Neutral current effects in nuclei

PHYSICS REPORTS (Review Section of Physics Letters) 50, No. 1 (1979) 1-85. NORTH-HOLLAND PUBLISHING COMPANY

NEUTRAL CURRENT EFFECTS IN NUCLEI* T.W. DONNELLY and R.D. PECCEIt Institute of Theoretical Physics and Department of Physics, Stanford University, Stanford, California 94305, U.S.A. Received 28 March 1978

Contents: Introduction Plan of this review and summary of results 1. Gauge theory models of weak and electromagnetic interactions: Construction of the minimal model 2. Gauge theory models of weak and electromagnetic interactions: Non-minimal models 3. Semi-leptonic weak and electromagnetic interactions in nuclei 3.1. General considerations 3.2. Nuclear reactions 3.3. Single-nucleon matrix elements 3.4. Nuclear many-body problem 3.5. Long wavelength limit 3.6. A simple example: The A = 3 system 4. Low-energy neutral current neutrino scattering from nuclei

3 4 6 16 27 27 31 34 36 38 41 44

4.1. Relationship among low-energy inelastic weak and electromagnetic processes 4.2. Relationship among low-energy elastic or isoelastic weak and electromagnetic processes 4.3. Single-particle matrix elements in the long wavelength limit 4.4. Selected examples 5. Intermediate-energy neutrino scattering from nuclei 5.1. General relationships among weak and electromagnetic processes at intermediate energies 5.2. Applications to specific nuclei 6. Neutral current effects in nuclei not induced by neutrinos Appendix A: Conventions Appendix B: Single-particle matrix elements References

44 50 54 56 64 65 67 75 80 81 82

Abstract: We review the effects of weak neutral currents in nuclei and show how different nuclear processes can sensitively test gauge theory models of the weak and electromagnetic interactions. Our attention is focused principally on neutral current neutrino interactions in nuclei, although one chapterof our review is devoted to weak neutral current effects in polarized electron scattering off nuclei.

Work supported in part by the National Science Foundation grant PHY 77—16188. t Permanent address after Sept. I, 1978: Max Planek Institut für Physik und Astrophysik, 8 München 40, West Germany.

*

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NEUTRAL CURRENT EFFECTS IN NUCLEI

T.W. DONNELLY and R.D. PECCEI Institute of Theoretical Physics and Department of Physics, Stanford University, Stanford, California 94305, U.S.A.

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

3

Introduction A great deal of our understanding of charge-changing weak interactions has come from careful studies of weak decays in nuclei. In this review we shall try to show that the study of neutral current processes in nuclei can play an analogous role in elucidating the structure of the weak neutral current. We want to discuss neutral current interactions in the context of spontaneously broken gauge field theories of the weak and electfomagnetic interactions. Although the prevailing opinion is that these field theories probably constitute the correct theoretical foundation for a theory of the weak interactions, there is not yet unanimous agreement on what is the “correct” gauge theory model.* Most of the present day data agree quite well with the original SU(2) x U(l) model, proposed sometime ago by Weinberg and Salam. However, there are some troublesome loose ends. For example, some atomic parity violation experiments, if taken at face value, appear to be in serious disagreement with the original model. Furthermore, recent experiments seem to indicate the need for more leptons and quarks which necessitate a further enlarging of the model. Because of these (real or apparent) difficulties of the SU(2) x U(l) model, theorists have proposed a variety of other gauge theory models, which need to be tested with existing data. All gauge theory models are built in such a way that they reproduce ordinary charge-changing reactions. Thus to select among them necessitates quite clear-cut experimental information on the structure of the weak neutral currents. This information is difficult to obtain experimentally, since one must either look at reactions involving neutrinos, and hence measure very low cross sections, or else measure very tiny asymmetry parameters. Furthermore, it is also difficult to unravel the experimental information so as to be able to distinguish among different gauge theory models. This is because, in most experiments, several pieces of the weak neutral current can contribute and models are then not sharply differentiated. Studying neutral current effects in nuclei is not substantially easier experimentally (nor is it orders of magnitude harder!). However, nuclei can act as marvelous “filters” in which, under suitable circumstances, only given components of the weak nuclear current are effective. Hence, they offer the advantage that they can more readily select among competing gauge models of the weak interactions. In writing this review we have been keenly aware of the large gap that exists between nuclear and particle physicists in their approach to weak processes. On the one hand, some nuclear physicists appear to have adopted the attitude that because in nuclei the effective weak interaction is well described by a current-current Hamiltonian all the gauge theory “theoretical baggage”, including new quark degrees of freedom, is largely irrelevant for their purposes. On the other hand, some particle physicists seem to have an innate belief that nuclear physics is “dirty” and that it therefore cannot possibly tell them anything of importance. We have strived hard to bridge this gap and expose as fallacies these prejudices. The structure of gauge theory models does influence markedly the nature of the weak neutral current in nuclei and hence is relevant for a nuclear physicist. Conversely, low-energy neutral current neutrino scattering processes in nuclei, or the measurement of certain polarization asymmetries in electron scattering, are processes which can be calculated in essentially a model-independent way once analog f3-decay and/or electron scattering data are known. Thus these processes should be relevant for particle physicists because they do provide clear tests of competing gauge theory models of the weak interactions. Indeed, as we particularly emphasize in section 4, low-energy neutral current neutrino excitation of nuclear states can provide evidence for the existence of quark degrees of freedom beyond charm. * Since this report was written, new evidence has appeared whichstrongly favors an SU(2) X U(l) gauge theory. We have summarized these recent developments in an addendum at the end of section 2.

4

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

Plan of this review and summary of results This review has been divided into six separate sections which, although interconnected, can also be profitably read independently. Sections 1 and 2 deal with the construction and principal properties of various gauge theory models of the weak and electromagnetic interactions. Section 3 describes the methods needed to analyze semi-leptonic weak and electromagnetic interactions in nuclei. Sections 4 and 5, which form the core of our report, apply the formalism of the preceding chapters to neutral current neutrino scattering in nuclei. Finally, section 6 discusses how the structure of the weak neutral current can be deduced from the study of polarized electron scattering off nuclei. In section 1 we detail, in a simple context, what are the principal ingredients that go into constructing a gauge theory model of the weak interactions. In this chapter we construct, what we have called, the minimal model. This model is just the one proposed originally by Weinberg and Salam in which one incorporates hadrons through the GIM mechanism. The gauge group is SU(2) x U(l). In section 2 we examine how the minimal model fares with experiment and discuss a variety of other gauge theory models of the weak interactions. These models have been proposed either to explain some experimental facts which are not so easily accounted for in the minimal model, or are enlargements of the minimal model to provide for new quark or lepton degrees of freedom. In this section we describe in some detail both models based on SU(2) x U(1) and models based on larger weak groups, notably SU(2)L x SU(2)R x U(l) and SU(3) x U(1). At the end of this section we summarize, for all the models discussed, what is the expected structure of the effective weak neutral current in nuclei. Table 2.1 indicates how widely the various models differ in the amounts of vector and axial-vector, isoscalar and isovector weak currents that they possess. Section 3 describes the general formalism needed for calculating weak and electromagnetic processes in nuclei. We discuss the decomposition of the hadronic current into multipoles and present formulas for the principal processes of interest in nuclei, namely, electron and neutrino scattering, yand a-decay and /.L-capture. The multipole matrix elements taken between single-particle states are displayed in section 3.3 and the nuclear many-body problem is addressed in section 3.4. We show here that, as long as the hadronic current operators are taken to be one-body operators and we do not truncate the space of single-particle basis states used, no approximations enter into the expressions for the matrix element of multipole operators taken between nuclear states. These matrix elements are given in terms of sums of single-particle matrix elements, known from section 3.3, weighted by numerical coefficients, the one-body density matrix elements. Although in a realistic case one is forced to truncate the single-particle basis to a manageable size, one can still accurately interrelate various processes. We discuss in particular how, with a truncated basis, one can hope to compute neutral current neutrino scattering to a 10—20% accuracy by first determining the relevant density matrix elements from electron scattering and/or f~-decay.An example of how this procedure works in the case of the A = 3 system is presented at the end of this section. These techniques are extensively used in section 5. Section 3.5 discusses the long wavelength limit of the multipole operators. This limit is important for studying low-energy neutrino scattering where one finds that effectively only a few of the multipoles contribute and that many processes are interrelated. This is a subject which we discuss in much more detail in section 4. Section 4 deals with low-energy neutrino scattering from nuclei. Here by low-energy neutrinos we mean either reactor neutrinos or those obtainable from stopped ir—~—efacilities such as those at LAMPF. In these circumstances the long wavelength limit of the multipole operators is an exceedingly good approximation. In section 4.1 we show how inelastic neutral current processes initiated by

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

5

low-energy neutrinos are essentially computable in a model-independent way from a knowledge of f3-decay rates and electron scattering (or y-decay) from given states or their isospin analogs. What contributes in these transitions is essentially only the axial-vector part of the weak neutral current and the use of different nuclear states can select either the isoscalar or isovector pieces of this current. In particular, for isovector transitions, we show that the total neutral current excitation cross section is directly proportional to the ~3-decayrate from an analog state, times a coefficient which varies depending on which gauge theory model one is dealing with. Similar results hold for isoscalar transitions. For example, for a t~J= 1 transition we find that the neutrino excitation cross section is directly proportional to a known y-decay rate times a coefficient which typifies a given gauge theory model. Other isoscalar transitions, as well as transitions containing both isovector and isoscalar pieces are slightly more model-dependent. It should be clear from these results that low-energy neutrino excitation in nuclei can serve as a natural selector of competing gauge theory models of the weak interactions. Section 4.2 is addressed to elastic neutral current processes initiated by neutrinos. Here the vector piece of the neutral current dominates. These processes are exceedingly difficult to measure experimentally, but important nevertheless in astrophysics. Section 4.3 is concerned with the long wavelength limit of single-particle matrix elements. The results presented here are of use in estimating the expected rates for neutrino scattering in cases where no electron scattering data or ~3-decay information is available. Finally in section 4.4 we present results for low-energy neutrino scattering for several typical cases. Here we discuss both our own work as well as work of others, notably the Russian school and the work of Lee on reactor anti-neutrinos. Specifically we present results for the 0.478 MeV 7Li; 1~0—~2~0, 7.028 MeV ‘4N; following transitions 1~0—*0~1, 3.562 MeV 6Li; 0~0—*1~1,15.110 MeV ‘2C and o~O—~ 1~0,12.710 MeV ‘2C. For the first two transitions we present rates expected with a typical flux of reactor antineutrinos. For all these transitions we also given rates expected for the neutrino facility at LAMPF. Although these rates are not large the experiments appear certainly feasible. Section 5 deals with neutrino reactions off nuclei at a somewhat higher energy than that considered in section 4. Here the cross sections are larger but there is some more dependence on the underlying nuclear physics. In section 5.1 we give general relationships expected between weak and electromagnetic processes at intermediate energy involving electron scattering, charge-changing neutrino reactions and neutral current neutrino scattering. We show that in general neutral current neutrino scattering is completely determinable for isovector transitions, but not so for isoscalar transitions (except under special circumstances, e.g. with the minimal Weinberg—Salam model for T = 0 states). In section 5.2 we discuss applications to selected nuclei. There we make use of material discussed in section 3 for dealing with the nuclear many-body problem in a truncated basis. We show how the set of many-body density matrix elements obtained from an analysis of electron scattering, $-decay and it-capture can be used to predict neutrino excitation cross sections. We study two cases in particular, involving the A = 6 and the A = 12 systems. We analyze in 6Li both the transitions l~0—~0~1 (3.562 MeV) and l~0—*2~0 (4.31 MeV). In 12C we study the 0~0~-~ 1~1(15.110 MeV) and briefly the 0~0—+l~0(12.7 10 MeV) transitions. We also present a brief summary of other work done in a few cases including isoelastic processes (in the A = 3 and A = 11 systems). In section 6 we discuss non-neutrino-induced ways of studying the weak neutral current. We concentrate in this chapter on how polarized electron scattering off nuclei can probe the structure of the weak neutral current. We quote general formulas, due to Feinberg and to Walecka, for the difference in cross sections between left- and right-polarized electron scattering off nuclei, and apply it to three simple examples. We show first that for elastic scattering from 0~0systems the cross section ~

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

6

difference divided by the cross section sum is independent of nuclear physics detail and is a sensitive function of the underlying gauge theory model. Next we consider inelastic abnormal parity excitations involving either isoscalar or isovector transitions. We show that in these cases the expected answer is again sensitive to the underlying gauge theory, but that different parameters are measured in these instances. For these transitions the results are not entirely free of nuclear physics uncertainties. However, the model-dependence in the nuclear physics calculations, which we discuss for two typical examples in ‘2C (considered in the previous two chapters) is small enough that these processes can also give critical information on gauge theory models of the weak interactions.

1. Gauge theory models of weak and electromagnetic interactions: Construction of the minimal model It is well-known that the standard current-current model for the charge-changing weak interactions is not renormalizable. This means, in effect, that higher-order processes cannot be calculated in a well-defined parameter-independent way. The replacement of the interaction Lagrangian density ~‘weak

=

(G/V2)~~

(1.1)

.

by one in which the current

~

couples directly to a massive spin-one vector boson: (1.2)

~

also does not lead to a renormalizable theory. This is because the massive vector propagator, for a boson of mass M~,has the form* ~

_~+q~qJM~ q

,..

~

(

.

and thus, for large momentum, it is not sufficiently damped. In the late 1960’s Weinberg [1.1] and Salam [1.21constructed models of the weak and electromagnetic interactions based on spontaneously broken gauge field theories. These models were shown by ‘t Hooft [1.3], in 1971, to be renormalizable. Here as well the weak currents couple to weak vector bosons; however, the masses of these weak bosons are acquired through spontaneous symmetry breaking. This circumstance, along with the underlying gauge structure, is sufficient (and necessary) to assure the renormalizability of the theory. In this chapter we shall describe how gauge theory models of the weak and electromagnetic interactions are constructed and shall present the ingredients necessary for a minimal model. There is a well-defined prescription for constructing gauge models of the weak and electromagnetic interactions [1.4]. The crucial ingredient for renormalizability, as we have mentioned, is to start with an underlying gauge symmetry which is then broken by the presence of scalar mesons with non-vanishing vacuum expectation values, the so-called Higgs bosons [1.5]. When the symmetry is broken in this way some of the gauge fields acquire masses, but the theory remains renormalizable [1.3]. It is useful to break down the construction of these gauge models into three main parts: (1) An invariance group, which we shall call G~,of the weak and electromagnetic interactions is selected. The fermions of the theory (leptons and quarks) are assumed to transform irreducibly under G~. *See Appendix A for a discussion of our conventions concerning four-vectors and Dirac y-matrices.

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

7

(2) The global invariance of the theory under G~is assumed to hold also locally and necessitates the introduction of a gauge field for each of the symmetry currents of G~. (3) The local invariance is spontaneously broken down to U(1) [or U(1) times a discrete symmetry]. This breakdown endows each of the gauge fields, save one, with masses. Let us now discuss in some more detail each of the above points. The choice of the group G~and of the fermion representations is done in such a way as to guarantee that two of the symmetry currents of G~can be associated with the usual charge raising and lowering weak currents. Furthermore, one more symmetry current is associated with the usual electromagnetic current. Thus G~must have at least three generators and the group structure will, in general, be non-Abelian. If we require that the invariance transformations of G~hold also at the local level we must introduce compensating gauge fields, which, because of the non-Abelian nature of G~,will be of the type first discussed by Yang and Mills [1.6]. Furthermore, it will be necessary to introduce into the theory “kinetic” terms for the gauge fields. The underlying gauge invariance, however, prevents the appearance of masses for these gauge fields. To endow the gauge fields, responsible for the weak interactions, with masses, and to preserve the renormalizability of the theory, it is necessary to appeal to spontaneous breakdown of the theory. The most efficient way to do this is to introduce into the theory scalar fields whose self-interaction causes them to have non-vanishing vacuum expectation values. The local invariance of the kinetic energy terms of the scalar fields necessitates quartic interaction terms involving two scalar fields and two gauge fields. When the scalar fields are replaced by their vacuum expectation values, these terms will generate mass terms for the gauge fields. Before we discuss realistic gauge models of the weak and electromagnetic interactions it will prove instructive to detail the above three steps within a simple context. All the features that will appear in the simple example given below carry through, with minor modifications, in the more realistic cases. Consider a non-interacting fermion field ~/‘with Lagrangian density 2’

~

(1.4)

where this theory clearly has a global invariance in which x)~*e~A4~,(x).

(1.5)

This global invariance has a symmetry current J~(x)= ifr(x)yAi(x),

(1.6)

and the generator of the transformation (1.5) is, as usual, given by T=

J

dxJ0(x, t).

(1.7)

We may make the global invariance under the U(l) transformation (1.5) local by introducing a gauge field. That is, we can construct a Lagrangian which is invariant under the local transformation (1.8)

ifr(x)—+ e”~i~(x),

provided we introduce the gauge field A~(x)such that a~A(x).

(1.9)

8

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

This locally invariant Lagrangian can be obtained from the globally invariant Lagrangian (1.4) by the replacement of the ordinary derivative, a,~r,by a covariant derivative D,~i(x)= (ô.~ igA~(x))i/i(x).

(1.10)

—

Indeed, it is easy to check that 111 ~

(.

is invariant under the transformations (1.8) and (1.9). Note that the demand for local invariance has produced an interaction term between the gauge field A,~,and the globally symmetric current J,~.That is g4iy,4A,.. = ~ (1.12) This is a general feature of the demand for local invariance. The Lagrangian (1.11) is not complete since it does not have any kinetic energy terms for the A~. field. This can be readily remedied by adding to (1.11) the usual gauge invariant kinetic energy (familiar from electromagnetism) ~int

=

(1.13)

Fp~Fp.,,

.~‘kin

where F,~= ~

—

~

(1.14)

Note, that the local gauge invariance of the A~.field, eq. (1.9), prevents the addition of a mass term for this field. We can, however, generate a mass for the A~.field via spontaneous symmetry breaking. This is done by adding to our model theory a complex self-interacting scalar field, where we will demand that this scalar field 4, also be invariant under local U(1) transformations: (lAS) Then it is easy to see that the Lagrangian —(o~. igA,~)4,(ô,.+ igA~)4,t V(4,4,~) —

~~scaIar

=

—

is locally invariant. Furthermore, if the potential V(4,4,~)possesses its minimum not at 4, 4, will have a non-vanishing vacuum expectation value. Consider, for instance t) = h(4,4,t—~v2)2. V(4,4, Clearly to lowest order (4,(x)) = v/V2.

(1.16) =

0, the field (1.17)

(1.18)

Hence, we must reparametrize 4,(x) in terms of fields with vanishing vacuum expectation values. A particularly convenient parametrization is 4,(x)

=

(1/\/2)(v

+

p(x))e~~,

(1.19)

where p(x) and ~(x) have vanishing vacuum expectation values. It is easy to check that the field ~(x)

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

9

does not enter into the Lagrangian. To see this note that (di.

—

igA~j4,= (e~/V2)[(d5. igA~, ia,~)(v+ p)]. —

(1.20)

—

We may define a gauge transformed field B.

=

A54 + (l/g)3,~,

(1.21)

so that all the ~ dependence in (1.20) is contained in an irrelevant phase: (3~ igA54)4, —

=

(e’~/V2)[(~,. igB54)(v + p)].

(1.22)

—

Using eq. (1.22) in the scalar Lagrangian yields 2 V(~(v+ p)2). 2’scaiar = ~I(3,. igB54)(v + p)1 We note in particular that, since v 0, 25sca1ar contains a mass term for the gauge field B —

—

(1.23)

—

54. This is the celebrated Higgs mechanism [1.5]. rn our example we find that 2B ~~‘mass = ~(vg) 54B5. —

(1.24)

and that the gauge field acquires a mass because of the spontaneous breakdown of symmetry, reflected in the fact that (4,) 0. We have illustrated in a simple context the three steps to be followed in the construction of gauge models of the weak and electromagnetic interactions. To wit: (1) Pick an invariance group and the representation of fermions under the group; (2) Make the invariance local by introducing gauge fields; (3) Break the invariance spontaneously so that some (or all) the gauge fields acquire masses. We are now ready to discuss realistic models of the weak and electromagnetic interactions. The minimal group G~,that could contain both the electromagnetic current and have two charge-changing weak currents is 0(3), and a model based on this group was proposed sometime ago by Georgi and Glashow [1.7]. However the experimental discovery of weak neutral currents [1.8, 1.9] essentially rules out this simple group structure. The next most complicated group structure, with four generators, is provided by G~= SU(2) x U(1), and this is the group chosen originally by Weinberg

[1.1]and Salam [1.2]in their pioneering work. The additional generator in SU(2) x U(1) allows for the existence of a weak neutral current. In this sense SU(2) x U(l) appears as the minimal weak group that can encompass the presently known weak and electromagnetic currents. Most gauge theory models for the weak and electromagnetic interactions that have been proposed in the last few years are based on SU(2) x U(1). Recently, however, there has been some interest in larger G~groups. Two examples are provided by G~= SU(2)L X SU(2)R x U(l) and by G~= SU(3) x U(1). The former group has seven symmetry currents while the latter has nine symmetry currents. We shall discuss these larger groups and models based upon them in section 2; however, for simplicity, and concreteness, we restrict our present discussion to the minimal group SU(2) x U(1). Having picked an invariance group one must next decide how the leptons and quarks transform under the group. We begin by discussing the lepton assignments. For SU(2) x U(1), Weinberg [1.1]and

Salam [1.2]assumed that the left-handed (negative helicity) component of the electron and its neutrino transformed as an SU(2) doublet while the right-handed (positive helicity) component of the electron was taken as an SU(2) singlet. A similar choice is made for the muon and its neutrino. Thus under the SU(2) piece of SU(2) x U(1) the leptons transform as: (xi~)

e

~

(Ps) LI.L

L

(1.25)

10

T.W. Donnelly and RD. Peccei, Neutral current effects in nuclei

Here the notation aL, a~stand for the helicity projections (see appendix A): aL

2(1 +

y5)a;

aR =

‘y5)a.

—

(1.26)

The transformation properties of the leptons under U( 1) are arranged so that the electromagnetic charge is given by the formula 3+Y, (1.27) Q=T where T3 is the third component of the SU(2) generators T’, with [T’, T’]

=

1,1, k

iI,kT,

=

1,2,3

(1.28)

and Y is the U(l) generator. The above choices for the lepton fields guarantee that the weak SU(2) currents J~= (l/\/2)(J 5~±iJ~)

(1.29)

are proportional to the usual charge raising and lowering weak currents of the leptons. Let us check this for the electrons, say. Since ~

= ç~[(ie~)i~

(Ye)

.i(~ ±ir2)

is a doublet under SU(2) we have (1.30)

(Ye)]

where the ii’s are the usual Pauli matrices. Thus J~= (1/2V2) iey,~(l+ ‘y5)e =

(l/2V2)~754(1

+

(1.31a) (1.31b)

Ys)11e,

which proves our assertion. In view of eq. (1.27) the electromagnetic current is just given by a linear combination of J~and J~— the U(1)3~JY symmetry current: (1.32) jern

J

In practice it proves convenient to turn this relation around and eliminate J~in favor of J~,and J~m. We shall do that shortly. It is clear that a Lagrangian density composed of the kinetic energy terms for the electron and its neutrino is SU(2) x U(l) invariant (idem for the muon and its neutrino). That is -~‘kin =

(je~)LY

8

54

(Pc)..

-~

854eR

(1.33)

is invariant under global SU(2) x U(l) transformations. (Note that mass terms for the electrons are not SU(2) x U(1) invariant.) Our next step toward construction of the model is to make (1.33) locally invariant by introducing compensating gauge fields. This is done by replacing the ordinary derivatives in (1.33) by covariant derivatives in much the same way as was illustrated in our simple example [4]. Obviously, for each symmetry current of G~we shall need a gauge field. The net effect of making (1.33) locally invariant under SU(2) x U(1) transformations is to produce interaction terms between the symmetry currents and the gauge fields. We find in this way that I

,

Y

~î~~=gW54J54+g Y54J54,

(1.34)

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

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where g and g’ are the coupling constants associated with SU(2) and U(1) transformations respectively and are in general distinct. It is useful to rewrite (1.34) by making use of the general relation (1.32). Thus

~

gW~J~+g1Y

(1.35)

54(Jm_J~).

This interaction Lagrangian must contain the usual electromagnetic interaction: m, (1.36) 2’em = eA54J~ where A 54 is the photon field. We may extract precisely such a term out of eq. (1.35) by writing W~ and V54 as linear combinations of the photon field and a new field 4. We shall see that this latter field is the one which couples to the neutral current. We take W~.=cosOw4+sinOw A54 V54

=

—

sinOw

(1.37a)

4 + cosO~A54,

(1.37b)

where ow (the Weinberg angle) typifies the admixture. Then eq. (1.35) becomes ~~~int =

g(W~J

+

W~J~)+ Z54[g cosOw J~ g’ sinow J~”+ g’ sinO~J~] m—g’cosO~ J~]. (1.38) —

+A54[gsinO~ J~.+g’cosO~ J~

Here, of course,

(1.39)

W~= (1/V2)(W. ±iW~). Now for the photon piece of eq. (1.38) to be of the standard form, (1.35), it follows that g sinO~= g’ cosO~= e.

(1.40)

With this identification we may write the interaction Lagrangian as ~

e{A

20wJ~m)+ 1 [W~J~+ W~J~]}. 54J~m+Z54

Slfl w C050w

(J~_sin

(1.41)

slflOw

The current =

—

Sjfl~O~yJ~,

(1.42)

which couples to 4, can now be identified as the neutral weak current in the model. For eq. (1.41) to represent a phenomenologically viable model of the weak interaction it must be that the gauge fields 4 and W~are very massive. Indeed the mass of the W~fields must be such that the effective Lagrangian obtained by exchanging W’s among charged currents should reduce at low momentum transfer to the phenomenological current-current form: $~‘phen =

(GI\/2)~

.

J~,

(1.43)

~h754(l + 73)1/w

(1.44)

where, for the the leptons 154

=

e 754(1 +

7 5)Pe

+

We have seen that gauge invariance forbids assigning the W’s and Z a direct mass term. However, their masses can be generated via the Higgs mechanism, which is the third, and final step, in the construction of the model. Assuming that the W’s have acquired a mass this way, then for M~~ q~

12

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

(with q,. being the four-momentum transfer) eq. (1.41) yields for W-exchange: (1.45) where the second line follows from eq. (1.31). Comparing the above with eq. (1.43) identifies (1.46) e2/8 sin2O~M~,. The mass of the W-meson is a function of the mixing angle O~,the Weinberg angle. It is bounded by =

~

~ \/2e2/8G

(38 GeV)2,

(1.47)

and hence is indeed very large. The mass of the Z-boson, which is also acquired through spontaneous symmetry breaking, need not in principle bear any numerical resemblance to that of the W-bosons. However, the simplest mechanism which generates a mass for the W-bosons and Z-bosons may give us an indication of what order-of-magnitude relationship one can expect between these masses. Let us explore this point. To provide a mechanism for symmetry breaking we introduce, as Weinberg [1] did, a complex Higgs doublet (1.48) We want to break SU(2) x U(l) spontaneously down to U(1), with the surviving U(1) being associated with electromagnetism. This will happen if we give 4,°a non-vanishing expectation value (that is, the vacuum expectation value preserves the charge). We want to focus on the kinetic energy of the 4, fields. Because we want these terms to be locally SU(2) x U(1) invariant, we must write then in terms of covariant derivatives. To wit: ~~kin

=

—

(D,.45)(D,.4,)t,

(1.49)

where D,.4s

=

(8,. —ig ~T 1

W’,. +i~g’Y,.)4,.

(1.50)

[The choices of phases made in (1.50) guarantees that the charge assignments indicated for 4, in eq. (1.48) are correct.] The mass terms for the W,~,and V,. fields, or alternatively for W~,4 and A,. follow from giving 4,°a non-zero vacuum expectation value. Let us take (4,°)= v/V2. Then the masses for the gauge fields can be deduced from eq. (1.49) with 4, replaced by its vacuum expectation value. This gives 2(l,0)[~.gr 1W’,.—~g’Y,.][~gr, W’~—~sg’V54](~j).

(1.51)

Using the relationahip (1.40) and the definition (1.37) we have 2v2/4sin2O~)[W. W. + W2,. Wa,. + Z,.4/cos2Ow.I ~~mass = ~(e

(1.52)

~~~mass

—~v

—

Hence, in this simple model, we deduce that M~= M~,/cos2O~ ~ M2~.

(1.53)

In general, for more complicated Higgs breaking M~will not be given by eq. (1.53). It is useful,

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

13

therefore, to introduce a parameter cos2O~

2W/M2Z

M

=

(1.54)

so that ~ = 1 corresponds to the simple symmetry breaking of Weinberg [1.1]. We should note that, as asserted, the Higgs mechanism has given mass only to the W~and

4

fields. The photon remains massless, as indeed it must. Furthermore, we may also use the Higgs mechanism to give mass to the electron and to the muon. Let us see how this happens in the case of the simple symmetry breaking chosen by Weinberg. Define (1.55) Clearly a term in which the electron interacts with the Higgs mesons, = Ge[(ic~)L

(_~4O.)

eR +

~)

~~(ç~f,_4,~)(

L]

(1.56)

is SU(2) x U(1) invariant. When 4i°gets replaced by its vacuum expectation value eq. (1.56) generates a mass term for the electron, ~~‘mass =

—

—(GevIV2)ëe,

(G~v/V2)(ëLeR+ eReL) =

(1.57)

and we identify me=Gev/”s/2.

(1.58)

Note that v is a large mass [compareeqs. (1.52) and (1.4~)].Thus the coupling of the Higgs meson to the electron (or to fermions in general) is very weak. The lepton assignments (1.25) under the weak SU(2) group are in some sense minimal. That is, they reproduce the known charged weak currents and the electromagnetic current without any additional

pieces. One could always augment this minimal structure by introducing other letpons. For example, one could assign eR not an SU(2) singlet, but to a doublet by introducing a heavy neutral lepton Ne. Such a modification would add terms to the charge currents of the type Jadd

=

e

y,.(l

—

(1.59)

75)Ne.

If Ne were sufficiently massive it is quite possible that the charged current (1.59) would have escaped

experimental detection. However, the modification of the lepton assignments have radical consequences for the neutral current. For example, if the electron assignments were as in eq. (1.25): (1.60)

(:~)Le~,

then for electrons, or

=

2Ow ~:m=

—

eL~y,.eL+

J~4 Sifl

sin2O~e y~e

(1.61)

—

(1.62) Hence the neutral current in the minimal model (Weinberg-Salam) has both vector and axial-vector pieces. =

~[ey

20~ 1)]. —

5y,.e+ ë’y,. e(4sin

14

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

If, on the other hand, the assignments for the electron and its partners were (1.63)

(~)L(~)R~’~,

then N

-

T3IPe’\

-

—

Ti”Ne’\

-

1~.e

1\e ~ = (Ve~)i~y,. ~ which, for electrons, gives (J~)eiectron =

—

eL2’y,.

eL

—

eR2y,.

)R

2 ~“

eR + 5~flO~ ëy,.e

em

O~J,.

(1.64)

,

=

y,. e{2 sin2O~ l}. —

(1.65)

Hence we see that in this latter model the electron’s piece of the neutral current is purely vectorial. We shall discuss in the next chapter some experimental results which may indicate the necessity for enlarging the assignment of leptons from that of the minimal (Weinberg-Salam) model. We turn now instead to the incorporation of the hadronic weak currents in the model. Because the hadrons interact strongly we cannot write down directly their currents in terms of hadronic fields. We may, however, write these currents in terms of effective quark degrees of freedom. Such currents incorporate correctly all known selection rules and for our purposes will suffice. Until quite recently it appeared that the charged hadronic weak current [corresponding to eq. (1.44) for leptons] contained only two pieces [1.101.In terms of quark degrees of freedom it could be written as

$,.=d y,.(l+y 5)ucosO~+~y,.(l+y5)usin6~.

(1.66)

here u, d, s are the usual Gell-Mann—Zweig quarks of charges 2/3, —1/3, —1/3 and O~, is the Cabibbo angle. Note that (1.66) is of the V—A form. However, this does not imply that for the hadronic matrix elements the axial-vector and the vector form factors will be identical. Because the quarks eventually bind strongly to make hadrons we expect that, among hadronic states, (hadrons ~

dl hadrons) (hadrons U-y,.’y5d~ hadrons)

(1.67)

aside from kinematical factors. Note further that the full current is a mixture of pieces that conserve strangeness and violate strangeness conservation. To incorporate the current (1.66) into the SU(2) x U(l) model is simple. One just assigns the quarks as follows under the weak SU(2) group: XL=

(dcosO~ssiflO)uRdRsR.

(1.68)

Note that, as this assignment amply demonstrates, the weak SU(2) assignment has nothing to do with the strong isospin behavior of the quarks (that is, under the usual strong isospin, the u and d quarks form a doublet while s is a singlet). The above assignment, however, is disastrous! Having fixed the fermion representations, besides the charged currents we may also compute the weak neutral current. Since 2O~J jN = j~,. sin 4”, (1.69) —

we see that J~will contain a piece which mixes the d and s quarks: =

—

(d1~y,.SL+ SL~Y,.dL) sinOc cos0~+...

(1.70)

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

15

There are stringent limits on the existence of strangeness-changing neutral currents. For instance the

rate [1.11] F(K~—+~s~ti/I’(K~.-+ all) = (1.0 ±0.3) x 10_8.

(1.71)

Given the strength of the observed strangeness-conserving currents [1.8.1.9],a term like (1.70) would

produce a rate for (1.71) many orders of magnitude greater than observed! A revolutionary solution to this conundrum was presented in 1970 by Glashow, Iliopoulos and Maiani [1.10].Their suggestion, in our more modern language, was to add an extra quark degree of

freedom and an extra weak doublet. This new quark the charmed quark c carries a new quantum number charm. It is an isospin singlet and has charge 2/3. The new weak doublet combines the —

—

negative helicity components of the charmed quark with the orthogonal combination of ths s and d

quarks. That is, one has (_dsin9c~scosgc)LcR

~L

(1.72)

under SU(2) x U(1). The existence of this new doublet adds new pieces to the charged currents and modifies, in a salubrious way, the neutral current. It is easy to see that (1.66) gets augmented by =

d y,.(1 + y5)c sinO~+ ~y,.(1+ y5)c cosO~.

—

(1.73)

More importantly, however, and the real raison d’être for the invention of this charmed quark, is that the weak neutral current now pieces. We have 29WJ~’ = XLhas no lowest order, strangeness-changing 2OWJ~Lm =

—

S1fl

= ~{ÜL7,.UL

754~SXL —

aL7,.dL} + {ëLy,.cL

+ ~L7,.~T3~L — SL7,.SL} —

—

S1fl

sin2Ow J~m.

(1.74)

The discovery of the J/i/i particles [1.13,1.14] and the subsequent discovery of mesons [1.151and baryons [1.16]which have all the characteristics of being charmed has by now amply justified Glashow, Iliopoulos and Maiani’s radical suggestion. The important lesson for our purposes is that the existence of charm was a necessity forced by gauge theories of the weak interactions. Thus its discovery, along with the discovery of neutral currents, provides added credibility to this approach to

the weak interactions. Let us recapitulate the structure of what we may call the minimal, or standard, gauge model of the weak and electromagnetic interactions. The gauge group is SU(2) x U(1) and the lepton and quark assignments under the weak SU(2) group are given by T

i~ptons. •

CO

SL —

(

11C -

I

II — ~L ~

54

-~

— — eR /hR.

\e,L

Quarks: XL = (d cos&+ ~ 5iflOjL

~L

=

(—d sinO~-~~ cosoC)L

UR cR

dR 5R•

(1.75)

In the next section we will discuss briefly how the standard model fares with experiment and indicate

possible alternatives.

16

1’. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

2. Gauge theory models of the weak and electromagnetic interactions: non-minimal models In the last few years an enormous variety of models of weak and electromagnetic interactions have appeared in the literature All these models have as a common feature that they add more quarks and/or more leptons and/or more currents to the standard model discussed in the preceding section. In this section we would like to consider some of these non-minimal models. Our motivation for doing this is three~fold:* (1) A variety of these non-minimal models are potentially serious candidates for a theory of the weak and electromagnetic interactions. Unfortunately, these models are not widely known outside of a limited group of particle physics specialists. In discussing them here we hope to remedy somewhat the situation. (2) The motivation for many of these non-minimal models has been the apparent failure of the standard model to explain certain experimental facts. These models are, in several instances, in better agreement with present day weak interaction data. (3) Non-minimal models have in general different predictions for weak processes in nuclei. Thus, in considering possible nuclear weak interaction experiments, it is extremely useful to keep in mind other options besides the standard model. Putting it another way, weak interaction experiments in nuclei can serve as a test of competing viable gauge models of the weak and electromagnetic interactions. [2.11. Having developed in the last section the general procedures for constructing gauge theory models of the weak and electromagnetic interactions we could just proceed to write down directly the structure of various non-minimal models. However, the rather complicated quark and lepton structure of these models might appear rather arbitrary to a reader that does not know the motivation behind a given model. Thus we shall devote most of this chapter to describing various successes and difficulties of the standard model and how certain non-minimal models modify them. The reader who is not interested in these details will find at the end of this section a summary of the models we shall consider in our discussion of neutral current effects in nuclei. In the minimal model, which we shall henceforth call W-S-GIM, one can readily compute neutral current neutrino scattering. Since M~is very heavy one has an effective current-current interaction. Using eq. (1.41), (1.46), and (1.56) it is easy to see that [2.2] (2.1) where J~’ is given by [see eq. (1.42)] 2J~= 2J3,. —2 sin2O~Jm.

(2.2)

It is apparent from eqs. (2.1) and (2.2) that to compute neutrino scattering with the W-S-GIM model one needs to know how to compute the matrix elements of J~ as well as to have a value for the (free) parameters 4~and sin2O~.There are three cases of interest (purely leptonic scattering, elastic neutrino scattering from hadrons and deep inelastic neutrino scattering from hadrons) for which we can determine these matrix elements. For purely leptonic scattering the current $~ is known explicitly in * After this report was completed further experimental information on neutral current processes as well as new theoreticalanalyses have become available. These matters are summarized in an Addendum at the end of this section. This new information is strongly in favor of the standard SU(2) x U( I)model, with fermions in left-handed doublets. Thus most of the particular models presented in this section appear to be only of academic interest. However, the overall structure of these models should still serve to illustrate the underlying principles needed for constructing gauge theory models of the weak interactions.

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

17

terms of leptonic fields and the answer for the T-matrix is immediate. For elastic neutrino hadron scattering we can compute the matrix element of 1~between nucleon states by strong isospin invariance. Finally for deep inelastic neutrino scattering we may write 1~[2.3]in terms of its quark degrees of freedom and make use of a quark-parton model to compute the neutrino cross-sections. (More sophisticated analyses [2.4]make use of operator product expansions [2.5] and asymptotically free gauge theories [2.6].) It is perhaps worthwhile to comment on the use of strong isospin invariance in evaluating the matrix element of fl~.Let us focus on the part of J~not due to the electromagnetic current. In terms of quark degrees of freedom we have 2J~.= [XLY,.TSXL +

~L7,.T3’~L]

=

~{u y,.(1 + y5)u

—

dy,.(1

+

‘y5)d}+ ~{e y,.(1 + y5)c

—

~

y,.(l

+

y~)s}. (2.3)

Now in a quark model we do not expect the last two terms in (2.3) to contribute very much between states which have no “valence” charmed or strange quarks. Hence, in computing nucleon matrix elements, we can effectively neglect the contribution of the pieces of ~ containing charmed and strange quarks. Thus if IN) is a nucleon state 3 3,. + A~LIN’). (2.4) (NI2J 54IN’) (NI~{uy(1+ 75)U dy,.(1 + .ys)d}~N’)= (NI V Here V~and A~are the third components of the strong isospin vector and axial-vector currents, —

whose matrix elements between nucleons are known from f3-decay. That is, we may calculate, say, (NI V~IN’)from (NI V~’2IN’)by a (strong) isospin rotation. We will return to this point in section 3. A fit to all existing neutral ~urrent neutrino scattering data on both leptonic and hadronic targets has been done by a number of authors [2.7,2.8].They find that the W-S-GIM model gives an adequate fit to the existing data with sin2Ow 0.3 and ~ = 1. (An analysis of some very recent experiments [2.9] performed at the CERN-SPS yield a somewhat lower value of the Weinberg angle, sin2Ow = 0.24± 0.02). There is, however, a problem looming on the horizon. In the W-S-GIM model the electron couples to the neutral current in a parity-violating way and it is easy to see that the effective Lagrangian describing this interaction is given by (see eq. (1.61)) /

-

N

‘~‘eff (G/v2)~[e[y,.(1+ y

5)—4 sin O~y,.]e].$,..

(2.5)

This interaction will cause parity violation in atomic physics [2.10,2.11]. Recent experiments carried out in Seattle and Oxford to search for these effects in heavy atoms report a value for the parity-violation parameter which is substantially smaller than the W-S-GIM prediction and which could

be consistent with zero [2.12,2.13]. The Seattle-Oxford experiments are sensitive to the axial vector coupling coming from the electron vertex. This is because, effectively, only the vector part of $~contributes in heavy atoms, since it is only this part which gives a coherent contribution. If these results are taken at face value then the W-S-GIM model is in trouble.* Two possible ways out exist: 1) There is some uncertainty in the atomic physics part of the calculation of the parity-violating effect and this could be adduced as the cause of the disagreement [2.14]. 2) The W-S-GIM leptonic assignments are not correct.

Clearly we are not prepared to comment on the first possibility. We can, however, discuss the *

Very recently, results from a similar experiment atN&eosibirsk (L.M. Barkov report at the XIX High Energy Physics Conference, Tokyo 19Th)

have become available. This group reports seeing a parity violating effect of a magnitude and sign in agreement with the w-S-GIM model. Thus the experimental situation now is also unclear!

18

T. W. Donnelly and RD. Peccei. Neutral current effects in nuclei

second point. Indeed, as we indicated at the end of section 1, if the electron assignments under SU(2) X U(1) were taken as

(~)L(~)R

(1.63)

NeL,

where Ne is a new neutral heavy lepton, then the neutral current due to the electrons is purely vectorial (see eq. (1.65)). In this case one would expect an essentially null result from the Seattle—Oxford experiments. Extension of the W-S-GIM model by adding to it as yet unseen neutral leptons has been contemplated by a number of authors [2.15]. The main purpose of these extended models is to reconcile theory with the null result of the Seattle—Oxford experiment. However, one must be careful how one modifies the theory, since it has repercussions on other experiments, like neutrino—lepton scattering. The modification indicated in eq. (1.63) however, fares as well with the neutrino scattering data as the W-S-GIM model [2.161. Clearly the best test of these leptonic modifications of the W-S-GIM model would be the discovery of a heavy neutral lepton. Other tests are possible. For example, several of the new leptonic models [2.17] that have been proposed have enough richness in them that they allow, in higher order, some processes like ~-~ey which are strictly forbidden in the W-S-GIM model. In section 6 we shall discuss some nuclear physics tests, involving electron scattering on nuclei, which can also test for additional leptonic structures. The possibility of having neutral heavy leptons is not the only “perturbation” that one can envisage for the W-S-GIM model. It appears now more certain that the correct interpretation of the anomalous ~e production at SPEAR is that new heavy leptons r~are being produced [2.18].Fortunately,, for our purposes we need not worry precisely how these states fit into the theory, because we shall be for the most part interested in neutral current neutrino scattering off nuclei. Then, as long as the group is SU(2) x U(l), and we put the neutrinos in left-handed doublets, the effective Lagrangian for these processes remains that given in eq. (2.1) and all we need to know is J~between nuclei. Clearly, this is only sensitive to the quark assignments in the model and not to the lepton assignments. The hadronic part of the W-S-GIM model can be tested by both elastic and deep inelastic neutrino and antineutrino scattering on nucleons. Until quite recently there existed an anomaly [2.191in high-energy antineutrino deep inelastic scattering the so-called y-anomaly whose most natural —

—

interpretation required a modification of the quark assignments in the W-S-GIM model. Recent data [2.20,2.21] from the CERN—SPS, however, show no such anomalous enhancement in antineutrino deep inelastic scattering and are in substantial agreement with the predictions of the W-S-GIM model. Although the y-anomaly is most probably an experimental artifact, it is worthwhile bo briefly discuss some of the models that were generated by theorists to “explain” this phenomena. Our motivation for doing so, even in the light of recent experiments, is two-fold: (1) These models can be made consistent with the new CERN data provided that the masses of the “new” quarks are sufficiently large; (2) These models illustrate in a simple fashion how modifications of the quark assignments generate different pieces of the weak neutral current, which can in principle be tested by low-energy nuclear physics experiments. To illustrate what is involved in high-energy deep inelastic neutrino experiments, it is useful to

adopt a simple quark-parton model in which all sea contributions and the Cabibbo angle are neglected (more sophisticated analyses can of course be done, and some of the results we shall quote are based

on these less naive approaches). In our simple model, and assuming that the correct weak interaction theory is that given by the W-S-GIM model, the cross-section for the processes vN j.e X, ,3N -~

-*

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

19

are given by [2.4]: do-~/dxdy

(2G2ME4,/sr) F(x)[1],

=

2ME,Jir)F(x)[(1

(2.6a) —

(2.6b)

y)2].

do~/dxdy = (2G Here x = q2 2,. being the four-momentum transfer and E~,E,. (E~ E,.)/E~with q being the lab54/2M(E~ energies E,.), of they =neutrino and muon respectively. The function F(x) can be determined from deep inelastic electron scattering [2.22].Clearly this model predicts crio~”= ~ and specific y distributions for the neutrino and antineutrino processes. The y-anomaly consisted in the appearance at high energy of an extra term in the antineutrino cross-sections, which modified the (1 y)2 distribution to (1 y)2 + 1. Such a term could arise if, in the W-S-GIM model, in addition to the usual —

—

—

—

X and ~ quark multiplets one had a new right-handed doublet and left-handed singlet [2.23]: =

(2.7)

~

involving another quark, the b quark, of charge

—~. Since only iu-~bj.t~ is allowed (by charge conservation) this new doublet only affects antineutrino scattering. Further, since the charged current coupling of u to b is right-handed, it is not hard to show that the y-distribution arising from this new doublet is proportional to 1 and not (1 y)2. If the b-quark is sufficiently massive, the effects of the ~ doublet will not be seen, in charge-changing processes, until it is kinematically possible to produce states involving the b-quark. Hence the effect of (2.7) is to modify the antineutrino distribution in eq. (2.6b) to —

do~’/dxdy

=

(2G2ME,,/ir)F(x) {(1

—

yf

+

t(E~)1},

(2.8)

where t(E~)is some threshold function which eventually asymptotes to 1. The data of Benvenuti Ct al. [2.19]necessitated the above modification with t(E~)having a threshold at around 50 GeV. The higher statistics CERN data [2.20,2.21] is consistent with no such term. This of course does not necessarily rule out an addition like (2.7) to the W-S-GIM model provided the mass of the b quark is high enough. As we shall explicitly show later in this paper the presence or not of a term like (2.7) can be tested, irrespective of the mass of the b-quark, in low-energy neutral current neutrino scattering experiments on nuclei. A great variety of models [2.24]were suggested to “explain” the now non-existent y-anomaly! As

we remarked above, these models, provided the masses of the new quarks introduced are high enough, can still be in accord with present day data. We shall select, for illustrative purposes, three such models. Their principal merits, at this stage in our knowledge of the weak interactions, is that they have interesting neutral current structures. All three models have “new” quarks beyond the charmed quark and have both V-A and V + A structures. We list below their quark structure setting for simplicity all mixing angles of the Cabibbo type to zero. (The mixing angles do not affect the neutral currents since in all these models the GIM mechanism [2.25]is in effect). We shall also only indicate the SU(2) x U(1) doublets in the models. (1) b-quark models [2.26,2.271 fu\ fc\ (u\ fc \ ~Al : ~\U/R \U/R ~U/L \S/L

(

‘~

b and b’ are heavy quarks of charge

—~.

)

20

T.W. Donnelly and RD. Peccei, Neutral current effects in nuclei

(2) q-quark models [2.241 /u\ fc\ fd\ fs\ ~d)L ~5)L ~q)~ ~q’)~y’

(2.10)

q and q’ are new heavy quarks of charge (3) Vector models [2.28]

—~.

fu\ (c\ (t\ (u\ (t\ (c\ ~d)js)L~b)L ~b)R~d)R~s)R’

(2.11)

Here b is a new heavy quark of charge t is a new heavy quark of charge ~. We should remark that the vector models are in serious, probably fatal, disagreement with elastic neutrino scattering, while the q-quark models do not fare too well with neutral current neutrino scattering [2.7,2.8]. Hence, from a phenomenological point of view only the b-quark model is credible, provided the b-quark is heavy enough. (The b’-quark, since it couples to the charmed quark and thus gives no y-anomaly, could be of low mass. We shall return to this point later.) The b-quark model, however, may be in theoretical trouble! We should explain briefly this last point. Recall that the mechanism invented by Glashow, Iliopolous and Maiani guaranteed that there be no strangenesschanging neutral currents to lowest order. It turns out that this mechanism also suppresses these processes to higher order so that there are no O(aG) strangeness-changing neutral currents [2.25].This is in accord with experiment and again attests to the inspired choice of the GIM mechanism. Glashow and Weinberg [2.29] have argued recently that models, like the b-quark model, in which quarks of the same charge are given different left (right) SU(2) assignments will fail to suppress strangenesschanging neutral current processes to O(aG). They argue that although one can arrange the coupling of quarks of the same charge to the Higgs mesons so there is no quark-mixing to lowest order in the weak interactions, this property cannot be maintained to higher order in the weak interactions. In turn, this means that it is not “natural” for models of this type to have also the O(aG) strangeness-changing terms absent. Although the Glashow—Weinberg argument [2.29] is compelling, we know so little about the mechanism that produces the weak symmetry breaking that it may be premature to rule out models, like the b-quark model, solely on theoretical grounds. Indeed, there are other theoretical restrictions on models of the weak interactions which we are ignoring. They include the requirement of not having any Adler—Bell—Jackiw anomalies [2.30] and the requirement of not inducing strong CP violation via instantons [2.311.These requirements can usually be satisfied by enlarging the number of fermions or Higgs bosons in the theory and we shall not worry unduly about them here. As far as the Glashow—Weinberg “naturalness” requirement goes, we could suppose, with Fritzsch [2.32], Ramond [2.33], and others, that there may well be some dynamical mechanism which prevents the mixing of —~,

light quarks (d, s) with heavy quarks like the (b, b’). We prefer to leave the matter open and let future experiments decide whether or not the b-quark models are viable.

There are recent experimental indications which may point to the presence of new quarks and new weak currents. These experiments (of course!) have spawned yet more weak interaction models. In a recent letter Herb et al. [2.34] have reported a high-mass resonance (M 10 GeV) in the di-muon spectrum in the reaction pp ~ This “resonance”, which in fact could be various resonances masked by resolution, is most likely a new hadron which is a q~composite, just like the J/4i was a cë composite. It is yet too early to tell whether this new quark q is of charge ~ or but preliminary theoretical analyses [2.35] seem to favor the choice of —~. If this interpretation is correct, then we -+

—~,

could identify this quark with the b’ quark appearing in the model displayed in eq. (2.9). (Since the b’

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

21

quark couples predominantly to the charmed quark c, its relatively low mass, m~, 5 GeV would not cause any unwarranted effects in deep inelastic neutrino scattering.) An alternative interpretation has been advocated by Ellis et al. [2.36]. They suggest that the quark responsible for the ~ enhancement in the experiment of Herb et al. [2.34] is part of yet another left-handed doublet present in a SU(2) x U(1) theory. This model had been suggested some time ago by Harari [2.37] and Kobayashi and Maskawa [2.38]. We detail below for future reference the quark and lepton assign(~)L

ments of this model (again we have not indicated mixing angles of the Cabibbo type and only write down the doublets in the theory). HKM-model (u\ (c\ (t\ \d)L ~S)L ~b)L’

(Pe\

(v,.~

(~,,\

\e)L

~¼i.L)L

\T)L

(2.12)

Here b with charge is presumably the quark involved in the Herb et al. experiment and f is the new heavy lepton found at SPEAR; v,, is its accompaning neutrino and t is a heavy charge ~ quark. —~

Before closing this section, we would like to discuss another class of models which have become very topical lately. These models use a weak group which is larger than SU(2) x U(1). Furthermore, in contrast to other schemes which also use larger groups [2.27],all components (or nearly all components) of the weak group have roughly the same strength. Thus these models have more than one effective charged and neutral currents. We shall discuss two specific examples. In the first model the weak group is G~= SU(2)L X SU(2)R x U(1), while for the second model G~= SU(3) x U(1). The motivation behind the first model was to avoid, in a different way, the null result of the atomic parity violation experiment. The second model, at least the Lee—Weinberg version [2.39],which we shall detail, was constructed to explain the recently observed trimuon events in deep inelastic neutrino scattering [2.40].This model is also in agreementwith the Seattle—Oxford atomic physics results. The first model, as far as neutrino interactions go, has the same structure as the W-S-GIM model and thus is in accord with present day experimental data. In the SU(2)L X SU(2)R X U(1) model the quarks and most of the leptons are arranged in doublets under one of the two SU(2) groups. The structure of the model is arranged so that no large right-handed charged currents appear between the known leptons and quarks so as to avoid experimental contradiction. This can be done either by picking the doublets under SU(2)R carefully or by introducing appropriate Higgs mesons which guarantee that the charged-boson which mediates all right-handed processes is very heavy ~ M2WL) [2.41]. If there are only four quarks the second option is the only one possible. With six quarkseither option is allowed. We shall detail fordefinitiveness a six quark model in which M2~~ = M~,L[2.42]. Under SU(2)L the quarks and leptons are arranged as ~‘

SU(2)L:

(~~) (C)

(t)

dLsLbL

(ye)

e

(u:).

(vs) L/.t

LT

(2.13)

L

Again Cabibbo angles are suppressed and only the SU(2)L doublets are indicated. The charges of the t and b quarks are ~and respectively. Under SU(2)R we have the doublets —~

SU(2)R:

(U)

(C)

(t)

bRSRdR

(Ne) (N e

R/L

54) (NT) R

T

(2.14)

.

R

Again Cabibbo angles are suppressed. However, to guarantee that no experimental troubles arise, the quark doublets in (2.14) must essentially be as shown since we want a negligible charged current. The leptons N~,N,., Nr are new heavy neutral leptons. In (2.14) we have only indicated the SU(2)R doublets, all other fields are SU(2)R singlets. (~)R

22

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

In this SU(2)L x SU(2)R x U(l) model there are seven gauge bosons. Four are charged and we shall call them W~,W~.Of the remaining three neutrals, one is identified with the photon and the other two mediate neutral current processes. If one assumes [2.41,2.421that there is a parity operation which interchanges WL’E-~WR then there are only two coupling constants in the theory: g for the SU(2) groups and g’ for the U(1) group. The electric charge is identified as Q=T~+T~+X,

(2.15)

where X is the U(l) generator. (Compare eq. (1.27).) Hence as in the SU(2) x U(l) case we may always eliminate all reference to the X-current in favor of the electromagnetic current. Thus the interaction Lagrangian for the model is (2.16)

We rewrite this in terms of the electric charge e and a mixing angle g

e/sinORL,

g

=

e/Vcos

°RL

by taking (2.17)

20RL~

In view of eq. (2.15) we take the photon field to be A,.

S1flORL

=

(W~,.+ W~,.)+ \/cos 2ORL X,.,

(2.18)

and define two (massive) neutral fields as orthogonal combinations of the above: cos 2ORL( Wi.,. + W~,.) V’2

=

ZA,.

=

—

5IflORL

X,.,

(2.19)

(l/V2)(W~,. W~,.).

(2.20)

—

In terms of these definitions the interaction Lagrangian of the model can be written as follows CD

I

—

+

A

Tern

+

-

\~2sin ORL

1 .

~

/

LV,.

5IflORL v cos 20RL

r~

i~

~2~JL,.+JR,.,—s1n

ZA,.’[J~,.—J~,.1+ 1

sinORL

~

2~ ~em ~7RL~,.

(2.21)

Equation (2.21) is the analog for SU(2)LX SU(2)RX U(l) of eq. (1.41). However, the fields Z~,ZA and W~,W~are not yet mass eigenstates. If we introduce Higgs bosons to give mass to the vector bosons, and to the fermions in the theory, we will in general mix Z~.with ZA and W~with W~.It is possible, however, to choose the Higgs bosons in such a way that Z~,ZA, W~and W~are mass eigenstates. The simplest such choice introduces Higgs bosons which transform according to the ~) representation of SU(2)L x SU(2)R [2.421and yields the following masses for the vector bosons: (~,

~

=

~

=

M~A= cos OR[~ M~ 5,

(2.22)

with (compare eq. (1.15)) 2/(8 sin2ORL M~,,). (2.23) G/V’2 = e Neutral current neutrino scattering in this model proceeds through both Z~and ZA exchange. It is easy to check, however, that in view of the relations (2.22) and (2.23), the effective current—current interaction is identical to the one obtained for the SU(2)xU(l) model, with ~= I and the identification of = O~.Thus, as far as neutrino scattering is concerned, the prediction of this model is the same as that of the HKM model of eq. (2.12) which was based on SU(2))< U(l). However, as can be readily checked, for weak interactions involving electrons there are important differences: °RL

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

23

the Z~gauge boson couples only to the vector part of the weak neutral current of electrons and quarks, while the ZA gauge boson couples only to the axial-vector part of the weak neutral current of electrons and quarks. Hence this model predicts no atomic physics parity violations. The last model we want to discuss is one based on a still larger group: G~= SU(3) X U(1). This

model has nine currents. Of these, four are charged and five are neutral, one of them being the electromagnetic current. We will describe briefly the version of this model due to Lee and Weinberg [2.39]. These authors impose on their model a discrete symmetry which guarantees that the theory

“naturally” conserve strangeness in the neutral current sector. The fermions in the theory transform as SU(3) triplets and are organized as follows: /u\ /c\ /t\ /g\ Quarks (d s) (b) (h) \b/L\h!L\d/R\s/R

(

/v,.\

/Pe\

(e j (

Leptons

(2.24)

0 /

\

\

(Ej

~t

M

\EJL\M/L\ejR

J

.

(2.25)

!,L’R

The u, c (t, g) quarks, of charge also have right- (left-) handed singlets. The E°and M°leptons have left-handed singlets. The b and h quarks are new quarks of charge —~. We have suppressed mixing angles of the Cabibbo type. The discrete symmetry assigns a quantum number ± to the triplets and ~,

singlets as follows:

(:) (I) L

(2.26)

(+)R(-)L.

+R

Thus Cabibbo-like mixing can only occur between fermions of the same quantum number. This

guarantees “naturalness” [2.39]. All eight weak currents in this model have approximately the same strength. This is because the model was built to explain the (apparently) copious production of trimuon events at Fermilab [2.40]. They arise in the model from the chain reaction v,. + N

-~

W+X L4 ~±

+

,a

+

X”,

(2.27)

with roughly the right branching ratio. The electromagnetic charge in the model is given by (compare eq. (1.27)) (2.28)

Q=T3+~=T8+T0,

where T’, i 0LW

angle

= I,..., 8 are the SU(3) generators and T° is the U(1) generator. Introducing a mixing via (g is the SU(3), g’ is the U(1) coupling constant):

g = (2/V3)e/sin

OLW,

g’ = e/cos.OLW,

(2.29)

we may eliminate the U(l) field, V°,in terms of the photon field and other neutral fields. Defining new fields Z,. = cosOLw(~A~.+ \/3 A~) sinOLW Y~, (2.30) —

24

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

A,. =

sinOLW (~ A~. +~\/3 A~)+

C050

5w

Y~,

Y,.=—~V3A~+~A~, where

(2.31) (2.32)

the photon field, we have ~inte{A,.J~~ I z,.. [(J~+_LJ~)_sin2oLWJ~J is

A,.

6LW

cosOLw

sin + (l/\/3 sinOLW) Y,. . (J~,. \/3 J~,.)+ (2/\/3 sinOLW) W~J~J~ -

(2.33)

where W’,.Jt,. runs over i = 1, 2, 4, 5, 6, 7. These last terms contain four charged current interactions and two neutral current interactions which are not diagonal. Equation (2.33) is not yet in a useful form, since the masses of the vector meson fields are unspecified. Lee and Weinberg [2.39]generate masses for these fields by introducing both a triplet and a complex octet of Higgs mesons. In this way they find degenerate masses for the charged intermediate bosons W~,U~.The masses of the neutral bosons are given in terms of these charged boson masses by M~= {4/(3 cos2 OLW(l

+

l))}M,~,~,

(2.34)

M~.= {41/(l + l)}M~,

(2.35)

M~= {21/(l

+

l)}(l + 8)M~,,,

(2.36)

+

1)}(l

(2.37)

M~< 2= {21/(l

—

8)M~,.

Here X1 and X2 are the non-diagonal neutral bosons (linear combinations of A6 and A7) and the parameters 6 and I (which are free) represent various ratios of the expectation values of the triplet Higgs and the octet Higgs. The theory reproduces ordinary charge-changing weak interactions with the identification of the Fermi constant 2/8M2~= e2/2\/~sin2 OLW M~y. (2.38) G/\/2 = g A fit to neutral current neutrino data gives [2.43] sin2 OLW

0.2,

6

0.

(2.39)

We should remark that this model also predicts a very small parity violation effect in heavy atoms because the axial neutral current coupling of electrons vanishes. For our purposes we need to extract from eq. (2.38) the effective Lagrangian for neutral current neutrino scattering. Now for neutrinos J~+(l/V3)J~ ~[1’e7,.(l+ 7s)Ve+ i,.y,.(l + y

5)v,.1,

(2.40)

while J~,.—V3J~,.=0.

(2.41)

Hence only Z exchange will contribute to neutral current neutrino scattering. Using eqs. (2.33), (2.34) and (2.40), we find the effective Lagrangian for neutrino scattering to be ~utrino

where

=

~[iy(l

+

Y5)Pe~~,.y,.(l

+

ys)~,.1 (I~)eff,

(2.42)

OLW

(2.43)

—

(I~)eff= ~~(l

+

l){2J~+

J~,. 2 sin

Jcm~

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

25

Table 2.1 Gauge theory coupling constants Model

ic

W-S-GIM b-quark q-quark vector HKM SU(2)LXSU(2)RXU(1)

1 ~ ~ 1

SU(3)x U(1) ~aV~—a,,,;

ai~°~ a~P a~

~

0

1

I I

I 1

1

0

0 0 0

2 1 1

0 0 0

0

1

~

—~

1 0 1 1

a,,,, 26~, 2 sin 2 sin28~ 2 sin2 O~ 2 Sm2Ow 2sin2Ow 2sin2ORL

I 511128LW

~=a~,~=O,i

Experimentally, sin2Ow = sin2O~ 0.3; ~~~9Lw 0.2 while ~ 1, 1

0.2.

We are actually interested only in the part of this current due to ordinary (u, d) quarks. Using eq. (2.24) we find {( 1 + l)/\/3} {~[Uy,.u dy,.d] + ~Uy,.y

2OLWJ~m}. (2.44) 5u ~ 51fl It is useful to summarize our brief discussion of alternate models of the weak interactions by indicating what is the effective current which couples to neutrinos. We shall write the effective interaction Lagrangian in the form =

($~)~

—

—

+7 5)Pe+ i,.y,.(1+y5)v,.] ($~)e~

(2.45)

and decompose ($~)effinto vector and axial vector currents which, under strong isospin, transform as isoscalars and isovectors. We then write 3,. + a ~ As,. + a ~!A3,. aCmJe,.rn. (2.46) (I~)eff = a ~ V’,. + a ~ V The currents V~,A3,. are the third components of the strong isospin currents, while V’,., As,. are their —

isoscalar counterparts. They are normalized to ~r

3and

~,

where here r~is the strong isospin. Thus the

electromagnetic current is just m=V~+Vs,.. J~ In terms of quark degrees of freedom we have

(2.47)

V~= ~{uy,.u dy,.d},

(2.48)

—

3

A,.

~{uy,.y 1—

=

—

5udy,.75d},

(2.49)

—

V. = ~{Uy,.u+ d’y54d}, S

1—

(2.50)

—

A,. = 5{uy,.y5u+dy,.y5d}.

In table 2.1 we present the values for K,

(2.51) a~, ~

a~,

and acm that follow for the various models

considered.

Having detailed the structure expected from various gauge models of the weak interactions, we are now ready to discuss how these models can be tested in nuclear physics experiments.

26

T. W. Donnelly and R.D. Peccei. Neutral current effects in nuclei

Addendum New data on deep inelastic neutral current neutrino scattering, as well as elastic scattering data and single pion product have become available very recently. All this data are summarized, with appropriate references, by C. Baltay in his report at the XIX International Conference of High Energy Physics, Tokyo Aug. 1978. From this data it is possible to make a model independent analysis of the neutrino neutral current coupling to u and d quarks. Such an analysis was done originally by P.Q. Hung and J.J. Sakurai (Phys. Lett. 72B (1977) 208) and extended to include the new information available by L.F. Abbott and R.M. Barnett (Phys. Rev. Lett. 40 (1978) 1303) and E. Paschos (BNL preprint). The result of these and other analyses are also included in Baltay’s report. It has become conventional to write the effective neutral current (J;N)eff as ULUY,.(l + 75)11 + URÜY,.(I y5)u + dLdy,.(l + y5)d + dRdy,.(l y5)d. 11L, uR, dL, dR are simply related to those displayed in table 2.1. To wit The parameters uL = ~sa~ + ~ ~Gem + ~ + =

—

—

—

(2.52) (2.53a)

I 0 I I I 10 I I uR—~av+4av—ya~m—4aA—4aA 1

0

I

I

I

1

0

I

I

dL—~yav—4av+6aem+4aA~aA

dR ~ya~

—

~

+

~aem ~aA+4aA.

(2.53d)

—

The results of the analyses discussed above yield the following values for these parameters uL =

0.35 ±0.07

(2.54a)

UR

—0.19 ±0.06

(2.54b)

dL = —0.40 ±0.07

(2.54c)

dR= 0.0 ±0.11.

(2.54d)

=

The W-S-GIM model with sin2 O~,= 0.23 is in excellent agreement with the above values:

W-S-GIM (sin2

Ow

0.23)

0.35

(2.55a)

—0.15

(2.55b)

dL=—0.42

(2.55c)

dR = 0.08.

(2.55d)

uL UR

=

The same values also apply for the HKM and SU(2)L x SU(2)R x U(l) models. However, this latter model at least in the version presented in this section is in conflict with the new parity violation data from Novosibirsk and the results of the SLAC e-d polarized scattering experiment (R. Taylorreport at the Tokyo Conference). All the other models appear to be ruled out by eqs. (2.54). For example the b-quark model has UL = UR, the vectormodel has UL = UR, dL = dR, the Lee—Weinberg version of SU(3) x U(1) has dL = dR,etc. As a last comment we should note that these analysesalso indicate that = 0.98 ±0.05, again —

—

K

in agreement with the standard model (J.J. Sakurai, Oxford Neutrino Conference Proceedings, July 1978).

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

27

3. Semi-leptonic weak and electromagnetic interactions in nuclei In this section we shall begin the study of weak and electromagnetic interactions in nuclei. Nuclear targets are of special interest because of the wide selection of quantum numbers made available to us by choosing specific nuclear transitions. This will allow us to concentrate on particular pieces of the weak/electromagnetic interaction and to see pronounced effects (for instance, in neutral current

neutrino scattering) which relate directly to the underlying gauge theory models discussed in the previous two chapters. Before turning to specific examples, we shall present in this section a summary of the relevant ideas needed in a description of the nuclear problem. More detail may be found in refs. [3.1—3.3]; here we wish to include only enough material to make the basic approach clear.

We shall begin with a general discussion of the form of the interaction Hamiltonian and the multipole decomposition of the hadronic current in section 3.1. Here we introduce some of th.e

notation commonly used in nuclear physics for the tensorial nature of the currents [3.1—3.3]. In section 3.2 we give expressions for a variety of semi-leptonic weak and electromagnetic interactions with nuclei which are of interest. Then, to make a connection with nuclear dynamics, the next two

subsections (3.3 and 3.4) contain descriptions of the single-nucleon matrix elements of the weak and electromagnetic current operators and of the nuclear many-body problem. Here we shall describe a unified approach to these processes in terms of one-body nuclear densities. Since one of our primary goals in integrating the material presented in this article is to study low-energy neutrino scattering from nuclei, we shall be interested in the long wavelength forms for the matrix elements under discussion and thus consider this special limit in section 3.5. Finally, for the reader who is unfamiliar

with the approach to nuclear physics summarized here, we shall briefly study a very simple special example (the 3H—3He system) in section 3.6.

3.1. General considerations In fig. 3.1. we show the general kinematics for the semileptonic processes under discussion. The leptons have 4-momenta k,. and k,’, with helicities A and A’ respectively, while the hadronic systems are labelled (K,.A) and (K,~A’)(other quantum numbers such as isospin are suppressed at present). The 4-momentum transferred in the processes is q,. = (k’ k),. = (K K’),. and we shall use the notation ~qj q and q 0 w for the 3-momentum and energy transfer respectively. We shall consider —

—

—

these processes only in the lowest-order approximation in the weak and electromagnetic interaction. Then the 2) interaction Hamiltonian will bewhere proportional (1Iq~.for 1/M2 for weak bosons, M is to thea gauge boson boson mass propagator and is assumed to photons; be large 1/(q~.+ M K’

LEPTO,,~~~DRONS

Fig. 3.1. Semi-leptonic weak and electromagnetic interactions.

28

T. W Donnelly and RD. Peccei, Neutral current effects in nuclei

compared to a typical nuclear momentum transfer) times the product of the leptonic current density operator j,. with the hadronic current density operator i,.. That is 3.l)

W—jJ,..

The structure of the leptonic current j,. for the particular process in question can be deduced from our discussion in sections 1 and 2 (see also refs. [3.1—3.31). In all cases this current will be a combination of vector and axial-vector currents. It proves convenient to write the 7-matrix structure of the leptonic current in the form y,.(av + aAys), where av and aA take on specific values depending on the process under discussion. For example, for electron scattering (via the electromagnetic interaction), the matrix elements of the leptonic current are given by

(j,.)

=

(3.2)

Ue(k’A’)y,.Ue~(kA),

with av = 1 and aA = 0. Similarly, in charge-changing neutrino reactions we have (j,.)

=

ü,-(k’A’)’y,.(l + y5)u~,(kA),

(3.3)

where 1 = e or ~aand av = aA = 1, yielding the standard V-A weak interaction. For neutral current weak interactions more complicated forms for av and aA emerge. If the leptons in question are neutrinos then again av = aA = 1 (see eq. (2.45)). If the leptons electrons or amuons aA 0v =are — I + 4 sin2O~,, A = —1then (see av eq.and (2.5)). areonmodel dependent. In leptonic the W-S-GIM model,isfor If the other hand the assignment as example, in eq. (1.63) then av = —2 + 4 sin2 Ow, aA = 0.

We now turn to a discussion of the hadronic current density operator ~,. The work of sections 1 and 2 has given us the structure of J,. in terms of quark degrees of freedom. Since we are only interested in the matrix elements oft,. in nuclei we need only retain the pieces of ~,. which involve the u and d quarks. Since, as far as strong isospin goes, the u and d quarks form an isodoublet, the effective hadronic current density operator J,. contains only strong isoscalar and isovector pieces. This is implicit in the form of the charge-changing weak current of eq. (1.66) [remember only the first piece is effective in nuclei] and was detailed explicitly in eq. (2.46) where we decomposed the effective

neutral weak current into strong isoscalar and isovector components. Of course, the electromagnetic current is also a mixture of isoscalar and isovector pieces. Furthermore, a second general property follows directly from the structure oft,. in terms of quark degrees of freedom: the total hadronic current in general has both vector and axial-vector pieces. In view of the above properties we shall decompose the hadronic current density operators,. [from now on we use the caret to indicate a second-quantized operator, operating in the nuclear Hilbert space] into irreducible tensors in strong isospin space. Only the tensors with Y = 0, ~ = 0 (isoscalar), .~T= 1, .4(~-= 0 (isovector, neutral current) and Y = 1, ~ = ± 1 (isovector, charge-changing current) will contribute (see eq. (3.28) for the explicit form assumed for the isospin dependence of the currents). Each of these tensors contains a vector and an axial-vector piece so that the full decomposition oft,. takes the form ~

_r ~

V

~

A

J/J~4~

Here J,.(f~)is the vector (axial-vector) piece of i,.. The constants /3~ and f3~are independent of Z.~-.This follows directly from our preceding discussion and embodies the, so-called, isovector triplet hypothesis. To be more specific, we have the following: 1. For electron scattering (via the electromagnetic interaction),

j,. =(J,.)~+(J,.)10,

(3.5)

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

29

which is purely vector, non-charge-changing (.A~= 0) with isoscalar and isovector parts (j3 ~ = 1, f3~=O, ~=0,l); 2. For charge-changing weak interactions such as f3~-decay,it-capture (or e-capture), neutrino and anti-neutrino reactions, (is,, l) and (i~,l~)respectively (1 = e or p.), =

(I,.)~. + (I~)I4~,

± 1,

A~-=

(3.6)

which is purely isovector, charge-changing (.A(~-= ±1) with vector and axial-vector parts (fl~ = = ~ (1) O 3. Neutral current weak interactions such as neutrino (v, and anti-neutrino (i, IY) scattering (either electron or muon neutrinos), ,

V

—

A

—

5,

is’)

=

~ki,.)~+ ~~I~)®

+ ~V154)1O +

~

(3.7)

which has only J~-= 0 pieces, but otherwise in general has both vector and axial-vector parts and both isoscalar and isovector components to each. The parameters I3~, ~f) are related to the parameters a~, a~ and acm of section 2 (see table 2.1) by ~

I3

(~_)_

—av

~acm,

(y)

A

We should note that (3.5) and (3.6) embody the conserved-vector-current hypothesis (CVC). This is a natural result of the underlying gauge theories. There is just one isovector conserved current and it is a part of the electromagnetic current. Since we have assumed that ‘the current operators transform as irreducible tensors in isospin space and shall assume that the nuclear states involved have good isospin quantum numbers, we may use the Wigner—Eckart theorem to write (T’MT’ I i’srA(.j TM,.) =

(

) T’-MT. (—T.L~T)

(T’II D~T),

(3.9)

where ‘P is either I,. or .I~.Thus in the most general case there are only four reduced matrix elements in isospin space to consider (T’II(J,.)25.IIT) and (T’II(J~,.)~-IIT)for Y = 0 and 1. Of course, by selecting the quantum numbers T and T’ we may examine the isoscalar and isovector current separately. We shall assume that the nuclear targets are sufficiently heavy compared to typical excitation energies and momenta so that we may neglect recoil effects in computing the nuclear transition matrix elements of the currents (we shall, however, include the effects of recoil on the density-of-states [3.1—3.3]). Having a common coordinate system for initial and final nuclear states, it is useful (since the states are specified by their angular momenta) to expand the current operators in multipole projections [3.1—3.3]: for the vector current, ~~~5~tj; ~~q)E

jdxM~1(qx)io(x)5~..

~0

(3.l0a)

L,;r~~(q)~Jdx(~-VM~1(qx))J(x)~-~,~0

(3.lOb)

t~t,;r~y(q)~Jdx (-i-VxM~(qx)) J(x)~~~, $~ 1

(3.lOc)

‘

J

dxM~j(qx) J(x)~~, ‘

1,

30

T. W Donnelly and RD. Peccei, Neutral current effects in nuclei

the Coulomb, longitudinal, transverse electric and transverse magnetic multipole respectively; and for the axial-vector current likewise

sf dxM~(qx) ~

~

~ 0,

l~(x~,~0,

L~~(q)~fdx(-~-VM~’(qx))

T~5~(q)~f dx (-!-vxM~(qx)) f~(x)~,~

f

(3.lla) (3.llb)

(3.llc)

1,

5, ~(q)~ dxM~,’(qx) JS(x)~, ~ 1. T~Z The projection functions in eqs. (3.10) and (3.11) may all be written in terms of

(3.lld)

‘

M~(qx)~j,(qx)Y~(1l~)

(3.l2a)

and M~ 2(qx)~j.~q(qx) ‘&~I(11X). (3.l2b) 85all have parity (~)~‘:the other four, ‘Pn~a~ A~,I~,and D~’ all have The multipoles ~ L,, T~T~ parity (_)~I~ Since we have assumed that the vector current is conserved (not so for the axial-vector current, however), we may use current conservation to eliminate one of the four vector multipoles in eq. (3.10); generally the longitudinal multipole is written in terms of the Coulomb multipole [3.1—3.31, = —

~ ~

(3.13)

and we are left with three vector and four axial-vector multipole operators. We may now use angular

momentum conservation and the Wigner—Eckart theorem to generalize eq. (3.9): (J’M~.;T’MT.!T~,..~IJMJ;TMT) =

(_ )J.~MJ~(~

~

~

(_

)T’~MT~ (-i~I:.

L

MT) X (J’;

T’

T 17

J;T),

(3.14)

where T is any one of the operators in eqs. (3.10) and (3.11), and where the symbols indicate matrix elements reduced both in angular momentum and isospin. Finally, the multipole projections of the full current operator may be constructed from these pieces: f3~jI)~ ~

j

~cI

~

— — =

(3’~)

~

i~r(q)+ ~Y) (T)

A

~eI

(~Y)

T,~,.7~(q)+~A

$v

,~-mag

— 0(5)~f.rnag

‘-‘

=

~ ~A(~./T4(,,k~?)

~

~

i

~

~

(9)

A~~,;3A~,(q) ‘5

j

~

j~j;~~q~ ‘e15

T~,;~5(q)

(3.15a) (3.lsb) (3.lSc)

&9)~f’mag~

.Mt~9.a~Q’~ ~A

~~

and in general contain both parities. Having considered the general multipole forms for the weak/electromagnetic currents, we shall proceedto a discussion ofthe specific semi-leptonic weak and electromagneticprocesses ofinterest in the next section.

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

31

3.2. Nuclear reactions In this section we summarize the basic expressions for the cross-sections of the semi-leptonic weak and electromagnetic processes of interest in the present work. Specifically, we shall wish to use electromagnetic electron scattering and the conventional (measured) weak processes, ~3-decayand p.-capture, to determine the nuclear physics aspects of the problem as well as possible. The more exotic neutrino reactions (proceeding via the charge-changing weak interaction) and neutrino scattering (proceeding via the neutral weak interaction) may then be predicted with confidence in the underlying nuclear many-body problem. The charge-changing and neutral current processes interrelate different charge substates for states with total isospins T and T’ (assumed to be good quantum numbers). Consequently, it is useful to write all of the following expressions in terms of matrix elements reduced in isospin, as well as in angular momentum. The precise isospin nature of the currents will be specified in the next section (eq. (3.28)). Details of the derivation of the following expressions may be found in references [3.l]—[3.3]. 1. For electron scattering from a nuclear state IJ; TMT) to state If’; T’MT’) via the electromagnetic interaction in lowest order (parity-violating effects are discussed in section 6) we have [3.1—3.3] dO’eeIf TMT

J’~T’M T ~

~

,,~

,

—

316

2 4ITcTMF(q,O)

1 + 2 sin2(~9)/MA’

where the electron is scattered from a state with 3-momentum k, energy e to a state with 3-momentum k’, energy through an angle 0. MA is the target mass, oM = (a cos ~0/2 sin2(~9))2is the Mott cross section and F2(q, 9) is the nuclear form factor. The last may be written in the Rosenbluth form in terms of longitudinal and transverse form factors, each independent of 9: ‘

2)2F2L(q)+ (~~+tan2(~0)) F2r(q),

F2(q, 9)= (q2

(3.17)

54/q

where F~(q)=

1

+

~

F~(q)= 2J+ 1

~ f ~ (

T’

~=O,1

..~i

I

~,,

~

T )(J’;T’~.~,;~(q)~f;T)J2,

0

r’

(—M~.’ ~ +

(3.18a)

T

M~)

~

‘P~(q)

T’

~

~~f’;T’~

t,m~(q)~f;T)~. (3.18b)

4

Only the vector multipoles enter and the longitudinal multipole has been eliminated as stated above. Note that the transition rate for real photon emission from state If’; T’M.) to state If; TMT) may be written [3.1]in terms of the transverse form factor (at q2,. = 0) for electroexcitation from If; TMT) to If’; T’MT’) given in eq. (3. 18b) w.,,(J’; T’MT’

—~

J; TM~)=

~

1j .8iTaqF~(q)].

(3.19)

2. The /3-decay rates where the nuclear system goes from state If’; T’M-) to state If ; TMT) may be

32

T. W Donnelly and RD. Peccei. Neutral current effects in nuclei

written similarly [3.3, 3.41: 4

W~ 2(W~—)2

de13

(df~1df~k 1 [(1 + ~ . (3)j(J’; T’

x

~~

T’

b . /3 + 2b 44 24 (b + f3)~e(J’;T’ .~ff,.

1(q)~J +

/

T)~+(1

—

.

I MT T ±1

2

2 .

P)I(J’;

T’,.1(q)~J

; T)~

.

1(q)~J; T)(J’; T’ 2’,~i(q)~J; T)*]

—

44

P)(RJ’ ; T’~Y~,(q)~ J; ~ 24 . (j, — /3)~e(J’;T’

T)~+~J’; T’~~q)

2)

ilJ; T)1

~

1(q) ~J; T)(J’: T’

~q)~ J; T)*]},

(3.20)

where G is the 2 Fermi constant (actually G cos 0~,where 0,, is the Cabbibbo angle; however, we use the value GM 0 = 1.023 x i0~determined by 0~.~0~Fermi /3-decays in nuclei and so the Cabibbo angle is effectively included [3.3—3.5]).The electron or positron has momentum k, energy (with maximum W~)and we let k = fJ, ~ = v/~i’~ where p is the neutrino momentum, = v~and 4 = q/~q~, where energy-momentum conservation requires w = + and q = k + v. The function F~(Z,) takes into account the distortion of the electron wave function by the Coulomb field of the daughter nucleus of charge Z and is given approximately by 2”~ 1), (3.21) F~(Z,) 2/(e where = ±Za/13, f3 = /3J. 3. For p.-capture (or e-capture) from the ls Bohr atomic orbit accompanied by a nuclear transition from state If; TM. 1.) to state If’: T’MT’) we have the rate [3.3, 3.41 is

is

—

i~

22 4Gis

1

,

w,.(J;TMT-~J;TMT)=l+/M2J+l~M

T

,

—l1

T

2

MT

2 +~J(J’; T’ (~T~

1(P)-

x

{~i
~

;~(“)-

~(v))~I~

J; T)J~J,

,(isfl~, J; T)1

(3.22) where the multipole operators are evaluated at neutrino energy as determined by energy conservation using the lepton atomic binding energy and the nuclear excitation energy. As indicated by the is

the two terms in eq. (3.22) include the lepton wave function in the radial integrals involved in evaluating the matrix elements. As a good approximation, since 4 ~

15(x) is a slowly-varying function

over the extent of the nuclear volume, we may extract this function from inside the matrix elements and use only the average value [3.3,3.4] 2= (R/1T)[{m,.MA/(m,.

~ R~?,,(O)I where R is a reduction factor taking into

(323)

+ MA)}Za13,

=

account the finite extent of the nuclear transition density. 11 and (i,, l~)respectively, with e or p., which can cause nuclear transitions from state If: TMT) to If’; T’MT.) have cross-sections

4. The charge-changing neutrino and anti-neutrino reactions,

1

=

(is,,

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

33

[3.3—3.5]

1

TIMT,)2G2!

1 T’

1 T ~2 ±1 MT)

61~

{

J~j;i(q) J; T)I2 (3 +2I’~44 P)I(f’; T’~~i(q)~f;T)I2 —24 (1 + (3)~e(f’; T’ A 1,(q) f; T)(f’; T’ .,~‘,;i(q) f; T)*] 2+I(f’; T’~~~~q)~1f; T)12) ::~;T)I ~ [(1— . 44. /3)(I(f’;T’::::~eIg;i(q)::.’ ±2j . (1’— p)~e(f’;T’ ~~ 1(q) ~f;T)(J’; T’~~ J; T)*]},

x

[(1+ j~’,

~

.

/3) x K.!’; T’

—

i~.

.

+

(3.24)

where the quantities are defined as in /3-decay above except that energy-momentum conservation requires that w = is and q = k v. Of particular interest is the extreme relativistic limit (ERL) where ~ m, (/3 = 1) and we have [3.2—3.5] —

—

dcr I ERL (f; TM.1--~f’;T’MT’) 1

_4G2! +

/

I(f’;

1 T \2I cos 20{~ T’ ~1 MT/ 2 ,~.°I

T’

—

L%,;i(q)+~~’,;i(q)!Lf; T)I

[(~+tan~)(KJ’;T’~~~1(q) ~J;T)I2+(f; T

~

~ 2tan ~

(~+tan2~) ~e(J’; 1/2

T’~

~~q)

J; T)j2)

,(q) ~f; T)(J’; T’~~

~

~f; T)*]},

(3.25)

where 0 is the angle between the leptons. 5. Finally, the cross sections for (elastic and inelastic) neutral current neutrino and anti-neutrino scattering [3.5—3.7], (v,, v) and (1,, i~),with I = e or p., inducing nuclear transitions from state If; TMT) to state If’; T’MT’) may be written [3.5] do 1 ~?I~:1TMT’T’MT~~

x{~ ~

+~

+

X

2(GK)2!~_ 1

( ~ (—MT1” ~

[(~+tan2~)(I

Itr~,iIT’

k—MT’

‘5~,

20

v2J+l’05 ~

2

~

.~

0

MT

~

/

T’

Y

T

1~—M~ 0

MT

.T T\ 0 MT)<~’ T’~ ~q)~!J;

IT’ ~ ~\—MT~0

)(f’;T’~r~’;

T\’T’

MTA—MT.

x(f’; T’~~~’(q)~!J; T)*]},

~‘

0

T

2

9i.(q)~!J;T)f

~

0 T)I ) ~ 2tan~(~-~+tan2~) 2

MT)~e(J;

2

1/2

T’~ ;sr(q)~J;T)

(3.26)

34

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

where 0 is the neutrino scattering angle, v(v’) is the initial (final) neutrino energy and q = v’ — v, where v(v’) is the initial (final) neutrino momentum and K is defined in table 2.1.

The problem is now reduced to computing the nuclear manybody matrix elements of the multipole operators listed in eqs. (3.15). As a step towards this goal, in the next section (3.3) we shall consider the one-body problem. The single-particle matrix elements may then be embedded in nuclear many-body problem by treating the multipole operators as one-body operators (section 3.4).

Single-nucleon matrix elements To proceed further in analyzing the multipole matrix elements required in our description of the processes discussed in the last section, we shall need a microscopic treatment of the nuclear currents. 3.3

We begin by considering the case of a single nucleon where we have the general form (based on Lorentz covariance, conservation of parity, time-reversal invariance and isospin invariance) [3.1—3.4] (K’A’; fm,. I (J,.(0))D~~ I KA; ~m1)= i ü(K’A’) x [Fr~7,. + F~o-,.~q~ +iF~q,.]u(KA)(~m,.~ I~ I~ m1), (K’A’; ~m,. I

(J~(0))~~ =

I

KA; ~m,)

1~~y

i U(K’A’)[F~y5y,. —i F

5q,. — ~

m,.I I~~r I~

(3.27a)

(3.27b)

Here the plane-wave single-nucleon states are labelled with three-momentum K(K’), helicity A(A’) and isospin ~ The momentum transfer is given by q,. = (K — K’),.. The isospin dependence of the single-nucleon currents is contained in 1 I~—~X T0T3 T±I~(Tl±iT2)

~=0,

A90

~7l,

iit.~s0

~=1,

L~,=±1

(3.28)

The Wigner—Eckart theorem in isospin yields immediately [3.8] / i! ~2m,.

i4&9

~

!2m,1—~—, \— ., I/2—m,’~

—

j

2

~—m,.

II ~

2

m,

where [x] \/2x + 1 and we have used the facts that (~flhII~) and= (~ITfI~) tX~~ = F~(q~),~ = = 0,\/2 1, X 1,2, 5,=A,\/6. P, T (vector (Dirac), The~single-nucleon form factors F vector (Pauli), scalar, axial, pseudoscalar, tensor) are all functions of q~.We shall continue to use the CVC hypothesis, in which case there are no induced scalar (second-class vector) currents, F~ = 0. Furthermore, in the present work we will assume that there are no induced tensor (second-class axial-vector) currents, F!~= 0. The vector form factors may be studied through electron scattering on the proton and deuteron; for our purposes it sufficies to take [3.1—3.41 F~(q~) = F~(0)fsN(q~), where F~(0)= 1, .~T = 0,1 and Fr(q~)+ 2MNF~(q~) with p. ~°~(0) = p.5 = 0.8795 and p.~)(Ø) = = 4.706 being the single-nucleon isoscalar and isovector magnetic moments respectively (in nuclear magnetons). We take the common single-nucleon form factor [3.4] to have the form fSN(~~) = [1+ q~/(855 MeV)2f2. For F~(q~j) we use F~(0)fSN(~~) as well, with F~(0)= —1.23 as determined by the Gamow—Teller part of the neutron /3-decay rate [3.3—3.5].For F~(q~) we use pion-pole dominance and the Goldberger-Treiman relation to write

T. W. Donnelly and RD.

Peccei, Neutral current effects in nuclei

35

[3.3—3.5] F~(q~) = 2MNF~(q~)/(q~ + m~).

We are then left with two unknown single-nucleon form factors F~°~(q~) and F~”(q~). For neutrinos, or ultra-relativistic leptons, the induced-pseudoscalar term cannot enter [3.3, 3.41 and so we shall omit it from our discussion. For lack of any better information we shall take F~(q~) equal to F~(q~). We should remark that a non-relativistic quark model [3.9]evaluation gives —F~(0):F~2(0):—F~(0)= ~:1: 1, so that our assumption should at least be in the right ball park. We proceed to make a non-relativistic reduction of these matrix elements of the single-nucleon current [3.3] and employ nuclear shell model single-particle states labelled with sets of quantum numbers a {na (Ia ~)j0,ma; m,0} { a; mi,, m~,,}.Here ~a is a principal quantum number characterizing the single-particle state of energy ~a, Ia is the orbital angular momentum, Ia the total angular momentum of projection m5,,, and the isospin ~has projection m,,,. In using the shell model basis set of single-particle wave functions rather than the plane wave basis appropriate to free single-nucleon ~,

kinematics, we do not satisfy the (free) on-shell energy-momentum relationship; however, a measure of how far off-shell we are is ELI/MN 4 1, and we shall assume that the free single-nucleon current may

be used unchanged for the single-particle shell model matrix elements of the current. Exactly as in eqs. (3.10) and (3.11) we make multipole decompositions of the vector and

axial-vector currents and so consider reduced matrix elements of the sort 1.

I..

1

.—

1

and ~ Rather(3.30) than where T is any one of the seven multipole operators M, T~, T dealing directly with these multipole operators which contain both the single-nucleon form factors F~(q ~) and the operators V, a, etc. (resulting from the non-relativistic reduction of the y’s), it is convenient to define seven new operators, T~(qx),~ = 1,.. . , 7 (independent of the form factors Fx(q~j)[3.4,3.5]: (a ii T~y(q) a)

(n~.(l~’s) Ia’; ~ii T~~(q) ii fla(la~)fa;~), m~,M5, L5, ~

1,

M,~’(qx),

(3.31a)

=

2,

L~’(qx) Mj(qx). (1/q)V,

(3.3lb)

=

3,

~J”(qx)~ _i[~VxM~(qx)]

~=

=

[5]’[

—

.

V~M~+

1V,

(3.31c)

1(qx) + \“~+1M~,(qx)].

4=4, 5,

4

qx)~M,~J(qx)a, ~,~qx)

—

(3.31d)

i[-~- VxMj(qx)] a

=

[,,~]~[

—

V$M~÷,(qx) + V~+l M~~ 1(qx)] . (3.3 le)

4

1[v~nM;~+ =

6,

!,~‘(qx) [-~-vM~’(qx)].

=

[J]

(3.31f)

1(qx) + V~ M,~j..,(qx)].

(3.31g) where M,~’and M,~are definedin eq. (3. 1~2).We may combine these operators with the isospin operators given in eq. (3.28) to write: 4

~

=

7,

11~’(qx) M,~’(qx)u (1/q)V, .

;~(qx)= p~~)(q ~)M,~’(qx)I~,

(3.32a)

T. W.

36

Donnelly and RD. Peccei, Neutral current effects in nuclei

t9t(q~

iT~

5~(qx)= ~

‘(qx)]I:~,

(3.32b)

+ ~p.

9~(qx)= (q/MN)[F~(q~J~’(qx) —~ ~

;~t~(qx) = (q/MN )[F~(q~)Q~’(qx) + ~(F~(q~) + q0F~(q~))~“‘(qx)]I~’~, 2/2M~)F~(q ~)]~“‘(qx)I~, = [F~(q ~) — (q —iT~,;~r~y(qx) = F~(q~)~‘~‘(qx)I~,

(3.32c) (3.32d)

(3.32e) (3.32f)

T 1.;y~(qx) = F~~(q ~)I’(qx)I~.

(3.32g)

(Note that, to the order we are working, qoIqI 4 1, we could justifiably drop the q0 term in eq. (3.32d). In situations where one wishes to exploit the full symmetries of the multipoles, such as in isoelastic processes [3.5], it is better to drop this piece.) The doubly-reduced matrix elements of these multipole operators are then composed of reduced matrix elements in isospin space of the operators I~ given in eq. (3.28):

(MI’~II~)[,9r]/\/~ =

(3.33)

and of reduced matrix elements in angular momentum of the seven basic nuclear operators ~ = 1, 7, given in eqs. (3.31). These latter matrix elements are of the form (n’1’j’IIT~IInlj)and are -.

.

,

listed in appendix B for completeness. It should be noted that the definitions in eqs. (3.31) have been chosen so that, with the phase conventions used for the single-particle states Inli, m1) (see appendix B), the reduced matrix elements of the Tw are all real. Thus for a given choice of single-particle basis wave functions {4~tj,mj(X)},the single-particle matrix elements are now specified functions of momentum transfer q. In the next section we show how the nuclear matrix elements of the currents may be expanded in terms of these (assumed) known quantities. 3.4. Nuclear many-body problem We shall take as given the single-particle reduced matrix elements of the currents discussed in the last section. We now proceed to the nuclear currents by making the basic assumption that all of the semi-leptonic weak and electromagnetic operators can be approximated to sufficient accuracy in the regions of 4-momentum transfer of interest by one-body operators (operating in the nuclear Hilbert space — that is, with nonexplicit meson degrees of freedom). The question of how significant two-body meson exchange-current effects are has been addressed to some extent [e.g. 3.10]. It appears that for electroexcitation of discrete nuclear levels at momentum transfers below q —~ 500 MeV meson exchange currents generally provide small modifications to the dominant one-body terms and that to perhaps the 10% level we need to deal only with these latter pieces of the current. Exchange-current effects in weak interaction processes are less well studied, but indications [3.4,3.5, 3.11, 3.12] are good for proceeding with the type of one-body analysis to be presented in the following. Of course, it is of

interest to find some specific nuclear transitions or special kinematic conditions where such an analysis clearly fails, thus signalling a major role for exchange currents. We shall then assume that any one of the weak and electromagnetic multipole operators can be written as a one-body operator [3.13]: =

~(a’IT,..I~,;fJ.~g,(q)Ia)c,,.cc,

(3.34)

T. W. Donnelly and RD.

Peccei, Neutral current effects in nuclei

37

where c,,, = 0(~ EF)a, + O(F a)Sab~-.LI, with a~a particle destruction operator and b~aa hole creation operator operating above and below the Fermi surface of energy respectively. Here again the notation a {a; mia, m11) with {a} {na(Ia~)ja;~} has been used for the single-particle quantum numbers and, in addition, we define —a ~{a; —rn,,,, —m,,,}. The phase factor S~ (...)Ja_mJ~(....)1/2_mt~ is included [3.13]to maintain the irreducible tensor character of c,, and its adjoint c ~,,.The one-body operator is thus expanded in a complete set of bilinear combinations of creation and annihilation operators with single-particle matrix elements as weighting factors. So long as complete sets {a} and {a’} are employed there is no approximation in eq. (3.34). The reduced matrix element of such an operator between nuclear many-body states (of arbitrary complexity), Is) with angular momentum J, isospin T and Is’) with f’, T’ may then quite generally be written [3.2,3.5, 3.12] —

F

ii~. ~,

:: ~ ~ 1. :: ~‘i,;9ji~qj~J

r~ 1 /

— ‘~‘ —

~

/\(2 ::

~(s’s)~, ~a,, 21~I.q,::a/~l~,;g-I~a ~

~

~

~

aa’

t

where (a’ T1~.(q) a) are(one-body the reduced single-particle matrix elements in the last section. The numerical coefficients density matrix elements) are givendiscussed by ~I/;:~,ka’a) = [IF’[~Ji’(f’; T’ ~~J;,9’)

(3.36) where ëa’a(5A~,;Y4t~-)is defined to be the tensor product [3.13]of ~ and S,,c_a. We shall consider our complete set of single-particle wave functions to consist in general of bound states and standing waves in the continuum, in which case, time-reversal invariance implies that the one-body density matrix is real. No matter how complicated the nuclear states are, the exact reduced matrix elements of any one-body multipole operator can be expressed as linear combinations of an infinite set of single-particle reduced matrix elements with admixture coefficients ~/i~,~(a’a).Of course generally to make any progress we must truncate the sums over a and a’ to a small subspace [3.4,3.5, 3.11, 3.12], in which case there are only a few numbers s/i~(a’a)needed to characterize the many-body matrix elements. Such truncations are also made in shell model calculations: for example, one may use some effective nuclear Hamiltonian and restrict one’s attention to just the ip shell or just the 2s-ld shell. However, even within these simple truncated model subspaces there are usually an enormous number of shell model configurations. The essence of the one-body density approach taken here is that, no matter how complicated the states within the subspace may be, there are only a few density matrix elements which can enter. Ideally, for a given nuclear transition s’s we would adopt the following procedure: (1) Use a measured process such as electron scattering to extract many-body matrix elements of a f; T),

set of multipole operators, including the q-dependence of these matrix elements. (2)

Expand these many-body matrix elements in terms of (known) single-particle matrix elements

as in eq. (3.35). Using the q-dependence of the single-particle matrix elements, we may deduce a set of one-body density matrix elements lfJ~’~(a’a). (3) Having a set of ifr’s we may turn the argument around. For some other multipole operator of the same rankj and 5~(e.g. as required in discussing the weak interactions), we may combine the *‘s and ‘When we come to employ this basic equation for the processes discussed in section 3.2, we shall include as a multiplying factor on the right-hand side of eq. (3.35) the center-of-mass form factor f,,, 4(q). We have suppressed it in the present section. In the special case of harmonic oscillator wave functions this factor has a particularly simple form: 2q2/4A1, (3.35a) exp[b parameter and A is the nucleon number. where fcM(~) b is the=oscillator

38

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

the appropriate single-particle matrix elements to predict the required many-body matrix elements. Our predictions are then to a large extent independent of nuclear physics uncertainties and we may concentrate on the detailed nature of the elementary interactions themselves.

Of course practically it frequently happens that the complete set of ~“s (even within a reasonable truncated subspace) cannot be determined fully through lack of experimental information. We may then combine shell model calculations and what data do exist to determine a set of i/i’s which are at least maximally constrained by experimental impact [3.4, 3.5, 3.11, 3.12]. The procedure outlined here will be made clear with the example given at the end of this chapter and with the analyses presented in

section

5.

However, before proceeding to that example, it is useful to discuss a special limit of the

many-body matrix elements of the multipole operators, namely where q—*0, the long wavelength limit. This is done in the next section and will be of relevance in discussing low-energy neutral current neutrino scattering in section 4. 3.5.

Long wavelength limit (LWL)

One of our prime objectives in this article is to discuss how information on the underlying neutral

weak current and the quark structure of the nucleon may be obtained by studying the process of nuclear excitation via inelastic neutrino scattering. In particular, it is of interest to study the situation where the neutrinos or anti-neutrinos are at relatively low energies. For example, at the LAMPF neutrino facility [3.14],pions are produced at the beam stop, come to rest and decay ~ p.~+ neutrino energy is = 29.79 MeV). Subsequently the muons decay at rest (p.~ e~+ ~e + i,,,, ~ = 53 MeV) and so this facility provides i,. and ~e at energies below 53 MeV. Note that the muon neutrinos and anti-neutrinos are below threshold for the charge-changing reactions, (is,., p.1 and (i~,.,p.~),whereas (isa, e) reactions are allowed. In contrast, neutral current neutrino and anti-neutrino scatterings, (is,, is) and (,7,, i~),are allowed both for electrons (1 = e) and muons (I = p.). The maximum amount of momentum transferred in these processes is ~max= 2is — as, where as is the nuclear excitation energy. Thus the momentum transfer is also small, where momentum transfer may be measured in terms of a typical nuclear momentum Q, with Q kF (the Fermi momentum) 250 MeV, or equivalently, a typical nuclear radius R Q~. The long wavelength limit, (q/Q) 4 1, is then of interest in discussing these processes and is considered in the present section. The multipole projections in eqs. (3.10) and (3.11) were made with the functions M,~(qx)and M~(qx)given in eqs. (3.12). When q is small (compared to Q) we may use the fact that 1)!!, (3.37) j.~(qx) (qx)~/(2~+ 1)!! =(q/Q)~(Qx)2/(2~’+ ...~

is,,,,

—*

is,,,,

to write M(qx)

(q/Q)’m(Qx)

+

O[(q/Q)’~2],

(3.38a)

q-.O

M,~(qx)

~

~ (q/Q)~m,~(Qx) + O[(q/Q)~2],

(3.38b)

where Y~(fl~),

(3.39a)

m~e(Qx)~(

2~~1),, ~

(339b)

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

39

Since Q was chosen to be a typical nuclear momentum, the product Qx is of order unity and so the projection functions m,~(Qx)and m,42(Qx) may be taken to have nuclear matrix elements of order unity.

The seven basic nuclear operators T~j~,(qx),4 in the long wavelength limit (qIQ 4 1) as follows: 4=1, 4

=

2,

4

=

3,

=

1,.. . , 7, given in eqs. (3.31) may then be written

M~’(qx)—*(q/Q)’[m~’(Qx)],

(3.40a)

‘(qx)-~(q/Q)[mj(Qx)~ (1/Q)V], ~~‘(qx)

4 = 4, 4: = 5, 4

=

...~

(3.40b)

(q/Q)’z[~]’V$nm~_i(Qx)

.

£,~‘(qx)-*(q/Q)’[m~~j(Qx) . a], qx)-~(qIQ)’’[$]’\/jimj_i(Qx)~

6,

(1IQ)V],

(3.40c) (3.40d)

a],

(3.40e)

‘(qx)-*(q/Q)’’[5]’V~m~j_1(Qx)~a],

47,

fl,~’(qx)-+(qIQ)’’[m,~”(Qx)a. (l/Q)V],

where in these equations we restrict our attention to point: in the long wavelength limit

,,~

(3.40g)

~ 1. Note that one relationship emerges at this

(3.41)

1”(qx)= \/~—j~’~’(qx), (q/Q)41. The cases where ~

(3.401)

0 must be handled separately: for the vector current 2(Qx)2/6 +~ . .), M°o(qx)= j0(qx) Y°0= jo(qx)/\/4ir—s. (1/\/4ir)(1 (q/Q) and so for elastic scattering, where state s’ equals state s in eq. (3.35), we have =

(3.42)

—

M°o(qx)

—*

i/V4

1r,

(elastic),

(3.43)

while for inelastic scattering where the leading term in the low-q expansion in eq. (3.42) vanishes, since

are orthogonal, we have 2[—(1/V4.rr)(Qx)2/6], (inelastic). (3.44) M°o(qx)—~(q/Q) Thus the low-q behavior of the inelastic Coulomb monopole (CO) is the same as for the Coulomb s’

and

s

quadrupole (C2) — both matrix elements are proportional to (q/Q)2. Finally for the axial-vector current in cases where J = 0 we have: ~‘o’°(qx)-~(q/Q)[m°o a] 1 and

1[(1/v4ir)a

ng(qx)—~(qIQ)—

(3.45) .

(3.46)

(1/Q)V1,

these being the only two multipole operators allowed (see eqs. (3.10), (3.11) and (3.32)). When these results are substituted in eq. (3.32), we obtain the long wavelength form of the complete multipole operators. Two special relationships can be proven: (1) from eq. (3.41) we find that T~~,;

.~/~J~ 1

5r~r(qx) =

~ ~ 1, (q/Q) 4 1

(3.47)

40

T. W. Donnelly and RD. Peccei. Neutral current effects in nuclei

and (2) from current conservation for the vector current we have (s’

t~9(q) s)

— (Es. —E,)

.~JLt...i

l~,yj(q) s),

(s’

(3.48)

where E,. and E, are the nuclear energies of the states Is’) and Is) respectively and as = E,. — E,. The leading orders for both vector and axial-vector multipole matrix elements are summarized in table 3.1. Of course these results are to be regarded as order-of-magnitude estimates only. When the detailed nuclear dynamics are taken into account, some matrix elements may be enhanced (for example in the case of giant resonances), either enhanced or diminished (for example when effective charges are called for) or diminished (for example in the case of isoscalar El transitions). The familiar ordering of the multipoles is clearly seen however: we have in y-decay, for instance, where q/Q is generally very small that ~

Ml~M2~M3~...

We see that the leading multipoles (i.e. those that are of order unity) are: (1) the M0 (elastic) multipole which enters in elastic electron scattering and constitutes the Fermi matrix element in /3-decay and (2) the L~and T~’~ multipoles (they are proportional through eq. (3.47)) which enter in Gamow-Teller /3-decay. For low-energy neutrino and anti-neutrino scattering from nuclei we are interested in these leading multipoles as they yield the largest cross sections. The M0 (elastic) multipole only enters in elastic scattering; it cannot cause inelastic transitions to take place. Consequently if our attention is directed to inelastic scattering, where the subsequent decay of the nuclear level that has been excited is used as a signal for the event, we will be most interested in the ~ = I multipoles. The axial-vector pieces (L~and Tv’) are of order unity, whilst the vector piece (T~”~) is of order (Q/MN)~ q/Q = q/MN, as can be seen in table 3.1. Since q 60 MeV typically in low-energy neutrino scattering, the axial-vector Table 3.! Order of multipole matrix elements in the long wavelength limit* Vector

Axia!.vector

I Q/MN

M0(elastic)’

w/Q

M, T~ 5 T~~a M,(inelastic), M

L~,T~” M~ L. L~.T~”.T?Is~ M

q/Qw/Q 2.Q//t4 (q/Q)

T1’ T~

qJQ

(q/Q).Q/MN (q/Q). q/Q

2

*The cross sections

w

L. Ti”. T~’ M~

are proportional to the squares of these

quantities. For an order-of-magnitude estimate of the amplitudes take Q 250 MeV and 10Mev, then QIMN 1/4, w/Q 1/25 and q/Q = 1/25 for q w (as in y-decay or $-decay), q/Q = 1/4 for q 60 Me’s’ (as in low-energy neutrino processes). ‘The isoscalar part of this is enhanced by an additional factor of A.

T.W.

Donnelly and RD. Peccei, Neutral current effects in nuclei

41

matrix elements dominate over the vector matrix element by an order-of-magnitude or more. (We will present detailed calculations in the fifth chapter which demonstrate this explicitly.) We are left with an important conclusion: using low-energy neutrino and anti-neutrino scattering from nuclei we may, by choosing ,9’~= 0 or 1 Ml transitions, study specifically the isoscalar or isovector axial-vector pieces of the neutral weak current, to the exclusion of all vector contributions [3.6, 3.7]. 3.6. A simple example: The A

3 system

Before embarking on a discussion of transitions where the full complexity of the nuclear many-body problem arises we shall consider a simple example in this subsection, the ground states of the f”T = ~1 system with A = 3, 3H(MT = —~)and 3He (MT = +~). This example is one of a general class of isoelastic processes [3.5] which involve elastic electron and neutrino scattering and the

(almost elastic) charge-changing processes between the members of a ground-state isodoublet. By making use of rotational invariance, isospin invariance, time-reversal invariance and parity, and using

hermitian operators the set of multipoles which can enter is considerably reduced [3.5].If secondclass currents are neglected only the following multipole operators contribute (f half-integral): ~

~=0,2,4,...,2f—1,

~f’mag I

~

~

ac—i .2 ~ — L~.)~ .J,.

J

4t~~4t~,

—

.

c 1~J,J,.

.,

.. ,

U,

,j = 1, 3, 5;. . . , 2f, or 1. Thus for the A=3 system we must consider A~,.

~

with ~=0

(3.49)

4=,

0, 1’~, and t~i. The multipole operators were written in terms of the seven basic nuclear operators in Eqs. (3.32) and we see that in general we require

M~”,

~=0,2,4,...

,2J—1,

J=l,3,5,...,2J,

~‘,

~=1,3,5,...,2f, J=1,3,5,...,2f,

~ ~4ti,

(3.50)

where these operators were defined in eqs. (3.31). For the A = 3 system we must consider M,...0, ~=1.

To describe the nuclear physics of4He(0~0) the A =to3 be system let 1sus make the following a closed 3H simple and 3Heapto proximation for the states: we consider 112-shell nucleus and be states with one hole in the filled ls,, 2 shell, namely 4He). (3.51) R.M~ ~.M’T)= b~j,2,Mj;(I,2)MTI Here b creates a hole with single-particle quantum numbers a {na(la~)ja;~ rn-s, m, 1}

{a; mia, m,1}.

(See the discussion in section 3.4.) Of course this is not the most general configuration allowed — there may also be one particle two hole (lp—2h), 2p—3h and 3p—4h configurations involving the complete —

single-particle basis. With our simple approximation the one-body-density matrix elements (eq. (3.36)) are easily calculated (the labels s = s’ = ground states of the A = 3 system are suppressed here): ~]‘[~]‘(~M~.; —

,-.~ 112—Mj.

— “

!

~MT’I~~SI/2ISI,2(J.A1,; /

I

I \

—

6

2

J

Al I lJ

II .111.5

I ~ \j.~ 11 1k IVIj/

~)I~Mj;

1/2—Mi-I

‘

~MT) I ~

Al \‘~‘T

~.

J II ~

2

Ad

IVITI

‘~ ~

2

j~t~,;~k1Si/2

42

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei =

[3rF~(~HeIbIsii~,Mjt1I2MrEm~m

[I]’

X C~ij,m.1/2,.(—

)h/2+m(

—

~

‘I~~.9AI2~)

)h12+~’c,s,s.~m,:I/2._,..]b~,s,M,;(I/2)MTI4He),

(3.52b)

which leads to —

2

t//J~(lSI/2) —

4’5,o3~)o—(—)

(3.53a)

A=3

fl~=0, ~T=O

—l

~=l,

(3.53b)

= E~T=l

Having obtained the one-body density matrix elements in this simple approximation, next we require the single-particle reduced matrix elements of the allowed operators to be able to reconstitute the many-body matrix elements using eq. (3.35). From the general expressions in appendix B we obtain \/4ir(ls11~~M’o(q)f~ls 1/2) = \/2(ls ,,2~j0(qx)f ls 1,2),

(3.54a)

\/4iT(ls1j2II~1(q)Ills112)= 0,

(3.54b)

\/4ir(ls I/2II~‘1(q)IIls 1,2)

2(1s112Ij0(qx)~ls1/2)~

(3.54c)

\/2(lsl,2I jo( qx)Jls I/2)~

(3.54d)

~

=

1/2) =

and thus all ls112-shell single-particle matrix elements are proportional to a single radial integral (lsii2Ijo(x)Ils112) =

J

2 dx j x

0(qx) R1~11,(x).

(3.55)

If we use harmonic oscillator radial wave functions, 4b 312e —(x/b)2/2 R~°(x)= 2i,~“ where b is the oscillator parameter, we have

(ls~jO(qx)Ils)H.o.= e~, where

y

F(q)

(3.56)

(3.57)

(bq/2)2. However we may be more general and define (ls 112Jjo(qx)~ls112).

(3.58)

Then, using eq. (3.33) for the isospin matrix element, we obtain the complete set of Is112-shell

single-particle reduced matrix elements of the multipole operators given in eqs. (3.32): 2F(q), (ls,,2~ M0~(q) ls112 ~) = F’~(q~j[~T](4ir)~’ (1st, t~(q~)[ff](2ir)~2F(q), 2 ~ —iT77(q) ls112~)= (q/2 N)p. (lsl, 2F(q), 2 ~ — iL~sr(q) lsl,2 ~) [F~(q~) 2~fN F’~(q~,,)][~J](4ir~” (is 1,2 ~ —iT~’~.(q)1s1,2 ~) = ~ =

(3.59a)

(3.59b) (3.59c)

—

(3.59d)

T.W. Donnelly and RD. Peccei, Neutral current effects in nuclei

43

These results combined with the density matrix given in eq. (3.53) yield the required many-body matrix elements of all of the multipole operators involved in our description of weak and electromagnetic processes for the A = 3 system (section 3.2). We shall postpone our discussion of detailed predictions [3.5]until sections 4 and 5 and conclude this section only with a brief treatment of the long wavelength limit for comparison with subsection 3.5. In the long wavelength limit (q —*0) we have F(0) = 1, regardless of the radial wave functions. We then find that (ls1,2 ~ M’o~(q) ls,,2 ~)

q—~O

(3.60a)

2, F~(0)[~](4ir)”

(Is,, 2 ~ —iT~~(q)ls1,~~)

‘

q

(ls,,2 ~ —iL~~(q)ls,,2 ~)

~

(3.60b) 2,

-.0

(3.60c)

F~’~(0)[~T1(4ir)”

(lsl,

2. (3.60d) 2 ~ —iT~~(q)1s~,~ ~) q-.O ‘ F(0)[,~](2i~)” These results should be compared with the general orders given in table 3.1. We may, for example, consider the elastic electron scattering form factors given in eqs. (

W—~T

F~(q)M~ =

3.l8)t: 0

~ J~o;o(q)~ ~ (—iL

MTX2

0

MTX2

~“

A~o;i(q)~ 1)(2 (3.6ia)

2F(q)+ (2M~/V6)x ~

1 /V2) X 3F~°~(q ~,.)(4irY” =

(4ir) ‘[~(3 + 2MT)]2f sN(q ~)f~M(q)F2(q)

=

(3.6id)

A/2 + MT, MT = —~for 3H and MT I

F~-(q)M~

=

=

+

~for 3He, 11

I

=~(—~T

+

(3.61c)

z2I4~

q-.0

where Z

(3.6 ib)

(—L

0 0

~

~~q)~

(3.62a)

1)J2

~I(1/V2) x (q/2M~)p.(O)(q~,,)(21ryh/2F(q) + (2M~/V6) x~

=

(8ir)

I(q/~~)2[~(p.S

—1

q-.0

—

(8ir) (q/MN)

~ 2

2 p.MT,

‘See eq. (3.35a) for the center-of-mass form factor fcM(q).

~,,)f~?~I(q)

(3.62b) F2(q)

(3.62c) .6

44

T. W’. Donnelly and R.D. Peccei, Neutral current effects in nuclei

where p.MT = — 2MTp.”) is the magnetic moment in nuclear magnetons (see also refs. [3.5]and [3.151). That is, for our simple nuclear model we have for 3H, p.~l/2 = p.p (the proton moment) and for 3He, p.÷112= p.n (the neutron moment) which are actually quite close to the experimental values. In a similar manner, if we wish to calculate the /3-decay rate 3H tHe + e + ie using eq. (3.20), in the long wavelength limit (which is certainly applicable for this process) we require only ~(p.5

-*

(~~ (~~

M~.,(0) it~i(0)

~

~) = (3/4i~)~’~ (Fermi decay) ~

~) = (~ —iT~i(0) ~

~)/\/2 (3/4ir)”2F~>(0) (Gamow—Teller decay). ~j;

(3.63a) (3.63b)

It is then a straightforward matter to compute the /3-decay rate or indeed to calculate the cross sections for all of the processes discussed in section 3.2 (see ref. [35]). This simple example has been included to serve as a bridge between the formalism of the preceeding sections and the detailed results to be discussed in the following two chapters. Here the nuclear physics aspects of the problem are particularly simple, whereas in the succeeding chapters the nuclear many-body problem will generally involve the ideas presented in section 3.4 in a less trivial manner.

4. Low-energy neutral current neutrino scattering from nuclei In this section we discuss the weak and electromagnetic processes considered in the last chapter

under conditions where the momentum transfer is sufficiently small that the long wavelength limit (LWL) of section 3.5 is applicable. Our aim is to be able to predict the rates for neutral current neutrino scattering from nuclei, thus providing a tool to study the gauge theory models discussed in sections 1 and 2. We begin by considering inelastic processes, where, as we saw in section 3.5, the 5 = 1, ~1T = no Gamow—Teller (GT) multipole dominates at low momentum transfers [4.1, 4.2]. In section 4.1 we derive several general relationships among the inelastic weak and electromagnetic processes at low energies. In section 4.2 we present a brief discussion of elastic scattering at low energies, where, in addition to the GT multipole, we must now also consider the $ = 0, ~- = no Fermi (F) multipole. While it is difficult to devise a practical scheme for measuring elastic neutrino scattering cross sections, they may be of relevance in astrophysics in considering very high density matter as in gravitational collapse [4.3] and so we include this section for completeness. In section 4.3 we obtain single-particle estimates for the required multipole matrix elements in the long wavelength limit. Finally we conclude this chapter with a discussion of the application of these general ideas to specific experiments, (a) involving nuclear reactors and (b) involving a stopped 1T-p.-e neutrino source such as the LAMPF neutrino facility. We use the examples given here to highlight the sensitivity of neutrino scattering from nuclei to the underlying quark structure of the hadrons and gauge theories of the weak

and electromagnetic interactions. 4.1. Relationships among low-energy inelastic weak and electromagnetic processes As we saw in section 3.5 (Table 3.1), in the long wavelength limit only three multipoles survive, M

0(elastic), L~,and T~”.The first (the Fermi multipole) is proportional to a combination of the

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

45

nuclear number operator and the third component of the total isospin operator (or equivalently to a combination of the proton and neutron number operators). Since the nuclear states are eigenstates of these operators, the monopole operator in the LWL is diagonal and cannot contribute to inelastic transitions. It can of course contribute in elastic cases (or more generally isoelastic processes [4.4])

and these are discussed in the next section. Furthermore, the remaining two operators are proportional in the LWL (eq. (3.47)), D?~,;~j~.y(qx)V~i-(qx), (q/Q41), (4.1) and so there is only a single even parity dipole operator required to describe inelastic neutral current neutrino scattering at low energies. Of special interest is the fact that only the axial-vector current enters and so we may preferentially study this specific part of the weak interaction. By selecting purely isoscalar or isovector transitions using the (approximately) good isospin quantum numbers of the nuclear states involved we may focus on the corresponding pieces of the weak neutral current. That is, the neutral current neutrino scattering cross sections involve directly the coefficient a = /3~, Y= 0 and I given in table 2.1 (see eq. (3.4)). In detail the isp’ and v~’cross sections are given by eq. (3.26) and yield in the long wavelength limit d LWL(J=1) (J;TMT -*J’; T’M~.) =

2(GK)2!_ (1 +

(~J’~ ~) /3~(J’;T’~L~.(o)~ J; T)~

sin2~)2J~1

(4.2)

or, integrating over q2 (that is, over all solid angles), we have LWL(J

cTJ~ =

1)

(J;TMT—*J’;T’Mr.)

12(GK)2(v_w)2 2~~

1

~,,

(-A;:

~ Mg.)

t~.~-(0)J;T)I,

(4.3a)

where the notation was defined in section 3.2. The isospin 3-j symbols may be evaluated [4.5]to obtain

I

LWL(J-’l)

~:

(I; TMT

-*

0)IJ1~ T?:u1~S ~ — —

~

fv—w\ MN )

p1,,,

+

TT

2 K

1

(2J+ 1)(2T+ 1)

MrMi-

~

O( T’TPA\

VT(T ~2(T’ MT +1) + ÔT’T+I( ) (T_T+l)/2f(T —

0~nj. ,

1;O~.

.,::

T

2—(2MT)2[”2

T + 1)(2T’ ++T+ 1)

+1) J

x/3~(J’;T’~]L~i(0)~J;T)f, (4.3b)

where the isovector 8T’T term must have T ~ ~ and where the scale is set by the factor S~ 12G2M~= 5.563 X i0~cm2. As a rough estimate we may take ((v w)/MN)2 i0~as a typical value and see that we are dealing with cross sections of order few x 10~° cm2 x nuclear factors. As we shall see below, these latter nuclear factors are generally at least an order-of-magnitude less than unity and typical (i.e. not excessively hindered for some reason) cross sections are of order lO_41 cm2. Before discussing the nuclear physics aspects of the problem in detail let us list here the cross —

46

T.W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

sections for other processes in the long wavelength limit. For charge-changing neutrino reactions (eq.

(3.24)) we have LWL(5 =

1)

(J; TM,----*J’; T’MT.) cos °~2J~1(~T.

2G2~(3/3

=

2 T’~L~.1(0)~J; T)1

±1 MT)I

(4.4)

LWL(51)

(J~TMT—*J’; T’MT.)

r

2

24G2fl(v_as)2

=

1

2~~

fv—as\2 ___________

~

MN)

(4.5a)

T’~L~,(0)~J; flI

±1 MT)

(~T.

(2J+l)(2T+l)6MT.MT~1

T(T+ 1) (T

±MT

T±MT+

(T’+ T+1)(2T’+ 1)

x (~~±

2,

1(T’ ±MT)(T’ ±MT + 1) + ~T.T~I(T where the

~ M~)(T’~ MT

+

1))}RJ’; T’~L~.~(0)~J; T)I (4.5b)

term must of course have T ~ ~. In a similar notation the Gamow—Teller /3-decay rates

~TT

(eq. (3.20)) for transitions J’: T’MT.—*J: TMT may be written 6 —G2

LWL(51)

wit

J

1 d/3e2(W~_)2F~(Z~) (2J’+ l)(2T+

x {~TT T(~+ 1)(T’~MT)(T’± MT

T +i)(2T’+ 1)(ö T..T±I(T

+

l)8MT.MT±1

1)

MT)(T _MT +

1) 2,

~ M~)(T’~ MT term as above. +

with T ~ ~ for the

8TT

~T.T-I(T

+

l))} x I(J’; T’~L~i(0)~J;T)1

(4.6)

Finally, we wish to consider electromagnetic processes in the long wavelength limit for the same = 1, ~ir = no (Ml) multipole. Upon examining table 3.1 we see that in the LWL the appropriate multipole, T7’~,is proportional to q/MN. Thus it is convenient to define mag

~

\(Al —

;

~Iv1N,q,1

\ 7.mag ~ I.U,:gJ4ti-~qX,,

in which case the transverse electron scattering form factor for the transition J; TMT —~J’;T’MT. (eq.

T. W Donnelly and R.D. Peccei, Neutral current effects in nuclei

47

(3.18b)) may be written LWL(M1)

F~-(q)~

=

2 1 (~—)2J+

_f —

T’ ~ (—MT. 0

~

S

\

q

(2.1+ 1)(2T+ 1)

kMN) /

kI

T 2 MT)~~’ T’~1~7(0)~J; T)~

Al

IVI7’ TT\/T(T+1)

(4.8a)

iJ~.Ts::imaa(0\::J.T Mi-Mr

OT’T\

,

::

rip’

1;O

k

1 I F I) VIVLT) i-2(T’+T+1)(2T’+1)

T’,T±1 — (T’—T+1)/2J k

T)I

x(J’; T’~I~(0)~J;

~j. T P

~

—

~

Al

\2~ 1/2

(4.8b)

where, as in eq. (4.3b), the isovector ~5T’T term must have T ~ ~. The Ml 7-decay rate for the transition J’; T’M’T J; TM~may be written —*

2J 1 LWL(M1) w~’~(J’; T1MT,—’.J; TMT)= [2J’+l .81TaqF~.(q)]

(4.9)

using eq. (3.19) and is proportional to w3 as expected. We may now proceed to interrelate these processes in the long wavelength limit. We begin by considering purely isovector transitions (.T = 1). To evaluate the inelastic neutrino scattering cross sectiongiven in eqs.(4.2) and (4.3)werequirethe reduced matrix elementof L~,~~(0),which is precisely the same matrix element that enters in the charge-changing neutrinoreactions (eq. (4.4) and (4.5)) and in the /3-decay rates (eq. (4.6)). For example, consider a nuclear system where J”T = 0~0and J’~’T’= 1~1, such as the ground states of ‘2C(040), ‘2B(1~1,MT’= —1) and ‘2N(F’i, MT’= + 1) and the 15.11 MeV isospin analog state in ‘2C(1~1,MT’ = 0). The /3-decay rates are then 4

LWLLJ=1)

= ~

G2f”(W~’)5J(1;

1 L~,

1(0)~0; 0)12,

(4.10)

where we define the dimensionless phase-space integral w~

1

±

I

i’ui~~ k~’OJ

1

2

±

i d/3 (W0 J

±

2

—

(4.11)

) F (Z, ).

From the experimental /3-decay rates we can then deduce a value for the one required nuclear matrix element and write the charge-changing neutrino reaction cross section 2col’~””’’~ a’

LWL(5=I)

fv—w\

~2 S~ = -~-

18~MN)

2f~(W~)5

(4.12)

G

and neutral current neutrino scattering cross section LWL(J=~1) —

0~

~,

—

2

2L 2S K

/ ~‘

Vk

—

~2

±LWL(5=1)

V 0 ~ j~(I)~2 OP MN kPAJ G2f~’(W~)5

)

4 13

directly in terms of the /3-decay rates. Hence a measurement of the neutral current process (eq. (4.13)) will yield a model-independent value for the isovector axial coupling constant K 1$~iand provide a test for the gauge theory models of the weak interaction which predict specffic values for this quantity (see table 2.1). Similar analyses can be applied in the general case for isovector transitions with

48

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

using the explicit isospin factors given in eqs. (4.2—4.6). We shall postpone arbitrary J’TT and ~ the discussion of specific nuclear transitions until section 4.4. Next we consider purely isoscalar transitions (~= 0). Of course now we do not have a measured

weak interaction process such as /3-decay to use to determine the nuclear matrix element required to predict the neutrino scattering rate (eqs. (4.2) and (4.3)). This process involves the reduced matrix elements of the operator L~ 1oo(0).The only other neutral current process at our disposal is electron scattering (with y-decay as a special limit, eqs. (4.8) and (4.9)) and this involves a different operator t7’~;oo(O) (eq. (4.7)). However the matrix elements of these two operators may be interrelated. The axial-vector operator involves only the basic multipole operator I~(eq. (3.32e)), which may be related to I’~in the long wavelength limit using eq. (3.41). The vector operator, on the other hand, involves both ~ and ~ (see eq. (3.32c)). However these two basic operators take on special values in the long wavelength limit: ~

—*—(1/’s/241T)(L)M

(4.14a)

\/2/3i~(S)~,

(4. l4b)

—* LWL

where L and S are the orbital and spin angular momentum operators respectively. The relations are obtained using the properties of spherical harmonics and vector spherical harmonics. Then, using the fact that the total angular momentum operator (whose matrix element is known) is J

=

L + S, we

obtain an equation relating matrix elements of ~ and I in the LWL to a known number. Specifically, computing reduced matrix elements between states s’: J’T’ and s: fT we find that (J’; T’~I?’~(0)~J; T)

=

{~~0)}J’; T’ fl;o(0) ~J; T) + iS ,.,VJ(J + 1)/96’rr[J][T].

(4.15)

For inelastic transitions (s’

s) the second term in eq. (4.15) does not contribute; that is, the off-diagonal matrix elements of J are zero. Thus the matrix element of L~required in inelastic neutrino scattering may be expressed in terms of the inelastic electron scattering cross section or 7-decay rate for the same pair of states. The isoscalar Ml 7-decay rate for a transition J’; TMT .1; TMT(~T = 0) is —*

given by eqs. (4.8) and (4.9): (M1.~=O)

— —

8ira /J.~T:: ~inag (2T + l)(2J’ + 1) i~i~’~ :: t i~ø

—

T’ 2 /

,

.

6a

s

A

—

::

‘siTa

j

Ct)

~.L

2

/

,.

.

(2T+ 1)(2J’+ 1)~iJl.F~°~(0)J \J T::Li;o(0) ::J, T, ,

(4.l6b)

.

Note that these rates tend to be suppressed because of the near cancellation of the isoscalar magnetic moment, — = 0.3795. However this does not imply that the matrix element of L~,which enters in the neutrino scattering cross section, is itself small. We may express the (is, is’) or (~,1’) cross section for the transition J; TMT J’; TMT(~T= 0) directly in terms of the 7-decay rate: ~‘

—*

LWL(J.—1,~=O)

a’

j.—,y, ~ ~, — ~U.I__+_I% ~ —

~

2

/ ~

‘s2J+1i4ira~w

\ ‘I

2

(0)

1~($A ) /

21( (O)IA\’12 A k’-’)l 1

i

1

i~s—-~j

1

I

o —

(Ml,6=O) Wi-

,

(4.17)

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

49

so that, given the isoscalar Ml 7-decay rate, a measurement of the neutrino scattering cross section will permit a model-independent determination of the isoscalar axial coupling constant KI/3~I.By examining table 2.1 we see that gauge theory models yield a wide variation of values for this quantity and thus a measurement of the sort we are considering would provide a stringent test for those models. To reiterate the point made above, note that, although the isoscalar Ml 7-decay rates are generally small, the isoscalar neutrino scattering cross sections may be comparable to or larger than

similar isovector cross sections (i.e. for non-zero values of /3 since they contain the enhancement factor {F~°>(0)/(~~)}2 = 10.5 for our chosen set of coupling constants. We shall return to specific examples in section 4.4. Finally in the general case where both isoscalar and isovector matrix elements enter for the neutral current processes we use all of the ideas presented above. That is, we employ both the analog /3-decay rates and the y-decay rates to obtain predictions for the neutrino scattering cross sections. Upon examining eq. (4.3b) for neutrino scattering we see that the only case of this sort occurs when T’ = T 0 (the T’ = T = 0 case must be purely isoscalar). The isovector matrix elements of L~may be obtained using the analog /3-decay rate as discussed above. For the Gamow—Teller (GT) transition ~,

S—

J’; TMT;—*J; TMT we obtain (using eqs. (4.6) and (4.11)) IJ~.

T~’L :: 5 0 i~i

::

::j.

r’/ =Xp —

T

— —

j’

r

LWLLJI)

~ I k~ ijiT 0p 1..6G2I~(W~)5(T ~ MT)(T ±MT pi’r.±.

U

l\

2

—

+

~1/2

4 18

l)JI.

where MT = MT ± 1 for /3~-decay.The GT rate may be separated from the Fermi (F) rate as discussed in the next section. The complication arises in determining the isoscalar matrix element of L~because the y-decay now involves a combination of isoscalar and isovector matrix elements of t?~.The device where J is used to relate matrix elements of tr~to matrix elements of L~works only for the isoscalar parts, since J is isoscalar, and we must resort to additional tactics. The Ml 7-decay rate for the transition J’; TMT J; TMT(~= 0 and 1) is (eqs. (4.8) and (4.9)) —*

o4M0(Mr)

=

(2T +1X2J

+

1) MNI

T~Ir 0~(0) J; T) + \/T(T+

1) (j’; T I~(0)~J;T)I,

(4.19) from which we find, using eq. (4.15) as before, that ::j. T MT 1) (J’; T :: 5i;o(o):: )I 1+ \/T(T+ (J’; TflI~(0)~J; T~f7~(0)~J; T)) J’. T”L

(O)in~ g

X~(MT)= [T][J’]MN

s

~L

k’-’) —~

MICA,

(U-,~ ,..

~‘~I/2

(4.20)

k1~’TJ~

4i7~aw

)

if we assume that ‘s/T~(J’;T~I~(0)~J; T)I >\/T+ 1I(J’; T~I~(0)~J; T)j, as is usually the case, since the isovector term involves ~ while the isoscalar term involves ~ s — ~ and j~v S then we may extract~the isoscalar matrix element of L~by comparing the analog 7-decay rates having V

MT

=

— ~,

±T:

KJ’; T tNote that we

L~o(0)~J;T)I = ~IXi-(+T) Xi-(—T)I. —

are not neglecting the isoscalar contribution, but only making a choice of sign here.

(4.21)

50

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

Finally we may write the neutrino scattering cross section for the transition J; TMT —* J’; TMT using eq. (4.3b): LWLLj=1) a’

/ =

—

Al

~2

S~K2(~’ \MN/ (0)

2

1~’T 1 /3~°~’(X(+T)X(T))+ (2J+l)(2T+l) VT(T+1)

(4.22)

Here ~ = ±I is the relative sign between the isoscalar and isovector matrix elements which we cannot determine using only unpolarized processes. We may however resort to single-particle estimates (see section 4.3) or more sophisticated nuclear models (see section 5) to determine relative signs. Since this is only a very weak requirement of the nuclear models employed, the neutrino scattering cross sections predicted in the present manner are virtually model-independent. Again we shall delay discussions of specific nuclear transitions until section 4.4.

4.2. Relationships among low-energy elastic or isoelastic weak and electromagnetic processes

We now turn briefly to a discussion of elastic neutral current neutrino scattering from nuclei. While such a process is extremely difficult to measure in the laboratory (with neutrinos in both initial and final states, and only the recoiling nucleus as a signal that the event occurred), it is of relevance in astrophysics [4.3]. For nuclear matter at extremely high densities, such as in gravitational collapse leading to supernovae, the elastic scattering of neutrinos may be significant in providing a means of

releasing their energy and resisting the collapse. Consequently in this section we include a treatment of low-energy elastic neutrino scattering along the lines of the previous section where inelastic processes were discussed. At the same time we shall derive some general relationships among a

variety of weak and electromagnetic processes at low energies, including elastic neutrino and elastic electron scattering and processes such as /3-decay and charge-changing neutrino reactions between

isospin analog states in neighboring nuclei (isoelastic processes [4.4]). The general expression for neutral current neutrino scattering is given in eq. (3.26). Using the results of section 3.5 we obtain the long wavelength limit of the elastic cross section:

d (J;TMT—*J;TMT) =

I~.,(—k.

2G2K2.~~

2~~ i {cos~

+

~

(1+sin2~)

(—CT

0

~

MT)’~

~

T

2

o;~(0)~J; T)1

TL~(o)~J;T)~}~

(4.23)

where we assume that the ground state s has quantum numbers J; TMT. Note that no multipole L

0

occurs. In fact using parity, time-reversal invariance, isospin invariance and the hermiticity of the operators we may prove some general results [4.4]:only the even-i Coulomb multipoles M, and odd-i multipoles T~, L~and T~’have non-zero matrix elements for isoelastic processes. Evaluating the 3-j symbols and integrating over all solid angles we obtain

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

51

LWL

a’

~

=

(J; TMT, elastic) (M)(2f+ 1)(2T+ l){31/3v(J; T~ICIo;o(0)~J; T)

sVK

~VT(T+

T)I+ ~/3~(J;Tj~L~o(0)~J; T)

1)/3~~(J T~I~Io.i(0)~J;

T~iL~i(o)~J; T)I},

(4.24)

where of course the isovector terms contribute only if T ~ ~and the axial-vector terms contribute only if J ~ ~. Thus in general we require four multipole matrix elements to predict the neutrino scattering cross sections. The Coulomb matrix elements are easily evaluated. Using the results of section 3 we can easily verify that in the LWL V4irA~oo;oo(0)=

~Jc

(4.25a)

\~Moo;io(0)= i’3, where

(4.25b)

~~Q’ is the total nucleon number operator with eigenvalue A and

7’~is the total isospin operator

(third component) with eigenvalue MT. Thus the Coulomb matrix elements are given by \/~~(J; T ~A~Io;o(0) ~J; T) = ~A[J][T],

(4.26a)

V~(J;T~ICfo;1(0)!~J; T)= VT(T+ l)[J][T],

(4.26b)

and, for a nucleus whose ground state has J = 0, we have completely defined the cross section in eq. (4.24). When J ~ we require in addition the axial-vector matrix elements. We may obtain information on these by considering the magnetic dipole moments of the isospin analog ground states, ~ (MT), where M~= J; TMT I i’?~o(q)+ f?~o(q)~J, M~= J; TMT)}

—i\/~urn

~ (MT)

=

\/~i~T[J

(4.27a)

~J~j{~

T~—iI~(0)~J; T)+VT~+ 1)(J; T~—iI7~(0)~J; r>}.

(4.27b) By considering members of the isomultiplet with MT = (J; Tll—iI~(0)~J; T)= ~

±T we

obtain

T~•l~(T)_I2(_T)), (J; T~—iI~(0)llJ; T)

~

(4.28a)

J J~,

T~.

~/ (4.28b)

52

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

We may then employ eq. (4.15) to write (J; T

i11~o(0) J; T)

{V2F~(0)}(J T~—iI~(0)~!J; T)— -1_-[T][J]\/J(J+ ,a V96ir

=

1)

—~

1

V2F~°’0 =

1 [T][J].~~JL~~— (/.L(T) + ~t(—T) J).

)} v~-

~

(4.29)

—

If T = 0, the elastic neutrino scattering cross section for an arbitrary value of J is defined at this point; if not, we are left with only the isovector L~matrix element in addition to evaluate. Note that elastic electron scattering in the LWL does not yield any new information, since from eqs. (3.18) we have

1

F~(q)LWL=

(J; T~ICIo. 0(0)~J; T)+

(2J+l)(2T+l)

.

MT

\/T(T+l)

(J; T~S~t0.1(0)~J; T) 2 (4.30a)

11

k—

(~

2

(4.30b)

2/4ir,

A + MT) = Z using eqs. (4.26), where Z is the proton number, and =

LWL

F2i

/

\

—

— (2J +

using eqs. (4.27).

=

\2

_______________ gq~

l)(2T + 1)

(1124ir){(J +

~,MN)

If.

T:: ~maAi ::

T

i~o0~nj. ~. j::

(.1, T~I~(0)~J; T)I 2~(MT)2, l)/J}(q/MN)

J~

~,

(4.31a) (4.31b)

For the case J ~ T ~ we turn to the analog (isoelastic) /3-decay transitions, J; TMT —~J: TMT, where the states differ only in their third components of isospin (MT = MT ±1 for f3~-decay). Following the same procedure as in the last section, employing eq. (3.20), we obtain ~,

=

X

I

LWL

(J; TMT

—*

J; TMT,

4 G2~(W~)5(T~ MT)(T

±MT + l)SMT,MT±I

T(T+ l)(2T+ l)(2J+ l)I~T~L~i(0)~J; T)12},

where we have used eq. (4.26b) to evaluate the isovector M

(4.32)

0 (Fermi) matrix element. Eq. (4.32) may then be inverted to yield an expression for (f; T L~t(0)~J; T)I in terms of the /3-decay rate. We lack only the relative sign between (J; T~L~o(0)~jJ; T) and (.1; T~iL~i(0)~1J; T) to completely specify the elastic neutrino scattering cross section (eq. (4.24)). Again we may resort to single-particle estimates (see the next section) or more sophisticated many-body treatements of the nuclear states involved to estimate this sign; indeed this is only a rather weak requirement of the nuclear dynamics and our predictions at this point are largely model-independent. Finally we note that the isoelastic charge-changing neutrino reaction cross sections (eq. (3.24)) are

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

53

given in the LWL by LWL

a’

(J; TMT

,,51~ — —

—~

J; TMT’, isoelastic)

2(T+MT)(T±MT+ l)SMT,,MT±1 fvw\ S~/3~ MN) (2.1 + l)(2T + l)T(T + 1)

! j. ~

(

,

T::A~f 0 ::j. r 2 ):: ) O;I(

+I(f; T~L~i(0)!~J; T)12}

,

(4.33)

and is completely defined using the above expressions for the isovector matrix elements. As usual, this is only relevant when T ~ ~ and has no axial-vector term if J = 0. To illustrate these ideas we consider the case where J = 0, which applies to the stable even-even nuclear ground states. The neutral current elastic neutrino scattering cross section (eq. (4.24)) is then

given by LWL

2

(J = 0: TMT, elastic) =

2

s~(~—) {~A$V~ + M~,/3~}2

~—

s~ (~—)2{~(z~+ N)/3 ~P+ ~(Z

= ~—

s~ (~_)2{Z(aV~+ a

—

(4.34b)

N)/3 (1)}2

—

2aem)

+ N(a ~

(4.34a)

—

a (I))}2

(4.34c)

where we have used the facts that A = Z + N, MT = (Z N)/2, where Z is the proton number, N is the neutron number, and /3S~= a~v~ — ar,,,, = 0,1 (see table 2.1). In table 4.1 we list the values of K, a ~ + a ¶2 2aemand a a ¶2 forthe various gauge theory models discussed in section2. We see that the elastic neutrino scattering cross sections (for J = 0 nuclei) show very marked differences for the gauge theory models considered as far as their N and Z dependences are concerned. The b-quark model, for example, does not depend on N, but depends strongly on Z. In table 4.2 we present elastic neutrino scattering cross sections for some J = 0 nuclei in various gauge theory models. Examples of odd nuclei can be found in ref. [4.4]. —

—

—

Table 4.1 Coupling constants for elastic neutrino scattering from J = 0 nuclei at low energies* Model

a~+aij1—2a,mt

abo)~am

W-S-GIM b-quark q-quark Vector

I —4 sin2Ow —0.2 3—4sin2Ow” 1.8 2—4sin29w~ 0.8 2—4sin28w~ 0.8

—1 0 1 —2

HKM

1—4sin2Ow~’—0.2

—1

SU(2)L X SIJ(2)~ X U(1) SU(3)xU(1)

1—4 sin 9RL —0.2 1—3sin2OLw’= 0.4

—

~Seetable 2.1.

K

=

I for all models except SU(3)xU(1) where

K0.7.

t We use sin28~ SIfl2ORL 0.3; =

—1

sin 8LW

0.2.

54

T. W Donnelly and RD. Peccei, Neutral current effects in nuclei Table 4.2. Elastic neutrino scattering cross sections for J

=

0 nuclei

2MeV2) 5,-2~i-LWI.~f =

Model*

4He

W-S-GIM, HKM and SU(2)L x SU(2)R x U(I) b-quark q-quark

0. elastic) (cm 2C

2.41 x 5.42x 5.42 x 2.41 x 0.30 X

Vector

SU(3) x U(I)

iO~ I0~’ 10 ~ l0~ I0~’

‘6Fe

208Pb

2.17 x I0-~~ 5.18 x 1042 4.88x ~ 9.16x 1Q42 4.88 x to-~’ 10.8 x 1O~ 2.17 x io—~~ 6.43 x 1042 0.25 x t0~~ 0.79 x 10.42

8.48 x 9.12x 15.4 x 14.5 x 1.78 x

tO 4! I0_41 lO_41

1O~’

*See table 2.1.

4.3. Single-particle matrix elements in the long wavelength limit

In practice the general relationships derived in the preceeding two sections may be inadequate for a given nuclear system due to the unavailability of precise experimental data and it may be necessary to employ specific nuclear models for the states involved to obtain predictions for the neutral current neutrino scattering cross sections. Alternatively we may have at our disposal calculations of the nuclear dynamics (see the discussions in section 5) which we may use to predict neutrino scattering cross sections at low energies. In either case we shall be interested in using the decomposition of many-body matrix elements into linear combinations of single-particle matrix elements given in eq. (3.35). The nuclear many-body problem is then focussed on determining the one-body density matrix elements i,t’~i(a’a). Consequently in this section we give expressions for the appropriate singleparticle matrix elements in the long wavelength limit. As we have seen in this chapter the only relevant operators in the LWL are M 0, L~,and T’~’~ (which are related, eq. (4.1)) and traK The first of these has matrix elements which are always known in the LWL, as discussed in section 4.2, and so need not be considered further in this section. Single-particle matrix elements in the LWL of the remaining operators may be written: (a’ —iL~pj(0) a) = (a’ _iT~~s3,.(0)a)/\/2 = [~T]F~?(0)(a’II~I~(0)IIa) (4.35a)

(a’ —itr~(0) a) =

~

—

2(a’II~i(0)JJa)),

(4.35b)

where we have employed eqs. (3.32), (3.33) and (4.1). Thus we require the matrix elements (reduced in angular momentum) of the two basic nuclear operators I’ and ~ evaluated in the long wavelength limit. Using the general expressions in appendix B we may write: (n’l’j’II~I’i(0)IInlj)=

..._i_.._ (_)1+i_1/2[j~][jI{J ~

—(n’l’j’II~i(0)IInhj)= ~

(—)‘~si’,

1

j’ = j, j ±I,

(_)1±iI2\/l(l+ l)/6[l][J’][J]{31 d~’,

(4.36a)

j’=j, j±1.

(4.36b) ~

(4.37a)

(4.37b)

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

55

By explicit evaluation of the 6— j symbols [4.5]we find: s~’= ([j]/\/3[l])(S,.3[2j — l]/\/2+ Sjj±I[j’]), d’,”

=

(\/l(l + 1)/\.”3[l])(543V21 — j + ~[2j— I]

(4.38a) (4.38b)

— 5,,~~~(_)1+1_t/2)•

Values for these quantities are shown in figs. 4.1 and 4.2 for 0 ~ I ~4. These results may then be employed in computing cross sections in the LWL for all the processes discussed in this chapter. As one example, we write the inelastic neutral current neutrino scattering cross section in the LWL (eq. (4.3b)) in terms of single-particle matrix elements using eq. (3.35): for the transition s: J; TMT s’:.1’; T’MT we have

—~

LWLLJ=1)

oj,

2

=

SVK2(~)

(2.1+ 1)(2T+ 1)

VT(T+ MT

I~

{sT.TPa’axa’~—iL~oo~a

l)+ST.T±1( —

2—(2MT)2l~’2 ) (T_T.s-1)1/2J(T’+ 12(T’+ T+ 1)(2T’+ 1)1 T+ l)

x /3(1)III(s;s)(ala)(a~ —iL~(O) a)}~ =

(v_~) (2.1 + l)(2T +

~

~

V3IST’T

\ X

T+ l)2—(2MT)2~’2 12(T’+ T+ l)(2T’+ 1)

(T_T+1)1/2f(T+

~s/T(T+ 1)

/3 ~F~](0)iIi~

nlj)

~

MT +

(4.39a)

+ST’T.-I()

~(nlj’,nIl)]

I,

—

where we have used the results of the present section to express the cross section directly in terms of the quantities sf’. As noted previously only values T’ = T ~ are allowed for the isovector 5TT term and we have only j’ = j, j ±1. As a specific result let us assume that we have purely single-particle states:

s: a nia’mjdmt,Ic)

s: a ntjmjdm,IC),

with j’ =

j, j ± 1

and rn

1 = m1 = ±~ where Ic) is a 0~0closed-shell core. Then, directly evaluating the one-body density matrix elements using eq. (3.36), we have ~~,~r)(nljl, nlj) = 1 for 9’ = 0 or 1.

Substitution into eq. (4.39b) 2S yields

a’I~

—

LWL(J=1)

2

1) (1IM(0) ~

=

+ 2m

2

(4.40a)

1/3~F~(0)) =

8ir(2j+ 1) (v—:)2( “)2(F~”(O))2(pt0~ + 2m:/3~)2

(4.40b)

under the assumption F~°~(0) = F~(0)which we employ in the present work. Noting from fig. 4.1 the fact that ~ 1, we have LWLfl=1)

a’jpp;

2

=3.35x l0_38cm2X

—

2

2,

21lC~1 (PM) ~~+2m1/3~)

which, for {(v w)/MN}2 —

(4.41)

l0~(typical for a stopped iT-j.~-eneutrino facility), yields cross sections on

56

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei 6

2

5

d~’

Fig. 4.1. The quantity sf~(defined in the text, eq. (4.38a)) given as a function oft for allowed values of land i’.

Fig. 4.2. The quantity d(’ (defined in the text, eq. (4.38b)) given as a 2”’~’2~= function of I for allowed values of land j’. Note that d7 —

the order of l0~’cm2, given that the last factor in eq. (4.41) is of order unity. Indeed, this last proviso may or may not be true, as is seen by examining table 2.1. 4.4.

Selected examples

In this subsection we illustrate the ideas presented in the rest of this section by applying them to a selected set of nuclear transitions. In these two subsections we discuss only inelastic neutral current neutrino scattering. We begin by considering the scattering of anti-neutrinos produced in fission reactors [4.1, 4.6]. An excellent review of practical target nuclei exists [4.7] and so we have selected only a few examples in the present work to bring out several features which go beyond that discussion. Next we consider neutrino scattering for neutrinos and anti-neutrinos from a stopped ir-~-efacility, such as exists at LAMPF [4.1,4.2]. Again, our aim here is to illustrate the basic ideas, not to provide an exhaustive list of nuclear candidates. 4.4.1. Inelastic scattering of reactor anti-neutrinos Fission reactors produce copious quantities of low-energy electron anti-neutrinos having an energy spectrum as shown in fig. 4.3 (ref. [4.8]). The rapid decrease with energy of this spectrum makes it important to find nuclear transitions with small excitation energies to make the event rate as large as

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

57

possible and so we shall restrict our attention at once to such cases. On the other hand, the signal for the neutral current excitation would most likely be the 7-decay of the nuclear state excited and having too low an energy for this 7-ray makes its detection difficult. Consequently, there is also a practical lower bound to the nuclear excitation energy (in ref. [4.7]w 600 keV was chosen for this bound). Of —

course, as is clear from the discussion in this chapter, we are interested in nuclear transitions which have large Ml matrix elements. Finally, the low nuclear abundance or lack of ease of producing selected isotopes may rule them out as potential candidates. We consider two examples in detail, 6Li and 7Li; the analysis for other cases proceeds in a similar manner.

6Li(1, iY)6Li*(1+0_+0+l, 3.562 MeV) This is a pure isovector~= 1, ~ir = no transition of the type discussed in the first part of section 4.1. We may employ the analog /3-decay of the ground state of 6He, 6He(0~1) 6Li( l~0)+ e + i~ (see the insert to fig. 4.3), in eq. (4.13) to obtain the neutrino scattering cross section. Note that eq. (4.13) was derived for a /3-decay transition J’ = 1 —~J= 0, whereas here we have the reverse and hence must have an additional factor of ~ on the right-hand side of the equation to allow for the different 2.1+ 1 factors (see eqs. (4.3b) and (4.6)). With Wp- = 0.858±0.002sec”’ [4.9] we obtain —~

oj 101

LWLLfl=~’=I)

—

2

1.833 x l038 ~

=

101

-

MT~

-

-

/7

‘~0

/

/

/

~

100

~‘

// //

WEIGHTED CROSS SECTI0N-~, (xIO46cm2MeV~/N/Ve)

—

SPECTRUM

/

1044~m21~

~‘

/ / /

: :--

......-.

4 111Mev)

i\

/

/~~vv

~..

I 2

~/

.

~/I.... ....

/

I 0

100

/

_/_~.

--

/

/

......-

.-.~‘

-‘.‘

I I

~~cT1~,11i(xlO44cm2/N)

......,....

—

/

“.,

/~xI0_46cm2Mev1/N/V~,_7”\\ CROSS SECTION

—I

o2

— — —

1(WEIGHTED

-

\~__Sfv~ 0 I I

lol—

(4.42)

+

6’

-

3.562 MeV).

/‘

3.562

6He

-

=

-

MT~O

-

-

(Li, cv

(/3~)2K2,

I 6

Fig. 4.3. Low.energy anti-neutrino scattering from ‘Li. The dotted curve is the i~,spectrum from a fission reactor [4.8].The solid curve shows the calculated vi” cross section (per nucleon) as a function of neutrino energy v and the dashed curve is this cross section when folded with the i~,spectrum.

//I/

0

7~e

fv

MT~-~ 0.478 -~ ,--‘ VL’IY./

/

‘I

1ô2 8

MT4 0.429 11 22

~Li

I 2

I 4 is 1Mev)

I

I 6

8

Fig. 4.4. Low-energy anti-neutrino scattering from 7Li. The solid curve shows the calculated vi” cross section (per nucleon) as a function of neutrino energy v and the dashed curve is this cross section when folded with the i~,spectrum (see fig. 4.3).

58

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

In fig. 4.3 we show this cross section as a function of anti-neutrino energy as well as the total effective cross section obtained by folding this result with the ie spectrum (also shown). Here we have employed Kfl~= 1, as is appropriate for three of the gauge theory models (including the W-S-GIM model)given in table 2.1 For the other models discussed in section 2, having /3 = ~or 0, the cross sections shown in fig. 4.3 must be multiplied by 0.25, 2.25 and 0 respectively. Note that there is no dependence on aem for such transitions, and that for the SU(3) x U(1) model K 0.7. Thus an immediate conclusion is obtained: measurement of the (i, iY) cross section for a transition such as this one in 6Li directly constitutes a determination of the isovector axial-vector coupling (modulo I/3~I.Even a rather crude measurement may be sufficient to eliminate one or more of the is

~

~,

K)

gauge theory models from contention. To set the scale for such reactor experiments we take the results given in fig. 4.3 with a flux of 2 x 1013i~cm2 sec’ [4.8]and obtain an integrated event rate of 0.6 y’s day~kg’ 6Li (consistent with the figure 300 y’s day’ m3 6Li obtained in ref. [4.7]).Thus, although certainly not large, the event rate is not impractical with high-flux reactors and large-scale targets. Isoscalar Ml transitions Next in order in our discussion in section 4.1 were isoscalar, 5 = 1, = no transitions. Such examples would be of very great interest in providing sensitive tests of the underlying gauge theory models. Unfortunately, the most likely candidates of this type have rather high excitation energies and so are unfavorable for reactor experiments. They are, however, very relevant for the stopped ir-p.-e facility experiments discussed below and at that point we shall introduce several potential candidates. ~

7Li(i~,i’)7Li*(~1~~~ ~1,0.478 MeV) Finally we have the general situation where both isoscalar and isovector neutral currents can enter as discussed at the end of section 4.1. Odd nuclei provide examples of such transitions and are potentially excellent candidates for reactor experiments, since the nuclear excitation energies involved may be quite small. Here we consider the case of 7Li which is one of the most favorable. The pertinent part of the A = 7 level scheme is shown as an insert in fig. 4.4. The ground state of 7Be may undergo electron capture to the 0.478 MeV level of 7Li(’r, 12 = (53.44 ±0.09) d with a branching ratio of 10.4%, see ref. [4.9]) and so yield the quantity X~defined in eq. (4.18). Actually the derivation in section 4.1 assumed that the charge-changing weak process involved was f3~-decay,however the reduction of the general electron capture rate (eq. (3.22)) to its long wavelength form proceeds in a totally analogous fashion and is not included here. We obtain a value 7Be(EC)7Li*(0.478 Mev)). (4.43) X~ = (~ ~ L~,(0) ~ ~ = 0.567, ( Next from the ~ 7-decay rates in 7Li(MT = —~)and 7Be(MT = +~)with corresponding lifetimes Tm = 105 ± 5fs and 192 ±2Ofs respectively [4.9],we obtain the quantities XY(MT) defined in eq. (4.20): ~

X~(—~)= =

—~

4.82±0.11

(7Li),

(4.44a)

4.19±0.22

(7Be).

(4.44b)

With the choice of sign discussed in section 4.1 we obtain from eq. (4.21) L~. 0(0) ~ ~

=

0.32

(4.45)

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

59

and hence from eq. (4.22) —

LWLLJ=1)

oj

=

2

6.954 x l0’~cm2(vM(~)K2

X

I0.32/3~ 0.3274)/3~I2, —

(7Li, cv

=

0.478 MeV),

(4.46)

“V

where 4) = ± 1 is a remaining relative sign between isoscalar and isovector pieces which cannot be determined with the present experimental input to the analysis. A pure single-particle prediction using the results of section 4.3 yields (lpl/

2~

iL~9r(0) 1p312 ~)= [ff’]F~’~(0)(lpi,2II~~’i(0)II1p3i2),

(4.47)

with (4.48a)

(1p1/2II~I(0)IIlp3,2)= (l/\/4ir)s 1/2 3/2 =

~\/2,

(4.48b)

where we have used eqs. (4.35a), (4.36b) and (4.38b). Thus we find (lpl,2 ~ Li;1(0) 1~3/~ ~ = 0.567,

(4.49a)

I(lpl/2; ~nLi;o(0)~ iP3/2 ~)I= 0.327,

(4.49b)

in excellent agreement with the values obtained above, indicating that this is a rather pure singleparticle transition. The relative phase in this case is

4)

=

+

1 and so we adopt this value in predicting

neutrino scattering cross sections. In fig. 4.4 we show the resulting anti-neutrino scattering cross section and the cross section folded with the reactor anti-neutrino spectrum. Here we have used the W-S-GIM values of the couplings, /3 = 0, /3 = I and K = 1. For the other gauge theory 2f~ ~models /3 discussed in section 2 (see table 2.1) we must multiply the results in fig. 4.4 by a factor K having the values listed in table 4.3. Again it is clear that even a rather crude measurement of the (13, 13’) cross section in 7Li would be sufficient to eliminate one or more of the gauge theory models from contention. With the W-S-GIM model we obtain an event rate of 3.8 y’s day’ kg’ 7Li (consistent with the figure 2000 ‘y’s day”’ m3 7Li obtained in ref. [4.7]).In fact of all the candidates for nuclear targets Table 4.4 Event rates for nuclear excitation via the scattering of reactor neutrinos followed by y-decay as calculated by Lee [4.7] Target Table 4.3 7Li(P, P’)SLi*(0.478 MeV) cross sections relative to the W.S-GIM prediction

p~

K2f(,p~)

-l

I ~

0

0

I 0.97 3.9 0

0

1

1

Model’ W-S-GIM b-quark q-quark vector

HKM

SU(2)LxSU(2)RxU(l) SU(3))( U(1) *

See Table 2.1.

0

0

1

1

i0~

Excitation energy Event rate* (MeV) (7’s day~m’’ target) ____________________________________________________________ 7Lit 0.478 2000 “Cu “Co “Cu ‘Lit “Mn 23Na

0.770 1.189 0.669 3.562 1.507 1.885 0.440

660 570 460 300 260 250 160

‘Assuming a flux of 2x 10” ~cm2sec’’ and employing the W-S-GIM model. t See also the analysis in the present work. _____________________________________________________

60

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

considered by Lee [4.71the rate in 7Li is the largest and so may be the best choice for neutral current neutrino excitation using reactor anti-neutrinos. In table 4.4 we list some of the targets considered by Lee with his calculated event rates.

In concluding this subsection several additional points should be mentioned. One is that nuclear systems where the (13, 13’) excitation leads to a level which subsequently decays via a cascade of two or more ‘V-rays is of particular interest [4.1]. By detecting the 7-rays in coincidence one may improve the signal-to-noise ratio. Lee considers a variety of nuclei with cascades and concludes that 55Mn and ‘9F provide two of the best candidates. Indeed other coincidences may be of interest such as n, a with

9Be as a target [4.7]. Decay modes other than 7-decay are also of interest in providing a signal for the neutral current excitation. One favored case is the disintegration of the deuteron via neutrino scattering with the unbound proton or neutron as the detected particle (see refs. [4.10, 4.11] and references contained therein). Lee [4.7] has repeated the calculation and obtains a rate of 820 events day’ m3 D 20, where the W-S-GIM model has been employed. The process involves a transition from the (predominantly) 15 = 0) deuteron to the 0(T’ = 1) neutron-proton continuum. Consequently the cross section is proportional to K 2(/3 ~))2 and so the above event rate must be multiplied by 1 for the W-S-GIM, HKM and SU(2)L x SU(2)R x U(1) models, by 0.25 for the b-quark model, by 0.12 for the SU(3) x U(1) model, by 2.25 for the q-quark model and by 0 for the vector model (see table 2.1). Again there is a high degree of sensitivity to the choice of underlying gauge model. Finally we direct the reader to other references on the subject of nuclear excitation via reactor anti-neutrinos [4.12,4.13]. 4.4.2. Inelastic scattering of neutrinos from stopped ir-~-eneutrino facilities Next we turn to a discussion of inelastic neutral current neutrino excitation with neutrinos and

anti-neutrinos from a stopped ir-~-eneutrino facility such as the one at LAMPF. Here the source is the beam stop for the 800 MeV proton linear accelerator. Copious quantities ofpions are produced, come to rest and decay via [4.14] 1T~*,U~t1,~

(Tm =

2.603 x 10’ sec,

is

=

29.79 MeV),

(4.50a)

following which the muons come to rest and decay via 4’—~e~v~13~.(‘Tm= 2.197x lO6sec, Vmax 52.83MeV). (4.50b) p Thus three species of neutrinos are present in the neutrino spectrum (the spectrum is shown in fig. 4.5, see ref. [4.15]). As the muon neutrino and anti-neutrinos are too low in energy to initiate the charge-changing reactions /Ll, (13k,, ~) only the electron neutrinos are active insofar as the charge-changing weak interaction is concerned through the reaction (ise, e). However all three species can participate in neutral current neutrino scattering. An intriguing possibility exists here: if an (EM,

experiment could be devised using a pulsed beam and one selected only events initiated by the prompt muon neutrinos (vp), that is for times less than 1 sec after the pulse, only neutral current neutrino scattering would be allowed with no (va, el signal. .t

We proceed to a discussion of several potential candidates for targets in such stopped ir-~-e

experiments. As in the preceeding subsection, the list is meant to be illustrative only and not exhaustive, although several of the examples presented appear to be among the most favored. Note that the restriction to low excitation energies imposed in the reactor neutrino case is not as important for the experiments discussed here, since the neutrinos are of higher energies. The signal in all but one

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

61

4 1)

Io_2 MeV

-

COla

p1Mev)

Fig. 4.5. The neutrino spectrum from a ~ neutrino facility. The spike represents neutrinos from the decay ~ the subsequent three-body decay ~ -~e~~ gives rise to the two curves, with the v~spectrum peaked at lower energies than the P,. spectrum.

case (the treatment of the 12.71 MeV level in 12C) for the (v, v’) excitation is the subsequent ‘V-decay

of the excited nucleus. 6Li(v, v1)6Li*, 6Li(13, 13’)6Li*(1+0_+0~1,3.562 MeV) This case was considered in the previous subsection (see eq. (4.42)) and also in refs. [4.1]and [4.13]. When the cross sections are folded with the neutrino spectrum (fig. 4.3) we obtain an event rate of 10 ‘V’s dayt Mg’ 6Li assuming a flux of 2 x i07 neutrinos of each species per cm2 per sec as is reasonable for the LAMPF neutrino facility. This is for gauge theory models with /3 = 1, sc = 1 such as the W-S-GIM model. As before, for the other models listed in table 2.1, this rate must be multiplied

by 0.25, 2.25 and 0 when /3 = ~, ~and 0 respectively. Such experiments appear to be feasible and are of considerable interest, since they constitute a direct measurement of ~/3 ¶~.(For the SU(3) x U(1) model there is an extra reduction of K2 1/2.) 7Li(v, v’)7Li*, 7Li(13, 13l)7Li*(~1—~~1,0.478 MeV) This case was also analyzed above and discussed in refs. [4.1]and [4.13].The neutral current cross section was given in eq. (4.46), where we argued that the choice of sign should be 4) = + 1. Folding these cross sections with the neutrino spectrum (fig. 4.5) yields an event rate of 47’S day’ Mg’ 7Li again assuming a flux of 2 X i07 v’s cm2 sec’ of each species and employing the W-S-GIM model. For the gauge theory models discussed in sections 1 and 2 this rate must be multiplied by the factor K2f($~,/3~)given in table 4.3. Again the sensitivity to the underlying gauge theory models is apparent. ‘4N(v, v’)’4N*, ‘4N(13, 13’)14N(1~0—’2~0, 7.028 MeV)

This case provides one of the best examples of an isoscalar (~ = 0) transition. It was not discussed in the previous subsection because of the rather high excitation energy and the resultant rather low rate with the rapidly falling reactor neutrino spectrum. However, for stopped ir-~&-e facility neutrinos

62

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

with maximum energy 53 MeV the rate is not drastically hindered. This transition has been considered previously in ref. [4.13] and more recently reanalyzed in ref. [4.16]. Here we present only the most

important details as the analysis is very straightforward using the formalism given above in section 4.1. The Ml ‘V-decay rate is measured to be (9.1 ±1.3)x 102eV with a branching ratio of (98.6±0.3)% to the ground state [4.17]. Using eq. (4.17) we obtain a neutrino scattering cross section of LWL(.,5~=I.3=O)

/

=

2.42 X

~2

(f3~)2K2, (‘4N. cv = 7.028 MeV), (4.51) MN and find an event rate after folding this with the neutrino spectrum in fig. 4.5 of 1.1 ‘y’s day’ Mg’ ‘4N. Again we have taken the neutrino flux to be 2 x l0~is’s cm2 sec’ (of each species). We have set 1/3 ¶~I= as is appropriate for the b-quark, q-quark and SU(3) x U(l) models (see table 2.1). For the SU(3) x U(l) model there is an extra reduction of K2 =~, while for the other models discussed in sections 1 and 2, the W-S-GIM, vector, HKM and SU(2)L X SU(2)R x U(1) models, the cross section from eq. (4.51) is zero. Of course there is a small contribution from the vector current which we are neglecting in the long wavelength limit and so the cross section is very small, but not zero. We will return to this point in the next section. An experiment performed with this target would be of great interest as it constitutes a direct measurement of the very interesting gauge coupling constant l/3~°i. A non-zero result signals the existence of new quark degrees of freedom above charm. 0~

10_38

cm2(P — W)

~,

‘2C(v, v’)’2C*, ‘2C(13, 13’)’2C*(0±0~~~ l~l,15.110MeV and 0~0—*l’O, 12.710MeV) As a final example in this subsection we consider the 1~1(15.110MeV) and l~0(12.710 MeV) levels in ‘2C. These have been discussed previously in refs. [4.2] and [4.13]. The 0~0—*l~1transition is another example of a purely isovector process and may be treated using the /3-decays of the ground states of 12B and ‘2N (analogs of the 15.110 MeV level in ‘2C). The lifetimes [4.18] are (20.41 ±0.06) msec with a branching ratio of 97. 13% for ‘2B /3-decay and (10.97 ±0.04) msec with a branching ratio of 94.25% for ‘2N ptdecay to the ground state of 12C. From eq. (4.10) we deduce values for the matrix element of L5:

(452 )

~ l~L5 :: i;i( 0 ):: ::0.O~_J’0.247/3~decay 10.235f3~-decay

.

~‘

and we may employ the average value 0.24 to predict the neutrino scattering cross section. From eq. (4.13) we obtain

cr1

LWLtJ=~=I) =

—

l.08x

2

l0_38cm2(11M~)) (/3~))2K2

(‘2C,co

=

15.110MeV)

(4.53)

and find for the event rate when this is folded with the neutrino spectrum as above, 1.2 ‘V’s day’ Mg’ ‘2C. Here we have set /3~ = K = 1 as is appropriate for the W-S-GIM model. For the other models given in table 2.1 having /3 = ~and 0 we must multiply the result here by 0.25, 2.25 and 0 respectively, and if K 1, by K2. We shall return to a much more detailed discussion of this transition in section 5. We conclude this subsection with an analysis of the 12.7 10 MeV level in ‘2C. A previous discussion by the authors [4.2]treated this state as an eigenstate of isospin with T’ = 0. However it is known that there is some isospin mixing present and so it is worthwhile to reanalyze this case. We write the state ~

~,

T. W. Donnelly and R.D. Peccei~Neutral current effects in nuclei

63

vector as

IJ’=

l,MJ;MT.=0)=\/l—l52~J’=l,MJ.;T’=0,MT=0)+ôjJ’

1,M~.;T’=l,MT’=0), (4.54)

where 8 measures the amount of the isospin mixing. A generalization of eq. (4.3) leads to the neutrino scattering cross section LWL(5=1)

2 =

S~(h1_w)

2

K21\/i

8/3~(l;0~f,~o(0)~0; 0)+~=8/3~](1;I

L

0)1

1(0)~0;

(4.55)

whereas generalizations of eqs. (4.8) and (4.9) lead to a ‘V-decay rate M1~(l~ MT. = 0—~’0~0; MT = 0) = w(~_)2I\/l 82(1; 0~I?~(O)~O;O) cv~,, +ç~8(l;1 i~~(0)~0; 0)12. —

(4.56)

We may relate the isoscalar matrix elements using eq. (4.15): (l;0!L~o(0)~0;0)= 2F~°~’O’ ‘ (1;0~I~~(0)~0;0),

(4.57)

/L

however we are then left with too many unknowns and no further information directly relating to this transition. Thus we must resort to some approximate treatment of the nuclear dynamics. First let us write an expression analogous to eq. (4.57) for the isovector matrix elements: (1; 1

V2F”~0

L~.

1(o) 0; 0)

=

{

~

)}
(4.58)

where ‘q is defined as twice the matrix element of the operator i~divided by the matrix element of ~‘, both at zero momentum transfer. The analogous quantity in the isoscalar case is 1/2 (see section 4.1). Following the approach taken in ref. [4.19]we the matrix deduced fromfor /3-the andisovector ‘V-decay 2C and use its analogs (seeelements above) as estimates involving the 1~1(15.11 Mev) level in ‘ matrix elements required in studying the 12.71 MeV level. As a result we find that eq. (4.58) can be satisfied quite well with = 0; that is, that the magnetization current dominates over the convection current in the ground state ‘V-decay of the 15.11 MeV level. Using this estimate for (1; 1 t?~(0)~]0; 0) in eq. (4.56) we are left with two unknowns, the isoscalar matrix element (1; 0 t~(0) 0; 0) and the isospin mixing parameter 6, and one datum, the 12.71 MeV 7-decay rate [4.14].if we resort to a shell model calculation [4.20]of the ‘V-decay rate assuming that the state is completely T’ = 0, yielding = 0.113 eV, we can deduce values for 8 and (1; 0~ t?~(0)~0; 0) and, through eq. (4.57), for (1; 0 L~o(0) 0; 0). Two solutions are found, having 8 = 0.054 and 0.194, however the former appears to be favored [4.19]and is chosen here (see also ref. [4.13]).The neutrino scattering cross section can now be written ,~“

—

LWLtJ=1)

u

=

2

0.92 x l0_38 cm2(1~M~)) (/3 ~ + 0.06/3 ~2~c2

(‘2C, cv

=

12.710 MeV).

(4.59)

if we set (8 ~ +0.06/3 ~2K2 = 1 we obtain an event rate upon folding with the neutrino spectrum of 1.3 events day’ Mgt ‘2C. The events in this case are predominantly a-decay to SBe* (2.90 MeV) followed by 8Be* -. 2a with a 98% branching ratio or, more unfrequently, ‘y-deexcitation of the 12.71 MeV level to the ground state or to the 2~0(4.439MeV) state of ‘2C. In table 4.5 we list values for the quantity

64

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei Table 4.5 2C(v, p’)~C*(12.710MeV) and Coupling constants for the reactions ‘ MeV) ‘2C(i, P’)’2C’(l2.710 Model’ w.S.GIM

b-quark q-quark Vector

HKM

SU(2)LXSU(2),~xU(l) SU(3)xU(1)

$~‘

$~ ($~+0.06~T)2pc2

0

1

0.0036

—l —1

1 1

0 0

0 1 1

0.22 0.17 0 0.0036

1

1

0

0.0036 0.14

*See table 2.1.

(/3 ~ + 0.06$ ~2K2

which multiplies this event rate. We note that, even with isospin mixing, the predictions of gauge theory models which have an isoscalar axial-vector current (ft~ 0) differ considerably from those which do not. 5. Intermediate-energy neutrino scattering from nuclei In this section we present a brief discussion of weak and electromagnetic processes at intermediate energies. Here the momentum transfers q may be on the order of, or greater than, Q, a typical nuclear momentum transfer (Q — k,~ 200—250 MeV). In this case the long wavelength limit (LWL) considered in the last chapter is no longer applicable and we must invoke the more general formalism presented in section 3. Of course, as the word “intermediate” indicates, we shall remain in the non-relativistic regime where q24M~,. In section 5.1 we state several general relationships which exist among the weak and electromagnetic processes at intermediate energies. In particular electron scattering and the charge-changing neutrino reactions (v,, l), (i~,l~)to isospin analog states may be used to deduce something about neutral current neutrino scattering, (v,, v) and (13,, 13,~).At the intermediate energies considered it is very important to have information for momentum transfers other than in the LWL (for example, the information provided by /3- and ‘V decay, see section 4). Electron scattering will play a major role in this regard with additional information coming from /L-capture at its low, but not too low, momentum transfer (see section 5.2). In section 5.2 we confront the nuclear many-body problem and use the one-body density matrix approach discussed in section 3. To illustrate these ideas and obtain predictions for neutrino scattering at intermediate energies, we apply the formalism to several specific transitions. We consider the A= 6 and A = 12 systems in some detail, drawing on past work for some of the analysis. Inelastic neutrino scattering cross sections for a few interesting states in these systems are obtained using the set of gauge theory models discussed in sections 1 and 2. As we shall see, these predictions indicate that rather unambiguous information on the underlying gauge theory can be obtained from measurements of neutrino scattering cross sections in nuclei. Such experiments appear to be feasible and, because of the importance of what may be learned from them, we feel should be seriously pursued. Finally we conclude this chapter with a brief summary of some of the other work done on weak and electromagnetic processes at intermediate energies.

65

T. W Donnelly and RD. Peccei, Neutral current effects in nuclei

5.1. General relationships among weak and electromagnetic processes at intermediate energies

Typical nuclear level schemes are illustrated in figs. 5.1. and 5.2, where we focus only on the isospin designations at present. The neutral current transitions are shown as double lines along with their analog charge-changing transitions (single lines with arrows). We shall discuss the various inelastic and isoelastic (i.e. elastic transitions and the analog charge-changing cross section) separately. We begin by recalling eq. (3.16) where the electron scattering cross section is written in terms of the form factor F2(q, 8), which is itself written in terms of longitudinal and transverse parts, F2L(q) and F~(q),in eqs. (3.17, 3.18). We may define analogous form factors for the charge-changing neutrino reactions (eq. (3.25)) and neutral current neutrino scattering cross sections (eq. (3.26)): ERL

(J; TMT—*J’; ~ (J; TMT

—~

J’; T’MT’)

(5.la)

8),

(5. ib)

2G2i~2 cos2 ~ H~(q,8), ~-

where the notation is defined in section 3.2. We consider only the extreme relativistic limit (ERL) for the neutrino reactions for simplicity; the generalization to the regime where the l~energy is small compared to its rest mass (involving eq. (3.24)) is straight-forward. Let us first consider inelastic transitions from a ground state having isospin T to excited states and their analogs having isospin ~4I = T + 1. These are mediated only by isovector (Y = 1) currents. By comparing the neutrino scattering form factors to the electron scattering and neutrino reaction form

:

T-iAT

T+I I + I 1+1

I

~1~

f7~0

~ M ‘~-T-I I

(T?-~)

f~r.I(T3j)

/

~

—

T2

T

T

N 1

N1 ~—T

M1I~—T+I

Fig. 5.1. Typical isospin structure for inelastic weak sod electromagnetic processes in nuclei. The neutral current neutrino scattering transitions are indicated by transitions aredouble indicated, lines. Both isoscalar (~0) and isovector (5~= I)

— I

+

I

—

2

N1 ~—T Fig. 5.2. As for fig. 5.1, except for isoelastic processes including3H—’He elastic neutrino isodoublet scattering the Mr (double ±1lines line). in the In cases figuresuch are as the reserved.

66

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

factors we obtain: H 2±~—

(1)2

V

—

(1)2~ 2 A J

(1(2! A 2

(

2

l\

~T+1

1)

1

2

~],

±f3~/3~[(2T+ l)G~+_T~

(.~=

1).

(5.2)

Thus, if we knew the electron scattering cross section and had measurements of the neutrino and anti-neutrino reaction cross sections, we could predict the neutral current neutrino scattering cross sections in a nuclear-model-independent manner for such isovector transitions. The only model dependence is on the underlying gauge theory (through the coupling constants /3 /3 ~ and K) and this is just the situation we desire. Perhaps it should be noted that many experiments may be conceived to measure the neutrino reaction and neutrino scattering cross sections simultaneously and so our relationship (5.2) could be applied immediately. Next we turn to transitions between states with the same isospin T where in general both isoscalar = 0) and isovector (~ = 1, except when T = 0) currents enter. These may be either inelastic (fig. 5.1) or isoelastic (fig. 5.2) transitions. We cannot write a general relationship such as eq. (5.2) in this case. Certain special cases are of interest however. If T = 0 we have only purely isoscalar (f 0) ~,

transitions to consider and may write the neutral current neutrino scattering form factors — A), (T 0), H2±= /3(0)2 F2 + f3~2 H2(A)±I3~13~H2(V =

as

(5.3)

where the last two terms contain all of the dependence on the axial-vector current (through matrix elements of M5, L5, TCIS and ~ see eq. (3.26)) and where the electron scattering form factor has all the purely vector current dependence. We have no model-independent way of determining H2(A) and H2(V—A), the purely axial-vector and the vector/axial-vector interference contributions respectively.

If we consider the sum of the neutrino and anti-neutrino scattering form factors, +

H2.)

= /3~2

F2+

/3(0)2

H2(A),

(T

=

0),

(5.4)

we may eliminate the V—A interference term and so deduce a special bound: ~(H2÷+H2.)~f3~052F2,

(T=0).

(5.5)

We note from table 2.1 that this is an equality for the W—S—GIM, Vector, HKM and SU(2)L X SU(2)~x U(1) gauge theory models studied there. Furthermore we may distinguish between these models with $~= 0 (but in general /3~ 0) and the other models with /3~ 0 (the b-quark, q-quark and SU(3) x U(1) models in table 2.1) by measuring the difference between the neutrino and anti-neutrino form factors: —

H2.I = I/3~/3~H2(V-A)I,(T

=

0).

(5.6)

If this difference vanishes, we may reasonably conclude that /3~ = 0 (since in general /3~ 0 and H2(V—A) 0) and that we have the first class of models, have equality in eq. (5.5) and may determine p(IJ)S

If the difference does not vanish, we have the second class of models and only the upper bound

in eq. (5.5). Thus T = 0—* T

=

0 transitions can help to decide amongst the various gauge theory

models in a nuclear-model-independent way. A special subcase of this T = 0 T = 0 situation occurs when we have only a = 0~ —*0k transition. Then, with the only natural parity monopole (5 = 0) operator being purely vector in nature, we have equality in eq. (5.5) for all of the gauge models and may directly determine the coupling —+

67

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

constant (0)2 =

(H2+ + H ~)/2F2

(0~0 —~0~0).

(5.6)

(In effect, we can only determine experimentally (K/3~)2.) Another special case, similar to the one above but not restricted to isospin T = 0, occurs for elastic electron and neutrino scattering. Again only the vector monopole operator i%~o enters, now with isoscalar and isovector parts in general. if we assume that the spatial distribution of protons and neutrons in the 0~ground state is the same, then there is only a single form factor and we can write H2±= ~{Z(a~

+a

—

2aem) + N(a ~

—

a t~”)}2F2/Z2,

(J

=

0, elastic),

(5.7)

where we have used the results of section. 4.2. The quantities a ~ + a — 2aem and a a are given for the various gauge theory models in table 4.1. This result depends only on the assumption of equal proton and neutron distributions, but is otherwise valid for all intermediate energies (i.e. not just in the LWL). In fact even when J 0 this result will be approximately true, at least for heavy nuclei where Z ~ 1 and N 1. In such cases the matrix elements of M —

~‘

0, which, being coherent, are proportional to Z

and N, will dominate over the other allowed multipole matrix elements which are not coherent. Only at large scattering angles where the monopole is suppressed relative to the higher multipolarity contributions to the cross section will the result not be approximately true (see the discussion in ref. [5.1]).

Thus we see that several important general relationships can be obtained, interrelating the neutral current processes, and their dependence on the underlying gauge theory models, to the chargechanging weak and electromagnetic processes, which are all of the standard form at the energies of interest here. We must be more specific however, since we do not yet have measured neutrino reaction cross sections at our disposal. To obtain predictions for the neutral current processes at intermediate energies we must in general enter into a discussion of the nuclear many-body problem. We shall do so in the rest of this section, employing a variety of nuclear models. 5.2. Applications to specific nuclei

The nuclear many-body problem was discussed briefly in section 3.4, insofar as it pertains to the present work. We emphasized there that, if we restrict our attention to one-body operators for the weak and electromagnetic processes considered, the entire content of the nuclear dynamics resides in the one-body density matrix elements (eq. (3.36)). Somehow or other we must obtain information on these quantities to be able to make predictions for the yet-to-be-measured processes such as neutral current neutrino scattering. In certain cases a specific nuclear model may be used, having a set of parameters whose values are adjusted to fit known information on electromagnetic (and perhaps conventional weak) processes involving the states of interest. From such models a set of one-body density matrices may be obtained and the more exotic6He—6Li. neutrino-induced sections the predicted. By havingcross constrained model toOne fit such example is the A = the 6 system, certain existingconsidered data (andbelow perhaps tested model by predicting other rates and comparing with experiment, see below), we hope that the neutrino reaction and neutrino scattering cross sections can be predicted with a high degree of confidence, possibly even at the 10% level. We may carry this one step further. In certain cases, such as the A = 12 system, ‘2B—’2Ci2N considered below, we may work directly in terms of the one-body density matrices. Within a

68

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

truncated model space we have only a relatively small number of matrix elements to determine and, using all of the relevant experimental information that is available, we can complete the analysis. We proceed to discussions of these two systems (A = 6 and A = 12) and then, at the end of the chapter, return to review other calculations performed on a variety of nuclei. 5.2.1. The A

=

6 system, 6He—6Li

The lowest-lying energy levels in the A = 6 system are indicated in fig. 5.3 (see ref. [5.2]).We shall focus on transitions between the 1~0ground state of 6Li and the 04i (3.562 MeV) state in 6Li (and its analog, the ground state of 6He), as well as the 2~0(4.31 MeV) state in 6Li. The 1~0~-*0’l (isovector) transition was discussed in detail in ref. [5.3] with a subsequent reanalysis in ref. [5.4] (see also ref s.[5.5] and [5.6]). In the present work we extend these ideas to include the 1~0*+2~0 (isoscalar) transition. We assume that the 1s 112-shell is arbitrary completely filled in all ofcases the extra two 4He are in the ip-shell, with combinations lPI/2 and and that 1P3/2. Under these nucleons above circumstances the states of interest can be written:

1~0)=AI(1p

2 l~0)+BI(lp 2 l~0), 312) 312lp112);L0)+ CJ(1p112) 0~1)= DI(1p 2 0~l)+EI(1p 20~1), 312) 1i2) j2~0)= F~(lp 3121p112);2~0),

(5.8a)

(5.8b) (5.8c)

where the individual terms are allconfigurations anti-symmetrized two-particle configurations. Note this regard 2 or (lpI,2)2 allowed for the 2~0 state. Since we haveintruncated to that there areand no have (1p3,2)treated the 1s the ip-shell 112-shell as being closed, we have the normalization conditions

2+B2+C2=D2+E2=F2=1. (5.9) A In ref. [5.3]the single-particle basis was chosen to consist of harmonic oscillator wave functions with oscillator parameter b, although more complicated basis sets have also been considered [5.3—5.6]. In our present discussion we shall also retain the oscillator basis for simplicity and thus have seven parameters to determine (A, B, C, D, E, F, b) with three normalization conditions (eq. (5.9)). Given these parameters the relevant one-body density matrices can be constructed and, having the necessary

single-particle matrix elements of the appropriate operators (see appendix B), we can use the formalism of section 3 to obtain the weak and electromagnetic cross sections of interest. In ref. [5.31 this procedure was followed. Using the magnetic dipole and electric quadrupole moments of the 6Li ground state, ~ and Q, and elastic magnetic electron scattering, and using inelastic electron scattering to the 01 state in 6Li the parameters A, B, C, D, E and b were determined. This required using eqs. (3.16—3.18) with the long wavelength limits which define and Q and employed the basic equation .t

0

N

~—I

4.31

2+0

Fig. 5.3. Partial level scheme for the A = 6 system

1’. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

69

(3.35). In the present work in discussing the 2~0state we retain the same valuet for b and, from the normalization condition, have taken F = 1. Thus all of the parameters are fixed; the values are given

below: A

D = 0.80±0.03,

0.810 ±0.001,

=

—0.581±0.001,

B =

b = 2.03 ±0.02 fm.

E=0.60±0.04,

C= 0.084±0.002,

F= 1.0,

The one-body density matrix elements are also fixed at this stage. By evaluating eq. (3.36) for the states involved we obtain the following expressions:

for s’ = s

1~0(indicated by “1”) (11) (~5) ~ = (A2 + ~B2)= 1.010, cf0;0 =

= (11)33

;o

2V’3 = 3.464;

=LA2+~B20.482,

(~ ~)

31 cfi;o(~) = (11)

11 1I’i;o(~) (11)

(11) 1 3 IPi;o

(~)= ~Q~AB

A2

+ ~B2

(11)31

(11)13 = (~)= “cf2;0 (~)

=

3 3

—0.308, AB—~BC= 0.171;

~

—

..LAD =

=

1 3 =~~LBE = —0.246, (~)

(21) 11

;o

( )=

(21) 3 3 cf2;O

— \/1OBF

(~)= ~~BF

=

=

(01)

3

1

(~)= —CE = —0.050;

(5.llb)

=

1~0

i/,o~_ 21)31

—“~~CF = 0094 . ,

cfi;o~~AF0.286, (21)13

0.459;

0.252,

AF = 0.665; (~)= _____ 2\/i~

(21) 13

I,/’2;o

0.145,

BF

BD = 0.232,

i/’i;i (~) (01) 11 t/s~

and for s’ = 2~0(indicated by “2”), s (21) 3 3

1~0

0.458,

(01) i/i~

*i;o(~)

(5.lla)

2V10

for s’ = 0~1 (indicated by “0”), s (~)

0.348,

2V2

(11) 33

(01) i/ill

BC =

__~~_~B2+\/2C2= —0.109;

cf2;o (~ ~)= cf2;0

(~ ~)= V3(~B2+ C2) = 0.305,

cf0;0 (11)

(21) 3 1

cf2;o (~) = ~CF

=

0.073,

(21)33 V’3 cf~~o (~)= _ç,=BF = —0.356,

(5.llc)

2

tIn using the same value for b a word of caution should be included considering the unbound nature of the excited state. The results should be considered as orientative, because there is no control from other processes.

70

T. W. Donnelly and RD. Peccei, Neutral current effects in nuclei

where the numerical values result from using the numbers in eq. (5.10). Note that in the case of the = = 0 density matrices we must include the contribution of the (closed) 1s 112-shell.state of 6He and 6He, js-capture to the ground Three tests of the model are available, the /3-decay of threshold photopion production 6Li(’V, ir~)6He(g.s.).The quality of the agreement with experiment for these predictions [5.3, 5.4] gives us confidence that we may proceed to predict the neutrino-induced processes with some reliability. The neutral current neutrino and anti-neutrino scattering cross sections are calculated using eq. (3.26) with the basic equation (3.35) and the results in appendix B. The predictions obtained using the gauge theory models of sections 1 and 2 (see table 2.1) are shown in figs. 5.4 and 5.5. Several things should be noted: (1) The total cross sections are larger at these intermediate energies than in the LWL, although, once the main strength of the nuclear form factor has been integrated over (i.e. for neutrino energies v — 250 MeV and hence momentum transfers up to qmax 500 MeV), the cross sections remain reasonably constant with increasing neutrino energy. (2) The neutrino and anti-neutrino cross sections in general differ at intermediate energies and provide us with a measure of the V-A interference terms (see eq. (3.26)). If in fact they do not differ, then, depending on whether the transition is isoscalar or isovector, likely /3 or /3 ~8~] vanishes and some of the gauge theory models can be eliminated (we assume here that /3 ~ 0 and that the nuclear matrix elements do not in general vanish).

12

Cx

)xlO —41 cm 2 )

cm2)

10

.

6L~)l~0~0~I,3.562)

/

/

~

/ \

/

10

/ I

“1’

2.0—

—

v

/

/

j

.

3

~——_.._

I

V

8

6L1 (l+0~2+0, 4.31)

/

0_Ill,,

/ //

o_Vhl’

6 I.0-

/

/

I/ I

I I

:0

~~—: 100

200 1/

300 CM

400

500

eV I

Fig. 5.4. Neutrino and anti-neutrino cross sections (solid and dashed lines respectively) versus neutrino energy v for the isovector transition in ‘Li. The numbers indicate the gauge theory models employed (see table 2.1): (1) W-S-GIM, HKM and SU(2)L x SU(2)~x U(1); (2) b-quark; (3) q-quark; (4) Vector; and (5) SU(3) x U(1). For curve (4) the v and C) cross sections are the same.

/

I

,—

200ii

)MeV)300

400

500

Fig. 5.5. Neutrino and anti-neutrino cross sections (solid and dashed lines respectively) versus neutrino energy v for the isoscalar transition in ‘Li. All multipoles (J = 1, 2 and 3) are included. The numbers indicate the gauge theory models employed (see table 2.1): (1) W-S-GIM, Vector, HKM and SU(2)L x SU(2)R x U(1); (2) bquark and q-quark; and (3) SU(3) x U(l). For curve (1) the v and C) cross sections are the same.

71

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

(3) Most importantly, the cross sections are seen to be sensitively dependent on the underlying gauge theory model and, given our believed confidence in the nuclear physics aspects of the predictions, even a relatively crude measurementof the cross sections should go a long waytowards narrowing down the possible choices of gauge models. We note in passing that a similar analysis can be applied to other nuclei which have two particles outside of (or two holes within) a closed shell. The A = 14 and A = 18 systems provide two such examples. 5.2.2. The A = 12 system, ‘2B—’2C—’2N Next we turn to the A = 12 system as an example of treating the nuclear many-body problem directly in terms of one-body density matrix elements. We consider the levels shown in fig. 5.6. We take the 1s 112-shell to be completely filled and allow the remaining eight nucleons to occupy only the ip-shell, so that, within this truncated model space we have and only four elements: fordensity iPI/2. Inmatrix fact rather than cfi;i(~), cfi;i(~), cfi;i(~) and cf~(~), where stands for using the latter two matrix elements, it is more convenient to work with cf~?defined by: 1P3/2

“~“

“~“

I/’~I~?

cfi;i(~~)±litf

(5.12)

11(~~).

Furthermore we use harmonic oscillators for the single-particle wave functions and hence have a fifth parameter, the oscillator parameter b. First we consider electron scattering involving the inelastic transition 0~0 1~1.We may write the form factor (see eqs. (3.16—3.18)) as follows: -‘*

i

r

12

2 1 I_,~,,,,, 2 —y I FT{q)=~—~ ~ fSN(q~)fCM(q)e ~(~)i ?ITLIVIN J

(5.13)

,

where we use eq. (3.35a) for the center-of-mass correction and use the results of section 3.3 for the single-nucleon form factor. The factor q/MN accounts for the LWL behavior of the Ml form 2. factor For (seespecific table 3.1). In writing eq. (5.13) in this way weip-shell have defined a function p (y) where the choice of harmonic oscillators in the we have the following form: y = (bq/2) p(y)

=

a

0— a1y (H.O., ip-shell).

(5.14)

Thus we have a function three parameters, and b to [5.7—5.14] fit to the experimental MI ee’inform 2 fitwith to the highest quality a0, dataa1 available was made resulting the factor.shown A minimum —x and in the values: curve in fig. 5.7 a

0 = 1.34,

a1

=

0.95,

b

=

1.70 fm.

(5.15)

The polynomial p(y) may be expressed in terms of the single-particle matrix elements of ~ M1~—I

M1-~0 15.110

M1~+I 2N

128

~

~

+

I

‘

12c Fig. 5.6. Partial level scheme for the A

=

12 system.

I

using

72

T.W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

0

0.2

-

0.4

0.6

-

2 q

0.8

(fm

.0

1.2

2

2C plotted in terms of the function p(y) defined in the text (eq. (5.13)). The data Fig. 5.7. Inelastic electron scattering (O’0—~i’l, 15.11 MeV) in ‘ are from refs. [5.7—5. 14]. The straight line represents a fit with lp-shell harmonic oscillator wave functions.

eq. (3.32c) and the results of appendix B with the one-body density matrix elements as multiplying coefficients (cf. the basic equation (3.35)). In fact, upon examining the nature of the P3/2P1/2 and P1/2P3/2 matrix elements for the operator T~, it can be shown that the polynomial p(y) depends only on cfi;i(~), cfi;i(~) and cft?. with no dependence on ~i’~?.We may use the values in eq. (5.15) for a 0 and a1 and the explicit forms of the single-particle matrix elements to write ~ ~) and çli~~ ~) in terms of ~/i~

i

;

(—I.

‘P j:~. =

0.683cf~—0.3 11,

t/hi;i(~) =

0.157cf~+ 0.036.

(5.16)

Next we evaluate the /3~-decayrates (eq. (3.20)) and es-capture rate (eq. (3.22)) as functions of cf~and explicit dependence on cb1~1(~~) and cfi;i(~~). By requiring that the calculated rates agree with experiment we obtain relationships between i/,~7~ and cf~,as shown in fig. 5.8. The lines correspond to fitting the /3- and ±decay rates and to fitting the maximum and minimum of the cf~,using eqs (5.16) to eliminate

p

measured p-capture rate. The shaded region represents reasonable mutually acceptable values for cf~? and hence, through eqs. (5.16), for cfi;i(~) and cfi;i(~) as well. Thus we have obtained a region in the space of one-body density matrix elements which yields a good compromise fit to electron scattering, /3-decay and n-capture. Note that a similar analysis was recently performed independently by Haxton [5.17].His density matrix elements are in qualitative agreement with the ones deduced here and yield a form factor p (y) (fig. 5.7) in excellent agreement with the present analysis (for example, he obtains a0 = 1.38, a1 = 1.02, b = 1.76fm compared with our values a0 = 1.34, a1 = 0.95, b = l.7Ofm). We are now in a position to predict the neutral current neutrino scattering cross sections using the formalism of section 3 along with the present one-body density matrix analysis. Results for the various gauge theory models considered in sections 1 and 2 are shown in fig. 5.9. For these predictions we have used the middle of the shaded region in parameter space in fig. 5.8. If we use any other point within the shaded region, the calculated vv’ cross sectibns vary by less than 5% and thus we may hope that these predictions are valid to perhaps the 5—10% level. As in the previous subsection where we discussed the A = 6 system, we see that significantly

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

-1.0

-1.5

~0.5(~)

0

73

0.5

Fig. 5.8. Determination ofthe one.body density matrix using conventional weak interaction rates. The density matrix elements i/ii~?are related by requiring that the calculated a-decay and es-capture rates agree with the experimental values. A choice for the mutually acceptable region of parameter space is indicated by the cross-hatching.

IC 41cm2)

(a 10

I2~(0~0—

I + 1,15.110)

81/

‘d•__•__

3

~

6-

ii:”

4-

2-

III

___

/7,’

II~/ ~

5--

~2

C’ ~

0

100

200

300

400

500

v(MeV) Fig. 5.9. Neutrino and anti-neutrino cross sections (solid and dashed lines respectively) versus neutrino energy v for the isovector transition in I2~The numbers indicate the gauge theory models employed (see table 2.1): (1) W-S-CIM, HKM and SU(2h.x SU(2)R X U(l); (2) b-quark; (3) q-quark; (4) Vector; and (5) SU(3)X U(l). For curve (4) the i’ and C) cross sections are the same.

74

T. W. Donnelly and RD. Peccei. Neutral current effects in nuclei

different results are obtained when the various gauge theory models are used and that even a reasonably crude determination of the cross section would be sufficient to help eliminate from contention one or more of the gauge theory models. To make contact with the previous chapter where we employed the long wavelength limit, these predictions were carried down in energy to the region v — 0—53 MeV. We find that the LWL is accurate to the 10—20% level; for example, at v = 30 MeV the LWL results are 15% higher than the corresponding present results, thus confirming the validity of the long wavelength approximation. As a final note, we compare the present results with a previous analysis [5.18, 5.19] in which the space of one-body density matrix elements was even more restricted (cf11(~) 0, lII1;1(~)= 1/’i:i(~~) = 0). That analysis had only two adjustable parameters, cf1~1(~ ~) and b. The i.’v’ cross sections obtained [5.19], however, agree very well with our present results and indicate that high quality predictions can be obtained even with a rather severely restricted density matrix. remark that in 2C considered in Section 4, we may We employ a similar the case of the 12.7 10 MeV 1~state in ‘ two-parameter model to obtain predictions for the vi” cross sections, whereas, with the lack of sufficient data for that transition, we cannot presently constrain the density matrix elements in the more complete five-parameter model. We have every reason to believe that the two-parameter predictions are reliable, however, and shall return to further discussion of this transition in the next chapter.

5.2.3. Other calculations of neutral current neutrino scattering at intermediate energies We conclude this section with a very brief discussion of several other calculations of neutral current neutrino scattering at intermediate energies. Only a few representative calculations are considered. First we mention the work of Gershtein et al. [5.20] in which a variety of nuclear transitions are discussed, including cases in 12C, ‘4N and 160 Similar conclusions regarding the general trends were reached by these authors, although in the present work we have considered a wider class of gauge theory models and have used the one-body density matrix approach to obtain relatively

model independent predictions. Subsequently their analysis of the 14N case was redone in ref. [5.211. In ref. [5.22] the Goldhaber—Teller model was used to describe the nuclear current densities for transitions in the giant resonance region. The main merit of the Goldhaber—Teller model lies in its simplicity, making possible the treatment of a variety of nuclei in that work. Again the results are similar to the more microscopic calculations presented here. Finally we wish to mention again the work on elastic neutrino scattering and the more general isoelastic processes [5.1]. In ref. [5.11two cases, the A = 3 (see also section 3.6) and A = 11 ground state systems, were considered in some detail. A rather involved one-body density matrix analysis was

undertaken for the latter case, thus providing another non-trivial example of these techniques. In this discussion only the W-S-GIM model was considered. In all cases which have been discussed of intermediate enejgy neutrino scattering from nuclei the same basic conclusions emerge: the cross sections, while small, are not so small as to make experiments impossible; even relatively crude measurements of these neutral current processes would suffice to allow a distinction to be made amongst the various gauge theory models which underlie the neutral weak interaction. In this regard, the vv’ cross sections show wide variations for the different

gauge theory models, almost certainly beyond the uncertainties in the nuclear structure aspects of the problem.

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

75

6. Neutral current effects in nuclei not induced by neutrinos Neutrino scattering off nuclear targets is just one of the processes that is sensitive to the weak neutral current. There are both other semi-leptonic and purely non-leptonic nuclear reactions which can serve equally well to study the weak neutral current. In the case of reactions not involving

neutrinos all measurements require the detection, in some fashion or other, of a parity violation effect. We shall concentrate in this section on the study of parity violation effects in electron scattering off nuclei. This process is the most amenable to study from a theoretical point of view, since it involves a

semi-leptonic weak interaction. Thus we shall be able to use, practically verbatim, the formalism we have developed in sections 3, 4 and 5. Before discussing this semi-leptonic example we should comment briefly on what might be called non-leptonic parity violation effects. There are a variety of experiments which have been performed, or are being performed, in this general area. They include parity violations in hadron—nucleus scattering (notably p-deuteron scattering [6.1]), the measurement of polarization assymetries in

selected nuclear y-decays (a relevant example is the ~ —~~ transition in 19F [6.2]),thermal neutron capture of polarized neutrons [6.3],purely forbidden nuclear decays like 160 (2; 8.88 MeV) -. a + ‘2C [6.4],etc. All these experiments have in common that what is measured is the effect of a purely non-leptonic weak interaction. The effective weak Hamiltonian which governs these processes necessarily involves the product of two weak currents and it is to be evaluated between two strongly interacting states. This circumstance makes the theoretical analysis of these processes more difficult than that which is employed in polarized electron scattering. To begin with, there is some question on how applicable is the usual short distance analysis commonly employed in dealing with non-leptonic processes [6.5].Secondly, and this is the principal disadvantage from our point of view, the effects of the weak neutral currents are not cleanly separated out. This is because the effective parity violating potential receives contributions from both the product of two neutral currents as well as the product of two charged currents. Non-leptonic parity violating effects have been reviewed a few years ago by Fischbach and Tadic

[6.6]and Gari [6.7].These reviews, however, do not include the effects of neutral weak currents. A recent series of papers by Gari and Reid [6.8],Koike and Komura [6.9]and Galic and collaborators [6.10] remedy this situation by calculating the contribution of the neutral currents in the W-S-GIM

model. A more recent preprint of Cung and Kim [6.11]examines the effects of neutral currents in a b-quark model. A different approach, which also involves neutral current effects, is contained in a paper by Desplanques and Missimer [6.12].Although the quality of agreement of these calculations with experiment vary, and the effect of the inclusion of the neutral currents is significant (especially in the Y = 1 channel), it appears premature to us to advocate using these analyses to select from among

alternative models of the weak interactions. Essentially the same conclusion has been arrived at by Weinberg [6.13]in his recent review presented at the Zurich conference. The uncertainties in these calculations, including matters of principle [6.5,6.13], appear to preclude their direct use in testing viable gauge models, at least at our present rudimentary stage of knowledge.

Let us concentrate therefore on parity violation effects in electron scattering. It is clear that these effects are in some sense complementary to the detection of parity violations in atoms. In both cases electron and nuclear vertices are involved and parity violations arise because the coupling at either vertex is not purely vectorial. In high Z atoms, if the axial-vector coupling to electrons is nonvanishing, these effects are enhanced because the neutral current coupling at the nucleus is coherent.

Similar coherence effects occur in nuclei. However, in nuclei it may be also possible to study

76

T.W. Donnelly and R.D. Peccei. Neutral current effects in nuclei

incoherent processes which still have a non-negligible cross section. This last point is of some interest since, as we have mentioned in section 2, if the neutral current coupling to electrons is purely vectorial all coherent parity violation effects vanish. To our knowledge, Feinberg [6.141was the first one who suggested that the study of parity violations in electron scattering off nuclei may prove fruitful as a means of examining the structure of the weak neutral current. (Earlier work on the asymmetry expected between deep inelastic scattering of left-handed or right-handed electrons off nucleons had been done by Derman [6.15]and by Berman and Primack [6.16].A recent paper by Cahn and Gilman [6.17] examines this subject anew with the view of testing different gauge models of the weak interactions.) Feinberg argued that the interference between the ordinary electromagnetic amplitude and the parity violating piece of the weak neutral current would produce a measurable asymmetry in polarized electron scattering experiments. He suggested in particular that one should measure the difference between the differential cross sections for elastic scattering of right-handed and left-handed incident electrons on a 0~0nucleus. In this case there is just one possible form factor and, if one normalizes this difference cross section by the total cross section, the answer is entirely independent of any nuclear physics detail. As we shall see, this is a coherent process and hence it is sensitive to the axial-vector coupling of the electrons to the neutral current. More recently Walecka [6.181has investigated in detail the question of parity violations in electron

scattering. He has derived a general formula for the difference in cross section for scattering of electrons of positive or negative helicity from a given initial state to a specified final state. Walecka’s formula can be written in terms of the reduced matrix elements of the vector multipole operators M, ~5 Dmag5 and i”~ j~ ~j.mag and i.d1 (see eqs. (3.10)) and of the axial-vector multipole operators J~~ (see eqs. (3.11)). It further depends, of course, on the coefficients ay and aA of the electron coupling to the neutral current, as well as the hadronic coefficients /3 ~, /3 ~ (given in table 2.1) which detail the strong isospin decomposition of the weak neutral current. The general formula is complicated, but it simplifies considerably in the relativistic regime where one can neglect the electron’s mass. One obtains in this limit the following formula for the cross-section difference: K,

\f

~~ (

2oM/cos2~O ( 4ir Gq~c I + 2 sin2 ~8IMA\2J + iI t2iraV2J

~~do~_ dfI d11

x ~e{a~

[(~cos2~

+

q

sin2~) ~

2cos2~~

a ~/3

sin

~ cos2 q

T’

\MT

~ 0

T

T’

~9’ T

M~I\M~

10

MT

TXJ’; T’~f~r’(q)~J; T)*

(J’; T’~f~’~-(q) ~J;T)(J’; T’~f~(q)!~J;T)*}

(J’; T’~ ,.~-(q)~J; T)(J’; T’ ~iC1~~r(q)~J;T)*] 1/2

2 —

I

{(J’; T’~~(q)~J; +

q +(~4)

~‘=0,

+

sin2

~ {(J’, T’ 1’~~-(q) J, T)(J’ T’ f~5(q) J, T)* +(J’; T’~~r(q)i~J;

T)(J’; T’~f~5~..(q)~J; T)*}}. (6.1)

T. W Donnelly and R.D. Peccei, Neutral current effects in nuclei

77

We should note for reference that the spin averaged cross section, ~(dcrt/dfI + dcr ~IdIi), is given by eqs. (3.16, 3.17 and 3.18).

We have seen in section 2 that the parameters /3 ~ and /3 ~ differ depending on what model of the weak interactions one considers. Similarly also av and aA are a function ofthe weak gauge model. For SU(2) x U(1), as we have discussed earlier, one has the choice of letting the right-handed electron be a singlet [eR]or be part of a doublet This leads respectively to av = —1 +4 sin2 Ow, aA = —1 and (~)R.

av = —2 + 4 sin2 O~,aA = 0. The version of the SU(2)L X SU(2)R x U(1) model that we presented in section 2 leads to no parity violating effects in electron scattering. This is because the two intermediate bosons Z~and ZA couple either vectorially or pseudo-vectorially to both electrons and nucleons. Hence effectively a~v= avf3A = 0. Finally in the SU(3) x U(1) Lee—Weinberg model only Z-exchange gives any parity violations in electron scattering, and one finds that in this case av = —l + 3 sin2 °Lw, aA = 0. We summarize in table 6.1 these results for the various models. The values of K and p~can be gleaned from table 2.1. We should note that only the first version of the SU(2) x U(1) model has non-vanishing coherent effects, while all models, except the SU(2)L X SU(2)R x U(1) model, have non-vanishing values for av/3A and thus give rise to incoherent parity

p~,

violations. We will examine three rather simple cases of the general formula (6.1). First consider elastic scattering off a 0~0nucleus. Then only the isoscalar piece of the vector neutral current enters and the cross section difference is proportional to a single nuclear form factor: (0~0~Mo;o(q)~0~0)I2. This

factor disappears on normalizing the difference cross section by the sum cross section. One finds then simply

do~—do~— Gq~ do~+du~ 2irav2 —

________

~—[

~_.

(0)

(6.2)

Ka~~j3~,].

Essentially this result was derived by Feinberg [6.14]except that he worked in the context of the original W-S-GIM model where aA = —1, p~ = —2 sin2 Ow, K = 1. Since this is a coherent process the difference vanishes if aA = 0. This is analogous to what happens in the case of atomic parity violating

transitions in heavy atoms where the incoherent pieces are subdominant. In both cases the asymmetry measures Ka43 ~ and hence provides, in different physical contexts, a measure of the same underlying couplings. If the theoretical calculations of the atomic parity violating effects are not in error, then the null result of the Seattle—Oxford experiment informs us that no parity violations would be seen in the above process. Nevertheless, given some of the present uncertainties in the atomic calculations it would appear worthwhile to try to study the analog nuclear process. The kinematical factor of q~will suppress the asymmetry at small momentum transfers. At moderate momentum transfers, q~ Table 6.1 Effective electron couplings Model

av

aA

SU(2)xU(l)~ SU(2)XU(l)t SU(2)L x SU(2)~)
—l+4sin29w —2+4sin2Ow

0

0

SU(3)xU(I)

—l+3sin2HLW

0

—l 0

*The electron assignment has ei~as an SU(2) singlet. tThe electron assignment has eR as a member of a

SU(2) doublet.

78

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

(400 MeV)2, the asymmetry is of the order of 3 x i0~x (—KaA/3~).For 160 for example, the elastic scattering cross section has a secondary maximum at around these momentum transfers and a measurement of an asymmetry of this order of magnitude does not appear to be entirely out of the question. We urge that such an experiment be done since it would provide in an unambiguous theoretical fashion a value for KaA/3~! As a second example we consider, with Walecka [6.18],the excitation by a polarized electron beam of abnormal parity T = 1 states from a T 0 ground state of spin-parity 0~.For the transitions i~,2, 3~,...clearly only the ~ and T~ 1multipoles contribute to the difference cross section (6.1). The total cross section itself depends only on T,~and thus the cross section ratio will depend on the nuclear structure only through the ratio (6.3) 2C, as we have by now repeatedly For specific states to like l~1 transition 15.11 MeV in ‘ (see sections 4 and 5). argued, one ought be the able0~0 to estimate ~ to aat10—15% accuracy Using eq. (6.1) as well as eqs. (3.16) and (3.18), along with the above definition, it is easy to show that for these abnormal parity transitions =

~e(I; 1

t~1(q) 0~0)/~J;1

I’~7(q)~0~ 0).

—~

J

do~~ du ~ Gq~ ~ [(q2/q2)cos2 ~O+ sin2 ~~]1/2 ~ ~ do~+do~ 2~aV2l’~~’ [(q~2q2)cos2~O+sin2~O]h/2 Kav/3A ~NS(~)J. —

—

(6.4)

Various points are worth remarking upon. First, we see that for these incoherent processes there is still an asymmetry even if aA 0. Second, because the structure function ~s(q) is the ratio of two different multipole operators, which in principle have different diffraction minima, it is possible to choose momentum transfers in eq. (6.4) for which the asymmetry is very large. This point was emphasized by Walecka [6.181.Unfortunately, as he also remarked, the maximum asymmetry occurs precisely near the diffraction minima for the process, since it corresponds to a zero, or near zero, of ~J; 1 T,~(q) 0~0). Thus it is not entirely clear whether going to the diffraction minima makes in any sense the experimental task of detecting the effect any easier! In fact one can argue that it is =

probably better to avoid working too near the diffraction minima because here the details of the nuclear physics may be murky. For example, it is quite likely that at the diffraction minima exchange current effects [6.19]are of some importance. Thus, from this point of view, it may be clearly better to try to detect the asymmetry in the neighborhood of the first diffraction maximum where the nuclear physics effects are better understood. In principle, however, it would be desirable to cover a rather broad range of momentum transfers as a check of eq. (6.4) and of the theoretical estimate of the

nuclear structure factor ~~(q). It is perhaps worth elaborating a bit on these points. If one works at near backward angles (which are experimentally favored anyway for detecting these inelastic magnetic processes) then the kinematical factor in eq. (6.4) goes to 1 and one has simply

fdo~—dcr~\

— —

\do~+do~i

Gq~

~i~_

,—{Ka43v

(1)

KavPA

(1)

~Ns(q)}.

(6.5)

9~~2irav2 One may estimate the nuclear structure factor near the first diffraction maximum by neglecting altogether the contribution of the convection current. In this case there is a relation between and TrnaS, namely (see eqs. (3.32)) ~‘

p(l)

($; 1 ~I’~1(q)~i0~0)= —~---4-(J; 1 ~I’~7(q) 0~0).

~

(6.6)

79

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

Hence we obtain, approximately, Idffr —d\~’~-~ Gq~—1IKa~8v (1) _________ Kav/3A(l)2MNFA \do-t+dcrj,e=,,. 2ij~aV2i q ~L

(6.7)

.

For the case of the 0~0-~1~1transition in ‘2C the above approximation works to about 10%. A more careful calculation by O’Connell, Donnelly and Walecka [6.20],using the dominant iP1/2lP3/2 singleparticle matrix element (see section 5) yields for the structure factor for this case, the expression ~

~1AI

y(l) ~NS

where y

~.(1)(

— LIVLNU A —

=

~

i.....1

J —

1 2)’ ~y 1/2f’ —

( qb/2)2 where b is the oscillator parameter. We note that near the first diffraction maximum ,

already the second term in eq. (6.7) dominates. For the above transition in ‘2C, ~~4.~(qmax) 2MNF~/qmaxp~

umax

100 MeV and

(6.9)

—5.

Hence, approximately we expect, numerically, i.i

.1

(UU.rUO~1

~ q~qmax

~1 1 S=IT \ucrt rucTV

~—5ç

~ -~ ~.‘

Lu ~

g~(1)~! A

V

This is a somewhat smaller asymmetry than the one predicted for the 0~0 0~0elastic scattering transition in 160 at its second diffraction maximum. However, it should be emphasized that these two —~

examples measure fundamentally different quantities. The elastic scattering transition measures the combination Ka4~~ while the magnetic inelastic transition is essentially sensitive to Kavf3 ~. The enhancement of the av/3A terms in comparison to the a~flvterms which occurs in the

abnormal parity transitions from 0~,0-+ l~,2,..., 1 is even more pronounced if one considers instead isoscalar transitions. This important point has been recently emphasized by Bernabeu and Eramzhyan [6.21].This can be readily understood as follows. For an isoscalar transition, again neglecting the convection current, one expects that 2M~F~°~/q~f. (6.11) Since F~A°~is comparable to ~ at least in a quark model estimate (see section 2), but the isoscalar magnetic moment is much smaller than the isovector magnetic moment, one has that ~ ~ Some care must be exercised, however. Because

~

is so small it is probably not a good ap-

proximation to neglect the convection current altogether. Indeed, at zero momentum transfer it is not hard to show that the convection current has a comparable contribution, but of opposite sign, to the magnetic term. Using conservation of angular momentum one finds, for the particular transition in ‘2C 0~0—~ 1~0at 12.71 MeV (see the developments in section 4 and also ref. [6.22]),that = 2MN F~/{q(~f — ~)}

(LWL).

(6.12) ‘P3/2 single-particle matrix element as in the isovector case

More generally we obtain, using the lP1/2 discussed above to describe the q-dependence of the many-body matrix element, ~~Lf — h~LY1N

~NS

q

c’(O)(

i_.i

A J 1 2Y ~sl~1_~y...1/2~s

1

We see that at moderate momentum transfers (q 100 MeV) the nuclear structure effect provides a very large enhancement factor indeed and that the detectable asymmetry there may be as large as

80

T. W Donnelly and R.D. Peccei, Neutral current effects in nuclei

iO~.More importantly, from a weak interaction point of view, this asymmetry is essentially dominated by the Kav/3~term. Hence it provides a sensitive test to the presence or absence of an isoscalar axial piece of the weak neutral current. That is, for an isoscalar abnormal parity transition we expect, in the backwards direction fdo~—do~\ do1+da~ ~

)

,— [

Gq~. —

—

(0)

(0)

Ka~/3~] ~Ns(q).

(6.14)

2iraV2

Hence these experiments can directly test for the presence of an isoscalar axial current in an analogous way to that provided by the low-energy neutrino scattering experiments discussed in section 4. We should, however, interject a note of caution. Because the enhancement factor ~~(q) is so large its calculation is not entirely devoid of uncertainties. As Bernabeu and Eramzhyan [6.21]have emphasized, one must be especially of possible admixtures in the final 12Ccareful has a slight isospinisospin 1 component (see section 4) atstate. q = 0 For the example if the 12.72 MeV state in structure factor ~ would now be given by (0)

~NS(0)

—

f

2MNF’A°~ 1 1 q 1—o)—~+8~f’

(6.15)

where ô characterizes the degree of admixture. Clearly, because p” is so large, even a few percent degree of admixture may cause 10—20% uncertainties in ~ However, if one is principally interested in finding out whether is zero or not, one may perhaps tolerate uncertainties of this magnitude.

p~

We have tried to emphasize in this section the utility of polarized electron scattering off nuclei for testing competing models of the weak interaction. By necessity we have only illustrated the main ideas by studying just a few selected examples. There may well be other nuclear transitions in which the polarization asymmetries, due to the weak neutral current, are larger than those presented in our

specific examples. We hope that the work presented here will stimulate a more thorough search of suitable nuclear targets.

Appendix A: Conventions We collect together here some of the conventions used in this work. Our main criterion for using the particular metric, y-matrices, notation, etc., chosen here is that they agree with our past work on

semi-leptonic weak and electromagnetic interactions in nuclei and should lead to the least confusion when we draw upon that work. 1. Weuseh=c=1. 2. For four-vectors we use a,~= (a, a 0), b,. = (b, b0), where three-vectors are denoted a and b (~= 1, 2, 3) and time components a0 and b0. The scalar product is taken to be a b

—

a0b0,

where the repeated index is summed over (~= 0, 1, 2, 3). We use for the magnitude of the three-vector the notation a 2 a~. a —

=

Iaj. For the scalar product of a four-vector with itself we write

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

51

3. The set of 7-matrices used are the following: 10

—iu\

11

.

“=k~~ o)’

74l70~O

0\ —i)’

1 0 —1 75~71727314~(~.l

0

Thus the helicity projection operators are AR = ~(I

—

(right-handed),

75)

AL = ~(1+ y~) (left-handed).

4. The Dirac equation is (iyMae. — m)u(p) = 0 with normalization utu = 1 and with ñ 5. A caret over a symbol denotes an operator in the nuclear Hilbert space with two exceptions, the unit vectors b v/v and 4 q/q used in section 3. Appendix B: Single-particle matrix elements

The single-particle reduced matrix elements of the seven basic nuclear operators defined in eqs. (3.31) are straightforward to calculate using angular momentum recoupling relations given in Edmonds [B.1] (see refs. [B.2],[B.3],[B.4]for more detail). For completeness we collect here the expressions necessary to compute any of the required matrix elements. With the definitions (eqs. (3.12)) M,~(qx)_=j,(qx)Yj~(fl1) 4 M,~(qx) j9(qx)Y, 21(fl~)

(B.la) (B.lb)

and using the notation [z] V~~I we have the following: ~+j+1/2

(n’l’j’fIIvI,(qx)~Inlj)=

~‘

!

i

~ 0)(n”i’Ii~(.P)Inhi)~

[l’][l][J’][i][J]{. ~

~

11’ 1 2 ~ 1 ~

I,

(n’l’j’IIM,.~(qx). u~nhj)=

1~7J= V’6 [I’][I][I’][1][5] [2]~ 4ir ~j,

1)(n’l’J’~J2(p)In1J)~ ~‘

B.2b)

~, 2+j+1/2

(n’i’j’

M,.~(qx).

~

11n11

=

—

i42

1

~‘

[1’][i][j’][5][2J{.

~

~[—Vfli~+ 1]{~’ ~ + Vi[!

(B.2a)

~

1) (r 2

~‘

1— 1)(n~!llPJ j

[i’][j’][j][2j—1115]

&_+!~±-i) Jnhj)J,

{~2j— I

~ [—aJ.l+1/2(n’1’J’Ii1~~)(~-_ .!-)pnlj) + a j,l_1/2(fl’1’J’IJ.,(p)(a~+ —

;}

1+

9(p)

(n’l’j’ M,(qx)o’ . -~-V nil) =

‘

(B.2c)

x (I’ I 2J— I)

~~)In’i]~

(B.2d)

82

T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

where p

qx and where these expressions are now written in terms of radial matrix elements,

(n’i’j’ I C(p) I nil) =

f x2 dxR~,~.(x)C(qx) R~,1(x).

(B.3)

For the special case of harmonic oscillator wave functions we may obtain analytic expressions for the radial matrix elements [B.2]. We use the nomenclature ni = is, lp, 2s id, 2p if, etc., and suppress the subscript j, since the radial wave functions are the same for I = I ±~in this case. First, for n or n’ greater than unity we may use recursion relations to rewrite the results in terms of wave functions with n or n’ equal to one only: —

R21(x) =

(2)_h/2([l +

1]R11(x)

R31(x) =

(8)_h/2([i +

1] [1+ 2] R11(x)

—

—

[I+ 21R11±2(x)) —

2[I

+ 212 R

(B.4a) 11±2(x)+ [i + 3] [1+ 4] R 11±4(x)).

(B.4b)

Second, the terms involving derivatives (eqs. (B.2c) and (B.2d)) may be rewritten using the formulas 2[i + 11 R (~4__1~) R11(x) = —(8y)~ 11+1(x) (B.5a)

(~4_+ ~_t_i) R11(x) = (8y)_1/2 where y

(2(1] R 11_1(x) [I + 1] R 11÷1(x)) 2. Finally, the resulting radial matrix elements (all with n —

=

(bq/2)

(i1’~j.~p(p)~il) (2y)212 (22 +i)!!((2l’+ =

l)!!(21+I)!!)h12

x F(~(2 i’ —

—

1);

,~

n’ = +~

(B.5b) 1) may be evaluated

y),

(B.6)

where the last factor is the confluent hypergeometric function F’ ~

‘—l+~ ~a(a+1)y2~ 13y /3(13+1)2!

(B7)

which is a polynomial of order —a in y when a is a negative integer. In other situations the radial matrix elements may have to be evaluated numerically. Acknowledgements We are both grateful to J.D. Walecka for helpful conversations and encouragement. One of us (R.D.P.) is indebted to J. Bernabeu for some illuminating discussions on polarized electron scattering. Finally, it is a pleasure to thank E. Hurd for her carful typing of the manuscript and B. Serot for help in proofreading this work at the preprint stage. References [1.1] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264. [1.21A. Salam, in: Elementary particle physics, ed. N. Svartholm (Stockholm 1968) p. 367. [1.3] G. ‘t Hooft, NucI. Phys. B33 (1971) 173; B35 (197l) 167. [1.4] E. Abers and B.W. Lee, Phys. Reports 9 (1973) I. [1.5]P.W. Higgs, Phys. Rev. Lett. 12 (1964) 132. [1.6] C.N. Yang and R. Mills, Phys. Rev. 96 (1954) 191.

T. W Donnelly and R.D. Peccei, Neutral current effects in nuclei

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83

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T. W. Donnelly and R.D. Peccei, Neutral current effects in nuclei

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