One thing I sometimes like to joke is that in physics, there are only three numbers: zero, one and infinity. By that I mean that you can get a decent rough estimate in many cases by treating the relevant parameters as one of those three values. The entire field of
dimensional analysis involves setting numbers to be one in the appropriate units; for example, consider atomic physics. We are in the quantum regime, so we need
Planck's constant h; the dominant force is electromagnetism, so we'll need the
vacuum permittivity ε
0; and the electrons form the "outside" of an atom, so let's also consider the
electron charge
e and mass
me. There's only one way to combine these objects to have the dimensions of energy:
$\frac{m_e e^4}{\epsilon_0^2 h^2}$
Up to an overall constant, this is the
Rydberg, which indeed characterises the energies of atomic physics, and which is normally derived after several weeks of quantum mechanics.
Setting things to be zero is fairly intuitive. Small things
normally have small effects, and can be ignored at first. Correcting for them being non-zero is then precisely a
perturbation series. Interestingly, setting numbers to infinity is pretty similar; there are plenty of situations where the mathematics can be exactly solved when a coupling
g goes to infinty, and then corrections come as a series in inverse powers of
g. A somewhat different example is in the strong interaction, which has three
colours (analogous to the single electric charge). Before I was born, Dutch physicist
Gerard 't Hooft was able to successfully analyse the strong interaction by setting the number of colours to be infinity. Despite three not being very big, the approximation was successful.
In a similar vein, we have the
paper I want to discuss today. Like 't Hooft, Bai and Torroba are approximating a number that equals three by infinity. Instead of gauge interactions and colour, they have chosen to look at flavour and the number of generations.