Cross-link relations between $\pi$ and $\rho$-meson channels and the QCD vacuum

We discuss cross-link relations between the $\pi$ and $\rho$-meson channels emerging from two different descriptions of the QCD vacuum: Instanton physics and QCD sum rules with nonlocal condensates (NLC). We derive in both schemes an intriguing linear relation between the $\pi$ and the $\rho^\|$-meson distribution amplitudes in terms of their conformal coefficients and work out the specific impact of the scalar NLC in these two channels. Using a simple model with Gaussian decay of the scalar NLC, we are able to relate it to the moments of the pion non-singlet parton distribution function measurable in experiment -- a highly nontrivial result. The implications for the pion and the $\rho^\|$-meson DAs entailed by the obtained cross-link relations are outlined in terms of two generic scenarios.


I. INTRODUCTION
Recently, Polyakov and Son [1] have employed the instanton approach in combination with dispersion relations for the two-pion distribution amplitude (DA) [2] to obtain model-independent relations for the ratio a (ρ) 2 /a (π) 2 . Within this approach, the Gegenbauer moments of the ρ-meson DA can be expressed in terms of the two-pion distribution amplitudes (2πDAs). Using the conformal expansion ϕ (tw-2) π(ρ) (x, µ 2 ) = ψ 0 (x) + ∞ n=2,4,... a n (µ 2 )ψ n (x) (1) with the Gegenbauer basis where ϕ asy π = ψ 0 (x) = 6x(1 − x) ≡ 6xx is the asymptotic pion DA, it was found that a (ρ) 2 and a (π) 2 are linearly related. It was argued that this is deeply rooted in chiral dynamics and the general properties of quantum field theory (QFT), such as unitarity, crossing, and dispersion relations. Going one step further, they used the soft pion theorem and the crossing symmetry [2] to relate the second Gegenbauer coefficient a (ρ) 2 of the ρ-meson DA to the third Mellin moment of the pion valence-quark parton distribution function (PDF) measured in DIS. Referring for the full derivation to [1], one obtains the following relation   (3) * Electronic address: mikhs@theor.jinr.ru † Electronic address: stefanis@tp2.ruhr-uni-bochum. de The main ingredient in the last equation is the measurable quantity while c is a low-energy subtraction constant in the dispersion relation for the generalized Gegenbauer moments B nl (W 2 ) used for the 2πDAs in [2] (see there for details). Fixing this constant by virtue of the instanton model of the QCD vacuum, one can determine the ratio a determined on the lattice with a next-to-leading order (NLO) matching to the MS scheme (adding the errors in quadrature) [3], with the value in the first line below M (π) 3 (µ = 2 GeV) = 0.114 ± 0.020 [4] 0.110(7)(12) [5] , obtained at NLO in the phenomenological analysis in [4] within the xFitter framework. The second line shows the result of a recent lattice calculation [5]. One observes that the lattice estimate is within the error margin of the first line. This way one gets [1] a (ρ) 2 a (π) 2 (µ = 2 GeV) = −(1.15 ± 0.86)(1.0 ± 0.1) .
The above results provide the motivation for the present investigation with the attempt to derive similar linear relations between the π and ρ-meson channels within the QCD sum-rules approach with nonlocal condensates (NLC-SR for short) used in [6]. The key ingredients in NLC-SR are the nonlocal condensates [7][8][9][10][11]. In particular the scalar NLC Φ S (x, M 2 ), where M 2 is the Borel parameter, and the scale λ 2 q = qD 2 q qq µ 2 0 ∼1GeV 2 = 0.40 ± 0.05 GeV 2 , (8) which defines the virtuality of vacuum quarks, are crucial for the determination of the shape of the pion DA employing NLC-SR [6,12].
This approach was originally used to derive a set of pion DAs (termed Bakulev-Mikhailov-Stefanis or BMS) for λ 2 q = 0.40 GeV 2 in agreement with the CLEO experimental data [13] on the pion-photon transition form factor (TFF). It also provides good agreement with more recent data from Belle [14] and BABAR(Q 2 9 GeV 2 ) [15], see [16]. The coefficients a (π) 2 and a (π) 4 encapsulate the main theoretical characteristics of the nonperturbative QCD vacuum. Later, this scheme was employed in [17] to construct another variant of the pion DA with a platykurtic profile. This DA intrinsically combines two key features: endpoint suppression of the pion DA due to the vacuum quark virtuality λ 2 q = 0.45 GeV 2 and unimodality of the DA at x = 1/2, as found in the context of Dyson-Schwinger equations to account for the dynamically generated mass of the confined quark propagator, see [18] for a review. At the scale µ 2 = 4 GeV 2 one has the coefficients (a (π) 2 , a (π) 4 ) µ 2 pk = (0.057, −0.013). This scheme [19] also provided a platykurtic DA for the longitudinally polarized ρ-meson: (a (ρ) 2 , a (ρ) 4 ) µ 2 pk = (0.017, −0.021), while previous attempts to construct the ρ-meson DA within this NLC-SR scheme were reported in [20,21].
The paper is structured as follows. In Sec. II we compare the predictions for the π-ρ-meson coefficients following from Eq. (3) in [1]. We then derive an analogous expression to (3) using the NLC-SR scheme and discuss the dynamical difference of the impact of the scalar NLC in the axial and the vector channel. We also work out a linear relation between functionals depending on ϕ ρ and ϕ π and extend the result to higher orders of the conformal expansion. In Sec. III, we establish the linear relation between the π and ρ-meson DAs in terms of the specific characteristics of the NLC-SR. The ultimate goal of this section is to relate the scalar NLC to the third/fifth Mellin moment of the pion PDF. The implications of the negative sign of a (ρ) 2 for the pion and ρ -meson DAs are addressed in Sec. IV. Our conclusions are presented in Sec. V. The expressions for the scalar NLC are provided in Appendix A, while the correspondence between the instanton approach and the NLC-SR scheme is shown in Appendix B.  [3] limited by the red endpoints. The range a (ρ) 2 = 0.132 ± 0.027 from [22] is shown as a short red segment at the upper end of the vertical axis. The NLC-SR fiducial regions are shown for the pion by the green segment [6] and for the ρ -meson by the blue segment along the vertical axis [20]. The projections of the endpoints of the green segment and its center (green dot) are denoted by blue points on the a (ρ) 2 -axis. The yellow points mark the locations of the platykurtic pion and ρ DAs [17,19], respectively. The black points a the cross-link relations following from Eq. (3). This is done in Fig. 1 in terms of the blue diagonal line which represents the linear relation between such pairs.
The following results are displayed. i) a (π) 2 values of the set of BMS DAs [6]: green segment. ii) a (ρ) 2 range from NLC-SR in [20]: blue segment bounded by blue points in good agreement with the projections of the endpoints of the green a (π) 2 segment. iii) a (π) 2 estimates from lattice QCD with NNLO and N 3 LO matching to the MS scheme [3]: red segment and interval between the smaller red endpoints, respectively. iv) a (ρ) 2 from the lattice calculation in [22]: red segment at the upper end of the vertical axis. v) NLC-SR platykurtic π and ρ DAs [17,19]: yellow dots on the horizontal and vertical axis, respectively. vi) The black dots within the green a  intervals show the locations of the pion [23] and the ρmeson [24] DAs, respectively, obtained from a Dyson-Schwinger equations (DSE) based approach- [23] (pion) and [24] (ρ-meson). One notices that the a  given above, cf. (6), we find A (π) 2 ≈ 0.1 (9) in agreement with Eq. (7). Let us also emphasize the good correspondence in i) and ii) between the segments of a (π) 2 and a (ρ) 2 derived from NLC-SR. This circumstance makes it tempting to derive a relation similar to Eq. (3) within the NLC-SR approach for the pion [6,20] and the ρ -meson [19][20][21]. We start with the generalized sum rules for the π and ρ-meson channels. Taking recourse to the sum rules for the pion in the axial channel and for the ρ-meson in the vector channel [20] (equations (7) and (8) there), we write The term ∆ A1ρ ′ is determined by the difference of the contributions of the higher resonances in the phenomenological parts of the QCD SR for the axial and vector channels: Here M 2 is the Borel parameter within a stability window to be determined later in accordance with the standard QCD SR practice [25] in such a way as to best reproduce the DA moments. Strictly speaking, Eq. (10) should be considered in the weak sense, i.e., for smooth convolutions on both sides within the stability domain of M 2 , see, for instance, [6,20]. The expression for Φ S (x, M 2 ) together with some explanations is given in Appendix A (see also [6,26]), while the decay constants of the next resonances have the values (13) and have been determined in [6] and [20], respectively. The dependence of the scalar NLC Φ S (x, M 2 ) on the Borel scale M 2 and the quark virtuality λ 2 q = 0.4 GeV 2 , is given graphically in Fig. 2 making use of the simplest Gaussian model for the QCD vacuum from [8,12].
Let us now consider the origin of the difference between ϕ ρ and ϕ π ensuing from the particular structure of Eq. (10). The term ∆ A1ρ ′ represents a simple phenomenological contribution that takes into account the difference of [0, 1] using the value λ 2 q = 0.4 GeV 2 , see [6]. The lighter shaded areas correspond to M 2 = 0.6 GeV 2 and the darker shaded areas to M 2 = 0.9 GeV 2 . An analogous notation is used for the thinner (green) and thicker (red) lines.
the higher resonances in the vector and axial channels as expressed by their decay constants, cf. (12). In contrast, the appearance of the term 2 f 2 π Φ S (x, M 2 ) represents an evident dynamical manifestation of the distinct character of interaction of the scalar vacuum condensate in these two channels. Moreover, it is usually the dominant contribution.
What is the physical reason for the dynamical difference, encoded in −2 Φ S , between ϕ ρ and ϕ π in Eq. (10)?
The pion DA ϕ π is extracted from the axial-axial correlator because it originates as the pion projection of the axial (nonlocal) current, while the ϕ ρ DA is extracted from the vector-vector correlator because it is the ρmeson projection of the (nonlocal) vector current.
The scalar NLC is the only vacuum condensate, among others included in the used approximation, whose contribution is affected by the gamma-matrix structure of the coefficient function. All other condensate contributions to the theoretical part of the QCD SR are the same for both channels, i.e., axial-axial and vector-vector, of these correlators and are accumulated in ϕ π . As a result, the repeated commutations of γ 5 with an axial vertex, change the sign of the coefficient function of the scalar condensate relative to a chain of commutations with vector vertices. We will discuss the implications of these effects for the ρ and π DAs in Sec. IV.
Mathematically, Eq. (10) generates a linear relation between any (linear) functionals depending on the DAs ϕ ρ and ϕ π . This means that if such a relation holds for the ρ-meson DA, this functional can be replaced by the right hand side (RHS) of Eq. (10), where the π DA and Φ S enter.
For our further considerations, it is useful to make use of the convolution Then we can employ the Gegenbauer expansion of the meson DA ϕ M in Eq. (1) to get expressions in terms of the coefficients a n of meson M , where a M n are the coefficients of the "conformal expansion" over the adjoint harmonics and N m = 3(m + 1)(m + 2)/2(2m + 3) are normalization constants.
Convoluting expression (10) with ψ m (x), we obtain in which the last term in the square brackets dominates over the second one. For example, for n = 2, the second "resonances" term contributes only a few % compared to the scalar NLC. The notation sr = means that one should take the average of the RHS over M 2 within the stability window in the Borel parameter, i.e., in order to obtain a certain numerical value of the left hand side (LHS).

III. CONNECTION BETWEEN SCALAR NLC AND PION PDF
In this section we work out relation (17) for the second and fourth meson moment in connection with the NLC-SR and its vacuum parameters. To this end, all scaledependent quantities are evolved from µ 2 = 4 GeV 2 to the typical scale of QCD SR µ 2 0 ≃ 1 GeV 2 using the NLO evolution procedure described in [16] (see also [19,21]).
where ψ 0 = 1. This equation can be used to estimate the "constant" C(M 2 ) = C which has a physical sense similar to c ∈ [0.7 − 0.9] calculated in [1]. It is useful to rewrite relation (17) by expressing the factor e C(M 2 )m 2 ρ through the LHS of (18) to obtain the more homogenous form is a normalization factor. It is worth noting that relation (19a) for n = 2 possesses a sufficient stability with respect to M 2 and finally yields for a 2 (µ 2 0 ) = 0.047(58) obtained from the original NLC-SR in [21] but is more accurate.
We are now in the position to relate the moment expressions, obtained within the two considered nonperturbative approaches to the QCD vacuum, using the scalar condensate. Evaluating Eq. (19) for n = 2 and averaging the RHS over M 2 , we obtain for convenience. Here we employed the notation where f denotes the average over the stability window of M 2 : In order to get the same outcome from expressions (20) and (3), we have to conclude that Evolving the estimate in (6) to the scale µ 2 0 , we obtain for the LHS of (22) the value 0.171 (22) [5], which is shown as a solid line in Fig. 3 together with its uncertainties denoted by dashed red lines. The quantity Φ 2 (M 2 ) together with its upper boundary is displayed by means of where in the RHS of (23b) we employed the mean value of the interval in (23a). Note that the uncertainties in the estimate of Φ 2 are entailed by the uncertainties of the value of the quark condensate √ α sq q 2 and the use of the single parameter λ 2 q , cf. (8), to describe correlations in the quark NLC, see Appendix A.
These considerations establish the approximate validity of Eq. (22). This equation represents an intriguing relationship between the measurable quantity M 3 (the third moment of the pion PDF) and the scalar NLC Φ S which parameterizes the nontrivial vacuum of QCD. Let us recall here that Φ S significantly dominates in the RHS of (22), while the LHS of this equation was obtained by appealing to the general properties of quantum field theory that gave rise to the nontrivial relation (3) derived in [1]. It involves no further assumptions or theoretical modeling.
From this perspective, the condition for the sign of a (ρ) 2 in (7) turns out to be directly related to the QCD vacuum characteristics in terms of the mean value of the scalar condensate: If a and vice versa. The same condition can be obtained directly from Eq. (3) using the replacement as illustrated in Fig. 1 (at the scale µ 2 = 4 GeV 2 ). = −0.129 (25a) [6,21] in the RHS, while the value of Φ 4 turns out to be also negative. This value of a (ρ,sr) 4 agrees well (with an even better accuracy) with a (ρ) 4 = −0.057(118) obtained in the standard NLC-SR analysis in [21]. The expressions for the conformal coefficients for any n in both discussed approaches are given in Appendix B.
In Table I we present pairs (a ρ 2 , a ρ 4 ) computed by using as input some favored pion DAs: (i) BMS model [6] and (ii) a new DA determined in [26] (denoted there by the symbol ▲). This DA belongs to the BMS family and complies with the latest lattice result [3] at N 3 LO. The dispersive approach elaborated in [2] gives a linear relation analogous to (3) also for the higher conformal coefficients. Based on the results obtained in [2] (see, also, the discussion in Appendix B) one derives the relation where the coefficient B 43 (0) was determined to be B 43 (0) ≈ −0.12, while the low-energy subtraction constant c 4 can only be poorly estimated from the instanton model [2] to have the value c 4 = 1 GeV −2 . Comparing (25b) with (25a), and assuming the approximate equality we claim the validity of the relation The numerical evaluation of the RHS of Eq. (26b) amounts to the mean value of Φ 4 = −0.069(50) within the fiducial window in M 2 (at the normalization scale µ 2 0 ≃ 1 GeV 2 ). In the LHS of this equation we can use the estimate for M (π) 5 ≡ x 4 = 0.027(2) from the Jefferson Lab Angular Momentum (JAM) PDF in [5] (at µ 2 = 4 GeV 2 ) and the estimate for B 43 (0) so that evolving the LHS to µ 2 0 , we obtain for Eq. (26b) the approximate numerical equality (within errors) LHS:0.049(4)−0.12 = −0.071(4) ≈ RHS:−0.069(50) (27) to complete the comparison. The reasonable numerical agreement of the estimates in (23) and (27) gives support to the claim that the nonlocal scalar condensate Φ S and the pion non-singlet PDF [q π (x) −q π (x)] are indeed linearly related to each other. This unexpected result provides the possibility to unravel the basic characteristics of the nonperturbative QCD vacuum using measurements in DIS.
Let us now consider the estimate c 4 = 1 GeV −2 obtained from the instanton model [2]. For this value, the proposed relation exp c 4 m 2 ρ ≃ N in Eq. (26a) cannot be realized. The RHS of Eq. (26b) becomes instead the weighted sum of Φ 4 and a (π) 4 so that Substituting a (π) 4 ≈ −0.129 (see Table I and the correlation between the estimates of B 43 and c 4 . From the above considerations, we conclude that Eq. (29) is approximately fulfilled even for this case.

IV. IMPLICATIONS FOR THE ρ -MESON AND PION DA
Let us now discuss the implications of the cross-link relations derived in the previous sections for the pion and ρ -meson DAs.
Polyakov and Son claimed in [1] that the different signs of a (ρ) 2 < 0 and a (π) 2 > 0 could be regarded as an indication that the corresponding DAs may differ significantly. The origin of this difference could be ascribed to their distinctive response to the nonperturbative structure of the QCD vacuum. To make this difference more explicit, we display below their nonperturbative content in terms of NLC's:  (a2, a4) plane. The large "rectangle" intersecting with the a2 = 0 line denotes the region of ρ DAs determined with NLC-SR. It encloses the shaded platykurtic regime, where ✪ is the DA determined in [19]. The symbol ★ refers to the bimodal DAs from NLC-SR in [21] and [20], respectively. The symbol ▼ represents the ρ DA from the lightfront model in [29] and ◆ marks the asymptotic DA. Outside this "rectangle" one has the following ρ DAs: [30] and ♦ ♦ ♦ [24]. The other large "rectangle" further to the right contains the domain of the BMS pion DAs [6] obtained with NLC-SR, with ✖ denoting the bimodal BMS pion DA. The pion DA represented by ⊛ ⊛ ⊛ was determined in [26] as the crossing point of the long middle line of this "rectangle" with the lattice result a (π) 2 = 0.116 +19 −20 at N 3 LO with three-loop matching to the MS scheme [3]. The dashed "rectangle", crossing the a (π) 4 = 0 line, shows the platykurtic range of pion DAs with ✜ marking the DA determined in [17]. The symbols △ and ▲ reproduce, respectively, the DAs obtained within holographic AdS/QCD [31] and a DSE-based approach [23], while • shows the pion DA from the instanton model in [32] after NLO evolution to the scale 4 GeV 2 . Similarly, • represents the instanton-based ρ DA from [33].
These contributions represent the (i) Φ S (scalar fourquark condensate), (ii) Φq Aq (quark-gluon-antiquark condensate), (iii) Φ V (vector quark condensate), and (iv) Φ G (gluon condensate), with explicit expressions given in Appendix A of [28]. One notices that the four-quark condensate enters the expression for the ρ -meson DA with the opposite sign relative to the π DA. As a consequence, it tends to reduce the relative weight of this condensate with the result that the DA moments (or conformal coefficients) become smaller. This may indeed entail different shapes for these DAs.
For instance, the ρ DA could have a unimodal shape close to the asymptotic form describing a more or less equal distribution of longitudinal momentum between the two valence quarks, while the nonperturbative effects in the pion could lead to a much broader unimodal or bimodal distribution which favors unequally distributed momentum shares away from x = 1/2, see, e.g., [6,34].
A negative sign prediction for a (ρ) 2 is not completely new. In fact, in [19] it was found that in the case of the two ρ DAs determined with NLC-SR, notably the platykurtic DA [19] and the DA from [21], the range of a (ρ) 2 can indeed be negative, see Fig. 4. However, this possibility was not considered any further. In contrast, the negative sign of a (ρ) 2 in [1] appears as a strict finding of rather general principles of QFT and the instanton model, giving rise to the condition a (π) 2 (µ 2 = 4 GeV 2 ) = 0.078 < 7/6 M (π) 3 , and, thus, deserves particular attention.
Let us discuss these issues in terms of Fig. 4, which shows the locations of various pion and ρ DAs in the (a 2 , a 4 ) plane at the scale µ 2 = 4 GeV 2 , and apply Eq.
(3) in conjunction with Fig. 1. Then, only two classes of ρ DAs are possible depending on the sign of a (ρ) 4 . If it is positive, then the corresponding DA will belong to the area above the a (ρ) 4 = 0 line and will have a profile close to the asymptotic DA. If a 4 is negative, the DA will be located below the a (ρ) 4 = 0 line and will have an almost asymptotic profile combined with a mild endpoint suppression. The broadest variant of these ρ DAs is a platykurtic, i.e., a unimodal distribution with suppressed endpoints [19], belonging to the embedded dashed "rectangle" in Fig. 4.
In all considered cases, these DAs will differ substantially from the endpoint-suppressed, but bimodal pion DAs within the BMS "rectangle" [6] shown in this figure in terms of solid lines. The difference between a quasiasymptotic ρ DA and the endpoint-enhanced bimodal Chernyak-Zhitnitsky pion DA [34], obtained with local condensates with a CZ 2 (2 GeV) = 0.42 [19] is even stronger and is located outside the displayed range of a (π) 2 . Similar considerations apply also to broad unimodal pion DAs as those obtained from the DSE based approach, see [18] for a recent review. Indeed, from Fig. 4 one sees that the corresponding pion DA, given by ▲, is a broad unimodal distribution differing strongly from the asymptotic one. A similar observation applies also to the pion DA derived in holographic QCD, shown by △. Hence, this scenario supports the assumption that the π and the ρ interact differently with the nonperturbative QCD vacuum giving rise to very different DAs.
The other possible scenario would allow both meson DAs to have similar shapes. In fact, the ρ -meson DA can be obtained from the pion bilocal correlator by the replacement γ µ γ 5 → γ µ [34] so that one may naively think that this does not cause significant changes. The simplest possibility would be that their shapes are both close to ϕ asy π , see Fig. 1 in [19]. But it has been shown by various authors that in this case, the pion-photon transition form factor will underestimate the data considerably, see, for instance, [16] for a recent state-of-the-art analysis based on lightcone sum rules and further references. This implies that the pion DA must be broader than the asymptotic form at experimentally accessible momenta.
The only remaining possibility for these mesons to have similar profiles with a (ρ) 2 < 0 and a (π) 2 > 0 in combination with pion-photon TFF predictions that agree with most experimental data, is that both have a < 0. In this case, their corresponding profiles will have to be platykurtic. As has been shown in [17,19], and also more recently in [26], the platykurtic pion DA reproduces the trend of all data supporting asymptotic scaling at high Q 2 from Q 2 ≥ 1 GeV 2 up to momenta ∼ 40 GeV 2 . The most appropriate numerical values of the conformal coefficients, can be extracted by imposing additional constraints from lattice simulations. It is worth noting that the latest lattice calculation of a (ρ) 2 in [22] gives the rather large positive value 0.132(±0.027) that is incompatible with the restriction a (ρ) 2 < 0. On the other hand, employing the most recent lattice constraint on a (π) 2 with NLO or NNLO accuracy from [3], one finds from Fig. 1  and a (π) 4 , from their overlapping region as shown in [16]. However, the new a (π) 2 = 0.116 +0.019 −0.020 lattice estimate of [3] at N 3 LO with three-loop matching to the MS scheme favors a pion DA with a moderate bimodal profile, like ⊛ ⊛ ⊛ [26], giving support to the first considered scenario. Further constraints are needed along the upper diagonal in Fig.  4 in order to resolve the fine details of the pion DA more reliably.

V. CONCLUSIONS
In this work we investigated cross-link relations between the Gegenbauer coefficients of the ρ -meson and pion DAs. We showed that the linear relation between a (ρ) 2 and a (π) 2 obtained recently in [1] on the basis of rather general assumptions in combination with estimates from the instanton vacuum can also be obtained using QCD sum rules with nonlocal condensates. In fact, we were able to derive an intriguing relation between the third Melin moment of the pion PDF measured in DIS and the scalar condensate (see Eq. (22)). We also extended this cross-link relation to the next order Gegenbauer coefficients establishing further the connection between these two nonperturbative descriptions of the QCD vacuum (see Eq. (26b)). These findings may contribute to a better understanding of the intrinsic structure of the QCD vacuum by measurements of the pion PDF. The COMPASS++/AMBER experiment at CERN may provide high-precision data for the pion structure function to extract such information.
Adopting a broader perspective, we discussed the general implications for the pion and ρ -meson DAs entailed by the strict application of Eq. (7) by formulating two different generic scenarios. With respect to the space (a 2 , a 4 ), we found that the imposition of a (ρ) 2 < 0 not only reduces the available range of the ρ -meson coefficients, it also contributes important constraints for the proper selection of the pion DA. We concluded that either these mesons may have (for whatever reason) very different DAs or, if these are assumed to be similar (for whatever reason), then they can only have a platykurtic, i.e., unimodal profile with suppressed tails [16,17,19,35]. The most recent lattice result from [3] for a (π) 2 seems to support at N 3 LO rather the first scenario while the analogous estimates at NLO and NNLO conform also with the second option. The possibility that both DAs are close to the asymptotic form seems to be excluded because the pion-photon TFF calculated with ϕ asy π underestimates the available data considerably demanding a broader distribution for the pion DA.
(ii) A similar relation can be obtained within the dispersive approach. Indeed, in [1,2] the coefficient a that can be compared with Eq. (B1).