Lattice perturbation theory

Lattice perturbation theory?

Lattice QCD is primarily a tool for understanding the nonperturbative regime of QCD—so why develop lattice perturbation theory?

In some ways, lattice QCD was born before its time and has had to wait for the development of computers - and algorithms - powerful enough to cope with the computational demands of QCD. Lattice perturbation theory, perhaps better thought of as "perturbation theory with lattice actions", has played an important role in the growth of lattice QCD from the very beginning. Broadly construed, there are four ways in which lattice perturbation theory when studying heavy quark systems.

Weak coupling

Lattice perturbation theory can serve as a direct cross-check of computations carried out in the weak-coupling regime. In practice this has been a less common use of lattice perturbation theory, although the references listed on this page share the spirit of this approach.

Renormalisation

We can use perturbative calculations to determine renormalisation parameters, both of the bare parameters of lattice actions, such as the mass and wavefunction renormalisation, and of matrix elements. The lattice serves as an ultraviolet regulator by excluding all momenta greater than π/a; therefore we can view lattice QCD as simply another regularisation scheme. The renormalised parameters can often be calculated relatively easily in perturbation theory. Please note the relatively in that statement. This should be taken to mean that these calculations are necessarily easy in some absolute sense.

Matching

Lattice perturbation theory provides a method for systematically matching regularisation schemes. Experimental results are typically expressed in the Modified Minimal Subtraction (MS-bar) scheme. To compare lattice simulations with experimental results, we must relate lattice parameters, such as the coupling constant and scheme-dependent renormalisation parameters, to the equivalent parameters in the MS scheme. This matching process is often performed perturbatively. Relating renormalisation schemes is of course intimately related to the process of renormalisation itself; the process of renormalising bare parameters does not, however, necessarily require matching to other schemes, so I tend to think of them as somewhat separate enterprises.

Improvement

Finally, lattice perturbation theory allows us to improve lattice gauge theories: by improving the bare operator coefficients of the lattice action; by improving lattice operators; and by improving the scale dependence of nonperturbative renormalisation parameters.

Improving bare operator coefficients forms an integral part of the Symanzik improvement program. For example, lattice NRQCD is an effective theory for heavy quarks at finite lattice spacing. Lattice NRQCD is an expansion in irrelevant operators with coefficients chosen by matching to continuum QCD in such a way that continuum results can be determined from lattice calculations. At tree-level, these coefficients are defined to be unity, but beyond tree-level radiative corrections modify the coefficients and the ultraviolet behaviour of the nonrelativistic action differs from that of continuum QCD. This difference introduces radiative errors. In principle we could tune the coefficients of these additional operators nonperturbatively to reproduce the correct continuum results. In practise, however, nonperturbative tuning is time-consuming and reduces the predictive power of the lattice theory. Lattice perturbation theory can therefore sometimes be a better method for determining the lattice action parameters.

But is it justified?

Lattice QCD is essentially a nonperturbative tool, so we may be inclined to ask: "Is it reasonable or appropriate to apply perturbative techniques to a nonperturbative formulation of QCD?" A justification for the use of lattice perturbation theory is provided in G.P. Lepage's "Redesigning lattice QCD" lectures: the perturbative renormalisation factors account for the momenta excluded by the lattice cutoff. This hard ultraviolet cutoff is typically at least 6 GeV for current lattice spacings of around 0.1 fm. The coupling constant is certainly small at these energies, α_s(π/a)=0.2, and perturbation theory is likely to be valid. We therefore expect the effects of the excluded momenta are to be well described by perturbation theory.

There are issues, of course. Matching renormalisation schemes via perturbation theory introduces perturbative truncation errors that one would generally rather avoid. It can be hard to estimate the size of higher-order perturbative contributions, leading to potentially important systematic uncertainties. Perturbative approximations also impinge on one of the more elegant conceptual advantages of the lattice formalism: that lattice QCD allows us to calculate the nonperturbative properties of QCD ab initio. This is not to say that lattice perturbation theory is not useful, reliable or powerful when used appropriately, far from it. But only that, in some regards, the use of lattice perturbation theory leads us slightly away from the ideal of true ab initio QCD calculations. Lattice QCD is necessarily a messy subject when it comes to the technical details, however, and, in a world of limited resources, unphysical pion masses, degenerate up and down quark masses and perhaps only a couple of flavours in the sea (although some of these days are passing), lattice perturbation theory may simply be another technical necessity, at least for certain calculations or processes.