Automatic calculations of Feynman integrals in the Euclidean region
|
|
- Simon Hancock
- 5 years ago
- Views:
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
arxiv: v1 [hep-ph] 19 Apr 2007
DESY 07-037 HEPTOOLS 07-009 SFB/CPP-07-14 arxiv:0704.2423v1 [hep-ph] 19 Apr 2007 AMBRE a Mathematica package for the construction of Mellin-Barnes representations for Feynman integrals Abstract J. Gluza,
More informationAMBRE a Mathematica package for the construction of Mellin-Barnes representations for Feynman integrals
DESY 07-037 -- DRAFT version -- 2007-03-26 14:59 AMBRE a Mathematica package for the construction of Mellin-Barnes representations for Feynman integrals J. Gluza, K. Kajda Department of Field Theory and
More informationEvaluating multiloop Feynman integrals by Mellin-Barnes representation
April 7, 004 Loops&Legs 04 Evaluating multiloop Feynman integrals by Mellin-Barnes representation V.A. Smirnov Nuclear Physics Institute of Moscow State University Mellin-Barnes representation as a tool
More informationNumerical Evaluation of Multi-loop Integrals
Numerical Evaluation of Multi-loop Integrals Sophia Borowka MPI for Physics, Munich In collaboration with G. Heinrich Based on arxiv:124.4152 [hep-ph] HP 8 :Workshop on High Precision for Hard Processes,
More informationNumerical multi-loop calculations: tools and applications
Journal of Physics: Conference Series PAPER OPEN ACCESS Numerical multi-loop calculations: tools and applications To cite this article: S. Borowka et al 2016 J. Phys.: Conf. Ser. 762 012073 Related content
More informationNumerical Evaluation of Multi-loop Integrals
Numerical Evaluation of Multi-loop Integrals Sophia Borowka MPI for Physics, Munich In collaboration with: J. Carter and G. Heinrich Based on arxiv:124.4152 [hep-ph] http://secdec.hepforge.org DESY-HU
More informationNumerical Evaluation of Loop Integrals
Numerical Evaluation of Loop Integrals Institut für Theoretische Physik Universität Zürich Tsukuba, April 22 nd 2006 In collaboration with Babis Anastasiou Rationale (Why do we need complicated loop amplitudes?)
More informationFeynman Integrals Mellin-Barnes representations Sums
T. Riemann CALC July 0-20, 2009, JINR, Dubna, Russia Feynman Integrals Mellin-Barnes representations Sums Helmholtz International School Calculations for Modern and Future Colliders July 0 20, 2009, JINR,
More informationTowards a more accurate prediction of W +b jets with an automatized approach to one-loop calculations
Towards a more accurate prediction of W +b jets with an automatized approach to one-loop calculations Laura Reina Loops and Legs in Quantum Field Theory Wernigerode, April 2012 Outline Motivations: W +b-jet
More informationFeynman Integrals Mellin-Barnes representations Sums
T. Riemann 30 March 03 April 2009 CAPP, DESY, Zeuthen 1 Feynman Integrals Mellin-Barnes representations Sums Computer Algebra and Particle Physics The DESY CAPP School 30 March 3 April, 2009, Zeuthen,
More informationReduction of Feynman integrals to master integrals
Reduction of Feynman integrals to master integrals A.V. Smirnov Scientific Research Computing Center of Moscow State University A.V. Smirnov ACAT 2007 p.1 Reduction problem for Feynman integrals A review
More informationSome variations of the reduction of one-loop Feynman tensor integrals
Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 1 / 6 Some variations of the reduction of one-loop Feynman tensor integrals Tord Riemann DESY, Zeuthen in
More informationNumerical multi-loop calculations with SecDec
Journal of Physics: Conference Series OPEN ACCESS Numerical multi-loop calculations with SecDec To cite this article: Sophia Borowka and Gudrun Heinrich 214 J. Phys.: Conf. Ser. 523 1248 View the article
More informationIntroduction to Loop Calculations
Introduction to Loop Calculations May Contents One-loop integrals. Dimensional regularisation............................... Feynman parameters................................. 3.3 Momentum integration................................
More information1 The Quantum Anharmonic Oscillator
1 The Quantum Anharmonic Oscillator Perturbation theory based on Feynman diagrams can be used to calculate observables in Quantum Electrodynamics, like the anomalous magnetic moment of the electron, and
More informationMultiloop integrals in dimensional regularization made simple
Multiloop integrals in dimensional regularization made simple Johannes M. Henn Institute for Advanced Study based on PRL 110 (2013) [arxiv:1304.1806], JHEP 1307 (2013) 128 [arxiv:1306.2799] with A. V.
More informationStructural Aspects of Numerical Loop Calculus
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,
More informationFeynman integrals and multiple polylogarithms. Stefan Weinzierl
Feynman integrals and multiple polylogarithms Stefan Weinzierl Universität Mainz I. Basic techniques II. Nested sums and iterated integrals III. Multiple Polylogarithms IV. Applications The need for precision
More informationThe Non-commutative S matrix
The Suvrat Raju Harish-Chandra Research Institute 9 Dec 2008 (work in progress) CONTEMPORARY HISTORY In the past few years, S-matrix techniques have seen a revival. (Bern et al., Britto et al., Arkani-Hamed
More informationSector Decomposition
Sector Decomposition J. Carter Institute for Particle Physics Phenomenology University of Durham Student Seminar, 06/05/2009 Outline 1 What is Sector Decomposition? Why is Sector Decomposition Important?
More informationFeynman Integrals Mellin-Barnes representations Sums
Tord Riemann 6 + 23 June 200 HUB Berlin Feynman Integrals Mellin-Barnes representations Sums Two lectures given at HUB, Berlin, June 200 Tord Riemann, DESY, Zeuthen for exercises and additional material:
More informationarxiv: v1 [hep-ph] 30 Dec 2015
June 3, 8 Derivation of functional equations for Feynman integrals from algebraic relations arxiv:5.94v [hep-ph] 3 Dec 5 O.V. Tarasov II. Institut für Theoretische Physik, Universität Hamburg, Luruper
More informationTord Riemann. DESY, Zeuthen, Germany
1/ v. 2010-08-31 1:2 T. Riemann Tensor reduction Corfu 2010, Greece Algebraic tensor Feynman integral reduction Tord Riemann DESY, Zeuthen, Germany Based on work done in collaboration with Jochem Fleischer
More informationFive-loop massive tadpoles
Five-loop massive tadpoles York Schröder (Univ del Bío-Bío, Chillán, Chile) recent work with Thomas Luthe and earlier work with: J. Möller, C. Studerus Radcor, UCLA, Jun 0 Motivation pressure of hot QCD
More informationEvaluating double and triple (?) boxes
Evaluating double and triple (?) boxes V.A. Smirnov a hep-ph/0209295 September 2002 a Nuclear Physics Institute of Moscow State University, Moscow 9992, Russia A brief review of recent results on analytical
More informationTwo-loop Heavy Fermion Corrections to Bhabha Scattering
Two-loop Heavy Fermion Corrections to Bhabha Scattering S. Actis 1, M. Czakon 2, J. Gluza 3 and T. Riemann 1 1 Deutsches Elektronen-Synchrotron DESY Platanenallee 6, D 15738 Zeuthen, Germany 2 Institut
More informationSchematic Project of PhD Thesis: Two-Loop QCD Corrections with the Differential Equations Method
Schematic Project of PhD Thesis: Two-Loop QCD Corrections with the Differential Equations Method Matteo Becchetti Supervisor Roberto Bonciani University of Rome La Sapienza 24/01/2017 1 The subject of
More informationarxiv:hep-th/ v1 2 Jul 1998
α-representation for QCD Richard Hong Tuan arxiv:hep-th/9807021v1 2 Jul 1998 Laboratoire de Physique Théorique et Hautes Energies 1 Université de Paris XI, Bâtiment 210, F-91405 Orsay Cedex, France Abstract
More informationUnitarity, Dispersion Relations, Cutkosky s Cutting Rules
Unitarity, Dispersion Relations, Cutkosky s Cutting Rules 04.06.0 For more information about unitarity, dispersion relations, and Cutkosky s cutting rules, consult Peskin& Schröder, or rather Le Bellac.
More informationFrom Tensor Integral to IBP
From Tensor Integral to IBP Mohammad Assadsolimani, in collaboration with P. Kant, B. Tausk and P. Uwer 11. Sep. 2012 Mohammad Assadsolimani From Tensor Integral to IBP 1 Contents Motivation NNLO Tensor
More informationN = 4 SYM and new insights into
N = 4 SYM and new insights into QCD tree-level amplitudes N = 4 SUSY and QCD workshop LPTHE, Jussieu, Paris Dec 12, 2008 Henrik Johansson, UCLA Bern, Carrasco, HJ, Kosower arxiv:0705.1864 [hep-th] Bern,
More informationarxiv:hep-ph/ v1 21 Jan 1998
TARCER - A Mathematica program for the reduction of two-loop propagator integrals R. Mertig arxiv:hep-ph/980383v Jan 998 Abstract Mertig Research & Consulting, Kruislaan 49, NL-098 VA Amsterdam, The Netherlands
More informationarxiv:hep-ph/ v2 23 Jan 2006
WUE-ITP-2005-014 arxiv:hep-ph/0511200v2 23 Jan 2006 Abstract Automatized analytic continuation of Mellin-Barnes integrals M. Czakon Institut für Theoretische Physik und Astrophysik, Universität Würzburg,
More informationarxiv:hep-lat/ v1 30 Sep 2005
September 2005 Applying Gröbner Bases to Solve Reduction Problems for Feynman Integrals arxiv:hep-lat/0509187v1 30 Sep 2005 A.V. Smirnov 1 Mechanical and Mathematical Department and Scientific Research
More information2P + E = 3V 3 + 4V 4 (S.2) D = 4 E
PHY 396 L. Solutions for homework set #19. Problem 1a): Let us start with the superficial degree of divergence. Scalar QED is a purely bosonic theory where all propagators behave as 1/q at large momenta.
More informationThe rare decay H Zγ in perturbative QCD
The rare decay H Zγ in perturbative QCD [arxiv: hep-ph/1505.00561] Thomas Gehrmann, Sam Guns & Dominik Kara June 15, 2015 RADCOR 2015 AND LOOPFEST XIV - UNIVERSITY OF CALIFORNIA, LOS ANGELES Z Z H g q
More informationFeynman Integral Calculus
Feynman Integral Calculus Vladimir A. Smirnov Feynman Integral Calculus ABC Vladimir A. Smirnov Lomonosov Moscow State University Skobeltsyn Institute of Nuclear Physics Moscow 119992, Russia E-mail: smirnov@theory.sinp.msu.ru
More informationNumerical evaluation of multi-scale integrals with SecDec 3
Numerical evaluation of multi-scale integrals with SecDec 3 Sophia Borowka University of Zurich Project in collaboration with G. Heinrich, S. Jones, M. Kerner, J. Schlenk, T. Zirke 152.6595 [hep-ph] (CPC,
More informationLoop-Tree Duality Method
Numerical Implementation of the Loop-Tree Duality Method IFIC Sebastian Buchta in collaboration with G. Rodrigo,! P. Draggiotis, G. Chachamis and I. Malamos 24. July 2015 Outline 1.Introduction! 2.A new
More informationFeynman integrals as Periods
Feynman integrals as Periods Pierre Vanhove Amplitudes 2017, Higgs Center, Edinburgh, UK based on [arxiv:1309.5865], [arxiv:1406.2664], [arxiv:1601.08181] Spencer Bloch, Matt Kerr Pierre Vanhove (IPhT)
More informationQCD Factorization and PDFs from Lattice QCD Calculation
QCD Evolution 2014 Workshop at Santa Fe, NM (May 12 16, 2014) QCD Factorization and PDFs from Lattice QCD Calculation Yan-Qing Ma / Jianwei Qiu Brookhaven National Laboratory ² Observation + Motivation
More informationarxiv: v2 [hep-th] 7 Jul 2016
Integration-by-parts reductions from unitarity cuts and algebraic geometry arxiv:1606.09447v [hep-th] 7 Jul 016 Institute for Theoretical Physics, ETH Zürich, 8093 Zürich, Switzerland E-mail: Kasper.Larsen@phys.ethz.ch
More informationThe Pentabox Master Integrals with the Simplified Differential Equations approach
The Pentabox Master Integrals with the Simplified Differential Equations approach Costas G. Papadopoulos INPP, NCSR Demokritos Zurich, August 25, 2016 C.G.Papadopoulos (INPP) 5box QCD@LHC 2016 1 / 36 Introduction
More informationSPLITTING FUNCTIONS AND FEYNMAN INTEGRALS
SPLITTING FUNCTIONS AND FEYNMAN INTEGRALS Germán F. R. Sborlini Departamento de Física, FCEyN, UBA (Argentina) 10/12/2012 - IFIC CONTENT Introduction Collinear limits Splitting functions Computing splitting
More informationColor-Kinematics Duality for Pure Yang-Mills and Gravity at One and Two Loops
Physics Amplitudes Color-Kinematics Duality for Pure Yang-Mills and Gravity at One and Two Loops Josh Nohle [Bern, Davies, Dennen, Huang, JN - arxiv: 1303.6605] [JN - arxiv:1309.7416] [Bern, Davies, JN
More information7 Veltman-Passarino Reduction
7 Veltman-Passarino Reduction This is a method of expressing an n-point loop integral with r powers of the loop momentum l in the numerator, in terms of scalar s-point functions with s = n r, n. scalar
More information1) The line has a slope of ) The line passes through (2, 11) and. 6) r(x) = x + 4. From memory match each equation with its graph.
Review Test 2 Math 1314 Name Write an equation of the line satisfying the given conditions. Write the answer in standard form. 1) The line has a slope of - 2 7 and contains the point (3, 1). Use the point-slope
More informationarxiv: v2 [hep-ph] 21 Sep 2015
FR-PHENO-2015-001, MPP-2015-27, ZU-TH 1/15 SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop arxiv:1502.06595v2 [hep-ph] 21 Sep 2015 S. Borowka a, G. Heinrich b, S. P. Jones b,c,
More informationNumerical evaluation of multi-loop integrals
Max-Planck-Institut für Physik, München, Germany E-mail: sjahn@mpp.mpg.de We present updates on the development of pyse CDE C, a toolbox to numerically evaluate parameter integrals in the context of dimensional
More informationReduction to Master Integrals. V.A. Smirnov Atrani, September 30 October 05, 2013 p.1
Reduction to Master Integrals V.A. Smirnov Atrani, September 30 October 05, 2013 p.1 Reduction to Master Integrals IBP (integration by parts) V.A. Smirnov Atrani, September 30 October 05, 2013 p.1 Reduction
More informationThe Feynman Propagator and Cauchy s Theorem
The Feynman Propagator and Cauchy s Theorem Tim Evans 1 (1st November 2018) The aim of these notes is to show how to derive the momentum space form of the Feynman propagator which is (p) = i/(p 2 m 2 +
More informationUnitarity-based methods and Integration-by-parts identities for Feynman Integrals
Università degli Studi di Padova DIPARTIMENTO DI FISICA ED ASTRONOMIA G. GALILEI Corso di Laurea Magistrale in Fisica Tesi di laurea magistrale Unitarity-based methods and Integration-by-parts identities
More informationFundamental equations of relativistic fluid dynamics
CHAPTER VI Fundamental equations of relativistic fluid dynamics When the energy density becomes large as may happen for instance in compact astrophysical objects, in the early Universe, or in high-energy
More informationNNLO antenna subtraction with two hadronic initial states
NNLO antenna subtraction with two hadronic initial states Institut für Theoretische Physik, Universität Zürich, Winterthurerstr. 190, 8057 Zürich, Switzerland E-mail: radja@physik.uzh.ch Aude Gehrmann-De
More informationS. Ghorai 1. Lecture XV Bessel s equation, Bessel s function. e t t p 1 dt, p > 0. (1)
S Ghorai 1 1 Gamma function Gamma function is defined by Lecture XV Bessel s equation, Bessel s function Γp) = e t t p 1 dt, p > 1) The integral in 1) is convergent that can be proved easily Some special
More informationRenormalization of the fermion self energy
Part I Renormalization of the fermion self energy Electron self energy in general gauge The self energy in n = 4 Z i 0 = ( ie 0 ) d n k () n (! dimensions is i k )[g ( a 0 ) k k k ] i /p + /k m 0 use Z
More informationFunctional equations for Feynman integrals
Functional equations for Feynman integrals O.V. Tarasov JINR, Dubna, Russia October 9, 016, Hayama, Japan O.V. Tarasov (JINR) Functional equations for Feynman integrals 1 / 34 Contents 1 Functional equations
More informationarxiv: v3 [hep-ph] 20 Apr 2017
MITP/14-76 A quasi-finite basis for multi-loop Feynman integrals arxiv:1411.7392v3 [hep-ph] 2 Apr 217 Andreas von Manteuffel, a Erik Panzer, b, c and Robert M. Schabinger a a PRISMA Cluster of Excellence
More informationFractal Geometry of Minimal Spanning Trees
Fractal Geometry of Minimal Spanning Trees T. S. Jackson and N. Read Part I: arxiv:0902.3651 Part II: arxiv:0909.5343 Accepted by Phys. Rev. E Problem Definition Given a graph with costs on the edges,
More informationRegularization Physics 230A, Spring 2007, Hitoshi Murayama
Regularization Physics 3A, Spring 7, Hitoshi Murayama Introduction In quantum field theories, we encounter many apparent divergences. Of course all physical quantities are finite, and therefore divergences
More informationSystems of differential equations for Feynman Integrals; Schouten identities and canonical bases.
Systems of differential equations for Feynman Integrals; Schouten identities and canonical bases. Lorenzo Tancredi TTP, KIT - Karlsruhe Bologna, 18 Novembre 2014 Based on collaboration with Thomas Gehrmann,
More informationScalar One-Loop Integrals using the Negative-Dimension Approach
DTP/99/80 hep-ph/9907494 Scalar One-Loop Integrals using the Negative-Dimension Approach arxiv:hep-ph/9907494v 6 Jul 999 C. Anastasiou, E. W. N. Glover and C. Oleari Department of Physics, University of
More informationProgress on Color-Dual Loop Amplitudes
Progress on Color-Dual Loop Amplitudes Henrik Johansson IPhT Saclay Nov 11, 2011 Amplitudes 2011 U. of Michigan 1004.0476, 1106.4711 [hep-th] (and ongoing work) Zvi Bern, John Joseph Carrasco, HJ Outline
More information4 4 and perturbation theory
and perturbation theory We now consider interacting theories. A simple case is the interacting scalar field, with a interaction. This corresponds to a -body contact repulsive interaction between scalar
More informationarxiv: v1 [hep-ph] 20 Jan 2012
ZU-TH 01/12 MZ-TH/12-03 BI-TP 2012/02 Reduze 2 Distributed Feynman Integral Reduction arxiv:1201.4330v1 [hep-ph] 20 Jan 2012 A. von Manteuffel, a,b C. Studerus c a Institut für Theoretische Physik, Universität
More informationAdvanced Quantum Mechanics
Advanced Quantum Mechanics Rajdeep Sensarma! sensarma@theory.tifr.res.in Lecture #22 Path Integrals and QM Recap of Last Class Statistical Mechanics and path integrals in imaginary time Imaginary time
More informationPHY 396 L. Solutions for homework set #20.
PHY 396 L. Solutions for homework set #. Problem 1 problem 1d) from the previous set): At the end of solution for part b) we saw that un-renormalized gauge invariance of the bare Lagrangian requires Z
More informationHopf algebras and factorial divergent power series: Algebraic tools for graphical enumeration
Hopf algebras and factorial divergent power series: Algebraic tools for graphical enumeration Michael Borinsky 1 Humboldt-University Berlin Departments of Physics and Mathematics Workshop on Enumerative
More informationHopf algebras and factorial divergent power series: Algebraic tools for graphical enumeration
Hopf algebras and factorial divergent power series: Algebraic tools for graphical enumeration Michael Borinsky 1 Humboldt-University Berlin Departments of Physics and Mathematics Workshop on Enumerative
More informationLinear reducibility of Feynman integrals
Linear reducibility of Feynman integrals Erik Panzer Institute des Hautes Études Scientifiques RADCOR/LoopFest 2015 June 19th, 2015 UCLA, Los Angeles Feynman integrals: special functions and numbers Some
More informationBessel Functions Michael Taylor. Lecture Notes for Math 524
Bessel Functions Michael Taylor Lecture Notes for Math 54 Contents 1. Introduction. Conversion to first order systems 3. The Bessel functions J ν 4. The Bessel functions Y ν 5. Relations between J ν and
More informationREDUCTION OF FEYNMAN GRAPH AMPLITUDES TO A MINIMAL SET OF BASIC INTEGRALS
REDUCTION OF FEYNMAN GRAPH AMPLITUDES TO A MINIMAL SET OF BASIC INTEGRALS O.V.Tarasov DESY Zeuthen, Platanenallee 6, D 5738 Zeuthen, Germany E-mail: tarasov@ifh.de (Received December 7, 998) An algorithm
More informationParallel adaptive methods for Feynman loop integrals. Conference on Computational Physics (CCP 2011)
Outline Elise de Doncker 1 Fukuko Yuasa 2 1 Department of Computer Science, Western Michigan University, Kalamazoo MI 49008, U. S. 2 High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki
More informationGaussian processes and Feynman diagrams
Gaussian processes and Feynman diagrams William G. Faris April 25, 203 Introduction These talks are about expectations of non-linear functions of Gaussian random variables. The first talk presents the
More informationOne-Loop Integrals at vanishing external momenta and appli. Higgs potentials reconstructions CALC Mikhail Dolgopolov. Samara State University
One-Loop Integrals at vanishing external momenta and applications for extended Higgs potentials reconstructions CALC 2012 Samara State University 1. Basic examples and an idea for extended Higgs potentials
More informationN=4 SYM in High Energy Collision and the Kalb-Ramond Odderon in AdS/CFT
N=4 SYM in High Energy Collision and the Kalb-Ramond Odderon in AdS/CFT Chung-I Tan Brown University Dec. 19, 2008 Miami R. Brower, J. Polchinski, M. Strassler, and C-I Tan, The Pomeron and Gauge/String
More informationQuantum Field Theory II
Quantum Field Theory II PHYS-P 622 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory Chapters: 13, 14, 16-21, 26-28, 51, 52, 61-68, 44, 53, 69-74, 30-32, 84-86, 75,
More informationSection 4: The Quantum Scalar Field
Physics 8.323 Section 4: The Quantum Scalar Field February 2012 c 2012 W. Taylor 8.323 Section 4: Quantum scalar field 1 / 19 4.1 Canonical Quantization Free scalar field equation (Klein-Gordon) ( µ µ
More informationMaster integrals without subdivergences
Master integrals without subdivergences Joint work with Andreas von Manteuffel and Robert Schabinger Erik Panzer 1 (CNRS, ERC grant 257638) Institute des Hautes Études Scientifiques 35 Route de Chartres
More informationNLO-QCD calculation in GRACE. - GRACE status - Y. Kurihara (KEK) GRACE Group LoopFest IV
NLO-QCD calculation in GRACE - GRACE status - Y. Kurihara (KEK) GRACE Group 19/Aug./2005 @ LoopFest IV GRACE Author list J. Fujimoto, T. Ishikawa, M. Jimbo, T. Kaneko, K. Kato, S. Kawabata, T. Kon, Y.
More information11 a 12 a 21 a 11 a 22 a 12 a 21. (C.11) A = The determinant of a product of two matrices is given by AB = A B 1 1 = (C.13) and similarly.
C PROPERTIES OF MATRICES 697 to whether the permutation i 1 i 2 i N is even or odd, respectively Note that I =1 Thus, for a 2 2 matrix, the determinant takes the form A = a 11 a 12 = a a 21 a 11 a 22 a
More informationRichard Williams C. S. Fischer, W. Heupel, H. Sanchis-Alepuz
Richard Williams C. S. Fischer, W. Heupel, H. Sanchis-Alepuz Overview 2 1.Motivation and Introduction 4. 3PI DSE results 2. DSEs and BSEs 3. npi effective action 6. Outlook and conclusion 5. 3PI meson
More informationSimplified differential equations approach for NNLO calculations
Simplified differential equations approach for NNLO calculations Costas. G. Papadopoulos INPP, NCSR Demokritos, Athens UCLA, June 19, 2015 Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, 2015 1 / 39
More informationConvergence of sequences and series
Convergence of sequences and series A sequence f is a map from N the positive integers to a set. We often write the map outputs as f n rather than f(n). Often we just list the outputs in order and leave
More informationDeep inelastic scattering and the OPE in lattice QCD
Deep inelastic scattering and the OPE in lattice QCD William Detmold [WD & CJD Lin PRD 73, 014501 (2006)] DIS structure of hadrons Deep-inelastic scattering process critical to development of QCD k, E
More informationGaussian integrals and Feynman diagrams. February 28
Gaussian integrals and Feynman diagrams February 28 Introduction A mathematician is one to whom the equality e x2 2 dx = 2π is as obvious as that twice two makes four is to you. Lord W.T. Kelvin to his
More informationA New Identity for Gauge Theory Amplitudes
A New Identity for Gauge Theory Amplitudes Hidden Structures workshop, NBI Sept 10, 2008 Henrik Johansson, UCLA Z. Bern, J.J. Carrasco, HJ arxive:0805 :0805.3993 [hep-ph[ hep-ph] Z. Bern, J.J. Carrasco,
More information1 Probability : Worked Examples
1 Probability : Worked Examples 1) The information entropy of a distribution {p n } is defined as S = n p n log 2 p n, where n ranges over all possible configurations of a given physical system and p n
More informationPhysics 217 FINAL EXAM SOLUTIONS Fall u(p,λ) by any method of your choosing.
Physics 27 FINAL EXAM SOLUTIONS Fall 206. The helicity spinor u(p, λ satisfies u(p,λu(p,λ = 2m. ( In parts (a and (b, you may assume that m 0. (a Evaluate u(p,λ by any method of your choosing. Using the
More informationSTA G. Conformal Field Theory in Momentum space. Kostas Skenderis Southampton Theory Astrophysics and Gravity research centre.
Southampton Theory Astrophysics and Gravity research centre STA G Research Centre Oxford University 3 March 2015 Outline 1 Introduction 2 3 4 5 Introduction Conformal invariance imposes strong constraints
More informationWard Takahashi Identities
Ward Takahashi Identities There is a large family of Ward Takahashi identities. Let s start with two series of basic identities for off-shell amplitudes involving 0 or 2 electrons and any number of photons.
More informationMathematics of Physics and Engineering II: Homework problems
Mathematics of Physics and Engineering II: Homework problems Homework. Problem. Consider four points in R 3 : P (,, ), Q(,, 2), R(,, ), S( + a,, 2a), where a is a real number. () Compute the coordinates
More informationWeek 5-6: Lectures The Charged Scalar Field
Notes for Phys. 610, 2011. These summaries are meant to be informal, and are subject to revision, elaboration and correction. They will be based on material covered in class, but may differ from it by
More informationRenormalizability in (noncommutative) field theories
Renormalizability in (noncommutative) field theories LIPN in collaboration with: A. de Goursac, R. Gurău, T. Krajewski, D. Kreimer, J. Magnen, V. Rivasseau, F. Vignes-Tourneret, P. Vitale, J.-C. Wallet,
More informationMarch Algebra 2 Question 1. March Algebra 2 Question 1
March Algebra 2 Question 1 If the statement is always true for the domain, assign that part a 3. If it is sometimes true, assign it a 2. If it is never true, assign it a 1. Your answer for this question
More informationReview of Power Series
Review of Power Series MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Introduction In addition to the techniques we have studied so far, we may use power
More informationLoop Integrands from Ambitwistor Strings
Loop Integrands from Ambitwistor Strings Yvonne Geyer Institute for Advanced Study QCD meets Gravity UCLA arxiv:1507.00321, 1511.06315, 1607.08887 YG, L. Mason, R. Monteiro, P. Tourkine arxiv:1711.09923
More informationSpectral Representation of Random Processes
Spectral Representation of Random Processes Example: Represent u(t,x,q) by! u K (t, x, Q) = u k (t, x) k(q) where k(q) are orthogonal polynomials. Single Random Variable:! Let k (Q) be orthogonal with
More informationThe path integral for photons
The path integral for photons based on S-57 We will discuss the path integral for photons and the photon propagator more carefully using the Lorentz gauge: as in the case of scalar field we Fourier-transform
More informationTwo-loop Remainder Functions in N = 4 SYM
Two-loop Remainder Functions in N = 4 SYM Claude Duhr Institut für theoretische Physik, ETH Zürich, Wolfgang-Paulistr. 27, CH-8093, Switzerland E-mail: duhrc@itp.phys.ethz.ch 1 Introduction Over the last
More information