Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

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

+ 3 like - 0 dislike
532 views

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?


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

asked Feb 23, 2015 in Theoretical Physics by gn0m0n (80 points) [ revision history ]
edited Feb 24, 2015 by gn0m0n

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOv$\varnothing$rflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...