Sommerfeld Enhancement

[Jay-qu]

So this aint your every day school physics/math question but maybe someone can help me :)

 

I am looking at the enhancement to the cross section of a particle annihilation via interactions from a scalar higgs particle.

 

This can be calculated by solving the Schrodinger equation with a Yukawa potential with all the relevant couplings and masses in place.

 

Playing around (and using the radial form of the equation) one arrives at:

[math]

\frac{\partial}{\partial r}^2\chi® = (k^2+\frac{ m \kappa^2}{\pi r}e^{-m_{H_1}r})\chi®

[/math]

m = mass of annihilating particles (~100GeV)

m_H1 = mass of higgs (~15GeV)

kappa = coupling between particles (~0.18)

k = momentum of annihilating particles (~0.0005)

 

So we have a second order equation to be solved with the boundary conditions:

[math]

\chi (0) -> 0

[/math]

and

[math]

\chi (\inf) -> sin(kr+\delta)

[/math]

delta is just an arbitrary phase.

 

To find the enhancement one needs to find the value of [math]\chi '(0)[/math].

 

This cannot be done analytically. I have attempted to do this numerically with mathematica, to no avail. So I am asking if anyone knows of an approach to solving this equation.

 

Cheers, Jay

[Erasmus00]

I'm confused by what you are trying to do- what is [imath]\chi[/imath] the wavefunction for? A bit of derivation would be enlightening. How are you handling annihilation with something like the Schroedinger equation, which has a fixed particle number?

 

As to the equation, I'd start by fourier transformin- the Yukawa potential has a nice analytic transform and the resulting equation can be solved algebraically. From there, you can numerically invert the transform.

[Qfwfq]

I suppose you are talking about the radial factor and leaving out the angular terms according to angular momenta, for an outgoing state, which perhaps could help with Erasmus' queries. I suspect you should have an arbitrary factor in front of the sine in the asymptotic boundary condition? Off hand, this ought to be proportional to the first derivative in the other boundary condition at 0, either one would wind up determining the other.

 

I have attempted to do this numerically with mathematica, to no avail. So I am asking if anyone knows of an approach to solving this equation.

I don't know how good that product is, though I've heard folks talk well of it. Were you trying iterative methods with it? Which ones? Have you tried Bessel, Henkel and Neuman functions? If there's nothing better you can always go (painstakingly!) by suitably small increments from r = 0, for an arbitrary value of first derivative there.

[Jay-qu]

I'm confused by what you are trying to do- what is [imath]\chi[/imath] the wavefunction for? A bit of derivation would be enlightening. How are you handling annihilation with something like the Schroedinger equation, which has a fixed particle number?

 

As to the equation, I'd start by fourier transformin- the Yukawa potential has a nice analytic transform and the resulting equation can be solved algebraically. From there, you can numerically invert the transform.

Sorry, I was posting in a hurry so avoided the physics in hope of a quick mathematical solution (a bit rude of me).

 

Specifically this is looking at dark matter annihilations (neutralinos), that attract each other by exchanging neutral scalars (higgs). The cross section is still calculated with QFT but the enhancement factor can be handled by Schrodinger because of the non-relativistic speeds the neutralinos move at. The interactions are modeled by a Yukawa potential and the enhancement is supposed to be proportional to the probability density of the wavefunction at r=0.

 

[math]\chi[/math] is a rescaling of the solution to the radial Schrodinger equation [math] R->\frac{\chi}{r}[/math]. If you are interested the derivation is in the appendix of this paper: [0810.0713] A Theory of Dark Matter

 

It turns out that the enhancement is factor is: [math]|\frac{\chi ' (r=0)}{k} |^2 [/math]