The motivation for relaxing Eq. (9) is to go beyond the approximation of infinite collisional energy and be able to describe observables such as the energy momentum tensor of the Glasma in a rapidity dependent manner. A simple generalization is given by
$$\begin{aligned} \rho _{({\tiny \mathrm {A}}, {\tiny \mathrm {B}})}(x) \approx \lambda (x^\pm ) \rho _{({\tiny \mathrm {A}}, {\tiny \mathrm {B}})}(\mathbf{x}_T), \end{aligned}$$
(16)
where \(\lambda \) is a normalized function, which determines the longitudinal shape of the charge density. The width of \(\lambda \) should be directly related to the Lorentz-contracted diameter of the nucleus. It should be noted that Eq. (16) is a special case where the color structure does not depend on the longitudinal coordinate \(x^\pm \) and more general color charge densities \(\rho _{({\tiny \mathrm {A}}, {\tiny \mathrm {B}})}(x^\pm , \mathbf{x}_T)\) are possible as well.
A direct consequence of allowing for finite longitudinal extent is that the collision event is not just a single point in the Minkowski diagram (see Fig. 2), but an extended space-time region in which the nuclei are able to interact. The non-perturbative nature of the color fields of the nuclei and the extended interaction time generally make analytical calculations in this space-time region intractable. Our approach to the 3+1D Glasma is therefore to simulate not just the evolution in the future light cone, i.e. region IV in Fig. 2, but the whole collision [35]. This setup is formulated in the laboratory frame using (t, z) coordinates and the time evolution is performed in the direction of t instead of proper time \(\tau \). The initial conditions of such a time evolution in t are specified in the following way: at some initial time \(t_0 < 0\) sufficiently far away from the collision time \(t = 0\), the color charge densities of the nuclei are non-overlapping in z. In LC gauge, the color field of each nucleus is given by
$$\begin{aligned} A^{i=x,y}_{({\tiny \mathrm {A}}, {\tiny \mathrm {B}})}(t_0, \mathbf{x}) = \frac{1}{ig} V_{({\tiny \mathrm {A}}{\tiny \mathrm {B}})}(t_0, \mathbf{x}) \partial ^i V^\dagger _{({\tiny \mathrm {A}}{\tiny \mathrm {B}})}(t_0,\mathbf{x}), \end{aligned}$$
(17)
where \(\mathbf{x} = (x, y, z)\) is a three-dimensional coordinate vector. Equation (17) solves the classical YM Eqs. (2) with the current given by Eq. (16) [35]. Since the fields vanish exponentially fast between the two nuclei, the superposition of both color fields
$$\begin{aligned} A^\mu (t_0, \mathbf{x}) = \delta ^\mu _{i=x,y} \left( A^i_{\tiny \mathrm {A}}(t_0, \mathbf{x}) + A^i_{\tiny \mathrm {B}}(t_0, \mathbf{x}) \right) , \end{aligned}$$
(18)
is a valid solution to the YM Eqs. (8) at time \(t_0\). Evolving this initial condition for the YM field to \(t > 0\) yields a genuinely 3+1D description of the rapidity dependent Glasma. Equations (17) and (18) are compatible with the temporal gauge condition \(A^0(t, \mathbf{x}) = 0\), \(\forall t\in \mathbb {R}\), which is a convenient gauge choice for an evolution along \(x^0 = t\).
One of the main differences to the standard 2+1D Glasma is that in the laboratory frame the system not only consists of the color fields but also the color currents of the nuclei. In order to solve the YM Eqs. which include color currents, we make use of the CPIC method [40,41,42,43,44]. CPIC allows for consistent simulations of color charged point particles coupled to color fields on a lattice. For a comprehensive description of our numerical methods we refer to [35, 39].
The numerical treatment of the fields follows the common real-time lattice gauge theory approach: by discretizing Minkowski space-time as a hypercubic lattice with spacings \(a^i\) and time-step \(\varDelta t\), the YM Eqs. can be written in the standard leapfrog scheme
$$\begin{aligned} E_i(t+ \frac{\varDelta t}{2}, \mathbf{x})= & {} - \sum _j \frac{\varDelta t}{(a^j)^2} \left[ U_{i,j}(t, \mathbf{x}) + U_{i,-j}(t, \mathbf{x}) \right] _\mathrm {ah} \nonumber \\&+ \varDelta t \, j_i(t, \mathbf{x}) + E_i(t - \frac{\varDelta t}{2}, \mathbf{x}), \end{aligned}$$
(19)
$$\begin{aligned} U_i(t + \varDelta t, \mathbf{x})= & {} \exp \bigg ( i \varDelta t \, E_{i}(t+\frac{\varDelta t}{2}, \mathbf{x}) \bigg ) U_i(t, \mathbf{x}), \end{aligned}$$
(20)
where \(U_{i}(t, \mathbf{x}) \simeq \exp \big ( i g a^i A_i(t, \mathbf{x}) \big )\) are the gauge links, \(E_i(t, \mathbf{x})\) is the chromo-electric field on the lattice and \(\left[ X \right] _\mathrm {ah}\) denotes the anti-hermitian, traceless part of a matrix X. The plaquette variables \(U_{i,j}(t, \mathbf{x})\) are defined as
$$\begin{aligned} U_{i,j}(t, \mathbf{x}) = U_i(t, \mathbf{x}) U_j(t, \mathbf{x} + \hat{a}_i) U^\dagger _i(t, \mathbf{x} + \hat{a}_i) U^\dagger _j(t, \mathbf{x}). \end{aligned}$$
(21)
The color currents of the nuclei \(j_i(t, \mathbf{x})\), which enter on the right hand side of Eq. (19), require careful treatment. The main idea of using the CPIC method to describe collisions in the CGC framework is to replace the continuous color charge distributions \(\rho \) of the nuclei by a large number of auxiliary particles with time-dependent color charges \(Q_k(t)\) such that the original color charge distribution is sufficiently well approximated on a lattice:
$$\begin{aligned} \rho (t, \mathbf{x}) \approx \sum _k Q_k(t) \delta ^{(3)}(\mathbf{x} - \mathbf{x}_k(t)), \end{aligned}$$
(22)
where k is the particle index. Similar to the boost invariant case, we assume these auxiliary particles to be recoil-less. Thus, the trajectories of the particles \(\mathbf{x}_k(t)\) are fixed and not part of the dynamics of the system. The time-dependence of the color charges \(Q_k(t)\) is determined from the discretized continuity Eq.
$$\begin{aligned}&\frac{\rho (t + \frac{\varDelta t}{2}, \mathbf{x}) - \rho (t - \frac{\varDelta t}{2}, \mathbf{x})}{\varDelta t} \nonumber \\&= \sum _i \frac{j_i(t, \mathbf{x}) - U^\dagger _{i}(t, \mathbf{x} - \hat{a}_i) j_i(t, \mathbf{x} - \hat{a}_i) U_{i}(t, \mathbf{x} - \hat{a}_i)}{a^i}, \end{aligned}$$
(23)
which is the discrete analogue of gauge covariant continuity equation
$$\begin{aligned} D_\mu J^\mu (t, \mathbf{x} ) = 0. \end{aligned}$$
(24)
In our setup, the color charge \(Q_k(t)\) of each particle is mapped to its nearest grid point on the spatial lattice in each time-step. Whenever the nearest grid point of a particular point charge changes from one lattice site \(\mathbf{x}\) to a neighbouring lattice site \(\mathbf{y}\), parallel transport is applied to the color charge accordingly:
$$\begin{aligned} Q_k\left( t + \frac{\varDelta t}{2}\right) = U^\dagger _{\mathbf{x} \rightarrow \mathbf{y}}(t) Q_k\left( t - \frac{\varDelta t}{2}\right) U_{\mathbf{x} \rightarrow \mathbf{y}}(t), \end{aligned}$$
(25)
where \(U_{\mathbf{x} \rightarrow \mathbf{y}}(t)\) is the appropriate gauge link connecting \(\mathbf{x}\) and \(\mathbf{y}\). At the same time, the movement of the particle generates a color current \(j_i(t, \mathbf{x})\) in accordance with Eq. (23). In CPIC, this treatment of particles is known as the nearest-grid-point scheme [40]. By evolving the color charges of the particles in this manner, the discretized field Eqs. (19) and (20) are solved consistently in the sense that the discrete Gauss law
$$\begin{aligned}&\sum _i \left( E_i(t + \frac{\varDelta t}{2}, \mathbf{x}) - \tilde{E}_i(t + \frac{\varDelta t}{2}, \mathbf{x} - \hat{a}_i) \right) = \nonumber \\&\qquad \rho (t+\frac{\varDelta t}{2}, \mathbf{x}) , \end{aligned}$$
(26)
remains satisfied throughout the simulation and gauge covariance on the lattice is retained. In Eq. (26) the parallel transported electric field is given by
$$\begin{aligned}&\tilde{E}_i\left( t + \frac{\varDelta t}{2}, \mathbf{x} - \hat{a}_i\right) =\nonumber \\&\qquad U^\dagger _{i}\left( t, \mathbf{x} - \hat{a}_i\right) E_i\left( t + \frac{\varDelta t}{2}, \mathbf{x} - \hat{a}_i\right) U_{i}\left( t, \mathbf{x} - \hat{a}_i\right) . \end{aligned}$$
(27)
We find that numerically stable 3+1D Glasma simulations using the leapfrog scheme Eqs. (19) and (20) require high lattice resolution with particularly fine lattice spacing along the beam axis z. In practice, this can be computationally prohibitive. This problem is related to a subtle numerical instability inherent to the leapfrog scheme known as the numerical Cherenkov instability [46], which also affects traditional electromagnetic plasma simulations. This unphysical instability is caused by lattice artifacts which modify the propagation of wave modes on the lattice: high frequency modes propagate at a lower phase velocity compared to low frequency modes (i.e. numerical dispersion). In contrast, the auxiliary particles move at the speed of light by design. This situation is reminiscent of charged particles moving through a medium in which the in-medium speed of light is lower than the particle velocity, which leads to the well-known phenomenon of Cherenkov radiation. The particular lattice discretization used in Eqs. (19) and (20) leads to a similar, although purely numerical generation of Cherenkov radiation. Due to the numerical instability the nuclei are not able to propagate stably unless a very small lattice spacing is chosen along z, which reduces the mismatch in propagation velocities. Fortunately, these numerical problems can be solved by modifying the lattice discretization of the color fields. In [38] we show how an improved numerical scheme restores the correct propagation of wave modes along the beam axis such that artificial Cherenkov radiation is effectively avoided. This modification mainly amounts to replacing specific spatial plaquette terms (see Fig. 3a) with time-averaged generalizations of these terms. Figure 3b shows one example of such a time-averaged term. In comparison to the leapfrog scheme Eqs. (19) and (20), which is an explicit finite difference method, our numerical method is in the form of a system of semi-implicit equations, which is solved in an iterative manner. Our semi-implicit method therefore trades computational performance for numerical stability and accuracy.