# White Paper

# White Paper

## About

## Section Authors

The major sections of the White Paper were written and edited by the section authors listed at the top of each section:

**Section 2: Data-driven calculations of HVP**

*M. Benayoun, C. M. Carloni Calame, H. Czyz ̇, M. Davier, S. I. Eidelman, M. Hoferichter, F. Jegerlehner, A. Keshavarzi, B. Malaescu, D. Nomura, M. Passera, T. Teubner, G. Venanzoni, Z. Zhang***Section 3: Lattice QCD calculations of HVP**

*T. Blum, M. Bruno, M. Ce, C. T. H. Davies, M. Della Morte, A. X. El-Khadra, D. Giusti, Steven Gottlieb, V. Guelpers, G. Herdoiza, T. Izubuchi, C. Lehner, L. Lellouch, M. K. Marinkovic, A. S. Meyer, K. Miura, A. Portelli, S. Simula, R. Van de Water, G. von Hippel, H. Wittig***Section 4: Data-driven and dispersive approach to HLbL**

*J. Bijnens, G. Colangelo, F. Curciarello, H. Czyz, I. Danilkin, F. Hagelstein, M. Hoferichter, B. Kubis, A. Kupsc, A. Nyffeler, V. Pascalutsa, E. Perez del Rio, M. Procura, C. F. Redmer, P. Sanchez-Puertas, P. Stoffer, M. Vanderhaeghen***Section 5: Lattice approaches to HLbL**

*N. Asmussen, T. Blum, A. Gerardin, M. Hayakawa, R. J. Hudspith, T. Izubuchi, L. Jin, C. Lehner, H. B. Meyer, A. Nyffeler***Section 6: The QED contributions to $a_\mu$**

*T. Aoyama, T. Kinoshita, M. Nio***Section 7: The electroweak contributions to $a_\mu$**

*D. Stoeckinger, H. Stoeckinger-Kim*

## Editorial Board

The writing of the White Paper by the section authors was coordinated by the editorial board, which also performed the final assembly into one document:

- Gilberto Colangelo (University of Bern)
- Michel Davier (University of Paris-Saclay and CNRS, Orsay)
- Simon Eidelman (BINP, Novosibirsk)
- Aida X. El-Khadra (University of Illinois)
- Martin Hoferichter (University of Bern)
- Christoph Lehner (University of Regensburg and Brookhaven National Lab)
- Tsutomu Mibe (KEK)
- Andreas Nyffeler (University of Mainz)
- Lee Roberts (Boston University)
- Thomas Teubner (University of Liverpool)

## Summary Table

Summary of the contributions to $a_\mu$. After the experimental
number from E821, the first block gives the main results for
the hadronic contributions from Secs. 2-5, as well as the combined result for
HLbL scattering from phenomenology and lattice QCD constructed
in Sec. 8.
The second block summarizes the quantities entering our recommended SM value, in particular, the total HVP contribution, evaluated from $e^+e^-$ data, and the total HLbL number. The construction of the total HVP and HLbL contributions takes into account correlations among the terms at different orders, and the final rounding includes subleading digits at intermediate stages.
In the Citation Refs. column the references upon which the
corresponding results are based are listed as downloadable
bib files together with the citation commands.

The HVP evaluation is mainly based on the experimental
references: bib,
cite.

In addition, the HLbL evaluation uses input from
the experimental references: bib, cite.

The lattice QCD calculation of the HLbL contribution builds
on crucial methodological advances from the referneces:
bib,
cite.

Finally,
the QED value uses the fine-structure constant obtained from
atom-interferometry measurements of the Cs atom:
bib,
cite

Contribution | Section | Equation | Value x 10^{11} |
Citation Refs |
---|---|---|---|---|

Experimental average (E821+E989) | [updated] | 116 592 061(41) | bib, cite | |

HVP LO ($e^+e^-$) | Sec. 2.3.7 | Eq. (2.33) | 6931(40) | bib, cite |

HVP NLO ($e^+e^-$) | Sec. 2.3.8 | Eq. (2.34) | -98.3(7) | bib, cite |

HVP NNLO ($e^+e^-$) | Sec. 2.3.8 | Eq. (2.35) | 12.4(1) | bib, cite |

HVP LO (lattice, udsc) |
Sec. 3.5.1 | Eq. (3.49) | 7116(184) | bib, cite |

HLbL (phenomenology) | Sec. 4.9.4 | Eq. (4.92) | 92(19) | bib, cite |

HLbL NLO (phenomenology) | Sec. 4.8 | Eq. (4.91) | 2(1) | bib, cite |

HLbL (lattice, uds) |
Sec. 5.7 | Eq. (5.49) | 79(35) | bib, cite |

HLbL (phenomenology + lattice) | Sec. 8 | Eq. (8.10) | 90(17) | bib, cite |

QED | Sec. 6.5 | Eq. (6.30) | 116 584 718.931(104) | bib, cite |

Electroweak | Sec. 7.4 | Eq. (7.16) | 153.6(1.0) | bib, cite |

HVP ($e^+e^-$, LO + NLO + NNLO) | Sec. 8 | Eq. (8.5) | 6845(40) | bib, cite |

HLbL (phenomenology + lattice + NLO) | Sec. 8 | Eq. (8.11) | 92(18) | bib, cite |

Total SM Value | Sec. 8 | Eq. (8.12) | 116 591 810(43) | bib, cite |

Difference: $\Delta a_\mu\colon= a_\mu^{\text{exp}}-a_\mu^{\text{SM}}$ | Sec. 8 | [updated] | 251(59) |