Speaker
Description
Let us remind that in the one-photon-exchange approximation (OPEA) the elastic e-d scattering amplitude is proportional to the contraction $T(ed \rightarrow e' d') = \varepsilon_\mu (e,e')\langle \mathbf q\, M' | J^\mu(0) | \mathbf 0\, M \rangle$ where we have introduced the notation $\varepsilon_\mu (e,e') = \bar u_{e'}(k') \gamma_\mu u_e(k)$. Here the operator $J^{\mu}(0)$ is the Nöther current density $J^{\mu}(x)$ at the point $x=(t,\mathbf x)=0$, sandwiched between the eigenstates of a "strong" field Hamiltonian $H$, viz., the deuteron states $| \mathbf P, M \rangle$. These states meet the eigenstate equation $P^{\mu} | \mathbf P, M \rangle = P^{\mu}_{d} | \mathbf P, M \rangle$ with $P^{\mu}_{d} = (E_d, \mathbf P)$, $E_d = \sqrt{\mathbf P^2 + m_d^2}$, $m_d = m_p + m_n - \varepsilon_d$, the deuteron binding energy $\varepsilon_d > 0$ and eigenvalues $M=(\pm 1,0)$ of the third component of the total (field) angular-momentum operator in the deuteron center-of-mass (details in [1]). Further, $u_e(k)$ ($u_e'(k')$) the Dirac spinor for incident (scattered) electron. In its general form, relativistic deuteron electromagnetic current $\langle \mathbf q\, M' | J^\mu(0) | \mathbf 0\, M \rangle$ can be expressed (see, e.g., survey [2]) through the charge monopole ($G_C$), magnetic dipole ($G_M$) and charge quadrupole ($G_Q$) form factors (FFs) of the deuteron. Such static quantities as the deuteron charge $e_d$, its magnetic moment $\mu_d$ and quadrupole moment $Q_d$ are given by $e_d = G_C(0)\,{e}$, $\mu_d = G_M(0)\,{e}/{2 m_d}$, $Q_d = G_Q(0)\,{e}/{m_d^2}$. Other our contribution is devoted to a fresh field-theoretical calculation of these moments. In parallel, for our attempts to ensure gauge independent treatment of similar electromagnetic (EM) processes we prefer to employ a generalization [3] of the Siegert theorem, in which the amplitude of interest is given by
$T(ed \rightarrow e' d') = \left[ \omega \mathbf \varepsilon(e', e) - \mathbf q \varepsilon_0(e', e) \right] \mathbf D(\mathbf q) + \left[ \mathbf q \times \mathbf \varepsilon(e', e) \right] \mathbf M(\mathbf q)$ with the so-called generalized electric
$\mathbf D(\mathbf q) = -i \omega^{-1} \int \limits_0^1 \frac{d\lambda}{\lambda} \nabla_{\mathbf q} \left\{ \left[ \sqrt{ \lambda^2 \mathbf q^2 + m_d^2} - m_d \right] \langle \lambda \mathbf q; M' | \rho(0) | \mathbf 0; M \rangle \right\}$
and magnetic
$\mathbf M(\mathbf q) = -i \int \limits_0^1 d\lambda \nabla_{\mathbf q} \times \langle \lambda \mathbf q; M' | \mathbf J(0) | \mathbf 0; M \rangle$
dipole moments. We will show the links between the deuteron FFs and these quantities. In addition, to be more constructive we consider the following expansion for the "clothed" current operator
$J^{\mu}(0)=WJ^{\mu}_{c}(0)W^{\dagger}=J^{\mu}_{c}(0)+[R,J^{\mu}_{c}(0)]+\frac{1}{2}[R,[R,J^{\mu}_{c}(0)]]+...,$
in the $R$-commutators (see Eq. (13) in [3]), where $J^{\mu}_{c}(0)$ is the initial current operator in which the bare operators $\{\alpha\}$ are replaced by the clothed ones $\{\alpha_{c}\}$ and $W=\exp R $ the corresponding unitary clothing transformation. This decomposition involves one-body, two-body and more complicated interaction currents. In case of the deuteron whose states belong to the clothed two-nucleon sector, our consideration leads to division $J^{\mu}(0)=J_{one-body}^{\mu}+J_{two-body}^{\mu}$. The operator $J_{two-body}^{\mu}$ is analogue of the meson exchange current in the conventional theory. Special attention is paid to finding such contributions to the deuteron form factors.
References
1. A. Shebeko and I. Dubovyk, Few Body Syst. 54 1513 (2013).
2. R. Gilman and F. Gross, J. Phys. G: Nucl. Part. Phys. 28 R37 (2002).
3. L. Levchuk and A. Shebeko, Phys. At. Nucl. 56 227 (1993).
