Decipher the width of the X (3872) via the QCD sum rules

In this work, we take the X (3872) as the hidden-charm tetraquark state with both isospin I = 0 and 1 components, then investigate the strong decays X (3872) → J/ψπ + π − , J/ψω , χ c 1 π 0 , D ∗ 0 ¯ D 0 and D 0 ¯ D 0 π 0 with the QCD sum rules. We take account of all the Feynman diagrams, and acquire four QCD sum rules based on rigorous quark-hadron duality. We obtain the total decay width about 1 MeV, which is in excellent agreement with the experiment data Γ X = 1 . 19 ± 0 . 21 MeV from the Particle Data Group, it is the ﬁrst time to reproduce the tiny width of the X (3872) via the QCD sum rules, which supports assigning the X (3872) as the hidden-charm tetraquark state with the J PC = 1 ++ .

In 2015, the LHCb collaboration studied the angular correlations in the B + → X(3872)K + decays with the subprocess X(3872) → ρ 0 J/ψ → π + π − µ + µ − to measure orbital angular momentum contributions and to determine the J P C of the X(3872) to be 1 ++ [28].The X(3872) state is probably the best known (and most enigmatic) representative of the X, Y and Z states.One important discriminant between different models is the width of the X(3872).In 2020, the LHCb collaboration updated the mass and width of the X(3872), and obtained the Breit-Wigner width Γ = 0.96 +0. 19 −0.18 ± 0.21 MeV [29], or 1.39 ± 0.24 ± 0.10 MeV [30], which indicates non-zero width of the X(3872) and leads to the average width Γ X = 1.19 ± 0.21 MeV listed in "The Review of Particle Physics" [31].As known, we cannot assign a hadron with the mass alone, and we should study the decays to obtain more robust interpretation.Up to today, only the decays X(3872) → J/ψπ + π − , J/ψω, J/ψγ, ψ ′ γ, χ c1 π 0 , D * 0 D0 and D 0 D0 π 0 are established [31].
In the present work, we will focus on the scenario of tetraquark states.In Ref. [16], we take the pseudoscalar, scalar, axialvector, vector, tensor (anti)diquarks as the basic constituents, and construct the scalar, axialvector and tensor tetraquark currents to study the mass spectrum of the ground state hidden-charm tetraquark states with the QCD sum rules in a comprehensive way, and observe that the X(3872) can be assigned to be the hidden-charm tetraquark state with the quantum numbers J P C = 1 ++ .According to the recent combined data analysis, the decays X(3872) → J/ψρ → J/ψπ + π − and X(3872) → J/ψω → J/ψπ + π − π 0 have almost the same branching fractions [32], the isospin breaking effects in the decays are large enough and beyond the naive ρ − ω mixing.In this work, we introduce the isospin breaking effects explicitly and study the decays X(3872) → J/ψπ + π − , J/ψω, χ c1 π 0 , D * 0 D0 and D 0 D0 π 0 with the QCD sum rules based on rigorous quark-hadron duality, and try to decipher the width of the X(3872).
The article is arranged as follows: we obtain the QCD sum rules for the hadronic coupling constants in section 2; in section 3, we present numerical results and discussions; section 4 is reserved for our conclusion.
As the operator product expansion is concerned, we calculate the vacuum condensates up to dimension 5, and obtain the QCD spectral densities through double dispersion relation, as In calculations, we neglect the gluon condensates due to their tiny contributions [34,35].We accomplish the integral over ds ′ firstly at the hadron side, then match the hadron side with the QCD side bellow the continuum thresholds s 0 and u 0 to obtain rigorous quark-hadron duality [34,35], In the following, we write down the hadron representation explicitly, where we introduce the parameters C ′ ρ/ω/π/D to stand for all the contributions concerning the higher resonances in the s ′ channel, where the densities ρ ) and ρD (s ′ , m 2 D * , m 2 D ) are complex and we have no knowledge about the higher resonant states, as the spectrum is vague.We take the unknown functions C ′ ρ/ω/π/D as free parameters and adjust the suitable values to obtain flat Borel platforms for the hadronic coupling constants G ′ XJ/ψρ , G ′ XJ/ψω , G ′ Xχπ and G ′ XD * D , respectively [34,35].In Ref. [14] (also Ref. [33]), Navarra and Nielsen approximate the hadron side of the correlation functions as then only match them with the QCD side below the continuum threshold s 0 , where the B ρ/ω stand for the pole-continuum transitions, and we have changed their notations (symbols) into the present form for convenience.Although Navarra and Nielsen take account of the continuum contributions by introducing a parameter s ′ 0 in the s ′ channel phenomenologically, they neglect the continuum contributions in the u channel at the hadron side by hand.It is the shortcoming of that work.While in this work, we match the hadron side with the QCD side bellow the continuum thresholds s 0 and u 0 to obtain rigorous quark-hadron duality, and we take account of the continuum contributions in the s ′ channel.
We set p ′2 = p 2 in the correlation functions Π H (p ′2 , p 2 , q 2 ), and perform double Borel transform in regard to P 2 = −p 2 and Q 2 = −q 2 , respectively, then we set T 2 1 = T 2 2 = T 2 to obtain four QCD sum rules, where In Ref. [14] (also Ref. [33]), Navarra and Nielsen set p ′2 = p 2 in the correlation functions Π H (p ′2 , p 2 , q 2 ), perform single Borel transform in regard to P 2 = −p 2 , and take the Q 2 = −q 2 as a free parameter to parameterize the off-shell-ness of the hadronic coupling constants G XJ/ψρ and G XJ/ψω , which are fitted into some functions of Q 2 , then extract them to the physical points q 2 = m 2 ρ/ω , and finally too large partial decay widths are obtained.The schemes are quite different, we should not be surprised that the predictions in Ref. [14] and in this work are also quite different.
In calculations, we factorize out the mixing angle θ in Eqs.( 22)-( 25) so as to facilitate determining the mixing effects, and redefine the hadronic coupling constants G and free parameters C, then it is easy to study the dependence on the mixing angle θ.

Numerical results and discussions
We take the conventional vacuum condensates, qq = −(0.24± 0.01 GeV) 3 , qg s σGq = m 2 0 qq , m 2 0 = (0.8 ± 0.1) GeV 2 at the energy scale µ = 1 GeV [36,37,38], and take the M S mass m c (m c ) = (1.275± 0.025) GeV from the Particle Data Group [31].We set m u = m d = 0 and take account of the energy-scale dependence from re-normalization group equation, where , Λ QCD = 210 MeV, 292 MeV and 332 MeV for the flavors n f = 5, 4 and 3, respectively [31,39], and we choose n f = 4, and evolve all the input parameters to the energy scale µ = 1 GeV.At the hadron side, we take m π ± = 0.13957 GeV, m π 0 = 0.13498 GeV, m J/ψ = 3.0969 GeV, m χc1 = 3.51067 GeV, m ρ = 0.77526 GeV, m ω = 0.78266 GeV, f π = 0.130 GeV from the Particle Data Group [31], 418 GeV [41], f χc1 = 0.338 GeV [42], 16] from the QCD sum rules, and f π m 2 π /(m u + m d ) = −2 qq /f π from the Gell-Mann-Oakes-Renner relation.In calculations, we fit the free parameters as GeV 2 and T 2 D = (4.0− 5.0) GeV 2 , where the subscripts ρ, ω, π and D denote the corresponding channels.We obtain uniform enough flat platforms T 2 max − T 2 min = 1 GeV 2 where the max and min denote the maximum and minimum, respectively.In Fig. 1, we plot the hadronic coupling constants G XJ/ψρ , G XJ/ψω , G Xχπ and G XD * D with variations of the Borel parameters at large intervals.In the Borel windows, there appear very flat platforms indeed, it is reliable to extract the hadron coupling constants.Now, we estimate the uncertainties in the following ways.For example, the uncertainties of an input parameter ξ, ξ = ξ + δξ, result in the uncertainties λ where we only present the central value of the G Xχπ due to the tiny partial decay width of the X(3872) → χ c1 π 0 .Now we take the hadron masses m X = 3.87165 GeV, m D * 0 = 2.00685 GeV, m D 0 = 1.86484GeV and m J/ψ = 3.09690 GeV from the Particle Data Group to calculate the partial decay widths [31].As the X(3872) lies near the thresholds of the final states J/ψρ, J/ψω and D * D, we should take account of the finite width effects of the ρ, ω and D * mesons, due to the decay cascades, Then we obtain the partial decay widths via trial and error, as there is an additional parameter θ, . The hadronic coupling constants from the QCD sum rules in Eqs.( 22)-( 25 to parameterize the "off-shell" effects due to the J/ψρ and J/ψω thresholds, as the X(3872) lies near the J/ψρ and J/ψω thresholds.At the mass-shell s = m 2 ρ and m 2 ω , they reduce to 1 to match with the zero width approximation in the QCD sum rules.At the thresholds, s = ∆ 2 2π and ∆ 2 3π , the available phase-spaces are very small, the decays ρ → ππ and ω → πππ only take place through the lower tails, which can be taken as some intermediate sates with the same quantum numbers as the ρ and ω except for the masses, and are greatly suppressed.On the other hand, the "off-shell" effects on the hadronic coupling constants are considerable, we should introduce some form-factors to parameterize them.
All in all, in this work, we reproduce the small width of the X(3872) via the QCD sum rules for the first time.

Conclusion
In this work, we take the X(3872) as the hidden-charm tetraquark state with both isospin I = 0 and 1 components, then investigate the hadronic coupling constants G ′ XJ/ψρ , G ′ XJ/ψω , G ′ Xχπ and G ′ XD * D with the QCD sum rules in details.We select the optimal tensor structures and take account of all the Feynman diagrams, then acquire four QCD sum rules based on the rigorous quark-hadron duality.After careful calculations, we obtain the hadronic coupling constants, then determine the mixing angle via trial and error, and obtain the partial decay widths for the X(3872) → J/ψπ + π − , J/ψω, χ c1 π 0 , D * 0 D0 and D 0 D0 π 0 .The total width is about 1 MeV, which is in excellent agreement with the experiment data Γ X = 1.19 ± 0.21 MeV from the Particle Data Group, it is the first time to reproduce the small width of the X(3872) via the QCD sum rules.The present calculations support assigning the X(3872) as the mixed hidden-charm tetraquark state with the quantum numbers J P C = 1 ++ .

Figure 1 :
Figure 1: The central values of the hadronic coupling constants with variations of the Borel parameters T 2 , where the (I), (II), (III) and (IV) denote the G XJ/ψρ , G XJ/ψω , G Xχπ and G XD * D , respectively.