QED and QCD calculations are generally done by truncating an infinite series of terms representing more and more byzantine paths by which a carrier boson can get from point A to point B.

The formula for determining uncertainty in the final result of a QED calculation (i.e. the Standard Model quantum theory of the electromagnetic force a.k.a. quantum electrodynamics) and the uncertainty in the final result of a QCD calculation (i.e. the Standard Model quantum the strong force that holds protons and neutrons together, and indirectly holds atomic nuclei together a.k.a. quantum chromodynamics) is, at an order of magnitude level, very similar.

The main differences are (1) the strength of the coupling constants of the respective theories, (2) the fact that in QED photons don't interact with photons while in QCD gluons interact with gluons, and (3) the fact that QED has only positive and negative electromagnetic charges, while QCD has red, antired, blue, antiblue, green and antigreen charges (which confusingly due to group theory produces eight rather than nine color-anticolor pair combinations for gluons).

Relative precision in a QED or QCD calculation equals one part per a^(2*L) which a is a constant that is about 12 for QED and about 2 for QCD, and L is the number of loops in the calculation. The factor should be roughly 5 for the weak force.

These factors primarily arise from the relative strengths of the respective coupling constants of the two theories, with a four loop calculation including terms with powers of eight of the coupling constant, although it isn't quite that simple, but the other factors primarily influence the number of calculations that have to be made per loop.

It other words, it takes more loops in QCD to get the same level of precision as one would in QED, and each new loop in QCD is harder to calculate than the corresponding one in QED.

In the case of QED:

* a one loop calculation is precise to about 1 part per 144.

* a two loop calculation is precise to about 1 part per 20,736.

* a three loop calculation is precise to about 1 part per 2,985,984 (a.k.a. about 3 parts per 10 million).

* a four loop calculation is precise to about 1 part per 429,981,696 (a.k.a. about 2 parts per billion).

* a five loop calculation is precise to about 1 part per 61,917, 364,224 (a.k.a. about 2 parts per 100 billion).

In the case of QCD:

* a one loop calculation is precise to about 1 part per 4.

* a two loop calculation is precise to about 1 part per 16.

* a three loop calculation is precise to about 1 part per 64.

* a four loop calculation is precise to about 1 part per 256 (a.k.a. about 4 parts per thousand).

* a five loop calculation is precise to about 1 part per 1024.

A seven loop calculations in QCD is less precise than a two loop calculation in QED. A one loop calculation in QED is almost as precise as a four loop calculation in QCD.

Higher loop calculations that can be ignored in QED can't be ignored in QCD, which is good because each additional loop is profoundly more cumbersome to calculate than the loop before it.

A deeper analysis can be found at Physics Stack Exchange. It also points out that the infinite series that are being approximated are not truly convergent, and that the errors start getting bigger rather than smaller after about five loops.

But, this currently doesn't matter much. Nobody can do measurement approaching the one part per 61 billion precision of a five loop QED calculation, so we don't need that calculation to be more precise. And, nobody can actually do a QCD calculation of more than five loops, so the loss of precision in going beyond that point is merely hypothetical.

It does illustrate, however, the limits of the current perturbative QCD techniques made using this method - which is about one part per thousand with the theoretically maximal accuracy of using this approach.

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