Equation linking superstring vibration with mass?

[R. E. Nettleton]

Is there an existing equation, or set of equations, that directly links the specific vibration of a superstring filament with the resultant particle mass?

A brief explanation of any mathematics would be helpful. Thank you.

 

[fzero]

You are probably thinking of an analogue to the formula for the bosonic string
$$ M^2 = \frac{4}{\alpha'} (N-1),$$
where $N$ is the oscillator level and $1/(2\pi \alpha')$ is the string tension. There are related formulas for the superstrings, but you really need to understand the worldsheet conformal field theory to understand where they come from. I couldn't give a brief explanation of it, but the lectures by Tong are extremely clear. The bosonic formula is arrived at the end of Ch. 1 and further developed in Ch. 2. For the superstring formulas, the book by Zweibach is probably the gentlest reference.

Also, the fact that $1/\sqrt{\alpha'}$ is usually of order of the Planck mass means that the massive states corresponding to higher vibration states have nothing to do with the elementary particles that we measure. The observed particles must come from the massless, lowest-level string states and acquire masses via some version of the Higgs mechanism at energies much lower than the string scale. So the formula above isn't very useful for phenomenology and you really have to understand how the fermion modes in the CFT are used to construct states.

I can't think of a way to give a brief explanation of this, since the explanation of the symbols that would appear in the equations would take several weeks of lectures in a string theory course..