Massive On-shell QED One-loop Box

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

DateString[]

$Version

Derivation of MB representation

<<MB.m

<<AMBREv1.2.m

(* The Exp[eps EulerGamma] is added later *)

boxB4l2m=Fullintegral[{1},{PR[k1,m,n1]PR[k1+p1,0,n2]PR[k1+p1+p2,m,n3]PR[k1-p3,0,n4]},{k1}];
invariants={(p1^2)->(m^2),(p2^2)->(m^2),(p3^2)->(m^2),p1 p2->-(m^2)+(s/2),p1 p3->-(m^2)+(t/2),p2 p3->-(-(m^2)+(s/2)+(t/2))};

IntPart[1]

repr=SubLoop[integral];

fin=repr/.{n1->1,n2->1,n3->1,n4->1}

(* Appling Barnes-Lemmas *)

fin=BarnesLemma[fin,1]

(* x=(m^2)/t, y=s/t *)

fin=fin/.((((m^2)))^z1) (((-s))^z2) (((-t))^(-2-eps-z1-z2))->(((-x))^z1) (((y))^z2) (((-t))^(-2-eps))

(* Finding rules and applying MBcontinue *)

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

integrals=MBcontinue[fin,eps->0,Rules ]//MBmerge//Simplify

(* next line, B4l2mMBrepres, is the general MB-representation at arbitrary, small eps for B4l2m *)

B4l2mMBrepres= integrals

?MBexpand

B4l2mMBexpand=MBexpand[B4l2mMBrepres,Exp[eps EulerGamma],{eps,0,1}]//MBmerge

?ToString

ToString[B4l2mMBexpand,InputForm,PageWidth->60]

Lk = {x->-1/10,y->1/50,t->-10};

MBintegrate[B4l2mMBexpand,Lk]

Derivation of sums

<<MBsums.v1.0.m

dim1int=B4l2mMBexpand[[1]]

dim1sum =MBIntToSum[dim1int,Lk,{z1->L}]

ToString[dim1sum,InputForm,PageWidth->60]

num1=MBintegrate[{dim1int},Lk]//N

num2=DoAllMBSums[dim1sum,50,Lk]//N

num1[[1]]-num2

dim2int=B4l2mMBexpand[[2]]

dim2sum =MBIntToSum[dim2int,Lk,{z2->L,z1->L}]

dim2sum =MBIntToSum[dim2int,{t->-10},{z1->L,z2->L}]

dim2sum =MBIntToSum[dim2int,Lk,{z1->L,z2->L}]

MBintegrate[{dim2int},Lk]

DoAllMBSums[dim2sum,50,Lk]//N

ToString[dim2sum,InputForm,PageWidth->60]