The problem is that the sigma calculation is only about rejecting the null model (ie, the model of "background noise"). That number is calculated without including info about any theory/evidence regarding the Higgs boson.
So, all of the confidence and confirmation regarding the Higg's boson better actually be coming from "how closely the data fits the model overall, a lack of competing hypotheses to explain said data, and to what degree other evidence exists to support the model".
As an example, I'd imagine detecting the Higgs boson at 1 eV energy levels is theoretically predicted to be even less likely than billions/trillions to one odds, therefore detector noise would be a more likely explanation for such results, despite the low p-value.
So, all of the confidence and confirmation regarding the Higg's boson better actually be coming from "how closely the data fits the model overall, a lack of competing hypotheses to explain said data, and to what degree other evidence exists to support the model".
As an example, I'd imagine detecting the Higgs boson at 1 eV energy levels is theoretically predicted to be even less likely than billions/trillions to one odds, therefore detector noise would be a more likely explanation for such results, despite the low p-value.