Numerology with scalar fields

Today is a good day to do some numerology, being March 14th and Einstein's birthday. Consider a quantum field theory in \(D\) spacetime dimensions with scalars \(\varphi\) and an interaction of the form \(g_{n}\varphi^{n}\). We are going to measure everything in units of mass and units of \( \hbar \). Taking the speed of light to be dimensionless leads to length, time, momentum and energy written in terms of mass and \( \hbar \). For example, the action has units of \( \hbar \). The scalar field has units\[ \left[ \varphi \right] =  \left( \frac{3 - D}{2} \right) \left[ \hbar \right] + \left( \frac{D - 2}{2} \right) \left[ M \right] \]The dimension of the coupling satisfies\[ \left[ g_{n} \right] = \left[ \hbar \right] - n \left[ \varphi \right] - D \left[ L \right] \]Using the dimension for \(\varphi\) and the fact that \(\left[ L \right] = \left[ \hbar \right] - \left[ M \right] \) we get\[\left[ g_{n} \right] = \left( \frac{(n - 2)D + 2 - 3n}{2} \right) \left[ \hbar \right] + \left( \frac{(2-n)D + 2n}{2} \right) \left[ M \right]\]Here comes the numerology. For \(n = 3\) one gets\[\left[ g_{3} \right] = \left( \frac{D - 7}{2} \right) \left[ \hbar \right] + \left( \frac{6 - D}{2} \right) \left[ M \right]\]For \(n = 4\) one gets\[\left[ g_{4} \right] = \left( D - 5 \right) \left[ \hbar \right] + \left( 4 - D \right) \left[ M \right]\]And finally, for \(n = 6\) one get\[\left[ g_{6} \right] = 2\left( D - 4 \right) \left[ \hbar \right] + 2\left( 3 - D \right) \left[ M \right]\]In all three cases the mass dimension of the coupling vanishes in a dimension lower than the \(\hbar\) dimension. For \(n = 3\) we have vanishing mass dimension in \(D = 6\) and vanishing \(\hbar\) dimension in \(D = 7\). For \(n = 4\) we have vanishing mass dimension in \(D = 4\) and vanishing \(\hbar\) dimension in \(D = 5\). And finally for \(n = 6\) we have vanishing mass dimension in \(D = 3\) and vanishing \(\hbar\) dimension in \(D = 4\). These three cases coincide with the three canonical cases of \(adS_{D+1} / CFT_{D}\). Indeed, \(\mathcal{N} = 4\) super Yang-Mills in four spacetime dimensions contains scalar fields that interact with a quartic vertex and ABJM theory in three spacetime dimensions has scalar fields that interact via a sextic vertex. I guess this means that the theory related to the \(\mathcal{N} = (2, \, 0)\) theory in six spacetime dimensions has scalars interacting via a cubic vertex.

Actually, the coefficient of \(\left[M\right]\) in \(D\) dimensions equals the negative of the coefficient of \(\left[ \hbar \right]\) in \(D+1\) dimensions for any value of \(n\). However, only for the three cases above we have integer solutions.

If one performs the same analysis on the coupling constant in Yang-Mills theory, the dimension comes out like\[\left[ g_{YM} \right] = \left( \frac{D - 3}{2} \right)\left[\hbar \right] + \left( \frac{4 - D}{2} \right) \left[ M \right]\]However, the coupling appears in the field theory action in the combination \(g_{YM} / \hbar\) and this leads to\[\left[ g_{YM} \right] - \left[\hbar \right] = \left( \frac{D - 5}{2} \right)\left[\hbar \right] + \left( \frac{4 - D}{2} \right) \left[ M \right]\]