Chiral perturbation theory (ChPT) is an effective
field theory constructed with a Lagrangian
consistent with the (approximate) chiral symmetry
of quantum chromodynamics (QCD), as well as
the other symmetries of parity and charge
conjugation. ChPT is a theory which allows
one to study the low-energy dynamics of QCD.
== Goals ==
In the low-energy regime of QCD, the degrees
of freedom are no longer quarks and gluons,
but rather hadrons. This is a result of confinement.
If one could "solve" the QCD partition function,
(such that the degrees of freedom in the Lagrangian
are replaced by hadrons) then one could extract
information about low-energy physics. To date
this has not been accomplished.
Because QCD becomes non-perturbative at low
energy, it is impossible to use perturbative
methods to extract information from the partition
function of QCD. Lattice QCD is an alternative
method that has proved successful in extracting
non-perturbative information.
== Method ==
According to Steven Weinberg, an effective
theory can be useful if one writes down all
terms consistent with the symmetries of the
parent theory. In general there are an infinite
number of terms which meet this requirement.
Therefore in order to make any physical predictions,
one assigns to the theory a power-ordering
scheme which organizes terms by some pre-determined
degree of importance. The ordering allows
one to keep some terms and omit all other,
higher-order corrections which can be safely,
temporarily ignored.
There are several power counting schemes in
ChPT. The most widely used one is the
p
{\displaystyle p}
-expansion. However, there also exist the
ϵ
{\displaystyle \epsilon }
,
δ
,
{\displaystyle \delta ,}
and
ϵ
′
{\displaystyle \epsilon ^{\prime }}
expansions. All of these expansions are valid
in finite volume, (though the
p
{\displaystyle p}
expansion is the only one valid in infinite
volume.) Particular choices of finite volumes
require one to use different reorganizations
of the chiral theory in order to correctly
understand the physics. These different reorganizations
correspond to the different power counting
schemes.
In addition to the ordering scheme, most terms
in the approximate Lagrangian will be multiplied
by coupling constants which represent the
relative strengths of the force represented
by each term. Values of these constants – also
called low-energy constants or LECs – are
usually not known. The constants can be determined
by fitting to experimental data or be derived
from underlying theory.
=== The model Lagrangian ===
The Lagrangian of the p-expansion is constructed
by writing down all interactions which are
not excluded by symmetry, and then ordering
them based on the number of momentum and mass
powers.
The order is chosen so that
(
∂
π
)
2
+
m
π
2
π
2
{\displaystyle (\partial \pi )^{2}+m_{\pi
}^{2}\pi ^{2}}
is considered in the first-order approximation,
where
π
{\displaystyle \pi }
is the pion field and
m
π
{\displaystyle m_{\pi }}
the pion mass. Terms like
m
π
4
π
2
+
(
∂
π
)
6
{\displaystyle m_{\pi }^{4}\pi ^{2}+(\partial
\pi )^{6}}
are part of other, higher order corrections.
It is also common to compress the Lagrangian
by replacing the single pion fields in each
term with an infinite series of all possible
combinations of pion fields. One of the most
common choices is
U
=
exp
⁡
{
i
F
(
π
0
2
π
+
2
π
−
−
π
0
)
}
{\displaystyle U=\exp \left\{{\frac {i}{F}}{\begin{pmatrix}\pi
^{0}&{\sqrt {2}}\pi ^{+}\\{\sqrt {2}}\pi ^{-}&-\pi
^{0}\end{pmatrix}}\right\}}
where
F
{\displaystyle F}
= 93 MeV.
In general different choices of the normalization
for
F
{\displaystyle F}
exist and one must choose the value that is
consistent with the charged pion decay rate.
=== Renormalization ===
The effective theory in general is non-renormalizable,
however given a particular power counting
scheme in ChPT, the effective theory is renormalizable
at a given order in the chiral expansion.
For example, if one wishes to compute an observable
to
O
(
p
4
)
{\displaystyle {\mathcal {O}}(p^{4})}
, then one must compute the contact terms
that come from the
O
(
p
4
)
{\displaystyle {\mathcal {O}}(p^{4})}
Lagrangian (this is different for an SU(2)
vs. SU(3) theory) at tree-level and the one-loop
contributions from the
O
(
p
2
)
{\displaystyle {\mathcal {O}}(p^{2})}
Lagrangian.)
One can easily see that a one-loop contribution
from the
O
(
p
2
)
{\displaystyle {\mathcal {O}}(p^{2})}
Lagrangian counts as
O
(
p
4
)
{\displaystyle {\mathcal {O}}(p^{4})}
by noting that the integration measure counts
as
p
4
{\displaystyle p^{4}}
, the propagator counts as
p
−
2
{\displaystyle p^{-2}}
, while the derivative contributions count
as
p
2
{\displaystyle p^{2}}
. Therefore, since the calculation is valid
to
O
(
p
4
)
{\displaystyle {\mathcal {O}}(p^{4})}
, one removes the divergences in the calculation
with the renormalization of the low-energy
constants (LECs) from the
O
(
p
4
)
{\displaystyle {\mathcal {O}}(p^{4})}
Lagrangian. So if one wishes to remove all
the divergences in the computation of a given
observable to
O
(
p
n
)
{\displaystyle {\mathcal {O}}(p^{n})}
, one uses the coupling constants in the expression
for the
O
(
p
n
)
{\displaystyle {\mathcal {O}}(p^{n})}
Lagrangian to remove those divergences.
== Successful application ==
=== Mesons and nucleons ===
The theory allows the description of interactions
between pions, and between pions and nucleons
(or other matter fields). SU(3) ChPT can also
describe interactions of kaons and eta mesons,
while similar theories can be used to describe
the vector mesons. Since chiral perturbation
theory assumes chiral symmetry, and therefore
massless quarks, it cannot be used to model
interactions of the heavier quarks.
For an SU(2) theory the leading order chiral
Lagrangian is given by
L
2
=
F
2
4
t
r
(
∂
μ
U
∂
μ
U
†
)
+
λ
F
3
4
t
r
(
m
q
U
+
m
q
†
U
†
)
{\displaystyle {\mathcal {L}}_{2}={\frac {F^{2}}{4}}{\rm
{tr}}(\partial _{\mu }U\partial ^{\mu }U^{\dagger
})+{\frac {\lambda F^{3}}{4}}{\rm {tr}}(m_{q}U+m_{q}^{\dagger
}U^{\dagger })}
where
F
=
93
{\displaystyle F=93}
MeV and
m
q
{\displaystyle m_{q}}
is the quark mass matrix. In the
p
{\displaystyle p}
-expansion of ChPT, the small expansion parameters
are
p
Λ
χ
,
m
π
Λ
χ
.
{\displaystyle {\frac {p}{\Lambda _{\chi }}},{\frac
{m_{\pi }}{\Lambda _{\chi }}}.}
where
Λ
χ
{\displaystyle \Lambda _{\chi }}
is the chiral symmetry breaking scale, of
order 1 GeV (sometimes estimated as
Λ
χ
=
4
π
F
{\displaystyle \Lambda _{\chi }=4\pi F}
).
In this expansion,
m
q
{\displaystyle m_{q}}
counts as
O
(
p
2
)
{\displaystyle {\mathcal {O}}(p^{2})}
because
m
π
2
=
λ
m
q
F
{\displaystyle m_{\pi }^{2}=\lambda m_{q}F}
to leading order in the chiral expansion.
=== Hadron-hadron interactions ===
In some cases, chiral perturbation theory
has been successful in describing the interactions
between hadrons in the non-perturbative regime
of the strong interaction. For instance, it
can be applied to few-nucleon systems, and
at next-to-next-to-leading order in the perturbative
expansion, it can account for three-nucleon
forces in a natural way
