Landau pole

In physics, the Landau pole is the energy scale (or the precise value of the energy) at which a coupling constant (the strength of an interaction) of a quantum field theory appears to become infinite. Such a possibility was pointed out by the physicist Lev Davidovich Landau. The dependence of coupling constants on the energy scale is one of the basic ideas behind the renormalization group.

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. This inconsistency appears only if the measured or renormalized coupling constant is nonzero, so the theory can be considered as suffering from quantum triviality

Since however the Landau pole is generally calculated using perturbative one-loop or two-loop calculations, it is possible that these calculations are incorrect 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^{[citation needed]}. Several calculations performed within the context of lattice gauge theory claim to show that the Landau pole is, in fact, only an artifact of the perturbative calculations.^{[citation needed]}

However, if they are true phenomena, and not artifacts of the perturbative calculation, Landau poles at high energy are viewed as problems. The Landau pole of QED can, in some cases, be removed if QED is embedded into a Grand Unified Theory. It has been speculated that these can be removed within superstring theory^{[citation needed]}.

An equation

In the 1950s, Landau started to research 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 naïvely insert into 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

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:

• 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 that has made some look for a better theory.^{[citation needed] See: [1]}

See also

Quantum triviality

References

1. ^ L. Landau, E. Lifshitz et L.Pitayevsky. Théorie Quantique Relativiste II Partie. Editions MIR Moscou 1973, p. 164.