Radial excitation of light mesons and the $\gamma \gamma^* \rightarrow \pi^0$ transition form factor

We investigate radial excitation of the quark-antiquark pair in the $\pi^0$ meson and its effects on the $\gamma \gamma^* \rightarrow \pi^0$ transition form factor in the framework of light-cone perturbative QCD. The existing constraints on the light-cone wave function of the lowest Fock state $|q { \bar q} \rangle$ in the $\pi^0$ meson allow a sizeable radial excitation of the quark-antiquark pair. We construct the light-cone wave function for the quark-antiquark pair in the first radially excited state (the 2S state) using a simple harmonic oscillator potential. The distribution amplitude obtained for the 2S state has two nodes in $x$ at low scale of $Q$ and thereby has a much strong scale dependence than the 1S state. Contributions from this radial excitation to the $\gamma \gamma^* \rightarrow \pi^0$ transition form factor exhibit different $Q^2$-dependence behavior from the ground state and thus can modify the prediction for the transition form factor in the medium-large region of $Q^2$.


I. INTRODUCTION
Light-cone perturbative QCD has been applied to the calculations for many inclusive and exclusive hadronic processes. The test of these calculations against available experimental data, particularly for many exclusive processes, is usually retarded by the possible higher order and higher twist contributions to the theoretical calculations at low and medium regions of momentum transfer (Q 2 ) and the limited availability of experimental data at high Q 2 . The measurements of the γγ * → π 0 transition form factor (TFF) [1,2], the simplest QCD process involving a hadron, posed a very interesting challenge to the theoretical calculations. While the results from the BaBar Collaboration [1] show a rapid growth for Q 2 > 15 GeV 2 and a significant deviation from the asymptotic prediction from perturbative QCD, the results from the Belle Collaboration [2] are much more in agreement with theoretical expectations. A lot of theoretical studies have been done (see, for example, [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]) in explaining these measurements for the pion TFF.
In this work, we investigate the possible radial excitation of the quark-antiquark (qq) pair in the π 0 and study its contribution to the γγ * → π 0 transition form factor. The excited qq pair, when in the first radially excited state with the lowest angular momentum (i.e. the 2S state), has the same quantum numbers as the ground state (the 1S state), i.e. J P C = 0 −+ ; such an excitation does not violate any fundamental principles of QCD and thus is allowed in the quark models of hadrons. In Section II, we construct the light-cone wave function for the qq pair in the 2S state using the wave function in the arXiv:2109.02856v1 [hep-ph] 7 Sep 2021 center-of-mass (CM) frame for the simple harmonic oscillator potential, and the Brodsky, Huang and Lepage (BHL) prescription [22,23] for connecting the equal-time (instant-form) wave function in the centre-of-mass frame and the light-front wave function. We exam the existing constraints on the pion light-cone wave function and the possibility for the qq pair in the 2S state. In Section III, we calculate the contribution of the possible 2S state to the γγ * → π 0 transition form factor. A summary is given in the last Section.

II. LIGHT-CONE WAVE FUNCTION AND DISTRIBUTION AMPLITUDE WITH RADIAL EXCITATION
Light-cone wave functions (LCWFs) are universal and contain all nonperturbative information of partons in the hadrons. The light-cone wave functions can be deduced from experimental data and/or from nonperturbative QCD computations [24]. In this study we adopt a widely used method which was first suggested by Brodsky, Huang and Lepage [22,23] in computing the LCWFs for the mesons. In this method, a connection between the equal-time (instant-form) wave function in the centre-of-mass (CM) frame, usually computed using effective potentials for the qq pair, and the light-front wave function is where q i is the momentum of constituent i in the CM frame, and x i and k ⊥i are the light-cone momentum fraction and transverse momentum of the constituent i in the light-cone frame. For a two-identical- and thereby can establish a relationship for the wave functions both in the light-cone form ψ(x, k ⊥ ) and in the equal-time form ψ(q), For the equal-time wave function in the CM frame, we employed the wave function computed with the harmonic oscillator potential for the quark-antiquark system, V (r) = 1 2 mω 2 r 2 , where r is the separation between the quark and antiquark. The wave functions for the ground state (1S) and the first radially excited state (2S) are [25], where B = 1 2 mω. The wave functions in the momentum space can be written in the form of Using relationship Eq. (2) one can obtain the spatial parts of light-cone wave functions for the 1S and 2S states. The obtained LCWF for the ground state is the well-known Gaussian form. In the widely-used factorized Gaussian form for the LCWF the dependence on the quark mass m is absorbed into the overall normalization factor for the wave function. Thus we use the following forms of the LCWFs for the qq pair in the 1S and 2S state, where the parameters a 1 , a 2 and B can be fixed, in principle, by considering existing constraints on the pion LCWF.
The pion distribution amplitude (DA) is defined in the light-front formalism as the integral of the valance qq LCWF in light-cone gauge A + = 0, where µ is an arbitrary scale. Thus one can define the distribution amplitudes for the 1S and 2S states, φ 1S (x, µ) and φ 2S (x, µ), using the LCWFs given by Eqs. (7) and (8), Both distribution amplitudes turn into the asymptotic form, φ(x) ∼ x(1−x), at large scale of µ. However, the exponential dependence on the scale µ, via the combination −µ 2 /[4Bx(1 − x)], means the DAs could be substantially different from the asymptotic form for small µ. Furthermore, the DA for the 2S state have two nodes, with negative values at the medium x region for µ 2 < 11.35B (see Fig. 2).
Four constraints have been identified in determining the parameters in the LCWF of the lowest Fock state of the pion [26]. The lepton decay of π → µν suggests where f π = 92.4 MeV is the pion decay constant. This constraint depends on the choice of the scale µ, although it is common practice in the literature to choose a high enough value for µ in aiming to reduce this dependence on the choice of scale for any calculations.
The second constraint is obtained [22] by relating chiral anomaly prediction for the π 0 → γγ decay width to the light-cone formalism prediction for the TFF at the limit Q 2 → 0, The prediction for the TFF at Q 2 → 0 in the light-cone formalism involves a divergent integral and relies on the expansion of the wave function under the condition of q ⊥ k ⊥ (with Q 2 = q 2 ⊥ ) [22]. This prescription could be invalid when the 2S state is included in the calculation since the wave function for the 2S state given by Eq. (8) has an additional dependence on transverse momentum apart from the exponential factor.
The third constraint is the result from requiring the probability of finding the qq Fock state in the pion to be smaller than 1, With the wave functions given by Eqs. (7) and (8), the probabilities are found to be P 1S qq = 12π 2 for the 1S and 2S states, respectively. The requirement of the probability being smaller than 1 puts constraints on the combination of parameters a 1 and a 2 with B.
The fourth constraint comes from considerations of the average quark transverse momentum of the pion which is defined as the root-mean-squared for the quark transverse momentum, k ⊥ = k 2 ⊥ qq , with It is known experimentally that the average quark transverse momentum of the pion should be about a few hundreds MeV. Thus the average quark transverse momentum of the pion in the lowest Fock state should be in the range of a few hundreds MeV. For the wave functions given by Eqs. (7) and (8) Equations (12), (13), (14) and (15) form a set of constraints on the pion LC wave function. Applying those constrains for the 1S state, we have Requiring k ⊥

1S
qq to be in the range of (300 ∼ 500) MeV together with those constraints we have a 1 = √ 3/f π and B = (0.0843 ∼ 0.156) GeV 2 , which suggests the probability P 1S qq is in the range of 0.25 ∼ 0.46 1 . The obtained probability suggests the higher Fock state components such as |qqg , |qqqq etc and/or the radial excitation of the lowest Fock state |qq in the pion are significant. While there are limited discussions on the former possibility, we are not aware of any discussion on the latter possibility and its significance as we mentioned in the previous section.
The parameter a 2 cannot be determined by using the first and the second constraints [Eq. (12) and Eq. (13)] since the DA for the 2S state have two nodes when µ is not very large and the contribution from the 2S state to the integral in Eq. (12) could vanish while it is not clear whether Eq. (13) is applicable for the 2S state. We will set a 2 to be such a value that the probability for the 2S state is 1/2 of that for the 1S state and study its effect on the π 0 → γγ transition from factor in the next section. Taking

III. THE EFFECTS ON THE γγ * → π 0 TRANSITION FORM FACTOR
The existence of radially excited components in the pion, though not at a very significant level, may have potential impacts on various processes. We study the effects on the γγ * → π 0 transition form factor in the framework of light-cone perturbative QCD. The transition form factor can be written as, when the involved momentum transfer Q 2 is large [27][28][29], where ψ π (x, k ⊥ ) is the pion light-cone wave function and T H is the hard scattering amplitude, The LCWFs for both the 1S and 2S states, Eqs. (7)and (8), depend on the transverse momentum through k 2 ⊥ . The transition form factor can be expressed in terms of the distribution amplitudes for the 1S and 2S sates, The first term in Eq. (22) represents the contribution from the 1S state while the second term represents the contribution from the 2S state. The ratio of the contribution to the transition form factor from the 2S state to that from the 1S state, R(Q 2 ) = F 2S πγ (Q 2 )/F 1S πγ (Q 2 ) is shown in figure 3. The contribution from the 1S state is positive for all Q 2 . Figure 3 shows that the contribution from the 2S state changes sign from negative to positive at Q 2 just above 6 GeV 2 . In terms of the magnitude, the contribution from the 2S state is about 40% of that from the 1S state at Q 2 = 1 Gev 2 and 25% at Q 2 = 40 GeV 2 .
It is the combinationxQ that sets the scale in the DAs, thus the TFF at a scale Q is determined by the DAs at all scales up to Q 2 , rather than just the DAs at that particular scale. The 2S state DA evolves more significantly than the 1S state DA. Thus including the contribution from the 2S state will introduce stronger dependence on Q 2 than including the 1S state only for a large range of Q 2 . In figure 4 we show the contributions from both the 1S and 2S states, and the total for the transition form factor, in comparison with the data from various experimental groups. Including the 2S state contribution in the calculations makes the TFF grow faster than including the 1S state only, but the growth is still far below what is suggested by the BaBar data. This reaffirms the conclusion made in [6,10,11] [10]. A better agreement with the Bell's data could be achieved by including the next-to-leading corrections and fine turning the probability for the 2S state to be in the range of 10 ∼ 15%. The calculations for the small Q 2 region (Q 2 < 10 GeV 2 ) are much smaller than the experimental data as the nonperturbative contributions are expected to be dominant in this region [5,21].

IV. SUMMARY
In the Fock state expansion for the light mesons, the probability of finding the qq pair in the 1S state is generally very small. Analyzing all constraints for the pion light-cone wave function and distribution amplitude from considering the lepton and two-photon decays and experimental information for the quark transverse momentum, we found the probability for the pion to be in the 1S state is in the range of 25% ∼ 46%. Apart from higher Fock states involving multiple quark-antiquark pairs and gluons, the qq pair could be in the radially excited states. We constructed the light-cone wave function for the first radially excited state (the 2S state) of the pion employing a phenomenological connection between the light-cone wave function and the equal-time wave function in the center-of-mass frame which is obtained using an effective simple harmonic oscillator potential. The distribution amplitude of the 2S state at low scale has two nodes in x due to the light-cone wave function having a node in the momentum space. This is significantly different from the distribution amplitude of the 1S state which is positive in the whole range of x at any scale. The soft QCD evolution has a much stronger impact on the distribution amplitude for the 2S state than the 1S state.
We calculate the contribution from the radially excited component (the 2S state) to the γγ → π 0 transition form factor. It is found that this contribution grows faster (when the TFF is expressed as Q 2 F πγ (Q 2 )) than the 1S state. A probability for the 2S state in the range of 10% ∼ 15% gives a good agreement with experimental data from the Belle Collaboration; however, it is still very difficult to describe the fast growth shown by the data from the BaBar Collaboration.
The existence of radial excitations in the light mesons will have impacts on predictions for many observables. A study on the TFFs for the other light mesons such as the η and η might provide some insights on the nature and extent of the radial excitations in the hadrons, although the η-η mixing might complicate the conclusions one can draw from such a study. Such an analysis and further studies for the other processes will be presented in future work.