Parity violation in electron-deuteron scattering. II. Break-up channels

ANNALS

OF PHYSICS

137, 378-440

(1981)

Parity Violation in Electron-Deuteron II. Break-up Channels* W-Y.

P. HWANG,

E. M. HENLEY,

Institute for Nuclear Theory, University of Washington, Received

Scattering.

AND GERALD

A. MILLER

Department of Physics, FM-IS, Seattle, Washingion 98195 June 25, 198 1

Parity violation in electron-deuteron inelastic scattering is decribed. An impulse approximation, modified to incorporate gauge invariance, is employed. Additional meson-exchange currents are included. Normal-parity and abnormal-parity wave function components are generated numerically with a Reid soft-core potential for the former and a general parityviolating weak potential with adjustable coupling constants for the latter. Numerical results for parity-conserving differential cross sections are in good agreement with existing data. For low n-p excitation energies and medium-energy electrons, we find that parity-violating asymmetries are dominated by contributions from neutral weak currents so that the Weinberg-Salam theory can be tested. For low-energy electrons, 5 MeV SE, 6 50 MeV, our results indicate that the asymmetry caused by nuclear parity violation is roughly as important as that due to neutral weak currents. The pion-nucleon parity-violating coupling,f,, as well as the rho- and omega-nucleon parity-violating couplings, may be determinable from such experiments. Further, it is possible to check the experiment of Lobashov et al., which detects circular polarization in the thermal-neutron capture reaction.

1. INTRODUCTION The feasibility of using the scattering of polarized electrons to elucidate the detailed structure of the electronic and hadronic weak neutral currents has been demonstrated by the well-known SLAC-Yale experiment [ 11. The parity violation observed in these experiments at large four-momentum transfers and for high-energy electrons is in agreement with the Weinberg-Salam (WS) theory [2]. But these findings do not help to determine the isoscalar axial neutral current, which is expected to be very small [3], nor do they tie down precisely the other three couplings: the vector isoscalar and isovector and the axial vector isovector ones. Nuclear targets and medium- and low-energy electrons can till in to determine these *This report was prepared as an account of work sponsored by Neither the United States nor the United States Department of Energy, any of their contractors, subcontractors, or their employees, makes any assumes any legal liability or responsibility for the product or process use would not infringe privately owned rights. The U.S. Government’s right to retain a nonexclusive royalty-free covering this paper, for governmental purposes, is acknowledged.

378 0003.4916/81/140378-63$05.00/O Copyright 0 198 1 by Academic Press, Inc. All rights of reproduction in any form reserved.

the United States Government. nor any of their employees, nor warranty, express or implied, or disclosed, or represents that its license

in and to the copyright

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weak couplings. At low energies, it is necessary to include asymmetries caused by nuclear parity violation, which is of some theoretical interest in itself (41. In a preceding paper (51, referred to as I henceforth (see Erratum to I at the end of this article), we have described a methodology for theoretically investigating parity violation in both the elastic and break-up channels of electron-deuteron scattering, restricting ourselves to energies in which non-relativistic expansions are allowed for nucleons. It was pointed out that the standard impulse approximation (NOIA) needs to be improved to incorporate gauge invariance (GI). More specifically, the application of the convection current in the NOIA leads to predictions which often grossly violate GI constraints. Thus, in order to study parity violation in deuteron electrodisintegration, an explicit modilication of the NOIA to incorporate GI is required. A model for the strong NN interaction, e.g., the Reid soft-core potential ] 6 ]. is needed for numerical predictions. The plan of this paper is as follows: In Section 2, we set up the notation for a description of deuteron electrodisintegration. We then describe a modification of the NOIA to restore GI. This modification includes some meson exchange currents (MEC) effects. Additional meson exchange currents, which are taken into account explicitly in our calculation, are briefly discussed as well. In Section 3, we compare the results obtained by our method for differential cross sections in the absence of parity violation with experiments. In Sections 4 and 5, we report separately our calculated asymmetries for parity violation caused by neutral weak currents and those induced by nuclear parity violation. Although the measured asymmetries are the sum of these two contributions, our results indicate that, in the medium energy regime [ 100 MeV 5 E, 5 500 MeV and E,,, 5 10 MeV with E, and E,,, the incident electron energy and (np) relative energy], the asymmetry is dominated by that due to neutral weak currents. Thus, such an experiment can be used to test the WS theory. In the low-energy regime [5 MeV 5 E, 5 50 MeV and E,,, 5 1 MeV], the asymmetry caused by nuclear parity violation [PV], is comparable to, but can, in principle. be separated from. that due to neutral weak currents. Our detailed analyses indicate that a very precise experiment can significantly aid in placing constraints on the individual coupling constants. In the low-electron-energy region it is also possible to check the experiment of Lobashov et al. [7], which detects circular polarization in the thermal-neutron capture reaction, n + p + D + y. Most of the technical details, in particular, those formulas which relate the various covariant form factors to wave functions, are collected in appendices. In contrast to the existing literature [8, 91, we have adopted a realistic and comprehensive approach to the problem. We generate numerically both the normalparity and abnormal-parity wave functions for the Reid soft-core potential together with a general parity-violating weak potential (with adjustable p.v. coupling constants). We then use these wave functions to compute the various covariant form factors. We believe that our results can be employed with some confidence in the planning of parity violation experiments in electron-deuteron scattering.

s95/131/2

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AND

MILLER

2. THEORY

For the deuteron electrodisintegration dp,, se> + D(P”‘,

reaction described by P) + e(pL, $1 + F(pw, P),

(1)

the double differential cross section da/dEL do, and the parity-violating asymmetry &’ for the scattering by electrons polarized in the direction n* parallel or antiparallel to the incident electron’s momentum are given, in the laboratory frame, by p(-J

G

2::.

= ($)*q,]p,.,]

(8E:sin’$)-‘D;

~ ~ Ll’2’u(n* * si = 1) - LP’o(n* . 6, = -1) d’2’o(n^.e^,=1)+d’2’o(n”.e^i=-1) &+a =

(2)

= ~(z, + I 9

Gs2 27ta2112 I

d(‘) = - 2i?/D,

(3)

with

Here qA s (p”’ - pW)A = (p: - pe)*, 6, z p,/] pe/, g, = p:/\ pi 1, and cos 6 = Si +t?,. a and G are, respectively, the tine-structure and Fermi coupling constants, g, and g, characterize the electron weak neutral currents, s,, sh,, cCi), and CW describe the internal degrees of freedom related to angular momentum (J), and pre, and L?,,, are,

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381

respectively, the relative three-momentum and the relative solid angle of the (np) system. Further, .F.i, .Px’, and ,PJp’, and ,?A are described by’

where Jn(x), N’,v’(x), and Ivli”‘(x) are, respectively, the hadronic electromagnetic current, the hadronic polar neutral weak current, and the hadronic axial neutral weak current. We write

Ay(x) = h,Z:3’(x) + h; Y.\(X), Njp’(x)

= hAz.y3’(X)

+ h’A Y;(x).

(6)

with Z’,(x), Y.3(x), I’j:‘(x), and V,(x) the isovector polar, isoscalar polar, isovector axial, and isoscalar axial currents, respectively. In the WS theory, the coupling constants g,. g,, h,, h:, h,, and hi are related to the Weinberg angle 8, (see 1I. We shall analyze the asymetry .cjcz’ caused by neutral weak currents as functions of these coupling constants with special emphasison tests of the WS theory. To calculate the double differential cross section and the asymmetry, we find it useful to decompose the calculation into two steps: For a given (np) final state of fixed angular momentum and parity, we first parametrize the matrix elements ./r,. 7 /‘(V’ and 7y’ in terms of Lorentz covariant form factors as in the so-called /.I’ Al “elementary-particle” treatment (EPT) [ 10, 111. Subsequently, we invoke a dynamical model, primarily the “nucleon-only” impulse approximation (NOIA) modified so that gauge invariance (GI) is satisfied, to compute these covariant form factors. As pointed out in I, GI gives rise to constraints which are generally violated by the NOIA 112, 131. We make use of Siegert’s theorem [ 141 to restore GI to the NOIA. This procedure is justifiable to the extent that meson-exchangecurrents (MEC) do not modify the NOIA charge density in the long-wavelength limit 11q/ --$0 1 115. 16I. We write

(J&)1 NOIA/GI = iJ&)

’ See I for further

notation.

twm

3

(7a)

382

HWANG,

HENLEY,

AND

MILLER

I d3x e-iq’x&~ 1 - {J(x)}~~~~~, 3 I

dSX e-‘9’“E* *I

. {J(~)hm

(7c)

Here fi and p are, respectively, the Hamiltonian and the kinetic energy operator, E, 3 3’ = q/]ql, and E*, = ~2-“‘(3 f ij?‘) with $‘,p’, z^’ three orthogonal unit vectors. To obtain Eq. (7b) we use E;/LePiqex = (i/]q 1)Ve-iq’” and apply the continuity equation after integration by parts. The derivation of Eq. (7~) has been given in Ref. [ 121. We refer to the scheme defined by Eqs. (7a)-(7c) as the “nucleon-only impulse approximation constrained by gauge invariance (NOIA/GI).” We can rewrite Eqs. (7a) and (7b) immediately as follows:

The CM coordinate is factored out automatically for the second term in the righthand side of Eq. (7~) if each constituent nucleon coordinate is expressed relative to the CM position. We then carry out the integration over d3x by making use of the Sfunctions in the NOIA charge density. For the (np) system, we can rewrite Eq. (7~) in the following form:

(8~)

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383

x /-f$[(L+l)jL(!$)-~jL+,j-k$)J + - 191 -(L + 3)j, (~)+(~)j~+~~~)]!.(8d) 4mP [

G,(K q,i I=

2;;+++l;’

)“‘(+

$

( x [L(L+

1) (q)j-FJ

jr.(y),

Ok)

@f) Here o = V, S, with e,(q2) and es(q2) the nucleon isovector and isoscalar charge form factors, respectively [see I]. v/& rci)) and v,(r’, 4”‘) are the deuteron and scattering-(np) wave functions with r’s YP- r7,. Further, the “conditional” summation over L, as denoted by L/o in Eq. (8c), is understood as a sum over even L when w = S and a sum over odd L when o = V. With the NOIA expressions characterized in detail in I, Eqs. (8a)-(8f) can be employed to derive the various covariant form factors. In general, a covariant form factor F(q’) [we suppress the dependence on (p”‘)’ and (pc”)‘] can be represented as a sum of several contributions: F(q2) = m*)lNOIA/G* lF(q*)/

NOIA/GI

=

(F(q2)/NOIA

+ F((12h4EC + 6F(q2).

+ im*NEM, (9)

One must take care to avoid double counting since {F(q2)}N0,AIG, includes some meson-exchange current contributions. In our work, the electromagnetic contribution is due solely to the magnetic interaction between the two nucleons, which ~m2Ll is GI and is not included in NOIA/GI. For the purpose of the present paper, ~F(q2NE31 is expected to be negligible. Further, in view of the dominance of the final (np) IS,, state (subchannel a), “additional” MEC are neglected in all other final-state channels. An advantage for the study of deuteron electrodisintegration is that there are three kinematic variables which can be selected-the incident electron energy E,, the relative energy E,,, of the (np) system, and the electron scattering angle B. For small E rely it is only necessary to include the IS, and 3P,.,.,(np)-scattering states [see I]. In

384

HWANG,HENLEY,AND

MILLER

this paper, we make and justify this approximation. Since the covariant form factors in these four channels are to be used frequently, we list their definitions in Appendix A. To summarize our formalism, we restrict our attention to sufficiently small E,,, so that the transition into the IS, state is dominant. We then use the NOIA/GI to calculate the contributions from the (np) ‘S, as well as those from the 3P0,1,2 states. Additional MEC contributions to the Q channel [‘S,] are included as in the work of Lock and Foldy [ 171. No double counting occurs since, in this channel, the NOIA is identical with the NOIA/GI [ i.e., SF”,(q2 ) = 01. In our formalism, the GI constraints derived in Z are satisfied and none of the form factors become singular in the limit of zero four-momentum transfer.

3. DIFFERENTIAL CROSS SECTIONS WITHOUT PARITY VIOLATION

In this section, we apply our formalism to calculate differential cross sections in the absence of parity violation. The purpose of this investigation is to compare our results with experimental measurements and to justify quantitatively the approximation that only the transitions into the ‘S,,(a) and 3PJ C/3, y, 8 for J = 0, 1,2, respectively) (np)-scattering states need to be included. (3a) Formulae To calculate the double differential cross section du/dE: dl2, the quantity D in Eq. (2) can be expressed in terms of the various covariant form factors (Appendix A) as follows: D 2 D(‘S,)

+ c

D(3PJ),

D(‘S,) = f $ lG(,2>12 P X [(E, + E:)2 -E,E:(l D(3PO) = f

[ 2 1F$(q’)[‘(

+ cos 8)](1 - cos 0),

1 - cos 0)

+ FtXq’> -+l+$)2]~l+cos69j, +lF%q2)12 ,“,: D(‘91)=$

I

2

I(

q(q’)

-$-

P

mz2>

(lob)

2(1)I

cose)

(1Oc)

PARITY

+

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G(q2)+Wq2)

+ I((1 +~)~(q’)+t~(q’)+~(q’)) WP2)

385

II

1;~] (1+cosQ[9 Clod)

+ I( G(q2)+m12)glQI2 2 ILlm12)12

= f

P )I

1912 +

f

G(q2>

+

I(

P )I

- 4 Re F’t*(q’)

+ ++

mq2)12 L I(

I((1

Fi(q’)

+

mq2>

+ [ I((1 tf

2

Gxq2)g

I(

m12)

tmq2)~

+ m2) Id2

lq12 g

2 )I

2 11 p(1 t cos e) )I I

t$-+xq%$-pxq.))~2 t&)(F6’(42)t$Gtq2))

2 -&l 1412 +yf&P mq’)+F3q2) +m12)) iI 1 P

tcos8) .

(1Oe)

i

The evaluation of the form factors appearing in the quantity D, including our wave functions for the deuteron and of the relevant (np)-scattering states, are given in Appendix B. After carrying out the angular momentum algebra, we express all form factors in terms of radial wave functions. For additional MEC in the a channel, we use the results of Lock and Foldy [ 171. Major differences between our formalism and that of Lock and Foldy [ LF] ] 17 ] are summarized below: (1)

LF include the ‘Pi, 3S,-3D,

[a, /I], and ‘Dz (np)-scattering

states. Indeed,

386

HWANG,

HENLEY,

AND

MILLER

their NOIA results can be used to justify the neglect of these states in the kinematic regime considered by us Inamely, small E,,,]. (2) LF do not include explicitly the contribution due to the 3P, state. However, we find that the contribution of the ‘Pz is more important than that due to the 3P, and ‘P, states. (3) We apply the dynamical model (the NOIA/GI plus MEC) only to evaluate the covariant form factors. The comparison is carried out in the Breit frame. In contrast, LF use the NOIA (plus MEC) in the Iaboratory frame. The frame dependence is reiated to relativistic effects and is not appreciable unless q2 is large [e.g., 220 fm-‘1. The application of the NOIA to the 3PJ channels may be questionable in certain kinematic regimes [ 12, 131. On the other hand, differences between our formalism and that of Fabian and Arenhovel [FA] [ 181 include: (1) FA make the truncation according to the multipolarity of the transition operator while we (as well as LF) make the truncation in terms of the final states; (2) FA evaluate explicitly MEC contributions and, for electric multipoles, subtract the components corresponding to the difference between the NOIA/GI and the NOIA; and (3) FA do not consider the frame dependence of the dynamical model. For the kinematic regime considered in this paper, these differences appear to be only of minor importance. (3b) Numerical Results The parity-conserving double differential cross section do(E,)/d& dLl is a function of three kinematic variables which are usually chosen as either {q*, E,,,, 0) or {E,, E,,,, 0} [all measured in the laboratory frame]. It is convenient to choose the (np) relative energy E,,, as a kinematic variable since one can choose it small enough so that final orbital angular momentum states can be truncated at I= 1. We illustrate in Figs. 1 and 2 how the observed data [ 19-2 l] is described by our formalism. For the parity-conserving (P.c.) nucleon-nucleon (NN) force, we employ the Reid soft-code potential [6]. The sensitivity of our predictions to the p.c. NN potential would need to be investigated if a parity violation experiment of extraor-. dinarily high precision could become feasible. In Fig. 1, the double differential cross section do/dE: d.l2 at 0= 1800 and Ere, = 3.0 MeV is shown as a function of q2 for 0 < q* 2 12 fm-*. The data of Rand et al. [ 191 and Ganichot [20] are fitted reasonably well by the curve designated by which represents the full calculation of our formalism. If the ‘so + 3po,1,29 3P,,,+,(np)-scattering states are neglected from the calculation, the resultant curve, designated by iSo, reproduces these data as well. In Fig. 2, da/dE: df2 at E, = 222.6 MeV and 0 = 157’ is shown as a function of E,,, for 05 E,,, 5 IO MeV. The tit to the recent data of Simon et al. [21] is impressive. The inclusion of states other than the ‘S, becomes increasingly important as Erel increases. In addition, we have considered the fit to the recent data of Bernheim et al. [22] and confirmed their claim that theoretical computations are too low for q*> 15 fm-*,

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q2 FIG. 1. The double differential cross section du/dEkdR a function of q* for 0 < q2< 12 fm-‘. The data points Ganichot et al. (Ref. 1201).

SCATTERING.

387

I1

(fm-2) at 0 = 180” and E,,, = 3.0 MeV is shown as are from Rand et al. (Ref. 1 191) and from

i

IO

8

6

4

E rel

(MeVl

FIG. 2. The double differential cross section du,/dEhdR as a function of E,,, for 0 < E,,, < 10 MeV. The data points

2

0

at E, = 222.6 MeV and 0 = 157” is shown are from Simon el al. (Ref. 121 I).

388

E,,,

HWANG,

HENLEY,

AND

MILLER

FIG. 3. The percentage contributions of the ‘S, and 3P,,,,2 = 3.0 MeV are plotted against 9’ for 0 < 9’ ,$ 14 fme2.

to du/dE:dQ

at 0= 180” and

as compared to their data. The discrepancy for q* 2 15 fm-* can be due to the neglect of p-meson-exchange currents [23], relativistic effects, etc. Since our primary concern is to apply the formalism to investigation of parity violation in e+D-+e+n+p for, e.g., q2s12fm-‘, our formalism is justified and compares well to other analysis [ 17, 18, 211. In order to justify the truncation in relative (n-p) angular momenta in our formalism, we illustrate in Figs. 3, 4a and 4b the point that, for a given q2 [E,] and 6, one can always keep Ere, sufficiently small to suppress even the contributions due to the 3P,,,,2 states. In Fig. 3, the percentage contributions of the-IS,, and 3P0,1,2 to do/dEh dS2 at 19= 180° and Ere, = 3.0 MeV are plotted against q* for 0 & q* < 14 fm-*. As already observed in our previous study [ 121 of y + D + n +p [for which q* = 01, the contribution due to states other than the ‘S,, may be important, as can be seen for small q* in Fig. 3. However, for small E,,, and except for small q*, the ‘S, channel is the dominant one. In Figs. 4a and 4b, the ratios D(3PJ)/D(1S,) are shown as a function of Ere, for (a) B= 180° and q2 = 5 fme2 and (b) 0= 180° and q2 = 10m4 fm-*. Calculations like these with those obtained by Lock and Foldy [ 171, help us to choose a kinematic regime for which the truncation is justified. In Fig. 5, various contributions to ReF;(q*) are plotted against q* for 0 < q* 2 14 fmW2. [Im F’$(q*)/Re F;(q*) = -1.70 since E,,, is fixed to be 3.0 MeV.] Our numerical results, except the pionic term {FnM(q2)}PION agree very well with those obtained by LF for the same kinematic conditions [see their Fig. 111. The double integration required in the evaluation of {Fg(q2)jpION [see Eqs. (B2e) and (B2f) in Appendix B] has been carried out with care and the result for )ql + 0 agrees with the standard limit [24]. The sign of our {F$(q2)}p,0N changes at q* less than, but close to, 11 fm-* and its magnitude for 10 fm-* 5 q25 14 fm-* is somewhat smaller than that

PARITY

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E rel (a) ‘. fm~

(b)

The ratios The ratios

D(zP,.,,2,)/D(‘S,) D(‘P,,,,.2)/D(‘S,)

e-D

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389

II

(MeV)

as a function as a function

of E,,, of E,,,

are are

shown shown

for for

0 = 180” 8= 180”

and and

‘.

obtained by LF. In both calculations, 1however, the (F$(q2)),,,,, appears to be negligible as compared to, e.g., the pair term {F~(q2))pAIR. To illustrate the importance of GI, we plot in Fig. 6 the ratios {D(3P,)},,,,/{D(3P,)}..,,,, as a function of q2 for Ere,= 3.0MeV and 0= 180”. For q2 2 1 fme2, the difference between the NOIA and the NOIA/GI predictions is moderate for the ‘P,, and 3P2 but very large for the 3P, channel, especially for small values of q2. However, in view of Figs. 3 and 4, this difference does not affect greatly the double differential cross section, To conclude our investigations on the double differential cross section, we present in Tables I and II do/dE’, df2 as a function of {E,, E,,,, e}, respectively, in the medium-energy regime [E,,, 5 10 MeV and 100 MeV 2 E, 5 500 MeV] and in the low-energy regime [E,,, 2 1 MeV and E, 5 50 MeV]. For each pair of energies, the

390

HWANG, Id

HENLEY, 1 I

1 1 1 I

----

s,gn flipped

q2

AND 0 ,

MILLER 1 I

5 I

1

(fm-2)

FIG. 5. The various contributions to ReF”,(q’) are plotted against q* for 0 ,< q26 14 fme2. The known phase shift at E,,, = 3.0 MeV can be used to calculate Im E;“M(9’) [i.e., Im F;(q’)/Re FG(q’)= -1.7041.

three entries in Table I correspond, respectively, to (1) the full calculation in our formalism, (2) ‘S, (NOIA + MEC) + 3Po,1,2(NOIA), and (3) ‘S,(NOIA + MEC) + ‘P,,,(NOIA). As indicated earlier, the third entries are expected to be close to those obtained by LF. The importance of the 3P, channel is reflected by the differences

q2

(fiG2)

for E,,, = 3.0 MeV and 0 = 180” are shown as a F*G.6. The ratios I~(‘~J)JNolAIl~(“~,))N014/CI

function of 9’ for 0 < 92 5 12 fmm2.

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TABLE d’o/dE:,dO

[in Units

of lo-”

100

B=

as a Function

of E, and E,,,”

180”

3.0 3.65 x 10” 4.15 x 102 3.67 x 10’

391

II

I

cm’/MeV] (a)

E, (MeV 1

SCATTERING.

6.0 3.52 x 10’ 4.92 x 10’ 3.84 x 10‘

9.0 3.22 x 10’ 5.26 x 10’ 3.96 x 10’

150

986 1026 956

1010 1151 946

1170 1169 1094

200

275 279 266

272 288 248

310 345 268

250

79.5 80.0 76.7

77.1 79.1 69.3

86.3 90.9 72.2

300

23. I 23.1 22.1

22.4 22.6 19.8

25.0 25.6 20.2

350

6.46 6.44 6.13

6.40 6.38 5.47

7.30 7.31 5.57

400

1.64 1.63 1.52

1.73 1.70 1.37

2.08 2.03 1.4 I

450

0.339 0.33 1 0.290

0.4 17 0.396 0.275

0.562 0.529 0.296

500

0.045 0.04 1 0.024

0.088 0.077 0.029

0.149 0.129 0.037

’ For each pair of energies, the three entries correspond, respectively, formalism. (2) ‘S,(NOIA + MEC) + ‘P n.,,2(NOIA), and (3) ‘S,(NOIA

to (1) the full calculation + MEC) + ‘P,,,(NOIA).

in our

between the second and third entries. On the other hand, the differences between the first and second entries measure the importance of including GI. In Table II, the two entries for a given pair of energies correspond to da/dEL df2 at t9 = 180” and at B = 90”, respectively. The difference between the NOIA and the NOIA/GI is found to be extremely important for cross sections at E,,, = 1.0 MeV, for which the contributions due to the

392

HWANG,

HENLEY,

AND

TABLE dZu/dE~dff

[in Units of 10ms6cmr/MeV] (b)

I as a Function of E, and Ere,

8=90°

3.0

E, WW 100

MILLER

6.0

2.55

2.14 x lo4 x lo4 1.71 x 104

2.17 x lo4 2.81 x lo4 1.75 x 104

150

5.71 x 10’ 6.36 x 10” 5.13 x 103

x 103 x 10’ 6.21 x 10’

200

1.88 x lo3 1.99 x 10’ 1.78 x 10’

2.52

7.50 9.32

2.18 x lo3 x 10’ 1.90 x 10’

9.0

1.55 x 104 2.21 x 104 1.39 x lo4 8.72 x 10’ 11.56 x lo3 7.21 x 10’ 2.79 3.47 2.32

250

700 722 672

730 801 651

881 1006 742

300

272 277 263

270 286 243

310 343 261

350

107 109 104

104 108 93.4

116 125 97.0

400

42.0 42.4 40.5

40.4 41.5 36.0

44.9 47.2 36.7

450

16.1 16.1 15.4

15.6 15.1 13.6

17.4 18.0 13.8

500

5.84 5.85 5.52

5.82 5.88 4.89

x 10’ x 10’ x 10’

6.69 6.82 4.96

’ For each pair of energies, the three entries correspond, respectively, to (1) the full calculation in our formalism, (2) ‘S,(NOIA + MEC) + ‘P,*,,,(NOIA), and (3) ‘S,(NOIA + MEC) + 3P,q,(NOIA). 3po

1 2

states are even more important

than those due to the ‘S,. For instance, our

do/dh: df2 at (E,, 6) = (5 MeV, 180’3 (5 MeV, 900), (50 MeV, 180“), and (50 MeV,

90*) should be multiplied respectively by 7.10, 5.11, 1.10, and 1.13 in order to yield the corresponding NOIA results. For Ere, 5 0.1 MeV, the cross section is dominated by the ‘S, contribution so that the deficiency of the NOIA related to GI is of less significance.

PARITY VIOLATION

IN e--D SCATTERING.

TABLE d’o/dE:dR

[in Units

II

of 1O-36 cm’/MeV]

E, (MeV)

1

5

393

I1

as a Function

of E, and E,,,”

0.1

0.01

3.50 x 10’ 1.37 x los

4.53 x IO4 7.65 x 10”

2.58 x 10J 4.02 x 10J

10

2.39 x lo4 9.92 x lo4

5.64 x lo4 9.49 x 10’

3.23 x 10J 5.02 x 10’

15

2.28 x IO’ 1.05 x IO’

5.77 x 10’ 9.94 x IO4

3.28 x 10’ 5.25 x 10

20

2.26 x lo4 1.06 x 10’

5.56 x 10’ 9.86 x lo4

3.13 x IO4 5.21 x IOJ

25

2.23 x 10J 1.03 x 10’

5.17 x IO’ 9.52 x 10’

2.88 x 10’ 5.03 x 10’

30

2.17 x 10’ 9.67 x 10’

4.71 x IO1 9.02 x lo4

2.60 x IO” 4.77 x 10

35

2.07 x lo4 8.95 x IO4

4.23 x 10’ 8.44 x 10’

2.32 x IO’ 4.47 x IOJ

40

1.95 x 104 8.16 x IO4

3.75 x 10’ 7.83 x 10’

2.04 x 10’ 4.15 x IO4

45

1.80 x 10’ 7.34 x IO4

3.30 x 10 7.20 x IO

1.79 x lOA 3.83 x 10”

50

1.65 x IO’ 6.55 x lo4

2.89 x 10’ 6.59 x 10”

1.56 x loJ 3.51 x IO“

” For each pair of energies. (2) 0 = 90”.

_____--~

the two entries

4. THE ASYMMETRY

correspond.

CAUSED

respectively.

BY NEUTRAL

(1 ) 8=

WEAK

180”

(upper

entry)

and

CURRENTS

(4a) Formulae We investigate the asymmetry caused by neutral weak currents, using the formalism of Section 2. Identifying the isovector neutral current with the isovector electromagnetic current, we have

(lla)

NV/D = h,.

This result follows since we include only the isovector final states in our formalism. Analogously, we write

N,ID = WV*,

E,,, , 0

(lib)

394

HWANG,

HENLEY,

AND

MILLER

Using Eq. (j), we have

d(Z) = g&2

i g, 4 + g, ww9

Ere,30)).

A determination of J/(‘) permits the study of both g,h, and gvh,. Electrodisintegration of the deuteron has the advantage that the separate dependence of .M(” on the kinematic variables q’, Ere,, and 0 can be studied. Some special feasures of the weak interaction may thus be obtainable. As indicated by Eq. (12) the size of L@” is approximately proportional to q ‘. For sufficiently large q2 [e.g., q2 2 1 fm-‘I, the asymmetric &’ is dominated by .MCz’. [The asymmetry dCE) caused by nutfear parity violation is important only in the low-energy regime; see the next section for details.] Therefore, it appears feasible that medium-energy electron-deuteron scattering can be used to study both gAh, and gvh,. Using Eq. (4) and Appendix A, we obtain, in the laboratory frame,

WI u%)

4 F;;E:) =3

(1 _ cos 6) Re FE*(q*) F”,(q*),

(13b)

(l _ cos 0) Re Ft*(q’)

(13c)

P

4 (Ee=+$:) ~*(3fs) = j-

@(q*),

P

xd3&)

= 34

(E;;EL)

(1 - cos e)

P

x

Re

R(u*~-&G(4'))*4(q*~ I(

+

(

191 am,

- wxq’)

P

* 1

KfI”,(q2)

+ fm*)

+ %(q*N*mq*)lI9

+ m*))*

a?*)

(134

PARITY

VIOLATION

IN e-D

SCATTERING.

II

395

+ml*) + (g)*Fxq’)] It is known that the MEC contribution to the time component of the axial weak current is important [25]. However, the axial form factors related directly to the time component of the axial weak current, namely, Fg(q*), FnlA,P(q2), and Fi(q*) do not appear in Eqs. (13b)-( 13e). It is also known (261 that the MEC contribution to the spatial components of the axial weak current is of the same order of magnitude as relativistic corrections, which are of less importance for the purpose of the present paper. Accordingly, we neglect MEC in the evaluation of those axial form factors which appear in Eqs. (13b)-( 13e). Explicit formulae for the form factors appearing in Eqs. (13) are given in Appendix C. (4b) Numerical Results In Fig. 7, the asymmetry &‘(‘) caused by neutral weak currents is plotted as a function of q* for E,,, = 3.0 MeV and 8 = 180° and for E,,, = 3.0 MeV and 9 = 90”. In the WS theory, the small and smooth shift of the curve from 8 = 180° to 0 = 90” indicates that the size of z&‘(‘) is determined primarily by the four-momentum transfer squared q*. If the values assumed by g, and g, are interchanged [27], i.e., g, = -1.0 and g, = 0.1, the resultant asymmetry JS?(‘) differs dramatically from that for the WS theory but is still insensitive to the electron scattering angle 8. Figure 7 suggests that

0

2

6

4 q*

8

IO

12

14

(fm-*)

FIG. 7. The asymmetry ,J”’ caused by neutral E,,, = 3.0 MeV and 0 = 180” and for I?,,, = 3.0 MeV and the Weinberg-Salam theory with sin* 8, = 0.225 and the vector-dominant solution [ g, = - 1.0, g, = 0.1, h, = 0.55,

weak currents is plotted against qz for 6’= 90”. The curve designated WS stands for curve designated by VD corresponds to the and h, = 11.

396

HWANG,

HENLEY,

AND

MILLER

-I -2 -3 -4 -5

W&d=

0

2

4

3.0MeV)

6

8

IO

12

14

FIG. 8. The asymmetry J&” caused by neutral weak currents is shown as a function of q* for 0 = 180° and E,,, = 3.0 MeV and for 0 = 180’ and Erc, = 9.0 MeV. The meaning of WS and VD is the same as in Fig. 7.

a decisive discrimination between the vector-dominant [ VD] solution [I g,l K I gv/ Z 1 ] and the axial-vector-dominant solution [I g, I CC I g, I z 1, e.g., in the WS theory] can be provided by a measurement of the asymmetry & as a function of q* for 1 fm-* ,$ q* $J 14 fm-*.* This last result is elaborated further by

FIG. 9. The asymmetry J&” shown as a function of g,.

caused by neutral weak currents at E,,, = 3.0 MeV and B = 180° is

* The asymmetry .d “) for any given g,hv and gv h, can be calculated from our WS and VD results.

PARITY VIOLATION

FIG. 10. The asymmetry ,rJ”’ shown as a function of g,.

IN e-D SCATTERING.

11

397

caused by neutral weak currents at E,,, = 3.0 MeV and f?= 180” is

,

,

VD( E,,, = 3 0 MN)

E,

(MeV)

FIG. 11. The asymmetry ~c/‘~’ caused by neutral weak currents is shown as a function of the incident electron energy, E, for E,,, = 3.0 MeV and 0 = 180’ and for E,,, = 9.0 MeV and .9= 180”. See Fig. 7 for the meaning of WS and VD.

398

2 5 -0.2 5 ,” -0.4 P 5L -0.6 5 E” -0.0 0” Y -1.0 Y

HWANG, HENLEY, AND MILLER

-

8.-,,,I

g40

00 0

(degrees

120

0

160

40 8

)

8

(degrees

00 (degrees)

120

160

)

FIG. 12. The asymmetry &“‘) caused by neutral weak currents for E,,, = 3.0MeV and E, = 100 MeV (a), 300 MeV (b) and 500 MeV (c) is shown as a function of the electron scattering angle 8. See Fig. 7 for the meaning of WS and VD.

Fig. 8, in which the asymmetry &(‘) as a function of q* is shown for B = 180° and two values of Ere,. According to Fig. 8, E,,, is also not a “controlling” parameter, leaving only q* in dictating the size of .M”~‘. The asymmetry ,c4(‘) due to nuclear parity violation can be neglected in the above kinematic region [see Section 51. In Figs. 9 and 10, we illustrate the sensitivity of &“), respectively, to g, and g,, keeping the remaining weak coupling constants the same as in the WS theory. For the

PARITY

VIOLATION

IN e-D TABLE

SCATTERING.

III

The Asymmetry -& (” [in Units of 10e5] E, and E,,, [in the Medium-Energy (a)

E, (MeV)

399

II

as a Function Regime]”

of

t?= 180”

3.0

6.0

9.0

loo

-0.185 -1.483

-0.171 -1.454

-0.173 -1.385

150

-0.5 19 -2.094

-0.502 -2.107

-0.485 -2.125

200

-0.994 -2.533

-0.971 -2.568

-0.947 -2.6 10

250

-1.601 -2.732

-1.570 -2.807

-1.540 -2.880

300

-2.335 -2.588

-2.293 -2.740

-2.252 -2.875

350

-3.202 -1.927

-3.140 -2.238

-3.082 -2.504

400

-4.227 -0.366

-4.121 -1.077

-4.03 1 -1.640

450

-5.505 +3.131

-5.262 +1.082

-5.095 -0.218

500

-7.204 + 10.347

-6.458 f3.291

-6.174 +0.857

’ For each pair of energies, the first (upper) entry corresponds to the WS theory with sin2 0, = 0.225 and the second entry corresponds to the vector-dominant solution described in the text ] g, = -1.0, g, = 0.1. and h,. h, as in the WS theory with sin2 9, = 0.2251. The asymmetry JZ”” at 0 = 180” for any other value of (g, h,, g,h,) can be computed easily from this table.

purpose of illustration, we have used E,,, = 3.0 MeV, 0 = 180°, and q2 = 3 fm --’ or 6 fm-*. Since an experiment is often planned with an incident electron beam of a definite energy (E,), it is also of some importance to express the asymmetry z”(~’ as a function of (E,, E,,,, 0) rather than of (q’, E,,,, 0). In this case, XJ’(‘) is expected to vary considerably with both E, and 0, since q* is a function of both E, and O[q’ = 2E,E:(l - cos Q]. In Fig. 11, we plot the asymmetry J/“) as a function of E, (instead of q*) for t9 = 180’ and two values of E,,,. In the WS theory with sin’ 8, = 0.225, the asymmetry 4’) at 8= 180’ is sensitive to E, [for 100 MeV 5 E, 5 500 MeV] but is rather insensitive to the value of E,,, [for E,,, 5 10 MeV]. If the vector-dominant solution (VD) (271 is adopted, the asymmetry

HWANG,

HENLEY,

TABLE

AND

MILLER

III

The Asymmetry XI(~) [in Units of lo-‘]

as a Function of

E, andE,,, [in the Medium-Energy Regime]” (b) 8=90”

3.0

6.0

9.0

loo

-0.125 -0.497

-0.137 -0.322

-0.137 -0.269

150

-0.271 -1.202

-0.293 -0.899

-0.300 -0.748

200

-0.505 -1.795

-0.515 -1.590

-0.52 1 -1.427

250

-0.829 -2.207

-0.824 -2.121

-0.820 -2.03 1

300

-1.234 -2.467

-1.219 -2.462

-1.206 -2.446

350

-1.717 -2.564

-1.694 -2.625

-1.671 -2.67 1

400

-2.278 -2.460

-2.245 -2.590

-2.214 -2.701

450

-2.918 -2.079

-2.873 -2.310

-2.830 -2.511

500

-3.648 -1.284

-3.582 -1.699

-3.523 -2.05 1

4 WV

a For each pair of energies, the first (upper) entry corresponds to the WS theory with sin’ 8, = 0.225 and the second entry corresponds to the vector-dominant solution described in the text [g, = -1.0, g, = 0.1, and h,, h, as in the WS theory with sin2 0, = 0.2251. The asymmetry .@’ at 0 = 90’ for any other value of (gvh,, g,h,) can be computed easily from this table.

J/(” as a function of E, is again drastically different from what is expected in the WS theory. Further, the VD results exhibit some sensitivity to Ere, for E, 2 300 MeV. The significant differences, shown in Figs. 7, 8, and 11, between the WS and VD theories should be useful in making precision tests of the WS theory. In Figs. 12a, 12b, and 12c, the asymmetry J(‘) as a function of the electron scattering angle 8 is shown for (a) E, = 100 MeV and E,,, = 3.0 MeV, (b) E, = 300 MeV and Ere, = 3.0 MeV, and (cc)E, = 500 MeV and Ere, = 3.0 MeV. These figures indicate that the difference between the WS theory and the vector-dominant solution persists at all electron scattering angles except in the forward direction [small $1.

PARITY

VIOLATION

IN e-D

TABLE

SCAITERJNG.

IV

The Asymmetry .d”‘[in Units of lo-‘] E, and E,,, [in the Low-Energy

E, WV)

401

II

1.0

as a Function Regime]”

of

0.1

0.01

5

0.113 0.020

0.523 0.3 11

0.552 0.365

10

0.742 0.111

1.160 0.740

1.177 0.827

15

1.237 0.077

1.571 1.046

1.584 1.172

20

1.508 -0.04 1

1.781 1.259

1.792 1.416

25

1.573 -0.228

1.797 1.384

1.805 1.561

30

1.442 -0.477

1.622 I .42 1

1.628 1.609

35

1.123 -0.780

1.260 1.371

1.264 1.561

40

0.619 -1.129

0.714 1.233

0.715 1.416

45

-0.068 -1.520

-0.015 1.005

-0.016 1.177

50

-0.936 -1.949

-0.923 0.686

-0.927 0.842

’ For each pair of energies, the first and second entries tively. The WS theory with sin’ 8, = 0.225 is assumed.

correspond

to 6 = 180” and 0 = 90°. respec-

To complete our presentation of numerical results on the asymmetry A@‘, we illustrate in Tables IIIa, IIIb, and IV the asymmetry Nz) as a function of E,. IT,,,. and 8. For each pair of energies in Tables IIIa [f?= 180°] and IIIb /l, = 90”1, the first entry is for the WS theory with sin’0, = 0.225 and the second entry is for the vectordominant solution described earlier. For Table IV, the first and second entries correspond, respectively, to 8 = 180’ and B= 90” and the WS theory with sin’ 0, = 0.225 is assumed.

402

HWANG,

HENLEY,

AND

MILLER

5. THE ASYMMETRY CAUSED BY NUCLEAR PARITY VIOLATION (5a) Formulae In the preceding section, we have confined ourselves to parity nonconservation (PNC) caused by neutral weak currents, analyzing the feasibility of using medium energy electron-deuteron scattering to study the neutral weak coupling constants. It has been assumed that for q2 > 1 fm-’ PNC caused by nuclear parity violation is negligible. This critical assumption is justified in the present section. There are more important reasons for investigating PNC caused by nuclear parity violation; namely, to better determine and understand the parity-violating nucleon-nucleon force. In particular, low-energy electrodisintegration of the deuteron has been suggested as a check of Lobashov’s experiment [7], in which he detected a large PNC effect in the thermal neutron capture reaction, n + p + D + yf To calculate the asymmetry A?(‘) as given by Eqs. (3) and (4), we write rnd(‘S,)

+ i hq3PJ). J=O

(144

Here we have

fl(lS,) = 9 (E;;EL) (1 -

cos 0) Re FG*(q*) G;(q*),

(14b)

P

(1 - cos 8) Re Fr(q*) lq3P,)

= f

(E;;ELJ

Gh(q*),

(14c)

(1 - cos 8)

P

X

Re

f

(g-)‘[

I(

JXq*)

=

f

X Re lF~*k*l

1

am)*

[t

‘%(q’) + (e)’

* GZXq*)

+ + e(q*)

GZAq*,]

(l-cos8)

(E;+,“’

+ +-

*GXq*) 2mp

(+-w)

- (+Xq*l+

jq3P,)

-

+ 4k’))

1,

,

(G%q’) -$ G;(q*)]

G%q*))

* G;(q*)

(144

PARITY

VIOLATION

IN e-D

SCATTERING.

II

403

+ (~)‘f;6*(9’)[ (Gtk’)- $ G:(q’)) +Gjt(q’) + (+$,’ Gf4q2)] + ($-)’ F:(q2)[- f (G:(q’)- $ Gh’)) -+G;(q2)

I!

.

(14e)

The evaluation of the p.v. form factors appearing in Eqs. (14a)-(14e) requires a determination of the abnormal-parity components in wave functions [see, e.g., Schrodinger Eq. (Bla) in Appendix B]. For this purpose, the inhomogeneous equations listed in Appendix D, together with the boundary condition that the solutions are well-behaved in the limit r + 0 and r -+ CO, are used. We use the NOIA/GI to calculate these p.v. form factors and write, for instance,

(15) The meaning of each tabulated in Appendix NOIA related to GI constraints derived in in the limit q2 -+ 0. If by themselves; such q2 << 1 fmV2J. (5b) Numerical

term in Eq. (15) can be understood readily from the formulas E. As can be seen from this appendix, the deficiency of the has been resolved consistently. In other words, all the GI I are satisfied and none of these form factors becomes singular more MEC need to be included, they must be gauge invariant contributions are likely to be small [particularly when

Results

To describe numerical results for the asymmetry ~8’) caused by nuclear parity violation, we use the parity-violating (p.v.) coupling constants which are calculated from the “best” values proposed by Desplanques et al. [28]. This set is described by Eq. (D2) [in Appendix D] and is to be abbreviated as DDH. Within an uncertainty of about a factor of two the resultant p.v. nucleon-nucleon (NN) potential is in accord with general theoretical expectations 1291. Recently, a reasonable tit to the p.v. data in “F, “F, “Ne, and 160 is found if the DDH “best” values are reduced by a factor of 2-4 [30]. Therefore, we use the DDH p.v. NN potential to calculate the asymmetry Ld(c) in the medium-energy regime so as to establish the result that the asymmetry in this kinematic region is dominated by that due to neutral weak currents. This makes a test of the WS theory feasible since a detailed knowledge of the p.v. NN potential is not essential.

404

HWANG,

HENLEY,

r

AND

MILLER

(fm)

FIG. 13. The abnormal-parity components d,,,(r) [‘s,] and v’,,,,(r) [‘P,] in the deuteron wave function. The p.v. NN potential of Desplanques et al. together with the Reid soft-core potential is used.

In the low-energy regime, the situation is complicated since the asymmetry ~8~) caused by nuclear parity violation is in general comparable to that due to neutral weak currents. Since only p.v. coupling constants f,, hz, hi, hi, h;‘, and h: enter the determination of abnormal-parity wave functions [see Appendix D], a detailed survey of the asymmetry as a function of E,, E,,, , and 8 should, in principle; enable us to determine the eight couplings g,h,, g, hv,f,, hi, hi , hi, h;‘, and hh. For purposes of illustration, we assume the WS theory for neutral weak currents so that gv h, and g,h, are given and the asymmetry ~4~‘) caused by neutral weak currents can be subtracted [see, e.g., Table IV].3 In the low-energy regime, we use the DDH p.v. NN potential to calculate J@), which in turn is decomposed into the contributions due to f,, hi, hi, hz, and h:, respectively [h;’ = 0 in the DDH p.v. NN potential 1. Our results indicate that, unless h: and hi are “extraordinarily” large, a determination of the asymmetry ,tP(‘) as a function of E,, Ere, , and 6 should help us to determine the three p.v. coupling constants f,, hz, and hz. This conclusion is of interest since the experiment of Laboshov et al. [ 71 is also sensitive to these p.v. coupling constants and low-energy electrondeuteron inelastic scattering appears to be less difficult than a repetition of the Lobashov experiment. In Figs. 13-15, the abnormal-parity components in wave functions are illustrated for the case of the DDH p.v. NN potential. The Reid soft-core NN potential is used to treat the p.c. strong interaction. For a (np)-scattering state with a definite relative energy Ere,, the p.c. component is generated and an inhomogeneous Schrodinger equation for the p.v. components [see Appendix D] is then solved numerically. These numerical wave functions are used to compute the various p.v. form factors [in 3 We make this assumption because g, h, and g, h, can also be determined by experiments other than low-energy electron4euteron inelastic scattering.

PARITY

VIOLATION

IN e-D

40xlo-~,"'~,,,",,,",",, r

SCATTERING.

E ,e, = IO-’ MeV ’

II

405

,,,,. (a)

30.Re y,,, (r)

O.,,,,I,,,,I,,,,I,,,,l,,,,i

0

5

IO

I5 (fm)

r

20

2i

,(,

30

14x10-*,,,,,,",,,,"',',,',',,' &I

-6

0

"',',1,,'I'II"rI"II~"' 5 IO

=I.0

15 r

(b)-

MeV

20

25

30

(fm)

FIG. 14(a). The abnormal-parity component Go,,(r) l’pO] in the ‘S, (np)-scattering E,,, = lo-~’ MeV. The p.v. NN potential of Desplanques et al. together with the Reid soft-core is used. (b) Same as (a). except that E,,, = 1.0 MeV is assumed instead of E,,, = 10 ’ MeV.

state at potential

Appendices A and E], which are required in the determination of the asymmetry .vf(c’. In Fig. 16, the asymmetry s#‘(‘) at E,,, = 3.0 MeV and 13= 180° for the DDH p.v. NV potential is shown as a function of q* for 0 < q* 5 14 fm-*. Comparing Fig. 16 with Figs. 7 and 8, we find that, except for very small q*, the asymmetry ~8’) caused by nuclear parity violation is in general only a few percent of that due to neutral

406

HWANG,

HENLEY,

AND

-,I.-

MILLER

30

r (fm) FIG. 15. The abnormal-parity component t70,,r,(r)[ ‘,I?~] in the ‘P, (up)-scattering state at E,,, = 1.0 MeV. The p.v. NN potential of Desplanques et al. together with the Reid soft-core potential is used.

weak currents. A similar conclusion can be reached if Table V, in which the asymmetry M’(~) (19= 180’) is shown as a function of E, and Ere, [in the medium-energy regime], is contrasted with the corresponding parts of Table IIIa. This percentage contribution of zP’ is probably even smaller because (1) the p.v. coupling constants may need to be reduced by factors of 24 in order to fit the p.v. data in, e.g., light nuclei (Ref. [30]) and (2) the q2 dependence of the various p.v. coupling constants

FIG. 16. The asymmetry ~8’~’ at E,,, = 3.0 MeV and 0= 180’ is shown as a function of q* for 0 5 q*5 14 fmm2. The DDH p.v. NN potential is used and only the results corresponding to the full calculation I’S, + ‘P,,,,,; NOIA/GI/MEC] are shown.

PARITY

VIOLATION

IN e-D

SCATTERING.

407

II

has been neglected in our approach. However, the ratio ST(~)/&‘(‘) does not fall off with q2 [E,] as rapidly as one might speculate. This is because in the Z-boson exchange mechanism all of the momentum transfer is taken up by a single nucleon, whereas the p.v. NN force allows two nucleons to share the momentum transfer. [An example of this momentum sharing idea is shown in Fig. 5, where it is seen that the meson-exchange current contributions (which involve two nucleons) to F;(q’) fall off much more slowly with q2 than the NOIA contribution.] For low-energy electrondeuteron scattering, E, 5 50 MeV, we describe in Table VI the asymmetry .d(‘) caused by nuclear parity violation as a function of E,, E Xl 7 and 6 for the DDH p.v. NN potential. An illustrative example of the angular dependence of the asymmetry &‘(‘) is shown in Fig. 17, together with asymmetry .cjcz) caused by neutral weak currents. Comparing Table VI with Table IV, we find that, in the low energy regime, the asymmetry ~8’) is in general comparable to that due to neutral weak currents. As mentioned earlier, a detailed survey of the asymmetry as a function of E,, Ere,, and 8 in the low-energy region should help to determine the eight couplings g, h, , g, h,, f,, hi, hi, hz ,hb’, and h:. For simplicity, we assume that the asymmetry &‘(‘) caused by neutral weak currents is computed reliably. The observed asymmetry [J@‘(‘)]~~,, as a function of E,, Ere,, and 0 can be expected to differ quantitatively from Table VI since the DDH p.v. NN potential is not yet established to be correct. The question of crucial importance is: What can we learn from a study of the observed asymmetry [.d(c’]exp as a function of E,. E,,,, and 8? TABLE The Asymmetry

,rw””

V

Caused by Nuclear Parity Violation at 0 = 180” [in Units of E, and E,,, [in the Medium-Energy Regime]”

E, WV 1

3.0

100

of 10~ ‘1 as a Function

6.0

9.0

-1.573

-1.390

-1.366

150

-2.327

-1.853

-1.522

200

-3.129

-2.577

-2.113

250

-3.969

-3.343

-2.776

300

-4.947

-4.181

-3.468

350

-6.241

400

-8.2

-4.215

-5.181 10

-6.46

1

-5.008

450

--I 1.69

-7.992

-5.592

500

-16.83

-7.664

-4.5

‘The p.v. coupling (Ref. 1281).

constants

are calculated

from

the “best”

values

proposed

10

by Desplanques

et al.

408

HWANG, HENLEY, AND MILLER

To answer the last question, we decompose in Tables VIIa-e the calculated asymmetry ~6’) for the DDH p.v. NN potential into the contributions due to the p.v. coupling constants f,, hi, hi, ht , and hk, respectively. We note that hi’ = 0 in the DDH p.v. NN potential; the role of a nonvanishing hb’ is described in Table VIII for which hb’ = -7 x 10e8 is used in view of a recent estimate [3 11. [We delete from these tables the column with Ere, = 0.01 MeV because it is hard to attain such highenergy resolution for the (np) relative energy E,,,.] Since the p.v. asymmetry is linear in the coupling constants f,, hz , hi, hz, h;, and hb’, these tables allow us to compute easily the asymmetry J&‘(‘) for any set of parameters. Conversely, if the observed asymmetry [JP]~~~ is known, these tables should help us determine f,, hz, h:, hz, hk, and hb’. In practice, the answer to the question posed above can be simplified considerably since these tables indicate that, unless hj, hk, and h;’ lie well outside the “reasonable” ranges proposed by Desplanques et al [28], the contributions to the asymmetry JP) from hi, hi, and hb’ are negligible. Therefore, we expect that only upper bounds on the p.v. coupling constants hi, hf, , and h;’ can be obtained from observed asymmetries A?(‘). To understand general characteristics of the contributions due to f,, hi, and hi, we illustrate in Figs. 18a and b the asymmetry J@) [0 = 180”] as a function of E, for Ere, = 1.0 MeV and E,,, = 0.1 MeV, respectively. For incident electron energies E, greater than about 30 MeV, the asymmetry L#‘) is dominated by the contribution due to f, if f, 2 lo-‘, so that a clean determination off, may be feasible. Once f, is determined from the observed asymmetry SS?(~)in the energy region E, = (30~(50) MeV, the asymmetry &‘(‘) due to hi and hi can be determined for 5 MeV 5 E, 5 20 MeV. However, a separate determination of

0

40

60 8

120

160

(degrees)

FIG. 17. The asymmetry Jgcf’ at E, = 5.0 MeV and E,,, = 0.1 MeV is shown as a function of 13for the DDH p.v. NN potential. The asymmetry d”’ caused by neutral weak currents is also shown for + .dCZ’(VD) is greater that both the WS and VD models. Note that #“(DDH) .d”‘(DDH) + .dCZ’(WS) by almost an order of magnitude.

PARITY

VIOLATION

IN e-D

SCATTERING.

E re, = l.OMeV

----------

20

30 E,

04x07

, '.

0

,

,

1 --

---i IO

/

=lEO”

8

40

50

60

(MeV) ,

,

,

I

/

2

/ (bi

-"-f_ _ _ -_---/

I

I 20

hoP

I

I 30 Ee

1 2

I

I,,,1 40

50

15

25

60

(MeV)

FIG. 18. The asymmetry of .ni”’ at B = 180” and E,,, = 1.0 MeV function of E, for 5 MeV 6 E, 2 50 MeV. The DDH p.v. NN potential components are indicated.

5

-

___------

-04:;+'-‘._

I IO

409

II

35

45

(a). 0. I MeV (b) is sown as a is assumed and the dominant

55

E, (Me’,‘) FIG. 19. The structure (E,,,. 8) = (1.0 MeV, IStY’),

function F,(E,, E,,,, 8) of Eq. (1.0 MeV, 900), (0.1 MeV, 180”),

(16) is shown and (0.1 MeV,

as a function of E, 900). respectively.

for

410

HWANG,

HENLEY,

TABLE

AND

MILLER

VI

The Asymmetry M’(‘) Caused by Nuclear Parity Violation [in Units of lo-‘] as a Function of E, and E,,, [in the Low-Energy Regime]’

1.0

0.1

0.01

5

-1.276 -0.646

-0.682 -0.766

-0.448 -0.521

10

-1.266 -0.580

-0.463 -0.446

-0.373 -0.35 1

15

-1.079 -0.419

-0.479 -0.399

-0.439 -0.359

20

-0.985 -0.357

-0.545 a.414

-0.53 1 -0.404

20

-0.949 -0.335

-0.630 -0.454

-0.632 -0.464

30

-0.945 -0.335

-0.724 -0.507

-0.735 -0.529

35

-0.960 -0.349

-0.823 -0.567

-0.837 -0.597

40

-0.990 -0.372

-0.923 -0.632

-0.939 -0.667

45

-1.032 -0.404

-1.022 -0.700

-1.039 -0.736

50

-1.083 -0.443

-1.120 -0.769

-1.137 -0.805

E, WW

’ The p.v. coupling constants are calculated from the “best” values proposed by Desplanques et al. (Ref. [28]). The first and second entries correspond to 8= 180° and 0= 90°, respectively.

h: and hi appears difficult since the behavior of the asymmetry due to a negative hi is similar to that of the asymmetry caused by a positive h:. Our next step is to present the results of Table VII in a manner which can be more readily applied to the extraction of weak coupling constants from experimental data. We do this by making a decomposition of S@ (E,, Ere,, 0) into its various contributions as

PARITY

VIOLATION

IN

TABLE The Asymmetry [in Units (a)

The Only

e-D

SCATTERING.

11

411

VII

S’ “’ Caused by Nuclear Parity Violation of lo-‘] as a Function of E, and E,,, [in the Low-Energy Regime]”

Nonvanishing to Bef,

4 WeV)

p.v. Coupling Constant = 4.56 x 10-l

Is Chosen

0.1

1.0 -__

~__

5

-1.107 -0.559

-0.332 -0.35 I

10

-1.111 -0.505

-0.302 -0.254

15

-0.96 1 -0.369

-0.370 --0.272

20

-0.89 1 -0.318

-0.459 -0.3 I7

25

-0.868 -0.302

-0.557 -0.373

30

-0.872 -0.305

-0.658 -0.435

35

-0.893 -0.319

-0.761 -0.502

40

-0.925 -0.343

-0.863 -0.571

45

-0.968 -0.374

-0.963 -0.64 1

50

-1.020 -0.413

-1.061 -0.713

’ The first and second respectively.

entries

correspond

to 0 = 180” and 0 = 90”.

Table

continued

The energy and angular dependenceof F,(E,, E,,,, 0) is shown in Fig. 19, and that of Rp(& Er,, 30) is shown in Fig. 20. The difference between Rp(E,, E,,,, 0) and a constant determines the ability to separate the pionic from the other contributions of shorter range. Indeed, R, varies significantly with E,. Plotted separately in Fig. 2 1 are the contributions of individual terms each of which have a range determined by the massof the p meson. There is significant variation among the various terms, so it appears that a very precise experiment can significantly aid in placing constraints on the individual coupling constants.

412

HWANG,

HENLEY,

TABLE

AND

MILLER

VII (continued)

(b) The Only Nonvanishing p.v. Coupling Constant Is Chosen to Be hz = -1.14 x 10m6

E, WeV)

1.0

0.1

5

-0.798 -0.409

-0.65 1 -0.776

10

-0.677 -0.338

-0.280 -0.348

15

-0.447 -0.213

-0.171 -0.217

20

-0.299 -0.151

-0.119 -0.155

25

-0.206 -0.114 -0.148 -0.090

-0.09 1 -0.119 -0.076 -0.096

35

-0.112 -0.074

-0.069 -0.08 1

40

-0.090 -0.063

-0.065 -0.07 1

45

-0.077 -0.054

-0.064 -0.066

50

-0.070 -0.049

-0.063 -0.062

30

Table

continued

To conclude this section, we relate the p.v. experiment via low-energy electron-deuteron scattering to the experiment of Lobashov et al. [7], which detects the circular polarization P, in the thermal neutron capture reaction, II + p + D + y. We find from Eqs. (3), (lob) and (14b) that, for small q* and small Ere,,

_OP(E)

z

-2

GXO)

2m,(E,

+

Eb)

FM(O) [Ez + EL* + 2E,E; sin’(B/2)] =P

where A[=2.2245 MeV]

4% + 3) y [E: + EL2 + 2E,EL sin*(8/2)]

is the deuteron binding

’

(17)

energy. We expect A@’ E P, for

PARITY

VIOLATION

TABLE (c) The Only

4 WW

IN

VII

e-D

SCATTERING.

II

(continued)

Nonvanishing p.v. Coupling Constant to Be hf, = -1.14 x lo-’

1.0

Is Chosen

0.1

+0.037 +0.019

tO.080

10

f0.03 1 +0.015

+0.033 +0.043

1s

to.019 to.010

+0.020 +0.026

20

to.012 to.007

i-o.012 10.018

25

+0.006 to.005

+0.007 +0.013

30

to.001 to.003

+0.003 +0.009

35

-0.003 to.002

+0.0004 +0.006

40

-0.006 to.001

-0.003 -to.003

45

-0.009 -0.00 1

-0.005 +0.002

50

-0.011 -0.00 1

-0.008 -0.001

5

413

+0.095

Table continued

deuteron electrodisintegration very close to threshold ]E, z A >> Ei], which is, however, difficult to carry out experimentally. The p.v, coupling constants can be adjusted to reproduce the Lobashov’s result, Py = -(1.30 f 0.45) x 10m6. However, the resultant p.v. NN potential implies an asymmetry .cJ(‘) which in general differs dramatically from what is anticipated from the DDH p.v. NN potential. As an illustrative example, we calculate the asymmetry .s+‘(“, using the p.v. coupling constants of Bowman et al. [32]. These coupling constants are chosen to best fit the currently existing data, including Lobashov’s result [7]. Our results are summarized in Table IX. A comparison of this table with Table VI indicates that the asymmetry JZZ”~) at E,= (5 - 15) and E,,, SO.1 MeV is larger than what is expected from the DDH p.v. NV potential by an order of magnitude. An experiment in this kinematic region is expected to be easier than

414

HWANG,

HENLEY,

TABLE (d)

The Only

AND

VII

MILLER

(continued)

Nonvanishing p.v. Coupling to Be /2:=-1.9x lo-’

4 WV

Constant

Is Chosen

1.0

0.1

-0.0005 -0.0002

-0.0001

10

-0.0005 -0.0003

-0.0002 -0.0001

15

-0.0006 -0.0002

-0.0003 -0.0002

20

-0.0006 -0.0002

-0.0004 -0.0002

25

-0.0007 -0.0002

-0.0005 -0.0004

30

-0.0008 -0.0002

-0.0006 -0.0004

35

-0.0008 -0.0002

-0.0007 -0.0005

40

-0.0008 -0.0003

-0.0008 -0.0005

45

-0.0009 -0.0003

-0.0009 -0.0006

50

-0.0009 -0.0004

-0.0010 -0.0007

5

-0.81

I 5

I 15

I

I 25

I

-0.0001

I 35

---I

90’ 180’ I 45

I 55

E, (MeV)

FIG. 9.

20.

The structure

function

R,(E,,

Ere,,

0) of Eq. (16)

is shown

along

the same line as in Fig.

PARITY

VIOLATION

TABLE (e) The Only

IN e-D

VII

SCATTERING.

(conGnrred)

Nonvanishing p.v. Coupling Constant to Be hi=-9.5~ 10.’

Is Chosen

E,,, (MeV) ~ _~~~. ~~ .% WV)

415

II

1.0

.~.

0.1

5

+0.593 +0.303

+0.22 1 +0.266

10

+0.492 +0.248

f0.086 -to. 1 13

15

f0.3 11 +0.153

+0.042 +0.064

20

+0.194 +0.105

10.02 I to.040

25

+0.120 +0.076

f0.012 -to.025

30

f0.075 +0.057

to.008 +0.015

35

to.049 to.042

+0.007 -to.01 1

40

to.032 -to.033

+0.009 f0.008

45

f0.023 to.025

+o.o 11 f0.006

50

to.019 to.020

to.01 3 to.008

E,,i

= 0 I MeV

I-

E, FIG.

E,,,=

(MeV)

21. The structure functions r:, rk, l.OMeV l8= 180”. 90”/ and (b) E,,,=O.l

rl.

ad r;’ MeV [f?=

are shown 180’. 9O”j.

as a function

of Es. for

(a)

416

HWANG,

HENLEY, TABLE

AND

MILLER

VIII

Same as Tables VIIa-e Except that the Only Nonvanishing p.v. Coupling Constant Is Chosen to Be hi’ = -7 x lo-* (Ref. [31])

1.0

0.1

5

0.0402 0.0207

0.0146 0.0178

10

0.0328 0.0168

0.0052 0.0073

15

0.0201 0.0102

0.0020 0.0038

20

0.0117 0.0068

0.0003 0.0020

25

0.0064 0.0048

-0.0007 +0.0008

30

0.003 1 0.0033

-0.0013 -0.0001

35

0.0010 0.0022

-0.0015 -0.0006

40

-0.0004 +0.0015

-0.0018 -0.0011

45

-0.0012 +0.0008

-0.0019 -0.0013

50

-0.0018 f0.0003

-0.0020 -0.0015

4 WV)

deuteron electrodisintegration very close to threshold. Thus, a check of the Lobashov’s result via low-energy electron-deuteron inelastic scattering appears feasible. 6.

SUMMARY

We have considered parity violation in electron-deuteron inelastic scattering. In Section 2, we described how the impulse approximation [NOIA] can be modified to take into account gauge invariance [GI]. The resultant approximation [NOIA/GI] together with “additional” meson-exchange current [MEC] is employed in the evaluation of covariant form factors. The GI constraints derived in a preceding paper [5] are fulfilled. In Section 3, very good tits to the parity-conserving differential cross section data have been obtained for the kinematic variables considered: q2 5 14 fm-2

PARITY

VIOLATION

IN

TABLE Same as Table

4 WV)

e-D

SCAITERING.

417

II

IX

VI Except that the p.v. Coupling Constants Those of Bowman ef al. (Ref. 1321)

1.0

0.1

5

-2.032 -1.032

-6.410 -7.589

10

-1.963 -0.909

-2.970 -3.523

15

-1.707 -0.648

-2.058 -2.353

20

-1.742 -0.562

-1.713 -1.843

25

-2.002 -0.565

-1.610 -1.590

30

-2.421 -0.628

-1.648 -1.472

35

-2.947 -0.745

-1.785 -1.439

40

-3.535 -0.914

-2.002 -1.468

45

-4.155 -1.135

-2.285 -1.545

50

-4.784 -1.407

-2.628 -1.663

Are

-__

and Ere, 2 10 MeV. Our results [Figs. 5, 6, and Table I, II 1 are also in good agreement with existing theoretical literature [ 17, 181. In Sections 4 and 5, we reported separately results for parity violation caused by neutral weak currents and those induced by nuclear parity violation. Model dependences have been examined in both cases. For medium-energy electrons and low excitation energies, the asymmetry is dominated by the contribution from neutral weak currents so that the Weinberg-Salam theory can be tested. For low-energy electrons, E, 2 50 MeV, the asymmetry caused by nuclear parity violation is comparable to but can be separated from that due to neutral weak currents. A survey of the asymmetry caused by nuclear parity violation as a function of E,, E,,, , and 0 can significantly aid in placing constraints on the individual coupling constants. The same experiment should also provide a direct check of the experiment of Lobashov et al. 171, which detects the circular polarization in the thermal-neutron capture reaction, n + p -+ D + y.

418

HWANG,HENLEY,AND APPENDIX

MILLER

A

In this appendix, we duplicate from I definitions of the various covariant form factors, which are used throughout the present paper. For the transition with F= ‘S, (subchannel a), we have, with qA E (p’” - pw>n grid Q, E (p”) + JPR)~, ‘?wm

IJa(O)l me

P’)>

= 2’/2&

(Ala)

('WP'") IJa@)l%J"', P'))(-) , ('&(P'")

IW'(O)l

W"',

P))

2’/2

=

For the transition

@lb)

WC)

with F= 3P,, (subchannel /3), we have

(‘m+-9

IJa(O)I Wi’7

P))

Qa q - tci’

F$(q2) - 2M ,,F$(q’)

,

(A24

P

=

("&(P?

2'12ic

"bcpv

I~y"(o>l =

$

P

we

$

Wb)

G!&'),

P>>

r(i) qo Qv 2’12i,5 awv It 2m,j+

B 2 1.

642~)

For the transition with F = 3P, (subchannel y), we have (‘mP?

P-9 IJa(O)l qP”‘,

4”‘))

mz2),

(A34

PARITY

+

if”

. r(i)

VIOLATION

IN

e-D

SCATTERING.

<” * + ir-----

*FJ(q’) P

4 rCi) P

* 9

2mP

II

4/t

F.(q2),

r,“,> IJJW

w”‘,

(A3c)

2mP

For the transition with F = 3P2 (subchannel 6), we introduce polarization tensor of second rank, namely <,“,. We have (34(Pm”,

419

a symmetric

P))

,z2m,

4*

F&*)

- $

F6,k’))

n +

p

4,

P

-I’ ----fl(q2) 2m,

* 9

2m, y (A44

p-l 4,

= ie .bna +

Oa

r(i)

2m*

iE ~Io,,~ppd!!L

i-its

$

Gd,(q’) - & P 6

,p - 9, 4,


pQ

2

2m, 2M Gx(q )

Q, fir . q Gs(qz)

r’p’K pm 2m, 2m, 2M2m,

+k

GiXq*)

q

2mP

’

rci, --4, Q, qn ’ 2m, 2M ( -2m, G:(q2) - &

G:(q2)) ,

(A4b)

420

HWANG,

HENLEY,

AND

MILLER

(A4c) We note in particular that, in favor of linear independence, the form factors Fi(q2), Gz(q2), and F;(q’) as introduced in I have been deleted.

APPENDIX

B

In this appendix, we give the definition for wave functions of the scattering-(np) states and tabulate the formulas for evaluating factors related to the calculation of the double differential Section 3). In the presence of a parity-violating (p.v.) interaction,

the deuteron and of the covariant form cross section (see we write

w,(F) - (r 13s, + 3, + ‘P, + ‘P,)

where r (=rp - rn) is the relative coordinate, j+PJ”ls are standard vector spherical harmonics, and vi, $ are isospin wave functions. Similar notation is to be used, respectively, for IS,, + ‘p,, , ‘P, + ‘go, ‘Pi + 3$, + 30”, , and ‘Ppz+ ‘d, + 362. For a given p.v. nucleon-nucleon (NN) potential pPp.,,, we can obtain the abnormal-parity components of the various wave functions by solving inhomogeneous Schrodinger equations (see Appendix D). The normalizations of normal-parity scattering-(np) states are fixed by the following illustrative example, F+rt~uol,(r) = i(47c)“23”2rjl(~p~

r + a,,,) eidall

(cos 6 - 2”‘i sin 8 sin #), Here (I p 1, 19,9) characterizes the relative momentum

@lb)

of the (np) system with respect

PARITY

VIOLATION

IN e-D

SCATTERING.

II

421

to the quantization axis of its total spin. However, the angular factor (-l/3”*) (COS e - 21’2 i sin 0sin 4) disappears as a result of the integration over the final, unobserved momentum. The EPT-NOIA connections have been tabulated in Appendix D of I. The method with minor modifications can also be employed to connect the EPT and the NOIA/GI (Eqs. (8aj(8f)). Carrying out angular integrations implies in the nuclear matrix elements [PI”” (see I) and those implied in Eqs. @a)--(8f), we can express the form fador V%2)\No,NGl (=VTs2)lN01A + &‘(q*)) in terms of radial wave functions. As regard meson-exchange currents (MEC) in the a channel, we follow Lock and Foldy ( 17 1 and consider only three dominant MEC contributions: pair excitation (PAIR), rescattering (RESC), and pion exchange (PION). Specifically, the cutoff factor given by their Eq. (134) is used to evaluate the PION term. In what follows. we tabulate the formulas for evaluating the various covariant form factors related to the calculation of the double differential cross section (Eqs. (2) and (10a j(lOe)). We have, for the a channel, i2”*c&?*)1N0,,4 = -(e&t)

+ i4q2))

j30 druZoo(r) 0

x jjo (9

U(T) - 2-‘lzj2 (J$) 4 2”V’n;(4’)]

w(r)/, = 0,

Wa) Wb)

=- 2 ($-)fA(qz) jr drY,(m,r) vko(r>j, (F) X (u(r) + 2-“*w(r)},

=25

(-)4n

m-GN&2)/o* w*

WC)

dr Wv-1

vtoo(r>

I SDvZoo(r) u(r) - 2-I’* j,

We)

422

HWANG,

Equations [17]:

(B2c)-(B2e)

HENLEY,

AND

MILLER

are to be used in conjunction

with the following notations

4x,y)= (F )I [(X2+y2+4m;)2-4x2y2][(X2+yZ+4mf)-’ 5 1 -1ln (x -Y>’ + 4mf, - (x2+y2+-m~+Tm~ 2 )I I(x + y)’ + 4m’, (x2 + y2 + mS, + mi)-’

+ [(X2 + y2 + rni + mz)2 - 4xzy2] -

5

x2+y2+-i-mi+Tm:

(

1

-1

[ ln

)I I

(x - y)’ + rnz + mi (x+y)2+m5,+mi /I ’

f2/(47c) = 0.080, MA = 0.95 GeV, w* 3 M(d) - M(N) = 293 MeV, GNyb(q2) = 2.353(1 + 9.0(q2)1’2)1’2 exp{-3.15(q2)1’2} (q2 in GeV2], -1 [m, = 776 MeV]. (W ( P1 For Eq. (B2a) and what follows, e,(q2) and &q2) are, respectively, the nucleon isovector electric charge and anomalous magnetic moment form factors (see I). MEC and have, with For the 3PJ (/I, y, 6) channels, we neglect “additional” P=lqld29 Fz(q2) =

1+ $

- 21’23 OW

PARITY

VIOLATION

IN e-D

-O” dr u&,(r)(u(r)

SCATTERING.

II

423

- 2%7(r))

0

-40(2j 191

’

@)-pj

*

@))+

4%

.‘X - e&*) 1 dr Ghl(r) 1 (f - 5) j,(p) + f (er) - f’zw(r’) -0 P

= -e,(q*)

(1 + c)

- e,(q’) x

2.I

2m,qo /q,2

drj,@)

v,*,,,(r)@(r)

- 2’12w(r))

-a j,

drj,@)

G,(r)

r d u(r) - 2”2w(r) r 1 mp dr i

+ 2- u*3w(r)/(m,r)

\q,jam dr.i,@) ~&,(r>W) + +(eds’) +h(q2)) 9o 2-“2wWl,

(B3c)

6( 2”vgq*)] 2m, go .a -- q? 3 /q(’ (q( .I0 drjl@)

u&,(r)(u(r)

&gq2)

- 2”*w(r)h

= 0,

(B3d)

Wb)

424 u?xq2)

HWANG,

HENLEY,

AND

MILLER

LJOIA

drU;l,(r)(U(r)

vx42hm4

=--p-

edq2)

m 10

9 WC)

+ 2-1’2W(r)),

(B4f)

+ 2-“2w(r))

= -1G(q2NNoIA

+ dq2)

(

f

‘I2 2m 1

drj,@) uT,k)(u(r)

m72NNoI*

44

ti

v-T(q2hx4

= 09

(JW

q(q2)

= 0,

Wh)

PARITY

+

(eds’)

+ rudq2))

VIOLATION

2m“’

IN e-D

61’2

lo. 3

SCATTERING.

(-21 dr[j,@)

425

11

+ j&)1

u;,,(r)

w(r),

Wa)

-0

P SJ’s,(q’>

= 7

e,(q*)

lrn

dr vfll(r)

(u(r) - 7

w(r))

-0

r 40 X I- /gl GA@) - Pj,cO>> + dr GIL (4 w(r)

X /, [qj2@)

- 6j4@)1wW/(m,r)

- (e&*) + dq2))

3” %lrn

+ 3U2cO) +j,@)l

k$

(F)

1

dr u&ll(r) 0

X j,h) (4rl+ 7 w(r))+-f-. 2”2j,@) w(r)1,

WC)

W%2) = e,(q2) $ (+)I’* x

I

- f$

[MP)

($)‘jr

dr t.$,1(r) w(r)

-k?i,@)l

+ Isl 4m, [-63X7)

+PY-.,@)~ 1

+e,tq2)+(+I”’ (2)’ x/m”

($-p)j,@)vf,l(r)$$

(F)

0

+ ev(q2) 6 “* (2)*

jr dr [+j:@)

1f’%*Nrw~~ = edq’) 3”* 211zpoo ,q, I o drG,W

-j,@,]

4&-)

WY(m,r)9

Wf)

PARITY

VIOLATION

IN

e-D

SCATTERING.

42-l

II

(Bk)

X irn dr ufll(r) -0

[j,(p)

(u(r)

-7

w(r))

- $ + 29,(p)

w(r)/

,

Wh)

(B5i)

It is important to note that, the four-momentum transfer (q, iq,) used in the EPTNOM and EPT-NOM/G1 connections is measured in the Breit frame, which has been chosen in the derivation of these connections, The transformation of (q, iq,) from the laboratory frame to the Breit frame is given by 4 Breit = dab * 2M/(2M + qbab), Breit = r(2Mq~b - q2)/(2M + qb”“,, 40 y= (1 - [Iq’“bl/(2M+qbab)]2}-“2.

(B6)

Therefore, (q, iq,) should always be understood implicitly as (qsreit, iqtreit) in this appendix as well as in Appendices C and E and as (qiab, iq:“) throughout the text.

APPENDIX

C

In this appendix, we tabulate the axial form factors related asymmetry due to Z-boson exchange (see Section 4). connections given in Appendix D of I and rewriting the matrix elements in terms of the normal-parity wave functions B), we ontain, with p = jq[ r/2,

to the calculation of the Using the EPT-NOVA various reduced nuclear (as defined in Appendix

428

HWANG,

HENLEY,

AND

MILLER

(CW

K&2*) INOIA

- 09

NOIA-

CC44

{F:tq2)iNOIA

(C4b) iFh2)

INO1

A

(C4c)

= -h,f,(q2)

3 ( 32 ) I/2 (1,1)3J: 2mp

drj,@)

APPENDIX

G!lltr> wtrb

WO

D

In this appendix, numerical values for the parity-violating (p.v.) coupling constants of Desplanques et al. [28] [DDH] are specified and the inhomogeneous Schrodinger equations satisfied by the abnormal-parity components of wave functions [Eqs. (Bla) and (B lb)] are tabulated. To use the parametrized form of the p.v. NN potential PP.,. introduced in I (Ref. [5]), we relate our p.v. coupling constants to those adopted by DDH, a, = &/0.880 = -g&&c, a; = &/4.706 = -g, hi/x,

PARITY

VIOLATION

IN e-D

SCATTERING.

429

II

aI = 4&J:, + &LhfJlK~ /I1 = -(4.706g,h; + 0.880gwh;)/rc, 6 = (g,hk - g,hLM, a, =/?,/4.706

= -g,,ht/(k),

k = -g,h;'/K; K s

-Gm;f,(O)/2”’

= -6.08 X lW6.

(Dl)

Here g,(=13.45), g,(=2.79), and g,(=3g,) are, respectively, the rrNN, pNN, and wNN p.c. coupling constants. If the “best” values proposed in the same reference are used for the various p.v. coupling constants, we obtain a0 - 3ab = 1.3 1, p, - 3& = 7.15, a, + a; - (i)"" a2 = -0.429 p, + #& - ($‘2 p* = -1.02 a; = -0.148, A =o, g&/2”*

= 4.34 x lo-“.

P2)

We proceed to tabulate the inhomogeneous Schrodinger equations which are needed in obtaining the abnormal-parity components of wave functions [Eqs. (Bla)-(Blb)]. Th e method to derive these equations is straightforward and has been explained briefly in Appendix A of I. For the purpose of the present paper, we use the realistic Reid soft-core potential [6] for the p.c. interaction. We find --

1 ( m1j&

q,,w

- 2ny) /@h-3&J

) + { V(‘P,) -E}

d,,,(r)

[2v,(r)u’(r)+u~(r)U(r)-ZULU]

- (PII - 3Pb) u;(r) u(r) 1 - 2 ($)"

16% - 34

[2v,(r)W'(r)+

v;(r)w(r)+

4v,(r)F] Pa)

IO

HWANG,

1

--

(

q,,(r)-

ml-2

($)“’

28y))+

HENLEY,

AND

{v(3P,)-E)

MILLER

d,,,(r)

1a; [ 20,(r) u’(r) + 2$(r) u(r) - 20,(r) - e-1 r

+Mr)+qmu(r) I --

2 a; 2uJr) 3”2 l [

+ M&9

+ hub)

w’(r) w(r)

+ v;(r)

I

w(r) + 4v,(r)

1

-w(r) r I

;

tD3b)

(D4)

- - 1 p 1o,(r) + ( Vct3S, - ‘D,) -E}

v’,,,(r)

+ 2 . 21’2VT(3S1 - ‘D,) d,,.(r)

mP

= -2

(t)“’

ia; [ %h9

611(r)

- @:W +WA-)> htr> 1T

+ ubG-1 uIll(r)

+ 20,(r)-

vllltr) r

1 CD64

PARITY

VIOLATION

IN e-D

SCATTERING.

43 1

II

-- 1 q*,(r)- 681;:(rJ ) + {Iq3s1- 30,) mP( - 2V,(3Sl - 3Dl) - 3VL,(3S, - ‘D,) -E) + 2 * 2”‘V,(3Sl =-3’/2

2

I

4

tTlz,(r)

- 30,) tTlol(r) [

2%(r)

~l,ll@)

+

q&9

Ulll@)

-

4u,(r)--- r vlll(r)

- @:W +WP>> uIll(r) I;

1 (Deb)

1 cgzo(r) - 6a2i:(r)) + 1WD2) - El d220(r> m, ( = 2 (i)‘l’

1 (a0 + ah -

+ Q-1 u2,,(r) - 4v,W -

--

1 mP (

(PO+&-

(+)l’ vzllW ~

a2) [20,(r)

v;,,(r)

I

Pa)

(~)~.8.)v:lr)v2,,(r)~,

z7T2,(r) - 6’2~~‘r) ) + { V(3D2) - E} filzI(r)

= -2 (-q’

Id, [ 2qk)4,,09

+ v;Wu211W

- (v’,(r)+ TV;) ~~~~(4 I. APPENDIX

-4v,W-

u211w

r

I

P7b) E

In this appendix, we tabulate the parity-violating electromagnetic form factors related to the calculation of the asymmetry due to nuclear parity violation (see Section 5). Using the EPT-NOIA connections given in Appendix D of I, deriving by a similar procedure the connections between the EPT and the NOIA/GI (Eqs. (8a)-(8f)), and carrying out the required angular integrations, we obtain, with P=191rP9

(El)

432

HWANG,

{2v”G~(q’;

$

+

‘D,

HENLEY,

AND

MILLER

-+ “&,h,,,

=fe,(p2)jrn dr6$,1(r) ][j,@) +j*@)] $$ (uw- ;vz”‘r’) 0

6{2”‘G:(q*;

P

3S, + 3D, -, ‘Fo)}

= - +- e,(q*) Jamdr u’& ,(r)(u(r) - 2”*w(r))

- 21’23 +.d [

- +j2W]

~trY(q7)

Wb)

1;

Pv2Wq2; ‘4 -+ ‘~ONNOI~

= $ edq*)j O”drUZ,Wijo@)+j,@)l$ 0

I

$ (*) P

WC) 6{2”*G;(q*;

‘P, --) IS,)}

= - 23V2 e,(q2) 1 Q1dr Goo(r) 6, 1oW 0

x I- $

WI(P)

-

Pj*@))

-- evtq2) m drv,*,,(r) 3”* I 0

+ JiL drn,(-4jA.P)+ Pj*@)) 1 [$-fp]jl(p)*$

1

(*)

W-9

PARITY

VIOLATION

IN e-D

SCATTERING.

II

433

We) 6{2V2G;(3P,

+ ‘So)} = 0.

Gf,(q’)

Wf)

= iGfAq*;

3& + 34 -+ ‘%,I,,,,

+ Pi,(q*;

‘p, --f 3Po)l,,,,

+ 1Gf,(q2; 36 -+ 3Po)ho,,,~

053)

/Note that 6G$(q*) = 01;

P”*G$(q*; 3S, +‘4-+‘$J},,~~ =(eds’> +k(q3)) jam drGo0 (rNjo@) 49-

w(r)13

2-“W>

Wa)

{2"*Gf,(q2; 'P, -+ 3PO))m, = $T (e&l') + iu,(q2)) !$ drIjoCP>+j,@)l

u,*,1(r) 3,10(r).

Wb)

{2"*G,&*; 3P,--) 3PO)}jv0,A = (3)"' (e&'> +cr&*)) jm drlj,@) - +j,@>l u&,(r) CIIW 0

-

3"*e,(q2)(2m,/lq0

Gil(r)

jmdultd 0 GL.&*)

=

F?;t;,~k*~ +

3S,

PX,,.(s*;

G,Aq*;

- $esh2) sGgq2;

2m m 2j drj,@) 191 0

3s, + 3D, +

C,(r)

33, + “d,) = 0;

+ 'P,

'PI

WC 1

LW.

-,

30,

-+

35‘,

--)

3J',ho,/,/c,

3~JLwm,ci~~

w(r)l(m,r)-

+

3~,)h,~,,,

(E5)

Wa) Q-1

434 E&12;

HWANG,

HENLEY,

Wo@)

+j,@)l

AND

MILLER

3p, + 3~,NNor*

= +

Mq2)

+ +e,(q’)

+Ps(41))jrn

2

0

1oy drj,@)

+,I@-)

fJTl103 v’, 1I (Mm,

~1116-1

(E6c)

97

dG;1;(q2;3P, + 3P,) = 0;

W4

Scy,(q$ ‘P, - 9,) = 0;

wf)

{c;y,(q2; 3& + 3D, -, 3$ + 3d,)},,,,

scy,(q*; ‘S, + 3D, + 3f, + “Dl) 2m * = - $ e,(q2) P WT2,09 191 jo

w(r) - 2v2(?ol(d w(r)

t fiT2,(r) 4r))l

4mpi-3,@)+~j,@)lI I I%@) -~j,@)l +JQJ -&&72) jrdr (+)j,@) lG2,Wr~(T) X - f$

--Zv2 [Cro,(r)rz

(T)

+ rY:,,(r)r$

W3.i2@) -ti3@)l[%(r)

(+)I

[

- 8”*qo,(r)l

W/r*;

Wb)

435

Wd)

We) Wf)

SQ*(q2; ‘P, -+ Y,) = 0. G”(q’)

= {G’(q’;

‘S, + 3Q -+ ‘8,)],,,A,,,

+ {GS(q2; ‘S, + 34 --t 3Who~w~, + iG”(q’;

3P, -+ 3Ww3,Nc3,

+ V%*;

*P, -, 3~2)}i.m.4,c3;

(W

G;(q*) = 0;

=-

15 2

“2

( ) --.- 73

(ev(q2) + ru&‘))

1

,,,2.Lb7)

w(r)1T

dGS,(q*; ‘S, + 3~, + lb,) = 0;

jom dr u;*zoW

(E9)

.A@)

8v2 4r) + 7 w(r)

1 (ElOa)

(ElOb)

436

HWANG,

HENLEY,

AND

MILLER

(ElOc)

6G:(q2; ‘S, + ‘D, -+ ‘6,) dr t7$21(r)(!.4(r) + 2-u2W(r)) x --. 5u2 3 +e,(q’)j;dr

FtXq2;

1oy2

.-

-2

I912

'4

=-f

[J+J]j2@)tTf21(r)r$(

dq2)

I

oO"

Wj,@)

2V2es(q2)G

2m

j

00 0

&

][j,@)+j,@)]$-$

- $9

(ElOd)

G221W w(W2;

-+ 3f'2>ho1~ *

x

-ti,@)l

u(r)+yV2w(r))

Mq2)

$1 l(r)

(*)-+[j,(P)-+j3(P)]

+ c1Aq2))

jaw W2@)

4,,V)lOv~j

G%(r)

(ElOe)

6 1 drh

6G;(q2; ‘PI -, “P,) 2m = $2 dq2) (9/j

m 0

drei,(r)

~,llW

x - f$ Pj2@)--Pj,@)l+ Jq( 4m, [-5j2@)+ d3@)l1 I

+6.~e,(~~)j~~~~[3i~@)-Pjl@)lu:ll(l)8;11(r)/1~; P%q2;

'P,

-,

3P2N,,,,

= 0,

(El0f-l WW

PARITY

VIOLATION

IN e--D

SCATTERING.

6Gs,(q*; ‘p, -+ ?,) P%(q2;

3S,

=-

+

‘D,

(+)v2

-+

@&“>

= 0.

(ElOh)

1D2)lNo1,4

+

dq*))

jm 0

dr

(j,@,

- y.i,@)

- ~j&l]

%

+

3D,

Cf20(r)

w(r),

(El la)

FP,(q *; 27, + 30, + ‘B,) = 0; iG;(q2;

437

II

(El lb)

--$ 3~2N,,,,

x d dr 3 - 1.7

1 (eAs*> +Mq*))

jm 0

dr G2,(r) (El ic)

6G;(q*;

3S, + 3D, -+ 362)

5 l/2

= 6

e&*)

2m * p j dr v’$21(r)(u(r) I91 0

+~.&e,(qz)!_dr 1ov* + j$es(q2)

+ 2-“*w(r))

(+-p)j,@,a,cr,r$( o 1 m dOj2@) -Pj,@)l

Ct21(r)WV?

u(r)+~v2w(r))

(El Id)

438

HWANG,

dG;(q’;

HENLEY,

AND

MILLER

‘S, + 3D, + ‘fi,) = 0;

V%(q*; 3& + ‘D, -,

(E12b)

‘D2hm

=-

(E12a)

X

6Gs,(q3; ‘S, + 3D, -a ‘D”J = 0; {G”y(q*; ‘S, + 3D, -, = -3

(+)“’

(E12b)

3D2ho,~

dq*)

(2)’

1,1’ drj3@) c?2,W w(r)/(q,r)

- 5 y2@s(q2> + c1A4*))

dr G21(r)

XI (

(E12c)

j2@>

(E12d)

6G;(q2; ‘S, + 3D, + “B,) = 0; {G;(q*;

3f’, --$ 3f’2)ho~,

=$(e,(q*) +,&*N (2)’ jrndrj,@) Gll(r> ~ll&Q7

(E12e)

0

6G,(q2; 3P, + 3P2) = 0;

(E12f)

PARITY VIOLATION Fdy(q*;

‘P,

+

IN e-D SCATTERING.

439

II

9’2N,,,,

2m

2.m

= 3(ev(q2) +~dq*))tIq Ii

dr.l-*@)

$1

IQ-1

4

(El%9

,1(41

0

(E12h)

6G;Jq2; ‘p’, + jP2) = 0.

ERRATUM

TO I

1. In Eq. (19d) [p. 531, the term proportional to u(‘) . q/(2m,) has the coefficient .fA(q2) i - (2m,q,/m~)f,(q*) instead offA C2. In Eq. (B7d) [p. 721, the factor [ should act only on the fA(q2) term. Accordingly, the same modification is understood for Eqs. (D31), (D3q), and (D4m) and thef,(q*) term in Eq. (Dlg) should be deleted. 3. On p. 78, Eq. (D3e) should read: FyK(q2)= -(l/A 4. 5. 6. 7. 8.

>mx2)1

e(q2) = 0.

On p. 78, the last term of Eq. (D3g) should be deleted. In Eq. (D3h) [p. 781. the two coefficients -l/5”* and -l/2”* should be interchanged. In Eq. (D3k) [p. 791, the arrow sign “-+” should be deleted. In Eq. (D4k) [p. 811, the last coefficient -i(s) 5 * “* should be replaced by --f(S)“‘. On p. 81, Eq. (D4n) should read:

1912 (5 1 F%12)

m+$F;(q*)P

= -hAfA(q2) 21’22

/;.r-

(+)“* [“I+&

,o,y(.

AKNOWLEDGMENT This work

was supported

in part by the U.S. Department

of Energy

Note added in proof: After this article was completed, we learned of the related work by M. Porrmann, Nucl. P~.w. A 360 (1981). 25 1. However, Porrmann omits gauge invariance constraints and exchange currents in both the parity-conserving and parity-violating amplitudes.

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AND

MILLER

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