|
|
| Line 1: |
Line 1: |
| {{About|the beta functions of theoretical physics|other beta functions|Beta function (disambiguation)}}
| | Amusement Center Manager Roman Sampley from Wingham, usually spends time with hobbies and interests for instance football, property developers in [http://www.epefania.com/new-property-cooling-measures-introduced/ singapore property agent] and bee keeping. Loves to see unfamiliar cities and places like Tabriz Historic Bazaar Complex. |
| | |
| {{quantum field theory}}
| |
| | |
| In [[theoretical physics]], specifically [[quantum field theory]], a '''beta function''', ''β(g)'', encodes the dependence of a [[Coupling constant|coupling parameter]], ''g'', on the [[energy scale]], ''μ'', of a given physical process described by [[quantum field theory]].
| |
| It is defined as
| |
| :: <math>\beta(g) = \frac{\partial g}{\partial \log(\mu)} ~,</math>
| |
| and, by dint of the underlying [[renormalization group]], it has no explicit dependence on ''μ'', so it only depends on ''μ'' implicitly through ''g''.
| |
| This dependence on the energy scale thus specified is known as the [[Coupling constant#Running coupling|running]] of the coupling parameter, a fundamental
| |
| feature of scale-dependence in quantum field theory, and its explicit computation is achievable through a variety of mathematical techniques.
| |
| | |
| ==Scale invariance==
| |
| If the beta functions of a quantum field theory vanish, usually at particular values of the coupling parameters, then the theory is said to be [[Scale invariance|scale-invariant]]. Almost all scale-invariant QFTs are also [[Conformal symmetry|conformally invariant]]. The study of such theories is [[conformal field theory]].
| |
| | |
| The coupling parameters of a quantum field theory can run even if the corresponding [[classical field theory]] is scale-invariant. In this case, the non-zero beta function tells us that the classical scale invariance is [[Conformal anomaly|anomalous]].
| |
| | |
| ==Examples==
| |
| Beta functions are usually computed in some kind of approximation scheme. An example is [[Perturbation theory (quantum mechanics)|perturbation theory]], where one assumes that the coupling parameters are small. One can then make an expansion in powers of the coupling parameters and truncate the higher-order terms (also known as higher [[Feynman graph|loop]] contributions, due to the number of loops in the corresponding [[Feynman graph]]s).
| |
| | |
| Here are some examples of beta functions computed in perturbation theory:
| |
| | |
| ===Quantum electrodynamics===
| |
| | |
| The one-loop beta function in [[quantum electrodynamics]] (QED) is
| |
| | |
| *<math>\beta(e)=\frac{e^3}{12\pi^2}~,</math>
| |
| | |
| or
| |
| | |
| *<math>\beta(\alpha)=\frac{2\alpha^2}{3\pi}~,</math>
| |
| | |
| written in terms of the [[fine structure constant]], ''α'' = ''e''<sup>2</sup>/4π .
| |
| | |
| This beta function tells us that the coupling increases with increasing energy scale, and QED becomes strongly coupled at high energy. In fact, the coupling apparently becomes infinite at some finite energy, resulting in a [[Landau pole]]. However, one cannot expect the perturbative beta function to give accurate results at strong coupling, and so it is likely that the Landau pole is an artifact of applying perturbation theory in a situation where it is no longer valid.
| |
| | |
| ===Quantum chromodynamics===
| |
| | |
| The one-loop beta function in [[quantum chromodynamics]] with <math>n_f</math> {{pad|.1em}}[[Flavour (particle physics)#Quantum chromodynamics|flavours]] is
| |
| | |
| *<math>\beta(g)=-\left(11-\frac{2n_f}{3}\right)\frac{g^3}{16\pi^2}~,</math>
| |
| | |
| or
| |
| | |
| *<math>\beta(\alpha_s)=-\left(11-\frac{2n_f}{3}\right)\frac{\alpha_s^2}{2\pi}~,</math>
| |
| | |
| written in terms of ''α<sub>s</sub>'' = <math>\frac{g^2}{4\pi}</math> .
| |
| | |
| If ''n''<sub>''f''</sub> ≤ 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.
| |
| | |
| ===SU(N) Non-Abelian gauge theory===
| |
| While the gauge group of QCD is <math>SU(3)</math>, we can generalize to any number of colors, <math>N_c</math>, with 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</math> of <math>G</math>, the one-loop beta function is
| |
| | |
| *<math>\beta(g)=-\left(\tfrac{11}{3}C_2(G)-\frac{4}{3}n_fC(R)\right)\frac{g^3}{16\pi^2}~,</math>
| |
| | |
| where <math>C_2(G)</math> is the [[Casimir invariant|quadratic Casimir]] of <math>G</math> and <math>C(R)</math> is another Casimir invariant defined by <math>Tr (T^a_RT^b_R) = C(R)\delta^{ab}</math> for generators <math>T^{a,b}_R</math> of the Lie algebra in the representation R. For gauge fields (''i.e.'' gluons) 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>C(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.
| |
| | |
| This famous result was derived nearly simultaneously in 1973 by [[H. David Politzer|Politzer]],<ref>
| |
| {{cite journal
| |
| | author=H.David Politzer
| |
| | year=1973
| |
| | title=Reliable Perturbative Results for Strong Interactions?
| |
| | journal=Phys. Rev. Lett.
| |
| | volume=30 | pages=1346–1349
| |
| | doi=10.1103/PhysRevLett.30.1346
| |
| | url=http://inspirehep.net/record/81351?ln=en|bibcode = 1973PhRvL..30.1346P }}</ref> [[David Gross|Gross]] and [[Frank Wilczek|Wilczek]],<ref>
| |
| {{cite journal
| |
| | author=D.J. Gross and F. Wilczek
| |
| | year=1973
| |
| | title=Asymptotically Free Gauge Theories. 1
| |
| | journal=Phys. Rev. D
| |
| | volume=8 | pages=3633–3652
| |
| | doi=10.1103/PhysRevD.8.3633
| |
| | url=http://inspirehep.net/record/81404 |bibcode = 1973PhRvD...8.3633G }}</ref> and [[Gerard 't Hooft|'t Hooft]],<ref>
| |
| {{cite journal
| |
| | author=G. 't Hooft
| |
| | year=1973
| |
| | title=
| |
| | postscript=. Unpublished.}}</ref> for which the first three were awarded the [[List of Nobel laureates in Physics|Nobel Prize in Physics]] in 2004.
| |
| | |
| === Minimal Supersymmetric Standard Model ===
| |
| {{Main|Minimal_Supersymmetric_Standard_Model#Gauge-Coupling Unification}}
| |
| | |
| ==See also==
| |
| *[[Callan-Symanzik equation]]
| |
| *[[Quantum triviality]]
| |
| *[[Banks–Zaks fixed point|Banks-Zaks fixed point]]
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| | |
| ==Further reading==
| |
| * Peskin, M and Schroeder, D.; ''An Introduction to Quantum Field Theory,'' Westview Press (1995). A standard introductory text, covering many topics in QFT including calculation of beta functions; see especially chapter 16.
| |
| * Weinberg, Steven; ''The Quantum Theory of Fields,'' (3 volumes) Cambridge University Press (1995). A monumental treatise on QFT.
| |
| * Zinn-Justin, Jean; ''Quantum Field Theory and Critical Phenomena,'' Oxford University Press (2002). Emphasis on the renormalization group and related topics.
| |
| | |
| ==External links==
| |
| | |
| [[Category:Quantum field theory]]
| |
| [[Category:Renormalization group]]
| |
| [[Category:Scaling symmetries]]
| |
Amusement Center Manager Roman Sampley from Wingham, usually spends time with hobbies and interests for instance football, property developers in singapore property agent and bee keeping. Loves to see unfamiliar cities and places like Tabriz Historic Bazaar Complex.