Table Of ContentQuark tensor and axial charges within the
4
1
0 Schwinger-Dyson formalism
2
n
a
J
9
2
] NodokaYamanaka∗,TakahiroM.Doi,ShotaroImai,HideoSuganuma
h
p DepartmentofPhysics,GraduateSchoolofScience,KyotoUniversity,
-
p Kitashirakawa-oiwake,Sakyo,Kyoto606-8502,Japan
e E-mail: [email protected]
h
[
1 We calculate the tensor and axial charges of the quark in the Schwinger-Dyson formalism of
v LandaugaugeQCD. Itis foundthatthedressedtensorandisovectoraxialchargesofthe quark
0
2 are suppressed against the bare quark contribution, and the result agrees qualitatively with the
4 experimentaldata.Weshowthatthisisduetothesuperpositionofthespinflipofthequarkarising
7
. from the successive emission of gluons which dress the vertex. For the isoscalar quark axial
1
0 charge,we haveanalyzedtheSchwinger-Dysonequationbyincludingtheleadingunquenching
4 quark-loopeffect. Itis foundthatthe suppressionis moresignificant, due to the axialanomaly
1
effect.
:
v
i
X
r
a
XVInternationalConferenceonHadronSpectroscopy-Hadron2013
4-8November2013
Nara,Japan
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
QuarktensorandaxialchargeswithintheSchwinger-Dysonformalism NodokaYamanaka
1. Introduction
Theanalysis ofthenucleon partonstructure playsanessential roleinclarifying thedynamics
ofthequantum chromodynamics [1]. Thespin-dependent quarkdistributions ofthenucleon inthe
leading twistaregivenbythehelicityandthetransversity distributions functions ofthequark,and
their firstmomentsarecalledtheaxialcharge (D q)andthetensorcharge (d q), respectively. Inthe
nonrelativistic constituent quark model, which considers three massive quarks in the nucleon, the
axialandtensorchargesofthequarkintheprotonbecometheirspin,sothatwehaveD u=d u= 4
3
for the u quark, and D d =d d =−1 for the d quark. The quark axial and tensor charges in the
3
nucleonarechiral-evenand-oddquantities, respectively, andtheirdifferenceprobeshowrelativis-
tic the polarized quarks are. Then, their study is important in understanding the structure of the
nucleon. Wealsonotethatthequarktensorchargerelatesthequarkelectricdipolemoment(EDM)
tothenucleonEDM,anobservablesensitivetotheCPviolationoftheelementaryinteractions, and
isthusanimportantquantity inthesearchofnewphysicsbeyondthestandard model[2].
The experimental study of these charges for the proton, however, gives smaller results com-
paredtothenaivequarkmodelprediction, namely[3,4,5]
DS = 0.32±0.03±0.03, (1.1)
g = −1.27590±0.00239+0.00331, (1.2)
A −0.00377
d u = 0.860±0.248, (1.3)
d d = −0.119±0.060, (1.4)
where DS ≈ D u+D d and g = d u−d d. Here, the quark tensor charges were renormalized at
A
m =2GeV.AlsothelatticeQCDstudiesofthequarkaxialandtensorcharges givesmallerresults
thanthenaivequarkmodelprediction [6],inqualitativeagreementwiththeexperimentaldata. We
should thereforetrytoclarifythesourceofthissuppression withsomenonperturbative method.
As a powerful nonperturbative way to investigate the dynamics of the quantum field theory
and in particular the low energy QCD, we have the Schwinger-Dyson (SD) formalism, and many
studiessuchasthedynamicalquarkmass,themesonmasses,theformfactors,etc,havebeendone
so far [7, 8, 9, 10]. In this paper, we will try to clarify the effect of the gluon vertex dressing and
analyze the source of the deviation of the quark charges. The effect in question, the vertex gluon
dressing, iswellwithintheapplicability oftheSDformalism.
2. Basics oftheSD Formalism
Inthiswork,weconsidertheSDformalismoftheLandaugaugeQCDwiththerainbow-ladder
approximation. Here, the SD formalism includes the infinite order of the strong coupling, i.e.
nonperturbative effects, and the quark-gluon vertex is also renormalization group (RG)-improved
attheone-loopwhichgivesthereplacement g2s Z (q2)g m ×G n (q,k)→a (q2)g m ×g n whereZ (q2)
4p g s g
is the gluon dressing function, and G n (q,k) is the dressed quark-gluon vertex. We use the RG-
improvedstrongcoupling a (p2)withinfrared (IR)regularization àlaHigashijima[7]
s
24p (p< p )
a s(p2)= 11N1c2−p2Nf 1 (p≥ pIR) . (2.1)
( 11Nc−2Nf ln(p2/L 2QCD) IR
2
QuarktensorandaxialchargeswithintheSchwinger-Dysonformalism NodokaYamanaka
HerewetakeN =N =3and p satisfyingln(p2 /L 2 )= 1. WeusetheQCDscaleparameter
c f IR IR QCD 2
L = 900 MeV. This large scale parameter is taken to reproduce the chiral quantities in this
QCD
framework [9]. With this setup, the chiral condensate (renormalized at m =2 GeV) is given by
hq¯qi=−(238MeV)3 and fp ≃70MeV.
TheSchwinger-Dyson equation (SDE)fortheaxialandtensor charges aredepicted inFig. 1.
The calculated dynamical charges give the contribution of the single quark to the corresponding
nucleoncharges. Fortheexactexpressions ofSDEanddetails,wereferthereadertoRefs. [9,10].
= + +
Figure1: TheSchwinger-Dysonequationforthequarkaxialandtensorchargesexpresseddiagrammati-
cally. Thegreyblobsrepresentthedynamicalcharges,andtheblackdotthebarecharge. Notethatthelast
unquenchingdiagramdoesnotcontributetothetensorandisovectoraxialcharges.
3. Analysis
Thesolutions of the quark tensor and axial SDEare shown in Figs. 2and 3, respectively. If
we associate the dressed dynamical quark with the constituent quark, our result can be combined
withthenonrelativistic constituent quarkmodelprediction. Forthequarktensorcharge, wehave
4
d u = S1(0)m ≃0.8,
3
1
d d = − S1(0)m ≃−0.2,, (3.1)
3
1
0.9
0.8
)
2
E
p 0.7
(
1
S
0.6
0.5
0.4 Tensor charge (L QCD=900MeV)
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
p (MeV)
E
Figure 2: The dressing function of the dynamical quark tensor charge S (p )s mn (not renormalized)
1 E
obtainedaftersolvingtheSDE.ThehorizontalaxisdenotestheEuclideanmomentum.
3
QuarktensorandaxialchargeswithintheSchwinger-Dysonformalism NodokaYamanaka
1.4
1.2
1
0.8
2) 0.6
E
p 0.4
(
1
G 0.2
0
-0.2
--00..64 UnQquueenncchheedd aaxxiiaall cchhaarrggee ((LL QQCCDD==990000MMeeVV))
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
p (MeV)
E
Figure 3: The dressing functionof the dynamicalquarkaxialchargeG (p )g m g obtainedafter solving
1 E 5
theSDE.ThehorizontalaxisdenotestheEuclideanmomentum.
where the above tensor charges are renormalized at m = 2 GeV. In the above derivation, it is
assumed that the nucleon is composed of three constituent valence quarks with negligible spin-
dependent many-body interactions. Thesuppression ofthetensor charge agrees qualitatively with
theresultsobtained fromtheextraction fromexperimental data(1.3)and(1.4)[9].
Forthequarkaxialcharge, wehave
5
g = G (0)≃1.43, (3.2)
A 1
3
DS = G1(0)m ≃−0.47, (3.3)
wherewehaveagaincombinedthesolutionoftheSDEwiththeconstituentquarkmodelargument.
For the isovector quark axial charge g , the dynamical axial charge is suppressed compared with
A
thebareone. TheresultofEq. (3.2)isinqualitativeagreement withtheexperimental value(1.2).
The suppression of the quark tensor and isovector axial charges can be explained by the spin
flip of the quark after each emission or absorption of the gluon (see Fig. 4). This mechanism is
consistentwiththeangularmomentumconservationsincethequarksandgluonshavespinonehalf
and one, respectively. By outputting the tensor and axial charges after each iteration of the SDE,
wehaveconfirmedthattheresultconvergesbyoscillating. Thisshowsthatthereversalofthequark
spinispreferred inthegluonemission/absorption, andisthusconsistent withouranalysis.
sqz=+21 sqz=(cid:0)12 sqz=+21 sqz=(cid:0)12 sqz=+21
(cid:16) (cid:17)
1
q sq = 2
g (sg =1)
Figure4: Theschematicpictureofthequarkspinflipwiththegluonemission/absorption.
As shown in Eq. (3.3), the dynamical isoscalar quark axial charge DS is also suppressed
against the bare one, and we see that DS is much more suppressed than g , well below zero.
A
4
QuarktensorandaxialchargeswithintheSchwinger-Dysonformalism NodokaYamanaka
The unquenching diagram of Fig. 1 actually gives the axial anomaly contribution, and this result
suggests that the axial anomaly has a significant effect in the suppression of the isoscalar quark
axialcharge.
4. Summary
In this paper, we have calculated the quark axial and tensor charges in the SD formalism of
the Landau gauge QCD with RG-improved rainbow-ladder truncation. For the quark tensor and
isovector axial charges, our result suggests that the gluon dressing of the vertex suppresses the
charges, and is in qualitative agreement with the experimental data. The isoscalar quark axial
charge receives a larger suppression due to the axial anomaly through the unquenching quark-
loop. Thisunquenching effectmaybelargelyoverestimatedduetothelargeuncertaintyintreating
the IR region. For, the axial anomaly also contributes to the many-body effect via the exchange
interaction. Tostudy theproblem ofthe proton spin quantitatively, wemusttherefore evaluate the
many-body effecttogether withthediscussion ofthispaper.
Acknowledgments
This work is in part supported by the Grant for Scientific Research [Priority Areas “New
Hadrons” (E01:21105006), (C) No.23540306] from the Ministry of Education, Culture, Science
andTechnology (MEXT)ofJapan.
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