A Novel Approach For Event-By-Event Early Gluon Fields

TL;DR

This paper introduces an analytical method for generating early-time gluon fields in high-energy nuclear collisions, offering a potentially more efficient alternative to numerical Yang-Mills solutions.

Contribution

It presents a novel analytical recursion relation approach for calculating classical gluon fields, differing from traditional numerical methods.

Findings

Preliminary results demonstrate the method's viability.

Analytical recursion provides reliable early-time gluon field estimates.

Potential for improved computational efficiency in event generation.

Abstract

We report on efforts to construct an event generator that calculates the classical gluon field generated at early times in high energy nuclear collisions. Existing approaches utilize numerical solutions of the Yang-Mills equations after the collision. In contrast we employ the analytically known recursion relation in the forward light cone. The few lowest orders are expected to lead to reliable results for times of up to the inverse saturation scale, tau_0 ~ 1/Q_s. In these proceedings we sketch some calculational details, and show some preliminary results.

Figures

Equations

$$A_{(n)}=$$

$$A_{\perp (n)}^i=$$

$$+ig\sum _{k+l+m=n-4}\left[A_{(k)},[D_{(l)}^i,A_{(m)}]\right]),$$

$$A_{\perp (0)}^i$$

$$A_{(0)}$$

$$G\left(\mathbf{r}\right)=\frac{1}{2{\pi }^2{\lambda }^2}\int d^{2}z{K}_0\left(m\mathbf{z}\right)e^{-\left(\mathbf{r}-\mathbf{z}\right)/{\lambda }^2}$$

$$A_k^i=-\frac{1}{ig}{U}_k\left(\mathbf{r}\right){\partial }^i{U}_k^{†}\left(\mathbf{r}\right)$$

$$U_k\left(\mathbf{r}\right)=\mathcal{P}\exp \left[ig\int d{z}^{\mp }{\alpha }_k\left(\mathbf{r},z^{\mp }\right)\right].$$

$$E_0=ig{\delta }^{ij}\left[A_1^i,A_2^j\right],B_0=ig{ϵ}^{ij}\left[A_1^i,A_2^j\right].$$

$$T^{0i}=\frac{\tau }{2}\left({\alpha }_i\cosh \eta +{\beta }_i\sinh \eta \right)+\mathcal{O}({\tau }^3)$$

$${\alpha }_i$$

$${\beta }_i$$

Full text

The initial phase of collisions of heavy nuclei at very high energies is believed to be be described by color glass condensate (CGC) as an effective theory [1]. The initial wave functions and the collision system just after the collision are dominated by classical gluon fields [2, 3, 4, 5, 6]. The density of gluons and their average transverse momentum in the initial wave functions is given by a new energy scale, the saturation scale $Q_{s}$. At times $\tau_{0}\sim 1/Q_{s}$ after the collision fast growing quantum fluctuations lead to a demise of the purely classical picture and drive the system toward kinetic equilibrium [7, 8, 9]. Eventually a quark gluon plasma close to local kinetic equilibrium emerges.

The boundary value problem for classical fields after the collision was first formulated in [6]. Several groups have since solved the problem numerically, see e.g. [10, 11, 12, 13]. More recently the IP-Glasma model [14, 15] combined numerical solutions of the Yang Mills problem with an event-by-event sampling of initial charge densities according to the IP-Sat model [16, 17] and a subsequent matching to viscous fluid dynamics [18]. In these proceedings we report on our efforts to build an even generator based on an alternative approach [19]. We use the recursive solutions to the Yang-Mills boundary value problem in the forward light cone discussed in [20, 21, 22]. For a gauge field $(x^{+}A,-x^{-}A,A_{\perp}^{i})$ in light cone coordinates in Fock-Schwinger gauge we use a power series $A=\sum_{n}A_{(n)}\tau^{n}$, and similar for the transverse field $A_{\perp}^{i}$, $=1,2$ . The recursion relations solving the Yang-Mills equations read

[TABLE]

with initial conditions

[TABLE]

where $A_{1}^{i}$ and $A_{2}^{i}$ ($i=1,2$) are the fields in the two nuclei before the collision. Using the few lowest orders of the recursion should give reasonable results up to the time $\tau_{0}$. The obvious advantage of our approach for an event-by-event calculation is that no differential equations need to be solved for the time evolution. The numerical effort simply goes into calculating coefficients of a power series and is reasonably fast. On the other hand, the convergence of the power series is limited to early times. In the preliminary results reported below we restrict ourselves to the second order. Systematic tests are under way to see how far the recursion can be pushed while mainting numerical stability and low run times.

In the following we discuss some of the steps implemented so far. Event averages are known analytically for a number of quantities that can be used to check the accuracy of the event-by-event Monte Carlo code. Of course the matching of analytic event-averaged results is only a necessary, not a sufficient, criterion. Eventually the code should be benchmarked against existing numerical solvers event-by-event. The three main steps in the code are

Sampling of the transverse color charge densities $\rho_{k}^{a}(x,y)$ for nuclei $k=1,2$ and $a=1,\ldots,N_{c}^{2}-1$. 2. 2.

Computation of the nuclear fields $A_{1}^{i}$, $A_{2}^{i}$ ($i=1,2$) from the nuclear charges, respectively. 3. 3.

Calculation of the field after the collision using the recursion relation.

Sampling. The sampling assumes the Gaussian charge distributions of the McLerran-Venugopalan model [2, 3], $\langle\rho^{a}_{k}\rangle=0$, $\langle\rho_{k}^{a}\rho_{k}^{b}\rangle\sim\delta^{ab}\mu_{k}(x,y)$ (see [22] for details). For the results shown here it is assumed that the functional dependence of the variance $\mu_{k}(x,y)$ of the charge distribution is given by the nuclear thickness function of the respective nucleus. More physical choices, including correlations from nucleons, or the IP-Sat model could be implemented. Fig. 1 shows a typical example for one sampled color component of a nuclear charge distribution.

Nuclear Fields. To arrive at gauge fields in Fock-Schwinger gauge we use the standard approach shown in [4]. We solve the Yang-Mills equation for a single nucleus first in covariant gauge to obtain a covariant potential $\alpha_{k}(x,y)$. Subsequently we apply a gauge transformation to the desired gauge. $\alpha_{k}(x,y)$ for each nucleus is the solution to a Poisson equation. We solve the Poisson equation by applying a pre-tabulated coarse-grained Greens function

[TABLE]

where $m$ is the infrared scale for the Poisson problem. The coarse graining scale $\lambda$ acts as a UV cutoff that can be conveniently chosen to rid the results of lattice artefacts, $1/Q_{s}\gg\lambda\gg a$ where $a$ is the lattice constant. Fig. 2 shows one color component for a typical covariant potential. Fig. 3 shows one necessary test of the numerical implementation. In this case the correlation function $\langle\alpha^{a}(\mathbf{r}_{1})\alpha^{a}(\mathbf{r}_{2})\rangle$ is computed numerically over 7800 events and compared to the analytic result. Both correlation functions agree very well. The gauge transformation requires us to compute

[TABLE]

from the path-ordered exponential

[TABLE]

The integral over the longitudinal direction is realized by summing over a discrete set of uncorrelated covariant gauge fields $\alpha_{k}$ along $z^{\mp}$ with the same $\mu_{k}$. Fig. 4 shows a typical example of one color component of the field $A^{1}$ in Fock-Schwinger gauge.

Collision. The initial longitudinal chromo-electric and chromo-magnetic fields after the collision can be computed readily from commutators of fields of the two nuclei

[TABLE]

We will focus here on the energy momentum tensor of the gluon field after the collision. In Ref. [22] its lowest order terms are given explicitly as commutators and covariant derivatives of the initial fields $E_{0}$ and $B_{0}$. The energy density at $\tau=0$ is $\varepsilon_{0}=(E_{0}^{2}+B_{0}^{2})/2$ and the initial transverse energy flow is

[TABLE]

($i=1,2$) where $\eta$ is the space-time rapidity and the rapidity-even and rapidity-odd flow terms are

[TABLE]

Fig. 5 shows an example for a central Pb+Pb event at large rapdity $\eta\gg 1$ with the transverse energy flow $T^{0i}$ superimposed on the initial energy density $\varepsilon_{0}$. The $\alpha$-flow follows gradients of the energy density, mimicking hydro-like behavior. In contrast the $\beta$-flow is determined by the underlying dynamics of the non-abelian gauge fields. It is, for example, responsible for the transport of angular momentum toward mid-rapidity in non-central collisions [23].

Fig. 6 shows the time evolution of the energy density for a typical collision between $\tau=0$ and $\tau=0.1$ fm/$c$ using the recursion relation to second order. The obvious effect of the brief time evolution is an expansion and diffusion of ”hot spots” in the energy density through the build up of flow.

In summary, we have shown preliminary results from an event generator for early-time classical gluon fields that is based on the recursive solution of the Yang Mills equations in the forward light cone. Obvious improvements can be made by (1) going to more realistic models of the average charge densities $\mu_{k}$, and by (2) pushing to higher orders in the recursion relation. Work on both issues is underway. Current checks rely on comparisons of event averages that are known analytically. Further checks of event-by-event results are desirable.

RJF would like to thank the organizers of Hard Probes 2016 for a wonderful conference. This work was supported by the US National Science Foundation under award no. 1516590 and award no. 1550221.