In physics, Landau pole is the energy scale (or the precise value of the energy) where a coupling constant (the strength of an interaction) of a quantum field theory becomes infinite. Such a possibility was pointed out by the eminent physicist Lev Davidovich Landau. The dependence of coupling constants on the energy scale is one of the basic ideas behind the renormalization group. A Superconductor demonstrating the Meissner Effect. ... In physics, energy scale is a particular value of energy determined with the precision of one order (or a few orders) of magnitude. ... In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... The word infinity comes from the Latin infinitas or unboundedness. It refers to several distinct concepts which arise in theology, philosophy, mathematics and everyday life. ... Lev Davidovich Landau (Ле́в Дави́дович Ланда́у) (January 22, 1908 – April 1, 1968) was a prominent Soviet physicist and winner of the Nobel Prize for Physics whose broad field of work included the theory of superconductivity and superfluidity, quantum electrodynamics, nuclear physics and particle physics. ... In physics, energy scale is a particular value of energy determined with the precision of one order (or a few orders) of magnitude. ... In theoretical physics, renormalization group (RG) refers to a set of techniques and concepts related to the change of physics with the observation scale. ...

Theories with asymptotic freedom have Landau poles at very low energies. However, the phrase "Landau pole" is usually used in the context of the theories that are not asymptotically free, such as quantum electrodynamics (QED) or a scalar field with a quartic interaction. The coupling constant grows with energy, and at some energy scale the growth becomes infinite and the coupling constant itself diverges. In physics, asymptotic freedom is the property of some gauge theories in which the interaction between the particles, such as quarks, becomes arbitrarily weak at ever shorter distances, i. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electromagnetism. ... In mathematics and physics, a scalar field associates a scalar to every point in space. ... In mathematics and elsewhere, the adjective quartic means fourth order, such as the function . ...

Landau poles at high energy are viewed as problems; more precisely, they are evidence that the theory (e.g. QED) is not well-defined nonperturbatively. The Landau pole of QED is removed if QED is embedded into a Grand Unified Theory or an even more powerful framework such as superstring theory. In quantum mechanics, perturbation theory is a set of approximation schemes for describing a complicated quantum system in terms of a simpler one. ... Grand unification, grand unified theory, or GUT is a theory in physics that unifies the strong interaction and electroweak interaction. ... Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings. ...

An equation

Everything started in the 1950s when Landau decided to understand the relation between the bare electric charge e and the renormalized electric charge e_R. He found the following equation:

This equation needs to be explained:

• e is the value of the electric charge that we naively insert to the Lagrangian, but it turns out that this number is actually not a constant, but rather an energy-dependent quantity
• e_R is the actual renormalized, measurable value of the charge (that determines how much the electrons attract each other at low energies), which is not quite the same thing as e
• N_f is the number of flavors; for "staggered" fermions we substitute N_f = 4
• Λ is the momentum cutoff i.e. the maximal value of the momentum that we allow to be taken into account
• m_R is the renormalized electron mass

Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interactions. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... Figure 1. ... In particle physics, flavor is a property of a fermion that identifies it, a label that specifies the name of the particle. ... In particle physics, fermions, (named after Enrico Fermi), are particles with semi-integer spin. ... In theoretical physics, cutoff usually represents a particular energy scale or length scale. ... Properties The electron is a lightweight fundamental subatomic particle that carries a negative electric charge. ... 1-loop vacuum polarization diagram This is the one loop contribution to the photon propagator Π due to vacuum polarization in QED. It causes a wave function renormalization for the photon, leading to a charge screening. ...

The right-hand side can be calculated from loops in Feynman diagrams (namely one-loop Feynman diagrams), i.e. as a contribution of quantum mechanics. It has a logarithmic form because the integral happens to be logarithmically divergent. Note that the equation has two obvious implications: In this Feynman diagram, electrons annihilate and become a quark-antiquark pair. ... In physics, a one-loop Feynman diagram is a connected Feynman diagram with only one cycle (unicyclic). ... A simple introduction to this subject is provided in Basics of quantum mechanics. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... In calculus, the integral of a function is a generalization of area, mass, volume and total. ...

• If the bare charge e is kept fixed, the theory (QED) has a trivial continuum () limit, namely
• When the renormalized charge e_R is kept fixed, the bare charge becomes singular (infinite) at

.

The latter singularity is the Landau pole. It does not affect the phenomenological success of perturbative calculations in QED because for all practical purposes, the cutoff Λ can be chosen much smaller than the huge scale Λ_Landau, comparable to the Planck scale, and it is still enough to describe all accessible experiments. Nevertheless, the Landau pole is an awkward theoretical feature of QED, a sufficiently awkward one to make us look for a better theory. Look up Phenomenology in Wiktionary, the free dictionary. ... In quantum mechanics, perturbation theory is a set of approximation schemes for describing a complicated quantum system in terms of a simpler one. ... In physics, Planck units are physical units of measurement originally proposed by Max Planck. ...

Since the Landau pole is calculated using one-loop or two-loop calculations but perturbation theory breaks down for large coupling constants, there has been some controversy as to whether the Landau pole really exists or if we have a UV fixed point instead. Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. ... A theory has a UV fixed point if its renormalization group flow runs towards a fixed point in the UV limit. ...