We have developed a new three-body method to compute the triple-alpha reaction rate at low temperatures where measurements are impossible and many numerical attempts have failed before. In this work, the triple-alpha is modeled as a three-body Borromean system in hyperspherical harmonics coordinates. In the low temperature region, the triple-alpha proceeds through a quadrupole transition from the 0+ continuum to the 2+1 bound state in 12C. The 2+1 bound state is obtained by solving a set of coupled channels equations in hyper-radius coordinates for negative energy with a boundary condition that requires the wavefunction to go to zero at large distances. The same approach can not be applied to the 0+ continuum state because it requires an exact boundary condition for the three charged particles. We therefore combine the R-matrix expansion, the R-matrix propagation method, and the screening technique in the hyperspherical harmonics basis to obtain a numerically stable three-body continuum wavefunction for the &alpha+&alpha+&alpha system. We employ the Ali-Bodmer potential for the alpha-alpha interaction which reproduces the low energy phase shifts as well as the 0+ resonance of 8Be. We add a three-body force to fit experimental data. Both the 2+1 bound state and the 0+2 resonant state in 12C are well reproduced in our framework. We find a dominant triangle three-alpha configuration for the Hoyle resonance by studying the density distribution function. The resonant and non-resonant continuum states of 12C( 0+) are obtained simultaneously, allowing us to include these two processes on the same footing. Long range Coulomb interactions show important effects especially in the low temperature regime. We also present a detailed convergence study of the triple-alpha reaction rate with respect to the screening radius and the size of the model space. The new rate agrees with the NACRE rate for temperatures greater than 0.07 GK, but a large enhancement at lower temperatures is found (&sim 1012 at 0.02 GK). We observe a transition between dominance of the non-resonant versus the resonant triple-alpha capture process around 0.06 GK. Our results are then compared to previous calculations where additional approximations were made. We also explore the impact in astrophysics namely, in stellar evolution, helium accreting white dwarfs, and helium accreting neutron stars.The presence of very narrow two-body and three-body resonances in addition to the strong, long-range Coulomb interaction makes the triple-alpha problem very challenging. Our framework which is the combination of various methods is a new approach to overcome the well known difficulty of the three charged-particle system. This method is particularly suited to the very low energy regime where measurements are impossible. In addition, the method we developed opens new opportunities in addressing three-body, low-energy reactions in other fields (e.g., atomic and molecular physics) where all three particles have charge.
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