In addition, QCD has - like all field theories - only an asymptotic perturbation series, which means that the series itself will also diverge if all terms are summed.
What does it mean? From what I know, if the sum over a series diverge, that means that the summation doesn't work, which means that the quantity you are trying to calculate, you can't get answer for that, for any quantity that comes back from your calculation must be of finite value.
But in QCD and QED things seem a lot more complicated, since some divergences are allowed:
This doesn't mean that QCD perturbation theory doesn't have ultraviolet divergences, it has those like any other unitary interacting field theory in 4d. These ultraviolet divergences though are not a sign of a problem with the theory, since the lattice definition works fine. This is in contrast to, say, QED, where the short lattice spacing limit requires the bare coupling to blow up, and it is likely that the theory blows up to infinite coupling at some small but finite distance. This is certainly what happens in the simplest interacting field theory, the quartically self-interacting scalar
- How many kinds of divergence there are in QCD and QED?
- And how do we know what kind of divergence is acceptable ( in the sense that we can still extract values out for prediction after some renormalization process)?
- If the sum diverge, then we won't be able to calculate the series's sum. Isn't that is defeating the purpose of the series? For any series, if the sum diverge after summing all the terms, then we know that the formula must be wrong or the series have no physical meaning. But why is it that for QCD series, the formula is still correct ( because it is used to extract coupling constants) and has physical meaning ( QCD series must correspond to something in reality)?
- The fact that QCD has non-convergent series means that it cannot be the fundamental theory of nature, right?