New Methods for Feynman Integrals V.A. Smirnov Nuclear Physics Institute of Moscow State University PSI, November 03, 2008 – p.1 V.A. Smirnov
Introduction. Methods of evaluating Feynman integrals Reduction to master integrals using IBP relations. Laporta algorithm and its implementations. FIRE Evaluating Feynman integrals by sector decompositions. FIESTA Evaluating Feynman integrals by the method of Mellin–Barnes representation. MBresolve.m A recent application: the three-loop quark static potential PSI, November 03, 2008 – p.2 V.A. Smirnov
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. d d k 1 d d k 2 . . . � � F Γ ( a 1 , a 2 , . . . ) = . . . 2 ) a 2 . . . ( p 2 1 − m 2 1 ) a 1 ( p 2 2 − m 2 d = 4 − 2 ǫ Besides usual propagators, one can have 1 ( v · k + i 0) a PSI, November 03, 2008 – p.3 V.A. Smirnov
Methods to evaluate Feynman integrals: analytical, numerical, semianalytical . . . An old straightforward analytical strategy: to evaluate, by some methods, every scalar Feynman integral generated by the given graph. PSI, November 03, 2008 – p.4 V.A. Smirnov
The standard modern strategy: to derive, without calculation, and then apply IBP identities between the given family of Feynman integrals as recurrence relations. A general integral of the given family is expressed as a linear combination of some basic (master) integrals. The whole problem of evaluation → constructing a reduction procedure evaluating master integrals PSI, November 03, 2008 – p.5 V.A. Smirnov
Solving reduction problems algorithmically: ‘Laporta’s algorithm’ [S. Laporta and E. Remiddi’96; S. Laporta’00; T. Gehrmann and E. Remiddi’01] A public version AIR [C. Anastasiou and A. Lazopoulos’04] Private versions [T. Gehrmann and E. Remiddi, M. Czakon, Y. Schröder, C. Sturm, P . Marquard and D. Seidel, V. Velizhanin, . . . ] Baikov’s method Gröbner bases. Suggested by O.V. Tarasov [O.V. Tarasov’98] An alternative approach: [A.V. Smirnov & V.A. Smirnov’05–07; A.G. Grozin, A.V. Smirnov and V.A. Smirnov’06 A.V. Smirnov, V.A. Smirnov, and M. Steinhauser’08 ] PSI, November 03, 2008 – p.6 V.A. Smirnov
FIRE = Feynman Integrals REduction [A.V. Smirnov’08] (implemented in Mathematica ) http://www-ttp.particle.uni-karlsruhe.de/ ∼ asmirnov Sectors 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 } Natural ordering. The goal of reduction: to make more non-positive indices. PSI, November 03, 2008 – p.7 V.A. Smirnov
Three different strategies in FIRE. 1. In sectors with a small number of non-positive indices, apply s -bases (generalizations of Gröbner bases). Constructing them automatically by a kind of Buchberger algorithm. SBases.m PSI, November 03, 2008 – p.8 V.A. Smirnov
2. In sectors with a large number of non-positive indices, integrate over a loop momentum explicitly and reduce the problem to a family of two-loop integrals where the index of one propagator is, possibly, shifted by ǫ or 2 ǫ . Consider the region a 2 , a 5 , a 10 ≤ 0 , a 7 , a 8 > 0 a 10 a 2 a 9 a 11 a ′ a ′ 6 7 a 1 a 3 a ′ a ′ 1 3 a 7 a 8 → a ′ 5 + ε a ′ a ′ 2 4 a 6 a 4 a 5 Apply s -bases (within FIRE) to such 2loop reduction problems with 7 indices. PSI, November 03, 2008 – p.9 V.A. Smirnov
Reduce indices a 2 , a 5 , a 10 , a 7 , a 8 to their boundary values, i.e. a 2 , a 5 , a 10 = 0 , a 7 , a 8 = 1 a 10 a 2 0 0 a 7 a 8 → 1 1 a 5 0 At these values, the transition to the 2loop problem because very simple (without multiple summations). PSI, November 03, 2008 – p.10 V.A. Smirnov
3. In ‘intermediate sectors’, the Laporta’s algorithm (implemented within FIRE) is applied. FIRE can be run in a ‘pure Laporta’ mode. ‘Lee ideas’ [R.N. Lee’08] In each sector one may find a single IBP that works for ‘most’ points in this sector. One might generate less IBPs because the other IBPs are naturally represented as a linear combination of these. QLink is used to access the QDBM database for storing data on disk from Mathematica. FLink allows to perform external evaluations by means of the Fermat program. Fermat speds up Together and GCD. http://www-ttp.particle.uni-karlsruhe.de/ ∼ asmirnov PSI, November 03, 2008 – p.11 V.A. Smirnov
Methods to evaluate master integrals: Feynman/alpha parameters Mellin–Barnes representation [V.A. Smirnov’99, J.B Tausk’99] method of differential equations [A.V. Kotikov’91, E. Remiddi’97, T. Gehrmann & E. Remiddi’00] PSI, November 03, 2008 – p.12 V.A. Smirnov
UV, IR and collinear divergences Regularization. Dimensional regularization. Formally, d 4 k = d k 0 � d d k k → where d = 4 − 2 ǫ Informally, use alpha parameters � ∞ i a 1 d α α a − 1 e i( k 2 − m 2 ) α ( − k 2 + m 2 − i 0) a = Γ( a ) 0 change the order of integration, take Gauss integrals over the loop momenta PSI, November 03, 2008 – p.13 V.A. Smirnov
� d 4 k e i( αk 2 − 2 q · k ) = − i π 2 α − 2 e − i q 2 /α → � d d k e i( αk 2 − 2 q · k ) = e i π (1 − d/ 2) / 2 π d/ 2 α − d/ 2 e − i q 2 /α PSI, November 03, 2008 – p.14 V.A. Smirnov
Graph Γ → i a + h (1 − d/ 2) π hd/ 2 F Γ ( a 1 . . . , a L ; d ) = � l Γ( a l ) � ∞ � ∞ U − d/ 2 e i V / U− i � m 2 l α l , � α a l − 1 d α 1 . . . d α L × l 0 0 l where h is the number of loops and � � U = α l , trees T l �∈ T q T � 2 � � � V = α l . 2 − trees T l �∈ T PSI, November 03, 2008 – p.15 V.A. Smirnov
i π d/ 2 � h � Γ( a − hd/ 2) F Γ ( q 1 , . . . , q n ; d ) = � l Γ( a l ) � ∞ � ∞ l α a l − 1 U a − ( h +1) d/ 2 � �� � l × d α 1 . . . d α L δ α l − 1 −V + U � m 2 � a − hd/ 2 � l α l 0 0 Sector decompositions. Hepp sectors [K. Hepp’66] α 1 ≤ α 2 ≤ . . . ≤ α L sector variables t l = α l /α l +1 , l = 1 , . . . , L − 1; t L = α L . Back: α l = t l . . . t L PSI, November 03, 2008 – p.16 V.A. Smirnov
Speer’s sectors [E. Speer’77] labelled by the elements of a UV forest F , � α l = t γ γ ∈F : l ∈ γ For Feynman integrals with the Euclidean external momenta ( ( � q i ) 2 < 0 for any subset of external momenta), Speer’s sectors are optimal for the resolution of the singularities of the integrand. PSI, November 03, 2008 – p.17 V.A. Smirnov
Recursively defined sector decompositions [T. Binoth and G. Heinrich’00] Primary sectors α i ≤ α l , l � = i = 1 , 2 , . . . , L , with new variables � if i � = l α i /α l t i = if i = l α l The contribution of a primary sector 1 1 ⎛ ⎞ � ⎠ U L − ( h +1) d/ 2 � � � ⎝� F l = . . . d t i � V L − hd/ 2 � � i � = l t l =1 0 0 PSI, November 03, 2008 – p.18 V.A. Smirnov
Next sectors are introduced in similar way. The goal is to obtain a factorization of U and V in final sector variables, i.e. to represent them as products of sector variables in some powers times a positive function. Strategies that are guaranteed to terminate [C. Bogner & S. Weinzierl’07] A, B, C, X Strategy S [A.V. Smirnov & M.N. Tentyukov’08] FIESTA (Feynman Integral Evaluation by a Sector decomposiTion Approach) http://www-ttp.particle.uni-karlsruhe.de/ ∼ asmirnov PSI, November 03, 2008 – p.19 V.A. Smirnov
The usage of Speer’s sectors within FIESTA. It turns out that, for Feynman integrals at Euclidean external momenta, Speer’ sectors are reproduced within Strategy S [A.V. Smirnov & V.A. Smirnov’08] (e.g. for four-loop propagator integrals) m62: 26304 sectors PSI, November 03, 2008 – p.20 V.A. Smirnov
✬✩ ✬✩ ✬✩ ✬✩ m61 m62 m63 m51 ✁ ◗ ✑ ✑ ✘✘✘ ◗ ✁ ✑ ◗ ✁❍❍ ✁ ✫✪ ✫✪ ✫✪ ✫✪ ❍ ✬✩ ✬✩ ✬✩ ✬✩ m41 m42 m44 m45 ❏ ✡ ❏ ✡ ❏ ✡ ❆❆ ✡ ✡ ✡ ✡ ❏ ❏ ✡ ❏ ✫✪ ❏ ✫✪ ❏ ✫✪ ✡ ✫✪ ❏ ✡ ✡ ✡ ✡ ❏ ✡ ❏ ✡ ✡ ❏ ✬✩ ✬✩ ✬✩ m34 m35 m36 m52 ✤✜ ✓✏ ❆ ❆ ✁ ❅ � ❆ ✁ ❆ ✒✑ ✣✢ � ❅ ✫✪ ❆ ✁ ✫✪ ✁ ✫✪ � ❅ ✁ ❆ ✁ ✬✩ ✬✩ ✓✏ m43 m31 m32 m33 ✤✜ ✤✜ ✓✏ ✓✏ ✓✏ ✓✏ ✓✏ ✒✑ ✓✏ ❅ � � ❅ ✒✑ ✒✑ ✒✑ ✒✑ ✒✑ ✣✢ ✣✢ ✫✪ ✫✪ ✒✑ ✬✩ ✬✩ ✬✩ m21 m22 m23 m24 ✤✜ ✤✜ ✓✏ ✓✏ ✓✏ ❅ � ❅ ❅ ✒✑ ✒✑ ✒✑ � ❅ ✣✢ ✣✢ ❍ ✫✪ ✫✪ ❅ ✫✪ ❍ ❍ ❅ ✬✩ ✬✩ ✓✏ ✬✩ ✬✩ m25 m26 m27 m11 ✓✏ ✓✏ ✓✏ ✒✑ ✂ ❇ ✒✑ ✒✑ ✒✑ ✂ ❇ ✂ ❇ ✫✪ ✫✪ ✫✪ ✫✪ ✬✩ ✗✔ ✬✩ ✤✜ m12 m13 m14 m01 ✤✜ ✤✜ ✤✜ ✓✏ ✓✏ ✑ ✖✕ ✑ ✑ ✒✑ ✒✑ ✣✢ ✣✢ ✣✢ ✫✪ ✫✪ ✣✢ Again, as in 3–loop case, ”glue–and–cut” relations provide with enough informa- tion to express coe fficient of expansion over e = 2 − D/ 2 of all these integrals 8
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