We now apply the formalism developed in the Chap. 6 to study the low-energy isoscalarS-wave and isovectorP-wave\(\pi \pi \) amplitudes. Their phase shifts are characterized by the presence of a broad increase and later plateau (\(\sigma \)) for the former and by a steep rise (\(\rho \)) for the later.

1. 1.

   The pole is located in the 2nd Riemann sheet. We discuss above in Chap. 7 how to change to different Riemann sheets.

2. 2.

   In hadron physics an appealing justification for the appearance of the\(\rho (770)\) in the spectrum is obtained by gauging the chiral symmetry in the nonlinear chiral Lagrangians [36–38].

3. 3.

   The Ref. [19] does not include inA(s) the tadpole contributions for calculating\(T_{I;\mathrm{Left}}(s)\). The idea followed by Ref. [19] is to keep only those contributions in the calculation of Ref. [42] that involve explicit LHC contributions, while the tadpoles are local ones. These tadpole contributions could also be removed by employing a different regularization scheme for the calculation.

4. 4.

   Our convention for the scattering amplitudes differs from that in Ref. [10] by a minus sign. There is also a reshuffling in the labeling of the states, so that we designate them with a label that increases as the associated threshold does.

Authors and Affiliations

1. Departamento de Física, Universidad de Murcia, Murcia, Spain

   José Antonio Oller

Corresponding author

Correspondence toJosé Antonio Oller.

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