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A NEW APPROACH TO THE THEORY OF ELECTROMAGNETIC
INTERACTIONS WITH BOUND SYSTEMS: CALCULATIONS OF
DEUTERON MAGNETIC AND QUADRUPOLE MOMENTS
A.V. Shebeko 1† , I.O. Dubovyk 2
(1) National Science Center ”Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
(2) Institute of Electrophysics and Radiation Technologies, Kharkov, Ukraine
† E-mail: shebeko@kipt.kharkov.ua
We would like to show one more application of the clothed particle representation
(CPR) (in particular, aimed at investigation of the e-d scattering [1, 2]. It is the case,
where one has to deal with the matrix elements 〈 P ⃗′ ,M ′ |J µ (0)| P ⃗ = 0,M〉 (to be definite in
the lab. frame). Here the operator J µ (0) is the Nöther current density J µ (x) at the point
x = 0, sandwiched between the eigenstates of a ”strong” field Hamiltonian H, viz., the
deuteron states | P ⃗ = 0,M〉 (| P ⃗′ = ⃗q,M ′ 〉) in the rest (the frame moving with the velocity
⃗v = ⃗q/m d , where ⃗q the momentum transfer). These
√
states meet the eigenstate equation
P µ | P,M〉 ⃗ = P µ d |⃗ P,M〉, with P µ d = (E d, P), ⃗ E d = ⃗P 2 +m 2 d , m d = m p + m n − ε d , the
deuteron binding energy ε d > 0 and eigenvalues M = (±1,0) 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 P 0 = K F +K I
and the boost operator B = B F + B I , where free parts K F and B F are ∼ b † c b c and
interactions K I and B I are ∼ b † cb † cb c b c , the deuteron eigenstate acquires the form
∫ ∫
|P,M〉 = dp 1 dp 2 C M ([P];p 1 µ 1 ;p 2 µ 2 )b † c(p 1 µ 1 )b † c(p 2 µ 2 )|Ω〉.
When finding the C-coefficients we use the relation |⃗q,M〉 = exp[i ⃗ β ⃗ B(α c )]|0,M〉, with
⃗β = β⃗n, ⃗n = ⃗n/n and tanhβ = v, that takes place owing to the property
exp(i ⃗ βB)P µ exp(−i ⃗ βB) = P ν L µ ν( ⃗ β), where L( ⃗ β) is the matrix of the corresponding
Lorentz transformation. Moreover, unlike the Bethe-Salpeter (BS) formalism, where the
matrix elements of interest can be evaluated in terms of the Mandelstam current sandwiched
between the deuteron BS amplitudes, our departure point is the expansion in the
R-commutators
J µ (0) = WJ µ c (0)W † = J µ c(0)+[R,J µ c (0)]+ 1 2 [R,[R,Jµ c (0)]]+...,(∗)
in which J c µ (0) is the initial current, where the bare operators {α} are replaced by the
clothed ones {α c } and W = expR the correspoding UCT. In its turn, the operator being
between the two-clothed-nucleon states contributes as J µ (0) = J µ one−body +Jµ two−body , where
the operator ∫
J µ one−body = dp ′ dpF p,n µ (p′ ,p)b † c (p)b c(p)
withF µ p,n(p ′ ,p) = eū(p ′ )F p,n
1 [(p ′ −p) 2 ]γ µ +iσ µν (p ′ −p) ν F p,n
2 [(p ′ −p) 2 ]u(p)thatdescribes
the virtual photon interaction with the clothed proton (neutron). Its appearance follows
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