CPT and effective Hamiltonians for neutral kaon and similar complexes*
University of Zielona Góra, Institute of Physics
Prof. Z. Szafrana 4a, 65-516 Zielona Góra, Poland
e-mail: K.Urbanowski@proton.if.uz.zgora.pl; K.Urbanowski@if.uz.zgora.pl
Received:
Rec. 17 June 2005
DOI: 10.12921/cmst.2005.11.01.63-71
OAI: oai:lib.psnc.pl:583
Abstract:
This paper begins with a discussion of the general form and general CP- and CPT- transformation properties of the Lee-Oehme-Yang (LOY) effective Hamiltonian for the neutral kaon complex. Next, the properties of the exact effective Hamiltonian determined by the properties of the exact transition amplitudes for this complex are discussed. Using the Khalfin Theorem we show that contrary to the standard result of the LOY theory, the diagonal matrix elements of the effective Hamiltonian governing the time evolution in the subspace of states of an unstable particle and its antiparticle need not be equal at for t > t0 (t0 is the instant of creation of the pair) when the total system under consideration is CPT invariant but CP noninvariant. The unusual consequence of this result is that, contrary to the properties of stable particles, the masses of the unstable particle “1” and its antiparticle “2” need not be equal for t o t0 in the case of preserved CPT and violated CP symmetries. We also show that there exists an approximation which is more accurate than the LOY, and which leads to an effective Hamiltonian whose diagonal matrix elements posses properties consistent with the conclusions for the exact effective Hamiltonian described above.
Key words:
approximate methods, CPT symmetry, neutral kaons, Weisskopf-Wigner approximation
References:
[1] W. Pauli, in: Niels Bohr and the Developmnet of Physics. ed. W. Pauli (Pergamon Press, London, 1955), pp. 30-51. G. Luders, Ann. Phys. (NY) 2, 1 (1957). R. Jost, Helv. Phys. Acta 30, 409 (1957). R. F. Streater and A. S. Wightman, CPT, Spin, Statistics and All That (Benjamin, New York, 1964). N. N. Bogolubov, A. A. Logunov and I. T. Todorov, Introduction to Axiomatic Field Theory (Benjamin, New York, 1975).
[2] V. F. Weisskopf and E. T. Wigner, Z. Phys. 63, 54 (1930); 65, 18 (1930).
[3] M. Nowakowski, Int. J. Mod. Phys. A 14, 589 (1999).
[4] J. Jankiewicz, Acta. Phys. Polon. B 36, 1901 (2005),; hepph/0506118.
[5] T. D. Lee, R. Oehme and C. N. Yang, Phys. Rev. 106, 340 (1957).
[6] T. D. Lee and C. S. Wu, Annual Review of Nuclear Science, 16, 471 (1966). Ed.: M. K. Gaillard and M. Nikolic, Weak Interactions, (INPN et de Physique des Particules, Paris, 1977); Chapt. 5, Appendix A. S. M. Bilenkij, Particles and nucleus, vol. 1. No 1 (Dubna 1970), p. 227 [in Russian].
P. K. Kabir, The CP-puzzle, Academic Press, New York 1968.
[7] J. W. Cronin, Rev. Mod. Phys. 53, 373 (1981). J. W. Cronin, Acta Phys. Polon., B 15, 419 (1984). V. V. Barmin, et al., Nucl. Phys. B 247, 293 (1984). L. Lavoura, Ann. Phys. (NY), 207, 428 (1991). C. Buchanan, et al., Phys. Rev. D 45, 4088 (1992). C. O. Dib and R. D. Peccei, Phys. Rev., D 46, 2265 (1992). R. D. Peccei, CP and CPT Violation: Status and Prospects, Preprint UCLA/93/TEP/19, University of California, June 1993.
[8] E. D. Comins and P. H. Bucksbaum, Weak interactions of Leptons and Quarks, (Cambridge University Press, 1983). T. P. Cheng and L. F. Li, Gauge Theory of Elementary Particle Physics, (Clarendon, Oxford 1984).
[9] Yu. V. Novozhilov, Introduction to the Theory of Elementary Particles (Nauka, Moskow 1972), (in Russian). W. M. Gibson and B. R. Pollard, Symmetry Principles in Elementary
Particle Physics (Cambridge University Press, 1976).
[10] C. B. Chiu and E. C. G. Sudarshan, Phys. Rev. D 42, 3712 (1990); E. C. G. Sudarshan, C. B. Chiu and G. Bhamathi, Unstable Systems in Generalized Quantum Theory, Preprint DOE-40757-023 and CPP-93-23, University of Texas, October 1993.
[11] L. Maiani, The Second Dafine Physics Handbook, vol. 1, Eds. L. Maiani, G. Pancheri and N. Paver, SIS-Pubblicazioni, INFN-LNF, Frascati, pp. 3- 26 (1995).
[12] K. Urbanowski and J. Piskorski, Improved Lee, Oehme and Yang approximation, Preprint of the Pedagogical University No WSP-IF 98-51, Zielona Góra, March 1998, physics/-/9803030; Found. Phys. 30, 839 (2000).
[13] Review of Particle Physics, The European Physical Journal, C 15, No 1-4 (2000).
[14] K. Urbanowski, Physics Letters B 540, 89 (2002); hepph/0201272.
[15] L. A. Khalfin, Preprints of the University of Texas at Austin: New Results on the CP-violation problem, (Report DOEER40200-211, Feb. 1990); A new CP-violation effect and
a new possibility for investigation of decay modes, (Report DOE-ER40200-247, Feb. 1991).
[16] P. K. Kabir and A. Pilaftsis, Phys. Rev. A 53, 66 (1996).
[17] L. A. Khalfin, Foundations of Physics, 27, 1549 (1997) and references one can find therein.
[18] L. P. Horwitz and J. P. Marchand, Helv. Phys. Acta 42, 801 (1969).
[19] K. Urbanowski, Bull. de L’Acad. Polon. Sci.: Ser. sci. phys. astron., 27, 155 (1979).
[20] K. Urbanowski, Acta Phys. Polon. B 14, 485 (1983).
[21] K. Urbanowski, Int. J. Mod. Phys. A 7, 6299 (1992). K. Urbanowski, Phys. Lett. A 171, 151 (1992).
[22] K. Urbanowski, Phys. Lett. B 313, 374 (1993).
[23] K. Urbanowski, Phys. Rev. A 50, 2847 (1994).
[24] K. Urbanowski, Int. J. Mod. Phys. A 8, 3721 (1993).
[25] K. Urbanowski, Int. J. Mod. Phys. A 10, 1151 (1995).
[26] W. Krolikowski and J. Rzewuski, Bull. Acad. Polon. Sci. 4, 19 (1956); W. Krolikowski and J. Rzewuski, Nuovo. Cim. B 25, 739 (1975) and refernces therein.
[27] J. Piskorski, Acta Phys. Polon., B 31, 773 (2000).
[28] K. Urbanowski, Int. J. Mod. Phys. A 13, 965 (1998).
[29] K. Urbanowski, Acta Phys. Polon. B 35, 2069 (2004); hepph/0202253.
This paper begins with a discussion of the general form and general CP- and CPT- transformation properties of the Lee-Oehme-Yang (LOY) effective Hamiltonian for the neutral kaon complex. Next, the properties of the exact effective Hamiltonian determined by the properties of the exact transition amplitudes for this complex are discussed. Using the Khalfin Theorem we show that contrary to the standard result of the LOY theory, the diagonal matrix elements of the effective Hamiltonian governing the time evolution in the subspace of states of an unstable particle and its antiparticle need not be equal at for t > t0 (t0 is the instant of creation of the pair) when the total system under consideration is CPT invariant but CP noninvariant. The unusual consequence of this result is that, contrary to the properties of stable particles, the masses of the unstable particle “1” and its antiparticle “2” need not be equal for t o t0 in the case of preserved CPT and violated CP symmetries. We also show that there exists an approximation which is more accurate than the LOY, and which leads to an effective Hamiltonian whose diagonal matrix elements posses properties consistent with the conclusions for the exact effective Hamiltonian described above.
Key words:
approximate methods, CPT symmetry, neutral kaons, Weisskopf-Wigner approximation
References:
[1] W. Pauli, in: Niels Bohr and the Developmnet of Physics. ed. W. Pauli (Pergamon Press, London, 1955), pp. 30-51. G. Luders, Ann. Phys. (NY) 2, 1 (1957). R. Jost, Helv. Phys. Acta 30, 409 (1957). R. F. Streater and A. S. Wightman, CPT, Spin, Statistics and All That (Benjamin, New York, 1964). N. N. Bogolubov, A. A. Logunov and I. T. Todorov, Introduction to Axiomatic Field Theory (Benjamin, New York, 1975).
[2] V. F. Weisskopf and E. T. Wigner, Z. Phys. 63, 54 (1930); 65, 18 (1930).
[3] M. Nowakowski, Int. J. Mod. Phys. A 14, 589 (1999).
[4] J. Jankiewicz, Acta. Phys. Polon. B 36, 1901 (2005),; hepph/0506118.
[5] T. D. Lee, R. Oehme and C. N. Yang, Phys. Rev. 106, 340 (1957).
[6] T. D. Lee and C. S. Wu, Annual Review of Nuclear Science, 16, 471 (1966). Ed.: M. K. Gaillard and M. Nikolic, Weak Interactions, (INPN et de Physique des Particules, Paris, 1977); Chapt. 5, Appendix A. S. M. Bilenkij, Particles and nucleus, vol. 1. No 1 (Dubna 1970), p. 227 [in Russian].
P. K. Kabir, The CP-puzzle, Academic Press, New York 1968.
[7] J. W. Cronin, Rev. Mod. Phys. 53, 373 (1981). J. W. Cronin, Acta Phys. Polon., B 15, 419 (1984). V. V. Barmin, et al., Nucl. Phys. B 247, 293 (1984). L. Lavoura, Ann. Phys. (NY), 207, 428 (1991). C. Buchanan, et al., Phys. Rev. D 45, 4088 (1992). C. O. Dib and R. D. Peccei, Phys. Rev., D 46, 2265 (1992). R. D. Peccei, CP and CPT Violation: Status and Prospects, Preprint UCLA/93/TEP/19, University of California, June 1993.
[8] E. D. Comins and P. H. Bucksbaum, Weak interactions of Leptons and Quarks, (Cambridge University Press, 1983). T. P. Cheng and L. F. Li, Gauge Theory of Elementary Particle Physics, (Clarendon, Oxford 1984).
[9] Yu. V. Novozhilov, Introduction to the Theory of Elementary Particles (Nauka, Moskow 1972), (in Russian). W. M. Gibson and B. R. Pollard, Symmetry Principles in Elementary
Particle Physics (Cambridge University Press, 1976).
[10] C. B. Chiu and E. C. G. Sudarshan, Phys. Rev. D 42, 3712 (1990); E. C. G. Sudarshan, C. B. Chiu and G. Bhamathi, Unstable Systems in Generalized Quantum Theory, Preprint DOE-40757-023 and CPP-93-23, University of Texas, October 1993.
[11] L. Maiani, The Second Dafine Physics Handbook, vol. 1, Eds. L. Maiani, G. Pancheri and N. Paver, SIS-Pubblicazioni, INFN-LNF, Frascati, pp. 3- 26 (1995).
[12] K. Urbanowski and J. Piskorski, Improved Lee, Oehme and Yang approximation, Preprint of the Pedagogical University No WSP-IF 98-51, Zielona Góra, March 1998, physics/-/9803030; Found. Phys. 30, 839 (2000).
[13] Review of Particle Physics, The European Physical Journal, C 15, No 1-4 (2000).
[14] K. Urbanowski, Physics Letters B 540, 89 (2002); hepph/0201272.
[15] L. A. Khalfin, Preprints of the University of Texas at Austin: New Results on the CP-violation problem, (Report DOEER40200-211, Feb. 1990); A new CP-violation effect and
a new possibility for investigation of decay modes, (Report DOE-ER40200-247, Feb. 1991).
[16] P. K. Kabir and A. Pilaftsis, Phys. Rev. A 53, 66 (1996).
[17] L. A. Khalfin, Foundations of Physics, 27, 1549 (1997) and references one can find therein.
[18] L. P. Horwitz and J. P. Marchand, Helv. Phys. Acta 42, 801 (1969).
[19] K. Urbanowski, Bull. de L’Acad. Polon. Sci.: Ser. sci. phys. astron., 27, 155 (1979).
[20] K. Urbanowski, Acta Phys. Polon. B 14, 485 (1983).
[21] K. Urbanowski, Int. J. Mod. Phys. A 7, 6299 (1992). K. Urbanowski, Phys. Lett. A 171, 151 (1992).
[22] K. Urbanowski, Phys. Lett. B 313, 374 (1993).
[23] K. Urbanowski, Phys. Rev. A 50, 2847 (1994).
[24] K. Urbanowski, Int. J. Mod. Phys. A 8, 3721 (1993).
[25] K. Urbanowski, Int. J. Mod. Phys. A 10, 1151 (1995).
[26] W. Krolikowski and J. Rzewuski, Bull. Acad. Polon. Sci. 4, 19 (1956); W. Krolikowski and J. Rzewuski, Nuovo. Cim. B 25, 739 (1975) and refernces therein.
[27] J. Piskorski, Acta Phys. Polon., B 31, 773 (2000).
[28] K. Urbanowski, Int. J. Mod. Phys. A 13, 965 (1998).
[29] K. Urbanowski, Acta Phys. Polon. B 35, 2069 (2004); hepph/0202253.
