Landau pole

In physics, Landau pole is the energy scale (or the precise value of the energy) where a coupling constant (the strength of an interaction) of a quantum field theory becomes infinite. Such a possibility was pointed out by the eminent physicist Lev Davidovich Landau. The dependence of coupling constants on the energy scale is one of the basic ideas behind the renormalization group.

Theories with asymptotic freedom have Landau poles at very low energies. However, the phrase "Landau pole" is usually used in the context of the theories that are not asymptotically free, such as quantum electrodynamics (QED) or a scalar field with a quartic interaction. The coupling constant grows with energy, and at some energy scale the growth becomes infinite and the coupling constant itself diverges.

Landau poles at high energy are viewed as problems; more precisely, they are evidence that the theory (e.g. QED) is not well-defined nonperturbatively. The Landau pole of QED is removed if QED is embedded into a Grand Unified Theory or an even more powerful framework such as superstring theory.

An equation

Everything started in the 1950s when Landau decided to understand the relation between the bare electric charge <math>e<math> and the renormalized electric charge <math>e_R<math>. He found the following equation:

<math>\frac{1}{e_R^2} - \frac{1}{e^2}=\frac{N_f}{6\pi^2}\ln\frac{\Lambda}{m_R}<math>

This equation needs to be explained:

  • <math>e<math> is the value of the electric charge that we naively insert to the Lagrangian, but it turns out that this number is actually not a constant, but rather an energy-dependent quantity
  • <math>e_R<math> is the actual renormalized, measurable value of the charge (that determines how much the electrons attract each other at low energies), which is not quite the same thing as <math>e<math>
  • <math>N_f<math> is the number of flavors; for "staggered" fermions we substitute <math>N_f=4<math>
  • <math>\Lambda<math> is the momentum cutoff i.e. the maximal value of the momentum that we allow to be taken into account
  • <math>m_R<math> is the renormalized electron mass

Missing image

The right-hand side can be calculated from loops in Feynman diagrams (namely one-loop Feynman diagrams), i.e. as a contribution of quantum mechanics. It has a logarithmic form because the integral happens to be logarithmically divergent. Note that the equation has two obvious implications:

  • If the bare charge <math>e<math> is kept fixed, the theory (QED) has a trivial continuum (<math>\Lambda\to\infty<math>) limit, namely <math>e_R\to 0<math>
  • When the renormalized charge <math>e_R<math> is kept fixed, the bare charge becomes singular (infinite) at
<math>\Lambda_{\mathrm{Landau}}=m_R\exp(6\pi^2/N_f e_R^2)<math>.

The latter singularity is the Landau pole. It does not affect the phenomenological success of perturbative calculations in QED because for all practical purposes, the cutoff <math>\Lambda<math> can be chosen much smaller than the huge scale <math>\Lambda_{\mathrm{Landau}}<math>, comparable to the Planck scale, and it is still enough to describe all accessible experiments. Nevertheless, the Landau pole is an awkward theoretical feature of QED, a sufficiently awkward one to make us look for a better theory.

Since the Landau pole is calculated using one-loop or two-loop calculations but perturbation theory breaks down for large coupling constants, there has been some controversy as to whether the Landau pole really exists.


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