Geodesic
Note on renormalization of quantum equation of motion
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 20, Tome 347 (2007), pp. 30-33

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Infinite part of one-prong vertex for matrix $\sigma$ model calculate in background field formalism in one-loop approach. Is shown that renormalizations of quantum equation of motion and effective action coincide if the additional factor has been introduced. 
@article{ZNSL_2007_347_a1,
     author = {A. A. Bagaev},
     title = {Note on  renormalization of quantum equation of motion},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {30--33},
     publisher = {mathdoc},
     volume = {347},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2007_347_a1/}
}
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A. A. Bagaev. Note on  renormalization of quantum equation of motion. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 20, Tome 347 (2007), pp. 30-33. http://geodesic.mathdoc.fr/item/ZNSL_2007_347_a1/