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How resonances synchronise on thresholds. D V Bugg, Queen Mary, London. See hep.arXiv: 0802.0934 J. Phys. G 35 (2008) 075005. Examples (MeV). f 0 (980) and a 0 (980) -> KK 991 f 2 (1565) -> ww 1566

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How resonances synchronise

on thresholds

D V Bugg, Queen Mary, London

See hep.arXiv: 0802.0934

J. Phys. G 35 (2008) 075005


Examples (MeV)

f0(980) and a0(980) -> KK 991

f2(1565) -> ww 1566

K0(1430) -> Kh’ ? 1453

X(3872) -> D(1865)D*(2007) 3872

Z(4430) -> D*(2007)D*1(2410) 4427

Y(4660) -> y’(3686)f0(980) 4466

Lc(2940) -> D*(2007)N 2945

P11(1710), P13(1720) ->wN 1720



Simple explanation:

phase space

D(s)= M2 – s – ig2r(s)

= M2 - s - Si Pi(s)

Im Pi = gi2ri(s)FFi (s) Q(s - thri)

Re Pi = 1 P ds’ Im Pi(s’)

  • (s’ – s)


(Im Pi arises from the pole at s = s’);

At threshold, Re P is positive definite.


f0(980) -> KK as an example

FF = exp(-3k2)

(R=0.8 fm)

Zero-point energy also helps attract the

resonance to threshold


Illustration with f0(980) parameters:

the KK threshold acts as an attractor.

Vary M2 of the Breit-Wigner denominator and keep g2 fixed.

M(MeV) Pole (MeV)

500 806 – i76

800 946 – i48

956 1004 – i17 (physical value)

990 1011 – i 4

1100 979 – i69


Incidentally, the dispersive term Re P is

equivalent to the loop diagrams for producing

the open channel:







(The Lamb shift is an example of the dispersive effects I am discussing).


A bit more algebra:

Re D(s) = M2 – s + g2 r

Above threshold, p r=-2r2 + . . . , r=2k/s1/2

Below, p r=p[(4m2K-s)/s]1/2 - 2v2 +. . .v=2|k|/s1/2

=p[(4m2K-s)/s]1/2 –2(4m2K-s)/s + . . .

(Flatte term)

i.e. the cusp contributes like a resonant term with

respect to the KK threshold; g2K and M(res) are very

strongly correlated. It is essential to have direct data

on the KK channel to break this correlation.


Tornqvist gives a formula for the KK components:

Y= |qqqq> + [(d/ds) Re P(s)]1/2 |KK>

1 + (d/ds) Re P(s)

For f0(980), KK intensity > 60%

For a0(980), > 35%


For f0(980):

g2(pp)=0.165 GeV2; g2(KK)=0.694 GeV2.

These are similar to f2(1270): g2(pp)=0.19 GeV2


Reminder on sigma and Kappa.

BES II data on J/Y -> wpp




pp elastic scattering:

f(elastic) = N(s)/D(s)

= K/(1 – iKr)

K = b(s – sA) in the simplest possible form, sA = m2p/2

-> b(s – sA) exp[-s/B] . . . . Bing Song Zou

-> (b1 + b2s)(s – sA)exp[-s/B] ….DVB for BES data

The CRUCIAL point is that the Adler zero appears in the

numerator, making the pp amplitude small near threshold.

But logically it MUST also appear in the denominator in the

term iKr.

The Kappa is similar: M = 750 MeV, G= 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.

Possible analogy between s and the Higgs boson:

  • In both cases the driving force rises linearly with s
  • Both relate to a Goldstone boson (the nearly massless p and the photon)
  • 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 s

I now want to argue that s, k, a0(980) and f0(980) are largely driven by meson exchanges.

  • They are very much lighter in mass than f0(1370), K0(1430), a0(1450) and f0(1710), which make a conventional nonet of similar mass to f2(1270) and a1(1260), i.e. qq 3P states.
  • Leutwyler et al can reproduce the s 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 pp S-wave is drivenlargely by r exchange.
  • Janssen, Speth et al (Julich) were able to predict the f0(980) and a0(980) using meson exchanges (r and K* exchanges) AND taking account of the dispersive effect at threshold.
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]


In the Mandelstam diagram, there are:






To state the obvious,

all three contribute to

resonance formation


i.e. the quark model is modified by decay channels

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 P33, D13,

D15 and F15 baryons. Suppose contributions to the

Hamiltionian are H11 and H22; the eigenvalue equation


H11 V Y = E Y

V H22

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


An amalgam of Confinement and Meson Exchanges:

Decay processes are not to be regarded as `accidental

couplings’ to qq states as in the 3P0 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.


As an example: we know the meson exchange

contributions to the `sigma’ in pp -> pp, pp -> KK,

pp -> 4p, etc. These should mix with qq states like

f0(1370) etc in the 1-2 GeV mass range. From my

analysis of pp -> 3p0 data on f0(1370) and Cern-Munich

data on pp -> pp, there is experimental evidence for

such mixing, Eur. Phys. J C 52 (2007) 55.




D D*

Pure cusp too wide


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 JP=1+, X(3872) is a molecule. The alternative is that X(3942) could be in the D D* P-wave with JP = 0- or 2-.


f2(1565) in pp, ww and rr





Re P

data needed on rr


Other examples

  • h(1405) and h(1475) are probably the same object.
  • The latter is almost purely KK*(890), with L=1 decays,
  • hence phase space rising as k3 from threshold, 1392
  • MeV. This phase space forces the KK* peak up in mass
  • by ~50 MeV. BES I fitted both with a single h(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.


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.


Polarisation data vital

I=0, C = +1 nn mesons;

I=1, C = -1 are nearly as complete, except 3S1 and 3D1


Experiments needed

  • pp -> hp, hhp, wp, wh from a polarised target, 300-2000 MeV/c (Flair)
  • 2)pp -> 2pN and 3pN from a polarised target, to 2.5 GeV/c (Belle at JPARC)
  • 3) diffraction dissociation of transversely polarised g, JLab, to separate 3S1 and 3D1 mesons
  • 4) J/Y radiative decays (BES 3).


  • 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.