1. Ted Barnes
Physics Div. ORNL
Dept. of Physics, U.Tenn.
Charmonium

2. Charmonium 1) Basic physics
2) Theoretical spectrum versus known states
3) NEW: open-flavor strong widths
4) E1 transitions
5) X(3872)
The numbers quoted in 2-4) will appear in
T.Barnes, S.Godfrey and E.S.Swanson (in prep.)
I will mainly quote cc potential model results, which provide a useful intuitive picture of charmonium. LGT (C.Morningstar) is not yet competitive for higher mass cc states but is of course the preferred technique and will eventually solve everything.

3. e
Small qq
separation g
Large qq
separation

4. LGT simulation showing the QCD flux tube
R = 1.2 [fm]
“funnel-shaped” VQQ(R)
Coul.
(OGE)
linear conft.
(str. tens. = 16 T)
The QCD flux tube
(LGT, G.Bali et al; hep-ph/010032)

5. Physically allowed hadron states (color singlets)_
Conventional quark model
mesons and baryons. qq q3
100s of e.g.s
ca. 106 e.g.s of (q3)n, maybe 1-3 others
(q3)n, (qq)(qq), (qq)(q3),…
nuclei / molecules
Basis state mixing may be
very important in some sectors.
”exotica” :
g2, g3,…
glueballs
maybe 1 e.g.
qqg, q3g,…
hybrids
maybe 1-3 e.g.s
q2q2, q4q,…
multiquarks
(q2q2),(q4q),…
multiquark clusters
controversial
e.g. Q(1542)

6. cc mesons states and spectrum
The nonrelativistic quark model treats conventional charmonia
as cc bound states.
Since each quark has spin-1/2, the total spin is
Sqq = ½ x ½ = 1 + 0
Combining this with orbital angular momentum Lqq gives states
of total
Jqq = Lqq spin singlets
Jqq = Lqq+1, Lqq, Lqq spin triplets
xxxxx
tot.

7. cc mesons quantum numbers
Parity Pqq = (-1) (L+1) C-parity Cqq = (-1) (L+S)
The resulting cc NL states N2S+1LJ have JPC =
1S: 3S ; 1S S: 23S ; 21S …
1P: 3P ; 3P ; 3P ; 1P P …
1D: 3D ; 3D ; 3D ; 1D D …
JPC forbidden to qq are called “JPC-exotic quantum numbers”.
; ; ; ; …
Plausible JPC-exotic candidates =
hybrids, glueballs (high mass), maybe multiquarks (fall-apart decays).

8. Charmonium
Theoretical spectrum versus known states

9. Charmonium (cc) A nice example of a QQ spectrum.
Expt. states (blue) are shown with the usual L classification.
Above 3.73 GeV:
Open charm strong decays
(DD, DD* …):
broader states
except 1D2 2- +, 2- -
3.73 GeV
Below 3.73 GeV:
Annihilation and EM decays.
(rp, KK* , gcc, gg, l+l-..):
narrow states.

10. Fitting cc potential model parameters.
as, b, mc, s fixed from 1P c.o.g. and all 1S and 2S masses.
blue = expt, red = theory.
as =
b = [GeV2]
mc = [GeV]
s = [GeV]

11. Predicted spin-dependent cc 1P multiplet splittings
(sensitive test of OGE)
Parameters as, b, mc, s fixed from 13PJ c.o.g. and all 1S, 2S masses, prev slide.
blue = expt, red = theory.
as =
b = [GeV2]
mc = [GeV]
s = [GeV]
OGE + lin. scalar conft.
1P1 (not shown) is 8 MeV below the 3PJ c.o.g.
Scalar conft. gives neg. L*S

12. Fitted and predicted cc spectrum
blue = expt, red = theory.
23F4 (4351)
23F3 (4355)
23F2 (4353)
21F3 (4353)
43S1 (4407)
41S0 (4387)
33P2 (4320)
33P1 (4272)
33P0 (4202)
31P1 (4281)
23D3 (4170)
23D2 (4161)
23D1 (4144)
21D2 (4160)
33S1 (4073)
31S0 (4047)
3F4 (4025)
3F3 (4032)
3F2 (4032)
1F3 (4029)
23P2 (3976)
23P1 (3927)
23P0 (3853)
21P1 (3936)
3D3 (3810)
3D2 (3803)
3D1 (3787)
1D2 (3802)
23S1 (3672)
21S0 (3635)
3P2 (3560)
3P1 (3507)
3P0 (3424)
1P1 (3517)
Previous fit (1S,2S,1Pcog.):
as =
b = [GeV2]
mc = [GeV]
s = [GeV]
as =
b = [GeV2]
mc = [GeV]
s = [GeV]
3S1 (3087)
1S1 (2986)

13. cc from the “standard” potential model S. Godfrey and N
cc from the “standard” potential model S.Godfrey and N.Isgur, PRD32, 189 (1985).

14. Godfrey-Isgur model cc spectrum (SG, private comm.)

15. cc from LGT <- 1- + exotic cc-H at 4.4 GeV What about LGT???
An e.g.: X.Liao and T.Manke,
hep-lat/ (quenched – no decay loops)
Broadly consistent with the cc potential model spectrum. No radiative or strong decay predictions yet.
cc from LGT
< exotic cc-H at 4.4 GeV
Small L=2 hfs.
oops…
1+ - cc has been withdrawn.

16. Charmonium
Open-flavor strong decays

17. Experimental R summary (2003 PDG)
How do strong decays happen at the QCD (q-g) level?
Experimental R summary (2003 PDG)
Very interesting open experimental question:
Do strong decays use the 3P0 model decay mechanism
or the Cornell model decay mechanism or … ?
e+e-, hence cc states only.
“Cornell” decay model:
(1980s cc papers)
(cc) <-> (cn)(nc) coupling from qq pair production by linear confining interaction.
Absolute norm of G is fixed! g0 br
vector confinement??? controversial

18. The 3P0 decay model: qq pair production with vacuum quantum numbers.
L I = g y y .
A standard for light hadron decays. It works for D/S in b1 -> wp.
The relation to QCD is obscure.

19. R and the 4 higher 1-- states
3770
4040
4160
4415
(plot from Yi-Fang Wang’s online BES talk, 16 Sept 2002)

20. What are the total widths of cc states above 3.73 GeV?
(These are dominated by open-flavor decays.)
43(15) MeV
78(20) MeV
52(10) MeV
< 2.3 MeV
23.6(2.7) MeV
PDG values

21. Strong Widths: 3P0 Decay Model
Parameters are g = 0.4 (from light meson decays), meson masses and wfns. 1D
3D [MeV] 3D
3D [MeV] 1D DD
23.6(2.7) [MeV]

22. Strong Widths: 3P0 Decay Model
33S [MeV]
31S [MeV] 3S DD
DD*
D*D*
DsDs
52(10) MeV

23. Y(4040) partial widths [MeV] (3P0 decay model): DD = 0.1 DD* = 32.9
Theor R from the Cornell model.
Eichten et al, PRD21, 203 (1980):
4040 DD
DD*
D*D*
4159
4415
Y(4040) partial widths [MeV]
(3P0 decay model):
DD = 0.1
DD* = 32.9
D*D* = 33.4 [multiamp. mode]
DsDs = 7.8
Y(4040) -> D*D* amplitudes
(3P0 decay model):
1P1 =
5P1 =
5F = 0
famous nodal suppression
of a 33S1 Y(4040) cc -> DD
std. cc and D meson SHO wfn. length scale

24. Strong Widths: 3P0 Decay Model
23D [MeV]
23D [MeV]
23D [MeV]
21D [MeV] DD
DD*
D*D*
DsDs
DsDs*
78(20) [MeV]

25. Y(4159) -> D*D* amplitudes: (3P0 decay model): 1P1 = + 0.081
Theor R from the Cornell model.
Eichten et al, PRD21, 203 (1980):
4040 DD
DD*
D*D*
4159
4415
Y(4159) -> D*D* amplitudes:
(3P0 decay model):
1P1 =
5P1 =
5F1 =
Y(4159) partial widths [MeV]
(3P0 decay model):
DD = 16.3
DD* = 0.4
D*D* = 35.3 [multiamp. mode]
DsDs = 8.0
DsDs* = 14.1
std. cc SHO wfn. length scale

26. Strong Widths: 3P0 Decay Model
23P [MeV]
23P [MeV]
23P [MeV]
21P [MeV] DD
DD*
DsDs

27. Strong Widths: 3P0 Decay Model1F
3F [MeV]
3F [MeV]
3F [MeV]
1F [MeV] DD
DD*
D*D*
DsDs

28. Charmonium n.b. I will discuss only E1 because of time limitations.
Yes, M1 is interesting too!
J/y -> ghc and y’ -> gh’c give mc,
and y’ -> ghc tests S*S corrections to
orthog. 1S-2S wfns.
Radiative transitions

29. E1 Radiative Partial Widths
Same model, wfns. and params as the cc spectrum.
Standard |<yf | r |yi >|2 E1 decay rate formula.
Expt. rad. decay rates from PDG 2002
2S -> 1P
23S1 -> 3P [keV]
23S1 -> 3P [keV]
23S1 -> 3P [keV]
21S0 -> 1P [keV]
18(2) [keV]
24(2) [keV] -
1P -> 1S
3P2 -> 3S [keV]
3P1 -> 3S [keV]
3P0 -> 3S [keV]
1P1 -> 1S [keV]
426(51) [keV]
288(48) [keV]
119(19) [keV] -

30. E1 Radiative Partial Widths
1D -> 1P
3D3 -> 3P [keV]
3D2 -> 3P [keV]
3P [keV]
3D1 -> 3P [keV]
3P [keV]
3P [keV]
1D2 -> 1P [keV]

32. E1 Radiative Partial Widths
3S -> 2P
33S1 -> 23P [keV]
33S1 -> 23P [keV]
33S1 -> 23P [keV]
31S0 -> 21P [keV]
3S -> 1P
33S1 -> 3P [keV]
33S1 -> 3P [keV]
33S1 -> 3P [keV]
31S0 -> 1P [keV]

33. E1 Radiative Partial Widths
2D -> 2P
23D3 -> 23P [keV]
23D2 -> P [keV]
23P [keV]
23D1 -> P [keV]
23P [keV]
23P [keV]
21D2 -> 21P [keV]
2D -> 1F
23D3 -> 3F [keV]
-> 3F [keV]
-> 3F [keV]
23D2 -> 3F [keV]
3F [keV]
23D1 -> 3F [keV]
21D2 -> 1F [keV]
2D -> 1P
23D3 -> 3P [keV]
23D2 -> 3P [keV]
3P [keV]
23D1 -> 3P [keV]
3P [keV]
3P [keV]
21D2 -> 1P [keV]

34. E1 Radiative Partial Widths
1F -> 1D
3F4 -> 3D [keV]
3F3 -> 3D [keV]
3D [keV]
3F2 -> 3D [keV]
3D [keV]
3D [keV]
1F3 -> 1D [keV]

35. X(3872) M = 3872.0 +- 0.6 +- 0.5 MeV M( Do + D*o) = 3871.5 +- 0.5 MeV
Belle Collab. S.-K.Choi et al, hep-ex/ ;
K.Abe et al, hep-ex/
X(3872)
B+ / - -> K+ / - ( p+p- J /Y )
y(3770) = 3D1 cc.
If the X(3872) is 1D cc,
an L-multiplet is split much more than expected assuming
scalar conft.
G < 2.3 MeV
Accidental agreement?
X = cc 2- + or 2- - or …,
or a molecular state?
M = MeV
M( Do + D*o) = MeV
n.b. M( D+ + D*-) = MeV

36. X(3872) from CDF G.Bauer, QWG presentation, 20 Sept. 2003. n.b.
most recent CDF II:
D.Acosta et al, hep-ex/ ,
5 Dec 2003.
M = pm 0.7 pm 0.4 MeV

37. The obvious guess if cc is 2 - + or 2 - -.
cc from the “standard” potential model S.Godfrey and N.Isgur, PRD32, 189 (1985).
2- +
2- -
(3D2 is a typo)
The obvious guess if cc is or
No open-flavor strong decays – narrow.

38. Charmonium Options for the X(3872)
T.Barnes and S.Godfrey, hep-ph/
Our approach:
Assume all conceivable cc assignments for the X(3872):
all 8 states in the 1D and 2P cc multiplets.
Nominal Godfrey-Isgur masses were
3D3(3849) P2(3979)
3D2(3838) P1(3953)
3D1(3.82) [y(3770)] P0(3916)
1D2(3837) P1(3956)
We assigned a mass of 3872 MeV to each state
and calculated the resulting strong and EM partial widths.

39. We cannot yet exclude 5 of the 8 1D and 2P cc assignments.
If X = 1D cc:
Total width eliminates only 3D1.
Large, ca. 300 – 500 keV E1 radiative partial widths to gcJ and ghc
are predicted for 1D assignments ( 3D3, 3D2 ) and 1D2.
If Gtot = 1 MeV these are 30% - 50% b.f.s!
The pattern of final P-wave cc states you populate identifies the initial cc state.
If X = 1D2 cc, you are “forced” to discover the hc!
If X = 2P cc:
23P1 and 21P1 are possible based on total width alone.
These assignments predict weaker but perhaps accessible radiative
branches to g J/y, gy’ and ghc, ghc’ respectively.
NOT to gcJ states. (E1 changes parity.)

40. DD* molecule options M(X) = 3872.0 +- 0.6 +- 0.5 MeV
(I prefer this assignment.)
This possibility is suggested by the similarity in mass,
M(X) = MeV
M( Do + D*o) = MeV
N.A.Tornqvist, PRL67, 556 (1991); hep-ph/
F.E.Close and P.R.Page, hep-ph/
C.Y.Wong, hep-ph/
E.Braaten and M.Kusunoki, hep-ph/
E.S.Swanson, hep-ph/
n.b. The suggestion of charm meson molecules dates back to 1976:
Y(4040) as a D*D* molecule;
(Voloshin and Okun; deRujula, Georgi and Glashow).

41. Interesting prediction of molecule decay modes:
E.Swanson, hep-ph/ : DoD*o molecule
with additional comps. due to rescattering.
J/yro
J/yw
Predicted total width ca. = expt limit (2 MeV).
Very characteristic mix of isospins: J/yro and J/yw decay modes expected.
Nothing about the X(3872) is input: this all follows from OpE and C.I. !!!

42. X(3872) summary: The X(3872) is a new state reported by Belle and CDF
in only one mode: J/y p+ p- . It is very narrow, G < 2.3 MeV.
The limit on gc1 is comparable to the observed J/y p+ p-.
The mass suggests that X is a deuteronlike DoD*o-molecule.
Naïvely, this suggests a narrow total X width of ca. 50 keV
and 3:2 b.f.s to DoDopo and DoDog.
However, internal rescatter to (cc)(nn) may be important.
This predicts G(X) = 2 MeV and remarkable, comparable
b.f.s to J/yro and J/yw [E.S.Swanson, hep-ph/ ].
The bleedin’ obvious decay mode J/y po po should be searched for, to test C(X) and establish whether p+ p- = ro.
Possible “wrong-mass” cc assignments to 1D and 2P levels can
be tested by their (often large) E1 radiative transitions to g(cc).

43. Charmonium: Summary
1) The spectrum fits a OGE + linear scalar conft. potential model reasonably well. More cc states will be useful to test this. (Pt. 4.)
2) Some cc states above 3.73 GeV in addition to and 2 - -
are expected to be relatively narrow, notably
3D3 ( G = 0.6 MeV) and 3F4 ( G = 9 MeV).
3) The multiamplitude strong decays y(4040), y(4159) -> D*D*
can be used to establish the dom. strong decay mechanism.
b.f.s to DD, DD*, Ds Ds … will be useful too.
[ 3) is my favorite new-age cc topic.]
4) E1 rad: y(3770) -> gc2 tests S-wave comp.
y(4040), y(4159) -> gDD search for new C=(+) cc states.
5) The X(3872) is likely a Do D*o molecule.
J/yro and J/yw decay modes?
X = cc options predict large E1 b.f.s to g + P-wave cc.