In the textbook, usually the fermion mass renormalization is introduced as follows: the mass shift [itex]\delta m[/itex] must vanish when [itex]m_0=0[/itex]. The mass shift must therefore be proportional to [itex]m_0[/itex]. By dimensional analysis, it can only depend logarithmically on [itex]\Lambda[/itex] (the ultraviolet cutoff): [itex]\delta m \sim \ln \Lambda[/itex]. Here I don't understand how the "dimensional analysis" is performed, can somebody explain it in detail? Thanks in advance.