Table Of ContentDynamical Structure of Baryons
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8 A. Aleksejevs∗
GrenfellCampusofMemorialUniversity
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h E-mail: [email protected]
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- S. Barkanova
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AcadiaUniversity
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h E-mail: [email protected]
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Comptonscatteringoffersauniqueopportunitytostudythedynamicalstructureofhadronsover
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3 a wide kinematicrange,with polarizabilitiescharacterizingthehadron’sactiveinternaldegrees
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offreedom. Wepresentcalculationsanddetailedanalysisofelectricandmagneticandthespin-
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1 dependentdynamicalpolarizabilitiesforthelowestinmassSU(3)octetofbaryons.Theseexten-
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1 sive calculations are made possible by the recent implementation of semi-automatized calcula-
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tionsinchiralperturbationtheorywhichallowsevaluatingpolarizabilitiesfromComptonscatter-
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1 inguptonext-to-the-leadingorder. Thedependenciesfortherangeofphotonenergiescovering
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v themajorityofthemesonphotoproductionchannelsareanalyzed.
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36thInternationalConferenceonHighEnergyPhysics,
July4-11,2012
Melbourne,Australia
∗Speaker.
©Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
DynamicalStructureofBaryons A.Aleksejevs
1. Introduction
One of the major goals of the low-energy QCDis the investigation of the baryon response to
theexternalelectromagneticfieldviamultipoleexcitationmechanism. Structureparameterswhich
describe that response are electric, magnetic and spin-dependent polarizabilities. In other words,
the polarizabilities are related to the deformability and stiffness of the baryon. Precise determi-
nation ofnucleon polarizabilities still requires substantial effort from both theory and experiment.
Forhyperons,thepolarizabilities areyettobemeasured. Inthiswork,westudythepolarizabilities
ofbaryons using theComptonscattering, whichisastraightforward process fromboththeoretical
and experimental points ofview. Ingeneral, thepolarizabilities inaverylowenergy region ofthe
Compton scattering are treated as static (with very little or no dependence on the photon energy),
butitcanbeassumed thatathigherenergies, andespecially nearmesonproduction threshold, this
static behavior will break and the polarizabilities will become dynamic. The main goal of this
work isastudy of dependence ofthe electric, magnetic and spin-dependent polarizabilities onthe
photon energy. Weusetherelativistic chiral perturbation theory (ChPTh)whileapplying themul-
tipole expansion approach for the Compton structure functions. The various versions of ChPTh
predict a rather broad spectrum of values for polarizabilities, but so far, it is only theory available
intheregimeofthenon-perturbative QCDandhave beenemployed hereusing ourcomputational
hadronic model(CHM[5]). CHMgivesusapossibility toavoid thelow-energy approximation in
theComptonstructure functions andretain allthepossible degrees offreedom arising fromSU(3)
chiralLagrangian. Weprovideashortdescriptionoftheformalismusedinthisworkinthesection
“Formalism”. Analysis of thedynamical behavior of polarizabilities along withtheir static values
ispresented inthe“Results”.
2. Formalism
In the presence of external electromagnetic field, induced electric and magnetic dipole mo-
ments of the baryon generate effective Hamiltonian H =−14pa E2−14pb H2. Here, propor-
eff 2 2
tionalityconstantsa andb arecalledelectricandmagneticpolarizabilities, respectively. Although
the polarizability values are quite small (10−4(fm3)), they were successfully measured by sev-
eral experimental groups using the Compton scattering and employing the dispersion sum rules
analysis to extract the polarizabilities from the cross section data. The current PDG [1] averaged
experimental valuesfortheelectricandmagneticpolarizabilities forprotonandneutronare:
a =(12.0±0.6)10−4(fm3); b =(1.9±0.5)10−4(fm3);
p p
a =(11.6±1.5)10−4(fm3); b =(3.7±2.0)10−4(fm3).
n n
For proton and neutron a and b values are approximately the same, and the positive value of the
magneticpolarizability pointstotheparamagnetic natureofthenucleon.
Ifthebaryonisplacedinthetime-varyingelectromagneticfield,anothersetofdipolemoments
isinducedandthefollowingeffectiveHamiltonianl describes thattypeofinteraction:
Hspin=−12pg s ·(E×E˙)−12pg s ·(B×B˙)−4pg s B E −4pg s E B
eff 2 E1E1 2 M1M1 M1E2 i j ij E1M2 i j ij
(2.1)
T = 1(¶ T +¶ T); T={E,B}.
ij 2 i j j i
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DynamicalStructureofBaryons A.Aleksejevs
The coefficients of the proportionality in the Eq.(2.1), g ,g ,g and g are called
E1E1 M1M1 M1E2 E1M2
spin-dependent polarizabilities and correspond to the dipole-dipole and dipole-quadrupole elec-
tric/magnetic transitions. Itisquitedifficulttomeasurethesespin-dependent polarizabilities sepa-
rately,butforthespecifickinematicsoftheforward/backwardscatteringthesestructureparameters
canbecombinedintoso-calledforwardg0andbackwardgp polarizabilities,g0=−gE1E1−gM1M1−
gM1E2−gE1M2 andgp =−gE1E1+gM1M1+gM1E2−gE1M2,andcanbeaccessed bytheexperiment.
Inordertoevaluatethepolarizabilities theoretically, onecanusetheComptonscatteringandrelate
theamplitude tothesetoftheComptonstructure functions R [3]inthefollowingway:
i
1 M(g B→g ′B)= R (e ′∗·e )+R (s′∗·s)+iR s ·(e ′∗×e )+iR s ·(s′∗×s)+
8p W 1 2 3 4
(2.2)
iR ((s ·kˆ)(s′∗·e )−(s ·kˆ′)(s·e ′∗))+iR ((s ·kˆ′)(s′∗·e )−(s ·kˆ)(s·e ′∗))
5 6
Here,W =w + w 2+m2 isthecenterofmassenergyandw istheenergyoftheincomingphoton.
q B
The unit magnetic vector (s=(kˆ×e )), polarization vector (e ) and unit momentum of the photon
k
(kˆ = )aredenoted bythe prime for thecase ofthe outgoing photon. Although the choice ofthe
k
basis for the invariant Compton amplitude is not unique [3], the basis in the Eq.(2.2) is the most
convenient for evaluating polarizabilities. Here, inEq.(2.2), the structure functions R aredirectly
i
related to the electric, magnetic and spin-dependent polarizabilities in the multipole expansion. If
we keep only dipole-dipole and dipole-quadrupole transitions in the multipole expansion of the
Comptonstructure functions [7],wehaverathersimpleconnecting formulastothepolarizabilities
ofthebaryon:
RNB=w 2a ; RNB=w 2b ; RNB=w 3(−g +g );
1 E1 2 M1 3 E1E1 E1M2
(2.3)
RNB=w 3(−g +g ); RNB=−w 3g ; RNB=−w 3g .
4 M1M1 M1E2 5 M1E2 5 E1M2
Although the polarizabilities used in Eq.(2.3) are defined as constants, it is essential to treat them
as energy-dependent quantities [4]. The reason behind this extension to the dynamical (energy-
dependent) polarizabilities is dictated by the fact that the Compton scattering experiments were
performed with 50 to 800 MeV photons and hence required additional theoretical information to
extrapolate theresultstozero-energy parameters. Itisalsowellknownthatthepolarizabilities can
becomeenergy-dependent duetotheinternalrelaxationmechanisms,resonances, andparticlepro-
duction thresholds. Accordingly, wekeep all orders inw fortheCompton structure functions (for
static polarizabilities we keep only order up to O(w 2) for R and up to O(w 3) for R ), and
1,2 3,4,5,6
hence determine the energy-dependent polarizabilities. A connection between dynamic and static
polarizabilities can be achieved by taking a limit to zero photon energy. The Compton structure
functionsuptoone-looporderarecalculatedusingCHM[5]basedontherelativisticchiralpertur-
bation theory. Inaddition, thestructure-dependent polecontribution tothenucleon polarizabilities
istakenintoaccountintheformofthenucleon D -resonance excitation.
3. Results
The polarizabilities calculated for the proton with the photon energies up to 300 MeV are
3
DynamicalStructureofBaryons A.Aleksejevs
Figure 1: Dependencies of the proton electric, magnetic and spin-dependent polarizabilities on photon
energyw (GeV)inthecenter-of-massreferenceframe. Thefirstrow(fromleft)correspondstotheelectric
andmagneticpolarizabilitiesin10−4(fm3).Thesecondrowarespin-dependentforwardg andbackward
0
gp polarizabilitiesin10−4(fm4).Green-dashedcurvecorrespondstothemeson-nucleonloopscontribution
onlyandsolid-redcurveistheresultwithD resonancepolecontributionadded.
shownonFig.1. Itisevidentthatbelow50MeVtheyhaveverysmallenergy dependence. Forthe
neutron, the energy dependencies of the dynamical polarizabilities have similar behavior except
the values are bigger on absolute scale. Here we will only provide a description for the proton
dynamical polarizabilities.
The electric polarizability of the proton has very strong, resonance-type dependence near the
pion production threshold. The D -pole contribution has a small effect while consistently reducing
a (w )values forallthe energies. Ofcourse, tomakefinalpredictions oftheChPThofthevalues
p
ofpolarizabilities, itisrequiredtoaddcontribution fromtheresonances intheComptonscattering
loops. Hence, in order to compare our results with experimental values, we have used resonance
loops results borrowed from the smallscale expansion (SSE)approach [6]. Ifno D -pole contribu-
tion is added, the magnetic polarizability in Fig.1 stays negative (diamagnetic) for almost all the
energies. The D -pole contribution is very large and shifts b (w ) from negative to positive (para-
p
magnetic) values for energies up to 250 MeV. This behavior is quite natural, since the pion loop
calculationsreflectmagneticpolarizabilitycomingfromthevirtualdiamagneticpioncloudandthe
D resonance contribution to b (w ) is driven by the strong paramagnetic core of the nucleon. The
p
spin-dependent polarizabilities, g0andgp ,havestrongdependencenearthepionproductionthresh-
oldandtheD -polecontributionisevidentneartheD productionthreshold. Ifwetakecontributions
of order O(p3) in ChPThpower counting, we get an excellent agreement with [8]. Our result for
theprotonpolarizabilities uptotheone-loop order, plusincluding D -poleandSSEcontribution is
thefollowing(inunitsof10−4(fm3)):
a =(7.38(p −loop)−0.95(D −pole)+4.2(SSE))=10.63;
p
b =(−2.20(p −loop)+3.0(D −pole)+0.7(SSE))=1.49.
p
Inthefollowingtable,welisttheresultsforthespin-dependent staticpolarizabilities:
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DynamicalStructureofBaryons A.Aleksejevs
Figure2: Electricandmagneticdynamicalpolarizabilitiesofhyperonsinunitsof10−4(fm3)asafunction
ofthephotonenergyw (GeV).Here,solidredlinerepresentstheelectricpolarizabilityanddashedgreenline
isthemagneticpolarizability.
10−4(fm4) O(p3)[9] O(p4)[10] O(e 3)[11] HYP.Dr. [12] Thiswork Exp.
g 4.6 -3.9 2.0 -1.1 1.1 -0.90±0.08±0.11
0(p)
gp (p) 4.6 6.3 6.8 7.8 6.1 8.0±1.8
The listed results have a broad spectrum of values, so clearly more work is needed in this area.
Our values in this table do not include the D resonance in the loops, but if we follow the trend of
the D -pole contribution into g0 and gp , wecansee that inclusion of resonance inthe loops for the
Comptonscattering willbringourresultscloser totheexperimental values. Thestaticelectric and
magnetic polarizabilities for hyperons have been calculated first in [13] and just recently calcula-
tions havebeen completed forthespin-dependent static polarizabilities in[14]. Bothgroups were
usingheavybaryonchiralperturbationtheory. Thedynamicalelectricandmagneticpolarizabilities
for hyperons first have been calculated in [15]. In Fig.2 we provide updated results for dynami-
cal electric and magnetic polarizabilities for hyperons using basis from Eq.(2.2) in the Compton
scattering amplitude.
Forallpolarizabilities listedinFig.(2),theelectricpolarizabilities haveverysimilarresonant-
typebehaviornearthemeson-production thresholdsandthemagneticpolarizabilities forallhyper-
ons have negative low energy (static) values. Once again it is important to include both pole and
loop resonance contributions foracomplete analysis. InFig.(3), wepresent thefirstresults onthe
forwardandbackwardspin-dependent dynamical polarizabilities forthehyperons.
As one can see from Fig.(3), for all hyperons, the spin-dependent backward polarizability
dominates theforwardpolarizability ontheabsolute scale. Simultaneously, theyallexhibit almost
static behavior in the very low energy region of the Compton scattering. For the all dynamical
polarizabilities of SU(3) octet of baryons, we find that their values are strongly governed by the
excitation mechanism reflected in the meson production peaks. Hence the study of these polariz-
abilities directly probes the internal degrees of freedom which govern the structure of baryons at
low energy. In this work, we have calculated the electric, magnetic and spin-dependent dynami-
5
DynamicalStructureofBaryons A.Aleksejevs
Figure3: Forward(g0)andbackward(gp )spin-dependentdynamicalpolarizabilitiesofhyperonsinunits
of10−4(fm4). Redsolidlineshowstheforwardspin-dependentpolarizabilityandgreendashedlinecorre-
spondstothebackwardspin-dependentpolarizability.
cal polarizabilities of the SU(3) octet of baryons using ChPTh implemented in CHM. We found
that predictions of the chiral theory derived from our calculations (up to one-loop order and not
including resonances in loop calculations) are somewhat consistent with the experimental results.
The calculations of the dynamical polarizabilities with baryon resonances in the loops is our cur-
rent goal. It is also evident that further experimental work is needed, especially for the hyperon
polarizabilities.
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