# Muon g-2

The g − 2 storage-ring magnet at Fermilab, which was originally designed for the Brookhaven g − 2 experiment. The geometry allows for a very uniform magnetic field to be established in the ring.

Muon g − 2 (pronounced "gee minus two") is a particle physics experiment at Fermilab to measure the anomalous magnetic dipole moment of a muon to a precision of 0.14 ppm,[1] which will be a sensitive test of the Standard Model.[2] It might also provide evidence of the existence of entirely new particles.[3]

The muon, like its lighter sibling the electron, acts like a spinning magnet. The parameter known as the "g factor" indicates how strong the magnet is and the rate of its gyration. The value of g is slightly larger than 2, hence the name of the experiment. This difference from 2 (the "anomalous" part) is caused by higher-order contributions from quantum field theory. In measuring g − 2 with high precision and comparing its value to the theoretical prediction, physicists will discover whether the experiment agrees with theory. Any deviation would point to as yet undiscovered subatomic particles that exist in nature.[4]

Four data-taking periods (Run 1,[5] Run 2,[6] Run 3, and Run 4[citation needed]) have been completed, with Run 5 currently ongoing. The results from the analysis of the Run 1 data were announced and published on April 7, 2021. The physicists reported that results from recent studies involving the particle challenged the Standard Model and, accordingly, may require an updating of currently understood physics.[7][8]

## Timeline

### Muon g − 2 at CERN

The storage ring of the muon g − 2 experiment at CERN

The first muon g − 2 experiments began at CERN in 1959 at the initiative of Leon Lederman.[9][10] A group of six physicists formed the first experiment, using the Synchrocyclotron at CERN. The first results were published in 1961,[11] with a 2% precision with respect to the theoretical value, and then the second ones with this time a 0.4% precision, hence validating the quantum electrodynamics theory.

A second experiment started in 1966 with a new group, working this time with the Proton-Synchrotron, still at CERN. The results were then 25 times more precise than the previous ones and showed a quantitative discrepancy between the experimental values and the theoretical ones, and thus required the physicists to recalculate their theoretical model. The third experiment, which started in 1969, published its final results in 1979,[12] confirming the theory with a precision of 0.0007%. The United States took over the g − 2 experiment in 1984.[13]

### Muon g − 2 at Brookhaven National Laboratory

The next stage of muon g − 2 research was conducted at the Brookhaven National Laboratory Alternating Gradient Synchrotron. The experiment was done similarly to the last of the CERN experiments with the goal of having 20 times better precision. The technique involved storing 3.094 GeV muons in a uniform measured magnetic field and observing the difference of the muon spin precession and rotation frequency via detection of the muon decay electrons. The advance in precision relied crucially on a much more intense beam than was available at CERN and the injection of muons into the storage ring, whereas the previous CERN experiments had injected pions into the storage ring, of which only a small fraction decay into muons that are stored. The experiment used a much more uniform magnetic field using a superferric superconducting storage ring magnet, a passive superconducting inflector magnet, fast muon kickers to deflect the injected muons onto stored orbits, a beam tube NMR trolley that could map the magnetic field in the storage region, and numerous other experimental advances. The experiment took data with positive and negative muons between 1997 and 2001. Its final result is aµ = (g − 2)/2 = 11659208.0(5.4)(3.3) × 10−10 obtained by combination of consistent results with similar precision from positive and negative muons.[14]

### Muon g − 2 at Fermilab

Fermilab is continuing the experiment conducted at Brookhaven National Laboratory[15] to measure the anomalous magnetic dipole moment of the muon. The Brookhaven experiment ended in 2001, but ten years later Fermilab acquired the equipment, and is working to make a more accurate measurement (smaller σ) which will either eliminate the discrepancy between Brookhaven's results and theory predictions or confirm it as an experimentally observable example of physics beyond the Standard Model.

The magnet was refurbished and powered on in September 2015, and has been confirmed to have the same 1.3 ppm basic magnetic field uniformity that it had before the move.

As of October 2016 the magnet has been rebuilt and carefully shimmed to produce a highly uniform magnetic field. New efforts at Fermilab have resulted in a three-fold improved overall uniformity, which is important for the new measurement at its higher precision goal.[16]

In April 2017 the collaboration was preparing the experiment for the first production run with protons – to calibrate detector systems. The magnet received its first beam of muons in its new location on May 31, 2017.[17] Data taking was planned to run until 2020.[18]

On April 7, 2021, the results of the experiment were published: aµ = 0.00116592040(54). The new experimental world-average results announced by the Muon g − 2 collaboration are: g-factor: 2.00233184122(82), anomalous magnetic moment: 0.00116592061(41). The combined results from Fermilab and Brookhaven show a difference with theory at a significance of 4.2 sigma, slightly under the 5 sigma (or standard deviations) that particle physicists require to claim a discovery, but still compelling evidence of new physics. The chance that a statistical fluctuation would produce equally striking results is about 1 in 40,000.[8] The same day lattice QCD results by the Budapest-Marseille-Wuppertal Collaboration (BMW) collaboration were published[19][20] which stood between the experimental value obtained at Fermilab and the theoretical value calculated by the Muon g-2 Theory Initiative,[21][22] subsequent works by the Coordinated Lattice Simulations (CLS) group[23][24] the European Twisted Mass Collaboration (ETMC)[25][26] have come closer each to the theoretical value suggesting there could be systematical errors in the estimation of the R-ratio of the hadronic vacuum polarization used by Fermilab.[27]

## Theory of magnetic moments

The g factor of a charged lepton (electron, muon, or tau) is very nearly 2. The difference from 2 (the "anomalous" part) depends on the lepton, and can be computed quite precisely based on the current Standard Model of particle physics. Measurements of the electron are in excellent agreement with this computation. The Brookhaven experiment did this measurement for muons, a much more technically difficult measurement due to their short lifetime, and detected a tantalizing, but not definitive, 3.7 σ discrepancy between the measured value and the prediction of the Standard Model (0.00116592089 versus 0.0011659180).[28]

## Design

The g − 2 ring arriving at its final destination – the experimental hall (MC1) at Fermilab – on July 30, 2014

Central to the experiment is a 50-foot (15 m)-diameter superconducting magnet with an exceptionally uniform magnetic field. This was transported, in one piece, from Brookhaven in Long Island, New York, to Fermilab in the summer of 2013. The move traversed 3,200 miles (5,100 km) over 35 days,[29] mostly on a barge down the East Coast and through Mobile, Alabama, to the Tennessee–Tombigbee Waterway and then briefly on the Mississippi. The initial and final legs were on a special truck traveling closed highways at night.

Sample 25 mm × 25 mm × 140 mm PbF2 crystals (bare and wrapped in Millipore paper) are pictured together with a 16 channel monolithic Hamamatsu SiPM.

### Detectors

The magnetic moment measurement is realized by 24 electromagnetic calorimetric detectors, which are distributed uniformly on the inside of the storage ring. The calorimeters measure the energy and time of arrival (relative to the injection time) of the decay positrons (and their count) from the muon decay in the storage ring. After a muon decays into a positron and two neutrinos, the positron ends up with less energy than the original muon. Thus, the magnetic field curls it inward where it hits a segmented lead(II) fluoride (PbF2) calorimeter read out by silicon photo-multipliers (SiPM).[30]

The tracking detectors register the trajectory of the positrons from the muon decay in the storage ring. The tracker can provide a muon electric dipole moment measurement, but not directly the magnetic moment measurement. The main purpose of the tracker is to measure the muon beam profile, as well as resolution of pile-up of events (for reduction of the systematic uncertainty in the calorimeter measurement).[30]

One of the 4 rows of 32 straws is shown. A straw (length of 100 mm, and diameter of 5 mm) acts like an ionisation chamber filled with 1:1 argon:ethane, with a central cathode wire at +1.6 kV.

### Magnetic field

To measure the magnetic moment to ppb level of precision requires a uniform average magnetic field to be of the same level precision. The experimental goal of g − 2 is to achieve an uncertainty level on the magnetic to 70 ppb averaged over time and muon distribution. A uniform field of 1.45 T is created in the storage ring using superconducting magnets, and the field value will be actively mapped throughout the ring using an NMR probe on a mobile trolley (without breaking the vacuum). Calibration of the trolley is referenced to the Larmor frequency of a proton in a spherical water sample at a reference temperature (34.7 °C), and is cross-calibrated to a novel helium-3 magnetometer.[30]

### Data acquisition

An essential component of the experiment is the data acquisition (DAQ) system, which manages the data flow from the detector electronics. The requirement for the experiment is to acquire raw data at a rate of 18 GB/s. This is accomplished by employing parallel data-processing architecture using 24 high-speed GPUs (NVIDIA Tesla K40) to process data from 12 bit waveform digitisers. The set-up is controlled by the MIDAS DAQ software framework. The DAQ system processes data from 1296 calorimeter channels, 3 straw tracker stations, and auxiliary detectors (e.g. entrance muon counters). The total data output of the experiment is estimated at 2 PB.[31]

## Collaboration

The following universities, laboratories, and companies are participating in the experiment:[32]

## References

1. ^ "Muon g − 2 Experiment" (main page). Fermilab. Retrieved April 26, 2017.
2. ^ Keshavarzi, Alex; Khaw, Kim Siang; Yoshioka, Tamaki (January 22, 2022). "Muon g − 2: A review". Nuclear Physics B. 975: 115675. arXiv:2106.06723. Bibcode:2022NuPhB.97515675K. doi:10.1016/j.nuclphysb.2022.115675. S2CID 245880824.
3. ^ Gibney, Elizabeth (April 13, 2017). "Muons' big moment could fuel new physics". Nature. 544 (7649): 145–146. Bibcode:2017Natur.544..145G. doi:10.1038/544145a. PMID 28406224. S2CID 4400589.
4. ^ "Muon g − 2 Collaboration to solve mystery". Muon g − 2 Experiment (Press release). Fermilab. Archived from the original on July 1, 2017. Retrieved April 30, 2017.
5. ^ "First results from the Muon g − 2 experiment at Fermilab" (Press release). Fermilab. March 7, 2021.
6. ^ "Muon g − 2 begins second run". phys.org. March 26, 2019.
7. ^ Overbye, Dennis (April 7, 2021). "Finding from particle research could break known laws of physics". The New York Times. Retrieved April 7, 2021. It's not the next Higgs boson – yet. But the best explanation, physicists say, involves forms of matter and energy not currently known to science.
8. ^ a b Marc, Tracy (April 7, 2021). "First results from Fermilab's Muon g − 2 experiment strengthen evidence of new physics" (Press release). Fermilab. Retrieved April 7, 2021.
9. ^ Farley, Francis (2004). "The dark side of the muon". In Álvarez-Gaumé, Luis (ed.). Infinitely CERN: Memories of fifty years of research, 1954–2004. Geneva, CH: Editions Suzanne Hurter. pp. 38–41. ISBN 978-2-940031-33-7. OCLC 606546795.
10. ^ "Archives of Muon g − 2 experiment". CERN Archive. 2007. Retrieved March 4, 2020.
11. ^ Charpak, Georges; Garwin, Richard L.; Farley, Francis J.M.; Müller, T. (1994). "Results of the g − 2 experiment". In Cabibbo, N. (ed.). Lepton Physics at CERN and Frascati. World Scientific. pp. 34 ff. ISBN 9789810220785.
12. ^ Combley, F.; Farley, F.J.M.; Picasso, E. (1981). "The CERN muon (g − 2) experiments". Physics Reports. 68 (2): 93–119. Bibcode:1981PhR....68...93C. doi:10.1016/0370-1573(81)90028-4. ISSN 0370-1573.
13. ^ "Enigma of the muon" (Press release). European Organization for Nuclear Research (CERN). Retrieved July 19, 2018.
14. ^ Bennett, G.W.; Bousquet, B.; Brown, H.N.; Bunce, G.; Carey, R.M.; Cushman, P.; et al. (Muon g − 2 Collaboration) (April 7, 2006). "Final report of the E821 muon anomalous magnetic moment measurement at BNL". Physical Review D. 73 (7): 072003. arXiv:hep-ex/0602035. Bibcode:2006PhRvD..73g2003B. doi:10.1103/PhysRevD.73.072003. S2CID 53539306.
15. ^ Farley, F. (2004). "The 47 years of muon g − 2". Progress in Particle and Nuclear Physics. 52 (1): 1–83. Bibcode:2004PrPNP..52....1F. doi:10.1016/j.ppnp.2003.09.004. ISSN 0146-6410.
16. ^ Holzbauer, J. L. (December 9, 2016). "The Muon g − 2 Experiment Overview and Status as of June 2016". Proceedings, 12th International Conference on Beauty, Charm, and Hyperons in Hadronic Interactions (BEACH 2016): Fairfax, Virginia, USA, June 12–18, 2016. XIIth International Conference on Beauty, Charm, and Hyperons in Hadronic Interactions. J. Phys. Conf. Ser. Vol. 770. p. 012038. arXiv:1610.10069. doi:10.1088/1742-6596/770/1/012038. "alt. source" – via inSPIRE.
17. ^ "Muon magnet's moment has arrived" (Press release). Fermilab. May 31, 2017.
18. ^ Gohn, W.; et al. (Muon g − 2 Collaboration) (November 15, 2016). "The muon g − 2 experiment at Fermilab". 18th International Workshop on Neutrino Factories and Future Neutrino Facilities Search (NuFact16) Quy Nhon, Vietnam, August 21–27, 2016. arXiv:1611.04964. "alt source" – via inSPIRE.
19. ^ Borsanyi, Sz.; Fodor, Z.; Guenther, J. N.; Hoelbling, C.; Katz, S. D.; Lellouch, L.; Lippert, T.; Miura, K.; Parato, L.; Szabo, K. K.; Stokes, F.; Toth, B. C.; Torok, Cs.; Varnhorst, L. (May 6, 2021). "Leading hadronic contribution to the muon magnetic moment from lattice QCD". Nature. 593 (7857): 51–55. arXiv:2002.12347. Bibcode:2021Natur.593...51B. doi:10.1038/s41586-021-03418-1. ISSN 0028-0836. PMID 33828303. S2CID 221151004.
20. ^ "Budapest-Marseille-Wuppertal Collaboration". www.bmw.uni-wuppertal.de.
21. ^ Aoyama, T.; Asmussen, N.; Benayoun, M.; Bijnens, J.; Blum, T.; Bruno, M.; Caprini, I.; Carloni Calame, C.M.; Cè, M.; Colangelo, G.; Curciarello, F.; Czyż, H.; Danilkin, I.; Davier, M.; Davies, C.T.H. (December 2020). "The anomalous magnetic moment of the muon in the Standard Model". Physics Reports. 887: 1–166. doi:10.1016/j.physrep.2020.07.006. S2CID 219559166.
22. ^ "Home | Muon g-2 Theory". muon-gm2-theory.illinois.edu. Retrieved March 14, 2023.
23. ^ Cè, M.; Gérardin, A.; von Hippel, G.; Hudspith, R. J.; Kuberski, S.; Meyer, H. B.; Miura, K.; Mohler, D.; Ottnad, K.; Paul, S.; Risch, A.; San José, T.; Wittig, H. (December 13, 2022). "Window observable for the hadronic vacuum polarization contribution to the muon \$g\ensuremath{-}2\$ from lattice QCD". Physical Review D. 106 (11): 114502. doi:10.1103/PhysRevD.106.114502. S2CID 56285714.
24. ^
25. ^ Alexandrou, Constantia; Bacchio, Simone; Dimopoulos, Petros; Finkenrath, Jacob; Frezzotti, Roberto; Gagliardi, Giuseppe; Garofalo, Marco; Hadjiyiannakou, Kyriakos; Kostrzewa, Bartosz; Jansen, Karl; Lubicz, Vittorio; Petschlies, Marcus; Sanfilippo, Francesco; Simula, Silvano; Urbach, Carsten (December 20, 2022). "Short \& intermediate distance HVP contributions to muon g-2: SM (lattice) prediction versus \$e^+e^-\$ annihilation data". arXiv:2212.10490 [hep-ph].
26. ^ "European Twisted Mass Collaboration". www-zeuthen.desy.de. Retrieved March 14, 2023.
27. ^ Alexandrou, Constantia; Bacchio, Simone; De Santis, Alessandro; Dimopoulos, Petros; Finkenrath, Jacob; Frezzotti, Roberto; Gagliardi, Giuseppe; Garofalo, Marco; Hadjiyiannakou, Kyriakos; Kostrzewa, Bartosz; Jansen, Karl; Lubicz, Vittorio; Petschlies, Marcus; Sanfilippo, Francesco; Simula, Silvano (December 16, 2022). "Probing the R-ratio on the lattice". arXiv:2212.08467 [hep-lat].
28. ^ "Physicists publish worldwide consensus of muon magnetic moment calculation" (Press release). Fermilab. June 11, 2020.
29. ^ Hertzog, David; Roberts, Lee (October 27, 2014). "Muon g − 2 storage ring starts a new life". CERN Courier. Retrieved April 26, 2017.
30. ^ a b c Grange, J.; Guarino, V.; Winter, P.; Wood, K.; Zhao, H.; Carey, R.M.; et al. (Muon g − 2 Collaboration) (January 27, 2015). Muon (g − 2) Technical Design Report (Report). arXiv:1501.06858. Bibcode:2015arXiv150106858G. "alt. source" – via inSPIRE.
31. ^ Gohn, W.; et al. (Muon g − 2 Collaboration) (November 15, 2016). "Data acquisition with GPUs: The DAQ for the muon g − 2 experiment at Fermilab". Proceedings, 38th International Conference on High Energy Physics (ICHEP 2016): Chicago, Illinois, USA, August 3–10, 2016. p. 174. arXiv:1611.04959. Bibcode:2016arXiv161104959G. doi:10.22323/1.282.0174. "alt. source" – via inSPIRE.
32. ^ "Muon g − 2 Collaboration". Muon g − 2 Experiment. Fermilab. Retrieved April 26, 2017.