Tetraquarks in the $1/N$ expansion: A new appraisal

We discuss the necessary, albeit not sufficient, conditions for tetraquark poles to occur in the $1/N$ expansion of QCD and find the minimum order at which such poles may appear. Assuming tetraquark poles, we find a new nonplanar solution with the minimal number of topologies and tetraquark species. The solution implies narrow states. Mixing with quarkonium states is allowed so that $P$-wave tetraquarks with $J^{\mathrm{P}\mathrm{C}}=1^{--}$ would couple to $e^{+}{e}^{-}$.

Figure 1

(a) Meson-meson scattering with a four-quark intermediate state [25]. (b) As in (a) but with the addition of an internal quark loop (see also Fig. 4). The handle is a necessary condition for the existence of genuine tetraquark cuts. Gray shaded surfaces indicate the full set of planar gluon interactions.

Figure 2

Meson-meson scattering with four-quarks intermediate states in [26]. (a) is topologically equivalent to Fig. 1 but (b) has no handles. Quark bilinear sources in (b) are inserted on both external and internal quark loops.

Figure 3

(c) A handle is added to the diagram in Fig. 2, changing the order in the $1/N$ expansion. In this way, we recover the diagram in Fig. 1, with a different insertions of the quark bilinear sources.

Figure 4

In (a) and (b) two equivalent drawings of diagram in Fig. 1 with the addition of one fermion loop.

Figure 5

(i) Fig 2 after the addition of one gluon next to the bilinear insertion in the internal loop. (ii) Planar graph equivalent to (i). (iii) Higher orders in $g$.

Figure 6

With quark bilinears all inserted on the external fermion loop, diagram (a) represents another contribution to meson-meson scattering, in addition to diagram in Fig. 3. The $s$-channel cuts correspond to the figure on the right and determine the mixing $q\overline{q}$-tetraquark. Diagram (b): the $s$-channel cuts, as indicated by the figure on the right, determine the product $fg$, in agreement with Eqs. (14) and (15).

Figure 7

Diagram to describe the transition of a tetraquark into a lower tetraquark with pion emission, e.g., $Y\to Z^{+}{\pi }^{-}$; see Eq. (16).