×

New results from a number operator interpretation of the compositeness of bound and resonant states. (English) Zbl 1398.81314

Summary: A novel theoretical approach to the problem of the compositeness \((X)\) of a resonance or bound state is developed on the basis of the expectation values of the number operators of the free particles in the continuum. This formalism is specially suitable for effective field theories in which the bare elementary states are integrated out but that give rise to resonance and bound states when implemented in nonperturbative calculations. We demonstrate that \(X=1\) for finite-range energy-independent potentials, either regular or singular. A non-trivial example for an energy-dependent potential is discussed where it is shown that \(X\) is independent of any type of cutoff regulator employed. The generalization of these techniques to relativistic states is developed. We also explain how to obtain a meaningful compositeness with respect to the open channels for resonances, even if it is complex in a first turn, by making use of suitable phase-factor transformations. Defining elementariness as \(X=0\), we derive a new universal criterion for the elementariness of a bound state. Along the same lines, a necessary condition for a resonance to be qualified as elementary is given. The application of the formalism here developed might be of considerable practical interest.

MSC:

81V25 Other elementary particle theory in quantum theory
81T99 Quantum field theory; related classical field theories
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Castillejo, L.; Dalitz, R. H.; Dyson, F. J., Phys. Rev., 101, 453, (1956)
[2] Vaughn, M. T.; Aaron, R.; Amado, R., Phys. Rev., 124, 1258, (1961)
[3] Salam, A., Il Nuovo Cim., XXV, 224, (1962)
[4] Weinberg, S., Phys. Rev., 130, 776, (1963)
[5] Weinberg, S., Phys. Rev., 137, B672, (1965)
[6] Weinberg, S., Phys. Rev., 131, 440, (1963)
[7] Lurié, D.; MacFarlane, A. J., Phys. Rev., 136, B816, (1964)
[8] Fritzsch, H.; Gell-Mann, M.; Leutwyler, H.; Gross, D. J.; Wilczek, F.; Politzer, H. D., Phys. Lett., Phys. Rev. Lett., Phys. Rev. Lett., 30, 1346, (1973)
[9] An authoritative account of the history of quantum theory of gauge fields, with emphasis on color confinement, is given in G. ’t Hooft, The Glorious days of physics: Renormalization of gauge theories, hep-th/9812203; An authoritative account of the history of quantum theory of gauge fields, with emphasis on color confinement, is given in G. ’t Hooft, The Glorious days of physics: Renormalization of gauge theories, hep-th/9812203
[10] Weinberg, S., The quantum field theory of fields. vol. I. foundations, (1995), Cambridge University Press New York, USA
[11] Henley, E. M.; Thirring, W., Elementary quantum field theory, (1962), McGray-Hill Book Company Inc. USA · Zbl 0111.43101
[12] Faddeev, L. D., Mathematical aspects of the three-body problem in the quantum scattering theory, (1965), Daniel Davey & Co. Inc. New York, USA · Zbl 0131.43504
[13] Gottfried, K.; Yan, T. M., Quantum mechanics: fundamentals, (2003), Springer New York · Zbl 1033.81003
[14] Weinberg, S., Phys. Lett. B, Nuclear Phys. B, 363, 3, (1991)
[15] Kaiser, N.; Siegel, P. B.; Weise, W., Nuclear Phys. A, 594, 325, (1995)
[16] Oller, J. A.; Oset, E., Nuclear Phys. A, Nucl. Phys. A, 652, 407, (1999), (erratum)
[17] Oller, J. A.; Oset, E., Phys. Rev. D, 60, (1999)
[18] Oller, J. A.; Meissner, U. G., Phys. Lett. B, 500, 263, (2001)
[19] Dobado, A.; Pelaez, J. R.; Oller, J. A.; Oset, E.; Pelaez, J. R., Phys. Rev. D, Phys. Rev. D, Phys. Rev. D, Phys. Rev. D, Phys. Rev. Lett., 80, 3452, (1998), (erratum)
[20] Caprini, I.; Colangelo, G.; Leutwyler, H., Phys. Rev. Lett., 96, (2006)
[21] Kang, X.-W.; Oller, J. A., Eur. Phys. J. C, 77, 399, (2017)
[22] Hernández, E.; Mondragón, A., Phys. Rev. C, 29, 722, (1984)
[23] Schweber, S., An introduction to relativistic quantum field theory, (1962), Row, Peterson and Company New York · Zbl 0111.43102
[24] Gell-Mann, M.; Low, F., Phys. Rev., 84, 350, (1951)
[25] Entem, D. R.; Oller, J. A., Phys. Lett. B, 773, 498, (2017)
[26] D.R. Entem, J.A. Oller, in preparation.; D.R. Entem, J.A. Oller, in preparation.
[27] Phillips, D. R.; Beane, S. R.; Cohen, T. D., Ann. Physics, 263, 255, (1998)
[28] van Kolck, U., Nuclear Phys. A, 645, 273, (1999)
[29] Hyodo, T.; Jido, D.; Hosaka, A., Phys. Rev. C, 85, (2012)
[30] Aceti, F.; Oset, E., Phys. Rev. D, 86, (2012)
[31] Nagahiro, H.; Hosaka, A., Phys. Rev. C, 90, (2014)
[32] Sekihara, T., Phys. Rev. C, 95, (2017)
[33] Bethe, H., Phys. Rev., 76, 38, (1949)
[34] Courant, R.; Hilbert, D., Methods of mathematical physics, (1953), Interscience New York · Zbl 0729.00007
[35] Tricomi, F. G., Integral equations, (1985), Dover Publications, Inc. New York, USA
[36] Agadjanov, D.; Guo, F.-K.; Ríos, G.; Rusetsky, A., J. High Energy Phys., 01, 188, (2015)
[37] Itzykson, C.; Zuber, J. B., Quantum field theory, (1980), McGraw-Hill Inc USA · Zbl 0453.05035
[38] Bogdanova, L. N.; Hale, G. M.; Marushin, V. E., Phys. Rev. C, 44, 1289, (1991)
[39] Baru, V., Phys. Lett. B, 586, 53, (2004)
[40] Morgan, D., Nuclear Phys. A, 543, 632, (1992)
[41] Albaladejo, M.; Oller, J. A., Phys. Rev. D, 86, (2012)
[42] Guo, Z.-H.; Oller, J. A., Phys. Rev. D, 93, (2016)
[43] Oller, J. A., Phys. Rev. D, 71, (2005)
[44] Aitala, E. M., Phys. Rev. Lett., 86, 770, (2001)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.