Equation of State of the Quark-Gluon Plasma in an External Magnetic Field to Fourth Order

Abstract: We extend a lattice-QCD determination of the response of the quark-gluon plasma to an external magnetic field using a Taylor expansion of the pressure at zero field. A zero-net-flux ``half-and-half'' magnetic-field configuration permits continuous differentiation with respect to the field amplitude on gauge ensembles generated at $B=0$. The electromagnetic vector potential is chosen to minimize stochastic noise in the Taylor operators. Previous results through $O(B<sup>2)$ are extended to $O(B<sup>4)$ using $2+1$-flavor HISQ/tree ensembles with $m_l/m_s=0.05$ and $N_t=8$. The renormalized quadratic coefficient confirms paramagnetic behavior above the QCD crossover. The fourth-order coefficient $C_4$ is small at high temperature and is enhanced near the crossover. At $T=167$ MeV the $C_4$ result at $48^3\times 8$ lies about two standard deviations above the corresponding $32<sup>3\times8$ one, indicating likely finite- volume effects. For $eB = 0.2$ GeV$<sup>2$, characteristic of magnetic fields relevant to heavy-ion phenomenology, the fourth-order contribution is small compared with the leading magnetic correction at moderate and high temperatures, but near the crossover the large-volume result is about half of the central value of the quadratic term, after the vacuum-pressure subtraction. We also place the method in the context of other lattice determinations of magnetic susceptibility and clarify the relevance of finite-volume effects for the thermal subtraction. This work was completed in 2014 and preliminary results were reported at Lattice 2014.