Recently, pion valence quark and u-quark distribution functions (π v(u)PDF) have become a hot topic in Hadron physics. π v(u)PDF has long history but we can say that people’s recent concern for π v(u)PDF follows two points. One point is asymptotic behavior and the other one is when this asymptotic behavior starts. Concern to asymptotic behavior maintains fairly long after showing the result that asymptotic behavior of E615 results [1] does not include NLO (next leading order) calculation and its reanalysis results [2] include NLO calculation, which are different. The former asymptotic behavior is exponent of ( 1 − x ) about 1 (contradict to QCD (quantum chromodynamics) prediction), but the latter one is about 2 (or more) (follow to QCD prediction). For theoretical side, the former asymptotic behavior is supported by constituent quark model [3], Nambu-Jona-Lasino model [4], duality argument [5] and recently, Pasquini et al. [6] and Xie et al. [7], while the latter behavior is supported by Dyson Schwinger equation [8]-[12] and recent Lattice QCD calculation [13][14]. For the second concern, it starts after phenomenological analysis such as JAM [15] and xFitter [16] (JAM and xFitter are working team’s name), which include NLO consideration, show that, in the range of medium x to close to 1, one half of their π vPDF, i.e., π uPDF, behave as close to that of E615 original analysis data, which means close to linear of ( 1 − x ) when x approaches 1. Note that both xFitter and JAM analysis are based on what form of π vPDF can generate cross-sections that agree with those of considered experiments. However, phenomenological analysis is not sued to argue asymptotic behavior because data set becomes unreliable when x approaches 1 (cross-section itself becomes obscure and number of data is small). In addition, even Dyson Schwinger model [11], which employs inhomogeneous Bethe-Salpeter equation, shows that the actual asymptotic behavior appears at really large Q, that is, x is far closer to 1 than commonly considered.

For kaon distribution functions case, in the absence of empirical information, to separate the quark flavor PDFs in the kaon, first-principle lattice QCD simulations can be used to complement the experimental measurement. For this approach (Lattice QCD), ETM collaboration [17] shows momentum fraction of valence quarks and that of gluons in both pion and kaon. There are only two available experiments for extracting kaon PDFs, i.e., NA3 data [18] and J/ψ production data [19] (these experiments did not show s-quark valence structure). To extract and compare to kaon PDFs from these experiments, we refer, for example, the article of Xu et al. [20] and statistical model by Bourrely et al. [21]. Recently, JAM collaboration [22] shows a first combined QCD analysis of the experimental cross-sections and lattice moments to extract the valence quark PDFs in the pion and kaon and their impact on the gluon momentum fractions. In this paper, in Section 2, we first show π v(u)PDF based on our approximate pion distribution amplitude derived in previous paper [23]. In Section 3, we derive an approximate kaon distribution amplitude and functions which can be compared to that of pion by using nonzero mass solutions for our ‘tHooft model (bound system in 1 + 1 dimensional QCD), same as pion case (zero mass solutions) as shown in [23]. Because s-quark is much heavier than u-quark, we employ the argument of massive quark case as shown in ref. [24], so that we cannot obtain exact solutions in coordinate space. Instead, we set up inhomogeneous integro-differential equations.

We showed a pion distribution amplitude in previous paper [23]. Here, we describe pion distribution functions based on our approximate pion distribution amplitude. Our approximate pion distribution amplitude is described as

where ρ = u 2 | α | , | α | = g 2 16 and x is fraction defined in the region as x ∈ [ 0 , 1 ] , and g 2 is a coupling constant.

The valence quark distribution functions f v ( x ) is defined as

Thus, in our case, our distribution functions is obtained as

Figure 1 shows our pion valence quark distribution functions obtained by Equation (3). Pion valence quark distribution functions with ρ = 0.5 and ρ = 3 correspond to twice larger than those shown in Figure 2(A) and Figure 2(B) in Raya et al.’s paper [25]. According to Raya et al., a distribution function (π uPDFs) shown in Figure 2(A) represents rest frame case and that shown in Figure 2(B) represents deep inelastic scattering case that is able to compare to E615 data. The reason why our peak values are twice larger than that of common pion u quark distribution functions is that our valence quark distribution functions represent q q ¯ system because they are derived from a charged pion wave function, which represents u d ¯ (or u ¯ d ). Because pion is the Nambu-Goldstone boson mode of QCD, its mass becomes zero at chiral limit (massless quark case). Our wave function is derived in this situation as shown in [23]. Twice larger value is shown both in xFittar [16] and JAM [15] analysis. Important point is that both xFittar and JAM derive a valence quark distribution function by fitting to cross-section data (xFittar used E615, NA10 (286 (GeV) and 194 (GeV)) and WA70 data and JAM used same data as xFittar except Hera data instead of WA70). Thus, in order to compare to common E615 data, we have to multiply a factor 1 2 . Pasquini et al. also show this procedure in ref. [6].

Figure 1. Pion valence quark distribution functions (blue curve ρ = 0.5 case and red curve ρ = 3 case).

Both xFittar and JAM show that peak value is around 0.7 to 0.8 and peak point is about x = 0.4. Our ρ = 3 valence quark distribution functions shows similar results. If we can consider that rest frame distribution functions corresponds to that of initial scale case, we can set 0.85 (GeV) for initial scale Q 0 of our ρ = 0.5 distribution functions and that value is adopted by Pasquini et al. [6]. Pasquini et al. describe an evolution pion u-quark distribution functions at Q 2 = 27 (GeV²), i.e., Q = 5.2 (GeV). Note that Pasquini et al. use notation μ 0 , μ 2 instead of Q. Thus, momentum acceleration is about 6 times as much. We use this consideration to obtain ρ = 3 valence quark distribution functions. In ‘tHooft model, mass square is propotional to g 2 / π . This means that dimension of g 2 is (GeV)², so that dimension ofuin ρ is same as momenta (GeV) because ρ is dimensionless and we set c = ℏ = 1 (c is velocity of light and ℏ is Plank constant divided by 2π).

Thus, momentum acceleration means that u value is 6 times as much. This dimensional consideration can be confirmed as follows. In previous paper [23], u is set after changing variable as x = e − i 3 π 4 (or e − i π 4 ) 1 | α | z (or z ¯ ) (this x is space coordinate), so that u is dimensionless. In this case, | q | u 2 | α | is dimensionless. Then, to obtain x 1 − x (this x is fraction), | q | is described as (Momentum dimensional value) momentum / Max momentum 1 − momentum / Max momentum = (momentum dimensional value) q ¯ 1 − q ¯ . Then, q ¯ is dimensionless and can be denoted as x (fraction). Now, u is multiplied by this (momentum dimensional value). Thus, dimension of u in ρ is momentum (GeV).

Note that common method to obtain an accelerated distribution function which can be compared to experimental data is adopting DGLAP evolution equation [26] from initial scale distribution function. Wu et al. [27] mention that, in the deep inelastic scattering, the hadron can be regarded as moving with an infinite momentum frame (IMF). In IMF, the hadron is moving with infinite four momentum (E, P) in the z direction. Then, they show that rest frame four-momentum ( p 0 , p ) transfers to four-momentum ( k 0 , k ) by inverse Lorenz boost and that rest frame p can be expressed by IMF variables ( ζ = k z P , k ⊥ ) when taking P → ∞ , where k z is longitudinal momentum of quarks in IMF (momentum consideration). Using this expression of wave function, they obtain distribution amplitude at initial momentum scale. Then, they obtain distribution function by DGLAP evolution equation (essentially coupling constant consideration). This means that they consider two processes that are momentum consideration and coupling constant consideration to obtain distribution functions from rest frame wave function. In our case, rest frame wave function is obtained in 1 + 1 dimensions. As shown in ref. [23], the proper distribution amplitude in rest frame is obtained by Fourier Transform with choosing ρ value that is choosing coupling constant g 2 value as fixing momentum u (coupling constant consideration). Distribution functions of deep inelastic scattering is expressed as in moving frame so that change of ρ value needs change of momentumubecause coupling constant g 2 is already chosen (momentum consideration). Thus, we also consider two processes that are momentum consideration and coupling constant consideration as same as Wu et al.

Figure 2 shows our pion u-quark distribution functions compared to E615 original analysis data [1]. Here, we use the following relation equation between valence quark distribution functions ( q q ¯ system) and u-quark distributions (q only).

Figure 2. Pion u quark distribution functions: pion u-quark (blue curve), E615 data (red dots).

In our case,

In order to derive kaon distribution functions, we need to obtain kaon distribution amplitude as shown in the case of that of pion [23]. For pion, as mentioned before, we used the fact that pion mass is zero in the chiral limit (massless quarks) because of pion is the Nambu-Goldstone boson mode. Thus, we used a wave function of our ‘tHooft model with zero mass case. For kaon, although kaon is also considered as the Nambu-Goldstone boson mode of QCD [28], we use the same consideration to construct kaon mass spectrum in 3 + 1 dimension massive quark case [24] instead of using chiral limit consideration. Main reason is following. We cannot express exact next mass besides zero-mass that should be kaon mass [29]. Thus, we prefer to representation of distribution amplitude and functions without explicit dependence of β . In fact, we can obtain distribution amplitude and functions without dependence of β as shown in later by using this consideration. In ref. [24], to obtain kaon mass spectrum, we used a wave function of f 0 ( 500 ) for massless quark case (chiral limit case) and applied the first order perturbation by considering quark mass term as perturbative Hamiltonian. Thus, in this time, we first construct an equation of motion with massive quark in 1 + 1 dimension case. Recalling the fact that distribution amplitude corresponds Fourier Transform of the space coordinate wave function [23], we do not have to consider the corresponding eigenvalue (mass) but need only space coordinate wave function. Thus, we can consider the quark mass term as inhomogeneous part of inhomogeneous second order integro-differential equation.

Before constructing an equation of motion with massive quark case in 1 + 1 dimensions, we check our equation motion in 3 + 1 dimensions by comparing to that of Suura. Suura’s definition of gauge invariant operator q η ξ ( 1 , 2 ) is described as [30]

where T r c denotes trace of color spin a and P denotes path ordering with straight line and η , ξ denote Dirac indices. Note that actual Suura’s notation of Dirac indices are α and β .

Note that Suura defined Dirac indices in QED case and Trace for color and path ordering were represented in QCD case. Note that Suura’s representation of path ordering is [ ⋯ ] + and Trace of color is explained by words.

Our case of gauge invariant operator q η ξ ( 1 , 2 ) is, for example, described as in ref. [31] as

Comparing Equation (6) and Equation (7), only difference is position of q ξ † ( 2 ) . If we move this quark field of Equation (7) to the same position of Equation (6), minus sign will appear because of anti-commutation of quark and anti-quark fields. This means that only difference between Equation (6) and Equation (7) is sign. Then, first we compare kinetic terms.

Suura’s definition of kinetic terms is

where α → L ( R ) means α → operating from left (right) on q ( 1 , 2 ) .

This means that kinetic terms become

When we consider relative coordinate as r → = x → ( 2 ) − x → ( 1 ) , Equation (6) becomes

where q s ( 1 , 2 ) and q k ( 1 , 2 ) denote Suura’s and ours of gauge invariant operator, respectively.

Equation (9) shows that our definition of kinetic terms is also derived from Suura’s one.

For massive quark case, Dirac equation becomes as

Note that normal notation of β ¯ is β .

Equation (10) and Equation (11) are obtained from the representation of Dirac equation as

Adopting metric is η 00 = − 1 , η 11 = η 22 = η 33 = 1 .

Adopting γ -matrices are

This description is following Weinberg’s one [32].

For Suura’s definition of gauge invariant operator case, massive quark terms become from Equation (10) and Equation (11) as

Thus, for our definition of gauge invariant operator case, massive quark terms become

Decomposition of gauge invariant operator is

Then, massive quark terms become

Thus, by using anti-particle argument mentioned in ref. [24], massive quark terms affecting our equation of motion are following:

This is exactly same form obtained in ref. [24].

Above results are important to construct an equation of motion with massive quark in 1 + 1 dimensions because we do not have proper Dirac equation in 1 + 1 (or 2) dimensions. We have to employ Dirac equation with massive quark in 3 + 1 dimensions to that in 1 + 1 dimension as an analogous form.

Thus, our equation of motion with massive quarks in 1 + 1 dimensions becomes

where

Note that E a ( x ) and A a ( x ) indicate E 1 a ( x ) and A 1 a ( x ) , respectively and that Equation (22) except mass terms is same as equation of motion adopting in previous paper [23].

We employ the metric system and γ -matrices in 1 + 1 dimensions as follows.

Note that sign of metric system is different from that in ref. [23] because of consideration of massive quark terms as mentioned before, and that we employed γ -matrices defined by Casher et al. [33].

The Bethe-Salpeter like amplitude is defined by sandwiching between vacuum state and physical state as

Because decomposition of gauge invariant operator in 1 + 1 dimensions is

After taking center of mass coordinate and relative coordinate as described in ref. [23], the Bethe-Salpeter like amplitude with massive quarks becomes

Note that we omit O ( 1 N ) term because we are considering large N limit case and that when P 1 = 0 (rest frame case), P 0 should be W 0 .

From Equation (27) and Equation (28), Χ 1 ( r ) becomes

Substituting Equation (31) into Equation (30), Equation (30) becomes

Using Equation (29) and Equation (31) and taking Χ ± ( r ) = Χ 3 ( r ) ± i Χ 2 ( r ) , we obtain following equations:

As described in ref. [23], we solve Equation (33) first. Because it is difficult to solve Equation (33) exactly, we use the following approximation. For the last term, Χ 3 ( r ) , (depending quark mass term), we take mass zero ( W 0 = 0 ( | β | = 0 )) solution derived in ref. [23] which is related to our pion wave function in 1 + 1 dimensions. Then, we consider the approximated last term of Equation (33) as an inhomogeneous term and after taking derivative with respect to r and some manipulation including Sokhotsky formula to singular integral equation [34] described in ref. [23], we can construct the inhomogeneous second order differential equation as

where λ = 1 2 i P 0 ( m 1 + m 2 ) 2 P 0 2 − P 1 2 , c 1 and c 2 are constants determined later.

Note that we change the notation of space coordinate rto x in Equation (35) and Equation (36) because we are working in the framework of 1 + 1 dimensions. From now on, we use this notation for space coordinate.

Note that Equation (35) and Equation (36) are composed by the quantities when x asymptotically approaches real axis in the upper-half hemisphere and in the lower-half hemisphere, respectively.

Homogeneous parts of equations for Equation (35) and Equation (36) are obtained by setting left-hand side of equations be zero for both cases as

One of the solutions is given in ref. [23] for both equations. These solutions are as follows:

As shown in Appendix C of ref. [23], the characteristic part of solution described in Equation (39) is Weber function D λ ¯ ( t ) type and that in Equation (40) is Weber function D − λ ¯ − 1 ( i t ) type.

Thus, the other solution for Equation (35) must be Weber function D − λ ¯ − 1 ( i t ) type and that for Equation (36) must be Weber function D λ ¯ ( t ) type. Because of λ ¯ = − 1 − 2 i π for Equation (35) and λ ¯ = − 2 i π for Equation (36), the other solutions for Equation (35) and Equation (36) become

Then, Equation (39) and Equation (41) are considered as basic solutions for homogeneous second order differential equation described in Equation (35) and Equation (40) and Equation (42) are considered as basic solutions for homogeneous second order differential equation described in Equation (36).

In order to construct particular solutions, we follow the argument of Ince [35]. Our case’s Wronskian are defined as

Equation (43) is Wronskian of Equation (35) and Equation (44) is Wronskian of Equation (36).

Using the following relation equation for derivative of Whittaker function W ′ κ , μ ( z ) [36]

the form of W r o n s ( 1 ) and W r o n s ( 2 ) are described as

where

where

According to Ince [35], when the basic solutions of homogeneous part are given as M 1 ( x ) and M 2 ( x ) and inhomogeneous part is given as f ( x ) , the particular solution is defined as

Lower limit of integral is chosen by boundary condition consideration.

Thus, our case of particular solutions becomes

Note that we omit coefficients of basic solutions in both Equation (48) and Equation (49) because these are inefficient for particular solutions.

Each Wronskian is described as

To construct distribution amplitude, that is a function in momentum space, we consider Fourier Transform of particular solution F p ( q ) as shown in ref. [23].