# Electroproduction of meson from proton

and the strangeness in the nucleon

###### Abstract

We analyze the process near the threshold within - cluster model as a probe of the strangeness content of proton. Our consideration is based on the relativistic harmonic oscillator quark model which takes into account the Lorentz-contraction effect of the hadron wavefunctions. We find that the knockout mechanisms are comparable to the diffractive production of vector-meson-dominance model when only (3–5)% strange quark admixture is assumed, which should be compared with (10–20)% of the nonrelativistic quark model prediction. The cross sections of the - and -knockout processes have qualitatively different dependence on the four-momentum transfer squared to the proton and may be distinguished experimentally. We also briefly discuss a way to determine the strangeness content of proton in 5-quark cluster model.

###### pacs:

^{†}

^{†}preprint: nucl-th/9612059

## I Introduction

Conventional phenomenological quark models widely used for describing low-energy properties of baryons treat nucleon as consisting of only up and down quarks; intrinsically, there is no strange quark in the nucleon. Considering the success of the constituent quark models [1], it may come as a surprise when some recent measurements and theoretical analyses indicate a possible existence of a significant strange quark content in the nucleon.

For example, analyses of the sigma term [2] in
pion-nucleon scattering suggest that about one third of the rest
mass of the proton comes from pairs inside the proton.
The EMC measurement of the proton spin structure functions in
deep-inelastic muon scattering [3, 4] has been interpreted
as an indication of the strange quark sea strongly polarized
opposite to the nucleon spin, leading to the conclusion that the
total quark spin contributes little to the total spin of the proton.
A similar conclusion^{1}^{1}1For other interpretations of these experiments, see, for
instance, Ref. [7].
has been drawn from the BNL elastic neutrino-proton scattering [5, 6].
This has stimulated a set of new experimental proposals [8] to
measure the neutral weak form factors of the nucleon which might be
sensitive to the strange quarks of the nucleon.

One of the intriguing idea associated with a direct probe of the strangeness content of proton is to use meson production from proton [9, 10, 11, 12]. Since meson is nearly 100% -state because of the - ideal mixing, its coupling to the proton is suppressed by the OZI rule. The main idea of this proposal, called OZI “evasion” process, is to determine the amount of the -admixture of nucleon, if any, by isolating its contribution in production processes. For example, admixture in the proton wavefunction allows contributions from “shakeout” and “rearrangement” diagrams in production from and collisions [12]. Another idea is to use -meson lepto- and electroproduction from proton target as advocated by Henley et al. [9, 10]. In this case, in addition to the diffractive production mechanism of the vector-meson–dominance model (VDM) we have contributions from the direct “knockout” mechanism.

If we consider admixture of proton, we can parameterize the Fock-space decomposition of the proton wavefunction as

(1) |

where denotes any combination of gluons and light quark pairs of
and quarks. Analyses of the production experiments can
give estimates on . For instance, investigation of
annihilation by connected quark line diagrams estimates that the
sea quark contribution to the proton wavefunction is
between 1% and 19% [13]. In Ref. [10], by using the
electroproduction its upper bound is estimated to be about
10-20%. To obtain this estimation, the authors used nonrelativistic
quark model (NRQM) and calculated the cross section of the
-knockout process. Because of the paucity of experimental
data [14, 15], it was compared with the VDM predictions^{2}^{2}2
The recent ZEUS experiment [16] was done at very high energy,
and is beyond the applicability of this work.
.
However, as was pointed out by our previous publication [17]
where a preliminary result was reported, the knockout contributions
are closely related to the hadron form factors and in the considered
kinematical region of production, the minimum value of
is about 3.6 GeV, where is the
three-momentum transferred to the hadron system. So it is clear that
the value of the momentum transfer in this process is too large to
use the NRQM because its predictions on hadron form factors are in
poor agreement with experiment at GeV.

In this paper, we improve the work of Ref. [10] by including relativistic effects based on the relativistic harmonic oscillator model (RHOM) [18, 19, 20, 21, 22, 23, 24] which describes successfully the proton form factors in a wide range of . We also carry out the calculations for the -knockout and its interference with the -knockout which were argued to be suppressed and left out in Ref. [10] as well as for the -knockout. The calculations are done both in NRQM and in RHOM for a comparison. As in Ref. [10], we will compare the cross sections of the knockout processes with the VDM predictions. However, this does not mean that the knockout mechanism should replace the VDM mechanism. The latter is present as a background of the knockout mechanism and our purpose is to determine a theoretical upper bound of using electroproduction process.

This paper is organized as follows. In the next section, we briefly review the general structure of the knockout and VDM differential cross sections and introduce the kinematical variables for meson electroproduction. Then in Sect. III we discuss the proton wavefunction of 5-quark cluster model. Section IV is devoted to the evaluation of the knockout process matrix elements within the non-relativistic harmonic oscillator quark model. In Sect. V we perform the calculations based on the RHOM which provides an explanation of the dipole-like dependence of the elastic nucleon form factor. This model, though probably it has not underlying physical significance, has the pleasant feature that basically all quantities of interest can be worked out analytically, and, in many cases, it allows understanding of the qualitative picture of the reaction. The nontrivial role of the relativistic Lorentz-contraction effect is also discussed. In Sec. VI we briefly discuss a way to extract out the strangeness content of proton in an extended quark model. Section VII contains a summary and some details in the calculation are given in Appendix.

## Ii Kinematics and cross sections

The one-photon exchange diagram for electroproduction is shown in Fig. 1. The four momenta of the initial electron and proton, final electron and proton, the produced meson, and the virtual photon are denoted by , , , , , and , respectively. In the laboratory frame, we write , , , where is the nucleon mass, , , and , respectively. The electron scattering angle is defined by . We also denote the electron mass and mass by and , respectively. The other invariant kinematical variables are , the minus of photon mass squared , the four-momentum transfer squared to the proton , the proton–virtual-photon center-of-mass energy , and the total energy squared in the CM system . We also use dimensionless invariant variables , and defined as

(2) |

So, in the laboratory frame we have

(3) |

In terms of the conventional -matrix elements , the differential cross section is given as

(4a) | |||

and | |||

(4b) | |||

where () and are the spin projections of the incoming and outgoing proton (electron) and outgoing meson, respectively, and |

(5) |

Upon integrating Eq. (4) over non-fixed kinematical variables we find the triple-differential cross section of the electroproduction in the laboratory frame in the form of

(6a) | |||

where | |||

(6b) | |||

and are the corresponding azimuthal angles, respectively. |

The tree diagrams that could contribute to electroproduction are illustrated in Fig. 2. In Fig. 2(a), the virtual photon turns into the meson and then scatters diffractively with the proton through the exchange of a Pomeron. This VDM of diffractive production has been widely used to describe vector-meson photo- and electro-productions. It generally reproduces well the dependence of the cross sections for fixed , but is not successful to account for the experimentally observed rapid decrease in the cross sections with increasing . The double differential cross section predicted by the VDM is [15]

(7) |

where

(8) |

which is related to the flux of transverse virtual photons in the laboratory frame (for fixed and ) by , with being the Jacobian . Here, is the virtual-photon polarization parameter

(9) |

As in Refs. [10, 15], we will work with the cross section , which is predicted by VDM as [15]

(10) |

where is the photoproduction cross section. The range of () can be obtained from . The first part of (10) represents the photoproduction cross section extrapolated to by the square of the propagator. The second represents a correction to the virtual photon flux where is the virtual photon momentum in CM frame. Explicitly it is written as [26]

(11) |

which can be approximated to unity in the large limit. This term is a measure of the ambiguity in the model predictions [15, 26], and comes from a choice made in the definition of the transverse photon flux. The term corrects the cross section for the longitudinal component which is missing at , and the exponential factor corrects for the fact that for a given the physical range of is smaller when than its range at . We fix the parameters following Refs. [14, 15] as GeV, with , and b, which is fitted for GeV.

The -dependence of the cross section can be obtained by assuming dependence. This gives

(12) |

provided that is negligible [10].

Figure 2(b) corresponds to the process where an pair is directly knocked out by the photon and Fig. 2(c) to the direct -knockout. It is also possible that the system would have some hadronic intermediate states like and before and after the meson is emitted as shown in Fig. 3. These diagrams represent some of the rescattering effects and deserves to be studied. However, we will focus only on the direct knockout mechanism of Fig. 2(b,c) in this paper and leave the other for future study. Instead of triple and double differential electroproduction cross sections and for the knockout process, respectively, we will work with the cross sections and defined as

(13) |

The knockout amplitude in the one-photon exchange approximation may be written in the most general form as

(14) |

where are the hadron and electron electromagnetic (e.m.) current matrix elements, respectively. The electron matrix element is given by

(15) |

where is the plane wave electron Dirac spinor ( denotes the spin projection) normalized as . The hadron e.m. current matrix element depends on the model for description of the initial and final hadron states and the form of the e.m. current operator . The additivity of the e.m. current in the quark model enables one to write the amplitude as

(16) |

where the first term describes the interaction of the e.m. field with the and quarks, i.e., the -cluster knockout, while the second one corresponds to the -knockout.

Then the squared amplitude consists of three terms which are the contributions of the - and -cluster knockout and the interference:

(17) |

## Iii Proton wavefunction

For simplicity and for our qualitative study, we approximate the proton wavefunction (1) as

(18) |

by absorbing the contributions from terms into the coefficient and keeping the leading order term in types. The -type terms must be present in the proton wavefunction. However, as we shall see, the cross sections of the knockout mechanism for electroproduction depends on , so that such an approximation may be justifiable in this process. We can further decompose the wavefunction as

(19) |

where the superscripts denote the spin of each cluster. Then is the strangeness admixture of the proton and and correspond to the spin-0 and spin-1 fractions of cluster, respectively. They are constrained to by the normalization of the wavefunction. The symbol represents a possible orbital angular momentum between the two clusters. In the simplest picture, the and quarks are in a relative state with respect to each other with negative intrinsic parity. The cluster is also in a relative state with respective to the CM of the cluster. We also neglect a possible hidden color components, which was shown to be negligible in SU(2) 5-quark model [25]. Therefore, the configuration (19) corresponds to and meson “in the air” in proton wavefunction, where (=) is the mixture of and .

Then to describe positive parity proton, the cluster should be in a relative -wave about the CM with the cluster that is the “bare” proton. More complicated configurations are possible by allowing complex combinations. But we expect that the above two components give major contribution to the electroproduction. Also excluded is the higher spin states of cluster. For example, we may include spin 3/2 cluster. However, since the isospin of is zero, the cluster should have isospin 1/2. Experimentally observed states with , are and at 1520 MeV and 1720 MeV, respectively. Because of their heavier mass, their role is expected to be small and excluded in our study as in Ref. [10].

The spin-orbital wavefunction can be obtained by noting that the total spin (with ) is

(20) |

where is the spin of the cluster , of the cluster (0,1), and the relative orbital angular momentum . Therefore, we can write the spin wavefunction as

(21) |

For , we have and

(22) |

And for , since can be either 1/2 or 3/2, we have

with . Finally, the spin-orbital wavefunction reads

(24) |

where denotes the bare proton wavefunction. When combined with the flavor, spatial and color wavefunctions, this completes our proton wavefunction.

## Iv Non-relativistic quark model

### iv.1 The model

For the spatial wavefunctions of hadrons, we use the nonrelativistic harmonic oscillator quark potential model in this Section. If we consider the proton wavefunction in 5 quark configurations, the spatial wavefunction is obtained from the Hamiltonian

(25) |

where the labels refer to the particles in the -cluster and to the -cluster, and is the mass of -th quark. In this work, we use MeV and MeV. This Hamiltonian can be diagonalized by introducing Jacobian coordinates as

(26) |

where .

Then the spatial wavefunction with momentum and the projection of the orbital angular momentum (=1), has the form

(27a) | |||||

where the normalized radial wavefunctions are | |||||

(27b) | |||||

where and represents . They are normalized as , and , . The dimensional parameters are | |||||

(27c) | |||||

with . The spatial wavefunctions of the bare proton and meson have the similar structure. |

We fix the dimensional parameters as in Ref. [10] by using the fact that they are related to the hadron rms radii. The wavefunctions (27) give

(28) |

and we assume that of the bare proton and of the are equal to and , respectively. By making use of the empirical values of ( fm) and ( fm) and by introducing scaling factor (=1.5) [10], we get

(29) |

To obtain we assumed that the spring constant of coordinate is the same as that of meson.

We also use the additive form of the NRQM e.m. current;

(30) |

where

(31) |

in momentum space, where and are the charge and mass of the -th quark and () is its final (initial) momentum.

### iv.2 -knockout

The -knockout process is depicted in Fig. 2(b). Because of the symmetric property of the spatial wavefunction and the current, it is manifest that only the part of the proton wavefunction (19) and the magnetic part of the e.m. current (31) can contribute. Then the -matrix is obtained as

(32) |

where and is a four-vector defined as

(33) |

The spatial overlap integral in NRQM is defined as

(34a) | |||||

where | |||||

(34b) | |||||

(34c) | |||||

(34d) |

By making use of

(35) |

for unpolarized case, we can obtain

(36) |

where

(37) |

The spatial integrals are calculated using (27) as

(38) |

This is the result derived in Ref. [10] for the knockout process. Since it depends on only through of , one can see that the cross section is maximum near by noting that increases as decreasing .

### iv.3 -knockout

As depicted in Fig. 2(c), the cluster is a spectator in this process. So only the part contribute contrary to the knockout. The relevant amplitude reads

(39) |

where

(40) |

with . The overlap integral reads

(41a) | |||||

where | |||||

(41b) | |||||

(41c) | |||||

(41d) |

Then we have the squared amplitude as

(42) |

where

(43) |

with

(44) |

where is defined in Eq. (5). The radial wavefunctions of NRQM give

(45) |

Then the cross section depends on only through of . This shows that in contrary to the -knockout, the -knockout process gives its main contribution near .

### iv.4 gauge invariance

The electron e.m. current in (15) satisfies the conservation condition . The gauge invariance implies the same condition for the hadronic current:

(46) |

In general, however, this condition is not satisfied in inelastic scattering [27]. In our case, this condition is satisfied only in the knockout process since only the magnetic part of the e.m. current contributes. In knockout process, however, the convection current takes part in the process and the relation (46) is not obeyed. The breakdown of the gauge invariance is crucial at small because the matrix elements of are proportional to as .

The most commonly used technique of enforcing gauge invariance [19, 27] in photoproduction is to project out the gauge non-invariant part as

(47) |

This modification, however, is not adequate for the electroproduction, where the electron e.m. current cancels the subtracted part.

A possible way for restoring the gauge invariance is to modify the longitudinal component of the current. Decomposition of the spatial component of the current (40) gives the longitudinal part as