Lattice simulations have demonstrated that a background (electro)magnetic field
reduces the chiral/deconfinement transition temperature of quantum chromodynamics
for
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0$ is the elementary charge),
which enter in the product with $B$. The quark masses are set to their
physical values along the line of constant physics~\cite{Borsanyi-ml-2010cj}.
The magnetic field is oriented along the positive $z$-direction and has
the quantized flux
\be
\Phi \equiv (aN_s)^2 \,eB = 6\pi N_b, \quad\quad\quad N_b\in \mathds{Z},\quad\quad\quad 0\le N_b
3.25\GeVt$. In addition, we can also gain some insight by considering the dimensionless combination $T_cw_0$. How close full QCD at $eB=3.25\GeVt$ is to the asymptotic limit can then be quantified by matching $T_cw_0$ with the anisotropic theory. Multiplying our full QCD results for $w_0$ by the transition temperature (here we take the definition of $T_c$ employing the inflection point of the strange quark number susceptibility, cf.\ figure~\ref{fig:pd}), $T_cw_0$ is shown in the right panel of figure~\ref{fig:w0}. Motivated by the scaling of $\beta_c$ (cf.\ the right panel of figure~\ref{fig:kappa1}), the results are plotted against $1/\sqrt{eB}$. Employing the result for $w_0$ from ref.~\cite{Borsanyi-ml-2012zs}, at zero magnetic field we have $T_c(B=0)\cdot w_0(B=0) = 0.174(3) \GeV\cdot 0.1755(19)\textmd{\,fm} = 0.155(3)$. \begin{figure}[t] \centering \includegraphics[width=7.49cm]{w0.pdf} \includegraphics[width=7.49cm]{tcw02.pdf} ]]>
0.5 \textmd{ GeV}^{-1}$) and in the anisotropic pure gauge theory ($1/\sqrt{eB}=0$). The dashed line indicates the $B=0$ limit.]]>
\tr\B_\perp^2=\tr\E_\perp^2>\tr\E_\parallel^2$ at low temperatures. The same hierarchy is also observed in the anisotropic gauge theory.\footnote{To see how the anisotropic di\emph{electric} constant affects the chromo\emph{magnetic} components, it is instructive to consider the gauge potential $A_\mu$. A large value of $\kappa$ implies a suppression of $\tr\E_\parallel^2$ and a corresponding suppression of the fluctuations in $A_z$ and in $A_t$. This suppression propagates into the magnetic sector and creates the anisotropy between $\tr\B_\perp^2$ and $\tr\B_\parallel^2$. Indeed, while the former contains $A_z$, the latter does not.} We note that the calculation leading to eq.~(\ref{eq:LagB}) can also be performed for nonzero temperatures. At $T>0$ an additional factor appears in the proper time integral due to the sum over Matsubara frequencies, containing an elliptic $\Theta$-function. This factor decouples from the $B$-dependence, implying that even for $T>0$, only the chromo-dielectric constant is affected. The coefficient $\kappa$ is, however, altered as \be \kappa(B,T) = \frac{1}{24\pi^2} \sum_f |q_f/e| \frac{|eB|}{m_f^2} \int \dd s \,e^{-s}\, \Theta_3\Big[\frac{\pi}{2},e^{-m_f^2/(4sT^2)}\Big]. \ee The integral over $s$ equals unity at $T=0$ and is reduced monotonously (and smoothly) as the temperature grows. Simulating the anisotropic gauge theory according to the Lagrangian~(\ref{eq:LagB}) on the lattice, we found that the theory exhibits a first-order phase transition. Thus, since the smooth $\kappa(T)$ dependence does not affect the discontinuous transition, in order to locate the critical temperature it suffices to simulate the theory at fixed (large) $\kappa$~values. ]]>