<< MB.m

<< AMBRE.m

Vertex with (k1.k1)^2 numerator

Fullintegral[{k1*k1, 
          k1*k1}, {PR[k1, m, n1]*PR[k1 + p1, 0, n2]*
            PR[k1 + p1 + p2, m, n3]}, {k1}];
invariants = {p1^2 -> m^2, p2^2 -> m^2, 
          p1*p2 -> s/2 - m^2};

IntPart[1]

repr = SubLoop[integral];

fin = Simplify[
            ARint[1, #] /. {n1 -> 1, n2 -> 1, n3 -> 1, 
                n4 -> 1}] & /@ repr

fin = BarnesLemma[#, 1] & /@ fin;

integrals = MBcontinue[#, eps -> 0, 
              MBoptimizedRules[#, eps -> 0, {}, {eps}], 
              Verbose -> False] & /@ fin;

ser = MBexpand[{#}, Exp[eps EulerGamma], {eps, 0, 0}] & /@ 
          integrals;

num = 
      Plus @@ (MBintegrate[#, {s -> -23, m -> 1, 
                  d -> 4}, Verbose -> False] & /@ 
            ser)

Numerator with general external momenta

Fullintegral[{k1*pa, 
          k1*pb}, {PR[k1, m, n1]*PR[k1 + p1, 0, n2]*
            PR[k1 + p1 + p2, m, n3]}, {k1}];
invariants = {p1^2 -> m^2, p2^2 -> m^2, 
          p1*p2 -> s/2 - m^2};

IntPart[1]

repr = SubLoop[integral];

fin = Simplify[
            ARint[1, #] /. {n1 -> 1, n2 -> 1, n3 -> 1, 
                n4 -> 1}] & /@ repr

fin = BarnesLemma[#, 1] & /@ fin;

fin = fin /.  {pa -> p1, pb -> p2} /. {p1 . p1 -> 
            m^2, p2 . p2 -> m^2, p1 . p2 -> s/2 - m^2, 
          p2 . p1 -> s/2 - m^2}

integrals = MBcontinue[#, eps -> 0, 
              MBoptimizedRules[#, eps -> 0, {}, {eps}], 
              Verbose -> False] & /@ fin;

ser = MBexpand[{#}, Exp[eps EulerGamma], {eps, 0, 0}] & /@ 
          integrals;

num = 
      Plus @@ (MBintegrate[#, {s -> -23, m -> 1}, 
                Verbose -> False] & /@ ser)

Numerator without external momenta

Fullintegral[{k1[mu1], k1[mu2], 
          k1[mu3]}, {PR[k1, m, n1]*PR[k1 + p1, 0, n2]*
            PR[k1 + p1 + p2, m, n3]}, {k1}];
invariants = {p1^2 -> m^2, p2^2 -> m^2, 
          p1*p2 -> s/2 - m^2};

IntPart[1]

repr = SubLoop[integral];

fin = Simplify[
            ARint[1, #] /. {n1 -> 1, n2 -> 1, n3 -> 1, 
                n4 -> 1}] & /@ repr

fin = BarnesLemma[#, 1] & /@ fin;

fin = Expand[p1[mu1]*g[mu2*mu3]*fin] //. 
                  g[mu1_*mu2_]*p_[mu2_] -> p[mu1] //. 
                p1_[n_]*p2_[n_] -> p1 . p2 /.  
              p1[_]^2 -> p1 . p1 /. p2[_]^2 -> p2 . p2 /. 
          g[__]^2 -> d /. {p1 . p1 -> m^2, p2 . p2 -> m^2, 
          p1 . p2 -> s/2 - m^2, p2 . p1 -> s/2 - m^2}

integrals = MBcontinue[#, eps -> 0, 
              MBoptimizedRules[#, eps -> 0, {}, {eps}], 
              Verbose -> False] & /@ fin;

ser = MBexpand[{#}, Exp[eps EulerGamma], {eps, 0, 0}] & /@ 
          integrals;

num = 
      Plus @@ (MBintegrate[#, {s -> -23, m -> 1, 
                  d -> 4}, Verbose -> False] & /@ 
            ser)