# Dispersive analysis of the pion transition form factor

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## Abstract

We analyze the pion transition form factor using dispersion theory. We calculate the singly-virtual form factor in the time-like region based on data for the \(e^+e^-\rightarrow 3\pi \) cross section, generalizing previous studies on \(\omega ,\phi \rightarrow 3\pi \) decays and \(\gamma \pi \rightarrow \pi \pi \) scattering, and verify our result by comparing to \(e^+e^-\rightarrow \pi ^0\gamma \) data. We perform the analytic continuation to the space-like region, predicting the poorly-constrained space-like transition form factor below \(1\,\text {GeV}\), and extract the slope of the form factor at vanishing momentum transfer \(a_\pi =(30.7\pm 0.6)\times 10^{-3}\). We derive the dispersive formalism necessary for the extension of these results to the doubly-virtual case, as required for the pion-pole contribution to hadronic light-by-light scattering in the anomalous magnetic moment of the muon.

## Keywords

Form Factor Transition Form Factor Dalitz Plot Partial Decay Width Chiral Anomaly## 1 Introduction

One of the biggest challenges of contemporary particle physics is the unambiguous identification of signs of beyond-the-standard-model physics. While high-energy experiments are mainly devoted to the search for new particles, high-statistics low-energy experiments can provide such a high precision that standard-model predictions can be seriously scrutinized. A particularly promising candidate for such an enterprise is the gyro-magnetic ratio of the muon; for a review see [1]. Since the muon is an elementary spin-1/2 fermion, the decisive quantity is the deviation of its gyro-magnetic ratio \(g\) from its classical value. This difference, caused by quantum effects, is denoted by \((g-2)_\mu \).

From the theory side the potential to isolate effects of physics beyond the standard model is limited by the accuracy of the standard-model prediction. Typically the limiting factor is our incomplete understanding of the non-perturbative sector of the standard model, i.e. the low-energy sector of the strong interaction, which is governed by hadrons as the relevant degrees of freedom instead of the elementary quarks and gluons. In fact, for \((g-2)_\mu \) the hadronic contributions by far dominate the uncertainties for the standard-model prediction. The largest hadronic contribution, hadronic vacuum polarization (HVP), enters at order \(\alpha ^2\) in the fine-structure constant \(\alpha =e^2/(4\pi )\) and can be directly related to *one* observable quantity, the cross section of the reaction \(e^+ e^- \rightarrow \,\)hadrons, by means of dispersion theory. In that way a reliable error estimate of HVP emerges from the knowledge of the experimental uncertainties in the measured cross section. At order \(\alpha ^3\) there are next-to-leading-order iterations of HVP as well as a new topology, hadronic light-by-light scattering (HLbL) [2]. It was recently shown in [3] that even next-to-next-to-leading-order iterations of HVP are not negligible at the level of accuracy required for the next round of \((g-2)_\mu \) experiments planned at FNAL [4] and J-PARC [5], while an estimate of next-to-leading-order HLbL scattering indicated a larger suppression [6].

With the increasing accuracy of the cross-section measurement for \(e^+ e^- \rightarrow \,\)hadrons that can be expected in the near future [7], the largest uncertainty for \((g-2)_\mu \) will then reside in the HLbL contribution. The key quantity here is the coupling of two (real or virtual) photons to any hadronic single- or many-body state. This quantity is not directly related to a single observable. However, it is conceivable to build up the hadronic states starting with the ones most dominant at low energies, in particular the light one- and two-body intermediate states. Based on a dispersive description of the HLbL tensor an initiative has recently been started to relate the one- and two-pion contributions for HLbL scattering to observable quantities [8, 9, 10].^{1}

For the dispersive treatment of the HLbL contribution to \((g-2)_\mu \) as envisaged in [8, 9, 10] one needs the pion transition form factor for arbitrary space-like virtualities \(q_1^2\) and \(q_2^2\) of the two photons. We will approach this aim in a multi-step process. In the present work we will formulate the dispersive framework for the general doubly-virtual transition form factor, but restrict the numerical analysis to the singly-virtual case, both in the space- and time-like regions. We will use data on \(e^+e^-\rightarrow 3\pi \) to fix the parameters and predict the cross section for \(e^+e^-\rightarrow \pi ^0\gamma \) as well as the space-like transition form factor to demonstrate the viability of the approach. While presently low-energy space-like data are scarce [31, 32], new high-statistics data can be expected in the near future from BESIII (see [33, 34]), which makes a calculation of the space-like singly-virtual form factor particularly timely. In a second step, the experimental information from \(e^+e^-\rightarrow \pi ^0\gamma \) both in space- and time-like kinematics will then serve as additional input for a full analysis of the doubly-virtual form factor.

The rest of the paper is organized as follows: in Sect. 2 we describe our framework for the determination of the \(\gamma ^*\rightarrow 3\pi \) amplitude. In Sect. 3 we formulate the general dispersion relation for the pion transition form factor with arbitrary virtualities for the two photons. In Sect. 4 we specialize the general framework to the case of one on-shell and one time-like photon. As a first application we will determine the cross section of the reaction \(e^+e^-\rightarrow \pi ^0\gamma \) and compare to the corresponding experimental results. Section 5 is devoted to the analytic continuation into the space-like region as well as the calculation of the slope of the form factor at zero momentum transfer. The Dalitz decay region is discussed in Sect. 6. We close with a summary and outlook in Sect. 7. An appendix is added to discuss the comparison of our results to the simple vector-meson-dominance picture.

## 2 The \(\gamma ^*\rightarrow 3\pi \) amplitude

### 2.1 Formalism

In contrast to this high accuracy the extractions of \(F_{3\pi }\) both from Primakoff measurements [53] (with chiral and radiative corrections from [54, 55, 56]) and \(\pi ^- e^-\rightarrow \pi ^- e^- \pi ^0\) [57] presently allow a test at the \(10\,\%\) level only. In [42] a dispersive framework (see also [55, 58, 59] for earlier work in this direction) was presented that provides a two-parameter description of the \(\pi ^-\gamma \rightarrow \pi ^-\pi ^0\) cross section valid up to \(1\,\text {GeV}\). This opens the possibility to profit from the high-statistics Primakoff data currently analyzed at COMPASS [60] concerning the extraction of \(F_{3\pi }\) to higher accuracy.

For fixed \(q^2\), the quantity \({\mathcal {F}}(s,q^2)\), given in (9), only has a right-hand cut starting at \(s=4M_\pi ^2\). The left-hand cut of the partial wave \(f_1(s,q^2)\) entirely resides in \(\hat{\mathcal {F}}(s,q^2)\). Furthermore, the amplitude develops a three-pion cut for \(q^2 > 9M_\pi ^2\), i.e. in kinematics allowing for the physical decay \(\gamma ^* \rightarrow 3\pi \). In this situation, the right- and left-hand cuts in \(s\) begin to overlap, which leads to a significant complication of the analytic structure, see the corresponding discussion in [61].

An important property of (13) concerns its linearity in the subtraction function \(a(q^2)\), which follows from the fact that \(\hat{\mathcal {F}}\) is defined in terms of the angular average of \({\mathcal {F}}\) itself (9). In this way, \(a(q^2)\) takes the role of a normalization, so that in practice (9) and (13) are solved by iteration for \(a(q^2)\rightarrow 1\), while the full solution is recovered by multiplying with \(a(q^2)\) in the end. However, since \(t\) as a function of \(s\) implicitly depends on \(q^2\), the subtraction function is not the only source of \(q^2\) dependence in the full solution.

^{2}In this case the respective subtraction constant \(a\) is fixed by the overall normalization of the Dalitz plot distribution and hence the corresponding partial decay width. The main complication when extending (13) to arbitrary virtualities \(q^2\) of the incoming photon arises from the fact that \(a\) depends on \(q^2\), a dependence that cannot be predicted within the dispersive framework itself, but has to be determined by different methods. Physically, \(a(q^2)\) contains the information how the isoscalar photon couples to hadrons. At low energies, this coupling is dominated by the three-pion state and can be accessed in \(e^+e^-\rightarrow 3\pi \). For the extraction of \(a(q^2)\) we need a representation that preserves analyticity and accounts for the phenomenological finding that the three-pion state is strongly correlated to the very narrow \(\omega \) and \(\phi \) resonances. We take

### 2.2 Fits to \(e^+e^-\rightarrow 3\pi \)

Before turning to the fit results, we first summarize the various uncertainty estimates that we have performed in the context of our fits to \(e^+e^-\rightarrow 3\pi \). First of all, in the calculation of \({\mathcal {F}}(s,q^2)\) we used three different \(\pi \pi \) phase shifts, the phases from [67, 68] and a version of [67] that includes the \(\rho '(1450)\) and the \(\rho ''(1700)\) resonances in an elastic approximation to try to mimic the possible impact of \(4\pi \) inelasticities [61]. In addition, we varied the cutoff \(\varLambda _{3\pi }\) in the dispersive integral (13) above which asymptotic behavior is assumed between \(1.8\) and \(2.5\,\text {GeV}\), see [41].

The prime source of \(e^+e^-\rightarrow 3\pi \) data below/above \(1.4\,\text {GeV}\) are the SND [69, 70] and CMD2 [71, 72]/the BaBar data sets [73], respectively. Restricting the fit (without \(\omega '\) and \(\omega ''\)) to the energy region below \(1.1\,\text {GeV}\), we observed that the SND data set can be described with a reduced \(\chi ^2\) close to \(1\), while the CMD2 scans can only be accommodated with a significantly worse \(\chi ^2\) (around \(2.4\)). We also checked if the respective fit reproduced the correct chiral anomaly by including \(\alpha \) in (15) as another fit parameter. For SND we indeed obtain \(\alpha =(1.5\pm 0.2)\alpha _{3\pi }\), while the fit to CMD2 even produces a negative value of \(\alpha \).

One explanation for this apparent tension could be provided by the fact that radiative corrections were not treated in exactly the same way in both experiments. Moreover, the CMD2 scans were restricted to a relatively narrow region around the \(\omega \) and \(\phi \) masses, limiting the sensitivity to the low-energy region (and thus particularly to the chiral anomaly). Such inconsistencies in the \(3\pi \) data base were already observed in [74] in the context of the HVP contribution to \((g-2)_\mu \), where the \(3\pi \) channel entered with a global reduced \(\chi ^2\) of \(3.0\). For the present study we will therefore consider two data sets: first, SND+BaBar and, second, the compilation from [74], in the following denoted by HLMNT. It includes all data sets mentioned so far as well as some older experiments [75, 76, 77, 78, 79, 80]. The rationale for doing so is that for the reasons explained above SND/BaBar appear to be the most comprehensive single data sets for low/high energies. Confronting the outcome of fits to the combination of both and to the comprehensive data compilation of [74] should allow for a reasonable estimate of the impact of the uncertainties in the \(e^+e^-\rightarrow 3\pi \) cross section on the prediction for the pion transition form factor.

\(\beta \ [\text {GeV}^{-5}]\) | \(c_\omega \ [\text {GeV}^{-1}]\) | \(c_\phi \ [\text {GeV}^{-1}]\) | \(c_{\omega '} \ [\text {GeV}^{-1}]\) | \(c_{\omega ''} \ [\text {GeV}^{-1}]\) | \(\chi ^2/\text {dof}\) | |
---|---|---|---|---|---|---|

SND+BaBar, \(1.1\,\text {GeV}\) | \(5.94\ldots 6.21\) | \(2.88\ldots 2.90\) | \(-(0.392\ldots 0.406)\) | – | – | \(1.01\ldots 1.04\) |

HLMNT, \(1.1\,\text {GeV}\) | \(5.92\ldots 6.18\) | \(2.81\ldots 2.83\) | \(-(0.374\ldots 0.387)\) | – | – | \(6.33\ldots 6.36\) |

SND+BaBar, \(1.8\,\text {GeV}\) | \(7.73\ldots 7.78\) | \(2.92\ldots 2.95\) | \(-(0.386\ldots 0.400)\) | \(-(0.27\ldots 0.43)\) | \(-(0.70\ldots 1.22)\) | \(3.18\ldots 3.48\) |

HLMNT, \(1.8\,\text {GeV}\) | \(7.78\ldots 7.82\) | \(2.88\ldots 2.90\) | \(-(0.366\ldots 0.378)\) | \(-(0.19\ldots 0.32)\) | \(-(0.53\ldots 1.02)\) | \(7.28\ldots 7.62\) |

## 3 Dispersion relations for the doubly-virtual \(\pi ^0\) transition form factor

## 4 Time-like form factor and \(e^+e^-\rightarrow \pi ^0\gamma \)

Reduced \(\chi ^2\) and \(\tilde{\chi }^2\) for the comparison of our result to the \(e^+e^-\rightarrow \pi ^0\gamma \) data of SND [85, 86] and CMD2 [87] as well as the combined data set. In each case, the upper line refers to the fit with \(\omega \) and \(\phi \) only, the lower line to the fit including \(\omega '\), \(\omega ''\). \(\chi ^2\) and \(\tilde{\chi }^2\) are calculated for all data points below \(1.1\,\text {GeV}\) (upper line) and \(1.4\,\text {GeV}\) (lower line), respectively

SND | CMD2 | SND+CMD2 | |
---|---|---|---|

\(\chi ^2/\text {dof}\) | \(1.74\) | \(4.50\) | \(3.12\) |

\(1.05\) | \(2.37\) | \(1.71\) | |

\(\tilde{\chi }^2/\text {dof}\) | \(0.71\) | \(1.42\) | \(1.06\) |

\(0.56\) | \(1.02\) | \(0.79\) |

Our result for the \(e^+e^-\rightarrow \pi ^0\gamma \) cross section is shown in Fig. 4. We repeat the calculation for each set of \(\pi \pi \) phase shifts and \(\varLambda _{3\pi }\), fitting the isoscalar part in each case both to SND+BaBar and HLMNT. The error band in Fig. 4 represents the uncertainty deduced from scanning over the input quantities in this way. Within uncertainties, the outcome agrees perfectly with the \(e^+e^-\rightarrow \pi ^0\gamma \) cross section measured by [85, 86, 87]. We would like to stress that this result is a prediction solely based on the input quantities described above, most prominently, \(e^+e^-\rightarrow 3\pi \) cross-section data, the \(\pi \pi \) \(P\)-wave phase shift, the pion vector form factor, and the low-energy theorems for \(F_{3\pi }\) and \(F_{\pi \gamma \gamma }\).

## 5 Slope parameter and space-like form factor

Slope parameter and chiral anomaly from the sum rules (38) and (39). For each fit and data set the upper line refers to the slope in units of \(10^{-3}\), while the lower line gives the sum-rule value for \(F_{\pi \gamma \gamma }\) normalized to (3). The ranges correspond to the uncertainty due to the \(\pi \pi \) phase shift and \(\varLambda _{3\pi }\)

SND+BaBar | HLMNT | |
---|---|---|

Fit below \(1.1\,\text {GeV}\) | \(30.4\ldots 31.2\) | \(30.1\ldots 30.9\) |

\(\varLambda _{\pi ^0}=1.1\,\text {GeV}\) | \(0.989\ldots 1.021\) | \(0.976\ldots 1.008\) |

Fit below \(1.8\,\text {GeV}\) | \(30.6\ldots 31.4\) | \(30.4\ldots 31.2\) |

\(\varLambda _{\pi ^0}=1.1\,\text {GeV}\) | \(0.992\ldots 1.026\) | \(0.985\ldots 1.019\) |

Fit below \(1.8\,\text {GeV}\) | \(30.4\ldots 31.2\) | \(30.3\ldots 31.1\) |

\(\varLambda _{\pi ^0}=1.4\,\text {GeV}\) | \(0.959\ldots 0.987\) | \(0.962\ldots 0.990\) |

Fit below \(1.8\,\text {GeV}\) | \(30.3\ldots 31.1\) | \(30.2\ldots 31.0\) |

\(\varLambda _{\pi ^0}=1.8\,\text {GeV}\) | \(0.944\ldots 0.966\) | \(0.947\ldots 0.970\) |

As expected, our prediction for the space-like form factor is very accurate at low energies (better than \(5\,\%\) for \(Q^2\le (1.1\,\text {GeV})^2\)), while the uncertainties become more sizable above \(1\,\text {GeV}\), reflecting the limited energy range used as input for the time-like calculation. The corresponding error band shown in Fig. 5 comprises the same uncertainty estimates already discussed in the context of the slope parameter (the energy region \(Q^2\ge (1.1\,\text {GeV})^2\), which is not reliably described any more in the time-like region, is indicated by the dashed lines in Fig. 5). At low energies the error band is dominated by the variation in the \(\pi \pi \) phase shift and \(\varLambda _{3\pi }\),^{3} whereas above \(1\,\text {GeV}\) the treatment of the high-energy region in the dispersive integral becomes increasingly important. The resulting curve is consistent with the existing data base, and will soon be tested by the forthcoming high-statistics low-energy data from BESIII.

## 6 Dalitz decay region \(\pi ^0 \rightarrow e^+ e^- \gamma \)

## 7 Summary and outlook

We presented the dispersive formalism to analyze the general doubly-virtual pion transition form factor. This includes all effects from elastic \(\pi \pi \) rescattering exactly through the respective phase shifts. To determine the isoscalar part that is dominated by \(3\pi \) intermediate states, we used data on \(e^+e^-\rightarrow 3\pi \). Furthermore, chiral low-energy theorems on the anomalies \(F_{3\pi }\) and \(F_{\pi \gamma \gamma }\) were implemented. As a first step, we carried out the phenomenological analysis of the singly-virtual case. We performed a detailed error analysis and verified our calculation in the time-like region by comparing to data for \(e^+e^-\rightarrow \pi ^0\gamma \), yielding very good agreement between theory and experiment. As further applications of the framework, we provided a precise value for the slope parameter, \(a_\pi =(30.7\pm 0.6)\times 10^{-3}\), as well as for the curvature term, \(b_\pi =(1.10\pm 0.02)\times 10^{-3}\). Finally, analytic continuation allowed for a prediction for the transition form factor in the low-energy space-like region that should be compared to the upcoming precise BESIII data.

To extend the calculation to higher energies requires additional input. One could for instance match to the predictions of quark counting rules [81], Regge theory [35], or light-cone sum rules [91, 92]. In the time-like region, with consistency between \(e^+e^-\rightarrow 3\pi \) and \(e^+e^-\rightarrow \pi ^0\gamma \) demonstrated, one could also fit simultaneously to both reactions to potentially decrease the uncertainties. The most important future extension will concern the generalization to the doubly-virtual case. This can be applied to predict the leptonic neutral pion decay \(\pi ^0\rightarrow e^+e^-\), but most importantly, will help pin down the pion-pole contribution to hadronic light-by-light scattering in \((g-2)_\mu \). Work in this direction is in progress.

## Footnotes

- 1.
- 2.
For a variant of this calculation see [63].

- 3.
At very low energies corrections to the low-energy theorem (3) will become relevant, since the transition form factor is normalized to \(F_{\pi \gamma \gamma }\). The corresponding uncertainties are not included in Fig. 5, but due to (37) can simply be recovered by adding a term \(Q^2\Delta F_{\pi \gamma \gamma }/e^2\).

## Notes

### Acknowledgments

We would like to thank Simon Eidelman, Denis Epifanov, Evgueni Goudzovski, Tord Johansson, Bastian Knippschild, Andrzej Kupść, Christoph Redmer, and Thomas Teubner for helpful discussions and correspondence, and Thomas Teubner for making the \(e^+e^-\rightarrow 3\pi \) data compilation from [74] available to us. Financial support by BMBF ARCHES, the Helmholtz Alliance HA216/EMMI, the Swiss National Science Foundation, the DFG (SFB/TR 16, “Subnuclear Structure of Matter”), by DFG and the NSFC through funds provided to the Sino-German CRC 110 “Symmetries and the Emergence of Structure in QCD,” and by the project “Study of Strongly Interacting Matter” (HadronPhysics3, Grant Agreement No. 283286) under the 7th Framework Program of the EU is gratefully acknowledged.

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