Heavy baryons with strangeness in a soliton model

(1)

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Heavy

baryons

with

strangeness

in

a

soliton

model

J.P. Blanckenberg,

H. Weigel

PhysicsDepartment,StellenboschUniversity,Matieland7602,SouthAfrica

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received26May2015

Receivedinrevisedform11September 2015

Accepted12September2015 Availableonline15September2015 Editor:J.-P.Blaizot

Keywords: Chiralsoliton

Collectivecoordinatequantization SU(3)symmetrybreaking Boundstateapproach Hyperﬁnesplitting Heavybaryonspectrum

We present resultsfrom achiralsolitonmodel calculationfor thespectrum ofbaryonswith asingle heavy quark (charmorbottom)and non-zerostrangeness.We treatthestrange componentswithina threeflavorcollectivecoordinatequantizationofthesolitonthatfullyaccountsforlightflavorsymmetry breaking.Heavybaryonsemergebybindingaheavymesontothesoliton. Thedynamicsofthisheavy mesonisdescribedbytheheavyquarkeffectivetheorywithfinitemasseffectsincluded.

1. Motivation

Baryonscontainingheavyquarkssuchascharmorbottomform anexcellentopportunitytostudythebindingofquarkstohadrons. Since there is no exact solution to quantum-chromo-dynamics (QCD), various models and approximations that focus on partic-ularfeaturesofQCDare relevant.Inthecontextofheavy baryons threeare ofparticularimportance. First,there istheheavy spin– ﬂavorsymmetry

[1]

1 _that _governs_the_dynamics _of_heavy _quarks.

Second,thereisthechiralsymmetrythat dictatestheinteractions among the light quarks. In addition to dynamical chiral symme-trybreaking, there issubstantial ﬂavor symmetry breakingwhen thestrange quarkis involved.Itisthusparticularlyinteresting to investigate baryons that, in the valence quark picture, are com-posed of a single heavy quark and two light, including strange, quarks. Thirdly, generalizing QCD from three to arbitrarily many colordegrees offreedom suggests toconsider baryons assoliton conﬁgurationsinaneffectivemesontheory

[3]

.

Ourpointofdepartureis achiralsolitonofmesonfieldsbuilt fromup anddownquarks [4]. Stateswithgoodbaryon quantum numbers are generated by quantizing the fluctuationsabout the soliton. The modes associated with (flavor) rotations have large non-harmoniccomponentsandconsequentlyaretreatedas

collec-E-mailaddress:weigel@sun.ac.za(H. Weigel). 1 _For_reviews_see_Ref._[2].

tive excitations. The Hamiltonian for thesecollective coordinates contains ﬂavor symmetry breaking terms that slightly suppress thenon-harmoniccontributions.Theimportantfeatureisthatthis Hamiltoniancan bediagonalizedexactly, i.e. thespectrumcan be determined beyond a perturbation expansion in the quark mass differences [5]. The resulting eigenvalues are associatedwith the strangenesscontribution tothe baryon masses.Forthe particular caseofkaon–nucleonscatteringthisapproachhasbeenveriﬁed

[6]

to yield the correct resonance position. Subsequently fields rep-resenting mesons witha single heavy quark are included. While theirheavyquarkcomponentsaresubjecttotheheavyspin–flavor symmetry,theirlightonescoupletothelightmesonfields accord-ing to chiralsymmetry such thatthe solitongeneratesan attrac-tive potential forthe heavy mesonfields [7]. Combined withthe soliton, a bound state in this potential builds the heavy baryon. (Thisisageneralization oftheso-calledboundstateapproach

[8]

that, in the harmonic approximation, describes hyperons in the Skyrme model [9].2₎ _The _strangeness _components _of _the _heavy

meson bound state are subject to thesame collective coordinate treatmentasthesoliton.

ShortlyaftertheboundstateapproachintheSkyrmemodelof pseudoscalarmesonswas appliedto hyperonsitwasextended to

2 _While _Ref._[10]_{comprehensively}_reviews_soliton_model_studies,_Ref._[6] thor-oughly discussesthe twoabovementioneddescriptionsofstrangenessinchiral solitonmodels,inparticularwithregardtothelargeNClimit.

http://dx.doi.org/10.1016/j.physletb.2015.09.026

(2)

heavierbaryons

[11]

.Inthose studies therelevance ofthe heavy spin–flavor symmetry was not yet recognized. Subsequently, also heavy vector meson fields were included [12]. More or less at the same time investigations were performedin the heavy limit scenario [13,14]. Those heavy limit studies included neither cor-rectionstotheheavyspin–flavorsymmetryfromfinitemassesnor strangenessdegreesoffreedom.Inthatcase,baryonslike

c

can-not be addressed. Strangeness was indeed included in Ref. [15], however,lightﬂavor symmetry breakingwas treatedina pertur-bationexpansion andﬁnitemass effectswere omitted.Thisdoes not distinguish betweeneven and oddparity or charm and bot-tombaryonsandtypicallyoverestimatesthebindingenergyofthe heavymeson

[7]

.Alsotheparametersofthefinal energyformula were fitted ratherthan calculated from a realistic solitonmodel. These widespread bound state studies derive a potential for the mesonfields from the soliton that is fixed in position. We note thatthispicture isstrictly validonlyin thelarge numberof col-orslimitwhenthesolitonismoremassivethantheheavymeson. Thoughthisapproachisasystematicandconsistentexpansionin thenumberofcolors,kinematicalcorrectionsshould beexpected intherealworldwiththreecolors.

Our soliton model calculation for the spectrum of heavy baryonswillimprovewithregardtothefollowingaspects:Wetake theparametersinthemassformulafromanactual solitonmodel calculation(weallow formoderateadjustmentof thelight ﬂavor symmetrybreaking strength), go beyondtheperturbation expan-sion in that symmetry breaking and construct the heavy meson bound state from a model that systematically incorporates ﬁnite masscorrections.Ourmodelcalculationwillproduceanextensive picture ofbaryons, fromthe nucleon up to the

b. We will not

considerdoubly-heavybaryons,though.

The spectrum of heavy baryons has been investigated in other approachesas well. Acomprehensive account of the (non-relativistic)quarkmodelapproachisgiveninRef. [16]withsome newerresults reported in Ref. [17]. Relativistic effects are incor-poratedwithin quark–diquarkmodels [18]. QCD sumrules were not only used to obtain the spectrum [19], but also to extract theheavy quarkmass poles

[20]

.LatticeQCDcalculationscan be tracedfromRef.[21]thatalsostudiesbaryonswithmorethanone heavyquark. Finally,Ref.[22] containscomprehensivereviewson baryonspectroscopythatdiscussavarietyofapproachesandmay beconsultedforfurtherreferences.

2. Thesolitonmodel

InchiralLagrangiansthe interactionterms are orderedby the numberofderivativesactingonthepseudoscalarﬁelds.Themore derivatives there are, the more unknown parameters appear in the Lagrangian. Replacing these higher derivatives by resonance exchangeterms isadvantageousbecausemoreinformationis avail-abletodeterminetheparameters.Wethusconsiderachiralsoliton thatisstabilizedby vectormesons

ρ

and

ω

[23] asareﬁnement oftheSkyrmemodel

[4,9]

.Othershortcomingsofthepseudoscalar soliton,liketheneutronprotonmassdifferenceortheaxialsinglet matrixelementofthenucleonarealsosolvedwhenincludinglight vectormesons

[10]

.

The basicbuilding block of the model is the chiral ﬁeld U

=

exp

i

8_a₌₁

φ

a

(

x

)λ

a

/

fa

,i.e. thenon-linearrealizationofthe pseu-doscalaroctet ﬁeld

φ

a

(

x

)

. Here fa are the respective decay

con-stants [ fπ

=

93 MeV (for a

=

1

,

2

,

3), fk

=

114 MeV (for a

=

4

,

. . . ,

7). The casea

=

8 requires additionalinput [24] butisnot relevanthere.]and

λ

a are theeightGell-MannmatricesofSU

(

3

)

.

Thestaticﬁeldconﬁgurationofthesolitonisthehedgehogansatz

U0

(

r

)

=

exp

τ

· ˆ

r F

(

r

)

,

ω

μ

(

r

)

=

ω

(

r

)

gμ0 and

ρ

(_im0)

(

r

)

=

ikmr

ˆ

k G

(

r

)

r

.

(1)

The isovector

τ

= (λ

1

, λ

2

, λ

3

)

comprises thethree Pauli matrices

fromtheisospinsubspaceofﬂavorSU

(

3

)

.Thespatialcomponents ofthe

ω

μ andthetimecomponentsofthe

ρ

μ ﬁeldsarezero.For thelatter,i isanisospin/ﬂavorindexandm

=

1

,

2

,

3 labelsits spa-tial components. The proﬁle functions F

(

r

)

,

ω

(

r

)

and G

(

r

)

enter theclassicalenergyfunctional,Ecl.Theproﬁlesaredeterminedby

the minimization of Ecl,subject to boundaryconditions that

en-sureunitbaryonnumber:

F

(

0

)

=

0

,

d

ω

(

r

)

dr

r=0

=

0 and G

(

0

)

= −2

.

(2)

Allprofilefunctionsvanishasymptotically.Configurationsthatare suitable forquantizationareobtainedbyintroducingtime depen-dentcollectivecoordinatesfortheflavororientation A

(

t

)

∈

SU

(

3

)

U

(

r

,

t

)

=

A

(

t

)

U0

(

r

)

A†

(

t

)

and

τ

· ρ

_μ

(

r

,

t

)

=

A

(

t

)

τ

· ρ

(_μ0)

(

r

)

A†

(

t

) .

(3)

In addition proﬁle functions are induced for the spatial compo-nents of

ω

μ and the time components of

ρ

μ [25,26]. Deﬁning eight angular velocities

a via the time derivative ofthe

collec-tivecoordinates i 2 8

a=1

a

λ

a

=

A†

(

t

)

d A

(

t

)

dt

,

(4)

allowsacompactpresentationoftheLagrangefunctionforthe col-lectivecoordinatesfromthelightmesonﬁelds

Ll

(

a

)

= −

Ecl

+

1 2

α

2 3

i=1

2_i

+

1 2

β

2 7

α=4

2_α

−

√

3 2

8

.

(5)

It isobtainedfromthe spatial integral overthe Lagrange density withtheabovedescribedfieldconfigurationsubstituted.Notethat the collective coordinates only appear via the angularvelocities; A doesnotappearexplicitly.Thelastterm,whichisonlylinearin thetime derivative,originatesfromtheWess–Zumino–Witten ac-tion [27] that incorporates theQCD anomaly.The coefficients

α

2

and

β

2 aremoments ofinertiaforrotations inisospace3 andthe strangenesssubspaceofﬂavorSU

(

3

)

,respectively.Thesemoments of inertia are functionals of the proﬁle functions and the varia-tionalprincipledeterminestheinduced componentsofthevector mesonﬁelds.ThestructureofthecollectivecoordinateLagrangian, Eq. (5) is generic to all chiral models that support soliton solu-tions.Theparticularnumericalvaluesfortheclassicalenergyand the moments of inertia are, of course, subject to the particular model.Here we employthe calculation described in Appendix A ofRef.[26]fortheentriesofEq.(5).

3. Heavymesonboundstate

Ineffectivemesontheories,theheavyﬂavorentersviaaheavy meson containing a single heavy quark (charm or bottom) of mass M.Thedynamicsoftheheavy mesonfollowstheheavy ﬂa-voreffectivetheory

[2]

thattreatsthepseudoscalar( P )andvector meson( Q μ)componentsequivalently.Thatis,inthelimitM

→ ∞

thesecomponentsarepartofasinglemultiplet(Theconstant four-velocity V μ characterizestheheavyquarkrestframe.)

(3)

H

=

1 2

1

+

γ

μVμ

i

γ

5P

+

γ

μQμ

where P

=

e−iM V·xP and Q_μ

=

e−iM V·xQμ

.

(6)

TheLagrangianthatdescribesthecouplingofthismultiplettothe light mesons includingthe vector mesons

ρ

and

ω

andrespects theheavyspin–ﬂavorsymmetryis

[28]

1 M

L

H

=

i VμTr

H DμH

¯

−

d Tr

H

γ

μ

γ

5pμH

¯

−

i

√

2c mρ Tr

H

γ

μ

γ

νFμνH

¯

+ . . . ,

(7)

whereH

¯

=

γ

0H†

γ

0.Wetakethecovariantderivativetobe4 Dμ

=

∂

μ

+

i vμ. The chiral currents of the light pseudoscalar mesons are vμ

,

pμ

=

2i

√

U

∂

μ

√

U†

_±

√

_U†

_∂

μ

√

U

and F μν is the ﬁeld strength tensor ofthe light vector mesons. The heavy–light cou-plingconstantsd

≈

0

.

53 andc

≈

1

.

60 weredeterminedfromheavy meson decays. A ﬁeld theory model that minimally extends to ﬁniteM and M∗ forthepseudoscalarandvectorcomponents, re-spectively,hasalsobeenconstructedinRef.[28]

L

H

=

D_μP

†DμP

−

1 2

Q_μν

†Qμν

−

M2P†P

+

M∗2Q_μ†Qμ

+

2iMd

P†pμQμ

−

Qμ†pμP

−

d 2

αβμν

₍

_Q να

)

†pμQβ

+

Q_β†pμQνα

−

2

√

2icM mV

2Q_μ†FμνQν

−

i M

αβμν

_D βP

† FμνQα

+

Qα†FμνDβP

,

(8)

sothat

L

H

→

L

H intheheavylimit.HereQ μν is theﬁeldstrength

tensor of the heavy vector mesons. The central feature is that, throughthecouplingtothelight mesonsoliton,solutionsforthe heavy meson ﬁelds emerge with energy 0

<

ω

<

M, i.e. bound states. (Negative energy bound states are also possible. Eventu-ally they build pentaquark baryons that will not be considered here.)The most strongly bound solutionhas P-wave structure in thepseudoscalarcomponent:

P

=

e iωt

√

4

π

(

r

)

r

ˆ

· ˆ

τ χ

,

Q0

=

eiωt

√

4

π

0

(

r

)

χ

and Qi

=

eiωt

√

4

π

i

1

(

r

)

ˆ

ri

+

1₂

2

(

r

)

i jkr

ˆ

j

τ

k

χ

.

(9)

Here P andQ μ arethreecomponentspinorswhoseﬂavorcontent is parameterized by the (constant) spinor

χ

. Since the coupling to the light mesonsoccurs via a solitonin the isospin subspace, only the ﬁrst two components of

χ

are non-zero. The four ra-dialfunctionsinEq.(9)coupletotheproﬁlesofthestaticsoliton, Eq.(1)inlineardifferentialequations.Normalizablesolutionsexist onlyforcertain valuesof

ω

.Thesesolutionsarethebound wave-functions.Theirconstruction,inparticularwithregardtoﬁnite M corrections, andtheir normalizationtocarry unit heavy charge is explainedinRefs.[7]and

[14]

,respectively.Aheavybaryonisthen a compound system of the soliton for the light ﬂavors and the bound state oftheheavy meson [8]. Thereare also boundstates

4 _Symmetry_allows_to_also_include_the _light_vector_meson_in_this_derivative_at theexpenseofanunknowncouplingconstant.Theboundstateenergiesonlyshow moderatesensitivityonthatconstant[7]soweomitithere.

inthe S-wavechannel inwhichtheheavy mesonﬁeldis param-eterized as(see Ref. [12] forparameterizations of higherangular momenta) P

=

e iωt

√

4

π

(

r

)

χ

,

Q0

=

eiωt

√

4

π

0

(

r

)

r

ˆ

· ˆ

τ χ

and Qi

=

eiωt

√

4

π

1

(

r

)

r

ˆ

ir

ˆ

· ˆ

τ

+

2

(

r

)

r

τ

· ∂

ir

ˆ

χ

.

(10)

They combine with the soliton to form negative parity heavy baryons [7,14]. Forconvenience we haveused equal symbolsfor the S and P -waveproﬁle functions but, of course, they are dif-ferent.Thecomputationoftheboundstateenergies

ω

from iden-tifying localized solutions to the equations of motions that arise by substitutingthe parameterizations, Eqs. (9)and (10),into the Euler–Lagrange equations ofEq. (8) isdetailed inAppendix A of Ref.[7].Thatreferencealsoprovidesﬁguresoftheresultingproﬁle functions.

Theheavymesonﬁeldsmustalsoaccountforthecollective ﬂa-vorrotationintroducedinEq.(3).Thisenforcesthesubstitution

P

−→

A

(

t

)

P and Qμ

−→

A

(

t

)

Qμ

,

(11)

wheretherighthandsidescontaintheﬁeldsintroducedinEqs.(9)

and

(10)

forP and S wavechannels, respectively.Thisgives non-zero strange components of the heavy mesons and couples the heavy meson strange quark to that of the soliton. Substituting thisflavorrotatingconfigurationintotheLagrangedensityand in-tegrating over space provides the collective coordinate Lagrange functionfromtheheavyfields

Lh

(

a

)

= −

ωχ

†

χ

+

1 2

√

3

χ

†

8

χ

+

ρχ

†

· τ

2

χ

.

(12)

Again,theﬂavorrotationmatrixA doesnotappearexplicitly.With the time dependenceofthecollective coordinates,termsthat in-volve

8_a₌₁

λ

a

a enter. In the heavy mesonsector the quadratic

termsprovidetheboundstatecontributionstothemomentsof in-ertia

α

2_and

_β

2_._Since_the_bound_state_{wave-functions}_are_strongly

localizedaroundthecenterofthesoliton5thelatterdominatesthe moments ofinertia.Itisthus safetoonlyretain thelinearterms in Eq. (12). At that order only a

=

1

,

2

,

3 and a

=

8 survive be-causethe boundstatesdonot haveanystrangenesscomponents. The normalizationof the bound state wave-function dictates the coefficients inthe firstandsecond terms. The hyperfinesplitting parameter

ρ

isafunctionalofallproﬁlefunctions,includingsome oftheinducedlightvectorﬁelds.Itsexplicitexpressionisgivenin Eqs. (B.1)–(B.4) of Ref. [14], whereit iscalled

χ

P and

χ

S for P

-andS-wavechannels,respectively.

4. QuantizationinSU

(

3

)

,symmetrybreakingandhyperﬁne splitting

Before weconstructa Hamiltonoperatorforthecollective co-ordinatesviaLegendretransformationoftheLagrangianLl

+

Lh we

recall that therotations introduced in Eq.(3) are not exact zero modesinanysensitivemodel.ThereasonisthatSU

(

3

)

ﬂavor sym-metryisexplicitlybrokenbydifferent(current)quarkmasses.This breakingismeasuredbytheratio

x

=

2ms

mu

+

md

,

(13)

5 _Their_asymptotic _behavior_is _e−|ω|r_∼_e−Mr _compared_to_e−mπr _of_the_chiral ﬁeld.

(4)

where the mq are the current quark masses of the respective

quarks.It can be estimatedfrom mesondata [24,29,30]. In early solitonmodelstudiesthisratio wasconsidered to bequite large, x

≈

30[24],orevenbigger

[31]

.Thiswasaccompaniedbysizable symmetrybreakingamongthehyperons

[26]

.Laterthisratiowas re-evaluatedandfoundtobesomewhatsmaller:20

≤

x

≤

25[30]. Thus it is appropriate to consider this ratiofor the (light) ﬂavor symmetrybreakingasatunableparameter.Thensymmetry break-ingaddstothecollectivecoordinateLagrangian

Lsb

(

A

)

= −

x 2

γ

[1

−

D88

(

A

)

]

,

(14) where Dab

=

12tr

λ

aA

λ

bA†

parameterizestheadjoint representa-tionofthe collectiverotations. The coeﬃcient

γ

isagaina func-tionalof theproﬁle functionsand acquiresits main contribution fromtheclassical ﬁelds, Eq.(1).It can be computedin any soli-tonmodel.(Intheliterature

γ

=

x

γ

is typicallyused.)The heavy mesons also contribute to the symmetry breaking parameter by appropriately substituting mass matricesin Eq.(8). For example, forthecharmheavymesoninthe P -wavechannelwehave

γ

=

γ

soliton

+

∞

0 drr2

m2_D

−

m2_D_s

2

+

m2_D∗

−

m2_D∗ s

−

2 0

+

12

+

1 2

2 2

.

(15)

Numericallythiscontributionissmallandcan easilybe compen-satedbyaslightchangeofx.

Wehave now collected all terms forthe collectivecoordinate LagrangianL

(

A

,

)

=

Ll

()

+

Lh

()

+

Lsb

(

A

)

andcanconstructthe

HamiltonoperatorbyLegendretransformation,

H

(

A

,

Ra

,

χ

)

=

Ecl

+

1 2

1

α

2

−

1

β

2

3 i=1 R2_i

+

1 2

β

2 8

a=1 R2_a

+

x 2

γ

[1

−

D88

(

A

)

]

−

3 8

β

2

1

−

1 3

χ

†

_χ

2

+ |

ω

|

χ

†

χ

+

Hhf

,

(16)

whereRa

=

_∂∂L_a deﬁnesthesaidLegendretransformation.The Ra

are the right generators of SU

(

3

)

since [ A

,

Ra]

=

A

(λ

a

/

2

)

upon

canonicalquantization.Thespinors

χ

containannihilationand cre-ationoperators fortheheavy mesonboundstate. Theyare quan-tized as ordinary harmonic oscillators. In particular

χ

†

_χ

_is _the

numberoperator fortheheavy meson boundstate. Since we are consideringhadronswithasingle heavyquark, contributionsthat arequarticin

χ

havebeenomittedforconsistency.(Inthesquare atermthat isexplicitlyofquarticorderis maintainedbecauseit cancelsa similartermin

_aR2

a,cf. subsectionbelow.)The

hyper-ﬁnesplittingpart,Hhf,thatemergesfromthelastterminEq.(12),

willbediscussedlater. 4.1. SU

(

3

)

diagonalization

The Hamiltonian, Eq. (16) is not complete without the con-straint

YR

=

√2₃R8

=

1

−

1₃

χ

†

χ

,

(17)

that arises from the terms linear in

8 in Eqs. (5) and (12).

Thustheheavybaryonshaverighthypercharge2

_/

₃_._Since_the_zero

strangenesscomponentsofanySU

(

3

)

representationhasequal hy-perchargeand right hypercharge, the SU

(

3

)

coordinates must be

quantized asdiquarks forheavy baryons [15].The most relevant diquarkrepresentationsaretheantisymmetricanti-tripletandthe symmetricsextet.

When symmetry breaking is included, elements of higher di-mensional representations withthesame ﬂavorand R1,2,3

quan-tumnumbersareadmixed.Weﬁrstdeterminethequantum num-ber r in the intrinsicspin

3_i₌₁R2

i

=

r

(

r

+

1

)

: In addition to its

dimensionality, an SU

(

3

)

representation is characterized by two sets of quantum numbers

(

I

,

I3

,

Y

)

for the ﬂavor and

(

r

,

r3

,

YR

)

for the Ra degrees of freedom, respectively. The ﬂavor

genera-tors are La

=

8b=1DabRb with L1,2,3

=

I1,2,3 and Y

=

√2₃L8

be-ingtheobservables.Low-dimensionalrepresentations(suchasthe anti-tripletandthesextet)arenon-degenerateandtheirelements with Y

=

YR have

|

I

|

= |

R

|

. Thus r equals the isospin (I) of

thezerostrangenesselementwithinan SU

(

3

)

representation:the anti-triplet has r

=

0 and the sextet has r

=

1. Symmetric and antisymmetric SU

(

3

)

representations do not mix under symme-try breaking. Hence r

=

0 and r

=

1 for a heavy baryon whose diquarkcomponentbuildsupfromtheanti-tripletandsextet, re-spectively. The admixture of higher dimensional representations hasbeenestimatedinaperturbationexpansionforhyperons

[32]

andheavybaryons

[15]

.Itcanalsobedoneexactlywithinthe so-calledYabu–Andoapproach

[5]

.ThestartingpointisanEulerangle representationofthecollectivecoordinates A inwhichthe conju-gate momenta Ra are differential operators. Then the eigenvalue

equation

₈

a=1 R2_a

+ (

x

γ

β

2

)

[1

−

D88

(

A

)

]

(

A

)

=

(

A

)

(18)

is cast into a set of coupled ordinary second order differential equations. The single variable is the strangeness changing angle in A. The particularsetting ofthe differentialequations depends onthe consideredﬂavor quantumnumbers.Forordinary baryons (YR

=

1) this treatment is reviewed in Ref. [10] andthe results

for diquark wave-functions that enter the heavy baryon wave-functions(YR

=

2

/

3)arereportedinRef.[33].Havingobtainedthe SU

(

3

)

-ﬂavoreigenvalue

fromthedifferentialequationswe sim-plifytheSU

(

3

)

partandwrite

H

(

A

,

Ra

,

χ

)

−→

H

(

χ

)

=

Ecl

+

1

α

2

−

1

β

2

r

(

r

+

1

)

2

+

2

β

2

−

3 8

β

2

1

−

1 3

χ

†

_χ

2

+ |

ω

|

χ

†

_χ

₊

_H hf

.

(19)

The dependence of the eigenvalues

on x varies with spin and isospin.Hencethereisimplicithyperﬁnesplitting,however,italso appearsexplicitlyaswediscussnext.

4.2. Hyperﬁnesplitting

The eigenstates ofthe Hamiltonian, Eq.(16) are combinations inwhicheach termisaproduct oftwofactors,one isafunction of A and the other of

χ

. The combinations are such that eigen-statesofflavorandtotalspinaregenerated.Theflavorinformation is completely containedin A because flavor transformations cor-respond to multiplying A by unitary matrices from the left. To construct total spin eigenstates we consider the effect of spatial rotations.Thesolitonisthehedgehogconfigurationandspatial ro-tationsare equivalenttomultiplying A byunitary SU

(

2

)

matrices fromthe right.For theheavy mesonbound state this multiplica-tion mustbecompensatedby an additionalﬂavor transformation ofthespinor

χ

.Thusthetotalspinis

(5)

Table 1

Modelresultsforthemassdifferencesofthecharmandbottombaryons:N=M−MN,c=M−Mc andb=M−Mb withtheM’scomputedfromEq.(22)in

comparisonwithavailableexperimentaldata.ThespinandisospinofaconsideredbaryonareI and j.TheSU(3)quantumnumberr isdeﬁnedinthetext.Alldataare in MeV.Seetextforexplanationofquestionmarkonc.

(I,j,r) x=25 x=30 Expt.[34]

Pos. par. Neg. par. Pos. par. Neg. par. Pos. par. Neg. par.

N c N c N c N c N c N c (0,1_/₂_,₀₎ _c ₁₂₃₀ ₀ ₁₄₇₉ ₂₄₉ ₁₂₃₃ ₀ ₁₄₈₂ ₂₄₉ ₁₃₄₇ ₀ ₁₆₅₃ ₃₀₆ (1,1_/₂_,₁₎ _c ₁₄₂₃ ₁₉₃ ₁₆₆₄ ₄₃₄ ₁₄₂₅ ₁₉₂ ₁₆₆₆ ₄₃₃ ₁₅₁₅ ₁₆₈ _– _– (1_/₂_,1_/₂_,₀₎ _c ₁₄₄₆ ₂₁₆ ₁₆₉₅ ₄₆₅ ₁₄₈₆ ₂₅₃ ₁₇₃₅ ₅₀₂ ₁₅₂₉ ₁₈₆ ₁₈₅₁ ₅₀₄ (0,1_/₂_,₁₎ _c ₁₆₉₃ ₄₆₃ ₁₉₃₄ ₇₀₄ ₁₇₅₆ ₅₂₃ ₁₉₉₇ ₇₆₄ ₁₇₅₆ ₄₀₉ _– _– (1_/2,1_/2,1) c 1557 328 1798 569 1588 355 1829 596 1637 290 – – (1,3_/₂_,₁₎ _c ₁₄₆₄ ₂₃₄ ₁₇₁₇ ₄₈₇ ₁₄₆₆ ₂₃₃ ₁₇₁₉ ₄₈₆ ₁₅₇₉ ₂₃₂ _– _– (1_/₂_,3_/₂_,₁₎ _c ₁₅₉₈ ₃₆₉ ₁₈₅₁ ₆₂₂ ₁₆₂₉ ₃₉₆ ₁₈₈₂ ₆₄₉ ₁₇₀₆ ₃₅₉ ₁₈₇₆ _529(?) (0,3_/₂_,₁₎ _c ₁₇₃₄ ₅₀₄ ₁₉₈₇ ₇₅₇ ₁₇₉₇ ₅₆₄ ₂₀₅₀ ₈₁₇ ₁₈₃₁ ₄₈₄ _– _– N b N b N b N b N b N b (0,1_/2,0) b 4391 0 4560 168 4394 0 4563 168 4681 0 4973 292 (1,1_/₂_,₁₎ _b ₄₆₀₁ ₂₁₀ ₄₇₇₁ ₃₈₀ ₄₆₀₃ ₂₀₉ ₄₇₇₃ ₃₇₉ ₄₈₇₂ ₁₉₁ _– _– (1_/₂_,1_/₂_,₀₎ _b ₄₆₀₈ ₂₁₆ ₄₇₇₆ ₃₈₅ ₄₆₄₇ ₂₅₃ ₄₈₁₆ ₄₂₁ ₄₈₅₅ ₁₇₄ _– _– (0,1_/₂_,₁₎ _b ₄₈₇₁ ₄₈₀ ₅₀₄₁ ₆₅₀ ₄₉₃₅ ₅₄₀ ₅₁₀₅ ₇₁₀ ₅₁₁₀ ₄₂₉ _– _– (1_/₂_,1_/₂_,₁₎ _b ₄₇₃₆ ₃₄₅ ₄₉₀₆ ₅₁₄ ₄₇₆₆ ₃₇₂ ₄₉₃₆ ₅₄₂ _– _– _– _– (1,3_/₂_,₁₎ _b ₄₆₁₇ ₂₂₆ ₄₇₈₅ ₃₉₃ ₄₆₁₉ ₂₂₅ ₄₇₈₇ ₃₉₂ ₄₉₈₃ ₂₁₂ _– _– (1_/₂_,3_/₂_,₁₎ _b ₄₇₅₁ ₃₆₀ ₄₉₁₉ ₅₂₈ ₄₇₈₂ ₃₈₇ ₄₉₅₀ ₅₅₅ ₅₀₀₆ ₃₂₅ _– _– (0,3_/₂_,₁₎ _b ₄₈₈₇ ₄₉₆ ₅₀₅₅ ₆₆₄ ₄₉₅₀ ₅₅₆ ₅₁₁₈ ₇₂₄ _– _– _– _– J

= −

R

−

χ

†

τ

2

χ

.

(20)

Calling j thespin oftheconsideredbaryon thisimplies R

· τ

=

j

(

j

+

1

)

−

r

(

r

+

1

)

−

3₄

∼

j

(

j

+

1

)

−

r

(

r

+

1

)

,wheretheexpectation valuereferstotheheavymesonboundstate.Intheapproximation wehaveagainomittedtermsthat formallyare quarticin

χ

.This scalarproductappearsintheLegendretransformationwithrespect to

,

∂

L

∂

· −

1 2

α

2

₋

_ρχ

†

_·

τ

2

χ

=

1 2

α

2R 2

₊

ρ

α

2R

· χ

†

τ

2

χ

.

(21)

Collectingpieceswegetthemassformula

M

=

1

α

2

−

1

β

2

r

(

r

+

1

)

2

+

2

β

2

−

3 8

β

2

1

−

N 3

2

+ |

ω

|

N

+

ρ

2

α

2[ j

(

j

+

1

)

−

r

(

r

+

1

)

] N

,

(22)

where N

=

0

,

1 counts the numberofheavy valencequarks con-tainedintheconsideredbaryon.Ithasbeenincludedinthe hyper-ﬁnesplittingtermsinceordinarybaryonshaver

=

j.Wehave col-lectedtheleadingcontributions tothebaryonenergyinthelarge numberofcolors(NC)expansion.However,a contribution

O(

N0C

)

is missing, the vacuum polarization energy Evac. It is the

quan-tumcorrectiontotheclassicalenergyEcl thatcannotberigorously

computedbecause thetheory is not renormalizable. Estimatesin theSkyrmemodelsuggestthat Evac considerablyreducesEcl[35].

Wecircumventthislimitationbyonlyconsideringmassdifferences forwhich Ecl and Evac cancelandconsequently omittheseterms

fromEq.(22).

Thisquantization schemepredicts two heavy

baryons with spin j

=

1

_/

₂_:_one_has_r

=

_{0 and} _the_other _r

=

_1._In_an_SU

₍

₃

₎

sym-metric world the former would be an anti-triplet state and the latterasextetstate.Thereisnomixingbetweenthesebaryons be-cause

H

,

R2

₌

_0._For _j

₌

3

_/

₂ _only_one_heavy

_baryon _emerges

in this scheme since then r

=

1 is required. For the

hyperon thereis alsoonly asingle optionwith j

=

1

_/

₂ _that _is_build _from

theoctetstate.Thiscountingsuggeststorelater tothe intermedi-atespin Jm deﬁnedinRef.[11].

5. Numericalresults

Asmentionedabove,weconsidermassdifferences,becausethe model predictions forthe absolute massesare subject to uncon-trollablequantumcontributions.

Weﬁndtheenergyeigenvalues

inEq.(18)forallbaryonsand then compute their energies accordingto Eq.(22).We adoptthe SU

(

3

)

parametersfromRef.[26]:

α

2

₌

₅

_.

₁₄₄

_/

_GeV,

_β

2

₌

₄

_.

₃₀₂

_/

_GeV

and6

γ

=

47MeV. Forthe heavy sector, the same solitonmodel was used in Ref. [14] to compute the bound state energies

ω

andhyperﬁneparameters

ρ

forboththe P - andS-wavechannels. From the modelcalculation described in Section 3 the following boundstateparameterswereobtained

ω

P

=

1326 MeV

,

ρ

P

=

0

.

140

,

ω

S

=

1572 MeV

,

ρ

S

=

0

.

181 (23)

and

ω

P

=

4494 MeV

,

ρ

P

=

0

.

053

,

ω

S

=

4663 MeV

,

ρ

S

=

0

.

046 (24)

in the charm and bottom sector, respectively (Ref. [14] lists the binding energies

ω

P,S

−

MD and

ω

P,S

−

MB.). Then we are left

with a single parameter, the effective symmetry breaking x de-ﬁned in Eq. (13), that is not fully determined. We present our results forthecharm andbottombaryon spectra inTable 1, that alsocontainsthedataforexperimentallyobservedcandidates

[34]

. WenotethatmostofthequantumnumberslistedinRef. [34]are adapted fromthequark modelandstressthat r isnot aphysical observable. Hence assigning the experimental results for

type baryons to a particular r value is a prediction. Ref. [34] further-morelists

c

(

2625

)

and

b

(

5920

)

withspin j

=

3

/

2 thatare not

containedinourapproach:Werequire

|

j

−

r

|

=

1

_/

₂_but_these

_’s

haveneitherstrangenessnorisospinsotheymusthaver

=

0 and j

=

1

_/

₂_._We _complete _the_picture _by _including_the _{corresponding}

resultsforthelow-lyingnon-heavybaryonsin

Table 2

.

Whencomparingourmodelresultstodatain

Table 1

and

Fig. 1

weseethatthemassdifferenceswithinagivenheavyquarksector

(6)

Fig. 1. (Coloronline.)Modelresultsandexperimentaldataforthemassdifferencesofpositiveparityheavybaryonsandthenucleon.Leftpanel:charmbaryons,rightpanel: bottombaryons.Theshadedareasarethemodelresultsfor x∈ [25,30]anddataareindicatedbylinesandthenumber(inMeV)iswrittenexplicitly.Asforordinary hyperons,theasterisksdenotetotalspinj=3

2.Notethedifferentscalesandoff-sets.Noexperimentaldatumfor∗bisavailable.

Table 2

Massdifferencesfornon-heavybaryonswithrespecttothenucleoninMeV.

∗ ∗ − 

x=25 134 218 320 324 438 551 661 101 x=30 162 253 404 323 461 601 740 151

expt. 177 254 379 293 446 591 733 125

isoverestimated. Forexample Mc

−

Mc

=

463MeV for x

=

25, whiletheempiricalvalueis409 MeV.Furtherincreaseofx wors-ensthepicture.Ontheotherhand,asizablevalue(x

∼

30)forthe symmetrybreakingisrequiredforagoodagreementfornon-heavy baryons.Simultaneouslythe splittingbetweendifferentsectorsis predictedonthelowside.The

c and

b areabout100 MeVand

300 MeV too low, respectively. This is inherited fromthe heavy ﬂavorcalculationwhichoverestimatesthebindingenergiesinthe sensethat it istooclose tothe estimatefromexact heavy ﬂavor symmetry.Thiscanalsobeseenfromtheparitysplittingwhichis underestimatedbyabout50 MeV(it vanishesintheheavy limit). Togetherwiththe effectofSU

(

3

)

symmetry breakingthe overes-timated binding combines to acceptable agreement for the mass differencesbetween the double strange baryons

c and

b and

thenucleon,atleastforx

=

30.Ithasbeenargued

[14]

that kine-matical correctionsdue to the solitonnot being inﬁnitely heavy changethepredictionsfor

ω

P,S appropriately.Andindeed,

replac-ing the heavy meson massesby the reducedmass builtin con-junctionwiththeclassicalsolitonenergyincreases

ω

P byroughly

100 MeVand

ω

S byalmost200 MeV.

For j

=

1

_/

₂ _and _positive _parity _there _is _an _interesting _effect

inthe

–

system. The observedmass difference decreasesand even changes sign when the heaviest ﬂavor turns from strange viacharmtobottom: M

−

M

=

125

,

14

,

−

17MeV.Partiallythe modelcalculation reproduces thiseffect. Forexample,for x

=

25 themass differences101, 23and6 MeV are predicted.Since the hyperﬁnesplittingonly hasa moderateeffect,themodelexhibits a similar scenario for the negative parity channel. Unfortunately, therearenodatatocomparewith.

Finally we discussourresults forthemassesof thosestrange heavybaryonsthathavepreviouslynotbeenconsideredinaheavy mesonsolitonmodelwithrealistic heavy mesonmasses:the

’s and

’s. Forthe positive parity heavy strange baryons we again observethatthemasssplittingswithinaheavymultipletare over-estimated.Amoderatereductionofthesymmetrybreakingratiox wouldbesuﬃcient tomatchtheexperimental data.Forthe

neg-ativeparity

c with j

=

1

/

2 thetoolarge binding ofthe S-wave

reverses thispicture.Thisisnotthecaseforitsspin 3

_/

₂

counter-part.Interestinglyenough,Ref. [34] assigns thequantum number ofthis resonanceby assuming itto join an SU

(

4

)

multiplet with thenegativeparity

c

(

j

=

3

/

2

)

.Wehavearguedabovethatthis

c

isnotcontainedinourapproachbutshouldbe associatedwitha D-waveheavymeson.Thus,asindicatedin

Table 1

,itis question-abletoidentify

(

I

,

j

,

r

,

p

)

= (

1

_/

₂

_,

3

_/

₂

_,

₁

_,

−)

_with

_c

₍

₂₈₁₅

₎

_._Rather

itisa predictionforanevenheavierresonancelikethe observed

c

(

2930

)

or

c

(

2980

)

whose quantum numbersstill need to be

determined

[34]

. 6. Conclusion

We have presented a model calculation for the baryon spec-trumthatcompriseslightandheavyflavors.Inparticularwehave focused on the role of light flavor symmetry breaking which is manifested by the strange quark being neither light nor heavy. Whenquantizingtheflavordegreesoffreedom,thecorresponding deviationsfromtheup-down sectorarehandled(numerically) ex-actly.Intheheavyflavorsectorthemodelisinspiredbytheheavy flavorsymmetry,withsubleadingeffectsarisingfromfinitemasses included.Theapproachalsoincludesthehyperfinesplittingforthe heavybaryons;amoderateeffectthatvanishesintheheavylimit. Themodelcalculationisall-embracing asitcontainsspin 1

_/

₂_and 3

_/

₂ _baryons_starting_from_the_lightest_baryon _(nucleon),_including

hyperonsandextendingtoheavybaryonsofeitherparitythathave the valencequark contentstrange–strange–bottom. Thespectrum iscomputedfroma singlemassformulawhereessentiallyall pa-rametersaredeterminedusingdatafromthebaryonnumberzero sector.Wehavealsocalculatedmassesforheavybaryonsthatare yet tobe observed.Thoughwe can only provide an estimate for theirmasses,weﬁndarealisticindication fortheirpositions rela-tivetoobservedbaryons.

Theoverallagreementwithdataisasexpectedforchiralsoliton modelestimates.As knownfromearlierstudies,themass predic-tionsfortheheavybaryonsareonthelowsidewhencomparedto thenucleon.Withinaheavybaryonmultipletthecomputedmass differencesarelargerthan theexperimentaldata.Thisappears to becausedby toostronga remnantoftheheavyspin–flavor sym-metryintheapproach.Anunderstandingthatgoesbeyond adopt-ing reducedmassesin theboundstate approachis required. Fur-thermorethefine-tuningofthesymmetrybreakingratiox aswell asothermodelparametersthatinfluencethesolitonproperties

(7)

ap-pearsasan obviousendeavor.Actually,acompleteanalysiswithin avectormesonsolitonmodel(butalsoachiralquarkmodelasthe Nambu–Jona-Lasinio model[36]) showsthat additionalsymmetry breakingoperators such as

3_i₌₁D8iRi [26] arise inthe

Hamilto-nian,Eq.(16).Theireffectsontheheavybaryonspectrumwillbe reportedinaforthcomingpaper.

Acknowledgement

Thisworksupportedinpart bytheNationalResearch Founda-tionNRF,grant 77454.

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Heavy baryons with strangeness in a soliton model

Physics

Letters

B

Heavy

baryons

with

strangeness

in

a

soliton

model

J.P. Blanckenberg,

H. Weigel

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

[1]

[3]

[6]

[8]

[11]

[7]

[20]

ρ

ω

[4,9]

[10]

=

φ

(

)λ

/

φ

(

)

=

=

,

,

=

=

,

. . . ,

=

λ

(

)

(

)

=



τ

· ˆ

(

)



,

ω

(

)

=

ω

(