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Stranger Still: Kaon Loops and Strange Quark

arXiv:hep-ph/9301229v1 11 Jan 1993Stranger Still: Kaon Loops and Strange QuarkMatrix Elements of the Nucleon*M. J. MusolfDepartment of PhysicsOld Dominion UniversityNorfolk, VA23529U.S.A.andCEBAF Theory Group, MS-12H12000 Jefferson Ave.Newport News, VA23606U.S.A.andM. BurkardtCenter for Theoretical PhysicsLaboratory for Nuclear Scienceand Department of PhysicsMassachusetts Institute of TechnologyCambridge, Massachusetts 02139U.S.A.CEBAF #TH-93-01January, 1993* This work is supported in part by funds provided by the U. S. Department of Energy(D.O.E.) under contracts #DE-AC02-76ER03069 and #DE-AC05-84ER40150.0

ABSTRACTIntrinsic strangeness contributions to low-energy strange quark matrix elements ofthe nucleon are modelled using kaon loops and meson-nucleon vertex functions taken fromnucleon-nucleon and nucleon-hyperon scattering. A comparison with pion loop contribu-tions to the nucleon electromagnetic (EM) form factors indicates the presence of significantSU(3)-breaking in the mean-square charge radii.

As a numerical consequence, the kaonloop contribution to the mean square Dirac strangeness radius is significantly smaller thancould be observed with parity-violating elastic ⃗ep experiments planned for CEBAF, whilethe contribution to the Sachs radius is large enough to be observed with PV electron scat-tering from (0+, 0) nuclei. Kaon loops generate a strange magnetic moment of the samescale as the isoscalar EM magnetic moment and a strange axial vector form factor havingroughly one-third magnitude extracted from νp/¯νp elastic scattering.

In the chiral limit,the loop contribution to the fraction of the nucleon’s scalar density arising from strangequarks has roughly the same magnitude as the value extracted from analyses of ΣπN. Theimportance of satisfying the Ward-Takahashi Identity, not obeyed by previous calculations,is also illustrated, and the sensitivity of results to input parameters is analyzed.1

1. IntroductionExtractions of the strange quark scalar density from ΣπN [1,2], the strange quark axialvector form factor from elastic νp/¯νp cross section measurements [3], and the strange-quarkcontribution to the proton spin ∆s from the EMC measurement of the g1 sum [4], suggest aneed to account explicitly for the presence of strange quarks in the nucleon in describing itslow- energy properties.

These analyses have motivated suggestions for measuring strangequark vector current matrix elements of the nucleon with semi-leptonic neutral currentscattering [5]. The goal of the SAMPLE experiment presently underway at MIT-Bates [6]is to constrain the strange quark magnetic form factor at low-|q2|, and three experimentshave been planned and/or proposed at CEBAF with the objective of constraining thenucleon’s mean square “strangeness radius” [7-9].

In addition, a new determination of thestrange quark axial vector form factor at significantly lower-|q2| than was obtained fromthe experiment of Ref. [3] is expected from LSND experiment at LAMPF [10].At first glance, the existence of non-negligible low-energy strange quark matrix el-ements of the nucleon is rather surprising, especially in light of the success with whichconstituent quark models account for other low-energy properties of the nucleon and itsexcited states.

Theoretically, one might attempt to understand the possibility of largestrange matrix elements from two perspectives associated, respectively, with the high- andlow-momentum components of a virtual s¯s pair in the nucleon. Contributions from thehigh-momentum component may be viewed as “extrinsic” to the nucleon’s wavefunction,since the lifetime of the virtual pair at high-momentum scales is shorter than the inter-action time associated with the formation of hadronic states [11].

In an effective theoryframework, the extrinsic, high-momentum contributions renormalize operators involvingexplicitly only light-quark degrees of freedom [5, 12]. At low-momentum scales, a virtualpair lives a sufficiently long time to permit formation strange hadronic components (e.g.,a KΛ pair) of the nucleon wavefunction [13].

While this division between “extrinsic” and“intrinsic” strangeness is not rigorous, it does provide a qualitative picture which suggestsdifferent approaches to estimating nucleon strange matrix elements.In this note we consider intrinsic strangeness contributions to the matrix elements⟨N|¯sΓs|N⟩(Γ = 1, γµ, γµγ5) arising from kaon-strange baryon loops. Our calculation isintended to complement pole [14] and Skyrme [15] model estimates as well as to quantify thesimple picture of nucleon strangeness arising from a kaon cloud.

Although loop estimateshave been carried out previously [16, 17], ours differs from others in several respects.First, we assume that nucleon electromagnetic (EM) and and weak neutral current (NC)2

form factors receive contributions from a variety of sources (e.g., loops and poles), so wemake no attempt to adjust the input parameters to reproduce known form factors (e.g.,GnE). Rather, we take our inputs from independent sources, such as fits to baryon-baryonscattering and quark model estimates where needed.

We compute pion loop contributionsto the nucleon’s EM form factors using these input parameters and compare with theexperimental values. Such a comparison provides an indication of the extent to whichloops account for nucleon form factor physics generally and strangeness form factors inparticular.

Second, we employ hadronic form factors at the meson nucleon vertices andintroduce “seagull” terms in order to satisfy the Ward-Takahashi (WT) Identity in thevector current sector.Previous loop calculations employed either a momentum cut-offin the loop integral [16] or meson-baryon form factor [17] but did not satisfy the WTIdentity. We find that failure to satisfy the requirements of gauge invariance at this levelcan significantly alter one’s results.

Finally, we include an estimate of the strange quarkscalar density which was not included in previous work.2. The calculation.The loop diagrams which we compute are shown in Fig.

1. In the case of vectorcurrent matrix elements, all four diagrams contribute, including the two seagull graphs(Fig.

1c,d) required by gauge invariance. For axial vector matrix elements, only the loopof Fig.

1a contributes, since ⟨M|Jµ5|M⟩≡0 for M a pseudoscalar meson. The loops 1aand 1b contribute to ⟨N|¯ss|N⟩.

In a world of point hadrons satisfying SU(3) symmetry,the coupling of the lowest baryon and meson octets is given byiLBBM = D Tr[(B ¯B + ¯BB)M] + F Tr[(B ¯B −¯BB)M](1)where√2B = Pa ψaλa and√2M = Pa φaλa give the octet of baryon and meson fields,respectively, and where D + F =√2gπNN = 19.025 and D/F = 1.5 according to Ref. [18].Under this parameterization, one has gNΣK/gNΛK =√3(F −D)/(D +3F) ≈−1/5, so thatloops having an intermediate KΣ state ought to be generically suppressed by a factor of∼25 with respect to KΛ loops.

Analyses of K + N “strangeness exchange” reactions,however, suggest a serious violation of this SU(3) prediction [19], and imply that neglectof KΣ loops is not necessarily justified. Nevertheless, we consider only KΛ loops since weare interested primarily in arriving at the order of magnitude and qualitative features ofloop contributions and not definitive numerical predictions.With point hadron vertices, power counting implies that loop contributions to themean square charge radius and magnetic moment are U.V.

finite. In fact, the pion loop3

contributions to the nucleon’s EM charge radius and magnetic moment have been computedpreviously in the limit of point hadron vertices [20]. Loop contributions to axial vectorand scalar density matrix elements, however, are U.V.

divergent, necessitating use of acut-offprocedure. To this end, we employ form factors at the meson-nucleon vertices usedin determination of the Bonn potential from BB′ scattering (B and B′ are members ofthe lowest-lying baryon octet) [18]:gNNMγ5 −→gNNMF(k2)γ5,(2)whereF(k2) =hm2 −Λ2k2 −Λ2i,(3)with m and k being the mass and momentum, respectively, of the meson.

The Bonn valuesfor cut-offΛ are typically in the range of 1 to 2 GeV. We note that this form reproducesthe point hadron coupling for on-shell mesons (F(m2) = 1).

An artifact of this choice isthe vanishing of the form factors (and all loop amplitudes) for Λ = m. Consequently, whenanalyzing the Λ-dependence of our results below, we exclude the region about Λ = m asunphysical.For Λ →∞(point hadrons), the total contribution from diagrams 1a and 1b to vectorcurrent matrix elements satisfies the WT Identity qµΛ(p, p′)µ = Q[Σ(p′) −Σ(p)], where Qis the nucleon charge associated with the corresponding vector current (EM, strangeness,baryon number, etc). For finite Λ, however, this identity is not satisfied by diagrams 1a+1balone; inclusion of seagull diagrams (1c,d) is required in order to preserve it.

To arriveat the appropriate seagull vertices, we treat the momentum-space meson-nucleon vertexfunctions as arriving from a phenomenological lagrangianiLBBM −→gBBM ¯ψγ5ψF(−∂2)φ,(4)where ψ and φ are baryon and meson fields, respectively. The gauge invariance of thislagrangian can be maintained via minimal substitution.

We replace the derivatives in thed’Alembertian by covariant derivatives, expand F(−D2) in a power series, identify theterms linear in the gauge field, express the resulting series in closed form, and convertback to momentum space. With our choice for F(k2), this prodecure leads to the seagullvertex−igBBMQF(k2)h(q ± 2k)µ(q ± k)2 −Λ2i(5)4

where q is the momentum of the gauge boson and where the upper (lower) sign corre-sponds to an incoming (outgoing) meson of charge Q (details of this procedure are givenin Ref. [21]).

With these vertices in diagrams 1c,d, the WT Identity in the presence ofmeson nucleon form factors is restored. We note that this prescription for satisfying gaugeinvariance is not unique; the specific underlying dynamics which give rise to F(k2) couldgenerate additional seagull terms whose contributions independently satisfy the WT Iden-tity.

Indeed, different models of hadron structure may lead to meson-baryon form factorshaving a different form than our choice. However, for the purposes of our calculation,whose spirit is to arrive at order of magnitude esitmates and qualitative features, the useof the Bonn form factor plus minimal substitution is sufficient.The strange vector, axial vector, and scalar density couplings to the intermediatehardrons can be obtained with varying degrees of model-dependence.

Since we are inter-ested only in the leading q2-behavior of the nucleon matrix elements as generated by theloops, we emply point couplings to the intermediate meson and baryon. For the vector cur-rents, one has ⟨Λ(p′)|¯sγµs|Λ(p)⟩= fV ¯U(p′)γµU(p) and ⟨K0(p′)|¯sγµs|K0(p)⟩= ˜fV (p + p′)µwith fV = −˜fV = 1 in a convention where the s-quark has strangeness charge +1.

Thevector couplings are determined simply by the net strangeness of the hadron, independentof the details of any hadron model.In the case of the axial vector, only the baryon coupling is required since pseudoscalarmesons have no diagonal axial vector matrix elements. Our approach in this case is touse a quark model to relate the “bare” strange axial vector coupling to the Λ to the bareisovector axial vector matrix element of the nucleon, where by “bare” we mean that theeffect of meson loops has not been included.

We then compute the loop contributions tothe ratioηs = G(s)A (0)gA,(6)where G(s)A (q2) is the strange quark axial vector form factor (see Eq. (10) below) andgA = 1.262 [22] is the proton’s isovector axial vector form factor at zero momen-tum transfer.Writing ⟨Λ(p′)|¯sγµγ5s|Λ(p)⟩= f 0A ¯U(p′)γµγ5U(p), one has in the quarkmodel f 0A =Rd3x(u2 −13ℓ2), where u (ℓ) are the upper (lower) components of a quarkin its lowest energy configuration [23, 24, 25].The quark model also predicts thatg0A ≡53Rd3x(u2 −13ℓ2).

In the present calculation, we take the baryon octet to be SU(3)symmetric (e.g., mN = mΛ), so that the quark wavefunctions are the same for the nucleon5

and Λ.† In this case, one has f 0A =35g0A. This relation holds in both the relativistic quarkmodel and the simplest non-relativistic quark model in which one simply drops the lowercomponent contributions to the quark model integrals.

We will make the further assump-tion that pseudoscalar meson loops generate the dominant renormalization of the bare axialcouplings. The Λ has no isovector axial vector matrix element, while loops involving KΣintermediate states are suppressed in the SU(3) limit as noted earlier.

Under this assump-tion, then, only the πN loop renormalizes the bare coupling, so that gA = g0A[1 + ∆πA(Λ)],where ∆πA(Λ) gives the contribution from the πN loop with the bare coupling to the in-termediate nucleon scaled out. In this case, the ratio ηs is essentially independent of theactual numerical predictions for f 0A and g0A in a given quark model; only the spin-flavorfactor35 which relates the two enters.For the scalar density,we require point couplings to both the intermediatebaryon and meson.We write these couplings as ⟨B(p′)|¯qq|B(p)⟩= f 0S ¯U(p′)U(p) and⟨M(p′)|¯qq|M(p)⟩= γM, where B and M denote the meson and baryon, respectively.

Ourchoices for f 0s and γM carry the most hadron model-dependence of all our input couplings.To reduce the impact of this model-dependence on our result, we again compute loopcontributions to a ratio, namely,Rs ≡⟨N|¯ss|N⟩⟨N|¯uu + ¯dd + ¯ss|N⟩. (7)In the language of Ref.

[2], on has Rs = y/(2 + y). Our aim in the present work is tocompute Rs in a manner as free as possible from the assumptions made in extracting thisquantity from standard Σ-term analyses.

We therefore use the quark model to computef 0S and γM rather than obtaining these parameters from a chiral SU(3) fit to hadron masssplittings. We explore this alternative procedure, along with the effects of SU(3)-breakingin the baryon octet, elsewhere [21].In the limit of good SU(3) symmetry for the baryon octet, the bare ¯ss matrix elementof the Λ is f 0S =Rd3x(u2−ℓ2).

Using the wavefunction normalization conditionRd3x(u2+ℓ2) = 1, together with the quark model expression for g0A, leads tof 0S = 12( 95g0A −1). (8)† We investigate the consequences of SU(3)-breaking in the baryon octet in a forthcom-ing publication [21].6

Neglecting loops, one has ⟨N|¯uu+ ¯dd+¯ss|N⟩= 3f 0S. We include loop contributions to boththe numerator and denominator of Eq.

(7). Although the latter turn out to be numericallyunimportant, their inclusion guarantees that Rs is finite in the chiral limit.Were the contribution from Fig.

1a dominant, the loop estimate of Rs would be nearlyindependent of f 0S. The contribution from Fig.

1b, however, turns out to have comparablemagnitude. Consequently, we are unable to minimize the hadron model-dependence in ourestimate of Rs to the same extent we are able with ηs and the vector current form factors.To arrive at a value for f 0S, we consider three alternatives: (A) Compute f 0S using the MITbag model value for g0A and use a cut-offΛ ∼ΛBonn in the meson-baryon form factor.

Thisscenario suffers from the conceptual ambiguity that the Bonn value for the cut-offmass inF(k2) allows for virtual mesons of wavelength smaller than the bag radius. (B) Computef 0S as in (A) and take the cut-offΛ ∼1/Rbag.

This approach follows in the spirit of so-called “chiral quark models”, such as that used in the calculation of Ref. [17], which assumethe virtual pseudoscalar mesons are Goldstone bosons that live only outside the bag andcouple directly to the quarks at the bag surface.

While conceptually more satisfying than(A), this choice leads to a form factor F(k2) inconsistent with BB′ scattering data. (C)First, determine g0A assuming pion-loop dominance in the isovector axial form factor, i.e.,gA = g0A[1+∆πA(Λ)].

Second, use this value of g0A to determine f 0S via Eq. (8).

Surprisingly,this procedure yields a value for g0A very close to the MIT bag model value for Λ ∼ΛBonnrather than Λ ∼1/Rbag as one might naively expect.Since all three of these scenarios are consistent with the bag estimate for g0A (andrenormalization constant, Z as discussed below) we follow the “improved bag” procedureof Ref. [24] to obtain γM.

The latter gives γM ≡⟨M(p′)|¯qq|M(p)⟩= 1.4/R, where R is thebag radius for meson M. Using Rπ ≈RK ≈3.5 GeV−1, one has γπ ≈γK ≈0.4 GeV. Thisprocedure involves a certain degree of theoretical uncertainty.

The estimate for γK (γπ)is obtained by expanding the bag energy to leading non-trivial order in mK and ms (mπand mu,d), and we have at present no estimate of the corrections induced by higher-orderterms in these masses.Using the above couplings, we compute the kaon loop contributions to the strangequark scalar density as well as vector and axial vector form factors. The latter are definedas⟨N(p′)|Jµ(0)|N(p)⟩= ¯U(p′)hF1γµ + iF22mNσµνqνiU(p)(9)⟨N(p′)|Jµ5(0)|N(p)⟩= ¯U(p′)hGAγµ + GPqµmNiγ5U(p)(10)7

where Jµ is either the EM or strange quark vector current and Jµ5 is the strange axialvector current. The induced pseudoscalar form factor, GP, does not enter semi-leptonicneutral current scattering processes at an observable level, so we do not discuss it here.In the case of the EM current, pion loop contributions to the neutron form factors arisefrom the same set of diagrams as contribute to the strange vector current matrix elementsbut with the replacements K0 →π−, Λ →n, and gNΛK →√2gπNN.

For the proton, onehas a π0 in Fig. 1a and a π+ in Figs.

1b-d. We quote results for both Dirac and Pauliform factors as well as for the Sachs electric and magnetic form factors [26], defined asGE = F1 −τF2 and GM = F1 + F2, respectively, where τ = −q2/4m2N and q2 = ω2 −|⃗q|2.We define the magnetic moment as µ = GM(0) and dimensionless mean square Sachs andDirac charge radii (EM or strange) asρsachs = dGE(τ)dττ=0(11)ρdirac = dF1(τ)dττ=0. (12)The dimensionless radii are related to the dimensionfull mean square radii by < r2 >sachs=6 dGE/dq2 = −(3/2m2N)ρsachs and similarly for the corresponding Dirac quantities.

Fromthese definitions, one has ρdirac = ρsachs + µ. To set the scales, we note that the Sachs EMcharge radius of the neutron is ρsachsn≈−µn, corresponding to an < r2n >sachs of about-0.13 fm2.

Its Dirac EM charge radius, on the other hand, is nearly zero. We note also thatit is the Sachs, rather than the Dirac, mean square radius which characterizes the spatialdistribution of the corresponding charge inside the nucleon, since it is the combinationF1 −τF2 which arises naturally in a non-relativisitc expansion of the time component ofEq.

(9).3. Results and discussionOur results are shown in Fig.

2, where we plot the different strange matrix elements asa function of the form factor cut-off, Λ. Although we quote results in Table I correspondingto the Bonn fit values for Λ, we show the Λ-dependence away from ΛBonn to indicatethe sensitivity to the cut-off.In each case, we plot two sets of curves corresponding,respectively, to m = mK and m = mπ, in order to illustrate the dependence on mesonmass as well as to show the pion loop contributions to the EM form factors.

The dashedcurves for the mean square radius and magnetic moment give the values for Λ →∞,corresponding to the point hadron calculation of Ref. [20].

We reiterate that the zeroes8

arising at Λ = m are an unphysical artifact of our choice of nucleon-meson form factor,and one should not draw conclusions from the curves in the vicinity of these points.In order to interpret our results, it is useful to refer to the analytic expressions forthe loops in various limits. The full analytic expressions will appear in a forthcomingpublication [21].In the case of the vector current form factors, the use of monopolemeson-nucleon form factor plus minimal substitution leads to the result thatF(i) = F point(i)(m2) −F point(i)(Λ2) + (Λ2 −m2) ddΛ2 F point(i)(Λ2),(13)where F point(i)(m2) is the point hadron result of Ref.

[20]. It is straightforward to show thatthe Λ-dependent terms in Eq.

(13) vanish in the Λ →∞limit, thereby reproducing thepoint hadron result. For finite cut-off, the first few terms in a small-m2 expansion of theradii and magnetic moment are given byρsachs = −13 g4π2(3 −5m2) ln m2Λ2 + · · · → g4π22 −13(3 −5m2) ln m2m2N+ · · ·(14)ρdirac = −13 g4π2(3 −8m2) ln m2Λ2 + · · · →−13 g4π2(3 −8m2) ln m2m2N+ · · ·(15)µ = g4π2m2 ln m2Λ2 + · · · → g4π2−2 + m2 ln m2m2N+ · · ·,(16)where m ≡m/mN.

Taking m = mπ and g =√2gπNN gives the neutron EM charge radiiand magnetic moment, while setting m = mK and g = gNΛK gives the strangeness radiusand magnetic moment. The expressions to the right of the arrows give the first few terms ina small-m2 expansion in the Λ →∞limit.

The cancellation in this limit of the logarithmicdependence on Λ arises from terms not shown explicitly (+ · · ·) in Eqs. (14-16).In the case of the axial form factor, we assume the pseudoscalar meson loops to givethe dominant correction to the bare isovector axial matrix element of the nucleon.

Thisassumption may be more justifiable than in the case of the vector current form factors,since the lightest pseudovector isoscalar meson which can couple to the nucleon is the f1with mass 1425 MeV. In this case one has gphysA≈g0A[1 + ∆πA] and ηs =35∆KA/[1 + ∆πA]where the35 is just the spin-flavor factor relating f 0A and g0A.

The loop contributions aregiven by the ∆π,KA, where∆K,πA= ± g4π2hΛ23m2 −Λ24 −Λ2+ 14(2m2 −Λ2) ln Λ2 + · · ·i(17)−→35 g4π2h−12 ln Λ2 + 54 + 12m2 + · · ·i,(18)9

where g = gNΛK or gπNN as appropriate, and where Λ = Λ/mN. The upper (lower) signcorresponds to the kaon (pion) loop.

The opposite sign arises from the fact that ∆πA receivescontributions from two loops, corresponding to a neutral and charged pion, respectively.The isovector axial vector coupling to the intermediate nucleon in these loops (n and p)have opposite signs, while the π±-loop carries an additional factor of two due to the isospinstructure of the NNπ vertex.For the scalar density, we obtain Rs = ˜∆KS /[3fS + ∆KS + ∆πS], where the loop contri-butions are contained in˜∆KS =gNΛK4π2hfSF as (mK, Λ) + γKF bs (mK, Λ)i(19)∆KS =gNΛK4π2h3fSF as (mK, Λ) + 2γKF bs (mK, Λ)i,(20)and ∆πS, where the expression for the latter is the same as that for ∆KS but with thereplacements ¯γK →¯γπ, mK →mπ, and gNΛK →√3gπNN, and where γπ,K = γπ,K/mN. Thefunctions F as and F bs , which represent the contributions from loops 1a (baryon insertion)and 1b (meson insertion), respectively, are given byF as (m, Λ) = 2Λ2 −m24 −Λ2+ 12m2 ln m2Λ2 + · · · →ln Λ2 −2 + 12m2 ln m2 + · · ·(21)F bs (m, Λ) = 1 −12Λ2m2 −m2 −Λ2Λ2 −m2ln m2Λ2 + · · · →1 + 12(1 −m2) ln m2 + · · · .

(22)The expressions in Eqs. (14-22) and curves in Fig.

2 lend themselves to a numberof observations. Considering first the vector and axial vector form factors, we note thatthe mean-square radii display significant SU(3)-breaking.

The loop contributions to theradii contain an I.R. divergence associated with the meson mass which manifests itself asa leading chiral logarithm in Eqs.

(14-15). The effect is especially pronounced for ρdirac,where, for Λ > 1 GeV, the results for m = mπ are roughly an order of magnitude largerthan the results for m = mK (up to overall sign).

In contrast, the scale of SU(3)-breakingis less than a factor of three for the magnetic moment and axial form factor over thesame cut-offrange. The chiral logarithms which enter the latter quantities are suppressedby at least one power of m2, thereby rendering these quantities I.R.

finite and reducingthe impact of SU(3)-breaking associated with the meson mass. Consequently, in a worldwhere nucleon strange-quark form factors arose entirely from pseudoscalar meson loops,one would see a much larger strangeness radius (commensurate with the neutron EM charge10

radius) were the kaon as light as the pion than one would see in the actual world. Thescales of the strange magnetic moment and axial form factor, on the other hand, wouldnot be appreciably different with a significantly lighter kaon.From a numerical standpoint, the aforementioned qualitative features have some inter-esting implications for present and proposed experiments.

Taking the meson-nucleon formfactor cut-offin the range determined from fits to BB′ scattering, 1.2 ≤ΛBonn ≤1.4 GeV,we find µs has roughly the same scale as the nucleon’s isoscalar EM magnetic moment,µI=0 =12(µp + µn). The loop contribution is comparable in magnitude and has the samesign as pole [14] and Skyrme [15] predictions.

While the extent to which the loop and polecontributions are independent and ought to be added is open to debate, the scale of thesetwo contributions, as well as the Skyrme estimate, point to a magnitude for µs that oughtto be observable in the SAMPLE experiment [6]. Similarly, the loop and Skyrme estimatesfor ηs agree in sign and rough order of magnitude, the latter being about half the valueextracted from the νp/¯νp cross sections [3].

Under the identification of the strange-quarkcontribution to the proton’s spin ∆s with G(s)A (0), one finds a similar experimental valuefor ηs from the EMC data [4].In contrast, predictions for the strangeness radius differ significantly between themodels. In the case of the Sachs radius, the signs of the loop and pole predictions differ.The sign of loop predictions corresponds to one’s naive expectation that the kaon, havingnegative strangeness, exists further from the c.m.

of the K −Λ system due to its lightermass. Hence, one would expect a positive value for ρsachss(recall that ρsachs and < r2 >sachshave opposite signs).

The magnitude of the loop prediction for ρsachssis roughly 1/4 to 1/3that of the pole and Skyrme models and agrees in sign with the latter. In the case of theDirac radius, the loop contribution is an order of magnitude smaller than either of the otherestimates.

From the standpoint of measurement, we note that a low-|q2|, forward-anglemeasurement of the elastic ⃗ep PV asymmetry, ALR(⃗ep), is sensitive to the combinationρsachss+ µs = ρdirac [28]. The asymmetry for scattering from a (Jπ, I) = (0+, 0) nucleussuch as 4He, on the other hand, is sensitive primarily to the Sachs radius [28].

Thus, werethe kaon cloud to be the dominant contributor to the nucleon’s vector current strangenessmatrix elements, one would not be able to observe them with the ALR(⃗ep) measurementsof Refs. [7,8], whereas one potentially could do so with the ALR(4He) measurements ofRefs.

[8,9]. Were the pole or Skyrme models reliable predictors of ⟨N|¯sγµs|N⟩, on theother hand, the strangeness radii (Dirac and/or Sachs) would contribute at an observablelevel to both types of PV asymmetry.

As we illustrate elsewhere [21], the scale of the pole11

prediction is rather sensitive to one’s assumptions about the asymptotic behavior of thevector current form factors; depending on one’s choice of conditions, the pole contributioncould be significantly smaller in magnitude than prediction of Ref. [14].

Given these results,including the difference in sign between the pole and both the loop and Skyrme estimates,a combination of PV asymmetry measurements on different targets could prove useful indetermining which picture gives the best description of nucleon’s vector current strangenesscontent.From Eqs. (19-22), one has that the loop contributions to the scalar density containboth U.V.

and chiral singularities. The U.V.

divergence arises from the ¯qq insertion in theintermediate baryon line, while the chiral singularity appears in the loop containing thescalar density matrix element of the intermediate meson. Despite the chiral singularity,loop contributions to the matrix elements mq⟨N|¯qq|N⟩are finite in the chiral limit due tothe pre-multiplying factor of mq.

The ratio Rs is also well-behaved in this limit as wellas in the limit of large Λ. For mπ →0 and mK →0 simultaneously, one has Rs ∼1/8,while for Λ →∞, the ratio approaches1/12.These limiting values are independentof the couplings fS and γπ,K and are determined essentially by counting the number oflogarithmic singularities (U.V.

or chiral) entering the numerator and denominator of Rs(note that we have not included η loops in this analysis). Consequently, the values forRs in the chiral and infinite cut-offlimits do not suffer from the theoretical ambiguitiesencountered in the physical regime discussed in Section 2.

It is also interesting to observethat the limiting results have the same sign and magnitude as the value of Rs extractedfrom the Σ-term.For Λ ≤ΛBonn and for mπ and mK having their physical values, the prediction for Rsis smaller than in either of the aforementioned limits and rather dependent on the choiceof f 0S and γM. The sensitivity to the precise numerical values taken by these couplings ismagnified by a phase difference between F as and F bs .

The range of results associated withscenarios (A)-(C) is indicated in Table I, with the largest values arising from choices (A)and (C). The change in overall sign arises from the sign difference between loops 1a and1b and the increasing magnitude of F bs relative to F as as Λ becomes small.

These resultsare suggestive that loops may give an important contribution to the nucleon’s strange-quark scalar density, though the predictive power of the present estimate is limited by thesensitivity to the input couplings. We emphasize, however, that our estimates of the vectorand axial vector form factors do not manifest this degree of sensitivity.12

We note in passing that scenario (C) gives a value for g0A = gA[1+∆πA(Λ)]−1 ≈g0A(bag)for Λ ∼ΛBonn. The value obtained for g0A in this case depends only on the assumption thatpseudoscalar meson loops give the dominant correction to g0A and involves no statementsabout the details of quark model wavefunctions.In contrast, for Λ ∼1/Rbag we findg0A ≈gA.

We also note that scenarios (A) and (C) are consistent with the bag value forthe scalar density renormalization factor, Z. The latter is defined as Z = ⟨H|¯qq|H⟩/Nq,where Nq is the number of valence quarks in hadron H [24, 27].

A test of consistency,then, is the extent to which the equality f 0S = Z is satisfied. When f 0S is computed usingEq.

(8), we find f 0S ≈0.5 in scenarios (A) and (C), while one has Zbag ≈0.5 [24]. Thesestatements would seem to support the larger values for Rs in Table I.As for the Λ-dependence of the form factors, we find that the radii do not changesignificantly in magnitude over the range ΛBonn ≤Λ ≤∞, owing in part to the importanceof the chiral logarithm.

The variation in the magnetic moment, whose chiral logarithm issuppressed by a factor of m2, is somewhat greater (about a factor of four for m = mK).The ratio ηs is finite as Λ →∞, with a value of ≈−1 in this limit. This limit is approachedonly for Λ >> the range of values shown in Fig.

2d, so we do not indicate it on the graph.The I.R. divergence (Λ << 1 f−1) in the vector current quantities is understandable fromEq.

(13), which has the structure of a generalized Pauli-Villars regulator. The impact ofthe monopole meson-nucleon form factor is similar to that of including additional loopsfor a meson of mass Λ.

From the I.R. singularity in the radii associated with the physicalmeson, one would expect a similar divergence in Λ, but with opposite sign.

The appearanceof a singularity having the same sign as the chiral singularity, as well as the appearanceof an I.R. divergence in the Λ-dependence of magnetic moment which displays no chiralsingularity, is due to the derivative term in Eq.

(13). In light of this strong Λ-dependenceat very small values, as well as our philosophy of taking as much input from independentsources (viz, BB′ scattering) we quote in Table 1 results for our loop estimates usingΛ ∼ΛBonn.It is amusing, nonetheless, to compare our results for m = mK and Λ ∼1 fm−1 withthose of the calculation of Ref.

[17], which effectively excludes contributions from virtualkaons having wavelength smaller than the nucleon size. Assuming this regime in the cut-offis sufficiently far from the artificial zero at Λ = mK to be physically meaningful, our resultfor ηs agrees in magnitude and sign with that of Ref.

[17]. In contrast, our estimates areabout a factor of three larger for the strangeness radii and a factor of seven larger forthe strange magnetic moment.

We suspect that this disagreement is due, in part, to the13

different treatment of gauge invariance in the two calculations. In the case of the axialvector form factor, which receives no seagull contribution, the two calculations agree.

Hadwe omitted the seagull contributions, our results for the Sachs radius would also haveagreed. For the Dirac radius our estimate would have been three times smaller and forthe magnetic moment three times larger than the corresponding estimates of Ref.

[17]. AtΛ ∼ΛBonn, the relative importance of the seagull for ρsachssand µs is much smaller (∼30% effect) than at small Λ, whereas omission of the seagull contribution to ρdiracswouldhave reduced its value by more than an order of magnitude.

We conclude that the extentto which one respects the requirements of gauge invariance at the level of the WT Identitycan significantly affect the results for loops employing meson-nucleon form factors. Wewould argue that a calculation which satsifies the WT Identity is likely to be more realisticthat one which doesn’t and speculate, therefore, that the estimate of Ref.

[17] representsan underestimate of the loop contributions to ρs and µs. An attempt to perform a chiral-quark model calculation satisfying the WT Identity in order to test this speculation seemswarranted.Finally, we make two caveats as to the limit of our calculation’s predictive power.

First,we observe that for Λ ∼ΛBonn, the pion-loop gives a value for µn very close to the physicalvalue, but significantly over-estimates the neutron’s EM charge radii, especially ρdiracn(seeFig. 2b).

One would conclude, then, that certainly in the case of mean-square radii, loopsinvolving only the lightest pseudoscalar mesons do not give a complete account of low-energy nucleon form factor physics. Some combination of additional loops involving heaviermesons and vector meson pole contributions is likely to give a more realistic descriptionof the strangeness vector current matrix elements.

In this respect, the work of Ref. [29] issuggestive.Second, had we adopted the heavy-baryon chiral perturbation framework of Ref.

[30],we would have kept only the leading non-analytic terms in m, since in the heavy baryonexpansion contributions of order mp, p = 1, 2, . .

. are ambiguous.

Removal of this ambiguitywould require computing order mp contributions to a variety of processes in order to tiedown the coefficients of higher-dimension operators in a chiral lagrangian. The presentcalculation, however, was not carried out within this framework and gives, in effect, a modelfor the contributions analytic in m. For m = mK, these terms contribute non non-negligiblyto our results.We reiterate that our aim is not so much to make reliable numericalpredictions as to provide insight into orders of magnitude, signs, and qualitative features ofnucleon strangeness.

Were one interested in arriving at more precise numerical statements,14

even the use of chiral perturbation theory could be insufficient, since it appears from ourresults that pole and heavier meson loops are likely to give important contributions tothe form factors. From this perspective, then, it makes as much sense to include termsanalytic in m as to exclude them, especially if we are to make contact with the previouscalculations of Refs.

[16, 17, and 20].In short, we view the present calculation as a baseline against which to compare future,more extensive loop analyses. Based on a simple physical picture of a kaon cloud, it offers away, albeit provisional, of understanding how strange quark matrix elements of the nucleonmight exist with observable magnitude, in spite of the success with which constituent quarkmodels of the nucleon account for its other low-energy properties.

At the same time, wehave illustrated some of the qualitative features of loop contributions, such as the impactof SU(3)-breaking in the pseudoscalar meson octet, the sensitivity to one’s form factor atthe hadronic vertex, and the importance of respecting gauge invariance at the level of theWT Identity. Finally, when taken in tandem with the calculations of Refs.

[14, 15], ourresults strengthen the rationale for undertaking the significant experimental investmentrequired to probe nucleon strangeness with semi-leptonic scattering.AcknowledgementsIt is a pleasure to thank S. J. Pollock, B. R. Holstein, W. C. Haxton, N. Isgur, andE. Lomon for useful discussions.References1.

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CaptionsFig. 1.

Feynman diagrams for loop contributions to nucleon strange quark matrix elements.Here, ⊗denotes insertion of the operator ¯sΓs where Γ = 1, γµ, or γµγ5. All four diagramscontribute to vector current matrix elements.Only diagam 1a enters the axial vectormatrix element.

Both 1a and 1b contribute to the scalar density.Fig.2.Strange quark vector and axial vector parameters as a function of nucleon-meson form factor mass, Λ.Here, ρs denotes the dimensionless Sachs (2a) and Dirac(2b) strangeness radii. The strange magnetic moment is given in (2c).

The axial vectorratio ηs is shown in (2d). Dashed curves indicate values of these parameters for Λ →∞.The ranges corresponding to the Bonn values for Λ are indicated by the arrows.Thestrong meson-nucleon coupling (g/4π)2 has been scaled out in (2a-c) and must multiplythe results in Fig.

2 to obtain the values in Table I.17

TablesTABLE ISourceρsachssµsηsRselastic νp/¯νp [3]−−−0.12 ± 0.07−EMC [4]−−−0.154 ± 0.044−ΣπN [1, 2]−−−0.1 →0.2kaon loops0.41 →0.49−0.31 →−0.40−0.029 →−0.041−0.007 →0.047poles [14]−2.12 ± 1.0−0.31 ± 0.009−−Skyrme (B) [15]1.65−0.13−0.08−Skyrme (S) [15]3.21−0.33−−Table I.Experimental determinations and theoretical estimates of strange-quark matrix elements of the nucleon. First two rows give experimental valuesfor ηs, where the EMC value is determined from the s-quark contribution to theproton spin, ∆s.

Third row gives Rs extracted from analyses of ΣπN. Final fourrows give theoretical estimates of “intrinsic’ strangeness contributions in varioushadronic models.

Loop values (row four) are those of the present calculation,where the ranges correspond to varying hadronic form factor cut-offover therange of Bonn values (see text). Final two rows give broken (B) and symmetric(S) SU(3) Skyrme model predictions.

First column gives dimensionless, meansquare Sachs strangeness radius.The dimensionless Dirac radius is given byρdiracs= ρsachss+ µs.18

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