Summary of charge and weak-charge formfactor
Discussion notes: FSU – ORNL – Erlangen Started 5. August 2012; updated 3. September 2012
Abstract
These notes summarize the evaluation of charge formfactor and weak-charge formfactor for SHF as well as for RMF. The influence of the magnetic contri- bution (also called spin-orbit contribution) to the intrinsic nucleon formfactor is discussed. The question of the model for the intrinsic nucleon formfactor is also addressed. Three different models are considered and compared.
1 Introduction
The recent work of [1] has emphasized the possible importance of magnetic contribu- tions to the weak-charge formfactor. The present notes carry forth these suggestions with a discussion in the Skyrme-Hartree-Fock model (SHF) as well as in the rela- tivistic mean-field model (RMF). Additionally, we address the question of a proper choice of the model for the intrinsic nucleon formfactor. Three models are considered:
first, the older Mainz fit from Simon & Walther [2, 3], abbreviated “S&W”, second, a simple dipole form fitted to recent knowledge on nucleon radii [1], abbreviated “dip”, and third, a recent Mainz analysis (for the proton formfactor) fitted by a polynomial form [4, 5], abbreviated “poly”.
The notes start in section 2 with the presentation and discussion of the final results for charge and weak-charge radii. The details of modeling and computation follow in the subsequent sections.
2 Results for r rms and surface thickness σ
We have extended the previously proposed evaluation of charge formfactors and weak- charge formfactors in two respects: first, we also include the magnetic contributions, and second, we consider three different models for the intrinsic nucleon formfactors.
For both extensions, we are interested on the effects on the form parameters of the
density: r.m.s. radius r rms , diffraction radius R diffr , and surface thickness σ. The
diffraction radius is found to be rather insensitive to all variants. We thus look only
at r rms and σ for charge and weak-charge distribution. The details of the evaluation
are explained in the subsequent section. We summarize here the net results for two
r.m.s. radius r surface thickness σ case ls dip poly S&W dip poly S&W SV-bas, 208 Pb
ρ C no 5.4925 5.4928 5.4887 0.90539 0.90705 0.90044 ρ C yes 5.4973 5.4976 5.4935 0.91029 0.91177 0.90518 ρ W no 5.6633 5.6638 5.6597 1.0991 1.1014 1.0950 ρ W yes 5.6717 5.6722 5.6682 1.1168 1.1189 1.1126 NL3, 208 Pb
ρ C no 5.3094 5.3098 5.3055 0.91101 0.91318 0.90665 ρ C yes 5.3059 5.3062 5.3020 0.90131 0.90295 0.89634 ρ W no 5.7354 5.7359 5.7319 1.0843 1.0876 1.0811 ρ W yes 5.7375 5.7379 5.7340 1.0919 1.0941 1.0876 SV-bas, 48 Ca
ρ C no 3.5090 3.5089 3.5031 0.91036 0.91406 0.90892 ρ C yes 3.4992 3.4994 3.4934 0.90885 0.91240 0.90715 ρ W no 3.6907 3.6909 3.6852 0.92106 0.92820 0.92021 ρ W yes 3.7137 3.7141 3.7083 0.93837 0.94466 0.93683 NL3, 48 Ca
ρ C no 3.3901 3.3935 3.3840 0.84836 0.85553 0.84999 ρ C yes 3.3707 3.3732 3.3646 0.83636 0.84237 0.83665 ρ W no 3.6752 3.6781 3.6697 0.86712 0.87124 0.86250 ρ W yes 3.6887 3.6907 3.6833 0.87040 0.87469 0.86598
Table 1: R.m.s. radii r rms and surface thickness σ from charge distribution ρ C and weak-charge distribution ρ W for two mean-field models (SHF with SV-bas, RMF with NL3) and two nuclei 208 Pb and 48 Ca. Results for the three considered models for the intrinsic nucleon formfactor are given: “dip” = recent dipole fit [1], “poly” = latest Mainz evaluation using polynomial form for the proton formfactor [5], “S&W” = old Mainz evaluation [2, 3] (for details see section 6). All values are given in units of fm.
typical nuclei, 208 Pb and 48 Ca, and for the two most widely used mean field models, SHF and RMF. The size of the effects from magnetic contributions and from varying the intrinsic formfactor has to be compared with the typical uncertainties which are for the charge radius r rms,C about 20 mfm and for the charge surface thickness σ C
about 40 mfm. Uncertainties are presently larger for weak-charge quantities. But in the long term, we aim at a similar precision.
Figure 1 summarizes the effect of the variations for the r.m.s. radii (upper block of three panels) and surface thicknesses (lower block of two panels). Table 1 provides the detailed values for radii and surface thicknesses for completeness.
We concentrate first on the r.m.s. radii (upper block in figure 1). The effect of magnetic contributions is small for 208 Pb, even ignorable for SHF. But it is sizable for
48 Ca. This finding complies with the results for the weak-charge skin in [1]. There
is only a weak dependence on the choice of the nucleon formfactor, again smaller
50 60 70 80 90
dip poly S+W dip poly S+W r
charge-r
proton[mfm]
48
Ca
208Pb
SV-bas, no ls SV-bas, ls NL3, no ls NL3, ls 150
200 250 300 350 400 450
dip poly S+W dip poly S+W
weak-charge skin [mfm]
48
Ca
208Pb
SV-bas, no ls SV-bas, ls NL3, no ls NL3, ls
70 80 90 100 110 120 130
dip poly S+W dip poly S+W r
weak-r
neutron[mfm]
48
Ca
208Pb
SV-bas, no ls SV-bas, ls NL3, no ls NL3, ls
100 105 110 115 120 125
dip poly S+W dip poly S+W σ
charge- σ
proton[mfm]
48
Ca
208Pb
105 110 115 120 125 130 135 140 145
dip poly S+W dip poly S+W σ
weak- σ
neutron[mfm]
48
Ca
208Pb
SV-bas, no ls SV-bas, ls NL3, no ls NL3, ls
Figure 1: Upper block: Effect of magnetic terms and intrinsic nucleon formfactor on r.m.s. radii for 208 Pb and 48 Ca. Differences of r.m.s. radii are shown to amplify the effects. These are: charge radius minus proton radius, weak-charge radius minus neutron radius, and weak-charge radius minus charge radius (the “weak skin”). Lower Block: The same for the surface thickness. Results for the three considered models for the intrinsic nucleon formfactor are given: “dip” = recent dipole fit [1], “poly” = latest Mainz evaluation using polynomial form for the proton formfactor [5], “S&W”
= old Mainz evaluation [2, 3] (for details see section 6).
for 208 Pb. The variations stay always below 10 mfm and thus below the general uncertainty in radii. Moreover, the difference between the two more recent models (dipole fit and Mainz polynomial form) is even smaller. Thus any one of these choices for the nucleon formfactors is acceptable what radii of 208 Pb is concerned. The case of
48 Ca clearly calls for inclusion of magnetic contributions. The choice of the intrinsic formfactor may also be an issue.
The surface thicknesses, shown in the lower block figure 1, show larger effects in both respects. Magnetic contributions are now sizable even for 208 Pb and the effect of the intrinsic nucleon formfactor is also larger, for 48 Ca and now also for 208 Pb. This is not surprising because the surface thickness is determined from the formfactor values at larger q where the models for the nucleon formfactor differ more (as we will see in section 6.4). The latest Mainz evaluation (parameterization “poly”) is probably the appropriate choice. Note, however, that a determination of surface thickness for the weak charge distribution is asking too much for the present status of experiment.
3 Basic parameters
The intrinsic nucleon formfactors are parameterized in terms of the Sachs formfactors G (S) E,p , G (S) E,n , G (S) M,p , and G (S) M,n , where E stands for electric and M for magnetic, p for proton and n for neutron. The actual parameterizations used for them in these notes are introduced in section 6. Before discussing the composition of formfactors in detail, we summarize a few useful abbreviations
Compton wavelng.: λ p = m ¯ h
p
c = 0.21030 fm , λ n = m ¯ h
p
c = 0.21001 fm , λ N = λ
p+λ 2
n= 0.21015 fm ,
Darwin factor: D = 1 4 λ 2 N = 0.01104 fm 2 , τ = D q 2
magnetic mom.: µ p = 2.79 , µ n = −1.91 .
(1)
A further crucial contribution is the c.m. correction to the formfactor F cm (q) = exp ¯h 2 q 2
8h ˆ P c.m. 2 i
!
. (2)
4 The charge formfactor
4.1 Case of the relativistic mean field model (RMF)
The particle density ρ is identified with the zeroth component of the vector current.
The next order contribution comes from the tensor current. These read in typical relativistic notation
ρ(r) ≡ ρ 0 (r) = X
i
¯
ϕ i (r)ˆ γ 0 ϕ i (r) , (3a) ρ (T) 00 (r) = i∇ · X
i
¯
ϕ i (r)ˆ α 0 ϕ i (r) , (3b)
where the sum runs over proton states, or neutron state respectively. Fourier-Bessel transformation yields the (radial) vector and tensor formfactors
F V,p/n (q) = Z
d 3 r e iq·r ρ 0,p/n (r) = 4π Z ∞
0
dr r 2 j 0 (qr)ρ 0,p/n (r) , (4) F T,p/n (q) =
Z
d 3 r e iq·r ρ (T) 00 (r) = 4π Z ∞
0
dr r 2 j 0 (qr)ρ (T) 00 (r) . (5)
The charge formfactor is then composed in analogy to [6] as F C (q) = X
t∈{p,n}
h G (C) 1,t (q)F V,t (q) + G (C) 2,t (q)F T,t (q) i
F cm (q) , (6a)
G (C) 1,p = h
G (S) E,p + µ p τ G (S) M i
(1 + τ ) −1 , (6b)
G (C) 1,n = h
G (S) E,n + µ n τ G (S) M i
(1 + τ ) −1 , (6c)
G (C) 2,p = λ N 2
h
µ p G (S) M − G (S) E,p i
(1 + τ ) −1 , (6d)
G (C) 2,n = λ N 2
h µ n G (S) M + G (S) E,n i
(1 + τ ) −1 . (6e)
These forms agree with those given in [1].
4.2 Case of non-relativistic models
The non-relativistic limit of the RMF formfactor in the previous section is rather involved and somewhat ambiguous. We report here the final recipe which is tuned to then range of q relevant for low-energy nuclear structure. Basic input quantities are now local density ρ and divergence of spin-orbit density ∇·J. By Fourier(-Bessel) transformation, we obtain the corresponding formfactors
F p/n (q) = Z
d 3 r e iq·r ρ(r) = 4π Z ∞
0
dr r 2 j 0 (qr)ρ p/n (r) , (7a) F ls,p/n (q) =
Z
d 3 r e iq·r ∇·J p/n (r) = 4π Z ∞
0
dr r 2 j 0 (qr)∇·J p/n (r) . (7b) These together with the intrinsic Sachs formfactors (13) constitute immediately the charge formfactor as
F C (q) = X
t∈{p,n}
h
G (C) E,t (q)F t (q) + G (C) M,t (q)F ls,n (q) i
F cm (q) . (8a)
The nucleons intrinsic charge formfactors are composed from the Sachs formfactors as
G (C) E,p = G (S) E,p (1 + τ ) −1/2 , (9a)
(9b)
G (C) E,n = G (S) E,n (1 + τ ) −1/2 , (9c)
G (C) M,p = ˜ µ p G (S) M − G (S) E,p D , µ ˜ p = − λ 2 N
2 µ p = −0.0616 , (9d) G (C) M,n = ˜ µ n G (S) M − G (S) E,n D , µ ˜ n = − λ 2 N
2 µ n = 0.0422 , (9e)
5 The weak-charge formfactor F W
The nuclear weak-charge formfactor is composed exactly as the charge formfactor, however employing now the nucleons intrinsic weak-charge formfactors. Many of the basic ingredients remain as before. This concerns the nuclear distributions and Sachs formfactors.
5.1 F W in case of the RMF
F W (q) = X
t∈{p,n}
h G (Z) 1,t (q)F t (q) + G (Z) 2,t (q)F T,t (q) i
F cm (q) (10a) G (Z) 1,p = Q p G (C) 1,p + Q n G (C) 1,n − G s , (10b) G (Z) 1,n = Q p G (C) 1,n + Q n G (C) 1,p − G s , (10c) G (Z) 2,p = Q p G (C) 2,p + Q n G (C) 2,n , (10d) G (Z) 2,n = Q p G (C) 2,n + Q n G (C) 2,p , (10e) Q p = N p 1 − 4 sin 2 (Θ W ) = 0.0721 , (10f)
Q n = −N n = −0.9878 , (10g)
sin 2 (Θ W ) = 0.23 , 1 − 4 sin 2 (Θ W ) = 0.08 , (10h) N p = 0.0721
1 − 4 sin 2 (Θ W ) = 0.9013 , N n = 0.9878 . . (10i) The N p/n are radiative corrections 1 . A possible magnetic contribution from the s- quark is ignored as well as the radiative correction thereof.
1
PGR2JP: The way the radiative corrections are applied in eq. (11) suggests that these are
associated with the intrinsic nucleon formfactors. The combination would differ in case that the
5.2 F W in case of non-relativistic models
F W (q) = X
t∈{p,n}
h G (Z) E,t (q)F t (q) + G Z M,t (q)F ls,n (q) i
F cm (q) (11a) G (Z) E,p = Q p G (C) E,p + Q n G (C) E,n − G s , (11b) G (Z) E,n = Q p G (C) E,n + Q n G (C) E,p − G s , (11c) G (Z) M,p = Q p G (C) M,p + Q n G (C) M,n , (11d) G (Z) M,n = Q p G (C) M,n + Q n G (C) M,p . (11e)
6 The nucleon Sachs formfactors
One finds in the literature various prescriptions for the Sachs formfactors of the nucleon. The three models which are used later on will be presented in the following three subsections. After that follows a comparison of these three models.
6.1 The Simon & Walther parameterization
The parameterization of Simon & Walther goes back to [2, 3]. It paremeterizes isoscalar and isovector formfactors as sum of dipole terms
G (S) typ (q) =
4
X
ν=1
a typ,ν
1 + q 2 /b typ,ν
(12) with typ∈ {“E,I=0”, “E,I=1”, “M ”} and with parameters as listed in table 2.
Proton and neutron Sachs formfactors are then obtained as G (S) E,p = 1
2 (G (S) E,I=0 + G (S) E,I=1 ) , (13a) G (S) E,n = 1
2 (G (S) E,I=0 − G (S) E,I=1 ) , (13b)
G (S) M,p = G (S) M , (13c)
G (S) M,n = G (S) M , (13d)
radiative corrections are related to the nuclear distribution. For then the composition should read G
(Z)1,p= N
ph Q
WG
(C)1,p− G
(C)1,n− G
si , G
(Z)1,n= N
nh Q
WG
(C)1,n− G
(C)1,p− G
si , G
(Z)2,p= N
ph Q
WG
(C)2,p− G
(C)2,ni , G
(Z)2,n= N
nh Q
WG
(C)2,n− G
(C)2,pi
,
Q
W= 1 − 4 sin
2(Θ
W) = 0.08 .
a 1 a 2 a 3 a 4 b 1 b 2 b 3 b 4
E,I=0 2.2907 -0.6777 -0.7923 0.1793 15.75 26.68 41.04 134.2 E,I=1 0.3681 1.2263 -0.6316 0.0372 5.00 15.02 44.08 154.2 M 0.6940 0.7190 -0.4180 0.0050 8.50 15.02 44.08 355.4
Table 2: Parameters of the model (12) for the nucleon formfactors. The b i are given in units fm −2 . The magnetic formfactors are taken from [2], the electric formfactors from [3].
n a E,n a M,n
1 -0.4980 +0.2472 2 +5.4592 -4.9123 3 -34.7281 +29.7509 4 +124.3173 -84.0430 5 -262.9808 +129.3256 6 +329.1395 -111.1068 7 -227.3306 +49.9753 8 +66.6980 -9.1659
Table 3: Parameters of the form- factor model (15). The coeffi- cients have units (GeV/c) −2n .
6.2 Simple dipole form
The study of the weak-charge formfactor in [1] employs for the Sachs formfactors a simple dipole form. It reads
G (S) E,p (Q 2 ) = G D (Q 2 ) =
1 + Q 2 12 r p 2
−2
, (14a)
G (S) E,n (Q 2 ) = −
Q 2 r 2 n /6 1 + Q 2 /M 2
−2
G D (Q 2 ) , (14b)
G (S) M,p (Q 2 ) = G (S) M,n (Q 2 ) = G D (Q 2 ) (14c) r p 2 = 0.769 fm 2 , r 2 p = −0.116 fm 2 , (14d) and it designed to reproduce the region of extremely low Q where the r.m.s. radius is defined. To that end, a rather recent value for r p 2 is employed. It complies with radius extracted from the extensive formfactor analysis of [4].
6.3 Up-to-date proton formfactor
The A1 collaboration of the IKP Mainz has recently performed a renewed analysis
of the proton formfactor which is summarized in [5]. From this analysis, we take the
model “dipole×polynomial” as an up to date parameterization whose proton radius
is r 2 p = 0.774 fm 2 , sufficiently close to the radius used in the dipole form (14) and to
S&W dip poly exp. [4]
r p 2 0.730 0.769 0.774 0.773±0.008 r n 2 -0.117 -0.116 -0.117
Table 4: Mean square proton and neutron radii in units of fm 2 as emerging from the different models for the nucleon Sachs formfactors, “S&W” = fit of Simon&Walther as given in section 6.1, “dip” = the simple dipole form as given in section 6.2, “poly”
= recent Mainz fit as given in section 6.3. The last column, “exp.”, shows the most recent result from the experimental compilation [4].
the recent evaluation [4]. The proton Sachs formfactor for this choice reads
G (S) E/M,p = 1 + Q 2 /a D −2
"
1 +
8
X
n=1
a E/M,n Q 2n
#
, a D = 0.71 (GeV/c) 2 , (15)
with parameters a E/M,n as given in table tab:poly. The work of [5] does not supply fresh neutron formfactors. We use the G (S) E/M,n as given in the Simon&Walther ansatz, see section 6.1. This form has a neutron radius which complies with the neutron radius of the dipole ansatz (14).
6.4 Comparison of nucleon formfactors
Table 4 compares the nucleonic mean square radii computed with the three different models for the intrinsic formfactor introduced in the previous subsections. The neu- tron radii are practically the same in all three models. Larger differences are seen for the protons. In particular, the older S&W form deviates substantially (about 5%) to a lower value. The other two forms agree with each other within 0.005 fm 2 which is below the present experimental uncertainty (last column). They also comply with the experimental data. The 5% deviation for S&W has to be put into perspective.
It is a deviation on a quantity which causes only a small correction on the overall nuclear r.m.s. radius. The net effect still remains small, at least for heavy nuclei.
Figure 2 compares the proton and neutron formfactors for the three models. The lower panels in each block show the formfactors as such. At this scale, differences for proton electric and neutron magnetic formfactors are small, somewhat larger for the other two. In order to compare with higher resolution, we show in the upper panels of each block the difference with respect to the formfactor of the latest Mainz compilation as introduced in section 6.3. It is obvious that differences build up with increasing q. The difference between old and new Mainz evaluation (green line) for the protons stays typically factor 1/2 smaller than the difference to the dipole fit.
This confirms what has been worked out earlier [2, 5] that one single dipole term does
not suffice to describe the whole nucleon formfactor at a broader scale of q. This may
become relevant when evaluating the form parameters beyond the r.m.s. radius. The
figure indicates the q values which are relevant for determining the surface thickness
σ for the case of 208 Pb and 48 Ca. While the differences look bearable for 208 Pb, they
0.8 0.9 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 proton electric F
Ep(q)
momentum q [fm
-1] Bernauer poly.
Simon+Walther Kelly dip.
-0.012 -0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004
F
Ep(q) diff. to Bernauer
208Pb
48Ca
0 0.01 0.02 0.03
0 0.2 0.4 0.6 0.8 1 1.2 1.4 momentum q [fm
-1]
Simon+Walther Kelly dip.
0 0.002 0.004 0.006
comparison of models for the nucleon electric formfactor
208
Pb
48Ca
0.8 0.9 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 proton electric F
Ep(q)
momentum q [fm
-1] Bernauer poly.
Simon+Walther Kelly dip.
-0.025 -0.02 -0.015 -0.01 -0.005 0
F
Ep(q) diff. to Bernauer
208Pb
48Ca
0.8 0.9 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 momentum q [fm
-1]
Simon+Walther Kelly dip.
-0.015 -0.01 -0.005 0
comparison of models for the nucleon magnetic formfactor
208