Well below 1%

When a theory is too hard to solve people try to consider lower dimensional cases. This also happened for Yang-Mills theory. The four dimensional case is notoriously difficult to manage due to the large coupling and the three dimensional case has been treated both theoretically and by lattice computations. In this latter case, the ground state energy of the theory is known very precisely (see here). So, a sound theoretical approach from first principles should be able to get that number at the same level of precision. We know that this is the situation for Standard Model with respect to some experimental results but a pure Yang-Mills theory has not been seen in nature and we have to content ourselves with computer data. The reason is that a Yang-Mills theory is realized in nature just in interaction with other kind of fields being these scalars, fermions or vector-like.

In these days, I have received the news that my paper on three dimensional Yang-Mills theory has been accepted for publication in the European Physical Journal C. Here is tha table for the ground state for SU(N) at different values of N compared to lattice data

N Lattice     Theoretical Error

2 4.7367(55) 4.744262871 0.16%

3 4.3683(73) 4.357883714 0.2%

4 4.242(9)     4.243397712 0.03%

∞ 4.116(6)    4.108652166 0.18%

These results are strikingly good and the agreement is well below 1%. This in turn implies that the underlying theoretical derivation is sound. Besides, the approach proves to be successful both also in four dimensions (see here). My hope is that this means the beginning of the era of high precision theoretical computations in strong interactions.

Andreas Athenodorou, & Michael Teper (2017). SU(N) gauge theories in 2+1 dimensions: glueball spectra and k-string tensions J. High Energ. Phys. (2017) 2017: 15 arXiv: 1609.03873v1

Marco Frasca (2016). Confinement in a three-dimensional Yang-Mills theory arXiv arXiv: 1611.08182v2

Marco Frasca (2015). Quantum Yang-Mills field theory Eur. Phys. J. Plus (2017) 132: 38 arXiv: 1509.05292v2