Study on the Mass Difference btw and using a Rel. 2B Model 2011. 08. 25 Jin-Hee Yoon ( ) Dept. of Physics, Inha University collaboration with C.Y.Wong

Study on the Mass Difference Study on the Mass Difference btw btw and and using a Rel. 2B using a Rel. 2B

ModelModel2011. 08. 25

Jin-Hee Yoon ( )

Dept. of Physics, Inha University

collaboration with C.Y.Wong and H. Crater

08/25/2011 APFB 2011 Inha Nuclear Theory Group

Introduction m = 0.140 GeV, m = 0.775 GeV

Why is the pion mass so small? Why is the pion mass not zero? Chiral symmetry breaking

What we will do? Trace the origin of the mass of using the 2B

relativistic potential model.


2-Body Constraint Dynamics

Two free spinless particles with the mass-shell constraint

→ removes relative energy & time0mpH 2

i2i

0i

0 )p,p(x,mp

MpH

21i2i

2i

2i

2ii

P. van Alstine et al., J. Math. Phys.23, 1997 (1982)

Two free spin-half particles with the generalized mass-shell constraint

→ potential depends on the space- like separation only & P ⊥ p


2-Body Constraint Dynamics

Pauli reduction + scale transformation

16-comp. Dirac Eq. → 4-comp. rel. Schrodinger Eq.

P. van Alstine et al., J. Math. Phys.23, 1997 (1982)

wmmm and

2wmmwε with mε(w)b where

(w)bS(r))A(r),,p,σ,(σΦpw

HεHεH

21w

22

21

2

w2w

2w

2

221w

22211


Why Rel.? Even within the boundary, their kinetic

energy can be larger than the quark masses. Already tested in e+e- binding system (QED)

[Todorov, PRD3(1971)] can be applied to two quark system( and )

(QCD) Can treat spin-dep. terms naturally


What's Good?

TBDE : 2 fermion basis

16-component dynamics 4-component Particles interacts through scalar and

vector interactions. Leads to simple Schrodinger-type equation. Spin-dependence is determined naturally.


QCD Potentials Common non-relativistic static quark

potential

Dominant Coulomb-like + confinement But asymptotic freedom is missing

Richardson[Phys. Lett. 82B,272(1979)]

FT

brr

αV(r) c

)/Λqln(1q1

2716π(q)V 222

~

27rr) f(Λ 8π

27r 8πV(r)

2

[Eichten et al., PRD 21 (1980)]


QCD Potentials Richardson potential in coord. space

For : asymptotic freedom

For : confinement

We will use fitting param.

27rr) f(Λ 8π

27r8πV(r)

2

r)ln(1-r)f( 0,r

1 r)f( ,r

)/( 22

2

r)eln(27r 16

27r8πV(r)

BK


QCD Potentials

Scalar Pot.

Vector Pot.

)/( 22

2

r)eln(27r 16

27r8πV(r)

BK

27r8πS(r)

2

r 4πee

r)eln(27r 16A(r) 21

22

)/(BK


What's Good again? TBDE : 2 fermion basis

16-component dynamics 4-component Particles interacts through scalar and vector

interactions. Yields simple Schrodinger-type equation. Spin-dependence is determined naturally.

No cutoff parameter No singularity


Formulation

SOX21SOD21

T2121SOT2121

SS21SO21D2

w2

ww

ΦLΦL

)Φ-r rΦL r r

ΦΦLΦAA2εSS2mΦ

)()(

ˆˆ3()(ˆˆ)(

i

central potentials + darwin + SO + SS + Tensor + etc.

wΦ

SOD & SOX =0

when m1=m2


Formulation


Formulation

SOD=0 and SOX=0 when m1=m2

For singlet state with the same mass, no SO, SOT contribution ←

Terms of ( D, SS, T ) altogether vanishes.

H=p2 + 2mwS + S2 + 2wA - A2

0S2LσσL 21


Formulation

For (S-state),

For (mixture of S & D-state),


FormulationOnce we find b2,

Invariant mass 22

221

2 mbmbw


MESON Spectra32 mesons


MESON SpectraL 0.4218 GeV

B 0.05081

K 4.198

mu 0.0557 GeV

md 0.0553 GeV

ms 0.2499 GeV

mc 1.476 GeV

mb 4.844 GeV

All 32 mesons


Wave Functions()X


Individual Contribution()terms magnitude Partial

sumTotal

<d2/dr2> 0.8508<2mwS>

<S2><2wA><-A2>

0.01030.0942-0.0598-0.4279

-0.3832

-0.8475<D>

<-3SS>-3.8043.340

-0.4643

b2 0.0033

W = 0.159 GeV-1


Individual Contribution()

W = 0.792 GeV-1


Summary Using Dirac’s rel. constraints, TBDE

successfully leads to the SR-type Eq. With Coulomb-type + linear potential,

By fitting to 32 meson mass spectra, determine 3 potential parameters and 6 quark masses

Small quark masses : mu=0.0557 GeV, md=0.0553 GeV

Non-singular (well-beaved) rel. WF is obtained. At small r, S-wave is proportional to D-wave.


Summary Can reproduce the huge mass split of In pion mass

Still large contributions from Darwin and Spin-Spin terms (3~4 GeV compared to 0.16 GeV)

But cancelled each other, remained ~20% Balanced with kinetic term resulting small pion mass

This Model inherits Chiral Symmetry Braking.

Thank you for your attention!

Backup slides


Overview of QQ Potential(1) Pure Coulomb :

BE=-0.0148 GeV for color-singlet=-0.0129 GeV for color-triplet(no convergence)

+ Log factor :

BE=-0.0124 GeV for color-singlet=-0.0122 GeV for color-triplet

+ Screening :

BE=-0.0124 GeV for color-singletNo bound state for color-triplet

0 and 2.0 S

rA

0 and 1ln

21

2.0

222

S

rer

A

0 and 1ln

21

2.0

222

S

rer

eAr


Overview of QQ Potential(2) + String tension(with no spin-spin

interaction)

When b=0.17 BE=-0.3547 GeVWhen b=0.2 BE=-0.5257 GeVToo much sensitive to parameters!

brS

rer

eAr

and 1ln

21

2.0

222


QQ Potential Modified Richardson Potential

2712

3211

4

f

s

n

222

1ln213

4

rer

eAr

s

rS 2

278

and

Parameters : m, And mass=m(T)A : color-Coulomb interaction with the screeningS : linear interaction for confinement


Work on process To solve the S-eq. numerically, We introduce basis functions

n(r)=Nnrlexp(-n2r2/2) Ylm

n(r)=Nnrlexp(-r/n) Ylm

n(r)=Nnrlexp(-r/√n) Ylm

… None of the above is orthogonal. We can calculate <p2> analytically, but all the other

terms has to be done numerically. The solution is used as an input again → need an

iteration Basis ftns. depend on the choice of quite

sensitively and therefore on the choice of the range of r.


Future Work Extends this potential to non-zero

temperature. Find the dissociation temperature and cross

section of a heavy quarkonium in QGP. Especially on Jto explain its suppression

OR enhancement. And more…

Documents

Study on the Mass Difference btw and using a Rel. 2B Model 2011. 08. 25 Jin-Hee Yoon ( ) Dept. of Physics, Inha University collaboration with C.Y.Wong