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Electroweak Reactions in the

arXiv:hep-ph/9303298v1 24 Mar 1993Electroweak Reactions in theNon-Perturbative Regime of QCDUlf-G. MeißnerUniversity of Berne, Institute for Theoretical Physics, CH-3012 Berne, Switzerland0 Contents1 Introduction12 Chiral Perturbation Theory with Nucleons22.1Chiral Symmetry in QCD22.2Chiral Perturbation Theory (Mesons)32.3Inclusion of Matter Fields52.4Threshold Pion Photo- and Electroproduction72.5Nucleon Compton Scattering93 The Quark Structure of the Nucleon123.1Currents, Sizes and Form Factors of the Nucleon123.2A Toy Model: Quark Distributions from Form Factors143.3Electroweak Currents153.4Parity–Violating Electron Scattering183.5Neutrino and Antineutrino Scattering offNucleons and Nuclei193.6Down and Dirty: QED, QCD and Heavy Quark Corrections214 Summary and Outlook251 IntroductionThese lectures are concerned with the structure of hadrons at low energies, wherethe strong coupling constant is large. Most of the hadronic world discussed here willbe made up of the light u, d and s quarks since these are the constituents of thelow-lying hadrons.

The best way to gain information about the strongly interactingparticles is thus the use of well-understood probes, such as the photon or the massiveweak gauge bosons. At very low energies, the dynamics of the strong interactionsis governed by constraints from chiral symmetry.

This leads to the use of effectivefield theory methods which in the present context is called baryon chiral perturbationtheory. In the first part of these lectures, I will briefly outline the basic frameworkof this effective field theory and use photo-nucleon processes to discuss the strengthsand limitations of it.

The basic degrees of freedom are the pseudoscalar GoldstoneLectures delivered at the XXXII. Internationale Universit¨atswochen f¨ur Kern- undTeilchenphysik, Schladming, Styria, Austria, February 24 - March 6, 1993.

2Ulf-G. Meißnerbosons chirally coupled to the matter fields like e.g. the nucleons.

The very low-energy face of the low-lying baryons is therefore of hadronic nature, essentially point-like Dirac particles surrounded by a cloud of Goldstone bosons. The informationabout the underlying quark structure is only indirect since one translates e.g.

quarkmasses into pseudoscalar meson masses. Nevertheless, certain aspects of the quarkstructure of the nucleon (or other baryons) can be extracted by making clever use ofthe well-known couplings of the quarks to the electroweak gauge bosons.

For that,it is important to understand the notion of hadron form factors, electromagneticor weak ones. These encode certain aspects of the quark structure of the nucleon.In addition, parity-violating electron scattering offnucleons or nuclei or neutrino-nucleon scattering allows one to measure the weak form factors related to the neutralcurrents which complement the information gained from the electromagnetic andweak charged current form factors.

This will be the theme of the second part of theselectures. However, since one is dealing with small effects like e.g.

left-right asymmetriesin polarized electron scattering offprotons at low energies, one also has to worryabout the effects of the heavy quarks c, b and t and about radiative corrections,which are nominally suppressed by powers of (α/4π) but sometimes enhanced bylarge logarithms or by some judiciously large factors. I will give a short state of theart summary on these topics at the end of these lectures.Naturally, I can only cover a small fraction of the many interesting phenomenarelated to low energy hadron physics.

I have chosen to mostly talk about the nucleonsince after all it makes up large chunks of the stable matter surrounding us and also isa good intermediary between the nuclear and the high energy physicists present at thisschool. Most of the methods presented here can easily be applied to other problems,and as it will become obvious at many places, we still have a long way to go tounderstand all the intriguing features of the nucleon in a systematic and controlledfashion.

Whenever possible, I will avoid to talk about models, with the exception ofsome circumstances where they offer some qualitative insight into certain aspects ofhadron structure.2 Chiral Perturbation Theory with Nucleons2.1 Chiral Symmetry in QCDConsider the QCD Lagrangian for the three light flavors u, d and s,LQCD = ¯q(iγµDµ −M)q + Lgluon(2.1)with q = (u, d, s) and I have suppressed all color indices. The quark fields can bechosen such that the quark mass matrix is diagonal,M =mumdms(2.2)The entries in (2.2) are the current quark masses.

At the typical hadronic scale ofΛ ≃1 GeV, these can be considered small. This holds certainly for the u and dquarks, the s quark is somewhat heavier (≃175 MeV) which makes it more difficult

Electroweak Reactions in the Non-Perturbative Regime of QCD3to deal with. To a good first approximation, it is, however, legitimate to set thecurrent quark masses to zero.

In that case, the QCD Lagrangian exhibits a flavorSU(3)L × SU(3)R × U(1)L+R × U(1)L−R symmetry. The vectorial U(1) is related tothe baryon number and the axial U(1) is afflicted by the anomaly which gives the η′a mass of 1 GeV.

The chiral group has therefore sixteen independent generators. Thisleads to sixteen conserved left- and right-handed currents,Jaµ,A = ¯qAγµλa2 qA(a = 1, .

. .

, 8 ; A = L, R)(2.3)or, equivalently, eight conserved vector and eight conserved axial-vector charges. Theseform an SU(3) algebra.

However, this symmetry of the Lagrangian is not present inthe ground state or the spectrum of the strongly interacting particles. Something verysimilar to the ferromagnet below the Curie temperature happens.

The magnet Hamil-tonian is rotationally invariant but below Tc spontaneous magnetization sets in andall spins align in one direction breaking the symmetry of the Hamiltonian. Similarlyin QCD only the vector charges are conserved allowing us to order the hadrons inmultiplets of certain isospin and hypercharge quantum numbers.

Nevertheless, theinformation about the axial charges is not lost. Goldstone’s theorem [1] tells us thatfor any broken generator there exists a massless boson, in this case of pseudoscalarnature.

This means that we should expect eight massless pseudoscalar mesons in theparticle data listings. This is not the case.

The fact of the matter is that the Goldstonebosons pick up a small mass related to the small current quark masses, like e.g.M 2π = (mu + md)B + O(M2)(2.4)where B = −< 0|¯qq|0 > /F 2π is the order parameter of the spontaneous symmetrybreaking and Fπ measures the strength of the non-vanishing transition amplitude< 0|Aiµ|πj >= iδijpµFπ . (2.5)In what follows, I will assume that B ∼1 GeV, i.e.

that the quark mass expansionof the pseudoscalar meson masses is indeed governed by the leading term linear inthe quark masses. It is interesting to observe that (2.4) gives us a recipe how totranslate quark into meson masses.

Evidently, at low energies the almost masslessGoldstone bosons are the dominant degrees of freedom. This has been the theme ofHeiri Leutwyler’s lectures [2] two years ago and I refer the reader for a more detaileddiscussion of these underlying ideas.

Also a recent review [3] is available which can beused as a first reading.2.2 Chiral Perturbation Theory (Mesons)The purest reaction of low-energy QCD is the elastic scattering of pions from pionsin the threshold region. I will use this as an example to discuss the principles under-lying the so-called chiral perturbation theory (CHPT).

It is convenient to collect theGoldstone bosons in a matrix-valued field U(x) = exp (iφ(x)/Fπ), in the standardnotation. To lowest order in the momenta and quark masses, the interactions of theGoldstone bosons e.g.

can be described by the gauged non-linear σ-model,

4Ulf-G. MeißnerL(2) = F 24 Tr[∇µU †∇µU] + F 2B4Tr[M(U + U †)] ,(2.6)with ∇µ a covariant derivative including the coupling to the external electroweakfields. The second term in (2.6) reflects the explicit symmetry breaking from thequark masses.

Both terms are of chiral dimension two since they contain either twoderivatives or one quark mass (which is equivalent to two derivatives, see (2.4)). Tolowest order, the strong interactions are therefore given in terms of two parameters.These are F, the pion decay constant in the chiral limit, and B related to the quarkcondensate (remember that I assume B to be substantially larger than F).

Calculatingtree diagrams from this effective Lagrangian leads to the celebrated current algebraresults. In particular, one recovers Weinberg’s prediction for the ππ scattering am-plitude, A(s, t, u) = (s −M 2π)/F 2π [4].

Clearly, this smells like an expansion in energysince close to threshold, the c.m. energy √s becomes very small and, consequently,the interaction between the Goldstone bosons weak.

This is a particular imprint of thespontaneously broken chiral symmetry which allows for a ”perturbative” expansionin the non-perturbative regime of QCD. Obviously, the effective Lagrangian (2.6) cannot be the whole story.

First, one would like to know how big the corrections fromhigher order terms (with more derivatives and/or quark mass insertions) are. Second,and more important, tree diagrams are always real.

Twenty years ago at this school,Lehmann [5] demonstrated that unitarity forces one to consider loop diagrams. Basedon the counting rules developed by Weinberg [6], which state that n-loop diagramsare suppressed by powers of q2n (with q a genuine small momentum), Gasser andLeutwyler [7] have systematized the whole approach for QCD.

Let us consider thecorrections which appear at next-to-leading order. First, there are the one-loop di-agrams mandated by unitarity.

Second, there are higher order contact terms withmore derivatives or quark mass insertions. These are accompanied by a priori un-known coupling constants, the so-called low-energy constants.

Their values can befixed from phenomenology [7] or understood to a high degree of accuracy from theQCD version of the vector-meson dominance principle [8]. It is at this next-to-leadingorder where the various equivalent lowest order approaches differ.

For example, thelinear σ-model is a perfect candidate to be used instead of (2.6), but at next-to-leadingorder it simply gives low-energy constants at variance with the data [7]. Let me returnto the example of ππ scattering.

The amplitude expanded to one loop accuracy takesthe formTππ(s, t, u) = A(s, t, u) + B(s, t, u) + C(s, t, u) ,(2.7)where B(s, t, u) includes the loop (unitarity) effects and C(s, t, u) the contributionfrom the higher dimensional operators. That this gives indeed a better descriptionof the ππ phase shifts than the tree level prediction A(s, t, u) can be seen in ref.

[9].In what follows, I will always confine myself to the one loop approximation. For adiscussion about its range of applicability, see e.g.

ref. [10].

Electroweak Reactions in the Non-Perturbative Regime of QCD52.3 Inclusion of Matter FieldsIn this section I will be concerned with adding the low-lying spin-1/2 baryons (N, Σ,Λ, Ξ) to the effective field theory. The inclusion of such matter fields is less straight-forward since these particles are not related to the symmetry violation.

However, theirinteractions with the Goldstone bosons is dictated by chiral symmetry. Let us denoteby B the conventional 3 × 3 matrix representing the spin-1/2 octet fields,B =1√2Σ0 +1√6ΛΣ+pΣ−−1√2Σ0 +1√6ΛnΞ−Ξ0−2√6Λ(2.8)It is most convenient to choose a non-linear realization of the chiral symmetry [11] sothat B transforms asB →KBK†(2.9)where K is a complicated function that does not only depend on the group elementsgL,R of the SU(3)L,R but also on the Goldstone boson fields collected in U(x), i.e.K(x) = K(gL, gR, U(x)) defines a local transformation.

Expanding K in powers of theGoldstone boson fields, one realizes that a chiral transformation is linked to absorptionor emission of pions, kaons and etas (which was the theme in the days of ”currentalgebra” techniques). These topics are discussed in more detail in [3].

The lowestorder meson-baryon Lagrangian is readily constructed. Let us restrict the discussionto processes with one incoming and one outgoing baryon, such as πN scattering,threshold pion photoprodcution or baryon Compton scattering (otherwise, we wouldhave to add contact n-fermion terms with n ≥4).

In that case, it takes the formL(1)MB = Tri ¯BγµDµB −m0 ¯BB + 12D ¯Bγµγ5{uµ, B} + 12F ¯Bγµγ5[uµ, B](2.10)with m0 the average octet mass (in the chiral limit) and Dµ the usual covariantderivative. There are two axial-vector type couplings multiplied by the conventionalF and D constants.

In the case of two flavors, there is only one such term propor-tional to gA = F + D. Notice that the lowest order effective Lagrangian containsone derivative and therefore is of dimension one as indicated by the superscript ’(1)’.In contrast to the meson sector (2.6), odd powers of the small momentum q are al-lowed (thus, to leading order, no quark mass insertion appears since M ∼q2). Itis instructive to expand (2.10) in powers of the Goldstone and external fields.

Fromthe vectorial term, one gets the minimal photon-baryon coupling, the two-Goldstoneseagull (Weinberg term) and many others. Expansion of the axial-vectors leads tothe pseudovector meson-baryon coupling, the celebrated Kroll-Rudermann term andmuch more.

However, as already stated, it is not sufficient to calculate tree diagramsfrom the lowest order effective theory. In the presence of baryons, the loop expansionis more complicated than discussed in the previous section.

First, since odd powers inq are allowed, a one-loop calculation of order q3 involves contact terms of dimensiontwo and three, i.e. combinations of zero or one quark mass insertions with zero tothree derivatives.

These terms are collected in L(2,3)MB and a complete list of them canbe found in Krause’s paper [12]. Second, the finiteness of the baryon mass in the chirallimit and that its value is comparable to the chiral symmetry breaking scale Λ ∼Mρ

6Ulf-G. Meißnercomplicates the low energy structure. This has been discussed in detail by Gasseret al.

[13]. Let me just give one illustrative example.

The one loop contribution tothe nucleon mass not only gives the celebrated non-analytic contribution proportionalto M 3π ∼M3/2 but also an infinite shift of m0 which has to be compensated by acounterterm of dimension zero. It is a general feature that loops produce analyticcontributions at orders below what one would naively expect (e.g.

below q3 from oneloop diagrams). Therefore, in a CHPT calculation involving baryons one has to worrymore about higher order contributions than it is the case in the meson sector.

Thereis one way of curing this problem, namely to go into the extreme non-relativisticlimit [14] and consider the baryons as very heavy (static) sources. Then, by a cleverdefinition of velocity-dependent fields, one can eliminate the baryon mass term fromthe lowest order effective Lagrangian and expand all interaction vertices and baryonpropagators in increasing powers of 1/m0.

This is similar to a Foldy-Wouthuysentransformation you all know from QED. In this limit one recovers a consistent deriva-tive expansion.

For example, the one loop contribution of the Goldstone bosons tothe baryon self-energy is nothing but the non-analytic M 3φ terms together with threecontact terms from L(2)MB. However, one has to be somewhat careful still.

The essenceof the heavy mass formalism is that one works with old-fashioned time-ordered per-turbation theory. So one has to watch out for the appearance of possible small energydenominators (infrared singularities).

This problem has been addressed by Weinberg[15] in his discussion about the nature of the nuclear forces. The dangerous diagramsare the ones were cutting one pion line (this only concerns pions which are not inthe asymptotic in- or out-states) separates the diagram into two disconnected pieces(one therefore speaks of reducible diagrams).

These diagrams should be inserted in aSchr¨odinger equation or a relativistic generalization thereof with the irreducible onesentering as a potential. So the full CHPT machinery is applied to the irreducible dia-grams.

This should be kept in mind. For the purposes I am discussing, we do not needto worry about these complications.

Being aware of them, it is then straightforwardto apply baryon CHPT to many nuclear and particle physics problems [3,16,17]. Iwill illustrate this on two particular examples in the next sections.

Before doing that,however, I would like to stress that all these calculations are only in their infancy.It is believed that for a good quantitative description one has to perform systematiccalculations to order q4, i.e. beyond next-to-leading order.

There are many indicationsof this coming from the baryon masses, hyperon non-leptonic decays and so on. Atpresent, only some rather unsystematic attempts have been done which incorporatesome q4 effects.

In particular, it has been argued that one should include the low-lyingspin-3/2 decuplet in the effective theory from the start [16,18]. A critical discussionof this approach can be found in ref.[19].

Clearly, more work is needed to clarify theseissues. Let me now proceed and discuss two applications of the framework outlinedhere.

These are calculations in flavor SU(2) were the loop corrections are considerablysmaller. As I will show, they shed some light on the non-perturbative structure of thenucleon.

Electroweak Reactions in the Non-Perturbative Regime of QCD72.4 Threshold Pion Photo- and ElectroproductionLet us consider the reaction γ(k) + N1(p1) →πa(q) + N2(p2) with N1,2 denotingprotons and/or neutrons and ’a’ refers to the charge of the produced pion. In the caseof real photons (k2 = 0) one talks of photoproduction whereas for virtual photons(radiated offan electron beam) the process is called electroproduction.

Of particularinterest is the threshold region where the photon has just enough energy to producethe pion at rest or with a very small three-momentum. In this kinematical regime, it isadvantegous to perform a multipole decomposition since at threshold only the S-wavessurvive.

These multipoles are labelled E (M)l±, with E (M) for electric (magnetic),l = 0, 1, 2, . .

. the pion orbital angular momentum and the ± refers to the total angularmomentum of the pion-nucleon system, j = l ± 1/2.

They parametrize the structureof the nucleon as probed with low energy photons. Let us first concentrate on thephotoproduction case.

At threshold, the differential cross section takes the form|k||q|dσdΩq→0= (E0+)2(2.11)i.e. it is entirely given in terms of the electric dipole amplitude E0+ (which at thresholdis real).

The quark mass expansion of E0+ is given byE0+ = −ie8πmµ(2 + µ)(1 + µ)3/2 f( ¯m/ΛQCD)(2.12)with µ = Mπ/m ≃1/7 and ¯m the renormalization group invariant quark masses.Modulo logarithms, one can expand f in powers of µ,f = 1µf−1 + f0 + µf1 + O(µ2). (2.13)For charged pions f−1 is non-vanishing and one is lead to the famous Kroll-Rudermannresult [20] which states that E0+ for γp →π+n and γn →π−p can be expressedin terms of the strong pion-nucleon coupling constant, the nucleon mass and somekinematical factors.

Consequently, in these cases E0+ is finite in the chiral limit.Matters are different for the case of neutral pions. Here, f−1 vanishes and thus theleading term in the expansion of E0+ goes like µ (µ2) for γp →π0p (γn →π0n).The amplitude is therefore sensitive to the chiral symmetry breaking which makesit a good object to study experimentally and theoretically.

If one performs a oneloop CHPT calculation, one can get unambigously the next-to-leading order terms inthe quark mass expansion of the eletric dipole amplitude. For the production of theproton, the threshold value is [21]E0+ = −egπN8πm µ1 −12(3 + κp) + ( m4Fπ)2µ + O(µ2),(2.14)with κp = 3.71 the anomalous magnetic moment of the proton.

The second term inthe square brackets is the new one found in the CHPT calculation. It stems from theso-called triangle diagram and its crossed partner.

If one expresses E0+ in terms of theconventional Lorentz invariant functions A1,2,3,4, one finds that these diagrams givea contribution δA1 = (egπN/32F 2π) µ, i.e. they are non-analytic in the quark masses

8Ulf-G. Meißnersince µ ∼√ˆm. Remember also that the electric dipole amplitude at threshold isproportional to µA1 and therefore one gets this novel contribution at next-to-leadingorder in the quark mass expansion.

Clearly, the expansion in µ is slowly converging,the coefficient of the term of order µ2 is so large that it compensates the leading termproportional to µ. Therefore, for a meaningful prediction one has to go further in theexpansion.

This has been done in the relativistic formalism which sums up some of thehigher order corrections [22]. One finds (at threshold) E0+ = −1.33 ± 0.09 using theconventional units of 10−3/Mπ+.

The error reflects solely the uncertainty in estimatingthe finite contact term contributing at order q3. This is somewhat below the generallyaccepted experimental value of −2.0 ± 0.2 (see e.g.

Bernstein and Holstein [23] orDrechsel and Tiator [24] and references therein). However, one should point out thatthe calculation was performed in the isospin limit with Mπ± = Mπ0 and mp = mn.

Itis a very tough problem to include isospin-breaking in a systematic fashion - I inviteyou to solve it. To get an idea about the effects related to it, one can perform a toycalculation and set the pion and nucleon masses by hand on their physical values inthe various diagrams.

Then, one finds E0+ = −1.97, which is an encouraging numberbut it has to be confirmed by a better calculation. The chief reason for the differenceto the isospin-symmetric case is that the contribution of the triangle diagram nowvanishes below the π+n threshold.

It is furthermore interesting to observe that theone-loop corrections to the electric dipole amplitudes of charged pion photoproductionmove the prediction closer to the data. These chiral corrections are suppressed bypowers of µ2 and µ2 ln µ with respect to the leading Kroll-Ruderman term (∼µ0) andtherefore a more accurate experimental determination of these quantities is called for.Let me now consider the electroproduction process.

First, the low-energy theoremsfor the two S-wave multipoles E0+ and L0+ (for virtual photons, there is also alongitudinal coupling to the nucleon) have been discussed in the light of CHPT inref.[25]. Let me stress that the LETs derived in refs.

[21,25] are the ones implied byQCD and that they can be tested experimentally. Notice that there still seems tobe some confusion concerning the meaning and the interpretation of these theorems[26].

However, I will not dwell on this topic here. Instead, I want to direct yourattention to some new data.

Welch et al. [27] have published the S-wave cross sectionfor the reaction γ⋆p →π0p very close to the photon point.

This measurement isa quantum step compared to previous determinations which mostly date back tothe seventies when pion electroproduction was still a hot topic in particle physics.In this experiment, k2 varied between -0.04 and -0.10 GeV−2 and the S-wave crosssection could be extracted with an unprecedented accuracy (see fig.2 in [27]). Thisis also the kinematical regime where a CHPT calculation might offer some insight.Indeed, in ref.

[28] it was shown that the k2-dependence of this cross section seems toindicate the necessity of loop effects. With conventional models including e.g.

formfactors and the anomalous magnetic moment coupling the trend of the data can notbe described. However, the corrections from the one loop diagrams to the tree levelprediction are substantial.

This gives further credit to the previously made statementthat a calculation beyond next-to-leading order should be performed. The last topic Iwant to address in this section concerns the determination of the nucleon axial radiusfrom charged pion electroproduction.

Let me briefly explain how the axial form factorcomes into play. The basic matrix element to be considered is the time-ordered productof the electromagnetic (vector) current with the interpolating pion field sandwiched

Electroweak Reactions in the Non-Perturbative Regime of QCD9between nucleon states. Now one can use the PCAC relation and express the pionfield in terms of the divergence of the axial current.

Thus, a commutator of the form[V,A] arises. Current algebra tells us that this gives an axial current between theincoming and outgoing nucleon fields and, alas, the axial form factor.

The isospinfactors combine in a way that they form a totally antisymmetric combination whichcan not be probed in neutral pion production. These ideas were formalized in thevenerable low-energy theorem (LET) due to Nambu, Luri´e and Shrauner [29] for theisospin–odd electric dipole amplitude E(−)0+ in the chiral limit,E(−)0+ (Mπ = 0, k2) = egA8πFπ1 + k26 r2A + k24m2 (κV + 12) + O(k3)(2.15)Therefore, measuring the reactions γ⋆p →π+n and γ⋆n →π−p allows to extract E(−)0+and one can determine the axial radius of the nucleon, rA.

This quantity measuresthe distribution of spin and isospin in the nucleon, i.e. probes the Gamov–Telleroperator σ · τ.

A priori, the axial radius is expected to be different from the typicalelectromagnetic size, rem ≃0.85 fm. It is customary to parametrize the axial formfactor GA(k2) by a dipole form, GA(k2) = (1 −k2/M 2A)−2 which leads to the relationrA =√12/MA.

The axial radius determined from electroproduction data is typicallyrA = 0.59±0.04 fm whereas (anti)neutrino-nucleon reactions lead to somewhat largervalues, rA = 0.65 ± 0.03 fm. This discrepancy is usually not taken seriously since thevalues overlap within the error bars.

However, it was shown in ref. [30] that pion loopsmodify the LET (2.15) at order k2 for finite pion mass.

In the heavy mass formalism,the coefficient of the k2 term reads16r2A +14m2 (κV + 12) +1128F 2π(1 −12π2 )(2.16)where the last term in (2.16) is the new one. This means that previously one hadextracted a modified radius, the correction being 3(1 −12/π2)/64F 2π ≃−0.046 fm2.This closes the gap between the values of rA extracted from electroproduction andneutrino data.

It remains to be seen how the 1/m suppressed terms will modify theresult (2.16). Such investigations are underway.2.5 Nucleon Compton ScatteringConsider low-energy (real) photons scattering offa spin-1/2 particle.

In forward di-rection, the scattering amplitude takes the formT1/2(ω) = f1(ω2) ǫ∗f · ǫi + iω f2(ω2) σ · (ǫ∗f × ǫi)(2.17)where ω is the photon frequency and the ǫi,f are the polarization vectors in the initialand final state, respectively. The energy expansion of the spin-dependent amplitudef1(ω2) readsf1(ω2) = −e2Z24πm + (α + β) ω2 + O(ω4)(2.18)where the first energy-independent term is nothing but the Thomson amplitude man-dated by gauge invariance.

Therefore, to leading order, the photon only probes some

10Ulf-G. Meißnerglobal properties like the mass or electric charge of the spin-1/2 target. At next-to-leading order, the non-perturbative structure is parametrized by two constants, theso-called electric and magnetic polarizabilities (more correctly, one might want totalk about ”magnetic susceptibility”, however, I will use the common language).

Adetailed discussion of the CHPT calculation of these fundamental two-photon observ-ables is given in ref. [3] (see also the references quoted there).

Here, I just want toadd a few educational remarks. For the proton, the Thomson term is non-vanishingand it is interesting to compare its magnitude to the one from the polarizability con-tribution using (α + β)p = 14.2 · 10−4 fm3.

The recent Illinois Compton scatteringexperiment [31] to determine αp and βp was performed at photon energies rangingfrom 32 to 72 MeV. While at the lowest ω the polarizability term amounts to a 31per cent correction to the Thomson term contribution (= −1.22 · 10−4 fm), at 72MeV the second term in the energy expansion is 1.5 times as large as the leadingterm.

Clearly, higher order (ω4) contributions have to be taken into account at suchenergies. Another remark concerns the behaviour of αp,n and βp,n in the chiral limit.These quantities diverge as 1/Mπ as the pion mass tends to zero.

This is expectedsince the two photons probe the long-ranged pion cloud, i.e. there is no more Yukawasuppression as in the case for a finite pion mass.

Now a well-known dispersion sumrule relates (α+β) to the total nucleon photoabsorption cross section. The latter is, ofcourse, also well-behaved in the chiral limit which at first sight seems to be at variancewith the behaviour of the expansion of the scattering amplitude.

But be aware thatthe general form of (2.18) has been derived under the assumption that there is a welldefined low-energy limit. Similar observations can also be made concerning the chiralexpansion of the ππ scattering amplitude discussed in section 2.

Let me now focus onthe spin-dependent amplitude f2(ω2) which has an expansion analogous to (2.18),f2(ω2) = f2(0) + γ ω2 + O(ω4)(2.19)By the optical theorem, the imaginary part of f2 is related to the photoabsorptioncross sections for circularly polarized photons on polarized nucleons,Im f2(ω2) = −18πσ3/2(ω) −σ1/2(ω)(2.20)where the indices 3/2 and 1/2 refer to the total γN helicities. We will come back tothis relation later on.

Let me now turn to the discussion of the Taylor coefficientsf2(0) and γ in eq.(2.19). There exists a celebrated LET for f2(0) due to Low, Gell-Mann and Goldberger [32].

Using very general principles like gauge invariance, Lorentzinvariance and crossing symmetry, they showed thatf2(0) = −e2κ28πm2 . (2.21)Here, κ denotes the anomalous magnetic moment of the particle the photon scattersoff.

In CHPT, one can easily derive the LET making use of the heavy baryon formalismas shown in ref.[33]. It stems from some contact terms which are nominally suppressedby powers of 1/m.

To one-loop order, one can give f2(ω2) in closed form since only afew loop diagrams contribute (in addition to the contact terms giving the LET),

Electroweak Reactions in the Non-Perturbative Regime of QCD11f2(ω2) = −e2κ28πm2 +e2g2A32π3F 2M 2ω2 arcsin2( ωM ) −1(2.22)From this the slope parameter γ follows immediately. Notice that γ is again the sumof an electric and magnetic piece, but in the absence of any data, I will only discussthe sum.

The em decomposition can be supplied upon request. To leading order, γ isidentical for the p and the n,γ =e2g2A96π3F 2πM 2π= 4.4 · 10−4 fm4 .

(2.23)In ref. [33] it was shown that including 1/m suppressed effects in the relativistic for-malism together with an estimate of the ∆(1232) resonance contribution leads toγp = −1.5 and γn = −0.5 (in conventional units), in fair agreement with the estimatebased on the dispersion sum ruleγ = −14π2Z ∞Mπdωω2σ3/2(ω) −σ1/2(ω)(2.24)which follows directly from eq.(2.20).

Using the helicity cross sections provided byWorkman and Arndt [34] one gets γempp= −0.89 and γempn= −0.53. Clearly, a directexperimental determination of these observables is called for.

To give an idea aboutthe size of the effect induced by the ”spin-polarizability” γ, let us compare the leadingorder term f2(0) = −5.19 ·10−4 fm2 with the correction from γω2 at various energies.For ω = 50 MeV, this term amounts to a 6 per cent reduction of the leading one. At100 MeV photon energy, the correction has grown to 22 per cent.

For comparison,evaluating the full one-loop amplitude (2.22), one finds corrections of 7 and 37 percentfor ω = 50 and 100 MeV, respectively. These latter numbers will, of course, be changedby higher loop corrections.

We notice that the influence of the next-to-leading orderterm is much less pronounced than in the case of the spin-independent amplitude andthus an accurate determination of γ and its electric and magnetic components will bemore difficult. The last topic I want to address in this section is the so-called Drell-Hearn-Gerasimov [35] sum rule and its extension to virtual photons.

For photons withfour-momentum k2 ≤0 the extended DHG sum rule takes the formI(k2) =Z ∞ωthrdωωσ1/2(ω, k2) −σ3/2(ω, k2)(2.25)For real photons, this reduces to the DHG sum rule, which has not yet been testedexperimentally. For the proton, one has I(0) = −πe2κ2p/2m2 = −0.53 GeV−2.

Atlarge k2 ≃−10 GeV2, the EMC [36] data tell us that I is positive and thus has to un-dergo a sign change. CHPT allows one to calculate the slope of I(k2) in the vicinity ofthe photon point [37].

While the most recent phenomenological analysis gives a kinkstructure at small k2 [38], in CHPT one finds that I(k2) increases monotonically. Atpresent, one can not decide upon this fine detail but has to wait for the experimentaldeterminations which are possible with CW machines and are planned (MAMI, CE-BAF, .

. .

). Finally, I wish to stress again that CHPT allows to systematically explorethe consequences of the spontaneously broken chiral symmetry of QCD.

It is not amodel as should have become clear from the above discussion but rather an exactnon-perturbative method.

12Ulf-G. Meißner3 The Quark Structure of the NucleonIn this section I will be concerned with the quark structure of the nucleon as re-vealed from measurements using electroweak probes. In particular, one can addressthe questions surrounding the possible admixtures of strange quark components intothe proton’s wave function.

Since these are mostly small effects, I will also discuss theeffects of heavy quark and radiative corrections to the processes which allow one todetermine the electroweak form factors.3.1 Currents, Sizes and Form Factors of the NucleonThe starting point of the discussion are the well-known electromagnetic and axialcurrents based on photon and charged vector boson exchanges. Let us first considerthe vector current Jµ stemming from the one-photon exchange.

Between nucleonstates of momentum p and helicity s (in general, I will suppress helicitiy indices)Lorentz-invariance, parity and charge conjugation allow us to write< N(p′, s′)|Jµ|N(p, s) >= e¯u(p′, s′)F1(q2)γµ + F2(q2)iσµνqν2mu(p, s)(3.1)with qµ = (p′ −p)µ the four-momentum transfer, m denotes the nucleon mass andu(p, s) a conventional nucleon spinor. The finite extension (non-perturbative struc-ture) is parametrized in terms of the Dirac and Pauli form factors (ffs), F1(q2) andF2(q2), respectively.

Gauge invariance of the electromagnetic interactions demands∂µJµ(x) = 0. The physical interpretation of these ffs is particularly transparent inthe Breit (brick wall) frame, in which the photon transfers no energy, qµ = (0, q).

Thetime component of Jµ is the charge density, J0 = ρ and the space components leadto the distribution of magnetism,< N(q/2, s′)|J0|N(−q/2, s) > = eGE(q2)δs′s= e[F1(q2) −τF2(q2)]δs′s< N(q/2, s′)|J|(−q/2, s) > =e2mGM(q2)χ†s′iσ × qχs=e2m[F1(q2) + F2(q2)]χ†s′iσ × qχs(3.2)with χs a two-component spinor and τ = −q2/4m2 = Q2/4m2 (Q2 > 0 for space-likephotons). Fourier-transformation of the electric charge density leads toZρ(r)j0(qr)d3r =Zρ(r)d3r −16Q2Zρ(r)r2d3r + .

. .= Z −16Q2 < r2 >E + .

. .

(3.3)where the first term is obviously the electric charge of the particle (here, the proton orthe neutron) and the second one defines the electric charge radius. It is conventionalto normalize these charge radii via< r2 >=6G(0)dG(q2)dq2q2=0(3.4)

Electroweak Reactions in the Non-Perturbative Regime of QCD13with the exception of the neutron charge radius since in that case G(0) = 0 so that< r2E >n is simply six times the slope of the neutron form factor. The electric (E)and magnetic (M) ffs are normalized as followsGpE(0) = 1,GnE(0) = 0,GpM(0) = µp = 2.793,GnM(0) = µn = −1.913 .

(3.5a)For the various radii, the presently available data are [39]< r2E >1/2p= 0.86 ± 0.01 fm,< r2M >1/2p= 0.86 ± 0.06 fm,< r2E >n= −0.12 ± 0.01 fm2,< r2M >1/2n = 0.88 ± 0.07 fm . (3.5b)A few remarks on these numbers are in order.

First, the typical electromagnetic (em)size is rem ≃0.85 fm and the negative value of the neutron charge radius squaredshows that a surplus of negative charge in the exterior region of the neutron mustexist. Most disturbing is the fact that in particular the magnetic charge radius ofthe neutron, which is a quantity as fundamental as say the magnetic moment, is onlypoorly known.

As pointed out by Arenh¨ovel and Schoch at this school [40] experimentsare under way to remedy the situation. At larger momentum transfers, it has becomecommon to use the so-called dipole fits, which proved to be embarrasingly good upto several GeV2 on typical logarithmic plots when elastic electron-proton scatteringexperiments were performed in the sixties and early seventies as discussed by Taylorhere [41].

The dipole fit readsGpE(Q2) = GpM(Q2)µp= GnM(Q2)µn= GD(Q2) =1(1 + Q2/0.71 GeV2)2 . (3.6)For the electric ffof the neutron, the ”trivial” dipole fit is GnE(Q2) = 0, however, innature this ffis small but non-vanishing.

It can be approximated by [42]GnE(Q2) = −µnτ1 + Q2/0.42 GeV2GD(Q2) . (3.7)In refs.

[43], you can take a look at the quality of these fits on a linear scale for Q2 upto 4 GeV2 - it is obvious that the dipole fit is at most a fair approximation. Also, it hasnever been given a sound theoretical foundation.

While the dipole mass MD = 0.843GeV is suggestive of vector meson exchange (like in the vector meson dominanceapproach to photon-hadron couplings), such a mechanism would naturally lead to amonopole fall-off. In case of the neutron electric ff, which is determined indirectly fromelastic scattering offthe deuteron and subtraction of the proton ff, a major uncertaintystems from the use of the underlying model of the nucleon-nucleon force as discussedin ref.[42].

Here, experiments using polarized electrons and polarized helium targetsare hoped to improve the theoretical uncertainties. Definitively, we need a betterdetermination of these fundamental nucleon properties.

It is also instructive to seewhat perturbative QCD tells us about the large Q2 behaviour of these ffs. Based on thequark counting rules [44], which state that to distribute the large photon momentumequally to three quarks in the nucleon, two gluon exchanges are necessary, one findsthat F1(Q2) ∼1/Q4 and F2(Q2) ∼1/Q6 since the gluon propagator goes like 1/Q2and in the case of F2 an extra 1/Q2 is needed for the helicity flip.

At which value ofQ2 this asymptotic behaviour sets in is not known, certainly way beyond 1 GeV2 as

14Ulf-G. Meißneroriginaly thought. Notice that the dipole fits to the nucleon ffs have the correct largeQ2 behaviour.

This is a further mystery surrounding these fits. It might be based onthe fact that a simple meson-cloud model of the nucleon also leads to such fall-offs(in the one-boson exchange approximation).

Let me now switch to the axial current(related to the exchange of charged vector bosons). Its nucleon matrix element reads< N(p′)|Aµi (0)|N(p) >= ¯u(p′)GA(q2)γµ + GP (q2)2mqµγ5τi2 u(p)(3.8)where GA(q2) is the axial ff.

The induced pseudoscalar ffGP (q2) is essentially poledominated and only accessible via scattering processes involving heavy leptons (likee.g. muons).

I will not consider it in what follows. GA(0) is nothing but the axial-vector coupling constant measured in neutron β-decay, gA = 1.26.

The Q2-dependenceof GA(Q2) follows again a dipole form,GA(Q2) =gA(1 + Q2/M 2A)2. (3.9)The value of the axial cut offmass was already discussed in section 2.4, it translatesinto a typical axial size of rA ≃0.65 fm.

This is considerably smaller than the typicalem size. Therefore, the size of the nucleon depends on the probe one uses.

In fact,the hierachy of these various nucleon sizes can nicely be understood in the topologicalsoliton model of the nucleon as spelled out in detail in ref.[45]. In a nutshell, theargument goes as follows.

The isoscalar baryon number current, whose space-integralis the baryon number, is given by the pion fields in the soliton with an extension ofrB = 0.5 fm. The isoscalar photon, however, sees in addition the virtual ω-mesoncontent of the nucleon.

This leads to an isoscalar charge radius of < r2E >I=0=r2B + 6/M 2ω = (0.8 fm)2 in nice agreement with the data. By a similar argument, oneexpects a somewhat smaller axial radius since the additional factor now carries theaxial vector mass of 1.1 GeV and one is thus naturally lead to an axial extension of0.65 fm.3.2 A Toy Model: Quark Distributions from Nucleon Form FactorsTo understand how the diferent electroweak properties as parametrized by the variousffs can give insight into the quark structure of the nucleon in the non-perturbativeregime, let us consider a simple (too simple) model.

The proton and the neutron aremade up of valence u and d quarks and a sea of quark-antiquark pairs. We assume thatthe p and the n are simply related by the interchange of one valence u with one valenced quark.

For the sake of simplicity, let us forget the gluons and just imagine someconfinement mechanism which keeps the current quarks within the typical hadronicvolume of 1 fm3. Denoting by Jjµ the em current associated to the quarks of flavorj = u, d, s, c, b, t, we can write [46]< N(p′, s′)|Jµ|N(p, s) > =< N(p′, s′)|XjJjµ|N(p, s) >= eXj¯u(p′, s′)QjγµF j1 + iσµνqν2mF j2u(p, s)(3.10)

Electroweak Reactions in the Non-Perturbative Regime of QCD15where Qj denotes the quark charges and the quarks are considered as point-like Diracparticles with no anomalous magnetic moment. In terms of the electric and magneticffs, this decomposition takes the formG(p,n)E,M =XjQjGj (p,n)E,M.

(3.11)These expressions are exact under the assumptions made as long as the sum extendsover all quark flavors. Furthermore, the Gj contain contributions from quarks as wellas from antiquarks.

Therefore, the q¯q sea does not contribute to the total charge sinceQj = −¯Qj. Since the p and the n as well as the u and the d quark are consideredto be essentially the same particles differing only by an isospin rotation, it should bepossible to separate the response of the various quark flavors [47].

For doing that,one has to make further assumptions. First, one postulates a local isospin invarianceof the type Ju,pµ (x) = Jd,nµ(x) and Jd,pµ (x) = Ju,nµ(x) (which has never been testedand is doubtful in the light of the recent structure function measurements).

Second,we assume that all other quark flavors do not contribute, i.e. Jq,pµ (x) = Jq,nµ (x) = 0for q = s, c, b, t. This allows to uniquely give the em u and d distributions in termsof appropriate combinations of the em ffs of the proton and the neutron.

The finalformulae are obtained by normalizing with respect to the total quark charges, e.g.pulling out a factor 4/3 for the u quark distribution in the proton. This leads to [48]GuE,M = GpE,M + 12GnE,M ,GdE,M = GpE,M + 2GnE,M .

(3.12)It is instructive to get some numbers from eq.(3.12). First, let us stick in the mag-netic moments.

We find GuM(0) = 1.836 and GdM(0) = −1.033, which is not verydifferent from the SU(3) result for constituent quarks. Notice, however, that herewe are considering current quarks.

The implications of this result are discussed inmore detail in Beck’s lecture [48]. Similarly, for the electric charge radii one finds< r2E >u= 0.68 ± 0.02 fm2 and < r2E >d= 0.51 ± 0.02 fm2 which is a 35 per centdifference.

To further separate the valence from the sea quarks, one needs additionalassumptions about the radial distributions as detailed in ref.[48]. In any case, thisdiscussion should only be considered illustrative, the model is too crude to attacha deep physical significance to these numbers.

However, in a very simple fashion itdemonstrates how a clever combination of measurable hadron form factors allows usto make statements about quark distributions. This problem will now be tackled in amore serious fashion.3.3 Electroweak CurrentsIn this section, I will define the electroweak currents related to the exchanges of thevarious vector bosons of the minimal standard model.

These spin-1 particles are themassless photon and the massive W ± and Z0 bosons. At tree level, the couplings ofthese to the various quark flavors are specified completely in terms of the U(1) gaugecoupling g and the SU(2) gauge coupling g′.

Equivalently, one can use the em couplinge and weak mixing angle θW since g = e sin θW = e s and g′ = g/4 cosθW = g/4 c.

16Ulf-G. MeißnerAlso, the vector boson masses are related at tree level via MZ = MW /c. For themost recent values, see Altarelli’s lectures at this school [49].

Let us consider againthe photon current which is purely vectorial and its couplings to the light quarks u,d and s. One hasJγµ =Xq=u,d,sQq ¯qγµq = 23(¯uγµu) −13( ¯dγµd + ¯sγµs)=Xq=u,d,s¯qγµ12λ3 + 1√3λ8q= 12(¯uγµu −¯dγµd) + 16(¯uγµu + ¯dγµd −2¯sγµs). (3.13)I have exhibited the two most frequently used conventions.

In the second line, thedecomposition in terms of Gell-Mann’s flavor SU(3) matrices is shown. One noticesthat the photon is blind to the singlet piece ∼λ0.

In the third line, the photoncurrent is decomposed into its isovector (IV) and isoscalar (IS) components. Thiscurrent leads to the em form factors GγE,M or F γ1,2 discussed before (the label ’γ’ isneeded to differentiate these ffs from the ones related to the weak neutral currentas discussed below).

The charged weak currents are flavor-changing and lead to theisovector axial ffdiscussed before. Further information comes from the weak neutralcurrent mediated by the Z0.

Its coupling to the quark has a vectorial and an axial-vector piece. Ignoring the overall coupling strength g′ and defining Vq = ¯qγµq for aquark of flavor q, the vector part at tree level readsV Zµ =14 −23s2Vu +−14 + 13s2(Vd + Vs)= 1212 −s2(Vu −Vd) + 1612 −s2(Vu + Vd −2Vs) −112(Vu + Vd + Vs)=12 −s212(Vu −Vd) + 16s2(Vu + Vd −2Vs) −14Vs(3.14)where again the second line gives the SU(3) components and the third line the de-composition into IS, IV and explicit strange pieces.

Similarly, with Aq = ¯qγµγ5q wehave for the axial-vector currentAZµ = −14Au + 14(Ad + As)= −14(Au −Ad) −112(Au + Ad −2As) + 112(Au + Ad + As)= −1212(Au −Ad)+ 14As(3.15)Two remarks are in order. First, the Z0 couples to the singlet component ∼λ0 ∼(u + d + s) and, second, there is no isoscalar axial coupling at tree level.

For thefollowing discussion, let me express the ffs related to the Z0-exchange in the flavorSU(3) basis following Kaplan and Manohar [50],

Electroweak Reactions in the Non-Perturbative Regime of QCD17F Z1,2 =Xα=0,3,8(bα −aαs2)F α1,2 ;F γ1,2 =Xα=0,3,8aαF α1,2 ,G1 = −Xα=0,3,8bαGαA = −Xα=0,3,8bαGα1 ,a0 = 0 ,a3 = 1 ,a8 =1√3 ;b0 = −14 ,b3 = 12 ,b8 =12√3 . (3.16)The value of the a’s and b’s are subject to radiative and heavy quark corrections asdiscussed below.

For completeness, I have also given the em ffs in this basis. Using forthe SU(3) generators Tr(T aT b) = δab/2 and supplying the isosinglet currents withan overall factor 1/3, the normalizations of the various ffs are given as follows.

Fromthe proton and the neutron charge, we haveF 31 (0) = 1/2 ,F 81 (0) =√3/2. (3.16a)Similarly, the p and n anomalous magnetic moments lead toF 32 (0) = (κp −κn)/2 ,F 82 (0) =√3(κp + κn)/2.

(3.16b)F 01 (0) is equal to unity since it gives the baryon number. However, this does notexclude a finite extension of s¯s pairs leading to a strange electric radius (in completeanalogy to the neutron charge radius).

The value of the singlet anomalous magneticmoment is not known. If there are no strange quarks in the proton, one has F 02 (0) =κp + κn = −0.12.

So a measurement of F 02 (0) will give information about the possiblestrange quark contribution to the anomalous magnetic moment, F 02 (0) = κp+κn+µs.The triplet and octet normalizations of the axial ffare given byG31(0) = gA = F + D = 1.26 ,G81(0) = (3F −D)/√3 = 0.32 . (3.16c)The isosinglet charge G01(0) is again sensitive to the strange quark content and cane.g.

be determined in polarized deep inelastic lepton scattering as discussed by Reyahere [51] (see also ref.[36]). Alternatively, neutrino scattering offnucleons or nucleimight be used to get a handle on G01(0).

This will be discussed below. At present,no experimental information is available about the Q2-dependence of the singlet andoctet axial ffs, so one either has to resort to some assumptions or models (see section3.5).

Clearly, the singlet axial charge and the singlet anomalous magnetic moment arethe best objects to study the strange quark content of the nucleon using electroweakprobes. The other obvious candidate is the so-called pion-nucleon σ-term which mea-sures the strength of the strange matrix element ms < p|¯ss|p >.

Combining πNdata, dispersion theory and constraints from chiral symmetry, Gasser et al. [52] havegiven the most precise determination of this quantity and its t-dependence.

From thisanalysis one deduces that ms < p|¯ss|p >≃130 MeV. This is a sizeable though notdramatic strange quark effect.

After defining the electroweak form factors, I will nowdiscuss some experiments which allow to extract them and the related strange quarkmatrix elements.

18Ulf-G. Meißner3.4 Parity-Violating Electron ScatteringConsider the scattering of polarized electrons on a nucleon or a nucleus. Due to the γ−Z0 interference, parity violation (pv) occurs and this allows to determine combinationsof the electromagnetic and weak neutral ffs.

The basic Feynman diagrams are simplythe one-photon exchange proprotional to F γ1,2 and the Z0-exchange proportional toF Z1,2 and G1. To get an idea about the magnitude of this interference effect, let usperform some dimensional analysis.

The elctromagnetic amplitude readsAem =< f|Jµem1Q2 jµ,em|i >∼αQ2 ,(3.17)where jµ,em is the well-known em lepton current, 1/Q2 the photon propagator andJµem the hadronic vector current parametrized by the em ffs. The parity-violating weakamplitude takes the formApvw =< f|V µaµ + Aµvµ|i >∼GF√2 ,(3.18)which is a product of the lepton (vµ, aµ) and hadron (Vµ, Aµ) vector and axial-vectorcurrents, the latter containing the weak form factors.

GF = 1.16 · 10−5 GeV−2 is theFermi constant. The total cross section can approximately be written asσ = |Aem + Apvw |2 ≃|Aem|2 + 2|A∗emApvw | .

(3.19)The observable of interest is the left-right asymmetry A,A = σR −σLσR + σL≃|A∗emApvw ||Aem|2∼GF m24παQ2m2 = 1.1 · 10−4 Q2GeV2 . (3.20)For a typical four-momentum of Q2 = 0.1 GeV2 in the non-perturbative regime wecan expect an asymmetry of 10−5.

This is a small number but within the reach ofpresent day technology. Historically, the first measurement of this asymmetry using adeuterium target at SLAC was performed at Q2 around 20 GeV2 [53].

To be specific,consider now as the target the proton. An elementary calculation givesAep = −GF Q2√2παǫGγEGZE + τGγMGZM −12(1 −4s2)√1 −ǫ2pτ(1 + τ)GγMG1ǫ(GγE)2 + τ(GγM)2= −GF Q2√2πα [AE + AM + AA](3.21)with 1/ǫ = [1 + 2(1 + τ) tan2(θ/2)] = 0 .

. .

1 given by the scattering angle of theelectron. In the second line, I have split the asymmetry into an electric, magneticand axial piece.

It is instructive to consider various kinematical regimes. At forwardangles (θ →0, ǫ →1), the electric part dominates at low Q2 and the asymmetryis thus sensitive to the strange electric ff.

At higher Q2, due to the τ prefactor, themagnetic part takes over and one can access GsM(Q2). Notice that in forward directionthe axial contribution is not only suppressed by 1 −4s2 ≃0.08 but also by√1 −ǫ2.In backward direction, matters are different since as θ →0, ǫ also tends to zero.

Soat low Q2 one can get GsM(Q2) with some contamination from the axial part. This

Electroweak Reactions in the Non-Perturbative Regime of QCD19is the kinematical regime where the SAMPLE experiment [54] hopes to determinethe strange anomalous magnetic moment. This warrants a closer look.

In a simplifiedanalysis, we set s2 = 1/4, θ = π and work at some low Q2, say 0.2 GeV2. Then,A(Q2) takes the simple form [55]A(Q2) ≃−GF Q2√2παGZMGγM≃−GF Q24√2πακp −F 02 (0)κp + 1= −GF Q24√2παR(3.22)assuming that the Q2-dependence is similar for GZM and GγM.

The value of R rangesform R = 1 to R = 0.28 for F 02 (0) = −1 . .

. 1.

For no strangeness contribution, wehave F 02 (0) = −0.12 and thus R = 0.64. To give an idea about what one can expect,let me consider some models.

The SU(3) Skyrme model with a proper treatment ofthe symmetry breaking leads to F 02 (0) = −0.17 . .

. −0.25 [56], the NJL model toF 02 (0) = −0.16 .

. .

−0.26 [57] and vector meson dominance supplemented with theφ-meson (i.e. explicit strange quarks) to F 02 (0) = −0.43 ± 0.09 [58].

Similarly, one canalso discuss the strange electric radius, i.e. the radius related to the operator ¯sγµs.A simple model based on a kaon cloud surrounding the nucleon leads to processeslike p →ΛK+, Σ0K+, .

. .

→p and lets us expect that < r2E >s is positive since thepositive charge sits in the cloud of the kaons. This is, however, a very crude estimate.In fact, most models tend to give a small and negative < r2E >s, like e.g.

VMD plus theφ-meson −0.14±0.04 fm2 [58] or the SU(3) Skyrmion −0.10±0.05 fm2 [56]. These arefairly small numbers and it appears doubtful that the strange electric radius can bedetermined accurately from a proton target in the near future (assuming, of course,that its value is as small as indicated by the models).

Here, nuclei might come inhandy. For isoscalar, spin-0 nuclei such as 4He or 12C, one finds for the asymmetry[59]A = GF Q24√2παs2 +GsE2(GpE + GnE),(3.23)which at very low Q2 allows to extract the weak mixing angle and at somewhat highermomenta the strange electric ff.

Of course, one has to take into account nuclearstructure issues like pv level mixing and so on. A lucid discussion of these topicscan be found in refs.[59,60].

Clearly, pv electron scattering is an interesting field andmany more aspects of it not covered here can e.g. be found in the proceedings of theCALTECH workshop [61].3.5 Neutrino and Antineutrino Scattering offNucleons and NucleiNeutrino scattering offnucleons or nuclei offers another possibility of exploring thestrange quark content of the nucleon.

The neutrino couples via the Z0 to the quarksand thus probes isosinglet, triplet and octet components. Let us first consider thedifferential cross section for elastic neutrino/antineutrino-proton scattering,dσdQ2 = G2F m28πE2νA ± B W + C W 2,(3.24)with Eν the neutrino energy, W = (4Eνm −Q2)/m2 and the ′±′ refers to the case ofneutrinos (antineutrinos).

The functions A, B and C depend on the ffs G1 and F Z1,2(I omit the superscript ’Z’ on F1 and F2),

20Ulf-G. MeißnerA = 4τG21(1 + τ) −4(1 −τ)(F 21 −τF 22 ) + 16τF1F2,B = −8τG1(F1 + F2) ,C = G21/4 + F 21 + τF 22 . (3.24a)To demonstrate this in more detail, let me simplify the analysis by assuming a Q2value of about 0.5 GeV2 and retaining only the terms linear in the strange ffs.

Thisallows to recast (3.24) in the form [62]dσdQ2 = σnon−strange1 −0.72F s1 (Q2) −0.78F s2 (Q2) −1.23Gs1(Q2). (3.24b)Obviously, to extract the strange matrix element < p|¯sγµγ5s|p >= Gs1(0), one hasto know something about F s1,2(Q2) or make some assumptions.

Ahrens et al. [63]and Kaplan and Manohar [50] have shown that setting F s1 = F s2 = 0 and assumingthat all three axial ffs have the same dipole mass (MA = 1.032 GeV), the extractedvalue of Gs1(0) is in agreement with the one obtained from polarized deep inelastic µpscattering [36].

However, as already stressed in ref. [63], this result is very sensitive tothe actual value of the diple mass MA.

This issue was further addressed by Bernardet al. [64].

They used the topological chiral soliton model of the nucleon to calculatethe isosinglet axial ffsince in the framework of this model the known triplet ffis welldescribed. It was found that the cut-offmass in the singlet channel is actually 20per cent larger than the one for the triplet and octet ffs.

Redoing the analysis withthese ffs, one finds that the value of Gs1(0) is reduced by a factor of three. A similaranalysis has recently been performed by Garvey et al.[65].

In conclusion, the oftenclaimed agreement concerning the value of Gs1(0) from the neutrino and EMC datacan only be considered accidental. To avoid the problems of extrapolating from thetypical Q2 of 0.5 to 1 GeV2 in elastic νp or ¯νp scattering, it was recently proposedthat (anti)neutrino induced quasi-free nucleon knock-out offnuclei might give a betterhandle on the strange matrix elements [66].

In the quasi-free region, one has smallmomentum transfers and typical neutrino energies of 0.2 GeV. The ratio of the protonto the neutron yield depends sensitively on Gs1 and F s2 ,R = Yp / Yn = F(Gs1, F s2 ) .

(3.25)For example, if F s2 (0) = −0.22, R¯ν varies between 0.85 and 1.9 for Gs1(0) between 0 and-0.2. For further details, see ref.[66].

Of course, it is mandatory to understand well thenuclear structure issues, in particular one assumes that the knock-out is involving onlyone nucleon. How good these assumptions are is still under debate.

What is importantis to realize that neutrino and pv electron scattering processes nicely complement eachother and that many different experiments have to be performed to ultimately pindown the strength of the strange matrix elements in the proton.

Electroweak Reactions in the Non-Perturbative Regime of QCD213.6 Down and Dirty: QED, QCD and Heavy Quark CorrectionsUp to now, I have entirely worked at tree level. At low energies, radiative correctionsand effects from heavy quarks are expected to be small.

However, in certain casesthe tree level couplings are suppressed or vanish (as it is the case with the isocalaraxial current) or one is trying to extract small numbers. Therefore, it is mandatoryto investigate the effects of QED and QCD corrections on the processes consideredbefore.

Since the neutrino has no direct QED and QCD couplings, let me start withthe discussion of corrections to neutrino-hadron (quark) scattering. To be specific,consider the induced isoscalar coupling.

At tree level it vanishes, but that does notmean that it is zero alltogether. The basic Feynman diagram which induces such acoupling is the famous triangle diagram.

On one end, the axial part (γµγ5) of theneutrino current couples and the other two corners are attached via gluon exchangesto the external light (u,d,s) quark. In the loop, quarks of all flavors run around.

Asimple one-gluon exchange is forbidden by color neutrality. This is the basic diagramto lowest order in the strong interactions.

The contribution of such a type of diagramhas to be finite since the sum over all quark flavors leads to anomaly cancellation,XiT i3 = 0 ,(3.26)where T3 denotes the weak isospin. However, we are interested at scales much belowthe intermediate vector boson and heavy quark masses.

In this energy regime, thetriangle diagram leads to an induced isoscalar current which is not suppressed byinverse powers of the heavy quark masses as naive decoupling would suggest. Indeed,a straightforward perturbative calculation of this two-loop diagram yields [67]AZµ = 12(Au −Ad) + (Au + Ad)14 g24π2 ln Πim2i+Πim2i−,(3.27)where mi± refers to the masses of the quarks with weak isospin ±1/2, i.e.

with charges2/3 and -1/3, respectively, and g is the strong coupling constant. Notice that theinduced isoscalar current (Au + Ad) depends logarithmically on the quark massesand not on inverse powers of them (as advertised).

Collins, Wilczek and Zee [67]have shown how to do this in a more systematic fashion. Their method is based onintegrating out quark doublets in succession.

Denote by H = At−Ab the axial currentof the heaviest doublet. The idea is to recast H in the formH =XiAi ¯Li +XiBi ¯Hi ,(3.28)where the light (¯Li) and the heavy ( ¯Hi) particle operators exhibit decoupling.

Thismeans that the matrix elements of the ¯Li see the heavy quarks only via power lawcorrections. Repeating this exercise for ˜H = Ac −As, the induced isoscalar currentreads∆AZµ = (Au + Ad)14αs(ms)παc(mc)πln m2cm2s≃0.05 (Au + Ad) ,(3.29)where I have neglected the much smaller contribution from the (t, b) doublet.

Ofcourse, this approach is too bold since at the scale of the strange quark mass λ =

22Ulf-G. Meißnerms ≃175 MeV, this perturbative treatment can not be justified any more. Therefore,Kaplan and Manohar [50] have generalized the method of ref.

[67] to integrate out thevarious quarks separately. Their argument is based on the assumption MZ > mt.

Thissimplifies the analysis but is not mandatory. In this case, integrating out the Z0 onehas an effective Lagrangian for neutrino-quark scattering,Leff= −GF√2Xu,d,s,c,b,t¯νγµ(1 −γ5)ν¯q(T3 −2s2Q)γµq −¯qT3γµγ5q.

(3.30)In principle, one should now calculate corrections to this effective Lagrangian. How-ever, due to the absence of direct QED or QCD couplings of the neutrino, one canrewrite these corrections as corrections to the hadronic current.

The procedure goesas follows. One uses the renormalization group to evolve (3.30) down to the scale ofthe top mass.

At that energy, one integrates out the t quark. This induces two typesof corrections, one class scaling like g2(mt)/m2t (i.e.

exhibiting decoupling). Since thet is removed from the theory, the Z0 axial current has an anomaly because (3.26) isnot fulfilled any more.

This leads to a multiplicative renormalization of the singletaxial current, which is the most important effect as we will see. One then proceeds byscaling down to λ = mb, integrating out the b and so on until one reaches the scaleλ = 1 GeV (which is a typical energy in the elastic νp scattering process and largeenough to justify the perturbative treatment).

Putting the important pieces from themultiplicative renormalizations together, the relation between the axial current atλ = MZ and at λ = 1 GeV reads [50]AZµ (λ = 1 GeV) = AZµ (λ = MZ) + 12(Au + Ad)∆A∆A = Λµc12 + 310ΛcbΛbt −1log Λ(mi, mj, Nf) = Nfαs(mi)παs(mj)πlog m2im2j,(3.31)which leads to ∆A = 0.02 [50]. Obviously one recovers the result (3.29) if one were tointegrate out the s quark too.

Notice that the finite pieces due to the renormalizationgroup evolution are negligible and that there is also an induced vector current with∆V < 10−4. The main effect of the photons comes from the penguin diagram leadingto a running of the weak mixing angle [68]s2(λ2) = s2(M 2W ) +Xjα3π lnM 2Wλ2Qj(T3j −2s2Qj) ,(3.32)where the index j extends over the quark flavours which are integrated out.

In addi-tion, there are pure next-to-leading order electroweak effects. These have been con-sidered in ref.[69].

The radiatively induced isoscalar axial current has a very smallcoefficient,−3α16πs21 + 12c21 −2s2 + 209 s4= −0.003 ,(3.33)which is negligible compared to the heavy quark effects. Therefore, if one restrcitsoneself to scales above λ ≃1 GeV, the QED and QCD corrections to neutrino-hadron

Electroweak Reactions in the Non-Perturbative Regime of QCD23scattering are known and under control. Matters become more messy if one attemptsto go to lower energies and also if one considers electron-hadron scattering as I willdo now.

In this case, we have in addition the direct photon exchange which gives riseto further complications. What we are after can be summarized as follows.

Denoteby C the tree level value of any coupling between the lepton and the hadron (quark)currents. Due to the radiative corrections, these couplings are modifiedC = C[1 + R] ,(3.34)so that R gives the ratio of the corresponding value including radiative correctionsto the tree level one (in case that C happens to vanish at tree level, one has tochoose another coupling as the reference point).

Naively, one expects the size of thesecorrections to be small, the typical scale beingGF m2 α4π ≤10−8 . (3.35)This rather small number is sometimes considerably enhanced and, furthermore, theradiative corrections can in some circumstances compete with the effects induced bythe strange quarks.

Holstein and Musolf [70] have given the most detailed evaluation ofthese effects. Clearly, the results of their calculations should be considered indicativesince considerable uncertainties are involved as will be discussed below.

A very basicand introductory presentation concerning the calculation of radiative corrections forthe pv processes under consideration has been given by Musolf [71] which the readernot familiar with the concepts of renormalization in the standard model certainlywill appreciate. In essence, there are two classes of Feynman diagrams contributingat one loop order.

The first class involves only one quark in the nucleon. Typicalrepresentatives are vertex or vector boson propagator corrections.

More complicatedare the diagrams involving two quarks, such as the γZ0 box or exchange current typediagrams where a massive vector bosons is exchanged between two quarks while thephoton couples to one of them (many pictorials are displayed in ref.[71]). Let me firstconsider the one quark type diagrams.

These depend on the not yet known massesof the t quark and the Higgs boson and, if there is physics beyond the standardmodel, implicetely on the parameters of this new physics. This latter dependence isparametrized in terms of S, T and U or ǫ1, ǫ2 and ǫ3 (as discussed by Altarelli here[49]).

There are two effects which can conspire to give much larger values for theseloop corrections than the estimate (3.35) would suggest. These are large logarithmsof the light fermion to the vector boson mass ratios and the suppression of treelevel couplings.

A particularly illustrative example is the scattering process νeµ →νeµ. While the tree level Z0 −µ coupling is suppressed, the W-vertex correctionto the Z0 −νe coupling induces a log(m2e/M 2W ), with me the electron mass.

Thecorresponding ratio follows to be [70]R(νeµ →νeµ) = α4π83ln(m2e/M 2W )4s2 −1≃0.5 . (3.36)This calculation is clean in that it involves only leptons.

For the case of electron-hadron scattering, the quarks are bound within a nucleon of a size of about 1 fm.Using the uncertainty principle, this allows one to get an idea about the typical quarkmomenta. However, this is a very crude estimate and induces some uncertainty (for

24Ulf-G. Meißnerdetails, see ref.[70]). Furthermore, it is not obvious which quark mass values one shouldinsert.

When one considers quark loop corrections to the vector boson propagators,one obviously deals with the current quarks. In contrast, in the various box diagramsit is more justified to use the so-called constituent masses ( ∼330 MeV for the uand d quarks) to account for the binding effects.

These subtleties are also discussedin refs.[70,71]. For processes of the type V (e) × A(N) (which means that the vectorcurrent of the electron couples to the nucleonic axial current), one finds correctionsRp = −0.65 .

. .

−0.28 and Rn = −0.56 . .

. −0.06 for the proton and the neutron,in order.

These are clearly larger than the dimensional argument (3.35) suggests.Furthermore, we also have to consider the many quark diagrams. Here, the situationis less transparent, i.e.

it is much more difficult to deal with this class. To proceed,one can resort to some kind of ”hadronic duality” [70,71,72].

This means that oneconsiders a meson cloud picture instead of the more complex quark diagrams. Asexamples, the excitation of quark-antiquark pairs with the quantum numbers of apion probed by the photon and with a successive vector boson exchange translatesto the photon coupling to a pion in flight, with one pion-nucleon coupling parity-conserving (strong) and the other pv (weak).

Similarly, other diagrams translate intoa γρ0 conversion followed by a pv ρN coupling. Such diagrams contribute to thenucleon axial current via [70,72]δAπN = gπNhπNm2π6mMπ+ lnMπm+ .

. .,δAρN = hρNgρN1m2ρ.

(3.37)The strong coupling constants (gπN, gρN) are fairly well known. Matters are differentfor the parity-violating couplings (hπN, hρN).

Their calculation has been pioneeredby Donoghue et al. [73], a recent update including additional experimental constraintsand chiral soliton model calculations can be found in refs.[74.75].

The pion cloudcontribution diverges in the chiral limit which is no surprise in the light of the dis-cussion presented in section 2. Ultimately, CHPT methods might shed some lighton these long-distance contributions.

In ref. [70], the estimates for processes of thetype V (e) × A(N) range from −0.07 .

. .

+ 0.37 for Rp and −0.07 . .

. + 0.24 for Rn.Combining these numbers with the ones from the one quark diagrams, one sees thatthere are potentially large corrections.

To make all this more transparent, let us con-sider again the determination of the strange anomalous magnetic moment from pvelectron-proton scattering. In backward direction and at small Q2 we have [76]Aep ≃−GF Q24√2πακpµp1+RMstrange+RMrad−2m(E′ + E)Q21 −4s2µpgA(1+RAp ), (3.38)where RMstrange, rad denotes the strangeness and radiative corrections to the magneticmoment and RAp the one to the proton axial current.

Two important observations arein order. First, the radiative corrections to the magnetic asymmetry are comparableto the strange quark induced ones,RMradRMstrange≃0.1µs.

(3.39)

Electroweak Reactions in the Non-Perturbative Regime of QCD25Second, there is a large correction to the background of the axial asymmetry whichmakes up roughly 30 percent of the signal for the SAMPLE kinematics. These twoeffects severely constrain the accuracy for determining the strange anomalous mag-netic moment.

To overcome these problems, it is therefore mandatory to performmany complementary experiments like e.g. backward-angle ed scattering, pv quasi-elastic scattering and so on.

For a nice overview about these topics, I refer to ref. [76].Finally, let me stress again that more theoretical effort is needed to further tightenthe limits on these radiative corrections so that an unambigous interpretation of theexperiments will be possible.4 Summary and OutlookThe standard model of the strong and electroweak interactions is an embarrasinglysuccessful theory.

It is least understood at its extreme energy limits. First, at very highenergies, it is not known what exactly triggers the spontaneoaus symmetry breaking ofSU(2)L × U(1)Y →U(1)em at a scale of < φ >= 250 GeV.

While the standard Higgsboson does the job, one is left with just too many free parameters to feel confident withsuch a scenario. At present, the physics beyond the standard model hides itself quiteeffectively as discussed by Altarelli at this school [49].

Future high-energy colliders likethe LHC or the SSC will hopefully shed light on the electroweak symmetry breakingsector. May be less spectacular, but as challenging is the problem of hadron structurein the non-perturbative regime.

Chiral perturbation theory is a method which allowsto work out systematically the consequences of the spontaneous chiral symmetrybreaking in QCD. It is based on a simultaneous expansion in the (small) externalmomenta and (light) quark masses (u,d,s) and enjoys considerable success in themeson sector.

In the first part of these lectures, I have mostly been concerned withthe problems surrounding the inclusion of matter fields in the chiral expansion. Whilethis is technically a straightforward procedure based on chiral counting rules, it isconceptually less transparent than in the meson sector due to the appearance of thenucleon mass term.

This is a scale of the order of the chiral symmetry breaking scalealso it is not related to the symmetry violation and, furthermore, it does not vanish inthe chiral limit. I have concentrated here on a few selected processes in the two-flavorsector like nucleon Compton scattering or pion photo- and electroproduction.

Sincethe u and d quarks are really light, the corresponding expansion parameters Mπ/4πFπand Epion/Fπ are small and one has a better change of a converging chiral expansion.However, as we have seen in the discussion of the LET (2.14), the appearance of thenucleon mass renders these dimensional arguments rather dangerous. It appears atthis point that in the heavy mass formulation of baryon CHPT such terms should notshow up since the baryon propagator does not include the nucleon mass.

However, inthat case the large new scale is hidden by means of the Goldberger-Treiman relation inlarge prefactors accompanying the axial-vector coupling strength gA. All calculationsperformed so far have been carried out in the one-loop approximation indicating thatit is mandatory to include at least the terms of order q4 (which are beyond next-to-leading order). This argument is further strengthened when one considers the three-flavor sector.

Here, one very often finds large kaon and eta loop corrections whichmake one feel uneasy about the validity of the chiral expansion [77]. The suggestion

26Ulf-G. Meißnerthat these large loop effects are largely cancelled by contributions from the spin-3/2decuplet in the intermediate states [16,18] is appealing but has not yet been puton a firm basis, which means that one has to perform a full order q4 (or higher)calculation. There are, however, on-going activities in this area and further progresscan be expected soon.The second topic of these lectures concerned the quark structure of the nucleon atlow energies.

I have discussed how particular combinations of the electromagnetic andweak form factors, which parametrize the structure of the nucleon and its flavor de-composition, allow to extract interesting matrix elements like the strange anomalousmagnetic moment, F s2 (0), or the strange axial nucleon charge, Gs1(0). Parity-violatingelectron scattering (γZ0-interference) and (anti)neutrino scattering offnucleons ornuclei are the tools to determine these and other matrix elements like e.g.

the exten-sion of the nucleon as given by the strange electric radius. However, and that wasa major topic here, since in most cases one tries to extract small numbers, one alsohas to account for radiative and heavy quark corrections.

While these are nominallysmall, they often tend to be enhanced by large factors way above the naive dimen-sional estimates. In the most extreme case (cf.

the isoscalar axial coupling), the treelevel couplings with which one works at leading order are vanishing. In neutrino-hadron scattering, if one does not go below a typical scale of say 1 GeV, these variouscorrections are under control.

As a particular example, I have discussed the inducedisoscalar axial coupling, which is not only empirically interesting but also shows howunder certain circumstances effective field theory methods can and should be used tocaluclate QED, QCD and heavy quark corrections [50]. When it comes to electron-hadron scattering, the situation is much less satisfactory.

First, as I discussed, inmany experiments one wants to work at much smaller energy scales and therefore hasto account for the effects of quark confinement. Second, the direct photon couplingto the hadrons (quarks) gives rise too much more diagrams than it is the case inneutrino scattering.

The most systematic analysis of radiative corrections to parity-violating electron-hadron scattering [70] translates the Feynman diagrams involvingmore than one quark in the nucleon into ”hadronic ones”, i.e. making use of the me-son cloud picture.

This induces uncertainties which are of the order of the calculatedeffects themselves. Another source of uncertainty is the momentum distribution of thebound quarks which one can only account for in an approximate fashion.

Obviously,the theorists are called for providing better calculations of this type so that the exper-imenters have a better change to extract the various strange matrix elements. Furthercomplications arise when one considers the scattering offnuclei [78].

As I pointed out,many complimentary experiments which are sensitive to different combinations of theform factors and have different next-to-leading order corrections have to and will beperformed at MAMI, CEBAF, . .

. [79].

These experiments are eagerly awaited. Tocome back to the first part of the lectures, once we will have learned how to treatthe three-flavor sector of baryon CHPT, one will also be able to discuss the strangematrix elements and confront the theoretical predictions with the data.

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