Quark 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. References [1] C.A.Aidala,S.D.Bass,D.Hasch,andG.K.Mallot,Thespinstructureofthenucleon,Rev.Mod. Phys.85(2013)655[arXiv:1209.2803]. [2] N.Yamanaka,AnalysisoftheElectricDipoleMomentintheR-parityViolatingSupersymmetric StandardModel,Springer,Berlin,2014. [3] M.G.Alekseevetal.(COMPASSCollaboration),Quarkhelicitydistributionsfromlongitudinalspin asymmetriesinmuon-protonandmuon-deuteronscattering,Phys.Lett.B693(2010)227. [4] B.Plasteretal.(UCNACollaboration),Measurementoftheneutronb -asymmetryparameterA with 0 ultracoldneutrons,Phys.Rev.C86(2012)055501[arXiv:1207.5887]. 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