Phase diagram of QCD in a magnetic field

Recent advances in our understanding of the phase structure and the phase transitions of hadronic matter in strong magnetic fields $B$ and zero quark chemical potentials ${\mu }_f$ are reviewed in detail. Many aspects of QCD are described using low-energy effective theories and models such as the bag model, the hadron resonance gas model, chiral perturbation theory ($\chi \mathrm{PT}$), the Nambu–Jona-Lasinio (NJL) model, the quark-meson (QM) model, and Polyakov-loop extended versions of the NJL and QM models. Their properties and applications are critically examined. This includes mean-field calculations as well as approaches beyond the mean-field approximation such as the functional renormalization group. Renormalization issues are discussed and the influence of the vacuum fluctuations on the chiral phase transition is pointed out. At $T=0$, model calculations and lattice simulations predict magnetic catalysis: The quark condensate increases as a function of the magnetic field. This is covered in detail. Recent lattice results for the thermodynamics of non-Abelian gauge theories with emphasis on $\mathrm{S}\mathrm{U}(2{)}_c$ and $\mathrm{S}\mathrm{U}(3{)}_c$ are also discussed. In particular, inverse magnetic catalysis around the transition temperature $T_c$ as a competition between contributions from valence quarks and sea quarks resulting in a decrease of $T_c$ as a function of $B$ is focused on. Finally, recent efforts to modify models in order to reproduce the behavior observed on the lattice are discussed.

Figure 1

Magnetic field as a function of proper time $\tau$ for three different values of the impact parameter $b$. From [114].

Figure 2

Critical temperature as a function of $|qB|/m_{\pi }^2$ in the MIT bag model. See the main text for details.

Figure 3

Normalized quark condensate at one and two loops in the chiral limit as a function of $T$ for $B=0$ and $|qB|=5m_{\pi }^2$. From [8].

Figure 4

One-loop graphs contributing to the self-energy of the neutral pion.

Figure 5

Constituent quark masses $M_u$ and $M_d$ as functions of $B$ measured in units of $m_{\pi }^2/e$ for $c=0$. From [23].

Figure 6

Constituent quark mass $M$ in the chiral limit as a function of $T$ for three different values of the magnetic field and for $c=1/2$. From [23].

Figure 7

Constituent quark mass $M$ at the physical point as a function of $T$ for four different values of the magnetic field and for $c=1/2$. From [23].

Figure 8

Normalized effective potential ${\mathcal{F}}_{0+1}/f_{\pi }^4$ in the chiral limit for $T=0$. Tree level (solid curve), one loop with $|qB|=0$ (dashed curve), one loop with $|qB|=5m_{\pi }^2$ (dash-dotted curve), and one loop with $|qB|=10m_{\pi }^2$ (double dot-dashed curve).

Figure 9

The transition temperature for the chiral transition as a function of $|qB|/m_{\pi }^2$. Black squares are the chiral limit and red circles are the physical point.

Figure 10

The effective potential in the chiral limit normalized by $f_{\pi }^4$ for four different temperatures and $|qB|=5m_{\pi }^2$, where we have included the vacuum fluctuations (bottom curve, blue: $T=0$, next-to-bottom curve, pink: $T=140\mathrm{MeV}$, next-to-top curve, orange: $T=T_c=177.9\mathrm{MeV}$, and top curve, green: $T=185\mathrm{MeV}$).

Figure 11

The effective potential in the chiral limit normalized by $f_{\pi }^4$ for four different temperatures and $|qB|=5m_{\pi }^2$, where we omitted the vacuum fluctuations (bottom curve, blue: $T=0$, next-to-bottom curve, pink: $T=120\mathrm{MeV}$, next-to-top curve, orange: $T=T_c=158.0\mathrm{MeV}$, and top curve, green: $T=176\mathrm{MeV}$).

Figure 12

Coefficient ${\alpha }_2$ normalized to $f_{\pi }^2$ as a function of temperature $T$ for different values of the magnetic field. From [148].

Figure 13

Coefficient ${\alpha }_4$ as a function of temperature $T$ for different values of the magnetic field. From [148].

Figure 14

Coefficient ${\alpha }_4$ as a function of magnetic field $B$ at $T_c^{*}$. From [148].

Figure 15

Individual contributions to the HRG pressure as a function of the temperature $T$ for $B=0$. From [53].

Figure 16

Individual contributions to the HRG pressure as a function of the temperature $T$ for $|qB|=0.2{\mathrm{GeV}}^2$. From [53].

Figure 17

Phase diagram in the $B\text{−}T$ plane. From [80].

Figure 18

Normalized constituent quark mass and Polyakov loop in the chiral limit as a function of $T/m_{\pi }$. From [45].

Figure 19

Transition temperatures for the deconfinement and the chiral transition in the PNJL model as a function of $|qB|/m_{\pi }^2$ in two-color QCD. From [45].

Figure 20

Diagrammatic representation of the exact flow equation for the effective action ${\mathrm{\Gamma }}_k[\varphi ]$ for the quark-meson model. The lines denote the exact field-dependent propagators for bosons (solid) and fermions (dashed), while the circles denote the insertion of the regulator functions $R_k^B(p)$ or $R_k^F(p)$.

Figure 21

Normalized critical temperature $T_{\varphi }(B)/T_{\varphi }(0)$ for the chiral transition as a function of the magnetic field $B$ with and without the inclusion of the Polyakov loop.

Figure 22

The transition temperature for the chiral transition as a function of $|qB|/m_{\pi }^2$ for three different approximations normalized to the transition temperature at $B=0$. From [107].

Figure 23

Normalized constituent quark mass as a function of the normalized temperature for the three different approximations and different values $|qB|$. From [107].

Figure 24

Wave-function renormalization terms $Z_{k=0}^{\parallel }$ and $Z_{k=0}^{\perp }$ as functions of the normalized temperature for various magnetic fields. From [107].

Figure 25

Magnetic catalysis in the quark-meson model at the physical point and at $T=0$. The vacuum expectation value of the field $\varphi$ as a function of the magnetic field scaled by the ultraviolet cutoff $\mathrm{\Lambda }$.

Figure 26

Relative increment of the $⟨\overline{u}u⟩$ (black circles) and $⟨\overline{d}d⟩$ (red up triangles) condensates as well as their average (blue down triangles) as a function of $b$, where $|qB|\sim b(180\mathrm{MeV}{)}^2$. From [47].

Figure 27

Comparison of the continuum limit of the change of the condensate with that of chiral perturbation theory ([43]; [8], [9]) and the (P)NJL model ([81]). From [18].

Figure 28

Dyson-Schwinger equation for the inverse gluon propagator.

Figure 29

Regularized $⟨\overline{u}u⟩$ (black) and $⟨\overline{d}d⟩$ quark condensates (red) together with the $⟨\overline{u}u⟩$ condensate in the quenched approximation (dashed). From [132].

Figure 30

Comparison of the up-quark relative increment with the lattice results of [47]. $\omega$ is a parameter of the gluon dressing function. From [168].

Figure 31

Transition temperature normalized to $T_c$ at $B=0$ for the deconfinement and the chiral transition as a function of $|qB|/T^2$. Here $T=1/N_{t}a$, where $N_t$ is the number of lattice points and $a$ is the lattice spacing. From [46].

Figure 32

Transition temperature for the deconfinement as a function of $|qB|$ for different lattice spacings (solid curves) and the continuum-extrapolated result (band). From [15].

Figure 33

Phase diagram in the $B\text{−}T$ plane. From [54].

Figure 34

Relative increments $r(B)$, $r^{\text{val}}(B)$, and $r^{\text{sea}}(B)$ at zero temperature See the main text for details. From [47].

Figure 35

Spectral density of the Dirac operator for three different values of the magnetic field $B$. From [31].

Figure 36

Scatter plot of the down-quark condensate as a function of $\mathrm{\Delta }{S}_f(B)$. The magnetic field strength is $|qB|\approx 0.5{\mathrm{GeV}}^2$ and $T\sim T_c$. From [31].

Figure 37

Mass dependence of the bare chiral condensate for three different values of the magnetic field and two different temperatures: 147 MeV (left) and 195 MeV (right). From [101].

Figure 38

Expectation values of the Polyakov loop (left) and the chiral condensate (right) vs $|qB|$ at $T=195\mathrm{MeV}$. From [101].

Figure 39

Conjectured phase diagram in the $B\text{−}T$ plane. The horizontal line is $T^{*}=195\mathrm{MeV}$. From [101].

Figure 40

Transition temperature $T_c$ for the chiral transition as a function of $T_0$ for different values of the magnetic field. From [68].

Figure 41

Normalized transition temperature $T_c$ for the chiral transition as a function of $|qB|$. The gray band is the lattice results from [18], From [68].

Figure 42

The average $(1/2)({\mathrm{\Sigma }}_u+{\mathrm{\Sigma }}_d)$ as a function of temperature $T$ for different values of the magnetic field. The data points are the lattice results from [18]. From [55].

Figure 43

Thermal susceptibility $\chi$ as a function of the temperature $T$ for different values of the magnetic field. From [55].

Figure 44

Pseudocritical temperature $T_{\mathrm{pc}}$ temperature as a function of the magnetic field $B$ in the NJL model with the $B$ and $T$-dependent coupling (199). From [55].

Figure 45

Normalized pseudocritical temperature $T_c^{\chi }/T_c^{\chi }(B=0)$ as a function of $|qB|$ in the NJL model with a $B$ and $T$-dependent coupling. From [58].

Figure 46

Pseudocritical temperatures for the chiral (solid line) and deconfinement (dashed line) transitions as functions of $|qB|$ in the PNJL model. From [58].

Figure 47

$T_c$ as a function of the Yukawa coupling $g$ for various values of $B$. $B=0$: black solid line, $|qB|=5m_{\pi }^2$: red dashed line, and $|qB|=10m_{\pi }^2$: blue dash-dotted line. From [68].

Figure 48

Critical temperature as a function of the Yukawa coupling $g$ for various values of the sharp cutoff as well as a renormalization scale of $\mathrm{\Lambda }=100\mathrm{MeV}$. From [11].