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