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CHIRAL SYMMETRY AND PARITY–VIOLATING MESON–NUCLEON VERTICES* #

arXiv:hep-ph/9211300v1 24 Nov 1992CHIRAL SYMMETRY AND PARITY–VIOLATING MESON–NUCLEON VERTICES* #Ulf-G. Meißner†Universit¨at BernInst. f¨ur Theoret.

PhysikSidlerstr. 5CH-3012 Bern SwitzerlandABSTRACTIn this lecture, I review progress made in the calculations of the parity-violating meson-nucleon interaction regions.

The underlying framework isthe topological chiral soliton model of the nucleon. Emphasis is put on thecomputation of theoretically and experimentally accessible nuclear parityviolating observables.I stress the importance of the interplay of strongand weak interactions which makes this field interesting and challenging.

Ialso discuss recent developments pointing towards the importance of strangequark admixtures in the proton wave function.INTRODUCTIONOur understanding of the hadronic weak interactions has progressed consider-ably in the last two decades. Still, the almost unique tool to study the non-leptonic,strangeness conserving part of the weak Hamiltonian are few-nucleon systems.

In gen-eral, nuclear parity-violating (pv) observables cannot be calculated reliably enoughso that we could deduce stringent limits on the standard model from them. Statedin another way: We are still far away from extracting e.g.

the Weinberg angle tosome decent precision from nuclear parity violation. On the other hand, there arenow very sophisticated parametrizations of the strong force between two nucleonsavailable which allow us to test our understanding of the hadronic weak interactionsin terms of meson exchanges.

Direct W- or Z-exchange between nucleons is wipedout by the hard core of the NN-force, but there still remains a long-range component* Work supported in part by Deutsche Forschungsgemeinschaft and by Schweiz-erischer Nationalfonds.# Invited talk presented at the Workshop on “Baryons as Skyrme Solitons”, Siegen,September, 1992.† Heisenberg fellow1

of the weak interactions between nucleons, which can be parametrized in terms ofpv meson-nucleon interaction vertices. One way to calculate these pv couplings isto make use of the quark model.1 There is, however, a considerable uncertainty inthese calculations which stems from the fact that the pertinent multiquark opera-tors have to be calculated at low energies (E ≪MW,Z).

Gluonic corrections arise,and unavoidably one enters the non-perturbative regime where the strong couplingconstant αs(q2) becomes larger than unity. These problems are most pronounced inthe case of the pion which dominates the long-range part of the pv potential.

TheGoldstone-boson-like character has always posed problems for quark-model practi-tioners. Quite contrary, the recently popular topological soliton models of the nu-cleon like the Skyrme model2 and generalizations thereof3 naturally incorporate thepseudoscalar as well as the low-lying vector multiplets.

Here we have reached ourstarting point for a calculation of the parity-violating meson-nucleon couplings andform factors.4, 5, 6, 7 The soliton approach to the nucleon is far from being perfect,but it has the conceptual advantage that it allows for a simultaneous calculation ofthe strong and weak interaction regions, a point which is generally overlooked byquark model enthusiasts. Furthermore, nuclear parity violation can also be used asa testing ground to find out the limitations of the soliton scenario — often more canbe learned from the failures of a model than from its successes.Another interesting aspect of nuclear parity violation is the quest for findingfew-nucleon systems which can be calculated with some reliability and where theexperimenters have a change of detecting a clear signal.

Here, I will focus on tworather different systems.In proton-proton scattering, one can observe longitudi-nal asymmetries of the order 10−7, which appear to be awfully small.However,progress in experimental techniques now allows for experiments with an accuracy ofδAL ≃±1.0 · 10−8 and therefore a fairly sensitive test of the meson-exchange pic-ture underlying the theoretical description of this process. A very different system isthe nucleus 18F, in which nuclear amplification takes place and the observed circularpolarization of emitted γ-rays is of the order 2 · 10−4.

Luckily for the theorists, theβ-decay of the daughter nucleus 18Ne allows one to gauge the rather involved shellmodel calculations,8 although sceptical minds tend to look at these calculations witha certain dose of disbelieve. As we will see, too few “good” nuclear systems are con-sidered at present and therefore the restrictions on the pv meson-nucleon couplingsare by far too soft.Finally, during the last year, effective field theory methods have been used to gainfurther insight into the strength of the pv meson–nucleon couplings.9 These resultsseem to indicate a large enhancement from operators involving strange quarks to2

various coupling constants. Furthermore, some couplings not considered so far (likee.g.

the pv πNγ vertex) might be of importance. I will discuss these topics in theend of this lecture.PV MESON-NUCLEON INTERACTION REGIONSIn the meson-exchange parametrization of the weak nuclear force, one usuallyonly considers the exchange of charged pions and the vector mesons ρ and ω. CPinvariance does not allow for the coupling of neutral scalar or pseudoscalar mesonsto nucleons, eliminating the infamous scalar mesons, the η, η′ and the π0 (the δ± areconsidered a form factor corrections to the π±-exchange).

Then there remains theφ(1020) — its coupling to the nucleon is generally supposed to be OZI-suppressedand not considered.10 This might, however, be a too simplistic approach in lightof the discussion surrounding the admixture of strange operators into the proton’swavefunction. At present no final conclusion can be drawn and I will make life easyon us and neglect the φ for the time being.

I will pick up this theme in the finalsection.Unavoidably I will have to define the basic couplings which parametrize the pvnuclear potential. For the pion, there is only a ∆I = 1 (isovector) coupling (to firstorder in the pion field)LpvπN = −Gπ(q2)√2EMNχ†f (⃗τ ×⃗π)3 χi(1)with χi,f denoting nucleon spinors, q2 the invariant momentum transfer squared atthe πN-vertex and I have (for simplicity) given the non-relativistic reduction of thisvertex.

In the case of the ω, ∆I = 0 and ∆I = 1 couplings are possible,LpvωN = χ†fh0ω(q2) + h1ω(q2)τ 3 EMN⃗σT +⃗σL· ⃗ωχi(2)with E =M 2N +⃗q2/41/2and ⃗σL,T the longitudinal and transverse spin-operator,respectively. For the ρ, one has isoscalar, isovector and isotensor verticesLpvρN = χ†f"h0ρ(q2)τ a + h1ρ(q2)δa3 + h2ρ(q2)2√63τ 3δa3 −τ a#× EMN⃗σT +⃗σL·⃗ρaχi −iE2M 2Nh′1ρ (q2)χ†f⃗σ ·⃗q⃗τ ×⃗ρ03 χi .

(3)3

Generally, the coupling h′1ρ is neglected,11 but I will not follow this historical pathhere. From Eqs.

(1) – (3) it becomes obvious that the pv interaction regions arecharacterized by coupling constants hM = hM(q2 = 0) and form factors F pvM (q2) =hM(q2)/hM (in case of the pion, I use Gπ ≡hπ).In the topological chiral soliton model12 underlying the calculation of the pvvertices, nucleons arise as solitons of a non-linear meson theory.This non-linearmeson theory is constructed in harmony with chiral symmetry and anomaly con-straints and all its parameters are fixed from mesonic reactions like e.g. (ρ0 →π+π−,ω →π+π−π0, ω →π0γ, .

. .

. The Lagrangian and its parameters are completelydetermined in the meson sector, and the calculation of nucleon properties proceedswithout any new parameters, i.e.

no fudging is possible! That is certainly an appeal-ing aspect of the soliton approach to the nucleon and it poses several restrictions.

Ofcourse, the model does not perfectly predict all nucleon properties.Now: How can we calculate the pv couplings appearing in Eqs. (1) – (3)?

Forthat, we consider the current × current form of the weak Hamiltonian with thecurrents being of (V –A)-type. To pick out the pv pieces, consider the ∆I = 0, 1 or2 components of products like VµAµ and IµAµ, with Vµ the vector, Iµ the isoscalarand Aµ the axial current.

These currents are already given in terms of the mesonfields which make up the soliton, and their explicit expressions can be found e.g. inRef.

[5]. One then makes use of the “background-scattering” method, which amountsto an expansion of the meson fields around the soliton background.

For the pion, wewrite13⃗π = ⃗πS +⃗πf(4)and similarly for ρ and ω. ⃗πS is the “hard” component of the pion field makingup the soliton and ⃗πf a small pionic fluctuation (“soft” component).

Inserting theexpressions (4) into the soliton currents and these into the weak Hamiltonian, all onehas to do is to find the terms linear in ⃗πf (or ⃗ρf or ωf). Quantizing the respectiveoperators which are given in terms of the collective variables (A, ˙A),2, 3 one canimmediately read offthe coupling constants and form factors for the meson underconsideration.

In particular, one cannot only construct pv meson-nucleon vertices,but also the equivalent pv N∆-transition couplings. I will come back to this pointlater on.For details, the interested reader should consult Refs.

[5,6].I will notgive any explicit formula here, but rather make a few comments on the results ofRef. [5].

First, the pv πN coupling is completely dominated by the neutral currentcontribution.4

KMDDHDZAHRR˜Fπ0.610.83.15.00.0→27.1F05.915.911.58.0–15.9→43.0F10.20.3–0.50.3–0.1→0.6F25.313.39.39.810.6→15.3H11.70.50.00.0—G024.58.016.327.0–23.9→43.1G14.14.89.210.0–3.3→8.0Table: Effective weak meson-nucleon coupling constants in units of 10−7.We present the result of the soliton model calculation of Kaiser andMeißner(KM) [5,6] together with the quark model results of Desplanqueset al. (DDH) [1] as well as Dubovik and Zenkin(DZ) [16].

The value for h′1ρin the column DDH is taken from Holstein’s calculation in ref.[11]. The“reasonable ranges”(RR) defined by DDH are also given.

The column AHgives the best fit values of Adelberger and Haxton.17The charged current contribution can be estimated in the factorization approxima-tion, GCCπ= cos2 θc < π|Aµ|0 >< p|Vµ|n >= GF cos2 θcfπ(Mn −Mp). The elec-tromagnetic mass difference of the neutron and the proton is well-reproduced inthe model,4 whereas a strong part of Mn −Mp is somewhat underestimated.14.Taking as an upper limit the empirical value Mn −Mp ≃1.3 MeV, we findA = GNCπGCC ∼13.5, consistent with previous estimates15 and the quark modelcalculations of Ref.

[1] (A ≃24). The numerical value for the effective pion-exchangecoupling, ˜Fπ = gπNNGπ√32 with gπNN the strong πN coupling, is considerablysmaller in the soliton model than in the quark model, we find ˜F solπ= 0.6 ve rsus˜F qπ = 10.8,8 or ˜F qπ = 3.1 16 (in units of 10−7).For the vector meson couplings, the results are less different.

Using the standarddefinitions Fi = −gρNNhiρ/2, and Gi = −gωNNhiω/2 and H1 = −gρNNh′1ρ /4, we find5

that the soliton and the quark model predict the following pattern for the ρ-couplings:F0 >∼F2 ≫F1. The absolute values of the constant s Fi, are, however, reduced inthe soliton approach.

For H1, the soliton model predicts a value three times as largeas the quark model.11 In the case of the ω, all calculations give G0 > G1, but G0 isconsiderably enhanced in the soliton approach and close to the “best-fit” estimate ofAdelberger and Haxton.17 These results are summarized in the table.As already stated, the calculation of pv N∆M (M = π, ρ, ω) transition verticesproceeds along the same lines and only differs at the step when one quantizes thecollective coordinates.In our model, the pv πN∆coupling has ∆I = 0 and 1components and reads non-relativisticallyLpv∆Nπ =E4M 3N δabh0∆(q2) + ǫ3abh1∆(q2)χ†∆⃗S ·⃗q⃗σ ·⃗qT aχNπb + h.c.(5)with ⃗S and ⃗T the conventional N∆transition spin and isospin operators, respectively.It is easy to convince oneself that h0∆(q2) = 0 in this model,6 naively, a non-vanishingisoscalar π∆N-vertex would lead to a non-zero CP-violating πN-vertex (a more fool-proof argument is given in Ref. [6]).

The isovector vertex does not vanish, and forthe “minimal” model18 we findh1∆(0)Gπ = 1.10 . (6)The presence of πρω-correlations in the effective action tends, however to decreasethis ratio.

For the V ∆N-couplings, we find (V = ρ or ω):hiρ∆N(q2) =3√2hiρ(q2)(i = 0,′ 1, 2) ,h1ω∆N(q2) =3√2h1ω(q2)(7)and h1ρ∆N(q2) = h0ω∆N(q2) ≡0. These predictions are insofar interesting since inthe seventies it was argued that e.g.

the ρ∆N-couplings are negligible19 — quite incontrast to our results. A recently performed quark model calculation by Feldmanet al.29 along the lines of DDH gives results rather different from the soliton modelpredictions.

The source of these discrepancies is not yet understood. In a similarfashion, one easily derive the corresponding M∆∆vertices,Gπ∆∆(q2) = Gπ(q2)hiρ∆∆(q2) = 15hiρ(q2) (i = 0,′ 1, 2) ; h1ρ∆∆(q2) = h1ρ(q2)h0ω∆∆(q2) = h0ω(q2) ; h1ω∆∆(q2) = 15h1ω(q2)(8)6

The calculation of the associated weak form factors proceeds in a straightforwardway.6 In Fig. 1 we show the weak πN form factor Gπ(q2) in comparison with theequivalent strong form factor GπNN(q2) as well as the monopole with cut-offΛ =1 GeV.

As it turns out, all form factors can be fitted by monopoles at low ⃗q2,F M(⃗q2) = hMΛ2Λ2 +⃗q2(9)Fig. 1:The weak πN form factor Gπ(q2) in comparison to its strong coun-terpart GπNN(q2) and a monopole fit with a cut–offΛ = 1 GeV (solid line).and are very similar to the respective strong form factors.This is the first timethat such a calculation has been performed and its result can be understood asfollows: The intrinsic scale of the meson-nucleon interaction regions is set by thetopological baryon charge radius, rB ≃0.5 fm.

From that, one can deduce a cut-offscale Λ ≃√6 /rB ≃1 GeV. It is, however, not that simple because the dynamicaltreatment of the vector mesons modifies this result.

Defining by RM the ratio of the(averaged) weak to strong MN cut-offs (all form factors of monopole type), we findRπ = 1.15 ,Rρ = 0.91 ,Rω = 0.77(10)7

which justifies within the accuracy of the model the assumption of taking the sameform factors for the strong and weak vertices as it was done e.g. by Driscoll andMiller20 in their study of the pv pp-interaction.Another topic which can be discussed in the framework of the chiral solitonmodel are the corrections of pv two–pion exchange.

One motivation to do this isthat correlated 2π–exchange gives rise to the intermediate range attraction of theparity–conserving NN interaction. Furthermore, recent investigations point towardsthe importance of pv 2π exchange even below production threshold.31 Using thesoliton model, Norbert Kaiser and I have shown that the inclusion of pion loopsgives the intermediate range attraction with just the right strength as compared tothe Paris potential.32 Similarly, we have worked out corrections to the various pvρN couplings.33 The effects of irreducible two–pion corrections are generally small,of the order of 10 .

. .20%.

This is in agreement with older dispersion–theoreticalinvestigations.34 So we finally have all the tools at hand to make contact to experi-ment.PARITY-VIOLATION IN PROTON-PROTON SCATTERINGThe simplest system in which one can probe certain components of the weakpv inter-nucleon force is the two nucleon system. By scattering polarized protonsoffa hydrogen target, parity violation shows itself in a non-vanishing longitudinalasymmetry,AL = σ+ −σ−σ+ + σ−(11)assuming a 100% longitudinal polarization of the beam and having taken care ofthe Coulomb-corrections σ± are the cross-sections for scattering positive/negativehelicity protons from an unpolarized target.

The calculation of this process in theDWBA as pioneered by Brown et al.21 goes as follows. One splits the total scatteringamplitude Fss′ into a strong and weak partFss′ = Fss′ + fss′(12)for total spins s and s′.

Now it is of utmost importance to take into account thestrong distortions, i.e.calculating the weak scattering amplitude with distortedwaves ψ(−)sand ψ(+)s, i.e. fss′ =< ψ(−)s|Vpv|ψ(+)s> with Vpv the pv one-meson-exchange potential.

It should be pointed out that the strong distortions govern theenergy-dependence of the analyzing powers AL. Recently, Driscoll and Miller20 havedone the most complete calculation based on the Bonn-Potential21 for the strong8

force and an equivalently constructed weak potential with the pv couplings takenfrom the quark model1 and using the same vertex functions for the weak and strongform factors.I should point out here that for obvious reasons there is no pioncontribution to this process and one essentially tests the vector-meson couplingshppρ = h0ρ + h1ρ + h2ρ√6 and hppω = h0ω + h1ω.Recently, Doug Driscoll and I have repeated this calculation22 by including thesoliton model predictions hppρ = −5.15 · 10−7 and hppω = −8.20 · 10−7. The resultingcurve for AL is shown in Fig.

2, for the quark1,16 and the soliton model.22 Theshape of the curve as predicted by the soliton model follows closer the empiricaltrend suggested by the low-energy data.23 In fact, a χ2 calculation for the threecurves shown in fig.2 gives χ2 = 34/3 (DDH), χ2 = 26/3 (DZ) and χ2 = 8/3 (KM)as disussed in ref.26 (at that time, the Bonn result was not available). Also, themaximum at plab = 0.95 GeV/c is flatter than in the calculation using the quarkmodel parameter.

Furthermore, the energy at which the asymmetry changes sign islarger than the quark model predicts, which can be traced back to the fact that in thesoliton model hppω > hppρ , in contrast to the quark model with hppω < hppρ . Of particularinterest is the value of AL at 222 MeV.

This is the energy selected for an upcomingpp parity violation measurement at TRIUMF because δ(1S0) + δ(3P0) = 0 at thisenergy and the j = 0 contribution to the analyzing power consequently vanishes. Themeasurement of the dominant j = 2 contribution gives a different combination of hppρand hppω than the j = 0 contribution to AL, which is already measured at 15 and45 MeV.23.

The predictions using the quark1 and soliton model5 weak parameters,respectively, differ by ∆AL = 4.6 · 10−8. To be more precise, the various predictionsare:AL(DDH) = 5.0 · 10−8 , AL(DZ) = 2.6 · 10−8 , AL(KM) = 3.7 · 10−9(13)The projected long-term accuracy of the upcoming TRIUMF experiment is(δAL)stat ≃±1 · 10−8,24 which should be sufficient to discriminate between thesetwo predictions.

Notice that a similar experiment is also planned at COSY.30 Thisexperiment should set rather stringent limits on some combinations of the pv ρN andωM couplings. To stress it again, the pp system is a particularly good example of theinterplay of weak and strong interactions and it is therefore mandatory to treat bothof them consistently (for further discussion, see Refs.

[17,20]). A possible loophole toall of this will be discussed in the last section.9

Fig. 2:Parity-violating asymmetry in pp-scattering.

The solid line givesthe prediction based on the weak couplings as given by the soliton model,22whereas the dashed and dashed-dotted lines are based on the quark modelcalculations of refs.1 and 16, respectively.PARITY VIOLATION IN 18F AND DEUTERON PHOTODISINTEGRATIONThe nucleus 18F is what I called a “good system” before. It exhibits “nuclearamplification” in that it has two close-by levels of opposite parity which are separatedby only 39 keV (the next level which could mix with these is approximately 2 MeVaway) and the dominant E1-transition from the level at 1.081 MeV to the groundstate is suppressed, which leads to |M1/E2| ≃112.

The M1-transition is, of course,only possible because of the mixing of the opposite parity-levels. Altogether, thisamounts to an amplification of approximately (2/0.039) ∗112 ≃6 · 103 (for furtherdetails, see Ref.

[17]). Theoretically, one can calibrate the shell-model calculationto extract the pv circular polarization from the β-decay of 18Ne, because the pion-exchange of this β-decay up to an overall isospin rotation,25 and therefore calculation10

and measurement of 18Ne (0+1) →18F (0−0) β-decay serves as a “gauge” for theaccuracy and amounts effectively to a large model-independent limit on the weak piondecay constant. The latter dominates completely this ∆I = 1 pv observable, and onecan deduce a limit on ˜Fπ, ˜Fπ ≤3.4 · 10−7.

Here, we have used the experimentalcircular polarization, |Pγ(18F)| = (0.17 ± 0.58) · 10−3. The quark model predictionof Ref.

[1], ˜Fπ = 10.8 · 10−7, is clearly in contradiction to this result.What does the soliton model give? Of course, ˜Fπ is considerably reduced, sowe expect a smaller asymmetry.

The vector meson contribution is enhanced, andtaking nuclear structure calculation from Ref. [17], we predict Pγ(18F) = 2.2 · 10−4,not far from the central value of the experiment.

We should, however, not put toomuch emphasis on this closeness of the experimental and theoretical number, butrather state that the strength of the pv πN coupling should still be considered asthe main theoretical puzzle. I am sure that ˜Fπ should come out smaller than inRef.

[1], but whether it is as small as predicted by the soliton model can only bechecked if more theoretical and experimental information on the ∆I = 1 part of thepv nuclear force are available. One particularly interesting candidate to study inmore detail would be the reaction ⃗n+p →d+γ or the inverse process ⃗γ +d →n+p.A calculation of the circular asymmetry as a function of the photon energy hasbeen performed some time ago by Oka.27 I have used this calculation in Ref.28 toinvestigate the sensitivity of the circular asymmetry AL to the various pv couplings.Considering photon energies below 30 MeV, AL(ω) increases linearly with energywhen one uses the quark model couplings of DDH or DZ, with the slope determinedby the strength of the pv πN coupling.

For the DDH-case the pion contributionis completely dominant for all energies, whereas for the DZ-parameters the reducedπNN strength leads to an overall decrease of AL(ω). For the soliton model, however,things are significantly different.First, between 1 and 20 MeV , AL(ω) shows aflat minimum at about ωL ≈12 MeV and only after ωL ≥20 MeV a gradual risein AL(ω) sets in.

Also, the overall magnitude of the effect is an order of magnitudesmaller for the weak parameters predicted by the soliton model. It would be worthwileto measure the asymmetry say at 10 and 20 MeV incident energy, although theeffect is small, the tremendously different slope of AL(ω) should be detectable in andedicated experiment.

Of course, as already mentioned, a more thorough theoreticalstudy has also to be done. First, a more consistent calculation employing e.g.

theBonn-potential and the equivalently constructed weak potential should be performed.Second, the effects of meson-exchange curents, which play an important role in theaccurate description of the deuteron properties have to be included. Therefore, theseresults should only be considered as a guide, but the trends exhibited will certainly11

not be wiped out by a more elaborate calculation. A more detailed discussion is givenin Ref.28.In the last section, I will discuss some medium renormalization effects whichmight come to the rescue of the large value for the pv pion–nucleon coupling aspredicted by DDH.

However, for the deuteron photodisintegration process just dis-cussed, such a renormalization cannot be operative since the deuteron is essentiallyan ensemble of two free nucleons.THE NUCLEON ANAPOLE MOMENTApart from the electric dipole moment, there is one other pv coupling of thephoton to nucleons (spin-1/2 fields), the so–called anapole moment. It has recentlyattracted new interest35 since its contribution might be enhanced considerably innuclei, similar to the case of 18F just discussed.

For on–shell nucleons, current con-servation and Lorentz invariance require that pv corrections to matrix elements ofthe electromagnetic current take the form< N(p′)|Jemµ,pv(0)|N(p) >= a(q2)M 2N¯u(p′)[γνqνqµ −q2γµ]γ5u(p)(14)with q2 = (p′ −p)2. In the Breit frame, where the photon transfers no energy, thismatrix element reads< N(⃗q/2)|Jemµ,pv|N(−⃗q/2) >= E⃗q 2M 3Na(⃗q 2)χ†f⃗σT χi(15)with ⃗σT = ⃗σ −ˆq⃗σ · ˆq the transverse spin operator and a(q2) the nucleon anapole formfactor.

The anapole moment has isoscalar and isovector components,a(0) = aS(0) + aV (0)τ3(16)In the soliton model, one can easily calculate the anapol moment and form factor.36For that, one identifies the matrix element in (15) with the Fourier transform of thepv electromagnetic current. For the usual hedgehog ans¨atze, its has the general form⃗Jpv(⃗r) = Γ1(r)⃗σ + Γ2(r) ˆr⃗σ · ⃗r(17)with Γ1,2(r) functions of the various meson profiles whose explicit form we do notneed here.

However, one immediately encounters a difficulty. Current conservationdemands Γ′1(r) + Γ′2(r) + 2Γ2(r)/r = 0, where the prime denotes differentiation12

with respect to r. This condition is not met. Interestingly, if one switches offthe ρ–meson fields and considers the so–called ω–stabilized Skyrmion37, then the divergencecondition is fulfilled.

This peculiar behaviour might be traced back to the fact thatin the isoscalar channel one has exact vector meson dominance (VMD) but not inthe isovector one (compare the discussion of Hwang and Nigoyi38 of VMD and gaugeinvariance for pv photon–nucleon couplings). To get an idea of the size of the anapolemoment, let me crudely rstore gauge invariance by subtracting the pieces whichviolate current conservation.

In that case, the ”minimal” model gives as(0) = 4·10−8,and the extension of the pv γN vertex is given by a mean square radius of about0.4 fm corresponding to a monopole form factor with a cut–offof Λ = 1.23 GeV.At present, I can not offer a solution to the problem concerning the violation ofcurrent conservation, but I suspect that is it related to the rather crude quantisationprocedure used (which is known to do harm to e.g. the chiral algebra of the charges38).RECENT DEVELOPMENTSThere are some recent developments (partly outside the soliton model) whichindicate some interesting new effects and might lead to a reconsideration of sometopics discussed so far.

The first one is due to a calculation of Dai, Savage, Liuand Springer.9 They calculate an effective Hamiltonian for ∆I = 1 nuclear parityviolation, including the effects of the heavy quarks s, c and b. At the scale of the W-boson mass, the pv ∆I = 1 Hamiltonian is, of course, well known and given in termsof eight four–quark operators with known Wilson coefficients.

Integrating out the band the c quark successively, one has a tower of effective theories. For each of these,the anomalous dimension matrix is calculated to one loop in the QCD corrections andthe effective field theories are matched.

By this procedure, one can finally go downto the hadronic scale of Λχ = 1 GeV and compare the Wilson coefficients Ci(Λχ)with the original ones, Ci(MW ). The important observation made in ref.9 is that theoperators involving strange quarks are substantially larger than the ones involvingonly the up and down quarks, approximatelyCstrangei(Λ)Cnon−strangei(Λ)∼1sin2 ΘW∼5(18)The authors of ref.9 did not compute hadronic matrix elements at the scale Λχ,but resorted to the meson–exchange picture and large Nc arguments.

In that case(Nc →∞), factorization can be justified and one finds for the ρ0N pv matrix element< ρ0N|H∆I=1pv|N >= 13GF sin2 ΘW fρǫ∗ρµ{ −0.95 < N|¯uγµγ5u + ¯dγµγ5d|N >+ 13.4 < N|¯sγµγ5s|N >}(19)13

Consider first the case were the strange matrix element vanishes (like in DDH). In thatcase, one finds h(1)ρ= −1.9 · 10−8, quite consistent with the DDH value.

However,the large relative factor in front of the new, un–colored strange contribution caneasily alter this result by an order of magnitude. Combining the EMC–data andhyperon decay rates, one has < N|¯uγµγ5u + ¯dγµγ5d|N >≃−< N|¯sγµγ5s|N >≃−(0.2 ± 0.1)Sµ with Sµ the nucleon spin vector.

In this case, h(1)ρ= −2.9 · 10−7,which is an enhancement of a factor 15. If that were true, all previous estimatesof pv meson–nucleon couplings can be offthe mark by large factors.

However, weshould not forget that in last years many of the matrix elements which indicated alarge contribution of the stramge quark sea to various nucleon properties have beentamed, the prime example being the famous πN Σ term.In a similar fashion, Kaplan and Savage40 have recently reanalyzed the pv pion–nucleon couplings making use of baryon chiral perturbation theory. They have de-rived the most general pv and CP–conserving effective pion–nucleon–photon La-grangian to first order in derivatives and first order in the photon field and to allorders in the pion field.

This effective Lagrangian is parametrized by a few couplingconstants, which are labelled h0,1,2V, h1,2Aand h1πNN = Gπ. Apart from the standardpv pion–nucleon coupling (discussed before), the authors of ref.40 mainly concen-trate on the novel pv γπNN and the pv ππNN vertices (the latter one has beenalready been considered by nuclear theorists in the seventies).

Three different meth-ods are used in ref.40 to estimate the strength of these coupling constants, namelyfactorization, dimensional analysis and relations to ∆S = 1 hyperon decay matrixelements. From these methods, the dimensional analysis is considered most reliable.The most interesting results of this are 1) a large contribution of the strange quarksto Gπ (together with a large value for this coupling), 2) a sizeable strangeness en-hancement for the pv ππNN coupling h1A and 3) a large value for the strength of thepv γπNN coupling.

Taking these estimates face value, drastic consequences wouldarise. First, in the case of the 18F experiment, interference between the one–pionexchange (considered so far) and the novel γπNN vertex might complicate the anal-ysis of the data and ultimately relax the bound on Gπ.

Similarly, for the plannedTRIUMF and COSY experiments measuring parity violation in pp scattering at 230MeV, one would have to consider two–pion exchange, not only the conventional onearising from e.g. intermediate ∆resonances, but also the one due to the large pvππNN coupling.

However, before jumping too far, one should not forget that theresults of ref.40 should be considered indicative – more elaborate calculations of thehadronic matrix elements are necessary (using e.g. lattice methods) and also morecomplex nuclear structure calculations involving these novel couplings have to be14

performed before one can draw a final conclusion. For more details on these topics,please consult ref.40.MEDIUM RENORMALIZATION OF Gπ ?There exist ample evidence that suggests scale changes of fundamenral propertiesof nucleons in nuclei.

Some pertinent examples are the first EMC effect, the quench-ing of the axial–vector coupling constant gA in nuclear β–decay or the behaviour ofthe longitudinal and transverse strength functions in quasi–elastic electron scatter-ing offnuclei. These effects are there and they are important, but thier origin stillremains to be explained in a consistent treatment of many–body effects and fun-damental scale changes of the nucleon properties.

The chiral soliton model allowsto systematically investigate the constraints from chiral symmetry on such possi-ble medium modifications.41,42 The basic idea is the following: In the soliton model,baryon properties are fixed once the mesonic input is determined. We know, however,that meson masses and coupling constants change in the baryon–rich environment.43This immediately leads to density or temperature–dependent nucleon properties.44For the meson sector, I will use here results from the Nambu–Jona-Lasinio modelwhich have been obtained in collaboration with V´eronique Bernard.42 For not toolarge densities ρ, one finds for the pion decay constant and the vector and scalarmeson masses (all other quantities are essentially unaffected)F ∗π = Fπ(0)[1 −Rπρρ0] , m∗V = mV (0)[1 −RVρρ0] , m∗σ = mσ(0)[1 −Rσρρ0](20)where the ’*’ denotes quantities in the medium and ρ0 is the nuclear matter density.The range of values for Rπ,V,σ is discussed in ref.42.

For simplicity, let me take anuniversal and equal value, Rπ = RV = Rσ = R. This is not a direct consequenceof the NJL model but compatible with it. For the sake of the argument I will makehere, this simplification is justified.

In ref.42, which is a widely overlooked paper, Ihave shown that most of the pv meson–nucleon couplings are very sensitive to suchmedium effects, quite in contrast to their strong counterparts. In particular, the mostimportant pion–nucleon couplings show the following medium renormalization (forR = 0.2 and at nuclear matter density)G∗π(ρ0)Gπ(0) = 0.65 ,g∗πNN(ρ0)gπNN(0) = 0.99(21)and similar results for the vector meson couplings.

One can understand this verydifferent behaviour if one takes a closer look at the expressions for the various coupling15

constants. Using the dimensionless variable x = gFπr, with g the universal vector-meson–pion coupling, one notices that the weak couplings depend one much higherpowers of Fπ than their strong counterparts and thus are more sensitive to mediummodifications.

A more detailed account of this can be found in ref.42.Finally, let me point out some recent work by Grach and Shmatikov45 whichconcerns yet another mechanism to bring down the value of the pv πN coupling inthe medium. The basic idea of thier work is that the rescattering of emitted pionsleads to a strong suppression of Gπ (the basic Feynman diagrams are shown in fig.3).Using monopole form factors with a cut offΛ ≃7Mπ ≃1 GeV to regulate thediverging loop integrals, they findFig.

3:Strong pion rescattering in the medium.45 The solid, double anddashed lines denote nucleons, ∆’s and pions, in order.Strong meson–nucleon vertices are depicted by open circles and weak vertices by the crossedcircles.G(r)π= Gπ1 + g2πNN8π2 I1 + g2π∆N8π21427I2= Gπ(1 −0.76 + 0.01)(22)which leads to˜F (r)π= G(r)π gπNN/√32 ≃2.9 · 10−7(23)which is below the bound from the 18F experiment. This is an interesting suggestion,but it definitively needs a better treatment (better regularization procedure) andshould also be applied to the other pv meson–nucleon couplings.

Also, one shouldunderstand the relation to the soliton model results discussed before.16

OPEN PROBLEMSInstead of rephrasing what I have said so far, let me just mention the two salientproblems which have to be addressed in the framework of the chiral soliton model toallow for a deeper understanding of the pv meson–nucleon interaction regions.• Realistic versions of the three flavor Skyrme model are now available.Theydo not indicate a large strange component in proton wave function. It wouldbe worthwhile and necessary to extend the analysis of the pv interaction re-gion discussed here.

This would also allow to addres such questions like thestrength of the φ–couplings and the relation to the ∆S = 1 hyperon decay ma-trix elements. Ultimately, such calculations will shed some light on the recentdevelopments concerning the possible enhancement of various weak couplingsdue to the strange color–singlet operators.• An old problem is whether the soliton model calculations should be supplementedby strong interaction enhancement factors or whether these are already containedin the non–perturbative soliton currents.This question was to some extentaddressed in ref.7, where it was argued that the inclusion/omission of thesefactors would at most lead to uncertainties of the order of 30 per cent, i.e.

leadto corrections within the accuracy of the model. To my opinion, this question isnot yet settled.

Its resolution will also bring about the answer to the questionof including operators which are not of the canonical VµAµ–type.ACKNOWLEDGEMENTSIt is my pleasure to thank the organizers for their excellent work which madethis meeting truely memorable. I would like to thank V. Bernard, D. Driscoll and N.Kaiser for many enjoyable collaborations.

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