Imaginary-time formalism for triple-$\alpha$ reaction rates

Using imaginary-time formalism, it is shown that the triple-$\alpha$ reaction rate can be reliably calculated without the need to solve scattering problems involving three charged particles. The calculated reaction rate is found to agree well with the empirical NACRE rate, which is widely adopted in stellar evolution calculations. The reason for this is explained using $R$-matrix theory. Extremely slow convergence is found to occur when a coupled-channel expansion is introduced, which helps to explain the very different reaction rates obtained using a coupled-channel approach.

Figure 1

Calculated triple-$\alpha$ reaction rates for different choices of radial cutoff distances, $r_{\max }$ and $R_{\max }$. The NACRE rate is also shown for comparison.

Figure 2

Calculated triple-$\alpha$ reaction rate for different choices of maximal angular momentum of the partial wave, $L_{\max }$.

Figure 3

Energy expectation value and variance as function of temperature. The insets show density distributions for three different temperatures.

Figure 4

Calculated triple-$\alpha$ reaction rate using coupled-channel expansion for different numbers of channels, $n_{\max }$. The energy eigenvalue of the $\alpha \mathrm{-}\alpha$ system corresponding to the $n_{\max }$ channel is indicated in the parentheses.