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QCD and chiral perturbation theory

The QCD Lagrangian has the form

where is the gluon field, is the quark field of the -th flavor
(
), is the covariant derivative, is the field strength tensor, and is the
color charge. The term describes pure gluon dynamics, the term corresponds to the quark
kinetic energy and quark-gluon interaction, and the term is responsible for the quark masses. In the massless
limit, the QCD Lagrangian
depends on only one dimensionless parameter . At the same time
the strong coupling constant becomes scale dependent due to renormalization. The scale parameter of QCD is
determined from experiment. Provided the quarks are massless, the chirality (helicity) of a quark is conserved and the QCD
Lagrangian is symmetric with respect to rotations in the flavor space independently for
right- and left-handed quarks, thus the massless QCD has the global symmetry
described by the group
. While the Lagrangian
is chirally symmetric, the ground state of the massless QCD (vacuum) does not
have the same property because the chiral symmetry is spontaneously broken:
The characteristic feature of this spontaneous symmetry breaking is the emergence
of massless pseudoscalar particles (Goldstone bosons) (in the case of two flavors, ,
they correspond to the triplet of pions). The essence of the chiral perturbation
theory is to consider the quark mass term as a perturbation [13].
The mass term explicitly breaks the chiral symmetry, so that the Goldstone bosons
get nonzero masses, and in the leading order the pion mass squared is proportional
to the quark mass: Chiral perturbation theory (CHPT) is an * effective* field theory constructed
as an expansion in momenta and masses of physical particles, which are considered
to be small on a hadronic scale of about 1 GeV. This approach is extended in
baryon chiral perturbation theory, so that the meson interaction with "heavy"
baryons can be treated as well [12]. Along this way the current algebra
derived long time ago gets a modern framework. The near-threshold pion-nucleon
scattering amplitude is of great interest for chiral perturbation theory. In
particular, the isospin-even part of the scattering amplitude at the
(unphysical) Cheng-Dashen point (, ) is related to
the sigma term which describes the measure of the explicit chiral symmetry breaking:

where denotes the proton state, is the proton mass, and
is the average mass of the and quarks. On the other hand, the sigma
term is related to the observed -breaking mass differences in the baryon
octet and the strangeness content of the nucleon. For detailed discussion of
these relations we refer to [11,14,15,16] and references therein.
Since chiral perturbation theory is built as a low-energy expansion, it naturally
concerns low energy parameters, such as scattering lengths and ranges, which
provide very important experimental input determining the parameters of CHPT.
Precise experimental data are therefore crucial for a critical test of the predicting
power of chiral perturbation theory. For a discussion of these topics we refer
to [17] and references therein.

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** Up:** Strong interaction
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*Pionic Hydrogen Collaboration*

*1998*