As far as I know, the spectrum of any (?) String Theory is of the form $$ M^2\propto N $$ where $N$ is the number operator. The lightest known particle being the electron, I am led to think that we should observe particles with masses $m_e,2m_e,3m_e,\dots$

In more general terms, the spectrum of ST seems to be harmonic, as the operators are always essentially oscillators. Cf. the Veneziano amplitude, with poles at $s=4(n-1)/\alpha'$. This brings me to my question: how does the phenomenology of ST deal with the (obviously?) unobserved tower of particles, with masses in harmonic progression? does the geometry of the extra dimensions have anything to do with the apparently erratic behaviour of low-energy physics?

  • $\begingroup$ Waitwaitwait. Neutrinos have masses, and smaller ones, at that. Not that this should matter for your question. The Veneziano amplitude pertained to hadrons, which broadly fall on Regge trajectories. You are not talking about the gravitational string, here? $\endgroup$ Sep 25, 2017 at 23:11
  • $\begingroup$ @CosmasZachos good point. I should replace $m_e\to m_\nu$ in the first paragraph. $\endgroup$ Sep 26, 2017 at 9:06

1 Answer 1


It's more complicated than that. To get the Standard Model, a standard technique is to use intersecting brane models, usually many D6s in the compact space, in which the spectrum depends on the angles between the branes and the particles comes from vibrational modes of the open strings stretched between the branes.

Roughly speaking, in superstring theory you will get:

$$\alpha' M^2 = N_B + N_F + f(\theta_i)$$

where $N_B$ and $N_F$ are the oscillator numbers and $\theta_i$ the angles between the branes in the compact spaces (think to factorized tori for simplicity), for some function f. While the angles are not completely arbitrary, due to supersymmetry constraints for instance, you can get a realistic spectrum in this way. The presence of multiple branes in each stack will give the different families of leptons and quarks.

A nice and simple introduction to the topic can be found in the classic book by Zwiebach.

  • $\begingroup$ But even this corrected formula doesn't apply to the electron mass, because the electron mass (in such models) comes from a yukawa term produced e.g. by a string instanton. It's not a string excitation of this sort... I guess your comment does address the remark about "unobserved tower of particles". $\endgroup$ Sep 26, 2017 at 1:32

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