# Light-cone QCD predictions for elastic -scattering in the intermediate energy region

###### Abstract

The contributions of helicity-flip matrix elements to the deuteron form factors are discussed in the light-cone frame. Normalized , , and are obtained in a simple QCD-inspired model. We find that plays an important role in . Our numerical results are consistent with the data in the intermediate energy region.

###### pacs:

13.40.Gp, 12.38.Bx, 24.85.+p, 27.10.+h^{†}

^{†}preprint: IHEP-TH-25

## I Introduction

The electromagnetic properties of the deuteron have received extensive attention since it can be used to explore the quark and gluon degrees of freedom in the simplest nuclei. Many approaches are suggested to explain its electromagnetic form factors. Among them, the conventional meson-nucleon picture [1, 2] gives a satisfying explanation in the low energy region, i.e. GeV. However, as mentioned by Arnold et al. [3], the experimental results are in sharp disagreement with the meson exchange calculations at higher momentum transfer (see Fig. 1). It means that quark and gluon degrees of freedom must be taken in account. Moreover, the success of dimensional counting rules [4] implies that perturbative QCD(pQCD) may give correct predictions at large momentum transfers. PQCD predicts [5] that, in the Light-cone frame(LCF), the helicity-zero to zero matrix element will be the dominant helicity amplitude at large for elastic ed-scattering. Carlson and Gross [6] have pointed out that LCF helicity-flip amplitudes, and , are suppressed by factors of and , respectively. It is argued [7] that the dominance of , thus the validity of perturbative QCD predictions, begins at GeV. Assuming the helicity-zero to zero dominance, A simple ratio of form factors for the deuteron is predicted

(1) |

with the kinematic factor .

Unfortunately, only to the lowest order, over 300 000 diagrams containing six fermion lines connected by five gluons are required to obtain the full elastic ed-scattering amplitude in pure pQCD approach. Such a calculation can be found in the work of Farrar, Huleihel, and Zhang [8], which shows that the direct application of pQCD to deuteron form factor at experimentally accessible momentum transfers has still a long way to go.

To make detailed predictions for deuteron electromagnetic form factors, we have suggested a model [9] in the intermediate energy region. Based on the reduced nuclear amplitude defined by Brodsky and Hiller [10], the model makes the points that a simple nuclear wave function can represent the data of the deuteron electromagnetic structure function and shows a scenario where is already dominant at of 1 GeV. Normalized can be extracted from fitting the data of . It is pointed out [11] that the helicity-flip matrix element can not be neglected in the expression of . In addition to that, we find that plays an important role in . Neglecting it will result in a contradiction with the data and the conventional meson-nucleon prediction in the low energy region. Due to the kinematical reason, the ratio of form factors in Eq. (1) should be modified even at extremely high . We will model the helicity-flip matrix elements and in the intermediate energy region according to the pQCD predictions. The normalized structure functions and , tensor polarization , and form factors and will be discussed on the basis of the normalized and the above model.

A QCD-inspired model for will be reviewed in Sec. II. Under a phenomenological consideration, the model is applied to obtain normalized , and in Sec. III, where is also discussed. The numerical results and summary are presented in Sec. IV and Sec. V, respectively.

## Ii a Qcd-Inspired Model

For electron scattering on a deuteron, the Rosenbluth cross section is

(2) |

where and are determined by [12]:

(3) | |||||

(4) |

with , where is the deuteron mass. The deuteron form factor is defined as .

In the standard LCF, defined by [13] , and , the above form factors can be obtained from the plus component of three helicity matrix elements:

(5a) | |||||

(5b) | |||||

(5c) |

To avoid the complicated pQCD calculation, a model for the deuteron form factor has been suggested [9]. The point is that the hard kernel at large is assumed to be the perturbative amplitude for the six quarks to scatter from collinear to the initial two-nucleon configuration to collinear to the final two-nucleon configuration, where each nucleon has roughly equal momentum. Since the gluon is a color octet, the single-gluon exchange between two color-singlet nucleon is forbidden. In the lowest order, the quark-interchange is necessary in addition to the one-gluon exchange between two nucleons. For the binding energy of the deuteron is small, we can divide roughly the kernel into two parts. One represents the interchange of quarks and the gluon exchange between two nucleons, which transfer about half of the transverse momentum of the virtual photon from the struck nucleon to the spectator nucleon. Another part is the inner evolution of two nucleons. The first part leads to the reduced form factor of the deuteron and the latter leads to the form factors of two nucleons. A vector boson(color singlet) with an effective mass is introduced to represent the quark-interchange and the one-gluon exchange effects.

Assuming the dominance in the structure function , we get [9]

(6) |

where is the hard scattering amplitude and is the body distribution amplitude of the deuteron defined by

(7) |

The argument for the nucleon form factor, , is since, in the limit of zero binding energy, each nucleon must change its momentum from to . Using Brodsky-Huang-Lepage prescription [14] from a harmonic oscillator wave function, , in the rest frame, the body wave function can be written as

(8) |

where is determined by the normalization of the wave function and by fitting the data of . A direct calculation of diagrams in Fig. 2 gives

(9) |

where is an effective coupling constant. We have taken the nucleon mass to be half of the deuteron mass in Eq. (9).

## Iii The Role of Helicity-Flip Matrix Elements

In the intermediate energy region, dominates the charge form factor , but not and , because the kinematic factor is still small. While , the contributions to both and are enhanced by a factor of . The contribution to is enhanced by . Although and are suppressed for dynamical reason, they may contribute significantly to and because of the kinematic enhancement. Since dominate while is small, the dominance works very well in determining . As for , the helicity-flip matrix element must be taken into account [11]. In , both and may be important.

As is well known, pQCD predicts that and are suppressed by factors and , respectively. The QCD scale is around 200 MeV. In order to explore the role of and , we assume the following relations phenomenologically :

(10a) | |||||

(10b) |

where and are two parameters. These relations are expected to be reasonable as GeV, where is a scale that determines the dominance [7]. Ref. [11] reveals that contributes significantly to . The contribution to was also discussed there. However, as mentioned above, should be taken into account in determining due to the same reason as . Detailed analysis shows that it plays an important role in the intermediate energy region. By assuming relation (10) we have picked up the contributions of and which may be comparable with that of to and in the intermediate energy region. Nonleading contributions to that may enter at the same order of are neglected for kinematic reason. Inclusion of these terms will change the magnitude of slightly.

Substituting Eq. (10) into Eq. (5c), becomes

(11) |

If is suppressed strongly, will be negative while the pQCD begin to be valid. Since it is positive at the origin, there must be a node in the region of GeV. It is in sharp disagreement with the meson-nucleon picture (e.g. see Ref. [1]) and the experimental data [15]. Another constraint on the parameters can be obtained from the data of , which reveals that changes its sign at . It turns out

(12) |

with . Combining Eq. (11) and Eq. (12), it is shown that will keep positive at any momentum transfers if . Different choices of will produce very different , but have little effect on and .

## Iv Numerical Results

At first, we assume the dominance in , from which we get Eq. (6). By fitting the data of (see Fig. 1) we obtain the parameters: and .

Then, given , we can get the expressions of , , and by substituting Eq. (10) into Eqs. (5). As a demonstration, we will show the case for simplicity. In this case is .

(13a) | |||||

(13b) | |||||

(13c) |

with . The inclusion of helicity-flip matrix elements has little effect on (see Fig. 1), but it changes the to 0.11. The normalized structure function is given in Fig. 3, where we have chosen the parameter GeV (i.e. ) from fitting the data. From Eq. (13c) it is easy seen that keeps positive for all momentum transfers for is negative. For different choices of , the normalized and are shown in Fig. 4 and Fig. 5, respectively. The corresponding magnitudes of are shown there, too. At very large momentum transfers, say , the ratio of form factors will be slightly modified to

(14) |

## V Summary

We have discussed the electromagnetic form factors of the deuteron in a QCD inspired model. Detailed kinematic analysis in LCF reveals that, provided the validity of pQCD, the helicity-zero to zero matrix element dominates the gross structure function in both the large and intermediate energy region, which is used to extract the normalized from a simple model for . Further analysis shows that and are also important to determine other form factors. In the present work, and are modeled according to some pQCD predictions at high . Normalized is obtained, whose vanishing at GeV is used to determine the extent that helicity-flip matrix elements are suppressed to. To which extent the is suppressed will strongly effect the behavior of in the intermediate energy region. We find that, If is suppressed strongly, will change its sign twice and one of its nodes lies in the GeV region. It is contrary to the meson-nucleon picture and experimental data. If , will keep positive for all momentum transfers in our model. Different choices of have little effect on and . As an example, we demonstrate a simple case by choosing , but a larger shows a better fit for and . Since the momentum transfers of available data are not high enough to determine the value of reliably, we just show a qualitative result. At very high , the ratio of form factors, , will be slightly modified. The ratio of and is the same as predicted by pQCD as [6]. It is apparent that is bigger than that predicted by Brodsky and Hiller [7]. The kinematic factor leads to the contributions of to in both the large and intermediate energy regions.

By assuming relation (10) we have picked up the leading contributions of and which may be comparable with that of to and in the intermediate energy region. There are corrections from the next to leading order contributions to which have the form similar to . For these corrections can be neglected because of the lack of kinematic enhancement as has. For they enter at the same order as . Inclusion of these terms will diminish the magnitude of parameter in the same way as increasing parameter . Since is not important in itself(i.e. is insensitive to ), our conclusions will not suffer from neglecting them. Precise can not be determined with these terms unknown. It is shown in Fig. 4 and Fig. 5 that is around 0.2.

Since we obtain the normalized by assuming its dominance in and relation (10) by the validity of pQCD, our conclusions, except the ratio, are valid only in the intermediate energy region. It is found that has a node [15] at GeV, which is predicted by conventional meson-nucleon method, too. Although our model can not predict the accurate position, it still shows that the node lies at somewhere GeV. At very large , the contributions of hidden-color states should be taken into account to get a normalized . To some extent, our model shows a smooth connection with pQCD predictions in the high energy region and with traditional nuclear physics conclusions in the low energy region. Thus it can be expected to unify the predictions for the deuteron form factors from the low energy to large energy region. The experiments at CEBAF will be crucial to build a more realistic model in the intermediate energy region.

###### Acknowledgements.

The authors would like to thank Professor T. Huang for his valuable discussions. This work was supported by National Science Foundation of China (NSFC) and Grant No. LWTZ-1298 of Academia Sinica.## References

- [1] M. Gari and H. Hyuga, Nucl. Phys. A 264, 409(1976).
- [2] P.L. Chung, F. Coester, B.D. Keister, and W.N. Polyzou, Phys. Rev. C 37, 2000(1988).
- [3] R.G. Arnold, B.T. Chertok, E.B. Dally, A. Grigorian, C.L. Jordan, W.P. Schütz, R. Zdarko, F. Martin, and B.A. Mecking, Phys. Rev. Lett. 35, 776(1975).
- [4] S.J. Brodsky and B.T. Chertok, Phys. Rev. Lett. 37, 269(1976); Phys. Rev. D 14, 3003(1976).
- [5] S.J. Brodsky and G.P. Lepage, Phys. Rev. D 22, 2157(1980).
- [6] C.E. Carlson and F. Gross, Phys. Rev. Lett. 53, 127(1984).
- [7] S.J. Brodsky and J.R. Hiller, Phys. Rev. D 46, 2141(1992).
- [8] G.R. Farrar, K. Huleihel, and H. Zhang, Phys. Rev. Lett. 74, 650(1995).
- [9] J. Cao and H. Wu, Phys. Rev. C 54, 1006(1996).
- [10] S.J. Brodsky and J.R. Hiller, Phys. Rev. C 28, 475(1983).
- [11] A.P. Kobushkin and A.I. Syamtomov, Phys. Rev. D 49, 1637(1994), Phys. At. Nucl. 58, 1477(1995).
- [12] R.G. Arnold, C.E. Carlson, and F. Gross, Phys. Rev. C 21, 1426(1980).
- [13] S.D. Drell and T.M. Yan, Phys. Rev. Lett. 24, 181(1970).
- [14] S.J. Brodsky, T. Huang, G.P. Lepage, in Particle and Fields, edited by A.Z. Capri and A.N. Kamal (Plenum Publishing Corporation, New York, 1983), p143; T. Huang, Proceedings of XX-th International Conference on high energy physics, Madison, Wisconsin, 1980, p1000.
- [15] M. Garcon et al., Phys. Rev. C 49, 2516(1994).
- [16] S. Platchkov et al., Nucl. Phys. A 510, 740(1990).
- [17] S. Auffret et al., Phys. Rev. Lett. 54, 649(1985); R.G. Arnold et al., Phys. Rev. Lett. 58, 1723(1987); P.E. Bosted et al., Phys. Rev. C 42, 38(1990).