Function definitions:

ϕ[c_,Λ_,x_,y_]:=κ/.FindRoot[((1+κ)/(1-κ))Exp[2 c κ]==(((((x-Λ))^2)+(y^2))/((((x+Λ))^2)+(y^2))),{κ,0.1}]
Ω[c_,Λ_,x_,y_]:=((4 (Λ^2))/(((1/(1-((ϕ[c,Λ,x,y])^2)))+c)((((x-Λ))^2)+(y^2))((((x+Λ))^2)+(y^2))))

(*Note that we have defined c=2π/e^2 here, and that we have set m=1.
We choose axes so that our partons sit at positions (Λ,0) and (-Λ,0).  *)

GraphicsRow[{Plot3D[Ω[0.0,2,x,y],{x,-6,6},{y,-6,6},BoxRatios->{1,1,1},AxesLabel->{"x","y","Ω"},BaseStyle->{FontFamily->"Helvetica",FontSize->14},PlotRange->All,PlotPoints->30,Mesh->False,Ticks->{Automatic,Automatic,{0,1}}],
Plot3D[Ω[0.5,2,x,y],{x,-6,6},{y,-6,6},BoxRatios->{1,1,1},AxesLabel->{"x","y","Ω"},BaseStyle->{FontFamily->"Helvetica",FontSize->14},PlotRange->All,PlotPoints->30,Mesh->False,Ticks->{Automatic,Automatic,{0,0.8}}],
Plot3D[Ω[1.5,2,x,y],{x,-6,6},{y,-6,6},BoxRatios->{1,1,1},AxesLabel->{"x","y","Ω"},BaseStyle->{FontFamily->"Helvetica",FontSize->14},PlotRange->All,PlotPoints->39,Mesh->False]},Spacings->Scaled[-0.1],ImageSize->1000]