# $d=2$ pole argument of quadratic divergences in Peskin & Schroeder's book

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? • I assume the Ward identity mentioned states that there should be no divergences somehow, and if this Ward identity is valid for$d=2$as well, the cancellation of divergences in$d=2$is a check on the result.$\lambda\phi^4$has fewer identities restricting divergences. Q2: That's the case when only considering the theory at$d=4\$. Sometimes it is useful to consider other dimensionalities to understand better what's going on. – David Vercauteren Dec 17 '14 at 7:07