An Introduction To Quantum Field Theory (Frontiers in Physics) by Michael E. Peskin & Daniel V. Schroeder

Author:Michael E. Peskin & Daniel V. Schroeder [Peskin, Michael E.]
Language: eng
Format: epub
ISBN: 9780813345437
Publisher: Westview Press
Published: 1995-10-02T07:00:00+00:00

c. Write the β functions for λ and ρ in 4—∈ dimensions. Show that there are nontrivial fixed points of the renormalization group flow at ρ/λ = 0, 1, 3. Which is the most stable? Sketch the pattern of coupling constant flows. This flow implies that the critical exponents are those of a symmetric two-component magnet.

Chapter 13

Critical Exponents and Scalar Field Theory

The idea of running coupling constants and renormalization-group flows gives us a new language with which to discuss the qualitative behavior of scalar field theory. In our first discussion of φ4 theory, each value of the coupling constant—and, more generally, each form of the potential and each spacetime dimension—gave a separate problem to be explored. But in Chapter 12, we saw that φ4 theories with different values of the coupling are connected by renormalization-group flows, and that the pattern of these flows changes continuously with the spacetime dimension. In this context, it makes sense to ask the very general question: How does φ4 theory behave as a function of the dimension? This chapter will give a detailed answer to this question.

The central ingredient in our analysis will be the Wilson-Fisher fixed point discussed in Section 12.5. This fixed point exists in spacetime dimensions d with d < 4; in those dimensions it controls the renormalization group flows of massless theory. The scalar field theory has manifest or spontaneously broken symmetry according to the sign of the mass parameter m2. Near m2 = 0, the theory exhibits scaling behavior with anomalous dimensions whose values are determined by the renormalization group equations. For d > 4, the Wilson-Fisher fixed point disappears, and only the free-field fixed point remains. Again, the theory exhibits two distinct phases, but now the behavior at the transition is determined by the renormalization group flows near the free-field fixed point, so the scaling laws are those that follow from simple dimensional analysis.

The continuation of these results to Euclidean space has important implications for the theory of phase transitions in magnets and fluids. As we discussed in the previous chapter, the ideas of the renormalization group imply that the power-law behaviors of thermodynamic quantities near a phase transition point are determined by the behavior of correlation functions in a Euclidean φ4 theory. The results stated in the previous paragraph then imply the following conclusions for critical scaling laws: For statistical systems in a space of dimension d > 4, the scaling laws are just those following from simple dimensional analysis. These predictions are precisely those of Landau theory, which we discussed in Chapter 8. On the other hand, for d < 4, the critical scaling laws are modified, in a way that we can compute using the renormalization group.

In d = 4, we are on the boundary between the two types of scaling behavior. This corresponds to the situation in which φ4 theory is precisely renormalizable. In this case, the dimensional analysis predictions are corrected, but only by logarithms. We will analyze this case specifically in Section 13.2.

Though it is not obvious, the case d = 2 provides another boundary.

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