Anomalous decay and scattering processes of the \(\eta \) meson
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Abstract
We amend a recent dispersive analysis of the anomalous \(\eta \) decay process \(\eta \rightarrow \pi ^+\pi ^\gamma \) by the effects of the \(a_2\) tensor meson, the lowestlying resonance that can contribute in the \(\pi \eta \) system. While the net effects on the measured decay spectrum are small, they may be more pronounced for the analogous \(\eta '\) decay. There are nonnegligible consequences for the \(\eta \) transition form factor, which is an important quantity for the hadronic lightbylight scattering contribution to the muon’s anomalous magnetic moment. We predict total and differential cross sections, as well as a marked forward–backward asymmetry, for the crossed process \(\gamma \pi ^\rightarrow \pi ^\eta \), which could be measured in Primakoff reactions in the future.
1 Introduction
The present article is built on the following observation. If we continue the amplitude (2) naively to negative t, we ought to observe a zero at or near \(t =  1/\alpha \approx 0.76\,\text {GeV}^2\). Such a kinematical regime is indeed accessible: in the crossed reaction \(\gamma \pi ^ \rightarrow \pi ^\eta \), which could be measured in a Primakofftype reaction, i.e., the scattering of a charged pion in the strong Coulomb field of a heavy nucleus, producing an additional \(\eta \). Such a Primakoff program is currently pursued by the COMPASS Collaboration (see, e.g., Ref. [11] for an overview), using a \(190\,\text {GeV}\) \(\pi ^\) beam and cutting on very small momentum transfers in order to isolate the photonexchange mechanism from diffractive background. In this way, COMPASS can investigate \(\gamma \pi ^\) reactions to various final states, in particular Compton scattering in order to extract the chargedpion polarizabilities [12, 13], \(\pi ^\pi ^0\) to investigate the chiral anomaly [14, 15], or three pions testing chiral predictions [16, 17]. In this paper, we want to provide the theoretical motivation to also measure the final state \(\pi ^\eta \), as well as a prediction for the cross sections that are to be expected.

its inclusion will demonstrate to what extent the feature expected from Eq. (2), a zero (or at least a pronounced minimum) in certain differential cross sections, can survive in a more complete description of the amplitude;

it will provide a characteristic breakdown scale in the \(\pi \eta \) invariant mass squared \(s = M_{\pi \eta }^2\), above which \(\pi \eta \) resonances dominate the cross section;

finally, we can use the \(a_2\) as the likely most important lefthandcut structure for the decay \(\eta \rightarrow \pi ^+\pi ^\gamma \), to study to what extent it affects the decay amplitude, and whether its effect is consistent with the experimental decay data available.
2 \(\eta \rightarrow \pi \pi \gamma \) with lefthand cuts
2.1 Amplitude, kinematics
2.2 Treelevel contribution of the \(a_2(1320)\)
2.3 Unitarization
Obviously, the \(a_2\) s and uchannel exchanges will also generate nonvanishing projections onto F and higher tchannel partial waves. These partial waves are real as long as we neglect pion–pion rescattering effects in those higher waves, which is entirely justified for \(\eta \rightarrow \pi ^+\pi ^\gamma \) (and even for \(\eta '\rightarrow \pi ^+\pi ^\gamma \)), given the smallness of the corresponding phases; compare the discussion in Ref. [32]. However, even the real part of the Fwave is entirely negligible: while in the chiral power counting, it is suppressed compared to the Pwave by another power of \(p^2/m_{a_2}^2\), we have checked that kinematical prefactors effectively suppress it by more than 3 orders of magnitude in the physical decay region of \(\eta \rightarrow \pi ^+\pi ^\gamma \), and still by 2 for \(\eta '\rightarrow \pi ^+\pi ^\gamma \). We will therefore discuss the comparison to decay data in the following section still in the approximation indicated in Eq. (9), using the Pwave only.
3 Comparison to decay data
3.1 \(\eta \rightarrow \pi ^+\pi ^\gamma \) decay spectrum
In this section, we compare the amplitude constructed in the previous section to the data on \(\text {d}\Gamma /\text {d}t\) as obtained by the KLOE Collaboration [10]. The decay distribution was measured with arbitrary normalization, which has to be fixed independently from the branching fraction \(\mathcal {B}(\eta \rightarrow \pi ^+\pi ^\gamma ) = (4.22\pm 0.08)\,\%\), as well as the total width of the \(\eta \) [26].
3.2 Impact on the \(\eta \) transition form factor
As far as the phenomenological description of the \(\eta \rightarrow \pi ^+\pi ^\gamma \) decay data of Ref. [10] is concerned, the two amplitudes, with and without \(a_2\) effects included, are clearly equivalent: they describe the data equally well, and in fact, the two fit curves displayed in Fig. 4 deviate from each other by less than \(1~\%\) in the whole decay region. This is different in the wider kinematic range of the similar decay \(\eta '\rightarrow \pi ^+\pi ^\gamma \), which we discuss in Appendix A. While the available data do not yet allow one to prefer one amplitude over the other in a statistically valid sense, the comparison of the extracted subtraction constants \(\alpha _\Omega \) and an \(\alpha '_\Omega \) defined in an analogous manner for \(\eta '\rightarrow \pi ^+\pi ^\gamma \) seems to favor somewhat the decay amplitude including the curvature effects induced by the \(a_2\).
However, we have emphasized in the introduction that the decay amplitude \(\eta \rightarrow \pi ^+\pi ^\gamma \) serves as a crucial input to a dispersive analysis of the \(\eta \) transition form factor [5], where the dispersion integral extends over a much larger range in energy (in principle, up to infinity). We therefore may expect to see a somewhat more significant deviation between the two amplitudes in there.
4 Phenomenology for \(\gamma \pi \rightarrow \pi \eta \)
For completeness, we also display the cross section with the relative sign of the \(a_2\) contribution, see Eq. (28), flipped (and all other parameters adjusted such as to best reproduce the \(\eta \rightarrow \pi ^+\pi ^\gamma \) decay data); we see that the transition from the nearthreshold to the resonance region looks quite different, for reasons that will become transparent below. The uncertainty in the resonance peak is obviously dominated by those in the \(a_2\) coupling constants \(c_Tg_T\), while near threshold, the errors coming from the total decay rate \(\Gamma (\eta \rightarrow \pi ^+\pi ^\gamma )\) as well as \(\alpha _\Omega \) are more important.
We wish to reemphasize that there is no fixed relation between the phase of our schannel partial waves to \(\pi \eta \) scattering phase shifts according to a finalstate theorem. As the corresponding \(\pi \eta \) phases are not theoretically determined in the way the \(\pi \pi \) [24, 36, 37] or \(\pi K\) [38] phases are, unitarization using model phases seems to offer no significant improvement. Furthermore, the \(a_2\) is a largely inelastic resonance with respect to \(\pi \eta \) scattering anyway, with the dominant decay channel being \(\pi \rho \) [see Eq. (30)], such that no simple version of Watson’s theorem applies, and any unitarization would have to implement a coupledchannel formalism.
5 Summary
In this article, we have studied the effects of the \(a_2\) tensor meson on the decay \(\eta \rightarrow \pi ^+\pi ^\gamma \) as well as the analytic continuation of the decay amplitude for the scattering process \(\gamma \pi ^\rightarrow \pi ^\eta \). We have included the Dwave \(\pi \eta \) resonance as a lefthand cut structure of a dispersive representation that obeys the correct finalstate phase relation for the \(\pi ^+\pi ^\) Pwave. While the decay spectra measured by the KLOE Collaboration can be described equally well with and without the \(a_2\) effects, there seems to be an indication for better consistency of the subtractions constants when comparing to the similar decay \(\eta '\rightarrow \pi ^+\pi ^\gamma \). The slope parameter of the resulting \(\eta \) transition form factor is reduced by about \(7~\%\) in the dispersive integral up to \(1\,\text {GeV}^2\) compared to a previous analysis [5].
We have predicted different observables for the \(\eta \) production reaction \(\gamma \pi ^\rightarrow \pi ^\eta \) at energies up to the \(a_2\) resonance. The peak cross section is predicted to be \((12\pm 2)~\mu \text {b}\), similar in size to the \(\gamma \pi ^\rightarrow \pi ^\pi ^0\) cross section in the \(\rho \) peak [15]. Fixing the relative sign of the \(a_2\) to the more likely solution from decay phenomenology, we find an interesting P–Dwave interference effect, leading to almost perfect zeros in the differential cross section, and a very strong forward–backward asymmetry in the energy region between threshold and the \(a_2\) peak. These predictions provide strong motivation to study the corresponding Primakoff reaction e.g. at COMPASS, which may help to further scrutinize the physics of light mesons relevant for hadronic corrections to the muon’s anomalous magnetic moment.
Footnotes
 1.
In fact, if we construct the Omnès function from the phase of the pion vector form factor instead of from the \(\pi \pi \) Pwave phase shift [24] as in Ref. [33], the central value of \(\alpha _\Omega \) reduces to \(1.37\,\text {GeV}^{2}\), rather close to Eq. (3). We disregard the effects of varying the \(\pi \pi \) phase input in the following: they are compensated by corresponding shifts in \(\alpha _\Omega \) to a very large extent, and they lead to insignificant uncertainties compared to other error sources.
 2.
 3.
As a side remark, we point out that fixing an effective \(a_2\rightarrow \pi \rho \) coupling constant from the known branching fraction \(\mathcal {B}(a_2\rightarrow \pi \rho )\) should also allow us to include \(a_2\) effects in the decays \(\eta '\rightarrow 4\pi \) [41, 42], thus going further beyond vectormeson dominance.
Notes
Acknowledgments
We would like to thank Christoph Hanhart, Martin Hoferichter, and Andreas Wirzba for useful discussions and comments on the manuscript, and Andrzej Kupść for supplying us with the acceptancecorrected data from Ref. [10]. Financial support by the DFG (SFB/TR 16, “Subnuclear Structure of Matter”) and the Bonn–Cologne Graduate School of Physics and Astronomy is gratefully acknowledged.
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