Pion and kaon valence-quark quasiparton distributions
Shu-Sheng Xu Lei Chang Craig D. Roberts Hong-Shi Zong











Department of Physics, Nanjing University, Nanjing, Jiangsu 210093, China


[]leichang@nankai.edu.cn
School of Physics, Nankai University, Tianjin 300071, China


[]cdroberts@anl.gov



Physics Division, Argonne National Laboratory, Argonne, Illinois
60439, USA



Department of Physics, Nanjing University, Nanjing, Jiangsu 210093, China
Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing, Jiangsu 210093, China






Algebraic Ansätze for the Poincaré-covariant Bethe-Salpeter wave functions of the pion and kaon are used to calculate their light-front wave functions (LFWFs), parton distribution amplitudes (PDAs), quasi-PDAs (qPDAs), valence parton distribution functions (PDFs), and quasi-PDFs (qPDFs).

The LFWFs are broad, concave functions; and the scale of flavour-symmetry violation in the kaon is roughly 15%, being set by the ratio of emergent masses in the s-and u-quark sectors.

qPDAs computed with longitudinal momentum P_z =1.75GeV provide a semiquantitatively accurate representation of the objective PDA; but even with P_z=3GeV, they cannot provide information about this amplitude's endpoint behaviour.

On the valence-quark domain, similar outcomes characterise qPDFs.  In this connection, however, the ratio of kaon-to-pion u-quark qPDFs is found to provide a good approximation to the true PDF ratio on 0.3≲ x ≲ 0.8, suggesting that with existing resources computations of ratios of quasi-parton-distributions can yield results that support empirical comparison.















Pion and kaon valence-quark quasiparton distributions
    Hong-Shi Zong
    22 February 2018
=====================================================






§ INTRODUCTION


Since the discovery of quarks in deep inelastic scattering (DIS) experiments at the Stanford Linear Accelerator Center <cit.>, parton distributions have occupied a central role in high-energy nuclear and particle physics; and today there is a vast international programme aimed at their measurement.  Such measurements are possible owing to the existence of factorisation theorems <cit.>, which entail that the cross-sections for various processes can be written as the convolution of a piece calculable using QCD perturbation theory and a parton distribution function (PDF), which is independent of the process used.  PDFs are therefore a characterising property of the chosen hadronic target.  This means they are also essentially nonperturbative, i.e. their calculation is a problem in strong QCD (sQCD), with sound results demanding the use of a nonperturbative method with a traceable connection to QCD.  The extent and importance of this computational challenge is canvassed elsewhere <cit.>.

An ab initio approach to sQCD is provided by the numerical simulation of lattice-regularised QCD (lQCD).   However, a given PDF is mathematically defined as an expectation value of some bilocal operator evaluated along a light-like line, an object which cannot be evaluated using the methods of lQCD.  This approach only provides access to the expectation value of local operators, i.e., in this context, to the Mellin moments of the PDF.  That would not be an issue if every moment were accessible because a probability distribution is completely determined once all its moments are known.  However, discretised spacetime does not possess the full rotational symmetries of the Euclidean continuum.  Hence, only the lowest three non-trivial moments can readily be calculated; and they are insufficient to support a model-independent reconstruction of the PDF.  A number of paths are being pursued to circumvent this problem <cit.>.

Herein, using continuum methods in quantum field theory, we explore some aspects of the large P_z (longitudinal-momentum) approach to the lQCD computation of parton distributions <cit.>, viz. since the maximum value of P_z is bounded in any lQCD simulation, what is the lowest value of P_z^ max for which the quasi-distribution provides a realistic sketch of the true distribution; and given a P_z^ max quasi-distribution, is it possible to extract reliable information about the true distribution?

These issues are considered, e.g. in Refs. <cit.> using spectator models of the proton.




We, on the other hand, choose to focus on the pion and kaon because there has been significant progress in the continuum computation of the distribution amplitudes and functions of these systems in recent years <cit.> and first lQCD results on some of their quasi-distributions are now available <cit.>.   Our discussion complements the analysis in Refs. <cit.>.

Sec. <ref> describes the framework used to represent π- and K-mesons as bound states in quantum field theory, specifying the elements at an hadronic scale, ζ_H ∼ 1GeV.  Herein, we do not consider perturbative QCD evolution <cit.>: such evolution doesn't affect our comparisons between quasi-distributions and their associated light-front distribution functions, which are all made at the same scale.  Section <ref> focuses on π and K leading-twist light-front wave functions, their derived parton distribution amplitudes and attendant quasi-distributions, introducing general formulae and providing numerical illustrations.  The kindred analysis of valence-dressed-quark quasi-parton-distribution-functions (qPDFs) is presented in Sec. <ref>.  We conclude in Sec. <ref>.






§ PION AND KAON BOUND STATES



Many insights into the character of the pion and kaon have been drawn using the following simple expressions for the relevant dressed-quark propagators and Bethe-Salpeter amplitudes <cit.>:



    S_f(k)     =[ -i γ· k + M_f] Δ(k^2,M_f^2) ,
    
     n_G Γ_G(k;P_G)     = i γ_5  ∫_-1^1 dw ρ_G(w) Δ̂(k_w^2,Λ^2) ,


where

M_f is the dressed-quark mass evaluated in the neighbourhood k^2≃ 0,  f=u,s (we work in the isospin symmetric limit, so M_u=M_d);

Δ(s,t) = 1/[s+t], Δ̂(s,t) = t Δ(s,t);

k_w = k+ (w/2) P, with P^2=-m_G^2, G=π, K;

and ρ_G=π,K(w) is a spectral weight whose form determines the pointwise behaviour of the associated meson's Bethe-Salpeter amplitude, with n_G the related canonical normalisation constant.

One of the strengths of these Ansätze is that they can be chosen to ensure that a primarily algebraic computation yields results which are pointwise similar to the most sophisticated predictions currently available for parton distribution amplitudes and functions, PDAs and PDFs.

Notably, our approach to the continuum bound-state problem is Poincaré-covariant and hence, with complete generality, we may write

    P_G = (0,0,P_z, i E_P) ,   E_P = [P_z^2 + m_G^2]^12.


One branch of our analysis will focus on the leading-twist two-dressed-parton distribution amplitudes of the π- and K-mesons, the computation of which requires a projection onto the light-front of the given meson's (unamputated) Bethe-Salpeter wave function.  Working with the K^+ meson as an illustration, this wave function is (k^K_- = k- P_K/2)

    χ_K(k^K_-;P_K)     = S_u(k) Γ_K(k^K_-;P_K) S_s(k-P) ;

and the part which contributes to the leading-twist (twist-two) PDA is readily computed:


    n_K χ_K^(2)(k^K_-;P_K)     =
     M(k;P_K)
    ∫_-1^1 dw ρ_K(w)  D(k;P_K) ,
    
     M(k;P_K)      = -γ_5[ γ· P_K  M_u+ γ· k (M_u-M_s) 
    
            + σ_μν k_μ P_Kν]  ,
    
     D(k;P_K)     =  Δ(k^2,M_u^2) Δ((k-P)^2,M_s^2) 
    
            ×Δ̂(k_w-1^2,Λ^2) .



One may now introduce two Feynman parameters, combine the denominators into a single quadratic form, and thereby arrive at:

    χ_K^(2)(k^K_-;P_K)     =  M(k;P_K) ∫_0^1  dα 2   X_K(α;σ^3(α)) ,

with σ = (k-α P_K)^2+ Ω_K^2,


    Ω_K^2     = v M_u^2 + (1-v)Λ^2 
    
        + (M_s^2-M_u^2)(α - 12[1-w][1-v]) 
    
         + ( α [α-1] + 14 [1-v] [1-w^2]) M_K^2 ,
    
    
     X_K(α;σ^3)     =
    [
    ∫_-1^1-2α dw ∫_1+2α/w-1^1 dv . 
    
         . + ∫_1-2α^1  dw ∫_w-1+2α/w+1^1 dv ]ρ_K(w) / n_K Λ^2/σ^3 .

Formulae for the π-meson are readily obtained by setting s→ d, m_K → m_π, and using isospin symmetry.

As has long been known <cit.> and is demonstrated for parton distributions in, e.g. Refs. <cit.>, distinctions between the K- and π-mesons are driven by dynamical chiral symmetry breaking (DCSB), expressed in Eq. (<ref>) by the difference between the dressed s- and u quark masses: (M_s^2-M_u^2).







§ LIGHT-FRONT WAVE FUNCTIONS AND QUASI-PDAS





 §.§ Algebraic Analysis



The pseudoscalar meson's leading-twist two-dressed-parton light-front wave function (LFWF) can be written:

    f_K ψ_K^↑↓(x,k_⊥^2)     =  tr_ CD∫_dk_∥δ_n^x(k_K)
    γ_5 γ· n χ_K^(2)(k^K_-;P_K) ,

where
f_K is the kaon's leptonic decay constant;
the trace is over colour and spinor indices;
∫_dk_∥ = (1/π)∫ dk_3 dk_4;
δ_n^x(k_K)= δ(n· k - x n· P_K);
and n is a light-like four-vector, n^2=0, n· P_K = -m_K.  The twist-two PDA follows immediately:


    φ_K(x)     = 1/16π^3∫ d^2k_⊥ψ_K^↑↓(x,k_⊥^2) ,
    
       ∫_0^1dx φ_K(x)  = 1 .



Consider now the following Mellin moments:


    ⟨ x^m    ⟩_Ψ^↑↓_K =   ∫_0^1 dx   x^m ψ_K^↑↓(x,k_⊥^2)
    
        = 1/f_K n· P∫_dk_∥[n· k/n· P]^m γ_5 γ· n χ_K^(2)(k^K_-;P_K)
    
        = 12/f_K∫_0^1 dα α^m  Y_K(α;σ^2) ,


where we have used Eqs. (<ref>)–(<ref>), and Y(α;σ_⊥^2) = [M_u(1-α) +M_sα ] X(α;σ_⊥^2), σ_⊥=k_⊥^2+Ω_K^2.  Comparing Eqs. (<ref>) and (<ref>), it is apparent that

    ψ_K^↑↓(x,k_⊥^2) = 12/f_K Y_K(x;σ_⊥^2) ,

where α→ x in Eqs. (<ref>), (<ref>).  The π-meson formula is obvious by analogy.  The compactness of these results is one merit of the algebraic Ansätze in Eqs. (<ref>).

Combining Eqs. (<ref>), (<ref>), a twist-two dressed-parton quasi-PDA (qPDA) is obtained via the replacement n →ñ, ñ=(0,0,1,0), viz.

    f_K φ̃_K(x) =  tr_ CD∫_dkδ_ñ^x(k_K)  γ_5γ·ñ χ_K^(2)(k^K_-;P_K) ,

where δ_ñ^x(k_K)= δ(ñ· k - x ñ· P_K); and ∫_dk is a Poincaré-invariant definition of the four-dimensional integral.  Following a series of steps similar to those used above, one arrives directly at the following result:

    φ̃_K(x)     = P_z/16π^3∫_0^1 dα∫_-∞^∞ dk  ψ_K^↑↓(α, k^2+(x-α)^2 P_z^2) .

The expression for φ̃_π(x) is obvious by analogy and matches Eq. (20) in Ref. <cit.>.



 §.§ Numerical Illustrations




  §.§.§ Wave Functions



It is now possible to study the P_z-evolution of the pointwise-form of meson qPDAs and chart their connection with the objective PDA.  To proceed, it is necessary to specify the parameters and spectral densities in Eq. (<ref>).  For the latter, we use

    u_G ρ_G(w)      = 1/2 b_0^G[
     sech^2([w-w_0^G]/[2 b_0^G]) . 
    
          . +  sech^2([w+w_0^G]/[2 b_0^G]) ] [1 +w  v_G ]  ,

where b_0^G, w_0^G, v_G, are parameters, and u_G is a derived constant that ensures unit normalisation of the density.  This form is compact and yet has sufficient flexibility to produce pion and kaon valence-quark PDAs and PDFs whose features are consistent with contemporary predictions.

Regarding the parameters, we choose M_u=0.31GeV, matching the infrared scale of the u-quark mass function obtained using modern gap-equation kernels <cit.>; set M_s= 1.2 M_u, which is typical of the size obtained in phenomenologically efficacious continuum analyses <cit.>; float Λ_π,K to fit the leptonic decay constants:



    f_π = 1/n· P_π tr_ CD∫_dkγ_5 γ· n χ_π (k_-^π;P_π) ,
    
        f_K = 1/n· P_K tr_ CD∫_dkγ_5 γ· n χ_K (k_-^K;P_K) ,


where m_π=0.14GeV, m_K=0.49GeV; and choose b_0^G, w_0^G  such that the meson PDAs are broad, concave functions whose lowest nontrivial Mellin moments match those obtained in modern analyses <cit.>:



    ⟨ (2x-1)^2⟩_φ_π    := ∫_0^1 dx  (2x-1)^2 φ_π(x) ≈ 0.25 ,
    
    ⟨ 2x-1⟩_φ_K   ≈ -0.04 , ⟨ (2x-1)^2⟩_φ_K≈ 0.25 .


With

    [   Λ_π b_0^π w_0^π   v_π   Λ_K b_0^K w_0^K   v_K;   M_u   0.1  0.73     0 2 Λ_π b_0^π  0.95  0.16 ]


we obtain f_π =0.092GeV, f_K = 0.11GeV, in agreement with experiment <cit.>, and satisfy Eqs. (<ref>).  Recall that in connection with quantities that undergo QCD evolution, our models should be understood as producing results valid at an hadronic scale, ζ_H ∼ 1GeV.



The pion and kaon leading-twist dressed-parton LFWFs, obtained using Eqs. (<ref>), (<ref>), (<ref>), (<ref>), are depicted in Fig. <ref>.

Considered as a function of x, with k_⊥^2 fixed, these wave functions are broad and concave.  Conversely, at fixed x, they fall as 1/k_⊥^4 on k_⊥^2≫Λ_G^2.


In QCD, the behaviour is 1/k_⊥^2 (up to ln k_⊥^2-corrections).  Our model's decay rate is amplified because Eq. (<ref>) retains only the γ_5 piece of the pseudoscalar meson Bethe-Salpeter amplitude.  Two additional “pseudovector” Dirac structures are prominent in symmetry-preserving solutions of the Bethe-Salpeter equations for light pseudoscalars <cit.>; and the omission of these components produces the 1/k_⊥^4 decay at ultraviolet momenta <cit.>.  This has a benefit: all integrals appearing herein are convergent.  Restoring the pseudovector components, the LFWFs recover the 1/k_⊥^2 decay characteristic of meson wave functions in QCD.  Consideration of regularisation and renormalisation is then necessary; but that is straightforward and has no material effect on our discussion, which is why we exploit the simplicity of Eq. (<ref>).

The lower panel of Fig. <ref> reveals that the K^+ LFWF is distorted, with its maximum located at (x=0.44,k_⊥^2=0), viz. displaced relative to that of ψ_π and thereby indicating that the dressed s̅-quark carries a larger fraction of the kaon's momentum than the u-quark.  As noted elsewhere <cit.>, the magnitude of this SU(3)-flavour-symmetry breaking shift (≃ 15%) is set by DCSB mass-scales, as expressed, e.g. in M_s/M_u = 1.2.

Having obtained the leading-twist LFWFs, one may compute the two-dressed-parton distribution amplitudes using Eq. (<ref>), with the results depicted in Fig. <ref>.  Consistent with the LFWFs, the PDAs are broad, concave functions.  Notably, although our models for the π and K are simple, they yield PDAs that agree qualitatively and semiquantitatively with results computed using more sophisticated approaches <cit.>.  The peak in the kaon PDA lies at x=0.44.





Notwithstanding the fact that a light-front Hamiltonian cannot in principle produce an eigenfunction of the form ψ(x,k_⊥^2) ∼ψ_1(x) ψ_2(k_⊥^2) for any system because, e.g. it violates momentum conservation, such a product Ansatz is often used to produce numerical estimates of various quantities.  Given a hadron G, it is typically introduced in the form:

    ψ_G(x,k_⊥^2)  factorised Ansatz=φ_G(x)  ψ_F(k_⊥^2) ,

with ψ_F, the k_⊥^2 profile function, often chosen to provide exponential decay.  In order to judge the accuracy of estimates obtained therewith, Fig. <ref> depicts the ratio:

    R^ψφ_G = ψ̂_G^↑↓(x,k_⊥^2)/φ_G(x) ,

evaluated at a number of k_⊥^2-values and normalised at each such that ∫ dx  ψ̂_G^↑↓(x,k_⊥^2) =1.  If a product Ansatz were a good approximation, then R^ψφ_G≡ 1.



The upper panel of Fig. <ref> depicts the ratio in Eq. (<ref>) obtained for the pion, G=π.  It is not unity; and the discrepancy grows with increasing momentum until k_⊥^2≈ 1GeV^2, whereafter the ratio has a fairly static profile.  On the other hand, the departure from unity is not great: the L_1-deviation saturates at approximately 2%.  One might therefore argue that Eq. (<ref>), with appropriate power-law behaviour for ψ_F(k_⊥^2), could be quantitatively useful for integrated properties of the pion and serve as a fair guide to the pointwise behaviour of ψ_π(x,k_⊥^2).

One should nevertheless bear in mind that any product Ansatz will be poorest on the domain of greatest correlation between the independent variables; and owing to momentum conservation, that domain is the neighbourhood of the endpoints, x=0,1, as evident in Fig. <ref>.

The situation is somewhat different for the kaon.  Depicted in the lower panel of Fig. <ref>, the ratio departs from unity by as much as 70%.  The L_1-deviation is 15% at k_⊥^2=0, initially drops with increasing k_⊥^2, but increases on k_⊥^2 ≳ 0.1GeV^2 to reach a limiting value of ≈ 20%.  In such circumstances, with a well-chosen power-law form for ψ_F,  Eq. (<ref>) might provide a fair indication of integrated kaon properties, but it can only at best be a sketchy guide to pointwise features of ψ_K(x,k_⊥^2).







  §.§.§ Quasi Parton Distribution Amplitudes


Eq. (<ref>) can now be used to compute pion and kaon qPDAs, with the results depicted in Fig. <ref>.

Focusing first on the pion (upper panel), it is evident that the result obtained with P_z=1.0GeV does not closely resemble φ_π(x): the L_1-difference between the two curves is 42% and the (2x-1)^2-moment obtained by integrating φ̃_π(x) on x∈ [0,1] is just 33% of the objective value.[The objective value for this moment is 0.25, Eq. (<ref>).  On physical grounds <cit.>, the pion's (2x-1)^2-moment should lie between the conformal limit value, 1/5, and the result obtained using φ_π(x)=constant, viz. 1/3.  Using φ̃_π(x;P_z=1  GeV), the moment defined here takes the value 0.22: (0.22-1/5)/(0.25-1/5) = 0.33.]

The step to P_z=1.75GeV brings material improvement, so that the φ̃_π(x) provides a qualitatively sound approximation to φ_π(x): the L_1-difference between the two curves is 18%, the (2x-1)^2-moment is 78% of the objective value, and one can reasonably conclude that the target PDA is a broad, concave function.

Further increments in P_z, however, do not bring much improvement.  For example, with P_z=3.0GeV, the L_1-difference between the qPDA and the PDA is 10% and the (2x-1)^2-moment is 85% of the objective value.  This outcome is a reflection of the fact that once the perturbative domain is entered, evolution in QCD is logarithmic.

It is noteworthy, too, that the pointwise forms of φ̃_π(x) leak significantly from the domain 0<x<1.  This prevents a determination of the target PDAs endpoint behaviour even with P_z=3GeV.  That behaviour is crucial because it fixes the magnitude of the leading-order, leading-twist perturbative QCD results for numerous observables <cit.> and hence sets the benchmark against which existing and foreseen experiments aimed at testing solid QCD predictions must be compared <cit.>.

Notably, in order to reach P_z ≈ 3GeV in a lQCD simulation, one would need a lattice with roughly 48 spatial sites and a spacing of 0.06fm.

Turning attention now to the kaon qPDAs, there are similarities with the pion case.  Using P_z=1.75GeV, φ̃_K(x) provides some reliable qualitative information about φ_K(x): the L_1-difference between the two curves is 28%, the qPDA peaks at x=0.45, and its (2x-1)^1 moment is 75% of the objective value.  On the other hand, the  (2x-1)^2-moment is just 29% of the goal.  Once again, incrementing P_z does not greatly improve the situation.  Using P_z=3.0GeV, the L_1-difference between φ̃_K(x) and φ_K(x) is 20%, the qPDA peaks at x=0.43, the (2x-1)^1 moment is 82% of the objective value, but the  (2x-1)^2-moment is only 35% of the goal.  In this case, reducing the L_1-difference between φ̃_K(x) and φ_K(x) to 10% would require P_z ≈ 20GeV.



In closing this subsection we return to SU(3)-flavour-symmetry violation in the kaon qPDAs, plotting [φ̃_K^-(x)-φ̃_K^+(x)]/2 in Fig. <ref>.  Evidently, using modest values of P_z, the qPDAs provide a fair pointwise description of the true difference on x∈[0.3,0.7].  Again, however, the behaviour on large domains near the endpoints is poorly represented.  The figure also displays



    δ M(x)     = (M_u - M_s) (1-x) + x (M_s - M_u)/M_u+M_s , 
    
       ≈ (f_π - f_K) (1-x) + x (f_K - f_π)/f_π+f_K .


The comparison of δ M(x) with [φ̃_K^-(x)-φ̃_K^+(x)] highlights that the scale of flavour-symmetry breaking in the kaon distribution amplitudes measures differences between the emergent masses of s- and u-quarks in the Standard Model.  The analogue of Eq. (<ref>) produced using Higgs-generated current-masses is an order-of-magnitude too large at the qPDAs' extrema.



§ QUASI PARTON DISTRIBUTION FUNCTIONS





 §.§ Algebraic Analysis: qPDFs


In describing valence-dressed-quark parton distribution functions at an hadronic scale, ζ_H, the  impulse-approximation (handbag diagram) is inadequate because it omits contributions from the gluons which bind valence-quarks into a hadron.  A remedy for this flaw is described and used to compute pion and kaon valence-quark distribution functions in Refs. <cit.>.  Using the kaon as an illustration:



    u_V^K(x)     =  tr_ CD∫_dkδ_n^x(P_K) 
    
         × [n·∂_k H_u(k;P_K)] H_s(k;P_K) ,
    
    s_V^K(x)     = u_V^K(1-x) ,


where n·∂_k = n_μ (∂ / ∂ k_μ),


    H_u(k;P_K)     = Γ̅_K(k_-^K;-P_K) S_u(k) ,
    
    H_s(k;P_K)     = Γ_K(k_-^K; P_K) S_s(k-P_K) ,


with Γ̅(k_-^K,P_K) = C^†Γ̅(-k_-^K,P_K)^ T C, where C is the charge conjugation matrix and (·)^ T denotes a transposed matrix.  Expressions for analogous distributions in the π are obtained by changing s→ d.

Canonical normalisation of the kaon's Bethe-Salpeter amplitudes ensures

    ∫_0^1 dx   u_V^K(x) = 1 = ∫_0^1 dx   s_V^K(x) .

Consequently, using Eq. (<ref>), one finds immediately:

    1 = ∫_0^1 dx   x [ u_V^K(x) + s_V^K(x)] .


In obtaining these results, one must use mathematical features of the matrix trace, properties of propagators and Bethe-Salpeter amplitudes under charge conjugation, and the following identity: for n^2=0,

    0 =  tr_ CD∫_dkδ_n^x(P_K) n·∂_k [H_u(k;P_K)   H_s(k;P_K)] .


Arriving at a quasi-PDF extension of Eqs. (<ref>) is almost as straightforward as making the transition from PDAs to qPDAs, described in Sec. (<ref>): one has



    ũ_V^K(x)     =  tr_ CD∫_dkδ_ñ^x(P_K) 
    
       × [ñ·∂_k H_u(k;P_K)] H_s(k;P_K) -  S(x)  ,
    
    s̃_V^K(x)     = ũ_V^K(1-x) ,
    
    
    
    
    
     S(x)     = 12 tr_ CD∫_dkδ_ñ^x(P_K)  
    
         ×ñ·∂_k [H_u(k;P_K) H_s(k;P_K)] .


Analogous to the procedure in Sec. (<ref>), the primary step is simply n→ñ in the PDF formulae.  However, the correction term, S(x), is also needed.  Its presence is suggested by the role of Eq. (<ref>) in ensuring momentum conservation; and it guarantees, inter alia, Eq. (<ref>).  Once again, analogous distributions in the π are obtained by replacing s→ d.

Using Eqs. (<ref>), one may readily establish


    ∫_-∞^∞    dx̃ ũ_V^K(x̃) = ∫_0^1dx u_V^K(x) = 1 ,
    
    ∫_-∞^∞    dx̃ s̃_V^K(x̃)  = ∫_0^1dx  s_V^K(x) = 1 , 
    
    ∫_-∞^∞    dx̃ x̃ [ ũ_V^K(x̃) + s̃_V^K(x̃)] 
          = ∫_0^1 dx   x [ u_V^K(x) + s_V^K(x)] = 1 .


Evidently, Eqs. (<ref>) define purely valence quark quasidistributions.



 §.§ Numerical Illustrations: qPDFs


We now use Eqs. (<ref>), (<ref>), (<ref>), (<ref>) to compute the pion and kaon qPDFs.  The calculation is straightforward, following the pattern in Sec. <ref>: one uses Feynman parametrisation to combine denominator products into a single quadratic form, Cauchy's theorem to evaluate the k_4 integral, direct evaluation for ∫ d^2k_⊥, and finally numerical integration over the Feynman parameters.  The results are depicted in Fig. <ref>.  (The objective PDFs were obtained using the approach described in Ref. <cit.> and checked using the overlap representation <cit.>.)



A cursory comparison between Figs. <ref> and <ref> reveals that a valence-quark qPDF is typically a better approximation to the objective result than a qPDA at any given value of P_z.  Looking closer at the pion (Fig. <ref>, upper panel), the L_1-differences are
19% (P_z=1GeV),
9% (P_z=1.75GeV),
5% (P_z=2.4GeV),
4% (P_z=3GeV).
This series indicates that even with P_z=1GeV, the pion's valence-quark qPDF delivers a qualitatively sound approximation to the true result; and the step to P_z=1.75GeV brings noticeable improvement; but, as with the qPDAs, improvement is slow on P_z > 1.75GeV.

Similar, too, is the pointwise behaviour of the valence-quark qPDFs in the neighbourhood of the endpoints: as with the qPDAs, the qPDFs leak   significantly from the domain 0<x<1.

This is important because one of the earliest predictions of the QCD parton model, augmented by features of perturbative QCD (pQCD), is that the valence-quark distribution function in a pseudoscalar meson behaves as follows <cit.>:

    q_V^G(x;ζ_H)  large x∼ (1-x)^2+γ,

where γ≳ 0 is an anomalous dimension.  Verification of Eq. (<ref>) is an important milestone on the path toward confirmation of QCD as the theory of strong interactions <cit.>.  In this connection we recall that Ref. <cit.> (the E615 experiment) reported a pion valence-quark PDF obtained via a leading-order pQCD analysis of their data, viz. u_V^π(x) ∼ (1-x), seemingly a marked contradiction of Eq. (<ref>).  Subsequent computations using continuum methods appropriate to QCD bound-states <cit.> confirmed Eq. (<ref>) and eventually prompted reconsideration of the E615 analysis, with the result that at next-to-leading order and including soft-gluon resummation <cit.>, the E615 data can be viewed as being consistent with Eq. (<ref>).  New data are essential in order to check this reappraisal of the E615 data and settle the controversy.  This goal is a focus of an approved tagged DIS experiment at the Thomas Jefferson National Accelerator Facility (JLab) <cit.>.  Such data could also be obtained with the common muon proton apparatus for structure and spectroscopy (COMPASS) detector at CERN <cit.> and at a future electron ion collider (EIC) <cit.>.



These observations emphasise that quantitatively reliable lQCD results bearing upon Eq.(<ref>) would be very valuable. However, the challenge to delivering such outcomes using qPDFs is highlighted by Fig. <ref>.  On a domain of valence-quark x, this figure compares the pion qPDFs in the upper panel of Fig. <ref> with the objective-PDF:

    u_V^π(x)x>0.95≈ 113  (1 - x)^2,

and another curve, whose x∈[0,1] L_1-difference from the objective valence-quark PDF is just 2%, but which is ∝ (1-x)^1  at large-x.  Evidently, even the P_z=3GeV qPDF is unable to distinguish between these two distinctively different results.  (That x>0.9 is required before (1-x)^2 behaviour is visible in the pion's valence-quark distribution was remarked upon earlier <cit.>.)



We now redirect our attention to kaon valence-quark qPDFs.  As with kaon qPDAs, there are similarities with the pion.  For instance, L_1-differences are
25% (P_z=1GeV),
12% (P_z=1.75GeV),
8% (P_z=2.4GeV),
6% (P_z=3GeV),
indicating, again, that even with P_z=1GeV, the kaon's qPDF delivers a qualitatively sound approximation to u_V^K(x); the step to P_z=1.75GeV brings noticeable improvement, but changes are slow thereafter.

The remarks made in connection with the pion qPDFs' large-x behaviour hold with equal force for the kaon.



It has been argued that the ratio u_V^K(x)/u_V^π(x) serves as a sensitive probe of the difference between the gluon distributions in the pion and kaon <cit.>, and that this difference can reveal much about the emergence of mass in the Standard Model <cit.>.  Experimental data on the ratio is available <cit.>, but one measurement is insufficient for complete confidence.  Newer data would be welcome, in which connection tagged DIS at JLab might also be useful <cit.>, as could the COMPASS detector at the CERN  <cit.> and a future EIC <cit.>.  With these things in mind, in Fig. <ref> we depict the ratio ũ_V^K(x)/ũ_V^π(x).

Evidently, for P_z≥ 1.75GeV, much as was the case with the qPDA asymmetry depicted in Fig. <ref>, the ratio of qPDFs is quantitatively a good approximation to the objective ratio on a material domain, viz.  0.3≲ x ≲ 0.8.  This domain almost covers that upon which empirical data is available.  We therefore anticipate that contemporary lQCD simulations could provide a sound prediction for this ratio before next generation experiments are completed.



§ SUMMARY AND PERSPECTIVE



Employing a continuum approach to bound-states in quantum field theory and practical algebraic Ansätze for the Poincaré-covariant Bethe-Salpeter wave functions of the pion and kaon, we computed the leading-twist two-dressed-parton light-front wave functions (LFWFs), ψ(x,k_⊥^2); parton distribution amplitudes (PDAs), φ(x); quasi-PDAs (qPDAs), φ̃(x); valence parton distribution functions (PDFs), u_V(x); and quasi-PDFs (qPDFs), ũ_V(x), for these systems.

The LFWFs are broad, concave functions, with power-law k_⊥^2-decay.  Whilst the pion's LFWF is symmetric about x=1/2, ψ_K(x,k_⊥^2) peaks at (x=0.44,k_⊥^2=0), expressing SU(3)-flavour-symmetry violation with a magnitude determined by differences between dynamical (not explicit) mass generation in the s- and u-quark sectors.  Looking closely at the LFWFs, we found that a carefully constructed product Ansatz, viz. ψ(x,k_⊥^2) ∼ψ_1(x) ψ_2(k_⊥^2), although flawed in principle, can provide fair estimates of integrated π, K properties.

The LFWFs provide direct access to π and K PDAs and qPDAs; and for each system the qPDF provides a semiquantitatively reliable representation of the associated PDA when computed using a longitudinal momentum P_z=1.75GeV.  However, improvements thereafter are slow; and, notably, even with P_z=3GeV, the qPDA cannot provide information about the true PDAs endpoint behaviour.

Regarding pion and kaon valence-quark PDFs and qPDFs, we found that at any given P_z, a qPDF delivers a better representation of the associated PDF than does a qPDA of the objective PDA.  In fact, even with P_z = 1GeV the qPDF provides a qualitatively clear picture of the PDF.  However, as with qPDAs, differences between qPDFs and PDFs diminish slowly on P_z>1.75GeV; and, similarly, even with P_z=3GeV, qPDFs cannot be used to determine the objective PDF's large-x behaviour.  On the other hand, the ratio ũ_V^K(x)/ũ_V^π(x) does provide a good approximation to u_V^K(x)/u_V^π(x) on 0.3≲ x ≲ 0.8, in consequence of which we expect that contemporary simulations of lattice-regularised QCD can deliver a reasonable prediction for this ratio before next generation experiments are completed.

It is natural to extend this analysis to the neutron and proton, for which analogous algebraic Ansätze for the bound-state Faddeev wave functions exist or can readily be developed <cit.>.  We anticipate that the outcome will be qualitatively similar: in particular, that even using modest values of P_z, a material valence-quark x-domain will exist upon which ratios of qPDFs may provide sound representations of the PDF ratios measured empirically <cit.>.








We are grateful for constructive suggestions from
J. Chen, T. Horn, L. Liu, C. Mezrag, J. Rodríguez-Quintero and S. Riordan.

Work supported by:

China Postdoctoral Science Foundation (under Grant No. 2016M591809);

the Chinese Government's Thousand Talents Plan for Young Professionals;

the Chinese Ministry of Education, under the International Distinguished Professor programme;

U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract no. DE-AC02-06CH11357;

and National Natural Science Foundation of China (under Grant Nos. 11475085, 11535005 and 11690030).






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