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A Primer for Chiral Perturbation Theory

Part of the book series: Lecture Notes in Physics ((LNP,volume 830))

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Abstract

The leading-order Lagrangians \({\fancyscript{L}}_2\) and \({\fancyscript{L}}_{\pi N}^{(1)}\) generate the results of current algebra and the PCAC hypothesis in a straightforward manner: one simply calculates the given process in terms of tree-level diagrams derived from \({\fancyscript{L}}_2\) and \({\fancyscript{L}}_{\pi N}^{(1)}.\) In the previous two chapters we developed techniques to also calculate higher-order (quantum, i.e. loop) corrections in a systematic way. To that end we made use of the EFT approach based on the momentum expansion of the most general Lagrangian compatible with the underlying symmetries of QCD. In the mesonic sector, the combination of dimensional regularization with the modified minimal subtraction scheme of ChPT led to a straightforward correspondence between the loop expansion and the chiral expansion in terms of momenta and quark masses at a fixed ratio, and provided a consistent power counting for renormalized quantities. Although the situation turned out to be more complex in the baryonic sector, we introduced several approaches resulting in consistent power-counting schemes. In this final chapter we will illustrate the methods developed in Chap. 4 by means of a few selected applications beyond tree level.

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Notes

  1. 1.

    The existence of a consistent power counting in HBChPT relies on specifying the renormalization scheme. See Sect. V of Ref. [53] for a discussion of this point.

  2. 2.

    The corrections of first order in \(1/m\) in Eq. 4.113 contain a piece of the type

    $$ \frac{1}{2m}{\bar{\mathcal{N}}}_v \left[(v\cdot D)^2-D^2 \right]{\mathcal{N}}_v. $$

    Using the field redefinition [65]

    $$ {\mathcal{N}}_v\to \left[1+\frac{iv\cdot D}{4m} -{\frac{{\mathsf{g}}_A S_v\cdot u}{4m}}\right] {\mathcal{N}}_v, $$

    the term containing \(v\cdot D\) can be eliminated. As in the case of the two-flavor mesonic Lagrangian at \({\fancyscript{O}}(q^4)\) (see Exercise 3.25), one finds equivalent parameterizations of \(\widehat{{\fancyscript{L}}_{\pi N}^{(2)}}\) (and also of the higher-order Lagrangians) in the baryonic sector.

  3. 3.

    In the remaining part of this section, we adopt the common practice of leaving out the projector \(P_{v+}\) in the propagator and (possibly) in vertices with the understanding that all operators act only in the projected subspace.

  4. 4.

    In our convention, \(k_3\) is larger by a factor of two than in Refs. [128, 162], because we use \(\ln(M/m)\) instead of \(\ln(M^2/m^2).\)

  5. 5.

    In the linear sigma model with explicit symmetry breaking (see Sect. 2.4), the double commutator \(\sigma_{11}(x)\) is proportional to the sigma field. This is the origin of the name sigma term.

  6. 6.

    The quark-model estimate of Eq. 4.202 yields \(g=1.075.\)

  7. 7.

    Note that not all two-loop integrals can be decomposed in this way. However, this special case is sufficient for our considerations.

  8. 8.

    Since we discuss the form factors in the space-like region, here we adopt the convention of taking \(Q^2=-q^2\) as the argument of the form factors as is common practice in the context of electron scattering.

  9. 9.

    The Lagrangian corresponds to the one of Ref. [72] with the replacements \({\tilde{F}}^+_{\mu\nu}\to f^+_{\mu\nu}\;\hbox{and}\;\hbox{Tr}(F^+_{\mu\nu})\to 4 v^{(s)}_{\mu\nu}.\)

  10. 10.

    Since we work in the isospin-symmetric limit we set \(m_p=m_n=m_N.\)

  11. 11.

    In the following, spin and isospin quantum numbers as well as isospinors are suppressed.

  12. 12.

    Note that different notations for the loop integrals are used in the literature.

  13. 13.

    The \(\pi^0\)-exchange graph driven by the WZW term of Sect. 3.5.3 results in an isovector contribution which is usually subtracted.

  14. 14.

    In principle, the VCS amplitude \(\gamma^\ast\pi\to\gamma\pi\) can be investigated in the reaction \(\pi e\to\pi e \gamma\) [179].

  15. 15.

    See Ref. [9] for a different approach to include pions perturbatively.

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Authors and Affiliations

  1. Johannes Gutenberg-Universität Mainz, Institut für Kernphysik, Johann-Joachim-Becher-Weg 45, 55099, Mainz, Germany

    Dr. Stefan Scherer

  2. Department of Physics and Astronomy, University of South Carolina, 712 Main Street, Columbia, SC, 29208, USA

    Dr. Matthias R. Schindler

  3. Department of Physics, The George Washington University, Washington, DC, USA

    Dr. Matthias R. Schindler

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Scherer, S., Schindler, M.R. (2011). Applications and Outlook. In: A Primer for Chiral Perturbation Theory. Lecture Notes in Physics, vol 830. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19254-8_5

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