Unification of guage couplings in the presence of extra dimensions

[karnten07]

Hi guys,

I'm reading about unification of gauge couplings in the presence of extra dimensions but I'm coming unstuck in my understanding of what a Kaluza-Klein mode/excitation/tower is. I've looked ont he net and in string theory books but have been unable to find mcuh that is helping my understanding, Can anyone point me in the right direction or give a simple (assuming 2nd year physics degree knowledge) definition for each. Much appreciated

karnten07

 

[nrqed]

Hi guys,

I'm reading about unification of gauge couplings in the presence of extra dimensions but I'm coming unstuck in my understanding of what a Kaluza-Klein mode/excitation/tower is. I've looked ont he net and in string theory books but have been unable to find mcuh that is helping my understanding, Can anyone point me in the right direction or give a simple (assuming 2nd year physics degree knowledge) definition for each. Much appreciated

karnten07

Consider a particle in a one-dimensional box of length "L". Then you know that it's momentum will be quantized in multiples proportional to 1/L. Now consider a particle in 4+1 dimensions let's say, with one spatial dimension curled up with a circumference L. Consider a nonrelativistic fee particle for simplicity. Then its total energy is proportional to p^2. Now, this can be decomposed into a 3D part and the extra dimension part as

E \simeq \frac{p^2}{2m} + \frac{n^2}{L^2}

I am not being careful here with factors of 2, Pi, etc.
The key point is that from the point of view of the ordinary 3+1 dimensions, we end up with what looks like an infinite number of massive particles with increasing masses which depends on the size of the extra dimension. This is the KK tower.

A nice elementary discussion is in Zwiebach's book on String theory.

 

[karnten07]

Consider a particle in a one-dimensional box of length "L". Then you know that it's momentum will be quantized in multiples proportional to 1/L. Now consider a particle in 4+1 dimensions let's say, with one spatial dimension curled up with a circumference L. Consider a nonrelativistic fee particle for simplicity. Then its total energy is proportional to p^2. Now, this can be decomposed into a 3D part and the extra dimension part as

E \simeq \frac{p^2}{2m} + \frac{n^2}{L^2}

I am not being careful here with factors of 2, Pi, etc.
The key point is that from the point of view of the ordinary 3+1 dimensions, we end up with what looks like an infinite number of massive particles with increasing masses which depends on the size of the extra dimension. This is the KK tower.

A nice elementary discussion is in Zwiebach's book on String theory.

Thanks for the explanation, its a little clearer now. I have Zwiebachs book here but can't find where KK towers, modes or excitations are introduced. I just checked on the course materials on the MIT courseopenware site as well as i thought it may be easier to do a pdf search for the terms but don't know which section it is most likely to be in. Do you or anyone know of where abouts in the book these ideas are first met (or in the lecture notes online)? Many thanks

karnten07

 

[jdstokes]

A good, albeit technical introduction to the subject can be found in

http://arxiv.org/abs/hep-ph/0503177

Keep in mind that the concept is really quite simple. When a particle enters the periodic extra dimension, it behaves like a particle in a box (with periodic boundary conditions) so its momentum becomes quantized (see 1st or 2nd or year quantum mechanics). There are an infinite number of quantized modes in a box, and you can take my word for it that the quantization number is proportional to the mass of each mode. Hence an infinite tower of massive KK modes.

This simple analogy does not explain how KK modes arise when the extra dimension is infinite, however. To understand that, you need a more general way to think about it.

Consider a 5D field \varphi(x^\mu,y) which depends on ordinary 4D spacetime x as well as the additional dimension y. The idea is to write this in `separated variable' form X(x^\mu) Y(y) where X depends only on x and Y only on y. But you can't do this for arbitrary fields unless you include and infinite summation (called a generalised Fourier series)

\varphi(x^\mu,y) = \sum_n X_n(x^\mu) Y_n(y)

Now you interpret each X_n as a 4D field with wavefunction profile Y_n along the extra dimension. So there you have it, out pops an infinite number of KK modes.

 

[nrqed]

Thanks for the explanation, its a little clearer now. I have Zwiebachs book here but can't find where KK towers, modes or excitations are introduced. I just checked on the course materials on the MIT courseopenware site as well as i thought it may be easier to do a pdf search for the terms but don't know which section it is most likely to be in. Do you or anyone know of where abouts in the book these ideas are first met (or in the lecture notes online)? Many thanks

karnten07

I had in mind section 2.9 in the book.

He does discuss winding modes vs kk modes a bit but I had not realized how little he talks explicitly about KK modes. You are better off with introductory papers on the archives such as the one suggested by jdstokes.