Abstract
We analyze the analytic structure of the Covariant Spectator Theory (CST) contribution to the self-energy amplitude for a scalar particle in a \({\phi^2\chi}\) theory. To this end we derive dispersion relations in 1+1 and in 3+1 dimensional Minkowski space. The divergent loop integrals in 3+1 dimensions are regularized using dimensional regularization. We find that the CST dispersion relations exhibit, in addition to the usual right-hand branch cut, also a left-hand cut. The origin of this “spectator” left-hand cut can be understood in the context of scattering for a scalar \({\phi^2\chi^2}\) -type theory. If the interaction kernel contains a linear confining component, its contribution to the self-energy vanishes exactly.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Jegerlehner F., Nyffeler A.: The muon g-2. Phys. Rep. 477, 1–110 (2009). doi:10.1016/j.physrep.2009.04.003
Friman, B., Höhne, C., Knoll, J., Leupold, S., Randrup, J., Rapp, R.: The CBM Physics Book: Compressed Baryonic Matter in Laboratory Experiments. Lecture Notes in Physics, Springer, Berlin (2011)
Maris P., Roberts C.D.: Dyson–Schwinger equations: a tool for hadron physics. Int. J. Mod. Phys. E12, 297–365 (2003). doi:10.1142/S0218301303001326
Wick G.: Properties of Bethe–Salpeter wave functions. Phys. Rev. 96, 1124–1134 (1954). doi:10.1103/PhysRev.96.1124
Zuilhof M., Tjon J.: Electromagnetic properties of the deuteron and the Bethe–Salpeter equation with one boson exchange. Phys. Rev. C22, 2369–2382 (1980). doi:10.1103/PhysRevC.22.2369
Carbonell J., Karmanov V.: Solutions of the Bethe–Salpeter equation in Minkowski space and applications to electromagnetic form factors. Few Body Syst. 49, 205–222 (2011). doi:10.1007/s00601-010-0133-5
Carbonell J., Karmanov V., Mangin-Brinet M.: Electromagnetic form factor via Bethe–Salpeter amplitude in Minkowski space. Eur. Phys. J. A 39, 53–60 (2009). doi:10.1140/epja/i2008-10690-6
Gross F.: Three-dimensional covariant integral equations for low-energy systems. Phys. Rev. 186, 1448 (1969)
Gross F.: Relativistic few-body problem. I. Two-body equations. Phys. Rev. C 26, 2203 (1982)
Gross F.: Relativistic few-body problem. II. Three-body equations and three-body forces. Phys. Rev. C 26, 2226 (1982)
Stadler A., Gross F.: Covariant spectator theory: foundations and applications. Few Body Syst. 49, 91–110 (2011)
Nieuwenhuis T., Tjon J.: Nonperturbative study of generalized ladder graphs in a \({\varphi^2 \chi}\) theory. Phys. Rev. Lett. 77, 814–817 (1996). doi:10.1103/PhysRevLett.77.814
Gross F., Milana J.: Covariant, chirally symmetric, confining model of mesons. Phys. Rev. D 43, 2401–2417 (1991). doi:10.1103/PhysRevD.43.2401
Gross F., Milana J.: Decoupling confinement and chiral symmetry breaking: an explicit model. Phys. Rev. D 45, 969–974 (1992). doi:10.1103/PhysRevD.45.969
Gross F., Milana J.: Goldstone pion and other mesons using a scalar confining interaction. Phys. Rev. D 50, 3332–3349 (1994). doi:10.1103/PhysRevD.50.3332
Savkli C., Gross F.: Quark-antiquark bound states in the relativistic spectator formalism. Phys. Rev. C 63, 035–208 (2001). doi:10.1103/PhysRevC.63.035208
Nambu Y., Jona-Lasinio G.: Dynamical model of elementary particles based on an analogy with superconductivity. I. Phys. Rev. 122, 345–358 (1961). doi:10.1103/PhysRev.122.345
Becher T., Leutwyler H.: Baryon chiral perturbation theory in manifestly Lorentz invariant form. Eur. Phys. J. C 9, 643–671 (1999)
Gross, F.: Construction of covariant effective field theories. Unpublished (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Henryk Witala at the occasion of his 60th birthday.
Rights and permissions
About this article
Cite this article
Biernat, E.P., Gross, F., Peña, T. et al. Scalar-Particle Self-Energy Amplitudes and Confinement in Minkowski Space. Few-Body Syst 54, 2283–2301 (2013). https://doi.org/10.1007/s00601-012-0491-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00601-012-0491-2