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
)
=
∂
g
∂
log
⁡
(
μ
)
,
{\displaystyle \beta (g)={\frac {\partial
g}{\partial \log(\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.
== 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 Higgs bosons is
β
(
g
)
=
−
(
11
−
n
s
3
−
2
n
f
3
)
g
3
16
π
2
,
{\displaystyle \beta (g)=-\left(11-{\frac
{n_{s}}{3}}-{\frac {2n_{f}}{3}}\right){\frac
{g^{3}}{16\pi ^{2}}}~,}
or
β
(
α
s
)
=
−
(
11
−
n
s
3
−
2
n
f
3
)
α
s
2
2
π
,
{\displaystyle \beta (\alpha _{s})=-\left(11-{\frac
{n_{s}}{3}}-{\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 }
.
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.
=== Minimal Supersymmetric Standard Model
===
== See also ==
Callan-Symanzik equation
Quantum triviality
Banks-Zaks fixed point
