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Chiral Perturbation Theory for Baryons

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A Primer for Chiral Perturbation Theory

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

Abstract

So far we have considered the purely mesonic sector of chiral perturbation theory, involving the interaction of Goldstone bosons with each other and with external fields. However, ChPT can be extended to also describe the dynamics of baryons at low energies. Here we will concentrate on matrix elements with a single baryon in the initial and final states. As in the mesonic sector, Green functions are calculated in an effective-Lagrangian approach in combination with a power counting. The symmetries of QCD and the pattern of their breaking again constrain the possible interaction terms appearing in the effective Lagrangian, and we will discuss several approaches to obtain a consistent power counting.

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Notes

  1. 1.

    Technically speaking the adjoint representation is faithful (one-to-one) modulo the center \(Z\) of SU(3), which is defined as the set of all elements commuting with all elements of SU(3) and is given by \(Z=\{\text{\small 1}{\kern-3.5pt}1, \exp(2\pi i/3){\text{\small 1}{\kern-3.5pt}1}, \exp (4\pi i/3) { \text{\small 1}{\kern-3.5pt}1 } \}\).

  2. 2.

    The power counting will be discussed below.

  3. 3.

    The quantity \(m_N-m\) is of \({\fancyscript{O}}(q^2)\) as we will see in Sect. 4.5.3.

  4. 4.

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

  5. 5.

    In fact, also the definition of the pion-nucleon form factor of Eq. 4.24 contains a sign opposite to the standard convention so that, in the end, the Goldberger-Treiman relation emerges with the conventional sign.

  6. 6.

    Using \(m_N=(m_p+m_n)/2=938.92\) MeV, \(g_A=1.2695(29)\), \(F_\pi=92.42(26)\) MeV, and \(g_{\pi N}=13.21^{+0.11}_{-0.05}\) [56], one obtains \(\Updelta_{\pi N}=(2.37^{+0.89}_{-0.51})\) %.

  7. 7.

    The terminology “first and second classes” refers to the transformation property of strangeness-conserving semi-leptonic weak interactions under \({\fancyscript{G}}\) conjugation [61] which is the product of charge symmetry and charge conjugation \({\fancyscript{G}}={{\fancyscript{C}}}\,\exp(i\pi Q_{V2})\). A second-class contribution would appear in terms of a third form factor \(G_T\) contributing as

    $$ G_T(t) \bar{u}(p^{\prime}) i\,{\frac{\sigma^{\mu\nu} q_\nu}{2 m_N}}\, \gamma_5 \,{\frac{\tau_i}{2}} \,u(p). $$

    Assuming perfect \({\fancyscript{G}}\)-conjugation symmetry, the form factor \(G_T\) vanishes.

  8. 8.

    One also finds the parameterization

    $$ T=\bar{u}(p^{\prime})\left(D-{\frac{1}{4m_N}} [\not\!{q}^{\prime},\not\!{q}]B\right)u(p) $$

    with \(D=A+\nu B\), where, for simplicity, we have omitted the isospin labels.

  9. 9.

    For easier comparison with the result of ChPT we have chosen the sign opposite to the standard convention of Eq. 1.70. See also Sect. 4.3.1.

  10. 10.

    Recall that we use the normalization \(\bar{u}u=2 m_N\).

  11. 11.

    The threshold parameters are defined in terms of a multipole expansion of the \(\pi N\) scattering amplitude [14]. The sign convention for the \(s\)-wave scattering parameters \(a_{0+}^{(\pm)}\) is opposite to the convention of the effective-range expansion.

  12. 12.

    We do not expand the fraction \(1/(1+\mu)\), because the \(\mu\) dependence is not of dynamical origin.

  13. 13.

    The result, in principle, holds for a general target of isospin \(T\) (except for the pion) after replacing 3/4 by \(T(T+1)\) and \(\mu\) by \(M_\pi/M_T\).

  14. 14.

    The calculations were performed in the heavy-baryon approach (see Sect. 4.6.1) in which the \(c_i\) are renormalization-scale independent.

  15. 15.

    This relation can be understood as follows: For each internal line we have a propagator in combination with an integration with measure \(d^4 k/(2\pi)^4\). Therefore, there are \(I_\pi+I_N\) integrations. However, at each vertex we have a four-momentum conserving delta function, reducing the number of integrations by \(N_\pi+N_N-1\), where the \(-1\) is related to the overall four-momentum conserving delta function \(\delta^4(P_f-P_i)\).

  16. 16.

    In the low-energy effective field theory there are no closed fermion loops. In other words, in the single-nucleon sector exactly one fermion line runs through the diagram connecting the initial and final states.

  17. 17.

    Note that we work with the basic Lagrangian.

  18. 18.

    For brevity, we use the expression “up to \({\fancyscript{O}}(q^n)\)” to mean “up to and including \({\fancyscript{O}}(q^n)\)” in the following.

  19. 19.

    Note that \(P_{v\pm}\) do not define orthogonal projectors in the mathematical sense, because they do not satisfy \(P^{\dagger}_{v\pm}=P_{v\pm}\), with the exception of the special case \(v^\mu=(1,0,0,0)\).

  20. 20.

    Because of Eq. 4.99, a partial derivative acting on \({\fancyscript{N}}_v\) produces a small four-momentum.

  21. 21.

    We include the combination \(\sigma^{\mu\nu}\gamma_5\) for convenience.

  22. 22.

    For notational convenience we suppress the subscripts \(m\) and \(n\) of Eq. 4.134.

  23. 23.

    \({\rm det}(\Uplambda)=1\) and \({\Uplambda^0}_0\geq 1\).

  24. 24.

    It is common practice to denote both Lorentz transformations and the tensor describing the \(\Updelta\) with the same symbol \(\Uplambda\).

  25. 25.

    Note that \(m_\Updelta\) denotes the leading-order contribution to the mass of the \(\Updelta\) in an expansion in small quantities.

  26. 26.

    The Cartesian notation is convenient for displaying final results in a compact form while the spherical notation is used to apply angular momentum coupling methods. Recall \(x_{-1}=(x_1-ix_2)/\sqrt{2}\), \(x_0=x_3\), and \(x_{+1}=-(x_1+ix_2)/\sqrt{2}\).

  27. 27.

    We now follow common practice in physics and omit the \(\otimes\) symbol.

  28. 28.

    We return to the repeated-index summation convention, because the ranges of summation should now be clear.

  29. 29.

    We have explicitly included the projection operator in the definition of the Lagrangian.

  30. 30.

    With this choice we associate a factor \(iS^{\mu\nu}_F(p)\) with an internal \(\Updelta\) line of momentum \(p\).

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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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  1. Dr. Stefan Scherer
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Correspondence to Stefan Scherer .

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Scherer, S., Schindler, M.R. (2011). Chiral Perturbation Theory for Baryons. 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_4

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