input_case nil$ in "loop.red"$ % Radiative correction to e+ e- -> hadrons % ---------------------------------------- % gamma* -> q anti-q mass p1=0,p2=0; mshell p1,p2; let p1.p2=s/2; index m,n; vector k; operator V,S; for all p let V(p)=g(f,p),S(p)=g(f,p)/p.p; % Matrix element squared in the tree approximation d0:=V(p1)*V(m)*V(-p2)*V(m); % One-loop correction d2:=V(p1)*V(n)*V(p1+k)*V(m)*V(-p2+k)*V(n)*V(-p2)*V(m)/d0; on gcd; % CF alphas/(4*pi) assumed d2:=-(4*pi)^2*i*Feyn(k,d2,{{p1},{-p2},{}}); Fint(Fint(exe(E*d2))); % ultraviolet divergence let (-s*x(1)*x(2))^E=ex(L*E)*x(1)^E*x(2)^E; d2:=d2$ clear (-s*x(1)*x(2))^E; % Calculation of the Feynman parameter integral operator f; d2:=d2*x(1)^(1+E)*x(2)^(1+E)*f(-1-E,-1-E)$ for all m,n let x(1)*f(m,n)=f(m+1,n),x(2)*f(m,n)=f(m,n+1); d2:=d2$ for all m,n clear x(1)*f(m,n),x(2)*f(m,n); for all m,n let f(m,n)=G1(1+m)*G1(1+n)/G1(3+m+n); d2:=d2; for all m,n clear f(m,n); on div; d2:=exe(d2); % expansion in e % Adding the conjugated contribution d2:=sub(L=L-i*pi,d2)$ d2:=d2+sub(i=-i,d2); off div; % gamma* -> q anti-q g mass k=0; mshell k; let p1.p2=s/2*(x1+x2-1),p1.k=s/2*(1-x2),p2.k=s/2*(1-x1); off gcd; d31:=-V(p1)*V(n)*S(p1+k)*V(m)*V(-p2)*V(m)*S(p1+k)*V(n)$ d32:=-V(p1)*V(n)*S(p1+k)*V(m)*V(-p2)*V(n)*S(-p2-k)*V(m)$ d3:=d31+d32$ clear d31,d32; d3:=d3+sub(x1=x2,x2=x1,d3)$ on gcd; d3:=d3/d0$ Ph2:=s^(-E)*ex(gam*E)*G0(-1)/(8*pi*G0(-2)*(1-2*E))$ x3:=2-x1-x2$ Ph3:=s^(1-2*E)*ex(2*gam*E)/(128*pi^3*(1-2*E)*G0(-2) *(1-x1)^E*(1-x2)^E*(1-x3)^E)$ let s^E=ex(L*E); d3:=(4*pi)^2*d3*Ph3/Ph2; clear s^E; clear Ph2,Ph3; % Calculation of the integral d3:=sub(x1=1-y1,x2=1-y2,d3)$ d3:=d3*y1^(1+E)*y2^(1+E)*(1-y1-y2)^E*f(-1-E,-1-E,-E)$ for all l,m,n let y1*f(l,m,n)=f(l+1,m,n),y2*f(l,m,n)=f(l,m+1,n); d3:=d3$ for all l,m,n clear y1*f(l,m,n),y2*f(l,m,n); for all l,m,n let f(l,m,n)=G1(1+l)*G1(1+m)*G1(1+n)/G1(3+l+m+n); d3:=d3; for all l,m,n clear f(l,m,n); on div; d3:=exe(d3); d:=d2+d3; off div; clear d2,d3,d; end;