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
D V Bugg, Queen Mary, London
See hep.arXiv: 0802.0934
J. Phys. G 35 (2008) 075005
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
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’)
(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)
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 + . . .
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%
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
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).
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
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
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.
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
data needed on rr
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