# RS 1, massless scalar field, and sep of vars.

1. Feb 29, 2008

### robousy

Hey, this is rather involved but I hope someone can help me out.

I am reading http://arxiv.org/abs/0704.3626 ( the casimir force in randall sundrum models) and am trying to get from equation 2.1 :

$$g^{\mu\nu}\partial_\mu\partial_\nu\Phi+e^{2ky}\partial_y(e^{-4ky}\partial_y\Phi)=0$$

to equation 2.2 ...'The general solution for the non-zero modes can be eppressed in terms of Bessel functions of the 1st and 2nd kind as:

$$\chi(y)=e^{2ky}(a_1 J_2(\frac{m_Ne^{ky}}{k}) +a_2Y_2(\frac{m_Ne^{ky}}{k}) )$$

I'll show you my attempts and if anyone has the patience can maybe help me.

First express field $$\Phi$$ as:
$$\Phi(x_\mu,y)=\phi(x_\mu)\chi(y)$$

Plug this into the original differential equation

$$(g^{\mu\nu}\partial_\mu\partial_\nu\phi)\chi+e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)\phi=0$$

Divide by $$\phi\chi$$ to obtain

$$\frac{ (g^{\mu\nu}\partial_\mu\partial_\nu\phi) }{ \phi}+ \frac{e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)}{\chi}=0$$

As the first term only depends on $$x_\mu$$ and the second on y, each term in the equation must be a constant, so for the $$\chi(y)$$ term we can write:

$$\frac{e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)}{\chi}=n^2$$

where $$n^2$$ is some constant.

Tidying this up a bit we get:

$$e^{-2ky}\partial^2_y\chi} -4ke^{-2ky}\partial_y\chi-n^2\chi=0$$

Make this easier to compare to Bessels equation by writing

$$e^{2ky} \rightarrow x^2, \chi\rightarrow y$$

My final equation now looks like:

$$x^2y''-4kx^2y'-n^2y=0$$

Whereas Bessels equation is

$$x^2y''+xy'+(x^2-n^2)y=0$$

So this is my problem, I obtain an expression that is not quite bessels equation. Can anyone see the error of my logic??

Thank you so much.

Last edited: Feb 29, 2008
2. Feb 29, 2008

### kdv

But I don't see how you can make that change of variable. In Bessel equation, the parameter "x" is the variable differentiated with respect to! So you must be more careful. I will look into it when I have some time tonight.

3. Mar 1, 2008

### robousy

Cool Kdv, any help appreciated, and I see what you are saying about the change of variable so thank you on that point.

4. Mar 2, 2008

### BenTheMan

Hmmm I seem to remember working this out before. I think the problem is that you have to add boundary terms to make it work right, that is, to get a good definition of von Neuman or Dirichlet B.C.'s.

Check out the Lecture notes by Gherghetta : http://arxiv.org/abs/hep-ph/0601213. I think he works it out in detail, but I could be wrong.

Also you have to be careful. The separation of variables that you should use is

$$\Phi(x_{\mu},y) = \sum_n \phi_n(x_{\mu})\chi_n(y)$$

up to some constant that gives you the correct mass dimension. This gives you the KK tower, etc.

5. Mar 2, 2008

### kdv

I can't get it to work out. But I suspect there is something more to the story. They mention treating separately the N=0 mode from the others and this does not show up at all in the steps you followed. See the post by BenTheMan too.

6. Mar 2, 2008

### George Jones

Staff Emeritus
I couldn't get it to work out either, so I glanced at the reference given in the original post's paper. This reference is the the link given in BenTheMan's post. As BenTheMan said, there's another term.

7. Mar 2, 2008

### robousy

Ok guys, I did have a look at the Les Houches Lectures last week but it didn't help much. I'll take another look this evening and see if I can make progress. Thanks all for having a look at this.

Cheers for the correction in my separation of vars Ben. Hope all is good with you!!

Rich

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook