Influence of Lorentz violation on the hydrogen spectrum

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)
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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.

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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.

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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
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Analysis of the influence of the vector
coupling term on the Dirac equation
? Modified Dirac Lagrangean
Where
Modified Dirac equation
Dispersion relation
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Energy eigenvalues
C - violation E ? E-
In order to obtain plane-wave solutions
The presence of the background implies
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Free Particle solutions
Eigenenergy
Eigenenergy
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Nonrelativistic limit
Dirac Lagrangean
External eletromagnetic field
Two coupled equations
Nonrelativistic limit
Implying
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Using the identity
We obtain the nonrelativistic Hamiltonian
Pauli Hamiltonian Lorentz-violating terms
Lorentz-violating Hamiltonian
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Evaluation of the corrections induced on the
hydrogen spectrum
First order Perturbation theory ?
1-particle wavefunction
In the absence of magnetic external field, (A0),
only the first term contributes
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Taking the background along the z-axis, we have
The integration possesses two contributions. The
first one is
A consequence of
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Second contribution
The angular integration is rewritten as
Considering the relations,
It implies
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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
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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
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Using
We obtain
Once
The magnetic external field does not yield any
new correction, unless the usual Zeeman effect.
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Analysis of the influence of the axial vector
coupling term
Modified Dirac Lagrangian
Modified Dirac equation
, we have
Multiplying by
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Multiplying again by
We attain the following dispersion relation
, ?
For
, ?
For
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Free particle solutions
Writing
Which implies
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Free particle spinors
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Nonrelativistic limit
Starting from
Implementing the conditions
and neglecting the term , we
obtain
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Nonrelativistic Hamiltonian
Lorentz-violating Hamiltonian
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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
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Relevant relations
For
For
With
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Taking into account the orthogonality relation
We obtain
sign () for j l1/2
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!
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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
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Contribution of the term
First order evaluation
The operator acts on the 1-particle
wavefunction
so that
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Considering
Only the terms in contribute to the result
The average of the momentum operator on an atomic
bound state is null.
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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.
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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.

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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.

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Non-minimal coupling
Mass dimension
Modified Dirac equation
Defining
Adopting Dirac representation
We have
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Nonrelativistic limit
For the strong spinor component
Canonical momentum
After some algebraic development, it results
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In the absence of magnetic field, the relevant
terms are
In the presence of magnetic field, the
contributions stem from
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Calculation of corrections in the absence of an
external magnetic field
First term
Hydrogen 1-particle wave function
and
Identity
So that
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In spherical coordinates
Considering
We have
- Such a correction implies breakdown of the
accidental degenerescence (regardless the
spin-orbit interaction).
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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
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Second term
In absence of BExt
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Outcome
Where it was used
Magnitude of the correction
Numerical value
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Third term
For a Coulombian field
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Ket Relations
For
For
With
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So we have
Magnitude of the correction
This result leads to the same bound of the latter
result
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In the presence of magnetic field
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First term
Second term
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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.
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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.

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Acknowledgments

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