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.Comment: 4 pages, 2 figure

Carlson, Carl E.

Vanderhaeghen, Marc

English

arXiv

C. E. Carlson

M. Vanderhaeghen

Springer - Publisher Connector

Empirical transverse charge densities in the deuteron

DOI 10.1140/epja/i2009-10800-0
Letter
Eur. Phys. J. A 41, 1–5 (2009) THE EUROPEAN
PHYSICAL JOURNAL A
Empirical transverse charge densities in the deuteron
C.E. Carlson1 and M. Vanderhaeghen1,2,a
1 Physics Department, College of William and Mary, Williamsburg, VA 23187, USA
2 Institut fu¨r Kernphysik, Johannes Gutenberg-Universita¨t, D-55099 Mainz, Germany
Received: 21 February 2009
Published online: 21 May 2009 – c© Societa` Italiana di Fisica / Springer-Verlag 2009
Communicated by M. Anselmino
Abstract. 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 mov-
ing towards the deuteron. Equivalently, this corresponds
with a frame where the deuterons have a large momen-
tum component along the z-axis chosen along the direc-
tion 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± ≡ a0 ± a3). We
furthermore choose a light front frame where the virtual
photon four-momentum q has q+ = 0, and has a trans-
verse component (lying in the xy-plane) indicated by the
transverse vector q⊥, satisfying q2 = −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 u¯(0)γ+u(0)− 1/3 d¯(0)γ+(0)d(0), (1)
considering only u and d quarks. Each term in the expres-
sion is a positive operator since q¯γ+q ∝ |γ+q|2.
In the following, we will use empirical information on
deuteron elastic FFs to study the deuteron quark charge
a e-mail: marcvdh@kph.uni-mainz.de
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, respec-
tively. 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 eˆx+
sinφq eˆy). Furthermore in eq. (2), the helicity form factors
G+λ′ λ are real (due to time reversal invariance) and depend
only on Q2.
We can then deﬁne transverse charge densities for a
deuteron in helicity states of λ = ±1 or λ = 0 as,
ρdλ(b)≡
∫
d2q⊥
(2π)2
e−iq⊥·b
1
2P+
〈
P+,
q⊥
2
, λ
∣∣∣∣J+
∣∣∣∣P+,
−q⊥
2
, λ
〉
=
∫ ∞
0
dQ
2π
QJ0(bQ)G+λλ(Q
2), (3)
where b ≡ b (cosφb eˆx + sinφb eˆy) denotes the position in
the xy-plane from the transverse c.m. of the deuteron. The
two independent helicity-conserving FFs G+1 1 and G
+
0 0 can
2 The European Physical Journal A
be expressed in terms of GC,M,Q as,
G+1 1 =
1
1 + η
{
GC + η GM +
η
3
GQ
}
,
G+0 0 =
1
1 + η
{
(1− η)GC + 2η GM − 2η3 (1 + 2η)GQ
}
,
(4)
with η ≡ Q2/(4M2d ), and Md is the deuteron mass.
The deﬁnitions and normalizations of GC , GM , and
GQ are the customary ones obtained from the matrix el-
ements of the electromagnetic current [7],
〈p2, λ2| Jμ |p1, λ1〉 = −(ε∗2 · ε1)2PμG1(Q2)
− (εμ1ε∗2 · q−εμ2 ∗ε1 · q
)
GM (Q2) + q · ε1 q · ε∗2
Pμ
M2d
G3(Q2),
(5)
with ε1,2 the polarization vectors of the initial and ﬁnal
deuteron. The charge and quadrupole FFs follow from,
GC = G1 +
2
3
η GQ,
GQ = G1 + (1 + η)G3 −GM , (6)
with normalizations GC(0) = 1, GM (0) = μd (magnetic
moment in units e/(2Md)), and GQ(0) = Qd (quadrupole
moment in units e/M2d ).
For numerical evaluation, we use the parameterization
of the deuteron form factor data given as ﬁt II by Ab-
bott et al. [8]. This parameterization is based on forms
suggested in [9], which read
GC =
G2(Q
2
4 )
2η + 1
[(
1− 2
3
η
)
g0+
8
3
√
2η g1 +
2
3
(2η − 1)g2
]
,
GM =
G2(Q
2
4 )
2η + 1
[
2g0 +
2(2η − 1)√
2η
g1 − 2g2
]
,
GQ =
G2(Q
2
4 )
2η + 1
[
−g0 +
√
2
η
g1 − η + 1
η
g2
]
, (7)
where the dipole form factor G(Q2) = (1 + Q2/δ2)−2 has
a (non-standard) mass parameter δ = 898.52MeV. The
reduced amplitudes are
g0=
4∑
i=1
ai
α2i + Q2
, g1=
4∑
i=1
Qbi
β2i + Q2
, g2=
4∑
i=1
Q2ci
γ2i + Q2
,
(8)
and the parameters from [8] are given in table 1.
Figure 1 shows the transverse charge densities for def-
inite helicity (longitudinally polarized) deuterons. Recall
that the transverse charge density is the charge density
projected onto a plane perpendicular to the line of sight,
which also deﬁnes the longitudinal direction. The charge
density for the λ = 1 state, in the upper panel, is smooth
and peaks in the center. The λ = 0 state, in the middle
panel, features a dip in the center, which can also be seen
Table 1. Parameters for the deuteron form factor ﬁt II of
ref. [8].
i = 1 i = 2 i = 3 i = 4
ai (fm
−2) 1.57057 12.23792 −42.04576 27.92014
bi (fm
−1) 0.07043 0.14443 −0.27343 0.05856
ci −0.16577 0.27557 −0.05382 −0.05598
α2i (fm
−2) 1.52501 8.75139 15.97777 23.20415
β2i (fm
−2) 43.67795 30.05435 16.43075 2.80716
γ2i (fm
−2) 1.87055 14.95683 28.04312 41.12940
-2 -1 0 1 2
-2
-1
0
1
2
bx [fm]
by [fm]
-2 -1 0 1 2
-2
-1
0
1
2
bx [fm]
by [fm]
-2 -1 1 2
0.05
0.1
0.15
0.2
ρ0d , ρ1d [1/fm  ]2
by [fm]
Fig. 1. Quark transverse charge densities in the deuteron. Up-
per panel: ρd1. Middle panel: ρ
d
0. The lightest (darkest) regions
correspond with largest (smallest) densities. Lower panel: the
density along the y-axis for ρd1 (dashed curve) and ρ
d
0 (solid
curve). For the deuteron FFs, we use the empirical parameter-
ization II of ref. [8], based on ref. [9].
in the lower panel, where the density along the y-axis is
plotted for both states.
C.E. Carlson and M. Vanderhaeghen: Empirical transverse charge densities in the deuteron 3
This dip —found directly from the data— reﬂects
the low charge density or “hole” in the center-of-the-
deuteron wave function. For the helicity-0 state, calculated
S- and D-state wave functions produce toroidal equiden-
sity surfaces in the central regions, with the axis along the
z-direction [10]. Hence looking along the z-axis for this
state, one can see the hole in the center, albeit the hole
be partly ﬁlled in because one must look though the outer
layers of the deuteron which are mostly S-state and spher-
ically symmetric. We emphasize that the dip seen in our
ﬁgure is based only on observation (codiﬁed in the form
factor ﬁts) and a light front interpretation of the data.
The result is consistent with but does not use calculations
based on models of nucleon-nucleon interactions. Further,
the light front viewpoint gives a two-dimensional charge
density that is relativistically correct, unlike charge den-
sities obtained from Fourier transforming FFs in an equal
time formalism. The light front viewpoint also gives a neu-
tron central charge density which is negative [4], because
it involves the Fourier transform of the Dirac FF F1 in-
stead of the Sachs FF GE . More physically, the negative
core may be understood [11] as quarks at small impact pa-
rameter correlating with high momenta in all directions,
noting that high longitudinal momentum quarks in a neu-
tron are dominantly down quarks.
We next consider the charge densities for a transversely
polarized deuteron, denoting the transverse polarization
direction by S⊥ = cosφS eˆx + sinφS eˆy. The transverse
charge densities can be deﬁned as
ρdT s⊥(b) ≡
∫
d2q⊥
(2π)2
e−i q⊥·b
1
2P+
×
〈
P+,
q⊥
2
, s⊥
∣∣∣∣J+
∣∣∣∣P+,
−q⊥
2
, s⊥
〉
, (9)
where s⊥ is the deuteron spin projection along the direc-
tion of S⊥. The transverse spin state can be expressed in
terms of the light front helicity spinor states as
|s⊥ = ±1〉 = 12
(
|λ = +1〉 ±
√
2eiφS |λ = 0〉
+ e2iφS |λ = −1〉
)
,
|s⊥ = 0〉 = 1√
2
(
|λ = +1〉 − e2iφS |λ = −1〉
)
. (10)
By working out the Fourier transform in eq. (9) for the
two cases where s⊥ = +1 and s⊥ = 0, one obtains
ρdT 1(b) =
∫ ∞
0
dQ
2π
Q
{
J0(bQ)
1
2
(
G+1 1 + G
+
0 0
)
+sin(φb − φS)J1(bQ)
√
2G+0 1
− cos 2(φb − φS)J2(bQ)12 G
+
−1+1
}
, (11)
ρdT 0(b) =
∫ ∞
0
dQ
2π
Q
{
J0(bQ)G+1 1
+cos 2(φb − φS)J2(bQ)G+−1 +1
}
. (12)
One notices from eqs. (11), (12) that the transverse charge
density ρdT 1 contains monopole, dipole and quadrupole
ﬁeld patterns, whereas ρdT 0 only contains monopole and
quadrupole ﬁeld patterns. The deuteron helicity FF with
one unit of helicity ﬂip, which governs the dipole ﬁeld pat-
tern in ρdT 1, can be expressed in terms of GC,M,Q as
G+0 1 = −
√
2η
1 + η
{
GC − 12(1− η)GM +
η
3
GQ
}
, (13)
whereas the deuteron helicity FF with two units of helicity
ﬂip, giving the quadrupole ﬁeld patterns, is,
G+−1+1 =
η
1 + η
{
GC −GM −
(
1 +
2η
3
)
GQ
}
. (14)
The four-deuteron helicity FFs given here are not inde-
pendent. The angular condition [12] relating them is,
(2η + 1)G+1 1 + 2
√
2η G+0 1 + G
+
−1+1 −G+0 0 = 0. (15)
Magnetic dipole moments in a rest frame appear, be-
cause of relativity, as electric dipole moments in a fast-
moving frame [13]. The electric dipole moment (EDM)
corresponding to the transverse charge densities ρdT s⊥ is
dds⊥ ≡ e
∫
d2b b ρdT s⊥(b). (16)
Equation (12) gives dd0 = 0, whereas eq. (11) yields
dd1 = − (S⊥ × eˆz) {GM (0)− 2}
(
e
2Md
)
. (17)
Expressing the spin-1 magnetic moment in terms of the
g-factor, i.e. GM (0) = g, one sees that the induced EDM
dd1 is proportional to g− 2. The same result was found for
the case of a spin-1/2 particle [5]. One thus observes that
for a particle without internal structure (corresponding
with g = 2 [14]), there is no induced EDM.
The electric quadrupole moment corresponding with
the transverse charge densities ρdT s⊥ is, for S⊥ = eˆx,
Qds⊥ ≡ e
∫
d2b (b2x − b2y) ρdT s⊥(b). (18)
From eqs. (11), (12) one obtains,
Qd1 = −
1
2
Qd0
=
1
2
{[GM (0)− 2] + [GQ(0) + 1]}
(
e
M2d
)
. (19)
For a spin-1 particle without internal structure, exempli-
ﬁed by W and Z gauge bosons in standard electroweak
theory, it is required that at tree level GM (0) = 2
and GQ(0) = −1; these values also allow satisfying the
Gerasimov-Drell-Hearn sum rule to lowest order in per-
turbation theory [15,16]. For elementary particles, any
deviations from these values would indicate new (beyond
4 The European Physical Journal A
-2 -1 1 2
0.05
0.1
0.15
0.2
ρT0
d , ρT1
d
0 1 2
0
1
2
0 1 2
0
1
2
-2 -1
-2
-1
bx [fm]
by [fm]
-2 -1
-2
-1
bx [fm]
by [fm]
by [fm]
[1/fm  ]2
Fig. 2. Quark transverse charge densities for a deuteron which
is polarized along the positive x-axis. Upper panel: ρdT 1. Middle
panel: ρdT 0. The light (dark) regions correspond with largest
(smallest) densities. The lower panel compares the density
along the y-axis for ρdT 1 (dashed curve) and ρ
d
T 0 (solid curve).
For the deuteron e.m. FFs, we use the empirical parameteriza-
tion II of ref. [8], based on ref. [9].
standard model) physics. (Current data from pp¯ collisions,
up to
√
s = 1.96TeV, are compatible with zero anomalous
WWγ and ZZγ couplings [17].) For composite particles,
it is the deviations from these benchmark values that in-
dicate deformations of the states.
It is thus interesting to observe from eq. (19) that
Qds⊥ is only sensitive to the anomalous parts of the spin-1
magnetic dipole and electric quadrupole moments, and
vanishes for a particle without internal structure. The
deuteron magnetic dipole moment GM (0) is 1.71 [18],
close to a spin-1 particle’s natural value. However, in con-
trast to the W and Z gauge bosons, the deuteron has a
large anomalous quadrupole moment. Its measured value
is GQ(0) = 25.84 (3) [19], highlighting the prominent role
of the pion exchange potential.
Pictorial results for the transverse charge density with
transverse deuteron polarization are shown in ﬁg. 2. The
upper panel shows the charge distribution for the state
with projection-1 in the x-direction. One sees the large ef-
fects of the quadrupole term together with a small overall
shift of the charge distribution along the y-axis. The lat-
ter contrasts to the large shift of the charge in the neutron
case [5], where the anomalous moment is far from its point-
like value. As noted [13,20], an object with a magnetic
dipole moment when stationary exhibits an electric dipole
moment when seen by a moving observer, proportional to
the vector product v × (magnetic moment). The middle
panel shows the charge density for a state with projection-
0 in the x-direction. One notices that the quadrupole term
stretches the charge along the y-axis but does not cause
any shift of the charge center. The two cases are com-
pared in the lower panel, which plots the density along
the y-axis.
In summary, we used recent empirical information on
the deuteron e.m. FFs to map out the transverse charge
densities in longitudinally and in transversely polarized
deuterons. Notably, one sees a dip in the center-of-the-
charge distribution for helicity-zero deuterons. This is in
harmony with nuclear force model calculations, which for
the zero helicity case predict toroidal equidensity surfaces
for higher densities, with the axis along the quantiza-
tion direction. Transversely polarized deuterons show di-
pole and quadrupole structure in the charge distributions.
Their electric dipole and quadrupole moments only de-
pend on the spin-1 particle’s anomalous magnetic dipole
moment and its anomalous electric quadrupole moment,
arising from its internal structure.
The work of C.E.C. is supported by the NSF under grant PHY-
0555600. The work of M.V. is supported in part by DOE grant
DE-FG02-04ER41302.
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