Examination of Possible Proton Magic Number <i>Z</i> = 126 with the Deformed Relativistic Hartree-Bogoliubov Theory in Continuum

Whether <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><mo>=</mo><mn>126</mn></mrow></semantics></math></inline-formula> is a proton magic number has been controversial in nuclear physics. The even-even <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mmultiscripts><mi>Ubh</mi><none></none><none></none><mprescripts></mprescripts><mn>126</mn><none></none></mmultiscripts></semantics></math></inline-formula> isotopes are calculated based on the DRHBc calculations with PC-PK1. The evolutions of quadrupole deformation and pairing energies for neutron and proton are analyzed to study the possible nuclear magicity. Spherical shape occurs and neutron pairing energy vanishes at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>258</mn></mrow></semantics></math></inline-formula> and 350, which are the results of possible neutron magicity, while the proton pairing energy never vanishes in Ubh isotopes, which does not support the proton magicity at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><mo>=</mo><mn>126</mn></mrow></semantics></math></inline-formula>. In the single-proton spectrum, there is no discernible gap at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><mo>=</mo><mn>126</mn></mrow></semantics></math></inline-formula>, while significant gaps appear at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><mo>=</mo><mn>120</mn></mrow></semantics></math></inline-formula> and 138. Therefore, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><mo>=</mo><mn>126</mn></mrow></semantics></math></inline-formula> is not supported as a proton magic number, while <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><mo>=</mo><mn>120</mn></mrow></semantics></math></inline-formula> and 138 are suggested as candidates of proton magic numbers.