This is the first of several post where I will try to explain why supersymmetry is so appealing. I am starting with what is known as the hierarchy problem. This is not my favorite reason for studying supersymmetry, but it is historically the first one that really compelled physicists to study supersymmetry. Here is a good reference.

Feynman diagrams are literally terms of a perturbation theory expansion. Loops in Feynman diagrams represent higher order quantum corrections to underlying physics.

To determine the mass of a particle loops have to be taken into account. Loops drive the mass of a particle to infinity for they represent divergent integrals. Even if you make an energy cutoff to avoid infinity, loops drive the particle mass all the way to your energy cutoff.

Given the above statement, the reason why the masses of particles are so small is because symmetries "cancel" loop contributions and drive loop corrections to zero.

Mass of the Photon(See picture above):

When you calculate loops for to determine the photon mass you find the loop contribution is zero. This is because QED has a U(1) symmetry that cancels the loops and drives the mass to zero. Hence, because of symmetries the photon remains massless.

Mass of the Electron (See picture above):

Electron mass is small. This is because QED has an approximate axial U(1) symmetry in addition to the U(1) spoken above. This symmetry is broken giving the electron a slight mass. Because, the symmetry is "almost" there, enough of the loop contributions are canceled, and the electron's mass is still small. (Compared to infinity or the Plank scale where you make your cutoff.)

Mass of Higgs:

Now we turn to Higgs. There is no symmetry in the SM that should keep the Higgs mass small. Because of this, the mass of the Higgs should be driven toward infinity and should therefore be as big as the cutoff scale.(Usually taken to be plank scale.) But if the standard model Higgs is real, it should have a small mass, around 1 TeV.

This is called the hierarchy problem. There is no symmetry in the standard model to cancel these loop contributions to give the Higgs a small mass, yet the Higgs mass should be small if it exists.

So now here is the golden question: What symmetry, if it existed, would drive the quadratic loops of the Higgs mass to zero?

It turns out a symmetry that transforms fermions to bosons and vice versa is exactly the symmetry that does the trick. It does the job perfectly and the "quadratic divergences" of the loops are exactly canceled.

This symmetry is called supersymmetry. This was the first "convincing" evidence that something like supersymetry needs to exists. If it does, it makes sense that the Higgs mass is small.

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