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NON-STANDARD PHYSICS AND NUCLEON STRANGENESS

arXiv:hep-ph/9212218v1 3 Dec 1992NON-STANDARD PHYSICS AND NUCLEON STRANGENESSIN LOW-ENERGY PV ELECTRON SCATTERING*M. J. Musolf** and T. W. DonnellyCenter for Theoretical PhysicsLaboratory for Nuclear Scienceand Department of PhysicsMassachusetts Institute of TechnologyCambridge, Massachusetts 02139U.S.A.Submitted to: Zeitschrift f¨ur Physik CCTP#2149September 1992* This work is supported in part by funds provided by the U. S. Department of Energy(D.O.E.) under contract #DE-AC02-76ER03069.

** Present address: Department of Physics, Old Dominion University, Norfolk, VA 23529and Physics Division MS 12H, Continuous Electron Beam Accelerator Facility, NewportNews, VA 23606.0

ABSTRACTContributions from physics beyond the Standard Model, strange quarks in the nucleon,and nuclear structure effects to the left-right asymmetry measured in parity-violating (PV)electron scattering from 12C and the proton are discussed. It is shown how lack of knowl-edge of the distribution of strange quarks in the nucleon, as well as theoretical uncertaintiesassociated with higher-order dispersion amplitudes and nuclear isospin-mixing, enter theextraction of new limits on the electroweak parameters S and T from these PV observables.It is found that a series of elastic PV electron scattering measurements using 4He could sig-nificantly constrain the s-quark electric form factor if other theoretical issues are resolved.Such constraints would reduce the associated form factor uncertainty in the carbon andproton asymmetries below a level needed to permit extraction of interesting low-energyconstraints on S and T from these observables.

For comparison, the much smaller scale ofs-quark contributions to the weak charge measured in atomic PV is quantified. It is likelythat only in the case of heavy muonic atoms could nucleon strangeness enter the weakcharge at an observable level.1

1. IntroductionIt has recently been suggested that measurements of the “left-right” helicity differenceasymmetry (ALR) in parity-violating (PV) elastic electron scattering from 12C nuclei andof the weak charge (QW) in atomic PV experiments using 133Cs are potentially sensitive tocertain extensions of the Standard Model at a significant level.1 In particular, these observ-ables carry a non-negligible dependence on the so-called S-parameter characterizing exten-sions of the Standard Model which involve degenerate multiplets of heavy fermions.2 It isargued that a 1% measurement of ALR(12C) or a 0.7% determination of QW(133Cs) (equiv-alent to a 1% determination of the weak mixing angle) would constrain S to |δS| ≤0.6, asignificant improvement over the present limit of δS = ±2.0 (exp’t) ± 1.1 (th’y) obtainedfrom QW(133Cs).1 The level of systematic precision achieved in the recently completedMIT-Bates measurement of ALR(12C),3 along with prospects for improving statistical pre-cision with longer run times at CEBAF or MIT-Bates, suggest that a 1% ALR(12C) mea-surement could be feasible in the foreseeable future.

Similarly, improvements in atomicstructure calculations4 have reduced the theoretical error in QW(133Cs) to roughly 1%,and the prospects for pushing the experimental uncertainty below this level also appearpromising.5 If such high-precision, low-energy measurements were achieved, the resultantconstraints on non-standard physics would complement those obtainable from measure-ments in the high-energy sector.The latter are generally equally sensitive to both Sand the T-parameter, where the latter characterizes standard model extensions involvingnon-degenerate heavy multiplets.1, 2In this work, we point out the presence of terms in ALR(12C) not considered in Ref. [1]involving nucleon and nuclear structure physics which must be experimentally and/or the-oretically constrained in order to achieve the limits on S suggested above.

Specifically, we2

consider contributions involving the distribution of strange quarks in the nucleon, multi-boson “dispersion corrections” to tree-level electromagnetic (EM) and weak neutral current(NC) amplitudes, and isospin impurities in the nuclear ground state. We show that lackof knowledge of ρs, the dimensionless mean square “strangeness radius”, introduces uncer-tainties into ALR(12C) at a potentially problematic level.

We further show how a series oftwo measurements of ALR for elastic scattering from 4He could constrain ρs sufficiently toreduce the associated uncertainty in ALR(12C) to below 1% . In addition, we observe thatan improved theoretical understanding of dispersion corrections and isospin impurities forscattering from (Jπ, I) = (0+, 0) nuclei is needed in order both to determine ρs at an inter-esting level and to constrain S to the level suggested in Ref.

[1]. For comparison, we alsodiscuss briefly the interplay of nucleon strangeness and non-standard physics in PV elastic⃗ep scattering and atomic PV.

In the former instance, a 10% determination of ALR(⃗ep)at forward-angles could yield low-energy constraints on S and T complementary to thoseobtained from either atomic PV or ALR(12C), if the strangeness radius were constrained tothe same level as appears possible with the aforementioned series of 4He measurements. Adetermination of ρs with PV ⃗ep scattering alone would not be sufficient for this purpose.In contrast, the impact of strangeness on the interpretation of QW(133Cs) is significantlysmaller, down by at least an order of magnitude from the dominant atomic theory uncer-tainties.

Only in the case of PV experiments with heavy muonic atoms might ρs enterat a potentially observable level. Other prospective PV electron scattering experiments– such as elastic scattering from the deuteron or quasielastic scattering – are discussedelsewhere.6−92.

Hadronic neutral current, new physics and strangeness3

The low-energy PV observables of interest here are dominated by the charge (µ = 0)component of the hadronic vector NC. In terms of quark fields, the nuclear vector NCoperator may be written in terms of the isoscalar and isovector EM currents and a strangequark current:6JNCµ= ξI=1VJEMµ(I = 1) +√3ξI=0VJEMµ(I = 0) + ξ(0)V V (s)µ,(1)where V (s)µ= ¯sγµs and the ξV ’s are couplings determined by the underlying electroweakgauge theory.

In writing Eq. (1), we have eliminated terms involving (c, b, t) quarks, sincetheir contributions to nuclear matrix elements of J NCµare suppressed (see below).

In theminimal Standard Model, one hasξ(0)V= −[1 + R(0)V ]√3ξI=0V= −4sin2 θW[1 + RI=0V](2)ξI=1V= 2(1 −2sin2 θW)[1 + RI=1V] ,where sin2 θW is the weak mixing angle and the R(a)Vare higher-order corrections to tree-level electron-nucleus NC amplitudes. In addition, one may define couplings which governthe low-|Q2| NC charge scattering from the neutron and proton:ξpV ≡12[√3ξI=0V+ ξI=1V] = (1 −4sin2 θW)[1 + RpV ]ξnV ≡12[√3ξI=0V−ξI=1V] = −[1 + RnV ] .

(3)At the operator level, the ξV ’s are determined entirely in terms of couplings of the Z0 tothe (u, d, s) quarks, including contributions from radiative corrections within or beyond the4

framework of the Standard Model, both of which may be included in the R(a)V .10 Upon tak-ing nuclear matrix elements of J NCµ , one must include in the R(a)Vadditional contributionsarising from strong interactions between quarks in intermediate states.10, 11 Further con-tributions arising from isospin impurities in the nuclear ground state are discussed below.Corrections owing to neglect of the (c, b, t) quarks in writing Eq. (1) have been estimatedin Ref.

[12] and may be included in the R(a)Vfor a = 0 and I = 0 as R(a)V→R(a)V (ewk)−∆V ,where ∆V ∼10−4. No such corrections enter RI=1V.The motivation for considering PV electron scattering as a probe of new physics maybe seen, for example, by noting the S- and T-dependencies of the R(a)V .

Following Ref. [1],in which MS renormalization was used in computing one-loop electroweak corrections, onehasRI=0V(new) = 0.016S −0.003TRI=1V(new) = −0.014S + 0.017T(4)RpV (new) = −0.206S + 0.152TRnV (new) = 0.0078T.Within the framework of Ref.

[1], a value of the top-quark mass differing from 140 GeVwould also generate a non-zero contribution to T. The different linear combinations ofS and T appearing in Eqs. (4) suggest that a combination of PV electron scatteringexperiments could provide interesting low-energy constraints on these two parameters.One such scenario is illustrated in Fig.

1, where the constraints attainable from a 1%measurement of ALR(12C) and a 10% determination of ξpV from a forward-angle ALR(⃗ep)measurement are shown. For comparison, the present constraints from QW(133Cs) are also5

shown. One expects these constraints to be tightened by a factor of two to three withfuture measurements.13 While QW(133Cs) is effectively independent of T, both ALR(12C)and the forward-angle ⃗ep asymmetry carry a non-negligible dependence on T. Hence, oneor both of the latter could complement the former as a low-energy probe of new physics.Inaddition, one might also consider PV electro-excitation of the ∆(1232) resonance as ameans of extracting RI=1V.

This quantity is relatively more sensitive to T than are RI=0V,R(0)V , and QW(133Cs), so that a determination of the former would further complementany low-energy constraints attained from the latter.14 It is unlikely, however, that theexperimental and theoretical uncertainties associated with ALR(N →∆) will be reducedto the level necessary to make such a measurement relevant as an electroweak test in thenear term.8, 14 Consequently, a combination of PV scattering experiments on 12C and/orthe proton, together with atomic PV, appear to hold the most promise for placing low-energy, semileptonic constraints on new physics. Before such a scenario is realized, however,other hadronic physics dependent terms entering the PV asymmetries must be analyzed.We now consider these additional contributions, focusing first on the simplest case of 12C.3.

PV elastic scattering from carbonIn the limit that the 12C ground state is an eigenstate of strong isospin, matrix el-ements of the isovector component of the current in Eq. (1) vanish.Moreover, sincethis nucleus has zero spin, only monopole matrix elements of the charge operator con-tribute.

In the absence of the strange-quark term in Eq. (1), one has ⟨g.s.

∥ρNC∥g.s.⟩=√3ξI=0V⟨g.s. ∥ρEM∥g.s.⟩, so t hat ALR(12C) ∝⟨g.s.

∥ρNC∥g.s.⟩/⟨g.s. ∥ρEM∥g.s.⟩=√3ξI=0V.In short, the asymmetry becomes independent of the nuclear physics contained in the EM6

and NC matrix elements15, 16 and carries a dependence only on the underlying gaugetheory coupling, ξI=0V. Upon including the strange-quark term one has6, 17ALR(12C) = A0Q2"4sin2 θW(1 + RI=0V) + G(s)E (Q2)GI=0E(1 + R(0)V )#,(5)where A0 = Gµ/(4√2πα) = 8.99 × 10−5GeV−2, Gµ is the Fermi constant measured inmuon decay, Q2 = ω2 −|⃗q|2 ≤0 is the four-momentum transfer squared, and G(s)E (Q2) andGI=0E(Q2) are the Sachs electric form factors18 appearing in single-nucleon matrix elementsof V (s)µand J EMµ(I = 0).

Note that at the one-body level, the strangeness and EM chargedensity operators, ˆρ(s) and ˆρEM(I = 0), respectively, are identical, apart from the singlenucleon form factors which enter multiplicatively. Consequently, any dependence on thenuclear wavefunction cancels from the asymmetry, leaving only the ratio of form factorsin the second term of Eq.

(5). For RI=0Vone hasRI=0V= RI=0V(st’d) + RI=0V(new) −RI=0V(QED) + RI=0V(had) + Γ −∆V,(6)where RI=0V(st’d) are Standard Model electroweak radiative correct ions to tree-level elec-tron quark PV NC amplitudes, RI=0V(new) denote contributions from extensions of theStandard Model as in Eqs.

(4),RI=0V(QED) are QED radiative corrections to the EMamplitude entering the denominator of ALR(12C) (hence, the minus sign in Eq. (6)),RI=0V(had) are strong-interaction hadronic contributions to higher-order electroweak am-plitudes, Γ is a correction due to isospin impurities in the 12C ground state,19 and ∆V is theheavy-quark correction discussed previously.

The correction R(0)Vappearing in the secondterm of Eq. (5) may be written in a similar form.

For fixed top-quark and Higgs masses,7

the RI=0V(st’d) and RI=0V(QED) can be determined unambiguously, up to hadronic uncer-tainties associated with quark loops in the Z0 −γ mixing tensor and two boson-exchange“box” diagrams ( see, e.g., Ref. [11]).Before discussing the remaining terms in Eq.

(6), we note here an additional featureof spin-0 nuclei which simplifies the interpretation of the PV asymmetry. In general, whenworking to one-loop order, one must also include bremsstrahlung contributions to thehelicity-dependent (-independent) cross sections entering the numerator (denominator) ofALR.

These contributions, although not loop corrections, enter the cross section at thesame order in α as one-loop amplitudes and should be formally included in the RI=0V(st’d)and RI=0V(QED).At low momentum transfer, one need only consider bremsstrahlungfrom the scattering electron (Fig. 2), since the target experiences very small recoil and isunlikely to radiate.

The contributions to the EM and EM-NC interference cross sectionsfrom the amplitudes of Fig. 2 aredσbremEM∝|Ma + Mb|2 = MaM ∗a + MaM ∗b + MbM ∗a + MbM ∗b(7a)dσbremINT ∝MaM ∗c + MaM ∗d + MbM ∗c + MbM ∗d + c.c.

,(7b)where the Mi are the amplitudes associated with the diagrams in Fig. 2.

For simplicity, weconsider only the first terms on the right side of Eqs. (7).

The arguments for the remainingterms are similar. For these terms one hasMaM ∗a = (4πα)3Q4˜LEMµν W µνEM(8a)MaM ∗c = −(4πα)2Q2Gµ2√2˜LINTµν W µνINT ,(8b)where the W µν are hadronic tensors formed from products of the hadronic electromagneticand weak neutral currents, and where the ˜Lµν are the corresponding tensors formed from8

the leptonic side of the diagrams in Fig. 2.

The W µν are identical to the tree-level hadronictensors, since the only differences between the diagrams of Fig. 2 and the tree-level graphsinvolve the lepton line.

For the leptonic tensors, one has after averaging over initial andsumming over final states20˜LEMµν = 12[(K′ + q)2 −m2e]−2Trnγλ(/K ′ + /q + me)γµ(1 + γ5/s)(9a)× (/K + me)γν(/K ′ + /q + me)γσ(/K + me)oελεσ˜LINTµν= 12[(K′ + q)2 −m2e]−2Trnγλ(/K ′ + /q + me)γµ× (geV + geAγ5)(1 + γ5/s)(9b)(/K + me)γν(/K ′ + /q + me)γσ(/K + me)oελεσ ,where Kµ (K′µ) are the initial (final) electron momenta, qµ is the momentum of the outgoingphoton having polarization εµ, sµ is the initial electron spin, and geV (geA) are the vector(axial vector) NC couplings of the electron.Taking the electron and radiated photon on-shell (K2 = K′ 2 = m2e, q2 = 0) andworking in the extreme relativistic limit (Ee/me >> 1) for which sµ →(h/me)Kµ, withh being the electron helicity, one hasMaM ∗a = (4πα)3Q41212K′ · q2× Trnγν(/K ′ + /q)γσ/K ′γλ(/K ′ + /q)γµ/KoελεσW µνEM(10a)MaM ∗c = −(4πα)2Q2Gµ2√2h212K′ · q2×−geV Trnγν(/K ′ + /q)γσ/K ′γλ(/K ′ + /q)γµ/Kγ5o(10b)+ geATrnγν(/K ′ + /q)γσ/K ′γλ(/K ′ + /q)γµ/KoελεσW µνINT.9

For elastic scattering from spin-0 nuclei, only the µ = ν = 0 components of the W µνare non-vanishing.Since the trace multiplying geV in Eq. (10b) is anti-symmetric in µand ν, this term does not contribute.

Adding Eqs. (10) to the absolute squares of thecorresponding tree-level amplitudes leads todσtreeEM + dσbremEM∼12(4πα)2Q4Trnγ0/K ′γ0/Ko+(4πα)Trnγ0(/K ′ + /q)γσ/K ′γλ(/K ′ + /q)γ0/KoελεσW 00EM(11a)dσtreeINT + dσbremINT ∼−h2(4πα)Q2Gµ2√2geATrnγ0/K ′γ0/Ko+(4πα)Trnγ0(/K ′ + /q)γσ/K ′γλ(/K ′ + /q)γ0/KoελεσW 00INT.

(11b)Since ALR = (dσ+INT −dσ−INT)/dσEM, and since the quantities inside the square bracketsin Eqs. (11a) and (11b) are identical, they cancel from the asymmetry.

It is straightfor-ward to show that this cancellation occurs even when the remaining terms in Eqs. (7) areincluded.

In short, the bremsstrahlung contributions drop out entirely from ALR, leavingthe expression of Eq. (5) unchanged.

One could, of course, attempt to be more rigorousand integrate bremsstrahlung cross sections over the detector acceptances, etc. In doingso, however, one would only modify the form of the expressions inside the square bracketsin Eqs.

(11) and not change the fact that they are identical in the two equations. Thecancellation of bremsstrahlung contributions to the asymmetry would still obtain in thiscase.

We note that this result does not carry over to nuclei having spin > 0. In the lattercase, ALR receives contributions from the leptonic vector NC (first term on the right sideof Eq.

(10)). There exists no term in dσbremEMto cancel the corresponding contribution fromdσbremINT .10

Returning to the remaining terms in Eq. (6), we emphasize that in contrast to the firstthree terms, the remaining terms are theoretically uncertain, due to the present lack oftractable methods for calculating low-energy strong interaction dynamics from first prin-ciples in QCD.

Of particular concern are multi-boson-exchange dispersion contributionsto RI=0V(had), such as those generated by the diagrams of Fig.3.We note that nei-ther the G(s)E -term of Eq. (5) nor the nuclear, many-body contributions to the dispersioncorrections were included in the discussion of Ref.

[1].We first consider the impact of strangeness on the extraction of S from ALR(12C).To that end, we employ an “extended” Galster parameterization21 for the single-nucleonform factors appearing in Eq. (5): GI=0E=12 [GpE + GnE], GpE = GVD, GnE = −µnτGVDξn,G(s)E= ρsτGVDξs, where µn is the neutron magnetic moment, τ = −Q2/4m2N, GVD =(1 + λVDτ)−2 is the standard dipole form factor appearing in nucleon form factors, andξn,s = (1 + λ(n,s)Eτ)−1 allow for more rapid high-|Q2| fall-offthan that given by the dipoleform factor.

From parity-conserving electron scattering, one has λVD ≈4.97 and λn ≈5.6.21It is possible that G(s)Efalls offmore rapidly at high-|Q2| than the 1/Q4 behavior exhibitedby this parameterization, but for the momentum transfers of interest here,22 this choice issufficient. The parameters ρs and λ(s)Echaracterize the low- and moderate-|Q2| behavior,respectively, of G(s)Eand are presently un-constrained.

Because the nucleon has no netstrangeness, G(s)E must vanish at Q2 = 0 = τ. Hence, like GnE, which also must vanish at thephoton point, the G(s)Ecarries a linear dependence on |Q2| near the photon point.

Whileno experimental information on G(s)Eexists, theoretical predictions for the mean-squarestrangeness radius (of which ρs is a dimensionless version) have been made using differentmodels.21−25, 14 Since these models generally predict qualitatively different behaviors of11

G(s)Eat moderate-|Q2|, we choose the simple and convenient Galster-like parameterizationin which variations in this moderate-|Q2| behavior are characterized by a single parameterλ(s)Eto be constrained by experiment.Under these choices, the strange-quark term in Eq. (5) induces a fractional shift inthe ALR(12C) asymmetry given by∆ALRALR=ρsτξs2sin2 θW[1 −µnτξn](12)neglecting R(0)V .

Taking the average value for ρs predicted in Ref. [22], choosing λ(s)E= λn,and working at the kinematics of the recent MIT-Bates ALR(12C) measurement (τ ≈0.007), Eq.

(12) indicates about a -3% shift in ALR(12C). Any uncertainty in G(s)Eon thisscale would weaken by a factor of three the limits on S predicted in Ref.

[1].From the standpoint of reducing the uncertainty in ALR(12C) Standard Model tests,as well as that of learning about the distribution of strange quarks in the proton, it isclearly desirable to constrain G(s)Eas tightly as possible. To that end, a combination oftwo measurements of ALR on a (0+, 0) target could constrain G(s)Esufficiently to reduce theG(s)E -induced error in a subsequent determination of S from ALR(12C) to below |δS| = 0.6.For this purpose, we consider 4He rather than 12C.

The statistical precision, δALR/ALR,achievable for either nucleus goes as F−1/2, where the figure of merit F = σALR2, withσ being the EM cross section.6 For both nuclei, δALR/ALR displays a succession of localminima as a function of |Q2|, corresponding to successive local maxima in the cross section.Since the relative sensitivity of G(s)Eto ρs and λ(s)Echanges with |Q2|, a measurements ofALR(0+, 0) in the vicinity of different local minima in δALR/ALR would impose somewhatdifferent joint constraints on ρs and λ(s)E . The EM cross section falls offmore gently with12

|Q2| for 4He than for 12C, so that for the former, the first two δALR/ALR minima aremore widely separated in |Q2| than for the latter. Consequently, the constraints on G(s)Eobtainable with two measurements carried out, respectively, at the first two δALR/ALRminima on 4He could be more restrictive than with a similar series involving 12C.To complete this analysis, we consider a combination of two such ALR(4He) experi-ments carried out roughly under conditions that are representative of what could be achiev-able with a moderate solid angle detector at CEBAF: luminosity L = 5 × 1038cm−2s−1,scattering angle θ = 10◦, solid angle ∆Ω= 0.01 steradians, beam polarization Pe = 100%,and run time T = 1000 hours.26 The constraints resulting from these two prospectivemeasurements are shown in Fig.

4. Since nothing at present is know experimentally aboutG(s)E , we assume two different models for illustrative purposes: (A) (|ρs|, λ(s)E ) = (0, λn)and (B) (|ρs|, λ(s)E ) = (2, λn).

The value of |ρs| in model (B) corresponds roughly to theaverage prediction of Ref. [22].

From these results, we find that for model (B), the uncer-tainty remaining in G(s)Eafter the series of 4He measurements would be sufficiently smallto keep the associated error in a lower-|Q2| Standard Model test with either 12C or 4Hebelow 1%. In the case of model (A), even though λ(s)Eis not constrained, the lower-|Q2|measurement appears to keep the G(s)E -induced error in a (0+, 0) Standard Model testbelow 1% , independent of the value of λ(s)E .Before such 4He constraints could be attained or a 1% Standard Model test per-formed, ambiguities associated with dispersion corrections in RI=0V(had) and with theisospin-mixing parameter Γ must be resolved.Turning first to the former, we focuson nuclear many-body contributions to the amplitudes associated with Fig.3.SinceALR(0+, 0) ∼M P VNC(I = 0)/M P CEM(I = 0), where M P VNC(I = 0) (M P CEM(I = 0)) are the13

isoscalar parity-violating (-conserving) scattering amplitudes, and since the dispersion cor-rections enter as O(α) corrections to the tree-level amplitudes, one has RI=0V(disp) ∼RV V ′V(I = 0) −RγγV (I = 0), where RV V ′Vis a dispersion correction to the tree-level Z0-exchange amplitude involving one or more heavy vector bosons and RγγVis the two-photoncorrection to the isoscalar electromagnetic amplitude. Although one might na¨ıvely hopefor some cancellation between these two corrections, the different Q2-dependences carriedby each makes such a possibility unlikely.

Whereas RγγV→0 as |Q2| →0, since the tree-level EM amplitude has a pole at Q2 = 0, RV V ′Vneed not vanish in this limit since thetree-level NC amplitude has a pole at Q2 = M 2Z.Generally speaking, one expects the scale of hadronic contributions to RI=0V(disp) tobe of O(α/4π). Indeed, theoretical estimates of such contributions to the 2-γ, PC, epscattering amplitud e indicate that RγγV (ep)<∼1% at intermediate energies.27, 28 However,experimental information on RγγVsuggests that the dispersion corrections for scatteringfrom nuclei can be significantly larger than the one-body (ep) scale.

Results from therecent MIT-Bates measurement of RγγV (I = 0) for 12C show that this correction couldbe as large as 20% in the first diffraction minimum and several percent in the regionsoutside the minimum where a (0+, 0) Standard Model test or G(s)E -determination mightbe undertaken.29 In the latter regions, the experimental error in RγγV (I = 0) is of thesame order as the correction itself, and the overall level of agreement between these resultsand theoretical calculations30 is rather poor. In short, experimentally and theoreticallyuncertain many-body effects appear to enhance the scale of RγγV (I = 0) to a level which isimportant for the interpretation of ALR(0+, 0).14

In the case of PV amplitudes, no experimental information exists on RV V ′V(I = 0). Itis unlikely that this quantity will be measured directly, so that one must rely on nuclearmodel-dependent theoretical estimates of its scale.

Of particular concern is the Z0 −γdispersion amplitude which, for elementary e −q scattering, contains logarithms involvingthe ratios |M 2Z/s| and |M 2Z/u|, where the scale of the invariant variables s and u is setby the incoming electron momentum and the typical momentum of the quark bound inthe target nucleus.10 This logarithmic scale mismatch suggests that contributions fromlow-energy intermediate states involving hard-to-calculate hadronic collective excitations(e.g., the nuclear giant resonance) could be important. Given the scale of the RγγV (I = 0)results, the discrepancy with theory, and the need for a theoretical estimate of RZγV (I = 0),significant progress in theoretical understanding of many-body contributions to the dis-persion corrections is needed in order to keep the corresponding uncertainty in ALR(0+, 0)below one percent.The quantity Γ(q) (q ≡| ⃗q |) in Eq.

(6) has been introduced to take into account thefact that nuclei such as 4He and 12C are not exact eigenstates of strong isospin with I = 0.Since, the EM interaction does not conserve isospin, one expects states having I ̸= 0 to bepresent as small [O(α)] components in the nuclear ground states. For nuclei whose majorconfigurations involve either the 1s shell (4He) or the 1p shell (12C) the isospin-mixingcorrection Γ(q) is likely to be quite small at low momentum transfer (|Γ|<∼1%).19 Thisspecial situation arises because of the difficulty of supporting isovector breathing modesin the relevant nuclear model spaces; since primarily a single type of radial wave functionplays a role, radial excitations are suppressed.15

We emphasize that this conclusion need not apply to spin-0 nuclei beyond the 1s −1p shell.For nuclei in the 2s −1d shell, for example, one has wave functions whichdisplay different radial distributions (viz., 2s and 1d), making it possible to have importantisovector breathing-mode admixtures introduced into the nuclear ground states. For nucleibeyond 40Ca an additional issue arises.Since in this region the stable 0+ nuclei haveN > Z and, thus, I ̸= 0 from the outset, both isoscalar and isovector matrix elementsof the monopole operators enter (even in the absence of isospin-mixing).

In this case,isospin-mixing effects appear in two ways: (1) several eigenstates of isospin can mix toform the physical states (as above) and (2) the mean fields in which the protons andneutrons in the nucleus move may be slightly different. This latter effect was exploredin Ref.

[19], where it was found that ALR for elastic scattering from 0+ N > Z nuclei israther sensitive to the difference between Rp and Rn, the radii of the proton and neutrondistributions in the nuclear ground state, respectively. The reason for this sensitivity isthat |ξnV | >> |ξpV |, making the NC “charge” densities for the neutron and proton roughlycomparable in magnitude.These observations imply that the extraction of interesting constraints on S, T, andG(s)Efrom measurements of ALR for spin-0 nuclei in this region is likely to be more difficultthan for spin-0 nuclei in the 1s-1p shell.

On the other hand, such measurements couldprovide a new window on certain aspects of nuclear structure. Since the EM charge radiuscan be determined quite precisely using, e.g., parity-conserving (PC) electron scattering,a measurement of ALR would provide a way to determine Rn.

A 1% determination of Rnappears to be achievable. For a nucleus such as 133Cs, with its importance for atomic PV, itmay prove useful to employ electron scattering to explore some of these issues.

The charge16

and neutron distributions could be studied, thereby helping to reduce Rn uncertaintiesappearing in QW(133Cs) (see Eq. (16) below), and some indication concerning the degreeof isospin-mixing [Eq.

(16)] could be obtained.4. PV elastic scattering from the protonAs illustrated in Fig.1, ALR(⃗ep) carries a stronger dependence on T than eitherQW(133Cs) or ALR(12C), so that a measurement of the former, in combination of one orboth of the latter, could provide an interesting set of low-energy constraints on S andT.

Na¨ıvely, one might expect the interpretation of ALR(⃗ep) to be simpler than that ofALR(12C), since one has no many-body nuclear effects to take into account.However,the spin and isospin quantum numbers of the proton allow for the presence of severalform factors in ALR(⃗ep) not appearing in the 12C asymmetry, with the result that theinterpretation of PV ⃗ep scattering is in some respects more involved than that of elasticscattering from (0+, 0) nuclei. A detailed discussion of PV elastic ⃗ep scattering can befound in Refs.

[6, 31, 32], and we focus here solely on scattering in the forward direction.At low momentum transfer and in the forward direction, the ⃗ep asymmetry has theform6ALR(⃗ep) ≈aoτhξpV −nGnE + G(s)E + τµp(GnM + G(s)M )oi+ O(τ 2) ,(13)where ao ≈3 × 10−4.The first term on the right side of Eq. (13) (containing ξpV ) isnominally independent of hadronic physics for essentially the same reasons as is the firstterm in the carbon asymmetry of Eq.

(5). The terms contained inside the curly bracketsall enter at O(τ), since both GnE and G(s)Evanish at the photon point.

From Eq. (13) onesees immediately the additional complexity of the proton asymmetry in comparison withthat of carbon.

The neutron EM form factors appear in ALR(⃗ep), since the isovector and17

isoscalar EM currents enter the hadronic neutral current (Eq. (1)) with different weightingsthan in the hadronic EM current.

The presence of these form factors introduces one sourceof uncertainty not present at the same level in ALR(12C). In addition, both the electric andmagnetic strangeness form factors contribute at O(τ), and their presence also complicatesthe interpretation of the asymmetry.As in the case of ALR(12C), the τ-dependence of the terms in Eq.

(13) suggests atwo-fold strategy of measurements: (a) a very low-τ measurement to determine ξpV , withan eye to obtaining the constraints indicated in Fig. 1, and (b) a moderate-τ measurementaimed at constraining the linear combination of form factors appearing in the second termof Eq.

(13).The second of these measurements could be of interest for a number ofreasons: to extract limits on the strangeness form factors, to constrain G(s)Efor purposesof interpreting ALR(12C) as a Standard Model test, or to constrain this term for the samepurpose but with a very low-τ ALR(⃗ep) measurement. Considering first scenario (a), wenote that it is not possible to perform a Standard Model test at arbitrarily low-τ, sincethe statistical uncertainty increases for decreasing momentum transfer.

For purposes ofillustration, then, we analyze a prospective measurement at the limits of τ and forwardscattering angle expected to be achievable at CEBAF Hall C. In order to achieve the 10%statistical uncertainty needed for the constraints in Fig. 1, a 1000 hour experiment wouldbe needed, assuming 100% beam polarization.

Under these conditions, the impact of formfactor uncertainties on a determination of ξpV is non-negligible. The dominant uncertaintyis introduced by G(s)E .

An uncertainty in the strangeness radius of δρs = ±2 (correspondingto the magnitude of the prediction in Ref. [22]) would induce nearly a 30% uncertainty inthe extracted value of ξpV , a factor of three greater than the uncertainty assumed in Fig.

1.18

Similarly, an uncertainty in the value of µs of ±0.3, also corresponding to the magnitudeof the prediction in Ref. [22], would generate roughly a 20% error in ξpV .These statements point to the need for better constraints on the strangeness formfactors if an interesting Standard Model test is to be performed with PV ⃗ep scattering.Turning, then, to strategy (b), we consider the constraints one might place on these formfactors with a moderate-τ ALR(⃗ep) measurement.The difficulty here is that it is notpossible to separate the form factors with ⃗ep scattering alone.

As discussed in Ref. [6], a“perfect” backward-angle ALR(⃗ep) measurement (0% experimental error) might ultimatelyallow a determination of µs with an error of ±0.12, thereby reducing the µs-induced un-certainty in a forward-angle Standard Model test below a problematic level.

A subsequentdetermination of the second term in Eq. (13) might then allow a determination of G(s)E .We show in Fig.

4 the constraints in (ρs, λ(s)E ) space such a measurement might achieve,assuming experimental conditions similar to those of recent CEBAF proposals.33−35 Wenote that these constraints would not be sufficient to permit either a 10% determination ofξpV from a low-τ ALR(⃗ep) measurement or a 1% Standard Model test with elastic scatteringfrom 12C. In the former case, the G(s)E -induced uncertainty in ξpV would still be on the orderof 20% .

In fact, as Fig. 4 illustrates, it appears that a series of ALR(4He) measurementscould place far more stringent limits on G(s)Ethan appears possible with PV ⃗ep scatteringalone.

Indeed, these limits would be sufficient to permit one to probe new physics withboth ALR(⃗ep) and ALR(12C) at the level assumed in Fig. 1.5.

Atomic PVOne should expect the impact of form factor uncertainties on the interpretation ofQW to be considerably smaller than for electron scattering asymmetries, due to the very19

small effective momentum-transfer associated with the interaction of an atomic electronwith the nucleus. Below, we quantify this statement with regard to the strangeness formfactors, and note that only in the case of PV experiments with heavy muonic atomsmight nucleon strangeness contribute at an observable level.

To that end, consider the PVatomic hamiltonian which induces mixing of opposite-parity atomic states and leads to thepresence of QW-dependent atomic PV observables:ˆHatomP V= Gµ2√2Zd3x ˆψ†e(⃗x)γ5 ˆψe(⃗x)ρNC(⃗x) + · · ·,(14)where ˆψe(⃗x) is the electron field and ρNC(⃗x) is the Fourier Transform of ρNC(⃗q), the matrixelement of the charge component of Eq. (1).

For simplicity, we have omitted terms involv-ing the spatial components of the nuclear vector NC as well as the nuclear axial vector NC.For a heavy atom, the leading term in Eq. (14) is significantly enhanced relative to the re-maining terms by the coherent behavior of the nuclear charge operator.

Consequently, onetypically ignores the contribution from all magnetic form factors. Following Ref.

[36], wewrite the matrix element of the leading term in ˆHatomP Vbetween atomic S1/2 and P1/2 statesin the form ⟨P| ˆψ†e(⃗x)γ5 ˆψe(⃗x)|S⟩= N Csp(Z)f(x), where N is a known overall normaliza-tion, Csp(Z) is an atomic structure-depende nt function, and f(x) = 1 −12(x/xo)2 + · · ·gives the spatial-dependence of the electron axial charge density. In a simple model wherea charge-Z nucleus is taken as a sphere of constant electric charge density out to radius R,one has xo = R/Zα neglecting small corrections involving the electron mass.

In this case,atomic matrix elements of Eq. (14) become⟨P| ˆHatomP V|S⟩= Gµ2√2N Csp(Z)hQ(0)W + ∆Q(n, p)W+ ∆Q(s)W + ∆Q(I)Wi+ · · ·,(15)20

whereQ(0)W =Z −N2ξI=1V+√3Z + N2ξI=0V(16a)∆Q(n, p)W= 12√3ξI=0V+ ξI=1V⟨I0 ∥AXk=112[1 + τ3(k)]h(xk)∥I0⟩+ 12√3ξI=0V−ξI=1V⟨I0 ∥AXk=112[1 −τ3(k)]h(xk)∥I0⟩(16b)∆Q(s)W = −ξ(0)V ρs4m2N⟨I0 ∥AXk=1∇2kh(xk)∥I0⟩(16c)∆Q(I)W = λξI=1Vh⟨I0 ∥AXk=1h(xk)τ3(k)∥I1⟩+ (I1 ↔I0)i+ · · ·,(16d)with h(x) = f(x) −1, and with ⟨I0 ∥ˆO∥I0⟩denoting reduced matrix elements of a nuclearoperator ˆO in a nuclear ground state having nominal isospin I0. The terms in Eq.

(16a)are those usually considered in analyses of QW. The term ∆Q(n, p)Wcarries a dependenceon the ground-state neutron radius, Rn.

The impact of uncertainties in Rn on the use ofQW for high-precision electroweak tests has been discussed in Refs. [36, 37].

Eqs. (16c)and (16d) give, respectively, the leading contributions to QW from G(s)Eand from isospinimpurities in the nuclear ground state.

In arriving at Eq. (16), we have kept terms in f(x)only up through quadratic order and employed R = roA1/3, ro ≈1 fm, for the nuclearradius.

We have shown explicitly only the contribution to ∆Q(I)W arising from the mixingof a single state of isospin I1 into the ground state of nominal isospin I0 with strengthλ. Additional contributions to QW arising from the single-nucleon EM charge radii arediscussed elsewhere.37According to Ref.

[1], neglect of all but Eq. (16a) leads to the prediction QW(133Cs) =−73.20−0.8 S −0.005 T, so that a 0.7% determination of QW(133Cs) would constrain S to21

|δS| ≤0.6. As noted in Ref.

[36], a 10% uncertainty in Rn would generate a 0.7% error inQW(133Cs). While hadron-nucleus scattering typically permits a 5 - 10% determination ofRn for heavy nuclei,19, 36 no experimental information on Rn for cesium isotopes presentlyexists.

A series of PC and PV electron scattering experiments on 133Cs could determineits neutron radius to roughly 1% accuracy.6 In the meantime, one must rely on nuclearmodel calculations of Rn. The scale of the associated theoretical uncertainty in QW(133Cs)is presently the subject of debate.37From Eq.

(16c), we find that an uncertainty in the strangeness radius induces an errorin the weak charge of δQW(133Cs) = −0.025δρs. For δρs on the order of the average valueof Ref.

[22], the corresponding uncertainty in QW(133Cs) is slightly less than 0.1% , morethan an order of magnitude below the dominant theoretical error associated with atomicstructure1, 4 and well below the level needed for an interesting QW(133Cs) Standard Modeltest. As expected, the situation differs sharply from that of PV electron scattering.

Indeed,a measurement of ALR(12C) would have to be carried out at | ⃗q | ≈30 MeV/c — roughlyan order of magnitude smaller than in the experiment of Ref. [3] – to be equally insensitiveto G(s)E .We close with observations on the possibility of observing G(s)Eusing PV experimentson muonic atoms.

It has been noted recently that 1 - 10% measurements of PV observ-ables for muonic boron may be feasible in the future at PSI.13, 38 Since the ratio of Bohrradii ae0/aµ0 = me/mµ ∼207, the muon in these atoms is more tightly bound for a givenset of radial and angular momentum quantum numbers. One might expect, then, an en-hanced sensitivity to short-range contributions to QW, such as those associated with Rn22

or ρs. To analyze the latter possibility, we solve the Dirac equation for a muon orbit-ing a spherically-symmetric nuclear charge distribution, keeping terms involving mµ.39The result of this procedure is to make the replacement xo = R/Zα →[3R/4mµZα]1/2in the function h(x) in Eq.

(16).The scale of ∆Q(s)Wis correspondingly enhanced by4mµR/3Zα ∼4mµroA1/3/3Zα over its magnitude for an electronic atom. In the case of133Cs, this enhancement factor is ≈8, making QW(µCs) roughly as sensitive to ρs as isALR(⃗ep).

The sensitivity of ∆Q(s)Wfor a muonic lead atom is roughly two times greaterthan ∆Q(s)W (µCs).For light muonic atoms, on the other hand, the ρs contribution isstill suppressed. In the case of muonic boron, for example, uncertainties associated withρs would not enter the parameters ξpV and ξnV at an observable level.

Consequently, onemust go to heavy muonic atoms. While the sensitivity of the latter to Rn-uncertaintiesis also enhanced, these uncertainties could be reduced through a combination of PC andPV elastic electron scattering experiments.6, 19 Given the simplicity of atomic structurecalculations for muonic Cs or Pb (essentially a one-lepton problem), the theoretical atomicstructure uncertainties entering QW-determinations should not enter at a level problematicfor G(s)Edeterminations.

Thus, an experiment of this type could complement PV electronscattering as a probe of strange quarks in the nucleon.The remaining obstacle is theexperimental one of achieving sufficient precision. To this end, it would be desirable tofind a heavy muonium transition for which the PV signal is enhanced by accidental neardegeneracies between opposite-parity atomic levels.6.

ConclusionsWith any attempt at a precision electroweak test involving a low-energy hadronicsystem, one must ensure that all sources of theoretical hadronic physics uncertainties fall23

below the requisite level. The situation contrasts with purely leptonic or high-energy elec-troweak tests.

In the former case, given a model of electroweak interactions, one can makeprecise and unambiguous predictions for different observables, up to uncertainties associ-ated with unknown parameters (e.g., mt and MH) and with hadronic loops. In the latterinstance, strong-interaction uncertainties are controllable through the use of a perturba-tive expansion and QCD.

In the non-perturbative low-energy regime, however, one mustrely on the use of symmetries as well as model estimates of, or independent experimentalconstraints on, hadronic effects. The scale of uncertainty in a low-energy semi-leptonicelectroweak test, then, is set by experimental input and, where such is lacking, any rea-sonable model estimate.

In the foregoing discussion, we have noted that completion of oneor more PV electron scattering experiments has the potential to complement atomic PVas a low-energy probe of new physics. At present, however, experimental limits on nucleardispersion corrections, as well as theoretical predictions for the nucleon’s strangeness formfactors, indicate that these two sources of hadronic physics uncertainty are too large tomake interesting electroweak tests possible with low-energy polarized electrons.

We haveshown how a series of PV elastic scattering experiments with 4He could reduce the uncer-tainty associated with the strangeness radius below a problematic level. Achieving a betterunderstanding of nuclear dispersion corrections remains a challenge for both experimentand theory.ACKNOWLEDGEMENTSIt is a pleasure to thank E.J.

Beise, S.B. Kowalski, S.J.

Pollock, and L. Wilets foruseful discussions.24

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These conditions differ somewhat from those expected to be attainable at CEBAF.The Hall A spectrometers would detect electrons at θ = 12.5◦with a solid angle of∆Ω= 0.016 sr. In an estimate of the statistical precision, the smaller solid angle iscompensated by an increase in the figure of merit in the more forward direction.

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FIGURE CAPTIONSFig. 1.

Present and prospective constraints on S, T parameterization of non-standard physicsfrom low- and intermediate-energy PV observables. Short-dashed lines give presentconstraints from cesium atomic PV.1, 4, 5 Solid lines give constraints from a 1%ALR(12C) measurement.

Long-dashed lines correspond to a 10% determination of ξpVfrom a forward-angle measurement of ALR(⃗ep). For simplicity, it is assumed that allexperiments agree on common central values for S and T, so that only the deviationsfrom these values are plotted.Fig.

2. Electron bremsstrahlung for electromagnetic (Fig.

2a,b) and weak neutral current(Fig. 2c,d) scattering from a hadronic target.

Target bremsstrahlung is assumed tobe negligible for low-energy (small recoil) processes.Fig. 3.

Dispersion corrections to tree-level EM and NC electron-nucleus scattering ampli-tudes. Here, V, V ′ are any one of the Z0, W ±, γ vector bosons and |i⟩(|f⟩) are initial(final) nuclear states.Fig.

4. Constraints imposed on G(s)Efrom prospective PV elastic scattering experiments.Dashed-dot curves and solid curves give, respectively, constraints from possible low-and moderate-|Q2| measurements of ALR(4He).

Dashed lines give constraints fromseries of forward- and backward-angle ALR(⃗ep) measurements.Panels (a) and (b)correspond to two models for G(s)Ediscussed in the text, where the canonical valuesof (|ρs|, λ(s)E ) are indicated by the large dot.28

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