Landau pole

According to Landau, Abrikosov, and Khalatnikov,[8] the relation of the observable charge g_obs to the “bare” charge g₀ for renormalizable field theories when Λ ≫ m is given by

${\displaystyle g_{\text{obs}}={\frac {g_{0}}{1+\beta _{2}g_{0}\ln \Lambda /m}}\qquad \qquad \qquad (1)}$

where m is the mass of the particle and Λ is the momentum cut-off. If g₀ < ∞ and Λ → ∞ then g_obs → 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 g_obs,

${\displaystyle g_{0}={\frac {g_{\text{obs}}}{1-\beta _{2}g_{\text{obs}}\ln \Lambda /m}}.\qquad \qquad \qquad (2)}$

As Λ grows, the bare charge g₀ = g(Λ) increases, to finally diverge at the renormalization point

${\displaystyle \Lambda _{\text{Landau}}=m\exp \left\{{\frac {1}{\beta _{2}g_{\text{obs}}}}\right\}.\qquad \qquad \qquad (3)}$

This singularity is the Landau pole with a negative residue,   g(Λ) ≈ −Λ_Landau /(β₂(Λ − Λ_Landau)).

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[9]

${\displaystyle {\frac {dg}{d\ln \mu }}=\beta (g)=\beta _{2}g^{2}+\beta _{3}g^{3}+\ldots \qquad \qquad \qquad (4)}$

which gives Eqs.1,2 if it is integrated under conditions g(μ) = g_obs 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 Bogoliubov and Shirkov,[10] there are three qualitatively different cases:

• (a) if β(g) has a zero at the finite value g^∗, then growth of g is saturated, i.e. g(μ) → g^∗ for μ → ∞;
• (b) if β(g) is non-alternating and behaves as β(g) ∝ g^α with α ≤ 1 for large g, then the growth of g(μ) continues to infinity;
• (c) if β(g) ∝ g^α with α > 1 for large g, 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 [11] 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 g_obs 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 g_obs = 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 [12] 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 g_obs < ∞, the theory is internally inconsistent. The only way to avoid it, is for μ₀ → ∞, which is possible only for g_obs → 0. It is a widespread belief that both QED and φ⁴ theory are trivial in the continuum limit.

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 pure academic interest, for the following reason. The role of g_obs 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 (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.[13]

A deeper understanding of the physical meaning and generalization of the renormalization process leading to Landau poles comes from condensed matter physics. Leo P. Kadanoff's paper in 1966 proposed the "block-spin" renormalization group.[14] The blocking idea is a way to define the components of the theory at large distances as aggregates of components at shorter distances. This approach was developed by Kenneth Wilson.[15] He was awarded the Nobel prize for these decisive contributions in 1982.

Assume that we have a theory described by a certain function ${\displaystyle Z}$ of the state variables ${\displaystyle \{s_{i}\}}$ and a set of coupling constants ${\displaystyle \{J_{k}\}}$. This function can be a partition function, an action, or a Hamiltonian. Consider a certain blocking transformation of the state variables ${\displaystyle \{s_{i}\}\to \{{\tilde {s}}_{i}\}}$, the number of ${\displaystyle {\tilde {s}}_{i}}$ must be lower than the number of ${\displaystyle s_{i}}$. Now let us try to rewrite the ${\displaystyle Z}$ function only in terms of the ${\displaystyle {\tilde {s}}_{i}}$. If this is achievable by a certain change in the parameters, ${\displaystyle \{J_{k}\}\to \{{\tilde {J}}_{k}\}}$, then the theory is said to be renormalizable. The most important information in the RG flow are its fixed points. The possible macroscopic states of the system, at a large scale, are given by this set of fixed points. If these fixed points correspond to a free field theory, the theory is said to exhibit quantum triviality, and possesses a Landau pole. Numerous fixed points appear in the study of lattice Higgs theories, but it is not known whether these correspond to free field theories.[3]

Solution of the Landau pole problem requires the calculation of the Gell-Mann–Low function β(g) at arbitrary g and, in particular, its asymptotic behavior for g → ∞. Diagrammatic calculations allow one to obtain only a few expansion coefficients β₂, β₃, ..., which do not allow one to investigate the β function in the whole. Progress became possible after the development of the Lipatov method for calculating large orders of perturbation theory:[16] One may now try to interpolate the known coefficients β₂, β₃, ... with their large order behavior, and to then sum the perturbation series.

The first attempts of reconstruction of the β function by this method bear on the triviality of the φ⁴ theory. Application of more advanced summation methods yielded the exponent α in the asymptotic behavior β(g) ∝ g^α, a value close to unity. The hypothesis for the asymptotic behavior of β(g) ∝ g was recently presented analytically for φ⁴ theory and QED.[17][18][19] Together with positiveness of β(g), obtained by summation of the series, it suggests case (b) of the above Bogoliubov and Shirkov classification, and hence the absence of the Landau pole in these theories, assuming perturbation theory is valid (but see above discussion in the introduction ).

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