Evidence of quantum phase transition in carbon-12 in a 3$\alpha$ model and the problem of hypothetical X17 boson.

Lowest energy spectrum of the $^{12}$C nucleus is analyzed in the 3$\alpha$ cluster model with a deep $\alpha\alpha$ potential of Buck, Friedrich and Wheatley with Pauli forbidden states in the $S$ and $D$ waves. The direct orthogonalization method is applied for the elimination of the 3$\alpha$ Pauli forbidden states. The effects of the first order quantum phase transition are shown in the lowest $^{12}$C($0_1^+)$ and $^{12}$C($2_1^+)$ states from "gas-like" phase to a deep "liquid" phase. It is found that the absolute value of the "liquid" phase energy in the $^{12}$C($2_1^+)$ state is very close to the mass of the hypothetical X17 boson. The energy defect of about 17 MeV in the experiments with the $^8$Be \cite{X17} and $^4$He \cite{X17a} nuclei could be explained as a background effect due to the quantum phase transition in the $^{12}$C nucleus.