FERMILAB-PUB-99/022-T. Ab Initio Study of Hybrid 6gb Mesons. K.J. ,Juge. Fermz National Accelerator Laboratory, P.O. Box 500, Batavia. IL GO510. J. Kuti and ...
m
Fermi National Accelerator
Laboratory FERMILAB-Pub-99/022-T
Ab Initio Study of Hybrid bgb Mesons
K.J. Juge
Fermi National Accelerator Laboratory P.0. Box 500, Batavia, Illinois 60510
J. Kuti and C.J. Morningstar
University of California at San Diego La Jolla, California 92093-0319
April 1999
Submitted to Physical Review A
Operated by Universities Research Association Inc. under Contract No. DE-AC02-76CH03000
with the United States Department of Energy
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Fermi National Accelerator
Laboratory FERMILAB-PUB-99/022-T
Ab Initio Study of Hybrid 6gb Mesons
K.J. ,Juge Fermz National
Accelerator
Laboratory,
P.O.
Box 500, Batavia.
IL GO510
J. Kuti and C.J. llorningstar Dept.
of Physics.
University
of California
at San Diego,
La Jolla,
Californza
92093-0319
Abstract Hybrid &b
molecules in which the heavy bb pair is bound together by the
excited gluon field g are studied using the Born-Oppenheimer quenched numerical
simulations.
The consistency
expansion and
of results from the two
approaches reveals a simple and compelling physical picture for heavy hybrid states. PACS number(s):
11.15.Ha, 12.38.Gc,
12.39.Mk
Typeset
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“perated
bv
UnlVerSltleS
ReSearCn
Assoclaoon Inc. under Contract No. DE-ACOZ-76CH03000 wllh the Umted
using REVTE;u
StateS
Depanment Of Energy
In addition states
to conventional
which contain
hadrons,
excited
the most amenable
QCD predicts
gluon fields.
to theoretical
Hybrid
treatment.
the BB threshold
hybrid excitations
not only directly
expansion
which is our primary
by numerical
picture
using hybrid
approach
smaller
final analysis quarks
the hybrid
on four distinct
NRQCD
Born-Oppenheimer The hybrid quarks
results meson
correspond
isotropic
hybrid
can be treated
higherof the
from simulations. with improved [7]. The hy-
are dramatically
in the nonrelativistic [8,9]. We report
formuhere our
the effects of dynamical
cgc state heavy
on their impact
was determined
quark
sea on
recently
and agrees with our
of Fig. 1).
analogous
to a diatomic
B as spatially-fixed
gluon field as a function
levels defines
and neglect
previously
molecule:
to the nuclei and the fast gluon field corresponds
energy
QCD in
lattices
we will comment
for the slowly moving
[3] (see caption
the energy levels of the excited excited
studies
Although
simulations,
First, one treats the quark Q and antiquark
these
lattice
bgb states.
in lattice
lattice technology
using the Wilson gauge action
in our quenched
expansion
results
have appeared
anisotropic
states
We test the accuracy
with high-precision
The mass of the lowest hybrid
spectrum.
[lo] without
in recent
QCD (NRQCD)
are not included
first calculated
effects.
picture.
of heavy hybrid
are the first to exploit anisotropic
with improved
than those obtained
lation of lattice
potentials
on some of our results
brid meson mass uncertainties
where the lowest
of a simple physical
for the description
and retardation
by comparison
reports
structure
Heavy hybrid mesons can
we work to leading order in the expansion
spin, relativistic,
preliminary
precisely
early
but also using the Born-Oppenheimer
for the development
was introduced
Our hybrid meson simulations actions;
simulation,
guidance
in Refs. [4,5] and was applied
Born-Oppenheimer
accessible:
[l,2] revealed a complex resonance
the masses of the lowest bgb states.
be studied
order terms involving
pairs are
[3].
In this work, we determine
Ref. [6]. In this new study,
and hybrid
with heavy hb quark
and 11.2 GeV in e+e- armihiliation,
are expected
The Born-Oppenheimer
mesons
of glueballs
They are also experimentally
results from the CUSB and CLEO collaborations between
the existence
an adiabatic
2
potential
the slow heavy
to the electrons
[4].
color sources and determines
of the $Q
separation
r; each of
Vaps(r).
The quark
motion
is
then restored quarkonia
by solving the Schrodinger
equation
are based on the lowest-lying
from the excited simulations),
potentials.
a distinct
Once the static
it is a simple matter
hybrid quarkonium
states
advantage
phenomenological
operations
the quark
lattice
of charge
States
operation
label for the C states;
glueball.
This is states,
and spatial
quantum
inversion
Following denoted
by the subscripts
by a superscript
about
J, of the under
the midpoint from molecular
by the capital
Greek letters
There
parity-
is an additional
in a plane containing
In Ref. [7], the potentials
ro [ll];
the curves
state
will eventually
are
in Fig. 1 assume
exceed the mass of the lightest
the excited levels will become unstable the LB0 spectrum
with a centrifugal momentum
scale parameter
+ (-).
gluon field
number
notation
g (u).
levels are
Note that as r becomes small (below 0.1 fm), the gaps between
Given these static potentials,
the orbital angular
the symmetry
momentum
which are even (odd) under the above-mentioned
levels and the Ci ground
equation
of the excited
of the total angular
of the QgQ system.
of the hadronic
When this happens,
Schrodinger
eigenstates
C states which are even (odd) under a reflection
= 450 MeV (see below).
the excited
and
yield valuable
[7]. The three lowest-lying
to energy
conjugation
are denoted
axis are denoted
in terms
studies
with A = 0, 1,2,. . . are typically
states
charge-conjugation
-1 I-0
(LBO) approximation.
the LB0 wave functions
axis, and by 77= fl,
and antiquark
c, n, a, . . .) respectively.
calculated
of conventional
which yield only the very lowest-lying
A of the projection
gluon field onto the molecular
the molecular
spectrum
(via lattice
and sizes of these states which should greatly facilitate
in previous
by the magnitude
spectroscopy,
the complete
In addition.
shown in Fig. 1. These levels correspond
between
states emerge
of the excited gluon field in the presence of a static quark-antiquark
pair has been determined
the combined
quarkonium
applications.
The energy spectrum
characterized
Conventional
have been determined
Born-Oppenheimer
uncertainties.
the structures
hybrid
potentials
over meson simulations
concerning
potential;
to determine
in the leading
often with large statistical information
static
in each of these potentials.
factor (L&)
of the quark-antiquark
For the II, and C; levels, we attribute
is easily obtained
against
glueball decay.
by solving the radial
= L(L + 1) - 2A2 + (Jz), where L,, pair. For the “9’ potential,
the lowest nonvanishing
is
(Ji) = 0.
value (Ji) = 2 to the excited
gluon
field.
angular
Let S be the sum of the spins of the quark
momentum
eigenvalues
of a meson is given by J = L + S.
and antiquark, In the LB0
L(L + 1) and S(S + 1) of L2 and S2 are good quantum
and charge conjugation
C of each meson is given in terms
and C = E 77 (-l)L+“-ts, A > 0. Note that radial quantum Results
the total
approximation,
numbers.
the
The parity
for each static
potential,
the LB0
P
of L and S by P = E (-l)L+iz-tl
where L >_ A and e = 1 for C+, t = -1 for C-, and E = fl
number
energies
depend
for
only on L and the
n.
for the LB0
spectrum
of conventional
6b and hybrid
in Fig. 1. The heavy quark mass Mb is tuned to reproduce
are insensitive
to small changes
5% change in Mb results in changes
bgb states
Y(lS)
state in the Ci potential.
in the heavy quark mass.
to the splittings
are shown
the experimentally-known
mass: MT = 2Mb + Eo, where Eo is the energy of the lowest-lying Level splittings
then
(with respect
For example,
to the 1s state)
a
ranging
from 0.1 - 0.8%. Below the BB averaged
threshold,
experimental
measurements
ment with experiment and possible
mixings
the lowest-lying other hybrid
results
between
hybrid
are in very good agreement
of bottomonium
is lost, suggesting
potentials
the conventional
the LB0
the states
significant
(from the II, potential)
is about
are significantly
lying.
1s and 1P states
higher
are compared
suppressed
For all of the hybrid
The applicability
of the excited-state
of the Born-Oppenheimer
effects. The difference
the lowest order NRQCD
between
Hamiltonian
can be tested
by comparing
The mass of
densities
of the lowest-lying
is large in comparison
here, the wave functions be affected
for
II, hybrid with the 1s are strongly
noticeably
by the
from bgb + bb + glueball decay. approximation
relies on the smallness
the leading Born-Oppenheimer
is the p’. -3; coupling
in motion and the gluon field. This retardation spectrum,
potentials.
10.9 GeV. Hybrid mesons from all
masses cannot
potentials
order effects
The radial probability
state
agree-
frorn higher
adiabatic
with that
studied
near the origin so that the hybrid
small-r instability
retardation
states
Above the threshold,
corrections
from different
state in Fig. 1. Note that the size of the hybrid and 1P states.
states.
with the spin-
between
Hamiltonian
of and
the quark color charge
effect, which is not included in the LB0
the LB0 mass splittings
with those determined
from
meson simulations
in NRQCD.
In order to obtain try channel, functions
the masses of the first few escited
we obtained
Monte Carlo estimates
crucial to use anisotropic than the spatial improve
for a matrix
Cij(t) = (O]Mi(t)M~(O)]O) at t wo different
the hybrid mesons are expected
lattice
put aspect
ratios
lattices spacing
our knowledge
gluons are described
to be rather
by the improved
the spatial plaquette. determined
gauge-field
lattice
spectrum action
simulations.
of Ref. [la].
details
the static-quark
potential
lattice [13], we have verified that radiative heavy quarks lattice.
are treated
The NRQCD
action
kinetic energy operator tivistic
corrections
within
depending
U,(x) on the lattice projected
final smeared
are smeared is replaced
of ra/as are
= < in all of our calculations. orientations
on the
as/at are small. The
[14], modified
for an anisotropic
derivative
and the leading
E and chromomagnetic
rela-
B fields,
on a given time-slice
using t,he algorithm
C$(x)Ct
as follows.
First,
is iterated
nc times,
by Uj(x). Next, let g(x) and x(x) denote and antiquark,
= ia,x*(x),
the
of Ref. [15] in which every spatial link
by itself plus C times the sum of its four neighboring
a heavy quark
define a smearedquark
were
to remove O(a,) and O(a,“)errors);
back into SU(3); this procedure
is defined such that
link u, from
the calculation
temporal
on spin, the chromoelectric
Mi(t) are constructed
link variables
which annihilate
p, in-
are not included.
Our meson operators spatial link variables
only a covariant
the
to each simulation
to the anisotropy
framework
(with two other operators
and higher derivatives
staples,
includes
The couplings
from Wilson loops in various
the NRQCD
a, is much smaller
the mean spatial
concerning
corrections
large, it is
are listed in Table I. Following
link ut = 1 and obtain
Further
fluctuations
[12]. In our simulations,
given in Ref. [12]. Note that we set the aspect ratio using a,/a, By extracting
Because the masses of
been used to dramatically
The values for r. in terms of a, corresponding
in separate
rneson correlation
spacing
have already
glueball
in a given symme-
of hybrid
spacings.
sizes for each simulation
Ref. [12], we set the mean temporal
states
high and the statistical
Such lattices
of the Yang-Mills
