# Dispersion relation for hadronic light-by-light scattering: two-pion contributions

## Abstract

In our third paper of a series dedicated to a dispersive treatment of the hadronic light-by-light (HLbL) tensor, we derive a partial-wave formulation for two-pion intermediate states in the HLbL contribution to the anomalous magnetic moment of the muon (g - 2) _{μ}, including a detailed discussion of the unitarity relation for arbitrary partial waves. We show that obtaining a final expression free from unphysical helicity partial waves is a subtle issue, which we thoroughly clarify. As a by-product, we obtain a set of sum rules that could be used to constrain future calculations of γ*γ* → ππ. We validate the formalism extensively using the pion-box contribution, defined by two-pion intermediate states with a pion-pole left-hand cut, and demonstrate how the full known result is reproduced when resumming the partial waves. Using dispersive fits to high-statistics data for the pion vector form factor, we provide an evaluation of the full pion box, a$$π-box\atop{μ}$$ =-15.9(2) × 10 ^{-11}. As an application of the partial-wave formalism, we present a first calculation of ππ-rescattering effects in HLbL scattering, with γ*γ* → ππ helicity partial waves constructed dispersively using ππ phase shifts derived from the inverse-amplitude method. In this way, the isospin-0 part of our calculation can be interpreted as the contribution of the f0(500) to HLbL scattering in (g - 2) _{μ}. We also argue that the contribution due to charged-pion rescattering implements corrections related to the corresponding pion polarizability and show that these are moderate. Our final result for the sum of pion-box contribution and its S-wave rescattering corrections reads a$$π-box\atop{μ}$$ + a$$ππ, π-pole LHC\atop{μ, J=0}$$ = -24(1) × 10 ^{-11}.

- Authors:

- Univ. of Bern (Switzerland). Inst. for Theoretical Physics
- Univ. of Washington, Seattle, WA (United States). Inst. for Nuclear Theory; Univ. of California, Santa Barbara, CA (United States). Kavli Inst. for Theoretical Physics
- European Organization for Nuclear Research (CERN), Geneva (Switzerland). Theoretical Physics Dept.
- Univ. of Bonn (Germany). Helmholtz Inst. for Radiation and Nuclear Physics, Bethe Center for Theoretical Physics; Univ. of California, San Diego, CA (United States). Dept. of Physics

- Publication Date:

- Research Org.:
- Univ. of California, San Diego, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1393483

- Grant/Contract Number:
- SC0009919

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2017; Journal Issue: 4; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Chiral Lagrangians; Effective Field Theories; Nonperturbative Effects; Precision QED

### Citation Formats

```
Colangelo, Gilberto, Hoferichter, Martin, Procura, Massimiliano, and Stoffer, Peter.
```*Dispersion relation for hadronic light-by-light scattering: two-pion contributions*. United States: N. p., 2017.
Web. doi:10.1007/JHEP04(2017)161.

```
Colangelo, Gilberto, Hoferichter, Martin, Procura, Massimiliano, & Stoffer, Peter.
```*Dispersion relation for hadronic light-by-light scattering: two-pion contributions*. United States. doi:10.1007/JHEP04(2017)161.

```
Colangelo, Gilberto, Hoferichter, Martin, Procura, Massimiliano, and Stoffer, Peter. Thu .
"Dispersion relation for hadronic light-by-light scattering: two-pion contributions". United States.
doi:10.1007/JHEP04(2017)161. https://www.osti.gov/servlets/purl/1393483.
```

```
@article{osti_1393483,
```

title = {Dispersion relation for hadronic light-by-light scattering: two-pion contributions},

author = {Colangelo, Gilberto and Hoferichter, Martin and Procura, Massimiliano and Stoffer, Peter},

abstractNote = {In our third paper of a series dedicated to a dispersive treatment of the hadronic light-by-light (HLbL) tensor, we derive a partial-wave formulation for two-pion intermediate states in the HLbL contribution to the anomalous magnetic moment of the muon (g - 2)μ, including a detailed discussion of the unitarity relation for arbitrary partial waves. We show that obtaining a final expression free from unphysical helicity partial waves is a subtle issue, which we thoroughly clarify. As a by-product, we obtain a set of sum rules that could be used to constrain future calculations of γ*γ* → ππ. We validate the formalism extensively using the pion-box contribution, defined by two-pion intermediate states with a pion-pole left-hand cut, and demonstrate how the full known result is reproduced when resumming the partial waves. Using dispersive fits to high-statistics data for the pion vector form factor, we provide an evaluation of the full pion box, a$π-box\atop{μ}$ =-15.9(2) × 10-11. As an application of the partial-wave formalism, we present a first calculation of ππ-rescattering effects in HLbL scattering, with γ*γ* → ππ helicity partial waves constructed dispersively using ππ phase shifts derived from the inverse-amplitude method. In this way, the isospin-0 part of our calculation can be interpreted as the contribution of the f0(500) to HLbL scattering in (g - 2)μ. We also argue that the contribution due to charged-pion rescattering implements corrections related to the corresponding pion polarizability and show that these are moderate. Our final result for the sum of pion-box contribution and its S-wave rescattering corrections reads a$π-box\atop{μ}$ + a$ππ, π-pole LHC\atop{μ, J=0}$ = -24(1) × 10-11.},

doi = {10.1007/JHEP04(2017)161},

journal = {Journal of High Energy Physics (Online)},

number = 4,

volume = 2017,

place = {United States},

year = {Thu Apr 27 00:00:00 EDT 2017},

month = {Thu Apr 27 00:00:00 EDT 2017}

}

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