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Advanced methods of multi-loop integrals calculations: status and perspectives
SUI ELSEVIER
Nuclear Physics B (Proc. Suppl.) 116 (2003) 378-381
L
S
www.elsevier.eom/Iocate/npe
Advanced methods of multi-loop integrals calculations: status and perspectives. P.A.Baikov a* a Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia A brief review of the methods for 3- and 4-loop integrals calculation in presented.
1. I n t r o d u c t i o n Higher orders calculations in the perturbative q u a n t u m field theory demand the evaluation of nmlti-loop integrMs which are functions of m a n y variables like masses and external momenta. As a rule, complete calculations are impossible and in order to get at least something, one needs to expand the integrals, finally leaving with so called "0-scale" (depending only on the regularization parameter, usually space-time dimension D), or "l-scale" (depending on one essential argument like some combination of masses and external momenta). In the following we briefly review several approaches dealing with 1-scale 3-loop and 0-scale 4-loop integrals. All these approaches include, sometimes indirectly, the idea of the algebraic reduction of the given integrals to the restricted set of the irreducible integrals using integration by part (IBP)identities [1,2]. Recently [3] it was shown that the problems of the algebraic reduction of integrals with the same total number of external and internal molnenta are equivalent. In particular, it means that algorithms for algebraic reduction of 5-loop vacuum integrals can be used for 4-loop propagators or 3-loop vertexes or 2-loop boxes; that is true "complexity" p a r a m e t e r of the problem is L L = ( n u m b e r of l o o p s ) + ( n u m b e r of legs), assuming vacuum integrals (no external momenta) correspond to l-leg. So "multi-loop" algorithms can be used for "few-loop multi-leg" integrals as well. *work was supported by INTAS grant 00-00313, by RFBR grant 01-02-16171 and by Volkswagen Foundation.
2.
Integration
by parts
Let us remind the origin of I B P identities. Let
dDpdDk I(n) = (q +p)2~l(q + k)2n2k2~ap2,,4(p _ k)2,~5 '
(1)
F(_n_) = / I(_n) = f(n_)(q2)D-~'~. Integration by part according to p~ gives f
0 = / O p , (p'I(n)) =
(K(n)
=
-
(4-
- 1)1 + -
(4-
- 3-)5+)f(_n),
where
K(_n)
_
D - 2n 4 - n 1 - n5,
l+f(n__)
=
n l / ( n 1 -- 1, ...),
1-f(_n)
-
f(nl-1,...).
IBP with various (Opy~)-insertions provides the whole set of identities. Combining them one can try to construct recursive algorithm which reduce the integral with given powers of propagators to the combination of the irreducible integrals. In particular, the integral f(1, 1, 1, 1, 1) can be reduced in the following way:
f ( 1 , 1 , 1, 1,1) = - 2(D D - 4- 3) f ( 1 , 1, 1, 1, 0) -~
2 ( 3 D - 8 ) ( 3 D - 1 0 ) f ( 1 , 0 ' 1,0,1). (D - 4) 2
0920-5632/03/$ - see front matter © 2003 Elsevier Science B.V. All rights reserved. doi: 10.1016/S0920-5632(02)02373-3
(2)
PA. Baikov/Nuclear Physics B (Proc. Suppl.) 116 (2003) 378-381
In many important cases such recursive algorithms were constructed: 3-loop vacuum with 1 m a s s [4,5]; propagator: 3-loop massless [1,2], 3loop with 1 mass [6], 3-loop in the heavy quark approximation [7], 2-loop with arbitrary masses IS]; 2-loop box (some sub-cases) [9,10]. Nevertheless, the extension of this approach for more complicated cases faces with severe technical problems: first, number of topologies to be considered increases significantly, and second, the construction of the algorithm for each topology became more difficult because of growing number of indexes to be reduced. As the result, although it seems to be possible in general, the working time is unacceptably large, at the scale of several man-years and more. And the last but not least, the performance and reliability of the resulting program will be under the big question. So let us consider some recent ideas which help in this situation. 3. 3 - l o o p 1-scale i n t e g r a l s
The 3-loop DIS structure functions demand calculation of the 3-loop box diagrams with the following kinematics: Pl -- P4 = P, P2 -- P3 = Q, p2 = 0; so the result should be the function of only one variable, say f ( p Q / Q 2 ) . The coefficients in the Taylor expansion of f ( p Q / Q 2 ) are 3-loop massless propagator diagrams; they were calculated up to the 14th power which fixes small argument behavior. To get the complete function, one should know the analytical dependence of these coefficients on the expansion power N. The calculation includes the following steps [11]: 1) Using IBP obtain recurrence relations for fN,fN-l,...,fN-k. 2) Solve them in terms of the Harmonic sums S ~ ...... ~(N) =
~
1 / ( i 7 ~ .. . ~
).
(a)
N~_il...~_i~ ~ l
3) Convert { f N } ~ f ( p Q / Q 2 ) , obtaining f as the combination of poly-logs and some new functions. Problems. Each of these steps includes nontrivial investigation which can not be fully automatized.
379
Status. About all 70 sub-topologies which are involved in the 3-loop DIS calculation are solved, the structure functions calculation are very close to finish. The extention of this approach to the 4-loop level critically depends on the possibility to automatize, which is now under the question.
4. S o l v i n g t h e f i n i t e s e t s o f r e c u r r e n c e relations Suppose the general reduction algorithm is unknown, but the particular physical problem demands the calculation of "middle-size" set of integrals, which is large enough for integrals being calculated individually, but not too large for keeping the result for each integral. In this case one can express most of these integrals through the set of more simple integrals by solving the system of the recurrence relations for partial values of indexes, that is without constructing the algorithm for arbitrary values [12]. In practice, to make the final set smaller, it is profitable to use relations not only for integrals of interest, but also for those with closest values of indexes. In fact, one should enlarge the system of equations until the size of the remaining set became stable (or until the limit of the computer resources). The success of the reduction also depends on the ordering definition, that is which integral should be considered as more simple. For example one can minimize the sum of absolute values of indexes; or only part of the indexes. Problems. As far as the reduction can be incomplete, the number of integrals remained can be large, probably several hundreds at 4-loop level. Status. This approach is used for long-term 4-loop (ge - 2) analytical calculation. The intermediate results shows that the CPU time for each particular topology is reasonable (order of hours), but setup for each case demands significant hand efforts, so at present difficult to estimate the total time of the project.
P.A. Baikov/Nuclear Physics B (Proc. Suppl.) 116 (2003) 378-381
380
5. Calculating irreducible integrals by difference equations The standard point of view is that Feynman integrals are solution of the IBP with fixed values for special sets of indexes (irreducible integrals). The alternative idea [13] is to fix large index asymptotic of the IBP solutions. In this case the values of the irreducible integrals can be calculated within IBP approach also. Lets consider
J
F(x)
=
dDpl • .. dDpg (Pl + m2)X(...)1 ... (...)1"
(4)
IBP relates F(x), F(x+1), ..., F(x+k) plus integrals with some lines shrunk. This difference equation (with information on large-x behavior) can be solved to infinite sum and evaluated numerically with high precision. Problems. 1) The difference equation for F(x) is a very nontrivial combination of IBP relations. There is no standard way to obtain, the author suggested to use the method described in the previous section (but keeping x arbitrary). 2) Large x behavior for Euclidean massive integrals can be calculated relatively easily, the other cases are more difficult. Status. 3-loop (g~ - 2) irreducible integrals are calculated [14], 4-loop are in progress [15]. 6. Direct calculation of coefficients near irreducible integrals
where P(xk) is the polinom over xk describing the topology [17]. The various complex integration contours in (6) provide with desirable variety of c (i) (n_n_). Note that the set of irreducible integrals for (5) can be constructed using the criterion of the irreducibility [18]. The auxiliary integrals (6) are still too complicated for direct calculation. But, as far as we know that c(i)(n) are rational over D, we can reconstruct their exact form by calculating sufficiently many coefficients in the expansion over 1/D. These coefficients are the combinations of f {dxipxkl I ... xkpVe -xAx and hence are rational numbers (in contrast to coefficients in D ~ 4 expansion, for example). Problems. Starting from the 4-loop level very large sets of Gauss integrals are necessary to calculate. For example, the calculation of the a~N~ contribution to R(s) in QCD [19] involves several billions of Gauss integrals. Status. All massless 4-loop propagators topologies are programmed. The nearest task is the full a 5 contribution to R(s) for scalar currents. Very previous estimate shows that total CPU time are too large (several CPU years), so further development is necessary. For example, in addition to the 1/D expansion one can use the information about the behavior for special rational values of D (residues in singular points). The same approach can be used for higher loop integrals also, the main limiting factor is the computer power available.
Let us assume that IBP reduction is possible. Then the given integral F(n) can be represented through the sum of irreducible integrals F(ni): = c
+...
+ c
(5)
where coefficients c (k) (n_) are rational over D. Let us calculate them directly, avoiding the construction of the reduction procedure. For that note that they are solutions of the IBP relations with initial conditions c(i)(n_k) = 5ik. So c(i)(n) have the same algebraic properties as the original integral F(n_), but are analytically simpler and can be calculated [16,17] as the combination of the auxiliary solutions of the IBP relations in the form 5(i)(n)
= / c dX~nl .
Xl
.
.
dXk p(xk)D/2_LL/2 Xk
(6)
7. S u m m a r y The recent progress in IBP-motivated multiloop integrals techniques allows to calculate 1scale 3-loop, 0-scale 4-loop integrals. In some cases independent methods are available. Several calculational projects (3-loop DIS, 4loop (g~ - 2), 5-1oop R(s)) are in progress. There is theoretical possibility to expand calculations to higher loops (depending on computer power and practical demands).
P.A. Baikov/Nuclear Physics B (Proc. Suppl.) 116 (2003) 378-381
REFERENCES
1. K. G. Chetyrkin and F. V. Tkachov, Nucl. Phys. B 192 (1981) 159. 2. F.V. Tkachov, Phys. Lett. B 100 (1981) 65. 3. P.A. Baikov and V. A. Smirnov, Phys. Lett. B 477 (2000) 367 [arXiv:hep-ph/0001192]. 4. D . J . Broadhurst, Z. Phys. C 54 (1992) 599. 5. L. V. Avdeev, Comput. Phys. Commun. 98 (1996) 15 [arXiv:hep-ph/9512442]. 6. K. Melnikov and T. van Ritbergen, Nucl. Phys. B 591 (2000) 515 [arXiv:hepph/0005131]. 7. A. G. Grozin, JHEP 0003 (2000) 013 [arXiv:hep-ph/0002266]. 8. O.V. Tarasov, Nucl. Phys. B 502 (1997) 455 [arXiv:hep-ph/9703319]. 9. V.A. Smirnov and O. L. Veretin, Nucl. Phys. B 566 (2000) 469 [arXiv:hep-ph/9907385]. 10. C. Anastasiou, T. Gehrmann, C. Oleari, E. Remiddi and J. B. Tausk, Nucl. Phys. B 580 (2000) 577 [arXiv:hep-ph/0003261]. 11. S. Moch, J. A. Vermaseren and A. Vogt, arXiv:hep-ph/0209100. 12. S. Laporta and E. Remiddi, Phys. Lett. B 379 (1996) 283 [arXiv:hep-ph/9602417]. 13. S. Laporta, Int. J. Mod. Phys. A 15 (2000) 5087 [arXiv:hep-ph/0102033]. 14. S. Laporta, Phys. Lett. B 523 (2001) 95 [arXiv:hep-ph/0111123]. 15. S. Laporta, arXiv:hep-ph/0210336. 16. P. A. Baikov, Phys. Lett. B 385 (1996) 404 [arXiv:hep-ph/9603267]. 17. P. A. Baikov, Nucl. Instrum. Meth. A 389 (1997) 347 [arXiv:hep-ph/9611449]. 18. P. A. Baikov, Phys. Lett. B 474 (2000) 385 [arXiv:hep-ph/9912421]. 19. P. A. Baikov, K. G. Chetyrkin and J. H. Kuhn, Phys. Rev. Lett. 88 (2002) 012001 [arXiv:hep-ph/0108197].
381
Nuclear Physics B (Proc. Suppl.) 116 (2003) 378-381
L
S
www.elsevier.eom/Iocate/npe
Advanced methods of multi-loop integrals calculations: status and perspectives. P.A.Baikov a* a Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia A brief review of the methods for 3- and 4-loop integrals calculation in presented.
1. I n t r o d u c t i o n Higher orders calculations in the perturbative q u a n t u m field theory demand the evaluation of nmlti-loop integrMs which are functions of m a n y variables like masses and external momenta. As a rule, complete calculations are impossible and in order to get at least something, one needs to expand the integrals, finally leaving with so called "0-scale" (depending only on the regularization parameter, usually space-time dimension D), or "l-scale" (depending on one essential argument like some combination of masses and external momenta). In the following we briefly review several approaches dealing with 1-scale 3-loop and 0-scale 4-loop integrals. All these approaches include, sometimes indirectly, the idea of the algebraic reduction of the given integrals to the restricted set of the irreducible integrals using integration by part (IBP)identities [1,2]. Recently [3] it was shown that the problems of the algebraic reduction of integrals with the same total number of external and internal molnenta are equivalent. In particular, it means that algorithms for algebraic reduction of 5-loop vacuum integrals can be used for 4-loop propagators or 3-loop vertexes or 2-loop boxes; that is true "complexity" p a r a m e t e r of the problem is L L = ( n u m b e r of l o o p s ) + ( n u m b e r of legs), assuming vacuum integrals (no external momenta) correspond to l-leg. So "multi-loop" algorithms can be used for "few-loop multi-leg" integrals as well. *work was supported by INTAS grant 00-00313, by RFBR grant 01-02-16171 and by Volkswagen Foundation.
2.
Integration
by parts
Let us remind the origin of I B P identities. Let
dDpdDk I(n) = (q +p)2~l(q + k)2n2k2~ap2,,4(p _ k)2,~5 '
(1)
F(_n_) = / I(_n) = f(n_)(q2)D-~'~. Integration by part according to p~ gives f
0 = / O p , (p'I(n)) =
(K(n)
=
-
(4-
- 1)1 + -
(4-
- 3-)5+)f(_n),
where
K(_n)
_
D - 2n 4 - n 1 - n5,
l+f(n__)
=
n l / ( n 1 -- 1, ...),
1-f(_n)
-
f(nl-1,...).
IBP with various (Opy~)-insertions provides the whole set of identities. Combining them one can try to construct recursive algorithm which reduce the integral with given powers of propagators to the combination of the irreducible integrals. In particular, the integral f(1, 1, 1, 1, 1) can be reduced in the following way:
f ( 1 , 1 , 1, 1,1) = - 2(D D - 4- 3) f ( 1 , 1, 1, 1, 0) -~
2 ( 3 D - 8 ) ( 3 D - 1 0 ) f ( 1 , 0 ' 1,0,1). (D - 4) 2
0920-5632/03/$ - see front matter © 2003 Elsevier Science B.V. All rights reserved. doi: 10.1016/S0920-5632(02)02373-3
(2)
PA. Baikov/Nuclear Physics B (Proc. Suppl.) 116 (2003) 378-381
In many important cases such recursive algorithms were constructed: 3-loop vacuum with 1 m a s s [4,5]; propagator: 3-loop massless [1,2], 3loop with 1 mass [6], 3-loop in the heavy quark approximation [7], 2-loop with arbitrary masses IS]; 2-loop box (some sub-cases) [9,10]. Nevertheless, the extension of this approach for more complicated cases faces with severe technical problems: first, number of topologies to be considered increases significantly, and second, the construction of the algorithm for each topology became more difficult because of growing number of indexes to be reduced. As the result, although it seems to be possible in general, the working time is unacceptably large, at the scale of several man-years and more. And the last but not least, the performance and reliability of the resulting program will be under the big question. So let us consider some recent ideas which help in this situation. 3. 3 - l o o p 1-scale i n t e g r a l s
The 3-loop DIS structure functions demand calculation of the 3-loop box diagrams with the following kinematics: Pl -- P4 = P, P2 -- P3 = Q, p2 = 0; so the result should be the function of only one variable, say f ( p Q / Q 2 ) . The coefficients in the Taylor expansion of f ( p Q / Q 2 ) are 3-loop massless propagator diagrams; they were calculated up to the 14th power which fixes small argument behavior. To get the complete function, one should know the analytical dependence of these coefficients on the expansion power N. The calculation includes the following steps [11]: 1) Using IBP obtain recurrence relations for fN,fN-l,...,fN-k. 2) Solve them in terms of the Harmonic sums S ~ ...... ~(N) =
~
1 / ( i 7 ~ .. . ~
).
(a)
N~_il...~_i~ ~ l
3) Convert { f N } ~ f ( p Q / Q 2 ) , obtaining f as the combination of poly-logs and some new functions. Problems. Each of these steps includes nontrivial investigation which can not be fully automatized.
379
Status. About all 70 sub-topologies which are involved in the 3-loop DIS calculation are solved, the structure functions calculation are very close to finish. The extention of this approach to the 4-loop level critically depends on the possibility to automatize, which is now under the question.
4. S o l v i n g t h e f i n i t e s e t s o f r e c u r r e n c e relations Suppose the general reduction algorithm is unknown, but the particular physical problem demands the calculation of "middle-size" set of integrals, which is large enough for integrals being calculated individually, but not too large for keeping the result for each integral. In this case one can express most of these integrals through the set of more simple integrals by solving the system of the recurrence relations for partial values of indexes, that is without constructing the algorithm for arbitrary values [12]. In practice, to make the final set smaller, it is profitable to use relations not only for integrals of interest, but also for those with closest values of indexes. In fact, one should enlarge the system of equations until the size of the remaining set became stable (or until the limit of the computer resources). The success of the reduction also depends on the ordering definition, that is which integral should be considered as more simple. For example one can minimize the sum of absolute values of indexes; or only part of the indexes. Problems. As far as the reduction can be incomplete, the number of integrals remained can be large, probably several hundreds at 4-loop level. Status. This approach is used for long-term 4-loop (ge - 2) analytical calculation. The intermediate results shows that the CPU time for each particular topology is reasonable (order of hours), but setup for each case demands significant hand efforts, so at present difficult to estimate the total time of the project.
P.A. Baikov/Nuclear Physics B (Proc. Suppl.) 116 (2003) 378-381
380
5. Calculating irreducible integrals by difference equations The standard point of view is that Feynman integrals are solution of the IBP with fixed values for special sets of indexes (irreducible integrals). The alternative idea [13] is to fix large index asymptotic of the IBP solutions. In this case the values of the irreducible integrals can be calculated within IBP approach also. Lets consider
J
F(x)
=
dDpl • .. dDpg (Pl + m2)X(...)1 ... (...)1"
(4)
IBP relates F(x), F(x+1), ..., F(x+k) plus integrals with some lines shrunk. This difference equation (with information on large-x behavior) can be solved to infinite sum and evaluated numerically with high precision. Problems. 1) The difference equation for F(x) is a very nontrivial combination of IBP relations. There is no standard way to obtain, the author suggested to use the method described in the previous section (but keeping x arbitrary). 2) Large x behavior for Euclidean massive integrals can be calculated relatively easily, the other cases are more difficult. Status. 3-loop (g~ - 2) irreducible integrals are calculated [14], 4-loop are in progress [15]. 6. Direct calculation of coefficients near irreducible integrals
where P(xk) is the polinom over xk describing the topology [17]. The various complex integration contours in (6) provide with desirable variety of c (i) (n_n_). Note that the set of irreducible integrals for (5) can be constructed using the criterion of the irreducibility [18]. The auxiliary integrals (6) are still too complicated for direct calculation. But, as far as we know that c(i)(n) are rational over D, we can reconstruct their exact form by calculating sufficiently many coefficients in the expansion over 1/D. These coefficients are the combinations of f {dxipxkl I ... xkpVe -xAx and hence are rational numbers (in contrast to coefficients in D ~ 4 expansion, for example). Problems. Starting from the 4-loop level very large sets of Gauss integrals are necessary to calculate. For example, the calculation of the a~N~ contribution to R(s) in QCD [19] involves several billions of Gauss integrals. Status. All massless 4-loop propagators topologies are programmed. The nearest task is the full a 5 contribution to R(s) for scalar currents. Very previous estimate shows that total CPU time are too large (several CPU years), so further development is necessary. For example, in addition to the 1/D expansion one can use the information about the behavior for special rational values of D (residues in singular points). The same approach can be used for higher loop integrals also, the main limiting factor is the computer power available.
Let us assume that IBP reduction is possible. Then the given integral F(n) can be represented through the sum of irreducible integrals F(ni): = c
+...
+ c
(5)
where coefficients c (k) (n_) are rational over D. Let us calculate them directly, avoiding the construction of the reduction procedure. For that note that they are solutions of the IBP relations with initial conditions c(i)(n_k) = 5ik. So c(i)(n) have the same algebraic properties as the original integral F(n_), but are analytically simpler and can be calculated [16,17] as the combination of the auxiliary solutions of the IBP relations in the form 5(i)(n)
= / c dX~nl .
Xl
.
.
dXk p(xk)D/2_LL/2 Xk
(6)
7. S u m m a r y The recent progress in IBP-motivated multiloop integrals techniques allows to calculate 1scale 3-loop, 0-scale 4-loop integrals. In some cases independent methods are available. Several calculational projects (3-loop DIS, 4loop (g~ - 2), 5-1oop R(s)) are in progress. There is theoretical possibility to expand calculations to higher loops (depending on computer power and practical demands).
P.A. Baikov/Nuclear Physics B (Proc. Suppl.) 116 (2003) 378-381
REFERENCES
1. K. G. Chetyrkin and F. V. Tkachov, Nucl. Phys. B 192 (1981) 159. 2. F.V. Tkachov, Phys. Lett. B 100 (1981) 65. 3. P.A. Baikov and V. A. Smirnov, Phys. Lett. B 477 (2000) 367 [arXiv:hep-ph/0001192]. 4. D . J . Broadhurst, Z. Phys. C 54 (1992) 599. 5. L. V. Avdeev, Comput. Phys. Commun. 98 (1996) 15 [arXiv:hep-ph/9512442]. 6. K. Melnikov and T. van Ritbergen, Nucl. Phys. B 591 (2000) 515 [arXiv:hepph/0005131]. 7. A. G. Grozin, JHEP 0003 (2000) 013 [arXiv:hep-ph/0002266]. 8. O.V. Tarasov, Nucl. Phys. B 502 (1997) 455 [arXiv:hep-ph/9703319]. 9. V.A. Smirnov and O. L. Veretin, Nucl. Phys. B 566 (2000) 469 [arXiv:hep-ph/9907385]. 10. C. Anastasiou, T. Gehrmann, C. Oleari, E. Remiddi and J. B. Tausk, Nucl. Phys. B 580 (2000) 577 [arXiv:hep-ph/0003261]. 11. S. Moch, J. A. Vermaseren and A. Vogt, arXiv:hep-ph/0209100. 12. S. Laporta and E. Remiddi, Phys. Lett. B 379 (1996) 283 [arXiv:hep-ph/9602417]. 13. S. Laporta, Int. J. Mod. Phys. A 15 (2000) 5087 [arXiv:hep-ph/0102033]. 14. S. Laporta, Phys. Lett. B 523 (2001) 95 [arXiv:hep-ph/0111123]. 15. S. Laporta, arXiv:hep-ph/0210336. 16. P. A. Baikov, Phys. Lett. B 385 (1996) 404 [arXiv:hep-ph/9603267]. 17. P. A. Baikov, Nucl. Instrum. Meth. A 389 (1997) 347 [arXiv:hep-ph/9611449]. 18. P. A. Baikov, Phys. Lett. B 474 (2000) 385 [arXiv:hep-ph/9912421]. 19. P. A. Baikov, K. G. Chetyrkin and J. H. Kuhn, Phys. Rev. Lett. 88 (2002) 012001 [arXiv:hep-ph/0108197].
381