Black-Hole Solutions with Scalar Hair in Einstein-Scalar-Gauss-Bonnet Theories

In the context of the Einstein-scalar-Gauss-Bonnet theory, with a general coupling function between the scalar field and the quadratic Gauss-Bonnet term, we investigate the existence of regular black-hole solutions with scalar hair. Based on a previous theoretical analysis, that studied the evasion of the old and novel no-hair theorems, we consider a variety of forms for the coupling function (exponential, even and odd polynomial, inverse polynomial, and logarithmic) that, in conjunction with the profile of the scalar field, satisfy a basic constraint. Our numerical analysis then always leads to families of regular, asymptotically-flat black-hole solutions with non-trivial scalar hair. The solution for the scalar field and the profile of the corresponding energy-momentum tensor, depending on the value of the coupling constant, may exhibit a non-monotonic behaviour, an unusual feature that highlights the limitations of the existing no-hair theorems. We also determine and study in detail the scalar charge, horizon area and entropy of our solutions.