Production of the bottom analogues and the spin partner of the X(3872) at hadron colliders

Using the Monte Carlo event generator tools Pythia and Herwig, we simulate the production of bottom/charm meson and antimeson pairs at hadron colliders in proton-proton/antiproton collisions. With these results, we derive an order-of-magnitude estimate for the production rates of the bottom analogues and the spin partner of the $X(3872)$ as hadronic molecules at the LHC and Tevatron experiments. We find that the cross sections for these processes are at the nb level, so that the current and future data sets from the Tevatron and LHC experiments offer a significant discovery potential. We further point out that the $X_b/X_{b2}$ should be reconstructed in the $\gamma \Upsilon(nS) (n=1,2,3)$, $\Upsilon(1S)\pi^+\pi^-\pi^0$, or $\chi_{bJ}\pi^+\pi^-$ instead of the $\Upsilon(nS)\pi^+\pi^-$ final states.

cays of the X(3872). An important aspect involves the discrimination of a compact multiquark configuration and a loosely bound hadronic molecule configuration. Recent calculations of the hadroproduction rates at the LHC based on nonrelativistic QCD indicate that the X(3872) could hardly be an ordinary charmonium χ c1 (2P ) [13,14], while there are sizable disagreements in theoretical predictions in the molecule picture [15][16][17][18][19].
To clarify the intriguing properties and finally decipher the internal nature, more accurate data and new processes involving the production and decays of the X(3872) will be helpful. For instance, one may obtain useful information on the flavor content of the X(3872) from precise measurements of decays of neutral/charged B mesons into the X(3872) associated with neutral/charged K * mesons.
On the other hand, it is also expedient to look for the possible analogue of the X(3872) in the bottom sector, referred to as X b following the notation suggested in Ref. [20]. If such a state exists, measurements of its properties would assist us in understanding the formation of the X(3872) as the underlying interaction is expected to respect heavy flavor symmetry. In fact, the existence of such a state was predicted in both the tetraquark model [21] and hadronic molecular calculations [22][23][24].
The mass of the lowest-lying 1 ++bq bq tetraquark was predicted to be 10504 MeV in Ref. [21], while the mass of the BB * molecule based on the mass of the X(3872) is a few tens of MeV higher [23,24]. In Ref. [23], the mass was predicted to be (10580 +9 −8 ) MeV for a typical cut-off, corresponding to a binding energy of (24 +8 −9 ) MeV. Notice that there is a big difference between the predicted X b and the X(3872). The distance of the mass of the X(3872) to the D 0D * 0 threshold is much smaller than the distance to the D + D * − threshold. This difference leaves its imprint in the wave function at short distances through the charmed meson loops so that a sizeble isospin breaking effect is expected. However, the mass difference between the charged and neutral B mesons is only (0.32 ± 0.06) MeV [8], and the binding energy of the BB * system may be larger than that in the charmed sector due to a larger reduced mass. In addition, while the isospin breaking observed in the X(3872) decays into J/ψ and two/three pions can be largely explained by the phase space difference between the X(3872) → J/ψρ and the X(3872) → J/ψω [25], the phase space difference between the Υρ and Υω systems will be negligible since the mass splitting between the X b and the Υ(1S) is definitely larger than 1 GeV. Therefore, we expect that the isospin breaking effects would be much smaller for the X b than that for the X(3872). Consequently, the X b should be an isosinglet state to a very good approximation, in line with the predictions in Refs. [22][23][24].
Since the mass of the X b is larger than 10 GeV and its quantum numbers J P C are 1 ++ , it is unlikely to be discovered at the current electron-positron collision facilities, though the prospect for an observation in the Υ(5S, 6S) radiative decays at the Super KEKB in future may be bright due to the expected large data sets, of order 50 ab −1 [26]. In this paper, we will follow the previous work on the search for exotic states at hadron colliders [16][17][18][19][27][28][29] and focus primarily on the production of the X b and its spin partner, a B * B * molecule with J P C = 2 ++ , denoted as X b2 , at the LHC and the Tevatron. Results on the production of the spin partner of the X(3872), X c2 with J P C = 2 ++ , will also be given. Notice that due to heavy quark spin symmetry, the binding energies of the X b2 and X c2 are similar to those of the X b and X(3872), respectively. This paper is organized as follows. We begin in Sec. II by discussing the factorization formula for the pp/p → X b /X b2 (both pp and pp will be written as pp for simplicity in the following) amplitudes in case that the X b /X b2 is a bound state not far from threshold. Our numerical results for the cross sections of the inclusive processes pp → B ( * )B * and pp → X b /X b2 are presented in Sec. III. The last section contains a brief summary.

II. HADROPRODUCTION
Consider an S-wave loosely bound state with a binding energy b. If the scattering length a = 1/ √ 2µb, where µ is the reduced mass of the constituents, is much larger than the range of forces, the system has universal properties determined by solely by the scattering length a [30].
In particular, the energy dependence of the S-wave scattering amplitude in the region close to the threshold should be described by (see, for instance, Ref. [30]) if there is no any other nearby resonance with the same quantum numbers. Here, E is the energy relative to the scattering threshold.
Based on the universal amplitudes 1 and the Migdal-Watson theorem, the authors in Refs. [18,19] have derived a formula which may be used as an estimate for the cross section of the inclusive production of an S-wave loosely bound hadronic molecule, and used it in the case of the process pp → X(3872). If the binding energy of the X b , E X b , is small so that the the binding momentum 2µE X b is much smaller than the mass of the pion, the lightest meson which can be exchanged between a pseudoscalar and a vector bottom mesons, one can use the same formula to estimate the production cross section of the X b , which reads where σ(pp → BB * [k < Λ]) is the cross section for the inclusive process pp → BB * with the relative momentum of the B in the meson pair rest frame smaller than a cutoff Λ ≈ M π . This quantity will be estimated using Monte Carlo event generators in the following section.
The above formula is derived on the basis of factorization. In order to form a molecule, the mesonic constituents must be produced at first and have to move collinearly with a small relative momentum. Such configurations originate from the inclusive QCD process which contains aQQ pair with a similar relative momentum in the final state. Thus, at least a third parton needs to be produced in the recoil direction, which corresponds to a 2 → 3 parton process. In our explicit realization, the 2 → 3 process can be generated initially through hard scattering, and the parton shower will produce more quarks via soft radiations.

III. RESULTS AND DISCUSSIONS
Following our previous work [29], we use Madgraph [31] to generate the 2 → 3 partonic events with a pair of a heavy quark and an antiquark (bb orcc) in the final states, and then pass them to the Monte Carlo event generators for hadronization. To improve the efficiency of the calculation, we apply the partonic cut for the transverse momentum p T > 2 GeV for heavy quarks and light jets, m bb < 11 GeV (k BB * = 1.5 GeV and k B * B * = 1.4 GeV at the hadron level), m cc < 5 GeV (k DD * = 1.6 GeV and k D * D * = 1.5 GeV) and ∆R(b,b) < 1 (∆R(c,c) < 1) where ∆R = ∆η 2 + ∆φ 2 (∆φ is the azimuthal angle difference and ∆η is the pseudo-rapidity difference of the bb). We choose Herwig [32] and Pythia [33] as our hadronization generators, whose outputs are analyzed using the Rivet library [34]. Notice that although the binding energies of the X b and X b2 are normally predicted at the order of a few tens of MeV, the uncertainty could be of the same order as discussed in Ref. [23].
Since none of them has been observed, we will attempt to use three benchmark values: for the X c2 with a 0.3 MeV binding energy) 2 . From the table, one sees that the cross sections for the X b2 is similar to those for the X b , and the ones for the X c2 are two orders of magnitude larger. Tevatron experiments (CDF and D0) at 1.96 TeV, we use |y| < 0.6; the rapidity range 2.0 < y < 4.5 is used for the LHCb. Since the binding energy is not measured, we use 1 MeV, 2 MeV and 5 MeV for illustration.
For the X c2 , we choose 0.3 MeV, 2 MeV and 5 MeV.   (20) Recently, the CMS Collaboration has presented results of a first search for new bottomonium states, with the main focus on the X b , decaying to Υ(1S)π + π − . The search is based on a data sample corresponding to an integrated luminosity of 20.7 fb −1 at √ s = 8 TeV [35]. No evidence for the X b is found, and the upper limit at a confidence level of 95% on the product of the production cross section of the X b and the decay branching fraction of X b → Υ(1S)π + π − has been set to be where the range corresponds to the variation of the X b mass from 10 to 11 GeV.
Using the current experimental data on the σ(pp → Υ(2S)), we can convert the above ratio into the cross section which can be directly compared with our results. Since the masses of the Υ(2S) and X b are not very different, it may be a good approximation to assume that the ratio given in Eq. (3) is insensitive to kinematic cuts. Using the CMS measurement in Ref. [36]: with the cuts p T <50 GeV and |y| < 2.4 for the Υ(2S), we get Since the branching ratio B(X b → Υ(1S)π + π − ) is expected to be tiny (see below), the above upper bound is much larger than our predictions.
As already discussed in the Introduction, the X b and X b2 are probably isosinglets. In contrast to the X(3872), the isospin breaking decays of these two states will be heavily suppressed. Thus, one cannot simply make an analogy to the X(3872) → J/ψπ + π − and try to search for the X b in the Υ(1S, 2S, 3S)π + π − channels, as the isospin of the Υ(1S, 2S, 3S)π + π − systems is 1 when the quantum numbers are J P C = 1 ++ . This could be the reason for the negative search result by the CMS Collaboration [35]. Possible channels which can be used to search for the X b and X b2 include the Υ(nS)γ (n = 1, 2, 3), Υ(1S)π + π − π 0 and χ bJ π + π − . The X b2 can also decay into BB in a D-wave, and the decays of the X c2 are similar to those of the X b2 with the bottom being replaced by its charm analogue. The isospin breaking decay X c2 → J/ψπ + π − through an intermediate ρ meson should be largely suppressed compared with the decay of the X(3872) into the same particles because the mass of the X c2 is about 140 MeV higher than that of the X(3872), and the phase space difference between the J/ψρ and J/ψω becomes tiny.
Compared with the pionic decays, the Υ(nS)γ (n = 1, 2, 3) final states are advantageous because no pion needs to be disentangled from the combinatorial background. The disadvantage is the low efficiency in reconstructing a photon at hadron colliders. Since the X(3872) meson has a sizable partial decay width into the J/ψγ [8] B(X(3872) → γJ/ψ) > 6 × 10 −3 , presumably the branching ratio for the X b → γΥ is of this order. If so, the cross section for the Apart from the production rates, the nonresonant background contributions can also play an important role in the search for these molecular states at hadron colliders since a signal could be buried by a huge background. To investigate this issue, we consider the X b as an example, which will be reconstructed in Υ + γ final states. In this process, the inclusive cross section σ(pp → Υ + anything) can serve as an upper bound for the background. It has been measured at √ s = 7 TeV by the ATLAS Collaboration as [43] σ(pp → Υ(1S)(→ µ + µ − ) + anything) = (8.01 ± 0.02 ± 0.36 ± 0.31) nb, where 2.6% is the branching fraction of the Υ(1S) → µ + µ − [8], and 10 −2 is a rough estimate for the branching fraction of the X b → Υ(1S)γ. The value of the signal/background ratio can be significantly enhanced in the data analysis by employing suitable kinematic cuts which can greatly suppress the background, and accumulating many more events based on the upcoming 3000fb −1 data [40].

IV. SUMMARY
In summary, we have made use of the Monte Carlo event generator tools Pythia and Herwig, and explored the inclusive processes pp/p → B 0B * 0 and pp/p → B * 0B * 0 at hadron colliders. Based on the molecular picture, we have derived an estimate for the production rates of the X b , X b2 and X c2 states, the bottom and spin partners of the X(3872), at the LHC and Tevatron experiments.
We found that the cross sections are at the nb level for the hidden bottom hadronic molecules X b and X b2 , and two orders of magnitude larger for the X c2 . Therefore, one should be able to observe them at hadron colliders if they exist in the form discussed here. The channels which can be used to search for the X b and X b2 include the Υ(nS)γ (n = 1, 2, 3), Υ(1S)π + π − π 0 , χ bJ π + π − and BB (the last one is only for the X b2 ), and the channels for the X c2 is similar to those for the X b2 (with the bottom replaced by its charm analogue). In fact, both the ATLAS and D0 Collaborations reported an observation of the χ b (3P ) [41,42], whose mass is (10534 ± 9) MeV [8], slightly lower than the X b and X b2 , in the Υ(1S, 2S)γ channels. A search for these states will provide very useful information in understanding the X(3872) and the interactions between heavy mesons. Especially, if the X b , which is the most robust among the predictions in Ref. [23] based on heavy quark symmetries, cannot be found in any of these channels, it may imply a non-molecular nature for the X(3872).