Hidden-charm Pentaquark Production at e⁺e⁻ Colliders^*

To study the properties and production mechanisms of the multiquark state is important to understand the quark model and the strong interactions deeply. The newest evidence for the existance of the pentaquark is from LHCb experiments. Recently, the LHCb Collaboration announced the observation of two charged hidden-charm resonances ${P}_{c}^{+}(4380)$ and ${P}_{c}^{+}(4450)$ produced in the process ${\Lambda }_{b}^{0}\to {K}^{-}J/\psi p$.^[1–3] The corresponding decay channel P_c → J/ψp indicates that their minimal quark content is $uudc\bar{c}$. The most important work at present is to confirm whether the new resonances are pentaquarks or not. Up to now, lots of theoretical investigations have been finished. Several possibilities for these new resonances to be baryon-meson molecules, two- or three-cluster compact states, etc., have been discussed.^[4–5] In fact, studies on hidden-charm pentaquarks have been started before the observation of the P_c states. In Ref. [6], the interaction between a charmed baryon and an anticharmed meson was studied and it was found that hadronic molecules with the mass above 4 GeV are possible. While in Refs. [7–18], the possibility for this kind of bound state to be the hidden-heavy pentaquark was studied.

In the mean time, the investigations on the pentaquark production in various collisions have also been performed. In Ref. [15], the discovery potential of hidden-charm pentaquarks in the photon-induced production was discussed. In Ref. [19], the J/ψ photoproduction of the two P_c states off the proton was proposed to understand their nature. References [20–24] also presented the pentaquark production in γN collision processes. In Refs. [17, 25–27], the hidden-charm pentaquark effects in the π⁻p reaction were considered. Discussions for the production in bottom baryon decays^[28–30] and in heavy-ion or pA collisions^[31–32] can also be found.

However, most of the discussions on production in the present literature are at hadron level and the understanding for the pentaquark structures needs more studies. Obviously, the ${P}_{c}^{+}(4380)$ is not a bound state of J/ψ and nucleon. A compact $uudc\bar{c}$ pentaquark state with colored $c\bar{c}$ may be formed through gluon-exchange interactions. The spectrum and qualitative decay properties of the compact pentaquarks^[33–34] indicate that such a configuration is not contradicted with the observed P_c states. In principle, various configurations of a five-quark system related to different production mechanisms are worthy to be studied in detail. For the hadron-hadron molecules, the produced quarks fragment firstly into various hadrons and then the residual strong interactions between these hadrons lead to possible hadronic molecules. One may study this kind of production mechanism at hadron level.^[35] For the compact $uudc\bar{c}$ pentaquark state with colored $c\bar{c}$, a feasible approach is the framework proposed in Ref. [36]. Since the gluon is easily converted to a colored charm-anticharm pair, the production rate of the considered compact pentaquark might be significant. We here discuss the production of such a type of pentaquark in the multiproduction process, which can help to better understand their structure as well as the strong interaction mechanism at the hadronization scale.^[37] The information of the cross section, rapidity and transverse momentum distributions, etc., of the relevant particles on a specific collider can help the experimentalists to set the proper triggers and cutoffs for the measurements.^[35,38] Among the high energy collisions, the e⁺ e⁻ annihilation process is of special advantage for its clean background and one can gain more clear pictures on the color and other structure evolution via the study of the production.

For this kind of hidden-charm pentaquark states, we can rely on the perturbative QCD (PQCD) to calculate the charm quark pair production. On the other hand, how to embed the PQCD result into the production amplitude of the pentaquark, depends on the structure of the state and the framework for the approximation. Here we ignore the consideration of it as hadron bound state, which has been studied in the general case.^[35] A benchmark framework could be the heavy quark effective theory, which provides the feasible factorization formulation to connect the PQCD process with the parameterization of the nonperturbative QCD process, i.e., the PQCD produced $c\bar{c}$ transiting to the pentaquark, in the rest frame of the bound state. For concrete, we employ the nonrelativistic QCD (NRQCD) factorization framework. The NRQCD factorization approach has been used to discuss the production of Ξ_cc in Ref. [36] and that of T_cc in Ref. [39] at various e⁺ e⁻ colliders. The aim of this paper is to study the hidden-charm pentaquark production via a color-octet $c\bar{c}$ pair fragmentation in the process

$$\begin{eqnarray}&&{e}^{-}({p}_{1})+{e}^{+}({p}_{2})\to {\gamma }^{* }/{Z}^{0}\to {P}_{c}(k)+X,\end{eqnarray} \tag{ 1 }$$

where p₁, p₂, and k denote the momenta of the related particles. The unobserved part X can always be divided into a perturbative part X_P and a nonperturbative part X_N, X = X_N + X_P. The corresponding invariant amplitude can be written as

$$\begin{eqnarray}\begin{array}{ll} {\mathcal M} = & \displaystyle \int \frac{{{\rm{d}}}^{4}{k}_{1}}{{(2\pi )}^{4}}{A}_{ij}({k}_{1},{k}_{2},p)\displaystyle \int {{\rm{d}}}^{4}{x}_{1}{{\rm{e}}}^{-{\rm{i}}{k}_{1}\cdot {x}_{1}}\langle {P}_{c}(k)\\ & +{X}_{N}|{\bar{Q}}_{i}({x}_{1}){Q}_{j}(0)|0\rangle, \end{array}\end{eqnarray} \tag{ 2 }$$

where both i and j take Dirac and color indices. Q(x) is the Dirac field for charm quark. k₁ and k₂ respectively represent the momentum of c and $\bar{c}$ in the pentaquark state. p denotes the total momentum of the partons appearing in the perturbative part. In this paper, we will discuss two cases for the perturbative part, one is ${e}^{+}{e}^{-}\to c\bar{c}g$, and the other is ${e}^{+}{e}^{-}\to c\bar{c}Q\bar{Q}$. For the latter case, the quark pair can be light (u, d, s) or heavy (c, b).

First we consider the $c\bar{c}g$ case. If ⟨P_c(k) + X_N| of Eq. (2) is replaced by the state of a free charm-anticharm quark pair with momenta k₁ and k₂ and a gluon with momenta k_g, is the amplitude for the process ${e}^{+}{e}^{-}\to {\gamma }^{* }/{Z}^{0}\to c\bar{c}g$, so the corresponding contribution is the production of the color-octet charm quark-antiquark pair and then this pair fragments into the pentaquark P_c.

For the process

$$\begin{eqnarray}&&{e}^{+}({p}_{1})+{e}^{-}({p}_{2})\to {\gamma }^{* }/{Z}^{0}\to c({k}_{1})+\bar{c}({k}_{2})\\ &&{\rm{\quad\quad}}+g({k}_{g})\to {P}_{c}(k)+X,\end{eqnarray} \tag{ 3 }$$

the corresponding cross section can be written as

$$\begin{eqnarray}\begin{array}{ll}{\rm{d}}\sigma = & \frac{1}{2s}\frac{1}{4}\frac{{{\rm{d}}}^{3}k}{{(2\pi )}^{3}}\frac{{{\rm{d}}}^{3}{k}_{g}}{{(2\pi )}^{3}2{E}_{g}}{(2\pi )}^{4}{\delta }^{4}({p}_{1}+{p}_{2}-k-{k}_{g})\displaystyle \int \frac{{{\rm{d}}}^{4}{k}_{1}}{(2{\pi }^{4})}\frac{{{\rm{d}}}^{4}{k}_{3}}{(2{\pi }^{4})}{A}_{ij}({k}_{1},{k}_{2},{k}_{g}){({\gamma }^{0}{A}^{\dagger }({k}_{3},{k}_{4},{k}_{g}){\gamma }^{0})}_{lm}\\ & \times \displaystyle \int {{\rm{d}}}^{4}{x}_{1}{{\rm{d}}}^{4}{x}_{3}{{\rm{e}}}^{-{\rm{i}}{k}_{1}\cdot {x}_{1}+{\rm{i}}{k}_{3}\cdot {x}_{3}}\langle 0|{\bar{Q}}_{l}(0){Q}_{m}({x}_{3})|{P}_{c}+{X}_{N}\rangle \langle {P}_{c}+{X}_{N}|{Q}_{i}({x}_{1}){\bar{Q}}_{j}(0)|0\rangle .\end{array}\end{eqnarray} \tag{ 4 }$$

Here we take nonrelativistic normalization for P_c. Both l and m take Dirac and color indices related to γ⁰A^†γ⁰. The spin average of initial leptons, spin summation of final P_c, and the polarization and color summation of gluon are implied. k₃ and k₄ respectively represent the momenta of c and $\bar{c}$ in γ⁰A^†γ⁰ as shown in Fig. 1 where the black box represents the Fourier transformed matrix element of the second line in Eq. (5). By using translational covariance one can eliminate the summation over X_N. Defining the creation operator a^†(k) for P_c with the three momentum k, we obtain

$$\begin{eqnarray}\begin{array}{ll}{\rm{d}}\sigma = & \frac{1}{2s}\frac{1}{4}\frac{{{\rm{d}}}^{3}k}{{(2\pi )}^{3}}\frac{{{\rm{d}}}^{3}{k}_{g}}{{(2\pi )}^{3}2{E}_{g}}\displaystyle \int \frac{{{\rm{d}}}^{4}{k}_{1}}{(2{\pi }^{4})}\frac{{{\rm{d}}}^{4}{k}_{3}}{(2{\pi }^{4})}{A}_{ij}({k}_{1},{k}_{2},{p}_{3},{p}_{4}){({\gamma }^{0}{A}^{\dagger }({k}_{3},{k}_{4},{p}_{3},{p}_{4}){\gamma }^{0})}_{lm}\\ & \times \displaystyle \int {{\rm{d}}}^{4}{x}_{1}{{\rm{d}}}^{4}{x}_{2}{{\rm{d}}}^{4}{x}_{3}{{\rm{e}}}^{-{\rm{i}}{k}_{1}\cdot {x}_{1}-{\rm{i}}{k}_{2}\cdot {x}_{2}+{\rm{i}}{k}_{3}\cdot {x}_{3}}\langle 0|{\bar{Q}}_{l}(0){Q}_{m}({x}_{3}){a}^{\dagger }(k)a(k){Q}_{i}({x}_{1}){\bar{Q}}_{j}({x}_{2})|0\rangle .\end{array}\end{eqnarray} \tag{ 5 }$$

Since heavy quarks move with a small velocity v_Q inside the pentaquark in its rest frame, the Fourier transformed matrix element can be expanded in v_Q with fields of NRQCD. The relation between NRQCD fields and Dirac field Q(x) in P_c's rest frame is

$$\begin{eqnarray}&&Q(x)={{\rm{e}}}^{-{\rm{i}}{m}_{Q}t}\bigg\{\begin{array}{c}\psi (x)\\ 0\end{array}\bigg\}+{{\rm{e}}}^{{\rm{i}}{m}_{Q}t}\bigg\{\begin{array}{c}0\\ \chi (x)\end{array}\bigg\}{\mathscr{O}}({v}_{Q})\end{eqnarray} \tag{ 6 }$$

where ψ(x)(χ(x)) denotes the Pauli spinor field that annihilates (creates) a heavy (anti-) quark. We will work at the leading order of v_Q. In order to express our results for the Fourier transformed matrix element in a covariant way, we employ the four-velocity of the pentaquark with v_μ = k_μ/M_Pc. The Fourier transformed matrix element is related to that in the rest frame:^[36]

$$\begin{eqnarray}&&{v}^{0}\displaystyle \int {{\rm{d}}}^{4}{x}_{1}{{\rm{d}}}^{4}{x}_{2}{{\rm{d}}}^{4}{x}_{3}{{\rm{e}}}^{-{\rm{i}}{k}_{1}\cdot {x}_{1}-{\rm{i}}{k}_{2}\cdot {x}_{2}+{\rm{i}}{x}_{3}\cdot {k}_{3}}\langle 0|{\bar{Q}}_{l}(0){Q}_{m}({x}_{3}){a}^{\dagger }({\boldsymbol{k}})a({\boldsymbol{k}}){Q}_{i}({x}_{1}){\bar{Q}}_{j}({x}_{2})|0\rangle \\ &&{\rm{\quad\quad}}=\displaystyle \int {{\rm{d}}}^{4}{x}_{1}{{\rm{d}}}^{4}{x}_{2}{{\rm{d}}}^{4}{x}_{3}{{\rm{e}}}^{-{\rm{i}}{k}_{1}\cdot {x}_{1}-{\rm{i}}{k}_{2}\cdot {x}_{2}+{\rm{i}}{x}_{3}\cdot {k}_{3}}\langle 0|{\bar{Q}}_{l}(0){Q}_{m}({x}_{3}){a}^{\dagger }({\boldsymbol{k}}=0)a({\boldsymbol{k}}=0){Q}_{i}({x}_{1}){\bar{Q}}_{j}({x}_{2})|0\rangle .\end{eqnarray} \tag{ 7 }$$

Using Eq. (6), one can expand the matrix element in Eq. (7) with ψ^†(x) and χ(x). The space-time of the matrix element with NRQCD fields is controlled by the scale m_Qv_Q. Hence at leading order of v_Q one can neglect the space-time dependence in ψ^†(x) and χ(x). With this approximation, the matrix element in Eq. (7) is

$$\begin{eqnarray}&&\langle 0|{\psi }_{{\lambda }_{3}}^{{a}_{3}\dagger }(0){\chi }_{{\lambda }_{4}}^{{a}_{4}}(0){a}^{\dagger }a{\chi }_{{\lambda }_{1}}^{{a}_{1}}(0){\psi }_{{\lambda }_{2}}^{{a}_{2}\dagger }(0)|0\rangle, \end{eqnarray} \tag{ 8 }$$

where we suppressed the notation k = 0 in a and a^† and it is always implied that NRQCD matrix elements are defined in the rest frame of P_c. The superscripts a_i (i = 1, 2, 3, 4) are for the color of quark fields, while the subscripts λ_i (i = 1, 2, 3, 4) for the quark spin indices. Within NRQCD, the matrix element (8) can be expressed as

$$\begin{eqnarray*}&&\langle 0|{\psi }_{{\lambda }_{3}}^{{a}_{3}\dagger }(0){\chi }_{{\lambda }_{4}}^{{a}_{4}}(0){a}^{\dagger }a{\chi }_{{\lambda }_{1}}^{{a}_{1}}(0){\psi }_{{\lambda }_{2}}^{{a}_{2}\dagger }(0)|0\rangle ={\delta }_{{\lambda }_{4}{\lambda }_{3}}{\delta }_{{\lambda }_{2}{\lambda }_{1}}{T}_{{a}_{3}{a}_{4}}^{c}{T}_{{a}_{1}{a}_{2}}^{c}{h}_{1}+{\sigma }_{{\lambda }_{4}{\lambda }_{3}}^{n}{\sigma }_{{\lambda }_{2}{\lambda }_{1}}^{n}{T}_{{a}_{1}{a}_{2}}^{c}{T}_{{a}_{3}{a}_{4}}^{c}{h}_{3}+\cdots, \end{eqnarray*}$$

where σⁱ (n = 1, 2, 3) are Pauli matrices, T^c (c = 1, . . ., 8) are Gell-Mann matrices, and we neglect the color-singlet term since it is irrelevant here. The parameters h₁ and h₃ are defined as:

$$\begin{eqnarray}&&{h}_{1}=\frac{1}{8}\langle 0|({\psi }^{\dagger }{T}^{c}\chi ){a}^{\dagger }a({\chi }^{\dagger }{T}^{c}\psi )|0\rangle, {\rm{\quad}}{h}_{3}=\frac{1}{16}\langle 0|({\psi }^{\dagger }{T}^{c}{\sigma }^{n}\chi ){a}^{\dagger }a({\chi }^{\dagger }{T}^{c}{\sigma }^{n}\psi )|0\rangle .\end{eqnarray} \tag{ 9 }$$

h₁(h₃) represents the probability for a color-octet $Q\bar{Q}$ pair in a ¹S₀(³S₁) state to transform into the pentaquark. With these results, the Fourier transformed matrix element in Eq. (7) can be expressed as

$$\begin{eqnarray}&&{v}^{0}\displaystyle \int {{\rm{d}}}^{4}{x}_{1}{{\rm{d}}}^{4}{x}_{2}{{\rm{d}}}^{4}{x}_{3}{{\rm{e}}}^{-{\rm{i}}{k}_{1}\cdot {x}_{1}-{\rm{i}}{k}_{2}\cdot {x}_{2}+{\rm{i}}{k}_{3}\cdot {x}_{3}}\langle 0|{\bar{Q}}_{l}(0){Q}_{m}({x}_{3}){a}^{\dagger }({\boldsymbol{k}})a({\boldsymbol{k}}){\bar{Q}}_{i}({x}_{1}){Q}_{j}({x}_{2})|0\rangle \\ &&{\rm{\quad\quad}}={(2\pi )}^{4}{\delta }^{4}({k}_{1}-{m}_{Q}v){(2\pi )}^{4}{\delta }^{4}({k}_{2}-{m}_{Q}v){(2\pi )}^{4}{\delta }^{4}({k}_{3}-{m}_{Q}v)({T}_{{a}_{3}{a}_{4}}^{c}{T}_{{a}_{1}{a}_{2}}^{c})\\ &&{\rm{\quad\quad}}{\rm{\quad}}\times [-{({P}_{-}{\gamma }_{5}{P}_{+})}_{ji}{({P}_{+}{\gamma }_{5}{P}_{-})}_{ml}{\rm{\quad}}{h}_{1}+{({P}_{-}{\gamma }^{\mu }{P}_{+})}_{ji}{({P}_{+}{\gamma }^{\nu }{P}_{-})}_{ml}({v}_{\mu }{v}_{\nu }-{g}_{\mu \nu }){\rm{\quad}}{h}_{3}],\end{eqnarray} \tag{ 10 }$$

where the spin projection operator is defined by P_± = (1 ± γ · v)/2, and i, j, l, m are used for Dirac indices. Substituting Eq. (10) into Eq. (5), we obtain the differential cross section as

$$\begin{eqnarray}\begin{array}{ll}{\rm{d}}\sigma = & \frac{1}{8s}\displaystyle \int \frac{{{\rm{d}}}^{3}k}{{(2\pi )}^{3}{v}_{0}}\displaystyle \int \frac{{{\rm{d}}}^{3}{k}_{g}}{{(2\pi )}^{3}2{E}_{g}}{A}_{ij}(k,{k}_{g}){[{\gamma }^{0}{A}^{+}(k,{k}_{g}){\gamma }^{0}]}_{lm}{(2\pi )}^{4}{\delta }^{4}({p}_{1}+{p}_{2}-k-{k}_{g}){\rm{\quad}}{T}_{{a}_{3}{a}_{4}}^{c}{T}_{{a}_{1}{a}_{2}}^{c}\\ & \times \{-{({P}_{-}{\gamma }_{5}{P}_{+})}_{ji}{({P}_{+}{\gamma }_{5}{P}_{-})}_{ml}{\rm{\quad}}{h}_{1}+[{({P}_{-}\rlap{/}{v}{P}_{+})}_{ji}{({P}_{+}\rlap{/}{v}{P}_{-})}_{ml}-{({P}_{-}{\gamma }_{\sigma }{P}_{+})}_{ji}{({P}_{+}{\gamma }_{\sigma }{P}_{-})}_{ml}]{h}_{3}\}.\end{array}\end{eqnarray} \tag{ 11 }$$

After integrating over the phase space, the total cross section for the pentaquark production in the process (3) can be written as

$$\begin{eqnarray}&&\sigma =\frac{128\pi {\alpha }_{s}{\sigma }_{0}}{{(4{m}_{c}^{2}+{M}_{P}^{2}-2s)}^{2}}\left(1-\frac{{M}_{P}^{2}}{s}\right)[{\mathscr{A}}\ {h}_{1}+ {\mathcal B} \ {h}_{3}],\end{eqnarray} \tag{ 12 }$$

where M_P is the pentaquark mass. α_s is the strong coupling constant. σ₀ = 4πα²/3s with α the fine structure constant.

$$\begin{eqnarray}&&{\mathscr{A}}=\frac{4{(s-{M}_{P}^{2})}^{2}}{{M}_{P}}\left\{\frac{{Q}_{c}^{2}}{s}+\frac{2{V}_{e}{V}_{c}{Q}_{c}(s-{M}_{Z}^{2})}{{(s-{M}_{Z}^{2})}^{2}+{\Gamma }_{Z}^{2}{M}_{Z}^{2}}+\frac{s({A}_{e}^{2}+{V}_{e}^{2}){V}_{c}^{2}}{{(s-{M}_{Z}^{2})}^{2}+{\Gamma }_{Z}^{2}{M}_{Z}^{2}}\right\},\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray} &&{\mathcal B} =\frac{3{A}_{c}^{2}({A}_{e}^{2}+{V}_{e}^{2})}{{M}_{P}[{(s-{M}_{Z}^{2})}^{2}+{\Gamma }_{Z}^{2}{M}_{Z}^{2}]}\bigg\{{[{M}_{P}^{3}-3s{M}_{P}+2{m}_{c}(s+{M}_{P}^{2})]}^{2}+\frac{2}{3}{(s-{M}_{P}^{2})}^{2}[2s-{(2{m}_{c}+{M}_{P})}^{2}]\bigg\},\end{eqnarray} \tag{ 14 }$$

where M_Z (Γ_Z) denotes the mass (width) of the Z⁰ boson, Q_f is the electric charge of the corresponding fermion.

$$\begin{eqnarray*}&&{V}_{f}=\frac{1}{\sin 2{\theta }_{w}}({T}_{3}^{f}-2{Q}_{f}{\sin }^{2}{\theta }_{w}),{\rm{\quad}}{A}_{f}=\frac{{T}_{3}^{f}}{\sin 2{\theta }_{w}},{\rm{\quad}}f=e,c,\end{eqnarray*}$$

where θ_w is the weak mixing angle, and we choose sin² θ_w = 0.23.

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Obviously, only the weak interactions contribute to the coefficient of h₃ due to Furry theorem. If h₁ and h₃ are at the same magnitude order, at B factory energy, the contribution to the pentaquark production from h₁ is dominant, while at Z⁰ pole, that from h₃ is dominant. In our numerical calculation, we set m_c = 1.6 GeV, M_Z = 91.2 GeV, Γ_Z = 2.5 GeV, M_P = 4.38 GeV, α(M_Z) = 1/128 and α_s(M_Z) = 0.118. The cross section of the process (3) is

$$\begin{eqnarray}&&\sigma =\bigg\{42433.3\bigg[\frac{{h}_{1}}{{({\rm{GeV}})}^{3}}\bigg]+3.3\bigg[\frac{{h}_{3}}{{({\rm{GeV}})}^{3}}\bigg]\bigg\}\ {\rm{fb}}{\rm{\quad}}{\rm{at\ }}B{\rm{\ factory\ energy}},\\ &&\sigma =\bigg\{393.5\bigg[\frac{{h}_{1}}{{({\rm{GeV}})}^{3}}\bigg]+2640.0\bigg[\frac{{h}_{3}}{{({\rm{GeV}})}^{3}}\bigg]\bigg\}\ {\rm{fb}}{\rm{\quad}}{\rm{at}}{\rm{\quad}}{Z}^{{\rm{0}}}{\rm{poleenergy}}.\end{eqnarray} \tag{ 15 }$$

Up to now, h₁ and h₃ are still unkown exactly. One can attempt to relate the nonperturbative factors h₁ and h₃ to the wave function of the $c\bar{c}$ in the pentaquark, e.g.,

$$\begin{eqnarray}&&{h}_{3}={c}_{3}\frac{1}{4\pi }|{R}_{c\bar{c}}(0){|}^{2},\end{eqnarray} \tag{ 16 }$$

where ${R}_{c\bar{c}}(\xi )$ represents the radial wave function of the $c\bar{c}$ quark pair in the pentaquark P_c and ξ is the distance between c and $\bar{c}$. In the heavy quark limit, h₁ and h₃ are identical up to the corrections of order ${v}_{Q}^{2}$, i.e., ${h}_{3}={h}_{1}[1+{\mathscr{O}}({v}_{Q}^{2})]$.^[40] The coefficient c₃ is due to the fact that h₃ is defined by a color-octet matrix element. In the non-perturbatuive process, the transition from a color-octet state to the final color-singlet pentaquark state introduces extra suppressions of certain power of the relative velocity between the heavy pair in their rest frame, v_Q, which we expand the amplitude around its zero value as done above. So the matrix element is suppressed by the small factor proportional to powers of v_Q.^[40] In the following numerical calculations we simply take c₃ as 10⁻¹ for the charm sector.

The radial wave function ${R}_{c\bar{c}}(\xi )$ can be calculated in potential models. As a first estimation for its value at origin, we consider a simple model where the Hamiltonian contains the kinetic term and the harmonic oscillator as the color confinement potential.^[39] With this model, the interaction between the heavy quarks can be easily separated out. This part of Hamiltonian reads

$$\begin{eqnarray}&&{\hat{H}}_{\xi }=-\frac{1}{{m}_{c}}{\nabla }_{\xi }^{2}+\frac{{k}_{\xi }}{2}{\xi }^{2},\end{eqnarray} \tag{ 17 }$$

where k_ξ is the strength constant. k_ξ = 0.33 GeV³, is determined by reproducing the center-of-mass for the ground $c\bar{c}$ states.^[39] After solving the Schrödinger equation, one gets

$$\begin{eqnarray}&&{R}_{c\bar{c}}(0)={\bigg(\frac{7}{16}\mu {k}_{\xi }\bigg)}^{3/8}{\bigg[\frac{4}{\sqrt{\pi }}\bigg]}^{1/2}.\end{eqnarray} \tag{ 18 }$$

Substituting this expression into Eq. (16), one estimates h₁ = h₃ = 0.0036 GeV³, which is used in our numerical calculations. So we obtain the P_c production cross section of the process (3). The results are listed in Table 1. To investigate the contributions from the spin-singlet and triplet color-octet $c\bar{c}$ fragmentation in detail, we study the angular distribution (1/σ) (dσ/d cos θ) of the pentaquark P_c. The results are shown in Fig. 2, where θ is the angle between the moving direction of the e⁻ beam and that of P_c. One can see clearly from the figure that the difference between the contribution from h₁ and that from h₃ is significant.

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Table 1.  Pentaquark proction cross section(in units of fb) of the process (3).

$\sqrt{s}=10.6$ GeV		$\sqrt{s}=91.2$ GeV
Singlet	Triplet	Singlet	Triplet
152.748	0.0119	1.404	9.504

Next we consider the pentaquark state that can also be produced from the corlor-octet $c\bar{c}$ fragmentation in the process

$$\begin{eqnarray}\begin{array}{ll}{e}^{+}({p}_{1}){e}^{-}({p}_{2}) & \to {\gamma }^{* }/{Z}^{0}\to c({k}_{1})\bar{c}({k}_{2})Q({p}_{3})\bar{Q}({p}_{4})\\ & \to {P}_{c}(k)+X,\end{array}\end{eqnarray} \tag{ 19 }$$

where Q = c, b, and X = X_P + X_N as before. p₃ and p₄ respectively denote the momentum of Q and $\bar{Q}$, etc. For this case, ⟨P_c(k) + X_N| of Eq. (1) is replaced by the state of four heavy quarks including at least one $c\bar{c}$ pair. The corresponding differential cross section can be written as

$$\begin{eqnarray}\begin{array}{ll}{\rm{d}}\sigma = & \frac{1}{2s}\frac{{{\rm{d}}}^{3}k}{{(2\pi )}^{3}}\displaystyle \int \frac{{{\rm{d}}}^{3}{p}_{3}}{{(2\pi )}^{3}2{E}_{3}}\frac{{{\rm{d}}}^{3}{p}_{4}}{{(2\pi )}^{3}2{E}_{4}}{(2\pi )}^{4}{\delta }^{4}({p}_{1}+{p}_{2}-k-{p}_{3}-{p}_{4})\\ & \times \displaystyle \int \frac{{{\rm{d}}}^{4}{k}_{1}}{(2{\pi }^{4})}\frac{{{\rm{d}}}^{4}{k}_{3}}{(2{\pi }^{4})}{A}_{ij}({k}_{1},{k}_{2},{p}_{3},{p}_{4}){({\gamma }^{0}{A}^{\dagger }({k}_{3},{k}_{4},{p}_{3},{p}_{4}){\gamma }^{0})}_{lm}\\ & \times \displaystyle \int {{\rm{d}}}^{4}{x}_{1}{{\rm{d}}}^{4}{x}_{3}{{\rm{e}}}^{-{\rm{i}}{k}_{1}\cdot {x}_{1}+{\rm{i}}{k}_{3}\cdot {x}_{3}}\langle 0|{\bar{Q}}_{l}(0){Q}_{m}({x}_{3})|{P}_{c}+{X}_{N}\rangle \langle {P}_{c}+{X}_{N}|{\bar{Q}}_{i}({x}_{1}){Q}_{j}(0)|0\rangle .\end{array}\end{eqnarray} \tag{ 20 }$$

Similar to the above derivation, we use translational covariance to eliminate the sum over X_N and obtain

$$\begin{eqnarray}\begin{array}{ll}{\rm{d}}\sigma = & \frac{1}{2s}\frac{{{\rm{d}}}^{3}k}{{(2\pi )}^{3}}\displaystyle \int \frac{{{\rm{d}}}^{3}{p}_{3}}{{(2\pi )}^{3}2{E}_{3}}\frac{{{\rm{d}}}^{3}{p}_{4}}{{(2\pi )}^{3}2{E}_{4}}\displaystyle \int \frac{{{\rm{d}}}^{4}{k}_{1}}{{(2\pi )}^{4}}\frac{{{\rm{d}}}^{4}{k}_{3}}{{(2\pi )}^{4}}{A}_{ij}({k}_{1},{k}_{2},{p}_{3},{p}_{4}){({\gamma }^{0}{A}^{\dagger }({k}_{3},{k}_{4},{p}_{3},{p}_{4}){\gamma }^{0})}_{lm}\\ & \times \displaystyle \int {{\rm{d}}}^{4}{x}_{1}{{\rm{d}}}^{4}{x}_{2}{{\rm{d}}}^{4}{x}_{3}{{\rm{e}}}^{-{\rm{i}}{k}_{1}\cdot {x}_{1}-{\rm{i}}{k}_{2}\cdot {x}_{2}+{\rm{i}}{k}_{3}\cdot {x}_{3}}\langle 0|{\bar{Q}}_{l}(0){Q}_{m}({x}_{3}){a}^{\dagger }({\boldsymbol{k}})a({\boldsymbol{k}}){\bar{Q}}_{i}({x}_{1}){Q}_{j}({x}_{2})|0\rangle .\end{array}\end{eqnarray} \tag{ 21 }$$

The factorized cross section is displayed in Fig. 3. With the help of Eq. (10), the differential cross section for this kind of processes can be further expressed as

$$\begin{eqnarray}\begin{array}{ll}{\rm{d}}\sigma = & \frac{1}{2s}\displaystyle \int \frac{{{\rm{d}}}^{3}k}{{(2\pi )}^{3}{v}_{0}}\displaystyle \int \frac{{{\rm{d}}}^{3}{p}_{3}}{{(2\pi )}^{3}2{E}_{3}}\frac{{{\rm{d}}}^{3}{p}_{4}}{{(2\pi )}^{3}2{E}_{4}}{(2\pi )}^{4}{\delta }^{4}({p}_{1}+{p}_{2}-k-{p}_{3}-{p}_{4})\\ & \times \{-{h}_{1}{({P}_{-}{\gamma }_{5}{P}_{+})}_{ji}{({P}_{+}{\gamma }_{5}{P}_{-})}_{ml}+{h}_{3}[{({P}_{-}\rlap{/}{v}{P}_{+})}_{ji}{({P}_{+}\rlap{/}{v}{P}_{-})}_{ml}-{({P}_{-}{\gamma }_{\sigma }{P}_{+})}_{ji}{({P}_{+}{\gamma }_{\sigma }{P}_{-})}_{ml}]\}\\ & \times ({T}_{{a}_{3}{a}_{4}}^{c}{T}_{{a}_{1}{a}_{2}}^{c}){A}_{ij}({k}_{1},{k}_{2},{p}_{3},{p}_{4}){({\gamma }^{0}{A}^{\dagger }({k}_{3},{k}_{4},{p}_{3},{p}_{4}){\gamma }^{0})}_{lm}.\end{array}\end{eqnarray} \tag{ 22 }$$

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The angular distribution (1/σ) (dσ/d cos θ) at Z⁰ pole is shown in Fig. 4, where θ₃ is the angle between the momentum of the c quark (p₃) and that of P_c. Because of mass effect, the cross section drops in the small angular range. From Fig. 4, our results show that for the case of spin-singlet (h₃ = 0), the cross section is suppressed when the angular θ₃ is near π/2. In this case, one of the free charm quarks takes a relatively large transverse momentum with respect to the c-quark to compensate that of the P_c, and P_c production via gluon fragmentation is prohibited because of the angular momentum conservation at the $gq\bar{q}$ vertex. Employing Eq. (22), we can get the total cross section of pentaquark production related to the process (19) at B factory energy and at Z⁰ pole. In our following discussions, we simply set h₁ = h₃ = 0.0036 GeV³. The numerical results of spin-singlet and spin-triplet contributions are given in Table 2. Our results show that at Z⁰ pole energy, the P_c production cross section contributed from the process (19) is much larger than that from the process (3).

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Table 2.  Pentaquark production cross section (in units of fb) of the process (19).

Associate $Q\bar{Q}$	$\sqrt{s}=10.6$ GeV		$\sqrt{s}=91.2$ GeV
	Singlet	Triplet	Singlet	Triplet
$c\bar{c}$	0.198	0.666	0.0756	302.407
$b\bar{b}$	–	–	0.0504	373.176

Table 3.  Pentaquark production cross section (in units of fb) of the process (23).

Associate $q\bar{q}$	$\sqrt{s}=10.6$ GeV		$\sqrt{s}=91.2$ GeV
	Singlet	Triplet	Singlet	Triplet
$u\bar{u}$	0.0	0.612	0.0	299.808
$d\bar{d}$	0.0	0.155	0.0	399.636
$s\bar{s}$	0.0	0.155	0.0	386.424

Finally, we investigate the P_c production in light quark q (u, d, s) jet fragmentation by studying the process

$$\begin{eqnarray}&&{e}^{+}{e}^{-}\to q\bar{q}{g}^{* }{\rm{\quad}}{\rm{with}}{\rm{\quad}}{g}^{* }\to c\bar{c}\to {P}_{c}+X.\end{eqnarray} \tag{ 23 }$$

The numerical results are presented in Table 3. Since in this process, P_c is produced via the gluon fragmentation, only the spin-triplet component contributes to the cross section. For this process at Z⁰ pole, the cross section contributed from the spin-triplet component is comparable to that of the process (19), which shows that for these two processes, the hidden-charm pentaquark produced from gluon fragmentation is dominant. The cross sections of processes (19) and (23) are both larger than that of the process (3) because of the momentum conservation. The hadron and gluon produced in the final state must recoil from each other in the process (3), which makes the gluon very hard so that it is suppressed.

To summarize, in this paper, we study the compact hidden-charm pentaquark production via the color-octet charm-anticharm pair fragmentation in e⁺e⁻ annihillation with clean backgrounds. The most straightforward application of our analysis is the B factory at present and in the future. Based on our above calculations, at B factory energies, the dominant production process is e⁺e⁻ → P_c + g. This means P_c is dominantly produced in a two-jet like event. Belle collaboration now has collected an integrated luminosity about 1000 fb⁻¹ and the events number of P_c production could be 10⁵. The future operation of B factory will accumulate even more events. So to set a jet algorithm trigger as suggested in Ref. [38] may help us to obtain a clear signal or upper limit for P_c production in e⁺e⁻ annihillation. Our method can be easily applied to investigate the production of the pentaquark state including a $b\bar{c}$ or $c\bar{b}$ at e⁺e⁻ colliders. In the future, at high energy e⁺e⁻ colliders, e.g., a high luminosity Z-factory, CEPC and Linear collider, a large number of the events related to our studies in this paper will be accumulated, so that the corresponding measurement with high precision is possible. Once the direct production of the hidden-charm pentaquark states is confirmed at e⁺e⁻ colliders, it will be very helpful to understand the quark model and the strong interactions.

Acknowledgements