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To probe the kaon structure in the high-energy
$ e-p $ collision, we exploit the abundant "kaon cloud" from proton dissociation due to the large coupling$g_{\rm N\Lambda K}$ . This type of electron-"meson cloud" scattering dominates in the t channel with one meson exchange, called the Sullivan process [45], which is shown in Fig. 1. To ensure that the electron beam hits the "kaon cloud," we need to tag the leading Λ of high energy and small transverse momentum. The Λ baryon acts as the spectator carrying a large fraction of the incoming proton's momentum and going far-forward. To measure the kaon structure, we also need to ensure that the virtual kaon is broken up by the high energy probe. In literature, the internal structure of the quasi-real kaon in the process resembles the internal structure of the real kaon, provided the momentum transfer is not large ($ \lesssim 0.9 $ GeV$ ^2 $ ) [46].Figure 1. The Sullivan process [45] for deep inelastic scattering with the production of a leading Λ baryon. The leading Λ carries a significant amount of the momentum of the beam proton.
The invariant kinematical variables describing the leading Λ tagged DIS are: the momentum square
$ Q^2 $ of the photon probe, Bjorken variable$x_{\rm B}$ , inelasticity y of the scattering, longitudinal momentum fraction$ x_{\rm L} $ carried by the Λ baryon, and square of the momentum transfer t from the proton to the virtual kaon. According to the momenta of the particles labeled in Fig. 1, these kinematical variables are defined as,$ \begin{aligned}[b] Q^2 \equiv &-q^2,\quad x_{\rm B} \equiv \frac{Q^{2}}{2P_{\rm p} \cdot q},\quad y \equiv \frac{P_{\rm p} \cdot q}{P_{\rm p} \cdot P_{\rm e}},\\ x_{\rm L} \equiv & \frac{P_{\rm \Lambda}\cdot q}{P_{\rm p}\cdot q},\quad t \equiv (P_{\rm p} - P_{\rm \Lambda})^2 = p_{\rm K}^2. \end{aligned} $
(1) In the definition,
$ x_{\rm L} $ denotes the longitudinal momentum fraction (energy fraction approximately) of the final Λ baryon to the incoming proton. In the DIS experiment, the leading Λ tagged process dominates in the large-$x_{\rm L}$ region ($ \gtrsim 0.5 $ ), hence a proper cut on the$x_{\rm L}$ variable efficiently selects the events that are sensitive to the kaon structure. In addition, t denotes the momentum square of the virtual kaon, which is an important variable for the extrapolation of the real kaon structure.To estimate the statistical error of the measurement, we need to know the number of events of the interests. Therefore, we first need to know the cross section of the reaction. With the azimuthal angle integrated, the four-fold differential cross section of the leading Λ tagged DIS is expressed as [34, 35, 47],
$ \begin{aligned}[b] \frac{{\rm d}^4\sigma({ep\rightarrow e\Lambda X})}{{\rm d}x_{\rm B}{\rm d}Q^2{\rm d}x_{\rm L}{\rm d}t} =& \frac{4\pi\alpha^2}{x_{\rm B}Q^4}\left(1-y+\frac{y^2}{2}\right)F_2^{\rm L\Lambda(4)}(Q^2, x_{\rm B}, x_{\rm L}, t)\\ =& \frac{4\pi\alpha^2}{x_{\rm B}Q^4}\left(1-y+\frac{y^2}{2}\right)\\&\times F_2^{\rm K}\left(\frac{x_{\rm B}}{1-x_{\rm L}} ,Q^2\right)f_{\rm K^+/p}(x_{\rm L},t).\\[-15pt] \end{aligned} $
(2) From this equation, it can be found that we can extract the four-fold leading-Λ structure function
$ F^{\rm L\Lambda(4)}_2 $ . In the kaon pole model, the leading-Λ structure function can be factorized into the product of the kaon structure function$ F_2^{\rm K} $ and kaon flux around the proton$ f_{\rm K^+/p} $ . In an effective theory of the kaon pole, the kaon flux is given by [34, 35, 47],$ \begin{aligned}[b] f_{\rm K^+/p}(x_{\rm L},t)= & \frac{1}{2\pi}\frac{g^2_{\rm N\Lambda K}}{4\pi}(1-x_{\rm L})\\& \times \frac{-t}{(m_{\rm K}^2-t)^2} {\rm exp}\left(-R^2_{\rm \Lambda K}\frac{t-m_{\rm K}^2}{1-x_{\rm L}}\right), \end{aligned} $
(3) in which the coupling is
$ g_{\rm N\Lambda K}^2/4\pi = 14.7 $ , and$ R_{\rm \Lambda K} = 1 $ GeV$ ^{-1} $ is a form-factor parameter representing the radius of the proton's$\Lambda-\rm K$ Fock state. With these formulae presented above, we can compute the cross section of the leading Λ baryon tagged DIS process. -
To compute the statistical error of the kaon structure function, we simply need to compute the statistical error of the cross section, because these two experimental observables are directly related. The statistical uncertainty of the cross section measurement depends on the number of events collected during the experiment. To estimate the number of events of an experiment, we need to know the cross section of the reaction (provided by the model described in above sections), the integrated luminosity of the experiment, and the event selection criteria of the reaction. For a year operation of good quality beams, EicC could accumulate approximately 50 fb
$ ^{-1} $ integrated luminosity of$ e-p $ collisions. Hence, we take the integrated luminosity of 50 fb$ ^{-1} $ for the simulation. To ensure that the collected events are mainly from electron-"kaon cloud" collisions, we take the following event selection criteria:$ x_{\rm L}>0.55 $ ,$ P_{\rm T}^{\rm \Lambda}<0.5 $ GeV,$M_{X} > 1$ GeV, and$ W>2 $ GeV.$ x_{\rm L}>0.55 $ and$ P_{\rm T}^{\rm \Lambda}<0.5 $ GeV ensure that the events are from the Sullivan process of the t channel, while$ W > 2 $ GeV is the conventional DIS criterium. Figure 8 presents the energy and pseudorapidity distributions of the Λ and its decays, after the event selection criteria, geometrical acceptance of the detectors, and low energy threshold of the calorimeters. The zero-degree calorimeter is suggested to cover the angle from 0 to 3 degrees around the beam, to collect more neutrons and photons.Figure 8. (color online) The energy and θ angle distributions of high momentum far-forward Λ baryon and its decay chains, with the geometric cut and energy threshold of electromagnetic calorimeters applied, for the simulation data of leading Λ baryon tagged DIS at EicC.
To obtain the cross section at each kinematical point, we need to count the number of events in different kinematical bins. The typical kinematical binning is presented in Fig. 9, for the events in the
$ Q^2 $ range of$ (3,5) $ GeV$ ^2 $ . We focus on the events at relatively small$ |t| $ ($ <0.85 $ GeV$ ^2 $ ), a condition suggested by DSE calculation to ensure that the extrapolation to the real kaon structure is valid and effective [46]. With the event selection criteria discussed in the above paragraph, we calculate the number of events in each bin, using the following formula,Figure 9. (color online) The binning scheme in
$ -t $ versus$ x_{\rm K} $ plane for the Monte-Carlo data in$ Q^2 $ range of$ (3,5) $ GeV$ ^2 $ .$ \begin{align} N_{i} = L \overline{\sigma}_{i} B_r \epsilon \Delta x_{\rm K} \Delta Q^2 \Delta x_{\rm L} \Delta t (1-x_{\rm L}), \end{align} $
(4) where L is the integrated luminosity of the suggested experiment,
$\overline{\sigma}_{i}$ is the averaged differential cross section in kinematical bin i,$ B_r $ is the branching ratio of Λ decaying into the neutron and two photons,$ \epsilon $ is the detector efficiency for collecting all the final states of the reaction, i.e.,$ \epsilon = \epsilon_n * \epsilon_{\gamma_1} * \epsilon_{\gamma_2} $ , the factor$(1-x_{\rm L})$ is the Jacobian coefficient for the transform from$ x_{\rm B} $ space to$ x_{\rm K} $ space, and together, the other factors express the size of the kinematical bin. For the detectors of common performance, we assume$ \epsilon_{\gamma} = $ 90% for detecting and identifying the photons from$ \pi^0 $ decay, and$ \epsilon_n = $ 50% for detecting and identifying the far-forward neutrons. Finally, the relative statistical error of the kaon structure function$ \delta(F_2^{\rm K}) / F_2^{\rm K} $ in each kinematical bin is estimated to be$1/\sqrt{N_{i}}$ .By counting the simulated events in each kinematical bin, we calculate the statistical uncertainty of the kaon structure function for the proposed experiment at EicC. Fig. 10 presents the relative statistical error of
$ F_2^{\rm K} $ in the kinematical bin of$ 3\; {\rm GeV^2}<Q^2<5\; {\rm GeV^2} $ . It can be observed in the plot that the statistical uncertainty increases with an increase in$ x_{\rm K} $ . For the data at$ x_{\rm K}<0.3 $ , the projected statistical uncertainty is smaller than 1%. With the$ x_{\rm K} $ increasing up to approximately 0.85, the statistical uncertainty is approximately 5%. In the future, these precise data will provide an excellent test of the predictions of lattice QCD and DSE. At higher$ Q^2 $ values up to 50 GeV$ ^2 $ , the statistical uncertainty projections are also projected and illustrated in Fig. 11 ($ Q^2\sim 25 $ GeV$ ^2 $ ) and Fig. 12 ($ Q^2\sim 40 $ GeV$ ^2 $ ). With wider kinematical bins and fewer data points, the estimated statistical precision of$ F_2^{\rm K} $ measurement remains optimal. For the data points in the region of$ x_{\rm K}<0.6 $ , the relative statistical uncertainties are less than 5%. In addition, for the data points in the region of$ x_{\rm K}<0.8 $ , the relative statistical uncertainties are less than 10%. These experimental data over a wide range of$ Q^2 $ will provide an interesting opportunity to test the QCD evolution equations in the kaon sector, as well as extract the gluon distribution in the kaon via the scaling violation.Figure 10. (color online) The statistical error projections of the kaon structure function at
$ Q^2 \sim 4 $ GeV$ ^2 $ . We calculate the statistical error at each bin center. The right axis is a scale indicating the size of the statistical error.
Tackling the kaon structure function at EicC
- Received Date: 2021-09-22
- Available Online: 2022-06-15
Abstract: Measuring the kaon structure beyond proton and pion structures is a prominent topic in hadron physics, as it is one way to understand the nature of the Nambu-Goldstone boson of QCD and observe the interplay between the EHM and HB mechanisms for hadron mass generation. In this study, we present a simulation of the leading Λ baryon tagged deep inelastic scattering experiment at EicC (Electron-ion collider in China), which is engaged to unveil the internal structure of kaon via the Sullivan process. According to our simulation results, the suggested experiment will cover the kinematical domain of