This is how to derive the beta functions of string theory sigma model!!
P.S.: if you’re interested in the Background Field Method as it relates to one-loop Yang-Mills beta functions and asymptotic freedom of QCD, see Sections 16.6 and 16.7 of Peskin and Schroeder.
(P.P.S.: If you’re a real masochist, you can also peruse some personal notes on the background field method that I made as a graduate student approximately a decade and a half ago…)
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