

 09 Dec 94

The Continuous

Electron Beam Accelerato

rF acilit

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CEBAF-TH-94-08 Hea vy Mesons

In A Relativi

sti cMo

del

J. Zeng,

J. W.

Van

Orden

and W. Rob

erts

\Lambda

Departmen tof Ph ysics

Old Dominion

Univ ersit y, Norfolk,

VA 23529

and

Con tin uous

Electron

Beam Accelerator

Facilit y

12000 Jefferson

Av enue,

Newp ort News,

VA 23606

Abstract

Motiv ated by the

presen

tin terest

in the

hea vy quark

effectiv etheory

,w euse

the

spectator equation to treat

the mesonic

bound states of hea

vy quarks.

The kernel

we use

is based

on scalar

confining

and vector

Coulom

bp oten

tials.

Wa ve functions are treated

to leading

order and energies

to order

1=m Qin

the hea vy-ligh

t

systems, and order

1=m

2Qin

hea vy-hea

vy systems.

Our results

are in reasonable

agreemen twith exp erimen

tal measuremen

ts. We

estimate

tw oof

the parameters

of the

hea vy quark

effectiv etheory

,and prop ose further

calculations

that ma yb e

undertak en in the

future.

\Lambda National Young Inv estigator

Hea vy Mesons

In A Relativistic

Mo del

J. Zeng,

J. W.

Van

Orden

and W. Rob

erts

y

Dep artment

of Physics,

Old Dominion

University, Norfolk, VA 23529

and Continuous Ele ctr on

Be am

Ac celer

ator Facility

12000 Jefferson

Avenue, Newp ort News,

VA 23606.

Motiv ated by the

presen

tin terest

in the

hea vy quark

effectiv etheory

,w euse

the

spectator equation to treat

the mesonic

bound states of hea

vy quarks.

The kernel

we

use is based

on scalar

confining

and vector

Coulom b poten

tials.

Wa ve functions

are

treated to leading

order and energies

to order

1=m Q

in

the

hea vy-ligh

tsystems,

and

order 1=m

2Q in

hea

vy-hea

vy systems.

Our results

are in reasonable

agreemen twith

exp erimen

tal measuremen

ts. We estimate

tw oof

the parameters

of the

hea vy quark

effectiv etheory

,and prop ose further

calculation

sthat ma yb eundertak

en in the

future.

I. INTR

ODUCTION

Recen tly ,there

has been

great theoretical

interest in hadrons

con taining

b

and cquarks.

This has stemmed

largely from the realization

that, in the

formal

limit when the mass

of one

of the

quarks

in ahadron

istak en to infinit

y,symmetries ab ove

and

bey ond

those

usually

asso ciated

with quan tum chromo

dynamics

(QCD) arise. This realization

has led to the

dev elopmen

tof the hea vy quark

effectiv etheory

(HQET) [1] [2] [3].

In the

framew

ork of this

effectiv

etheory

,corrections to the

formal

limit can be systematically

included. One very imp ortan

t

phenomenological consequence of this

has been

an um ber

of attempts

to extract

Vcb from

exp erimen

tal data,

with little mo del

dep endence

in the

result.

Despite the po wer

inheren

tin HQET,

there is still

muc hthat

this effectiv

e

theory can not tell us ab out

the prop

erties

of hea

vy hadrons.

As an example,

HQET allo ws us to infer

the absolute

normalization

of some

of the

form

factors necessary

for describing

the deca ys of hadrons

with beaut yto those

with

charm. We also

kno w ho w to

include,

in asystematic

wa y, corrections

to these

normalizations due to the

finite

masses

of the

band

cquarks,

as well

as those

due to perturbativ

eQCD effects. We can even

deduce

bounds on the

slop es of

these form factors

at aparticular

kinematic poin t. Ho wev

er, we kno w nothing

ab out

the exact

dep endence

of these

form factors

on kinematic

inv arian

ts. As

asecond example, HQET leads us to the

conclusion

that the spectra

of B and

D mesons

should be very

muc halik

e, mo

dulo

1=m band

1=m ceffects.

Ho wev

er,

this effectiv

etheory

tells us nothing

ab out

the details

of the

spectra,

suc has

the

exact ordering

of states,

or their

masses.

In essence,

HQET pro vides

aframew

ork

for systematically

extracting symmetry relations and the corrections

to the

formal hea vy-quark

limit but can predict

neither the spectra

of the

hea vy mesons

nor the approac

hto the hea vy-quark

limit. Un til we

kno w ho w to

solv

enonperturbativ eQCD, the details

men tioned

ab ove,

along

with man yothers,

are the

realm of mo

dels:

suc hmo

dels con tin ue to pla

ya

crucial

role in our

understanding

of QCD.A

mo del

that

isquite

successful

in predicting

the mesonic

spectra isthe relativised constituen

tquark mo del

of Go dfrey

and Isgur

[4]. Indeed,

it was

this

mo del and

its applications

to weak

deca ys that

originally

suggested the existence

of hea

vy-quark

symmetries

whic hin

turn led to HQET.

This mo del

pro vides

relativistic kinematic corrections

to the

standard

nonrelativistic

quark mo del

using alinear

confining

poten tial and acolor

Coulom bin teraction.

Meson spectra

calculated with this mo del

are remark

ably close

to exp

erimen

tal masses

in all

fla vor

sectors.

Ho wev

er, since

one of the

ob jectiv

es of hea

vy quark

theory

isthe

calculation of weak

deca yamplitudes

and form

factors,

it is necessary

to use

a

relativistically cov arian

tmo del.

A cov

arian

textension

to the

Go dfrey-Isgur

mo del

can

be constructed

using

the spectator

or Gross

equation

[5], whic

hhas

been used with some

success

in

mo dels

of the

nucleon-n

ucleon interaction

[6], as well

as in quark

mo dels

of mesons

comp osed of equal

mass quarks

and an tiquarks

[7]. This

equation

can be related

to the

Bethe-Salp

eter equation

by placing

one of the

intermediate-state

particles

on the

positiv

e-energy

mass-shell.

This has the adv an tages

that the prescrib

ed

constrain ton the relativ

eenergy

ismanifestly

cov arian

tand

that in the

limit

that

the mass

of one

constituen

tgo es to infinit

y(the

static limit),

the wa ve equation

reduces to the

Dirac

equation

for the

ligh tparticle

[8]. This

isa prop ert yof

the

2

full Bethe-Salp

eter equation

that is lost

when

the infinite

sum of con

tributions

to the

kernel

istruncated.

Clearly ,the prop erties

of the

spectator

equation mak e

itideal for studying

the prop

erties

of hea

vy mesons

at finite

mass.

In this

article

we use

the spectator

equation to construct

aconstituen

tquark

mo del

of hea

vy mesons.

In particular,

we will

use the spectator

equation as

abasis for construction

and expansion

of the

hea vy meson

spectra

and wa ve

functions in 1=m

Q,

where

m Q

isthe

hea vy quark

mass. This allo ws us to study

the hea vy meson

spectra

in the

approac

hto the hea vy quark

symmetry

limit. By

cho osing

areasonable

set of mo

del parameters

we are

able

to obtain

aresp ectable

fit to the

observ

ed hea vy meson

masses and to predict

the appro

ximate

masses

of hea

vy mesons

whic hha ve not

yet been

observ

ed.

This article

isorganized

as follo

ws. In the

next

section,

we describ

ethe mo del

that we use

for hea vy mesons,

including

the deriv

ation

of aw

ave

equation

from

the spectator

equation. In Section

III, three

metho ds of obtaining

solutions of

the wa ve equation

are describ

ed, while

in Section

IV we displa

your results.

In

Section V, we presen

tsome

conclusions. II.

THE

MODEL

A. Q_q and

q_Q mesons

The spectator

equation is most

easily

understo

od in relation

to the

BetheSalp eter equation.

The Bethe-Salp

eter vertex

function

for tw ob

ound

fermions

is

represen ted by Fig.

1and

can be written

as

\Gamma ( p;P

)= i Z

d4k(2ss )4V

(p; k; P)

S(1)F

(k 1;m

1)S

(2)F (k 2;m

2)\Gamma (

k; P)

;

(1)

where p =

12(p 1\Gamma

p2),

k =

12(k 1\Gamma

k2),

V is the

Bethe-Salp

eter kernel

and

S(

i)F( ki;

m i)

is the

free

Dirac

propagator

for particle

i. The

Dirac

indices

are

suppressed for simplicit

y.

The spectator

vertex function

can be obtained

from the Bethe-Salp

eter vertex

function by placing

one of the

fermions

on its positiv

e-energy

mass-shell.

For our

mo del the

hea vy quark

(particle

2) isplaced

on shell

while

the ligh tquark

(particle

1) remains

off shell.

This isac hiev ed by areplacemen

tof the propagator

S(2)F (k 2;m

2)!

\Gamma 2 ssi

m 2

E( k 2;m

2) ffi `

k0\Gamma

P

02 +

E( k 2;m

2) '

\Lambda +

(2) (k 2;m

2);

(2)

where

pp 12 \Gamma

P

P k2 k1

V

p1p2

\Gamma

FIG. 1. Feynman

diagrams represen ting the equation

for the

Bethe-Salp

eter vertex

function. \Lambda +

(2) (k 2;m

2)

= X

s

02 u

(2) (k 2;s

02;m

2)_u

(2) (k 2;s

02;m

2);

(3)

and replacing

p, k and

k1 by the

corresp

onding

quan tities

^p,

^kand

^k1 with

particle 2on mass shell. The on-shell

energy isgiv en by E( p; m)

= p

p2 + m

2.

The spectator

vertex function

isthen

\Gamma ( ^p;P

)= Z

d3 k

(2ss )3

m 2

E( k 2;m

2) V

(^p;

^k; P)

S(1)F

(^k 1;m

1)\Lambda

+(2)

(k 2;m

2)\Gamma (

^k; P)

: (4)

Defining the spectator

wa ve function

as

s 2(^p;

P) = S(1)F

(^p 1;m

1)_u

(2) (p 2;s

2;m

2)\Gamma (

^p;P );

(5)

the wa ve function

satisfies the wa ve equation

S(1)F

\Gamma 1 (^p 1;m

1) s2(

^p;P

)=

Z d3k(2ss )3

m 2

E( k 2;

m 2) X

s

02 V

s2 ;s

02(^p;

^k;P ) s

02(^k

;P );

(6)

where Vs 2;s

02(^p

;^k; P) = _u(2)

(p 2;s

2;m

2)V

(^p;

^k; P)

u(2)

(k 2;

s02;

m 2):

(7)

This wa ve equation

isco varian

tand can be easily

bo osted

from frame

to frame.

It is generally

easier to solv

ethe

wa ve

equation

in the

bound-state

rest frame

where the angular

expansions

of the

wa ve function

and poten

tial are defined.

In

the rest

frame

P = (W

;0 ), p 1=

\Gamma p 2=

p, k 1=

\Gamma k 2=

k, p01 = W\Gamma

E( p; m 2),

p02 = E(

p; m 2),

k01 = W\Gamma

E( k; m 2),

and

k02 = E( k; m 2)

where

W isthe

boundstate mass.

The wa ve equation

can be written

as

hfl (1)

0(W

\Gamma E(

p; m 2))\Gamma

fl(1) \Delta p\Gamma

m 1 i

s 2(p

;W )=

Z d3k(2ss )3

m 2

E( k; m 2) X

s

02 V

s2 ;s

02(p

;k ;W

) s

02(k

;W

);

(8)

3 4

where Vs 2;s

02(p

;k ;W

)=

_u(2) (\Gamma p; s2;

m 2)V

(p ;k ;W

)u

(2) (\Gamma

k; s02;

m 2):

(9)

Since we wish

to examine

the approac

hto the limit

m 2!

1, it is useful

to

rewrite this equation

in anonco

varian tform

by defining

\Psi s2(

p)j r

m 2

E( p; m 2)

s2(

p; W )

(10)

and Us 2;s

02(p

;k ;W

)j r

m 2

E( p; m 2) V

s2 ;s

02(p

;k ;W

) r

m 2

E( k; m 2)

(11)

to giv e

hfl (1)

0(W

\Gamma E (p ;m

2))\Gamma

fl(1) \Delta p\Gamma

m 1 i

\Psi s2(

p) =

Z d3k(2ss )3 X

s

02 U

s2 ;s

02(p

;k ;W

)\Psi s

02(k

):

(12)

Itis necessary

to assume

some form for the

kernel

V in order

to expand

ab out

the infinite

mass limit.

Here we assume

that the kernel

is of

the

simplest

form

whic hcan

be reduced

to that

used in ref.

[4]. We cho ose

the kernel

to be

V( p;k

;P )=

Vs( Q

2)

+ fl(1)

\Delta fl

(2) Vv (Q

2);

(13)

where Q

2=

(k\Gamma

p)

2\Gamma

[E (k;

m 2)\Gamma

E( p; m 2)]

2:

(14)

Vv (Q

2) isa

vector

poten tial whic

his acolor

Coulom bin teraction

and the confining force

is the

result

of the

scalar

poten tial Vs( Q

2).

This

choice

of interaction

assumes that the Loren

tz gauge

isused

in the

color

Coulom

bin teraction.

Using the explicit

form of the

Dirac

spinors

in (12)

and the Dirac

fl-matrices

to reduce

particle

2to the Pauli

spin space,

and defining

aw ave

function

whic h

isan op erator

in the

Dirac

space of particle

1and the Pauli

space of particle

2,

\Psi =P

s

02 O/

s

02\Psi

s

02,

(12)

becomes

ifl (1)

0(W

\Gamma E (p ;m

2))\Gamma

fl(1) \Delta p\Gamma

m 1 j

\Psi ( p)

=

Z d3k(2ss )3 `

(E (p ;m

2)

+ m 2)(

E( k; m 2)

+ m 2)

4E (p ;m

2)E

(k; m 2)

'

12

\Theta ae` 1\Gamma

oe(2) \Delta p oe(2)

\Delta k

(E (p ;m

2)

+m

2)(

E( k; m 2)

+ m 2) '

Vs( Q

2)

+fl

(1)

0 `

1+

oe(2) \Delta p oe(2)

\Delta k

(E (p ;m

2)

+ m 2)(

E( k; m 2)

+ m 2) '

Vv (Q

2)

+ fl(1)

\Delta `

oe(2) \Delta p oe(2)

(E (p ;m

2)

+ m 2)

+

oe(2) oe(2)

\Delta k

(E (k;

m 2)

+ m 2) '

Vv (Q

2) oe

\Psi ( k);

(15)

Expanding eq. (15)

to order

1=m 2,

we

find

`fl (1)

0(W

\Gamma m 2\Gamma

p22m 2 )\Gamma

fl(1) \Delta p\Gamma

m 1 '

\Psi ( p)

=

Z d3k(2ss )3 i

Vs( q2 )+

fl(1)

0V v(q

2)

+

12m2 fl(1)

\Delta ( oe(2)

oe(2)

\Delta k + oe(2)

\Delta p oe(2)

)V v(q

2) '

\Psi ( k):

(16)

where q= k\Gamma p.

Eq. (16)

can be Fourier

transformed

to coordinate

space, multiplied

from the

left by fl(1)

0and

then rearranged

to giv ethe

wa ve equation

H\Psi ( r) = W \Psi ( r);

(17)

where the hermitian

hamiltonian

is H

= H 0+

H 1with

H 0=

ff

(1)\Delta

1i r + fi(1)

m 1+

fi(1) Vs( r) +V

v(r

)+

m 2;

(18a)

H 1=

12m2 n \Gamma r

2\Gamma i n Vv (r)

;ff

(1)\Delta

r o + ff

(1)\Delta

oe(2)

\Theta ^ rV

0v(r ) o ;

(18b)

where ^ris the unit

vector

in the

radial

direction.

Eq. (18a)

isthe Dirac equation

for particle

1with scalar and vector

poten tials

plus the mass

of the

hea vy quark,

particle

2. The

solutions

of the

Dirac

equation

with suc ha

poten

tial ha ve

been

extensiv

ely studied.

The op erators

nj (1)

2;j

(1)z ;K

;S

(2)z o

(19)

5 6

TABLE I. Values

of `and

_`for various

values of ^

`

_`

^1 ! 0

j1 \Gamma

12

j1 +

12

^1 ? 0

j1 +

12

j1 \Gamma

12

are aset

of mutually

comm uting op erators

whic hcomm

ute with

H 0,where

j(1) =

L+ S(1) ,S

(1) =

12\Sigma

(1) =

12fl

(1)5 ff

(1)

,K

(1) = fi(1)

(\Sigma

(1)\Delta

j(1)\Gamma

12) and

S(2) =

12oe (2) .

The eigenstates

of H 0can

then be lab elled

by the

corresp

onding set of quan

tum

num bers fn; j1; m j1;

^1 ;s 2g.

The

wa ve equation

asso ciated

with H 0can

then be

written as H 0\Psi

(0)n^ 1j 1m

j1

s2(

r) = W

(0)n^ 1j 1\Psi

(0)n^ 1j 1m

j1

s2(

r);

(20)

where \Psi

(0)n^

1j 1m

j1

s2(

r) = 0@

G n`j

1(

r)

r

Y

m j1 `12

j1 (\Omega )

iF n`j

1(

r)

r

Y

m j1_ `12

j1 (\Omega )

1A O/s 2;

(21)

with Y

m j1

`12

j1 (\Omega )

= X m `;s

1 o/

`m `; 12

s1 fifififi j1m

j1 AE

Y`m `(\Omega )

O/s 1;

(22)

and O/s 1and

O/s 2are

the Pauli

spinors

for particles

1and 2, resp

ectiv

ely .The

eigen value

^1 =\Sigma

(j 1+

12) can

be an ynonzero

integer. The values

of `and

_`

asso ciated

with various

values of ^1

are

displa

yed in Table

I.

Note that the zeroth

order inv arian

tmass

W

(0)n^ 1j 1is

determined

by n, ^1

and

j1, or equiv

alen tly by n, j1,

and

`. The

parit yof the Q_q bound

state isgiv en by

P = (\Gamma 1)

`+1

.

The first term

on the

righ thand

side of (18b)

isthe kinetic

energy of particle

2. Both

the first

and second

terms on the

righ thand

side of (18b)

comm ute with

the set of op erators

giv en in (19).

Ho wev

er, the

third

term do es

not

comm

ute

with an yof

these

op erators,

but instead

comm utes with

\Phi J 2;J

z;P\Psi

(23)

where J= j(1) + S(2)

and P isthe

parit yop erator.

The eigenstates

of the

total

hamiltoni an H = H 0+

H 1can

then be lab elled

by the

set of quan

tum num bers

fn; J;M

J;

Pg

.

The eigenstates

and eigenenergies

of the

hamitonian

H can

be calculated

directly .Ho wev er, the

ob jectiv

eof the calculations

presen ted here

isto pro duce

wa ve

functions

whic hcan

be used

in the

calculation

of form

factors

and deca y

constan ts as an

expansion

in po wers

of the

inv erse

of the

hea vy quark

mass m 2.

In order

to main

tain consistency

in this

expansion,

the masses

and wa ve functions

should be calculated

perturbativ ely .The

first order

correction

to the

quark

bound

state mass isgiv en by

W

(1)nJ P= Z

d3r \Psi

(0)

y n^ 1j 1J

M J(r

)H 1\Psi

(0)n^ 1j 1J

M J(r

);

(24)

where \Psi

(0)n^

1j 1J

M J(r

)= X

m j1

;s 2 o/

j1m

j1;

12 s 2 fifififiJ

M J AE

\Psi

(0)n^

1j 1m

j1

s2(

r):

(25)

The bound

state mass to first

order

is

W nJ P=

W

(0)n^ 1j 1+

W

(1)nJ P:

(26)

The scalar

and vector

poten tials in the

calculations

presen ted here

ha ve

the

form Vs( r) = br +c;

(27)

Vv (r)

=\Gamma

43

3Xi=1 ffir erf

(fl ir)

:

(28)

The vector

poten tial is, as in ref.

[4], based

on aparametrization

of the

running

QCD coupling

constan t.

B. Q

_Q mesons

The situation

for mesons

made of ahea

vy quark

and the corresp

onding antiquark is somewhat

more complicated.

The problem

is that

the prescription

of placing

particle 2on mass shell in the

Bethe-Salp

eter vertex

equation

(1) to

obtain the spectator

vertex equation

(4) is clearly

asymmetrical.

This results

in asp

ectator

vertex function

whic his no longer

an eigenfunction

of the

charge

conjugation op erator.

The solution

of this

problem

is to

construct

aset of coupled equations

for the

vertex

functions

whic hha ve either

particle

1or particle

7 8

2on mass shell [7]. These

equations

ha ve

been

solv ed in ref.

[7] for q_q-systems

con taining

only ligh tquarks.

Ho wev

er, since

we are

interested

in expanding

ab out

the infinite

mass limit,

this additional

complication

is not

necessary

and ahamiltonian

with leading

1=m Q

corrections

can be constructed

from (4). The starting

poin tis the spinor

decomp osition of the

Dirac

propagator

of particle

1in the meson

rest frame

S(1)F (^k 1;m

Q)

=

m Q

E( k; m Q) X

s

01 ^

u(1) (k; s01; m Q)

_u(1)

(k ;s

01;m

Q)

W\Gamma 2E (k ;m

Q)

+ ij

+

v(1)

(\Gamma k; s01;

m Q)

_v(1)

(\Gamma k; s01;

m Q)

W\Gamma ij

*: (29)

Using eqs. (29) and (3) in eq.

(4),

we can

write

[9]

\Gamma ( ^p;P

)= X s

01s

02 Z

d3k(2ss )3

m Q

E( k; m Q)

V( ^p;^k

;P )

iu (1) (k;

s01; m Q)

u(2)

(\Gamma k; s02;

m Q)\Psi

(+)s 01;s

02 (k

)

+v

(1) (\Gamma k; s01;

m Q)

u(2)

(\Gamma k; s02;

m Q)\Psi

(\Gamma ) s

01;s

02 (k

) j ;

(30)

where \Psi

(+)s 01;s

02 (k

)=

m Q

E( k; m Q)

_u(1)

(k; s01; m Q)

_u(2)

(\Gamma k; s02;

m Q)\Gamma (

^k;P )

W\Gamma 2E (k ;m

Q)

; (31)

and \Psi

(\Gamma

) s

01;s

02 (k

)=

m Q

E( k; m Q)

_v(1)

(\Gamma k; s01;

m Q)

_u(2)

(\Gamma k; s02;

m Q)\Gamma (

^k;P )

W

: (32)

Multiplying the terms

of (30)

to the

left resp ectiv

ely by

m Q

E( p; m Q)

_u(1)

(p ;s 1;m

Q)

_u(2)

(\Gamma p; s2;

m Q)

(33)

and m Q

E( p; m Q)

_v(1)

(\Gamma p; s1;

m Q)

_u(2)

(\Gamma p; s2;

m Q)

;

(34)

means that eq. (12)

can be rewritten

as the

pair

of coupled

integral equations

(W\Gamma 2E (p ;m

Q))

\Psi

(+)s1 ;s 2(p

)= X

s

01;s

02 Z

d3k(2ss )3 h

U

++s1;s

2;s

01;s 02 (p

;k ;W

)\Psi

(+)s 01;s

02 (k

)

+ U

+\Gamma s1;s

2;s

01;s 02 (p

;k ;W

)\Psi

(\Gamma ) s

01;s

02 (k

) i ;

(35)

and W \Psi

(\Gamma

) s1 ;s 2(p

)= X

s

01;s

02 Z

d3k(2ss )3 h

U

\Gamma +s1;s

2;s

01;s 02 (p

;k ;W

)\Psi

(+)s 01;s

02 (k

)

+ U

\Gamma \Gamma s1;s

2;s

01;s 02 (p

;k ;W

)\Psi

(\Gamma ) s

01;s

02 (k

) i ;

(36)

where U

++s1;s

2;s

01;s 02 (p

;k ;W

)=

m

2Q

E( p; m Q)

E(

k; m Q)

\Theta _u

(1) (p ;s 1)_u

(2) (\Gamma p; s2)

V( p; k; W )u

(1) (k;

s01) u(2)

(\Gamma k; s02)

;

(37)

U

+\Gamma s1;s

2;s

01;s 02 (p

;k ;W

)=

m

2Q

E( p; m Q)

E(

k; m Q)

\Theta _u

(1) (p ;s 1;m

Q)

_u(2)

(\Gamma p; s2)

V( p; k; W )v

(1)

(\Gamma k; s01)

u(2)

(\Gamma k; s02)

;

(38)

U

\Gamma +s1;s

2;s

01;s 02 (p

;k ;W

)=

m

2Q

E( p; m Q)

E(

k; m Q)

\Theta _v

(1) (\Gamma p; s1)

_u(2)

(\Gamma p; s2)

V( p; k; W )u

(1) (k;

s01) u(2)

(\Gamma k; s02)

;

(39)

U

\Gamma \Gamma s1;s

2;s

01;s 02 (p

;k ;W

)=

m

2Q

E( p; m Q)

E(

k; m Q)

\Theta _v

(1) (\Gamma p; s1;

m Q)

_u(2)

(\Gamma p; s2)

V( p; k; W )v

(1)

(\Gamma k; s01)

u(2)

(\Gamma k; s02)

:

(40)

These coupled

equations

can then

be reduced

to the

Pauli

spin space

and

expanded in po wers

of 1=m

Q.

In this

case,

only U

++

\Psi

(+)

con tributes

to order

1=m

2Q. Defining

aw ave

function

whic his an op erator

in the

spin

spaces

of both

particles as

\Psi = X s

01;s

02 O/

s

01O/

s

02\Psi

(+)s 01;s

02 ;

(41)

eq. (35)

becomes`

W\Gamma 2m Q\Gamma

p2mQ ' \Psi ( p)

= Z

d3 k

(2ss )3U

(p ;k )\Psi (

k);

(42)

9 10

whereU( p; k)

= Vs(

q2 )+

Vv (q

2)\Gamma

14m2Q h\Gamma V

0s(q

2) +V

0v(q 2) \Delta \Gamma

k2\Gamma

p2\Delta

2

+V s(q

2) i p2 + k2

+ oe(1)

\Delta p oe(1)

\Delta k + oe(2)

\Delta p oe(2)

\Delta k j

+V v(q

2) i p2 + k2\Gamma

oe(1) \Delta p oe(1)

\Delta k\Gamma

oe(2) \Delta p oe(2)

\Delta k j

\Gamma Vv (q

2) i

oe(1)

oe(1)

\Delta k + oe(1)

\Delta p oe(1)j

\Delta i oe(2)

oe(2)

\Delta k + oe(2)

\Delta p oe(2)ji

: (43)

Eq. (42) can then

be Fourier

transformed

to coordinate

space to extract

the

hamiltoni an

H = H 0+

H 1;

(44)

with H 1=

H c+

H hyp

+ H so+

H SR

+ H VR

;

(45)

where H 0=

\Gamma r

2mQ + Vs( r) + Vv (r)

+2 m Q;

(46a)

H c=

1m2Q ae 14 \Theta r 2V s(r

) \Lambda \Gamma

[V v(r

)\Gamma

Vs( r)]r

2+ [V

0s(r )\Gamma

V

0v(r

)]

@@r oe

; (46b)

H hyp

=

1m2Q ae

12 ^ 1r V 0v(r )\Gamma

V

00v(r

) *`

S\Delta

^rS\Delta

^r\Gamma

13 S 2 '

+\Theta r

2V v(r

) \Lambda `

13 S 2\Gamma

12 'oe

;

(46c)

H so

=

1 2m

2Qr

[3V

0v(r )\Gamma

V

0s(r

)]S\Delta

L;

(46d)

H S(V)R

=\Gamma

14m2Q \Theta r

2; \Theta

r

2;F

S(V)R

(x) \Lambda \Lambda ;

(46e)

and S = S(1)

+ S(2)

. Here

FS(V)R (x) is the

Fourier

transformation

of

dV s(v )(q

2)=d

q2 .F

or our

choices

of Vs(

r) and

Vv (r),

we find

H SR

=

bm2Q `

L22r \Gamma

3 @@ r \Gamma

r @

2@r2\Gamma

1r '\Gamma

cm2Q ` 1r @@ r +

12 @2@r 2\Gamma

L22r 2 '

;

(46f )

H VR

=

Vv (r)

2m

2Qr 2L

2

\Gamma

1 3m

2Q p

ss X i

ffi fli e\Gamma

fl2i r2 `

10 fl2i\Gamma

4fl

4ir 2+

8fl

2ir

@@r \Gamma

8r @@ r \Gamma

4 @

2@r2 '

;(46g)

Eq. (46a)

is the

nonrelativistic

hamiltonian for equal

mass quarks

in scalar

and vector

poten tials. H ccon

tains

cen tral

and orbital

con tributions.

H hyp

isthe

hyp erfine

interaction

consisting of atensor-force

term and aspin-spin

interaction.

H so

isthe

spin-orbit

interaction.

H SR

and

H VR

are scalar

and vector

retardation

terms asso ciated

with the third

term on the

righ t-hand

side of (43).

Note that

our spin-dep

enden tin teractions

H hyp

and

H so

ha ve

the

same

forms

as those

in

man yother

quark mo dels

(see for example:

[4,10,11]), but the spin-indep

enden t

interactions do not.

The spin-indep

enden tcorrection

includes H c,

H SR

and

H VR

.In

these

contributions, H SR

,H

VR

and

the term

[V

0s(r )\Gamma

V

0v(r

)]

@@r in H care

gauge

dep enden t. H SR

and

H VR

are from

the second

term in the

expansion

of V( Q

2)

=

V( q2 )\Gamma

14m2Q V

0(q

2) \Gamma

k2\Gamma

p2\Delta

2+O

(1=m

3Q). Had

we chosen

the Coulom

bgauge,

these terms would not exist.

Most other quark mo dels

do not

include

retarded

interactions. (Ref. [12] giv es another

expression

for the

retardation

effect.) We

will sho wthat

with the scalar

and vector

poten tials in (27)

and (28),

retardation

con tributions

are comparable

with the spin-dep

enden tin teractions.

The op erators

fH 0;L

2;S 2;J 2;J zg

where

J =

L+

S, are

aset

of mutually

comm uti ng

hermitian

op erators.

The eigenstates

of H 0can

then be lab elled

by

the corresp

onding set of quan

tum num bers

fn; L; S; J; M Jg

.The

wa ve equation

asso ciated

with H 0can

then be written

as

H 0\Psi

(0)nLS

JM J(r

)=

W

(0)nL \Psi

(0)nLS

JM J(r

);

(47)

where \Psi

(0)nLS

JM J(r

)=

unL (r)r Y

M J LS

J(\Omega )

;

(48)

11 12

and Y

M J LS

J(\Omega )

= X M L;M

S h

LM

LS

M Sj

JM

Ji

YLM

L(\Omega )

jSM Si

(49)

isthe spin spherical

harmonic.

The hyp erfine

interaction

(46c) mixes

states with \Delta L =\Sigma 2for

S =

1.

As

aresult, L is no

longer

ago od quan

tum num ber for solutions

of the

complete

hamiltoni an. Ho wev

er, these

states

ha ve

the

same

parit yand

charge quan tum

num bers since

P = (\Gamma

1)

L+1

and C = (\Gamma 1)

L+

Sfor

\Psi

(0)

.The

first-order

correction

to the

mass

can then

be written

as

W

(1)nJ PC

= Z

d3r \Psi

(0)

y nLS

JM J(r

)H 1\Psi

(0)nLS

JM J(r

)

= E c+

E hyp

+E

so+

E SR

+E

VR

:

(50)

where P = (\Gamma

1)

L+1

and C = (\Gamma 1)

L+

S.

The

bound

state mass to first

order

is

W nJ PC

= W

(0)nL

+ W

(1)nJ PC

(51)

One ma yalso

include

an annihilation

term in the

hamiltonian.

Ho wev

er, this

term first app ears

at order

ff

2s

m

2Q

[13]

[4], while

in our

mo del

the leading

spindep enden

teffects

are of order

ff s

m

2Q.

Since

ffs is small

in the

hea vy quark

system

(ff s(m

2c),

0:35 and ffs (m

2b),

0:22),

we exp ect the annihilation

effects on Q_Q

spectra to be small.II I. SOLUTION

OF THE

W AVE

EQUA

TIONS

A. Q_q sector

The Dirac

equation

(20) can be reduced

by using

the explicit

forms of the

zeroth order wa ve function

(21) and the Dirac

matrices

ff and

fialong

with the

iden tit y

oe(1) \Delta ^rY

m j1 `12

j1 (\Omega )

=\Gamma Y

m j1_

`12

j1 (\Omega )

(52)

to extract

the coupled

radial wa ve equations

[14]

dG n`j 1(r

)

dr

+

^1r G n`j

1(r

)=

(m 1+

Vs( r)\Gamma

Vv (r)

+ E

(0)n`j

1)F

n`j 1(r

);

(53)

dF n`j 1(r

)

dr

\Gamma

^1r Fn`j

1(r

)=

(m 1+

Vs( r) + Vv (r)\Gamma

E

(0)n`j

1)G

`j 1(r

);

(54)

where

E

(0)n`j

1=

W

(0)n^ 1j 1\Gamma

m 2:

(55)

We ha ve

obtained

three separate

numerical

solutions of these

coupled

equations using tw o differen

ttec hniques,

direct integration

and the matrix

diagonalization-v ariati onal tec hnique.1.

Dir ect Inte gration

This approac

huses stepping

tec hniques

to obtain

solutions

to the

differential equations.

Suc htec

hniques

are muc hmore

efficien tif an ylarge

asymptotic

damping of the

radial

wa ve

functions

can be extracted

and reduced

radial wa ve

equations can then

be integrated.

The scale

of the

asymptotic

variation of the

radial wa ve functions

isdetermined

by the

string

tension

bapp earing

in the

scalar

poten tial (27).

Defining

adimensionless

radial variable

ae= b1

=2r

,and

determining the asymptotic

beha vior of the

radial

wa ve

functions,

the reduced

wa ve

functions g(ae )and

f(ae )are

defined

in terms

of G and

F by

G( r) = g(ae

)e

\Gamma

12( ae2 +fl

ae);

F( r) = f(ae

)e

\Gamma

12( ae2 +fl

ae);

(56)

where fl= 2(m 1+

c)=b

1= 2,

and

cis the constan

tshift in the

scalar

poten tial.

Coupled equations

for the

reduced

wa ve functions

that result

are

`ddae \Gamma ae\Gamma

fl2 +

^1ae ' g(ae )=\Gamma

ff+ +ae

\Gamma V v(ae

) \Delta f(ae

);

(57)

`ddae \Gamma ae\Gamma

fl2 \Gamma ^1ae ' f(ae )=\Gamma

ff\Gamma +ae

+ V v(ae

) \Delta g(ae

);

(58)

where V v(ae

)=

Vv (r)

=b

1= 2,

ff\Sigma

=

fl2\Sigma

"and

"= E

(0)n`j

1=b

1= 2.

In order

to integrate

the differen

tial equations

it is necessary

to kno

w the

values of the

functions

and their

deriv ativ es at some

poin tand

then to ha ve

a

stepping algorithm

that predicts

the values

of the

functions

and their

deriv ativ es

at subsequen

tp oin ts.

The

values

of the

functions

and their

first deriv ativ es

at ae=

0are

obtained

by construction

of aseries

solution for the

functions

for

small ae. An

adaptiv

eRunge-Kutte

routine [15] isused

to integrate

the differen

tial

equations for increasing

values of ae.

Energy

eigen values

can be found

by adjusting

the value

of the

energy

un til

the

functions

ha ve

the

correct

asymptotic

beha vior

13 14

as determined

by an asymptotic

expansion of the

functions

at some

large finite

ae. This

pro cess

of finding

the eigenenergies

is called

the sho oting

metho d[15].

In the

calculations

sho wn

here,

the accuracy

of the

eigen

values

is increased

by

integrating up from

ae= 0and

do wn

from

some large finite aeto some

intermediate

poin twhere

the values

of g(ae

)and

f(ae )are

required

to matc

h.

Asecond variation on this

metho

dis to use

the reduced

radial wa ve equations

(57) and (58) to eliminate

f(ae )to

obtain

asecond

order differen

tial equation

for

g(ae ). This

equation

can then

be integrated

in amanner

similar to the

Shr" odinger

equation for the

Q_Q

sector.

2. Variational

Metho d

The starting

poin tfor

the `variational'

solution of eqs.

(53, 54) isthe

pair of

equations E

(0)n`jG

nj`( r) = (m

1+

Vs + Vv )G

nj`( r) +

^1\Gamma

1 r

F

nj_`(

r)\Gamma

dF

nj_`( r)

dr

;

E

(0)n`jF

nj_`( r) = (V v\Gamma

m 1\Gamma

Vs)F

nj_`( r) +

^1

+1r G

nj`( r) +

dG

nj`( r)

dr

;

F

nj_`(

r) =

Fn`j

(r)r ;G

nj;`( r) =

G n`j

(r)r

:

(59)

The functions

F and

G are

expanded

in aset

of orthonormal

basis functions

OEi`( r=%

)

Gnj `(r

)=

NXi=1 ffni OEi`(

r=% );

F

nj_`(

r) =

NXi=1 fini OEi_`(

r=% );

(60)

with Z 10 dr r2OE

i\Lambda `( r=%

)OE

k`(r

=%)

= ffii;k

:

(61)

%is the size

parameter

of the

wa ve

functions,

and is used

as the

variational

parameter in this

calculation.

Substituting the expansion

of eq.

(60)

into eq. (59),

multiplying

by OEk

\Lambda `(_`)(

r=%

)

and integrating,

leads to the

set of equations

E

(0)n`j

ffnk

=

NXi=1 o/

m 1+

Vs( r) + Vv (r) AE

k`;i`

ffni

+

NXi=1 o/

^1\Gamma 1 r AE

k`;i

_` fi

ni\Gamma

NXi=1 o/

ddr AE k`;i

_` fi

ni;

E

(0)n`j

fink

=

NXi=1 o/

Vv (r)\Gamma

m 1\Gamma

Vs( r) AE

k_` ;i_`

fini

+

NXi=1 o/

^1 +1r AE

k_`;i`

ffni +

NXi=1 o/

ddr AE k_` ;i`

ffni

;

(62)

where we use

the sym bolic

notation

o/ (r) AE

k` 1;i`

2 = Z

10 dr r2OE

k\Lambda `1( r=%

) (r)

OEi` 2(r

=%) :

(63)

The tw osets

of equations

represen ted by eq.

(62)

can be com

bined

into the single

eigen value

equation `\Omega m + Vs(

r) + Vv (r)\Gamma

Eff

\Omega ^ 1\Gamma

1 r\Gamma

ddr ff

\Omega ^ 1+1r

+

ddr ff

\Omega V v(r

)\Gamma

m\Gamma

Vs( r)\Gamma

Eff '`

fffi ' = 0:

(64)

The size of the

matrix

in eq.

(64) is 2N

\Theta 2N

. Solutions

to the

eq.

(59)

are obtained

by varying

the wa ve

function

size parameter

%, diagonalising

the

matrix in eq.

(64)

for eac hv alue

of %, and

searc hing for stationary

poin ts in the

eigen values

as functions

of %.

In principle,

ifthe size of the

expansion

basis N is

tak en to1

,solutions

obtained in this

wa yw

ould

be exact

and indep

enden tof

%. In practice,

the pro cedure

outlined

ab ove

iscarried

out for finite

N, and

N is

increased un til

the

eigen

values

are largely

indep enden tof %, for

some

reasonable

range in %.

With

this metho

d, the

low er N eigen

values

obtained

corresp ond to

negativ eenergy

states, while the higher

N eigen

values

are those

of interest

for

this problem.For this

problem

we ha ve

used

harmonic

oscillator wa ve

functions

for the

expansion, with N = 10

and

N = 20.

We compare

the numerical

solutions

that we obtain

using this pro cedure

with those

that are obtained

using the other

previously describ ed metho

ds. As exp ected,

the variational

solutions are better

for N = 20,

and

the eigen

values

are within

1% of those

obtained

by solving

the

equations by the

metho

ds describ

ed in the

previous

subsection.

15 16

0.0 0.5 1.0 1.5 2.0 2.5

1/ (Ge

V)

0.0 0.5 1.0 1.5 2.0 2.5 3.0En (GeV)

n=5 n=4 n=3 n=2 n=1 0.0 0.5 1.0 1.5 2.0 2.5

1/ (Ge

V)

0.0 0.5 1.0 1.5 2.0 2.5 3.0En (GeV)

n=5 n=4 n=3 n=2 n=1

FIG. 2. Energy

eigen values

as afunction

of

1%, for

N = 10

and

N = 20

.

B. Q

_Q sector

Using eq. (48)

in eq.

(47)

and defining

ae= b1

=2r

,the

differen

tial equation

for

the radial

wa ve function

is

^\Gamma 1_ `

d2dae 2\Gamma

L( L+

1) ae2

'+ V v(ae

)+

ae * unL

(ae) = "u nL

(ae)

;

(65)

where _ =

m Q=b

1= 2,

"=

(W

(0)nL\Gamma

2m Q\Gamma

c)=b

1= 2and

V v(ae

)=

Vv (r)

=b

1= 2.

Determining the asymptotic

beha vior of the

radial

wa ve

function,

the reduced

radial wa ve function

g(ae )can

be defined

by

unL (ae) = g(ae

)e

\Gamma _

12( 23ae

32\Gamma

"ae

12)

(66)

The app earance

of fractional

po wers

of aein

the argumen

tof the exp onen

tial

function in (66)

leads

to coefficien

ts with

fractional

po wers

of aein

the differen

tial

equation for g(ae ). This

complicates

the expansion

of the

reduced

radial wa ve

functions for small

and large

values

of ae.

It is,

therefore,

con venien

tto define

the

variable ,= ae1

=2.

The

differen

tial equation

for g(, )can

then be written

as

^\Gamma ,2

d2d, 2+ i

,\Gamma 2"_

12, 2+

4_,

4 j

dd,

+ i 4L (L +1)

+_

12(",

+2 ,3)\Gamma

_"

2, 2+

4_,

4V v(,

2) j*

g(, )=

0

(67)

This equation

can be used

to dev

elop

expansions

for small

and large

,to pro vide

boundary conditions for numerical

integration

of the

differen

tial equation.

Since the Runge-Kutte

metho dis designed

to integrate

systems of coupled

first-order differen tial equations

itis necessary

to reexpress

the differen

tial equation (67) as the

coupled

pair dd, g(,

)=

f(, );

(68)

and^ \Gamma ,

2dd,

+ i ,\Gamma 2"_

12, 2+

4_,

4 j* f(, )

+ i 4L (L +1)

+_

12(",

+2 ,3)\Gamma

_"

2, 2+

4_,

4V v(,

2) j

g(, )=

0:

(69)

This system

can then

be solv

ed by Runge-Kutte

integration and intermediatepoin tsho

oting

tec hniques.

IV. RESUL

TS

Once the zeroth-order

solutions are found,

the perturb

ed energies

can be

calculated using (24) and (50).

The masses

asso ciated

with the bound

states are

giv en by (26)

and (51).

These

dep end

on the

quark

masses

m u,

m s,

m cand

m b

as applicable

for eac hmeson;

the parameters

of the

scalar

poten tial (27)

band

c; and

the parameters

of the

vector

poten tial (28)

ffi and

fli for

i=

1;2 ;3.

The

mo del con tains

atotal

of tw elv

eparameters.

In obtaining

the results

sho wn

here,

the vector

poten tial parameters ff2 = 0:15

;

ff3 = 0:2

;

fl1 = 0:5

;

fl2 = 1:581

; fl3 = 15 :81

;

(70)

are fixed

at the

same

values

as giv en in ref.

[4]. The

remaining

vector poten tial

parameter ff1 isreexpressed

asff1= ffcr it\Gamma

ff2\Gamma

ff3 :

(71)

17 18

TABLE II. Parameters

of the

mo del.

parameter value commen ts

ff cr

it

0.674 limiting value of ff s

b 0.180 GeV

2

string tension

c 0.02 GeV

see eq. (27)

m u

0.258 GeV

m s

0.400 GeV

m c

1.53 GeV

m b

4.87 GeV

TABLE III. Fitted

meson spectra

for Q_q mesons.Mass

(GeV)

Meson J

P

theory

exp erimen

ta

D 0

\Gamma

1.85

1.87

D

\Lambda

1

\Gamma

2.02

2.01

D 1

1+ 2.41

2.42

D

\Lambda 2

2+ 2.46

2.46

B 0

\Gamma

5.28

5.28

B

\Lambda

1

\Gamma

5.33

5.33

D s

0

\Gamma

1.94

1.97

D

\Lambda s

1

\Gamma

2.13

2.11

B s

0

\Gamma

5.37

5.38

B

\Lambda s

1

\Gamma

5.43

5.43

aExp erimen

tal values

are quoted

[16] to the

nearest

10 MeV

due to am biguities

in

assigning the calculated

values to specific

charge states.

where ffcr it

is the

value

of the

running

coupling constan tat Q

2

=

0 as

parametrized in ref.

[4].

ffcr itand

the remaining

mo del

parameters

are adjusted

to fit the

masses

of a

selection of mesons.

The resulting

values are listed

in Table

II. The

fitted

meson

spectra for the

Q_q sector

are listed

in Table

III and

the fitted

meson

spectra

for

the Q

_Qare

listed in Table

IV. Additional

states whic hw ere not

used

in the

fitting pro cedure

were calculated

and adetailed

discussion of the

results

for the

Q_q and

Q_Q ispresen

ted in the

follo

wing

tw osubsections.

TABLE IV. Fitted

meson spectra

for Q

_Qmesons.

Mass

(GeV)

Meson J

PC

theory

exp erimen

t

jc 0

\Gamma +

3.00

2.98

J= (1S )

1

\Gamma \Gamma

3.10

3.10

O/ c0

0++ 3.44

3.42

O/ c1

1++ 3.50

3.51

O/ c2

2++ 3.54

3.56

J= (2S )

1

\Gamma \Gamma

3.73

3.69

\Upsilon (1 S)

1

\Gamma \Gamma

9.46

9.46

O/ b0(1

P)

0++

9.85

9.86

O/ b1(1

P)

1++

9.87

9.89

O/ b2(1

P)

2++

9.89

9.92

\Upsilon (2 S)

1

\Gamma \Gamma

10 .02

10 .02

O/ b0(2

P)

0++

10 .24

10 .24

O/ b1(2

P)

1++

10 .26

10 .26

O/ b2(2

P)

2++

10 .28

10 .27

\Upsilon (3 S)

1

\Gamma \Gamma

10 .39

10 .36

A. Q_q sector

For the Q_q sector,

the zeroth-order

eigenenergy E

(0)n`j

1=

W

(0)n^ 1j 1\Gamma

m 2is

indep enden tof the hea vy quark

mass, as would

be exp

ected

in the

hea vy quark

limit, where the hea vy quark

should

act as astatic

source. The zeroth-order

spectrum dep ends

only on the

ligh tquark

mass. The first-order

correction to the

mass W

(1)nJ P

is prop

ortional

to 1=m

2and

splits eac hof

the unp erturb

ed states.

These features

are illustrated

in Fig.

3whic

hsho ws W

(0)n^ 1j 1\Gamma

m 2for

a_u quark

as

solid lines and W nJ P\Gamma

m 2=

W

(0)n^ 1j 1+

W

(1)nJ P\Gamma

m 2with

ac quark

as the

hea vy

quark (dotdashed

lines) and with

ab quark

as the

hea vy quark

(dashed

lines).

Fig. 4is asimilar

spectrum where the ligh tquark

isno w an

_squark.

Note that to zeroth

order the ordering

of the

j1 = `\Sigma

1=2

states

isrev ersed

for the

`= 2states

in comparison

to the

`= 1states.

This phenomenon,

called

multiplet inv ersion,

has been

predicted

[17] for Q_q mesons

with m 2AE

m 1.

It

results from the dominance

of the

Thomas-precession

over the spin-dep

enden t

forces in this

limit.

For the states

presen ted here,

the root

mean

square

momen tum of the

zerothorder wa ve function

isappro ximately

0:9 GeV.

Clearly

,b oth

uand

squarks

are

very relativistic.

In addition,

itis possible

to obtain

some sense of the

con vergence

19 20

0- 1-

0- 1-

0- 1- 0- 1- 0+ 1+

0+ 1+

1+ 2+

1+ 2+

2-1-

2-1-

3-2- 3-2-

l=0 ,j=1

/2

l=1 ,j=1

/2

l=1 ,j=3

/2

l=2 ,j=3

/2

l=2 ,j=5

/2

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4GeV

W bum2

W cum2

W

(0)-m

2fo

rm eso nsw

ith uq uar k

FIG. 3. This

figure

sho ws W \Gamma m 2for

b_u and

c_u to the

zeroth

order and to the

first

order. l1 and

j1 are

the quan

tum num bers

for orbital

angular

momen tum and total

angular momen tum of the

_uquark.

The states

ha ve

been

lab elled

as J

P.

0- 1-

0- 1-

0- 1- 0- 1-0+ 1+ 0+ 1+

1+ 2+

1+2+

2-1- 2-1-

3-2- 3-2-

l=0 ,j=1

/2

l=1 ,j=1

/2

l=1 ,j=3

/2

l=2 ,j=3

/2

l=2 ,j=5

/2

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4GeV

W bsm2

W csm2

W

(0)-m

2fo

rm eso nsw

ith squ

ark

FIG. 4. This

figure

sho ws W \Gamma m 2for

b_s and

c_s to the

zeroth

order and to the

first

order. l1 and

j1 are

the quan

tum num bers

for orbital

angular

momen tum and total

angular momen tum of the

_uquark.

The states

ha ve

been

lab elled

as J

P.

21 22

1.85 2.50 2.98 3.37

2.02 2.

62 2.71 3.07 3.13

2.74 2.76 3.16 3.17

2.78 3.19 3.24

3.24 3.26

2.27 2.78 3.20

2.40 2.41 2.89 3.29 3.30

2.46 2.

94 3.00 3.34 3.38

3.01 3.03 3.39 3.41

3.03 3.41 3.46

0- 1- 2- 3- 4- 0+ 1+ 2+ 3+ 4+

1.60 2.00 2.40 2.80 3.20 3.60

this wor k

dat a

FIG. 5. c_u spectrum.

In this

figure,

solid lines represen

tthe results

of our

calculation

for the

masses

of c_u

mesons,

W ,to

the first

order

in the

perturbation;

dotted lines

represen tthe data.

of the

p=m

expansion

for the

corrections

to the

infinite-hea

vy-quark-mass

limit

since

pr msmc ,

12while

pr msmb ,

15. Therefore,

the higher-order

correction that

are neglected

here should

be considerably

larger for the

the cquark

than the

bquark. Indeed, this problem

will become

worse with increasing

nsince pr ms

should increase

with increasing

n. This

isseen

in the

shift

of the

0\Gamma states

relativ e

to the

unp erturb

ed states

whic hincreases

with n.

Figs. 5to 9sho w predictions

for the

masses

of Q_q

mesons,

W ,to

first

order

in the

perturbation

(solid lines).

In the

spectra

for mesons

with _uand

_squarks,

the available

data are plotted

for comparison

as dotted

lines. Ref. [16] has also

listed states D J(2

:440)

and D sJ(2

:573)

with uncertain

quan tum num bers.

We

believ ethey

corresp ond to the

state

1+ (2: 41)

in Fig.

5and

the state

2+ (2: 58)

in Fig.

6resp

ectiv ely .F

or the

b_c mesons,

calculated

masses from [4] are

plotted

because no data

exist at presen

t. For

the b_c mesons,

pr msmc ,

1. This

sho ws that

although the mass

of the

cquark

isrelativ ely large

itis quite

relativistic

in this

case.

1.94 2.61 3.09

2.13 2.73 2.82 3.19 3.25

2.86 2.88 3.28 3.29

2.90 3.31 3.36

3.37 3.39

2.38 2.90 3.32

2.51 2.52 3.00 3.01 3.41 3.42

2.58 3.06 3.12 3.46 3.50

3.13 3.15 3.52 3.53

3.16 3.54 3.58

0- 1- 2- 3- 4- 0+ 1+ 2+ 3+ 4+

1.60 2.00 2.40 2.80 3.20 3.60

this wor k

dat a

FIG. 6. c_s spectrum.

See caption

of Fig.

5.

23 24

5.28 5.

83 6.21 6.52

5.33 5.

87 5.97 6.24

5.96 5.98 6.31 6.32

5.97 6.32 6.38

6.36 6.39

5.65 6.06 6.39

5.69 6.10 6.42 6.43

5.71 6.12 6.19 6.44 6.49

6.18 6.20 6.48 6.50

6.18 6.49 6.55

0- 1- 2- 3- 4- 0+ 1+ 2+ 3+ 4+

5.00 5.40 5.80 6.20 6.60 7.00

this wor k

dat a

FIG. 7. b_u spectrum.

See caption

of Fig.

5.

5.37 5.

93 6.31 6.62

5.43 5.

97 6.07 6.34

6.07 6.08 6.41 6.42

6.08 6.42 6.49

6.47 6.49

5.75 6.17 6.50 5.79 5.80 6.20 6.21 6.52 6.53

5.82 6.22 6.30 6.54 6.60

6.28 6.31 6.59 6.61

6.29 6.60 6.65

0- 1- 2- 3- 4- 0+ 1+ 2+ 3+ 4+

5.00 5.40 5.80 6.20 6.60 7.00

this wor k

dat a

FIG. 8. b_s spectrum.

See caption

of Fig.

5.

25 26

6.26 6.85 7.

24 7.55

6.34 6.90 7.

01 7.28 7.35 7.58 7.64

7.02 7.03 7.36 7.65

7.04 7

.37 7.43 7.66 7.70

7.43 7.44 7.70 7.71

6.68 7.10 7.43 7.71 6.73 6.74 7.14 7.15 7.46 7.47 7.74

6.76 7.

16 7.24 7.48 7.54 7.76 7.80

7.24 7.25 7.54 7.55 7.80 7.81

7.25 7.55 7.60 7.81 7.85

0- 1- 2- 3- 4- 0+ 1+ 2+ 3+ 4+

6.00 6.40 6.80 7.20 7.60 8.00

this wor k

God frey and Isgu r

FIG. 9. b_c spectrum.

See caption

of Fig.

5.

In these

figures,

the results

are in go od

agreemen

twith the data,

whic hvindicates our choices

of poten

tials and parameters.

Ho wev

er, the

calculated

hyp erfine

splittings are all larger

than in the

data.

The agreemen

tis muc hb etter

in the

b-fla vored

mesons

than in the

c-fla vored

mesons.

There are three

possible

reasons

for this

discrepancy

.First, as has

been

men tioned

earlier,

this mo del isexp

ected

to work

better

for b-fla

vored

mesons

than for c-fla

vored

mesons

due to the

more

rapid con vergence

of the

nonrelativistic

expansion applied to the

hea vy quark.

Secondly ,these calculations

do not

include

an yeffects

asso ciated

with possible

strong deca yof the hea vy mesons.

The coupling

to these

strong

deca yc hannels

will result

in shifts

in the

meson

masses as well

as deca

ywidths

for hea vy mesons

ab ove

deca ythresholds.

These shifts will be greatest

near the deca ythresholds.

The third

possible

reason for the

large

hyp erfine

splittings

ma yha

ve its

origin

in the

parametrization

of ffs (r),

particularly

at small

r. While

man yfunctional

forms ma yb eused

for this

parametrization,

eac hform

ma yb eexp

ected

to lead

to quite

differen

t1 =m Q

con

tributions,

esp ecially

in the

hyp erfine

term.

This

question iscurren tly under

inv estigation.

The third

term on the

righ thand

side of (18b)

has off-diagonal

matrix elemen ts bet

ween

states

with j1 differing

by unit

yand

with `differing

by either

0

or 2. These

mixings

do not

affect

the spectrum

to order

1mQ but

should

result

in shifts

in some

states

at higher

order in all

of these

systems.

This should

be

particularly apparen tfor the 1+ states

whic hare

nearly

degenerate

to order

1mQ

for all Q_q

mesons

calculated

here.

One very interesting

asp ect

of this

calculation

is the

mapping

of our

mo del

on to

the

hea vy quark

effectiv etheory

,with aview to evaluating

some of the

parameters and dynamical

quan tities

(suc has

univ ersal

form factors)

of the

effectiv etheory

.While we do not

endea

vor to perform

suc ha

calculation

for all

suc hquan

tities here, some commen

ts are

merited.

Although we ha ve

included

all of the

1=m

Qterms

that arise from the spectator

equation, itis not clear

that these

corresp

ond to all

of the

1=m

Qterms

of HQET.

In particular,

in the

spectator

equation, the hea vy quark

is treated

as being

exactly on its mass

shell.

In con trast,

in HQET,

the hea vy quark

is allo

wed

to be

sligh

tly off its mass

shell (via the equation

p_ = m Qv

_+

k_ ), and

this

leads to terms

that ma yb eabsen

tfrom the form

ulation

presen ted here.

The full

ramifications of this

are also

under

inv estigation.

Un til this

question

is resolv

ed, we dare

not examine

quan tities

that are intimately bound up in the

1=m

Qstructure

of the

effectiv

etheory

or the

mo del.

We

can, ho wev

er, examine

quan tities

that dep end

only

on the

leading-order

structure

of the

mo del,

as we

believ

ethat

this is areasonably

accurate represen tation of

the effectiv

etheory

.In particular,

in the

effectiv

etheory

,one exp ects

that the

27 28

hea vy quark

should

act as astatic

color source.

This very imp ortan

tfeature

is

repro duced

in the

mo del,

as the

leading

dynamical

beha vior isdescrib

ed in terms

of aDirac

equation

for the

ligh tquark.

Tw oquan

tities of interest

in HQET

are

_\Lambda and

*1 ,whic

hare defined

by

M M

= m Q+

_\Lambda + O `

1mQ ' ;

\Omega M (v) fifi

_hQ (iD

)2h Q fifi

M (v) ff

= 2M

M* 1:

_\Lambda is crucial

for the

effectiv

etheory

,as it app

ears

as the

coefficien

tin the 1=m Q

expansion: the expansion

coefficien tis written

as

_\Lambda =m

Q.

_\Lambda is,

in essence,

the

con tribution

to the

mass

of the

meson

from the mass

and kinetic

energy of the

"bro wn muc

k". The

left hand

side of the

second

expression

ab ove

isprop

ortional

to the

kinetic

energy of the

hea vy quark.

The meson

states in the

bra and

ket

ab ove

are the leading

order represen

tation, and so corresp

ond to our

zerothorder calculation.

From our mo del,

we obtain

_\Lambda =

0:45

GeV for the

ground

state pseudoscalar/v

ector doublet,

and *1 = 0:67

GeV

2. These

values are in

reasonable agreemen twith other values

in the

literature

[3]. Further

asp ects

of

the relationship

of our

mo del

to HQET

are discussed

in the

conclusions.

B. Q

_Q sector

Figs. 10 and

11 sho w the

spectra

for c_c and

b_b mesons

as calculated

with

eqs. (44)-(51).

As before,

the calculated

masses are sho wn

as solid

lines and the

exp erimen

tal masses

as dotted

lines. The D

_Dand

B

_Bthresholds

are sho wn

as

horizon tal dotdashed

lines across

the Figs.

10 and

11 resp

ectiv

ely .Ref.

[16] has

also listed

states hc(1 P) with

mass 3:526 GeV and jc(2 S) with

mass 3:590 GeV.

We believ

ethey

corresp ond to the

states

21S 0(3

:67)

and 11P 1(3

:51)

in Fig.

10

resp ectiv

ely .

The b_b spectrum

isin quite

go od

agreemen

twith the data

for the

states

lying

belo wthe

BB threshold.

The agreemen

tdeteriorates

as the

masses

approac hand

cross the BB

threshold.

As argued

in the

previous

section, this ma yb ethe

result

of the

absence

of coupling

to strong

deca yc hannels.

The agreemen

tfor the c_c is

less satisfactory

.This ma yb ean

indication

of the

inadequacy

of the

truncation

of

the nonrelativistic

expansion at order

1m2Q. In both

cases the hyp erfine

splitting

of the

spin

triplet

states isto olarge.

Since the hyp erfine

tensor interaction

has non-zero

off diagonal

matrix elemen ts for

states

with spin 1and

with L differing

by 0or

2, there

should

be

mixings of states

suc has

3S 1with

3D 1and

3P 2with

3F 2.

These

mixings

do

11S 0(3.

00)

21S 0(3.

67)

31S 0(4.

13)

13S 1(3.

10)

23S 1(3.

73)

13D 1(3.

80)

33S 1(4.

18)

23D 1(4.

22)

43S 1(4.

56)

33D 1(4.

59)

11P 1(3.

51)

21P 1(3.

99)

13P 0(3.

44)

23P 0(3.

94)

13P 1(3.

50)

23P 1(3.

99)

13P 2(3.

54)

23P 2(4.

02)

13F 2(4.

06)

11D 2(3.

82)

21D 2(4.

24)

13D 2(3.

82)

23D 2(4.

24)

13D 3(3.

83)

23D 3(4.

24)

11F 3(4.

06)

13F 3(4.

06)

13F 4(4.

06)

DD thre sho ld

0-+ 1-- 1+

-

0+

+

1+

+

2+

+

2-+

2-- 3-- 3+

-

3+

+

4+

+

2.80 3.20 3.60 4.00 4.40 4.80Gev this wor k

dat a

FIG. 10. c_c spectra.

See caption

of Fig.

5.

not affect

the spectrum

to order

1m2Q but

should

result in shifts

in some

states

at

higher order in both

the b_b and

c_c spectra.

Table V sho

ws the

individual

con tributions

to the

masses

W of an

um ber

of

b_b states

from W

(0) ,E

c,E

hyp

,E so,

E SR

and

E VR

.The

retardation

con tributions

E SR

and

E VR

are clearly

gauge dep enden

tsince

they would

not app ear in the

Coulom bgauge.

E cis

also

gauge

dep enden

t. These

con tributions

ma yalso

be

sensitiv eto the choice

of quasip

oten tial prescription.

To this

order

E hyp

,E

so

should be indep

enden tof these

factors.

Note that the scalar

and vector

retardation con tributions

are of opp

osite

sign and therefore

tend to cancel.

Ho wev

er the

sum of these

con tributions

is comparable

with E hyp

and

E so.

The

assumption

that the scalar

retardation

poten tial dep ends

only on the

square

of the

exc hanged

four-mom entum Q

2is

uncon

trolled

and it is possible

to prop

ose forms

for this

retardation poten tial whic

hw ould

eliminate

the scalar

term altogether.

29 30

11S 0(9.

41)

21S 0(10

.00)

31S 0(10

.37) 13S 1(9.

46)

23S 1(10

.02)

13D 1(10

.14)

33S 1(10

.39)

23D 1(10

.46)

43S 1(10

.68)

33D 1(10

.73)

53S 1(10

.93)

63S 1(11

.15)

11P 1(9.

88)

21P 1(10

.27) 13P 0(9.

85)

23P 0(10

.24)

13P 1(9.

87)

23P 1(10

.26)

13P 2(9.

89)

23P 2(10

.28)

13F 2(10

.35) 11D 2(10

.15)

21D 2(10

.47)

13D 2(10

.15)

23D 2(10

.47)

13D 3(10

.15)

23D 3(10

.47) 11F 3(10

.36)

13F 3(10

.36)

13F 4(10

.36)

BB thr esh old

0-+ 1-- 1+

-

0+

+

1+

+

2+

+

2-+

2-- 3-- 3+

-

3+

+

4+

+

9.20 9.60 10.00 10.40 10.80 11.20Gev

this wor k

dat a

FIG. 11. b_b spectra.

See caption

of Fig.

5.

TABLE V. Zeroth

order and various

first order

interaction

energies in the

b_b spectrum

(GeV)

State W W

(0)

Ec Eh yp

Eso

ESR EVR ESR + EVR

11 S0

9.41

9.5315

-0.0602 -0.0367 0.0000 0.0072 -0.0297

-0.0224

13 S1

9.46

9.5315

-0.0602

0.0122 0.0000 0.0072 -0.0297

-0.0224

21 S0

10 .00

10 .0892

-0.0708

-0.0192 0.0000 0.0175 -0.0192

-0.0017

23 S1

10 .02

10 .0892

-0.0708

0.0064 0.0000 0.0175 -0.0192

-0.0017

31 S0

10 .37

10 .4511

-0.0839

-0.0146 0.0000 0.0302 -0.0160

0.0142

33 S1

10 .39

10 .4511

-0.0839

0.0049 0.0000 0.0302 -0.0160

0.0142

41 S0

10 .66

10 .7411

-0.0992

-0.0125 0.0000 0.0447 -0.0144

0.0303

43 S1

10 .68

10 .7411

-0.0992

0.0042 0.0000 0.0447 -0.0144

0.0303

51 S0

10 .91

10 .9928

-0.1162

-0.0113 0.0000 0.0608 -0.0135

0.0473

53 S1

10 .93

10 .9928

-0.1162

0.0038 0.0000 0.0608 -0.0135

0.0473

61 S0

11 .14

11 .2202

-0.1345

-0.0105 0.0000 0.0781 -0.0128

0.0653

63 S1

11 .15

11 .2202

-0.1345

0.0035 0.0000 0.0781 -0.0128

0.0653

11 P1

9.88

9.9438

-0.0610 -0.0023 0.0000 0.0126 -0.0169

-0.0043

13 P0

9.85

9.9438

-0.0610 -0.0074 -0.0243 0.0126 -0.0169

-0.0043

13 P1

9.87

9.9438

-0.0610

0.0049 -0.0121

0.0126 -0.0169

-0.0043

13 P2

9.89

9.9438

-0.0610 -0.0001 0.0121 0.0126 -0.0169

-0.0043

21 P1

10 .27

10 .3321

-0.0752

-0.0016 0.0000 0.0244 -0.0143

0.0101

23 P0

10 .24

10 .3321

-0.0752

-0.0056 -0.0182 0.0244 -0.0143

0.0101

23 P1

10 .26

10 .3321

-0.0752

0.0036 -0.0091

0.0244 -0.0143

0.0101

23 P2

10 .28

10 .3321

-0.0752

-0.0001 0.0091 0.0244 -0.0143

0.0101

11 D 2

10 .15

10 .2072

-0.0637

-0.0008 0.0000 0.0186 -0.0139

0.0047

13 D 1

10 .14

10 .2072

-0.0637

-0.0011 -0.0097 0.0186 -0.0139

0.0047

13 D 2

10 .15

10 .2072

-0.0637

0.0016 -0.0032

0.0186 -0.0139

0.0047

13 D 3

10 .15

10 .2072

-0.0637

-0.0001 0.0064 0.0186 -0.0139

0.0047

21 D 2

10 .47

10 .5277

-0.0792

-0.0006 0.0000 0.0315 -0.0125

0.0190

23 D 1

10 .46

10 .5277

-0.0792

-0.0009 -0.0080 0.0315 -0.0125

0.0190

23 D 2

10 .47

10 .5277

-0.0792

0.0013 -0.0027

0.0315 -0.0125

0.0190

23 D 3

10 .47

10 .5277

-0.0792

-0.0001 0.0053 0.0315 -0.0125

0.0190

11 F3

10 .36

10 .4164

-0.0717

-0.0004 0.0000 0.0250 -0.0124

0.0126

13 F2

10 .35

10 .4164

-0.0717

-0.0004 -0.0047 0.0250 -0.0124

0.0126

13 F3

10 .36

10 .4164

-0.0717

0.0008 -0.0012

0.0250 -0.0124

0.0126

13 F4

10 .36

10 .4164

-0.0717

-0.0001 0.0035 0.0250 -0.0124

0.0126

31 32

V. CONCLUSION

AND OUTLOOK

We ha ve

constructed

this mo del

for hea vy mesons

based on arelativistic

bound state equation,

namely the spectator

equation. The calculated

spectra are

in quite

go od

agreemen

twith the exp erimen

tal data.

The parameter

values we

ha ve

are

reasonable,

and comparable

to other

mo dels

of similar

typ e. The

mo del

isderiv ed by expanding

the spectator

equation in 1=M

Q,

where

M Qis

the mass

of

the hea vy quark.

This treatmen

tis exp ected

to work

better

for b-fla

vored

mesons

than for c-fla

vored

mesons

since in c-fla

vored

mesons,

v,

12c, but

in b-fla

vored

mesons, v,

15c, and

our results

confirm

this exp ectation.

The retardation

con tribution

to the

Q_Q

mesons,

whic his missing

in other

quark mo dels,

has anoticeable

effect. Annihilation

effects ha ve

been

neglected,

as

they are suppressed

by additional

po wers

of ffs (M

Q),

whic

his asmall

parameter.

In addition

to the

questions

curren tly being

inv estigated

(parametrization

of ffs (r),

1=m Q

terms),

this work

op ens

up man

ya ven ues

of inv estigation.

Of

primary imp ortance

isthe application

of the

mo del

to deca

ypro

cesses

of hea

vy

mesons. In particular,

the calculation

of the

Isgur-Wise

functions that describ

e

the semileptonic

deca ys, not

only

for deca

ys to pseudoscalars

and vectors,

but also

to excited

states, are of great

interest.

In HQET,

these form factors

are essen

tially

the overlaps

of the

appropriately

bo osted

wa ve

functions.

It will

be interesting

to see

ifthis

relationship

bet ween

the form

factors

and the wa ve functions

arises

in the

presen

tmo del, and ifso,

ho w.

In addition,

the slop eof

the Isgur-Wise

function for the

elastic

deca ys ma yalso

be calculated,

and various

HQET sum

rules chec ked.

The strong

and electromagnetic

deca ys ma yalso

be treated

with the wa ve

functions that we ha ve.

These

are particularly

interesting for the

D

\Lambda and

D

\Lambda s

states, as the

former

lie so close

to the

Dss threshold,

while the latter

lie belo

w

the DK

threshold,

and thus

deca yradiativ

ely .In

addition,

quan tities

suc has

meson deca yconstan

ts ma

yalso

be evaluated.

ACKNO WLEDGEMENTS

The authors

would lik eto

ackno

wledge

man yuseful

con versations

with Franz

Gross and Nathan

Isgur. This work was supp orted

by the

Departmen

tof Energy

under con tracts

DE-A C05-84ER40150

and DE-F

G05-94ER40832

,and by the

National Science

Foundation

under the National

Young Inv estigator

program.

[1] N. Isgur

and M. B. Wise,

Ph ys.

Lett.

232B ,113 (1989);

237B ,527 (1990).

[2] N. Isgur

and M. B. Wise,

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,W orld

Scien tific (1992),

Sheldon

Stone, ed.,

p. 158.

[3] M. Neub

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Rep.

245 ,259

(1994).

[4] S. Go

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and N. Isgur,

Ph ys.

Rev.

D 32 ,189

(1985).

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Ph ys.

Rev.

186 ,1448

(1969).

[6] F. Gross,

J. W.

Van

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and K. Holinde,

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C 41

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D 43 ,2401

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Ph ys.

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C 26 ,2203

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W. Buc k, Phd.

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33 34

