since at low energy scales $v'/c < 1$, indicating a strong renormalization of the effective group velocity of the propagating quasi-particles due to the presence of the magnetic background fluctuations. In contrast, for larger energy scales the effect becomes weaker, thus recovering the asymptotic limit $v'/c \to 1$ as $q_0/m \gg 1$.

On the other hand, our results show that the effective quasi-particle charge experiences a strong renormalization $z_3 < 1$ at low energies $q_0/m$, while the effect becomes negligible a large energy scales $q_0/m \gg 1$, since $z_3 \to 1$ as an asymptotic limit. We interpret this as a charge screening due to the spatial magnetic fluctuations in the background. On the other hand, $z_3$ tends to decrease as a function of the average background magnetic field intensity, achieving an asymptotic limit $z_3 \to 1/3$ as shown in Eq. (7.59). We remark that, in a semi-classical picture the relevant length-scale is the Landau radius $\ell_B = 1/\sqrt{eB}$, representing the typical size of the “cyclotron radius" of the helicoidal trajectories that propagate along the magnetic field axis. Therefore, the stronger the magnetic field, the smaller the Landau radius, and hence the quasi-particle propagator is modulated towards higher momentum and energy components that, as previously discussed, are more sensitive to the magnetic noise renormalization effects, as is verified by the trend observed both in $z_3$ and in $v'/c$, that strongly decrease as the average magnetic field intensity increases $eB/m^2 \gg 1$.

The analysis and results presented in this work only concern the study of the quasi-particle fermion propagator in the noisy magnetic field background. However, the effective model obtained via the replica method and its consequences can be extended towards the study of other physical quantities, such as the photon polarization tensor.

# 8. QED Fermions in a noisy magnetic field background from the perspective of the effective action approach

## 8.1. Introduction

In this section, we discuss an extended analysis of the work carried out in Ref. [306]. The present approach explores the noisy magnetic background scenario, this time from the perspective of a non-perturbative approach. The basic idea is that the gauge field $A_{mu}$ can be decomposed according to

$$A^{\mu}(x) \to A^{\mu}(x) + A_{\text{BG}}^{\mu}(x) + \delta A_{\text{BG}}^{\mu}(x) \qquad (8.1)$$

where the first term on the right side in the previous equation represents the quantum photon field, whereas BG refers a classical background associated to the presence of an external magnetic field, which was assumed to be static and uniform. An additional novelty of the present analysis is the consideration of space-time fluctuations whereas in [306] only spatial dependent fluctuations were taken into account. The last term in the previous equation, $\delta A_{BG}^{\mu}(x)$, encodes the white-noise fluctuations with respect to the mean value $A_{BG}^{\mu}(x)$ that satisfy

$$\begin{aligned} \langle \delta A_{\text{BG}}^{j}(x) \delta A_{\text{BG}}^{k}(x') \rangle_{\Delta} &= \Delta_{B} \delta_{j,k} \delta^{(4)}(x - x'), \\ \langle \delta A_{\text{BG}}^{\mu}(x) \rangle_{\Delta} &= 0. \end{aligned} \qquad (8.2)$$

If we consider the case of relativistic heavy-ion collisions it is quite natural to incorporate the presence of spatio-temporal fluctuations in the background field, corresponding to the most general case. In our previous discussion [306], only spatial fluctuations as source of the noisy effects were considered. Here we extend such scenario. The source for such noisy gauge field will be a classical current $J_{cl}^{\mu}(x) + \delta J_{cl}^{\mu}(x)$. The noisy gauge fields are solutions of $\Box(A_{BG}^{\mu} + \delta A_{BG}^{\mu}) = J_{cl}^{\mu}(x) + \delta J_{cl}^{\mu}(x)$,
and the background noisy magnetic field $\mathbf{B}_{BG}+\delta\mathbf{B}_{BG} = \nabla \times (\mathbf{A}_{BG}+\delta\mathbf{A}_{BG})$ satisfies the Gauss law $\nabla \cdot (\mathbf{B}_{BG}+\delta\mathbf{B}_{BG}) = 0$. These statistical properties are represented by a Gaussian functional distribution of the form

$$dP[\delta A_{BG}^{\mu}] = \mathcal{N}e^{-\int d^4x \frac{[\delta A_{BG}^{\mu}(x)]^2}{2\Delta_B}} \mathcal{D}[\delta A_{BG}^{\mu}(x)], \qquad (8.3)$$

with the corresponding statistical average over background fluctuations defined by

$$\langle \hat{O} \rangle_{\Delta} = \int dP[\delta A_{BG}^{\mu}] \hat{O}[\delta A_{BG}^{\mu}]. \qquad (8.4)$$

There is a number of important physical scenarios where the presence of strong magnetic fields determine the dynamics of relativistic particles. The quark-gluon plasma [290, 291, 292, 293] and heavy-ion collisions [288, 289, 172, 171] are among them. For technical reasons, as we said, the theoretical analysis of such systems was simplified to the configuration of a constant (both static and uniform), such that the Schwinger proper time formalism can be applied [304, 303, 307]. However, in a more realistic description of such phenomena, the electromagnetic field may develop spatio-temporal patterns [289, 288] that will then in principle modify the associated physical predictions. We discussed such possibility in our recent work [306], in the frame of a perturbative analysis to conclude that such magnetic noise effects may indeed be relevant. In this work, we emphasize this point again, we revisit the problem from a non-perturbative point of view, in order to shed further light into such effects over a broader range of magnetic noise intensities. Following the analysis presented in our previous work [306], we shall consider a physical scenario where a classical and static magnetic field background, possessing local random fluctuations, modifies the quantum dynamics of a system of fermions. For this purpose, we shall assume the standard QED theory involving fermionic fields $\psi(x)$, as well as gauge fields $A^{\mu}(x)$. In the latter, we shall distinguish three physically different contributions [306]

$$A^{\mu}(x) \rightarrow A^{\mu}(x) + A_{BG}^{\mu}(x) + \delta A_{BG}^{\mu}(x). \qquad (8.5)$$

Here, $A^{\mu}(x)$ represents the dynamical photon quantum field, while BG stands for “background”, thus capturing the presence of a classical external field $\mathbf{B} = \langle \nabla \times \mathbf{A}_{BG}(x) \rangle_{\Delta}$, assumed to be static and uniform as imposed by the experimental conditions. Moreover, for this BG contribution, we consider the effect of white noise fluctuations $\delta A_{BG}^{\mu}(x)$ with respect to the mean value $A_{BG}^{\mu}(x)$, satisfying the statistical properties

$$\begin{aligned} \langle \delta A_{BG}^{j}(x) \delta A_{BG}^{k}(x') \rangle_{\Delta} &= \Delta_{B} \delta_{j,k} \delta^{(4)}(x-x'), \\ \langle \delta A_{BG}^{\mu}(x) \rangle_{\Delta} &= 0. \end{aligned} \qquad (8.6)$$

We remark that the sources of the noisy gauge field are, in the context of heavy-ion collisions, the anisotropic and incoherent distribution of ionic beams in the initial stages prior to thermalization, that are modeled here as a classical current $J_{cl}^{\mu}(x) + \delta J_{cl}^{\mu}(x)$. Therefore, our noisy gauge fields are solutions of $\Box(A_{BG}^{\mu} + \delta A_{BG}^{\mu}) = J_{cl}^{\mu}(x) + \delta J_{cl}^{\mu}(x)$, and hence by construction the background noisy magnetic field $\mathbf{B}_{BG} + \delta\mathbf{B}_{BG} = \nabla \times (\mathbf{A}_{BG} + \delta\mathbf{A}_{BG})$ satisfies the Gauss law $\nabla \cdot (\mathbf{B}_{BG} + \delta\mathbf{B}_{BG}) = 0$.

We also remark that in contrast with our previous work [306], where we assumed only spatial fluctuations, here we shall assume spatio-temporal fluctuations in the background gauge fields. These statistical properties are represented by a Gaussian functional distribution of the form

$$dP[\delta A_{BG}^{\mu}] = \mathcal{N}e^{-\int d^4x \frac{[\delta A_{BG}^{\mu}(x)]^2}{2\Delta_B}} \mathcal{D}[\delta A_{BG}^{\mu}(x)], \qquad (8.7)$$
with the corresponding statistical average over background fluctuations defined by

$$ \langle \hat{O} \rangle_{\Delta} = \int dP [\delta A_{\text{BG}}^{\mu}] \hat{O} [\delta A_{\text{BG}}^{\mu}]. \qquad (8.8) $$

In heavy-ion collisions (HIC), strong magnetic fields $\mathbf{B} = \nabla \times \mathbf{A}_{\text{BG}}$ are generated locally within a small spatial region whose characteristic length scale is $L \sim \sqrt{\sigma}$, with $\sigma$ being the effective cross-section. In these collisions, the dominant component of the magnetic field is along the axial z-direction, such that on average we have $\langle \mathbf{B} \rangle = \hat{e}_3 B$. There are also smaller transverse components $\delta B_x$ and $\delta B_y$ such that we can estimate the fluctuation of the field within the small collision region to be on the order of $(\delta B)^2 \sim (\delta B_x)^2 + (\delta B_y)^2$. Since many such collisions occur at different points in space and their time-span $\delta\tau \sim L'/c$, an approximate model for this physical scenario is provided by the magnetic random noise Eq. (8.6). By dimensional analysis, the magnitude of $\Delta_B$ is of the order:

$$ \Delta_B \sim (\delta B)^2 L^5 L' \sim (\delta B)^2 \sigma^{5/2} L'. \qquad (8.9) $$

The effective cross-section for a nuclear collision can be estimated as the fraction $f$ of the area of perfectly central collisions between two nuclei, each with a radius of $r_A$

$$ \sigma = f \pi r_A^2. \qquad (8.10) $$

Here, $f$ represents the fraction of the geometrical cross-section $\sigma_{\text{geom}}$, which is defined as the area of the circle with a radius of $r_1 + r_2 = 2R$ in a maximum peripheral collision, and the cross-section $\sigma_b$ for a peripheral collision with impact parameter $b$ [308, 302]:

$$ f = \frac{\sigma_b}{\sigma_{\text{geom}}} = \left( \frac{N_{\text{part}}}{2N} \right)^{2/3}, \qquad (8.11) $$

where $\sigma_b$ describes an effective nucleus of radius $b$. The nuclear radius is always written as $r_A = r_0 N^{1/3}$, where $N$ is the number of nucleons per ion and $r_0 \sim 10^{-3} \text{ MeV}^{-1}$. Here, $N_{\text{part}}$ is the number of participants corresponding to the effective nucleus.

Therefore, under these considerations, we have

$$ \Delta_B \sim \pi^{5/2} (\delta B)^2 r_0^5 N^{5/3} \left( \frac{N_{\text{part}}}{2N} \right)^{5/3} L'. \qquad (8.12) $$

In peripheral heavy-ion collisions, the magnetic field fluctuations along the transverse plane are approximately $|e \delta B| \sim m_{\pi}^2/4$, where $m_{\pi}$ is the pion mass [301, 302]. For an Au+Au collision with $N = 197$, and if $N_{\text{part}}/N = 1/2$, we obtain (for $L' \sim r_A$)

$$ e^2 \Delta_B \sim 1.6 \times 10^{-6} \text{ MeV}^{-2}, \qquad (8.13) $$

or for less central collisions with $N_{\text{part}}/N = 1/8$:

$$ e^2 \Delta_B \sim 1.6 \times 10^{-7} \text{ MeV}^{-2}. \qquad (8.14) $$

In general, for many technical details we will refer in what follows to Ref. [103], the original article were we presented our analysis. As we shall later show, the effects of magnetic noise are controlled by the dimensionless parameter $\Delta = e^2 \Delta_B m_f^2$, where $m_f$ is the fermion mass. If one considers the mass of the proton, then the relevant dimensionless scale would be

$$ \Delta_{\text{proton}} \sim 0.16 - 1.6, \qquad (8.15) $$

whereas if one considers the mass of the constituent quark species
$$ \Delta_{\text{quark}} \sim 0.018 - 0.18. \qquad (8.16) $$

These considerations allow us to write the Lagrangian for this model as a superposition of two terms

$$ \mathcal{L} = \mathcal{L}_{\text{FBG}} + \mathcal{L}_{\text{NBG}}. \qquad (8.17) $$

The first term represents the system of fermions (and photons) immersed in the deterministic background field (FBG)

$$ \mathcal{L}_{\text{FBG}} = \bar{\psi} (i \not\partial - e \not A_{\text{BG}} - e \not A - m_f) \psi - \frac{1}{4} F_{\mu\nu} F^{\mu\nu}, \qquad (8.18) $$

where $F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$ is the strength tensor for the dynamical quantum gauge fields (photons). The second term in the lagrangian Eq. (8.17) represents the interaction between the Fermions and the classical noise (NBG)

$$ \mathcal{L}_{\text{NBG}} = \bar{\psi} (-e \delta A_{\text{BG}}) \psi. \qquad (8.19) $$

The generating functional (in the absence of sources) for a given realization of the noisy fields is given by

$$ Z[A] = \int \mathcal{D}[\bar{\psi}, \psi] e^{i \int d^4 x [\mathcal{L}_{\text{FBG}} + \mathcal{L}_{\text{NBG}}]}. \qquad (8.20) $$

We need the generator of connected correlation functions to extract the physics of this system. However, as the presence of disorder is modeled by a statistical ensemble of different realizations of the magnetic background noise $\delta A_{\text{BG}}^{\mu}(x)$, we need to calculate the disorder-averaged generator of connected correlation functions $\langle \ln Z \rangle_{\Delta}$. For this purpose, we apply the replica method, which is based on the following identity [299]

$$ \langle \ln Z[A] \rangle_{\Delta} = \lim_{n \to 0} \frac{\langle Z^n[A] \rangle_{\Delta} - 1}{n}. \qquad (8.21) $$

Here, we defined the statistical average according to the Gaussian functional measure of Eq. (8.7) as in Eq. (8.8), and $Z^n$ is obtained by incorporating an additional “replica" component for each of the Fermion fields, i.e. $\psi(x) \to \psi^a(x)$, for $1 \le a \le n$. The “replicated" Lagrangian has the same form as Eqs. (8.18) and (8.19), but with an additional sum over the replica components of the Fermion fields. Therefore, the averaging procedure leads to

$$ \begin{aligned} \langle Z^n[A] \rangle_{\Delta} &= \int \prod_{a=1}^{n} \mathcal{D}[\bar{\psi}^a, \psi^a] \int \mathcal{D}[\delta A_{\text{BG}}^{\mu}] e^{-\int d^4 x \frac{[\delta A_{\text{BG}}^{\mu}(x)]^2}{2\Delta_B}} \\ & \times e^{i \int d^4 x \sum_{a=1}^{n} (\mathcal{L}_{\text{FBG}}[\bar{\psi}^a, \psi^a] + \mathcal{L}_{\text{DBG}}[\bar{\psi}^a, \psi^a])} \\ &= \int \prod_{a=1}^{n} \mathcal{D}[\bar{\psi}^a, \psi^a] e^{i \bar{S}[\bar{\psi}^a, \psi^a; A]}, \end{aligned} \qquad (8.22) $$

where in the last step we explicitly performed the Gaussian integral over the background noise, leading to the definition of the effective averaged action for the replica system

$$ \begin{aligned} i \bar{S}[\bar{\psi}^a, \psi^a; A] &= i \int d^4 x \left( \sum_a \bar{\psi}^a (i \not\partial - e \not A_{\text{BG}} - e \not A - m_f) \psi^a - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} \right) \\ & \quad - \frac{e^2 \Delta_B}{2} \int d^4 x \int d^4 y \sum_{a,b} \sum_{j=1}^3 \bar{\psi}^a(x) \gamma^j \psi^a(x) \bar{\psi}^b(y) \gamma_j \psi^b(y). \end{aligned} \qquad (8.23) $$

Clearly, we end up with an effective interacting theory between vector currents corresponding to different replicas, with a coupling constant proportional to the fluctuation amplitude $\Delta_B$ that characterizes the magnetic noise, as defined in Eq. (8.6).
The “free” part of the action corresponds to Fermions in the average background classical field $A_{\text{BG}}^{\mu}(x)$. We choose this background to represent a uniform, static magnetic field along the z-direction $\mathbf{B} = \hat{e}_3 B$, using the gauge [303]

$$A_{\text{BG}}^{\mu}(x) = \frac{1}{2}(0, -Bx^2, Bx^1, 0). \qquad (8.24)$$

In addition, as we shall focus on the analysis for the fermion propagator, we shall not consider the photons in this scenario $A^{\mu}(x) = 0$, and hence its corresponding strength tensor $F_{\mu\nu} = 0$.

## 8.2. Auxiliary boson fields

Let us now introduce a Hubbard-Stratonovich transformation via a set of complex boson fields $Q_j(x)$, by means of the Gaussian integral identity

$$ \begin{aligned} & e^{-\frac{e^2 \Delta_B}{2} \int d^4 x \int d^4 y \sum_{a,b}^{n} \sum_{j=1}^{3} \bar{\psi}^a(x) \gamma^j \psi^a(x) \bar{\psi}^b(y) \gamma_j \psi^b(y)} \\ &= \mathcal{N} \left[ \prod_{j=1}^{3} \int \mathcal{D}Q_j(x) \mathcal{D}Q_j^*(x) \right] e^{-\frac{e^2 \Delta_B}{2} \int d^4 x |Q_j(x)|^2 + i e \int d^4 x Q_j(x) \sum_{a=1}^{n} \bar{\psi}^a(x) \gamma^j \psi^a(x) - i e \int d^4 x Q_j^*(x) \sum_{a=1}^{n} \bar{\psi}^a(x) \gamma^j \psi^a(x)} \end{aligned} \qquad (8.25) $$

With this transformation into the averaged, replicated $n^{th}$ power of the generating functional Eq. (8.22), we obtain the equivalent form

$$ \begin{aligned} \langle Z^n[A_{\text{BG}}] \rangle_{\Delta} &= \mathcal{N} \left[ \prod_{j=1}^{3} \int \mathcal{D}Q_j(x) \mathcal{D}Q_j^*(x) \right] e^{-\frac{e^2 \Delta_B}{2} \int d^4 x |Q_j(x)|^2} \\ & \quad \times \left[ \prod_{a=1}^{n} \int \mathcal{D}[\bar{\psi}^a, \psi^a] \right] e^{i \int d^4 x \left\{ \sum_{a=1}^{n} \bar{\psi}^a(x) (i \not\partial - e A_{\text{BG}} - m_f - e \gamma^j (Q_j - Q_j^*)) \psi^a(x) \right\}} \\ &= \mathcal{N} \left[ \prod_{j=1}^{3} \int \mathcal{D}Q_j(x) \mathcal{D}Q_j^*(x) \right] e^{-\frac{e^2 \Delta_B}{2} \int d^4 x |Q_j(x)|^2} \left[ \det (i \not\partial - e A_{\text{BG}} - m_f - e \gamma^j (Q_j - Q_j^*)) \right]^n \\ &= \mathcal{N} \left[ \prod_{j=1}^{3} \int \mathcal{D}Q_j(x) \mathcal{D}Q_j^*(x) \right] e^{-\frac{e^2 \Delta_B}{2} \int d^4 x |Q_j(x)|^2 + n \text{Tr} \ln [i \not\partial - e A_{\text{BG}} - m_f - e \gamma^j (Q_j - Q_j^*)]} \end{aligned} \qquad (8.26) $$

Based on this identity, and combined with Eq. (8.21), we obtain the effective action

$$ \begin{aligned} i S_{\text{eff}}[A_{\text{BG}}] - i S_0[A_{\text{BG}}] &= \langle \ln Z[A_{\text{BG}}] \rangle_{\Delta} - \ln Z_0[A_{\text{BG}}] \\ &= \mathcal{N} \left[ \prod_{j=1}^{3} \int \mathcal{D}Q_j(x) \mathcal{D}Q_j^*(x) \right] e^{-\frac{e^2 \Delta_B}{2} \int d^4 x |Q_j(x)|^2} \lim_{n \to 0} \frac{1}{n} \left[ e^{n \text{Tr} \ln [i \not\partial - e A_{\text{BG}} - m_f - e \gamma^j (Q_j - Q_j^*)]} - 1 \right] - \ln Z_0[A_{\text{BG}}] \\ &= \langle \text{Tr} \ln (i \not\partial - e A_{\text{BG}} - m_f - e \gamma^j (Q_j - Q_j^*)) \rangle_{\Delta} - \text{Tr} \ln (i \not\partial - e A_{\text{BG}} - m_f), \end{aligned} \qquad (8.27) $$

where $\langle (\cdot) \rangle_{\Delta}$ represents the average over the Gaussian functional measure of the complex fields $Q_j(x)$. Let us define the inverse fermion propagator, including the classical background field, as follows

$$ S_{\text{F}}^{-1}(x-y) = (i\not\partial - eA_{\text{BG}} - m_f)_x \delta^{(4)}(x-y) \qquad (8.28) $$
Therefore, for the effective action we have

$$
\begin{align}
i S_{\text{eff}}[A_{\text{BG}}] &= \langle \text{Tr} \ln (i\partial\!\!/ - eA_{\text{BG}} - m_f - e\gamma^j(Q_j - Q_j^*)) \rangle_{\Delta} \nonumber \\
&= \langle \text{Tr} \ln (S_F^{-1} - e\gamma^j(Q_j - Q_j^*)) \rangle_{\Delta} \nonumber \\
&= \text{Tr} \ln S_F^{-1} \nonumber \\
&\quad + \langle \text{Tr} \ln (\mathbf{1} - eS_F\gamma^j(Q_j - Q_j^*)) \rangle_{\Delta}. \tag{8.29}
\end{align}
$$

By noticing that i S₀[A BG] = Tr ln S⁻¹F, we have

$$
\begin{align}
i S_{\text{eff}}[A_{\text{BG}}] & - i S_0[A_{\text{BG}}] \nonumber \\
& = \langle \text{Tr} \ln (\mathbf{1} - e S_{\text{F}} \gamma^j (Q_j - Q_j^*)) \rangle_{\Delta} \tag{8.30}
\end{align}
$$

where the right hand side contains all the effects of the noise.

8.3. Saddle-point approximation (Mean-field)

The result in Eq. (8.30) can be expressed in the explicit functional integral form

$$
i S_{\text{eff}}[A_{\text{BG}}] - i S_0[A_{\text{BG}}] = \mathcal{N} \left[ \prod_{j=1}^{3} \int \mathcal{D}Q_j(x) \mathcal{D}Q_j^*(x) \right] e^{-\frac{\delta}{\Delta_B} \int d^4x |Q_j(x)|^2 + \ln[\text{Tr} \ln(\mathbf{1} - eS_F \gamma^j(Q_j - Q_j^*))]} . \quad (8.31)
$$

In order to study the effects of the background noise, we shall adopt a mean-field approximation, by searching for the
saddle-point of the exponent in Eq. (8.26)

$$
\begin{equation}
\begin{aligned}
& \frac{\delta}{\delta Q_j(x)} \left\{ -\frac{2}{\Delta_B} \int d^4 y Q_l(y) Q_l^*(y) + \ln \left[ \text{Tr} \ln \left[ \mathbf{1} - e S_F \gamma^l (Q_l - Q_l^*) \right] \right] \right\} = 0, \\
& \frac{\delta}{\delta Q_j^*(x)} \left\{ -\frac{2}{\Delta_B} \int d^4 y Q_l(y) Q_l^*(y) + \ln \left[ \text{Tr} \ln \left[ \mathbf{1} - e S_F \gamma^l (Q_l - Q_l^*) \right] \right] \right\} = 0.
\end{aligned}
\tag{8.32}
\end{equation}
$$

This condition leads to the equation (assuming homogeneous solutions of the form $Q_j(x) \equiv Q_j$)

$$
Q_j^* = -e \frac{\Delta_B \text{Tr} \left[ S_F \gamma^j (\mathbf{1} - e S_F \gamma^l (Q_l - Q_l^*))^{-1} \right]}{2 \text{Tr} \ln [\mathbf{1} - e S_F \gamma^l (Q_l - Q_l^*)]} \quad (8.33)
$$

From the second equation in Eq. (8.32), we obtain the additional condition

$$
Q_j^* = -Q_j. \tag{8.34}
$$

We can combine both equations, by noticing that

$$
\begin{align}
Q_j + Q_j^* &= 0 \notag \\
q_j &= Q_j - Q_j^* = 2i \operatorname{Im} Q_j \neq 0, \tag{8.35}
\end{align}
$$

where the second line implies that $q_j$ is a pure imaginary number, and it satisfies the non-linear equation

$$
q_j = e \Delta_B \frac{\text{Tr} \left[ S_F \gamma^j (\mathbf{1} - e S_F \gamma^l q_l)^{-1} \right]}{\text{Tr} \ln [\mathbf{1} - e S_F \gamma^l q_l]} . \quad (8.36)
$$
<figure><img src="image_34.png" /><figcaption>**Figure 8.1:** Non-trivial solutions of Eq. (8.56) for the Case 1 as a function of $\mathcal{B}$ for $\Delta \in [0.1, 0.04]$. The dashed line represents the smooth envelope connecting the discrete non-trivial solutions.</figcaption></figure>

The numerator of this equation can be expanded as an infinite geometric series as follows

$$ 
\begin{aligned}
\mathrm{Tr} \left[ S_F \gamma^j (\mathbf{1} - e S_F \gamma^l q_l)^{-1} \right] &= \sum_{l=0}^{\infty} e^l \mathrm{Tr} \left[ S_F \gamma^j (S_F \gamma^k q_k)^l \right] \\
&= \sum_{l=1}^{\infty} e^l \left[ \Pi_{\alpha=1}^l q_{k_\alpha} \right] \mathrm{Tr} \left[ S_F \gamma^j S_F \gamma^{k_1} S_F \gamma^{k_2} \dots S_F \gamma^{k_l} \right].
\end{aligned} 
\quad (8.37) $$

On the other hand, the denominator can also be expanded by means of the Taylor series for the natural logarithm, as follows

$$ 
\begin{aligned}
\mathrm{Tr} \ln [\mathbf{1} - e S_F \gamma^l q_l] &= \sum_{l=1}^{\infty} \frac{e^l}{l} \mathrm{Tr} \left[ (S_F q) \right]^l \\
&= \sum_{l=1}^{\infty} \frac{e^l}{l} \left[ \Pi_{\alpha=1}^l q_{k_\alpha} \right] \mathrm{Tr} \left[ S_F \gamma^{k_1} S_F \gamma^{k_2} \dots S_F \gamma^{k_l} \right].
\end{aligned} 
\quad (8.38) $$

From the exact expression Eq. (8.36), we can extract the leading contribution by retaining terms up to third order in the $q_j$ coefficients in the numerator, while retaining up to second order in the denominator

$$ q_j = e \Delta_B \frac{e \mathcal{M}^{jl} q_l + e^3 \mathcal{M}^{jlmn} q_l q_m q_n}{\frac{e^2}{2} \mathcal{M}^{mn} q_m q_n}, \quad (8.39) $$

where we defined the matrix coefficients

$$ 
\begin{aligned}
\mathcal{M}^{jl} &= \mathrm{Tr} [S_F \gamma^j S_F \gamma^l] \\
&= \int \frac{d^4 k}{(2\pi)^4} \mathrm{tr} [S_F(k) \gamma^j S_F(-k) \gamma^l],
\end{aligned} 
\quad (8.40) $$

and

$$ 
\begin{aligned}
\mathcal{M}^{jlmn} &= \mathrm{Tr} [S_F \gamma^j S_F \gamma^l S_F \gamma^m S_F \gamma^n] \\
&= \int \frac{d^4 k}{(2\pi)^4} \mathrm{tr} [S_F(k) \gamma^j S_F(k) \gamma^l S_F(k) \gamma^m S_F(k) \gamma^n].
\end{aligned} 
\quad (8.41) $$

Here, tr[·] stands for trace over the space of Dirac matrices. The Schwinger propagator in Fourier space is defined by Eq. (7.15) and, more importantly for calculation purposes, by its alternative form Eq. (7.17).
<figure><img src="image_35.png" /><figcaption>Figure 8.2: Non-trivial solutions for $q_3$ of Eq. (8.56) as a function of $eB$ for $\Delta_B \in [0.1, 4]$.</figcaption></figure>

We can analyze the possible solutions to Eq. (8.39), by casting it into the form of a quasi-linear system

$$ (\Delta_B \mathcal{M} + \tilde{\mathcal{M}}[\mathbf{q}]) \mathbf{q} = 0, \qquad (8.42) $$

where we defined

$$ [\tilde{\mathcal{M}}[\mathbf{q}]]^{jl} = \left( -\frac{1}{2} \delta^{jl} \mathcal{M}^{mn} + e^2 \Delta_B \mathcal{M}^{jlmn} \right) q_m q_n. \qquad (8.43) $$

There is always a trivial solution $\mathbf{q} = 0$ to Eq. (8.42). However, nontrivial solutions $\mathbf{q}$ may exist provided that the (nonlinear) matrix coefficient is singular, i.e.

$$ \det (\Delta_B \mathcal{M} + \tilde{\mathcal{M}}[\mathbf{q}]) = 0. \qquad (8.44) $$

In order to analyze this second condition, we need to evaluate the matrix coefficients explicitly. For this purpose, we first discuss the mathematical representation of the Schwinger propagator in the next section.

## 8.4. The Schwinger propagator

As discussed in the previous section, the matrix coefficients depend on traces and integrals involving products of the fermion propagator immersed in the constant background magnetic field. Therefore, this allows us to use directly the Schwinger proper-time representation of the free-Fermion propagator dressed by the background field, whose direction is chosen along the z-axis, $\mathbf{B} = \hat{e}_3 \mathbf{B}$, as follows [304, 303]

$$ [S_F(k)]_{a,b} = -i\delta_{a,b} \int_0^\infty \frac{d\tau}{\cos(eB\tau)} e^{i\tau(k_\parallel^2 - k_\perp^2 \frac{\tan(eB\tau)}{eB\tau} - m_f^2 + i\epsilon)} \\ \times \left\{ \begin{aligned} & [\cos(eB\tau) + \gamma^1\gamma^2 \sin(eB\tau)] (m_f + k_\parallel) \\ & + \frac{k_\perp}{\cos(eB\tau)} \end{aligned} \right\}, \qquad (8.45) $$

which is clearly diagonal in replica space. Here, as usual, we separated the parallel from the perpendicular directions with respect to the background external magnetic field by splitting the metric tensor as $g^{\mu\nu} = g_{\parallel}^{\mu\nu} + g_{\perp}^{\mu\nu}$, with

$$ \begin{aligned} g_{\parallel}^{\mu\nu} &= \operatorname{diag}(1, 0, 0, -1), \\ g_{\perp}^{\mu\nu} &= \operatorname{diag}(0, -1, -1, 0), \end{aligned} \qquad (8.46) $$
thus implying that for any 4-vector, such as the momentum $k^\mu$, we write

$$k = k_{\perp} + k_{\parallel}, \qquad (8.47)$$

and

$$k^2 = k_{\parallel}^2 - k_{\perp}^2, \qquad (8.48)$$

respectively. In particular, we have $k_{\parallel}^2 = k_0^2 - k_3^2$, while $k_{\perp} = (k^1, k^2)$ is the Euclidean 2-vector lying in the plane perpendicular to the field, such that its square-norm is $k_{\perp}^2 = k_1^2 + k_2^2$. The Schwinger propagator can alternatively be expressed as [306]

$$ \begin{aligned} [S_F(k)]_{a,b} &= -i\delta_{a,b} [(m_f + k) A_1 \\ &\quad + (ieB)\gamma^1\gamma^2 (m_f + k_{\parallel}) \frac{\partial A_1}{\partial k_{\perp}^2} + (ieB)^2 k_{\perp} \frac{\partial^2 A_1}{\partial(k_{\perp}^2)^2}] \\ &= -i\delta_{a,b} \left[ (m_f + k_{\parallel}) A_1 + \gamma^1\gamma^2 (m_f + k_{\parallel}) A_2 + A_3 k_{\perp} \right] \end{aligned} \qquad (8.49) $$

Here, we defined the function

$$A_1(k, B) = \int_0^{\infty} d\tau e^{i\tau(k_{\parallel}^2 - m_f^2 + i\epsilon)} - i \frac{k_{\perp}^2}{eB} \tan(eB\tau), \qquad (8.50)$$

that clearly reproduces the scalar propagator (with Feynman prescription) in the zero-field limit

$$\lim_{B \to 0} A_1(k, B) = \frac{i}{k^2 - m_f^2 + i\epsilon}, \qquad (8.51)$$

and its derivatives

$$ \begin{aligned} A_2(k, B) &\equiv \int_0^\infty d\tau \tan(eB\tau) e^{i\tau(k_\parallel^2 - t_B(\tau)k_\perp^2 - m_f^2 + i\epsilon)} \\ &= ieB \frac{\partial A_1}{\partial(k_\perp^2)}, \end{aligned} \qquad (8.52a) $$

$$ \begin{aligned} A_3(k, B) &\equiv \int_0^\infty \frac{d\tau}{\cos^2(eB\tau)} e^{i\tau(k_\parallel^2 - t_B(\tau)k_\perp^2 - m_f^2 + i\epsilon)} \\ &= A_1 + (ieB)^2 \frac{\partial^2 A_1}{\partial(k_\perp^2)^2}. \end{aligned} \qquad (8.52b) $$

As we showed in detail in our previous work [306], an exact representation for the function (8.50) is given by

$$ \begin{aligned} A_1(k) = & \frac{i e^{-\frac{k_{\parallel}^2}{eB}} e^{-\frac{ie\pi}{2eB} (k_{\parallel}^2 - m_f^2 + i\epsilon)}}{2eB} \Gamma\left(-\frac{k_{\parallel}^2 - m_f^2 + i\epsilon}{2eB}\right) \\ & \times U\left(-\frac{k_{\parallel}^2 - m_f^2 + i\epsilon}{2eB}, 0, \frac{2k_{\perp}^2}{eB}\right), \end{aligned} \qquad (8.53) $$

where $\Gamma(z)$ is the Gamma function, while $U(a, b, z)$ represents the Tricomi's Confluent Hypergeometric function.

## 8.5. Results and Discussion

For the analysis of our numerical results, it is convenient to define the following dimensionless groups,

$$ \begin{aligned} \Delta &= e^2 \Delta_B m_f^2, \\ \mathcal{B} &= \frac{eB}{m_f^2}, \end{aligned} \qquad (8.54) $$
<figure><img src="image_36.png" /><figcaption>Figure 8.3: Critical magnetic field $B_c$ of Fig. 8.2 as a function of $\Delta$, in two different ranges of such parameter.</figcaption></figure>

respectively. The matrices $M^{ij}$ and $M^{ijkl}$, as defined by Eq. (8.40) and Eq. (8.41) respectively, are calculated by tracing over the Dirac matrix space. All technical details concerning the evaluation of traces as well as the resulting momentum integrals are explained in appendices A and B in [103]. The interested reader is referred to those Appendices for further mathematical details.

In order to analyze the existence and features of non-trivial solutions for the order parameter $q = (q_1, q_2, q_3)$, we solve the secular equation Eq. (8.44), by assuming two different symmetry conditions, according to the directions orthogonal ($\perp$) and parallel ($\parallel$) to the magnetic field, respectively.

### 8.6. Case 1: $q_3^2 \equiv q_\parallel^2$, with $q_1 = q_2 = 0$

If we impose the condition $q_1 = q_2 = 0$ into Eq. (8.43), we have

$$ \begin{aligned} [\tilde{\mathcal{M}}[q_\parallel]]^{jl} &= \left(-\frac{1}{2}\delta^{jl}\mathcal{M}^{33} + e^2\Delta_B\mathcal{M}^{jl33}\right) q_\parallel^2 \\ &\equiv C_\parallel^{jl} q_\parallel^2. \end{aligned} \qquad (8.55) $$

Therefore, substituting this reduced expression into the secular equation Eq. (8.44), we obtain

$$ \det (\Delta_B \mathcal{M} + q_\parallel^2 C_\parallel) = 0. \qquad (8.56) $$

Furthermore, using elementary matrix properties and given that the matrix $\mathcal{M}$ is non-singular, we can manipulate the expression above as follows

$$ \begin{aligned} \det (\Delta_B \mathcal{M} + q_\parallel^2 C_\parallel) &= \\ \det (\Delta_B \mathcal{M}) \cdot \det (\mathbf{1}_3 + q_\parallel^2 \Delta_B^{-1} \mathcal{M}^{-1} C_\parallel) &= 0. \end{aligned} \qquad (8.57) $$

Given that $\mathcal{M}$ is non-singular, the above expression implies the secular condition

$$ \det (\mathbf{1}_3 + q_\parallel^2 \Delta_B^{-1} \mathcal{M}^{-1} C_\parallel) = 0. \qquad (8.58) $$

Our analysis at this point is consistent, up to second order powers in the components of the order parameter. Therefore,
<figure><img src="image_37.png" /><figcaption>**Figure 8.4:** Discrete solutions for the order parameter (normalized by its asymptotic limit) as a function of $\Delta$, for fixed $\mathcal{B}$ (filled squares). The continuous line represents the envelope function defined by Eq. (8.59) before the conditions of Eq. (8.35) have been applied.</figcaption></figure>

applying the elementary identity $\det(1 + \epsilon X) = 1 + \epsilon \operatorname{tr} X + O(\epsilon^2)$, we expand Eq. (8.58) to obtain

$$ 
\begin{aligned}
q_{\parallel}^2 &= -\frac{\Delta_B}{\operatorname{tr}(\mathcal{M}^{-1}\mathcal{C}_{\parallel})} \\
&= -\frac{\Delta_B}{-\frac{1}{2}\mathcal{M}^{33}\operatorname{tr}(\mathcal{M}^{-1}) + e^2\Delta_B\operatorname{tr}(\mathcal{M}^{-1}\tilde{\mathcal{M}}^{(33)})} \\
&= \frac{\chi_{\parallel}\Delta_B}{1 + \mathcal{K}_{\parallel}\Delta_B},
\end{aligned} 
\qquad (8.59) $$

where we defined the parameters

$$ 
\begin{aligned}
\chi_{\parallel} &= \frac{2}{\mathcal{M}^{33} \operatorname{tr}(\mathcal{M}^{-1})} \\
\mathcal{K}_{\parallel} &= \frac{-2e^2 \operatorname{tr}(\mathcal{M}^{-1} \tilde{\mathcal{M}}^{(33)})}{\mathcal{M}^{33} \operatorname{tr}(\mathcal{M}^{-1})},
\end{aligned} 
\qquad (8.60) $$

as well as the reduced matrix

$$ [\tilde{\mathcal{M}}^{(33)}]^{jl} \equiv \mathcal{M}^{jl33}. \qquad (8.61) $$

Figure 8.1 illustrates the non-trivial solutions of Eq. (8.56) for Case 1, as a function of the external magnetic field and various values of $\Delta$. It can be observed that there exists a region where the discrete solutions exhibit a monotonically increasing pattern with a smooth envelope, abruptly terminated at a point beyond which only the trivial solution $q_{\parallel} = 0$ exists. We refer to this point as the *critical magnetic field* $\mathcal{B}_c$. A similar scenario arises when the magnitude of $\Delta$ is increased, as demonstrated in Fig. 8.2. Furthermore, it is worth noting that for higher values of $\Delta$, the solutions become approximately identical, and the $\mathcal{B}_c$ converges toward a specific limit as is shown in Fig. 8.3.

In line with the results presented above, it is worth emphasizing that a specific combination of parameters $(\Delta, \mathcal{B})$ plays a pivotal role in giving rise to a discrete set of non-trivial solutions characterized by $q_{\parallel} \neq 0$, or in the context of
<figure><img src="image_38.png" /><figcaption>Figure 8.5: Non-trivial solutions of Eq. (8.56) for the Case 2 as a function of B for $\Delta \in [0.1, 0.04]$. The dashed line is the smooth envelope connecting those discrete non-trivial solutions.</figcaption></figure>

Eq. (8.35), resulting in purely imaginary values for $q_j$. This behavior is depicted in Figure 8.4, offering valuable insights into the system's dynamics. Figure 8.4 illustrates a spectrum of these parameters, revealing that for fixed values of B not all the values of $\Delta$ produce non-trivial solutions (those are discrete). We have also shown in Figure 8.4, by a dashed line, the smooth envelope of those discrete points. Nevertheless, for higher values of $\Delta$ the solutions become a quasi-continuum and saturate to the value $q_{\parallel}^2(\infty)$ defined as

$$q_{\parallel}^2(\infty) = \lim_{\Delta \to \infty} q_{\parallel}^2 = \frac{\chi_{\parallel}}{\mathcal{K}_{\parallel}}, \qquad (8.62)$$

where $q_{\parallel}^2$ is given in Eq. (8.59). In the opposite limit, for very small values of $\Delta$, Eq. (8.59) shows that the order parameter follows an approximately linear trend with a slope defined by

$$\chi_{\parallel} = \lim_{\Delta_B \to 0} \frac{\partial}{\partial \Delta_B} q_{\parallel}^2. \qquad (8.63)$$

## 8.7. Case 2: $q_1^2 = q_2^2 \equiv q_{\perp}^2$, with $q_3 = 0$

In this case, the matrix Eq. (8.43) reduces to

$$ \begin{aligned} [\tilde{M}[q_{\perp}]]^{jl} &= \left( -\frac{1}{2}\delta^{jl}(M^{11} + M^{22} + M^{12} + M^{21}) \right. \\ & \qquad \left. + e^2 \Delta_B (M^{jl11} + M^{jl22} + M^{jl12} + M^{jl21}) \right) q_{\perp}^2 \\ & \equiv C_{\perp}^{jl} q_{\perp}^2. \end{aligned} \qquad (8.64) $$

For this case, the secular Eq. (8.44) reduces to

$$\det (\Delta_B \mathcal{M} + q_{\perp}^2 \mathcal{C}_{\perp}) = 0, \qquad (8.65)$$

and repeating the same procedure as described in Case 1, we obtain

$$\det (\mathbf{1}_3 + q_{\perp}^2 \Delta_B^{-1} \mathcal{M}^{-1} \mathcal{C}_{\perp}) = 0. \qquad (8.66)$$
<figure><img src="image_39.png" /><figcaption>Figure 8.6: Critical magnetic field $B_c$ of Fig. 8.5 as a function of $\Delta$ for the Case 2.</figcaption></figure>

Finally, just as in Case 1, we retain only second order powers of $q_{\perp}$ in Eq. (8.66) to arrive at the explicit algebraic solution

$$q_{\perp}^2 = - \frac{\Delta_B}{\text{tr}(\mathcal{M}^{-1}\mathcal{C}_{\perp})} = \frac{\chi_{\perp}\Delta_B}{1 + \mathcal{K}_{\perp}\Delta_B}, \qquad (8.67)$$

where we defined the parameters

$$\begin{aligned} \chi_{\perp} &= \frac{2}{\left(\sum_{m,n=1,2} \mathcal{M}^{mn}\right) \text{tr}(\mathcal{M}^{-1})}, \\ \mathcal{K}_{\perp} &= \frac{-2e^2 \sum_{m,n=1,2} \text{tr}(\mathcal{M}^{-1}\tilde{\mathcal{M}}^{(mn)})}{\left(\sum_{m,n=1,2} \mathcal{M}^{mn}\right) \text{tr}(\mathcal{M}^{-1})}, \end{aligned} \qquad (8.68)$$

and the reduced matrices

$$[\tilde{\mathcal{M}}^{(mn)}]^{jl} = \tilde{\mathcal{M}}^{jlmn}. \qquad (8.69)$$

As already discussed in Eq. (8.35), the order parameter $q_j$ (for $j = 1, 2, 3$) is a pure imaginary number.

Figure 8.5 illustrates the non-trivial solutions for $q_{\perp}$ in Case 2, considering various values of $\Delta$. Similar to Case 1, these solutions are discrete and highly dependent on the magnetic field and noise parameter. Notably, the critical magnetic field in this scenario is lower compared to Case 1. Furthermore, Fig. 8.6 demonstrates the behavior of the critical magnetic field $B_c$, which exhibits a distinct functional form from that shown in Fig. 8.3. The relationship between $q_{\perp}^2$ and $\Delta$, with an envelope given by Eq. (8.67), is depicted in Fig. 8.7, where discrete non-trivial solutions at particular values of $\Delta$ are permissible for a constant magnetic field. Here, we can also identify the value at which the sigmoidal Eq. (8.67) saturates as a function of the noise $\Delta_B$,

$$q_{\perp}^2(\infty) = \lim_{\Delta \to \infty} q_{\perp}^2 = \frac{\chi_{\perp}}{\mathcal{K}_{\perp}}. \qquad (8.70)$$

In the opposite limit, for very small values of $\Delta$, Eq. (8.67) shows that the order parameter follows an approximately linear trend with a slope defined by

$$\chi_{\perp} = \lim_{\Delta_B \to 0} \frac{\partial}{\partial \Delta_B} q_{\perp}^2. \qquad (8.71)$$
<figure><img src="image_40.png" /><figcaption>**Figure 8.7:** Discrete solutions for the order parameter (normalized by its asymptotic limit) as a function of $\Delta$, for fixed $\mathcal{B}$ (filled squares). The continuous line represents the envelope function defined by Eq. (8.67) before the conditions of Eq. (8.35) have been applied.</figcaption></figure>

Consequently, we can infer that the underlying physics in both cases is closely related, and the mechanism leading to the emergence of a non-trivial value of the order parameter maintains a consistent nature and interpretation, regardless of the specific values of $(q_1, q_2, q_3)$.

## 8.8. Physical interpretation of the order parameter $Q_j$

The physical interpretation of the order parameter components can be obtained from the functional representation Eq. (8.26), before we integrate out the fermions, to obtain, via saddle-point, the mean expectation value

$$Q_j = ie\Delta_B \langle\langle \bar{\psi}\gamma_j\psi \rangle\rangle_{\Delta} = -Q_j^*, \qquad (8.72)$$

where here the double bracket stands for the statistical average over the classical noise in the background field, as well as the quantum expectation value of the corresponding observable, which clearly is a component of the vector current for the fermions, $j = 1, 2, 3$.

It is important to remark the physical effect that this order parameter, at the mean-field level, produces in the disorder-averaged fermion propagator, that is given by

$$iS_{F,\Delta}^{-1}(x-y) = (i\delta - eA_{BG} - m_f - e\delta)_x \delta^{(4)}(x-y), \qquad (8.73)$$

where we defined $\delta = \gamma^j q_j$, for $q_j = Q_j - Q_j^*$ the order parameter. Clearly then, the differential equation for the propagator in the presence of the magnetic noise is given by

$$(i\delta - eA_{BG} - m_f - e\delta)_x S_{F,\Delta}(x-y) = i\delta^{(4)}(x-y). \qquad (8.74)$$
It is straightforward to verify that, if $S_F(x - y)$ represents the Schwinger propagator in the presence of the average background magnetic field, then the function

$$S_{F,\Delta}(x-y) = e^{-ieq \cdot (x-y)} S_F(x-y) \qquad (8.75)$$

is a solution to Eq. (8.74). Indeed, by direct substitution, we have

$$ \begin{aligned} (i\not\partial - eA_{BG} - m_f - e\not q)_x S_{F,\Delta}(x-y) &= (i\not\partial - eA_{BG} - m_f - e\not q)_x [e^{-ieq \cdot (x-y)} S_F(x-y)] \\ &= e^{-ieq \cdot (x-y)} (i\not\partial - eA_{BG} - m_f)_x S_F(x-y) = i\delta^{(4)}(x-y), \end{aligned} \qquad (8.76) $$

where in the last step we applied the definition Eq. (8.28) for the Schwinger propagator in the absence of noise. Therefore, we conclude that at the level of the disorder-averaged propagator, Eq. (8.75) introduces an exponential damping effect given that the order parameter $q_j$ is a pure imaginary number.

## 8.9. Summary

In this section, we considered a system of QED fermions submitted to an external, classical magnetic field. In particular, we studied the effects of white noise in this magnetic field with respect to an average uniform value $\langle \mathbf{B} \rangle_{\Delta} = \hat{e}_3 B$, as a function of the standard deviation $\Delta_B$, over the fermion propagator. As discussed in the introduction to this section, this represents a statistical model for the actual scenario in heavy-ion collisions, where strong magnetic fields emerge for very short times within small spatial regions, whose size is of the order of the scattering cross-section. Since several such collisions occur at different points in space, the physical situation can be represented by a statistical ensemble, for different realizations of the magnetic field fluctuations, which are then described as a random variable.

We analyzed our model by applying the replica formalism, that led us to an effective action in terms of auxiliary boson fields. A mean field analysis of the corresponding effective action reveals that the magnetic noise effects can be captured by an order parameter, whose physical interpretation is the statistical ensemble average of the expectation value of the fermion vector current components. Therefore, non-trivial solutions where this order parameter acquires a non-zero value break the U(1) gauge symmetry in the system, as a consequence of the statistical noise in the background magnetic field. An interesting feature of such non-trivial solutions is that they exist only for certain discrete values of the average background magnetic field. Such discrete values can be identified to be in correspondence with the quantized Landau levels associated to the average background field. This feature is then consistent with the interpretation of the order parameter as the ensemble average of the fermion current. In addition, for a fixed value of the disorder strength characterized by $\Delta_B$, we find an upper critical value of the average background magnetic field $B_c$, beyond which the non-trivial solutions cease to exist in favour of a vanishing order parameter. This region of parameter space is then characterized by a dominance of the average background field over noise, whose effect then becomes negligible. In contrast, in the limit of very strong magnetic noise $\Delta_B \to \infty$, we observe that the order parameter asymptotically saturates to a constant finite value $q_{\parallel,\perp}^2(\infty)$ that depends on the field, but is independent of $\Delta_B$, as can be clearly seen from Eq. (8.59) and Eq. (8.67), respectively.

Remarkably, in the context of the fermion propagator, we showed that the order parameter, which is strictly imaginary, represents a finite screening length that leads to weak localization effects. Our present analysis is restricted to the fundamental level of the fermion propagator, but its consequences could manifest themselves in physical observables, such as effective collision rates for certain processes.
# 9. Photon mass generation from magnetic fluctuations

## 9.1. Introduction

The importance of the role of magnetic fields in processes related to high-energy physics has been garnering attention in recent years. In particular, there is growing theoretical evidence indicating that during the early stages of a heavy-ion collision (HIC), very intense magnetic fields are created, which must be included in current calculations and data analysis [309, 310]. Moreover, these magnetic fields are expected to have a direct impact on the final observables. Therefore, it is crucial to understand their role and the modifications of the observables that arise from magnetic-related contributions. For instance, the emission of prompt photons during these early stages has been studied, revealing that the inclusion of external magnetic fields in the participant's region introduces an additional source of photons carrying a characteristic elliptic flow [171, 311, 312, 313]. This phenomenon arises from the breaking of spatial symmetry due to the presence of a privileged direction marked by the magnetic field.

On the other hand, the creation of intense magnetic fields in HICs coincides with the Color Glass Condensate (CGC) stage, where the medium is purely described by a high-density gluon occupation [314, 315, 316, 317]. Indeed, the prompt photon production by gluon fusion or splitting processes is enabled by the presence of magnetic fields which are otherwise not allowed. Therefore, it is important to ask: How can the magnetic fields modify the dispersive relations of gauge fields, such as gluons and photons? In an effort to answer that question, the one-loop polarization tensor for photons and gluons in the presence of an external magnetic field has been computed, both for strong and weak limits [318]. Those works showed that the presence of constant magnetic fields doesn't modify the dispersion relation of such particles, i.e., they don't generate an associated magnetic mass. The latter is achieved by computing the poles of the inverse propagator at the limit of vanishing four momenta, which is crucial for preserving the $U(1)$ symmetry and ensuring the fulfillment of the Ward-Takahashi identity [319].

Clearly, assuming the existence of a constant magnetic field in HICs is not very realistic. In fact, simulations of collisions for various centralities, energies, and species indicate that although the magnetic field in the collision center is predominantly oriented in one spatial direction, there are fluctuations in other directions [302]. In some cases, these fluctuations are of the order of 1/4 of the highest value of the magnetic field. Following this scenario, in previous works, we proposed to study the impact of classical background magnetic fields, which possess stochastic fluctuations, on the properties of a QED medium [320, 321]. By employing the so-called replica trick [322], we derived an effective interaction term for QED fermions in the presence of such a noisy magnetic field. This approximation leads, at the perturbative level, to a renormalization of the fermion propagator, which now describes quasi-particles propagating in a dispersive medium. The results from both perspectives revealed that magnetic fluctuations break the $U(1)$ symmetry, and thus, the dispersion relations for gluons and photons may be modified.

In this section, by computing the one-loop polarization tensor in the limit of vanishing four momenta, we show that gauge fields may acquire a magnetic mass due to fluctuations in the external magnetic field. The structure of the paper is as follows: in Sec. 9.2, we propose the expression for the one-loop polarization tensor that incorporates white-noise stochastic fluctuations. In Sec. 9.3, we summarize the results in the limit $p^\mu \to 0$, where the expressions are analytically computed. In Sec. 9.4, we present the analysis to determine the magnetic masses for the gauge fields. Finally, in Sec. 9.5, we summarize and present our conclusions. This manuscript is based on the published version that can be found in Ref. [323].
## 9.2. The one-loop polarization tensor

To proceed with our analysis, we express the one-loop contribution to the photon/gluon polarization tensor, as illustrated in Fig. 9.1, as follows

$$i\Pi_{\Delta}^{\mu\nu} = -\frac{1}{2} \int \frac{d^4 k}{(2\pi)^4} \mathrm{Tr} \{ i q_f \gamma^{\nu} i S_{\Delta}^{(-)} (k) i q_f \gamma^{\mu} i S_{\Delta}^{(-)} (k-p) \} - \frac{1}{2} \int \frac{d^4 k}{(2\pi)^4} \mathrm{Tr} \{ i q_f \gamma^{\nu} i S_{\Delta}^{(+)} (-k+p) i q_f \gamma^{\mu} i S_{\Delta}^{(+)} (-k) \}, \quad (9.1)$$

where $q_f$ represents the electric charge of the fermion within the loop, accounting for both particle and antiparticle contributions.

<figure><img src="image_41.png" /><figcaption>**Figure 9.1:** Feynman diagrams contributing to the one-loop photon polarization tensor depict arrows in the propagators indicating the direction of charge flow, while dashed arrows represent momentum flux.</figcaption></figure>

To perform an analytical computation of Eq. (9.1), we will utilize the renormalized fermion propagator in the presence of static (quenched) white noise spatial fluctuations, focusing particularly on the regime of a strong external magnetic field [320, 321]. Specifically, the effective interaction between fermions arises from averaging over the background magnetic noise. We account for the effects of magnetic noise-induced interaction by modifying the Schwinger propagator with a self-energy, as illustrated diagrammatically in the Dyson equation shown in Fig. 7.1. It is noteworthy that in this theory, the skeleton diagram for the self-energy is depicted in Fig. 7.2, and the dressed propagator is determined by

$$iS_{\Delta}^{(\pm)}(p) = C(p) \left( m + \gamma_0 p^0 + \frac{\gamma_3 p^3}{z(p)} \right) \mathcal{P}^{(\pm)}(p), \qquad (9.2)$$

where $m$ is the fermion mass and

$$C(p) = i \frac{z(p) e^{-p_{\perp}^2 / |q_f B|}}{p_{\parallel}^2 - z^2(p) m^2}, \qquad (9.3a)$$

$$z(p) = 1 + \frac{3 \Delta |q_f B| e^{-p_{\perp}^2 / |q_f B|}}{4 \pi \sqrt{p_0^2 - m^2}}. \qquad (9.3b)$$

$$z_3(p) = \frac{1}{z(p)} \left( 1 + \frac{\Delta |q_f B| e^{-p_{\perp}^2 / |q_f B|}}{4\pi \sqrt{p_0^2 - m^2}} \right), \qquad (9.3c)$$

$$\mathcal{P}^{(\pm)}(p) = \frac{1}{2} [1 \pm \mathrm{sign}(q_f B) i z_3(p) \gamma^1 \gamma^2]. \qquad (9.3d)$$

As outlined in Ref. [320], the computation of the fermion propagator self-energy was carried out to order $\Delta \equiv q_f^2 \Delta_B$. Thus, to preserve consistency with this approximation level, we expand Eq. (9.2) as follows

$$iS_{\Delta}^{(\pm)}(p) = iS_{0}^{(\pm)}(p) + i\Delta \left( \frac{|q_f B|}{2\pi} \right) \left[ \Theta_1(p)(\not p_{\parallel} + m)\mathcal{O}^{(\pm)} - \Theta_2(p)\gamma^3\mathcal{O}^{(\pm)} \pm \Theta_3(p)i\gamma^1\gamma^2(\not p_{\parallel} + m) \right] + O(\Delta^2), \quad (9.4)$$
where

$$iS_0^{(\pm)}(p) = 2i \frac{e^{-\mathbf{p}_{\perp}^2 / |q_f B|}}{p_{\parallel}^2 - m^2} (\not p_{\parallel} + m) \mathcal{O}^{(\pm)} \quad (9.5)$$

is the fermion propagator in the presence of an intense magnetic field, the spin-projection operator is given by

$$\mathcal{O}^{(\pm)} = \frac{1}{2} [1 \pm \mathrm{sign}(q_f B) i \gamma^1 \gamma^2], \quad (9.6)$$

and we defined the functions

$$\Theta_1(p) \equiv \frac{3(p_{\parallel}^2 + m^2)e^{-2\mathbf{p}_{\perp}^2 / |q_f B|}}{(p_{\parallel}^2 - m^2)^2 \sqrt{p_0^2 - m^2}}, \quad (9.7a)$$

$$\Theta_2(p) \equiv \frac{3p_3 e^{-2\mathbf{p}_{\perp}^2 / |q_f B|}}{(p_{\parallel}^2 - m^2) \sqrt{p_0^2 - m^2}}, \quad (9.7b)$$

$$\Theta_3(p) \equiv \frac{e^{-2\mathbf{p}_{\perp}^2 / |q_f B|}}{(p_{\parallel}^2 - m^2) \sqrt{p_0^2 - m^2}}. \quad (9.7c)$$

Here, we distinguished between the parallel (‖) and perpendicular (⊥) Minkowski subspaces, delineated by their respective orientations relative to the background external magnetic field, by decomposing the metric tensor as

$$g^{\mu\nu} = g_{\parallel}^{\mu\nu} + g_{\perp}^{\mu\nu}, \quad (9.8a)$$

where

$$\begin{aligned} g_{\parallel}^{\mu\nu} &= \mathrm{diag}(1, 0, 0, -1) \\ g_{\perp}^{\mu\nu} &= \mathrm{diag}(0, -1, -1, 0). \end{aligned} \quad (9.8b)$$

The latter implies that for any four-vector

$$p^{\mu} = p_{\parallel}^{\mu} + p_{\perp}^{\mu}, \quad (9.9a)$$

we get

$$p^2 = p_{\parallel}^2 - \mathbf{p}_{\perp}^2, \quad (9.9b)$$

with

$$\begin{aligned} p_{\parallel}^2 &= p_0^2 - p_3^2 \\ \mathbf{p}_{\perp}^2 &= p_1^2 + p_2^2. \end{aligned} \quad (9.9c)$$

### 9.3. The one-loop polarization tensor in the limit $p^{\mu} \to 0$

From Eq. (9.1), the one-loop polarization tensor takes the form

$$i\Pi_{\Delta}^{\mu\nu} = i\Pi_{0}^{\mu\nu} + i \frac{q_f^2 |q_f B| \Delta}{4\pi} \sum_{i=1}^{3} T_i^{\mu\nu}, \quad (9.10)$$
where $i\Pi_0^{\mu\nu}$ is the one-loop polarization tensor in the strong field limit and in the absence of fluctuations [324]:

$$i\Pi_0^{\mu\nu} = \frac{i q_f^2 |q_f B|}{4\pi^2} e^{-\mathbf{p}_{\perp}^2 / 2|q_f B|} p_{\parallel}^2 \mathcal{I}_0(p_{\parallel}^2) \left( g_{\parallel}^{\mu\nu} - \frac{p_{\parallel}^{\mu} p_{\parallel}^{\nu}}{p_{\parallel}^2} \right), \qquad (9.11)$$

with

$$\mathcal{I}_0(p_{\parallel}^2) = \int_0^1 dx \frac{x(1-x)}{x(1-x)p_{\parallel}^2 - m^2}. \qquad (9.12)$$

Hence, in the limit $p^\mu \to 0$, the tensor is given by

$$\lim_{p^\mu \to 0} i\Pi_\Delta^{\mu\nu} = i \frac{q_f^2 |q_f B| \Delta}{4\pi} \sum_{i=1}^3 \lim_{p^\mu \to 0} T_i^{\mu\nu}, \qquad (9.13)$$

where

$$\lim_{p^\mu \to 0} T_1^{\mu\nu} = -\frac{i|q_f B|}{32\pi m} g_{\parallel}^{\mu\nu}, \qquad (9.14a)$$

$$\lim_{p^\mu \to 0} T_2^{\mu\nu} = -\frac{i|q_f B|}{2\pi m} (g_{\parallel}^{\mu\nu} + 2\delta_3^\mu \delta_3^\nu), \qquad (9.14b)$$

and

$$\lim_{p^\mu \to 0} T_3^{\mu\nu} = \frac{i|q_f B|}{3\pi m} \left[ g_{\perp}^{\mu\nu} - \frac{1}{4} g_{\parallel}^{\mu\nu} \right]. \qquad (9.14c)$$

Further details of the calculation can be found in Ref. [323].

## 9.4. Mass of gauge fields by magnetic fluctuations

To understand how magnetic field fluctuations might influence the potential generation of mass in the gauge fields, we identify the poles of the propagator. Utilizing the Dyson equation, the inverse of the propagator is given by

$$[D^{\mu\nu}(p)]^{-1} = [D_0^{\mu\nu}(p)]^{-1} - i\Pi^{\mu\nu}(p), \qquad (9.15)$$

where the “free” photon propagator, in the Feynman gauge is

$$D_0^{\mu\nu}(p) = \frac{-i g^{\mu\nu}}{(p^2 + i\epsilon)} \qquad (9.16)$$

Adopting the method detailed in Ref. [325], the poles corresponding to the dynamic mass appear as the coefficients of $g_{\parallel}^{\mu\nu}$ and $g_{\perp}^{\mu\nu}$, respectively, when evaluating the limits $p_0 \to 0$ and $\mathbf{p} \to 0$ in $i\Pi^{\mu\nu}(p)$. So then, from Eqs. (9.14):

$$[D^{\mu\nu}(p)]^{-1} = ig_{\parallel}^{\mu\nu} (p^2 + iM_{\parallel}^2 + i\epsilon) + ig_{\perp}^{\mu\nu} (p^2 - iM_{\perp}^2 + i\epsilon) + 3iM_{\perp}^2 i\delta_3^{\mu} \delta_3^{\nu} + \dots, \qquad (9.17)$$

where we defined the magnetic effective masses in both parallel and transverse projections by the coefficients

$$\begin{aligned} M_{\parallel}^2 &\equiv \frac{59\alpha_{\text{em}}B^2\tilde{\Delta}}{96\pi}m^2, \\ M_{\perp}^2 &\equiv \frac{\alpha_{\text{em}}B^2\tilde{\Delta}}{3\pi}m^2. \end{aligned} \qquad (9.18)$$

We emphasize the physical interpretation of these effective masses by comparing them with the poles of the photon propagator along each polarization direction, given by

$$p^2 + iM_{\perp, \parallel}^2 = p_0^2 - \mathbf{p}^2 + iM_{\perp, \parallel}^2 \qquad (9.19)$$
which indicates an effective photon dispersion relation for each polarization direction as

$$ \omega_{\perp, \parallel}(\mathbf{p}) = \sqrt{\mathbf{p}^2 - iM_{\perp, \parallel}^2}, \qquad (9.20) $$

where the corresponding effective mass is complex

$$ m_{\perp, \parallel} \equiv (-iM_{\perp, \parallel}^2)^{1/2} = \frac{1 - i}{\sqrt{2}} M_{\perp, \parallel}. \qquad (9.21) $$

The real part signifies the typical damping effect, while the imaginary part introduces an oscillatory component. This characterization aligns with our understanding of random magnetic fluctuations creating an effective dispersive medium for both fermions and photons (or gluons) equally.

## 9.5. Summary

In this section, we focused on the one-loop contribution to the gluon and photon polarization tensor when these gauge fields are in a medium subjected to a strong, external, and classical magnetic field with white-noise stochastic fluctuations. These fluctuations are incorporated into the fermion propagators by correcting the *free* fermion propagators using the so-called *replica trick*, which averages the fluctuations over the QED Lagrangian. This approach implies the existence of an effective model with fermion-fermion interactions, which is perturbatively expanded to find the propagator at order $\Delta$. By explicitly computing the tensor, we showed that it doesn't satisfy the Ward-Takahashi identity, indicating a break in the $U(1)$-QED symmetry. This finding is supported by previous works, using both perturbative and effective Lagrangian methods [320, 321].

The lack of $U(1)$ symmetry suggests a possible generation of magnetic mass for the considered gauge fields. In fact, by following the standard process to find the poles of the photon's propagator, we analytically computed the factor associated with these magnetic masses. The results show that the generated masses possess imaginary parts, indicating that in a fluctuating magnetic medium, a refractive index may appear. Importantly, the gauge field mass differs in each direction with respect to the magnetic field, also indicating birefringent effects due to the violation of Lorentz symmetry.

## 9.6. Conclusions and outlook

In this Part we have discussed recent theoretical developments whose aim is to elucidate properties of strongly interacting matter under the influence of magnetic fields. We have tackled the problem from the perspective of hadron as well as of fundamental QCD degrees of freedom. From the hadron perspective, special attention has been devoted to the magnetic driven change of meson masses, interactions as well as the relation of the topological cumulants with the chiral condensate in a magnetic background for infinite as well as finite volume. It is found that light meson masses are influenced by the mixing between pseudoscalar, vector and axial vector mesons induced by the magnetic field. Arguments to explore the quark-gluon structure using relatively low magnetic fields have been presented by discussing the calculation of meson form factors in meson exchange processes. It was also shown that finite volume corrections to thermodynamic observables and condensates in a magnetic background are substantial since magnetic effects enter into the chiral perturbation theory description at the same order than finite size effects.

From the point of view of fundamental QCD degrees of freedom, the properties of hot and dense strongly interacting matter in a very strong magnetic field have also been explored using pQCD. It is found that medium loop corrections are negligible, as compared to the free term, for very large magnetic fields and physical choices of the renormalization running
scale. Also, possible effects of the presence of a strong magnetic field at pre-equilibrium during a heavy-ion reaction have been studied in terms of the photon production channel. It is shown that both the yield and $v_2$ get enhanced allowing for an extra contribution that could explain the slight differences found when applying state-of-the-art descriptions to photon production in a relativistic heavy-ion collision environment. Finally, the consequences of a non-uniform, but instead fluctuating field, such as the one that would be more likely to be produced in physical situations, has been also explored. The presence of a noisy background modifies the particle propagators and shows up as an energy-dependent effective index of refraction causing screening for charged fermions as well as the generation of a photon/gluon mass.

The review discussed in this Part is clearly non-exhaustive, but it provides a picture of some of the state-of-the-art efforts to explore the strong interaction using as a probe a magnetic field. Important aspects of these studies such as the properties of screening masses and their relation to IMC [89, 106, 107, 120, 121, 326], the enhancement of particle emission induced by a magnetic background [327], effects on the Mott transition [56, 328], among others, have not been explored but certainly constitute avenues that promise to produce a clearer picture of the properties of strongly interacting matter probed by magnetic fields.
## Anomalous transport and non-perturbative
phenomena on and off the lattice

### 10. Introduction

Background electromagnetic fields source a variety of fascinating phenomena in non-Abelian quantum field theories like Quantum Chromodynamics (QCD) or the electroweak theory. For the strong interactions, all characteristic non-perturbative features of the QCD medium, like chiral symmetry breaking, confinement or topology, are affected by background fields in a substantial and non-trivial manner. Topological effects in particular, when combined with background electromagnetic fields or vorticity, can lead to novel transport phenomena. Due to the prominent role played by the axial anomaly for them, these are referred to as anomalous transport phenomena and are also found to arise in various condensed matter physics systems. Similarly intricate non-perturbative effects are generated by background fields in the electroweak theory.

In this Part, we discuss recent developments pertaining to the impact of background magnetic fields on non-perturbative phenomena in the realm of QCD and the electroweak theory. A special emphasis is put on the study of anomalous transport phenomena in QCD as well as on low-dimensional QED, relevant for certain condensed matter systems. Most of these effects are deeply non-perturbative and require either first-principles lattice field theory simulations or effective approaches like hydrodynamics.

We begin the discussion with lattice QCD simulations of anomalous transport phenomena. In Sec. 11, the leading-order conductivities are determined on the lattice for the chiral magnetic effect (CME) and the chiral separation effect (CSE) in equilibrium. The CSE coefficient is observed to be strongly suppressed in the chiral symmetry broken region and to approach its perturbative massless value as the temperature grows. In turn, the in-equilibrium CME coefficient is found to vanish for all temperatures. The chiral separation effect is investigated for nonzero chemical potentials in two-color QCD in Sec. 12 in the presence of heavy flavors. The latter are found to enhance the CSE conductivity and, simultaneously, also lead to a reduction of the electric conductivity. This type of behavior is discussed in relation to the Kondo effect.

Next, novel anomalous transport phenomena are studied in a hydrodynamic approach in Sec. 13, highlighting their potential phenomenological applications for various physical systems. Hydrodynamics is based on symmetries, while the quark-gluon plasma generated in heavy-ion collisions, neutron stars and astrophysical plasma is subject to symmetry-breaking, e.g., due to the chiral anomaly of QCD and strong external magnetic fields, which break parity and isotropy, respectively. Thus, in Sec. 13, the construction of the proper hydrodynamic description of such systems is reviewed taking these broken symmetries into account, which gives rise to novel hydrodynamic transport effects. One well-known hydrodynamic transport effect arising from the breaking of parity in a strong magnetic field is the CME. Sec. 13 discusses this effect far away from equilibrium in a holographic model, focusing on the dependence of the time-evolution of the CME currents on initial conditions.

Moving from the QCD setting to non-relativistic condensed matter systems, analogous anomalous transport phenomena can be found. The discovery of Dirac and Weyl materials, in which the charge carriers obey a relativistic-like equation
of motion, allowed for close analogies between high energy and condensed matter phenomena. While materials of this
kind in 3-space dimensions have been successfully explored, presenting an Abelian version of the CME, in Sec. 14 the
yet poorly explored (2+1)-dimensional case is discussed. The motivation for considering these materials is presented and
new analogies that include these systems are proposed. As a specific example, a new Hall effect, whose construction was
inspired by the symmetry patterns of the CME, is discussed in detail for planar Weyl materials.

Besides anomalous transport effects, background magnetic fields affect further non-perturbative features of quantum
field theories. In Sec. 15, the QCD phase diagram is investigated at nonzero baryon density in the presence of magnetic
fields. By determining the fluctuations of conserved charged in this magnetized and dense environment, the equation of
state of the system is constructed. Magnetic fields are expected to affect the phase structure not just in QCD, but also in
the electroweak theory. Sec. 16 reports about the latest developments in the study of a superconducting phase emerging
for magnetic fields comparable to the electroweak scale. This novel phase is found to involve inhomogeneous condensates
of both W and Z bosons with a vortex structure that has features of both fluids and solids. The transition between the
normal and superconducting phases is found to be an analytic crossover, which is followed, at a higher field, by another
crossover to a symmetric phase with restored electroweak symmetry.

# 11. Anomalous transport coefficients from lattice QCD

## 11.1. Introduction

Anomalous transport phenomena provide a very interesting testing ground for the properties of the QCD vacuum. The interplay between electromagnetic fields and vorticities with quantum anomalies yields a wide variety of these transport effects, whose properties have been subject of intense study in the last decades.

The most celebrated of these phenomena is the chiral magnetic effect (CME) [329], the generation of a vector current in the presence of a magnetic field and chiral imbalance. Its possible existence is not only interesting from a purely theoretical point of view, but it is also currently subject to an intense experimental effort. It was detected in condensed matter systems [330] and it is actively searched for in heavy-ion collisions [331, 332, 333].

Another example of these effects is the chiral separation effect (CSE) [334, 335], which corresponds to the existence of an axial current in dense and magnetized matter. Although often regarded as a “dual” of CME, the equilibrium properties of these two phenomena greatly differ, as it will be discussed in this work. Together, the CME and the CSE are expected to form the chiral magnetic wave (CMW) [336], a collective excitation that would arise in these extreme environments.

The main objective of this work is to review the analytic calculation of the CME/CSE conductivities (precisely defined below) for non-interacting fermions, highlighting the importance of proper UV regularization, as well as in comparison to lattice calculation results. We also present the results for the conductivities in the presence of interactions using full lattice QCD simulations. A detailed discussion of all these results can be found in [337, 338]. For a recent review on lattice investigations of anomalous transport phenomena and, in general, on effects of background electromagnetic fields in QCD, see [339].
## 11.2. Conductivities

Let us consider a system in the presence of a constant background magnetic field $B$, which without the loss of generality we choose to point in the third spatial direction. We can define the charged vector and axial currents

$$J_{\nu}^{Q} = \sum_{f} \frac{q_f}{e} \frac{T}{V} \int d^4 x \bar{\psi}_f(x) \gamma_{\nu} \psi_f(x), \quad J_{\nu 5}^{Q} = \sum_{f} \frac{q_f}{e} \frac{T}{V} \int d^4 x \bar{\psi}_f(x) \gamma_{\nu} \gamma_5 \psi_f(x), \qquad (11.1)$$

where $f = u, d, s, \dots$ labels the quark flavors and $q_f$ are the corresponding electric charges. Analogously, we can consider the baryon vector and axial currents

$$J_{\nu}^{\beta} = \frac{1}{3} \sum_{f} \frac{T}{V} \int d^4 x \bar{\psi}_f(x) \gamma_{\nu} \psi_f(x), \quad J_{\nu 5}^{\beta} = \frac{1}{3} \sum_{f} \frac{T}{V} \int d^4 x \bar{\psi}_f(x) \gamma_{\nu} \gamma_5 \psi_f(x). \qquad (11.2)$$

To parameterize the zeroth component of these currents (the density and the chiral density), we use chemical potentials, which can either couple to the electric charge $\mu^Q$, $\mu_5^Q$ or to the baryon number $\mu^B$, $\mu_5^B$. In the following we consider the physically motivated case of a charged vector current and a baryon chiral chemical potential for the CME, as well as a charged axial current and a baryon chemical potential for the CSE.<fnref n="2" /> For clarity, we drop these indices in the rest of the manuscript.

For studying both effects, we will consider the first coefficient in the Taylor expansion of the conductivities, i.e. the linear response to weak $B$ and $\mu_5/\mu$

$$\langle J_3 \rangle = C_{\text{CME}} \mu_5 eB + \mathcal{O}(\mu_5^3, B^3), \qquad (11.3)$$

$$\langle J_{35} \rangle = C_{\text{CSE}} \mu eB + \mathcal{O}(\mu^3, B^3). \qquad (11.4)$$

We will refer to $C_{\text{CME}}/C_{\text{CSE}}$ interchangeably as conductivity coefficients or simply conductivities. We can study these coefficients via derivatives of the currents with respect to the appropriate chemical potential

$$\left. \frac{\partial \langle J_3 \rangle}{\partial \mu_5} \right|_{\mu_5=0} = C_{\text{CME}} eB, \qquad (11.5)$$

$$\left. \frac{\partial \langle J_{35} \rangle}{\partial \mu} \right|_{\mu=0} = C_{\text{CSE}} eB. \qquad (11.6)$$

We note that the expectation values on the left-hand side are to be evaluated at zero $\mu/\mu_5$ but finite magnetic field $B$. This enables a lattice QCD study of $C_{\text{CSE}}$, since simulations at finite chemical potential suffer from a sign problem. Using the analogous approach when studying CME is motivated by the possibility of using the same ensembles and that although simulations at $\mu_5 \neq 0$ are possible, an implementation with staggered fermions is technically challenging [337].

To understand the precise nature of these effects in equilibrium, we present in the next section the study of $C_{\text{CME}}$ and $C_{\text{CSE}}$ for a system of free fermions.

## 11.3. Non-interacting fermions

The seminal paper [329] established CME as a non-dissipative, stationary effect, whose conductivity is fixed by the axial anomaly and hence topologically protected. It was further argued that it would survive in an equilibrium setup, with the

<footnote n="2">²For non-interacting fermions, there is no ambiguity in the choice of any possible combination. However, in full QCD different combinations change the result in the case of CSE [338], but not for CME [337]. In the different setups there is an overall factor $C_{\text{dof}}$ which we use to rescale all our results (see [338] for a more detailed discussion).</footnote>
same conductivity as the real-time effect. However in thermal equilibrium, the focus of our current investigation, Bloch's theorem forbids the global flow of conserved currents and so the $C_{CME}$ must vanish [340]$^3$. The role of UV regularization in the vanishing result has been widely raised in the CME literature, see [342, 343, 344, 345, 346] and references therein. We will now also illustrate it by the calculation of $C_{CME}$ for a system of free fermions. In addition, we present an analogous calculation of $C_{CSE}$ where we reproduce the known results for this conductivity [334, 335].

### 11.3.1. Analytic results

We consider one flavor of a colorless, non-interacting fermion of charge $q$ and mass $m$, in the presence of a constant and homogeneous magnetic field $B$. We start our discussion with the calculation of $C_{CME}$ at $T = 0$, which accounts to the calculation of the two-point function

$$ \left. \frac{\partial \langle J_3 \rangle}{\partial \mu_5} \right|_{\mu_5=0} = \frac{T}{V} \int d^4 x \int d^4 y \langle \bar{\psi}(x) \gamma_3 \psi(x) \bar{\psi}(y) \gamma_0 \gamma_5 \psi(y) \rangle. \qquad (11.7) $$

We regularize the UV behavior using the Pauli-Villars method (PV) in a textbook manner [347]. Unlike a momentum cutoff, this regulator respects gauge invariance in QED, which turns out to be a very important ingredient in the study of the CME. We introduce three new regulator fields with coefficients $c_s$ and masses $m_s$, reserving $s = 0$ for the physical field (with physical mass $m$). The parameters are then

$$ c_0 = c_1 = 1, \quad c_2 = c_3 = -1, \qquad (11.8) $$

$$ m_0^2 = m^2, \quad m_1^2 = m^2 + 2\Lambda^2, \quad m_2^2 = m_3^2 = m^2 + \Lambda^2, \qquad (11.9) $$

with $\Lambda \to \infty$ to be taken at the end of the calculation.

Using Wick's theorem we can rewrite the right hand side of Eq. (11.7) to obtain

$$ C_{\text{CME}} qB = \frac{iT}{V} \sum_{s=0}^{3} c_s \int d^4 x \int d^4 y \text{Tr} [\gamma_3 S_s(x, y) \gamma_0 S_s(y, x)], \qquad (11.10) $$

where the PV fields are already taken into account under the sum over $s$, and $S_s$ is the fermion propagator for the field $s$ in a homogeneous magnetic background. The latter reads [348]

$$ S_s(x, y) = \Phi(x, y) \int \frac{d^4 p}{(2\pi)^4} e^{-ip(x-y)} \tilde{S}_s(p). \qquad (11.11) $$

Here

$$ \Phi(x, y) = \exp [iqB(x_1 + y_1)(x_2 - y_2)/2], \qquad (11.12) $$

is the Schwinger phase, and

$$ \tilde{S}_s(p) = \int_0^\infty dz e^{iz(p_0^2 - m_s^2 - p_3^2) - i \frac{p_1^2 + p_2^2}{|qB|} \tan(z|qB|)} [\not p + m_s + (p_1\gamma_2 - p_2\gamma_1) \tan(zqB)] [1 - \gamma_1\gamma_2 \tan(zqB)]. \qquad (11.13) $$

Carrying out the traces we obtain

$$ C_{\text{CME}} qB = -4 \sum_{s=0}^{3} c_s \int \frac{d^4 p}{(2\pi)^4} \int_0^\infty dz_1 dz_2 e^{i(z_1+z_2)(p_0^2-m_s^2-p_3^2)} (m_s^2 - p_0^2 - p_3^2) e^{-i \frac{p_1^2+p_2^2}{qB} [\tan(z_1qB)+\tan(z_2qB)] [\tan(z_1qB)+\tan(z_2qB)]}. \qquad (11.14) $$

<footnote>$^3$Notice that Bloch's theorem allows local currents to flow, which have been discussed very recently in [341].</footnote>
Now we Wick rotate the momentum components $p_0 = ip_4$ and the Schwinger parameters $iz_1 = Z_1, iz_2 = Z_2$. The momenta integrals are simple Gaussians, and evaluating them factorizes the magnetic field dependence, uncovering an explicitly linear behavior in B. The formula is simplified to

$$C_{\text{CME}} = \sum_{s=0}^{3} c_s \frac{m_s^2}{4\pi^2} \int_{0}^{\infty} dZ_1 \int_{0}^{\infty} dZ_2 \frac{e^{-m_s^2(Z_1+Z_2)}}{Z_1 + Z_2}. \qquad (11.15)$$

Performing the integrals, we arrive at

$$C_{\text{CME}} = \frac{1}{4\pi^2} \sum_{s=0}^{3} c_s = 0. \qquad (11.16)$$

In this expression, the relevance of the regulator is clear. The PV fields cancel the contribution from the physical field, yielding a vanishing $C_{\text{CME}}$ in equilibrium.

It is straightforward to generalize this calculation to finite temperatures $T$, replacing the frequency component of the integral by a sum over fermionic Matsubara frequencies

$$\int \frac{dp_0}{2\pi} \rightarrow iT \sum_{n=-\infty}^{\infty}, \qquad (11.17)$$

while also replacing $p_0 \rightarrow i\omega_n = i2\pi T(n + 1/2)$ in $\tilde{S}_s(p)$. This does not change the result for $C_{\text{CME}}$, showing that the current vanishes for temperature independently.

An analogous calculation for $C_{\text{CSE}}$ leads, however, to a non-vanishing result. We start from

$$\left. \frac{\partial \langle J_{35} \rangle}{\partial \mu} \right|_{\mu=0} = \frac{T}{V} \int d^4 x \int d^4 y \langle \bar{\psi}(x) \gamma_0 \psi(x) \bar{\psi}(y) \gamma_3 \gamma_5 \psi(y) \rangle. \qquad (11.18)$$

Now

$$C_{\text{CSE}} qB = 4i \sum_{s=0}^{3} c_s T \sum_{n} \int \frac{d^3 p}{(2\pi)^3} \int_{0}^{\infty} dz_1 dz_2 e^{i(z_1+z_2)(-\omega_n^2-m_s^2-p_3^2)} (m_s^2 - \omega_n^2 + p_3^2) e^{-i \frac{p_1^2+p_2^2}{qB} [\tan(z_1qB)+\tan(z_2qB)]} [\tan(z_1qB)+\tan(z_2qB)]. \qquad (11.19)$$

The $p_{1,2}$ integrals can be carried out and the expression simplifies to

$$C_{\text{CSE}} = -\frac{1}{2\pi^2} \sum_{s=0}^{3} c_s T \sum_{n} \int_{-\infty}^{\infty} dp_3 \int_{0}^{\infty} dz_1 dz_2 e^{i(z_1+z_2)(-\omega_n^2-m_s^2-p_3^2)} (m_s^2 - \omega_n^2 + p_3^2), \qquad (11.20)$$

allowing the evaluation of the $z_1$ and $z_2$ integrals as well. We find

$$C_{\text{CSE}} = -\frac{1}{2\pi^2} \sum_{s=0}^{3} c_s T \sum_{n=-\infty}^{\infty} \int_{-\infty}^{\infty} dp_3 \frac{m_s^2 - \omega_n^2 + p_3^2}{(\omega_n^2 + p_3^2 + m_s^2)^2}. \qquad (11.21)$$

The Matsubara sum then evaluates to

$$C_{\text{CSE}} = -\frac{1}{2\pi^2} \sum_{s=0}^{3} c_s \int_{-\infty}^{\infty} dp_3 \frac{dn_F(E_p^{(s)})}{dE_p^{(s)}}, \qquad (11.22)$$

where $n_F(x) = (e^{x/T} + 1)^{-1}$ is the Fermi-Dirac distribution and $E_p^{(s)} = \sqrt{p_3^2 + m_s^2}$. The derivative of the Fermi-Dirac distribution vanishes for infinite masses, therefore only the $s = 0$, physical term remains from the sum over PV fields,

$$C_{\text{CSE}} = -\frac{1}{\pi^2} \int_{0}^{\infty} dp_3 \frac{dn_F(E_p)}{dE_p} = \frac{1}{2\pi^2} \int_{0}^{\infty} dp_3 \left[ 1 + \cosh\left(\sqrt{p_3^2 + (m/T)^2}\right) \right]^{-1}. \qquad (11.23)$$

We plot the result of the integral in Fig. 11.1, which shows that $C_{\text{CSE}}$ is a non-trivial function of $m/T$. Note that Bloch's theorem does not forbid the existence of CSE, since the axial current is not conserved due to the finite mass and the chiral anomaly.
<figure><img src="image_42.png" /><figcaption>**Figure 11.1:** Leading coefficient in the CSE conductivity for non-interacting fermions as a function of $m/T$.</figcaption></figure>

### 11.3.2. Lattice results

With these analytical results at hand, we now turn to the lattice formulation. First of all, we check the results in the non-interacting case to benchmark our lattice setup. For a detailed discussion on the lattice observables and the definition of the currents, see [337, 338]. For free fermions we use two distinct discretizations: Wilson and staggered quarks. In both formulations it is possible to define vector and axial currents which fulfill the Ward identities (see [349] for staggered and [350] for Wilson), which are point-split, meaning that they connect fermions at nearest neighbor sites.

For comparison, in the Wilson formulation it is also possible to define a local vector current, involving fermion fields at the same point, which however is not conserved on the lattice.

Nevertheless, it is commonly used in the study of hadron properties, since it has the correct quantum numbers. It was also used to study CME with Wilson fermions in QCD and in the quenched approximation [351, 352], obtaining $C_{\text{CME}} \neq 0$. It is therefore important to study the effect of using the non-conserved current, in order to clarify its role in the non-vanishing CME result.

<figure><img src="image_43.png" /><figcaption>**Figure 11.2:** Lattice spacing dependence of the CME (left panel) and the CSE (right panel) coefficients for non-interacting fermions. Besides the correct, conserved vector currents, we also use a local, non-conserved current for Wilson quarks – this gives an unphysical result for both effects. A comparison to the quenched study [352], which used a non-conserved vector current for the CME, is also included.</figcaption></figure>

On the lattice, the coefficients are obtained by calculating the expectation values (11.5) and (11.6) at different values of $eB$, and then performing a linear fit. The slope of the fit (rescaled by the appropriate factor accounting for the charges
of different flavors) corresponds to the desired conductivities. In Fig. 11.2, we show the results for $C_{\text{CME}}$ and $C_{\text{CSE}}$ for free fermions with a mass of $m/T = 4$. The calculation is performed on a four-dimensional lattice with $N_s$ spatial points and $N_t$ temporal points, with $a$ the lattice spacing. The spatial volume is $V = L^3 = (aN_s)^3$ and the temperature is $T = (aN_t)^{-1}$.

On the left panel of Fig. 11.2, we can see the results for the CME. Both Wilson and staggered fermions, when using a conserved vector current, yield a vanishing $C_{\text{CME}}$. This is in agreement with the arguments presented earlier, since the Euclidean lattice formulation is inherently in equilibrium and uses a gauge invariant regularization. However, when a local vector current is used with Wilson fermions, the continuum limit deviates from zero, which is an indication that the $C_{\text{CME}} \neq 0$ in [352] is very likely linked to the use of a non-conserved vector current. In particular our free results suggestively approach the continuum limit estimation of [352] in the quenched theory.

It is important to emphasize at this point that the absence of the CME is not due to the lack of chirality in the system. It can be shown, both analytically and on the lattice, that the axial susceptibility, which corresponds to the linear response of the chiral density to a weak $\mu_5$, is not vanishing [337].

The right panel of Fig. 11.2 shows the result for the CSE. Similarly to the CME, when the appropriate currents are used, the results in the continuum limit using either discretization agree with eq. (11.1) for $m/T = 4$, while the non-conserved Wilson current shows a deviation. The same behavior was found for other values of $m/T$.

From these results we learn the importance of using the proper discretization of the current on the lattice, as well as a positive cross-check of our setup. This enables us to proceed to the full QCD – this is what we discuss next.

## 11.4. Results

### 11.4.1. Quenched theory

We now present the results on how interactions affect the CME and the CSE. We start with an intermediate step: the quenched theory. This setup is not only less computationally challenging, but also enables a direct comparison with existing lattice studies of these effects in the literature.

We perform simulations with Wilson and staggered fermions as valence quarks, while the configurations are generated using the plaquette action (these configurations were already used in [119, 32]). The pion mass is tuned to be $M_\pi \approx 415$ MeV in the staggered case, while for Wilson fermions we use $M_\pi \approx 710$ MeV. The latter is to best compare to the CME study [352]. Besides zero temperature simulations we do a temperature scan in the range from 260 MeV to 520 MeV.

In our setup, $C_{\text{CME}}$ was found to be zero in the whole range of temperatures, see [337] for details. This is a first confirmation that interactions do not affect the arguments presented before, and will be further reinforced by the results in dynamical QCD.

However, there is another important aspect that can be studied. As already mentioned, we can directly compare our setup with the one used in [352], where the vector current was directly measured at finite $B$ and $\mu_5$. We show the comparison in Fig. 11.3. As it can be seen, when a non-conserved vector current is used in our setup, we reproduce the results from [352], deviating from zero. However, when a conserved current is used, $C_{\text{CME}}$ vanishes. We conclude that the $C_{\text{CME}} \neq 0$ result is just an artifact of using a local vector current, and it shows the importance of using the appropriate discretization of currents to study anomalous transport effects on the lattice.

For the quenched study of the CSE, there is a subtlety related to the effect of non-trivial Polyakov loop sectors, see [338] for details. Once this is taken into account, the conductivity was found to be suppressed at low temperatures
<figure><img src="image_44.png" /><figcaption>**Figure 11.3:** Lattice spacing dependence of the CME coefficient in quenched QCD, using conserved (blue) and non-conserved (black) vector currents, including a comparison to [352].</figcaption></figure>

and approach the value corresponding to the free massless fermion value for high $T$. The latter behavior is to be expected due to asymptotic freedom, and is in qualitative agreement with a quenched study with overlap fermions with massless valence quarks [353].

### 11.4.2. Full QCD

In this last subsection we summarize our results for full dynamical QCD using 2 + 1 flavors of stout-improved staggered fermions with physical quark masses. The conductivities are calculated for a wide range of temperatures, from $T \approx 0$ MeV to 400 MeV.

In the left panel of Fig. 11.4 we present the results for the CME. The coefficient vanishes within errors for all temperatures, which confirms the arguments presented throughout this work about the absence of the CME in thermal equilibrium, generalizing it to QCD. Analogously as in the case of free fermions, it can be shown that the nonzero value of $C_{\text{CME}}$ is not due to the absence of chirality – the axial susceptibility in full QCD is also nonzero. In particular, this observable shows a gradual increase around the crossover temperature $T_c$ towards the free massless value for high $T$ [337].

<figure><img src="image_45.png" /><figcaption>**Figure 11.4:** Left panel: continuum extrapolation of the CME coefficient in full QCD [337]. Right panel: continuum extrapolation of the CSE coefficient in full QCD, with a comparison to a model involving a free gas of $p$ and $\Sigma^{\pm}$ hadrons at low temperature [338].</figcaption></figure>
Finally, we present the results for $C_{\text{CSE}}$ in the right panel of Fig. 11.4. Unlike $C_{\text{CME}}$, it shows a highly non-trivial structure in equilibrium. $C_{\text{CSE}}$ is highly suppressed at low $T$, with a sharp increase around the chiral transition temperature, approaching the analytical value for free fermions $C_{\text{CSE}}(m/T = 0) = 1/2\pi^2$ at high temperatures [338]. The suppression at low $T$ is in agreement with a two-color QCD study [354], and it can be further cross-checked with a hadron resonance gas type calculation. The results are also in qualitative agreement with the recent results obtained using holographic QCD [355].

## 11.5. Summary and outlook

In this section, we have studied the equilibrium aspects of two anomalous transport phenomena, the chiral magnetic effect and the chiral separation effect. We have shown an analytic calculation of the conductivity for these effects, pointing out the importance of UV regularization. While the CSE shows a non-trivial behavior as a function of $m/T$ even in thermal equilibrium, the CME may only be present out-of-equilibrium. We cross-checked these results with a lattice evaluation with Wilson and staggered fermions, which showed complete agreement with the analytical results if the appropriate discretization of currents was used. The latter turns out to be the most crucial aspect of the simulation.

We then discussed interacting lattice QCD results, in the quenched approximation and in the full dynamical theory. In the quenched theory for the CME, we were able to directly compare to the simulations of [352], where a non-zero result for $C_{\text{CME}}$ was obtained, and show that the reason behind this is the use of a non-conserved vector current on the lattice. If a conserved one is used, the CME vanishes within errors for all temperatures. In turn, the CSE conductivity was found to be suppressed at low $T$ and approach the free massless fermion limit of $C_{\text{CSE}}$ at high $T$.

These results from the quenched theory are confirmed in full dynamical QCD with staggered fermions tuned to the physical point, where the CME was found to vanish for a wide range of temperatures. Regarding the CSE, its conductivity vanishes for temperatures lower than the crossover temperature, with a sharp increase over it, finally approaching the non-interacting massless fermion limit. Although directly detecting $C_{\text{CSE}}$ in experiments is very challenging since it amounts to an axial current, the structure of the temperature dependence of the conductivity could play a role in the detection of the chiral magnetic wave.

An important open question is the out-of-equilibrium behavior of $C_{\text{CME}}$ in QCD [356, 357]. Lattice QCD simulations do not have direct access to real-time properties, since only Euclidean correlators can be calculated. However, it is possible to learn about these properties by inferring the spectral function from the Euclidean correlator [358] through spectral reconstruction. This is a mathematically ill-posed problem and therefore a complicated and delicate procedure, where small uncertainties in the original correlator may impact the final result substantially. However, there is a wide variety of tools that alleviate the difficulties of this problem. As an outlook, we plan to use spectral reconstruction methods to calculate the out-of-equilibrium $C_{\text{CME}}$ in QCD, which promises a deeper understanding of the effect of interactions for the CME.
# 12. Enhancement of Chiral Separation Effect due to heavy quark flavours: a signature of QCD Kondo effect?

## 12.1. Introduction

Chiral Separation Effect (CSE) is one of the macroscopic transport phenomena generated by microscopic quantum anomalies - violation of classical symmetries at the quantum level. CSE originates in Adler-Bell-Jackiw axial anomaly and leads to the generation of a macroscopic axial current $j^A$ in the direction of an external magnetic field $B$ [359, 334, 335]:

$$j_k^A = \bar{q} \gamma_5 \gamma_k q = \sigma_{CSE}(\mu, T) B_k, \qquad (12.1)$$

where $k = \{x, y, z\}$ enumerates spatial dimensions, $\bar{q}, q$ are fermion (quark) fields, $\gamma_5$ and $\gamma_k$ are the Dirac $\gamma$-matrices, $\mu$ is the fermion chemical potential and $T$ is the temperature. For free non-interacting fermions with unit electric charge, the corresponding transport coefficient $\sigma_{CSE}(\mu, T)$ takes a particularly simple form

$$\sigma_{CSE}(\mu) = \frac{\mu}{2\pi^2}, \qquad (12.2)$$

where the coefficient $1/(2\pi^2)$ is often identified with the universal coefficient of the $U(1)_A$ axial anomaly.

In combination with other anomalous transport phenomena, CSE can lead to observable signatures in heavy-ion collision experiments, such as electric quadrupole moment of quark-gluon plasma [360], and novel hydrodynamical excitations such as the Chiral Magnetic Wave [336].

As anomalous transport phenomena became the subject of intense theoretical research in the past decades, it was quickly realized that CSE can get both perturbative [361] and non-perturbative [362, 363] corrections in Quantum Chromodynamics (QCD) and Quantum Electrodynamics (QED) where interactions between fermions are mediated by gauge bosons. In view of potential experimental detection, in particular, in the data obtained during the RHIC isobar run [333], it becomes important to quantify the strength of possible corrections to CSE in QCD. Currently, the only way to quantify such corrections entirely from first principles is provided by lattice gauge theory simulations.

CSE is particularly suitable for lattice gauge theory simulations, as it exists in a state of thermal equilibrium and involves only well-defined lattice observables. The Chiral Vortical Effect (CVE) is another anomalous transport phenomenon which exists in a thermal equilibrium state. However, lattice signatures of CVE involve expectation values of the energy-momentum tensor [364, 365], which is one of the most challenging quantities to define and measure on the lattice. In contrast, the Chiral Magnetic Effect (CME) is known to vanish in thermal equilibrium state [343], which makes it conceptually difficult to simulate CME on the lattice.

A particular challenge for studying CSE in first-principle lattice QCD simulations is the infamous fermionic sign problem at finite baryon density $\mu$, which is necessary for the CSE to exist. For this reason, first lattice studies of CSE [353] were performed in quenched $SU(3)$ lattice gauge theory with overlap valence quarks, where no statistically significant deviations from free quark result were found on both sides of the confinement-deconfinement transition.

The next step is to consider CSE in one of the QCD-like gauge theories where the fermionic sign problem is absent due to some special symmetry, like $SU(2)$ gauge theory with $N_f = 2$ light dynamical quarks. However, background magnetic field breaks the time-reversal invariance which ensures the positive-definiteness of path integral weight in $SU(2)$ gauge theory. This does not allow to study CSE in simulations with finite background magnetic field, as in [353]. A way to resolve this difficulty is to use the linear response approximation and expand the expectation value of electric current
in powers of magnetic field. While this expansion does not work for uniform magnetic fields by virtue of total flux quantization in a finite volume, one can consider magnetic fields with nonzero wavelength $k$ and zero total flux. This leads to the following Kubo formula for the CSE coefficient $\sigma_{CSE}(\mu, T)$ [366]:

$$ \sigma_{CSE}(\mu, T, k_z) = \langle j_x^A(k_z) j_y^V(-k_z) \rangle / k_z. \qquad (12.3) $$

The expectation value in this formula is taken with respect to the thermal equilibrium state without background magnetic field. $j_x^A(k_z)$ and $j_y^V(k_z)$ are the Fourier transforms of the local axial and vector current densities $j_k^A(r) = \bar{q}(r) \gamma_5 \gamma_k q(r)$ and $j_k^V(r) = \bar{q}(r) \gamma_k q(r)$ with momentum that is directed along the $z$ axis (that is, the other components $k_x$ and $k_y$ are equal to zero). The relevant limit for hydrodynamical description of CSE is $k_z \to 0$.

The Kubo formula (12.3) was used in [354] to study CSE in finite-density $SU(2)$ LGT with dynamical quarks. In this work, a suppression of the CSE response was found at low temperatures and densities. Phenomenologically, this suppression can be described by the formula $\sigma_{CSE}(\mu, T) \sim \rho_V(\mu, T)$, where $\rho_V(\mu, T)$ is the electric charge density induced by finite chemical potential $\mu$ [363].

Subsequent effort to measure CSE in full lattice QCD simulations with $SU(3)$ used linear expansion in the chemical potential $\mu$ in fixed background magnetic field $B$ to circumvent the fermionic sign problem [367, 338], confirming the suppression of CSE at low temperatures.

## 12.2. CSE and the QCD Kondo effect

The suppression of anomalous transport phenomena in low-temperature QCD makes perfect sense, as non-Abelian gauge fields tend to confine chiral fermions, thus preventing them from participating in collective transport responses. This common-sense expectation makes it particularly exciting to find physical mechanisms that could actually enhance anomalous transport responses.

In this section we demonstrate one such mechanism related to the presence of heavy quarks. As demonstrated in [368, 369], the amplitude of scattering of light quarks off heavy quarks of mass $m_Q$ is enhanced as $\log(m_Q)$ in the vicinity of the Fermi surface of light quarks. This phenomenon is analogous to Kondo effect in condensed matter physics, and has been termed “the QCD Kondo effect”. The enhancement of heavy-light quark interactions near the Fermi surface leads to the formation of the Kondo condensate $\langle \bar{Q}q \rangle$, where $\bar{Q}$ is the heavy quark field and $q$ stands for light quarks [370]. At the mean-field level, this effect can be described by an additional term in the Lagrangian [371]:

$$ \mathcal{L} \to \mathcal{L} + \Delta \left( \bar{q}Q + \bar{q}\gamma_{\mu} \frac{k_{\mu}}{|k|} Q \right). \qquad (12.4) $$

This additional coupling between heavy and light quarks induces a divergence of the density of states of the light quarks $q$ in the vicinity of the Fermi surface [371]. The dominant contribution to the axial-vector current-current correlator (12.3) comes from momenta in the vicinity of the Fermi surface (see Fig. 1 in [343]), thus an enhanced density of states in this region due to the Kondo effect is likely to lead to an enhancement of the Chiral Separation Effect. This is precisely what was found in the mean-field calculation in [371].

## 12.3. Lattice setup: $SU(2)$ lattice gauge theory with $N_f = 2 + 1$ light and heavy quark flavours

A beautiful mean-field prediction for the enhancement of the CSE response due to the presence of heavy quarks clearly calls for a first-principle check in lattice gauge theory simulations. Since the full lattice QCD with $SU(3)$ gauge group
suffers from the fermionic sign problem at finite chemical potential $\mu$, in this Contribution we study the enhancement of CSE in SU(2) lattice gauge theory with $N_f = 2 + 1$ light and heavy quark flavours.

In our study we use gauge field configurations generated using the standard HMC algorithm with $N_f = 2$ mass-degenerate rooted staggered fermions (with bare mass $m_q^{\text{stag}} = 0.005$), one extra flavour of heavy rooted staggered fermion, and a tree-level improved Symanzik gauge action. We use ensembles with heavy fermion masses equal to $a m_Q = 1$, $a m_Q = 0.5$ and $a m_Q = 0.2$, with numbers of gauge configurations ranging from 200 to 1000. We use lattices with spatial size $L_s = 24$ and variable temporal sizes ranging between $L_t = 10$ and $L_t = 22$. We work at fixed gauge coupling $\beta = 1.7$ and hence at fixed lattice spacing (which we do not determine), changing the temperature by changing $L_t$. The chiral crossover occurs at $L_t \approx 16$. Overall, our lattice setup is similar to the one used in the works [372, 354].

To ensure the absence of fermionic sign problem because of the extra heavy flavour, chemical potential is only introduced for light flavours. This is justified because, physically, only the existence of the Fermi surface for light quarks is crucial for the Kondo effect. Furthermore, because of large quark masses, heavy flavours are much less sensitive to chemical potential.

For valence quarks, we use Wilson-Dirac fermions with HYP-smeared gauge links [373]. The main reason to use Wilson-Dirac valence quarks is that the axial current operator entering (12.3) is easy to implement. We use the conventional point-split definition of $j^A$, introduced in [374], as well as the conserved lattice vector current operator.

The Wilson-Dirac bare mass for light flavours is tuned to $m_{WD} = -0.21$ to achieve the same pion mass $m_\pi \approx 0.16$ as for staggered quarks that are used as sea quarks. The $\rho$-meson mass on the same ensemble is approximately $m_\rho \approx 0.4$, thus our pion mass is still quite large. On the other hand, large $m_\pi$ ensures the smallness of finite-volume artifacts on our moderately large lattices.

In order to work in the QCD-like regime of the SU(2) lattice gauge theory, we work at a small value of the chemical potential $a\mu = 0.05 < a m_\pi / 2$. The reason to use such a small value is that at $\mu > m_\pi / 2$ the theory enters the diquark condensation regime, where physics is quite different.

<figure><img src="image_46.png" /><figcaption>**Figure 12.1:** The CSE transport coefficient $\sigma_{\text{CSE}}(\mu, T, k_z)$ as a function of lattice momentum $k_z$ at $a\mu = 0.05$ and at two different temperatures: $T > T_c$ on the left ($L_t = 14$) and $T < T_c$ on the right ($L_t = 22$).</figcaption></figure>
<figure><img src="image_47.png" /><figcaption>**Figure 12.2:** Heavy quark mass dependence of electric conductivity and chiral condensate.</figcaption></figure>

## 12.4. CSE enhancement in $N_f = 2 + 1$ SU(2) gauge theory

As in the work [354], we use the axial-vector current-current correlator (12.3) at finite momentum $k_z$ to extract the momentum-dependent CSE coefficients $\sigma_{CSE}(\mu, T, m_Q, k_z)$. It is important to stress that we only consider the contribution of light quarks into this correlator, in order to isolate the trivial enhancement of CSE due to increased number of flavours from the nontrivial physics that might be associated with the Kondo effect.

The lattice results for $\sigma_{CSE}(\mu, T, m_Q, k_z)$ are shown on Fig. 12.1 for different values of the heavy quark mass. We use two values of temperature above and below the chiral crossover temperature which corresponds to temporal lattice size $L_t \approx 1$ ($aT_c \approx 16$). For comparison, we also present the free-quark result obtained on the same lattice and the same fermion action (with bare Wilson-Dirac mass $m_{WD} = 0.005$), as well as numerical results for $N_f = 2$ light flavours taken from [354]. The latter results formally correspond to the limit $m_Q \to +\infty$, where heavy quarks completely decouple.

At the temperature $1/(aT) = L_t = 14$ (left plot on Fig. 12.1), the presence of heavy quarks results in just a minor modification of the CSE transport coefficient, although the trend towards enhancement is already visible. More or less all lattice results are reasonably close to the corresponding free quark result.

On the other hand, at the lower temperature $1/(aT) = L_t = 22$ (left plot on Fig. 12.1), CSE enhancement becomes obvious, and the addition of heavy quark with bare mass $a m_Q = 0.2$ changes the low-momentum limit of $\sigma_{CSE}(\mu, T, m_Q, k_z)$ by almost a factor of two in comparison with the result for two light flavours (blue line/data points). However, we never observe an enhancement that would go beyond the corresponding free-fermion result.

One possible explanation for CSE enhancement is that adding more quark flavours drives the theory away from the confining regime with spontaneous chiral symmetry breaking. We check to what extent is the chiral symmetry breaking for light quarks because of the presence of heavy quarks, on Fig. 12.2a we also present lattice results for the chiral condensate of light quarks, $\bar{q}q$, at different heavy quark masses and different temperatures. We indeed see that the chiral condensate decreases as the heavy quark mass goes down and they influence the dynamics more and more. However, this effect is not very large. For example, for $1/(aT) = L_t = 22$ and $m_Q = 0.2$, the chiral condensate is only around 25% smaller than for the two-flavour case ($m_Q \to +\infty$). This has to be compared with a factor of two increase in the CSE
transport coefficient $\sigma_{CSE}$. It seems therefore that the relatively mild tendency towards restoration of spontaneously broken chiral symmetry with more quark flavours cannot explain much more significant increase of the CSE response.

A paradigmatic signature of the Kondo effect in condensed matter physics is a non-monotonous dependence of the electric DC conductivity $\sigma$ on temperature, expressed by the famous formula [375]

$$ \frac{1}{\sigma(T)} = \frac{1}{\sigma_0} + a T^2 + c_m \log (T^{-1}) + b T^5. \qquad (12.5) $$

While the $bT^5$ term is related to electron-phonon interactions, the first three terms are universal and from the Fermi liquid behavior of fermion degrees of freedom. This formula predicts a maximum of electric conductivity (equivalently, a minimum of resistivity) at some intermediate temperature. Similar results were also obtained in the context of QCD Kondo effect using renormalization group techniques [376].

In order to check whether we can observe such a non-monotonous dependence of conductivity on temperature, we also calculated electric conductivity for our ensemble of gauge configurations. To this end we calculated Euclidean correlators of vector currents (again, considering only the light quark contributions) and using the Backus-Gilbert algorithm to extract conductivity, as in [372].

Our results for the DC limit of electric conductivity are shown on Fig. 12.2b. We only observe a monotonous decrease of $\sigma(T)$ towards lower temperatures, without any statistically significant dependence on the heavy quark mass.

## 12.5. Discussion and conclusions

Our lattice results qualitatively confirm the enhancement of the CSE response in the low-temperature regime in the presence of heavy quarks, first predicted in the mean-field approximation in [371]. However, a complementary study of the temperature dependence of electric conductivity, which is a “smoking gun” for Kondo physics, suggests that we are not in the Kondo regime yet. In particular, the value of the chemical potential $a\mu = 0.05$ used in our study may be too small for the formation of a well-defined Fermi surface. This is definitely true in the free quark case, where only the few lowest discrete lattice momenta $k_i = \frac{2\pi m_i}{L_s}$ would contribute to the would-be Kondo effect. However, the fact that we do observe a significant enhancement of CSE in the presence of heavy quarks suggests that the mechanism of this enhancement may be even more general than the Kondo effect as such.

Two ways to reach the true Kondo regime for SU(2) lattice gauge theory would be:

* To use very large spatial lattice sizes $L_s$, where there would be sufficiently many discrete lattice momenta within the Fermi surface $\epsilon(k) < \mu$. If we want to stay in the QCD-like phase of SU(2) lattice gauge theory with $\mu < m_\pi/2$, $L_s$ should be of the order of few hundred sites, which is unfeasible with currently available computational resources.

* To use very large values of $\mu \gg m_\pi/2$. Even though such large values are in the diquark condensation phase, Fermi surface can still be formed [377, 378]. There are good reasons to expect that this weak-coupling, low-temperature, high-density regime should be similar in both SU(2) gauge theory and in full QCD. This direction seems more promising for future exploration.

Another important feature of the Kondo effect is that formally the Kondo coupling is enhanced towards heavier fermions $\log(m_Q)$ [375]. However, in lattice QCD simulations of the grand canonical ensemble the density of heavy fermions is suppressed as $e^{-\mu m_Q}$, which is a much stronger factor. Qualitatively, this explains why in our case the CSE response increases towards smaller $m_Q$. A cleaner way to study Kondo physics as such would be simulations of the
canonical ensemble with fixed density of heavy quarks, rather than at fixed chemical potential [379]. In the limit of $m_Q \to +\infty$, heavy quarks can also be introduced as insertions of Polyakov loops in lattice observables. We leave these refinements for future work.

Experimentally, the enhancement of CSE in the presence of heavy quarks can be used to construct refined experimental probes, where signatures of CSE or the closely related Chiral Magnetic Wave/electric quadrupole moment [360] would be correlated with the abundance of heavy mesons or baryons (e.g. charmonium/bottomonium).

# 13. Novel hydrodynamic transport coefficients in extreme magnetic fields & the CME far from equilibrium

## 13.1. Introduction

Relativistic hydrodynamics is a state-of-the art framework for describing plasmas generated in heavy-ion collisions [380], astrophysical plasmas [381, 382], and neutron star mergers [383, 384]. Hydrodynamics is based on the symmetries of the system to be described, in particular, standard hydrodynamics assumes parity-invariance and spatial isotropy. However, it is experimentally established [385, 386] that the chiral anomaly of QCD, responsible for the fast pion decay, breaks the parity invariance [387, 388]. In addition, strong magnetic fields, $B$, of leading order in the hydrodynamic derivative expansion (yet small compared to the temperature scale $T$, i.e. $B \ll T^2$) break spatial isotropy. Consequently, the hydrodynamic description of QCD-plasma subject to strong magnetic fields needs to be constructed under these restricted symmetry constraints (anisotropy and parity-violation).

As a major result, novel transport effects arise [389, 390, 391, 1]. Among those novel coefficients there is the non-dissipative shear-induced Hall conductivity [1]. This coefficient measures the response of a charge current (for example, in $x$-direction) to a shear in the fluid velocity perpendicular to it (for example, in the $(y, z)$-plane), see Fig. 13.1. There are also novel susceptibilities in equilibrium, for example, the perpendicular magnetic vorticity susceptibility [1], which measures the shift in the equilibrium energy density and pressure when the plasma is subject to a vortical magnetic field, see Fig. 13.1. How these novel effects are derived and interpreted is outlined in section 13.2, see [1] for a detailed interpretation.

Another major result due to the anisotropy is that known transport coefficients such as the shear viscosity take distinct values along the plane containing the magnetic field (parallel) versus the plane perpendicular to it [392, 1]. At large magnetic fields, the parallel shear viscosity divided by the entropy density can drop below the holographic KSS value [393] and can even vanish [392, 1]. Remarkably, transport coefficients known from 2+1-dimensional systems, such as Hall viscosities, emerge due to anisotropy [1]. These altered shear and Hall viscosities can affect the evolution of elliptic flow and thus may have phenomenological consequences, since analyses like [394] need to be refined.

One transport process arising from parity-violation through the chiral anomaly already at weak magnetic fields, i.e. at sub-leading order in the hydrodynamics gradient expansion, is the chiral magnetic effect (CME) [395, 396, 397, 398]. This effect at strong magnetic fields appears at leading order in the derivative expansion. Thus, the CME at strong magnetic fields becomes a potential equilibrium phenomenon, as explained in the main text. The CME predicts an electric charge current in response to an external magnetic field in presence of an axial charge imbalance [397, 398], see Fig. 13.1. The CME was experimentally verified in Weyl semi-metals, see for example [330], while results from heavy-ion experiments are being debated [333]. Two major insights into the CME stemming from holographic plasma are (i) the rigorous time
<figure><img src="image_48.png" /><figcaption>**Figure 13.1:** Parity-violating transport effects arising in strong magnetic fields: the shear-induced Hall current is a novel effect requiring strong magnetic fields, just like the novel energy/pressure shift induced by a strong vortical magnetic field in equilibrium. The chiral magnetic effect (CME) already appears at weak and persists at strong magnetic fields. All three depicted transport effects require a nonzero chemical potential in addition to parity-violation and strong magnetic field. There are 27 transport coefficients with Kubo formulae derived in [1].</figcaption></figure>

evolution of charged plasma states in magnetic fields [399, 400, 401, 402, 2, 3], see section 13.3, and (ii) the dynamical generation of axial charge imbalance [403].

## 13.2. Novel transport coefficients in extreme magnetic fields from chiral hydrodynamics [1]

Hydrodynamic constitutive relations can be derived either from a generating functional [404, 405, 406, 389, 407, 408, 409, 410, 411, 412] or constructed directly [413, 414]. For the chiral hydrodynamic constitutive relations, a hybrid approach turns out to be most efficient, deriving their equilibrium part from a generating functional and constructing their non-equilibrium part directly.

**Equilibrium generating functional & constitutive relations.** Considering hydrodynamics as an effective field theory, one notes that the hydrodynamic framework is constrained by the existence of an equilibrium generating functional. For systems with finite correlation lengths that are in equilibrium, and with sources that vary on length scales much larger than the correlation length, the generating functional $W_s = -i \ln Z[g, A]$ is a local functional of the Killing vector $V$ and the sources [404, 405]. The generating functional can then be expanded in a derivative expansion. The traditional zero derivative terms are the temperature $T$, fluid velocity $u^\mu$ and chemical potential $\mu$, which can all be written in terms of the Killing vector $V$ and the external sources $A$ and $g$. In addition, for systems in strong magnetic fields $B^\mu = \frac{1}{2}\epsilon^{\mu\nu\rho\sigma}u_\nu F_{\rho\sigma}$, one can take $B^\mu$ and $B^2 = B^\mu B^\nu g_{\mu\nu}$ to be order zero in derivatives [389, 390, 1]. In this case, the generating functional contains corrections at first order in the derivative expansion, $W_s \supset \int d^4x \sqrt{-g} M_n(T, \mu, B^2)s_n$ where $M_n$ are thermodynamic susceptibilities and $s_n$ are scalars containing only one derivative and that do not vanish in equilibrium such as $B^\mu \partial_\mu (\frac{B^2}{T^2})$ and $\epsilon^{\mu\nu\rho\sigma} F_{\mu\nu} F_{\rho\sigma}$, see table 1 of [1].

The equilibrium constitutive relations which relate the one point functions of the stress energy tensor $T^{\mu\nu}$ and the current $J^\mu$ to the hydrodynamic variables can be derived by varying the equilibrium generating functional with respect to the sources, and they depend on the susceptibilities. For example,

$$ \text{pressure} \sim T^{\mu\nu}(g_{\mu\nu} + u_{\mu}u_{\nu}) \sim M_2\epsilon_{\mu\nu\rho\sigma}B_{\mu}u_{\nu}\partial_{\rho}B_{\sigma}, \quad \text{energy} \sim T^{\mu\nu}u_{\mu}u_{\nu} \sim M_2\epsilon_{\mu\nu\rho\sigma}B_{\mu}u_{\nu}\partial_{\rho}B_{\sigma}, \quad (13.1) $$

indicates that there is an equilibrium response, a shift in the energy and pressure, to a non-uniform magnetic field.
**The chiral anomaly.** For systems with a chiral anomaly, the generating functional is not gauge invariant, $\delta_\alpha W_{cons} = \frac{C}{24} \int d^4x \sqrt{g} \alpha \epsilon^{\mu\nu\rho\sigma} F_{\mu\nu} F_{\rho\sigma}$, where C is the anomaly coefficient.<fnref n="4" /> The consistent current found by varying the generating functional $W_{cons}$ with respect to the sources is not gauge invariant, so it is useful to work with a covariant current $J_{cov}^\mu = J_{cons}^\mu - \frac{1}{6}C\epsilon^{\mu\nu\rho\sigma}A_\nu F_{\rho\sigma}$. Expanding the generating functional order by order in derivatives, and varying it with respect to the sources we find the equilibrium constitutive relations for fluids with a chiral anomaly

$$T_A^{\mu\nu} = T_s^{\mu\nu} + v_T u^{(\mu} \Omega^{\nu)} + v_{TB} u^{(\mu} B^{\nu)}, \quad J_{cov}^{\mu} = J_s^{\mu} + v \Omega^{\mu} + v_B B^{\mu}. \qquad (13.2)$$

In the above, $T_s^{\mu\nu}$ and $J_s^\mu$ are the equilibrium stress tensor and current for a system without an anomaly, and

$$v = v_{TB} = \frac{C}{2} \mu^2 + c_1 T^2, \quad v_B = C \mu, \quad v_T = \frac{C}{3} \mu^3 + 2c_1 T^2 \mu, \qquad (13.3)$$

are the chiral transport coefficients. The chiral magnetic conductivity $v_B$ parametrizes the response of the current to a magnetic field (the CME, see Fig. 13.1), the chiral vortical heat conductivity $v_T$ the response of the heat current to the fluid vorticity, and the chiral vortical conductivity $v = v_{TB}$ the response of the current to the fluid vorticity, and of the heat current to a magnetic field. The coefficient $c_1$ is related to the mixed gauge-gravitational anomaly [415], and there is an additional coefficient $c_2$ [1] (it vanishes for CPT invariant systems), which is set to zero in this section.

**Equilibrium Kubo formulae.** Varying the equilibrium constitutive relations twice with respect to the sources we find the static Kubo formulae for thermodynamic coefficients. For example, for $\mathbf{k} = k_z \hat{\mathbf{z}}$, and a magnetic field $\mathbf{B} = B_0 \hat{\mathbf{z}}$

$$\langle T^{xz}(\mathbf{k})T^{yz}(-\mathbf{k}) \rangle = -2iB_0^2 M_2 k_z, \quad \langle J^x(\mathbf{k})J^y(-\mathbf{k}) \rangle \sim -iv_B k_z. \qquad (13.4)$$

Note that Eq. (13.4) together with Eq. (13.3) implies that the chiral magnetic effect can manifest itself on the level of two-point functions in an inhomogenous equilibrium state in presence of a strong magnetic field.<fnref n="5" />

**Out-of-equilibrium constitutive relations.** The non-equilibrium terms in the constitutive relations can be constrained by symmetry arguments.<fnref n="6" /> For a relativistic fluid in strong magnetic fields, there are 14 independent hydrodynamic transport coefficients, 10 of which are dissipative and 4 are non-dissipative. An example of a non-dissipative transport coefficient is the novel shear-induced Hall conductivity $\tilde{\sigma}_\Sigma$ (a dissipative one is the novel shear-induced conductivity $\sigma_\Sigma$),<fnref n="7" />

$$J^\mu \sim \sigma_\Sigma \Sigma^\mu + \tilde{\sigma}_\Sigma \tilde{\Sigma}^\mu, \quad \Sigma^\mu = (\Delta^\mu \nu - B^\mu B^\nu / B^2) \sigma_{\nu\rho} B^\rho / \sqrt{B^2}, \quad \tilde{\Sigma}^\mu = \epsilon^{\mu\nu\rho\sigma} u_\nu B_\rho \Sigma_\sigma / \sqrt{B^2}. \qquad (13.5)$$

where $\sigma^{\mu\nu} = \Delta^{\mu\alpha} \Delta^{\nu\beta} (\nabla_\alpha u_\beta + \nabla_\beta u_\alpha - \frac{2}{3} g_{\alpha\beta} \nabla \cdot u)$ is the fluid shear tensor and $\Delta^{\mu\nu} = g^{\mu\nu} + u^\mu u^\nu$.

<footnote n="4">⁴For simplicity, a microscopic theory with one single anomalous U(1)-symmetry is considered here.</footnote>

<footnote n="5">⁵This is because the factor of momentum $k_z$ (after a Fourier transform) stems from a spatial derivative acting on the chemical potential $\mu$. Thus, in momentum space one obtains $\langle J^x(k_z) J^y(-k_z) \rangle \sim -iC\mu k_z$, if an equilibrium state with inhomogenous $\mu(\mathbf{x})$ is subject to a magnetic field of zeroth order in the derivative expansion. Note, however, the restrictions imposed by the generalized Bloch theorem [340]. In addition, we note that lattice computations involving massive fermions report not to find any CME in equilibrium states [337], see this topic reviewed in [339].</footnote>

<footnote n="6">⁶It is possible to formulate a non-equilibrium generating functional which sources the non-equilibrium constitutive relations [404, 405, 406, 389, 407, 408, 409, 410, 411, 412]. However, doing so significantly increases the complexity of the formalism so we approach the non-equilibrium terms with the standard phenomenological approach [413, 414] for simplicity.</footnote>

<footnote n="7">⁷The shear-induced conductivity is labelled $c_8$ and the shear-induced Hall conductivity is labelled $c_{10}$ in [1].</footnote>
**Hydrodynamic Kubo formulae.** Finding the Kubo formulae for the hydrodynamic transport coefficients requires solving the linearized hydrodynamic equations. For example, for a magnetic field $\mathbf{B} = B_0\hat{\mathbf{z}}$, one finds [1]

$$ \lim_{\omega \to 0} \frac{1}{\omega} \mathrm{Im} G_{T^{\mathrm{tx}}T^{\mathrm{xz}}} (\omega, \mathbf{k} = 0) \sim \sigma_{\Sigma}, \quad \lim_{\omega \to 0} \frac{1}{\omega} \mathrm{Im} G_{T^{\mathrm{tx}}T^{\mathrm{yz}}} (\omega, \mathbf{k} = 0) \sim \tilde{\sigma}_{\Sigma}. \qquad (13.6) $$

**Eigenmodes.** With the hydrodynamic equations and the constitutive relations, one can study the eigenmodes of small perturbations around equilibrium [1]. For example, for momenta $\mathbf{k} \parallel \mathbf{B}$, we find two “sound” waves and the chiral magnetic wave

$$ \omega = v_{\pm}k - \frac{i\Gamma_{\pm}}{2}k^2, \quad \omega = v_0k - iD_{\parallel}k^2. \qquad (13.7) $$

The velocities $v_0$ and $v_{\pm}$ and damping coefficients $\Gamma_{\pm}$ and $D_{\parallel}$ depend on the transport coefficients in the constitutive relations. Remarkably, the chiral magnetic wave only propagates when there is an anomaly, i.e., $v_0 \propto C$, and the “sound” waves propagate at different speeds depending on the direction they travel in, i.e., $|v_+ - v_-| \propto C$.

**Physics of transport coefficients.** The hydrodynamic framework described above uncovers 14 non-equilibrium transport coefficients (5 more are related through Onsager relations), 4 chiral transport coefficients and 5 thermodynamic transport coefficients [1]. An example of a thermodynamic transport coefficient is the magnetic vortical susceptibility $M_2$ appearing in Eq. (13.1). It parametrizes the response of the energy and pressure to a non-uniform magnetic field, see Fig. 13.1. It also paramterizes the shear response to a non-zero Poynting vector $\mathbf{B} \times \mathbf{E}$. For example, for $\mathbf{B} = B_0\hat{\mathbf{z}}$ and $\mathbf{E} = E_0\hat{\mathbf{y}}$, there is a shear term $T^{xz} \sim \frac{\partial M_2}{\partial \mu} B_0^2 E_0$. The chiral conductivities are completely fixed in terms of the anomaly coefficients $C$ and $c_1$, see Eq. (13.3). They parametrize the response of the charge current and heat current to a magnetic field and fluid vorticity (see Eq. (13.2)), and are essential for propagation of the chiral magnetic wave in (13.7). Thermodynamic and chiral transport coefficient are equilibrium quantities, and they can be computed or experimentally observed in equilibrium configurations, see Eq. (13.4). The shear induced conductivities $\sigma_{\Sigma}$ and $\tilde{\sigma}_{\Sigma}$ are examples of non-equilibrium transport coefficients. They measure the current response to a shear in the fluid velocity. For example, for a system with shear in the $(y, z)$-plane subject to a magnetic field $B = B_0\hat{\mathbf{z}}$, the current responds as $J^x \sim \tilde{\sigma}_{\Sigma} (\partial_y u_z + \partial_z u_y)$, see Fig. 13.1 and $J^y \sim \sigma_{\Sigma} (\partial_y u_z + \partial_z u_y)$. Note that the $\tilde{\sigma}_{\Sigma}$ response is orthogonal to the perturbation, and constitutes a Hall-like response of the current to the shear, which turns out to be dissipationless. Non-equilibrium transport coefficients require time dependent perturbations to compute or measure experimentally, see for example the hydrodynamic Kubo formulae in Eq. (13.6). For a detailed discussion of all the transport coefficients, see section 2.5 of [1].<fnref n="8" />

**A holographic model displaying chiral hydrodynamic behavior.** As an explicit example that the above transport coefficients are non-zero for a microscopic theory with a chiral anomaly, a simple holographic model was used in [1] to compute these coefficients for a charged plasma in quantum field theory with a chiral anomaly at strong coupling and in presence of a (strong) magnetic field. The model in question is a simplified version of the action (13.8) with the vector field $V$ (and $F$), the Stückelberg field $\theta$ and the gauge field mass $m_s$ all set to zero. The dual theory corresponds to a sector of $\mathcal{N} = 4$ Super-Yang-Mills (SYM) with gauge group $SU(N_c)$ at large $N_c$ and large 't Hooft coupling $\lambda$.<fnref n="9" /> The

<footnote n="8">⁸A relation between the conformal anomaly and the Nernst effect was first observed in [416] and shown generally using hydrodynamics in [1].</footnote>

<footnote n="9">⁹The regime of applicability of hydrodynamics in strong magnetic fields for these systems has recently been considered in [417].</footnote>
<figure><img src="image_49.png" /><figcaption>**Figure 13.2:** Transport coefficients in the presence of the chiral anomaly (parametrized by $\gamma = 2/\sqrt{3} \sim C$). Without chiral anomaly or for vanishing chemical potential, all three transport coefficients are zero. Left: The dimensionless thermodynamic susceptibility $M_2T$ as a function of the axial chemical potential $\mu/T$. Different curves correspond to magnetic fields $\tilde{B} = B/T^2 = \{0.05, 12.5, 30\}$ (red, blue, purple). Middle: The dimensionless shear-induced conductivity $\sigma_\Sigma/T^3$. Right: The dimensionless shear-induced Hall conductivity $\tilde{\sigma}_\Sigma/T^2$.</figcaption></figure>

transport coefficients can be computed by first computing holographic two point functions numerically, and using the Kubo formulae (13.4) and (13.6). See Fig. 13.2 for the plots of $M_2$, $\sigma_\Sigma$ and $\tilde{\sigma}_\Sigma$ in this model.<fnref n="10" />

## 13.3. The chiral magnetic effect far from equilibrium [2, 3]

**Holographic CME model.** The holographic model used in [2, 3], as far as we know originally due to [418], is given by

$$S = \frac{1}{2\kappa_5^2} \int_M d^5x \sqrt{-g} \left[ R + \frac{12}{L^2} - \frac{1}{4}(F^2 + F_{(5)}^2) - \frac{m_s^2}{2}(A_\mu - \partial_\mu \theta)^2 + \frac{\alpha}{3} \epsilon^{\mu\nu\rho\sigma\tau} (A_\mu - \partial_\mu \theta) (3F_{\nu\rho}F_{\sigma\tau} + F_{\nu\rho}^{(5)}F_{\sigma\tau}^{(5)}) \right] + S_{\text{GHY}} + S_{\text{ct}} \quad (13.8)$$

where $S_{\text{GHY}}$ is the Gibbons-Hawking-York boundary term to make the variational problem well defined, $L$ is the anti-de Sitter (AdS) radius, $\kappa_5^2$ is the Newton constant, $\alpha$ the Chern-Simons coupling and $m_s$ the mass of the gauge field. The Levi-Civita tensor is defined as $\epsilon^{\mu\nu\rho\sigma\tau} = \epsilon(\mu\nu\rho\sigma\tau)/\sqrt{-g}$. The Stückelberg field is denoted as $\theta$ whereas the field strengths are defined as $F = dV$ and $F_{(5)} = dA$. $\theta(x^\mu)$ couples the operator $\text{Tr}\{G \wedge \tilde{G}\}$ ($G$ is the gluon field strength) thus playing the role of the $\theta$ angle [419]. Note that the gluon field strength does not appear explicitly but is mediated through $\theta$. Moreover, the axial gauge field couples to $\theta$ through the mass term and hence the dual axial current is non-conserved due to the non-Abelian anomaly. Finally, the axial gauge field and the vector gauge field are coupled through the Chern-Simons term accounting for the Abelian anomaly.

**CME in holographic Bjorken flow.** In both [2, 3] time dependent solutions for the metric, the two gauge fields and the Stückelberg field were obtained working with generalized Eddington-Finkelstein coordinates and a metric ansatz consistent with having the dual plasma expanding homogeneously and uniformly in a direction perpendicular to the magnetic field. This produces a vector magnetic field whose magnitude is given by $B^1 = \frac{B}{\tau}$ as seen by a co-moving observer at the boundary. In addition, the gauge field ansatz allows for choosing nonzero axial charge density and for a resulting vector current (CME current) along the magnetic field.

Solving the Einstein-Maxwell equations of motion arising from (13.8), one obtains the expectation value of the energy-

<footnote n="10">¹⁰Note the different conventions for the Chern-Simons factor $\gamma$ compared to section 13.3.</footnote>
momentum tensor, the vector and axial currents via the holographic correspondence [2, 3]

$$ \kappa_5^2 \langle T_{ab} | T_{ab} \rangle = \mathrm{diag}(\epsilon(\tau), P_1(\tau), P_2(\tau), P_3(\tau)), \quad 2\kappa_5^2 \langle J_5^a | J_5^a \rangle = (Q_5(\tau), 0, 0, 0), \quad 2\kappa_5^2 \langle J^a | J^a \rangle = (0, J_{CME}(\tau), 0, 0). \qquad (13.9) $$

From the Ward identity $\partial_a \langle T^{ab} | T^{ab} \rangle = 0$, the evolution equation for the energy density follows

$$ \partial_{\tau} \epsilon(\tau) - \frac{P_1(\tau)}{\tau} - \frac{P_2(\tau)}{\tau} - \frac{B^2}{8\tau^3} + \frac{2\epsilon(\tau)}{\tau} = 0. \qquad (13.10) $$

When $B/\tau^3$ can be ignored and $P_1 \sim P_2 \sim \epsilon(\tau)/3$, this reduces to $\tau \partial_{\tau} \epsilon(\tau) + 4/3 \epsilon(\tau) = 0$ which is the evolution equation describing a conformal fluid undergoing Bjorken expansion [420]. This asymptotic behavior of the dual fluid is highly useful for making instructive choices of the initial data, and parameters, so that the results of the calculation can be compared, as closely as possible, to collider experiments which generate the QGP (and potentially produce chiral magnetic phenomena). Holographic time evolution [421, 422] requires an initial choice of the spatial metric components and the field $V$ on the initial time slice as well as an initial choice of: energy density, axial charge density and magnetic field [2, 3]. In addition, one must fix the gravitational coupling $\kappa_5$ and Chern-Simons coupling $\alpha$. Fixing the initial energy density, axial charge density etc. is typically done by ensuring that the evolution of these flow through values associated with those used in hydrodynamic modeling of the QGP. For instance, for the energy density, initial data is chosen by appealing to the asymptotic behavior of Eq. (13.10), while for the Chern-Simons coupling $\alpha$, one matches the value of the anomaly coefficient of three-flavor QCD. For further information and different matching choices, we refer the interested reader to the detailed discussions given in [2, 3].

<figure><img src="image_50.png" /><figcaption>**Figure 13.3:** Left: The holographic CME current $\langle J_{CME} \rangle$ along the magnetic field is displayed for different beam energies (initial energy densities). The inset image displays the same, but for initial data which holds fixed the choice of the initial axial charge density $\langle J_{(5)}^0 \rangle = 0.00032 \text{ GeV}^3$, and the initial magnetic field $eB \approx m_\pi^2$. Right: The time integrated CME current, yielding the accumulated charge as a result of the CME current. The purple dashed line corresponds to the main figure on the left, the yellow line corresponds to the inset.</figcaption></figure>

**Physical questions.** With a holographic setup capable of mimicking basic aspects of a heavy-ion collision (HIC), the goals of [2, 3] are stated as: “Given reasonable assumptions on the initial conditions, what is the magnitude and evolution of the CME current during a holographic HIC? How does the collision energy alter the CME current? Does enough CME signal build up if one incorporates the fact that chiral charge is generated and dissipates dynamically via topological processes?”

**Holographic CME results.** The results of these calculations are best separated by $m_s = 0$ [2] and $m_s \neq 0$ [3]. Beginning with $m_s = 0$, a sample of the CME currents obtained from the numerical solutions is displayed in the left
<figure><img src="image_51.png" /><figcaption>**Figure 13.4:** Left: $\langle J_{\text{CME}} \rangle / \epsilon_{\infty}^{3/4}$ (blue), $n_5 / e / \epsilon_{\infty}^{3/4 + \Delta/4}$ (black) and $A_v(1)$ (purple). The dashed lines correspond to $\Delta = 1.25 \cdot 10^{-7}$ ($m_s \approx 0$) and the solid lines to $\Delta = 0.3$ ($m_s \neq 0$). The red lines are fits outlined in the main text. Right: Double logarithmic plot showing the late time behavior of the chiral magnetic current for different values of $\Delta$ (black lines). The green dashed lines are formula (13.11) for the corresponding values. The plots are for the $\sqrt{s} = 200$ GeV initial conditions (taken from [3]).</figcaption></figure>

image of Fig. 13.3. Both the image and inset image show the CME current generated during the evolution initially grow but quickly begin to decrease as the axial charge is diluted by the expansion of the medium. The inset image shows the result of choosing the same initial value for the magnetic field and the same initial axial charge density for each choice of initial energy density. The main image displays the CME current generated when allowing for the initial values of the axial charge density and magnetic field to depend on the initial energy density.

To better characterize the size of the CME response, consider the time integrated response, yielding the charge accumulated per area $q_V/\tilde{A}$, displayed as a function of beam energy in the right image of Fig. 13.3. The yellow line corresponds to CME current in the inset image of the left figure in Fig. 13.3 (yellow line), the purple dashed line is the result for the left main image of Fig. 13.3 (purple dashed line). Interestingly these two curves in Fig. 13.3 (right), have opposite behavior as a function of beam energy, $s^{1/2}$. The more realistic choice of allowing for the energy dependence of the axial charge and magnetic field (purple curve) results in a larger CME signal at larger beam energies. In the case of $m_s \neq 0$, the model incorporates axial charge generation and relaxation by means of the non-Abelian anomaly thus incorporating the effect of topological dynamics. This has the advantage that axial charge does not have to be artificially introduced but rather is generated dynamically. The fact that the axial current is non-conserved for $m_s \neq 0$ is reflected in the fact that it acquires an anomalous dimension (like the singlet $U(1)_A$ current in QCD) $\dim(\langle J_5 \rangle) = 3 + \Delta$, where $\Delta = -1 + \sqrt{1 + m_s^2}$. In the expanding non-Abelian plasma, the Chern-Simons diffusion rate and hence the axial charge relaxation rate are time dependent due to the falloff of the magnetic field with time ($B \sim 1/\tau$). For small values (or for $\alpha = 0$) of the Abelian anomaly, the axial charge relaxation rate increases for increasing the strength of the magnetic field. If the Abelian anomaly is strong enough, the opposite is true and the axial charge relaxation rate decreases [403]. The specific value of the Abelian anomaly considered in [3], lies in the latter regime and hence axial charge relaxation is accelerated towards later times. The modified relaxation dynamics also impacts the chiral magnetic current which relies on the axial charge and hence is decaying faster. Overall, we find that for $m_s \neq 0$, the chiral magnetic current decays faster and reaches lower peak values compared to the left figure in Fig. 13.3 i.e. when $m_s = 0$. The falloff follows $\langle J_{\text{CME}} \rangle / \epsilon_{\infty}^{3/4} \sim A e^{B\tau+C/\tau} \tau^D$ where the exponents $B, C, D$ are monotonically increasing functions of $m_s$, see left side in Fig. 13.4. Note that for $m_s = 0$, the late time decay follows $\langle J_{\text{CME}} \rangle \sim 1/\tau^{4/3}$. For details and values see [403]. Remarkably, the late time behavior of the chiral magnetic current may be captured in terms of the horizon value of the
temporal component of the axial gauge field $A_v(\tau, 1)$, the magnetic field, the strength of the Abelian anomaly $\alpha$ and the strength of the non-Abelian anomaly $m_s$

$$ \langle J_{\text{CME}} \rangle = \frac{\alpha}{3(1 - \Delta)} A_v(\tau, 1) B(\tau), \qquad (13.11) $$

where $A_v(\tau, 1)$ mimicks the role of an axial chemical potential in our system out-of-equilibrium with explicitly broken $U(1)_A$ symmetry. The excellent agreement can be seen in the right side of Fig. 13.4. Note the structural similarity of (13.11) to the hydrodynamic formula $J_{\text{CME}} \sim v_B B \sim C \mu B$, see Eq. (13.2) and Eq. (13.3).

## 13.4. Discussion and Conclusions

**Chiral hydrodynamics in strong magnetic fields.** The first line of major progress reviewed in Sec. 13.2 stems from developing hydrodynamics with magnetic fields as an effective field theory: In strong magnetic fields, a large number of novel transport coefficients and susceptibilities can arise [1, 389, 390], see also Fig. 13.1. The relevance of these novel terms in QCD plasma has to be estimated. If they turn out to be relevant, they will influence for example the time evolution of elliptic flow (since one of the shear viscosities divided by entropy density can vanish), the time evolution of the magnetic field itself, and the detectability of the chiral magnetic effect. Consequently, relevant effects should be included in hydrodynamic codes used for the analysis of heavy-ion collision data or for the simulation of neutron star evolution, especially for magnetars. Already at weak magnetic fields, the chiral magnetic effect is sufficient [423] to account for the observed [424] neutron star kicks accelerating proto neutron stars to approximately 1,000 km per second. In strong magnetic field hydrodynamics, this kick mechanism may be amplified. In addition to what was discussed here, there are several noteworthy transport coefficients, such as novel conductivities, novel Hall conductivities, novel Hall viscosities, various bulk viscosities, the Nernst effect, and novel susceptibilities, see section 2.5 of [1] for a detailed interpretation.

**Relations to lattice QCD.** Utilizing the chiral hydrodynamics constructed in [1], it may be possible to compute the eight thermodynamic coefficients, including novel susceptibilities such as $M_2$, in lattice QCD with magnetic fields [13, 14, 15]. Comparison of lattice QCD data in various strong magnetic fields suggests that the ratio of transverse to longitudinal pressure as function of $T^2/B$ matches that of $N = 4$ SYM from fairly small to large values of $T^2/B$ [425]. This motivates that holographic models provide realistic estimates for QCD in strong magnetic fields, especially in the limit $T^2 \gg B$ considered in this contribution. Holographic results can be compared to lattice results, see for example [425, 426].

**Chiral magnetohydrodynamics.** The hydrodynamic section 13.2 considered external (non-dynamical) magnetic fields. An extension of the presented framework to include dynamical magnetic fields into magnetohydrodynamics (MHD) was demonstrated for non-chiral hydrodynamics [390, 427]. A subset of the possible terms of chiral MHD was derived in [428]. A full derivation of all possible terms in chiral MHD is an open problem, but straightforward with the methods outlined in this contribution. Correspondingly, on the gravity side of holographic duals, the standard boundary conditions can be modified in order to obtain dynamical gauge fields in the boundary gauge theory [429, 430, 431]. This technique allows to holographically model the interaction of dynamical electric and magnetic fields with charged plasma, which should be applied to model the generation and evolution of electric and magnetic fields in HICs. This will help estimate the CME in HICs, which is one goal of the AdS4CME Collaboration [432].

**Extending chiral hydrodynamics to vector and axial symmetry.** The chiral hydrodynamic framework discussed in section 13.2 only involves one single $U(1)$, which is broken by an anomaly [1]. This suffices to tell what type of transport effects can appear in either a conserved vector or an anomalous axial current. In order to match QCD more closely, in
future work, two U(1)-symmetries, one global axial and one vector gauge symmetry, should be considered. Similarly, the
non-Abelian anomaly should be included in the construction of chiral hydrodynamics.

**Holographic models of the CME generation.** The second line of major progress reviewed in Sec. 13.3 is based on the holographic modeling of dynamically evolving plasma in search for the CME [2, 3]: *Collectively, the time evolution of all the considered holographic CME currents very strongly depends on the choice of initial conditions and parameter choices, see Fig. 13.3 and 13.4.* One model [2] states that the energy dependence of the charge accumulated due to the CME current hinges on the initial conditions of the Bjorken-like flow, compare the yellow solid curve to the dashed purple curve in Fig. 13.3. In the latter case, the accumulated charge due to the CME is increasing monotonically with the collision energy $s^{1/2}$, indicating that LHC would have a better chance of seeing the CME than RHIC. In an extension of [2], axial charge is dynamically generated [3].<sup>11</sup> This leads to a faster decay of the CME current and to lower peak values compared to [2]. A fully dynamical study including dynamical magnetic field and dynamically generated axial charges is a future goal for holographic models [430].

**Main conclusions.** In conclusion, the novel hydrodynamic effects arising in extreme magnetic fields, see Sec. 13.2 have the potential to heavily impact various disciplines, ranging from heavy-ion collisions and neutron star physics, to the description of astrophysical plasma. Holographic models including magnetic fields, serve as discovery tools for new physics, such as the hydrodynamic chiral vortical effect [434, 435, 398], and as testing grounds for proposed hydrodynamic effects [1]. They can even be pushed to investigate plasma dynamics far from equilibrium, as illustrated in the CME case study [400, 401, 402, 2, 3], see Sec. 13.3.

# 14. Anomalous transport in low dimensions

## 14.1. Introduction

The chiral magnetic effect (CME) is a remarkable mechanism based on anomalous properties of the quark-gluon plasma (QGP), predicted almost two decades ago [329, 436]. Due to its potential impact in different areas of physics, an intense activity - on both theoretical and experimental fronts - has been carried out aiming to achieve a robust observation of it. The proposal developed by Kharzeev and collaborators beautifully describes the microscopical theory behind it, connecting abstract concepts like the topological properties of the configurations of non-Abelian gauge fields to tangible quantities that may be manifest in the final state of the particles that reach the detectors in large accelerators colliding heavy-ions.

As it is well-known in the community, the mechanism predicts the generation of an electric current driven by the chiral anomaly associated to a magnetic field. After hadronization of the quarks and gluons, this would result in a difference in the number of positively charged pions and negatively charged pions in opposite hemispheres. Unfortunately, it is very hard to have control over the two elements that are fundamental for the emergence of the CME in the QGP. On one hand, the topological configurations of the gauge fields are confirmed by lattice QCD, but its rate, size of domains, etc, in the

<footnote n="11"><sup>11</sup>A holographic model recently predicted the dynamically generated spatial distribution of axial charge and its time evolution [403]. The spatial extent of correlations of electric currents which are responsible for the experimental manifestations of the chiral magnetic effect in heavy-ion collisions are quite large (~1 fm), grow with time, and behave consistent with sphaleron dynamics. Such results should be compared to lattice QCD for the sphaleron rate in three flavor QCD [433]. Axial charge profiles dynamically generated from holographic models [403] should be used as input for hydrodynamic codes modeling HICs.</footnote>
actual plasma are extremely hard to predict, being possible that the signal is so small that it will be completely washed
in experiments. On the other hand, only recently the first experimental evidence of the huge magnetic field generated in
non-central collisions predicted more than a decade ago was achieved. The most reliable results involve ultra-peripheric
collisions, which is not a suitable environment for the CME [102]. More dramatic than that, it is not clear if the field
lasts long enough to really affect the system. This is under investigation and a consensus has not been reached.

Besides that the fundamental ingredients of the CME are far from being under control, the dynamics of the system involving large thermal fluctuations is very likely to interfere in the signal. All this factors have represented obstacles hard to overcome during the past decade in the run for obtaining an unambiguous signal of the CME. The community is currently tuned to new analysis on isobar collisions [437], hoping to catch the evidence of this important phenomenon.

Nevertheless, the intense work accomplished during the last couple of decades has build a solid and sophisticated
theoretical framework based mainly on quantum field theory to describe the CME and other transport phenomena in the
QGP. A few years ago the theoretical knowledge developed in the context of the QGP gained an alternative application.
It has been shown that in certain materials the interaction of the charge carriers with the underlying lattice produces
quasi-particles whose dynamics is described by Dirac equation, mimicking the behavior of relativistic particles [438]. This
behavior was predicted for honeycomb lattices before the experimental discovery of Dirac materials and it was one of the
first properties to be tested in graphene after its synthesis.

This connection with relativistic fermions allowed for the search of phenomena predicted in the context of high energy
physics, in Dirac/Weyl material. This is specially fruitful on what concerns transport phenomena, including the CME and
other effects proposed to be observed in the QGP. A few years ago, an Abelian version of the CME effect was observed
in ZrTe₅ [330]. The successful setup produces the chiral anomaly by turning on colinear electric and magnetic fields,
responsible for generating the necessary chiral imbalance. More details on how the chiral anomaly operates in these
systems can be found in [439]. The core of the observation is on the quadratic dependence of the electric current on the
magnetic field. This happens because the magnetic field plays a double role, inducing the chiral chemical potential and
aligning the spins. Therefore, the raise of chiral imbalance in this case is controlled since the fields are external and their
intensity can be tuned. This represents an important advantage on what concerns feasibility, in contrast to the CME
in the QGP. On the other hand, in materials the particles suffer scattering and the chirality is not preserved, existing a
relaxation time that dictates the typical scale for which the effect fades. After the first observation other groups have
reported similar results in other materials [440, 441, 442, 443].

All the materials where macroscopic manifestation of the chiral anomaly was observed were in (3+1) dimensions. This
is expected since it is not possible to define chirality in odd dimensions. In this work we propose to extend the analogy,
exploring materials in (2+1)D, but considering mechanisms based on the parity anomaly rather than the chiral anomaly.
Considering a symmetry pattern totally analogous to the CME, we were able to predict a novel anomalous Hall effect in
planar materials. From this point we followed two avenues. The first was searching for lattice configurations where the
symmetry pattern arises naturally. The second, was checking if the symmetry breaking could be generated by interaction
with external fields, in a similar fashion as the CME in ZrTe₅.

14.2. Honeycomb lattices

We aim to explore planar materials containing relativistic-like fermions. We work with honeycomb lattices since in this
kind of structure it is very clear how the relativistic behavior appears. Although the pristine honeycomb does not satisfied
<figure><img src="image_52.png" /><figcaption>Figure 14.1: Honeycomb lattice represented as two triangular sublattices.</figcaption></figure>

the required conditions, we start by considering its corresponding Lagrangian and on top of that we add the necessary interventions.

A pure honeycomb lattice, like in graphene, can be represented as in Fig.14.1. It is usual to describe it using two triangular sublattices, represented in Fig.14.1 in different colors. When calculating the band structure one can check that there are two inequivalent points where the energy vanishes. Those are called Dirac points. Expanding the Hamiltonian around these points, one obtains a linear dispersion relation, what is a clear signature of a relativistic-like behavior. It is also usual to consider sublattice and Dirac points as degrees of freedom of the fermionic system. If one represents the fermions in 4-component spinors, in addition to the linear dispersion relation another feature typical from relativistic systems emerges: the equation of motion is written in terms of Dirac matrices and taking the continuous limit the Lagrangian that comes out is totally analogous to the Lagrangian of massless Quantum Electrodynamics in (2+1)D (QED$_3$), [444, 445],

$$ \mathcal{L} = \sum_s \bar{\psi}_s (i\gamma^0 \hbar D_t + i\hbar v_F \gamma^x D_x + i\hbar v_F \gamma^y D_y) \psi_s. \qquad (14.1) $$

Here the index s labels the (electron) spin and the covariant derivative $D_\alpha = \partial_\alpha - (ie/\hbar c)A_\alpha$. $A_\alpha$ is the gauge field associated to the electromagnetic interaction and the fields $\psi_s \equiv \psi_s(t, \mathbf{r})$ denote four-component spinors that account for both Dirac point (also called valley) and sublattice (also called pseudospin) index. Note that based on the structure of Dirac matrices, the valley index is analogous to chirality quantum number in systems in (3+1)D.

To make the valley degree of freedom more transparent we rewrite our Lagrangian using the pseudo-chiral projections. We decompose the fermion field as

$$ \psi = \psi_k + \psi_{k'} = \frac{1}{2}(1 + \gamma^5)\psi + \frac{1}{2}(1 - \gamma^5)\psi. \qquad (14.2) $$

Doing this we can write the Lagrangian in terms of each valley separately,

$$ \mathcal{L} = \sum_{s, \chi=k, k'} \bar{\psi}_{\chi, s} [i\gamma^0 \hbar \partial_t + i\hbar v_F \gamma^x D_x + i\hbar v_F \gamma^y D_y] \psi_{\chi, s}, \qquad (14.3) $$

Up to this point, this description is applicable to honeycomb lattice whose lattice points are all identical, as it happens in graphene. The band structure around the Dirac points can be schematically represented in Fig.14.2(a). There is no gap between the valence and conduction band, and they touch in one point. The two species, analogous to the chiralities, can be represented at this stage by two symmetric pairs of cones.
<figure><img src="image_53.png" /><figcaption>**Figure 14.2:** Blue cones represent valence bands and red cones represent conduction band. Honeycomb lattice generates gapless band structure, where the bands touch in one point. From the six points of vanishing energy, only two are non-equivalent. The Dirac structure that comes out naturally in the Lagrangian allows us to build an analogy between the two pairs of cones and the chirality degree of freedom.</figcaption></figure>

<figure><img src="image_54.png" /><figcaption>**Figure 14.3:** The presence of a mass in the Lagrangian signals the opening of a gap between the valence and conducting bands. We propose a configuration where the gaps opened in each pair of cones is different, mimicking the imbalance of chirality in the CME. This situation is represented in the left panel of the figure, where we also added a chemical potential indicated by the red line, controlling the Fermi level. On the right, besides opening a gap, we also lift the spin degeneracy in order to avoid cancellation of the current of particles with opposite spin.</figcaption></figure>

Following a close analogy with the CME, we want now to induce a specific symmetry breaking that will play the role of the imbalance in chirality responsible for the CME. We achieve this, exploring the connection of the valley degree of freedom and chirality of the quarks in Quantum Chromodynamics (QCD), described above. In order to mimic the imbalance in chirality responsible for inducing the CME in the QGP, we need to break the symmetry between the two components in Eq. (14.3). The way we choose to achieve this is by adding masses to the Lagrangian, where the magnitude of the masses are different for each projection in Eq. (14.1). The resulting Lagrangian is [446]

$$ \mathcal{L} = \sum_{s,\chi=k,k'} \bar{\psi}_{\chi,s} [i\gamma^0\hbar\partial_t + \mu\gamma^0 + i\hbar v_F \gamma^x D_x + i\hbar v_F \gamma^y D_y + m_{s,\chi}\gamma^z] \psi_{\chi,s}, \quad (14.4) $$

where we also included the chemical potential, $\mu$. Without loss of generality, we may assume $m_k m'_k > 0$. This is the final Lagrangian for the fermions we employ in our procedure. In Fig.14.3 we represent the Dirac cones after gap opening, considering different gaps for each specie of fermion. The red line represents the chemical potential, strategically placed in a way that only one pair of cones are contributing to the current. This point will be clarified in a moments.
## 14.3. Reduced-QED

In our analysis we aim to explore reactions of the system to external stimuli of electromagnetic fields. On doing this we must be careful since the electrons in the systems of interest are constrained to the plane while the electromagnetic fields are not. Neither the Quantum Electrodynamics in (3+1)D (QED$_4$) nor Quantum Electrodynamics in (2+1)D (QED$_3$) are suitable to describe it and we need to use a framework that consider systems with different dimensionality interacting. We adopt the theory known as reduced-QED (RQED) [447] or pseudo-QED (PQED) [448] as our framework. This is a scale invariant theory [449], where the gauge fields living in (3+1)D suffer a dimensional reduction in order to obtain their effective action on the plane where the fermions are constrained. For a comprehensive review on this theory, including formal aspects and applications, see [450].

After the dimensional reduction, fermions in (2+1)D are added and a new effective theory completely defined in (2+1)D is obtained. The action is given by

$$S_{eff} = -\frac{e^2}{8} \int d^3x d^3x' j^\mu(x) \frac{1}{\sqrt{-\Box_E}} j^\mu(x'). \qquad (14.5)$$

Here, since the fermions are constrained to the plane, the third space component of the current, $j^3 = 0$. It can be easily verified that the action in Eq. (14.5) can be obtained from the Lagrangian

$$\mathcal{L}_{RQED} = -\frac{1}{4} F^{\mu\nu} \left[ \frac{2}{\sqrt{\Box}} \right] F_{\mu\nu} - e j^\mu A_\mu + \mathcal{L}_M + \mathcal{L}_{GF}, \qquad (14.6)$$

where $\mathcal{L}_M$ is the fermion Lagrangian given by Eq. (14.4) and $\mathcal{L}_{GF}$ is a gauge fixing term. This is the Pseudo or Reduced-QED Lagrangian. Note that the main consequence of this modified action compared to the usual QED$_4$ is that the dependence of the photon propagator on momentum will appear as $1/q$ rather than the inverse of the square dependence obtained in QED$_4$. This softens the IR behavior of the theory.

## 14.4. Anomalous Hall effect

After defining the Lagrangian, the next step is to investigate the transport coefficients. We are particularly interested in the perpendicular conductivity, and work under the assumption that the system responds linearly to external fields,

$$\langle j \rangle = \sigma_{xy} E e_x, \qquad (14.7)$$

where $j$ is an electric current also in-plane but perpendicular to the applied $E$. The transverse current is usually associated to Hall effect and we expect it to be topologically protected, that is the motivation of our interest. Of course one could also calculate the direct current $j_{xx}$, but this would give us the usual Ohm law. Also, we are interested in the DC current, for which $\omega \to 0$, nevertheless we will use the trick to consider a time dependent field and vector potential and take the limit until the end,

$$\mathbf{E} = E e^{-i\omega t} \mathbf{e}_y, \qquad \mathbf{a} = -\frac{iE}{\omega} e^{-i\omega t} \mathbf{e}_y. \qquad (14.8)$$

To calculate the conductivity we apply Kubo formula of linear response theory. We follow the classical derivation described in [451] to obtain

$$\langle j(t) \rangle = \frac{i}{\hbar} \langle 0 | \int_{-\infty}^{t} d\tau [\Delta H(\tau), j(t)] | 0 \rangle. \qquad (14.9)$$

getting the Kubo relation for the DC anomalous conductivity,

$$\sigma_{xy} = \lim_{\omega \to 0} \frac{1}{\hbar\omega} \left\{ \int_{0}^{\infty} d\tau e^{i\omega\tau} \langle 0 | [j_{y}(0), j_{x}(\tau)] | 0 \rangle \right\}. \qquad (14.10)$$
The detailed development of the calculation of the conductivity can be found in [452]. The current-current correlator is closely connected to the polarization tensor and can be written in terms of the latter as

$$ \sigma = \lim_{\omega \to 0} \frac{1}{\hbar\omega} \Pi^{yx}. \qquad (14.11) $$

Kubo formula is particularly interesting in this context, since it was proved that the 1-loop contribution for the photon self energy is exact in this case and higher order contributions are identically zero [453], which means that Coleman-Hill theorem applies to PQED. Finally, calculating the polarization tensor [452] the result for the conductivity is given by

$$ \sigma = - \lim_{\omega \to 0} \frac{1}{\hbar\omega} \frac{m}{|m|} \frac{e^2}{4\pi} \theta(m^2 - \mu^2) \varepsilon_{yx0}\omega = \frac{e^2}{4\pi\hbar} \frac{m}{|m|} \theta(m^2 - \mu^2) \qquad (14.12) $$

for each one of the two-component spinor. Note that the result is independent of the magnitude of the mass and depends only on its signal. The theta function controls the contribution from each valley, in a way that for a fixed gap, one can turn on and off the conductivity by adjusting the chemical potential. Summing up the contributions from the two valleys,

$$ \sigma_{xy} = \frac{e^2}{4\pi\hbar} \left( \frac{m_k}{|m_k|} \theta(m_k^2 - \mu^2) - \frac{m_{k'}}{|m_{k'}|} \theta(m_{k'}^2 - \mu^2) \right). \qquad (14.13) $$

The expression above shows that if the gaps for both pair of cones is the same, the conductivity and consequently the current $j_{xy}$ vanishes. Conversely, if we are able to tune the chemical potential such that $m_k > 0$, $m_{k'} > 0$ and $m_{k'}^2 < \mu^2 < m_k^2$, only one specie of electrons, coming from one pair of cones, will contribute to the current and the cancellation does not occur. In addition, we verify that considering the spin as an additional degree of freedom, may also cause a cancellation. The symmetries involved in a spin flip (particularly how it reacts to time reversal symmetry) are translated in the Lagrangian as a flip of the mass signal. Therefore, if non-polarized electrons are playing the game, the conductivity vanishes again and no transverse current is generated by applying an electric field. We thus need to lift the spin degeneracy and adjust the chemical potential between the conduction bands of associated to each spin type, as showed in Fig.14.3.

To obtain this result, we assumed that asymmetric gaps could be produced without caring about how to generate it. Now it is time to investigate how this could be produced in real setups. We followed two avenues. The first one was looking for materials whose structure naturally yields to the desired symmetry pattern. We have mapped features and symmetries and from that we have proposed a few candidates to harbor the effect. Details and candidates can be found in [452]. The other possibility we explored, was to check if interaction with external fields could lead to dynamical mass generation. This investigation is described in the next subsection.

## 14.5. External fields generating anomalous transport

As we discussed in the last couple of subsections, a gap opening in the band structure is translated in the continuous limit as a mass term for the fermions in the Lagrangian. As a possibility to obtain the mass structure required to activate the Hall effect described in the last subsection, we investigated if interaction with external fields can drive dynamical mass generation [454].

In (3+1)D parallel electric and magnetic fields are the responsible for the chiral anomaly and flip of the chirality of some of the electrons [330]. In our analogy, we seek for mechanisms that can break the symmetry between the pair of cones that are connected by parity transformation. We choose to verify if the same configuration can generate this asymmetry, since this combination of fields is odd under parity transformations. In terms of the gauge fields, the presence
of parallel electric and magnetic fields can be encoded in a Chern-Simons term,

$$ \mathcal{L}_{CS} = \frac{\kappa}{2} \varepsilon^{\mu\nu\rho} A_{\mu} \partial_{\nu} A_{\rho} - A_{\mu} J^{\mu}. \qquad (14.14) $$

Following the line we described previously, we apply PQED in order to deal with the difference in dimensionality of fermion and gauge fields, and add the Chern-Simons term to obtain the total Lagrangian

$$ \begin{aligned} \mathcal{L}_{RQED}^{CS} = & -\frac{1}{4} F^{\mu\nu} \frac{2}{(-\Box)^{1/2}} F_{\mu\nu} + \bar{\psi}(i\gamma^{\mu} \partial_{\mu} + e\gamma^{\mu} A_{\mu}) \\ & + m_e + \tau m_o \psi + \frac{1}{2\zeta}(\partial \cdot A)^2 + \frac{-i\theta}{4} \varepsilon^{\mu\nu\rho} A_{\mu} \partial_{\nu} A_{\rho}, \end{aligned} \qquad (14.15) $$

Note that in this Lagrangian we included a specific combination of mass terms that may open a gap in the system. Specifically, we allow for the general case where an ordinary Dirac mass $m_e\bar{\psi}\psi$ can emerge. On top that, there is another mass term that potentially may be generated and is particularly interesting to us, known as Haldane mass $m_o\bar{\psi}\tau\psi$, with $\tau = [\gamma_3, \gamma_5]/2$. This term does not break chiral symmetry, but rather violates parity (and time-reversal) and because the CS is parity violating we expect a relation between them. This choice is justified because working with chiral projectors, it yields to different magnitude of mass for each chirality.

As stated before, we are interested in dynamical mass generation driven by quantum effects. For this reason, we adopt the Schwinger-Dyson formalism to verify if the interaction with the external fields generates a mass term non-perturbatively. In this approach the bare propagators are corrected by the self-energy where all the propagators involved receive corrections, generating an infinite tower of coupled equations to be solved. In practice it is necessary to truncate the system of equations at a certain level. The corresponding equations for the two-point functions are

$$ S^{-1}(p) = S_{0}^{-1}(p) - \Xi(p), \qquad (14.16) $$

$$ \Delta_{\mu\nu}^{-1}(p) = \Delta_{0\mu\nu}^{-1}(p) - \Pi_{\mu\nu}(p) \qquad (14.17) $$

The bare photon propagator derived from the Lagrangian Eq. (14.16) can be written as [453]:

$$ \begin{aligned} \hat{\Delta}_{\mu\nu}(\mathbf{q}) = & \frac{1}{2q} \frac{1}{(1+\theta^2)} \left( \delta_{\mu\nu} - \frac{q_{\mu}q_{\nu}}{q^2} \right) \\ & + \frac{\zeta}{q^2} \frac{q_{\mu}q_{\nu}}{q^2} - \frac{1}{2q^2} \frac{\theta}{(1+\theta^2)} \epsilon_{\mu\nu\rho} q^{\rho}. \end{aligned} \qquad (14.18) $$

where the influence of the Chern-Simons term is quantified by the parameter $\theta$. Note that in the limit $\theta \to 0$, the photon propagator recover the usual form within PQED. To account for non-perturbative corrections, we assume a general form for the fermion propagator:

$$ S_F^{-1}(p) = A_e(p)\rlap{\slash} + A_o(p)\tau\rlap{\slash} - B_e(p) - B_o(p)\tau, \qquad (14.19) $$

and for the sake of simplicity, we adopt the rainbow-ladder approximation, where a bare photon-fermion vertex is considered. A study of more a sophisticated vertex applied to PQED can be found in [455]. Given that, we start from the gap equation

$$ S_{\pm}^{-1}(p) = S_{0\pm}^{-1}(p) + 4\pi\alpha \int \frac{d^3k}{(2\pi)^3} \gamma^{\mu} S_{\pm}(k) \gamma^{\nu} \Delta_{\mu\nu}(q), \qquad (14.20) $$

where

$$ \begin{aligned} S_{\pm}^{-1}(p) &= (\rlap{\slash} p + M_{\pm}(p))\chi_{\pm} \\ S_{0}^{-1}(p) &= \rlap{\slash} p\chi_{\pm} \\ S_{\pm}(k) &= -\frac{\rlap{\slash} k + M_{\pm}(k)}{k^2 + M_{\pm}^2(k)}\chi_{\pm}. \end{aligned} \qquad (14.21) $$
<figure><img src="image_55.png" /></figure>

<figure><img src="image_56.png" /><figcaption>Figure 14.4: M+ and M− extracted from the gap equation as a function of external momentum for different values of the parameter θ. Here θ = ηθc with 0 ≤ η ≤ 1 and fixed value of the coupling α = 1 . 07 αc</figcaption></figure>

Here M_+ = m_e + m_o, M_- = m_e - m_o and Δ_μν is the photon propagator given by Eq. (14.18). The details of this calculation can be followed in [454] and the solution within our approach is given by:

$$
\begin{align}
M_{\pm}(p) &= p^{-\frac{1}{2}} \sqrt{1-\frac{\alpha}{\alpha_c}} - \frac{1}{2} \left( c_2 p \sqrt{1-\frac{\alpha}{\alpha_c}} + c_1 \right) \nonumber \\
&\quad + f(\theta, p) \nonumber \\
&= \Lambda e^{-\pi/\sqrt{\alpha/\alpha_c - 1}} + f(\theta, p), \tag{14.22}
\end{align}
$$

where

$$
\alpha_c = \frac{\pi}{8}(1 + \theta^2) \tag{14.23}
$$

and

$$
\begin{equation}
\begin{split}
f(\theta, p) = \left[ \right. & \left( \pi (\theta^2 + 1) \left( \mp \frac{2\alpha\theta}{\theta^2 + 1} \right) \left( \kappa (\alpha + 3\pi (\theta^2 + 1)) \right. \right. \\
& \left. \left. \left( \Lambda^3 + 14p^3 \right) - 54p^4 (\alpha + \pi (\theta^2 + 1)) \right) \right] \\
& \left. \left. \times \frac{1}{18\kappa p^2 (\alpha + \pi (\theta^2 + 1)) (\alpha + 3\pi (\theta^2 + 1))} \right].
\end{split}
\tag{14.24}
\end{equation}
$$

The first term in Eq. (14.22) presents the so called Miransky scaling and corresponds to the solution in the absence of the Chern-Simons term. As usual, one can notice from Eq. (14.22) that there is a critical value of the coupling above which the solutions exists. It means that provided the coupling has a suitable value, both the Dirac and Haldane masses can be generated by interaction with external field in the configuration considered here. The solutions of Eq. (14.22) are represented in Fig.14.4 as a function of external momentum for different values of the parameter θ. In Fig.14.5 the values of M+ and M− are represented in the static limit. We can clearly see that from a certain critical value of θ the masses associated to different chiralities assume different values, indicating non-vanishing value of Dirac and Haldane mass.

14.6. Summary

Here we summarize results connecting the chiral magnetic effect and analogues in condensed matter in low dimensions. Although the CME itself is forbidden in odd dimensions, we show that a close analogy is possible exploring the parity anomaly rather than the chiral anomaly to produce a conserved current. We explore the well-known fact that the valley degree of freedom in honeycomb lattices corresponds to chirality on what concerns their group theory and propose to open different gaps in the two pair of Dirac cones as a way to mimic the imbalance of chirality that is a fundamental element in the CME in the quark-gluon plasma. The result is a topologically protected current that resembles a valley-Hall effect.
<figure><img src="image_57.png" /><figcaption>Figure 14.5: $M_\pm$ re-scaled by their $\theta = 0$ value. The full line corresponds to $M_+$ and dashed line $M_-$. The red dots are the absolute value of $M_+$ above the critical $\theta$.</figcaption></figure>

We then proceeded to investigate how to physically generate the gaps we need and follow two lines. The first is to perform ab initio simulations of electronic structure to identify candidates that may intrinsically harbor the effect. The second track we follow is that the gaps may be open by interaction with external fields. To investigate non-perturbative effects, we calculated the Schwinger-Dyson equations for the system and showed that for a given critical coupling, two types of mass terms emerge in the Lagrangian. These combination is precisely what is needed to achieve the gaps required. Along all the procedure we adopt the framework of pseudo-QED, that accounts for gauge fields in (3+1)D interacting with fermions constrained to move in planar space.

# 15. Magnetic fields, nonzero densities and QCD on the lattice

## 15.1. Introduction

Basic ingredients for chiral effects are in general some external field and density of some charge. The most renowned example, chiral magnetic effect [329], appears in systems with external magnetic field and chiral density. External magnetic field and nonzero baryon density are expected to lead to an axial current, it is an essence of the Chiral Separation Effect (CSE)[334, 335].

It is also expected that large magnetic fields and densities can be created in peripheral heavy ion collisions [436, 95], possibly leading to the emergence of chiral effects. For this reason in the current section we would like to discuss in general the properties of QCD in external magnetic field with nonzero baryon density with a particular attention to the bulk properties, including the phase diagram and the Equation State.

QCD thermodynamics is described by several conserved charges, including baryon number $B$, electric charge $Q$ and strangeness $S$, which in grand canonical description are coupled to the corresponding chemical potentials $\mu_B$, $\mu_Q$, $\mu_S$. In the lattice study one typically works with chemical potentials $\mu_u$, $\mu_d$, $\mu_s$ coupled to the density of $u$, $d$ and $s$ quarks. Formulae connecting the chemical potentials in two different bases read:

$$ \begin{aligned} \mu_u &= \mu_B/3 + 2\mu_Q/3 & \mu_B &= \mu_u + 2\mu_d \\ \mu_d &= \mu_B/3 - \mu_Q/3 & \mu_Q &= \mu_u - \mu_d \\ \mu_s &= \mu_B/3 - \mu_Q/3 - \mu_S & \mu_S &= \mu_d - \mu_s \end{aligned} \qquad (15.1) $$

In general one could study the properties of QCD as a function of all three chemical potentials $\mu_B$, $\mu_Q$, $\mu_S$. However, in this section we mainly discuss and present the results for the case $\mu_s = 0$, $\mu_u = \mu_d = \mu_B/3 = \mu$, which approximately
corresponds to the zero strangeness $\langle S \rangle \approx 0$ and might approximate the conditions realized in heavy ion collisions. More
detailed tuning of the chemical potentials to meet the conditions created in heavy ion collisions (given by zero strangeness
$S = 0$ and electric-to-baryon charge ratio $Q/B \approx 0.4$) can be found in [456] (without magnetic field) and in [457] (with
magnetic field).

15.2. Phase diagram

First we would like to summarize our knowledge about the QCD phase diagram as a function of three variables: temper-
ature, external magnetic field and density.

At zero magnetic field and vanishing baryon density the phase with the broken chiral symmetry at low temperatures
and the chirally restored phase are known to be separated by a wide crossover [458].

The phase diagram of QCD in the plane temperature - magnetic field was studied extensively on the lattice [13, 12, 459, 460, 202, 203]. It is currently known that nonzero magnetic field decreases the pseudo-critical temperature of the chiral phase transition - the phenomenon is called inverse magnetic catalysis effect. The transition becomes sharper at larger magnetic field, possibly becoming a first-order phase transition at very large magnetic fields $eB \sim 5 - 10 \text{ GeV}^2$ [202, 203].

The phase diagram of QCD in the plane baryon density (or chemical potential) - temperature has very important
phenomenological meaning and is under active study by multiple groups (for recent reviews see, e.g., [461, 462]). The
main difficulty for lattice studies of the phase diagram with nonzero baryon density is the sign problem [463]. Standard
approaches to overcome this problem include analytical continuation from imaginary $\mu_B$ or Taylor expansion around
$\mu_B = 0$. Both these methods are based on the fact that the chiral phase transition at zero and small values of $\mu_B$ is a
crossover, thus for not very large values of the baryon chemical potential physical observables are analytic functions of
the system parameters in general and in particular of the $\mu_B$. In this case one can obtain results for QCD at nonzero $\mu_B$
either by performing simulations at imaginary $\mu_B$, which is free from the sign problem, and then analytically continue to
real values of $\mu_B$, or by calculating the Taylor coefficients of various quantities at $\mu_B = 0$ and then performing a numerical
summation of the Taylor series. For this reason the precise data on the phase diagram of QCD at nonzero baryon density
are only known for small and moderate values of the chemical potential $\mu_B/T \le 2.5 - 3$ [464, 465].

Much less is known about the phase diagram of QCD as a function of all three parameters: temperature, nonzero
baryon density and magnetic field. Lattice QCD studies of this problem were carried out in [466]. To overcome the sign
problem, in this paper the authors performed the simulations at the imaginary values of the baryon chemical potential
and analytically continued the results to the real values of $\mu_B$. This approach is justified since the phase transition in
QCD is a crossover for small and moderate values of magnetic field and chemical potential.

In the paper [466] two different phase transitions were discussed: the chiral, which was determined from the chiral
observables: the chiral condensate and the chiral susceptibility, and the deconfinement phase transition, which was
determined using the single quark entropy [467]. Since the finite temperature phase transition in QCD is a crossover
rather than a genuine phase transition, different observables can follow different behaviour and predict slightly different
transition temperature and transition width.

The final sketch of the phase diagram from [466] is presented in Fig.15.1. In this Figure the phase digram of QCD
in the plane $T-eB$ is shown for several values of the baryon chemical potential $\mu_B = 0, \mu_B = 250$ MeV and $\mu_B = 500$
MeV, the data for the latter two values of the magnetic field were obtained using the method of analytical continuation.
<figure><img src="image_58.png" /><figcaption>**Figure 15.1:** (From [466]). The tentative phase diagram in the plane temperature-magnetic field $T - eB$ for several values of the baryon chemical potential $\mu_B = 0, 250$ and 500 MeV.</figcaption></figure>

We would like to stress, that the method of analytical continuation might be questionable for the largest studied value of $\mu_B = 500$ MeV, however we present it here in order to enhance the effect of the corresponding parameters on the phase transition. We can learn from this plot that both magnetic field and baryon density decrease the pseudo-critical temperature of the chiral and deconfining phase transition and when both these parameters are turned on their effect is just combined. Note, however, that the width of the phase transition has a nontrivial dependence on the parameters - with small magnetic field it decreases with $\mu_B$ and at large magnetic fields it increases with $\mu_B$. For the parameters under study no clear signatures of the first-order phase transition were observed.

## 15.3. Equation of State

Equation of State of plays a pivotal role in studies of the quark-gluon plasma, providing an important input for hydro-dynamical simulations. It describes how thermodynamic quantities, including pressure $p$, energy density $\epsilon$ and others depend on the parameters of the system. In this Section we discuss, how magnetic field changes the QCD Equation of State.

The basic quantity for the Equation of State is pressure $p$, which is given by the logarithm of the partition function $Z$:

$$p = \frac{T}{V} \ln Z(\mu_B, \mu_S, \mu_Q, T, V, eB), \qquad (15.2)$$

here $T$ is the system temperature and $V$ is its volume. Since the partition function $Z$ and its logarithm cannot be determined in lattice calculations, the standard way which is used for the determination of the pressure $p$ is a calculation of its derivatives. For example, baryon density is given by the derivative with respect to the baryon chemical potential: $n_B = \frac{\partial p}{\partial \mu_B}$ and can be directly calculated on the lattice. Given that simulations at nonzero $\mu_B$ are hampered by the sign problem, one typically studies the Taylor expansion of the pressure $p$ in terms of the chemical potential $\mu_B$:

$$\frac{p}{T^4} = c_0 + c_2 \left(\frac{\mu_B}{T}\right)^2 + c_4 \left(\frac{\mu_B}{T}\right)^4 + c_6 \left(\frac{\mu_B}{T}\right)^6 + O\left(\left(\frac{\mu_B}{T}\right)^8\right). \qquad (15.3)$$

It naturally leads to the following Taylor expansion of $n_B$ in terms of $\mu_B$:

$$\frac{n_B}{T^3} = 2c_2 \frac{\mu_B}{T} + 4c_4 \left(\frac{\mu_B}{T}\right)^3 + 6c_6 \left(\frac{\mu_B}{T}\right)^5 + O\left((\mu_B/T)^7\right) \qquad (15.4)$$
<figure><img src="image_59.png" /><figcaption>**Figure 15.2:** (From [470]). The dependence of the coefficients $c_2, c_4, c_6$ on the temperature for several values of magnetic field. Data for $eB = 0$ were calculated based on results of [471], data for nonzero magnetic field are from [470]. Dashed lines correspond to the ideal gas approximation.</figcaption></figure>

<figure><img src="image_60.png" /><figcaption>**Figure 15.3:** (From [470]). Excess of the pressure $\Delta p = p(\mu_B) - p(\mu_B = 0)$ due to nonzero baryon density $\mu_B$ for three different values of magnetic field $eB = 0.3 \text{ GeV}^2$, $eB = 0.6 \text{ GeV}^2$ and $eB = 1.2 \text{ GeV}^2$. Dashed lines correspond to the ideal gas approximation.</figcaption></figure>

Without external density $\mu_B = 0$ the Equation of State is given by the coefficients $c_0$ and its dependence on the temperature and magnetic field. Coefficient $c_0$ was extensively studied in [468, 469, 15], where it was found that magnetic field leads to a large additional contribution to the thermodynamic observables at magnetic fields $eB \sim 0.5 \text{ GeV}^2$.

The behaviour of the coefficients $c_2, c_4$ and $c_6$ and their dependence on the magnetic field was studied in [470]. In Fig. 15.2 we present the dependence of these coefficients on the temperature for several values of magnetic field $eB = 0.0$ (from [471]), $eB = 0.3$, $eB = 0.6$, $eB = 1.2 \text{ GeV}^2$. The main conclusion is that the coefficients $c_2, c_4$ and $c_6$ and, consequently, the pressure $p$ are significantly enhanced by the external magnetic field. At large magnetic fields their temperature dependence exhibits a peak structure. One can also expect that at high temperatures the system is described by an (almost) ideal gas approximation. Its predictions are also presented in Fig. 15.2 by dashed lines, which seem to agree with the lattice results at high temperatures above the chiral phase transition. Finally, in Fig. 15.3 we present an additional contribution to the pressure (15.3) coming from the baryon density $\Delta p = p(\mu_B) - p(0)$ for several values of magnetic field $eB = 0.3, 0.6$ and $1.2 \text{ GeV}^2$. From Fig. 15.2 and 15.3 we can see that nonzero magnetic field not only significantly increases the pressure, but also modifies its dependence on the temperature and the baryon chemical potential $\mu_B$.

In Ref. [457] the first results for the QCD Equation of State beyond the approximation $\mu_s = 0, \mu_u = \mu_d$ are presented. The authors tuned the chemical potentials to maintain the conditions $n_S = 0, n_Q/n_B = 0.4$, which correspond to the initial conditions in heavy ion collision. The authors also determine the leading order coefficient $c_2$ for these conditions, which is also significantly enhanced by the magnetic field.
## 15.4. Fluctuations of the conserved charges

Fluctuations of the conserved charges: baryon density (B), electric charge (Q) and strangeness (S) are closely related to the QCD phase diagram and its Equation of State, probing different degrees of freedom. Moreover, these quantities (more precisely, their combinations) can be easily accessible experimentally, building a bridge between lattice calculations and heavy ion collision experiments.

Coefficients that describe the fluctuations, are defined as:

$$ \chi_{ijk}^{BQS} = \left. \frac{\partial^{i+j+k}(p/T^4)}{\partial^i(\mu_B/T) \partial^j(\mu_Q/T) \partial^k(\mu_S/T)} \right|_{\mu_B=\mu_Q=\mu_S=0} \qquad (15.5) $$

Due to the sign problem in QCD, on the lattice the Equation of State is computed as the Taylor expansion of the pressure $p$ in $\mu_B$, $\mu_S$ and $\mu_Q$, thus naturally producing the values for the coefficients $\chi_{ijk}^{BQS}$. Without magnetic field the behaviour of the fluctuations $\chi_{ijk}^{BQS}$ was extensively studied on the lattice [464, 465].

Behaviour of the fluctuations up to the second order in QCD with the external magnetic field was studied in [472, 473, 457]. In [472] quadratic fluctuations and correlations of the conserved charges were calculated on the lattice at heavier than physical pion mass $m_\pi \approx 220$ MeV. It was found that magnetic field significantly changes the properties of the system. First, values of the coefficients $\chi_{ijk}^{BQS}$ notably grow with increasing magnetic field. Second, the temperature dependence of these coefficients at high magnetic fields starts to be non-monotonous, exhibiting a peak around the pseudo-critical temperature. Finally, the position of an inflection point or a peak in the temperature dependence of $\chi_{ijk}^{BQS}$ is shifted towards lower temperatures, indicating that the change in the degrees of freedom occurs at lower temperatures. In [457] the quadratic fluctuations and the conditions corresponding to the strange neutrality were studied for nonzero values of magnetic field and at the physical value of the pion mass. Results presented in [457] seem to exhibit the same qualitative behaviour as in [472]. Note that the results for the Equation of State of QCD in magnetic field discussed in Sec. 15.3 also show the same trend. In [473] the authors studied specific combinations of observables: the correlation between the baryon charge and the electric charge $\chi_{11}^{BQ}$ and the ratio of the chemical potentials $\mu_Q/\mu_B$ (tuned to the conditions corresponding to the heavy ion collision experiments). Based on their results which indicate that these quantities significantly grow at sufficiently large magnetic field, it was proposed that one can use these combination $\chi_{11}^{BQ}$ and $\mu_Q/\mu_B$ to estimate the values of the magnetic field produced in real heavy-ion collision experiments (the so-called magnetometer).

## 15.5. Summary

Bulk properties of QCD under such pairs of external parameters as density-temperature and magnetic field-temperature were extensively studied in lattice simulations. However, much less is known about the properties of QCD in the three-dimensional space temperature-density-magnetic field. In this section we reviewed current status of the lattice studies of this system: phase diagram, Equation of State and related to it conserved charge fluctuations. Our knowledge is limited due to the sign problem for QCD at nonzero density, for this reason, the studies are restricted mainly to the Taylor expansion of the corresponding quantities and small values of the baryon chemical potential $\mu_B/T < 2.5 - 3$. However, we see that magnetic fields and baryon density significantly modify the properties of QCD, the critical temperature and possibly the nature of the phase transition. The results of the presented studies also suggest some observables that can quantify the value of the magnetic field in experiments (magnetometer).

Presented studies do not go to larger values of magnetic field, in particular, they even do not come close to the CEP, which is predicted to exist at very large values of magnetic field. Possible existence of this CEP and its connection to
the chiral CEP in QCD in the $\mu-T$ plane could be a subject of possible future studies. Among other possible directions one could consider studies of the phase diagram and other properties of QCD with other fields and other parameters, including the chiral chemical potential $\mu_5$, electric field $eE$, non-homogeneous fields and others.

# 16. Superconductivity of vacuum in an extreme magnetic field: evidence from the Electroweak sector of the Standard model and implications for QCD

## 16.1. Introduction and Motivation

### 16.1.1. Strong magnetic fields in Nature

An intense magnetic field can drastically affect various properties of physical systems and can even create new states of matter. For example, the strongest steady fields in Earth laboratories reach values of about a few tens of Tesla (T), which is enough to observe the quantum Hall effect in graphene at room temperature [474]. In plasma physics, a magnetic field of a similar scale is designed to hold the thermonuclear plasma of temperature $1.5 \times 10^8$ K at ITER Tokamak [475]. The strained graphene sheets may generate synthetic magnetic fields up to enormous values of about 300 T [476]. These are, perhaps, the strongest steady fields that we can reach in Earth conditions.

More powerful, persistent magnetic fields can, of course, exist in astrophysical environments. The fields with the strengths of the order $10^8...11$ T are supposed to be present at the surfaces of highly magnetized neutron stars, called magnetars [477]. The magnetar SGR 1806-20 possesses the strongest recorded value of about $10^{11}$ T which has been confirmed to from investigation of bright flares of short-duration $\gamma$-ray bursts of this star [478]. In the cores of the magnetars, one expects to encounter even more powerful steady fields up to $10^{14}$ T [479, 480].

According to theoretical estimates, much stronger but transient magnetic fields of the order of $10^{14...16}$ T appear for a short time of $10^{-(23...24)}$ s, in quark-gluon plasma which is routinely created in non-central relativistic heavy-ion collisions at RHIC and LHC [95, 481]. The field of $10^{14}$ T has been recently confirmed by the STAR experiment at the Relativistic Heavy Ion Collider [482].

The cosmological electroweak phase transition might have developed a very strong magnetic field of the electroweak strength, $10^{20}$ T, which could have influenced the subsequent evolution of the Universe [483, 484]. While this field bears the word “weak” in its name, it is among the strongest fields in the Standard Model of particles. Similarly, immense fields can possibly exist even in the modern Universe in the vicinity of the magnetized black holes [485, 486].

The presence of a background magnetic field can alter the phase diagram of fundamental field theories, impacting the transitions between various states of matter. This phenomenon bears significant relevance to the early Universe, as, during the cooling stages of its evolutionary stages, the Universe underwent a sequence of high-temperature transitions. Notably, the electroweak symmetry breaking occurred at temperature $T^{\text{EW}} = 159.5(1.5)$ GeV $\simeq 2 \times 10^{15}$ K via a smooth crossover transition [487]. It has been immediately followed by the color confinement (hadronization) transition, that was accompanied by the chiral symmetry restoration at $T^{\text{QCD}} = 156.5(1.5)$ MeV $\simeq 2 \times 10^{12}$ K [488]. The confining and chiral transitions, described by quantum chromodynamics (QCD), also have a crossover nature: they overlap and appear at the same temperature range. While the electroweak crossover does not experience a qualitatively noticeable modification at moderate (in the electroweak scale) $10^{20}$ T magnetic fields [489], the magnetic-field background of the strength of a few
GeV squared (corresponding to $10^{17}$ T) can generate a new end-point thus turning the finite-temperature QCD crossover into a first-order phase transition [203]. Surprisingly, the magnetic field background of sufficiently powerful strength can also influence the properties of the vacuum itself, which, by definition, contains no matter et al. Below, we will discuss the magnetic properties of a cold, empty vacuum devoid of matter.

### 16.1.2. Characteristic scales of magnetic field in particle physics

The impact of a magnetic field on a particular phenomenon is set by comparison of its strength with the relevant mass or length scales involved. For example, for electromagnetic interactions described by quantum electrodynamics, the relevant intensity of the magnetic field is set by the Schwinger limit,<fnref n="12" />

$$B^{\text{QED}} = m_e^2/e \approx 4 \times 10^9 \text{T}, \qquad (16.1)$$

determined by the electron mass $m_e$. At this strength – which is already bypassed by the fields near the surface of magnetars [477] – the vacuum acquires optical birefringence properties [490] and can act as a “magnetic lens” which is able to distort and magnify images [491] similarly to the galactic-scale gravitational lens.

Strong interactions, described by quantum chromodynamics, become sensitive to the presence of the magnetic field when it reaches the strength of the hadronic mass scale,

$$B^{\text{QCD}} \sim m_p^2/e \sim 10^{16} \text{T}, \qquad (16.2)$$

where $m_p$ is the proton mass. Such fields lead to the effect of magnetic catalysis [492, 493, 348], which implies, in particular, a persistent enhancement of the chiral symmetry breaking in QCD vacuum as the external magnetic field strengthens.

Electroweak interactions are influenced by much stronger magnetic fields,

$$B_W^{\text{EW}} = m_W^2/e \approx 1.1 \times 10^{20} \text{T}, \qquad (16.3)$$

determined by the W-boson mass $m_W \approx 80.4$ GeV, which sets a typical mass scale in the electroweak sector. More than thirty years ago, it was suggested that such fields generate instability of vacuum, which proceeds via an onset of the condensation of the W bosons. The presence of the instability can be deduced from the classical equations of motion of the electroweak model [494, 495, 496, 497, 498, 499]. Much later, it became clear that the transition into this exotic electroweak vacuum state marks a radical change in the transport properties of the vacuum, which acquires anisotropically superconducting [500] and superfluid [501] properties.

For completeness, we also mention that the electroweak vacuum should possess the second phase transition at an even higher magnetic field:

$$B_H^{\text{EW}} = m_H^2/e \approx 2.7 \times 10^{20} \text{T}, \qquad (16.4)$$

which is determined by the Higgs mass $m_H = 125.1$ GeV. Above this field, $B_{c2}$, the electroweak symmetry is expected to be restored [502, 503, 499].

<footnote n="12">¹²We use the units $\hbar = c = 1$.</footnote>
### 16.1.3. Vacuum instability in the electroweak sector and the puzzle in QCD

The W boson plays a crucial role in the instability of the electroweak vacuum because this particle has three important properties: it is an (i) electrically charged (ii) vector particle that (iii) possesses a large gyromagnetic ratio (the Landé factor), $g = 2$. As a result, the W mesons strongly couple to the magnetic field via their large magnetic moments. The importance of these properties can be understood from the energy spectrum of a free particle of a mass $m$ placed in the background of an external steady uniform magnetic field $B$:

$$ \epsilon_{n,s_z}^2(p_z) = p_z^2 + (2n - gs_z + 1)eB + m^2. \qquad (16.5) $$

Here $n$ is a nonnegative integer number that labels the Landau levels, $s_z = 0, \pm 1$ is the spin projection on the axis of the field $z$, and $p_z$ is the particle momentum along the $z$ axis. We assume that $eB > 0$.

With an increase of the magnetic field, the mass of the W polarization branch (16.5) with the quantum numbers $n = 0, s_z = +1, p_z = 0$ decreases, $M^2(B) = m^2 - eB$, and becomes a purely imaginary quantity if the magnetic field exceeds the critical value $B_c = m^2/e$. Notice that this effect arises due to the big gyromagnetic ratio $g = 2$. For W bosons, the critical field is given by Eq. (16.3), above which the vacuum experiences a tachyonic instability associated with a second-order phase transition [495, 496, 497, 498, 499]. The vacuum turns into an inhomogeneous vortex-dominated phase which will be discussed in more detail below. In realistic quantum field theories, the W bosons interact with other fields, so that this mechanism should be further scrutinized to ascertain its robustness in the presence of interactions.

One can reasonably wonder whether analogous phenomena can exist in other physical systems that also host electrically charged vector particles. A similar mechanism has indeed been suggested to operate at the level of QCD that possesses electrically charged mesonic bound states with vector quantum numbers, the $\rho$-meson excitations [500]. In this case, the $\rho$-meson mass $m_\rho \simeq 770$ MeV determines the free-limit value of the critical field, $B_c = m_\rho^2/e$, that appears to be of the order of the characteristic field in QCD, $B_c \sim B^{QCD}$ (16.2). If this mechanism works, then the vacuum may also acquire electromagnetic superconducting properties mediated by the condensation of $\rho^+$ mesons.

The existence of this intriguing superconducting state of the QCD vacuum is supported by low-energy models [122] and holographic approaches [504, 81]. At the same time, it is questioned by symmetry arguments [125] and effective field theory calculations [74]. Furthermore, this exotic phase avoids a direct first-principle detection in numerical simulations [125, 119], presumably because of numerical difficulties associated with the inhomogeneity of the charged condensate and the smooth crossover nature of the suspected superconducting transition [505, 506, 507]. Moreover, the first-principle lattice simulations of QCD do not display any signature of a tachyonic instability in a zero-temperature QCD vacuum exposed to high magnetic fields [125, 118, 67, 117, 119, 508]. In other words, the masses of vector mesons do not vanish at the range of the relevant fields (16.2) in the apparent contradiction to the classical analysis within effective models. While these results would be sufficient to exclude the presence of the superconducting phase in QCD vacuum at strong fields, we show below that the same questions can also be asked at the level of the electroweak sector there the tachyonic instability associated with a second-order phase transition, was also predicted by theoretical analysis [496, 497, 498, 499]. Remarkably, despite the absence of any tachyonic instability, the electroweak vacuum still has a superconducting electroweak phase [509]. This observation gives hope for a search for a similar phase in QCD, addressed in more detail in this paper.

The key argument in favor of the existence of the new QCD phase, which is consistent with available numerical results, is that the transition from the usual hadron phase to the superconducting phase at the field strengths of the order of the QCD scale (16.2) may proceed via a smooth crossover, which prevents any instability. While this statement sounds
questionable in view of the simple analysis of the free-particle spectrum (16.5), the first-principle numerical simulations show that no instability occurs in the electroweak model, where the vacuum enters a similar W-condensed state via a magnetic-field-induced crossover. In other words, in the absence of a second-order phase transition, the instability should never appear in the electroweak model. Still, the magnetic field produces the superconducting phase in the electroweak model.

Thus, a QCD transition to the new phase can also occur at a nonvanishing $\rho$-meson mass in the absence of any thermodynamic singularity [506]. The latter scenario has a hypothetical character since it requires confirmation from a first-principle simulation. In this respect, the electroweak model provides us with a compelling playground, backed by the similarity of the superconducting mechanisms in both QCD and electroweak systems.

Notice that even in the electroweak model, where the condensation of the W bosons has robustly been predicted at the classical level, the existence of the magnetic-field-induced phase was questioned in the theoretical literature before the numerical first-principle results [509] became available. At the classical level, the formation of the periodic vortex lattice in the background magnetic field has been established not only analytically but also numerically [510]. However, this classical-level scenario, together with the arguments based on loop computations [511], has been questioned in Ref. [512] where it was shown that quantum corrections could add a radiative term to the classical W mass in such a way that the mass does not vanish at the critical field $B_c = B_W^{EW}$. Accordingly, it was concluded that no thermodynamic instability should appear in the electroweak model. Moreover, earlier numerical simulations of the electroweak model in the background magnetic field did not reveal the presence of the vortex-dominated phase around the finite-temperature electroweak crossover [489], which could be explained by a destructive role of strong thermal fluctuations. Therefore, the very existence of the superconducting phase in an electroweak model is a rather nontrivial fact. Below, we review the first-principle numerical results obtained in the electroweak model which demonstrate the existence [509].

Before closing this section, it is worth mentioning that the mechanisms that drive the vacuum superconductivity at QCD [500] and electroweak [500, 501] scales are very similar to reentrant superconductivity that was suggested to occur in very clean superconducting materials in high—in the condensed matter standards—magnetic fields [513]. This phenomenon is suggested to be catalyzed by magnetic-field-induced condensation of Cooper pairs of electrons in a p-wave bound state. The p-wave Cooper pair is a condensed-matter counterpart of the $\rho^+$ meson, the latter being an electrically charged bound state of a quark and an anti-quark. The strength of the magnetic field required for this phenomenon to occur is suggested to be in the range of a few dozen Teslas. [513]

## 16.2. Magnetic-field induced vortex phase in the electroweak model

We consider the bosonic sector of the Electroweak model with the Lagrangian

$$ \mathcal{L}_{\text{EW}} = -\frac{1}{2} \text{Tr} (W_{\mu\nu} W^{\mu\nu}) - \frac{1}{4} Y_{\mu\nu} Y^{\mu\nu} + (D_{\mu} \phi)^{\dagger} (D^{\mu} \phi) - \lambda \left(\phi^{\dagger}\phi - \frac{v^2}{2}\right)^2, \quad (16.6) $$

where $W_{\mu\nu}^a = \partial_{\mu}W_{\nu}^a - \partial_{\nu}W_{\mu}^a + ig\epsilon^{abc}W_{\mu}^bW_{\nu}^c$ and $Y_{\mu\nu} = \partial_{\mu}Y_{\nu} - \partial_{\nu}Y_{\mu}$ are the field strengths of the SU(2) gauge field $W_{\mu}^a$ and U(1)$_Y$ hypercharge gauge field $Y_{\mu}$, respectively. These fields interact with the complex scalar Higgs doublet $\phi \equiv (\phi_1, \phi_2)^T$ via the covariant derivative, $D_{\mu} = \partial_{\mu} + \frac{i}{2}gW_{\mu}^a\sigma^a + \frac{i}{2}g'Y_{\mu}$, where $\sigma^a$ ($a = 1, 2, 3$) are the Pauli matrices. The ratio of the U(1) and SU(2) gauge couplings, $g'/g = \tan\theta_W$, defines the electroweak mixing (Weinberg) angle $\theta_W$ with $\sin^2\theta_W \equiv 1 - m_W^2/m_Z^2 = 0.22290(30)$ [514]. The dimensionless coupling $\lambda$ controls the self-interaction of the Higgs field, and the only dimensionful parameter $v$ gives, at low temperature, the vacuum expectation value to the Higgs field,