A study of ccc̄c̄ tetraquark decays in 4 muons and in D(∗)D̄(∗) at LHC

Abstract We perform a quantitative analysis of the decays of ccc̄c̄ tetraquarks with JPC = 0++, 2++ into 4 muons and into hiddenand open-charm mesons and estimate, for the first time, the fully charmed tetraquark decay width. The calculated cross section upper limit is ∼ 40 fb for the 4 muons channel, and ∼ 28 nb for the D(∗)D̄(∗) → eμ channel. On the basis of our results, with the present sensitivity LHCb should detect both signals, for 0++ and 2++ fully-charmed tetraquarks.


I. INTRODUCTION
In this note we consider production and decay at proton colliders of the fully charmed tetraquarks, T = cccc. In particular, we consider the 4µ and meson-meson decays, the latter revealed through the eµ signature of their weak decays. We focus on the ground states with J P C = 0 ++ , 2 ++ . We shall use the method applied recently to production and decay of fully bottom tetraquarks, bbbb in [1], briefly described in the following.
Preliminary evidence for a 4µ resonance has been presented in a recent seminar of the LHCb Collaboration [2], which is in line with our estimates and indicates that 4µ and meson-meson channels may be the key to the study of these truly exotic hadrons.
The hypothetical existence of hadronic states with more than minimal quark content (qq or qqq) was proposed by Gell-Mann in 1964 [3] and Zweig [4], followed by a quantitative model by Jaffe [5] for the lightest scalar mesons, described as diquark anti diquark pairs.
Recent years have seen considerable growth in the observations of four valence quark states that cannot be included in the well-known systematics of qq mesons, like Z(4430) [6,7] and Z(4248) [8]. Similar particles have also been found in the bottom sector, Z b (10610) and Z b (10650), observed by the Belle collaboration [9] (see [10,11] for recent reviews).
Theoretically, J P C = 0 ++ is expected for the cccc ground-state. Following Ref. [1] we present a calculation of decay widths and branching ratios of the main, hidden-and opencharm channels of cccc tetraquarks. Our estimates apply to tetraquarks close (below or above) to the 2J/Ψ threshold.
Our results are as follows.
Decay rates are proportional to the ratio of overlap probabilities of the annihilating cc pairs in T and J/Ψ: Branching ratios are uniquely determined and reported in Table I, which is the basis of our results. In particular, we find B(T → 4µ) = 2.7 · 10 −6 (J P C = 0 ++ ); B(T → 4µ) = 16 · 10 −6 (J P C = 2 ++ ). (2) The total width is expressed as: With two alternative definitions of the coordinates, as explained later, the model in [19] gives the two estimates ξ = (1.8 or 5.1) The same model gives in excellent agreement with the value obtained from charmonium decay into a muon pair, |Ψ J/Ψ (0)| 2 = 0.13 GeV 3 , see also [23].
To estimate the T width, we take for ξ the geometric average of the two estimates and Our best estimate is then We extend the calculation to the J P C = 2 ++ , fully-charmed tetraquark. J = 2 tetraquarks are produced in p + p collisions with a statistical factor of 5 with respect to the spin 0 state. The decay T → η c + light hadrons is suppressed but annihilations into meson pairs take place at a greater rate.
[cccc] With (6), we find: The results of Tab. I combined with the recent determination by LHCb of the cross section for 2J/Ψ production at 13 TeV [22], give ecouraging upper bounds to the production of T → 4µ at LHC

II. DETAILS OF THE CALCULATION
We give here a brief description of our method, the reader may consult Ref. [1] for more details. The starting point is the Fierz transformation, which brings cc together [10]: quark bilinears are normalised to unity, subscripts denote the dimension of colour representations, and superscripts the total spin. Similarly, for the J = 2 tetraquark, one finds: We describe T decay as due to individual decays of one of the cc pairs in (10), described as follows (see [1] for details).
1. The colour singlet, spin 0 pair decays into 2 gluons, which are converted into confined, light hadrons with a rate of order α 2 S ; taking the spectator cc pair into account, this decay leads to: T → η c + light hadrons. 4. The colour octet, spin 0 pairs annihilate into a pair of light quarks (necessary to neutralize the colour of the spectator cc pair) with amplitude of order α 2 S and rate of the order of α 4 S , which we neglect.
Total T decay rate is the sum of individual decay rates, obtained from the simple for- We normalise the overlap probabilities to |Ψ J/Ψ (0)| 2 , derived from the J/Ψ decay rate into lepton pairs. Eq. (12) applied to this case gives: In terms of the Vector Meson Dominance parameter [25] defined by with f a pure number, one obtains [26]: Numerical results. The contribution to the T decay rate of the colour singlet, spin 0 decay is 1 Our method of calculation is borrowed from the theory of K electron capture, where an atomic electron reacts with a proton in the nucleus to give a final nucleus and a neutrino, see [1].
We have used the spectroscopic coefficient in (10) and have set Similarly Finally we consider the annihilation of (cc) 1 8 into light quark pairs, Fig. 1. The numerical factor associated to the traces of the colour matrices along fermion closed paths, C (the Chan-Paton factor [27]) gives the effective coupling constant of the process, α ef f = Cα S , which is what replaces Q c α in Eq. (13). From Fig. 1 we read C = √ 2/3 and find 2 : Using Eq. (15), α S = 0.3 and massless q, we obtain 2 The factor 2 arises from the two choices of the annihilating bilinear: given the symmetry of the tetraquark, we may call c 1 the annihilating c quark and pair it to eitherc 1 orc 2 ; the spectroscopic factor is from (10); in parenthesis vσ(cc → qq) and Γ(T ) = Γ 0 + Γ 1 + 3Γ 5 = 21 · ξ MeV (21) Eq. (20) gives the total decay rate into pseudoscalar and vector meson pairs. It is easy to see that the rate is shared between pseudoscalar and vector mesons in the ratio 1 : 3 [1].

III. THE VALUE OF ξ
To minimize systematic errors, it is desirable to estimate ξ = |Ψ T (0)| 2 /|Ψ J/Ψ (0)| 2 with the same method for numerator and denominator. In the so-called gaussian approximation, constituents wave functions in charmonia and tetraquarks are taken as simple products of gaussians, with the shape parameters obtained by minimising the expectation of the QCD Hamiltonian given in [19,28], see [1] for more details and references.
Constituent coordinates in the tetraquark are defined as [19] x, y : antiquarks; z, 0 : quarks and one introduces the Jacobi coordinates ξ ξ ξ 1 = x x x − y y y; ξ ξ ξ 2 = z z z; ξ ξ ξ 3 = x x x + y y y − (z z z + 0 0 0) A probability function describing cc separation can be obtained in two ways: (i) integrating |Ψ| 2 over y and z gives the probability distribution of the distance of one c from thec sitting in the origin; (ii) integrating over ξ 1 and ξ 2 gives the probability distribution of the distance of the diquark center of mass to the di-antiquark one, which we indicate by = (ξ 3 /2). We obtain ξ x = |Ψ T (x = 0)| 2 |Ψ J/Ψ (0)| 2 = 1.8; geometrical mean, with the previous results used as an error estimate: Branching ratios do not depend on ξ.

IV. TETRAQUARK CROSS SECTIONS
Combining Eqs. (18) and (21) we obtain: corresponding to the cross section upper bound where σ(pp → 2J/Ψ) 15.2 nb is the two-J/Ψ production cross section measured by LHCb at 13 TeV [22].  We focus on the eµ inclusive channel and give in Tab The largest part of the signal (the total signal for J P C = 2 ++ ) arises from the decay of T into a pair of vector mesons. Vector particles decay promptly into a pseudoscalar plus a soft pion or photon(s) and contribute to the signal on the same basis as the pseudoscalars.
In conclusion, production in the 4µ channel and decay rates that we estimate for the cccc tetraquarks are tantalizingly similar to the preliminary results presented by the LHCb Collaboration [2]. The meson-meson channel with the eµ signature may provide an additional, complementary tool to identify and study the spectacular, exotic cccc tetraquarks.
We thank Sheldon Stone for an enlightening discussion and advice on a preliminary, March 2020, version of these notes.