Xin-Jian Wen ( 温新建 ) CCNU Shanxi University Efrain J. Ferrer & Vivian de la Incera University of Texas at El Paso Anisotropic structure of the running coupling constant in a strong magnetic field

Outline  Vacuum polarization and running coupling without magnetic field  QCD coupling constant in a strong magnetic field  Anisotropic pressure and critical temperature  summary

J.D.Griffiths, Introduction to Elementary particle Screening effect by a dielectric medium 1 vacuum polarization and running coupling without magnetic field

QED Comparization of QED and QCD QCD Electron and positron are half-integer spin and ruled by Pauli exclusion principle. QED vacuum has normal diamagnetic properties and is screening. Gluons are integer. Gluon behave like a paramagnetic medium. And this implies antiscreening. But if the test charges are close together, they can penetrate each others’ particle cloud and will not feel any screening or antiscreening.

Renormalized running coupling constant Color electric permitivity QCD asymptotic freedom and antiscreening are dominated by the one-loop contribution in vacuum polarization tensor

(a) Vacuum polarization (b) Fermion self energy (c) Vertex correction Renormalized QCD coupling constant —dimensional regularization

Our motivation is to explain how the magnetic field enters into the coupling constant and to investigate the anisotropic structure of QCD matter under strong magnetic field. 2 QCD coupling constant in a strong magnetic field

Preceding work on coupling constant depending on magnetic field In 2002, Miransky, hep-ph/ In 2013, Andreichikov, Orlovsky, Simonov, PhysRevLett Meson mass 1) Analytical expression at ultra-strong magnetic field Magnetic field is of the order of the energy scale of the Fermions

Since there is no LQCD data for, they do the fit to reproduce 2) Fit the Lattice QCD result for any value of magnetic field In 2014, Ferreira, PRD89, (2014) Preceding work on coupling constant depending on magnetic field

Quark-loop contribution to gluon self energy Coordinate space Transform it into momentum space with the help of Ritus eigenfunction method

Ritus eigenfunctions method the parabolic cylinder functions The fermion self-energy operator is a function of the operators and

Quark loop to the gluon self-energy In the low energy region, fermions in the LLL only contribute to the longitudinal components of the polarization tensor.

Approximation To satisfy the asymptotic limits Quark loop contribution will produce the anisotropic structure. Quarks in the LLL will contribute to the coupling constant in longitudinal direction.

Ferrer, Incera, Wen PRD,91 (2015) Anisotropic structure of coupling constant

3 Anisotropic pressure and critical temperature Anisotropic pressure in nuclear matter or quark matterreflects the breaking of the rotational symmetry by the magnetic field. Isayev & Yang PLB 707 (2012) 163Wen PRD (2013) Ferrer et.al, PRD82 (2010)

Ferrer, Incera, Wen PRD,91 (2015) Magnetic field dependent coupling constant is used to interpret the inverse magnetic catalysis. is the spin operator.

Summary 1.A magnetic field affects the color Coulomb potential through quark loops with gluon external legs. 2. If the field is strong enough to force the quarks to remain in the LLL. The degeneracy factor is propotional to eB. The dynamics is dimensionally reduced to D-2. 3.The loops of these LLL quarks will lead to a significant anisotropy in the gluon self-energy and hence in the coupling because these loops only contribute to the longitudinal components of the self-energy. Thank you!