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QCD and chiral perturbation theory

The QCD Lagrangian has the form

$\displaystyle {\cal L}_{QCD}$ $\textstyle =$ $\displaystyle {\cal L}_{g} + {\cal L}_{q} + {\cal L}_{m}$  
$\displaystyle {\cal L}_{g}$ $\textstyle =$ $\displaystyle -\frac{1}{4g^2} {\rm Tr}\; (G_{\mu\nu} G^{\mu\nu})$  
$\displaystyle {\cal L}_{q}$ $\textstyle =$ $\displaystyle \sum_{f=1}^{N_f} \bar{\psi}_f iD_{\mu}\gamma^{\mu} \psi_f$  
$\displaystyle {\cal L}_{m}$ $\textstyle =$ $\displaystyle \sum_{f=1}^{N_f} m_f \bar{\psi}_f \psi_f$  
$\displaystyle D_{\mu}$ $\textstyle =$ $\displaystyle \partial_{\mu} - i g A_{\mu}$  
$\displaystyle G_{\mu\nu}$ $\textstyle =$ $\displaystyle [ D_{\mu}, D_{\nu}  

where $A_{\mu}$ is the gluon field, $\psi_f$ is the quark field of the $f$-th flavor ( $\psi_f=(u,d,c,s,t,b)$), $D_{\mu}$ is the covariant derivative, $G_{\mu\nu}$ is the field strength tensor, and $g$ is the color charge. The term ${\cal L}_{g}$ describes pure gluon dynamics, the term ${\cal L}_{q}$ corresponds to the quark kinetic energy and quark-gluon interaction, and the term ${\cal L}_{m}$ is responsible for the quark masses. In the massless limit, the QCD Lagrangian $({\cal L}_{g}+{\cal L}_{q})$ depends on only one dimensionless parameter $g$. At the same time the strong coupling constant $\alpha_s(\mu)$ 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 $SU(N_{F})_R\times SU(N_{F})_L$. 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, $N_{F}=2$, they correspond to the triplet of pions). The essence of the chiral perturbation theory is to consider the quark mass term ${\cal L}_{m}$ 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 $\pi N$ scattering amplitude at the (unphysical) Cheng-Dashen point ($\nu=(s-u)=0$, $t=2m_{\pi}^2$) is related to the sigma term which describes the measure of the explicit chiral symmetry breaking:
$\displaystyle \sigma$ $\textstyle =$ $\displaystyle \frac{\hat{m}}{2M} \langle p \vert \bar{u}u+\bar{d}d \vert p \rangle$ (3)

where $\vert p\rangle$ denotes the proton state, $M$ is the proton mass, and $\hat{m}=(m_u+m_d)/2$ is the average mass of the $u$ and $d$ quarks. On the other hand, the sigma term is related to the observed $SU(3)$-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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Pionic Hydrogen Collaboration