ESTING CPT AND LORENTZ INVARIANCE WITH THE ANOMALOUS SPIN PRECESSION OF THE MUO



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The Muon g-2 Experiment at BNL
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The muon g-2 experiment determines the $g$ value anomaly $a_{\mu} = (g - 2) / 2$ of the muon by measuring the difference $\omega_a$ between the spin precession angular frequency $\omega_s$ and the cyclotron angular frequency $\omega_c$ of highly polarized muons of momentum 3.09GeV/$c$ in a 14.2m diameter storage ring with a homogeneous magnetic field of 1.45T strength. The field is determined from the NMR frequency of protons in water, calibrated relative to the free proton NMR frequency $\omega_{p}$. The anomaly $a_{\mu}$ is determined via $$a_{\mu} = \frac{\omega_a / \omega_p}{\mu_\mu / \mu_p
- \omega_a / \omega_p}$$ where the ratio $\mu_\mu / \mu_p$ is taken from the measurement by W. Liu et al. @lambda For muons with $\gamma = 29.3$, $\omega_a$ is not affected by the electrostatic field from the quadrupoles used for vertical focussing. Experimental details and the result for $a_{\mu^{+}}$ based on the data of 1999 with an uncertainty of 1.3ppm are described elsewhere. @prl

Predicted Effects of CPT/Lorentz Violation on $\mathbf{\omega_a}$
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Kostelecký et al. @smextension suggested an extension of the Standard Model which retains SU(3)$\times$SU(2)$\times$U(1) gauge invariance, renormalizability and energy conservation, but allows for violation of CPT and Lorentz invariance. The additional muon terms in the Lagrangian are given by @kostbluhm $$\label{eqn_lagrange}
\begin{array}{rcl}
{\mathcal L'} &=& - a_{\kappa} \bar{\psi} \gamma^{\kappa} \psi
- b_{\kappa} \,\bar{\psi} \gamma_{5} 
\gamma^{\kappa} \psi 
 - \frac{1}{2} H_{\kappa \lambda} \,
\bar{\psi} \sigma^{\kappa \lambda} \psi \\
&&+ \frac{1}{2} i c_{\kappa \lambda} \bar{\psi} \gamma^{\kappa} 
\stackrel{\leftrightarrow}{D^{\lambda}} \psi
+ \frac{1}{2} i \, d_{\kappa \lambda} \,
\bar{\psi} \gamma_{5} \gamma^{\kappa} 
\stackrel{\leftrightarrow}{D^{\lambda}} \psi \quad .
\end{array}$$ All terms violate Lorentz symmetry whereas CPT is only broken by the terms involving $a_{\kappa}$ and $b_{\kappa}$. The additional terms are expected to be very small, based on established experimental bounds on CPT violation for other particles like $K^{0}$ – $\bar{K}^{0}$ @pdg or the electron. @dehmelt From Eq. (\[eqn\_lagrange\]) the correction to $\omega_{a}$ can be calculated. @kostbluhm The result is $$\label{eqn_shifts_cel}
\delta \omega_{a}^{\mu^{\pm}} = 2 \check{b}_{Z}^{\mu^{\pm}} \cos \chi + 
2 (\check{b}_{X}^{\mu^{\pm}} \cos \Omega t + 
   \check{b}_{Y}^{\mu^{\pm}} \sin \Omega t) \sin \chi$$ where $$\label{eqn_defcheck}
\check{b}_{J}^{\mu^{\pm}} \equiv \pm \frac{b_{J}}{\gamma} + m_{\mu} d_{J0} 
+ \frac{1}{2} \varepsilon_{JKL} H_{KL} \quad .$$ Here, the model parameters are expressed in terms of a non-rotating celestial frame of reference whose $Z$-direction is oriented along the Earth's rotational axis. The angle $\chi$ is the geographic colatitude at the experiment, i.e. the angle between the vertical direction of the laboratory frame and the Earth's axis. Due to the rotation of the Earth, $\delta \omega_{a}^{\mu^{\pm}}$ has a component oscillating with the sidereal frequency $\Omega \equiv 2 \pi / 23{\rm h} 56{\rm min}$. Eq. (\[eqn\_shifts\_cel\]) predicts two experimental signatures of CPT/Lorentz violation, discussed below.

Sidereal time dependence of $\omega_{a}^{\mu^{+}}$ and $\omega_{a}^{\mu^{-}}$
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A g-2 experiment measuring $\omega_{a}$ for $\mu^{+}$ or $\mu^{-}$ over a period of several days is sensitive to the oscillating terms in Eq. (\[eqn\_shifts\_cel\]). From the oscillation amplitude $\hat{\omega}_{a}^{\mu^{+}}$ or $\hat{\omega}_{a}^{\mu^{-}}$ a combination of model parameters can be extracted: $$\sqrt{(\check{b}_{X}^{\mu^{\pm}})^{2} + (\check{b}_{Y}^{\mu^{\pm}})^{2}}
%= \sqrt{(\pm \frac{b_{X}}{\gamma} + m_{\mu} d_{X0} 
%+ H_{YZ})^{2} + (\pm \frac{b_{Y}}{\gamma} + m_{\mu} d_{Y0} + H_{ZX})^{2}}
= \frac{\hat{\omega}_{a}^{\mu^{\pm}}}{2 \, |\sin \chi|}$$ Since $\omega_{a}$ is proportional to the magnetic field $B$, the test has to be performed at constant $B$.

The parameter combination probed with this test is very similar to the one determined from muonium spectroscopy. @muonium The only difference is that for muonium the quantities $\check{b}_{J}^{\mu^{\pm}}$, defined in Eq. (\[eqn\_defcheck\]), are replaced by their non-relativistic limits with $\gamma = 1$. @kostbluhm Therefore the sensitivity of g-2 tests to the basic parameter components $b_{X}$ and $b_{Y}$ is suppressed by the factor $\gamma=29.3$ with respect to muonium, for a given sensitivity to $\check{b}_{X}^{\mu^{\pm}}$ and $\check{b}_{Y}^{\mu^{\pm}}$.

An analysis procedure for sidereal variations is currently being developed using the $\mu^{+}$ data of 1999 whose uncertainty for the time average of $\omega_{a}$ is $\sigma(\langle\omega_{a}\rangle) = 1.3\,$ppm. The error on the sidereal oscillation amplitude $\hat{\omega}$ is determined analytically from the second derivatives of $\chi^{2}$ for the fit function (\[eqn\_fit\]) discussed in Section \[sec\_analysis\]. A simple calculation yields $\sigma(\hat{\omega}_{a}) \approx \sqrt{2} \sigma(\langle\omega_{a}\rangle) =
1.8$ppm for the 1999 data. This leads to an achievable uncertainty for $\sqrt{(\check{b}_{X}^{\mu^{\pm}})^{2} + (\check{b}_{Y}^{\mu^{\pm}})^{2}}$ of about 1.2$\times 10^{-24}$ GeV. Taking into account the above-mentioned suppression by $\gamma=29.3$, the parameters $b_{X}$ and $b_{Y}$ can be probed with a sensitivity of the order 3.5$\times 10^{-23}$GeV, whereas muonium tests achieve a limit of 2$\times 10^{-23}$GeV.

To quantify the level of CPT/Lorentz violating effects, the dimensionless figure of merit $r_{\hat{\omega}_{a}} \equiv \frac{\hat{\omega}_{a}}{m_{\mu}}$ was introduced @kostbluhm, interpreting $\delta \omega_{a}$ as an energy shift of the muon and comparing its amplitude with the rest energy $m_{\mu}$. With the 1999 data set $r_{\hat{\omega}_{a}}$ can be probed down to a level of 0.19$\times 10^{-22}
\ll \frac{m_{\mu}}{m_{\rm Planck}}$, implying sensitivity to physics beyond the Planck scale. With the total data set for $\mu^{+}$ the sensitivity will be significantly better since in 2000 approximately four times more data were recorded. The final statistics for $\mu^{-}$ are expected to be similar.

Comparison of $\omega_{a}$ for positive and negative muons
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From Eq. (\[eqn\_shifts\_cel\]) follows that two experiments at the same colatitude $\chi$ and at the same magnetic field will observe the time-averaged $\omega_{a}$-difference $$\label{eqn_mu+mu-}
\Delta \omega_{a} \equiv
\langle\omega_{a}^{\mu^{+}}\rangle - \langle\omega_{a}^{\mu^{-}}\rangle \nonumber = 
4 \frac{b_{Z}}{\gamma} \cos \chi \quad.$$ Thus the measurement of $\Delta \omega_{a}$ sets a bound on $b_{Z}$ and is therefore complementary to the measurement of sidereal variations which involve the parameters $b_{X}$ and $b_{Y}$. The CERN g-2 experiment @cern which measured both $\omega_{a}^{\mu^{+}}$ and $\omega_{a}^{\mu^{-}}$ with an uncertainty of 10ppm, obtained $\Delta \omega_{a} = 2\pi (-5.5 \pm 3.3)\,$Hz, corresponding to $b_{Z} = (-2.3 \pm 1.4)\times10^{-22}$GeV, or a figure of merit @kostbluhm $r_{\Delta \omega_{a}} \equiv \frac{\Delta \omega_{a}}{m_{\mu}} = 
(-2.2 \pm 1.3) \times 10^{-22}$. With the final statistics of the g-2 experiment at BNL the sensitivity is expected to improve by roughly a factor 15.

One can also compare $\omega_{a}^{\mu^{+}}$ from BNL with $\omega_{a}^{\mu^{-}}$ from CERN. However, one needs to consider that the two experiments are situated at different geographic colatitudes $\chi_{1}$ and $\chi_{2}$ and that the magnetic fields $B_{1}$ and $B_{2}$ were slightly different. Using Eq. (\[eqn\_shifts\_cel\]), the measured frequencies can be written as $$\begin{aligned}
\langle\omega_{a}^{\mu^{+}}\rangle = \frac{e}{m} a B_{1} + 
\langle\delta\omega_{a}^{\mu^{+}}\rangle \\
\langle\omega_{a}^{\mu^{-}}\rangle = \frac{e}{m} a B_{2} + 
\langle\delta\omega_{a}^{\mu^{-}}\rangle\end{aligned}$$ To cancel the B-field dependent terms, we scale the CERN result $\langle\omega_{a}^{\mu^{-}}\rangle$ by $\frac{B_{1}}{B_{2}}$ and subtract it from $\langle\omega_{a}^{\mu^{+}}\rangle$: $$\begin{aligned}
\label{eqn_mu+mu-cernbnl}
&&\Delta \omega_{a} =
\langle\omega_{a}^{\mu^{+}}\rangle - \frac{B_{1}}{B_{2}} 
\langle\omega_{a}^{\mu^{-}}\rangle 
\nonumber \\
&&= 2 \frac{b_{Z}}{\gamma} \left(\cos \chi_{1} + \frac{B_{1}}{B_{2}} 
\cos \chi_{2}\right) +
2 \left(m_{\mu} d_{Z0} + H_{XY}\right) \left(\cos \chi_{1} - \frac{B_{1}}{B_{2}}
\cos \chi_{2}\right)~~.~\end{aligned}$$ The usefulness of such a test is its sensitivity to a parameter combination involving not only $b_{Z}$ but also $d_{Z0}$ and $H_{XY}$: From the 1999 result @prl for $\omega_{a}^{\mu^{+}}$ and the CERN value @cern for $\omega_{a}^{\mu^{-}}$ we obtain $ 
b_{Z} + \gamma \frac{\cos \chi_{1} - \frac{B_{1}}{B_{2}}\cos \chi_{2}}
{\cos \chi_{1} + \frac{B_{1}}{B_{2}} \cos \chi_{2}} (m\, d_{z0} + H_{xy})
= (-1.4 \pm 1.0)\times10^{-22}\,{\rm GeV,}
$ where $\chi_{1}$(BNL) = 49.1$^{\rm o}$, $\chi_{2}$(CERN) = 43.8$^{\rm o}$ and $\frac{B_{1}}{B_{2}} = 0.984$. This corresponds to $r_{\Delta \omega_{a}} = (-1.3 \pm 0.9)\times10^{-22}$.

Proposed Analysis Procedure for Sidereal Variations of $\mathbf{\omega_a}$ {#sec_analysis}
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At present a search for sidereal variations of $\omega_{a}$ in the data set of 1999 for positive muons is underway. These data with a statistical uncertainty of 1.25ppm in $\omega_{a}$ were collected in 806 runs distributed over 25 days. The typical length of a run was half an hour. While in the g-2 analysis @prl the decay positron time spectra of all runs were summed before the fit, the search for time variations in $\omega_{a}$ requires analyzing each run individually. After applying a correction for pulses arriving with a time separation smaller than the resolution of the pulse-finding algorithm, the g-2 analysis fits the time spectra with the function $$N(t) = N_0\, {\rm e}^{-\frac{t}{\tau}} [1 + A \cos(\omega_a t + \phi)] \cdot
f_{\rm CBO}(t) \cdot f_{\rm ml}(t)$$ The data of single runs are only sensitive to the exponential decay modulated by the g-2 oscillation. The parameters pertaining to perturbations due to coherent betatron oscillations, $f_{\rm CBO}(t)$, and muon losses, $f_{\rm ml}(t)$, are fixed to the values found in the g-2 analysis. This is a valid approach as long as $f_{\rm CBO}(t)$ and $f_{\rm ml}(t)$ are not affected by sidereal variations from CPT/Lorentz violating effects.

The values $\omega_{a\,i}$ determined for each run at a time $t_{i}$ are fitted with the function $$\label{eqn_fit}
\omega_{a}(t_{i}, \omega_{p\,i}) = K\, \omega_{p\,i} + \hat{\omega}_{a}
\cos(\Omega t_{i} + \Phi)$$ taking into account the magnetic field monitored with NMR probes in terms of the free proton precession frequency $\omega_{p}$. $K$, $\hat{\omega}$ and $\Phi$ are free parameters whereas the sidereal frequency $\Omega$ is fixed. Including the magnetic field into the fit is necessary because its variations of up to 0.5ppm cause variations in $\omega_{a}$ of equal relative size. Particularly dangerous are day-night field variations caused by temperature changes which could fake a sidereal oscillation.

Another systematic concern involves possible effects of CPT/Lorentz violation on the time and frequency references used in the experiment. They could exhibit their own sidereal variations and hence either mask the variation for muons or introduce false signatures. A sidereal change of $\omega_{p}$ – the basis of the NMR field measurement – could counteract with the muon-related change in $\omega_{a}$. However, atomic clock comparisons @atomic provide an upper bound on the shift of $\omega_{p}$ on the mHz level, well below a ppb of the measured frequency. Finally, the Loran-C frequency standard used for the timing of the positron pulse readout and for the NMR measurement is based on Cesium hyperfine transitions with $m_{F} = 0$, which are insensitive to any preferential direction in space and will therefore not induce any sidereal variations.

Conclusions
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Tests based on muon g-2 experiments are sensitive to the muon sector of the CPT/Lorentz violating extension (\[eqn\_lagrange\]) of the Standard Model. The search for sidereal variations of $\omega_{a}$ and the comparison of $\omega_{a}$ for $\mu^{+}$ and $\mu^{-}$ probe complementary parts of the parameter space of the extension. An analysis searching for sidereal variations of $\omega_{a}$ is currently being done.

Acknowledgements {#acknowledgements .unnumbered}
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This work was supported by the U.S. Department of Energy, the U.S. National Science Foundation, the German Bundesminister für Bildung und Forschung, the Russian Ministry of Science, and the US-Japan Agreement in High Energy Physics. Mario Deile acknowledges support by the Alexander von Humboldt Foundation.

References {#references .unnumbered}
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<span>99</span> W. Liu et al., . Muon (g-2) Collaboration: and references therein. D. Colladay and V.A. Kostelecký: ; . R. Bluhm, V.A. Kostelecký and C.D. Lane: ; R. Bluhm, V.A. Kostelecký and N. Russell: . D.E. Groom et al.: Review of Particle Physics, Eur. Phys. J. <span>**C15**</span>, 1 (2000). R. K. Mittleman et al.: . V.W. Hughes et al.: . J. Bailey et al., . In this publication the comparison between $\mu^{+}$ and $\mu^{-}$ is given in terms of $r_{g} \equiv \Delta g/g$ rather than $r_{\Delta \omega_{a}}$. Both figures are equivalent, but see the discussion in Ref (@kostbluhm). V.A. Kostelecký and C.D. Lane: and references therein.

{'source': 'arXiv_out/hep-ex0110044_extract_cptproc'}