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In this work we study the Higgs boson decays into two photons and into onephoton and one $Z$ gauge boson within the context of the non-linear EffectiveField Theory called the Electroweak Chiral Lagrangian. We present a detailedcomputation of the corresponding amplitudes to one-loop level in the covariant$R_\xi$ gauges. We assume that the fermionic loop contributions are as in theStandard Model and focus here just in the computation of the bosonic loopcontributions. Our renormalization program and the anatomy of the variouscontributions participating in the $R_\xi$ gauges are fully explored. With thispresent computation we demonstrate the gauge invariance of the EChL result, notonly for the case of on-shell Higgs boson, but also for the most general andinteresting case of off-shell Higgs boson. We finally analyse and conclude onthe special relevance of the Goldstone boson loops, in good agreement with theexpected chiral loops behaviour in Chiral Lagrangians. We perform a systematiccomparison with the corresponding computation of the Standard Model in the$R_\xi$ gauges and with the previous EChL results in the unitary gauge. Thiswork represents the first computation within the EChL of these Higgsobservables to one-loop in the most general $R_\xi$ gauges and with a fullrenormalization program description, not yet fully explored in the previousliterature and which is different to the most frequently used in the linearEffective Field Theory (SMEFT).

Anatomy of Higgs decays into $\gamma \gamma$ and $\gamma Z$ within the EChL in the $R_\xi$ gauges