279. Sommerfeld enhancement
ASTRO, COND-MAT, HEP-TH/PH — By Roberto Iengo on February 21, 2009 at 4:05 pm
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This is a brief guest blog post by Roberto Iengo from SISSA (Trieste) who is going to explain us what is Sommerfeld enhancement. Dmitry.
Consider two particles which collide and then undergo some reaction as the result of the collision. An interesting case is the annihilation of two dark matter particles, resulting in the emission of gamma rays or electron positron pairs which could be a long-awaited signal on the dark matter nature. The rate for this process is the product of the cross-section times the incident flux. In the case of the dark matter particles the estimated rate seems to be too low to support the above interpretation of the events which have been observed in some astroparticle experiments.
Here the Sommerfeld enhancement (so named after an early study in the 30′s of the last century) could play a role: if the two colliding particles attract each other, the rate is enhanced.
Classically it is clear that the incident flux is enhanced due to the focusing effect of the attraction; on the other hand also the velocity at which the collision-annihilation takes place is increased. Now the dependence of the cross-section on the velocity varies with the angular momentum: in general it increases with the velocity, except that for zero angular momentum it goes like the inverse of the velocity. Nevertheless, a classical computation for an attractive Coulomb force shows a rate enhancement in all the cases (the classical computation needs a specific assumption on when the collision-annihilation occurs: here it is assumed that it occurs when the two particles are at minimal relative distance).
A more precise computation requires quantum mechanics. Although the applications so far are non-relativistic, it is convenient to describe the phenomenon with the formalism of Quantum Field Theory: two particles propagate in the space-time and interact by exchanging vector bosons (which mediate the attraction) before annihilating. The Feynman graph description of this process gives an integral equation in momentum space relating the full amplitude 
(i.e. including the effect of the attraction) to the bare (i.e. ignoring the attraction) annihilation amplitude 
. This equation already incorporates the focusing effect and therefore the full annihilation rate is as usual given by 
times the phase space.
In the non-relativistic limit the integral equation can be rewritten as an in-homogeneous Schroedinger equation, containing the attractive potential, relating the full and bare amplitudes in the coordinate space. Its solution gives the full amplitude in terms of a convolution of a solution of the homogeneous equation with the bare amplitude . Explicit results depend on the behavior of on the momentum and on the parameters of the attractive potential (which has in general a Yukawa form), namely its coupling constant and its range; they can be obtained numerically in the general case, and also analytically for a Coulomb potential, in which case they are quite similar qualitatively to the results of the classical description.
Some references:
L.D.Landau,E.M.Lifshitz “Quantum Mechanics”, Pergamon Press 1958, chapt. XV par. 118.
R.Iengo “Sommerfeld effect: general results from field theory diagrams” arXiv:0902.0688 [hep-ph].
2 Comments
Dear Roberto,
I’m a graduate student trying to understand this from a more concrete standpoint. When we learn QFT one of the first realistic calculations we often do is the QED [i.e. bar{psi} i slashed{D} psi - (1/4) F^2] process e+e- -> e+e- or something similar.
Do you mean the standard result we get is not correct in the non-relativistic limit? How does the integral equation you mentioned above look like for the specific case of QED? And can you point us to a place where we can learn about Sommerfeld enhancement in a familiar setup like QED?
Thanks!
Dear Wanderer,
the standard computations in QED, like the one
you mention, are done using the perturbation expansion:
one computes first the leading order and if necessary
the corrections which are assumed to be small.
The point is that in some cases the corrections are not small
and the perturbation expansion cannot be applied.
A typical case is when a bound state is formed, for instance,
related to your example, the positronium.
The binding energy is of the order of (m times alpha^2).
Since the kinetic energy is of the same order,
this means that the velocity is v ~ alpha.
Therefore the nonperturbative phenomenon
(here the bound state) occurs for alpha/v ~ 1.
This is the case also for the Sommerfeld enhancement
which is important when alpha/v is of the order of,
or higher, than 1.
As for the bibliography on the Sommerfeld enhancement in a contest
of QED, I do not think that there is much.
There is some related discussion in the book of Peskin and Scroeder pag.148:
they discuss the formation of a mu+mu- bound state from a e+e- initial state.
They say: the Coulomb attraction of muons is an important effect
(which is not included in the perturbative formula for e+e- -> mu+mu-).
They find that, to take into account it, one has to multiply
the perturbative formula (which is S wave) by Psi(r=0),i.e. the bound state wave function at the origin.
This is very similar the the Sommerfeld enhancement for the S wave,
with the difference that here the enhancement is in the initial rather than in the final state,
and that it appears the scattering wave function at the origin, rather than the bound state one.
Best wishes
Roberto Iengo
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