Q1: My question is, in the context of dimensional regularisation(DREG, in dimension $d$), why do they mention the absence of $d=2$ pole in the gauge theory cases[1], whereas the $d=2$ pole is not discussed about in $\lambda\phi^4$ theory[2]?
For details, refer to Peskin & Schroeder's "An introduction to Quantum Field Theory":
in [1], page 251 (Section 7.5), in calculation of the vacuum polarisation diagram in QED,
$$...\:\sim\:\Gamma(2-\frac{d}{2})g^{\mu\nu}...$$ We would have expected a pole at $d=2$, since the quadratic divergence in $ dimensions becomes a logarithmic divergence in 2 dimensions. But the pole cancels. The Ward identity is working.
also page 525 (Section 16.5) , One-Loop divergences of Non-Abelian Gauge Theory--The gauge Boson Self-Energy:
Now we are ready to put these results together. In the sum of the three diagrams, The coefficient of $\Gamma(1-\frac{d}{2})g^{\mu\nu}...$ is
$$...=(1-\frac{d}{2})(d-2).$$ The first factor cancels the pole of the gamma function at $d=2$. Thus, the sum of the three diagrams has no quadratic divergence and no gauge boson mass renormalisation.
in [2]. However, in page 328 (Section 10.2), calculating the self-energy diagram in $\lambda\phi^4$ theory, there is the contribution:
$$\Gamma(1-\frac{d}{2})...$$
which has a pole at $d=2$, and no comment is made there.
Q2: I've always thought that in DREG, as long as the $d=4$ divergences are canceled, everything is fine, so what's the significance in talking about $d=2$ divergence?
Q3: Is there any serious problem to overlook the $d=2$ pole, for example, in $\lambda\phi^4$ theory?
Q4: By the way, is the "fine-tunning problem" related to the $d=2$ pole in DREG, if yes, how?
