Transcript Theoretical description of the charmonium structure in QCD

Theoretical description of the
charmonium structure in QCD
Gabi Hoffmeister
06.12.2007
Summary
1. Introduction
2. Charmonium spectroscopy and theoretical
potential models
3. Transitions and decays of cc
4. New states above the DD-threshold
5. Conclusion
2
1. Introduction
Charmonium production

Color-suppressed b  c decay

Predominantly from B-meson decays

e+e- annihilation/Initial State Radiation (ISR)

e+e- collision below nominal cm energy

JPC = 1--

Double charmonium production

Typically one J/Y or Y, plus second cc state

Two-photon production

Access to C = +1 states

pp annihilation

All quantum numbers available
J/Y, Y(2S)
hc,
K
,
cc,...
0
K ,K*, Kresonances…
Untagged gg:
Charmonium states with
JPC = 0+, 2+
J = 0,2
J=1
3
1. Introduction
History of discovered charmonium states

1974: first charmonium state J/Y with mJ/Y = 3096 MeV discovered (SLAC: e+e→ Y → e+e-, m+m-, hadrons and BNL: p + Be → J → e+e- + X)

1974: discovery of Ψ´ (excited 3S1 state) with mΨ´ = 3.686 GeV and Γ ≤ 2.7 MeV
at SLAC

Studying of radiative decays of Ψ´: BR (Ψ´ → J/Ψ + p- + p+) = 0.32
BR (Ψ´ → J/Ψ → neutrals) = 0.25
No other narrow resonances found from reactions e+e- → hadrons


1976: cc,1,2,3 (triplet states 3P0,1,2) discovered from radiative decays of Y´→ g
cc,J

1980: discovery of 1S0 singlet hc with mass m = 2.98 GeV in decay Y´→ g
hc
1982: hc´ (excited state of hc) seen at Crystall Ball (SPEAR) with mhc = 3594
MeV


1977: Discovery of upsilon meson U (bottonium bb with JPC = 1--) at Fermi Lab
with mU ≈ 9.46 GeV via p-Cu interaction again with very narrow width ~52 keV

Many excited states of the U like in case of J/Ψ (similar energy levels)

Bound state tt non observed: top-quark decays before building a bound state
(t → W+ + b)
4
2. Charmonium spectroscopy and
theoretical potential models
Y´
ηc‘
hc
cc
J/Ψ
ηc
Charmonia:
• Singlet S-states (spin 0):
• Triplet S-states (spin 1):
hc, hc´
J/Y, Y´,Y´´,…
Singlet P-states (spin 0): hc
Triplet p-states (spin 1): c1,2,3
5
2. Charmonium spectroscopy and
theoretical potential models
Charmonium states
Experimental data can be used to compare results to the
expected values of different theoretical potential models
www.e18.physik.tu-muenchen.de/teaching/struktur-dynamik-hadronen/ charmonium_1.pdf –
6
2. Charmonium spectroscopy and
theoretical potential models



c,b are heavy quarks
can be treated in nonrelativistic approximations
(Schrödinger equation + static potential) because relativistic corrections are small
At small distances: one-gluon exchange dominates (asymptotic freedom): V ~ 1/r
At large distances confining potential:
Coulomb + linear potential: V  
„Cornell-Potential“
4 s
 kr
3 r
Vector
part Vv
Scalar
part Vs
=> Fits to the data show that Vv is small
Contributions to the cc-potential:
k = (2pa´) -1 ≈ 0.18 GeV² is the string tension (energy density of qq pair in
string model of hadrons) with typical slope a´= 1 GeV-² of a hadronic Regge
trajectory
7
2. Charmonium spectroscopy and
theoretical potential models
Fine structur splitting (spin orbit interaction):
Vs: scalar part from confining term
Vv: vector part from one-gluon (vector boson) exchange
Spin spin (splitting of singlet and triplet states):
→ no contribution from Vs
Tensor term:
By computating the various expectation values one obtains mass splitting relations:

J2 4
 
LS 

2
2  Sten  
 
12 L  S
1
(3P2)
-1
(3P1)
-2
(3P0)
2

 

 6 L  S  4 S 2 L2
2(2 L  1)(2 L  3)

2 S2 3
 
S1  S 2 

4
¼ (3S1)
-¾ (1S0)
-1/5 (3P2)

0
(3P1)
-2
(1P0)
8
2. Charmonium spectroscopy and
theoretical potential models
The resulting mass relations for the triplet are:
1
m( 3P2 )  m  mso  mten
m(3P1 )  m  mso  mten m(0P1 )  m  2mso  2mten
5
→ testing if long-range potential transforms as a 4-scalar Vs or a 4-vector Vv considering a
modification of the Cornell model (V = br - a/r):
Vs  (1   )br,
Vv  br 
a
r
with
0   1
Inserting this potential model and setting  
m(3P2 )  m(3P1 ) 2 16  19  5
R


m(3P1 )  m(3P0 ) 5 8  5  
b r 1
a r 3
0.66
(hb(1P))
0.70
(hb(2P))
experimental data
on U-P-wave
for vector confinement (x ≈ 1) formula in accord with experimental data only for  ≈ 0,
whereas scalar confinement (x ≈ 0) larger range 0.4 ≤  ≤ 1.0 in accord with exp. values
Conclusion: confinement produced by a long-range 4-scalar interaction
9
3. Transitions and decays of cc

Annihilation:

Generally suppressed for bound state

Decay to leptons is a clean experimental signal

Strong interaction:

Dominant above ~3.72 GeV (D mesons)

Suppressed below this mass threshold

Radiative transition:

EM radiative transition emitting photon

Emission of gluons producing light quarks
Features:

Suppression of strong decays leads to (relatively) long lifetimes, narrow widths

Radiative decays are competitive; often most accessible transitions

Selection rules:
 Conservation of J
 Conservation of P,C in strong and electromagnetic decays
10
3. Transitions and decays of cc
All quarkonia are unstable and decay through: 1) annihilation processes and
2) radiative transitions
1) Annihilation processes (electromagnetic and hadronic decays):
4 2
2
for a bound state with wavefunction Yn(x)
( S0 )  v    n (0) 

(
0
)
n
2
in electromagnetic decay
m

2
( em) 1
Including QCD radiative corrections and substituting the electric charge by
ec = (2/3)e for the charm-quark charge and a color factor of 3:
192 2 n (0) 
 s (mc ) 
cc  (n S0 ) 
1

3
.
4
2

 
81 mc

2
cc  gg (n1S0 ) 
8 (mc ) n (0) 
 s (mc ) 
1

10
.
6
2

 
3  mc

f
g
c
2
2
s
c
c
el.mag.
decay
1
c
c
*
f
hadronic
decay
gg or
gg
c
ggg or
ggg
for 3S1 state
11
3. Transitions and decays of cc

Decays from the 3S1-system with 3 final particles or a lepton pair including QCD radiative
corrections :
2
64 2 n (0)  16  s ( mc ) 
3
electromagnetic decay
cc ll ( n S1 ) 
1
2


9  M cc
3



40( 2  9) s3 ( mc ) n (0) 
 s (mc ) 
cc 3 g ( n S1 ) 
1

4
.
9


81 mc2



2
3
1024( 2  9) 3 n (0) 
 s (mc ) 
cc 3 ( n S1 ) 
1

12
.
6


2187 mc2



hadronic decay
2
3
el.mag. decay
128( 2  9) s2  (0) 
 s (mc ) 
cc  gg ( n S1 ) 
1

0
.
9


81 mc2



2
3
Problems:
- factor lΨn(0)l² comes from non relativistic approximation, can be modified by
relativistic corrections
- second order terms O(as²) could play an important role
12
3. Transitions and decays of cc

hadronic transitions: J/Ψ, Ψ´→ PV, PP, VV
(P: pseudoscalar and V: vector mesons)
with quark flavor basis:
el.magn. J/Y and Y´ decays into
meson pairs
G-parity and isospin
violating transitions with BR
~ 10-4 - 10-3, supressed by
factor ~10-2 - 10-1 compared
to G-parity and isospin
allowed J/Ψ decays
mixing mechanism for charmonium
decays into meson pairs
Charmonium state possesses Fock
components of light quarks, can
therefore decay through these by a
soft mechanism; node in 2S radial
function leads to suppression of
mechanism in Ψ´decays
mixings:
13
3. Transitions and decays of cc
branching ratios of decays of J/Ψ and Ψ´ into
meson pairs from experimental data (Beijing
Electron Spectrometer Collaboration)
BR(´ f )
BR(´ l l  )

 (0.124  0.003)
BR( J /   f ) BR( J /   l l  )
„12%-rule“
G parity violating transition
flavor symmetry breaking
mixing
Isospin violating transition
14
3. Transitions and decays of cc

2) radiative transitions (M1 and E1 dipole transitions):
H
int
fi
 e
i
Qi
f
2mi
Dipole approximation:



 
 
1
ik ri
e
 2 * pˆ i  i i  (k   * ) i eit
2V
e

 ik ri
 
 1  ik  ri  ...
w = Ei -
with
Ef
for E1, M1
Schrödinger wave function for charmonium: Y(r) = Yspin·Ylm (J, f)
Rnl(r)
M1 transitions (no parity change, spin flip: DL = 0, DS = 1):
J/Ψ
→ hc + g,
Ψ´ →
hc +ie g,
→  hc ´ + g
  *
 Ψ´ Q
Qi 
M
it
where    i  i
H fi →J/Ψ + f g
 i  (k   ) i e

2V
i 2mi
i 2mi
1
mag
V
M1

 2dd (mi  E f   ) f H int
i
2 
(2 )
2
V 2 E f

(2 )2 mi
M1
f H int
i
2
d mi  E f 
2p·(phase space)
mag
4  ec2 3
2 J f  1 f j0  r  i

2
3mc
 2 
2
where j0 is the spheric Bessel function
(jo(x) = sin(x)/x )
Relativistic corrections and anomalous magnetic moment for quarks are neglected!
15
3. Transitions and decays of cc
E1 transitions (parity changes, no spin flip: DL = 1, DS = 0):
Ψ´ → g + cc,J → J/Ψ

+ g

ipˆ i
1
ie   Q1 Q2  
E
r i    *  eit
H fi  ie
f Qi
i    *eit 
 f  
mi
2V
2V   m1 m2 
i

1
4  ec2 3
2 J f  1 f r i 2 S fi
el 
27
Jf : spin of the final state and Sfi =

where
f r i   r 2 dr{R f (r )rRi (r )}
1 for spin singlet
transition
3 for spin triplet
transitions
0
3,h
estimation of decay width by building ratios: (hc   c )  3
 (  c , J  J /  )
 , 
c
c ,J
Determination of as(mc):


10( 2  9) s3 (m 3 S )
 3 S1  3 g  hadrons
1

3
*
 
2 2
 S1    l l
81 Qi


from experimental decay width one gets:
0.18
81 2Qi2 (V  hadrons)
 (mV ) 
10( 2  9) (V  l l  )
3
s
as(mF) ≈ 0.44
as(mJ/Ψ) ≈ 0.21, as(mU) ≈
16
3. Transitions and decays of cc
Higher multipole contributions in charmonium



Magnetic quadrupole (M2) amplitudes provide indirect measure of charmed quark´s anomalous
magnetic moment and are sensitive to D-wave mixtures in S-wave states (Ψ´´ – Ψ´)
Affect angular distributions in decays Ψ´→ cc,J g and cc,J → J/Ψ g (experimentally
accessible through interference with dominant E1 amplitudes)
Radiative widths given by helicity amplitudes A, A´ with  labelling the projection of the spin
of cc,J parallel (A) or antiparallel (A´) to the photon
setting e
ax·Eg/(4mc) where x = 1 for Ψ´→ g cc,J and x = +1 for
cc,J → J/Ψ
kc: quark anomalous magnetic moment
(deviation from Dirac magnetic moment mc = ⅔ ec/(2mc))
Searching for interferences with dominant E1 amplitudes (cc,J → g J/Y): expected
normalized M2/E1 ratios a2:
17
3. Transitions and decays of cc
Hadronic transitions [QQ → (QQ)´+ light hadrons]

examples:

above DD-threshold: C(3872) → J/Ψ p+p- and Y(3940) →
J/Ψ w
theoretical
description uses multipole expansion for gluon emission, very similar to usual
multipole expansion for photonic transitions:
(color electric and color magnetic emission from a heavy quark)
ta: generator of color SU(3),(a = 1,…,8)


Single interaction of HI changes color singlet QQ initial state i into some color octett QQ
state, second interaction HI is required to return to a color singlet QQ final state (f) -> at least
two gluons have to be emitted
Ordering of amplitudes in powers of velocity with leading contribution from color electric gluon
k b
j a
emissions:
 
sum over all allowed QQ octett
intermediate states nO
lowest mass light
hadron state: p

nO
i x t nO nO x t f
Ei  EnO
0 Eaj Ebk H
S-wave 2p-system
D-wave 2p-system
18
3. Transitions and decays of cc
Properties of Ψ(2S) → g cc,J E1 radiative transition with Gtot [Ψ(2S)] = 33713
keV
Properties of transitions cc,J → g J/Ψ
Partial widths and BR
for spin-singlet states,
E1 radiative transition
O = r (GeV-1) for E1
and O = j0(kr/2) for M1
transitions
Ψ´→ g hc´/
hc
hc´ → g
hc
Ψ´→ g J/Ψ
Phys. Rev. D 66,
014012 (2002)
hc → g
hc
19
3. Transitions and decays of cc
Ψ(2S)
Decay to g hc(1S):
•
forbidden M1 transition (would vanish in the limit of Eg = 0 because of orthogonality of 1S
and 2S wave functions) at photon energy of 638 MeV →
≠0
-3
averaged BR = (3.00.5)·10 => G[Ψ(2S) → g hc(1S)] = (1.00 0.16) keV
Decay to g hc(2S):
•
allowed M1 transition characterized by
≈ 1 for small photon energies
•
Assumption: matrix elements for Y(2S) → ghc´(2S) and J/Y(1S) → ghc(1S) are equal
=> (2S-2S)-rate =
times (1S-1S)-rate leading to a BR =
-4
(2.6 0.7)·10 and therefore G[Ψ(2S) → g hc´(2S)] = (87 25) eV
Hadronic transitions to J/Ψ:
•
Via electric dipole emission of gluon pair followed by its hadronization into pp
dominating decay
mode in pions
20
3. Transitions and decays of cc
Ψ(2S) → p0 hc → p0 g hc
hc
CLEO data with
MeV
background function plus signal
21
4. New states above the DD-threshold


Discovery of a new signal X(3872) in B+X K+, XJ/Ψ p+p- at Belle in
2003 with narrow width G < 2.3 MeV and mass mX = 3871.20.6 MeV
Confirmed by CDF, D0 and BaBar
• X  J/Ψ g radiative decay confirmed by BaBar determines C =
+1
• Belle/CDF dipion angular analysis in XJ/Ψp+p- favours JPC =
1++
22
4. New states above the DD-threshold
Interpretation of X(3872)
•
•
•
•
•
•
similar to charmonium: narrow state decaying to J/Ψp+pabove DD threshold should be wide and XDD dominant
Quantum numbers established: 1++
It does not fit into the charmonium model!
m(X) ≈ m(D) + m(D*0) => X could be a bound state of 2 D mesons, a D0D*0 molecule
assumption supported by predictions of mass, decay modes, JPC, branching fractions and
small binding energy (deuteron like)
Other exotic predictions: - “tetraquark” 4 quark bound state
- “glueball” gluon bound state, charmonium-gluon hybrid ccg
PRL 98, 082001 (2007)
Further new states discovered:

X(3940):
- discovered by Belle in double charmonium
production e+e-J/Ψ X(3940)
- Decays to DD* but not DD and J/Ψ w
- Likely excited charmonium state (hc’’’ or cc1’)
- JPC = 0-+,1++ ?
XDD*
23
4. New states above the DD-threshold

Z(3930)
- observed in the two-photon decays
gg  Z(3930)  DD
- Predicted mass and width match
charmonium assignment of cc2’
- JPC = 2++

-
Y(3940)
ZDD
m(Y )  (3943 11 13) MeV/c2
YJ/y w
- discovered by Belle the decay BKY, Y (w J/Ψ)
m(Y )  (3943 11 13) MeV/c2
(Y )  (87  22  26) MeV
- Possible cc1’ charmonium state but
requires further investigation
- not found in DD or DD* final states
- JPC = 1++, …
If X=Y, difficult to explain absence of Y  open charm => Hybrid?
24
4. New states above the DD-threshold

Y(4260)

new peak in ISR events discovered at
Babar, found in decay
Y(4260)J/Ψp+pm(Y )  (4259 826 ) MeV/c2
(Y )  (88  2364 ) MeV





e+e- requires quantum numbers JPC = 1-However, all of the 1-- charmonium states
have already been discovered!
Very difficult to accommodate as cc, unless
previous assignments are wrong
for Y(4260)J/Ψ p+p-, Belle reproduces
17
2
BaBar’s signal: m(Y )  (4247 1226 ) MeV/c
8
(Y )  (1081910
) MeV
Broad second peak at slightly lower mass:
2
m(?)  (4008 4072
28 ) MeV/c
(?)  (226 4487
79 ) MeV
25
4. New states above the DD-threshold
Candidates for
hybrids
Ψ(4415)
: IG(JPC)
= 0-(1-)
Ψ(4160)
: IG(JPC)
= 0-(1-)
Ψ(4040)
: IG(JPC)
= 0-(1- )
26
5. Conclusion






Charmonium states and decay widths can be calculated quite well in NRQCD
but in order to obtain a higher precision relativistic corrections have to be
included
Determination of as(mc) from various rations of decay widths
4 s
Charmonium model with V  
 kr has great success below the DD3 r
threshold
Above DD threshold, several states remain undiscovered or need further study
A recent flood of experimental results from the B-factories is challenging our
understanding of the strong force:
- What is the nature of the new “Y” states?
 Meson molecules? Tetraquarks? Hybrids? Glueballs? Something else?
 Rich new spectroscopy?
What excited unknown states do exist? => waiting for data of (upgraded) Bfactories like Babar, Belle, CLEO, BES
searching for resonances with non-quarkonium JPC (1-+, …)
27
Thanks for your attention!
28
References












„A modern introduction to Particle Physics“, chapter 8, Fayyazuddin Riazuddin, World
Scientific
„Dynamics of the Standard Model“, chapter 13, J.F. Donoghue E. Golowich B.R. Holstein,
Cambridge Monographs on particle physics, Nuclear physics and cosmology
Lecture notes Università di Pisa, Prof. V. Cavasinni, Particelle Elementari I, „modello a quark“,
2006/07
www.e18.physik.tu-muenchen.de/teaching/struktur-dynamik-hadronen/ charmonium_1.pdf –
www.e18.physik.tu-muenchen.de/teaching/struktur-dynamik-hadronen/ charmonium_2.pdf –
theory.gsi.de/~leupold/lecture-1-13_7_07.pdf
„Implications of light-quark admixtures on charmonium decays into meson pairs“, Phys. Rev.
D, Vol. 62, 074006, T. Feldmann, P. Kroll
http://uk.arxiv.org/PS_cache/arxiv/pdf/0711/0711.1927v2.pdf
http://arxiv.org/abs/hep-ph/0701208
www-rnc.lbl.gov/ISMD/talks/Aug9/1130_Fulsom.ppt
„Production of singlet P-wave cc and bb states“, Phys. Rev. D 66, 014012 (2002), S. Godfrey,
J.L. Rosner
„Two-pion transitions in quarkonium revisited“. Phys. Rew. D 74, 05022 (2006), M.B. Voloshin
29
Determinating as(mc)

Extraction from partial decay widths ratios from J/Ψ:
 s (mc2 )  0.1900..10
05

Extraction from hc:
 s (mc2 )  0.3000..03
05

Extraction from cc,J:
s (mc2 )  0.29600..016
019

≈ 1.8 => large correction => caution with the value
for
and
cc,2
 s (mc2 )  0.3200..02
02
for
cc,0
Extraction from J/Ψ:
 s (mc2 )  0.17500..08
08
30
New charmonium states
Further resonances observed in e+e-  YgISR (certainly JPC=1--)
Most of these 1-- states should
preferentially decay into D(*)D(*)
states.
Ψ(3770), Ψ(4040),
Ψ(4415)
[regular charmonia]
clearly visible, nothing else
Y(4350)y2S
y(4
160
y(4
)
040)
y’
y(37
70)
J
/
y
4660
4350
4260
Only seen
in
Ψ(2S)pp
4
Ψs to place!
4008
Y(4260) can be fit by a tetraquark model
(decaying into J/yf0 …) or a hybrid (with
31
gpp)
Multipole expansion in QCD

Chromo-electric dipole transition:
For Y2
→Y1p+pwhere
so with effective Hamiltonian
S-wave

D-wave
mixing of 3D1 – 3S1 in 3S1 states
Chromo-magnetic dipole transition:
for 3S1-states
for 1S0-states
where
supressed by (v/c)²
F1, F2 : coordinate parts of Swave functions
32