Pis'ma v ZhETF, vol. 99, iss. 11, pp. 715-720 © 2014 June 10
Nucleon form factors for the elastic electron-deuteron scattering at high
momentum transfer
A. V. Bekzhanov+*1\ S. G. Bondarenko+, V. V. Burov+
+ Joint Institute for Nuclear Research, 141980 Dubna, Russia *Far Eastern Federal University, 690950 Vladivostok, Russia Submitted 19 March 2014
The reaction of the elastic electron-deuteron scattering at high momentum transfer is investigated within the Bethe-Salpeter approach. The relativistic covariant Graz II separable kernel of nucleon-nucleon interactions is used to analyze the deuteron structure functions, form factors, and tensor of polarization components. The modern data for the electromagmetic nucleons structure from the double polarization experiments as well as some other models of the nucleon form factors are considered.
DOI: 10.7868/S0370274X14110010
1. Introduction. The deuteron being the simplest two-nucleon bound system is a powerful instrument to study strong interactions. The reaction of elastic electron-deuteron scattering provides information not only on NN interaction but also on the electromagnetic structure of nucleons. Such investigations at high energies are of great interest nowadays, especially in the context of future experiments at being upgraded JLab facilities.
It is necessary to note that in order to describe the elastic form factors of the deuteron at high momentum transfer (Q2 = —q2 > 2 (GeV/c)2) the relativistic properties of the strong interactions should be taken into account. Here, properties of core nuclear forces play a very important role. From the physical point of view, elastic electron-deuteron scattering at transfer momentum up to 6 (GeV/c)2 is an amazing phenomenon taking into account that binding energy of the deuteron is very small (2.2 MeV). So the subject of investigation has a great significance for the nuclear and particle physics.
Some approaches, based on the Bethe-Salpeter (BS) equation [1], satisfy this condition, among them are the light-front dynamics [2], the equal-time equation [3], BS approach with separable interaction [4] and so on. In the last approach, one has to solve the system of linear integral equations for both the NN scattered states and the bound state - the deuteron. In order to find a solution of a system of integral equations, it is a good idea to use a separable ansatz [4] for the interaction kernel in the BS equation. Then, one can transform the integral equations into a system of algebraic linear ones which
-'-'e-mail: bekzhanov@jinr.ru
can be solved. Parameters of the interaction kernel are extracted from an analysis of phase shifts for respective partial-wave states and low-energy parameters as well as deuteron properties (bound state energy, magnetic moment, elastic form factors etc.). In the Refs. [5] and [6] the latter approach was developed and applied to the reaction of the deuteron electrodisintegration.
The electromagnetic (EM) structure of nucleons at high momentum transfer is another topic of interest. In the paper, four models for the nucleon form factors are used. First of them - the dipole fit (DFF) [7] - was widely used. The main feature of this model is that the ratio of electric (G^) and magnetic (G£?) proton form factors is constant. Another one - the relativistic harmonic oscillator model (RHOM) [8] - is the quark model with a relativistic harmonic oscillator potential.
However, recently there was an intensive discussion that the ratio obtained by the Rosenbluth separation technique differs from the one obtained by the recoil polarization method [9, 10]. To describe the results of the latter method, it is necessary to use a certain parametrization of the ratio as some linear function of the transfer momentum squared. The model with described ratio for the electric proton form factor and the Galster parametrization [11] for the neutron electric form factor - modified dipole fit (MDFF1) - is also considered (see also, [12] and [6]).
Recently the Unitary and Analytic (U&A) approach has been used to develop new nine-resonance model [13]. This model which includes new experimental data on the nucleon EM form factors as well as a new method of introducing the asymptotic behavior for the EM form factors also used in calculations.
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In contrast to the Ref. [14], the influence of the new parametrization of proton electric form factor is investigated, and in the development of our previous papers [15] and [16], the deuteron form factors are calculated at high energies where the analytic structure of the vertex functions should be taken into account.
Also we have considered high-energy dynamics of the poles contributions which arise from the analytic structure of the separable kernel.
The paper is organized as follows: in Sec. 2 we describe the models of the nucléon form factors and consider analytic structure of the deuteron EM current in the Sec. 3. The obtained results are discussed in Sec. 4. In Sec. 5 the conclusion is given.
2. Models of the nucléon form factors. We calculate elastic electron-deuteron scattering in the rela-tivistic impulse approximation within the BS approach with the covariant Graz II (rank III) kernel of the NN interaction [17] and [14].
Details of the calculations of the deuteron structure functions A(q2), B(q2), charge Fc(q2), quadrupole Fq(q2), and magnetic Fm{<12) form factors and tensor polarization components T20, T21 of the final deuteron can be found in [15].
We use four models of the electromagnetic nucléon form factors (see also [7, 8,12,13]):
• the original dipole fit for the proton and neutron form factors (DFF) is
Fd = (1 + Q2/0.71)
-2
GPE — Fd,
GnE= 0,
r<p m
Hp
r<P
G
m
{¿n G E:
(1)
the modified dipole fit 1 (MDFF1) is
r<p
^ e
G
[1 -0.13(Q2 -0.04)]fd, Mnr
E 1 + 5.6t ^M = Mp-frfj
Fd
G'1
m
: UnFd-
(2)
In MDFF1 we take into account the latest JLab data [9] for the proton electric form factor by the following ratio /i,pGpe/Gpm = 1 - 0.13(Q2 - 0.04), while for the neutron electric form factor we use the Galster [11] parametrization;
• the proposed nine-resonance U&A model of the nucleon has 12 free parameters. Their values were obtained from the analysis of the existent experimental data and additionally new one measured
recently in Mainz. All details and formulas can be found in Ref. [13];
• the relativistic harmonic oscillator is
1
/(3)
X
x exp
" (l + Q2/2m2)2
1 -Q2 2 • 0.42 1 + Q2/2m2 '
gpe = №,
GnE = Q2/2m2I(-s\
Or ,
r<p
TM
= №.
(3)
The relativistic harmonic oscillator model is based on the quark model with the relativistic oscillator potential. All the FFs calculated in this model have the correct asymptotic behavior. The only free parameter in the model is the oscillator parameter which was found from fitting of the experimental data.
Above Hp = 2.7928 and /xn = —1.9130 are the magnetic moments of the nucleons, Q2 = —q2 > 0 is the transfer momentum squared, r = Q2/4m2, m is the nucleon mass, and all dimensional parameters are in (GeV/c)2.
3. Analytic structure. After the partial-wave decomposition the matrix element of the deuteron current has the following form
{D'M'\3»\DM) =lttM(q2) F[S\q2) +
= |p|24p| d(cose) x
X <i>L,{p'o, \v'\)4>l{vo, IpI) X
li,l=0,2
„2\
(4)
x A,2 M'M M^0'
|p|, cos e, q2
where the function F
l',l 1,2 M'M m
(po, |p|, cos 0, q2) is the re-
sult of the trace calculations. The radial part of the amplitude is
4>l(po, IpI) = s++(p0, \p\)9l(po, IpI
(5)
with ghipo, |p|) being the radial part of the vertex function and
s++(po,|p|) = -F /r)-^T)(6)
(Md/2 + po - Ep){Md/2 - po - Ep)
being the positive energy part of the propagators and the energy Ep = \Jm2 + p2.
Analyzing the analytic structure of expressions (5) and (6) we can write the following expression for the poles in the po complex plane:
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• initial deuteron for propagator S++(po, |p|):
po = ±Md/2 TEP± ie, (7)
for functions Ql{po, |p|):
po = ±El3k =F ie; (8)
• final deuteron for propagator S++(p'0, |p'|):
Po = -(1+4 n)Md± ± sjEl + 4ÇMd\p\ cos e + 4^2M2 T ie, (9)
for functions gL'i'Po, |p'|):
po = -i]Md ± ± + 2£Md|p| cos 9 + eMj T ie, (10)
with the energy Epk = ^J+ p2, = Q2/4Mj, and
C= vWT^J.
To calculate the matrix elements (4) we should perform the Wick rotation procedure. However, during the used procedure, the poles (9) and (10) can get into the contour of the po integration. Additionally, the residue in these poles should be taken into account. All contributions from the poles have the threshold value on Q2 which have the following form:
for the propagator S++(p'0, |p'|):
Qo = Md(2m - Md), for the functions guip'o, |p'|):
Ql = 4 Md/3k. The Wick rotation procedure can be written as:
V
i J fdpo = j f dp a -
—oo —oo
- 2nJ2HQ2-Ql) Resfc(/,po =P§), (11) k
where the threshold values Q% for the Graz II kernel are in Table.
It is seen that calculations with the Q2 > 1.182 (GeV/c)2 must take into account contribution from the poles of vertex function.
4. Results and discussion. Figs. 1-6 show the influence of the considered models of the nucléon electromagnetic form factors to the elastic electron-deuteron scattering at high momentum transfer.
Threshold values for the poles of the kernel
k Ql (GeV/c)2
0 0.004
1 1.182
2 1.736
3 3.915
4 5.965
g2 (GeV/c)2
Fig. 1. (Color online) The deuteron structure function A(q2) as a function of the transfer momentum squared. Calculations with DFF (black solid line), MDFF1 (dashed red line), [13] (gray dotted line) and RHOM (blue dashed dotted line) nucleon form factors are shown. Experimental data are taken from [18]
In Fig. 1 the deuteron structure function A(q2) is shown. It is seen that the difference between considered models is significant and at the Q2 = 10 (GeV/c)2 it reaches the value of about 2 orders for the Ref. [13] and RHOM models. We can also see that the best model up to Q2 =
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