Lattice Yang-Mills

Question

I have often heard statements like, Continuum limit of Lattice Yang-Mills doesnot exist in dimensions higher then four. Is there a neat way to see this? or could some one point me to some relevant literature where this has been shown. thanks

Answer 1

This may look like a deep question and in some sense, it is. But the explanation is completely trivial, doesn't really depend on the "lattice" at all, and boils down to dimensional analysis.

The action of Yang-Mills theory $$ S =\int d^D x \frac{1}{2 g^2} {\rm Tr} \, F_{\mu\nu} F^{\mu\nu} $$ Now, $F_{\mu\nu}\sim [D_\mu,D_\nu]$ has dimension of squared mass - because the (covariant) derivatives have the dimension of mass in these conventions. That's why $F^2$ has dimension $mass^4$. Because $d^D x$ has $mass^{-D}$, it's clear that the dimension of $g$ has to be $$ mass^{(4-D)/2}$$ For $D=4$, the coupling constant is classically dimensionless. For $D>4$, the coupling constant has the dimension of a negative power of mass (positive power of length); for $D<4$, it's the other way around.

The strength of interactions - e.g. the probability of scattering - goes like a positive power of $g$. However, to get e.g. the probability I mentioned, we must make it dimensionless. Dimensional analysis tells us that the right "dimensionless coupling" is $$g \cdot L^{(4-D)/2}$$ where $L$ is a characteristic distance scale of the experiment. For $D>4$, the interactions are therefore getting stronger - and infinitely strong in the $L\to 0$ limit. In particular, if $L=a$ is the lattice spacing which is send to zero, you will need to adjust the coupling constant at the lattice scale to be infinite if you want to get a finite $g \cdot L^{(4-D)/2}$ for a finite $L$. However, there's really no way to define a gauge theory with an infinite coupling, even at the level of a lattice.

In fact, for $D>4$, the gauge theory action above only works as an effective action for long enough distances $L$ where the "dimensionless coupling" is small enough. This is the only starting point where "a theory" exists and if you try to extrapolate it to much shorter distances, you will find out that the theory becomes ill-defined once the dimensionless coupling constant exceeds one or so - at the distance scale corresponding to a power of $g$.

So no theory that could be identified with the gauge theory exists at supershort distances if $D>4$. Alternatively, you may naively define a lattice gauge theory for $D>4$ with a finite coupling constant. However, the scaling laws guarantee that at distances that are infinitely longer - as in the continuum limit - the interactions will vanish. At any rate, you can't get finite yet non-vanishing interactions at both distances $a$ that are sent to zero as well as finite distances $L$.

The case of $D<4$ is different because the interactions become stronger at longer distances. So one may start with a lattice with "infinitesimal" interactions and they go stronger at longer distances.