A Comparative Study of Nucleon Structure in Light-Front Quark Models in AdS/QCD

0.26 MB·

by  Chandan Mondal

Few-Body Systems manuscript No.
(will be inserted by the editor)
Chandan Mondal · Dipankar Chakrabarti
A Comparative Study of Nucleon Structure in
Light-Front Quark Models in AdS/QCD
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0 Received:date / Accepted:date
2

n
a Abstract We presenta comparativestudy of nucleonstructure suchas electromagnetic formfactors,
J transverse charge and magnetization densities in three different models within AdS/QCD framework.

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Keywords Electromagnetic form factor Transverse densities AdS/QCD Quark model
· · ·
]
h
p 1 Introduction
-
p
e Inrecentyears,AdS/QCDhasemergedasoneofthemostpromisingtechniquestounravelthestructure
h of mesons and nucleons. AdS/CFT conjecture relates a strongly coupled gauge theory in d space-
[ time dimensions by a dual weak coupling gravity theory in AdS space. To exploit this duality to
d+1
addressthe problemsin QCD,the conformalinvarianceneeds to be broken.Inthe literaturethere are
1
v two methods to achieve this goal, one is called hard wall model where a sharp cut-off is put in the
2 hologhaphic direction in the AdS space where the wave functions are made to vanish and the other is
5 called the soft wall model in which a confining potential is introduced in the AdS space which breaks
2 the conformal invariance and allows the QCD mass scale.
1
Electromagnetic form factors provide us insights into the structure of the nucleons and have been
0 measured in many experiments. In the light-cone frame with q+ = q0 + q3 = 0, the charge and
.
1 anomalous magnetization densities in the transverse plane can be identified with the two-dimensional
0 Fourier transform(FT) of the electromagnetic form factors. The contributions of individual quark to
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the nucleon charge and magnetization densities are obtained from the flavor decompositions of the
1
transverse densities.
:
v Here we present a detailed analysis of the nucleon form factors in AdS/QCD soft-wall models
i [1; 2; 3; 4]. The flavor form factors are obtained by decomposing the Dirac and Pauli form factors for
X
nucleons using the charge and isospin symmetry. We also present a comparative study of the nucleon
r
a as wellasthe flavorcontributionsto the nucleonchargeand anomalousmagnetizationdensities in the
transverse plane [5].
2 Nucleon and flavor form factors in ADS/QCD models
Model I : Model I refers to the AdS/QCD model for nucleon form factors proposed by Brodsky and
T´eramond [2]. The relevant AdS/QCD action for the fermion field is written as
i i
S = d4xdz√g Ψ¯eMΓAD Ψ (D Ψ¯)eMΓAΨ µΨ¯Ψ V(z)Ψ¯Ψ , (1)
Z (cid:16)2 A M − 2 M A − − (cid:17)
Chandan Mondal and DipankarChakrabarti
Department of Physics, Indian Instituteof Technology Kanpur,Kanpur208016, India
E-mail: [email protected], E-mail: [email protected]
2
where eM = (z/R)δM is the inverse vielbein and V(z) is the confining potential which breaks the
A A
conformalinvarianceandRistheAdSradius.Ind=4dimensions,Γ = γ , iγ .TomaptheDirac
A µ 5
{ − }
equation in AdS space with the light front wave equation, one identifies z ζ (light front transverse
impactvariable)andsubstitutesΨ(x,ζ)=e−iP·xζ2ψ(ζ)u(P)andsets µR→=ν+1/2whereν =L+1
(more details can be found in [2]). For linear confining potential U(ζ|) = (|R/ζ)V(ζ) = κ2ζ, one gets
thelightfrontwaveequationforthebaryonwhichleadstothe AdSsolutionsofnucleonwavefunctions
ψ+(z) and ψ−(z) corresponding to different orbital angular momentum Lz =0 and Lz =+1 [2]
√2κ2 κ3
ψ+(ζ)∼ψ+(z)= R2 z7/2e−κ2z2/2, ψ−(ζ)∼ψ−(z)= R2z9/2e−κ2z2/2. (2)
The Dirac form factors in this model are obtained by the SU(6) spin-flavor symmetry and given by
dz 1 dz
F1p(Q2)=R4Z z4V(Q2,z)ψ+2(z), F1n(Q2)=−3R4Z z4V(q2,z)(ψ+2(z)−ψ−2(z)). (3)
Aprecisemappingforthespin-flipnucleonformfactorusingtheactioninEq.(1)isnotpossible.Thus,
the Pauli form factors for the nucleons are modeled in this model as
dz
F2p/n(Q2)=κp/nR4Z z3ψ+(z)V(Q2,z)ψ−(z). (4)
The Pauli form factors are normalized to Fp/n(0) = κ where κ are the anomalous magnetic
2 p/n p/n
moment of proton/neutron. V(Q2,z) is the bulk-to-boundary propagator[2]. Here we use the value
κ = 0.4GeV which is fixed by fitting the ratios of Pauli and Dirac form factors for proton with the
experimental data [3; 6].
4 1.4
Gayou 01
1.2 Gayou 02
Punjabi 05
2V] 3 1 PAurrcinkgettot n1 007
2ppQ F/F [Ge2112 GGPPAuuraarnciyynkjooageuubtto ti00 n0112 5007 pp−1κ/F F12p000...468 MMLFoo dddeeiqll uIIIark
Model I 0.2
Model II
LF diquark
0 0 1 2 3 4 5 6 7 8 9 0 0 1 2 3 4 5 6 7 8 9
(a) Q2 [GeV2] (b) Q2 [GeV2]
0.7
Passchier 99
0.1 Herberg 99 0.6 ZWhaur r0e1n 04
Glazier 05 Herberg 99
nGE00..0068 PBRWPZPBMhleallioaaaoaursdsssrrms drttte0eece ua0l1nh rrtnI5 h i00 e0 1660r40 39nH9e nR000...345 GPRBGMMLFleoleooaa rhiddsdzmseeetii eqe0u ll0 rur8tII h5Ia00 r540k3
0.04 Model II
LF diquark 0.2
0.02 0.1
(a) 00 0.5 1 1.5Q2 [G2eV2]2.5 3 3.5 4 (b) 00 0.2 0.4 Q0 .6[GeV]0.8 1 1.2
Fig. 1 Nucleon form factors (a) the ratio of Dirac and Pauli form factors for proton is multiplied by Q2 =
−q2 = −t, (b) the ratio is divided by κp; (c) electric Sach form factor for neutron GnE(Q2), (d) the ratio
Rn= µn GnE. Thered dashed and pinkdash dot lines represent the Model I and Model II and the solid black
Gn
M
lines represent thequark-diquarkmodel [4]. The references of theexperimental data can be found in Ref.[3].
Model II:Theothermodelofthe nucleonformfactorswasformulatedbyAbidin andCarlson[1].
SincetheactiondefinedinEq.(1)cannotgeneratethePauliformfactors,theyintroducedanadditional
gauge invariant non-minimal coupling term
d4x dz √g Ψ¯eA eB [Γ ,Γ ]FMNΨ. (5)
Z M N A B
3
0.5 0.2
Diehl 13 Diehl 13
Cates 11 Cates 11
0.4 MMLFoo dddeeiqll uIIIark 0.15 MMLFoo dddeeiqll uIIIark
2GeV] 0.3 2GeV]
2uQF [10.2 2dQF [10.1
0.05
0.1
00 0.5 1 1.5 2 2.5 00 0.5 1 1.5 2 2.5
(a) Q [GeV] (b) Q [GeV]
0.35 0.4
Diehl 13 Diehl 13
0.3 CMaotdeesl 1I1 CMaotdeesl 1I1
2u2QF [GeV]20000..12..1255 MLFo ddeiql uIIark 2d2−QF [GeV]2000...123 MLFo ddeiql uIIark
0.05
00 0.5 1 1.5 2 2.5 00 0.5 1 1.5 2 2.5
(c) Q [GeV] (d) Q [GeV]
Fig. 2 Flavordependentform factors for uandd quarks.Theexperimentaldata aretakenfrom [9; 10]. The
lines are same as in Fig.1.
1.4
2 Kelly Kelly
Model I 1.2 Model I
Model II Model II
1.6 LF−diquak 1 LF−diquark
p−2ρ(b) [fm]ch01..82 p−2ρ (b) [fm]m00..68
0.4
0.4
0.2
00 0.2 0.4 0.6 0.8 1 1.2 00 0.4 0.8 1.2 1.6 2
(a) b [fm] (b) b [fm]
0.1 0
0
−0.3
−0.1
n−2ρ(b) [fm]ch−−−000...432 n−2ρ (b) [fm]m−−00..96
Kelly Kelly
−0.5 Model I −1.2 Model I
Model II Model II
−0.6 LF−diquark LF−diquark
(c) 0 0.2 0.4 0b.6 [fm] 0.8 1 1.2 (d) −1.50 0.4 0.8b [fm]1.2 1.6 2
Fig. 3 Transverse charge and anomalous magnetization densities for nucleon. (a) and (b) represent ρ and
ch
ρm fortheproton.(c)and(d)sameasprotonbutforneutron.Lineswith circlerepresenttheparametrization
of Kelly [11].
This additional term also provides an anomalous contribution to the Dirac form factor. In this model
the form factors are given by[1]
Fp(Q2)=C (Q2)+η C (Q2), Fn(Q2)=η C (Q2), (6)
1 1 p 2 1 n 2
Fp(Q2)=η C (Q2), Fn(Q2)=η C (Q2). (7)
2 p 3 2 n 3
The functions C (Q2) are defined as C (Q2) = a+6 , C (Q2) = 2a(2a−1) , and
i 1 (a+1)(a+2)(a+3) 2 (a+1)(a+2)(a+3)(a+4)
C (Q2) = 48 , where a = Q2/(4κ2). The value of κ = 0.350GeV is fixed by simultaneous
3 (a+1)(a+2)(a+3)
fit to proton and rho meson masses. The other parameters are determined from the normalization
conditions of the Pauli form factor at Q2 = 0 and are given by η = 0.224 and η = 0.239 [1]. The
p n
−
PauliformfactorsinModelIandModelIIareidentical,themaindifferenceisintheDiracformfactor
where we have an additional contribution in Model II coming the extra term added.
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1.8 2
0.6 1.6 0.6 1.8
1.4 1.6
0.3 1.2 0.3 1.4
1.2
m] 1 m]
b [fy 0 0.8 b [fy 0 01.8
−0.3 0.6 −0.3 0.6
0.4 0.4
−0.6 0.2 −0.6 0.2
−0.6 −0.3 b [0fm] 0.3 0.6 −0.6 −0.3 b [0fm] 0.3 0.6 0
(a) x (b) x
0 0.3
0.6 −0.05 0.6 0.2
−0.1 0.1
0.3 −0.15 0.3 0
b [fm]y 0 −−00..225 b [fm]y 0 −−00..21
−0.3 −0.3 −0.3 −0.3
−0.4
−0.6 −0.35 −0.6 −0.5
−0.4
−0.6 −0.3 b [0fm] 0.3 0.6 −0.6 −0.3 b [0fm] 0.3 0.6
(c) x (d) x
Fig. 4 Unpolarized and transversely polarized charge densities for proton(upper) and neutron(lower) in LF
diquark model.
Table 1 Electromagnetic radii of thenucleons
Quantity Model I Model II LF diquark Measured data
rp (fm) 0.810 0.980 0.786 0.877±0.005
rEp (fm) 0.782 0.921 0.772 0.777±0.016
M
hr2in (fm2) −0.088 −0.123 −0.085 −0.1161±0.0022
E
rn (fm) 0.796 0.937 0.7596 0.862+0.009
M −0.008
Quark-diquarkmodelinAdS/QCD:Hereweconsideralightfrontscalarquark-diquarkmodel
for nucleon[7] where the 2-particle wavefunction is modeled from the soft-wall AdS/QCD solution. In
the light front overlapformalism, the electromagnetic form factors in this model are given by
F1q(Q2)= Z 1dxZ d126kπ⊥3 (cid:20)ψ++q∗(x,k′⊥)ψ++q(x,k⊥)+ψ−+q∗(x,k′⊥)ψ−+q(x,k⊥)(cid:21), (8)
0
F2q(Q2)= −q12Minq2 Z 1dxZ d126kπ⊥3 (cid:20)ψ++q∗(x,k′⊥)ψ+−q(x,k⊥)+ψ−+q∗(x,k′⊥)ψ−−q(x,k⊥)(cid:21), (9)
− 0
where k′⊥ = k⊥ +(1−x)q⊥. ψλλqNq(x,k⊥) are the LFWFs with specific nucleon helicities λN = ±
and for the struck quark λ = , where plus and minus correspond to +1 and 1 respectively. The
q ± 2 −2
LFWFs are specified at an initial scale µ =313 MeV [7] :
0
k1+ik2
ψ++q(x,k⊥) =ϕ(q1)(x,k⊥), ψ−+q(x,k⊥)=− xM ϕ(q2)(x,k⊥),
n
k1 ik2
ψ+−q(x,k⊥) = x−M ϕ(q2)(x,k⊥), ψ−−q(x,k⊥)=ϕ(q1)(x,k⊥), (10)
n
4π log(1/x) k2 log(1/x)
ϕ(qi)(x,k⊥) =Nq(i) κ r 1 x xaq(i)(1−x)bq(i)exp(cid:20)− 2κ⊥2 (1 x)2(cid:21). (11)
− −
For aq(i) = b(qi) = 0, ϕ(qi)(x,k⊥) reduces to the AdS/QCD prediction [2]. κ is the AdS/QCD scale
parameter which is taken to be 0.4 GeV [6; 3]. The parameters a(i) and b(i) with the constants N(i)
q q q
are fixed by fitting the electromagnetic properties of the nucleons [8].
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Flavor decompositionsof the nucleon form factors :Underthechargeandisospinsymmetry
it is straightforwardto write down the flavor decompositions of the nucleon form factors as [9]
Fp =e Fu+e Fd and Fn =e Fu+e Fd, (i=1,2) (12)
i u i d i i d i u i
wheree ande arechargeofuanddquarksrespectively.Thenormalizationsoftheflavorformfactors
u d
are Fu(0) = 2,Fu(0) = κ and Fd(0) = 1,Fd(0) = κ where the anomalous magnetic moments for
1 2 u 1 2 d
the up and down quarks are κ =2κ +κ =1.673 and κ =κ +2κ = 2.033.
u p n d p n
−
In Fig.1 we comparethe results for electromagnetic formfactors of nucleons calculated in different
AdS/QCD models. The flavor form factors are shown in Fig.2. The figures show that the results of
the Model I and the quark-diquark model are in good agreement with experimental data whereas the
Model II deviates from the data. Only for Fd, Model I deviates at higher Q2 from the data and also
1
for Gn(Q2) Model II is better than Model I. The fitted results for the electromagnetic radii of the
E
nucleons are listed in Table 1.
3 Transverse charge and magnetization densities
The transverse charge density inside the nucleons is given by
ρ (b)= d2q⊥F (q2)eiq⊥.b⊥ = ∞ dQQJ (Qb)F (Q2), (13)
ch Z (2π)2 1 Z 2π 0 1
0
where b represents the impact parameter and J is the cylindrical Bessel function of order zero. One
0
candefine the magnetizationdensity(ρ (b))in the similarfashionwithF is replacedby F ,whereas,
M 1 2
ρ (b)= b(∂ρ (b)/∂b)canbeinterpretedasanomalousmagnetizationdensity.Weevaluatethequark
m M
contribut−ions to the nucleon transversee densities using charge and isospin symmetry (see [5; 4]).
For transveersely polarized nucleon, the charge density is given by
1
ρ (b)=ρ sin(φ φ ) ρ . (14)
T ch b s m
− − 2M b
n
ThetransversepolarizationofthenucleonisgivenbyS⊥ =(cosφsxˆ+sinφsyˆ)andthetransverseimpact
parameterb⊥ =b(cosφbxˆ+sinφbyˆ).Withoutlossofgenerality,thepolarizationofthenucleonistaken
alongx-axisie., φ =0.ThesecondterminEq.(14),providesthe deviationfromcircularsymmetryof
s
the unpolarized charge density.The nucleon charge and anomalous magnetization densities presented
inFig.3suggestthatthe quark-diquarkmodelagreeswiththe phenomenologicalparametrizations[11]
much better than the Model I and the Model II and Model I is better compare to Model II. The
charge densities between unpolarized and transversely polarized nucleon in the quark-diquark model
are compared in Fig.4. The unpolarized densities are axially symmetric in transverse plane while for
thetransverselypolarizednucleonstheybecomedistorted.Fornucleonpolarizedalongxdirection,the
densities get shifted towards negative y-direction. Due to large anomalous magnetization density, the
distortion in neutron charge density is found to be stronger than that for proton.
Acknowledgements CMthankstheSciencceandEngineeringResearchBoard(SERB),GorernmentofIndia
forsupportingthetravelgrantundercontractno.ITS/3444/2015-2016toattendtheconferenceLightCone2015
where thework was presented.
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