The light front wave functions and diffractive electroproduction of vector mesons.

The leading Fock-state configuration of a vector meson state is expressed with the nonperturbative LFWFs $\Phi^{\Lambda}_{\lambda,\lambda^{\prime}}$

	$\displaystyle|M\rangle^{\Lambda}$	$\displaystyle=\sum_{\lambda,\lambda^{\prime}}\int\frac{d^{2}\boldsymbol{k}_{T}}{(2\pi)^{3}}\,\frac{dx}{2\sqrt{x\bar{x}}}\,\frac{\delta_{ij}}{\sqrt{3}}$
		$\displaystyle\hskip 28.45274pt\Phi^{\Lambda}_{\lambda,\lambda^{\prime}}(x,\boldsymbol{k}_{T})\,b^{\dagger}_{f,\lambda,i}(x,\boldsymbol{k}_{T})\,d_{f,\lambda^{\prime},j}^{\dagger}(\bar{x},\bar{\boldsymbol{k}}_{T})|0\rangle.$		(1)

Here the $\boldsymbol{k}_{T}=(k^{x},k^{y})$ is the transverse momentum of the quark $f$, and $\bar{\boldsymbol{k}}_{T}=-\boldsymbol{k}_{T}$ for antiquark $\bar{f}$. The longitudinal momentum fraction carried by quark is $x=\frac{k^{+}}{P^{+}}$, and $\bar{x}=1-x$ for antiquark. The $i$ and $j$ are color indices. The quark helicity $\lambda$ runs through $\uparrow$ and $\downarrow$, while the meson helicity $\Lambda$ runs through $0$ and $\pm 1$. The $\Phi(x,\boldsymbol{k}_{T})$’s can be further expressed with amplitudes $\psi(x,\boldsymbol{k}_{T}^{2})$’s which contain only scalars arguments $x$ and $\boldsymbol{k}_{T}^{2}$ Ji et al. (2003). Denoting the quark helicity $\uparrow=+$ and $\downarrow=-$, and omitting the function arguments, one finds for longitudinally polarized mesons

$\displaystyle\Phi_{\pm,\mp}^{0}$

$\displaystyle=\psi^{0}_{(1)},\ \ \ \ \ \Phi_{\pm,\pm}^{0}=\pm k_{T}^{(\mp)}\psi^{0}_{(2)},$

(2)

with $k_{T}^{(\pm)}=k^{x}\pm ik^{y}$, and for transversely polarized mesons ($\Lambda=\pm 1$)

	$\displaystyle\Phi_{\pm,\pm}^{\pm 1}$	$\displaystyle=\psi^{1}_{(1)},$	$\displaystyle\Phi_{\pm,\mp}^{\pm 1}$	$\displaystyle=\pm k_{T}^{(\pm)}\psi^{1}_{(2)},$
	$\displaystyle\Phi_{\mp,\pm}^{\pm 1}$	$\displaystyle=\pm k_{T}^{(\pm)}\psi^{1}_{(3)},$	$\displaystyle\Phi_{\mp,\mp}^{\pm 1}$	$\displaystyle=(k_{T}^{(\pm)})^{2}\psi^{1}_{(4)}.$		(3)

Note the $\Lambda=-1$ meson can be obtained from $\Lambda=+1$ with a $\hat{Y}$ transform, which consists a parity operation followed by a 180°  rotation around the y axis Ji et al. (2004). With the help of $\hat{Y}$ and charge parity, we find the constraints

$\displaystyle\psi_{(i)}^{\Lambda}(x,\boldsymbol{k}_{T}^{2})=\psi_{(i)}^{\Lambda}(1-x,\boldsymbol{k}_{T}^{2}),$

(4)

with one exception

$\displaystyle\psi^{1}_{(2)}(x,\boldsymbol{k}_{T}^{2})$

$\displaystyle=-\psi^{1}_{(3)}(1-x,\boldsymbol{k}_{T}^{2}).$

(5)

In the end, for $\rho^{0}$ (with isospin symmetry) or $J/\psi$, there are totally five independent $\psi^{\Lambda}_{(i)}$’s at leading Fock-state.

Within the DS-BSEs framework, the vector mesons can be solved with their covariant Bethe-Salpter wave functions. In practice, this is achieved by taking the rainbow-ladder (RL) truncation and aligning the quark’s DSE for full quark propagator $S(p)$ and meson’s BSE for BS wave functions $\Gamma_{\mu}^{M}(k,P)$ Maris and Tandy (1999), i.e.,

	$\displaystyle S(p)^{-1}$	$\displaystyle=$	$\displaystyle Z_{2}\,(i\gamma\cdot p+m^{\rm bm})+Z_{2}^{2}\int^{\Lambda}_{\ell}\!\!{\cal G}(\ell)\ell^{2}D_{\mu\nu}^{\rm free}(\ell)\frac{\lambda^{a}}{2}\gamma_{\mu}S(p-\ell)\frac{\lambda^{a}}{2}\gamma_{\nu},\rule{10.00002pt}{0.0pt}$		(6)
	$\displaystyle\Gamma^{M}_{\mu}(k;P)$	$\displaystyle=$	$\displaystyle-Z_{2}^{2}\int_{q}^{\Lambda}\!\!{\cal G}((k-q)^{2})\,(k-q)^{2}\,D_{\mu\nu}^{\rm free}(k-q)\frac{\lambda^{a}}{2}\gamma_{\mu}S(q_{+})\Gamma^{M}_{\mu}(q;P)S(q_{-})\frac{\lambda^{a}}{2}\gamma_{\nu},$		(7)

Here $\int^{\Lambda}_{q}$ implements a Poincaré invariant regularization of the four-dimensional integral, with $\Lambda$ the regularization mass-scale. $D^{\rm{free}}_{\mu\nu}$ is the free gluon propagator. $m^{\rm bm}(\Lambda)$ is the current-quark bare mass. $Z_{2}$ is the quark wave function renormalisation constants at renormalization point $\mu$. Here a factor of $1/Z_{2}^{2}$ is picked out to preserve multiplicative renormalizability in solutions of the gap and Bethe-Salpeter equations Bloch (2002). The Bethe-Salpeter amplitudes are eventually normalized canonically (see, e.g., Eq. (25) in Ref. Maris and Tandy (1999)).

The modeling function ${\cal G}(l^{2})$ absorbs the strong coupling constant $\alpha_{s}$, as well as dressing effect from quark-gluon vertex and full gluon propagator. Popular models include the earlier Maris-Tandy (MT) model, and the later Qin-Chang (QC) model Qin et al. (2012)

$${\cal G}_{QC}(s)=\frac{8\pi^{2}}{\omega^{4}}D\,{\rm e}^{-s/\omega^{2}}+\frac{8\pi^{2}\gamma_{m}}{\ln[\tau+(1+s/\Lambda_{\rm QCD}^{2})^{2}]}{\cal F}(s).$$

(8)

The first term models the infrared behavior, and the second term is perturbative QCD result Maris and Roberts (1997); Qin et al. (2012). The QC model improves the infrared part of MT model to be in concert with modern gauge sector study, while in hadron study the two are equally good. Combined with the RL truncation, the MT and/or QC model well describes a range of hadron properties, including the pion and $\rho$ meson masses, decay constants and various elastic and transition form factors Maris et al. (1998); Maris and Tandy (1999, 2000a); Jarecke et al. (2003); Bhagwat and Maris (2008); Xu et al. (2019). The success also extends to nucleon by solving the Faddeev equation Eichmann et al. (2010); Eichmann (2011). These achievements owe greatly to the nice property of the RL truncation by preserving the (near) chiral symmetry of QCD (respecting the axial vector Ward-Takahashi identity) Maris et al. (1998). It is therefore capable of simultaneously describing the almost massless pion as a Goldstone boson and the much more massive $\rho$ and nucleon, reflecting different aspects of the DCSB. Here we will explore the prediction of RL DS-BSEs on the vector meson LF-LFWFs.

Having solved Eqs. (6-7) and obtain $S(p)$ and $\Gamma_{\mu}^{M}(q;P)$, we then project the vector meson BS wave function $\chi^{M}_{\mu}(k,P)=S(k+P/2)\,\Gamma_{\mu}^{M}(k,P)\,S(k-P/2)$ onto the light front to obtain the LF-LFWFs using

$\displaystyle\Phi^{\Lambda}_{\lambda,\lambda^{\prime}}(x,\boldsymbol{k}_{T})$

$\displaystyle=-\frac{1}{2\sqrt{3}}\int\frac{dk^{-}dk^{+}}{2\pi}\delta(xP^{+}-k^{+})\textrm{Tr}\left[\Gamma_{\lambda,\lambda^{\prime}}\gamma^{+}\chi^{M}(k,P)\cdot\epsilon_{\Lambda}(P)\right].$

(9)

This can be derived by generalizing the projection method for pseudo-scalar meson in Liu and Soper (1993); Burkardt et al. (2002). Here the $\epsilon_{\Lambda}(P)$ is the meson polarization vector. The $\Gamma_{\pm,\mp}=I\pm\gamma_{5}$ and $\Gamma_{\pm,\pm}=\mp(\gamma^{1}\mp i\gamma^{2})$ projects out certain (anti)quark helicity configuration. The $\chi^{M}_{\mu}(k,P)$ can be expressed with the dressed quark propagator $S(k)$ and BS amplitude $\Gamma^{M}_{\mu}(k,P)$ as . The trace is taken over Dirac, color and flavor indices. An implicit color factor $\delta_{ij}$ is associated with $\Gamma_{\lambda,\lambda^{\prime}}$, as well as a flavor factor diag$(1/\sqrt{2},-1/\sqrt{2})$ for $\rho^{0}$. Then we calculate the $(2x-1)$-moments of $\psi^{\Lambda}_{(i)}(x,\boldsymbol{k}_{T}^{2})$ (which are equivalent to $\Phi_{\lambda,\lambda^{\prime}}^{\Lambda}$ through Eqs. (2-II.1)) at every $|\boldsymbol{k}_{T}|$, i.e.,

$\displaystyle\langle(2x-1)^{m}\rangle_{|\boldsymbol{k}_{T}|}^{(i)}$

$\displaystyle=\int dx(2x-1)^{m}\psi^{\Lambda}_{(i)}(x,\boldsymbol{k}_{T}^{2}),$

(10)

with $m=0,1,2,..$. From these moments we reconstruct the LFWFs. For practical reasons, we treat the $\rho$ and $J/\psi$ with somewhat different techniques.

In solving the $\rho$ DS-BSEs, we only take the infrared part of the QC model, i.e., the first term on the right hand side of Eq. (8). We refer to it as QC-IR model. Since the support of light quark propagator and BS amplitude are dominated by low relative momentum, the ultraviolet term of QC model has relatively small effect. Such treatment was also employed in other light quark sector studies Fischer and Williams (2009); Chang and Roberts (2009). Adopting the well-determined parameters $\omega=0.5\,$GeV, $D=(0.82\,{\rm GeV})^{3}/\omega$ Qin et al. (2012); Shi et al. (2014); Xu et al. (2019) and the current quark mass $m_{u/d}=5$ MeV, we reproduce $m_{\pi}=131$ MeV and $f_{\pi}=90$ MeV, as well as $m_{\rho}=717$ MeV and $f_{\rho}=140$ MeV comparing to experimental values $m_{\rho}=775$ MeV and $f_{\rho}=156$ MeV Zyla et al. (2020). We choose QC-IR model rather than QC model as it renders an exponentially $k^{2}-$suppressed $\Gamma^{M}_{\mu}(k,P)$. This allows us to directly compute up to ninth-moment with Eq. (10), with the numerical noises heavily suppressed. Note with QC model, only the first two or three moments can be directly computed for now. We then fit the moments with a flexible parameterization Shi et al. (2014, 2015)

$\displaystyle\psi_{(i)}^{\Lambda}(x,\boldsymbol{k}_{T}^{2})$

$\displaystyle\approx[x(1-x)]^{\alpha-1/2}\!\!\!\sum_{j=0,2}^{\ }a_{j}^{\alpha}C_{j}^{\alpha}(2x-1)+[x(1-x)]^{\alpha^{\prime}-1/2}\,\sum_{j^{\prime}=1,3}a_{j^{\prime}}^{\alpha^{\prime}}C_{j^{\prime}}^{\alpha^{\prime}}(2x-1),$

(11)

where the $\alpha$, $\alpha^{\prime}$, $a_{j}^{\alpha}$ and $a_{j^{\prime}}^{\alpha^{\prime}}$ are fitting parameters. They implicitly depend on the $\boldsymbol{k}_{T}^{2}$, $\Lambda$ and $i$. The $C_{j}^{\alpha}(x)$ is the Gegenbaur polynomial of order $\alpha$, so the first term on the right hand side of Eq. (11) is symmetric in $x$ with respect to $x=1/2$, and the second term is anti-symmetric. They are devised to fit the even and odd (2x-1)-moments separately. Using Eq. (11), we well reproduce all the moments with deviations less than 1%.

For $J/\psi$, we choose the parameters $\omega=0.7$ GeV and $D=(0.6\ \textrm{GeV})^{3}/\omega$ from a recent global analysis on heavy meson spectrum involving both charm and bottom quarks Chen et al. (2020). They are a bit different from that in the light sector, as the the DCSB dressing effects they mimic are quantitatively different between light and heavy sectors. In principle, this deviation can be reduced by going beyond RL truncation. Meanwhile we keep the ultraviolet term of the QC model as it is more relevant for heavy quarks. With running quark mass $m_{c}(\mu=m_{c})=1.33$ GeV, we get $m_{J/\psi}=3.09$ GeV and $f_{J/\psi}=300$ MeV, as compared to PDG data $m_{c}(\mu=m_{c})=1.28$ GeV, $m_{J/\psi}=3.096$ GeV and $f_{J/\psi}=294$ MeV by leptonic decay $\Gamma(J/\psi\rightarrow e^{+}e^{-})=5.53$ keV Zyla et al. (2020). To compute the $J/\psi$ LFWFs, we adopt the technique used in Shi and Cloët (2019); Shi et al. (2020): by fitting the meson BS amplitude with Nakanishi-like representation Nakanishi (1963) and the quark propagator with pairs of complex conjugate poles form which is particularly accurate in heavy sector Souchlas (2010), we are able to compute point-wisely accurate LFWFs. More details can be found in the appendix.

We then obtain all five LF-LFWFs of $\rho$ and $J/\psi$, as displayed in Fig. 1 and Fig. 2. These LF-LFWFs satisfy all the general requirements of Eqs. (2-5). Noticeably, the $\rho$ and $J/\psi$ LFWFs are very different in profile. At small and moderate $\boldsymbol{k}_{T}^{2}$, the $J/\psi$ LFWFs are distributed closer to $x=1/2$, while the $\rho$ LFWFs are more broadly distributed. This is qualitatively consistent with the phenomenological $\rho$ LFWFs fitted to diffractive $\rho$ production HERA data Forshaw and Sandapen (2010) and the AdS/QCD prediction Forshaw and Sandapen (2012).

A quick comparison of our LF-LFWFs with other theoretical calculations is to look into the twist-2 distribution amplitude (DA) $\phi^{V}_{\parallel}(x;\mu)$, defined as the $\boldsymbol{k}_{T}-$integrated LFWF $\phi_{\parallel}^{V}(x;\mu)=\frac{\sqrt{6}}{f_{V}}\int^{|\boldsymbol{k}_{T}|=\mu}\frac{d^{2}\boldsymbol{k}_{T}}{(2\pi)^{3}}\psi_{(1)}^{0}(x,\boldsymbol{k}_{T}^{2}).$ For the DA moment $\langle\xi^{2}\rangle=\langle(2x-1)^{2}\rangle$, we obtain $\langle\xi^{2}\rangle^{\rho}=0.269$ as compared to sum rule results 0.251(24) Ball et al. (2007), 0.216(21) Pimikov et al. (2014) and 0.241(28) Fu et al. (2016) at the scale of about $1$ GeV. Note that we determine our scale for $\rho$ to be $\mu\approx 2\omega=1$ GeV, as it is an implicit cutoff within QC-IR model. Meanwhile the lattice QCD gives $\langle\xi^{2}\rangle^{\rho}=0.268(54)$ Arthur et al. (2011) and $0.245(9)$ Braun et al. (2017) at a higher scale of $2$ GeV. For $J/\psi$, we obtain $\langle\xi^{2}\rangle^{J/\psi}=0.093$ as compared to sum rule results $0.083(12)$ Fu et al. (2018) and $0.070(7)$ Braguta (2007) and light-front holography prediction $0.096(20)$ Li et al. (2017) at the scale of $\mu=m_{c}$, indicating a significantly narrower DA.

Beside the broadness, another difference between the $\rho$ and $J/\psi$ LF-LFWFs is their contribution to Fock-states normalization. The meson’s LFWFs of all Fock-states should normalize to unity in general, i.e.,

	$\displaystyle 1$	$\displaystyle=\sum_{\lambda,\lambda^{\prime}}N_{\lambda,\lambda^{\prime}}^{\Lambda}+N_{\textrm{HF}},$		(12)
	$\displaystyle N_{\lambda,\lambda^{\prime}}^{\Lambda}$	$\displaystyle=\int_{0}^{1}dx\int\frac{d\boldsymbol{k}_{T}^{2}}{2(2\pi)^{3}}|\Phi^{\Lambda}_{\lambda,\lambda^{\prime}}(x,\boldsymbol{k_{T}})|^{2}.$		(13)

The HF refers to higher Fock-states. Our result is listed in Table. 1. The $N_{HF}$ is obtained by subtracting unity with the leading Fock-state contribution. As the DS-BSEs incorporate many higher Fock-states by summing up infinitely many Feynman diagrams, one can see the higher Fock-states contribute considerably to $\rho$ as compared to $J/\psi$. Combining that our $\rho$ LFWFs start with a small current quark mass $m_{f}=5$ MeV, they hence direct to the parton nature of light quarks inside $\rho$.