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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 = gi2 ri(s)FFi (s) Q(s - thri)
Re Pi = 1 P         ds’ Im Pi(s’)
                (s’ – s)
thri

(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:
                     
K

          K           

(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,      r=-2r2 + . . . , r=2k/s1/2
Below,  r=[(4m2K-s)/s]1/2 - 2v2 +. . .v=2|k|/s1/2
=[(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()=0.165 GeV2; g2(KK)=0.694 GeV2.
These are similar to f2(1270): g2()=0.19 GeV2
Reminder on sigma and Kappa.

BES II data on J/Y -> w
f2(1270)

s

b1(1235)
 elastic scattering:

f(elastic) = N(s)/D(s)
= K/(1 – iKr)
K = b(s – sA) in the simplest possible form, sA = m2/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  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).
Possible analogy between s 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 s
I now want to argue that s, k, a0(980) and f0(980) are largely
driven by meson exchanges.

1) 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.
2) 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  S-wave is driven
largely by r exchange.
3) 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:
t-channel
---------------------------------------------

u-channel

s-channel
To state the obvious,                      resonances
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
is
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  -> ,  -> KK,
 -> 4, etc. These should mix with qq states like
f0(1370) etc in the 1-2 GeV mass range. From my
analysis of pp -> 30 data on f0(1370) and Cern-Munich
data on  -> , there is experimental evidence for
such mixing, Eur. Phys. J C 52 (2007) 55.
X(3872)

J/y r   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 , ww and rr
Intensity
rr
ww


Re P

data needed on rr
Other examples
1) 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.
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.
Polarisation data
vital

I=0, C = +1 nn mesons;
I=1, C = -1 are nearly as complete, except 3S1 and 3D1
Experiments needed
1)pp -> h, hh, w, wh from a polarised target, 300-
2000 MeV/c (Flair)
2)p -> 2N and 3N 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).
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.

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