Speaker
Description
We continue our explorations [1] of the electromagnetic properties of the deuteron with help of the method of unitary clothed transformations (UCTs) [2,3]. It is the case, where one has to deal with the matrix elements $\langle \mathbf P', M'| J^{\mu}(0) | \mathbf P = \mathbf 0, M \rangle$. Here the operator $J^{\mu}(0)$ is the Nöther current density $J^{\mu}(x)$ at the point $x=0$, sandwiched between the eigenstates of a "strong" field Hamiltonian $H$, viz., the deuteron states $| \mathbf P = \mathbf 0, M \rangle$. 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 $M=(\pm1,0)$ of the third component of the total (field) angular-momentum operator in the deuteron center-of-mass (details in [3]). In the subspace of the two-clothed-nucleon states 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 $\sim b_{c}^{\dagger}b_{c}^{\dagger}b_{c}b_{c}$, the deuteron eigenstate gets the form $|\mathbf{P},M\rangle=\int d\mathbf{p}_{1}\int d\mathbf{p}_{2}C_{M}([\mathbf{P}];\mathbf{p}_{1}\mu_{1};\mathbf{p}_{2}\mu_{2}) b_{c}^{\dagger}(\mathbf{p}_{1}\mu_{1})b_{c}^{\dagger}(\mathbf{p}_{2}\mu_{2})|\Omega\rangle $ and we will show how one can find the $C$-coefficients within the clothed particle representation (CPR). 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 primary 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 between the two-clothed-nucleon states contributes as $J^{\mu}(0)=J_{one-body}^{\mu}+J_{two-body}^{\mu}$, where the operator
$J_{one-body}^{\mu}=\int d\mathbf{p}'d\mathbf{p}F_{p,n}^{\mu}(\mathbf{p}',\mathbf{p})b_{c}^{\dagger}(\mathbf{p}')b_{c}(\mathbf{p})$
with $ F_{p,n}^{\mu}(\mathbf{p}',\mathbf{p})=e\bar{u}(\mathbf{p}'){F_{1}^{p,n}[(p'-p)^{2}]\gamma^{\mu}+ i\sigma^{\mu\nu}(p'-p)_{\nu}F_{2}^{p,n}[(p'-p)^{2}]}u(\mathbf{p})$ that describes the virtual photon interaction with the clothed proton (neutron). 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). Since, as before in [1], we start with the following formula
$\mu_d = \frac{1}{m_d} \langle \mathbf 0; M^{\prime}=1 | \frac{1}{2} \left[ \mathbf B \times \mathbf J(0) \right]^z | \mathbf 0; M=1 \rangle$
for the magnetic moment of the deuteron, special attention has been paid to finding a relativistic correction due to the interaction part $\mathbf{B}_{I}$ of the RIA results obtained in [1].
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).
