The Barrel Organ

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The Barrel Organ

Data tables for Lorentz and CPT violation

V. Alan Kostelecky´, REVIEW OF MODERN PHYSICS, VOLUME 83, JANUARY–MARCH 2011

Physics Department, Indiana University, Bloomington, Indiana 47405, USA

Physics Department, Northern Michigan University, Marquette, Michigan 49855, USA

(Received 6 March 2010; published 10 March 2011)

This work tabulates measured and derived values of coefficients for Lorentz and CPT violation in

the standard-model extension. Summary tables are extracted listing maximal attained sensitivities

in the matter, photon, and gravity sectors. Tables presenting definitions and properties are also

DOI: 10.1103/RevModPhys.83.11 PACS numbers: 11.30.Cp, 11.30.Er

I. Introduction 11

II. Summary Tables 12

III. Data Tables 15

IV. Properties Tables 20

A. Minimal QED extension 22

B. Minimal SME 28

C. Nonminimal photon sector 29

I. INTRODUCTION

Recent years have seen a renewed interest in experimental

tests of Lorentz and CPT symmetry. Observable signals of

Lorentz and CPT violation can be described in a modelindependent

way using effective field theory (Kostelecky´ and

Potting, 1995).

The general realistic effective field theory for Lorentz

violation is called the standard-model extension (SME)

(Colladay and Kostelecky´, 1997; 1998; Kostelecky´, 2004).

It includes the standard model coupled to general relativity

along with all possible operators for Lorentz violation. Both

global and local Lorentz violation are incorporated. Since

CPT violation in realistic field theories is accompanied by

Lorentz violation (Greenberg, 2002), the SME also describes

general CPT violation. Reviews of the SME can be found

in Kostelecky´ (1999), (2002), (2005), Bluhm (2006),

Kostelecky´ (2008), (2011).

Each Lorentz-violating term in the Lagrange density of the

SME is constructed as the coordinate-independent product of

a coefficient for Lorentz violation with a Lorentz-violating

operator. The Lorentz-violating physics associated with any

operator is therefore controlled by the corresponding coefficient,

and so any experimental signal for Lorentz violation

can be expressed in terms of one or more of these coefficients.

The Lorentz-violating operators in the SME are systematically

classified according to their mass dimension, and

operators of arbitrarily large dimension can appear. At any

fixed dimension, the operators are finite in number and can in

principle be enumerated. A limiting case of particular interest

is the minimal SME, which can be viewed as the restriction

of the SME to include only Lorentz-violating operators of

mass dimension 4 or less. The corresponding coefficients for

Lorentz violation are dimensionless or have positive mass

The results summarized here concern primarily but not

exclusively the coefficients for Lorentz violation in the minimal

SME.We compile data tables for these SME coefficients,

including both existing experimental measurements and some

theory-derived limits. Each of these data tables provides

information about the results of searches for Lorentz violation

for a specific sector of the SME. For each measurement or

constraint, we list the relevant coefficient or combination

of coefficients, the result as presented in the literature, the

context in which the search was performed, and the source

citation. The tables include results available from the literature

up to 31 July 2010, with updates provided by Kostelecky´

and Russell (2011).

The scope of the searches for Lorentz violation listed in the

data tables can be characterized roughly in terms of depth,

breadth, and refinement. Deep searches yield great sensitivity

to a small number of SME coefficients. Broad searches cover

substantial portions of the coefficient space, usually at a lesser

sensitivity. Searches with high refinement disentangle combinations

of coefficients. In the absence of a compelling signal

for Lorentz violation, all types of searches are necessary to

obtain complete coverage of the possibilities.

As a guide to the scope of the existing searches, we extract

from the data tables three summary tables covering the

sectors for matter (electrons, protons, neutrons, and their

antiparticles), photons, and gravity. These summary tables

list our best estimates for the maximal attained sensitivities to

the relevant SME coefficients in the corresponding sectors.

Each entry in the summary tables is obtained under the

assumption that only one coefficient is nonzero. The summary

tables therefore provide information about the overall

search depth and breadth, at the cost of masking the search

In addition to the data tables and the summary tables, we

also provide properties tables listing some features and definitions

of the SME and the coefficients for Lorentz violation.

The Lagrange densities for the minimal QED extension in

Riemann spacetime, the minimal SME in Riemann-Cartan

spacetime, and the nonminimal photon sector in Minkowski

REVIEW OF MODERN PHYSICS, VOLUME 83, JANUARY–MARCH 2011

0034-6861= 2011=83(1)=11(21) 11 _ 2011 American Physical Society

spacetime are provided in tabulated form. The mass dimensions

of the operators for Lorentz violation and their properties

under the various discrete spacetime transformations are

displayed. Standard combinations of SME coefficients that

appear in the literature are listed. Along with the data tables

and the summary tables, the properties tables can be used to

identify open directions for future searches. Among these are

first measurements of unconstrained coefficients, improved

sensitivities to constrained coefficients, and studies disentangling

combinations of coefficients.

The order of the tables is as follows. Table I contains a list

of all tables. The three summary tables are presented next,

Tables II, III, and IV. These are followed by the data tables,

Tables V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, and XV.

The properties tables appear last, Tables XVI, XVII, XVIII,

XIX, XX, XXI, XXII, XXIII, and XXIV.

A description of the summary tables is given in Sec. II.

Information about the format and content of the data tables is

presented in Sec. III, while Sec. IV provides an overview of

the properties tables. The bibliography for the text and all the

tables follows Sec. IV.

II. SUMMARY TABLES

Three summary tables are provided (Tables II, III, and IV),

listing maximal experimental sensitivities attained for coefficients

in the matter, photon, and gravity sectors of the

minimal SME. To date, there is no compelling experimental

evidence supporting Lorentz violation. A few measurements

suggest nonzero coefficients at weak confidence levels. These

latter results are excluded from the summary tables but are

listed in the data tables.

In these three summary tables, each displayed sensitivity

value represents our conservative estimate of a 2_ limit,

given to the nearest order of magnitude, on the modulus of

the corresponding coefficient. Our rounding convention is

logarithmic: A factor greater than or equal to 100:5 rounds

to 10, while a factor less than 100:5 rounds to 1. In a few cases,

tighter results may exist when suitable theoretical assumptions

are adopted; these results can be found in the data tables

Where observations involve a linear combination of the

coefficients appearing in the summary tables, the displayed

sensitivity for each coefficient assumes for definiteness that

no other coefficient contributes. Some caution is therefore

advisable in applying the results in these summary tables to

situations involving two or more nonzero coefficient values.

Care in applications is also required because under some

circumstances certain coefficients can be intrinsically unobservable

or can be absorbed into others by field or coordinate

redefinitions as described in Sec. IV.A.

In presenting the physical sensitivities, we adopt natural

units with ℏ ¼ c ¼ _0

¼ 1 and express mass units in GeV.

Our values are reported in the standard Sun-centered inertial

reference frame (Bluhm et al., 2002) widely used in the

literature. This frame is illustrated in Fig. 1. The origin of the

time coordinate T is at the 2000 vernal equinox. The Z axis is

directed north and parallel to the rotational axis of the Earth at

T ¼ 0. The X axis points from the Sun toward the vernal

equinox, while the Y axis completes a right-handed system.

Some further details about this frame, including transformations

to other frames, can be found in Sec. III A and

Appendix C of Kostelecky´ and Mewes (2002).

Table II lists the maximal attained sensitivities involving

electrons, protons, neutrons, and their antiparticles. For each

distinct massive spin-half Dirac fermion in the minimal SME

in Minkowski spacetime, there are 44 independent observable

combinations of coefficients for Lorentz violation in the

nonrelativistic limit. Of these, 20 also control CPT violation.

The 44 combinations are conventionally chosen as the tilde

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

12 V. Alan Kostelecky´ and Neil Russell: Data tables for Lorentz and CPT violation

Rev. Mod. Phys., Vol. 83, No. 1, January–March 2011

coefficients shown. The definitions of these 44 tilde coefficients

in terms of coefficients in the minimal SME are listed

in Table XVIII. All the definitions appear elsewhere in the

literature (Bluhm et al., 2003) except the four combinations

J and ~cTT. The three tilde coefficients ~b_

J are the antimatter

equivalent of the tilde coefficients ~bJ. They appear in nonrelativistic

studies of antimatter properties, such as the hyperfine

transitions of antihydrogen (Bluhm et al., 1999). The

tilde coefficient ~cTT is a simple scaling of the coefficient cTT

in the minimal SME, introduced here to ensure completeness

of the set of tilde coefficients. All tilde coefficients have

dimensions of GeV in natural units. In Table II, a superscript

indicating the particle species of relevance is understood on

all coefficients. For example, the first line of the table presents

limits on three different tilde coefficients, ~be

table, a dash indicates that no sensitivity to the coefficient has

been identified to date. A few maximal sensitivities listed in

the electron column are obtained by applying the inverse of

the definitions in Table XVIII to the electron-sector data in

Table III displays the maximal attained sensitivities to

coefficients for Lorentz violation in the photon sector of the

minimal SME. There are 23 observable coefficient combinations

for photons, of which four also control CPT violation.

The 19 tilde coefficients listed in the table are conventional

TABLE III. Maximal sensitivities for the photon sector.

Coefficient Sensitivity

ð~_eþÞXY 10_32

ð~_eþÞXZ 10_32

ð~_eþÞYZ 10_32

ð~_eþÞXX _ ð~_eþÞYY 10_32

ð~_eþÞZZ 10_32

ð~_o_ÞXY 10_32

ð~_o_ÞXZ 10_32

ð~_o_ÞYZ 10_32

ð~_o_ÞXX _ ð~_o_ÞYY 10_32

ð~_o_ÞZZ 10_32

ð~_e_ÞXY 10_17

ð~_e_ÞXZ 10_17

ð~_e_ÞYZ 10_17

ð~_e_ÞXX _ ð~_e_ÞYY 10_17

ð~_e_ÞZZ 10_16

ð~_oþÞXY 10_13

ð~_oþÞXZ 10_14

ð~_oþÞYZ 10_14

ðVÞ00 10_43 GeV

ðVÞ10 10_42 GeV

ðVÞ11 10_42 GeV

ðVÞ11 10_42 GeV

TABLE II. Maximal sensitivities for the matter sector.

Coefficient Electron Proton Neutron

~bX 10_31 GeV 10_31 GeV 10_32 GeV

~bY 10_31 GeV 10_31 GeV 10_32 GeV

~bZ 10_29 GeV – –

~bT 10_26 GeV – 10_26 GeV

J; ðJ ¼ X; Y; ZÞ 10_22 GeV – –

~c_ 10_18 GeV 10_24 GeV 10_27 GeV

~cQ 10_17 GeV 10_21 GeV 10_10 GeV

~cX 10_19 GeV 10_25 GeV 10_25 GeV

~cY 10_19 GeV 10_25 GeV 10_25 GeV

~cZ 10_19 GeV 10_24 GeV 10_27 GeV

~cTX 10_18 GeV 10_20 GeV –

~cTY 10_18 GeV 10_20 GeV –

~cTZ 10_20 GeV 10_20 GeV –

~cTT 10_18 GeV 10_11 GeV 10_11 GeV

~dþ 10_27 GeV – 10_27 GeV

~d_ 10_26 GeV – 10_26 GeV

~dQ 10_26 GeV – 10_26 GeV

~dXY 10_26 GeV – 10_27 GeV

~dYZ 10_26 GeV – 10_26 GeV

~dZX 10_26 GeV – –

~dX 10_22 GeV 10_25 GeV 10_28 GeV

~dY 10_22 GeV 10_25 GeV 10_28 GeV

~dZ 10_19 GeV – –

XT 10_26 GeV – 10_26 GeV

YT 10_26 GeV – 10_26 GeV

ZT 10_26 GeV – 10_27 GeV

~gT 10_27 GeV – 10_27 GeV

~gc 10_26 GeV – 10_27 GeV

~gQ – – –

~g_ – – –

~gTJ; ðJ ¼ X; Y; ZÞ – – –

~gXY 10_17 GeV – –

~gYX 10_17 GeV – –

~gZX 10_18 GeV – –

~gXZ 10_17 GeV – –

~gYZ 10_17 GeV – –

~gZY 10_18 GeV – –

~gDX 10_22 GeV 10_25 GeV 10_28 GeV

~gDY 10_22 GeV 10_25 GeV 10_28 GeV

~gDZ 10_22 GeV – –

TABLE IV. Maximal sensitivities for the gravity sector.

Coefficient Electron Proton Neutron

__ aT 10_11 GeV 10_11 GeV 10_11 GeV

__ aX 10_6 GeV 10_6 GeV 10_5 GeV

__ aY 10_5 GeV 10_5 GeV 10_4 GeV

__ aZ 10_5 GeV 10_5 GeV 10_4 GeV

__ eT 10_8 10_11 10_11

__ eX 10_3 10_6 10_5

__ eY 10_2 10_5 10_4

__ eZ 10_2 10_5 10_4

Coefficient Sensitivity

_sXX _ _sYY 10_9

_sXX þ _sYY _ 2_sZZ 10_7

V. Alan Kostelecky´ and Neil Russell: Data tables for Lorentz and CPT violation 13

Rev. Mod. Phys., Vol. 83, No. 1, January–March 2011

TABLE V. Electron sector.

Combination Result System Ref.

ð_0:9 _ 1:4Þ _ 10_31 GeV Torsion pendulum Heckel et al. (2008)

ð_0:9 _ 1:4Þ _ 10_31 GeV Torsion pendulum Heckel et al. (2008)

~bZ ð_0:3 _ 4:4Þ _ 10_30 GeV Torsion pendulum Heckel et al. (2008)

þ 4 ~dþ _ ~dQ

Þ ð0:9 _ 2:2Þ _ 10_27 GeV Torsion pendulum Heckel et al. (2008)

þ 4~dþ _ ~d_ _ ~dQ

þ tan_ð~dYZ _ ~H XTÞ

ð_0:8 _ 2:0Þ _ 10_27 GeV Torsion pendulum Heckel et al. (2008)

ð2:8 _ 6:1Þ _ 10_29 GeV K/He magnetometer Kornack et al. (2008)

ð6:8 _ 6:1Þ _ 10_29 GeV K/He magnetometer Kornack et al. (2008)

~bX ð0:1 _ 2:4Þ _ 10_31 GeV Torsion pendulum Heckel et al. (2006)

~bY ð_1:7 _ 2:5Þ _ 10_31 GeV Torsion pendulum Heckel et al. (2006)

ð_29 _ 39Þ _ 10_31 GeV Torsion pendulum Heckel et al. (2006)

~b? <3:1 _ 10_29 GeV Torsion pendulum Hou, et al. (2003)

j ~bZj <7:1 _ 10_28 GeV Torsion pendulum Hou, et al. (2003)

re <3:2 _ 10_24 Hg/Cs comparison Hunter et al. (1999)

j~bj <50 rad=s Penning trap Dehmelt et al. (1999)

a ;diurnal <1:6 _ 10_21 Penning trap Mittleman et al. (1999)

j ~bJj; ðJ ¼ X; YÞ <10_27 GeV Hg/Cs comparison Kostelecky´ and

cTT ð_4 to 2Þ _ 10_15 Collider physics Altschul (2010)b*

cðTXÞ ð_30 to 1Þ _ 10_14 Collider physics Altschul (2010)b*

cðTYÞ ð_80 to 6Þ _ 10_15 Collider physics Altschul (2010)b*

cðTZÞ ð_11 to 1:3Þ _ 10_13 Collider physics Altschul (2010)b*

0:83cðTXÞ þ 0:51cðTYÞ þ 0:22cðTZÞ ð4 _ 8Þ _ 10_11 1S-2S transition Altschul (2010)a*

ð_2:9 _ 6:3Þ _ 10_16 Optical, microwave

Mu¨ ller et al. (2007)*

2 cðXYÞ ð2:1 _ 0:9Þ _ 10_16 Optical, microwave

Mu¨ ller et al. (2007)*

2 cðXZÞ ð_1:5 _ 0:9Þ _ 10_16 Optical, microwave

Mu¨ ller et al. (2007)*

2 cðYZÞ ð_0:5 _ 1:2Þ _ 10_16 Optical, microwave

Mu¨ ller et al. (2007)*

ð_106 _ 147Þ _ 10_16 Optical, microwave

Mu¨ ller et al. (2007)*

_ZZ ð13:3 _ 9:8Þ _ 10_16 Optical, microwave

Mu¨ ller et al. (2007)*

2 cðYZÞ ð2:1 _ 4:6Þ _ 10_16 Optical, microwave

Mu¨ ller (2005)*

2 cðXZÞ ð_1:6 _ 6:3Þ _ 10_16 Optical, microwave

Mu¨ ller (2005)*

2 cðXYÞ ð7:6 _ 3:5Þ _ 10_16 Optical, microwave

Mu¨ ller (2005)*

ð1:15 _ 0:64Þ _ 10_15 Optical, microwave

Mu¨ ller (2005)*

_ 0:25ð~_e_ÞZZj <10_12 Optical, microwave

Mu¨ ller (2005)*

2 cðXYÞj <8 _ 10_15 Optical resonators Mu¨ller et al. (2003)b*

j <1:6 _ 10_14 Optical resonators Mu¨ller et al. (2003)b*

jcXX þ cYY _ 2cZZj <10_5 Doppler shift Lane (2005)*

jcTJ þ cJTj; ðJ ¼ X; Y; ZÞ <10_2 Doppler shift Lane (2005)*

ð_3 to 5Þ _ 10_15 Astrophysics Altschul (2006)b*

ð_0:7 to 2:5Þ _ 10_15 Astrophysics Altschul (2006)b*

cZZ ð_1:6 to 2:5Þ _ 10_15 Astrophysics Altschul (2006)b*

cðYZÞ ð_2:5 to 1:8Þ _ 10_15 Astrophysics Altschul (2006)b*

ð_7 to 4Þ _ 10_15 Astrophysics Altschul (2006)b*

ð_0:5 to 1:5Þ _ 10_15 Astrophysics Altschul (2006)b*

c0Z ð_4 to 2Þ _ 10_17 Astrophysics Altschul (2006)b*

þ 0:16cðXYÞ _ 0:14cðXZÞ _ 0:47cðYZÞ

þ 0:22cð0XÞ þ 0:74cð0YÞ _ 0:64cð0ZÞ þ c00

<1:3 _ 10_15 Astrophysics Altschul (2007)c*

j0:58cXX þ 0:04cYY þ 0:38cZZ

_ 0:14cðXYÞ _ 0:47cðXZÞ þ 0:12cðYZÞ

þ 0:76cð0XÞ _ 0:19cð0YÞ _ 0:62cð0ZÞ þ c00

<2:5 _ 10_15 Astrophysics Altschul (2007)c*

14 V. Alan Kostelecky´ and Neil Russell: Data tables for Lorentz and CPT violation

Rev. Mod. Phys., Vol. 83, No. 1, January–March 2011

combinations of the 19 dimensionless coefficients in the

minimal SME. The remaining entries in the table concern

combinations of the four coefficients controlling CPT violation,

which have dimensions of GeV in natural units. The

definitions of all 23 combinations are taken from the literature

(Kostelecky´ and Mewes, 2002, 2007) and are provided in

Table IV displays the maximal attained sensitivities to

certain coefficients for Lorentz violation involving the gravity

sector of the minimal SME. Two classes of coefficients can be

distinguished in this context: ones appearing in the matter

sector, and ones appearing in the pure-gravity sector. For the

first class, Table IV contains results for the 24 coefficients _ ae

_ involving the electron, proton, and

neutron sectors. These observables are associated with

CPT-odd operators and have dimensions of GeV in natural

units. The prefactor _ is a model-dependent number

(Kostelecky´ and Tasson, 2009). For the second class, the table

displays nine combinations of the nine dimensionless coefficients

for Lorentz violation s__. Additional sets of coefficients

involving the gravity sector exist, but no sensitivities to

them have been identified to date.

III. DATA TABLES

We present 11 data tables compiled from the existing

literature. Of these, 10 tables include results for various

sectors of the minimal SME: the electron sector (Table V),

the proton sector (Table VI), the neutron sector (Table VII),

the photon sector (Table VIII), the charged-lepton sector

(Table IX), the neutrino sector (Table X), the meson sector

(Table XI), the electroweak sector (Table XII), the gluon

sector (Table XIII), and the gravity sector (Table XIV). The

remaining table (Table XV) lists existing bounds on nonminimal

coefficients for Lorentz violation in the photon

Combination Result System Ref.

__ ð_13 to 2Þ _ 10_16 Astrophysics Stecker and Glashow (2001)*

ð0:1 _ 1:8Þ _ 10_27 GeV Torsion pendulum Heckel et al. (2008)

ZT ð_4:1 _ 2:4Þ _ 10_27 GeV Torsion pendulum Heckel et al. (2008)

YT _ ~dZX ð_4:9 _ 8:9Þ _ 10_27 GeV Torsion pendulum Heckel et al. (2008)

Þ ð1:1 _ 9:2Þ _ 10_27 GeV Torsion pendulum Heckel et al. (2008)

j <2 _ 10_14 Astrophysics Altschul (2007)b*

jdYYj; jdZZj <3 _ 10_15 Astrophysics Altschul (2007)b*

jdðXYÞj <2 _ 10_15 Astrophysics Altschul (2007)b*

jdðXZÞj <2 _ 10_14 Astrophysics Altschul (2007)b*

jdðYZÞj <7 _ 10_15 Astrophysics Altschul (2007)b*

jdTXj <5 _ 10_14 Astrophysics Altschul (2007)b*

j <5 _ 10_15 Astrophysics Altschul (2007)b*

j <8 _ 10_17 Astrophysics Altschul (2007)b*

j ~dJj; j~gD;Jj; ðJ ¼ X; YÞ <10_22 GeV Hg/Cs comparison Kostelecky´ and Lane (1999)*

TABLE V. (Continued)

TABLE VI. Proton sector.

Combination Result System Ref.

~b? <6 _ 10_32 GeV K/He magnetometer Brown et al. (2010)

ð6:0 _ 1:3Þ _ 10_31 GeV K/He magnetometer Kornack et al. (2008)

~bY ð1:5 _ 1:2Þ _ 10_31 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffififfiffiffiffiffiffiffiffiffi GeV K/He magnetometer Kornack et al. (2008)

ð3 _ 2Þ _ 10_27 GeV H maser Humphrey et al. (2003)

j ðJ ¼ X; YÞ <2 _ 10_27 GeV H maser Phillips et al. (2001)

j ðJ ¼ X; YÞ <10_27 GeV Hg/Cs comparison Kostelecky´ and Lane (1999)*

ð_0:3 _ 2:2Þ _ 10_22 GeV Cs fountain Wolf et al. (2006)

~c_ ð_1:8 _ 2:8Þ _ 10_25 GeV Cs fountain Wolf et al. (2006)

ð0:6 _ 1:2Þ _ 10_25 GeV Cs fountain Wolf et al. (2006)

~cY ð_1:9 _ 1:2Þ _ 10_25 GeV Cs fountain Wolf et al. (2006)

ð_1:4 _ 2:8Þ _ 10_25 GeV Cs fountain Wolf et al. (2006)

ð_2:7 _ 3:0Þ _ 10_21 GeV Cs fountain Wolf et al. (2006)

~cTY ð_0:2 _ 3:0Þ _ 10_21 GeV Cs fountain Wolf et al. (2006)

~cTZ ð_0:4 _ 2:0Þ _ 10_21 GeV Cs fountain Wolf et al. (2006)

j <10_11 Doppler shift Lane (2005)*

j; ðJ ¼ X; Y; ZÞ <10_8 Doppler shift Lane (2005)*

!c <4 _ 10_26 Penning trap Gabrielse et al. (1999)

j; ðJ ¼ X; YÞ <10_25 GeV Hg/Cs comparison Kostelecky´ and Lane (1999)*

V. Alan Kostelecky´ and Neil Russell: Data tables for Lorentz and CPT violation 15

Rev. Mod. Phys., Vol. 83, No. 1, January–March 2011

Each of these 11 data tables contains four columns. The

first column lists the coefficients for Lorentz violation or their

relevant combinations. Results for coefficients of the same

generic type are grouped together. Certain results involve

combinations of coefficients across more than one sector;

each of these has been listed only once in the table deemed

most appropriate. Some minor changes in notation or format

have been introduced as needed, but for the most part the

results are quoted as they appear in the cited references.

Definitions for standard combinations of coefficients are

provided in the properties tables that follow. A few authors

use unconventional notation; where immediate, the match to

the standard notation is shown. Parentheses enclosing a pair

of indices on a coefficient indicate symmetrization without a

The second column contains the measurements and

bounds, presented in the same form as documented in the

literature. For each generic type of coefficient, the results are

listed in reverse chronological order. If no significant figures

appear in the quoted limit on an absolute value, the order of

magnitude of the limit is given as a power of 10. Where both

statistical and systematic errors appear in a given result, they

are quoted in that order.

The third column contains a succinct reminder of the

physical context in which the bound is extracted, while the

fourth column contains the source citations. The reader is

referred to the latter for details of experimental and theoretical

procedures, assumptions underlying the results, definitions

of unconventional notations, and other relevant

information. Results deduced on theoretical grounds are distinguished

from those obtained via direct experimental measurement

by an asterisk placed after the citation.

Tables V, VI, and VII contain data for the electron, proton,

and neutron sectors, respectively. Each table is divided into

sections focusing sequentially on combinations involving

the coefficients b_, c__, H__, d__, and g___. A superscript

indicating the particle species of relevance is understood on

all coefficients in these three tables. Standard definitions for

the coefficients and their combinations are provided in

Tables XVI and XVIII. Some results depend on _ ’ 23:5_,

which is the angle between the equatorial and ecliptic planes

in the solar system. Note that existing bounds on observables

_ are obtained from gravitational

experiments and are listed with the gravity-sector

results in Table XIV.

Table VIII presents the photon-sector data. Most of the

combinations of coefficients for Lorentz violation appearing

in the first column are defined in Tables XVI and XIX. The

ðBÞjm arise from analyses

(Kostelecky´ and Mewes, 2007, 2008, 2009) using spinweighted

spherical harmonics. The factor of _ appearing

in some places is the speed of the Earth in the standard Suncentered

reference frame, which is about 10_4 in natural

Tables IX, X, and XI list measurements and bounds on

coefficients for Lorentz violation involving second- and thirdgeneration

fermions in the minimal SME. Results for muons

and tau leptons are in Table XI, while those for neutrinos are

TABLE VII. Neutron sector.

Combination Result System Ref.

~b? <3:72 _ 10_32 GeV He/Xe magnetometer Tullney et al. (2010)

ð0:1 _ 1:6Þ _ 10_33 GeV K/He magnetometer Brown et al. (2010)

~bY ð2:5 _ 1:6Þ _ 10_33 GeV K/He magnetometer Brown et al. (2010)

j ~b?j <3:7 _ 10_33 GeV K/He magnetometer Brown et al. (2010)

b? <2 _ 10_29 GeV Ultracold neutrons Altarev et al. (2009)

i 2_ð53 _ 45Þ nHz Xe/He maser Flambaum et al. (2009)*

j; ðJ ¼ X; YÞ <10_28 GeV Maser/magnetometer Altschul (2009)a*

~bX ð_3:7 _ 8:1Þ _ 10_32 GeV K/He magnetometer Kornack et al. (2008)

ð_9:0 _ 7:5Þ _ 10_32 GeV K/He magnetometer Kornack et al. (2008)

ð8:0 _ 9:5Þ _ 10_32 GeV Xe/He maser Cane` et al. (2004)

_~bX _ 0:0034~dX _ 0:0034~gDX ð2:2 _ 7:9Þ _ 10_32 GeV Xe/He maser Cane` et al. (2004)

_ cos_ð~gT _ 2 ~dþ þ 1

~dQÞ þ sin_ð~dYZ _ ~HXTÞ

ð_1:1 _ 1:0Þ _ 10_27 GeV Xe/He maser Cane` et al. (2004)

_~H ZT ð0:2 _ 1:8Þ _ 10_27 GeV Xe/He maser Cane` et al. (2004)

2 ~gTÞ _ ð~gT _ 2 ~dþ þ 1

~dQÞ ð_1:8 _ 1:9Þ _ 10_27 GeV Xe/He maser Cane` et al. (2004)

ð_1:1 _ 0:8Þ _ 10_27 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GeV Xe/He maser Cane` et al. (2004)

ð~bXÞ2 þ ð~bYÞ2

ð6:4 _ 5:4Þ _ 10_32 GeV Xe/He maser Bear et al. (2000)

rn <1:5 _ 10_30 Hg/Cs comparison Hunter et al. (1999)

j; ðJ ¼ X; YÞ <10_30 GeV Hg/Cs comparison Kostelecky´ and Lane (1999)*

j; jcðTJÞj; ðJ ¼ X; Y;ZÞ <5 _ 10_14 Astrophysics Altschul (2008)b*

jÞ <1:7 _ 10_8 Pulsar timing Altschul (2007)a*

j; ðJ ¼ X; YÞ <10_25 GeV Be/H comparison Kostelecky´ and Lane (1999)*

j~c_j; j~cZj <10_27 GeV Hg/Hg & Ne/He

Kostelecky´ and Lane (1999)*

2 _JKLmgKLTj; ðJ ¼ X; YÞ <10_28 GeV Maser/magnetometer Altschul (2009)a*

jdðXZÞj; jdðTZÞj <5 _ 10_14 Astrophysics Altschul (2008)b*

j; ðJ ¼ X; YÞ <10_28 GeV Hg/Cs comparison Kostelecky´ and Lane (1999)*

16 V. Alan Kostelecky´ and Neil Russell: Data tables for Lorentz and CPT violation

Rev. Mod. Phys., Vol. 83, No. 1, January–March 2011

TABLE VIII. Photon sector.

Combination Result System Ref.

ð~_e_ÞXY ð0:8 _ 0:6Þ _ 10_16 Rotating microwave resonators Hohensee et al. (2010)

ð~_e_ÞXY ð_0:31 _ 0:73Þ _ 10_17 Rotating optical resonators Herrmann et al. (2009)

ð~_e_ÞXY ð0:0 _ 1:0 _ 0:3Þ _ 10_17 Rotating optical resonators Eisele, et al. (2009)

ð~_e_ÞXY ð_0:1 _ 0:6Þ _ 10_17 Rotating optical resonators Herrmann et al. (2008)

ð~_e_ÞXY ð_7:7 _ 4:0Þ _ 10_16 Optical, microwave resonators Mu¨ ller et al. (2007)*

ð~_e_ÞXY ð2:9 _ 2:3Þ _ 10_16 Rotating microwave resonators Stanwix et al. (2006)

ð~_e_ÞXY ð_3:1 _ 2:5Þ _ 10_16 Rotating optical resonators Herrmann et al. (2005)

ð~_e_ÞXY ð_0:63 _ 0:43Þ _ 10_15 Rotating microwave resonators Stanwix et al. (2005)

ð~_e_ÞXY ð_1:7 _ 1:6Þ _ 10_15 Optical, microwave resonators Mu¨ ller (2005)*

ð~_e_ÞXY ð_5:7 _ 2:3Þ _ 10_15 Microwave resonator, maser Wolf et al. (2004)

ð~_e_ÞXY ð1:7 _ 2:6Þ _ 10_15 Optical resonators Mu¨ ller et al. (2003)a

ð~_e_ÞXY ð1:4 _ 1:4Þ _ 10_13 Microwave resonators Lipa et al. (2003)

ð~_e_ÞXZ ð1:5 _ 1:3Þ _ 10_16 Rotating microwave resonators Hohensee et al. (2010)

ð~_e_ÞXZ ð0:54 _ 0:70Þ _ 10_17 Rotating optical resonators Herrmann et al. (2009)

ð~_e_ÞXZ ð0:4 _ 1:5 _ 0:1Þ _ 10_17 Rotating optical resonators Eisele et al. (2009)

ð~_e_ÞXZ ð_2:0 _ 0:9Þ _ 10_17 Rotating optical resonators Herrmann et al. (2008)

ð~_e_ÞXZ ð_10:3 _ 3:9Þ _ 10_16 Optical, microwave resonators Mu¨ ller et al. (2007)*

ð~_e_ÞXZ ð_6:9 _ 2:2Þ _ 10_16 Rotating microwave resonators Stanwix et al. (2006)

ð~_e_ÞXZ ð5:7 _ 4:9Þ _ 10_16 Rotating optical resonators Herrmann et al. (2005)

ð~_e_ÞXZ ð0:19 _ 0:37Þ _ 10_15 Rotating microwave resonators Stanwix et al. (2005)

ð~_e_ÞXZ ð_4:0 _ 3:3Þ _ 10_15 Optical, microwave resonators Mu¨ ller (2005)*

ð~_e_ÞXZ ð_3:2 _ 1:3Þ _ 10_15 Microwave resonator, maser Wolf et al. (2004)

ð~_e_ÞXZ ð_6:3 _ 12:4Þ _ 10_15 Optical resonators Mu¨ ller et al. (2003)a

ð~_e_ÞXZ ð_3:5 _ 4:3Þ _ 10_13 Microwave resonators Lipa et al. (2003)

ð~_e_ÞYZ ð1:7 _ 1:3Þ _ 10_16 Rotating microwave resonators Hohensee et al. (2010)

ð~_e_ÞYZ ð_0:97 _ 0:74Þ _ 10_17 Rotating optical resonators Herrmann et al. (2009)

ð~_e_ÞYZ ð_0:6 _ 1:4 _ 0:5Þ _ 10_17 Rotating optical resonators Eisele et al. (2009)

ð~_e_ÞYZ ð_0:3 _ 1:4Þ _ 10_17 Rotating optical resonators Herrmann et al. (2008)

ð~_e_ÞYZ ð0:9 _ 4:2Þ _ 10_16 Optical, microwave resonators Mu¨ ller et al. (2007)*

ð~_e_ÞYZ ð2:1 _ 2:1Þ _ 10_16 Rotating microwave resonators Stanwix et al. (2006)

ð~_e_ÞYZ ð_1:5 _ 4:4Þ _ 10_16 Rotating optical resonators Herrmann et al. (2005)

ð~_e_ÞYZ ð_0:45 _ 0:37Þ _ 10_15 Rotating microwave resonators Stanwix et al. (2005)

ð~_e_ÞYZ ð0:52 _ 2:52Þ _ 10_15 Optical, microwave resonators Mu¨ ller (2005)*

ð~_e_ÞYZ ð_0:5 _ 1:3Þ _ 10_15 Microwave resonator, maser Wolf et al. (2004)

ð~_e_ÞYZ ð3:6 _ 9:0Þ _ 10_15 Optical resonators Mu¨ ller et al. (2003)a

ð~_e_ÞYZ ð1:7 _ 3:6Þ _ 10_13 Microwave resonators Lipa et al. (2003)

ð~_e_ÞXX _ ð~_e_ÞYY ð0:2 _ 1:0Þ _ 10_16 Rotating microwave resonators Hohensee et al. (2010)

ð~_e_ÞXX _ ð~_e_ÞYY ð0:80 _ 1:27Þ _ 10_17 Rotating optical resonators Herrmann et al. (2009)

ð~_e_ÞXX _ ð~_e_ÞYY ð0:8 _ 2:0 _ 0:3Þ _ 10_17 Rotating optical resonators Eisele, et al. (2009)

ð~_e_ÞXX _ ð~_e_ÞYY ð_2:0 _ 1:7Þ _ 10_17 Rotating optical resonators Herrmann et al. (2008)

ð~_e_ÞXX _ ð~_e_ÞYY ð_12 _ 16Þ _ 10_16 Optical, microwave resonators Mu¨ ller et al. (2007)*

ð~_e_ÞXX _ ð~_e_ÞYY ð_5:0 _ 4:7Þ _ 10_16 Rotating microwave resonators Stanwix et al. (2006)

ð~_e_ÞXX _ ð~_e_ÞYY ð5:4 _ 4:8Þ _ 10_16 Rotating optical resonators Herrmann et al. (2005)

ð~_e_ÞXX _ ð~_e_ÞYY ð_1:3 _ 0:9Þ _ 10_15 Rotating microwave resonators Stanwix et al. (2005)

ð~_e_ÞXX _ ð~_e_ÞYY ð2:8 _ 3:3Þ _ 10_15 Optical, microwave resonators Mu¨ ller (2005)*

ð~_e_ÞXX _ ð~_e_ÞYY ð_3:2 _ 4:6Þ _ 10_15 Microwave resonator, maser Wolf et al. (2004)

ð~_e_ÞXX _ ð~_e_ÞYY ð8:9 _ 4:9Þ _ 10_15 Optical resonators Mu¨ ller et al. (2003)a

ð~_e_ÞXX _ ð~_e_ÞYY ð_1:0 _ 2:1Þ _ 10_13 Microwave resonators Lipa et al. (2003)

ð~_e_ÞZZ ð143 _ 179Þ _ 10_16 Rotating microwave resonators Hohensee et al. (2010)

ð~_e_ÞZZ ð_0:04 _ 1:73Þ _ 10_17 Rotating optical resonators Herrmann et al. (2009)

ð~_e_ÞZZ ð1:6 _ 2:4 _ 1:1Þ _ 10_17 Rotating optical resonators Eisele et al. (2009)

ð~_e_ÞZZ ð_0:2 _ 3:1Þ _ 10_17 Rotating optical resonators Herrmann et al. (2008)

ð~_e_ÞZZ ð223 _ 290Þ _ 10_16 Optical, microwave resonators Mu¨ ller et al. (2007)*

ð~_e_ÞZZ ð143 _ 179Þ _ 10_16 Rotating microwave resonators Stanwix et al. (2006)

ð~_e_ÞZZ ð_1:9 _ 5:2Þ _ 10_15 Rotating optical resonators Herrmann et al. (2005)

ð~_e_ÞZZ ð21 _ 57Þ _ 10_15 Rotating microwave resonators Stanwix et al. (2005)

ð~_e_ÞZZ ð_2:9 _ 2:2Þ _ 10_14 Optical resonators Antonini et al. (2005)

jð~_e_ÞðklÞj <4 _ 10_18 Astrophysics Klinkhamer and Risse

V. Alan Kostelecky´ and Neil Russell: Data tables for Lorentz and CPT violation 17

Rev. Mod. Phys., Vol. 83, No. 1, January–March 2011

Combination Result System Ref.

ð~_oþÞXY ð_1:5 _ 1:2Þ _ 10_12 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffififfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rotating microwave resonators Hohensee et al. (2010)

½2cTX _ ð~_oþÞYZ_2 þ ½2cTY _ ð~_oþÞZX_2

<1:6 _ 10_14 Compton scattering Bocquet et al. (2010)

_ð~_oþÞXY ð_0:14 _ 0:78Þ _ 10_17 Rotating optical resonators Herrmann et al. (2009)

ð~_oþÞXY ð1:5 _ 1:5 _ 0:2Þ _ 10_13 Rotating optical resonators Eisele et al. (2009)

_ð~_oþÞXY ð_2:5 _ 2:5Þ _ 10_17 Rotating optical resonators Herrmann et al. (2008)

ð~_oþÞXY ð1:7 _ 2:0Þ _ 10_12 Optical, microwave resonators Mu¨ ller et al. (2007)*

ð~_oþÞXY ð_0:9 _ 2:6Þ _ 10_12 Rotating microwave resonators Stanwix et al. (2006)

ð~_oþÞXY ð_2:5 _ 5:1Þ _ 10_12 Rotating optical resonators Herrmann et al. (2005)

ð~_oþÞXY ð0:20 _ 0:21Þ _ 10_11 Rotating microwave resonators Stanwix et al. (2005)

ð~_oþÞXY ð_1:8 _ 1:5Þ _ 10_11 Microwave resonator, maser Wolf et al. (2004)

ð~_oþÞXY ð14 _ 14Þ _ 10_11 Optical resonators Mu¨ ller et al. (2003)a

ð~_oþÞXZ ð1:7 _ 0:7Þ _ 10_12 Rotating microwave resonators Hohensee et al. (2010)

_ð~_oþÞXZ ð_0:45 _ 0:62Þ _ 10_17 Rotating optical resonators Herrmann et al. (2009)

ð~_oþÞXZ ð_0:1 _ 1:0 _ 0:2Þ _ 10_13 Rotating optical resonators Eisele et al. (2009)

_ð~_oþÞXZ ð1:5 _ 1:7Þ _ 10_17 Rotating optical resonators Herrmann et al. (2008)

ð~_oþÞXZ ð_3:1 _ 2:3Þ _ 10_12 Optical, microwave resonators Mu¨ ller et al. (2007)*

ð~_oþÞXZ ð_4:4 _ 2:5Þ _ 10_12 Rotating microwave resonators Stanwix et al. (2006)

ð~_oþÞXZ ð_3:6 _ 2:7Þ _ 10_12 Rotating optical resonators Herrmann et al. (2005)

ð~_oþÞXZ ð_0:91 _ 0:46Þ _ 10_11 Rotating microwave resonators Stanwix et al. (2005)

ð~_oþÞXZ ð_1:4 _ 2:3Þ _ 10_11 Microwave resonator, maser Wolf et al. (2004)

ð~_oþÞXZ ð_1:2 _ 2:6Þ _ 10_11 Optical resonators Mu¨ ller et al. (2003)a

ð~_oþÞYZ ð0:2 _ 0:7Þ _ 10_12 Rotating microwave resonators Hohensee et al. (2010)

_ð~_oþÞYZ ð_0:34 _ 0:61Þ _ 10_17 Rotating optical resonators Herrmann et al. (2009)

ð~_oþÞYZ ð_0:1 _ 1:0 _ 0:4Þ _ 10_13 Rotating optical resonators Eisele et al. (2009)

_ð~_oþÞYZ ð_1:0 _ 1:5Þ _ 10_17 Rotating optical resonators Herrmann et al. (2008)

ð~_oþÞYZ ð_2:8 _ 2:2Þ _ 10_12 Optical, microwave resonators Mu¨ ller et al. (2007)*

ð~_oþÞYZ ð_3:2 _ 2:3Þ _ 10_12 Rotating microwave resonators Stanwix et al. (2006)

ð~_oþÞYZ ð2:9 _ 2:8Þ _ 10_12 Rotating optical resonators Herrmann et al. (2005)

ð~_oþÞYZ ð0:44 _ 0:46Þ _ 10_11 Rotating microwave resonators Stanwix et al. (2005)

ð~_oþÞYZ ð2:7 _ 2:2Þ _ 10_11 Microwave resonator, maser Wolf et al. (2004)

ð~_oþÞYZ ð0:1 _ 2:7Þ _ 10_11 Optical resonators Mu¨ ller et al. (2003)a

ð~_oþÞYX _ 0:432ð~_oþÞZX ð4:0 _ 8:4Þ _ 10_9 Microwave resonators Lipa et al. (2003)

ð~_oþÞXY _ 0:209ð~_oþÞYZ ð4:0 _ 4:9Þ _ 10_9 Microwave resonators Lipa et al. (2003)

ð~_oþÞXZ _ 0:484ð~_oþÞYZ ð1:6 _ 1:7Þ _ 10_9 Microwave resonators Lipa et al. (2003)

ð~_oþÞYZ þ 0:484ð~_oþÞXZ ð0:6 _ 1:9Þ _ 10_9 Microwave resonators Lipa et al. (2003)

jð~_oþÞðijÞj <2 _ 10_18 Astrophysics Klinkhamer and Risse

~_tr ð_1:5 _ 0:74Þ _ 10_8 Rotating microwave resonators Hohensee et al. (2010)

ð_0:3 _ 3Þ _ 10_7 Microwave interferometer Tobar et al. (2009)

j <5 _ 10_15 Collider physics Altschul (2009)b*

ð_5:8 to 12Þ _ 10_12 Collider physics Hohensee et al. (2009)a,

00 <6 _ 10_20 Astrophysics Klinkhamer and Schreck

_ <9 _ 10_16 Astrophysics Klinkhamer and Schreck

~_tr <1:4 _ 10_19 Astrophysics Klinkhamer and Risse

j <8:4 _ 10_8 Optical atomic clocks Reinhardt et al. (2007)

j~_trj <2:2 _ 10_7 Heavy-ion storage ring Hohensee et al. (2007)*

j <2 _ 10_14 Astrophysics Carone et al. (2006)*

_ 2 Carone et al. (2006)*

j~_trj <1:6 _ 10_5 Sagnac interferometer Cotter and Varcoe (2006)

ð98:2_; 182:1_Þðk

Þj &10_37 Astrophysical birefringence Kostelecky´ and Mewes

ð87:3_; 37:3_Þðk

Þj &10_37 Astrophysical birefringence Kostelecky´ and Mewes

TABLE VIII. (Continued)

18 V. Alan Kostelecky´ and Neil Russell: Data tables for Lorentz and CPT violation

Rev. Mod. Phys., Vol. 83, No. 1, January–March 2011

Combination Result System Ref.

Þ _ 10_31 CMB polarization Kostelecky´ and Mewes

Þ _ 10_31 CMB polarization Kostelecky´ and Mewes

ffiPffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffififfiffiffiffiffiffiffiffi (2007)*

<5 _ 10_32 Astrophysical birefringence Kostelecky´ and Mewes

(2002)*, (2009)*

jkaj for somea <2 _ 10_37 Astrophysical birefringence Kostelecky´ and Mewes

jkaj for a ¼ 1; . . . ; 10 <2 _ 10_32 Astrophysical birefringence Kostelecky´ and Mewes

j <16 _ 10_21 GeV Schumann resonances Mewes (2008)*

j <12 _ 10_21 GeV Schumann resonances Mewes (2008)*

ffiffiffiffiffiffiffi

Þ _ 10_43 GeV CMB polarization Kostelecky´ and Mewes

j ð15 _ 6Þ _ 10_43 GeV CMB polarization Kostelecky´ and Mewes

(2007)*, (2009)*

_ð3 _ 1Þ _ 10_42 GeV CMB polarization Kostelecky´ and Mewes

Þ _ 10_43 GeV CMB polarization Kostelecky´ and Mewes

j ð0:57 _ 0:70ÞH0 Astrophysical birefringence Carroll and Field (1997)*

j 10_41GeV Astrophysical birefringence Nodland and Ralston

j <6 _ 10_43 GeV Astrophysical birefringence Carroll et al. (1990)*,

Kostelecky´ and Mewes

j <14 _ 10_21 GeV Schumann resonances Mewes (2008)*

ð1:1 _ 1:3 _ 1:5Þ _ 10_43 GeV CMB polarization Komatsu et al. (2010)

ð0:04 _ 0:35Þ _ 10_43 GeV CMB polarization Xia et al. (2010)*

ð_0:64 _ 0:50 _ 0:50Þ _ 10_43 GeV CMB polarization Brown et al. (2009)

ð4:3 _ 4:1Þ _ 10_43 GeV CMB polarization Pagano et al. (2009)

ð_1:4 _ 0:9 _ 0:5Þ _ 10_43 GeV CMB polarization Wu et al. (2009)

ð2:3 _ 5:4Þ _ 10_43 GeV CMB polarization Kostelecky´ and Mewes

ðVÞ00 <2:5 _ 10_43 GeV CMB polarization Kahniashvili et al.

Kostelecky´ and Mewes

ð1:2 _ 2:2Þ _ 10_43 GeV CMB polarization Komatsu et al. (2009),

Kostelecky´ and Mewes

ð12 _ 7Þ _ 10_43 GeV CMB polarization Kostelecky´ and Mewes

ð2:6 _ 1:9Þ _ 10_43 GeV CMB polarization Xia et al. (2008)*,

Kostelecky´ and Mewes

ð2:5 _ 3:0Þ _ 10_43 GeV CMB polarization Cabella et al. (2007)*,

Kostelecky´ and Mewes

ð6:0 _ 4:0Þ _ 10_43 GeV CMB polarization Feng et al. (2006)*,

Kostelecky´ and Mewes

ð1:1 _ 1:4ÞH0 Astrophysical birefringence Carroll and Field

ðVÞ00 <2 _ 10_42 GeV Astrophysical birefringence Carroll et al. (1990)*,

Kostelecky´ and Mewes

TABLE VIII. (Continued)

V. Alan Kostelecky´ and Neil Russell: Data tables for Lorentz and CPT violation 19

Rev. Mod. Phys., Vol. 83, No. 1, January–March 2011

in Table X. For both these tables, many of the coefficients

appearing in the first column are specified in the lepton sector

of Table XX. The neutrino results in Table X are obtained in

the context of various simplified models, as discussed in the

references. Experimental sensitivities to coefficients for operators

involving second- and third-generation quark fields

are presently limited to mesons and are presented in Table XI.

The coefficients appearing in this table are composite quantities

defined in the corresponding references. They are effective

coefficients for which complete analytical expressions

are as yet unknown, formed from certain quark-sector coefficients

appearing in Table XX and from other quantities

arising from the quark binding in the mesons.

Tables XII and XIII concern coefficients in the gauge

sectors of the minimal SME. Results for the electroweak

sector are listed in Table XII, while those for the gluon sector

are in Table XIII. The coefficients for the electroweak sector

are defined in the gauge and Higgs sections of Table XXI. The

gluon-sector coefficient is the analog of the corresponding

photon-sector coefficient defined in Table XIX. To date, all

results for the gauge sector are deduced from theoretical

considerations.

Table XIV presents measurements and bounds concerning

the gravity sector of the minimal SME. The specific combinations

of coefficients in the pure-gravity sector that appear

in the first column are defined in the references. They are

expressed in terms of the coefficients for Lorentz violation

listed in the gravity section of Table XXI.

The final data table, Table XV, contains a compilation of

some measurements and bounds on coefficients for Lorentz

violation in the nonminimal SME. Attention is restricted

to the photon sector, in which results are available for a

variety of nonrenormalizable operators of dimensions 5, 6,

7, 8, and 9. A convenient basis for classifying operators of

dimension d is given by the spin-weighted spherical harmonics

(Kostelecky´ and Mewes, 2009). The corresponding coefficients

are listed in Table XXIV. Some constraints have been

obtained for the vacuum coefficients for Lorentz violation,

ðBÞjm for even d and k

ðVÞjm for odd d,

where the subscripts jm label the angular-momentum quantum

numbers. In the first column of Table XV, the usual

spherical harmonics 0Yjm are evaluated at specified angles,

which are the celestial coordinates of certain astrophysical

sources. None of the vacuum-orthogonal coefficients for

Lorentz violation have been measured to date.

IV. PROPERTIES TABLES

Nine properties tables are provided, listing various features

and definitions related to Lorentz violation. Four tables

concern the terms in the restriction of the minimal SME

to quantum electrodynamics (QED) in Riemann spacetime.

For this theory, which is called the minimal QED extension,

the tables include information about the operator structure

(Table XVI), the action of discrete symmetries (Table XVII),

and some useful coefficient combinations (Tables XVIII and

XIX). Two tables contain information about the matter sector

(Table XX) and the gauge and gravity sectors (Table XXI)

of the minimal SME in Riemann-Cartan spacetime. Another

table (Table XXII) summarizes some features of the coefficients

for Lorentz violation in the neutrino sector. The two

remaining tables (Tables XXIII and XXIV) provide information

about the operator structure and the spherical coefficients

for Lorentz violation in the nonminimal photon sector.

For these properties tables, our primary conventions are

those of Kostelecky´ (2004). Greek indices _; _; _; . . . refer to

curved-spacetime coordinates and Latin indices a; b; c; . . . to

local Lorentz coordinates. The vierbein formalism (Utiyama,

1956; Kibble, 1961)), which relates the two sets of coordinates,

is adopted to facilitate the description of spinors on the

spacetime manifold. The determinant e of the vierbein e_

related to the determinant g of the metric g__ by e ¼

ffiffiffiffiffiffiffiffi

The conventions for the Dirac matrices _a are given in

Appendix A of Kostelecky´ (2004). The Newton gravitational

constant GN enters as the combination _ _ 8_GN, and it has

dimensions of inverse mass squared.

In the Minkowski-spacetime limit, the metric g__ is

written ___ with diagonal entries ð_1; 1; 1; 1Þ. For decompositions

into time and space components, we adopt the

Sun-centered frame of Fig. 1 and use indices J; K; L; . . . to

denote the three spatial components X; Y; Z. The sign of the

antisymmetric tensor _____ is fixed via the component

¼þ1, and the antisymmetric symbol in three spatial

dimensions is defined with _XYZ

¼þ1. Note that some of

the literature on the SME in Minkowski spacetime adopts a

metric ___ of opposite sign, following the common present

TABLE IX. Charged-lepton sector.

Combination Result System Ref.

_ð1:0 _ 1:1Þ _ 10_23 GeV BNL g_

_ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Bennett et al. (2008)

<1:4 _ 10_24 GeV BNL g_

_ 2 Bennett et al. (2008) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<2:6 _ 10_24 GeV BNL g_ _ 2 Bennett et al. (2008) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<2 _ 10_23 GeV Muonium spectroscopy Hughes et al. (2001)

Þ ð_1:4 _ 1:0Þ _ 10_22 GeV BNL, CERN g_

_ 2 data Deile et al. (2002)

ð_2:3 _ 1:4Þ _ 10_22 GeV CERN g_

_ 2 data Bluhm et al. (2000)*, Deile et al. (2002)

ð1:8 _ 6:0Þ _ 10_23 GeV BNL g_ _ 2 Bennett et al. (2008)

jc_j <10_11 Astrophysics Altschul (2007)c*

jcj <10_8 Astrophysics Altschul (2007)c*

20 V. Alan Kostelecky´ and Neil Russell: Data tables for Lorentz and CPT violation

Rev. Mod. Phys., Vol. 83, No. 1, January–March 2011

TABLE X. Neutrino sector.

Combination Result System Ref.

ð_3:1 _ 0:9Þ _ 10_20 GeV MiniBooNE Katori (2010)

ð0:6 _ 1:9Þ _ 10_20 GeV MiniBooNE Katori (2010)

ð_0:9 _ 1:8Þ _ 10_20 GeV MiniBooNE Katori (2010)

e_ ð_4:2 _ 1:2Þ _ 10_20 GeV MiniBooNE Katori (2010)

ð7:2 _ 2:1Þ _ 10_20 MiniBooNE Katori (2010)

ð_0:9 _ 2:8Þ _ 10_20 MiniBooNE Katori (2010)

ð1:3 _ 2:6Þ _ 10_20 MiniBooNE Katori (2010)

ð5:9 _ 1:7Þ _ 10_20 MiniBooNE Katori (2010)

e_ ð_1:1 _ 3:7Þ _ 10_20 MiniBooNE Katori (2010)

e_ ð1:7 _ 3:4Þ _ 10_20 MiniBooNE Katori (2010)

e_ ð2:6 _ 0:8Þ _ 10_19 MiniBooNE Katori (2010)

j <5:9 _ 10_23 GeV MINOS FD Adamson et al. (2010)

j <6:1 _ 10_23 GeV MINOS FD Adamson et al. (2010)

j <3:0 _ 10_20 GeV MINOS ND Adamson et al. (2008)

j <0:5 _ 10_23 MINOS FD Adamson et al. (2010)

j <2:5 _ 10_23 MINOS FD Adamson et al. (2010)

j <2:4 _ 10_23 MINOS FD Adamson et al. (2010)

j <1:2 _ 10_23 MINOS FD Adamson et al. (2010)

_j <0:7 _ 10_23 MINOS FD Adamson et al. (2010)

j <9 _ 10_23 MINOS ND Adamson et al. (2008)

j <5:6 _ 10_21 MINOS ND Adamson et al. (2008)

j <5:5 _ 10_21 MINOS ND Adamson et al. (2008)

j <2:7 _ 10_21 MINOS ND Adamson et al. (2008)

j <1:2 _ 10_21 MINOS ND Adamson et al. (2008)

j <1:3 _ 10_21 MINOS ND Adamson et al. (2008)

ð0:2 _ 1:0Þ _ 10_19 GeV LSND Katori (2010)

ð4:2 _ 1:5Þ _ 10_19 GeV LSND Katori (2010)

ð_1:7 _ 1:8Þ _ 10_19 GeV LSND Katori (2010)

ð1:0 _ 5:4Þ _ 10_19 GeV LSND Katori (2010)

ð0:3 _ 1:8Þ _ 10_18 LSND Katori (2010)

ð_5:2 _ 1:9Þ _ 10_18 LSND Katori (2010)

ð2:1 _ 2:2Þ _ 10_18 LSND Katori (2010)

ð1:3 _ 6:7Þ _ 10_18 LSND Katori (2010)

ð_2:7 _ 1:0Þ _ 10_17 LSND Katori (2010)

ð1:1 _ 1:2Þ _ 10_17 LSND Katori (2010)

ð_1:1 _ 5:9Þ _ 10_18 LSND Katori (2010)

j2 ð10:7 _ 2:6 _ 1:3Þ _ ð10_19 GeVÞ2 LSND Auerbach et al. (2005)

j2 ð9:9 _ 2:3 _ 1:4Þ _ ð10_19 GeVÞ2 LSND Auerbach et al. (2005)

ð10:5 _ 2:4 _ 1:4Þ _ ð10_19 GeVÞ2 LSND Auerbach et al. (2005)

00 <2 _ 10_11 Cosmic rays Altschul (2009)c*

a cos_ Excluded Multiple Barger et al. (2007)*

a sin_n^ Excluded Multiple Barger et al. (2007)*

c Excluded Multiple Barger et al. (2007)*

b <1:6 _ 10_23 GeV Atmospheric Messier (2005)

c <1:4 _ 10_26 Atmospheric Messier (2005)

<5 GeV Atmospheric Messier (2005)

V. Alan Kostelecky´ and Neil Russell: Data tables for Lorentz and CPT violation 21

Rev. Mod. Phys., Vol. 83, No. 1, January–March 2011

usage in quantum physics instead of the one in relativity.

Under this alternative convention, terms in the Lagrange

density with an odd number of index contractions have

opposite signs to those appearing in this work. The numerical

results for the SME coefficients in the tables are unaffected

by the convention.

A. Minimal QED extension

Table XVI concerns the minimal QED extension, for

which the basic nongravitational fields are a Dirac fermion

c and the photon A_. The electromagnetic field-strength

_ @_A_. The pure-gravity sector involves

the Riemann tensor R____, the Ricci tensor R__,

the curvature scalar R, and the cosmological constant _.

The spacetime covariant derivative D_ corrects local

Lorentz indices using the spin connection, corrects spacetime

indices using the Cartan connection, and contains the

usual gauge field A_ for the photon. The notation D

abbreviation for the difference of two terms, the first with

derivative acting to the right and the second acting to

the left. Note that Table XVI is restricted to the zero-torsion

limit of the minimal SME. The general case (Kostelecky´,

TABLE XI. Meson sector.

Combination Result System Ref.

ð_6:3 _ 6:0Þ _ 10_18 GeV K oscillations Di Domenico (2010)

ð2:8 _ 5:9Þ _ 10_18 GeV K oscillations Di Domenico (2010)

ð2:4 _ 9:7Þ _ 10_18 GeV K oscillations Di Domenico (2010)

ð0:4 _ 1:8Þ _ 10_17 GeV K oscillations Di Domenico (2010), Di Domenico (2008)

ð_1 _ 4Þ _ 10_17 GeV K oscillations Di Domenico (2008)

j <9:2 _ 10_22 GeV K oscillations Nguyen (2002)

j <9:2 _ 10_22 GeV K oscillations Nguyen (2002)

j <5 _ 10_21 GeV K oscillations Kostelecky´ (1998)*, Kostelecky´ and Van Kooten (2010)*

Þ ð_2:8 to 4:8Þ _ 10_16 GeV D oscillations Link et al. (2003)

ð_7 to 3:8Þ _ 10_16 GeV D oscillations Link et al. (2003)

ð_7 to 3:8Þ _ 10_16 GeV D oscillations Link et al. (2003)

Þ ð_3:0 _ 2:4Þ _ 10_15 GeV Bd oscillations Aubert et al. (2008)

ð_22 _ 7Þ _ 10_15 GeV Bd oscillations Aubert et al. (2008)

ð_27 to _ 4Þ _ 10_15GeV Bd oscillations Aubert et al. (2008)

Þ _ð5:2 _ 4:0Þ _ 10_15 GeV Bd oscillations Aubert et al. (2006)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð37 _ 16Þ _ 10_15 GeV Bd oscillations Kostelecky´ and Van Kooten (2010)*

ð3:7 _ 3:8Þ _ 10_12 GeV Bs oscillations Kostelecky´ and Van Kooten (2010)*

_ ð_1:5 to 200Þ _ 10_11 Astrophysics Altschul (2008)a*

jc_j <10_10 Astrophysics Altschul (2007)c*

jcKj <10_9 Astrophysics Altschul (2007)c*

jcDj <10_8 Astrophysics Altschul (2007)c*

jcBd j; jcBs j <10_7 Astrophysics Altschul (2007)c*

TABLE XII. Electroweak sector.

Combination Result System Ref.

j <3 _ 10_16 Cosmological birefringence Anderson et al. (2004)*

jðk_BÞ__j <0:9 _ 10_16 Cosmological birefringence Anderson et al. (2004)*

j <1:7 _ 10_16 Cosmological birefringence Anderson et al. (2004)*

ÞZZj <10_27 Clock comparisons Anderson et al. (2004)*

j <10_27 Clock comparisons Anderson et al. (2004)*

ÞYZj <10_25 Clock comparisons Anderson et al. (2004)*

j <4 _ 10_13 H_ ion, p_ comparison Anderson et al. (2004)*

j <10_31 Xe-He maser Anderson et al. (2004)*

j <2:8 _ 10_27 Xe-He maser Anderson et al. (2004)*

jkWj <10_5 Astrophysics Altschul (2007)c*

TABLE XIII. Gluon sector.

Combination Result System Ref.

j <2 _ 10_13 Astrophysics Carone et al. (2006)*

22 V. Alan Kostelecky´ and Neil Russell: Data tables for Lorentz and CPT violation

Rev. Mod. Phys., Vol. 83, No. 1, January–March 2011

TABLE XIV. Gravity sector.

Combination Result System Ref.

j <10_7 GeV Free-fall weak equivalence principle (WEP) Kostelecky´ and Tasson (2010)*

ÞTj <10_10 GeV Force-comparison WEP Kostelecky´ and Tasson (2010)*

j <10_7 GeV Free-fall WEP Kostelecky´ and Tasson (2010)*

ÞTj <10_10 GeV Force-comparison WEP Kostelecky´ and Tasson (2010)*

j <10_8 GeV Combined WEP Kostelecky´ and Tasson (2010)*

3mpðceþp_nÞTT

<10_8 GeV Free-fall WEP Kostelecky´ and Tasson (2010)*

<10_11 GeV Force-comparison WEP Kostelecky´ and Tasson (2010)*

j <10_6 GeV Solar System Kostelecky´ and Tasson (2010)*

Þ_j <10_6 GeV Solar System Kostelecky´ and Tasson (2010)*

j <1 _ 10_11 GeV Torsion pendulum Kostelecky´ and Tasson (2009)*

jðcnÞQj <10_8 Combined WEP Kostelecky´ and Tasson (2010)*

j <10_8 Solar System Kostelecky´ and Tasson (2010)*

jðcnÞ_j <10_7 Solar System Kostelecky´ and Tasson (2010)*

_XX _ _YY ð4:4 _ 11Þ _ 10_9 Atom interferometry Chung et al. (2009)

_XY ð0:2 _ 3:9Þ _ 10_9 Atom interferometry Chung et al. (2009)

_XZ ð_2:6 _ 4:4Þ _ 10_9 Atom interferometry Chung et al. (2009)

_YZ ð_0:3 _ 4:5Þ _ 10_9 Atom interferometry Chung et al. (2009)

_TX ð_3:1 _ 5:1Þ _ 10_5 Atom interferometry Chung et al. (2009)

_TY ð0:1 _ 5:4Þ _ 10_5 Atom interferometry Chung et al. (2009)

_TZ ð1:4 _ 6:6Þ _ 10_5 Atom interferometry Chung et al. (2009)

_XX _ _YY ð_5:6 _ 2:1Þ _ 10_9 Atom interferometry Mu¨ ller et al. (2008)

_XY ð_0:09 _ 79Þ _ 10_9 Atom interferometry Mu¨ ller et al. (2008)

_XZ ð_13 _ 37Þ _ 10_9 Atom interferometry Mu¨ ller et al. (2008)

_YZ ð_61 _ 38Þ _ 10_9 Atom interferometry Mu¨ ller et al. (2008)

_TX ð5:4 _ 4:5Þ _ 10_5 Atom interferometry Mu¨ ller et al. (2008)

_TY ð_2:0 _ 4:4Þ _ 10_5 Atom interferometry Mu¨ ller et al. (2008)

_TZ ð1:1 _ 26Þ _ 10_5 Atom interferometry Mu¨ ller et al. (2008)

sXX _ sYY ð_1:2 _ 1:6Þ _ 10_9 LLR & atom interferometry Battat et al. (2007)*,

Chung et al. (2009)*

sXX þ sYY _ 2sZZ ð1:8 _ 38Þ _ 10_9 LLR & atom interferometry Battat et al. (2007)*,

Chung et al. (2009)*

sXY ð_0:6 _ 1:5Þ _ 10_9 LLR & atom interferometry Battat et al. (2007)*,

Chung et al. (2009)*

sXZ ð_2:7 _ 1:4Þ _ 10_9 LLR & atom interferometry Battat et al. (2007)*,

Chung et al. (2009)*

sYZ ð0:6 _ 1:4Þ _ 10_9 LLR & atom interferometry Battat et al. (2007)*,

Chung et al. (2009)*

sTX ð0:5 _ 6:2Þ _ 10_7 LLR & atom interferometry Battat et al. (2007)*,

Chung et al. (2009)*

sTY ð0:1 _ 1:3Þ _ 10_6 LLR & atom interferometry Battat et al. (2007)*,

Chung et al. (2009)*

sTZ ð_0:4 _ 3:8Þ _ 10_6 LLR & atom interferometry Battat et al. (2007)*,

Chung et al. (2009)*

s11 _ s22 ð1:3 _ 0:9Þ _ 10_10 Lunar laser ranging (LLR) Battat et al. (2007)*

s12 ð6:9 _ 4:5Þ _ 10_11 LLR Battat et al. (2007)*

s01 ð_0:8 _ 1:1Þ _ 10_6 LLR Battat et al. (2007)*

s02 ð_5:2 _ 4:8Þ _ 10_7 LLR Battat et al. (2007)*

s__c ð0:2 _ 3:9Þ _ 10_7 LLR Battat et al. (2007)*

ð_1:3 _ 4:1Þ _ 10_7 LLR Battat et al. (2007)*

10_9 Perihelion precession Bailey and Kostelecky´ (2006)*

js_j 10_8 Perihelion precession Bailey and Kostelecky´ (2006)*

j 10_13 Solar-spin precession Bailey and Kostelecky´ (2006)*

V. Alan Kostelecky´ and Neil Russell: Data tables for Lorentz and CPT violation 23

Rev. Mod. Phys., Vol. 83, No. 1, January–March 2011

TABLE XV. Nonminimal photon sector.

Combination Result System Ref.

ð98:2_; 182:1_Þk

j <7 _ 10_33 GeV_1 Astrophysical birefringence Kostelecky´ and Mewes (2009)*

j <2 _ 10_32 GeV_1 Astrophysical birefringence Kostelecky´ and Mewes (2009)*

ð87:3_; 37:3_Þk

j <4 _ 10_33 GeV_1 Astrophysical birefringence Kostelecky´ and Mewes (2009)*

j <1 _ 10_32 GeV_1 Astrophysical birefringence Kostelecky´ and Mewes (2009)*

ð3:2 _ 2:1Þ _ 10_20 GeV_1 CMB polarization Gubitosi et al. (2009)*

ð3 _ 2Þ _ 10_20 GeV_1 CMB polarization Kostelecky´ and Mewes (2007)*

Þ _ 10_20 GeV_1 CMB polarization Kostelecky´ and Mewes (2007)*

Þ _ 10_20 GeV_1 CMB polarization Kostelecky´ and Mewes (2007)*

_ð10 _ 3Þ _ 10_20 GeV_1 CMB polarization Kostelecky´ and Mewes (2007)*

Þ _ 10_20 GeV_1 CMB polarization Kostelecky´ and Mewes (2007)*

_ð8 _ 3Þ _ 10_20 GeV_1 CMB polarization Kostelecky´ and Mewes (2007)*

ð116_; 334_Þc

ðIÞjm <3:9 _ 10_22 GeV_2 Astrophysical dispersion Vasileiou (2010)

ðIÞ00 <1:4 _ 10_21 GeV_2 Astrophysical dispersion Vasileiou (2010)

ð147_; 120_Þc

ðIÞjm <3:2 _ 10_20 GeV_2 Astrophysical dispersion Abdo et al. (2009),

Kostelecky´ and Mewes (2009)*

ðIÞ00 <1:1 _ 10_19 GeV_2 Astrophysical dispersion Abdo et al. (2009),

Kostelecky´ and Mewes (2009)*

ð120_; 330_Þc

j <7:4 _ 10_22 GeV_2 Astrophysical dispersion Aharonian et al. (2008),

Kostelecky´ and Mewes (2009)*

j <2:6 _ 10_21 GeV_2 Astrophysical dispersion Aharonian et al. (2008),

Kostelecky´ and Mewes (2009)*

ð50:2_; 253_Þc

_ 10_22 GeV_2 Astrophysical dispersion Albert et al. (2008),

Kostelecky´ and Mewes (2009)*

_ 10_22 GeV_2 Astrophysical dispersion Albert et al. (2008),

Kostelecky´ and Mewes (2009)*

ð99:7_; 240_Þc

j <1 _ 10_16 GeV_2 Astrophysical dispersion Boggs et al. (2004),

Kostelecky´ and Mewes (2009)*

j <4 _ 10_16 GeV_2 Astrophysical dispersion Boggs et al. (2004),

Kostelecky´ and Mewes (2009)*

ð98:2_; 182:1_Þðk

Þj &10_29 GeV_2 Astrophysical birefringence Kostelecky´ and Mewes (2009)*

ð87:3_; 37:3_Þðk

Þj &10_29 GeV_2 Astrophysical birefringence Kostelecky´ and Mewes (2009)*

Þ _ 10_10 GeV_2 CMB polarization Kostelecky´ and Mewes (2007)*

Þ _ 10_10 GeV_2 CMB polarization Kostelecky´ and Mewes (2007)*

Þ _ 10_10 GeV_2 CMB polarization Kostelecky´ and Mewes (2007)*

ð98:2_; 182:1_Þk

j <2 _ 10_24 GeV_3 Astrophysical birefringence Kostelecky´ and Mewes (2009)*

j <7 _ 10_24 GeV_3 Astrophysical birefringence Kostelecky´ and Mewes (2009)*

ð87:3_; 37:3_Þk

j <5 _ 10_25 GeV_3 Astrophysical birefringence Kostelecky´ and Mewes (2009)*

j <2 _ 10_24 GeV_3 Astrophysical birefringence Kostelecky´ and Mewes (2009)*

ð116_; 334_Þc

ðIÞjm <2:1 _ 10_25 GeV_4 Astrophysical dispersion Vasileiou (2010)

ðIÞ00 <7:6 _ 10_25 GeV_4 Astrophysical dispersion Vasileiou (2010)

ð147_; 120_Þc

ðIÞjm <2:6 _ 10_23 GeV_4 Astrophysical dispersion Abdo et al. (2009),

Kostelecky´ and Mewes (2009)*

ðIÞ00 <9:2 _ 10_23 GeV_4 Astrophysical dispersion Abdo et al. (2009),

Kostelecky´ and Mewes (2009)*

ð99:7_; 240_Þc

j <3 _ 10_13 GeV_4 Astrophysical dispersion Boggs et al. (2004),

Kostelecky´ and Mewes (2009)*

j <9 _ 10_13 GeV_4 Astrophysical dispersion Boggs et al. (2004),

Kostelecky´ and Mewes (2009)*

ð98:2_; 182:1_Þðk

Þj &10_20 GeV_4 Astrophysical birefringence Kostelecky´ and Mewes (2009)*

24 V. Alan Kostelecky´ and Neil Russell: Data tables for Lorentz and CPT violation

Rev. Mod. Phys., Vol. 83, No. 1, January–March 2011

2004) involves additional operators constructed with the

torsion tensor T_

__. The Minkowski-spacetime limit of

QED with nonzero torsion contains terms that mimic

Lorentz violation, so searches for Lorentz violation can be

used to bound components of the torsion tensor (Kostelecky´

In Table XVI, each line specifies one term in the Lagrange

density for the QED extension in Riemann spacetime. Both

conventional QED terms and ones with Lorentz violation

are included. The first column indicates the sector to which

the term belongs. The second column lists the coefficient

controlling the corresponding operator. Note the standard

use of an uppercase letter for the coefficient H__, which

distinguishes it from the metric fluctuation h__. The third

column shows the number of components for the coefficient.

The next three columns list the operator, its mass dimension,

and the vierbein factor contracting the coefficient and the

operator. The final two columns list the properties of the

term under CPT and Lorentz transformations. A CPT-even

operator is indicated by a plus sign and a CPT-odd one by a

minus sign, while terms violating Lorentz invariance are

identified by a check mark.

Combination Result System Ref.

ð87:3_; 37:3_Þðk

Þj &10_20 GeV_4 Astrophysical birefringence Kostelecky´ and Mewes (2009)*

ð98:2_; 182:1_Þk

j <6 _ 10_16 GeV_5 Astrophysical birefringence Kostelecky´ and Mewes (2009)*

j <2 _ 10_15 GeV_5 Astrophysical birefringence Kostelecky´ and Mewes (2009)*

ð87:3_; 37:3_Þk

j <1 _ 10_16 GeV_5 Astrophysical birefringence Kostelecky´ and Mewes (2009)*

j <4 _ 10_16 GeV_5 Astrophysical birefringence Kostelecky´ and Mewes (2009)*

TABLE XV. (Continued)

TABLE XVI. Lagrange density for the minimal QED extension in Riemann spacetime.

Sector Coeff. No. Operator Dim. Factor CPT L.V.

Fermion m c_ c 3 _e þ

a_ 4 _c _ac 3 _ee_

b_ 4 _c _5_ac 3 _ee_

H__ 6 _c _abc 3 _ee_

c__ 16 _c _bD $

d__ 16 _c _5_bD $

g___ 24 _c _bcD $

Photon F__F__ 4 _1

Þ_ 4 A_F__ 3 1

ðkFÞ____ 19 F__F__ 4 _1

Gravity R 2 e=2_ þ

u 1 R 2 _e=2_ þ

s__ 9 R__ 2 e=2_ þ !

t____ 10 R____ 2 e=2_ þ !

TABLE XVII. C, P, T properties of operators for Lorentz violation in QED.

Coefficient C P T CP CT PT CPT

þ þ þ þ þ þ þ

bJ; gJTL; gJKT; ðkAF

þ þ _ þ _ _ _

bT; gJTT; gJKL; ðkAFÞT þ _ þ _ þ _ _

cTJ; cJT; ðkFÞTJKL þ _ _ _ _ þ þ

_ þ þ _ _ þ _

_ þ _ _ þ _ þ

HTJ; dTT; dJK _ _ þ þ _ _ þ

aJ; eJ; fT _ _ _ þ þ þ _

V. Alan Kostelecky´ and Neil Russell: Data tables for Lorentz and CPT violation 25

Rev. Mod. Phys., Vol. 83, No. 1, January–March 2011

As an example, consider the fourth row of Table XVI. This

concerns the term in the fermion sector with coefficient a_

for Lorentz violation. The coefficient has four independent

components, which control the four Lorentz-violating operators

The gravitational couplings of this operator are

contained in the vierbein product ee_

a. The corresponding

term in the Lagrange density for the minimal QED extension

in Riemann spacetime is La

dimension 3 and is CPT odd. The Minkowski-spacetime limit

of this term can be obtained by the vierbein replacement

a. The number of index contractions in La is

two, one each for the _ and a indices, so the overall sign

of La is unaffected by the choice of convention for the

Minkowski metric.

The properties listed in Table XVI are those of the operators

in the Lagrange density rather than those associated with

observables. The issue of observability of a given coefficient

can be subtle because experiments always involve comparisons

of at least two quantities. The point is that in certain tests

a given coefficient may produce the same effect on two

or more quantities and so may be unobservable, or it may

produce effects indistinguishable from those of other coefficients.

This situation can often be theoretically understood

via a field redefinition that eliminates the coefficient from

the relevant part of the Lagrange density without affecting

the dynamics of the experiment in question. For example,

a constant coefficient a_ in the minimal QED extension in

Minkowski spacetime is unobservable in any experiment

involving a single fermion flavor because it can be absorbed

as a phase shift in the fermion field (Colladay and Kostelecky´,

1997, 1998). The situation changes in Riemann spacetime,

where three of the four components of a_ become observables

affecting the gravitational properties of the fermion

(Kostelecky´, 2004). Another example is provided by the

coefficient f_ in the minimal QED extension in Minkowski

spacetime, which can be converted into a coefficient of the

c__ type via a change of spinor basis (Altschul, 2006a).

Additional subtleties arise because any experiment must always

choose definitions of clock ticking rates, clock synchronizations,

rod lengths, and rod isotropies. This involves 10

free coordinate choices and implies the unobservability

of 10 combinations of coefficients for Lorentz violation

(Kostelecky´ and Tasson, 2010).

Table XVII lists the properties under discrete-symmetry

transformations of the Lorentz-violating operators in the

minimal QED extension (Kostelecky´ et al., 2002). The seven

transformations considered are charge conjugation C, parity

inversion P, time reversal T, and their combinations CP, CT,

PT, and CPT. The first column specifies the operator by

indicating its corresponding coefficient. Each of the other

columns concerns one of the seven transformations. An even

operator is indicated by a plus sign and an odd one by a minus

sign. The table contains eight rows, one for each of the eight

possible combinations of signs under C, P, and T.

Table XVIII lists the definitions of the 44 combinations of

coefficients for Lorentz violation that frequently appear in

experimental analyses involving the fermion sector of the

minimal QED extension in Minkowski spacetime in the

nonrelativistic limit. These combinations are conventionally

denoted by tilde coefficients, listed in the first column of the

table. Note that six of these combinations, ~cX, ~cY, ~cZ, ~gTX,

~gTY, and ~gTZ, are denoted as ~cQ;Y, ~cQ;X, ~cXY, ~gQ;Y, ~gQ;X, and

TABLE XIX. Definitions for the photon sector of the minimal

Symbol Combination Components

ð~_eþÞJK _ðkF

ð~_e_ÞJK _ðkF

2 _KPQðkFÞTJPQ _ 1

2 _JPQðkFÞTKPQ 3

k2 ðkFÞTXYZ 1

k7 ðkFÞTYTZ þ ðkFÞXYXZ 1

k9 ðkFÞTXXZ _ ðkFÞTYYZ 1

ffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffi

TABLE XVIII. Definitions for the fermion sector of the minimal

Symbol Combination Components

XT HXT þ mðdZY _ gXTT _ gXYYÞ 1

~d_ mðdXX _ dYYÞ 2

~dQ mðdXX þ dYY _ 2dZZ _ gYZX _ gZXY þ 2gXYZÞ 1

~dZX mðdZX þ dXZ _ gYZZ þ gYXXÞ 1

~gQ mðgXTX þ gYTY _ 2gZTZÞ 1

~gJK mðgJTT þ gJKKÞ; ðno K sum; J _ KÞ 6

~cQ mðcXX þ cYY _ 2cZZÞ 1

~cTJ mðcTJ þ cJTÞ 3

26 V. Alan Kostelecky´ and Neil Russell: Data tables for Lorentz and CPT violation

Rev. Mod. Phys., Vol. 83, No. 1, January–March 2011

TABLE XXI. Lagrange density for the boson sector of the minimal SME in Riemann-Cartan

Sector Coeff. Operator Dim. Factor CPT L.V.

Higgs _2 _y_ 2 e þ

_ ð_y_Þ2 4 _ 1

ðD__ÞyðD__Þ 4 _e þ

Þ_ _yD__ þ H:c: 3 ie _ !

Þ__ ðD__ÞyðD__Þ þ H:c: 4 1

Þ__ _yW___ 4 _1

Þ__ _y_B__ 4 _1

Gauge TrðG__G__Þ 4 _1

TrðW__W__Þ 4 _1

ðk1Þ_ B_B__ 3 e_____ _ !

Þ 3 e_____ _ !

ðk3Þ_ TrðG_G__ þ 2

3 ig3G_G_G_Þ 3 e_____ _ !

____ TrðG__G__Þ 4 _1

____ TrðW__W__Þ 4 _1

ðkBÞ____ B__B__ 4 _1

Gravity R 2 e=2_ þ

ðkTÞ___ T___ 1 e=2_ þ !

s__ R__ 2 e=2_ þ !

t____ R____ 2 e=2_ þ !

Þ_____ T__T___ 2 e=2_ þ !

Þ____ D_T___ 2 e=2_ þ !

TABLE XX. Lagrange density for the fermion sector of the minimal SME in Riemann-Cartan

Sector Coeff. Operator Dim. Factor CPT L.V.

Lepton _LA_aD $

A_aD $ _LB 4 _1

A_aD $ _RB 4 _1

A_aD $ _QB 4 _1

A_aD $ _UB 4 _1

A_aD $ _DB 4 _1

þ H:c: 4 _e þ

þ H:c: 4 _e þ

A_DB þ H:c: 4 _e þ

A_c_abUB þ H:c: 4 _1

V. Alan Kostelecky´ and Neil Russell: Data tables for Lorentz and CPT violation 27

Rev. Mod. Phys., Vol. 83, No. 1, January–March 2011

~gXY, respectively, in some early publications. The definitions

in the table are given for a generic fermion of mass m. Most

applications in the literature involve electrons, protons, neutrons,

and their antiparticles, for which the corresponding

mass is understood. The final column lists the number of

independent components of each coefficient. For matter involving

electrons, protons, neutrons, and their antiparticles,

there are therefore 132 independent observable coefficients

for Lorentz violation in the minimal QED sector of the SME

in Minkowski spacetime.

Table XIX presents definitions for certain combinations of

the 23 coefficients for Lorentz violation in the photon sector

of the minimal QED extension in Minkowski spacetime. This

table has three sections. The first section consists of five rows

listing 19 widely used combinations of the 19 coefficients for

CPT-even Lorentz violation. The second section provides 10

alternative combinations involving the 10 CPT-even Lorentzviolating

operators relevant to leading-order birefringence

(Kostelecky´ and Mewes, 2002). The third section lists four

combinations of the four coefficients for CPT-odd Lorentz

violation. These combinations appear when a basis of spinweighted

spherical harmonics is adopted.

Table XX concerns the fermion-sector terms in the

Lagrange density of the minimal SME in Riemann-Cartan

spacetime. The column headings are similar to those in

Table XVI. In the lepton sector, the left- and right-handed

TABLE XXII. Coefficients in the neutrino sector.

Coeff. Dim. Oscillation CPT L.V.

AB 3 _ $ _; _ _ $ _ _ þ

AB 3 _ $ _; _ _ $ _ _ _ !

½H___AB 3 _ $ _ _ þ !

½ðcLÞ___AB 4 _ $ _; _ _ $ _ _ þ !

AB 4 _ $ _ _ _ !

TABLE XXIII. Quadratic Lagrange density for the nonminimal photon sector in Minkowski spacetime.

Coeff. No. Operator Dim. Factor CPT L.V.

Þ_ _ ðkAFÞ_ 4 A_F__ 3 1

_1_2 36 A_@_1@_2F__ 5 1

_1_2_3_4 120 A_@_1@_2@_3@_4F__ 7 1

ðd þ 1Þðd _ 1Þðd _ 2Þ A_@_1

@_ðd_3ÞF__ odd d 1

Þ____ _ ðkFÞ____ 19 þ 1 F__F__ 4 _1

Þ_____1_2 126 F__@_1@_2F__ 6 _1

Þ_____1_2_3_4 360 F__@_1@_2@_3@_4F__ 8 _1

Þ_____1_ðd_4Þ ðd þ 1Þdðd _ 3Þ F__@_1

@_ðd_4ÞF__ even d _1

TABLE XXIV. Spherical coefficients for the nonminimal photon sector in Minkowski spacetime.

Type Coeff. Dim. n j No.

ðIÞjm Even, _ 4 – 0; 1; . . . ; d _ 2 ðd _ 1Þ2

ðEÞjm Even, _ 4 – 2; 3; . . . ; d _ 2 ðd _ 1Þ2 _ 4

ðBÞjm Even, _ 4 – 2; 3; . . . ; d _ 2 ðd _ 1Þ2 _ 4

ðVÞjm Odd, _ 3 – 0; 1; . . . ; d _ 2 ðd _ 1Þ2

Vacuum orthogonal ðc :ðdÞ

njm Even, _4 0; 1; . . . ; d _ 4 n; n _ 2; n _ 4; . . . ;_ 0

ðd_1Þðd_2Þðd_3Þ

njm Even, _6 1; 2; . . . ; d _ 4 n; n _ 2; n _ 4; . . . ;_ 0

ðd_1Þðd_2Þðd_3Þ

njm Even, _6 1; 2; . . . ; d _ 4 n þ 1; n _ 1; n _ 3; . . . ;_ 1

ðd_4Þðd2þdþ3Þ

njm Even, _6 2; 3; . . . ; d _ 4 n; n _ 2; n _ 4; . . . ;_ 2

ðd_4Þðd2_2d_9Þ

njm Even, _6 1; 2; . . . ; d _ 4 n; n _ 2; n _ 4; . . . ;_ 1 dðd_2Þðd_4Þ

njm Even, _6 1; 2; . . . ; d _ 4 n þ 1; n _ 1; n _ 3; . . . ;_ 2

ðdþ3Þðd_2Þðd_4Þ

njm Odd, _5 0; 1; . . . ; d _ 4 n; n _ 2; n _ 4; . . . ;_ 0

ðd_1Þðd_2Þðd_3Þ

njm Odd, _5 0; 1; . . . ; d _ 4 n þ 1; n _ 1; n _ 3; . . . ;_ 1

ðdþ1Þðd_1Þðd_3Þ

njm Odd, _5 1; 2; . . . ; d _ 3 n; n _ 2; n _ 4; . . . ;_ 1

ðdþ1Þðd_1Þðd_3Þ

28 V. Alan Kostelecky´ and Neil Russell: Data tables for Lorentz and CPT violation

Rev. Mod. Phys., Vol. 83, No. 1, January–March 2011

leptons are denoted by LA and RA, where A is the generation

index. The SU(2) doublet LA includes the three neutrino

fields _e, __, and _ and the left-handed components of the

three charged leptons e, _, and . The SU(2) singlet RA

contains the right-handed components of e, _, and . The

derivative D_ is both spacetime and SUð3Þ _ SUð2Þ _ Uð1Þ

covariant. The quark fields are denoted UA, DA, and QA,

where A is the generation index. The right-handed components

of the u, c, and t quarks are the SU(2) singlets UA, while

the right-handed components of d, s, and b are the SU(2)

singlets DA. The six left-handed quark fields are contained in

the SU(2) doublet QA. The Yukawa sector involves terms

coupling the Higgs doublet _ to the leptons and to the quarks.

The conventional Yukawa-coupling matrices are denoted

AB. The Hermitian conjugate of

an operator is abbreviated H.c. in the table.

Table XXI presents information about the Higgs, gauge,

and pure-gravity sectors for the Lagrange density of the

minimal SME in Riemann-Cartan spacetime. The structure

of the table is the same as that of Table XX. As before, D_ is

both a spacetime and an SUð3Þ _ SUð2Þ _ Uð1Þ covariant

derivative. The complex Higgs field is denoted _, the

SU(3) color gauge fields and the SU(2) gauge fields are the

Hermitian adjoint matrices G_ and W_, respectively, while

the U(1) hypercharge gauge field is the singlet B_. Each

gauge field has an associated field strength, denoted G__

for the strong interactions, W__ for the weak interactions,

and B__ for the hypercharge. The pure-gravity sector of

Table XXI differs from that in Table XVI only in the addition

of terms involving the torsion field T_

The minimal SME in Riemann-Cartan spacetime described

in Tables XX and XXI can be reduced to the minimal QED in

Riemann spacetime described in Table XVI as follows. For

the gauge sector, including the covariant derivatives, remove

all the gauge fields except the charge U(1) field in the photon

! A_, and remove all the Higgs terms. For the

gravity sector, remove all the torsion terms. For the fermion

sector, restrict the lepton generation index to a single value,

remove all quark and neutrino terms, and replace the Yukawacoupling

terms with the relevant fermion mass terms.

Table XXII concerns the neutrino sector of the SME,

including both neutrino masses and Lorentz-violating terms.

For the latter, we restrict our attention to terms of mass

dimension 4 or less that involve three generations of active

neutrinos and antineutrinos, allowing for possible violations

of SUð3Þ _ SUð2Þ _ Uð1Þ gauge symmetry and lepton number

(Kostelecky´ and Mewes, 2004). In the table, the first row

involves the usual neutrino mass matrix em

indices A; B take values e, _, and , while the other rows

concern coefficients for Lorentz violation. The first column

lists the coefficients, and the second column gives the dimension

of the corresponding operators in the Lagrange density.

The third column indicates generically the type of neutrino

oscillations controlled by the coefficients. The final two

columns list the properties of the operators under CPT and

Lorentz transformations.

C. Nonminimal photon sector

Table XXIII provides information on the nonminimal photon

sector of the full SME in Minkowski spacetime. The

relevant part of the Lagrange density includes operators of

arbitrary dimension d that are both gauge invariant and

quadratic in the photon field A_ (Kostelecky´ and Mewes,

2009). The structure of the table is similar to that adopted for

Tables XVI, XX, and XXI, with each row associated with a

term in the Lagrange density. The first column lists the

coefficient for Lorentz violation, while the second column

counts its independent components. The next three columns

provide the corresponding operator appearing in the Lagrange

density, its mass dimension, and the factor contracting the

coefficient and the operator. The last two columns list the

properties of the operator under CPT and Lorentz transformations,

using the same conventions as Table XVI.

Three sections appear in Table XXIII, separated by horizontal

lines. The first section concerns the conventional

Lorentz-preserving Maxwell term in the Lagrange density

for the photon sector. The second and third sections concern

coefficients associated with operators of odd and even dimensions

d, respectively. Each of these sections has three rows for

the lowest three values of d, along with a final row applicable

to the case of general d. The notation for the coefficients

incorporates a superscript specifying the dimension d of the

corresponding operator. Note that the mass dimension of the

coefficients is 4 _ d. In each section, the first row describes

terms in the minimal SME, and the match is provided between

the general notation for nonminimal coefficients and

the standard notation used for the minimal SME in

Table XVI. In the case of mass dimension 4, there are 19

independent Lorentz-violating operators. However, for this

case the number in the second column is listed as 19 þ 1 to

allow for an additional Lorentz-preserving trace term, which

maintains consistency with the expression for general d

in the last row.

Table XXIV summarizes properties of spherical coefficients

for Lorentz violation in the nonminimal photon sector

of the full SME in Minkowski spacetime (Kostelecky´ and

Mewes, 2009). The spherical coefficients are combinations of

the coefficients listed in Table XXIII that are of particular

relevance for observation and experiment. They can be separated

into two types. One set consists of vacuum coefficients

that control leading-order effects on photon propagation in

FIG. 1. Standard Sun-centered inertial reference frame (Bluhm

V. Alan Kostelecky´ and Neil Russell: Data tables for Lorentz and CPT violation 29

Rev. Mod. Phys., Vol. 83, No. 1, January–March 2011

the vacuum, including birefringence and dispersion. The

complementary set contains the vacuum-orthogonal coefficients,

which leave photon propagation in the vacuum unaffected

at leading order. The two parts of Table XXIV reflect

this separation, with the part above the horizontal line involving

the vacuum coefficients and the part below involving the

vacuum-orthogonal ones.

In Table XXIV, the first column of the table identifies the

type of spherical coefficients, while the second column lists

the specific coefficient. The coefficient notation reflects properties

of the corresponding operator. Coefficients associated

with operators leaving unchanged the leading-order photon

propagation in the vacuum are distinguished by a negation

diacritic :. A symbol k denotes coefficients for birefringent

operators, while c denotes nonbirefringent ones. The superscript

d refers to the operator mass dimension, while the

subscripts n, j, and m determine the frequency or wavelength

dependence, the total angular momentum, and the z component

of the angular momentum, respectively. The superscripts

E and B refer to the parity of the operator, while the numerals

0, 1, or 2 preceding E or B refer to the spin weight. Note that

the photon-sector coefficients in the minimal SME correspond

to the vacuum coefficients with d ¼ 3; 4. The third,

fourth, and fifth columns of Table XXIV provide the allowed

ranges of the dimension d and of the indices n and j. The

index m can take values ranging from _j to j in unit increments.

The final column gives the number of independent

coefficient components for each operator of dimension d.

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he Barrel Organ