input_case nil$ in "loop.red"$ % photon self energy % ------------------ on gcd; operator V,r; for all p let V(p)=g(f,p),r(p)=g(f,p)+m; vector p,k; index n; let p.p=-m^2*s; z:=-i*4*pi*4/(d-1)/p.p*ex(-E*log(2))* Feyn(k,V(n)*r(k+p)*V(n)*r(k),{{p,m},{0*p,m}})$ z:=Fint(exe(z))$ let atan(s/(sqrt(s)*sqrt(-s-4)))=sqrt(s+4)/sqrt(-s-4)/2 *log((sqrt(s+4)+sqrt(s))/(sqrt(s+4)-sqrt(s))); factor log; on div; 3*pi*z; clear atan(s/(sqrt(s)*sqrt(-s-4))); remfac log; % electron self energy % -------------------- nospur f; let abs(m)=m; z:=-i*(4*pi)^2*Feyn(k,V(n)*r(k+p)*V(n),{{p,m},{}})$ remind n; spur f; z1:=z*V(n)/p.n$ z2:=z/m$ z1a:=sub(s=-1,z1)$ z2a:=sub(s=-1,z2)$ % on mass shell z1b:=sub(s=-1,-df(z1,s))$ z2b:=sub(s=-1,-df(z2,s))$ % derivatives z1:=Fint(exe(z1))$ j!!:=1$ z2:=Fint(exe(z2))$ let log(m^2*s+m^2)=log(s+1)+log(m^2); z1:=z1$ z2:=z2$ clear log(m^2*s+m^2); factor log; on div; z1; z2; remfac log; % Exact calculation of integrals in d dimensions operator f; z1a:=z1a*x(1)^(2*E)*f(-2*E)$ z2a:=z2a*x(1)^(2*E)*f(-2*E)$ z1b:=z1b*x(1)^(1+2*E)*f(-1-2*E)$ z2b:=z2b*x(1)^(1+2*E)*f(-1-2*E)$ for all n let x(1)*f(n)=f(n+1); z1a:=z1a$ z2a:=z2a$ z1b:=z1b$ z2b:=z2b$ for all n clear x(1)*f(n); for all n let f(n)=1/(n+1); z1a:=z1a; z2a:=z2a; z1b:=z1b; z2b:=z2b; for all n clear f(n); z1a:=exe(z1a); z2a:=exe(z2a); z1b:=exe(z1b); z2b:=exe(z2b); clear z1,z2,z1a,z2a,z1b,z2b; % vertex function % --------------- vector p1,p2; index l; nospur f; off gcd; off div; let p1.p1=-m^2*s1,p2.p2=-m^2*s2,p1.p2=m^2/2*(t-s1-s2); % General vertex function (not printed to save space) z:=-i*(4*pi)^2* Feyn(k,V(l)*r(k+p2)*V(n)*r(k+p1)*V(l),{{p1,m},{p2,m},{}})$ s1:=-1$ s2:=-1$ % both initial and final electrons are on shell % Projectors to single out form factors operator x1,x2; v1:=-x1(2)/(2*m)*(p1+p2).n+(x1(1)+x1(2))*g(f,n)$ v2:=sub(x1=x2,v1)$ spur f; index n; vv:=v1*r(p1)*v2*r(p2)$ matrix mm(2,2),mv(2,1); for i:=1:2 do for j:=1:2 do mm(i,j):=df(vv,x1(i),x2(j)); nospur f; remind n; for i:=1:2 do mv(i,1):=df(v1,x1(i)); on gcd; mv:=mm^(-1)*mv$ Pf1:=mv(1,1); Pf2:=mv(2,1); clear mm,mv; clear v1,v2,vv; % Form factors spur f; index n; off gcd; z1:=r(p1)*Pf1*r(p2)*z$ z2:=r(p1)*Pf2*r(p2)*z$ clear Pf1,Pf2; on gcd; z1:=z1; z2:=z2; % Ultraviolet divergence of F1 z1:=exe(E*z1)$ off gcd; z1:=Fint(Fint(z1))$ on div; z1; z2:=exe(z2); z0:=sub(t=0,z2)$ % F2 j!!:=2$ z2:=Fint(z2)$ let atan((x(1)*t+2)/(sqrt(t)*sqrt(-t-4)*x(1)))= sqrt(t+4)/sqrt(-t-4)/2 *log((2/x(1)+t+sqrt(t)*sqrt(t+4)) /(2/x(1)+t-sqrt(t)*sqrt(t+4))), atan((t+2)/(sqrt(t)*sqrt(-t-4)))=sqrt(t+4)/sqrt(-t-4)/2 *log((2+t+sqrt(t)*sqrt(t+4))/(2+t-sqrt(t)*sqrt(t+4))); z2:=z2$ clear atan((x(1)*t+2)/(sqrt(t)*sqrt(-t-4)*x(1))), atan((t+2)/(sqrt(t)*sqrt(-t-4))); z2:=Fint(z2)$ let atan(t/(sqrt(t)*sqrt(-t-4)))=sqrt(t+4)/sqrt(-t-4)/2 *log((sqrt(t+4)+sqrt(t))/(sqrt(t+4)-sqrt(t))), log((-sqrt(t)*sqrt(t+4)-t-2)/(sqrt(t)*sqrt(t+4)-t-2))= 2*log((sqrt(t+4)+sqrt(t))/(sqrt(t+4)-sqrt(t))); z2:=z2; % F2 clear atan(t/(sqrt(t)*sqrt(-t-4))), log((-sqrt(t)*sqrt(t+4)-t-2)/(sqrt(t)*sqrt(t+4)-t-2)); % anomalous magnetic moment j!!:=2$ z0:=Fint(Fint(z0)); clear z1,z2,z0; % axial anomaly Fint(exe((d-4)/d*Feyn(k,k.k,{{0*k,a,3}}))); end;