Running of the Higgs mu term (or: running of individual mass terms in a complicated mass matrix)

Question

I am wondering how to calculate the (one-loop) beta function for an individual mass term that appears in combination with a number of other mass terms in the coefficients of a number of fields. What I mean might be best illustrated by talking about some term in the Higgs potential in the MSSM, such as $\mu$ or $m_{H_u}$. Does one just go about the usual calculation using the Callan-Symanzik equation? If so, does one have to solve the equation for a high number of different correlation functions?

For instance, if

1. the term I'm interested in is $m$
2. $m$ appears as a coefficient of 4 (mass eigenstate) fields' quadratic terms
3. each of those 4 fields has one other quantity added with $m$ (so that each mass term looks like $\frac12(m^2+c_i^2)\phi_i^2$)

then does that mean I need to calculate at least $1+4\times1=5$ Green's functions?

More generally, if the mass for every particle multiplying $m$ is given by $m_i^2= m^2 + c_{ij} d_j^2$, where the $d_j$ are the other mass quantities, and $j$ runs from 1 to $n$, then do we necessarily need to calculate at least $n+1$ Green's functions? (Here $i$ counts the number of fields that appear multiplied by $m$, but this number doesn't matter, I think.) I am basically thinking that in calculating all the diagrams that will be needed to find $m$'s beta function, all of the $d_j$ will also appear, and that therefore we will need at least max$(j)+1$ equations.

Is there a reference where something like the RGE for the $\mu$ term in a supersymmetric model is worked out in some detail? Or the $A$ terms or explicit soft SUSY breaking masses? Or more generally, where the RGEs for some masses appearing in complicated combinations are worked out?

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More details:

I am interested in calculating the (one-loop) beta function for a 'vector mass' in a SUSY model. By vector mass, I mean something like $\mu_Q$ in a superpotential term

$\mu_Q \bar{Q} Q$,

where $Q, \bar{Q}$ are superfields in the fundamental and conjugate representations of the gauge group, respectively. This is obviously completely analogous to the supersymmetric Higgs mass term

$\mu H_u H_d$.

Like the Higgs $\mu$ term in the MSSM, in this model the $\mu_Q$ term will be found in combination with other mass terms in the relevant mass matrices. For instance, eq. 8.1.2 in Martin's A Supersymmetry Primer (http://arxiv.org/abs/hep-ph/9709356) shows the coefficients of the quadratic terms of the scalar fields $H_u^0, H_d^0$ are combinations of $\mu, m_{H_u}, m_{H_d}, b, g,g'$. And then there are the Higgsino mass matrices (or neutralino and chargino mass matrices, more accurately, of which the Higgsinos are part), in which it also appears.

In my model, the fact that the $\mu_Q$ term is found in combination with other mass terms follows from the existence of other terms in the superpotential, like the Yukawa term

$y_u Q \bar{U} H_u$,

as well as from soft SUSY breaking terms.

Any insights or references that address this topic would be most helpful.

This post imported from StackExchange Physics at 2015-02-23 09:16 (UTC), posted by SE-user gn0m0n