Let G be a simple Lie group and let G(ℂ((t,t ^-1))) be its loop group.

The Lie algebra g((t,t ^-1)) has a well known central extension (see e.g. Wikipedia) given by the cocycle
c(f,g) = Res₀ < f dg >. Here, < > : g⊗g→ℂ denotes some invariant bilinear form on g, and f dg is the (g⊗g)-valued differential given by multiplying f and dg.

Question: It there a similarly concrete cocycle for the central extension of G(ℂ((t,t ^-1))) by ℂ*?

To give you an idea of what I'm looking for, let me show you a cocycle for central extension by S¹ of the smooth loop group $LG = \mathit{Map} _ {C^\infty} (S^1,G)$ of a compact Lie group $G$.

Pick a bounding disc D_γ : D² → G for each element γ ∈ LG. The cocycle is then given by

$$c(\gamma,\delta) = exp\big(i\cdot\big(\quad\int \langle D_\gamma^*\theta_L,D_\delta^*\theta_R\rangle +\int H^*\eta\quad \big)\big)$$

where $\theta_L,\theta_R\in\Omega(G,\mathfrak{g})$ are the Maurer-Cartan 1-forms, $\eta\in\Omega^3(G)$ is the Cartan 3-form,
and $H:D^3\to G$ in a homotopy between $D_\gamma D_\delta$ and $D _ {\gamma\delta}$.

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Reference:
The cocycle for the smooth loop group can be found on page 19 of the paper
From Loop groups to 2-groups, by Baez, Crans, Schreiber, and Stevenson.