Correction to four-loop RG functions in the two-dimensional lattice O(n)   $\sigma$-model
  We report the result of our evaluation of the Feynman diagrams appearing in the determination of the four-loop renormalization group functions in the two-dimensional lattice O($n$) $\sigma$-model by Caracciolo and Pelissetto. In the list of the integrals used for the determination of those functions, we find that three entries were not correctly evaluated. We give the values for them corrected by us including those for all other integrals which we computed with higher precision. These results are then applied to revise the determination of the second analytic correction to correlation length $\xi$ and spin susceptibility $\chi$ by Caracciolo et al. as well as our determination of the mass gap by means of a finite volume technique where we explicitly made use of the four-loop $\beta$-function. In both cases we find sizeable changes in predictions. In the meantime there appeared a paper by Alles et al. where they revised one finite integral in the list of our corrected integrals. After having taken the new revised value into consideration, we found that there are no noticeable changes in the perturbative predictions of the present paper including the final conclusions. 
Dong-Shin Shin