Test of CPT and Lorentz Invariance from Muonium Spectroscopy
V.W. Hughes, M. Grosse Perdekamp† , D. Kawall and W. Liu
Yale University, Department of Physics, New Haven, CT 06520-8121, USA

K. Jungmann and G. zu Putlitz

arXiv:hep-ex/0106103v1 29 Jun 2001

Universität Heidelberg, Physikalisches Institut, D-69120 Heidelberg, Germany
Following a suggestion of Kostelecký et al. we have evaluated a test of CPT and Lorentz invariancefrom the microwave spectroscopy of muonium. Hamiltonian terms beyond the standard model
violating CPT and Lorentz invariance would contribute frequency shifts δν12 and δν34 to ν12 andν34 , the two transitions involving muon spin flip, which were precisely measured in ground statemuonium in a strong magnetic field of 1.7 T. The shifts would be indicated by anti-correlated
oscillations in ν12 and ν34 at the earth’s sidereal frequency. No time dependence was found in ν12or ν34 at the level of 20 Hz, limiting the size of some CPT and Lorentz violating parameters at thelevel of 2 × 10−23 GeV, representing Planck scale sensitivity and an order of magnitude improvement
in sensitivity over previous limits for the muon.
11.30.Er, 11.30.Cp, 12.20.Fv, 36.10.Dr

Much current theoretical work is devoted to finding amore fundamental and general underlying theory fromwhich the standard model of particle physics could bededuced as the low energy limit. String theory is a central candidate, with the feature that the assumptions underlying the CPT theorem are invalid for strings, which
are extended objects. CPT violation then becomes a possible signature of more fundamental underlying theories,
which can be probed by sensitive experimental tests of
CPT invariance.
Some years ago Kostelecký and coworkers [1–3] developed a plausible extension of the standard model basedon spontaneous breaking of Lorentz and CPT symmetry in an underlying theory without gravity. Their low
energy effective theory provides a theoretical basis for establishing quantitative bounds on CPT invariance. The
analysis was done in the context of conventional relativistic quantum mechanics and quantum field theory
in four dimensions, retaining the usual gauge structure
and renormalizability. The Lorentz and CPT violating
additions to the standard model Lagrangian are highly
suppressed to remain compatible with established experimental bounds.
As applied to muonium (µ+ e− bound state), the mostrelevant terms in the extension to the standard model
are those in the QED limit, involving only muons, electrons and photons. The additional terms in the Lorentzviolating Lagrangian lead to a modified Dirac equation
and are given by [2]L = −alα ψ̄l γ α ψl − blα ψ̄l γ5 γ α ψl↔
11− Hαβ ψ̄l σ αβ ψl + icαβ ψ̄l γ α D β ψl22↔
1+ idαβ ψ̄l γ5 γ α D β ψl .
2violating, while a and b are CPT odd, and H, c andd are CPT even. The lepton number violating terms allowed in the extension are not relevant here and havebeen omitted.
To predict the perturbations to muonium, we consider
the energy level diagram of the ground state of muonium
in a magnetic field (Fig. 1). The transition frequencies
ν12 and ν34 have been measured [4] with high precision
and are used to determine the hyperfine structure interval∆ν and the ratio µµ /µp of the muon magnetic momentto the proton magnetic moment.
Leading-order Lorentz violating energy shifts δν12 andδν34 due to the new terms in the Lagrangian can be calculated using perturbation theory and relativistic twofermion techniques. For our observed transitions at thestrong magnetic field of 1.7 T, dominantly only muon
spin flip occurs so these energy shifts are characterized
by the muon parameters alone of the extended theory.
This approach results in :δν12 ≈ − δν34 ≈ b̃µ3 /π,
where b̃µ3bµ3dµ30 mµ(4)µH12[5] are laboratory++≡
frame parameters. High precision experiments on muonium (M ) can measure or set limits on the parametersof these symmetry violating terms, which are sensitive at
the Planck scale level [5].
Predictions of the values of ν12 and ν34 from standard
theory - dominantly the QED terms - requires values formany atomic constants including mµ , µµ , and ∆ν as wellas the calculation of higher order QED radiative corrections. The relevant constants and calculations are notknown to as high accuracy as the experimental determinations of ν12 and ν34 . Indeed several of these constants- ∆ν and µµ /µp - are obtained from the muonium experiment. Comparing predictions for ν12 and ν34 (basedon independent determinations of the required atomicconstants) with the experimental results results in poor

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The lepton fields are denoted by l = e− , µ− , and iDα =i∂α − qAα where q = −|e|. All terms are Lorentz1

sensitivity to the non-standard model energy shifts δν12and δν34 .
However, the theory with CPT and Lorentz violation
involves spatial components in a celestial frame of reference, and since the laboratory rotates with the earth,
these spatial components vary with time, and consequently the experimentally observed ν12 and ν34 may
oscillate about a mean value at the earth’s sidereal frequency Ω = 2π/23 hr 56 m with amplitudes δν12 andδν34 . No such signal would be obtained from the standard model. In the non-rotatingcelestial frame of refer
ence with equatorial axes X̂, Ŷ , Ẑ where Ẑ is oriented
along the earth’s rotational North Pole, an experimental
constraint on δν12 implies [5]q221|sinχ| b̃µX + b̃µY ≤ δν12(5)πin which one standard deviation errors include both statistical and systematic errors. To search for a time dependence of ν12 and ν34 , we have employed the following
algorithm. Data from each resonance line run (each lasting about half an hour) are fit at the measured magnetic
field strength and Kr pressure to determine provisionalline centers for ν12 and ν34 . These line centers were thentransformed to their values in a magnetic field strength
corresponding to a free proton precession frequency of
72.320 000 MHz. The data were taken at Kr pressuresof 0.8 and 1.5 atm, so the line centers were corrected fora small quadratic pressure shift, and then were extrapolated linearly to their values at zero pressure, using apressure shift coefficient determined from the data. Linecenters for ν12 and ν34 obtained in this way were thengrouped as a function of sidereal time, where time zerohas been set as the time in 1995 when we obtained ourfirst data.
Non-zero values for δν12 and δν34 could arise from systematic effects which lead to variations in the parameters
affecting the line centers - particularly variations having
a period of ≈ 24 hr. Principal concerns are possible daynight variations of the magnetic field strength, and of thedensity and temperature of the Kr gas.
Day-night variations of several ◦ C in the temperatureof the experimental hall lead to oscillations in the magnetic field strength of the persistent-mode superconducting solenoid of ≤ 1 ppm. Changes of 0.05 ppm in thefield strength were easily resolved, and the oscillation’s
effects on the line centers were accounted for in extracting
the line centers. Temperature changes also affected thediamagnetic shielding constant of the water in the NMR
probes used to monitor the field (the probes were nottemperature-stabilized, but were in good thermal contact with the microwave cavity which was temperaturestabilized to 0.1 ◦ C). The maximum conceivable 2 ◦ C
day-night changes in water temperature would change
the NMR frequencies by 0.02 ppm, leading to errors in
the line centers of about 2.5 Hz; of opposite sign for ν12and ν34 . This potential effect is well below the statisticalsensitivity for sidereal variations of 12 to 15 Hz.
The effect on these data of the variation of Kr pressure
with time has been evaluated. The front end window tothe Kr stopping target was of 3 mil mylar, and flexed with
day-night variations of the external atmospheric pressure.
This induced fractional day-night oscillations in the Kr
gas target pressure which were measured to be about
2.5 × 10−4 . Through pressure shift coefficients of about-16.5 kHz/atm for ν12 and -19.5 kHz/atm for ν34 , theresulting shifts in the line centers (typically 7.5 Hz inν34 and 6 Hz in ν12 ) were automatically accounted for in
performing the linear extrapolation to zero pressure, andshould not contribute any significant time variation toνij . The pressure shift coefficients depend on the averagevelocities of the atoms, and so are functions of temperature. The fractional changes in the transition frequencies

in which χ ∼ 90◦ is the angle between Ẑ and the quantization axis defined by the laboratory magnetic field atLos Alamos where the muonium experiment was performed. The transformation from the lab frame quantity b̃µ3 to the non-rotating celestial frame quantities b̃µJ(where J = X, Y, Z) is given byb̃µ3 = b̃µZ cos χ + b̃µX cos Ωt + b̃µY sin Ωt sin χ. (6)The non-rotating frame quantities, b̃µJ are defined by b̃µJ ≡µ[5]. The experiment has poorbµJ + mµ dµJ0 + 12 ǫJKL HKL
sensitivity to the celestial frame parameter b̃µZ , but ideal
sensitivity to the pair b̃µX and b̃µY .
The sum of the transition frequencies, ν12 + ν34 , isequal to the ground state hyperfine splitting, ∆ν, andsince we expect (see Eqn.4) δν12 + δν34 ≈ 0, no sidereal
variation is expected in the hyperfine interval. At thehigh field strengths of this experiment, the difference intransition frequencies ν12 − ν34 is almost proportional tothe magnetic moment of the muon. By looking for asidereal variation in the difference δν12 − δν34 ≈ 2b̃µ3 /π,
we are essentially probing a possible sidereal variation of
the magnetic moment of the muon (while the g factor
remains constant).
The accurate measurements of ν12 and ν34 were donein a microwave magnetic resonance experiment [4]. Muonium was formed by electron capture by muons stoppingin a krypton gas target. Resonance lines were observed
by varying the magnetic field with fixed microwave frequency and by varying the microwave frequency with
fixed magnetic field. A line narrowing technique wasused involving observation of a transition signal only
from M atoms which have lived considerably longer thanτµ ∼ 2.2 µs (Fig. 2). The values reported for ν12 andν34 at a magnetic field strength corresponding to a free
proton precession frequency of 72.320 000 MHz wereν12 (exp) = 1 897 539 800 (35) Hz (18 ppb),
ν34 (exp) = 2 565 762 965 (43) Hz (17 ppb)

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field dependent, and the latter is not a fundamental property of the muon. It is natural to take the muon massas the basic energy scale since it is an input parameterof the standard model from which the atomic propertiesare derivable. The ratio of Lorentz and CPT violating
deviations of muonium energy levels to the muon mass
might be expected to be suppressed by the ratio of thelow-energy (standard model) scale to the Planck scaleof the underlying theory [9]. In this context the result
achieved here, 5 × 10−22, is comparable to the dimensionless scaling factor mµ /MP ∼ 10−20 .
Bounds on a different linear combination of the muon
µparameters bµJ , dµJ0 , and HKL
than from muonium canalso be obtained from precision measurements of theanomalous magnetic moment of the muon aµ = (gµ −2)/2[5], where one can search for sidereal variations of theanomalous precession frequency ωaµ (the difference between spin and cyclotron frequencies). Also, the parameter b̃µZ , which is not tested in the latest muonium experiment, is bound at the level of 10−22 GeV by the measurements of the anomalous precession frequencies ωaµ ofµ+ and µ− measured at CERN and BNL [5,10]. Further
improvements in the Lorentz and CPT violating parameters for the muon, coming from muonium, will require
higher intensity muon sources, as the uncertainties arepredominantly statistical.
In conclusion, no unambiguous violation of CPT orLorentz invariance has been observed although tests havebeen done on many systems since 1960 [11]. The resultspresented here represent Planck scale sensitivity, and are
an order of magnitude improvement in sensitivity overprevious limits on the muon parameters of the theory.
They also represent the first test for sidereal variationsin the 2nd generation of leptons, where Lorentz and CPT
violating effects may be enhanced [5]. Higher sensitivity
tests as well as tests on different physical systems will
surely be made [3,12].
We thank V.A. Kostelecký for helpful discussions, andthe U.S. DOE and BMBF (Germany) for supporting thisresearch.

with temperature (measured in hydrogen and its isotopes[6]) are roughly 1×10−11 ◦ C−1 Torr−1 . Given the temperature stability of the Kr gas of about 0.1 ◦ C, temperaturedependent errors in the extrapolation of the line centersto their vacuum values would be limited to a few Hz, well
below the statistical sensitivity of our test.
Other potential concerns involve the two frequency references used in the experiment - the proton precessionfrequency forming the basis of the magnetic field determination, and the Loran-C 10 MHz frequency reference
used for the NMR and microwave frequency synthesizers. The Loran-C standard is based on hyperfine transitions in Cs with mF =0, and so is insensitive to any
preferred spatial orientation, and would not introduce
a signature for Lorentz violation into the spectroscopic
measurements. Time dependent gravitational red or blue
shifts of the frequency standard, caused by tidal distortions of the earth, are below the 10−16 level and too
small to be seen. Finally, bounds on clock comparisons of199
Hg and 133 Cs [7,8] place limits on the Lorentz violating energy shifts in the precession frequency of a protonof 10−27 GeV, which imply the NMR measurements arefree of shifts well below the Hz level.
All the data obtained in 1995 and 1996 are plotted asa function of time measured as a function of a sidereal
day in Fig. 3, where twelve points at ≈ 2 hr. intervals
are plotted, and the vertical scale is in Hz. The data forν12 and ν34 were fit separately by the functions
νij (t) = hνij i + δνij sin (2πt + φij )(9)where t is the time in sidereal days, and the fit parameters
are hνij i, the amplitude of the possible time variation
δνij , and the phase φij (where no phase relation was
assumed between ν12 and ν34 ). The amplitudes for δν12and δν34 are consistent with zero, −4±13 Hz and −13±15Hz respectively.
As stated above, the theory being tested requiresδν12 ≈ − δν34 . A plot of ν12 − ν34 versus siderealtime is shown in Figure 4, and fit for a sinusoidal variation (as in Eqn.9) where a common phase is assumed
between ν12 and ν34 . The data exhibit no variation withtime within ± 20 Hz, which corresponds to a 68% confidence level (one sigma) limit on the non-rotating framecomponents (see Eqn. 4)q2
2b̃µX + b̃µY ≤ 2 × 10−23 GeV.
(10)†The figure of merit of these results as a test of CPTviolation is taken asq
2
2< 2π | δν12 |≈ 5 × 10−22 . (11)2 b̃µX + b̃µY /mµ ∼mµThe limits on δν34 and δ(ν34 − ν12 ) yield similar values.
Other choices for the denominator in the figure of merit
such as νij or ∆ν are inappropriate since the former are B
3Current Address : Riken-BNL Research Center, Upton,
NY 11973, USA[1] V.A. Kostelecký and R. Potting: Nucl. Phys. B359, 545(1991); Phys. Lett. B381, 389 (1996)[2] D. Colladay and V.A. Kostelecký: Phys. Rev. D55, 6760(1997); Phys. Rev. D58, 116002 (1998)[3] CPT and Lorentz Symmetry, ed. V.A. Kostelecký (WorldScientific, Singapore, 1999)[4] W. Liu et al.: Phys. Rev. Lett. 82, 711 (1999)[5] R. Bluhm, V.A. Kostelecký, and C.D. Lane: Phys. Rev.
Lett. 84, 1098 (2000)

[6] C.L. Morgan and E.S. Ensberg: Phys. Rev. A7, 1494
(1973)[7] V.A. Kostelecký and C.D. Lane: Phys. Rev. D 60, 116010(1999)[8] J.D. Prestage et al.: Phys. Rev. Lett. 54, 2387 (1985);
C.J. Berglund et al.: Phys. Rev. Lett. 75, 1879 (1995)
and references therein[9] R. Bluhm, V.A. Kostelecký, and N. Russell, Phys. Rev.
Lett. 79, 1432 (1997).
[10] H.N. Brown et al., Phys. Rev. Lett. 86, 2227 (2001); J.
Bailey et al., Nucl. Phys. B150, 1 (1979).
[11] V.W. Hughes, H.G. Robinson and V. Beltran-Lopez,Phys. Rev. Lett. 4, N504 (1960), R.W.P. Drever, Philos. Mag. 6, 683 (1961).
[12] S. Coleman and S.L. Glashow, Phys. Rev. D 59, 116008(1999).

FIG. 3. Two years of data on ν12 and ν34 are shown binned
versus sidereal time and fit for a possible sinusoidal variation.
The amplitudes are consistent with zero.

FIG. 1. Breit-Rabi energy level diagram of ground statemuonium. At high fields, the indicated transitions, ν12 andν34 are essentially muon spin flip transitions.

3.09→4.04µs

403.12→4.07µs
3020Signal (%)

20

100

6.89→7.84µs

50

6.92→7.87µs

02025
100200300

400

νp - 72000 (kHz)

400

500

600

ν - 1897000 (kHz)700

0FIG. 4. Two years of data on ν12 -ν34 and ν12 + ν34 = ∆νare shown binned versus sidereal time and fit for a possible
sinusoidal variation. The amplitudes are consistent with zero.

FIG. 2. Muonium resonance lines (data and fit) for ν12taken using magnetic field sweep on the left, and microwave
frequency sweep on the right, for muonium atoms which havedecayed in selected time intervals after formation.

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