𝒑𝒉𝒊 meson electroproduction at low transferred momentum

The electroproduction of phi meson is analyzed. The amplitude consist of two terms, the soft and the hard pomeron terms. it is assumed, that both pomerons are Regge poles. In the present approach, the two pomeron amplitude for the photoproduction processes is extended to construct an amplitude for the electroproduction processes by introducing the photon virtual mass ( 𝑄 ! ) . The residues of the two poles become functions of ( 𝑄 ! ) . These functions are extracted by fitting the differential cross section ( #$ #% ) of the extended model at constant energy ( 𝑤 ) to the experimental data using the chi-square method. At each 𝑄 ! , the hard and the soft terms are multiplied by the factors 𝐶 & and 𝐶 ’ to match the data. The vertices of the pomeron coupling to the interacting particles are calculated by the dimensions of the vertices which are related to the masses of the interacting particles at each vertex.


Introduction
Diffractive electroproduction of a vector meson ( ) is represented by the process * → p (1) where = , , , /Ѱ…vector mesons The main controlling variables are: ! , ) , with is the squared of the transferred momentum and ) is the meson mass. The hard region is reached if one of these variables is large [1,2]. In that case, the QCD calculations should be used [1][2][3][4]. Most of the qualitative properties of the data can be reproduced by such calculations. However, the calculations suffer from uncertainties in the gluon distribution and the vector meson wave function which introduces uncertainty in the overall normalization. Different models introduced ! dependence using different approaches. Some of these models assume that the coupling of pomeron to the photon-vector meson vertex is responsible for this dependence [1,5]. Some other models used the ! in the parameters of pomeron trajectory [5][6].
*Corresponding author at Physics Department, College of Science, University of Anbar. Ramadi ,Anbar ,Iraq; E-mail addresssc.af.ak33@uoanbar.edu.iq As the process in present case is elastic, the variable is not a hard scale. Furthermore, the phi mass is about 1.02 , then the ! is the only hard variable [3]. The two pomeron model involves contributions from the hard and the soft terms. A transition from a soft regime to a hard regime in the process is set by ! . Data on the total cross section for the photoproduction at high is parameterized as * [7][8][9]. The parameter [10][11]increases as the mass scale ( ) ! ) increases while, for electroproduction [3] increases with the hard scale ( ) ! + ! ). However, the rate of increase of decreases with increasing the hard scale [3,4]. The values of can be obtained by fitting the total cross section using * at each ! . Experimentally, The differential cross section is parameterized as +,|%| , with decreases from about 12 +! at small value of the hard scale to a universal scale value of about 5 +! as the hard scale increases [12].
To derive an amplitude for vector meson production by a virtual photon in eq.(1), we extend the two pomeron amplitude for the photoproduction by introducing the virtual photon mass ( ! ) to the amplitude [3,4 , 13-15], then: with = ! is the total center of mass of energy squared. & and ' are normalization constants in the photoproduction amplitude . The soft and the hard trajectories of the pomeron are taken here as:  [1,17] i.e: These radii are related to the inverse of the masses of the interacting particles at each vertex. Then, the radius of the proton vertex is given by: ! Ge +! is a fitting parameter , < ! is the meson mass. The largest radius of interaction is that of the real photon interaction, while the smallest radius is when ! is very large as given by eq. (7).
in eq.(2) introduces the variation of the cross section with ! at a constant energy [4]. The value of may be deduced from the data. Extracting the slopes of the pomeron trajectories in the exponent of in eq.(2), then the forward slope including eq.(5) should take the following forms for the soft and hard terms [1,4]: Squaring eq.( 2) we get: introduces the contribution from the longitudinal photons. The differential cross section for meson photoproducton can be obtained by setting ! = 0 in eq. (9). For electroproduction the soft and the hard terms eq.(9 ) should be multiplied by the factors & and ' for the normalization with the data, then: Using the minimum ( ! ) method, these factors are adjusted by comparing eq.(10) with the experimental data [18][19][20]. It is found that all the ! values are less than one. The resulting & and ' factors are denoted by the weights. The initial guessed for the value of in eq.(10) may be motivated by the form of the ! dependence of the experimental data. The obtained values for meson are ( =1.36 for ! ⩽ 6.6 and =1.4 for otherwise). It is clear that the values of weights for photoproduction are & = ' =1.0

Results
In the following the results of using the two pomeron model in calculating the electroproduction of vector meson is discussed. .

meson electroproduction
Fitting the  The weights & as a function of ! are plotted in fig.1 (a) , while the weights & are plotted in fig. 1(b). Fig.(1) (a) Hard pomeron and (b) (12) It is clear that the hard pomeron weights in fig.1(a) increases with ! , but the rate of the increase is decreasing with ! . Therefore, the curve may scale at large ! . As ! increases the hard weight dominates. On the other hand, the contribution from the soft pomeron is decreasing.

t dependence
The data [18] on the differential cross section ( #$ #% ( * → )) as a function of at = 125 and for different Q ! values (Q ! =2.4, 3.6, 5.2, 6.9, 12.6 and 19.7 GeV ! ) are shown in fig.(2). The data are compared with the results of the two pomeron model in eq. (10). The dashed curves are the results of the model using eqs. (11,12). A reasonable fit for the data in terms of the model is clear.

dependence
The total elastic cross section is obtained by integrating eq. (10) over from = 0 to =1.13. at different values of ! as shown in fig.(3). The results of the calculation are compared with the data [19] on the total cross section as a function of in fig.(3) at values of different ! (2.4,3.8,6.5 and 13. ! ). We notice the steep increase of the cross section with energy. As expected, the data rise like * . this form can be used to fit the data and to find the value of at each ! . An agreement between the model and the data is clear in fig.(3). The steep increase with energy at a constant ! is attributed to the high intercept of the hard pormeron trajectory.

Summery and Conclusions
The two pomeron model is used to calculate the electroproduction of phi meson. The photoproduction amplitude is extended by introducing the virtual photon mass to construct the electroproduction amplitude .In addition to the fitting factors found in photoproduction amplitude, the hard and soft pomeron terms must be multiplied by certain fitting factors known as weights. The minimal chi square approach may be used to progressively change these weights until the electroproduction data is well-fitted. Tables (1) for the Phi meson tabulate the weight values as a function of ! . The hard weight rises with ! and is predicted to scale large ! values. The model shows agreements with the data on the differential and total cross sections in figs. (2,3).