Ground state pseudoscalar mesons on the light front: from the light to heavy sector

We extract the leading Fock-state light front wave functions (LF-LFWFs) of both the light and heavy pseudoscalar mesons, e.g., the pion (at masses of 130 MeV, 310 MeV and 690 MeV), $\eta_c$ and $\eta_b$, from their covariant Bethe-Salpeter wave functions within the rainbow-ladder (RL) truncation. It is shown that the LF-LFWFs get narrower in $x$ (the longitudinal momentum fraction of meson carried by the quark) with the increasing current quark mass, and the leading twist parton distribution amplitudes (PDAs) inherit this feature. Meanwhile, we find in the pion the LF-LFWFs only contribute around 30\% to the total Fock-state normalization, indicating the presence of significant higher Fock-states within. In contrast, in the $\eta_c$ and $\eta_b$ the LF-LFWFs contribute more than $90$\%, suggesting the $Q\bar{Q}$ valence Fock-state truncation as a good approximation for heavy mesons. We thus study the 3-dimensional parton distributions of the $\eta_c$ and $\eta_b$ with the unpolarized generalized parton distribution function (GPD) and the transverse momentum dependent parton distribution function (TMD). Through the gravitational form factors in connection with the GPD, the mass radii of the $\eta_c$ and $\eta_b$ in the light-cone frame are determined to be $r_{E,{\rm LC}}^{\eta_c} =0.150$ fm and $r_{E,{\rm LC}}^{\eta_b} =0.089$ fm respectively.


I. INTRODUCTION
Originating in the Higgs mechanism [1,2], huge current quark mass difference resides in the quantum chromodynamics (QCD), which yield a diverse hadronic phenomenon across the light and heavy sectors. In hard hadronic processes, the quarks' parton nature, which is directly associated with their current masses, gets exposed. The parton structure of the hadrons is thus of great interest by revealing the substructure of hadrons, and meanwhile experimentally accessible through various hard exclusive and/or inclusive processes.
Theoretically, the hadrons' parton structure are formulated in terms of different sorts of parton distributions. The parton distribution amplitude (PDA), for instance, is an important quantity that incorporates the internal nonperturbative dynamics within the QCD bound state. It serves as a soft input for the factorization of various hard exclusive processes, such as deeply virtual meson production [3,4], B meson decay [5][6][7] and exclusive charmonium / + pair production in + − annihilation [8,9]. The determination of the PDAs rely heavily on nonpertubative QCD methods. In the light quark sector, phenomenological models and methods, i.e., QCD sum rule [10,11], light-front holographic QCD [12] and the Dyson-Schwinger/Bethe-Salpeter equations method (DS-BSEs) [13][14][15] gave their predictions. Meanwhile, the lattice QCD has predicted it first one or two nontrivial moments [16][17][18]. Recently, with the help of large momentum effective theory (LaMET) [19], the lattice QCD is giving much more information by charting the pointwise behavior of the PDAs [20,21]. Notably, agreement * cshi@nuaa.edu.cn between the lattice QCD and DS-BSEs is found for pion chiral-extrapolated to physical mass [22]. On the other hand, there is no lattice QCD result on PDAs in the heavy sector yet. Recently, it is proposed that the B meson PDA can be determined from heavy quark effective theory (HQET), by combining the LaMET and the Euclidean lattice simulation techniques [23]. As compared to light mesons, heavy mesons are arguably simpler as the quarks move much slower inside, so light front potential models [9,24] and nonrelativistic QCD (NRQCD) are applicable [25,26]. Meanwhile, QCD sum rule and DSEs also extend from the light sector to the heavy sector and give their predictions on heavy meson DAs [7,[27][28][29][30].
On the other hand, the parton distribution functions (PDF) of hadrons, and in particular their 3-dimensional extension as the generalized parton distribution functions (GPDs) [3,31,32] and transverse momentum dependent parton distributions functions (TMDs) [33] draw much attention in recent years. The GPDs provide a unified description of the parton distribution in the longitudinal momentum and transverse spatial coordinates [34,35], while the TMDs incorporate the transverse motion of partons and their spin-orbit correlations [36,37]. Meanwhile, the GPDs are connected with hadron matrix elements of the energy-momentum tensor through -weighted moments, which provide valuable information of the spin, energy, and pressure distributions within hadrons [32,[38][39][40][41]. The GPDs and TMDs thus provide much more abundant information on hadron's parton structure. Although present focus on hadrons lies mostly in the light sector, e.g., the nucleon and pion/kaon mesons that are relatively stable [42][43][44], it is of theoretical interest to look into the 3D structure of the heavy hadrons with the help of GPD and TMD.
As light-cone quantities, the PDA, GPD and TMD are interconnected by the light-front wave functions (or light-cone wave functions). The PDA is the LF-LFWF integrated over transverse momentum [45], and the GPD and TMD can be calculated with overlap representations in terms of LFWFs [46][47][48][49]. The standard way to obtain the LFWFs is by diagonalizing the light-cone Hamiltonian. Challenges lie in the construction of lightcone Hamiltonian in connection with QCD, as well as its diagonolization when more Fock-states are involved [50]. Recently, with the help of basis light-front quantization (BLFQ) technique, the |¯ state is included in pion and the calculation thus goes beyond the leading Fock-state truncation [51]. On the other hand, an alternative approach exists by extracting the LFWFs from hadrons' covariant wave functions in the ordinary spacetime frame, namely the instant form [48,[52][53][54][55]. Using this approach, we obtained the LF-LFWFs of the pion and kaon [56,57], and later the vector mesons and / [58]. A unique advantage of this approach is that it circumvents the light-cone Hamiltonian construction and diagonalization, and allows the extraction of LF-LFWFs from many Fock-states embedded. In this work, based on the study of the pion at physical mass in [57], we will predict LF-LFWFs of fictitious pion at masses of 310 MeV and 690 MeV which are directly accessible by lattice QCD, as well as and from the heavy sector. The pseudoscalar meson sits in a special position as in the chiral limit it is the Goldstone boson of dynamical chiral symmetry breaking (DCSB). The pion is thus dominated by the DCSB phenomenon while the Higgs mechanism is almost irrelevant [59]. However, in the heavy mesons, the situation is the opposite: the Higgs mechanism generates most of the quark masses (and consequently the hadron mass) but the DCSB effect weakens. The pseudoscalar mesons across the light and heavy sectors thus provide a good window to observe how the LF-LFWFs evolve with the strength shift between the DCSB and Higgs mechanism. The present study is therefore motivated in several directions: In the light quark sector, we study the LF-LFWFs of the pion at the physical mass (≈ 130MeV) and make predictions at testing masses of = 310 MeV and 690 MeV. The later two cases are chosen as they are directly accessible in lattice simulation [22]. Note that lately the authors of [60] have proposed a way to extract the hadron LFWFs through lattice QCD based on LaMET, so lattice results can be expected. In the heavy sector, we report the and LF-LFWFs determined for the first time from the DS-BSEs. Using these LFWFs, we analyze the PDAs of and and investigate their 3D parton structure with the help of GPD and TMD. This paper is organized as follows: In Sec. II we introduce the DS-BSEs formalism and extract from the covariant BS wave functions the LF-LFWFs. In Sec. III, we show the calculated LF-LFWFs of pion (at different masses), and , as well as their PDAs. A comparison is made between the light and heavy mesons. In Sec. IV, the parton structure of and are studied by means of the GPD (at zero skewness) and unpolarized TMD. Finally we conclude in Sec. V

II. FROM BETHE-SALPETER WAVE FUNCTIONS TO LF-LFWFS
Within the DS-BSEs framework, the mesons are treated as bound states and described by their covariant BS wave functions. Within the rainbow-ladder (RL) truncation, which is taken throughout this work, the pseudoscalar mesons can be solved by aligning the quark's DSE for full quark propagator ( ) and meson's BSE for BS amplitude Γ ( , ) [61], i.e., Here ∫ Λ implements a Poincaré invariant regularization of the four-dimensional integral, with Λ the regularization mass-scale. The free is the free gluon propagator in the Landau gauge. The quark momentum partition ± = ± /2. The ( ) is the current-quark mass renormalized at scale of . The 2 and 4 are the quark wave function and mass renormalization constants respectively. Here a factor of 1/ 2 2 is picked out to preserve multiplicative renormalizability in solutions of the DSE and BSE [62]. The Bethe-Salpeter amplitudes are eventually normalized canonically withΓ ( , − ) = −1 Γ (− , − ) and = 2 4 . Notably, the RL truncation preserves the (near) chiral symmetry of QCD by respecting the axial vector Ward-Takahashi identity [63]. The DCSB is therefore faithfully reflected and the Goldstone nature of light pseudoscalar mesons are manifested. Meanwhile, the RL truncation also applies to the heavy mesons [64][65][66]. We therefore solve the pion and heavier and mesons in the same RL truncation. The modeling function G( 2 ) in Eqs. (1,2) absorbs the strong coupling constant , as well as the dressing effect in both the quark-gluon vertex and full gluon propagator. Popular models include the Maris-Tandy (MT) model, and the later Qin-Chang (QC) model [67] The second term in Eq. 4 is the perturbative QCD result [61,67] that describes the UV behavior, while the first term incorporates essentially nonperturbative dressing effects at low and moderate momentum. As compared to the MT model, the QC model improves the far infrared behavior of gluon propagator to be in line with lattice QCD [68,69] and modern DSE study [70], while in hadron study the two are equally good. Historically, combined with the RL truncation, the MT and/or QC models well describe a range of hadron properties, including the pseudoscalar and vector meson masses, decay constants and various elastic and transition form factors [63,[71][72][73][74][75].
The model parameters in this work are set up as follows. For pion, we consider three cases, i.e., = 130 MeV, = 310 MeV and = 690 MeV. The first one is physical while the later two are for exploratory purpose, but directly accessible in lattice QCD simulations. In Eq. (4), we employ the well determined parameters = 0.5 GeV, . Meanwhile, we omit the UV term of Eq. (4), which determines the UV behavior of pion's BS amplitude. Physically, this means we will only focus on the low and moderate part of the pion's LF-LFWFs, but discard the their UV part. In this case, Eqs. (1,2) are super-renormalizable and the renormalization constants 2 and 4 can be set to 1. The only remaining parameter is the current quark mass and we take / = 5 MeV, 27 MeV and 119 MeV, which produce = 130 MeV, 310 MeV and 690 MeV respectively. Note that / = 119 MeV already reaches the strange quark mass.
Using these parameters, the ( ) and Γ ( ; ) can be numerically solved with Eqs. (1,2). Note that the ( ) takes the general decomposition The F = , , and are scalar functions of 2 , · and 2 . In the end, we get the numerical solutions to , and F 's.
On the other hand, in the light-cone frame, the pseudoscalar meson with valence quark and valence antiquarkh at the leading Fock-state is given by [50,81] is the transverse momentum of the quark , and = + + is the light-cone longitudinal momentum fraction of the active quark. The rest variables are¯= 1 − and¯= − . The = (↑, ↓) denotes the quark helicity and / √ 3 is the color factor. The † and † are the creation operators for quark and antiquark respectively. The Φ 1 , 2 ( , ) are the LFWFs that encode the nonperturbative internal dynamical information.

III. LF-LFWFS OF PSEUDOSCALAR MESONS
The LF-LFWFs for the pion are shown in Fig. 1. From the top row to the bottom, we display the results of = 130 MeV, 310 MeV and 690 MeV respectively. Comparing these LF-LFWFs, a prominent feature is that as the current quark mass increases, the LF-LFWFs get narrower in . Fig. 1 therefore suggests that the DCSB tends to broaden the −distribution of pseudoscalar mesons LF-LFWFs while the explicit chiral symmetry breaking brought by Higgs mechanism does the opposite. The tendency continues to the heavy sector. In Fig. 2 we show the LF-LFWFs of and mesons, which are significantly narrower than those in the light quark sector. On the other hand, the light meson LF-LFWFs decrease much faster than the heavy meson in . This indicates the transverse motion of quarks in heavy mesons are more active than that in the light system.
Aside from the profile, the magnitude of the LF-LFWFs also provide important information of the pseudoscalar mesons' parton structure. Rigorously speaking, the meson's LFWFs of all Fock-states should normalize to unity, i.e., with , = The HF refers to higher Fock-states contribution. The 's are simply the overlap between the LFWFs and their complex conjugate, so they are positive definite. Here we infer the value of HF by subtracting unity with the LF-LFWFs contribution. Our result is listed in Table. I. Apparently, as the system gets heavier, the higher Fockstates contribution diminishes. This suggests i) heavy systems as¯and¯could be well approximated with their LF-LFWFs and ii) in light system such as pion where the Higgs Mechanism is almost irrelevant, there are lots of higher Fock-states generated in association with the DCSB. We therefore believe this is another novel property of the parton structure of mesons in connection with the DCSB.
Finally, there is a shift in the relative strength between 0 ( , 2 ) and 1 ( , 2 ) as the quark mass changes. Defining the ratio we find = 0.56, 0.16 and 0.04 for pion, and respectively. Therefore as the spin anti-parallel (S-wave) LFWF 0 ( , 2 ) provides more contribution, there is also considerable P-wave component in the pion. In the heavy quarkonium, the S-wave component becomes dominant as the system gets non-relativistic.  We next look at the leading-twist parton distribution amplitudes (PDA) of the mesons. The twist-2 PDA of pseudoscalar meson was originally defined as theintegrated LFWF [45], i.e., with the normalization condition In Fig. 3, we show our results on pion DAs for three masses, i.e., 130 MeV, 310 MeV and 690 MeV, denoted by solid, dashed and dotted curves respectively. The green, red and blue bands are adopted from lattice QCD calculation that employed same pion masses respectively, while at a hadron momentum of = 1.75 GeV in the LaMET approach [22]. General agreement between the two calculations is found, including the evolution of PDA with increasing current quark mass (or pion mass). For instance, the PDAs of pion with mass 130 MeV and 310 MeV do not differ much, i.e., they are broad and concave functions. On the other hand, at the mass of 690 MeV, the PDAs get significantly narrower and is close to the asymptotic form 6 (1 − ) (gray dot-dash-dashed curve), as first pointed out in DS-BSEs [28] and confirmed later in [22]. We note that the our PDAs are associated with a scale around 1.0 GeV: since we only took the first term in Eq. (4), the model implements a soft momentum cutoff ≈ Λ ≈ 2 = 1.0 GeV. Meanwhile the lattice QCD result is at the scale of 2 GeV. Under the Efremov-Radyushkin-Brodsky-Lepage (ERBL) evolution [45,85], all curves would evolve very slowly toward the asymptotic form. ( , ≈ ) (purple dot-dashed) from sum rule calculation [7]. One can see the PDAs of agree very well. For , there is some deviation between the two curves. However, the deviation reduces if the uncertainties are considered. For instance, in [7] the moments (2 − 1) with m=2,4,6 are 0.067 ± 0.007, 0.011 ± 0.002,0.003 ± 0.001 respectively, while our red dashed curve gives 0.059, 0.0125 and 0.0047. On the other hand, covariant light front model [86] and BLFQ approach [24] all yield semi-quantitatively similar results. In the absence of lattice QCD calculation, they all suggest that the PDAs of heavy pseudoscalar mesons are narrowly distributed in as compared to asymptotic form 6 (1 − ) (gray dot-dash-dashed curve).

IV. SPATIAL AND MOMENTUM TOMOGRAPHY OF THE HEAVY PSEUDOSCALAR MESONS.
In the light-cone gauge, the unpolarized quark GPD of pseudoscalar meson is defined as The is the parton's averaged light-cone momentum fraction and = − Δ + 2 + is the skewness. The momentum transfer is = Δ 2 = − The GPD has two distinct domains, where | | < | | is the ERBL region and 1 > | | > | | is the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) region, named after their evolution with factorization scale . Here we will be focusing on the GPD of and at zero skewness, i.e., the ( , 0, ). It gives rise to many interesting quantities, e.g., the onedimensional collinear parton distribution function (PDF), the impact parameter dependent parton distributions (IPDs) and the gravitational form factor (GFF). The overlap representation of ( , 0, ) in terms of LFWFs reads [46,47,83,87]  Here we rescale the LF-LFWFs of and so that the PDF, which relates to the GPD as ( ; 0 ) = ( , 0, 0; 0 ), is normalized to unity. This approximation is based on the finding that higher Fock-states contribute little to the normalization, as shown in Table. I. However, the value of 0 is priorly unknown. Phenomenologically, it is usually determined by comparing the first moment of PDF with experiment or lattice QCD [88,89], but this is infeasible in the heavy sector due to the lack of experiment data. In this paper, we follow [90] and presume that 0 to be around 2 . That corresponds to 0 ≈ 2.6 GeV for and 0 ≈ 8.6 GeV for , close to the natural and mass scales. We remind that the various parton distributions studied below are all implicitly associated with such a scale, which will not be written explicitly.
The two-dimensional Fourier transform of ( , 0, Δ 2 ) gives the IPD GPD which characterizes the probability density of partons on the transverse plane with the light-cone momentum fraction and the impact parameter [35]. The is defined as the separation between the active parton and the origin of transverse center of momentum, i.e., ≡ ,1 = ,1 − . In a two-body picture, = ,1 + (1 − ) ,2 , with , the transverse position of -th parton. We show our results in Fig. 6, with the left column devoted to charm quark in , and the right column for bottom quark in . The charm quark in is more broadly distributed in both and impact parameter as compared to bottom quark in at their hadronic scales. We then integrate the −dependence in the IPDs and obtain their spatial parton distribution ,(0) ( ) = ∫ 1 0 ( , 2 ). The result is plotted in the upper panel of Fig. 7 and the lower panel display their density plots. We determine their mean squared impact parameters By definition, the 2 can be regarded as the square of quark distribution radius in the light-cone frame. Within the leading Fock-state truncation and for a charged pseudoscalar meson, for instance the + , the light-cone charge radius of + is 2 , less than 15%, suggesting roughly the error brought by leading Fock-state truncation in . We anticipate the deviation would be even smaller for heavier . The gravitational form factor of and can be connected with the −moment of the GPD at = 0, i.e., The 2,0 ( ) denotes the quark's contribution to the hadron's gravitational form factor ( ), which enters the general decomposition of the matrix element of energymomentum tensor (EMT) of spin-0 states [92][93][94] ( )| (0)| ( ) = can be defined as the mean value of 2 ⊥ weighted by EMT in the light-cone frame, namely 1 It is thus related to the gravitational form factor ( ) by [95] 2 ,LC = −4 = (0.089 fm) 2 respectively. It's interesting to compare them with the quark distribution radius 2 = (0.157 fm) 2 and 2 = (0.092 fm) 2 above. The two radii are very close, with the lightcone energy radius a bit smaller. Such behaviors are in agreement with the finding of [24] in heavy mesons.
We finally investigate the transverse momentum distribution of quarks within and . The unpolarized leading-twist TMD of pseudoscalar meson is defined as with the gauge link omitted. At hadronic scale, its overlap representation reads [49] We show in Fig. 9 their density plots. The result of is on the left while that of on the right. These TMDs share the characteristics of the LF-LFWFs, i.e., they peek at the center = 0.5 and are narrowly distributed in . Comparing between the and , we find the heavier bottom quarks are more centered around = 0.5 while more broadly distributed in . Their mean transverse momentum of the valence quarks = ∫ 2 1 ( , 2 )| | is = 0.65 GeV and = 1.02 GeV. It's also interesting to look into the form of the transverse momentum dependence within the TMD, which we demonstrate with Fig. 10. For the past years, the Gaussian and/or Gaussian-based | |-dependent models have been very popular in parameterizing the TMDs of pion and nucleon in the light quark sector [56,[96][97][98][99][100][101][102].
Here we explore its validity in the heavy sector with the Gaussian form G ( , 2 ) = − 2 / 2 ( ) . We find the fitting curves (gray dotted) largely overlap with the original curves. The fitting parameters were determined to be

V. CONCLUSION
We study the leading Fock-state light front wave functions of the light and heavy pseudoscalar mesons, using a unified framework of rainbow-ladder DS-BSEs and light front projection method. The LF-LFWFs of the pion at masses of 310 MeV and 690 MeV, and those of the heavy and are reported for the first time within DS-BSEs. The LF-LFWFs of pion at physical mass and 310 MeV are broadly distributed in and close to each other, while at the mass of 690 MeV which sits at the strange quark point, i.e., the fictitious¯, the LF-LFWFs get distinctly narrower. This trend continues in the heavy sector for the and . Such property is reflected in the twist-2 PDA which is the −integrated LF-LFWF. General agreement is found between our pion PDAs at different masses and those from lattice QCD [22]. In the heavy sector, where lattice QCD is absent, our PDAs of and are comparable with those from sum rule prediction [7].
The contribution of the LF-LFWFs to meson states on the light front is further analyzed. As listed in Table. I, the LF-LFWFs of pion contribute only 30% to the Fock-state normalization, while in and they contribute more than 90%. This indicates the existence of considerable higher Fock-states in pion, and meanwhile the dominant role of LF-LFWFs in determining heavy mesons. In particular, the later strongly suggests the leading Fockstate truncation as a reasonable means in dealing with and .
We thus take the leading Fock-state truncation, and study the unpolarized quark GPD and TMD of and using the light front overlap representation given in Eq. (20) and Eq. (27). We associate the resolution scale to the sum of consistuent valence (anti)quark masses, i.e., 2 , and study the the spatial distribution of valence quarks with IPD GPD. It is found that heavier quarks are spatially more centered in heavier mesons. The study on gravitational form factor also reveals that the heavier meson has smaller energy radius in the light-cone frame. In the transverse momentum space, we find the heavier quark is more broadly distributed in , but more centered around longitudinal momentum fraction = 0.5. We also find the unpolarized TMD PDF of and can be approximated with −independent Gaussian functions, as an extension to rudimentary phenomenological parameterizations of hadron TMD in the light sector.