Data Tables for Lorentz and CPT Violation

0.37 MB·English

by  Alan Kostelecky

IUHET 538, January 2010
Data Tables for Lorentz and CPT Violation
V. Alan Kostelecky´a and Neil Russellb
aPhysics Department, Indiana University, Bloomington, IN 47405
bPhysics Department, Northern Michigan University, Marquette, MI 49855
(Dated: January 2010 edition)
ThisworktabulatesmeasuredandderivedvaluesofcoefficientsforLorentzandCPTviolationin
theStandard-ModelExtension. Summarytablesareextractedlistingmaximalattainedsensitivities
in the matter, photon, and gravity sectors. Tables presenting definitions and properties are also
compiled.
0
1
0 I. INTRODUCTION narymatter(electrons, protons, andneutrons), photons,
2 andgravity. Thetablesincluderesultsavailablefromthe

n literature up to December 31, 2009. More recent papers
Recent years have seen a renewed interest in experi-
a can be found online [7].
J mental tests of Lorentz and CPT symmetry. Observable
signalsofLorentzandCPTviolationcanbedescribedin Theorderofthetablesisasfollows. TableIcontainsa
5
listofalltables. Thethreesummarytablesarepresented
a model-independent way using effective field theory [1].
next, Tables II, III, and IV. These are followed by the
] The general realistic effective field theory for Lorentz
h data tables, Tables V to XV. The properties tables ap-
p violation is called the Standard-Model Extension (SME)
pear last, Tables XVI to XXIV.
- [2,3]. ItincludestheStandardModelcoupledtoGeneral
p A description of the summary tables is given in Sec.
Relativity along with all possible operators for Lorentz
e II.Informationabouttheformatandcontentofthedata
h violation. Both global [2] and local [3] Lorentz violation
tables is presented in Sec. III, while Sec. IV provides an
[ are incorporated. Since CPT violation in realistic field
overview of the properties tables. The bibliography for
theoriesisaccompaniedbyLorentzviolation[4],theSME
3 the text and all the tables follows Sec. IV.
v also describes general CPT violation. Reviews of the
7 SME can be found in Refs. [5, 6].
8
Each Lorentz-violating term in the Lagrange den- II. SUMMARY TABLES
2
sity of the SME is constructed as the coordinate-
0
. independentproductofacoefficientforLorentzviolation Threesummarytablesareprovided(TablesII,III,IV),
1
withaLorentz-violatingoperator. TheLorentz-violating
0 listingmaximalexperimentalsensitivitiesattainedforco-
physics associated with any operator is therefore con-
8 efficients in the matter, photon, and gravity sectors of
0 trolled by the corresponding coefficient, and so any ex- the minimal SME. To date, there is no compelling ex-
: perimental signal for Lorentz violation can be expressed
v perimentalevidencesupportingLorentzviolation. Afew
in terms of one or more of these coefficients.
i measurements suggest nonzero coefficients at weak con-
X
The Lorentz-violating operators in the SME are sys- fidence levels. These latter results are excluded from the
ar tematically classified according to their mass dimension, summary tables but are listed in the data tables.
and operators of arbitrarily large dimension can appear. In these three summary tables, each displayed sen-
At any fixed dimension, the operators are finite in num- sitivity value represents our conservative estimate of a
ber and can in principle be enumerated. A limiting case 2σ limit, given to the nearest order of magnitude, on
of particular interest is the minimal SME, which can be the modulus of the corresponding coefficient. In a few
viewed as the restriction of the SME to include only cases, tighter results may exist when suitable theoreti-
Lorentz-violating operators of mass dimension four or cal assumptions are adopted; these results can be found
less. The corresponding coefficients for Lorentz violation in the data tables that follow. Where observations in-
are dimensionless or have positive mass dimension. volve a linear combination of coefficients in the tables,
Theresultssummarizedhereconcernprimarilybutnot the displayed sensitivity for each coefficient assumes for
exclusively the coefficients for Lorentz violation in the definiteness that no other coefficient contributes. Some
minimalSME.WecompiledatatablesfortheseSMEco- caution is therefore advisable in applying the results in
efficients, including both existing experimental measure- thesesummarytablestosituationsinvolvingtwoormore
mentsandsometheory-derivedlimits,andweprovideta- nonzero coefficient values. Care in applications is also
bles listing some relevant definitions and properties. We required because under some circumstances certain coef-
alsoextractsummarytableslistingourbestestimatesfor ficients can be intrinsically unobservable or can be ab-
the maximal attained sensitivities in three sectors: ordi- sorbed into others by field or coordinate redefinitions.
2
gravity sector of the minimal SME. Two classes of co-
efficients can be distinguished in this context: ones ap-
pearing in the matter sector, and ones appearing in the
pure-gravity sector. For the first class, Table IV lists the
12 observables involving the coefficients ae, ap, and an
µ µ µ
for electrons, protons, and neutrons, respectively. These
observables are associated with CPT-odd operators and
havedimensionsofGeVinnaturalunits. Theprefactorα
is a model-dependent number [11]. For the second class,
the table displays nine combinations of the nine dimen-
sionless coefficients for Lorentz violation sµν. Additional
sets of coefficients involving the gravity sector exist, but
FIG. 1: Standard Sun-centered inertial reference frame [8].
no sensitivities to them have been identified to date.
III. DATA TABLES
In presenting the physical sensitivities, we adopt nat-
ural units with ~ = c = ǫ0 = 1 and express mass units We present 11 data tables compiled from the existing
in GeV. Our values are reported in the standard Sun- literature. Of these, 10 tables include results for various
centered inertial reference frame widely used in the liter- sectors of the minimal SME: the electron sector (Table
ature. This frame is illustrated in Fig. 1. It has Z axis V),theprotonsector(TableVI),theneutronsector(Ta-
directed north and parallel to the rotational axis of the ble VII), the photon sector (Table VIII), the charged-
Earth. The X axis originates from the Sun and points lepton sector (Table IX), the neutrino sector (Table X),
towards the vernal equinox, while the Y axis completes the meson sector (Table XI), the electroweak sector (Ta-
a right-handed system. The time coordinate T is fixed ble XII), the gluon sector (Table XIII), and the gravity
withoriginatthe2000vernalequinox. Detailsaboutthis sector(TableXIV). Theremainingtable(TableXV)lists
frame, including transformations to other frames, can be existing bounds on nonminimal coefficients for Lorentz
found in Section IIIA of Ref. [9]. violation in the photon sector.
TableIIliststhemaximalattainedsensitivitiesinvolv- Each of these 11 data tables contains four columns.
ing protons, neutrons, and electrons. For each distinct The first column lists the coefficients for Lorentz vio-
massive spin-half Dirac fermion in the minimal SME in lation or their relevant combinations. Results for coef-
Minkowski spacetime, there are 44 independent observ- ficients of the same generic type are grouped together.
able combinations of coefficients for Lorentz violation in Certainresultsinvolvecombinationsofcoefficientsacross
the nonrelativistic limit. Of these, 20 also control CPT more than one sector; each of these has been listed only
violation. The44combinationsareconventionallychosen once in the table deemed most appropriate. Some mi-
as the tilde coefficients shown. The definitions of these nor changes in notation or format have been introduced
44tilde coefficients in termsof minimal SMEcoefficients as needed, but for the most part the results are quoted
arelistedinTableXVIII. Allthedefinitionsappearelse- as they appear in the cited references. Definitions for
where in the literature [8] except the four combinations standard combinations of coefficients are provided in the
˜b∗J andc˜TT. InTableII,alltildecoefficientshavedimen- properties tables that follow. A few authors use uncon-
sions of GeV in natural units. A dash indicates that no ventional notation; where immediate, the match to the
sensitivity to the coefficient has been identified to date. standard notation is shown.
Table III displays the maximal attained sensitivities The second column contains the measurements and
to coefficients for Lorentz violation in the photon sector bounds, presented in the same form as documented in
of the minimal SME. There are 23 observable coefficient the literature. For each generic type of coefficient, the
combinationsforphotons,ofwhichfouralsocontrolCPT resultsarelistedinreversechronologicalorder. Ifnosig-
violation. The 19 tilde coefficients listed in the table nificantfiguresappearinthequotedlimitonanabsolute
are conventional combinations of the 19 dimensionless value, the order of magnitude of the limit is given as a
coefficients in the minimal SME. The remaining entries power of 10.
in the table concern combinations of the four coefficients The third column contains a succinct reminder of the
controllingCPTviolation,whichhavedimensionsofGeV physical context in which the bound is extracted, while
in natural units. The definitions of all 23 combinations the fourth column contains the source citations. The
are taken from the literature [9, 10] and are provided in reader is referred to the latter for details of experimen-
Table XIX. tal and theoretical procedures, assumptions underlying
Table IV displays the maximal attained sensitivities the results, definitions of unconventional notations, and
to certain coefficients for Lorentz violation involving the other relevant information. Results deduced on theoret-
3
ical grounds are distinguished from those obtained via references. They are expressed in terms of the coeffi-
direct experimental measurement by an asterisk placed cients for Lorentz violation listed in the gravity section
after the citation. of Table XXI.
Tables V, VI, and VII contain data for the electron, The final data table, Table XV, contains a compila-
proton, and neutron sectors, respectively. Each table tion of some measurements and bounds on coefficients
is divided into sections focusing sequentially on combi- for Lorentz violation in the nonminimal SME. Atten-
nations involving the coefficients b , c , H , d , and tion is restricted to the photon sector, in which results
ν µν µν µν
g . Standarddefinitionsforthesecoefficientsandtheir are available for a variety of nonrenormalizable opera-
µνλ
combinations are provided in Tables XVI and XVIII. tors of dimensions 5, 6, 7, 8, and 9. A convenient ba-
Some results depend on η 23.5◦, which is the angle sis for classifying operators of dimension d is given by
≃
between the equatorial and ecliptic planes in the solar the spin-weighted spherical harmonics [13]. The corre-
system. Notethattheexistingboundonthecombination spondingcoefficientsarelistedinTableXXIV. Somecon-
of observables involving ae, ap, and an is obtained from straints have been obtained for the vacuum coefficients
ν ν ν
gravitational experiments and is listed with the gravity- for Lorentz violation, which are c(d) , k(d) , k(d)
(I)jm (E)jm (B)jm
sector results in Table XIV. for even d and k(d) for odd d, where the subscripts
(V)jm
Table VIII presents the photon-sector data. Most of
jm label the angular-momentum quantum numbers. In
the combinations of coefficients for Lorentz violation ap-
the first column of Table XV, the usual spherical har-
pearing in the first column are defined in Tables XIX
monics Y are evaluated at specified angles, which are
and XVI. The alternative combinations k((V3))jm, k((E4))jm, the celes0tijaml coordinates of certain astrophysical sources.
and k(4) arise from analyses [10, 12, 13] using spin- None of the vacuum-orthogonal coefficients for Lorentz
(B)jm
weighted spherical harmonics. The factor of β⊕ appear- violation have been measured to date.
ing in some places is the speed of the Earth in the stan-
dard Sun-centered reference frame, which is about 10−4
in natural units. IV. PROPERTIES TABLES
Tables IX, X, and XI list measurements and bounds
oncoefficientsforLorentzviolationinvolvingsecond-and Ninepropertiestablesareprovided,listingvariousfea-
third-generation fermions in the minimal SME. Results tures and definitions related to Lorentz violation. Four
formuonsandtauleptonsareinTableIX,whilethosefor tables concern the terms in the restriction of the min-
neutrinos are in Table X. For both these tables, many of imal SME to quantum electrodynamics (QED) in Rie-
thecoefficientsappearinginthefirstcolumnarespecified mann spacetime. For this theory, which is called the
in the lepton sector of Table XX. The neutrino results minimal QED extension, the tables include information
in Table X are obtained in the context of various simpli- about the operator structure (Table XVI), the action of
fiedmodels,asdiscussedinthereferences. Experimental discrete symmetries (Table XVII), and some useful coef-
sensitivitiestocoefficientsforoperatorsinvolvingsecond- ficient combinations (Table XVIII and Table XIX). Two
andthird-generationquarkfieldsarepresentlylimitedto tables contain information about the matter sector (Ta-
mesons and are presented in Table XI. The coefficients ble XX) and the gauge and gravity sectors (Table XXI)
appearing in this table are composite quantities defined of the minimal SME in Riemann-Cartan spacetime. An-
in the corresponding references. They are effective co- other table (Table XXII) summarizes some features of
efficients for which complete analytical expressions are the coefficients for Lorentz violation in the neutrino sec-
asyetunknown, formedfromcertain quark-sectorcoeffi- tor. The two remaining tables (Table XXIII and Table
cients appearing in Table XX and from other quantities XXIV)provideinformationabouttheoperatorstructure
arising from the quark binding in the mesons. and the spherical coefficients for Lorentz violation in the
Tables XII and XIII concern coefficients in the gauge nonminimal photon sector.
sectors of the minimal SME. Results for the electroweak For these properties tables, our primary conventions
sector are listed in Table XII, while those for the gluon are those of Ref. [3]. Greek indices µ,ν,λ,... refer to
sector are in Table XIII. The coefficients for the elec- curved-spacetimecoordinatesandLatinindicesa,b,c,...
troweak sector are defined in the gauge and Higgs sec- to local Lorentz coordinates. The vierbein formalism
tions of Table XXI. The gluon-sector coefficient is the [14],whichrelatesthetwosetsofcoordinates,isadopted
analogue of the corresponding photon-sector coefficient to facilitate the description of spinors on the spacetime
defined in Table XIX. To date, all results for the gauge manifold. The determinant e of the vierbein e a is re-
µ
sector are deduced from theoretical considerations. latedtothedeterminantgofthemetricg bye=√ g.
µν
−
Table XIV presents measurements and bounds con- The conventions for the Dirac matrices γa are given in
cerning the gravity sector of the minimal SME. The spe- Appendix A of Ref. [3]. The Newton gravitational con-
cific combinations of coefficients in the pure-gravity sec- stant G enters as the combination κ 8πG , and it
N N
≡
tor that appear in the first column are defined in the has dimensions of inverse mass squared.
4
In the Minkowski-spacetime limit, the metric g transformations considered are charge conjugation C,
µν
is written η with diagonal entries ( 1,1,1,1). For parity inversion P, time reversal T, and their combina-
µν
−
decompositions into time and space components, we tions CP, CT, PT, and CPT. The first column speci-
adopt the Sun-centered frame of Fig. 1 and use in- fies the operator by indicating its corresponding coeffi-
dices J,K,L,... to denote the three spatial components cient. Each of the other columns concerns one of the
X,Y,Z. The sign of the antisymmetric tensor ǫ is seven transformations. An even operator is indicated by
κλµν
fixed via the component ǫ =+1, and the antisym- a plus sign and an odd one by a minus sign. The table
TXYZ
metricsymbolinthreespatialdimensionsisdefinedwith contains eight rows, one for each of the eight possible
ǫ =+1. NotethatsomeoftheliteratureontheSME combinations of signs under C, P, and T.
XYZ
in Minkowski spacetime adopts a metric ηµν of opposite Table XVIII lists the definitions of the 44 combina-
sign, following the common present usage in quantum tions of coefficients for Lorentz violation that frequently
physics instead of the one in relativity. Under this alter- appear in experimental analyses involving the fermion
nativeconvention,termsintheLagrangedensitywithan sectoroftheminimalQEDextensioninMinkowskispace-
oddnumberofmetriccontractionshaveoppositesignsto time in the nonrelativistic limit. These combinations are
those appearing in this work. conventionally denoted by tilde coefficients, listed in the
Table XVI concerns the minimal QED extension, first column of the table. Note that six of these com-
for which the basic nongravitational fields are a Dirac binations, c˜X, c˜Y, c˜Z, g˜TX, g˜TY, and g˜TZ, are denoted
fermion ψ and the photon Aµ. The electromagnetic as c˜Q,Y, c˜Q,X, c˜XY, g˜Q,Y, g˜Q,X, and g˜XY, respectively,
field-strength tensor is F = ∂ A ∂ A . The pure- in some early publications. The definitions in the table
µν µ ν ν µ
gravity sector involves the Riemann−tensor R , the are given for a generic fermion of mass m. Most applica-
κλµν
RiccitensorR ,thecurvaturescalarR,andthecosmo- tions in the literature involve electrons, protons, or neu-
µν
logical constant Λ. The spacetime covariant derivative trons, for which the corresponding mass is understood.
D corrects local Lorentz indices using the spin connec- Thefinalcolumnliststhenumberofindependentcompo-
µ
tion, corrects spacetime indices using the Cartan con- nents of each coefficient. For ordinary matter involving
nection, and contains the usual gauge field A for the protons, electrons, and neutrons, there are therefore 132
µ
↔ independent observable coefficients for Lorentz violation
photon. The notation D is an abbreviation for the dif-
µ in the minimal QED sector of the SME in Minkowski
ference of two terms, the first with derivative acting to
spacetime.
the right and the second acting to the left. Note that
Table XIX presents definitions for certain combina-
Table XVI is restricted to the zero-torsion limit of the
tions of the 23 coefficients for Lorentz violation in
minimal SME. The general case [3] involves additional
the photon sector of the minimal QED extension in
operators constructed with the torsion tensor Tα . The
µν Minkowski spacetime. This table has three sections.
Minkowski-spacetime limit of QED with nonzero torsion
The first section consists of five rows listing 19 widely
contains terms that mimic Lorentz violation, so searches
used combinations of the 19 coefficients for CPT-even
for Lorentz violation can be used to bound components
Lorentz violation. The second section provides 10 alter-
of the torsion tensor [15].
nativecombinations involvingthe10CPT-evenLorentz-
In Table XVI, each line specifies one term in the La- violating operators relevant to leading-order birefrin-
grangedensityfortheQEDextensioninRiemannspace- gence [9]. The third section lists four combinations
time. Both conventional QED terms and ones with of the four coefficients for CPT-odd Lorentz violation.
Lorentz violation are included. The first column indi- Thesecombinationsappearwhenabasisofspin-weighted
cates the sector to which the term belongs. The second spherical harmonics is adopted.
column lists the coefficient controlling the corresponding
TableXXconcernsthefermion-sectortermsintheLa-
operator. Note the standard use of an upper-case let-
grange density of the minimal SME in Riemann-Cartan
ter for the coefficient H , which distinguishes it from
µν spacetime. The column headings are similar to those
the metric fluctuation h . The third column shows the
µν in Table XVI. In the lepton sector, the left- and right-
numberofcomponentsforthecoefficient. Thenextthree handed leptons are denoted by L and R , where A is
A A
columns list the operator, its mass dimension, and the the generation index. The SU(2) doublet L includes
A
vierbein factor contracting the coefficient and the opera- the three neutrino fields ν , ν , ν and the left-handed
e µ τ
tor. Thefinaltwocolumnslistthepropertiesoftheterm componentsofthethreechargedleptonse,µ,andτ. The
under CPT and Lorentz transformations. A CPT-even SU(2) singlet R contains the right-handed components
A
operator is indicated by a plus sign and a CPT-odd one of e, µ, and τ. The derivative D is both spacetime and
µ
byaminussign,whiletermsviolatingLorentzinvariance SU(3) SU(2) U(1)covariant. Thequarkfieldsarede-
are identified by a check. notedU× ,D ,×andQ ,whereAisthegenerationindex.
A A A
Table XVII lists the properties under discrete- The right-handed components of the u, c, and t quarks
symmetry transformations of the Lorentz-violating op- are the SU(2) singlets U , while the right-handed com-
A
erators in the minimal QED extension [16]. The seven ponents of d, s, and b are the SU(2) singlets D . The
A
5
six left-handed quark fields are contained in the SU(2) thecoefficientforLorentzviolation,whilethesecondcol-
doublet Q . The Yukawa sector involves terms coupling umncountsitsindependentcomponents. Thenextthree
A
the Higgs doublet φ to the leptons and to the quarks. columns provide the corresponding operator appearing
The conventional Yukawa-coupling matrices are denoted intheLagrangedensity,itsmassdimension,andthefac-
(G ) , (G ) , and (G ) . The hermitian conju- torcontractingthecoefficientandtheoperator. Thelast
L AB U AB D AB
gate of an operator is abbreviated h.c. in the table. two columns list the properties of the operator under
Table XXI presents information about the Higgs, CPT and Lorentz transformations, using the same con-
gauge, and pure-gravity sectors for the Lagrange den- ventions as Table XVI.
sity of the minimal SME in Riemann-Cartan spacetime. Three sections appear in Table XXIII, separated by
The structure of the table is the same as that of Ta- horizontal lines. The first section concerns the conven-
ble XX. As before, D is both a spacetime and an tional Lorentz-preserving Maxwell term in the Lagrange
µ
SU(3) SU(2) U(1)covariantderivative. Thecomplex density for the photon sector. The second and third
× ×
Higgsfieldisdenotedφ,theSU(3)colorgaugefieldsand sections concern coefficients associated with operators of
the SU(2) gauge fields are the hermitian adjoint matri- odd and even dimensions d, respectively. Each of these
cesG andW ,respectively,whiletheU(1)hypercharge sections has three rows for the lowest three values of d,
µ µ
gauge field is the singlet B . Each gauge field has an along with a final row applicable to the case of general
µ
associated field strength, denoted G for the strong in- d. Thenotation forthecoefficientsincorporatesasuper-
µν
teractions, W for the weak interactions, and B for scriptspecifyingthedimensiondofthecorrespondingop-
µν µν
the hypercharge. The pure-gravity sector of Table XXI erator. Note that the mass dimension of the coefficients
differs from that in Table XVI only in the addition of is 4 d. In each section, the first row describes terms
−
terms involving the torsion field Tα . in the minimal SME, and the match is provided between
µν
The minimal SME in Riemann-Cartan spacetime de- the general notation for nonminimal coefficients and the
scribedinTablesXXandXXIcanbereducedtothemin- standard notation used for the minimal SME in Table
imalQEDinRiemannspacetimedescribedinTableXVI XVI. In the case of mass dimension four, there are 19
as follows. For the gauge sector, including the covariant independent Lorentz-violating operators. However, for
derivatives, remove all the gauge fields except the charge this case the number in the second column is listed as
U(1) field in the photon limit B A , and remove all 19+1toallowforanadditionalLorentz-preservingtrace
µ µ
→
the Higgs terms. For the gravity sector, remove all the term, which maintains consistency with the expression
torsion terms. For the fermion sector, restrict the lepton for general d in the last row.
generation index to a single value, remove all quark and Table XXIV summarizes properties of spherical coef-
neutrino terms, and replace the Yukawa-coupling terms ficients for Lorentz violation in the nonminimal photon
with the relevant fermion mass terms. sector of the full SME in Minkowski spacetime [13]. The
Table XXII concerns the neutrino sector of the SME, spherical coefficients are combinations of the coefficients
including both neutrino masses and Lorentz-violating listed in Table XXIII that are of particular relevance for
terms. For the latter, we restrict attention to terms of observationandexperiment. Theycanbeseparatedinto
massdimensionfourorlessthatinvolvethreegenerations two types. One set consists of vacuum coefficients that
of neutrinos and antineutrinos, allowing for possible vi- control leading-order effects on photon propagation in
olations of SU(3) SU(2) U(1) gauge symmetry and the vacuum, including birefringence and dispersion. The
× ×
lepton number [17]. In the table, the first row involves complementary set contains the vacuum-orthogonal co-
the usual neutrino mass matrix me , where the indices efficients,whichleavephotonpropagationinthevacuum
AB
A,B take values e, µ, and τ, while the other rows con- unaffectedatleadingorder. ThetwopartsofTableXXIV
cern coefficients for Lorentz violation. The first column reflectthisseparation,withthepartabovethehorizontal
lists the coefficients, and the second gives the dimension lineinvolvingthevacuumcoefficientsandthepartbelow
of the corresponding operators in the Lagrange density. involving the vacuum-orthogonal ones.
The third column indicates generically the type of neu- In Table XXIV, the first column of the table identi-
trino oscillations controlled by the coefficients. The final fies the type of spherical coefficients, while the second
two columns list the properties of the operators under column lists the specific coefficient. The coefficient no-
CPT and Lorentz transformations. tation reflects properties of the corresponding operator.
Table XXIII provides information about the nonmini- Coefficientsassociatedwithoperatorsleavingunchanged
mal photon sector of the full SME in Minkowski space- the leading-order photon propagation in the vacuum are
time. The relevant part of the Lagrange density includes distinguished by a negation diacritic . A symbol k de-
¬
operators of arbitrary dimension d that are both gauge notes coefficients for birefringent operators, while c de-
invariant and quadratic in the photon field A [13]. The notes nonbirefringent ones. The superscript d refers to
µ
structure of the table is similar to those adopted for Ta- the operator mass dimension, while the the subscripts n,
bles XVI, XX, and XXI, with each row associated with j,mdeterminethefrequencyorwavelengthdependence,
a term in the Lagrange density. The first column lists the total angular momentum, and the z-component of
6
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8
TABLE I: List of tables
Type Table Content
Summary II Maximal sensitivities for the matter sector
III Maximal sensitivities for the photon sector
IV Maximal sensitivities for the gravity sector
Data V Electron sector
VI Proton sector
VII Neutron sector
VIII Photon sector
IX Charged-lepton sector
X Neutrino sector
XI Meson sector
XII Electroweak sector
XIII Gluon sector
XIV Gravity sector
XV Nonminimal photon sector
Properties XVI Lagrange density for the minimal QED extension in Riemann spacetime
XVII C, P, T, properties for operators for Lorentz violation in QED
XVIII Definitions for the fermion sector of the minimal QED extension
XIX Definitions for the photon sector of the minimal QED extension
XX Lagrange density for the fermion sector of the minimal SME in Riemann-Cartan spacetime
XXI Lagrange density for the boson sector of the minimal SME in Riemann-Cartan spacetime
XXII Coefficients in the neutrino sector
XXIII Quadratic Lagrange density for the nonminimal photon sector in Minkowski spacetime
XXIV Spherical coefficients for the nonminimal photon sector in Minkowski spacetime
9
TABLE II: Maximal sensitivities for the matter sector
Coefficient Proton Neutron Electron
˜b 10−27 GeV 10−31 GeV 10−31 GeV
X
˜b 10−27 GeV 10−31 GeV 10−31 GeV
Y
˜b – – 10−30 GeV
Z
˜b – 10−27 GeV 10−27 GeV
T
˜b∗ (J =X,Y,Z) – – –
J
c˜ 10−25 GeV 10−27 GeV 10−19 GeV
−
c˜ 10−22 GeV – 10−19 GeV
Q
c˜ 10−25 GeV 10−25 GeV 10−19 GeV
X
c˜ 10−25 GeV 10−25 GeV 10−19 GeV
Y
c˜ 10−24 GeV 10−27 GeV 10−19 GeV
Z
c˜ 10−20 GeV – 10−18 GeV
TX
c˜ 10−20 GeV – 10−18 GeV
TY
c˜ 10−21 GeV – 10−20 GeV
TZ
c˜ – – 10−18 GeV
TT
d˜ – 10−27 GeV 10−27 GeV
+
d˜ – 10−27 GeV 10−27 GeV
−
d˜ – 10−27 GeV 10−27 GeV
Q
d˜ – 10−27 GeV 10−27 GeV
XY
d˜ – 10−26 GeV 10−27 GeV
YZ
d˜ – – 10−26 GeV
ZX
d˜ 10−25 GeV 10−29 GeV 10−22 GeV
X
d˜ 10−25 GeV 10−28 GeV 10−22 GeV
Y
d˜ – – 10−19 GeV
Z
H˜ – 10−26 GeV 10−27 GeV
XT
H˜ – 10−27 GeV 10−27 GeV
YT
H˜ – 10−27 GeV 10−27 GeV
ZT
g˜ – 10−27 GeV 10−27 GeV
T
g˜ – 10−27 GeV 10−27 GeV
c
g˜ – – –
Q
g˜ – – –
−
g˜ (J =X,Y,Z) – – –
TJ
g˜ – – –
XY
g˜ – – –
YX
g˜ – – –
ZX
g˜ – – –
XZ
g˜ – – –
YZ
g˜ – – –
ZY
g˜ 10−25 GeV 10−29 GeV 10−22 GeV
DX
g˜ 10−25 GeV 10−28 GeV 10−22 GeV
DY
g˜ – – –
DZ
10
TABLE III: Maximal sensitivities for the photon sector
Coefficient Sensitivity
(κ˜ )XY 10−32
e+
(κ˜ )XZ 10−32
e+
(κ˜ )YZ 10−32
e+
(κ˜ )XX (κ˜ )YY 10−32
e+ e+
−
(κ˜ )ZZ 10−32
e+
(κ˜ )XY 10−32
o−
(κ˜ )XZ 10−32
o−
(κ˜ )YZ 10−32
o−
(κ˜ )XX (κ˜ )YY 10−32
o− o−
−
(κ˜ )ZZ 10−32
o−
(κ˜ )XY 10−16
e−
(κ˜ )XZ 10−16
e−
(κ˜ )YZ 10−16
e−
(κ˜ )XX (κ˜ )YY 10−17
e− e−
−
(κ˜ )ZZ 10−17
e−
(κ˜ )XY 10−13
o+
(κ˜ )XZ 10−13
o+
(κ˜ )YZ 10−13
o+
κ˜tr 10−14
k(3) 10−43 GeV
(V)00
k(3) 10−42 GeV
(V)10
Rek(3) 10−42 GeV
(V)11
Imk(3) 10−42 GeV
(V)11