1. Topics in meson spectroscopy for Compass
David Bugg, Queen Mary, London
(i) Look for the I=0 JPC=1-+ partner of p1(1600/1662)
(ii) Measure s(4p)/s(2p) from MeV for JP=0+,1-,2+.
Sort out f0(1370), f2(1565) and r(1450).
Check if f2(1810) exists or is really f0(1790).
(iii) Check if p1(1405) is resonant or just a threshold cusp.

2. The p1(1600/1660) looks a good hybrid candidate.
It ought to have an I=0 partner of similar mass.
The only likely decay mode is to a1(1260)p:
JPC = = 1-+.
This is NOT easy because no angular dependence: there are only the r and a1 signals to go on, and the a1 is broad. However, other JPC will have angular dependence.
There exist possible partners with JPC = 2-+:
h2(1860) and p2(1880); also p(1800), 0-+

3. I = 0, C = +1 states from Crystal Barrel in flight; 10 sets of data fitted simultaneously.
See Phys. Rep. 397 (2004) 257.
f2(1810) dubious
f0(1790) not f2(1810)
h2(1860) extra state; not ss

4. J/y -> wKK J/y -> wpp
1710 in pp <11% at 95% CL
J/y -> fKK J/y -> fpp
No > KK
1790 -> pp

5. Crystal Barrel in flight also observe f0(1770 +- 12) in (hh)p0, but the PDG lists it as f0(1710)!
BES II also observe a striking peak in
wf at MeV with G = 105 MeV
and JP = 0+. But the PDG lists it under
X(1835) (recently confirmed) which is
observed in hpp; this cannot possibly have JP = 0+!

6. Next topic.
Branching ratios to rr, ww and ss compared with pp over the mass range 1200 to 2200 MeV are important, for the simple reason that they rise rapidly with mass and are the dominant cross sections. This is not very glamorous, but is presently the limitation in understanding f0(1300/1370), f0(1500), r(1450) and f2(1565).
It is necessary to isolate the OPE contribution to all of these from their t-dependence. The problem with exchanges at large t is that their spin content is not well understood, making the interpretation of results obscure.
Detail : rr and ww channels are related by SU(2)
g^2(rr)/g^2(ww) = 3; but their phase space factors differ after folding their widths into calculations.

7. Words of advice on parametrising amplitudes.
Do NOT use the K-matrix. It has many problems (which experts know). The main problem is that it ASSUMES all channels are known, but that is not usually true. If strong channels like 4p and not known well – chaos!
2) K-matrix elements do NOT necessarily add (as is usually assumed). Phases could add instead! K = tan d; why not K = (KA + KB)/(1 – KAKB)?
3) Extended Unitarity is experimentally disproved, see Eur. Phys. J C (2008) 73.

8. f0(980) and a0(980) -> KK 991 f2(1565) -> ww 1566
SHARP THRESHOLDS.
Many resonances are attracted to thresholds.
MeV
f0(980) and a0(980) -> KK
f2(1565) -> ww
f0(1790/1812) -> wf
X(3872) -> D(1865)D*(2007)
Y(4660) -> y’(3686)f0(980)
Lc(2940) -> D*(2007)N
K0(1430) -> Kh’ ?
K1(1420) -> KK*

9. D(s)= M2 - s - Si Pi(s) Im Pi = gi2 ri(s)FFi (s)
BW = N(s)/D(s) where D(s)=M2 - s – iMGtot(s)
phase space
D(s)= M2 - s - Si Pi(s)
Im Pi = gi2 ri(s)FFi (s)
Re Pi = 1 P ds’ Im Pi(s’)
(s’ – s)
The full form of the BW is
thri
D(s)=M2 - s – Re P(s) – i Im P(s)

10. f0(980) -> KK as an example
cusp
FF = exp(-3k2)
(R=0.8 fm)
Re P acts as an effective attraction pulling the resonance to the threshold.

11. The parameters of f0(980) are accurately known
from BES 2 data on J/y -> fpp and fKK. One can
play the game of varying M of the Breit-Wigner and
evaluating the pole position; illustrative numbers -
M(MeV) Pole (MeV)
– i76
– i59
– i31
– i21 i4
– i28
– i69

12. f0(1370) as an example:
The peak in pp is given by the Breit-Wigner folded with phase space. Likewise for 4p. The peak in pp is at MeV, but in 4p is at ? MeV: Euro. Phys. J C52 (2007). This explains the large variation of M,G in PDG tables from differing analyses.
NB: Phase variation horribly wrong with constant G

13. The f2(1565) is an even more serious case
The f2(1565) is an even more serious case. It is attracted to the ww threshold by the sharp cusp. The f2(1640) is the SAME resonance when the Breit-Wigner lineshape is folded with ww phase space, Baker et al., Phys. Lett. B467 (1999) 147. There is a second serious problem: in the Flatte formula, f a 1/[M2 – s – i g2 r(s)],
the phase space factor r is analytically continued below the ww threshold. Unless it is cut off strongly by a form factor, it creates strong interferences with f2(1270). Those interferences account for f2(1430) of the PDG.
The r(1450) has VERY uncertain parameters if treated as a Breit-Wigner of constant width. Could have a mass as low as 1250 MeV??

14. The PDG lists two closely separated h(1405) and h(1475)
The PDG lists two closely separated h(1405) and h(1475). The h(1405) decays to hs, a0(980)p and kK with L=0 and is unaffected by centrifugal barrier effects. The h(1475) decays to K*K, whose threshold is at 1394 MeV. This decay has L=1, so the intensity ~ p3, shifting the peak up strongly. All decays can be fitted well with a single h(1440) including dispersive effects.
See arXiv: and

15. The cusp effect may explain away p1(1405)
The cusp effect may explain away p1(1405). It lies very close to sharp thresholds for b1(1235)p and f1(1285)p, which both have quantum numbers of p1(1600/1660). It would be a valuable exercise for a phenomenologist to see if data over this whole mass range can be fitted in terms of (a) coupling constants of p1(1600), (b) their phase space with (c) appropriate form factors v. k (centre of mass momentum for each channel). Essentially the opening of these channels introduces inelasticities at the b1(1235)p and f1(1285)p thresholds and forces the hp? and rp amplitudes to move on the Argand diagram in a similar way to a resonance.

16. General remark: Oset, Oller et al find they can generate MANY states from meson exchanges. This is along the lines of Hamilton and Donnachie, who found in 1965 that meson exchanges have the right signs to generate P33, D13, D15 and F15 baryons.
Suppose contributions to the Hamiltionian are H11 and H22; the eigenvalue equation is
H V Y = E Y
V H22
H11 refers to q-q; H22 to s,t,u exchanges.
V is the mixing element between them.
Two solutions: E= (E1+E2)/2 + [(E1-E2)2 - |V|2]1/2
Two KEY points: mixing LOWERS the ground state, hence increasing the binding. The eigenstate is a linear combination of qq and meson-meson.

17. There is in fact an exact mathematical corresponance between this mixing and the covalent bond in chemistry, where states line benzene are well-known linear combinations of more than one configuration.
see hep-ph/ ; J.Phys. G37 (2010)
X(3872) most likely has JPC=1++, and is a linear combination of cc and DD*; very recently, arXiv , I have stumbled upon evidence that X(3915), observed in decays to ww, also has JPC = 1++. If this is confirmed, it is the first example of strong mixing generating two eigenstates close in mass, but with orthogonal combinations of cc and meson-meson. The same evidence explains Y(4140) of CDF decaying to fJ/Y.

18. CONCLUSION: there is still lots to do and learn in meson spectroscopy
CONCLUSION: there is still lots to do and learn in meson spectroscopy. I doubt that heavy meson spectroscopy can go beyond narrow states, but it is important to complete light meson spectroscopy and baryon spectroscopy and learn what we can about hybrids and glueballs. Good luck!
The spectroscopy of I=1, C = +1 states and C=-1 states with I=0 and 1 could be completed by measuring polarisation in pp -> 3p0, hhp, hp; wp and whp0; wh and wp0p0 at the forthcoming Flair facility. It just needs an extracted p beam of 5 x 104 p/s.