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We briefly describe the cosmological lithium problems followed by a summary of our recent theoretical work on the magnitude of the effects of electron screening, thepossible existence of dark matter parallel universes and the use of non-extensive (Tsal-lis) statistics during big bang nucleosynthesis. Solutions within nuclear physics are also discussed and recent measurements of cross-sections based on indirect experimental techniques are summarized.

Asymptotic Normalization Coefficients method is one of the indirect methods used in nuclear astrophysics. The method allows to deduce a direct part of a radiative capture cross section from the measurement of a two body reaction under specific conditions. The experimental works started in collaboration of NPI CAS, TAMU and INFN LNS from 90-ties and continue till the present. The introduction to this method is presented and experimental experiences from NPI CAS and the collaboration are shared.

The 13C(α, n)16O reaction is the neutron source for the main component of the s-process. It is is active inside the helium-burning shell of asymptotic giant branch stars, at temperatures ≲ 108 K. In this temperature region, corresponding to an energy interval of 140 − 230 keV, the 13C(α, n)16O cross section is dominated by the −3 keV sub-threshold resonance due to the 6.356 MeV level in 17O. Direct measurements could not establish its contribution owing to the Coulomb barrier between interacting nuclei, strongly reducing the cross section at astrophysical energies. Similarly, indirect measurements and extrapolations yielded inconsistent results, calling for further investigations. The Trojan Horse Method was applied to the 13C(6Li, n16O)d quasi-free reaction to access the low as well as the negative energy region of the 13C(α, n)16O reaction. By using the generalized R-matrix approach, the asymptotic normalization coefficient (C̃17O(1/2+)α13C)2 of the 6.356 MeV level was deduced. For the first time, the Trojan Horse Method and the asymptotic normalization coefficient were used in synergy. Our indirect approach lead to (C̃17O(1/2+)α13C)2 = 7.7−1.5+1.6 fm−1, slightly larger than the values in the literature, determining a 13C(α, n)16O reaction rate slightly larger than the one in the literature at temperatures lower than 108 K, with enhanced accuracy.

The reaction 17O(n, α)14C was studied using virtual neutrons coming from the quasi-free deuteron break-up in the three body reaction 17O+d → α+14C+p. This technique, called virtual neutron method, extends the Trojan Horse method to neutron-induced reactions allowing to study the reaction cross section avoiding the suppression effects coming from the penetrability of the centrifugal barrier. For incident neutron energies from thermal up to a few hundred keV, direct experiments have shown the population of two out of three expected excited states at energies 8213 keV and 8282 keV and the influence of the sub-threshold level at 8038 keV. In the present experiment the 18O excited state at E* = 8.125 MeV, missing in the direct measurement, is observed. The angular distributions of the populated resonances have been measured for the first time. The results unambiguously indicate the ability of the method to overcome the centrifugal barrier suppression effect and to pick out the contribution of the bare nuclear interaction.

In the last decades the Trojan Horse method has played a crucial role for the measurement of several charged particle induced reactions cross sections of astrophysical interest. To better understand its cornerstones and its applications to physical cases many tests were performed to verify all its properties and the possible future perspectives. The Trojan Horse nucleus invariance for the binary d(d,p)t reaction was therefore tested using the quasi free 2H(6Li, pt)4He and 2H(3He,pt)H reactions after 6Li and 3He break-up, respectively. The astrophysical S(E)-factor for the d(d,p)t binary process was then extracted in the framework of the Plane Wave Approximation applied to the two different break-up schemes. The obtained results are compared with direct data as well as with previous indirect investigations. The very good agreement confirms the applicability of the plane wave approximation and suggests the independence of binary indirect cross section on the chosen Trojan Horse nucleus also for the present case.

Most of the nuclei in the mass range 90 ≲ A ≲ 208 are produced through the so-called s-process, namely through a series of neutron capture reactions on seed nuclei followed by β-decays. The 13C(α, n)16O reaction is the neutron source for the main component of the s-process. It is active inside the helium-burning shell of asymptotic giant branch stars, at temperatures ≲ 108 K, corresponding to an energy interval of 140 − 230 keV. In this region, the astrophysical S(E)-factor is dominated by the −3 keV sub-threshold resonance due to the 6.356 MeV level in 17O. Direct measurements could not soundly establish its contribution owing to the cross section suppression at astrophysical energies determined by the Coulomb barrier between interacting nuclei. Indirect measurements and extrapolations yielded inconsistent results, calling for further investigations. The Trojan Horse Method turns out to be very suited for the study of the 13C(α, n)16O reaction as it allows us to access the low as well as the negative energy re- gion, in particular in the case of resonance reactions. We have applied the Trojan HorseMethod to the 13C(6Li, n16O)d quasi-free reaction. By using the modified R-matrix approach, the asymptotic normalization coefficient (C˜α13 C17O(1/2+))2${\left( {\tilde C_{{\alpha ^{13}}{\rm{C}}}^{17{\rm{O(1/}}{{\rm{2}}^{\rm{ + }}}{\rm{)}}}} \right)^2}$ of the 6.356 MeV level has been deduced as well as the n-partial width, allowing to attain an unprecedented accuracy for the 13C(α, n)16O astrophysical factor. A preliminary analysis of a partial data set has lead to (C˜α13C17O(1/2+))2 = 6.7−0.6+0.9 fm−1,${\left( {\tilde C_{{\alpha ^{13}}{\rm{C}}}^{17{\rm{O(1/}}{{\rm{2}}^{\rm{ + }}}{\rm{)}}}} \right)^2}\, = \,6.7_{ - 0.6}^{ + 0.9}\,{\rm{f}}{{\rm{m}}^{ - 1}},$ slightly larger than the values in the literature, determining a 13C(α, n)16O reaction rate in agreement with the most results in the literature at ∼ 108 K, with enhanced accuracy thanks to this innovative approach.

We present main features of a surface-integral approach to the Coulomb few-body scattering problem. This approach is valid for both short-range and Coulombic longe-range interactions. We give new general definitions for the potential scattering amplitude. For the Coulombic potentials the generalized amplitude gives the physical on-shell amplitude without recourse to a renormalization procedure. New post and prior forms for the amplitudes of breakup, direct and rearrangement scattering in a Coulomb three-body system are also presented. The Green’s functions and formal solutions of the Schrödinger equation in integral form are not used. Therefore, for the purpose of defining the scattering amplitudes the knowledge of a complicated analytic structure of the Green’s function in the complex-energy plane is not required.

Astrophysical reactions proceeding through compound states represent one of the crucial part of nuclear astrophysics. However, due to the presence of the Coulomb barrier, it is often very difficult or even impossible to obtain the astrophysical S (E) factor from measurements in the laboratory at astrophysically relevant energies. The Trojan Horse method (THM) provides a unique tool to obtain the information about resonant astrophysical reactions at astrophysically relevant energies. Here the theory and application of the THM for the resonant reactions is addressed.