Abstract
We consider the one-particle sector of the spinless Yukawa model, which describes the interaction of a nucleon with a real field of scalar massive bosons (neutral mesons). The nucleon as well as the mesons have relativistic dispersion relations. In this model we study the dependence of the nucleon mass shell on the ultraviolet cut-off \({\Lambda}\) . For any finite ultraviolet cut-off the nucleon one-particle states are constructed in a bounded region of the energy-momentum space. We identify the dependence of the ground state energy on \({\Lambda}\) and the coupling constant. More importantly, we show that the model considered here becomes essentially trivial in the limit \({\Lambda \to \infty}\) regardless of any (nucleon) mass and self-energy renormalization. Our results hold in the small coupling regime.
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References
Bachmann, S., Deckert, D., Pizzo, A.: The Mass Shell of the Nelson Model without Cut-Offs. http://arXiv.org/abs/1104.3271v2 [math.ph], 2011
Eckmann J.-P.: A model with persistent vacuum. Commun. Math. Phys. 18(3), 247–264 (1970)
Fröhlich J.: On the infrared problem in a model of scalar electrons and massless, scalar bosons. Ann. Inst. H. Poincaré Sect. A (N.S.) 19, 1–103 (1973)
Fröhlich J.: Existence of Dressed One Electron States in a Class of Persistent Models. Forts. der Phys. 22(3), 159–198 (1974)
Gross L.: Existence and uniqueness of physical ground states. J. Funct. Anal. 10(1), 52–109 (1972)
Gross L.: The relativistic polaron without cutoffs. Commun. Math. Phys. 31(1), 25–73 (1973)
Hainzl C., Seiringer R.: Mass renormalization and energy level shift in non-relativistic QED. Adv. Theor. Math. Phys. 6(5), 847–871 (2002)
Hiroshima F., Spohn H.: Mass renormalization in nonrelativistic quantum electrodynamics. J. Math. Phys. 46, 042302 (2005)
Könenberg, M., Matte, O.: The mass shell in the semi-relativistic Pauli-Fierz model. http://arXiv.org/abs/1204.5123v1 [math.ph], 2012
Lieb E., Loss M.: A bound on binding energies and mass renormalization in models of quantum electrodynamics. J. Stat. Phys. 108(5–6), 1057–1069 (2002)
Lieb, E., Loss, M.: Self-Energy of Electrons in Non-Perturbative QED. In: Walter, T. (ed.) The Stability of Matter: From Atoms to Stars, Selecta of Elliott H. Lieb. Berlin/Heidelberg: Springer, 2005, pp. 607–623
Mastropietro, V.: Non-perturbative renormalization. Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., 2008
Pizzo A.: One-particle (improper) States in Nelson’s Massless Model. Ann. H. Poincare. 4(3), 439–486 (2003)
Spohn, H.: Dynamics of Charged Particles and their Radiation Field. Cambridge: Cambridge University Press, October 2004
Takaesu, T.: Ground States of the Yukawa models with Cutoffs. http://arXiv.org/abs/0912.4951v1 [math.FA], 2009
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Communicated by H. Spohn
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Deckert, D.A., Pizzo, A. Ultraviolet Properties of the Spinless, One-Particle Yukawa Model. Commun. Math. Phys. 327, 887–920 (2014). https://doi.org/10.1007/s00220-013-1877-9
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DOI: https://doi.org/10.1007/s00220-013-1877-9