Pseudoscalar meson decay constants and distribution amplitudes up to the twist-4 in the light-front quark model

In the light-front quark model (LFQM) amenable to the simultaneous study of both the mass spectroscopy and the wave function related observables, we examine the decay constants and distribution amplitudes (DAs) up to the twist-4. The analysis of the heavy pseudoscalar mesons is carried out both in the $1S$ and $2S$ states. This investigation involves calculating the local and nonlocal matrix elements $⟨0|\overline{q}\mathrm{\Gamma }q|P⟩$ using three distinct current operators $\mathrm{\Gamma }=({\gamma }^{\mu }{\gamma }_5,i{\gamma }_5,{\sigma }^{\mu \nu }{\gamma }_5)$. Considering a general reference frame where ${\mathbf{P}}_{\perp }\ne 0$ and investigating all available current components, we examine not only the frame independence but also the component independence of the decay constants. The explicit findings from our study provide the evidence for the equality of the three pseudoscalar meson decay constants obtained from the three distinct current operators $\mathrm{\Gamma }$. The notable agreement in decay constants is attained by imposing the Bakamjian-Thomas construction of the LFQM, namely, the meson state is constructed by the noninteracting quark and antiquark representations while the interaction is added to the mass operator, which provides the self-consistency condition replacing the physical mass $M$ with the invariant mass $M_0$ for the noninteracting quark-antiquark representation of the meson state. In addition to obtaining the process-independent pseudoscalar meson decay constant, regardless of the choice of current operators $\mathrm{\Gamma }$, we further demonstrate its explicit Lorentz and rotation invariance. In particular, we present the analysis conducted on the twist-4 DA derived from the minus component of the axial-vector current. Finally, we discuss the various twist DAs and their $\xi$-moments associated with the $1S$ and $2S$ heavy pseudoscalar mesons.

Figure 1

Two-dimensional plot of LFWF of (a) $D$ and (b) ${\eta }_c$ mesons for each helicity configuration. Note that the longitudinal momentum $x$ is carried by the light quark.

Figure 2

Helicity contributions to decay constant for 1S and 2S heavy pseudoscalar mesons. The pattern histogram represents subtracted contribution. Since the contribution from $H_{\uparrow \downarrow }=H_{\downarrow \uparrow }$, we sum them up for simplicity. It also applies to $H_{\uparrow \uparrow }=H_{\downarrow \downarrow }$. Here we set ${\mathbf{P}}_{\perp }=0$ for the case of ${\gamma }^{-}{\gamma }_5$. For the ${\sigma }^{\mu \nu }{\gamma }_5$ case, the helicity contribution depends on the choice of assigning $x$ to light or heavy quark, see Sec. 5c for details.

Figure 3

The $P_{\perp }$ dependence of the helicity contribution of ${\gamma }^{-}{\gamma }_5$ for 1S and 2S pseudoscalar heavy mesons with $|{\mathbf{P}}_{\perp }|=P_{\perp }$. The solid line represents the contributions from $H_{\uparrow \downarrow }+H_{\downarrow \uparrow }$, while the dashed line represents $H_{\uparrow \uparrow }+H_{\downarrow \downarrow }$.

Figure 4

The 3D plots of wave functions for $D(1S)$ mesons defined by $f_A=\int {\mathrm{d}}^3\overrightarrow{k}{\psi }^{\mu }(\overrightarrow{k})$: (a) ${\psi }^{\mu }(\overrightarrow{k})$ with various current component, (b) ${\psi }^{\mu }(\overrightarrow{k})$ without the $k_z$ term, and (c) $\tilde{\psi }(\overrightarrow{k})={\psi }^{+,\perp }(\overrightarrow{k})-{\psi }^{-}(\overrightarrow{k})$. Displayed in (d)–(f) are those of $D(2S)$ mesons.

Figure 5

Two-particle DAs with various twists for 1S and 2S heavy pseudoscalar mesons with various quark flavor contents where the longitudinal momentum $x$ is carried by the light quark.

Figure 6

Two-dimensional plot of the DAs of $D(2S)$ for various twist. Here we define ${\mathbf{k}}_{\perp }^2=y/(1-y)$ to make a rectangular domain. The longitudinal momentum $x$ is carried by the light quark.

Figure 7

Helicity contribution to DAs with various twist. Although the total DAs of twist-2, twist-3, and twist-4 are the same, they consist of different helicity contributions. We note that the total DA of twist-3 with ${\sigma }^{+-}{\gamma }_5$ is different from the others.

Figure 8

The difference in the helicity contribution depending on the choice of assigning the LF longitudinal momentum fraction $x$ to the light or heavy quark. The dashed and dotted lines represent the ordinary and higher helicity, respectively.

Figure 9

Comparison of ${\psi }_{3;P}(x)$ with and without the inclusion of $(1-{\rho }_{+})$ factor.