# Hadronic-vacuum-polarization contribution to the muon’s anomalous magnetic moment from four-flavor lattice QCD

## Abstract

We calculate the contribution to the muon anomalous magnetic moment hadronic vacuum polarization from {the} connected diagrams of up and down quarks, omitting electromagnetism. We employ QCD gauge-field configurations with dynamical $u$, $d$, $s$, and $c$ quarks and the physical pion mass, and analyze five ensembles with lattice spacings ranging from $a approx 0.06$ to~0.15~fm. The up- and down-quark masses in our simulations have equal masses $$m_l$$. We obtain, in this world where all pions have the mass of the $pi^0$, $$10^{10} a_mu^{ll}({ m conn.}) = 637.8,(8.8)$$, in agreement with independent lattice-QCD calculations. We then combine this value with published lattice-QCD results for the connected contributions from strange, charm, and bottom quarks, and an estimate of the uncertainty due to the fact that our calculation does not include strong-isospin breaking, electromagnetism, or contributions from quark-disconnected diagrams. Our final result for the total $$mathcal{O}(alpha^2)$$ hadronic vacuum polarization to the muon-s anomalous magnetic moment is~$$10^{10}a_mu^{ m HVP,LO} = 699(15)_{u,d}(1)_{s,c,b}$$, where the errors are from the light-quark and heavy-quark contributions, respectively. Our result agrees with both {it ab-initio} lattice-QCD calculations and phenomenological determinations from experimental $e^+e^-$-scattering data. It is $1.3sigma$ below the "no new physics" value of the hadronic-vacuum-polarization contribution inferred from combining the BNL E821 measurement of $$a_mu$$ with theoretical calculations of the other contributions.

- Authors:

- Publication Date:

- Research Org.:
- Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States); Univ. of Colorado, Boulder, CO (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP)

- Contributing Org.:
- Fermilab Lattice, HPQCD, and MILC Collaborations

- OSTI Identifier:
- 1600067

- Alternate Identifier(s):
- OSTI ID: 1529343; OSTI ID: 1764432

- Report Number(s):
- FERMILAB-PUB-19-064-T

Journal ID: ISSN 2470-0010; PRVDAQ; 034512

- Grant/Contract Number:
- AC02-07CH11359; FG02-13ER41976; SC0009998; SC0010005; SC0010120; SC0015655; TG-MCA93S002

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 101 Journal Issue: 3; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Lattice field theory; Quantum chromodynamics; Muons; Magnetic moment; Ab initio calculations; Lattice QCD

### Citation Formats

```
Davies, C. T. H., DeTar, C., El-Khadra, A. X., Gámiz, E., Gottlieb, Steven, Hatton, D., Kronfeld, A. S., Laiho, J., Lepage, G. P., Liu, Yuzhi, Mackenzie, P. B., McNeile, C., Neil, E. T., Primer, T., Simone, J. N., Toussaint, D., Van de Water, R. S., Vaquero, A., and Fermilab Lattice, HPQCD, and MILC Collaborations. Hadronic-vacuum-polarization contribution to the muon’s anomalous magnetic moment from four-flavor lattice QCD. United States: N. p., 2020.
Web. https://doi.org/10.1103/PhysRevD.101.034512.
```

```
Davies, C. T. H., DeTar, C., El-Khadra, A. X., Gámiz, E., Gottlieb, Steven, Hatton, D., Kronfeld, A. S., Laiho, J., Lepage, G. P., Liu, Yuzhi, Mackenzie, P. B., McNeile, C., Neil, E. T., Primer, T., Simone, J. N., Toussaint, D., Van de Water, R. S., Vaquero, A., & Fermilab Lattice, HPQCD, and MILC Collaborations. Hadronic-vacuum-polarization contribution to the muon’s anomalous magnetic moment from four-flavor lattice QCD. United States. https://doi.org/10.1103/PhysRevD.101.034512
```

```
Davies, C. T. H., DeTar, C., El-Khadra, A. X., Gámiz, E., Gottlieb, Steven, Hatton, D., Kronfeld, A. S., Laiho, J., Lepage, G. P., Liu, Yuzhi, Mackenzie, P. B., McNeile, C., Neil, E. T., Primer, T., Simone, J. N., Toussaint, D., Van de Water, R. S., Vaquero, A., and Fermilab Lattice, HPQCD, and MILC Collaborations. Tue .
"Hadronic-vacuum-polarization contribution to the muon’s anomalous magnetic moment from four-flavor lattice QCD". United States. https://doi.org/10.1103/PhysRevD.101.034512.
```

```
@article{osti_1600067,
```

title = {Hadronic-vacuum-polarization contribution to the muon’s anomalous magnetic moment from four-flavor lattice QCD},

author = {Davies, C. T. H. and DeTar, C. and El-Khadra, A. X. and Gámiz, E. and Gottlieb, Steven and Hatton, D. and Kronfeld, A. S. and Laiho, J. and Lepage, G. P. and Liu, Yuzhi and Mackenzie, P. B. and McNeile, C. and Neil, E. T. and Primer, T. and Simone, J. N. and Toussaint, D. and Van de Water, R. S. and Vaquero, A. and Fermilab Lattice, HPQCD, and MILC Collaborations},

abstractNote = {We calculate the contribution to the muon anomalous magnetic moment hadronic vacuum polarization from {the} connected diagrams of up and down quarks, omitting electromagnetism. We employ QCD gauge-field configurations with dynamical $u$, $d$, $s$, and $c$ quarks and the physical pion mass, and analyze five ensembles with lattice spacings ranging from $a approx 0.06$ to~0.15~fm. The up- and down-quark masses in our simulations have equal masses $m_l$. We obtain, in this world where all pions have the mass of the $pi^0$, $10^{10} a_mu^{ll}({ m conn.}) = 637.8,(8.8)$, in agreement with independent lattice-QCD calculations. We then combine this value with published lattice-QCD results for the connected contributions from strange, charm, and bottom quarks, and an estimate of the uncertainty due to the fact that our calculation does not include strong-isospin breaking, electromagnetism, or contributions from quark-disconnected diagrams. Our final result for the total $mathcal{O}(alpha^2)$ hadronic vacuum polarization to the muon-s anomalous magnetic moment is~$10^{10}a_mu^{ m HVP,LO} = 699(15)_{u,d}(1)_{s,c,b}$, where the errors are from the light-quark and heavy-quark contributions, respectively. Our result agrees with both {it ab-initio} lattice-QCD calculations and phenomenological determinations from experimental $e^+e^-$-scattering data. It is $1.3sigma$ below the "no new physics" value of the hadronic-vacuum-polarization contribution inferred from combining the BNL E821 measurement of $a_mu$ with theoretical calculations of the other contributions.},

doi = {10.1103/PhysRevD.101.034512},

journal = {Physical Review D},

number = 3,

volume = 101,

place = {United States},

year = {2020},

month = {2}

}

https://doi.org/10.1103/PhysRevD.101.034512

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