Massive photons and Lorentz violation

0.26 MB·English

by  Mauro Cambiaso

Massive photons and Lorentz violation
Mauro Cambiaso,1 Ralf Lehnert,2 and Robertus Potting3
1Universidad Andres Bello, Departamento de Ciencias Fisicas,
Facultad de Ciencias Exactas, Avenida Republica 220, Santiago, Chile
2Indiana University Center for Spacetime Symmetries, Bloomington, IN 47405, USA
3CENTRA, Departamento de F´ısica, Universidade do Algarve, 8005-139 Faro, Portugal
(Dated: January 13, 2012)
All quadratic translation- and gauge-invariant photon operators for Lorentz breakdown are in-
cluded into the Stueckelberg Lagrangian for massive photons in a generalized R gauge. The cor-
ξ
responding dispersion relation and tree-level propagator are determined exactly,and some leading-
order results are derived. The question of how to include such Lorentz-violating effects into a
perturbative quantum-field expansion is addressed. Applications of these results within Lorentz-
2 breakingquantumfieldtheoriesincludetheregularization ofinfrareddivergencesaswellasthefree
1 propagation of massive vector bosons.
0
2 PACSnumbers: 11.30.Cp,11.30.Er,11.30.Qc,12.20.-m,12.60.Cn

n
a
J I. INTRODUCTION terspacebasictheoreticalresults,suchasexpressionsfor
thedispersionrelationandpropagator,arecurrentlystill
4
1 Recent yearshave witnesseda growinginterestin pre- lacking. Thepresentinvestigationreportsonprogressto-
cisiontestsofLorentzandCPTinvariance. Thisinterest wards filling this gap.
h] canbe partly attributed to the availability ofnew obser- MassivevectorparticleswithLorentzandCPTbreak-
t vational data and the development of ultra-sensitive ex- ing are not only of interest for phenomenological stud-
p- perimental techniques [1]. Moreover, minute departures ies of the heavy gauge bosons Z0 and W±. They have
e fromLorentzandCPT symmetrycanbe accommodated previously served a valuable tool for investigations of
h invarioustheoreticalideasbeyondestablishedphysics[2] the mass-dimension three Lorentz- and CPT-violating
[ providing a phenomenological opportunity to search for Maxwell–Chern–Simons term [18]. More importantly,
1 novel effects possibly arising at the Planck scale. massive vector fields play a key role for theoretical stud-
v At presently attainable energies, Lorentz- and CPT- ies involving the photon because they provide a popular
5 violating effects are expected to be governed by effec- methodforregularizinginfrareddivergencesinperturba-
4 tivequantum-fieldcalculations. Infact,this latterappli-
tive field theory [3]. The general framework based on
0 cationrepresentstheprimaryfocusofthisstudy. Butwe
this premise is known as the Standard-Model Exten-
3
anticipate that our results for the expression of the dis-
sion (SME) [4–6]. This framework contains the usual
.
1 Standard Model of particle physics and general relativ- persion relation and the propagatorare equally valid for
0 the quadraticpartofthe fullSME’sZ0 andW± sectors.
ity as limiting cases and therefore permits the identifi-
2 cation and analysis of essentially all currently feasible Theoutlineofthepresentworkisasfollows. SectionII
1
Lorentz and CPT tests. To date, the SME has been provides the basic ideas behind the construction of the
:
v employed for phenomenological studies involving cosmic modelwearestudying. Afewremarksonthestructureof
Xi radiation [7], meson factories [8] and other particle col- the resulting modified Maxwell–Stueckelberg equations
liders[9],resonancecavities[10],neutrinos[11],precision arecontainedinSec.III.SectionIVdeterminestheexact
r
a spectroscopy [12], and gravity [13]. dispersion relation and propagator for our model. Sec-
tion V discusses some leading-order results that are ex-
The SME has also served as the basis for various the-
pectedtobeusefulforpracticalcalculations,andSec.VI
oretical investigations of Lorentz and CPT symmetry.
gives a brief summary of our results. Some supplemen-
These investigations have have shed light on the SME’s
tary material is collected in various appendices.
mathematicalstructure[14],spontaneousLorentzbreak-
down and Nambu–Goldstone modes [15], classical lim-
its of Lorentz- and CPT-violating physics [16], quan-
tumcorrectionsandrenormalizability[17,18],featuresof II. MODEL BASICS
non-renormalizable contributions [19], etc. At the same
time,theseanalyseshavesolidifiedvariousaspectsofthe Ourprimarygoalistointroduceaphotonmassforreg-
SME’s theoretical foundation. ularizing infrared divergences in perturbative quantum-
Onetopicthathasremainedcomparativelyunexplored field calculations in a general R gauge. This requires a
ξ
concerns Lorentz and CPT violations in massive vec- smooth behavior of the internal-symmetry structure in
tor particles in the SME: published work [20] has been the massless limit. Thus, the usual Proca term by it-
focused on phenomenological analyses confined to the self is insufficient, and the mass needs to be introduced
CPT-evenZ0andW± sectorsoftheminimalSME.How- via,e.g.,the Stueckelbergmethod[21]. Theoriginalver-
ever,evenforthissmallregioninthefullSME’sparame- sion of this method is appropriate for U(1) gauge theo-
2
ries. The method needs modifications for photons em- ple as possible. For example, the conventional Lorentz-
bedded in the U(1)×SU(2) gauge structure of the Stan- symmetric Stueckelberg expression would likely suffice.
dard Model [22], and it generally fails for non-Abelian However, we introduce an additional set of Lorentz-
vectorfields. The Lorentz-violatinggeneralizationof the breakingcoefficientsforreasonsoutsidethe presentU(1)
Stueckelberg method, which is discussed in this section context: certain aspects of the Stueckelberg model, such
and the subsequent one, therefore applies solely to the as the dispersion relation and propagator, will turn out
SME’s QED limit. We do, however, expect the remain- to be equally valid for the Lorentz-violatingZ0 and W±
ing partof our study, containedin Secs.IV and V, to be bosons. These particles contain not only the equiva-
applicable to more general massive vector fields. lent of the kˆ and kˆ coefficients, but also, e.g., a
F AF
Ourstartingpointistheusualfree-photonLagrangian (k )µν mass-type term. For wider applicability in this
φφ
minimally coupled to an external conserved current jµ. electroweakcontext,we therefore alsoinclude a (k )µν-
φφ
ThegeneralizationofthisLagrangiantoincludearbitrary typecontributionintoourStueckelbergmassterm. With
local, coordinate-independent, translation- and gauge- these considerations in mind, we implement the Stueck-
invariant physics with Lorentz- and CPT-symmetry elberg method [21] by introducing a scalar field φ in the
breakdown can be cast into the following form [6]: following way:
Lγ=−41F2−A·j δLm = 12(∂µφ−mAµ)ηˆµν(∂νφ−mAν), (3)
− 14Fκλ(kˆF)κλµνFµν + 12ǫκλµνAλ(kˆAF)κFµν. (1) where m denotes the photon mass and
Here, the field strengths and potentials are real-valued ηˆµν =ηµν +Gˆµν. (4)
and obey the conventional relation F =∂ A −∂ A ,
µν µ ν ν µ
andǫκλµν denotesthetotallyantisymmetricsymbolwith Here, the Gˆµν represents the full-SME generalization of
ǫ0123 = +1. Lorentz and CPT breakdown is controlled the minimal-SME’s (kφφ)µν coefficient and is given by:
by the quantities kˆF and kˆAF, which are givenexplicitly ∞
by the following expressions [6] Gˆµν = (G(2n))µνα1...α2n−4∂ ...∂ , (5)
α1 α2n−4
∞ nX=2
(kˆ )κλµν= (k(2n))κλµνα1...α2n−4∂ ...∂
F F α1 α2n−4 whereeach(G(2n))µνα1...α2n−4 iscontractedwithaneven
nX=2
number of derivatives [23], is spacetime constant, sym-
∞
(kˆ ) = (k(2n+1)) α1...α2n−2∂ ...∂ . (2) metricinµandν,andtotallysymmetricinα1...α2n−4.
AF κ AF κ α1 α2n−2 This definition still contains some Lorentz-symmetric
nX=1
pieces at each mass dimension, which can be eliminated
Each k(d) coefficient as well as each k(d) coefficient is if necessary. For example, the Lorentz-covariant contri-
F AF
taken as nondynamical, spacetime constant, and totally bution ∼ ηµν contained in (G(4))µν can be removed by
symmetricinitsαindices. Thesuperscript(d)labelsthe taking this coefficient as traceless.
massdimensionofthecorrespondingphotonoperator,so The inclusion of Gˆµν does not invalidate the Stueckel-
that the unit of the actual coefficient is [GeV]4−d. bergmethod. Atthis point,weleaveGˆµν undetermined.
The next step is to add a mass-type term δL for the We only require it to be small, so that in the limit of
m
photonto the aboveLagrangian(1). Inthe conventional vanishing Lorentz violation, ηˆµν approachesηµν without
case, such a contribution is restricted by Lorentz sym- a change in signature and rank [24]. In the present U(1)
metry. Inthepresentsituation,thisrestrictionisabsent, context, where δLm is intended to serve as a regulator,
and more freedom in the choice of δLm exists. This ad- specific regions in (kF(d), kA(dF)) parameter space may only
ditionalfreedompartlydependsonthetypeofphysicsto be compatible withcertaindefinite choicesfor Gˆµν, such
be described. For instance, one may wish to model gen- as Gˆµν = 0, as discussed above. For applications in the
eral Lorentz violation for hypothetical massive photons. heavy-bosoncontext, Gˆµν represents an arbitrary physi-
Alternatively, the aim may be to regularize infrared di- cal parameter that can only be fixed by observation.
vergences that often arise in quantum-field calculations Asintheordinarycase,theresultingLagrangianL ≡
m
involving massless photons governed by Lagrangian (1). L +δL changes under a local gauge transformation
γ m
In these two examples, the former may allow more addi-
tionalfreedom for δLm than the latter: consider a situa- δAµ =∂µǫ(x), δφ=mǫ(x) (6)
tion in which the coefficients in Lagrangian (1) are such
by total-derivative terms. However, in the absence of
that a subgroup of the Lorentz group remains intact. A
topological obstructions and with the usual boundary
regulator breaking this residual symmetry may be prob-
conditions, the action and thus the physics remain un-
lematic,sothatviolationsoftheremaininginvariantsub-
changedunder the transformation(6). The next natural
group may have to be excluded from δL .
m stepthenistoselectagauge-fixingconditionF[A,φ]. As
Inthepresentwork,theprimarypurposefortheintro-
usual, a multitude of choices for F[A,φ] are acceptable.
duction of a photon-mass term is to regularize potential
We take
infrared divergences in quantum-field contexts. In prin-
ciple, the structure of δL can then be chosen as sim- F[A,φ]=∂ ηˆµνA +ξmφ, (7)
m µ ν
3
a choice that will turn out to be convenient for our pur- Employing Eq. (12), it is apparent that ∂ ηˆµνAph = 0
µ ν
poses. Application of the usual Gaussian smearing pro- on shell. With Eq. (13) at hand, we can now substi-
cedure leads to the following gauge-fixing term to be in- tute the decomposition of the vector potential A =
ν
cluded into the Lagrangian: Aph−∂ (∂ ηˆαβA )/(ξm2) into the field equations (11).
ν ν α β
Being a gradient, the auxiliary excitation does not con-
1
Lg.f. =− (∂µηˆµνAν +ξmφ)2, (8) tribute to Fαβ. By virtue of its equation of motion (12),
2ξ this component also disappears from the A term in
µ
Eq.(11). ThezerodivergenceofAph,onthe otherhand,
where ξ is an arbitrary gauge parameter, as usual. The ν
associated Faddeev–Popov determinant det δF in the implies thatitvanishes whencontractedwith the ξ term
δǫ in Eq. (11). Our decomposition then gives
path integral results in the ghost term (cid:0) (cid:1)
ηµαηνβ∂ +(kˆ ) ǫµναβ +(kˆ )µναβ∂ F
L =−c¯(∂ ηˆµν∂ +ξm2)c, (9) µ AF µ F µ αβ
F.P. µ ν h i
+ m2ηˆµν Aph=jν (14)
µ
where c and c¯ are anticommuting scalars. We mention
(cid:2) (cid:3)
that the possibility of introducing additional Lorentz vi- for the field equations. Note that the auxiliary compo-
olation into the ghost Lagrangian has been studied [25]. nent has disappeared entirely (Fαβ only involves Apνh)
We disregard this option in what follows. and that the source jν excites the physical degrees of
We are now in the position to present our model La- freedom in a ξ-independent way.
grangian L=L +δL +L +L explicitly:
γ m g.f. F.P.
1 IV. FEATURES OF THE GENERAL SOLUTION
L=−1F2−A·j+ 1m2A ηˆµνA − (∂ ηˆµνA )2
4 2 µ ν 2ξ µ ν
− 1φ(∂ ηˆµν∂ +ξm2)φ−c¯(∂ ηˆµν∂ +ξm2)c Inthissection,westudygeneralpropertiesofthesolu-
2 µ ν µ ν
tionsoftheequationofmotion(11). Thisequationholds
− 14Fκλ(kˆF)κλµνFµν + 12ǫκλµνAλ(kˆAF)κFµν. (10) exactlyforourLorentz-violatingStueckelbergphotonsat
the classicallevel. Butthe modelalsoexhibitsnumerous
It is apparent that the scalar φ and the ghosts c and c¯
similaritiestoheavygaugebosonswithLorentzviolation.
arenowuncoupledandcanbeintegratedoutofthepath For example, the Z0 and W± sectors of the SME also
integralyieldinganunobservablenormalizationconstant.
contain operators of the type k , k , and G; examples
F AF
Wewillthereforedisregardthesefieldsinoursubsequent
of these are k , k , and k , respectively. Although the
W 2 φφ
analysis.
linearequation(11)cannothold exactlyfor non-Abelian
gauge bosons, the types of operators quadratic in the
fieldsdoagreewiththoseforthe photon. Onecanthere-
III. STRUCTURE OF THE FIELD EQUATIONS
fore anticipate that Eq. (11) does govern most aspects
of the tree-level free behavior of Z0 and W± within the
The equations of motion for our Lagrangian(10) read
SME.Inparticular,the dispersionrelationandpropaga-
tor derived below are expected to hold not only for our
ηµαηνβ∂ +(kˆ ) ǫµναβ +(kˆ )µναβ∂ F
µ AF µ F µ αβ modified Stueckelberg photons, but also for the heavy
h i
SME gauge bosons at tree level.
1
+ m2ηˆµν + ηˆµαηˆνβ∂α∂β Aµ=jν. (11) We begin with the plane-wave dispersion relation. To
(cid:20) ξ (cid:21)
this end, we Fourier transform Eq. (11), which yields
Owing to its underlying antisymmetric structure, the
terminvolvingFαβ inEq.(11) hasvanishing divergence. p2ηµν −pµpν −m2ηˆµν + 1ηˆµαηˆνβp p
α β
Contraction of the field equations with ∂ therefore re- (cid:20) ξ
ν
moves the F term and places the constraint
αβ
+2(kˆ )αµβνp p −2i(kˆ ) ǫαβµνp A¯ =−¯µ (15)
F α β AF α β ν
(cid:21)
ηˆµν∂ ∂ +ξm2 (∂ ηˆαβA )=0 (12)
µ ν α β
(cid:0) (cid:1) for the equations of motion in pµ-momentum space.
on the Aµ term, where we have used our earlier assump- Here, a bar denotes the Fourier transform, and it is un-
tionofaconservedsourcejν. Asperdefinition, ηˆαβ isof derstood that the replacement ∂ →−ip has been imple-
rank four, so (∂αηˆαβAβ) projects out one of the degrees mented in the Lorentz-violating quantities kˆF, kˆAF, and
of freedom contained in Aµ. The ξ-dependent Eq. (12) Gˆ. Letus brieflypause atthis point to introducea more
showsthatthesourcejν doesnotexcite(∂αηˆαβAβ). This concise notation that will enable us present many of our
degree of freedom therefore is an auxiliary mode. subsequent results in a more compact form. We define
Continuing with this decomposition, it is natural to
define a component Kˆµν ≡(kˆ )αµβνp p , Eˆµν ≡(kˆ ) ǫαβµνp , (16)
F α β AF α β
1 because in the dispersion relation and the propagator,
Apνh ≡Aν + ξm2∂ν(∂αηˆαβAβ). (13) kˆF and kˆAF will always appear in this form. Moreover,
4
whenconvenientweabbreviatethecontractionofasym- remind the reader that most of these are artifacts of our
metric tensor with a 4-vector by placing the vector as a effective-Lagrangian approach and must be eliminated.
super- or subscript on the tensor, e.g., Gˆµ ≡ Gˆµνp or Only those wave momenta that represent perturbations
p ν
(Gˆ2)p ≡ GˆµGˆαpµpν, etc. We now rewrite the equations oftheusualLorentz-symmetricsolutionsshouldbeinter-
p α ν
of motion (15) simply as S(p)µ A¯ν(p)=−¯µ(p), where preted as physical. In any case, a determination of the
ν
exactrootsofthegeneraldispersionrelation(19)appears
1 to be unfeasible. However, an exact discrete symmetry
Sµν ≡(p2−m2)ηµν − 1− pµpν −2iEˆµν +2Kˆµν
(cid:18) ξ(cid:19) of the plane-wave solutions is discussed in Appendix C,
the massless limit is studied in Appendix D, and some
1
−m2Gˆµν + Gˆµpν +Gˆνpµ+GˆµGˆν (17) leading-order results are presented in Sec. V.
ξ p p p p
(cid:0) (cid:1) In the more general case of non-vanishing sources, the
is the expression of the modified Stueckelberg operator constructionofsolutionscanbeachievedwithpropagator
in our new notation. functions. Paralleling the ordinary Lorentz-symmetric
The plane-wavedispersionrelationgovernssource-free case, we implicitly define the pµ-momentum space prop-
motion¯µ =0andcanthereforebestatedastheusualre- agatorP(p)µ viaP(p)µ S(p)ν ≡−iηµ,wherewehave
ν ν λ λ
quirementthatthedeterminantofS(p)µν vanishes. With employed the usual quantum-field convention by includ-
theresultsderivedinAppendixA,thistranslatesintothe ing a factor of (−i). It is thus evident that the propaga-
equation tor is given by P(p)µ = −iS−1(p)µ . With Eq. (A8),
ν ν
we obtain an exact, explicit expression for the modified
[S]4−6[S]2[S2]+3[S2]2+8[S][S3]−6[S4]=0. (18)
Stueckelberg propagator in momentum space:
Here, Sn is the nth matrix power of S(p)µ , and the
ν
−i
square brackets denote the matrix trace. P = 1[S3]11+ 1[S]311− 1[S2][S]11
The various trace expressions in Eq. (18) can be cast det(S) 3 6 2
(cid:0)
into a factorized form: − 1[S]2S+ 1[S2]S+[S]S2−S3 . (21)
2 2
(cid:1)
ξdet(S)=(ηˆµνpµpν −ξm2)Q(p)=0. (19) As before, Sn denotes the nth matrix power of S(p)µν,
and the matrix trace is abbreviated by square brackets.
The (ηˆµνp p −ξm2) piece is associated with the aux-
µ ν We mention that restricting each of the infinite sums in
iliary mode described by Eq. (12). The ξ independent Eq. (2) to their first term and setting both m and Gˆ to
factor Q(p) governs the three physical degrees of free-
zeroyieldsthelimitinwhichpreviouspropagatorexpres-
dom. Both factors in the dispersion relation (19) can
sions have been considered [27].
alsocontainunphysicalOstrogradski-typedegreesoffree-
When the exact tree-level propagator (21) is Fourier-
dom [26], which are introduced because our Lagrangian
transformed to position space, an integration contour
containshigherderivatives[6]. Inthepresumedunderly-
mustbeselected. Asintheconventionalcase,thischoice
ingtheory,forwhichthe effectivefieldtheory(10)repre-
depends on the boundary conditions (e.g., retarded, ad-
sents the low-energy limit, these modes must be absent.
vanced, Feynman, etc.). In the present case, a further
Consequently, they should also be eliminated from our
issue arises. As per our earlier assumption in Sec. II,
low-energy model (10).
the Lorentz-violating terms are to be treated as pertur-
An explicit calculation shows that
bations of the usual Lorentz-invariant solutions, and we
Q=(p2−m2)3+r (p2−m2)2+r (p2−m2)+r , (20) already commented on the need to eliminate spurious
2 1 0
modes. Inthepresentcontextconcerningthepropagator,
where the coefficients r = r (p) are momentum- this may, for example, be achieved with a careful choice
j j
dependent coordinate scalars determined by traces of of (counter)clockwise integration contours that encircle
combinations of the various Lorentz-violating tensor ex- only the desired poles for each of the two orientations.
pressionsappearinginEq.(15). Theyvanishinthelimit We remark that these issues are absent when the model
kˆ ,kˆ ,Gˆ → 0. The explicit expressions for the r , is restricted to terms of mass dimension three and four.
F AF j
which can be found in Appendix B, are not particularly
transparent. Notethatingeneralthephysicaldispersion
relation (20) does not represent a true cubic equation in V. LEADING-ORDER RESULTS
the variable (p2−m2) because the r (p) are momentum
j
dependent. The absence of compelling observational evidence for
The dispersion relation (19) restricts the set of all departures from Lorentz and CPT symmetry in nature
possible Fourier momenta pµ ≡ (ω,p~) to those associ- implies that if the coefficients kˆ and kˆ are nonzero,
F AF
ated with plane-wave solutions of the free model. Since theymustbeextremelysmall. Thesamereasoningholds
in general kˆ , kˆ , and Gˆ contain high powers of pµ, true for Gˆ, at least in the context of the Z0 and W±
F AF
there can be a corresponding multitude of plane-wave bosons [28]. This fact is consistent with the theoreti-
frequencies ω(p~) for any givenwave 3-vectorp~, a fact re- cal expectation that potential Lorentz and CPT viola-
flected in the momentum dependence of the r (p). We tion in nature would be heavily suppressed, for exam-
j
5
ple by at least one power of the Planck scale. For most opposed to the spurious Ostrogradski modes) the phys-
phenomenological studies and many theoretical investi- ical roots are characterized by small R that approach
j
gations it is therefore justified to drop higher orders in zero in the limit of vanishing Lorentz breaking. This ob-
kˆ , kˆ , and Gˆ. This section contains a brief discussion servation suggests solving Eq. (22) perturbatively by in-
F AF
of a few leading-order results. troducingaparameterλmultiplyingtheRj,andwriting
We begin by considering the physical piece Q of the a Taylor expansion for the plane-wave frequencies:
dispersion relation given by Eq. (20). We remind the
ωj,± =ωj,±+λωj,±+λ2ωj,±+... , (24)
reader that Q = 0 cannot be considered a true cubic p~ p~,0 p~,1 p~,2
equationinthevariable(p2−m2)duetothepdependence
wherewehaveidentifiedthelabelj oftherootexpression
of the coefficients r , r , and r . We can nevertheless
0 1 2
with the label a of the plane-wave frequency. Substitut-
employ the expressions R , j ∈ {1,2,3}, for the roots of
j
ing(24)onbothsidesofEq.(22)andexpandinginpow-
a cubic to transform the single equation Q = 0 into an
ers of λ yields an expression from which the coefficients
equivalent set of three equations:
of the Taylor series (24) can be determined by matching
p2−m2 =R (ωa,±,~p,kˆ ,kˆ ,Gˆ). (22) theappropriateterms. SolutionstoEq.(22)arethenob-
j p~ AF F tained by taking λ = 1, and the Lorentz-invariant roots
ωj,± = ωj,± are recovered for λ = 0. Thus, this proce-
JustasfortheoriginalexpressionQ=0,itseemsunfeasi- p~ p~,0
dure continuously connects six exact dispersion-relation
ble to determine the exact dispersion-relationroots from
solutions—two for each j in Eq. (22)—to the Lorentz-
the above three equations: the plane-wave frequency
ωa,± contained in the 4-momentum pµ = (ωa,±,p~) still symmetric mass shell. This behavior is consistent with
p~ p~ thatofthe physicalroots,sothatthe proceduresimulta-
appears on both sides of Eq. (22). In particular, this set
neouslyservesasafilterforrejectingthespuriousmodes,
of equations (22) will in general possess more than six
which cannot remain finite in the λ→0 limit.
solutions, which is consistent with the fact that Q = 0
A useful practical way to solve for the coefficients in
fails to be a true cubic and can still contain a multitude
Expansion(24)isiteration: substituteasafirstiteration
of spurious Ostrogradskimodes. ωj,± = ωj,± on the right-hand side of Eq. (22), which
Before continuing, one may ask whether the six physi- p~ p~,0
calsolutionscanexhibitundesirablefeatures. Adetailed determines the improved value ωj,± for the plane-wave
p~,1
analysisofthisquestionwouldbeinterestingbutliesout- frequencyontheleft-handsideofthis equation. Repeat-
side our present scope. However, continuity implies that ing this process yields
small Lorentz violation leads to small deviations from
theconventionaldispersion-relationbranches,sothatthe ωj,± =± p~2+m2+R (ωj,±,~p,kˆ ,kˆ ,Gˆ), (25)
p~,n+1 j p~,n AF F
solutions must in generalbe well behaved within the va- q
lidity range of the our effective field theory. Exceptions where we have denoted the iterative step by the sub-
fromthis expectationrequirenon-genericcircumstances, script n on ωj,±. One can check that the nth iterative
p~,n
such as particular parameter combinations. For exam-
step possesses the correct value for (at least) the first n
ple, in the minimal SME only a single coefficient—the
coefficients in Eq. (24). In this process, there is actually
timelike k(3)—is known to be problematic under some no need to introduce explicitly the parameter λ, as we
AF
circumstances[4]. At the levelofthe dispersionrelation, can take it equal to unity immediately.
this is reflected in the presence of complex-valued solu- Note that particular care may be required for spe-
tions. Forexample,ifm=0andk(3) istheonlynon-zero cial regions in momentum space. One example concerns
AF
Lorentz-violating coefficient, we have very small m and p~. The exact branches associated to
theconventionalpositive-andnegative-valuedmassshell
R =−2(k(3))2−2 (k(3))4+(p·k(3))2. (23) could then lie closely together, so that convergence to
2 AF q AF AF the correct branch must be ensured. Moreover, the ex-
act solution for a positive branch may briefly dip into
WhenkA(3F) istimelike,thej =2contributioninEqs.(22) the negative spectrum and vice versa leading to an un-
cannothaverealsolutionsωp~forsmallenoughp~: Suppose equalnumberofpositiveandnegativerootsinthis small
there were real solutions ωp~ ∈ R, then the square root momentum-space region. The signs of the square root
would be real, and R2 would be real and negative. For in the recursive equation (25) may then have to be ad-
the special value p~ = ~0, this would give ω~02 = R2(p~ = justed correspondingly. Other issues, such as isolated
~0) < 0, which contradicts our assumption ω ∈ R. In momentum-space points at which the λ expansion of s
p~ 1
what follows, we disregard such isolated regions in SME in the Eq. (B7) starts at O(λ), may be resolved by em-
parameter space. ploying a different perturbation scheme.
The physical solutions have to represent small per- Since j ∈{1,2,3}andsincefor largeenoughp~thereis
turbations to the Lorentz-symmetric dispersion relation apositiveandanegativesquarerootforeachj,thereare
p2−m2 = 0. Equation (22) establishes that the depar- sixequationscorrespondingtosixseeminglyindependent
tures from the conventional case are controlled by the physical plane-wave frequencies for a given ~p. However,
three R . In a concordantframe [5], this means that (as the behavior of the modified Stueckelberg operator S(p)
j
6
underthereplacementp→−pleadstoacorrespondence siblyallowingprocessesthatareforbiddenintheLorentz-
between the positive- and negative-root solutions. This symmetric case. Second, the numerator governs features
implies that there are in fact only three independent po- associated with the polarization states. For example,
larization modes, as expected for a massive spin-1 field. Lorentz-violatingnumerator corrections could yield sup-
Moredetailsregardingthislineofreasoningcanbefound pressed processes that would be forbidden by angular-
in Appendix C. momentum conservation in the corresponding Lorentz-
Next, we consider the momentum-space propagator invariant situation.
given by Eq. (21). This propagator describes two phys- If corrections to both of these features are to be re-
ical features. First, the poles of the denominator select tained, the denominator and the numerator in Eq. (21)
the plane-wave momenta. In a Feynman-diagram con- should be expanded separately to leading order. With
text, for instance, Lorentz-violating corrections to the suchanapproximation,thedominantcontributiontothe
usualpoles cancause the propagatorto go on-shell,pos- propagator can be separated into the two pieces:
p2−m2+2[Kˆ]+Gˆp−m2[Gˆ] ηµν −2Kˆµν +2iEˆµν +m2Gˆµν +pµGˆν +pνGˆµ
Pµν =−i(p2−m2) p p p
(cid:0) 3 p2−m2−(cid:1) R ( p~2+m2,p~,kˆ ,kˆ ,Gˆ)
j=1 j AF F
(p2Q−m2(cid:2)) p2−m2+2[Kˆp]−m2[Gˆ] (cid:3)
+i(1−ξ) pµpν +higher order. (26)
(ηˆµνp p −ξm2) 3 p(cid:0)2−m2−R ( p~2+m2,(cid:1)p~,kˆ ,kˆ ,Gˆ)
µ ν j=1 j AF F
Q (cid:2) p (cid:3)
In the denominators, we have used the leading-order ex- propagator is given by [29]:
pressions for the R implementing our previous results
j ηµν −pµpν/m2 pµpν/m2
for the dispersion relation. P (p)µν =−i −i . (28)
0 p2−m2 p2−ξm2
We note that the first piece of the propagator expres-
sion is independent of ξ and only exhibits poles corre- Starting from these zeroth-order expressions, we may
sponding to the three physical modes. The second piece now decompose the full Stueckelberg operator as
is ξ dependent and contains the additional pole associ-
S(p)µν =S (p)µν −δS(p)µν, (29)
atedtotheauxiliarymode. Asexpected,thesecondpiece 0
is without physical effects: contraction with a source ¯ν with
yieldsanoverallfactorp·¯, whichvanishesforconserved
δS(p)µν =2iEˆµν −2Kˆµν +m2Gˆµν
currents. In any case, all contributions from the sec-
ond piece would necessarily have to be proportional to − 1 Gˆµpν +Gˆνpµ+GˆµGˆν . (30)
(p·¯)pµ,whichispuregauge. Moreover,thesecondterm ξ p p p p
(cid:0) (cid:1)
vanishes identically in Feynman–’t Hooft gauge ξ = 1.
SuppressingLorentzindicesandthe dependence onp for
We further remark that the expression (26) does not
brevity, we can now expand the full propagatorP about
propagatethe additionalmultitude ofunphysicalhigher-
its Lorentz-symmetric value P as follows:
0
derivative modes discussed in the previous section. This
is required for a smooth behavior in the limit of van- P =−i(S0−δS)−1
ishing Lorentz breakdown. Some remarks regarding the =−i 1−δS S−1 S −1
0 0
massless limit of the propagator (26) are contained in
Appendix D. =−iS(cid:2)(cid:0)0−1 1−δS S(cid:1)0−1 (cid:3)−1
In many situations, Lorentz-violating corrections to (cid:0) ∞ (cid:1)
the pole structure of the propagatorcan be disregarded. =P0 1+ (iδS P0)n
(cid:18) (cid:19)
Then, the expression for propagator can be simplified nX=1
even further with an approximationthat essentially cor- =P0+P0 iδS P0+P0 iδS P0 iδS P0+... , (31)
responds to treating the Lorentz-violating pieces in the where we used P = −iS−1. The infinite geometric se-
Lagrangian as interaction terms. This alternative ap- 0 0
ries should be convergent for small enough Lorentz vio-
proximation is determined next.
lation δS. In the context of quantum-field perturbation
We beginbyconsideringthe Lorentz-invariantpartS0 theory, the expansion (31) can be represented diagram-
of Stueckelberg operator matically with a propagator insertion iδS into internal
photon lines, as shown in Fig. 1. This result appears
S (p)µν =(p2−m2)ηµν − 1− 1 pµpν. (27) to be consistent with Matthews’ theorem [30]. Like the
0
(cid:18) ξ(cid:19) previousexpression(26), the expansion(31)ofthe prop-
agatorarounditsLorentz-symmetricvalueautomatically
The expression for the corresponding momentum-space accomplishes the elimination of the unphysical poles.
7
Appendix A: Matrix relations
Thisappendixprovidesacollectionofvarioustextbook
linear-algebra results necessary for the line of reasoning
inthemaintext. Forcompleteness,wehavesketchedthe
derivation of these results.
FIG. 1. Lorentz-violating propagator insertion. The wavy Consider an arbitrary 4×4 matrix Λ and denote its
lines represent the usual Lorentz-symmetric Stueckelberg four (not necessarily distinct) eigenvalues by λj, where
propagator given by Eq. (28). The square box denotes the j = 1,...,4. The characteristic polynomial C(λ) of Λ
theLorentz-breaking insertion determined by Eq. (30). can then be expressed as
C(λ)=(λ−λ )(λ−λ )(λ−λ )(λ−λ ). (A1)
1 2 3 4
VI. SUMMARY
BytheCayley–Hamiltontheorem,thematrixΛmustsat-
isfyC(Λ)=0. Expandingtheright-handsideofEq.(A1)
In this work, we have employed the Stueckelberg and replacing λ→Λ then yields
method to introduce a mass term into the full SME’s
photon sector in an R -type gauge, such that the usual Λ4+c3Λ3+c2Λ2+c1Λ+c011=0, (A2)
ξ
smooth behavior of the model in the m → 0 limit is
where the coefficients c are given by
j
maintained. Wehavediscussedtheresultingequationsof
motioninthepresenceofexternalconservedsources,and c0=λ1λ2λ3λ4,
studiedtheirsolutions. Ourresultsincludetheexactdis- c =−λ λ λ −λ λ λ −λ λ λ −λ λ λ ,
1 1 2 3 1 4 3 2 4 3 1 2 4
persionrelationandpropagatorfor free massivephotons
c =λ λ +λ λ +λ λ +λ λ +λ λ +λ λ ,
2 1 2 3 2 4 2 1 3 1 4 3 4
incorporating all possible Lorentz-violating, translation-
c =−λ −λ −λ −λ . (A3)
and gauge-invariant Lagrangian contributions at arbi- 3 1 2 3 4
trarymassdimension. Fromthese exactresults,we have The next task is to express the c in terms of man-
j
extracted leading-order expressions for the dispersion- ifestly component-independent expressions involving Λ.
relation roots and the propagator. The method for ob- To this end, we transform Λ to its Jordan normal form
taining the roots is recursive; it also yields higher-order Λ′ =M−1ΛM, where M is an appropriate 4×4 matrix.
corrections, and it naturally separates the physical solu- Since such a similarity transformation leaves unchanged
tions fromspuriousmodes arisingfromhigher-derivative the eigenvalues, it is straightforwardto verify that
terms in the Lagrangian. Finally, we have verified that
′
det(Λ)=λ λ λ λ ,
the number of physical modes is three, as expected for a 1 2 3 4
massive vector particle. Tr(Λ′n)=λn+λn+λn+λn, (A4)
1 2 3 4
Our results are intended primarily to provide a flexi- where n∈N. Inspection now establishes that
ble tool for regularizinginfrared divergencesin quantum ′
c =det(Λ),
electrodynamicsinthe presenceofgeneralLorentzviola- 0
c =1Tr(Λ′2)Tr(Λ′)− 1Tr(Λ′3)− 1Tr(Λ′)3,
tion. Afollow-upworkinthiscontextisalreadyinprepa- 1 2 3 6
ration[31]. Inaddition,ourexpressionsforthedispersion c =1 Tr(Λ′)2−Tr(Λ′2) ,
2 2
relation and propagator also hold for the quadratic part c =−T(cid:2)r(Λ′). (cid:3) (A5)
of the SME’s heavy-boson sector, and are therefore ex- 3
pectedtofindapplicationsintheoreticalandphenomeno- Since both the determinant as well as the trace of a
logicalstudiesofelectroweakphysicswithLorentzbreak- square matrix remain unchanged under similarity trans-
ing. In the m→0 limit, our results yield the previously formations, the relations in Eqs. (A4) and (A5) hold, in
unknown massless photon propagator in the full SME, fact,alsoforouroriginal4×4matrixΛ. Employingthese
a quantity indispensable for many Lorentz-symmetryin- results in Eq. (A2) gives
vestigations in electrodynamics.
Λ4−[Λ]Λ3+ 1 [Λ]2−[Λ2] Λ2
2
+ 1[Λ2][Λ]− 1(cid:0)[Λ3]− 1[Λ]3(cid:1) Λ+det(Λ)11=0. (A6)
2 3 6
(cid:0) (cid:1)
Here, we have abbreviated the matrix trace by square
brackets. This result establishes that the set of matrices
ACKNOWLEDGMENTS
{11,Λ,Λ2,Λ3,Λ4} is linearly dependent.
Equation (A6) can be employed to determine further
This work has been supported in part by the Por- relations involving the 4 × 4 matrix Λ. For example,
tuguese Funda¸ca˜o para a Ciˆencia e a Tecnologia, by the taking the trace of Eq. (A6) yields the expression
Mexican RedFAE, by Universidad Andr´es Bello under
det(Λ)= 1 [Λ]4− 1[Λ4]+ 1[Λ2]2
Grant No. UNAB DI-27-11/R,as well as by the Indiana 24 4 8
University Center for Spacetime Symmetries. + 1[Λ][Λ3]− 1[Λ]2[Λ2] (A7)
3 4
8
for the determinant of Λ. Another relation can be ob- It is also convenient to extract the traceless part Gˆµν
tained by multiplying Eq. (A6) with the inverse matrix from the full Gˆµν:
Λ−1. The resulting equationcanthen be solvedfor Λ−1:
Gˆµν ≡Gˆµν − 1[Gˆ]ηµν. (B2)
4
1
Λ−1 = 1[Λ3]11+ 1[Λ]311− 1[Λ2][Λ]11 Here, we have again denoted the matrix trace by square
det(Λ)(cid:0)3 6 2 brackets [Gˆ] ≡ Gˆµµ. Inspection of our model La-
− 1[Λ]2Λ+ 1[Λ2]Λ+[Λ]Λ2−Λ3 . (A8) grangian (10) reveals that for the photon the decompo-
2 2
(cid:1) sition (B2) can be implemented by the replacements
In Eqs. (A7) and (A8) we have again abbreviated the
Gˆµν →Gˆµν
matrix trace by square brackets.
m→mˆ ≡m 1+ 1[Gˆ]
4
ξ−1 →ξˆ−1 ≡ξ(cid:0)−1 1+ 1(cid:1)[Gˆ] 2, (B3)
4
Appendix B: Dispersion relation (cid:0) (cid:1)
whereitisunderstoodthatξˆ−1/2 hastobeplacedatthe
appropriate position in the gauge-fixing term. We also
To present r , r , and r appearing in the physical
0 1 2 remind the reader that we denote the contraction of a
dispersionrelation(20), we introduce the followingmore
4-vectorwitha symmetric2-tensorbywritingthe vector
efficientnotation. We suppressthe AF subscriptonkˆAF asasuperscriptorasubscriptonthetensor: forexample
and drop the caret: (Gˆ2)k ≡GˆαGˆβp (kˆ )γ, etc.
p β γ α AF
With this notation, the r coefficients in the physical
(kˆ )µ ≡kµ. (B1) dispersion relation (20) take the following form:
AF
r =4[Kˆ]3+ 8[Kˆ3]−4[Kˆ][Kˆ2]+8mˆ2Kˆk+2[Kˆ]2Gˆp−2[Kˆ2]Gˆp−4(k·p)2Gˆp−4mˆ2[GˆKˆ2]+4mˆ2[Kˆ][GˆKˆ]
0 3 3 k p p p
−2mˆ2(GˆKˆGˆ)p+2mˆ2[Kˆ](Gˆ2)p+2mˆ2[GˆKˆ]Gˆp+8mˆ2(k·p)Gˆk+2mˆ4[Gˆ2Kˆ]−mˆ4[Kˆ][Gˆ2]−4mˆ4Gˆk
p p p p k
+mˆ4(Gˆ3)p− 1mˆ4[Gˆ2]Gˆp− 1mˆ6[Gˆ3]−8[GˆKˆ3]+8[Kˆ][GˆKˆ2]−4[Kˆ]2[GˆKˆ]+4[Kˆ2][GˆKˆ]+8KˆkGˆp
p 2 p 3 k p
+4mˆ2[Gˆ2Kˆ2]−2mˆ2[GˆKˆ]2+2mˆ2[(GˆKˆ)2]+mˆ2[Kˆ]2[Gˆ2]−mˆ2[Kˆ2][Gˆ2]−4mˆ2[Kˆ][Gˆ2Kˆ]+4mˆ2(Gˆk)2
p
−4mˆ2GˆkGˆp−2mˆ4[Gˆ3Kˆ]+mˆ4[GˆKˆ][Gˆ2]+ 2mˆ4[Kˆ][Gˆ3]− 1mˆ6[Gˆ2]2+ 1mˆ6[Gˆ4],
k p 3 8 4
r =2[Kˆ]2−2[Kˆ2]−4(k·p)2+4mˆ2k2+2[Kˆ]Gˆp+2mˆ2[GˆKˆ]+mˆ2(Gˆ2)p− 1mˆ4[Gˆ2]+8Kˆk+4k2Gˆp−4mˆ2Gˆk,
1 p p 2 k p k
r =2[Kˆ]+Gˆp+4k2. (B4)
2 p
Theabovedispersionrelationcanbe shownto reduceinthe appropriatelimittomass-dimensionthreeresultsquoted
in Ref. [18].
TodeterminethequantitiesR appearinginEq.(22),weapplytheusualTschirnhaus-typetransformationforcubic
j
equations to Expression (20). This yields the equivalent depressed form of the physical dispersion relation Q=0:
3
p2−mˆ2+ 2[Kˆ]+ 1Gˆp+ 4k2 +s p2−mˆ2+ 2[Kˆ]+ 1Gˆp+ 4k2 +s =0, (B5)
3 3 p 3 1 3 3 p 3 0
(cid:16) (cid:17) (cid:16) (cid:17)
where coefficients s and s are given by
0 1
s =16[Kˆ]3+ 8[Kˆ3]− 8[Kˆ][Kˆ2]+ 8(k·p)2[Kˆ]+8mˆ2Kˆk− 8mˆ2k2[Kˆ]+ 8[Kˆ]2Gˆp− 4[Kˆ2]Gˆp− 2[Kˆ](Gˆp)2
0 27 3 3 3 k 3 9 p 3 p 9 p
− 8(k·p)2Gˆp+ 2 (Gˆp)3−4mˆ2[GˆKˆ2]+ 8mˆ2[Kˆ][GˆKˆ]−2mˆ2(GˆKˆGˆ)p+ 4mˆ2[Kˆ](Gˆ2)p+ 4mˆ2[GˆKˆ]Gˆp
3 p 27 p 3 p 3 p 3 p
− 1mˆ2Gˆp(Gˆ2)p− 4mˆ2k2Gˆp+8mˆ2(k·p)Gˆk+2mˆ4[Gˆ2Kˆ]− 2mˆ4[Kˆ][Gˆ2]−4mˆ4Gˆk− 1mˆ4[Gˆ2]Gˆp
3 p p 3 p p 3 k 3 p
+mˆ4(Gˆ3)p− 1mˆ6[Gˆ3]+ 8k2[Kˆ]2+ 8k2[Kˆ2]− 16[Kˆ]Kˆk+ 16k2(k·p)2− 4k2(Gˆp)2+8[Kˆ][GˆKˆ2]
p 3 9 3 3 k 3 9 p
−8[GˆKˆ3]−4[Kˆ]2[GˆKˆ]+4[Kˆ2][GˆKˆ]+ 16KˆkGˆp− 16k2[Kˆ]Gˆp−4mˆ2[Kˆ][Gˆ2Kˆ]+2mˆ2[(GˆKˆ)2]
3 k p 9 p
+4mˆ2[Gˆ2Kˆ2]−2mˆ2[GˆKˆ]2+mˆ2[Kˆ]2[Gˆ2]−mˆ2[Kˆ2][Gˆ2]− 8mˆ2k2[GˆKˆ]+4mˆ2(Gˆk)2− 8mˆ2GˆkGˆp
3 p 3 k p
− 16mˆ2k4+ 8mˆ2[Kˆ]Gˆk− 4mˆ2k2(Gˆ2)p−2mˆ4[Gˆ3Kˆ]+mˆ4[GˆKˆ][Gˆ2]+ 2mˆ4[Kˆ][Gˆ3]+ 2mˆ4k2[Gˆ2]
3 3 k 3 p 3 3
− 1mˆ6[Gˆ2]2+ 1mˆ6[Gˆ4]+ 64k4[Kˆ]− 32k2Kˆk− 16k4Gˆp+ 16mˆ2k2Gˆk+ 128k6,
8 4 9 3 k 9 p 3 k 27
s =2[Kˆ]2−2[Kˆ2]−4(k·p)2+4mˆ2k2+ 2[Kˆ]Gˆp− 1(Gˆp)2+2mˆ2[GˆKˆ]+mˆ2(Gˆ2)p− 1mˆ4[Gˆ2]+8Kˆk
1 3 3 p 3 p p 2 k
− 16k2[Kˆ]+ 4k2Gˆp−4mˆ2Gˆk− 16k4. (B6)
3 3 p k 3
9
With s and s at hand, the usual formulae for the roots of a cubic determine explicit expressions for the R
0 1 j
quantities. TheseexpressionscanthenbeusedfortheiterationEq.(25): Supposewewanttocalculatethefirst-order
frequencies from the zeroth-order solution ( p~2+m2,p~), which corresponds to the first iteration step in Eq. (25).
With ( p~2+m2,~p) and Eqs. (B6), we canpin principle check the sign of the discriminant 4s 3 +27s 2, at least
1 0
to leadping order. A positive discriminant is associated with complex-valued Rj, which would appear to lead to
unphysical roots. We therefore take the discriminant to be non-positive in what follows. The R can then be
j
expressed trigonometrically:
s 1 3s 3 2π
R =2 − 1 cos cos−1 0 − − j − 2[Kˆ]− 1Gˆp− 4k2, j ∈{1,2,3}. (B7)
j r 3 (cid:20)3 (cid:18)2s r s (cid:19) 3 (cid:21) 3 3 p 3
1 1
These explicit R can now, for example, be employed in even in p even and thus real. Complex conjugation of
j
the context of Eq. (25). Sµν(p)thereforeonlychangesthesignofitskˆ piece. It
AF
We remarkthat forthe specialcaseofa vanishing dis- thus becomes evident that the momentum-space Stueck-
criminant, Eq. (B7) determines a multiple root, so that elberg operator is left unaffected under the combination
at least two physical excitations propagatewithout bire- of the two operations:
fringence. We also note that for non-positive discrim-
inant the argument of the cosine function in Eq. (B7) Sµν(+p,+kˆAF,kˆF,Gˆ)=Sµν(−p,+kˆAF,kˆF,Gˆ)∗
is real, and its magnitude therefore remains bounded by =Sνµ(−p,+kˆ ,kˆ ,Gˆ), (C2)
AF F
one. The limit of zero Lorentz violation is then deter-
mined by s1, [Kˆ], Gˆpp, and k2. These coefficients will where the Hermiticity of Sµν has been used. Since any
generically be momentum dependent. In particular, the determinant is invariant under transposition, we find
zerocomponentsfortheunphysicalsolutionsmaydiverge
in this limit, so that the corresponding behavior of the detS(+p,kˆ ,kˆ ,Gˆ)=detS(−p,kˆ ,kˆ ,Gˆ), (C3)
AF F AF F
R is unclear a priori. However, for the physical solu-
j
tions the plane-wave frequencies must remain bounded i.e., p → −p represents a discrete symmetry of the dis-
because they are defined to approach the conventional persionrelation. Thisinvariancealsobecomesclearwhen
expressions in the Lorentz-symmetric limit. For vanish- Eqs. (19), (20), and (B4) are combined.
ing Lorentz breaking, the physical Rj are therefore seen Continuing our analysis in a concordant frame [5], we
to approach zero, as required by self-consistency. fix a wave 3-vector p~ and imagine solving the disper-
sion relation (19). Following the reasoning presented in
Sec.V,wecaneliminatethespuriousOstrogradskimodes
Appendix C: Discrete symmetries and focus on the six physicalbranches of the dispersion-
relation solution with exact ωj,±. In these expressions,
p~
An interesting question concerns the number of inde- the ± label must correspond to the sign of the solution,
pendent plane-wave solutions. Some insight into this is- a property that follows immediately from the required
sue can be gained by investigating discrete symmetries proximitytotheLorentz-symmetricsheets[32]. Without
of both the plane-wave polarization vectors and the dis- lossofgenerality,wenowconsiderthespecificplane-wave
persion relation (19). 4-momentum with positive energy (pj+)µ ≡ (ω−j,+p~,−p~).
SincetheLagrangian(10)isreal,themomentum-space The p → −p symmetry of the dispersion relation im-
Stueckelberg operator must be hermitian plies that (pj−)µ ≡ (−ω−j,+p~,~p) also represents a solution,
butone withnegativefrequency. Moreover,this solution
Sµν(p)∗ =Sνµ(p), (C1) must be physical: since pj is close to the conventional
+
positivesheet,pj mustbeaperturbationofthenegative
−
which can also be established directly by inspection of sheet. With a suitable selection of the j labels, we may
Eq. (17). Next, we consider Sµν(p) under p→−p. Only thereforeconcludeωj,− =−ωj,+. Insummary,the setof
p~ −p~
the kˆ piece reversesits signbecause it is oddin p. All physical plane-wave momenta is given by
AF
other terms are p even and remain unchanged. We also
need the behavior of Sµν(p) under complex conjugation. (pj )µ =(ωj,+,~p) , (pj )µ =(−ωj,+,~p) , (C4)
+ p~ − −p~
Since the position-space Stueckelberg operator is purely n o
real, complex-valued contributions to the momentum-
andisthusdeterminedentirelybytheexpressionsforthe
spaceSµν(p)canonlyarisefromtheFourierreplacement three positive-frequency solutions ωj,+.
∂ → −ip. We will focus on real-valued plane-wave mo- p~
In addition to the plane-wavemomenta (pj )µ, the as-
menta p since otherwiseaninterpretationas a propagat- ±
ing solution is difficult. It then follows that the p-odd sociatedpolarizationvectors(εj±)µ(p~)areneededtocon-
kˆ term is purely imaginary, while all other terms are structaplanewave. Giventheεj (p~)associatedwiththe
AF +
10
positive-frequency roots, we can construct the remain- of the Lorentz-violating kˆ . One can show that the co-
F
ing negative-frequency εj (p~) as follows. Up to normal- ordinate scalar Iˆobeys the following identities:
−
ikzeartnioelno,ftthheepSotulaercikzaeltbioenrgvoepcetorarstoarr:eSb(yωjd,±efi,p~n)itεijon(p~i)n=th0e. 112IˆGˆpp =2[GˆKˆ3]−2[GˆKˆ2][Kˆ]+[GˆKˆ] [Kˆ]2−[Kˆ2]
ThisdefinitioncontainsS(ω−j,+p~,−p~)εj+(−p~p~)=0±asaspe- 41Iˆp2 =3[Kˆ][Kˆ2]−[Kˆ]3−2[Kˆ3]. (cid:0) ((cid:1)D2)
cial case. Complex conjugation combined with Eq. (C2) With these identities, the full massless dispersion rela-
then yields S(−ωj,+,p~)εj ∗(−p~) = 0. Together with the tion (19) can be cast into the form
−p~ +
plane-wave momentum property (C4), this equation is ξˆdetS =(ηˆµνp p )2 p4+2[Kˆ]p2+8Kˆk− 1Iˆ
identified as the defining relation for εj−(p~). Using suit- µ ν (cid:16) k 3
able normalizations, we arrive at −4(k·p)2+4k2p2+2[Kˆ]2−2[Kˆ2] . (D3)
(cid:17)
(εj )µ(p~)=(εj ∗)µ(−p~). (C5) It thus becomes apparent that in the massless limit the
− +
expression (ηˆµνp p ) can be factored out of Q.
µ ν
We remark that this relation continues to hold for The next step is to identify the physical part of the
dispersion-relation roots with higher multiplicities when massless dispersion relation (D3). Clearly, one of the
the bases of KerS(−ω−j,+p~,p~) are chosen appropriately. (ηˆµνpµpν)factorsoriginatesfromtheauxiliarymodeand
Anarbitrary,free,real-valued,physicalsolutionAµ(x) remains unphysical. To gain insight into the other fac-
tors,itisinstructivetorecallthemodelLagrangian(10).
canberepresentedbythefollowingFouriersuperposition
of plane waves [33]: In the zero-mass limit, Gˆ enters the equations of motion
for the photon only via the gauge-fixing term L and
g.f.
Aµ(x)=Z (2dπ3p~)3 Xj=31haj+(p~)(εj+)µ(p~)e−ipj+·x psfohhlolyouswilcdsatlthhbaeertceatfhuoerseeseintcooctnondleta(aηˆdinµνstopηˆµo=pbνs)ηe+fravcGaˆtbo.lreWmeeffucesotcntascl.lsuoIdtbeetthuhenant-
+aj−(p~)(εj−)µ(p~)e−ipj−·x +c.c. (C6) thelastfactorinEq.(D3),whichisfreeofGˆ,governsthe
physical excitations for m = 0. This factor is in agree-
i
ment with the general photon dispersion relation given
This expression suggests that six Fourier coefficients
in Ref. [6], as required by consistency.
{aj (p~),aj (p~)} with j = 1,2,3 must be specified inde-
+ − The exact massive propagator (21) as well as its
pendently to determine a general solution. However, a
leading-order approximation (26) exhibit six physical
change of integration variable p~ → −p~ in the expression
polesthatmustbe encircledbythe integrationcontours.
associatedwiththenegative-frequencysolutionstogether
Theaboveargumentidentifiesthosetwoofthesesixpoles
with Eqs. (C4) and (C5) yields
thatshouldbecomeunphysicalinthemasslesscase. The
question then arises as to whether the unphysical char-
Aµ(x)= d3~p 3 bj(p~)(εj )µ(p~)e−ipj+·x+c.c., (C7) acterofthesetwomodesisreflectedinthepolestructure
Z (2π)3 + of the propagator in the m → 0 limit. This may be es-
Xj=1 tablished by demonstrating that the (ηˆµνp p ) factor in
µ ν
thedenominatorofthepropagatoriscancelledbyacorre-
where we have defined bj(p~) ≡ aj+(p~)+aj−∗(−p~). This sponding(ηˆµνpµpν)factorinthenumeratorforvanishing
establishesthatthe six physicaldispersion-relationroots
mass. An investigation of this cancellation for the exact
are associated with three degrees of freedom.
expression (21) would be interesting but lies outside the
scope of the present work. We instead focus on the on
the leading-order piece (26) of the propagator, which is
Appendix D: Massless limit appropriate for the majority of practical applications.
In the discussionof the leading-orderpropagator(26),
itwasalreadymentionedthatthe (1−ξ)contributionin
In the context of regularizing infrared divergences in
the second line leaves unaffected observable quantities.
quantum-field perturbation theory, the massless limit of
Wemaythereforefocusonthephysicalcontributioncon-
our Lorentz-violating Stueckelberg model is of particu-
tained in the first line of Eq. (26). In the m → 0 limit,
lar importance. The photon mass was introduced into
the numerator −iNµν of this first piece has the form
our Lagrangian via the Stueckelberg method, which en-
sures the proper behavior in this limit. Nevertheless, it Nµν =p2 2[Kˆ]ηµν −2Kˆµν +2iEˆµν +pµGˆν +pνGˆµ
p p
is interesting to study explicitly the dispersion relation +(cid:0)p2 p2+Gˆp ηµν. ((cid:1)D4)
and the propagator of our model for vanishing m, for p
example, to compare with previous results for massless The ηµν term(cid:0) in the s(cid:1)econd line of the above Eq. (D4)
Lorentz-violating photons. contains the factor p2+Gˆp =ηˆµνp p . All terms in the
p µ ν
We begin by defining the contraction first line represent leading-order corrections, which re-
mainunaffectedbythe additionofhigher-orderLorentz-
Iˆ≡ǫ ǫ p p p p (kˆ )αβκρ(kˆ )λστγ(kˆ )ιδµν (D1) violating terms. A judicious choice for such higher-order
αβγδ κλµν ρ σ τ ι F F F