The transverse structure of the pion in momentum space inspired by the AdS/QCD correspondence

We study the internal structure of the pion using a model inspired by the AdS/QCD correspondence. The holographic approach provides the light-front wave function (LFWF) for the leading Fock state component of the pion. We adopt two different forms for the LFWF derived from the AdS/QCD soft-wall model, with free parameters fitted to the available experimental information on the pion electromagnetic form factor and the leading-twist parton distribution function. The intrinsic scale of the model is taken as an additional fit parameter. Within this framework, we provide predictions for the unpolarized transverse momentum dependent parton distribution (TMD), and discuss its property both at the scale of the model and after TMD evolution to higher scales that are relevant for upcoming experimental measurements.

construct a spin-dependent LFWF, which is not naturally present in the original AdS/QCD approach (see for example the phenomenological pion LFWF of [12] and [14]).
A crucial ingredient of the calculation is to identify the energy scale where the model is valid. Light-front holographic QCD describes the nonperturbative regime of QCD and therefore is expected to be valid at low energies, approximately of the order of hadron masses. As soon as the energy increases, a gradual transition to the regime of perturbative QCD takes place (see [6,[52][53][54][55] for more details on the transitions from one description to the other). In our work, we assume that the transition between the nonperturbative regime (where the model is applicable) and the perturbative regime (where pQCD is applicable) occurs at a precise scale, which we define as the model scale. We fix this scale by fitting the pion PDF to available phenomenological parametrizations, after applying DGLAP evolution equations [56,57]. Once the initial scale of the model is fixed, we also discuss the application of pQCD-based TMD evolution equations [58] to the unpolarized pion TMD.
The outline of this work is as follows: in Section II we introduce the explicit expressions for two forms of the LFWFs in the AdS/QCD soft-wall model [8,9]. After introducing the quark mass in a Lorentz invariant way and deriving analytical expressions for the relevant hadronic matrix elements, we fix the free parameters of the LFWFs and the scale of the model by fitting simultaneously the data of the form factor [59][60][61][62] and phenomenological parametrizations of the pion PDF [63]. With these sets of parameters, we provide predictions for the unpolarized TMD at the model scale in Section III. In Section III A we discuss TMD evolution [58]. We estimate the effects of the evolution for the broadening of the width and the change in the shape of the distribution, providing predictions to be tested with upcoming experimental data from COMPASS [51]. In Section IV we draw our conclusions.

II. THE PION LFWF
Thanks to the AdS/QCD correspondence it is possible to relate the gravitational theory defined in the fivedimensional AdS space to the Hamiltonian formulation of QCD on the light-front. This allows one to obtain a suitable first approximation of the valence wave function for mesons. In particular, the direct comparison of the expression for the form factors derived in both formalism offers the possibility of identifying the spinless string modes in five-dimensional AdS space with the meson LFWFs.
In Ref. [8], inspired by [64,65], the correspondence is performed by using the expression for the transition matrix element of the free electromagnetic current propagating in the AdS space, evaluated between five-dimensional AdS modes that correspond to the incoming and outgoing hadron states in a soft-wall model effective potential. Taking into account only the two-parton valence component, the explicit expression for the pion LFWF reads where the superscript V indicates that we are considering the LFWF for the "pure-valence" state of the pion. The quark masses in the pion LFWF are included following the prescription suggested in [66], i.e. by completing the invariant mass of the system as where m = m 1 = m 2 and, from momentum conservation, k ⊥ = k ⊥1 = −k ⊥2 and x = x 1 = 1 − x 2 . As a result, the expression (1) becomes where A is a normalization constant fixed by the condition Using the LFWF overlap representation of the PDF and form factor, we obtain where |q ⊥ | 2 = Q 2 . The condition (4) implies that is always consistently understood and we discuss results for the π + hadron, as the distributions for the π 0 and π − can be related by isospin and charge conjugation symmetry.
An alternative expression for the LFWF has been derived in [9], considering the mapping of the matrix element of a confined electromagnetic current propagating in a warped AdS space to the LFWF overlap representation of the pion form factor. In this case, one obtains a LFWF which incorporates the effects due to non-valence higher-Fock states generated by the "dressed" confined current, and therefore represents an "effective" two-parton state of the pion. It reads where the superscript E indicates that we are considering an "effective-valence" component of the LFWF. At variance with the pure-valence LFWF, the effective-valence LFWF is not symmetric in the longitudinal variables x and 1 − x of the active and spectator quark, respectively. Introducing the quark mass dependence as outlined above, the effective-valence LFWF becomes where the parameter A is once more fixed by demanding the validity of (4). The corresponding expressions for the PDF and the form factor are given by We fix the parameters of the LWFs (3) and (9) by fitting the available experimental data for the pion electromagnetic form factor [59][60][61][62] and the parametrization of the pion PDF in [63]. For the fit of the PDF, we apply the DGLAP evolution equations at next-to-leading-order (NLO) to evolve the PDF from the (low) scale of the model Q 0 to the scale Q = 5 GeV of the parametrization, using the HOPPET code [67]. We leave in the initial scale Q 0 as an additional free parameter to be fitted with the data. Starting from the functional form of the parametrization [63], we select 61 equally-spaced points from x = 0.2 to x = 0.8 and for each of them we construct error bars by propagation of the errors on the individual parameters. Summing the PDF points and the 58 form factor points, we perform the fit using in total 119 points. In the case of the pure-valence LFWF we consider two different fitting strategies: either we fix the quark mass to a constant value ("current quark" mass m = 0.005 GeV and "constituent quark" mass m = 0.2 GeV) or, alternatively, we let the quark mass entering as an additional fit parameter. For the effective-valence LFWF, we fix the quark mass to the same values as before, but we include also the limit of massless quarks (leaving the quark mass as a free parameter in this case leads anyway to a vanishing mass). The results of the fit are summarized in Tab. II. In the following we discuss the results for two sets of parameters in Tab. II corresponding with the lowest value of the total χ 2 d.o.f. for non-vanishing quark mass. In Fig. 1 we show the results for the form factor of the pure-valence (solid curve) and effective-valence (dashed curve) LFWF. The corresponding results for the PDF are shown in Fig. 2(a) and 2(b), respectively. The solid curves show the results at the hadronic scale, and the dashed curves are obtained after NLO evolution to Q = 5 GeV. The shaded band corresponds to the results from the parametrization at Q = 5 GeV of Ref. [63]. The results from the pure-valence LFWF are in good agreement with the available experimental and phenomenological information, while a worst comparison, especially for the form factor, is obtained in the case of the effective-valence LFWF.
The mass parameter κ plays a very important role, as it is originally the only free parameter of the theory and it is related to the strength of the confining harmonic potential in the soft-wall model [2,68]. The value κ ≈ 0.37 GeV obtained in the pure-valence LFWF case is similar to what was obtained in Ref. [8], whereas in the study of the effective-valence LFWF we obtain smaller values, κ ≈ 0.26 GeV, compared to previous analyses [12,16]. Moreover, we point out that a larger value, namely κ = 0.54 GeV, is needed in order to describe the hadronic mass spectra and the Regge trajectories [69][70][71] and this value has been quite extensively used (see [5,14] for a more complete overview). Recent works [72,73] quote a value of κ = 0.523 GeV to reproduce Regge slopes for mesons and baryons and to realize the transition from the non-perturbative (described by light-front holography) and the perturbative regimes, which occurs at an energy scale of about 1 GeV.
Our result for the initial scale is Q 0 ∼ 0.5 GeV in the pure-valence case and is consistent with the values obtained in different phenomenological quark models [41,42], where the scale is fixed by requiring that the model results for the momentum carried by the valence quarks match the experimental value, after DGLAP evolution. We also notice that the fit of the quark mass provides a value that is quite close to the average effective light-quark mass obtained in LF holographic QCD from the meson spectrum [5]. In the case of the effective-valence LFWF, we expect that the inclusion of the effects of higher-order Fock state components should correspond to a higher hadronic scale. This is the case when comparing the results between the effective-valence and pure-valence LFWF with m = 200 MeV and similar values of κ. However, for the other quark-mass scenarios we find similar values of Q 0 in the two models, which are compensated by much lower values for the parameter κ in the case of the effective-valence LFWF. Both the values of κ and the initial scale Q 0 differ with respect to [72,73].

III. TMD ANALYSIS
The unpolarized TMD f 1 (x, k 2 ⊥ ) can be obtained from the following LFWF overlap [42] which reduces to the PDF in Eq. (5) after integration over k ⊥ . Using the expressions in Eqs. (3) and (8), one finds that the TMD in both models is a Gaussian distribution in k ⊥ , with an x-dependent mean square transverse momenta, i.e.
where k ⊥ = |k ⊥ |. In Fig. 3 we show the results for the TMD in the two models, as function of x and k 2 ⊥ . As in the case of the PDF, the pure-valence model is symmetric under the exchange of x → 1 − x, while this symmetry is lost when including effects beyond the valence sector in the effective-valence LFWF. The fall-off in k 2 ⊥ is Gaussian in both models.
The width of the distribution k 2 ⊥ (x) is shown as function of x in Fig. 4. It is slightly larger in the pure-valence model, with a maximum at x = 0.5 and the characteristic symmetric behaviour around the maximum. Integrating over x, one obtains k 2 ⊥ V = 0.023 GeV 2 . In the case of the effective-valence LFWF the maximum is shifted at lower values of x, i.e. x = 0.28, and the result after x-integration is k 2 ⊥ E = 0.020 GeV 2 .

A. TMD evolution
As explained before, the AdS/QCD LFWF and the resulting TMDs are obtained at a scale of about 0.5 GeV. In order to be able to compare with data or extractions, TMDs need to be evolved according to TMD evolution equations (see, e.g., Ref. [74]). These equations describe the broadening of the initial TMD due to gluon radiation.
Even though TMD evolution equations are based on perturbative QCD calculations, their implementation requires the introduction of some prescriptions to avoid extending the calculations outside their region of validity. In general, such prescriptions have the effect of inhibiting perturbative gluon radiation at low transverse momentum and at low Q, but must be complemented with an additional component of gluon radiation, usually referred to as nonperturbative component of TMD evolution [75]. This component cannot be predicted by perturbative QCD, but has to be extracted from experimental measurements, taking advantage of the fact that it is highly universal (i.e., it is independent of the quark's flavor and spin, the parent hadron, the type of process, and whether one considers TMD distribution and fragmentation functions). It may be possible to use AdS/QCD correspondence also to compute the nonperturbative components of TMD evolution, but we leave this issue to future studies (for a recent example of a computation of the behavior of the nonperturbative component of TMD evolution see Ref. [76]) .
Several prescriptions have been proposed in the literature (see, e.g., Refs. [75,[77][78][79][80]). In principle, if complemented with the appropriate nonperturbative components, they should lead to compatible results for the evolved TMDs. However, there is still considerable uncertainty on the nonperturbative components and systematic studies of these uncertainty are still lacking. We therefore choose a specific implementation of TMD evolution equations, which has been successfully applied to the description of data in the range 1.2 GeV Q 80 GeV. Details of this implementation are discussed in Ref. [81]. We summarize here the most important points.
TMD evolution is implemented in the space Fourier-conjugate to k ⊥ . Therefore, we first define the Fouriertransformed TMDs and we use the following form for the evolved TMDs in b ⊥ space (see Refs. [58,82]) where the label a indicates the parton type. We consider the above equation at Next-to-Leading Logarithmic (NLL) approximation and at leading order in α S . In this case, the convolution at the beginning of the evolved formula reduces simply to i=q,q,g and the expression for the Sudakov form factorS(b * ; µ b , µ) can be found, e.g., in Ref. [83,84]. We further use We introduced the following variableb * ≡ b max The above choice guarantee that at the initial scale Q = Q 0 any effect of TMD evolution is absent. The model results are thus preserved and in particular the relation between TMD and collinear PDF is maintained. The value of the g 2 parameter should be extracted from experimental data, keeping all other choices fixed. In a recent analysis, the parameter was found to be 0.13 ± 0.01 GeV 2 in combination with a b max that was half of the value we assume here. Since b max and g 2 are in general anticorrelated, we choose for the present analysis the following three values g 2 = 0.09, 0.11, 0.13 GeV 2 .
After evolution to 1 GeV, as already observed, the width of the TMD increases to about 0.3 GeV in both versions of the model. The x dependence of the TMD width is rather flat. The symmetry about x = 0.5 of the pure-valence model is lost. The two models become quite similar to each other: the position of the peak is the same within a 5% error. At 5 GeV, the width of the TMD increases to about 0.7 GeV at x = 0.5 and increases at low x, and is again very similar in the two versions of the model. In summary, TMD evolution from the model scale (0.5 GeV) to a typical experimental scale of 5 GeV increases the width of the TMD of almost one order of magnitude and leads to an x dependence of the width that is different from the original one, with no strong difference between the two versions of the model.

IV. CONCLUSIONS
We have performed a study of the pion using light-front holographic QCD, which allows us to construct pion LFWFs. We took into consideration two different versions of the pion LFWFs: pure-valence and effective-valence. For each version, the model contains three free parameters: the mass parameter κ (expressing the strength of the confining harmonic potential that breaks conformal invariance), the quark mass, and the scale of the model. We fix the parameters by comparison to experimental information on pion form factors and PDFs.
We obtain a value of κ in agreement with previous estimate [8], for the pure-valence version of the model. For the effective-valence version, we obtain a smaller value [12,16]. The best agreement with data in the case of massive quarks is obtained for a quark mass m = 0.05 GeV for the pure-valence version, and m = 0.005 GeV for the effectivevalence version. In order to achieve a fair agreement with the pion PDF at 5 GeV, the model scale has to be set to about 0.5 GeV. This turns out to be true both for the pure-valence and the effective-valence LFWF.
The sets of parameters obtained have then been used to study the unpolarized TMD of the pion. At the model scale, the resulting TMD has a Gaussian shape with a width (defined as the position of the peak of the distributions k ⊥ f 1 (x, k 2 ⊥ )) of about 0.1 GeV at x = 0.5. The x dependence of this width is different in the two versions of the model: in the pure-valence model the TMD attains its maximal width at x = 0.5; in the effective model, this happens at x = 0.28. After the TMD is evolved to a typical experimental scale of about 5 GeV, its width increases by almost one order of magnitude. The x dependence is different from the one at the model scale: the width grows monotonically as x decreases, and the differences between the two versions of the model fade away.