Landau pole
Template:Short description In physics, the Landau pole (or the Moscow zero, or the Landau ghost)<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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.<ref name="LAKh">Template:Cite journal</ref><ref>Lev Landau, in Template:Cite book</ref> 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 Template:Math 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,<ref name="TrivPurs"> Template:Cite journal</ref> 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.<ref> Template:Cite journal</ref><ref> Template:Cite journal</ref><ref> Template:Cite journal</ref><ref> Template:Cite journal</ref>
Brief historyEdit
According to Landau, Abrikosov, and Khalatnikov,<ref name="LAKh"></ref> the relation of the observable charge Template:Math to the "bare" charge Template:Math for renormalizable field theories when Template:Math is given by Template:NumBlk where Template:Mvar is the mass of the particle and Template:Math is the momentum cut-off. If Template:Math and Template:Math then Template:Math and the theory looks trivial. In fact, inverting Template:EquationNote, so that Template:Math (related to the length scale Template:Math) reveals an accurate value of Template:Math, Template:NumBlk{1-\beta_2 g_\text{obs} \ln \Lambda/m}. </math>|Template:EquationRef}}
As Template:Math grows, the bare charge Template:Math increases, to finally diverge at the renormalization point Template:NumBlk \right].</math>|Template:EquationRef}}
This singularity is the Landau pole with a negative residue, Template:Math.
In fact, however, the growth of Template:Math invalidates Template:EquationNote, Template:EquationNote in the region Template:Math, since these were obtained for Template:Math, so that the nonperturbative existence of the Landau pole becomes questionable.
The actual behavior of the charge Template:Math as a function of the momentum scale Template:Mvar is determined by the Gell-Mann–Low equation<ref name="GellMannLow" >Template:Cite journal</ref> Template:NumBlk which gives Eqs. Template:EquationNote, Template:EquationNote if it is integrated under conditions Template:Math for Template:Math and Template:Math for Template:Math, when only the term with Template:Math is retained in the right hand side. The general behavior of Template:Math depends on the appearance of the function Template:Math.
According to the classification of Bogoliubov and Shirkov,<ref>Template:Cite book</ref> there are three qualitatively different cases: Template:Ordered list Landau and Pomeranchuk<ref>L.D.Landau, I.Ya.Pomeranchuk, Dokl. Akad. Nauk SSSR 102, 489 (1955); I.Ya.Pomeranchuk, Dokl. Akad. Nauk SSSR 103, 1005 (1955).</ref> tried to justify the possibility (c) in the case of QED and Template:Math theory. They have noted that the growth of Template:Math in Template:EquationNote drives the observable charge Template:Math to the constant limit, which does not depend on Template:Math. 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 Template:Math, it is all the more valid for Template:Math of the order or greater than unity: it gives a reason to consider Template:EquationNote to be valid for arbitrary Template:Math. Validity of these considerations at the quantitative level is excluded by the non-quadratic form of the Template:Mvar-function.Template:Citation needed
Nevertheless, they can be correct qualitatively. Indeed, the result Template:Math can be obtained from the functional integrals only for Template:Math, while its validity for Template:Math, based on Template:EquationNote, may be related to other reasons; for Template:Math 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 <ref>Template:Cite journal</ref> 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 Template:Math, the theory is internally inconsistent. The only way to avoid it, is for Template:Math, which is possible only for Template:Math. It is a widespread belief Template:By whom that both QED and Template:Math theory are trivial in the continuum limit.
Phenomenological aspectsEdit
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 believedTemplate:Cn 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 Template:Math group of electroweak theory also has a Landau pole which is usually consideredTemplate:By whom 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 Template:Math in Template:EquationNote, Template:EquationNote is played by the fine structure constant Template:Math and the Landau scale for QED is estimated as Template:Val, 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 Template:Val, while the Planck scale, at which quantum gravity becomes important and the relevance of quantum field theory itself may be questioned, is Template:Val. The energy of the observable universe is on the order of Template:Val.
The Higgs boson in the Standard Model of particle physics is described by Template:Math 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. This can even lead to a predictable Higgs mass in asymptotic safety scenarios. Lattice calculations have also been useful in this context.<ref>For example, Template:Cite journalTemplate:Cite journal, which suggests Template:Math.</ref>
Connections with statistical physicsEdit
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.<ref>L.P. Kadanoff (1966): "Scaling laws for Ising models near Template:Math", Physics (Long Island City, N.Y.) 2, 263.</ref> 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.<ref>K.G. Wilson(1975): The renormalization group: critical phenomena and the Kondo problem, Rev. Mod. Phys. 47, 4, 773.</ref> He was awarded the Nobel prize for these decisive contributions in 1982.
Assume that we have a theory described by a certain function Template:Math of the state variables Template:Math and a set of coupling constants Template:Math. This function can be a partition function, an action, or a Hamiltonian. Consider a certain blocking transformation of the state variables Template:Math, the number of Template:Math must be lower than the number of Template:Math. Now let us try to rewrite Template:Math only in terms of the Template:Math. If this is achievable by a certain change in the parameters, Template:Math, 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.<ref name="TrivPurs"/>
Large order perturbative calculationsEdit
Solution of the Landau pole problem requires the calculation of the Gell-Mann–Low function Template:Math at arbitrary Template:Mvar and, in particular, its asymptotic behavior for Template:Math. Diagrammatic calculations allow one to obtain only a few expansion coefficients Template:Math, which do not allow one to investigate the Template:Mvar function in the whole. Progress became possible after the development of the Lipatov method for calculating large orders of perturbation theory:<ref>L.N.Lipatov, Zh.Eksp.Teor.Fiz. 72, 411 (1977) [Sov.Phys. JETP 45, 216 (1977)].</ref> One may now try to interpolate the known coefficients Template:Math with their large order behavior, and to then sum the perturbation series.
The first attempts of reconstruction of the Template:Math function by this method bear on the triviality of the Template:Math theory. Application of more advanced summation methods yielded the exponent Template:Mvar in the asymptotic behavior Template:Math, a value close to unity. The hypothesis for the asymptotic behavior of Template:Math was recently presented analytically for Template:Math theory and QED.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref><ref>Template:Cite journal</ref> Together with positiveness of Template:Math, 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 ).