Stars as Laboratories for Fundamental Physics - MPP Theory Group

Processes in a Nuclear Medium 155
4.9 Novel Phases of Nuclear Matter
4.9.1 Pion Condensate
Free particles cannot radiate because of energy-momentum constraints
which can be overcome by exchanging momentum with “bystander”
particles (bremsstrahlung). No bystander is required if the momentum
is taken up by the pion field directly which in Fig. 4.1 only mediated the
nucleon interaction (Bahcall and Wolf 1965a,b). This possibility is particularly
important if a pion condensate develops so that the medium
is characterized by a macroscopic, classical pion field. The pion dispersion
relation can be such that the lowest energy state involves a
nonvanishing momentum k π (Baym 1973; Kunihiro et al. 1993; Migdal
et al. 1990). Nucleons can exchange pions with the condensate and
thereby pick up a momentum k π which then allows for the radiation of
axions or other particles (Fig. 4.12).
Fig. 4.12. Axion emission by nucleons scattering off a pion condensate.
There is another amplitude with the axion attached to N 1 .
The pion condensate causes a periodic potential for the nucleons
which thus must be described as quasi-particles or Bloch states with
a main momentum component p and admixtures p ± k π . Because an
eigenstate of energy is no longer an eigenstate of momentum, these
quasi-particles can emit radiation without violating energy-momentum
conservation. Therefore, the process of Fig. 4.12 can be equally described
as a decay Ñ1 → Ñ2 a.
The rate for this reaction was calculated by Muto, Tatsumi, and
Iwamoto (1994) who found that the dominant contribution was from a
π ◦ condensate. It forms a periodic potential A sin(k π · r) where A is a
dimensionless amplitude which is small compared to unity for a weakly
developed condensate. The Bloch states are to lowest order in A
Ñ ± p = N ± p ∓ Aκ 0
2
( N
±
p+kπ
+
N ± )
p−k π
, (4.78)
E p+kπ − E p E p−kπ − E p