# What does QCD look like in higher dimensions?

It was pointed out as a comment on my question on atomic physics in higher dimensions that that question implicitly rests on an assumption that QCD, and thus the structure of atomic nuclei, is pretty much unchanged in higher dimensions. That seems a reasonable assumption to me, based on my admittedly extremely sketchy knowledge of QCD (which might reasonably be summarized as "gluon-mediated forces between quarks act like springs, so dimensionality doesn't matter"), but it is indeed an assumption. While there is plenty of internet-accessible discussion on the effects on electromagnetism when moving into higher dimensions, cursory searching on my part was only able to turn up two papers on higher-dimension QCD, neither of which address the structure of nucleons or nuclei.

So, what does happen to QCD in higher dimensions? Do we still get neutrons and protons in 4+1, 5+1, or higher-dimensional spacetimes, or does stuff get Weird?

• To be fair, we don't even know if $3+1$ QCD really does predict protons and neutrons. It is an educated guess, backed up by indirect data, but we have no ab-initio proof. – AccidentalFourierTransform Jul 16 at 19:27
• @AccidentalFourierTransform Is it therefore reasonable to assume that nuclear structure in higher dimensions can be analogous to that of our universe, in the absence of a proof that it should be otherwise? – Logan R. Kearsley Jul 16 at 19:29
• Reasonable? who's to say. On the one hand, a Coulomb-like effective description suggests that no bound states exist in $d>4$ (cf. physics.stackexchange.com/q/255961/84967). On the other hand, the generalized Coleman-Mermin-Wagner theorem suggests that the center can be broken in $d\ge4$, so one may expect confinement. But I don't think we have tools powerful enough to settle the question in $d=4$, let alone in $d>4$. Heck, I don't even know what the beta function looks like in $d>4$ (but this is probably easy to find online if one wants to). – AccidentalFourierTransform Jul 16 at 19:34
• Related: Would Color Confinement apply in higher dimensions?. Although that other question is related, the only currently-posted answer (mine) is about lattice QCD, rather than QCD in continuous spacetime. The answer posted here by @HansMoleman addresses the situation in continuous spacetime, which is presumably what the question is really about. – Chiral Anomaly Jul 16 at 23:22

Yang-Mills theory is IR free in $$D \geq 5$$ dimensions. So at low energies any such theory is non-interacting: in particular there are no bound states of quarks. Moreover, it's not clear how to "construct" $$D \geq 5$$ nonabelian gauge theory at short distances - you can't just take the continuum limit. This just means that any such theory can only arise as a low-energy approximation of a UV theory that has more degrees of freedom. If anything, QCD in $$D=4$$ is qualitatively pretty close to the situation in $$D=3$$, and to a lesser extent to QCD in $$D=2$$.