Renormalization group running of the parity operator in Lorentz-violating quantum field theory

In conventional relativistic quantum field theory, the discrete operators <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi>C</a:mi> </a:mrow> </a:math> , <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mrow> <c:mi>P</c:mi> </c:mrow> </c:math> , and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mrow> <e:mi>T</e:mi> </e:mrow> </e:math> are matrix operators with no renormalization scale dependence. However, in a Lorentz-violating theory with a fermion <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msup> <g:mi>f</g:mi> <g:mi>μ</g:mi> </g:msup> </g:math> term in the action, these operators may acquire nontrivial renormalization group behavior. Because the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msup> <i:mi>f</i:mi> <i:mi>μ</i:mi> </i:msup> </i:math> term may actually be exchanged in the action for an equivalent <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msup> <k:mi>c</k:mi> <k:mrow> <k:mi>ν</k:mi> <k:mi>μ</k:mi> </k:mrow> </k:msup> </k:math> term, the scale dependence depends explicitly on the renormalization scheme, even at one-loop order. The scheme dependence means it is always possible to set the scale dependence parameter <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:mi>X</m:mi> </m:math> to zero, but for analyses of some high-energy electron-photon processes, using a scheme with <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>X</o:mi> <o:mo>=</o:mo> <o:mn>0</o:mn> </o:math> —and thus definite scale dependences for <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>C</q:mi> </q:math> , <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>P</s:mi> </s:math> , and <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>T</u:mi> </u:math> —may nonetheless be more convenient.