A non-perturbative effect of gluons for scalar diquark in the Schwinger-Dyson formalism

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by  Shotaro Imai

A non-perturbative effect of gluons for scalar
4 diquark in the Schwinger-Dyson formalism
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6 ShotaroImai∗,HideoSuganuma

] DepartmentofPhysics,GraduateSchoolofScience,KyotoUniversity,
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Kitashirakawa-oiwake,Sakyo,Kyoto606-8502,Japan
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- E-mail: [email protected]
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Thediquarkhasbeenconsideredtobeimportanteffectivedegreeoffreedominhadronphysics,
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especially for multi-quark physics and the structure of heavy hadronic states. Using the
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v Schwinger-Dyson formalism, we investigate the non-perturbative effect of gluons for a scalar
9 diquarkwithrenormalization-groupimprovedcouplingintheLandaugauge. Here,wetreatthe
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0 scalardiquarkasaneffectivedegreeoffreedomwithapeculiarsize,whilethediquarkisorigi-
3 nallya bound-state-likeobjectof two quarks. Since the diquarkhasa non-zerocolorcharge,it
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1 stillstronglyinteractswithgluons.Weevaluatethegluonicnon-perturbativeeffecttothediquark,
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4 consideringthesizeeffectofthediquark.Weinvestigatethemassfunctionofthediquarkinboth
1 caseswith aconstantbarediquarkmassandtwiceofthe runningquarkself-energy. Itisfound
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v thatthediquark,especiallythesmalldiquark,obtainsalargeeffectivemassbythegluonicdress-
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X ing effect. The scalar diquarkmassseemsto be dynamicallygeneratedby the non-perturbative
r effect,althoughitdoesnothavechiralsymmetryexplicitly.
a
XVInternationalConferenceonHadronSpectroscopy-Hadron2013
4-8November2013
Nara,Japan
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Anon-perturbativeeffectofgluonsforscalardiquarkintheSchwinger-Dysonformalism ShotaroImai
1. Introduction
Recentexperimentshavediscoveredmanyheavyhadronicstatesincludingexoticstates,which
cannot beunderstood asordinary hadrons [1]. Inordertodescribe thestructure oftheirstates, the
diquarkisconsideredasaneffectivedegreeoffreedom. Forthediquark,themostattractivechannel
bytheone-gluon exchange isthecolorandflavoranti-triplet 333¯ andspinsingletwithevenparity
c,f
0+. If the diquark correlation is developed in a hadron such as a heavy baryon (Qqq), this scalar
diquark channel would be favored. The diquark is made by two quarks with gluonic interaction,
and it still strongly interacts with gluons because of its non-zero color charge. The dynamics of
diquark andgluonsmayaffectthestructureofahadron.
In quark-hadron physics, the Schwinger-Dyson (SD) formalism is often used to evaluate the
non-perturbative effect based on QCD[2, 3, 4]. Weapply the SDformalism to the scalar diquark
withitspeculiar size, andinvestigate theeffective diquark mass, which reflectsanon-perturbative
dressing effect by gluons. The diquarks are sometimes treated as point-like object or local boson
fields. However,thediquarkmusthaveaneffectivesize,sinceitisabound-state-like objectinside
ahadron. Therefore, weinvestigate alsothesizeeffectofthediquark foritsdynamics.
2. The Schwinger-Dyson Equation forthe ScalarDiquark
We consider the scalar diquark as an effective degree of freedom with a peculiar size like a
meson,assumingittobeanextendedfundamentalfieldf (x)[5]. Thescalardiquarkinteractswith
gluons sinceithasnon-zero colorchargeandisaffected bynon-perturbative gluonic effects[6].
To evaluate the non-perturbative effect, we take the Schwinger-Dyson (SD) formalism with
the rainbow-ladder truncation with the Higashijima-Miransky approximation a ((p −k )2) ≈
s E E
a (max(p2,k2)) in the Landau gauge. Here, p denotes the Euclidean momentum. We use a
s E E E
renormalization-group(RG)-improved coupling inthecaseofN =3andN =3,
c f
a (p2)= g2(p2E) =11N1c2−p2Nf ln(p2E/1L 2QCD) (p2E ≥ p2IR) , (2.1)
s E 4p  12p 1 (p2 ≤ p2 )
11Nc−2Nf ln(p2IR/L 2QCD) E IR
with an infrared regularization of asimple cut at p ≃640 MeV which leads to ln(p2 /L 2 )=
IR IR QCD
1/2, and theQCDscale parameter L =500 MeV[3,4]. Toinclude the size effect ofdiquark,
QCD
weintroduce asimpleformfactorinthe4DEuclideanspaceas
L 2 2
fL (p2E)=(cid:18)p2 +L 2(cid:19) , (2.2)
E
where the momentum cutoff L corresponds to the inverse of the diquark size R, and we here set
R≡L −1. Thesizeeffectofthediquark canbeincluded inthevertexasa s(p2)→a s(p2)fL (p2).
The SD equation for the scalar diquark with the bare diquark mass mf is diagrammatically
expressed asFig.1andiswrittenby
8 ¥
S 2(p2E)=m2f +p dkEkEa s(kE2)fL (kE2)
Z0
−2a s(p2E)fL (p2E) pEdk kE5 − 2 ¥ dk a s(k2)fL (kE2)kE. (2.3)
p p2E Z0 EkE2 +S 2(kE2) p ZpE E kE2 +S 2(kE2)
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Anon-perturbativeeffectofgluonsforscalardiquarkintheSchwinger-Dysonformalism ShotaroImai
p p p k p p p
Σ = + Σ Σ + Σ
m
φ
p−k k
Figure1: TheSchwinger-Dysonequationforthescalardiquark. Theshadedblobistheself-energyS (p2),
the dashed line denotes the scalar diquark propagator and the curly line the gluon propagator. The last
diagraminRHSispeculiartermofscalartheory.
3. Numerical Result
3.1 TheParametersintheDiquarkTheory
Thebaremassmf andthesizeR(inverseofcutoffL )arefreeparametersofthediquarktheory.
Sincethediquarkisoriginallymadeoftwoquarks,thebarediquarkmasscanbesimplyconsidered
as twice of the quark mass. Inthis paper, weconsider twocases of the bare diquark mass. One is
twice of the constituent quark mass, i.e., mf =600 MeV. The other is twice of the running quark
self-energy, i.e., mf (p2E)=2S q(p2E), where S q(p2E) is determined by the SD equation for single
quark [2, 3, 4]. We also consider two cases of the diquark size R corresponding to the cutoff L .
OneisthetypicalsizeofabaryonR=1fm,i.e.,L =200MeV,whichgivestheupperlimitofthe
size(thelowerlimitofthecutoff). ThesecondisthetypicalsizeofaconstituentquarkR≃0.3fm,
i.e.,L =600MeV,whichgivesthelowerlimitofthesize(theupperlimitofthecutoff).
3.2 TheConstantBareMassCase
We first show in Fig. 2 the case of the constant bare mass mf =600 MeV with dependence
on the diquark size R. The diquark self-energy S (p2E) is always larger than the bare mass mf and
almostconstantexceptforasmallbumpstructureataninfraredregion. Thevalueoftheself-energy
strongly depends onthesizeR,e.g.,the“compactdiquark” withR≃0.3fmhasalargemass.
700 Σ(p2) 1130
] 704 ]
V V
e e
M 703 M
[ [ 1125
) 702 ) 1100
2p 0 500 1000 2p 1122
(Σ m Σ( 1119
φ
0 1000 2000 3000
600
1070
0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000
p [MeV] p [MeV]
(a) R=1fm (i.e.,L =200MeV) (b) R≃0.3fm (i.e.,L =600MeV)
Figure2: Thescalar-diquarkself-energyS (p2)as a functionofthe momentumin the constantbare-mass
case ofmf =600MeV with (a)R=1fm and(b)R≃0.3fm. Inbothcases, there appearsa smallbump
structure,whichisdisplayedinthesmallwindow. Intheleftfigure,theoriginalbaremassmf isplottedfor
comparison.
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Anon-perturbativeeffectofgluonsforscalardiquarkintheSchwinger-Dysonformalism ShotaroImai
3.3 TheRunningBareMassCase
WeshowinFig.3thecaseoftherunningbaremassmf (p2E)=2S q(p2E)withdependenceonR.
Thediquarkself-energy S (p2)alsostronglydependsonthediquarksizeR. Inthelow-momentum
E
region,thebehaviorofS (p2)reflectstherunningpropertyofthebaremass,especiallyintheR=1
E
fmcase,thegluoniceffectseemstobesmall,becauseofS (p2)≈2S (p2). Inthehigh-momentum
E q E
region,thediquarkself-energykeepsalargefinitevalue,whilethebaremassmf (p2E)goestozero.
Thissuggests themassgeneration ofthescalardiquark bythegluonic radiativecorrection [6].
1400
1000
V] 800 V] 1050 Σ(p2)
e e
M 600 M
[ Σ(p2) [ 700

) )
2 400 2
p p
Σ( 200 2Σq(p2) Σ( 350 2Σq(p2)
0 0
0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000
p [MeV] p [MeV]
(a) R=1fm (i.e.,L =200MeV) (b) R≃0.3fm (i.e.,L =600MeV)
Figure 3: The scalar-diquarkself-energyS (p2) as a functionof the momentumin the runningbare-mass
casewith(a)R=1fmand(b)R≃0.3fm. Thediquarkbaremassmf (p2)=2S q(p2)isalsoplottedwith
thedottedlineforcomparison.
4. DynamicalMass Generationwithout ChiralSymmetry
Finally,weconsiderthezerobare-masscaseofdiquarks,mf ≡0. TheresultisshowninFig.4
forthetwocasesofR=1fmandR≃0.3fmonthediquarkeffectivesize. Theself-energy S (p2)
E
isalways finiteandtakes alarge value evenformf ≡0. Themassgeneration mechanism inQCD
is usually considered in the context of spontaneous chiral symmetry breaking. On the other hand,
our scalar-diquark theory is composed by an effective scalar-diquark field f (x) and does not have
thechiralsymmetryexplicitly, although theoriginaldiquark isconstructed bytwoquarks.
WeconsiderthatQCDhasseveraldynamicalmass-generationmechanism,evenwithoutchiral
symmetrybreaking. Forexample,whilethecharmquarkhasnochiralsymmetry, somedifference
seems to appear between current and constituent masses for charm quarks: the current mass is
m ≃ 1.2 GeVat the renormalization point m =2GeV [1], and the constituent charm quark mass
c
isM ≃1.6GeVinthequarkmodel. Furthermore thegluonismoredrastic case. Whilethegluon
c
massiszeroinperturbationQCD,thenon-perturbativeeffectoftheself-interactionofgluonsseems
to generate a large effective mass of 0.6 GeV [7], and the lowest glueball mass is about 1.6GeV.
Although the heavy quark and gluons donot have the chiral symmetry, they obtain large effective
massbynon-perturbativeeffects. Inthissense,thescalardiquarkmasscanbealsogeneratedbythe
gluonic effect. Thus, weconsider thatthemassgeneration mechanism isageneral property ofthe
stronginteractingtheory,andonetypicalmassgenerationisgivenbyspontaneouschiral-symmetry
breaking, whichisalsocaused bythenon-perturbative interaction.
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Anon-perturbativeeffectofgluonsforscalardiquarkintheSchwinger-Dysonformalism ShotaroImai
380 980
V] 360 V] 960
e e
M M
[ 340 [ 940

) )
2 2
p p
Σ( 320 Σ( 920
300 900
0 1000 2000 3000 0 1000 2000 3000
p [MeV] p [MeV]
(a) R=1fm (i.e.,L =200MeV) (b) R≃0.3fm (i.e.,L =600MeV)
Figure 4: The scalar-diquark self-energy S (p2) as a function of the momentum in the massless case of
mf =0. Theself-energyS (p2)isfiniteinbothcases.
5. Summary
We have investigated the gluonic dressing effect to the scalar diquark, considering the size
effectofthediquarkinahadron. Thenon-perturbative effectisevaluated intheSchwinger-Dyson
(SD)formalismintheLandaugauge. Sincethediquarkislocatedinsideahadron,thediquarksize
R must be smaller than the hadron (∼1fm) and larger than the constituent quark (∼0.3 fm). We
have considered two cases of the constant bare mass mf =600 MeV and the running bare mass
mf (p2E)=2S q(p2E). Thediquark self-energy strongly depends on the size R=L −1 in both cases,
especially thesmalldiquark (R≃0.3fm,i.e.,L =600MeV)obtains alargeeffectivemassbythe
gluonic dressing effect.
Wefindthattheeffectivediquarkmassisfiniteandlargeevenforthezerobare-masscase,and
thevaluestronglydependsonthesizeR,whichisanexampleofdynamicalmassgenerationbythe
gluonic dressing effect,withoutchiralsymmetrybreaking.
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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