Nuclear interactions and the space-like structure of the pion
Proceedings of 6th International Workshop on Chiral Dynamics — PoS(CD09)
Three instances are discussed in which results produced by chiral perturbation theory can be reliably pushed to high space-like values of transferred momenta. 1. nuclear interactions: At present, expansions are available for about 20 components of both two-and three-nucleon forces, and the vast majority of them follows the patterns predicted by chiral symmetry. The outstanding exception is V + C , the isospin independent central potential. Standard calculations show that this O(q 3 )
... O(q 3 ) contribution is about 10 times larger than the leading O(q 2 ) isospin dependent term V − C . In spite of defying counting rules, these results are quite well supported by phenomenology up to distances smaller than 1 fm (→ |t| ∼ 20 M 2 π ). nucleon sigma-term: The configuration space nucleon scalar form factorF s (r) is an important substructure of V + C , and its integration over the entire volume yields σ N , the nucleon σ -term. Perturbative results based on diagrams involving N and ∆ intermediate states vanish at large distances, and increase monotonically as one approaches the nucleon center, where they can become arbitrarily large. Assuming that the pion cloud of the nucleon is constructed at the expenses of the surrounding condensate, an upper limit forF S (r) can be set at a critical radius R ≃ 0.6 fm (→ |t| ∼ 40 M 2 π ), where a phase transition takes place. This mechanism excludes the problematic region and yields 43 MeV< σ N < 49 MeV, in agreement with the empirical value 45±8 MeV. 3. space-like structure of the pion: The extension of the model for σ N to the pion describes it as a Goldstone boson at large distances, surrounded by a quark-antiquark condensate. As one moves towards its center, the condensate is gradually destroyed and a phase transition occurs at a distance R ≃ 0.6 fm (→ |t| ∼ 40 M 2 π ). When only pion loops are considered, the model depends on just M π and F π , and yields r 2 π S = 0.50 fm 2 andl 4 = 3.9. The inclusion of a scalar resonance of mass 980 MeV, with two known coupling constants, improves these values to r 2 π S = 0.59 fm 2 andl 4 = 4.3, well within the error bars of the precise estimates r 2 π S = 0.61 ± 0.04 fm 2 and l 4 = 4.4 ± 0.2, produced in 2001 by Colangelo, Gasser and Leutwyler. In both cases, results are given in terms of simple analytic expressions.