Empirical transverse charge densities in the deuteron

Using the recent empirical information on the deuteron electromagnetic form factors we map out the transverse charge density in the deuteron as viewed from a light front moving towards the deuteron. The charge densities for a transversely polarized deuteron are characterized by monopole, dipole and quadrupole patterns. PACS. 13.40.Gp Electromagnetic form factors – 21.10.Ft Charge distribution – 21.45.Bc Two-nucleon system Electromagnetic form factors (FFs) of the deuteron have received a lot of attention in recent years; for recent reviews see, e.g., refs. [1,2]. Based on the large amount of precise deuteron form factor data, it is therefore of interest to exhibit the spatial information on the quark charge distributions in the deuteron. In this letter, we develop the general formalism to extract charge densities for a spin-1 particle and apply it to the case of the deuteron. In the following we will consider the electromagnetic deuteron elastic FFs when viewed from a light front moving towards the deuteron. Equivalently, this corresponds with a frame where the deuterons have a large momentum component along the z-axis chosen along the direction of P = (p + p′)/2, where p (p′) are the initial (final) deuteron four-momenta. We indicate the deuteron light front + component by P (defining a± ≡ a ± a). We furthermore choose a light front frame where the virtual photon four-momentum q has q = 0, and has a transverse component (lying in the xy-plane) indicated by the transverse vector q⊥, satisfying q = −q⊥ 2 ≡ −Q2. In such a light front frame, the virtual photon only couples to forward-moving partons and the + component of the electromagnetic current J has the interpretation of the quark charge density operator. It is given by J(0) = +2/3 ū(0)γu(0)− 1/3 d̄(0)γ(0)d(0), (1) considering only u and d quarks. Each term in the expression is a positive operator since q̄γq ∝ |γ+q|2. In the following, we will use empirical information on deuteron elastic FFs to study the deuteron quark charge a e-mail: [email protected] densities in the transverse plane. It is customary to denote the three deuteron elastic e.m. FFs by GC , GM , and GQ, corresponding to the Coulomb monopole (GC), magnetic dipole (GM ), and Coulomb quadrupole (GQ) FFs, respectively. Similar relations between nucleon densities and FFs can be found in [3–6]. We start by expressing the matrix elements of the J(0) operator between deuteron states as 〈 P, q⊥ 2 , λ′ ∣∣∣ J(0) ∣∣∣P+,−q⊥ 2 , λ 〉 = (2P+)ei(λ−λ )φq ×G+λ′ λ(Q2), (2) where λ = ±1, 0 (λ′ = ±1, 0) denotes the initial (final) deuteron light front helicity, and where q⊥ = Q(cosφq êx+ sinφq êy). Furthermore in eq. (2), the helicity form factors G+λ′ λ are real (due to time reversal invariance) and depend only on Q. We can then define transverse charge densities for a deuteron in helicity states of λ = ±1 or λ = 0 as, ρλ(b)≡ ∫ dq⊥ (2π)2 e−iq⊥·b 1 2P+ 〈 P, q⊥ 2 , λ ∣∣∣J+ ∣∣∣P+, −q⊥ 2 , λ 〉