A dispersive analysis of η ' → π + π - γ and η ' → ℓ + ℓ - γ.

We present a dispersive representation of the η ' transition form factor that allows one to account, in a consistent way, for the effects of ρ - ω mixing in both the isoscalar and the isovector contributions. Using this formalism, we analyze recent data on η ' → π + π - γ to constrain the isovector part of the form factor, individually and in combination with data for the pion vector form factor, which suggests a tension in the ρ - ω mixing parameter. As a first application, we use our results, in combination with the most recent input for the isoscalar part of the form factor, to predict the corresponding spectrum of η ' → ℓ + ℓ - γ , in particular we find the slope parameter b η ' = 1.455 ( 24 ) GeV - 2 . With forthcoming data on the latter process, our results establish the necessary framework to improve the evaluation of the η ' -pole contribution to the anomalous magnetic moment of the muon using experimental input from both η ' decay channels.

Introduction

The transition form factors (TFFs) of pseudoscalar mesons, with , describe the interaction with two (virtual) photons,where is the electromagnetic current, are the photon momenta, and . For both photons on-shell, these form factors determine the di-photon decays governed by the chiral anomaly [1, 2]with . For the pion, the corresponding normalization was experimentally [3] found to be close to the prediction from the low-energy theorem [4-6] in terms of the pion decay constant , to the extent that higher-order corrections [7-10] thwart agreement with experiment. For the analog relations depend on the details of – mixing [11-17], see Ref. [18] for a review. Here, we will use the outcome of the PDG global fit,  [19], i.e.,Beyond the normalization, understanding the momentum dependence of the TFFs is critical to be able to calculate the pseudoscalar-pole contributions to hadronic light-by-light scattering (HLbL) in the anomalous magnetic moment of the muon. Currently, the main uncertainty in the Standard-Model prediction [20-48]originates from hadronic vacuum polarization, see, e.g., Refs. [20, 49–54] for further discussion, but to match the final projected precision of the Fermilab experiment, which will improve upon the current world average  [55-59]by more than another factor of two, also the uncertainties in the subleading HLbL contribution, in Ref. [20] estimated as  [33–46, 60–65]need to be reduced accordingly. Recent progress includes a second complete lattice calculation [66], while on the phenomenological side the role of higher intermediate states and the implementation of short-distance constraints are being scrutinized [67-75]. Besides these subleading contributions, the and poles are currently estimated using Canterbury approximants alone [37] – as opposed to the pole, for which independent calculations from dispersion relations [40, 41], lattice QCD [42], and Canterbury approximants all give a coherent picture – so that a full dispersive analysis is called for to corroborate the corresponding uncertainty estimates. Several steps in this direction have already been taken in previous work [76-79], in particular, towards a better understanding of the role of factorization-breaking terms [79].

In this paper, we address another subtlety that is related to the interplay of isoscalar and isovector contributions. In principle, since , have isospin , both photons need to be either isoscalar or isovector, leading to a simple vector-meson-dominance (VMD) picture of decays proceedings either via two mesons or some combination of and . However, isospin-breaking effects are resonance enhanced just as in the pion vector form factor (VFF), so that both the admixture of an into the isovector contribution and, vice versa, of intermediate states into the isoscalar component can become phenomenologically relevant, at least in the vicinity of the resonance. This issue becomes particularly important when the two-pion cut in the isovector contribution is constrained via data for , since also in this case isoscalar corrections will enter. Based on Ref. [80] we develop a coupled-channel formalism that allows one to disentangle these effects in a consistent manner, apply the result to recent data from BESIII [81], and then calculate the resulting singly-virtual TFF to predict the spectrum for . A summary of the formalism is given in Sect. 2, with a detailed derivation in the appendix. Fits to the data are presented in Sect. 3, followed by the prediction for in Sect. 4 and our conclusions in Sect. 5.

Formalism

The singly-virtual TFF in the definition (1.1) determines the spectrum for according towhere s is the invariant mass of the lepton pair and the phase-space variable. Once is known, the di-lepton spectrum can thus be predicted.

However, the differential decay width (2.1) scales as in the fine-structure constant , leading to a challenging experimental signature. Additional information on the energy dependence can be obtained by combining the decay with the pion VFF ,which determines the discontinuity of the dominant intermediate states. Experimentally, the decay is more easily accessible, given that in this case the differential decay width only scales as , and information on the spectrum can then be used to reconstruct the isovector part of the TFF. This strategy is straightforward as long as isospin violation is neglected, while the transition from to becomes more intricate once such corrections are included. In particular, – mixing is enhanced by the presence of the resonance propagator, and therefore needs to be included to obtain a realistic line shape. In such a situation, one cannot consider the channel in isolation anymore, since also depends on – mixing, leading to a spectral function in which the double discontinuities of and intermediate states, corresponding to and , respectively, no longer cancel. In Appendix A–Appendix D we systematically develop a formalism that avoids such inconsistencies, allowing for a meaningful consideration of – mixing in both the isoscalar and isovector contributions. The central result is given bya dispersion relation constructed from a spectral function whose double discontinuity vanishes, see Eq. (D.16). Equation (2.3) is expressed in terms of the – mixing parameter and the weights , defined in Eq. (D.8), which determine the isoscalar contribution to the slope of the TFF, as well as the second-order polynomialintroduced to describe the decay spectrum below.

Fits to and

For the pion VFF, defined in Eq. (2.2), we employ a dispersive representationwhereis the Omnès function [82] and denotes the P-wave scattering phase shift in the isospin channel. As input for the phase shift we use the solution of the Roy-equation analysis optimized for fits to pion VFF data of Ref. [27]. The term multiplying the Omnès function in Eq. (3.1) takes the effects of inelastic contributions, such as , as well as our ignorance about the high-energy behavior of the phase shift into account, where the constant is left as a free parameter to be constrained by the fit. The isospin-breaking effect of – mixing in is parameterized via(in line with in Eq. (D.4)), where will be determined by the fit.

In our formalism, the differential decay spectrum of is described by [83]see Appendix D.2 for the derivation. The appearance of the pion VFF herein is due to the universality of P-wave final-state interactions [76, 84]. The constants A, , and in P(s), see Eq. (2.4), are used as fit parameters, and we refer to Eqs. (C.4) and (D.10) for the definitions of the coupling constants and , respectively.

We fit to several time-like pion VFF data sets: provided by the energy-scan experiments SND [85, 86] and CMD-2 [87-90], where in both cases diagonal errors were given, as well as from radiative-return measurements BaBar [91, 92] (below ) and KLOE [93-96], where in both cases statistical and systematic covariance matrices were provided. Furthermore, the recent data set for the spectrum measured by BESIII [81] is used in the fit, which largely supersedes older data in statistical accuracy [97]. In order to avoid the d’Agostini bias [98] in the minimization of a with naively constructed covariance matrices when dealing with normalization uncertainties, an iterative fit procedure, proposed in Ref. [99] and applied to in Ref. [27], is employed. Since the pion VFF is defined as a pure QCD quantity in Eq. (2.2), the pion VFF data sets have been undressed of vacuum-polarization (VP) effects. As noted in Ref. [27], a minor rescaling of energy for each individual data set leads to a significant improvement of the fit quality. Equivalently, we take the mass for individual data sets as a fit parameter. In the case of fitting the combined KLOE data set, we follow Ref. [27] and assign a global mass with individual mass shifts to each of the three underlying data sets, from 2008 [93], 2010 [94], and 2012 [95]. Furthermore, the mass shifts are constrained by penalties , with calibration uncertainty  [100]. These terms are counted as additional data points in the number of degrees of freedom. Finally, for BaBar and KLOE the observables are cross sections weighted over energy bins, an effect that is also included in our fit in the same way as in Ref. [27]. Further data sets are available from BESIII [101] and SND [102], but not yet included for consistency: since we use the phase shift from Ref. [27], we restrict our analysis to the same data sets used therein, in particular, since the results for the decays are insensitive to the precise choice of VFF data sets.

The BESIII data set for includes the number of selected events, the number of background events, the detection efficiency, and the detector resolution root mean square in the respective energy bin. The latter is taken into account by taking the convolution of the theoretical spectrum in Eq. (3.4) with a Gaussian distribution of mean corresponding to the respective bin center and standard deviation given by the respective value for the energy resolution. Efficiency and number of background events are subsequently included in the fit function. Since the data are given in the form of an event rate, the physical values of the fit parameters A and are extracted by utilizing the constraintin the single fit to the BESIII decay spectrum, where the partial width is given by the corresponding quantities in Table 1. The couplings and have been extracted by means of VMD models [83] to bewhere the experimental input quantities are listed in Table 1 as well. Note that the definition of the coupling differs from the one in Ref. [83] by a factor of the elementary charge e, with such factors always factorized throughout this work. Also note that the given value for the coupling has been corrected for VP in order to be consistent with the definition of the pion VFF as a pure QCD quantity. The correction amounts to a factor of , where is the Standard Model VP function of Ref. [26].

Table 1

Input parameters used in this work. Note that becomes a free parameter for the VFF fits, to account for the uncertainty in energy calibration in each data set as well as the tension with determinations from and . The quoted numbers for , are VP subtracted, to ensure consistency with the bookkeeping as defined in the appendix. We also show the analog quantities for the , which enter the isoscalar part of the TFF (and are consistent with analogous determinations from  [104]). The entries for , , from Ref. [103] are consistent with but more precise than the current PDG averages

Quantity	Value	Reference
M_η^′	957.78(6)MeV	[19]
Γ_η^′	0.188(6)MeV	[19]
F_η^′γγ	0.3437(55)GeV^-1	[19]
Br(η^′→π^+π^-γ)	29.5(4)%	[19]
M_ω	782.607(23)MeV	[31]
Γ_ω	8.69(4)MeV	[31]
Γ(ω→e^+e^-)Br(ω→3π)	0.5698(31)(82)keV	[103]
Br(ω→3π)	89.2(7)%	[19]
Br(η^′→ωγ)	2.50(7)%	[19]
M_ϕ	1019.197(20)MeV	[31]
Γ_ϕ	4.22(5)MeV	[31]
Γ(ϕ→e^+e^-)Br(ϕ→3π)	0.1841(21)(80)keV	[103]
Br(ϕ→3π)	15.24(33)%	[19]
Br(ϕ→η^′γ)	6.22(21)×10^-5	[19]

In principle, the provided data would need to be fit by the binned maximum likelihood strategy, i.e., by minimizing the log-likelihood functionwhere represents the number of events in bin i and is the value of the fit function in this energy bin. This proves to be disadvantageous for our purpose, since we aim at a combined fit of the pion VFF data sets and the spectrum in order to extract a common – mixing parameter . In Table 2, the fit parameters to the BESIII data stemming from the binned maximum likelihood method are compared to those of a fit, by minimizingwhere Poissonian errors have been assumed. The fit gives a and the errors of the fit parameters have been inflated by a scale factor . While the central values all agree with each other within error margins, the error estimates of the fit are slightly lower than those of the binned maximum likelihood fit, but sufficiently close to justify a fit in combination with VFF data. The quoted errors for the extracted values of A and include the errors of all fit parameters and their correlations through Eq. (3.5) as well as the errors of the experimental input parameters.

Table 2

Comparison of the fit outcome of the differential decay width in Eq. (3.4) to the BESIII spectrum [81] of the binned maximum likelihood and minimum strategies

Quantity	Likelihood	χ^2
A[GeV^-3]	17.10(35)	17.07(32)
β[GeV^-2]	0.589(53)	0.599(51)
γ[GeV^-4]	-0.204(53)	-0.214(50)
ϵ_ρω×10^3	1.624(73)	1.622(55)
M_ω[MeV]	783.07(33)	783.08(31)

In the combined fit of the spectrum and the pion VFF data sets, the – mixing parameter is a shared parameter. In comparison to the single fit, we are confronted with the problem that we fit the physical value of and not the one including the event rate . Therefore, needs to be calculated at every step of the fit iteration. Defining , Eq. (3.5) can be written aswhereand notation as in Eq. (3.4), in order to determine with input of in physical units. Hence, the main difference to the individual fit is that to be able to perform a combined fit with the VFF data sets, we need to ensure physical units already at each step in the fit iteration. The linear parameter in the pion VFF is used as a shared parameter for all data sets in the combined fit as well.

The result of the combined fit is listed in Table 3, where the errors are inflated by the overall . The error of the physical value of A takes into account the correlations and inflated errors of the fit parameters as well as the errors of all input quantities. The results for and are in good agreement with Ref. [27], validating the simplified fit form (3.1) as a convenient way to compare – mixing in the pion VFF and . The outcome of the combined fit for the case of the spectrum is shown in Fig. 1, whereis shown to emphasize differences in the vicinity of the – mixing effect.

Table 3

Combined fit to several pion VFF data sets (BaBar, KLOE, CMD-2, SND) and spectrum (BESIII) with overall . In the row for KLOE, the three values for refer to the combinations of the global KLOE mass and the corresponding mass shifts of the three underlying data sets from 2008, 2010, 2012, respectively. See main text for details

Fig. 1

Fit to the differential decay rate of (individually or combined with the VFF). To highlight differences in the – region, we show the associated function , as defined in Eq. (3.11), compared to the experimental data from BESIII [81]. The function is calculated in units that still contain the event rate

Both in Fig. 1 and by comparing Tables 2 and 3, one sees that both processes are not fully consistent, with the tension in the most apparent one: the data prefer a value , while the combined fit, statistically dominated by the VFF data sets, produces a value close to , with negligible uncertainty at the level of the difference. This mismatch is highlighted by the presentation as in Fig. 1, and comes out substantially larger than any effect that could be obtained by variation of fit strategy and choice of input parameters. In none of the variants we studied did the combination of experimental errors and external input suggest an uncertainty beyond , which leads us to conclude that this tension is significant.

A minor tension also occurs in the mass of the .1 Here, the results from come out significantly below the preferred value from  [28, 31], see Table 1, but extractions from are known to be sensitive to a phase in  [92], with the corresponding difference at least partially explained by radiative intermediate states that can generate such an isospin-breaking effect [105]. In contrast, the data favor a central value that deviates about in the opposite direction, albeit with moderate significance. Since also the recent data from BESIII [106] suggest a similar increase compared, e.g., to Refs. [88, 103, 107], as does Ref. [101] compared to other data sets, it is difficult not to see a pattern in BESIII determinations of .

Predicting

In order to determine the TFF according to Eq. (2.3), the – mixing parameter , the parameter associated with inelastic contributions to the pion VFF , as well as the polynomial parameters A, , and of the spectrum from the combined fit serve as input. Additionally, from the quantities listed in Table 1, the mass and width, the TFF normalization , together with the associated couplings in Eq. (3.6) enter the description of the isovector part of the TFF. The isoscalar part, consisting of the contributions of the and resonances, depends on the couplings in Eq. (D.9), which were calculated from the quantities in Table 1. In the time-like regime, the resulting TFF then appears as shown in Fig. 2, where it is compared to experimental data [108].

Fig. 2

Determination of the TFF by means of Eq. (2.3) in comparison to data from BESIII [108] (statistical and systematic errors added in quadrature) scaled with from Table 1 and the VMD model of Eq. (D.23) with resonance contribution according to Eq. (D.22); for the kinematic range accessible in decays (left) and a larger time-like region including the resonance with inset magnifying the low-s region (right). See also Ref. [109] for an earlier version of this figure

Since the form of polynomial P(s) and the linear polynomial multiplying the Omnès function in the representation of the pion VFF are governed by physics below , their values at high energies do not bear much meaning. Therefore, and in order to improve the convergence properties of the dispersion integrals in Eq. (2.3) at the same time, they are led to constant values and above a certain cutoff value . This cutoff is varied between , the point where P(s) reaches its maximum ( with and from the combined fit) and the point where P drops below its value at (). Together with the errors of the input quantities in Table 1, the errors of the parameters from the combined fit, and their correlations, this procedure is used to generate the error band of the dispersive TFF representation shown in Fig. 2. In this figure, the dispersive curve displays the broad peak from the isovector part of the TFF. Around the mass, the – mixing effect is overlaid by the narrow peak of the resonance from the isoscalar part of the TFF. Slightly below an invariant mass square of the isoscalar contribution begins to set in. The available experimental data are in excellent agreement with the dispersive prediction, but not accurate enough to distinguish it from the VMD model. Figure 2 also shows the peak of the resonance, which, however, is not accessible in decays, but could be scanned in near threshold.

The low-energy properties of the TFF are described by the slope parameter , which is defined byIt can be calculated via a sum rule, following from Eq. (2.3), and disentangled into its isovector contributionand isoscalar contributionwith . Numerical values are listed together with the evaluation of the sum rule for the normalization (D.21) in Table 4. As expected from VMD, the isovector part dominates both the normalization and the slope, saturating the former by and producing about of the latter. In combination with the isoscalar part the sum rule for the normalization is fulfilled by , suggesting a slightly faster convergence than similar sum rules in Refs. [40, 41, 110, 111]. The two isospin-breaking corrections to the normalization, in the isovector and isoscalar part, combine to about and thus prove negligible compared to the uncertainty in the sum-rule evaluation. Accordingly, these corrections become most relevant in the resonance region, where even subleading effects are enhanced by the small width of the resonance, see Fig. 2. The resonance peak would be about increased in magnitude if .

Table 4

Contributions from the various components of the TFF to the sum rules of the normalization and the slope parameter

	(I=1)_ϵ_ρω=0	Δ(I=1)_ϵ_ρω≠0	(I=0)ϵ_ρω=0ω	Δ(I=0)ϵ_ρω≠0ω	(I=0)^ϕ	Total
Norm [%]	71.85(93)	-0.1433(20)	7.20(23)	-0.1389(46)	15.85(61)	94.6(1.2)
b_η^′[GeV^-2]	1.185(23)	0	0.1176(32)	0	0.1526(53)	1.455(24)

Our result for the slope,is consistent with, but considerably more precise than the experimental determinations  [108] (from ) and  [112],  [113] (via ), and also agrees very well with  [16] extracted via Padé approximants. Moreover, the improved formalism and combined fit allowed us to substantially reduce the uncertainties compared to the previous dispersion-theoretical analysis  [77], also reflecting the improved input on from Ref. [81] in comparison to the earlier measurement [97]. A close-up of the low-s region of the TFF is included in Fig. 2 as well.

Conclusions

This work presents progress in a dispersive calculation of the TFF of the , as required for future studies of the -pole contribution to HLbL scattering in the anomalous magnetic moment of the muon. First, we established a formalism that enables a consistent implementation of isospin-breaking – mixing effects, both in the isoscalar and isovector part of the form factor. The technical derivation is presented in a self-contained way in the appendices of the paper, leading to our main result (2.3) for the final TFF representation. Moreover, we worked out how the information on the isovector TFF contained in the spectrum for can be combined, in terms of the decomposition (3.4).

These results form the basis for the application studied in the main part of the paper, an analysis of recent data from BESIII on and subsequent prediction of the full TFF. A combined fit of the spectrum and data on the pion VFF reveals a tension in the – mixing parameter, with the former suggesting a substantially smaller value than extracted from isospin breaking in , see Fig. 1. Given the modest fit quality already in the individual fit as well as another minor tension in the mass, it would be desirable to scrutinize these observations with an independent measurement of the spectrum, e.g., from CLAS at Jefferson Lab [114].

As a final step, we used the parameters determined in the global fit to and , in combination with up-to-date input on the isoscalar resonances, to predict the full TFF at low energies, see Fig. 2 and Table 4. Deviations from VMD are most visible in the vicinity of the resonance, where the dispersive implementation of the meson via intermediate states changes the line shape compared to the strict VMD limit. Our results should prove valuable for the analysis of future data on and  [115, 116], in particular towards improving the calculation of the -pole contribution to HLbL scattering.