Electromagnetic and hadronic decay of fully heavy tetraquark

In this study, we compute the electromagnetic and hadronic decay widths of the S-wave fully heavy tetraquark $T_{4Q}$ ($Q=c$ or $b$) at lowest order in $\alpha_s$ and $v$, in the framework of nonrelativistic QCD. The short-distance coefficients are determined through the standard procedure of matching. The nonperturbative long-distance matrix elements are related to the phenomenological four-body Schr\"odinger wave functions at the origin, whose values are taken from literature. The branching fractions are predicted to be around $10^{-3}$ and $10^{-6}$ for the $T_{4c}$ hadronic decay and electromagnetic decay, respectively. Combing our results with the $T_{4c}$ production cross sections at the LHC, we also predict the event numbers for various decay channels. With integrated luminosity $\mathcal{L}=100 \,{\rm fb}^{-1}$, it is expected that the event numbers can reach $10^3- 10^4$ for $T_{4c}\to \gamma\gamma$, and $10^5- 10^6$ for $T_{4c}\to {\rm LH}$, at the LHC. The detecting prospect is promising.


I. INTRODUCTION
Searching for exotic states is full of challenges and opportunities.Compared with conventional hadrons, exotic states may provide a better environment to reveal the nonperturbative nature of QCD.In 2020, LHCb collaboration reported a narrow resonance around 6.9 GeV with the significance larger than 5σ, dubbed as X(6900), a broad structure with the mass ranging from 6.2 GeV to 6.8 GeV, and a hint for another structure around 7.2 GeV in the di-J/ψ invariant mass spectrum [1].The narrow structure contains at least cccc configuration and therefore is naturally considered as a fully charmed tetraquark.By assuming that the nonresonant single-parton scattering (NRSPS) continuum is not disturbed, the mass and width of X(6900) are determined to be M = 6950 ± 11 ± 7 MeV and Γ = 80 ± 19 ± 33 MeV, respectively, while, they become M = 6886 ± 11 ± 11 MeV and Γ = 168 ± 33 ± 69 MeV when assuming the NRSPS continuum interferes with a broad structure above the di-J/ψ mass threshold.
While the spectra have been widely studied, the decay property and production mechanism of the fully-heavy tetraquarks are relatively less investigated, particularly based on model independent methods.In Ref. [43], a factorization formula for S-wave T 4c inclusive production is presented in the framework of nonrelativistic QCD (NRQCD) [44] (the similar idea can be also found in [45,46]).A key observation is that, prior to hadronization, two charm quarks and two anticharm quarks have to be created at a rather short distance < 1/m c , thus one can invoke asymptotic freedom to factorize the production rate as the product of the perturbative calculable short-distance part and the nonperturbative longdistance part.By employing this factorization formula, the T 4c production at the LHC was studied in Refs.[43,47], and production at the B factory was investigated in Refs.[48,49].
Analogous to the case in quarkonium sector, one anticipates that the NRQCD factorization ansatz should hold for the fully heavy tetraquark inclusive and electromagnetic decay.This work aims to compute the decay widths for T 4Q → γγ and T 4Q → LH, where LH denotes the light hadrons.Note that, by assigning T 4Q to be an S-wave tetraquark, the feasible J P C quantum number of T 4Q can be 0 ++ , 1 +− and 2 ++ .
The paper is organized as follows.In Sec.II, we present the factorization formulas for T 4Q hadronic and electromagnetic decay, and sketch the procedure to match the shortdistance coefficient (SDC).In Sec.III, we describe the computing techniques, and present the analytic expressions for the SDCs.Sec.IV is devoted to phenomenological predictions and discussions.We make a summary in Sec.V.

II. THEORETICAL FRAMEWORK
A. NRQCD factorization for T 4Q decay In Ref. [43], the NRQCD factorization formula for T 4Q inclusive production was proposed.It is straightforward to convert the factorization formula into that for the T 4Q electromagnetic and hadronic decay.Specifically, we can express the decay widths of T 4Q → LH at the lowest order in velocity as follows 1 where the superscript in T 4Q denotes the quantum number J P C of the T 4Q , m Q and m H signify the masses of the heavy quark Q and the T 4Q respectively, and c represent the SDCs.The long-distance matrix elements (LDMEs) in ( 1) are defined via where J = 0, 2, and the T 4Q state is nonrelativistically normalized.Note that, in Eq. ( 2), we have made use of the "vacuum-saturation approximation" to transfer the hadronic matrix elements into the electromagnetic matrix elements [44].The operators in (2) are constructed in the diquark-antidiquark basis, where the spin configuration and color configuration of the diquark are correlated due to Fermi statistics.Specifically, the S-wave spin-singlet 1 Due to the Landau-Yang theorem, the T 1 +− 4Q into double photons or double gluons is strictly forbidden.Moreover, T 0 ++ 4Q or T 1 +− 4Q decay into a pair of light quarks is also forbidden according to the helicity conservation of light quark and the angular momentum conservation.The T 1 +− 4Q can decay into triple gluons or a gluon associated with a pair light quarks, which however is suppressed by the strong coupling constant α s .Thus, we will not consider the decay of diquark must be a color-sextet, while the S-wave spin-triplet diquark must be a color-triplet.Explicitly, these operators read where ψ and χ † are Pauli spinor fields that annihilate the heavy quark and antiquark, respectively, σ i denotes Pauli matrix, and ǫ H denotes the polarization tensor of the T 2 ++ 4Q .The Latin letters i, j, k = 1, 2, 3 signify the Cartesian indices, whereas a, b, c, d = 1, 2, 3 denote the color indices.The symmetric traceless tensor is The color projection tensors in (3) are given by It is straightforward to convert the factorization formula (1) into that for T 4Q → γγ.Correspondingly, we replace the subscript LH with γγ in the SDCs to denote the SDCs for T 4Q → γγ.

B. Procedure to determine the SDCs
With the spirit of factorization, the SDCs in (1) are insensitive to the hadronization effects in the tetraquark, thus they can be deduced with the aid of the standard perturbative matching techniques.That is, by replacing the physical T 4Q state in (1) with a fictitious "tetraquark" state composed of a pair of heavy quarks and a pair of heavy antiquarks, carrying the same quantum number as the physical T 4Q .Conveniently, we label the fictitious state with T 4Q .After this replacement, we can compute both sides of (1) order by order in α s , thus the SDCs can be solved at desired order of α s .
To calculate the left-hand side of (1), we first write down the amplitude of the free QQ Q Q decay, then enforce the QQ Q Q in the desired spin, total angular momentum, and color quantum numbers.In a shortcut, we employ the spin-singlet projector Π 0 and spintriplet projector Π 1 to enforce the diquark in S = 0 and S = 1 respectively, and we make use of the color projectors C ab;cd 6⊗ 6 and C ab;cd 3⊗3 to extract the color-sextet and color-triplet parts of the diquark, respectively.For the case of the diquark and anti-diquark in the spin-triplet configuration, we apply the covariant projectors to pick out the total angular momentum number of T 4Q .Concretely, we ensure the QQ Q Q in J P C = 0 ++ through the replacement for the spin-singlet diquark configuration, and ensure the QQ Q Q in J P C = 0 ++ , 2 ++ through the replacement for the spin-triplet diquark configuration, where u and v denote the Dirac spinors of heavy quarks.In ( 6) and ( 7), the spin projectors are and the covariant projectors J µν 0,2 are where P denotes the momentum of the T 4Q , and η µν ≡ −g µν + P µ P ν P 2 .Furthermore, the squared amplitude can be obtained by multiplying the amplitude with its complex conjugate, summing the polarizations of the final states, and averaging the polarizations of the T 4Q .It is worth noting that, in order to match c mixing , one must utilize the replacement (7) for the quark-level amplitude, and the replacement (6) for the complex conjugate of the amplitude.
For the sake of completeness, we present the results for the perturbative LDMEs that involve where the heavy quark states are relativistically normalized, which is consistent with the convention adopted for the spin projectors (8).Now, we collect all the essential ingredients to evaluate the SDCs.

III. ANALYTIC EXPRESSIONS FOR THE SDCS
We use the package FeynArts [50] to generate quark-level Feynman diagrams and the corresponding amplitudes.There are 40, 4, and 62 nonvanishing Feynman diagrams for T 4Q → γγ, T 4Q → q q, and T 4Q → gg, respectively.Some representative diagrams are illustrated in Fig. 1.After implementing the replacements ( 6) and (7) to ensure QQ Q Q in the correct quantum numbers, we use the packages FeynCalc [51] and FormLink [52,53] to conduct the trace over Dirac and SU(N c ) color matrices, and the contraction over Lorentz indices. (a) FIG. 1: Some representative Feynman diagrams for T 4Q → γγ, T 4Q → q q, and T 4Q → gg.
Following the matching procedure sketched in Sec.II, it is straightforward to deduce the SDCs for T 4Q → γγ, and for T 4Q → LH, where n L = 3 represents the number of light quark flavors.Note that the n L term in c (2) LH corresponds to the contribution from T 4Q → q q.It is worth emphasizing that the m H in the SDCs, originating from the prefactor of the formula of decay width, cancels the same factor in (1), thus the final decay widths are free of the m H .

IV. PHENOMENOLOGICAL PREDICTIONS
Prior to making phenomenological predictions, we need to fix the various input parameters. 2 We take the charm quark mass to be m c = 1.5 GeV.We take the fine structure coupling constant α = 1/137.The QCD running coupling constant at µ R = 2m c is evaluated with the aid of the package RunDec [54].We have varied µ R from m c to 4m c to estimate the theoretical uncertainty.
We should further choose the NRQCD LDMEs.These nonperturbative LDMEs can be related to the phenomenological four-body Schrödinger wave functions at the origin [47] In this work, we adopt two phenomenological models to evaluate the LDMEs [22,35].In both models, Cornell-type potentials with spin-dependent corrections are assumed, and Gaussian basis method is utilized to solve for the four-body tetraquark wave functions.However unlike the Model I [22], which is based on nonrelativistic quark model, there involves a relativistic kinetic term in Model II [35].We enumerate the values of the LDMEs for T 4c in Table I.Note that there is a sign difference for the value of O (0) mixing from the two models.[22] and Model II [35], in unit of GeV 9 .If assuming T 4c decay is saturated by its decay into double J/ψ, we can approximate the total decay width of T 4c through where 0.12 GeV corresponds to the central value of the decay width determined by the ATLAS collaboration [5], and is roughly the average of the two fitting values from the LHCb measurement [1].This value is also consistent with the no-interference fitting value by the CMS collaboration [3].Now, we collect all ingredients to make phenomenological predictions.The theoretical results of the decay widths as well as the corresponding branching fractions for T 4c → γγ and T 4c → LH are tabulated in Table II.We observe that, the branching fractions for T 4c → LH are about three order-of-magnitude larger than these for T 4c → γγ, which is attributed to enhancement from the strong coupling constant, i.e., α 2 s /α 2 ≈ 10 3 .By comparing the predictions from the two phenomenological models, we find the theoretical results for 2 ++ tetraquark are insensitive to the models, while the predictions for 0 ++ from Model I are more than two times larger than from Model II.As can be seen from Table I, the values of O (0) mixing in the two models take different signs.Therefore, the interfering term is constructive in Model I, while destructive in Model II, which explains why the branching fractions for 0 ++ from Model I are larger.Finally, it is worth noting that our prediction for the Br[T 0 ++ 4c → γγ] is roughly consistent with the value (2.77±0.36)×10−6 estimated based on the vector meson dominance [55].Combing our predictions with the T 4c production cross sections at the LHC [47], we can further estimate the event yields for the T 4c hadronic and electromagnetic decay at the LHC.The event numbers for various channels are tabulated in Table III.For reference, we also copy the T 4c cross sections from Ref. [47] in the table.It is expected that there will be plenty of T 4c → LH signals accumulated at the LHC.In spite of potentially copious background events, it seems that the observation prospects for the T 4c hadronic decay are promising.On the other hand, the event numbers for the electromagnetic decay T 4c → γγ are much smaller, nevertheless, it is hopeful to probe these channels, thanks to a clean final state topology in which the diphoton invariant mass can be reconstructed with high precision.

V. SUMMARY
By applying the NRQCD factorization formalism, we compute the decay widths for the S-wave fully heavy tetraquark T 4Q hadronic as well as electromagnetic decay at lowest order in α s and v.The SDCs are determined through the procedure of perturbative matching.The LDMEs are related to the phenomenological four-body Schrödinger wave functions at the origin, whose values are taken from Refs.[22,35].To obtain the branching fractions for T 4c decay, we approximate the total decay width of T 4c with Γ[T 4c → J/ψJ/ψ].The latter value has been determined by the LHCb, CMS, and ATLAS collaborations.We find the branching fractions are around 10 −3 and 10 −6 for the T 4c hadronic decay and electromagnetic TABLE III: Estimation on the event yields at the LHC, where the T 4c production cross sections at the LHC σ LHC are taken from Ref. [47] and the integrated luminosity L = 100 fb −1 is chosen.The two uncertainties in N events are transferred from the uncertainties in the cross sections and branching fractions, respectively.are insensitive to the phenomenological models, while the predictions for T 0 ++ 4c from Ref. [22] are more than two times larger than these from Ref. [35].This feature is quite similar to the case for the T 4c production at the LHC.Combing the T 4c production cross sections with the branching fractions for T 4c decay, we estimate the event numbers for various decay channels at the LHC.The observation prospect seems to be promising for both the T 4c hadronic and electromagnetic decay.

TABLE I :
Numerical values of the LDMEs for T 4c in Model I

TABLE II :
Theoretical predictions on the decay widths and branching fractions.We estimate the uncertainty by sliding µ R from m c to 4m c .