The energy-momentum tensor at the earliest stage of relativistic heavy-ion collisions

Abstract

Nuclear collisions at high energies produce a gluon field that can be described using the Colour Glass Condensate (CGC) effective theory at proper times $\tau \lesssim 1$ fm. The theory can be used to calculate the gluon energy-momentum tensor, which provides information about the early time evolution of the chromo-electric and chromo-magnetic fields, energy density, longitudinal and transverse pressures, and other quantities. We obtain an analytic expression for the energy-momentum tensor using an expansion in the proper time, and working to sixth order. The calculation is technically difficult, in part because the number of terms involved grows rapidly with the order of the $\tau$ expansion, but also because of several subtle issues related to the definition of event-averaged correlators, the method chosen to regulate these correlators, and the dependence of results on the parameters introduced by the regularization and nuclear density profile functions. All of these issues are crucially related to the important question of the extent to which we expect a CGC approach to be able to accurately describe the early stages of a heavy-ion collision. We present some results for the evolution of the energy density and the longitudinal and transverse pressures. We show that our calculation gives physically meaningful results up to values of the proper time which are close to the regime at which hydrodynamic simulations are initialized. In a companion paper [1] we give a detailed analysis of several other experimentally relevant quantities that can be calculated from the energy-momentum tensor.

Appendices

Appendix A: Notation

We use at different times three different coordinate systems. Minkowski, light-cone, and Milne (or co-moving) coordinates. The collision axis is defined to be the z-axis. The two transverse coordinates are always the last two elements in the position 4-vector and will be denoted $\vec {x}_\perp$. We will write the position 4-vector as

$$\begin{aligned}&x^\mu _{\mathrm{mink}} = (t,z,\vec {x}_\perp ) \nonumber \\&x^\mu _{\mathrm{lc}} = (x^+,x^-,\vec {x}_\perp ) \nonumber \\&x^\mu _{\mathrm{milne}} = (\tau ,\eta ,\vec {x}_\perp ) \end{aligned}$$

(A1)

with the usual definitions

$$\begin{aligned}&x^+=\frac{t+z}{\sqrt{2}} ~~\text {and}~~ x^-=\frac{t-z}{\sqrt{2}} \end{aligned}$$

(A2)

$$\begin{aligned}&\tau = \sqrt{t^2-z^2}=\sqrt{2x^+ x^-} ~~\text {and}~~ \eta = \frac{1}{2} \ln \left( \frac{x^+}{x^-} \right) \,. \end{aligned}$$

(A3)

We define the relative and average transverse coordinates

$$\begin{aligned}&\vec {r} = \vec {x}_\perp -\vec {y}_\perp ~~\text {and}~~ \vec {R} = \frac{1}{2}\left( \vec {y}_\perp +\vec {x}_\perp \right) \,. \end{aligned}$$

(A4)

We will write unit vectors as ${\hat{r}}=\vec {r}/|\vec {r}|=\vec {r}/r$ and ${\hat{R}}=\vec {R}/|\vec {R}|=\vec {R}/R$ and use standard notation for derivatives like

$$\begin{aligned} \partial ^i_{x} \equiv -\frac{\partial }{\partial x_\perp ^i} ~~\text {and}~~ \partial ^i_{R} \equiv -\frac{\partial }{\partial R^i}\,. \end{aligned}$$

(A5)

In light-cone coordinates we have

$$\begin{aligned} \partial ^{+} = \frac{\partial }{\partial x^-} ~~\text {and}~~ \partial ^{-} = \frac{\partial }{\partial x^+}\,. \end{aligned}$$

(A6)

We note that the chain rule gives

$$\begin{aligned}&-\partial ^i_{x} = \frac{\partial }{\partial r^i} +\frac{1}{2} \frac{\partial }{\partial R^i} \nonumber \\&-\partial ^i_{y} = -\frac{\partial }{\partial r^i} +\frac{1}{2} \frac{\partial }{\partial R^i} \,. \end{aligned}$$

(A7)

The metric tensors in these three coordinate systems are $g_{\mathrm{mink}} = (1,-1,-1,-1)_{\mathrm{diag}}$ and

$$\begin{aligned} g_{\mathrm{lc}} = \left( \begin{array}{cccc} 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 \\ \end{array} \right) \,,\qquad g_{\mathrm{milne}}= \left( \begin{array}{cccc} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} -\tau ^2 &{} 0 &{} 0 \\ 0 &{} 0 &{} -1 &{} 0 \\ 0 &{} 0 &{} 0 &{} -1 \\ \end{array} \right) \,.\nonumber \\ \end{aligned}$$

(A8)

The coordinate transformations are

$$\begin{aligned} x^\mu _{\mathrm{mink}}= & {} M^\mu _{~\nu } x^\nu _{\mathrm{lc}}\,,\qquad M^\mu _{~\nu } = \frac{dx^\mu _{\mathrm{mink}}}{dx^\nu _{\mathrm{lc}}} = \left( \begin{array}{cccc} \frac{1}{\sqrt{2}} &{} \quad \frac{1}{\sqrt{2}} &{} \quad 0 &{}\quad 0 \\ \frac{1}{\sqrt{2}} &{}\quad -\frac{1}{\sqrt{2}} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ \end{array} \right) \nonumber \\ x^\mu _{\mathrm{mink}}= & {} M^\mu _{~\nu } x^\nu _{\mathrm{milne}}\,,\qquad M^\mu _{~\nu } = \frac{dx^\mu _{\mathrm{mink}}}{dx^\nu _{\mathrm{milne}}} \nonumber \\= & {} \left( \begin{array}{cccc} \cosh (\eta ) &{} \quad \tau \sinh (\eta ) &{}\quad 0 &{}\quad 0 \\ \sinh (\eta ) &{}\quad \tau \cosh (\eta ) &{}\quad 0 &{} \quad 0 \\ 0 &{}\quad 0 &{}\quad 1 &{} \quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ \end{array} \right) \,. \end{aligned}$$

(A9)

We define a 4-dimensional gradient operator where the transverse components are derivatives with respect to the average coordinate $\vec {R}$ defined in Eq. (A4). We can transform this gradient operator from Milne to Minkowski coordinates by taking the inverse of the transpose of (A9). This gives

$$\begin{aligned} \partial _\mu ^{\mathrm{mink}} = \left( \begin{array}{c} \cosh (\eta )\frac{\partial }{\partial \tau } - \frac{\sinh (\eta )}{\tau }\frac{\partial }{\partial \eta } \\ -\sinh (\eta )\frac{\partial }{\partial \tau } + \frac{\cosh (\eta )}{\tau }\frac{\partial }{\partial \eta } \\ -\partial _R^1\\ -\partial _R^2 \end{array} \right) \,. \end{aligned}$$

(A10)

The generators $t_a$ of SU$(N_c)$ satisfy

$$\begin{aligned}&[t_a,t_b] = i f_{abc} t_c \nonumber \\&\text {Tr}(t_a t_b) = \frac{1}{2}\delta _{ab} \nonumber \\&f_{abc} = -2i \text {Tr}\big (t_a[t_b,t_c]\big ) \,. \end{aligned}$$

(A11)

Functions like $A_\mu$, $J_\mu$, $\rho$ and $\Lambda$ are SU$(N_c)$ valued functions and can be written as linear combinations of the SU$(N_c)$ generators. In the adjoint representation we write the generators with a tilde as $({\tilde{t}}_a)_{bc} = -i f_{abc}$.

The covariant derivative is defined as $D_\mu = \partial _\mu - i g A_\mu$. In the adjoint representation this becomes $D_{\mu \,ab} = \delta _{ab}\partial _\mu - g f_{abc}A_{\mu \,c}$. Gauge transformations are written

$$\begin{aligned}&U(x) = \text {exp}[i t_a \theta _a(x)]\nonumber \\&\Psi (x) \rightarrow U^\dagger (x)\Psi (x) \nonumber \\&A^\mu (x) \rightarrow \frac{i}{g} U^\dagger (x) \partial ^\mu U(x) + U^\dagger (x) A^\mu (x) U(x) \, \end{aligned}$$

(A12)

$$\begin{aligned}&D_\mu (x) \rightarrow U^\dagger (x) D_\mu (x) U(x) \end{aligned}$$

(A13)

$$\begin{aligned}&F_{\mu \nu }(x) \rightarrow U^\dagger (x) F_{\mu \nu }(x)U(x) \,. \end{aligned}$$

(A14)

We will use two specific gauge transformations (see equations (D8, D10))

$$\begin{aligned} U_1(x^-,\vec {x}_\perp ) = {{\mathcal {P}}}\mathrm{exp}\left[ i g \int _{-\infty }^{x^-} dz^- \Lambda _{1\,a}(z^-,\vec {x}_\perp ) \,t_a\right] \,\nonumber \\ U_2(x^+,\vec {x}_\perp ) = {{\mathcal {P}}}\mathrm{exp}\left[ i g \int _{-\infty }^{x^+} dz^+ \Lambda _{2\,a}(z^+,\vec {x}_\perp )\,t_a\right] \, \end{aligned}$$

(A15)

where we use the “left later” convention for path ordering. In the adjoint representation we write

$$\begin{aligned} W_1(x^-,\vec {x}_\perp ) = {{\mathcal {P}}}\mathrm{exp}\left[ i g \int _{-\infty }^{x^-} dz^- \Lambda _{1\,a}(z^-,\vec {x}_\perp ) \,{\tilde{t}}_a\right] \,\nonumber \\ W_2(x^+,\vec {x}_\perp ) = {{\mathcal {P}}}\mathrm{exp}\left[ i g \int _{-\infty }^{x^+} dz^+ \Lambda _{2\,a}(z^+,\vec {x}_\perp )\,{\tilde{t}}_a\right] \,. \end{aligned}$$

(A16)

These matrices satisfy the usual identity

$$\begin{aligned} U^\dagger t_a U = W_{ab}t_b = t_b W^\dagger _{ba}\,. \end{aligned}$$

(A17)

Covariant derivatives in Milne coordinates include both the gauge field contribution and curvature terms. Products of covariant derivatives acting on a scalar function $\phi$ are written

$$\begin{aligned}&\nabla _\mu \, \phi =D_\mu \, \phi \nonumber \\&\nabla _\mu \nabla _\nu \,\phi = (D_\mu D_\nu -\Gamma ^\lambda _{\mu \nu }D_\lambda ) \, \phi \nonumber \\&\nabla _\alpha \nabla _\mu \nabla _\nu \,\phi = (D_\alpha \nabla _\mu \nabla _\nu -\Gamma ^\tau _{\alpha \mu } \nabla _\tau \nabla _\nu -\Gamma ^\tau _{\alpha \nu }\nabla _\mu \nabla _\tau )\,\phi \,.\nonumber \\ \end{aligned}$$

(A18)

The connection $\Gamma ^\lambda _{\mu \nu }$ can be calculated from the metric tensor (A8) and one easily shows that the only non-zero components are

$$\begin{aligned} \Gamma ^0_{11} = \tau \text {~~ and~~} \Gamma ^1_{01} = \Gamma ^1_{10} = \frac{1}{\tau }\,. \end{aligned}$$

(A19)

Appendix B: Energy-momentum tensor in terms of fields

Our result for the energy-momentum tensor in Eq. (16) can be written in terms of electric and magnetic fields. To obtain this expression we transform the field-strength tensor to Minkowski space and then extract the components of the electric and magnetic fields. We remind the reader that in our notation a Minkowski space 4-vector is written $v^\mu _{\mathrm{mink}} = (v^0,v^z,v^x,v^y)= (v^0,v^z,\vec {v}_\perp )$ and the field-strength tensor therefore takes the form

$$\begin{aligned} F^{\mu \nu }_{\mathrm{mink}} = \left[ {\begin{array}{cccc} 0 &{}\quad -E^z &{}\quad -E^x &{}\quad -E^y \\ E^z &{}\quad 0 &{}\quad -B^y &{}\quad B^x \\ E^x &{}\quad B^y &{}\quad 0 &{}\quad -B^z \\ E^y &{}\quad -B^x &{}\quad B^z &{}\quad 0 \\ \end{array} } \right] \,. \end{aligned}$$

(B1)

The energy-momentum tensor can then be written in terms of field components using Eqs. (14, B1). Since the energy-momentum tensor is symmetric we only need to give the components on the upper half triangle which are

$$\begin{aligned}&T_{\mathrm{mink}}^{00}=\frac{1}{2}\left( E^x_a E^x_a+ E^y_a E^y_a+E^z_a E^z_a+ B^x_a B^x_a\right. \nonumber \\&\qquad \qquad \left. + B^y_a B^y_a+B^z_a B^z_a\right) \nonumber \\&T_{\mathrm{mink}}^{01} = E^y_a B^z_a-E^z_a B^y_a\nonumber \\&T_{\mathrm{mink}}^{02} =E^z_a B^x_a-E^x_a B^z_a\nonumber \\&T_{\mathrm{mink}}^{03} = E^x_a B^y_a-E^y_a B^x_a \nonumber \\&T_{\mathrm{mink}}^{11} = -\frac{1}{2}\left( E^x_a E^x_a-E^y_a E^y_a-E^z_a E^z_a+B^x_a B^x_a\right. \nonumber \\&\qquad \qquad \left. -B^y_a B^y_a-B^z_a B^z_a \right) \nonumber \\&T_{\mathrm{mink}}^{12} = -E^x_a E^y_a-B^x_a B^y_a \nonumber \\&T_{\mathrm{mink}}^{13} = -E^x_a E^z_a-B^x_a B^z_a \nonumber \\&T_{\mathrm{mink}}^{22} = \frac{1}{2}(E^x_a E^x_a-E^y_a E^y_a+E^z_a E^z_a+B^x_a B^x_a\nonumber \\&\qquad \qquad -B^y_a B^y_a+B^z_a B^z_a) \nonumber \\&T_{\mathrm{mink}}^{23} = -E^y_a E^z_a-B^y_a B^z_a \nonumber \\&T_{\mathrm{mink}}^{33} = \frac{1}{2}(E^x_a E^x_a+E^y_a E^y_a-E^z_a E^z_a+B^x_a B^x_a\nonumber \\&\qquad \qquad +B^y_a B^y_a-B^z_a B^z_a)\,. \end{aligned}$$

(B2)

We then transform our result for the field-strength tensor in Milne coordinates into Minkowski coordinates, and then extract the field components. All field components at $\tau >0$ can be written in terms of the lowest order components in Eq. (10). We remind the reader that our notation is $E \equiv E^z(0,\vec {x}_\perp )$, $B\equiv B^z(0,\vec {x}_\perp )$ and ${{\mathcal {D}}}^i = \partial ^i-ig \alpha _\perp ^i(0,\vec {x}_\perp )$. The transverse field components $\vec {E}_\perp$ and $\vec {B}_\perp$ have contributions only at orders that correspond to odd powers of $\tau$ and the longitudinal components $E^z$ and $B^z$ get contributions only from even powers of $\tau$. Our results are

$$\begin{aligned}&E_{(1)}^i(\vec {x}_\perp ) = -\frac{1}{2} \sinh (\eta )[{{\mathcal {D}}}^i,E] -\frac{1}{2}\epsilon ^{ij}\cosh (\eta )[{{\mathcal {D}}}^j,B] \nonumber \\&B_{(1)}^i(\vec {x}_\perp ) = \frac{1}{2}\epsilon ^{ij}\cosh (\eta )[{{\mathcal {D}}}^j,E] -\frac{1}{2} \sinh (\eta )[{{\mathcal {D}}}^i,B] \nonumber \\&E_{(2)}^z = \frac{1}{4}[{{\mathcal {D}}}^i,[{{\mathcal {D}}}^i,E]] \nonumber \\&B_{(2)}^z = \frac{1}{4}[{{\mathcal {D}}}^i,[{{\mathcal {D}}}^i,B]] \nonumber \\&E^i_{(3)} = -\frac{1}{16}\sinh (\eta )[{{\mathcal {D}}}^i,[{{\mathcal {D}}}^j,[{{\mathcal {D}}}^j,E]]]\nonumber \\&\qquad \qquad - \frac{1}{16}\cosh (\eta )\epsilon ^{ij}[{{\mathcal {D}}}^j,[{{\mathcal {D}}}^k,[{{\mathcal {D}}}^k,B]]] \nonumber \\&\qquad \qquad -\frac{ig}{16}\cosh (\eta )\Big ( [E,[{{\mathcal {D}}}^i,E]] - [B,[{{\mathcal {D}}}^i,B]] \Big )\nonumber \\&\qquad \qquad -\frac{ig}{8}\sinh (\eta )\epsilon ^{ij}[E,[{{\mathcal {D}}}^j,B]]\nonumber \\&B^i_{(3)} = \frac{1}{16}\cosh (\eta )\epsilon ^{ij}[{{\mathcal {D}}}^j,[{{\mathcal {D}}}^k,[{{\mathcal {D}}}^k,E]]]\nonumber \\&\qquad \qquad -\frac{1}{16} \sinh (\eta )[{{\mathcal {D}}}^i,[{{\mathcal {D}}}^j,[{{\mathcal {D}}}^j,B]]] \nonumber \\&\qquad \qquad +\frac{ig}{16}\sinh (\eta )\epsilon ^{ij} \Big ( [E,[{{\mathcal {D}}}^j,E]] - [B,[{{\mathcal {D}}}^j,B]] \Big )\nonumber \\&\qquad \qquad -\frac{ig}{8}\cosh (\eta )[E,[{{\mathcal {D}}}^i,B]]\nonumber \\&E^z_{(4)} = \frac{1}{64}[{{\mathcal {D}}}^i,[{{\mathcal {D}}}^i,[{{\mathcal {D}}}^j,[{{\mathcal {D}}}^j,E]]]]\nonumber \\&\qquad \qquad + \frac{ig}{16} \epsilon ^{ij}[[{{\mathcal {D}}}^i,E],[{{\mathcal {D}}}^j,B]] \nonumber \\&B^z_{(4)} = \frac{1}{64}[{{\mathcal {D}}}^i,[{{\mathcal {D}}}^i,[{{\mathcal {D}}}^j,[{{\mathcal {D}}}^j,B]]]]\nonumber \\&\qquad \qquad + \frac{3ig}{64} \epsilon ^{ij}[[{{\mathcal {D}}}^i,B],[{{\mathcal {D}}}^j,B]] \nonumber \\&\qquad \qquad - \frac{ig}{64} \epsilon ^{ij}[[{{\mathcal {D}}}^i,E],[{{\mathcal {D}}}^j,E]] - \frac{g^2}{64} [E,[E,B]]\,.\nonumber \\ \end{aligned}$$

(B3)

We comment that these results have somewhat different form compared to Ref. [15] but are equivalent.

Appendix C: Initial conditions

1.1 1. Preliminaries

In this appendix we will derive the initial conditions for the differential equations that give the gauge potentials in the forward light-cone. These conditions were originally obtained by working with sources with zero width across the light-cone that are represented with delta functions, by matching singular terms [35, 36]. We work with sources with a small but finite width, and therefore the boundary conditions should be obtained by integrating the YM equation across the light-cone. We start from the YM equation in the adjoint representation which has the form

$$\begin{aligned}&\partial _\mu \partial ^\mu A^\nu _a-\partial _\mu \partial ^\nu A^\mu _a +g f_{abc}\left( (\partial _\mu A^\mu _b)A^\nu _c \right. \nonumber \\&\quad \left. - 2(\partial _\mu A^\nu _b)A^\mu _c + (\partial ^\nu A^\mu _b)A_{\mu c} \right) \nonumber \\&\quad +g^2 f_{abc}f_{cmn}A_{\mu b} A^\mu _m A^\nu _n - J^\nu _a = 0 \,. \end{aligned}$$

(C1)

We can find boundary conditions that relate the ansatz functions in different regions of spacetime by integrating the YM equation across the lines that separate the different regions.

1.2 2. First boundary condition

In the case of singular sources, the first boundary condition is obtained by matching singular terms in the YM equation at the point $x^+=x^-=0$ at the tip of the light-cone. To obtain the corresponding condition for sources of finite width, we consider the integral of the YM equation over a small diamond shaped area centered on the tip of the light-cone. Taking $\nu =i$ (one of the transverse spatial indices) we calculate the integral

$$\begin{aligned} \lim _{\mathrm{w}\rightarrow 0}\int _{-\mathrm{w}/2}^{\mathrm{w}/2} dx^-\, \int _{-\mathrm{w}/2}^{\mathrm{w}/2} dx^+\,(\text {YM equation}) = 0 \end{aligned}$$

(C2)

where the zero on the right of the equation is from the fact that the potentials in all regions of spacetime satisfy the YM equation. The contribution to the left side of (C2) from most terms in the YM equation is trivially zero, but there are some terms that do not automatically give zero. The conditions that force the sum of these terms to be zero are the boundary conditions we are looking for.

The densities $\rho _1(x^-,\vec {x}_\perp )$ and $\rho _2(x^+,\vec {x}_\perp )$ diverge as $1/\mathrm{w}$ when $\mathrm{w}\rightarrow 0$, but the pre-collision potentials in light-cone gauge remain finite, which can be seen from Eqs. (D8–D11). It is straightforward to show that there is only one term in the integrand that gives a non-zero contribution to the integral. This term is $\partial ^+\partial ^- A^i_a(x^+,x^-,\vec {x}_\perp )$ and gives

$$\begin{aligned} 0= & {} \int _{-\mathrm{w}/2}^{\mathrm{w}/2} dx^-\int _{-\mathrm{w}/2}^{\mathrm{w}/2} dx^+ \, \partial ^+\partial ^- A^i_a(x^+,x^-,\vec {x}_\perp ) \\= & {} A^i_a(\mathrm{w}/2,\mathrm{w}/2,\vec {x}_\perp ) - A^i_a(\mathrm{w}/2,-\mathrm{w}/2,\vec {x}_\perp ) \\&- A^i_a(-\mathrm{w}/2,\mathrm{w}/2,\vec {x}_\perp ) + A^i_a(-\mathrm{w}/2,-\mathrm{w}/2,\vec {x}_\perp )\,. \end{aligned}$$

Taking the limit $\mathrm{w}\rightarrow 0^+$ and using equation (6) we obtain

$$\begin{aligned} \alpha _{\perp \,a}^i(0,\vec {x}_\perp ) = \lim _{\mathrm{w}\rightarrow 0}\left( \beta _{1\,a}^i(x^-,\vec {x}_\perp ) + \beta _{2\,a}^i(x^+,\vec {x}_\perp )\right) \, \end{aligned}$$

(C3)

which is the first boundary condition in Eq. (17), written in the adjoint representation.

1.3 3. Second boundary condition

When singular sources are used, the second boundary condition is obtained by matching singular terms in the YM equation across the positive branch of one of the light-cone variable axes. To obtain the corresponding condition for sources of finite width, we consider the integral of the YM equation across a small strip centered on the positive $x^-$ axis. We set the free index $\nu$ in the YM Eq. (C1) to $\nu =-$ and calculate

$$\begin{aligned} \lim _{\mathrm{w}\rightarrow 0}\int _{-\mathrm{w}/2}^{\mathrm{w}/2} dx^+\,(\text {YM equation}) = 0\,. \end{aligned}$$

(C4)

The only non-zero contributions come from terms with a derivative with respect to $x^+$. Two of these terms that give zero are

$$\begin{aligned} t_1= & {} \int _{-\mathrm{w}/2}^{\mathrm{w}/2} dx^+\, A^+_a(x^+,x^-,\vec {x}_\perp ) \partial ^- A_b^-(x^+,x^-,\vec {x}_\perp ) \\\le & {} {\bar{A}}_a^+ \int _{-\mathrm{w}/2}^{\mathrm{w}/2} dx^+\, \partial ^- A_b^-(x^+,x^-,\vec {x}_\perp )\\ t_2= & {} \int _{-\mathrm{w}/2}^{\mathrm{w}/2} dx^+\, A_a^-(x^+,x^-,\vec {x}_\perp ) \partial ^- A_b^+(x^+,x^-,\vec {x}_\perp ) \\\le & {} {\bar{A}}_a^- \int _{-\mathrm{w}/2}^{\mathrm{w}/2} dx^+\, \partial ^- A_b^+(x^+,x^-,\vec {x}_\perp )\, \end{aligned}$$

where ${\bar{A}}_a^+$ and ${\bar{A}}_a^-$ indicate the maximum value of the corresponding potential on $x^+\in [-\mathrm{w}/2,\mathrm{w}/2]$. For the first term, the integral is finite but the prefactor goes to zero, using (6). For the second term, the the prefactor is finite but the integral goes to zero, again using (6). Collecting the non-zero terms Eq. (C4) becomes

$$\begin{aligned} 0= & {} \int _{-\mathrm{w}/2}^{\mathrm{w}/2} dx^+\,\bigg [2\,\partial ^+\partial ^- A_a^-(x^+,x^-,\vec {x}_\perp ) +\partial ^-\partial ^i A_a^i(x^+,x^-,\vec {x}_\perp ) \nonumber \\&+ g f_{abc} A^i_b(x^+,x^-,\vec {x}_\perp ) \partial ^- A^i_c(x^+,x^-,\vec {x}_\perp ) - J^-_a(x^+,\vec {x}_\perp )\bigg ]\,.\nonumber \\ \end{aligned}$$

(C5)

The first term is straightforward to integrate and gives

$$\begin{aligned} \lim _{\mathrm{w}\rightarrow 0}\text {term}_1= & {} 2\lim _{\mathrm{w}\rightarrow 0}\left( \partial ^+ A^-_a(\mathrm{w}/2,x^-,\vec {x}_\perp ) \right. \nonumber \\&\left. - \partial ^+ A^-_a(-\mathrm{w}/2,x^-,\vec {x}_\perp ) \right) = -2\alpha (0,\vec {x}_\perp )\,\nonumber \\ \end{aligned}$$

(C6)

where we used (6) in the last step. The results of the previous section tell us that

$$\begin{aligned} A^i_a(x^+,x^-,\vec {x}_\perp )= & {} \beta _{1\,a}^i(x^-,\vec {x}_\perp ) + \beta _{2\,a}^i(x^+,\vec {x}_\perp ) + {{\mathcal {O}}}(\mathrm{w}) ~~\nonumber \\&\text {for}~~ x^+\in [-\mathrm{w}/2,\mathrm{w}/2]\, \end{aligned}$$

(C7)

from which we find that the second term of (C5) gives

$$\begin{aligned} \lim _{\mathrm{w}\rightarrow 0} \text {term}_2 = \lim _{\mathrm{w}\rightarrow 0} \int _{-\mathrm{w}/2}^{\mathrm{w}/2} dx^+ \partial ^- \partial ^i \beta ^i_{2\,a}(x^+,\vec {x}_\perp )\,. \end{aligned}$$

(C8)

The limit of the third term in (C5) can be written using Eq. (C7) as

$$\begin{aligned}&\lim _{\mathrm{w}\rightarrow 0}\text {term}_3\nonumber \\&\quad = gf_{abc}\lim _{\mathrm{w}\rightarrow 0}\bigg (\beta ^i_{1\,b}(x^-,\vec {x}_\perp ) \int _{-\mathrm{w}/2}^{\mathrm{w}/2} dx^+\, \partial ^- A^i_c(x^+,x^-,\vec {x}_\perp ) \nonumber \\&\qquad + \int _{-\mathrm{w}/2}^{\mathrm{w}/2} dx^+\,\beta ^i_{2\,b}(x^+,\vec {x}_\perp ) \partial ^- A^i_c(x^+,x^-,\vec {x}_\perp ) \bigg )\,. \end{aligned}$$

(C9)

The first term on the right side of (C9) is

$$\begin{aligned} \lim _{\mathrm{w}\rightarrow 0}\text {term}_{3a}= & {} gf_{abc}\lim _{\mathrm{w}\rightarrow 0}\beta ^i_{1\,b}(x^-,\vec {x}_\perp )\left( A^i_c(\mathrm{w}/2,x^-,\vec {x}_\perp )\right. \nonumber \\&\left. -\beta ^i_{1\,c}(x^-,\vec {x}_\perp )\right) \,\nonumber \\= & {} g f_{abc} \,\lim _{\mathrm{w}\rightarrow 0} \beta ^i_{1\,b}(x^-,\vec {x}_\perp )\beta ^i_{2\,c}(x^+,\vec {x}_\perp )\, \end{aligned}$$

(C10)

where we have used Eq. (6) in the first step and Eq. (C7) to obtain the second line. The second term on the right side of (C9), using (C7), is

$$\begin{aligned}&\lim _{\mathrm{w}\rightarrow 0}\text {term}_{3b} \nonumber \\&\quad = gf_{abc}\lim _{\mathrm{w}\rightarrow 0} \int _{-\mathrm{w}/2}^{\mathrm{w}/2} dx^+\, \beta ^i_{2\,b}(x^+,\vec {x}_\perp ) \partial ^- \beta ^i_{2\,c}(x^+,\vec {x}_\perp )\,.\nonumber \\ \end{aligned}$$

(C11)

Collecting these results Eq. (C4) with $\nu =-$ now has the form

$$\begin{aligned}&\lim _{\mathrm{w}\rightarrow 0}\left[ \text {term}_1 + \text {term}_2 + \text {term}_{3a} + \text {term}_{3b} \right. \nonumber \\&\quad \left. - \int _{-\mathrm{w}/2}^{\mathrm{w}/2} dx^+ J^-(x^+,\vec {x}_\perp )\right] = 0\,. \end{aligned}$$

(C12)

It is straightforward to show that the piece $\lim _{\mathrm{w}\rightarrow 0}$[$\text {term}_2 + \text {term}_{3b} - \int _{-\mathrm{w}/2}^{\mathrm{w}/2} dx^+ J^-(x^+,\vec {x}_\perp )$] is just the integral of the YM equation for the pre-collision potential $\beta _2$ in the absence of the source corresponding to ion 1, and can therefore be set to zero. Using Eqs. (C6, C10) the surviving terms give

$$\begin{aligned} \alpha _a(0,\vec {x}_\perp ) = \frac{g}{2} f_{abc} \,\lim _{\mathrm{w}\rightarrow 0^+} \beta ^i_{1\,b}(x^-,\vec {x}_\perp )\beta ^i_{2\,c}(x^+,\vec {x}_\perp )\,\nonumber \\ \end{aligned}$$

(C13)

which is the second boundary condition in Eq. (17), written in the adjoint representation.

Appendix D: 2-potential correlation function

In this appendix we give the derivation of the 2-point correlation function defined in Eq. (21), the result for which is given in Eqs. (22, 23, 24). The result has appeared previously in the literature, and we present it here for completeness, and to explain our notation.

The pre-collision potentials can be expressed in terms of the ion sources by solving the YM equation in the pre-collision region. This is done most easily by making a gauge transformation. The ansatz (6) together with the boundary condition (17) expresses the pre-collision potentials in terms of the transverse components $\beta ^i_1$ and $\beta ^i_2$ which are conventionally called light-cone gauge potentials. We can transform these potentials without violating our chosen gauge condition (5) by exploiting residual gauge freedom. Furthermore, since the two pre-collision regions to the left and right of the forward light-cone are not causally connected, we can work in different gauges in each of these regions.

First we discuss ion 1. The pre-collision potential can be transformed so that the light-cone gauge form

$$\begin{aligned}&\beta _1^-(x^-,\vec {x}_\perp ) = \beta _1^+(x^-,\vec {x}_\perp )=0~\text {and}~ \beta _1^i(x^-,\vec {x}_\perp )\ne 0\nonumber \\ \end{aligned}$$

(D1)

becomes

$$\begin{aligned} \beta _{1\,\mathrm{cov}}^-(x^-,\vec {x}_\perp )= & {} \beta _{1\,\mathrm{cov}}^i(x^-,\vec {x}_\perp )=0\text {~and~}\nonumber \\ \beta _{1\,\mathrm{cov}}^+(x^-,\vec {x}_\perp )\equiv & {} \Lambda _1(x^-,\vec {x}_\perp )\,. \end{aligned}$$

(D2)

The new potential satisfies $\partial _\mu \beta _{1\,\mathrm{cov}}^\mu = \partial ^-\beta ^+_{1\,\mathrm{cov}}=0$ and is conventionally called the covariant gauge potential. In covariant gauge the YM equation has the simple form

$$\begin{aligned}&\nabla _\perp ^2 \Lambda _1(x^-,\vec {x}_\perp ) = -\rho _1(x^-,\vec {x}_\perp ) \end{aligned}$$

(D3)

which can be easily solved to obtain

$$\begin{aligned} \Lambda _1(x^-,\vec {x}_\perp ) = \int d^2z_\perp \,G(\vec {x}_\perp - \vec {z}_\perp )\,\rho _1(x^-,\vec {z}_\perp ) \end{aligned}$$

(D4)

with

$$\begin{aligned} G(\vec {x}_\perp ) = \frac{1}{2\pi } K_0(m |\vec {x}_\perp |)\,. \end{aligned}$$

(D5)

The function $K_0$ is a modified Bessel function of the second kind, and m is an infra-red regulator whose definition is discussed in Sect. 4.1. In exactly the same way we obtain the covariant gauge solution for the second ion

$$\begin{aligned} \Lambda _2(x^+,\vec {x}_\perp ) = \int d^2z_\perp \,G(\vec {x}_\perp - \vec {z}_\perp )\,\rho _2(x^+,\vec {z}_\perp )\,. \end{aligned}$$

(D6)

Next we must find the residual gauge transformation that allows us to obtain the light-cone gauge potentials from our covariant gauge solutions. For ion 1 we must solve $\beta _1^+(x^-,\vec {x}_\perp )=0$ with

$$\begin{aligned} \beta _1^+(x^-,\vec {x}_\perp )= & {} \frac{i}{g} U_1^\dagger (x^-,\vec {x}_\perp ) \partial ^+ U_1(x^-,\vec {x}_\perp ) \nonumber \\&+ U_1^\dagger (x^-,\vec {x}_\perp ) \beta _{1\,\mathrm{cov}}^+(x^-,\vec {x}_\perp ) U_1(x^-,\vec {x}_\perp ) \,.\nonumber \\ \end{aligned}$$

(D7)

The solution is

$$\begin{aligned} U_1(x^-,\vec {x}_\perp ) = {{\mathcal {P}}}\mathrm{exp}\big [i g \int _{-\infty }^{x^-} dz^- \Lambda _1(z^-,\vec {x}_\perp )\big ]\, \end{aligned}$$

(D8)

where the lower limit on the integral is chosen to give retarded boundary conditions. The transverse components in light-cone gauge therefore satisfy

$$\begin{aligned} \beta _1^i(x^-,\vec {x}_\perp ) = \frac{i}{g} U_1^\dagger (x^-,\vec {x}_\perp ) \partial ^i U_1(x^-,\vec {x}_\perp ) \,. \end{aligned}$$

(D9)

For ion 2 we proceed in the same way. The covariant gauge potential is defined as

$$\begin{aligned} \beta _{2\,\mathrm{cov}}^-(x^+,\vec {x}_\perp )= & {} \beta _{2\,\mathrm{cov}}^i(x^+,\vec {x}_\perp )=0\text {~and~}\\ \beta _{2\,\mathrm{cov}}^+(x^+,\vec {x}_\perp )\equiv & {} \Lambda _2(x^+,\vec {x}_\perp )\,, \end{aligned}$$

the corresponding residual gauge transformation is

$$\begin{aligned} U_2(x^+,\vec {x}_\perp ) = {{\mathcal {P}}}\mathrm{exp}\big [i g \int _{-\infty }^{x^+} dz^+ \Lambda _2(z^+,\vec {x}_\perp )\big ]\,, \end{aligned}$$

(D10)

and the light-cone gauge transverse potential is obtained from the covariant potential as

$$\begin{aligned} \beta _2^i(x^+,\vec {x}_\perp ) = \frac{i}{g} U_2^\dagger (x^+,\vec {x}_\perp ) \partial ^i U_2(x^+,\vec {x}_\perp )\,. \end{aligned}$$

(D11)

A more convenient expression for the light-cone gauge potential can be constructed from these results. For ion 1 we use Eqs. (D1, D2) to obtain

$$\begin{aligned}&F_1^{+i}(x^-,\vec {x}_\perp )=\partial ^+\beta _1^i(x^-,\vec {z}_\perp ) \nonumber \\&F^{+i}_{1\mathrm cov}(x^-,\vec {x}_\perp ) = - \partial _x^i \Lambda _1(x^-,\vec {x}_\perp ) \end{aligned}$$

(D12)

which gives

$$\begin{aligned}&\beta _1^i(x^-,\vec {x}_\perp ) \nonumber \\&\quad = \int _{-\infty }^{x^-} dz^- \; F_1^{+i}(z^-,\vec {x}_\perp ) \nonumber \\&\quad = \int _{-\infty }^{x^-} dz^- U_1^\dagger (z^-,\vec {x}_\perp ) F_{1\mathrm cov}^{+i}(z^-,\vec {x}_\perp ) U_1(z^-,\vec {x}_\perp ) \nonumber \\&\quad = -\int _{-\infty }^{x^-} dz^- \; U_1^\dagger (z^-,\vec {x}_\perp ) \partial ^i_x\Lambda _1(z^-,\vec {x}_\perp ) U_1(z^-,\vec {x}_\perp )\nonumber \\&\quad = -\int _{-\infty }^{x^-} dz^- \; \partial ^i_x\Lambda _{1\,a}(z^-,\vec {x}_\perp ) (W_1)_{ab}(z^-,\vec {x}_\perp ) t_b\,, \nonumber \\ \end{aligned}$$

(D13)

where we use W to denote the Wilson line in the adjoint representation (see Appendix A). An analogous result for the light-cone gauge potential from the second ion can be obtained in the same way and we do not write it explicitly.

Now we calculate the correlator in the first line of Eq. (21), and we suppress the index 1 that indicates that potentials and sources are those of the first ion. The calculation of correlators for the second ion is exactly analogous. Using Eq. (D13) we have that the correlator we want to calculate can be written

$$\begin{aligned}&\langle \beta ^i(x^-,\vec {x}_\perp ) \beta ^j(y^-,\vec {y}_\perp )\rangle \nonumber \\&\quad = \int _{-\infty }^{x^-} dz^- \int _{-\infty }^{y^-} dw^- \; t_ct_d \langle W_{ec}(z^-,\vec {x}_\perp ) \nonumber \\&\qquad W_{fd}(w^-,\vec {y}_\perp ) \rangle \partial _x^i\partial ^j_y \langle \Lambda _e(z^-,\vec {x}_\perp ) \Lambda _f(w^-,\vec {y}_\perp ) \rangle \,. \end{aligned}$$

(D14)

We define the function $\gamma$ using the equation

$$\begin{aligned} \delta _{ab}\,g^2\, \delta (x^- - y^-)\gamma (x^-,\vec {x}_\perp ,\vec {y}_\perp ) \equiv \langle \Lambda _a(x^-,\vec {x}_\perp )\Lambda _b(y^-,\vec {y}_\perp )\rangle \nonumber \\ \end{aligned}$$

(D15)

and $\gamma$ is obtained from equations (D4, 19) as

$$\begin{aligned}&\gamma (x^-,\vec {x}_\perp ,\vec {y}_\perp ) = \int d^2 \vec {z}_\perp \, \lambda (x^-,\vec {z}_\perp )\, G(\vec {x}_\perp -\vec {z}_\perp )\,G(\vec {y}_\perp - \vec {z}_\perp )\,. \nonumber \\ \end{aligned}$$

(D16)

The correlator of Wilson lines has been calculated in Ref. [38] where it is shown that

$$\begin{aligned}&\delta _{cd}\,\langle W_{a c}(x^-,\vec {x}_\perp ) W_{b d}(y^-,\vec {y}_\perp ) \rangle \nonumber \\&\quad = \delta _{ab}\mathrm{exp}\left[ \frac{g^4 N_c}{2}\int ^{x^-}_{-\infty } dz^-\big (2\gamma (z^-,\vec {x}_\perp ,\vec {y}_\perp )\right. \nonumber \\&\quad \left. - \gamma (z^-,\vec {x}_\perp ,\vec {x}_\perp ) - \gamma (z^-,\vec {y}_\perp ,\vec {y}_\perp )\big )\right] \,. \end{aligned}$$

(D17)

Using Eqs. (D15, D17) we find that Eq. (D14) can be written in the adjoint representation as

$$\begin{aligned}&\langle \beta _a^i(x^-,\vec {x}_\perp ) \beta _b^j(y^-,\vec {y}_\perp )\rangle \nonumber \\&\quad = \delta _{ab} \,g^2 \int ^{x^-}_{-\infty }dz^-\int ^{y^-}_{-\infty }dw^- \,\delta (z^--w^-) \;\partial _x^i \partial _y^j \gamma (z^-,\vec {x}_\perp ,\vec {y}_\perp ) \nonumber \\&\qquad \mathrm{exp}\bigg [ \frac{g^4 N_c}{2}\int ^{z^-}_{-\infty } dv^- \big (2\gamma (v^-,\vec {x}_\perp ,\vec {y}_\perp ) - \gamma (v^-,\vec {x}_\perp ,\vec {x}_\perp )\nonumber \\&\qquad - \gamma (v^-,\vec {y}_\perp ,\vec {y}_\perp )\big ) \bigg ] \,. \end{aligned}$$

(D18)

Next we take the limit that the width of the source current $\rho (x^-,\vec {x}_\perp )$ across the light-cone goes to zero. Using Eq. (20) we rewrite (D16) as

$$\begin{aligned}&\gamma (x^-,\vec {x}_\perp ,\vec {y}_\perp ) = h(x^-) \,{\tilde{\gamma }}(\vec {x}_\perp ,\vec {y}_\perp )\, \end{aligned}$$

(D19)

where

$$\begin{aligned}&{\tilde{\gamma }}(\vec {x}_\perp ,\vec {y}_\perp ) = \int d^2 z_\perp \, \mu (\vec {z}_\perp )\, G(\vec {x}_\perp -\vec {z}_\perp )\,G(\vec {y}_\perp - \vec {z}_\perp ) \,. \nonumber \\ \end{aligned}$$

(D20)

We define the functions $\Gamma (\vec {x}_\perp ,\vec {y}_\perp )$ and ${\tilde{\Gamma }} (\vec {x}_\perp ,\vec {y}_\perp )$ as

$$\begin{aligned}&\Gamma (z^-,\vec {x}_\perp ,\vec {y}_\perp ) = 2\gamma (z^-,\vec {x}_\perp ,\vec {y}_\perp ) - \gamma (z^-,\vec {x}_\perp ,\vec {x}_\perp )\nonumber \\&\quad - \gamma (z^-,\vec {y}_\perp ,\vec {y}_\perp ) \nonumber \\&{\tilde{\Gamma }} (\vec {x}_\perp ,\vec {y}_\perp ) = 2{\tilde{\gamma }}(\vec {x}_\perp ,\vec {y}_\perp ) - {\tilde{\gamma }}(\vec {x}_\perp ,\vec {x}_\perp ) - {\tilde{\gamma }}(\vec {y}_\perp ,\vec {y}_\perp ) \,. \nonumber \\ \end{aligned}$$

(D21)

Using these definitions we rewrite the exponential in (D18) as

$$\begin{aligned} \mathrm{exp}\left[ ~ \cdots ~\right] = \mathrm{exp}\left[ \frac{g^4 N_c}{2}\;{\tilde{\Gamma }}(\vec {x}_\perp ,\vec {y}_\perp ) \int ^{z^-}_{-\infty } dv^- h(v^-) \right] \,. \nonumber \\ \end{aligned}$$

(D22)

We need to substitute (D22) into (D18) and take the limit that the width of the source distributions goes to zero. The function $h(z^-)$ behaves like a delta function in this limit, and it appears therefore that the calculation should be simple. However, we must proceed carefully to be sure that the delta functions in the integrand have support. We define

$$\begin{aligned} f(x^-)\equiv & {} \int ^{x^-}_{-\infty } dz^- h(z^-) ~~\text {and}~~ g(\vec {x}_\perp ,\vec {y}_\perp ) \nonumber \\\equiv & {} \frac{g^4 N_c}{2}\;{\tilde{\Gamma }}(\vec {x}_\perp ,\vec {y}_\perp ) \end{aligned}$$

(D23)

so that Eq. (D18) becomes

$$\begin{aligned}&\langle \beta _a^i(x^-,\vec {x}_\perp ) \beta _b^j(y^-,\vec {y}_\perp )\rangle \nonumber \\&\quad = \delta _{ab}\,g^2\, I(x^-,y^-,\vec {x}_\perp ,\vec {y}_\perp ) \, \partial _x^i \partial _y^j {\tilde{\gamma }}(\vec {x}_\perp ,\vec {y}_\perp ) \end{aligned}$$

(D24)

where

$$\begin{aligned}&I(x^-,y^-,\vec {x}_\perp ,\vec {y}_\perp ) \nonumber \\&\quad = \int ^{x^-}_{-\infty }dz^-\int ^{y^-}_{-\infty }dw^-\,\delta (z^--w^-) h(z^-) e^{g(\vec {x}_\perp ,\vec {y}_\perp ) f(z^-)} \nonumber \\&\quad = \frac{1}{g(\vec {x}_\perp ,\vec {y}_\perp )} \int ^{x^-}_{-\infty }dz^-\int ^{y^-}_{-\infty }dw^-\,\delta (z^--w^-)\frac{\partial }{\partial z^-} e^{g(\vec {x}_\perp ,\vec {y}_\perp ) f(z^-)}\nonumber \\&\quad = \frac{1}{g(\vec {x}_\perp ,\vec {y}_\perp )} \int ^{\mathrm{min}(x^-,y^-)}_{-\infty }dz^- \frac{\partial }{\partial z^-} e^{g(\vec {x}_\perp ,\vec {y}_\perp ) f(z^-)} \nonumber \\&\quad = \frac{1}{g(\vec {x}_\perp ,\vec {y}_\perp )}\big [ e^{g(\vec {x}_\perp ,\vec {y}_\perp ) f(\mathrm{min}(x^-,y^-))}-1\big ] \,.\nonumber \\ \end{aligned}$$

(D25)

Taking the limit that the width of the function $h(z^-)$ goes to zero so that $f\big (\mathrm{min}(x^-,y^-)\big )\rightarrow 1$ [see Eqs. (20, D23)] gives

$$\begin{aligned} \lim _{\mathrm{w}\rightarrow 0}I(x^-,y^-,\vec {x}_\perp ,\vec {y}_\perp ) = \frac{1}{g(\vec {x}_\perp ,\vec {y}_\perp )}\big [ e^{g(\vec {x}_\perp ,\vec {y}_\perp ) }-1\big ] \,. \nonumber \\ \end{aligned}$$

(D26)

Combining Eqs. (D23, D24,  D26) we write

$$\begin{aligned} \lim _{\mathrm{w} \rightarrow 0} \langle \beta _{a}^i(x^-,\vec {x}_\perp ) \beta _{b}^j(y^-,\vec {y}_\perp )\rangle \equiv \delta _{ab} B^{ij}(\vec {x}_\perp ,\vec {y}_\perp ) \, \end{aligned}$$

(D27)

with

$$\begin{aligned}&B^{ij}(\vec {x}_\perp ,\vec {y}_\perp ) = \frac{2 }{g^2 N_c {\tilde{\Gamma }}(\vec {x}_\perp ,\vec {y}_\perp )} \; \left( \mathrm{exp}[\frac{g^4 N_c}{2}\;{\tilde{\Gamma }}(\vec {x}_\perp ,\vec {y}_\perp ) ]-1\right) \;\nonumber \\&\qquad \qquad \qquad \quad \partial _x^i \partial _y^j {\tilde{\gamma }}(\vec {x}_\perp ,\vec {y}_\perp ) \,. \end{aligned}$$

(D28)

The calculation of correlators for the second ion is exactly analogous and Eqs. (D27, D28) can be used for either the first or the second ion by using the charge density $\mu _1(\vec {x}_\perp )$ or $\mu _2(\vec {x}_\perp )$ in Eqs. (D20, D21).