Anomalous decay and scattering processes of the eta meson

We amend a recent dispersive analysis of the anomalous $\eta$ decay process $\eta\to\pi^+\pi^-\gamma$ by the effects of the $a_2$ tensor meson, the lowest-lying 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 light-by-light 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 $\gamma\pi^-\to\pi^-\eta$ that could be measured in Primakoff reactions in the future.


Introduction
The decay η → π + π − γ is one of the processes driven by the chiral anomaly [1,2]. The reduced scalar decay amplitude (to be defined below) in the SU(3) chiral limit and at vanishing momenta is given entirely in terms of the electromagnetic coupling e and the pion decay constant F π , Higher-order corrections to the anomaly can be evaluated in chiral perturbation theory [3], and pion-pion rescattering in the final state resummed effectively using dispersion theory [4]. Besides thus being an interesting decay in its own right to test our understanding of the interaction of light pseudoscalar mesons with photons, this decay is particularly noteworthy as a fundamental ingredient in a dispersive analysis of the η transition form factor η → γγ * [5]. This quantity is a crucial input necessary for the ongoing program to analyze the hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon, combining as many pieces of experimental information as possible in a model-independent fashion [6,7]. A similar analysis has also been pursued for the π 0 transition form factor [8].
As pointed out in Ref. [4], the decays of η and η ′ into π + π − γ pose a beautiful and simple example to demonstrate the universality of final-state interactions. Neglecting (tiny) contributions of F and higher partial waves for the pion pair, the authors show that the reduced decay amplitude can be written as where t = M 2 ππ is the squared invariant mass of the pion pair, F V π (t) denotes the pion vector form factor as measured in e + e − → π + π − , and P (t) is a polynomial. Comparison to experimental data obtained by the WASA-at-COSY [9] and KLOE [10] collaborations demonstrated that within experimental accuracy, the polynomial can be assumed to be linear, P (t) = A(1 + αt), with [10] α = (1.32 ± 0.13) GeV −2 . ( This result gives rise to several interesting questions. Obviously, Eq. (2) is only an approximation, tested successfully in the physical decay region, 4M 2 π ≤ t ≤ M 2 η . The universality of final-state interactions expressed therein is only valid in the region of elastic pion-pion rescattering, which is phenomenologically a good approximation up to roughly t ≈ 1 GeV 2 . From generic considerations about the asymptotic behavior of the decay amplitude, one would rather expect P (t) to become constant for large t, such that the decay amplitude falls like 1/t similar to the asymptotic behavior of F V π (t). The continuation beyond the physical regime is interesting in particular with regard to the application within a dispersive integral to obtain the η transition form factor [5], as in principle that integral covers all energies.
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/α ≈ −0.76 GeV 2 . Such a kinematical regime is indeed accessible: in the crossed reaction γπ − → π − η, which could be measured in a Primakoff-type reaction, i.e., the scattering of a charged pion in the strong Coulomb field of a heavy nucleus, producing an additional η. Such a Primakoff program is currently pursued by the COMPASS collaboration (see, e.g., Ref. [11] for an overview), using a 190 GeV π − beam and cutting on very small momentum transfers in order to isolate the photon-exchange mechanism from diffractive background. In this way, COMPASS can investigate γπ − reactions to various final states, in particular Compton scattering in order to extract the charged-pion polarizabilities [12,13], π − π 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 π − η, as well as a prediction for the cross sections that are to be expected.
For this purpose, beyond using crossing symmetry, we need to amend the amplitude (2) for the following reason. The assumption underlying Eq. (2) is the neglect of so-called left-hand cuts: the two pions undergoing final-state interactions are assumed to originate from a point source, such that the amplitude is of form factor type, and any interaction (resonant or nonresonant) in the πη channel is neglected. This approximation can be justified at low energies by appealing to chiral perturbation theory: the πη P -wave is chirally suppressed (as well as all higher partial waves) [18,19], an imaginary part only appears at three-loop order, any phase shift is therefore expected to be very small. Furthermore, the πη P -wave has exotic quantum numbers J P C = 1 −+ , and the search for possible resonances in this channel is not fully conclusive so far [20,21]. The first well-established resonance that is therefore going to be important in the process γπ → πη is the D-wave tensor meson a 2 (1320). To investigate its influence is important for several reasons: 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 πη invariant mass squared s = M 2 πη , above which πη resonances dominate the cross section; -finally, we can use the a 2 as the likely most important left-hand-cut structure for the decay η → π + π − γ, to study to what extent it affects the decay amplitude, and whether its effect is consistent with the experimental decay data available.
The outline of this article is as follows. In Sect. 2, we recapitulate the dispersive representation of the η → π + π − γ decay amplitude of Ref. [4], before calculating contributions of the a 2 tensor meson first at tree level, then including pion-pion rescattering effects dispersively. Section 3 compares the resulting observables to the measured η → π + π − γ decay spectrum and briefly discusses the possible impact on the η transition form factor. In Sect. 4, we give our predictions for the crossed process γπ − → π − η, discussing total and differential cross sections, the leading partial waves, as well as the resulting pronounced forward-backward asymmetry. We close with a summary. A brief discussion of the related decay η ′ → π + π − γ is relegated to an Appendix.

Amplitude, kinematics
We write the decay amplitude for the process η(q) → π + (p 1 )π − (p 2 )γ(k) (4) in terms of a scalar function F (s, t, u) according to with the Mandelstam variables given as s = (q − p 1 ) 2 , t = (p 1 + p 2 ) 2 , and u = (q − p 2 ) 2 . F (s, t, u) in the chiral limit fulfills the low-energy theorem F (0, 0, 0) = F ηππγ . The cosine of the t-channel center-of-mass angle is given by The t-channel partial-wave expansion is of the form where P ′ l (z t ) denote the derivatives of the standard Legendre polynomials. Due to the strong suppression of F η γ π − π + π + π − P Fig. 1 Graphical illustration of the discontinuity equation (11). The gray circle denotes the t-channel P -wave projection of the η → π + π − γ decay amplitude, whereas the white circle stands for the P -wave pion-pion scattering amplitude. and higher partial waves at low energies, we will almost exclusively be concerned with the P -wave, which is obtained by angular projection according to The differential decay rate with respect to the pionpion invariant mass squared is given by where the ellipsis in the second line represents neglected higher partial waves.
In the absence of left-hand cuts and ignoring inelasticities, the P -wave should obey the following representation [4]: where Ω(t) is the Omnès function [22] given in terms of the pion-pion P -wave phase shift δ(t) ≡ δ 1 1 (t), and P (t) is a polynomial. The representation (10) is a solution to the discontinuity relation as obtained from elastic pion-pion rescattering, see Fig. 1. It obviously fulfills Watson's final-state interaction theorem [23]: the phase of f 1 (t) agrees with the elastic scattering phase δ(t). In the following, we will take δ(t) from the representation given in Ref. [24]. As already pointed out in the introduction, comparison with data [9,10] suggested that the polynomial P (t) is linear in the decay region, Fig. 2 Tree-level contributions of the a 2 (1320) resonance to η → π + π − γ in the s-(left) and u-channel (right).
to very good accuracy. In fact, in Ref. [4], the Omnès function was replaced by the pion vector form factor F V π (t), which is a phenomenologically attractive representation insofar as the latter is itself directly experimentally observable. Both representations are equivalent modulo a moderate shift in the parameter α → α Ω due to the observation that the form factor is in turn proportional to the Omnès function up to a linear polynomial below 1 GeV, with a slope of the order of 0.1 GeV −2 [5].

Tree-level contribution of the a 2 (1320)
We begin by calculating the tree-level contribution of the a 2 tensor meson to the amplitude η → π + π − γ as shown in Fig. 2. For the formalism of coupling tensor mesons to Goldstone bosons, we follow Ref. [25]. The single necessary interaction term required to describe the decay of a tensor meson into two pseudoscalars is given by where . denotes the trace in flavor space. For simplicity, we only display the nonstrange SU(2) part of the tensor field relevant to our calculation explicitly, The Goldstone bosons are encoded in the field From Eq. (13), we can calculate the decay width for a 2 → πη, employing the polarization sum of the a 2 [25] pol ǫ µν (l)ǫ * ρσ (l) = P µν,ρσ (l), and find where λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + ac + bc) denotes the usual Källén function. Equation (17), with the total width Γ a2 = (107 ± 5) MeV and the branching fraction B(a 2 → πη) = (14.5 ± 1.2)% [26], leads to the coupling strength in perfect agreement with the number obtained in Ref. [25] from the decay f 2 → ππ (compare also Ref. [27]), thus confirming SU(3) symmetry in this channel.
The coupling of the a 2 to pion and photon can be deduced from a Lagrangian (compare Refs. [28,29]) where f µν . .) the quark charge matrix, and we have neglected additional currents. Equation (19) leads to the radiative decay width which, compared to B(a 2 → πγ) = (2.68 ± 0.31)× 10 −3 , leads to If we combine the Lagrangians (13) and (19) with the tensor propagator iP µν,ρσ (l)/(m 2 a2 − l 2 ), we can calculate the a 2 -exchange contribution F a2 (s, t, u) to η → π + π − γ. We find which is completely fixed by experimental information up to an overall sign.
A few remarks are in order concerning Eq. (22). First, we can also perform an s-channel partial-wave expansion according to which is the natural partial-wave expansion for γπ − → π − η in terms of the scattering angle θ s . The partialwave expansion of the s-channel a 2 exchange amplitude G(s, t, u) then reads Phrased differently, G(s, t, u) contains a nonresonant Pwave contribution (that has no a 2 propagator) in addition to the expected resonant D-wave. This is a wellknown problem of higher-spin propagators, see e.g. the discussion in Ref. [25]. We cannot easily subtract the P -wave and use the D-wave alone, as Eq. (24) shows that both partial waves individually display an artificial pole ∝ 1/s, which is not present in the full amplitude (22). While a pole at s = 0 is not kinematically accessible in either of the two processes we consider in this article, it precludes a dispersive reconstruction of t-channel rescattering as discussed in the following section. We therefore retain the P -wave part in Eq. (24); its effect turns out to be numerically small. Second, we fix the sign of c T g T in the following way. As pointed out in Ref. [4], the vector-meson contributions to η → π + π − γ determined in Ref. [3] can be rewritten, using the limit of a large number of colors (i.e., neglecting loop effects) and expanding the ρ propagators to leading order in the spirit of resonance saturation of chiral low-energy constants, as where we have used the approximation In other words, Eq. (25) predicts α ρ Ω ≈ 1/(2m 2 ρ ) = 0.84 GeV −2 , a little more than half of the phenomenological value α Ω ≈ 1.52 GeV −2 (when using Eq. (10) for the definition of α Ω and not the pion vector form factor for α). We can now similarly expand Eq. (22) to leading order in inverse powers of m 2 a2 . If we neglect the induced quark mass renormalization of the anomaly (proportional to M 2 π , M 2 η ), we find the following estimate for the a 2 contribution to α Ω : We shall see below that the true effect when including the a 2 in a new extraction of the slope parameter α Ω from data is significantly smaller, mainly due to curvature effects in the induced amplitude. Still, while effects in particular of excited ρ ′ resonances can be nonnegligible, we take the discrepancy between the ρ-induced slope and the experimentally determined value as an indication that the sign of the a 2 contribution ought to be positive, We will work on from this hypothesis, and give further hints below that data indeed suggests this to be the more likely solution.
As a final remark, we will later insert a nonvanishing, energy-dependent width in the a 2 propagator in Eq. (22) by hand, using the parametrization [30] which explicitly takes into account the a 2 decays into final states πη and πρ with relative branching fractions p η = 0.17, p ρ = 0.83, using the barrier factor R = 5.2 GeV −1 . The a 2 is sufficiently far from the πρ "threshold" that it seems a justifiable approximation to treat the ρ as a stable particle in this case. In contrast to using a constant width, this parametrization provides the correct threshold behavior of the imaginary part, as well as a reasonable phase above the resonance.

Unitarization
It is obvious that simply adding the tree-level a 2 contribution (22) to the original amplitude (10)  relation in the t-channel P -wave is of the form G(t) is the projection of the a 2 exchange graphs onto the t-channel P -wave. It is given explicitly bŷ G(t) contains a square-root singularity at t = 0, signaling the onset of the left-hand cut. AsĜ(t) approaches a constant for large arguments t → ∞, two subtractions in Eq. (31) are sufficient, as the Omnès function behaves asymptotically as Ω(t) ∼ 1/t for δ(t) → π. The number of subtractions therefore exactly reflects the original form in Eqs. (10), (12). It is easy to see that the full t-channel P -wave resulting from Eq. (31), has the correct phase, while F (t) alone is subject to the inhomogeneous discontinuity relation The representation (31), using the inhomogeneityĜ(t) as input to the dispersive integral, preserves unitarity in the t-channel in the presence of left-hand cuts, which are approximated by resonance (here: a 2 ) contributions. This is closely related to the methods used e.g. in Ref. [29] for γγ → ππ, or in Ref. [31] for semileptonic B-decays. We cannot easily apply an iterative procedure to determine left-hand cuts from right-hand cuts and vice versa, as done e.g. in the analysis of the closely related Primakoff process γπ → ππ [15], as we do not have independent information on πη scattering phases at our disposal. Obviously, the a 2 s-and u-channel exchanges will also generate nonvanishing projections onto F -and higher t-channel 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 η → π + π − γ (and even for η ′ → π + π − γ), given the smallness of the corresponding phases; compare the discussion in Ref. [32]. However, even the real part of the F -wave is entirely negligible: while in the chiral power counting, it is suppressed compared to the P -wave by another power of p 2 /m 2 a2 , we have checked that kinematical prefactors effectively suppress it by more than 3 orders of magnitude in the physical decay region of η → π + π − γ, and still by 2 for η ′ → π + π − γ. We will therefore discuss the comparison to decay data in the following section still in the approximation indicated in Eq. (9), using the P -wave only.
3 Comparison to decay data 3.1 η → π + π − γ decay spectrum In this section, we compare the amplitude constructed in the previous section to the data on dΓ/dt 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 B(η → π + π − γ) = (4.22 ± 0.08)%, as well as the total width of the η [26].
We first (re)fit the representation (10), (12). We obtain where the error is only due to the statistical uncertainty in the data and neglects all the systematic effects discussed in Ref. [10]. The difference in the central value compared to α in Eq. (3) is due to employing the Omnès function instead of the pion vector form factor, as discussed above. 1 The quality of the fit is excellent, 1 In fact, if we construct the Omnès function from the phase of the pion vector form factor instead of from the ππ P -wave with a χ 2 per degree of freedom of χ 2 /ndof = 0.94. The subtraction constant A that, in this case, serves as an overall normalization of the amplitude, is A = (5.43 ± 0.12 ∓ 0.04) GeV −3 , where the first error is due to the uncertainty in the integrated partial width, and the second due to the uncertainty in α Ω , almost perfectly anticorrelated with the latter. A thus seems well compatible with F ηππγ , see Eq. (1). In Fig. 4, we plot the following observable, obtained from the data from Ref. [10]: i.e. within the accuracy of the amplitude representation without left-hand cuts, we expect to findP (t) = P (t)/P (0) = 1 + α Ω t. As the quality of the fit suggests, the linear curve (blue dashed) describes the data perfectly.
Including the effects of a 2 exchange (properly unitarized in the t-channel), the subtraction constant α Ω in Eq. (31) has to be refitted to the data. We obtain with χ 2 /ndof = 0.90. The uncertainty of the a 2 coupling constants induces an additional error in α Ω of ±0.01 GeV −2 , which we will neglect in the following.
phase shift [24] as in Ref. [33], the central value of α Ω reduces to 1.37 GeV −2 , rather close to Eq. (3). We disregard the effects of varying the ππ phase input in the following: they are compensated by corresponding shifts in α Ω to very large extent, and lead to insignificant uncertainties compared to other error sources.
The resulting fit is also shown in Fig. 4. The reduction in the value of α Ω compared to Eq. (35) may seem surprisingly small, given the estimate of the a 2 contribution to this parameter in Eq. (27). The reason is the curvature inP (t): in fact, the derivativeP ′ (t) (that equals the constant α Ω in the simple fit) varies fromP ′ (4M 2 π ) = 1.69 GeV −2 toP ′ (M 2 η ) = 1.30 GeV −2 within the decay phase space; outside phase space, we find e.g.P ′ (1 GeV 2 ) = 0.46 GeV −2 , and naive continuation to yet higher energies makes the derivative vanish and change sign around √ t = 1.25 GeV. It finally diverges at t = 0 due to the square-root singularity.

Impact on the η transition form factor
As far as the phenomenological description of the η → π + π − γ 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 η ′ → π + π − γ, which we discuss in Appendix A. While the available data do not yet allow to prefer one amplitude over the other in a statistically valid sense, the comparison of the extracted subtraction constants α Ω and an α ′ Ω defined in an analogous manner for η ′ → π + π − γ 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 η → π + π − γ serves as a crucial input to a dispersive analysis of the η 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.
We refer e.g. to Ref. [5] for all pertinent definitions concerning the singly virtual η transition form factor F ηγ * γ (Q 2 , 0), which at small photon virtualities can be expanded according to The slope parameter b η is divided into an isovector I = 1 and an isoscalar I = 0 piece. The isoscalar part is small: employing ω + φ dominance together with data input on ω, φ → ηγ yields b (I=0) η ≈ −0.022 GeV −2 [5]. The slope is therefore almost entirely given by the isovector contribution, which in turn is dominated by π + π − intermediate states; see Fig. 5. The correspond-η γ π + π − γ * Fig. 5 Two-pion cut contribution to the isovector part of the (singly virtual) η transition form factor. Here, the gray circle denotes the t-channel η → π + π − γ P -wave, while the white circle is the pion vector form factor.
ing sum rule can be written as [5] b (I=1) where F V π (t) is the standard pion vector form factor, and we have written the dispersion integral with a cutoff Λ 2 instead of integrating to infinity. The η → γγ amplitude A η γγ is obtained from the corresponding partial width by Following Ref. [5], we vary the cutoff in the range With the decay amplitude (10), (12), we find b (I=1) η = {2.04 . . . 2.22} ± 0.04 α ± 0.02 B ± 0.01 F V π , (41) where the indicated range follows the range of cutoffs, and the errors are due to uncertainties in α Ω , 2 the branching ratios for η → π + π − γ and η → γγ, and the pion vector form factor. For the latter, we employ the pion vector form factor parametrizations of Refs. [33,34] (or approximations thereof). Using however the partial wave f 1 (t) as in Eq. (33), the result reduces to with the additional error due to the uncertainty in the a 2 coupling constants. That is, the slope is reduced by about 7%, a bit more than the combined error cited in Ref. [5], for a cutoff Λ 2 ≈ 1 GeV 2 ; this reduction is increased for higher cutoffs (due to the increasingly stronger curvature effects). A more detailed investigation of a 2 effects on the η (and η ′ ) transition form factor(s), beyond the value of the slope at the origin, should still be pursued. η γ π − π − Fig. 6 Radiative correction to γπ − → π − η due to photonexchange diagram.

Phenomenology for γπ → πη
In the previous section, we have constructed an η → π + π − γ decay amplitude including the leading lefthand-cut contribution, and have shown that this amplitude describes the available decay data very well. As this representation includes the lightest resonance that can contribute in the πη system, we are well-equipped to now consider the crossed process which is described by the same amplitude as the decay process in Sect. 2 withp 1 = −p 1 (using time-reversal invariance). The Mandelstam variables are defined as before, e.g. s = (p 1 + q) 2 denotes the total energy squared in the center-of-mass system, t = (p 2 −p 1 ) 2 is related to the pion momentum transfer etc. In particular, Eq. (23) is the natural partial-wave expansion in scattering kinematics. The (polarization-averaged) differential cross section is given by from which one obtains for the total cross section where we have inserted the s-channel partial-wave expansion (23) up to F -waves in the second step. As a cautionary side remark, we wish to point out that it has been emphasized in Ref. [35] for the similar process γπ − → π − π 0 that there is one significant effect due to radiative corrections, which is due to photon exchange in the t-channel, compare  Total cross section σ(s) for γπ − → π − η. The blue band shows the cross section obtained from crossing the decay amplitude of Ref. [4]; the red band corresponds to the full amplitude including a 2 effects. Finally, the yellow band displays the full cross section for the relative sign of the a 2 contribution flipped. The insert magnifies the near-threshold region. See main text on the error bands.
the process under investigation here, the inclusion of this effect amounts to correcting the scattering amplitude in the form Strictly speaking, the photon-exchange amplitude would have to be amended by form factor effects, including both the η transition as well as the pion vector form factor; however, these corrections were shown to be very small in γπ − → π − π 0 [35]. The inclusion of the correction (46) may be desirable if experimental data on γπ − → π − η become sufficiently precise in the future; we still neglect it for the following investigation. We show the total cross section in Fig. 7. We compare the cross section obtained from the decay amplitude in Ref. [4] by crossing to the full cross section including a 2 effects. We find that dominance of t-channel dynamics holds roughly up to √ s = 1 GeV, while above, the tensor resonance begins to dominate. We predict a peak cross section of about (12 ± 2)µb, which is of a similar order of magnitude as the cross section of γπ − → π − π 0 at the ρ peak [15]. 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 η → π + π − γ decay data); we see that the transition from the near-threshold 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 First three partial waves for γπ − → π − η. The moduli are shown in the normalization 1 2 l(l + 1)|g l (s)| for P -wave (blue bands), D-wave (red), and F -wave (green); bands with dashed borders refer to the analytic continuation of the decay amplitude in Ref. [4], while the full bands show the full result including a 2 effects. The phase of the complete D-wave is represented by the red-striped band. All indicated bands combine the uncertainties in Γ (η → π + π − γ), α Ω , and the a 2 couplings c T g T . c T g T , while near threshold, the errors coming from the total decay rate Γ (η → π + π − γ) as well as α Ω are more important.
In the introduction, we pointed out that a naive continuation of the η → π + π − γ decay amplitude Eqs. (10), (12) would lead to a zero in the scattering amplitude γπ − → π − η at t = −1/α Ω . As s increases, this zero first appears in the differential cross section dΓ/dz s in backward direction, i.e. for z s = −1. Given the form of the partial-wave expansion (23), F (s, t, u) = g 1 (s) + 3z s g 2 (s) + . . . , and assuming F -and higher partial waves are small, this will occur once the D-wave is one third the size of the P -wave, as long as relative phases are small. In our amplitude representation, the only imaginary part stems from the energy-dependent width of s-channel a 2 exchange; the P -wave phase is neglected, and all partial waves induced by t-channel exchange are obviously real. For better comparison and due to we display the first three partial waves multiplied with 1 2 l(l + 1) in Fig. 8; the intersection of P -and D-wave curves then gives an indication at the energy at which an additional zero in the angular distribution will occur, with the precise position slightly modified by the small, but nonnegligible F -wave. We compare the full amplitude including the a 2 to the continuation of the decay amplitude from Ref. [4]. The decisive observation is that including the a 2 , the D-wave becomes more important than the P -wave at even lower energies, around √ s = 0.9 GeV, where the phase is still tiny-we therefore indeed expect to observe an almost perfect vanishing of the amplitude. To demonstrate that this is not trivially so, Fig. 8 also shows what would happen with the opposite sign of the a 2 contribution: negative interference of s-channel a 2 and t-channel exchange leads to a near-vanishing of the D-wave around 1.1 GeV (which is the cause for the rapid phase variation at that energy), and its rise towards the a 2 peak only overtakes the P -wave once the phase is significant. As a consequence, no near-complete cancellation ever occurs at any energy.
We wish to re-emphasize that there is no fixed relation between the phase of our s-channel partial waves to πη scattering phase shifts according to a final-state theorem. As the corresponding πη phases are not theoretically determined in the way the ππ [24,36,37] or πK [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 πη scattering anyway, with the dominant decay channel being πρ (see Eq. (30)), such that no simple version of Watson's theorem applies, and any unitarization would have to implement a coupled-channel formalism.
For illustration, we also show the resulting angular distribution at three sample energies √ s = 0.9 GeV, 1.0 GeV, and 1.1 GeV in Fig. 9, indicating the transition between the threshold (P -wave dominance, dσ/dz s ∝ (1 − z 2 s )) and the resonance region (D-wave dominance, dσ/dz s ∝ (1 − z 2 s )z 2 s ). The very different features of the different signs of the a 2 are clearly visible. An observable that allows one to capture the key features of the effect discussed even in a comparably low-statistics The color code is as in Fig. 9. measurement is to characterize the behavior in terms of a forward-backward asymmetry, where in the second line, we have neglected all partial waves beyond F -waves, as well as imaginary parts of P -and F -wave. Both the continuation of the decay amplitude without a 2 [4] and our full model with the preferred relative sign for the a 2 display a very large positive asymmetry, peaked just below √ s = 1 GeV for the full model; for the opposite a 2 sign, the asymmetry is small near threshold, and subsequently even turns negative. An experimental verification of this asymmetry would therefore confirm that our description of the decay amplitude including the a 2 , and the resulting con-sequences for the η transition form factor, are indeed reasonable.

Summary
In this article, we have studied the effects of the a 2 tensor meson on the decay η → π + π − γ as well as the analytic continuation of the decay amplitude for the scattering process γπ − → π − η. We have included the D-wave πη resonance as a left-hand cut structure of a dispersive representation that obeys the correct finalstate phase relation for the π + π − P -wave. 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 η ′ → π + π − γ. The slope parameter of the resulting η transition form factor is reduced by about 7% in the dispersive integral up to 1 GeV 2 compared to a previous analysis [5].
We have predicted different observables for the η production reaction γπ − → π − η at energies up to the a 2 resonance. The peak cross section is predicted to be (12±2)µb, similar in size to the γπ − → π − π 0 cross section in the ρ peak [15]. Fixing the relative sign of the a 2 to the more likely solution from decay phenomenology, we find an interesting P -D-wave 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. P η ′ (t) Fig. 11 Representation of the decay distribution η ′ → π + π − γ from Ref. [39]; see main text for details. The blue curve shows the linear fit, including the gray band for the fit uncertainty. The red curve with the yellow band includes the effects of a 2 exchange in addition. The vertical dashed lines represent the limits of phase space at 4M 2 π and M 2 η ′ .
we can explain the ratio of the couplings by a mixing angle of θ = (−12.4 ± 2.7) • , which is somewhat smaller than the standard value θ ≈ −20 • , but close enough that we are confident the difference can be explained by higher-order terms. In particular, we can safely conclude that the sign of g ′ T in Eq. (A.2) agrees with the one of g T . 3 In Fig. 11, we display the observableP η ′ (t) defined in strict analogy to Eq. (36), comparing the data of Ref. [39] to fits with a linear parametrization, as well as including effects of the a 2 . Due to the rather large error bars, we show the fit results as bands, not just the best fit. The linear fit leads to a slope parameter α ′ Ω = (0.6 ± 0.2) GeV −2 , (A.5) with a reduced χ 2 of χ 2 /ndof = 1.23. It was argued in Ref. [4] that in the limit of a large number of colors, α Ω = α ′ Ω should be expected, so the slopes of the polynomial would agree for η and η ′ decay. Comparing Eqs. (35) and (A.5), phenomenology seems rather at odds with this prediction. However, including the effects of the a 2 in the amplitude representation, we find much stronger curvature effects than for the η decay as anticipated, with the residual slope fitted to be now with χ 2 /ndof = 1.38. The additional uncertainty due to the a 2 couplings is ±0.1 GeV −2 . In this case, the fit quality becomes slightly worse (overall better fits are essentially precluded by the third-to-last data point at √ t = [800, 825] MeV); however, α ′ Ω is now in markedly better agreement with the value found for α Ω in Eq. (37).P η ′ (t) can be approximated in the decay region 4M 2 π ≤ t ≤ M 2 η ′ by a quadratic polynomial to about 1% accurcy. The a 2 contribution predicts the curvature to beβ ′ Ω = (−1.0 ± 0.1) GeV −4 . As a side remark, we can also take this result as another strong indication on the correctness of the sign of c T g T and c T g ′ T : a negative sign would lead to a residual slope α ′ Ω of (0.06 ± 0.12) GeV −2 .
A more rigorous test of the decay spectrum predicted here, with higher-statistics data from BESIII, would be extremely welcome.