Symmetry-preserving calculation of pion light-front wave functions

Abstract

Poincaré-covariant Bethe-Salpeter wave functions are used to calculate light-front wave functions (LFWFs) of the pion, $π$, and an analogue state, $π_{s\bar s}$. The current masses of the degenerate valence constituents in the $π_{s\bar s}$ are around $25$-times larger than those of the pion's valence constituents. Both valence spin-antialigned ($\mathcal L=0$) and valence spin-aligned ($\mathcal L=1$) components are obtained and combined to produce the complete LFWF for each system. Comparing predictions delivered by two distinct Bethe-Salpeter kernels, the impact of nonperturbative dynamical effects contained in the more sophisticated (bRL) kernel are seen to be significant; and contrasts between $π$, $π_{s \bar s}$ results reveal the interplay between emergent hadron mass and mass effects owing to Higgs-boson couplings. Amongst the results, one finds that for $π$, $π_{s\bar s}$, the LFWFs can be approximated by a separable form, with that representation being pointwise reliable in the bRL cases. Moreover, the $\mathcal L=1$ component is important; so a LFWF obtained after omission of this piece is typically a poor representation of the system. These features are naturally expressed in $π$, $π_{s\bar s}$ transverse momentum dependent parton distribution functions (TMDs). In this connection, it is found that a Gaussian \textit{Ansatz} can only provide a rough guide to TMD pointwise behaviour: magnitude deviations between \textit{Ansatz} and prediction exceed a factor of two on $k_\perp^2 \gtrsim 0.55\,$GeV$^2$. One should therefore be cautious in interpreting conclusions drawn from phenomenological analyses based upon Gaussian \textit{Ansätze}.

Figures (5)

• Figure 1: Mellin moments, $m=0,\ldots, 5$, of $\pi$ LFWF, defined by Eq. \ref{['PionMMs']}: the magnitudes decrease with increasing $m$. Panel A. RL, spins antialigned, $\langle x^0 \rangle_\pi^{\uparrow\downarrow}(0)=55.9/\Lambda_I$. Panel B. RL, spins aligned, $\langle x^0 \rangle_\pi^{\uparrow\uparrow}(0)=(9.61/\Lambda_I)^2$. Panel C. bRL, spins antialigned, $\langle x^0 \rangle_\pi^{\uparrow\downarrow}(0)=72.5/\Lambda_I$. Panel D. bRL, spins aligned. $\langle x^0 \rangle_\pi^{\uparrow\uparrow}(0)=(12.8/\Lambda_I)^2$. (Recall $\Lambda_I=1\,$GeV, Eq. \ref{['defcalG']}.)
• Figure 2: Ratios of LFWF Mellin moments, Eq. \ref{['eq:ratios']}. In each panel, the solid lines mark the $k_\perp^2$-independent $m$ moment of the DA in Eq. \ref{['DefineDA']}. Panel A. RL, spins antialigned. Panel B. RL, spins aligned. Panel C. bRL, spins antialigned. Panel D. bRL, spins aligned.
• Figure 3: LFWFs reconstructed from their Mellin moments, expressed by Eq. \ref{['RealSep']} and central values of the coefficients in Table \ref{['tab:paramspi']}. Each curve is normalised by its peak size, determined by the associated values of $\rho$, $a_{0}^{{\mathpzc L}}+c_{0}^{{\mathpzc L}}$ in Table \ref{['tab:paramspi']}. Panel A. $\pi$ RL, spins antialigned. Panel B. $\pi$ RL, spins aligned. Panel C. $\pi$ bRL, spins antialigned. Panel D. $\pi$ bRL, spins aligned. Panel E. $\pi_{s\bar{s}}$ RL, spins antialigned. Panel F. $\pi_{s\bar{s}}$ RL, spins aligned. Panel G. $\pi_{s\bar{s}}$ bRL, spins antialigned. Panel H. $\pi_{s\bar{s}}$ bRL, spins aligned.
• Figure 4: LFWFs for the $\pi$ and $\pi_{s\bar{s}}$. Panel A -- RL truncation; and Panel B -- bRL truncation. Legend, both panels: solid purple -- $\psi_\pi^0$ and dashed purple -- $\psi_{\pi_{s\bar{s}}}^0$; and dot-dashed green -- $|k_\perp|\psi_\pi^1$ and dotted green -- $|k_\perp|\psi_{\pi_{s\bar{s}}}^1$. Further, in each panel, the LFWFs are normalised by the appropriate value of $\psi^0(1/2,0)$, so the results are dimensionless and the $k_\perp^2$-dependence is directly comparable. This is the meaning of the subscript $n$ on the ordinate labels.
• Figure 5: Helicity-independent TMDs obtained from LFWFs in Fig. \ref{['fig:LFWF']}. In each case, normalisation guarantees Eq. \ref{['EqBNcons']}. Panel A. $\pi$ RL. Panel B. $\pi$ bRL. Panel C. $\pi_{s \bar{s}}$ RL. Panel D. $\pi_{s \bar{s}}$ bRL.