Effective Field Theories

Effective field theories encode the predictions of a quantum field theory at low energy. The effective theory has a fairly low ultraviolet cutoff. As a result, loop corrections are small, at least if the effective action contains a term which is quadratic in the fields, and physical predictions can be read straight from the effective Lagrangean. Methods will be discussed how to compute an effective low energy action from a given fundamental action, either analytically or numerically, or by a combination of both methods. Basically,the idea is to integrate out the high frequency components of fields. This requires the choice of a "blockspin",i.e. the specification of a low frequency field as a function of the fundamental fields. These blockspins will be the fields of the effective field theory. The blockspin need not be a field of the same type as one of the fundamental fields, and it may be composite. Special features of blockspins in nonabelian gauge theories will be discussed in some detail. In analytical work and in multigrid updating schemes one needs interpolation kernels \A from coarse to fine grid in addition to the averaging kernels $C$ which determines the blockspin. A neural net strategy for finding optimal kernels is presented. Numerical methods are applicable to obtain actions of effective theories on lattices of finite volume. The constraint effective potential) is of particular interest. In a Higgs model it yields the free energy, considered as a function of a gauge covariant magnetization. Its shape determines the phase structure of the theory. Its loop expansion with and without gauge fields can be used to determine finite size corrections to numerical data.Comment: 45 pages, 9 figs., preprint DESY 92-070 (figs. 3-9 added in ps format

Effective Field Theories and Inflation

We investigate the possible influence of very-high-energy physics on inflationary predictions focussing on whether effective field theories can allow effects which are parametrically larger than order H^2/M^2, where M is the scale of heavy physics and H is the Hubble scale at horizon exit. By investigating supersymmetric hybrid inflation models, we show that decoupling does not preclude heavy-physics having effects for the CMB with observable size even if H^2/M^2 << O(1%), although their presence can only be inferred from observations given some a priori assumptions about the inflationary mechanism. Our analysis differs from the results of hep-th/0210233, in which other kinds of heavy-physics effects were found which could alter inflationary predictions for CMB fluctuations, inasmuch as the heavy-physics can be integrated out here to produce an effective field theory description of low-energy physics. We argue, as in hep-th/0210233, that the potential presence of heavy-physics effects in the CMB does not alter the predictions of inflation for generic models, but does make the search for deviations from standard predictions worthwhile.Comment: 19 pages, LaTeX, no figures, uses JHEP

Renormalons in Effective Field Theories

We investigate the high-order behavior of perturbative matching conditions in effective field theories. These series are typically badly divergent, and are not Borel summable due to infrared and ultraviolet renormalons which introduce ambiguities in defining the sum of the series. We argue that, when treated consistently, there is no physical significance to these ambiguities. Although nonperturbative matrix elements and matching conditions are in general ambiguous, the ambiguity in any physical observable is always higher order in $1/M$ than the theory has been defined. We discuss the implications for the recently noticed infrared renormalon in the pole mass of a heavy quark. We show that a ratio of form factors in exclusive $\Lambda_b$ decays (which is related to the pole mass) is free from renormalon ambiguities regardless of the mass used as the expansion parameter of HQET. The renormalon ambiguities also cancel in inclusive heavy hadron decays. Finally, we demonstrate the cancellation of renormalons in a four-Fermi effective theory obtained by integrating out a heavy colored scalar.Comment: Minor changes mad

Effective Field Theories from QCD

We present a method for extracting effective Lagrangians from QCD. The resulting effective Lagrangians are based on exact rewrites of cut-off QCD in terms of these new collective field degrees of freedom. These cut-off Lagrangians are thus ``effective'' in the sense that they explicitly contain some of the physical long-distance degrees of freedom from the outset. As an example we discuss the introduction of a new collective field carrying the quantum numbers of the $\eta'$-meson. (Contribution presented by R. Sollacher at the workshop ``QCD'94'', Montpellier, France, July 7-13, 1994. To appear in those proceedings.)Comment: 6 pages, GSI-94-5

Effective Field Theories for Heavy Quarkonium

We briefly review how nonrelativistic effective field theories give us a definition of the QCD potentials and a coherent field theory derived quantum mechanical scheme to calculate the properties of bound states made by two or more heavy quarks. In this framework heavy quarkonium properties depend only on the QCD parameters (quark masses and \als) and nonpotential corrections are systematically accounted for. The relation between the form of the nonperturbative potentials and the low energy QCD dynamics is also discussed.Comment: Invited Plenary talk at The 20th European Conference on Few-Body Problems in Physics. September 10-14 2007. Pisa, Italy. To be published on Few-Body System