Influence of Lorentz- and CPT-violating terms on the Dirac equation

0.22 MB·English

by  Manoel M. Ferreira Jr

Influence of Lorentz- and CPT-violating terms on the Dirac equation
Manoel M. Ferreira Jr and Fernando M.O. Moucherek
Universidade Federal do Maranh˜ao (UFMA), Departamento de F´ısica,
∗
Campus Universit´ario do Bacanga, S˜ao Luiz - MA, 65085-580 - Brasil
The influence of Lorentz- and CPT-violating terms (in ”vector” and ”axial vector” couplings)
ontheDiracequationisexplicitlyanalyzed: planewavesolutions,dispersionrelationsandeigenen-
ergies are explicitly obtained. The non-relativistic limit is worked out and the Lorentz-violating
Hamiltonian identified in both cases, in full agreement with the results already established in the
literature. Finally, the physical implications of this Hamiltonian on the spectrum of hydrogen are
evaluated both in the absence and presence of a magnetic external field. It is observed that the
7
0 fixed background, when considered in a vector coupling, yields no qualitative modification in the
0 hydrogen spectrum, whereas it does provide an effective Zeeman-like splitting of the spectral lines
2 whenever coupled in the axial vector form. It is also argued that the presence of an external fixed
field does not imply new modifications on the spectrum.
n
a
J PACSnumbers: 11.30.Er,11.10.Kk,12.20.Fv

9

3
v
8 I. INTRODUCTION
1
0
Lorentz covariance, as is well-known, is a good symmetry of the fundamental interactions comprised in the
1
0 traditional framework of a local Quantum Field Theory, from which the Standard Model is derived. However,
6 since the beginning 90´s, Lorentz-violating theories have been proposed as a possible candidate of signature
0 of a more fundamental physics defined in a higher scale of energy, not accessible to the present experiments.
/
h A pioneering work due to Carroll-Field-Jackiw[1] has proposed a CTP-odd Chern-Simons-like correction term
t (ǫµνκλv A F )totheconventionalMaxwellElectrodynamics,thatpreservesgaugeinvariancedespitebreaking
- µ ν κλ
p Lorentz and parity symmetries. Some time later, Colladay & Kostelecky [2], [3] adopted a quantum field
e
theoretical framework to address the issue of CPT- and Lorentz-breakdown as a spontaneous violation [4]-[6].
h
: In this sense, they constructed the extended Standard Model (SME), an extension to the Standard Model
v
whichmaintainsunaffectedthe SU(3)×SU(2)×U(1)gaugestructure ofthe usualtheoryandincorporatesthe
i
X CPT-violationas an active feature of the effective low-energybrokenaction. In the brokenphase, the resulting
r effective action exhibits breakdown of CPT and Lorentz symmetries at the particle frame, but conservation of
a
covariance under the perspective of the observer inertial frame. The parameters representing Lorentz violation
are obtained as the vacuum expectation values of some tensor operators belonging to the underlying theory.
The SME incorporates all the tensor terms that yield scalars (by contracting standard model operators with
Lorentz breaking parameters) in the observer frame.
Timely, it is worthwhile to point out the existence of alternative mechanisms that bring about equivalent
Lorentz-breaking effects. Indeed, noncommutative field theories [7]-[11] also generate Lorentz-violating terms
of equal structure, able to imply similar effects to the ones of the SME phenomenology. Another mechanism is
varying fundamental couplings [12]-[14] which amounts to the incorporation of Lorentz-violating terms in the
action as well. In fact, varying couplings leads to the breakingof temporaland spatial translations,which may
be seen as a particular case of Lorentz breakdown. In a cosmological environment, this issue may be used to
investigatecandidatefundamentaltheoriescontainingascalarfieldwithaspacetime-varyingexpectationvalue,
once the associated Lorentz-breaking effects may be taken as a signature for an underlying theory. Further,
Lorentzviolationstill appears inother theoreticalcontexts, involvingthe considerationof loopgravity[15]-[16]
and spacetime foam [17]-[18].
∗Electronicaddress: [email protected],[email protected]
2
The gaugesector of the SME model has been extensively studied in severalworksboth in (1+3)and (1+2)-
dimensions [19]-[60], with many interesting results. Concerning the fermion sector, in the context of the SME,
Colladay& Kostelecky[2], [3] have devised Lorentz-violatingterms compatible with U(1) gaugesymmetry and
renormalizability. These terms are explicitly written as below:
1 i ←→ i ←→
L=−v ψγµψ−b ψγ γµψ− H ψσµνψ+ c ψγµDνψ+ d ψγ γµDνψ, (1)
µ µ 5 µν µν µν 5
2 2 2
where the Lorentz-breakingcoefficients v ,b ,H ,c ,d arise as vacuum expectation values of tensor quan-
µ µ µν µν µν
tities defined in an underlying theory. The first two terms are CPT-odd and the others are CPT-even. Firstly,
the fermionsectorofthe SME modelhasbeen investigatedin ageneralway(by discussing dispersionrelations,
plane-wave solutions, and energy eigenvalues). Later, it has been addressed in connection with CTP-violating
probing experiments , which involve comparative studies of cyclotron frequencies of trapped-atoms [61]-[62],
clock-comparison tests [63], spectroscopic comparison of hydrogen and antihydrogen [64], analysis of muon
anomalous magnetic moment [65], study of macroscopic samples of spin-polarized solids [66], and so on.
The interest of the present work lays only on the two CPT-odd terms, linked to the fermion field by an
assigned”vector”and ”axialvector” coupling, respectively. The main objective is to examine the effects of the
Lorentz-violating background on the Dirac equation and solutions, focusing on its nonrelativistic regime and
possible implications on the hydrogen spectrum. Some results concerning this study were already discussed in
the literature. Indeed, the nonrelativistic Hamiltonian associated with Lagrangian (1) was already evaluated
by means of a Foldy-Wouthuysen expansion in refs. [67]-[68]. Moreover, the corresponding shifts of atomic
levelswereperturbativelycarriedoutinabroadperspectiveinref. [64],[69]. Inthe presentpaper,however,the
analysisofthehydrogenspectruminthepresenceCPT-oddtermsisdoneinadifferentway(morespecific,direct
and simpler), also including the action of an external constant magnetic field. The starting point is the Dirac
Lagrangiansupplemented by Lorentz and CPT-violating terms. The dispersion relations, plane-wave solutions
and eingenenergies are carried out for each one of the considered couplings. In the sequel, the investigation
of the nonrelativistic limit is performed. This is a point of interest due to its connection with real systems of
Condensed Matter Physics, a true environment where the presence of a background may be naturally tested.
The effect of the background on the spectrum of hydrogen atom is then evaluated, initially for the case of the
vectorcoupling,forwhichitisreportedno correctiononthe hydrogenspectrum. Inthe caseofthe axialvector
coupling, the spinor solutions come out to be cumbersome and the nonrelativistic limit altered. The Pauli
equationis supplementedbyterms thateffectivelymodify the spectrumofthe hydrogenina similarwayasthe
usual Zeeman effect. This sort of theoretical modification may be combined with fine spectral analysis to set
up precise bounds on the magnitude of the corresponding Lorentz-violating coefficient. It is still shown that
the presence of an external fixed magnetic field does not lead to new Lorentz-violating effects.
This paper is outlined as follows. In Sec. II, it is considered the presence of the term v ψγµψ in the Dirac
µ
Lagrangian. The modified Dirac equation, dispersion relations, plane-wave solutions and energy eingenvalues
are evaluated. The nonrelativistic limit is analyzed and the corresponding Hamiltonian worked out. In a first
orderevaluation,itisshownthattheLorentz-violatingtermsdonotmodifythehydrogenspectrum. InSec. III
the presence of the axial vector term, b ψγ γµψ, in the Dirac sector is considered. Again, the modified Dirac
µ 5
equation, dispersion relations, plane-wave solutions and eingenvalues are carried out. Finally, the low-energy
limit is studied and the Hamiltonian evaluated. A first order computation shows that the Lorentz-violating
terms contribute to the spectrum hydrogen, causing a Zeeman splitting of the spectral lines. In Sec. IV, one
presents the Conclusion and final remarks.
II. LORENTZ-VIOLATING DIRAC LAGRANGIAN (”VECTOR” COUPLING)
Themostnaturalandsimplewaytocoupleafixedbackground [vµ =(v ,−→v)]toaspinorfieldisdefining
0
a vector coupling, given as follows:
L´=L −v ψγµψ, (2)
Dirac µ
3
←→
where L is the usual Dirac Lagrangian (L = 1iψγµ ∂ ψ−m ψψ) and v is one of the CPT-odd
Dirac Dirac 2 µ e µ
parameters that here represents the fixed background responsible for the violation of Lorentz symmetry in the
frame of particles. In true, the term v ψγµψ behaves as a scalarjust in the observerframe, in which v is seen
µ µ
as a genuine 4-vector. The Euler-Lagrange equation applied on this Lagrangian provides the modified Dirac
equation:
(iγµ∂ −v γµ−m )ψ =0, (3)
µ µ e
which corresponds to the usual Dirac equation supplemented by the Lorentz-violating term associated with
the background. The initial task is to investigate the plane-wave solutions, which may be attained by writing
the spinor in terms of a plane-wave decomposition, ψ =Ne−ix·pw(p ), where N is the normalization constant
µ
and w(p ) is the (4×1) spinor written in the momenta space. Taking it into account, eq. (3) is rewritten in
µ
momentum space:
(γµp −v γµ−m )w(p)=0. (4)
µ µ e
ItispossibletoshowthateachcomponentofthespinorwsatisfiesachangedKlein-Gordonequationwhichrepre-
sentsthedispersionrelationofthismodel. Infact,multiplyingthisequationontheleftby(γµp −v γµ+m ),
µ µ e
it results:
p·p−2p·v+v·v−m2 w(p)=0, (5)
e
whoseenergysolutionsare: E± =v0±(cid:0) m2e+(−→p −−→v)2. Here,(cid:1)onehastwodifferentenergyvalues,onepositive
(E+), another negative (E−). The nepgative solution should be reinterpreted as positive-energy anti-particles.
Even after the reinterpretation, the eigenenergies remain different. This is an evidence of charge conjugation
breakdown, as it will be properly discussed ahead.
Now, the spinors w(p) compatible with such equation should be achieved. Adopt an explicit representation
for the Dirac matrices1 and writing w(p) in terms of two 2×1 spinors (w and w ), the following spinor
A B
equations are obtained:
1 −→ −→ −→
w = σ ·(p − v)w , (6)
A B
(E−v −m )
0 e
1 −→ −→ −→
w = σ ·(p − v)w . (7)
B A
(E−v +m )
0 e
In order to attain a simple solution, a usual procedure for construction of plane-wave spinors is followed: a
1 0
starting form, or , for one of them is proposed, so that the other is straightforwardly derived by
0 1
(cid:18) (cid:19) (cid:18) (cid:19)
means of eqs. (6), (7). These two 2×1 spinors must then be grouped in a single normalized (4×1) spinor.
Following this procedure, after reinterpretation2, four independent (4×1) spinors, u (particle solutions) and
i
v (anti-particle solutions), are attained:
i
1 0
0 1
u1(p)=N (pz−vz)  , u2(p)=N (px−vx)+i(py−vy)  , (8)
 (px−EEv++x)mm+eei(−−pvvy00−vy)   EE−++(pmmz−ee−−vzvv)00 
   
1 Here,oneadoptstheDiracrepresentationforγ−matrices: γ0=„ I0 −0I «, γi=„−0σi σ0i «,γ5=iγ0γ1γ2γ3=„ 0I I0«,
withσi=(σx,σy,σz)beingthewell-knownPaulimatrices.
2 It shouldbejustrememberedthat thereinterpretation procedureconsists inturninganegative-energy solutionintoa positive-
→− →−
energyanti-particle(forwhichtheenergyandmomentum mustbereverted: E→−E, p →−p).
4
(pz+vz) (px+vx)+i(py+vy)
E+me+v0 E+me+v0
v1(p)=N (px+Ev+x)m+ei(+pvy0+vy)  , v2(p)=N −E(+pzm+ev+zv)0 , (9)
1 0
   
 0   1 
   
   
where N is the normalization constant. In the solutions (8), (9), one of the effects of the background is
−→ −→ −→
manifest: to shift the energy and momentum by a constant: E →E−v , p →(p − v). It is also instructive
0
to exhibit the energy eigenvalues associated with the four solutions above. In this case, one can write two
eigenvalue equations: Hu =E(u)u ,Hv =E(v)v , with i=1,2,and E(u) =v + m2+(−→p −−→v)2 1/2,E(v) =
i i i i i i i 0 e i
m2+(−→p +−→v)2 1/2−v . Here, E(u) stands for the particle energy whereas E(v) represents the anti-particle
e 0 i i (cid:2) (cid:3)
energy. In the reinterpretation procedure, it was obviously assumed that the magnitude background is minute
(cid:2) (cid:3)
near the electron mass (v << m ), regarded as a correction effect. This must be so once many experiments
0 e
demonstrate the validity of Lorentz covariance with high precision. It should still be pointed out that these
energy values are in agreement with the similar ones obtained in refs. [2]-[3], [67]-[68]. The attainment of
different energies for particle and anti-particle E(v) 6=E(u) is an evidence that the charge conjugation (C)
i i
symmetry has been broken. Indeed, the term v(cid:16)ψγµψ is C-o(cid:17)dd and PT-even, that is, it implies breakdown of
µ
charge conjugation, and conservation of combined PT operation. An ease way to demonstrate such a violation
is to apply the charge conjugation operator C = iγ0γ2 on the modified Dirac equation, as given in eq. (11).
This procedure will lead to the correspondingDirac equation for the chargeconjugate spinor (Ψ =CΨ∗) with
c
an opposite sign for the term v ψγµψ, which implies breaking of C-symmetry3.
µ
One should now enquire about the spin interpretation of these solutions. Obviously, such solutions will not
present the same spin projection as the usual Dirac free-particle solutions. But in some particular cases, it is
possible to show that such solutions exhibit the same spin projection. For instance, whenever the background
and the momentum are aligned along the z-axis, the spinors take the form:
1 0 (pz+vz) 0
0 1 E+m0e+v0 −(pz+vz)
u1 =N (pz−vz)  , u2 =N 0  , v1 =N 1 , v2 =N E+m0e+v0 . (10)
 E+m0e−v0   −(pz−az)   0   1 
   E+me−v0     
       
−→ σ 0
Such solutions are eigenstates of the helicity operator, S ·p=S = 1Σ , with: Σ = z . Thus, the
z 2 z z 0 σ
(cid:18) z (cid:19)
spinors u and v have eigenvalue +1 (spin up) whereas the spinors u and v have eigenvalue −1 (spin down).
1 1 2 2
b
Hence, the presence of the fixed backgrounddoes not suffice in principle to change the spin polarization of the
new states. A detailed study of the spin projections may only be obtained by constructing the spin projector
operators. This point is addressed by Lehnert in ref. [67].
A. Nonrelativistic limit
Every good relativistic theory must exhibit a sensible low-energy limit whose predictions may be compared
with the results of other correlated nonrelativistic theories. Such a requirement sets up the correspondence
between an intrinsically relativistic theory and a nonrelativistic one. In a well-known case, the nonrelativistic
3 OnetakesasstartingpointtheDiracequation(iγµ∂µ−eγµAµ−vµγµ−m)ψ=0,whichforananti-particlemustberewritten
withoppositechargesign: (iγµ∂µ+eγµAµ−vµγµ−m)ψc=0,beingψc theanti-particlespinor. Inthecasethe C-symmetry
holdson,thisexactequationmightbealsoobtainedbyapplyingthechargeconjugationoperatorC=iγ0γ2 ontheinitialDirac
equation. Making it, one attains: (iγµ∂µ+eγµAµ+vµγµ−m)ψc = 0, where one notes the opposite sign of the term vµγµ.
ThisputsinevidencetheC-breakdown. AsimilarproceduremaybeemployedtodemonstratetheconservationofPTsymmetry.
5
limit of the Dirac theory yields the Pauli equation, which consists of the Schr¨odinger equation supplemented
with the spin-magnetic interaction. Hence, to work in the nonrelativistic limit allows to investigate quantum
mechanical features of a system without losing relativistic effects (like spin) of the original theory. In the
present case, where the Dirac theory is being corrected by a Lorentz-violating coupling term, one expects that
the nonrelativisticregimebe welldescribedbythe PauliequationincorporatingLorentz-violatingterms. Itwill
be shown that this is exactly the case.
To correctly analyze the nonrelativistic limit of Lagrangian (2), this model is considered in the presence of
an external electromagnetic field (A ), so that Lagrangian(2) is rewritten in the form:
µ
1 ←→
L= iψγµD ψ−m ψψ−v ψγµψ, (11)
µ e µ
2
where D = ∂ +ieA . The external field is implemented into our previous equations by means of the direct
µ µ µ
substitution: pµ →pµ−eAµ. Replacing it into eqs. (6) and (7), there follows:
1 −→ −→ −→ −→
w = σ ·(p −eA − v)w , (12)
A (E−eA0−m −v0) B
e
1 −→ −→ −→ −→
w = σ ·(p −eA − v)w . (13)
B (E−eA0+m +v0) A
e
In the low-velocity limit, it obviously holds (−→p)2 ≪ m2, eA ≪ m , conditions that impose the smallness
e 0 e
of kinetic and potential energy before the relativistic rest energy (m ). With it, the energy of the system is
e
written as E = m +H, where H represents the nonrelativistic Hamiltonian. From eqs. (12) and (13), the
e
spinors w ,w are readas the large and the small components, once the magnitude of w is much larger than
A B A
w . By replacing eq. (13) into eq. (12) and implementing the low-energy conditions, one should retain only
B
the equation for the strong component (w ),
A
H −eA0−v0 w = 1 −→σ ·(−→p −e−→A −−→v)−→σ ·(−→p −e−→A −−→v)w , (14)
A (2m +v0) A
e
(cid:0) (cid:1)
−→ −→ −→ −→ −→ −→ −→ −→ −→
whichdescribesthephysicsofthenonrelativisticlimit. Usingtheidentity,(σ·a)(σ· b)= a· b +iσ·(a× b),
eq. (14) is reduced to the form,
Hw = (−→p −e−→A −−→v)2 + 1 −→σ ·[∇×(−→A −−→v)]+(eA0+v0) w , (15)
A A
2m 2m
(cid:26) e e (cid:27)
where H is the nonrelativistic Hamiltonian. Specifically, concerningthe spin-orbitinteraction, one cansee that
−→
such background does not yield any modification, once ∇× v = 0. Now, comparing Eq. (15) with the Pauli
equation, the Hamiltonian takes a more familiar form:
H = (−→p −e−→A)2 − e~ −→σ ·−→B +eA0 + −2(−→p −e−→A)·−→v +v0+ −→v2 . (16)
2m 2m 2m 2m
(cid:26)" e # " e e#(cid:27)
Thefirsttermintobracketscontainsthewell-knownPauliHamiltonian,whereasthesecondoneisthecorrection
Hamiltonian arising from the Lorentz-violating background. This specific term, object of our attention, is
rewritten below:
2i−→v ·−→∇ 2e−→A ·−→v −→v2
H = + +v + . (17)
LV 0
2m 2m 2m
e e e
Here, note that the breakdown of charge conjugation is no more manifest, once the relativistic dispersion
relationhasdegeneratedin asingle expressionfor particlesandanti-particles. Lookingateq. (17), the lasttwo
terms changethe nonrelativistic Hamiltonianonly by a constant,whichdoes notrepresentany physicalchange
6
(it just shifts the levels as a whole, not modifying the transition energies). Thus, just the first and the second
are able to induce modifications on a physical system. The purpose now is to investigate the contribution of
these two terms on the 1-particle wave functions (Ψ) of the hydrogen. It should be taken into account only
−→
the first term, once the hydrogen atom is initially regarded as a free system (A = 0). This contribution is
expectedtobenull,onceitrepresentsanaverageofthelinearmomentumonanatomicboundstate. Explicitly,
this energy quantity is correctly worked out as a first order perturbation on the corresponding 1-particle wave
functions, namely: ∆E = i hnlm|−→v ·−→∇|nlmi, where n,l,m are the usual quantum numbers that label the
me
1-particle wave function for the hydrogen atom, Ψ (r,θ,φ)=R (ρ)Θ (θ)Φ (φ). Replacing such a form in
nlm nl lm m
∆E, with the gradient operator written in spherical coordinates, it implies
∆E = i R (r)∗ ∂Rnl(r)|Θ (θ)|2|Φ (φ)|2−→v ·r+ |Rnl(r)|2|Φm(φ)|2Θ (θ)∗ ∂Θlm(θ)−→v ·θ
nl lm m lm
m ∂r r ∂θ
e Z (cid:26)
+im|Rnl(r)|2|Θlm(θ)|2|Φm(φ)|2−→v ·φ d3r. b b(18)
rsinθ
(cid:27)
For explicit calculation, the vector −→v can bebplaced along the z-axis, so that: −→v · r = v cosθ,−→v · θ =
z
−→
−v sinθ, v ·φ=0. Thus, one notes that the first two terms exhibit the presence of angular additional factors,
z
cosθ and sinθ, respectively. The first term is explicitly written as: b b
b
∆E = ivz R (r)∗ ∂Rnl(r)|Θ (θ)|2cosθ r2sinθdrdθ =0. (19)
1 nl lm
m ∂r
e Z (cid:20) (cid:21)
This null result is a consequence of π |Θ (θ)|2cosθ sinθdθ = 0, which holds for the associated Legendre
0 lm
functions. Following, the second term h i
R
∆E =−ivz |Rnl(r)|2Θ (θ)∗ ∂Θlm(θ)sinθ r2sinθdrdθ, (20)
2 lm
m r ∂θ
e Z " #
is now analyzed. The involved angular integration reads as πΘ (θ)(∂Θ /∂θ)sin2θdθ =
0 lm lm
1 [Θ (z)∂Θ /∂z](z2 −1)dz = 0, which comes out null as consequence of the recurrence relation, (z2 −
−1 lm lm R
R1)ddzΘlm(z)=lzΘlm(z)−(l+m)Θl−1,m(z),andofthefollowingorthogonalityrelation: 0πΘlm(z)Θpm(z)dz =
0, for l 6= p. Therefore, the total energy correction is null, that is: ∆E = 0. This means that the presence of
R
the Lorentz-violating background does not imply any energy shift in the hydrogen spectrum. It is instructive
to claim that this null correction is in full accordance with the role played by the term v ψγµψ: it only brings
µ
about a 4-momentum shift, pµ → pµ−vµ, without any physical consequence on the spectrum of the system.
It is also possible to understand it by reading the effect of the background as a gauge transformation. Indeed,
making use of a field redefinition, ψ → Ψ(x) = ψ(x)e−iv·x, it is possible to remove the background from the
theory,sothatLagrangian(2)takesontheusualfreeform(writtenintermsofthefieldΨ),namely: L´=L .
Dirac
This is true in any theory containing only one fermion field. For this result to remain valid in the case of a
multifermion theory, the fermions families should be uncoupled with each other (no interacting fermions) and
be coupled to the same Lorentz-violating parameter (v ) [2]-[3].
µ
Thegeneralresultprovidedbytherelativisticspectrumofthehydrogenmaybeattainedbytheexactsolution
of the modified Dirac equation (3), taken in the presence of the Coulombian potential. This solution, however,
will yield nothing new, once it corresponds exactly to the conventional relativistic solution shifted according
to pµ → pµ − vµ. Finally, it should be noted that this null outcome is not due to the specific choice of
the background spatial orientation, vµ = (v ,0,0,v ); by adopting a background along an arbitrary direction,
0 z
−→
v =(v ,v ,v ), identical calculations straightforwardly yield the same null result for ∆E.
x y z
So far, the hydrogen spectrum has been investigated only in the absence of external field. In the presence of
−→ −→
afixedmagneticfield,onenotesthatthetermeA· v/m ofeq. (17)maycontributetoafirstordercalculation
e
by the quantity:
∆EA·v = e Ψ∗ −→A ·−→v Ψd3r. (21)
m
e Z (cid:16) (cid:17)
7
−→ −→ −→ −→ −→
Knowing that A = −r × B/2, for a fixed magnetic field along the z-axis, B = B z, it results: A =
0
−B0(y/2,−x/2,0). This implies ∆EA·v = −(eB0/2me) Ψ∗(yvx−xvy)Ψd3r, whose explicit calculation leads
to ∆EA·v =0. Therefore, one concludes that the presenRce of a fixed external magnetic fieldbdoes not yield any
Lorentz-violating contribution to hydrogen spectrum besides the usual Zeeman effect.
It is instructive to remark that these calculations hold equivalently for the case of a positron, for which the
modified Pauli equation stems from (iγµ∂ +eγµA +v γµ−m)ψ = 0. In comparison with eq. (16), the
µ µ µ c
positron nonrelativistic Hamiltonian exhibits opposite charge and opposite vµ parameter, implying a Lorentz-
violating Hamiltonian in the form H = [−i−→v ·−→∇/m +e(−→A ·−→v)/m −v +−→v2/2m ]. However, as in the
LV e e 0 e
electron case, this Hamiltonian yields no physically detectable energy shift. This issue is obviously related to
the analysis of the hydrogen and antihydrogen spectroscopy, realized in wide sense in ref. [64]. In this work, it
isalsotakenintoaccountthe effectofthe Lorentz-violatingbackgroundonthe hyperfine structure(considering
the proton spin).
III. LORENTZ-VIOLATING DIRAC LAGRANGIAN (”AXIAL VECTOR” COUPLING)
Amongst the possible coefficients involved with the breaking of Lorentz symmetry in the fermion sector of
the SME, shown in eq. (1), our interest rest in one that is also CPT-odd, b ψγ γµψ. This torsion-like term [?
µ 5
]-[74]is linkedwiththe fixedbackgroundbymeansofanassignedaxialvectorcoupling. Takingitinto account,
one writes:
1 ←→
L= iψγµ ∂ ψ−m ψψ−b ψγ γµψ. (22)
µ e µ 5
2
The first step is to determine the new Dirac equation stemming from the above Lagrangian,namely:
(iγµ∂ −b γ γµ−m )ψ =0. (23)
µ µ 5 e
This modified equation is then rewritten in the momentum space,
(γµp −b γ γµ−m )w(p)=0, (24)
µ µ 5 e
provided a plane-wave solution is proposed. In order to obtain the dispersion relation associated with
such an equation, it should be multiplied by (γµp −b γ γµ+m ), so that one obtains: [p2 − m2 − b2 +
µ µ 5 e e
γ (p/b/−b/p/)]w(p)=0. This expression presents contributions out of the main diagonal of the spinor space. In
5
order to achieve an expression totally contained in the main diagonal, equally valid for each component of the
spinor w, the preceding equation shall be multiplied by (p2−m2−b2−γ (p/b/−b/p/), which yields the following
e 5
dispersion relation:
(p2−m2−b2)2+4p2b2−4(p·b)2 =0.
e
This is a fourth order relation for the energy that can be exactly solved only in special cases. In the case of
−→
a purely timelike background, bµ = (b ,0), and a purely spacelike background, bµ = (0, b), one respectively
0
achieves:
−→ −→
E =± p2+m2+b2±2b |p|, (25)
e 0 0
q
−→ −→ −→ −→ −→ 1/2
E =± p2+m2+ b2±2 m2 b2+(b · p)2 . (26)
e e
r
h i
Notice that there is no breakdown of charge conjugation in this case. In fact, after usual reinterpretation both
particle and anti-particle exhibit the same energy values, that is, the positive roots given in eqs. (25), (26).
8
Therefore, Lagrangian (22) does not imply C-violation. This may be explicitly demonstrated by means of the
procedure employed in Footnote 2.
Taking into account the γ−matrices definition, given at footnote 1, eq.(23) gives rise to two coupled spinor
equations for w and w :
A B
E−−→σ ·−→b −m w + b0−−→σ ·−→p w =0, (27)
e A B
−→σ(cid:16) ·−→p −b0 w + (cid:17)−E+−→(cid:0)σ ·−→b −m (cid:1)w =0, (28)
A e B
(cid:0) (cid:1) (cid:16) (cid:17)
leading to the following spinor relations:
w = 1 (E−m )(−→σ ·−→p)−(E−m )b0−b0(−→σ ·−→b)+−→b ·−→p +i−→σ ·−→c w , (29)
A E2 e e B
2(cid:26) (cid:27)
w = 1 (E+m )(−→σ ·−→p)−(E+m )b0−b0(−→σ ·−→b)+−→b ·−→p +i−→σ ·−→c w , (30)
B E2 e e A
1(cid:26) (cid:27)
where: −→c =−→b ×−→p,E2 = E+m )2−b·b ,E2 = E−m )2−b·b .
1 e 2 e
To construct the plane-wave solutions, one follows the general procedure adopted in the preceding section.
(cid:2)(cid:0) (cid:1)(cid:3) (cid:2)(cid:0) (cid:1)(cid:3)
The resulting 4×1 spinor solutions are given below:
1
0
u1 =N (E+m ) p −b0 −b0b +−→b ·−→p +ic /E2  , (31)
e z z z 1
 
 (Eh+me)(px+(cid:0) ipy)−(cid:1)b0(bx+iby)+i(cx+iicy) /E12 
 
(cid:2) 0 (cid:3)
1
u2 =N (E+m )(p −ip )−b0(b −ib )+i(c −ic ) /E2 , (32)
e x y x y x y 1
(cid:2) −(E+me) pz +b0 +b0bz+−→b ·−→p −icz /E(cid:3)12 
 
 h (cid:0) (cid:1) i 
(E+m )(p +b0)+b0b +−→b ·−→p −ic /E2
e z z z 2
v1 =N (Eh+me)(px+ipy)−b0(bx+iby)−i(cy+iicz) /E22  , (33)
1
 (cid:2) (cid:3) 
 0 
 
 (E+m )(p −ip )+b0(b −ib )−i(c −ic ) /E2
e x y x y x y 2
−(E+m ) p −b0 −b0b +−→b ·−→p +ic /E2
v =N (cid:2) e z z z (cid:3)2 , (34)
2
 h (cid:0) (cid:1) 0 i 
 1 
 
 
where N is the normalization constant. The eigenvalues of energy are the ones evalu-
ated in eqs. (25), (26) that are now exhibited in the following eigenenergy relations:
Hu = E(u)u , with E(u) = −→p2+m2+b2+(−1)i2b |−→p| 1/2, for bµ = (b ,0), and E(v) =
i i i i e 0 0 0 i
−→p2+m2+−→b2+(−1)i2[m2−→b2+(−→b(cid:2)·−→p)2]1/2 1/2, for bµ = (0,(cid:3)→−b), and i = 1,2. Here, E(u) stands for
e e i
hthe particle and anti-particle energy. Despite theicumbersome form of these spinors,it is possible to show that
in the case of bµ = (b ,0,0,b ) and p = (0,0,p ), such solutions are eigenstates of the spin operator Σ with
0 z z z
eigenvalues ±1, in much the same way as observed in the foregoing section.
9
A. Nonrelativistic limit
The nonrelativistic limit of the model described by Lagrangian (22) is now worked out in much the same
way of the previous section. The objective is to identify the corrected Hamiltonian and possible energy shifts
induced on the spectrum of hydrogen in the presence and absence of an external magnetic field. Considering
the presence of an external electromagnetic field minimally coupled to the spinor field:
1 ←→
L= iψγµD ψ−m ψψ−b ψγ γµψ, (35)
µ e µ 5
2
where D =∂ +ieA . Taking into account the external field, eqs. (27) and (28) take on the form:
µ µ µ
E−−→σ ·−→b −m −eA w + b0−−→σ ·(−→p −e−→A) w =0, (36)
e 0 A B
h−→σ ·(−→p −e−→A)−b0 w i− E−h−→σ ·−→b +m −eA iw =0. (37)
A e 0 B
h i h i
The low-energy limit is implemented by the following conditions: (−→p)2 ≪ m2, eA ≪ m , E = m +H.
−→ −→ e 0 e e
Furthermore, one still assumes that the factor σ · b must be neglected in eq. (37), once the background is
supposed to be small whenever compared with the electron mass. Implementing all these conditions, it holds
for the strong component:
Hw = −→σ ·(−→p −e−→A)−→σ ·(−→p −e−→A)−2b −→σ ·(−→p −e−→A)+b2 /2m +eA +−→σ ·−→b w , (38)
A 0 0 e 0 A
(cid:26)h i (cid:27)
After some algebraic calculations, one achieves:
H =H + −→σ ·−→b −2b −→σ ·(−→p −e−→A)/2m +b2/2m . (39)
Pauli 0 e 0 e
h i
This is the modified full Hamiltonian,composedby the Paulianda Lorentz-violatingpart(H ), wherein lies
LV
our interest. Provided that H has two interesting new terms (the third one is constant), one should try to
LV
figure out whether these terms imply real corrections to the spectrum of hydrogen. Taking into account these
−→ −→ −→ −→
informations, the effective Lorentz-violating Hamiltonian assumes the form: H = σ · b −2b (σ · p)/2m ,
LV 0 e
−→ −→ −→
where it was taken A =0. One then starts analyzing the term σ · b, whose first order contribution is:
−→ −→
∆Eσ·b =hnljmjms|σ · b|nljmjmsi. (40)
Here, n,l,j,m are the quantum numbers suitable to address a situation where occurs addition of angular
j
momenta (L and S). To solve this calculation, it is necessary to write the |jm i kets in terms of the spin
j
eigenstates |mm i, which is done by means of the general expression: |jm i = hmm |jm i |mm i, where
s j s j s
mX,ms
hmm |jm i are the Clebsch-Gordon coefficients. Evaluating such coefficients for the case j = l+1/2,m =
s j j
m+1/2,one has: |jm i=α |m↑i+α |m+1↓i;onethe other hand,forj =l−1/2,m =m+1/2,itresults:
j 1 2 j
|jm i = α |m ↑i−α |m+1 ↓i, with: α = (l+m+1)/(2l+1),α = (l−m)/(2l+1). Now, taking into
j 2 1 1 2
accountthe orthonormalizationrelationhm′mp′s|mmsi=δm′mδm′sms,itisppossibleto showthateq.(40)reduces
simply to ∆Eσ·b =hjmj|σzbz|jmji, whose explicit calculation leads to:
b m
z j
∆Eσ·b =± , (41)
2l+1
10
where the positive and negative signs correspond to j = l+1/2 and j = l−1/2, respectively. Thus, in this
first order evaluation the energy turns out correctedby a quantity depending on ±m , in a very similar way to
j
the well-known Zeeman effect. Indeed, each line of the spectrum is split into (2j+1) lines, with a b /(2l+1)
z
linear separation. This correction was also obtained in ref. [70]. Once the magnitude of such splitting depends
directly on the modulus of the background,this theoretical outcome may be used to set up an upper bound on
the breaking parameter (bµ).
Next,oneevaluatesthefirstordercontributionofthesecondtermofH tothehydrogenspectrum,namely:
LV
ib0 −→ −→
∆Eσ·p = hnljmjms|σ ·∇|nljmjmsi, (42)
m
e
The 1-particle wave function,Ψ = ψ (r,θ,φ)χ , now contains a spin function, χ . In order to
nljmjms nljmj sms sms −→
solve eq. (42), one should note that the gradient operator acts on the spatial function ψ , whereas σ
nljmj
operates on the spin function, so that it reads:
= ib0 R (r)∗ ∂Rnl(r)|Θ (θ)|2|Φ (φ)|2hjm |−→σ ·r|jm i+ |Rnl(r)|2|Φm(φ)|2Θ (θ)∗
nl lm m j j lm
m ∂r r
e Z (cid:26)
∂Θlm(θ)hjm |−→σ ·θ|jm i+ |Rnl(r)|2|Θlm(θ)|2|Φm(φ)|2bhjm |(−→σ ·φ)|jm i d3r. (43)
j j j j
∂θ rsinθ
(cid:27)
b −→b
Writing the sphericalversorsintermsofthe Cartesianones,oneobtains:σ ·r =sinθcosφσ +sinθsinφσ +
x y
−→ −→
cosθσ , σ ·θ =cosθcosφσ +cosθsinφσ −sinθσ , σ ·φ=−sinφσ +cosφσ .Itisclearthatonlythe terms
z x y z x y
proportional to σz yield non-null expectation values on the kets |jmji, which ibmplies:
b b
±ib m ∂R |R |2 ∂Θ
∆Eσ·p = (2l+01)mj Rn∗l(r) ∂rnl |Θlm(θ)|2cosθ− nrl Θ∗lm(θ) ∂θlm sinθ d3r. (44)
e Z (cid:26) (cid:27)
These are exactly the same integrals involved in the expressions of ∆E and ∆E , already evaluated in the
1 2
previous section. So, it is obvious that: ∆Eσ·p = 0. Hence, the sole non-null first order effect on the hydrogen
−→ −→
spectrum is a Zeeman-like splitting stemming from the correction term σ · b.
−→ −→
Another point that deserves attention is related to the correction term 2eb σ · A, present in eq. (39). This
0
−→
term is obviously null for the ”free” hydrogen atom (once A = 0). For the case the atom is subjected to the
influence of an external magnetic field, however, this term must be taken into account. For a fixed magnetic
−→ −→
field along the z-axis, B =B z, one has A =−B (y/2,−x/2,0), so that the correction may be written as:
0 0
b0e −→ −→ B0b0e
∆Eσ·A = hnljbmjms|σ · A|nljmjmsi=− hnljmjms|yσx−xσy|nljmjmsi. (45)
m 2m
e e
Considering the effect of the spin operators on the kets |jmji, a null correction (∆Eσ·A =0) turns out. One
should remark that this result remains null even for an arbitrary orientation of the magnetic field. Therefore,
theconclusionisthatanexternalfixedfielddoesnotimplyanyadditionalcorrectiontothewell-knownZeeman
effect. In this case, the Lorentz-violating effect of eq. (41) corrects the usual Zeeman splitting just by a small
−→
quantity proportionalto |b|. The general result provided by the relativistic spectrum of the hydrogen may be
examinedby the exactsolutionofthe modified Dirac equation(3)in the presenceofthe Coulombianpotential.
This case implies qualitative modifications on the usual relativistic hydrogen spectrum, both in the case of a
purely timelike or purely spacelike background. It is now under development.
IV. CONCLUSION
Inthiswork,theeffectsofCPT-andLorentz-violatingbackgroundterms(stemmingfromamorefundamental
theory) on the Dirac equation have been studied. This analysis has considered two different ways of coupling
the fermion field to the background. One has started with the vector coupling, for which the modified Dirac