Recoil effects on nucleon electromagnetic form factors
Alessandro Dragoa, Manuel Fiolhaisb and Ubaldo Tambinia
arXiv:nucl-th/9703053v1 25 Mar 1997
a Dipartimento
di Fisica, Università di Ferrara, and INFN, Sezione di Ferrara,
Via Paradiso 12, Ferrara, Italy 44100
b Departamento de Fı́sica da Universidade
and Centro de Fı́sica Teórica, P-3000 Coimbra, Portugal
Abstract
The electromagnetic form factors are computed using eigenstates of linear momentum for the nucleon. The latter is described in the framework of the chiral
color-dielectric model, projecting the hedgehog ansatz on eigenstates of angular
momentum and isospin. Form factors are well reproduced, with the exception of the
magnetic one for the proton, up to q 2 ≃ 0.5 GeV2 . The effect of the removal of the
spurious center-of-mass contributions shows up mainly in the electric form factor of
the proton. A noticeable improvement is obtained with respect to the calculation
without linear momentum projection.
1
Introduction
We report on a theoretical calculation of the electromagnetic form-factors of
the nucleon in the space-like region, performed in the framework of an effective
model — the chiral color-dielectric model (CDM) [1,2] — in which the nucleon
is described as a chiral soliton. The model contains quark and meson degrees
of freedom and a phenomenological scalar field which is responsible for quark
confinement.
In the previous calculations of the form factors in the framework of the chiral
soliton models of the nucleon, as the linear sigma model [3], the Nambu-JonaLasinio model [4], the Skyrme model [5] and the CDM [6], it was always
assumed that the nucleon is at rest before and after the interaction with
the virtual photon, the so-called static approximation. In the present work
we overcome, at least in part, the technical difficulties associated with the
computation of the form factors when the nucleon initial and final states are
eigenfunctions of linear momentum, at least non relativistically. Our formalism
is a generalization of the one presented in a work by Neuber et al. [7,8], where
static properties of the nucleon have been computed in the framework of the
Preprint submitted to Elsevier Science
30 July 2018
CDM. In their case therefore it was enough to build eigenstates of the angular
momentum having zero linear momentum.
The main technical problem in our calculation is due to the non-commutativity
of the projectors on linear and on angular momentum. It will be shown that
the problem can be solved by taking a suitable average on the
direction of the
R
transfered momentum, as it is suggested by the equation dq̂ [Pq , PJM ] = 0.
Morover it will be shown that the Fourier transform of the matrix elements of
the electromagnetic current does not depend on the direction of the momentum
transfered, if nucleon states are considered. Therefore the integration on q̂
does not imply any approximation. The use of the effective commutativity of
the two projectors simplifies the computation of the doubly projected form
factors. These mathematical aspects are applicable in general to any quarkmeson chiral soliton model.
The numerical results shown in this paper refer to the electric and the magnetic
proton and neutron form factors computed in a particular version of the chiral
CDM. This is the so-called “single minimum” version which gives good results
in quark matter calculations [9]. The model contains two parameters that we
adjust in order to reproduce the average ∆–N mass and the isoscalar nucleon
radius. All other results are obtained without any further parameters’ fitting.
This paper is organized as follows. In Section 2 the electromagnetic formfactors are defined. In Section 3 we review the chiral color-dielectric model
and the way model states representing a nucleon with definite momentum are
obtained. Section 4 is devoted to the formalism to compute the electric and the
magnetic form factors of the nucleon in the projected hedgehog state. Finally,
in Section 5, the results are presented and discussed.
2
The electromagnetic form factors of the nucleon
Let |Nα (pµ )i represent a nucleon state of mass MN , with spin and isospin
described by α. The standard definition of the electromagnetic form factors is
given by:
µ
hNf (p′µ ) |Jem
(0)| Ni (pµ )i
=
ūf (p′µ )
"
F1 (qρ2 )γ µ
F2 (qρ2 ) µν
+i
σ qν ui (pµ ). (1)
2MN
#
They only depend on the modulus of the momentum transfer of the virtual
photon qµ = p′µ − pµ , being real functions of qµ q µ = (q 0 )2 − q 2 . We work in
the Breit frame where the photon 4-momentum is q µ = (0, q), i.e. the energy
transfer is zero. Our results will be presented as a function of q = |q|.
2
In the Breit frame, Eq. (1) reads explicitly
q
q
q
F2 (q 2 ) µν
q
µ
hNf ( ) |Jem
σ qν ui (− ). (2)
(0)| Ni (− )i = uf ( ) F1 (q 2 )γ µ + i
2
2
2
2MN
2
#
"
Normalizing the Dirac spinors so that uu = 1,
u(p) =
s
E + MN
2MN
1
σ·p
E+MN
| χi,
(3)
the matrix element (2) can be worked out yielding
q
q
µ
hNf ( ) | Jem
(0)| Ni (− )i
2
2
3
X
hχf |[σ × q]j | χi i
δµj
= F1 (q 2 ) hχf |χi iδµ0 + i
2MN
j=1
3
X
q2
hχf |[σ × q]j | χi i
− F2 (q ) hχf |χi i
δµj .
δ
−
i
µ0
2
4MN
2MN
j=1
2
(4)
From F1 (q 2 ) and F2 (q 2 ) it is usual to define the so-called Sachs form factors,
GE (q 2 ) and GM (q 2 ), where E and M stand for “electric” and “magnetic”
respectively, which are expressed by
q2
F2 (q 2 )
4MN2
GM (q 2 ) = F1 (q 2 ) + F2 (q 2 ).
GE (q 2 ) = F1 (q 2 ) −
(5)
(6)
Using these definitions and Eq. (4) one obtains explicit formulas for the electric
and the magnetic form factors:
q
q
0
GE (q 2 )hχf |χi i = hNf ( ) |Jem
(0)| Ni (− )i
2
2
i
3
GM (q 2 )
q
q
hχf |σ × q| χi i = hNf ( ) |J em (0)| Ni (− )i .
2MN
2
2
(7)
(8)
The projected chiral color-dielectric model
In this work we shall use a chiral version of the CDM, whose Lagrangian reads
[1,2,7]
3
1
g
q̄ (σo + iγ5 τ · π) q + (∂µ χ)2 − V (χ)
χ
2
1
1
+ (∂µ σo )2 + (∂µ π)2 − U (σo , π) .
2
2
L = iq̄γ µ ∂µ q +
(9)
This is a model with interacting quarks, chiral mesons σo and π and also a
chiral singlet scalar field χ responsible for confinement. The potential U (σo , π)
in (9) for the chiral mesons is the so-called ‘mexican-hat’ potential
U (σo , π) =
2
λ2 2
σo + π 2 − ν 2 + cσo + d.
4
(10)
The chiral symmetry SU(2) × SU(2) of L is explicitly broken by the small
term cσo in (10); the last term in the same expression is a constant fixed in
order to have min U = 0. The parameters λ and ν are related to the sigma
and the pion masses and to the pion decay constant:
λ2 =
m2σ − m2π
;
2fπ
ν 2 = fπ2 −
m2π
.
λ2
(11)
For the χ field potential we consider the quadratic form
1
V (χ) = M 2 χ2 ,
2
(12)
where M is the χ mass. It is well known that the chiral CDM allows for soliton
solutions in which the quarks are absolutely confined [1,2,7]. In such solutions
the χ mean field is a decreasing function of the distance, approaching zero
in the limit r → ∞. This generates a raising dynamical mass for the quarks
and confines them. In previous works an exhaustive study of the model with
a quartic (or ‘double minimum’) potential was carried out [6,7]. In this work
we consider just the quadratic (‘single minimum’) potential for the confining
field. We recently showed [9] that for ‘double minimum’ potentials and for all
sets of parameters fitting nucleon properties, the equation of state for quark
matter turns out to be unrealistic. Indeed even at very low density the energy
per baryon number for quark matter turns out to be smaller than that for
nuclear matter. Using instead a quadratic potential for the χ field a realistic
equation of state is obtained.
Altogether, the parameters of the model defined by (9) are the pion and sigma
masses (fixed at mπ = 0.139 GeV and mσ =1.2 GeV), the pion decay constant
(fπ = 0.093 GeV), and g and M, the quark-meson-χ coupling constant and
the χ-mass, respectively.
In order to obtain model states representing the nucleon we used the procedure explained in great detail in Ref. [7] which, for the sake of complete4
ness, is sketched here. We consider three valence quarks in the hedgehog state
| hi = √12 (|u ↓ i − |d ↑ i), all occupying the same lowest positive energy sorbital and surrounded by clouds of χ, sigmas and pions, described by coherent states ( | Πi, for pions; | Σi, for sigmas; and | χi, for the confining field).
The meson mean fields are the expectation values of the field operators in the
corresponding coherent states.
We can write the quark single particle states and the meson mean fields as
1
u(r)
h r|q i = √
| hi
4π ıv(r)σ · r̂
σo (r) = hΣ |σ| Σi − fπ = σ(r) − fπ
r
π(r) = hΠ |π| Πi = φ(r)
r
χ(r) = hχ |χ| χi = χ(r) ,
(13)
(14)
(15)
(16)
where σ is just the fluctuating part of the σo field. Altogether, the hedgehog
ansatz reads
|ψhh i = Ch†
3
| Πi | Σi | χi,
(17)
where Ch† creates a particle in the single quark state (13).
Of course, solitons described by the hedgehog | ψhh i cannot represent physical
baryons because they are not eigenstates of angular momentum or isospin.
In addition, (17) represents a localized object and therefore the translational
symmetry of the model hamiltonian is also broken in such states. In particular
they contain spurious centre-of-mass components which contribute to the energy and to the other observables. However, a nucleon at rest can be obtained
by applying the projector onto linear momentum q = 0 together with the
projector onto angular momentum-isospin. The linear momentum projector is
given by [10]
Pq =
1
2π
3 Z
da eıa·q U(a),
(18)
where U(a) is the translation operator. It is well known that due to the symmetry of the hedgehog it is enough to perform a single projection (e.g. onto
spin), since this automatically projects onto the same value of isospin. The operator which projects out from the hedgehog a state with angular momentum
J and isospin T = J is
J
PM
MT
= (−1)
J+MT
2J + 1 Z 3
J∗
d Ω DM,−M
(Ω)R(Ω),
T
2
8π
5
(19)
J
where Ω = (α, β, γ) are the three Euler angles, DM,K
(Ω) are the Wigner
functions and R(Ω) is the rotation operator. In the following we will consider
J
MT = −M and use the shorthand notation PJM ≡ PM
−M .
The radial functions in (13)-(16) are determined using an approximate variationafter-projection method firstly suggested by Leech and Birse [11]. They are
computed by minimizing the expectation value of the (normal-ordered) model
hamiltonian in the model baryon state with quantum numbers J = T = 21
and linear momentum zero:
Pq=0 PJM |Ψhh i = |J, T, M, q = 0 i .
(20)
In the model there are two parameters, g and M, yet to be fixed. However,
because of the smoothness of the χ-field in a typical soliton solution and of
the relative weakness
of the chiral meson clouds, the relevant parameter turns
√
out to be G = gM. In the quark matter sector this is indeed the only free
parameter of the model [12]. If G is fixed to reproduce the isoscalar radius
of the nucleon one obtains G = 0.2 GeV. For this parameter the nucleondelta average mass is around 1.13 GeV (experimental value 1.085 GeV). It is
interesting to observe that if g and M are changed, keeping G fixed, the static
properties of the nucleon are essentially unchanged. For example, for G = 0.2
GeV, changing the mass of the χ field in the range 0.8–2.0 GeV, affects the
results by less than 1%.
In the present version of the CDM, the nucleon-delta mass splitting results
only from the quark-pion interaction. Due to the weakness of the pionic field,
the nucleon delta mass splitting obtained is too small. The experimental value
of the splitting could be recovered if, in addition, a color-magnetic interaction
(like in the MIT bag model or in the cloudy bag model) was considered [13].
We will come back to this point in the conclusions.
4
Electromagnetic form factors in the projected hedgehog
In order to compute the electromagnetic nucleon form factors one has to evaluate matrix elements of the electromagnetic current operator. In the CDM
the latter is given by
µ
Jem
(x)
=:
3
X
a=1
q̄a (x)γ
µ
1 τ3
qa (x) + [π(x) ∧ ∂ µ π(x)]3 : .
+
6
2
(21)
As mentioned in the Introduction, in the past the matrix elements of this operator have been computed in the static approximation (the nucleon is assumed
6
to be at rest before and after the interaction with the photon). In the present
work we go beyond this approximation, since we compute the matrix elements
using momentum eigenstates. In principle these should be obtained by boosting [14] the nucleon zero momentum eigenstate (20). However, the technical
difficulties associated with boosting are prohibitive. We approximate this operation by a Peierls-Yoccoz projection, i.e. we consider our model state with
momentum q to be given by
|N(q) i →
q
s
(2π)3 δ 3 (0)
Pq |ΨJM i
E
q
,
MN h Pq ΨJM |Pq ΨJM i
(22)
where the square roots are just normalization factors and
|ΨJM i = PJM |Ψhh i.
(23)
The approximation involved in assuming projected instead of boosted states
is valid for small q.
Before presenting the formalism to compute the form factors as matrix elements of the electromagnetic current (21) taken between nucleon states, let
us recall that
Z
dz [F (z)U(z), R(Ω)] = 0,
(24)
if F (z) is a scalar function of z [15]. This can be seen writing explicitly the
commutator, rotating the argument of F and U and redefining z ′ = R(Ω)z.
The previous commutation relation will be very useful in the following.
Another important point is that, due to the symmetry of the hedgehog, rotations of this state in spin or isospin space are equivalent. Therefore, as it
was already pointed out in the previous Section, projecting the hedgehog onto
spin J implies a simultaneous projection onto isospin T = J, and the two
projections are equivalent. Hence, the following relations hold:
Pq PJM |Ψhh >= Pq PT M |Ψhh >= PT M Pq |Ψhh >6= PJM Pq |Ψhh >,
(25)
where the commutation between operators working in isospin space and operators working in ordinary space has been used. The projector PT M is defined
similarly to the projector PJM (Eq. (19)), but with the rotation operator R
acting in isospin space, replacing the rotation operator in spin space. We shall
exploit relations (25) in the evaluation of the magnetic form factors.
7
4.1 Electric form factor
From the definition of the electric form factor (7) and using the correspondence
(22), the electric form-factor is given by
E
GE (q ) =
MN
2
R
0
(0)|P− q2 ΨJM i
dx hP q2 ΨJM |Jem
hP q2 ΨJM |P q2 ΨJM i
,
(26)
and using the explicit form of the linear momentum projector (18) one obtains
E
GE (q ) =
MN
2
R
q
′
0
dx db db′ e−i(b+b )· 2 hU(b′ )ΨJM |Jem
(0)|U(b)ΨJM i
.
R
′ −i(b−b′ )· q2
′
db db e
hU(b )ΨJM |U(b)ΨJM i
(27)
In principle the form factors should be functions of q 2 only. However, due to
the approximate treatment of the center-of-mass motion this is in general no
longer guaranteed and there is a spurious dependence on the angle between
q and the quantization direction. However, if J = 1/2 states are considered,
it is possible to show that the form factor is indeed a function of q 2 only. In
fact, due to parity, the Fourier transform of the matrix element of the current
has to be a function of (q · J)2 which, for J = 1/2, is proportional to q 2 . We
can therefore integrate the direction q̂, both in the current matrix element
and in the normalization factor at the denominator. After the expansion of
the exponentials in spherical waves, only the ℓ = 0 wave contributes and one
gets
E
GE (q 2 ) =
MN
R
′
0
|q hU(b′ )ΨJM |Jem
(0)|U(b)ΨJM i
dx db db′ j0 | b+b
2
R
′
db db′ j0 | b−b
|q hU(b′ )ΨJM |U(b)ΨJM i
2
. (28)
The integration on q̂ allows for further simplifications. In fact, we can now
prove that the translation operator U(b) and the rotation operator R(Ω),
which enters the projector on angular momentum [Eq. (19)], can be exchanged
in the previous formula, although they don’t commute. Let us, first of all, note
that
†
0
0
U (b′ )Jem
(0)U(b) = Jem
(b′ )U(b − b′ )
(29)
and expand the spherical Bessel functions j0 in power series of |b ± b′ |q/2.
Notice that the integration on the direction of the momentum transfer has
eliminated the dependence of the integrand in Eq. (28) on the angle between
q and b ± b′ . After the introduction of the new variables
z = b − b′
,
y = b′
(30)
8
we can define the scalar function
Fs,m (z) =
Z
0
dy y 2m+s (cos θzy )s Jem
(y)
(31)
where θzy is the angle between the directions z and y. The l.h.s. of Eq. (28)
can be written as a sum of terms of the form:
Z
dz hΨhh |PJM Fs,m (z)U(z)PJM |Ψhh i z 2k+s .
(32)
We can now apply Eq. (24) to each of the previous terms so that one of the
two projectors on angular momentum in Eq. (28) can be eliminated both in
the numerator and in the denominator. The electric form factor can finally be
written as
GE (q 2 ) =
1 E
N MN
Z
da
Z
dr
Z
∗
J
d3 Ω j0 (qr)DM
M (Ω)
a
a
0
× < U(− )Ψhh |Jem
(r)|U( )R(Ω)Ψhh >
2
2
(33)
where the new variables a and r are related to the previous b and b′ through
b′ = r − a/2
(34)
b = r + a/2
(35)
and the normalization factor reads
Z
Z
qa J ∗
N = da d3 Ω j0 ( ) DM
M (Ω) < Ψhh |U(a)R(Ω)|Ψhh > .
2
(36)
The numerator of Eq.(33) is the sum of three pieces [see Eq. (21)]: the isoscalar
quark, the isovector quark and the isovector pion contributions.
4.2 Magnetic form factor
From the definition of the magnetic form factor (8) and taking again the
correspondence (22) we can write
E
GM (q 2 )
(α × q) =
i
2MN
MN
R
dx hP q2 ΨJM |J em (0)|P− q2 ΨJM i
hP q2 ΨJM |P q2 ΨJM i
where we have introduced α = hσi.
9
(37)
In the case of the magnetic form factors, where the space components of the
electro-magnetic current appear, it is not possible to define a scalar function
as we did in Eq. (31). The projection operators will therefore always rotate
in a non-trivial way the current. To simplify the expression of the magnetic
form factor we will instead make use of the relations (25). The current matrix
element can therefore be rewritten as
hΨhh |PJM P q2 J em (0)P− q2 PJM |Ψhh i = hΨhh |PT M P q2 J em (0)P− q2 PT M |Ψhh i
= hΨhh |P q2 PT M J em (0)PT M P− q2 |Ψhh i .
(38)
The current contains an isoscalar and an isovector piece. The first one commutes with the current, while the isovector one transforms as [16]:
iv
T
T
PM
M J em ,0 PM M =
+1
X
Q=−1
T
CQ (T, M)J iv
em, Q PM −Q,M
(39)
where J iv
em ,Q stands for the spherical isospin component Q of the isovector
part of the vector electromagnetic current and
CQ (T, M) = h10; T M|T Mih1Q; T M −Q|T Mi.
(40)
To extract the magnetic form factor out of Eq. (37) we multiply both terms
of the equation by α × q and integrate over q̂. As for the electric form factors,
this integration is trivial because the Fourier transform of the matrix element
depends only on q 2 .
After a straightforward algebraic derivation on obtains the isoscalar (is) part
of the magnetic form factor:
2
1 E
Gis
M (q )
=
2MN
N MN
3j1 (qr) J ∗
DM M (Ω)
2qr
X
a
a
×
ǫ3jk hU(− )Ψhh | rj [J is
em (r)]k |U( )R(Ω)Ψhh i,
2
2
jk
Z
da
Z
dr
Z
d3 Ω
(41)
where [J is
em ]k stands for the cartesian k component of the vector electromagnetic current (isoscalar part) and the normalization factor N is given by Eq.
(36), as before.
For the isovector (iv) part of the magnetic form factor we get
10
Z
Z
Z
2
Giv
1 E X
3j1 (qr) J ∗
M (q )
=
DM −Q,M (Ω)
CQ da dr d3 Ω
2MN
N MN Q
2qr
X
a
a
×
ǫ3jk hU(− )Ψhh | rj [J iv
em ,Q (r)]k |U( )R(Ω)Ψhh i .
2
2
jk
5
(42)
Results and discussion
The nucleon electric and magnetic form factors are presented in Figs. 1-4.
The experimental data shown were taken from Refs. [17–19]. We present our
results in the space like region 0 ≤ q 2 ≤ 0.5 GeV2 .
All the results in Figures 1–4 were obtained using G = 0.2 GeV (actually
g = 0.024 GeV and M = 1.7 GeV but,√as mentioned before, the results
basically depend on the combination G = gM). No parameter was fitted to
reproduce the experimental form factors.
For the sake of comparison, in Figs. 1-4 we also present the results for the
form factors computed in the static approximation, i.e. without performing the
linear momentum projection. This is the traditional approximation considered
in previous calculations of the form factors in the framework of soliton models
(see Ref. [3] for the linear sigma model and Ref. [6] for the double hump version
of the chiral CDM).
As it can be seen from Figs. 1-2, the electric form factors are rather satisfactory.
One has to take into account large incertitudes in the experimental analysis of
the electric form factor of the neutron, which is obtained from scattering on
the deuteron and depends hence on the wave function of the latter. The effect
of the projection on linear momentum is particularly relevant in the proton
electric form factor.
The magnetic form factors are less satisfactory. This is probably due to the
weakness of the spin-spin interaction obtainable in this model, at least working
within the projected mean-field approximation.
As it appears from the figures, all the computed form factors underestimate
the data for large q 2 . It can be interesting to note that, also studying structure
functions, one sees that in the region of large x = Q2 /2Mν, where the momentum carried by the quarks is large, the computed quantities underestimate the
data [20]. These problems are probably due to the approximate treatment of
translational invariance and are therefore not so much related to the specific
model used in this paper.
The chiral CDM has now been used to compute many different quantities.
11
We can try to summarize the results. The chiral CDM gives good results if
problems not involving the spin are considered. This can be seen also from
the computation of the unpolarized structure functions [20]. Also the study
of the transition from nuclear to quark matter within this model seems very
promising, suggesting a smooth transition between the two phases [9] and
giving interesting results for neutron stars [21].
On the other hand, since the model does not contain enough tensor force, it
provides poor results for observables which involve the spin. The magnetic
form factors are therefore not totally satisfactory and the polarized structure
functions overestimate the data [20], indicating that most of the spin is carried
by the quarks because the pion is very weak. It is not yet clear whether
the weakness of the pionic field is intrinsic to the model and therefore other
degrees of freedom have to be considered, or stems from the approximations
used to solve the field equations. Concerning this second possibility, there are
indications that a large ∆-N mass splitting could be obtained using the same
ingredients considered here but allowing the scalar and the vector diquarks to
have different radii [22].
Acknowledgement
We thank L. Caneschi for many useful discussions and for a careful reading of
the manuscript. M.F. is indebted to T. Neuber for many conversations. The
computer codes used in the present work were based on those developed by T.
Neuber et al. [7] for the nucleon static properties. This work was supported in
part by INFN Section of Ferrara and by the Calouste Gulbenkian Foundation
(Lisbon).
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[19] T. Eden et al., Phys. Rev. C 50 (1994) R1749.
[20] V. Barone, A. Drago and M. Fiolhais, Phys. Lett. B 338 (1994) 433.
[21] A. Drago, U. Tambini and M. Hjorth-Jensen, Phys. Lett. B 380 (1996) 13.
[22] I.L. Grach and I.M. Narodetskii, Few Body Syst. 16 (1994) 151.
13
Fig. 1 Electric form factor of the neutron as a function of momentum transfer
in the space like region. The dashed line corresponds to the static approximation. The full curve corresponds to the full calculation, i.e. combined linear
and angular momentum projections.
14
Fig. 2 Electric form factor of the proton. Dashed and full curves as in Fig. 1.
15
Fig. 3 Magnetic form factor of the neutron. Dashed and full curves as in Fig
1.
16
Fig. 4 Magnetic form factor of the proton. Dashed and full curves as in Fig 1.
17
Alessandro Drago