Matter-gravity couplings and Lorentz violation

0.72 MB·English

by  Alan Kostelecky

Matter-gravity couplings and Lorentz violation
V. Alan Kostelecky´ and Jay D. Tasson
Physics Department, Indiana University, Bloomington, IN 47405, U.S.A.
(Dated: IUHET544, June 2010; version published in Phys.Rev. D 83, 016013 (2011))
The gravitational couplings of matter are studied in the presence of Lorentz and CPT violation.
At leading order in the coefficients for Lorentz violation, the relativistic quantum hamiltonian is
derivedfromthegravitationallycoupledminimalStandard-ModelExtension. Forspin-independent
effects,thenonrelativisticquantumhamiltonianandtheclassicaldynamicsfortestandsourcebodies
are obtained. A systematic perturbative method is developed to treat small metric and coefficient
fluctuationsaboutaLorentz-violatingandMinkowskibackground. Thepost-newtonianmetricand
the trajectory of a test body freely falling under gravity in the presence of Lorentz violation are
established. Anillustrative exampleispresentedfor abumblebeemodel. Thegeneral methodology
is used to identify observable signals of Lorentz and CPT violation in a variety of gravitational
1
experimentsand observations, includinggravimetermeasurements, laboratory and satellite tests of
1
theweak equivalenceprinciple, antimatter studies, solar-system observations, and investigations of
0
thegravitational properties oflight. Numeroussensitivities tocoefficientsfor Lorentzviolation can
2 be achieved in existing or near-future experiments at the level of parts in 103 down to parts in
n 1015. Certain coefficients are uniquely detectable in gravitational searches and remain unmeasured
a todate.
J

4
2 I. INTRODUCTION theequationforthetrajectoryofatestbodymovingun-
der gravityin the presenceofLorentz violation,allowing

] alsofor Lorentz-violatingeffects fromthe compositionof
c General Relativity (GR) is known to provide an accu-
the test and source bodies and for effects from possible
q ratedescriptionofclassicalgravitationalphenomenaover
- additional long-range modes associated with Lorentz vi-
a wide rangeof distance scales. A foundational property
r olation. We also seek to explore the implications of our
g of the gravitational couplings of matter in GR is local
[ analysis in a wide variety of experimental and observa-
Lorentz invariance in freely falling frames. The realiza-
2 tion that a consistent theory of quantum gravity at the tional scenarios, identifying prospective measurable sig-
nals and thereby enabling more complete searches using
v Planck scale mP ≃ 1019 GeV could induce tiny man- matter-gravity couplings.
6 ifestations of Lorentz violation at observable scales [1]
0 hasrevivedinterestinstudiesofLorentzsymmetry,with Despite the current lack of a satisfactory quantum
1
numerous sensitive searches for Lorentz violation being theory of gravity, established gravitational and particle
4
undertaken in recent years [2]. phenomena at accessible energy scales can successfully
.
6 Gravitational signals of Lorentz violation are more be analyzed using the field-theoretic combination of GR
0
challenging to study than ones in Minkowski spacetime and the Standard Model (SM). This combination there-
0
1 for several reasons, including the comparative weakness fore serves as a suitable starting point for a comprehen-
: of gravity at the microscopic level and the impossibility sive effective field theory describing observable signals
v of screening gravitational effects on macroscopic scales. of Planck-scale Lorentz and CPT violation in gravity
i
X Bothforpurelygravitationalinteractionsandformatter- and particle physics [9]. The present paper adopts this
r gravitycouplings,Lorentzviolationscanbeclassifiedand generalframework,knownasthegravitationalStandard-
a enumerated in effective field theory [3]. Several searches ModelExtension(SME)[3],toanalyzeLorentzviolation
for purely gravitational Lorentz violations in this con- inmatter-gravitycouplings. EachtermviolatingLorentz
text have recently been performed [4–6] using a treat- symmetry in the SME Lagrange density is a scalar den-
ment in the post-newtonian regime [7]. These results sity under observer general coordinate transformations
enlarge and complement the impressive breadth of tests and consists of a Lorentz-violating operator multiplied
ofGRperformedinthecontextoftheparametrizedpost- by a controlling coefficient for Lorentz violation. Under
newtonian (PPN) formalism [8]. mildassumptions,CPTviolationineffective fieldtheory
comes with Lorentz violation [10], so the SME also de-
Althoughthecouplingbetweenmatterandgravityhas
scribes generalbreaking of CPT symmetry. This feature
historically been a primary source of insights into the
playsacrucialroleforcertainsignalsofLorentzviolation
properties of the gravitational field, a general study of
in what follows.
matter-gravity couplings allowing for Lorentz violation
in the context of effective field theory has been lacking In this work, our focus is on gravitational Lorentz vi-
todate. Inthiswork,weaddressthisgapintheliterature olationin matter-gravitycouplings, both with and with-
andinvestigatethe prospects for searchesfor Lorentz vi- out CPT violation. These couplings introduce operator
olation involving matter-gravity couplings. Our goal is structures offering sensitivity to coefficients for Lorentz
to elucidate both theoretical and experimental aspects violationthatareintrinsicallyunobservableinMinkowski
of the subject. We seek a post-newtonian expansion for spacetime. In fact, comparatively large gravitational
2
Lorentz violation in nature could have remained unde- field theory. One technical issue is extracting a mean-
tected in searches to date because gravity can provide ingful quantum theory in the presence of gravitational
a countershading effect [11], so this line of investigation fluctuations. We resolve this issue via a judicious field
has a definite discovery potential. Several searches for redefinition, which yields a hamiltonian that is hermi-
gravitationalLorentzviolationhaveledtoconstraintson tian with respect to the usual scalar product for wave
SME coefficients for Lorentz violation with sensitivities functions and that reduces correctly to known limiting
downtopartsin109 [4–6,12],andadditionalconstraints cases. We construct the relativistic quantum hamilto-
can be inferred by reanalysis of data from equivalence- nian at leading order in Lorentz violation and gravity
principle tests [11]. fluctuations. For the spin-independent terms, we per-
ThenatureoftheLorentzviolationplaysacrucialrole form a Foldy-Wouthuysen transformation to obtain the
in determining the physics of matter-gravity couplings. nonrelativistic hamiltonian. Section IV treats the classi-
In Riemann geometry, externally prescribing the coef- caldynamics correspondingto the quantum theory. The
ficients for Lorentz violation as fixed background con- pointparticleactionispresentedandthestructureoftest
figurations is generically incompatible with the Bianchi and source bodies is discussed. The equations of motion
identities and hence problematic [3]. This issue can be foratestparticleandthemodifiedEinsteinequationsare
avoidedviaspontaneousLorentzbreaking[1], inwhicha derived. We describethemethodologyforhandlingcoef-
potential term drives the dynamical development of one ficientandmetricfluctuations. Combiningtheresultsde-
or more nonzero vacuum values for a tensor field. This termines the Lorentz-violating trajectory of a test body.
mechanism implies the underlying Lagrange density is The results are illustrated in Sec. V in the context of a
Lorentzinvariant,sothecoefficientsforLorentzviolation special class of models of spontaneous Lorentz violation
are expressed in terms of vacuum values and can there- known as bumblebee models.
fore serve as dynamically consistent backgrounds satis-
The remaining research sections of the paper, Secs.
fying the Bianchi identities. The presence of a potential
VI-XI, concern implications of our theoretical analysis
driving spontaneous Lorentz violation implies the emer-
for experiments and observations. Section VI contains
genceofmasslessNambu-Goldstone(NG)modes[13]as-
basic facts concerning frame choices and outlines the
sociatedwith field fluctuations along the brokenLorentz
canonical Sun-centered frame used for reporting mea-
generators[14]. Ifthepotentialissmooth,massivemodes
surements. We also consider the sensitivities to coef-
can also appear [15]. Some features of the NG and mas-
ficients for Lorentz violation that can be attained in
sivemodesaregeneric,whileothersarespecifictodetails
practical situations. Section VII treats laboratory tests
ofthemodelbeingconsidered. Inanycase,thenatureof
near the surface of the Earth using neutral bulk mat-
these modes plays a key role in determining the physical
ter, neutral atoms, or neutrons. The theoretical descrip-
implications of spontaneous Lorentz violation.
tion of these tests is presented to third post-newtonian
For the purposes of the present work, the presence of order, and some generic features of the test-body mo-
NG modes is particularly crucial because they can cou- tion are discussed. A wide variety of gravimeter and
ple to matter and can transmit a long-range force. A equivalence-principle tests is analyzed for sensitivities to
careful treatment of these modes is therefore a prereq- coefficients for Lorentz violation that are presently un-
uisite for studies of Lorentz violation in matter-gravity constrained. Satellite-based searches for Lorentz viola-
couplings. Inwhatfollows,we developamethodologyto tion using equivalence-principle experiments are studied
extractthedominantLorentz-violatingeffectsinmatter- in Sec. VIII. A generic situation is analyzed, and the
gravity couplings irrespective of the details of the un- resultsareappliedtomajorproposedsatellitetests. Sec-
derlying model for spontaneous Lorentz violation. In ef- tion IX treats gravitational searches using charged par-
fect,theNGmodesaretreatedviaaperturbationscheme ticles, antihydrogen, and particles from the second and
that takes advantage of symmetry properties of the un- third generations of the SM. Estimated sensitivities in
derlying Lagrange density to eliminate them in favor of future tests are provided, and illustrative toy models for
gravitationalfluctuationsandbackgroundcoefficientsfor antihydrogenstudies are discussed. Searches for Lorentz
Lorentzviolation. ThistreatmentallowsleadingLorentz- violation using solar-system observations are described
violating effects from a large class of plausible models to in Sec. X. We consider measurements of coefficients for
be handled in a single analysis. Lorentz violation accessible via lunar and satellite rang-
The portionofthis paperdevelopingtheoreticalissues ing and via studies of perihelion precession. Section XI
spansSecs.II-V. ItbeginsinSec.II withareviewofthe addresses various tests involving the effects of gravita-
SME framework. We present the field-theoretic action, tional Lorentz violation on the properties of light. We
describeitslinearization,anddiscussobservabilityissues analyzetheShapirotimedelay,thegravitationalDoppler
forthecoefficientsforLorentzviolation. Wealsodescribe shift, and the gravitational redshift, and we consider
the two notions of perturbative order used in the subse- the implications for a variety of existing and proposed
quent analysis, one involving Lorentz and gravitational searches of these types. Finally, in Sec. XII we summa-
fluctuationsandtheotherbasedonapost-newtonianex- rize the paper and tabulate the various estimated actual
pansion. SectionIII concerns the relativistic andnonrel- andattainablesensitivitiestocoefficientsforLorentzand
ativistic quantum mechanics arising from the quantum CPT violation obtained in the body of this work.
3
Throughout the paper, we follow the conventions of Inthepresentcontext,theactionofthecovariantderiva-
Refs. [3] and [7]. In particular, the Minkowski metric is tive D on ψ is
µ
diagonalwithentries( 1,+1,+1,+1). Greekindicesare
− D ψ ∂ ψ+ 1iω abσ ψ. (4)
used for spacetime coordinates, while Latin indices are µ ≡ µ 4 µ ab
used for local Lorentz coordinates. Appendix A of Ref. It is convenient to introduce the symbol
[3] provides a summary of most other conventions. Note
that parentheses surrounding index pairs in the present (ψDµ)≡∂µψ− 41iωµabψσab (5)
work denote symmetrization with a factor of one-half.
for the action of the covariant derivative on the Dirac-
conjugate field ψ. The action (3) contains the covariant
derivative in a combination defined by
II. FRAMEWORK
↔
χΓa D ψ χΓaD ψ (χD )Γaψ. (6)
µ µ µ
≡ −
The focus of this work is the study of relativity viola- ThesymbolsΓa andM appearingintheaction(3)are
tions in realistic matter-gravity interactions. The basic
defined by
fieldtheoryofrelevanceconcernsasinglefermionfieldψ
coupled to dynamical gravity and incorporating Lorentz Γa ≡ γa−cµνeνaeµbγb−dµνeνaeµbγ5γb
and CPT violation. In this section, we summarize the e eµa if eµaγ 1g eνaeλ eµ σbc (7)
action for the model, describe the linearization proce- − µ − µ 5− 2 λµν b c
dure, discuss conditions for the observability of effects, and
and present the perturbation scheme developed for the M m+a eµ γa+b eµ γ γa+ 1H eµ eν σab. (8)
analysis to follow. ≡ µ a µ a 5 2 µν a b
The first term of Eq. (7) leads to the usual Lorentz-
invariant kinetic term for the Dirac field, while the first
A. Action termofEq.(8)leadstoaLorentz-invariantmass. Aterm
of the form im γ could also appear in M, but here we
5 5
suppose it is absorbed into m via a chiral field redefini-
The theory of interest is a special case of the gravi-
tion. The coefficient fields for Lorentz violation a , b ,
tationally coupled SME [3]. The action can be written µ µ
c , d , e , f , g , H typically vary with space-
as µν µν µ µ λµν µν
time position. The coefficient field H is antisymmet-
µν
S =S +S +S′. (1) ric, while gλµν is antisymmetric in λµ. Note the use of
G ψ
anuppercaseletterforH ,whichavoidsconfusionwith
µν
the metric fluctuation h . The CPT-odd operators for
ThefirstterminthisexpressionistheactionS contain- µν
G
Lorentzviolationareassociatedwiththecoefficientfields
ingthe dynamicsofthe gravitationalfield, includingany
a , b , e , f , and g .
coefficientsfor Lorentzviolationinthatsector. The geo- µ µ µ µ λµν
The form of the action (3) implies the torsion Tλ
metricframeworkisaRiemann-Cartanspacetime,which µν
allowsboththeRiemanncurvaturetensorRκ andthe entersthefermionactiononlyviaminimalcoupling. This
λµν couplinghasthesameformasthatofthecoefficientfield
torsion tensor Tλ . To incorporate fermion-gravity in-
µν b , so the effects of minimal torsion can be incorporated
µ
teractions, the vierbein formalism [16] is adopted, with
into a matter-sector analysis by replacing b with the
the vierbein eµ and the spin connection ω ab taken as µ
a µ effective coefficient field
the fundamental gravitational objects. In the limit of
zero torsion and Lorentz invariance, SG reduces to the (beff)µ ≡bµ+ 81Tαβγǫαβγµ. (9)
Einstein-Hilbert action of General Relativity,
Note that nonminimal torsion couplings can be incorpo-
rated into the more general coefficient fields appearing
1
S d4x (eR 2eΛ), (2) inthe full SME.Nonminimaltorsioncouplingsandtheir
G
→ 2κ −
Z experimental constraints are discussed in Ref. [17].
ThefinalterminEq.(1)istheactionS′containingthe
where κ 8πG , e is the vierbein determinant, and Λ
N
≡ dynamicsassociatedwiththecoefficientfieldsforLorentz
is the cosmological constant.
violation. Addressing possible contributions from this
The second term in Eq. (1) is the action S for the
ψ sector is the subject of Sec. IVC.
fermion sector of the SME. In this work, we limit at-
tention to terms in this sector with no more than one
derivative, which is the gravitationallycoupled analogue
B. Linearization
of the minimal SME in Minkowski spacetime. In this
limit, the action S for a single Dirac fermion ψ of mass
ψ
For the purposes of this work, it suffices to consider
m can be written as
weakgravitationalfieldsinaMinkowski-spacetimeback-
↔ ground. Under these circumstances, the Latin local in-
S = d4x(1ieeµ ψΓa D ψ eψMψ). (3)
ψ 2 a µ − dices can be replaced with Greek spacetime indices, so
Z
4
the weak-field forms of the metric, vierbein, and spin C. Observability
connection can be written as
A given coefficient for Lorentz violation can lead to
g = η +h ,
µν µν µν observable effects only if it cannot be eliminated from
eµν = ηνσeµσ ≈ ηµν + 12hµν +χµν, the Lagrangedensity viafield redefinitionsorcoordinate
ω = η η ω ρσ choices [3, 18–25]. In this subsection, we outline some
λµν µρ νσ λ implications of this fact relevant to the present work.
∂ χ 1∂ h + 1∂ h
≈ λ µν − 2 µ λν 2 ν λµ
+1(T +T T ). (10)
2 λµν µλν − νλµ
1. Field redefinitions
The quantities χ contain the six Lorentz degrees of
µν
freedom in the vierbein. One result of key interest here is that matter-gravity
ThecoefficientfieldsforLorentzviolationareexpected couplings can obstruct the removal of some coefficients
to acquire vacuum values through spontaneous Lorentz that are unphysical in the Minkowski-spacetime limit.
breaking. An arbitrary coefficient field tλµν... can there- For example, in the single-fermion theory in Minkowski
fore be expanded about its vacuum value tλµν..., spacetime, the coefficient aµ aµ for Lorentz and CPT
≡
violation in Eq. (8) is unobservable because it can be
tλµν... = tλµν...+t7λµν..., (11) eliminated by the spinor redefinition
wherethefluctuationt7λµν... includeesmasslessNGmodes ψ(x) exp[if(x)]ψ(x) (14)
→
and massive modes [14, 15]. The vacuum value t
λµν...
tλµν... iscalledthecoeefficientforLorentzviolation. On≡e with f(x) = aµxµ. However, in Riemann or Riemann-
chaninsiteadchoosetoexpandthecontravariantcoefficient Cartan spacetimes we have aµ aµ+a7 µ, so this redefi-
≡
field tλµν..., nitiontypicallyleavesthe four componentsofthe fluctu-
ationa7 µ inthetheory. Instead,theredeefinition(14)with
tλµν... = tλµν...+tλµν..., (12) anappropriatef(x)canbe usedtomoveonecomponent
ofthecoefficientfielda intotheotherthree,unlessa is
e µ µ
wheretλµν... isrelatedtotλµν... byeraisingwithηµν. The constantor the total derivative ofa scalar[3]. Note that
inthepresenceofgravitythisfreedommaybeinsufficient
reader is cautioned that the relation between t7λµν... and toeliminateanycomponentsofthecoefficienta because
the index-lowered version t of tλµν... involves terms µ
λµν... the components of the fluctuation a7 µ can depend on all
containing contractions of tλµν... with hµν. Tehis paper four components of a through the equations of motion.
µ
usestheexpansion(11)anedgivesexepressionsintermsof This line of reasoningshows that gravitationalcouplings
e
t7λµν.... provide a unique sensitivity to the coefficient a , which
µ
To provide a smooth match between our analysis and
can be exploited in various experiments [11].
eprevious work on the matter sector of the SME in Anothertypeoffieldredefinitioncanbe writteninthe
Minkowski spacetime and on the gravitational sector in
generic form [3]
asymptotically Minkowski spacetime, we make two as-
sumptions about the coefficients for Lorentz violation. ψ(x) [1+v(x) Γ]ψ(x), (15)
First, we assume they are constant in asymptotically in- → ·
ertial cartesian coordinates, where Γ represents one of γa, γ γa, σab and v(x) is a
5
complex function carrying the appropriate local Lorentz
∂αtλµν... =0. (13) indices. Thiscanbeviewedasaposition-dependentcom-
ponentmixinginspinorspace. Fieldredefinitionsofthis
This preserves translation invariance and hence
typecanbeusedtodemonstratetheleading-orderequiv-
energy-momentum conservation in the asymptoti- alenceofobservablephysicaleffects due tocertaincoeffi-
cally Minkowski regime. It also ensures that our barred
cients for Lorentz violation. An example relevant in the
coefficients correspond to the usual coefficients for present context is a redefinition involving a real vector
Lorentz and CPT violation investigated in the minimal
v (x). Together with assumption (13), this redefinition
a
SME in Minkowski spacetime [18]. Second, we assume can be used to show that at leading order in Lorentz vi-
thatthe vacuumvaluest aresufficientlysmalltobe
λµν... olation the coefficients a and e always appear in the
µ µ
treated perturbatively. This is standard and plausible,
combination
since any Lorentz violation in nature is expected to
be small. These two assumptions suffice for most (a ) a me , (16)
eff µ µ µ
of the analysis that follows. To obtain the leading ≡ −
Lorentz-violating corrections to h without specifying up to derivatives of fluctuations. Combining this result
µν
a dynamical model for the coefficient fields for Lorentz with the above discussion of the redefinition (14) shows
violation, one further assumption is required, which is that observables involving gravitational couplings offer
presented in Sec. IVC. the prospect of measuring (a ) .
eff µ
5
Related ideas can be used to simplify the weak-field Alternatively,onecouldperformthecoordinatetransfor-
limit of the theory (3). In particular, the antisymmet- mation
ric part χ of the vierbein can be removed everywhere
except forµνpossible contributions to fluctuations of the xµ xµ′ =xµ+cµ xν, (20)
→ ν
coefficient fields, by applying the field redefinition
which instead redefines the fermion-sector background
ψ(x) → exp[−14iχµν(x)σµν]ψ(x) mfecettirvice tpohobteonη-µsνecatonrdcaoteffilecaideinntg2ocrde+r p(rkod)uαces .thTeheifs-
1 1iχ σµν 1 χ χ σµνσαβ ψ(x). µν F µαν
≈ − 4 µν − 32 µν αβ coordinatechoiceis equally validforanalysis,andasbe-
(cid:0) (cid:1) (17) fore the orthogonal combination 2cµν −(kF)αµαν is un-
observable.
Note that this redefinition takes the form of a Lorentz
Similar results apply for experimental searches for
transformation on ψ but that the other fields in the La-
Lorentz violation involving comparisons of different
grange density remain unaffected. Note also that in the
fermion species. Labeling the species by w, each has
absenceofLorentzviolationχµν canberemovedentirely, a coefficient cw . Then, for example, the effective metric
a fact compatible with the interpretation in Ref. [14] of µν
for any one species X can be reduced to η by a co-
µν
the role of χµν in Lorentz-violating theories. In the re- ordinate transformation with cX of the form (20). The
mainderofthiswork,weassumetheredefinition(17)has µν
resulting effective coefficients for the remaining species
been performed on the Lagrange density, so that quan- involvethedifferencescw cX ,whichinthiscoordinate
tities such as Γa, M, and eµ are understood to acquire µν− µν
a scheme become the relevant observable combinations of
no contributions from χ except possibly through the
µν coefficients.
fluctuations ofthe coefficientfields forLorentzviolation.
In the gravity sector, the situation is more involved
because a geometrically consistent treatment of Lorentz
violation generically requires the incorporationof effects
2. Coordinate choices
from the NG modes to ensure the Bianchi identities are
satisfied [3]. It turns out that the 10 relevant coefficient
The observability of certain combinations of coeffi- components in the gravity sector are the vacuum values
cientsforLorentzviolationisalsoaffectedbythefreedom s ofthe coefficientfieldss inthegravity-sectorSME
µν µν
to make coordinate choices. Intuitively, the key point is Lagrange density
thatanyonesectorofthe SMEcanbe usedtodefinethe
scales of the four coordinates, to establish the meaning 1 esµνR . (21)
of isotropy, and to set the synchronization scheme. The Lgravity ⊃ 16πG µν
N
freedomthereforeexiststochoosethesectorinwhichthe
effective background spacetime metric takes the form of We find that at leading order the transformation (19)
the Minkowski metric ηµν. This implies that in any ex- generates an accompanying shift sµν →sµν −(kF)αµαν,
perimental configuration there are always 10 unobserv- while the transformation (20) produces the shift sµν
→
able combinations of coefficients for Lorentz violation. sµν +2cµν.
As an illustration, consider the SME restricted to the One way to obtain these results is to consider the
single-fermionandphotonsectors[18,22]. Inthefermion leading-order effect of a metric shift on the equations of
sector,the10relevantcoefficientcomponentsarethevac- motion, and then to match to the known results [7] for
uum values c of the coefficient fields c in Eq.(7) be- observable effects in a post-newtonian expansion. Con-
µν µν
cause these coefficients enter S in the same way as the sider, for example, the restriction of the SME to the
ψ
metric. In the photon sector,the SME Lagrangedensity Einstein-Hilbert action and the single-fermion action Sψ
contains a term with nonzero cµν only. At leading order, the coordinate
transformation(20)removesc fromthe fermionaction
µν
Lphoton ⊃−14e(kF)κλµνFκλFµν, (18) atthecostofintroducingametricshiftgµν →gµν−2cµν
in the Einstein-Hilbert term. The resulting equations
andthe 10relevantcoefficientcomponentscanbe shown of motion involve the Einstein tensor G (g 2c) with
µν
−
to be the trace (k )α . At leading order, the coordi- shifted argument, which can be written in terms of the
F µαν
nate transformation Einstein tensor G (g) for the originalmetric and an ef-
µν
fective energy-momentum tensor φc . We find
xµ xµ′ =xµ 1(k )αµ xν (19) µν
→ − 2 F αν
G (g 2c) = G (g) φc ,
redefines the background metric to take the form η in µν − µν − µν
µν φc 2(η cαβR 2cα R + 1c R
the photon sector. The effective metric in the fermion µν ≈ µν αβ − (µ ν)α 2 µν
sector is then also changed, with the observable coeffi- +cαβR ). (22)
αµνβ
cient combination becoming c +(k )α /2. The or-
µν F µαν
tahbolegoinnaalncyoemxbpienraimtioennt2sciµnνv−olv(iknFg)αonµαlyνtihsetsheutswuonsoebcsteorrvs-. Tnehwetotrnaiacen-rteevremrseΦdsforamrisΦincµgν forfomφcµνEqm.a(t2ch1)esatnhde gpiovsetn-
µν
6
explicitly as Eq. (24) of Ref. [7] with the combination initialstage wouldcomplicate the ensuing analysis,so in
s replaced by 2c . The transformation (20) there- what follows we often write results in terms of h while
µν µν µν
fore produces the shift s s + 2c , as claimed. commenting as needed on the post-newtonian counting.
µν µν µν
→
A similar line of reasoning verifies the claimed shift To preserve a reasonable scope, this work focuses on
s s (k )α for the coordinate transformation dominant perturbative effects involving both Lorentz vi-
µν → µν − F µαν
(19). olationandgravity. We nextdiscussthe relevantpertur-
Tokeepexpressionscompactthroughoutthiswork,we bative orders required to achieve this goal.
choose to work with coordinates satisfying Considerfirstcontributionsfromthefluctuationt7λµν...
(kF)αµαν =0. (23) odfepthenedcsooenfficthieentnafiteulrdes.ofTthheedaecttaioilnedS′stinruEctqu.r(e1)o.fIet7nλµtνh..e.
To obtain results valid for arbitrary coordinate choices, scheme adopted here, t7λµν... can be viewed as a sereies in
t and h of the form
the following substitutions can be applied throughout: λµν... µν
e
cwµν → cwµν + 12(kF)αµαν, t7λµν... = t7(λ0µ,ν0)...+t7(λ0µ,ν1)...+t7(λ1µ,ν0)...+t7(λ1µ,ν1)...+.... (25)
sµν → sµν −(kF)αµαν. (24) Foer spontaneeous Loreentz breaeking, t7λeµν... includes mas-
sivemodesandmasslessNGmodes[15,26]. Inthiswork,
We emphasize that all coordinate choices are equivalent we suppose the massive modes eitheer are frozen or have
for theoretical work or for data analysis, with the choice negligibledegreeofexcitation. Incorporatingtheirpossi-
(23) adopted here being purely one of convenience. bleeffectsintotheanalysisofmatter-gravityphenomena
wouldbeofpotentialinterestbutliesbeyondourpresent
scope. In contrast, the massless NG modes play a key
D. Perturbation scheme roleinwhatfollows. Theirfatecanincludeidentification
withthephotoninEinstein-Maxwelltheory[14],withthe
Inthiswork,weareinterestedinexperimentalsearches gravitonin GR [27, 28], or with a new force [11, 29, 30],
for Lorentz violation involving gravitational effects on or they can be absorbed in the torsion via the Lorentz-
matter. Many of these searches involve test particles Higgs effect [14]. In some models the NG modes can be
moving in background solutions to the equations of mo- interpreted as composite photons [31] or gravitons [32].
tion for gravity and for the coefficient fields. Since the We consider below the perturbative orders required for
gravitational fields involved are weak and since no com- the various possibilities.
pelling evidence for Lorentz violation exists to date, any SupposefirsttheNGmodescorrespondtophotons,or
effects are expected to be small. We can therefore focus more generally to a known force field other than grav-
attention on perturbative modifications to the behavior ity. The termt7(0,0) then contains conventionalLorentz-
λµν...
oftestparticles. Thissubsectiondescribestheschemewe invariant terms describing this field in Minkowskispace-
usetotrackperturbativeordersintheconstructionofthe (0,1)
time, while t7 e contains conventional leading-order
relativistic quantum hamiltonian and in the subsequent λµν...
gravitational interactions with the field. Effects from
developments.
both these teerms are therefore part of the conventional
Perturbative effects on physical observables can arise descriptionofthe force. The termt7(1,0) describespossi-
throughmodificationstothebackgroundcoefficientfields λµν...
ble Lorentz violations in Minkowski spacetime involving
t forLorentzviolationandtothebackgroundgravi-
λµν... theknownfield,manyofwhichareetightlyconstrainedby
tationalfieldg ,ordirectlythroughmodificationstothe
µν experiments[2]. For the purposesofthis work,whichfo-
equation of motion for the test particle. The analysis of
cuses on Lorentz violationinvolvinggravity,we can take
these effects is simplified by introducing an appropriate
this term as experimentally negligible. The dominant
notion of perturbative order. Several ordering schemes
term of interest is therefore t7(1,1) , which lies at O(1,1).
arepossible. Inthiswork,weadoptaschemethattracks λµν...
If the NG modes correspond to gravitons as, for ex-
the orders in the coefficients for Lorentz violation t
λµν...
and in the metric fluctuation h . The overall pertur- ample, in the cardinal modeel [27], then the expansion
µν
bative order of a given term in an equation is denoted (25) contains no terms at O(m,0). The term t7(0,1) cor-
λµν...
as O(m,n), where m represents the order in tλµν... and responds to the gravitational fluctuations hµν, and its
n the order in h . Within this scheme, the fluctua- effectsarepartoftheconventionaldescriptioneofgravity.
µν
tions t7λµν... of the coefficient fields for Lorentz violation The dominant term of interest is therefore again t7λ(1µ,ν1)....
are viewed as secondary quantities that are determined IfinsteadtheNGmodescorrespondtoapresentlyun-
via theeir equations of motion in terms of the coefficients observed force field, then t7(0,0) and t7(0,1) descriebe un-
λµν... λµν...
forLorentzviolationandthegravitationalfield. Thereis observed Lorentz-invariant effects in Minkowski space-
alsoasubsidiarynotionoforderassociatedwiththeusual time and in leading-ordergeravitationaelcouplings. These
post-newtonianexpansionofh itself. Wedenoteapth- modes must therefore be eliminated from the analy-
µν
orderterminthis latterexpansionasPNO(p). However, sis prior to interpretation of observations, via solving
performing an explicit post-newtonian expansion at the the equations of motion or otherwise. In what follows,
7
we suppose this elimination has been performed where Lorentz-invariant part of the hamiltonian can therefore
needed. The term t7(1,0) describes possible Lorentz vi- be truncated at O(0,2), while the Lorentz-violating part
λµν...
olation in Minkowski-spacetime involving the unknown canbetruncatedatO(1,1). Tomaintainconsistentpost-
field. Forpresentpuerposes,wetakethistermtobeexper- newtoniancounting,theO(0,2)termsmustbelimitedto
imentallynegligible,althoughinprincipleitmightoffera PNO(5), while the O(1,1) terms are limited to PNO(3).
novelwaytoaccesscertaintypesofpresentlyunobserved Next, note that for laboratory and solar-system tests
interactions at exceptional sensitivities. The remaining the variations in hµν over the experimental scale L are
termt7(λ1µ,ν1)... displayedin Eq.(25)describesthe dominant spmlea,ltlhceotmyppiacraeldvatoluheµoνf,t|h∂eαghrµaνv|it≪ati|ohnµaνl/aLc|c.elFeroartieoxnamon-
Lorentz-violating gravitational effects involving the un-
the surface of the Earth is g 10−32 GeV, which is
knowne interaction. As discussed in Sec. IIC1, certain ≃
tiny compared to the ratio of the gravitational poten-
Lorentz-violating effects are observable only in the pres-
tial h 10−9 and the size of a typical laboratory
ence of gravity, and so under suitable circumstances ob- µν
servableexperimentalsignalsfromt7(1,1) couldarise[11] expe|rime|nt≃L ≃ 1015 GeV−1. Terms in the relativis-
λµν... tic hamiltonian proportional to derivatives of h can
µν
despite the tight existing experimental constraints [33]
therefore be limited to O(0,1). Finally, note that prod-
on the direct observation of additieonal interactions due
ucts of Lorentz-violating terms lead to higher-order ef-
(0,0) (0,1)
to the lower-order terms t7 and t7 . In scenarios
λµν... λµν... fects with operatorstructuresmatching onesalreadyap-
with an unobserved force, we must therefore also allow
pearinginthefermionsectoroftheMinkowski-spacetime
for O(1,1) effects involvineg t7(λ1µ,ν1).... e SME.Thesearealreadyaccessibleinnongravitationalex-
The remaining possibility is that NG modes are ab- periments. Itthereforesufficesforourpurposetorestrict
sorbed into the torsion. Toehandle this case, note that attention to terms at leading order in Lorentz violation.
the matter-sector role of minimal torsion can be treated To summarize,the constructionof the perturbativerela-
in parallel with the coefficient field bµ for Lorentz vio- tivistic quantum-mechanical hamiltonian can be limited
lation according to Eq. (9). Existing experimental con- to terms at perturbative orders O(0,1), O(1,0), O(1,1),
straints on minimal torsion components are tight, lying and O(0,2), except for terms involving derivatives of the
below 10−27-10−31 GeV [17, 34, 35], so effects involving gravitationalfields, which can be limited to O(0,1).
minimal torsioncanbe treatedin the same way as those
corresponding to a presently unobserved force field. We
can therefore limit attention to O(1,1) effects as before.
III. QUANTUM THEORY
In principle, any nonminimal torsioncomponents canbe
treatedinasimilarfashionbecausetheyplayaroleanal-
Thissectionstudiesthequantummechanicsassociated
ogous to coefficient fields in the nonminimal SME, but
with the fermion action S in Eq. (3). We begin in Sec.
effects of this type lie beyond our present scope. ψ
IIIA by addressing the issue of the unconventional time
Consider next the metric fluctuation h . For appli-
µν
dependencearisingfromtheDiracequationderivedfrom
cations to gravitational tests with matter, h can be
µν
S . The relativisticquantum-mechanicalhamiltonianH
treated as a background field obtained by solving the ψ
is then obtained in Sec. IIIB, and the relevant parts of
appropriate equation of motion, which is the modified
the nonrelativistic limit H are extracted in Sec. IIIC.
EinsteinequationinthepresenceofLorentzviolation. It NR
canthereforebeviewedasthesumofaLorentz-invariant
piece h(0,1) with a series of corrections of increasing per-
µν
turbative order in t , A. Time dependence
λµν...
h =h(0,1)+h(1,1)+h(2,1)+.... (26) In the weak-field limit, the Lagrange density for
µν µν µν µν Lψ
the action (3) takes the schematic form
When we specify the perturbativeorderof anexpression
ccoornrteacitnitnegrmhsµνfrionmwthhaetafoblolovwess,eirtieissaurnedienrcsltuodoedd.that the L= 21i[ψ(γµ+Cµ)∂µψ−(∂µψ)(γµ+Cµ)ψ]−ψDψ, (27)
For a given expression, establishing the relevant per- where Cµ and D represent spacetime-dependent opera-
turbative orderfor ouranalysis typicallyinvolvesa com- tors without derivatives acting on ψ. These operators
binationofexperimentalrestrictionsandtheoreticalcon- satisfy the conditions
siderations. As an illustration, we outline here the rea-
soning establishing the appropriate perturbative orders (γ0Cµ)† =γ0Cµ, (γ0D)† =γ0D, (28)
in the construction of the relativistic quantum hamilto-
nian. First, note that terms quadratic in h involve and Cµ is perturbative.
µν
PNO(4) and higher. Since the sensitivity of current lab- The Euler-Lagrangeequations obtained from Eq. (27)
oratory and solar-system tests lies at the PNO(4) level, yield a Dirac equation with unconventional time depen-
we must keep these terms but can discard terms cu- dence,
bic in h and ones involving the product of coefficients
µν
for Lorentz violation with terms quadratic in h . The i(γ0+C0)∂ ψ =[ i(γj+Cj)∂ 1i∂ Cµ+D]ψ. (29)
µν 0 − j− 2 µ
8
This equation differs from the standard Dirac form by 2. Parker method
the presence of C0, which impedes the interpretation of
the operator acting on ψ on the right-hand side as the Another method has been presented by Parker [39] in
hamiltonian. Inthissubsection,twoapproachesaddress- the context of field theory in curved spacetime. It in-
ing this issue at first order in Cµ are discussed. volves multiplying the Dirac equation by a suitable fac-
tor that removes the unconventionaltime dependence to
the desired order in Cµ. The resulting hamiltonian is
1. Field-redefinition method
hermitian with respect to a modified scalar product.
ApplyingthismethodatfirstorderinCµ requiresleft-
Onemethodforconstructingthehamiltonianhasbeen multiplyingbothsidesofEq.(29)withγ0(1 C0γ0). The
developedinthecontextoftheSMEinMinkowskispace- ensuing hamiltonian can be written as −
time [36–38]. It uses an appropriate field redefinition
at the level of the action to ensure the Dirac equation Hˆ =H(0)+H(1), (36)
ψ ψ
emerges with conventional time dependence. In typical
applications, the field redefinition is defined perturba- where H(0) is given in Eq. (32) and the first-order cor-
tively at the desired order in Cµ. rection in Cµ is
For present purposes, it suffices to work at first order
in Cµ. The appropriate field redefinition is Hψ(1) =−iγ0Cj∂j+iγ0C0γ0γj∂j−21iγ0∂µCµ−γ0C0γ0D.
(37)
ψ = Aχ, A 1 1γ0C0. (30)
≡ − 2 Thesubscriptψ indicatesanoperatoractingontheorig-
The resulting hamiltonian can be written as inal spinor ψ.
H =H(0)+H(1), (31) In this method, the modified Dirac equation implies a
χ χ modified continuity equation and hence requires a mod-
whereH(0) isthehamiltonianintheabsenceofCµ,given ified scalar product. At first order in Cµ, the continuity
by equation is
H(0) = iγ0γj∂j +γ0D. (32) ∂ [ψ†(1+γ0C0)ψ]+∂ [ψ†γ0(γj +Cj)ψ]=0, (38)
− 0 j
The correction H(1) is first order in Cµ and takes the
χ andthe probabilitydensity canbe identifiedas the com-
form binationψ†(1+γ0C0)ψ. Thecorrespondingscalarprod-
H(1) = iγ0(Cj 1C0γ0γj + 1γ0γjC0)∂ uct is
χ − − 2 2 j
1i(γj∂ C0+γ0∂ Cj)
−12γ0(C0jγ0D+Dγj0C0). (33) hψ1,ψ2iP = d3x ψ1†(1+γ0C0)ψ2. (39)
−2 Z
The subscript χ serves as a reminder that the operator Provided H(1) is time independent, the hamiltonian Hˆ
ψ ψ
acts on the spinor χ. Note that the hamiltonian Hχ is is hermitian with respect to this modified scalar prod-
hermitianwithrespecttotheusualscalarproductinflat uctandsoquantum-mechanicalcalculationscanproceed.
space, When C0 is time dependent, hermiticity with respect to
theproduct(39)canberestoredbyaddinganextraterm
χ ,χ = d3x χ†χ . (34)
h 1 2if 1 2 [40]. We thereby obtain the hermitian hamiltonian
Z
Thisimplies,forexample,thatenergiescanbecalculated H = 1iγ0∂ C0+Hˆ (40)
in the usual way. ψ 2 0 ψ
Somephysicalinsightintothefield-redefinitionmethod at first order in C0.
isobtainedbynotingthatthecombinationγµ+Cµ takes
the generic form
γµ+Cµ =Eµ γa, (35) 3. Comparison
a
whereEµ canbeinterpretedasaneffectiveinversevier-
a The hamiltonians H in Eq. (31) and H in Eq. (40)
bein. It reduces to the conventional inverse vierbein eµ χ ψ
a typically have different forms. For our present purposes,
in the purely gravitational case but includes coefficients
however,theyarephysicallyequivalentbecausetheygive
for Lorentz violation when Lorentz symmetry is broken.
rise to the same eigenenergies at first order.
This suggests that the field-redefinition method can be
To demonstratethis, firstnote thatthe difference ∆H
interpreted as transforming the problematic situation of
between the two hamiltonians can be written as
a fermion on an effective manifold with vierbein com-
ponents E0 into the physically equivalent but tractable ∆H =H(1) H(1) =H(0)A AH(0), (41)
a χ − ψ −
theoryofadifferentfermionfieldonamanifoldwithvier-
bein components E0 =δ0 , in which the hamiltonian is where A is givenby Eq.(30). This canbe shownby ma-
a a
hermitian with respect to a conventionalscalar product. nipulationofthefieldredefinitionandtheDiracequation
9
or verified by direct calculation. The physical quantities The first-ordercorrectionto the conventionalhamilto-
of interest are the eigenenergies. These are obtained as nianH(0,0) arisingfromLorentzviolationcanbe written
perturbations Eχ(1,ψ) to the unperturbed values E(0), cal- H(1,0) = a0 me0+2ic(j0)∂j (mc00 iej∂j)γ0
culable at first order as − − −
f ∂jγ0γ +[a +i(c η +c )∂k]γ0γj
j 5 j 00 jk jk
−
E(1) = ψ(0),H(1)ψ(0) , (42) +( b 2id ∂j)γ +(iH +2g ∂k)γj
ψ,χ h ψ,χ if − 0− (j0) 5 0j j(k0)
[b +i(d ∂k+d ∂ ) 1mgkl0ǫ ]γ γ0γj
where ψ(0) are solutions to the unperturbed Schr¨odinger − j jk 00 j − 2 jkl 5
(1Hklǫ +md )γ γj
equation H(0)ψ(0) =E(0)ψ(0). However, the expectation − 2 jkl j0 5
value of ∆H is zero in this scalar product, so the first- −iǫjlm(gl00ηkm+ 21glmk)∂kγ5γj. (46)
orderperturbationstothe energiesareidenticalforboth
This result matches the one previously obtained for the
hamiltonians.
SME in Minkowski spacetime [37] when the change in
The above first-order result suffices for the present
metric signature is incorporated. When minimal torsion
work, although we anticipate equivalence also holds at
is included in the analysis, its background value enters
higher orders in a complete analysis. For our purposes,
Eq. (46) through the replacement b (b ) specified
µ eff µ
the field-redefinition method proves technically and con- →
byEq.(9). Itcanbe constrainedthroughareinterpreta-
ceptually easier because the hamiltonian is hermitian
tion of experiments searchingfor nonzero b [17, 34, 35].
µ
with respect to the usual scalar product. We therefore
Aninterestingissueis the extenttowhichthe gravita-
adopt the field-redefinition method in the remainder of
tional and inertial effects in Eq. (45) mimic the Lorentz-
this work,and all references to H implicitly refer to H .
χ violating effects in Eq. (46). For example, in a rotat-
ing frame ofreference the term ∂jh0kǫ γ γ0γl/4 in Eq.
jkl 5
(45) contains a coupling of the rotation to the spin of
B. Relativistic hamiltonian the particle with the same operator structure as the
term b γ γ0γj in Eq. (46). At this order, a frame
j 5
−
rotation can therefore mimic potential signals arising
The relativistic hamiltonian for the action S in Eq.
ψ from a nonzero coefficient b for Lorentz and CPT vi-
j
(3)canbe obtainedviathe field-redefinitionmethod. At
olation. This effect has been observed in tests with a
the appropriate perturbative order, we find the operator
spin-polarized torsion pendulum [41]. The same term in
A of Eq. (30) takes the form
Eq.(45)alsocontainsgravitomagneticeffects thatarein
principle observable in tests searching for b if sufficient
µ
A = 1 1γ0 ee0 Γa γ0 3e(0,1)e(0,1)γ0 sensitivity is reached. Certain Lorentz-violating effects
− 2 a − − 4
can be separated from gravitational and inertial effects
h− 43e(0,1)h0µγµ− 136h0µh0νγµγ0γν because the former generate time-varying signals due to
3e(0,1)Γ0 3h Γ0γ0γµ+γµγ0Γ0 . themotionoftheEarthandcanhaveflavordependence,
− 2 − 8 0µ but a complete separation may be problematic.
(cid:0) (4(cid:1)3i) The O(1,1) contribution to H can be separated as
H(1,1) = H(1,1)+H(1,1)+H(1,1)+H(1,1)+H(1,1)
Implementingthefieldredefinitionandvaryingthetrans- h a b c d
formedLagrangedensityresultsinaDiracequationfrom +H(1,1)+H(1,1)+H(1,1)+H(1,1). (47)
e f g H
which the hamiltonian H can be identified.
The hamiltonian H can be split into pieces according Here, the term H(1,1) arises from Lorentz-violating cor-
h
to perturbative order, rections to the metric fluctuation hµν. It is given by
H(1,1) = ih(1,1)∂j 1mh(1,1)γ0
H =H(0,0)+H(0,1)+H(1,0)+H(1,1)+H(0,2). (44) h j0 − 2 00
+1i(h(1,1)+h(1,1)η )γ0γk∂j. (48)
2 jk 00 jk
The component H(0,0) is the conventional hamiltonian
The other terms in H(1,1) are labeled according to the
for the Lorentz-invariant Minkowski-spacetime limit of
type of coefficient for Lorentz violation involved. The
the theory. The first-order Lorentz-invariantpiece is
contributions involving the four-component coefficients
H(0,1) = 1i(h +h η )γ0γk∂j +ih ∂j 1mh γ0 aµ and bµ of mass dimension one are
2 jk 00 jk j0 − 2 00
+41∂jh0kǫjklγ5γ0γl+ 21i∂jhj0 Ha(1,1) = a7 0−ajhj0+ a7 j − 12ajh00− 12akhjk γ0γj
+1i(∂ h +∂kh )γ0γj. (45) (cid:0) (cid:1) (49)
4 j 00 jk e e
and
Itrepresentsthefirst-ordercorrectiontotheconventional H(1,1) = b7 +bjh γ
hamiltonianH(0,0) arisingfromgravitationalandinertial b − 0 j0 5
effects. +(cid:0) b7ej − 12bjh00(cid:1)− 21bkhjk γ0γ5γj, (50)
(cid:0) (cid:1)
e
10
respectively. In the latter equation, effects fromminimal O(1,1). Formostpurposes,itisnecessarytofindexpres-
torsion are included via the replacement b (b ) sionsforthesefluctuationspriortousingthehamiltonian
µ eff µ
→
given in Eq. (9). H in a given analysis. This issue is addressed further in
The contributions involving the dimensionless coeffi- Sec. IVC, where the spin-independent coefficient fluctu-
cients cµν and dµν are ations (a7 eff)µ and c7µν are considered in more detail.
The remaining piece of the hamiltonian H lies at per-
Hc(1,1) = i c700ηjk+c7kj + 21c00(h00ηjk−hjk) turbativee order O(e0,2) and represents the second-order
(cid:2)+2c(l0)hl0ηjk+ 41ck0hj0− 41cj0hk0 Lerotriaenltezff-eincvtsa.riIatnttackoenstrtihbeuftoiormnfromgravitationalandin-
e e
1c h ηl +hl c hl γ0γk∂j
− 2 lj 00 k k − kl j H(0,2) = i 1h h η + 1h hl0η 1h h
−−m2i(cid:0)ecc77(0j00)+(cid:0)+12cc(0k00h)0h0jk++2(cid:1)cc(jjk0h)hk0j0(cid:1)∂γj(cid:3)0 (51) −(cid:0)m8−038810hhj00l00hhl0kjk0(cid:1)γ+0γ122kh∂jl0j0h−j0ihjγkk00−.hjk4∂j00 jk(57)
and (cid:0) (cid:1)
e (cid:0) (cid:1)
H(1,1) = 2i d7 (j0)+d hjk+d(jk)h γ ∂ C. Nonrelativistic hamiltonian
d (k0) k0 5 j
+i(cid:0)ed7 00+ 21d00h00+2d(k0)hk0(cid:1)γ0γ5γj∂j Most experimental tests of interest in this work
+i(cid:0)de7 kj + 41dk0hj0− 41dj0hk0−(cid:1) 21d00hjk are nonrelativistic. In this section, we use a Foldy-
Wouthuysen transformation [42] to extract from the rel-
(cid:2)e− 21dlj h00ηlk+hlk −dklhlj γ0γ5γk∂j ativistic hamiltonian H the parts of the nonrelativistic
+m d7 j0ηjk(cid:0)− 21dj0hjk−(cid:1)dkjhj0 γ5(cid:3)γk hamiltonian HNR relevant for the subsequent analyses.
The Foldy-Wouthuysen transformation is a system-
−41i(cid:0)medj0hk0ǫjklγl, (cid:1) (52) aticprocedurefordeterminingthenonrelativisticcontent
of certain relativistic quantum-mechanicalhamiltonians.
respectively. The dimensionless four-component coeffi-
For a massive four-component Dirac fermion, the trans-
cients e and f generate the expressions
µ µ
formation generates a series expansion in powers of the
He(1,1) = i e7 j + 41e0hj0− 21ejh00−ekhjk γ0∂j fmeramtiioonncmanombeenitmumpl.emIennttehdeapsreusseunatl,cabsuet, ctahreetmraunsstfobre-
−(cid:0)me7 0+mejhj0+ 14me0hj0γ0γj(cid:1) (53) taken to keep track of both the order in momentum and
e
the perturbativeorderO(m,n)incoefficients forLorentz
and
e violation and in gravitationalfluctuations.
H(1,1) = f7 + 1f h 1f h fkh γ0γ ∂j, Performing the Foldy-Wouthuysen transformation for
f − j 4 0 j0− 2 j 00− jk 5 the complete hamiltonian H of Eq. (44) is cumbersome
(cid:0) (cid:1) (54) andalsounnecessaryforourscopebecausemostattained
e
sensitivitiestospincouplingsareunlikelytobeimproved
while the dimensionless coefficient g leads to
λµν bystudyingthesuppressedeffectsfromgravitationalcou-
plings. However,only limited sensitivity currently exists
Hg(1,1) = 2g7 k(j0)− 41gl00hl0ηjk−gl(j0)hlk−2gk(l0)hlj to spin-independent effects controlled by the coefficients
(cid:0)+2gk(jl)hl0 γk∂j aµandeµbecausetheseareunobservableforbaryonsand
e charged leptons in Minkowski spacetime. In the remain-
+i g7 l00ηjk−(cid:1)12g7 klj − 21gk00 h00ηjl−hjl derofthisworkwefocusongeneralspin-independentef-
(cid:2)− 21gn00hlnηjk− 14gjk0h(cid:0)l0 (cid:1) fects, which are associated with the coefficients aµ, cµν,
e+2gl(n0)heηjk 1gkl0hj0+ 1gkljh and eµ. Since the minimal torsion coupling also involves
n0 − 8 4 00 spin, this focus implies also disregarding nonrelativistic
+ 21gklnhjn− 21gknjhln ǫklmγ5γm∂j effects due to torsion, effectively restricting attention to
−21m g7 jk0+gkm0hjm+gjk(cid:3)mhm0 ǫjklγ0γ5γl tchuerrelinmtitsicnogpeR,iaemFaonldny-gWeoomutehturyy.senAlathnoaulygshisbinecyoornpdoroautr-
+21mi(cid:0)gjk0hk0γ0γj + 81mgjk0hl0ǫjk(cid:1)lγ5. (55) ing spin-dependent effects could lead to additional tor-
e sion sensitivities beyond those obtained via searches for
Finally,theantisymmetriccoefficientHµν ofmassdimen- bµ [3, 17, 34, 43].
sion one contributes Intherelativisticquantumtheory,theuppertwocom-
ponents of the four-component wave function describe
HH(1,1) = −21 H7 jk−Hjmhkm− 21Hjkh00 ǫjklγ5γl the particle while the lower two describe the antiparti-
−i(cid:0)H7ej0+ 21Hk0hjk+Hjkhk0 γ(cid:1)j. (56) pclaer.t ThecohnataminilitnognitaenrmHs tchaantbmeixsetphaerautpedpeirntaondanloowdedr
Intheaboveex(cid:0)epressions,thefluctuations(cid:1)ofthevarious compOonents and an even part that involves no mix-
E
coefficient fields appear in H only at perturbative order ing. The idea of the Foldy-Wouthuysen method is to