Long-Range Azimuthal Correlation, Entanglement, and Bell Inequality Violation by Spinning Gluons at the Large Hadron Collider

We apply the recently developed concept of the nucleon energy-energy correlator (NEEC) for the gluon sector to investigate the long-range azimuthal angular correlations in proton-proton collisions at the Large Hadron Collider. The spinning gluon in these collisions will introduce substantial nonzero <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>cos</mml:mtext> <mml:mo>(</mml:mo> <mml:mrow><mml:mn>2</mml:mn> <mml:mi>ϕ</mml:mi></mml:mrow> <mml:mo>)</mml:mo></mml:math> asymmetries in both Higgs boson and top quark pair productions, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ϕ</mml:mi></mml:math> is the azimuthal angle between the forward and backward energy correlators in the NEEC observables. The genesis of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>cos</mml:mtext> <mml:mo>(</mml:mo> <mml:mrow><mml:mn>2</mml:mn> <mml:mi>ϕ</mml:mi></mml:mrow> <mml:mo>)</mml:mo></mml:math> correlation lies in the intricate quantum entanglement. Owing to the substantial <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>cos</mml:mtext> <mml:mo>(</mml:mo> <mml:mrow><mml:mn>2</mml:mn> <mml:mi>ϕ</mml:mi></mml:mrow> <mml:mo>)</mml:mo></mml:math> effect, the NEEC observable in Higgs boson and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi> <mml:mover><mml:mi>t</mml:mi> <mml:mo>¯</mml:mo></mml:mover> </mml:math> production emerges as a pivotal avenue for delving into quantum entanglement and scrutinizing the Bell inequality at high-energy colliders.