PHYSICAL REVIEW C VOLUME 48, NUMBER 6 DECEMBER 1993

Neutrino-nucleus quasifree neutral current reactionsand the nucleon strange quark content

C. J. Horowitz and Hungchong KimNuclear Theory Center and Department of Physics, Indiana University, Bloomington, Indiana $7/08

D. P. MurdockDepartment of Physics, Tennessee Technological University, Cookeville, Tennessee 88505

S. PollockInstitute for Nuclear Theory, University of Washington, Seattle, Washington 98195

and National Institute for Nuclear Physics and High Energy Physics, Section K (NIKHEF K), -

P.O. Box $1882, 1009DB Amsterdam, The Netherlands(Received 1 June 1993)

Differential cross sections for neutrino- (and antineutrino-) induced nucleon knockout are calcu-lated for neutral-current reactions from nuclei. A relativistic Fermi gas model with binding energycorrections is used. We examine the accuracy with which strange quark axial and vector currentparameters can be extracted from both existing and (possible) future experiments. Both high (above1 GeV) and low (near 150 MeV) neutrino energies and the knockout of neutrons and protons areconsidered.

PACS number(s): 25.30.Pt, 24.85.+p

I. INTRODUCTION

There is considerable interest in possible contributionsof strange quarks to form factors of the nucleon. Informa-tion on these would provide insight into @CD and placeconstraints on phenomenological models of hadron struc-ture. Strange quark contributions can be investigated inparity violating electron scattering [1,2] and in neutralcurrent neutrino scattering. Neutrino scattering appearsto provide the most accurate information on strange ax-ial currents. In addition, as we examine in Sec. IV, neu-trino scattering can also provide important constraintson strange vector currents.

Experiments can be performed on free protons or onneutrons and protons bound in nuclei. Quasifree scatter-ing from nuclei has two important advantages: (1) Nu-clei provide convenient targets. Indeed, most neutrinodetectors contain nuclei in addition to hydrogen. Hencequasifree nucleons are an important source of statisticsand or background. (2) Nuclei are the only usable neu-tron targets. The ratio of ejected protons to neutrons issensitive to strange quark information [3].

However, before quasifree scattering data can be com-bined with free proton experiments, two questions mustbe addressed. First, are the strange quark matrix el-ements in a nucleus difFerent from those in a nucleon?Second, are there uncertainties from conventional nu-clear structure efI'ects which complicate the extractionof strange quark matrix elements from quasifree data?

The first question is very interesting. What is the den-sity dependence of the strange quark content of the nu-cleons The old EMC [4,5] efFect showed that up anddown quark distribution functions were about 10 percentdiferent in a nucleus. Furthermore, there are general

arguments which suggest that the strange quark con-tent should increase with density. Basically, the increas-ing quark Fermi energies and the interactions betweenhadrons should help to counteract the large strange quarkmass. In the limit of very high density, a transition fromnuclear to quark matter is expected, and the ground stateof quark matter (in weak equilibrium) is believed to bestrange matter with a large strange quark content [6].

The same conclusion can be reached in hadronic mod-els. Here strange quark contributions can be modeled asloops involving kaons (or other strange mesons) and hy-perons [7]. Calculations of neutron matter suggest thatat high densities there will be significant contributionsfrom real hyperons. Indeed, if a kaon condensate [8] wereto take place at high densities, this could imply a largestrange quark content.

Unfortunately, not much is known about strange quarkcontributions to the nucleon, much less their density de-pendence. This is an important area for future work.However, for the remainder of this paper we will simplyassume that strange quark matrix elements are the samein a nucleus as in a free nucleon.

The second question is more straightforward but stillvery important. Will complications from conventionalnuclear structure eKects impede the extraction of strangequark matrix elements? Understanding nuclear structurecorrections is a clear prerequisite before a large experi-ment can be mounted.

In a BNL experiment [9] data were collected at rel-atively high momentum transfers (Q = 0.5—1 GeV )where it was hoped that nuclear structure efI'ects weresmall. These will, no doubt, be large in a proposed lowenergy (E —150 MeV) Los Alamos experiment [10].However, one can measure a ratio of ejected protons toneutrons which may be less sensitive to nuclear struc-

0556-2813/93/48(6)/3078(10)/$06. 00 3078 1993 The American Physical Society

NEUTRINO-NUCLEUS QUASIFREE NEUTRAL CURRENT. . . 3079

ture [3]. We examine both of these experiments below.There has been considerable work on charged-current

quasifree scattering (see, for example, Ref. [11]). How-ever, neutral-current reactions have important kinematicdifferences from charged-current scattering. Often forneutral-current reactions, one integrates over the unob-served outgoing neutrino.

In this paper a relativistic Fermi gas model is usedfor calculating neutral-current quasifree scattering. Thismodel is easy to deal with, includes relativistic kinemat-ics, and can be extended to more sophisticated treat-ments of the nuclear target. The formalism is presentedin Section II, while Sec. III presents results for nucleonenergy spectra (both difFerential and angle integrated) fora variety of neutrino incident energies. Section IIIB dis-cusses the sensitivity of neutron and proton cross sectionsto various strange quark parameters and other theoreticaluncertainties (such as the mass in the axial-vector formfactor). Sec. IV examines the limits that can be placedon strange quark parameters from both existing data anda variety of proposed experiments. Finally Sec. V is asummary.

II. FORMALISM

In this section we develop a relativistic Fermi gas modelfor a nuclear target. This nuclear matter calculation isexpected to be appropriate at the high excitation ener-gies involved in quasifree scattering. Indeed, Fermi gasmodels have provided an excellent qualitative descrip-tion of quasielastic electron scattering [2,12]. Our simplemodel omits a number of nuclear effects. Final state in-teractions (FSI) between the outgoing nucleon and thenucleus have now been examined by Garvey et al. [13].

They find FSI do not appreciably change the cross section(except in the high-energy tail) or the ratio of neutronto proton yields. Therefore our results neglecting FSIshould provide a useful first orientation.

Relativistic e8'ects from a strong Lorentz scalar po-tential were found to be important in parity violatingquasielastic electron scattering [14,15]. This is becausethe parity violating scattering is very sensitive to thesmall isoscalar transverse response (which is enhancedby a scalar potential). However, neutrino scattering hasa large contribution from the pure axial-vector responseand is not very sensitive to the isoscalar transverse re-sponse. Therefore, we do not expect large relativisticcorrections. (Note this will be checked explicitly in fu-ture work. )

Corrections from the residual NN interaction couldmodify the neutrino cross sections. These can be es-timated in a random phase approximation (RPA) andare not expected to be important at the relatively highmomentum transfers (0.5 ( Q & 1 GeV ) of theBrookhaven experiment [9] (because the residual interac-tion is small at high Q ). Corrections could be somewhatmore important for the low-energy LAMPF neutrino [16].However, nonrelativistic RPA calculations [3] do not findlarge efFects. This will be explicitly checked in our rela-tivistic formalism in future work.

We do not expect large corrections from these nuclearefFects. Nonetheless, we caution that final conclusionson extracting strange quark parameters from neutrinoscattering may require further theoretical calculations.These should verify that the corrections are indeed small.

We begin with the phase space expression for theneutrino-proton elastic scattering cross section in the lab-oratory frame [17]:

Here the recoiled proton has energy E' (momentum p')and is detected in solid angle 0', while the incident (final)neutrino has energy k (k'). Equation (1) can be appliedto a free proton at rest by taking the initial proton energyto be E = M. The invariant matrix element M will bediscussed below.

We now consider the simplest modifications of Eq. (1)to describe quasifree scattering froin a nucleus (hereE g M). First, we average Eq. (1) over a Fermi gasmomentum distribution for the struck proton using

ky

d @=1

and multiply by the number of protons Z in the target.We discuss values for the Fermi momentum k~ below.Second, we modify the energy-conserving delta functionin Eq. (1) to include a possible average binding energy Bof the struck nucleon.

The cross section for a nucleus is then

d2o Zp'E' 3dA'dE' (2vr) 47rk

kF

, , 6(k + E —B —k' —E') iM i', (3)

where the bar over ~ indicates an average over initialspins and a sum over final spins in the usual way. Theinvariant matrix element M depends on the weak neutralcurrent of a proton J~,

JW. = ri(k')p" (1 —ps) v(k) U(p') J„U(p) .2

(4)

Here U(p) is a spinor for the initial proton (normalized

3080 HOROWITZ, KIM, MURDOCK, AND POLLOCK 48

to UtU = 2E), v is a spinor for the neutrino, and G~ isthe Fermi constant for beta decay.

The neutral current has vector and axial parts thatdepend on the nucleon form factors,

(5)

with the vector current having Dirac (Fq) and Pauli (F2)contributions,

Vj2 —— 4—F~F2k k'(p' —p*) . (k —k'), (14)

Q2V» —— ', k . k' [p* . kp' k + p* k'p' . k'+ M'k . k'],

(15)

A = 4G'„p* . kk' p' + p' kk' p* + M'k . k'], (16)

J = yp~+ tE22M

while the axial current is

(6)Vgi ——+8GgFi p . kp k —k pp . k], (17)

J„=—G5

F; = —(2sin O~ —2)F,"—2F,"+ 2F,', (8)

for i = 1 (Dirac) and 2 (Pauli), while for neutron knock-out one has

The vector current form factors (F, ) for i = 1 and 2 arelinear combinations of the electromagnetic form factorsfor protons (F,")and n. eutrons (F; ). In addition we in-clude possible strange quark contributions (F,') which weassume to be isoscalar. For proton knockout one has

V~2 ——+4G~F2k k'(k + k') (p* + p').

Here p„* = (po + B,pj with pe = E' = po + By p~ + M2. The plus sign in Eqs. (17) and (18) ap-plies for neutrino scattering while the minus sign is forantineutrinos. The integral in Eq. (3) is evaluated nu-merically.

The quasifree cross section depends on the nuclear pa-rameters Z, the Fermi momentum k~, and the averagebinding B. For the ejection of neutrons, the form factorsF; are changed as in Eqs. (8) and (9) and in the Appendixand, of course, the proton number Z is replaced by theneutron number ¹

F; = —(2sin Og —2)F,"—2F,"+2F,' . . (9) III. B.ESULTS

1G~ = —(g~ —g~) GD

2(10)

for proton knockout and

1Gg = —(—g~ —g )GD

for neutron knockout. Here g~ ——1.26, g& describes pos-sible strange quark contributions, and GD describes themomentum transfer dependence (which we assume to bea simple dipole and the same for g~ and g&); see theAppendix. Note, we assume strange quarks provide theonly isoscalar axial current. In principle, radiative correc-tions can also produce isoscalar contributions; however,we expect these to be small [18]. Finally, a pseudoscalarcurrent (q„p ) will not contribute to neutrino scattering.

The square of the invariant matrix element is readilyevaluated (although the kinematic dot products do notsimplify as much as for the free nucleon cross sectionbecause of the binding energy B):

Our parametrization of the electromagnetic form factorsis that of Refs. [2,12] and is included in the Appendix forcompleteness. We parametrize strange quark contribu-tions in terms of a magnetic moment (p, ) and an electricradius (p, ) (the Appendix). The axial form factor is as-sumed to be

k~ ——225 MeV

and we simply adopt this value for neutrino scattering.An average Fermi momentum for a heavier nucleus mightbe somewhat larger than Eq. (19) (because of a largervolume to surface ratio) but still less than = 260 MeV.This should have only a very small e8'ect on our results.

The average binding energy B is zero for a free Fermigas. For nuclear matter we employ the minimal value,

B = (k~2+ M')'~' —M (20)

which insures that all occupied states are bound (haveenergies less than M). For our choice of k~ this cor-responds to 27 MeV and agrees well with quasielasticelectron scattering results.

A. Cross sections

We now present results for quasifree scattering fromC. The Fermi momentum in nuclear matter (at normal

density) is of order 260 MeV. However, a nucleus also hasa surface region of lower density so an appropriate aver-age k~ is somewhat below this value. Fermi gas modelsof electron scattering from ~2C have used [12,14]

l' = 4G&(V» + V» + V22+ A + V» + V») (»)

V» —4F,' [p* . kk'. p'+ p' kk' p* —M'k . k'], (13)

A sample di8'erential cross section is shown in Fig. 1 forprotons ejected from ~2C at 20' and 40' (from the beamdirection). The incident neutrino energy is 500 MeV. Forthis figure, all strange quark contributions are zero. The

NEUTRINO-NUCLEUS QUASIFREE NEUTRAL CURRENT. . . 3081

I i I

f

I I I I

f

I I I If

I15—a ~

1.0

0.5W

b'd

0.00 100 200

T, (MeV)300

0.6I

CO

0.4t

W

O.a'aV

0.00 40 60

8 (deg)80

I I I

100

FIG. 1. Differential cross section for proton knockout at20 and 40 from a 500 MeV neutrino beam. The solid(dashed) line includes (neglects) binding energy corrections.

cross section at 20 shows an interesting double humpedstructure. It is kinematically allowed (depending on thenucleon's initial momentum) to produce either a low- orhigh-energy nucleon. However, it is impossible to pro-duce a medium-energy nucleon. These two regions mergeinto one for larger angles. This is seen in the 40 results.Binding energy corrections (dashed curves) have only amodest effect for these neutrino and proton energies.

A Brookhaven experiment has been performed usingwide band neutrino and antineutrino beams of averageenergy near 1.3 GeV I9]. Eighty percent of the eventscorrespond to quasielastic proton knockout from Cand 20 percent to elastic scattering from free protons.We average our results over the neutrino spectrum n(k)[j n(k)dk = I] by performing another numerical inte-gration,

FIG. 2. The cross section averaged over BNL neutrinospectrum versus the recoil angle for the kinetic energy of therecoil proton T„=250 MeV. The dotted line is for free scat-tering. The other two lines are for the quasifree scattering(solid, B = 0; dashed, R g 0).

Q = 2MT„. (22)

Of course, quasifree scattering will involve a range of mo-rnentum transfers to the nucleon around Eq. (22) for agiven observed T„. Nevertheless, Eq. (22) can be used todefine do /d "Q " in terms of do /dE,

d 2 2M dQdE (23)

Thus (do/dQ ) involves integrations over the Fermi gasmomentum distribution, the neutrino energy spectrumand the proton scattering angle.

Figure 3 shows our predictions for (do /dQ ) compared

dOdEdk n(k)

I

)

l I If

I ~ C if

I I I if

I I

1.5—

In general there are only small differences between freeproton and our quasielastic Fermi gas results at these rel-atively high energies. Furthermore, binding energy cor-rections tend to be small. However, Fig. 2 shows the an-gular distribution for 250 MeV ejected protons. (Note, inorder to compare free and quasifree results we have plot-ted the cross section per proton. ) The Fermi gas is seento have a broader angular distribution than the free re-sults. Unfortunately, the acceptance of the Brookhavendetector is falling rapidly for angles near 60 . This isprecisely the region where the free and quasifree distri-butions differ. Therefore one must carefully consider thedifference between free and quasifree angular distribu-tions when determining the detector acceptance for 250MeV protons. This problem becomes much less severefor higher energies.

The Brookhaven group has presented differential crosssections per effective momentum transfer squared "Q ."For a free proton, this is simply related to the recoil en-ergy T~ in the laboratory frame,

o 1.0

A 0.5—CY

bV

0.00.4 0.6 0.8

Q (GeV )1.0

FIG. 3. Differential cross section averaged over BNL neu-trino spectrum versus q = 2M~T~ The four upper cur. vesare for neutrinos and the lower four curves are for antineu-trinos. The solid curves include strange quarks (gA

———0.19)while the dashed curves do not (gz ——0). Immediately un-der these curves are dot-dashed and dotted curves which alsoinclude binding energy corrections.

HOROWITZ, KIM, MURDOCK, AND POLLOCK 48

10- I I I

i

I

ProtonsI I I

i

I I I I

[1 I I I

i

I I I

to the BNL data. Note, our calculations are for quasifreescattering and include binding energy corrections whilethe experimental "data" is for "free protons. " The au-thors of Ref. [9] tried to take corrections from Fermimotion and binding energy out of the measured data.Thus it is slightly inconsistent to compare our calcula-tions with the published cross sections. However, thereare no published "raw" cross sections. Furthermore atBNL energies these effects are small so these ambiguitiesare not important.

We also show results in Fig. 3 including strange quarkcontributions with g& ———0.19 (see below). The cal-culation with strange quarks is closer to the data. Thisagrees with the original analysis of Ref. [9] and is the bestdirect evid. ence we have for nonzero strange quark con-tributions. Note, the calculations in Fig. 3 used a dipoleaxial form factor with a cutoK mass of M~ ——1.032 GeV(see the Appendix). We discuss the important sensitivityof our results to M~ below.

Next, the Los Alamos neutrino beam (from pion decayin flight) has a range of energies with an average near150 MeV. At these low energies, binding energy correc-tions reduce the cross section by about 40 percent com-pared to a free Fermi gas. The Los Alamos experimentdoes not have angular resolution; therefore, we integratethe diBerential cross section over all scattering angles.Figure 4 shows proton spectra (der/dE') for a 150 MeVincident neutrino. The upper dotted line shows the spec-trum for elastic scattering from free protons (multipliedby Z = 6 to compare with later 2C results). This linehas an abrupt cutoA' near 35 MeV, which is the maximumobtainable recoil energy. The dashed and dot-dashedcurves are for quasifree scattering with B = 0 while thesolid and lower dotted curves describe quasifree scatter-ing including binding energy corrections. Note that thequasifree cross sections extend substantially beyond the

elastic cutofF.We now examine strange quark contributions. Two ex-

periments have roughly determined g&. The EMC grouphas measured As = —0.19 + 0.03 + 0.05 [19], while theBNL group has measured g& ———0.15 + 0.09 [9]. As hasa weak scale dependence (due to the anomalous dimen-sion associated with the axial singlet anomaly [20]), andwas measured at (Q2) 10.7 GeV . If this scale depen-dence is assumed to be negligible, Ls and g& would bethe same. Therefore we show results in Fig. 4 with

g~ = —0.19

(and p, = p, = 0). Strange quark contributions tothe axial current are seen to enhance both the free andquasifree cross sections by about 40 percent. Note,strange quark contributions to the vector current couldcancel some of this increase (see below).

The cross section with binding and using g&———0.19

is close to the calculation without binding for g&0. Therefore, there could be some confusion betweenstrange quark and nuclear binding eKects. Thus it willbe diKcult to extract g& from a measurement of quasifreescattering only. This would require excellent agreementamong theorists on the size of binding eKects and thedocumentation of a small theoretical uncertainty in thenuclear calculations. At this time, we do not know if thiscan be done.

However, it may be possible (assuming enough statis-tics and energy resolution) to fit a theoretical shape forthe quasifree spectrum to high energy events (which can-not be elastic scattering). Then this spectrum couldbe used to subtract the quasifree contribution from thelower-energy elastic scattering events (and Anally use theelastic scattering to determine g&). Alternatively, thiscould provide a check on other methods of quasifree back-ground subtraction.

The spectrum for ejected neutrons is shown in Fig. 5.(Again, the elastic neutrino-neutron cross section is mul-tiplied by six to compare with the six neutrons in ~2C. )

]0Neutrons

N

b

00 20 40 60

T (MeV)80 i00

FIG. 4. Di8'erential cross section for E„= 150 MeV andthe proton knockout case. The plot with big dots is forfree scattering with g& ———0.19 and the small dots is forg& ——0. The other four curves are for quasifree scattering.The dot-dashed and dashed curves are for H = 0 and theother two are for B g 0. The dotted and dot-dashed are for

gA ———0.19 while the dashed and solid curves use gz ——0.

00 20 40 60

T (Me V)

J

80 100

FIG. 5. The same as Fig. 4 but for the neutron knockoutcase.

NEUTRINO-NUCLEUS QUASIFREE NEUTRAL CURRENT. . . 3083

1.4—TABI E II. Diff'erential cross sections and logarithmic

derivatives with respect to various parameters for a 150 MeVneutrino to knockout a 30 MeV neutron or proton.

G4

0 1.0

0.8—

0.6—

Free

I I I I I I I I I

g~=0.0

do adE'„

1 Oo bOM~

1 OCT

&gx~ ~ps~ Ops

p0.79

0.23

—1.22

—0.11

0.0

0.64

0.18

1.21

0.11

0.0

p0.48

0.28

—1.58

0.13

0.0

0.32

0.19

1.48

—0.12

0.0

0 80 40T (MeV)

60 80 10 fm /MeV.

FIG. 6. The ratio der/dE' for protons over that for neutronsfor E = 150 MeV versus nucleon kinetic energy. The dashedlines are for quasifree scattering with B g 0 while the solidlines have B = 0. The dotted lines are for free scattering.

TABLE I. DifFerential cross section averaged over the BNLneutrino spectrum [9] and logarithmic derivatives with respectto various parameters for proton and neutron knockout atQ = 0.75 GeV .

(')OCr b

OM~1 OQ

~/As

0.65

2.27

—1.02

—0.25

0.0

0.58

1.63

0.98

0.26

0.03

0.23

3.63—1.60

0.16

0.02

0.16

1.88

0.06

0.13

10 "cm'/Gev'CeV

The neutron cross sections are somewhat larger than theproton ones because ~c„~ = ~Ei(q2 = 0)[ is much larger.The binding energy corrections reduce the cross sectionby about 40 percent (almost exactly as they did for pro-tons). However now, strange quark axial currents decrease the cross section by 40 percent. Although thestrange quark contribution is assumed to be isoscalar,the dominant effect is the interference of g& with theisovector g~. Thus, strange quarks decrease the neutronand increase the proton cross section (by almost the sameamount).

The ratio do/dE' for neutrons over that for protonsis shown in Fig. 6. This ratio is insensitive to bindingenergy corrections and may be insensitive to other the-oretical uncertainties as well. For example, absorptionis expected to be similar for neutrons and protons andsimply reduce both by about the same amount. Further-more, the ratio may be insensitive to many sources ofexperimental error. For example, uncertainties in the in-cident neutrino fIux should cancel in the ratio. Finally,

the ratio is very sensitive to g& and changes by about 80percent from g&

——0 to g& ———0.2. Therefore, the ra-tio might allow an accurate determination of g&. Theseresults are very similar to those of Garvey et al. [3].

B. Linear error analysis

We now examine the sensitivity of cross sections tochanges in various parameters. For simplicity we firstgive a linear error analysis. This should give qualitativeestimates of the sensitivity to changes in a single param-eter. Later we give more sophisticated error estimatesfor multiple parameters.

First we consider the BNL experiment, or an experi-ment like it at high energies. In Table I we calculate thelogarithmic derivatives of the cross section with respectto the strange quark parameters (g&, p„and p, ) andto the axial form factor cutoff mass M~. For simplicitywe consider an average momentum transfer of Q = 0.75GeV. We consider both neutrino and antineutrino beamsand the possibility of detecting either protons or neu-trons. (The neutrons would have to be in a future exper-iment. )

Table I allows us to clear up some confusion concern-ing the results of the BNL experiment. First, some thinkthat the combination of neutrino and antineutrino mea-

ParameterM~9&PsI-I s

sin (Ow)

Statistical+0.01+0.02+2.0+0.1+0.01

With systematics+0.03+0.07+3.0+0.4+0.02

TABLE III. Uncertainties (67 percent confidence) in singlevariable extractions from our fictitious data set (as describedin text), assuming standard values for the remaining param-eters. The first column ignores all systematic errors, the sec-ond column allows for a +10 percent uncertainty in the normof both v and v's separately. Results are for a high-energyexperiment such as the one performed at BNL [9].

3084 HOROWITZ, KIM, MURDOCK, AND POLLOCK

TABLE IV. As in Table I, but for simultaneous two parameter extractions. (All systematicerrors are ignored here. )

Parameter 1MA

QA

PsPs

sin (Hw)

Parameter 2

QA

psPsPs

MA

Spread in param. 1+0.1+0.04+0.02+3.0+0.2+0.05

Spread in param. 2

+0.2+4.0+0.2+0.2+0.01+0.7

do /dM~do /dg~

(25)

For example, an uncertainty of 0.036 GeV in the valueof M~ leads to a change in g& of 6 0.08 (if we use theantineutrino numbers in Table I). This is of the sameorder of magnitude as the total error estimated by theauthors of Ref. [9] for statistical and other systematicerrors. However, it is somewhat less than the magnitudeof the extracted g&. Therefore, we conclude that, if thetheoretical uncertainty in M~ is of order 0.036 GeV orless, there is significant evidence for a nonzero g&. Thisis important because there is no other direct evidence (atthis time) for a nonzero strange quark contribution.

However, if the uncertainty in M~ is larger than this,the experiment provides essentially no evidence for anonzero g&. Indeed, this is the case if the BNL groupuses only its data and tries to determine both M~ andg&. However, M~ is reasonably determined from othercharged-current neutrino scattering experiments. Thereis no reason not to use this information. Clearly, one canuse charged currents to determine M~ (since charged cur-rents are insensitive to g&) and then analyze the neutralcurrents for g&. The present error in the extracted M~from the world supply of charged-current data is of or-der 0.036 GeV or perhaps somewhat smaller with more

surements in the BNL experiment greatly reduced thesystematic error. While this may be true for extractingother quantities such as the Weinberg angle, it does notappear to be true for extracting g&. Instead, most ofthe sensitivity to g& comes from the antineutrino dataalone since the logarithmic derivative of the cross sectionwith respect to g& is larger for antineutrinos than forneutrinos. Furthermore, the derivatives have the samesign. Therefore a systematic error, such as a change inthe common overall normalization of the neutrino Aux,does not cancel when v and v data are combined.

Second, there is much confusion over the correlationin the extracted value of g& with the assumed value ofM~. We can make a simple linear estimate of how achange in M~ will afFect the extracted g&. Note, thisis meant to be a qualitative guide and should not besubstituted for a more sophisticated complete analysis ofan experiment. Furthermore, we are not attempting toredo the experimental analysis of Ref. the author of [9]and we have no quarrel with their conclusions.

The change in the extracted g& from a change in M~is estimated as

modern data [21]. We conclude that the BNL data doindeed provide some evidence for a nonzero g&.

Tables III and IV also examine the sensitivity of thecross section to the vector strange quark parameters p,and p, . As expected, this is less than for g&. Never-theless, the experiment does provide some informationon these previously unconstrained parameters. This isanalyzed further in Sec. IV.

Next we consider a low-energy experiment such as theone proposed at Los Alamos. We calculate in Table IIthe logarithmic derivative of the cross section da/dE fora 150 MeV neutrino to eject a 30 MeV nucleon. Wealso calculate the logarithmic derivatives of the protonto neutron ratio. We see that the cross section at lowerenergies is less sensitive to M~. Furthermore, some ofthe remaining sensitivity cancels if the proton to neutronratio is taken. Now there is almost no sensitivity to p, .However, there is some sensitivity to p, either in a crosssection or ratio measurement.

Using these simple linear estimates we calculate in Ta-bles III and IV the contributions of various uncertaintiesto the error in a g& measurement from either a cross sec-tion or ratio experiment. We arbitrarily assume an ex-perimental error of 10 percent. We see that a 10 percentcross section measurement (assuming everything else isknown) could determine g& to +0.085 while a 10 per-cent ratio measurement could yield an error of +0.039if everything else were known. An uncertainty of 0.036GeV in M~ will not make a significant contribution tothese errors. However, if p, is completely unconstrainedby other experiments then it could be a problem for theratio measurement of g&. Indeed, a +0.5 uncertainty in

p, would lead to a 0.059 uncertainty in the extractedg&. Of course, any nuclear structure or other theoreticaluncertainty could only increase these error estimates.

IV. LIMITS ON STRANGENESS PARAMETERS

In this section, we reconsider the sensitivity of elas-tic neutrino-nucleon scattering cross sections to strangequark parameters. For simplicity, we consider onlyfree nucleon scat tering, rather than folding into thefull quasielastic nuclear cross sections discussed above.The goal is to examine possible limits on the nucleonstrangeness content, and to consider what kind of sepa-ration of the various form factors (electric, magnetic, andaxial) might be possible [2]. This section is for kinemat-ics and measurements at moderate neutrino and antineu-

48 NEUTRINO-NUCLEUS QUASIFREE NEUTRAL CURRENT. . . 3085

trino energies ( 1 GeV, as in Ref. [9]).As discussed earlier, we do not attempt to reanalyze

the results of Ref. [9], as indeed was done recently byGarvey et al. [22], but we wish to point out the gen-eral sensitivity of such data to the strange matrix ele-ments. For this purpose, we run simplified "thought ex-periments, " generating fictitious "data" by assuming thestandard model, zoith no strangeness in the nucleon, andthen adding random statistical errors of the same size asobserved at BNL. To estimate the effects of systematicerrors, in certain cases we also allow Aux normalizationto Goat by +10 percent.

Given such a sample "data" set, and some numberof desired parameters, we minimize y and then gridsearch for all parameter values yielding (y2 —y2,.„)( %,where N is determined by the number of degrees of free-dom [23].i This should yield an estimate of the scale ofuncertainty that a more complete analysis of real dataprovides in detail. Our approach has the virtue of beingeasily extendible to examine possible other experimentalsituations, including future low q2 experiments, and p/nratios. It goes beyond the linear analysis of Sec. IIIB,providing a method to examine correlations between var-ious sets of vector and axial strange quark parameters.Our results are in rough agreement with Ref. [22].

Table III gives the expected statistical error in vari-ous parameters of interest, assuming that all the othersare precisely known from other experiments. The secondcolumn shows the effect of including a +10 percent sys-tematic uncertainty. In both columns, the uncertaintiesare often clear underestimates due to correlations. As anexample, Fig. 7 shows the contour corresponding to 67percent confidence for extracting both g& and M~ simul-taneously from the data, assuming all other parametersare known. The narrowness of the curve at fixed M~,which corresponds to the spread in Table III, does notgive a full representation of the uncertainty in extractingg&. From Fig. 7, we see that if M~ is not independentlyconstrained, Lg& is larger, by almost an order of magni-tude, than its value in Table III (with fixed M~). Notethat Fig. 37 in Ref. [9], corresponding to our Fig. 7, in-cludes systematic uncertainties, and is thus larger thanwhat is shown here. We have checked that the shape ofthe contour in Fig. 7 is insensitive to the assumed centralvalues, so the uncertainties extracted should be reason-able estimates even if nature yields g& g 0.

In ¹ig.8, we show the correlations between p, andg&. Notice the relative scales the experiment is moresensitive to the axial term than the vector. In princi-ple p,, is completely unknown from previous experiments.In Fig. 9, we show the correlations between p, and p„

0.2

0.1

—0. 1

—0.20.8 0.9 1 1.1 1.2 1.3

Mq

FIG. 7. y contour showing 67 percent confidence region inextraction of il = gz/g~ and M~ only from our fictitious dataset (Q 0.5—1 GeV ). Systematic errors are not included,and standard model values (with 0 strangeness) are assumedto generate the data, as described in the text. Data havebeen integrated over an incident neutrino spectra similar tothat given in Ref. [9], and the y minimum is roughly 13 (/14DOF).

I I I I

I

I I

0.01

0.00

with statistical errors only included. The bounds arenot unreasonably large, implying that with the other pa-rameters (namely, g& and M~) pinned down, such ex-periments do indeed have some sensitivity to the vectorstrangeness parameters.

An alternative way to demonstrate the relative sensi-tivity of this experiment to the vector strangeness param-eters is to plot contours of the cross sections themselves(not y ) as a function of, e.g. , p, and p, . In Fig. 10 weshow this at fixed Q = 0.75 GeV, and fixed M~ = 1.03GeV, for both v-p and v-p cross sections. The centralsolid curve is a line of constant neutrino cross sectionwhile the left and right solid curves show +10%%uo varia-tions. The dashed curves are similar results for antineu-trinos. It is clear that neither v-p nor v-p alone constrainsthese vector form factors well. However, the two experi-ments are nicely orthogonal for these degrees of freedom,and the intersection region is surprisingly small. Thisfigure, however, does not show the effects of uncertainty

—0.01

With this procedure, the contours of (y —y;„) are notnecessarily elliptical, because the cross section (and thus y )is not linear in all parameters. In this case, we estimate theuncertainties in extracted parameters from the projected fullwidth of the y contour.

I I i i i I

—0.2 -0.1 0 P. 1. 0.2

FIG. 8. As Fig. 7, but showing correlations of p, with g.

3086 HOROWITZ, KIM, MURDOCK, AND POLLOCK 48

I

~

I I I I

~

I

0—

-0.2 —0. 1 0 0.1 0.2

FIG. 9. As Fig. 7, but showing correlations of p, with p, .

6

in other input quantities, especially M~, which signif-icantly spreads the uncertainty in the extracted vectorstrangeness parameters.

Table IV lists the projections of two-dimensional con-tours for several choices of parameter pairs. Note thatthe sensitivity to p, is rather weak, but nonzero. Asobserved in the linear analysis of Sec. IIIB, there arealso strong correlations between, e.g. , g& and M~. Theexperimental BNL result of g&

———0.15, assuming nostrange vector form factors, could thus in fact be consis-tent with, e.g. , no axial strangeness, but M~ increasedfrom its nominal value by roughly 7 percent. However,as discussed in Sec. IIIB, M~ is measurable in chargechanging cross sections, and thus in principle is indepen-dently determined.

The values in Table IV are still not reasonable esti-mates for the uncertainties of the extracted parameters,because in fact more than two are currently unknown.Fixing all parameters but p„p„and g&, we construct athree-dimensional contour, and project the edges of the67 percent confidence volume onto the three parameters.This yields, with MA fixed, Ap, = +5, Ap, = +0.3, andDg& +0.08. There remains a strong correlation with

M~, as can be seen by doing the same with g& fixed. Thisyields Lp, = +5, Ap, —+0.3, and AM~ —+0.06 GeV.These spreads are somewhat insensitive to the values in-put to generate the cross sections. The addition of sys-tematic errors from the data will of course further weakenthese limits. Allowing all four parameters to vary yieldsLp, —+6, Lp, = +0.5, Lg~ = +0.7, AM~ —+0.2GeV. However these numbers are overestimates of theuncertainties because M~ is known to better than +0.2GeV from charged-current data. This charged. -currentdata should be included in the fits. Instead, our largenumbers show the difhculty in extracting both M~ andg& using only the neutral-current data. In summary, theextraction of g& in high-energy experiments depends onthe value of M~.

Furthermore, high-energy experiments contain someinformation on p, . We have obtained an error band of+0.3 for p, in a fit neglecting systematic errors and theerror in M~. Including these effects will increase (per-haps double) this error band. Nevertheless, this crude er-ror band is still useful because presently p, is completelyunconstrained.

The use of neutron information (e.g. , with deuterontargets) would help significantly in reducing these errorbands. As an example, assuming M~ is independentlyconstrained, a fictitious data set including n/p cross sec-tion ratios with similar statistical uncertainties, in thesame kinematic regime considered above, reduces the er-ror range for g&, p„and p, by roughly factors of 2.

V. SUMMARY AND CONCLUSIONS

We have calculated neutral-current neutrino-nucleusquasielastic cross sections using a relativistic Fermi gasmodel. With this model we have examined the sensitivityof results to strange quark and axial current parameters.Calculations were presented for the existing BNL exper-iment (which measured relatively high-energy protons).Also we examined possible future experiments at bothhigh and low energies measuring either proton spectra orproton to neutron ratios.

4 —-

0

Note, actually the cross section is sensitive to

~tot gw s 2 (26)

—6—2 —1.5 —1 —0.5 0 0.5 1

Thus if MA is accurately known (from charged-current exper-iments), then the neutral-current data can determine Gz atthe q„of the experiment. It is convenient to parametrize G&as

FIG. 10. Cross section contours versus p, and p, Crosssections are evaluated at Q = 0.75 GeV, averaging overBNL neutrino and antineutrino spectra. Solid curves showneutrino cross sections, dashed curves are antineutrinos. Thecontours show +10 percent variation in the cross section.

gw

(1 —q~/M'2)2 (27)

If M& ——M&, then the experiment can directly determine g&.Nevertheless, the experiment can still determine the combi-nation of gz and Mz which yields the same Gz(q„) even ifM~ g M~.

48 NEUTRINO-NUCLEUS QUASIFREE NEUTRAL CURRENT. . . 3087

Measuring the proton to neutron ratio at low energiesmay substantially reduce nuclear structure uncertaintiesin the extracted value of g&. However, possible correc-tions from RPA, meson exchange currents, or other ef-fects should be examined further before final conclusionscan be drawn. One should also consider measuring thisproton to neutron ratio at higher energies. This mayincrease the precision with which strange quark informa-tion can be extracted. Uncertainties from errors in thedipole mass M~ can be minimized by determining M~from charged-current data. Finally, neutral current neu-trino experiments may provide useful, although crude,limits on the vector strange quark parameter p, .

F(") = 7A„(1 —rl)G/(I+ r),

and

A„= 1.793, A„= —1.913,

rl = (1+5.6r)

Here the anomalous moments are,

(A3)

(A4)

(A5)

(Ao)

(A7)

ACKNOWLEDGMENTS

Vile would like to thank David Griegel for useful discus-sions. This research was supported by U.S. Departmentof Energy under Grant No. DE-FG02-87ER-40365.

This parametrization is good for the neutron form fac-tors provided w &( 1. The strange quark contributionsto the isoscalar are parametrized in terms of p, whichdescribes the strange quark contributions to the anoma-lous moment, and p, which describes the strange quarkcontributions to the electric form factor

APPENDIX

We adopt the form factor parametrization used inRef. [2]. First the electromagnetic form factors are writ-ten in terms of a simple dipole,

Fi = (p. + p, )rG/(I + ~),

F2 = (&. —p, r )G/(1+ 'r).

G = (1 + 4.97'), ~ = q„/4M, — ( 1) The G~ in the axial form factor [See Eqs. (10) and (11)]1s

(A2) GD = (1 —q„/M )~ (A10)

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