Hyperfine Structure of Ground-State Nucleon in Chiral Quark Model

Duojie JIA@Northwest Normal University

The 7th International Symposium on Chiral Symmetry in Hadrons and Nuclei (Behang Univ. Beijing,Sept.,27-30,Oct. 2013 ) 1

Suported partially by NNSF of China (No. 10965005) NNSF of China (No. 11265014)

Collaborated with RuiBin Wan, WenBo Dang, YuBin Dong; Thanks for discussions with A. Jarah, J. He, X. Liu 

Outline

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Motivation Quark masses in QCD and models Mass role and pion role in models Chromo-magnetic interaction in ChQM Hyperfine splitting of nucleon spectrums Summary

QCD is very different at long and short distances ( ＜ ΛQCD): q Condensing or melting, depending the scale(momentum) at which you see it!

Motivation In explaining neclear force and hadron

structures, QCD is still challenging due to its complex nonpertubative nature ：

（ 1 ） gluon/quark condensate vacuum（ 2 ） absence of confining dimension (by itself)（ 3 ） Complicated phases

1/30R r A

CondensateQCD vacuum

3

3

3

(0.25 )

(0.21 )s

uu dd GeV

m ss GeV

q

2 4G 0.005s GeV

2G ,s qq

Condensate

Homogeneousat long distance

Inhomogeneous at short distance

Motivation

QCD(continuum): Nontrivial vacuum, Lack of unified degrees of

freedom at long and short distances( ＜ ΛQCD) ; Few parameters : mi(current masses) ; g(quark-gluon coupling) : running; μ (energy scale )

4

Hard to model the hadrons

Besides Lattice QCD, ChPT, with appro. global symmetry (ChSymmetry) of QCD, and spontaneously broken, gives the pion-octet pseudoscalar(pseudo-Nambu-Goldstone), a Chiral Lagrangian (pion octet +other SU(3)V hadron multiplets )

• Compute hadron observables at low E• fix the light quark masses by extraploting Lattice QCD

Motivation First-principle QCD(Lattice): The parameters :

mi(current masses) ; g(quark-gluon coupling) : running a (the lattice spacing )

5

What does a quark mass mean when free quarks don’t exist? adjust the bare quark masses in doing a lattice calculation to match physical hadron properties. For the continuum limit, the bare quark masses flow along the renormalization group, ---- extract a renormalized quark mass (asymptotic freedom + a renormalization scheme).

In real world, isospin broken by the non-degeneracy of u and d, and by electromagnetism, both comparable order to the hadron spectrumts:

u

d

-The light quark masses, important parameter for hadron physics and nuclear physics,-is of interest to determine them, and to see their effects

-The light quark masses, important parameter for hadron physics and nuclear physics,-is of interest to determine them, and to see their effects

Quark masses in QCD and models

Lattice prediciton: mud=3.5MeV,ms=95MeV[Budapest–Marseille–Wuppertal Collaboration / PLB 701 (2011) 265–268]

The precision below 2% level; ms/mud = 27.53(20)(08), which

is scheme independent( better than 1%).

Quark masses in QCD and models

Quark models use the concept of constituent mass, not well-defined in QCD, model-dependent.

It used in Chiral quark model(ChQM). [A.Manohar, H. Georgi, NPB234(1984)189]

may come from chiral rotation [P. Simic, PRL. 55, 40--43 1985]

Quark mass varys, depending on models

For nonrelativistic QM(NRp), mu=md=0.2-0.3,ms=0.5GeV

For RQM, quite smaller, e.g.PChQM, mu=md=7MeV,ms=175MeV

bag model : mu=md ～0,ms=300MeV

Quark masses inbag models

In the bag model,With degrees of freedom (quark,gluon):Masses:

mud=0;ms=0.3GeV

Mass role and pion role in models

Quark current

Diag-gluon

[ ][ ]QCDZ dG dqdq 4 1exp ( )

2a ai d xq i m q TrG G J G

( / 2)a aJ gq q

†q ( ) ,q ( )U UL L R RV x q V x q

† †( )[ ]( )U U U UL R L Rq V q V i m V q Vq

55[ - ]U Uq i i V i A mU q

UUQiU aa )1(2

1)1(

2

1]exp[ 555

5

The pion(NG particles) entersas requiring local Ch.SymmetryIn the effective field theory of Quark-meson----Chiral rotation

† †1{ , }

2V A V V V V

Why QM works?

Mahohar-Georgi model in ChQT: 2 scales occurs(2 phases)

(250MeV)ΛQCD ＜ Q ＜ Λχ (1GeV),

Confining Mixing Free

Strong coupling (s) weak due to the presence of constituent mass

10

5

2†

q[ ig ]

1( )

4 2

MGA

F

i g G iV A M q

fTr U U TrG G

L

2U V 5( ) ( )D U i V A

/ 2a aG G Constitiuent quark

Constituent Mass(soft mass), corresponding to CS breaking .

Gluons

2 350qq

QM g MeV

M

invarinat under chiral SU(3)

L×SU(3)R

Non-renormalizable terms suppressed by

MGL( / )numberf

22U=V = exp(i /f)π τ

Quark mass role in Baryons

The simplest fit for baryon masses:

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Fine fit, so why RQM? (1) More constraints on

models(including ChSymmetry)(2) Less parameters for spin-interaction

The NRp estimate for baryon masses:[PRD12(1975)147]

1 2 3 ' i j

i j i j

M m m m Am m

mu=363 ms=583

0

2 2

1 1

2 1 1 4( )

3 3

ii i P

i ji j s

i j i j i j i j

M M m am m

cQQ b d

m m m m m m

mu=300mP=336

2' ( / ) 50uA m MeV

1-4%

Smaller

Quark mass in bag pictures

The MIT bag-RQM, degrees (quark and/or gluons), Confinement put in by Bag boundary condition/effective mass

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[ ( ( )) ]MIT q i m r q B L

0 ,( )

,

m r Rm r

M r R

m0

00

01

( )( )

ˆ( )

iEtljm F C

E mj kr

Eq N x y e

E mi j kr r

E

2 2 1/ 20 0( , ) [ ( / ) ]M E m R m x R

13 00 3 34 2.04 4

| |3 3Bag

M n d xT n nE R B n B RR

Mass

scale

300MeV2-5MeV

The consititent mass( ～ 1/R) mainlyfrom BC while the current mass contributes

a few MeV: A mechanism for mass splitting

2 3

,

4

3

4[ ( 6) ( 1) 3 ( 1)]

3

ija a cM i j i jBag

a j j i j

ijc

E g d xR

n n S S I IR

B B

The MIT bag prediction with chromo-electric and magnetic interation:

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Reasonable except for pion

mud=0

Allowed mass splitting for nucleons:1) Kinetic energy splitting

i=u,d,s2) Chromo-magnetic splitting

(i,j=u,d,s)

2 2 1/ 20( , ) [ ( / ) ]i iM E m R m x R

4[ ( 6) ( 1) 3 ( 1)]

3ij

M cE n n S S I IR

Mass splitting in bag pictures

Data:Δmud=2.5MeV, Δmsu=100MeVWhen mq changes so do the kineticand chromo-magnetic energies slightly.

Mass role in ChQM

The consti. mass varying Δmi and the EM effects breaks the flavor SU(3) mainly;

The mass varying dominates for p-n spliting, in ground states

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05[ ( ( )) ]ChQM

i i c i

U

q i m S x U V q

L

L

2† † † 2

2

1[ ] [ , ] ,

4 32U f

tr U U tr U U U Ue

L

20

0

( ) /

( ) / , . .,

( ) tanh(( / ) ),

c

n

S x S r a

V r r e g

r r d

a=0.3/0.2;L=0.75/0.2; M=0.3;S0=0;kappa=-1; d=1.0;alpha0=0.8;

Vc

Chromo-magnetic interaction in ChQM

The plot of effective mass for n

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The chromo-magnetic interaction is similar to that of bag models

The mass-term [mi+S(r)U5] contributes to (1)Confinement through the S-potential (2)Quark wavefuntion and pion

configurations(3) Magnetic moment via quark magnetic

moment;(4) Hadron mass and its splitting

2 3

,

7

0 0

4

3

2 ( , ) ( , ),

( , ) ' ( '), ( ) ' ( ')

a a cM c i j i j i j ijBag

a j j i j

ij i i

r s si i i

E g d x I

drI m r m r

r

m r dr r m dr r

B B

2'( ') ( ') ( ')

3

i j i j

s rr G r F r N

( )

ˆ( )iEt

ljm F C

G rNq y e

iF r rr

0.0740

Chromo-magnetic interaction in ChQM

The radial Eq. of Motion of a quark: With Y determined by Y equations

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, 1/ 2

, 1/ 2

( )

( )l M

l M

G rNq

F rr

cos

cos

q

q

dGG LE LM Y LVc F

dx xdF

F LE LM Y LVc Gdx x

2 2 2

42 2 2

2

2( )cos ( )

4( ) (1 cos( )) ,

x x

c

H N dx FG GF GF L M S Y G Fx

L BLV G F x Y E

LN

2 22

2 2

2 2 22 4

2

sin ( ) 2sin(2 ) sin ( )1 1

( ) ( ) ( )sin

4

xx x

Y Y YY Y

x x x

LN M x G Fe L B Y

x

/ , .,xY dY dx etc

The Y profile determined by a dynamics, eg., the Coupled Skyme lagrangian here, it can be set by comparing with ChPT

Quark configurationin ChQM Length scale L=3.75GeV-1

Chromo-magnetic interaction in ChQM D. Jia, L.Yu,R. Wan, arXiv:1308.0700v1

Hyperfine splitting in nucleon masses

⊿m (MeV) u/d mass

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0.360 300/302.5

0.723 300/305.0

1.011 300/307.0

1.293 300/309.0

1.436 300/310.0

1.723 300/312.0

Hyperfine splitting in nucleon masses The CSB explained by NJL model, with quark pair condensation The lator is fixed by the gap equation

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1.293n pm m

md (MeV)

ΔM=Mn-Mp (MeV)

mu=300

309=md

Summary

•The hyperfine structure of ground-state nucleon is studied in chiral quark model with nonlinear pion interaction in which quarks move in the potential of Coulomb-like plus linear form.

•The mass splitting of ground-state nucleon is given by taking into account the colour magnetic interaction between quarks and found to be in agreement with data. •The connection of the model with the bag models is discussed

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Thanks !!!