We determine the equation of state of 2+1-flavor QCD with physical quark masses, in the presence of a constant (electro)magnetic background field on the lattice. To determine the free energy at nonzero magnetic fields we develop a new method, which is based on an integral over the quark masses up to asymptotically large values where the effect of the magnetic field can be neglected. The method is compared to other approaches in the literature and found to be advantageous for the determination of the equation of state up to large magnetic fields. Thermodynamic observables including the longitudinal and transverse pressure, magnetization, energy density, entropy density and interaction measure are presented for a wide range of temperatures and magnetic fields, and provided in ancillary files. The behavior of these observables confirms our previous result that the transition temperature is reduced by the magnetic field. We calculate the magnetic susceptibility and permeability, verifying that the thermal QCD medium is paramagnetic around and above the transition temperature, while we also find evidence for weak diamagnetism at low temperatures.

Article funded by SCOAP3

0$. Note that the total energy of the system, $\epsilon^{\rm total}=\epsilon+\epsilon^{\rm field}$, includes the energy of the medium $\epsilon$ as well as the work necessary to maintain the constant external field, $\epsilon^{\rm field}=\eBM$~\cite{kittel2004elementary}. The two expressions in eq.~(\ref{eq:fundeq}) thus correspond to two different conventions for the definition of the energy density. The entropy density and the magnetization can be obtained as \be \frac{1}{V}\fracp{\F}{T} = -s,\quad\quad\quad \frac{1}{V}\fracp{\F}{(eB)} = -\M. \label{eq:diffrels} \ee The corresponding differential relation for the pressure is somewhat more involved. Since the magnetic field marks a preferred direction, the pressures $p_i$ in the transverse (perpendicular to $\mathbf{B}$) and in the longitudinal (parallel to $\mathbf{B}$) directions may be different. In ref.~\cite{Bali-ml-2013esa} we have shown that this possible anisotropy depends on the precise definition of $p_i$. Writing the volume as the product of linear extents $V=L_x L_y L_z$, the pressure components are related to the response of the system to compressions along the corresponding directions, i.e. \be p_i = -\frac{1}{V} L_i \frac{\partial \F}{\partial L_i}. \label{eq:pidef} \ee In order to unambiguously define $p_i$, we have to specify the trajectory in parameter space, along which the partial derivative is evaluated. In ref.~\cite{Bali-ml-2013esa}, we have distinguished between a setup where the magnetic field $B$ is kept fixed during the compression (the ``$B$-scheme''), and a setup where the magnetic flux $\Phi=eB \cdot L_xL_y$ is kept fixed (the ``$\Phi$-scheme''). The $B$-scheme results in \emph{isotropic} pressures, whereas the $\Phi$-scheme gives \emph{anisotropic} pressures: \be p_x^{(B)}=p_y^{(B)}=p_z,\quad\quad\quad p_x^{(\Phi)}=p_y^{(\Phi)}=p_z-\eBM. \label{eq:BPHI} \ee The difference in the transverse components for the $\Phi$-scheme is due to the fact that the compressing force in this case also acts against the magnetic field. Note that the definition of the pressures as spatial diagonal components of the energy-momentum tensor exhibits the $\Phi$-scheme anisotropy~\cite{Ferrer-ml-2010wz}. This is due to the fact that the energy-momentum tensor is usually defined through the variation of the action with respect to the metric at fixed $\Phi$, see ref.~\cite{Bali-ml-2013esa}. In contrast to $p_{x,y}$, the longitudinal pressure is independent of the scheme, and in the thermodynamic limit $V\to\infty$ simplifies to \be p_z=-f. \label{eq:pzdef} \ee Note that the appropriate scheme to be used depends on the physical situation that one would like to describe. In particular, it is specified by the trajectory $B(L_i)$, along which the compression perpendicular to the magnetic field proceeds. As will be explained below, in lattice regularization it is natural to keep the flux fixed, and thus, the lattice measurements correspond directly to the $\Phi$-scheme. However, this does not represent a limitation of the lattice approach, since one can easily translate from one scheme into another. The pressure components for a general $B(L_i)$ trajectory (``general scheme'') can be found by combining our results for the longitudinal pressure and for the magnetization (both are contained in online resources on the paper's page), \be p_x^{\rm (general)} = p_z + \M \cdot L_x\frac{\partial (eB)}{\partial L_x}. \label{eq:generalscheme} \ee This relation reproduces the $B$- and $\Phi$-schemes, eq.~(\ref{eq:BPHI}), for the trajectories $B(L_i)=B$ and $eB(L_i)=\Phi/(L_xL_y)$, respectively. Another important observable for the EoS is the interaction measure (trace anomaly), \be I\equiv\epsilon-p_x-p_y-p_z, \label{eq:Idef} \ee which contains the energy of the medium and the three pressures. Thus, $I$ also depends on the scheme:\footnote{One may understand the scheme-dependence of $I$ as follows. The trace anomaly represents the response to a rescaling of the length scale $\xi$ in the system. To define this rescaling unambiguously, the trajectory $B(\xi)$ has to be specified, i.e.\ a scheme has to be chosen. Hence, $I$ becomes scheme-dependent. As a simple example, consider the magnetic field in the absence of particles. Taking into account the energy $B^2/2$ of the magnetic field, one obtains $I^{(\Phi)}=0$, while $I^{(B)}=2B^2$. Notice that in the $\Phi$-scheme a dimensionless number characterizes the magnetic field, whereas in the $B$-scheme we introduced a dimensionful parameter into the system. This is reflected by the vanishing of the trace anomaly in the former case, and the nonzero value of $I$ in the latter.} \be I^{(B)} = \epsilon - 3p_z, \quad\quad\quad I^{(\Phi)} = \epsilon - 3p_z + 2\eBM, \label{eq:Idef2} \ee whereas the energy density (like $p_z$) is by construction scheme-independent, \be \epsilon = I^{(B)} + 3p_z = I^{(\Phi)} + 3p_z -2\eBM. \ee Eqs.~(\ref{eq:fundeq}),~(\ref{eq:pzdef}) and~(\ref{eq:Idef2}) reveal that the entropy density can also be calculated as \be s = \frac{\epsilon + p_z}{T}. \ee Finally, the derivative of the magnetization with respect to $B$ at vanishing magnetic field gives the magnetic susceptibility, \be \chi_B = \left.\fracp{\M}{(eB)}\right|_{B=0} = -\frac{1}{V} \left.\frac{\partial^2 \F}{\partial (eB)^2}\right|_{B=0}. \label{eq:defsusc} \ee ]]>

0$ to extract the quadratic part, resulting in smaller errors. Increasing instead the statistics at $B=0$ for the half-half method, we could also improve the signal-to-noise ratio of the latter, however finite volume effects should also be studied carefully in this case. Since the results at $eB>0$ will be in any case necessary for calculating the higher order contributions to the renormalized pressure, it is advantageous to use the generalized integral method to extract the quadratic term as well. We proceed by discussing the dependence of $\P[\Delta p_z]$ on $a$. First we perform the $B\to0$ extrapolation separately for each of our five lattice spacings. The resulting values are expected to lie on the curve $\QEDb(a)\cdot \log(\Lambda_{\rm H} a)$, see eq.~(\ref{eq:T0fB}). We consider the universal one-loop QCD corrections to $\QEDb(a)$ --- i.e.\ terms up to $i=1$ in eq.~(\ref{eq:betacorr}). The strong coupling in the lattice scheme is defined as $g^2(1/a)=6/\beta(a)$. The so obtained function $\QEDb(a)\cdot \log(\Lambda_{\rm H}a)$ is fitted to the data with $\Lambda_{\rm H}$ considered as a free parameter. The result is indicated by the orange band in the right panel of figure~\ref{fig:pertfit}. For comparison we also carry out a similar fit in the free case, which corresponds to a simple linear fit with fixed slope $\QEDb^{\rm free}=0.0169$ (which is obtained using eq.~(\ref{eq:betafree}) for three flavors). The two fits agree within errors for the whole range, indicating that for our lattice spacings, the QCD corrections to the QED $\beta$-function in eq.~(\ref{eq:betacorr}) are smaller than our statistical errors. \begin{figure} \centering \includegraphics[width=0.49\textwidth]{charge_renorm.pdf} \includegraphics[width=0.49\textwidth]{massdep.pdf} \vspace*{-.3cm} ]]>

0$), cf.\ eqs.~(\ref{eq:diffrels}) and~(\ref{eq:pzdef}). The two sides are compared in the right panel of figure~\ref{fig:splitting} for a zero-temperature lattice at $\beta=3.55$ (corresponding to $a=1.09 \textmd{ GeV}^{-1}$), showing nice agreement. Note that the relation eq.~(\ref{eq:consistency}) is only valid at vanishing temperature and is subject to corrections as $T$ grows. ]]>

0$, the magnetic field changes the thermal distribution and is expected to modify $s$. \begin{figure} \centering \includegraphics[width=0.49\textwidth]{cont_norm_entropy.pdf} \includegraphics[width=0.49\textwidth]{cont_norm_Dentropy.pdf} \vspace*{-.3cm} ]]>

0$) around and above the transition region. On the other hand, there appears to be a weakly diamagnetic region ($\chi_B<0$) at low temperatures $T\lesssim 100 \textmd{ MeV}$, where pions dominate. \item {\bf Validity of HRG and of PT.} For the susceptibility, the Hadron Resonance Gas model breaks down already at $T\approx 120 \textmd{ MeV}$. However, perturbation theory successfully describes the lattice data at suprisingly low temperatures $T\approx 200-300 \textmd{ MeV}$. \end{itemize} The presence of the background magnetic field necessitates the renormalization of the electric charge. This has several implications: \begin{itemize} \item {\bf Renormalization.} The pressure undergoes additive renormalization at $T=0$. The divergent term that needs to be subtracted is logarithmic in the lattice spacing and its coefficient equals the lowest-order QED $\beta$-function (with QCD corrections at the scale $1/a$). \item {\bf Magnetic catalysis.} At zero temperature, the phenomenon of magnetic catalysis (the enhancement of the quark condensate by the magnetic field) to quadratic order in $eB$ is related to the positivity of the QED $\beta$-function. \item {\bf Susceptibility.} For high temperatures, the magnetic susceptibility increases logarithmically with $T$, at a rate given by the QED $\beta$-function (with QCD corrections at the scale $T$). \item {\bf Adler function.} The second derivative of the entropy density with respect to the magnetic field is related to the perturbative Adler function at high temperatures. \end{itemize} In addition, we considered characteristic points of a few observables to explore the QCD phase diagram in the $B-T$ plane. This analysis indicates that the transition temperature decreases as $B$ grows, in agreement with our previous results where other thermodynamic observables (light quark condensate, strange quark number susceptibility and Polyakov loop) were used~\cite{Bali-ml-2011qj,Bruckmann-ml-2013oba}. Lattice results indicating this tendency have also been obtained using overlap fermions in $N_f=2$ QCD~\cite{Bornyakov-ml-2013eya} and in two-color QCD with four equally charged staggered quark flavors~\cite{Ilgenfritz-ml-2013ara}. The reduction of $T_c(B)$ has been reproduced within the bag model~\cite{Fraga-ml-2012fs} and is also supported by large $N_c$ arguments~\cite{Fraga-ml-2012ev}. Nevertheless, this feature remains a property of the chiral/deconfinement transition that most low-energy effective theories or models cannot reproduce, or only for a limited range of magnetic fields, see, e.g., refs.~\cite{Fayazbakhsh-ml-2010bh,Fraga-ml-2013ova,Andersen-ml-2013swa}. Recent studies of the Nambu-Jona-Lasinio model, however, indicate that taking into account a $B$-dependent Polyakov loop scale parameter~\cite{Ferreira-ml-2013tba}, or the magnetic field-induced running of the strong coupling~\cite{Farias-ml-2014eca,Ferreira-ml-2014kpa,Ayala-ml-2014iba} might resolve this discrepancy. ]]>