b_{1} technical note 2013-05 May 2013

Diluting Asymmetries in Extracting b_{1}^{d} from a measurement of A_{zz}

E. Long (UNH), O. Rondon (INPP-UVA)

Abstract

The PR12-13-011 proposal to seeks to extract the deuteron structure function b_{1}^{d} from a measurement of A_{zz}. To leading order with an unpolarized electron beam, A_{zz} can be

measured by taking a ratio of electrons scattered from a tensor-polarized deuterium target and electrons scattered from an unpolarized target. However, higher order asymmetry terms come into effect when the beam is polarized. This document includes these higher

order terms into the equations used in the b_{1}^{d} proposal.

1 Asymmetries

The measurement that the PR12-13-001 proposal is promoting is a ratio of the number of electrons scattered off of tensor-polarized deuterium. Assuming an unpolarized beam, to leading order this ratio takes the form of

\frac{N_{P ol}}{N_{u}} − 1 = f \frac{1}{2}A_{zz}P_{zz},  (1)

where N_{P ol} is the number of events scattered from a tensor-polarized target, N_{u} is the number of events scattered from an unpolarized target, f is a dilution factor, A_{zz} is the quantity being extracted, and P_{zz} is the tensor polarization.

A number of other quantities come into play when the beam is polarized, such that its helicity is parallel to the deuteron spin direction, and when higher order effects are considered. This includes A_{V}^{d}, A_{∥} = D(A_{1} + ηA_{2}), and A_{T}^{ed} from equations 25-28 in [1] as well as the

parity-violating asymmetry A_{EW}, where η = ^{\frac{ϵ\sqrt{Q^{2}}}{E−E′ϵ}}, ϵ is the longitudinal virtual photon polarization, D = (1 − ϵ\frac{E′}{E})/(1 + ϵR), and R = \frac{σ_{L}}{σ_{T}}. This gives the form of the counted events when the target is tensor-polarized as

N_{P ol} = A[L_{He}σ_{He}^{u}+L_{N}σ_{N}^{u}+L_{D}σ_{D}^{u} (1 + A_{∥}P_{b}P_{z} + A_{T}^{ed}P_{zz}P_{b} + A_{V}^{d} P_{z} + \frac{1}{2}A_{zz}P_{zz}) ]t_{P ol}, (2)

and when the target is unpolarized as

N_{u} = A (L_{He}σ_{He}^{u} + L_{N}σ_{N}^{u} + L_{D}σ_{D}^{u}) (1 + A_{EW} P_{b}) t_{u}.  (3)

Taking the ratio of these types of events, which will be measured during the experiment, leads to a derivation of

\frac{N_{P ol}}{N_{u}} = \frac{A [L_{He}σ_{He}^{u} + L_{N}σ_{N}^{u} + L_{D}σ_{D}^{u} (1 + A_{∥}P_{b}P_{z} + A_{T}^{ed}P_{zz}P_{b} + A_{V}^{d} P_{z} + \frac{1}{2}A_{zz}P_{zz})] t_{P ol}}{A (L_{He}σ_{He}^{u} + L_{N}σ_{N}^{u} + L_{D}σ_{D}^{u}) (1 + A_{EW} P_{b}) t_{u}}

(4)

\frac{N_{P ol}}{N_{u}} = (\frac{t_{P ol}}{t_{u}} ) [\frac{L_{He}σ_{He}^{u} + L_{N}σ_{N}^{u} + L_{D}σ_{D}^{u} (1 + A_{∥}P_{b}P_{z} + A_{T}^{ed}P_{zz}P_{b} + A_{V}^{d} P_{z})}{(L_{He}σ_{He}^{u} + L_{N}σ_{N}^{u} + L_{D}σ_{D}^{u}) (1 + A_{EW} P_{b})}

+ \frac{L_{D}σ_{D}^{u}}{(L_{He}σ_{He}^{u} + L_{N}σ_{N}^{u} + L_{D}σ_{D}^{u}) (1 + A_{EW} P_{b})} \frac{1}{2}A_{zz}P_{zz}] .

(5)

We can condense this by introducing a dilution factor,

f = \frac{L_{D}σ_{D}^{u}}{L_{He}σ_{He}^{u} + L_{N}σ_{N}^{u} + L_{D}σ_{D}^{u}} ,  (6)

and assuming that t_{P ol} ≈ t_{u}, such that

\frac{N_{P ol}}{N_{u}} = \frac{1}{(1 + A_{EW} P_{b})}[1 + f (A_{∥}P_{b}P_{z} + A_{T}^{ed}P_{zz}P_{b} + A_{V}^{d} P_{z}) + f \frac{1}{2}A_{zz}P_{zz}].  (7)

We can use this to determine A_{zz} by

A_{zz} = \frac{2}{fP_{zz}} [(1 + A_{EW} P_{b}) (\frac{N_{P ol}}{N_{u}} ) − 1 − f (A_{∥}P_{b}P_{z} + A_{T}^{ed}P_{zz}P_{b} + A_{V}^{d} P_{z})] . (8)

If an unpolarized electron beam is available, P_{b} = 0 and A_{zz} is greatly simplified to

A_{zz} = \frac{2}{fP_{zz}} [\frac{N_{P ol}}{N_{u}} − 1 − fA_{V}^{d} P_{z}] .  (9)

References

[1] W. Leidemann, E.L. Tomusiak, and H. Arenhovel. Inclusive deuteron electrodisintegration with polarized electrons and a polarized target. Phys.Rev., C43:1022–1037, 1991.