O. Moreno and T. W. Donnelly MITweb.mit.edu/lns/PEB_Workshop/talks/donnelly_peb.pdf · MIT...

O. Moreno and T. W. DonnellyMIT

2MIT Workshop: March 2013

Two basic references were used for most of this study:

T. W. Donnelly, J. Dubach and I. Sick, Nucl. Phys. A503 (1989) 589 (DDS)

This was where isospin mixing was explored and where theidea of using PV electron scattering to measure the radius ofthe neutron distribution was introduced (e.g., PREX, CREX, etc.).

O. Moreno, P. Sarriguren, E. Moya de Guerra, J. M. Udias, T. W. Donnellyand I. Sick, Nucl. Phys. A828 (2009) 306 (Moreno et al.)

Here the original ideas of DDS were extended using moresophisticated nuclear modeling, specifically self-consistent deformed Skyrme HF mean field theory with BCS pairing,and strangeness (which was absent in DDS) was included.


Given the possibility of making PV asymmetry measurements at levels of uncertainty better than 1%, the purpose of thepresent study is to provide some evaluation of the impact ofnuclear structure uncertainties on the problem.

These include (at least) the following:

• Effects from Coulomb distortion• Effects from isospin mixing• Impact of MEC effects• Electric and magnetic strangeness effects• Elastic versus inelastic scattering differences

when measurements are performedat low resolution


The PV asymmetry has the form:

where the hadronic responses are

The notation AV and VA denotes the leptonic-hadronic WNC couplings

SM values:

aA = -1aV = 4sin2 θW – 1

= - 0.08


Let us start by considering elastic scattering from 12Cwhere, since the ground state is 0+, only C0 multipoles can occur.

In the absence of isospin mixing (assuming the ground state of carbonhas good isospin T=0 and therefore no isovector C0 matrix elements) andin the absence of strangeness contributions (and therefore no strangenessC0 matrix elements) one has the following:

… and hence the hadronic ratio is just a constant

Feinberg (1975)

WNC form factor


Clearly, in the kinematic region studied, namely energy 150 MeVand momentum transfers from 0.2 to 0.8 fm-1, including the rangeof angles 25o to 45o which may be of interest for future measurements,variations in the predicted PV asymmetry itself are hard to see

… and hence the results to follow are all given in the following form.When examining the asymmetry with some effect present versus whenthat effect is absent one can write

… to focus on the deviation from unity, Γ.


Elastic PC electron scattering yields (with some level of uncertainty)the ground-state charge distribution, from which one can obtain thestatic Coulomb potential. The electron scattering states should becomputed by solving the Dirac equation using this potential (DWBA).

To assess the size of this Coulomb distortion effect, as an approximation,we used the 3-parameter Fermi distribution

… and, maintaining the integral, varied the parameters c or d so that the measured rms charge radius fell in the accepted experimental range

Let us begin with Coulomb distortion.


Next, we looked at isospin mixing coming from having a Coulombinteraction between protons in the nucleus

(Note: there could be other sources of isospin mixing)

[For references see DDS and Moreno et al.]

Several variations of the nuclear ground-state modeling were considered:

•Various frequently used Skyrme interactions•Effects from changing the amount of pairing•Different amounts of deformation of the 12C ground state


(1) MEC contributions of non-strange type will cancel in asituation where there is no isospin mixing and no strangeness.

(2) With isospin mixing but no strangeness, the isoscalar and isovector nuclear matrix elements are modified in differentways, typically at the 10% level or less at low momentum transfers, and so the curves above may change by roughlythis amount (which is small at low-q, since the isospin mixingeffects amount to a few parts in 1000).

(3) With isospin mixing and strangeness in the nucleons(i.e., 1-body strangeness) the dominant uncertainties comefrom strangeness, as discussed below.

(4) There can be exotic MEC effects involving strangenesscontent not in the nucleons, but in the exchanged mesons(i.e., 2-body strangeness), see M. J. Musolf, R. Schiavillaand T. W. Donnelly, Phys. Rev. C50 (1994) 2173)… these are hard to evaluate, since that study was for 4He;it was found that these exotic MEC effects are expected to besmaller than the conventional MEC effects at low-q.

MEC effects


The electric and magnetic strangeness form factors were parametrizedin the following way:

Note: magnetic strangeness is a small effect at low-q for Coulombmultipoles, since it only enters in the relativistic spin-orbit term;however, for inelastic transitions where transverse magnetizationcurrents are important, it can be very important.

Next, strangeness was considered.

… and for the strangeness parameters we used a recent analysis ofPV ep scattering in R. Gonzalez-Jimenez, J. A. Caballero and T. W. Donnelly, Physics Reports 524 (2013) 1.


δA/A = 1%


Magnetic strangeness from therelativistic spin-orbit contribution


… and also variations from using different EM form factors… here Mainz data


Finally, consider what happens when the resolution is not good enoughto resolve the ground state, and thus where a sum over some set ofexcited states must be included.

One has the following for the PV asymmetry, summing over a set of states:

with weighting factors involving the EM cross sections for the individual states

where

and thus


12C


Consider a situation where all states below the T=1 15.11 MeV state must be summed

Starting point: Given (1) no isospin mixing (no isoscalar/isovector differences)(2) no strangeness (implying only non-strange isoscalar currents enter)(3) tree level WNC couplings (no isoscalar axial-vector coupling, i.e. in T’)

… then all of the states (being T=0) have the same asymmetry as the groundstate and so the answer for the total is the same

However, with strangeness (and isospin mixing) the dependences in the L and T contributions (still assuming no T’ contributions) are different.

For example, in the T responses there are both convection and magnetization currents and so both electric and magnetic strangeness effects occur at leading order


A(el+inel with s)/A(el with s) – 1

inel = sum over 1p-shell ph T=0 statesusing extremes of strangeness


Brief Summary

Effect at low-q Size of effect Uncertainty(%) (%)

Elastic scattering from 12C

Coulomb distortion 3 0.01Isospin (EM) mixing 0.4 0.05 ?MEC contributions < 0.1 ? < 0.1 ?Electric strangeness 0 - 1 1Magnetic strangeness 0 - 0.05 0.05

Inclusion of excited states

Inelastic effects (T=0) 0 - 1 ? ?(= 0, if no isospin mixing and no strangeness at tree level)

Energy: 150 MeVAngle: 35o

… clearly electric strangeness currently has the largest uncertainty,and going to lower values of q (perhaps using higher energies and smaller angles) would lower this.

O. Moreno and T. W. Donnelly MITweb.mit.edu/lns/PEB_Workshop/talks/donnelly_peb.pdf · MIT...

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