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Chiral perturbation theory

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                                    Heinrich Leutwyler (2012), Scholarpedia, 7(10):8708.
   [3]doi:10.4249/scholarpedia.8708      revision #138476 [[4]link to/cite this article]
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   Curator: [7]Heinrich Leutwyler
   Contributors:


   0.50 -

   [8]Riccardo Guida

   [9]Johan Bijnens
     * [10]Prof. Heinrich Leutwyler, ITP, Bern University, CH

   $ \newcommand{\qbar}{\bar{q}} \newcommand{\ubar}{\bar{u}}
   \newcommand{\dbar}{\bar{d}} \newcommand{\sbar}{\bar{s}}
   \newcommand{\lvac}{\langle 0|\,} \newcommand{\rvac}{\,|0\rangle}
   \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}}
   \newcommand{\bdm}{\begin{displaymath}} \newcommand{\edm}{\end{displaymath}}
   \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}}
   \newcommand{\no}{\nonumber \\} \newcommand{\fs}{\,.} \newcommand{\co}{\,,}
   \newcommand{\al}{&\!\!\!} \newcommand{\eff}{{\mathrm{\it eff}}}
   \newcommand{\indR}{{\mathrm{\it R}}} \newcommand{\indL}{{\mathrm{\it L}}}
   \newcommand{\QCD}{{\mathrm{\it QCD}}} \newcommand{\Lo}{\stackrel{\mathrm{\small
   LO}}{=}} \newcommand{\NLo}{\stackrel{\mathrm{\small NLO}}{=}}
   \newcommand{\dx}{d^{\hspace{0.1mm}4}\hspace{-0.5mm}x} \newcommand{\U}{\cal U}
   \newcommand{\fnlab}[1]{\langle#1\rangle} $ Chiral Perturbation Theory
   ($\chi\hspace{-0.1em}$PT) is a model-independent method for the analysis of the
   low energy properties of the strong interaction, the prototype of what is called
   an effective field theory. The name derives from the fact that chiral symmetry
   plays a central role in this context.

Contents

     * [11]1 Strong interaction at low energies
     * [12]2 Symmetry
     * [13]3 Spontaneous symmetry breakdown
     * [14]4 Nambu-Goldstone bosons
     * [15]5 Pion pole dominance
     * [16]6 Effective field theory
     * [17]7 Foundations of Chiral Perturbation Theory
     * [18]8 Green functions and external fields
     * [19]9 Generating functional, gauge invariance
     * [20]10 Effective Lagrangian at leading order
     * [21]11 Discussion
     * [22]12 Higher orders
     * [23]13 Mass of the pion to one loop
     * [24]14 Renormalization
     * [25]15 Values of the low energy constants
     * [26]16 $\pi\pi$ scattering
          + [27]16.1 Tree approximation
          + [28]16.2 Beyond leading order
          + [29]16.3 Experimental results
          + [30]16.4 Lattice results
     * [31]17 Extensions
          + [32]17.1 Expansion in the mass of the strange quark, SU(3)$\times$SU(3)
          + [33]17.2 $\chi\hspace{-0.1em}$PT on the lattice
          + [34]17.3 Baryons, nuclear forces
          + [35]17.4 Thermodynamics of QCD, finite size effects, $\epsilon$-regime
     * [36]18 Conclusions
     * [37]19 Acknowledgment
     * [38]20 References
     * [39]21 Further reading
     * [40]22 See Also

Strong interaction at low energies

   In the framework of the Standard Model, the strong interaction is described by
   Quantum Chromodynamics (QCD). The reason why an effective field theory is needed
   to analyze the low energy properties of this theory is that, in the case of QCD,
   the standard method of analysis (expansion in powers of the coupling constant)
   only works at high energies, where [41]asymptotic freedom ensures that the
   interaction can be treated as a perturbation. At low energies, the relation
   between the degrees of freedom present in the [42]Lagrangian of QCD ([43]gluons,
   quarks) and those visible in the spectrum of physical states (mesons, baryons)
   cannot be analyzed in terms of an expansion in powers of the coupling constant.
   Many models that resemble QCD in one respect or the other have been proposed to
   understand this relation: constituent quarks, Nambu-Jona-Lasinio-model, linear
   $\sigma$ model, hidden local symmetry, holography, Anti-de-Sitter-space/Conformal
   Field Theory and many others. Some of these may be viewed as simplified versions
   of QCD that do catch some of the salient features of the theory at the
   semi-quantitative level, but none provides a basis for a coherent approximation
   scheme that would allow us, in principle, to solve QCD. Nonperturbative methods
   are required to analyze the low energy end of the spectrum. There are two methods
   that do not rely on an expansion in powers of the QCD coupling constant:
   effective field theory and simulation on a lattice.

   At low energies, the most important property of the system is the energy gap,
   i.e. the difference between the energies of ground state and first excited state.
   In quantum field theory, the ground state is the vacuum, while the first excited
   state contains a single particle at rest, the lightest particle occurring in the
   spectrum of the theory. Since the lightest particle in QCD is the pion, the
   energy gap is given by $M_\pi c^2 $. At low energies, the characteristic
   properties of QCD all derive from the fact that the pion is remarkably light, so
   that the energy gap is small.

Symmetry

   Already in 1960, more than a decade before the discovery of QCD, Nambu found out
   why the pion is so light: the strong interaction has a hidden, approximate
   symmetry [44](Nambu, 1960). At the time when Nambu proposed the idea, the strong
   interaction was not understood at all and the occurrence of approximate
   symmetries looked mysterious, but with the discovery of QCD, the mystery
   disappeared: in this theory, approximate symmetries do occur naturally, because
   the fermions come in several flavours. The interaction with the gluons is the
   same for all flavours - within QCD, the only difference between a $u$- and a
   $d$-quark, for instance, is that $m_u$ differs from $m_d$. If the masses of the
   two lightest quarks were the same, QCD would have an exact isospin symmetry:
   invariance under rotations in the internal space spanned by the two lightest
   flavours: \begin{equation}\tag{1} q=\left\{ \hspace{0.1cm}
   \begin{array}{c}u\\d\end{array} \hspace{0.1cm}\right\}, \hspace{0.5cm} q'=V\cdot
   q, \hspace{0.5cm}V\in \mathrm{SU(2)\hspace{1cm}isospin \,\,rotation}
   \end{equation} Heisenberg had introduced a symmetry group with this structure
   into nuclear physics, already in 1932 [45](Heisenberg, 1932). In the meantime,
   the evidence for the strong interaction to be approximately invariant under
   isospin rotations is overwhelming. In particular, the mesons and baryons do occur
   in nearly degenerate isospin multiplets. Disregarding the electromagnetic
   interaction, the splitting within these multiplets is due entirely to the
   difference between $m_u$ and $m_d$. For isospin to represent an approximate
   symmetry, the difference $m_u-m_d$ must be small and vice versa: if the
   difference is small, then QCD is approximately symmetric under the group SU(2) of
   isospin rotations.

   Small compared to what? QCD has an intrinsic scale, $\Lambda_{QCD}$, which is
   independent of the quark masses and carries the dimension of an energy.
   Expressing the quark masses in energy units, the condition for isospin to be an
   approximate symmetry can be written as $|m_u - m_d|$