In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, specifically
quantum field theory, a beta function, ''β(g)'', encodes the dependence of a
coupling parameter, ''g'', on the
energy scale
In physics, length scale is a particular length or distance determined with the precision of at most a few orders of magnitude. The concept of length scale is particularly important because physical phenomena of different length scales cannot aff ...
, ''μ'', of a given physical process described by
quantum field theory.
It is defined as
::
and, because of the underlying
renormalization group
In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the ...
, 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
Running is a method of terrestrial locomotion allowing humans and other animals to move rapidly on foot. Running is a type of gait characterized by an aerial phase in which all feet are above the ground (though there are exceptions). This is ...
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
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versio ...
. The study of such theories is
conformal field theory
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometime ...
.
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
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...
, 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 graph
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduce ...
s).
Here are some examples of beta functions computed in perturbation theory:
Quantum electrodynamics
The one-loop beta function in
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
(QED) is
*
or, equivalently,
*
written in terms of the
fine structure constant in natural units, .
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
In physics, the Landau pole (or the Moscow zero, or the Landau ghost) is the momentum (or energy) scale at which the coupling constant (interaction strength) of a quantum field theory becomes infinite. Such a possibility was pointed out by the phy ...
. 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
flavours and
scalar colored bosons is
:
or
:
written in terms of ''α
s'' =
.
If ''n''
''f'' ≤ 16, the ensuing beta function dictates that the coupling decreases with increasing energy scale, a phenomenon known as
asymptotic freedom
In quantum field theory, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become asymptotically weaker as the energy scale increases and the corresponding length scale decreases.
Asymptotic fre ...
. 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
, and determines 3 colors, we can generalize to any number of colors,
, with a gauge group
. Then for this gauge group, with Dirac fermions in a
representation of
and with complex scalars in a representation
, the one-loop beta function is
:
where
is the
quadratic Casimir of
and
is another Casimir invariant defined by
for generators
of the Lie algebra in the representation R. (For
Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is ass ...
or
Majorana fermions, replace
by
, and for real scalars, replace
by
.) For gauge fields (''i.e.'' gluons), necessarily in the
adjoint
In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type
:(''Ax'', ''y'') = (''x'', ''By'').
Specifically, adjoin ...
of
,
; for fermions in the
fundamental
Fundamental may refer to:
* Foundation of reality
* Fundamental frequency, as in music or phonetics, often referred to as simply a "fundamental"
* Fundamentalism, the belief in, and usually the strict adherence to, the simple or "fundamental" idea ...
(or anti-fundamental) representation of
,
. Then for QCD, with
, 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
)
, image = Nobel Prize.png
, alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then " ...
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
The top quark, sometimes also referred to as the truth quark, (symbol: t) is the most massive of all observed elementary particles. It derives its mass from its coupling to the Higgs Boson. This coupling y_ is very close to unity; in the Standard ...
's Yukawa coupling. These Yukawa couplings change their values depending on the energy scale at which they are measured, through ''
running
Running is a method of terrestrial locomotion allowing humans and other animals to move rapidly on foot. Running is a type of gait characterized by an aerial phase in which all feet are above the ground (though there are exceptions). This is ...
''. The dynamics of Yukawa couplings of quarks are determined by the
renormalization group equation:
,
where
is the
color
Color (American English) or colour (British English) is the visual perceptual property deriving from the spectrum of light interacting with the photoreceptor cells of the eyes. Color categories and physical specifications of color are assoc ...
gauge
Gauge ( or ) may refer to:
Measurement
* Gauge (instrument), any of a variety of measuring instruments
* Gauge (firearms)
* Wire gauge, a measure of the size of a wire
** American wire gauge, a common measure of nonferrous wire diameter, ...
coupling (which is a function of
and associated with
asymptotic freedom
In quantum field theory, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become asymptotically weaker as the energy scale increases and the corresponding length scale decreases.
Asymptotic fre ...
) and
is the Yukawa coupling. This equation describes how the Yukawa coupling changes with energy scale
.
The Yukawa couplings of the up, down, charm, strange and bottom quarks, are small at the extremely high energy scale of
grand unification,
GeV. Therefore, the
term can be neglected in the above equation. Solving, we then find that
is increased slightly at the low energy scales at which the quark masses are generated by the Higgs,
GeV.
On the other hand, solutions to this equation for large initial values
cause the ''rhs'' to quickly approach smaller values as we descend in energy scale. The above equation then locks
to the QCD coupling
. 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.
The value of the quasi-fixed point is fairly precisely determined in the Standard Model, leading to a predicted
top quark
The top quark, sometimes also referred to as the truth quark, (symbol: t) is the most massive of all observed elementary particles. It derives its mass from its coupling to the Higgs Boson. This coupling y_ is very close to unity; in the Standard ...
mass of 230 GeV. The observed top quark mass of 174 GeV is slightly lower than the standard model prediction by about 30% which suggests there may be more Higgs doublets beyond the single standard model Higgs boson.
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
In physics, the Callan–Symanzik equation is a differential equation describing the evolution of the Correlation function (quantum field theory), ''n''-point correlation functions under variation of the energy scale at which the theory is define ...
*
Quantum triviality
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.
Renormalization group
Scaling symmetries