Ab Initio Study of Hybrid bgb Mesons

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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 ...
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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

4%

“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