16
Lattice Field Theory, and the
Renormalization Group Revisited
16.1 Introduction
Throughout this book, thus far, we have relied on perturbation theory as the
calculational tool, justifying its use in the case of QCD by the smallness of the
coupling constant at short distances; note, however, that this result itself required the summation of an infinite series of perturbative terms. As remarked
in section 15.3, the concomitant of asymptotic freedom is that α s really does
become strong at small Q
2 , or at long distances of order Λ
−1
∼ 1 fm. Here we
MS
have no prospect of getting useful results from perturbation theory: it is the
non-perturbative regime. But this is precisely the regime in which quarks bind
together to form hadrons. If QCD is indeed the true theory of the interaction
between quarks, then it should be able to explain, ultimately, the vast amount
of data that exists in low energy hadronic physics. For example: what are
the masses of mesons and baryons? Are there novel colourless states such as
glueballs? Is SU(2) f or SU(3) f chiral symmetry spontaneously broken? What
is the form of the effective interquark potential? What are the hadronic form
factors, in electromagnetic (chapter 9) or weak (chapter 20) processes?
After more than 30 years of theoretical development, and machine advances, numerical simulations of lattice QCD are now yielding precise answers
to many of these questions, thereby helping to establish QCD as the correct
theory of the strong interactions of quarks, and also providing reliable input
needed for the discovery of new physics. Lattice QCD is a highly mature
field, and many technical details are beyond our scope. Rather, in this chapter we aim to give an elementary introduction to lattice field theory in general,
including some important insights that it generates concerning the renormalization group. We return to QCD in the final section, with some illustrative
results.
In thinking about how to formulate a non-perturbative approach to quantum field theory, several questions immediately arise. First of all, how can we
regulate the ultraviolet divergences, and thus define the theory, if we cannot
get to grips with them via the specific divergent integrals supplied by perturbation theory? We need to be able to regulate the divergences in a way
which does not rely on their appearance in the Feynman graphs of pertur151
Lattice Field Theory, and the
Renormalization Group Revisited
16.1 Introduction
Throughout this book, thus far, we have relied on perturbation theory as the
calculational tool, justifying its use in the case of QCD by the smallness of the
coupling constant at short distances; note, however, that this result itself required the summation of an infinite series of perturbative terms. As remarked
in section 15.3, the concomitant of asymptotic freedom is that α s really does
become strong at small Q
2 , or at long distances of order Λ
−1
∼ 1 fm. Here we
MS
have no prospect of getting useful results from perturbation theory: it is the
non-perturbative regime. But this is precisely the regime in which quarks bind
together to form hadrons. If QCD is indeed the true theory of the interaction
between quarks, then it should be able to explain, ultimately, the vast amount
of data that exists in low energy hadronic physics. For example: what are
the masses of mesons and baryons? Are there novel colourless states such as
glueballs? Is SU(2) f or SU(3) f chiral symmetry spontaneously broken? What
is the form of the effective interquark potential? What are the hadronic form
factors, in electromagnetic (chapter 9) or weak (chapter 20) processes?
After more than 30 years of theoretical development, and machine advances, numerical simulations of lattice QCD are now yielding precise answers
to many of these questions, thereby helping to establish QCD as the correct
theory of the strong interactions of quarks, and also providing reliable input
needed for the discovery of new physics. Lattice QCD is a highly mature
field, and many technical details are beyond our scope. Rather, in this chapter we aim to give an elementary introduction to lattice field theory in general,
including some important insights that it generates concerning the renormalization group. We return to QCD in the final section, with some illustrative
results.
In thinking about how to formulate a non-perturbative approach to quantum field theory, several questions immediately arise. First of all, how can we
regulate the ultraviolet divergences, and thus define the theory, if we cannot
get to grips with them via the specific divergent integrals supplied by perturbation theory? We need to be able to regulate the divergences in a way
which does not rely on their appearance in the Feynman graphs of pertur151