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Landau pole
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.
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.
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Landau pole
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.
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.