Schoonschip, 68000 version of June 27, 1991. Public version.
Date: Fri Aug  9 1991 00:35:29.  Memory: start 0001B84C, length 476348.

Command line: +1 Anomaly.e Xax


C Anomaly 1. Evaluation of the coefficients C(i,j), i=1,2, in terms
     of C0 and B0.

> P ninput

Begin. Time 1 sec.
 
c11 = 
 + Det^-1*c0
  * ( 1/2*pDp*pDk + 1/2*pDp*kDk )
 
 - c0
 
 + B0(pDp,m)*Det^-1
  * ( - 1/2*pDp )
 
 + B0(kDk,m)*Det^-1
  * ( - 1/2*pDk )
 
 + B0(qDq,m)*Det^-1
  * ( 1/2*pDp + 1/2*pDk )
 
c12 = 
 + Det^-1*c0
  * ( - 1/2*pDp*kDk - 1/2*pDk*kDk )
 
 + B0(pDp,m)*Det^-1
  * ( 1/2*pDk )
 
 + B0(kDk,m)*Det^-1
  * ( 1/2*kDk )
 
 + B0(qDq,m)*Det^-1
  * ( - 1/2*pDk - 1/2*kDk ) + 0.
 
 
c24 = 
 + 1/4*i*Pi^2
 
 + Det^-1*c0
  * ( - 1/4*pDp*pDk*kDk - 1/8*pDp*kDk^2 - 1/8*pDp^2*kDk )
 
 + c0
  * ( - 1/2*m^2 )
 
 + B0(pDp,m)*Det^-1
  * ( 1/8*pDp*pDk + 1/8*pDp*kDk )
 
 + B0(kDk,m)*Det^-1
  * ( 1/8*pDp*kDk + 1/8*pDk*kDk )
 
 + 1/4*B0(qDq,m)
 
 + B0(qDq,m)*Det^-1
  * ( - 1/8*pDp*pDk - 1/4*pDp*kDk - 1/8*pDk*kDk ) + 0.
 
 
c21 = 
 + i*Pi^2*Det^-1
  * ( - 1/4*pDp )
 
 + Det^-2*c0
  * ( 3/4*pDp^2*pDk*kDk + 3/8*pDp^2*kDk^2 + 3/8*pDp^3*kDk )
 
 + Det^-1*c0
  * ( 1/2*m^2*pDp - pDp*pDk - pDp*kDk - 1/4*pDp^2 )
 
 + c0
 
 + B0(pDp,m)*Det^-2
  * ( - 3/8*pDp^2*pDk - 3/8*pDp^2*kDk )
 
 + B0(pDp,m)*Det^-1
  * ( pDp )
 
 + B0(kDk,m)*Det^-2
  * ( - 3/8*pDp*pDk*kDk - 3/8*pDp^2*kDk )
 
 + B0(kDk,m)*Det^-1
  * ( 1/4*pDp + 3/4*pDk )
 
 + B0(qDq,m)*Det^-2
  * ( 3/8*pDp*pDk*kDk + 3/8*pDp^2*pDk + 3/4*pDp^2*kDk )
 
 + B0(qDq,m)*Det^-1
  * ( - 5/4*pDp - 3/4*pDk )
 
c23 = 
 + i*Pi^2*Det^-1
  * ( 1/4*pDk )
 
 + Det^-2*c0
  * ( - 3/8*pDp*pDk*kDk^2 - 3/8*pDp^2*pDk*kDk - 3/4*pDp^2*kDk^2 )
 
 + Det^-1*c0
  * ( - 1/2*m^2*pDk + pDp*kDk + 1/2*pDk*kDk )
 
 + B0(pDp,m)*Det^-2
  * ( 3/8*pDp*pDk*kDk + 3/8*pDp^2*kDk )
 
 + B0(pDp,m)*Det^-1
  * ( - 1/8*pDp - 1/2*pDk )
 
 + B0(kDk,m)*Det^-2
  * ( 3/8*pDp*pDk*kDk + 3/8*pDp*kDk^2 )
 
 + B0(kDk,m)*Det^-1
  * ( - 5/8*kDk )
 
 + B0(qDq,m)*Det^-2
  * ( - 3/4*pDp*pDk*kDk - 3/8*pDp*kDk^2 - 3/8*pDp^2*kDk )
 
 + B0(qDq,m)*Det^-1
  * ( 1/8*pDp + 1/2*pDk + 5/8*kDk )
 
c22 = 
 + i*Pi^2*Det^-1
  * ( - 1/4*kDk )
 
 + Det^-2*c0
  * ( 3/4*pDp*pDk*kDk^2 + 3/8*pDp*kDk^3 + 3/8*pDp^2*kDk^2 )
 
 + Det^-1*c0
  * ( 1/2*m^2*kDk - 1/4*kDk^2 )
 
 + B0(pDp,m)*Det^-2
  * ( - 3/8*pDp*pDk*kDk - 3/8*pDp*kDk^2 )
 
 + B0(pDp,m)*Det^-1
  * ( - 1/4*pDk + 1/4*kDk )
 
 + B0(kDk,m)*Det^-2
  * ( - 3/8*pDp*kDk^2 - 3/8*pDk*kDk^2 )
 
 + B0(qDq,m)*Det^-2
  * ( 3/8*pDp*pDk*kDk + 3/4*pDp*kDk^2 + 3/8*pDk*kDk^2 )
 
 + B0(qDq,m)*Det^-1
  * ( 1/4*pDk - 1/4*kDk ) + 0.
 
> P input

C Check: must be zero.

	Z Diff23=c23-C23p
	*end
 
Diff23 = 0. + 0.
 

End run. Time 2 sec.
Schoonschip, 68000 version of June 27, 1991. Public version.
Date: Fri Aug  9 1991 00:35:33.  Memory: start 0001B84C, length 476348.


C Anomaly 2. The axial current triangle graphs. Vertex:  i*G5*G(al).

C The triangle anomaly.
  Computing the axial current.
  The expressions for the C(i,j) have been computed separately,
  and are contained in the block CFU.

> P ninput
> P input

	V p,r,q,k
	I al,mu,nu,L1,L2,L3
	A N,N_,Pi,m,Ax,Det
	F Fx,Cx,C,c,C0,Bx,B0,B1

C  Triangle graphs.
   (k,al) => (p,mu),(q,nu) with all momenta pointing inwards.
   There are two graphs, differing with respect to each other by
   reversal of the fermion direction (or by the interchange p <=> q
   and mu <=> nu).

	Z A(al,mu,nu) =
	 - i*Cx(m)*G5(1,2)*G(2,3,al)*
			   (i*G(3,4,k) + i*G(3,4,r) + m*Gi(3,4))
	 		  *G(4,5,mu)*
			   (i*G(5,6,r) + i*G(5,6,k) + i*G(5,6,p) + m*Gi(5,6))
			  *G(6,7,nu)*
			   (i*G(7,1,r) + m*Gi(7,1))*Ax

	 - i*Cx(m)*G5(1,2)*G(2,3,al)*
			   (- i*G(3,4,r) + m*Gi(3,4))
	 		  *G(4,5,nu)*
			   (- i*G(5,6,r) - i*G(5,6,k) - i*G(5,6,p) + m*Gi(5,6))
			  *G(6,7,mu)*
			   (- i*G(7,1,r) - i*G(7,1,k) + m*Gi(7,1))*Ax

L 2	Id,Gammas,"C
	*yep

C Try to work out terms with three r, to avoid the C(3,i).
  Such terms have more than one r inside the trace.
  Move them towards each other, to produce rDr
  Do this by moving them to the right.

L 1	Id,G(1,"t,"4,G5,al,r,L1~,L2~,L3~,L4~) =
	 - G(1,"t,"4,G5,al,L1,r,L2,L3,L4)
	 + 2*D(L1,r)*G(1,"t,"4,G5,al,L2,L3,L4)
L 2	Id,G(1,"t,"4,G5,al,L2~,r,L1~,L3~,L4~) =
	 - G(1,"t,"4,G5,al,L2,L1,r,L3,L4)
	 + 2*D(L1,r)*G(1,"t,"4,G5,al,L2,L3,L4)
L 3	Id,G(1,"t,"4,G5,al,L2~,L3~,r,L1~,L4~) =
	 - G(1,"t,"4,G5,al,L2,L3,L1,r,L4)
	 + 2*D(L1,r)*G(1,"t,"4,G5,al,L2,L3,L4)
L 4	Id,G(1,"t,"4,G5,al,L2~,L3~,L4~,r,L1~) =
	 - G(1,"t,"4,G5,al,L2,L3,L4,L1,r)
	 + 2*D(L1,r)*G(1,"t,"4,G5,al,L2,L3,L4)
L 5	Id,Gammas,"C
L13	Id,G(1,"t,"4,G5,al,r,L1~,L2~,L3~,L4~) =
	 - G(1,"t,"4,G5,al,L1,r,L2,L3,L4)
	 + 2*D(L1,r)*G(1,"t,"4,G5,al,L2,L3,L4)
L14	Id,G(1,"t,"4,G5,al,L2~,r,L1~,L3~,L4~) =
	 - G(1,"t,"4,G5,al,L2,L1,r,L3,L4)
	 + 2*D(L1,r)*G(1,"t,"4,G5,al,L2,L3,L4)
L15	Id,G(1,"t,"4,G5,al,L2~,L3~,r,L1~,L4~) =
	 - G(1,"t,"4,G5,al,L2,L3,L1,r,L4)
	 + 2*D(L1,r)*G(1,"t,"4,G5,al,L2,L3,L4)
L16	Id,G(1,"t,"4,G5,al,L2~,L3~,L4~,r,L1~) =
	 - G(1,"t,"4,G5,al,L2,L3,L4,L1,r)
	 + 2*D(L1,r)*G(1,"t,"4,G5,al,L2,L3,L4)
L17	Id,Gammas,"C
	*yep

C There may still be some traces with two r's left.

L 1	Id,G(1,"t,"4,G5,al,r,L1~,L2~) =
	 - G(1,"t,"4,G5,al,L1,r,L2)
L 2	Id,Gammas,"C
	*yep

C Work out the rDr terms, writing rDr = rDr+m^2 - m^2.
  In the resulting two point function one has
  1/((r+k)^2+m^2)*((r+p+k)^2+m^2). Shift r => r-k.

L 1	IF Cx(m)*rDr=Bx(m)*Shift - Cx(m)*m^2
L 2	IF Shift=1
L 3	Id,r(mu~)=r(mu)-k(mu)
L 3	Al,Func,r(mu~)=r(mu)-k(mu)
L 3	Al,Dotpr,r(mu~)=r(mu)-k(mu)
	ENDIF
	ENDIF
L 4	Id,Gammas,"C
	*yep

C Now do the integration over the loop momentum r.

L 1	Id,All,r,N,Fx

L 2	Id,Adiso,Cx(m)*Fx(L1~,L2~)=
	 + k(L1)*k(L2)*C(2,1,m) + p(L1)*p(L2)*C(2,2,m)
	 + k(L1)*p(L2)*C(2,3,m) + p(L1)*k(L2)*C(2,3,m)
	 + D(L1,L2)*C(2,4,m)
L 3	Id,Adiso,Cx(m)*Fx(L1~) = k(L1)*C(1,1,m) + p(L1)*C(1,2,m)
L 3	Al,Adiso,Bx(m)*Fx(L1~) = p(L1)*B1(pDp,m)
L 4	Id,Cx(m)= C0(m)
L 4	Al,Bx(m)= B0(pDp,m)

L 5	Id,Gammas
	*yep

C Work out the C(i,j).

> CFU{}
> P ninput
> P input

L 3	Id,B1(x~,m) = -0.5*B0(x,m)

L 4	Id,p(al)*Epf(mu,nu,la~,ka~)=
	    p(mu)*Epf(al,nu,la,ka)
	  + p(nu)*Epf(mu,al,la,ka)
	  + p(la)*Epf(mu,nu,al,ka)
	  + p(ka)*Epf(mu,nu,la,al)
L 4	Al,k(al)*Epf(mu,nu,la~,ka~)=
	    k(mu)*Epf(al,nu,la,ka)
	  + k(nu)*Epf(mu,al,la,ka)
	  + k(la)*Epf(mu,nu,al,ka)
	  + k(ka)*Epf(mu,nu,la,al)

L 5	Id,Multi,pDk^2=kDk*pDp-Det
L 6	Id,Det=kDk*pDp-pDk^2

	*yep

C Write the result, i.e. the axial current, in the more standard form.

L 1	Id,k(mu~)=-q(mu)-p(mu)
L 2	Id,Func,k(mu~)=-q(mu)-p(mu)
L 3	Id,Dotpr,k(mu~)=-q(mu)-p(mu)
L 4	Id,Multi,pDq^2=qDq*pDp-Det
> P output
	*yep
 
A(al,mu,nu) = 
 + Epf(al,mu,nu,p)
  * ( 4*i*Pi^2*Ax )
 
 + Epf(al,mu,nu,q)
  * ( - 4*i*Pi^2*Ax )
 
 + Epf(al,mu,p,q)*p(nu)
  * ( - 4*i*Pi^2*Ax*Det^-1*pDq )
 
 + Epf(al,mu,p,q)*q(nu)
  * ( 4*i*Pi^2*Ax*Det^-1*pDp )
 
 + Epf(al,nu,p,q)*p(mu)
  * ( - 4*i*Pi^2*Ax*Det^-1*qDq )
 
 + Epf(al,nu,p,q)*q(mu)
  * ( 4*i*Pi^2*Ax*Det^-1*pDq )
 
 + C0(m)*Epf(al,mu,nu,p)
  * ( - 8*m^2*Ax + 2*Ax*Det^-1*pDp*qDq^2 - 2*Ax*Det^-1*pDp^2*qDq )
 
 + C0(m)*Epf(al,mu,nu,q)
  * ( 8*m^2*Ax + 2*Ax*Det^-1*pDp*qDq^2 - 2*Ax*Det^-1*pDp^2*qDq )
 
 + C0(m)*Epf(al,mu,p,q)*p(nu)
  * ( 8*m^2*Ax*Det^-1*pDq + 6*Ax*Det^-2*pDp*pDq*qDq^2 + 6*Ax*Det^-2
  *pDp^2*pDq*qDq + 12*Ax*Det^-2*pDp^2*qDq^2 - 8*Ax*Det^-1*pDp*qDq )
 
 + C0(m)*Epf(al,mu,p,q)*q(nu)
  * ( - 8*m^2*Ax*Det^-1*pDp - 12*Ax*Det^-2*pDp^2*pDq*qDq - 6*Ax*Det^-2
  *pDp^2*qDq^2 - 6*Ax*Det^-2*pDp^3*qDq + 8*Ax*Det^-1*pDp*pDq
  + 8*Ax*Det^-1*pDp*qDq + 4*Ax*Det^-1*pDp^2 )
 
 + C0(m)*Epf(al,nu,p,q)*p(mu)
  * ( 8*m^2*Ax*Det^-1*qDq + 12*Ax*Det^-2*pDp*pDq*qDq^2 + 6*Ax*Det^-2
  *pDp*qDq^3 + 6*Ax*Det^-2*pDp^2*qDq^2 - 8*Ax*Det^-1*pDp*qDq
  - 8*Ax*Det^-1*pDq*qDq - 4*Ax*Det^-1*qDq^2 )
 
 + C0(m)*Epf(al,nu,p,q)*q(mu)
  * ( - 8*m^2*Ax*Det^-1*pDq - 6*Ax*Det^-2*pDp*pDq*qDq^2 - 6*Ax*Det^-2
  *pDp^2*pDq*qDq - 12*Ax*Det^-2*pDp^2*qDq^2 + 8*Ax*Det^-1*pDp*qDq )
 
 + B0(pDp,m)*Epf(al,mu,nu,p)
  * ( 2*Ax*Det^-1*pDp*pDq - 2*Ax*Det^-1*pDp*qDq )
 
 + B0(pDp,m)*Epf(al,mu,nu,q)
  * ( 2*Ax*Det^-1*pDp*pDq - 2*Ax*Det^-1*pDp*qDq )
 
 + B0(pDp,m)*Epf(al,mu,p,q)*p(nu)
  * ( - 6*Ax*Det^-2*pDp*pDq*qDq - 6*Ax*Det^-2*pDp^2*qDq + 2*Ax*Det^-1
  *pDp )
 
 + B0(pDp,m)*Epf(al,mu,p,q)*q(nu)
  * ( 6*Ax*Det^-2*pDp^2*pDq + 6*Ax*Det^-2*pDp^2*qDq - 8*Ax*Det^-1*pDp )
 
 + B0(pDp,m)*Epf(al,nu,p,q)*p(mu)
  * ( - 6*Ax*Det^-2*pDp*pDq*qDq - 6*Ax*Det^-2*pDp*qDq^2 + 4*Ax*Det^-1
  *pDq + 4*Ax*Det^-1*qDq )
 
 + B0(pDp,m)*Epf(al,nu,p,q)*q(mu)
  * ( 6*Ax*Det^-2*pDp*pDq*qDq + 6*Ax*Det^-2*pDp^2*qDq - 2*Ax*Det^-1
  *pDp )
 
 + B0(qDq,m)*Epf(al,mu,nu,p)
  * ( 2*Ax*Det^-1*pDp*qDq - 2*Ax*Det^-1*pDq*qDq )
 
 + B0(qDq,m)*Epf(al,mu,nu,q)
  * ( 2*Ax*Det^-1*pDp*qDq - 2*Ax*Det^-1*pDq*qDq )
 
 + B0(qDq,m)*Epf(al,mu,p,q)*p(nu)
  * ( - 6*Ax*Det^-2*pDp*pDq*qDq - 6*Ax*Det^-2*pDp*qDq^2 + 2*Ax*Det^-1
  *qDq )
 
 + B0(qDq,m)*Epf(al,mu,p,q)*q(nu)
  * ( 6*Ax*Det^-2*pDp*pDq*qDq + 6*Ax*Det^-2*pDp^2*qDq - 4*Ax*Det^-1
  *pDp - 4*Ax*Det^-1*pDq )
 
 + B0(qDq,m)*Epf(al,nu,p,q)*p(mu)
  * ( - 6*Ax*Det^-2*pDp*qDq^2 - 6*Ax*Det^-2*pDq*qDq^2 + 8*Ax*Det^-1
  *qDq )
 
 + B0(qDq,m)*Epf(al,nu,p,q)*q(mu)
  * ( 6*Ax*Det^-2*pDp*pDq*qDq + 6*Ax*Det^-2*pDp*qDq^2 - 2*Ax*Det^-1
  *qDq )
 
 + B0(kDk,m)*Epf(al,mu,nu,p)
  * ( - 2*Ax*Det^-1*pDp*pDq + 2*Ax*Det^-1*pDq*qDq )
 
 + B0(kDk,m)*Epf(al,mu,nu,q)
  * ( - 2*Ax*Det^-1*pDp*pDq + 2*Ax*Det^-1*pDq*qDq )
 
 + B0(kDk,m)*Epf(al,mu,p,q)*p(nu)
  * ( 12*Ax*Det^-2*pDp*pDq*qDq + 6*Ax*Det^-2*pDp*qDq^2 + 6*Ax*Det^-2
  *pDp^2*qDq - 2*Ax*Det^-1*pDp - 2*Ax*Det^-1*qDq )
 
 + B0(kDk,m)*Epf(al,mu,p,q)*q(nu)
  * ( - 6*Ax*Det^-2*pDp*pDq*qDq - 6*Ax*Det^-2*pDp^2*pDq - 12*Ax*Det^-2
  *pDp^2*qDq + 12*Ax*Det^-1*pDp + 4*Ax*Det^-1*pDq )
 
 + B0(kDk,m)*Epf(al,nu,p,q)*p(mu)
  * ( 6*Ax*Det^-2*pDp*pDq*qDq + 12*Ax*Det^-2*pDp*qDq^2 + 6*Ax*Det^-2
  *pDq*qDq^2 - 4*Ax*Det^-1*pDq - 12*Ax*Det^-1*qDq )
 
 + B0(kDk,m)*Epf(al,nu,p,q)*q(mu)
  * ( - 12*Ax*Det^-2*pDp*pDq*qDq - 6*Ax*Det^-2*pDp*qDq^2 - 6*Ax*Det^-2
  *pDp^2*qDq + 2*Ax*Det^-1*pDp + 2*Ax*Det^-1*qDq ) + 0.
 

C Take any one of the following three options.

L 1	Id,Ax=k(al)		! Taking the divergence of the axial current.
C Id,Ax=p(mu)		! Must be zero: gauge invariance.
C Id,Ax=q(nu)		! Must be zero: gauge invariance.
L 2	Id,k(mu~)=-q(mu)-p(mu)
L 3	Id,Func,k(mu~)=-q(mu)-p(mu)
L 4	Id,Dotpr,k(mu~)=-q(mu)-p(mu)
L 5	Id,Multi,pDq^2=qDq*pDp-Det
	*end
 
A(al,mu,nu) = + Epf(mu,nu,p,q)
  * ( 8*i*Pi^2 )
 
 + C0(m)*Epf(mu,nu,p,q)
  * ( - 16*m^2 ) + 0.
 

End run. Time 7 sec.
Schoonschip, 68000 version of June 27, 1991. Public version.
Date: Fri Aug  9 1991 00:35:42.  Memory: start 0001B84C, length 476348.


C Anomaly 3. The axial current triangle graphs. Vertex:  i*G5*G(al).
     An expansion in terms of the external momenta is done
     (assuming them to be small with respect to the loop mass m).
     The result shows that the axial current is at least of third
     order in the momenta.

C The triangle anomaly.
  Computing the axial current to order 3 in the external momenta.

	V p,r,q,k
	I al,mu,nu,L1,L2,L3,L4
	A N,N_,Pi,m,Ax,ax,Det,Den
	F Ln,Fx,Cx,C,c,C0,Bx,B0,B1
	X DY(L1,L2,L3,L4)=D(L1,L2)*D(L3,L4)+D(L1,L3)*D(L2,L4)+D(L1,L4)*D(L2,L3)
	X DZ(L1,L2,L3,L4,L5,L6)=
	   D(L1,L2)*DY(L3,L4,L5,L6)
	 + D(L1,L3)*DY(L2,L4,L5,L6)
	 + D(L1,L4)*DY(L3,L2,L5,L6)
	 + D(L1,L5)*DY(L3,L4,L2,L6)
	 + D(L1,L6)*DY(L3,L4,L5,L2)

C  Triangle graphs.
   (k,al) => (p,mu),(q,nu) with all momenta pointing inwards.
   There are two graphs, differing with respect to each other by
   reversal of the fermion direction (or by the interchange p <=> q
   and mu <=> nu).

	Z A(al,mu,nu) =
	 - i*Cx(m)*G5(1,2)*G(2,3,al)*
			   (i*G(3,4,k) + i*G(3,4,r) + m*Gi(3,4))
	 		  *G(4,5,mu)*
			   (i*G(5,6,r) + i*G(5,6,k) + i*G(5,6,p) + m*Gi(5,6))
			  *G(6,7,nu)*
			   (i*G(7,1,r) + m*Gi(7,1))*Ax

	 - i*Cx(m)*G5(1,2)*G(2,3,al)*
			   (- i*G(3,4,r) + m*Gi(3,4))
	 		  *G(4,5,nu)*
			   (- i*G(5,6,r) - i*G(5,6,k) - i*G(5,6,p) + m*Gi(5,6))
			  *G(6,7,mu)*
			   (- i*G(7,1,r) - i*G(7,1,k) + m*Gi(7,1))*Ax

L 2	Id,Gammas,"C

C Cx(m) is the three point function:
   1 / (r^2+m^2)*((r+k)^2+m^2)*((r+k+p)^2+m^2)
  Expand the second and third denominator, use k+p = -q. Below Den stands
  for 1/(r^2+m^2).

L10	Id,Cx(m)=Den^3*Dev2*Dev3
L11	Id,Dev2= 1 - (2*kDr+kDk)*Den + (2*kDr+kDk)^2*Den^2 - (2*kDr+kDk)^3*Den^3
L11	Al,Dev3= 1 - (-2*qDr+qDq)*Den + (-2*qDr+qDq)^2*Den^2 - (-2*qDr+qDq)^3*Den^3
L13	Id,Multi,Den^7=0
	*yep
L 1	Id,All,r,N,Fx

C Keep up to and including third order.

L 2	Id,Count,ax,"F,G,"F,Fx,k,1,q,1,p,1
L 3	Id,Multi,ax^4=0
L 4	Id,ax=1

C Doing the momentum integrals.

L 5	Id,Fx(L1~,L2~,L3~,L4~,L5~)=0
L 5	Al,Fx(L1~,L2~,L3~)=0
L 5	Al,Fx(L1~)=0

L 6	Id,Fx(L1~,L2~,L3~,L4~,L5~,L6~)*Den^6 =
	 DZ(L1,L2,L3,L4,L5,L6)*i*Pi^2/960/m^2
L 6	Al,Fx(L1~,L2~,L3~,L4~,L5~,L6~)*Den^5 =
	 DZ(L1,L2,L3,L4,L5,L6)*(i*DEL/192-i*Pi^2/192*Ln(m))
L 6	Al,Fx(L1~,L2~,L3~,L4~,L5~,L6~)*Den^4 =
	 DZ(L1,L2,L3,L4,L5,L6)*m^2*(-i*DEL/48+i*Pi^2/48*(-1+Ln(m)))

L11	Id,Fx(L1~,L2~,L3~,L4~)*Den^6 = i*Pi^2/480/m^4*DY(L1,L2,L3,L4)
L11	Al,Fx(L1~,L2~,L3~,L4~)*Den^5 = i*Pi^2/96/m^2*DY(L1,L2,L3,L4)
L11	Al,Fx(L1~,L2~,L3~,L4~)*Den^4 =
	 DY(L1,L2,L3,L4)*(i*DEL/24 - i*Pi^2/24*Ln(m))
L11	Al,Fx(L1~,L2~,L3~,L4~)*Den^3 =
	 DY(L1,L2,L3,L4)*m^2*(-i*DEL/8 + i*Pi^2/8*(-1 + Ln(m)))

L15	Id,Fx(L1~,L2~)*Den^5 = i*Pi^2/48/m^4*D(L1,L2)
L15	Al,Fx(L1~,L2~)*Den^4 = i*Pi^2/12/m^2*D(L1,L2)
L15	Al,Fx(L1~,L2~)*Den^3 = D(L1,L2)*(i*DEL/4 - i*Pi^2/4*Ln(m))

L17	Id,Den^5=i*Pi^2/12/m^6
L17	Al,Den^4=i*Pi^2/6/m^4
L17	Al,Den^3=i*Pi^2/2/m^2

	*yep

C Take the trace.

	B DEL,i,Pi
L 1	Id,Gammas
	*yep

L 1	Id,Multi,m^-4=0
L 2	Id,N=4+N_
L 3	Id,N_*DEL=-2*Pi^2
L 4	Id,N_=0
L 5	Id,k(mu~)=-q(mu)-p(mu)
L 5	Al,Func,k(mu~)=-q(mu)-p(mu)
L 5	Al,Dotpr,k(mu~)=-q(mu)-p(mu)
	*yep

L 1	Id,p(al)*Epf(mu,nu,la~,ka~)=
	    p(mu)*Epf(al,nu,la,ka)
	  + p(nu)*Epf(mu,al,la,ka)
	  + p(la)*Epf(mu,nu,al,ka)
	  + p(ka)*Epf(mu,nu,la,al)
L 1	Al,q(al)*Epf(mu,nu,la~,ka~)=
	    q(mu)*Epf(al,nu,la,ka)
	  + q(nu)*Epf(mu,al,la,ka)
	  + q(la)*Epf(mu,nu,al,ka)
	  + q(ka)*Epf(mu,nu,la,al)

> P output
	*yep
 
A(al,mu,nu) = 
 + Epf(al,mu,nu,p)*i*Pi^2
  * ( 2/3*m^-2*Ax*pDq + 4/3*m^-2*Ax*qDq )
 
 + Epf(al,mu,nu,q)*i*Pi^2
  * ( - 4/3*m^-2*Ax*pDp - 2/3*m^-2*Ax*pDq )
 
 + Epf(al,mu,p,q)*p(nu)*i*Pi^2
  * ( 2/3*m^-2*Ax )
 
 + Epf(al,mu,p,q)*q(nu)*i*Pi^2
  * ( 4/3*m^-2*Ax )
 
 + Epf(al,nu,p,q)*p(mu)*i*Pi^2
  * ( - 4/3*m^-2*Ax )
 
 + Epf(al,nu,p,q)*q(mu)*i*Pi^2
  * ( - 2/3*m^-2*Ax ) + 0.
 

C Take any one of the following three options.

C Id,Ax=k(al)		! Taking the divergence of the axial current.
C Id,Ax=p(mu)		! Must be zero: gauge invariance.
L 1	Id,Ax=q(nu)		! Must be zero: gauge invariance.
L 2	Id,k(mu~)=-q(mu)-p(mu)
L 3	Id,Func,k(mu~)=-q(mu)-p(mu)
L 4	Id,Dotpr,k(mu~)=-q(mu)-p(mu)
	*end
 
A(al,mu,nu) = 0. + 0.
 

End run. Time 13 sec.
Schoonschip, 68000 version of June 27, 1991. Public version.
Date: Fri Aug  9 1991 00:35:58.  Memory: start 0001B84C, length 476348.


C Anomaly 4. The pseudo scalar graphs. Vertex: 2*m*G5.

C The triangle anomaly.
  Computing the pseudo scalar triangle diagrams.

	V p,r,q,k
	I al,mu,nu,L1,L2,L3
	A N,N_,Pi,m,Ax,Det
	F Fx,Cx,C,c,C0,Bx,B0,B1

C  Triangle graphs.
   (k) => (p,mu),(q,nu) with all momenta pointing inwards.
   There are two graphs, differing with respect to each other by
   reversal of the fermion direction (or by the interchange p <=> q
   and mu <=> nu).

	Z A(al,mu,nu) =
	 -   Cx(m)*G5(1,3)*2*m*
			   (i*G(3,4,k) + i*G(3,4,r) + m*Gi(3,4))
	 		  *G(4,5,mu)*
			   (i*G(5,6,r) + i*G(5,6,k) + i*G(5,6,p) + m*Gi(5,6))
			  *G(6,7,nu)*
			   (i*G(7,1,r) + m*Gi(7,1))*Ax

	 -   Cx(m)*G5(1,3)*2*m*
			   (- i*G(3,4,r) + m*Gi(3,4))
	 		  *G(4,5,nu)*
			   (- i*G(5,6,r) - i*G(5,6,k) - i*G(5,6,p) + m*Gi(5,6))
			  *G(6,7,mu)*
			   (- i*G(7,1,r) - i*G(7,1,k) + m*Gi(7,1))*Ax

L 2	Id,Gammas,"C
	*yep

C Now do the integration over the loop momentum r.

L 1	Id,All,r,N,Fx

L 2	Id,Adiso,Cx(m)*Fx(L1~,L2~)=
	 + k(L1)*k(L2)*C(2,1,m) + p(L1)*p(L2)*C(2,2,m)
	 + k(L1)*p(L2)*C(2,3,m) + p(L1)*k(L2)*C(2,3,m)
	 + D(L1,L2)*C(2,4,m)
L 3	Id,Adiso,Cx(m)*Fx(L1~) = k(L1)*C(1,1,m) + p(L1)*C(1,2,m)

L 4	Id,Cx(m)= C0(m)

L 5	Id,Gammas

C Write the result in the more standard form.

L13	Id,k(mu~)=-q(mu)-p(mu)
L14	Id,Func,k(mu~)=-q(mu)-p(mu)
L15	Id,Dotpr,k(mu~)=-q(mu)-p(mu)
L16	Id,Multi,pDq^2=qDq*pDp-Det
	*end
 
A(al,mu,nu) = + C0(m)*Epf(mu,nu,p,q)
  * ( - 16*m^2*Ax ) + 0.
 

End run. Time 0 sec.
