Beta function (physics)

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-invariant. Almost all scale-invariant QFTs are also 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 anomalous.

Beta functions are usually computed in some kind of approximation scheme. An example is 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 loop contributions, due to the number of loops in the corresponding Feynman graphs).

Here are some examples of beta functions computed in perturbation theory:

{{Main|Quantum electrodynamics}}

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

• $\backslash beta(e)=\backslash frac\{e^3\}\{12\backslash pi^2\}~,$

or, equivalently,

• $\backslash beta(\backslash alpha)=\backslash frac\{2\backslash alpha^2\}\{3\backslash pi\}~,$

written in terms of the fine structure constant in natural units, {{math|α {{=}} e²/4π}}.{{cite book |last1=Srednicki |first1=Mark Allen |title=Quantum field theory |date=2017 |publisher=Cambridge Univ. Press |location=Cambridge |isbn=978-0-521-86449-7 |page=446 |edition=13th printing}}

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and so it is likely that the Landau pole is an artifact of applying perturbation theory in a situation where it is no longer valid.

{{Main|Quantum chromodynamics}}

The one-loop beta function in quantum chromodynamics with $n\_f$ flavours and $n\_s$ scalar colored bosons is

:$\backslash beta(g)=-\backslash left(11-\; \backslash frac\{n\_s\}\{6\}\; -\; \backslash frac\{2n\_f\}\{3\}\backslash right)\backslash frac\{g^3\}\{16\backslash pi^2\}~,$

or

:$\backslash beta(\backslash alpha\_s)=-\backslash left(11-\; \backslash frac\{n\_s\}\{6\}-\backslash frac\{2n\_f\}\{3\}\backslash right)\backslash frac\{\backslash alpha\_s^2\}\{2\backslash pi\}~,$

written in terms of α_s = $g^2/4\backslash pi$ .

Assuming n_s=0, 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.

While the (Yang–Mills) gauge group of QCD is $\backslash mathrm\{SU\}(3)$, and determines 3 colors, we can generalize to any number of colors, $N\_c$, with a gauge group $G=\backslash mathrm\{SU\}(N\_c)$. Then for this gauge group, with Dirac fermions in a representation $R\_f$ of $G$ and with complex scalars in a representation $R\_s$, the one-loop beta function is

:$\backslash beta(g)=-\backslash left(\backslash frac\{11\}\{3\}C\_2(G)-\backslash frac\{1\}\{3\}n\_sT(R\_s)-\backslash frac\{4\}\{3\}n\_f\; T(R\_f)\backslash right)\backslash frac\{g^3\}\{16\backslash pi^2\}~,$

where $C\_2(G)$ is the quadratic Casimir of $G$ and $T(R)$ is another Casimir invariant defined by $Tr\; (T^a\_RT^b\_R)\; =\; T(R)\backslash delta^\{ab\}$ for generators $T^\{a,b\}\_R$ of the Lie algebra in the representation R. (For Weyl or Majorana fermions, replace $4/3$ by $2/3$, and for real scalars, replace $1/3$ by $1/6$.) For gauge fields (i.e. gluons), necessarily in the adjoint of $G$, $C\_2(G)\; =\; N\_c$; for fermions in the fundamental (or anti-fundamental) representation of $G$, $T(R)\; =\; 1/2$. Then for QCD, with $N\_c\; =\; 3$, the above equation reduces to that listed for the quantum chromodynamics beta function.

This famous result was derived nearly simultaneously in 1973 by Hugh David Politzer,

{{cite journal

| author=H.David Politzer

| year=1973

| title=Reliable Perturbative Results for Strong Interactions?

| journal=Phys. Rev. Lett.

| volume=30 | issue=26

| pages=1346–1349

| doi=10.1103/PhysRevLett.30.1346

| url=http://inspirehep.net/record/81351?ln=en|bibcode = 1973PhRvL..30.1346P | doi-access=free

}} David Gross and Frank Wilczek,

{{cite journal

| author=D.J. Gross and F. Wilczek

| year=1973

| title=Asymptotically Free Gauge Theories. 1

| journal=Phys. Rev. D

| volume=8 | issue=10

| pages=3633–3652

| doi=10.1103/PhysRevD.8.3633

| url=http://inspirehep.net/record/81404 |bibcode = 1973PhRvD...8.3633G | doi-access=free

}}. for which the three were awarded the Nobel Prize in Physics in 2004.

Unbeknownst to these authors, Gerard 't Hooft had announced the result in a comment following a talk by Kurt Symanzik at a small meeting in Marseille in June 1972, but he never published it.

{{cite journal

| author=G. 't Hooft

| year=1999

| title= When was Asymptotic Freedom discovered?

| journal=Nucl. Phys. B Proc. Suppl.

| volume=74 | issue=1

| pages=413–425

| doi=10.1016/S0920-5632(99)00207-8

|arxiv = hep-th/9808154 |bibcode = 1999NuPhS..74..413T | s2cid=17360560

}}

{{Main|Infrared fixed point}}

In the Standard Model, quarks and leptons have Yukawa couplings to the Higgs boson. These determine the mass of the particle. Most all of the quarks' and leptons' Yukawa couplings are small compared to the top quark's Yukawa coupling. These Yukawa couplings change their values depending on the energy scale at which they are measured, through running. The dynamics of Yukawa couplings of quarks are determined by the renormalization group equation:

$\backslash mu\; \backslash frac\{\backslash partial\}\{\backslash partial\backslash mu\}\; y\; \backslash approx\; \backslash frac\{y\}\{16\backslash pi^2\}\backslash left(\backslash frac\{9\}\{2\}y^2\; -\; 8\; g\_3^2\backslash right)$,

where $g\_3$ is the color gauge coupling (which is a function of $\backslash mu$ and associated with asymptotic freedom) and $y$ is the Yukawa coupling. This equation describes how the Yukawa coupling changes with energy scale $\backslash mu$.

The Yukawa couplings of the up, down, charm, strange and bottom quarks, are small at the extremely high energy scale of grand unification, $\backslash mu\; \backslash approx\; 10^\{15\}$ GeV. Therefore, the $y^2$ term can be neglected in the above equation. Solving, we then find that $y$ is increased slightly at the low energy scales at which the quark masses are generated by the Higgs, $\backslash mu\; \backslash approx\; 100$ GeV.

On the other hand, solutions to this equation for large initial values $y$ cause the rhs to quickly approach smaller values as we descend in energy scale. The above equation then locks $y$ to the QCD coupling $g\_3$. This is known as the (infrared) quasi-fixed point of the renormalization group equation for the Yukawa coupling.{{cite journal|last1=Pendleton|first1=B.|last2=Ross|first2=G.G.|title=Mass and Mixing Angle Predictions from Infrared Fixed points|journal=Phys. Lett.|date=1981|volume=B98|issue=4 |page=291|doi=10.1016/0370-2693(81)90017-4|bibcode = 1981PhLB...98..291P }}{{cite journal|last1=Hill|first1=C.T.|title=Quark and Lepton masses from Renormalization group fixed points|journal=Phys. Rev.|date=1981|volume=D24|issue=3 |page=691|doi=10.1103/PhysRevD.24.691|bibcode = 1981PhRvD..24..691H }} No matter what the initial starting value of the coupling is, if it is sufficiently large it will reach this quasi-fixed point value, and the corresponding quark mass is predicted.

{{Main|Minimal Supersymmetric Standard Model#Gauge-Coupling Unification}}

Renomalization group studies in the minimal supersymmetric Standard Model (MSSM) of grand unification and the Higgs–Yukawa fixed points were very encouraging that the theory was on the right track. So far, however, no evidence of the predicted MSSM particles has emerged in experiment at the Large Hadron Collider.

• Banks–Zaks fixed point
• Callan–Symanzik equation
• Quantum triviality

{{Reflist}}

• 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.

{{Authority control}}

Category:Renormalization group

Category:Scaling symmetries