• Source: Beta function (physics)

In theoretical physics, specifically quantum field theory, a beta function, β(g), encodes the dependence of a coupling parameter, g, on the energy scale, μ, of a given physical process described by quantum field theory.
It is defined as




β
(
g
)
=
μ



∂
g


∂
μ



=



∂
g


∂
ln
⁡
(
μ
)




,


{\displaystyle \beta (g)=\mu {\frac {\partial g}{\partial \mu }}={\frac {\partial g}{\partial \ln(\mu )}}~,}


and, because 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 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. The concept of Beta function was Introduced by Ernst Stueckelberg and André Petermann in 1953.




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-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.




Examples


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:




= Quantum electrodynamics

=

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




β
(
e
)
=



e

3



12

π

2






,


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


or, equivalently,




β
(
α
)
=



2

α

2




3
π




,


{\displaystyle \beta (\alpha )={\frac {2\alpha ^{2}}{3\pi }}~,}


written in terms of the fine structure constant in natural units, α = e2/4π.
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




n

f




{\displaystyle n_{f}}

flavours and




n

s




{\displaystyle n_{s}}

scalar colored bosons is




β
(
g
)
=
−

(

11
−



n

s


6


−



2

n

f



3



)




g

3



16

π

2






,


{\displaystyle \beta (g)=-\left(11-{\frac {n_{s}}{6}}-{\frac {2n_{f}}{3}}\right){\frac {g^{3}}{16\pi ^{2}}}~,}


or




β
(

α

s


)
=
−

(

11
−



n

s


6


−



2

n

f



3



)




α

s


2



2
π




,


{\displaystyle \beta (\alpha _{s})=-\left(11-{\frac {n_{s}}{6}}-{\frac {2n_{f}}{3}}\right){\frac {\alpha _{s}^{2}}{2\pi }}~,}


written in terms of αs =




g

2



/

4
π


{\displaystyle g^{2}/4\pi }

.
Assuming ns=0, if nf ≤ 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 (Yang–Mills) gauge group of QCD is



S
U
(
3
)


{\displaystyle SU(3)}

, and determines 3 colors, we can generalize to any number of colors,




N

c




{\displaystyle N_{c}}

, with a gauge group



G
=
S
U
(

N

c


)


{\displaystyle G=SU(N_{c})}

. Then for this gauge group, with Dirac fermions in a representation




R

f




{\displaystyle R_{f}}

of



G


{\displaystyle G}

and with complex scalars in a representation




R

s




{\displaystyle R_{s}}

, the one-loop beta function is




β
(
g
)
=
−

(



11
3



C

2


(
G
)
−


1
3



n

s


T
(

R

s


)
−


4
3



n

f


T
(

R

f


)

)




g

3



16

π

2






,


{\displaystyle \beta (g)=-\left({\frac {11}{3}}C_{2}(G)-{\frac {1}{3}}n_{s}T(R_{s})-{\frac {4}{3}}n_{f}T(R_{f})\right){\frac {g^{3}}{16\pi ^{2}}}~,}


where




C

2


(
G
)


{\displaystyle C_{2}(G)}

is the quadratic Casimir of



G


{\displaystyle G}

and



T
(
R
)


{\displaystyle T(R)}

is another Casimir invariant defined by



T
r
(

T

R


a



T

R


b


)
=
T
(
R
)

δ

a
b




{\displaystyle Tr(T_{R}^{a}T_{R}^{b})=T(R)\delta ^{ab}}

for generators




T

R


a
,
b




{\displaystyle T_{R}^{a,b}}

of the Lie algebra in the representation R. (For Weyl or Majorana fermions, replace



4

/

3


{\displaystyle 4/3}

by



2

/

3


{\displaystyle 2/3}

, and for real scalars, replace



1

/

3


{\displaystyle 1/3}

by



1

/

6


{\displaystyle 1/6}

.) For gauge fields (i.e. gluons), necessarily in the adjoint of



G


{\displaystyle G}

,




C

2


(
G
)
=

N

c




{\displaystyle C_{2}(G)=N_{c}}

; for fermions in the fundamental (or anti-fundamental) representation of



G


{\displaystyle G}

,



T
(
R
)
=
1

/

2


{\displaystyle T(R)=1/2}

. Then for QCD, with




N

c


=
3


{\displaystyle 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 Politzer, Gross and Wilczek, for which the three were awarded the Nobel Prize in Physics in 2004.
Unbeknownst to these authors, G. 't Hooft had announced the result in a comment following a talk by K. Symanzik at a small meeting in Marseilles in June 1972, but he never published it.




= Standard Model Higgs–Yukawa Couplings

=

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:




μ


∂

∂
μ



y
≈


y

16

π

2






(



9
2



y

2


−
8

g

3


2



)



{\displaystyle \mu {\frac {\partial }{\partial \mu }}y\approx {\frac {y}{16\pi ^{2}}}\left({\frac {9}{2}}y^{2}-8g_{3}^{2}\right)}

,
where




g

3




{\displaystyle g_{3}}

is the color gauge coupling (which is a function of



μ


{\displaystyle \mu }

and associated with asymptotic freedom) and



y


{\displaystyle y}

is the Yukawa coupling. This equation describes how the Yukawa coupling changes with energy scale



μ


{\displaystyle \mu }

.
The Yukawa couplings of the up, down, charm, strange and bottom quarks, are small at the extremely high energy scale of grand unification,



μ
≈

10

15




{\displaystyle \mu \approx 10^{15}}

GeV. Therefore, the




y

2




{\displaystyle y^{2}}

term can be neglected in the above equation. Solving, we then find that



y


{\displaystyle y}

is increased slightly at the low energy scales at which the quark masses are generated by the Higgs,



μ
≈
100


{\displaystyle \mu \approx 100}

GeV.
On the other hand, solutions to this equation for large initial values



y


{\displaystyle y}

cause the rhs to quickly approach smaller values as we descend in energy scale. The above equation then locks



y


{\displaystyle y}

to the QCD coupling




g

3




{\displaystyle g_{3}}

. This is known as the (infrared) quasi-fixed point of the renormalization group equation for the Yukawa coupling. 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.




= Minimal Supersymmetric Standard Model

=

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.




See also


Banks–Zaks fixed point
Callan–Symanzik equation
Quantum triviality




References






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.

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