Exotic states and their properties from large Qcd
Abstract
The analysis of twoordinarymeson scattering amplitudes in the limit of a large number, of the colour degrees of freedom of quantum chromodynamics, with suitably decreasing strong coupling and all quarks transforming according to the gauge group’s fundamental representation, enables us to establish a set of rigorous consistency conditions for the emergence of a tetraquark (i.e., a bound state of two quarks and two antiquarks) as a pole in these amplitudes. For genuinely flavourexotic tetraquarks, these constraints require the existence of two tetraquark states distinguishable by their preferred couplings to two ordinary mesons, whereas, for cryptoexotic tetraquarks, our constraints may be satisfied by a single tetraquark state, which then, however, may mix with ordinary mesons. For elucidation of the tetraquark features, the consideration of the subleading contributions proves to be mandatory: for both variants of tetraquarks, their decay widths fall off like for large
Exotic states and their properties from large QCD
Hagop Sazdjian
Institut de Physique Nucléaire, CNRSIN2P3, Université ParisSud, Université ParisSaclay, 91406 Orsay Cedex, France
Email: sazdjian@ipno.in2p3.fr
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1 Incentive: Implications of the Large Limit of QCD on Polyquark Bound States
Tetraquark mesons are exotic bound states of two quarks and two antiquarks hypothesized to be predicted by quantum chromodynamics, the quantum field theory governing the strong interactions. We extract information on general features of tetraquarks by considering fourpoint Green functions of bilinear quark currents serving as interpolating operators of mesons composed of antiquark and quark , , where represent the flavour quantum numbers of the quarks and the generalized Dirac matrix fits to the interpolated meson’s parity and spin quantum numbers, within that generalization of QCD given by the limit of a large number of colour degrees of freedom, that is, by a quantum field theory based on the gauge group , with fermions transforming according to the dimensional fundamental^{1}^{1}1For the sake of simplicity, in particular, in order to deal with a unique limit, let us disregard the other logical possibility of fermions transforming according to the dimensional antisymmetric representation of . representation of [1]; the strong finestructure coupling is assumed to decrease, for , like .
Systematic application of the limit puts us in a position to discriminate unambiguously two classes of hadrons according to their large behaviour: those that survive the large limit as stable bound states and hence may be dubbed as “ordinary”, and the others, i.e., those that do not but disappear for [2]. However, although tetraquarks can only appear at an subleading order [3], in order to enable observability by experiment their decay widths should not grow with [4]. It is straightforward to prove that at large the meson decay constants rise like [1].
2 Analysis of Tetraquark Poles in Meson–Meson Scattering Amplitudes at Large
For welldefiniteness, let’s base this large QCD study on a variety of plausible assumptions:

For the analysis of tetraquarks, the large limit makes sense and works well; the application of the expansion to tetraquarks is justified and allows us to arrive at reliable conclusions.

In the large limit, poles interpretable as tetraquark bound states exist in the complex plane.

In the series expansions in powers of of those point Green functions which potentially accommodate tetraquark poles, the tetraquark states arise at the lowest possible order.

The masses, , of the tetraquark states, , do not grow with but remain finite for .
Before embarking on elucidating the dynamics of the formation of a tetraquark bound state, the main issue is to single out, in the expansion of a fourpoint Green function in powers of and , those (large) Feynman diagrams that might develop tetraquark poles. To this end, we impose, for tetraquarks supposedly consisting of (anti)quarks of masses , , and and, with respect to the channel, incoming external momenta and the following set of basic selection criteria [5]:

A tetraquarkphile Feynman diagram depends nonpolynomially on its variable .

A tetraquarkphile Feynman diagram supports appropriate fourquark intermediate states and exhibits the corresponding branch cuts starting at the branch points .
Only Feynman diagrams complying with both criteria can contribute to the physical tetraquark pole.
Under these premises, we derive, for various classes of tetraquarks , the large behaviour of • the tetraquarkphile fourpoint Green functions, identified by a subscript , • the amplitudes for transitions between tetraquark and two ordinary mesons, and • the tetraquark decay rate [5]:

For genuinely exotic tetraquarks , involving four different quark flavours, the correlators without (Fig. 1) and with (Fig. 2) a flavour reshuffle behave differently at large :
This observation forces us to conclude that there exist, at least, two different tetraquark states, called and , each with a preferred twomeson decay channel, but with parametrically the same decay rate of order . Phrased in other words, “always two there are, … no less” [6]:
The tetraquarks and may mix, with mixing parameter decreasing at least as fast as .

For cryptoexotic tetraquarks , with quark flavour of the ordinary mesons , the correlators without (Fig. 3) and with (Fig. 4) a flavour reshuffle have similar behaviour:
The implied constraints may be solved by a single tetraquark state decaying according to
and mixing with ordinary mesons with mixing strength dropping not slower than .
3 Summary: Insights on Minimum Numbers and Decay Rates of Tetraquark Types
(Crypto) exotic tetraquarks are narrow: their decay widths vanish in the limit . Unlike earlier claims [7], they have widths of order . If exotic, they come in two versions, with dependent branching ratios. Our results [5] generalize ones got for special cases or channels [8].
Acknowledgement. D.M. was supported by the Austrian Science Fund (FWF), project P29028N27.
References
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 [6] Yoda, JM, in Star Wars — Episode I: The Phantom Menace (1999, in a galaxy far, far away), scene 197.
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