# Studying Antimatter Gravity with Muonium

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## Abstract

**:**

## 1. Introduction

## 2. Method

- Development of improved low-velocity muon and muonium beams;
- Development of a sufficiently precise interferometer; and
- Development of a sufficiently precise interferometer alignment and calibration technique.

#### 2.1. Interferometer

#### 2.2. Muonium Beam

#### 2.3. Interferometer Alignment and Calibration

## 3. Systematic Uncertainties

## 4. Prospects

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

CNM | the Center for Nanoscale Materials at Argonne National Laboratory |

FWHM | full-width at half maximum |

$\overline{g}$, g | the gravitational acceleration at the earth’s surface of antimatter and matter, respectively |

GR | general relativity |

IIT | Illinois Institute of Technology |

IPRO | Inter-Professional Project |

IR | infrared |

MAGE | the Muonium Antimatter Gravity Experiment |

MCP | microchannel plate |

MDPI | Multidisciplinary Digital Publishing Institute |

Mu | muonium |

muCool | cooled muon beam R&D program at PSI |

PBS | polarizing beam splitter |

PSI | Paul Scherrer Institute |

RMS | root-mean-square |

SFHe | superfluid helium |

TFG | tracking frequency gauge |

UNCD | ultrananocrystalline diamond |

vdWE | van der Waals effect |

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1. | |

2. | The only published direct test so far [4] has yielded the limit $-65<\overline{g}/g<110$. |

3. | |

4. | This also suggests a solution to what has been called “the worst prediction in physics”: that the gravitational zero-point energy of the universe seems to exceed the size of the cosmological constant by a factor $\sim {10}^{120}$ [52]. |

5. | |

6. | An alternative derivation in an accelerated reference frame, in which the beam travels in a straight, horizontal line and the interferometer accelerates upwards at g, may be somewhat easier to follow and leads to the same conclusion. |

7. | For overlapped interferometer beams as in our case, the optimal open fractions have been shown to be (0.60, 0.43, 0.37) [54], which can feasibly be fabricated in our proposed approach. |

8. | Note that the sensitivities of Figure 7 are somewhat more pessimistic than that of Equation (1), due to inclusion of estimated decay losses from the Mu source to the first grating. The size of this effect will depend on the final source–interferometer distance, which will depend on cryostat engineering details yet to be determined. |

9. | The “obvious” solution of comparing muonium and antimuonium beams is unfortunately not feasible, since it is impractical to produce a sufficiently positron-rich ${\mu}^{-}$ stopping medium. |

10. | If necessary, to eliminate the possible ambiguity between Mu and X-ray events, the ${e}^{-}$ accelerating potential can be turned off during calibration runs. |

**Figure 1.**Principle of three-grating muonium-gravity interferometer, shown schematically in elevation view, with gravitational deflection and phase shift $\Delta \varphi $ exaggerated for clarity. Muonium beam enters from left, slow-electron detector is at right. Not shown: ring electrodes, which accelerate slow electrons onto their detector, starting downstream of grating 3 and continuing within (helically wound) scintillating-fiber-barrel positron-tracking detector; and scintillating-bar hodoscope surrounding positron-tracking detector.

**Figure 2.**Muonium interferometer support scheme in channel-cut single-crystal silicon optical bench. (

**a**) Optical bench with gratings, showing mirrors at upper grating corners for alignment interferometers; (

**b**) Section A-A detail showing mounting scheme of each grid within its silicon frame.

**Figure 3.**Semiconductor-laser tracking frequency gauge (TFG) precision in picometers vs. averaging time in seconds, determined by measuring the half-meter optical path-length difference of a Michelson interferometer using two TFGs simultaneously, operated at 1560 nm with a common optical path (from [73]). The demonstrated stability (Allan deviation less than 3 pm for averaging times ranging from about 1 s to about 1000 s) implies that X-ray calibrations need not be repeated more often than every 10 to 20 min; given operation in a cavity of modest finesse (as in Figure 4), the calibration interval extrapolates to 10 days.

**Figure 4.**Optical layout concept of the two TFG measurement interferometers observing the position of grating 3, mounted on bottom outer surface of optical bench, in two orthogonal views, with one light path indicated in red. (Similar arrangements on the outer sides of the optical bench can monitor the alignment of the other two gratings.) The TFG is shown operating in a cavity, which enhances the performance over operation in a Michelson interferometer.

**Figure 5.**3D drawing of 2-layer barrel scintillating-fiber tracker, surrounded by outer scintillator-bar hodoscope used for trigger purposes and to break reconstruction ambiguity.

**Figure 6.**Muonium production vs. temperature in liquid helium (from [76]); open circles are results in pure ${}^{4}$He, filled circles, ${}^{4}$He + 0.2% ${}^{3}$He, and triangles, pure ${}^{3}$He. (Quantity plotted is observed muon decay asymmetry; an observed asymmetry of 0.105 corresponds to 100% muonium formation.)

**Figure 7.**Representative MAGE sensitivity estimates vs. grating separation for beam options described in text, with 0.5 $\mathsf{\mu}$m-thick gratings of 100 nm pitch, assuming 10% contrast and that statistical uncertainties dominate over systematics; shown is beam time required for 5$\sigma $ determination of the sign of $\overline{g}$ (i.e., $\delta \overline{g}/g=0.4$).8

Cause | Size | Parameter | Value | Effect (g) |
---|---|---|---|---|

E gradient | 100 V/m${}^{2}$ | ${\alpha}_{\mathrm{Mu}}$ | $7\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-31}$ m${}^{3}$ | 0.04 * |

B gradient | <1 nT/mm | ${\mu}_{\mathrm{Mu}}$ | $9\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-24}$ J/T | <0.005 ${}^{\u2020}$ |

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## Share and Cite

**MDPI and ACS Style**

Antognini, A.; Kaplan, D.M.; Kirch, K.; Knecht, A.; Mancini, D.C.; Phillips, J.D.; Phillips, T.J.; Reasenberg, R.D.; Roberts, T.J.; Soter, A. Studying Antimatter Gravity with Muonium. *Atoms* **2018**, *6*, 17.
https://doi.org/10.3390/atoms6020017

**AMA Style**

Antognini A, Kaplan DM, Kirch K, Knecht A, Mancini DC, Phillips JD, Phillips TJ, Reasenberg RD, Roberts TJ, Soter A. Studying Antimatter Gravity with Muonium. *Atoms*. 2018; 6(2):17.
https://doi.org/10.3390/atoms6020017

**Chicago/Turabian Style**

Antognini, Aldo, Daniel M. Kaplan, Klaus Kirch, Andreas Knecht, Derrick C. Mancini, James D. Phillips, Thomas J. Phillips, Robert D. Reasenberg, Thomas J. Roberts, and Anna Soter. 2018. "Studying Antimatter Gravity with Muonium" *Atoms* 6, no. 2: 17.
https://doi.org/10.3390/atoms6020017