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 23:11
last modified 18 Jun 2010
last executed on 

In[3]:= 
In[3]:= MBresolve 1.0
by Alexander Smirnov
more info in arXiv:0901.0386
last modified 4 Jan 09

In[4]:= 
In[4]:= 
In[5]:= 
In[5]:= 
In[6]:= 
In[7]:= 
In[7]:= >>External momenta = p1[mu1] p1[mu2] p1[mu3]
>>Starting LoopByLoop calculation
--iteration nr: 1 with momentum: k2
  Run ?INT to see description of below output 

 
>   {INT[{k2[mu3]}, 1, PR[k2, 0, n2] PR[-k1 + k2, 0, n3] PR[k2 + p1, 0, n5], 
 
>     N/A]}
  F polynomial during this iteration 

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

 
>   {INT[{k1[mu1], k1[mu2], k1[mu3]}, 
 
           2 - eps - z2     z2
>     ((-1)             (-s)   Gamma[2 - eps - n2 - n3 - z1] Gamma[-z1] 
 
>        Gamma[3 - eps - n2 - n5 - z2] Gamma[-z2] Gamma[n2 + z1 + z2] 
 
>        Gamma[-2 + eps + n2 + n3 + n5 + z1 + z2]) / 
 
>      (Gamma[n2] Gamma[n3] Gamma[5 - 2 eps - n2 - n3 - n5] Gamma[n5]), 
 
>     PR[k1, 0, n1 - z1] PR[k1 + p1, 0, 
 
>       -2 + eps + n2 + n3 + n4 + n5 + z1 + z2], N/A], 
 
                                    2 - eps - z2     z2
>    INT[{k1[mu1], k1[mu2]}, -(((-1)             (-s)   
 
>          Gamma[3 - eps - n2 - n3 - z1] Gamma[-z1] 
 
>          Gamma[2 - eps - n2 - n5 - z2] Gamma[-z2] Gamma[n2 + z1 + z2] 
 
>          Gamma[-2 + eps + n2 + n3 + n5 + z1 + z2] p1[mu3]) / 
 
>        (Gamma[n2] Gamma[n3] Gamma[5 - 2 eps - n2 - n3 - n5] Gamma[n5])), 
 
>     PR[k1, 0, n1 - z1] PR[k1 + p1, 0, 
 
>       -2 + eps + n2 + n3 + n4 + n5 + z1 + z2], N/A]}
  F polynomial during this iteration -(s X[1] X[2])

>>Contracting and finalizing output
--contracting...
--finalizing output...
>>Checking Barnes 1-st lemma...
>> Barnes 1st Lemma will be checked for: {z2, z1} <<                  2
                                                    Starting with dim=
 
>    representation...

1. Checking z2
2. Checking z1>> Representation after 1st Barnes Lemma:  <<
   Could not apply Barnes-Lemma
>> Barnes 1st Lemma will be checked for: {z2, z1} <<                  2
                                                    Starting with dim=
 
>    representation...

1. Checking z2
2. Checking z1>> Representation after 1st Barnes Lemma:  <<
   Could not apply Barnes-Lemma
>> Barnes 1st Lemma will be checked for: {z2, z1} <<                  2
                                                    Starting with dim=
 
>    representation...

1. Checking z2
2. Checking z1>> Representation after 1st Barnes Lemma:  <<
   Could not apply Barnes-Lemma
>> Barnes 1st Lemma will be checked for: {z2, z1} <<                  2
                                                    Starting with dim=
 
>    representation...

1. Checking z2
2. Checking z1>> Representation after 1st Barnes Lemma:  <<
   Could not apply Barnes-Lemma
>> Barnes 1st Lemma will be checked for: {z2, z1} <<                  2
                                                    Starting with dim=
 
>    representation...

1. Checking z2
2. Checking z1>> Representation after 1st Barnes Lemma:  <<
   Could not apply Barnes-Lemma
>> Barnes 1st Lemma will be checked for: {z2, z1} <<                  2
                                                    Starting with dim=
 
>    representation...

1. Checking z2
2. Checking z1>> Representation after 1st Barnes Lemma:  <<
   Could not apply Barnes-Lemma

In[8]:= 
In[8]:= CREATING RESIDUES LIST0.162 seconds
EVALUATING RESIDUES..........0.0002 seconds
CREATING RESIDUES LIST..........0.196 seconds
EVALUATING RESIDUES..........0.0037 seconds
CREATING RESIDUES LIST..........0.1938 seconds
EVALUATING RESIDUES..........0.0037 seconds
CREATING RESIDUES LIST..........0.1981 seconds
EVALUATING RESIDUES..........0.0038 seconds
CREATING RESIDUES LIST0.1579 seconds
EVALUATING RESIDUES..........0.0002 seconds
CREATING RESIDUES LIST..........0.1996 seconds
EVALUATING RESIDUES..........0.0038 seconds

                     -5 - 2 eps  7
Out[8]= {{MBint[((-s)           s  Gamma[-eps - z1] Gamma[-z1] 
 
>         Gamma[4 - eps + z1] Gamma[1 - eps - z2] Gamma[-2 eps - z1 - z2] 
 
>         Gamma[-z2] Gamma[1 + 2 eps + z2] Gamma[1 + z1 + z2] 
 
>         Gamma[1 + eps + z1 + z2]) / 
 
>       (Gamma[2 - 2 eps] Gamma[1 - z1] Gamma[4 - 3 eps - z2] 
 
>         Gamma[2 + eps + z1 + z2]), 
 
>      {{eps -> 0}, {z1 -> -0.234828, z2 -> -0.576688}}]}, 
 
               2
>    {MBint[-(s  Gamma[1 - eps] Gamma[2 eps] Gamma[1 - 2 eps - z1] 
 
>          Gamma[-eps - z1] Gamma[-z1] Gamma[1 + z1] Gamma[3 - eps + z1] 
 
>          Gamma[1 + eps + z1]) / 
 
               2 eps
>       (2 (-s)      Gamma[4 - 3 eps] Gamma[2 - 2 eps] Gamma[1 - z1] 
 
>         Gamma[2 + eps + z1]), {{eps -> 0}, {z1 -> -0.163093}}], 
 
                 -5 - 2 eps  7
>     MBint[((-s)           s  Gamma[-eps - z1] Gamma[-z1] 
 
>         Gamma[3 - eps + z1] Gamma[1 - eps - z2] Gamma[1 - 2 eps - z1 - z2] 
 
>         Gamma[-z2] Gamma[2 eps + z2] Gamma[1 + z1 + z2] 
 
>         Gamma[1 + eps + z1 + z2]) / 
 
>       (2 Gamma[2 - 2 eps] Gamma[1 - z1] Gamma[4 - 3 eps - z2] 
 
>         Gamma[2 + eps + z1 + z2]), 
 
>      {{eps -> 0}, {z1 -> -0.576391, z2 -> 0.947163}}]}, 
 
               2
>    {MBint[-(s  Gamma[1 - eps] Gamma[2 eps] Gamma[1 - 2 eps - z1] 
 
>          Gamma[-eps - z1] Gamma[-z1] Gamma[1 + z1] Gamma[3 - eps + z1] 
 
>          Gamma[1 + eps + z1]) / 
 
               2 eps
>       (2 (-s)      Gamma[4 - 3 eps] Gamma[2 - 2 eps] Gamma[1 - z1] 
 
>         Gamma[2 + eps + z1]), {{eps -> 0}, {z1 -> -0.163093}}], 
 
                 -5 - 2 eps  7
>     MBint[((-s)           s  Gamma[-eps - z1] Gamma[-z1] 
 
>         Gamma[3 - eps + z1] Gamma[1 - eps - z2] Gamma[1 - 2 eps - z1 - z2] 
 
>         Gamma[-z2] Gamma[2 eps + z2] Gamma[1 + z1 + z2] 
 
>         Gamma[1 + eps + z1 + z2]) / 
 
>       (2 Gamma[2 - 2 eps] Gamma[1 - z1] Gamma[4 - 3 eps - z2] 
 
>         Gamma[2 + eps + z1 + z2]), 
 
>      {{eps -> 0}, {z1 -> -0.576391, z2 -> 0.947163}}]}, 
 
               2
>    {MBint[-(s  Gamma[1 - eps] Gamma[2 eps] Gamma[1 - 2 eps - z1] 
 
>          Gamma[-eps - z1] Gamma[-z1] Gamma[1 + z1] Gamma[3 - eps + z1] 
 
>          Gamma[1 + eps + z1]) / 
 
               2 eps
>       (2 (-s)      Gamma[4 - 3 eps] Gamma[2 - 2 eps] Gamma[1 - z1] 
 
>         Gamma[2 + eps + z1]), {{eps -> 0}, {z1 -> -0.163093}}], 
 
                 -5 - 2 eps  7
>     MBint[((-s)           s  Gamma[-eps - z1] Gamma[-z1] 
 
>         Gamma[3 - eps + z1] Gamma[1 - eps - z2] Gamma[1 - 2 eps - z1 - z2] 
 
>         Gamma[-z2] Gamma[2 eps + z2] Gamma[1 + z1 + z2] 
 
>         Gamma[1 + eps + z1 + z2]) / 
 
>       (2 Gamma[2 - 2 eps] Gamma[1 - z1] Gamma[4 - 3 eps - z2] 
 
>         Gamma[2 + eps + z1 + z2]), 
 
>      {{eps -> 0}, {z1 -> -0.576391, z2 -> 0.947163}}]}, 
 
                 -5 - 2 eps  7
>    {MBint[((-s)           s  Gamma[1 - eps - z1] Gamma[-z1] 
 
>         Gamma[3 - eps + z1] Gamma[-eps - z2] Gamma[-2 eps - z1 - z2] 
 
>         Gamma[-z2] Gamma[1 + 2 eps + z2] Gamma[1 + z1 + z2] 
 
>         Gamma[1 + eps + z1 + z2]) / 
 
>       (Gamma[2 - 2 eps] Gamma[1 - z1] Gamma[3 - 3 eps - z2] 
 
>         Gamma[2 + eps + z1 + z2]), 
 
>      {{eps -> 0}, {z1 -> -0.430071, z2 -> -0.146351}}]}, 
 
               2
>    {MBint[-(s  Gamma[eps] Gamma[2 eps] Gamma[1 - eps - z1] Gamma[-z1] 
 
>          Gamma[1 - 2 eps + z1] Gamma[1 - eps + z1]) / 
 
               2 eps
>       (2 (-s)      Gamma[2 - 2 eps] Gamma[3 - eps]), 
 
>      {{eps -> 0}, {z1 -> -0.859981}}], 
 
                 -5 - 2 eps  7
>     MBint[((-s)           s  Gamma[1 - eps - z1] Gamma[-z1] 
 
>         Gamma[2 - eps + z1] Gamma[-eps - z2] Gamma[1 - 2 eps - z1 - z2] 
 
>         Gamma[-z2] Gamma[2 eps + z2] Gamma[1 + z1 + z2] 
 
>         Gamma[1 + eps + z1 + z2]) / 
 
>       (2 Gamma[2 - 2 eps] Gamma[1 - z1] Gamma[3 - 3 eps - z2] 
 
>         Gamma[2 + eps + z1 + z2]), 
 
>      {{eps -> 0}, {z1 -> -0.285712, z2 -> -0.233645}}]}}

In[9]:= 
In[10]:= 
In[11]:= 
In[11]:=              2                                       2
res={MBint[(s  Gamma[1 - z1] Gamma[-z1] Gamma[1 + z1]  
 
                                2          3
>        (-12 - 42 eps - 105 eps  - 196 eps  + 48 eps EulerGamma + 
 
                  2                     3                    2           2
>          168 eps  EulerGamma + 420 eps  EulerGamma - 96 eps  EulerGamma  - 
 
                  3           2          3           3
>          336 eps  EulerGamma  + 128 eps  EulerGamma  + 24 eps Log[-s] + 
 
                 2                  3                 2
>          84 eps  Log[-s] + 210 eps  Log[-s] - 96 eps  EulerGamma Log[-s] - 
 
                  3                             3           2
>          336 eps  EulerGamma Log[-s] + 192 eps  EulerGamma  Log[-s] - 
 
                 2        2         3        2
>          24 eps  Log[-s]  - 84 eps  Log[-s]  + 
 
                 3                   2         3        3
>          96 eps  EulerGamma Log[-s]  + 16 eps  Log[-s]  + 
 
                3                     3
>          2 eps  PolyGamma[0, 1 - z1]  + 
 
                 2
>          27 eps  (-2 - 7 eps + 8 eps EulerGamma + 4 eps Log[-s]) 
 
                                2         3                     3
>           PolyGamma[0, 1 + z1]  + 54 eps  PolyGamma[0, 1 + z1]  + 
 
                2                     2
>          3 eps  PolyGamma[0, 1 - z1]  
 
>           (-2 - 7 eps + 8 eps EulerGamma + 4 eps Log[-s] + 
 
                                                 2
>             6 eps PolyGamma[0, 1 + z1]) - 6 eps  PolyGamma[1, 1 - z1] - 
 
                 3
>          21 eps  PolyGamma[1, 1 - z1] + 
 
                 3
>          24 eps  EulerGamma PolyGamma[1, 1 - z1] + 
 
                 3
>          12 eps  Log[-s] PolyGamma[1, 1 - z1] - 
 
                 2                               3
>          30 eps  PolyGamma[1, 1 + z1] - 105 eps  PolyGamma[1, 1 + z1] + 
 
                  3
>          120 eps  EulerGamma PolyGamma[1, 1 + z1] + 
 
                 3
>          60 eps  Log[-s] PolyGamma[1, 1 + z1] + 
 
>          9 eps PolyGamma[0, 1 + z1] 
 
                                2                             2
>           (4 + 14 eps + 35 eps  - 16 eps EulerGamma - 56 eps  EulerGamma + 
 
                    2           2                         2
>             32 eps  EulerGamma  - 8 eps Log[-s] - 28 eps  Log[-s] + 
 
                    2                           2        2
>             32 eps  EulerGamma Log[-s] + 8 eps  Log[-s]  + 
 
                   2                              2
>             2 eps  PolyGamma[1, 1 - z1] + 10 eps  PolyGamma[1, 1 + z1]) + 
 
>          3 eps PolyGamma[0, 1 - z1] 
 
                                2                             2
>           (4 + 14 eps + 35 eps  - 16 eps EulerGamma - 56 eps  EulerGamma + 
 
                    2           2                         2
>             32 eps  EulerGamma  - 8 eps Log[-s] - 28 eps  Log[-s] + 
 
                    2                           2        2
>             32 eps  EulerGamma Log[-s] + 8 eps  Log[-s]  + 
 
>             6 eps (-2 - 7 eps + 8 eps EulerGamma + 4 eps Log[-s]) 
 
                                            2                     2
>              PolyGamma[0, 1 + z1] + 18 eps  PolyGamma[0, 1 + z1]  + 
 
                   2                              2
>             2 eps  PolyGamma[1, 1 - z1] + 10 eps  PolyGamma[1, 1 + z1]) - 
 
                 3                         3
>          18 eps  PolyGamma[2, 1] - 16 eps  PolyGamma[2, 2] - 
 
                3                        3
>          2 eps  PolyGamma[2, 3] + 2 eps  PolyGamma[2, 1 - z1] + 
 
                 3                                 2
>          18 eps  PolyGamma[2, 1 + z1])) / (96 eps ), 
 
>     {{eps -> 0}, {z1 -> -0.859981}}], 
 
              2           2              2
>    MBint[-(s  Gamma[-z1]  Gamma[1 + z1]  Gamma[3 + z1] 
 
                                2                              2
>         (12 + 90 eps + 435 eps  - 48 eps EulerGamma - 360 eps  EulerGamma + 
 
                  2           2        2   2
>           96 eps  EulerGamma  - 8 eps  Pi  - 24 eps Log[-s] - 
 
                   2                 2
>           180 eps  Log[-s] + 96 eps  EulerGamma Log[-s] + 
 
                  2        2         2                     2
>           24 eps  Log[-s]  + 24 eps  PolyGamma[0, 1 - z1]  + 
 
                 2                  2
>           6 eps  PolyGamma[0, -z1]  + 12 eps PolyGamma[0, 1 + z1] + 
 
                  2
>           90 eps  PolyGamma[0, 1 + z1] - 
 
                  2
>           48 eps  EulerGamma PolyGamma[0, 1 + z1] - 
 
                  2
>           24 eps  Log[-s] PolyGamma[0, 1 + z1] + 
 
                 2                     2
>           6 eps  PolyGamma[0, 1 + z1]  - 12 eps PolyGamma[0, 2 + z1] - 
 
                  2
>           90 eps  PolyGamma[0, 2 + z1] + 
 
                  2
>           48 eps  EulerGamma PolyGamma[0, 2 + z1] + 
 
                  2
>           24 eps  Log[-s] PolyGamma[0, 2 + z1] - 
 
                  2
>           12 eps  PolyGamma[0, 1 + z1] PolyGamma[0, 2 + z1] + 
 
                 2                     2
>           6 eps  PolyGamma[0, 2 + z1]  - 12 eps PolyGamma[0, 3 + z1] - 
 
                  2
>           90 eps  PolyGamma[0, 3 + z1] + 
 
                  2
>           48 eps  EulerGamma PolyGamma[0, 3 + z1] + 
 
                  2
>           24 eps  Log[-s] PolyGamma[0, 3 + z1] - 
 
                  2
>           12 eps  PolyGamma[0, 1 + z1] PolyGamma[0, 3 + z1] + 
 
                  2
>           12 eps  PolyGamma[0, 2 + z1] PolyGamma[0, 3 + z1] + 
 
                 2                     2
>           6 eps  PolyGamma[0, 3 + z1]  + 
 
>           6 eps PolyGamma[0, -z1] 
 
>            (-2 - 15 eps + 8 eps EulerGamma + 4 eps Log[-s] - 
 
>              2 eps PolyGamma[0, 1 + z1] + 2 eps PolyGamma[0, 2 + z1] + 
 
>              2 eps PolyGamma[0, 3 + z1]) + 
 
>           12 eps PolyGamma[0, 1 - z1] 
 
>            (-2 - 15 eps + 8 eps EulerGamma + 4 eps Log[-s] + 
 
>              2 eps PolyGamma[0, -z1] - 2 eps PolyGamma[0, 1 + z1] + 
 
>              2 eps PolyGamma[0, 2 + z1] + 2 eps PolyGamma[0, 3 + z1]) + 
 
                  2                             2
>           24 eps  PolyGamma[1, 1 - z1] + 6 eps  PolyGamma[1, -z1] + 
 
                 2                             2
>           6 eps  PolyGamma[1, 1 + z1] - 6 eps  PolyGamma[1, 2 + z1] + 
 
                 2
>           6 eps  PolyGamma[1, 3 + z1])) / (96 eps Gamma[2 + z1]), 
 
>     {{eps -> 0}, {z1 -> -0.163093}}], 
 
               2           2
>    MBint[(3 s  Gamma[-z1]  Gamma[3 + z1] Gamma[1 - z2] Gamma[1 - z1 - z2] 
 
                                                2
>        Gamma[-z2] Gamma[z2] Gamma[1 + z1 + z2]  
 
>        (-1 - 2 eps + 2 eps Log[-s] + eps PolyGamma[0, -z1] + 
 
>          eps PolyGamma[0, 3 + z1] + eps PolyGamma[0, 1 - z2] - 
 
>          3 eps PolyGamma[0, 4 - z2] + 2 eps PolyGamma[0, 1 - z1 - z2] - 
 
>          2 eps PolyGamma[0, z2] - eps PolyGamma[0, 1 + z1 + z2] + 
 
>          eps PolyGamma[0, 2 + z1 + z2])) / 
 
>      (2 Gamma[1 - z1] Gamma[4 - z2] Gamma[2 + z1 + z2]), 
 
>     {{eps -> 0}, {z1 -> -0.576391, z2 -> 0.947163}}], 
 
             2                                                    2
>    MBint[(s  Gamma[-z1] Gamma[3 + z1] Gamma[-z1 - z2] Gamma[-z2]  
 
                                         2
>        Gamma[1 + z2] Gamma[1 + z1 + z2]  
 
>        (-1 - 2 eps + 2 eps Log[-s] + eps PolyGamma[0, 1 - z1] + 
 
>          eps PolyGamma[0, 3 + z1] - 3 eps PolyGamma[0, 3 - z2] + 
 
>          2 eps PolyGamma[0, -z1 - z2] + eps PolyGamma[0, -z2] - 
 
>          2 eps PolyGamma[0, 1 + z2] - eps PolyGamma[0, 1 + z1 + z2] + 
 
>          eps PolyGamma[0, 2 + z1 + z2])) / 
 
>      (Gamma[3 - z2] Gamma[2 + z1 + z2]), 
 
>     {{eps -> 0}, {z1 -> -0.430071, z2 -> -0.146351}}], 
 
             2                                                       2
>    MBint[(s  Gamma[-z1] Gamma[2 + z1] Gamma[1 - z1 - z2] Gamma[-z2]  
 
                                     2
>        Gamma[z2] Gamma[1 + z1 + z2]  
 
>        (-1 - 2 eps + 2 eps Log[-s] + eps PolyGamma[0, 1 - z1] + 
 
>          eps PolyGamma[0, 2 + z1] - 3 eps PolyGamma[0, 3 - z2] + 
 
>          2 eps PolyGamma[0, 1 - z1 - z2] + eps PolyGamma[0, -z2] - 
 
>          2 eps PolyGamma[0, z2] - eps PolyGamma[0, 1 + z1 + z2] + 
 
>          eps PolyGamma[0, 2 + z1 + z2])) / 
 
>      (2 Gamma[3 - z2] Gamma[2 + z1 + z2]), 
 
>     {{eps -> 0}, {z1 -> -0.285712, z2 -> -0.233645}}], 
 
             2           2
>    MBint[(s  Gamma[-z1]  Gamma[4 + z1] Gamma[1 - z2] Gamma[-z1 - z2] 
 
                                                    2
>        Gamma[-z2] Gamma[1 + z2] Gamma[1 + z1 + z2]  
 
>        (-1 - 2 eps + 2 eps Log[-s] + eps PolyGamma[0, -z1] + 
 
>          eps PolyGamma[0, 4 + z1] + eps PolyGamma[0, 1 - z2] - 
 
>          3 eps PolyGamma[0, 4 - z2] + 2 eps PolyGamma[0, -z1 - z2] - 
 
>          2 eps PolyGamma[0, 1 + z2] - eps PolyGamma[0, 1 + z1 + z2] + 
 
>          eps PolyGamma[0, 2 + z1 + z2])) / 
 
>      (Gamma[1 - z1] Gamma[4 - z2] Gamma[2 + z1 + z2]), 
 
>     {{eps -> 0}, {z1 -> -0.234828, z2 -> -0.576688}}]}

In[12]:= 
In[12]:= Shifting contours...
Performing 7 lower-dimensional integrations with NIntegrate...1...2...3...4...5...6...7Higher-dimensional integrals
Preparing MBpart1eps1 (dim 2)
Preparing MBpart2eps1 (dim 2)
Preparing MBpart3eps1 (dim 2)
Preparing MBpart4eps1 (dim 2)
Preparing MBpart5eps0 (dim 2)
Preparing MBpart6eps0 (dim 2)
Preparing MBpart7eps0 (dim 2)
Preparing MBpart8eps0 (dim 2)
Running MBpart1eps1
Running MBpart2eps1
Running MBpart3eps1
Running MBpart4eps1
Running MBpart5eps0
Running MBpart6eps0
Running MBpart7eps0
Running MBpart8eps0

Out[12]//InputForm= 
{-178.1799706900294 - 
  7.56250000000172435545742075614406256568`15.954589770191005/eps^2 - 
  20.45058399892981858711725974504767900446`15.868670493443355/eps + 
  18.36451375075856*eps, {0.01450302313340337 + 0.013348548934498048*eps, 0}}

In[13]:= 
In[13]:= 
17.41user 0.32system 0:17.76elapsed 99%CPU (0avgtext+0avgdata 0maxresident)k
0inputs+0outputs (0major+125682minor)pagefaults 0swaps
