Landau pole
Encyclopedia
In physics
, the Landau pole 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 co-workers
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
theory, i.e. 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.
Recent developments give a new insight into this problem.
, Khalatnikov
, the relation of the observable charge
with the “bare” charge
for renormalizable field theories is given by expression
where
is the mass of the particle, and 
is the momentum cut-off. For finite 
and 
the observed charge
tends to zero and the theory looks trivial.
In fact, the proper interpretation of Eq.1 consists in its inversion, so that
(related to the length scale 
is chosen to give a correct value of 
:
With growth of
the bare charge 
increases and turn to infinity at the point
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, the Landau pole is the momentum (or energy) scale at which the coupling constant
Coupling constant
In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...
(interaction strength) of a quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
becomes infinite. Such a possibility was pointed out by the physicist Lev Landau
Lev Landau
Lev Davidovich Landau was a prominent Soviet physicist who made fundamental contributions to many areas of theoretical physics...
and his co-workers
The fact that coupling constants depend on the momentum (or length) scale is one of the basic ideas behind the renormalization group
Renormalization group
In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...
.
Landau poles appear in theories that are not asymptotically free
Asymptotic freedom
In physics, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become arbitrarily weak at energy scales that become arbitrarily large, or, equivalently, at length scales that become arbitrarily small .Asymptotic freedom is a feature of quantum...
, such as quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
(QED) or
theory, i.e. a scalar fieldScalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...
with a quartic
Quartic function
In mathematics, a quartic function, or equation of the fourth degree, is a function of the formf=ax^4+bx^3+cx^2+dx+e \,where a is nonzero; or in other words, a polynomial of degree four...
interaction, such as may describe the Higgs boson
Higgs boson
The Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...
. 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
Vacuum polarization
In quantum field theory, and specifically quantum electrodynamics, vacuum polarization describes a process in which a background electromagnetic field produces virtual electron–positron pairs that change the distribution of charges and currents that generated the original electromagnetic...
). This is a case of quantum triviality
Quantum triviality
In a quantum field theory, charge screening can restrict the value of the observable "renormalized" charge of a classical theory. Ifthe only allowed value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting...
, 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
Perturbation theory (quantum mechanics)
In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an...
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
Lattice field theory
In physics, lattice field theory is the study of lattice models of quantum field theory, that is, of field theory on a spacetime that has been discretized onto a lattice. Although most lattice field theories are not exactly solvable, they are of tremendous appeal because they can be studied by...
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.
Recent developments give a new insight into this problem.
Brief history
According to Landau, AbrikosovAlexei Alexeyevich Abrikosov
Alexei Alexeyevich Abrikosov is a Soviet and Russian theoretical physicist whose main contributions are in the field of condensed matter physics. He was awarded the Nobel Prize in Physics in 2003.- Biography :...
, Khalatnikov
Isaak Markovich Khalatnikov
Isaak Markovich Khalatnikov is a leading Soviet physicist, well known for his role in developing the BKL conjecture in general relativity.Khalatnikov was born in Dnipropetrovsk and graduated from Dnipropetrovsk State University with a degree in Physics in 1941. He has been a member of the...
, the relation of the observable charge
with the “bare” charge
for renormalizable field theories is given by expression
where
is the mass of the particle, and is the momentum cut-off. For finite and the observed charge
tends to zero and the theory looks trivial.In fact, the proper interpretation of Eq.1 consists in its inversion, so that
(related to the length scale is chosen to give a correct value of :
With growth of
the bare charge increases and turn to infinity at the point-
The latter singularity is the Landau pole. In fact, the growth of
invalidates Eqs.1,2 in the region
(since they were obtained for 
) and reality of the Landau pole becomes doubtful.
The actual behavior of the charge
as a function of the
momentum scale
is determined by the Gell-MannMurray Gell-MannMurray Gell-Mann is an American physicist and linguist who received the 1969 Nobel Prize in physics for his work on the theory of elementary particles...
–LowFrancis E. LowFrancis Eugene Low was an American theoretical physicist. He was an Institute Professor at MIT, and served as provost there from 1980 to 1985.-Early career:...
equation
which gives Eqs.1,2 if it is integrated under conditions
for
and 
for
, when only the term with 
is
retained in the right hand side. The general behavior of
depends on the appearance of the function 
. According to classification by Bogoliubov and ShirkovDmitry ShirkovDmitry Vasil'evich Shirkov is a Russian theoretical physicist known for his contribution to quantum field theory and to the development of the renormalization group method.-Biography:...
, there are three qualitatively different situations:
(a) if
has a zero at the finite value 
, then growth of 
is saturated, i.e. 
for
;
(b) if
is non-alternating and behaves as
with 
for large
, then the growth of 
continues to infinity;
(c) if
with 
for large
, then 
is divergent at finite value 
and the real Landau pole arises: the theory is internally inconsistent due to
indeterminacy of
for 
.
Landau and PomeranchukIsaak PomeranchukIsaak Yakovlevich Pomeranchuk was a Soviet physicist, who was the founder and first head of the theory division at ITEP. The particle pomeron is named in his honour. For his work, Pomeranchuk was twice awarded Stalin Prize .-External links:**...
tried to justify the possibility (c)
in the case of QED and
theory. They have noted that the growth of 
in Eq.1 drives the observable charge
to the constant limit, which does not depend on
. 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
, it is all the more
valid for
of the order or greater than unity : it gives a reason to consider Eq.1 to be valid for arbitrary 
.
Validity of these considerations on the quantitative level is
excluded by non-quadratic form of the
-function.
Nevertheless, they can be correct qualitatively.
Indeed, the result
can be obtained
from the functional integrals only for
, while its validity for 
, based on Eq.1, may be related with other reasons; for 
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 CarloMonte Carlo methodMonte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
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 trivialityQuantum trivialityIn a quantum field theory, charge screening can restrict the value of the observable "renormalized" charge of a classical theory. Ifthe only allowed value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting...
in full theory (beyond its perturbation context), as can be seen by a reductio ad absurdumReductio ad absurdumIn logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...
. Indeed, if
is finite, the theory is internally inconsistent. The only way to avoid it, is to tend 
to infinity, which is possible only for 
.
It is a widespread opinion, that QED and
theory are
trivial in the continuum limit. In fact, available information
confirms only “Wilson triviality”, which is equivalent to
positiveness of
for 
and can be considered as firmly established. Indications of “true” quantum triviality are not numerous and allow different interpretation.
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 the Landau pole could be a sign of new physics entering via its 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
in Eqs.1,2 is played by the fine structure constant α ≈ 1/137 and the Landau scale for QED is estimated as 10283keV/c2, which is far beyond any energy scale relevant to observable physics. For comparison, the maximum energies accessible at the Large Hadron ColliderLarge Hadron ColliderThe Large Hadron Collider is the world's largest and highest-energy particle accelerator. It is expected to address some of the most fundamental questions of physics, advancing the understanding of the deepest laws of nature....
are of order 1013 eV, while the Planck scalePlanck scaleIn particle physics and physical cosmology, the Planck scale is an energy scale around 1.22 × 1019 GeV at which quantum effects of gravity become strong...
, at which quantum gravityQuantum gravityQuantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
becomes important and the relevance of quantum field theoryQuantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
itself may be questioned, is only 1028 eV.
The Higgs bosonHiggs bosonThe Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...
in the Standard ModelStandard ModelThe Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
of particle physicsParticle physicsParticle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
is described by
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 needs calculation of the Gell-Mann–Low function
at arbitrary 
and, in particular, its asymptotic
behavior for
. This problem is very difficult and
was considered as hopeless for many years: by diagrammatic calculations one can obtain only few expansion coefficients
, which do not allow to investigate the 
function in the whole. The progress became possible
after development of the LipatovLev LipatovLev Nikolaevich Lipatov is a Russian physicist, well known for his contributions to nuclear physics. He is the head of Theoretical Physics Division at St. Petersburg's Nuclear Physics Institute of Russian Academy of Sciences in Gatchina...
method for calculation of large orders
of perturbation theory : now one can try to interpolate the known
coefficients
with their large order behavior and
to sum the perturbation series. The first attempts of reconstruction of the
function witnessed on triviality of 
theory. Application of more advanced summation methods gave the exponent
in the asymptotic behavior 
a value close to unity. The hypothesis for the asymptotics 
was recently confirmed analytically for 
theory and QED
.
Together with positiveness of
, 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.
-
