Dispersive analysis of the $\gamma\gamma^{*} \to \pi \pi$ process

We present a theoretical study of the $\gamma\gamma^{*} \to \pi^+\pi^-, \pi^0\pi^0$ processes from the threshold through the $f_2(1270)$ region in the $\pi\pi$ invariant mass. We adopt the Omn\`es representation in order to account for rescattering effects in both s- and d-partial waves. For the description of the $f_0(980)$ resonance, we implement a coupled-channel unitarity. The constructed amplitudes serve as an essential framework to interpret the current experimental two-photon fusion program at BESIII. They also provide an important input for the dispersive analyses of the hadronic light-by-light scattering contribution to the muon's anomalous magnetic moment.


Introduction
It is still an open question whether a current ultra-precise (g − 2) µ measurement can probe the physics beyond the Standard Model. The presently observed 3 − 4 σ deviation between theory [1,2,3] and experiment [4] has a potential to get more significant once results from new measurements both at FER-MILAB [5] as well as at J-PARC [6] will be available. On the other hand, the current theoretical error entirely results from hadronic contributions. The hadronic uncertainties mainly originate from the hadronic vacuum polarization (HVP) and the hadronic light-by-light (HLbL) processes. Forthcoming data from the high luminosity e + e − colliders, in particular from the BESIII and Belle-II Collaborations will further reduce the uncertainty in the HVP over the next years to make it commensurate with the experimental precision on (g − 2) µ . The remaining hadronic uncertainty results from HLbL, where apart from the pseudo-scalar pole contribution, a further nontrivial contribution comes from the two-particle intermediate states such as ππ, πη and KK.
The rescattering of ππ and πη are responsible for the contribution from f 0 (500), f 0 (980), f 2 (1270) and a 0 (980) which can be taken into account in a dispersive framework. Among those, only f 2 (1270) can be attributed as a genuine QCD state, i.e., state that does not originate from long-range interactions [7]. Given the fact that it is relatively narrow, its contribution to the (g − 2) µ can be accounted in two ways: using a pole contribution as it is given in [8] (updated in [9] using recent data from the Belle Collaboration [10]), or through fully dispersive formalisms [11] and [12] with the input from γ * γ * → ππ. The comparison will shed light into the effective resonance description of other resonances such as axial-vector contributions [1,8].
In this letter, we present an analysis of the double virtual photon fusion reaction with pions in the final state. Our approach relies on the modified Muskhelishvili-Omnès formalism, which proves to be efficient in the description of the real photon data [13]. Within the maximal analyticity assumption [14], all the non-analytic behavior of the amplitude should be coming from the unitarity and crossing symmetry constraints. Therefore in order to write the dispersion-integral representation for the partial wave helicity amplitudes, one needs to make sure that they are free from kinematic constraints at thresholds or pseudothresholds. The critical step in finding these constraints is the decomposition of the amplitude into Lorentz structures and invariant amplitudes [15]. The latter are expected to satisfy the Mandelstam dispersion-integral representation [16]. Once a suitable set of Lorentz structures is found, the rest is straightforward. Our work is a continuation of a previous work where, for the first time, the single virtual case for the d-wave has been studied [17]. In the double virtual photon case, there is an additional complication related to the anomalous threshold behavior as it was pointed out in [18]. We will show an alternative way of taking this contribution into account using an appropriate contour deformation.

Kinematic constraints
The two-photon fusion reaction γ * γ * → ππ is a subprocess of the unpolarized double tagged process e + (k 1 )e − (k 2 ) → e + (k 1 )e + (k 2 )π(p 1 )π(p 2 ) which is given (in Lorenz gauge) as with q 1 ≡ k 1 − k 1 , where the momenta of leptons k 1 and k 2 are detected. This corresponds with the kinematical situation where the photons with momenta q 1 and q 2 have finite spacelike virtualities, q 2 1 = −Q 2 1 and q 2 2 = −Q 2 2 . By contracting the hadronic tensor H µν with polarization vectors, one defines helicity amplitudes H λ 1 λ 2 which can be further decomposed into partial waves is a Wigner rotation function and θ is the c.m. scattering angle. In Eq. (2), N = 1 for γ * γ * → ππ and N = 1/ √ 2 for γ * γ * → KK to ensure the same unitarity relations for the identical and non-identical particles in the case of I = 0.
It is well known that p.w. amplitudes h (J) λ 1 λ 2 may have kinematic singularities or obey kinematic constraints [19,20]. Therefore it is important to find a transformation to a new set of amplitudes which are more appropriate to use in partial-wave dispersion relations. The key step is to decompose the scattering amplitude into a complete set of invariant amplitudes [15] where i satisfies a gauge invariance constraint, i.e. q 1µ L µν i = q 2ν L µν i = 0. The numbering of the Lorentz structures is chosen such that in the single virtual case only L µν 1,2,3 contribute to the process [17] while in the real photon case only L µν 1,2 are relevant, which coincide with the tensor structures used in [21,22]. The invariant amplitudes F i are free from kinematic singularities or constraints and depend on the Mandelstam variables, which we choose as, s = (q 1 + q 2 ) 2 , t = (p 1 − q 1 ) 2 and u = (p 1 − q 2 ) 2 . The prefactor (t − u) in front of the tensor L µν 3 is chosen so as to make all five amplitudes F i even under pion and photon crossing symmetry (t ↔ u) [23,24]. We emphasize that the basis (4) is minimal and nondegenerate only for the Born subtracted amplitudes, while the Born terms itself possess a double pole structure in the softphoton limit, as a manifestation of Low's theorem [25]. The kinematic constraints can be obtained by analyzing projected helicity amplitudes in terms of the quantities, which are free of any singularities due to the properties of the Legendre polynomials [20]. In Eq. (5), q and p are initial and final relative momenta in the c.m. frame. Due to specifics of our basis (4) all the results below are given for the Born subtracted p.w. amplitudesh (J) where for s-wave it holds [11] h (0) with 2). Note that in the single virtual or real photon cases these constraints are required by the soft-photon theorem [25] and have been implemented already in [23,26]. The kinematically uncorrelated amplitudes for the s-wave can be written ash In [17] the kinematically unconstrained basis of the partial wave amplitudes were derived for the single virtual case. Below we extend this result for the double-virtual case for J = 2, with where λ is the Källén triangle function andh +,−,0 were introduced for conveniencē We emphasize that in addition to the s (±) kin points the p.w. amplitudes for J 0 exhibit a so-called centrifugal barrier factor at 4 m 2 π . The new set of amplitudes can be obtained similar to (8) using a (5 × 5) transformation matrix K i j where i = 1, .., 5 and j ≡ λ 1 λ 2 = {++, +−, +0, 0+, 00} 1 . We emphasize, that Eq.(9) shows the correlation of the p.w helicity amplitudes explicitly, as compared to the result based on the Roy-Steiner equations [27,18], where kinematic constraints are hidden in the integral kernels. The full set of these off-diagonal kernels is given in [18], and the final solution is obtained by diagonalization of the kernel matrix.

Dispersion relations
The new set of amplitudesh (J) 1−5 contains only dynamical singularities. These are right and left-hand cuts and one can write a dispersion relation in the following form where we used that Disch (J) i (s) = Disc h (J) i (s) along the righthand cut. The latter is determined by the unitarity condition and in the elastic approximation is given by where ρ(s) is a two-body phase space factor and t (J) (s) is the hadronic scattering amplitude, which is normalized as Im(t (J) ) −1 = −ρ. For the energy region above 1 GeV, it is necessary to take into account the inelasticity. The first relevant inelastic channel is KK which is required to capture the dynamics of the f 0 (980) scalar meson. For the coupled-channel case, the phase-space function ρ(s) and the amplitude t (J) (s) turn into (2 × 2) matrices, while h (J) i will be written in the (2 × 1) form with elements h (J) i and k (J) i which correspond to γ * γ * → ππ and γ * γ * → KK amplitudes, respectively. The solution to Eq. (13) is given by the well known Muskhelishvili-Omnès (MO) method for treating the final-state interactions [28]. It is based on writing a dispersion relation forh (J) i (Ω (J) ) −1 [13], where Ω (J) is the Omnès function which satisfies a similar unitarity constraint Disc In result we obtain, which can be straightforwardly generalized for the coupledchannel case. The Born subtracted amplitudes along the lefthand cut (second term inside the brackets) are given by multipion exchanges in the t and u channels which in practice can 1 Note that when Q 2 1 = Q 2 2 (and pions in the final state) special care is required. In that case, H +0 = −H 0+ and only four Lorentz tensors in (4) are independent. Therefore one needs to reshuffle (9,12) in such a way that only four amplitudesh (J) i survive. We checked that numerically the results for Q 2 1 ≈ Q 2 2 given by (9) are consistent with the strict Q 2 1 = Q 2 2 limit. be approximated by resonance (R) exchanges [13]. The dominant contribution is generated by vector mesons ω and ρ. The contribution from other heavier resonances will be absorbed in an effective way by allowing for a slight adjustment of the V Pγ coupling [17].
Here we note that there is a freedom of writing the DR. In principle, one could write a dispersion relation for the combi- (Ω (J) ) −1 as it was done for γγ * → ππ in [23]. However in this case the nonphysical asymptotic behaviour of any Lagrangian based approach (Re h (J),R ∼ s J R ) will significantly limit the range of applicability of the dispersion result. In contrast, the discontinuity along the left-hand cut (at least for vector mesons exchanges) is asymptotically bounded at high energy and does not have any polynomial ambiguities [13].

Left-hand cuts
The generalization of the Born contribution to the case of off-shell photons is performed by multiplying the scalar QED result by the electromagnetic pion (kaon) form factors [29,24] which lead to the following invariant amplitudes where i = π (K) for γ * γ * → ππ (KK). We note that the double pole structure of the Born amplitudes does not bring an extra complication to Eq. (16), since its singularities lie outside of the physical region. The electromagnetic spacelike pion and kaon form factors in the region Q 2 1 GeV 2 are parameterized by simple monopole forms yielding the following mass parameters Λ π = 0.727 (5)  The vector-meson exchange left-hand cuts are obtained by the effective Lagrangian which couples photon, vector (V) and pseudoscalar (P) meson fields, where

F Vexch
where in the following we will use g V Pγ C ρ ±,0 π ±,0 γ C ωπ 0 γ /3 as the only fit parameter, as discussed in [17], yielding g V Pγ = 0.33 GeV −1 . This value lies within 10% with the PDG average g PDG V Pγ = 0.37(2) [4], thus justifying the approximation of left-hand cuts by vector mesons. The slight difference accounts for the contribution from other heavier left-hand cuts, which in general should be taken into account by imposing Regge asymptotics. Such a study is however beyond the scope of the present analysis. In Eq. (19) f V,π (Q 2 i ) are vector meson transition form factors. For the ω, we use the dispersive analysis from [30] (see also [31]), while for the ρ (sub-dominant) contribution we use the VMD model [32]. We note, that the form factors are well defined only for the pole contribution. Using the fixed-s Mandelstam representation, one can show that the vector pole contribution corresponds to replacing t and u by m 2 V in the numerators of Eq. (19). This is different compared to Eq. (17) where the pion pole contribution coincides exactly with the scalar QED Born contribution multiplied by the electromagnetic pion form factors [29,24]. We emphasize that for the DRs written in the form (16) only Disc h (J),V λ 1 λ 2 (s) is required as input, which is unique for the vector-pole contribution.

Analytic structure of the left-hand cuts
In order to find a solution of the dispersion relations given in (16), one needs to understand the singularity structure of the p.w. amplitudes h (J) i as a function of the complex variable s. For the space-like photons the p.w. Born amplitudes are real functions above the threshold and do not bring any complexity. On the other hand, the vector-meson exchange left-hand cut is determined by four branching points: s = 0, s = −∞ and When one photon is real, the cut consists of two pieces: (−∞, s (−) L ] and [s (+) L , 0]. However, when both photons carry a space-like virtuality, one has to be careful, since for Q 2 1 Q 2 2 > (m 2 V −m 2 π ) 2 the left-hand branch point s (−) L moves to the right and reaches the pseudo-threshold point s (+) kin and only then moves to the left (see Fig.1). In this case the integration along the cut acquires an additional piece [s (−) L , s (+) kin ] which is related to an "anomalous" discontinuity [33]. In addition, the integral around s (+) kin , in general, is non zero and requires a special care [18]. Indeed, according to (9), the J = 2 p.w. amplitude schematically . Splitting the contour path into an integral up to s (+) kin − and a circular integral of radius around s (+) kin (dashed curve in Fig.1) produces the cancellation of two singular pieces and require accurate numerical implementation [18]. We follow here a different strategy and enlarge the contour around s (+) kin such that one stays away from possible numerical issues related to the anomaly piece (see Fig.1). We propose to present h V (s) in the physical region as The location of s j is determined by the condition that the imaginary part of the logarithm in (21) changes sign and therefore requires a proper choice of the Riemann sheet which we want to avoid. The merit of (22) is such that it works for both anomaly and non-anomaly cases, so one can use it for any space-like Q i including the "transition" line when In addition, it is independent on the degree of singularity and can be used equally well for higher p.w. with J > 2. The generalization to the physical case with Omnès functions (16) is then straightforward since all of the quantities are well defined at complex energies.
For time-like virtualities (which are not of interest in the present work) we refer the reader to [23,34] where different cases of overlapping left and right hand cuts are considered.

Hadronic input
For the s-wave isospin I = 0 (I = 2) amplitude we use the coupled-channel (single channel) Omnès function from a dispersive summation scheme [19,36] which implements constraints from analyticity and unitarity. The method is based on the N/D ansatz [37], where the set of coupled-channel (single channel) integral equations for the N-function are solved numerically with the input from the left-hand cuts which we present in a model-independent form as an expansion in a suitably constructed conformal mapping variable. These coefficients in principle can be matched to χPT at low energy [38].
For the d-wave I = 0, 2 amplitudes we use the single-channel Omnès function in terms of the corresponding phase shifts, Its numerical evaluation requires a high-energy parametrization of the phase shifts. We use a recent Roy analysis [39] below 1.42 GeV, and let the phase smoothly approach π (0) for I = 0 (I = 2) respectively.

Discussion and results
In Figs. 2 and 3 we plot the γ * γ * → ππ cross sections which involve either two transverse (T T ) photon polarizations or two longitudinal (LL) photon polarizations or one transverse and one longitudinal (T L) photon polarization defined by where for the the neutral pions one has to include a symmetry factor of 1/2. The quantities σ T T , σ T L , σ LT and σ LL enter the cross section for the process e + e − → e + e − ππ given in Refs. [42,43]. It sets the convention for the flux factor, while the convention for the wave functions of the longitudinally polarized photons is chosen as This convention reproduces continuously the real photon limit. Using unsubtracted dispersion relations, we postdict the cross-sections for the real photon case and give predictions for finite virtualities. We implement rescattering in s and d -waves, while the partial waves beyond are approximated by the Born terms. Including Born left-hand cuts alone predicts a reasonable description of the f 0 (500) and f 0 (980) regions, however fails to describe f 2 (1270) resonance. For the latter, the inclusion of heavier left-hands cuts is necessary [13]. Following our previous work [17], we approximate them with only vector mesons exchanges and slightly adjust the coupling g V Pγ in Eq. (19) to reproduce the f 2 (1270) peak in the γγ → π 0 π 0 cross-section. We emphasize that this is the only parameter that we adjust to the real photon data to get a nice overall agreement (see Fig.2). We also note that the convergence of the unsubtracted dispersive integrals for J = 2 is in general better than for J = 0 due to the centrifugal barrier factor. Therefore, including vector meson left-hand cuts in the s-wave requires adding at least one subtraction, which can be fixed from χPT. We checked that for relatively small Q 2 , the results of the two solutions are very similar. Since the finite Q 2 prediction from χPT are expected to get large corrections for Q 2 > 0.25 GeV 2 we decided to stay with the unsubtracted DR. In the present letter we show a selected result 2 for a fixed value Q 2 1 = 0.5 GeV 2 for one photon virtuality and different values Q 2 2 = 0.25, 0.5, 0.75, 1.0 GeV 2 for the second photon virtuality (see Fig.3). The last two Q 2 2 points are above the anomaly point. For the σ T T and σ LL we emphasize the importance of the unitarization, which significantly increases the pure Born prediction at low energy. For σ T L , we notice that the helicity-1 contribution increases with increasing virtualities.
It is instructive to compare our approach with dispersive studies based on the Roy-Steiner equations [18]. Apart from a different strategy of treating kinematic singularities and anomalous thresholds, the latter analyses suggests that s-wave couples to d-wave, with a strength related to the high energy assumption. Moreover, an extra subtraction produces a 1/s singular behavior [18]. We agree that in general, the hyperbolic DR is supposed to a more fundamental starting point than a p.w. dispersion relation, because it implements a crossing symmetry exactly. However, this not the case when it is solved using MO analysis for s and d-waves. Any truncated p.w. expansion introduces an incorrect dependence on the cross-channel energy variable, and the high energy asymptotic on the p.w. helicity amplitude can not be rigorously connected to the asymptotic behavior of the invariant amplitudes. Mathematically, the truncation of p.w. expansion translates into an arbitrariness in choosing the boundary condition for the solution of an integral equation and should be implemented by hand. Under an assumption 2 The preliminary plots for Q 2 1 = Q 2 2 = 0.5 GeV 2 shown in [3] suffered from a numerical instability in the calculation of one of the five dispersive integrals, which led to an overestimation of σ LL in the f 2 (1270) region, leaving the predictions for σ T T and σ T L mainly unchanged. γ * γ * → π + π -   γ * γ * → π + π -Q 1 2 =0.5 γ * γ * → π 0 π 0 Figure 3: Predictions for σ T T , σ T L , σ LL cross sections for γ * γ * → π + π − (left panels) and γ * γ * → π 0 π 0 (right panels) for Q 2 1 = 0.5 GeV 2 and Q 2 2 = 0.25, 0.5, 0.75, 1.0 GeV 2 and for full angular coverage | cos θ| ≤ 1. The Born results are shown by dotted curves. of maximal analyticity, the p.w. dispersion relation proves to be a transparent approach where an additional subtraction or the treatment of many coupled channels is straightforward. Nevertheless, getting an agreement of two independent analyses will show the robustness of the input to the (g − 2) µ and at the same time shed the light on possible systematic uncertainties [44].

Conclusion
In this work, we have presented a dispersive analysis of the γ * γ * → ππ reaction from the threshold up to 1.5 GeV in the ππ invariant mass. For the s-wave, we used a coupled-channel dispersive approach in order to simultaneously describe the scalar f 0 (500) and f 0 (980) resonances, while for the d-wave a single channel Omnès approach is adopted. The obtained results will serve as one of the relevant inputs to constrain the hadronic piece of the light-by-light scattering contribution to the muon's a µ [11,12]. Especially it allows to estimate the contributions from f 0 (500), f 0 (980) and f 0 (1270), where the latter can be compared with the narrow resonance result [8,9].
There are still few open issues before it can implemented in a (g − 2) µ calculation. First, one needs to validate a current treatment of left-hand cuts by forthcoming BESIII data on the γγ * → π + π − and γγ * → π 0 π 0 reactions [45]. This is a prerequisite for a data-driven approach in quantifying the uncertainty of the HLbL contribution to a µ . Second, for higher Q 2 one has to incorporate constraints from perturbative QCD for the vector transition from factors f V,π (Q 2 ) which is the driving force governing the Q 2 dependence of the f 2 (1270) resonance [3]. This will be investigated in a future work.