The Schwinger model is the 2d QED with massless fermions. An important result about it (which I would like to understand) is that this is a gauge invariant theory which contains a free massive vector particle.
The original article by Schwinger Gauge invariance and mass, II, Phys. Review, 128, number 5 (1962), is too concise for me.
QUESTION: Is there a more detailed/modern exposition of the above result?
Best Answer
You can take a look at Zinn-Justin, "Quantum field theory and critical phenomena", section 31.4 in 3rd edition.
Ok after some days of thinking i belive i've solved both problems, so i tought i just answer my self for future reference. (verba volant scripta manent)
For the practical problem: When i wrote the parametrization for the four momentums of the particles involved in the scattering i did wrongly assume that the gluon's momentum $K_A$ should have been corrected in order to account for it's virtuality. What did evade me was that the virtuality of the gluon was already codified in the quark and antiquark momentums $K_B$ and $K_C$ via the term $p^2_\perp$. Infact let's write $K_A=(p,0,0,p)$ (as if it was real) and the other two as in the question above. Then from conservation of energy and by taking the square we have: $$K^2_A=(K_B+K_C)^2=\frac{p^2_\perp}{2z(1-z)}+O(p^4_\perp)$$ so, since $K^2_A\neq0 $ we see that we have already accounted for the gluon having a small virtuality which comes from the parametrization of the quarks four momenta. There is no need then to further modify $K_A$ in order to account for it's vituality. Furthermore the substitution i made in the question effectively cancelled the virtuality of the gluon! (like i added it in $K_B$ and $K_C$ and subtracted it in $K_A$)
If we try to do the calculation of: (the prevoius relationships of the momenta are mostly false now!) $$\frac{8}{K_A \cdot n}[(K_C \cdot K_A)(K_B \cdot n)-(K_C \cdot K_B)(K_A \cdot n)+(K_B \cdot K_A)(K_C \cdot n)]$$
now we get, knowing that:
$$K_A \cdot n=p\\K_B \cdot n=zp+\frac{p^2_\perp}{2zp}\\K_C \cdot n=(1-z)p+\frac{p^2_\perp}{2(1-z)p}\\K_A \cdot K_B=\frac{p^2_\perp}{2z}\\K_A \cdot K_C=\frac{p^2_\perp}{2(1-z)}\\K_B \cdot K_C=p^2_\perp+\frac{z \ p^2_\perp}{2(1-z)}
+\frac{(1-z) \ p^2_\perp}{2z}$$
exactly the result it should have come: $$\frac{8}{K_A \cdot n}[(K_C \cdot K_A)(K_B \cdot n)-(K_C \cdot K_B)(K_A \cdot n)+(K_B \cdot K_A)(K_C \cdot n)]=-8p^2_\perp+O(p^4_\perp)$$
To the conceptual side: In non abelian gauge theory the non physical degrees of freedom of the gluon do not cancel themeselves when calculating scattering amplitudes as in QED. A way to say this is: it depends upon the fact that the underyling gauge symmetries are different and that causes a modification to the generating functional and so to the ward identities. Indeed that's why there are the ghost fields that have the exact role of eliminating the non physical degrees of freedom.
Now then in order to obtain the correct physical amplitude we should account for the ghost contributions to the process. This isn't very convenient when calculating such an easy process as a tree level amplitude, it's much much easier to manually cut the non physical degrees of freedom by replacing the full polarization sum with only the transverse one.
Best Answer
You can take a look at Zinn-Justin, "Quantum field theory and critical phenomena", section 31.4 in 3rd edition.