Документ взят из кэша поисковой машины. Адрес оригинального документа : http://theory.sinp.msu.ru/~smirnov/Ex2S2.pdf Дата изменения: Sat Oct 25 23:02:01 2008 Дата индексирования: Mon Oct 1 20:02:20 2012 Кодировка: Windows-1251

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MB 1.1

by Michal Czakon

more info in hep-phк0511200 last modified 06 Mar 08

<< MBкMB.m

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H The integrand of the MB integral for the one-loop massless box diagram with p2^2=p3^2=p4^2=0 L Box1@a1_, a2_, a3_, a4_D := IS2-a1-a2-a3-a4-ep-z Tz Gamma@a1 + a2 + a3 + a4 - 2 + ep + zD Gamma@a2 + zD Gamma@a4 + zD HGamma@a1D Gamma@a2D Gamma@a3D Gamma@a4D Gamma@4 - a1 - a2 - a3 - a4 - 2 epDL; L Gamma@2 - a1 - a2 - a4 - ep - zD Gamma@2 - a2 - a3 - a4 - ep - zD Gamma@- zD M л

p1^2=

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H S

H Notation: s=Hp1+p2L^2=-S, t=Hp1+p2L^2=-T; I Pi^Hdк2L is pulled out, as always Box1@1, 1, 1, 1D
-2-ep-z

The box with the powers of the propagators equal to one L Tz Gamma@- 1 - ep - zD2 Gamma@- zD Gamma@1 + zD2 Gamma@2 + ep + zD Gamma@- 2 epD Gamma@- 2 epD

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B1 = % к. 8S 1, T x<

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B1Rules = MBoptimizedRules@B1, ep 0, 8<, 8ep>

xz Gamma@- 1 - ep - zD2 Gamma@- zD Gamma@1 + zD2 Gamma@2 + ep + zD

MBrules::norules : no rules could be found to regulate this integral
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Level 1

Taking -residue in z = - 1 - ep Level 2 Integral 81<
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2 integralHsL found ::MBintB- 2 x- x-
1-ep 1-ep

EulerGamma x

-1-ep

MBintB

xz Gamma@- 1 - ep - zD2 Gamma@- zD Gamma@1 + zD2 Gamma@2 + ep + zD Gamma@- 2 epD

Gamma@- epD2 Gamma@1 + epD PolyGamma@0, 1 + epD Gamma@- 2 epD

Gamma@- epD2 Gamma@1 + epD PolyGamma@0, - epD Gamma@- 2 epD

Gamma@- 2 epD

Gamma@- epD2 Gamma@1 + epD

-

x

-1-ep

+

Gamma@- epD2 Gamma@1 + epD Log@xD Gamma@- 2 epD

-

, 88ep 0<, 8<,

, :8ep 0<, :z -

1 2

>>F>


2

Ex2S2.nb

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B1select = MBpreselect@B1cont, 8ep, 0, 0 -1-ep -1-ep

EulerGamma x-

1-ep

Gamma@- epD2 Gamma@1 + epD Log@xD Gamma@- 2 epD Gamma@- 2 epD

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B1select = MBpreselect@B1cont, 8ep, 0, 1 -1-ep -1-ep

Gamma@- epD2 Gamma@1 + epD PolyGamma@0, 1 + epD EulerGamma x-
1-ep

Gamma@- 2 epD

Gamma@- epD2 Gamma@1 + epD - 2x
-1-ep

- Gamma@- 2 epD

Gamma@- epD2 Gamma@1 + epD PolyGamma@0, - epD , 88ep 0<, 8<

+

Gamma@- epD2 Gamma@1 + epD Log@xD Gamma@- 2 epD Gamma@- 2 epD

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B1exp = MBexpand@B1select, E ^ HEulerGamma epL, 8ep, 0, 1
MBintB

xz Gamma@- 1 - ep - zD2 Gamma@- zD Gamma@1 + zD2 Gamma@2 + ep + zD Gamma@- 2 epD

Gamma@- epD2 Gamma@1 + epD PolyGamma@0, 1 + epD

Gamma@- 2 epD

Gamma@- epD2 Gamma@1 + epD - 2x
-1-ep

- Gamma@- 2 epD

Gamma@- epD2 Gamma@1 + epD PolyGamma@0, - epD , 88ep 0<, 8<>F>

+

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res1 = B1exp@@1DD@@1DD 4 B1exp@@2DD ep2 x - 4
2

MBintB- 2 ep xz Gamma@- 1 - zD2 Gamma@- zD Gamma@1 + zD2 Gamma@2 + zD, :8ep 0<, :z - 3x - 2 Log@xD ep x + 7 ep 2 Log@xD 6x + ep Log@xD 3x
3

17 ep PolyGamma@2, 1D 3x

, 88ep 0<, 8<>F>

+

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17 ep PolyGamma@2, 1D 3x

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int1 = B1exp@@2DD@@1DD

MBintB- 2 ep xz Gamma@- 1 - zD2 Gamma@- zD Gamma@1 + zD2 Gamma@2 + zD, :8ep 0<, :z - - 2 ep xz Gamma@- 1 - zD2 Gamma@- zD Gamma@1 + zD2 Gamma@2 + zD - 2 ep xz Gamma@- zD3 Gamma@1 + zD 1+z
3

1 2

>>F

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Simplify@% кк. 8 Gamma@- 1 - zD Gamma@- zD к H- 1 - zL, Gamma@2 + zD Gamma@1 + zD H1 + zL +z

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H

2 ep 3 xz Csc@ zD3 1+z

Now we take residues at z=0,1,2,.,.. Csc@ Hn + zLD3

L

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1+n+z


Ex2S2.nb

3

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% к. 8Csc@ Hn + zLD H- 1L ^ n Csc@ zD < % к. 9 H- 1L3 2 H- 1L
3n

ep 3 xn

+z

1+n+z
n

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-

- Residue@%, 8z, 0 3

2 H- 1Ln ep 3 xn H1 + nL3 1

1+n+z

H- 1Ln =
+z

Csc@ zD3

Csc@ zD3

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Apart@% к. n n - 1, nD
1+n

H- 1Ln ep xn I2 +2 + 2 n 2 + n2 2 - 2 Log@xD - 2 n Log@xD + Log@xD2 + 2 n Log@xD2 + n2 Log@xD2 M - 2 H- 1Ln ep x n
-1+n

2

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H

res2 = Sum@%, 8n, 1, Infinity 13

Log@xD

+

H- 1Ln ep x-

1+n

n

I2 + Log@xD2 M

I- ep 2 Log@1 + xD - ep Log@xD2 Log@1 + xD - 2 ep Log@xD PolyLog@2, - xD + 2 ep PolyLog@3, - xDM Numerical check L

res1 + res2 4 ep2 H S x - 4

NIntegrate@int1 к ep к. 8x 0.76, ep 0.3, z - 0.5 + I y1<, 8y1, - Infinity, Infinity 2

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H% к. x T к S L S
-2-ep

I- ep 2 Log@1 + xD - ep Log@xD2 Log@1 + xD - 2 ep Log@xD PolyLog@2, - xD + 2 ep PolyLog@3, - xDM This is our result Hup to I Pi^Hdк2L L L
-2-ep

3x

-

+

7 ep 2 Log@xD 6x

+

ep Log@xD 3x

3

+

17 ep PolyGamma@2, 1D 3x

+

1 x

4S

ep2 T

-

4 2 S 3T

-

ep S LogA T E S 3T

3

+

T2 T T T T ep LogB F LogB1 + F - 2 ep LogB F PolyLogB2, - F + 2 ep PolyLogB3, - F S S S S S

17 ep S PolyGamma@2, 1D 3T

2 S LogA T E S ep T

+

7 ep 2 S LogA T E S 6T + 1 T

+

S - ep 2 LogB1 +

T S

F-