Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://theory.sinp.msu.ru/~smirnov/ll10.pdf Äàòà èçìåíåíèÿ: Mon May 10 21:20:22 2010 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 19:58:47 2012 Êîäèðîâêà:

Dimensional recurrence relations: an easy way to evaluate higher orders of expansion in
V.A. Smirnov Nuclear Physics Institute of Moscow State University and Institut fur Theoretische Teilchenphysik, KIT ¨



in collaboration with R.N. Lee and A.V. Smirnov
V.A. Smir nov Loops and Legs'10 ­ p.1


Applications of the Roman Lee's method based on the use of dimensional recurrence relations.

V.A. Smir nov

Loops and Legs'10 ­ p.2


Applications of the Roman Lee's method based on the use of dimensional recurrence relations. Evaluating master integrals for the three-loop form factors

V.A. Smir nov

Loops and Legs'10 ­ p.2


Applications of the Roman Lee's method based on the use of dimensional recurrence relations. Evaluating master integrals for the three-loop form factors Evaluating master integrals for the three-loop static potential (in progress)

V.A. Smir nov

Loops and Legs'10 ­ p.2


Applications of the Roman Lee's method based on the use of dimensional recurrence relations. Evaluating master integrals for the three-loop form factors Evaluating master integrals for the three-loop static potential (in progress) Non-planar massless propagator diagram (as a by-product)

V.A. Smir nov

Loops and Legs'10 ­ p.2


Applications of the Roman Lee's method based on the use of dimensional recurrence relations. Evaluating master integrals for the three-loop form factors Evaluating master integrals for the three-loop static potential (in progress) Non-planar massless propagator diagram (as a by-product) Conclusion

V.A. Smir nov

Loops and Legs'10 ­ p.2


IBP

[K.G. Chetyrkin & F.V. Tkachov'81]

V.A. Smir nov

Loops and Legs'10 ­ p.3


IBP

[K.G. Chetyrkin & F.V. Tkachov'81]

The whole problem of the evaluation of a given family of Feynman integrals

V.A. Smir nov

Loops and Legs'10 ­ p.3


IBP

[K.G. Chetyrkin & F.V. Tkachov'81]

The whole problem of the evaluation of a given family of Feynman integrals constructing a reduction procedure using IBP evaluating master integrals

V.A. Smir nov

Loops and Legs'10 ­ p.3


IBP

[K.G. Chetyrkin & F.V. Tkachov'81]

The whole problem of the evaluation of a given family of Feynman integrals constructing a reduction procedure using IBP evaluating master integrals Theorem
[A. Smirnov & A. Petukhov'10]

The number of master integrals is finite

V.A. Smir nov

Loops and Legs'10 ­ p.3


Solving IBP relations by Lapor ta's algorithm
[S. Lapor ta & E. Remiddi'96; S. Lapor ta'00; T. Gehrmann & E. Remiddi'01]

V.A. Smir nov

Loops and Legs'10 ­ p.4


Solving IBP relations by Lapor ta's algorithm
[S. Lapor ta & E. Remiddi'96; S. Lapor ta'00; T. Gehrmann & E. Remiddi'01]

Three public versions:

V.A. Smir nov

Loops and Legs'10 ­ p.4


Solving IBP relations by Lapor ta's algorithm
[S. Lapor ta & E. Remiddi'96; S. Lapor ta'00; T. Gehrmann & E. Remiddi'01]

Three public versions: AIR
[C. Anastasiou & A. Lazopoulos'04]

V.A. Smir nov

Loops and Legs'10 ­ p.4


Solving IBP relations by Lapor ta's algorithm
[S. Lapor ta & E. Remiddi'96; S. Lapor ta'00; T. Gehrmann & E. Remiddi'01]

Three public versions: AIR
[C. Anastasiou & A. Lazopoulos'04] [A.V. Smirnov'08]

FIRE (in Mathematica; a C++ version is private)

V.A. Smir nov

Loops and Legs'10 ­ p.4


Solving IBP relations by Lapor ta's algorithm
[S. Lapor ta & E. Remiddi'96; S. Lapor ta'00; T. Gehrmann & E. Remiddi'01]

Three public versions: AIR
[C. Anastasiou & A. Lazopoulos'04] [A.V. Smirnov'08]

FIRE (in Mathematica; a C++ version is private) Reduze

[C. Studerus'09]

V.A. Smir nov

Loops and Legs'10 ­ p.4


Solving IBP relations by Lapor ta's algorithm
[S. Lapor ta & E. Remiddi'96; S. Lapor ta'00; T. Gehrmann & E. Remiddi'01]

Three public versions: AIR
[C. Anastasiou & A. Lazopoulos'04] [A.V. Smirnov'08]

FIRE (in Mathematica; a C++ version is private) Reduze Private versions

[C. Studerus'09]

[T. Gehrmann & E. Remiddi, S. Lapor ta, M. Czakon, Y. SchrÆder, A. Pak, C. Sturm, P. Marquard & D. Seidel, V. Velizhanin, . . . ]

V.A. Smir nov

Loops and Legs'10 ­ p.4


Solving IBP relations by Lapor ta's algorithm
[S. Lapor ta & E. Remiddi'96; S. Lapor ta'00; T. Gehrmann & E. Remiddi'01]

Three public versions: AIR
[C. Anastasiou & A. Lazopoulos'04] [A.V. Smirnov'08]

FIRE (in Mathematica; a C++ version is private) Reduze Private versions

[C. Studerus'09]

[T. Gehrmann & E. Remiddi, S. Lapor ta, M. Czakon, Y. SchrÆder, A. Pak, C. Sturm, P. Marquard & D. Seidel, V. Velizhanin, . . . ]

Solving IBP relations in other ways
[Baikov, Tarasov, Lee]

V.A. Smir nov

Loops and Legs'10 ­ p.4


A new method of evaluating master integrals is based on the use of dimensional recurrence relations [O. Tarasov'96] and analytic proper ties of Feynman integrals as functions of d.
[R. Lee'09]

V.A. Smir nov

Loops and Legs'10 ­ p.5


A new method of evaluating master integrals is based on the use of dimensional recurrence relations [O. Tarasov'96] and analytic proper ties of Feynman integrals as functions of d.
[R. Lee'09]

Master integrals for three-loop form factors of the photon-quark and the effective gluon-Higgs boson ver tex originating from integrating out the heavy top-quark loops.
[P. Baikov, K. Chetyrkin, A. and V. Smirnovs, & M. Steinhauser'09; G. Heinrich, T. Huber, D. Kosower and V. Smirnov'09]

V.A. Smir nov

Loops and Legs'10 ­ p.5


A new method of evaluating master integrals is based on the use of dimensional recurrence relations [O. Tarasov'96] and analytic proper ties of Feynman integrals as functions of d.
[R. Lee'09]

Master integrals for three-loop form factors of the photon-quark and the effective gluon-Higgs boson ver tex originating from integrating out the heavy top-quark loops.
[P. Baikov, K. Chetyrkin, A. and V. Smirnovs, & M. Steinhauser'09; G. Heinrich, T. Huber, D. Kosower and V. Smirnov'09]

Evaluating up to transcendentality weight six.

V.A. Smir nov

Loops and Legs'10 ­ p.5


A new method of evaluating master integrals is based on the use of dimensional recurrence relations [O. Tarasov'96] and analytic proper ties of Feynman integrals as functions of d.
[R. Lee'09]

Master integrals for three-loop form factors of the photon-quark and the effective gluon-Higgs boson ver tex originating from integrating out the heavy top-quark loops.
[P. Baikov, K. Chetyrkin, A. and V. Smirnovs, & M. Steinhauser'09; G. Heinrich, T. Huber, D. Kosower and V. Smirnov'09]

Evaluating up to transcendentality weight six. The missing finite par ts of A9,2 and A9,4 were recently analytically evaluated [R. Lee, A. and V. Smirnovs'10]
V.A. Smir nov Loops and Legs'10 ­ p.5


E

Q s

E

Q s

E

Q s

E

Q ¢ ¢ f f s

A5,

1

A5,

2

A6,
Q e e e e s

1

A6,

2

E

Q t t t t t s

E

E

Q e e e e s

A6,

3

A7,

1

A7,

2

E

Q t t t t t s

E

Q t t t t t s

E

Q t t t t t s

A7,

3

A7,
Q 7 7 7 s

4

A7,
Q s

5

E

E

A8
Q 7 7 7 s

A9,

1

E

E

Q t t s

V.A. Smir nov

A9,

2

A9,

4

Loops and Legs'10 ­ p.6


A9,4 and lower master integrals

A4

A5

,1

A5

,2

A6

,1

A6

,2

A6

,3

A7

,4

A7

,2

A7

,5

A9
V.A. Smir nov

,4
Loops and Legs'10 ­ p.7


Recursive evaluation.

V.A. Smir nov

Loops and Legs'10 ­ p.8


Recursive evaluation. The complexity level of a given master integral is the maximal number of nested lower master integrals.

V.A. Smir nov

Loops and Legs'10 ­ p.8


Recursive evaluation. The complexity level of a given master integral is the maximal number of nested lower master integrals. The diagrams of the first row can be expressed in terms of gamma functions at general d (complexity level 0).

V.A. Smir nov

Loops and Legs'10 ­ p.8


Recursive evaluation. The complexity level of a given master integral is the maximal number of nested lower master integrals. The diagrams of the first row can be expressed in terms of gamma functions at general d (complexity level 0). Star t from the diagrams of the second row (complexity level 1), then turn to the complexity level 2 etc.

V.A. Smir nov

Loops and Legs'10 ­ p.8


Recursive evaluation. The complexity level of a given master integral is the maximal number of nested lower master integrals. The diagrams of the first row can be expressed in terms of gamma functions at general d (complexity level 0). Star t from the diagrams of the second row (complexity level 1), then turn to the complexity level 2 etc. Consider, for example, A7,2 .

V.A. Smir nov

Loops and Legs'10 ­ p.8


Recursive evaluation. The complexity level of a given master integral is the maximal number of nested lower master integrals. The diagrams of the first row can be expressed in terms of gamma functions at general d (complexity level 0). Star t from the diagrams of the second row (complexity level 1), then turn to the complexity level 2 etc. Consider, for example, A7,2 . Four lower master integrals, A4 , A5,1 , A5,2 , and A6,3 .

V.A. Smir nov

Loops and Legs'10 ­ p.8


Recursive evaluation. The complexity level of a given master integral is the maximal number of nested lower master integrals. The diagrams of the first row can be expressed in terms of gamma functions at general d (complexity level 0). Star t from the diagrams of the second row (complexity level 1), then turn to the complexity level 2 etc. Consider, for example, A7,2 . Four lower master integrals, A4 , A5,1 , A5,2 , and A6,3 .
A4 , A
5,1

,A

5,2

are of complexity level 0.

V.A. Smir nov

Loops and Legs'10 ­ p.8


A6,3 is of complexity level 1. Solving dimensional recurrence relation


A A A

6,3

(d) = A

1,1 6,3

(d)
k =0

A

1,2 6,3

(d + 2k ) + A 4 21
d 2

2 6,3

(d) ,
3 d 2 -5 2

1,1 6,3

1 -3 d

(d) = - sin( d)A (d) =

2 6,3

(d) =
d 2

cs c

cs c

(3d - 10) d -3 -1
3

d 2 d -1 2

,

1,2 6,3

(7d - 18) sin
2


3d 2

3 (d - 3)

.

V.A. Smir nov

Loops and Legs'10 ­ p.9


After A

6,3

turn to A7,2 .

V.A. Smir nov

Loops and Legs'10 ­ p.10


After A6,3 turn to A7,2 . The dimensional recurrence relation gives
A7,2 (d + 2) = c7,2 (d)A7,2 (d) +c6,3 (d)A6,3 (d) + c5,2 (d)A5,2 (d) + c

5,1

(d)A

5,1

(d) + c4 (d)A4 (d)

where cn are rational functions.

V.A. Smir nov

Loops and Legs'10 ­ p.10


After A6,3 turn to A7,2 . The dimensional recurrence relation gives
A7,2 (d + 2) = c7,2 (d)A7,2 (d) +c6,3 (d)A6,3 (d) + c5,2 (d)A5,2 (d) + c

5,1

(d)A

5,1

(d) + c4 (d)A4 (d)

where cn are rational functions. ~ Turn to A7,2 (d) = (d)A7,2 (d) where the summing factor (d) is chosen as
(d) = (d - 3) cos
d 2

cos

6

-

d 2

cos
2

d 2

+

6



5d 2



d 2

-9

.

-2

V.A. Smir nov

Loops and Legs'10 ­ p.10


~ For A7,2 (d), the recurrence relation is simpler: ~ ~ ~ ~ ~ ~ A7,2 (d + 2) = A7,2 (d) + A6,3 (d) + A5,2 (d) + A5,1 (d) + A4 (d) , ~ where An (d) = (d + 2)cn (d)An (d).

V.A. Smir nov

Loops and Legs'10 ­ p.11


~ For A7,2 (d), the recurrence relation is simpler: ~ ~ ~ ~ ~ ~ A7,2 (d + 2) = A7,2 (d) + A6,3 (d) + A5,2 (d) + A5,1 (d) + A4 (d) , ~ where An (d) = (d + 2)cn (d)An (d).

The general solution:


~ A7,2 (d) =
l =0

~ ~ ~6 A5,2 (d - 2 - 2l) + A5,1 (d - 2 - 2l) + A2,3 (d - 2 - 2l)


-

~ A1,1 (d + 2l) 6,3
l =0 k =0

A

1,2 6,3

(d + 2l + 2k ) -

~ A4 (d + 2l) + (z ) ,
l =0

where z = exp[i d].
V.A. Smir nov Loops and Legs'10 ­ p.11


To fix (z ), we need information about analytical proper ties in the basic stripe chosen as S = {d| Re d (4, 6]}.

V.A. Smir nov

Loops and Legs'10 ­ p.12


To fix (z ), we need information about analytical proper ties in the basic stripe chosen as S = {d| Re d (4, 6]}. Sector decompositions
[T. Binoth & G. Heinrich'00,04; C. Bogner & S. Weinzierl'07, J. Gluza (talk on Wednesday)]

V.A. Smir nov

Loops and Legs'10 ­ p.12


To fix (z ), we need information about analytical proper ties in the basic stripe chosen as S = {d| Re d (4, 6]}. Sector decompositions FIESTA
[T. Binoth & G. Heinrich'00,04; C. Bogner & S. Weinzierl'07, J. Gluza (talk on Wednesday)] [A. Smirnov and M. Tentyukov'08, A. and V. Smirnovs and M. Tentyukov'09]

V.A. Smir nov

Loops and Legs'10 ­ p.12


To fix (z ), we need information about analytical proper ties in the basic stripe chosen as S = {d| Re d (4, 6]}. Sector decompositions FIESTA
[T. Binoth & G. Heinrich'00,04; C. Bogner & S. Weinzierl'07, J. Gluza (talk on Wednesday)] [A. Smirnov and M. Tentyukov'08, A. and V. Smirnovs and M. Tentyukov'09]

SDAnalyze[U,F,h,degrees,dmin,dmax]

where U and F are the basic functions, h is the number of loops, degrees are the indices, and dmin and dmax are values of the real par t of d that determine the basic stripe.

V.A. Smir nov

Loops and Legs'10 ­ p.12


To fix (z ), we need information about analytical proper ties in the basic stripe chosen as S = {d| Re d (4, 6]}. Sector decompositions FIESTA
[T. Binoth & G. Heinrich'00,04; C. Bogner & S. Weinzierl'07, J. Gluza (talk on Wednesday)] [A. Smirnov and M. Tentyukov'08, A. and V. Smirnovs and M. Tentyukov'09]

SDAnalyze[U,F,h,degrees,dmin,dmax]

where U and F are the basic functions, h is the number of loops, degrees are the indices, and dmin and dmax are values of the real par t of d that determine the basic stripe. For A7,2 , FIESTA says that there can be simple poles at d = 14/3, 5, 16/3, 6

V.A. Smir nov

Loops and Legs'10 ­ p.12


To fix (z ), we need information about analytical proper ties in the basic stripe chosen as S = {d| Re d (4, 6]}. Sector decompositions FIESTA
[T. Binoth & G. Heinrich'00,04; C. Bogner & S. Weinzierl'07, J. Gluza (talk on Wednesday)] [A. Smirnov and M. Tentyukov'08, A. and V. Smirnovs and M. Tentyukov'09]

SDAnalyze[U,F,h,degrees,dmin,dmax]

where U and F are the basic functions, h is the number of loops, degrees are the indices, and dmin and dmax are values of the real par t of d that determine the basic stripe. For A7,2 , FIESTA says that there can be simple poles at d = 14/3, 5, 16/3, 6
(z ) is fixed up to the function a1 + a2 cot
2

(d - 6)

V.A. Smir nov

Loops and Legs'10 ­ p.12


To fix the two constants, an MB representation can be used
-2 1 - 5 (2d - 7) (2 )2 -1
2

A7,2 (d) = â
d 2 3d 2

d 2

- z1 - 2
2

(d - 2)

d 2 3d 2

(d - 3)

(-z1 )(-z2 ) (d - z1 - 4)

- z1 - 5

3d - z2 - 6 (z1 + z2 + 1)(d - z1 - z2 - 5) 2

â(z2 + 1)

3d 3d - z 1 - z 2 - 6 - + z 1 + z 2 + 7 dz 1 dz 2 . 2 2

V.A. Smir nov

Loops and Legs'10 ­ p.13


To fix the two constants, an MB representation can be used
-2 1 - 5 (2d - 7) (2 )2 -1
2

A7,2 (d) = â
d 2 3d 2

d 2

- z1 - 2
2

(d - 2)

d 2 3d 2

(d - 3)

(-z1 )(-z2 ) (d - z1 - 4)

- z1 - 5

3d - z2 - 6 (z1 + z2 + 1)(d - z1 - z2 - 5) 2

â(z2 + 1)

3d 3d - z 1 - z 2 - 6 - + z 1 + z 2 + 7 dz 1 dz 2 . 2 2

MB tools at http://projects.hepforge.org/mbtools/ MB.m [M. Czakon'05] MBresolve.m [A. Sm barnesroutines.m
irnov'09] [D. Kosower'08]

V.A. Smir nov

Loops and Legs'10 ­ p.13


To fix the two constants, an MB representation can be used
-2 1 - 5 (2d - 7) (2 )2 -1
2

A7,2 (d) = â
d 2 3d 2

d 2

- z1 - 2
2

(d - 2)

d 2 3d 2

(d - 3)

(-z1 )(-z2 ) (d - z1 - 4)

- z1 - 5

3d - z2 - 6 (z1 + z2 + 1)(d - z1 - z2 - 5) 2

â(z2 + 1)

3d 3d - z 1 - z 2 - 6 - + z 1 + z 2 + 7 dz 1 dz 2 . 2 2

MB tools at http://projects.hepforge.org/mbtools/ MB.m [M. Czakon'05] MBresolve.m [A. Sm barnesroutines.m
A
V.A. Smir nov

irnov'09] [D. Kosower'08]

7,2

(6 - 2) = -

41 1 5 5 5 2

+ O ( ),

0

A

7,2

(5 - 2) = -

5/2 24

+ O ( 0 )
Loops and Legs'10 ­ p.13


3 tan (z ) = 20 5 3 - tan 20 5 3 + cot3 60 3 + cot 20 5

d - 10 2 d + 2 10 d 2

3 d - tan - 36 6 2 3 d + tan + 36 2 6 d 2 d + 2 5 .

13 3 + cot 180

d - 5 2

3 - cot 20 5

V.A. Smir nov

Loops and Legs'10 ­ p.14


3 tan (z ) = 20 5 3 - tan 20 5 3 + cot3 60 3 + cot 20 5

d - 10 2 d + 2 10 d 2

3 d - tan - 36 6 2 3 d + tan + 36 2 6 d 2 d + 2 5 .

13 3 + cot 180

d - 5 2

3 - cot 20 5

A result is then expressed in terms of a double series with factorized expressions. Calculating it with very high precision and using PSLQ
V.A. Smir nov

[H.R.P. Ferguson & D.H. Bailey'91]

Loops and Legs'10 ­ p.14


A

9,4

(4 - 2) = e

-3E

1 8 43 2 1 - 6 - 5 + 1+ 9 9 108 4
2

109 (3) 14 53 + + + 9 9 27

1 3

608 (3) 311 2 481 4 1 + - 17 - - 9 108 12960 2 949 (3) 2975 2 (3) 3463 (5) 11 2 85 4 1 +- - + + 84 + + 9 108 45 18 108
[P. Baikov, K. Chetyrkin, A. and V. Smirnovs, & M. Steinhauser'09] [ G. Heinrich, T. Huber, D. Kosower and V. Smirnov'09]

434 (3) 299 2 (3) 3115 (3)2 7868 (5) + - - + - 339 9 3 6 15 77 2 2539 4 247613 - - + 4 2592 466560
V.A. Smir nov

6

+ O ( )

[R. Lee, A. and V. Smirnovs'10]

Loops and Legs'10 ­ p.15


A9,2 and lower master integrals

A4

A5

,1

A5

,2

A5

,3

A5

,4

A6

,2

A6

,3

A7

,1

A7

,3

A7

,4

A7

,2

A7

,5

A8

A9
V.A. Smir nov

,2
Loops and Legs'10 ­ p.16


A

9,2

(4 - 2) = e

-3E

2 5 20 17 - 6- 5+ + 9 6 9 54
2

2

1 4

31 (3) 50 181 + - + 3 9 216

1 3
4

347 (3) 110 17 2 119 + + - + 18 9 9 432

1 2
4

514 (3) 341 2 (3) 2507 (5) 170 19 2 163 +- - + - + + 9 36 15 9 6 960
[P. Baikov, K. Chetyrkin, A. and V. Smirnovs, & M. Steinhauser'09] [ G. Heinrich, T. Huber, D. Kosower and V. Smirnov'09]

1

1516 (3) 737 2 (3) 2783 (5) 130 2 + - - 29 (3) + - 9 24 6 9 2 943 4 195551 + +- 2 1080 544320
V.A. Smir nov

6

+ O ( )

[R. Lee, A. and V. Smirnovs'10]

Loops and Legs'10 ­ p.17


Evaluating three-loop quark and gluon form factors
[Baikov, Chetyrkin, A. and V. Smirnovs, and Steinhauser'09]

V.A. Smir nov

Loops and Legs'10 ­ p.18


Evaluating three-loop quark and gluon form factors
[Baikov, Chetyrkin, A. and V. Smirnovs, and Steinhauser'09]

Completely analytic form of the results
[R. Lee, A. and V. Smirnovs'10]

V.A. Smir nov

Loops and Legs'10 ­ p.18


Evaluating three-loop quark and gluon form factors
[Baikov, Chetyrkin, A. and V. Smirnovs, and Steinhauser'09]

Completely analytic form of the results
[R. Lee, A. and V. Smirnovs'10]

An independent calculation (published one week ago)
[Gehrmann, Glover, Huber, Ikizlerli &Studerus'10]

V.A. Smir nov

Loops and Legs'10 ­ p.18


Evaluating three-loop quark and gluon form factors
[Baikov, Chetyrkin, A. and V. Smirnovs, and Steinhauser'09]

Completely analytic form of the results
[R. Lee, A. and V. Smirnovs'10]

An independent calculation (published one week ago)
[Gehrmann, Glover, Huber, Ikizlerli &Studerus'10]

Also the subleading O () terms for the fermion-loop type contributions were calculated.

V.A. Smir nov

Loops and Legs'10 ­ p.18


Evaluating three-loop quark and gluon form factors
[Baikov, Chetyrkin, A. and V. Smirnovs, and Steinhauser'09]

Completely analytic form of the results
[R. Lee, A. and V. Smirnovs'10]

An independent calculation (published one week ago)
[Gehrmann, Glover, Huber, Ikizlerli &Studerus'10]

Also the subleading O () terms for the fermion-loop type contributions were calculated. It is possible to evaluate the whole O () par t of the form factors :)

V.A. Smir nov

Loops and Legs'10 ­ p.18


A9,1 (4 - 2) = e 1 +2 +

-3E

1 1 1 - + 185 24 3
2

53 29 2 + 18 216

35 (3) 29 149 - - 18 2 216

1 +

307 (3) 129 139 2 5473 4 - + + + 18 2 72 25920

793 (5) 871 2 (3) 1153 (3) 3125 4 19 2 537 + + - - - 10 216 18 5184 8 2

287 (3) 2969 2 (3) 5521 (3)2 8251 (5) + - + + - 2 216 36 30 2133 97 2 4717 4 761151 - + + + 2 8 28115 186624 +
2 6

195 (3) 5887 2 (3) 138403 4 (3) 799 (3)2 22487 (5) - + + + 2 72 25920 4 30
6

11987 2 (5) 228799 (7) 8181 969 2 1333 4 4286603 + - + - - - 10115 126 2 8 320 6531840

+ ...

V.A. Smir nov

Loops and Legs'10 ­ p.19


Most complicated master integrals for the three-loop static quark potential [A. and V. Smirnovs & M. Steinhauser'0

9]

I

1

I

2

I

3

I

4

I

5

I

6

I

7

I

8

+i0

+i0

+i0

I

9

I

10

I

11

I

12

I

13

I

14

I

15

I

16

I

17

I

18

V.A. Smir nov

Loops and Legs'10 ­ p.20


V.A. Smir nov

Loops and Legs'10 ­ p.21


N

8,3

P6

,5

P5

P5
,3

,4

P4

V.A. Smir nov

Loops and Legs'10 ­ p.22


N

8,3

(d - 4) (d + 2) = N 8(d - 2)(d - 1)(2d - 7)(2d - 5) 4 5d2 - 28d + 38 + P 2 (d - 2)(d - 1)(2d - 5) 5 (d - 4)
,3

8,3

â 37d3 - 313d2 + 858d - 752 P6
2

[4(d - 4)(d - 2)(d - 1)(2d - 7)(2d - 5)(3d - 8)]-
,5

1

â 43d4 - 478d3 + 1963d2 - 3530d + 2352 P5
3 2

- 2(d - 4) (d - 3)(d - 2)(d - 1)(2d - 7)(2d - 5)
,4

-1

- (d - 4) (d - 3) (d - 2)(d - 1)(2d - 7)(3d - 8) â 401d6 - 7251d5 + 54491d4 - 217784d3 +489064d2 - 581248d + 287232 P4

-1

V.A. Smir nov

Loops and Legs'10 ­ p.23


10 6 e-3E 20 (5) + 68 (3)2 + 1 - 2 189 + + + +
4 2

[Chetyrkin, Kataev & Tkachov'80, Kazakov'84]

34 4 (3) - 5 2 (5) + 450 (7) 15

[Becavac'06]

3

9072 6487 8 2 2 - (5, 3) - 2588 (3) (5) - 17 (3) + 5 10500

5

4897 6 (3) 6068 (3)3 13063 4 (5) 225 2 (7) 88036 (9) - + - + - 630 3 120 2 9 2268 2 (5, 3) + 42513 (8, 2) - 145328 (3) (7) 5
10 2 2

11813 4 (3)2 28138577 + -73394 (5) + 647 (3) (5) - 120 9355500

+ ...

V.A. Smir nov

Loops and Legs'10 ­ p.24


Pull out another factor to kill terms with pure
[Chetyrkin, Kataev & Tkachov'80, D. Broadhurst'99]

2

(1 - 2) + + + +
2

2

34 4 (3) + 450 (7) 15

(1 - )2 (1 + ) (2 - 2)

3

10 6 20 (5) + 68 (3)2 + 189

3

12072 8519 8 - (5, 3) - 2448 (3) (5) + 5 13500 1292 6 (3) 4640 (3)3 1202 4 (5) 88036 (9) - + + - 189 3 5 9 42513 (8, 2) - 142178 (3) (7) - 73022 (5)
2 0

4 5

232 4 (3)2 593053 1 - + 3 187120

+ ...
Loops and Legs'10 ­ p.25

V.A. Smir nov


Pull out another factor to kill terms with pure
1 (1 - 2)(1 - ) + + +
2 3

2

[Kazakov'84]

10 6 20 (5) + 68 (3)2 + 189

34 4 (3) + 450 (7) 15

3

9072 8519 8 - (5, 3) - 2568 (3) (5) + 5 13500 1352 6 (3) 5864 (3)3 542 4 (5) 88036 (9) - + + - 189 3 5 9 1466 4 (3)2 592063 1 + -73382 (5)2 - 15 187110
0

4

+5 (42513 (8, 2) - 144878 (3) (7)

+ ...

V.A. Smir nov

Loops and Legs'10 ­ p.26


The coefficients in the -expansion of planar massless propagator diagrams up to five loops should be expressed in terms of multiple zeta values, while the non-planar graphs may contain, in addition, multiple sums with 6th roots of unity.
[Brown'08]

V.A. Smir nov

Loops and Legs'10 ­ p.27


The full color dependence of the 4-loop 4-gluon amplitude in N=4 SUSY YM in terms of 50 4-loop 4-point integrals.
[Bern, Carrasco, Johansson & Roiban'10]

The critical dimension at which the amplitude first diverge. For 4 loops, this is d=11/2. The subleading-color par ts of the divergence require the three-loop propagator non-planar master integral.

V.A. Smir nov

Loops and Legs'10 ­ p.28


The full color dependence of the 4-loop 4-gluon amplitude in N=4 SUSY YM in terms of 50 4-loop 4-point integrals.
[Bern, Carrasco, Johansson & Roiban'10]

The critical dimension at which the amplitude first diverge. For 4 loops, this is d=11/2. The subleading-color par ts of the divergence require the three-loop propagator non-planar master integral.
-6.198399226750

V.A. Smir nov

Loops and Legs'10 ­ p.28


The full color dependence of the 4-loop 4-gluon amplitude in N=4 SUSY YM in terms of 50 4-loop 4-point integrals.
[Bern, Carrasco, Johansson & Roiban'10]

The critical dimension at which the amplitude first diverge. For 4 loops, this is d=11/2. The subleading-color par ts of the divergence require the three-loop propagator non-planar master integral.
-6.198399226750

confirmed by FIESTA (-6.1977)

V.A. Smir nov

Loops and Legs'10 ­ p.28


The full color dependence of the 4-loop 4-gluon amplitude in N=4 SUSY YM in terms of 50 4-loop 4-point integrals.
[Bern, Carrasco, Johansson & Roiban'10]

The critical dimension at which the amplitude first diverge. For 4 loops, this is d=11/2. The subleading-color par ts of the divergence require the three-loop propagator non-planar master integral.
-6.198399226750

confirmed by FIESTA (-6.1977) We have
-6.1983992267494959168200925479819368763478987989679152 . . .
V.A. Smir nov Loops and Legs'10 ­ p.28


Conclusion

The method is powerful :)

V.A. Smir nov

Loops and Legs'10 ­ p.29


Conclusion

The method is powerful :) Results at general d are expressed in terms of multiple well convergent series. They can applied at any d to obtain Laurent expansions in = (d0 - d)/2.

V.A. Smir nov

Loops and Legs'10 ­ p.29


Conclusion

The method is powerful :) Results at general d are expressed in terms of multiple well convergent series. They can applied at any d to obtain Laurent expansions in = (d0 - d)/2. Higher powers of the expansion in can easily be evaluated.

V.A. Smir nov

Loops and Legs'10 ­ p.29


Conclusion

The method is powerful :) Results at general d are expressed in terms of multiple well convergent series. They can applied at any d to obtain Laurent expansions in = (d0 - d)/2. Higher powers of the expansion in can easily be evaluated. Combination with other methods: IBP reduction via Lapor ta algorithm, PSLQ, sector decomposition (FIESTA), MB, ...

V.A. Smir nov

Loops and Legs'10 ­ p.29


Conclusion

The method is powerful :) Results at general d are expressed in terms of multiple well convergent series. They can applied at any d to obtain Laurent expansions in = (d0 - d)/2. Higher powers of the expansion in can easily be evaluated. Combination with other methods: IBP reduction via Lapor ta algorithm, PSLQ, sector decomposition (FIESTA), MB, ... to be fur ther developed
V.A. Smir nov Loops and Legs'10 ­ p.29