Quantum chromodynamics (QCD) [1–3] describes the production of hadrons by incorporating strong interactions among quarks, antiquarks, and gluons. One of the most significant problems and an avenue for improving our comprehension of QCD and confinement is to decipher the multidimensional structure of a hadron, thus providing insight into several nonperturbative aspects of QCD. Hadrons possess partons inside them, and the respective partonic states can be described by six-dimensional phase-space distributions called Wigner distributions [4,5]. Wigner distributions are the quantum-mechanical constructions that are closest to a classical probability density in phase space. The probability density of discovering a parton (gluon or quark) carrying the parent hadron’s light-front (LF) longitudinal momentum fraction x is described by the parton distribution function (PDF) [4–11]. For the description of observables that are also sensitive to the transverse kinematics of a parton, the concept of PDFs has been extended to transverse-momentum-dependent parton distributions (TMDs) [12–19] and generalized parton distributions (GPDs) [20–27] to include the information of transverse momentum and transverse coordinate distributions, respectively. After a few phase-space reductions, Wigner distributions reduce to TMDs and GPDs. Both TMDs and GPDs present a three-dimensional visualization of the hadron. The Wigner distributions integrated over the transverse momenta reduce to the GPDs at zero skewness (ζ=0) [28]. On the other hand, integrating it over the transverse impact parameter, with zero momentum transfer, they reduce to the TMDs. TMDs can be measured in certain reactions like semi-inclusive deep inelastic scattering [29,30], Drell-Yan processes [16,31–34], and Z0/W± production [35–37]. GPDs are extracted from the QCD description of hard exclusive reactions like deeply virtual Compton scattering [21,38–41] and deeply virtual meson production [42,43]. Analyzing the GPDs can reveal details about the quarks’ spatial distributions. Further, Wigner distributions are Fourier transforms of the generalized transverse-momentum-dependent distributions (GTMDs) [44,45] from the transverse momentum transfer Δ⊥ to the impact parameter b⊥, without integration over the LF energy. GTMDs are functions of the LF three momentum of the parton as well as the momentum transfer to the hadron [28].

The spin-orbit correlations (SOCs) between a hadron and a quark can be explained on the basis of the phase-space average of Wigner distributions. Comprehending spin composition of hadrons has been a fascinating subject of research lately [46–49], and understanding the multidimensional structure makes it possible to analyze characteristics such as SOCs, spin-spin correlations, quark-gluon correlations, and other such interactions. It would be interesting to explore the connection of the GPDs and GTMDs of spin-0 and spin-1 mesons with their spin-orbital angular momentum correlations and proceed to calculate their analytical results. In particular, the correlations between the hadron spin and the orbital motion of partons inside the hadron can bring much bigger insight into the spin structure of the hadrons [50]. Given that a parton’s orbital angular momentum (OAM) and spin contributions have intrinsic negative parity, the only nonvanishing single-parton (a=q, G) correlations allowed by parity invariance are Sa·SN, La·SN, and Sa·La. Here, Lq,G represents the quark (or gluon) OAM, SN is the spin of the hadron, and Sq,G portrays the spin of the constituent quark (or gluon) [50]. The initial two types of correlations are commonly referred to as OAM and spin contributions of parton a to the spin of the hadron, whereas the third and final type is the parton’s SOC. We use the difference between the right-handed and left-handed contributions of the quark longitudinal OAM to describe the quark longitudinal SOC, which is expressed by Czq.

Dirac, in 1949 [51], recognized that one may set up a dynamical theory in which the dynamical variables refer to the physical conditions on a front x+=0. The resulting dynamics is called the LF dynamics, which Dirac referred to as “front-form” for brevity. The LF dynamics [30,52–54] is a beneficial model framework, which helps us study the internal structure of hadrons [30,51] and has direct applications in the Minkowski space [55]. LF quantization provides a framework to describe the perturbative and nonperturbative regimes of QCD. LF dynamics can be realized by a number of different models, and for this work we have adopted the LF quark model (LFQM) [56–58]. The LFQM is based on the algebra of generators of the Lorentz group in the LF dynamics [59]. The component quark and antiquark in a bound state must be on-mass shell in conventional LFQM [2,60–62]. The spin-orbit wave function is derived from the conventional time-independent spin-orbit wave function supplied by the quantum numbers JPC [63] using the well-known Melosh transformation, which is independent of interactions [64]. The LFQM is primarily concerned with the valence quarks of hadrons, which are among the primary elements responsible for the overall composition and properties of hadrons. With accurate parameter choices, the model describes several hadron characteristics, including form factors (FFs), distribution amplitudes, decay constants, etc., thus establishing a phenomenological link between hadron properties and the wave function of the quark constituents, which has been successful in many instances.

Quark SOCs have been studied earlier for the spin-12 hadrons [65] and also investigated for the case of spin-0 hadrons like pions and kaons [50,66]. However, no work has been reported for the remaining members of the spin-0 mesons. Further, the spin-1 mesons (light and heavy) have also remained unexplored in this regard. We may obtain special insights into the orbital motion of quarks and their intrinsic longitudinal spin inside spin-0 and spin-1 mesons, owing to the quark SOC [50]. In light of the successes of the LFQM and the importance of the quark SOCs, it becomes essential to extend this work across all members in both the spin-0 and spin-1 meson spectrum. To make the application of this work broader, we have included the light as well as heavy mesons for both the spin-0 and spin-1 cases. We have utilized the conventional definition of the leading-twist GTMD G1,1 in our calculations and have solved the correlators for the leading-twist GPD case. To derive the outcome for the SOC, we have integrated the GTMD G1,1 twice in terms of transverse momentum k⊥ and fraction of momentum transfer to the active quark x involving the entire wave function of the LFQM. We have visualized the behavior of the spin-orbit correlator Czq in our chosen model via two-dimensional plots with respect to the longitudinal momentum fraction x and transverse momenta of quark k⊥. We have also presented the model-dependent results for the pion and kaon. Further, the physical implications of the SOC for both the spin-0 and spin-1 light and heavy mesons have been discussed.

This paper is arranged as follows. In Sec. II, we have presented the correlation between the quark spin and OAM inside the hadron. In Sec. III, we have quantitatively discussed LFQM: the model employed to define our LF wave functions (LFWFs). In Sec. IV, we have shown how the spin-OAM correlations can be methodically derived from the GPDs and the GTMDs. These relations have been presented for both the light and heavy spin-0 and spin-1 mesons. Further, in Sec. V, we have defined our model parameters and presented our model results for the SOC. Finally, we have summarized our results in Sec. VI.

The local gauge-invariant LF operators for the quark longitudinal spin and OAM have been of unique interest since they enter Ji’s decomposition of the total angular momentum operator in QCD [46,50], which is given as J^z=S^zq+L^zq+J^zG.(1)Here, L^zq refers to the gauge-invariant LF quark longitudinal OAM, which can be further decomposed into right-handed and left-handed quark contributions as L^zq=∫d3x12ψ¯γ+(x×iD↔)zψ=L^zqR+L^zqL,(2)where the symmetric covariant derivative is defined by D↔=∂←-∂→-2igA [50]. ψR,L=12(I±γ5)ψ and d3x=dx-d2x⊥. The knowledge and understanding of quark SOCs give us a complete characterization of the hadron’s internal structure. The local gauge-invariant correlation is described by Czq^=∫d3x12ψ¯γ+γ5(x×iD↔)zψ=L^zqR-L^zqL.(3)The quark OAM operator may be represented in terms of the gauge-invariant energy-momentum tensor as follows [65]: L^zq=∫d3x(x1Tq+2^-x2Tq+1^),(4)where Tμν^ is the energy-momentum tensor operator given by Tqμν^=12ψ¯γμiD↔νψ(5)=TqRμν^-TqLμν^Fq,(6)and TR,Lμν^=12ψ¯R,LγμiD↔νψR,L. Further, the quark SOC operator is given as follows: Czq^=∫d3x(x1Tq5+2^-x2Tq5+1^),(7)where Tq5μν^ may be regarded as the parity-odd partner of Tμν^ and can be expressed as [65] Tq5μν^=12ψ¯γμγ5iD↔νψ(8)=TqRμν^+TqLμν^.(9)The nonforward matrix components of Tq5μν^ inserted between two meson states may be parametrized as a sum of two form factors C˜q(t) and F˜q(t) [67–70], which can be expressed as ⟨k′|Tq5μν^(0)|k⟩=-P[μiεν]+ΔP2P+(C˜q(t)-2F˜q(t))+iεμνΔPF˜q(t)+O(Δ2).(10)Substituting Eq. (10) into the matrix elements of Eq. (7), within the symmetric LF frame (P⊥=0⊥), we get Czq=⟨p|C^zq|p⟩⟨p|p⟩=C˜q(0).(11)Thus, we just need to compute the form factor C˜q(t) in order to determine the quark SOCs for the mesons.

In the LF technique, a sequence of LFWFs in the Fock-state basis are used to define the wave functions of the meson describing a composite state at a certain LF time [71]. The meson eigenstate |M(P+,P⊥,Sz)⟩ can be expressed in terms of its component eigenstate |n⟩ using the LF Fock-state expansion and can be expressed as [28,66,72] |M(P+,P⊥,Sz)⟩=∑n,λi∫∏i=1ndxid2k⊥ixi16π316π3δ(1-∑i=1nxi)δ(2)(∑i=1nk⊥i)|n;xiP+,xiP⊥+k⊥i,λi⟩Ψλ1,λ2Λ(xi,k⊥i).(12)Here, we denote P=(P+,P-,P⊥) as the meson’s total momentum and Sz as the longitudinal spin projection of the target. The LF momentum coordinates and relative momentum fractions of the mesonic components are denoted by k⊥i and xi=ki+/P+, respectively. The quark’s transverse and longitudinal momentum fractions are represented by the symbols k⊥ and x, respectively. In contrast, 1-x and -k⊥, respectively, describe the same for the antiquark spectator. λi is the helicity, and mi is the mass of the ith constituent correspondingly. In Eq. (12), xiP⊥+k⊥i=p⊥i is the physical transverse momentum. Ψλ1,λ2Λ(xi,k⊥i) is the LFWF with different spin and helicity projections, and λ1(2) describes the helicity of the quark (antiquark) in the meson. Also, Λ=T refers to the transverse spin projections of the mesons. To simplify our calculations, we have considered the minimal Fock-state description of the meson in the form of a quark-antiquark pair and is expressed as [73] |M(P,S)⟩=∑λ1,λ2∫dxd2k⊥16π3x(1-x)|x,k⊥,λ1,λ2⟩Ψλ1,λ2Λ(x,k⊥).(13)The momenta of the meson (P), constituent quark (k1), and antiquark (k2) in the LFQM are given as P=(P+,M2P+,0⊥),(14)k1=(xP+,k⊥2+mq2xP+,k⊥),(15)k2=((1-x)P+,k⊥2+mq¯2(1-x)P+,-k⊥).(16)Here, mq(mq¯) refers to the boost invariant mass of the quark (antiquark), and M refers to the mass of the meson, which is given by M2=k⊥2+mq2x+k⊥2+mq¯21-x.(17)The LF meson wave function is expressed as [74,75] Ψλ1,λ2Λ(x,k⊥)=ϕ(x,k⊥)Xλ1,λ2Λ(x,k⊥).(18)Here, Xλ1,λ2Λ(x,k⊥) represents the spin wave function, and ϕ(x,k⊥) is the momentum-space wave function of the meson, respectively. Let us begin our discussions with spin-0 meson. The spin wave functions are derived through the Melosh-Wigner rotation [76–78], and for spin-0 mesons, they are expressed as [79] XSP0(x,k⊥)=∑λ1,λ2K0(x,k⊥,λ1,λ2)X1λ1X2λ2,(19)where SP0 stands for the spin-0 meson. K0(x,k⊥,λ1,λ2) is the coefficient of the spin wave function. The different helicity combinations are described as K0(x,k⊥,↑,↓)=[(xM+mq)((1-x)M+mq¯)-k⊥2]/2w1w2,(20)K0(x,k⊥,↓,↑)=-[(xM+mq)((1-x)M+mq¯)-k⊥2]/2w1w2,(21)K0(x,k⊥,↑,↑)=[(xM+mq)k2L-((1-x)M+mq¯)k1L]/2w1w2,(22)K0(x,k⊥,↓,↓)=[(xM+mq)k2R-((1-x)M+mq¯)k1R]/2w1w2.(23)Here, the subscripts ↑ and ↓ denote the transverse polarizations of the quark along the directions e^x and -e^x, respectively, and w1=[(xM+mq)2+k⊥2]12,(24)w2=[((1-x)M+mq¯)2+k⊥2]12.(25)

For the spin-1 mesons, the spin wave functions occurring in Eq. (18) take the form for Λ=T(+) as [80] X+,+T(+)(x,k⊥)=mq(M+2m)+k⊥2(M+mq)mq2+k⊥2,(26)X-,+T(+)(x,k⊥)=-kR((1-x)M+mq)(M+mq)mq2+k⊥2,(27)X+,-T(+)(x,k⊥)=kR(xM+mq)(M+mq)mq2+k⊥2,(28)X-,-T(+)(x,k⊥)=-kR2(M+mq)mq2+k⊥2.(29)Similarly, the spin wave functions occurring in Eq. (18) take the form for Λ=T(-) as X+,+T(-)(x,k⊥)=-kL2(mq+M)mq2+k⊥2,(30)X-,+T(-)(x,k⊥)=-(xM+mq)kL(M+mq)mq2+k⊥2,(31)X+,-T(-)(x,k⊥)=((1-x)M+mq)kL(M+mq)mq2+k⊥2,(32)X-,-T(-)(x,k⊥)=mq(M+2m)+k⊥2(M+mq)mq2+k⊥2,(33)where kR(L)=kx±iky.The momentum-space wave function can be described using the Brodsky-Huang-Lepage method as [81] ϕ(x,k⊥)=Aexp[-mq2+k⊥2x+mq¯2+k⊥21-x8β2-(mq2-mq¯2)28β2(mq2+k⊥2x+mq¯2+k⊥21-x)].(34)Here, β refers to the harmonic oscillator (HO) scale parameter and A to the normalization constant. For mq=mq¯=m, we have pionlike mesons, and for mq≠mq¯, we have kaonlike mesons.

The energy-momentum tensor operator Tμν^ has no fundamental probe that can couple to its parity-odd partner Tq5μν^ in high energy physics. However, by connecting the respective form FFs to the exact moments of the GTMDs or GPDs, we can obtain a representation of Tq5μν^ [50]. The relationship between the FFs may be derived employing the following QCD relation: ϕ¯γ[μγ5iD↔ν]ϕ=2mϕ¯iσμνγ5ϕ-εμναβ∂α(ψ¯γβϕ).(35)

For the numerical calculations, we have taken the input parameters of the LFQM for different quark masses (mq,mb,mc,ms) with q=(u,d) and different variational HO parameters (βqq¯,βqc¯,βqb¯,βqs¯,βsc¯,βbb¯,βcc¯,βsb¯,βcb¯) from Refs. [2,50]. These parameters have been presented in Table I and obtained by reproducing the mass spectra using the variational principle, which has been successful in computing various physical properties such as decay constants, electromagnetic form factors, and distribution amplitudes [2].

In order to compute the spin-orbit correlators using the G1,1 GTMD for the respective meson, we have used Eq. (44) along with the quark masses and HO parameters from Table I. We have summarized the calculated results of the SOC inside various spin-0 mesons in Table II. From the table, we observe that the sign of the correlation is negative for all spin-0 mesons, which clearly implies that the quark longitudinal spin and quark OAM tend to be antialigned inside the respective spin-0 meson. This correlation between the quark spin and OAM takes into account the effective number of quarks inside a parent hadron [5]. Therefore, it would be interesting to compare the absolute value of the spin-0 mesons to that of the nucleons. It is clear from the results that the magnitude of SOC for the mesons is less than that of the nucleon (Czu/n=-0.9 and Czd/n=-0.53 [65]) pointing toward a weaker correlation inside the mesons, which seems to be due to more effective quarks inside the nucleons in comparison to those in the mesons.

In order to show the dependence of quark SOC on the range of longitudinal momentum fraction x, we integrate Czq over k⊥ and show the variation of Czq(x) with respect to x for various spin-0 mesons in Fig. 1. In Fig. 1(a), we present the most commonly studied mesons: the pion and the kaon. Here, it is observed that the largest contribution for the pion and kaon comes from the region where the longitudinal momentum fraction x is around 0.4 and 0.38, respectively. In Fig. 1(b), we present the ηb and ηc mesons and their highest x contribution comes at 0.5 and 0.5, respectively. The plots in this case are symmetric, which is due to a negligible difference in quark and antiquark masses. However, the Czq values for ηb and ηc are extremely low when compared to those of the pion and kaon. Similarly, in Fig. 1(c), we have considered the B mesons. The largest contribution for the B mesons B+, B0, Bs0, and Bc+ comes from regions where x is around 0.18, 0.18, 0.21, and 0.5, respectively. The peaks shift toward higher x values, which is due to the increasing inequality in the quark and antiquark masses inside the B mesons having the quark contents as B+(ub¯), B0(db¯), Bs0(sb¯), and Bc+(cb¯). For the case where the quark is lighter than the other antiquark in the meson, a smaller longitudinal momentum fraction x is carried by the quark, hence, leading to the distribution peak at lower values of x. Further, in Fig. 1(d) the x dependence of Czq(x)) for all of the D mesons has been presented. The peak of the distribution for the D-mesons D+, D0, and Ds+ has been observed at longitudinal momentum fractions of x equal to 0.7, 0.7, and 0.62, respectively. For D+ and D0, the quark (c) being heavier than the antiquark (u or d) carries a larger longitudinal momentum fraction x. This shifts the peak of the distribution to higher values of x, and the curve is shifted to the right. For the case of Ds+, the difference between the quark (c) and the antiquark (s) is less compared to that of D+ or D0, and the distribution peaks at a comparatively lower x value.

Further, in Fig. 2 we have presented the dependence of the spin-orbit contribution with transverse momentum k⊥ at fixed values of longitudinal momentum fraction x. Figures 2(a), 2(c), and 2(e) show the dependence of SOC on the k⊥ using the GTMD approach, whereas Figs. 2(b), 2(d), and 2(f) show the dependence using the GPD approach. Figure 2(a) presents the variation of π+, K+, K0, ηb, ηc at x=0.5, Fig. 2(c) presents B+, B0, Bs0, Bc+ at x=0.7, and Fig. 2(e) presents D+, D0, Ds+ at x=0.3. On the other hand, Fig. 2(b) presents the variation of π+, K+, K0, ηb, ηc at x=0.4, Fig. 2(d) presents B+, B0, Bs0, Bc+ at x=0.4, and Fig. 2(f) presents D+, D0, Ds+ at x=0.6. It would be important to mention here that we have taken different values of x for each set to project the difference between different mesons. Keeping the same x values will not affect the dependence but will only affect the amplitude of the SOC. It is observed that even though we have the same Czq value for both the GTMD and GPD approaches, the transverse momentum dependence is different for them [50]. For the GTMD case, the Czq(x,k⊥) is negative over the whole region of k⊥ for all the mesons, which is in agreement with the pion case detailed in Ref. [50]. We notice that for the mesons with light quarks, the peaks occur at lower values of k⊥ but with a comparatively large amplitude. The peaks are narrow and sharp for the light quark mesons, but as we increase the value of k⊥, they diminish and tend to zero. For the case of heavy quark mesons, the peaks appear at higher values of transverse momentum and are broader. The amplitudes also become quite small. Further, using the GPD approach, Czq(x,k⊥) comes out to be positive for higher values of k⊥ in the case of π+, K+, K0, ηb, ηc presented in Fig. 2(b) for x=0.4. As the k⊥ values decrease, the Czq(x,k⊥) value first decreases and then increases for the case of π+. For ηc, it increases with decreasing k⊥, for K+ and K0, it decreases with decreasing k⊥, and for ηb there is a negligible increase in value with decreasing k⊥. These results are in agreement with the results in Ref. [50]. For the case of B mesons B+, B0, Bs0, Bc+ in Fig. 2(d), the variation of Czq(x,k⊥) has been presented for x=0.4. In these cases, the results are negative throughout the k⊥ region but tend to zero for higher values of k⊥. The SOCs for the D mesons presented in Fig. 2(f) for x=0.6 remain positive but tend to zero beyond k⊥=1.00  GeV. This opposite behavior of the B mesons and D mesons is due to the difference in the quark distributions having light and heavy masses, respectively.

We now compute the SOC for the case of spin-1 mesons. We consider Eq. (44) and replace the G1,1 relation with the expression g1(x,k⊥) TMD from Eq. (64). The numerical results of Czq for the spectrum of spin-1 mesons having definite quark contents have been presented in Table III. The sign of the correlation comes out to be positive for spin-1 mesons implying that the quark OAM and the quark longitudinal spin tend to be directly aligned inside the respective spin-1 mesons. The difference between the alignment of SOC for the spin-1 meson and the spin-0 meson is because of the spin density term in the energy-momentum tensor defined in Eq. (35). There are only two possibilities of the value of Czq, which can be either positive or negative depending on the alignment of the quark longitudinal spin and quark OAM. For the case of spin-1 mesons, positive values are obtained.

In Fig. 3, we have presented the dependence of Czq on longitudinal momentum fraction x for different spin-1 mesons. We discuss the cases of ρ+, K*0, and K*+ in Fig. 3(a) where the maximum contribution of Czq(x) for ρ+, K*0, and K*+ is at 0.50, 0.48, and 0.48 respectively. In Fig. 3(b), we present the SOCs for J/ψ, ϕ, and ϒ mesons. In this case, a symmetry is observed because of a similar quark content in the mesons. The largest contributions come approximately around 0.50 for all the mesons in this plot. In Fig. 3(c), we consider the B mesons, and the largest contribution for B*+, B*0, Bs*0, and Bc*+ mesons is for the values of x at 0.10, 0.10, 0.12, and 0.30 for the respective mesons. Similar to the case of the spin-0 mesons, in the case on spin-1 mesons the quarks carry a smaller longitudinal momentum fraction x when the quark mass is lighter than its corresponding antiquark. This results in a peak at lower values of x, and the curve shifts toward the left. Finally, in Fig. 3(d), we present the D mesons. The highest x contribution for the D mesons D+, D0, and Ds+ is found to be at x=0.22, 0.22, and 0.28, respectively. The shifting of the peak is again due to the difference in the quark and antiquark masses giving a peak at higher values of longitudinal momentum fraction x when this difference is small.

In Fig. 4, we have shown the dependence of the spin-orbit contribution with transverse momentum k⊥ at fixed values of longitudinal momentum fraction x for the spin-1 mesons using the GTMD approach. Figure 4(a) presents the variation of ρ+, K*0, and K*+ at x=0.5, Fig. 4(b) presents J/ψ and ϒ at x=0.5, Fig. 4(c) presents B*+, B*0, Bs*0, and Bc*+ at x=0.3, and Fig. 4(d) presents D+, D0, and Ds+ at x=0.7. As discussed earlier, the peak of the quark distribution depends on the quark content of the meson. Mesons having similar quark and antiquark masses tend to have a higher amplitude for its distribution. For the case of mesons having a difference in the quark and antiquark masses, the amplitude varies in proportion to the mass difference. It is also observed that Czq(x,k⊥) remains positive across the entire k⊥ region.

In this work, we studied the correlation between the quark’s orbital angular motion and the quark’s longitudinal spin inside the light and heavy mesons with spin-0 and spin-1. We started by defining the gauge-invariant LF quark longitudinal OAM and decomposing it into its constituent left-handed and right-handed quark contributions. The quark SOC is described by the difference between the right- and left-handed quark contributions of this longitudinal OAM. We defined T^q5μν and further decomposed it into two FFs out of which one form factor is the SOC ascertained by the form factor Czq. We considered two approaches in order to calculate Czq. One is the GTMD approach, where Czq is defined by the leading-twist correlator G1,1 for the spin-0 mesons and g1(x,k⊥) for the case of spin-1 mesons. The alternative way is the GPD technique, in which the first x moments of F1q(x,ζ,t) and H1q(x,ζ,t) at ζ=0, t=0  GeV2 combine to yield the correlation’s expectation value. We calculated the analytical results for Czq by considering the overlap representation of the GPDs and the GTMDs in LFQM. We listed the numerical results of Czq for both the spin-0 and spin-1 mesons. There is a small variation in our results on comparing them with the SOCs of the pion and kaon in the LFQM [50,66]. This variation is due to the different HO parameters and normalization constants. The negative sign in the case of spin-0 mesons indicates that the OAM and quark spin tend to be anticorrelated. The positive sign in the case of spin-1 mesons indicates that OAM and quark spin tend to be directly correlated. We presented the dependence of longitudinal momentum fraction x and the transverse momentum k⊥ for the longitudinal SOC where we considered k⊥ dependence for both the GPDs and GTMDs while dealing with spin-0 mesons and only the GTMDs in the context of spin-1 mesons. Since the spin-1 meson has an added spin wave part, we found that the dependence of longitudinal momentum fraction varies from the case of the spin-0 case. We presented fresh insights into the SOC inside the spin-0 and spin-1 mesons with our analysis of the quark longitudinal spin, which has not been investigated before except for the case of pions and kaons. This work helps us understand the spin structure of different mesons.

Future experimental data from SPD at the NICA collider at JINR (Dubna, Russia) [86,87] will provide deep insight into the spin structure of the hadrons. The upcoming new and upgraded experiments at JLab, DESY, EIC (electron-ion collider) [88] will in the future come up as a valuable sources in accessing the spin physics data, which will give extensive information to probe the multidimensional structure of hadrons.