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Electromagnetic Moments of the

arXiv:hep-ph/9308317v1 22 Aug 1993Electromagnetic Moments of theBaryon DecupletMalcolm N. Butler†Department of Physics, Stirling HallQueen’s University, Kingston, Canada, K7L 3N6Martin J. Savage †† ∗Department of Physics, U.C.

San DiegoLa Jolla, CA 92093-0319Roxanne P. SpringerDepartment of PhysicsDuke University, Durham, NC 27708AbstractWe compute the leading contributions to the magnetic dipole and electric quadrupolemoments of the baryon decuplet in chiral perturbation theory. The measured value for themagnetic moment of the Ω−is used to determine the local counterterm for the magneticmoments.

We compare the chiral perturbation theory predictions for the magnetic mo-ments of the decuplet with those of the baryon octet and find reasonable agreement withthe predictions of the large–Nc limit of QCD. The leading contribution to the quadrupolemoment of the ∆and other members of the decuplet comes from one–loop graphs.

Thepionic contribution is shown to be proportional to Iz (and so will not contribute to thequadrupole moment of I = 0 nuclei), while the contribution from kaons has both isovectorand isoscalar components.The chiral logarithmic enhancement of both pion and kaonloops has a coefficient that vanishes in the SU(6) limit. The third allowed moment, themagnetic octupole, is shown to be dominated by a local counterterm with correctionsarising at two loops.

We briefly mention the strange counterparts of these moments.UCSD/PTH 93-22, QUSTH-93-05, Duke-TH-93-56August 1993hep-ph/9308317† Address as of September 1, 1993: Department of Astronomy and Physics, Saint Mary’sUniversity, Halifax, NS, Canada B3H 3C3††Address as of September 1, 1993: Department of Physics, Carnegie Mellon University,Pittsburgh, PA, USA 15213∗SSC Fellow1

Static electromagnetic moments are a valuable tool for understanding internal struc-ture. In nuclear physics, static moments played a crucial role in understanding the strongtensor interaction arising from one-pion exchange which lead to significant deviations fromspherical symmetry in simple nuclei like the deuteron.

In the same way, hadronic structurecan be investigated using static moments. The magnetic moments of the octet baryonshave been understood in the context of SU(3) for many years [1].

Leading, model inde-pendent corrections to these SU(3) relations have been computed in chiral perturbationtheory [2][3] and have been found to improve agreement with experimental data.The electrostatic properties of the ∆and other members of the baryon decuplet havereceived little theoretical attention since experimental data has been scarce until recently.The Ω−magnetic moment was found to be µΩ= −1.94 ± 0.17 ± 0.14 µN [4], and arecent measurement for the ∆++ moment using pion bremsstrahlung (a model dependentextraction) found µ∆++ = 4.5 ± 0.5 µN [5] (we note that this result is not used by thePDG in their estimate [4]). The magnetic moments have been examined in the cloudy-bagmodel [6], in quark models [4]–[9], in a Bethe-Salpeter model [10], in the Skyrme model[11]–[13], using QCD sum rules [14], and also recently in quenched lattice gauge theory[15][16].

Our goal here is to understand the magnetic moments in a model-independent,systematic way, using chiral perturbation theory.The decuplet baryons also have electric quadrupole and magnetic octupole moments.These moments have been studied recently using quenched lattice QCD [15][16]. In chiralperturbation theory, the pion one–loop graphs tend to dominate over either kaon one–loopgraphs or the local counterterm because of the presence of the (calculable) chiral logarithm,log(M 2π/Λ2χ).

This was seen in the calculation of the electric quadrupole matrix elementfor the decay ∆→Nγ [17] [18]. For the decuplet quadrupole moments, however, we findthat the pion one–loop contributions are proportional to the third component of isospin,with the result that baryons with Iz = 0 receive no contribution from the lowest orderpion loop, and Iz = 12 baryons receive approximately equal contributions from kaon andpion loops at lowest order.

These are still formally dominant over the dimension six localcounterterm for the quadrupole moment. We show that the octupole moment is dominatedby a dimension seven local counterterm with corrections occurring first at two loops.Heavy baryon chiral perturbation theory uses the chiral symmetry of QCD to constructan effective low–energy theory to describe the dynamics of the goldstone bosons associatedwith the spontaneous breaking of chiral symmetry.The baryons can be included in aconsistent manner, as shown in Ref.

[19]. For a review of the simplifications available2

in calculating with this formalism see Ref. [20].

The decuplet of resonances as explicitdegrees of freedom has been shown to be important for most physical observables, and forconsistency of the perturbative expansion [19][21]. Further, Ref.

[22] shows that includingthe decuplet (in fact, the entire tower of I = J baryons) is required in order for a lowenergy theory of pions and nucleons to be unitary in the large–Nc limit of QCD (whereNc is the number of colours). The decuplet appears with couplings to the pions satisfyinga contracted SU(2Nf) algebra (where Nf is the number of light flavours in the theory).Ref.

[23] shows that the corrections to the relations arising from this SU(2Nf) symmetryoccur first at order 1/N 2c .The leading SU(3) invariant local counterterm for the decuplet magnetic moment isgiven by a dimension five operator [3]LCTM1 = −i eMNµcqkT µvkT νvkFµν ,(1)where T µ is the decuplet field and qk is the charge of the kth baryon of the decuplet. Wehave normalised the coefficient of the operator so that the magnetic moment of the kthbaryon is qkµc nuclear magnetons.

A simple tree-level fit to the magnetic moment of theΩ−hyperon gives µc = 1.94±0.22 µN (where we have added the systematic and statisticalerrors of µΩin quadrature). The leading corrections to the magnetic moments arise fromthe one-loop diagrams shown in fig.

1. A computation gives the following matrix elements:MT T T = ie16π2 H2(T v · kTvµ −T vµTv · k)Aµ 23Xiαif 2MiF(∆mi, Mi)(2)andMT BT = ie16π2 C2(T v · kTvµ −T vµTv · k)Aµ Xiβif 2MiF(∆mi, Mi) ,(3)whereF(∆m, M) = ∆m logM 2Λ2χ+p∆m2 −M 2 log∆m +√∆m2 −M 2 + iǫ∆m −√∆m2 −M 2 + iǫ,(4)and Aµ is the electromagnetic gauge field.

The superscripts TTT and TBT denote thecontribution from graphs with intermediate decuplet and octet baryons respectively. Themass splitting between the external baryon and the baryon in the loop is ∆mi, the massof the relevant pseudogoldstone boson is Mi (i = π or K), the chiral symmetry breakingscale is Λχ, and the decay constant of the meson in the loop is fMi (fπ = 132MeV and3

fK = 1.22fπ). The decuplet-octet-meson coupling constant is C and the decuplet-decuplet-meson coupling constant is H. The constants αi and βi are the product of the electriccharge of octet meson i and SU(3) Clebsch–Gordan coefficients (explicit values are givenin the appendix).

The one–loop corrections to the multipole moments depend only on thecoupling constants C and H. Using C = −1.2 ± 0.1, H = −2.2 ± 0.6 [17] and the measuredvalue of the Ω−magnetic moment to fix µc, we predict the magnetic moments of the othermembers of the baryon decuplet. These results are shown in table 1 and also graphicallyin fig.

2. It is clear from fig.

2 that the SU(3) violating corrections induced by the one-loopgraphs (dominated by the contribution from kaons) is small and that the tree level relation(where the magnetic moment is proportional to the electric charge of the baryon) is notbadly broken. The one–loop chiral perturbation theory prediction for the ∆++ magneticmoment of µ∆++ = 4.0±0.4 (the tree-level result is µ∆++ = 5.8±0.7) agrees within errorswith the recent measurement (but model dependent extraction) of µ∆++ = 4.5 ± 0.5 [5],and the results from quenched lattice QCD µ∆++ = 4.9 ± 0.6 [15], but is significantlysmaller than the prediction of the naive quark model µ∆++ ∼5.6 [9].

For the chargedmembers of the decuplet we agree with the quenched lattice computations [15] but differin the predictions for the magnetic moments of the neutral baryons. Note that the neutralbaryon magnetic moments do not depend on the local leading counterterm that appearsfor the charged baryons, making these predictions independent of the measured value ofµΩ−.As mentioned earlier, the axial matrix elements in I = J baryons such as N and ∆must obey a contracted SU(2Nf) algebra in the large Nc limit of QCD [22].

This resultsfrom the need for the low energy theory of baryons and goldstone bosons to be unitaryin the large Nc limit. Further, it was shown that this requires the 1/Nc correction tothe axial matrix elements be proportional to the leading term.

Therefore, relationshipsbetween axial matrix elements have vanishing 1/Nc corrections, but are corrected at order1/N 2c and higher [23]. A similar argument can be constructed for the matrix elements ofthe isovector magnetic moment operator.

We expect that they satisfy the relations of acontracted SU(2Nf) algebra up to corrections arising from terms 1/N 2c and higher in the1/Nc expansion. In this limit the isovector magnetic moments satisfyµ∆++ −µ∆−µp −µn= 95 + O( 1N 2c)(5)andµ∆+ −µ∆0µp −µn= 35 + O( 1N 2c).

(6)4

Our analysis of magnetic moments is a non-trivial test of these relations. The local coun-terterm given in (1) has both isoscalar and isovector components, since it is proportionalto the electric charge operator.

At tree–levelµ∆++ −µ∆−µp −µntree= −3µΩµp −µn∼1.2(7)andµ∆+ −µ∆0µp −µntree= −µΩµp −µn∼0.4,(8)which are about 2/3 the values expected in the large Nc limit. Including the one–loopgraphs improves the situation somewhat and we find thatµ∆++ −µ∆−µp −µnone−loop= 1.35 ± 0.15(9)andµ∆+ −µ∆0µp −µnone−loop= 0.45 ± 0.05 .

(10)Despite the fact that both quantities are still smaller than the numbers expected from large-Nc QCD, the one-loop corrections tend to reduce the discrepancy in each case. There aremodifications to the large Nc relations from terms subleading in the 1/Nc expansion andalso corrections at the 25% level from terms higher order in the chiral expansion that mayimprove the agreement.The quadrupole moment for each of the decuplet baryons receives a contribution fromboth long–distance physics in the form of pion and kaon loops, and from short distancephysics in the form of a local counterterm with an unknown coefficient.

This dimensionsix counterterm has the formLCTE2 = QCTeΛ2χqi(TµviT νvi + TνviT µvi −12gµνTσviTviσ)vα∂µFνα . (11)The contribution to the quadrupole moment from the diagrams in fig.

1 are formallyenhanced over the naive contribution from the local counterterm by a chiral logarithm,log(M 2/Λ2χ), and we will neglect the contributions from the local counterterm, takingQCT ∼0 for the rest of this discussion. The explicit contributions from the graphs in fig.

1areQT T T = −ie16π229H2ω(T v · kTvµ + T vµTv · k −12kµT v · Tv)Aµ Xiαif 2MiG(∆mi, Mi) (12)5

andQT BT = ie16π2 C2 ω6 (T v · kTvµ + T vµTv · k −12kµT v · Tv)Aµ Xiβif 2MiG(∆mi, Mi) ,(13)whereG(∆m, M) = logM 2Λ2χ+∆m√∆m2 −M 2 log∆m +√∆m2 −M 2 + iǫ∆m −√∆m2 −M 2 + iǫ. (14)As before, ∆mi is the mass splitting between the external and loop baryon, Mi is the massof the goldstone boson in the loop, and fMi is the meson decay constant.

The notation forα and β is the same as for the magnetic moment equations. We can extract quadrupolemoments from this calculation by using the definition of the quadrupole interaction energy,HQ = −16XijQij∂Ei∂xj ,(15)where E is the electric field and Qij is the quadrupole tensor which is symmetric andtraceless.

The quadrupole moment is defined to be Qzz, and can be extracted from (12)and (13). The results for the various members of the decuplet, neglecting the formallysubdominant counterterm of (11), are shown in Table 2 and graphically in fig.

3. Thesemoments are large, and comparable to the moments of light nuclei such as the deuteron(QD = 2.8 × 10−27e −cm2).

With such large moments, the presence of constituent ∆’sin nuclei might have a significant effect on nuclear quadrupole moments.Naively, thepion loop graphs should be logarithmically enhanced over the kaon loop graphs. Yet, theClebsch-Gordan coefficients (given in the appendix) are such that the quadrupole momentgenerated by the pion loops depend only upon the Iz quantum number of the baryon (thekaon loops have both isovector and isoscalar dependence).

This is distinctly different fromthe dependence of the local counterterm which depends on the charge of the baryon. Theimportance of this result becomes apparent when considering the quadrupole moment ofa nucleus, in particular an I = 0 nucleus such as the deuteron.

One might imagine thatthe intrinsic quadrupole moment of the ∆would contribute to the quadrupole moment ofa nucleus through virtual ∆states. However, as most of the intrinsic quadrupole momentof the ∆depends on Iz, this contribution to the quadrupole moment of an I = 0 nucleusvanishes.

Hence, the ∆contribution to the quadrupole moment of the deuteron is greatlysuppressed over naive expectations, appearing first from the kaon loop contribution.6

Our values for the quadrupole moments are not always consistent with the valuesfound in quenched lattice computations [15], though all but the neutral baryons agreewithin errors. In particular, where we find that the dominant component behaves as Iz,lattice computations find behaviour more consistent with dependence upon the baryoncharge.Another interesting, perhaps more mysterious result that can be found by examin-ing the Clebsch-Gordan coefficients in the appendix is that the coefficient of both thelog(M 2π/Λ2χ) and log(M 2K/Λ2χ) terms are proportional to 49H2 −C2.

This vanishes whenH/C = 3/2, which is exactly the relationship satisfied in the SU(6) limit. In this SU(6)limit, the contribution to the quadrupole moment from these one-loop graphs arises en-tirely from the mass splittings amongst the baryons.

We can reconcile our results withthat of quenched lattice QCD if indeed the axial couplings are very close to their SU(6)values. The quadrupole moments would then receive a non-negligible, and possibly dom-inant, contribution from the incalculable local counterterm (which we have neglected forour discussions), giving the characteristic dependence on the baryon charge that the lat-tice calculations find.

Our central value predictions would then be substantially smaller inmagnitude than those obtained using the experimentally fit values of H and C.In dealing with the magnetic moments of the decuplet, we saw that the large Nclimit of QCD gave results consistent with those of chiral perturbation theory calculations.For the quadrupole moments, the large Nc limit of the one-loop contribution approaches aconstant value. This is because the relationships between axial coupling constants F, D, C,and H approach their SU(6) values [22][23], and the hyperfine mass splittings between thebaryons vanish as 1/Nc [24].

The coefficient of the quadrupole counterterm and the 1/N 2ccorrections to the hyperfine mass splittings are needed in order to make a more explicitcomparison between the large Nc predictions and chiral perturbation theory results for thequadrupole moments.Finally, the decuplet baryons could also have a magnetic octupole moment. We canconstruct a dimension seven local counterterm for this moment, of the formLCTM3 = e ΘΛ3χqi(TµviSνv T αvi + TνviSαv T µvi + TαviSµv T νvi)ǫαβλσvβ∂µ∂νF λσ ,(16)where Θ is an unknown coefficient.

This tensor structure, in particular the three derivativesof the electromagnetic field, does not appear in the one–loop graphs shown in fig. 1.Therefore, the magnetic octupole moment will be dominated by the local counterterm andcorrections can first occur from two–loop diagrams.7

In addition to the electrostatic moments of these baryons we can examine their strangemoments. Strange moments of the nucleons as suggested in [25] have been the subject ofan immense amount of both theoretical and experimental interest.

Estimates of the size ofthese moments have been made for the octet baryons in the context of different hadronicschemes [26] [27]. The strange moments of the decuplet baryons may never be measured,yet we are able to see what form they will have in the language of chiral perturbation theory.The strange magnetic and quadrupole moments could be substantially different from theirelectromagnetic counterparts.

Since the strange charge operator has both flavour octetand singlet components, there are two unknown counterterms for each strange moment,with SU(3) structureL ∼S TabcQ(s)dcTabd+σ TabcTabcQ(s)αα,(17)where the strange charge matrix is Q(s)=diag(0,0,1) and S and σ are unknown coeffi-cients. For investigating the baryon sea, however, we are most interested in looking atthe strange moments of the non-strange baryons, namely the ∆’s.

The first term, S, doesnot contribute, which leaves one unknown counterterm, σ, that contributes equally to allbaryons in the decuplet, yet is unclear how to determine experimentally. We can computecorrections to the strange magnetic moments and also the dominant contribution to thestrange quadrupole moment just as we did in the electrostatic case.

For these strangemoments the pion loops do not contribute (they do not carry strange charge) but bothcharged and neutral kaons will contribute. Therefore, we expect the strange quadrupolemoment to be much smaller than the electrostatic counterpart for large Iz baryons, withthe other quadrupole moments perhaps comparable in size to the electrostatic ones.

Thestrange magnetic moment may be the same size as the electrostatic magnetic moment.Since the strange charge is an isoscalar, the moments of each of the ∆’s are identical. Theone–loop induced quadrupole moments are proportional to 49H2 −C2 (up to isospin break-ing mass differences), a quantity that vanishes in the SU(6) limit.

(Unlike the case forthe electromagnetic quadrupole moment, the intermediate baryons contributing to thesequadrupole moments are all isospin degenerate.) If the axial couplings are near the SU(6)point, as there is strong evidence to suggest, then the strange moments of the ∆’s are eachdominated by one incalculable local counterterm.

These quantities are of theoretical inter-est as the appearance of possibly another non-zero strange matrix element in non-strangehadrons.8

In conclusion, we have discussed the electrostatic properties of the ∆and other mem-bers of the baryon decuplet.Using chiral perturbation theory we have computed theleading non-analytic contributions to the magnetic dipole and electric quadrupole mo-ments, and shown that the leading contribution to the octupole moment is from a localdimension seven counterterm with corrections arising at two-loops. We have compared ourprediction for the ∆++ magnetic moment with its recent model dependent extraction frompion bremsstrahlung data and found it to be in good agreement.

The one-loop computa-tion moves the isovector magnetic moments into better agreement with the predictions oflarge-Nc QCD compared to the tree-level results. Although the quadrupole moments ofthe decuplet have not been measured yet, there may be some hope for such measurementsat CEBAF.

We computed the leading contribution to the quadrupole moments from long-distance pion and kaon loops, which are formally dominant over the dimension six localcounterterm. The pion contribution depends only on Iz and hence the contribution fromthe intrinsic quadrupole moment of ∆’s to that of an I = 0 nucleus from the ∆componentsin the nuclear wavefunction is suppressed.

This is an important result particularly for thedeuteron since the magnitude of the ∆quadrupole moments are comparable to that of thedeuteron. Further, the formally dominant terms of the form log(M 2π/Λ2χ) and log(M 2K/Λ2χ)vanish when the axial couplings approach their SU(6) limit.We have compared our results to those obtained in quenched lattice QCD [15] and findthat the magnetic moments of the charged baryons agree well.

This is not unexpected sincethey are dominated by the local counterterm that is fixed experimentally. This agreementdoes not exist for the neutral baryons, which have no counterterm dependence.Thepredictions for the charged baryon electric quadrupole moments also agree within errors,yet have a different dependence on the baryon isospin.

Again, there is not agreement forthe neutral baryons.Our leading contribution for the quadrupole moment arises frompion loops and depends on Iz only. The lattice computation indicates that the quadrupolemoment depends on the charge of the baryon.

These two results can be reconciled if theaxial coupling constants C and H satisfy SU(6) relations as required in the large Nc limitand also approximately found experimentally [17][20].In this scenario the quadrupolemoments receive a non-negligible and potentially dominant contribution from the local(incalculable) counterterm.We expect that some of our predictions will be tested at CEBAF, and that measure-ments of the quadrupole moments in particular may help test the validity of the heavy9

baryon chiral perturbation theory approach in understanding low–energy QCD. In addi-tion, we expect the comparison to help determine if the contracted SU(2Nf) algebra is auseful symmetry for describing low–energy hadronic properties.MNB acknowledges the support of the Natural Science and Engineering ResearchCouncil (NSERC) of Canada.

MJS acknowledges the support of a Superconducting Su-percollider National Fellowship from the Texas National Research Laboratory Commissionunder grant FCFY9219, and the hospitality of the Aspen Institute for Physics where muchof this work was done. RPS acknowledges the support of DOE grant DE-FG05-90ER40592.10

AppendixSU(3) Clebsch-Gordan coefficients α and β used for the evaluation of decuplet elec-tromagnetic moments. For brevity, only the intermediate baryon state is given as a label.The boson index is implicit.∆++α∆+ = 13βp = 1αΣ∗+ = 13βΣ+ = 1∆+α∆0 = 49βn = 13α∆++ = −13αΣ∗0 = 29βΣ0 = 23∆0α∆−= 13α∆+ = −49βp = −13αΣ∗−= 19βΣ−= 13∆−α∆0 = −13βn = −1Σ∗+αΣ∗0 = 29βΣ0 = 16βΛ = 12αΞ∗0 = 49βΞ0 = 13α∆++ = −13Σ∗0αΣ∗−= 29βΣ−= 16αΣ∗+ = −29βΣ+ = −16αΞ∗−= 29βΞ−= 16α∆+ = −29βp = −16Σ∗−αΣ∗0 = −29βΣ0 = −16βΛ = −12α∆0 = −19βn = −1311

Ξ∗0αΞ∗−= 19βΞ−= 13αΩ−= 13αΣ∗+ = −49βΣ+ = −13Ξ∗−αΞ∗0 = −19βΞ0 = −13βΛ = −12αΣ∗0 = −29βΣ0 = −16Ω−αΞ∗0 = −13βΞ0 = −112

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Table 1Magnetic moments of the baryon decuplet from chiral perturbation theory.Uncertainties reflect uncertainties in the couplings C and H, and in the magneticmoment of the Ω−used to constrain the local counterterm.µ (µN)µ (µN)∆++4.0 ± 0.4Σ∗+2.0 ± 0.2∆+2.1 ± 0.2Σ∗0−0.07 ± 0.02∆0−0.17 ± 0.04Σ∗−−2.2 ± 0.2∆−−2.25 ± 0.25Ξ∗00.10 ± 0.04Ω−(input)−1.94 ± 0.22Ξ∗−−2.0 ± 0.2Table 2Quadrupole moments of the baryon decuplet arising from one–loop graphsin chiral perturbation theory. Uncertainties reflect uncertainties in the couplingsC and H.Q (10−27e −cm2)Q (10−27e −cm2)∆++−0.8 ± 0.5Σ∗+−0.7 ± 0.3∆+−0.3 ± 0.2Σ∗0−0.13 ± 0.07∆00.12 ± 0.05Σ∗−0.4 ± 0.2∆−0.6 ± 0.3Ξ∗0−0.35 ± 0.2Ω−0.09 ± 0.05Ξ∗−0.2 ± 0.114

Figure CaptionsFig. 1.Leading one–loop graphs contributing to the multipole moments of the decupletbaryons.

The dashed lines correspond to charged goldstone bosons and the wigglyline to photons. T is a decuplet baryon and B is an octet baryon.Fig.

2.The magnetic moments of the decuplet baryons, in units of nuclear magnetons.The dark points are the moments derived from the central values of C and H andthe lighter lines are the associated uncertainties.Fig. 3.The quadrupole moments of of the decuplet baryons.The contribution fromthe local counterterm is subleading and we have set it to zero.

The quadrupolemoments here come from one-loop graphs only. The dark points are the momentsderived from the central values of C and H and the lighter lines are the associateduncertainties.15

BTTTTT

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246810-4-224420-2-4∆++∆+∆0∆−Σ∗+Σ∗−Ξ∗0Ξ∗−Ω−Σ∗0µ

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-2-1.5-1-0.500.511.51.00-1.0∆++∆+∆0∆−Σ∗+Σ∗0Σ∗−Ξ∗0Ξ∗−Ω−Q

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