Stars as Laboratories for Fundamental Physics - MPP Theory Group

154 Chapter 4
If the triangle condition is not satisfied the URCA processes require
bystander particles to absorb momentum, leading to the “modified
URCA process” shown in Fig. 4.11 (Chiu and Salpeter 1964). Of
course, as below in Sect. 4.9.1, the missing momentum can be provided
by pions or a pion condensate (Bahcall and Wolf 1965a,b).
An explicit result for the modified URCA rate in the OPE model
for the nucleon interactions is (Friman and Maxwell 1979)
Q mod. URCA = 11513π
120960 α2 π G 2 F cos 2 θ C C 2 A p F,e T 8 , (4.77)
ignoring factors of order unity to account for a nonzero m π and nucleon
correlations. As in the νν processes, only the axial-vector coupling
contributes. Some details of the phase-space integration can be found
in Shapiro and Teukolsky (1983). 24
The modified URCA reactions (Fig. 4.11) are closely related to νν
bremsstrahlung and the inelastic scattering process νNN → NNν discussed
earlier. It is interesting that the rate for the e p n → n n ν e process
does not diverge, in contrast with νNN → NNν where multiplescattering
effects had to be invoked to obtain a sensible result. The
divergence was due to the intermediate nucleon going on-shell for a
vanishing energy transfer. In e p n → n n ν e the electron has the energy
E F,e , the neutrino T , and because E F,e ≫ T (degenerate electrons!), the
minimum energy transfer to the leptons is E F,e and thus never zero. Of
course, this is the reason why the modified URCA reaction was invoked
in the first place: if the intermediate nucleon line is cut, the two
sub-processes are suppressed by energy-momentum constraints.
The direct and modified URCA process should be expressed in terms
of a common structure function applicable to charged-current processes.
One may expect that at high density, spin and isospin fluctuations may
suppress these reactions in analogy to the neutral-current processes
discussed in the previous section. However, the URCA processes have
not been discussed in the literature from this particular perspective.
The equivalent of the modified URCA process for quark matter was
calculated by Iwamoto (1980, 1982), more recently by Goyal and Anand
(1990), and numerically by Ghosh, Phatak, and Sahu (1994) who claim
that Iwamoto’s phase-space approximations can lead to substantial errors
in the emission rate.
24 Shapiro and Teukolsky’s phase-space volume involves a factor p 3 F,e because in
their treatment of the nuclear matrix element there is no cancellation of p 2 F,e with
1/ω 2 from the nucleon propagator which occurs in a bremsstrahlung calculation.