Tetraquarks, pentaquarks and dibaryons in the large N QCD
Abstract
We study the multiquark hadrons in large N QCD under the ’tHooft limit, extending Witten’s picture of the baryons. We explore the decay widths of tetraquarks, pentaquarks and dibaryons. Based on the decay behaviors, we point out that in the \(N\rightarrow \infty \) limit decay widths of tetraquarks stay constant, while those of pentaquarks and dibaryons above certain thresholds can diverge. In the large N limit, we find that the ground states of the three spectroscopic series are stable or narrow and that the excited states of pentaquarks and dibaryons above the indicated thresholds are not observables. We compare our results with those obtained in previous large N generalizations of tetraquarks.
1 Introduction
In a world of two colours, the above structures disappear: \(N=2\) QCD is made only of mesons, \(q{{\bar{q}}}\), “baryons”, qq, and molecules thereof. The new spectroscopic series start to appear at \(N=3\), (our world!) and can be extended to N colours.
As we will show, we find that decay amplitudes from the ground states may diverge at large N. However such decays are generally forbidden by phase space and the divergent amplitudes do not affect the observability of such particles. At \(N=\infty \), ground states of multiquark hadrons are narrow or stable, particularly in the case of the dibaryon.
Results for the decay amplitudes of excited states are summarised in Table 1. For tetraquarks, we find decay amplitudes of the excited states that stay constant or decrease with N, thus confirming that the corresponding hadrons are observables. For excited pentaquarks and dibaryons, the deexcitation amplitudes into the ground state plus one meson are limited. However, at \(N=\infty \), there are modes which have divergent amplitudes, namely \(P^* \rightarrow B+B+{{\bar{B}}}\), \(D^*\rightarrow NB+{{\bar{B}}}\) and \(D^*\rightarrow (N1)B\). Taken literally, these results would imply sharp thresholds at \(2B+{{\bar{B}}}\) and \((N1)B\) respectively, below which we expect observable pentaquarks and dibaryons, and above which we expect large, unobservable widths: a situation similar with charmonia above and below the open charm mesonantimeson threshold.
Decay amplitudes for the decay of multiquark hadrons from states with only one quark excited, \(A^*\). Amplitudes are normalised so that \(A^*^2=\Gamma \), thus including a \(1/\sqrt{N!}\) factor from the phase space of \({{{\mathcal {O}}}}(N)\) indistinguishable particles, see text. Entries in the Table give the leading N dependence. The first two columns refer to decays obtained from Fig. 1 by cutting one or more QCD strings with a \(q{{\bar{q}}}\) pair, the third column to the decay of an excited into the ground state by one meson emission. For multimeson or multibaryon decay amplitudes, polynomial in N appearing as prefactors of the exponentials are omitted. When giving the results for dibaryons, the diquarks inside are treated, at large N, as quasi classical particles
\(T^*\rightarrow \)  \(B+ {{\bar{B}}}\)  \(T+\) Meson  Mesons  
\(A^*\propto \)  \(N^{0}\)  \(N^{0}\)  \( < e^{\frac{N}{2}}\)  
\(P^*\rightarrow \)  \(B+ T\)  \(B+B+{{\bar{B}}}\)  \(P+\) Meson  \(B+\) Mesons 
\(A^*\propto \)  \(N^ {0}\)  \(N^{1/2}\)  \(N^{0}\)  \( < e^{\frac{N}{2}}\) 
\(D^*\rightarrow \)  \(NB+{{\bar{B}}}\)  \(D+\) Meson  \((N1)B\)  
\(A^*\propto \)  \(> e^{+\frac{N}{2}\log N}\)  \(N^{0}\)  \(> e^{+\frac{N}{2}\log N}\) 
2 Mesonbaryon couplings
As suggested by Witten, baryons become very simple in the limit \(N\rightarrow \infty \): quarks move independently from each other in an effective, HartreeFock, potential which is N independent. In the ground state, all quarks are in the same wavefunction, \(\phi _0(x)\), with an N independent baryon radius.
3 Tetraquark decays
Consider in Fig. 3 the case where quark lines in the initial diquark correspond to \(B_1\). Then \(i_{N}=1\) and the pair created by the gluon interaction provides the missing \(q^1\) and \({{\bar{q}}}_1\) to the product \(B_1{{\bar{B}}}^1\). We need to add all diagrams where the gluon is emitted by the other quark lines, which gives a factor \(N1\). Finally, summing over a gives a factor of N, since all the terms of the sum are equal to the one with \(a=1\).
Adding the diagrams in which gluon emission occurs from an antiquark line gives a factor two to the final result.
Finally, as a way to check our method, we compute the amplitude for the decay of a tetraquark to a complex \(T_{2}=(N2)q+(N2){{\bar{q}}}\), using the diagram in Fig. 5 restricted to the exchange of one gluon.
4 Pentaquarks
5 Dibaryons
Suppose that in the dibaryon, diquarks are in single diquark wave functions \(w_0(x),w_1(x) \dots \). We use a different symbol for the wave functions, to distinguish the motion of diquarks, which are quasi classical particles, with energy levels that vanish at \(N=\infty \), from the motion of quarks in the diquark, which are fully quantum with energy level spacing of order \({{{\mathcal {O}}}}(\Lambda _{QCD})\).
The 3! in the numerator arises because each monomial, say \(B(x_1)^\dagger B(x_2)^\dagger B(x_3)^\dagger w_a(x_1)w_b(x_2)w_c(x_3)\), when contracted with the external baryons gives rise to the full Slater determinant \({{\tilde{F}}}(k_1,k_2,k_3~~a,b,c)\) with the appropriate sign.

a factor \(\sqrt{N1}\) for the gluon emitted by a quark in \(B_1\) with the transition \(w_r \rightarrow w_0\);

a factor \(\sqrt{N}\) for the meson;

a factor \(\lambda ^2/N\) for gluon exchange.
In these scenarios, the amplitude (58) is ineffective, the ground state decays weakly and the deexcitation amplitudes of lightly excited dibaryons go to a constant value at large N, Eq. (59).
6 Discussions
In summary, we have extended Witten’s description of baryons in large N QCD [19] to the multiquark hadrons generated by replacing one or more antiquarks in an antibaryon with the generalised diquark made by \(N1\) quarks in the antisymmetric colour combination. The first step reproduces the generalised tetraquark considered by Rossi and Veneziano [21, 22]. Successive substitutions produce the large N generalisation of multiquark configurations considered for \(N=3\): pentaquarks [26] and dibaryons [27, 28].
We have studied in the present paper the decays into conventional baryons and mesons of ground and lowlying excited states, namely states with a finite energy difference with respect to the ground state, in the limit \(N\rightarrow \infty \).
Decay amplitudes for the ground states generally diverge with N. However, we have argued that the final configurations with divergent amplitudes are forbidden by phase space. In this case, we would expect ground states with narrow widths, for tetra and pentaquarks where multimeson states are available, or decay by weak interactions, in the case of dibaryons where also N baryon states are expected to be phasespace forbidden.
The decay amplitudes of multiquark low lying excited states are reported in Table 1. The first two columns refers to decays obtained from Fig. 1 by cutting one or more QCD strings with a \(q{{\bar{q}}}\) pair, the third column to the decay of an excited into the ground state by meson emission. The last column refers to decays obtained by reorganising the quarkantiquark pairs of the initial state into a multimeson state or in redistribuiting the quarks of one diquark to the other diquarks, to form a set of \(N1\) baryons.

Excited tetraquarks The amplitudes for the decay of the excited states do vanish or remain constant for \(N\rightarrow \infty \), therefore leading to observable states in this limit;

Tetraquark deexcitation amplitudes and \(B{{\bar{B}}}\) decay amplitudes are of the same order;

for \(N=3\) and flavour composition \([cu][{{\bar{c}}}{{\bar{u}}}]\) the threshold for twobaryon decay is \(2M(\Lambda _c)\sim 4570\) MeV; in Ref. [35] it is argued that X(4660) is a Pwave tetraquark decaying predominantly into \(\Lambda _c {{\bar{\Lambda }}}_c\) in addition to the mode into \(\psi (2S) \pi \pi \) [36].

Tetraquarkcharmonium mixing is exponentially suppressed;

excited pentaquarks and dibaryons: deexcitation amplitudes into the ground state and a meson remain limited for large N;

at \(N=\infty \) there are modes which give divergent amplitudes, namely \(P^* \rightarrow B+B+{{\bar{B}}}\) and \(D^*\rightarrow NB+{{\bar{B}}}\) or \((N1)B\);

taken literally, these results, imply sharp thresholds at \(2B+{{\bar{B}}}\) and \((N1)B\) respectively, below which we expect observable pentaquarks and dibaryons, and above which we expect large, unobservable widths: a situation similar to charmonia above and below the open charmanticharm meson threshold.

For \(N=3\) and pentaquark with flavour composition: \([cu][ud]{{\bar{c}}}\), corresponding to the states observed by LHCb [25], the threshold for “nonobservabilty” would be \(2M(\Lambda _c) + M(P)\sim ~5510\) MeV, for a double charmed dibaryon with flavour [cu][cd][ud] the threshold would be at: \(2M(\Lambda _c)\).
We have added the decay into \(B+{{\bar{B}}}\) which brings in a divergent behaviour for the ground state. The divergence at large N is not relevant for the width and the observability of the ground state, which is below threshold for the decay, but it makes it dominant as intermediate state in elastic \(B+ {{\bar{B}}}\) scattering, see Eq. (24). The 1 / N behaviour we find for the latter amplitude is in agreement with the result given in [21, 22].
Also novel is the result about the deexcitation of a tetraquark to the ground state plus a meson, expected to have a constant limit for \(N\rightarrow \infty \) and to be phenomenologically important.

a suppressed decay amplitude of the ground state into two mesons, of order \(N^{2}\); for large N this is larger that the exponentially suppressed amplitude in Table 1, but it takes \(N\ge 6\) for the power suppression to win over the exponential suppression (see [37] for a further suppression of two meson decay due to the potential barrier opposing \(q{{\bar{q}}}\) annihilation in the diquarkantidiquark picture);

amplitude of order \(N^{1/2}\) for the deexcitation into the ground state by meson emission;

tetraquarkcharmonium mixing occurs to order \(N^{3/2}\);

the decay of an excited tetraquark into \(B{{\bar{B}}}\) cannot be computed.
Footnotes
Notes
Acknowledgements
We thank A. Polosa for many interesting discussions, G. C. Rossi and G. Veneziano for a useful exchange on tetraquak decays, C. Liu as well as T. Cohen, F. J. LlanesEstrada, J. R. Pelaez and J. Ruiz de Elvira for useful correspondence on their works. V. R. thanks Prof. Xiangdong Ji for hospitality at the T. D. Lee institute, where this work has been done. This work is supported in part by National Natural Science Foundation of China under Grant Nos. 11575110, 11735010, 11747611, Natural Science Foundation of Shanghai under Grant No. 15DZ2272100.
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