# Regge trajectory and Kaluza Klein tower

The mass of hadrons in the Regge trajectory scales as

$m=\sqrt{\frac{J}{\alpha}-\alpha_0}=\sqrt{\frac{n}{\alpha}-\alpha_0}\propto \sqrt{n}$,

where $J=n$ is the spin of the particle (in natural units, for zero angular momentum) and $\alpha$ is the inverse QCD string tension.

In the Kaluza Klein (KK) model, the mass scales as

$m=\frac{n}{R}\propto n$

where $R$ is the radius of the KK compactified dimension.

Can we obtain any relation between these two mass formulas, perhaps yielding a relation between the KK radius $R$ and the inverse string tension $\alpha$ in the Regge trajectory?

• I'm not sure what you're asking. In the first formula, $n$ is related to spin/angular momentum. In the second formula, it just labels the $n$-th scalar in the Kaluza-Klein tower. There's no related between the two $n$, so what's the question? – ACuriousMind Nov 2 '15 at 14:31
• What "number of fermion"? The Kaluza-Klein states are scalars, their spin is known. Also, Regge trajectories are relations for resonance states, I have no idea where you see the connection to the mass of a free particle state such as the Kaluza-Klein states. – ACuriousMind Nov 2 '15 at 14:49
• Is there really no possible connection between the masses? Can't the resonance states be described as elementary free particles at some specific energy scale? – Thomas Nov 2 '15 at 15:01
• there is a connection between string theory and Regge poles,books.google.gr/… , and between string theory and kaluza klein superstringtheory.com/experm/exper5a1.html . Maybe you can extrapolate to a connection between KK and Regge – anna v Nov 2 '15 at 15:02
• see also this post here physics.stackexchange.com/questions/13728/… – anna v Nov 2 '15 at 15:08