gr.group theory - Explicit cocycle for the central extension of the algebraic loop group G(C((t))).

Let G be a simple Lie group and let G(ℂ((t))) 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))) 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

$$c(gamma,delta) = expbig(icdotbig(quadint langle D_gamma^*theta_L,D_delta^*theta_Rrangle
+int H^*etaquad big)big)$$

where $theta_L,theta_RinOmega(G,mathfrak{g})$ are the Maurer-Cartan 1-forms, $etainOmega^3(G)$ is the Cartan 3-form,
and $H:D^3to G$ in a homotopy between $D_gamma D_delta$ and $D _ {gammadelta}$.

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References:
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,
and also on page 8 of Mickelsson's paper From Gauge anomalies to Gerbes and Gerbal actions.