Baryons with functional methods

1.3 MB·

by  Christian S. Fischer

Baryons with functional methods
ChristianS.Fischer1,2,(cid:63)
1InstitutfürTheoretischePhysik,Justus-Liebig–UniversitätGiessen,35392Giessen,Germany
2HICforFAIRGiessen,35392Giessen,Germany
Abstract. We summarise recent results on the spectrum of ground-state and excited baryons and their form
7
factors in the framework of functional methods. As an improvement upon similar approaches we explicitly
1
takeintoaccounttheunderlyingmomentum-dependentdynamicsofthequark-gluoninteractionthatleadsto
0
dynamicalchiralsymmetrybreaking. Forlightoctetanddecupletbaryonswefindaspectruminverygood
2
agreement with experiment, including the level ordering between the positive- and negative-parity nucleon

n states. Comparingthethree-bodyframeworkwiththequark-diquarkapproximation,wedonotfindsignificant
a differences in the spectrum for those states that have been calculated in both frameworks. This situation is
J
differentintheelectromagneticformfactorofthe∆,whichmayservetodistinguishbothpicturesbycomparison
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withexperimentandlatticeQCD.

]
h
p
1 Introduction sharewithfairlysmallerrorbarsandverygoodagreement
-
p between different lattice groups and approaches for most
e In these proceedings we summarise our progress on the ground-state masses. Differences, however, occur for the
h
properties of baryons as obtained from the functional ap- excitedstates,whereanumberoftechnicalsubtletiesstill
[
proach (Dyson-Schwinger and Bethe-Salpeter equations) obscurethepicture.
1 toQCD.Alltopicsdiscussedinthiscontributionarecov-
v
ered in much more detail in a very recent review on
4
baryonsascovariantthree-quarkboundstates[1].
0 Inthepastyearsweexploredanothernon-perturbative
0 UnderstandingthebaryonexcitationspectrumofQCD approach to QCD, namely the functional methods of
1 andtheinternalstructureofbaryonsusingelectromagnetic
Dyson-Schwinger and Bethe-Salpeter equations (DSEs
0 andotherprobesisoneofthekeyelementsinunravelling and BSEs). They can be used on different levels of so-
.
1 the structure of the strong interaction. In the past years, phistication,rangingfromNJL-liketruncationswithcon-
0 significant experimental progress has been made by the
tact interactions, quark-diquark models using ansätze for
7 analysisofdatafromphoto-andelectroproductionexper-
propagators and wave functions as input, rainbow-ladder
1
: iments at JLAB, ELSA and MAMI [2–4]. As a result, a truncations using effective quark-gluon interactions, to
v numberofnewradialandorbitalexcitationsoftheground-
beyond-rainbow-ladder approaches explicitly solving a
Xi statebaryonshavebeenaddedtothePDG[5]. rangeofunderlyingDSEsfortheQCDGreen’sfunctions;
On the theoretical side, a number of non-perturbative
r see [1] for an overview. In the work summarised here
a methodscontributeanumberofinterestingaspectstoour
wehaveusedtherainbow-laddertruncationasaminimal
understandingofbaryons,whicharenotalwaysinagree- baselinebutalsoexploredeffectsbeyondrainbow-ladder.
ment with each other. In general, the quark model still
In addition, we do not rely on the widely used quark-
serves as a possible standard by which one may define
diquarkapproximationbutprovidedfirstsolutionsforthe
and distinguish the ‘expected’ from the ‘unexpected’. Its
masses of ground and selected excited states as well as
discussion along the years has created a list of standard
baryon form factors from the three-body Faddeev equa-
problems in baryon physics [6, 7]. Amongst these are
tion.Wearethereforeinapositiontobeabletosystemati-
theproblemofmissingresonancesand,related,theques-
callycomparebothapproachesusingthesameunderlying
tionwhetherbaryonsshouldbetterbedescribedasquark-
assumptionsforthequark-gluoninteraction.
diquark rather than three-quark states, thus reducing the
number of possible internal excitations. On the other
hand, dynamical coupled-channel calculations have sug-
Our results for the light baryon spectrum will be dis-
gested that parts of the spectrum may even be generated
cussed in Sec. 3. Corresponding results for form factors
purely dynamically, i.e. without a ‘bare’ three-quark or
will be the topic of Sec. 4, with a focus on the ∆ elec-
quark-diquarkseed[8,9].LatticeQCDcontributesitsown
tromagnetic form factors. First, however, we outline the
(cid:63)e-mail:christian.fi[email protected] frameworkinthenextsection.
= + + +
-1 -1
= + = + +
Figure1.Three-quarkFaddeevequation.
= + = + ++ + + + +
(cid:1836)(cid:2078) (cid:1915) (cid:945) (cid:1846)(cid:1965) (cid:1913) (cid:1846)(cid:1965) (cid:1914)
(cid:1914) (cid:1985)(cid:1913) (cid:1915) (cid:1985)(cid:1914) (cid:1913) (cid:1985)(cid:1915) (cid:2004) (cid:2004)
-1 -1
(cid:1913) (cid:1914) = (cid:1915) + = + +
-1 -1 = –1= + + + + +
= + = + +
= = + + + + +
Figure2. SimplificationoftheFaddeevequationinFig.1(topleft)tothequark-diquarkBethe-Salpeterequation(topright). The
bottom panel shows the ingredients that enter in the equation and are calculated beforehand: the quark propagator, diquark Bethe-
Salpeteramplitudesanddiquarkpropagators.
2 The functional approach quarks in addition [11]. Thus in the diquark calculation
ofthespectrumdiscussedbelowallfourtypesofdiquarks
In functional frameworks the masses and wave functions
havebeentakenintoaccount.
of baryons are extracted from their gauge-invariant poles
Asystematiccomparisonbetweenthethree-bodyand
in the (gauge-dependent) quark six-point Green function.
diquark-quark approach is only possible once the same
This procedure is related to the corresponding one in lat-
underlying quark-gluon interaction is chosen [12]. This
tice QCD, see Ref. [1] for more detailed explanations.
distinguishes a quark-diquark approach based on the dy-
An exact equation that can be derived in this approach is
namics of the underlying QCD from quark-diquark mod-
the covariant three-body Faddeev equation in Fig. 1. Its
els [13–15]. In our case we choose the effective interac-
ingredients are the fully dressed quark propagator (solid
tion introduced by Maris and Tandy in Ref. [16]; details
line with open circle) as well as the quark two-body
are discussed e.g. in [12, 17]. In this rainbow-ladder
and irreducible three-body interactions. The latter ones
framework the two-body interaction kernel between two
have been neglected so far. The equation determines the
quarks is approximated by the exchange of an effective
baryon’sFaddeevamplitude(theshadedhalf-sphere)and
gluon,whichcontainsthegluonpropagatorsupplemented
thebaryonbound-statemass.
bythoseeffectsofthequark-gluonvertexthatcanbecap-
A considerable simplification is obtained once the
tured by a function depending on the gluon momentum
quark-diquark approximation is chosen. To this end, one
only.
expands the quark-quark scattering matrix in terms of
Intherainbow-ladderapproximationonearrivesatan
separable diquark contributions, resulting in the quark-
interaction entirely of vector-vector type. In the heavy
diquark Bethe-Salpeter equation in the top-right corner
quark sector it is known that such types of interactions
of Fig. 2. Its main ingredients are again the dressed
are not sufficient to account for the different effects due
quarkpropagator,thediquarkpropagator(doublelinewith
to spin-spin and spin-orbit dependent forces between the
open circle) as well as the diquark Bethe-Salpeter ampli-
quarks,andhencecannotdescribeallmesonchannelssuf-
tude. In turn, the latter needs to be calculated from the
ficientlywell. Inpractice,itturnsoutthatthemesonspec-
diquark Bethe-Salpeter equation (bottom center diagram
truminthepseudoscalar,vectorandsometensorchannels
of Fig. 2). The dressed quark propagator is determined
arewellrepresentedinrainbow-ladder,whileotherssuffer
from its Dyson-Schwinger equation (bottom left diagram
fromtoomuchbinding[18,19]. In[12,20]asimplepro-
ofFig.2).
cedurehasbeendevisedtoremedythisproblemwithinthe
Animportantquestioninconnectionwiththediquark
quark-diquark approach. Scalar and axialvector diquarks
approximationishowmanydiquarkstoconsider. Itturns
arewellrepresentedinrainbow-ladder,whereasthepseu-
outthatthediquarkswithsmallestmassarethescalarand
doscalar and vector ones are too light. Reducing the in-
axialvector ones [10]. While both of these contribute to
teraction strength in these channels by a common factor
thenucleon,the∆baryoncanonlybemadeofanaxialvec-
gaugedbytheρ−a splittingremediesthisproblem.
tordiquarkduetothesymmetrypropertiesofthediquark 1
wavefunctions. Thus,scalarandaxialvectordiquarksto-
gether constitute a minimal set of diquarks necessary to 3 Light baryon spectrum
consider. However, as it turns out, the parity partners of
thenucleonandthethe∆cannotbedescribedadequately In Fig. 3 we show the resulting spectra from the three-
withoutincludingtheheavierpseudoscalarandvectordi- body calculation (left diagram, open boxes), the quark-
2.0 2.0
𝑀 [𝐺𝑒𝑉] 𝑀 [𝐺𝑒𝑉]
𝑁 𝛥 𝑁 𝛥
1.8 1.8
1.6 1.6
1.4 1.4
1.2 1.2
qqq
q-dq q-dq
PDG ** PDG **
1.0 PDG *** 1.0 PDG ***
PDG **** PDG ****
0.8 0.8
1+ 1− 3+ 3− 3+ 3− 1+ 1− 1+ 1− 3+ 3− 3+ 3− 1+ 1−
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
Figure3. Left: Nucleonand∆baryonspectrumfor JP = 1/2± and3/2± statesdeterminedwithinrainbow-ladder. Thethree-body
results(openboxes)arecomparedtothequark-diquarkspectrumwithfulldiquarkcontent(filledboxes),togetherwiththePDGvalues
andtheirexperimentaluncertainties[5].Thewidthsrepresentanestimateofthesystematicerrorofourresultsbasedontheextrapolated
eigenvaluecurvesoftheBSE,see[12]fordetails.Right:Nucleonand∆spectrumwithreducedstrengthinthepseudoscalarandvector
diquarkchannels;seetextforadetaileddiscussion.
diquark approximation without correction (left diagram, oneparameterthatisadjustedbytheρ−a splitting;every-
1
filled boxes) and the quark-diquark approximation with thingelseisunchanged. Theinfluenceofthisadjustment
improved pseudoscalar and vector diquarks (right dia- dependsonthequantumnumbersofthebaryonduetodif-
gram,filledboxes)[12]andcomparethemwiththetwo-, ferentweightsofthecorrespondingdiquarkcontributions
three-andfour-starstatesgivenbythePDG[5]. ascomparedtothescalarandaxialvectorone. Itisthere-
Letusfirstdiscussthenucleonchannel. Intheleftdi- fore highly non-trivial that the states arrange themselves
agramweobservegoodagreementofthethree-bodywith in the one-to-one correspondence with the experimental
the quark-diquark approach for the ground-state nucleon ones as observed in the right diagram of Fig. 3. Without
as well as the first radially excited state. The mass of furthercorrections,thelevelorderingbetweenthepositive
the latter agrees well with the Breit-Wigner mass of the andnegativenucleonparitysectorcomesoutcorrectlyon
Roper shown as a red shaded box1. The next two ex- aquantitativelevel.Notonlythefirstradialexcitationsbut
citedstatesare(numerically)onlyaccessibleinthequark- also the second and third ones are close to experimental
diquark framework and lie in the same ballpark as the states. Wethereforearriveataconsistentandquantitative
PDG’sN(1710)andN(1880). Inthenucleonchannelwith descriptionofthelightbaryonspectrumbelow2GeV.
negative parity, both approaches still agree very well but
thestatesaregenerallymuchtoolow. Clearly,thisreflects
Of course this does not mean that beyond rainbow-
the above discussed deficiency of the rainbow-ladder ap-
ladder effects, like pion-cloud and coupled-channel ef-
proximation.
A similar picture is observed in the various ∆ chan- fects,orfurthernon-Abeliancorrectionsonthegluonlevel
are absent [21]. Our results merely suggest that these ef-
nels. The quantitative agreement between the quark-
fectsareoppositeinsignandlargelycanceleachotherin
diquarkandthethree-bodyapproachissomewhatlesspro-
thenucleonand∆channels,whereasintheotherchannels
nounced than in the nucleon case but still satisfied on a
there is a net effect that can be absorbed by the extra pa-
semi-quantitative level. Taken at face value, this could
rameter for the pseudoscalar and vector diquarks that we
mean that interactions beyond the diquark approximation
could be more important for ∆ states than for baryons of introduced.Infact,inthemesonsectortherearestrongin-
dications that cancellations between non-Abelian correc-
nucleontype. Thishypothesisneedstobetestedinthefu-
ture. Again, the (s-wave) ∆-channel delivers results for tions [22] and pion cloud effects [23] occur. Both effects
havealreadybeenexploredforbaryonsonanexploratory
the ground and first radially excited state in agreement
level[24,25]andworkwillcontinueinthisdirection.
with experiment, while its parity partners and the other
twochannelsarepoorlyrepresented.
The picture changes when the deficiencies of the
Also very encouraging are the results for the strange
rainbow-ladder approximation are accounted for by ad-
baryon sector reported for the three-body framework
justingthebindinginthepseudoscalarandvectordiquark
in [29]. Here the masses of ground-state octet and de-
channels. Asexplainedabove,thisisdonebyintroducing
cuplet baryons have been reproduced on the few-percent
1ThemassevolutionoftheRoperwithvaryingpionmassisdiscussed level, whilethecalculationofexcitedstatesstillneedsto
in[12]andcomparedwithresultsfromlatticeQCD. beperformed.
1.2 4.0
1.0 DSE/BSE Lattice
3.0
0.8 qqq
0.6 (cid:1833)(cid:1935)(cid:1181)((cid:1843)(cid:1165)) 2.0 (cid:1833)(cid:1943)(cid:1182)((cid:1843)(cid:1165)) q-dq (cid:1865)(cid:2140) [6(cid:1839)91(cid:1857)(cid:1848)]
509
0.4 384
1.0 353
0.2
0.0 0.0
0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5
0.5 (cid:1843)(cid:1165)/ (cid:1839)(cid:2102)(cid:1165) 0.8 (cid:1843)(cid:1165)/ (cid:1839)(cid:2102)(cid:1165) 5
4
0.0
0.4
3
–0.5 2 (cid:1833)(cid:1943)(cid:1182)(0)
0.0
1
–1.0
–0.4 0
–1.5 (cid:1833)(cid:1943)(cid:1184)(0)
-1
–2.0 –0.8 -2 (cid:1833)(cid:1935)(cid:1183)(0)
0.0 0.5 (cid:1833)(cid:1935)(cid:1183)(1(cid:1843).0(cid:1165)) 1.5 0.0 0.5 (cid:1833)(cid:1943)(cid:1184)(1.(cid:1843)0 (cid:1165)) 1.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6
(cid:1843)(cid:1165)/(cid:1839)(cid:2102)(cid:1165) (cid:1843)(cid:1165)/(cid:1839)(cid:2102)(cid:1165) (cid:1865)(cid:2140)(cid:1165) [(cid:1833)(cid:1857)(cid:1848)(cid:1165)]
Figure4.Electricmonopole,magneticdipole,electricquadrupoleandmagneticoctupoleformfactorsforthe∆+baryon.Wecompare
latticedata[26]withtheDyson-Schwingerthree-quark[27]andquark-diquarkcalculations[28].ThebandsfortheDSEresultsreflect
thesystematicuncertainty.Theplotonthebottomrightshowsthecurrent-massevolutionofthestaticquantitiesfromthequark-diquark
calculation[28],wherestarsdenotetheexperimentalmagneticmomentsofthe∆+ andΩ− baryon. (Thevaluem2 ≈ 0.47wherethe
π
magneticmomentoftheall-strangeΩ−baryonisplottedcorrespondstoapionmassevaluatedwithconstituentstrangequarks.)
4 Selected form factors: electromagnetic dipole results agree with each other (apart from a small
∆ form factor regionatverylowQ2),theoctupolecomponenthasadif-
ferentsignforall Q2. Thus, highprecisionlatticeresults
Formfactorsinthefunctionalapproacharecalculatedby are clearly able to distinguish between them. Given the
coupling external currents to equations for Green func- presently available set of data for the electric quadrupole
tions. In the three-body approach, electromagnetic form component, there may be a slight tendency towards the
factors[30],axialformfactors[31]andhyperonformfac- three-body results. However, the accuracy is not good
tors [32] have been calculated and reviewed in [1]. Here enough to make a definite statement. Thus, it would be
wediscussaspectsoftheelectromagneticformfactorsfor very interesting to repeat the lattice calculation with to-
the ∆, which have been determined in the three-body ap- day’stechnicalandnumericalresources.
proach [27] as well as in the quark-diquark approxima- While experimental results for the ∆ form factors are
tion [28]. In Fig. 4 we display corresponding results for not available, static quantities like its magnetic moment
theelectricmonopoleandquadrupoleformfactoraswell canbeextracted. Theseareshowninthebottomrightplot
asthemagneticdipoleandoctupoleformfactor. Wecom- of Fig. 4. Compared are the bands for the quark-diquark
parethemwithresultsfromlatticeQCD[26]usingdiffer- results with the experimental data for the ∆+ and the Ω−
entvaluesfortheup/downquarkmassescorrespondingto baryon. The agreement with the central value of the ex-
differentpionmasses. perimentalerrorbarisverygood. Notethatthemagnetic
It is interesting to compare the three-body with the momentsarereal,sinceintherainbow-ladderapproxima-
quark-diquarkresults.Fortheelectricmonopoleformfac- tion used in the present approach decay channels are not
tor both approaches agree very well. The limit of van- included, see the review Ref. [1] for a more detailed dis-
ishing photon momentum, Q2 → 0, is constrained by cussionofthisaspect. Asdiscussedbefore, themagnetic
chargeconservation,whichisguaranteedautomaticallyin momentofthe∆+extractedfromthemagneticdipoleform
our framework due to gauge invariance. For all Q2 val- factor shows a small discrepancy between the three-body
ues the agreement with lattice results is furthermore on andthequark-diquarkresults. Thusalsomorepreciseex-
a quantitative level. The electric quadrupole form factor, perimental results have the potential to discriminate be-
however, is more sensitive to the internal structure of the tweenthetwo.
∆ baryon, as can be seen in the bottom left diagram of Ingeneral, weconfirmthe∆baryon’sdeformationas
Fig.4.Heretheresultsfromthethree-bodyandthequark- encoded in its electric quadrupole and magnetic octupole
diquarkapproachdisplayadifferentmomentumbehaviour form factors. Non-relativistically, a negative sign for the
andevenadifferentsignatlargemomenta. Thesamesig- electric quadrupole moment G (0) indicates an oblate
E2
nature is found for the magnetic form factors. While the chargedistributionforthe∆intheBreitframe[26]. From
the measurement of the γN → ∆ transition one can infer References
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E2 C
comparablevaluesarepredictedbyarangeofconstituent-
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Acknowledgement
D93,034026(2016),1512.00455
Theworkreportedherewouldnothavebeenpossiblewithoutthe
[22] C.S. Fischer, R. Williams, Phys. Rev. Lett. 103,
support from the Deutsche Forschungsgemeinschaft within the
SFB/TR16.IthankthemembersofSFB/TR16formanyfruitful 122001(2009),0905.2291
[23] C.S. Fischer, R. Williams, Phys. Rev. D78, 074006
discussionsandthespokespersonUlrikeThomafortheexcellent
(2008),0808.3372
leadership. IamfurthermoreverygratefultoGernotEichmann,
HeliosSanchis-AlepuzandRichardWilliamsforthesuccessful [24] H. Sanchis-Alepuz, C.S. Fischer, S. Kubrak, Phys.
andpleasantcollaborationonthetopicssummarizedinthispro- Lett.B733,151(2014),1401.3183
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addition I would like to thank Reinhard Alkofer (together with 592(2015),1504.07776
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