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Influence of Lorentz violation on the hydrogen spectrum

Influence of Lorentz violation on the hydrogen spectrum. Manoel M. Ferreira Jr (UFMA- Federal University of Maranhão - Brazil). Colaborators: Fernando M. O. Moucherek (student - UFMA) Dr. Humberto Belich – UFES

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Influence of Lorentz violation on the hydrogen spectrum

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  1. Influence of Lorentz violation on the hydrogen spectrum Manoel M. Ferreira Jr (UFMA- Federal University of Maranhão - Brazil) Colaborators:Fernando M. O. Moucherek (student - UFMA) Dr. Humberto Belich – UFES Dr. Thales Costa Soares – UFJF Prof. José A. Helayël-Neto -CBPF

  2. Outline: Part 1) Results of the Paper:“Influence of Lorentz- and CPT-violating terms on the Dirac equation”, Manoel M. Ferreira Jr and Fernando M. O. Moucherek, hep-th/0601018, to appear in Int. J. Mod. Phys. A (2006). Part 2) Results of the Paper:“Lorentz-violating corrections on the hydrogen spectrum induced by a non-minimal coupling”, H. Belich, T. Costa Sores, M. M. Ferreira Jr, J. A. Helayel-Neto, F. M. O. Moucherek, hep-th/0604149, to appear in Phys. Rev. D (2006)]

  3. Standard Model Extension –SME • Conceived by Colladay & Kostelecky as an extension of the Minimal Standard Model. [PRD 55,6760 (1997); PRD 58, 116002 (1998).] • The underlying theory undergoes spontaneous breaking of Lorentz symmetry • Conceived as a speculation for probing a fundamental model for describing the Planck scale physics. • The low-energy effective model incorporates Lorentz-violating terms in all sectors of interaction. • Lorentz covariance is broken in the frame of particles but is preserved in the observer frame. • The renormalizability, gauge invariance and energy-momentum conservation of the effective model are preserved.

  4. First part: Results of the Paper:“Influence of Lorentz- and CPT-violating terms on the Dirac equation”, Manoel M. Ferreira Jr and Fernando M. O. Moucherek, hep-th/0601018, to appear in Int. J. Mod. Phys. A (2006). • It includes: • Dirac plane wave solutions, dispersion relations, eigenenergies; • Nonrelativist limit and nonrelativistic Hamiltonian; • First order energy corrections on the hydrogen spectrum; • Setting of an upper bound on Lorentz-violating parameter.

  5. SME Lorentz-violating Dirac sector: → Lorentz-violating coefficients (generated as v.e.v. of tensor terms of the underlying theory) → CPT- and Lorentz-odd coefficients → CPT- and Lorentz-even coefficients

  6. Analysis of the influence of the “vector coupling” term on the Dirac equation: → Modified Dirac Lagrangean Where: Modified Dirac equation: Dispersion relation:

  7. Energy eigenvalues: C - violation: E+ ≠ E- In order to obtain plane-wave solutions: The presence of the background implies:

  8. Free Particle solutions: Eigenenergy: Eigenenergy:

  9. Nonrelativistic limit Dirac Lagrangean: External eletromagnetic field: Two coupled equations: Nonrelativistic limit: Implying:

  10. Using the identity: We obtain the nonrelativistic Hamiltonian: Pauli Hamiltonian + Lorentz-violating terms: Lorentz-violating Hamiltonian:

  11. Evaluation of the corrections induced on the hydrogen spectrum First order Perturbation theory → 1-particle wavefunction: In the absence of magnetic external field, (A=0), only the first term contributes:

  12. Taking the background along the z-axis, we have: The integration possesses two contributions. The first one is: A consequence of:

  13. Second contribution: The angular integration is rewritten as: Considering the relations, It implies:

  14. Result: The presence of the background in vetor coupling does not induce any correction on the hydrogen spectrum . This result reflects the fact that this coupling yields just a momentum shift: The effect of the background may be seen as a gauge transformation: In such a transformation, the background may be “absorbed”, so that the lagrangean of the system recovers its free form:

  15. Analysis in the presence of an external magnetic field: In this case, the a contribution may arise from the A-term: For an external field along the z-axis: So we have:

  16. Using: We obtain: Once: The magnetic external field does not yield any new correction, unless the usual Zeeman effect.

  17. Analysis of the influence of the “axial vector” coupling term: Modified Dirac Lagrangian: Modified Dirac equation: , we have: Multiplying by:

  18. Multiplying again by: We attain the following dispersion relation: , → For , → For

  19. Free particle solutions: Writing: Which implies:

  20. Free particle spinors:

  21. Nonrelativistic limit Starting from: Implementing the conditions: and neglecting the term , we obtain:

  22. Nonrelativistic Hamiltonian : Lorentz-violating Hamiltonian:

  23. Evaluation of corrections on the hydrogen spectrum: In the absence of magnetic field: Contribution associated with: where n,l,j,mj, ms are the quantum numbers suitable to address a system with spin addition:

  24. Relevant relations: For: For: With:

  25. Taking into account the orthogonality relation: We obtain: sign (+) for j = l+1/2 Which implies: sign (-) for j = l-1/2 The energy is corrected by an amount proportional to ± mj, implying a correction similar to the usual Zeeman effect. This correction is attained in the absence of an external magnetic field!

  26. Upper bound on the Lorentz-violating parameter Regarding that spectroscopic experiments are able to detect effects of 10-10 eV, the following bound is set up:

  27. Contribution of the term : First order evaluation: The operator acts on the 1-particle wavefunction: so that:

  28. Considering: Only the terms in contribute to the result: The average of the momentum operator on an atomic bound state is null.

  29. Evaluation in the presence of na external magnetic field Magnetic field along the z-axis: So that: The external magnetic field does not induce any additional correction effect.

  30. Conclusions: • The Dirac nonrelativistic limit was assessed; the nonrelativistic Hamiltonian was evaluated. • The corrections induced on the hydrogen spectrum were evaluated in the presence and absence of external magnetic field. • For the coupling , no correction is reported. • For the case of the coupling , a Zeeman-like splitting is obtained (in the absence of BEXt.). • An upper bound of 10-10(eV) is set up on the magnitude of the background.

  31. Second Part:“Lorentz-violating corrections on the hydrogen spectrum induced by a non-minimal coupling” [H. Belich, T. Costa Sores, M. M. Ferreira Jr, J. A. Helayel-Neto, F. M. O. Moucherek, hep-th/0604149, to appear in Phys. Rev. D (2006)] Main goal:To evaluate the corrections induced on the hydrogen spectrum induced by a non-minimal coupling with the Lorentz-violating background. • It includes: • Dirac nonrelativist limit and nonrelativistic Hamiltonian; • First order energy corrections on the hydrogen spectrum; • Setting of upper bounds on Lorentz-violating parameter.

  32. Non-minimal coupling: Mass dimension: Modified Dirac equation: Defining: Adopting Dirac representation: We have:

  33. Nonrelativistic limit: For the strong spinor component: Canonical momentum: After some algebraic development, it results:

  34. In the absence of magnetic field, the relevant terms are: In the presence of magnetic field, the contributions stem from:

  35. Calculation of corrections in the absence of an external magnetic field First term: Hydrogen 1-particle wave function and Identity: So that:

  36. In spherical coordinates: Considering: We have: - Such a correction implies breakdown of the accidental degenerescence (regardless the spin-orbit interaction).

  37. Where: → Bohr radius Magnitude of this correction: Numerically: Regarding that spectroscopic experiments are able to detect effects as smaller than 10-10 eV, the following bound is set up:

  38. Second term: In absence of BExt:

  39. Outcome: Where it was used: Magnitude of the correction: Numerical value:

  40. Third term: For a Coulombian field:

  41. Ket Relations: For: For: With:

  42. So we have: Magnitude of the correction: This result leads to the same bound of the latter result:

  43. In the presence of magnetic field:

  44. First term: Second term:

  45. Third term: Magnitude of correction: Regarding that such a correction is undetectable for a magnetic strength of 1 G, we have: 1G ≈ 10-10 (eV)2 →Lorentz violation is more sensitively probed in the presence of an external magnetic field.

  46. Conclusions: • The nonrelativistic limit of the Dirac equation was assessed and the Hamiltonian evaluated. • The corrections on the hydrogen spectrum were properly carried out. • Such correction may be used to set up an upper bound of 10-25 (eV)-1 on the Lorentz-violating product. • Lorentz violation in the context of this model is best probed in the presence of an external magnetic field.

  47. Acknowledgments: • We express our gratitude to CNPq and FAPEMA (Fundação de Amparo à Pesquisa do Maranhão) for financial support.

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