Narrow-width tetraquarks in large-$N_{\mathrm{c}}^{}$ QCD

The properties of possibly existing tetraquarks are studied in the large-$N_{\mathrm{c}}^{}$ limit of QCD by means of four-point correlation functions of meson currents.The necessity of a detailed analysis of the singularities of Feynman diagrams, by means of the Landau equations, to recognize those diagrams that might contribute to the formation of tetraquark states, is emphasized. It is found, in general, that tetraquarks, if they exist, should have narrow widths of the order of $N_{\mathrm{c}}^{-2}$.


Line of approach
We study the properties of exotic and cryptoexotic tetraquarks through the analysis of meson-meson scattering amplitudes. Exotic tetraquarks are defined as containing four different quark flavours. Cryptoexotic tetraquarks contain three different quark flavours or less.
We consider four-point correlation functions of colour-singlet quark bilinears, having a coupling with a meson M ab , made of an antiquark a and quark b: where the large-N c behaviour of the coupling constant has also been outlined [2]. Spin and parity are not indicated in the above formulas and will be ignored in the subsequent analyses, since they are not relevant for the qualitative aspects of the problem. We consider all possible s-channels where a tetraquark may be present. To be sure that a QCD Feynman diagram may contain, through a pole term, a tetraquark contribution, one has to check that it receives a four-quark (more precisely, two-quark and two-antiquark) contribution in its s-channel singularities, plus additional gluon singularities that do not modify the N c -behaviour of the diagram.
If the tetraquark contains quarks and antiquarks with masses m j , j = a, b, c, d, then the diagram should have a four-particle cut starting at s = (m a + m b + m c + m d ) 2 . Its existence is checked with the use of the Landau equations [8,9].
Diagrams that do not have s-channel singularities, or have only two-particle singularities (quarkantiquark), cannot contribute to the formation of tetraquarks at their N c -leading order. They should not be taken into account for the N c -behaviour analysis of the tetraquark properties.
An account of the present work can be found in [10].

Exotic tetraquarks
We consider the case of four distinct quark flavours, denoted 1,2,3,4, with meson currents and the corresponding scattering processes: called "direct channel I", "direct channel II" and "recombination channel", respectively. In the case of "direct" channels, the corresponding four-point correlation functions are The leading and subleading Feynman diagrams for Γ (dir) I are represented in Fig. 1. Similar diagrams also exist for Γ (dir) II . It is understood that to each diagram there corresponds an infinite number of diagrams with insertions of gluon exchanges not changing the topology of the diagram and having the same N c -behaviour. It is the sum of such diagrams that may create singularities at the hadronic level, such as meson and/or tetraquark poles or two-meson cuts. Figure 1. Leading and subleading diagrams in the direct channel I of (7). Full lines represent quarks, curly lines gluons.
The leading behaviour of the direct-channel correlator functions is O(N 2 c ), while that of the subleading diagrams is O(N 0 c ). However, the leading diagrams (a) of Fig. 1 are disconnected and describe the propagation of two free ordinary mesons. It is only diagrams of the type (b) that may contribute to the scattering amplitude. On the other hand, analyzing, with the aid of the Landau equations, the structure of the singularities of diagrams (b) of Fig. 1, one finds that they have s-channel fourquark singularities, indicating that they may participate in the formation of tetraquark poles. One then deduces the behaviour of that part of the scattering amplitude that may come from a tetraquark intermediate state: For the "recombination" channel, the four-point correlation function is The leading and subleading diagrams are represented in Fig. 2. In spite of appearences, diagrams (a) and (b) of Fig. 2 do not have s-channel singularities. Their singularities appear in the u-and t-channels and correspond there to one-meson intermediate-state contributions. Therefore, they cannot contribute to the formation of tetraquark states. Only diagram (c) has s-channel (four-quark) singularities and thus may receive contributions from tetraquark states. The contribution of a tetraquark to the correlation function is then The fact that the direct and recombination amplitudes have different behaviours in N c [Eqs. (8) and (10)] , implies that two different tetraquarks, T A and T B , each having different couplings to the meson pairs, are needed to accommodate both types of behaviour.
Factorizing in the correlation functions the external meson propagators and the related couplings with the currents [Eqs. (1) and (2)], one obtains for the tetraquark-two-meson transition amplitudes the following behaviours: The total widths of the tetraquarks are The meson-meson scattering amplitudes at the tetraquark poles (leading contributions) are represented in Fig. 3. Figure 3. Couplings of the tetraquarks to two-meson states in meson-meson scattering.

Cryptoexotic tetraquarks
We now consider the case of three distinct quark flavours, denoted 1,2,3, with meson currents The following scattering processes are considered: M 12 + M 23 → M 13 + M 22 , recombination channel.
The direct channel four-point functions are The leading and subleading diagrams of Γ (dir) II are similar to those of the exotic case [Eq. (7)] and are represented in Fig. 5. Diagram (b) may receive contributions from tetraquark intermediate states. Figure 5. Leading and subleading diagrams of the direct channel II of (18).
One deduces the related contribution to the corresponding correlation function: The recombination-channel four-point correlation function is Its leading and subleading diagrams are represented in Fig. 6. As in the exotic case, diagrams (a) and (b) do not have s-channel singularities and cannot contribute to the formation of tetraquark poles. Diagrams (c) and (d) do have s-channel four-quark singularities and thus may receive contributions from tetraquark intermediate states. One then deduces the related contribution to the correlator function: Figure 6. Leading and subleading diagrams of the recombination channel of (21).
In the present case, direct and recombination diagrams have the same N c -behaviour. A single tetraquark T may accommodate all channels. The tetraquark-two-meson transition amplitudes are The total width of the tetraquark is The meson-meson scattering amplitudes at the tetraquark pole are represented in Fig. 7. Possible mixings of tetraquarks with one-meson states, when allowed by the existing quantum numbers, are found to be of the order of N −1/2 c and do not alter, at leading order, the results previously found.
The case of cryptoexotic channels with two quark flavours can be treated in a similar way as for three. One finds here additional diagrams to those in the case of three flavours; they do not modify, however, the main qualitative features of the tetraquarks obtained above.
In the case of three distinct quark flavours, denoted 1,2,3, one may also have the situation where the quark flavour 2, say, appears in two quark fields, rather than in a quark and an antiquark field. The meson currents are now j 12 = q 1 q 2 , j 32 = q 3 q 2 .
The following scattering process is then considered: Here, the direct and recombination channels are identical. The corresponding four-point correlation function is The leading and subleading diagrams are represented in Fig. 8. Only diagram (c) contains s-channel four-quark singularities and may receive contributions from tetraquark intermediate states. The tetraquark-two-meson transition amplitude and the tetraquark total width have, respectively, the following behaviours: The meson-meson scattering amplitude at the tetraquark pole is represented in Fig. 9.  Figure 9. Coupling of the tetraquark to two-meson states in meson-meson scattering.

Conclusion
The analysis of the s-channel singularities of Feynman diagrams is crucial for the detection of the possible presence of tetraquark intermediate states in correlation functions of meson currents. If, due to the operating confining forces, tetraquarks exist as stable bound states of two quarks and two antiquarks in the large-N c limit, with finite masses, then they should have narrow decay widths, of the order of N −2 c , much smaller than those of the ordinary mesons, which are of order N −1 c . For the fully exotic channel, with four different quark flavours, two different tetraquarks are needed to accommodate the theoretical constraints of the large-N c limit. In this case, each tetraquark has one predominant decay channel.