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While studying SUSY in 4D, I noticed the dynamical chiral superfield has dimension [GeV], whereas the dynamical vector superfield (for gauge theories) is unitless. Because I was introduced to the chiral and vector superfields as being irreducible components of a more general superfield, it seemed strange to me that they could have different mass dimensions.

But then I realized that this is not alien to me: in the Standard Model of electroweak interactions, the Dirac spinors which have left-chiral and right-chiral irreducible components carry different weak charges!

So is it reasonable to think of the mass dimension of a field to be yet another quantum number? For example, the chiral superfield is charge +1 and vector superfield is charge 0. Could I write down a transformation law? Can I do something as perverse as make it local?

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There are scale transformations (dilatations), which are a part of the conformal group. If you make it local you'll get a conformal gravity theory. The bosons associated to the scale transformations are called dilatons. In generic quantum field theories the scale symmetry is anomalous, so even if you don't have any parameters with mass dimension in the Lagrangian the symmetry is broken by the RG flow. – Michael Brown Jan 21 '13 at 2:38
The conformal dimension of a field is equivalent (up to a calculable additive shift) to the energy of the state one gets by creating a particle with this field in a radial quantization. So if you consider energy a "quantum number", you should do the same with the mass dimension of a field. In a supermultiplet, the different component fields have dimensions that differ by multiples of $1/2$ because the supercharge itself also carries a mass dimension and changes the mass dimension of fields it acts upon. – Luboš Motl Jan 21 '13 at 7:07

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