THE MESON SPECTRUMnorbertl/talks/ect.pdfshift stat shift width shift W(m) W(E)dE E-m-*width is...
Transcript of THE MESON SPECTRUMnorbertl/talks/ect.pdfshift stat shift width shift W(m) W(E)dE E-m-*width is...
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