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Page 1

arXiv:nucl-th/9912038
v2 6 Jan 2000

Parity Violating
Measurements of Neutron Densities

C. J.
Horowitz

∗

Dept. of Physics
and Nuclear Theory Center

Indiana
University

Bloomington,
Indiana 47405 USA

S. J.
Pollock

†

Dept. of Physics
, CB 390

University of
Colorado

Boulder, CO 80309
USA

P. A.
Souder

‡

Dept. of
Physics

Syracuse
University

Syracuse, N.Y.,
USA

R.
Michaels

§

Thomas Jefferson
National Accelerator Facility

Newport News, VA,
USA

(June 6,
2005)

∗

email:
[email-masked]

†

email:
[email-masked]

‡

email:
[email-masked]

§

email:
[email-masked]

1

Page 2

Abstract

Parity violating
electron nucleus scattering is a clean and powerful tool for

measuring the
spatial distributions of neutrons in nuclei with unprecedented

accuracy. Parity
violation arises from the interference of electromagnetic and

weak neutral
amplitudes, and the Z

0

of the Standard
Model couples primarily

to neutrons at
low Q

2

. The data can
be interpreted with as much confidence

as
electromagnetic scattering. After briefly reviewing the present
theoretical

and experimental
knowledge of neutron densities, we discuss possible par-

ity violation
measurements, their theoretical interpretation, and applications.

The experiments
are feasible at existing facilities. We show that theoretical

corrections are
either small or well understood, which makes the interpreta-

tion clean. The
quantitative relationship to atomic parity nonconservation

observables is
examined, and we show that the electron scattering asymme-

tries can be
directly applied to atomic PNC because the observables have

approximately
the same dependence on nuclear shape.

I.
INTRODUCTION

The size of a
heavy nucleus is one of its most basic properties. However, because of
a

neutron skin of
uncertain thickness, the size does not follow from measured charge radii
and

is relatively
poorly known. For example, the root mean square neutron
radius in

208

Pb,
R

n

is thought to be
about 0.25 fm larger then the proton radius
R

p

≈ 5.45 fm. An
accurate

measurement of
R

n

would provide
the first clean observation of the neutron skin. This is

thought to be an
important feature of all heavy nuclei.

The interior
baryon density of a heavy nucleus is closely related to its size. The
saturation

density of
nuclear matter ρ

0

is a fundamental
concept central to nuclear structure, the nature

of the
interactions between nucleons, models of heavy ion collisions and applications
of dense

matter in
Astrophysics. The value of ρ

0

is inferred from
the central density of heavy nuclei,

most
notably

208

Pb. One then
corrects for the effects of surface tension and Coulomb

interactions
(which tend to cancel) and deduces the saturation density of an infinite
system.

However, present
estimates of ρ

0

are based only
on the known proton density. Thus ρ

0

is

uncertain
because we do not have accurate information on the central neutron density.
An

accurate
measurement of the neutron radius
R

n

will constrain
the average interior neutron

density and help
refine our knowledge of ρ

0

.

Ground state
charge densities have been determined from elastic electron scattering,
see

for example ref.
[1].
Because the densities are both accurate and model independent they

2

Page 3

have had a great
and lasting impact on nuclear physics. They are, quite literally, our
modern

picture of the
nucleus.

In this paper we
discuss future parity violating measurements of neutron densities.
These

purely
electro-weak experiments follow in the same tradition and can be both
accurate

and model
independent. Neutron density measurements may have implications for
nuclear

structure,
atomic parity nonconservation (PNC) experiments, isovector interactions,
the

structure of
neutron rich radioactive beams, and neutron rich matter in astrophysics.
It

is remarkable
that a single measurement has so many applications in atomic, nuclear
and

astrophysics.

Donnelly, Dubach
and Sick [2]
suggested that parity violating electron scattering can

measure neutron
densities. This is because the Z−boson couples primarily to the
neutron

at low
Q

2

. Therefore one
can deduce the weak-charge density and the closely related neutron

density from
measurements of the parity-violating asymmetry in polarized elastic
scattering.

This is similar
to how the charge and proton densities are deduced from unpolarized
cross

sections.

Of course the
parity violating asymmetry is very small, of order a part per
million.

Therefore
measurements were very difficult. However, a great deal of experimental
progress

has been made
since the Donnelly et. al. suggestion, and since the early SLAC
experiment

[3].
This includes the Bates

12

C experiment [4],
Mainz

9

Be experiment [5],
SAMPLE [6] and

HAPPEX [7].
The relative speed of the HAPPEX result and the very good helicity
correlated

beam properties
of CEBAF show that very accurate parity violation measurements are

possible. Parity
violation is now an established and powerful tool.

For example, the
HAPPEX result suggests that strange quarks do not make large con-

tributions to
the nucleon’s electric form factor. Clearly additional experiments should
(and

will) be done to
further measure strange quarks. However, it is important to also
apply

parity violation
to other physics objectives such as neutron densities. This will allow one
to

take maximum
advantage of parity violation.

It is important
to test the Standard Model at low energies with atomic PNC, see for

example the
Colorado measurement in Cs [8,9].
These experiments can be sensitive to new

parity violating
interactions such as additional heavy Z−bosons. Furthermore, by
comparing

atomic PNC to
higher Q

2

measurements,
for example at the Z pole, one can study the

momentum
dependence of Standard model radiative corrections. However, as the
accuracy

of atomic PNC
experiments improves they will require increasingly precise information
on

neutron
densities [10,11].
This is because the parity violating interaction is proportional

to the overlap
between electrons and neutrons. In the future the most precise low
energy

Standard Model
test may involve the combination of an atomic PNC measurement and

parity violating
electron scattering to constrain the neutron density.

Unfortunately,
atomic PNC suffers from atomic theory uncertainties in the electron
den-

sity at the
nucleus. This motivates future atomic experiments involving isotope ratios
where

the atomic
theory cancels. However, these ratios may require even more nuclear
structure

information on
isotope differences of neutron densities. Parity violating electron
scattering

3

Page 4

measurements of
isotope differences is beyond the scope of this paper. Instead we focus
on

simpler
measurements of the neutron density in a single closed (sub)shell isotope.
These

measurements
should provide an important first step for later work on isotope
differences.

There have been
many measurements of neutron densities with strongly interacting
probes

such as pion or
proton elastic scattering, see for example ref. [12].
We discuss some of these in

section II.
Unfortunately, all such measurements suffer from potentially serious
theoretical

systematic
errors. As a result no hadronic measurement of neutron densities has
been

generally
accepted by the field. Because of the uncertain systematic errors, modern
mean

field
interactions are typically fit without using any neutron density information,
see for

example refs.
[13,14].

An electro-weak
measurement of the neutron density in a nucleus such
as

208

Pb
may

allow the
calibration of strongly interacting probes. By requiring that the hadronic
reaction

theory reproduce
the electro-weak measurement one should reduce theoretical errors.
This

is analogous to
using beta decay to calibrate (p, n) probes of Gamow Teller strength.
Once

proton nucleus
elastic scattering is calibrated it should be possible to study neutron
densities

in a variety of
other nuclei including radioactive beams.

Finally, there
is an interesting complementarity between neutron
radius measurements

in a finite
nucleus and measurements of the neutron radius
of a neutron star. Both provide

information on
the equation of state of dense matter. In a nucleus,
R

n

is sensitive to
the

surface symmetry
energy or the symmetry energy at low densities while the neutron
star

radius depends
on the symmetry energy at high densities.

In the future we
expect a number of improving radius measurements for nearby
isolated

neutron stars.
For example, from the measured luminosity and surface temperature one
can

deduce an
effective surface area and radius from thermodynamics. Candidate stars
include

Geminga [15]
and RX J185635-3754 [16].

This paper is
organized as follows. In section II we discuss the present theoretical and
ex-

perimental
knowledge of neutron densities. In section III we present general
considerations

for neutron
density measurements and include some experimental issues in section IV.
Sec-

tion V discusses
many possible theoretical corrections and shows that the interpretation
of

a measurement is
very clean. The relationship between neutron density measurements
and

atomic parity
nonconservation experiments is discussed in section VI. Finally we
conclude

in section
VII.

II. PRESENT
KNOWLEDGE OF NEUTRON DENSITIES

In this section
we discuss our present knowledge of neutron densities.
Unfortunately,

neutron density
uncertainties have not been extensively discussed in the literature.
Fortson

et al. [17]
give some discussion on the present uncertainties in neutron densities and how
this

uncertainty
impacts atomic PNC. They claim a relatively large error in the neutron radius

R

n

of order
10%.

4

Page 5

We believe the
most accurate information comes from theory. As we discuss below,

mean field
models predict a relatively small spread in neutron densities once the
effective

interaction is
constrained to reproduce observed charge densities and binding energies.
We

also discuss
neutron density measurements with elastic magnetic electron scattering
and

strongly
interacting probes.

A. Neutron
Density Theory

Mean field
models have been very successful at reproducing many features of
nuclear

charge densities
including measured charge radii. Figure 1,
adapted from Ring et al. [14],

shows
differences between neutron R

n

and proton radii
R

p

for a range of
nuclei for calcu-

lations based on
two typical interactions. A nonrelativistic zero range Skyrme force
gives

R

n

−R

p

≈ 0.1Fm
for

208

Pb while a
relativistic mean field calculation gives R

n

−R

p

≈
0.3Fm.

We don’t claim
that these two calculations represent extreme values. Rather they
represent

the range in
R

n

−
R

p

for two typical
classes of interactions.

In order for the
theoretical error to be less then the spread in Figure
1 one must demon-

strate that one
(or both) of the calculations is unrealistic. While this may be possible
in

the future, it
has not yet been demonstrated since both calculations are in common
use.

Of course, the
real uncertainty could be larger then the spread in Figure 1
if the two cal-

culations do not
probe the full parameter space of possible mean field interactions. In
the

absence of more
precise uncertainties, one can take this spread ≈ 0.2Fm as some measure
of

the present
uncertainty in R

n

. This compares
to a charge radius of 5.51 Fm. A one percent

measurement of
R

n

with an accuracy
of about 0.05Fm can clearly distinguish between the

two forces.
Furthermore, it can distinguish either prediction for
R

n

−
R

p

from zero
thus

cleanly
observing the neutron skin.

We note that the
relativistic mean field calculation with its larger
R

n

predicts a
signifi-

cantly smaller
central neutron density then does the Skyrme interaction. What gives rise
to

the differences
between the two calculations? Unfortunately, there is not much discussion
in

the literature.
We speculate that some of the difference arises because the relativistic
mean

field
calculation uses a finite range force while the Skyrme interaction is zero
range. There is

an attractive
interaction between the neutrons and the protons. A finite range force
allows

the neutrons to
sit at a slightly larger radius and still feel much of the attraction.
Therefore

we expect finite
range interactions to predict slightly larger R

n

−
R

p

then for zero
range

forces.

Once mean field
interactions are constrained to reproduce a neutron
radius measurement

in a stable
nucleus such as

208

Pb they can make
improved predictions for a variety of unstable

nuclei. We note
that the nonrelativistic Skyrme force SKX [18]
is designed for use in both

normal and
exotic nuclei such as

48

Ni,

68

Ni
and

100

Sn. One example
of relativistic mean field

calculations for
neutron rich nuclei is ref.
[19]. The structure of exotic nuclei is important

for Astrophysics
and for radioactive beams.

5

Page 6

FIGURES

FIG. 1. The
difference between neutron radii R

n

=
r

n

and proton radii
R

p

=
r

p

for
several

nuclei of
different mass number A. The filled symbols are for the relativistic mean field
NL1

interaction
while the open symbols are for the nonrelativistic zero range Skyrme skiii
interaction.

This figure is
taken from calculations of Ring et.al
[14]. A possible 1% measurement in

208

Pb
is

indicated by the
error bar which has been arbitrarily placed at R

n

−
R

p

=
0.

B. Neutron
Density Measurements

There have been
many measurements sensitive to neutron densities. Originally
neutron

radii were
extracted from Coulomb energy differences [20].
However, it is now thought these

measurements are
sensitive to isospin violating interactions. Next (p, d) and (d, t)
stripping

reactions are
sensitive to the tail in the neutron density at very large radius [21].
However,

stripping
reactions are not directly sensitive to the interior density. Because the
interior

density is much
larger then that in the tail it contributes significantly to
R

n

. Therefore
R

n

can not be
extracted from stripping experiments without making model
assumptions.

Proton nucleus
elastic scattering is sensitive to both the surface and interior
neutron

density [12].
Typically this data is analyzed in an impulse approximation where a
nucleon-

nucleon
interaction is folded with the nucleon density. Unfortunately, there are
corrections

to the impulse
approximation from for example multiple scattering and medium
modifica-

tions to the NN
interaction whose uncertainties are difficult to quantify. Limitations
in

the theoretical
analysis can show up as an unphysical dependence of the extracted
neutron

density on the
beam energy.

Future work on
extracting neutron densities from proton scattering would be very
useful.

This could take
advantage of advances in full folding calculations. Furthermore,
neutron-

nucleus elastic
scattering data would be very helpful. Comparing proton- and
neutron-

nucleus
scattering could help constrain the effective proton-neutron interaction.
Finally,

6

Page 7

if
proton-nucleus scattering can be calibrated to accurately reproduce a neutron
density

measurement in a
stable nucleus then it could be applied to a wide variety of other
nuclei.

For example, it
is possible to measure proton-nucleus scattering from radioactive beams
with

a hydrogen
target in inverse kinematics.

Finally, data
comparing the elastic scattering of positive and negative pions from
nuclei

exist [22],
but again there are uncertainties in the analysis [10]. These methods are
not

really directly
sensitive to the neutron density.

Elastic magnetic
electron scattering is an established tool for nuclear structure.
Fur-

thermore,
magnetic scattering is directly sensitive to the neutron magnetic moment.
Thus

information
about valence neutron radii can be extracted. Note, more calculations of
the

effects of
Coulomb distortions on magnetic scattering from heavy nuclei would be
useful.

See for example
[23].
However most of the neutrons in a heavy nucleus are coupled to spin

zero and make no
contribution to the magnetization. Therefore, magnetic scattering
can

not directly
determine R

n

.

We conclude that
no existing measurement of neutron densities or radii has an
established

accuracy of one
percent. While some conflicting claims may have been made, all
hadronic

probes of
R

n

suffer from some
reaction mechanism uncertainties. As a result there is no

agreement in the
community that any measurement has the requisite accuracy. Even if
it

is possible to
reach one percent accuracy with a hadronic probe, this accuracy has not
yet

been
established.

III. GENERAL
CONSIDERATIONS

In this section
we illustrate how parity violating electron scattering measures the
neutron

density. For
simplicity, this section uses the plane-wave Born approximation and
neglects

nucleon form
factors. The effects of Coulomb distortions and form factors are included
in

section V. These
are necessary for a quantitative analysis but they do not invalidate
the

simple
qualitative picture presented here.

The electron
interacts with a nucleus by exchanging either a photon or a
Z

0

boson.
The

propagator
involved in the interaction is of the form

1

Q

2

+
M

2

B

(3.1)

where
M

B

is the mass of
the exchanged boson. For the photon M

B

= 0, whereas for
the Z

0

the mass term
dominates. Since for elastic scattering from nuclei,
M

2

Z

≫
Q

2

, the
photon

term is much
larger than the Z

0

term. Note, we
use the convention Q

2

=
−q

µ

2

>
0.

Another
difference between the exchange of the photon and the
Z

0

is the couplings
to

both the
electron and the nucleons. The photon has purely vector couplings, and
couples

only to protons
at Q

2

= 0. We note
that for the spinless nuclei considered here, the magnetic

7

Page 8

moments cannot
contribute. The Z

0

has both vector
and axial vector couplings. Since the

nuclei being
considered are spinless, the net axial coupling to the nucleus is absent.
In

contrast to the
case for photons, the Z

0

has a much
larger coupling to the neutron than

the proton. In
addition, the Z

0

has a large
axial coupling to the electron that results in a

parity-violating
amplitude. The effect of the parity-violating part of the weak
interaction

may be isolated
by measuring the parity-violating asymmetry

A

LR

=

σ

R

−
σ

L

σ

R

+
σ

L

,

(3.2)

where
σ

L(R)

is the cross
section for the scattering of left(right) handed electrons. In
contrast

to the cross
section, the asymmetry is sensitive to the distribution of the neutrons in
the

nucleus. The
Z

0

also has a
vector coupling to the electron, but this term is neglected because

the contribution
cannot be isolated from the dominant photon amplitude.

The implication
of the above is that the potential between an electron and a nucleus
to

a good
approximation may be written

V (r) = V (r) +
γ

5

A(r)

(3.3)

where the usual
electromagnetic vector potential is

V (r)
=

d

3

r′Zρ(r′)/|r − r
′|

(3.4)

and where the
charge density ρ(r) is closely related to the point proton density
ρ

p

(r)
given

by

Zρ

p

(r)
=

p

ψ

†

p

(r)ψ

p

(r)
.

(3.5)

The axial
potential A(r) depends also on the neutron density:

Nρ

n

(r)
=

p

ψ

†

n

(r)ψ

n

(r)
.

(3.6)

It is given
by

A(r)
=

G

F

2

3/2

[(1 −
4sin

2

θ

W

)Zρ

p

(r) −
Nρ

n

(r)]

(3.7)

The axial
potential has two important features:

1. It is much
smaller than the vector potential, so it is best observed by measuring
parity

violation. It is
of order one eV while V (r) is of order MeV.

8

Page 9

2. Since
sin

2

θ

W

∼ 0.23, (1−4
sin

2

θ

W

) is small and
A(r) depends mainly on the neutron

distribution
ρ

n

(r).

The
electromagnetic cross section for scattering electrons with momentum transfer q
=

(Q

2

)

1/2

is given
by

dσ

dΩ

=

dσ

dΩ

Mott

|F

p

(Q

2

)|

2

(3.8)

where

F

p

(Q

2

)
=

1

4π

d

3

rj

0

(qr)ρ

p

(r)

(3.9)

is the form
factor for protons, where j

0

is the zero’th
spherical Bessel function. From F

p

(Q

2

),

one may
determine R

p

. One can also
define a form factor for neutrons

F

n

(Q

2

)
=

1

4π

d

3

rj

0

(qr)ρ

n

(r)

(3.10)

Thus
R

n

may be
determined if F

n

(Q

2

) is
known.

In Born
approximation the parity-violating asymmetry involves the interference
between

V (r) and A(r).
It is,

A

LR

=

G

F

Q

2

4πα

√

2

4
sin

2

θ

W

− 1
+

F

n

(Q

2

)

F

p

(Q

2

)

(3.11)

The asymmetry
is proportional to Q

2

/M

2

Z

(since
G

F

∝
M

−2

Z

) which is just
the ratio of the

propagators of
Eq. 3.1.
Since 1-4sin

2

θ

W

is small and
F

p

(Q

2

) is known we
see that A

LR

directly
measures F

n

(Q

2

). Therefore,
A

LR

provides a
practical method to cleanly measure

the neutron
form factor and hence R

n

.

IV.
EXPERIMENTAL ISSUES

The
experimental techniques for measuring small asymmetries of order 1 ppm
have

been
successfully deployed in parity experiments at electron scattering facilities
[3]-
[7]. The

following
general considerations applies to these experiments.

1) Often a
compromise

must be chosen
between optimizing the parity violating signal and the signal to noise
ratio.

The asymmetry
generally increases with Q

2

while the cross
section decreases, which leads

to an optimum
choice of kinematics.

2) A major
challenge for these measurements is

to maintain
systematic errors associated with helicity reversal at the ≈
10

−8

level.
There

must be at
least one, and preferably several, methods to reverse the helicity. Many
reversals

are needed
during an experiment, and they should follow a rapid and random
sequence

9

Page 10

to avoid any
correlation with noise. The helicity reversals should be uncoupled to
other

parameters
which affect the cross section. Experiments must measure the sensitivity
of

the cross
section to these parameters, as well as the helicity correlated differences in
them.

3) Electronic
pickup of the helicity correlated signals can cause a false asymmetry, as
can

helicity
correlated deadtime. 4) In a count rate limited experiment in which the
detected

particles must
be integrated in order to get the desired accuracy in a reasonable time,
the

linearity of
the detection system and the susceptibility to backgrounds are important
issues.

For the
high-rate experiments considered here, the radiation hardness of the detectors
is

also an issue.
5) The beam polarization must be measured with high precision.
Online

monitoring is
possible using a Compton polarimeter which is cross-calibrated using
Møller

and Mott
polarimeters whose absolute calibrations may be
≈1%.

A. Choices of
Target and Kinematics

There are two
nuclei which are of interest for a measurement of the neutron radius to

1%
accuracy,

208

Pb
and

138

Ba. They are
equally accessible experimentally. Pb has the

advantage that
it has the largest known splitting to the first excited state (2.6 MeV) of
any

heavy nucleus,
and thus lends itself well to the use of a flux integration technique. Also
Pb

has been very
well studied, and with its simple structure is a good first test case for
nuclear

theory. Ba has
the advantage that it is one of the nuclei being used for an atomic
physics

test of the
Standard Model.

The choice of
kinematics for a first measurement is guided by the objective of
minimizing

the running
time required for a 1% accuracy in R

n

. Figure 2
shows for the case of

208

Pb
the

three
ingredients which enter into this optimization: the cross section dσ/dΩ, the
parity

violating
asymmetry A, and the sensitivity to the neutron
radius ǫ = dA/A = (A1 − A)/A

where A is the
asymmetry computed from a mean field theory (MFT) calculation [24]
and

A1 is the
asymmetry for the MFT calculation in which the neutron
radius is increased by

1%. These three
ingredients, which each vary with energy and angle, are plotted in figure 2

for a beam
energy of 0.85 GeV. As we will show below, 0.85 GeV turns out to be the
energy

which minimizes
the running time for a 1% R

n

determination.
Using magnetic spectrometers

with high
resolution to isolate elastically scattered electrons, the optimal kinematics
can be

determined from
the allowable settings for angle and momentum of the spectrometer
by

searching for
the the minimum running time, which is equivalent to maximizing the
product

FOM ×
ǫ

2

= R ×
A

2

×
ǫ

2

(4.1)

where R is the
detected rate and is proportional to dσ/dΩ, and “FOM” is the
conventionally

defined figure
of merit for parity experiments, FOM = R × A

2

. Note that
rather than only

maximizing the
conventional FOM, parity violating neutron density measurements take
into

account the
sensitivity (ǫ) to R

n

which varies
with kinematics.

As an example,
we have performed the optimization for the Jefferson Lab Hall A
high

resolution
spectrometers supplemented by septum magnets that allow to reach
6

◦

scattering

10

Page 11

angle. The
calculations take into account the averaging over the finite acceptance
and

the energy
resolution needed to discriminate inelastic levels. Figure
2 shows the product

FOM ×
ǫ

2

for

208

Pb which peaks
at E = 0.85 GeV. Similar calculations for

138

Ba shows
an

optimum at 1.0
GeV (figure 3).
For both these nuclei the running time T in days to reach

a 1% accuracy
in R

n

is
approximately T ≈ 7/(P

2

IΩ) days, where
P is the polarization

(P ≈ 0.8 is
achievable), I is the average beam current in µA (I ≈ 50µA is
achievable)

and Ω is the
solid angle acceptance of the spectrometer in steradians. This optimum
point

corresponds to
q = 0.45 fm

−1

and 0.53
fm

−1

for Pb and Ba
respectively. In the plots

of FOM ×
ǫ

2

one can see a
secondary ridge where one might want to perform a second

measurement at
higher Q

2

to check the
shape dependence. Here the experimental running

time becomes
longer but the required accuracy in R

n

can be reduced.
As an example, for

208

Pb at E = 1.3
GeV, θ = 8

◦

, corresponding
to q = 0.92 fm

−1

the running
time to reach

2% in
R

n

is T ≈
19/(P

2

IΩ)
days.

q
(fm

-1

)

d

σ

/d

Ω

(mbarn)

Cross
Section

at E = 0.85
GeV

q
(fm

-1

)

A
(ppm)

Asymmetry

q
(fm

-1

)

ε

=
dA/A

Sensitivity

to
R_n

E
(GeV)

0.5

1

6

10

Θ

FOM x
ε

2

Pb

208

FIG. 2. Cross
section, parity violating asymmetry, and sensitivity to
R

n

for

208

Pb
elastic

scattering at
0.85 GeV. The fourth plot shows the variation of
FOM×ǫ

2

with energy and
angle,

showing an
optimum at 0.85 GeV for a 6

◦

scattering
angle which corresponds to q = 0.45 fm

−1

.

To reduce the
running time, a thick target is needed; the main issues are: 1) For
a

given energy
resolution required to discriminate excited states, there is an optimum
target

11

Page 12

thickness (≈
10% radiation length) that maximizes the rate in the detector. As the
target

becomes thicker
the radiative losses decrease the rate.

2) If at the
low-Q

2

where
the

experiments run
the rates from some low level inelastic states are sufficiently small
and

understood
theoretically, one may tolerate accepting them into the detector, thus
allowing

to integrate
more of the radiative tail, typically up to 4 MeV. 3) To improve the
heat

load capability
of the target, one may use various “cooling agents”, such as laminations
of

diamond
interleaved with the target material. One must have sufficient knowledge of
the

effect on the
parity signal. For example, if one accepts 2% rate
from

12

C and the
theory is

understood to
1%, the systematic error is only 0.02%. The theoretical error is discussed
in

section
V.

q
(fm

-1

)

d

σ

/d

Ω

(mbarn)

Cross
Section

at E=1.0
GeV

q
(fm

-1

)

A
(ppm)

Asymmetry

q
(fm

-1

)

ε

=
dA/A

Sensitivity

to
R_n

E
(GeV)

0.5

1

6

10

Θ

FOM x
ε

2

Ba

138

FIG. 3. Same as
figure 2
except for

138

Ba at 1.0
GeV

V. CORRECTIONS
TO THE ASYMMETRY

In this section
we document a number of corrections to the parity violating
asymmetry

and show that
they have small uncertainties. Therefore the interpretation of a
measurement

should be
clean. We consider coulomb distortions, strangeness and the neutron
electric

12

Page 13

form factor,
parity admixtures, dispersion corrections, meson exchange currents, shape
de-

pendence,
isospin admixtures, role of excited states and the effect of target
impurities.

A. Coulomb
distortions

By far the
largest known correction to the asymmetry comes from coulomb
distortions.

By coulomb
distortions we mean repeated electromagnetic interactions with the
nucleus

remaining in
its ground state. All of the Z protons in a nucleus can contribute
coherently

so distortion
corrections are expected to be of order Zα/π. This is 20 %
for

208

Pb.

Distortion
corrections have been accurately calculated in ref. [24].
Here the Dirac equa-

tion was
numerically solved for an electron moving in a coulomb and axial-vector
weak

potentials.
From the phase shifts, all of the elastic scattering observables including
the

asymmetry can
be calculated.

There are many
checks on the numerics of this calculation. First, known cross
sections

including those
at large angles are reproduced. Second, the code reproduces known
plane

wave
asymmetries. Finally, the sensitivity to the subtraction between helicities is
checked

by varying the
strength of the weak potential. We note that the forward angle
asymmetry

is much easier
to calculate then the backward angle cross section because the cross
section

involves
extreme cancellations in the sum over partial waves. It is expected that the
numer-

ical accuracy
in the asymmetry is significantly better then 1%. However, the code
neglects

terms involving
the electron mass over the beam energy. These are of order 0.1%.
There

are now a
number of independent codes which calculate the effects of coulomb
distortions

[19,25,26]
and verify the accuracy of ref. [24].

In summary,
distortion corrections are larger then the experimental error.
Furthermore,

they modify the
sensitivity to the neutron radius. However,
they have been calculated with

an accuracy
significantly better then the expected 3% experimental error.

Finally, since
the charge density is known it should be possible to “invert” the
coulomb

distortions and
deduce from the measured asymmetry the value of a Born
approximation

equivalent weak
form factor at the momentum transfer Q

2

of the
experiment. Thus, the main

result of the
measurement is the weak form factor F

W

(Q

2

) which is the
Fourier transform

of the weak
charge density ρ

W

(r),

F

W

(Q

2

)
=

d

3

rj

0

(qr)ρ

W

(r).

(5.1)

This can be
directly compared to mean field or other theoretical calculations. Note,
F

W

will not be
determined by comparing plane wave calculations to data. Instead, for
example,

a range of
model weak densities could be adjusted until full distorted wave
calculations

reproduce the
experimental asymmetry. Then, Eq. 5.1
is used to calculate F

W

(Q

2

).
In

principle this
procedure is slightly model dependent because full distorted wave
calculations

need some
information on F

W

(Q

2

) for q
different from the single measurement. However this

13

Page 14

model
dependence is expected to be very small and can be explored by studying a
variety

of model
densities.

B. Strangeness
and neutron electric form factors

From
F

W

the root mean
square radius of the weak charge distribution R

W

can be
deter-

mined, see
section VF. The weak radius, in turn, can be related to the radius of the
neutron

distribution
after making appropriate corrections. We emphasize that the experiment
mea-

sures a well
defined form factor of the weak charge distribution and that this can be
directly

compared to
mean field models without any additional corrections. However, if one
wishes

to go further
and extract a point neutron radius one must
correct for possible strange quark

contributions
and other nucleon form factors. We discuss these here. In addition,
there

could be meson
exchange currents which we discuss in a later subsection.

The electric
form factors for the coupling of a Z

0

to the proton
G

Z

p

and neutron
G

Z

n

are,

G

Z

p

=

1

4

(G

p

−
G

n

) −
sin

2

ΘG

p

−

1

4

G

s

,

(5.2)

G

Z

n

=

1

4

(G

n

−
G

p

) −
sin

2

ΘG

n

−

1

4

G

s

.

(5.3)

We are only
interested in electric form factors since magnetic form factors make no
contribu-

tion for a spin
zero target. Therefore we omit the label E for clarity. The proton
(neutron)

electromagnetic
form factor is G

p

(G

n

). The strange
quark form factor is G

s

and this
is

assumed the
same for neutrons and protons.

We fold these
form factors with point proton ρ

p

and neutron
ρ

n

densities to
obtain the

weak charge
density ρ

W

,

ρ

W

(r) = 4
d

3

r

′

[G

Z

n

(r

′

)Nρ

n

(|r −
r

′

|) +
G

Z

p

(r

′

)Zρ

p

(|r −
r

′

|)].

(5.4)

The densities
are normalized,

d

3

rρ

p

(r) =
1,

(5.5)

d

3

rρ

n

(r) =
1,

(5.6)

and

d

3

rρ

W

(r) =
Q

W

,

(5.7)

14

Page 15

where the weak
charge of the nucleus is,

Q

W

= −N + (1 −
4sin

2

Θ)Z.

(5.8)

The proton
R

p

, neutron
R

n

and weak
R

W

radii are
defined,

R

2

p

=

d

3

rr

2

ρ

p

(r),

(5.9)

R

2

n

=

d

3

rr

2

ρ

n

(r),

(5.10)

and

R

2

W

=

1

Q

W

d

3

rr

2

ρ

W

(r).

(5.11)

It is a simple
matter to calculate the weak radius from Eq. 5.4,

−Q

W

R

2

W

=
NR

2

n

+
(4sin

2

Θ −
1)ZR

2

p

+ [N +
(4sin

2

Θ −
1)Z]r

2

p

+ [Z +
(4sin

2

Θ −
1)N]r

2

n

+ (N +
Z)r

2

s

.

(5.12)

Here
r

2

p

is the mean
square charge radius of the proton, r

2

n

the square of
the neutron charge

radius and
r

2

s

is the mean
square strangeness radius.

Assuming the
neutron radius is much larger then
R

n

−
R

p

and
r

p

the above
reduces to

R

W

≈
R

n

+

Z(1 −
4sin

2

Θ)

N +
(4sin

2

Θ −
1)Z

(R

n

−
R

p

)

+

1

2R

n

{r

2

p

+

Z +
(4sin

2

Θ −
1)N

N +
(4sin

2

Θ −
1)Z

r

2

n

+

N +
Z

N +
(4sin

2

Θ −
1)Z

r

2

s

}.

(5.13)

For

208

Pb, assuming
R

n

≈ 5.50 fm and
sin

2

Θ = 0.23, we
have

R

W

≈
R

n

+
0.055(R

n

−
R

p

) +
0.061(±0.002) − 0.0089(±0.0003) − 0.011ρ

s

(5.14)

in fm. The
0.061 is from the charge radius of the proton and the -0.0089 from the
charge

radius of the
neutron [27].

The last term
in Eq. 5.14
is from strange quark contributions. The strange quark form

factor
G

s

has been
parameterized with ρ

s

,

15

Page 16

G

s

(Q

2

) =
ρ

s

τ/(1 +
4.97τ)

2

,

(5.15)

and τ =
Q

2

/4M

2

. We aim to
measure R

W

to 1% or about
0.055 fm. Therefore, strange

quarks will
contribute less then 1% as long as,

|ρ

s

| <
5.

(5.16)

This is a very
mild requirement which is already strongly supported by existing
experiments.

For example,
the HAPPEX measurement [7],

G

s

E

+
0.39G

s

M

(Q

2

=
0.48GeV

2

) = 0.023 ±
0.043,

(5.17)

and the SAMPLE
measurement [6],

G

s

M

(Q

2

=
0.1GeV

2

) = 0.23 ±
0.44,

(5.18)

yield,

0.011ρ

s

= −0.0043 ±
0.021fm.

(5.19)

The errors
quoted are combined statistical and systematic and we have assumed the
form

of Eq. 5.15
for the Q

2

dependence of
the strange form factor. Equation 5.19
limits the

strangeness
contributions to R

n

to 0.4%. Note,
if one assumes a different Q

2

dependence

than Eq. 5.15,
it may be possible to somewhat weaken this limit. However,
additional

measurements in
the near future will significantly tighten the constraints on strange
quarks

and clearly
rule out |ρ

s

| >
5.

Likewise, the
neutron electric form factor contributes far less then 1% to
R

W

.
Theoretical

models have
R

n

−
R

p

≤ 0.3 fm, so
the second term in Eq. 5.13
is also less then 1%. Indeed

to 1%, the
neutron radius directly follows from the
measured weak radius,

R

n

≈
R

W

−
0.06fm

(5.20)

We conclude
that the contribution of strange quarks or the neutron electric form factor
are

not issues for
a neutron radius measurement. The radius of the
neutron density of a heavy

nucleus can be
accurately determined from the measured weak
radius.

C. Parity
Admixtures

The spin zero
ground state of

208

Pb need not be
a parity eigenstate. There is probably

some small
admixture of 0

−

. However, so
long as the initial and final states are spin zero,

this parity
admixture can not produce a parity violating asymmetry in Born
approximation

16

Page 17

[28].
A multipole decomposition of the virtual photon has a
0

+

coulomb but no
0

−

multipole.

So long as the
exchanged virtual photon is spin zero, there is no parity violating
interference

because there
is only a single operator. This statement is true regardless of the parity of
the

initial or
final states or if the photon coupling involves a parity violating meson
exchange

current.
Therefore, parity admixtures should not be an issue for elastic scattering from
a

spin zero
nucleus.

D. Meson
Exchange Currents

Meson exchange
currents MEC can involve parity violating meson couplings. These
are

not expected to
be important for a spin zero target, see the subsection on parity
admixtures

above. Meson
exchange currents could also change the distribution of weak charge in
a

nucleus.
However, mesons are only expected to carry weak charge over a distance
much

smaller then
R

n

. This should
not lead to a significant change in the extracted neutron

radius. Let
r

2

MEC

be the square
of the average distance weak charge is moved by MEC.

Then following
Eq. 5.13
the correction to the weak radius will be of order
r

2

MEC

/R

n

.
This

is expected to
be very small because R

n

is
large.

This same
result can be viewed another way. Figure 4
shows a schematic diagram of

the weak charge
density. In the interior region the density is more or less constant.
In

this region,
MEC have very little effect. The density is simply the conserved weak
charge

divided by the
volume. It does not matter if the weak charge resides on the nucleons
or

on mesons going
between the nucleons. The only effect of MEC is to slightly change
the

surface
thickness. This is indicated by the dotted line in Fig. 4.
This change in surface

thickness will
only lead to a very small change in the weak radius. We conclude that
MEC

are unlikely to
be an issue for the interpretation of the weak
radius.

E. Dispersion
corrections

By dispersion
corrections we mean multiple electromagnetic or weak interactions
where

the nucleus is
excited from the ground state in at least one intermediate state. At the
low

momentum
transfers considered here, the elastic cross section involves a coherent sum
over

the Z protons
and is of order Z

2

. In contrast,
the incoherent sum of all inelastic transitions

is only of
order Z. Therefore we expect dispersion corrections to be of order α/Z. This
is

negligible.

F. Shape
dependence and Surface Thickness

In principle,
the weak radius follows from the derivative of a form factor evaluated
at

zero
Q

2

. In practice,
the measurement will be carried out at a small but nonzero
Q

2

.
Thus

17

Page 18

the extraction
of the weak radius from the measured form factor may depend slightly on
the

assumed surface
thickness.

We emphasize,
one primary use of a measurement is to calibrate mean field models
of

neutron
densities. One can simply calculate the weak form factor, Eq. 5.1,
and directly

compare theory
and experiment without any model dependence or the need to extract
a

neutron radius. However, if one wishes to extract a
neutron radius one must address
the

dependence of
the radius on the shape of the neutron distribution. One is most sensitive
to

the surface
thickness.

For example, if
the weak density of

208

Pb is modeled
as a Wood Saxon with radius

parameter
c,

ρ

W

(r) =
ρ

0

/{exp[(r −
c)/z] + 1},

(5.21)

then the
surface thickness parameter z ≈ 0.55 fm must be known to ±0.14 fm in order
to

extract
R

W

to 1% from an
asymmetry measurement at the proposed Q

2

= 0.008
GeV

2

.

Thus the
surface thickness or z must be known to only 25 % in order to extract
R

n

.

We believe the
surface thickness of the weak density is known to much better then
25%

for at least
two reasons. First, the surface thickness is strongly constrained by the
known

surface and
single nucleon separation energies. At large r the weak density is dominated
by

the most weakly
bound neutron. This decays exponentially with a known separation
energy.

Therefore the
large r behavior of the weak density is known. At somewhat smaller radii
the

density is
controlled by the surface energy. The very abrupt change in density, necessary
for

a small surface
thickness, implies a very high surface energy. Any model with a very
small

surface
thickness will fail to reproduce the known binding energies of a range of
nuclei. As

a result, all
mean field models, that we are aware of, have only a small spread in
surface

thickness –much
less then 25%– if they reproduce binding energies .

Second, the
surface thickness of the weak density is constrained by the measured
surface

thickness of
the charge density. Perhaps the easiest way to change the surface thickness
of

the neutron
density is to change the thickness of both the protons and neutrons.
However,

this will
quickly conflict with the measured charge density. Therefore, one has to try
and

change the
surface thickness of the neutrons without changing the proton density. This
will

necessitate
large separations in both energy and position between protons and neutrons.
To

accomplish
this, one will need energetic isovector interactions which in turn will change
the

binding
energies of nuclei as a function of N or Z and ruin agreement with known
masses.

Note, present
mean field models do an excellent job reproducing the surface thickness
of

measured charge
densities. This is a nontrivial check. Although one or more parameters
of

mean field
forces are often fit to charge radii, the detailed form of the surface density
is not

fit. Therefore
the excellent agreement between theory and experiment in the surface
region

demonstrates
both the power and basic correctness of these arguments that the
surface

thickness is
strongly constrained by measured binding energies.

We illustrate
the above points in Fig. 5
This shows the charge density in

208

Pb.
A

figure for the
weak density would be similar. Conventional mean field models, thin
dashed

18

Page 19

and dotted
curves, agree very well with the measured surface thickness (region beyond
r=5

fm). (We note
that a low Q

2

measurement is
insensitive to the interior density.) In contrast

the thick
dashed curve shows a relativistic mean field model with a very incorrect
surface

energy [29].
This calculation has an incompressibility (which is closely related to the
surface

energy) more
then a factor of two too large. This error is well outside of present
uncertainties.

Therefore the
surface properties of this calculation can be ruled out. Nevertheless even
with

this large
error, the surface thickness disagrees with data and other calculations by
only

10%. Since this
is less then 25% there would be no problem using this incorrect surface
to

extract the
neutron radius to 1%.

We state the
results of this section in a slightly different language. This measurement
is

sensitive to
the surface thickness at only the 25% level, while it is sensitive to the radius
at

the 1% level.
Since 25% is much larger then the present spread in surface thickness of
mean

field models
one will not learn new information about the surface. Instead a 1%
constraint

on the radius
does provide important new information on the radius because present
models

have a larger
spread then 1%.

Finally,
uncertainties from the surface thickness are even less important in
extracting

weak charge
information for atomic parity experiments. This is because the atomic
experi-

ments depend on
the surface thickness in somewhat similar ways to the electron
scattering

asymmetry. As a
result, some of the error from the unknown surface thickness cancels
in

comparing the
two experiments. Therefore, one could tolerate an uncertainty in the
surface

thickness of
more then 25% and still interpret the atomic experiment. This is discussed
in

section
VI.

We don’t
believe the dependence on the surface thickness is a problem. Nevertheless,
if

one wanted to
reduce the sensitivity there are two options. First, measure at a lower
Q

2

.

Unfortunately
this reduces the magnitude of the asymmetry and its sensitivity to the neutron

radius. More beam
time will be required and one may be more sensitive to systematic
errors.

Alternatively,
one could measure a second asymmetry at a higher
Q

2

. Within a
given model

of the shape of
the weak charge density this second point provides information on the
surface

thickness. For
example if one assumes a Wood Saxon neutron density, changing the
surface

thickness z
from 0.5 fm to 0.6 fm (at fixed mean square radius) decreases the asymmetry
by

8% at 850 MeV
and 12 degrees while the asymmetry is decreased by only 1.5% at 6
degrees.

Thus the large
angle point is much more sensitive to the surface thickness and in
principle

could help
constrain it. However, this second point will require considerable extra
beam

time.
Furthermore, the high Q

2

point is
sensitive to other features of the shape in addition

to the surface
thickness. Therefore, the interpretation of the high
Q

2

point may be
model

dependent.

In summary, one
only needs very mild information about the shape of the weak charge

density to
extract a radius from the measured asymmetry. One needs to know the
surface

thickness to
about 25%. We believe this is well within the accuracy of present mean
field

models. We
emphasize, even this small ambiguity does not effect the direct comparison
of

the measured
form factor to theoretical models.

19

Page 20

G. Inelastic
Contributions

In principle,
one could measure with enough energy resolution to avoid excited
state

contributions.
However in practice, there may be a gain in rate by running with
lower

resolution and
allowing a small contamination from excited states. For example, one
can

use a thicker
target with a larger energy loss. This contamination is expected to be
small

because
inelastic cross sections at low momentum transfer are typically much smaller
then

the elastic
cross sections.

It is useful to
estimate the inelastic asymmetry. The first excited state
in

208

Pb is at
2.6

MeV and has
spin and parity 3

−

. This is a
collective density oscillation
[30]. We expect the

longitudinal to
dominate over the transverse or axial responses (at forward angles). In
plane

wave Born
approximation the asymmetry for a natural parity spin J excitation is then [2],

A
=

GQ

2

4πα

√

2

4sin

2

Θ − 1
+

F

J

n

(Q

2

)

F

J

p

(Q

2

)

,

(5.22)

with G the
Fermi constant. Here the neutron transition form factor is,

F

J

n

(Q

2

) =
N

r

2

drj

J

(qr)ρ

tr

n

(r),

(5.23)

in terms of the
neutron transition density ρ

tr

n

(r) and a
similar expression for the proton

transition form
factor F

J

p

(Q

2

) in terms of
the proton transition density ρ

tr

p

(r).

The collective
density oscillation can be modeled as a deformation of the ground
state

density [30].
If the elastic neutron density is characterized by a radius
R

0

n

then the
excited

state has a
density parameter R

0

n

(θ),

R

0

n

(θ) ≈
R

0

n

[1 +
α

n

J

Y

J0

(θ)],

(5.24)

where the small
amplitude α

n

J

can be adjusted
to reproduce the magnitude of the cross

section.
Likewise the proton density is characterized by
R

0

p

(θ),

R

0

p

(θ) ≈
R

0

p

[1 +
α

p

J

Y

J0

(θ)],

(5.25)

with amplitude
α

p

J

. We assume the
radius parameter R

0

n

is proportional
to the root mean

square radius
R

n

and
R

0

p

is proportional
to R

p

, see Eqs. 5.9-
5.10.

The transition
density is then,

ρ

tr

n

(r) ≈
−α

n

J

R

0

n

d

dr

ρ

n

(r).

(5.26)

The experiment
is at a low Q

2

well below the
maximum in the inelastic form factor so one

can expand the
spherical Bessel function and integrate by parts to obtain,

20

Page 21

F

J

n

(Q

2

)

F

J

p

(Q

2

)
≈

α

n

J

N

α

p

J

Z

R

n

R

p

J

.

(5.27)

The
3

−

state has the
neutrons and protons oscillating primarily in phase (“isoscalar”),

α

n

J

≈
α

p

J

.

(5.28)

We will discuss
this in more detail below. With Eq. 5.28
the asymmetry is,

A
≈

GQ

2

4πα

√

2

4sin

2

Θ − 1
+

N

Z

R

n

R

p

J

.

(5.29)

In the limit
R

n

≈
R

p

this reduces
to,

A
≈

GQ

2

4πα

√

2

4sin

2

Θ − 1
+

N

Z

.

(5.30)

In the same
limits, plane wave and R

n

≈
R

p

, the elastic
asymmetry also reduces to Eq. 5.30.

Therefore, the
asymmetry for collective natural parity “isoscalar” excited states is similar
to

the elastic
asymmetry. This reduces the effect of the inelastic contamination.

Collective
“isovector” excitations where the neutrons oscillate out of phase from
the

protons,

α

p

J

≈
−α

n

J

,

(5.31)

have a
different asymmetry. In principle, these could be a concern. However, we believe
it

is possible to
use existing (e, e

′

) and (p,
p

′

), (p, n), etc.
cross section data to rule out large

“isovector”
strength.

This subsection
makes plane wave estimates of the asymmetry. Unfortunately, there
are

at present no
distorted wave calculations of the inelastic asymmetry. We expect
coulomb

distortion
effects to be similar for inelastic and elastic scattering because the electron
wave

functions are
the same and the plane wave asymmetries are comparable. Therefore, our
final

estimate of the
asymmetry of the 3

−

state
in

208

Pb
is,

A =
D

GQ

2

4πα

√

2

4sin

2

Θ − 1
+

N

Z

R

n

R

p

3

.

(5.32)

Here the
correction factor for Coulomb distortions D is taken from the elastic
calculations

of ref.
[24]. At 850 MeV and 6 degrees this is,

D ≈ 0.74 ±
0.26.

(5.33)

21

Page 22

We arbitrarily
assigned a 100% error to the 26% reduction from distortions because there
is

no explicit
inelastic calculation. Evaluating Eq. 5.32
for a realistic R

n

≈
R

p

+ 0.2 fm
yields,

A(3

−

) ≈
1.25A(elastic).

(5.34)

This 25 %
enhancement can be understood as follows. The neutron elastic form factor
is

reduced because
R

n

>
R

p

. Therefore the
elastic asymmetry is about 10% reduced from

Eq. 5.30

1

.
In contrast R

n

>
R

p

means that the
inelastic form factor will peak at a lower

Q

2

then the proton
form factor. Thus the Q

2

of the
measurement is slightly closer to the

neutron peak
then the proton. As a result A(3

−

) is increased
from Eq. 5.30.

At q =
0.45fm

−1

Eq. 5.32
yields,

A(3

−

) ≈ 0.83 ± 0.29
± 0.03 × 10

−6

.

(5.35)

Here the first
error is from the distortions and the second error assumes
R

n

is known to
1%.

The 0.83
magnitude is based on arbitrarily assuming R

p

= 5.5 and
R

n

=
5.7fm.

In summary, the
asymmetry for the first excited state in

208

Pb is
qualitatively similar to

the elastic
asymmetry. This reduces the size of the inelastic correction. There is
some

uncertainty
because no inelastic distorted wave calculations have been done.
However,

inelastic
contaminations are likely to be small, less then 1% of the rate, because the
elastic

cross section
at low Q

2

is large.
Therefore, the interpretation of a measurement is unlikely

to be a problem
even if one assumes very large errors for the inelastic
A.

H. Isospin
Violation

One uses
assumptions about isospin symmetry to go from various quark weak
currents

to nucleon and
eventually nucleus weak current matrix elements. First, our formalism,
and

that used by
most others, assumes good isospin in the nucleon. For example, Eqs. 5.2-
5.3

assume an up
quark matrix element in the proton is the same as a down quark
matrix

element in the
neutron. This is, no doubt, violated at some level. However,
calculations

such as ref. [10]
suggest only very small corrections. Thus we don’t expect our results to
be

impacted by
isospin violation in the nucleon.

Second there
is, of course, isospin violation in a heavy nucleus. For example the proton

radius is
different from the neutron radius. In a light
or medium mass nucleus it is often

convenient to
use an isospin formalism. This might start out with equal proton and
neutron

radii. In this
case, one must explicitly include corrections to the asymmetry from
isopin

violation.

1

In plane
wave

22

Page 23

In contrast
isospin symmetry is not very good for a heavy nucleus with N > Z.
There-

fore, in the
present paper we use a formalism which treats protons and neutrons
separately

and does not
assume good nuclear isospin. We simply allow the proton and neutron
den-

sities to be
independent. The weak charge density is calculated in Eq. 5.4
by separately

adding proton
and neutron contributions. As a result, we do not need to include any
further

corrections for
isospin violation in the nucleus.

Note,
corrections to the assumption that the weak charge density is the sum of
proton

and neutron
contributions, for example from density dependent form factors, can be
con-

sidered meson
exchange currents. Meson exchange currents have been discussed in
section

V.D and are
expected to be small on quite general grounds.

I. Target
Impurities

A practical
experiment could use a backing material to help support and cool the
target.

This could
allow a higher beam current and reduce the required beam time. However,
such

an impurity may
complicate the interpretation of the experiment. The parity
violating

asymmetry of
the impurity may not be known exactly. In addition, the impurity
may

introduce
additional problems such as low lying excited states. In this section we
discuss

one possible
composite target.

It may be
possible to make a

208

Pb target with
one or more Pb foils sandwiched between

thin diamond
foils. This may also work for Ba. The large Pb cross section will ensure
that

only a small
fraction of the counts, less then say 5%, are from Carbon. If such a sandwich
is

feasible, it
may be an elegant solution for several reasons. First the high thermal
conductivity

of the diamond
will efficiently transfer the beam power and keep the target from
melting.

Second,

12

C has a very
high first excited state, above 4 MeV, so one should not have to

worry about
excited states. Finally,

12

C is a light
nucleus with N = Z so the parity violating

asymmetry is
very simple and well known with only very small uncertainties.
Indeed,

12

C

has been used
for a Standard Model test where it was assumed the asymmetry is so
well

known that any
deviation tests the Standard Model [4].

We have
calculated the parity violating asymmetry for

12

C at the
proposed Pb kinematics

(850 MeV and
six degrees),

A(

12

C) = 0.660 ×
10

−6

,

(5.36)

this includes a
0.4% increase from Coulomb distortions and another 0.4% increase from
dif-

ferences
between the neutron and proton radii in Carbon. Equation 5.36
used relativistic

mean field
densities [29]
where the proton radius is slightly larger
because of Coulomb inter-

actions. The
Coulomb distortion correction is both small and has a very small
uncertainty.

The uncertainty
in the neutron radius correction is larger.
Nevertheless, it is very unlikely

to be orders of
magnitude larger then 0.4%. Note, a 0.4% error in a 5% impurity is
over

two orders of
magnitude away from impacting a 3% asymmetry measurement. We
conclude

23

Page 24

that the
uncertainty in the asymmetry of a

12

C backing is
two orders of magnitude smaller

then needed.
Thus such a backing should pose no problems to the interpretation of
an

experiment.

This
theoretical argument that the carbon backing does not pose a problem is
very

strong. In the
remote chance that there is still a question we note that the asymmetry
of

12

C has been
directly measured in an older Bates experiment [4].
This measurement has

a somewhat
crude error of 20 to 25 % and is at a somewhat higher
Q

2

. However, on
very

general grounds
one expects the asymmetry to scale approximately with
Q

2

. Therefore
it

is straight
forward to extrapolate the Bates measurement to our
Q

2

with an
extrapolation

error that is
probably significantly smaller then the experimental error. We note that
even

a 25%
measurement of a 5% impurity only contributes about 1% and this is smaller
then

the goal of 3%
for a

208

Pb
measurement.

If one
considers drastic deviations of the

12

C neutron
density then the Bates experiment

should be more
sensitive because its higher q is closer to one over
the

12

C radius. A
neutron

experiment
in

208

Pb will be run
at a Q

2

which is much
less sensitive to the

12

C
density

because Pb is
much bigger then C. Therefore the Bates measurement can help rule out
even

theoretically
unimaginable

12

C neutron
densities and insure that a carbon backing is not a

problem.

J. Asymmetry
Correction Conclusions

In this section
we have tried to discuss all known corrections to the parity
violating

asymmetry. We
find that an experiment determines a well defined form factor of the
weak

charge density
that can be directly compared to theoretical models. A point neutron
density

can be
determined from the measured weak charge density. Finally, the dependence on
the

surface
thickness is small so one can extract a neutron root mean square radius to 1%
from

a single
moderately low Q

2

point.

On a practical
side, contributions of excited states or target impurities should not
pose

a threat to the
interpretation of a measurement. Therefore an experiment could be
run

allowing some
inelastic contamination and using a target backing. This should reduce
the

beam time
needed.

The physics
data analysis of an experiment is summarized in Fig.
6. From the measured

asymmetry one
can deduce the weak form factor. This is the Fourier transform of the
weak

charge density
at the momentum transfer of the experiment. To deduce the weak
density

one must
correct for Coulomb distortions. This can be done accurately because the
charge

density is
known. There are now a number of independent Coulomb distortion
codes.

The weak charge
density can be directly compared to predictions of mean field or
other

theoretical
calculations. This will allow isovector interactions to be constrained. The
weak

charge density
can also be applied to atomic PNC experiments. As discussed in the
next

section this
application to atomic PNC is almost insensitive to the neutron density
surface.

24

Page 25

From the
measured weak charge density one can deduce a point neutron density by
mak-

ing small
corrections for known nucleon form factors. The uncertainty in these
corrections

from strange
quarks, the neutron electric form factor and MEC is small.

Finally from
this low Q

2

measurement of
the point neutron density one can deduce R

n

.

Because the
measurement is at low but not zero Q

2

one needs some
very mild information

on the shape of
the neutron density. The surface thickness must be known to about 25%
to

extract
R

n

to 1%. All
reasonable mean field models have a spread in surface thickness
much

less then 25%.
Therefore any mean field shape can be used to extract an equivalent
R

n

.

The physics
results of the experiment are the weak charge density, the point
neutron

density and
R

n

. The single
number R

n

accurately
summarizes the other information. How-

ever, if there
is ever a question about the very mild assumptions on the surface thickness,
or

if one wishes
to consider truly drastic changes in the surface thickness which are well
outside

the range of
present theory then one can use the neutron or weak density information
rather

then
R

n

.

FIG. 4. The
weak charge density of a heavy nucleus (schematic). Meson exchange
currents

and or nucleon
form factors can only change the density in the surface region. This is
indicated by

the dotted
line. The density in the interior is insensitive to
MEC.

25

Page 26

FIG.
5.

The charge
density in

208

Pb. The dotted
and thin dashed curves are mean field

calculations
with reasonable surface energies while the thick dashed curve is a mean field
calculation

with a very
high surface energy [29].
The experimental charge density is the solid
curve.

26

Page 27

Weak density
ρ

W

(Q

2

)

Neutron Density
ρ

n

(Q

2

)

Neutron Radius R

n

“Undo” Coulomb
distortions

(Charge density
known)

Measured
Asymmetry

G

E

n

Neutron form
fac. (ok)

G

E

s

Strange quarks
(|ρ

s

|<5)

MEC (Small
given R

n

large)

Assume surface
thickness good to 25%

Constrained by
measured:

surface
energies, charge density.

Mean Field
theory has very small spread

Atomic

PNC

Exp.

Mean

Field

+

Other

Models

Direct

application

Direct

comparison

FIG. 6.
Flowchart of the physics data analysis of a neutron
radius experiment, see text.

VI. CONNECTION
TO ATOMIC PARITY NONCONSERVATION

Parity
violating electron scattering (PVES) measurements of the neutron density
will

have an
important impact on atomic parity nonconservation (PNC) measurements. In
the

27

Page 28

future, the
most precise low energy tests of the Standard Model will require a
combined

knowledge of
neutron densities and atomic PNC observables. In this section we discuss
the

quantitative
relationship between PVES and atomic PNC. As an instructive illustration,
we

use approximate
parametrizations of the neutron density and calculate the relative
sensitivity

of both PVES
and atomic PNC to the shape of the nuclear distribution. For both
PVES

and atomic PNC,
these simplified analytical approximations agree well with the more
precise

numerical
solutions [24,10].
The sensitivity to the neutron distribution shape parameters is

found to be
approximately the same for PVES and atomic PNC, at the kinematics
where

PVES is most
feasible.

Atomic parity
violation experiments can measure the weak charge of a nucleus [9],
which

at tree level
in the Standard Model is Q

St.Mod

W

= (1 − 4
sin

2

θ

W

)Z − N. The
effect of finite

nuclear extent
is to modify N and Z to q

n

N and
q

p

Z respectively
[10],
where

q

n(p)

=

f(r)ρ

n(p)

(r)d

3

r.

(6.1)

This nuclear
structure correction involves an overlap integral somewhat similar to the
weak

form factor of
Eq. 5.1,
but here f(r) is a q-independent folding function determined from

the radial
dependence of the electron axial transition matrix element inside the nucleus.
If

the opposite
parity atomic states which mix are labelled s and p then f(r) ∝
ψ

†

p

(r)γ

5

ψ

s

(r).

We avoid
computing the absolute normalization of the electronic wavefunctions, a
calcula-

tion requiring
full many-body atomic wave function correlations, by setting f(0)=1.
The

approximations
we have already made are as follows: We treat the nucleons
nonrelativis-

tically,
ignoring weak nuclear magnetism effects, and we neglect non-nucleonic degrees
of

freedom. We
neglect terms involving the vector-electron interaction, and thus axial-
or

anapole-
nuclear interactions (Experimentally, this requires properly averaging over
hyper-

fine
transitions). The required electron s- and p- wave functions can be computed by
solving

the single
electron Dirac Equation in the presence of the nuclear charge distribution.
By

doing this, we
neglect effects of electron shielding in the vicinity of the nucleus, and
the

effects of
electronic binding energies (and thus many-body correlations) because these
are

small in
comparison with the nuclear Coulomb potential at short (fm) distances.
level

Atomic
theorists make predictions for atomic observables including a complete
many

body
computation of the axial matrix elements with proper norm [31].
To date, they have

generally
assumed an isoscalar nuclear density distribution
ρ

n

(r) =
ρ

p

(r), and
factored the

effect of
finite nuclear size into a coefficient of Q

W

. The fact that
ρ

n

=
ρ

p

means that
there

will then be a
small additive correction which should be made:

Q

expt

w

=
Q

St.Mod

w

+
∆Q

n−p

w

,

(6.2)

where
Q

expt

w

is the weak
charge extracted from atomic experiments, using atomic theory

calculations
which ignore any neutron-proton differences, and to a good
approximation,

∆Q

n−p

w

= N(1 −
q

n

/q

p

).

(6.3)

∆Q

n−p

W

is zero if
neutron and proton distributions are identical. There are additional
small

corrections [32]
to ∆Q

n−p

w

arising from
Standard Model radiative corrections, as well as

28

Page 29

additive
corrections arising from e.g. internal structure of the nucleon, but these can
be

safely
neglected since ∆Q

n−p

w

is itself so
small. Given nuclear structure model predictions

for
ρ

p

(r) and
ρ

n

(r), the
calculation of f(r) and ∆Q

n−p

w

is reasonably
straightforward, re-

quiring a
numerical solution of the Dirac Equation in the vicinity of the nucleus.
Results

from various
nuclear structure model distributions for several nuclei are given in Table I,

which shows
that neutron-proton distribution differences can affect measurements of
the

weak charge at
marginally measurable levels. The model dependent uncertainty in
∆Q

n−p

w

appears to be
comparable to the value itself, and exceeds Standard Model radiative
correc-

tion
uncertainties. This motivates improved knowledge of the neutron distributions,
since

charge
distributions are generally well measured experimentally.

For the
specific case of atomic Cs, recent measurements of transition polarizabilities
[9],

coupled with
previous measurements of parity nonconservation (PNC) [8]
have significantly

reduced
uncertainties associated with the extraction of
Q

w

(Cs). The
latest result [9],
Q

expt

w

=

−72.06(.28)

expt

(.34)

atomic
theory

is in mild
disagreement, at the 2.5σ level, with the Standard

Model
prediction of Q

St.Mod.

w

=
−73.20(.13)

theory

. [32]
The experimental number uses input

from atomic
theory calculations [33,34]
which incorporate finite nuclear size effects, but do

not include the
modification due to neutron-proton differences,
∆Q

n−p

w

. Using a
relatively

naive
approximation described below we find that, in order to reduce the contribution
of

nuclear
structure uncertainty to below the level of present atomic theory levels, one
would

need to know
the neutron radius to around ± 6%. To reduce
nuclear structure uncertainties

well below the
level (±0.13) of Standard Model radiative correction uncertainties
requires

knowledge of
the neutron radius to around ±2%. Due to the
more complex nature of

the Cs nucleus
(e.g. it is not spin 0) it is unlikely that a PVES experiment will
measure

the neutron radius of Cs directly, but measurement in a
nearby nucleus with comparable

N/Z (such as
Ba) should provide significantly improved confidence in the ability of
nuclear

structure
models to predict neutron radii in general, and hence reduce the nuclear
model

dependence of
these important Standard Model tests.

Precise
numerical codes have computed the effect of the neutron density on PVES [24].

It is
instructive to recover this result with a simpler analytical approximation, and
then to

connect this to
atomic PNC. To this end we first approximate f(r) and
∆Q

n−p

w

by
assuming

a uniform
nuclear distribution, i.e. ρ(r) a constant out to some radius C. In this case,
the

nuclear
potential energy is just

V (r) = (Zα)
∗

(−3 +
r

2

/C

2

p

)/2C

p

if r <
C

p

−1/r

if r >
C

p

,

(6.4)

neglecting
small contributions of neutrons to the nuclear charge distribution. The
single

electron Dirac
Equation can be solved in the presence of this potential by expanding
in

powers of (Zα),
making a power series for the Dirac wave functions inside the
nucleus.

2

2

We have also
calculated the (Zα)

4

corrections, as
well as terms of order (m

e

C

p

)(Zα) which
arise if

the electron
mass is left in the Dirac Equation. Including these small corrections, our
approximate

formulas
reproduce detailed numerical results for q

n

at around the
1% level.

29

Page 30

The result
is

f(r) = 1 −
(Zα)

2

1

2

(r/C

p

)

2

−

1

10

(r/C

p

)

4

+

1

150

(r/C

p

)

6

+
O(Zα)

4

(6.5)

One further
simplifying approximation can be made, that r

2

n

≈
r

2

p

,
characterizing the

difference by a
single small parameter, r

2

n

/
r

2

p

≡ 1+ǫ. The
result, after applying Eq. 6.1

is

q

p

≈

1 −
(Zα)

2

(.26),

(6.6)

q

n

≈ 1 −
(Zα)

2

(.26 +
.221ǫ),

(6.7)

∆Q

n−p

w

≈

N(Zα)

2

(.221ǫ)/q

p

.

(6.8)

This can be
compared with the PVES asymmetry in elastic electron-nucleus
scattering

for a spin-0
nucleus. In the Born approximation (and in the absence of isospin
violation)

this asymmetry
is given by Eqn. 3.11
Using the same approximations as above (uniform

distributions,
C

n

≈
C

p

) and defining
A

nom

≡
A

0

lr

(Q

2

) (1 − 4
sin

2

θ

W

) − N/Z , we
find

(A −
A

nom

)

A

nom

=

−N/Z

(1 − 4
sin

2

θ

w

−
N/Z)

F

n

(Q

2

)

F

p

(Q

2

)
−

1

(6.9)

≈ [1.06]ǫ
−

3

2

+

1

2

(qC

p

)

2

sin(qC

p

)

(sin(qC

p

) −
(qC

p

)
cos(qC

p

))

+
O(ǫ

2

)

(6.10)

For extremely
small momentum transfer, the expression in large parentheses can be
ex-

panded,
yielding −(qC

p

)

2

/10 +
O(qC

p

)

4

. Knowing
C

n

to ±1% means,
according to Eq. 6.8

that the weak
charge for lead would have an uncertainty due to neutron structure of ≈
±0.2,

to be compared
with Q

St.Mod

W

= −118.7 ± 0.2
(rad. corr). According to Eq. 6.10,
at q=0.45

fm

−1

, measuring
C

n

to 1% requires
an asymmetry measurement with errors around the 3%

level, in
agreement with the numerical results obtained in ref [24].

The
approximation scheme described above can be extended to include rough effects
of

the nuclear
shape using the method of Sandars [35],
adding a “thin edge” to the uniform

distribution,
parameterized by a new skin-thickness parameter η, defined for any
arbitrary

distribution
by

η
=

21
r

4

25
r

2
2

−
1.

(6.11)

This form is
chosen so η = 0 for a uniform distribution. Typically,
η

proton

≈ 0.10 for
a

nucleus such as
lead. The presence of a thin edge changes all the moments:

r

n

≈

3

n +
3

C

n

1 +
η

n(n +
3)

8

,

(6.12)

from which
r

2

=
C

2

(3/5)(1 +
1.25η) serves to define C for any distribution. Adding such

a “thin skin”
to the protons, the charge distribution is unchanged except in a small
region

30

Page 31

near C, and our
approximation for f(r) is still fairly accurate. The presence of the
thin

skin does
slightly modify the potential inside the nucleus, which in turn modifies f(r)
in

a well-defined
way. Adding a thin skin to the neutrons as well, assuming the
difference

∆η ≡
η

n

−
η

p

≪ 1, Eq. 6.8
is then modified to

∆Q

n−p

w

≈
N(Zα)

2

.22 (ǫ −
0.16∆η)/q

p

.

(6.13)

The
insensitivity of the atomic observable to higher moments beyond the rms radius
is seen

from the small
relative coefficient of ∆η.

To connect to a
more familiar measure of skin thickness, consider a Wood-Saxon form

for the
density, with radius and thickness parameters c and z, as given in equation 5.21.

An analytic
series expansion as z/c → 0 gives η ≈

4

3

π

2

(z

2

/c

2

)(1 −
π

2

z

2

/c

2

) + ...,
r

2

=

3

5

c

2

(1
+

7

3

π

2

z

2

/c

2

), and
C

2

≈
c

2

(1
+

2

3

π

2

z

2

/c

2

+ . . .) In
this way, we could eliminate ∆η in

favor of ǫ and
∆z/z, where e.g. ∆z ≡ z

n

−
z

p

is a difference
in Wood-Saxon neutron and

proton
thickness parameters, and as stated above, ǫ = r

2

n

/
r

2

p

− 1. Evaluating
f(r)

numerically in
the presence of this potential using the analytic series expansion for
moments

and linearizing
in ǫ and ∆z,

∆Q

n−p

w

(Pb) ≈ 10.6ǫ −
.37∆z/z,

(6.14)

while for
barium

∆Q

n−p

w

(Ba) ≈ 3.3ǫ −
0.13∆z/z.

(6.15)

In these
expressions, higher order effects in (Zα), as well as finite surface thickness,
have been

taken into
account numerically. (We use as nominal inputs [36]
c

P
b

= 6.624,
z

P
b

=
0.549,

c

Ba

= 5.700,
z

Ba

=
0.5314)

The PVES
asymmetry also gets modified by a thin skin, and Eq. 6.10
gets an additional

correction
term,

(A −
A

nom

)

A

nom

≈ Eq. 6.10 +
[1.06]∆η

15

8

+
(qC

p

)

2

qC

p

cos(qC

p

) − 6
sin(qC

p

)

8(sin(qC

p

) −
(qC

p

)
cos(qC

p

))

(6.16)

(neglecting any
O(η

p

) corrections
here.) In the limit of small momentum transfer, the term

in large
parenthesis goes to (qC

p

)

4

/280. The PVES
asymmetry thus becomes completely

insensitive to
∆η at small Q

2

, as one would
expect, but at larger Q

2

the surface
shape

becomes
relatively more important. Table II
shows the ratio of the coefficients of ∆η to

ǫ in Eq 6.16
as a function of momentum transfer. (Note that these numbers
implicitly

incorporate the
approximation η

p

→ 0.)
Incorporating finite skin thickness with a Wood-

Saxon for the
nucleon distributions, the asymmetry can be calculated using
asymptotic

expansion
formulas in c and z, and the result linearized in the small quantities ǫ and
∆z.

The resulting
formula is not especially illuminating, but the coefficients of this
expansion

are shown for
lead as a function of momentum transfer in Table III.

31

Page 32

Note that there
is a unique momentum transfer where the two observables are
sensitive

to the same
linear combination of neutron radius and
surface shape. For lead, assuming

a thin edge
(Table II),
and comparing with Eq. 6.13, this “matchup” is around q = 0.34

fm

−1

. Using the
Wood-Saxon form to incorporate the effects of finite skin thickness,
and

comparing with
Eq. 6.14,
the relative coefficients of ∆z/z and ǫ are matched for electron

scattering and
atomic PNC at q ≈ 0.32 fm

−1

. At this
kinematics point, a measurement of

PVES is
“optimized” to provide the direct information desired for the atomic
observable.

In the absence
of other constraints, this would be the optimal momentum transfer for
a

PVES
measurement if the goal is the most direct measurement of atomic weak
charge

corrections,
rather than a desire to extract and measure the details of the neutron
shape

distribution.
Of course, this is only true to the extent that still higher moments
(shape

differences
beyond the simple thin edge approximation) are not important, which is a
decent

approximation
for the atomic observable, but not so good for the PVES asymmetry. A
brute

force estimate
obtained by assuming a generalized three parameter Gaussian form for
the

nuclear
distributions, allowing the 3 parameters to vary but constraining them to
produce

values of A
within some small window of A

nom

, and then
calculating the corresponding

spread in
∆Q

n−p

w

, we find an
optimal q value of 0.32 fm

−1

. (All these
results, however, have

been calculated
at tree level, without Coulomb corrections.)

Another way of
understanding the above result is to compare the function
multiplying

ρ

n

(r) in the form
factor integral of Eq. 5.1
(i.e. j

0

(qr)) and in
the convolution integral of

Eq. 6.1
(i.e. f(r)). Since we are not interested in the volume integral of the weak
charge

density, which
is well known, we can subtract both f(r) and j

0

(qr) from 1,
and plot the

remainder to
study the relative sensitivity to the radius and to the surface thickness.
This

is shown in
figure 7.
For q = 0.30 fm

−1

, 1 −
j

0

(qr) is nearly
proportional to 1 − f(r), and

thus at this
Q

2

one is
sensitive to the same ratio of surface and radius in a JLAB parity

experiment and
in an atomic experiment. The curves for q = 0.45
fm

−1

are also shown,
in

this case the
curves are not identical but are similar enough so much of the error from
an

unknown surface
thickness cancels when comparing the two integrals.

For lead,
ignoring the effects of skin thickness, we found above that the rms neutron

radius should be
known at roughly the ±1% level to ensure that neutron structure
uncertain-

ties are
smaller than present Standard Model radiative corrections uncertainties
(roughly

±0.2, or about
±0.16%, for the weak charge.) Again ignoring thickness, table II
shows that

this would
require e.g. a PVES asymmetry measurement at q=0.45
fm

−1

at the ±3%
level.

Including the
effect of thickness, the linear combination of ǫ and ∆z/z required for the
weak

charge is not
quite the same as the linear combination measured in PVES at arbitrary q,
but

a linear error
propagation at q = 0.45 shows that as long as the relative uncertainty in
z

n

is

less than ≈
±50%, the additional uncertainty due to including skin thickness is
negligible.

Thus, a single
PVES measurement taken even at a kinematics point which is not
perfectly

optimized for
the atomic observable will still be sufficient to eliminate nuclear
structure

effects from
the atomic observable at levels below present Standard Model
uncertainties.

32

Page 33

TABLES

TABLE I.
Nuclear structure model predictions for ∆Q

n−p

q

. HFB stands
for Hartree Fock

Bogolyubov, Skl
stands for a Skyrme “SLy4” parameter set. G1 is a parameterization from
a

relativistic
nuclear model. The Standard Model value and uncertainty are from Ref. [32].

Model

Element

Q

St.Mod

w

∆Q

n−p

w

Gogny [37]

Pb

-118.70(0.19)

0.47

HFB-Skl [38]

“

“

0.57

G1 [39]

“

“

1.0

Gogny

Ba

-77.07(.13)

0.14

HFB-Skl

“

“

0.18

G1 [39]

“

“

0.30

TABLE
II.

Table of the
coefficients occurring in Eq. 6.16,
written in the form

(A −
A

nom

)/A

nom

=
c

1

(ǫ +
c

2

∆η), as a
function of momentum transfer, q, for a

208

Pb
target.

(ǫ measures the
n-p rms radius difference, and ∆η measures the n-p surface shape
differences)

Note that Eq.
6.16
is derived assuming η

p

→
0.

q
(fm

−1

)

c

1

c

2

0.2

-.21

-.068

0.25

-.33

-.11

0.3

-.50

-.16

0.35

-.72

-.22

0.4

-1.0

-.30

0.45

-1.4

-.40

0.5

-2.1

-.52

33

Page 34

TABLE
III.

Table of the
coefficients a

1

and
a

2

occurring in
the expression

(A −
A

nom

)/A

nom

=
a

1

ǫ +
a

2

∆z/z, as a
function of momentum transfer, q, for a

208

Pb
target.

The
coefficients of ǫ do not match those in Table II
for two reasons: finite thickness is incorporated

here
numerically, using the same nominal inputs as for Eq. 6.14.
Also, note that ∆z/z is itself a

function of
both ǫ and ∆η.

q
(fm

−1

)

a

1

a

2

a

2

/a

1

0.2

-.23

.003

-0.013

0.25

-.37

.008

-0.021

0.3

-.56

.017

-0.031

0.35

-.80

.034

-0.042

0.4

-1.1

.064

-0.056

0.45

-1.6

.12

-0.073

0.5

-2.3

.21

-0.094

34

Page 35

0

2

4

6

8

10

r
(fm)

0

0.5

1

1.5

1-j

0

(qr)

10ρ

1-j

0

(qr),
q=.3

1-j

0

(qr),
q=.45

6(1-f(r))

4.1(1-f(r))

FIG. 7.
Approximate nuclear weak density ρ(r) for Pb, along with the function
multiplying

ρ(r) in the
integrals for the weak form factor (namely, j

0

(qr)) and in
the atomic correction factor q

n

(namely f(r)).
In both cases the function is subtracted from one to eliminate the volume
integral

of the weak
charge density. Curves for 1 − j

0

(qr) are shown
for two different values of q in fm

−1

.

Note
4.1(1-f(r)) is almost identical to 1 − j

0

(qr) for
q=0.30fm

−1

.

VII.
CONCLUSION

With the advent
of high quality electron beam facilities such as CEBAF, experiments
for

accurately
measuring the weak density in nuclei through parity violating electron
scattering

(PVES) are
feasible. The measurements are cleanly interpretable, analogous to
electromag-

netic
scattering for measuring the charge distributions in elastic scattering. From
parity

violating
asymmetry measurements in elastic scattering, one can extract the weak
density

in nuclei after
correcting for Coulomb distortions, which have been accurately
calculated

[24].

By a direct
comparison to theory, these measurements test mean field theories
and

other models
that predict the size and shape of nuclei. They therefore can potentially
have

a fundamental
and lasting impact on nuclear physics.

Furthermore,
PVES measurements have important implications for atomic parity
non-

conservation
(PNC) experiments which in the future may become the most precise tests
of

the Standard
Model at low energies. We have shown that to a good approximation,
suffi-

cient for
testing the Standard Model, the dependence on nuclear shape parameters enters
the

35

Page 36

PVES and PNC
observables the same way; therefore, the PVES measurements are
directly

applicable to
the interpretation of atomic PNC if measured on the same nucleus.

Measurements of
the weak density lead to the neutron density distribution with
unprece-

dented
accuracy. As we have discussed in this paper, PVES yield significant
improvement

in the accuracy
of neutron densities compared to hadronic probes or magnetic
scattering.

We have shown
that the corrections due to strange quarks, neutron electric form
factors,

parity
admixtures, dispersion corrections, meson exchange currents, and several other
possi-

ble effects in
realistic experiments are all small. Further, an asymmetry measurement
from

a heavy nucleus
with 3% accuracy will both establish the existence of the neutron skin
and

characterize
its thickness. The neutron skin is an important, qualitative feature of
heavy

nuclei which
has never been cleanly established in a stable nucleus.

36

Page 37

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38