FERMILAB–PUB–94/032–T

hep-ph/9402210

Mesons with Beauty and Charm: Spectroscopy

Estia J. Eichten∗ and Chris Quigg†

Theoretical Physics Department

Fermi National Accelerator Laboratory

P.O. Box 500, Batavia, Illinois 60510

(January 21, 1998)

Abstract

Applying knowledge of the interaction between heavy quarks derived from

the study of cc and bb bound states, we calculate the spectrum of cb mesons.

We compute transition rates for the electromagnetic and hadronic cascades

that lead from excited states to the 1S0 ground state, and briefly consider the

prospects for experimental observation of the spectrum.

PACS numbers: 14.40Lb, 14.40Nd, 13.40Hq, 13.25-k

Typeset using REVTEX

∗Internet address: eichten@fnal.gov

†Internet address: quigg@fnal.gov

1



I. INTRODUCTION

The copious production of b quarks in Z0 decays at the Large Electron-Positron collider

(LEP) and in 1.8-TeV proton-antiproton collisions at the Fermilab Tevatron opens for study

the rich spectroscopy of mesons and baryons beyond B+u and B
0
d. In addition to B

0
s and Λ

0
b,

which have already been widely discussed, a particularly interesting case is the spectrum of

cb states and its ground state, the B+c meson [1].

Even more than their counterparts in the J/ψ and Υ families, the cb states that lie below

the (BD) threshold for decay into a pair of heavy-flavored mesons are stable against strong

decay, for they cannot annihilate into gluons. Their allowed decays, by E1 or M1 transitions

or by hadronic cascades, lead to total widths that are less than a few hundred keV. All

decay chains ultimately reach the 1S0 ground state Bc, which decays weakly. It may be

possible, in time, to map out the excitation spectrum by observing photons or light hadrons

in coincidence with a prominent decay of the Bc [2]. This would test our understanding of

the force between heavy quarks.

The weak decays of the cb ground state will be of particular interest because the influence

of the strong interaction can be estimated reliably [3]. The deep binding of the heavy quarks

within the Bc means that the spectator picture is misleading. Taking proper account of

binding energy, we expect a rather long lifetime that implies easily observable secondary

vertices. The deep binding also affects the Bc branching fractions and leads us to expect

that final states involving ψ will be prominent. The modes ψπ+, ψa+1 , ψρ
+, ψD+s , and ψ`

+ν`

will serve to identify Bc mesons and determine the Bc mass and lifetime.

In this Article, we present a comprehensive portrait of the spectroscopy of the Bc meson

and its long-lived excited states. In Section II, we estimate the mass of the Bc in the

framework of nonrelativistic quarkonium quantum mechanics and calculate the spectrum of

cb states in detail. In Section III, we compute rates for the prominent radiative decays of the

excited states and estimate rates and spectra of the hadronic cascades (cb)i → ππ+(cb)f and

(cb)i → η + (cb)f. Using this information, we outline a strategy for partially reconstructing

2



the cb spectrum. A brief summary appears in Section IV.

II. THE SPECTRUM OF Bc STATES

A. The Mass of Bc

Both in mass and in size, the mesons with beauty and charm are intermediate between

the cc and bb states. Estimates of the Bc mass can, consequently, be tied to what is known

about the charmonium and Υ families. To predict the full spectrum and properties of cb

states, we rely on the nonrelativistic potential-model description of quarkonium levels. The

interquark potential is known rather accurately in the region of space important for the J/ψ

and Υ families [4–6], which spans the distances important for cb levels. This region lies

between the short-distance Coulombic and long-distance linear behavior expected in QCD.

We consider four functional forms for the potential that give reasonable accounts of the cc

and bb spectra: the QCD-motivated potential [7] given by Buchmüller and Tye [8], with

mc = 1.48 GeV/c
2 mb = 4.88 GeV/c

2 ; (2.1)

a power-law potential [9],

V (r) = −8.064 GeV + (6.898 GeV)(r · 1 GeV)0.1 , (2.2)

with

mc = 1.8 GeV/c
2 mb = 5.174 GeV/c

2 ; (2.3)

a logarithmic potential [10],

V (r) = −0.6635 GeV + (0.733 GeV) log (r · 1 GeV) , (2.4)

with

mc = 1.5 GeV/c
2 mb = 4.906 GeV/c

2 ; (2.5)

3



and a Coulomb-plus-linear potential (the “Cornell potential”) [4],

V (r) = −
κ

r
+

r

a2
, (2.6)

with

mc = 1.84 GeV/c
2 mb = 5.18 GeV/c

2 (2.7)

κ = 0.52 a = 2.34 GeV−1 . (2.8)

We solve the Schrödinger equation for each of the potentials to determine the position

of the 1S center of gravity for cc, cb, and bb. The 3S1 –
1S0 splitting of the i̄ ground state

is given by

M(3S1) −M(
1S0) =

32παs|Ψ(0)|
2

9mimj
. (2.9)

The hyperfine splitting observed in the charmonium family [1],

M(J/ψ) −M(ηc) = 117 MeV/c
2 , (2.10)

fixes the strong coupling constant for each potential. We neglect the variation of αs with

momentum and scale the splitting of cb and bb from the charmonium value (2.10). The

resulting values of vector and pseudoscalar masses are presented in Table I. Predictions

for the cb ground-state masses depend little on the potential. The Bc and B
∗
c masses and

splitting lie within the ranges quoted by Kwong and Rosner [11] in their survey of techniques

for estimating the masses of the cb ground state. They find

6.194 GeV/c2 <∼ MBc
<
∼ 6.292 GeV/c

2 , (2.11)

and

6.284 GeV/c2 <∼ MB∗c
<
∼ 6.357 GeV/c

2 , (2.12)

with

65 MeV/c2 <∼ MB∗c −MBc
<
∼ 90 MeV/c

2 . (2.13)

4



We take

MBc = 6.258 ± 0.020 GeV/c
2 (2.14)

as our best guess for the interval in which Bc will be found [12].

We shall adopt the Buchmüller-Tye potential [8] for the detailed calculations that follow,

because it has the correct two-loop short-distance behavior in perturbative QCD.

B. Excited States

The interaction energies of a heavy quark-antiquark system probe the basic dynamics of

the strong interaction. The gross structure of the quarkonium spectrum reflects the shape

of the interquark potential. In the absence of light quarks, the static energy explicitly ex-

hibits linear confinement at large distance. Further insight can be obtained by studying

the spin-dependent forces, which distinguish the electric and magnetic parts of the interac-

tions. Within the framework of quantum chromodynamics, the nature of the spin-dependent

forces was first studied nonperturbatively by Eichten and Feinberg [13,14]. Gromes [15] sub-

sequently added an important constraint that arises from boost-invariance of the QCD forms

[16]. One-loop perturbative QCD calculations for the spin-dependent interactions in a meson

composed of two different heavy quarks have also been carried out [17–19].

The spin-dependent contributions to the cb masses may be written as

∆ =
4∑
k=1

Tk , (2.15)

where the individual terms are

T1 =
〈~L ·~si〉

2m2i
T̃1(mi,mj) +

〈~L ·~sj〉

2m2j
T̃1(mj,mi)

T2 =
〈~L ·~si〉

mimj
T̃2(mi,mj) +

〈~L ·~sj〉

mimj
T̃2(mj,mi) (2.16)

T3 =
〈~si ·~sj〉

mimj
T̃3(mi,mj)

T4 =
〈Sij〉

mimj
T̃4(mi,mj) ,

5



and the tensor operator is

Sij = 4 [3(~si · n̂)(~sj · n̂) −~si ·~sj] . (2.17)

In Eq. (2.16) and (2.17), ~si and ~sj are the spins of the heavy quarks, ~L is the orbital angular

momentum of quark and antiquark in the bound state, and n̂ is an arbitrary unit vector.

The total spin is ~S = ~si + ~sj.

The leading contributions to the T̃k have no explicit dependence on the quark masses. As-

suming that the magnetic interactions are short-range (∝〈r−3〉) and thus can be calculated

in perturbation theory, we have

T̃1(mi,mj) = −

〈
1

r

dV

dR

〉
+ 2T̃2(mi,mj)

T̃2(mi,mj) =
4αs
3
〈r−3〉 (2.18)

T̃3(mi,mj) =
32παs

9
|Ψ(0)|2

T̃4(mi,mj) =
αs
3
〈r−3〉 .

The connection between T̃1 and T̃2 is Gromes’s general relation; the other equations reflect

the stated approximations.

For quarkonium systems composed of equal-mass heavy quarks, the total spin S is a good

quantum number and LS coupling leads to the familiar classification of states as 2S+1LJ,

where ~J = ~L + ~S [20]. The calculated spectra are compared with experiment in Table II

(for the ψ family) and Table III (for the Υ family). Overall, the agreement is satisfactory.

Typical deviations in the charmonium system are less than about 30 MeV; deviations in

the upsilon system are somewhat smaller. The differences between calculated and observed

spectra suggest that the excitation energies in the cb̄ system can be predicted within a few

tens of MeV.

The leptonic decay rate of a neutral (QQ̄) vector meson V 0 is related to the Schrödinger

wave function through [23,24]

Γ(V 0 → e+e−) =
16πNcα

2e2Q
3

|Ψ(0)|2

M2V

(
1 −

16αs
3π

)
, (2.19)

6



where Nc = 3 is the number of quark colors, eQ is the heavy-quark charge, and MV is the

mass of the vector meson. The resulting leptonic widths, evaluated without QCD corrections,

are tabulated in Tables II and III. Within each family, the leptonic widths are predicted in

proper proportions, but are larger than the observed values. The QCD correction reduces

the magnitudes significantly; the amount of this reduction is somewhat uncertain, because

the first term in the perturbation expansion is large [25].

For unequal-mass quarks, it is more convenient to construct the mass eigenstates by jj

coupling, first coupling ~L+~sc = ~Jc and then adding the spin of the heavier quark, ~sb+ ~Jc = ~J.

The level shifts ∆(J) for the L = 1 states with (Jc =
3
2
,J = 2) and (Jc =

1
2
,J = 0) are

∆(2) =

(
1

4m2c
+

1

4m2b

)
T̃1 +

1

mbmc
T̃2 −

2

5mbmc
T̃4 (2.20)

∆(0) = −

(
1

2m2c
+

1

2m2b

)
T̃1 −

2

mbmc
T̃2 −

4

mbmc
T̃4 .

For a given principal quantum number, the two (L = 1,J = 1) cb states with Jc =
1
2

and 3
2

are mixed in general. The elements of the mixing matrix are

∆
(1)
3
2

3
2

=

(
1

4m2c
−

5

12m2b

)
T̃1 −

1

3mbmc
T̃2 +

2

3mbmc
T̃4

∆
(1)
3
2

1
2

= ∆
(1)
1
2

3
2

= −

√
2

6m2b
T̃1 −

√
2

3mbmc
T̃2 +

2
√

2

3mbmc
T̃4 (2.21)

∆
(1)
1
2

1
2

=

(
−

1

2m2c
+

1

6m2b

)
T̃1 −

2

3mbmc
T̃2 +

4

3mbmc
T̃4 .

Two limiting cases are familiar.

(i) With equal quark masses mb = mc ≡ m, the level shifts become

∆(2) =
1

2m2
T̃1 +

1

m2
T̃2 −

2

5m2
T̃4 (2.22)

∆(0) = −
1

m2
T̃1 −

2

m2
T̃2 −

4

m2
T̃4 ,

while the mixing matrix becomes

∆(1) =


 1

√
2

√
2 2



(
−T̃1 − 2T̃2 + 4T̃4

6m2

)
. (2.23)

7



The mass eigenstates are the familiar 1P1 and
3P1 states of the LS coupling scheme. In this

basis, they may be written as

|1P1〉 = −
√

2
3
|Jc =

3
2
〉 +

√
1
3
|Jc =

1
2
〉 (2.24)

|3P1〉 =
√

1
3
|Jc =

3
2
〉 +

√
2
3
|Jc =

1
2
〉

with eigenvalues 
 λ(

1P1)

λ(3P1)


 =


 0

3



(
−T̃1 − 2T̃2 + 4T̃4

6m2

)
. (2.25)

The position of the 1P1 level coincides with the centroid [5∆
(2) + 3λ(3P1) + ∆

(0)]/9 of the

3PJ levels.

(ii) In the heavy-quark limit, mb →∞, the level shifts of the J = 0, 2 levels become

∆(2) =
1

4m2c
T̃1 (2.26)

∆(0) = −
1

2m2c
T̃1 ,

while the mixing matrix becomes

∆(1) =


 1 0

0 −2



(
T̃1

4m2c

)
. (2.27)

The Jc =
3
2

and Jc =
1
2

states separate into degenerate pairs, as expected on the basis of

heavy-quark symmetry [26].

In the cb system, we label the mass eigenstates obtained by diagonalizing the matrix

(2.21) as n(1+) and n(1+′). For the 2P1 levels, the mixing matrix is

∆(2P) =


 −1.85 −2.80
−2.80 −4.23


 MeV , (2.28)

with eigenvectors

|2(1+)〉 = 0.552|Jc =
3
2
〉 + 0.833|Jc =

1
2
〉 (2.29)

|2(1+′)〉 = −0.833|Jc =
3
2
〉 + 0.552|Jc =

1
2
〉

8



and eigenvalues

λ2 = −6.09 MeV (2.30)

λ′2 = 0.00057 MeV .

For the 3P1 levels, the mixing matrix is

∆(3P) =


 −0.13 −2.54
−2.54 −6.91


 MeV , (2.31)

with eigenvectors

|3(1+)〉 = 0.316|Jc =
3
2
〉 + 0.949|Jc =

1
2
〉 (2.32)

|3(1+′)〉 = −0.949|Jc =
3
2
〉 + 0.316|Jc =

1
2
〉

and eigenvalues

λ3 = −7.76 MeV (2.33)

λ′3 = 0.711 MeV .

For the 4P1 levels, the mixing matrix is

∆(4P) =


 0.71 −2.44
−2.44 −8.31


 MeV , (2.34)

with eigenvectors

|4(1+)〉 = 0.245|Jc =
3
2
〉 + 0.969|Jc =

1
2
〉 (2.35)

|4(1+′)〉 = −0.969|Jc =
3
2
〉 + 0.245|Jc =

1
2
〉

and eigenvalues

λ4 = −8.93 MeV (2.36)

λ′4 = 1.32 MeV .

The calculated spectrum of cb states is presented in Table IV and Figure 1. Our spectrum

is similar to others calculated by Eichten and Feinberg [14] in the Cornell potential [4], by

9



Gershtĕın et al. [27] in the power-law potential (2.2), and by Chen and Kuang [28] in their

own version of a QCD-inspired potential. Levels that lie below the BD flavor threshold, i.e.,

with M < MD +MB = 7.1431±0.0021 GeV/c
2, will be stable against fission into heavy-light

mesons.

C. Properties of cb Wave Functions at the Origin

For quarks bound in a central potential, it is convenient to separate the Schrödinger

wave function into radial and angular pieces, as

Ψn`m(~r) = Rn`(r)Y`m(θ,φ) , (2.37)

where n is the principal quantum number, ` and m are the orbital angular momentum and

its projection, Rn`(r) is the radial wave function, and Y`m(θ,φ) is a spherical harmonic [29].

The Schrödinger wave function is normalized,

∫
d3~r|Ψn`m(~r)|

2 = 1 , (2.38)

so that

∫ ∞
0

r2dr|Rn`(r)| = 1 . (2.39)

The value of the radial wave function, or its first nonvanishing derivative at the origin,

R
(`)
n` (0) ≡

d`Rn`(r)

dr`

∣∣∣∣∣
r=0

, (2.40)

is required to evaluate pseudoscalar decay constants and production rates through heavy-

quark fragmentation [30]. The quantity |R
(`)
n` (0)|

2 is presented for four potentials in Table

V. The stronger singularity of the Cornell potential is reflected in spatially smaller states.

The pseudoscalar decay constant fBc, which will be required for the discussion of anni-

hilation decays cb̄ → W + → final state, is defined by

〈0|Aµ(0)|Bc(q)〉 = ifBcVcbqµ , (2.41)

10



where Aµ is the axial-vector part of the charged weak current, Vcb is an element of the

Cabibbo-Kobayashi-Maskawa quark-mixing matrix, and qµ is the four-momentum of the Bc.

The pseudoscalar decay constant is related to the ground-state cb wave function at the origin

by the van Royen-Weisskopf formula [23] modified for color,

f2Bc =
12|Ψ100(0)|

2

M
=

3|R10(0)|
2

πM
. (2.42)

In the nonrelativistic potential models we have considered to estimate MBc and MB∗c , we

find

fBc =




500 MeV (Buchmüller-Tye potential [8])

512 MeV (power-law potential [9])

479 MeV (logarithmic potential [10])

687 MeV (Cornell potential [4]).

(2.43)

Even after QCD radiative corrections of the size suggested by the comparison of computed

and observed leptonic widths for J/ψ and Υ, fBc will be significantly larger than the pion

decay constant, fπ = 131.74 ± 0.15 MeV [1]. The compact size of the cb̄ system enhances

the importance of annihilation decays.

III. TRANSITIONS BETWEEN cb STATES

As in atomic physics, it is the spectral lines produced in cascades from excited states to

the readily observable Bc ground state that will reveal the cb̄ level scheme. As in the J/ψ

and Υ quarkonium families, the transitions are mostly radiative decays. A few hadronic

cascades, analogs of the 23S1 → 1
3S1ππ transition first observed in charmonium, will also

be observable.

A. Electromagnetic Transitions

Except for the magnetic-dipole (spin-flip) transition between the ground-state B∗c and

Bc, only the electric dipole transitions are important for mapping the cb̄ spectrum.

11



1. Electric Dipole Transitions

The strength of the electric-dipole transitions is governed by the size of the radiator and

the charges of the constituent quarks. The E1 transition rate is given by

ΓE1(i → f + γ) =
4α <eQ>

2

27
k3(2Jf + 1)|〈f|r|i〉|

2Sif , (3.1)

where the mean charge is

<eQ>=
mbec −mceb
mb + mc

, (3.2)

k is the photon energy, and the statistical factor Sif = Sfi is as defined by Eichten and

Gottfried [31]. Sif = 1 for
3S1 →

3PJ transitions and Sif = 3 for allowed E1 transitions

between spin-singlet states. The statistical factors for d-wave to p-wave transitions are

reproduced in Table VI for convenience. The E1 transition rates and photon energies in the

cb̄ system are presented in Table VII.

2. Magnetic Dipole Transitions

The only decay mode for the 13S1 (B
∗
c ) state is the magnetic dipole transition to the

ground state, Bc. The M1 rate for transitions between s-wave levels is given by

ΓM1(i → f + γ) =
16α

3
µ2k3(2Jf + 1)|〈f|j0(kr/2)|i〉|

2 , (3.3)

where the magnetic dipole moment is

µ =
mbec −mceb

4mcmb
(3.4)

and k is the photon energy. Rates for the allowed and hindered M1 transitions between

spin-triplet and spin-singlet s-wave cb states are given in Table VIII. The M1 transitions

contribute little to the total widths of the 2S levels. Because it cannot decay by annihilation,

the 13S1 cb level, with a total width of 135 eV, is far more stable than its counterparts in

the cc and bb systems, whose total widths are 68 ± 10 keV and 52.1 ± 2.1 keV, respectively

[1].

12



B. Hadronic Transitions

A hadronic transition between quarkonium levels can be understood as a two-step process

in which gluons first are emitted from the heavy quarks and then recombine into light

hadrons. Perturbative QCD is not directly applicable, because the energy available to the

light hadrons is small and the emitted gluons are soft. Nevertheless, the final quarkonium

state is small compared to the system of light hadrons and moves nonrelativistically in the

rest frame of the decaying quarkonium state. A multipole expansion of the color gauge field

converges rapidly and leads to selection rules, a Wigner-Eckart theorem, and rate estimates

for hadronic transitions [32]. The recombination of gluons into light hadrons involves the

full strong dynamics and can only be modeled. The general structure of hadronic-cascade

transitions and models for the recombination of gluons into light hadrons can be found in a

series of papers by Yan and collaborators [33–36].

The hadronic transition rates for an unequal-mass QQ̄′ system differ in some details

from the rates for an equal-mass QQ̄ system with the same reduced mass. The relative

strengths of various terms that contribute to magnetic-multipole transitions are modified

because of the unequal quark and antiquark masses. The electric-multipole transitions are

only sensitive to the relative position of the quark and antiquark and will be unchanged in

form.

As in the cc̄ and bb̄ systems, the principal hadronic transitions in the cb̄ system involve

the emission of two pions. Electric-dipole contributions dominate in these transitions, and

so the equal-mass results apply directly. The initial quarkonium state is characterized by

its total angular momentum J′ with z-component M′, orbital angular momentum `′, spin

s′, and other quantum numbers collectively labelled by α′. The corresponding quantum

numbers of the final quarkonium state are denoted by the unprimed symbols. Since the

transition operator is spin-independent, the initial and final spins are the same: s′ = s.

Because the gauge-field operators in the transition amplitude do not depend on the heavy-

quark variables, the transition operator is a reducible second-rank tensor, which may be

13



decomposed into a sum of irreducible tensors with rank k = 0, 1, 2. The differential rate [33]

for the E1–E1 transition from the initial quarkonium state Φ′ to the final quarkonium state

Φ and a system of n light hadrons, denoted h, is given by

dΓ

dM2
(Φ′ → Φ + h) = (2J + 1)

2∑
k=0



k `′ `

s J J′




2

Ak(`
′,`) , (3.5)

where M2 is the invariant mass squared of the light hadron system, { } is a 6-j symbol,

and Ak(`
′,`) is the contribution of the irreducible tensor with rank k. The Wigner-Eckart

theorem (3.5) yields the relations among two-pion transition rates given in Table IX.

The magnitudes of the Ak(`
′,`) are model-dependent. Since the A1 contributions are

suppressed in the soft-pion limit [33], we will set A1(`
′,`) = 0. For some of the remaining

rates we can use simple scaling arguments from the measured rates in QQ̄ systems [37]. The

amplitude for an E1–E1 transition depends quadratically on the interquark separation, so

the scaling law between a QQ̄′ and the corresponding QQ̄ system states is given by [32,33]:

Γ(QQ̄′)

Γ(QQ̄)
=
〈r2(QQ̄′)〉2

〈r2(QQ̄)〉2
, (3.6)

up to possible differences in phase space. The measured values for the ψ′ → ψ + ππ,

Υ′ → Υ + ππ, and ψ(3770) → ψ + ππ transition rates allow good scaling estimates for the

2S → 1S + ππ and 3D → 1S + ππ transitions in the cb̄ system. We have estimated the

remaining transition rates by scaling the bb̄ rates calculated by Kuang and Yan [34] in their

Model C, which is based on the Buchmüller-Tye potential [8]. The results are shown in

Table X.

Chiral symmetry leads to a universal form for the normalized dipion spectrum [41],

1

Γ

dΓ

dM
= Constant ×

|~K|

M2Φ′
(2x2 − 1)2

√
x2 − 1 , (3.7)

where x = M/2mπ and

|~K| =

√
M2Φ′ − (M + MΦ)

2
√
M2Φ′ − (M−MΦ)

2

2MΦ′
(3.8)

14



is the three-momentum carried by the pion pair. The normalized invariant-mass distribution

for the transition 23S1 → 1
3S1 + ππ is shown in Figure 2 for the cc̄, cb̄, and bb̄ families. The

soft-pion expression (3.7) describes the depletion of the dipion spectrum at low invariant

masses observed in the transitions ψ(2S) → ψ(1S)ππ [42] and Υ(2S) → Υ(1S)ππ [43], but

fails to account for the Υ(3S) → Υ(1S)ππ and Υ(3S) → Υ(2S)ππ spectra [44]. We expect

the 3S levels to lie above flavor threshold in the cb̄ system.

By the Wigner-Eckart theorem embodied in Eq. (3.5), the invariant mass spectrum

in the decay Bc(2S) → Bc(1S) + ππ should have the same form (3.7) as the B
∗
c (2S) →

B∗c (1S) + ππ transition. Braaten, Cheung, and Yuan [30] have calculated the probability

for a high-energy b̄ antiquark to fragment into the cb̄ s-waves as 3.8 × 10−4 for b̄ → Bc(1S),

5.4 × 10−4 for b̄ → B∗c (1S), 2.3 × 10
−4 for b̄ → Bc(2S), and 3.2 × 10

−4 for b̄ → B∗c (2S).

Given the excellent experimental signatures for Bc(1S) decay and the favorable prospects

for Bc(2S) production in high-energy proton-antiproton collisions, it may be possible to

observe the 0 → 0 transition for the first time in the Bc family.

The 23S1 → 1
3S1 + η transition has been observed in charmonium. This transition pro-

ceeds via an M1–M1 or E1–M2 multipole. In the cb̄ system the E1–M2 multipole dominates

and the scaling from the cc̄ system should be given by

Γ(cb̄)

Γ(cc̄)
=

(mb + mc)
2

4m2b

〈r2(cb̄)〉

〈r2(cc̄)〉

M3ψ′

M3Φ′

[M2Φ′ − (MΦ + Mη)
2]1/2[M2Φ′ − (MΦ −Mη)

2]1/2

[M2ψ′ − (Mψ + Mη)
2]1/2[M2ψ′ − (Mψ −Mη)

2]1/2
, (3.9)

where MΦ′ and MΦ are the masses of the 2
3S1 and 1

3S1 cb̄ levels, respectively. Because

of the small energy release in this transition, the slightly smaller level spacing in the Bc

family compared to the J/ψ family (562 MeV vs. 589 MeV) strongly suppresses η-emission

in the cb̄ system. The observed rate of Γ(ψ′ → ψ + η) = 6.6 ± 2.1 keV [1] scales to

Γ(Bc(2S) → Bc(1S) + η) = 0.25 keV.

C. Total Widths and Experimental Signatures

The total widths and branching fractions are given in Table XI. The most striking

feature of the cb̄ spectrum is the extreme narrowness of the states. A crucial element in

15



unraveling the spectrum will be the efficient detection of the 72-MeV M1-photon that, in

coincidence with an observed Bc decay, tags the B
∗
c . This will be essential for distinguishing

the Bc(2S) → Bc(1S) + ππ transition from B
∗
c (2S) → B

∗
c (1S) + ππ, which will have a nearly

identical spectrum and a comparable rate. Combining the branching fractions in Table XI

with the b-quark fragmentation probabilities of Ref. [30], we expect the cross section times

branching fractions to be in the proportions

σB(Bc(2S) → Bc(1S) + ππ) ≈ 1.2 ×σB(B
∗
c (2S) → B

∗
c (1S) + ππ) . (3.10)

A reasonable—but challenging—experimental goal would be to map the eight lowest-

lying cb̄ states: the 1S, 2S, and 2P levels. A first step, in addition to reconstructing the

hadronic cascades we have just discussed, would be the detection of the 455-MeV photons

in coincidence with Bc, and of 353-, 382-, and 397-MeV photons in coincidence with B
∗
c →

Bc + γ(72 MeV). This would be a most impressive triumph of experimental art.

IV. CONCLUDING REMARKS

A meson with beauty and charm is an exotic particle, but prospects are good that it

will be discovered in the near future. As soon as Bc has been identified, the investigation of

competing weak-decay mechanisms, b̄ → c̄W + (represented by ψπ+, ψ`+ν, etc.), c → sW +

(represented by Bsπ
+, Bs`

+ν, etc.), and cb̄ → W + (represented by ψD+s , τ
+ντ , etc.), can

begin. The issues to be studied, and predictions for a wide variety of inclusive and exclusive

decays, are presented in a companion paper [3]. Before the end of the decade, it should

prove possible to map out part of the cb̄ spectrum by observing γ- and ππ-coincidences with

the ground-state Bc or its hyperfine partner B
∗
c .

ACKNOWLEDGMENTS

Fermilab is operated by Universities Research Association, Inc., under contract DE-

AC02-76CHO3000 with the United States Department of Energy. C.Q. thanks the Cultural

16



Section of the Vienna municipal government and members of the Institute for Theoretical

Physics of the University of Vienna for their warm hospitality while part of this work was

carried out.

17



REFERENCES

[1] We follow the nomenclature of the Particle Data Group, in which B mesons contain b̄

antiquarks. See Particle Data Group, Phys. Lett. B 239, 1 (1990); Phys. Rev. D 45, S1

(1992).

[2] The CDF Collaboration has reconstructed the χc states by observing photons in coin-

cidence with leptonic decays of J/ψ. See F. Abe et al. (CDF Collaboration), Phys. Rev.

Lett. 71, 2537 (1993).

[3] E. Eichten and C. Quigg, in preparation. For a brief preliminary account, see C. Quigg,

FERMILAB-CONF-93/265–T, contributed to the Workshop on B Physics at Hadron

Accelerators, Snowmass.

[4] E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, T.-M. Yan, Phys. Rev. D 17, 3090

(1978); ibid. 21, 313(E) (1980); ibid. 21, 203 (1980).

[5] C. Quigg and J. L. Rosner, Phys. Rev. D 23, 2625 (1981).

[6] W. Kwong, J. L. Rosner, and C. Quigg, Ann. Rev. Nucl. Part. Sci. 37, 325 (1987).

[7] J. Richardson, Phys. Lett. B 82, 272 (1979).

[8] W. Buchmüller and S.-H. H. Tye, Phys. Rev. D 24, 132 (1981).

[9] A. Martin, Phys. Lett. B 93, 338 (1980); in Heavy Flavours and High Energy Collisions

in the 1-100 TeV Range, edited by A. Ali and L. Cifarelli (Plenum Press, New York,

1989), p. 141.

[10] C. Quigg and J. L. Rosner, Phys. Lett. B 71, 153 (1977).

[11] W. Kwong and J. L. Rosner, Phys. Rev. D 44, 212 (1991).

[12] M. Baker, J. S. Ball, and F. Zachariasen, Phys. Rev. D 45, 910 (1992), use a dual-QCD

potential to calculate MBc = 6.287 GeV/c
2 and MB∗c = 6.372 GeV/c

2. We note that their

prediction for the (cc̄) 1S level lies at 3.083 GeV/c2, 16 MeV/c2 above the observed value.

18



R. Roncaglia, A. R. Dzierba, D. B. Lichtenberg, and E. Predazzi, Indiana University

preprint IUHET 270 (January 1994, unpublished), use the Feynman-Hellman theorem

to predict MB∗c = 6.320±0.010 GeV/c
2. S. Godfrey and N. Isgur, Phys. Rev. D 32, 189

(1985), estimate MBc = 6.27 GeV/c
2 and MB∗c = 6.34 GeV/c

2.

[13] E. J. Eichten and F. Feinberg, Phys. Rev. Lett. 43, 1205 (1979).

[14] E. J. Eichten and F. Feinberg, Phys. Rev. D 23, 2724 (1981).

[15] D. Gromes, Z. Phys. C 26, 401 (1984).

[16] General reviews of the Eichten-Feinberg-Gromes (EFG) formulation have been given by

M. Peskin, in Dynamics and Spectroscopy at High Energy, Proceedings of the Eleventh

SLAC Summer Institute on Particle Physics, SLAC Report No. 267, edited by P. M.

McDonough (Stanford Linear Accelerator Center, Stanford, CA, 1983), p. 151; E. J.

Eichten, in The Sixth Quark, Proceedings of the Twelfth SLAC Summer Institute on

Particle Physics, SLAC Report No. 281, edited by P. M. McDonough (Stanford Linear

Accelerator Center, Stanford, CA, 1985), p. 1. In Eq. (2.13), V2(R) should read
dV2(R)

dR
;

D. Gromes, in Spectroscopy of Light and Heavy Quarks, edited by Ugo Gastaldi, Robert

Klapisch, and Frank Close (Plenum Press, New York and London, 1987), p. 67; D.

Gromes, in The Quark Structure of Matter, Proceedings of the Yukon Advanced Study

Institute, edited by N. Isgur, G. Karl, and P. J. O’Donnell (World Scientific, Singapore,

1985), p. 1.

[17] W. Buchmüller, Y. J. Ng, and S.-H. Henry Tye, Phys. Rev. D 24, 3003 (1981).

[18] S. N. Gupta, S. F. Radford, and W. W. Repko, Phys. Rev. D 25, 3430 (1982); ibid. 26,

3305 (1982).

[19] For the unequal-mass case relevant to the cb̄ system, Y. J. Ng, J. Pantaleone, and S.-H.

Henry Tye, Phys. Rev. Lett. 55, 916 (1985) [see also J. Pantaleone, S.-H. H. Tye, and

Y. J. Ng, Phys. Rev. D 33, 777 (1986)], introduced a new, spin-dependent form factor.

19



However, these changes can be absorbed into the EFG formalism without introducing

any new form factors by allowing the existing form factor to depend (logarithmically) on

the heavy-quark masses. Recently, Yu-Qi Chen and Yu-Ping Kuang, “General relations

of heavy quark-antiquark potentials induced by reparametrization invariance,” China

Center of Advanced Science and Technology (World Laboratory) preprint CCAST-93-

37 (unpublished), extended the Gromes analysis [15] to show in general that no new

spin-dependent structures appear to order 1/m2.

[20] The expectation values of spin and orbital angular momentum operators are con-

veniently evaluated using 〈~si · ~sj〉 =
1
2
S(S + 1) − 3

4
, 〈~L · ~si〉 = 〈~L ·~sj〉 =

1
2
〈~L · ~S〉,

〈~L · ~S〉 = 1
2
[J(J + 1) − L(L + 1) − S(S + 1)]. The tensor operator can be written

as Sij = 2
[
3(~S · n̂)(~S · n̂) − ~S2

]
, for which [W. Kwong and J. L. Rosner, Phys. Rev. D

38, 279 (1988)] 〈Sij〉 = −[12〈~L · ~S〉
2 + 6〈~L · ~S〉−4S(S + 1)L(L + 1)]/[(2L−1)(2L + 3)].

[21] T. A. Armstrong et al. (E-760 Collaboration), Phys. Rev. Lett. 68, 1468 (1992); Nucl.

Phys. B 373, 35 (1992).

[22] T. A. Armstrong et al. (E-760 Collaboration), Phys. Rev. Lett. 69, 2337 (1992).

[23] Up to the color factor, this relation is due to R. Van Royen and V. F. Weisskopf, Nuovo

Cim. 50, 617 (1967); 51, 583 (1967).

[24] The QCD radiative correction factor is obtained by transcription from QED. See, for

example, R. Barbieri et al., Nucl. Phys. B105, 125 (1976); W. Celmaster, Phys. Rev.

D 19, 1517 (1979).

[25] If, for example, we interpret the factor (1 − 16αs/3π) as the beginning of an expansion

for (1 + 16αs/3π)
−1 with αs = 0.36, then the predictions for the ψ family agree with

experiment, within errors, while those for the Υ family are about 20% low.

[26] E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, T.-M. Yan, Phys. Rev. D 21,

203 (1980), observed in the context of potential models that, in the heavy-quark limit,

20



the state with Jlight quark =
3
2

is degenerate with the 3P2 level, while the state with

Jlight quark =
1
2

is degenerate with the 3P0 level. (See their Appendix on charmed mesons.)

See also J. L. Rosner, Comments Nucl. Part. Phys. 16, 109 (1986). The observation in

the heavy-quark limit of QCD is due to N. Isgur and M. B. Wise, Phys. Rev. Lett. 66,

1130 (1991).

[27] S. S. Gershtĕın, V. V. Kiselev, A. K. Likhoded, S. R. Slabospitskĭı, and A. V. Tkabladze,

Yad. Fiz. 48, 515 (1988); [Sov. J. Nucl. Phys. 48, 327 (1988)].

[28] Yu-Qi Chen and Yu-Ping Kuang, Phys. Rev. D 46, 1165 (1992). See also Yu-Qi Chen,

“The Study of Bc(B̄c) Meson and Its Excited States,” Institute for Theoretical Physics,

Academia Sinica, Ph.D. thesis, October 24, 1992 (unpublished).

[29] We adopt the standard normalization,
∫
dΩ Y ∗`m(θ,φ)Y`′m′(θ,φ) = δ``′ δmm′. See, for

example, the Appendix of Hans A. Bethe and Edwin E. Salpeter, Quantum Mechanics

of One- and Two-Electron Atoms (Springer-Verlag, Berlin, 1957).

[30] Eric Braaten, Kingman Cheung, and Tzu Chiang Yuan, Phys. Rev. D 48, R5049 (1993);

Kingman Cheung, “Bc Meson Production at Hadron Colliders by Heavy Quark Frag-

mentation,” NUHEP-TH-93-19 (unpublished); Yu-Qi Chen, “Perturbative QCD Pre-

dictions for the Fragmentation Functions of the P -Wave Mesons with Two Heavy

Quarks,” China Center for Advanced Science and Technology (World Laboratory)

preprint CCAST-93-4 (unpublished).

[31] E. Eichten and K. Gottfried, Phys. Lett. B 66, 286 (1977).

[32] K. Gottfried, in Proceedings of the International Symposium on Lepton and Photon

Interactions as High Energies, edited by F. Gutbrod, DESY, Hamburg (1978); Phys.

Rev. Lett. 40, 598 (1978); M. B. Voloshin, Nucl. Phys. B 154, 365 (1979).

[33] T.-M. Yan, Phys. Rev. D 22, 1652 (1980).

[34] Y.-P. Kuang and T.-M. Yan, Phys. Rev. D 24, 2874 (1981).

21



[35] Y.-P. Kuang, S. F. Tuan and T.-M. Yan, Phys. Rev. D 37, 1210 (1988).

[36] Y.-P. Kuang and T.-M. Yan, Phys. Rev. D 41, 155 (1990).

[37] It must be remembered in applying these relations to the cb̄ system that the physical

eigenstates with J = ` 6= 0 are linear combinations of the equal-mass spin-singlet and

spin-triplet states.

[38] R. A. Partridge, Ph. D. thesis, Caltech Report No. CALT-68-1150 (1984, unpublished).

[39] R. H. Schindler, Ph. D. thesis, Stanford Linear Accelerator Laboratory Report No.

SLAC-219 (1979, unpublished).

[40] J. Adler, et al. (Mark III Collaboration), Phys. Rev. Lett. 60, 89 (1988).

[41] L. S. Brown and R. N. Cahn, Phys. Rev. Lett. 35, 1 (1975).

[42] G. S. Abrams et al. (Mark I Collaboration), Phys. Rev. Lett. 34, 1181 (1975); G. S.

Abrams, in Proceedings of the 1975 International Symposium on Lepton and Photon

Interactions at High Energies, edited by W. T. Kirk (SLAC, Stanford, 1975), p. 25.

[43] D. Besson et al. (CLEO Collaboration), Phys. Rev. D 30, 1433 (1984).

[44] T. Bowcock et al. (CLEO Collaboration), Phys. Rev. Lett. 58, 307 (1987); I. C. Brock

et al. (CLEO Collaboration), Phys. Rev. D 43, 1448 (1991).

22



TABLES

TABLE I. Quarkonium ground-state masses (in GeV/c2) in three potentials.

Observable QCD, Ref. [8] Power-law, Ref. [9] Logarithmic, Ref. [10] Cornell, Ref. [4]

(cc̄) 1S 3.067 3.067 3.067 3.067

ψ 3.097 3.097 3.097 3.097

ηc 2.980 2.980 2.980 2.980

ψ −ηc 0.117
a 0.117b 0.117c 0.117d

(cb̄) 1S 6.317 6.301 6.317 6.321

B∗c 6.337 6.319 6.334 6.343

Bc 6.264 6.248 6.266 6.254

B∗c −Bc 0.073 0.071 0.068 0.089

(bb̄) 1S 9.440 9.446 9.444 9.441

Υ 9.464 9.462 9.460 9.476

ηb 9.377 9.398 9.395 9.335

Υ −ηb 0.087 0.064 0.065 0.141

aInput value; determines αs = 0.36.
bInput value; determines αs = 0.43.

cInput value;

determines αs = 0.37.
dInput value; determines αs = 0.31.

23



TABLE II. Charmonium masses and leptonic widths in the Buchmüller-Tye potential.

Level Mass (GeV/c2) Leptonic Width (keV)

Calculated Observeda Calculated Observeda

11S0 (ηc) 2.980 2.9788 ± 0.0019

13S1 (ψ/J) 3.097 3.09688 ± 0.00001 ± 0.00006
b 8.00 4.72 ± 0.35

23P0 (χc0) 3.436 3.4151 ± 0.0010

23P1 (χc1) 3.486 3.51053 ± 0.00004 ± 0.00012
b

23P2 (χc2) 3.507 3.55615 ± 0.00007 ± 0.00012
b

21P1 (hc) 3.493 3.5262 ± 0.00015 ± 0.0002
c

21S0 (η
′
c) 3.608

23S1 (ψ
′) 3.686 3.68600 ± 0.00010 3.67 2.14 ± 0.21

aSee Ref. [1]. bSee Ref. [21]. cSee Ref. [22].

24



TABLE III. bb̄ masses and leptonic widths in the Buchmüller-Tye potential.

Level Mass (GeV/c2) Leptonic Width (keV)

Calculated Observeda Calculated Observeda

11S0 (ηb) 9.377

13S1 (Υ) 9.464 9.46032 ± 0.00022 1.71 1.34 ± 0.04

23P0 (χb0) 9.834 9.8598 ± 0.0013

23P1 (χb1) 9.864 9.8919 ± 0.0007

23P2 (χb2) 9.886 9.9132 ± 0.0006

21P1 (hb) 9.873

21S0 (η
′
b) 9.963

23S1 (Υ
′) 10.007 10.02330 ± 0.00031 0.76 0.586 ± 0.029

33D1 10.120

33D2 10.126

33D3 10.130

31D2 10.127

33P0 (χb0) 10.199 10.2320 ± 0.0007

33P1 (χb1) 10.224 10.2549 ± 0.0006

33P2 (χb2) 10.242 10.26835 ± 0.00057

31P1 (hb) 10.231

31S0 10.298

33S1 10.339 10.3553 ± 0.0005 0.55 0.44 ± 0.03

41S0 10.573

43S1 10.602 10.5800 ± 0.0035

aSee Ref. [1].

25



TABLE IV. cb̄ masses (in GeV/c2) in the Buchmüller-Tye potential.

Level Calculated Mass Eichten & Feinberga Gershtĕın et al.b Chen & Kuangc

11S0 (Bc) 6.264 6.243 6.246 6.310

13S1 (B
∗
c ) 6.337 6.339 6.329 6.355

23P0 6.700 6.697 6.645 6.728

2 1+′ 6.736 6.740 6.741 6.760

2 1+ 6.730 6.719 6.682 6.764

23P2 6.747 6.750 6.760 6.773

21S0 6.856 6.969 6.863 6.890

23S1 6.899 7.022 6.903 6.917

33D1 7.012

33D2 7.012

33D3 7.005 (7.008)

31D2 7.009

33P0 7.108 7.067 7.134

3 1+′ 7.142 7.129 7.159

3 1+ 7.135 7.099 7.160

33P2 7.153 7.143 7.166

31S0 7.244 (7.327)

33S1 7.280

41S0 7.562

43S1 7.594

aSee Ref. [14]. bSee Ref. [27]. cSee Ref. [28]; the masses correspond to Potential I with

ΛMS = 150 MeV.

26



TABLE V. Radial wave functions at the origin and related quantities for cb̄ mesons.

Level |R
(`)
n` (0)|

2

QCD, Ref. [8] Power-law, Ref. [9] Logarithmic, Ref. [10] Cornell, Ref. [4]

1S 1.642 GeV3 1.710 GeV3 1.508 GeV3 3.102 GeV3

2P 0.201 GeV5 0.327 GeV5 0.239 GeV5 0.392 GeV5

2S 0.983 GeV3 0.950 GeV3 0.770 GeV3 1.737 GeV3

3D 0.055 GeV7 0.101 GeV7 0.055 GeV7 0.080 GeV7

3P 0.264 GeV5 0.352 GeV5 0.239 GeV5 0.531 GeV5

3S 0.817 GeV3 0.680 GeV3 0.563 GeV3 1.427 GeV3

TABLE VI. Statistical Factor Sif for
3PJ →

3DJ′ + γ Transitions.

J J′ Sif

0 1 2

1 1 1/2

1 2 9/10

2 1 1/50

2 2 9/50

2 3 18/25

27



TABLE VII. E1 Transition Rates in the cb̄ System.

Transition Photon energy (MeV) 〈f|r|i〉 (GeV−1) Γ(i → f + γ) (keV)

23P2 → 1
3S1 + γ 397 1.714 112.6

2(1+) → 13S1 + γ 382 1.714 99.5

2(1+) → 11S0 + γ 450 1.714 0.0

2(1+′) → 13S1 + γ 387 1.714 0.1

2(1+′) → 11S0 + γ 455 1.714 56.4

23P0 → 1
3S1 + γ 353 1.714 79.2

23S1 → 2
3P2 + γ 151 −2.247 17.7

23S1 → 2(1
+) + γ 167 −2.247 14.5

23S1 → 2(1
+′) + γ 161 −2.247 0.0

23S1 → 2
3P0 + γ 196 −2.247 7.8

21S0 → 2(1
+) + γ 125 −2.247 0.0

21S0 → 2(1
+′) + γ 119 −2.247 5.2

33D3 → 2
3P2 + γ 258 2.805 98.7

33D2 → 2
3P2 + γ 258 2.805 24.7

33D2 → 2(1
+) + γ 274 2.805 88.8

33D2 → 2(1
+′) + γ 268 2.805 0.1

33D1 → 2
3P2 + γ 258 2.805 2.7

33D1 → 2(1
+) + γ 274 2.805 49.3

33D1 → 2(1
+′) + γ 268 2.805 0.0

33D1 → 2
3P0 + γ 302 2.805 88.6

31D2 → 2(1
+′) + γ 268 2.805 92.5

33P2 → 1
3S1 + γ 770 0.304 25.8

33P2 → 2
3S1 + γ 249 2.792 73.8

33P2 → 3
3D3 + γ 142 −2.455 17.8

33P2 → 3
3D2 + γ 142 −2.455 3.2

33P2 → 3
3D1 + γ 142 −2.455 0.2

3(1+) → 13S1 + γ 754 0.304 22.1

3(1+) → 23S1 + γ 232 2.792 54.3

3(1+) → 33D2 + γ 125 −2.455 9.8

3(1+) → 33D1 + γ 125 −2.455 0.3

3(1+′) → 13S1 + γ 760 0.304 2.1

3(1+′) → 23S1 + γ 239 2.792 5.4

3(1+′) → 33D2 + γ 131 −2.455 11.5

3(1+′) → 33D1 + γ 131 −2.455 0.4

33P0 → 1
3S1 + γ 729 0.304 21.9

33P0 → 2
3S1 + γ 205 2.792 41.2

33P0 → 3
3D1 + γ 98 −2.455 6.9

28



TABLE VIII. M1 Transition Rates in the cb̄ System.

Transition Photon energy (MeV) 〈f|j0(kr/2)|i〉 Γ(i → f + γ) (keV)

23S1 → 2
1S0 + γ 43 0.9990 0.0289

23S1 → 1
1S0 + γ 606 0.0395 0.1234

21S0 → 1
3S1 + γ 499 0.0265 0.0933

13S1 → 1
1S0 + γ 72 0.9993 0.1345

TABLE IX. The relative rates for the allowed two-pion E1–E1 transitions between spin-triplet

states and spin-singlet states. The reduced rates are denoted by Ak(`
′,`) where k is the rank of

the irreducible tensor for gluon emission and `′ and ` are the orbital angular momenta of the initial

and final states respectively.

Transition Rate cb̄ Estimate (keV)a

33P2 → 2
3P2 + ππ A0(1, 1)/3 + A1(1, 1)/4 + 7A2(1, 1)/60 1.4

33P2 → 2
3P1 + ππ A1(1, 1)/12 + 3A2(1, 1)/20 0.03

33P2 → 2
3P0 + ππ A2(1, 1)/15 0.01

33P1 → 2
3P2 + ππ 5A1(1, 1)/36 + A2(1, 1)/4 0.05

33P1 → 2
3P1 + ππ A0(1, 1)/3 + A1(1, 1)/12 + A2(1, 1)/12 0.02

33P1 → 2
3P0 + ππ A1(1, 1)/9 0

33P0 → 2
3P2 + ππ A2(1, 1)/3 0.07

33P0 → 2
3P1 + ππ A1(1, 1)/3 0

33P0 → 2
3P0 + ππ A0(1, 1)/3 1.4

33DJ′ → 1
3S1 + ππ A2(2, 0)/5 32 ± 11

23S1 → 1
3S1 + ππ A0(0, 0) 50 ± 7

31P1 → 2
1P1 + ππ A0(1, 1)/3 + A1(1, 1)/3 + A2(1, 1)/3 1.4

31D2 → 1
1S0 + ππ A2(2, 0)/5 32 ± 11

21S0 → 1
1S0 + ππ A0(0, 0) 50 ± 7

aSum of π+π− and π0π0.

29



TABLE X. Estimated rates for two-pion E1–E1 transitions between cb̄ levels, scaled from cc̄

and bb̄ measurements and calculations.

Transition (QQ̄) rate (keV) 〈r2(cb̄)〉/〈r2(QQ̄)〉 Reduced rate (cb̄) (keV)

(bb̄) : 11.7 ± 2.2a 1.99 A0(0, 0) = 40 ± 8

23S1 →1
3S1 + ππ (cc̄) : 141 ± 27

a 0.70 A0(0, 0) = 69 ± 13

Mean A0(0, 0) = 50 ± 7

(cc̄) : 37 ± 17 ± 8b A2(2, 0) = 137 ± 70

33D1 →1
3S1 + ππ (cc̄) : 55 ± 23 ± 11

c 0.72 A2(2, 0) = 204 ± 94

Mean: 43 ± 15 A2(2, 0) = 160 ± 56

33P0 → 2
3P0 + ππ (bb̄) : 0.4

d 1.88 A0(1, 1) = 4.2

33P2 → 2
3P1 + ππ (bb̄) : 0.01

d 1.88 A2(1, 1) = 0.2

aParticle Data Group average [1]. bMeasured by the Crystal Ball [38] and Mark II [39]

Collaborations. cMeasured by the Mark III Collaboration [40]. dCalculated by Kuang and

Yan [34] using the Buchmüller-Tye potential [8].

30



TABLE XI. Total widths and branching fractions of cb̄ levels.

Decay Mode Branching Fraction (percent)
13S1: Γ = 0.135 keV

11S0 + γ 100
21S0: Γ = 55 keV

11S0 + ππ 91
2(1+′) + γ 9

23S1: Γ = 90 keV
13S1 + ππ 55
23P2 + γ 20
2(1+) + γ 16
23P0 + γ 9

23P0: Γ = 79 keV
13S1 + γ 100

2(1+): Γ = 100 keV
13S1 + γ 100

2(1+′): Γ = 56 keV
11S0 + γ 100

23P2: Γ = 113 keV
13S1 + γ 100

33D1: Γ = 173 keV
13S1 + ππ 18
23P2 + γ 2
2(1+) + γ 29
23P0 + γ 51

33D2: Γ = 146 keV
13S1 + ππ 22
23P2 + γ 17
2(1+) + γ 61

33D3: Γ = 131 keV
13S1 + ππ 24
23P2 + γ 76

31D2: Γ = 124 keV
11S0 + ππ 26
2(1+′) + γ 74

33P0: Γ = 71 keV
a

23P0 + ππ 2
13S1 + γ 31
23S1 + γ 57
33D1 + γ 10

3(1+): Γ = 86 keVa

13S1 + γ 26
23S1 + γ 63
33D2 + γ 11

3(1+′): Γ = 21 keVa

2(1+′) + ππ 7
13S1 + γ 10
23S1 + γ 26
33D2 + γ 55

33P2: Γ = 122 keV
a

23P2 + ππ 1
13S1 + γ 21
23S1 + γ 60
33D3 + γ 15
33D2 + γ 3

aShould this state lie above flavor threshold, dissociation into BD will dominate over the
tabulated decay modes.

31



FIGURES

FIG. 1. The spectrum of cb̄ states.

FIG. 2. Normalized dipion mass spectrum for the transition 23S1 → 1
3S1 +ππ in the ψ (dashed

curve), Bc (solid curve), and Υ (dotted curve) families.

32





0

2

4

6

8

0.28 0.36 0.44 0.52 0.60

23S
1
→13S

1
 + ππ

(1
/Γ

)d
Γ

/d
  

  
  

(G
e

V
)–

1

(GeV/c2)M

M