Pion Parton Distribution Function in Light-Front Holographic QCD

The pion parton distribution function, $u^{\pi}(x)$, is reexamined by a universal reparametrization function, $w_\tau(x)$, in the light-front holographic QCD (LFHQCD) approach. We show that, owing to the flexibility of $w_\tau(x)$, the large-$x$ behavior $u^{\pi}(x)\sim (1-x)^{2}$ can be contained within the LFHQCD formalism. From this fact, augmented by perturbative QCD and recent lattice QCD results, we state that such behavior cannot be excluded.

Motivation -During the rise of parton models, around the 1970s, a connection between the proton electromagnetic form factors (obtained via exclusive process) and its structure functions (inferred from deep inelastic scattering) was realized by Drell-Yann [1] and West [2]. Their findings yielded the so-called Drell-Yan-West relation (DYW), which entails that, when the momentum transfer (−t = Q 2 ) becomes asymptotically large, the proton electromagnetic form factor (EFF) falls as while the corresponding parton distribution function (PDF) behaves, at large-x (i.e., x → 1), as Here, x is the longitudinal momentum fraction carried by the parton -or Bjorken-x [3] -and τ , called twist, denotes the number of τ -components of the hadron state. In a subsequent work by Ezawa [4], it was shown that the pion violates the DYW relation. This can be attributed to the different number of constituents and spin. It is seen that, while the EFF exhibits the same asymptotic profile for both mesons, Eq. (1), the pion parton distribution function adopts the large-x form The leading-twist (τ = 3 for proton, τ = 2 for pion) entails the well-known 1/(−t) 2 and 1/(−t) falls of the proton and pion EFFs [5], respectively, and the x → 1 behavior of the PDFs is driven by Those patterns are further supported by perturbative Quantum Chromodynamics (pQCD) [5][6][7]. In fact, assuming a theory in which the quarks interact via the exchange of a vector-boson, asymptotically damped as (1/k 2 ) β , Eq. (5) generalizes as [8]: Hence, the large-x behavior of the valence-quark PDF is a direct measure of the momentum-dependence of the underlying interaction [6][7][8][9].
In the novel approach of light-front holographic QCD (LFHQCD) [10,11], it is suggested that the DYW relation is preserved for both the proton and pion [12]. Thereby, it predicts a valence pion PDF that, from the leading-twist-2 term, falls as feeding the controversy provoked by the E615-Experiment leading order (LO) analysis [13], which favors a large-x exponent of "1", in apparent contradiction with the parton models and pQCD. Many theoretical and phenomenological approaches have been participants in this debate, e.g. [8,9,12,[14][15][16][17][18][19][20][21][22][23][24][25]. Playing a key role in this controversy, the analysis of Aicher et al. [15] shows that, if a next-to-leading order (NLO) treatment of the data is performed and soft-gluon resummation is considered, it is possible to recover the pQCD prediction. On different grounds, the x → 1 profile of Eq. (5) is also favored by a recent lattice QCD (lQCD) result [20], in which a novel "Cross Section" (CS) technique [20,22] is employed to obtain the pointwise shape of the pion PDF. Furthermore, it is important to unravel the proton and pion properties together. Expose, for example, the origin and difference of their masses: if we accept QCD as the fundamental, underlying theory of the strong interactions (and we do), it is necessary to simultaneously explain the masslessness of the pion and the much larger size of the proton mass [26][27][28]. Similarly, it is vital to obtain a clear picture of the proton and pion parton distributions in the same approach. QCD predicts the profiles of Eqs. (4)-(5), thus we need to explain how those behaviors can (or cannot) take place.
In this letter, we revisit Ref. [12]. There, the authors present an appealing way to parametrize the PDFs and generalized parton distributions (GPDs), from an integral representation of the EFFs, but they claim that the falloff of the pion PDF at x → 1 is an unresolved issue. Our aim is to show that the large-x behavior of Eq. (3) can be perfectly accommodated within the same LFHQCD formalism, while also maintaining the correct counting rules for the proton.

arXiv:2001.07352v1 [hep-ph] 21 Jan 2020
Counting rules in LFHQCD -Following Ref. [12], the form factor is expressed in an integral representation as where N τ = √ π Γ(τ − 1)/Γ(τ − 2) and √ λ = 0.548 GeV; B(u, v) corresponds to the Euler Beta Function. The universal scale, λ, is fixed by the ρ meson mass [10,29]. Under the change of variable y = w τ (x) one can write, more generally: where the reparametrization function, w τ (x), is constrained by the conditions: Notice that we have introduced a τ -dependence in w τ (x). This is a key difference, with respect to [12], that we will exploit later. At zero skewness, the valence-quark GPD is conveniently expressed as where we identify the PDF and profile function, q τ (x) and f τ (x) respectively, as Then, a simple form for w τ (x) is suggested: with g(τ ), a τ > 0. The adopted profile of w τ (x) preserves the desired Regge behavior at small-x [11,12], while also satisfying the constraints of Eqs. (11). Thus, owing to the reparametrization invariance of the Euler Beta Function, F τ (t) exhibits the large-t falloff: which implies that the correct asymptotic behavior of the form factor [4][5][6] is faithfully reproduced. On the other hand, the x → 1 leading power of q τ (x) will exhibit the τ -dependence as follows: with h(τ ) = (τ − 1)g(τ ) − 1. Due to the arbitrariness on the choice of g(τ ), LFHQCD cannot predict its precise form, and so the exact counting rules. However, it is this flexibility what allows us to recover the corresponding counting rules for both pion and proton, Eqs. (4)- (5). Given the simplicity of Eq. (17), we propose the following for the PDFs: Rule-II : Thus, it follows from (17) that the spin− 1 2 relation (2) can be satisfied if Rule-I is chosen, while the spin−0 counterpart (3) holds if Rule-II is selected instead. Focusing on the pion, we will perform a numerical test to contrast the above rules against the phenomenology and analyze under which circumstances are these rules feasible. Pion valence-quark PDF -Consider the twist-4 pion valence-quark PDF as with normalization 1 0 dx u π (x; ζ) = 1 and γ = 0.125. The latter, twist-4 component, represents the meson cloud contribution determined in [10]. The PDF is defined at an intrinsic scale ζ = ζ 1 , which is set as ζ 1 = 1.1±0.2 GeV to keep in line with previous works [12,30]. Then, continuum analyses [18,19] are employed as benchmarks to estimate the value such that the a 2 coefficient in Eq. (15) can be determined. This is additionally cross-checked from the value <x> ζ2 ≈ 0.24, obtained at ζ 2 := 2 GeV after NLO evolution, as compared to the lQCD estimates from Refs. [21,[23][24][25]. To account for the impact of the twist-4 term, we also vary the ratio a 4 /a 2 from 0.1 to 1. Only mild effects at intermediate values of x are observed. Figure 1 displays the valence-quark PDFs, evolved to ζ 5 := 5.2 GeV, and its comparisson with experimental and lattice data [13,15,20]. For contrast, we have also included a recent Dyson-Schwinger equations (DSEs) result [18,19]. The t-dependence of the valence-quark GPD, for Rule-II, is presented in Figure 2. It is clear that Rule-I produces a PDF that is closer to the original experimental data [13], while the analogous for Rule-II matches the rescaled data from Ref. [15]. Either rule will give the correct large-t fall of the EFF in Eq. (16), but only in the second case one obtains the x → 1 behavior predicted by pQCD. This is readily achieved in the DSE formalism [17][18][19]: its direct connection with QCD ensures that perturbation theory is recovered, and so the connection of the asymptotic behavior of the gluon with the large-x behavior of the valence-quark PDF [8]. Moreover, state-of-the-art lQCD results [20] also establish that the asymptotic form of Eq. (5) is preferred. It is noteworthy that, even though the pion PDF Obtained NLO results at ζ5 = 5.2 GeV, from the rules in (18). The corresponding (blue and red) error bands account for the uncertainty in the initial scale, ζ1 = 1.1 ± 0.2 GeV and the variation of a4/a2 = 0.1 to 1. The broadest, gray band, corresponds to the novel lQCD "CS" result from [20] and the dashed-line depicts the DSE result [18,19]. Data points: (triangles) LO extraction "E615-Original" [13] and (circles) the NLO analysis "E615-Rescaled" of Ref. [15].

FIG. 2.
Valence-quark pion GPD. t-dependence of the valence-quark pion GPD at zero skewness. The plot above corresponds to Rule-II in Eq. (18), at the initial scale ζ1.
obtained from Rule-I differs from that of [12], the evoluted results are compatible. This is unsurprising since the corresponding reparametrization function is not dramatically different from its counterpart in [12]. Thus, although it is not be included in the present letter, we expect Rule-I to produce a congruent picture for the valence-quark PDF of the proton. These observations encourage us to select Rule-I for the case of the proton and Rule-II when studying pions, for an internally consistent description based on the LFHQCD formalism. Summary and conclusions -We have reanalized the LFHQCD approach of Ref. [12] to study the valencequark PDF of the pion. It has been proven that, given the flexibility of the universal reparametrization function, w τ (x), it is in fact possible to accomodate a large-x behavior of u π (x) ∼ (1 − x) 2τ −2 within this framework. Besides the agreement with the rescaled experimental data [15], this makes it compatible with the Ezawa findings [4] and the predictions from pQCD [5][6][7]. Recent continuum [18,19] and sophisticated lQCD studies [20] also favor this endpoint form. Due to this confluence of vastly different approaches, and given our observations, we state that the u π (x) ∼ (1 − x) 2 profile can not only be contained within the LFHQCD formalism, but also cannot be excluded. Besides, we sketched how a simultaneous description of the proton and pion distribution functions, that agrees with pQCD, can be achieved if the counting rules are chosen accordingly: we encourage the use of Rule-I for proton and Rule-II for pion.
We acknowledge helpful conversations with Yuan Sun. This work is supported by: the Chinese Government Thousand Talents Plan for Young Professionals.