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

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/ , 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/ , to appear in Phys. Rev. D (2006)]

Standard Model Extension –SME  Conceived by Colladay & Kostelecky as an extension of the Minimal Standard Model. [PRD 55,6760 (1997); PRD 58, (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.

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/ , 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. First part:

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

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

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

Free Particle solutions: Eigenenergy :

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

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

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:

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

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

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: Result :

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:

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

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

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

Free particle solutions: Writing: Which implies:

Free particle spinors:

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

Nonrelativistic Hamiltonian : Lorentz-violating Hamiltonian:

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

Relevant relations: For: With:

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

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

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

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

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.

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 B EXt. ).  An upper bound of (eV) is set up on the magnitude of the background.

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

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

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

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

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

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

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

Second term: In absence of B Ext :

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

Third term: For a Coulombian field:

Ket Relations: For: With:

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

In the presence of magnetic field:

First term: Second term:

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

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 (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.

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