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I'm currently working on the renormalisation part in Peskin, Schroeder QFT. There it is stated that non-logarithmic UV divergences give a mass renormalisation and thus are forbidden, e.g. for the propagator of the Photon in QED or the gauge boson in YM by gauge invariance. Another example is that chirality in QED implies that the electron propagator has only log divergence.

While I fully understand this argument, I am not aware of how the divergence structure of the corresponding diagrams affect the mass renormalisation, i.e. how non-log divergences give a real mass renormalisation and why log-divergences don't affect the physical mass.

Given for example the Photon renormalisation, there the structure of the renormalised propagator is

\begin{equation} \frac{i}{q^2 \left(1-\Pi(q^2)\right)}\left(\eta^{\mu \nu}q^2 - \frac{q^{\mu}q^{\nu}}{q^2}\right) \end{equation}

which can be derived alone from Lorentzinvariance and the Ward identity. Here it is absolutely clear why the physical mass is not affected. But how can I see that the propagator will have this form solely from the fact that it is only log divergent ?

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A mass term in the photon self-energy tensor would look like $Ag^{\mu\nu}$, where $A$ approaches a constant as $q^{2}\rightarrow0$. What is important is not the degree of divergence of $A$, but the Lorentz structure of the term. In particular, a term of the form $Bq^{2}g^{\mu\nu}$ is not a mass term; when contracted with the external fields $A^{\mu}$ and $A^{\nu}$ and transformed back to real space, it gives a term in the effective action that looks like $A^{\mu}\partial^{2}A_{\mu}$ (whereas a mass term would involve $\frac{1}{2}m_{\gamma}^{2}A^{2}$).

The Ward identity ensures that the the self-energy is proportional to $C(q^{2}g^{\mu\nu}-q^{\mu}q^{\nu})$. It is this Lorentz structure (with no $q$-independent term), not the degree of divergence, that ensures there is no photon mass. Sandwiching this self-energy tensor between external fields, transforming back to position space, and integrating by parts gives the contribution of this term to the effective action, which has the same $F^{\mu\nu}F_{\mu\nu}$ form as the usual Maxwell action.

However, the structure guaranteed by the Ward identity also ensures that there is no quadratic divergence. For dimensional reasons, each factor of the external momentum $q$ that appears in the self-energy means that the possible factor of the loop momentum did not appear. Having a self-energy with two powers of $q$ means that two possible powers of the loop momentum $\ell$ do not appear in the numerator of the integral, reducing the degree of divergence of the integration by two.

There is one caveat. $C$ is a function of $q^{2}$, and if it has a pole at $q^{2}=0$, both of the above arguments break down. As Peskin and Schroeder state, it is possible to prove that the pole does not exist, but the argument is not simple, and it is specific to 3+1 dimensions. (In 1+1 dimensions, with massless fermions, there is a pole, and so there is a radiatively generated photon mass.)

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