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The structure of the pion and effective electroweak

arXiv:hep-ph/9208252v1 26 Aug 1992DOE/ER/40322-170U. of MD PP #93-001The structure of the pion and effective electroweakcurrents in soliton models of the nucleon.Wojciech Broniowski∗and Thomas D. CohenDepartment of Physics and Astronomy, University of MarylandCollege Park, Maryland 20742-4111Nonminimal substitution terms in electroweak currents are studied in effectivechiral soliton models.

It is found that the terms describing the structure of the pionlead to sizable effects in form factors and polarizabilities of the nucleon.PACS numbers: 12.38.Lg, 12.40.Aa, 14.20.Dh, 14.60.FzTypeset Using REVTEX1

Much of what is known about the structure of the nucleon and other baryons has beenlearned from electromagnetic and weak probes. For well over a decade a variety of chiralsoliton models have been constructed to explain the structure of the baryon.

Most of theeffort in building these models has gone into the description of the strong interaction dynam-ics. It is generally assumed that the electromagnetic and weak properties can be determinedvia the study of currents obtained by gauging the lagrangian of the model.

In this notewe investigate this assumption and show that a good description of the strong interactiondynamics is not sufficient. One also needs a good description of the effective electroweakcurrents, and the minimal substitution obtained by gauging the effective theory does notnecessarily produce a good description of the effective currents.

In particular, we point outthat electroweak properties of the nucleon (e.g. form factors, polarizabilities) in chiral soli-ton models are affected significantly by terms in the effective lagrangian which describe thestructure of the pion.In principle, there are an infinite number of terms describing how a photon can couplewith pions in a nonminimal way.

Here we investigate the effects of the L9 and L10 termsintroduced in the standard chiral expansion Ref. [1].

We concentrate on these two terms forthe following reason — the coefficients associated with them can be obtained by experimentsin the meson sector, and are known with reasonable accuracy. We derive simple expressionsrelating nonminimal-coupling contributions to nucleon observables with the pion rms chargeradius and pion polarizability We find that these nonminimal coupling terms make sizablecontributions to a number of observable quantities.The starting point in construction of effective models of the nucleon structure is the choiceof an effective lagrangian with low-energy degrees of freedom.

In the large-Nc approach[2,3] baryons arise as solitons of these effective lagrangians, and semiclassical methods canbe used to describe their properties [4,5]. Popular models based on this idea range frompurely mesonic skyrmions [6,7], through models with quarks and mesons (chiral quark-mesonmodels [8], hybrid bag models [9], chiral color-dielectric models [10]), to the Nambu–Jona-Lasinio model in the solitonic treatment [11], which involves quark degrees of freedom only.In the usual approach, electroweak interactions are introduced to an effective theory bygauging the effective lagrangian.

This minimal coupling, however, by no means guaranteesthat the resulting currents correspond to currents of the underlying QCD. In fact, the“bosonization” procedure, i.e.

the procedure of obtaining an effective theory from QCD, andgauging, do not necessarily commute. Gauging QCD, and then “bosonizing” the currents ingeneral leads to a different result from first bosonizing QCD, and then gauging the effectivelagrangian.In recent years numerous attempts [12] were made to “derive” an effectivelagrangian from QCD.

Upon expanding in powers of pion fields and their derivatives, acomparison with the low-energy chiral perturbation theory lagrangian [1] can be made. Thequality of the fit in the corresponding low-energy constants depends on the model [12].

Forour present purpose the relevant terms are those describing the structure of the pion, namelyL9 and L10 in the notation of Ref. [1]:L9 = −˙ıL9 Tr[F LµνDµUDνU† + F RµνDµU†DνU],(0.1)L10 = L10 Tr[F LµνUF µν,RU†],(0.2)where F L,Rµν= Vµν ∓Aµν are the left and right chiral field strength tensors, and Vµν and Aµνdenote the vector and axial field strength tensors:2

Vµν = 12τa(∂µV aν −∂νV aµ ),Aµν = 12τa(∂µAaν −∂νAaµ). (0.3)The chiral field is parameterized asU = exp(˙ıτ · φ) = F −1π (σ + ˙ıτ · π),(0.4)where σ = Fπ cos φ, π = Fπ bφ sin φ.

The constants L9 and L10 in Eqs. (0.1,0.2) can be ex-pressed through measurable quantities [1,13], namely the pion isovector mean square radius,⟨r2⟩I=1π, and the pion polarizability απ = −βπ [1,13]:L9 = F 2π⟨r2⟩I=1π12,(0.5)L10 = mπF 2παπ4αQED−L9 = mπF 2παπ4αQED,(0.6)where αQED = 1/137.

Although the pion polarizability has not been measured reliably, onecan use chiral identities to re-express this quantity in terms of the hA constant in the decayπ+ →γνl+ [14].Effective models used to describe the nucleon have not stressed the physics in Eq. (0.2).For example, the lagrangians of the original Skyrme model [6,4], or the chiral quark model[15] do not include these terms at all, while the constants L9 and L10 in the NJL model[11] disagree to some extent with the experimental values [12].

For definiteness, below weconcentrate on the original Skyrme model [4] and the chiral quark model [16–18]. Othermodels, specifically models with explicit vector meson degrees of freedom, will be discussedat the end.One remark about the coefficients in the chiral expansion should be made before pro-ceeding.

In the conventional chiral perturbation theory approach in the meson sector, oneconsiders pionic loops, and through renormalization L9 and L10 acquire chiral logarithms.In the soliton approach one treats the effective lagrangian in the mean-field approximation(tree level), replacing the meson field operators by classical fields. Therefore it is appropriateto compare low-energy constants in soliton models with the full (renormalized) constantsfrom the chiral expansion.The lagrangians used in Refs.

[6,4,16–18] have the generic formLeff = L2 + ...,L2 = 14F 2πTr[∂µU∂µU†],(0.7)where the ellipses denote the Skyrme’s fourth-order interaction (skyrmion), or the couplingto quarks (chiral quark model), as well as the explicit chiral symmetry breaking term. Theprocedure of gauging replaces the derivatives in Eq.

(0.7) by covariant derivatives:∂µ →Dµ = ∂µ −˙ı[V µ, .] −˙ı{Aµ, .

}. (0.8)As a result, minimal substitution terms in the lagrangian, Lmin, appear.

In the skyrmion,they involve pionic terms arising from the quadratic and quartic terms, Lmin = L2min + L4min.In chiral quark model of Ref. [16–18] Lmin = L2min + Lquark, where Lquark is the contribution3

from the valence quarks. In both types of models terms L9 and L10 are missing, and weshould add them by hand in order for the theory to describe properly the properties of thepion.

The resulting lagrangian has the formL = L2 + Lmin + L9 + L10 + ...,(0.9)We first analyze the effect of the L9 term on the nucleon electromagnetic and axialform factors. The minimal substitution term L2min generates the following vector and axialcurrents:V ν,a2= ǫabcπb∂νπc, Aν,a2= σ∂νπa −πa∂νσ.

(0.10)Additional vector and axial source currents generated by the L9 term, V9 and A9, have theformV ν,a9= −4L9F −2π ∂µǫabc[(∂µπb)(∂νπc)],Aν,a9= −4L9F −2π ∂µ[(∂µσ)(∂νπa) −(∂νσ)(∂µπa)]. (0.11)Note that since these currents are total derivatives, they do not induce additional charges.One can also show that the magnetic moments are not changed, since r × V a9 is also a totalderivative.In the semiclassical treatment of hedgehog soliton models, a projection method is nec-essary to restore good quantum numbers of baryons.

In projection via cranking [4,5] onearrives at semiclassical expressions, which can be obtained from field-theoretic expressionsby a substitutionσ →σh, πa →cabπbh, ∂0πa →−ǫabcλbπch. (0.12)where σh = σ(r) and πa = braπh(r) are the usual hedgehog fields.

Collective operators, λaand cab have the following matrix elements in the nucleon state⟨N | λa | N⟩= 1Θ⟨N | Ja | N⟩,⟨N | cab | N⟩= −13⟨N | τ aσb | N⟩,(0.13)where J is the spin operator, and Θ is the moment of inertia.In the Breit frame, the form factors are defined in the usual way:⟨Nf(q2) | Jem0 (0) | Ni(−q2 )⟩= GE(q2)χ†fχi,(0.14)⟨Nf(q2) | J em(0) | Ni(−q2 )⟩= GM(q2)2MNχ†f(˙ıσ × q)χi,(0.15)⟨Nf(q2) | Aa(0) | Ni(−q2 )⟩= χ†fτ a2 EMNGA(q2)σ⊥+GA(q2) −q24M2NGP(q2)!σ∥#χi,(0.16)4

where χi and χf are the two-component nucleon spinors, σ∥= bq(σ · bq), σ⊥= σ −σ∥, andE =qM2N + q2/4.Using Eqs. (0.10-0.16) we obtain straightforwardly the following relations for the addi-tional contributions to the nucleon isovector form factors GE, GM and GA, which come fromthe nonminimal currents, Eq.

(0.11):G9,I=1X(q2) = −2L9F −2π q2G2,I=1X(q2),(0.17)where X = E, M or A, superscript 2 denotes the form factors corresponding to the currents(0.10), and superscript 9 denotes the new terms. The relation for G9P follows from PCAC,which implies thatMN GA(q2) −q24M2NGP(q2)!= Fπm2πGπNN(q)(m2π + q2).

(0.18)Since the introduction of terms L9 and L10 obviously does not influence GπNN(q2), thefollowing relation holds:G9A(q2) −q24M2NG9P(q2) = 0,(0.19)In fact, Eq. (0.17) can not be trusted for higher moments than the rms radii, since higherorder effects have not been included (this will be discussed later).

Using expressions for themoment of inertia, magnetic moments, and gA from Refs. ( [4,5]), and Eqs.

(0.5, 0.6), wearrive at the following corrections to the mean squared radii induced by the L9 term:⟨r2⟩9,I=1E=Θ2Θtotal⟨r2⟩I=1π,⟨r2⟩9,I=1M= µI=12µI=1total⟨r2⟩I=1π,⟨r2⟩9A =(gA)2(gA)total⟨r2⟩I=1π,(0.20)where the subscript 2 in the moment of inertia Θ, isovector magnetic moment µI=1, and gA,denotes contributions to mean square radii of the nucleon from the pionic currents (0.10).For example,Θ2 = 23Zd3x π2h. (0.21)In the Skyrme model of Ref.

[4,19] we find Θ2/Θtotal = µI=12/µI=1total ≃0.4 −0.6,(gA)2/(gA)total ≃0.5. In the chiral quark model one finds [5] Θ2/Θtotal ≃µI=12/µI=1total ≃0.6,(gA)2/(gA)total ≃0.5.

Since in experiment ⟨r2⟩π = 0.44fm2, Eqs. (0.20) lead to ∼0.2fm2contributions to the mean squared radii.Experimental values for the nucleon radii are [20]:⟨r2⟩I=1E= 0.82fm2, ⟨r2⟩I=1M=0.73fm2, ⟨r2⟩A = 0.65fm2.Therefore the corrections from Eqs.

(0.20) are substantial:∼25%, ∼30% and ∼30% for the three radii squared, respectively. For a quantity whichinvolves cancellations between isoscalar and isovector properties, namely the electric radius5

of the neutron, the L9 effect is of the order of 100% ! This shows that predictions for theneutron form factor are extremely sensitive to nonminimal coupling terms.Now we turn to the nucleon electromagnetic polarizabilities.

This issue has been dis-cussed in detail in Ref. [21].

We assume that the electric and magnetic fields are constant.Using our semiclassical methods we can easily identify pieces in the lagrangian which arequadratic in the E and B fields. The L9 and L10 terms leads to the following semiclassicalexpression:Zd3x L9 = 2 αQEDL9F −2π (E2 −B2)Zd3x 23π2h,Zd3x L10 = 2 αQEDL10F −2π (E2 −B2)Zd3x 23π2h.

(0.22)The coefficient of E2 (B2) in the lagrangian has the interpretation of twice the electric(magnetic) polarizability [21,22]. Recognizing the moment of inertia Θ2 in the integrals inEqs.

(0.22), and using Eq. (0.5, 0.6), we obtain the following expression the contributionsdue to the L9 and L10 terms to the electric (απN) and magnetic (βπN) polarizabilities on thenucleon:απN = −βπN = mπΘ2απ.

(0.23)Note the opposite signs of the electric and magnetic polarizabilities in (0.23), reflectingthe fact that απ = −βπ. Numerical values for mπΘ2 are of the order ∼0.5 −0.7 in thediscussed models.The value of απ can be determined experimentally, however existingexperimental data [23] do not seem reliable, and are in contradiction with a low-energytheorem [24,14,13], which gives απ = 2.8 × 10−4fm3.

With this value we getαπN = −βπN ∼1.3 × 10−4fm3,(0.24)which is a few times smaller compared to the minimal substitution terms, [21] but non-negligible, especially for the magnetic case, where large cancellations are expected. If ex-perimental numbers for απ were used [23], then a two-three times larger result follows.The physical interpretation of the results in Eqs.

(0.20, 0.23) is clear. Since the pionhas electromagnetic structure, it gives rise to an additional contribution to the nucleonelectromagnetic properties in models which have a pion cloud.

As an example, let us considerthe electric isovector form factor. Suppose ρcloud(r) describes the distribution of the classicalpion cloud around the nucleon, and ρπ(r) describes the distribution of charge in the pion.Folding of these two distributions results in the charge form factor of the nucleon of theform ˜ρcloud(q)˜ρπ(q).

In fact, Eqs. (0.17) can be viewed as a term in the expansion of thisproduct of form factors.

From a different point of view, namely from a description in termsof a field theory at the hadronic level, the electromagnetic current can couple to the nucleonvia pionic loop. The coupling to the pion involves a form factor, and this obviously modifiesthe properties of the nucleon.Up to this point we have not discussed how the the L9 and L10 terms included in ouranalysis behave in terms of Nc counting or chiral counting.

It is easy to verify that since⟨r2⟩I=1π, απ and mπ are of the order 1 in Nc-counting, and Θ is of the order Nc, the additionalcontributions to radii, Eq. (0.20), are of the order 1, and for polarizabilities, Eq.

(0.23) of6

the order Nc. These are the same orders as for the contributions resulting from the minimalcoupling, hence the L9 and L10 effect occur at the leading Nc level.In chiral counting these terms are assigned power 4 in momenta, whereas the minimalcoupling terms have the chiral order 2.

The difference in the application of chiral lagrangiansto the nucleons rather than mesons is that the terms with more gradients are not necessarilysuppressed. The gradients in a soliton are of the order of 1GeV , which is the same orderas the chiral scale (∼4πFπ, or mρ).

In our opinion there is no reason to expect why termssuch as L9 and L10 should be suppressed in soliton models.A more worrying issue is that one the L9 and L10 are just two of an infinite class ofterms which can modify the electroweak currents. For example, one could add a term as L9multiplied by Tr[∂σU∂σU†]n, (where n is an arbitrary positive integer), which has the sameNc-counting as L9.

Although such a term is a higher dimensional operator and as such issuppressed in chiral counting in the meson sector, as mentioned above it is not suppressedin a soliton. In principle, one can fix the coefficients for these terms by from experimentalstudies of interactions involving virtual photons and several pions.

In practice, however, wewill never be able to determine the phenomenological low-energy constants for any but a fewof such terms, and thus predictive power is lost. A priori there is no reason to believe thatthe effect of these higher dimensional operators on the effective currents should be small.In two classes of chiral models the physics of L9 and L10 terms is included.

One is the classof models with explicit vector meson degrees of freedom [25]. Since L9 basically describesthe physics of vector meson dominance, such models are expected to give reasonable valueof the constant L9.

The other class are the NJL models [11]. Upon expanding one-quarkloop in presence of external sources, these models generate low-energy constants.

Certainly,the quality of the fit depends on the details of the model (cut-offscheme, etc.). In the caseof models which generate terms L9 and L10, the corrections of Eqs.

(0.20,0.23) involve notthe phenomenological constants L9 and L10, but the differences of phenomenological andmodel-predicted valuesTo summarize, the effective electromagnetic and weak currents in chiral soliton modelsneed not be the ones obtained by minimal substition. We have demonstrated importanteffects of the nonminimal terms in the effective lagrangian on electroweak properties of thenucleon, associated with the L9 and L10 terms.

Effects of the order of 20 −30% are foundfor various nucleon mean radii squared in the original Skyrme model, or simplest versionof the simple chiral quark model. These corrections are clearly substantial.

In addition tothe effects which we know how to estimate, however, there are infinite number of additionalterms, with unknown constants, which can alter the effective currents. The contributionsfor these additional terms are not a priori small, and they enter at leading order in the 1/Ncexpansion.We should also add that the effects of the L9 and L10 terms are large in models in whichthe pion dominates.

In models where the pion is not the key dynamical factor, e.g. themodels with confinement [10], these effects are weaker, since the ratios in Eqs.

(0.20) aresignificantly less than 1.Support of the the National Science Foundation (Presidential Young Investigator grant),and of the U.S. Department of Energy is gratefully acknowledged. We thank Manoj Banerjeefor many helpful discussions.

One of us (WB) acknowledges a partial support of the PolishState Committee for Scientific Research (grants 2.0204.91.01 and 2.0091.91.01).7

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