INFN - Laboratori Nazionali di Frascati, May 1995
From: The Second DAFNE Physics Handbook
Chapter 11: Light Quark Spectroscopy
Eds. L. Maiani, G. Pancheri, N. Paver
Supported by the INFN, by the EC under the HCM contract number CHRX-CT920026 and by the authors home institutions.
Rutherford Appleton Laboratory,
Chilton, Didcot, Oxon,
OX11 OQX, England,
and
G.J. Gounaris
Department of Theoretical Physics, University of Thessaloniki,
Gr-54006, Thessaloniki, Greece.
can probe the nature of the
vector mesons directly produced in 
annihilation,
and also the C=+ states produced in the
radiative decays of these vector mesons. In the 
energy range, these latter states may have spin up to two and any parity.
Provided the background can be controlled and the spin of the
produced state is known, a measurement of the angular distributions in these
decays can reveal both the parity of the produced meson and
the ratios of its independent production helicity amplitudes. These ratios
provide sensitive tests for quark model classification of the states,
in particular the enigmatic 
. Thus, they can be used to
indicate whether the mesons involved
are consistent with the usual
interpretation, or whether other interpretations
e.g. hybrid, glueball or multiquark, are more
favorable for some of them.
In its first phase, the 
[1] 
factory will
concentrate on direct production of the 
at 1.02 GeV in the c.m.
After this initial programme, there is the possibility for DA
NE to
access electron positron annihilation
at energies above the 
mass, reaching to 1.5 or even
possibly 2 GeV in the c.m. This will enable the
study of the dynamical
nature of mesons lying
in a mass region that is
central to the quest for gluonic
hadrons [2], as well as enabling the test of radical
ideas on the nature
of confinement [3,4,5,6].
Thus, we may be able to identify the conventional 
mesons in the 1-2 Gev region, and to investigate the nature of certain
controversial or unconventional states that do not appear to fit
naturally in
the quark model description of hadrons [2,7].
The most direct application of 
is in the production of
vector mesons in
annihilation. Its primary role is, of course, to
concentrate on the
(1020), but 
may also help to clarify
the existence and couplings of states such as
[8],
[9],
and 
which have been claimed at various times in the
literature
[10,2,11].
The I=0,1 vector mesons 
,
,
and 
, have found their way to the PDG list
[2] and should be considered to some extent as
(containing at least) true resonances. The latest analysis
of their properties is given in [11]. In the past,
they have been most simply interpreted
as 
and 
quarkonia states [12],
however, more recent analyses by Clegg Donnachie and Kalashnikova
[11,13] conclude that this interpretation is probably
inadequate. One possibility for
a consistent description of the vector mesons in the 1.2 to 1.7 GeV
range arises if 
-glue hybrids are being manifested,
of which 
and 
are specific examples [13].
An alternative picture is that one does without hybrids, but
allows for mixing with 
states. In this latter
case one requires
further 
states, both 
and 
-like,
which are relatively low lying [13].
Donnachie, Clegg and Kalashnikova [14]
argue that the controversial 
and 
mentioned above, may be such states. Thus,
their existence is pivotal in
discriminating between the hybrid and the 
picture.
Specifically, in the hybrid picture theses states have no place
whereas they may be accommodated in the 
interpretation.
Important experiments attempting at clarifying their existence
could therefore be possible at 
.
The state
was once claimed to
have been seen in Bethe-Heitler interference [8].
The mass of this state was found to be 
,
while its total and electronic widths are given as
and 
. From this we would
conclude that in the absence of any other contributions at
the peak of the resonance
This is 1% of the 
signal and may be separable
from the 
with good statistics and resolution.
Hence a careful study on the high mass side of
the 
can decide whether 
and/or 
really exist.
may also be used to study the nature of C=+
states. The most fruitful way to do so is by measuring at the
peak of 
, whose mass is most recently determined to
be 
, while the electronic
and hadronic widths are 
and 
[11]. These results imply
an increase of the 
annihilation cross
section at its peak corresponding to
With a luminosity of 
, some months of
running at the higher energies
at 
will enable the study of radiative
decays of vector mesons e.g. 
into C=+ states. Several of these states, having spin up to
two and either parity, are particularly interesting. To motivate
our more detailed discussion we present first some illustrative examples.
The easiest interpretation of these
measurements arises if
is a 
quarkonium state, (as expected in the Godfrey
and Isgur picture of the quark model spectrum [15]), which could
be tested by looking at its radiative decay to the well
established 
states 
, 
,
and 
. In such a case the independent helicity
amplitudes satisfy certain constraints (to be discussed later, see
eqs (16) and (23)): if any of these relations is violated
then we would conclude that 
is not simply an ordinary 
quarkonium. Further useful
knowledge may subsequently be acquired by searching for the 
radiative decay to the controversial
, which has been claimed in 
radiative decays,
(see p.1486 in ref.[2])
and compare the ratios of its
various helicity production amplitudes to those for the well
established 
quarkonium state 
[2].
The interpretation of this latter measurement would of course
depend on whether the previously mentioned measurement supports
the quarkonium assignment of 
.
As another notable example we could measure the single amplitude
determining the production of the 
states 
or 
. This way we can test whether these mesons are
the 
analogs of the well established 
quarkonium
state 
, or whether one of them at least is exotic
[4,5,6]. The peculiarities of the 
sector can also be studied
at DA
NE. The interest here stems from the fact that the PDG
list already includes ten axial mesons in this region, not all of
which can be quarkonia. The odd one
out appears to be the 
. This state has also been seen in
and may be a 
molecule or even a
hybrid state [16,17]. There exist also some
questions concerning its parity so that the quark model exotic
is not fully excluded [2].
In the present paper, we consider
ways of elucidating such possibilities by looking at the
exclusive production of such states in the radiative decays of
(1460) or in the 
continuum. We always assume that
we know the spin J of the produced state and that we somehow
understand the background, on the treatment of which we present
some ideas below. These processes
enable us to measure the ratios of the helicity amplitudes
determining the production of these states, and thereby test
their parity. Moreover, since the quark
model makes unambiguous predictions for these ratios, very strong
constraints on possible deviations
from the quarkonium-like structure could be imposed.
The radiative decay for
(where 
denotes any C=+ state with 
), is described by
Here 
measures the projection of the spin of the initial
state along the
beam taken as the z axis in the 
c.m.
frame. Its only allowed values
are 
. The production angle of the
state with respect the same 
axis is denoted by 
,
while 
and 
describe the helicities of the
state and the photon respectively.
The various partial wave helicity
amplitudes for the radiative decay, (to lowest order in 
),
of the initially formed 
state, are given by
,
where the argument of 
emphasizes that the final photon in
process (3) is on its mass shell; i.e. the photon momentum
satisfies 
.
These amplitudes depend on the dynamics
responsible for the existence of the
initial 
and the final 
states. For 
,
where more than one amplitude contributes, the 
distribution is determined by
the ratios of these amplitudes and provides therefore
a strong test of the quark model.
More information on these ratios may be obtained
by looking also at the emission of a virtual photon decaying
to a 
pair in a process such as
Now the final state distribution depends also
on the angle 
, defined as the angle between the
production plane and
the decay plane of 
. By definition
. The final state distribution depends also on
the
angle 
and on the squared momentum
of 
(where 
) such that
where again 
is the 
helicity, 
or 
denote the helicity of 
, and
is the 
density matrix.
Denoting by 
the mass of the final leptons and
integrating out the 
polar angle in the 
decay plane, we find
where the rows and columns correspond to 
helicities +1,0,-1.
Of course, process (5) is suppressed with respect to (3) by an
extra power of 
and it is useful only so far as the
distribution is needed.
Since in 
cannot be very large, 
is very close to its
mass shell and behaves almost like a real photon. Therefore
we can neglect all amplitudes involving
a longitudinal 
, and conclude therefore that the same
kind of helicity amplitudes,
involving only transverse photons, contribute in both (3,4)
and (5,6). After integrating over 
in (6,7),
becomes essentially the
unit matrix, which means that the
distribution for on and off-shell photons is given by
the same expression in terms of the helicity amplitudes; compare
(4, 6). Parity and time inversion invariance for such
amplitudes imply respectively
In the following we apply (4, 6, 8, 9) to the production of
resonances with specific spin J and parity 
.
J=0 States. We start from the radiative production of a J=0 state with parity P, for which there is only one independent helicity amplitude taken to be
This leads to
for process (3) and
for (5). Eq (12) indicates that the sign of the 
coefficient discriminates between the two different parities
(
) of
the produced J=0 state; thus for example the 
coefficients
for 
and 
should be opposite to
those of 
and 
[18]; as we shall see below,
this is also true for higher
spin resonances. The overall production of 
mesons
involves an E1 transition and therefore is described by one
amplitude only. Thus in the quark model the
production of the 
quarkonium state is fully determined by
the radiative production of the well established 
quarkonia 
and 
; see below[19,21].
Therefore, means
of testing whether 
or 
are the 
analogs
of the well established 
, 
are supplied. The same process may also be used to study the
quark content of the candidates for radially excited 
states
namely 
, 
[2] and 
claimed by
MARK III in the 
and 
modes
[22].
J=1 States.
We next consider the J=1 case. This has the extra
interest that a quarkonium structure for the final
state meson is only possible in the 
configuration, and not the
one. Parity and time inversion invariance imply that
there exist only two independent amplitudes which we take to be
Substituting in (4,6), using (8,9), we get
for process (3) and
for (5). We note in (15) that the sign of the 
coefficient determines again the parity of the produced state.
Thus, provided J=1 for the produced state is established,
a positive sign for the 
term will be
an unambiguous signature that an
exotic 
state is formed. Taking into account
the fact that the 
nonet is
already complete and that 
appears as a tenth state,
it will be very interesting to use (15) in order to measure
its parity and discriminate
between a vector or axial vector assignment
[2,16]. The fact that 
has already been
seen in 
fusion argues in favor of the
feasibility of such a search [16].
Leaving this aside, we now consider the production of an 
state
through process (3). According to the quark model, if this state
is a 
quarkonium and the initial decaying one is
an ordinary 
vector meson, then the amplitudes are determined mainly
by E1 and M2 transitions
which, in an obvious notation, imply [19,20]
for real photons in the final state; (compare (13)).
Let us now specialize to the case that the initial
energy is around the 
mass, and that we
are interested in studying the structure of a state in the
region.
In such a case the momentum of the outgoing photon is about 40
MeV, which, according to the quark model implies strong
E1 dominance and 
.
For such a case (14) gives 
[19,21].
For process (5) on the other hand, where 
is non-vanishing,
the ratio 
will grow dramatically with 
[23]. Such a rise
has already been seen in baryons and has important implications for
the spin dependent electroproduction [24].
In the DA
NE kinematical region however, the final
is only slightly off shell, implying again
and a small 
value.
Thus, the basic quark model prediction which can be tested through
(14) and (15), is that the
production of a true 
quarkonium state through radiative
decay of a 
vector meson, is characterized by
in the DA
NE kinematical region.
This prediction is quite strong and safe. The background can
probably also be handled for the radiative production of a narrow state,
since then there should be no resonance-background interference in the decay
products of the radiatively produced state. Thus it should be
sufficient simply to subtract the off-resonance background in
the 
and 
distributions and then fit the remainder
using (14) or (15) respectively. Therefore, if the 
quantum numbers
of the produced state are
established, we can conclude that any indication from (14-16)
implying that 
is large and thus 
very different
from +1,
would either mean that the initial state is
(which is unlikely in this mass region, though 
mixing may be revealed, see refs
[15] and [25], or that
at least one of the participating mesons is exotic
[19,21,23].
J=2 States.
As a final example, we consider the production of a 
state
through process (3) and (5). The number of independent
amplitudes after parity and time inversion invariance is taken
into account, become now three. For these amplitudes we take
Substituting in (4,6), using (8,9) we get in this case
with
for process (3), and
with
for (5). Again the sign of the 
distribution
determines the parity of the produced 
state. The
results (19, 21) imply also
as a consequence of parity and time inversion invariance and the
approximation to neglect amplitudes involving longitudinal 
in
the DA
NE kinematical region for process (5).
In the remaining we restrict to the production of a 
which may be more interesting for DA
NE. If this state
happens to be a quarkonium 
, then standard quark
model considerations imply [19,21]
where the 
dependence is suppressed. Combining (23) with
(19) and (21) we get
which should be satisfied if the produced state is a 
quarkonium. This quarkonium prediction can be made even more
restrictive if E1 dominance for the amplitudes in (23) is taken
into account, which gives
Provided that the production of
a 
state is established, any violation of (24)
will be a proof that an exotic 
state has been
identified. The same conclusion could also be drawn if a
sufficiently strong violation of (25) is observed , implying an
unacceptably large value for 
.
Apart from a comment in the discussion after (12), up to now
we have not used the less general quark model
predictions relating
the amplitudes for producing P=C=+ mesons with different J.
If such relations are used, then connections between the 
,
and 
amplitudes are found. These connections stem
from the E1 dominance found in the quark model for the 
and
amplitudes, which combined with the fact that the 
amplitude for the 
production is purely E1 induced,
leads to (modulo phase space effects)
These relations may be used to verify the tacit assumption that the
axial meson 
is the
quarkonium partner of the established 
, and to explore the nature of 
.
Furthermore, we can also study whether either of 
and 
is a conventional 
quarkonium partner of 
,
or whether they have some other dynamical structure.
The 
is particularly interesting in this regard.
According to (26), if it is the 
partner of
, its production
strength is fixed. Conversely, if different production rates
are seen, it may be
possible to deduce whether 
is a ``small" (e.g. [4]) or
``large" (e.g. [5]) system; since the
E1 transition amplitude is in proportion to
(
), where 
and 
are the
charge and vector displacement of the charged constituents
(see [6]).
We have shown in the present paper that
the radiative production of narrow C=+ states in
annihilation at DA
NE can be a very helpful tool
in studying the hadronic structure in this mass range. In more detail
it can be used to indicate whether the mesons involved
are consistent with the usual
interpretation, or whether other interpretations
such as e.g. hybrid, glueball or multiquark, are more
favorable for some of them.
Acknowledgements: GJG would like to thank the RAL Theory Group for the hospitality offered to him during his visit Rutherford Appleton Laboratory where most of this work was done.