Solving the Bethe–Salpeter Equation in Euclidean Space
Tóm tắt
Different approaches to solve the spinor–spinor Bethe–Salpeter (BS) equation in Euclidean space are considered. It is argued that the complete set of Dirac matrices is the most appropriate basis to define the partial amplitudes and to solve numerically the resulting system of equations with realistic interaction kernels. Other representations can be obtained by performing proper unitary transformations. A generalization of the iteration method for finding the energy spectrum of the BS equation is discussed and examples of concrete calculations are presented. Comparison of relativistic calculations with available experimental data and with corresponding non relativistic results together with an analysis of the role of Lorentz boost effects and relativistic corrections are presented. A novel method related to the use of hyperspherical harmonics is considered for a representation of the vertex functions suitable for numerical calculations.
Tài liệu tham khảo
Rupp G., Tjon J.A.: Bethe–Salpeter calculation of three-nucleon observables with multirank separable interactions. Phys. Rev. C45, 2133–2142 (1992)
Fleischer J., Tjon J.A.: Bethe–Salpeter equation for J = 0 nucleon–nucleon scattering with one-boson exchange. Nucl. Phys. B84, 375–396 (1975)
Zuilhof M.J., Tjon J.A.: Electromagnetic properties of the deuteron and the Bethe–Salpeter equation with one-boson exchange. Phys. Rev. C22, 2369–2382 (1980) and references therein
Umnikov A.Yu., Kaptari L.P., Khanna F.C.: Deep inelastic scattering on the deuteron in the Bethe–Salpeter formalism. II. Realistic NN interaction. Phys. Rev. C56, 1700–1719 (1997)
Holl A., Krassnigg A., Maris P., Roberts C.D., Wright S.V.: Electromagnetic properties of ground-state and excited-state pseudoscalar mesons. Phys. Rev. C71, 065204–065216 (2005)
Gross F., Van Orden J.W., Holinde K.: Relativistic one-boson-exchange model for the nucleon–nucleon interaction. Phys. Rev. C45, 2094–2132 (1992)
Puzynin I.V. et al.: The generalized continuous analog of Newton’s method for the numerical study of some nonlinear quantum-field models. Phys. Part. Nucl. 30, 87–110 (1999)
Mangin-Brinet M., Carbonell J., Karmanov V.A.: Two-fermion relativistic bound states in light-front dynamics. Phys. Rev. C68, 055203–055232 (2003)
Mangin-Brinet M., Carbonell J., Karmanov V.A.: Relativistic bound states in the Yukawa model. Phys. Rev. D64, 125005–125022 (2001)
Frederico T., Sales J.H.O., Carlson B.V., Sauer P.U.: Light-front time picture of few-body systems. Nucl. Phys. A737, 260–264 (2004)
Efimov G.V.: On the Ladder Bethe–Salpeter Equation. Few Body Syst. 33, 199–217 (2003)
Bakker B.L.G., van Iersel M., Pijlman F.: Comparison of relativistic bound-state calculations in front-form and instant-form dynamics. Few Body Syst. 33, 27–47 (2003)
Carbonell J., Desplanques B., Karmanov V.A., Mathiot J.-F.: Explicitly covariant light-front dynamics and relativistic few-body systems. Phys. Rept. 300, 215–347 (1998)
Nieuwenhuis T., Tjon J.A.: O(4) Expansion of the Ladder Bethe–Salpeter Equation. Few Body Syst. 21, 167–185 (1996)
Lev F.M., Pace E., Salme G.: Poincaré covariant current operator and elastic electron-deuteron scattering in the front-form Hamiltonian dynamics. Phys. Rev. C62, 064004–064024 (2000)
Lev F.M., Pace E., Salme G.: Deuteron magnetic and quadrupole moments with a Poincaré covariant current operator in the front-form dynamics. Phys. Rev. Lett. 83, 5250–5253 (1999)
de Melo J.P.B.C., Frederico T., Pace E., Salme G.: Electromagnetic form factor of the pion in the space- and time-like regions within the front-form dynamics. Phys. Lett. B581, 75–81 (2004)
Salpeter E.E., Bethe H.A.: A relativistic equation for bound-state problems. Phys. Rev. 84, 1232–1242 (1951)
Wick G.C.: Properties of Bethe–Salpeter wave functions. Phys. Rev. 96, 1124–1134 (1954)
Mandelstam S.: Dynamical variables in the Bethe–Salpeter formalism. Proc. Roy. Soc. Lond. A233, 248–266 (1955)
Nakanishi N.: A general survey of the theory of the Bethe–Salpeter equation. Prog. Theor. Phys. Suppl. 43, 1–81 (1969)
Kaptari L.P., Umnikov A.Yu., Bondarenko S.G., Kazakov K.Yu., Khanna F.C., Kämpfer B.: Bethe–Salpeter amplitudes and static properties of the deuteron. Phys. Rev. C54, 986–1005 (1996)
Ciofi degli Atti C., Kaptari L.P.: Calculations of the exclusive processes 2 H(e,e′p)n,3 He(e,e′p)2 H, and 3 He(e,e′ p)(pn) within a generalized eikonal approximation. Phys. Rev. C71, 024005–024024 (2005)
Ciofidegli Atti C., Kaptari L.P., Treleani D.: Effects of the final state interaction in the electrodisintegration of the deuteron at intermediate and high energies. Phys. Rev. C63, 044601–044611 (2001)
Kubis J.J.: Partial-wave analysis of spinor Bethe–Salpeter equations for single-particle exchange. Phys. Rev. D6, 547–564 (1972)
Ciofidegli Atti C., Faralli D., Kaptari L.P., Umnikov A.Yu.: Relativistic structure of the deuteron: electrodisintegration and y scaling. Phys. Rev. C60, 034003–034020 (1999)
Kaptari L.P., Kämpfer B., Grosse E.: Solar proton burning process revisited within a covariant model based on the Bethe–Salpeter formalism. J. Phys. G26, 1423–1432 (2000)
Kaptari L.P., Kämpfer B., Dorkin S.M., Semikh S.S.: Elastic proton–deuteron backward scattering: relativistic effects and polarization observables. Phys. Rev. C57, 1097–1111 (1998)
Kaptari L.P., Kämpfer B., Semikh S.S., Dorkin S.M.: Exclusive charge exchange reaction pD → n(pp) within the Bethe–Salpeter formalism. Eur. Phys. J. A17, 119–132 (2003)
Kaptari L.P., Kämpfer B., Semikh S.S., Dorkin S.M.: Tagging the reaction pn rarr dphgr by backward protons in \({pd \to d \phi p_{sp}}\) processes. ibid J. Phys. G30, 1115–1126 (2004)
Kaptari L.P., Kämpfer B., Semikh S.S., Dorkin S.M.: Relativistic effects in proton induced deuteron breakup at intermediate-energies with forward emission of a fast proton pair. Eur. Phys. J. A19, 301–306 (2004)
Semikh S.S., Dorkin S.M., Beyer M., Kaptari L.P.: Solving the inhomogeneous Bethe–Salpeter equation. Phys. Atom. Nucl. 68, 2022–2033 (2005)
Dirac P.A.M.: Forms of relativistic dynamics. Rev. Mod. Phys. 21, 392–399 (1949)
Nakamura A., Satta L.: The tensor analyzing power in backward pd elastic scattering and its relation to pp → πd at intermediate energies. Nucl. Phys. A445, 706–716 (1985) and references therein
Berthet P. et al.: Elastic proton–deuteron backward scattering at energies from 0.6 to 2.7 GeV. J. Phys. G8, L111–L117 (1982)
Bonner B.E. et al.: nd Scattering at 180° for neutron energies from 200 to 800 MeV. Phys. Rev. Lett. 39, 1253–1256 (1977)
Dubal L. et al.: Measurements of dp backward elastic scattering at 3.43, 4.50, 5.75, and 6.60 \({\frac{GeV}{c}}\) incident deuteron momentum. Phys. Rev. D9, 597–603 (1974)
Carbonell J., Karmanov V.A.: Relativistic deuteron wave function in the light-front dynamics. Nucl. Phys. A581, 625–653 (1995)
Bondarenko S.G., Burov V.V., Beyer M., Dorkin S.M.: Isovector meson exchange currents in the Bethe–Salpeter approach. Phys. Rev. C58, 3143–3152 (1998)
Craigie N.S., Wilkin C.: Elastic proton–deuteron scattering at large angles. Nucl. Phys. B14, 477–488 (1969)
Nakamura A., Satta L.: The tensor analyzing power in backward pd elastic scattering and its relation to pp → πd at intermediate energies. Nucl. Phys. A445, 706–716 (1985)
Komarov V.I. et al.: Proton-induced deuteron breakup at GeV energies with forward emission of a fast proton pair. Phys. Lett. B553, 179–185 (2003)
Dorkin S.M., Kaptari L.P., Semikh S.S.: On the energy spectrum of the Bethe–Salpeter equation. Phys. Atom. Nucl. 60, 1629–1642 (1997)
Dorkin S.M., Beyer M., Semikh S.S., Kaptari L.P.: Two-Fermion Bound States within the Bethe–Salpeter Approach. Few Body Syst. 42, 1–32 (2008)