Weak interactions in polarized electron-nucleus scattering

Nuclear Physics A360 (1981) 251-290 © North-Holland Publishing Company

IN POLARIZED

WEAK INTERACTIONS ELECTRON-NUCLEUS

SCATTERING

M. PORRMANN* Institut fiir Theoretische Physik II, Ruhr-Universit?it, D-4630 Bochurn, Germany Received 26 September 1980 Abstract: Scattering of longitudinally polarized electrons from nuclei is analysed in view of information on weak interactions. Emphasis is given to the competition of contributions due to Z-boson and photon exchange. In some kinematical domains parity mixing of nuclear states and similar processes become predominant. The asymmetry of elastic deuteron scattering is well suited for studying the polar isoscalar current, even independently of the nuclear structure. At deuteron disintegration in principle all parts of the neutral weak currents may be observed. Both cases allow for clean tests of weak interaction theories.

1. Introduction T h e t h e o r y of weak interactions has achieved considerable progress in the last d e c a d e by application of the f r a m e w o r k of gauge theories, which at the same time resulted in a unified description of electromagnetic and w e a k processes. T h e minimal m o d e l of this kind of t h e o r y was p r o p o s e d by W e i n b e r g and Salam in 1967 [ref. 1)]. A n essential c o n s e q u e n c e of this and succeeding models is the prediction of the interaction of w e a k neutral currents. With the energies available presently, it is only this prediction by which the new theories can be tested experimentally. In the d o m a i n of h i g h - e n e r g y physics neutral weak currents have b e e n d e t e c t e d since 1973 in the scattering of neutrinos f r o m electrons and nuclei 2). T h e observations are in f a v o u r of the W e i n b e r g - S a l a m model, e x t e n d e d to h a d r o n s by G l a s h o w et al. 3), and seem to disagree with most others 4). T h e r e have b e e n m a n y o t h e r processes p r o p o s e d to get information on neutral w e a k currents, especially in the d o m a i n of nuclear and atomic physics 5). O n e of these possibilities is scattering electrons with longitudinal polarization f r o m light nuclei. C h a n g i n g the helicity of the electrons influences this process in the same way that a parity o p e r a t i o n does. T h e r e f o r e all parity violating contributions to the scattering, interfering with the electromagnetic amplitude, m a y be o b s e r v e d by an * This work is part of a thesis submitted for the degree of Ph.D., partly supported by Deutsche Forschungsgemeinschaft (Ga 153/7). 251

252

M. Porrmann / Weak interactions

asymmetry of the cross section for both helicities, which allows for separation of the weak interactions from the dominant electromagnetic ones. This method of searching for neutral weak currents in nuclear physics has been proposed since the early 1970's. Reya and Schilcher examined elastic scattering of polarized electrons from free nucleons 6), Feinberg estimated the scattering from spinless nuclei 7). Later Walecka calculated the asymmetry by weak current interactions of electrons and nuclei in general 8). Subsequently there have been some papers on this process for different models of weak neutral currents, partially including radiative corrections 9). However, besides the semileptonic weak interactions there are also purely hadronic processes between the nuclear constituents, which give rise to a parity violation in the electromagnetic amplitude and thus to an asymmetry too. These interactions have so far been looked for in parity-forbidden c~-decays and circularly polarized ,/-emission of some nuclei. However, these experiments allowed for but approximate results due to considerable uncertainties both in theoretical and experimental evaluation 10). Polarized electron scattering should provide more accurate information both on hadronic as well as semileptonic weak interactions. In 1977 the asymmetry due to these hadronic processes, especially by parity admixtures to nuclear states, was analysed by Porrmann and Gari and calculated in the limit of small momentum transfer for excitations of 18F and 19F [ref. 11)]. Recently there have been other calculations on this contribution for deuteron disintegration at threshold 12), scattering from 18F [ref. 13)] and from 12C and 13C [ref. 14)]. The subject of the present work is a systematic analysis of all processes contributing to the asymmetry up to medium energies, especially with regard to parity admixtures to nuclear states. Sect. 2 contains the general discussion of this asymmetry. Because of the competition of weak and electromagnetic amplitudes it is of interest to look for kinematical regions where these parts can be measured separately. Another aspect is the discussion of gauge invariance of electromagnetic currents under the influence of weak interactions. The general analysis is then applied to scattering off deuterons and evaluated numerically. This work is motivated first by proposed experiments 15) and secondly by the theoretical advantages, which reduce the uncertainties involved with calculations for other nuclei. In sect. 3 there is a detailed investigation of the competition of weak and electromagnetic amplitudes contributing to the asymmetry of elastic scattering, with emphasis on the part due to parity admixtures. One question is to determine those kinematical regions most suited for measuring the asymmetry. Sect. 4 contains the calculations for deuteron disintegration by polarized electrons up to pion threshold. Parity admixtures are not included. Emphasis is given to the influence of isospin mixing of weak polar currents.

M. Porrmann / Weak interactions

253

2. Polarized electron scattering otf light nuclei 2.1. G E N E R A L C O N S I D E R A T I O N S

The a s y m m e t r y of the cross section for right- and left-handed electrons is defined as

A -~

(dcr/dg2)R - (dcr/d/2)L (do-/d~2)R + (dcr/d/2)L '

(1)

It measures the ratio of parity violating and regular contributions to the scattering. This method therefore only demands for relative m e a s u r e m e n t s of the events for both electron helicities. The Lagrange densities for electromagnetic and weak interactions, fig. 1, are .LPvN= eJV.At` ,

&eve

~'ZN

.Z ~('Ze = -Clt`Zt`

= c J tZ` Z t ` ,

=

-ejV.At` ,

(2)

(The notation is that of Lurie and Leon 2o). h, c and e0 serve as units.) e and c denote the corresponding coupling constants 16) e 2 = 41ra .... = 0.30282 ,

(3)

C2 = ~/~ Gvm2z ~ 0.35 z

o

b

Fig. 1. Electron-nucleus interactions by (a) electromagnetic photon exchange and (b) weak neutral Z-boson exchange.

m z is the Z - b o s o n mass. The electron currents are • --

i?, = ijevt`4,o,

1

]z = ~4,o~,t`~(~ +/~s)#,o

•

(4)

The nuclear currents are classified by their isospin and vector character

s~ = jZ

=

v~ Y

IV

+ vt` ,

IS.rlS

,

I / t . -t-'y

IVTrIV

t~ISAIS

I/tz + v

..t`

. ~IV--IV

5-0

(5)

z~t` .

(IS and IV denote isoscalar and isovector parts, Vt` and A~, polar and axial vectors.) The polar vector currents V ~ and V~v are identical for electromagnetic and weak interactions because of CVC, which is strongly supported experimentally. The weak

M. Porrmann / Weak interactions

254

lagrangians of eq. (2) are of quite general form not restricted to the W S - G I M model. They only assume the vector-axial vector structure of weak currents and the property that there is just one neutral weak field Z , , which couples both to polar and axial currents. The W S - G I M model predicts 16) a = 1 - 4 sin 2 Ow, y IS = - 2 s i n 2 0 w , y

IV

=1-2sin 20w,

/3 = 1 , ~ IS =

O,

~IV

1.

(6)

(The value of 6 TM in general is restricted to one-nucleon currents.) The V - A mixture of the weak neutral currents of electron and nucleus causes parity violation of the Z-boson part of the scattering amplitude. In the Born approximation for the scattering off light nuclei, the only other sources for an asymmetry are pieces of the electromagnetic scattering amplitude, describing some complicated parity violating processes. Thus the photon effectively couples to a small axial vector current, fig. 2. The axial form factors of electromagnetic currents too are strictly restricted by gauge invariance to be at least of second order of the m o m e n t u m transfer at low energies 17). N611e et al. investigated some of these processes for electron and nucleon currents is). The resulting parity violating amplitudes are roughly three orders of magnitude smaller than the corresponding amplitude of Z-boson exchange. Other axial nucleon form factors may be due to the photon coupling to a p 0 or w meson by vector-meson dominance, where this meson has a parity violating vertex with the nucleon. However, Fischbach and Tadic demonstrated that these processes completely cancel with corresponding seagull contributions, necessary to maintain gauge invariance 19).

M. Porrmann / Weak interactions

255

final-state interactions, described by a parity violating n u c l e o n - n u c l e o n potential (figs. 3a, 3b), or during the p h o t o n coupling, described by parity violating m e s o n e x c h a n g e currents (figs. 3c, 3d . . . . ).

ca

÷

b

÷

÷

c

°°,

d

Fig. 3. Parity-violating processes in the meson-nucleon system, yielding parity admixtures and axial electromagnetic meson-exchange currents.

2.2. DERIVATION OF THE ASYMMETRY AND DISCUSSION

Fig. 4 is the d i a g r a m for the process considered. (pe)~, = (Pe, iEe) and (p'e), = (p'~, iE'¢) are the electron m o m e n t a in the initial and final state, respectively, (Pi), = (Pi, iEi) and (Pf)~, = (Pf, iEf) are those of the nucleus. Ji and Jf d e n o t e the nuclear spins, m e and mT the masses of electron and target nucleus, q,, --- (p~ - p'~ ),, = (q, iqo) the m o m e n t u m transfer to the nucleus.

P$~PfJf P,

Pi

Ji

Fig. 4. Kinematics of electron-nucleus scattering by exchange of photon or Z-boson.

T h e a m p l i t u d e s for e l e c t r o m a g n e t i c and w e a k electron-nucleus interactions, eq. (2), and the cross section for right- and l e f t - h a n d e d electrons are calculated by s t a n d a r d m e t h o d s 20) in the B o r n a p p r o x i m a t i o n . T h e electron mass is neglected for energies a b o v e s o m e 10 M e V , w e a k processes are t a k e n into account in first o r d e r of the F e r m i constant GF. B e c a u s e the scattering will be considered only up to m e d i u m energies, a nuclear physics t r e a t m e n t is used to evaluate the currents by r e p r e s e n tation in c o o r d i n a t e space and multipole expansion of the current matrix

elements 21,22).

256

M. Porrmann / Weak interactions

The Coulomb, longitudinal and the transverse magnetic and electric multipoles C I, L I, M J and E J are functionals of the nuclear electromagnetic and weak currents J, = (J, ip) [refs. 21.22)]

C-'(q) =- i J I d3x ]j(qx) YJ(;)p(x), L~(q) ==-(iJ+'/q) f d3x V[h(qx)YJ(5)] • J(x), (7)

MJ(q) =_i j f d3x[]1(qx) ¥J~Ja)(;)] . J ( x ) , El(q) =- (iJ/q) I d3x V × []j(qx) YJ~'tl)(.~)] • J(x). They are defined in the system where the momentum transfer q fixes one axis and thus depend on q---]q] only. The equation of continuity for the electromagnetic current and, by CVC, for the polar vector part of the weak c u r r e n t s 16) gives the relation

LJ (q) = - q ° cJ (q) . q

(8)

Thus there is a longitudinal multipole only for axial weak currents. Averaging and summing the absolute square of the total scattering amplitude over all polarizations except the initial electron spin yields the cross section for definite energy of the final nuclear state

1

RL

~

Mottl+2(E~/mT) sin2102 2 2 e mz

with do")

2 Ol e.m.

c°s 210

d - ~ Mott

4E 2 sin410

(/2 is the solid angle of the outgoing electron, Ote.m.the fine structure constant.) The alternating signs correspond to both helicities. The F are the form factors of the different processes involved, F:~g for the regular electromagnetic interaction and the additional terms FiVrreg, F~,z and F~,Z for the interference of the regular amplitude with those of irregular axial electromagnetic currents as well as polar and axial weak

M.

257

P o r r m a n n / W e a k interactions

currents, respectively. In terms of multipoles defined above, these form factors read F~¢g - 2 Z + ~

Firreg

=

4~" 2Ji + 1

Vc J=O ~ VA

I=1

Iz 12)

-{- VT J = l

IcJ

'

(MJlc:], + L- -v]M- v- J , )

(lO) 41r

2.]','+1 Vc

,,-J,.-,J*

J = O 1~'"r'l"~Z

i~rJ A.~J* + E ~ , E z )

q - V T S=i IJwt"/'t"lZ

'

( ~ArJ ~,J:~ ~ J • .J:~ l v l v L ; Z -t- IZ v M z ) . ,

2.//+1 VAs=I

The subscripts y and Z indicate which currents are contained in the multipoles. [The multipoles in eq. (10) are reduced matrix elements in spin space in the notation of E d m o n d s 23).] The Vc, VT and VA are part of the electron contribution to the amplitudes. 4 vc

=_

,

q 2

q,, 21~ VT-~ ~q2 + tan ~ ,

(11)

Vh--= tan ½0~/tan 2 10 + q ~2 _ Ee +E'e tan 210.

q

Iql

These results are identical to those of refs. 8,11). The form factors F vz are weighted by a term which represents the differences in the electronic currents, the total coupling constants and the propagators of weak and electromagnetic interactions. Because of the extreme short range of weak interactions, expressed by a high Z - b o s o n mass, contributions of weak currents to the cross section are suppressed at low m o m e n t a q., but increase monotonely with increasing q2. 2 2

c q,, e

2m z 2

GF

2

2 ",/~ "JTOCe.m . q " ' = 1 . 5 8 2 x 10 - 4 q ~2, / m p2, = 7.00 x 10 -6 fm 2 q2,

(12)

mp is the proton mass. The polar or axial character of the weak electron current does not influence the cross section except for the different dependence on the helicities. In eq. (10), neutral weak multipoles interfering with electromagnetic ones of the same type do always contain the polar vector part of the weak nuclear current. Therefore, first, there can be no contribution of an axial weak charge or an axial weak

258

M. Porrmann / Weak interactions

longitudinal multipole to electron scattering in the Born approximation, and secondly, axial weak currents only occur in interference of electric with magnetic multipoles. Neglecting all parity-conserving weak contributions, the asymmetry of polarized electron-nucleus scattering takes the form A = dq 2,a F ~ ~ - / 3 F ~ ~ F[eg

FiVrreg FrVeg ,

(13)

with the approximations mentioned above, d denotes the numerical factor of eq. (12), last line. The part proportional to c~ consists of the parity-violating coupling of a polar weak -VZ electronic current to an axial nuclear one ( F A ) , the part with/3 contains an axial ,vZ electronic and a polar nuclear current ( F v ) , both in relation to the regular electromagnetic form factor Fr~eg. Because of the differences in coupling strength and range of weak and electromagnetic interactions, these ratios of form factors are weighted additionally with the factor dq~. The second contribution to the asymmetry consists of the parity admixtures to nuclear states and parity-violating mesonexchange currents, which both effectively behave like axial electromagnetic nuclear currents. In the following the asymmetry will be discussed in a few simple cases and different kinematical regions in order to investigate in which way the different nuclear currents may be observed in the asymmetry. The vertex factors of the electronic part, expressed by the functions Vc, VT and VA, eq. (11), do not allow for interference of electric and magnetic multipoles at small scattering angles and for Coulomb terms at large ones. Thus, polar weak charge densities only contribute at low and medium angles. Axial nuclear currents should be observed at backscattering, whereas polar currents will be present at all angles, in general. Therefore, in order to avoid contributions from parity mixtures, one should measure the asymmetry at small angles. In this case, however, one cannot observe axial weak nuclear currents. There is a complication about axial weak nuclear currents. Experimentally there is some evidence for only very small polar parts in the neutral weak current of the electron, which is indicated by a fitted value of Weinberg's angle 0w in the region of 0.22 to 0.30 [ref. 4)]. For sin 2 0w = 0.25, the coefficient c~ vanishes, eq. (6). Therefore, all contributions of axial weak nuclear currents to the asymmetry will be significantly decreased by a small factor a, remaining important only in cases of relative large matrix elements. Because of CVC, the weak polar current is a linear combination of the isoscalar and isovector electromagnetic one (5). The same is true for the corresponding form factors F~Z

IS

T

IV

~/

=3' Freg(IS)+3" F r ~ ( I V )

•

(14)

M. Porrmann / Weak interactions

259

Thus for nuclear transitions of fixed isospin character, the matrix elements of polar currents cancel and this contribution to the asymmetry reduces to the global factor dq 2 times the corresponding coefficient y~s or ~ v . At small scattering angles, axial nuclear currents vanish in the asymmetry, which in the pure isospin case then takes a simple analytic form, A , ~ IS,IV~ 2 -tJY aq,, , (15) 0 << "n"

independent of details of the nuclear configuration. In the model of Weinberg and Salam, this asymmetry reads for sin 2 0w ~ 0.25 A

, ±3.5 x 10 -6 fm 2 q~

(16)

8<
(+ for the isoscalar, - for the isovector case). This offers a very favourable possibility of measuring independently both isospin parts of the neutral weak polar hadronic currents in nuclei, together with the amplitude/3 of the axial electronic one. In general cases, where only polar nuclear currents contribute, the asymmetry measures the isospin mixing of these currents. If the ratio of isoscalar and isovector matrix elements is known (14), this allows for determination of 3/s and y~v At backscattering, the asymmetry vanishes between states of zero spin in general. In a very rough approximation, all multipoles should decrease in a similar way at high momentum transfer, for their momentum dependence is mainly determined by integrals over Bessel functions. Therefore, the ratio of form factors of the asymmetry will in general be a slowly varying function of Iql. Because of the additional factor dq~, the Z-boson exchange yields the dominant contribution to the asymmetry at high momenta. Thus there may be considerable corrections due to axial electromagnetic currents only at low and medium energies. At values of qx << 1, for a typical distance x in the nucleus, the Coulomb and magnetic multipoles of order J are proportional to q~. Electric multipoles are of order qJ+l at elastic and q~ 1 at inelastic scattering for long wavelength 21). Especially, the lowest multipoles, which in general only contribute significantly in this case, are of order C°~l

, (17)

M ~ ~ ]ql/mN, Ea

Jto/mN [qZ/m2

forta>O for ~o = 0 ,

apart from the other factors, which roughly cancel in the asymmetry, w denotes the intrinsic nuclear excitation. [In deriving eq. (17) the same low-energy dependence is assumed for elastic electric multipoles both of axial and polar currents.) At forward scattering, the m o m e n t u m dependence of the multipoles of F v and FrYe, cancels at low energies and therefore the asymmetry is of order 2 a - - / 3 y d q,,,

[ql~O, 0<< rr ;

(18)

260

M. Porrmann / W e a k interactions

y denotes the ratio of the weak and electromagnetic currents, generalizing ,yISAV f o r isospin mixing. Therefore, considering the asymmetry at small angles and low energies provides no new information beyond the general case (15). Moreover, the a s y m m e t r y is strongly suppressed at these momenta. Considering backscattering at low energies, the asymmetry is much more complicated. Depending on the difference of parity ATr of both nuclear states, the a s y m m e t r y approaches to

. 2[

1

1

1

Ez

~Mz\

E~

M1

E 1.

1

aq~ a~-2r-.~-PvTr.1 + 2 A

Iql~0 ~

o~

z

dq ~, Ce-E-~- fl

Art=0 (19)

+2E~

Axial electromagnetic multipoles are of global order ~, which describes the amplitude of the parity mixture in nuclear states and parity-violating meson-exchange currents with respect to the regular ones. Typically, ~ is of order 10 7 t o 1 0 _6 and in very resonant mixing of order 10 -5 and above 10,19.~1). Defining C ~ d r n 2, with C ---- 1.58 X 10 -4, eq. (12), the a s y m m e t r y takes the following order of magnitude at large angles in the elastic, 2

A ~ -[3ycq~+ mN

~ Iql ,

(20)

mN

and inelastic case (o~ > 0)

mN

A-

Iql

l c q2~ ( a 6 [ q l _ f l y l + @lql ' mN \

09

/

(21) A'w=I

.

¢0

Although the multipoles leave their low-energy limit already at a few tens of MeV, the approximation above is expected to be valid up to higher energies because of partial cancellations of deviations from this limit in the asymmetry. The products of weak current amplitudes fly typically are of order 0.5, if the model of Weinberg is taken for an estimate; ~6 will be of order 0.1 or smaller for isovector transitions and much smaller for isoscalar processes. The factor Iq. I/mN is very small at low energies, whereas Iql/~o and o~/Iql are of the order one at threshold. Therefore, at regular parity changing transitions, even a small amplitude ~ suffices for predominance of the parity-violating electromagnetic contribution over a wide range of low and medium momenta. This predominance increases for decreasing excitation energies to. For parity conserving nuclear excitations, there also will be a significant contribution F~rre~ for quite large amplitudes ~,

M. Porrmann / Weak interactions

261

which however decreases with increasing m o m e n t u m transfer. Contrary to the case Art = 1, this effect will be large for large energies to. In elastic scattering there is no enhancement of irregular electromagnetic contributions. The part, F 71rreg, is suppressed by a factor Iql/rnN, although not as large as the part of weak nuclear currents. This means F l7r r e g contributes significantly to the asymmetry only at very low momenta, which on the other hand requires quite large amplitudes ~. Therefore it is the elastic scattering where the smallest contributions due to an irregular electromagnetic amplitude will be present in the asymmetry. Because of probably small values of a, axial nuclear currents are suppressed in the Z-boson part of the asymmetry, which therefore will predominantly measure weak polar currents at large angles too. It is only in regular electric dipole excitations that the multipoles favour weak axial nuclear currents by a factor [q l/to, which then may become dominant in the Z-boson part especially at low energies to. However, in this case there will also be a large contribution of an axial electromagnetic term, depending on the amplitude ~. To observe an asymmetry due to Z-boson exchange exceeding 10 7, the electrons should at least have energies of order 20 MeV. For nuclear excitations, however, there will be a contribution of photon exchange of order ~ at already very low electron energies. Experimentally, the asymmetry is measured by a tiny difference of counting rates NR, NL for right- and left-handed electrons. This difference must exceed at least the purely statistical fluctuations. Therefore the ratio

S - W R - Nd x/NR + NL

(22)

determines the statistical significance of a specific measurement. The square is $2 = ~ A t A ~ A

2 dcr

dO

(23)

(~, At a n d / t O denote the experimental luminosity, total counting time and counting solid angle.) Therefore, besides the experimental set up, the suitability of specific kinematical regions for an asymmetry measurement is not determined by its amplitude alone, but b y the combined theoretical quantity A 2 do'/d/2, the statistical sensitivity. This was named "figure of merit" by, e.g., Ellis and McKellar 5). At high energies the asymmetry at best increases with q2, whereas the cross section decreases very rapidly beyond a distinct maximum at low or medium momentum transfer, specific for the nuclear transition. Therefore, a measurement should be performed in the region of this maximum and at slightly larger momenta. The range of scattering angles most suited for a measurement depends on the angular variation of the asymmetry and the regular form factor in relation to the strong decrease of the Mott cross section at increasing angles (9).

M. Porrmann / Weak interactions

262 2.3. C U R R E N T S

AND

GAUGE

INVARIANCE

The multipoles, eq. (7), contain the spatial distribution of the nuclear current matrix elements J~(x). The current operators, restricted to one- and two-nucleon contributions, will be extracted from a field theoretical description in momentum space; the wave functions are calculated in the coordinate representation. The necessary transformation of the current density operator J~ (q, p) in the momentum representation, t

I

J.(q, p)=--(plp2lJ.]Plp2) , 1

!

!

p --=~(Pl - P 2 - P l

(24)

+P2),

(pi and Pl denote the nucleon momenta) to the coordinate space is performed in a general way, described by Gari and Hyuga 24). The resulting multipole operators in the representation by relative and c.m. coordinates r and R of the two nucleon subsystems are

(rRICJ(q)lrR)

= 2¢r 2

d3p

d12qY (q)o(q, P) exp (i(q • R +p • r)),

(relL'(q)lre)=-27r2fd3p d.Qq{ ~ / 2 ~ + 1 [ vJ--l(q)xJ(q, p)]J /J + l J+l (q) A xJ(q, p)]j} exp (i(q. R + p • r)) , - V-~-~-~[Y (25)

(rRlMJ(q)lrR) = 2~r2 f d3p dg2q[YJ(4) xJ(q, p)]1 exp (i(q • R + p . r)), (rR[E z (q)lrR): - 2 - 2 [ d3p daq{ ]~[_~_~[ J + 1 yJ-l(q) x J(q, p)]J + ~2j~

[ VJ+l(O ) xJ(q, p)]'} exp (i(q. R + p" r)).

The sums of the one-nucleon momenta pi +P~ of the current operators are evaluated as gradients acting on the wave functions. The one-nucleon currents of the electromagnetic and neutral weak interactions read, in covariant form 16.20),

J"~' - (2~) 3 E.G, ~i(p')(Fiy. +F2o-.~(p'-p)~)u(p), jz Ix

i

--

[

-"---2"mN _,

(26) ,,,,, o

(2~r) 3 V-~-~--~ utp )tgvY.

o • , + g A T ~ ' Y 5 + i~v(p +p).

,

- i h ° (p - p ) . y s ) u ( p ) .

The form factors in general contain both isoscalar and isovector parts. Consistent

263

M . P o r r m a n n / W e a k interactions

with the use of non-relativistic wave functions, the current operators are approximated by their non-relativistic limit to the lowest order of p/mN:

Jv=a3(tGM(2~') \'2mNtr×q

+ 2~N

(p + P')) ,

1 PV = (-2~)3 G E ,

(27)

jz_

1 [. gO

(27r ~ ~ t 2---~Nt r × q z l0

1 ~" ( - ~ ) 3

[ ~o

+S~__1 (p+p,)_gOAtr, x) 2ran

gO

+ p @}

~kf I -- 2 ~ N O' " ( ~

,

with 2

GE(q2)=-F1--~NF2, 2

GM(q2)--Fx--2mNFz,

0

fi(q~,)-=-- gv --2mN~v, FS(0) =FV(0) = ~ ,

OSM(0)= ½(1 +/Zp+tZ.),

o v (0) = ½(1 +/Zp-lZ.)

(/£p a n d / z , denote the anomalous magnetic moments of protons and neutrons). The weak nucleonic form factors depend on the interaction model. In the WeinbergSalarn model they are related to the electromagnetic form factors and those of the charged weak isovector currents gv, gA and fv [ref. 16)] gO 1 = ~ g v - 2 sin 2 OwGM,

f°v = ½fv + 2 sin 20wF2, o

1

gA = ~gA,

(28)

gA(0) = 1.26.

The neutral axial form factor gO is purely isovector in this model and the relative weights of the isospin parts of the polar currents are given by eq. (6). The equation of continuity for a system with total hamiltonian H reads in non-covariant form q • Jr~ = [H, pv].

(29)

It provides a relation between the hamiltonian of a given system and the corresponding electromagnetic current in its part not perpendicular to q. In case of free nucleons, the electromagnetic current of eq. (27) obeys eq. (29). Because of general commutation rules, the solution to eq. (29),

1 GE l* = i[H, p~'x]

(277") 3

mN

agrees with the convection part of eq. (27).

(30)

264

M. Porrmann/ Weak interactions

In the case of an interacting nucleonic system, the coupling of the nucleons to each other is mediated by meson exchange in a microscopic description. Therefore a photon may be absorbed by hucleons and mesons as well, which gives rise to additional parts of the electromagnetic current. Formally, the necessity for such new terms can be derived from eq. (29). In meson-theoretical potentials there are isospin functions ('rl • "re) for two nucleons, which in the commutation relations with the isovector part of the charge density result in isospin functions (rl x x2)~ for the electromagnetic current. This piece is part of a two nucleon contribution to J~, which cannot be accounted for by eq. (27). At very low energies the Taylor series of the electromagnetic interaction energy mainly consists of the electric dipole part. The Siegert theorem states that this static part (Iql = 0) can be expressed by an effective two-particle current J~ = i[H, D ] ,

D= I d3xxp~(x),

(31)

with the exact A-nucleon charge density 0 v, which however may be approximated at low energies by its single-nucleon contribution 25). Eq. (31) generalizes eq. (30) for a system of interacting nucleons. Due to the interaction energy in H, there are additionally two nucleon currents, each one corresponding to one piece of the potential. However, these currents are expected to be valid only in case of very low photon momenta. Magnetic contributions of two nucleon currents cannot be derived from the Siegert current at all. General methods to derive meson exchange currents are, e.g., the evaluation of properly chosen Feynman graphs 26) or, in a systematic way, calculation of effective nucleonic operators of the total meson-nucleon system after projection to the purely nucleonic states 27). The resulting currents, e.g. those due to pion exchange, obey the equation of continuity (29), with the corresponding potential in general 26). Comparing these currents with the approximation (31), the Siegert current can only account for those terms remaining at ]ql = 0. Both forms already differ in parts linear in q/m,~ (m,~ is the pion mass), which stresses the static character of eq. (31). The physical reason for this limited validity is the local nature of the meson-nucleon coupling, whereas the photon-meson coupling is not restricted to these points. This freedom cannot be accounted for by the form of the commutator in eq. (31). Therefore, the total meson-exchange current Jexc consists of the Siegert approximation with the potential V and a dipole operator Do due to single-nucleon charge densities plus parts at least linear in q/mi, where mi is the mass of the different mesons exchanged

Jexc = i[ V, Do]+ ~ ( q / m , , q / m o. . . . ).

(32)

M. Porrmann / Weak interactions

265

The term ~ may also include exchange contributions to the magnetic part of the electromagnetic current. Now, if the hamiltonian contains parity-violating weak contributions Wweakto be considered, the gauge invariance of the electromagnetic interaction demands the existence of corresponding parity-violating exchange currents. Therefore, parity violating potentials and parity violating axial currents must both be considered. The Siegert theorem states, that these axial currents, in the lowest order in the photon momentum, may be described by 10) Jaxial = i[gweak, Do].

(33)

Certainly this axial Siegert current only contains parts of all parity-violating exchange currents. However, it is sufficient to account for gauge invariance of the electromagnetic coupling at low energies. Considering one nucleon currents (27), only, yields a q J+l behaviour of elastic electric multipoles (25), at low photon momenta in those parts resulting from spin currents. However, those parts due to the nucleonic convection currents give rise to contributions of order qJ 1 in the matrix elements, which violate gauge invariance 17,21,28). This can be verified substituting the convection currents by the Siegert form (31). Using the Siegert current, which explicitly preserves gauge invariance in the lowest order of q, the gauge invariance violating contributions to elastic electric matrix elements are completely cancelled. This result may be obtained by application of the hamiltonian of eq. (31) to the nuclear states. As the whole matrix element vanishes without parity violation, the hamiltonian necessarily contains the weak potential Vweak. Therefore, the cancellation of gauge invariance violating contributions to these multipoles involves both polar and axial electromagnetic exchange currents. Summarizing, this means that in contrast to all other multipoles, where meson exchange currents are known to contribute only moderately at low photon momenta, e.g., refs. 24,26,29,30),the impulse approximation completely violates gauge invariance of elastic electric multipoles. Inversely, meson-exchange processes are essential for these matrix elements. This result has been achieved by Henley et al. in a somewhat different way, too 17). Moreover, the exchange currents considered by the Siegert current cancel the contributions of the nuclear convection current to elastic electric matrix elements not only in the lowest order in q~-l, but in all orders, because time reversal invariance makes these matrix elements of the Siegert current vanish in general 21,28). Using the form of electric multipoles given by Partovi31), which explicitly assumes gauge invariance, one can then derive the property that there are only spin currents and other terms perpendicular to q which contribute to elastic electric matrix elements in order q~+l. All longitudinal exchange currents must be of higher order.

M. Porrmann / Weak interactions

266

3. Elastic scattering off deuterons 3.1. G E N E R A L

PROPERTIES

OF THE

ASYMMETRY

The elastic scattering from deuterons is an example for other nuclei with the same configuration symmetry. This symmetry allows for only pure isoscalar currents except those terms due to isovector parity-violating admixtures to the wave function. Therefore, first, the polar weak nuclear current and the electromagnetic one only differ by the constant 7 Is, cf. subsects. 2.1 and 2.2, and the corresponding part of the asymmetry takes the form of eq. (15), which does not depend on the inner structure of the deuteron and thus allows for a very distinct measurement of this constant. Secondly, the axial neutral weak currents are very unlikely to contain isoscalar contributions both for theoretical and experimental reasons 4,32), and they will not be considered here. Therefore, the neutral weak current of the deuteron only consists of a polar part. Axial currents do only occur in the electromagnetic interaction, resulting from parity-violating potentials as well as isoscalar meson exchange terms. The asymmetry of elastic scattering from deuterons, as well as other isoscalar nuclei with spin, then reads t-J I S .

1 2VAM~E~

2

1

a =-lot aq,-4 Vc(JCOI2+IC~I2)+ VTjM1vl2.

(34)

The part due to Z-boson exchange measures thestrength of the axial weak electron current fl and of the isoscalar polar weak nuclear current yis together with the effective low-energy coupling of these currents to the neutral weak field d. At fixed momentum transfer this does not depend on the scattering angle, whereas the part due to photon exchange is restricted to large scattering angles. According to the discussion of subsect. 2.2, it is only at low electron energies and large angles that the axial electromagnetic current has to be considered in the total asymmetry. In case of backscattering, the ratio of irregular and regular electromagnetic form factors approaches "y 1 Firreg ~, 2 E~ (35) FG

o-~

M1 "

The long-wavelength behaviour of the multipoles, subsect. 2.3, results in the following character of the photon contribution t D I cos2 sin ~0 ~--~--~q3 for 0 << rr, Firreg

(36)

),

FrOg ,~0

1

Oz~q

for 0 = ~r.

At large angles the total asymmetry therefore takes the form, eq. (17), 2

)-[3"ylSCq-~-+D2mN q .

A q+0

O+rr

mN

mN

(37)

M. Porrmann/ Weak interactions

267

Both parts vanish at zero momentum transfer, but in different orders. For values of the constant D2mN not too small in relation to /33,XSc, the asymmetry therefore measures the axial electromagnetic nuclear currents at small momenta. In this connection, the fact is stressed that the long wavelength character of the multipoles used in eq. (37), properly accounts for gauge invariance of the irregular electromagnetic nuclear processes. Yet the photon part in the asymmetry at low energies and large angles behaves differently from the Z-boson part. This conclusion disagrees with one of the results of ref. : ) , parity-violating electromagnetic processes in elastic scattering to be suppressed quadratic in q,, too. The argument of ref. 17) fails in the case where the momentum dependence of regular currents has to be considered, which means magnetic scattering becomes predominant. The following part of this section is concerned with the calculation of the electromagnetic contribution to the asymmetry, in order to determine to what extent these processes can be seen in the asymmetry, and, on the other hand, to what degree the asymmetry exclusively measures the polar weak nuclear current. Because these processes will be interesting at low energies only, the current operators are approximated by the one-nucleon currents, eq. (27). Meson-exchange currents will be considered but in the form of the Siegert current for the multipole E l , to account for gauge invariance in the lowest order of q, cf. subsect. 2.3. Axial one-nucleon currents are not used at all, cf. subsect. 2.1. Further approximations are the use of a dipole form of the nucleonic form factors and the restriction to regular wave functions due to the Reid soft-core potential 33). 3.2. D E U T E R O N

E1 M U L T I P O L E

The internal deuteron wave function reads (rid)=

ul(r) Yl~m(P)xT=°,

Y I =0,2

(38)

F

u~ is the regular radial wave function, Y the vector spherical harmonics coupling orbit and spin parts and X T-° is the isospin contribution. Using the integrals G jt~t,(q) =- f

1 urn(r) dr ut,(r) D• (~qr)

(39)

the regular multipoles (25) of the deuteron in the impulse approximation (27), after some algebraic manipulations with irreducible tensors, take the f o r m 24.34)

C°(q)= ~/~-~2GS (G°o + G~2) , C2(q) = - 2

2 a S (G22 - 4 5 G 2 2 ) ,

M'(q)= - ,L2GSq(GoOo V4rr 3

44~

, o +~ - ~G22

mN 2G s q (GO2+G~2). mN

(40) , 2 Go22 + ~G22)

268

M. Porrmann / Weak interactions

Now, evaluating the photon exchange contribution to the deuteron asymmetry reduces to the calculation of the irregular E1 matrix element, which results from parity violation of the wave functions as well as of meson-exchange currents. The existence of parity violating nucleon-nucleon potentials yields an admixture ]aT) to the deuteron state, which may, be described by a Schr6dinger equation to first order in GF [ref. 3s)]

v, roo01& =

(E0- T-

Vweakld),

(41)

with appropriate boundary conditions. (E0 is the deuteron binding energy, T and Vstro~g denote the kinetic and regular potential energy operators.) The corresponding wave function consists of a singlet and a triplet p-wave (rid) =

Z

ws(r) y l ( l S ) ( ~ ) X T

S=0,1 T S

(42)

r

Due to the characteristics of the weak potentials, the p-waves are purely imaginary. These potentials are described by a rather simple model, an improved version of that of Gari and Reid 36), because the main purpose is to get an estimate for the role of these contributions. In the model, the parity-violating potentials are mediated by exchange of p-, ~o- and charged rr-mesons, interacting parity violating with one nucleon and regularly with the other one, figs. 3a, b. Except for some corrections 37), these processes should describe the weak potential down to the two-pion range 3s). The actual form of the potentials used in solving eq. (41) are

Vweak(r)-

8x/~rrmN(O'l+O'2)'

Pl--P2,

r

rl2,

e .,,.r] GFgA(O)m 2 COS2 0C (1 +Xv)(i~I X~2)' [ p,--P2,--7--] VOweak (r) ------

8~f2 q'rmN

{

+(~1-~2)"

2

V~eak (r) =

Pl-P2. .

2

e'r}) r

To'

(43)

(

GFgA(O)m,.,(--2Sm 0W) (I+xs)(icq×(~2)" 8~/2 rrmN

[

e ,,j] r~+r; Pl--p2,~----J 4

e

r

J

4

with the isospin functions T~2 =-- r~r2

+

± rl r2 , 1 -

To=- T~12 +

2 sin 2 0w 1 z z c o s 2 0c

~7"17"2 •

(44)

M. Porrmann / Weak interactions

269

0c denotes the Cabibbo angle, xs and ~v the anomalous isoscalar and isovector magnetic moments of the nucleons, g,~ the strong pion-nucleon coupling constant. The parity-violating pion-nucleon coupling constant f,, takes the value Lf,~[= 4.5 x 10 8 with only charged currents involved 10). Considering neutral currents too, the potential VTeak is enhanced by the factor A,~ which in simplest approximation within the Weinberg model is of f o r m 36) 2 sin 2 0w 3 sin 2 0 c "

A,~=I

(45)

The sign of the product A=f,~g,~ is assumed to be positive 38). Eqs. (43)-(45) present a model for weak potentials including neutral currents with a simple SU(3) treatment of the pion-nucleon coupling and a bare factorization of the vector meson-nucleon coupling. More complicated contributions to the parityviolating vertices may therefore modify the coupling strength as well as the isospin structure of the form of eq. (43). Other calculations of A,, are described in ref. 38), e.g., yielding values of this factor reaching up to 20 and more. However, the most probable value due to these investigations is a factor of about A,, = 10, which is still compatible with data from 19F and ~8F experiments 38). On the other hand, concerning the weak vector meson exchange, there effectively are just minor differences in the range of a few tens of percent comparing the forms of eq. (43) with results of more involved considerations using a simple parton model 38). It is only for rather complicated many-quark contributions, that there arise significant changes from the values of eq. (43), which are yet very uncertain. In subsect. 2.3, the essential role of gauge invariance for elastic electric multipoles has been pointed out. Therefore, instead of the expressions eqs. (7) and (25), the formally equivalent definition of Partovi 31) is used for the deuteron E1 multipole, which explicitly assumes gauge invariance for the currents. This a priori assumption cancels all static contributions of the one-nucleon convection currents. After Fourier transformation of the current operators (27), the deuteron E1 operator in internal coordinate representation in the impulse approximation reads

--

• q

[,

1

1A

(

S

z z 1 . T 1 -- "/'2"~ "]

11(~qr)Y ( r ) x GM(Crl--~r2)+ GV(~rl + ~ r 2 ) ~ } J

i

+4ran

l+r

jl(½qr)Yl(¢) r l - r ; 2

The term ('5~Or - -O/Or) describes the non-local part due to the convection current and acts as derivatives on the radial wave functions of the final and initial state, respectively. The other electric term proportional to ( E f - U i ) / q 2, describing the

M. Porrmann/ Weakinteractions

270

charge density part, vanishes forming the matrix element due to time reversal invariance, because it is essentially equivalent to an elastic C1 multipole. With use of the integrals

G~s~(q) =-I drul,(r)b(~qr) .1

Im ws~(r),

(47)

with regular and parity-violating radial wave functions, the E1 matrix element in the impulse approximation takes the form EX(q) = -

~_~2GSM q ( ~ 0 ~ 0 + ~ / ~ 0 ) + mN

-

v47r

mN

~ G0, -1 - G 2- ,1 )

~/-ff- G'~ q f drrh(~qr)[d(uo(r)+,,/ru2(r))im wdr) -4-~--~mN

-(uo(r)+~/~u2(r)~--~Im

wt(r)] .

(48)

The spin current yields isoscalar as well as isovector contributions, whereas the non-local convection part is purely isovector, because it acts but on the angular wave function. The form eq. (48) accounts for the proper q2 behaviour of E1 at low energies. However, for the reasons already given, the convection current has to be substituted at least by the Siegert current (31), to preserve gauge invariance to the lowest order of the currents, and that means of order q2 of E l . The matrix element of the Siegert current vanishes totally, cf. subsect. 2.3. Therefore the convection part of eq. (48) is completely cancelled by the lowest approximation of both polar and axial meson exchange currents. That means that only the spin current part contributes to E l . On the other hand, the convection part of eq. (48) measures the influence of mesonic processes to E1 to order q2 at low m o m e n t a .

3.3. NUMERICAL RESULTS AND DISCUSSION The amplitudes of the deuteron p-waves which solve eq. (41) with the Reid soft-core potential and the weak potentials of eq. (43) are given in table 1. They are calculated as square roots of the norm of these waves. Besides the Cabibbo model, two values for the e n h a n c e m e n t of the weak pion exchange by neutral currents have been used, that of eq. (45) for sin 2 0w = 0.25 and the factor A,~ = 10, which in some sense gives an upper bound from the experimental point of view. For the quantity Xv, which constitutes the tensor part of the regular p-nucleon coupling, the value 3.7059 of the vector-dominance model as well as the value 6.0 indicated by fits to experimental data 39) is used. The main features of the deuteron p-waves are the values of their amplitudes, which are significantly smaller than those of suitable states of other nuclei lo), due to

271

M. Porrmann / Weak interactions TABLE 1 Amplitudes of deuteron p-waves for different weak potentials, eq. (43)

Model for weak potential

1P1 3p1

xv = 3.7 ×v=6.0

Weinberg-Salam

Cabibbo A,~ = 1

A,, =2.15

A,, = 10

3.43 × 1 0 - 9 6.84×10 9 9.17×10 9

4.34 × 1 0 - 9 8.65×10 9 1.92×10 8

4.34 × 10 9 8.65×10 9 9.22×10 8

t h e a b s e n c e of small r e s o n a n t e n e r g y d e n o m i n a t o r s , a n d the a d d i t i o n a l s u p p r e s s i o n of t h e i s o s c a l a r c o m p o n e n t b e c a u s e of t h e large d - w a v e . T h e w e a k t o - p o t e n t i a l o n l y gives small c o n t r i b u t i o n s . T h e r e f o r e , i n c l u d i n g n e u t r a l currents, t h e p a r i t y a d m i x t u r e to t h e d e u t e r o n is m a i n l y d e t e r m i n e d by t h e w e a k p i o n - e x c h a n g e p o t e n t i a l , a n d t h e c o r r e s p o n d i n g i s o v e c t o r p - w a v e p r a c t i c a l l y is p r o p o r t i o n a l to A,~. T h e d e u t e r o n E1 m a t r i x e l e m e n t (48), in the i m p u l s e a p p r o x i m a t i o n with t h e C a b i b b o m o d e l v e r s i o n of the w e a k p o t e n t i a l (43), (~v = 3.7) is s h o w n in fig. 5, specified b y t h e isoscalar a n d i s o v e c t o r spin p a r t a n d the i s o v e c t o r c o n v e c t i o n part. T h e scale of d i f f e r e n c e s b e t w e e n t h e s e c o n t r i b u t i o n s is given b y t h e c o r r e s p o n d i n g n u c l e o n f o r m f a c t o r s a n d p - w a v e s . A t low m o m e n t a , the t o t a l m a t r i x e l e m e n t a n d t h e

let1 16 9

f-..~ /,' '\.

10-10 f

- ~ ..

"\

: :.'""'""...... ....

0

if:

..'".. .... ........y ........\ /

' " /

"\........

:\

.

I 10

20

3'0

40

q~//fm2

Fig. 5. Elastic deuteron E1 matrix element in modified impulse approximation, eq. (48), with isoscalar (dashed curve) and isovector spin contribution (dot-dashed curve) and convectidn part (dotted

curve), for p-waves to the Cahihbo model, ~v = 3.7. At low momenta, E1 and the isovector spin part are positive, the others negative.

272

M. Porrmann / Weak interactions

dominating isovector spin part are positive, the other ones are negative. The convection part at low energies contributes to about 3% to the matrix element. According to the discussion above, this indicates only little influence of gauge invariance violation, which may be cancelled by use of the Siegert current. A sudden increase of the contribution of longitudinal exchange currents does not seem probable, and therefore the E1 matrix element including the Siegert current, that means only the spin part, should provide a reasonable description up to moderate values of a few fm 1 m o m e n t u m transfer, which is used in the following.

IEII~

NNX N•NN\ ~\ \N\\\ 16_ 11 0

x ~ 10

2'0

i

'~/ II /-,0 IIq ~2//fni2

30

Fig. 6. Elastic deuteron E1 matrix element in the impulse approximation plus Siegert current, for different weak pion-nucleon couplings A= and regular p-nucleon tensor couplings x v = 3.7 (dashed lines) and ~v = 6.0 (solid lines). The matrix elements are positive up to the zeros.

The corresponding E1 multipoles for different p-waves are presented in fig. 6, the respective asymptotic coefficients at low m o m e n t a in table 2. The variation with A= demonstrates the predominance of the isovector spin contribution and the isovector p-wave. With neutral currents included, the isoscalar admixture only is of minor importance at low m o m e n t a . The influence of meson exchange terms will only change little with the weak potential, for it contributes to the isovector part to lowest order. Concluding, the parity-violating E1 matrix element of elastic deuteron scattering up to a few fm -1 m o m e n t u m transfer is mainly determined by the isovector spin current and the isovector weak pion potential. Meson-exchange currents exactly

M. Porrmann / Weak interactions

273

TABLE 2 Long-wavelength limit of elastic deuteron E1 matrix elements for different p-waves, in the impulse approximation plus Siegert current, including the isoscalar, So, and isovector spin current contribution Sx Model for weak potential

Cabibbo Weinberg-Salam A,,=2.15 Weinberg-Salam A~=10

xv

E l / q 2 ( f m 2)

So/q~(fm 2)

Sffq2(fm 2)

3.7 6.0 3.7 6.0 3.7 6.0

9 . 7 6 × 1 0 lo 8.50 x 10 -1° 2.27 x 10 -9 2.11x10 9 1.10x 10 s 1.08x10 s

_ 1 . 3 0 x 10-1o - 2 . 5 6 x 10 - l ° -1.65 x 10 -1° - 3 . 2 4 x 1 0 lo -1.65 x 10 m - 3 . 2 4 x 1 0 ~o

1.11x 10-9 1.11 × 10 -9 2.43 × 10 -9 2.43x10 9 1.11 x 10 s 1.11xl0 8

cancel all convection currents to order qO a n d q2. The qO part is essential, in o r d e r q2 however there only is a small contribution. The contributions of Z - b o s o n and photon exchange to the deuteron asymmetry, eq. (34), from now on will be denoted by A z and A v, respectively. The Z-boson part is described by the W e i n b e r g - S a l a m model, which yields the value -fly~Sd = 3.5 x'10 6 f m 2 for the corresponding coefficient at sin 2 0 w = a1, eq. (6). Variations of the Weinberg p a r a m e t e r in the range indicated by present experiments allow for a change of A z up to about 20% in both directions. Numerical results for the electromagnetic part A v of the asymmetry are shown in figs. 7 and 8. In these and all following figures in this section, the Reid soft-core potential, empirical dipole fit of nucleonic form factors and the empirical value x v = 6.0 for the weak 0-potential are used. The Weinberg p a r a m e t e r is sin 2 0w = ¼. Fig. 7 presents Av at the interesting angle 0 = ~- with three versions of the weak potential. The Z-boson part A z is included for comparison. A v is negative at low m o m e n t a and changes sign twice at zeros of M1 and E l . The extrapolation of the long-wavelength limits of the asymmetry (36), table 3, up to 1 fm 1 agrees with the actual values within 20%, because of partial cancellation of the deviation of single matrix elements from their low-energy behaviour. The pole of the a s y m m e t r y caused by the zero of M1 is of no physical importance, for it is not observable due to a zero of the cross section at the same point. Already the abscissa of the zeros of M1 and E1 are not quite significant, for they will be modified by other processes like mesonic currents, which have not been accounted for. In any case, at higher m o m e n t a , the values of Av in fig. 7 should only be considered as an estimate. However, it does not vary strongly in absolute scale beyond some fm ~ in contrast to A z , which agrees with the theoretical expectation of subsect. 2.2. Fig. 8 pictures the variation of A v at four scattering angles for a weak pion coupling A,~ = 10. For other weak potentials it changes correspondingly. A v is negative except for the region of pole and zero about 40 fm -2. The curves show up the different low-energy behaviour and the distinct e n h a n c e m e n t for backscattering. This

274

M. Porrmann / Weak interactions

Az l(J 4lAd

I&l

1[~ 5 -

Ay

16 6 •

A~ =10 An =10 A~=2.15 A~=I

10 7 .

6

Arc= 2.15

ib

3b

Fig. 7. Elastic deuteron asymmetryby Z-boson {Az) and photon exchange (Av) at 180%for three weak pion-nucleon couplings A= and ;-v = 6.0, in the impulse approximation plus Siegert current for A.,, sin2 0w = ¼.Az is positive, Av negative at low momenta and beyond 42 fm-2. character results first from the function VA, but it is significantly increased by the large difference of the longitudinal and transverse regular form factor of the deuteron, especially at low m o m e n t a 34,40). Table 3 shows the coefficients of the long wavelength limits of A~ for various p-waves. Comparing the Z-boson and photon contribution to the asymmetry at 180 °, fig. 7, A z exceeds A , about two orders of magnitudes at medium m o m e n t u m transfer even for a very large weak pion potential. This difference increases for larger values of q~ and for other angles. At very low m o m e n t a for 180% both contributions converge. In this range, far below 1 fm 1, Av may be expressed by its asymptotic form. Because of its linear variation with q, it always exceeds A z at arbitrary low momenta. This case, however, is of no interest because of tiny values of the asymmetry. Av and A z equal each other for that q equal to the ratio of the coefficients D2 and 3.5 × 10 6 fm 2, that is 0.0097 fm 1 without neutral currents (Cabibbo model) and 0.024 fm -1 and 0.12 fm 1 with A . = 2.15 and 10, respectively (Xv = 6.0). Up from these abscissa the ratio of both amplitudes of the asymmetry decreases inversely to q. At q = 1 fm i the photon part contributes but to about 0.8%, 2% and 11% of the Z-boson part for the three weak potentials.

275

M. Porrmann / Weak interactions

lay[

so° Y

lff 6 -

180" 150 °

4 /7

ld 7 -

~ 3\

\

104 -

\ 16e

15

Ib

2b

3b

~b

q~/f~z

Fig. 8. A , as in fig. 7 at four scattering angles, A,~ = 10, xv = 6.0.

TABLE 3 Long-wavelength limit coefficients of the elastic deuteron asymmetry by photon exchange for different p-waves, in the impulse approximation plus Siegert current, cf. eq. (36) Model for weak potential Cabibbo Weinberg-Salam A,~ = 2.15 Weinberg-Salam A,, = 10

Xv

Dl(fm 3)

D2(fm)

3.7 6.0 3.7 6.0 3.7 6.0

-4.09 × 10 -1° - 3 . 5 6 × 1 0 lO -9.51 × 1 0 - 1 ° - 8 . 8 4 x 10 -1° -4.59 × 1 0 - 9 -4.52 x 10 9

-3.91 x 10 -8 -3.40x10 8 -9.08 × 10 -8 -8.45 × 10 8 -4.38 x 1 0 - 7 -4.32 × 10 7

C o n c l u d i n g , t h e Z - b o s o n e x c h a n g e c a u s e s t h e p r e d o m i n a n t p a r t of t h e a s y m m e t r y of e l a s t i c s c a t t e r i n g f r o m d e u t e r o n . It is o n l y at 0 = 180 ° a n d e n e r g i e s b e l o w 100 M e V t h a t an a x i a l e l e c t r o m a g n e t i c n u c l e a r c u r r e n t h a s to b e c o n s i d e r e d f o r a v e r y s m a l l a s y m m e t r y . T h i s s m a l l e f f e c t o f a n axial c u r r e n t r e s u l t s f r o m a v e r y s m a l l a m p l i t u d e of p a r i t y a d m i x t u r e s , p a r t i c u l a r f o r t h e d e u t e r o n . O n t h e o t h e r h a n d , this

M. Porrmann / Weak interactions

276

specific situation allows for a very clean measurement of a weak electron-deuteron interaction which almost exclusively measures the polar weak isoscalar current. In order to find the kinematical region most suited for a measurement, the statistical sensitivity A 2 do'/dO should be considered, accounting both for a large physical effect and a sufficient statistical significance, cf. subsect. 2.2. It is given for the Z-boson contribution to the asymmetry in fig. 9 at four angles with sin 2 0w = ~. The strong decrease at higher momenta is caused by the cross section, the increase at very low momenta by the q~ behaviour of the asymmetry. The asymptotic relations are t6.5 A2 do" , dS2 q~O

× 10 -12 -t x- bc o t

2 ½0.q~_~ for 0<< 77", 2

sr

Ira-

(49)

4 6 . 9 x 1 0 1 4 / x b qu sr fm 4

for O~rr

2d

10 '1

1(~12

1013 . 30 °

ld u'

90*

150 °

6

~b

2b

3b

~

q~/f~

Fig. 9. Statistical sensitivity for the Z-boson part of elastic deuteron asymmetry at four scattering angles, sin 2 0w = ~.

M. Porrmann / Weak interactions

277

At fixed angle the maxima of A2z do-/dO are about 0.9 fm 2 for small and medium 0 and increase to 3 fm -2 for backscattering. Therefore, the asymmetry by Z-boson exchange should be measured at small scattering angles and a momentum transfer of about 1 to 10 fm -z. However, for extremely small angles the energy of the electrons necessary to achieve a given m o m e n t u m transfer will increase rapidly.

1(~ 18

10-~9

10-20

10-zl

lb

2'0

a'o

4b

Fig. 10. Statisticalsensitivityfor the photon part of elastic deuteron asymmetryat four scattering angles, A~ as in fig. 8. The same quantity A~ do'/dO for the photon contribution to the asymmetry is given in fig. 10. The enhancement factor used is A~ = 10. The decrease of the statistical sensitivity at higher momenta is more rapid than for the Z-boson part, because A v does not increase any longer. The long-wavelength limit is for 0<< zr

D3~-y~q.

l cos1 2v 4 A? do, ~d,(2 q~O [ D 1 ]

4~q~x

(50) 2

for O=zr.

M. Porrmann / Weak interactions

278

TABLE 4 Long-wavelength limit coefficients of the statistical sensitivity of the asymmetry of table 3, cf. eq. (50)

weak potential

Cabibbo Weinberg-Salam A.=2.15 Weinberg-Salam A==10

nv

3.7 6.0 3.7 6.0 3.7 6.0

3\~--

•

,m4)

8.89x 10-20 6.74 × 10-2o 4.81 × 10 -19 4.16×10 19 1.12 × 10 17 1.09×10 17

8.48× 10 18 6.43 × 10-18 4.59 × 10 17 3.97x10 17 1.07 x 10-15 1.04×10 15

The coefficients D3 and D4 are given in table 4 for different parity admixtures. Despite the very strong decrease of the cross section with increasing scattering angle, the sensitivity achieves its maximum at 0 ~ 7r and a m o m e n t u m transfer of about 0.7 fm -2. As in most other cases the photon contribution causes but a correction to the asymmetry by weak electron-deuteron interaction in this domain. However, the quantity A 2 d o ' / d O indicates that this correction to A should best be measured at backscattering angles and a m o m e n t u m transfer between 0.02 and 1 fm -2. With a weak pion coupling of A,~ -- 10, the photon contribution should here cause from 90 to 10% of the neutral weak currents part, but of opposite sign according to the positive sign of A,~f~g~ [ref. 38)]. There are two other figures, which show the angular distribution of the asymmetry, fig. 11, and the corresponding statistical sensitivity, fig. 12, for incident energies of 20, 100, 300 and 1000 MeV, for the Z-boson part A z (solid lines) and the photon part Av (dashed lines), with A,, = 10. A z is always positive, A v negative except for the right branch of the curve with Ee -- 1000 MeV. With increasing scattering angle, the m o m e n t u m transfer monotonously increases up to a value determined by the electron energy. Therefore, figs. 11 and 12 demonstrate the same characteristics as the discussion above: • 2 (a) the implicit dependence of A z on the scattering angle is only caused by its q , behaviour, (b) the asymmetry of photon exchange is strongly enhanced at 180 °, at 1000 MeV interrupted by a zero of the transverse multipole, (c) the small difference between A z and Av at 180 ° at small energies, that means very small m o m e n t u m transfer, and its strong increase for larger energies, (d) the different weight of the cross section for the statistical sensitivity of A z and A v and the consequence in measuring the Z-boson contribution at small angles, (e) the range most suited to measure the photon part of the asymmetry at about 180 ° and 100 MeV, that means q~ = 1 fro-2; however, there is little probability of measuring a correction of about 10% to an asymmetry of some 10 6.

279

M. Porrmann / Weak interactions

IA, I

IA I

1000 MeV

104.

300MeV 10.5 .

100 MeV

11~6-

/

300 MeV

.

,,--1000 MeV //t

/

10s .

//

/

,I .,

/ ld 9.

/

/ 1000NeV~

10-7 .

/t

\ /I "X,// /

~

I

/

/ / l

I I

o"

3b"

20Mev20 MeV

ii ,, I

I

/ f

100 MeV

.'

,I I

/

I

I

// /

I

/ /

/

I

,/

/

6(3°

9'0"

120"

e

Fig. 11. Elastic deuteron asymmetry by Z-boson (solid lines) and photon exchange (dashed lines) at four incident energies Ee; A z and Av as in figs. 8 and 9. A z is positive, Av negative except for the right branch at 1000 MeV.

The approximations made in deriving the asymmetry by weak electron-deuteron, interactions are not expected to have any considerable effect. Axial isoscalar weak currents are very unlikely. If there are any tiny currents of this type, they would only give very small contributions to the asymmetry at large angles. The uncertainty of the prediction of the Weinberg-Salam model for the exact strength of the polar isoscalar nuclear current and the axial electronic current may be accounted for by a variation of the parameter sin 2 0w. Concerning the contribution of the electromagnetic electron-deuteron interaction, only corrections up to 10 fm -2 are interesting for the asymmetry. Thus, the use of a certain nucleonic form factor and the neglect of mesonic exchange currents, which become relevant only for high momentum transfer, is unimportant. This justifies the approximation of longitudinal mesonic currents by the Siegert form. Contributions of transverse mesonic currents should not exceed 10% of the matrix elements. The quantity most uncertain for A~ then

280

M. Porrmann / Weak interactions

A•,z 1(~ 12

1¢ 3

100 MeV 300 MeV

/ 10~

I00 MeV

/

20 MeV

/

/

i 20 M e V

/

Id '7

/ I000

1

,

o*

MeV

-

-

/

\

/

/

/,~

60 °

~ . ~ \~

N

/

~

,/

90*

~ -4- 300MeV

-/*'-

//

~./

ao*

-

I

I

120°

' ~ 1000 MeV

lSOO lgOO

e

Fig. 12. Statistical sensitivity of the asymmetries of fig. 11. r e m a i n s t h e e x a c t f o r m of the w e a k p o t e n t i a l , e s p e c i a l l y t h e s t r e n g t h of the p a r i t y - v i o l a t i n g p i o n - n u c l e o n coupling.

4. Deuteron disintegration 4.1. DIFFERENCE FROM THE ELASTIC SCATTERING In inelastic s c a t t e r i n g , t h e d e u t e r o n passes to a s c a t t e r i n g s t a t e of a c o n t i n u o u s s p e c t r u m of r e l a t i v e e n e r g i e s E ' , w h i c h causes a m o d i f i c a t i o n of t h e cross section (9). D e n o t i n g t h e m o m e n t u m of the r e l a t i v e p r o t o n - n e u t r o n s t a t e b y p ' a n d the n o n - r e l a t i v i s t i c k i n e t i c e n e r g y of the r e l a t i v e a n d t h e c e n t e r of m o m e n t u m s y s t e m b y E' and E ..... pt2

E'------

mN ,

q 2 E ....

~4--Nm '

(51)

t h e d i f f e r e n t i a l cross s e c t i o n for s c a t t e r i n g to a c o n t i n u u m state of fixed r e l a t i v e

281

M. Porrmann/ Weak interactions

energy E ' reads ( dcr

)

~ e2. m .

d~d-E; R-3E~

cos2 10

mNIp'I

sin4~0 I + ( E

....

+E')/mg

L 2

2

x F, e g + F i r r e g + ( a ~ ) e m z

[see refs. 29.41) for the electromagnetic part]. The form factors F~ are the same as in eq. (10) except for an additional summation over all partial components 1O{s(E')) of the scattering state with energy E'. Thus the products of multipoles AJ(q) and B j (q) are to be understood as A JBJ * = E ( O { s ( E ' ) l l A J l l d ) ( O { s ( E ' ) l l B q l d )

* •

(53)

Isi

AJ and/~J are the corresponding operators. There are no interferences of multipoles of different tensor character because of the summation over all nucleonic degrees of freedom. In contrast to elastic scattering, energy and momentum transfer may in principle be varied independently. Therefore the factors q ,2 / q 2 in the kinematical functions may significantly differ from the order of magnitude 1. The asymmetry of the deuteron disintegration by polarized electrons takes the form of eq. (13) with the modified form factors. Because of the superposition of different spins and isospins in the scattering state, all parts of the nuclear currents may contribute to the asymmetry. In addition to the elastic scattering, there are polar isovector as well as axial nuclear currents, and the different kinematical behaviour of these currents and the polar isoscalar one will cause another dependence of the asymmetry on the momentum transfer besides the global q~ factor. The part of axial electromagnetic nuclear currents in the asymmetry has not been considered in the following numerical calculations. According to the results of Lee 12), parity admixtures at the disintegration threshold provide an asymmetry of order 10 7 to 10-8, and this effect is very unlikely to increase rapidly for larger energy transfer. Such an asymmetry certainly cannot be observed at present. 4.2. NUMERICAL RESULTS AND DISCUSSION The multipoles for the deuteron disintegration are again calculated in the impulse approximation of the nuclear currents (27). For the form of the matrix elements see ref. 29). Wave functions due to the Reid soft-core potential and the empirical dipole fit for the nucleonic form factors are used. The multipoles are considered up to order J = 5 and partial waves up to total spin / = 6. There is a simple relation between the transverse magnetic (electric) multipoles with axial currents and the spin current part of polar electric (magnetic) ones. Because of general vector properties, they differ by only a factor (gA/GM)2mN/[ql where gA and GM are the axial and polar form factors. Therefore the nuclear form

M. Porrmann / Weak interactions

282

E ' = 3 MeV 10 0 .

,o'4 /

\\

9 's T,{V . IV A

162

o

2b

3b

/rm-2

Fig. 13. Contributions to the deuteron disintegration form factors at an energy of the continuum state E ' = 3 M e V without nucleonic form factors; O, Tv and TA denote C o u l o m b and transverse polar and axial terms, IS and IV the isoscalar and isovector parts, respectively.

factor of interference with axial nucleonic currents F~,Z can be calculated from that of purely polar isovector spin currents. In figs. 13, 14, the products of matrix elements of the disintegration form factors are given for energies of the continuum state E ' = 3 M e V (fig. 13) and E ' = 100 M e V (fig. 14). These matrix elements do not yet contain the m o m e n t u m dependence of the single-nucleon form factors, which causes an additional decrease of all terms at higher m o m e n t u m transfer. O Isav denote the Coulomb parts of isoscalar and IS,IV isovector currents, respectively, T*V.A the transverse parts of purely polar currents (index V) and of interference of polar and axial currents (index A). The weights of the weak currents are also not contained in these terms. Thus, the form factors F; read FrOg -

4 ~ { Vc(O IS +O I V )+VT(Tv+TbV)} IS 2J; + 1-

F~Z

2 J4rr i+l{VC(ylSolS+yIvOIv)+

=

F~=

4~ 2./;+1

VA(6tST~+8,VT~AV).

,,;

VT~ ~

is~1s IV

+ yXVTIvV )}

,

(54)

M. Porrmann / Weak interactions

10 0

283

E ' : 100 MeV

lO-1

15z

103

1(~4

0

10

20

30 ~'/fm-2

Fig. 14. Same as in fig. 13 at E ' = 100 MeV.

Isoscalar axial currents are not considered again. The characteristic property of all matrix elements is a distinct m a x i m u m in the region of quasielastic scattering. The asymptotic decrease to zero at high m o m e n t a is only interrupted for the isovector transverse terms at low energies due to a zero of the p r e d o m i n a n t M1 transition to the 1S0 continuum state at about 12 fm -2. At low energies E ' all matrix elements except transverse isoscalar terms are of similar order in the region of the maximum, with a slight p r e d o m i n a n c e of the axial part. At m e d i u m m o m e n t u m transfer, the isoscalar C o u l o m b contribution QZS becomes the strongest one because of the minimum of the transverse isovector parts, whereas at large m o m e n t a q2 the transverse polar isovector term T~vv dominates at all energies, followed by the axial contribution T ~v. Increasing the relative energy E ' , C o u l o m b and transverse i s o ' e c t o r parts remain of similar order at low m o m e n t a . However, in the quasielastic domain, which now takes larger values of q2, polar and axial transverse isovector terms b e c o m e the leading contributions. Evaluating the a s y m m e t r y these interference terms are weighted differently by the kinematical functions and the relative strength of neutral weak currents of electrons

M. Porrrnann / Weak interactions

284

a n d n u c l e o n s . In the m o d e l of W e i n b e r g a n d Salam, it takes the form A = 7.00x

10 -6

q~R,

a VA T TM - Vc( y'S orS + ylV o R

TM)-

VT( ytSTIvs 't y I V , X- rlIvV )

V c ( o l S + oXV)+ V v ( T ~ + T TM)

--=

a = 1 - 4 sin z 0 w ,

yls = - 2 sin 2 0 w ,

,

(55)

yTv = 1 - 2 sin 2 0w •

T h e d e p e n d e n c e of the factor R on the W e i n b e r g angle does n o t simplify to a u n i q u e factor like in the elastic case, for a v a r i a t i o n of sin z 0w varies the linear c o m b i n a t i o n 1 of the matrix e l e m e n t s . A t sin 2 0w = ~ axial n u c l e a r c u r r e n t s do n o t c o n t r i b u t e (a vanishes) a n d ylS a n d Try are e q u a l besides an opposite sign. I n c r e a s i n g the •

W e i n b e r g angle causes o p p o s i t e a b s o l u t e changes of y

IS

a n d y~v a n d a rapid increase

of a ; the influence of axial a n d p o l a r isoscalar c u r r e n t s is increased, that of polar isovector t e r m s decreases. F o r sin 2 0w -> 1 all isoscalar terms yield positive c o n t r i b u tions to the a s y m m e t r y , those of axial a n d p o l a r isovector terms are negative. Figs. 15 a n d 16 p r e s e n t the a s y m m e t r y of d e u t e r o n d i s i n t e g r a t i o n (55) at energies of the c o n t i n u u m state E ' = 3 M e V (fig. 15) a n d E ' -- 100 M e V (fig. 16), scattering

IAI 10-4

165

10 6

id 7 0

10

20

30 ~Tfrfi 2

Fig. 15. Deuteron disintegration asymmetry at E' = 3 MeV, eq. (55). Curves: a for R = 1, b, c, d at 180° I 1 2 for sin2 0w = X, g, _~,and e, f, g the same at 10°. Curves e, f, g have to be multiplied by a factor 10. All curves are to be taken negative except a, g and the left branches of e, f.

M. Porrmann / Weak interactions

IAI

E' = 100 MeV

285

Q

-4

lO

l°sl

c

i07J 0

I

10

2'0

30 -~2/frri2

Fig. 16. Same as in fig. 15 at E' = 100 MeV. All curves are to be taken negative except a. angles 0 = 180 ° (curves b, c, d) and 0 = 10 ° (e, f, g), and for the value R = 1 (a). T h e r e are three values used for the W e i n b e r g p a r a m e t e r , sin 2 0w = 1 (b, e), .~ (c, f) and 2 (d, g) for different c o m b i n a t i o n s of weak currents. T h e three curves for 0 = 10 ° have to be multiplied by a factor 10. T h e a s y m m e t r y is negative in all cases except curves a and g and the left branches of curves e and f in fig. 15. A t 3 M e V relative e n e r g y (fig. 15) and 0 = 10 ° the a s y m m e t r y reveals a distinct positive m a x i m u m in the region of 1 2 - 1 6 f m -2 with values a b o u t ( 4 - 6 ) × 10 `5 d e p e n d i n g on sin 2 0w. This structure is mainly caused by the contribution of the isoscalar C o u l o m b part. A t higher m o m e n t a the transverse isovector terms b e c o m e p r e d o m i n a n t and change the sign of the a s y m m e t r y . T h e position of this zero strongly d e p e n d s on the W e i n b e r g p a r a m e t e r . O t h e r processes such as mesonic currents, which have b e e n neglected in this w o r k will p r o b a b l y give rise to some considerable contributions in this region 29) and therefore diminish the large p r e p o n d e r a n c e of the isoscalar C o u l o m b terms in the asymmetry. A t larger angles the axial terms contribute and the C o u l o m b parts decrease. T h e r e f o r e the region of positive a s y m m e t r y decreases, and it changes sign at low m o m e n t a too. A t backscattering the C o u l o m b terms vanish and the very small transverse isoscalar part is the only positive contribution to A, which yields a n a r r o w

286

M. Porrmann / Weak interactions

region of positive a s y m m e t r y around 12fro 2, not contained in flae figure. For the same reasons as mentioned above, the results for this region are rather uncertain. However, at all other m o m e n t a q2 the a s y m m e t r y should significantly be negative, and it exceeds that at 10 ° by roughly a factor 2 in absolute value for sin 2 0w = ¼. The approximate independence of the asymmetry from the Weinberg angle at backscattering in the range about 2 0 - 3 0 fm 2 results from the properties of ~ and y I V and the numerical relation of T~v and T~v. Varying sin 2 0w changes the contributions of axial and polar isovector terms in opposite direction with a factor 2 in favour of the axial part. Since this axial part is approximately half of the polar one, both variations cancel to a large extent. At 100 M e V relative energy (fig. 16) the a s y m m e t r y is negative at all m o m e n t a due to the p r e d o m i n a n c e of isovector terms. However, at forward scattering they are considerably weakened by isoscalar C o u l o m b terms. This also explains the strong dependence on the Weinberg angle. Varying sin 2 0w changes isoscalar and isovector contributions in the same direction. This kinematical region may therefore be suited to measuring the Weinberg parameter. For backscattering the a s y m m e t r y differs little from that at low energies, because the differences of the form factors in absolute value cancel calculating the ratio R. For sin 2 0w = ] the asymmetry at 180 ° is about 3 0 - 5 0 % larger than that for 10 ° . Concluding this discussion, at large scattering angles the asymmetry of deuteron disintegration does only vary little with the energy of the continuum state. Moreover, at medium m o m e n t u m transfer it is practically independent of the free p a r a m e t e r of the W e i n b e r g - S a l a m model. The a s y m m e t r y measures the polar isovector current together with a small amount of the axial nucleonic current. This supplements the case of elastic scattering, where only the polar isoscalar current can be observed. In forward scattering, there is a distinct dependence on the Weinberg angle. At low and medium m o m e n t u m transfer the a s y m m e t r y measures the isoscalar current at small energies and both isospin parts of the polar current at higher energies. However, the theoretical prediction for this angular region is not very certain, because the a s y m m e t r y is rather sensitive to small corrections. On the other hand, it is just this region where the isospin structure of the weak polar current may be observed. In order to determine a kinematical region suited for a measurement, the statistical sensitivity A 2 do-/dO has to be considered again. According to the discussion of subsect. 2.2, the a s y m m e t r y should be measured in the region of the quasielastic peak and at m o m e n t a little beyond it. The angular variation of the sensitivity is less than in the elastic case for there is only a minor difference between the longitudinal and transverse regular form factor. For fixed m o m e n t u m transfer and scattering energy the sensitivity increases with decreasing scattering angle. The variation of the factor sin -4 ½0, however, is weakened by the fact that the electron energy needed for the other conditions increases too. Therefore, by these qualitative arguments, at variable

M, Porrmann / Weak interactions

287

electron energy the a s y m m e t r y should be measured at small angles around the quasielastic peak. Finally, the angular distribution of the a s y m m e t r y and the corresponding statistical sensitivity are given in table 5 for fixed initial energies of Ee = 20 M e V and 150 M e V and three values of the Weinberg p a r a m e t e r . The cross sections for right- and left-handed electrons are first integrated over the whole energy range for the continuum states before evaluating the ratio of eq. (55). All scattering states considered are purely nucleonic. The four m o m e n t u m transfer qZ possible at these incident energies is rather small. Therefore the asymmetry mainly contains those contributions with small energy transfer, that means small continuum energy E', and it significantly increases with the scattering angle. The negative sign is due to the p r e d o m i n a n c e of isovector currents in all form factors at these kinematical conditions, except the region of 90 ° for E e - - 150 MeV, where the isoscalar Coulomb term becomes larger than the isovector Coulomb part. TABLE 5 A n g u l a r d i s t r i b u t i o n of t h e a s y m m e t r y of d e u t e r o n d i s i n t e g r a t i o n f o r t h r e e W e i n b e r g a n g l e s a n d t h e corresponding statistical sensitivity

A2~(p,b

A forsin 20w=

Ee

• sr - t ) f o r sin 2 0 w =

(MeV) 0.25 20

150

10 ° 50 ° 90 ° 135 ° 180 ° 10 ° 50 ° 90 ° 135 ° 180 °

-6.9×10 -1.6x10 -4.3×10 -6.8×10 -7.6x10 -5.1×10 -7.5×10 -1.2×10 -3.7x10 -5.2x10

0.33 -1° -8 -8 _8 -8 -8 _7 _6 -6 -6

-9.6x10 -2.4x10 -1.1×10 -2.5x10 -4.6×10 -3.7x10 -5.8x10 -2.0x10 -6.9x10 -1.0X!0

0.40 lo -8 7 -7 -7 8 -7 -6 -6 5

_1.2×10-9 -3.0×10 8 _1.6x10-7 -4.0×10 7 - 7 . 7 × 1 0 -7 _2.7×10-8 -4.5×10 7 - 2 . 6 × 1 0 -6 - 9 . 5 x 1 0 -6 _l.4xl0-S

0.25 1.8×10 3.8×10 5.8×10 5.3×10 3.5×10 3.2x10 5.6×10 3.5×10 6.9x10 9.5x10

17 16 16 -16 16 -14 -13 13 13 -13

0.33

0.40

3 . 4 × 1 0 17 8.6×10-16 3.3×10-15 7 . 5 × 1 0 15 1.3×10-14 1 . 7 × 1 0 14 3 . 4 x 1 0 -13 8.9×10-13 2.4x10-12 3 . 6 x 1 0 t2

5.2×10-17 1 . 4 × 1 0 15 7.1x10-15 1.9x10-14 3 . 7 x 1 0 1,* 8.9x10-15 2 . 0 × 1 0 13 1 . 5 × 1 0 12 4.5x10-12 7.0x10-12

T h e s e d a t a r e p l a c e t a b l e 1 o f ref. 42), w h i c h c o n t a i n s a m i s p r i n t at 2 0 M e V a n d 9 0 °.

Axial weak nuclear currents are rather important for the a s y m m e t r y especially at very low energies, if the weak electron current contains polar pieces at all, that means sin 2 0w # ¼. This yields a strong dependence of the asymmetry on the Weinberg p a r a m e t e r even at small angles and an average value of the ratio R which significantly exceeds 1 for Ee = 20 MeV. The rather large influence of the interference with axial weak nuclear currents results from the predominance of the axial E1 transition to the s-wave components of the scattering state at threshold and low m o m e n t a q, which e.g. in n e u t r i n o - d e u t e r o n disintegration determines the cross section rather completely 43).

288

M. Porrmann / Weak interactions

At fixed momentum transfer q2, the asymmetry only increases little at the transition from small to large scattering angles, figs. 15 and 16. In contrast, at fixed initial energy Ee this difference yields about two orders of magnitude due to the global q~ dependence. Because the cross section does not decrease to this extent, the statistical sensitivity takes its maximum at 180 ° . Therefore the asymmetry should be observed at backscattering and a fixed initial energy of about 150 MeV e.g. It mainly measures the isovector part of the polar nucleonic current together with the axial nucleonic current, if the weak current of the electron contains the corresponding polar term. For very small energies such as 20 MeV, however, an asymmetry measurement seems to be extremely difficult. Additionally, at this scale there certainly are contributions of axial electromagnetic processes to be considered 12), which have not been included here.

5. Conclusion

The aim of the present work has been to analyse which information on weak interactions is provided by scattering of longitudinally polarized electrons from light nuclei in general and from deuterons in special. Parity-violating amplitudes of weak electron-nucleus interactions mediated by Z-boson exchange and of axial electromagnetic nuclear currents induced by weak meson-nucleon interactions interfere with the regular electromagnetic amplitudes. They result in additional form factors of the cross section, which give rise to an asymmetry of the cross section for both electron helicities. The contribution of weak electron-nucleus scattering to the asymmetry is of order 2 7 x 10 6 fm 2 q~, that of electromagnetic electron-nucleus interaction is of order ~ of the amplitude of parity mixtures of nuclear states and parity-violating mesonexchange currents. This global structure is modified by the ratio of matrix elements of the different currents involved. At small scattering angles, the polar weak nuclear currents are predominant in the asymmetry, which measures the isospin mixing of the transition current, and, in suited cases, the relative strength of both isotensor currents without involving the nuclear structure in detail. Axial nuclear currents in the asymmetry are restricted to backward angles. The main difference of contributions of weak and electromagnetic nuclear currents to the asymmetry is their momentum dependence. At large momentum transfer, the Z-boson part always dominates. The relation at small energies depends on the transition symmetry and the amplitude ~. Especially for small electric excitations of the nucleus, the photon part always dominates. Gauge invariance is of essential importance calculating transverse electric multipoles of axial electromagnetic currents in elastic scattering. The asymmetry of elastic deuteron scattering almost exclusively measures the isoscalar polar current. It has a high statistical sensitivity at small angles and a m o m e n t u m transfer about 1 fm 1. Electromagnetic processes only contribute at

M. Porrmann / W e a k interactions

289

large angles a n d low energies. T h e d e u t e r o n is just an e x a m p l e , w h e r e there is practically n o c o m p e t i t i o n of Z - b o s o n a n d p h o t o n c o n t r i b u t i o n s . It allows for a clean test of netitral w e a k i n t e r a c t i o n theories p r e d i c t i n g a specific k i n e m a t i c a l d e p e n dence, i n d e p e n d e n t of the d e u t e r o n structure. T h e a s y m m e t r y of d e u t e r o n d i s i n t e g r a t i o n up to m e d i u m energies c o n t a i n s all pieces of the w e a k currents. By v a r i a t i o n of the k i n e m a t i c a l ranges it allows for i n d e p e n d e n t i n f o r m a t i o n s o n different c o m b i n a t i o n s of these currents. T h e a s y m m e t r y m e a s u r e s the isospin s t r u c t u r e of the polar n u c l e a r c u r r e n t s at small angles, a n d p o l a r a n d axial isovector c u r r e n t s at large ones. B a c k s c a t t e r i n g the a s y m m e t r y also e n a b l e s a distinct test of the W e i n b e r g - S a l a m m o d e l at m e d i u m m o m e n t u m transfer. T h e a u t h o r wishes to t h a n k M. G a r i for the i n s p i r a t i o n of this work a n d m a n y fruitful discussions. I also t h a n k W. MiJller for discussions a n d B. S o m m e r for the c o l l a b o r a t i o n on d e u t e r o n d i s i n t e g r a t i o n .

References

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