In this part has this identity and I could not understand, why factor 4, for me
$$k^{\mu}k^{\nu}=g^{\mu\nu}k²$$
Since $$k²=k_{\mu}k^{\mu}.$$
Unless the notation is the same but the product is different,Could someone explain to me why(the context is anomalous magnetic moment of electron)
Reference: M.D. Schwartz, Quantum Field Theory and the Standard Model; page 338.
Schwartz is using the fact that under the integration sign $$\int_{\mathbb{R}^d}\! \frac{d^dk}{(2\pi)^d} f(k^2) k^{\mu}k^{\nu} ~=~\int_{\mathbb{R}^d}\! \frac{d^dk}{(2\pi)^d} f(k^2) k^2 g^{\mu\nu}/d,$$ where $d$ is the spacetime dimension, cf. e.g. Ref. 2, this related Phys.SE post, and above comment by Oбжорoв.
References:
M.D. Schwartz, QFT & the standard model, 2014; section 17.2.
M.E. Peskin & D.V. Schroeder, An Intro to QFT; eq. (6.46).