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/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)]

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.

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.

First part:

SME Lorentz-violating Dirac sector:

,a v

→ 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

, , , ,a v H c d

, ,H c d

Analysis of the influence of the “vector coupling” term on the Dirac equation:

0

0

pmp

mi

e

e

→ Modified Dirac Lagrangean

Where:

Modified Dirac equation:

Dispersion relation:

v

A

B

w

w

Energy eigenvalues:

C - violation: E+ ≠ E-

In order to obtain plane-wave solutions:

The presence of the background implies:

Free Particle solutions:

2 20 ( )eE v m p v

2 20( )eE m p v v

Eigenenergy:

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:

*nml LV nmlE H dV

In the absence of magnetic external field, (A=0), only the first term contributes:

Taking the background along the z-axis, we have:

sin ;zv v

cos ;zv r v

The integration possesses two contributions. The first one is:

A consequence of:

0.v z

1 0E

sin ;zv v

Second contribution:

The angular integration is rewritten as:

Considering the relations,

It implies: 2 0E

p p v

( ) ( ) ( ) iv xx x x e

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:

1 2 0E E E Result:

( )L i m

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:

sin cos ,

sin sin ,

cos .

x r

y r

z r

We obtain:

The magnetic external field does not yield any new correction, unless the usual Zeeman effect.

Using:

Once: * 3

* 3

0,

0,

nml nml

nml nml

x d r

y d r

Analysis of the influence of the “axial vector” coupling term: 5b

Modified Dirac Lagrangian:

Modified Dirac equation:

Multiplying by: , we have:

Multiplying again by:

0( ,0)b b

We attain the following dispersion relation:

For , →

(0, )b b

, →For

Free particle solutions:

A

B

w

w

Writing:

Which implies:

Free particle spinors:

Nonrelativistic limit

Starting from:

Implementing the conditions:

and neglecting the term , we obtain:b

Nonrelativistic Hamiltonian :

Lorentz-violating Hamiltonian:

02 ( ) / 2LV eH b b p eA m

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:

Relevant relations:

For:

For:

With:

Taking into account the orthogonality relation:

We obtain:

Which implies:

The energy is corrected by an amount proportional to ± mj, 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

1010bE eV

Regarding that spectroscopic experiments are able to

detect effects of 10-10 eV, the following bound is set up:

Contribution of the term : 0

e

bp

m

First order evaluation:

The operator acts on the 1-particle wavefunction:

so that:

Considering:

Only the terms in contribute to the result:z

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:

5b

v

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.

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/0604149, 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

nlm 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 10-10 eV, the following

bound is set up:

Second term:

In absence of BExt:

Outcome:

Where it was used:

Magnitude of the correction:

Numerical value:

Third term:

For a Coulombian field:

Ket Relations:

For:

For:

With:

Magnitude of the correction:

So we have: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 ≈ 10-10 (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 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.

Acknowledgments: We express our gratitude to CNPq and

FAPEMA (Fundação de Amparo à Pesquisa do Maranhão) for financial support.