Production of axial-vector mesons at $e^+ e^-$ collisions with double-tagging as a way to constrain the axial meson LbL contribution to muon g-2 and/or hyperfine splitting of muonic hydrogen

We calculate cross sections for production of axial-vector $f_1(1285)$ mesons for double-tagged measurements of the $e^+ e^- \to e^+ e^- f_1(1285)$ reaction. Different $\gamma^* \gamma^* \to f_1(1285)$ vertices from the literature are used. Both integrated cross section as well as differential distributions are calculated. Predictions for a potential measurement at Belle II are presented. Quite different results are obtained for the different vertices proposed in the literature. Future measurements at $e^+ e^-$ colliders could test and/or constrain the $\gamma^* \gamma^* \to f_1 (a_1, f_1')$ vertices and associated form factors, known to be important ingredients for calculating contributions to anomalous magnetic moment of muon and hyperfine splitting of levels of muonic atoms.


I. INTRODUCTION
The coupling of neutral mesons to two photons is an important ingredient of mesonic physics. In Ref. [1] tensorial coupling was discussed for different types of mesons (pseudoscalar, scalar, axial-vector and tensor). In general, the amplitudes can be expressed in terms of functions of photon virtualities often called transition form factors. They were tested in details for pseudoscalar mesons (π 0 , η, η ′ ). Recently there was discussion how to calculate such objects for peseudoscalar [2] and scalar [3] quarkonia.
The axial vector mesons and in particular their coupling to photons [4] are very important in the context of their contribution to anomalous magnetic moment of muon [5][6][7][8][9][10].
The anomalous magnetic moment of muon is one of the most fundamental quantities in particle physics (see e.g. [11,12]). A first calculation of QED corrections to anomalous magnetic moment was performed long ago [13]. Recent state of art can be found e.g. in [11,12,14]. The current precision of QED calculation is so high that hadronic contributions to muon anomalous moment must be included. The so-called light-by-light (LbL) contributions are very important but rather uncertain. The coupling γ * γ * → f 1 (1285) is one of the most uncertain ingredients. Different couplings have been suggested in the literature.
Recently the contribution of the γ * γ * → f 1 (1285) coupling was identified and included in calculating hyperfine splitting of levels of muonic hydrogen, and turned out to be quite sizeable [15]. These are rather fundamental problems and better contraints on γ * γ * coupling are badly needed.
In calculating δa f 1 µ one often writes: where ρ f 1 µ (Q 2 1 , Q 2 2 ) is the density of the f 1 contribution to the muon anomalous magnetic moment. The integrand of (1.1) (called often density for brevity) peaks at Q 2 1 , Q 2 2 ∼ 0.5 GeV 2 and gives almost negligible contribution for Q 2 1 , Q 2 2 > 1.5 GeV 2 , see e.g. [7]. The γ * γ * f 1 (1285) coupling can be also quite important for hyperfine splitting of levels of muonic hydrogen [15]. It is also very important to calculate rare decays such as [16,17]. There both space-like and time-like photons enter corresponding loop integral(s) so one tests both regions simultaneously. The corresponding branch- ing fraction is very small (BF ∼ 10 −8 ). The same loop integral enters the production of f 1 in electron-positron annihilation [16,17]. There is already a first evidence of such a process from the SND collaboration at VEPP-2000 [20]. The f 1 (1285) was also observed in γp → f 1 (1285)p reaction by the CLAS collaboration [21]. The experimental results do not agree with theoretical predictions [22][23][24]. processes. The square (0, Q 2 0 )x(0, Q 2 0 ) close to the origin shows the region where the dominant contributions to g − 2 comes from. The square (Q 2 0 , ∞)x(Q 2 0 , ∞) marked in red represents the region which can be tested in double-tagging experiments. The short diagonal (Q 2 1 = Q 2 2 ) line represents region important for hyperfine splitting of levels of muonic hydrogen. The narrow strips along the x and y axis shows a possibility to study production of f 1 (1285) in e + A collisions at EIC. Marked is also the region of photon virtualities which contributes to f 1 → e + e − or to the production of f 1 (1285) in e + e − annihilation.

II. SOME DETAILS OF THE MODEL CALCULATIONS
sions. The small circle in the middle represent the γ * γ * → AV vertex tested in doubletagging experiment.
In the formalism presented e.g. in [4] the covariant matrix element for γ * γ * → f 1 (1285) is written as: where The generic diagram for e + e − → e + e − AV and kinematical variables used in this paper. 1 The same is true for other axial-vector (a 1 , f ′ 1 ) mesons. 4 In Ref. [15] the vertex was written as: In the nonrelativistic model We use the normalization of form factors In [15] the vertex was supplemented by the following factorized dipole form factor (2.7) The Λ D ≈ 1 GeV was suggested as being consistent with the L3 collaboration data [29].
We will ascribe also the name NQM (nonrelativistic quark model) to this vertex.

OPV2018 vertex
In Ref. [6] the vertex function for γ * γ * → f 1 was constructed based on an analysis of the f 1 (1285) → ρ 0 γ decay and vector meson dominance picture. The corresponding vertex for two-photon coupling there reads The value of g ρ is explicitly given in [6]. We supplemented this vertex with one common for all terms form factor of the VDM type: (2.10) consistent with the philosophy there.

LR2019 vertex
Finally we consider also the vertex used very recently in [9]. In this approach the vertex is The normalization was also given there. It was pointed out that the A(Q 2 1 , Q 2 2 ) function does not need to be symmetric under exchange of Q 2 1 and Q 2 2 . Actually asymmetric form factors calculated from the hard wall and Sakai-Sugimoto models were used there. In our evalution here we will use Hard Wall (HW2) form factors as well as factorized dipole symmetric/asymmetric form factors as specified below to illustrate the effect of the holographic approach. The HW2 form factor can be sufficiently well represented as: where Λ L > Λ S . We show the HW2 form factor and its factorized dipole approximate representation as a function of (log 10 (Q 2 1 ), log 10 (Q 2 2 )) in Fig.3.
Above we have denoted: The c A is defined in [8].
The reader is asked to note vanishing of F RS at Q 2 1 = Q 2 2 . This, as will be discussed below, has important consequences for the double tagged measurements.
The form factor used in RS2019 are antisymmetric. Additional symmetric form factors arising at higher order were discussed in a revised version of [8] (see Appendix C there).
In the following we will use the lower order result to illustrate the situation.
It was ascertained recently in [19] that the RχT approach provides only purely transverse axial-vector meson contributions.

MR2019 vertex
In Ref. [17] the following vertex was used (we change a bit notation to be consistent with our previous formulae) to the production of f 1 (1285) in the e + e − annihilation. Since in this case both space-like and time-like virtualities enter the calculation of the relevant matrix element the form 7 factors had to be generalized. In [17] the form factors were parametrized in the spirit of vector meson dominance approach as: One can see the characteristic ρ meson propagators. The F(q 2 1 , q 2 2 ) form factor is asymmetric with respect to q 2 1 and q 2 2 exchange to assure Bose symmetry of the amplitude. An extra q in the denominator was attached to the VDM-like vertex to assure "correct" behaviour of the form factors at large photon virtualities [1]. Of course, it is not obvious that such a correction should enter in the multiplicative manner. The coupling constant was found in [17]. It was allowed in [17] for g 2 to be complex. It was argue that |g 1 | ∼ g 2 to describe the first e + e − → f 1 (1285) data from VEPP-2000 [20]. We shall show in this paper how important is the interference of both terms in the DT case.

B. General requirements
Any correct formulation of the γ * γ * → f 1 (1285) vertex must fulfill at least three general requirements: • Gauge invariance requires:
Some vertices fulfil also where p is four-momentum of the axial-vector meson. This automatically guarantees that only spin-1 particle f 1 is involved and unphysical states are ignored. A related discussion can be found e.g. in [26].

C. Form factors
Some of the F(Q 2 1 , Q 2 2 ) form factors can be constraint from the so-called decay width into transverse and longitudinal photon, some are poorly know as they can not be obtained as they do not enter the formula for the radiative decay width. The radiative decay width is known [27] and isΓ γγ = 3.5 keV . (2.25) Then some of the form factors are parametrized as: (2.29) Both monopole and dipole parametrizations of form factors will be used in the following.
We will call the first two as factorized Ansatze and the next two as pQCD inspired powerlike parametrizations. 9 In general, the form factors in Eqs.(2.11) do not need to be symmetric with respect to Q 2 1 and Q 2 2 exchange [9]. For example in Ref. [9] asymmetric form factor A(Q 2 1 , Q 2 2 ) obtained in Hard Wall and Sakai-Sugimoto models were used to calculate contribution to anomalous magnetic moment of muon. Here we shall take a more phenomenological approach and try to parametrize the form factors in terms of simple functional forms motivated by physical arguments such as vector dominance model or asymptotic pQCD behaviour of transition form factors (see e.g. [28]).
The behaviour of transition form factors at asymptotia may be another important issue [18]. Where the pQCD sets in is interesting but still an open issue. It was discussed in [2] that for γ * γ * η c coupling this happens at very high virtualities. We leave this issue for the γ * γ * f 1 coupling for a future study.
The amplitude for the e + e − → e + e − f 1 reaction (see Fig.2) in high-energy approximation can be written as: (2.30) Above e 2 = 4πα em . The four-momenta are defined in Fig.2. The T ν 1 ν 2 α vertex function responsible for the γ * γ * → f 1 coupling was discussed in detail in the previous subsection.
The square of the matrix element, summed over polarizations of f 1 , can be obtained as: where P is spin-projection operator for spin-1 massive particle: The cross section for the 3-body reaction e + e − → e + e − f 1 (1285) can be written as The three-body phase space volume element reads The phase-space for the pp → pp f 1 reaction has four independent kinematical variables. In our calculation we integrate over ξ 1 = log 10 (p 1t ), ξ 2 = log 10 (p 1t ), azimuthal angle between positron and electron and rapidity of the produced axial-vector meson (four-dimensional integration). Here p 1t and p 2t are transverse momenta of outgoing positron and electron, respectively.
In the case of holographic approach first the A(Q 2 1 , Q 2 2 ) form factor entering the central vertex function (see Eq.(2.30)) is calculated on a two-dimensional grid and then the grid is used for interpolation for each phase space point (see (2.33)).

III. NUMERICAL PREDICTIONS
In Fig.4 we show a two-dimensional distribution (ξ 1 , ξ 2 ) of the full phase space cross section. Quite large cross sections are obtained for small ξ 1 and/or ξ 2 . In addition, the different models of the γ * γ * f 1 couplings lead to very different results for the total cross section. The measurement of the total cross section is, however, rather difficult.
In Fig.5 we show distributions in (t 1 , t 2 ) (four-momenta squared of the virtual photons as shown in Fig.2). Clearly some couplings generate strongly enhanced cross section at small t 1 , t 2 .
Clearly those different vertices lead to different cross sections even for very small photon virtualities where the cross section is relatively large. Could one measure inclusive cross section for production of axial-vector meson without tagging ? Is then γ * γ * → f 1 (1285) the dominant mechanism ? If yes, such measurements would verify the different vertices used in calculating δa µ (axial-vector meson contribution to a µ ).
Small Q 2 1 and Q 2 2 means small transverse momenta of f 1 (1285). Can one then identify f 1 (1285). Which channel is the best ? This requires further Monte Carlo studies. The resonant e + e − → f 1 (1285) production is very small [17] and important only at resonance in (2.30). To illustrate and explore the effect of Landau-Yang vanishing of T µνα vertex function for γ * γ * → f 1 in Fig.6 we plot the following quantity: The arbitrary scale M 0 is chosen to be M 0 = 1 GeV in the following.
One can clearly see vanishing of the special quantity (3.1) at Q 2 1 → 0 and Q 2 2 → 0 which reflects Landau-Yang theorem. Slightly different approach patterns to zero can be observed for the different couplings. For the RS coupling we observe deep valley arround Q 2 1 = Q 2 2 which is a direct consequence of the specific form factor used there. In this case Ω LY is much smaller than for other vertices in the limited range of Q 2 1 and Q 2 2 shown in the figure.

B. Double-tagging case
In Table 1  The results are also strongly dependent on the form factor used in the calculation which is discussed below. In Table 2  Now we wish to show several differential distributions for the double-tagged mode.
In Fig.7 we show distributions in rapidity and transverse momentum of f 1 (1285), t 1 or t 2 , azimuthal angle between outgoing electrons, averaged virtuality Q 2 a = (Q 2 1 + Q 2 2 )/2 (3.2) and the asymmetry parameter   The Bose symmetry requires that:

IV. CONCLUSIONS
In this paper the results of calculations of cross sections and differential distributions for the e + e − → e + e − f 1 (1285) have been performed using different γ * γ * → f 1 (1285) couplings known from the literature. These couplings were used previously to calculate hadronic light-by-light axial meson contributions to anomalous magnetic moment of muon as well as for hyperfine splitting of the muon hydrogen.
We have presented predictions relevant for future double-tagged experiments for Belle II. The results strongly depend on the details of calculation (type of tensorial coupling and/or form factors used). The form factor cannot be reliably calculated at present. We have presented several diferential distributions in photon virtualities, transverse momentum of f 1 (1285), distribution in azimuthal angle between outgoing electron and positron and so-called asymmetry of virtualities (ω). Especially the latter observable (asymmetry) seems promissing for verifying the quite different models of the γ * γ * AV coupling. The results strongly depend on details of the coupling(s). The double tagged measurement would therefore be very valueable to constrain the couplings and form factors and in a consequence would help to decrease uncertainties of their contribution to anomalous magnetic moment of muon and hyperfine splitting of muonic hydrogen.
In the present paper we concentrated on production of f 1 (1285) meson. A similar analysis could be performed for other axial-vector mesons such as a 1 (1260) or f 1 (1420).
Then coupling constants and some form factors must be changed in the calculation. On the experimental side, decay channels specific for a given meson must be selected.
The production of isoscalar axial-vector mesons is very interesting also in the context of central exclusive processes pp → pp f 1 . There the unknown ingredient is pomeronpomeronf 1 vertex. This will be discussed elsewhere [36].