Comparisons of electric charge and axial charge meson cloud distributions in the PCQM

0.53 MB·

by  X. Y. Liu

Comparisons of electric charge and axial charge meson cloud distributions
in the PCQM
X. Y. Liu,1,2,∗ K. Khosonthongkee,1 A. Limphirat,1 P. Suebka,1 and Y. Yan1,†
1School of Physics, Institute of Science, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand
2School of Mathematics and Physics, Bohai University, Liaoning 121013, China
(Dated: April 12, 2016)
The meson cloud distributions in r-space are extracted from the nucleon electromagnetic and
axialformfactorswhicharederivedintheperturbativechiralquarkmodel. Thetheoreticalresults
indicatethattheelectricchargeandaxialchargedistributionsofthethree-quarkcorearethesame,
the magnetic charge distributions of the meson cloud and three-quark core are more or less in the
sameregionandpeakatdistancesofaround2GeV−1,buttheaxialchargemesonclouddistributes
mainly inside the three-quark core.
6
1 PACSnumbers: 12.39.Ki,14.20.-c,14.40.-n
0
2
I. INTRODUCTION II. PERTURBATIVE CHIRAL QUARK MODEL
r
p
A The meson cloud of the nucleon, undoubtedly, plays a In the framework of the PCQM, baryons are consid-
relevantroleinthestudyoflowenergyelectroweakprop- ered as the bound states of three relativistic valence
1
erties of the nucleon. The meson cloud model, where quarks moving in a central Dirac field with V (r) =
1 eff
the nucleon is considered as a system of three valence S(r)+γ0V(r), while a cloud of pseudoscalar mesons, as

] quarks surrounded by a meson cloud [1–10], has recently the sea-quark excitations, is introduced for chiral sym-
h
been employed to study the generalized parton distribu- metryrequirements,andtheinteractionsbetweenquarks
p
tion [11, 12], nucleon electroweak form factors [13–18], andmesonsareachievedbythenonlinearσ modelinthe
-
p nucleon strangeness [19, 20], etc. In Refs. [21–23], me- PCQM. The Weinberg-type Lagrangian of the PCQM
e son cloud contributions to the neutron charge form fac- under an unitary chiral rotation [14, 15] is derived as,
h
tor have been studied and discussed in the meson cloud
[ LW(x)=L (x)+LW(x)+o((cid:126)π), (1)
model, while the effects of the meson cloud on electro- 0 I
2 magnetic transitions have been estimated in Refs. [24– L (x)=ψ¯(x)(cid:2)i∂/−γ0V(r)−S(r)(cid:3)ψ(x)
0
28v 2a6n]d. aIxniaolufrorpmrefvaicotuosrswaosrwksel[l2a7s,e2l8ec],trtohweeealkecptrroopmeargtineestoicf −12Φi(x)(cid:0)(cid:3)+MΦ2(cid:1)Φi(x), (2)
octet baryons have been studied in the perturbative chi-
14 ral quark model (PCQM). The theoretical results in the LWI (x)= 21F∂µΦi(x)ψ¯(x)γµγ5λiψ(x)
PCQM with predetermined quark wave functions are in
0 f
1. good agreement with the experimental data and lattice +4Fijk2Φi(x)∂µΦj(x)ψ¯(x)γµλkψ(x), (3)
QCD values. In addition, Ref. [28] reveals that the me-
0
son cloud plays an important role in the axial charge of where f are the totally antisymmetric structure con-
6 ijk
1 octetbaryons,contributing30%–40%tothetotalvalues, stant of SU(3), the pion decay constant F =88 MeV in
: and the similar effects have been also observed in other the chiral limit, Φ are the octet meson fields, and ψ(x)
v i
frameworks [29, 30]. Recently, the π-meson cloud distri- is the triplet of the u, d, and s quark fields taking the
i
X butions of the nucleon EM form factors in r-space have form
r beenevaluatedinthechiralperturbationtheory[31,32].  
a The results in Refs. [31, 32] show that the EM distribu- u(x)
tions peak at distances of around r =0.3 fm and fall off ψ(x)= d(x) . (4)
smoothly with increasing distance. In this work, we at- s(x)
tempttofurtherstudyandcomparethemesonclouddis-
tributions of the nucleon electromagnetic and axial form The quark field ψ(x) could be expanded in
factors in r-space in the framework of the PCQM.
The paper is organized as follows. In section II, we ψ(x)=(cid:88)(cid:0)bαuα((cid:126)x)e−iEαt+d†αυα((cid:126)x)eiEαt(cid:1), (5)
brieflydescribethebasicnotionsofthePCQM.Thecom- α
parison and discussion between EM and axial form fac-
where b and d† are the single quark annihilation and
tors of nucleon are given in section III. α α
antiquark creation operators. The ground state quark
wave function u ((cid:126)x) may, in general, be expressed as
0
(cid:18) (cid:19)
∗ lxy [email protected] g(r)
u ((cid:126)x)= χ χ χ , (6)
† [email protected] 0 i(cid:126)σ·xˆf(r) s f c
2
2.5 1.0 chargemesonclouddistributionsofthenucleonfromthe
�� ��
EM and axial form factors as shown in Fig. 1. Shown in
2.0 ���� 0.8 ����
Fig. 2 are the LO and meson cloud contributions to the
proton magnetic form factor Gp (Q2) (left panel) and
1.5 0.6 M
p2GQ()M 2ΝGQ()Α the nucleon axial form factor GNA(Q2) (right panel) in
1.0 0.4 r-space. It is found in Fig. 2 that the magnetic charge
distributionsofthethree-quarkcoreandthemesoncloud
0.5 0.2 are almost in the same region, but the loop diagrams
contributetotheGN(Q2)inaclearlysmallerregionthan
0.0 0.0 A
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 theLOdiagram,whichmayindicatethattheaxialcharge
Q2(GeV2) Q2(GeV2)
meson cloud distributes mainly inside the three-quark
core.
FIG. 1. Leading order (solid) and loop (dashed) contri- We also find that the magnetic charge distribution
butions to the proton magnetic (left panel) form factor and shownintheleftpanelofFig.2presentasignificantpeak
neutron axial (right panel) form factor. around r (cid:39)2 GeV−1 and fall off smoothly when the dis-
tance increases. The results turned out to be similar to
the ones of Refs. [31, 32].
where χ , χ and χ are the spin, flavor and color quark
s f c
Furthermore, we compare the LO contributions to the
wave functions, respectively.
Gp (Q2) and GN(Q2) in r-space, as presented in the left
In our previous works [27, 28], the ground state M A
panel of Fig. 3. It is clear that the LO Gp (r2) and
quark wave functions have been determined by fitting M
GN(r2)showasimilarr-dependence,whichmayindicate
the PCQM theoretical result of the proton charge form A
factor Gp(Q2) to the experimental data [27], and the that the electric charge and axial charge distributions of
E the constituent quarks are the same. The meson cloud
electromagnetic and axial form factors as well as elec-
Gp (r2) and GN(r2) in the right panel of Fig. 3 show
troweak properties of octet baryons in low energy region M A
that the axial charge distribution of the meson cloud is
have been studied in the PCQM based on the prede-
narrower and the peak is closer to the origin.
termined quark wave functions. The PCQM theoretical
results are in good agreement with experimental data
and lattice QCD values. More details could be found in
0.030 0.014
Refs. [27, 28]. �� ��
0.025 ���� 0.012 ����
0.010
0.020
IIDI.ISTERLIEBCUTTRIOICNSAONFDMAXESIAOLNCCHLAORUGDE p2Gr()M0.015 N2Gr()A0.008
r r0.006
0.010
0.004
Following our previous works [27, 28], we present in
0.005 0.002
Fig.1theprotonmagneticandnucleonaxialformfactors
separately in leading order (LO) and loop Feymann dia- 0.0000 2 4 6 8 10 0.0000 2 4 6 8 10
gram contributions. The PCQM results shown in Fig. 1 r[GeV-1] r[GeV-1]
clearlyrevealthattheLOdiagramresultsinadipole-like
form factor while the meson cloud leads to a flat contri- FIG. 2. Comparisons between the LO and meson cloud
bution to the magnetic and axial form factors. The flat distributionsforprotonmagnetic(leftpanel)andaxial(right
contribution may indicate that the meson cloud of the panel) form factors in r-space.
nucleon may distribute mainly in a very small region.
In general, the form factor F(q2) is the Fourier trans- 0.030 0.014
formationofchargedistributioninr-spaceandtakesthe ���(��) ���(��)
form, 0.025 ���(��) 0.012 ���(��)
0.010
(cid:90) 0.020
F(q2)= ρ((cid:126)r)e−iq(cid:126)·(cid:126)rd3(cid:126)r, (7) 2rGr()LO0.015 2rGr()Loop00..000068
where ρ((cid:126)r) is the charge density, and (cid:126)q is the three- 0.010 0.004
momentum transfer. If F(q2) has been determined, in 0.005 0.002
principle, the charge distribution ρ(r) could be obtained
0.000 0.000
by the inverse Fourier transformation, 0 2 4 6 8 10 0 2 4 6 8 10
r[GeV-1] r[GeV-1]
1 (cid:90)
ρ((cid:126)r)= F(q2)eiq(cid:126)·(cid:126)rd3(cid:126)q. (8)
(2π)3
FIG. 3. Comparisons between the magnetic and axial dis-
In this work, we extract, based on the inverse Fourier tributions in r-space for the LO (left panel) and loop (right
panel) diagrams.
transformationinEq.(8), themagneticchargeandaxial
3
In summary, one may conclude that the similar r- ACKNOWLEDGMENTS
dependence of the magnetic and axial form factors re-
sulted from the LO diagrams may indicate that the elec- This work is supported by Suranaree University of
tric charge and axial charge distributions of the con- Technology and Bohai university. XL and AL ac-
stituent quarks are the same. The magnetic charge dis- knowledge support by SUT-CHE-NRU (Project No.
tributions of the meson cloud and three-quark core are NV12/2558). This work is supported also by Na-
more or less in the same region and peak at distances tionalNaturalScienceFoundationofChina(ProjectNo.
of around 2 GeV−1, but the axial charge meson cloud 11547182),andtheDoctoralScientificResearchFounda-
distributes mainly inside the three-quark core. tion of Liaoning Province (Project No. 201501197).
[1] S. Th´eberge, A. W. Thomas, and G. A. Miller, Phys. (2011).
Rev. D 22, 2838 (1980). [18] G.Ramalho,K.Tsushima, andA.W.Thomas,J.Phys.
[2] A. W. Thomas, S. Th´eberge, and G. A. Miller, Phys. G: Nucl. Part. Phys. 40, 015102 (2013).
Rev. D 24, 216 (1981). [19] F. Carvalho, F. S. Navarra, and M. Nielsen, Phys. Rev.
[3] S. Chin, Nucl. Phys. A 382, 355 (1982). C 72, 068202 (2005).
[4] E. Oset, R. Tegen, and W. Weise, Nucl. Phys. A 426, [20] H. Chen, F. G. Cao, and A. I. Signal, J. Phys. G: Nucl.
456 (1984). Part. Phys. 37, 105006 (2010).
[5] T. Gutsche and D. Robson, Phys. Lett. B 229, 333 [21] D. Lu, K. Tsushima, A. Thomas, A. Williams, and
(1989). K. Saito, Phys. Lett. B 441, 27 (1998).
[6] Z. Dziembowski, H. Holtmann, A. Szczurek, and [22] L.GlozmanandD.Riska,Phys.Lett.B459,49 (1999).
J. Speth, Ann. Phys. 258, 1 (1997). [23] J. A. Rinehimer and G. A. Miller, Phys. Rev. C 80,
[7] J.SpethandA.Thomas,Adv.Nucl.Phys.24,83(2002). 025206 (2009).
[8] A. Faessler, T. Gutsche, V. E. Lyubovitskij, and [24] G. Ramalho, D. Jido, and K. Tsushima, Phys. Rev. D
K. Pumsa-ard, Phys. Rev. D 73, 114021 (2006). 85, 093014 (2012).
[9] B. Julia´-D´ıaz and D. Riska, Nucl. Phys. A 780, 175 [25] G.RamalhoandK.Tsushima,Phys.Rev.D88,053002
(2006). (2013).
[10] D. Chen, Y. Dong, M. Giannini, and E. Santopinto, [26] G. Ramalho and M. T. Pen˜a, Phys. Rev. D 89, 094016
Nucl. Phys. A 782, 62 (2007). (2014).
[11] B.PasquiniandS.Boffi,Phys.Rev.D73,094001(2006). [27] X. Y. Liu, K. Khosonthongkee, A. Limphirat, and
[12] B. Pasquini and S. Boffi, Nucl. Phys. A 782, 86 (2007). Y.Yan,J.Phys.G:Nucl.Part.Phys.41,055008(2014).
[13] V. E. Lyubovitskij, T. Gutsche, and A. Faessler, Phys. [28] X.Y.Liu,K.Khosonthongkee,A.Limphirat,P.Suebka,
Rev. C 64, 065203 (2001). and Y. Yan, Phys. Rev. D 91, 034022 (2015).
[14] V.Lyubovitskij,T.Gutsche,A.Faessler, andR.V.Mau, [29] J. Franklin, Phys. Rev. D 66, 033010 (2002).
Phys. Lett. B 520, 204 (2001). [30] G.RamalhoandK.Tsushima,arXiv:1512.01167[hep-ph]
[15] K. Khosonthongkee, V. E. Lyubovitskij, T. Gutsche, (2016).
A. Faessler, K. Pumsa-ard, S. Cheedket, and Y. Yan, [31] H. W. Hammer, D. Drechsel, and U. G. Meißner, Phys.
J. Phys. G: Nucl. Part. Phys. 30, 793 (2004). Lett. B 586, 291 (2004).
[16] B.PasquiniandS.Boffi,Phys.Rev.D76,074011(2007). [32] U. G. Meißner, AIP Conf. Proc. 904, 142 (2007).
[17] G.RamalhoandK.Tsushima,Phys.Rev.D84,054014