Reduction of Feynman integrals to master integrals

Transcription

1 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

2 Reduction problem for Feynman integrals A review of algorithmic approaches The description of our approach Examples and results New ideas and modifications Conclusion Work done in collaboration with V.A. Smirnov [A.V. Smirnov & V.A. Smirnov, JHEP 0601 (2006) 001; A.V. Smirnov, JHEP 0604 (2006) 026; V.A. Smirnov, hep-ph/ ; A.V. Smirnov, V.A. Smirnov, hep-ph/ (short reviews); A.G. Grozin, A.V. Smirnov and V.A. Smirnov, JHEP 0611 (2006) 022 A.V. Smirnov, V.A. Smirnov, and M. Steinhauser, work in progress] A.V. Smirnov ACAT 2007 p.2

3 Reduction problem for Feynman integrals Reduction problem for Feynman integrals A given Feynman graph Γ tensor reduction various scalar Feynman integrals that have the same structure of the integrand with various distributions of powers of propagators F(a 1,...,a n ) = d d k 1...d d k h E a Ea n n d = 4 2ǫ; the denominators E r are either quadratic or linear with respect to the loop momenta p i = k i, i = 1,...,h or to the independent external momenta p h+1 = q 1,...,p h+n = q N of the graph.. A.V. Smirnov ACAT 2007 p.3

4 Reduction problem for Feynman integrals An old analytical strategy: to evaluate, by some methods, every scalar Feynman integral generated by the given graph. A.V. Smirnov ACAT 2007 p.4

5 Reduction problem for Feynman integrals An old analytical strategy: to evaluate, by some methods, every scalar Feynman integral generated by the given graph. And what is already a traditional strategy: to derive, without calculation, and then apply integration by parts (IBP) relations between the given family of Feynman integrals as recurrence relations. [K.G. Chetyrkin and F.V. Tkachov 81] A general integral from the given family is expressed as a linear combination of some basic (master) integrals. A.V. Smirnov ACAT 2007 p.4

6 Reduction problem for Feynman integrals The whole problem of evaluation falls apart into two parts A.V. Smirnov ACAT 2007 p.5

7 Reduction problem for Feynman integrals The whole problem of evaluation falls apart into two parts constructing a reduction procedure A.V. Smirnov ACAT 2007 p.5

8 Reduction problem for Feynman integrals The whole problem of evaluation falls apart into two parts constructing a reduction procedure evaluating master integrals A.V. Smirnov ACAT 2007 p.5

9 Reduction problem for Feynman integrals The whole problem of evaluation falls apart into two parts constructing a reduction procedure evaluating master integrals The talk is devoted to the first part of the problem. A.V. Smirnov ACAT 2007 p.5

10 Reduction problem for Feynman integrals Required notation F(a 1,...,a n ) are functions of integer variables a 1,...,a n N n F is an infinitely dimensional linear space. The simplest basis: H a1,...,a n (a 1,...,a n) = δ a1,a 1...δ a n,a n Feynman integrals form a point in this vector space. All relations we use are linear functionals. A.V. Smirnov ACAT 2007 p.6

11 Reduction problem for Feynman integrals Types of relations A.V. Smirnov ACAT 2007 p.7

12 Reduction problem for Feynman integrals Types of relations Most commonly used relations: IBP relations. IBP: [K.G. Chetyrkin and F.V. Tkachov 81]... d d k 1 d d k 2... k i ( 1 p j E a Ea n n ) = 0, A.V. Smirnov ACAT 2007 p.7

13 Reduction problem for Feynman integrals Types of relations Most commonly used relations: IBP relations. IBP: [K.G. Chetyrkin and F.V. Tkachov 81]... d d k 1 d d k 2... k i ( 1 p j E a Ea n n ) = 0, αi F(a 1 + b i,1,...,a n + b i,n ) = 0, A.V. Smirnov ACAT 2007 p.7

14 Reduction problem for Feynman integrals Types of relations Most commonly used relations: IBP relations. IBP: [K.G. Chetyrkin and F.V. Tkachov 81]... d d k 1 d d k 2... k i ( 1 p j E a Ea n n ) = 0, αi F(a 1 + b i,1,...,a n + b i,n ) = 0, Lorentz-invariance (LI) identities [T. Gehrmann and E. Remiddi 00] A.V. Smirnov ACAT 2007 p.7

15 Reduction problem for Feynman integrals Types of relations symmetry relations, e.g., F(a 1,...,a n ) = ( 1) d 1a d n a n F(a σ(1),...,a σ(n) ), A.V. Smirnov ACAT 2007 p.8

16 Reduction problem for Feynman integrals Types of relations symmetry relations, e.g., F(a 1,...,a n ) = ( 1) d 1a d n a n F(a σ(1),...,a σ(n) ), Boundary conditions: F(a 1,a 2,...,a n ) = 0 when a i1 0,...a ik 0 for some subsets of indices i j ; A.V. Smirnov ACAT 2007 p.8

17 Reduction problem for Feynman integrals Types of relations symmetry relations, e.g., F(a 1,...,a n ) = ( 1) d 1a d n a n F(a σ(1),...,a σ(n) ), Boundary conditions: F(a 1,a 2,...,a n ) = 0 when a i1 0,...a ik 0 for some subsets of indices i j ; parity conditions,... A.V. Smirnov ACAT 2007 p.8

18 Reduction problem for Feynman integrals All those relations a possibility to express given Feynman integrals in terms of other Feynman integrals. We have to name certain integrals irreducible (master) and aim to reduce any other integral to those. An attempt to formalize the definition of master integrals was made in [A.V. Smirnov, JHEP 0604 (2006) 026]. The notion of the master (irreducible) integral A.V. Smirnov ACAT 2007 p.9

19 Reduction problem for Feynman integrals All those relations a possibility to express given Feynman integrals in terms of other Feynman integrals. We have to name certain integrals irreducible (master) and aim to reduce any other integral to those. An attempt to formalize the definition of master integrals was made in [A.V. Smirnov, JHEP 0604 (2006) 026]. The notion of the master (irreducible) integral a priority between the points (a 1,...,a n ) A.V. Smirnov ACAT 2007 p.9

20 Reduction problem for Feynman integrals All those relations a possibility to express given Feynman integrals in terms of other Feynman integrals. We have to name certain integrals irreducible (master) and aim to reduce any other integral to those. An attempt to formalize the definition of master integrals was made in [A.V. Smirnov, JHEP 0604 (2006) 026]. The notion of the master (irreducible) integral a priority between the points (a 1,...,a n ) an ordering. A.V. Smirnov ACAT 2007 p.9

21 Reduction problem for Feynman integrals How to choose an ordering? A.V. Smirnov ACAT 2007 p.10

22 Reduction problem for Feynman integrals How to choose an ordering? Feynman integrals are simpler, from the analytic point of view, if they have more non-positive indices. A.V. Smirnov ACAT 2007 p.10

23 Reduction problem for Feynman integrals How to choose an ordering? Feynman integrals are simpler, from the analytic point of view, if they have more non-positive indices. Solving IBP relations by hand reducing indices to zero A.V. Smirnov ACAT 2007 p.10

24 Reduction problem for Feynman integrals How to choose an ordering? Feynman integrals are simpler, from the analytic point of view, if they have more non-positive indices. Solving IBP relations by hand reducing indices to zero we come to the notion of sectors A.V. Smirnov ACAT 2007 p.10

25 Reduction problem for Feynman integrals Sectors ( topologies ): 2 n regions labelled by subsets ν {1,...,n}: σ ν = {(a 1,...,a n ) : a i > 0 if i ν, a i 0 if i ν} A.V. Smirnov ACAT 2007 p.11

26 Reduction problem for Feynman integrals Sectors ( topologies ): 2 n regions labelled by subsets ν {1,...,n}: σ ν = {(a 1,...,a n ) : a i > 0 if i ν, a i 0 if i ν} A sector is σ ν said to be lower than a sector σ µ if ν µ A.V. Smirnov ACAT 2007 p.11

27 Reduction problem for Feynman integrals Sectors ( topologies ): 2 n regions labelled by subsets ν {1,...,n}: σ ν = {(a 1,...,a n ) : a i > 0 if i ν, a i 0 if i ν} A sector is σ ν said to be lower than a sector σ µ if ν µ F(a 1,...,a n ) F(a 1,...,a n) if the sector of (a 1,...,a n) is lower than the sector of (a 1,...,a n ). A.V. Smirnov ACAT 2007 p.11

28 Reduction problem for Feynman integrals Sectors ( topologies ): 2 n regions labelled by subsets ν {1,...,n}: σ ν = {(a 1,...,a n ) : a i > 0 if i ν, a i 0 if i ν} A sector is σ ν said to be lower than a sector σ µ if ν µ F(a 1,...,a n ) F(a 1,...,a n) if the sector of (a 1,...,a n) is lower than the sector of (a 1,...,a n ). To define an ordering completely introduce it in some way inside the sectors (to be discussed later). A.V. Smirnov ACAT 2007 p.11

29 Reduction problem for Feynman integrals A review of algorithmic approaches The description of our approach Examples and results New ideas and modifications Conclusion A.V. Smirnov ACAT 2007 p.12

30 A review of algorithmic approaches A review of algorithmic approaches Initially the relations were being solved by hand but with the growth of the complexity of the problems it turns to be impractical. Laporta s algorithm [S. Laporta and E. Remiddi 96; S. Laporta 00; T. Gehrmann and E. Remiddi 01] When increasing the total power of the denominator and numerator, the total number of IBP equations grows faster than the number of independent Feynman integrals. Therefore this system of equations sooner or later becomes overdetermined, and one obtains the possibility to perform a reduction to master integrals A.V. Smirnov ACAT 2007 p.13

31 A review of algorithmic approaches Various implementations: one public version AIR several private versions [C. Anastasiou and A. Lazopoulos 04] [T. Gehrmann and E. Remiddi, M. Czakon, P. Marquard and D. Seidel, Y. Schröder, C. Sturm, A. Onishchenko,... ] A.V. Smirnov ACAT 2007 p.14

32 A review of algorithmic approaches Various implementations: one public version AIR several private versions [C. Anastasiou and A. Lazopoulos 04] [T. Gehrmann and E. Remiddi, M. Czakon, P. Marquard and D. Seidel, Y. Schröder, C. Sturm, A. Onishchenko,... ] Independent methods. A.V. Smirnov ACAT 2007 p.14

33 A review of algorithmic approaches Various implementations: one public version AIR several private versions [C. Anastasiou and A. Lazopoulos 04] [T. Gehrmann and E. Remiddi, M. Czakon, P. Marquard and D. Seidel, Y. Schröder, C. Sturm, A. Onishchenko,... ] Independent methods. Baikov s method Applications: [P.A. Baikov 96 07; V.A. Smirnov and M. Steinhauser 03] [P.A. Baikov, K.G. Chetrykin and J.H. Kühn 98 06; V.A. Smirnov and M. Steinhauser 03; B.A. Kniehl, A. Onishchenko, J.H. Piclum and M. Steinhauser 06] A.V. Smirnov ACAT 2007 p.14

34 A review of algorithmic approaches Reduction using Gröbner bases Historically, suggested by O.V. Tarasov [O.V. Tarasov 98] Reduce the problem to differential equations by introducing a mass for every line, a i i + m 2 i The most recent application: Massless two-loop off-shell vertex diagrams [F. Jegerlehner and O.V. Tarasov 05] A.V. Smirnov ACAT 2007 p.15

35 A review of algorithmic approaches Reduction using Gröbner bases Historically, suggested by O.V. Tarasov [O.V. Tarasov 98] Reduce the problem to differential equations by introducing a mass for every line, a i i + m 2 i The most recent application: Massless two-loop off-shell vertex diagrams [F. Jegerlehner and O.V. Tarasov 05] Direct application of Groebner bases (without the use of differential equations) [V.P. Gerdt 04, 05] This algorithm works (at the moment) only for Feynman graphs with three lines. A.V. Smirnov ACAT 2007 p.15

36 Reduction problem for Feynman integrals A review of algorithmic approaches The description of our approach Examples and results New ideas and modifications Conclusion A.V. Smirnov ACAT 2007 p.16

37 The description of our approach s-bases one more approach initially based on Gröbner bases [A.V. Smirnov and V.A. Smirnov 05] The description of this method Suppose first that we are interested in expressing any integral in the positive sector σ {1,...,n} as a linear combination of a finite number of integrals in it.... d d k 1 d d k 2... k i ( p j 1 E a Ea N N ) = 0 ci F(a 1 + b i,1,...,a n + b i,n) = 0 A.V. Smirnov ACAT 2007 p.17

38 The description of our approach The left-hand sides of IBP relations can be expressed in terms of operators of multiplication by the indices a i and shift operators Y i = i +,Yi = i, where (Y ± i F)(a 1,...,a n ) = F(a 1,...,a i ± 1,...,a n ) A.V. Smirnov ACAT 2007 p.18

39 The description of our approach The left-hand sides of IBP relations can be expressed in terms of operators of multiplication by the indices a i and shift operators Y i = i +,Yi = i, where (Y ± i F)(a 1,...,a n ) = F(a 1,...,a i ± 1,...,a n ) Thus, one can choose certain elements f i corresponding to IBP relations and write (f i F)(a 1,...,a n ) = 0 The choice is not unique, we will get rid of Y i A.V. Smirnov ACAT 2007 p.18

40 The description of our approach The algebra A 0 generated by shift operators Y i + and multiplication operators A i. It acts on the field of functions F of n integer variables. The ideal I of IBP relations generated by the elements f i. For any element X I we have (XF)(1, 1,...,1) = 0. A.V. Smirnov ACAT 2007 p.19

41 The description of our approach The algebra A 0 generated by shift operators Y i + and multiplication operators A i. It acts on the field of functions F of n integer variables. The ideal I of IBP relations generated by the elements f i. For any element X I we have Also we have (XF)(1, 1,...,1) = 0. F(a 1,a 2,...,a n ) = (Y a Y a n 1 n F)(1, 1,...,1) A.V. Smirnov ACAT 2007 p.19

42 The description of our approach The algebra A 0 generated by shift operators Y i + and multiplication operators A i. It acts on the field of functions F of n integer variables. The ideal I of IBP relations generated by the elements f i. For any element X I we have Also we have (XF)(1, 1,...,1) = 0. F(a 1,a 2,...,a n ) = (Y a Y a n 1 n F)(1, 1,...,1) The idea of the algorithm is to reduce the element in the right-hand side of the equation using the elements of the ideal I. A.V. Smirnov ACAT 2007 p.19

43 The description of our approach Suppose we are interested in an integral F(a 1,a 2,...,a n ) = (Y a Y a n 1 n F)(1, 1,...,1) A.V. Smirnov ACAT 2007 p.20

44 The description of our approach Suppose we are interested in an integral F(a 1,a 2,...,a n ) = (Y a Y a n 1 n F)(1, 1,...,1) The reduction problem reduce the monomial Y a 1 1 the IBP relations 1...Y a n 1 n modulo the ideal of Y a Y a n 1 n = r i f i + c i1,...,i n Y i Y i n 1 n A.V. Smirnov ACAT 2007 p.20

45 The description of our approach Suppose we are interested in an integral F(a 1,a 2,...,a n ) = (Y a Y a n 1 n F)(1, 1,...,1) The reduction problem reduce the monomial Y a 1 1 the IBP relations 1...Y a n 1 n modulo the ideal of Y a Y a n 1 n = r i f i + c i1,...,i n Y i Y i n 1 n Apply to F at a 1 = 1,...,a n = 1 to obtain F(a 1,a 2,...,a n ) = c i1,...,i n F(i 1,i 2,...,i n ), where integrals on the right-hand side are master integrals. A.V. Smirnov ACAT 2007 p.20

46 The description of our approach To do the reduction we need an ordering of monomials of operators Y i or, similarly, an ordering of points (a 1,...,a n ) in the sector: For two monomials M 1 = Y i Y i n 1 n and M 2 = Y j Y j n 1 n (M 1 F)(1,...,1) (M 2 F)(1,...,1) if and only if M 1 M 2 Then the reduction procedure becomes similar to the division of polynomials. But one needs to introduce a proper ordering. A.V. Smirnov ACAT 2007 p.21

47 The description of our approach Orderings on the algebra of operators Monomials degrees (sets of n non-negative integers). We require the following properties: i) for any a N n not equal to (0,...0) one has a (0,...0) ii) for any a,b,c N n one has a b if and only if a + c b + c. A.V. Smirnov ACAT 2007 p.22

48 The description of our approach Orderings on the algebra of operators Monomials degrees (sets of n non-negative integers). We require the following properties: i) for any a N n not equal to (0,...0) one has a (0,...0) ii) for any a,b,c N n one has a b if and only if a + c b + c. E.g., lexicographical ordering: A set (i 1,...,i n ) is higher than a set (j 1,...,j n ), (i 1,...,i n ) (j 1,...,j n ) if there is l n such that i 1 = j 1, i 2 = j 2,..., i l 1 = j l 1 and i l > j l. A.V. Smirnov ACAT 2007 p.22

49 The description of our approach Orderings on the algebra of operators Monomials degrees (sets of n non-negative integers). We require the following properties: i) for any a N n not equal to (0,...0) one has a (0,...0) ii) for any a,b,c N n one has a b if and only if a + c b + c. E.g., lexicographical ordering: A set (i 1,...,i n ) is higher than a set (j 1,...,j n ), (i 1,...,i n ) (j 1,...,j n ) if there is l n such that i 1 = j 1, i 2 = j 2,..., i l 1 = j l 1 and i l > j l. Degree-lexicographical ordering: (i 1,...,i n ) (j 1,...,j n ) if ik > j k, or i k = j k and (i 1,...,i n ) (j 1,...,j n ) in the sense of the lexicographical ordering. A.V. Smirnov ACAT 2007 p.22

50 The description of our approach An ordering can be defined by a matrix. Lexicographical, degree-lexicographical and reverse degree-lexicographical ordering for n = 5: , , A.V. Smirnov ACAT 2007 p.23

51 The description of our approach But the reduction does not always lead to a reasonable number of irreducible integrals one has to build a special basis first. A.V. Smirnov ACAT 2007 p.24

52 The description of our approach But the reduction does not always lead to a reasonable number of irreducible integrals one has to build a special basis first. Having elements with lowest possible degrees obtaining master integrals with minimal possible degrees. A.V. Smirnov ACAT 2007 p.24

53 The description of our approach But the reduction does not always lead to a reasonable number of irreducible integrals one has to build a special basis first. Having elements with lowest possible degrees obtaining master integrals with minimal possible degrees. One needs to build special bases an algorithm based on the Buchberger algorithm - S-polynomials, reductions. A.V. Smirnov ACAT 2007 p.24

54 The description of our approach But the reduction does not always lead to a reasonable number of irreducible integrals one has to build a special basis first. Having elements with lowest possible degrees obtaining master integrals with minimal possible degrees. One needs to build special bases an algorithm based on the Buchberger algorithm - S-polynomials, reductions. Moreover, we must keep in mind that we are interested in integrals not only in the positive sector. A.V. Smirnov ACAT 2007 p.24

55 The description of our approach Our algorithm [A.S.& V.S 05] : to build a set of special bases of the ideal of IBP relations (s-bases). sectors σ ν = {(a 1,...,a n ) : a i > 0 if i ν, a i 0 if i ν} In the sector σ {1,...,n}, consider Y i as basic operators. In the sector σ ν, consider Y i for i ν and Yi for other i as basic operators. Construct sector bases (s-bases), rather than Gröbner bases for all the sectors. An s-basis for a sector σ ν is a set of elements of a basis which provides the possibility of a reduction to master integrals and integrals whose indices lie in lower sectors, i.e. σ ν for ν ν. (It is most complicated to construct s-bases for minimal sectors.) A.V. Smirnov ACAT 2007 p.25

56 The description of our approach Many sectors seemingly the problem becomes harder. But the important simplification is that one is not trying to solve the reduction problem in each sector separately but allows to reduce the integrals in a given sector to lower sectors similarly to the by hand solutions. The construction close to the Buchberger algorithm but it can be terminated when the current basis already provides us the needed reduction. The basic operations are the same, i.e. calculating S-polynomials and reducing them modulo current basis, with a chosen ordering. After constructing s-bases for all non-trivial sectors one obtains a recursive (with respect to the sectors) procedure to evaluate F(a 1,...,a n ) at any point and thereby reduce a A.V. Smirnov given integral to master integrals. ACAT 2007 p.26

57 Reduction problem for Feynman integrals A review of algorithmic approaches The description of our approach Examples and results New ideas and modifications Conclusion A.V. Smirnov ACAT 2007 p.27

58 Examples and results Reduction of a family of Feynman integrals relevant to the three-loop static quark potential A family of Feynman integrals with 9 indices [A.V. Smirnov and V.A. Smirnov 05] [A.V. Smirnov & A.G. Grozin, A.V. Smirnov and V.A. Smirnov, JHEP 0611 (2006) 022] A.V. Smirnov ACAT 2007 p.28

59 Examples and results Symmetry: (1 2, 3 4, 7 8) Boundary conditions: F(a 1,...,a 9 ) = 0 if one of the following sets of lines has non-positive indices: {5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}, {3, 4, 6}. Master integrals: I 1 = F(1, 1, 0, 1, 1, 1, 1, 0, 0), I 2 = F(1, 1, 1, 1, 0, 0, 1, 1, 0), I 3 = F(1, 1, 0, 0, 0, 1, 1, 1, 0), I 4 = F(0, 1, 1, 0, 1, 1, 0, 1, 0), Ī 4 = F( 1, 1, 1, 0, 1, 1, 0, 1, 0), I 5 = F(0, 0, 0, 1, 1, 1, 1, 0, 0), I 6 = F(0, 1, 0, 0, 0, 1, 1, 1, 0), I 7 = F(0, 1, 0, 0, 1, 1, 1, 0, 0), Ī 7 = F(0, 2, 0, 0, 1, 1, 1, 0, 0), I 8 = F(0, 0, 0, 0, 0, 1, 1, 1, 0). A.V. Smirnov ACAT 2007 p.29

60 Examples of reduction: F(1,..., 1, 0) = + 3(d 4)(3d 10) 8(d 5)(2d 9) I 1 (d 3)(3d 10)(3d 8) 8(d 5)(3d 13)(3d 11) I 3 9(d 4)(d 2)(3d 10)(3d 8) 64(d 5)(2d 9)(2d 7)(3d 13) I 5 3(d 4)(3d 10) 16(d 5)(2d 9) I 2 3(d 2)(3d 11)(3d 10)(3d 8) 64(d 5)(2d 9)(2d 7)(3d 13)Ī4 3(3d 10)(3d 8) 32(d 5)(2d 9)(2d 7)Ī7, F(1,..., 1, 1) = 3(d 3)(3d 11) 16(d 5)(d 4)(2d 9) I 4 (d 2)(2d 7)(2d 5) 8(d 3)(2d 9)(3d 13) I 6 3(2d 7)2 (2d 5)(3d 11)(3d 7) 256(d 4) 2 I 7. (d 3)(2d 9) A.V. Smirnov ACAT 2007 p.30

61 Reduction problem for Feynman integrals A review of algorithmic approaches The description of our approach Examples and results New ideas and modifications Conclusion A.V. Smirnov ACAT 2007 p.31

62 New ideas and modifications New ideas and modifications The algorithm has been used to construct the reduction in JHEP 0611 (2006) 022. Now it is being applied with M. Steinhauser and V. Smirnov in a three-loop calculation. A.V. Smirnov ACAT 2007 p.32

63 New ideas and modifications New ideas and modifications The algorithm has been used to construct the reduction in JHEP 0611 (2006) 022. Now it is being applied with M. Steinhauser and V. Smirnov in a three-loop calculation. Constructing a basis requires a certain skill to find a proper ordering. A.V. Smirnov ACAT 2007 p.32

64 New ideas and modifications New ideas and modifications The algorithm has been used to construct the reduction in JHEP 0611 (2006) 022. Now it is being applied with M. Steinhauser and V. Smirnov in a three-loop calculation. Constructing a basis requires a certain skill to find a proper ordering. Automatic construction in some sectors. A.V. Smirnov ACAT 2007 p.32

65 New ideas and modifications New ideas and modifications The algorithm has been used to construct the reduction in JHEP 0611 (2006) 022. Now it is being applied with M. Steinhauser and V. Smirnov in a three-loop calculation. Constructing a basis requires a certain skill to find a proper ordering. Automatic construction in some sectors. Not a problem of cpu time or memory, but an algorithmical one. A.V. Smirnov ACAT 2007 p.32

66 New ideas and modifications Using regions instead of sectors and reduction to problems with less integrals Experience: s-bases are constructed easily if the number of non-positive indices in a given sector is small. A.V. Smirnov ACAT 2007 p.33

67 New ideas and modifications Using regions instead of sectors and reduction to problems with less integrals Experience: s-bases are constructed easily if the number of non-positive indices in a given sector is small. What can help: If the number of non-positive indices is large, there is usually a possibility to perform integration over some loop momentum explicitly in terms of gamma functions for general d. A.V. Smirnov ACAT 2007 p.33

68 New ideas and modifications Using regions instead of sectors and reduction to problems with less integrals Experience: s-bases are constructed easily if the number of non-positive indices in a given sector is small. What can help: If the number of non-positive indices is large, there is usually a possibility to perform integration over some loop momentum explicitly in terms of gamma functions for general d. However, to derive the corresponding explicit formulae, with multiple finite summations, for all necessary cases turns out to be impractical. A.V. Smirnov ACAT 2007 p.33

69 New ideas and modifications In this situation, there is another alternative. Let us distinguish the propagators (and irreducible numerators) involved in such an explicit integration formula. They are associated with a subgraph of the given graph. Let us solve IBP relations for the corresponding subintegral in order to express any such subintegral in term of some master integral. A.V. Smirnov ACAT 2007 p.34

70 New ideas and modifications In this situation, there is another alternative. Let us distinguish the propagators (and irreducible numerators) involved in such an explicit integration formula. They are associated with a subgraph of the given graph. Let us solve IBP relations for the corresponding subintegral in order to express any such subintegral in term of some master integral. Then, after using this reduction procedure, it will be sufficient to use explicit integration formulae only for some boundary values of the indices. This replacement is very simple, without multiple summations. A.V. Smirnov ACAT 2007 p.34

71 New ideas and modifications Coefficients in these reduction formulae depend not only on d and given external parameters but also on the propagators of the given graph which are external for the subgraph. A.V. Smirnov ACAT 2007 p.35

72 New ideas and modifications Coefficients in these reduction formulae depend not only on d and given external parameters but also on the propagators of the given graph which are external for the subgraph. Within our algorithm, it turned out possible to implement the solution of the recursive problem for the subgraph in terms of the Feynman integrals for the whole graph. In this reduction, pure powers of the parameters which are external for the subgraph transform naturally into the corresponding shift operators and their inverse. A.V. Smirnov ACAT 2007 p.35

73 New ideas and modifications Coefficients in these reduction formulae depend not only on d and given external parameters but also on the propagators of the given graph which are external for the subgraph. Within our algorithm, it turned out possible to implement the solution of the recursive problem for the subgraph in terms of the Feynman integrals for the whole graph. In this reduction, pure powers of the parameters which are external for the subgraph transform naturally into the corresponding shift operators and their inverse. Integrals which are obtained from initial integrals by an explicit integration over a loop momentum in terms of gamma function usually involve a propagator with a regularization by an amount proportional to ǫ. Our code is applicable also in such situations. A.V. Smirnov ACAT 2007 p.35

74 New ideas and modifications Combining with the Laporta algorithm Even if we have not constructed the s-bases for all sectors we can still run the reduction procedure. However, in some sectors there will be an infinite number of irreducible integrals left. A.V. Smirnov ACAT 2007 p.36

75 New ideas and modifications Combining with the Laporta algorithm Even if we have not constructed the s-bases for all sectors we can still run the reduction procedure. However, in some sectors there will be an infinite number of irreducible integrals left. One of the ideas is to use the Laporta algorithm in those sectors. Still it leads to several problems in linking two algorithms together. One of the disadvantages is that the reductions works much slower if the number of irreducible integrals goes high. A.V. Smirnov ACAT 2007 p.36

76 New ideas and modifications Combining with the Laporta algorithm Even if we have not constructed the s-bases for all sectors we can still run the reduction procedure. However, in some sectors there will be an infinite number of irreducible integrals left. One of the ideas is to use the Laporta algorithm in those sectors. Still it leads to several problems in linking two algorithms together. One of the disadvantages is that the reductions works much slower if the number of irreducible integrals goes high. Another idea is to combine two algorithms inside one code. The s-bases might give the reductions for the integrals the can work with. And for the remaining integrals we use systems of initial IBP s reduced by the s-bases part of the algorithm. A.V. Smirnov ACAT 2007 p.36

77 Conclusion A.V. Smirnov ACAT 2007 p.37

78 Conclusion The s-bases approach seems to be a good addition to the methods of automatic integral reduction. A.V. Smirnov ACAT 2007 p.38

79 Conclusion The s-bases approach seems to be a good addition to the methods of automatic integral reduction. The bases construction part of the algorithm should still be improved, but even now it is capable of constructing bases for quite complicated cases. A.V. Smirnov ACAT 2007 p.38

80 Conclusion The s-bases approach seems to be a good addition to the methods of automatic integral reduction. The bases construction part of the algorithm should still be improved, but even now it is capable of constructing bases for quite complicated cases. A constructed basis in this sector one needs no more real solving of equations the system of linear equations becomes upper-triangle. A.V. Smirnov ACAT 2007 p.38

81 Conclusion The s-bases approach seems to be a good addition to the methods of automatic integral reduction. The bases construction part of the algorithm should still be improved, but even now it is capable of constructing bases for quite complicated cases. A constructed basis in this sector one needs no more real solving of equations the system of linear equations becomes upper-triangle. Probably, the future is in combining the s-bases and the Laporta algorithm the bases can be constructed where possible and provide better input than standard IBP s in those sectors. A.V. Smirnov ACAT 2007 p.38