^{3}.

QED first-order radiative corrections to the differential cross section for elastic scattering of deuterons on electrons at rest are calculated. Radiative corrections due to soft- and hard-photon bremsstrahlung from the electron lines, the electron vertex correction, and the electron vacuum polarization are considered. The model-dependent contributions due to the bremsstrahlung from the deuteron and the deuteron vertex correction, depending on its internal structure, are not included. We consider an experimental setup where the final particles are recorded in coincidence and their energies are determined within some uncertainties. Formulas for the relevant kinematical variables, the cross section, and the radiative corrections are derived and numerical results are presented.

Polarized and unpolarized scattering of electrons off protons and light nuclei has been widely studied since these experiments give information on the internal structure of these particles (for recent reviews, see Refs.

The recent determination of the proton electromagnetic form factors, using the polarization method

In the region of small

Recently, the determination of the proton

Different sources of possible systematic errors of the muonic experiments have been discussed. However, no definite explanation of this difference has been given yet (see Refs.

The deuteron form factors have been also extensively investigated during recent years; see the reviews

The CREMA Collaboration has just published a value of the radius

As was noted in Ref.

In order to reach the smallest values of the transferred momentum to constrain the extrapolation to

To our knowledge, no experiment was performed yet on proton and deuteron scattering on atomic electrons. Such experiment is in principle possible at the Nuclotron accelerator, in Dubna, where polarized deuteron beams up to 13 GeV energy can be accelerated

In this paper, we extend to

Let us consider the reaction

A general characteristic of all reactions of elastic and inelastic hadron scattering by atomic electrons (that can be considered at rest) is the small value of the momentum transfer square, even for relatively large energies of the colliding particles. The electron mass cannot be neglected in the kinematics and dynamics of the reaction, even when the beam energy is of the order of GeV. The details of the inverse kinematics are given in Ref.

In the one-photon exchange (Born) approximation, the matrix element

These form factors are related to the standard deuteron form factors:

The matrix element squared is written as

The hadronic tensor

The differential cross section, as a function of the four-momentum transfer squared, is

Lastly, the differential cross section over the scattered-electron solid angle has the following expression:

Let us consider the QED radiative corrections which arise due to the soft- and hard-photon bremsstrahlung by the electrons, the electron vertex correction, and the electron vacuum polarization. The corresponding diagrams are shown in Fig.

Feynman's diagrams corresponding to the Born approximation, first-order vertex and photon self-energy corrections (top), initial and final real photon bremsstrahlung,

In this section, we use the standard expressions for the UV finite parts of the electron vertex and the vacuum polarization as well soft-photon corrections written in the initial electron rest frame. The electron mass is always taken explicitly into account.

We use the Lorentz and gauge-invariant Pauli-Willars subtraction procedure

The UV divergent part of the photon self-energy (vacuum polarization) is included into the renormalization of the electric charge. The UV divergent part of the electron self-energy (mass operator)

As the UV finite part of the mass operator

The virtual and soft corrections are calculated by the standard method (see Ref.

We separate the contribution

The electron structure function method can be applied if the condition

In this section, we calculate the radiative correction due to the hard-photon bremsstrahlung

In the experimental setup when only energies of the scattered deuteron and electron are measured, we use formalism developed in Ref. ^{1}

There is a misprint in our paper

Coordinate system and definition of the angles used for the integration over the variables of the final state.

Let us discuss the integration region in the right-hand side of Eq.

The shaded area represents the kinematically allowed region within the experimental setup in the plane of the variables

Usually the uncertainties

We consider the experimental setup where no angles are measured and therefore the orientation of the photon momentum

In the first case, we get, as experimental limit, the isotropic condition

The integration region over the variables

Maximum energy of the photon,

(a) Integration region over the variables

So, the integration region in

In this section, the conditions for the experimental uncertainties are set to

In our calculation, we use four different parametrizations of the deuteron form factors, and since the four-momentum transfer squared is rather small in this reaction, we can approximate these form factors by a Taylor series expansion with a good accuracy. On the Born level and when calculating the soft-photon bremsstrahlung, electron vertex, and vacuum polarization contributions, we can use also unexpanded expressions, but, in order to perform the analytical integrations in Eqs.

Therefore, we use the expansion over the variable

In this approach, we expand the quantity

In this approach, the deuteron form factors are saturated from the contribution of the isoscalar vector mesons,

The terms

The experimental data for

Values of the parameters

In this approach, the deuteron form factors are consistent with the results from popular

Parameters in Eq.

The three deuteron electromagnetic form factors have been determined by fitting directly the all existing measured differential cross section and polarization observables, according to the following expressions

Parameters corresponding to Eq.

In our calculations, we restrict ourselves to values of

(a)

To illustrate the dependence of the recoil-electron distribution on the deuteron beam energy, the Born cross section is shown in Fig.

Born differential cross section, defined by Eq.

The sensitivity (in percent) of this cross section to different form factor parametrizations is shown in Fig.

Difference of the recoil-electron distributions, Eq.

The hard-photon correction depends on the parameter

as a function of the recoil-electron energy for the

The quantity

The bell form of the curves in Figs.

Figure

Top: modified soft and virtual (

To calculate

At small values of the squared momentum transfer (small recoil electron energy

If the deuteron form factors are determined independently with high accuracy from other experiments, the measurement of the cross section

In Fig.

Sensitivity of the total radiative corrections in percent to the choice of the form factor parametrization, Eq.

In this paper, we studied deuteron elastic scattering on electron at rest. We derived the differential cross section including QED radiative corrections to the leptonic part of the interaction, in the case of a coincidence experimental setup. The recoil-electron energy distribution in elastic deuteron-electron scattering is illustrated. The detection of the recoil electron, in the energy range from a few MeV up to 10 GeV, allows us to collect small-

The measurements in the region of small-

The nontrivial part of the radiative corrections is the hard-photon bremsstrahlung contribution which takes place due to the uncertainty in the measurement of the deuteron (electron) energy,

We assumed that uncertainties in the final particle energies are proportional to their energies and we showed that the effect due to the nonzero quantity

The total correction

In our paper, we do not consider the radiative corrections involving deuteron as well as the background due to the Coulomb and strong interactions between deuteron and atomic nuclei. These questions require additional detailed investigations due to the particular kinematical conditions and will be object of a different work.

This work was partially supported by the Ministry of Education and Science of Ukraine (Projects No. 0115U000474 and No. 0117U004866), and by the National Academy of Sciences of Ukraine (Project No. Ts-3/53-2018). The research is carried on in the frame of the France-Ukraine IDEATE International Associated Laboratory (LIA). Thanks are due to Nancy Paul-Hupin for a careful reading of the paper.