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Coupling constant divergence at high energies
In physics, the Landau pole (or the Moscow zero, or the Landau ghost) is the momentum (or energy) scale at which the coupling constant (interaction strength) of a quantum field theory becomes infinite. Such a possibility was pointed out by the physicist Lev Landau and his colleagues. The fact that couplings depend on the momentum (or length) scale is the central idea behind the renormalization group. Landau poles appear in theories that are not asymptotically free, such as quantum electrodynamics (QED) or φ^(4) theory—a scalar field with a quartic interaction—such as may describe the Higgs boson. In these theories, the renormalized coupling constant grows with energy. A Landau pole appears when the coupling becomes infinite at a finite energy scale. In a theory purporting to be complete, this could be considered a mathematical inconsistency. A possible solution is that the renormalized charge could go to zero as the cut off is removed, meaning that the charge is completely screened by quantum fluctuations (vacuum polarization). This is a case of quantum triviality, which means that quantum corrections completely suppress the interactions in the absence of a cut off. Since the Landau pole is normally identified through perturbative one loop or two loop calculations, it is possible that the pole is merely a sign that the perturbative approximation breaks down at strong coupling. Perturbation theory may also be invalid if non adiabatic states exist. Lattice gauge theory provides a means to address questions in quantum field theory beyond the realm of perturbation theory, and thus has been used to attempt to resolve this question. Numerical computations performed in this framework seem to confirm Landau's conclusion that in QED the renormalized charge completely vanishes for an infinite cutoff.

In a theory intended to represent a physical interaction where the coupling constant is known to be non zero, Landau poles or triviality may be viewed as a sign of incompleteness in the theory. For example, QED is usually not believed to be a complete theory on its own, because it does not describe other fundamental interactions, and contains a Landau pole. Conventionally QED forms part of the more fundamental electroweak theory. The U(1)Y group of electroweak theory also has a Landau pole which is usually considered to be a signal of a need for an ultimate embedding into a Grand Unified Theory. The grand unified scale would provide a natural cutoff well below the Landau scale, preventing the pole from having observable physical consequences. The problem of the Landau pole in QED is of purely academic interest, for the following reason. The role of gobs in ,  is played by the fine structure constant α ≈ 1/137 and the Landau scale for QED is estimated as e=286, which is far beyond any energy scale relevant to observable physics. For comparison, the maximum energies accessible at the Large Hadron Collider are of order e=13, while the Planck scale, at which quantum gravity becomes important and the relevance of quantum field theory itself may be questioned, is e=28. The Higgs boson in the Standard Model of particle physics is described by φ^(4) theory (see Quartic interaction). If the latter has a Landau pole, then this fact is used in setting a "triviality bound" on the Higgs mass. The bound depends on the scale at which new physics is assumed to enter and the maximum value of the quartic coupling permitted (its physical value is unknown). For large couplings, non perturbative methods are required. Lattice calculations have also been useful in this context.