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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 theorySuperstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings. It is considered one of the most promising candidate theories of quantum gravity.

An equation

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

This equation needs to be explained:

• is the value of the electric chargeElectric charge is a fundamental property of some subatomic particles, which determines their electromagnetic interactions. It is one of the quantum numbers. Matter that possesses a charge is influenced by, and produces, electromagnetic fields. The intera that we naively insert to the LagrangianA Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a functional of the dynamical variables which concisely describes the equations of motion of the system. The equations of motion are obtained by means of an action principle, writte, but it turns out that this number is actually not a constant, but rather an energy-dependent quantity
• is the actual renormalizedIn quantum field theory (QFT) and the statistical mechanics of fields, renormalization refers to a collection of techniques used to express physical calculations in terms of observable quantities that already include some field effects. Renormalization ar, 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
• is the number of flavorIn particle physics, flavor is a property of a fermion that identifies it, a label that specifies the name of the particle. According to the Standard Model, quarks exist in six flavors: up, down, charm, strange, top and bottom (indicated with the symbolss; for "staggered" fermionFermions named after Enrico Fermi, are particles which form totally-antisymmetric composite quantum states. As a result, they are subject to the Pauli exclusion principle and obey Fermi-Dirac statistics. The spin-statistics theorem states that fermions has we substitute
• is the momentum cutoffIn theoretical physics, cutoff usually represents a particular energy scale or length scale. The phrase denotes the maximal or minimal value of energy, momentum, or length (or something along these lines) such that the objects with even larger (or smaller i.e. the maximal value of the momentum that we allow to be taken into account
• is the renormalized electronThe electron (also called negatron commonly represented as e&minus is a subatomic particle. In an atom the electrons surround the nucleus of protons and neutrons in an electron configuration. Electrons have the smallest electrical charge and when they mov mass

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 is kept fixed, the theory (QED) has a trivial continuum () limit, namely
• When the renormalized charge is kept fixed, the bare charge becomes singular (infinite) at

.

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 can be chosen much smaller than the huge scale , 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.