Deciphering the $X(3872)$ via its polarization in prompt production at the CERN LHC

Based on the hypothesis that the $X(3872)$ exotic hadron is a mixture of $\chi_{c1}(2P)$ and other states and that its prompt hadroproduction predominately proceeds via its $\chi_{c1}(2P)$ component, we calculate the prompt-$X(3872)$ polarization at the CERN LHC through next-to-leading order in $\alpha_s$ within the factorization formalism of nonrelativistic QCD, including both the color-singlet $^3\!P_1^{[1]}$ and color-octet $^3\!S_1^{[8]}$ $c\bar c$ Fock states. We also consider the polarization of the $J/\psi$ produced by the subsequent $X(3872)$ decay. We predict that, under ATLAS, CMS, and LHCb experimental conditions, the $X(3872)$ is largely longitudinally polarized, while the $J/\psi$ is largely transversely polarized. We propose that the LHC experiments perform such polarization measurements to pin down the nature of the $X(3872)$ and other $X$, $Y$, $Z$ exotic states with non-zero spin.

Based on the hypothesis that the X(3872) exotic hadron is a mixture of χc1(2P ) and other states and that its prompt hadroproduction predominately proceeds via its χc1(2P ) component, we calculate the prompt-X(3872) polarization at the CERN LHC through next-to-leading order in αs within the factorization formalism of nonrelativistic QCD, including both the color-singlet and color-octet 3 S [8] 1 cc Fock states. We also consider the polarization of the J/ψ produced by the subsequent X(3872) decay. We predict that, under ATLAS, CMS, and LHCb experimental conditions, the X(3872) is largely longitudinally polarized, while the J/ψ is largely transversely polarized. We propose that the LHC experiments perform such polarization measurements to pin down the nature of the X(3872) and other X, Y , Z exotic states with non-zero spin. The discovery of the X(3872) by Belle in 2003 [1], which soon afterward was confirmed by CDF [2], D0 [3], and BaBar [4], triggered the renaissance of hadron spectroscopy. Ever since then, many charmonium or charmonium-like states, named X, Y, Z, were discovered, and numerous theoretical studies were devoted to reveal their nature. We refer to Refs. [5,6] as the latest reviews of experimental results and theoretical approaches, respectively. Among the X, Y, Z hadrons, the X(3872) is the top highlighted state. CDF [7] and LHCb [8] have established the J P C = 1 ++ quantum numbers of the X(3872), and the very precise world average of its mass is m X = 3871.69 ± 0.17 MeV [9]. On the theory side, however, we are still far away from a convincing, overall picture to explain all the measurements. The popular models on the market include charmonium [10], D * 0D0 /D 0D * 0 molecule [11], tetraquark [12], hybrid [13], or some quantum-mechanical mixtures thereof; see, e.g., Ref. [14] for reviews. At hadron colliders, the X(3872) is most frequently produced promptly, as has been observed by CDF [2,15] at the Fermilab Tevatron and by LHCb [16], CMS [17], and ATLAS [18] at the CERN LHC. Besides mass spectrum and decay modes, prompt production provides a complementary source of information on the nature of the X(3872). For example, in our previous work [20], we showed that the pure χ c1 (2P ) option of the X(3872) can be excluded by analyzing its prompt hadroproduction rates in the framework of nonrelativistic-QCD (NRQCD) factorization [21], and we predicted the χ c1 (2P ) component to be around 30% under the assumption that the X(3872) state is a quantum mechanical mixture of a χ c1 (2P ) and a D * 0D0 /D 0D * 0 molecule as proposed in Ref. [22]. The study of X(3872) prompt production within NRQCD factorization was pioneered by Artoisenet and Braaten [23], who considered the color octet (CO) contribution, due to the cc Fock state 3 S [8] 1 , at leading order (LO). Prompt X(3872) hadroproduction was also studied in the molecular picture, and the cross section was found to greatly undershoot CDF data [24]. Although this problem could be remedied [25] by properly taking into account the rescattering mechanism [23], it is still under debate if the molecular picture can adequately describe all the experimental data [26]. Very recently, X(3872) plus soft-pion production has been proposed to settle this issue [27].
Despite concerted experimental and theoretical endeavor during the past decade, the quest for the ultimate classification of the X(3872) and other X, Y, Z states remains one of the most tantalizing challenges of hadron spectroscopy at the present time. Since the total spin of the X(3872) is 1, rather than 0, its polarization in prompt production is expected to be rather sensitive to its production mechanism and its internal structure. Moreover, the X(3872) has a considerable branching fraction to decay into the J/ψ. Under the assumption that the total spin of the charm quark pair is preserved during the decay process, the polarization of the J/ψ from the decay of the prompt X(3872) will help us to analyze the role of the cc pair inside the X(3872). To decipher the as-yet inscrutable nature of the X(3872), we thus propose, in this Letter, to measure at the LHC its polarization and that of the J/ψ that springs from it. Working in NRQCD factorization at next-to-leading order (NLO) in α s , we provide here, for the first time, the respective theoretical predictions under the likely assumption that the prompt hadroproduction of the X(3872) proceeds predominately via the χ c1 (2P ) component of its wave function at short distances. By doing so, we correct a reproducible error, common to existing literature on the NLO NRQCD treatment of P-wave heavy-quark pair polarization, related to the implementation of phase space slicing [28,29].
The observation that NRQCD factorization at NLO fails to yield a coherent description of the world data on J/ψ yield and polarization [28,30] may not affect our present analysis. In contrast to the χ cJ case relevant here, the color singlet (CS) contribution to direct J/ψ hadroproduction has not yet unfolded its leading power, proportional to 1/p 4 T , at NLO [31]. This will only happen at NNLO, where a new dominant CS production channel will open up, which will dynamically create an enhancement that is likely to exceed the parametric O(α s ) by orders of magnitude, with the potential to reconcile the J/ψ world data with NRQCD factorization. Furthermore, the 3 P [8] J CO channel, which contributes to S, but not P wave quarkonium production, is potentially very sensitive to NNLO corrections, owing to a cancellation between LO and NLO corrections [28,32]. Finally, the J/ψ polarization problem [28,30] is reduced to a tolerable level if one takes the point of view that data with p T < 10 GeV should be disregarded to suppress contributions violating NRQCD factorization [33], the more so if the leading logarithms in p 2 T /m 2 c are resummed [34]. Adopting the collinear parton model of QCD and NRQCD factorization, the spin density matrix of the differential cross section of prompt X(3872) hadroproduction can be evaluated as where f k/A (x) is the density function (PDF) of parton k with longitudinal-momentum fraction x inside hadron A, dσ ij (kl → cc[n] + anything) with i, j = 0, ±1 is the spin density matrix element of the respective partonic cross section, and O X(3872 being the long-distance matrix element (LDME) of cc Fock state n inside the χ c1 (2P ) and χ c1 (2P )|X(3872) being the overlap of the physical X(3872) and χ c1 (2P ) wave functions. At LO in v 2 , where v is the relative momentum in the motion of the cc pair, we have n = 3 P 1 , where 2S+1 L J refers to the spectroscopic notation and the label in brackets indicates CS and CO configurations. The evaluation of dσ ij (kl → cc[n] + anything) at NLO in NRQCD proceeds as in Ref. [28] upon the correction mentioned above. The production of polarized J/ψ's via the feed-down of promptly hadroproduced X(3872)'s may be treated in one sweep adopting the formalism outlined in Ref. [35]. NLO NRQCD polarization studies for promptly hadroproduced χ c1 's and χ c2 's may be found in Refs. [34,36].
The polarizations of the X(3872) and J/ψ can be measured by analyzing the angular distributions of their decay products. The J/ψ is easily reconstructed by its decay to a lepton pair l + l − , and the distribution in the polar angle θ of the l + flight direction in the J/ψ rest frame reads [37] where the polarization parameter λ ψ θ = (σ ψ 11 −σ ψ 00 )/(σ ψ 11 + σ ψ 00 ) takes the values 0, ±1 if the J/ψ is unpolarized and totally transversely/longitudinally polarized, respectively. The definition of θ depends on the choice of coordinate frame. The helicity (HX) frame, in which the polar axis is chosen to point along the J/ψ flight direction in the center-of-mass (CM) frame of the collision, and the Collins-Soper frame, in which the polar axis is defined as the bisector of the two beam directions [38], are most frequently used experimentally. For definiteness, we will use the HX frame throughout this Letter. The counterpart of Eq. (2) for the X (3872) is not yet available and will be derived in the following. So far, all X(3872) events of prompt hadroproduction have been reconstructed through the J/ψπ + π − decay channel. CMS [17] and ATLAS [18] found that almost all the π + π − pairs originate from ρ vector meson decay. Due to this ρ dominance, the partial decay amplitude of X(3872) → J/ψ + π + π − can be approximated by where f ρππ is a hadronic coupling constant. On the other hand, CPT conservation and Lorentz covariance restrict A µ (X(3872) → J/ψρ) to be a linear combination of A 1 µ = ε µαβγ ǫ α X ǫ * β ψ p γ ρ /m ρ and A 2 µ = ε µαβγ ǫ α X ǫ * β ψ p γ ψ /m ψ , if the J/ψ and ρ are in the S-wave channel. 1 We thus make the ansatz with the relative-weight factor g to be fitted to experimental data. Adopting the masses m X = 3.8717 GeV, m ψ = 3.097 GeV, m ρ = 0.7753 GeV, m π ± = 0.1396 GeV, and the total decay width Γ ρ = 0.149 GeV from Ref. [9], using the functional form BW ρ (p 2 ρ ) = (p 2 ρ − m 2 ρ + iΓ ρ p 2 ρ − 4m 2 π ) −1 , and integrating out the π + π − phase space numerically, we find the X(3872) decay distribution, W X (θ), to have the same form as in Eq. (2), with θ now being the polar angle of the J/ψ flight direction in the X(3872) rest frame, and the polarization parameter therein to be where R = σ X 00 /σ X 11 and f = (−0.56 + 1.28g + 3.12g 2 )/(13.7 + 30.6g + 18.2g 2 ). Finally, we determine g by fitting to the distributions of the X(3872) → J/ψπ + π − partial decay width in the π + π − invariant mass m ππ , normalized to unity, as measured by CMS [17] in the range 0.5 < m ππ < 0.78 GeV and by ATLAS [18] in the range 0.28 < m ππ < 0.79 GeV. We thus obtain g = −0.51 ± 0.10 with χ 2 /d.o.f. = 35.3/22 = 1.60. The goodness of the fit can also be judged from Fig. 1, which also contains the predictions evaluated with either A 1 µ or A 2 µ alone. The latter results are somewhat worse, yielding χ 2 /d.o.f. values of 45.9/23 = 2.00 and 80.0/23 = 3.48, respectively. Other realistic functional forms of BW ρ (p 2 ρ ) yield very similar results, albeit with slightly larger χ 2 /d.o.f. values. Inserting our fit result for g in Eq. (5) and setting in turn σ X 00 = 0 and σ X 11 = 0, we obtain the allowed corridor −0.066 ≤ λ X θ ≤ 0.141, where the lower bound f /(2−f ), upper bound −f , and 0 correspond to totally transversely, totally longitudinally, and unpolarized X(3872)'s, respectively. Our result for W X (θ) is new. We caution the reader that the functional form of f depends on the π + π − phase space integrated over, so that λ X θ does depend on the experimental acceptance cuts applied. This must be taken into account in the extraction of polarization parameters from experimental data of X(3872) → J/ψ + π + π − .
In our NLO NRQCD calculations, we use the on-shell mass m c = 1.5 GeV and the two-loop formula for α (n f ) s with n f = 4 active quark flavors. As for the proton PDFs, we adopt the CTEQ6M set [40], which comes with asymptotic scale parameter Λ (4) QCD = 326 MeV. We choose the MS renormalization, factorization, and NRQCD scales to be µ r = µ f = ξm T and µ Λ = ηm c , respectively, where m T = p 2 T + 4m 2 c is the transverse mass, and independently vary ξ and η by a factor of two up and down about their default values ξ = η = 1 to estimate the scale uncertainty.
The branching fraction B of X(3872) → J/ψπ + π − is not yet known, so that we can only determine the products O X [n] B. In our previous fit [20], we in-  cluded CDF [2,15], LHCb [16], and CMS [17] data of prompt X(3872) hadroproduction. Here, we perform an update by also including the recent ATLAS data [18].
1 ] B = 0.83 +0.12 −0.16 × 10 −4 GeV 3 , in good agreement with both our previous two-parameter fits [20], including and excluding the LHCb [16] data, lying inbetween them. The fit quality is excellent, with χ 2 /d.o.f. = 7.25/9 = 0.81. This is also evident from Fig. 2, where the cross sections of X(3872) prompt hadroproduction, differential in p X T , as measured by CMS [17] and ATLAS [18] are compared with our NLO NRQCD results. Also the integrated cross sections σ prompt (pp → X(3872) + anything)B = (3.1±0.7) nb and σ prompt (pp → X(3872)+ anything)B = (4.26 ± 1.23) nb measured by CDF [2,15] and LHCb [16] are compatible with our respective NLO NRQCD results, (2.2 ± 0.8) nb and (5.8 ± 1.5) nb. Here and in the following, the theoretical uncertainties are evaluated by combining the scale and fit errors in quadrature. Excluding the CDF [2,15] and LHCb [16] data from our fit and so imposing the cut p T > 10 GeV, we obtain O X [ 3 P 1 ] , we can also extract valuable information on the χ cJ (2P ) LDMEs, where we have exploited heavy-quark spin symmetry relations valid to LO in v 2 among the LDMEs for J = 0, 1, 2. This will in turn allow for new predictions of χ cJ (2P ) hadroproduction. In this connection, it is interesting to observe that our central value r = 0.055 is consistent with the result for the χ cJ (1P ) case found in Ref. [41], 0.045 ± 0.010.
With our new LDMEs, we are now in a position to predict λ X θ and λ ψ θ at NLO in NRQCD. We consider three LHC setups as for pp CM energy √ S and X(3872) rapidity y X , corresponding to LHCb [16], CMS [17], and ATLAS [18] experimental conditions: (i) √ S = 7 TeV and 2.0 < y X < 4.5; (ii) √ S = 7 TeV and |y X | < 1.2; and (iii) √ S = 8 TeV and |y X | < 0.75. In Fig. 3, λ X θ is presented as a function of p X T for these three setups. From there, we observe that the line shapes are very similar for the three setups and that λ X θ ranges between 0.04 and 0.10, which is in the upper part of the allowed corridor. Thus, on the basis of our hypothesis, the X(3872) is predicted to be predominantly longitudinally polarized in all three setups, which is now up to experimental verification. Unfortunately, a reliable extraction of the X(3872) polarization from the measurement of λ X θ is hampered by the narrowness of its allowed corridor, which necessitates high experimental precision. We note that Eq. (5) is restricted to the decay channel X(3872) → J/ψπ + π − and so is Fig. 3. However, the reader can easily extract the process independent quantity R from Fig. 3 using the relationship R = (1 + 15.2 λ X θ )/(1 − 7.09 λ X θ ) following from Eq. (5) with our fit value of g. R is found to monotonically fall with increasing p X T , from about 5 at p X T = 10 GeV down to its asymptotic value of about 2.2. Fortunately, the experimental disadvantage of λ X θ may be circumvented by measuring λ ψ θ for the J/ψ from X(3872) decay instead. In fact, λ ψ θ is much more sensitive to the J/ψ polarization than λ X θ is to the X(3872) polarization. Figure 3 also shows λ ψ θ as a function of p ψ T for the three setups. From there, we observe that the J/ψ is predicted to be largely transversely polarized, especially in the lower p ψ T range, where λ ψ θ > ∼ 0.5. Throughout the full p ψ T range considered, λ ψ θ is so distinctly separated from zero that this should be well discernible experimentally with reasonable statistics.
In the molecular picture, X(3872) is a loosely bound S-wave state of D * 0D0 + c.c. Since D 0 (D 0 ) is a pseudoscalar, all the information on the X(3872) polarization is carried by the D * 0 (D * 0 ) vector. At hadron colliders, prompt D * 0 's (D * 0 's) arise from the nonperturbative evolution of perturbatively produced c's (c's), and we are not aware of a mechanism that leads to polarized D * 0 's (D * 0 's). In fact, this argument is supported by several experimental measurements of D * 0 (D * 0 ) polarization in e + e − annihilation at different CM energies, for example by ARGUS [42]. We thus infer that, in the molecular picture, the prompt X(3872)'s would be unpolarized and so would the J/ψ's from their decays.
We now argue that the proposed X(3872) and J/ψ polarization measurements are feasible using the LHC signal events already on tape by now and thus are not subject to delay by the ongoing Long Shutdown 2, which will impede proton physics before May 2021. CMS managed to perfom a full-fledged ψ(2S) polarization measurement using 262 k ψ(2S) → µ + µ − events in the range 14 < p ψ ′ T < 50 GeV and |y ψ ′ | < 1.2 [43]. On the other hand, they collected 11.91 k X(3872) → J/ψπ + π − → µ + µ − π + π − events in almost the same kinematic range, 10 < p X T < 50 GeV and |y X | < 1.2, using an integrated luminosity of 4.8 fb −1 [17]. At present, the integrated luminosity accumulated by CMS is 29.3 fb −1 from Run 1 and 160 fb −1 from Run 2 [44]. Assuming that acceptance and efficiency have been approximately steady during the data taking periods, this translates into 11.91 k × 29.3/4.8 = 72.7 k and 397 k prompt X(3872) events waiting to be analyzed with regard to their X(3872) and J/ψ polarizations. The data sample from Run 2 alone is more than 50% more copious than the one underlying the ψ ′ polarization measurement [43] and should thus conveniently allow for the proposed polarization measurements. A similar conclusion can be drawn for LHCb on the basis of Refs. [16,45].
In summary, we studied the prompt hadroproduction of the mysterious X(3872) and its subsequent decay to J/ψ + π + π − in the NRQCD factorization framework [21] at NLO in α s , under the likely assumption that the creation of the X(3872) proceeds chiefly through the χ c1 (2P ) component of its short-distance wave function. We updated our previous fits of the X(3872) LDMEs [20] by including the latest ATLAS data [18], scoring an excellent goodness, as low as χ 2 /d.o.f. = 0.81 (see also Fig. 2). This also allowed us to predict the CO to CS LDME ratio of the χ cJ (2P )'s. Exploiting our fit results, we presented the first predictions of the polarization parameters λ X θ and λ ψ θ , in the HX frame for LHCb-, CMS-, and ATLAS-like setups. These imply that the X(3872) and J/ψ polarizations are largely longitudinal and transverse, respectively. Comparing the sizes of available LHC data sets on the X(3872) prompt yield with those on the ψ ′ polarization, we concluded that meaningful measurements of λ X θ and λ ψ θ should be feasible already now, during the LHC Long Shutdown 2. While the reliable interpretation of such measurements of λ X θ will be aggravated by the fact that the theoretically allowed λ X θ window is only 0.21 wide, there is no such limitation for λ ψ θ . Our predictions are distinctly different from those of the molecular picture [11], in which the X(3872) and J/ψ vectors are expected to be both unpolarized, as we argued on the basis of Ref. [42]. Their experimental verification would simultaneously confirm both the validity of NLO NRQCD for χ cJ polarization and the χ c1 (2P ) dominance hypothesis in X(3872) prompt production. Should the χ c1 (2P ) be discovered distinguishably from the X(3872), then the χ c1 (2P ) dominance hypothesis could be tested, regardless of the validity of NRQCD factorization, by comparing the J/ψ polarizations measured in χ c1 (2P ) and X(3872) decays. We conclude by urging the LHC collaborations to extract λ X θ and λ ψ θ from available and future prompt-X(3872) data and to perform similar analyses also for other X, Y, Z states with non-zero spin, which will allow us to distinguish between different models and so to take a major step in pinning down the nature of the X, Y, Z states.
We thank Eric Braaten and Xiao-Rui Lyu for very useful discussions. This work was supported in part by BMBF Grant No. 05H18GUCC1.