1. How resonances synchronise on thresholds See hep.arXiv: 0802.0934 J. Phys. G 35 (2008) 075005 D V Bugg, Queen Mary, London

2. f 0 (980) and a 0 (980) -> KK 991 f 2 (1565) ->  1566 K 0 (1430) -> K  ’ ? 1453 X(3872) -> D(1865)D*(2007) 3872 Z(4430) -> D*(2007)D* 1 (2410) 4427 Y(4660) ->  ’(3686)f 0 (980) 4466  c (2940) -> D*(2007)N 2945 P 11 (1710), P 13 (1720) ->  N 1720 Examples (MeV)

3. Simple explanation: D(s)= M 2 – s – ig 2  s ) = M 2 - s -  i  i (s) Im  i  g i 2  i (s)FF i (s)  (s - thr i ) Re   = 1 P ds’ Im  i  s ’  2  s’ – s) thr i phase space (Im  i arises from the pole at s = s’); At threshold, Re  is positive definite.

4. f0(980) -> KK as an example FF = exp(-3k 2 ) (R=0.8 fm) Zero-point energy also helps attract the resonance to threshold

5. Illustration with f 0 (980) parameters: the KK threshold acts as an attractor. Vary M 2 of the Breit-Wigner denominator and keep g 2 fixed. M(MeV) Pole (MeV) 500 806 – i76 800 946 – i48 956 1004 – i17 (physical value) 990 1011 – i 4 1100 979 – i69

6. Incidentally, the dispersive term Re  is equivalent to the loop diagrams for producing the open channel: K K     (The Lamb shift is an example of the dispersive effects I am discussing).

7. A bit more algebra: Re D(s) = M 2 – s + g 2 r Above threshold,  r=-2   +...,  =2k/s  Below,  r=  [(4m 2 K -s)/s] 1/2 - 2v 2 +...v=2|k|/s 1/2 =  [(4m 2 K -s)/s] 1/2 –2(4m 2 K -s)/s +... (Flatte term) i.e. the cusp contributes like a resonant term with respect to the KK threshold; g 2 K and M(res) are very strongly correlated. It is essential to have direct data on the KK channel to break this correlation.

8. Tornqvist gives a formula for the KK components:  = |qqqq> + [(d/ds) Re  (s)] 1/2 |KK> 1 + (d/ds) Re  (s) For f 0 (980), KK intensity > 60% For a 0 (980), > 35%

9. For f 0 (980): g 2 (  )=0.165 GeV 2 ; g 2 (KK)=0.694 GeV 2. These are similar to f 2 (1270): g 2 (  )=0.19 GeV 2

10. Reminder on sigma and Kappa. BES II data on J/  ->   f 2 (1270 ) b 1 (1235 )

11.  elastic scattering: f(elastic) = N(s)/D(s) = K/(1 – iK  ) K = b(s – s A ) in the simplest possible form, s A = m 2  /2 -> b(s – s A ) exp[-s/B].... Bing Song Zou -> (b 1 + b 2 s)(s – s A )exp[-s/B] ….DVB for BES data The CRUCIAL point is that the Adler zero appears in the numerator, making the  amplitude small near threshold. But logically it MUST also appear in the denominator in the term iK . The Kappa is similar: M = 750 MeV,  = 685 MeV In Production Reactions, N(s) need not be the same as for elastic scattering. Experimentally, N(s) = constant, and f(production) = constant/D(s). The basic mechanism is rho or Kstar exchange. The effective range form of LASS is a linear approximation in q but does not contain the Adler zero.

14. Possible analogy between  and the Higgs boson: 1)In both cases the driving force rises linearly with s 2)Both relate to a Goldstone boson (the nearly massless  and the photon) 3)In both cases, the unitarity limit is reached or close (2 TeV for weak interactions). Warning: bb, WW, ZZ, tt and jet-jet thresholds will affect the line-shape of the Higgs boson, just like the 

15. I now want to argue that , a 0 (980) and f 0 (980) are largely driven by meson exchanges. 1)They are very much lighter in mass than f 0 (1370), K 0 (1430), a 0 (1450) and f 0 (1710), which make a conventional nonet of similar mass to f 2 (1270) and a 1 (1260), i.e. qq 3 P states. 2)Leutwyler et al can reproduce the  pole using the Roy equations, which are founded on crossing symmetry, i.e. a left- hand cut related to the s-channel; this is the conventional way of treating meson exchanges. In fact, the  S-wave is driven largely by  exchange. 3)Janssen, Speth et al (Julich) were able to predict the f 0 (980) and a 0 (980) using meson exchanges (  and K* exchanges) AND taking account of the dispersive effect at threshold.

16. 4) Rupp, van Beveren and I have modelled all 4 states with a short-range confining potential coupled at r~0.65 fm to outgoing waves. Adler zeros are included in all cases. This successfully fits data for all four states with a universal coupling constant, except for SU3 coefficients, confirming they make a nonet. [Phys.Lett. 92 (2006) 265]

17. In the Mandelstam diagram, there are: s-channel resonances ----------------------------------------------------- --------------------------------------------- t-channel u-channel To state the obvious, all three contribute to resonance formation i.e. the quark model is modified by decay channels

18. Oset, Oller et al find they can generate many states from meson exchanges (including Adler zeros). Hamilton and Donnachie found in 1965 that meson exchanges have the right signs to generate P 33, D 13, D 15 and F 15 baryons. Suppose contributions to the Hamiltionian are H 11 and H 22 ; the eigenvalue equation is H 11 V  E  V H 22 The Variational Principle ensures the minimum E is the Eigenstate. Most non qq states are pushed up and become too broad to observe. There is an analogy to the covalent bond in chemistry

19. An amalgam of Confinement and Meson Exchanges: Decay processes are not to be regarded as `accidental couplings’ to qq states as in the 3 P 0 model: They are to be taken from meson exchanges (left-hand cut) and contribute to the formation of resonances on an equal basis with short-range `colour’ forces. What people refer to as qqqq components of the wave function are meson-meson. For example, NN forces are almost pure meson exchanges.

20. As an example: we know the meson exchange contributions to the `sigma’ in ,  -> KK, , etc. These should mix with qq states like f0(1370) etc in the 1-2 GeV mass range. From my analysis of pp -> 3   data on f 0 (1370) and Cern-Munich data on , there is experimental evidence for such mixing, Eur. Phys. J C 52 (2007) 55.

21. J/   D D* Pure cusp too wide X(3872)

22. The X(3872) could be a regular cc state captured by the D D* threshold. There is an X(3942) reported by Belle in D D* (but with only 25 events !) It is important to find its quantum numbers. If it has J P =1 +, X(3872) is a molecule. The alternative is that X(3942) could be in the D D* P-wave with J P = 0 - or 2 -.

23. f 2 (1565) in  and  Intensity    Re  data needed on 

24. Other examples  (1405) and  (1475) are probably the same object. The latter is almost purely KK*(890), with L=1 decays, hence phase space rising as k 3 from threshold, 1392 MeV. This phase space forces the KK* peak up in mass by ~50 MeV. BES I fitted both with a single  (1425). BES II data confirm this (to be published). 2) I have done full dispersive treatments of f0(1370), f0(1500), f0(1790) and a0(1450) – this clarifies details without changing results much.

25. Generalities I suspect there is a close relation between the Higgs boson and the sigma – just a difference in energy scale. The unitarity limit dictates where things happen. General Relativity cannot possibly be correct close to the unitarity limit, because of renormalisation effects (as yet incalculable). A bold extrapolation is that dark matter and dark energy are unitarity effects when gravity reaches the unitarity limit (a black hole). Accordingly, I believe understanding Confinement is very fundamental and deserves more experimental study. 4 experiments could settle most of the issues in meson and baryon spectroscopy.

26. I=0, C = +1 nn mesons; I=1, C = -1 are nearly as complete, except 3 S 1 and 3 D 1 Polarisation data vital

27. Experiments needed 1)pp -> ,  from a polarised target, 300- 2000 MeV/c (Flair) 2)  p -> 2  and 3  from a polarised target, to 2.5 GeV/c (Belle at JPARC) 3) diffraction dissociation of transversely polarised , JLab, to separate 3 S 1 and 3 D 1 mesons 4) J/  radiative decays (BES 3).

28. Conclusions 1)Dispersive effects due to rapidly opening thresholds are important, particularly in the mass range 1-1.7 GeV. 2) At sharp thresholds, the cusp in the real part of the amplitude can attract resonances over a mass range of at least 100 MeV. Zero point energy also helps to stabilise resonances at thresholds. 3) More work is needed allowing for these dispersive effects. 4) I suggest that Confinement is due to a combination of short-range colour forces and meson exchanges.