How to handle Yukawa contractions in calculating SUSY beta functions?

Question

I'm reading Chapter 6 of Martin's introduction to SUSY http://arxiv.org/abs/hepph/9709356, which is about RGEs in the MSSM. I tried to convince myself of some of the calculations, and I was particularly stuck with anything that has Yukawas in it. I don't know how to deal with them. Take for example the anomalous dimension formula, $$\gamma^i_j =\frac{1}{16 \pi^2} [\frac{1}{2} Y^{imn}Y^*_{jmn} - 2 g_a^2 C_a(i)\delta^i_j]$$ where, $g_a$ is a gauge coupling corresponding to a group $a$, and $C_a(i)$ is the Casimir invariant. And $Y^{imn}$ denotes Yukawas.

For example, $$\gamma^{H_u}_{H_u} =\frac{1}{16 \pi^2} [3 Y^*_t Y_t - \dots]$$ $$\gamma^{\bar{d}_3}_{\bar{d}_3} =\frac{1}{16 \pi^2} [2 Y^*_b Y_b - \dots]$$ $$\gamma^{\bar{Q}_3}_{\bar{Q}_3} =\frac{1}{16 \pi^2} [Y^*_t Y_t + Y^*_b Y_b - \dots]$$

the dots are for the second term which I know how to calculate. But I don't know how the results for the Yukawas were obtained!

My understanding is that the indices $i,j,k$ are family indices for chiral superfields. Explicitly, $$ Y^{ijk} = Y^{\Phi_i \Phi_j \Phi_k}$$ So if I wanted to calculate, the Yukawa part of $\gamma^{H_u}_{H_u}$, I would try: $$Y^{H_u mn}Y^*_{H_u mn} = Y^{H_u \bar{u}_m Q_n}Y^*_{H_u \bar{u}_m Q_n} + Y^{H_u Q_m \bar{u}_n}Y^*_{H_u Q_m \bar{u}_n} + \text{other permutations in superfields?} $$ If I chose third family approximation like in Martin's notes, then $m=n=3$ But I'm not sure this is correct or where it leads.

Could someone help me with this. How should I handle $Y^{imn}Y^*_{jmn}$ to get those results?

Answer 1

The three indices in $Y^{imn}$ go over all chiral superfields in the theory. In all the formulae, the factor $1/(16\pi^2)$ is copied from the master formula for $\gamma^i_j$. The factor $1/2$ in front of $Y^{imn} Y^*_{jmn}$ always cancels against the fact that there are two terms with $m,n$ interchanged – assuming that the indices $i,j$ represent an electroweak doublet, not singlet. It's because $m,n$ must be a doublet and singlet respectively, and they're therefore effectively distinguishable.

The factor $3$ in front of $Y^*_t Y_t$ appears because $m,n$ go over the doublet, singlet top quark chiral superfields but these superfields also have an extra 3-valued color index so the term is tripled. The factor $2$ in front of $Y^*_b Y_b$ is there because the remaining indices $m,n$ are chiral superfields that are electroweak doublets and there are two equal contributions $up/down$ and $down/up$ to get a singlet out of them. In the last expression for the $Q_3$ anomalous Yukawa dimensions, the factors are $1$ because the remaining indices refer to electroweak doublet+singlet fields but the up/down index for the doublet has to coincide with the same index of $Q_3$ on the left hand side; and because only the quark singlet field among $m,n$ carries a color but it's bound to be the same as the color of $Q_3$ from the left hand side, too. So there is no multiplicity of terms to sum.

Various other terms that could have contributed have been neglected for various reasons and decisions.