Multimeson model for the $D^{+}\to K^{+}{K}^{-}{K}^{+}$ decay amplitude

We propose an approach to describe the $D^{+}\to K^{-}{K}^{+}{K}^{+}$ decay amplitude, based on chiral effective Lagrangians, which can be used to extract information about $K\overline{K}$ scattering. It relies on factorization and its main novel feature is the role played by multimeson interactions characteristic of chiral symmetry. Our trial function is an alternative to the widely used isobar model and includes both nonresonant three-body interactions and two-body rescattering amplitudes, based on coupled channels and resonances, for S- and P-waves with isospin 0 and 1. The latter are unitarized in the $K$-matrix approximation and represent the only source of complex phases in the problem. The nonresonant component, given by chiral symmetry as a real polynomium, is an important prediction of the model, which goes beyond the ($2+1$) approximation. Our approach allows one to disentangle the two-body scalar contributions with different isospins, associated with the $f_0(980)$ and $a_0(980)$ channels. We show how the $K\overline{K}$ amplitude can be obtained from the decay $D^{+}\to K^{-}{K}^{+}{K}^{+}$ and discuss extensions to other three-body final states.

Figure 1

Amplitude $T$ for $D^{+}\to K^{-}{K}^{+}{K}^{+}$: (a) primary weak vertex; (b) weak vertex dressed by final state interactions; the full line is the $D$, dashed lines are pseudoscalars.

Figure 2

Competing topologies for the process $D^{+}\to P_a{P}_b{K}_3^{+}$; the solid green lines are the $D$ and dashed lines are $SU(3)$ pseudoscalars and the wavy line is the explicit W-boson; in (a) the $K_3^{+}$ is produced in a multimeson process, and in (b) in isolation.

Figure 3

Quark content of topologies (a) and (b) in Fig. 2.

Figure 4

Decay amplitude for $D^{+}\to K^{-}{K}^{+}{K}^{+}$; the weak vertex proceeds through the intermediate steps $D^{+}\to W^{+}$ and $W^{+}\to K^{-}{K}^{+}{K}^{+}$ and strong final state interactions are encompassed by the scattering amplitude $A$ (full red blob).

Figure 5

Dynamical structure of triangle vertices in Fig. 4; the wavy line is the $W^{+}$, dashed lines are mesons, continuous lines are resonances and the full red blob represents meson-meson scattering amplitudes described in Fig. 6; all diagrams within square brackets should be symmetrized by making $2\leftrightarrow 3$.

Figure 6

(a) Tree-level two-body interaction kernel ${\mathcal{K}}_{ab\to cd}^{(J,I)}$, a NLO $s$-channel resonance added to a LO contact term. (b) Structure of the unitarized scattering amplitude.

Figure 7

$S$-wave sector—top left: the continuous black curve (SW) is the modulus of the decay amplitude $T^S$, Eq. (34), in arbitrary units, whereas other curves are moduli of partial contributions; top right: moduli of the $K\overline{K}$ scattering amplitudes $A^{(0,1)}$, red curve, and $A^{(0,0)}$, blue curve; bottom: the continuous black curve (SW) is the phase of the decay amplitude $T^S$, Eq. (34), and other continuous curves are phases of partial contributions; the dashed curves represent the phases of the $K\overline{K}$ scattering amplitudes $A^{(0,1)}$ (red) and $A^{(0,0)}$ (blue).

Figure 8

$P$-wave sector—top left: the continuous black curve (PW) is the modulus of the decay amplitude $T^P$, Eq. (35), in arbitrary units, whereas other curves are moduli of partial contributions; top right: moduli of the $K\overline{K}$ scattering amplitudes $A^{(1,1)}$, red curve, and $A^{(1,0)}$, blue curve; bottom: the continuous black curve (PW) is the phase of the decay amplitude $T^P$, Eq. (35), and other continuous curves are phases of partial contributions; the dashed curves represent the phases of the $K\overline{K}$ scattering amplitudes $A^{(1,1)}$ (red) and $A^{(1,0)}$ (blue).

Figure 9

Phase shifts $\delta$ and inelasticity parameter $\eta$ for $K\overline{K}$ scattering—top: $S$-waves; bottom: $P$-waves; blue and red curves correspond, respectively, to isospin $I=0$ and $I=1$.

Figure 10

Results for $|A^{(0,0)}|$: the kaon mass is artificially lowered to $M_K=0.4\mathrm{GeV}$ and the dynamics is implemented just by a single $f_0(980)$; the black vertical line indicates the actual $K\overline{K}$ threshold. Left: modulus; right: phase.

Figure 11

Results for $|A^{(0,0)}|$: piecemeal construction of the amplitude, following steps given in Table 1; the continuous blue line (A) corresponds to the tail of the $f_0(980)$; the dashed blue curve (B) includes the contact chiral term; the red continuous curve (C) represents the unitarized $f_0(980)$ and $f_0(1370)$ joint contributions; the dashed red curve accounts for the coupling to $\pi \pi$ intermediate states; the continuous black curve (Triple-M) includes coupling to $\eta \eta$ intermediate states; top: modulus; bottom: phases; the latter also includes conventional phase shifts ${\delta }^{(0,0)}$, indicated by the dotted curves.

Figure 12

Results for $|A^{(0,0)}|$ inelasticities; conventions are the same as in Fig. 11.

Figure 13

Toy Dalitz plot for Triple-M in $D^{+}\to K^{-}{K}^{+}{K}^{+}$ decay with arbitrary normalization.

Figure 14

Intermediate $\pi \rho$ contribution to the $\varphi$ self-energy.