Perturbative and Nonperturbative Aspects of Quantum Field Theory Pro
The book addresses graduate students as well as scientists interested in applications of the standard model for strong and electroweak interactions to experimentally determinable quantities. Computer simulations and the relations between various approache
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ization in effective field theories, decoupling of heavy particles, power counting, and naive dimensional analysis. Effective Lagrangians are used to study the AS -- 2 weak interactions and chiral perturbation theory.
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Introduction
An important idea that is implicit in all descriptions of physical phenomena is that of an effective theory. The basic premise of effective theories is that dynamics at low energies (or large distances) does not depend on the details of the dynamics at high energies (or short distances). As a result, low energy physics can be described using an effective Lagrangian that contains only a few degrees of freedom, ignoring additional degrees of freedom present at higher energies. One of the main purposes of these lectures is to make these qualitative statements quantitative. First a simple example: The energy levels of the Hydrogen atom are calculated in textbooks using the SchrSdinger equation for an electron bound to a proton by a Coulomb potential. To a good approximation, the only properties of the proton that are relevant for the computation are its mass and charge. An understanding of the quark substructure of the proton (let alone quantum gravity) is not necessary to compute the energy levels of the Hydrogen states. This is true provided an answer which has some theoretical uncertainty is sufficient. A more accurate calculation of the energy levels, for example including the hyperfine splitting, requires that we also know that the proton has spin-I/2, and a magnetic moment of 2.793 nuclear magnetons. An even more accurate calculation of the energy levels requires some knowledge of the proton charge radius, etc. More details of the proton structure are needed as we require a more accurate answer for the energy levels. When we discuss effective theories, we will frequently talk about momentum scales characteristic of a given problem. The typical length scale characteristic of the Hydrogen atom is the Bohr radius ao = 1 / ( m e a ) , and the typical momentum scale is of order t t / a o ,,, 1 / a o = m e a , using units in which h = 1. The typical energy scale characteristic of Hydrogen is the Rydberg ,-~ m e a 2 , and the typical time scale is 1/(mea2). The Hydrogen atom is more complicated than many relativistic bound states because it has two characteristic scales, rnea and meOt2. We can now give a quantitative estimate of the error caused by neglected
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Aneesh V. Manohar
interactions: the energy levels of Hydrogen can be computed by ignoring all dynamics on momentum scales A much larger than men, with an error of order mea/A. As the desired accuracy increases, the scale A of the interactions that can be ignored, also increases. The relevant interactions in an effective theory also depend on the question being studied. In the Hydrogen atom, the energy levels can be computed to an accuracy (meo~/Mw) 2 while ignoring the weak interactions, but if we are interested in atomic parity violation, the weak interactions are the leading contribution since strong and electromagnetic inte
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