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 in 1954. 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 φ4 theory—a scalar field with a quartic interaction—such as ones that 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. However, lattice gauge theory provides a means to address questions in quantum field theory beyond the realm of perturbation theory, and numerical computations performed in this framework seem to confirm Landau's conclusion that in QED the renormalized charge completely vanishes for an infinite cutoff.
Brief history According to Landau, Alexei Abrikosov, and Isaak Khalatnikov, the relation of the observable charge gobs to the "bare" charge g0 for renormalizable field theories when Λ ≫ m is given by
where m is the mass of the particle and Λ is the momentum cut-off. If g0 < ∞ and Λ → ∞ then gobs → 0 and the theory looks trivial. In fact, inverting Eq. 1, so that g0 (related to the length scale Λ−1) reveals an accurate value of gobs,
As Λ grows, the bare charge g0 = g(Λ) increases, to finally diverge at the renormalization point
This singularity is the Landau pole with a negative residue, g(Λ) ≈ −ΛLandau / (β2(Λ − ΛLandau)). In fact, however, the growth of g0 invalidates Eqs. 1, 2 in the region g0 ≈ 1, since these were obtained for g0 ≪ 1, so that the nonperturbative existence of the Landau pole becomes questionable. The actual behavior of the charge g(μ) as a function of the momentum scale μ is determined by the Gell-Mann–Low equation (named after Murray Gell-Mann and Francis E. Low)
which gives Eqs. 1, 2 if it is integrated under conditions g(μ) = gobs for μ = m and g(μ) = g0 for μ = Λ, when only the term with β2 is retained in the right hand side. The general behavior of g(μ) depends on the appearance of the function β(g). According to the classification of Nikolay Bogolyubov and Dmitry Shirkov, there are three qualitatively different cases:
Landau and Isaak Pomeranchuk tried to justify the possibility (c) in the case of QED and φ4 theory. They have noted that the growth of g0 in Eq. 1 drives the observable charge gobs to the constant limit, which does not depend on g0. The same behavior can be obtained from the functional integrals, omitting the quadratic terms in the action. If neglecting the quadratic terms is valid already for g0 ≪ 1, it is all the more valid for g0 of the order or greater than unity: it gives a reason to consider Eq. 1 to be valid for arbitrary g0. Validity of these considerations at the quantitative level is excluded by the non-quadratic form of the β-function. Nevertheless, they can be correct qualitatively. Indeed, the result gobs = const(g0) can be obtained from the functional integrals only for g0 ≫ 1, while its validity for g0 ≪ 1, based on Eq. 1, may be related to other reasons; for g0 ≈ 1 this result is probably violated but coincidence of two constant values in the order of magnitude can be expected from the matching condition. The Monte Carlo results seems to confirm the qualitative validity of the Landau–Pomeranchuk arguments, although a different interpretation is also possible. The case (c) in the Bogoliubov and Shirkov classification corresponds to the quantum triviality in full theory (beyond its perturbation context), as can be seen by a reductio ad absurdum. Indeed, if gobs < ∞, the theory is internally inconsistent. The only way to avoid it, is for μ0 → ∞, which is possible only for gobs → 0. It is a widespread belief that both QED and φ4 theory are trivial in the continuum limit.
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