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 last modified 18 Jun 2010 In[3]:= In[3]:= MBnum v.0.1, last modified: 18.06.09 In[4]:= In[4]:= In[5]:= In[5]:= >>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, m, n5] PR[k2 + p1 + p2, m, n6] > PR[k2 + p1 + p2 + p4, 0, n7], N/A]} F polynomial during this iteration 2 2 > m FX[X[2] + X[3]] - PR[k1, m] X[1] X[2] - > PR[k1 + p1 + p2, m] 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 - z1 - z4 2 z1 z4 > {INT[{1}, ((-1) (m ) (-s) Gamma[-z1] Gamma[-z2] > Gamma[-z3] Gamma[2 - eps - n5 - n6 - n7 - z1 - z4] > Gamma[2 - eps - n4 - n5 - n6 - z1 - z2 - z3 - z4] Gamma[-z4] > Gamma[-2 + eps + n4 + n5 + n6 + n7 + z1 + z2 + z3 + z4] > Gamma[n6 + 2 z1 + z3 + z4 - z5] Gamma[-z5] Gamma[-2 z1 + z5] > Gamma[n5 + z2 + z4 + z5]) / > (Gamma[n4] Gamma[n5] Gamma[n6] Gamma[4 - 2 eps - n4 - n5 - n6 - n7] > Gamma[n7] Gamma[-2 z1]), > PR[k1, m, n1 - z2] PR[k1 + p1, 0, n2] PR[k1 + p1 + p2, m, n3 - z3] > PR[k1 + p1 + p2 + p4, 0, > -2 + eps + n4 + n5 + n6 + n7 + z1 + z2 + z3 + z4], N/A]} F polynomial during this iteration 2 2 > m FX[X[1] + X[3]] - s X[1] X[3] - t X[2] X[4] >>Contracting and finalizing output --contracting... --finalizing output... >>Checking Barnes 1-st lemma... n1 + n2 + n3 + n4 + n5 + n6 + n7 2 z1 + z6 z4 + z7 Out[5]= {((-1) (m ) (-s) -2 eps - n1 - n2 - n3 - n4 - n5 - n6 - n7 - z1 - z4 - z6 - z7 4 > (-t) t > Gamma[-z1] Gamma[-z2] Gamma[-z3] > Gamma[2 - eps - n5 - n6 - n7 - z1 - z4] > Gamma[2 - eps - n4 - n5 - n6 - z1 - z2 - z3 - z4] Gamma[-z4] > Gamma[n5 + z2 + z4] Gamma[n6 + z3 + z4] > Gamma[n5 + n6 + 2 z1 + z2 + z3 + 2 z4] Gamma[-z6] > Gamma[2 - eps - n1 - n2 - n3 + z2 + z3 - z6 - z7] > Gamma[4 - 2 eps - n1 - n3 - n4 - n5 - n6 - n7 - z1 - z4 - z6 - z7] > Gamma[-z7] Gamma[n1 - z2 + z7] Gamma[n3 - z3 + z7] > Gamma[-4 + 2 eps + n1 + n2 + n3 + n4 + n5 + n6 + n7 + z1 + z4 + z6 + > z7] Gamma[n1 + n3 - z2 - z3 + 2 z6 + 2 z7]) / > (Gamma[n2] Gamma[n4] Gamma[n5] Gamma[n6] > Gamma[4 - 2 eps - n4 - n5 - n6 - n7] Gamma[n7] Gamma[n1 - z2] > Gamma[n3 - z3] Gamma[6 - 3 eps - n1 - n2 - n3 - n4 - n5 - n6 - n7 - > z1 - z4] Gamma[n5 + n6 + z2 + z3 + 2 z4] > Gamma[n1 + n3 - z2 - z3 + 2 z7])} In[6]:= In[6]:= In[7]:= In[7]:= repr={((-1)^(n1 + n2 + n3 + n4 + n5 + n6 + n7)*(m^2)^(z1 + z6)*(-s)^(z4 + z7)*(-t)^(-2*eps - n1 - n2 - n3 - n4 - n5 - n6 - n7 - z1 - z4 - z6 - z7)*t^4*Gamma[-z1]*Gamma[-z2]*Gamma[-z3]*Gamma[2 - eps - n5 - n6 - n7 - z1 - z4]*Gamma[2 - eps - n4 - n5 - n6 - z1 - z2 - z3 - z4]*Gamma[-z4]*Gamma[n5 + z2 + z4]*Gamma[n6 + z3 + z4]*Gamma[n5 + n6 + 2*z1 + z2 + z3 + 2*z4]*Gamma[-z6]*Gamma[2 - eps - n1 - n2 - n3 + z2 + z3 - z6 - z7]*Gamma[4 - 2*eps - n1 - n3 - n4 - n5 - n6 - n7 - z1 - z4 - z6 - z7]*Gamma[-z7]*Gamma[n1 - z2 + z7]*Gamma[n3 - z3 + z7]*Gamma[-4 + 2*eps + n1 + n2 + n3 + n4 + n5 + n6 + n7 + z1 + z4 + z6 + z7]*Gamma[n1 + n3 - z2 - z3 + 2*z6 + 2*z7])/(Gamma[n2]*Gamma[n4]*Gamma[n5]*Gamma[n6]*Gamma[4 - 2*eps - n4 - n5 - n6 - n7]*Gamma[n7]*Gamma[n1 - z2]*Gamma[n3 - z3]*Gamma[6 - 3*eps - n1 - n2 - n3 - n4 - n5 - n6 - n7 - z1 - z4]*Gamma[n5 + n6 + z2 + z3 + 2*z4]*Gamma[n1 + n3 - z2 - z3 + 2*z7])} Length=1 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 z1 + z6 -2 - z1 - z6 3 Out[7]= {2.748907, {MBint[-((m ) (-s) Gamma[-z1] 3 > Gamma[1 + z1] Gamma[-z6] Gamma[1 + z6] 2 2 2 2 > (-6 + 6 eps EulerGamma + 7 eps Pi - 2 2 2 > 12 eps EulerGamma Log[-s] + 6 eps Log[-s] + 12 eps Log[-t] + 2 2 > 12 eps EulerGamma Log[-t] - 12 eps Log[-s] Log[-t] - 2 2 2 2 > 6 eps Log[-t] + 12 eps PolyGamma[0, -2 z1] + 2 2 > 12 eps PolyGamma[0, -z1] - 3 eps PolyGamma[0, 1 + z1] + 2 > 6 eps EulerGamma PolyGamma[0, 1 + z1] - 2 > 6 eps Log[-s] PolyGamma[0, 1 + z1] + 2 > 12 eps Log[-t] PolyGamma[0, 1 + z1] + 2 2 > 3 eps PolyGamma[0, 1 + z1] - 2 > 12 eps PolyGamma[0, -z1] > (EulerGamma - Log[-s] + Log[-t] + PolyGamma[0, 1 + z1]) + 2 > 12 eps PolyGamma[0, -2 z1] > (EulerGamma - Log[-s] + Log[-t] - 2 PolyGamma[0, -z1] + > PolyGamma[0, 1 + z1]) + 2 > 12 eps EulerGamma PolyGamma[0, -2 z6] - 2 > 12 eps Log[-s] PolyGamma[0, -2 z6] + 2 > 12 eps Log[-t] PolyGamma[0, -2 z6] + 2 2 > 12 eps PolyGamma[0, -2 z6] - 2 > 12 eps EulerGamma PolyGamma[0, -z6] + 2 > 12 eps Log[-s] PolyGamma[0, -z6] - 2 > 12 eps Log[-t] PolyGamma[0, -z6] - 2 > 24 eps PolyGamma[0, -2 z6] PolyGamma[0, -z6] + 2 2 > 12 eps PolyGamma[0, -z6] + 3 eps PolyGamma[0, 1 + z6] + 2 > 6 eps EulerGamma PolyGamma[0, 1 + z6] - 2 > 6 eps Log[-s] PolyGamma[0, 1 + z6] + 2 > 12 eps PolyGamma[0, -2 z6] PolyGamma[0, 1 + z6] - 2 > 12 eps PolyGamma[0, -z6] PolyGamma[0, 1 + z6] + 2 2 2 > 3 eps PolyGamma[0, 1 + z6] - 12 eps PolyGamma[1, -2 z1] + 2 2 > 6 eps PolyGamma[1, -z1] + 3 eps PolyGamma[1, 1 + z1] - 2 2 > 12 eps PolyGamma[1, -2 z6] + 6 eps PolyGamma[1, -z6] + 2 > 3 eps PolyGamma[1, 1 + z6])) / 2 > (12 eps t Gamma[-2 z1] Gamma[-2 z6]), 1 21 > {{eps -> 0}, {z1 -> -(-), z6 -> -(--)}}], 2 64 2 z1 + z6 -2 - z1 - z6 > MBint[((m ) (-s) Gamma[-z1] Gamma[1 + z1] > Gamma[-z1 - z2] Gamma[-z2] Gamma[z2] Gamma[-z1 + z2] > Gamma[-z2 - z6] Gamma[z2 - z6] Gamma[-z6] Gamma[1 + z6] > (-1 + 2 eps Log[-t])) / > (2 eps t Gamma[-2 z1] Gamma[1 - z2] Gamma[1 + z2] Gamma[-2 z6]), 1 1 21 > {{eps -> 0}, {z1 -> -(-), z2 -> -(-), z6 -> -(--)}}], 2 4 64 2 z1 + z6 z4 - z6 -2 - z1 - z4 > MBint[((m ) (-s) (-t) Gamma[-z1] 2 2 > Gamma[-1 - z1 - z4] Gamma[-z4] Gamma[1 + z4] 3 > Gamma[2 (1 + z1 + z4)] Gamma[2 + z1 + z4] Gamma[-z6] Gamma[1 + z6]) > / (s Gamma[2 + 2 z4] Gamma[-2 z6]), 1 5 21 > {{eps -> 0}, {z1 -> -(-), z4 -> -(--), z6 -> -(--)}}], 2 32 64 2 z1 + z6 -2 - z1 + z7 -2 - z6 - z7 3 > MBint[((m ) (-s) s (-t) Gamma[-z1] 2 > Gamma[1 + z1] Gamma[-z6] Gamma[-1 - z6 - z7] Gamma[-z7] 2 > Gamma[1 + z7] Gamma[2 (1 + z6 + z7)] Gamma[2 + z6 + z7]) / > (Gamma[-2 z1] Gamma[2 + 2 z7]), 1 21 17 > {{eps -> 0}, {z1 -> -(-), z6 -> -(--), z7 -> -(--)}}], 2 64 64 2 z1 + z6 -2 - z1 - z6 > MBint[((m ) (-s) Gamma[-z1] Gamma[-z2] > Gamma[-1 - z1 - z4] Gamma[-1 - 2 z1 - z2 - z4] Gamma[-z4] > Gamma[1 + z1 + z4] Gamma[1 + z2 + z4] Gamma[2 + 2 z1 + z2 + 2 z4] > Gamma[-1 - z1 - z2 - z4 - z6] Gamma[1 + z1 + z2 + z4 - z6] > Gamma[-z6] Gamma[2 + z1 + z4 + z6]) / > (2 t Gamma[-2 z1] Gamma[1 - z2] Gamma[3 + 2 z1 + z2 + 2 z4] > Gamma[-2 z6]), {{eps -> 0}, 1 1 5 21 > {z1 -> -(-), z2 -> -(-), z4 -> -(--), z6 -> -(--)}}]}} 2 4 32 64 In[8]:= In[8]:= before={2.748907`6.890705040698474, > {MBint[-((m^2)^(z1 + z6)*(-s)^(-2 - z1 - z6)*Gamma[-z1]^3*Gamma[1 + z1]* > Gamma[-z6]^3*Gamma[1 + z6]* > (-6 + 6*eps^2*EulerGamma^2 + 7*eps^2*Pi^2 - > 12*eps^2*EulerGamma*Log[-s] + 6*eps^2*Log[-s]^2 + > 12*eps*Log[-t] + 12*eps^2*EulerGamma*Log[-t] - > 12*eps^2*Log[-s]*Log[-t] - 6*eps^2*Log[-t]^2 + > 12*eps^2*PolyGamma[0, -2*z1]^2 + 12*eps^2*PolyGamma[0, -z1]^2 - > 3*eps*PolyGamma[0, 1 + z1] + > 6*eps^2*EulerGamma*PolyGamma[0, 1 + z1] - > 6*eps^2*Log[-s]*PolyGamma[0, 1 + z1] + > 12*eps^2*Log[-t]*PolyGamma[0, 1 + z1] + > 3*eps^2*PolyGamma[0, 1 + z1]^2 - > 12*eps^2*PolyGamma[0, -z1]* > (EulerGamma - Log[-s] + Log[-t] + PolyGamma[0, 1 + z1]) + > 12*eps^2*PolyGamma[0, -2*z1]* > (EulerGamma - Log[-s] + Log[-t] - 2*PolyGamma[0, -z1] + > PolyGamma[0, 1 + z1]) + > 12*eps^2*EulerGamma*PolyGamma[0, -2*z6] - > 12*eps^2*Log[-s]*PolyGamma[0, -2*z6] + > 12*eps^2*Log[-t]*PolyGamma[0, -2*z6] + > 12*eps^2*PolyGamma[0, -2*z6]^2 - > 12*eps^2*EulerGamma*PolyGamma[0, -z6] + > 12*eps^2*Log[-s]*PolyGamma[0, -z6] - > 12*eps^2*Log[-t]*PolyGamma[0, -z6] - > 24*eps^2*PolyGamma[0, -2*z6]*PolyGamma[0, -z6] + > 12*eps^2*PolyGamma[0, -z6]^2 + 3*eps*PolyGamma[0, 1 + z6] + > 6*eps^2*EulerGamma*PolyGamma[0, 1 + z6] - > 6*eps^2*Log[-s]*PolyGamma[0, 1 + z6] + > 12*eps^2*PolyGamma[0, -2*z6]*PolyGamma[0, 1 + z6] - > 12*eps^2*PolyGamma[0, -z6]*PolyGamma[0, 1 + z6] + > 3*eps^2*PolyGamma[0, 1 + z6]^2 - 12*eps^2*PolyGamma[1, -2*z1] + > 6*eps^2*PolyGamma[1, -z1] + 3*eps^2*PolyGamma[1, 1 + z1] - > 12*eps^2*PolyGamma[1, -2*z6] + 6*eps^2*PolyGamma[1, -z6] + > 3*eps^2*PolyGamma[1, 1 + z6]))/ > (12*eps^2*t*Gamma[-2*z1]*Gamma[-2*z6]), > {{eps -> 0}, {z1 -> -1/2, z6 -> -21/64}}], > MBint[((m^2)^(z1 + z6)*(-s)^(-2 - z1 - z6)*Gamma[-z1]*Gamma[1 + z1]* > Gamma[-z1 - z2]*Gamma[-z2]*Gamma[z2]*Gamma[-z1 + z2]* > Gamma[-z2 - z6]*Gamma[z2 - z6]*Gamma[-z6]*Gamma[1 + z6]* > (-1 + 2*eps*Log[-t]))/ > (2*eps*t*Gamma[-2*z1]*Gamma[1 - z2]*Gamma[1 + z2]*Gamma[-2*z6]), > {{eps -> 0}, {z1 -> -1/2, z2 -> -1/4, z6 -> -21/64}}], > MBint[((m^2)^(z1 + z6)*(-s)^(z4 - z6)*(-t)^(-2 - z1 - z4)*Gamma[-z1]* > Gamma[-1 - z1 - z4]^2*Gamma[-z4]*Gamma[1 + z4]^2* > Gamma[2*(1 + z1 + z4)]*Gamma[2 + z1 + z4]*Gamma[-z6]^3*Gamma[1 + z6] > )/(s*Gamma[2 + 2*z4]*Gamma[-2*z6]), > {{eps -> 0}, {z1 -> -1/2, z4 -> -5/32, z6 -> -21/64}}], > MBint[((m^2)^(z1 + z6)*(-s)^(-2 - z1 + z7)*s*(-t)^(-2 - z6 - z7)* > Gamma[-z1]^3*Gamma[1 + z1]*Gamma[-z6]*Gamma[-1 - z6 - z7]^2* > Gamma[-z7]*Gamma[1 + z7]^2*Gamma[2*(1 + z6 + z7)]*Gamma[2 + z6 + z7] > )/(Gamma[-2*z1]*Gamma[2 + 2*z7]), > {{eps -> 0}, {z1 -> -1/2, z6 -> -21/64, z7 -> -17/64}}], > MBint[((m^2)^(z1 + z6)*(-s)^(-2 - z1 - z6)*Gamma[-z1]*Gamma[-z2]* > Gamma[-1 - z1 - z4]*Gamma[-1 - 2*z1 - z2 - z4]*Gamma[-z4]* > Gamma[1 + z1 + z4]*Gamma[1 + z2 + z4]*Gamma[2 + 2*z1 + z2 + 2*z4]* > Gamma[-1 - z1 - z2 - z4 - z6]*Gamma[1 + z1 + z2 + z4 - z6]* > Gamma[-z6]*Gamma[2 + z1 + z4 + z6])/ > (2*t*Gamma[-2*z1]*Gamma[1 - z2]*Gamma[3 + 2*z1 + z2 + 2*z4]* > Gamma[-2*z6]), {{eps -> 0}, > {z1 -> -1/2, z2 -> -1/4, z4 -> -5/32, z6 -> -21/64}}]}} In[9]:= In[9]:= Shifting contours... -2 Power::infy: Infinite expression 0. encountered. FindMinimum::nrgnum: The gradient is not a vector of real numbers at {z1, z6} = {-0.5, -0.328125}. -2 Power::infy: Infinite expression 0. encountered. FindMinimum::nrgnum: The gradient is not a vector of real numbers at {z1, z6} = {-0.5, -0.328125}. -2 Power::infy: Infinite expression 0. encountered. General::stop: Further output of Power::infy will be suppressed during this calculation. FindMinimum::nrgnum: The gradient is not a vector of real numbers at {z1, z6} = {-0.5, -0.328125}. General::stop: Further output of FindMinimum::nrgnum will be suppressed during this calculation. ReplaceAll::reps: 1 {z1, -(-)} is neither a list of replacement rules nor a valid dispatch 2 table, and so cannot be used for replacing. ReplaceAll::reps: 21 {z6, -(--)} is neither a list of replacement rules nor a valid dispatch 64 table, and so cannot be used for replacing. Minimum search failed -2 - z1 - z6 3 3 MBint[(5 Gamma[-z1] Gamma[1 + z1] Gamma[-z6] Gamma[1 + z6] 2 2 2 > (6 EulerGamma + 7 Pi - 12 EulerGamma Log[5] + 6 Log[5] + 2 > 12 EulerGamma Log[7] - 12 Log[5] Log[7] - 6 Log[7] + 2 2 > 12 PolyGamma[0, -2 z1] + 12 PolyGamma[0, -z1] + > 6 EulerGamma PolyGamma[0, 1 + z1] - 6 Log[5] PolyGamma[0, 1 + z1] + 2 > 12 Log[7] PolyGamma[0, 1 + z1] + 3 PolyGamma[0, 1 + z1] - > 12 PolyGamma[0, -z1] (EulerGamma - Log[5] + Log[7] + > PolyGamma[0, 1 + z1]) + > 12 PolyGamma[0, -2 z1] > (EulerGamma - Log[5] + Log[7] - 2 PolyGamma[0, -z1] + > PolyGamma[0, 1 + z1]) + 12 EulerGamma PolyGamma[0, -2 z6] - > 12 Log[5] PolyGamma[0, -2 z6] + 12 Log[7] PolyGamma[0, -2 z6] + 2 > 12 PolyGamma[0, -2 z6] - 12 EulerGamma PolyGamma[0, -z6] + > 12 Log[5] PolyGamma[0, -z6] - 12 Log[7] PolyGamma[0, -z6] - 2 > 24 PolyGamma[0, -2 z6] PolyGamma[0, -z6] + 12 PolyGamma[0, -z6] + > 6 EulerGamma PolyGamma[0, 1 + z6] - 6 Log[5] PolyGamma[0, 1 + z6] + > 12 PolyGamma[0, -2 z6] PolyGamma[0, 1 + z6] - > 12 PolyGamma[0, -z6] PolyGamma[0, 1 + z6] + 2 > 3 PolyGamma[0, 1 + z6] - 12 PolyGamma[1, -2 z1] + > 6 PolyGamma[1, -z1] + 3 PolyGamma[1, 1 + z1] - > 12 PolyGamma[1, -2 z6] + 6 PolyGamma[1, -z6] + 3 PolyGamma[1, 1 + z6] > )) / (84 Gamma[-2 z1] Gamma[-2 z6]), 1 21 > {{eps -> 0}, {z1 -> -(-), z6 -> -(--)}}] 2 64 Performing 0 lower-dimensional integrations with NIntegrateHigher-dimensional integrals Preparing MBpart1eps0 (dim 4) Preparing MBpart2eps0 (dim 3) Preparing MBpart3eps0 (dim 3) Preparing MBpart4eps0 (dim 3) Preparing MBpart5eps0 (dim 2) Preparing MBpart6eps-1 (dim 3) Preparing MBpart7eps-1 (dim 2) Preparing MBpart8eps-2 (dim 2) Running MBpart1eps0 Running MBpart2eps0 Running MBpart3eps0 Running MBpart4eps0 Running MBpart5eps0 Running MBpart6eps-1 Running MBpart7eps-1 Running MBpart8eps-2 0.023524 0.0635884 Out[9]= {37.863253, {0.22464 - -------- + ---------, 2 eps eps -6 -6 2.09158 10 7.68737 10 > {0.0000206495 + ------------ + ------------, 0}}} 2 eps eps In[10]:= In[10]:= 125.30user 2.70system 0:41.64elapsed 307%CPU (0avgtext+0avgdata 0maxresident)k 0inputs+0outputs (2major+134216minor)pagefaults 0swaps