Kaon Boer-Mulders function using a contact interaction

Abstract

Using a symmetry preserving treatment of a vector*vector contact interaction (SCI), results are delivered for the four kaon transverse momentum dependent parton distribution functions (TMDs), viz. helicity-independent (HI) and Boer-Mulders (BM) TMDs for the kaon's $u$, $s$ valence degrees of freedom. In completing this analysis, we are able to deliver insights into, amongst other things, the role played by emergent hadron mass (EHM) phenomena in producing these TMDs; the EHM modulating effect of the Higgs-boson coupling that produces the strange quark current mass; the impact of gauge link models on whether predictions satisfy the positivity constraint that bounds the BM function relative to the HI TMD; and the size of the BM shift and effects thereupon of off-diagonal terms in the associated scale-evolution kernel.

Figures (7)

• Figure 1: Number density interpretation of the Boer-Mulders function. Legend. Vertical blue arrows -- transverse polarisation of the quark, $\vec{S}$; oblique brown vectors -- quark transverse momentum vector, $\vec{k}_\perp$.
• Figure 2: In order to obtain a nonzero Boer-Mulders function, one must require, at least, that the (valence) dof involved in the forward scattering event subsequently/initially interacts with the spectator via (multiple) gluon exchanges. Legend. Double line -- leading eikonal approximation to quark propagation under the influence of the gauge link, $1/n\cdot q$; small green circle -- eikonal quark-gluon vertex, $[-g n_\mu]$, where $g$ is the strong coupling; large purple circle -- kaon (pseudoscalar meson) Bethe-Salpeter amplitude, $\Gamma_K(k_{\pm q,0};P)$, with $k_{\pm q}=k\pm q-P_K/2$, $k_0=k-P_K/2$; thin solid line -- valence quark propagators, $S(k)$; spring-like line -- gluon propagator, $D_{\mu\nu}(q)$; and dashed lines -- incoming/outgoing mesons. The relative negative sign between the two diagrams expresses the sign-change between initial- and final-state eikonal-quark interactions.
• Figure 3: Panel A. Hadron scale SCI result for the helicity-independent $u$-in-$K$ TMD drawn from Eq. \ref{['uinKTMD']}. The $s$-in-$K$ TMD is obtained via Eq. \ref{['TMDSymmetry']}. Panel B. Analogous $u$-in-$\pi$ TMD.
• Figure 4: Panel A. Hadron scale SCI result for the $u$-in-$K$ BM function drawn from Eqs. \ref{['KaonMDBMFFX']}, \ref{['alphaL']}. The $s$-in-$K$ BM function is obtained via Eq. \ref{['BMSymmetry']}. Panel B. Analogous $u$-in-$\pi$ BM function.
• Figure 5: Panel A. $(x=1/2,k_\perp^2)$-dependence of kaon BM functions: solid purple curve -- Eq. \ref{['KaonMDBMFFX']}, momentum-dependent gauge link completion; and dashed blue curve -- Eq. \ref{['KaonMIBMFFX']}, pure-SCI result. Panel B. Checking the positivity bound, Eq. \ref{['EqPositive']}: the bound is violated by any result that crosses the horizontal dotted line. Legend: green curves, SCI pion results -- dotted, momentum-independent gluon; dot-dashed, momentum-dependent gauge link completion; otherwise as in Panel A.