Pion and kaon parton distribution functions from basis light front quantization and QCD evolution

TL;DR

This paper combines basis light front quantization and QCD evolution to compute pion and kaon PDFs, achieving results consistent with multiple experimental data sets and providing insights into meson structure.

Contribution

It introduces a novel approach integrating light front wave functions with QCD evolution to calculate meson PDFs, aligning well with experimental measurements.

Findings

Pion valence-quark PDF matches Fermilab experiment after QCD evolution.

Pion structure function agrees with DESY-HERA data at large x.

Kaon to pion up-quark PDF ratio aligns with CERN experiment.

Abstract

We investigate the parton distribution functions (PDFs) of the pion and the kaon by combining quantum chromodynamics (QCD) evolution with the basis light front quantization. The initial PDFs result from the light front wave functions obtained by diagonalizing the effective Hamiltonian consisting of the holographic QCD confinement potential, a complementary longitudinal confinement potential, and the color-singlet Nambu--Jona-Lasinio interactions. The valence-quark PDF of the pion, after QCD evolution, is consistent with the result from the E-0615 experiment at Fermilab. Meanwhile, the pion structure function calculated from the PDFs agrees with the ZEUS and the H1 experiments at DESY-HERA for large $x$. Additionally, the ratio of the up quark PDF of the kaon to that of the pion is in agreement with the NA-003 experiment at CERN. We also present the cross section for the pion-nucleus induced Drell-Yan process with the obtained pion PDFs supplemented by the PDFs of the target nuclei.

Figures

Tables

Table 1. Table 1: Dependence of the PDF fitting parameters on the longitudinal basis cutoff L max subscript 𝐿 max L_{\mathrm{max}} . With N max = 8 subscript 𝑁 max 8 N_{\mathrm{max}}=8 , the extrapolations are carried out by fitting to quadratic functions of L max − 1 superscript subscript 𝐿 max 1 L_{\mathrm{max}}^{-1} .

$L_{\max }$	$8$	$12$	$16$	$20$	$24$	$28$	$32$	Extrapolated to $+\infty$
${\pi }^{+}a=b$	$0.8045$	$0.6978$	$0.6549$	$0.6351$	$0.6249$	$0.6195$	$0.6163$	$0.5961$
$K^{+}a$	$0.7415$	$0.6823$	$0.6611$	$0.6500$	$0.6500$	$0.6403$	$0.6414$	$0.6337$
$K^{+}b$	$1.0002$	$0.9193$	$0.8907$	$0.8757$	$0.8761$	$0.8625$	$0.8643$	$0.8546$

Table 2. Table 2: Initial scales and the χ 2 superscript 𝜒 2 \chi^{2} /(d.o.f.) at the first three orders of the DGLAP equation. The χ 2 superscript 𝜒 2 \chi^{2} are defined as the sum of square-difference of our results with respect to the center values of the FNAL-E-0615 experiment Conway  et al. ( 1989 ) and the CERN-NA-003 experiment NA3 Collaboration  et al. ( 1983 ) , both at the respective experimental scales.

Order	Initial scale of pion	Initial scale of kaon	E-0615 ${\chi }^2/$(d.o.f.)	NA-003 ${\chi }^2/$(d.o.f.)
LO	$0.120\pm 0.012{\mathrm{GeV}}^2$	$0.133\pm 0.013{\mathrm{GeV}}^2$	$6.71$	$0.88$
NLO	$0.205\pm 0.020{\mathrm{GeV}}^2$	$0.210\pm 0.021{\mathrm{GeV}}^2$	$4.67$	$0.56$
NNLO	$0.240\pm 0.024{\mathrm{GeV}}^2$	$0.246\pm 0.024{\mathrm{GeV}}^2$	$3.64$	$0.50$

Table 3. Table 3: Comparison of the lowest four moments of the valence quark PDF in the pion based on the initial PDF from BLFQ-NJL model with the results from the global fit, lattice QCD, and phenomenological models at various scales. Results tabulated here at μ 2 ≥ 4 superscript 𝜇 2 4 \mu^{2}\geq 4 GeV 2 are also presented in Fig.  4 (a).

	${\mu }^2$ GeV2	$⟨x⟩$	$⟨x^2⟩$	$⟨x^3⟩$	$⟨x^4⟩$
DSE-RL (2018) Bednar et al. (2018)	1.69	0.268	0.125	0.076	0.054
WI-An (2018) Bednar et al. (2018)		0.268	0.114	0.059	0.037
JAM global fit (2018) JAM Collaboration et al. (2018)		0.268	0.127	0.074	0.048
BLFQ-NJL		${0.271}_{-0.020}^{+0.020}$	${0.124}_{-0.014}^{+0.014}$	${0.069}_{-0.009}^{+0.009}$	${0.044}_{-0.007}^{+0.007}$
Sutton (1992) Sutton et al. (1992)	4	$0.24\pm 0.01$	$0.10\pm 0.01$	$0.058\pm 0.004$
Hecht (2001) Hecht et al. (2001)		$0.24$	0.098	0.049
Chen (2016) Chen et al. (2016)		$0.26$	$0.11$	$0.052$
BSE (2018) Shi et al. (2018)		0.24
BSE (2019) Ding et al. (2019)		$0.24\pm 0.02$
QCDSF/UKQCD (2007) [lattice QCD] QCDSF-UKQCD Collaboration et al. (2007)		$0.27\pm 0.01$	$0.13\pm 0.01$	$0.074\pm 0.010$
DESY (2016) [lattice QCD] Abdel-Rehim et al. (2015)		$0.214\pm 0.015$
ETM (2018) [lattice QCD] ETM Collaboration et al. (2019)		$0.207\pm 0.011$	$0.163\pm 0.033$
JAM global fit (2018) JAM Collaboration et al. (2018)		$0.245\pm 0.005$	$0.108\pm 0.003$
BLFQ-NJL		${0.245}_{-0.018}^{+0.018}$	${0.106}_{-0.012}^{+0.012}$	${0.057}_{-0.008}^{+0.008}$	${0.035}_{-0.005}^{+0.005}$
Detmold (2003) [lattice QCD] Detmold et al. (2003)	5.76	$0.24\pm 0.01$	$0.09\pm 0.03$	$0.043\pm 0.015$
BLFQ-NJL		${0.236}_{-0.018}^{+0.018}$	${0.101}_{-0.011}^{+0.011}$	${0.054}_{-0.007}^{+0.007}$	${0.032}_{-0.005}^{+0.005}$
Watanabe (2018) Watanabe et al. (2018a)	27	0.23	0.094	0.048
Nam (2012) Nam (2012)		${0.214}_{-0.030}^{+0.016}$	${0.087}_{-0.019}^{+0.010}$	${0.044}_{-0.011}^{+0.006}$	${0.026}_{-0.008}^{+0.004}$
Wijesooriya (2005) Wijesooriya et al. (2005)		$0.217\pm 0.011$	$0.087\pm 0.005$	$0.045\pm 0.003$
BLFQ-NJL		${0.210}_{-0.016}^{+0.016}$	${0.084}_{-0.009}^{+0.009}$	${0.043}_{-0.006}^{+0.006}$	${0.025}_{-0.004}^{+0.004}$
Sutton (1992) Sutton et al. (1992)	49	$0.200\pm 0.015$	$0.080\pm 0.007$
Martinell (1988) [lattice QCD] Martinelli and Sachrajda (1988)		$0.23\pm 0.05$	$0.090\pm 0.035$
BLFQ-NJL		${0.202}_{-0.015}^{+0.015}$	${0.079}_{-0.009}^{+0.009}$	${0.040}_{-0.005}^{+0.005}$	${0.023}_{-0.003}^{+0.003}$

Table 4. Table 4: Lowest four moments of valence quark distributions in the kaon based on the initial PDF from BLFQ-NJL model. Comparisons are made with results from Refs.  Chen  et al. ( 2016 ); Watanabe  et al. ( 2018b ) .

flavor		${\mu }^2$ GeV2	$⟨x⟩$	$⟨x^2⟩$	$⟨x^3⟩$	$⟨x^4⟩$
$s^K$	BLFQ-NJL	1	${0.320}_{-0.024}^{+0.024}$	${0.158}_{-0.017}^{+0.018}$	${0.093}_{-0.012}^{+0.013}$	${0.061}_{-0.009}^{+0.010}$
$u^K$			${0.282}_{-0.021}^{+0.021}$	${0.128}_{-0.014}^{+0.014}$	${0.071}_{-0.010}^{+0.010}$	${0.044}_{-0.007}^{+0.007}$
$s^K$	BLFQ-NJL	4	${0.266}_{-0.020}^{+0.020}$	${0.119}_{-0.013}^{+0.013}$	${0.066}_{-0.009}^{+0.009}$	${0.041}_{-0.006}^{+0.006}$
$u^K$			${0.235}_{-0.018}^{+0.017}$	${0.097}_{-0.011}^{+0.011}$	${0.050}_{-0.007}^{+0.007}$	${0.030}_{-0.005}^{+0.005}$
$s^K$	BLFQ-NJL	16	${0.237}_{-0.018}^{+0.018}$	${0.100}_{-0.011}^{+0.011}$	${0.052}_{-0.007}^{+0.007}$	${0.031}_{-0.005}^{+0.005}$
$u^K$			${0.209}_{-0.016}^{+0.015}$	${0.081}_{-0.009}^{+0.009}$	${0.040}_{-0.005}^{+0.006}$	${0.023}_{-0.003}^{+0.004}$
$s^K$	Chen (2016) Chen et al. (2016)	27	0.36	0.17	0.092
	Watanabe (2018) Watanabe et al. (2018b)		0.24	0.096	0.049
	BLFQ-NJL		${0.228}_{-0.017}^{+0.017}$	${0.094}_{-0.010}^{+0.010}$	${0.049}_{-0.007}^{+0.007}$	${0.029}_{-0.004}^{+0.005}$
$u^K$	Chen (2016) Chen et al. (2016)		0.28	0.11	0.048
	Watanabe (2018) Watanabe et al. (2018b)		0.23	0.091	0.045
	BLFQ-NJL		${0.201}_{-0.015}^{+0.015}$	${0.077}_{-0.008}^{+0.009}$	${0.037}_{-0.005}^{+0.005}$	${0.021}_{-0.003}^{+0.003}$

Equations

$$H_{\mathrm{eff}}|\Psi ⟩=M^2|\Psi ⟩,$$

$$H_{\mathrm{eff}}$$

$$\displaystyle\quad-\frac{\kappa^{4}}{(m_{q}+m_{\bar{q}})^{2}}\partial_{x}\big{(}x(1-x)\partial_{x}\big{)}+H^{\rm eff}_{\rm NJL},$$

$$H_{\mathrm{NJL},\pi }^{\mathrm{eff}}$$

$$+{\bar{u}}_{\mathrm{u}s{1}^{\prime }}(p_1^{\prime }){\gamma }_5{u}_{\mathrm{u}s1}(p_1){\bar{v}}_{\mathrm{d}s2}(p_2){\gamma }_5{v}_{\mathrm{d}s{2}^{\prime }}(p_2^{\prime })$$

$$\displaystyle\quad+2\,\bar{u}_{\mathrm{u}s1^{\prime}}(p_{1}^{\prime})\gamma_{5}v_{\mathrm{d}s2^{\prime}}(p_{2}^{\prime})\,\bar{v}_{\mathrm{d}s2}(p_{2})\gamma_{5}u_{\mathrm{u}s1}(p_{1})\big{\}},$$

$$H_{\mathrm{NJL},K}^{\mathrm{eff}}$$

$$\displaystyle\quad+2\,\bar{u}_{\mathrm{u}s1^{\prime}}(p_{1}^{\prime})\gamma_{5}v_{\mathrm{s}s2^{\prime}}(p_{2}^{\prime})\,\bar{v}_{\mathrm{s}s2}(p_{2})\gamma_{5}u_{\mathrm{u}s1}(p_{1})\big{\}},$$

$$\displaystyle\big{|}\Psi(P^{+},\vec{P}^{\perp})\big{\rangle}=$$

$$\times {\psi }_{rs}(x,{\vec{\kappa }}^{\perp })b_r^{†}(x{P}^{+},{\vec{\kappa }}^{\perp }+x{\vec{P}}^{\perp })$$

$$\times d_s^{†}((1-x)P^{+},-{\vec{\kappa }}^{\perp }+(1-x){\vec{P}}^{\perp })|0⟩,$$

$${\psi }_{rs}(x,{\vec{\kappa }}^{\perp })$$

$$=\sum _{nml}\psi (n,m,l,r,s){\varphi }_{nm}\left(\frac{{\vec{\kappa }}^{\perp }}{\sqrt{x(1-x)}}\right){\chi }_l(x),$$

$${\varphi }_{nm}\left({\vec{q}}^{\perp };b\right)$$

$$\times L_n^{|m|}\left(\frac{{\vec{q}}^{\perp 2}}{b^2}\right)e^{im\phi },$$

$${\chi }_l(x;\alpha ,\beta )$$

$$=\sqrt{4\pi (2l+\alpha +\beta +1)}\sqrt{\frac{\Gamma (l+1)\Gamma (l+\alpha +\beta +1)}{\Gamma (l+\alpha +1)\Gamma (l+\beta +1)}}$$

$$\times x^{\beta /2}(1-x{)}^{\alpha /2}{P}_l^{(\alpha ,\beta )}(2x-1),$$

$$0\le n\le N_{\max },-2\le m\le 2,0\le l\le L_{\max },$$

$$f(x)$$

$$=\frac{1}{4\pi x(1-x)}\sum _{rs}\int \frac{d{\vec{\kappa }}^{\perp }}{(2\pi {)}^2}{\psi }_{rs}^{\ast }(x,{\vec{\kappa }}^{\perp }){\psi }_{rs}(x,{\vec{\kappa }}^{\perp })$$

$$=\frac{1}{4\pi }\sum _{n,m,l^{\prime },l,r,s}{\psi }^{\ast }(n,m,l^{\prime },r,s)\psi (n,m,l,r,s){\chi }_{l^{\prime }}(x){\chi }_l(x),$$

$${\int }_0^{1}xf(x)dx+{\int }_0^{1}xf(1-x)dx=1,$$

$$f(x)=x^a(1-x{)}^b/B(a+1,b+1),$$

$$⟨x^n⟩={\int }_0^{1}dx{x}^n{f}_v^{\pi /K}(x,{\mu }^2),n=1,2,3,4.$$

$$F_2^{\pi }(\beta ,{\mu }^2)=$$

$$\displaystyle\times\left[C_{q,2}\otimes(f_{q}^{\pi}+f_{\bar{q}}^{\pi})+2C_{g,2}\otimes f_{g}^{\pi}\right]\Big{\}},$$

$$C_{q,2}[z]=\frac{4}{3}{\left[\frac{1+z^2}{1-z}\left(\ln \frac{1-\mathrm{z}}{\mathrm{z}}-\frac{3}{4}\right)\right]}_{+},$$

$$C_{g,2}[z]=\frac{1}{2}\left[\left(z^2+(1-z{)}^2\right)\ln \frac{1-\mathrm{z}}{\mathrm{z}}-1+8z(1-\mathrm{z})\right],$$

$$C\otimes f^{\pi }={\int }_{\beta }^1\frac{dy}{y}C\left(\frac{\beta }{y}\right)f^{\pi }(y,{\mu }^2).$$

$$s=(p_1+p_2{)}^2,q=l+l^{\prime },$$

$$m^2=q^2,Y=\frac{1}{2}\ln \frac{q_0+q_3}{q_0-q_3},$$

$$x_{\mathrm{F}}=x_1-x_2,\tau =\frac{m^2}{s},$$

$$z=\frac{m^2}{\hat{s}}=\frac{\tau }{x_1{x}_2},y=\frac{\frac{x_1}{x_2}{e}^{-2Y}-z}{(1-z)(1+\frac{x_1}{x_2}{e}^{-2Y})},$$

$$x_1=\sqrt{\frac{\tau }{z}\frac{1-(1-y)(1-z)}{1-y(1-z)}}{e}^Y,$$

$$x_2=\sqrt{\frac{\tau }{z}\frac{1-y(1-z)}{1-(1-y)(1-z)}}{e}^{-Y}.$$

$$\frac{m^3{d}^2\sigma }{dmdY}=\frac{8\pi {\alpha }^2}{9}\frac{m^2}{s}\sum _{ij}\int d{x}_{1}d{x}_2$$

$$\times {\tilde{C}}_{ij}(x_1,x_2,s,m,{\mu }^2)f_{i/\pi }(x_1,{\mu }^2)f_{j/N}(x_2,{\mu }^2),$$

$${\tilde{C}}_{ij}(x_1,x_2,s,m^2,{\mu }^2)=\left|\frac{dzdy}{d{x}_{1}d{x}_2}\right|\frac{C_{ij}(z,y,m^2,{\mu }^2)}{[1-y(1-z)][1-(1-y)(1-z)]},$$

$$\frac{C_{q\bar{q}}}{e_q^2}=$$

Full text

I Introduction

Parton distribution functions (PDFs) encode the nonperturbative structure of a hadron by specifying how its longitudinal momentum is distributed to quarks and gluons. The determination of PDFs from the analysis of hard scattering processes is one of the main topics of hadron physics NA10 Collaboration et al. (1987); Freudenreich (1990); Sutton et al. (1992); Glück et al. (1999); Wijesooriya et al. (2005); Conway et al. (1989); NA3 Collaboration et al. (1983); Aicher et al. (2010); Watanabe et al. (2018a); Hecht et al. (2001); Nam (2012); Detmold et al. (2003); Holt and Roberts (2010); Pumplin et al. (2002); Ball et al. (2017); Alekhin et al. (2017); Dulat et al. (2016); Harland-Lang et al. (2015); Bednar et al. (2018). The structure of hadrons including their PDFs is expected to be described by quantum chromodynamics (QCD) in the low energy region where quarks are confined. In addition to color confinement, the explicit and the dynamical breaking of the global chiral dynamics leads to pions having a small mass when compared to other hadrons, taking the role of the pseudoscalar Goldstone bosons. In a chiral perturbation theory, the dynamics of which preserves local chiral symmetry, the pions dress the constituent quarks of an isolated nucleon Thomas (2007); Theberge et al. (1980); Thomas (1984). Meanwhile, the pseudoscalar kaons are the counterparts of the pions with one strange valence quark, the structure of which is crucial to our understanding of CP symmetry violation Woods et al. (1988); NA31 Collaboration et al. (1993); Gibbons et al. (1993). In this article, we are interested in explaining the partonic structure of the pions and the kaons in terms of their PDFs.

One of the available experiments with access to the pion PDFs is the Drell-Yan dilepton production in $\pi^{-}$-tungsten reactions NA10 Collaboration et al. (1987); Freudenreich (1990); Sutton et al. (1992). Several next-to-leading order (NLO) analyses of this Drell-Yan process have been performed by Refs. Sutton et al. (1992); Glück et al. (1999); Wijesooriya et al. (2005). The subsequent determination of the nucleon and the light meson PDFs with associated uncertainties from the experiment is available in Refs. Aicher et al. (2010); Watanabe et al. (2018a); Hecht et al. (2001); Nam (2012); Detmold et al. (2003); Holt and Roberts (2010); Pumplin et al. (2002); Ball et al. (2017); Alekhin et al. (2017); Dulat et al. (2016); Harland-Lang et al. (2015); JAM Collaboration et al. (2018). The pion PDF has also been the subject of detailed analyses in the phenomenological models in Refs. Frederico and Miller (1994); Shigetani et al. (1993, 1993), also including the chiral quark model Broniowski et al. (2008) and anti-de Sitter (AdS)/QCD models Gutsche et al. (2015a, b); Ahmady et al. (2018); HLFHS Collaboration et al. (2018). The pion PDFs have also been investigated within lattice QCD QCDSF-UKQCD Collaboration et al. (2007); Martinelli and Sachrajda (1988); Detmold et al. (2003); Abdel-Rehim et al. (2015); ETM Collaboration et al. (2019); Sufian et al. (2019). See Ref. Lin et al. (2018) for the corresponding review of lattice QCD results. Additionally, the first global fit analysis of PDFs in the pion has been performed in Ref. JAM Collaboration et al. (2018).

Although meson PDFs are expected to be universal, tension exists regarding the behavior of the pion valence PDF. On the one hand, from the analyses of the Drell-Yan data Sutton et al. (1992); Wijesooriya et al. (2005), the large-$x$ behavior of the pion valence PDF is expected to fall off linearly or slightly faster, which is supported by the constituent quark models Frederico and Miller (1994); Shigetani et al. (1993), the Nambu–Jona-Lasinio (NJL) model Shigetani et al. (1993), and duality arguments Melnitchouk (2003). This observation disagrees with perturbative QCD where the behavior of the same function has been predicted to be ${(1-x)^{2}}$ Farrar and Jackson (1979); Berger and Brodsky (1979); Brodsky and Yuan (2006); Yuan (2004), a behavior further supported by the Bethe-Salpeter equation (BSE) approach Hecht et al. (2001); Ding et al. (2019). However, the reanalysis of the data for the Drell-Yan process Aicher et al. (2010) including the next-to-leading logarithmic threshold resummation effects shows a considerably softer valence PDF at high $x$ when compared to the NLO analysis Sutton et al. (1992); Wijesooriya et al. (2005).

Information from experiments on the light-quark PDF of the kaons exists in the form of the ratio of the up (u) quark valence PDF in the kaon to that in the pion NA3 Collaboration et al. (1983); Conway et al. (1989). Theoretically, the kaon’s valence PDF from the BSE approach has been investigated in Ref. Nguyen et al. (2011). A more recent study of the pion and kaon valence PDFs from the BSE with a beyond-rainbow-ladder truncation of QCD shows a good agreement with the experimental valence PDF of the pion Shi et al. (2018). The kaon’s PDF has also been studied in several quark models such as the gauge-invariant nonlocal chiral-quark model Nam (2012), the dressed quark model Chen et al. (2016), and the NJL model Hutauruk et al. (2016); Davidson and Ruiz Arriola (2002). Meanwhile, the quasi-PDFs for the pion and the kaon have been given in Refs. Xu et al. (2018); Broniowski and Arriola (2017); Radyushkin (2017).

Our theoretical calculation of meson structures is based on the basis light front quantization (BLFQ) approach, which has been developed for solving many-body bound state problems in quantum field theories Vary et al. (2010); Wiecki et al. (2015); Li et al. (2016). BLFQ is a Hamiltonian-based formalism incorporating the light front dynamics Brodsky et al. (1998). This formalism has been successfully applied to the quantum electrodynamics (QED) systems including the electron self-energy Zhao et al. (2014) and positronium with strong coupling Wiecki et al. (2015). It has also been applied to heavy quarkonia Li et al. (2017) and $B_{c}$ mesons Tang et al. (2018) both as QCD bound states. Recently, the BLFQ approach using a Hamiltonian that includes the color-singlet NJL interaction to account for the chiral dynamics has been applied to the light mesons Jia and Vary (2019). Furthermore, the BLFQ formalism has been extended to time-dependent strong external field problems such as those in non-linear Compton scattering Zhao et al. (2013a). (For the reviews of BLFQ and its application, see Refs. Leitão et al. (2017); Adhikari et al. (2016); Li et al. (2017); Zhao et al. (2014, 2013b); Chen et al. (2017); Wiecki et al. (2015); Li et al. (2018a, 2016, b); Vary et al. (2010); Zhao et al. (2013a); Chakrabarti et al. (2014); Adhikari et al. (2019).) With light front kinematics, the PDFs can also be calculated using the microcanonical ensemble Jia and Vary (2018).

In this work, we elaborate on Ref. Lan et al. (2019) in the determination of the valence quark PDFs of the pion and the kaon at independent initial scales using the light front wave functions (LFWFs). These wave functions were obtained within the framework of BLFQ by diagonalizing the effective light front Hamiltonian whose interactions include the light front holographic QCD (LFHQCD) confinement potential in the transverse direction Brodsky et al. (2015), a longitudinal confinement potential Li et al. (2016), and the NJL interactions Klimt et al. (1990). These LFWFs have been successfully applied to compute the parton distribution amplitudes and the electromagnetic form factors for the pion and the kaon Jia and Vary (2019). We then evolve our initial valence quark PDFs of the pion and the kaon utilizing the next-to-next-to-leading order (NNLO) Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations Dokshitzer (1977); Gribov and Lipatov (1972); Altarelli and Parisi (1977) to the relevant scales in order to compare with the result of PDFs from the E-0615 experiment at Fermilab, with the pion structure function from the ZEUS and the H1 experiments at DESY-HERA, and with the ratio $u^{K}_{\rm v}(x,\mu^{2})/u^{\pi}_{\rm v}(x,\mu^{2})$ from the NA-003 experiment at CERN. Using our pion PDFs in conjunction with the nuclear PDFs from the nuclear Coordinated Theoretical-Experimental Project on QCD (nCTEQ) 2015 global fit Kovarik et al. (2016), instead of the collection of free nucleon PDFs used in Ref. Lan et al. (2019), we further calculate the cross section for the pion-nucleus induced Drell-Yan process, to show that our PDFs consistently describe the measured cross section data from a variety of experiments.

The paper is organized as follows. Section I is the introduction. The valence PDFs for the pion and the kaon from the BLFQ-NJL model are given in Sec. II. Section III discusses results of these PDFs following DGLAP evolution. Specifically, in Sec. III.1, we present the pion and the kaon PDFs at various scales as well as the implied structure function for the pion. Based on these pion PDFs, the cross section for the unpolarized Drell-Yan process is calculated in Sec. III.2. Section IV is the summary.

II BLFQ-NJL model for the light mesons

II.1 The light front confinement and NJL interactions for the light mesons

Let us start with an overview of the BLFQ-NJL model for the light mesons following Ref. Jia and Vary (2019). In the approach of BLFQ, the structures of the bound states are embedded in the LFWFs obtainable as the solutions of the time-independent light front Schrödinger equation

[TABLE]

where $H_{\mathrm{eff}}$ is the effective Hamiltonian of the system with the mass squared $M^{2}$ being the eigenvalue of $|\Psi\rangle$. In general, $|\Psi\rangle$ is the vector in the Hilbert space spanning into all Fock sectors. In the valence Fock sector, the effective Hamiltonian for the light mesons with non-singlet flavor wave functions is given by Jia and Vary (2019)

[TABLE]

where $m_{q}$ ($m_{\bar{q}}$) is the mass of the quark (antiquark), and $\kappa$ is the strength of the confinement. $\vec{\zeta}_{\perp}\equiv\sqrt{x(1-x)}\vec{r}_{\perp}$ is the holographic variable Brodsky et al. (2015), with $\vec{k}_{\perp}$ being the conjugate variable of $\vec{r}_{\perp}$. The $x$-derivative is defined as $\partial_{x}f(x,\vec{\zeta}_{\perp})=\partial f(x,\vec{\zeta}_{\perp})/\partial x|_{\vec{\zeta}}$. The first two terms in Eq. (2) are the light front kinetic energy for the quark and the antiquark. The third and the fourth terms are the confining potential in the transverse direction based on the LFHQCD Brodsky et al. (2015) and a longitudinal confining potential Li et al. (2016) that reproduces 3D confinement in the nonrelativistic limit. Additionally, the $H_{\mathrm{NJL}}^{\mathrm{eff}}$ is the color-singlet NJL interaction to account for the chiral dynamics Klimt et al. (1990).

The NJL interaction for the positively-charged pion is given by Jia and Vary (2019),

[TABLE]

which can be derived from the NJL Lagrangian after the Legendre transform in the two-flavor NJL model Klimt et al. (1990); Vogl et al. (1990); Vogl and Weise (1991); Klevansky (1992). Here, only the combinations of Dirac bilinears relevant to the valence Fock sector LFWFs of the $\pi^{+}$ in valence Fock sector are included. For the positively charged kaon, the interaction is given by

[TABLE]

obtained similarly from the Lagrangian of the three-flavor NJL model. Here ${u_{\mathrm{f}s}(p)}$ and ${v_{\mathrm{f}s}(p)}$ are solutions of the free Dirac equation, with the nonitalic subscripts representing the flavors while the italic subscripts designate the spins. Meanwhile, $p_{1}$ and $p_{2}$ are the momenta of the valence quark and the valence antiquark, respectively Jia and Vary (2019). The coefficients $G_{\pi}$ and $G_{K}$ are independent coupling constants of the theory. We have ignored the instantaneous terms due to the NJL interactions in deriving Eqs. (3) and (4). Explicit expressions and the detailed calculations of the matrix elements of the NJL interactions in the basis function representation we adopt can be found in Ref. Jia and Vary (2019).

Parameters in the BLFQ-NJL model are adjusted to reproduce the ground state masses of the pseudoscalar and vector mesons with light-light and light-strange nonsinglet flavor components. Meanwhile, the confining strengths are determined by the experimental charge radii of the $\pi^{+}$ and the $K^{+}$ Jia and Vary (2019).

II.2 Valence quark PDFs in the pion and the kaon from BLFQ

The LFWFs of the valence quarks in the $\pi^{+}$ meson and the $K^{+}$ meson have been solved in the BLFQ framework using the NJL interactions discussed in the previous subsection Jia and Vary (2019). In the leading Fock sector, the LFWF for the mesons is written as

[TABLE]

where $P=k+p$ is the light front $3$-momentum of the meson, $x=k^{+}/P^{+}$ is the longitudinal momentum fraction carried by the valence quark, and $\vec{\kappa}^{\perp}=\vec{k}^{\perp}-x\vec{P}^{\perp}$ is the relative transverse momentum. The valence wave function is then expanded in the following orthonormal basis:

[TABLE]

where $\phi_{nm}$ is the two-dimensional (2D) harmonic oscillator (HO) function, and $\chi_{l}$ is the longitudinal basis function. Here $n,\,m,$ and $l$ are basis quantum numbers corresponding to the radial excitation, the orbital angular momentum projection, and the longitudinal excitation, respectively. Explicitly, $\phi_{nm}$ is given by

[TABLE]

with $\tan(\varphi)=q^{2}/q^{1}$ and $L_{n}^{|m|}$ being the associated Laguerre function. Meanwhile, the longitudinal basis $\chi_{l}(x)$ is defined as

[TABLE]

with $P_{l}^{(\alpha,\beta)}(z)$ being the Jacobi polynomial and ${\alpha=2m_{\bar{q}}(m_{q}+m_{\bar{q}})/\kappa^{2}}$, ${\beta=2m_{q}(m_{q}+m_{\bar{q}})/\kappa^{2}}$. Here $m_{q}$ and $m_{\bar{q}}$ are the masses of the valence quark and the valence antiquark, respectively.

In order to numerically diagonalize $H_{\text{eff}}$, the infinite dimensional basis must be truncated. Because the NJL interactions do not couple to $|m|\geq 3$ basis states, we have a natural truncation for $m$ Jia and Vary (2019). Specifically, we apply the following truncation to restrict the quantum numbers Li et al. (2016, 2017); Jia and Vary (2019):

[TABLE]

where $L_{\text{max}}$ determines the basis resolution in the longitudinal direction whereas $N_{\text{max}}$ controls the transverse momentum covered by 2D HO functions. Notice that our definition of $N_{\text{max}}$ in Eq. (9) is different from that in Refs. Li et al. (2017); Tang et al. (2018).

The probability of finding a quark inside the meson carrying the momentum fraction $x$ is then given by Li et al. (2017)

[TABLE]

which is interpreted as the PDF for the valence quark. Correspondingly, the PDF for the valence antiquark is given by ${f(1-x)}$. In obtaining Eq. (10b) from Eq. (10a), the transverse integrals are evaluated exactly using the orthonormal property of the 2D HO functions. Equation (10) implies the following momentum sum rule:

[TABLE]

which indicates that, at our model scale, the valence quarks carry the entire momentum of the meson. Our normalization of the LFWF ensures that the normalizations of the PDFs for both valence quarks are $1$.

We then substitute the valence wave functions given by Eq. (6) obtained from Ref. Jia and Vary (2019) into Eq. (10b) to calculate the valence PDFs for the $\pi^{+}$ and the $K^{+}$. We show in the upper panels of Fig. 1 that with a fixed $L_{\mathrm{max}}$ the numerical PDFs oscillate about a single-peaked function, with the amplitude of the oscillation decreasing with increasing $L_{\mathrm{max}}$. Because the physical PDFs do not depend on the longitudinal cutoff, these oscillations are numerical artifacts. To remove such artifacts, we fit the resulting PDFs using the function

[TABLE]

for each $L_{\mathrm{max}}\in\{8,\,12,\,16,\,20,\,24,\,28,\,32\}$. Here ${B(a+1,b+1)}$ is the Euler Beta function that ensures the normalization of Eq. (12). Subsequently, we fit the $L_{\mathrm{max}}$ dependence of these fitting parameters by quadratic functions on $L^{-1}_{\mathrm{max}}$ and extrapolate to $L_{\mathrm{max}}\rightarrow+\infty$. The resulting fitting parameters and their extrapolations are given in Table 1 and the input PDFs of the pion and kaon corresponding to the extrapolations of the fitting parameters are shown in Fig. 1.

III PDFs, structure function, and cross sections

III.1 PDFs and structure function

By performing the QCD evolution, the valence-quark PDFs at high $\mu^{2}$ scale can be determined with the initial input using Eq. (12) with parameters extrapolated to the infinite longitudinal basis cutoff as given in the last column of Table 1. Specifically, we evolve our input PDFs to the relevant experimental scales ${\mu^{2}=16~{}\mathrm{GeV}^{2}}$ and ${\mu^{2}=20~{}\mathrm{GeV}^{2}}$ with independently adjustable initial scales of the pion and the kaon PDFs using the DGLAP equations Dokshitzer (1977); Gribov and Lipatov (1972); Altarelli and Parisi (1977). Here, we use the higher order perturbative parton evolution toolkit (HOPPET) to numerically solve the DGLAP equations Salam and Rojo (2009). We find that the initial scales increase when we progress from the leading order (LO) to NNLO. Meanwhile the evolved PDFs fit better to the experimental result demonstrated by smaller values of $\chi^{2}$ per degree of freedom (d.o.f.) at higher orders, as shown in Table 2. Since the results from the higher order DGLAP equation appear more reliable due to higher initial scales, only the results for the PDFs at NNLO are presented in this paper.

We adopt ${\mu_{0\pi}^{2}=0.240\pm 0.024~{}\rm{GeV}^{2}}$ for the initial scale of the pion PDF and ${\mu_{0K}^{2}=0.246\pm 0.024~{}\rm{GeV}^{2}}$ for the initial scale of the kaon PDF which we determine by requiring the results after NNLO DGLAP evolution to fit both the pion PDF results from the FNAL-E-0615 experiment Conway et al. (1989) and the ratio $u^{K}_{\rm v}/u^{\pi}_{\rm v}$ result from the CERN-NA-003 experiment NA3 Collaboration et al. (1983). At our central value of the initial scales, the $\chi^{2}$ per d.o.f. for the fit of the pion PDF is 3.64, whereas for the ratio $u^{K}_{\rm v}/u^{\pi}_{\rm v}$, the corresponding value is 0.50. The initial scales are the only adjustable parameters in this work and we assign them both a $10\%$ uncertainty. We interpret the initial scales associated to our model as effective scales where the structures of the mesons are described by the motion of the valence quarks only. While applying the DGLAP equations, we impose the condition that the running coupling $\alpha_{s}(\mu^{2})$ saturates in the infrared at a cutoff value of max $\{\alpha_{s}\}=1$. Note that the sea quark and the gluon distributions are absent in the initial scales of our model. The scale evolution allows quarks to emit and absorb gluons, with the emitted gluons allowed to create sea quarks as well as additional gluons.

In Fig. 2, we show our result for the valence-quark PDF of the pion. We compare the valence-quark distribution after QCD evolution with the result from the FNAL-E-0615 Conway et al. (1989) and with the reanalysis of the same result including soft gluon resummation Chen et al. (2016). The error band in the valence-quark distributions is due to the spread in the initial scale $\mu_{0}^{2}=0.240\pm 0.024$ GeV2 propagated by the QCD evolution. Our result favors the slower falloff in the large-$x$ region in the original analysis of the FNAL-E-0615 experiment. While in the intermediate region of $x$, our result is in agreement with the reanalysis of the FNAL-E-0615 result. The pion valence PDF from our model falls off at large $x$ as $(1-x)^{1.44}$, so there is a tension with the results obtained from the BSE Chen et al. (2016) and with the analysis in Ref. Chen et al. (2016) that incorporated the $(1-x)^{2}$ perturbative QCD falloff at large $x$ from the threshold resummation effects. We note, however, that there has been a recent fit to the FNAL-E-0615 result in LFHQCD which supports a linear falloff at high-$x$ HLFHS Collaboration et al. (2018).

Another comparison can be made for the pion PDFs at the initial scale of Ref. HLFHS Collaboration et al. (2018) at $\mu_{0}^{2}=(1.12\pm 0.32)~{}\mathrm{GeV}^{2}$. We find that at this scale the valence quarks carry 57% of the pion’s momentum from our model, close to the 54% given by Ref. HLFHS Collaboration et al. (2018). At the same scale, in contrast to the absence of gluon contributions in Ref. HLFHS Collaboration et al. (2018), our model allocates 35% of the pion’s momentum to the gluons and 8% to the sea quarks.

In Fig. 3 (a), we present the ratio of the u quark distributions in the kaon to that in the pion, with the valence-quark PDFs in the kaon shown in Fig. 3 (b). We observe that at $\mu^{2}=20~{}\mathrm{GeV}^{2}$, our center value of $u_{\rm{v}}^{K}/u_{\rm{v}}^{\pi}$ is in good agreement with the result from CERN-NA-003 experiment NA3 Collaboration et al. (1983) as well as with a phenomenological quark model (GRS, NLO) Glück et al. (1998) and the BSE approach Nguyen et al. (2011). One notices that the ratio decreases as $x$ increases. This phenomena is rooted in the results shown in Fig. 3 (b) where we compare the valence quark distributions of the kaon and the pion. We find that at the scale of $\mu^{2}=20~{}\mathrm{GeV}^{2}$ the distribution of the u quark PDF at high $x$ in the pion is above that in the kaon. This can be understood since $m_{\mathrm{s}}>m_{\mathrm{u}}$ the peak of the $\overline{\mathrm{s}}$ quark distribution in the kaon appears at higher $x$ compared to the u quark distribution. Therefore the $\overline{\mathrm{s}}$ quark carries larger momentum than the u quark does, reducing the probability of finding a u quark with high $x$ in the kaon. Specifically, the u quark PDF in the kaon falls off at large $x$ as $(1-x)^{1.60}$ whereas the same behavior in the pion is $(1-x)^{1.49}$. We also observe that in our model, the $\overline{\mathrm{s}}$ quark PDF in the kaon falls off as $(1-x)^{1.32}$.

We further evaluate the lowest four nontrivial moments of the valence quark PDF defined as

[TABLE]

The corresponding moments of the pion PDF at different scales are shown in Fig. 4 (a), together with the results from the global fit JAM Collaboration et al. (2018), lattice QCD QCDSF-UKQCD Collaboration et al. (2007); Martinelli and Sachrajda (1988); Detmold et al. (2003); Abdel-Rehim et al. (2015); ETM Collaboration et al. (2019), and several phenomenological models Nam (2012); Sutton et al. (1992); Wijesooriya et al. (2005). Our predictions are in good agreement with Refs. Martinelli and Sachrajda (1988); Detmold et al. (2003); Sutton et al. (1992); Wijesooriya et al. (2005); Nam (2012); JAM Collaboration et al. (2018). The numerical values of the lowest four moments of the pion PDF at various scales are presented in Table 3.

The scale dependence of the lowest four moments of the pion valence quark PDF is presented in Fig. 4 (b). These moments decrease uniformly as the scale $\mu^{2}$ increases, compensated by the increase in contributions from sea quarks and gluons. We find good agreement of our calculated $\langle x\rangle$ with the Jefferson Lab Angular Momentum Collaboration (JAM) global fit JAM Collaboration et al. (2018) over nearly 3 decades of the $\mu^{2}$ scale within our uncertainty and close to the central value. Additionally, the numerical values of the lowest four moments of the valence quark PDFs in the kaon at various scales are presented in Table 4.

With the pion PDFs known over a wide range of scales, we proceed to calculate the pion structure function $F^{\pi}_{2}(x_{\pi}=\beta,\mu^{2})$ using the parton model. Specifically in the NLO in perturbative QCD, the structure function can be expressed in terms of the PDFs as Gluck et al. (1992, 1995)

[TABLE]

with

[TABLE]

and

[TABLE]

Here $q$ is the flavor index and $e_{q}$ is the electric charge of the quark flavor $q$ in the units of the elementary charge while $g$ stands for the gluon. Here we have included heavy flavor contributions relevant to the scale of the structure functions. Our results for the structure function $F_{2}^{\pi}(x_{\pi}=\beta,\mu^{2})$ are shown in Fig. 5 and Fig. 6 in comparison with DESY-HERA-ZEUS ZEUS Collaboration et al. (2002) and DESY-HERA-H1 data Aaron et al. (2010) at the respective experimental scales. Both the ZEUS and the H1 data were determined from the neutron production in $ep$ collisions, $ep\rightarrow e^{\prime}Xn$ process. The $\beta$ on the horizontal axes in these figures is the parton momentum fraction relative to the pion which is defined as $\beta=x_{p}/(1-x_{\rm{L}})$, where $x_{p}$ is the parton momentum fraction relative to the proton. The momentum fraction carried by the neutron relative to the proton is $x_{\rm{L}}=0.73$ ZEUS Collaboration et al. (2002); Aaron et al. (2010). The two different sets of ZEUS data in Fig. 5 correspond to different pion fluxes used to determine $F_{2}^{\pi}$. One of them was obtained using the additive quark model (AQM) whereas the other is obtained using the effective one-pion-exchange flux (EF) in hadron-hadron charge-exchange reactions. The difference between these two results are attributed to the model dependence of the experimental analysis. Despite expecting that both AQM and EF are only valid when $x_{\rm{L}}\rightarrow 1$, our result appears to favor the AQM for $\mu^{2}\leq 240~{}\mathrm{GeV}^{2}$.

We notice from Fig. 5 and Fig. 6 that our results deviate from the data at very low $x$. We expect that at low initial scale the DGLAP evolution with leading twist is not sufficient at low $x$ Zhu et al. (2013); Boroun (2010a), and one needs to take into account of the higher twist corrections Boroun and Zarrin (2013); Boroun (2010b); Devee and Sarma (2014); Phukan et al. (2017); Rezaei and Boroun (2010); Boroun (2009); Lalung et al. (2017). On the other hand, our $F_{2}^{\pi}(x,\mu^{2})$ shows better agreement with data as $x$ increases. The component contributions from the valence quarks, sea quarks, and gluons to the total structure function $F_{2}^{\pi}$ of the pion at $55$ GeV2 are shown in Fig. 7. We observe that at low $x$ the sea quark contribution dominates. However, at large $x$ the distribution is mostly accounted for by the valence quarks.

III.2 Cross section of the unpolarized Drell-Yan process

In this section we present the cross section of the Drell-Yan process using our BLFQ-NJL model for the pion PDF. The momenta of the incoming hadrons are denoted by $p_{1,2}$. We define $l$ and $l^{\prime}$ as the momenta of the two outgoing leptons. The kinematics of the process are described by the invariant mass of the lepton pair $m$, center of mass energy square $s$, rapidity $Y$ or the Feynman variable $x_{\mathrm{F}}$, and the variable $\tau$, $z$ and $y$. These variables are defined and related to each other by Becher et al. (2008)

[TABLE]

where, $\hat{s}=x_{1}x_{2}s$. In the parton model, the $x_{i}$ denotes the fraction of the hadron momentum $p_{i}$ carried by the annihilating parton (or antiparton) and is given by

[TABLE]

Explicitly in the NLO in perturbative QCD, the cross section in terms of the PDFs is given by Becher et al. (2008); Anastasiou et al. (2003, 2004); JAM Collaboration et al. (2018)

[TABLE]

where $\widetilde{C}_{ij}$ are the hard-scattering kernels, which can be expanded in powers of the strong coupling $\alpha_{s}$. The sums extends over all possible partonic channels contributing at a given order in the expansion of $\widetilde{C}_{ij}$. At leading order, only the channels $(ij)=(q\bar{q})$ and $(\bar{q}q)$ contribute, whereas at NLO ($\sim\alpha_{s}$), we must include $(ij)=(\bar{q}q),(q\bar{q}),(gq),(qg),(\bar{q}g),(g\bar{q})$ in the sum. The expressions of the hard-scattering kernels at NLO are given in the appendix. In order to evaluate Eq. (17), we adopt the nuclear PDFs from the nCTEQ 2015 Kovarik et al. (2016) at the experimental scale $\mu^{2}=16$ GeV2 in conjunction with our pion PDFs at the same scale. While the PDFs for the tungsten and the beryllirum nuclei are readily available in Ref. Kovarik et al. (2016), we approximate the bound nucleon PDFs in the platinum nucleus by the corresponding bound nucleon PDFs in the gold nucleus.

After integrating out the $Y$ dependence of the differential cross section ${m^{3}d^{2}{\bf\sigma}/dm\,dY}$, we obtain our results plotted as a function of either $\sqrt{\tau}$ in Fig. 8 or $\tau$ in Fig. 9 to compare with the experimental data. The FNAL-E-0615 and the CERN-NA-003 data in Fig. 8(a) correspond to a tungsten and a platinum targets, respectively, whereas the data in Fig. 8(b) correspond to a tungsten target. In Fig. 8(a), we show the results evaluated using the tungsten and the platinum nuclear PDFs. We employ the tungsten nuclear PDF to compute the cross shown section in Fig. 8(b). We find that the cross sections per nucleon obtained by considering the tungsten and the platinum nuclear PDFs are very close. In Fig. 9, the CERN-WA-011 data represent a beryllium target and the same target nuclear PDF has been used by our approach to evaluate the cross section. In Fig. 10 we show the cross section $d{\bf\sigma}/dm$ as a function of $m$ and compare with the data of the FNAL-E-0326 experiment Greenlee et al. (1985) and the FNAL-E-0444 experiment Anderson et al. (1979) with 225 GeV pions, as well as with the data of CERN-WA-039 experiment with 39.5 GeV pions Corden et al. (1980). Notice the FNAL-E-0444 data correspond to a carbon target whereas the FNAL-E-0326, and the CERN-WA-039 data represent a tungsten target. We use the corresponding target nuclear PDFs to calculate the cross section displayed in Fig. 10. Based on Figs. 8-10, we find that our results are in acceptable agreement with data from widely different experimental conditions Conway et al. (1989); NA3 Collaboration et al. (1983); NA10 Collaboration et al. (1985); Barate et al. (1979); Corden et al. (1980); Greenlee et al. (1985); Anderson et al. (1979); Stirling and Whalley (1993).

IV Summary and conclusion

We calculated the valence-quark PDFs of the pion and the kaon in the framework of the basis light front quantization from their light front wave functions. These wave functions were obtained as the eigenfunctions of the effective Hamiltonian, consisting of confinement potentials and the color-singlet Nambu–Jona-Lasinio interactions. The parameters in the BLFQ-NJL model were adjusted to reproduce the experimental mass spectrum and the charge radii of the light mesons Jia and Vary (2019). The initial scales of our PDFs, the only adjustable parameters in this work, have been obtained by consistently fitting both the evolved valence pion PDFs to the FNAL-E-0615 experiment Conway et al. (1989) and the evolved ratio of the up quark PDFs in the kaon to that in the pion to the CERN-NA-003 experimental result NA3 Collaboration et al. (1983). The moments of the pion PDF have been found in agreement with the JAM global fit JAM Collaboration et al. (2018), with lattice QCD Martinelli and Sachrajda (1988); Detmold et al. (2003), as well as with phenomenological quark models Sutton et al. (1992); Wijesooriya et al. (2005); Nam (2012); Bednar et al. (2018); Hecht et al. (2001); Chen et al. (2016); Shi et al. (2018); Ding et al. (2019); Nam (2012); Wijesooriya et al. (2005) across various scales.

We have subsequently calculated the structure function $F_{2}^{\pi}(x,Q^{2})$ for the pion, the large $x$ behavior of which is consistent with the DESY-HERA experiment ZEUS Collaboration et al. (2002); Aaron et al. (2010). However, the discrepancies at small $x$ for the structure function suggest the need to include the higher-twist corrections and a non-vanishing initial gluon distribution to the DGLAP evolution. We have also studied the cross sections of the pion-nucleus induced Drell-Yan process in comparison with Refs. Conway et al. (1989); NA3 Collaboration et al. (1983); Stirling and Whalley (1993); Barate et al. (1979); Greenlee et al. (1985); Anderson et al. (1979); Corden et al. (1980), finding reasonable agreement with these various experimental data. These comparisons affirm the robustness of the BLFQ-NJL model with QCD evolution as a theoretical method to describe the structures of the pion and the kaon in the language of parton distribution functions.

Acknowledgements.

We thank Y. Li, S. Xu, J. Huston, W. Zhu, J. Gao, X. Chen, T. Liu, and R. Wang for many insightful discussions. C. M. is supported by the National Natural Science Foundation of China (NSFC) under the grants No. 11850410436 and No. 11950410753. X. Z. is supported by Key Research Program of Frontier Sciences, CAS, Grant No. ZDBS-LY-7020. S. J. and J. P. V. are supported by the Department of Energy under grants No. DE-FG02-87ER40371 and No. DE-SC0018223 (SciDAC4/NUCLEI).

Appendix A

The expressions of $\widetilde{C}_{ij}(x_{1},x_{2},s,m^{2},\mu^{2})$ are given by Anastasiou et al. (2003)

[TABLE]

[TABLE]

[TABLE]

respectively, where the plus prescription is defined as

[TABLE]