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