- #1
jys34
- 7
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Hi, I'm studying about Renormalization group.
I have a question about mass beta-function.
Usually, when we perform one-loop calculation to get counter-term coefficients,
resulting RG coefficient for mass scaling is given by
[itex]\mu \frac{d m}{d \mu} = a_1 (e ) m [/itex]
and [itex]a_1(e)[/itex] is come from divergent loop diagrams (with 1/epsilon divergence)
but mass itself is dimensionful, and I thought that it must scale not only at quantum-level but also tree-level, too.
so I expect that
[itex]\mu \frac{d m}{d \mu} = m + a_1 (e ) m [/itex]
because mass has +1 mass dimension and irrelevant under momentum scaling.
but, on the other hand, it is weird that masses in non-interacting theory also should be scaled.how do we treat RG scaling (or calculation of beta-functions) of dimensionful parameters in dimensional regularization? (I think that I have certain mistakes in above procedure..)
p.s.
Is it possible to change the result if we consider about some derivative-type background field
[itex]A_\mu^{ext} = \partial_\mu \phi(x) [/itex]
instead of mass? (here phi(x) is some dimensionless non-dynamical parameter)
I have a question about mass beta-function.
Usually, when we perform one-loop calculation to get counter-term coefficients,
resulting RG coefficient for mass scaling is given by
[itex]\mu \frac{d m}{d \mu} = a_1 (e ) m [/itex]
and [itex]a_1(e)[/itex] is come from divergent loop diagrams (with 1/epsilon divergence)
but mass itself is dimensionful, and I thought that it must scale not only at quantum-level but also tree-level, too.
so I expect that
[itex]\mu \frac{d m}{d \mu} = m + a_1 (e ) m [/itex]
because mass has +1 mass dimension and irrelevant under momentum scaling.
but, on the other hand, it is weird that masses in non-interacting theory also should be scaled.how do we treat RG scaling (or calculation of beta-functions) of dimensionful parameters in dimensional regularization? (I think that I have certain mistakes in above procedure..)
p.s.
Is it possible to change the result if we consider about some derivative-type background field
[itex]A_\mu^{ext} = \partial_\mu \phi(x) [/itex]
instead of mass? (here phi(x) is some dimensionless non-dynamical parameter)
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