Massive On-shell Planar Double Box

(* Time of evaluation and the version of the Mathematica kernel used *)

DateString[]

$Version

Derivation of MB representation

(* fin from http://prac.us.edu.pl/~gluza/ambre/examples/example7.nb *)

<<MB.m

fin=-(((((m^2)))^(z1+z6)) (((-s))^(z4+z7)) (((-t))^(-3-2 eps-z1-z4-z6-z7)) Gamma[-z1] Gamma[-z2] Gamma[-z3] Gamma[-1-eps-z1-z4] Gamma[-1-eps-z1-z2-z3-z4] Gamma[-z4] Gamma[1+z2+z4] Gamma[1+z3+z4] Gamma[2+2 z1+z2+z3+2 z4] Gamma[-z6] Gamma[-1-eps+z2+z3-z6-z7] Gamma[-2-2 eps-z1-z4-z6-z7] Gamma[-z7] Gamma[1-z2+z7] Gamma[1-z3+z7] Gamma[3+2 eps+z1+z4+z6+z7] Gamma[2-z2-z3+2 z6+2 z7])/(Gamma[-2 eps] Gamma[1-z2] Gamma[1-z3] Gamma[-1-3 eps-z1-z4] Gamma[2+z2+z3+2 z4] Gamma[2-z2-z3+2 z7])

rules=MBoptimizedRules[fin,eps->0,{},{eps}]

integrals=MBcontinue[fin,eps->0,rules,Verbose->False];

ser=MBexpand[{integrals},Exp[2*eps EulerGamma],
{eps,0,-2}]//MBmerge

Derivation of sums

<<MBsums.v1.0.m

int1=ser[[1]]/.((((m^2)))^(z1+z6)) (((-s))^(-2-z1-z6))-> (((-x))^(z1+z6)) (((-s))^(-2))(* (m^2)/s=x *)

Lk = {s->-1/5,x->-5,t->-1/10};

s1 =MBIntToSum[int1,Lk,{z1->R,z6->R}]

sx=Sum[(((((-1))^(-n1-n2)) (((-x))^(-n1-n2)) (((n1!))^2) (((n2!))^2))/(2 (eps^2) (s^2) t (x^2) (1+2 n1)! (1+2 n2)!)),{n1,0,Infinity},{n2,0,Infinity}]

MBintegrate[{int1},Lk]

DoAllMBSums[s1,5,Lk]//N

sx/.Lk//N

(* Numerical check with V.A.Smirnov (B0, compare Eq. 5 in hep-ph/0111160v1) *)

(* x = m^2/s *)

x0=1/Sqrt[1-4*x]

mx=Log[1-x0]
px=Log[1+x0]

b2=2*(mx-px)^2

B0=-x0^2/(s^2*(-t)^(1+2*eps))*(b2/eps^2+rest)

B0S=Series[B0,{eps,0,-2}]//Normal

B0S/.Lk//N

sx/.Lk//N

ToString[int1,InputForm,PageWidth->60]

ToString[s1,InputForm,PageWidth->60]