Speaker
Description
We continue our explorations [1] of the electromagnetic properties of the deuteron with help of the method of unitary clothing transformations (UCT) [2,3]. It is the case, where one has to deal with the matrix elements $\langle \mathbf P', M'| J^{\mu}(0) | \mathbf P, M \rangle$ (to be definite in the lab. frame). 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 $| \mathbf P, M \rangle$. Latter belong to the two-clothed-nucleon subspace with the Hamiltonian $H=P^{0}= K_{F} + K_{I}$ and the boost operator $\mathbf{B}= \mathbf{B}_{F} + \mathbf{B}_{I}$, where free parts $K_{F}$ and $\mathbf{B}_{F}$ are $\sim b_{c}^{\dagger}b_{c}$ and interactions $K_{I}$ and $\mathbf{B}_{I}$ are $\sim b_{c}^{\dagger}b_{c}^{\dagger}b_{c}b_{c}$. Further, we use the expansion in the $R$-commutators
$J^{\mu}(0)=WJ^{\mu}_{c}(0)W^{\dagger}=J^{\mu}_{c}(0)+[R,J^{\mu}_{c}(0)]+\frac{1}{2}[R,[R,J^{\mu}_{c}(0)]]+...,$
where $J^{\mu}_{c}(0)$ is the initial current in which the bare operators $\{\alpha\}$ are replaced by the clothed ones $\{\alpha_{c}\}$ and $W=\exp R $ the corresponding UCT. In its turn, the operator being sandwiched between the two-clothed-nucleon states contributes as $J^{\mu}(0)=J_{one-body}^{\mu}+J_{two-body}^{\mu}$. By keeping only the one-body contribution we arrive to certain off-energy-shell extrapolation of the so-called relativistic impulse approximation (RIA) in the theory of e.m. interactions with nuclei (bound systems). Of course, the RIA results [1] should be corrected including more complex mechanisms of e-d scattering (see other our contribution). Starting from the operator of the magnetic dipole moment $\mathbf \mu = \frac{1}{2} \int d \mathbf x \, \mathbf x \times \mathbf J (x)$ (reminiscent of the Biot-Savart formula from magnetostatics) one can show that the magnetic dipole moment of the deuteron being defined after Sachs as $z$-component of the matrix elements between narrow wave packets of the vector $\mathbf \mu$ for the stretched configuration, looks as
$\mu_d = \lim \limits_{\mathbf q \rightarrow 0} \left[
-\frac{i}{2} curl_{\mathbf q} \langle \frac{\mathbf q}{2};1 |\mathbf J (0)| - \frac{\mathbf q}{2};1 \rangle \right]^z,$
where the matrix elements $\langle \frac{\mathbf q}{2}; M_J |\mathbf J (0)| - \frac{\mathbf q}{2}; M'_J \rangle$ ($M_J=(\pm 1,0)$ projection of the total angular momentum) determine the corresponding current in the Breit frame. In this way the deuteron magnetic dipole moment can be expressed as
$\mu_d = \frac{1}{m_d} \langle \mathbf 0; 1 | \frac{1}{2} \left[ \mathbf B \times \mathbf J(0) \right]^z | \mathbf 0; 1 \rangle$
with the deuteron state $| \mathbf 0; 1 \rangle$ in the rest frame. In parallel, considering interaction energy of the system with the charge density $\rho (x) = J^0(x)$ in static external electric field and expanding it in the Cartesian electric moments one encounters the quadrupole moment tensor $Q_{ij} = \int d \mathbf x \, [ 3 x_i x_j - \delta_{ij} \mathbf x^2 ] \rho (\mathbf x)$ $(i,j=1(x),2(y),3(z))$. Then repeating the same trick with wave packets one gets the matrix elements
$\langle J M'_J| Q_{ij} | J M_J \rangle = - \lim \limits_{\mathbf q \rightarrow 0}\left[ \left\{ 3 \frac{\partial^2}{\partial q_i \partial q_j} - \delta_{ij} \frac{\partial^2}{\partial q_l^2 } \right\}\langle \frac{\mathbf q}{2} | \rho (0)| - \frac{\mathbf q}{2} \rangle \right]$
to introduce electric quadrupole moment $Q = \langle J J | Q_{33} | J J \rangle$. These formulae have been a departure point for our preceding RIA calculations [1]. Here we will show our results with the meson exchange currents and boost ($\mathbf B_I$ part) contributions included.
References
1. A. Shebeko and I. Dubovyk, Few Body Syst. 54 1513 (2013).
2. A. Shebeko, Chapter 1 In: Advances in Quantum Field Theory, ed. S. Ketov, 2012 InTech, pp. 3-30.
3. I. Dubovyk and A. Shebeko, Few Body Syst. 48 109 (2010).
