The SUSY F and D terms, and the scalar potential

[shirosato]

Hello all. Again, thank you for the help so far. Forgive the lack of tex in this post, it somehow was creating errors no matter what I tried.

My question this time involves understanding the F and D terms in SUSY theories. From what I understood, they were introduced as auxiliary fields (EOM: $$F=F^{\dagger}=0$$ and $$D^{a}=0$$) to ensure closure of the SUSY algebra on and off-shell.

At some point though when working out the superpotential for a supersymmetric gauge theory, the auxiliary terms are re-expressed in terms of scalar fields. When discussing EWSB in the MSSM, the D term gives the Higgs quartic self-interaction while the F term give: \mu(H_1)(H_2)

- Are the F and D terms thus physical then? What happened to the interpretation that they are fields integrated out and play no role in the EOM's? Why are these terms so important? I'm having a bit of trouble with these guys.

- From what I understand, the four relevant parameters in EWSB are tan{\beta}, \mu, m_1, m_2 which can be written in terms of free parameters in the SUSY breaking soft terms? - Shiro

 

[fzero]

For the F and D-terms there's a short answer and then a longer one because the terminology is used a bit more broadly in the literature.

The short answer is that the F and D components of chiral and vector superfields, respectively, are, as you say, auxiliary fields. They are not propagating degrees of freedom, since they don't have a canonical kinetic energy term. They appear in the action algebraically, without derivatives so they are integrated out. However the effect of integrating them out generates the effective potential for the other fields, so we wouldn't consider them unphysical.

The longer answer comes in because when we study the vacuum structure of SUSY and the conditions for SUSY to be broken, it's often possible to actually add terms to the Lagrangian that depend linearly on the F and D components. These terms are also called F and D-terms, and their effect is extremely physical, since they can often be used to guarantee that SUSY is spontaneously broken in the vacuum.

As for the electroweak parameters, there are various models of soft SUSY breaking, and I'd imagine that they all make slightly different predictions. For example, the scale of SUSY breaking can be different from the scale of EWSB. What is true is that the EWSB breaking cannot occur at a scale larger than SUSY breaking in the MSSM. So the EW Higgs sector will be determined by running the bare couplings of the MSSM along with the soft breaking parameters down to the EW scale. Martin's Supersymmetry Primer, hep-ph/9709356 is one reference that discusses some of this apparently, but there are probably more recent references like Dine's book too.