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THE NUCLEON “TENSOR CHARGES” AND THE SKYRME MODEL ⋆

arXiv:hep-ph/9207274v1 30 Jul 1992THE NUCLEON “TENSOR CHARGES” AND THE SKYRME MODEL ⋆James M. Olness †Center for Theoretical PhysicsLaboratory for Nuclear Scienceand Department of PhysicsMassachusetts Institute of TechnologyCambridge, MA 02139Submitted to: Physical Review D (Brief Reports)CTP#2122July 1992⋆This work is supported in part by funds provided by the U.S. Department of Energy(D.O.E.) under contract #DE-AC02-76ER03069.† National Science Foundation Graduate Fellow1

ABSTRACTThe lowest moment of the twist-two, chiral-odd parton distribution h1(x) of the nucleoncan be related to the so-called “tensor charges” of the nucleon.We consider the tensorcharges in the Skyrme model, and find that in the large-Nc, SU(3)-symmetric limit, themodel predicts that the octet isosinglet tensor charge, g8T , is of order 1/Nc with respect to theoctet isovector tensor charge, g3T . The predicted F/D ratio is then 1/3, in the large-Nc limit.These predictions coincide with the Skyrme model predictions for the octet axial charges, g8Aand g3A.

(The prediction F/D = 1/3 for the axial charges differs from the commonly quotedprediction of 5/9, which is based on an inconsistent treatment of the large-Nc limit.) Themodel also predicts that the singlet tensor charge, g0T , is of order 1/Nc with respect to g3T .2

The nucleon has three parton distributions at lowest twist, that is, at twist-two. Twoof these, f1(x) and g1(x), have been studied extensively, and have been measured in deep-inelastic scattering experiments [1].

The remaining distribution, h1(x), is relatively new, andhas only recently begun to receive attention in the literature [2]. It has not been measured−since it is chiral-odd, it is inaccessible to inclusive deep-inelastic scattering experiments.Ralston and Soper [3] have shown, however, that h1(x) plays an important role in polarizedDrell-Yan processes.

More recently, Collins [4] showed that it emerges naturally in the fac-torization of a general hard process into soft and hard sub-processes. In the parton modellanguage, h1(x) can be interpreted loosely as counting transversely polarized (valence) quarksin a transversely polarized nucleon.⋆The Q2 evolution of h1(x) has been calculated by Ko-daira et al.

[5], and by Artru and Mekhfi[6].Sum rules relating the moments of h1(x) to the nucleon matrix elements of local operatorscan be formulated in the standard way. The low moments are particularly interesting becausethey can be used to determine the gross features of the distribution.

The lowest moment, forexample, is referred to as the “tensor charge,” δq :δq(Q2) =Z ∞−∞dx h1(x, Q2) =Z 10dxh1(x, Q2) −h1(x, Q2). (1)Note that h1(x) carries flavor indices, which have been suppressed; there is an independenttensor charge, δq, for each quark flavor.

The tensor charge derives its name from its relationto the following nucleon matrix element:⟨PS| qσµνiγ5qQ2|PS⟩= 2M (SµPν −SνPµ)δq(Q2)(2)where q = u, d, s, etc. (The parameter Q2 appearing in equations (1) and (2) is a renormaliza-tion scale label, necessary to render h1(x) and the tensor charges well-defined in perturbativeQCD.

The tensor charges do not mix with other operators under renormalization, and aretherefore characterized by a single anomalous dimension, which has been calculated at oneloop [5,6]: γ = 2αs/3π. Henceforth, we will suppress all dependence on Q2.) It is conve-nient to arrange the δq’s into octet and singlet combinations, having definite flavor SU(3)transformation properties:g3T = 2(δu −δd)g8T =2√3(δu + δd −2δs)g0T = 2r23(δu + δd + δs) .

(3)The reason for the peculiar normalization will become clear shortly. For future reference, wealso define the axial charges, gaA:⟨PS|Ψλaγµγ5Ψ|PS⟩= SµgaA .

(4)⋆A more precise interpretation involves the transversely projected Pauli-Lubanski opera-tor; see reference [2].3

This is just the conventional definition of the axial charges. In the non-relativistic quarkmodel, the tensor charges are equal to the corresponding axial charges.

(We refer here tothe octet and singlet combinations, gaT and gaA, not the individual quark components, δq,and their axial charge counterparts, ∆q. According to the conventional definition of ∆q, i.e.,⟨PS|qγµγ5q|PS⟩= Sµ∆q, we find that ∆q = 2δq, in the non-relativistic quark model.

In thispaper, we will work almost exclusively with gaT and gaA.) Jaffe and Ji [2] considered the tensorcharges in the bag model, and found the tensor charges to be somewhat larger in absolutemagnitude than the corresponding axial charges.

It should be remembered, however, thatthis comparison involves an implicit, unknown scale, of the order of 1 GeV (i.e., the “bag”scale at which the operators are renormalized).The purpose of this note is to report the results of a Skyrme model analysis of the tensorcharges. ⋆In the large-Nc, SU(3)-symmetric limit, the model predicts that the octet isosinglettensor charge, g8T , is of order 1/Nc with respect to the (octet) isovector tensor charge, g3T :g8Tg3T= O 1Nc.

(5)This can also be phrased in terms of a prediction for the F/D ratio:FD = 13 + O 1Nc. (6)These predictions coincide with the large-Nc, SU(3)-symmetric Skyrme model predictions forthe octet axial charges, g8A and g3A.

(The prediction F/D = 1/3 for the axial charges differsfrom the commonly quoted prediction of 5/9, which is based on an inconsistent treatment ofthe large-Nc limit.) The model also predicts that the singlet tensor charge, g0T , is of order1/Nc with respect to g3T :g0Tg3T= O 1Nc.

(7)(In the case of the axial charges, the model predicts g0A/g3A = O 1N2c[7].) These predictionsare independent of the renormalization scale in QCD, since the tensor charges have a commonanomalous dimension.Before considering the Skyrme model, it will be useful to discuss the non-relativisticquark model (NQM).

As mentioned above, the tensor charges are equal to the correspondingaxial charges in the NQM. This is true for any value of Nc.

The explicit calculation of theaxial charges (and thus, the tensor charges) for arbitrary Nc has been carried out by Karland Paton [8], using the fact that the axial charges measure the spin carried by the quarks.Transcribing their results, we haveg3T = 13(Nc + 2)g0T =√2 g8T =r23 . (8)⋆We work with the Skyrme model in order to be concrete.However, the conclusionsare valid in the large-Nc limit of any chiral soliton model in which the soliton is quantizedsemi-classically, through the use of collective coordinates.4

In particular, the NQM predicts that g8T /g3T and g0T /g3T are of order 1/Nc.The knowncorrespondence [9] between the large-Nc NQM and the large-Nc Skyrme model suggests thatwe should expect similar results in the Skyrme model.The Skyrme model [10] describes the interactions of an octet of fundamental meson fields.In its simplest form, the effective action is given byΓ =Zd4xF 2π16 Tr (∂µU∂µU †) +132e2 Tr [(∂µU)U †, (∂νU)U †]2+ NcΓW Z ,(9)where U(x) = exp2iλaφa(x)/Fπ, and the φa’s are meson fields.Fπ is the pion decayconstant, and e is a phenomenological parameter. ΓW Z is the so-called Wess-Zumino term,which incorporates the effects of anomalies.

As is well known, the classical field equationsassociated with this lagrangian admit stable soliton solutions, the prototypical example ofwhich is the hedgehog solution:U0(⃗x) =expiF(r)⃗τ · ˆx001(10)This is quantized by introducing collective coordinates to describe the orientation of thesoliton in the group space SU(3)flavor (and thus, simultaneously, SU(2)spin):U(⃗x, t) = A(t)U0(⃗x)A−1(t) ,(11)where A(t) is a time-dependent SU(3) matrix. The resulting quantum states have the quantumnumbers of baryons.

By making a suitable choice of the parameters Fπ and e, the spectrumof low-lying baryons can be modeled approximately.In order to apply the Skyrme model in estimating nucleon matrix elements, one must firstidentify the appropriate Skyrme model operators, constructed in terms of meson fields. Thetensor charges are not associated with any symmetry of the QCD lagrangian, so their Skyrmemodel analogs cannot be generated by means of the standard Noether’s theorem route.

Othermethods of constructing equivalent operators in the Skyrme model were considered, such asthe method of Goldstone and Wilczek [11], but these apparently lead to ambiguous results. Atleading order in the 1/Nc expansion, however, the analysis simplifies considerably, and we canlearn about the flavor structure of the tensor charges simply by considering the transformationproperties of the operators appearing in equation (2).Consider the following matrix element:⟨PS|Ψλaσµνiγ5Ψ|PS⟩= 1M (SµPν −SνPµ)gaT.

(12)For a nucleon at rest, the tensor operator appearing in this equation yields non-vanishingmatrix elements only for (µ, ν) = (i, 0), i = 1, 2, 3. Denote these components of the operatorby Oai , and denote the equivalent Skyrme model operators by eOai .

The eOai transform as athree-vector under spatial rotations, and as an octet under SU(3) transformations (or as anSU(3) singlet, in the case of a = 0). In the large-Nc limit, the soliton rotates slowly, so asa first approximation, we ignore operators that involve time-derivatives, or in other words,factors of the canonical momenta.

(The large-Nc approximation is standard in applications5

of the Skyrme model; indeed, it is assumed implicitly in the quantization of the hedgehogsolution.) Then in direct analogy with the quantum mechanics of a non-relativistic pointparticle, we can evaluate the matrix elements of the eOai in terms of SU(3) Clebsch-Gordancoefficients, and a single (unknown) constant.

Standard Skyrme model formulas [9] give thefollowing results for the octet charges:⟨N| eOam|N⟩= kXn=1,2R8NaRnN R8NbRnN(13)The label N represents a nucleon state, which we take to have definite polarization in thez-direction, for simplicity. The index m indicates the J3 spin quantum number of the operatoreOa.

The label R denotes the SU(3) representation containing the nucleon −we will say moreabout this shortly. The direct product R × 8 contains two copies of the representation R,which are denoted by Rn in equation (13).

The label b represents a state with hyperchargezero, and isospin quantum numbers that are determined by the transformation properties ofeOam under spatial rotations: I = J = 1 and I3 = −J3 = −m.The constant k is unknown, but the scale-dependence of the tensor charges in QCDmeans that precise knowledge of k would be of questionable value, since we do not know howto determine the renormalization scale at which k applies. Note that k may depend on Nc.Indeed, as we will see, correspondence with the NQM suggests that k = O(Nc).The corresponding formula for the singlet charge yields a vanishing result, since there isno state b having I = 1 in the singlet representation.

It should be emphasized that equation(13) and its singlet counterpart are valid only when we neglect the contributions of operatorsinvolving time-derivatives.The corrections that result from such operators are discussedbelow. For now, we point out that these corrections do not necessarily vanish as Nc becomeslarge.We now discuss the identification of the large-Nc analog of the nucleon.

In a symmetricquark model, a baryon containing Nc = 2n+1 quarks must belong to the (1,n) representationof SU(3) if it is to have spin 12. (The (1,n) representation consists of the traceless, symmetrictensors having one upper index and n lower indices.) Furthermore, for Nc = 2n +1, the (1,n)representation is the only representation which can be projected out of the soliton solutionand which contains states with zero strangeness and I = J = 12 [9].

A consistent treatmentof the large-Nc limit therefore seems to require that we choose R = (1, n) in equation (13). (For a more extensive discussion on this issue, see reference [12].) With this choice, equation(13) yields the following results:g3T = 2k (Nc −1)(Nc + 6)2 + 9(Nc + 7)3(Nc + 7)(Nc + 3)2−→Nc→∞2k3g8T = 2k2√3(Nc + 3)(Nc + 7)−→Nc→∞4√3kN 2c.

(14)Thus, the ratio g8T /g3T vanishes in the large-Nc limit, although we will see that it is expectedto vanish linearly in 1/Nc, rather than quadratically. For comparison with the axial charges,it is useful to translate this result into a statement about the F/D ratio.

According to the6

standard definition of the reduced matrix elements, F and D, the isosinglet tensor charge isgiven by:g8T ∼1√3(3F −D) . (15)Thus, the large-Nc prediction is F/D = 1/3.As mentioned previously, this prediction isindependent of the renormalization scale.We now consider corrections to these results within the framework of the Skyrme model.Equation (13) is valid only if the operators that describe the tensor charges do not involvetime-derivatives of the collective coordinates.

Presumably, the quark operators appearing inequation (12) are matched onto an entire family of Skyrme model operators, and in general,this family will include operators with time-derivatives. We can argue on dimensional grounds,however, that the contributions of these operators are suppressed in the large-Nc limit.

In theprocess of reducing meson operators to collective coordinate operators (by integrating overall space), a space-derivative brings in a mass parameter of the order of the inverse protonradius. In contrast, a time-derivative brings in a mass parameter of the order of the rotationalkinetic energy of the soliton.

Thus, we consider the ratioL2/2I1/R∼L2MR(16)where I is the moment of inertia, and L is the angular momentum of the spinning soliton.The nucleon mass grows as Nc, while the radius remains constant in the large-Nc limit [13].Thus, we conclude that corrections arising from operators with time-derivatives should besuppressed by 1/Nc.In order to apply these ideas to the tensor charges, it is useful to consider the individualquark components, δu, δd, and δs. (This can be done unambiguously, since the octet andsinglet tensor charges are characterized by the same anomalous dimension.) As suggested bythe NQM, we will assume that δu and δd are of order Nc.

Equation (14) then requires that δsbe at most of order Nc. Recalling the considerations of the preceding paragraph, we find thatoperators with time-derivatives are expected to contribute to the δq’s at order unity.

Equation(13) and its singlet counterpart then imply that g3T is of order Nc, while g0T and g8T are oforder unity (and thus, finite, in the large-Nc limit). These conclusions, of course, depend onthe validity of our assumption that δu and δd are of order Nc.

However, the conclusion thatg8T /g3T and g0T /g3T are of order 1/Nc is readily seen to be independent of this assumption, andcan be taken as a legitimate prediction of the Skyrme model. From g8T /g3T = O 1Nc, we alsoobtain F/D = 1/3 + O 1Nc.Now consider the axial charges.

For a nucleon at rest, only the space-components of theaxial-vector current operator appearing in equation (4) yield non-vanishing matrix elements.If we denote the space-components by Aai , then the Aai transform in exactly the same wayas the Oai under SU(3)flavor × SU(2)spin. Thus, the analysis outlined above also applies tothe axial charges, and we draw the same conclusions as before: The ratio g8A/g3A is of order1/Nc, and F/D = 1/3 + O 1Nc; the ratio g0A/g3A vanishes in the large-Nc limit, although,as mentioned previously, this ratio is actually expected to vanish like (1/Nc)2, rather than1/Nc.

(This last result does not contradict the arguments above, which are only meant toplace crude limits on the large-Nc behavior. )7

Note that the prediction F/D = 1/3 for the axial charges differs from the commonlyquoted prediction of 5/9 (see, for example, reference [14]). The latter prediction results fromchoosing R=8 in equation (13).

Although this value (i.e., 5/9) is in good agreement with theexperimental result of 0.58±0.05 [15], it is based on an inconsistent treatment of the large-Nclimit, so the agreement should perhaps be regarded as fortuitous. The proper prediction,F/D = 1/3 + O 1Nc, clearly is not very precise for the realistic case of Nc = 3, although it isat least consistent with the experimental result.

In the case of the tensor charges, where noexperimental information is available, it would seem that the only reasonable prediction wecan make is F/D = 1/3 + O 1Nc.The experimental result for the ratio g0A/g3A is in fact small. Using the experimental valuesg0A = 0.098 ± 0.076 ± 0.113 and g3A = 1.254 ± 0.006 [16], we obtain g0A/g3A = 0.078 ± 0.109.Unfortunately, this result does not offer a useful standard against which to compare theprediction g0T /g3T = O 1Nc, since g0A/g3A is expected to be of order O 1N2c.In the above analysis, we stated that δs is at most of order unity with respect to δuand δd.

In fact, since g8T and g0T are both of order 1/Nc with respect to g3T , δs is actuallyof order 1/Nc with respect to δu and δd. (It also follows that δu + δd is of order 1/Nc withrespect to δu and δd, as in the NQM.) The same conclusion can be drawn in the case of theaxial charges.

If ∆q is the axial charge counterpart of the tensor charge δq, then ∆s is oforder 1/Nc with respect to ∆u and ∆d. Since ∆u and ∆d are of order Nc [10] in the Skyrmemodel, ∆s, the fraction of the nucleon spin carried by strange quarks, is of order unity, andnot of order Nc, as previously reported [7].

This is consistent with the suppression of the qqsea that is expected on general grounds in large-Nc QCD [13]. (The previous prediction that∆s = O(Nc) in the Skyrme model was based on an “Nc-independent” F/D ratio of 5/9.

Theargument used in that case breaks down for F/D = 1/3. )Finally, we discuss the effects of SU(3) symmetry violation on the above results.

The roleof the strange quark in the nucleon is poorly understood in general. In the particular case ofthe tensor charges, it does not seem possible to say anything rigorous about the effects of asymmetry-breaking mass term for the strange quark.

Within the framework of the Skyrmemodel, for example, we might approach this problem from the point of view of dimensionalanalysis. Unfortunately, dimensional analysis appears to be completely unreliable for SU(3)symmetry-breaking effects in the Skyrmion, since the dimensionful parameter that sets thescale for symmetry-breaking effects is of order 250 MeV (i.e., less than the kaon mass, atypical measure of symmetry violation in the chiral lagrangian) [12].

We simply cannot ruleout the possibility of large corrections stemming from SU(3) violation. For what it is worth,we point out that SU(3) appears to be a very good symmetry in the case of the axial charges,although again, the reasons for this are not well understood.SUMMARYWe consider the nucleon tensor charges in the Skyrme model.

We show that in the large-Nc, SU(3)-symmetric limit, the model predicts that g8T /g3T is of order 1/Nc. This is equivalentto the prediction F/D = 1/3+O 1Nc, which, unfortunately, is not very precise for the realisticcase of Nc = 3.

These predictions are identical to the Skyrme model predictions for the octetaxial charges. The model also predicts that g0T /g3T is of order 1/Nc.

All of these predictionsare in agreement with the predictions of the NQM. Unlike the NQM, the Skyrme modeldoes not readily indicate the large-Nc behavior of, say, g3T , although it is not inconsistent to8

assume that g3T is of order Nc in the Skyrme model. We point out that the commonly quotedprediction for the axial charge F/D ratio (i.e., 5/9) is based on an inconsistent treatment ofthe large-Nc limit.

A consistent treatment yields the prediction quoted above, and shows thatthe strange quark contribution to the nucleon spin is of order unity in the large-Nc Skyrmemodel.ACKNOWLEDGEMENTSI thank Robert Jaffe for suggesting this work, and for many useful discussions. I wouldalso like to thank Matthias Burkhardt and Eric Sather for useful discussions.REFERENCES[1] For a review and references, see A. Manohar, UCSD/PTH 92-10 (1992).

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