Prompt production of the hidden charm pentaquarks in the LHC

Motivated by the observation of the first hidden charm pentaquarks by the LHCb collaboration in 2015 and the updated analysis with an order-of-magnitude larger data set in 2019, we estimate their cross sections for the prompt production as well as their heavy quark spin partners, in the $\Sigma_c^{(*)}\bar{D}^{(*)}$ hadronic molecular picture, at the center-of-mass energy $7~\mathrm{TeV}$ in the $pp$ collision. Their cross sections are several $\mathrm{nb}$ and we would expect several tens hidden charm pentaquark events in the LHC based on its current integrated luminosity. The cross sections show a sizable deviation of the cross sections for hidden charm pentaquarks with the third isospin component $I_z=+\frac{1}{2}$ ($P_c^+$) from those with $I_z=-\frac{1}{2}$ ($P_c^0$). The cross sections decrease dramatically with the increasing transverse momentum. Our study can also tell where to search for the missing hidden charm pentaquarks. The confirmation of the complete hidden charm pentaquarks in the heavy quark symmetry would further verify their $\Sigma_c^{(*)}\bar{D}^{(*)}$ molecular interpretation. In addition, the relative strength among these cross sections for pentaquarks can help us to identify the quantum numbers of the $P_c(4440)$ and $P_c(4457)$.

in the pp collision. Their cross sections are several nb and we would expect several tens hidden charm pentaquark events in the LHC based on its current integrated luminosity. The cross sections show a sizable deviation of the cross sections for hidden charm pentaquarks with the third isospin component I z = + 1 2 (P + c ) from those with I z = − 1 2 (P 0 c ). The cross sections decrease dramatically with the increasing transverse momentum. Our study can also tell where to search for the missing hidden charm pentaquarks. The confirmation of the complete hidden charm pentaquarks in the heavy quark symmetry would further verify their Σ

I. INTRODUCTION
The successful prediction of the Ω baryon has set a milestone of the conventional quark model, which hints the existence of the color degree of freedom and leads to the fundamental theory of the strong interaction, i.e. quantum chromodynamics (QCD). It is also a typical example of the connection between spectroscopy and underlying dynamics. The multiquark was first quantitatively studied by Jaffe in 1976 [1] at the budding period of quark model. In the following tens of years, both theorists and experimentalists were committed to searching for the missing states and those beyond the conventional quark model, namely exotic states. The enthusiasm of studying multiquark states, however, was limited by the negative results and the low statistics of the experimental data. The situation broke until 2003 by the observation of the D * s0 (2317) [2] and the X(3872) [3]. As both the masses of the D * s0 (2317) and the X(3872) are significantly lower than its quark model expectation, they seriously challenged the conventional quark model and served as strong candidates for the exotic states. Up to now, tens of exotic candidates [4,5] have been observed and various proposals were put forward about their configurations from theoretical side [6][7][8][9][10][11][12][13][14][15][16].
In a more general concept, all the boson (fermion) hadrons are classified to meson (baryon).
The statistics of the experimental data for baryon sector is usually lower than that for the meson sector due to the number of quarks. Accordingly, the experimental data for exotic baryons are more scarce. Research enthusiasm is rekindled by the observation of the first hidden charm pentaquarks P c (4380) and P c (4450) in the J/ψp invariant mass distribution of the Λ b → J/ψpK − process [22].
As they strongly decay into J/ψp, they contain as least ccuud five quarks unambiguously. Even before their observation, the hidden charm pentaquarks were proposed analogous to the excited baryons in light sector, e.g. in Refs. [23][24][25][26][27][28][29]. An updated analysis [30] of the LHCb Collaboration with an order-of-magnitude larger data set combined Run-I and II shows that the P c (4450) splits into two narrow peaks P c (4440) and P c (4457), and a third pentaquark P c (4312) emerges. Various interpretations follow this analysis, such as hadronic molecules , compact pentaquarks [56,[62][63][64][65][66][67][68], hadro-charmonia [69][70][71], and cusp effects [56]. Among these interpretations, the Σ  [53,72] that the Σ * cD molecule should correspond to a new narrow P c (4380) which leaves a hint for its existence on the J/ψp distributions. However, the reason for almost invisibility of the Σ * cD * molecules remains to be understood. It could be caused by the small production for the Σ * cD * channel in the Λ 0 b decay compared to other channels. Therefore, searching for the missing pentaquarks in different processes is a demanding task to complete the full spectrum and shed light on the underlying dynamics. On the other side, their closeness to the Σ ( * ) cD ( * ) also indicates a large isospin violation in their decay rate, for instance the P c (4457) → J/ψ∆ + process [33], and a significant deviation of the cross sections of the P + c pentaquarks with the third isospin component I z = + 1 2 from those of their isospin partners P 0 c with I z = − 1 2 . Besides the Σ ( * ) cD ( * ) hadronic molecular picture, the peaking structures in J/ψp distributions may be caused by kinematical effects, e.g. triangle singularities or cusps [30,56,75]. For instance, the P c (4457) might be generated by the Λ + c (2590)D 0 D * * s triangle diagram [30]. The triangle singularities arise when all three hadrons in the triangle-diagrams are nearly on mass shells and are manifested as peaks in the mass distributions [15]. Their manifestation are particularly sensitive to the momenta of incoming and outgoing particles. One of the conditions for the triangle singularities is that the mass of the decaying particle, i.e. Λ 0 b , should be very close to the threshold of the connected two internal particles. Once the deviation is larger, the triangle singularity condition will not be satisfied and the corresponding peaks will disappear. Thus searching for the hidden pentaquark states in the prompt production with a large incoming energy region will help to exclude the potential triangle singularity interpretation. Motivated by the above arguments, we use Madgraph5 [76] and Phythia8 [77] to stimulate the prompt production rate of hidden charm pentaquarks in the hadronic molecular picture. Our framework is presented in Sec. II. Results and discussions follows. A brief summary and outlook are given in the last section.

II. FRAMEWORK
The inclusive production of a loosely bound S-wave hadronic molecule in hadron collision can be separated into a long-range part and a short-range part [78][79][80][81][82] 1 , which is based on the universal scattering amplitude for the low energy scattering. This kind of separation allows for estimating the 1 The calculation in this work is based on the Σ ( * ) cD ( * ) molecular picture. The analogous calculation can also be done for the compact pentaquarks similar to the production of the X(3872) in pp collision [83].
cross section of the inclusive production for a given hadronic molecule, for instance the production of the X(3872) [78,79] and its bottom analogs [82], the D s0 (2317) [80], the charged Z ( )± c and Z ( )± b [81]. We employ the formula presented in Refs. [81,82] to estimate the cross sections of inclusive prompt productions of hidden charm pentaquarks observed by LHCb in 2019.

A. Factorization
The production amplitude for the inclusive production of hidden charm pentaquarks, in the hadronic molecular picture, with small binding energy can be factorized as [81,82] as illustrated by interaction and a long-range interaction, respectively. This idea is proposed in Refs. [78,79] and has been used to estimate the production of heavy quarkonium-like states in Refs. [81,82,[84][85][86][87] and charm-strange molecules in Ref. [80] 2 . In this work, since we only aim at an estimate of order-of-magnitude, only the production through S-wave Σ + all] is the production amplitude of the corresponding constituent for a given 2 The debate of the yield of the X(3872) at high pT can be found in Refs. [83,85,88] pentaquark. T α is the amplitude for the Σ ( * ) cD ( * ) to the P c pentaquark.
is the intermediated two-body propagator, with p α the non-relativistic three momentum of the αth channel, extracting from below equation Here µ α , m α th and E are the reduced mass, threshold of the αth channel and the total energy, respectively. In low energy, we are interested in, to the leading order, the amplitudes M[ Σ and T α could be treated as constants [81,82], leaving the equation reduced into an algebraic equation. As the result, we only need to calculate the integration which is linearly divergent and needs to be regularized. To that end, a hard cut-off Λ is introduced to render the integral well defined The value of Λ is determined by the effectiveness of the low energy theorem which inherits the non-perturbative mechanism of strong interaction and is of order of 1 GeV. In our case, we take values Λ = [0.7, 1.3] GeV to estimate the cross sections 3 .
Before going into the production of the P c states, the cross section of the inclusive Σ production should be estimated by Monte Carlo (MC) simulation and reads as with dφ Σ ( * ) cD ( * ) +all the phase space of the Σ ( * ) cD ( * ) and all the other particles. k is the relative momentum between Σ ( * ) cD ( * ) in its center-of-mass frame. K Σ ( * ) cD ( * ) is a normalization factor to compensate the difference between the MC and experimental data, and is taken the value 1 as an order-of-magnitude estimate. In total, the cross section of the inclusive P c production can be written as where |g Pc means the fraction of the αth channel events to a given P c with spin J Pc . The sum i in the denominator runs all the pentaquark states which couple to the αth channel. g α is the coupling of the αth channel to a given P c state and the values can be found in App. A.

B. Monte Carlo Simulation
The production of hidden charm pentaquarks in the hadronic molecular picture should follow the production of the corresponding constituents, i.e. the heavy quark pair cc in the parton level. Considering the other produced particles in the inclusive process, a third parton should be produced simultaneously. As the result, the 2 → 3 parton process should be generated through hard scattering and hadronized into final hadrons via non-perturbative mechanism.
Similar to those in Refs. [81,82], we generate the 2 → 3 process via Madgraph5 [76] and use Phythia8 [77] for the hadronization. As the two constituents Σ ( * ) cD ( * ) should be collinear and with relative small momentum, the cut p T > 3.5 GeV and |y| < 2 are implemented for the heavy quark pair. In principle, all the parton level 2 → 3 processes will contribute. However, we demonstrate numerically, as shown in Fig. 2, that the gg → gcc 4 is the most important parton process similar to its contribution in the X(3872) production [78,79]. Accordingly, to improve the efficiency of the code, only the gg → gcc process is considered in Madgraph5. In addition, the dependence on the relative momentum k is [81,82] dσ[Σ at low energy without considering the final-state-interaction (FSI).
As the hadronization process is still unclear and model-dependent, the hadronization implemented in Phythia8 [77] is incomplete and we might underestimate the yield of heavy hadron 4 To estimate the uncertainty of this approximation, we compare the Σ + cD 0 inclusive cross sections within these two frameworks and find that the deviation is under 5%.
3. MC simulation of the cross section of the inclusive D 0 (left) and Λ + c (right) production in the pp collision at √ s = 7 TeV comparing with the experimental data from Refs. [89,90]. The rapidity cut |η| < 0.5 is implemented for charm quark to compare with the experimental data. The black box and blue triangle points are for the experimental data and MC results, respectively. pairs. A comparison between the MC simulation results, with all the parton processes considered, of the D 0 meson and the Λ + c charm baryon 5 with the experimental data are presented in Fig. 3. Although the experimental data and the MC simulation result are of the same order, the deviation is still sizable, especially for the charmed baryon. The deviation is because of the missing dynamics in Phythia8, for instance the feed-down charm meson/baryon from bottom meson/baryon [91]. As we only make an order-of-magnitude estimate for the cross sections, this deviation can be accepted.
in the hadronic molecular picture. To study the deviation quantitatively, the ratio between the cross sections of the P + c and P 0 c is also defined, where the statistic uncertainties have been cancelled out. The cross sections and the corresponding ratios for the inclusive production production rates decrease dramatically with the increasing p T , which stems from the decreasing behavior of the fragmentation functions [78]. The significant deviations of the cross sections of the charged P + c s from those of the neutral ones P 0 c s can also be seen directly from cross section 6 The third isospin components of P + c and P 0 c are Iz = + 1 2 and Iz = − 1 2 , respectively. In what follows, if the charged property of the hidden charm pentaquarks is not specified, the argument works for both of them.            Table I  The result of two solutions for the P c productions are collected in Table I and Fig. 5. In both solutions, the cross section for P c (4380) is the largest one due to the largest production of Σ * cD channel (Eq. (B1)) and its strong coupling to P c (4380), which makes the prompt production in pp collision an ideal platform for the search of the narrow P c (4380) [53,72]. For J P = 1/2 − channel, with subindexes A and B for the two solutions. As they mainly couple to the Σ cD , Σ cD * , Σ * cD * channels, respectively, the relations Eqs. (13) and (14) are largely determined by the cross sections of their constituents For the Σ * cD * relevant P c states, a further suppressed factor comes from the denominator of Eq. (8) in both solutions. That is because the cross section of a state is proportional to the absolute square of its coupling to the constitutes and thus its binding energy for shallow bound states [11].
From Table I Table I  coupling. An important observation is that the three Σ * cD * molecular states exhibit different patterns in two solutions. For Solution B, the state with higher spin has a larger binding energy, and thus effective coupling. Combined with the enhancing factor (2J + 1) in Eq. (8) for higher spins, it leads to a significant relation, see e.g. in Fig. 5, However, the enhancing factor for the higher spin is roughly balanced by the smaller effective coupling in Solution A, which makes the cross sections for the three states comparable. We stress that this difference is independent on the production of Σ and the corresponding potential is The case for the J P = 3 2 − channel, the dynamical channels are Σ cD * , Σ * cD , Σ * cD * and the corresponding potential is There is only one channel, i.e. Σ * cD * for the J P = 5 2 − channel and the potential is In the whole manuscript, we use the scattering amplitudes of pure contact results in Ref. [72] as inputs. The inclusion of the OPE and higher order contact potentials will not change the results significantly. When fit to the J/ψp invariant mass distribution of the Λ b → J/ψpK − process, two solutions can be found [53,72], i.e. Solution A and Solution B as denoted in Refs. [34,48]. In Solution A, the P c (4440) and the P c (4457) are assigned as 1 2 − and 3 2 − pentaquarks, respectively.
In Solution B, they are interchanged. In the whole manuscript, we use the effective couplings with hard cutoff 1 GeV in Ref. [72] as inputs. In the following, the effective couplings are collected in   FIG. 6. The cross section of the inclusive production of the Σ cD pairs in pp collisions at center-of-mass energy √ s = 7 TeV. The black inverted triangle, blue triangle, red box points are for the relative three momentum smaller than 0.5m π , m π and 2m π , respectively. As shown in the figures, there are significant isospin deviation for the inclusive productions of the Σ cD pairs. In our calculation, we use m π to estimate the prompt cross section.  FIG. 9. The caption is analogous to that of Fig. 6, but for the Σ * cD * channel.