ECT, March 2003

THE MESON SPECTRUM

by

Norbert Ligterink

Department of Physics and Astronomy

University of Pittburgh

Pittsburgh

NJLQM

InstantonModelBS

PS−exchModel

BetterModel

my

Beautiful Models Everywhere!!!

model

Quark Models Comparison

results

excited states

mesonsbaryonsstructure

S.S L.S

relativistickinetic

interactionmodel

self−energyquark +

chiralsymmetry

(Capstick,...)

Instanton BSE(Metsch,....)

Coulomb BCS (Swanson,....)

CQM

PS exchange(Plessas,...)

Lattice

π

SDE

MBX

MBX

MB(X)

No

partly

yes

meson

meson loops

partly

No

Nosure(?)quark

quark yes π

point form

yes

yes

No

# parameters

MBX

MXπquarkyes

yes (inst) partly partly No

some kin.

M(BX)

fixedspin

4

2

2

4

20

7

(roughly)

(Morningstar,...)

(Roberts,....)

issues

QCD in the Coulomb gauge

(Christ&Lee, Zwanziger, Swanson, Szczepaniak, Cotanch, ...)

• Separation of dynamics (massive gluons) and confining forces

(“scalar” potential)

• Massive quarks through chiral symmetry breaking a la Bogoliubov-

Valatin (minimizing the one-body energy).

• Perturbative expansion in massive quarks and gluons.

Coulomb ConfiningTransverse

The potential (Swanson and Szczepaniak)

Correlated VACUUM:

Vacuum energy. . . .

theory

to be minimized

dressing the potential

theory

0 0.5 1 1.5Momentum [GeV]

0

50

100

Pote

ntia

l [G

ev]

The result: — q2V (q) [GeV].

theory

The result in configuration space: — V (R).

issues

Why non-relativistic approach fails

E =3µ

mq+ 2mq − α

√2πµ + σ

√π

2µ

where µ is the wave function scale, e.g., exp{−r2/(2µ)}.

Bounded by either µ or mq or σ/√

µ.

u (m(p),p)u(m(p),p)Σs s s

Quark lines

Gap equation

dressedbare

Vacuum Energy:

(minimizing the vacuum energy by varying m(p))

theory

theory

0 0.5 1 1.5Momentum [GeV]

0

50

100

Pote

ntia

l [G

ev]

Dyn

amic

al M

ass

[MeV

]

The results: — q2V (q) [GeV] and — m(q) [MeV].

ψ λψa

ψ λψa

ψ λψa

ψ λψa

ψ λψa

Ψ∗ Ψ

(SELFENERGY (2x))

Ψ∗ Ψ ΨΨ∗

(TDA)

*

*

*

(RPA(chiral symmetry))

ψ λψa

Mesons

theory

issues

BEYOND (what else, what next)

• More complicated vacuum content: ggg, g4, qq, etc.

• Higher order vacuum diagrams. The transverse gluon?

• Short range interactions, spin-spin and renormalization (1S states)

• Coupling to decay channels: widths and shifts (data comparison?)

• Flavor mixing

WIDTHS and SHIFTS

W = <decay state|H|hadron>

threshold

tail

shift shiftstat

width

shift

W(m)W(E)dE

E−m−

*width is proportional to the coupling to the decay state*shift depends on width, tail, threshold, and mass.

coupling between the continuum and the hadron determines the width

(Comparing to data is not that straightforward)

issues

u

u

d

d

u,d,s

u,d,s

m+x

m+x

M+x

x

x

x

x

x

x

xm

M

3x

(2M+m)/3

(iso−scalars)

m

π π0 + −π

?

{ρ, ω, φ }{π, η, η }

likewise:

* How do they stay degenerate?

* How do they split?Flavor Mixing

H =f

remain degenerate

η,η ρ,ω

π

issues:

D( )θ φ,1/2

θ

φ

, π+φ)π−θD(1/2

−invariant

internal rotation

spin

z−axis

fourstates

momentum

D (x) = <jm’|R(x)|jm>m’m

j

Helicity basis:

calculation

calculation

2⊗ 2 spin states, 4 cases (J(PC)):

J(−[J],[J]) : (Singlet) = ↑↓ + ↓↑J(−[J],−[J]) : (Triplet) = ↑↑ ⊗Ylm

J([J],[J]) : (Triplet2) = ↑↑ Yl′(m−1)+ ↓↓ Yl(m+1)

The last set corresponds the “S-D” mixing throughtensor interaction.

calculation

PURE STATESJ−[J][J] [1JJ , J ≥ 0]:

KJ(p, k) = VJ(1+m(p)

E(p)

m(k)

E(k))+

(VJ−1

J

2J + 1+ VJ+1

J + 1

2J + 1

)p

E(p)

k

E(k)

0++:

K(p, k) = V0p

E(p)

k

E(k)+ V1(1 +

m(p)

E(p)

m(k)

E(k))

J−[J]−[J] [3JJ , J ≥ 1]:

KJ(p, k) = VJ(1+m(p)

E(p)

m(k)

E(k))+

(VJ−1

J + 1

2J + 1+ VJ+1

J

2J + 1

)p

E(p)

k

E(k)

calculation

MIXED STATESJ [J][J] [3(J − 1)J ,3 (J + 1)J , J ≥ 1]:

K11(p, k) = VJp

E(p)

k

E(k)+(VJ−1

J

2J + 1+ VJ+1

J + 1

2J + 1

)(1 +

m(p)

E(p)

m(k)

E(k))

K22(p, k) = VJp

E(p)

k

E(k)+(VJ−1

J + 1

2J + 1+ VJ+1

J

2J + 1

)(1 +

m(p)

E(p)

m(k)

E(k))

K12(p, k) =(VJ−1 − VJ+1

) √J(J + 1)

2J + 1

(m(p)

E(p)+

m(k)

E(k)

)

calculationResolving the implicit spin structure:

u†s(p + q/2)ut(p− q/2) =(E′ + m′)(E + m) + p2 − q2/4√

4EE′(E′ + m′)(E + m)δst

+χ∗si(~p × ~q) · ~σχt

2√

4EE′(E′ + m′)(E + m)

q � p � E:

V ∼ |u†sut|2 ∼ δst

(1−

q2

8M2

)+

χ∗si(~p × ~q) · ~σχt

4M2

p � q � E (hard gluon):

V ∼ |u†sut|2 ∼ δst

(1−

5q2

8M2

)+

χ∗si(~p × ~q) · ~σχt

4M2

0(−+) 0(++) 1(−−) 1(+−) 1(++) 2(−−) 2(−+) 2(++) 3(−−) 3(+−) 3(++) 4(−−) 4(−+) 4(++) 5(−−) 5(+−) 5(++)

0.5

1.0

2.5

2.0

1.5

LIGHT MESON SPECTRUM

0(−+) 0(++) 1(−−) 1(+−) 1(++) 2(−−) 2(−+) 2(++) 3(−−) 3(+−) 3(++) 4(−−) 4(−+) 4(++) 5(−−) 5(+−) 5(++)

0.5

1.0

2.5

2.0

1.5

LIGHT MESON SPECTRUM (isovector)

0(−+) 0(++) 1(−−) 1(+−) 1(++) 2(−−) 2(−+) 2(++) 3(−−) 3(+−) 3(++) 4(−−) 4(−+) 4(++) 5(−−) 5(+−) 5(++)

0.5

1.0

2.5

2.0

1.5

LIGHT MESON SPECTRUM (isovector)(+Anisovich et al (2002) Crystal Barrel data)

0(-+) 0(++) 1(--) 1(+-) 1(++) 2(--) 2(-+) 2(++) 3(--) 3(+-) 3(++) 4(--) 4(-+) 4(++) 5(--)0

1

2

3

Mas

s [G

eV]

πρ

<r> = 5.9 GeV-1

<r> = 6.0 GeV-1

The radii√〈r2〉 of the light mesons.

(Through Gaussian Fit Fourier Transform.)

0(−+) 0(++) 1(−−) 1(+−) 1(++) 2(−−) 2(−+) 2(++) 3(−−) 3(+−) 3(++) 4(−−) 4(−+) 4(++) 5(−−) 5(+−) 5(++)

3

3.5

4

4.5

5

CC Spectrum

Mass fit

0(−+) 0(++) 1(−−) 1(+−) 1(++) 2(−−) 2(−+) 2(++) 3(−−) 3(+−) 3(++) 4(−−) 4(−+) 4(++) 5(−−) 5(+−) 5(++)

confirmed Jcalculation

BB Spectrum

mass fit unconfirmed J9.5

10

10.5

11

11.5

CONCLUSIONS

• Coulomb gauge: when a potential makes sense

• The chiral pion

• Full spectrum, singular approach, minimal parametric fitting

• Meson properties available

• Meson interaction and decay under investigation