I've been reading a section in this thesis (pp 25-27) which reviews Duff's paper (pp 6-7 in particular) in which he calculates the tree-level vacuum expectation value (vev) of the graviton field (upon expanding around a flat background, i.e. $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$). I'm quite confused on a couple of steps though. Here is a sketch of the part of the calculation that I'm stuck on. He starts by considering the S-matrix for the interaction: $$ S_{J}=T\bigg\lbrace\text{exp}\bigg(i\int\mathrm{d}^{4}x\big[\mathcal{L}_{\text{int}}+\mathcal{L}_{J}\big]\bigg)\bigg\rbrace$$ where $\mathcal{L}_{\text{int}}=\kappa\mathcal{L}^{(1)}_{G}$ contains the cubic self-interaction terms of the graviton field, and we also include a classical source term $\mathcal{L}_{J}=\frac{\kappa}{2}h^{\mu\nu}J_{\mu\nu}$ with $J_{\mu\nu}=\sqrt{-g}T_{\mu\nu}$, and $T\lbrace\rbrace$ denoting the time ordering operator.

The author reviewing Duff's calculation starts by expanding the two exponential contributions, up to second order in the coupling $\kappa$, as follows: $$\text{exp}\bigg(i\int\mathrm{d}^{4}x\:\mathcal{L}_{J}\bigg) =1+\frac{i\kappa}{2}\int\mathrm{d}^{4}x\:h^{\mu\nu}(x)J_{\mu\nu}(x)-\frac{\kappa^{2}}{4}\int\mathrm{d}^{4}x\:\mathrm{d}^{4}y\:h^{\mu\nu}(x)h^{\alpha\beta}(y)J_{\mu\nu}(x)J_{\alpha\beta}(y)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad (1)$$ and $$\text{exp}\bigg(i\int\mathrm{d}^{4}x\:\mathcal{L}_{\text{int}}\bigg)=1+i\kappa\int\mathrm{d}^{4}x\:\Gamma_{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(x)h^{\mu_{1}\nu_{1}}(x)h^{\mu_{2}\nu_{2}}(x)h^{\mu_{3}\nu_{3}}(x)\qquad\qquad\qquad (2)$$ where $\Gamma_{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(x)$ is the 3-point vertex function.

Now, this is the part I don't understand. The first of these two expansions $(1)$ makes sense, however, I really can't understand the second one $(2)$. First of all, I assume that the author is stopping at first-order in the coupling $\kappa$ as higher order terms would correspond to loop corrections, and they're only considering tree-level in this calculation? Secondly, how does one arrive at the right-hand side of equation $(2)$? Shouldn't the 3-point vertex function be dependent on 3 space-time points, i.e. $\Gamma_{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}=\Gamma_{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(x_{1},x_{2},x_{3})$? How does one expand the interacting part of the graviton action $S_{\text{int}}=\int\mathrm{d}^{4}x\:\mathcal{L}_{\text{int}}$ to arrive at the expansion of the exponential on the right-hand side of $(2)$?

Apologies if this is trivial, but I'm having a real mental block on it. Any help would be much appreciated.


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