Beta function (physics): Difference between revisions

Revision as of 17:15, 28 October 2023

Revision as of 10:52, 13 December 2023

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The one-loop beta function in [[quantum electrodynamics]] (QED) is

*<math>\beta(e)=\frac{e^32}{12\pi^2}~,</math>

or, equivalently,

*<math>\beta(\alpha)=\frac{2\alpha^2}{3\pi}~,</math>

written in terms of the [[Fine-structure constant#In non-SI units|fine structure constant]] in natural units, {{math|''α'' {{=}} ''e''²/4π}}.

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{{Main|Quantum chromodynamics}}

The one-loop beta function in [[quantum chromodynamics]] with <math>n_f</math> [[Flavour (particle physics)#Quantum chromodynamics|flavours]] and <math>n_s</math> scalar colored bosons is

:<math>\beta(g)=-\left(11- \frac{n_s}{6} - \frac{2n_f}{3}\right)\frac{g^32}{16\pi^2}~,</math>

or

:<math>\beta(\alpha_s)=-\left(11- \frac{n_s}{6}-\frac{2n_f}{3}\right)\frac{\alpha_s^2}{24\pi} ~,</math>

written in terms of ''α<submath>s</sub>'' \alpha_s= <math>g^2/4\pi</math> .

If ''n''_''f'' ≤ 16, the ensuing beta function dictates that the coupling decreases with increasing energy scale, a phenomenon known as [[asymptotic freedom]]. Conversely, the coupling increases with decreasing energy scale. This means that the coupling becomes large at low energies, and one can no longer rely on perturbation theory.

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===SU(N) Non-Abelian gauge theory===

While the (Yang–Mills) gauge group of QCD is <math>SU(3)</math>, and determines 3 colors, we can generalize to any number of colors, <math>N_c</math>, with a gauge group <math>G=SU(N_c)</math>. Then for this gauge group, with Dirac fermions in a [[Representations of Lie groups|representation]] <math>R_f</math> of <math>G</math> and with complex scalars in a representation <math>R_s</math>, the one-loop beta function is

:<math>\beta(g)=-\left(\frac{11}{3}C_2(G)-\frac{1}{3}n_sT(R_s)-\frac{4}{3}n_f T(R_f)\right)\frac{g^32}{16\pi^2}~,</math>

where <math>C_2(G)</math> is the [[Casimir invariant|quadratic Casimir]] of <math>G</math> and <math>T(R)</math> is another Casimir invariant defined by <math>Tr (T^a_RT^b_R) = T(R)\delta^{ab}</math> for generators <math>T^{a,b}_R</math> of the Lie algebra in the representation R. (For [[Weyl]] or [[Majorana fermions]], replace <math>4/3</math> by <math>2/3</math>, and for real scalars, replace <math>1/3</math> by <math>1/6</math>.) For gauge fields (''i.e.'' gluons), necessarily in the [[Adjoint representation of a Lie group|adjoint]] of <math>G</math>, <math>C_2(G) = N_c</math>; for fermions in the [[Fundamental representation|fundamental]] (or anti-fundamental) representation of <math>G</math>, <math>T(R) = 1/2</math>. Then for QCD, with <math>N_c = 3</math>, the above equation reduces to that listed for the quantum chromodynamics beta function.