Automatic calculations of Feynman integrals in the Euclidean region

Transcription

1 Automatic calculations of Feynman integrals in the Euclidean region Krzysztof Kajda University of Silesia 9 dec 2008 Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

2 Outline Outline 1 Motivation 2 The starting point Feynman parameters 3 Numerical calculation using sector decomposition method CSectors.m 4 Mellin-Barnes approach 5 AMBRE-Automatic Mellin-Barnes REpresentation (new features) 6 Numerical cross-checks 7 Summary Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

3 Motivation Motivation Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

4 Motivation Motivation Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

5 Motivation Motivation Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

6 Motivation Motivation Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

7 Motivation Motivation Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

8 Feynman integral The starting point General Feynman integral L l=1 d d k a µ1 k 1... k a µm m l D ν1 n 1... Dνn d = 4 2ɛ L-number of loops n-number of internal lines m-rank of the numerator D-propagator, q 2 m 2 Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

9 Feynman parameters The starting point Feynman parameters for general integral G(k µ1 a 1... k µm a m ) = ( 1) Nν Γ(ν 1 )... Γ(ν n ) n Γ ( N ν d 2 L r 2 ( 2) r 2 r m ) dx j x νj 1 j δ ( 1 U Nν d 2 (L+1) m F Nν d 2 L r 2 ) n x i i=1 { Ar P m r} [µ 1,...,µ m] N ν = ν ν n U, F are polynomials and depend on diagram {A r P m r } [µ1,...,µm] is a tensor structure which depends on diagram and numerator Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

10 The starting point F & U polynomials M = det(m)m 1 U = det(m) F = det(m)j + Q MQ M - is L L matrix containing Feynman parameters Q - is an L dimensional vector composed of external momenta and Feynman parameters J - contains kinematic invariants and Feynman parameters More detailed description can be found in literature... Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

11 The starting point F & U polynomials Other method U = x 1 + x 2 + x 3 + x 4 F = tx 1 x 3 + sx 2 x 4 U: (i) every vertex is still connected to every other vertex by a sequence of uncut lines; (ii) no further cuts without violating (i) F : (iii) divide the graph into two disjoint parts such that within each part (i) and (ii) are obeyed and such that at least one external momentum line is connected to each part Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

12 Tensor structure The starting point The object: {A r P m r } [µ1,...,µm] is used to introduce tensor structure: Example m=2 { Ar P 2 r} [µ 1µ 2] r 2 = { A 0 P 2 + A 1 P 1 + A 2 P 0} [µ 1µ 2] = P µ1 P µ2 + g µ1µ2 m=3 { Ar P 3 r} [µ 1µ 2µ 3] = { A 0 P 3 + A 1 P 2 + A 2 P 1 + A 3 P 0} [µ 1µ 2µ 3] r 3 = P µ1 P µ2 P µ3 + g µ1µ2 P µ3 + g µ2µ3 P µ1 + g µ3µ1 P µ2 A 0, P 0 is one. A r is zero for r odd, and A r = g [µ1µ2 g µr 1µr] for r even. Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

13 The starting point Tensor structure P µi and g µiµj... P µi l [ M al Q l ] µi g µiµj ( M 1 ) ab g µiµj Example G(k µ1 1 kµ2 2 ) P µ1 P µ2 + g µ1µ2 l [ M 1l Q l ] µ1 [ M 2l Q l ] µ2 + ( M 1 ) 12 g µ1µ2 Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

14 Numerical calculation using sector decomposition method Sector decomposition method,,sector decomposition is a method to isolate divergencies from parameter integrals occurring in perturbative quantum field theory More detailed information can be found in hep-ph/ by G. Heinrich Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

15 Numerical calculation using sector decomposition method Sector decomposition method Example Two-dimensional parameter integral 1 1 I = dx dyx 1 aɛ y bɛ 1 x + (1 x)y 0 0 Division of the integration region into two sectors 1 1 I = dx dyx 1 aɛ y bɛ 1[Θ(x x + (1 x)y y) + Θ(y x)] 0 0 Substitution of y = xt in sector (1) and x = yt in sector (2) to remap the integration range to the unit square I = dxx 1 (a+b)ɛ dyy 1 (a+b)ɛ 0 0 dtt bɛ (1 x)t dtt 1 aɛ (1 y)t Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

16 Numerical calculation using sector decomposition method CSectors.m What is CSectors.m? a MATHEMATICA interface which uses c++ libraries from: sector-decomposition, Ch.Bogner, S.Weinzierl (arxiv: [hep-ph]) can build m-rank tensor structure for L-loop integrals process of numerical calculation of integrals is fully automatized Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

17 CSectors.m Numerical calculation using sector decomposition method How CSectors.m works 1 Integral 2 Feynman parameters L Y l=1 d d k µ 1 a k 1... k µm am l D ν Dν N N... Y n! nx dx jx ν j X 1 j δ 1 x i... U Nν d (L+1) m 2 na m ro [µ 1,...,µm] rp i=1 r m F Nν d 2 L r 2 3 Calculating: U and F polynomials, generating tensor structure (also polynomials) 4 Dividing expression into sum of integrals in respect of r 5 We end up with sum of the integrals of the type: I r = C n Y dx jx ν j 1 j δ 1! nx h x i U(x) a+ɛbi h F (x) c+ɛdi [Q r(x)] 6 C++ linked with GiNaC libraries is used to calculate I r integrals sector-decomposition, Ch.Bogner, S.Weinzierl (arxiv: [hep-ph]) i=1 Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

18 CSectors.m Numerical calculation using sector decomposition method How CSectors.m works 1 Integral 2 Feynman parameters L Y l=1 d d k µ 1 a k 1... k µm am l D ν Dν N N... Y n! nx dx jx ν j X 1 j δ 1 x i... U Nν d (L+1) m 2 na m ro [µ 1,...,µm] rp i=1 r m F Nν d 2 L r 2 3 Calculating: U and F polynomials, generating tensor structure (also polynomials) 4 Dividing expression into sum of integrals in respect of r 5 We end up with sum of the integrals of the type: I r = C n Y dx jx ν j 1 j δ 1! nx h x i U(x) a+ɛbi h F (x) c+ɛdi [Q r(x)] 6 C++ linked with GiNaC libraries is used to calculate I r integrals sector-decomposition, Ch.Bogner, S.Weinzierl (arxiv: [hep-ph]) i=1 Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

19 CSectors.m Numerical calculation using sector decomposition method How CSectors.m works 1 Integral 2 Feynman parameters L Y l=1 d d k µ 1 a k 1... k µm am l D ν Dν N N... Y n! nx dx jx ν j X 1 j δ 1 x i... U Nν d (L+1) m 2 na m ro [µ 1,...,µm] rp i=1 r m F Nν d 2 L r 2 3 Calculating: U and F polynomials, generating tensor structure (also polynomials) 4 Dividing expression into sum of integrals in respect of r 5 We end up with sum of the integrals of the type: I r = C n Y dx jx ν j 1 j δ 1! nx h x i U(x) a+ɛbi h F (x) c+ɛdi [Q r(x)] 6 C++ linked with GiNaC libraries is used to calculate I r integrals sector-decomposition, Ch.Bogner, S.Weinzierl (arxiv: [hep-ph]) i=1 Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

20 CSectors.m Numerical calculation using sector decomposition method How CSectors.m works 1 Integral 2 Feynman parameters L Y l=1 d d k µ 1 a k 1... k µm am l D ν Dν N N... Y n! nx dx jx ν j X 1 j δ 1 x i... U Nν d (L+1) m 2 na m ro [µ 1,...,µm] rp i=1 r m F Nν d 2 L r 2 3 Calculating: U and F polynomials, generating tensor structure (also polynomials) 4 Dividing expression into sum of integrals in respect of r 5 We end up with sum of the integrals of the type: I r = C n Y dx jx ν j 1 j δ 1! nx h x i U(x) a+ɛbi h F (x) c+ɛdi [Q r(x)] 6 C++ linked with GiNaC libraries is used to calculate I r integrals sector-decomposition, Ch.Bogner, S.Weinzierl (arxiv: [hep-ph]) i=1 Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

21 CSectors.m Numerical calculation using sector decomposition method How CSectors.m works 1 Integral 2 Feynman parameters L Y l=1 d d k µ 1 a k 1... k µm am l D ν Dν N N... Y n! nx dx jx ν j X 1 j δ 1 x i... U Nν d (L+1) m 2 na m ro [µ 1,...,µm] rp i=1 r m F Nν d 2 L r 2 3 Calculating: U and F polynomials, generating tensor structure (also polynomials) 4 Dividing expression into sum of integrals in respect of r 5 We end up with sum of the integrals of the type: I r = C n Y dx jx ν j 1 j δ 1! nx h x i U(x) a+ɛbi h F (x) c+ɛdi [Q r(x)] 6 C++ linked with GiNaC libraries is used to calculate I r integrals sector-decomposition, Ch.Bogner, S.Weinzierl (arxiv: [hep-ph]) i=1 Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

22 CSectors.m Numerical calculation using sector decomposition method How CSectors.m works 1 Integral 2 Feynman parameters L Y l=1 d d k µ 1 a k 1... k µm am l D ν Dν N N... Y n! nx dx jx ν j X 1 j δ 1 x i... U Nν d (L+1) m 2 na m ro [µ 1,...,µm] rp i=1 r m F Nν d 2 L r 2 3 Calculating: U and F polynomials, generating tensor structure (also polynomials) 4 Dividing expression into sum of integrals in respect of r 5 We end up with sum of the integrals of the type: I r = C n Y dx jx ν j 1 j δ 1! nx h x i U(x) a+ɛbi h F (x) c+ɛdi [Q r(x)] 6 C++ linked with GiNaC libraries is used to calculate I r integrals sector-decomposition, Ch.Bogner, S.Weinzierl (arxiv: [hep-ph]) i=1 Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

23 CSectors.m Numerical calculation using sector decomposition method How CSectors.m works 1 Integral 2 Feynman parameters L Y l=1 d d k µ 1 a k 1... k µm am l D ν Dν N N... Y n! nx dx jx ν j X 1 j δ 1 x i... U Nν d (L+1) m 2 na m ro [µ 1,...,µm] rp i=1 r m F Nν d 2 L r 2 3 Calculating: U and F polynomials, generating tensor structure (also polynomials) 4 Dividing expression into sum of integrals in respect of r 5 We end up with sum of the integrals of the type: I r = C n Y dx jx ν j 1 j δ 1! nx h x i U(x) a+ɛbi h F (x) c+ɛdi [Q r(x)] 6 C++ linked with GiNaC libraries is used to calculate I r integrals sector-decomposition, Ch.Bogner, S.Weinzierl (arxiv: [hep-ph]) i=1 Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

24 CSectors.m Numerical calculation using sector decomposition method How CSectors.m works 1 Integral 2 Feynman parameters L Y l=1 d d k µ 1 a... k µm k 1 am l D ν Dν N N... Y n! nx dx jx ν j X 1 j δ 1 x i... U Nν d (L+1) m 2 na m ro [µ 1,...,µm] rp i=1 r m F Nν d 2 L r 2 3 Calculating: U and F polynomials, generating tensor structure (also polynomials) 4 Dividing expression into sum of integrals in respect of r 5 We end up with sum of the integrals of the type: I r = C n Y dx jx ν j 1 j δ 1! nx h x i U(x) a+ɛbi h F (x) c+ɛdi [Q r(x)] 6 C++ linked with GiNaC libraries is used to calculate I r integrals sector-decomposition, Ch.Bogner, S.Weinzierl (arxiv: [hep-ph]) Steps from 1) to 6) are done automatically in MATHEMATICA! i=1 Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

25 Numerical calculation using sector decomposition method Using CSectors.m Example k µ 1 1 kµ 2 1 d d k 1d d k 2 [(p 1 k 1 + k 2) 2 m 2 ] 2 k 2 1 [(p2 k2)2 m 2 ]k 2 2 p 2µ1 p 1µ2 Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

26 Gamma function Mellin-Barnes approach Basic function in M-B method Γ(z) = 0 t z 1 e t dt Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

27 Mellin-Barnes method Mellin-Barnes approach 1) M-B formula 1 (A A n ) λ = 1 1 Γ(λ) (2πi) n 1 +i i... +i i A λ z2... zn 1 Γ(λ + z z n ) 2) Integration over Feynman parameters 1 n ( n ) dx j x νj 1 j δ 1 x i 0 i=1 i=1 dz 2... dz n n i=2 n Γ( z i ) i=2 = Γ(ν 1)... Γ(ν n ) Γ(ν ν n ) A zi i Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

28 Mellin-Barnes approach Mellin-Barnes approach Loop by loop approach 1 define kinematic which depends on external legs (invariants) 2 make decision about the order in which n 1-loop subloops will be worked out in a sequence 3 construct Feynman integral for the chosen subloop, make manipulations on the F polynomial to make it the most suitable for using MB representations G(k µ1... k µm ) = ( 1) Nν n! Y nx dx jx ν j 1 j δ 1 x i Γ(ν 1)... Γ(ν n) i=1 X Γ `N ν d 2 r 2 1 na m ro [µ 1,...,µm] rp r m ( 2) r 2 F Nν d 2 r 2 4 use the basic MB-relation 5 make integration over Feynman parameters 6 go back to the point (3) and repeat the steps till F in the last n subloop will be changed to the M-B integral Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

29 Mellin-Barnes approach Mellin-Barnes approach Loop by loop approach 1 define kinematic which depends on external legs (invariants) 2 make decision about the order in which n 1-loop subloops will be worked out in a sequence 3 construct Feynman integral for the chosen subloop, make manipulations on the F polynomial to make it the most suitable for using MB representations G(k µ1... k µm ) = ( 1) Nν n! Y nx dx jx ν j 1 j δ 1 x i Γ(ν 1)... Γ(ν n) i=1 X Γ `N ν d 2 r 2 1 na m ro [µ 1,...,µm] rp r m ( 2) r 2 F Nν d 2 r 2 4 use the basic MB-relation 5 make integration over Feynman parameters 6 go back to the point (3) and repeat the steps till F in the last n subloop will be changed to the M-B integral Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

30 Mellin-Barnes approach Mellin-Barnes approach Loop by loop approach 1 define kinematic which depends on external legs (invariants) 2 make decision about the order in which n 1-loop subloops will be worked out in a sequence 3 construct Feynman integral for the chosen subloop, make manipulations on the F polynomial to make it the most suitable for using MB representations G(k µ1... k µm ) = ( 1) Nν n! Y nx dx jx ν j 1 j δ 1 x i Γ(ν 1)... Γ(ν n) i=1 X Γ `N ν d 2 r 2 1 na m ro [µ 1,...,µm] rp r m ( 2) r 2 F Nν d 2 r 2 4 use the basic MB-relation 5 make integration over Feynman parameters 6 go back to the point (3) and repeat the steps till F in the last n subloop will be changed to the M-B integral Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

31 Mellin-Barnes approach Mellin-Barnes approach Loop by loop approach 1 define kinematic which depends on external legs (invariants) 2 make decision about the order in which n 1-loop subloops will be worked out in a sequence 3 construct Feynman integral for the chosen subloop, make manipulations on the F polynomial to make it the most suitable for using MB representations G(k µ1... k µm ) = ( 1) Nν n! Y nx dx jx ν j 1 j δ 1 x i Γ(ν 1)... Γ(ν n) i=1 X Γ `N ν d 2 r 2 1 na m ro [µ 1,...,µm] rp r m ( 2) r 2 F Nν d 2 r 2 4 use the basic MB-relation 5 make integration over Feynman parameters 6 go back to the point (3) and repeat the steps till F in the last n subloop will be changed to the M-B integral Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

32 Mellin-Barnes approach Mellin-Barnes approach Loop by loop approach 1 define kinematic which depends on external legs (invariants) 2 make decision about the order in which n 1-loop subloops will be worked out in a sequence 3 construct Feynman integral for the chosen subloop, make manipulations on the F polynomial to make it the most suitable for using MB representations G(k µ1... k µm ) = ( 1) Nν n! Y nx dx jx ν j 1 j δ 1 x i Γ(ν 1)... Γ(ν n) i=1 X Γ `N ν d 2 r 2 1 na m ro [µ 1,...,µm] rp r m ( 2) r 2 F Nν d 2 r 2 4 use the basic MB-relation 5 make integration over Feynman parameters 6 go back to the point (3) and repeat the steps till F in the last n subloop will be changed to the M-B integral Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

33 Mellin-Barnes approach Mellin-Barnes approach Loop by loop approach 1 define kinematic which depends on external legs (invariants) 2 make decision about the order in which n 1-loop subloops will be worked out in a sequence 3 construct Feynman integral for the chosen subloop, make manipulations on the F polynomial to make it the most suitable for using MB representations G(k µ1... k µm ) = ( 1) Nν n! Y nx dx jx ν j 1 j δ 1 x i Γ(ν 1)... Γ(ν n) i=1 X Γ `N ν d 2 r 2 1 na m ro [µ 1,...,µm] rp r m ( 2) r 2 F Nν d 2 r 2 4 use the basic MB-relation 5 make integration over Feynman parameters 6 go back to the point (3) and repeat the steps till F in the last n subloop will be changed to the M-B integral Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

34 Mellin-Barnes approach Mellin-Barnes approach Loop by loop approach 1 define kinematic which depends on external legs (invariants) 2 make decision about the order in which n 1-loop subloops will be worked out in a sequence 3 construct Feynman integral for the chosen subloop, make manipulations on the F polynomial to make it the most suitable for using MB representations G(k µ1... k µm ) = ( 1) Nν n! Y nx dx jx ν j 1 j δ 1 x i Γ(ν 1)... Γ(ν n) i=1 X Γ `N ν d 2 r 2 1 na m ro [µ 1,...,µm] rp r m ( 2) r 2 F Nν d 2 r 2 4 use the basic MB-relation 5 make integration over Feynman parameters 6 go back to the point (3) and repeat the steps till F in the last n subloop will be changed to the M-B integral Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

35 Loop by loop approach Mellin-Barnes approach Example Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

36 Mellin-Barnes approach Mellin-Barnes approach It is very important......to calculate subloops in specific order (in some cases), for example: starting with k 1 leads to F polynomial with huge number of terms over 20dim final representation will appear Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

37 AMBRE-Automatic Mellin-Barnes REpresentation AMBRE.m......is the MATHEMATICA package for creating Mellin-Barnes representations. Comput.Phys.Commun.177: ,2007 (J.Gluza, K.K, T.Riemann) Features M-B representation for: L-loop Feynman scalar integral 1-loop m-rank tensor integral New Features - under developement M-B representation for L-loop and m-rank tensor integral... Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

38 AMBRE-Automatic Mellin-Barnes REpresentation AMBRE.m new features How to calculate L-loop, m-rank tensor integral using loop-by-loop approach? Let s start with simple two loop example... Example (k 1 p 1)(k 1 p 1)(k 2 p 1) [k 2 1 ]ν 1 [(k 2 k 1) 2 ] ν 2 [(k 1 + p 1) 2 ] ν 3 [k 2 2 ]ν 4 [(k 2 + p 1) 2 ] ν 5 dd k 1d d k 2 Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

39 AMBRE-Automatic Mellin-Barnes REpresentation AMBRE.m new features Example 1 Iteration nr1 (first subloop) (k 1 p 1)(k 1 p 1)(k 2 p 1) [k 2 1 ]ν 1 [(k 2 k 1) 2 ] ν 2 [(k 1 + p 1) 2 ] ν 3 [k 2 2 ]ν 4 [(k 2 + p 1) 2 ] ν 5 dd k 1d d k 2 2 Calculating F polynomial for propagators of first subloop. F contains new propagators F = [k 2 2 ]x1x2 sx1x3 [(k2 + p1)2 ]x 2x 3 3 Generating tensor structure for first subloop, and separating expression into separate integrals P µ1 P µ2 + g µ 1 µ 2 Q µ1 Q µ2 + g µ 1 µ 2 (k µ 1 2 x2 pµ 1 1 x3)(kµ 2 2 x2 pµ 2 1 x3) + gµ 1 µ 2 {k µ 1 2 kµ 2 2 x2 2, kµ 2 2 pµ 1 1 x2x3, kµ 1 2 pµ 2 1 x2x3, pµ 1 1 pµ 2 1 x2 3, gµ 1 µ2 } 4 Apllying M-B formula integrating over x, we obtain M-B representation 5 Iteration nr2 (second subloop). We have to work on five integrals p 1µ1 p 1µ2 (k 2 p 1) [k 2 2 ] ν 4 [(k 2 + p 1) 2 ] ν 5 {kµ 1 2 kµ 2 2 MB1, kµ 2 2 pµ 1 1 MB2, kµ 1 2 pµ 2 1 MB3, pµ 1 1 pµ 2 1 MB4, gµ 1 µ2 MB 5}d d k 2 Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

40 AMBRE-Automatic Mellin-Barnes REpresentation AMBRE.m new features Example 1 Iteration nr1 (first subloop) (k 1 p 1)(k 1 p 1)(k 2 p 1) [k 2 1 ]ν 1 [(k 2 k 1) 2 ] ν 2 [(k 1 + p 1) 2 ] ν 3 [k 2 2 ]ν 4 [(k 2 + p 1) 2 ] ν 5 dd k 1d d k 2 2 Calculating F polynomial for propagators of first subloop. F contains new propagators F = [k 2 2 ]x1x2 sx1x3 [(k2 + p1)2 ]x 2x 3 3 Generating tensor structure for first subloop, and separating expression into separate integrals P µ1 P µ2 + g µ 1 µ 2 Q µ1 Q µ2 + g µ 1 µ 2 (k µ 1 2 x2 pµ 1 1 x3)(kµ 2 2 x2 pµ 2 1 x3) + gµ 1 µ 2 {k µ 1 2 kµ 2 2 x2 2, kµ 2 2 pµ 1 1 x2x3, kµ 1 2 pµ 2 1 x2x3, pµ 1 1 pµ 2 1 x2 3, gµ 1 µ2 } 4 Apllying M-B formula integrating over x, we obtain M-B representation 5 Iteration nr2 (second subloop). We have to work on five integrals p 1µ1 p 1µ2 (k 2 p 1) [k 2 2 ] ν 4 [(k 2 + p 1) 2 ] ν 5 {kµ 1 2 kµ 2 2 MB1, kµ 2 2 pµ 1 1 MB2, kµ 1 2 pµ 2 1 MB3, pµ 1 1 pµ 2 1 MB4, gµ 1 µ2 MB 5}d d k 2 Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

41 AMBRE-Automatic Mellin-Barnes REpresentation AMBRE.m new features Example 1 Iteration nr1 (first subloop) (k 1 p 1)(k 1 p 1)(k 2 p 1) [k 2 1 ]ν 1 [(k 2 k 1) 2 ] ν 2 [(k 1 + p 1) 2 ] ν 3 [k 2 2 ]ν 4 [(k 2 + p 1) 2 ] ν 5 dd k 1d d k 2 2 Calculating F polynomial for propagators of first subloop. F contains new propagators F = [k 2 2 ]x1x2 sx1x3 [(k2 + p1)2 ]x 2x 3 3 Generating tensor structure for first subloop, and separating expression into separate integrals P µ1 P µ2 + g µ 1 µ 2 Q µ1 Q µ2 + g µ 1 µ 2 (k µ 1 2 x2 pµ 1 1 x3)(kµ 2 2 x2 pµ 2 1 x3) + gµ 1 µ 2 {k µ 1 2 kµ 2 2 x2 2, kµ 2 2 pµ 1 1 x2x3, kµ 1 2 pµ 2 1 x2x3, pµ 1 1 pµ 2 1 x2 3, gµ 1 µ2 } 4 Apllying M-B formula integrating over x, we obtain M-B representation 5 Iteration nr2 (second subloop). We have to work on five integrals p 1µ1 p 1µ2 (k 2 p 1) [k 2 2 ] ν 4 [(k 2 + p 1) 2 ] ν 5 {kµ 1 2 kµ 2 2 MB1, kµ 2 2 pµ 1 1 MB2, kµ 1 2 pµ 2 1 MB3, pµ 1 1 pµ 2 1 MB4, gµ 1 µ2 MB 5}d d k 2 Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

42 AMBRE-Automatic Mellin-Barnes REpresentation AMBRE.m new features Example 1 Iteration nr1 (first subloop) (k 1 p 1)(k 1 p 1)(k 2 p 1) [k 2 1 ]ν 1 [(k 2 k 1) 2 ] ν 2 [(k 1 + p 1) 2 ] ν 3 [k 2 2 ]ν 4 [(k 2 + p 1) 2 ] ν 5 dd k 1d d k 2 2 Calculating F polynomial for propagators of first subloop. F contains new propagators F = [k 2 2 ]x1x2 sx1x3 [(k2 + p1)2 ]x 2x 3 3 Generating tensor structure for first subloop, and separating expression into separate integrals P µ1 P µ2 + g µ 1 µ 2 Q µ1 Q µ2 + g µ 1 µ 2 (k µ 1 2 x2 pµ 1 1 x3)(kµ 2 2 x2 pµ 2 1 x3) + gµ 1 µ 2 {k µ 1 2 kµ 2 2 x2 2, kµ 2 2 pµ 1 1 x2x3, kµ 1 2 pµ 2 1 x2x3, pµ 1 1 pµ 2 1 x2 3, gµ 1 µ2 } 4 Apllying M-B formula integrating over x, we obtain M-B representation 5 Iteration nr2 (second subloop). We have to work on five integrals p 1µ1 p 1µ2 (k 2 p 1) [k 2 2 ] ν 4 [(k 2 + p 1) 2 ] ν 5 {kµ 1 2 kµ 2 2 MB1, kµ 2 2 pµ 1 1 MB2, kµ 1 2 pµ 2 1 MB3, pµ 1 1 pµ 2 1 MB4, gµ 1 µ2 MB 5}d d k 2 Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

43 AMBRE-Automatic Mellin-Barnes REpresentation AMBRE.m new features Example 1 Iteration nr1 (first subloop) (k 1 p 1)(k 1 p 1)(k 2 p 1) [k 2 1 ]ν 1 [(k 2 k 1) 2 ] ν 2 [(k 1 + p 1) 2 ] ν 3 [k 2 2 ]ν 4 [(k 2 + p 1) 2 ] ν 5 dd k 1d d k 2 2 Calculating F polynomial for propagators of first subloop. F contains new propagators F = [k 2 2 ]x1x2 sx1x3 [(k2 + p1)2 ]x 2x 3 3 Generating tensor structure for first subloop, and separating expression into separate integrals P µ1 P µ2 + g µ 1 µ 2 Q µ1 Q µ2 + g µ 1 µ 2 (k µ 1 2 x2 pµ 1 1 x3)(kµ 2 2 x2 pµ 2 1 x3) + gµ 1 µ 2 {k µ 1 2 kµ 2 2 x2 2, kµ 2 2 pµ 1 1 x2x3, kµ 1 2 pµ 2 1 x2x3, pµ 1 1 pµ 2 1 x2 3, gµ 1 µ2 } 4 Apllying M-B formula integrating over x, we obtain M-B representation 5 Iteration nr2 (second subloop). We have to work on five integrals p 1µ1 p 1µ2 (k 2 p 1) [k 2 2 ] ν 4 [(k 2 + p 1) 2 ] ν 5 {kµ 1 2 kµ 2 2 MB1, kµ 2 2 pµ 1 1 MB2, kµ 1 2 pµ 2 1 MB3, pµ 1 1 pµ 2 1 MB4, gµ 1 µ2 MB 5}d d k 2 Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

44 AMBRE-Automatic Mellin-Barnes REpresentation AMBRE.m new features Example 1 Iteration nr1 (first subloop) (k 1 p 1)(k 1 p 1)(k 2 p 1) [k 2 1 ]ν 1 [(k 2 k 1) 2 ] ν 2 [(k 1 + p 1) 2 ] ν 3 [k 2 2 ]ν 4 [(k 2 + p 1) 2 ] ν 5 dd k 1d d k 2 2 Calculating F polynomial for propagators of first subloop. F contains new propagators F = [k 2 2 ]x1x2 sx1x3 [(k2 + p1)2 ]x 2x 3 3 Generating tensor structure for first subloop, and separating expression into separate integrals P µ1 P µ2 + g µ 1 µ 2 Q µ1 Q µ2 + g µ 1 µ 2 (k µ 1 2 x2 pµ 1 1 x3)(kµ 2 2 x2 pµ 2 1 x3) + gµ 1 µ 2 {k µ 1 2 kµ 2 2 x2 2, kµ 2 2 pµ 1 1 x2x3, kµ 1 2 pµ 2 1 x2x3, pµ 1 1 pµ 2 1 x2 3, gµ 1 µ2 } 4 Apllying M-B formula integrating over x, we obtain M-B representation 5 Iteration nr2 (second subloop). We have to work on five integrals p 1µ1 p 1µ2 (k 2 p 1) [k 2 2 ] ν 4 [(k 2 + p 1) 2 ] ν 5 {kµ 1 2 kµ 2 2 MB1, kµ 2 2 pµ 1 1 MB2, kµ 1 2 pµ 2 1 MB3, pµ 1 1 pµ 2 1 MB4, gµ 1 µ2 MB 5}d d k 2 Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

45 AMBRE-Automatic Mellin-Barnes REpresentation AMBRE.m new features Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

46 Numerical cross-checks Cross-checks So far the following cross-checks have been made between CSectors.m, AMBRE.m with MB.m (M. Czakon; hep-ph/ ) and/or analytic result obtained after IBP reduction to known master integrals also Self-Energy diagrams up to: rank five, two loop Vertex up to: rank five, two loop Boxes up to: rank two, three loop Scalar four loop tadpoles CSectors.m and AMBRE.m (R. Boughezal, M. Czakon; hep-ph/ ) Five and six point diagrams up to rank four checked between CSectors.m and hexagon.m (6pt and 5pt reduction) (T. Diakonidis, J. Fleischer, J. Gluza, K.K, T. Riemann, J.B. Tausk; Nucl.Phys.Proc.Suppl.183: ,2008) Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24

47 Summary Summary Summary and present status... two methods were presented: sector decomposition and Mellin-Barnes CSectors.m is able to calculate m-rank and L-loop integral using sectordecoposition s GiNaC libraries AMBRE.m is able to build Mellin-Barnes for m-rank and 2-loop (at the moment) integrals. Some further work must be done to L-loop version... Krzysztof Kajda (University of Silesia) Automatic calculations of Feynman integrals in the Euclidean region 9 dec / 24