# Landau pole

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In physics, the Landau pole or the Moscow zero 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 coupling constants depend on the momentum (or length) scale is one of the basic ideas 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 may describe the Higgs boson. In these theories, the renormalized coupling constant grows with energy. A Landau pole appears when the coupling constant becomes infinite at a finite energy scale. In a theory intended 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 calculated using 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. Lattice field 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 seems to confirm Landau's conclusion that QED charge is completely screened for an infinite cutoff.

## Brief history

According to Landau, Abrikosov, Khalatnikov, the relation of the observable charge gobs with the “bare” charge g0 for renormalizable field theories when Λ ≫ m is given by expression

$g_{obs}={\frac {g_{0}}{1+\beta _{2}g_{0}\ln \Lambda /m}}\qquad \qquad \qquad (1)$ where Template:Mvar 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:

$g_{0}={\frac {g_{obs}}{1-\beta _{2}g_{obs}\ln \Lambda /m}}.\qquad \qquad \qquad (2)$ As Template:Mvar grows, the bare charge g0 = g(Λ) increases, to diverge at the renormalization point

$\Lambda _{Landau}=m\exp \left\{{\frac {1}{\beta _{2}g_{obs}}}\right\}.\qquad \qquad \qquad (3)$ 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 exact reality of the Landau pole becomes doubtful.

The actual behavior of the charge g(μ) as a function of the momentum scale Template:Mvar is determined by the Gell-MannLow equation

${\frac {dg}{d\ln \mu }}=\beta (g)=\beta _{2}g^{2}+\beta _{3}g^{3}+\ldots \qquad \qquad \qquad (4)$ 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 standard classification, there are three qualitatively different cases:

• (a) if β(g) has a zero at the finite value g, then growth of Template:Mvar is saturated, i.e. g(μ) → g for μ → ∞;
• (b) if β(g) is non-alternating and behaves as β(g) ∝ gα with α ≤ 1 for large Template:Mvar, then the growth of g(μ) continues to infinity;
• (c) if β(g) ∝ gα with α > 1 for large Template:Mvar, then g(μ) is divergent at finite value μ0 and the real Landau pole arises: the theory is internally inconsistent due to indeterminacy of g(μ) for μ > μ0.

Landau and 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 on the quantitative level is excluded by non-quadratic form of the Template:Mvar-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 with 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, though 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 → ∞. It is a widespread belief that both QED and φ4 theory are trivial in the continuum limit. In fact, available information confirms only “Wilson triviality”, which just amounts to positivity of β(g) for g ≠ 0 and can be considered as firmly established. Indications of “true” quantum triviality are not numerous and allow different interpretations.

## Phenomenological aspects

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 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 as a need for an 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 pure academic interest. The role of gobs in Eqs. 1, 2 is played by the fine structure constant α ≈ 1/137 and the Landau scale for QED is estimated as 10286 eV, 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 1013 eV, while the Planck scale, at which quantum gravity becomes important and the relevance of quantum field theory itself may be questioned, is 1028 eV.

The Higgs boson in the Standard Model of particle physics is described by φ4 theory. 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.

## Recent developments

Solution of the Landau pole problem requires calculation of the Gell-Mann–Low function β(g) at arbitrary Template:Mvar and, in particular, its asymptotic behavior for g → ∞. This problem is very difficult and was considered as hopeless for many years: by diagrammatic calculations one can obtain only few expansion coefficients β2, β3, ..., which do not allow to investigate the Template:Mvar function in the whole. The progress became possible after development of the Lipatov method for calculation of large orders of perturbation theory: now one can try to interpolate the known coefficients β2, β3, ... with their large order behavior and to sum the perturbation series. The first attempts of reconstruction of the Template:Mvar function witnessed on triviality of φ4 theory. Application of more advanced summation methods gave the exponent Template:Mvar in the asymptotic behavior β(g) ∝ gα a value close to unity. The hypothesis for the asymptotics β(g) ∝ g was recently confirmed analytically for φ4 theory and QED. Together with positiveness of β(g), obtained by summation of series, it gives the case (b) of the Bogoliubov and Shirkov classification, and hence the Landau pole is absent in these theories. Possibility of omitting the quadratic terms in the action suggested by Landau and Pomeranchuk is not confirmed.