Double Polarization Observables in Pentaquark Photoproduction

TL;DR

This paper explores polarization observables in J/psi photoproduction to understand the properties of hidden charm pentaquarks, providing predictions for upcoming experiments at Jefferson Lab.

Contribution

It introduces the first estimates of polarization transfer and helicity correlation sensitivities to pentaquark properties in exclusive photoproduction.

Findings

Sensitivity estimates for polarization transfer KLL to pentaquark couplings.

Predictions for initial state helicity correlation ALL in polarized target experiments.

Benchmark results for the SBS experiment at Jefferson Lab.

Abstract

We investigate the properties of the hidden charm pentaquark-like resonances first observed by LHCb in 2015, by measuring the polarization transfer KLL between the incident photon and the outgoing proton in the exclusive photoproduction of J/psi near threshold. We present a first estimate of the sensitivity of this observable to the pentaquark photocouplings and hadronic branching ratios, and extend our predictions to the case of initial state helicity correlation ALL, using a polarized target. These results serve as a benchmark for the SBS experiment at Jefferson Lab, which proposes to measure for the first time the helicity correlations ALL and KLL in J/psi exclusive photoproduction, in order to determine the pentaquark photocouplings and branching ratios.

Tables

Table 1. Table 1: Parameters of the fits for different J P superscript 𝐽 𝑃 J^{P} assignments for the P c ​ ( 4450 ) subscript 𝑃 𝑐 4450 P_{c}(4450) state. Uncertainties are at the 68% confidence level, except for the branching ratio, whose upper limit is quoted at 95%.

$J^P$	${\frac{3}{2}}^{-}$	${\frac{5}{2}}^{+}$	${\frac{3}{2}}^{+}$	${\frac{5}{2}}^{-}$
$A$	$0.379\pm 0.051$	$0.380\pm 0.053$	$0.378\pm 0.049$	$0.381\pm 0.053$
${\alpha }_0$	$0.941\pm 0.047$	$0.941\pm 0.049$	$0.942\pm 0.045$	$0.941\pm 0.048$
${\alpha }^{\prime }$ (${\mathrm{GeV}}^{-2}$)	$0.364\pm 0.037$	$0.367\pm 0.039$	$0.363\pm 0.035$	$0.365\pm 0.037$
$b_0$ (${\mathrm{GeV}}^{-2}$)	$0.12\pm 0.14$	$0.13\pm 0.15$	$0.12\pm 0.14$	$0.13\pm 0.15$
${ℬ}_{\psi p}^{\left(4450\right)}$ (95%)	$\le 4.3\%$	$\le 1.4\%$	$\le 1.8\%$	$\le 0.71\%$

Table 2. Table 2: Values of the experimentally projected beam current I e subscript 𝐼 𝑒 I_{e} , length of the radiator in terms of radiation length X 0 subscript 𝑋 0 X_{0} and thickness times density of the target ρ free ⋅ l ⋅ subscript 𝜌 free 𝑙 \rho_{\text{free}}\cdot l .

	$I_e$ [$\mu \text{A}$]	$X_0$	${\rho }_{\text{free}}\cdot l$ [$\text{g}/{\text{cm}}^2$]
${\text{K}}_{\text{LL}}$ (SBS)	5.0	6$\cdot$10-2	1.08
${\text{A}}_{\text{LL}}$ (SBS)	0.1	10$\cdot$10-2	0.32

Equations

$$\text{matrixelement}{\lambda }_{\psi }{\lambda }_{p^{\prime }}{T}_P{\lambda }_{\gamma }{\lambda }_p$$

$$\times [{\epsilon }^{\mu }(p_{\gamma },{\lambda }_{\gamma })q^{\nu }-{\epsilon }^{\nu }(p_{\gamma },{\lambda }_{\gamma })q^{\mu }]$$

$$\times {\epsilon }_{\nu }^{\ast }(p_{\psi },{\lambda }_{\psi }).$$

$$F(s,t)=iA\;\bigg{(}\frac{s-s_{\textrm{th}}}{s_{0}}\bigg{)}^{\alpha(t)}\frac{e^{b_{0}(t-t_{\textrm{min}})}}{s},$$

$$\begin{array}{rl}\text{matrixelement}{\lambda }_{\psi }{\lambda }_{p^{\prime }}{T}_R{\lambda }_{\gamma }{\lambda }_p& =\\ =f_{\text{th}}(s)& \frac{\text{matrixelement}{\lambda }_{\psi }{\lambda }_{p^{\prime }}{T}_{\text{dec}}{\lambda }_R\text{matrixelement}{\lambda }_R{T}_{\text{em}}^{†}{\lambda }_{\gamma }{\lambda }_p}{M_R^2-s-i{\Gamma }_R{M}_R}.\end{array}$$

$$f_{\textrm{th}}(s)=\bigg{(}\frac{s-s_{\textrm{th}}}{s}\frac{M_{R}^{2}}{M_{R}^{2}-s_{\textrm{th}}}\bigg{)}^{\beta},$$

$$\text{matrixelement}{\lambda }_{\psi }{\lambda }_{p^{\prime }}{T}_{\text{dec}}{\lambda }_R=g_{{\lambda }_{\psi }{\lambda }_{p^{\prime }}}(p)d_{{\lambda }_R,{\lambda }_{\psi }-{\lambda }_{p^{\prime }}}^{J_R}(\cos {\theta }_{\text{CM}}),$$

$${\Gamma }_{\psi p}=$$

$$=$$

$$\text{matrixelement}{\lambda }_{\gamma }{\lambda }_p{T}_{\text{em}}{\lambda }_R=\frac{1}{M_R}\sqrt{\frac{8s{M}_p{M}_R{\overline{p}}_i}{4\pi \alpha }}\sqrt{\frac{{\overline{p}}_i}{p_i}}{A}_{{\lambda }_R}.$$

$${\Gamma }_{\gamma }=\frac{{\bar{p}}_i^2}{\pi }\frac{2M_p}{(2J_R+1)M_R}\left[{\left|A_{1/2}\right|}^2+{\left|A_{3/2}\right|}^2\right].$$

$$⟨{\lambda }_{\gamma }{\lambda }_p|T_{\text{em}}|{\lambda }_R⟩=\frac{\sqrt{4\pi \alpha }{f}_{\psi }}{M_{\psi }}⟨{\lambda }_{\psi }={\lambda }_{\gamma },{\lambda }_p|T_{\text{dec}}|{\lambda }_R⟩.$$

$$\frac{{\Gamma }_{\gamma }}{{\Gamma }_{\psi p}}=4\pi \alpha {\left(\frac{f_{\psi }}{M_{\psi }}\right)}^2{\left(\frac{{\bar{p}}_i}{{\bar{p}}_f}\right)}^{2\ell +1}{\mathcal{P}}_t.$$

$$|A_{1/2}{|}^2+|A_{3/2}{|}^2=$$

$$\times {\left(\frac{{\bar{p}}_i^2}{\pi }\frac{2M_p}{(2J_R+1)M_R}\right)}^{-1},$$

$$R=\frac{A_{1/2}}{\sqrt{|A_{1/2}{|}^2+|A_{3/2}{|}^2}},$$

$${\text{A(K)}}_{\text{LL}}=\frac{1}{2}\left[\frac{d\sigma (++)-d\sigma (+-)}{d\sigma (++)+d\sigma (+-)}-\frac{d\sigma (-+)-d\sigma (--)}{d\sigma (-+)+d\sigma (--)}\right],$$

$$d\sigma ({\lambda }_{\gamma }{\lambda }_p)$$

$$d\sigma ({\lambda }_{\gamma }{\lambda }_{p^{\prime }})$$

$$\frac{d\sigma }{dt}$$

$$\sigma$$

$$\Delta {\text{A}}_{\text{LL}}$$

$$\Delta {\text{K}}_{\text{LL}}$$

Full text

††preprint: JLAB-THY-19-3004

I Introduction

The LHCb data on $\Lambda_{b}\to J/\psi\,p\,K^{-}$ decay potentially indicate the existence of baryon resonances in the $J/\psi\,p$ spectrum Aaij et al. (2015, 2016, 2019) that do not fit predictions of the valence quark model. These would indeed have the minimum constituent quark content of $uudc\bar{c}$, i.e. that of compact hidden-charm pentaquarks or meson-baryon molecules. The first partial-wave analysis of the LHCb data favored two resonance structures, which were labeled $P_{c}(4380)$ and $P_{c}(4450)$. The asymmetries in the angular distributions suggest that these two $P_{c}$ states Jurik (2016) have opposite parity and the preferred assignments are $J^{P}=3/2^{-}$ and $J^{P}=5/2^{+}$ for the lighter and heavier state, respectively, but other assignments were not ruled out.

The most recent LHCb results Aaij et al. (2019), however, indicate that the narrower $P_{c}(4450)$ peak may represent two interfering states, labeled $P_{c}(4440)$ and $P_{c}(4457)$, having widths of $<49$ and $<20{\mathrm{\,Me\kern-1.00006ptV}}$, respectively. In addition, another narrow resonance was identified at $4312{\mathrm{\,Me\kern-1.00006ptV}}$ (see also the discussion in Ref. Fernández-Ramírez et al. (2019)). Given that the identification of newer peaks came from the analysis of the $J/\psi\,p$ mass spectrum alone, the spin-parity of these states is not known yet. For the same reason, the latter fits do not shed more light upon the broader $P_{c}(4380)$ state.

In addition to being compact five-quark states Maiani et al. (2015); Lebed (2015); Anisovich et al. (2015); Ali and Parkhomenko (2019), alternative structures are possible. In particular as the peaks appear close to open meson-baryon thresholds it is likely that they are due to attractive interactions between the two hadrons. For example, in the region of the $P_{c}(4450)$ there could be weakly bound $\bar{D}^{*}\Sigma_{c}$ and $\bar{D}^{*}\Sigma_{c}^{*}$ states  Chen et al. (2015a, b); Roca et al. (2015); Guo et al. (2019); Guo and Oller (2019); Xiao et al. (2019); Liu et al. (2019), and even in the absence of resonance or bound states it is possible to generate peaks from nearby cross-channel exchanges Szczepaniak (2016); Meißner and Oller (2015); Mikhasenko (2015); Guo et al. (2016). Such ambiguities in the interpretation highlight the need for additional measurements especially with different beam-on-target configurations. The use of photoproduction Kubarovsky and Voloshin (2015); Wang et al. (2015); Karliner and Rosner (2016); Hiller Blin et al. (2016); Wang et al. (2019) is especially appealing since it reduces the role of kinematic effects and minimizes model dependence of the partial-wave analysis. Furthermore, photoproduction at high energies is an efficient process for charm production Chekanov et al. (2002); Aktas et al. (2006), while production near threshold has long been advertised as a tool for studies of the residual QCD interactions between charmonium and the nucleon Brodsky et al. (2001); Perevalova et al. (2016).

The search for the $P_{c}(4450)$ –the narrower of the first two LHCb candidates– through a scan of the photoproduction cross sections has been proposed by the Hall C, CLAS12, and GlueX experiments at JLab Meziani et al. (2016); Stepanyan et al. (2017); Ali et al. (2019). The first results from GlueX are already available, and there is no evidence of narrow peaks Ali et al. (2019). Recently, an update on photoproduction studies based on the most recent LHCb results has been performed Wang et al. (2019); Cao and Dai (2019); Wu et al. (2019), albeit using the spin-parity assignment of the older LHCb amplitude analysis. Furthermore, the use of polarization observables has been recently proposed for an experiment at the Super BigBite Spectrometer (SBS) in Hall A at JLab Fanelli et al. (2018). It has been argued that these may reach higher signal-to-background ratios than the usual study of differential cross sections, at least in certain parts of the parameter space, and the discovery of a double-peak structure in the $P_{c}(4450)$ region makes these experiments even more relevant.

In this paper we detail the study of polarization observables to access the pentaquark signals. The polarization observables are sensitive to the interference between resonant and nonresonant contributions as well as between different resonance states. Polarization observables are determined by the photoproduction amplitudes of different helicities for the initial photons, while the unpolarized cross sections are determined by the squared absolute values of the photoproduction amplitudes. Therefore, the polarization data offers new information that is relevant in the evaluation of the resonance photo- and hadronic couplings and it is helpful in accessing the contributions from overlapping resonances. The polarization observables extend our capabilities to validate the mechanisms of the reaction models used in the data analyses through a combined fit of unpolarized cross sections and polarization measurements. Here we specifically focus on accessing the sensitivity needed to investigate the properties of the pentaquarks, by studying the helicity correlations between the polarized photon beam and the polarized target ($\textup{A}_{\textup{LL}}$) or recoil ($\textup{K}_{\textup{LL}}$) proton. The latter can be assessed by measuring the polarization transfer with the one-arm polarimeter in Hall A at JLab Fanelli et al. (2018). Given that there is no spin-parity assignment for the new $P_{c}(4440)$ and $P_{c}(4457)$ states, which is essential for making photoproduction predictions, and that the resolution might prevent the distinction between the two, we use the previous $P_{c}(4450)$ information in this feasibility study. In the following, by $P_{c}(4450)$ we refer to the collective effect of both $P_{c}(4440)$ and $P_{c}(4457)$ peaks. We also use the information on the broad $P_{c}(4380)$ state, while disregarding the new $P_{c}(4312)$, since its spin-parity is unknown, although a similar study can be applied in this lower mass region. If photoproduction experiments prove to be successful in identifying the $P_{c}$ signals, an amplitude analysis of spin-dependent observables will be mandatory, for which this paper lays the groundwork.

The paper is organized as follows. In Section II, we describe the reaction model for $J/\psi$ photoproduction off the proton. In Section III, we show the fits to the data and the predictions for the $\textup{K}_{\textup{LL}}$ and $\textup{A}_{\textup{LL}}$ asymmetries for different $P_{c}$ spin-parity assignments and values of the photocouplings. Section IV focuses on sensitivity studies for measuring these asymmetries at Hall A of JLab. Finally, Section V summarizes our conclusions.

II Reaction model

Starting from the reaction model of Ref. Hiller Blin et al. (2016) we incorporate spin-dependent interactions at energies near the threshold for $J/\psi\,p$ production. Furthermore we incorporate both a narrow peak, compatible with the original $P_{c}(4450)$ state, and the broader $P_{c}(4380)$.

II.1 Background contribution

The dominant nonresonant contribution, as shown in Fig. 1, is assumed to be that of diffractive photoproduction of the $J/\psi$ off the proton target. This is taken as the main background to the $P_{c}$ signals and it is realized by an effective $t$-channel Pomeron exchange model Close and Schuler (1999). The kinematic factors and spin dependence in the model correspond to a vector exchange, to enforce that the Pomeron has an intercept which is close to unity Lesniak and Szczepaniak (2003). The resulting covariant amplitude is given by

[TABLE]

Here, $u(p_{i},\lambda_{p})$ and $u(p_{f},\lambda_{p^{\prime}})$ are the Dirac spinors for the target and recoil protons, respectively, and $q$ is the photon 4-momentum. The initial and final nucleon momenta are denoted by $p$ and $p^{\prime}$ and their helicities by $\lambda_{p}$ and $\lambda_{p^{\prime}}$, respectively. The vectors $\varepsilon$ determine polarization of the photon and the $J/\psi$. As expected, in the high-energy limit the amplitude in Eq. (1) is proportional to $s$, the center-of-mass energy squared (see also Appendix C of Mathieu et al. (2018)). To account for the full dependence on the Mandelstam variables $s$ and $t$ associated with the Pomeron trajectory, the amplitude in Eq. (1) contains the function

[TABLE]

where $\alpha(t)=\alpha_{0}+\alpha^{\prime}t$ is the Pomeron trajectory (see e.g. Ref. Gribov (2012)). We fix the energy scale parameter to $s_{0}=1{\mathrm{\,Ge\kern-1.00006ptV}}^{2}$ and $s_{\textrm{th}}$ to the physical threshold, $(M_{\psi}+M_{p})^{2}$, where $M_{\psi}$ and $M_{p}$ are the masses of the $J/\psi$ and the proton, respectively. We note that $F(s,t)$ exponentially falls when moving away from the forward direction $t=t_{\textrm{min}}$. With this parametrization, the background can be computed for the entire range of $t$ (i.e. center-of-mass scattering angle $0^{\circ}\leq\theta_{\textrm{CM}}\leq 180^{\circ}$), but it has been derived and is more reliable in the forward region.

II.2 Pentaquark Resonances

Following Refs. Hiller Blin et al. (2016, 2018), we parametrize the pentaquark candidate contributions to $\gamma p\rightarrow P_{c}\rightarrow J/\psi\,p$ using Breit-Wigner amplitudes, as shown in Fig. 1. This parametrization was successfully used in studies of nucleon resonance photo- and electroexcitation amplitudes from the CLAS data Mokeev et al. (2012); Golovatch et al. (2019); Mokeev (2018). In terms of helicity amplitudes,

[TABLE]

We use the resonance mass $M_{R}$ and decay width $\Gamma_{R}$ for either pentaquark state as extracted from the original LHCb fit Aaij et al. (2015). Since Eq. (3) is finite at threshold for an $S$-wave resonance decay, but the background  in Eq. (2) vanishes, we include an additional factor,

[TABLE]

to reproduce the physical behavior of the resonant amplitude near threshold. Specifically, we choose $\beta=3/2$, which allows the resonance signal to fall off sufficiently fast from the peak towards threshold. The numerator in Eq. (3) is the product of two amplitudes. The first one,

[TABLE]

describes the coupling of the resonant state, with spin $J_{R}$ and helicity $\lambda_{R}$, to the $J/\psi\,p$ final state. The helicity amplitudes $g_{\lambda_{\psi}\lambda_{p^{\prime}}}\!(p)$ have a near-threshold behavior $\propto p^{l}$ for a given decay moment $p$ and orbital angular momentum $\ell$. In general, these amplitudes depend on the final-state helicities. However, since nothing is known about their behavior for any of the pentaquark states, we consider them to be equal in magnitude for any helicity: $g_{\lambda_{\psi}\lambda_{p^{\prime}}}\!(p)\equiv gp^{\ell}$ for $\lambda_{\psi}-\lambda_{p^{\prime}}>0$, or $g_{\lambda_{\psi}\lambda_{p^{\prime}}}\!(p)\equiv\eta gp^{\ell}$ for $\lambda_{\psi}-\lambda_{p^{\prime}}<0$, where $\eta=\pm 1$ corresponds to the naturality of the resonance. We assume that the amplitude of either $P_{c}$ state is dominated by the lowest partial wave, such that $\ell=0$ for a $J^{P}_{R}=3/2^{-}$ resonance, $\ell=1$ for $J^{P}_{R}=3/2^{+}$ or $5/2^{+}$, and $\ell=2$ for $J^{P}_{R}=5/2^{-}$. The magnitude of the couplings $g$ for either pentaquark state is then constrained by the partial width $\Gamma_{\psi p}$ through

[TABLE]

where $\mathcal{B}_{\psi p}$ is the branching ratio of the given $P_{c}$ state into $J/\psi$ and the final proton with momentum $\bar{p}_{f}$ evaluated at the resonance peak. We note that, in general, the hadronic couplings of the different $P_{c}$ states are independent.

The second amplitude describes the photoexcitation of the pentaquark resonance, parametrized in the usual way in terms of two independent photocouplings, $A_{1/2}$ and $A_{3/2}$,

[TABLE]

Here again, $\overline{p}_{i}$ is the momentum $p_{i}$ of the initial proton evaluated at the central mass of the resonance. The photocouplings are related by $A_{-\lambda_{R}}=\eta\,A_{\lambda_{R}}$. For the electromagnetic decay width $\Gamma_{\gamma}$ one then obtains

[TABLE]

As in Ref. Hiller Blin et al. (2016), the overall size of the photocouplings is estimated with a vector-meson dominance (VMD) model Kubarovsky and Voloshin (2015); Wang et al. (2015); Karliner and Rosner (2016), which relates the transverse $J/\psi$ helicity amplitudes and the electromagnetic decay amplitudes through

[TABLE]

The VMD model is expected to be a robust approximation, since the quantum numbers and mass of the resonance strongly suppress resonant contributions to the electromagnetic decay other than the $J/\psi$. Using Eqs. (6) and (9), we obtain

[TABLE]

The factor $\mathcal{P}_{t}$ is introduced to take into account that in Eq. (9) only the transverse polarizations of the $J/\psi$ contribute. The value of $\mathcal{P}_{t}$ then depends on the spin-parity assignment of the pentaquark resonance, reading $\mathcal{P}_{t}=2/3$ for $J^{P}_{R}=3/2^{-}$, $\mathcal{P}_{t}=3/5$ for $J^{P}_{R}=5/2^{+}$ or $3/2^{+}$, and $\mathcal{P}_{t}=1/3$ for $J_{R}^{P}=5/2^{-}$ Kubarovsky and Voloshin (2015). Combining the VMD assumption of Eq. (10) with Eq. (8), we obtain an expression for the quadrature sum

[TABLE]

which we use for the computation of the observables. Due to Eq. (6), it relates the photocouplings to the hadronic branching fraction size $\mathcal{B}_{\psi p}$.

In the fits of this work, the VMD condition is used such that $A_{1/2}=A_{3/2}$. In order to study the behavior at different relative photocoupling sizes, we then relax the equality condition, keeping the quadrature sum $|A_{1/2}|^{2}+|A_{3/2}|^{2}$ and the size of the hadronic couplings $g$ unchanged. Thus, we define the ratio

[TABLE]

and treat it as a free parameter.

III Results

While there are some quark model studies on the pentaquark photocouplings Wang et al. (2016); Ortiz-Pacheco et al. (2019), the experimental study of both the pentaquark hadronic and photocouplings is terra incognita. The spin-dependent observables $\textup{A}_{\textup{LL}}$ ($\textup{K}_{\textup{LL}}$) describe correlations between the helicities of the incoming photon and the incoming (outgoing) proton, given in terms of differential cross sections as

[TABLE]

with $d\sigma\equiv d\sigma/d\cos\theta_{\text{CM}}$, where the first helicity refers to the incident photon, and the second refers to the target ($\textup{A}_{\textup{LL}}$) or recoil ($\textup{K}_{\textup{LL}}$) proton in the center-of-mass frame Kroll and Passek-Kumerički (2018). For the computation of $\textup{A}_{\textup{LL}}$, the differential cross sections are obtained from the helicity amplitudes as follows

[TABLE]

The unpolarized differential and total cross sections are given by

[TABLE]

where as customary $t_{\text{min(max)}}=M_{\psi}^{2}-2p_{i}\left(\sqrt{p_{f}^{2}+M_{\psi}^{2}}\mp p_{f}\right)$. In Eq. (15a), $T=T_{P}+T_{R}(4380)+T_{R}(4450)$ is the coherent sum of the Pomeron background in Eq. (1) and the resonant contribution from the pentaquarks given by Eq. (3). Since the differential cross section is not very sensitive to a broad $P_{c}(4380)$, we set $\mathcal{B}_{\psi p}^{(4380)}=0$ in the fits to reduce the parameter space. The resonant parameters of the $P_{c}(4450)$ are fixed to the LHCb best values Aaij et al. (2015), while $\mathcal{B}_{\psi p}^{(4450)}$ is free. The background parameters $\alpha_{0}$, $\alpha^{\prime}$, $b_{0}$, and $A$ in Eq. (1) are also fitted. We fit $J/\psi$ photoproduction data from GlueX Ali et al. (2019) and SLAC Camerini et al. (1975). Unlike our previous works Hiller Blin et al. (2016, 2018), we ignore the very high-energy data from HERA and ZEUS Chekanov et al. (2002); Aktas et al. (2006), as well as the old unpublished data close to threshold Ritson (1976); Anderson (1976). This is done in order to have a better description of the region of interest which is now better constrained thanks to the GlueX data. For the background model to best reproduce the data, we include in the fit both the energy- and $t$-dependent information from Tables I and II of Ali et al. (2019). However, since the points come from the same data set and correlations are not reported, our statistical estimates must be considered with care. The curves are integrated over the (large) bin size, while the energy resolution is neglected. The mean fit parameters and their uncertainties for each spin-parity assignment of the $P_{c}(4450)$ have been calculated employing the bootstrap technique (see Hiller Blin et al. (2016) for details), and the results are shown in Tables 1 and 2. The results for $\alpha_{0}$ are compatible with unity. The values of $\alpha^{\prime}$ are a bit higher, but marginally compatible with the ones extrapolated from the SPS energies Erhan and Schlein (2000). We use Eqs. (1), (3), and (13) to give the predicted values of $\textup{A}_{\textup{LL}}$ and $\textup{K}_{\textup{LL}}$ for a given beam energy $E_{\gamma}$ and center-of-mass scattering angle $\theta_{\text{CM}}$. Note that the beam energy corresponding to the $P_{c}(4450)$ peak is $E_{\gamma}\approx 10.6{\mathrm{\,Ge\kern-1.00006ptV}}$, while for the $P_{c}(4380)$ it is $E_{\gamma}\approx 9.8{\mathrm{\,Ge\kern-1.00006ptV}}$. Some predictions for the polarization observables are shown in Figs. 3, 4 and 5.

IV Sensitivity studies

A measurement of the polarization observables was recently proposed for Hall A at JLab Fanelli et al. (2018). It will take advantage of the SBS setup developed for the GEp/SBS experiment in Hall A Gnanvo et al. (2015), which consists of a hadron (SBS) and an electron arm (ECAL). To measure the initial helicity state correlation $\textup{A}_{\textup{LL}}$, the experiment will scatter a circularly polarized photon beam onto a longitudinally polarized (NH3) target. The polarization transfer $\textup{K}_{\textup{LL}}$ will be measured by a proton polarimeter of the GEp setup in the SBS arm, used in combination with an unpolarized liquid hydrogen target. With this experimental setup, all three final-state particles $\gamma p\to J/\psi(\to e^{+}e^{-})p$ will be reconstructed, allowing for a clean signature of the candidate events with suppressed backgrounds Fanelli et al. (2018); Ali et al. (2019). An example of the predictions for the polarization observables in the SBS acceptance is given in Fig. 6.

We provide a sensitivity study of the main parameters of the LHCb pentaquark states based on toy Monte Carlo simulations of $\textup{A}_{\textup{LL}}$ and $\textup{K}_{\textup{LL}}$ experiments at JLab. As we said, this study relies on the two states seen in 2015, and the actual sensitivity is susceptible to change when more information about the new $P_{c}(4440)$ and $P_{c}(4457)$ becomes available. The code we used to calculate the observables is publicly available on the JPAC website JPAC Collaboration , and can be extended to other spin-parities and resonance parameters. The statistical uncertainty on $\textup{A}_{\textup{LL}}$ and $\textup{K}_{\textup{LL}}$ can be approximated according to Fanelli et al. (2018)111In the low-statistics limit, we imposed that these uncertainties cannot exceed 1.

[TABLE]

where $N_{\psi}$ is the total number of exclusive $J/\psi$ events expected to be detected. The factor $R$ is the rescaling due to the (small) background underneath the $J/\psi$ peak,222The background comes mainly from Bethe-Heitler continuum $e^{+}e^{-}$ production. and can be safely assumed $\approx 1$. It is worth recalling that the pentaquark signals and the Pomeron background are summed at the amplitude level, and for this reason in the equations above we do not define the number of pentaquark events, but rather use the number of measured $J/\psi\,p$ candidates. The photon beam polarization is approximately $P_{\gamma}\sim 0.8$,333This comes from an initial electron beam polarization $P_{e}\sim 0.85$ and the Maximon-Olsen formula Olsen and Maximon (1959) as a function of the incident $E_{\gamma}$, ranging from the $J/\psi$ threshold to the end-point energy $E_{e}$. while $P_{p}\sim 0.75$ is the average target proton polarization. In the case of $\textup{K}_{\textup{LL}}$, one considers the polarization transferred to the recoil proton. The average effective figure of merit $\left\langle F_{\text{eff}}\right\rangle=\sqrt{\epsilon_{\text{pol}}}A_{y}\sin\chi_{\text{prec}}$ includes the polarimeter efficiency $\epsilon_{\text{pol}}$, the polarimeter analyzing power $A_{y}$, and the spin precession angle in the SBS magnet $\chi_{\text{prec}}$. Following Ref. Fanelli et al. (2018), this figure of merit is approximated as $\left\langle F_{\text{eff}}\right\rangle\sim 0.07$.

Following the experimental design concept of Ref. Fanelli et al. (2018), SBS and ECAL are located at the right and left of the beamline, respectively, with central polar angles of 17*∘* and 22*∘*. We consider the experimental signatures that provide the best energy and mass resolutions, $\sigma(E_{\gamma})\sim 125{\mathrm{\,Me\kern-1.00006ptV}}$ and $\sigma(M_{J/\psi})\sim 20{\mathrm{\,Me\kern-1.00006ptV}}$ at an electron beam energy of $10{\mathrm{\,Ge\kern-1.00006ptV}}$, which is where one of the two leptons is reconstructed along with the proton in the hadronic arm, and the other lepton is detected in the electromagnetic calorimeter. Our minimal requirements are for the proton and the lepton in the hadronic arm to have an energy of $2$ and $1{\mathrm{\,Ge\kern-1.00006ptV}}$, respectively, and the other lepton to deposit an energy of $1{\mathrm{\,Ge\kern-1.00006ptV}}$ in the calorimeter. We refer to Ref. Fanelli et al. (2018) for further details on the experimental settings and selection criteria which have also been used for the simulation studies of the present paper. The final acceptance with these cuts is $\sim 1\%$. The expected yields are calculated based on the experimental conditions of Table 2 and by requiring the events to be within the detector acceptance. In particular, for a given photon energy range $(E_{1},E_{2})$ and time interval $\Delta t$, the yield is estimated as $N_{\psi}\approx I_{e}\cdot\left(\int_{E_{1}}^{E_{2}}f(E_{\gamma})\,\sigma(E_{\gamma})\,dE_{\gamma}\right)\cdot(\rho\cdot l)\cdot\epsilon\cdot\mathcal{B}(\psi\to e^{+}e^{-})\cdot\Delta t$, where $I_{e}$ is the electron beam current, $\sigma$ is the photoproduction cross section in Eq. (15b) as a function of the incident photon energy, $f(E_{\gamma})$ is the bremsstrahlung photon flux calculated for a radiator with $X_{0}$ radiation lengths according to Ref. Mo and Tsai (1969), $\rho\cdot l$ is the product of the target density and length, $\epsilon$ is the detection acceptance, and $\mathcal{B}\!\left(\psi\to e^{+}e^{-}\right)=5.94\%$ Tanabashi et al. (2018). The values of $I_{e}$, $X_{0}$ and $\rho\cdot l$ are given in Tab. 2. These values propagate into the statistical uncertainties defined by Eqs. (16) and (17). A fictitious systematic uncertainty of 2% was taken into account in the toy model. The spectrum of the bremsstrahlung photons is calculated as in Ref. Mo and Tsai (1969). The incident photon energy of interest ranges from about the $J/\psi$ production threshold to the end-point energy coinciding with the electron beam.

The proxy used to estimate the sensitivity to the $P_{c}$ states is based on the log-likelihood difference $\Delta\log{\mathcal{L}}$ between the background-only hypothesis, and the hypothesis that two $P_{c}$ resonances interfere with it. Wilks’ theorem then relates the value of $-2\Delta\log{\mathcal{L}}$ to a $\chi^{2}$ distribution with degrees of freedom equal to the difference in dimensionality between the two hypotheses Wilks (1938).

We focus here on a particular scenario, where the broad $P_{c}(4380)$ has a photocoupling ratio $R^{(4380)}=1/\sqrt{2}$ (corresponding to equal photocouplings $A_{1/2}^{(4380)}=A_{3/2}^{(4380)}$), the hadronic branching ratio is equal to that of the $P_{c}(4450)$, and the mass and the width of the two states are fixed to the best values measured in Ref. Aaij et al. (2015).

Pseudodata are generated in a 2D grid of points, varying the photocoupling ratio and the hadronic branching ratio of the $P_{c}(4450)$. For each point of the grid, multiple $\mathcal{O}(10^{3})$ toy models are computed producing binned data of $\textup{A}_{\textup{LL}}$ and $\textup{K}_{\textup{LL}}$ as a function of the incident energy.

The results of the sensitivity studies can be found in Fig. 7 for the exemplary spin-parity assignments $5/2^{+}$ and $3/2^{-}$ of the $P_{c}(4450)$, assuming that the $P_{c}(4380)$ havs opposite signature as explained above. They have been estimated assuming 250 days of collected data, both for $\textup{K}_{\textup{LL}}$ and $\textup{A}_{\textup{LL}}$, and a 80% live time. The effective efficiency includes the geometrical acceptance and a conservative detection efficiency $\epsilon_{\text{reco}}\sim 50\%$ to reconstruct the channel. We find that there is a projected sensitivity of more than $5\sigma$ in a larger region than the one already excluded by the cross section measurements only, in particular for $\textup{K}_{\textup{LL}}$.

V Summary

We presented for the first time a study of the polarization observables in hidden charm pentaquark photoproduction close to threshold. This was motivated by a recent Letter of Intent for the SBS experiment at Hall A of JLab, which proposed to study the polarization observables $\textup{K}_{\textup{LL}}$ and $\textup{A}_{\textup{LL}}$   due to their higher sensitivity to the signal when compared to data on differential cross sections.

We thus analyzed the possibility of observing these exotic structures, treating the $P_{c}(4440)$ and $P_{c}(4457)$ states as one combined $P_{c}(4450)$ peak, since there is as of now no information on the quantum numbers of the individual states. We updated the model in Hiller Blin et al. (2016), considering a Pomeron-like background added coherently to the two resonances $P_{c}(4450)$ and $P_{c}(4380)$, and fit to the available data on $J/\psi$ photoproduction close to threshold Camerini et al. (1975); Ali et al. (2019), including the new GlueX results.

If photoproduction experiments prove to be successful in pinning down the $P_{c}$ signals, more refined and systematic analyses on the differential cross section and the spin-parity properties of the pentaquarks will be mandatory, for which this work serves as a benchmark. We show that 250 days of collected data with the SBS experiment will give more than $5\sigma$ sensitivity to the $P_{c}$ signals in large regions of the parameter space, in particular for $\textup{K}_{\textup{LL}}$.

In conclusion, the polarization observables showed an excellent sensitivity to both photo- and hadronic couplings. Therefore, they provide a way to study the nature and properties of the exotic resonances.

The code to calculate the observables and generate the Monte Carlo toy data is publicly available on the JPAC website JPAC Collaboration .

Acknowledgements.

We thank A. Deur and M. Williams for useful discussions. We also thank L. Pentchev and B. Wojtsekhowski for useful comments on the manuscript. This work was supported by the U.S. Department of Energy under Grants No. DE-AC05-06OR23177, No. DE-FG02-87ER40365, and No. DE-FG02-94ER40818, PAPIIT-DGAPA (UNAM, Mexico) Grant No. IA101819, CONACYT (Mexico) Grants No. 251817 and No. A1-S-21389. This work was also supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), in part through the Collaborative Research Center [The Low-Energy Frontier of the Standard Model, Projektnummer 204404729 - SFB 1044], and in part through the Cluster of Excellence [Precision Physics, Fundamental Interactions, and Structure of Matter] (PRISMA+ EXC 2118/1) within the German Excellence Strategy (Project ID 39083149). V.M. acknowledges support from Comunidad Autónoma de Madrid through Programa de Atracción de Talento Investigador 2018 (Modalidad 1).