Multiplying terms with index notation

Question

I am trying to expand the flat-space action

$$ S_{BI} = -T_p \int{d^{p+1}} \sigma \ \mathrm{Tr}\left( e^{-\phi} \sqrt{ -\det(\eta_{ab} + 4\pi^2\alpha^2 \partial_a\Phi^i\partial_b\Phi^i + 2\pi \alpha F_{ab}) \det(Q^{i}_{j}) } \right).\tag{1} $$

After some manipulation, I want to use the power series expansion of the natural log to expand the term

$$ \text{ln}[\delta^{c}_{b} + \lambda^2\eta^{cd} \partial_d\Phi^i\partial_b\Phi^i ]\tag{2} $$ to the fourth power in $\eta^{cd} \partial_d\Phi^i\partial_b\Phi^i.$

My problem is, I don't understand how to compute powers of $\eta^{cd} \partial_d\Phi^i\partial_b\Phi^i $, i.e. $(\eta^{cd} \partial_d\Phi^i\partial_b\Phi^i)^2$. How do I treat the indices?

Answer 1

If you want to expand the Born-Infeld action$^1$ using $$\det(\mathbb{1}+M)~=~\exp({\rm tr}\ln(\mathbb{1}+M)),\tag{A} $$ then you need the logarithm of a whole matrix $$L~:=~\ln(\mathbb{1}+M)~=~-\sum_{n=1}^{\infty}\frac{1}{n} (-M)^n,\tag{B} $$ as opposed to the logarithm of a single matrix element (2). Then $$ L^a{}_b~=~M^a{}_b -\frac{1}{2}M^a{}_cM^c{}_b +{\cal O}(M^3),\tag{C}$$ and so forth. Here $a,b,c$ are world-volume indices.

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$^1$Additional complications arise from the (symmetrized) color-trace ${\rm Tr}$ in a non-abelian Dirac-Born-Infeld action (1). We will here assume for simplicity an abelian gauge group $U(1)$.