1. Introduction
The geometric structure of the proton is of fundamental interest in the field of high-energy physics. It is known that the proton consists of quarks and gluons, collectively called partons. In collider physics, it is important to understand how partons distribute within the proton since almost all observables require knowledge of parton distribution functions (PDFs) in quantum field theory. However, we are still far from a deep understanding of the behavior of partons inside the proton. Traditionally, the lepton-proton colliders are used to study PDFs, such as Hadron-Elektron Ring Anlage (HERA) at Deutsches Elektronen-Synchrotron (DESY). A point-like particle, like an electron, is used as a probe to resolve the internal structure of the proton, in which the electron emits a virtual photon that fluctuates into a quark-antiquark ($q\bar{q}$) dipole and then interacts with the target proton. Among the lepton-proton deep inelastic scattering (DIS) processes, the exclusive diffractive vector meson production process is an extraordinary tool to probe the proton wave function at high energies (equivalently small Bjorken-x).
The exclusive diffractive process requires at least two gluons to be exchanged between the $q\bar{q}$ dipole and proton target, thus the process is proportional to the squared parton density, which renders this process very sensitive to the structure of the proton. In addition, the exclusive diffractive process can provide access to obtain the transverse spatial distribution of partons in the proton at small x, since this process is the only process in which the transverse momentum transfer t is experimentally accessible, and the t is the Fourier conjugate of the impact parameter profile of the proton. Finally, the property of the exclusive diffractive scattering requires a large rapidity gap between the produced vector meson and scattered target proton, due to the color singlet exchange in the diffractive DIS process. The large rapidity gap feature offers an effective way to identify the diffractive events in the experimental measurements, which then provides the cleanest data and signals to investigate the spatial structure of the proton. Consequently, the exclusive diffractive vector meson production processes not only play an important role in the HERA experiments at DESY, but also are indispensable in the future electron-ion facilities at EIC [1], LHeC [2], and EicC [3] when studying the internal structure of the proton or nucleus. In this paper, we use J/Ψ production in electron-proton collisions at HERA energies to study the stringy structure of the proton.
There are two types of exclusive diffractive processes in the DIS, known as coherent and incoherent processes, see figure 1. For the coherent process, the target proton remains intact after the scattering, the differential cross-section of this process is proportional to the square of the first moment of the scattering amplitude averaged over the initial state configuration, and contains the information on the average transverse spatial structure of the target proton. In the incoherent process, the target proton breaks up after the scattering, the differential cross-section of the incoherent process varies directly with a variance of the scattering amplitude squared and, as such probes the detailed structure of the target proton. So, we will mainly use the incoherent process to investigate the fine spatial structure of the proton in this work.
Figure 1. The vector meson productions in coherent (left) and incoherent (right) Υ* + p exclusive diffractive DIS in dipole picture. |
There have been a lot of efforts in the studies of the proton shape over the past decade [4-17]. One track of these studies is based on the Color Glass Condensate (CGC) effective field theory, which has been proven be to a convenient and efficient method for calculating the quantities, such as cross-sections and structure functions, in both the inclusive and diffractive DIS. In CGC calculations, the scattering amplitude of the exclusive diffractive vector meson production in a virtual photon + proton (Υ* + p) DIS can be factorized into the convolution of three parts, the wave function of a virtual photon, $q\bar{q}$ dipole-proton scattering amplitude (dipole amplitude), and vector meson wave function. The key ingredient is the dipole amplitude since it includes all the QCD dynamics. In particular, it also contains information about the profile density of the proton. In CGC, one of the widely used dipole amplitudes is the impact parameter dependent saturation (IPsat) model [18], since the IPsat model contains information about the impact parameter and gives a rather good description of the experimental data, for instance, proton structure function (F2) at HERA [19], and charged hadron multiplicity distribution at LHC [20, 21]. More specifically, the IPsat model includes the proton profile density function Tp, thus it is a considerably suitable model to study the geometric structure of the proton.
Inspired by the constituent quark picture, a hot spot model was proposed by Mantysaari and Schenke on top of the IPsat model [5, 6]. The hot spot model supposes that the hot spots are inspired by the gluon emission of the respective three constituent quarks, and the transverse positions of these hot spots vary from event to event. The hot spot model provides a rather successful description of the exclusive diffractive vector meson production data at HERA and LHC energies [5, 22-24], and gives a hint about the lumpy structure of the proton at small x. Moreover, it shows that the proton shape fluctuates event-by-event. Thus, it gives an effective initial condition to explain the flow phenomenon in a small collision system. Since then, several other models based on the hot spot model have been introduced to study the detailed properties of the proton [10, 12, 15-17], see a review [25] and references therein.
To explore the possible structure of the proton, we construct a stringy proton model beyond the smallest distance approximation, which is inspired by the quenched lattice QCD calculations in [26]. In our improved stringy proton (ISP) model, the three constituent quarks are connected by gluon tubes which are merged at the Fermat point of the quark triangle, and the profiles of the gluon tubes initiated by the up and down quarks are treated separately. The transverse profile of the gluon tubes is assumed to follow the Gaussian distribution with width Bt (Bu and Bd), specifically, the Bu for the width of the up quark-initiated gluon tube and the Bd for the width of the down quark-initiated gluon tube. The positions of the constituent quarks also follow the Gaussian distribution with width Bp. We consider that both the profile of gluon tubes and the positions of the constituent quarks vary from event to event, which can be used to study the proton shape fluctuations.
The ISP model is used to calculate the J/Ψ production in Υ* + p diffractive DIS and compare it to the measurements at HERA. We find that the calculations are in good agreement with the data. In particular, at small t our results of J/Ψ production in the incoherent process are better matching with the data points than the ones from the hot spot model. Note that a method with the smallest distance approximated stringy proton (SD-SP) was used to evaluate the differential cross-section of J/Ψ production, however, it was found that the results at small t have large deviations from the measurements at HERA [6]. Meanwhile, we find that the radius of the proton calculated by the width parameters of the ISP model is consistent with the one from fitting to the recent data from the GlueX collaboration at Jefferson Lab [27, 28], which solves a tension that the proton radius extracted from the SD-SP is incompatible with the measured radius. Moreover, we study the fine structure of the gluon tubes which connect constituent quarks and the Fermat point of the quark triangle. We compute the differential cross-section of J/Ψ production in two cases, Bu ≥ Bd and the reverse (Bu < Bd), and compare the results to the HERA data. An interesting result is found that it seems the width of the up quark-initiated gluon tube is always larger than the width of the down quark-initiated gluon tube in the event-by-event cases. This finding is consistent with our previous outcomes obtained in the hot spot model [15].
2. Exclusive diffractive vector meson production in CGC
A brief review of the formalism of calculating the exclusive diffractive vector meson production in Υ* + p DIS is present in this section. All the calculations are based on the CGC framework and dipole picture. Generally, there are two types of exclusive diffractive processes in terms of whether the scattered target proton is dissociated or not, see figure 1. If the target proton remains intact after the scattering, we call it the coherent process. While the target proton is dissociated after the interaction, it is known as the incoherent process. The Good-Walker picture is a widely used approach to describe the exclusive diffractive process [29]. In this picture, the differential cross-section of the coherent process is proportional to the square of the first moment of the diffractive scattering amplitude and can be written as [30]1 ), we take into account the real part correction of the diffractive scattering amplitude by multiplying (1 + β2), since the diffractive scattering amplitude was derived from the assumption that it is purely imaginary. β is the ratio of the real to imaginary parts of the scattering amplitude 1 ) is called the skewness correction [31], 3 ), which is introduced to account for the imbalance momentum fractions x and $x^{\prime} $ of the two exchanged gluons between the $q\bar{q}$ dipole and proton target, see figure 1.
$\begin{eqnarray}\frac{{\rm{d}}{\sigma }^{{\gamma }^{* }+p\to V+p}}{{\rm{d}}t}=\frac{(1+{\beta }^{2}){R}_{g}^{2}}{16\pi }\left|\right.\left\langle \right.{{ \mathcal A }}^{{\gamma }^{* }+p\to V+p}(x,{Q}^{2},{\boldsymbol{\Delta }})\rangle {\left|\right.}^{2},\end{eqnarray}$
where $\langle$ ... $\rangle$ refers to the average over possible configurations of the target proton, ${{ \mathcal A }}^{{\gamma }^{* }+p\to V+p}$ is the diffractive scattering amplitude which will be discussed in detail below, Q2 is the virtuality of the virtual photon, ∆ is the momentum transfer between the incoming and outgoing proton, which has the relationship with momentum transfer square $t={({P}_{out}-{P}_{in})}^{2}$ as ${\boldsymbol{\Delta }}\equiv \sqrt{-t}$. Υ*, p and V denote the virtual photon, proton, and vector meson, respectively. In equation ( $\begin{eqnarray}\beta =\tan \left(\frac{\pi \lambda }{2}\right)\end{eqnarray}$
with $\begin{eqnarray}\lambda =\frac{{\rm{\partial }}\mathrm{ln}\left(\right.{{ \mathcal A }}^{{\gamma }^{* }+p\to V+p}\left)\right.}{{\rm{\partial }}\mathrm{ln}(1/x)}.\end{eqnarray}$
The factor Rg in equation ( $\begin{eqnarray}{R}_{g}=\frac{{2}^{2\lambda +3}}{\sqrt{\pi }}\frac{{\rm{\Gamma }}(\lambda +5/2)}{{\rm{\Gamma }}(\lambda +4)}\end{eqnarray}$
with λ defined in equation (In the Good-Walker picture, the differential cross-section of the incoherent process is defined as the difference between the second moment and first moment squared of the diffractive scattering amplitude and can be written as
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{d}}{\sigma }^{{\gamma }^{* }+p\to V+p^{\prime} }}{{\rm{d}}t} & = & \displaystyle \frac{(1+{\beta }^{2}){R}_{g}^{2}}{16\pi }\\ & & \times \,\left(\left\langle \right.\left|\right.{{ \mathcal A }}^{{\gamma }^{* }+p\to V+p}(x,{Q}^{2},{\boldsymbol{\Delta }}){\left|\right.}^{2}\rangle \right.\\ & & \left.\times \,-\left|\right.\left\langle \right.{{ \mathcal A }}^{{\gamma }^{* }+p\to V+p}(x,{Q}^{2},{\boldsymbol{\Delta }})\rangle {\left|\right.}^{2}\right),\end{array}\end{eqnarray}$
which is a variance. Thus, it can be used to probe the amount of fluctuations in the target wave function.To clearly see the function of coherent and incoherent cross-sections in the study of the spatial structure of the proton, we compare equation (1 ) with equation (5 ), one can see that the coherent cross-section depends on the average over the diffractive scattering amplitude and as such probes the average shape of the proton. The incoherent cross-section relies on the variance of the proton, which leads to it being sensitive to the internal structure of the proton. Therefore, these two cross-sections make it possible to extract the overall and detailed information about the proton structure.
Now turn to introduce the diffractive scattering amplitudes in equation (1 ) and equation (5 ). In the CGC framework, the amplitude of the Υ* + p diffractive scattering can be factorized into three subprocesses (see figure 1): Firstly, the virtual photon fluctuates into a $q\bar{q}$ dipole, where the virtual photon wave function Ψ can be precisely calculated by perturbative QED; Then, the $q\bar{q}$ dipole scatters off the target proton with the dipole-proton cross-section dσdip/d2b; Finally, the scattered dipole combines into a final state vector meson with wave function ΨV. The scattering amplitude of the exclusive diffractive vector meson production can be obtained by the convolution of the overlap wave function and dipole cross-section [30]6 ) refers to the longitudinal momentum fraction of the quark. The dipole-proton cross-section, dσdip/d2b, is a key ingredient of the diffractive scattering amplitude, it includes all the QCD dynamics of the interactions ($q\bar{q}$-target scattering processes via gluon exchange and rapidity evolution of the scattering amplitude) and also the structure information of the proton. We will discuss it in detail together with the stringy proton model in the next section. The $\left(\right.{{\rm{\Psi }}}^{* }{{\rm{\Psi }}}_{V}{\left)\right.}_{T,L}$ in equation (6 ) is the overlap wave function between the photon and the vector meson and can be written as [30]
$\begin{eqnarray}\begin{array}{l}{{ \mathcal A }}_{T,L}^{{\gamma }^{* }+p\to V+p}(x,{Q}^{2},{\boldsymbol{\Delta }})={\rm{i}}\displaystyle \int {{\rm{d}}}^{2}{\boldsymbol{r}}\displaystyle \int {{\rm{d}}}^{2}{\boldsymbol{b}}\displaystyle \int \frac{{\rm{d}}z}{4\pi }\left(\right.{{\rm{\Psi }}}^{* }{{\rm{\Psi }}}_{V}{\left)\right.}_{T,L}\\ \times \,\exp \left\{\right.-{\rm{i}}\left[\right.{\boldsymbol{b}}-\left(\right.\frac{1}{2}-z\left)\right.{\boldsymbol{r}}\left]\right.\cdot {\boldsymbol{\Delta }}\left\}\right.\frac{{\rm{d}}{\sigma }^{{\rm{dip}}}}{{{\rm{d}}}^{2}{\boldsymbol{b}}},\end{array}\end{eqnarray}$
where the two-dimensional vector ∆ is the Fourier conjugate to the center-of-mass of the dipole b - (1/2 - z)r with b being the impact parameter of the $q\bar{q}$ dipole with respect to the proton target, r is the transverse size of the $q\bar{q}$ dipole, the subscript T and L stand for the transverse and longitudinal contributions. The z in equation ( $\begin{eqnarray}\begin{array}{rcl}\left(\right.{{\rm{\Psi }}}^{* }{{\rm{\Psi }}}_{V}{\left)\right.}_{T} & = & {\hat{e}}_{f}\sqrt{4\pi {\alpha }_{em}}\frac{{N}_{c}}{\pi z(1-z)}\left\{\right.{m}_{f}^{2}{K}_{0}(\epsilon r){\phi }_{T}(r,z)\\ & & -\,[{z}^{2}+{(1-z)}^{2}]\epsilon {K}_{1}(\epsilon r){\partial }_{r}{\phi }_{T}(r,z)\left\}\right.,\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}\left(\right.{{\rm{\Psi }}}^{* }{{\rm{\Psi }}}_{V}{\left)\right.}_{L} & = & {\hat{e}}_{f}\sqrt{4\pi {\alpha }_{em}}\frac{{N}_{c}}{\pi }2Qz(1-z){K}_{0}(\epsilon r)\\ & & \times \left[\right.{M}_{V}{\phi }_{L}(r,z)+\delta \frac{{m}_{f}^{2}-{{\rm{\nabla }}}_{r}^{2}}{{M}_{V}z(1-z)}{\phi }_{L}(r,z)\left]\right.,\end{array}\end{eqnarray}$
where δ = 1, ${\hat{e}}_{f}$ is the vector meson effective charge (e.g. ${\hat{e}}_{f}=2/3$ for J/Ψ), mf are the mass of a quark with flavor f. MV is the mass of the produced vector meson, Nc = 3 is the number of colors, K0 and K1 are the modified Bessel functions of the second kind with ${\epsilon }^{2}\equiv z(1-z){Q}^{2}+{m}_{f}^{2}$, ${{\rm{\nabla }}}_{r}^{2}$ is a differential operator and defined as ${{\rm{\nabla }}}_{r}^{2}\equiv (1/2){\partial }_{r}+{\partial }_{r}^{2}$. The ΦT,L(r, z) are the scalar part of the vector meson wave functions, which cannot be precisely calculated and need to be modeled. In this work, we will use the boosted Gaussian scalar wave function, since it has been successfully used to describe the variety of diffractive measurements at HERA energies. In boosted Gaussian formalism, the scalar wave functions are given by [30] $\begin{eqnarray}\begin{array}{rcl}{\phi }_{T,L}(r,z) & = & {{ \mathcal N }}_{T,L}z(1-z)\\ & & \times \exp \left(-\frac{{m}_{f}^{2}{{ \mathcal R }}^{2}}{8z(1-z)}-\frac{2z(1-z){r}^{2}}{{{ \mathcal R }}^{2}}+\frac{{m}_{f}^{2}{{ \mathcal R }}^{2}}{2}\right).\end{array}\end{eqnarray}$
In this work, we will use J/Ψ production as a candidate signal to probe the stringy structure of the proton, since its mass is larger enough to make the perturbative calculation of its photoproduction cross-section reliable, and its mass is relatively small to make it having high statistics in the measurements. The relevant parameters of J/Ψ’s scalar wave functions, ${{ \mathcal N }}_{T,L}$, mf, ${ \mathcal R }$, and MV, are given in table 1.Table 1. Parameters of the ’boosted Gaussian’ wave function for J/Ψ [30]. |
| Meson | MV/GeV | mf/GeV | ${{ \mathcal N }}_{T}$ | ${{ \mathcal N }}_{L}$ | ${{ \mathcal R }}^{2}$/GeV-2 |
|---|---|---|---|---|---|
| J/ψ | 3.097 | 1.4 | 0.578 | 0.575 | 2.3 |
3. Dipole-proton cross-section
The dipole-proton cross-section in equation (6 ), is related to the forward elastic scattering amplitude N. According to the optical theorem, the dipole-proton cross-section can be written as 11 ) is a scale that relates to the r as
$\begin{eqnarray}\frac{{\rm{d}}{\sigma }^{{\rm{dip}}}}{{{\rm{d}}}^{2}{\boldsymbol{b}}}({\boldsymbol{b}},{\boldsymbol{r}},x)=2N({\boldsymbol{b}},{\boldsymbol{r}},x).\end{eqnarray}$
In the CGC framework, the rapidity (or x) evolution of the dipole scattering amplitude is governed by the JIMWLK4 evolution equation [32-36], which can be reduced to a closed Balitsky-Kovchegov (BK) equation in the mean field approximation [32, 37]. Both the JIMWLK and BK equations are extremely complicated integral differential equations, it is very difficult to solve them analytically in the full dynamics regions, although there are some analytic solutions in the saturation regime [38-42]. Meanwhile, several numerical methods have been developed to solve the leading order (LO) and next-to-leading order (NLO) impact parameter-independent BK equations, the reasonable solutions are obtained, and they can describe some measurements at HERA and LHC energies, e.g. proton structure functions [43-45]. It is known that there are some tensions once the impact parameter is included in the numerical solutions of the BK equations [46]. However, the impact parameter plays a key role in the study of the proton spatial structure, since the experimental measurable quantity (transverse momentum transfer t) is the Fourier conjugate to the impact parameter. Thus, we need to have the impact parameter information in the dipole scattering amplitude. Unfortunately, it is known that the numerical solution of the BK equation exhibits a strong Coulomb tail, once the impact parameter is included. Based on the reasons mentioned above, we choose to use the impact parameter dependent saturation (IPsat) model as the dipole scattering amplitude, which is widely used in the literature and successfully used in describing the data at HERA, and LHC energies [20, 21, 47]. In the IPsat model, the dipole scattering amplitude is given by [18] $\begin{eqnarray}N({\boldsymbol{b}},{\boldsymbol{r}},x)=1-\exp \left[-\frac{{\pi }^{2}{{\boldsymbol{r}}}^{2}}{2{N}_{c}}{\alpha }_{s}({\mu }^{2})xg(x,{\mu }^{2}){T}_{p}({\boldsymbol{b}})\right],\end{eqnarray}$
where the xg(x, μ2) is the gluon distribution whose evolution satisfies the DGLAP evolution equation. The μ in equation ( $\begin{eqnarray}{\mu }^{2}=\frac{4}{{{\boldsymbol{r}}}^{2}}+{\mu }_{0}^{2},\end{eqnarray}$
and the gluon distribution xg(x, μ2) at the initial scale ${\mu }_{0}^{2}$ is $\begin{eqnarray}xg(x,{\mu }_{0}^{2})={A}_{g}{x}^{-{\lambda }_{g}}{(1-x)}^{5.6},\end{eqnarray}$
where the parameters μ0, Ag, and λg are determined by the fit to the inclusive DIS reduced cross-section data at HERA [19].The transverse profile function, Tp(b), in equation (11 ) includes the spatial structure information of the proton. We use a color string-inspired picture to construct the transverse profile function in order to explore the possible structure of the proton. The idea of the stringy proton model is based on the quenched latticed QCD calculations, the three constituent quarks (two up and one down quarks) are connected by the gluon tubes (color fields) which merge at the Fermat point of the quark triangle, see figure 2. We sample the constituent quark positions by a three-dimensional Gaussian distribution with width Bp. The density profile of the gluon tube, which connects the constituent quarks and Fermat point of the quark triangle, also has a Gaussian distribution and can be obtained by integration over the longitudinal direction [5, 6]
$\begin{eqnarray}{T}_{t}({\boldsymbol{b}})=\frac{1}{2\pi {{\rm{B}}}_{{\rm{t}}}}\int {\rm{d}}{b}_{z}\exp \left(-\frac{{{\boldsymbol{b}}}^{2}+{b}_{z}^{2}}{2{{\rm{B}}}_{{\rm{t}}}}\right),\end{eqnarray}$
with width Bt. Then, the profile function of a proton, Tp(b), is established by summing over the three gluon tubes $\begin{eqnarray}{T}_{p}({\boldsymbol{b}})=\frac{1}{{N}_{t}}\displaystyle \sum _{i=1}^{{N}_{t}}{T}_{t}({\boldsymbol{b}}-{{\boldsymbol{b}}}_{i}),\end{eqnarray}$
where Nt = 3 is the number of gluon tubes. Note that an approach with the smallest distance approximated stringy proton was used to study the proton shape fluctuations. However, the results from the aforementioned approach show some deviations to the J/Ψ production data at HERA, although some extra fluctuations are included, such as saturation momentum (Qs) and dipole size fluctuations [6]. In this paper, our ISP model fully considers all the contributions from the gluon tubes and fluctuations. We find that the ISP model is good enough to reproduce the J/Ψ data at small t.
Figure 2. Selected proton density profiles from the hot spot model (left) and stringy proton model (right) at high-energy. |
Let’s turn to introduce the Qs fluctuations. It has been shown that the Qs fluctuations are significant in the description of the J/ψ production data at small t at HERA energies [5, 6]. We shall consider the Qs fluctuations followed by [6], where the saturation scale satisfies a log-normal distribution
$\begin{eqnarray}P\left(\right.\mathrm{ln}{Q}_{s}^{2}/\langle {Q}_{s}^{2}\rangle \left)\right.=\frac{1}{\sqrt{2\pi }\sigma }\exp \left[-\frac{{\mathrm{ln}}^{2}{Q}_{s}^{2}/\langle {Q}_{s}^{2}\rangle }{2{\sigma }^{2}}\right].\end{eqnarray}$
In terms of above distribution, the expectation of ${Q}_{s}^{2}/\langle {Q}_{s}^{2}\rangle $ is $\begin{eqnarray}E\left[\right.{Q}_{s}^{2}/\langle {Q}_{s}^{2}\rangle \left]\right.=\exp \left[\right.{\sigma }^{2}/2\left]\right..\end{eqnarray}$
If one calculates the $\langle {Q}_{s}^{2}\rangle $, which shows it about 13% (for σ = 0.5) larger than the one without including the Qs fluctuations. So, one has to normalize the log-normal distribution to ensure that the desired expectation remains the same. Note that we include the Qs fluctuations by the way that we let the Qs of each constituent quark-initiated tube fluctuate independently.Inspired by our previous findings based on the HS model [15], where we found that the up quark-inspired hot spot is different from the down quark-inspired hot spot at small x. Specifically, each hot spot has a different distribution width. In this work, we shall use equations (14 ) and (15 ) to investigate the fine structure of the gluon tube. We let the up quark-initiated two gluon tubes have the same width Bu, while the width of the down quark-initiated gluon tube is Bd. We compare our model calculations with HERA measurements. It shows that the width of the up quark-initiated gluon tube is always larger than or equal to the width of the down quark-initiated gluon tube, Bu ≥ Bd. This outcome is coincident with the results in [15] where the up quark-inspired hot spot width is larger or equal to the down quark-inspired hot spot width.
Besides the stringy proton model mentioned above, we would like to remind readers that the HS model is a popular candidate for studying the spatial structure of the proton in the literature [10-17], please see a review and also the references therein for details of the HS model [25]. The HS model assumes that the proton consists of several hot spots formed by the gluon emission of constituent quarks (see figure 2), the positions of the hot spots fluctuating from event to event, which leads to the proton shape fluctuating event-by-event. In hot spot model, the profile function of proton is written as [6]
$\begin{eqnarray}{T}_{p}({\boldsymbol{b}})=\frac{1}{{N}_{q}}\displaystyle \sum _{i=1}^{{N}_{q}}{T}_{q}({\boldsymbol{b}}-{{\boldsymbol{b}}}_{i}),\end{eqnarray}$
where Nq is the number of hot spot, and the profile density of each hot spot is assumed to be Gaussian $\begin{eqnarray}{T}_{q}({\boldsymbol{b}})=\frac{1}{2\pi {{\rm{B}}}_{{\rm{q}}}}\exp \left[-\frac{{{\boldsymbol{b}}}^{2}}{2{{\rm{B}}}_{{\rm{q}}}}\right]\end{eqnarray}$
with width parameter Bq, where the subscript q denotes u (up) or d (down) quark. The HS model provides a reasonable description of the vector meson production at HERA energies, especially for the incoherent measurements. However, the current data at HERA cannot exclude other possible topological structures of the proton due to statistics and limited kinematic regions, e.g. stringy gluon tube configuration. This work shall give an alternative view of the spatial structure of the proton.4. Numerical results
In this section, we will give the numerical results of the coherent and incoherent J/Ψ production by using the ISP model. Firstly, we will show comparisons of the J/Ψ differential cross-sections between the ISP model and the HS model. Then, the differential cross-sections of J/Ψ production in the cases of Bu ≥ Bd and Bu < Bd are present. The results presented below are obtained for 10000 configurations of the proton.
4.1. Comparisons between improved stringy proton and hot spot models
It is known that the original HS model assumes that the proton consists of three hot spots which are formed by the gluon emission of the constituent quarks, and the positions of the constituent quarks fluctuate from event to event. We would like to note that it has been found that the hot spots can grow in number at sufficiently high energies [10], but we focus on the physics at HERA energies (W ~ 100GeV) in this paper, so the energy dependence of the hot spot number can be neglected. In this study, we assume that the proton consists of three hot spots as has been done in [6]. The positions of the constituent quarks are sampled randomly in terms of Gaussian distribution. The correlations between constituent quark positions are omitted in the HS model. However, the correlations between the partons could have a significant impact on the proton shape, which is similar to the correlations between the nucleons in a nucleus. As is known that the nucleons’ correlation plays an important role in the initial condition of the hydrodynamic evolution of the hot and dense QCD medium. It is difficult to describe the small system flow phenomenon in high-energy heavy ion collisions without considering the correlations [48]. The stringy proton model supposes that the constituent quarks are correlated by gluon tubes (fields). The gluon tubes start from the constituent quarks and end up at the Fermat point of the quark triangle. Thus, in the stringy proton model, the spatial structure of the proton is modified by the correlations as compared to the HS model. To clearly see the role of the correlation between the constituent quarks, we calculate the differential cross-sections of J/Ψ production by using the HS model (uncorrelated) and ISP model (correlated). We compare the numerical results in order to show the significance of the correlations.
The coherent and incoherent differential cross-sections of the J/Ψ production as a function of transverse momentum transfer t in Υ* + p DIS are present in figure 3, where the experimental data are taken from H1 and ZEUS Collaborations at HERA [49-52]. The left panel of figure 3 gives results calculated at W = 75 GeV, while the right panel of figure 3 shows the calculations at W = 100 GeV. The sold curves denote the numerical results of the coherent differential cross-section, and the dashed curves represent the numerical results of the incoherent differential cross-section (similarly hereafter in the following figures). Note that the red curves are the results from the HS model, the green curves are the predictions from our ISP model, and the blue curves are the results computed by the SD-SP method. One can see that all the models can give a qualitative description of the coherent and incoherent data at W = 75 GeV and W = 100 GeV. This outcome seems reasonable due to the fact that all the models include the proton shape fluctuation effect which is a key ingredient in describing the vector meson productions. Although there is a visible difference in the coherent differential cross-sections of J/Ψ production between the HS and ISP models, especially at large t, it is hard to see which model is preferred by the data at W = 75 GeV (see the left panel of figure 3), since the measurements were only performed to the momentum transfer less than 1 GeV2. Fortunately, the measurements of the coherent J/Ψ production at W = 100 GeV provide the data up to t ~ 1.5 GeV2, see the right panel of figure 3. From the coherent process in the right panel of figure 3, one can clearly see that the ISP model (χ2/d. o. f ~ 1.8) is more favored by the data than the HS model (χ2/d. o. f ~ 3.9) at large t.
As is known that the incoherent differential cross-section is defined by the variance of the diffractive scattering amplitude, so it is more sensitive to the detailed structure of the proton, namely the details of the model of the proton profile density. We use this feature to show the difference between HS and ISP models in figure 3. At a glance, it seems that the dashed curves in figure 3 successfully match the data of the incoherent differential cross-section of the J/Ψ production in the full t range. However, there are remarkable deviations between the data and calculations from the HS model (χ2/d. o. f ~ 1.7) and SD-SP model (χ2/d. o. f ~ 1.8) at small t (see the left panel of figure 3), although the supplementary saturation scale fluctuations are taken into account. Interestingly, the ISP model (χ2/d. o. f ~ 1.1) gives a rather good description of the incoherent differential cross-section data not only at large t but also at small t. Note that we do not compute the χ2 of the incoherent process at the right panel of figure 3 due to few experimental data points resulting in too many uncertainties. We trace the reason why our ISP model exhibits a better performance than the other two models, it can be attributed to two aspects, correlation and fluctuation. Firstly, the ISP model uses the color string tube to consider the correlations between the constituent quarks, while the HS model ignores the quark correlations. Secondly, our ISP model goes beyond the smallest distance approximation and fully considers the contributions from the gluon tubes.
Finally, we would like to mention that the width parameters (Bp = 3.0 GeV-2, Bu = 1.0 GeV-2, and Bd = 1.0 GeV-2) resulting from the ISP model are consistent with the ones coming from the HS model [6]. When one roughly estimates the root mean square radius of the proton with ${r}_{p}=\sqrt{2({{\rm{B}}}_{{\rm{p}}}+{{\rm{B}}}_{{\rm{u}}/{\rm{d}}})}$, and gets rp = 0.55fm, which is coincidental with the one obtained by fitting to the data from the GlueX Collaboration at Jefferson Lab [27, 28]. However, the proton radius (rp = 0.61 fm) resulting from the SD-SP method does not match the one resulting from fitting to the GlueX data.
4.2. The fine structure of the proton with varied distribution width of the gluon tube
In recent lattice QCD studies [53], it has been shown that the up quark has a different density distribution from the down quark in the unpolarized proton, and the distortions between the up and down quarks are also different in the polarized proton. These findings indicate that the up quark and down quark in the proton have different structures at small x. Moreover, our recent studies in [15] also found that the distribution widths of the up and down quarks are different, and it seems that the distribution width of the up quark is always larger than or equal to the one of the down quark, which are favored by the HERA data. Inspired by the aforementioned outcomes, we plan to study the fine structure of the gluon tubes which connect the constituent quarks and the Fermat point of the quark triangle (see figure 2), and to see whether the up quark-initiated gluon tube has a different distribution from the down quark-initiated gluon tube.
We calculate the coherent and incoherent differential cross-sections of the J/Ψ production with varied distribution width of the gluon tube in Υ* + p DIS at W = 75 GeV and W = 100 GeV. The relevant results are shown in figure 4. To see the distribution difference between the up and down quarks initiated gluon tubes, we vary the gluon tube width parameters Bu and Bd, but keeping the average width of gluon tubes Bt = (2Bu + Bd)/3.0 = 1.0 GeV-2 unchanged, since it has been shown that Bt = 1.0 GeV-2 is an optimal parameter to reproduce the data points [15]. Note that in figure 4 the green curves are calculated by the ISP model with parameters Bp = 3.0 GeV-2 and Bu=Bd=1.0 GeV-2, which are used as the reference to evaluate the compatibility of the other possible distribution widths of the gluon tube since the green curves give a rather good description of the experimental data. The left (right) panels of figure 4 demonstrate the numerical results computed by the widths of the up quark-initiated gluon tubes larger than or equal to (smaller than) the widths of the down quark-initiated gluon tubes, Bu ≥ Bd (Bu < Bd). One can see that all the numerical results calculated with Bu ≥ Bd are compatible with the HERA data in both coherent and incoherent processes, while the predictions computed with Bu < Bd have some tensions with HERA data(green curves). Especially, the deviations are remarkable in the incoherent processes (see the upper right panel of figure 4, W = 75 GeV) with χ2/d. o. f about 1.1 ~ 1.6 for Bu < Bd as compared to χ2/d. o. f about 1.1 ~ 1.3 for Bu ≥ Bd. Note that we do not compute the χ2/d. o. f of the incoherent process at W = 100 GeV due to only two data points available rendering large uncertainties. However, one can clearly see the deviations at |t| ~ 0.6 GeV2 with a large gap between the purple and the green curves on the lower right panel of figure 4 in the incoherent process. These outcomes are consistent with the findings in our previous studies in [15] where we found that the width of the hot spot inspired by the gluon emission of the up quark is larger than or equal to the one from the gluon emission of the down quark.
Figure 4. The coherent (solid curves) and incoherent (dashed curves) differential cross-section of J/Ψ production as a function of |t| at W = 75 GeV and W = 100 GeV as compared to the data from H1 and ZEUS collaborations [49-52]. The numerical results of the left (right) panel are calculated by using Bu≥Bd (Bu < Bd). |
5. Conclusions and discussions
In order to study the possible geometric structure of the proton, we go beyond the smallest distance approximation to establish an improved stringy proton model. We present detailed event-by-event calculations of the exclusive diffractive J/Ψ production with our ISP model in the framework of Color Glass Condensate. We find very interesting outcomes that the numerical results of the incoherent differential cross-section of J/Ψ production computed by the ISP model at small t are more favored by the HERA data compared to ones resulting from the HS model and the SD-SP method. It shows that the correlation and fluctuation are the reasons why the ISP model has a better performance than the HS model and SD-SP method. Moreover, we find that the width parameters Bp, Bu, and Bd resulting from the ISP model are compatible with those from the HS model, and the radius of the proton computed by the width parameters of the ISP model is coincidental with the one obtained by fitting to the data from the GlueX Collaboration at Jefferson Lab.
We study the fine structure of the gluon tube within the ISP model. The differential cross-sections of exclusive diffractive J/Ψ production are calculated by letting Bu be different from Bd. We find that for the incoherent process, all the numerical results of the incoherent differential cross-section of J/Ψ production computed with Bu ≥ Bd are almost consistent with each other at W = 75 GeV and W = 100 GeV, respectively. The numerical results of the incoherent differential cross-section of J/Ψ production calculated with Bu < Bd are incompatible with the HERA data. This outcome seems to indicate the possibility that the width of the up quark-initiated gluon tubes is always larger than or equal to the width of the down quark-initiated gluon tube, in event-by-event cases. This finding is consistent with the result obtained by the HS model in [15] where the up quark-inspired hot spot width can have any times of size larger than or equal to the down quark-inspired hot spot width at small x.
All the numerical results presented in this paper are calculated by assuming that the positions of the constituent quarks and the profile of the gluon tubes have Gaussian distribution. In fact, they can be other distributions. It has been found if one uses the exponential distribution as the position distribution of the constituent quarks, the HS model predictions of the coherent differential cross-section of the J/Ψ production are improved at large t [54]. The studies of the influence of the position distribution of the constituent quarks and the profile distribution of the gluon tube on the exclusive diffractive J/Ψ production in the DIS process will be our next work [55], which shall provide further insights on the proton structure.