Mathematica 7.0 for Linux x86 (64-bit)
Copyright 1988-2008 Wolfram Research, Inc.

In[1]:= MB 1.2
by Michal Czakon
improvements by Alexander Smirnov
more info in hep-ph/0511200
last modified 2 Jan 09

In[2]:= 
In[2]:= AMBRE by K.Kajda   ver: 2.0 21 . 06 . 2010 at 23:13
last modified 18 Jun 2010
last executed on 

In[3]:= 
In[3]:= MBnum v.0.1, last modified: 18.06.09

In[4]:= 
In[4]:= 
In[5]:= 
In[5]:= 
In[6]:= 
In[6]:= >>External momenta = N/A
>>Starting LoopByLoop calculation
--iteration nr: 1 with momentum: k2
  Run ?INT to see description of below output 

 
>   {INT[{1}, 1, PR[k1 - k2, 0, n4] PR[k2, 0, n5] PR[k2 + p1 + p2, 0, n6] 
 
>      PR[k2 + p1 + p2 + p4, 0, n7], N/A]}
  F polynomial during this iteration 

 
>   -(PR[k1, 0] X[1] X[2]) - PR[k1 + p1 + p2, 0] X[1] X[3] - s X[2] X[3] - 
 
>    PR[k1 + p1 + p2 + p4, 0] X[1] X[4]
--iteration nr: 2 with momentum: k1
  Run ?INT to see description of below output 

 
                   2 - eps - z3     z3
>   {INT[{1}, ((-1)             (-s)   Gamma[-z1] Gamma[-z2] 
 
>        Gamma[2 - eps - n5 - n6 - n7 - z3] 
 
>        Gamma[2 - eps - n4 - n5 - n6 - z1 - z2 - z3] Gamma[-z3] 
 
>        Gamma[n5 + z1 + z3] Gamma[n6 + z2 + z3] 
 
>        Gamma[-2 + eps + n4 + n5 + n6 + n7 + z1 + z2 + z3]) / 
 
>      (Gamma[n4] Gamma[n5] Gamma[n6] Gamma[4 - 2 eps - n4 - n5 - n6 - n7] 
 
>        Gamma[n7]), PR[k1, 0, n1 - z1] PR[k1 + p1, 0, n2] 
 
>      PR[k1 + p1 + p2, 0, n3 - z2] 
 
>      PR[k1 + p1 + p2 + p4, 0, -2 + eps + n4 + n5 + n6 + n7 + z1 + z2 + z3], 
 
>     N/A]}
  F polynomial during this iteration -(s X[1] X[3]) - t X[2] X[4]

>>Contracting and finalizing output
--contracting...
--finalizing output...
>>Checking Barnes 1-st lemma...

In[7]:= 
In[7]:= 
In[8]:= 
In[8]:= repr={((-1)^(n1 + n2 + n3 + n4 + n5 + n6 + n7)*(-s)^(z3 + z4)*(-t)^(-2*eps - n1 - n2 - n3 - n4 - n5 - n6 - n7 - z3 - z4)*t^4*Gamma[-z1]*Gamma[-z2]*Gamma[2 - eps - n5 - n6 - n7 - z3]*Gamma[2 - eps - n4 - n5 - n6 - z1 - z2 - z3]*Gamma[-z3]*Gamma[n5 + z1 + z3]*Gamma[n6 + z2 + z3]*Gamma[2 - eps - n1 - n2 - n3 + z1 + z2 - z4]*Gamma[4 - 2*eps - n1 - n3 - n4 - n5 - n6 - n7 - z3 - z4]*Gamma[-z4]*Gamma[n1 - z1 + z4]*Gamma[n3 - z2 + z4]*Gamma[-4 + 2*eps + n1 + n2 + n3 + n4 + n5 + n6 + n7 + z3 + z4])/(Gamma[n2]*Gamma[n4]*Gamma[n5]*Gamma[n6]*Gamma[4 - 2*eps - n4 - n5 - n6 - n7]*Gamma[n7]*Gamma[n1 - z1]*Gamma[n3 - z2]*Gamma[6 - 3*eps - n1 - n2 - n3 - n4 - n5 - n6 - n7 - z3])}
Length=1


MBresidues::contour: 
   contour starts and/or ends on a pole of Gamma[-1 - 2 eps - z2 - z3]

MBresidues::contour: 
   contour starts and/or ends on a pole of Gamma[-1 - eps - z1 - z2 - z3]

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

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

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

General::stop: Further output of MBrules::norules
     will be suppressed during this calculation.
ETA's will be aplied on positions: {}
1. Calculating 'no eta' parts...
   Running MBcontinue...
   Running MBexpand...
2. Calculating 'eta' parts...
   No 'eta' parts found!!!

                                      2   2        4   4        3        3
Out[8]= {2.188838, {MBint[(24 - 14 eps  Pi  - 6 eps  Pi  + 4 eps  Log[-s]  - 
 
               4        4        2            2   2         2
>         2 eps  Log[-s]  - 2 eps  (-6 + 7 eps  Pi ) Log[-t]  - 
 
               4        4        3        2
>         2 eps  Log[-t]  - 4 eps  Log[-s]  
 
                  2                            2          3
>          (eps Pi  + 3 Log[-t] - 3 eps Log[-t] ) + 71 eps  PolyGamma[2, 1] + 
 
                                  2   2                   3   2
>         eps Log[-s] (-18 + 9 eps  Pi  + (36 eps - 10 eps  Pi ) Log[-t] - 
 
                   2        2        3        3         3
>            24 eps  Log[-t]  + 8 eps  Log[-t]  - 48 eps  PolyGamma[2, 1]) + 
 
                                   3   2         4
>         Log[-t] (-30 eps + 19 eps  Pi  - 94 eps  PolyGamma[2, 1])) / 
 
              4  2
>       (6 eps  s  t), {{eps -> 0}, {}}], 
 
                       2
>     MBint[(Gamma[-z1]  Gamma[z1] 
 
>         (6 eps Gamma[-z1] (4 eps Gamma[z1] 
 
                     z1         z1
>               ((-t)   + 2 (-s)   Gamma[1 - z1] Gamma[1 + z1]) - 
 
                   z1
>              (-t)   Gamma[1 + z1] 
 
>               (1 + 4 eps EulerGamma - 2 eps Log[-s] + 
 
>                 3 eps PolyGamma[0, -z1] + eps PolyGamma[0, z1])) + 
 
                z1
>           (-t)   Gamma[1 - z1] Gamma[1 + z1] 
 
                                            2           2        2   2
>            (6 + 12 eps EulerGamma + 12 eps  EulerGamma  - 4 eps  Pi  - 
 
                     2        2
>              12 eps  Log[-s]  - 12 eps Log[-t] - 
 
                     2                            2
>              24 eps  EulerGamma Log[-t] + 24 eps  Log[-s] Log[-t] - 
 
                    2                  2
>              9 eps  PolyGamma[0, -z1]  + 
 
>              6 eps (1 + 2 eps EulerGamma - 2 eps Log[-t]) 
 
                                        2                 2
>               PolyGamma[0, z1] + 3 eps  PolyGamma[0, z1]  + 
 
                     2
>              24 eps  Log[-s] PolyGamma[0, 1 + z1] - 
 
                     2
>              24 eps  Log[-t] PolyGamma[0, 1 + z1] - 
 
                     2                     2
>              12 eps  PolyGamma[0, 1 + z1]  + 
 
>              6 eps PolyGamma[0, -z1] 
 
>               (1 + 2 eps EulerGamma - 4 eps Log[-s] + 2 eps Log[-t] + 
 
>                 eps PolyGamma[0, z1] + 4 eps PolyGamma[0, 1 + z1]) - 
 
                    2                           2
>              9 eps  PolyGamma[1, -z1] - 21 eps  PolyGamma[1, z1] - 
 
                     2
>              12 eps  PolyGamma[1, 1 + z1]))) / 
 
               2  2     z1                                         5
>       (12 eps  s  (-t)   t Gamma[1 - z1]), {{eps -> 0}, {z1 -> -(-)}}], 
                                                                   8
 
                       2
>     MBint[(Gamma[-z2]  Gamma[z2] 
 
                 2
>         (24 eps  Gamma[-z2] Gamma[z2] 
 
                  z2         z2
>            ((-t)   + 2 (-s)   Gamma[1 - z2] Gamma[1 + z2]) + 
 
                z2
>           (-t)   Gamma[1 - z2] Gamma[1 + z2] 
 
                                            2           2        2   2
>            (6 + 12 eps EulerGamma + 12 eps  EulerGamma  - 4 eps  Pi  - 
 
                     2        2
>              12 eps  Log[-s]  - 12 eps Log[-t] - 
 
                     2                            2
>              24 eps  EulerGamma Log[-t] + 24 eps  Log[-s] Log[-t] - 
 
                    2                  2
>              9 eps  PolyGamma[0, -z2]  + 
 
>              6 eps (1 + 2 eps EulerGamma - 2 eps Log[-t]) 
 
                                        2                 2
>               PolyGamma[0, z2] + 3 eps  PolyGamma[0, z2]  + 
 
                     2
>              24 eps  Log[-s] PolyGamma[0, 1 + z2] - 
 
                     2
>              24 eps  Log[-t] PolyGamma[0, 1 + z2] - 
 
                     2                     2
>              12 eps  PolyGamma[0, 1 + z2]  + 
 
>              6 eps PolyGamma[0, -z2] 
 
>               (1 + 2 eps EulerGamma - 4 eps Log[-s] + 2 eps Log[-t] + 
 
>                 eps PolyGamma[0, z2] + 4 eps PolyGamma[0, 1 + z2]) - 
 
                    2                           2
>              9 eps  PolyGamma[1, -z2] - 21 eps  PolyGamma[1, z2] - 
 
                     2
>              12 eps  PolyGamma[1, 1 + z2]))) / 
 
               2  2     z2                                         1
>       (12 eps  s  (-t)   t Gamma[1 - z2]), {{eps -> 0}, {z2 -> -(-)}}], 
                                                                   8
 
                   -1 + z3     -2 - z3               2
>     MBint[(2 (-s)        (-t)        Gamma[-1 - z3]  Gamma[-z3] 
 
                       2
>         Gamma[1 + z3]  Gamma[2 + z3] 
 
>         (1 + eps EulerGamma + eps Log[-s] - 3 eps Log[-t] - 
 
>           4 eps PolyGamma[0, -1 - z3] + 2 eps PolyGamma[0, 1 + z3] + 
 
                                                                      3
>           3 eps PolyGamma[0, 2 + z3])) / eps, {{eps -> 0}, {z3 -> -(--)}}], 
                                                                      16
 
                   z4     -2 - z4               2                         2
>     MBint[(2 (-s)   (-t)        Gamma[-1 - z4]  Gamma[-z4] Gamma[1 + z4]  
 
>         Gamma[2 + z4] (-1 + eps EulerGamma + eps Log[-s] + eps Log[-t] + 
 
>           2 eps PolyGamma[0, 1 + z4] - eps PolyGamma[0, 2 + z4])) / (eps s)\
 
                                59
>       , {{eps -> 0}, {z4 -> -(--)}}], 
                                64
 
>     MBint[(Gamma[-z1] Gamma[z1 - z2] Gamma[-z2] 
 
>         (Gamma[1 - z1] Gamma[1 + z1] Gamma[-z2] Gamma[z2] + 
 
>           Gamma[-z1] Gamma[z1] Gamma[1 - z2] Gamma[1 + z2]) Gamma[-z1 + z2])
 
              2
>         / (s  t Gamma[1 - z1] Gamma[1 - z2]), 
 
                             5           1
>      {{eps -> 0}, {z1 -> -(-), z2 -> -(-)}}], 
                             8           8
 
                   -1 + z1 + z3     -2 - z1 - z3
>     MBint[-2 (-s)             (-t)             Gamma[-z1] Gamma[z1] 
 
                           2
>       Gamma[-1 - z1 - z3]  Gamma[-z3] Gamma[1 + z3] Gamma[1 + z1 + z3] 
 
                                                  5           3
>       Gamma[2 + z1 + z3], {{eps -> 0}, {z1 -> -(-), z3 -> -(--)}}], 
                                                  8           16
 
                   -1 + z2 + z3     -2 - z2 - z3
>     MBint[-2 (-s)             (-t)             Gamma[-z2] Gamma[z2] 
 
                           2
>       Gamma[-1 - z2 - z3]  Gamma[-z3] Gamma[1 + z3] Gamma[1 + z2 + z3] 
 
                                                  1           3
>       Gamma[2 + z2 + z3], {{eps -> 0}, {z2 -> -(-), z3 -> -(--)}}]}}
                                                  8           16

In[9]:= 
In[9]:= Shifting contours...
Performing 10 lower-dimensional integrations with NIntegrate...1...2...3...4...5...6...7...8...9...10Higher-dimensional integrals
Preparing MBpart1eps0 (dim 2)
Preparing MBpart2eps0 (dim 2)
Preparing MBpart3eps0 (dim 2)
Running MBpart1eps0
Running MBpart2eps0
Running MBpart3eps0

                                 0.0228571   0.0831878   0.00965743
Out[9]= {20.206313, {-0.103415 - --------- + --------- - ---------- - 
                                      4           3            2
                                   eps         eps          eps
 
       0.109074             -6
>      --------, {8.80801 10  , 0}}}
         eps

In[10]:= 
In[10]:= 
22.59user 0.19system 0:22.75elapsed 100%CPU (0avgtext+0avgdata 0maxresident)k
0inputs+0outputs (0major+79639minor)pagefaults 0swaps
