0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.