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I'm studying the chapter 16 of Peskin and Schroeder's An Introduction to Quantum field theory and I don't quiet understand the contraction rule used in this chapter. For example, for eq(16.63) $$\frac{1}{2}\int\frac{d^4p}{(2\pi)^4}\frac{-ig_{\rho\sigma}}{p^2}\delta^{cd}(-ig^2)\times[f^{abe}f^{cde}(g^{\mu\rho}g^{\upsilon\sigma}-g^{\mu\sigma}g^{\upsilon\rho})+f^{ace}f^{bde}(g^{\mu\upsilon}g^{\rho\sigma}-g^{\mu\sigma}g^{\upsilon\rho})+f^{ade}f^{bce}(g^{\mu\rho}g^{\upsilon\sigma}-g^{\mu\rho}g^{\upsilon\sigma})]$$which is the four-gauge-boson vertex gauge boson self-energy feynman diagram to eq(16.64)$$-g^2C_2(G)\delta^{ab}\int\frac{d^4p}{(2\pi)^4}\frac{1}{p^2}\cdot g^{\mu\upsilon}(d-1),$$ I think the trick used here for the second and third combination of structure constants are $$\delta^{cd}f^{ace}f^{bde}=f^{ade}f^{bde}=C_2(G)\delta^{ab}.$$

Here is my problem, according to my knowlege about contraction, the right form should be $$\delta^{d}_cf^{ace}f^{bde}=f^{ade}f^{bde}=C_2(G)\delta^{ab}$$ Why the book used $\delta$ with two superscript?

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    $\begingroup$ See knzhou’s comments on this related question: “The [Lie algebra] indices can be thought of as being raised or lowered by the [Lie algebra] metric, aka the Killing form. Conventionally the generators are chosen so that the Killing form is Euclidean, so raising or lowering indices does nothing.” If you ever used index notation for classical mechanics in Euclidean space, you probably wrote all indices in the down position, including when two are being contracted. $\endgroup$
    – Ghoster
    Commented Sep 12, 2022 at 4:03
  • $\begingroup$ @Ghoster Thanks, it really solved my problem $\endgroup$
    – David Shaw
    Commented Sep 12, 2022 at 4:17

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