In Weinberg's "The Quantum Theory of Fields Vol III" page 267 we're told that $n_f = 2N_f$. Where $n_f$ are the number of flavours and $N_f$ is the number of left chiral quark superfields (or the number of corresponding left chiral anti-quark superfields). This implies that the number of left chiral quark superfields is half the number of quark flavours.

I've become a little reliant on the above relation to explain why beta functions are sometimes written differently for the same theory. For example I use it to explain the equivelance between equation \begin{align*} b_0 = 3N_C - N_f \end{align*} in Notes on Seibergology page 117 and equation \begin{align*} b_0 = 3N_C - (1/2)N_f \end{align*} in Supersymmetry and Beyond page 3 (where in the latter $C_2 (G)$ has been substituted for $N_C$). Both these equations are supposed to describe $N=1$, $SU(N_C)$ SQCD beta functions but they look different. The reason for this, I decided, was that in the first equation we're using $N_f$ as the number of left chiral quark superfields whilst in the latter equation $N_f$ corresponds to $n_f$ in the first paragraph of this question. In other words $N_f$ in the second equation actually means the number of flavours. Substituting $n_f$ in place of $N_f$ in the second equation then using $n_f = 2N_f$ shows that these two equations are actually the same.

This seemed to tie things up. However, Duality in Supersymmetric Yang-Mills Theory page 17 has the equation \begin{align*} b_0 = 3N_C - N_f \end{align*} where it explains on page 15 that $Nf$ is the number of flavours of quarks (we should substitute $N_f$ for $n_f$ using the notation established in the first paragraph of this question). It also explains on this page that we're indeed working with $N=1$, $SU(N_C)$ SQCD. So this seems to mess things up. Has the last paper just forgotten to divide by two and explained $N_f$ in an incorrect manner or am I missing something fundamental? I realise it's most likely the latter.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.