Pole Landau

The Landau pole (or “Moscow Zero”) in quantum field theory is a feature depending on the traveling coupling constant on the energy scale, which prevents the coupling constant from being renormalized beyond a certain finite energy (or scattering momentum). From a physical point of view, this means that on the scale of energy on which the Landau pole is observed, the theory from which the renormalization group equation was obtained ceases to be applicable, and some new theory is required.

Typical renormalization group equation in which the Landau pole arises

{\frac {d\lambda }{d\ln \epsilon }}=\beta (\lambda ),

{\ displaystyle {\ frac {d \ lambda} {d \ ln \ epsilon}} = \ beta (\ lambda),}

{\ displaystyle {\ frac {d \ lambda} {d \ ln \ epsilon}} = \ beta (\ lambda),}

where the beta function has the following form

\beta (\lambda )=a\lambda ^{2}.

{\ displaystyle \ beta (\ lambda) = a \ lambda ^ {2}.}

{\ displaystyle \ beta (\ lambda) = a \ lambda ^ {2}.}

The solution to this renormalization group equation

\lambda (\epsilon )={\frac {\lambda (\epsilon _{0})}{1-a\lambda (\epsilon _{0})\ln {\frac {\epsilon }{\epsilon _{0}}}}}.

{\ displaystyle \ lambda (\ epsilon) = {\ frac {\ lambda (\ epsilon _ {0})} {1-a \ lambda (\ epsilon _ {0}) \ ln {\ frac {\ epsilon} {\ epsilon _ {0}}}}}.}

{\ displaystyle \ lambda (\ epsilon) = {\ frac {\ lambda (\ epsilon _ {0})} {1-a \ lambda (\ epsilon _ {0}) \ ln {\ frac {\ epsilon} {\ epsilon _ {0}}}}}.}

Depending on the sign of the constant a, this solution is determined either for sufficiently low energies ( a > 0, for example, in quantum electrodynamics ), or for sufficiently large energies ( a <0, as in asymptotically free theories, such as quantum chromodynamics ). This solution has a pole at energy $\epsilon _{0}\exp {\frac {1}{a\lambda (\epsilon _{0})}}$ ${\ displaystyle \ epsilon _ {0} \ exp {\ frac {1} {a \ lambda (\ epsilon _ {0})}}}$ ${\ displaystyle \ epsilon _ {0} \ exp {\ frac {1} {a \ lambda (\ epsilon _ {0})}}}$ , this pole is called the Landau pole.

Pole Landau

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