Axial-vector transition form factors and e ` e ´ Ñ f 1 π ` π ´

: We study the transition form factors (TFFs) of axial-vector mesons in the context of currently available experimental data, including new constraints from e ` e ´ Ñ f 1 p 1285 q π ` π ´ that imply stringent limits on the high-energy behavior and, for the first time, allow us to provide an unambiguous determination of the couplings corresponding to the two antisymmetric TFFs. We discuss how these constraints can be implemented in a vector-meson-dominance picture, and, in combination with contributions from the light-cone expansion, construct TFFs as input for the evaluation of axial-vector contributions to hadronic light-by-light scattering in the anomalous magnetic moment of the muon.


Introduction
The transition form factors (TFFs) of axial-vector mesons are crucial for estimating the hadronic light-by-light (HLbL) contribution to the anomalous magnetic moment of the muon a µ , in particular, for intermediate photon virtualities and the transition to shortdistance constraints (SDCs).At present, the axial-vector contribution included in the Standard-Model prediction for a µ  is responsible for a large fraction of the final uncertainty, a HLbL µ " 92p19qˆ10 ´11 [1,14,[18][19][20][21][22][23][24][25][26][29][30][31][32][33][34], especially, when taking into account the interplay with SDCs.In view of the expected experimental improvements beyond the current world average [35][36][37][38][39], the uncertainty in the HLbL contribution should be reduced by another factor of 2 to ensure that it does not play a role in the interpretation of the experiment [40,41].Such improvements are ongoing, both in lattice QCD [42][43][44][45][46] and with data-driven methods, including the derivation of higher-order SDCs [47][48][49], their implementation [50][51][52][53][54][55][56], and dispersion relations [57][58][59][60].In particular, to evaluate the axial-vector contributions, one needs robust input for their TFFs.To this end, constraints on the asymptotic behavior have been derived from the light-cone expansion (LCE) [61] and available data evaluated in a vector-meson-dominance (VMD) inspired parameterization [62], but, due to scarcity of data, no sufficiently complete and reliable solutions have been obtained so far, with ambiguities mostly affecting the determination of the two antisymmetric TFFs.This is in large part because the interaction of an axial-vector resonance A with two electromagnetic currents is suppressed by the Landau-Yang theorem [63,64]-stating that a spin-1 particle cannot decay into two on-shell photons-so that all observables require at least one non-zero virtuality.Some data are available for the space-like process e `e´Ñ e `e´A , for A " f 1 " f 1 p1285q and A " f 1  1 " f 1 p1420q [65][66][67][68][69][70], allowing one to extract the equivalent two-photon decay widths r Γ γγ and, thereby, the mixing angle between the two f 1 states.Accordingly, if U p3q symmetry is assumed, it suffices to determine the TFFs for the f 1 p1285q to be able to estimate the effect of the entire triplet including the a 1 p1260q.Focusing, therefore, on the f 1 resonance, for which most data are available, we compiled the constraints that follow from the radiative decays f 1 Ñ ργ and f 1 Ñ ϕγ, as well as f 1 Ñ 4π and f 1 Ñ e `e´i n Ref. [62], improving on previous work [71,72] by employing parameterizations that ensure the absence of kinematic singularities, include SDCs, and incorporate the spectral functions of the isovector resonances.We found that, unfortunately, the decay f 1 Ñ 4π does not provide any meaningful input for the TFFs, since dominated by f 1 Ñ a 1 π Ñ ρππ Ñ 4π, while f 1 Ñ e `e´w ould, in principle, be a very interesting observable, yet not at the current level of precision [73].
Ultimately, the currently available data were not sufficient to identify a unique solution for all three TFFs, especially the normalizations of the two antisymmetric TFFs and the momentum dependence of all three TFFs were only poorly determined.In the future, these limitations could be overcome by better data for e `e´Ñ e `e´f 1 and f 1 Ñ e `e´; in this paper, we instead propose to study existing data for e `e´Ñ f 1 π `π´ [ 74, 75]. 1 This process is also sensitive to all three TFFs, for one photon virtuality centered at the ρ mass and the other one determined by the center-of-mass energy of the e `e´p air.Phenomenologically, the reaction displays prominent resonance features from excited ρ resonances [88], primarily the ρp2150q, but, when interpreted as a limit on the non-resonant contribution, entails powerful constraints on the TFFs of the f 1 , both on the asymptotic behavior and the respective normalizations.
The outline of the paper is as follows: we first review the basic formalism for the axial-vector TFFs in Sec. 2, and then define improved VMD parameterizations in Sec. 3 that implement the asymptotic behavior observed in e `e´Ñ f 1 π `π´.In Sec. 4, we 1 The measurement of the e `e´Ñ 2pπ `π´q η [74] and e `e´Ñ KSK ˘π¯π`π´ [ 75] cross sections, in which the f1 peak can be identified, is partly motivated by hadronic vacuum polarization (HVP).The impact of such high-multiplicity channels on the HVP contribution to aµ, though, is much smaller than the current tensions observed between data-driven evaluations [6][7][8][9][10][11][12][76][77][78][79][80] and lattice QCD [81][82][83][84][85], e.g., a HVP summarize the previous observables and present the formalism in which we will analyze e `e´Ñ f 1 π `π´.The phenomenological analysis, including a review of the data base, a global fit, and a summary of the resulting TFF parameterizations, will be presented in Sec. 5, before concluding in Sec. 6.Finally, in App.A, we collect constants and parameters used throughout this work.

Axial-vector transition form factors
The helicity amplitudes for the decay of an axial-vector meson into two virtual photons, ApP, λ A q Ñ γ ˚pq 1 , λ 1 qγ ˚pq 2 , λ 2 q, are given by [61,62] where, following the Bardeen-Tung-Tarrach procedure [89,90], the tensor matrix element M µνα pq 1 , q 2 q can be decomposed into three independent Lorentz structures and form factors F i pq 2 1 , q 2 2 q that are free of kinematic singularities according to Here, m A is the mass of the respective axial-vector meson and the structures are completely antisymmetric (a) or symmetric (s) under photon crossing (µ Ø ν and q 1 Ø q 2 ); similarly, the associated form factors obey the indicated symmetry properties under the exchange of momenta, q 2 1 Ø q 2 2 .Furthermore, the prefactor i{m 2 A in Eq. (2.2) is chosen to obtain dimensionless TFFs with real-valued normalization and the Levi-Civita tensor is used in the convention ϵ 0123 " `1.For the formulation of SDCs and the analysis of the L3 data for e `e´Ñ e `e´f 1 , see Sec. 4.1, it is also useful to consider the basis defined by Since the Landau-Yang theorem [63,64] forbids the decay into two on-shell photons, one commonly defines the so-called equivalent two-photon decay width as [67] r Γ γγ " lim with one longitudinal quasi-real photon γ L, the spin-averaged-longitudinal-transversal (LT)-width and the differential decay width for fixed polarization where Ω is the center-of-mass solid angle and λpa, b, cq " a 2 `b2 `c2 ´2ab ´2ac ´2bc is the Källén function.In terms of the above form factors, one finds where α " e 2 {p4πq is the fine-structure constant.
The asymptotic behavior of the axial-vector TFFs can be analyzed by means of the LCE, leading to [61] F a 1 pq 2 1 , q 2 2 q " Op1{Q 6 q, where denote the average photon virtuality and asymmetry parameter, respectively.In the above, we furthermore introduced the effective decay constant with the decay constants F a A defined via (2.12) The Gell-Mann matrices λ a and the conveniently normalized unit matrix λ 0 " a 2{3 1 determine the flavor decomposition, with the flavor weights C a in the effective decay constant given by C 0 " 2{p3 ?6q, C 3 " 1{6, and C 8 " 1{p6 ?3q.In the symmetric doubly-virtual direction, the Op1{q 4 q limits become (λ « 1) [62] kpλq " 3λ 2 ´pλ 2 `4λ `1q log λ ´3 pλ ´1q 4  " Opλ ´1q, (2.13) but all singly-virtual limits of F a 2 {s pq2 1 , q 2 2 q diverge in the symmetrized basis.However, physical helicity amplitudes depend on linear combinations of the TFFs in such a way that only well-defined limits contribute to observables [61,62], which implies that F 2 pq 2 , 0q and F 3 p0, q 2 q have finite limits, while the opposite cases diverge.

Vector-meson dominance
The parameterization of the TFFs inspired by VMD is based on their decomposition into isovector and isoscalar components.Using U p3q symmetry, one can deduce the ratio [62] of isoscalar to isovector contributions for the f 1 γ ˚γ˚c oupling, where we inserted the L3 mixing angle θ A " 62p5q ˝[69, 70], 2 derived from for the corresponding J P C " 1 ``axial-vector nonet with the mixing pattern ˜f1 Hence, it is the isovector channel that dominates the process, with small isoscalar corrections at the level of 5%.
The minimal particle content necessary for a VMD construction of TFFs that individually obey the asymptotic constraints summarized in Sec. 2 requires the inclusion of three multiplets.More specifically, we will use ρ " ρp770q, ρ 1 " ρp1450q, and ρ 2 " ρp1700q for the isovector contributions and ω " ωp782q, ω 1 " ωp1420q, ω 2 " ωp1650q as well as ϕ " ϕp1020q, ϕ 1 " ϕp1680q, ϕ 2 " ϕp2170q for the isoscalar contributions.The introduction of a third multiplet, as required to obtain the correct asymptotic behavior for the antisymmetric TFFs, goes beyond the parameterizations of Ref. [62], ultimately, because the data on e `e´Ñ f 1 π `π´d emand such a steep decrease, including in kinematic configurations in which one virtuality is kept fixed at a finite but non-zero value.

Isovector contributions
In the space-like region, q 2 i ă 0, we propose to extend the isovector parameterizations from Ref. [62] as follows: leading to The resulting asymptotic behavior of the TFFs becomes with q 2 1 fixed to a finite value distinct from q 2 2 (left) and in the doubly-virtual direction (right).Crucially, the singly-virtual asymptotics now match the LCE result from Sec. 2 for arbitrary fixed q 2 1 , which, for q 2 1 " M 2 ρ , is mandatory for a realistic description of the e `e´Ñ f 1 π `π´d ata (the opposite case with fixed q 2 2 follows from symmetry). 3For time-like applications, the replacements M 2 Ñ M 2 ´iM Γ apply in the denominators of Eq. (3.6), i.e., after imposing the asymptotic behavior of the TFFs; due to the large widths of the ρ-like mesons, a narrow-width approximation, M 2 Ñ M 2 ´iϵ in the denominators, in general becomes insufficient here.A consequence of the faster decrease in the singly-virtual directions concerns an even faster decrease in the doubly-virtual case, much below the LCE expectation.Accordingly, in the final representation for the TFFs, we add the asymptotic contribution [62] where s m is a parameter that determines the scale of the transition.The implementation of these asymptotic contributions, or their variant including mass effects [62], becomes relevant for the axial-vector contributions in the HLbL loop integral.Here, we focus on the determination of the low-energy couplings in the VMD component of the parameterization, as can be obtained from e `e´Ñ f 1 π `π´.

Isoscalar contributions
In complete analogy to the above, the isoscalar parts of the form factors are parameterized according to 4 with the same asymptotic properties as in Eq. (3.8).Again, time-like applications imply the replacements M 2 Ñ M 2 ´iM Γ in the denominators, since the large widths of the excited isoscalar resonances do not allow for a narrow-width approximation.Finally, under the assumption of U p3q symmetry, the isoscalar coupling constants can be related to the 3 For F I"1 a 1 pq 2 1 , q 2 2 q, the LCE predicts an even faster decrease in the singly-virtual direction, but we do not consider yet another multiplet for the following reasons: (i) information from the LCE on this TFF is limited, i.e., no non-vanishing contribution survives at Op1{Q 4 q, in such a way that, in contrast to the other TFFs, we cannot add an LCE term to repair the behavior in the doubly-virtual direction and thus need to choose a compromise; (ii) the fit to e `e´Ñ f1π `π´p roduces a small coupling Ca 1 , in line with the LCE suppression; (iii) another multiplet would have a mass already in the energy range in which the data are to be described, so that no meaningful suppression could be generated even when introducing another state. 4We assume ideal mixing for the vector mesons, which prevents crossed terms involving ω and ϕ states.

Observables
4.1 e `e´Ñ e `e´f The equivalent two-photon decay width r Γ f 1 γγ " 3.5p6qp5q keV, as measured by the L3 collaboration [69], determines the normalization of the symmetric TFF, see Eq. (2.8).Taking into account the isoscalar contributions, where we followed the sign convention of Ref. [62].
The singly-virtual VMD limits can be further constrained by matching the L3 parameterization onto the full description of the e `e´Ñ e `e´f 1 cross section [61], is the dipole ansatz assumed in Ref. [69] and the form factors are given in the basis of Eq. (2.4).While the normalization agrees by construction, matching the slopes at q 2 " 0 leads to where the factor N ωϕ " 1 `Rω `Rϕ accounts for the isoscalar terms in the normalization.
From the procedure outlined in Ref. [62], it is straightforward to obtain the branching ratio of f 1 Ñ V γ, V " ρ, ω, ϕ, in the form where we defined and the couplings The normalizations R V , V " ω, ϕ, are given by Eq. (3.11), and R ρ " 1.
In addition, for V " ρ, information on the helicity amplitudes is available.The spinaveraged amplitude squared of the corresponding process f 1 Ñ ργ Ñ π `π´γ , with the subsequent decay of an on-shell ρ meson, is of the generic form where θ π `γ is the angle between the final-state π `and the photon.Similarly to the branching ratio of f 1 Ñ ργ, the ratio of helicity amplitudes for f 1 Ñ ργ Ñ π `π´γ results from a straightforward modification of the result presented in Ref. [62], e follow Ref. [62] and write the decay rate for f 1 Ñ e `e´a s where the scalar amplitude A 1 is implicitly defined by Mpf 1 pP q Ñ e `pp 2 qe ´pp 1 qq " e 4 ϵ µ pP qu s pp 1 qγ µ γ 5 A 1 v r pp 2 q (4.12) and further decomposes into terms proportional to the three VMD couplings (with isoscalar and isovector coefficients D I i ) and an asymptotic contribution D asym , For the case of products of narrow-resonance propagators as in Eq. (3.4), with squared masses x and y, the integral representation  [93]).The total isoscalar one follows as , and D asym is given for the matching points ?s m P t1.0, 1.3, 1.5, 1.7u GeV.The left column gives the reference point for which the widths of all vector mesons are neglected and the right column the more realistic case that includes the spectral function of the ρ (used as input in Table 3).
applies, with analytic expressions for the functions f i px, y, z, m f 1 q given in Ref. [62].In the same reference, we also provided evaluations of the asymptotic contribution D asym and studied in detail the sensitivity of the integrals to the spectral functions assumed for ρ and ρ 1 resonances when going beyond a narrow-resonance picture, to avoid unphysical imaginary parts in the loop integrals [94][95][96].In the decay region of the f 1 , by far the most important correction arises from the width of the ρ.Here, we provide a simple evaluation of Bpf 1 Ñ e `e´q for the representations constructed in Sec. 3 that captures this main effect, to ensure that our final solutions do not conflict with the SND measurement [73].To this end, we replace in Eq. (3.6), where the normalizations N a 1{2 , N s of the spectral function ρpxq (taken from Refs.[62,97,98]) are determined by demanding that the meaning of the couplings C a 1{2 , C s remain unaltered compared to the zero-width limit; numerical results for the coefficients D i are collected in Table 1. 5 Once improved data on Bpf 1 Ñ e `e´q become available, more refined analyses can be performed along the lines of Refs.[62,107].
The scattering process e `e´Ñ f 1 π `π´p robes the f 1 TFFs in the time-like region via e `e´Ñ γ ˚Ñ f 1 ρ Ñ f 1 π `π´, see Fig. 1.For our analysis, we thus define amputated f 1 Ñ ργ ˚form factors, which are related to γ ˚Ñ f 1 ρ via crossing symmetry, according to where and a width has been inserted into the denominators for the time-like application.As a first approximation, we consider the case in which the decay e `e´Ñ f 1 π `π´i s described by e `e´Ñ f 1 ρ, see Fig. 2, whose amplitude can be constructed from Eq. (2.2) ϵ μpq 1 qϵ ν pq 2 qϵ αpP q (4.18) ˆ"T µνα a 1 p´q 1 , q 2 q s F a 1 pq 2 2 q `T µνα a 2 p´q 1 , q 2 q s F a 2 pq 2 2 q `T µνα s p´q 1 , q 2 q s F s pq 2 2 q ı , and the prefactor follows in analogy to the derivation of f 1 Ñ V γ in Ref. [62].In particular, we have taken over the definition to establish the connection to g V γ , which, in a narrow-width approximation, is related to the dilepton decay or, more rigorously, to the residue at the pole [108].In order to determine the amplitude for e `e´Ñ γ ˚Ñ f 1 ρ, we calculate the diagram shown in Fig. 2, leading to where we dropped an unobservable overall phase.Spin-averaging the squared amplitude and performing the angular integration, we find for the total cross section, with s " q 2 2 , initial-and final-state momenta and the kinematic functions T a 2 ,a 2 pq 2 1 , sq " m 6 f 1 ps `q2 1 q ´m4 f 1 ps 2 `q4 1 ´6sq 2 1 q ´m2 f 1 ps ´q2 1 q 2 ps `q2 1 q `ps ´q2 1 q 4 , T s,s pq 2  1 , sq " m 2 f 1 ps `q2 1 q `m4 In a compact way, the cross section can be expressed in terms of the amputated helicity amplitudes [61] s H ``;0 pq 2 1 , sq " with polarization-vector normalizations ξ 2 1 " q 2 1 , ξ 2 2 " s, leading to and the sum extends over the three amplitudes in Eq. (4.25).
4.5 e `e´Ñ f 1 π `πT o obtain a reasonable threshold behavior, it is mandatory to go beyond the approximation of a narrow ρ and instead consider the full amplitude e `e´Ñ f 1 π `π´.To this end, we use Mpρ Ñ π `π´q " g ρππ ϵ µ pp ρ qpp ´´p `qµ (4.27) to calculate the diagram shown in Fig. 1, leading to where we again dropped an unobservable phase.From the spin-averaged squared matrix element and after carrying out the angular integrations, we obtain the differential cross

Quantity
Value Reference Bpf 1 Ñ ργq 4.2p1.0q%[62] r ργ 3.9p1.3q[110] Bpf 1 Ñ ϕγq 0.74p26q ˆ10 ´3 [111,112] Table 2: Data for e `e´Ñ e `e´f 1 and f 1 Ñ V γ used in our analysis. section with the kinematic functions T i,j pq 2 1 , sq as in Eq. (4.24).In terms of the amputated helicity amplitudes, we obtain (4.30)In general, the remaining integration over q 2 1 needs to be performed numerically, but it is instructive to consider the limit of a narrow resonance [109] 1 In this approximation, together with the q 2 1 integration of Eq. (4.30) indeed reproduces Eq. (4.26).For the phenomenological analysis of the e `e´Ñ f 1 π `π´d ata, we will use the full expression given in Eq. (4.29).

Data input for e
he process e `e´Ñ f 1 π `π´h as been measured in two different decay channels, f 1 Ñ ηππ [74] and f 1 Ñ K Kπ [75].The data for the cross section from both reconstruction methods are well compatible, indicating that systematic errors are smaller than the statistical uncertainties of the measurements.In the following, we will therefore assume that the data are indeed dominated by statistics.
Next, around ? s » 2 GeV, the cross section displays resonance structures [88], most prominently the ρp2150q, and, potentially, further excited ρ states.This implies that we cannot expect our theoretical description based on Eq. (4.29) to provide an adequate fit to the data, because ρ excitations beyond the ρp1700q are not included.However, the data still provide a valuable upper bound for the background contributions that our TFF parameterizations do describe; in fact, this constraint proves extremely stringent, immediately ruling out, by at least an order of magnitude, parameterizations that do not implement the doubly-virtual asymptotic behavior of Eq. (3.8).Even more, writing the cross section in terms of the couplings C a 1{2 , C s , one observes that moderate cancellations among the different terms are required to obey the upper limit implied by the e `e´Ñ f 1 π `π´d ata.With C s reasonably well determined from the L3 equivalent two-photon decay width, this thus implies a valuable constraint on the antisymmetric TFFs.Table 3: Best-fit results for the three VMD couplings C s , C a 1 , and C a 2 .The fit includes the constraints from the normalization and slope measured by L3 in e `e´Ñ e `e´f 1 , from Bpf 1 Ñ ργq, r ργ , and σpe `e´Ñ f 1 π `π´q , as well as, in the right column, from Bpf 1 Ñ ϕγq.All uncertainties are inflated by the scale factor S " a χ 2 {dof.The table also shows the correlations ρ ij among the three couplings and the values of Bpf 1 Ñ V γq, V " ω, ϕ, and Bpf 1 Ñ e `e´q implied by the fit result (the latter for ?s m " 1.3 GeV).The uncertainties for Bpf 1 Ñ V γq include the fit errors and ∆R ϕ , but no additional estimate of U p3q uncertainties.The predictions for Bpf 1  1 Ñ ϕγq and Bpf 1 1 Ñ ργq use the U p3q relations from Eq. (5.5).
To quantify this constraint, we proceed as follows: we first define the χ 2 function ) where n BaBar " 52 is the combined number of data points from Refs.[74,75], σ exp i and ∆σ exp i are central value and error at center-of-mass energy ?s i , respectively, and the Heaviside function demands that contributions to χ 2 BaBar only arise when the theoretical model exceeds the central value of the data, thus not penalizing a potential excess of the latter due to excited ρ resonances.Interpreting this χ 2 function in the usual statistical sense, however, puts an undue emphasis on the e `e´Ñ f 1 π `π´d ata, especially in view of the uncertainties from the contamination of resonant contributions.For this reason, we instead study contours in the C a 1 -C a 2 plane for which χ 2 BaBar {dof " 1 at a given value of C s , which Figure 3: Left: in the C a 1 -C a 2 plane for the respective best-fit value of C s (see Table 3), from L3 normalization and slope, Bpf 1 Ñ ργq, r ργ , and σpe `e´Ñ f 1 π `π´q .The gray ellipse represents the result of the global fit.Right: same figure for the global fit including, in addition, Bpf 1 Ñ ϕγq.
should provide a reasonable measure of the consistency of the encompassed values of C a 1{2 with the experimental constraints.We repeat this procedure for the relevant range of C s and formulate the resulting constraint on C a 1{2 in terms of an ellipse whose parameters are interpolated as a function of C s .The final constraint is then written as where ∆y a " ˜Ca 1 ´Cp0q with central values C p0q a 1{2 and covariance matrix C a , determined via the χ 2 BaBar {dof " 1 contour ellipse, implicitly depending on C s .This effective χ 2 function defined in Eq. (5.2) is then used as input in the global fit, counted as two data points in the number of degrees of freedom.This procedure is further motivated by the fact that the constraints imposed by the cross-section measurements at different energies will be highly correlated, since, if the upper limit is fulfilled at some point s i for a set of couplings C a 1 , C a 2 , C s , the smoothness of the cross section makes it likely that the same holds true at neighboring points as well.

Global fit
The results of the global fit are summarized in Table 3, Fig. 3, and Fig. 4, for variants with and without the U p3q constraint from Bpf 1 Ñ ϕγq.Without this input, we observe reasonable consistency among the various constraints, with a final value for C s close to the L3 value in Eq. (4.1).The coupling C a 1 comes out consistent with zero, while a non-zero value of C a 2 is obtained at 2σ significance.Crucially, owing to the inclusion of the BaBar data on e `e´Ñ f 1 π `π´ [ 74, 75], we are now able to provide an unambiguous solution for all three TFFs, including the two antisymmetric ones encoded in C a 1{2 .The best-fit point lies within the ellipse from σpe `e´Ñ f 1 π `π´q and, accordingly, the central line in Fig. 4 respects the bound for almost all data points, leaving a deficit that could be well explained by a ρp2150q-resonance signal.Moreover, the resulting prediction for Bpf 1 Ñ e `e´q is consistent with SND [73], suggesting a potential signal at the lower end of their range.In contrast, the prediction for Bpf 1 Ñ ϕγq comes out slightly too large in comparison to Ref. [111], in tension at the level of 1.5σ.The same tension is visible in the global fit including Bpf 1 Ñ ϕγq, as the χ 2 {dof deteriorates appreciably.Including the resulting scale factor S " 2.1 in the error estimates, all three couplings are consistent with the global fit without Bpf 1 Ñ ϕγq, but C s decreases compared to L3 and the central value of C a 2 moves much closer to zero.Within the sizable uncertainties, the cross section for e `e´Ñ f 1 π `π´i s still consistent, but the central line exceeds the data above the ρp2150q, in accordance with the best-fit point in Fig. 3 lying slightly outside the σpe `e´Ñ f 1 π `π´q ellipse.The resulting prediction for Bpf 1 Ñ e `e´q is still consistent with SND, and Bpf 1 Ñ ϕγq now agrees by construction.Table 3 also includes the predictions for Bpf 1 Ñ ωγq, Bpf 1  1 Ñ ϕγq, and Bpf 1 1 Ñ ργq, the latter two being related to the already determined couplings via the U p3q arguments in Sec.5.4.

Final representations
To summarize, we propose that the low-energy contributions to the TFFs of the f 1 be described by the parameterizations 2 q pq 2 see Eq. (3.6), Eq. (3.7), Eq. (3.10), and Eq.(3.11), with couplings C s , C a 1 , and C a 2 as determined in Table 3 (and R ρ " 1).These low-energy contributions are then to be supplemented by the asymptotic contributions from the LCE, see Sec. 2, to arrive at a complete description.
In order to estimate the impact of f 1 1 and a 1 , we also quote the corresponding expressions that follow from U p3q symmetry.For the f 1  1 , the analogous results are obtained by replacing where the errors only refer to the uncertainties propagated in θ A , cf.Eq. (3.2) and Eq.(3.11).
The coefficients in Eq. (5.5) show that isoscalar contributions will become much more important for the f 1 1 than for the f 1 , especially the ϕ.This observation is reflected by some evidence for a signal in the decay to the ϕγ final state, Bpf 1 1 Ñ ϕγq " 3p2q ˆ10 ´3 [92], which, within uncertainties, agrees with the predictions from Table 3 for the fit including Bpf 1 Ñ ϕγq, while the fit without Bpf 1 Ñ ϕγq predicts a larger branching fraction.The same reference also gives a limit Bpf 1 1 Ñ ργq ă 0.02 (95% C.L.), in agreement with both fits from Table 3.
The TFFs of the a 1 display a different isospin structure, with one isoscalar and one isovector photon each.Moreover, for ideal mixing, there is no contribution from the ϕ and its excitations, so that only contributions of the ρ-ω type survive.Accordingly, the overall scaling compared to the couplings in the f 1 TFFs is measured relative to the sum of all isovector and isoscalar contributions, leading to Choosing a symmetric decomposition of ζ V onto the ρ and ω contributions, we obtain

Conclusions
The transition form factors (TFFs) of axial-vector mesons are key input quantities for a data-driven evaluation of hadronic light-by-light (HLbL) scattering in the anomalous magnetic moment of the muon, yet they are notoriously poorly determined from experiment.Here, we performed a global analysis of all experimental constraints available for the f 1 p1285q and outlined how the f 1 p1420q and a 1 p1260q contributions can be estimated from U p3q symmetry.A crucial role is played by data for the cross section of e `e´Ñ f 1 π `π´, which provide valuable input on the asymptotic behavior and allowed us to find an unambiguous solution also for the antisymmetric TFFs.
The process e `e´Ñ f 1 π `π´p robes all three TFFs at one photon virtuality determined by the center-of-mass energy and the other one by the π `π´i nvariant mass, which in turn is dominated by the ρp770q.Accordingly, the data extending from threshold up to about 4.5 GeV are sensitive to the asymptotic behavior for one virtuality fixed at the ρ mass.The corresponding constraint demonstrates that the asymptotic behavior predicted by the light-cone expansion needs to set in early, for otherwise, the cross section exceeds data by an order of magnitude.We implemented this conclusion using a vector-mesondominance ansatz, leading to the parameterizations summarized in Sec.5.4.To account for contributions from even higher excited ρ resonances, such as the ρp2150q, we formulated the quantitative analysis as an upper limit, which still entails valuable constraints especially on the otherwise poorly determined couplings characterizing the antisymmetric TFFs.The global fit, see Sec. 5.3, shows good consistency with data for e `e´Ñ e `e´f 1 and f 1 Ñ ργ, predicting a branching fraction for f 1 Ñ e `e´a t the lower end of the signal strength reported by SND.Some tension is observed with f 1 Ñ ϕγ, which might point towards limitations of U p3q symmetry and/or the data base.
The final parameterizations describe the TFFs at low and intermediate virtualities, to be supplemented by an additional term from the light-cone expansion [61,62] that ensures the correct asymptotic behavior also in the doubly-virtual direction.Using this combined input, work is ongoing to evaluate the axial-vector contributions both in the HLbL basis of Ref. [55] and in the formalism of Ref. [60].In combination with the short-distance constraints from Refs.[47][48][49], the results presented here will thus be instrumental to arrive at a complete data-driven evaluation of HLbL scattering and to reduce the uncertainties to the level required by the final precision expected from the Fermilab experiment.[112] as used in this work (first panel), in comparison to the ρp770q, ρp1450q, and ρp1700q parameters from Refs.[124,125].

Figure 4 :
Figure 4: Comparison of our global fit results to the BaBar data[74,75] for σpe `e´Ñ f 1 π `π´q , without (left) and including (right) the constraint from Bpf 1 Ñ ϕγq.The red line denotes the central result and the band reflects the uncertainties propagated from C s , C a 1 , and C a 2 .

Table 1 :
Numerical values for the coefficients D I i in Eq. (4.13) (obtained using the Cuba library ϕγ

Table 4 :
Masses and decay widths from Ref.