x a 3 x + a 4, and where X, x 0, a 0, a 4 are functions of p 2 and of internal masses. Physicist s elliptic polylogarythms? Examples: 1 dx ln(1 x 2 ) k = ln 0 (1 x 2 ) 1/2 (1 k 2 x 2 ) 1/2 k K(k) π 2 K(k ), k = (1 k 2 ) 1/2.

17 For all one is worth although it s fun Sunset S112 into a combinations of inte- With some effort we can cast the grals X 0 X 0 dx y {Li 1 2 ; ln(1 1 x x )}, dx (x x 0 ) y {Li 1 2 ; ln(1 1 x x )}, where y 2 = a 0 x a 1 x a 2 x a 3 x + a 4, and where X, x 0, a 0, a 4 are functions of p 2 and of internal masses. Physicist s elliptic polylogarythms? Examples: 1 dx ln(1 x 2 ) k = ln 0 (1 x 2 ) 1/2 (1 k 2 x 2 ) 1/2 k K(k) π 2 K(k ), k = (1 k 2 ) 1/2. But is it useful to introduce a new class of functions new diagram? G. Passarino, L & L 04

18 General approach for numerical evaluation of arbitrary FD G = [ 1 B G S dx G(x) ], Smoothness requires that the kernel in and its first N derivatives should be continuous functions with N as large as possible. However, in most cases we will be satisfied with absolute convergence, e.g. logarithmic singularities of the kernel. This is particularly true around the zeros of B G where the large number of terms obtained by requiring continuous derivatives of higher order leads to large numerical cancellations.

19 x General approach for numerical evaluation of arbitrary FD G = [ 1 B G S dx G(x) ], is a vector of Feynman parameters, Smoothness requires that the kernel in and its first N derivatives should be continuous functions with N as large as possible. However, in most cases we will be satisfied with absolute convergence, e.g. logarithmic singularities of the kernel. This is particularly true around the zeros of B G where the large number of terms obtained by requiring continuous derivatives of higher order leads to large numerical cancellations.

20 x General approach for numerical evaluation of arbitrary FD G = [ 1 B G S dx G(x) ], is a vector of Feynman parameters, S is some simplex, Smoothness requires that the kernel in and its first N derivatives should be continuous functions with N as large as possible. However, in most cases we will be satisfied with absolute convergence, e.g. logarithmic singularities of the kernel. This is particularly true around the zeros of B G where the large number of terms obtained by requiring continuous derivatives of higher order leads to large numerical cancellations.

21 x General approach for numerical evaluation of arbitrary FD G = [ 1 B G S dx G(x) ], is a vector of Feynman parameters, G is an integrable function (in the limit ɛ 0+) and S is some simplex, Smoothness requires that the kernel in and its first N derivatives should be continuous functions with N as large as possible. However, in most cases we will be satisfied with absolute convergence, e.g. logarithmic singularities of the kernel. This is particularly true around the zeros of B G where the large number of terms obtained by requiring continuous derivatives of higher order leads to large numerical cancellations.

22 x General approach for numerical evaluation of arbitrary FD G = [ 1 B G S dx G(x) ], is a vector of Feynman parameters, G is an integrable function (in the limit ɛ 0+) and BG is a function of masses and external momenta whose zeros correspond S is some simplex, to true singularities of G, if any. Smoothness requires that the kernel in and its first N derivatives should be continuous functions with N as large as possible. However, in most cases we will be satisfied with absolute convergence, e.g. logarithmic singularities of the kernel. This is particularly true around the zeros of B G where the large number of terms obtained by requiring continuous derivatives of higher order leads to large numerical cancellations. G. Passarino, L & L 04

23 Algorithms of smoothness, example The irreducible two-loop vertex diagrams flowing inwards. V K. External momenta are p 2 6 P V K 1 4 p 1 V K 0 = 1 0 dx x 0 dy linear combination of {x, y} dependent C 0. No attempt is made to define a new class of HTF

24 Algorithms of smoothness, example The irreducible two-loop vertex diagrams flowing inwards. V K. External momenta are p 2 6 P V K 1 4 p 1 V K 0 = 1 0 dx x 0 dy linear combination of {x, y} dependent C 0. No attempt is made to define a new class of HTF d{x} 1 P ({x}) ln(1 + P ({x}) Q({x}) ) G. Passarino, L & L 04

25 Parameter dependent C 0 functions C 0 (λ ; a... f) = 1 0 dx x 0 dy V 1 λ ɛ (x, y), V (x, y) = ax 2 + by 2 + cxy + dx + ey + f i δ. The total result (no problem for ho in ɛ) reads as follows: C 0 = C λ ɛ C 01 + O ( ɛ 2),

26 Parameter dependent C 0 functions C 0 (λ ; a... f) = 1 0 dx x 0 dy V 1 λ ɛ (x, y), V (x, y) = ax 2 + by 2 + cxy + dx + ey + f i δ. Step 1 Define α to be a solution of b α 2 + c α + a = 0, introduce ( TV trick) A(y) = (c + 2 αb) y + d + e α, B(y) = b y 2 + e y + f. The total result (no problem for ho in ɛ) reads as follows: C 0 = C λ ɛ C 01 + O ( ɛ 2), G. Passarino, L & L 04

27 continued... Next Transform y y + α x, V (x, y) = A(y) x + B(y), split 1 dx x dy 1 dx αx dy = α dy 1 dx α dy 1 dx, α = 1 α, αx 0 y/α 0 y/α transform again : y = α y or y = α y, use 1 λ Aɛ x (A x + B) λɛ = (A x + B) 1 λɛ, and integrate by parts. Introduce A 1 (y) = A(α y), A 2 (y) = A( α y), B 1 (y) = B(α y), B 2 (y) = B( α y), and also Q 1,2 (y) = A 1,2 (y) + B 1,2 (y), Q 3,4 (y) = A 1,2 (y) y + B 1,2 (y), Q 5,6 (x, y) = A 1,2 (y) x + B 1,2 The result is C 0,n = 1 0 dy C 0,n(y), C 0,n = α A 1 [ln n+1 Q 1 ln n+1 Q 3 ] + α A 2 [ln n+1 Q 2 ln n+1 Q 4 ]. with Subtracted logarithmslnn Q1,2(y) = lnn Q1,2(y) lnn B1,2(y) G. Passarino, L & L 04

28 Recovering the anomalous threshold Suppose that we are considering a one-loop C 0 -function with p 2 1 = p 2 2 = m 2 and m 1 = m 3 = m, m 2 = M. Consider now one of the terms in the result, say ln Q 1 /A 1 ; we have a singularity when the zero of A 1, i.e. α y = d + e α c + 2 b α, is also a zero of B 1, which may occur only if s (s 4 m 2 + M 2 ) = 0, the anomalous threshold for this configuration. In the standard analytical approach C0 combination of 12 di-logarithms,

29 Recovering the anomalous threshold Suppose that we are considering a one-loop C 0 -function with p 2 1 = p 2 2 = m 2 and m 1 = m 3 = m, m 2 = M. Consider now one of the terms in the result, say ln Q 1 /A 1 ; we have a singularity when the zero of A 1, i.e. α y = d + e α c + 2 b α, is also a zero of B 1, which may occur only if s (s 4 m 2 + M 2 ) = 0, the anomalous threshold for this configuration. In the standard analytical approach C0 combination of 12 di-logarithms, the approach here is different and aimed to put the integrand in a form that is particular convenient for direct numerical evaluation.

30 Recovering the anomalous threshold Suppose that we are considering a one-loop C 0 -function with p 2 1 = p 2 2 = m 2 and m 1 = m 3 = m, m 2 = M. Consider now one of the terms in the result, say ln Q 1 /A 1 ; we have a singularity when the zero of A 1, i.e. α y = d + e α c + 2 b α, is also a zero of B 1, which may occur only if s (s 4 m 2 + M 2 ) = 0, the anomalous threshold for this configuration. In the standard analytical approach C0 combination of 12 di-logarithms, the approach here is different and aimed to put the integrand in a form that is particular convenient for direct numerical evaluation. The coefficients a,..., f are usually expressed in terms of masses and momenta which, however, may depend on additional Feynman parameters.

31 Recovering the anomalous threshold Suppose that we are considering a one-loop C 0 -function with p 2 1 = p 2 2 = m 2 and m 1 = m 3 = m, m 2 = M. Consider now one of the terms in the result, say ln Q 1 /A 1 ; we have a singularity when the zero of A 1, i.e. α y = d + e α c + 2 b α, is also a zero of B 1, which may occur only if s (s 4 m 2 + M 2 ) = 0, the anomalous threshold for this configuration. In the standard analytical approach C0 combination of 12 di-logarithms, the approach here is different and aimed to put the integrand in a form that is particular convenient for direct numerical evaluation. The coefficients a,..., f are usually expressed in terms of masses and momenta which, however, may depend on additional Feynman parameters. So, how do we get rid of apparent singularities?

32 Recovering the anomalous threshold Suppose that we are considering a one-loop C 0 -function with p 2 1 = p 2 2 = m 2 and m 1 = m 3 = m, m 2 = M. Consider now one of the terms in the result, say ln Q 1 /A 1 ; we have a singularity when the zero of A 1, i.e. α y = d + e α c + 2 b α, is also a zero of B 1, which may occur only if s (s 4 m 2 + M 2 ) = 0, the anomalous threshold for this configuration. In the standard analytical approach C0 combination of 12 di-logarithms, the approach here is different and aimed to put the integrand in a form that is particular convenient for direct numerical evaluation. The coefficients a,..., f are usually expressed in terms of masses and momenta which, however, may depend on additional Feynman parameters. So, how do we get rid of apparent singularities? G. Passarino, L & L 04

33 Integrable singularities and sector decomposition One of the main problems in numerical multidimensional integration is to handle integrable singularities lying in arbitrary regions of the integration volume; Extensions of standard techniques are to be preferred to procedures that automatically adapt themselves to the rate of variation of the integrand at each point ( Quasi - Semi - Analytical approach ). Example: I = 1 dx 1 dy x ln[1 + x ], a > 0. a x + y

34 Integrable singularities and sector decomposition One of the main problems in numerical multidimensional integration is to handle integrable singularities lying in arbitrary regions of the integration volume; Extensions of standard techniques are to be preferred to procedures that automatically adapt themselves to the rate of variation of the integrand at each point ( Quasi - Semi - Analytical approach ). Example: I = 1 dx 1 dy x ln[1 + x ], a > 0. a x + y The integral is well defined as it can be seen after performing the x- integration, I = 1 dy [Li 0 2 a Li 2 a + 1 ], y y

35 Integrable singularities and sector decomposition One of the main problems in numerical multidimensional integration is to handle integrable singularities lying in arbitrary regions of the integration volume; Extensions of standard techniques are to be preferred to procedures that automatically adapt themselves to the rate of variation of the integrand at each point ( Quasi - Semi - Analytical approach ). Example: I = 1 dx 1 dy x ln[1 + x ], a > 0. a x + y The integral is well defined as it can be seen after performing the x- integration, I = 1 dy [Li 0 2 a Li 2 a + 1 ], y y however a source of numerical instabilities is connected to the region where x y 0, since N/D are vanishing small in the argument of the logarithm.

36 G. Passarino, L & L 04

37 continued... A nice solution is to adopt a sector decomposition to factorize their common zero. We obtain I = 1 dx dy [ln (1 + 1 a + y ) + 1 x ln (1 + x a x + 1 )].

38 continued... A nice solution is to adopt a sector decomposition to factorize their common zero. We obtain I = 1 dx dy [ln (1 + 1 a + y ) + 1 x ln (1 + x a x + 1 )]. one can gain several orders of magnitude improvement in the returned error. For special values of external parameters a singularity may develop, for instance J(a) = 1 0 dx 1 0 dy 1 x ln[1 + x x + a y ] = 1 0 dx 1 0 dy [ln ( a y ) + 1 x ln (1 + x x + a )],

39 continued... A nice solution is to adopt a sector decomposition to factorize their common zero. We obtain I = 1 dx dy [ln (1 + 1 a + y ) + 1 x ln (1 + x a x + 1 )]. one can gain several orders of magnitude improvement in the returned error. For special values of external parameters a singularity may develop, for instance J(a) = 1 0 dx 1 0 dy 1 x ln[1 + x x + a y ] = 1 0 dx 1 0 dy [ln ( a y ) + 1 x ln (1 + x x + a )], which, after the sector decomposition shows that a = 0 is a singularity of J.

40 continued... A nice solution is to adopt a sector decomposition to factorize their common zero. We obtain I = 1 dx dy [ln (1 + 1 a + y ) + 1 x ln (1 + x a x + 1 )]. one can gain several orders of magnitude improvement in the returned error. For special values of external parameters a singularity may develop, for instance J(a) = 1 0 dx 1 0 dy 1 x ln[1 + x x + a y ] = 1 0 dx 1 0 dy [ln ( a y ) + 1 x ln (1 + x x + a )], which, after the sector decomposition shows that a = 0 is a singularity of J. A more realistic example: H = 1 0 dx 1 0 dy 1 x ln [1 + x a x + χ(y) ], χ(y) = h (y y ) (y y + ) i δ, δ 0 +.

41 G. Passarino, L & L 04

42 continued... Suppose that 0 < y < y + < 1, Split [0, 1] [0, y ] [y, y + ] [y +, 1], Transform y = y y, y = (y + y ) y + y, y = (1 y + ) y + y +, In this way all the zeros of N/D are located at the corners of [0, 1] 2 and we can apply a sector decomposition to obtain 7 sectors giving the following result: H = 1 dx 1 dy [H x H 2], 1 H 1 = y ln [1 + a h y + y (y (1 xy) y + ) ] + y ln [1 + 1 a h ( y) 2 (1 xy)y ] 1 + y ln [1 + a h ( y) 2 y ] + (1 y 1 +) ln [1 + a + h (1 y + ) 2 x y 2 + h y (1 y + )y ], x x H 2 = y (1 y) ln [1 + ] + y ln [1 + a x h y (y y y + ] a x h ( y) 2] x + (1 y + ) ln [1 + a x + h (1 y + ) ((1 y + ) y + y) ], y = y + y > 0.

43 G. Passarino, L & L 04

44 Something difficult: IR configurations p 2 P M m p 1 G. Passarino, L & L 04

45 Sector decomposition for pedestrians In multi-loop diagrams the IR singularities are often overlapping. Procedure: Place the singular points at the hedge of the parameters space and remap variables to the unit cube. Example: I = 1 0 dx dy P 2 ɛ (x, y), P (x, y) = a (1 y) x 2 + b y Decompose the integration domain x 2 x 2 x x1 0 1 x x dx 1 0 dy = 1 0 dx x 0 dy dy y 0 dx G. Passarino, L & L 04

46 continued... Remap the variables to the unit cube I = 1 0 dx dy x P 2 ɛ (x, xy) dx dy y P 2 ɛ (xy, y), Factorize I = 1 0 dx dy [ x 1 ɛ P 2 ɛ 1 (x, y) + y 1 ɛ P 2 ɛ 2 (x, y) ], P 1 = a (1 x y) x + b y, P 2 = a (1 y) x 2 + b Iterate the procedure until Perform a extract the IR poles: all polynomials are free from zeros. Taylor expansion in the factorized variables and integrate to I 2 = 1 ɛ ɛ dx P2 (x, 0) + 1 dx P2 (x, y) P2 2 (x, 0) dy y. If a, b > 0 we can integrate numerically, but this doesn t work

47 continued... Remap the variables to the unit cube I = 1 0 dx dy x P 2 ɛ (x, xy) dx dy y P 2 ɛ (xy, y), Factorize I = 1 0 dx dy [ x 1 ɛ P 2 ɛ 1 (x, y) + y 1 ɛ P 2 ɛ 2 (x, y) ], P 1 = a (1 x y) x + b y, P 2 = a (1 y) x 2 + b Iterate the procedure until Perform a extract the IR poles: all polynomials are free from zeros. Taylor expansion in the factorized variables and integrate to I 2 = 1 ɛ ɛ dx P2 (x, 0) + 1 dx P2 (x, y) P2 2 (x, 0) dy y. If a, b > 0 we can integrate numerically, but this doesn t work G. Passarino, L & L 04

48 Infrared singularities and hypergeometric functions In most of the two-loop cases sector decomposition has drawbacks: with the consequence that one cannot find any adeguate smoothness algorithm to handle the final integration. A second procedure The polynomial V of the IR C 0 can be rewritten as This transformation is designed to make V linear in x and a for all IR diagrams we seek for a transformation that make the Feynman integrand linear in some variable.

49 Infrared singularities and hypergeometric functions In most of the two-loop cases sector decomposition has drawbacks: the number of sectors tends to increase considerably; with the consequence that one cannot find any adeguate smoothness algorithm to handle the final integration. A second procedure The polynomial V of the IR C 0 can be rewritten as This transformation is designed to make V linear in x and a for all IR diagrams we seek for a transformation that make the Feynman integrand linear in some variable.

50 Infrared singularities and hypergeometric functions In most of the two-loop cases sector decomposition has drawbacks: the number of sectors tends to increase considerably; the procedure creates polynomials of very high degree; with the consequence that one cannot find any adeguate smoothness algorithm to handle the final integration. A second procedure The polynomial V of the IR C 0 can be rewritten as This transformation is designed to make V linear in x and a for all IR diagrams we seek for a transformation that make the Feynman integrand linear in some variable.

51 Infrared singularities and hypergeometric functions In most of the two-loop cases sector decomposition has drawbacks: the number of sectors tends to increase considerably; the procedure creates polynomials of very high degree; with the consequence that one cannot find any adeguate smoothness algorithm to handle the final integration. A second procedure The polynomial V of the IR C 0 can be rewritten as V (x, y) = m 2 x 2 + m 2 y 2 + (s 2 m 2 ) x y 2 m 2 x (s 2 m 2 ) y + m 2 ; transformation y = y + α x with m 2 α 2 + (s 2 m 2 ) α + m 2 = 0. This transformation is designed to make V linear in x and a for all IR diagrams we seek for a transformation that make the Feynman integrand linear in some variable. G. Passarino, L & L 04

52 continued... C 0 = C C2 0 C0 1 = (1 α) 1 dy y ɛ/2 dx V1 (x, y), C0 2 = α 1 dy y 0 0 dx V 1 ɛ/2 2 (x, y), with polynomials V 1 = [s (1 + α) y s + 2 (1 α) m 2 ] x α s y 2 2 (1 α) m 2 y + m 2, V 2 = {[α s + 2 (1 α) m 2 ] x [α s + (2 α 1) m 2 ]} y. The x-integration has the general form I i (y) = y dx[b 0 i(y) A i (y) x] 1 ɛ/2, i = 1, 2, = y B 1 ɛ/2 i 2F 1 (1 + ɛ/2, 1 ; 2 ; A i y). B i Using well-known properties 2F 1 (1 + ɛ 2, 1 ; 2 ; z) = 2 ɛ [ 2F 1 (1, 1 + ɛ 2 ; 1 + ɛ 2 ; 1 z) + (1 z) ɛ/2 2F 1 (1, 1 ɛ 2 ; 1 ɛ 2 ; 1 z)], G. Passarino, L & L 04

53 continued... we derive I i (y) = 2 A i ɛ B ɛ/2 i [1 (1 A i y) ɛ/2 ]. B i For i = 1 we can simply expand around ɛ = 0 obtaining C0 1 = 1 dy 1 0 A 1 (y) ln[1 A 1(y) B 1 (y) y], for i = 2 we find A 2 (y) = a(s, m 2 ) y, B 2 (y) = b(s, m 2 ) y 2, a(s, m 2 ) = α s + 2 (1 α) m 2, b(s, m 2 ) = α s + (2 α 1) m 2. It follows that C 2 0 = a 1 (s, m 2 ) b ɛ 2(s, m 2 ) 1 0 dy y 1+ɛ ln[1 a(s, m2 ) b(s, m 2 ) ] {1 ɛ 4 ln[1 a(s, m2 ) b(s, m 2 ) ]} = a 1 (s, m 2 ) ln[1 a(s, m2 ) b(s, m 2 ) ] {1 ɛ 1 4 ln[1 a(s, m2 ) b(s, m 2 ) ] 1 2 ln b(s, m2 )}. G. Passarino, L & L 04

54 The infrared pole has been isolated and infrared residue and infrared finite part are already in a form that allows for direct numerical integration. In general they are polynomilas in the residual Feynman parameters not more complicated than the original one and smothness algorithms become available.

55 The infrared pole has been isolated and infrared residue and infrared finite part are already in a form that allows for direct numerical integration. In general they are polynomilas in the residual Feynman parameters not more complicated than the original one and smothness algorithms become available. Another difficult problem: the collinear region Smoothness algorithms give integrals I = 1 0 dx 1 1 P (x) ln[1 + P (x) Q(x) ],

56 The infrared pole has been isolated and infrared residue and infrared finite part are already in a form that allows for direct numerical integration. In general they are polynomilas in the residual Feynman parameters not more complicated than the original one and smothness algorithms become available. Another difficult problem: the collinear region Smoothness algorithms give integrals I = 1 0 dx 1 1 P (x) ln[1 + P (x) Q(x) ], When one or more masses are vanishingly small instabilities may occur; example: I = 1 0 dx 1 0 dy 1 x χ ln[1 + x χ y ], χ = m2 + p 2 y,

57 The infrared pole has been isolated and infrared residue and infrared finite part are already in a form that allows for direct numerical integration. In general they are polynomilas in the residual Feynman parameters not more complicated than the original one and smothness algorithms become available. Another difficult problem: the collinear region Smoothness algorithms give integrals I = 1 0 dx 1 1 P (x) ln[1 + P (x) Q(x) ], When one or more masses are vanishingly small instabilities may occur; example: I = 1 0 dx 1 0 dy 1 x χ ln[1 + x χ y ], χ = m2 + p 2 y, where, for m p 2 there are instabilities around y = 0 G. Passarino, L & L 04

58 Solution Expand in masses via (multiple) Mellin - Barnes transforms;

59 Solution Expand in masses via (multiple) Mellin - Barnes transforms; if necessary, perform sector decomposition in the {x} - dependent polynomials;

60 Solution Expand in masses via (multiple) Mellin - Barnes transforms; if necessary, perform sector decomposition in the {x} - dependent polynomials; perform integration over the Feynman parameters;

61 Solution Expand in masses via (multiple) Mellin - Barnes transforms; if necessary, perform sector decomposition in the {x} - dependent polynomials; perform integration over the Feynman parameters; a typical result:

62 Solution Expand in masses via (multiple) Mellin - Barnes transforms; if necessary, perform sector decomposition in the {x} - dependent polynomials; perform integration over the Feynman parameters; a typical result: I = 1 2 i π +i i ds B(s, 1 s) F (s), F (s) = 5 n=1 I n (1 s) n + F reg(s),

63 Solution Expand in masses via (multiple) Mellin - Barnes transforms; if necessary, perform sector decomposition in the {x} - dependent polynomials; perform integration over the Feynman parameters; a typical result: I = 1 2 i π +i i ds B(s, 1 s) F (s), F (s) = 5 n=1 I n (1 s) n + F reg(s), after closing the integration s-contour over the right-hand complex halfplane one obtains I including all terms O (L n ) with n 0 (and more if it is needed)

64 Solution Expand in masses via (multiple) Mellin - Barnes transforms; if necessary, perform sector decomposition in the {x} - dependent polynomials; perform integration over the Feynman parameters; a typical result: I = 1 2 i π +i i ds B(s, 1 s) F (s), F (s) = 5 n=1 I n (1 s) n + F reg(s), after closing the integration s-contour over the right-hand complex halfplane one obtains I including all terms O (L n ) with n 0 (and more if it is needed) Remember: looking for singular points is not a blind procedure, solve first Landau equations

65 Solution Expand in masses via (multiple) Mellin - Barnes transforms; if necessary, perform sector decomposition in the {x} - dependent polynomials; perform integration over the Feynman parameters; a typical result: I = 1 2 i π +i i ds B(s, 1 s) F (s), F (s) = 5 n=1 I n (1 s) n + F reg(s), after closing the integration s-contour over the right-hand complex halfplane one obtains I including all terms O (L n ) with n 0 (and more if it is needed) Remember: looking for singular points is not a blind procedure, solve first Landau equations G. Passarino, L & L 04

66 Conclusions

67 Conclusions There exist Smoothness Algorithms that bring multi-loop Feynman integrals to a form generalizing Nielsen-Goncharov multiple polylogarithms,

68 Conclusions There exist Smoothness Algorithms that bring multi-loop Feynman integrals to a form generalizing Nielsen-Goncharov multiple polylogarithms, However, in moving from one-loop to multi-loops we move from quadratic forms to higher polynomials loosing all the corresponding tools (fractional trasformations, rationalization, etc.)

69 Conclusions There exist Smoothness Algorithms that bring multi-loop Feynman integrals to a form generalizing Nielsen-Goncharov multiple polylogarithms, However, in moving from one-loop to multi-loops we move from quadratic forms to higher polynomials loosing all the corresponding tools (fractional trasformations, rationalization, etc.) Nevertheless, we can use Numerical Loop-Calculus which, after Smoothness Algorithms, is based on

70 Conclusions There exist Smoothness Algorithms that bring multi-loop Feynman integrals to a form generalizing Nielsen-Goncharov multiple polylogarithms, However, in moving from one-loop to multi-loops we move from quadratic forms to higher polynomials loosing all the corresponding tools (fractional trasformations, rationalization, etc.) Nevertheless, we can use Numerical Loop-Calculus which, after Smoothness Algorithms, is based on a priori knwoledge of true singularities through analysis of Landau equations,

71 Conclusions There exist Smoothness Algorithms that bring multi-loop Feynman integrals to a form generalizing Nielsen-Goncharov multiple polylogarithms, However, in moving from one-loop to multi-loops we move from quadratic forms to higher polynomials loosing all the corresponding tools (fractional trasformations, rationalization, etc.) Nevertheless, we can use Numerical Loop-Calculus which, after Smoothness Algorithms, is based on a priori knwoledge of true singularities through analysis of Landau equations, Sector Decompositions for integrable singularities,

72 Conclusions There exist Smoothness Algorithms that bring multi-loop Feynman integrals to a form generalizing Nielsen-Goncharov multiple polylogarithms, However, in moving from one-loop to multi-loops we move from quadratic forms to higher polynomials loosing all the corresponding tools (fractional trasformations, rationalization, etc.) Nevertheless, we can use Numerical Loop-Calculus which, after Smoothness Algorithms, is based on a priori knwoledge of true singularities through analysis of Landau equations, Sector Decompositions for integrable singularities, Linearization and Hypergeometric function techniques for infrared configurations (classified a priori via Landau equations),

73 Conclusions There exist Smoothness Algorithms that bring multi-loop Feynman integrals to a form generalizing Nielsen-Goncharov multiple polylogarithms, However, in moving from one-loop to multi-loops we move from quadratic forms to higher polynomials loosing all the corresponding tools (fractional trasformations, rationalization, etc.) Nevertheless, we can use Numerical Loop-Calculus which, after Smoothness Algorithms, is based on a priori knwoledge of true singularities through analysis of Landau equations, Sector Decompositions for integrable singularities, Linearization and Hypergeometric function techniques for infrared configurations (classified a priori via Landau equations), Mellin - Barnes expansions for collinear configurations,

74 Conclusions There exist Smoothness Algorithms that bring multi-loop Feynman integrals to a form generalizing Nielsen-Goncharov multiple polylogarithms, However, in moving from one-loop to multi-loops we move from quadratic forms to higher polynomials loosing all the corresponding tools (fractional trasformations, rationalization, etc.) Nevertheless, we can use Numerical Loop-Calculus which, after Smoothness Algorithms, is based on a priori knwoledge of true singularities through analysis of Landau equations, Sector Decompositions for integrable singularities, Linearization and Hypergeometric function techniques for infrared configurations (classified a priori via Landau equations), Mellin - Barnes expansions for collinear configurations, Tensor reduction to generalized scalar functions (not presented here)

75 Conclusions There exist Smoothness Algorithms that bring multi-loop Feynman integrals to a form generalizing Nielsen-Goncharov multiple polylogarithms, However, in moving from one-loop to multi-loops we move from quadratic forms to higher polynomials loosing all the corresponding tools (fractional trasformations, rationalization, etc.) Nevertheless, we can use Numerical Loop-Calculus which, after Smoothness Algorithms, is based on a priori knwoledge of true singularities through analysis of Landau equations, Sector Decompositions for integrable singularities, Linearization and Hypergeometric function techniques for infrared configurations (classified a priori via Landau equations), Mellin - Barnes expansions for collinear configurations, Tensor reduction to generalized scalar functions (not presented here) Now that Basics are ready next talk will be on Physical Observables. G. Passarino, L & L 04