#### Revisiting radiative decays of \(1^{+-}\) heavy quarkonia in the covariant light-front approach

Eur. Phys. J. C
Revisiting radiative decays of 1+− heavy quarkonia in the covariant light-front approach
Yan-Liang Shi 0
0 C. N. Yang Institute for Theoretical Physics, Stony Brook University , Stony Brook, NY 11794 , USA
We revisit the calculation of the width for the radiative decay of a 1+− heavy Q Q¯ meson via the channel 1+− → 0−+ +γ in the covariant light-front quark model. We carry out the reduction of the light-front amplitude in the nonrelativistic limit, explicitly computing the leading and nextto-leading order relativistic corrections. This shows the consistency of the light-front approach with the non-relativistic formula for this electric dipole transition. Furthermore, the theoretical uncertainty in the predicted width is studied as a function of the inputs for the heavy-quark mass and wave function structure parameter. We analyze the specific decays hc(1 P) → ηc(1S) + γ and hb(1 P) → ηb(1S) + γ . We compare our results with experimental data and with other theoretical predictions from calculations based on non-relativistic models and their extensions to include relativistic effects, finding reasonable agreement.
1 Introduction
Heavy-quark Q Q¯ bound states play a valuable role in
elucidating the properties of quantum chromodynamics (QCD).
Since the discoveries of the J /ψ in 1974 [
1,2
] and other
cc¯ charmonium states, and the ϒ in 1977 [
3,4
] and other bb¯
states, we now have a very substantial set of data on the
properties and decays of these quarkonium states. Some reviews
include [
5–15
]. The goal of understanding these data
motivates theoretical studies, in particular, studies of the decays
of Q Q¯ states.
Among various decay channels, radiative decays are a very
good testing ground for models, since the emitted photon
is directly detected and the electromagnetic interaction is
well understood. An electric dipole (E1) transition is one
the simplest types of radiative decays. Here we consider E1
transitions of the form
(1)
1 P1 → 1 S0 + γ ,
where a spin-singlet P-wave Q Q¯ quarkonium state decays to
a spin-singlet S-wave Q Q¯ state. In terms of the spin J and
the charge and parity quantum numbers P and C, indicated
as J PC , this has the form 1+− → 0−+ + γ .
Several theoretical analyses of these E1 transition rates
have been carried out, using various models [
16–26
]. A
number of these models utilize the non-relativistic quantum
mechanics formula for an E1 transition, involving the
calculation of the overlap integral of the quarkonium wave
functions of the initial and final states. The quarkonium wave
function is obtained from the solution of the Schrödinger
equation with non-relativistic potentials, such as the Cornell
potential, V = −(4/3)αs(mQ)/r + σ r . The first term in
this potential is a non-Abelian Coulomb potential
representing one-gluon exchange at short distances, where αs(mQ) =
gs(μ)2/(4π ) is the strong coupling evaluated at the scale of
the heavy-quark mass, mQ, and the second term is the linear
confining potential, where σ = 0.18 GeV2 is the string
tension. Current data yield a fit to αs(μ) such that αs 0.33 at
the scale μ = 1.5 GeV relevant for cc¯ states and αs 0.21
at the scale μ = 4.7 GeV relevant for bb¯ states [27].
Relativistic corrections have also been calculated by replacing
the Schrödinger equation by the Dirac equation, and
computing corrections in powers of v/c, where v is the velocity
of the heavy (anti)quark in the rest frame of the Q Q¯ bound
state.
It is of interest to study the radiative decays (1) with
a fully relativistic approach, namely the light-front quark
model (LFQM) [28–41]. This approach naturally includes
relativistic effects of quark spins and the internal motion of
the constituent quarks. Another advantage of the light-front
quark model is that it is manifestly covariant. Hence it is easy
to boost a hadron bound state from one inertial Lorentz frame
to another one when the bound state wave function is known
in a particular frame [32]. The light-front approach has been
used to study semileptonic and nonleptonic decays of
heavyhc(1 P) → ηc(1S) + γ ,
and
hb(1 P) → ηb(1S) + γ .
We present several new results here. We carry out the
reduction of the light-front amplitude to the non-relativistic limit,
explicitly computing the leading and next-to-leading order
relativistic corrections. This shows the consistency of the
light-front approach with the non-relativistic formula for this
electric dipole transition. Furthermore, we investigate the
theoretical uncertainties in the predicted widths as functions
of the inputs for the heavy-quark mass and wave function
structure parameters. As in Ref. [45], we compare our
numerical results for these widths with experimental data and with
other theoretical predictions from calculations based on
nonrelativistic models and their extensions to include relativistic
effects, extending [45] with further study of the theoretical
uncertainties in our calculations. Specifically, we compare
our numerical results with results from [
9,11,20–23,25,26
]
as well as latest experimental data [27].
The paper is organized as follows: In Sect. 2, we review
the formulas for the radiative decay 1+− → 1−+ + γ .
In Sect. 3, we analyze the reduction of light-front
formulas when applied to heavy-quarkonium systems to
nonrelativistic limit and compare these with non-relativistic
quantum mechanical electric dipole transition formula. In
Sect. 4, we present our numerical results for the decay widths
of hc(1 P) → ηc(1S)+γ and hb(1 P) → ηb(1S)+γ
including an extended analysis of the theoretical uncertainties. Our
conclusions are given in Sect. 5.
flavor D and B mesons and also to evaluate radiative decay
rates of heavy mesons [42–46].
In this paper, we extend our previous work with Ke and
Li in Ref. [45] on the study of the radiative decays
(2)
(3)
(4)
Consider a decay of a Q Q¯ meson consisting of two
constituent particles (quark and antiquark). The momentum of
the parent meson is denoted as P = p1 + p2, where p1 and
p2 are the momenta of the constituent quark and antiquark,
with mass m1 and m2, respectively. The momentum of the
daughter Q Q¯ meson is written as P = p1 + p2, where p1
is the momentum of the constituent quark, with mass m1.
Here we have m1 = m2 = m = mQ. The four-momentum
of the parent meson with mass M can be expressed as P =
( P −, P +, P⊥), where P 2 = P + P − − |P ⊥|2 = M 2.
Similarly, for the daughter meson with mass M , one has
P 2 = M 2, as shown in Fig. 1 below. (Vector signs on
transverse momentum components are henceforth taken to
be implicit.)
The momenta of the constituent quark and antiquark ( p1,
p1 and p2) can be described by internal variables (x2, p )
thus: ⊥
p1+ = x1 P +, p2+ = x2 P +,
p1⊥ = x1 P⊥ + p⊥, p2⊥ = x2 P⊥ − p⊥,
x1 = ee11 −+ ep2z , x2 = ep1z ++ ee22 ,
where e1, e1 and e2 are the energy of the quark (antiquark)
with momenta p1, p1 and p2:
e1 =
e1 =
e2 =
m 2 ⊥ + pz2,
1 + p 2
m 2 ⊥ + pz 2,
1 + p 2
m2 ⊥ + pz2 .
2 + p 2
p⊥ = p⊥ − x2q⊥.
With the external momentum of the photon given as q =
P − P , p⊥ can be expressed as
(6)
(7)
(8)
(9)
2 Light-front formalism for the decays 1+− → 0−+ + γ
2.1 Notation
Here pz and pz can also be expressed as functions of internal
vAaprpiaebnldeisx(Bx2., p⊥), and explicit expressions can be found in
We first define some notation, retaining the conventions of
[39,41]. In light-front coordinates, the four-momentum p is
2.2 Form factors
pμ = ( p−, p+, p⊥).
where p± = p0 ± p3 and p⊥ = ( p1, p2). Hence, the Lorentz
scalar product p2 = pμ pμ is
p2 = ( p0)2 − |p|2 = ( p0)2 − ( p3)2 − |p⊥|2
= p+ p− − |p⊥|2.
Define external momentum variables to be P = P + P , q =
P − P , where q is the four-momentum of the photon that
is emitted in the radiative transition. The general amplitude
of the radiative decay (1) of the axial vector 1+− 1 P1 meson,
denoted as A, to the pseudoscalar 0−+ 1 S0 meson, denoted
as P, can be written as [41]:
(5)
i A
A( P ) → P( P )γ (q) = εμ∗(q)εν ( P )i A˜μν ,
(10)
where
i A˜μν = f1(q2)gμν + Pμ f+(q2) Pν + f−(q2)qν . (11)
In the above expression, we have used the condition εμ∗(q)qμ
= 0 to eliminate terms that are proportional to qμ. This
expression can be simplified further by using the
transversality property of axial vector polarization vector:
i A μν = f1(q2) gμν
1
− ( P · q)
where f2(q2) is linear combination of f+(q2) and f−(q2):
f2(q2) = − f+(q2) + f−(q2) .
Notice that in Eq. (13), f1(q2) and f2(q2) are not
independent. Because of electromagnetic gauge invariance, they are
related by the following equation:
qμA μν = 0 →
f1(q2) + f2(q2)( P · q) = 0 .
So the amplitude can be parameterized by f1(q2), which is
and the vertex function of the pseudoscalar meson P (0−+,
1 S0) is given by
HP γ 5 ,
− p2μ( p1 · p1 ) + m1m2 p1μ + m1m2 p1μ + m1m1 p2
μ
− (m1 − m2)2 − (m1 − m2)2 + (m1 − m1)2
+ qμ q2 − 2M 2 + N1 − N1 + 2N2 + 2(m1 − m2)2
− (m1 − m1)2 + Pμ q2 − N1 − N1 − (m1 − m1)2
p1μ( p1 · p2) + p1μ( p1 · p2)
2 p1μ
M 2 + M 2 − q2 − 2N2
Taking the physical value q2 → 0 in the form factor f1(q2)
and averaging initial state polarizations, the radiative
transition width of 1+− → 0−+ + γ is given by
polar.
1 |q|
= 3 · 8π M 2
|q|
|A |2 = 12π M 2 · | f1(0)|2 ,
where the energy of the emitted photon is related to the
masses of mesons as |q| = (M 2 − M 2)/(2M ).
2.3 Calculation of radiative decay amplitude
In the covariant light-front quark model, the vertex function
of the axial vector meson A (1+−, 1 P1) is given by
1
WA
−i HA
( p1 − p2)μ γ 5 ,
(20)
(21)
(22)
where HA and HP are functions of p1 and p2, and WA can
be reduced to a constant, which we will discuss later in this
subsection.
In the light-front framework that we use [39,41], at leading
order there are two diagrams that contribute to the A →
P + γ transition amplitude. These give the corresponding
contributions to this amplitude
i A μν ( A → P + γ ) = i A μν (a) + i A μν (b)
where i A μν (a) and i A μν (b) correspond to the left and right
diagram in Fig. 1, respectively. The contribution to the
amplitude from the right diagram can be obtained by taking the
charge conjugation of left diagram (see also [46]). So we
discuss the left-hand diagram, where the corresponding
transition amplitude is given by
e Ne1 Nc
(2π )4
HA HP
1 N1 N1 N2
Saμν ,
i A μν (a) = i
where
Saμν = Tr γ 5(/p1 + m1 )γ μ(/p1 + m1)γ 5(−/p2 + m2)
Here Ne1(e2) represents the electric charge of quark with
fourmomentum p1 ( p2). Here we have Ne1(e2) = eQ. In Eq.(23),
we have already applied the following relations:
p1 = p1 − q,
p2 = ( P + q)/2 − p1,
2 p1 · p2 = M 2 − N1 − m12 − N2 − m22,
2 p1 · p2 = M 2 − N1 − m12 − N2 − m22,
2 p1 · p1 = −q2 + N1 + m12 + N1 + m12.
Then we integrate over p1− by closing the contour in the
upper complex p1− plane, which amounts to the following
replacement [39,41]:
d4 p
−i π
where
In the above expressions, ϕ p( p , x2) is the light-front
⊥
momentum space wave function for initial P-wave meson
(1 P1), and ϕ( p⊥, x2) is the wave function for the final S-wave
meson, 1 S0. Some details concerning the wave functions are
given in Appendix A. The explicit forms of M0, M , M˜ 0 and
0
˜ 0 are listed in Appendix B. The definitions of εˆ∗, εˆ and
M
εˆρ∗ are given in [39,41].
After the integration over p1−, we have the following
replacement for p1μ and Nˆ 2 in Sˆaμν in the integral [39,41]:
pˆ1μ → Pμ A(11) + qμ A(21) ,
pˆ1μ pˆ1ν → gμν A(12) + Pμ Pν A(22)
+ ( Pμqν + qμ Pν ) A(32) + qμqν A(42),
pˆ1μ Nˆ 2 → qμ
pˆ1μ pˆ1ν Nˆ 2 → gμν A(12) Z2
A(21) Z2 + qq·2P A(12) ,
+ qμqν A(42) Z2 + 2 q · P A(21) A(2) ,
q2 1
where the explicit expressions for A(ji)(i, j = 1 ∼ 4) and Z2
are listed in Appendix B.
Combining Eqs. (26), (27) and (28), we get Saμν → Sˆaμν ,
where the explicit form can be found in Ref. [45]. Finally, we
obtain i A μν (a) as a function of the external four-momenta
P and q with the following parameterization:
i A μν (a) = f1a (q2) gμν
where the form factor f1a (q2) is given by
eeQ Nc
f1a (q2) = 16π 3
dx2d2 p 4
x2 Nˆ 1 Nˆ 1⊥ h Ah P wA
1
− ( P · q)
Pμqν ,
(29)
×
A(12)[M 2 + M 2 − q2 − (m1 − m2)2
− (m1 − m2)2 + (m1 − m1)2] − 2 A(12) Z2
eeQ Nc
= 16π 3
×
− p⊥2 −
dx2d2 p 4
x2 Nˆ 1 Nˆ 1⊥ h Ah P wA
( p⊥ · q⊥)2
q2
× [M 2 + M 2 −q2 −(m1 −m2)2 −(m1 − m2)2
1
− ( P · q)
This can be obtained from the result of the left-hand diagram
with the replacements m1 ↔ m2, m1 ↔ m2, m2 ↔ m1,
Ne1 ↔ Ne2 . The total form factor f1(q2) is the sum of
contribution from two diagrams:
2.4 Comments on effects of zero modes
As discussed in Refs. [39, 41], there are two classes of form
factors for the amplitude discussed in this section. One class
of form factors like f (q2) is associated with zero modes and
another class of form factors is free of zero-mode
contributions. In this paper, the form factor that contributes to the
radiative transition 1+− → 0−+ +γ , namely f1(q2), belongs
to the first class and contains zero-mode contributions. In
this case, the substitution in Eq. (28) is not exact and
contains residual spurious terms that are proportional to a
lightlike four-vector ω = (2, 0, 0). These terms are not Lorentz
covariant. As discussed in Refs. [39, 41], the zero-mode
contribution cancels away the residual spurious ω terms.
Furthermore, form factors like f (q2) receive additional residual
contributions, which can be expressed in terms of the B((nm))
and C((nm)) functions defined in Appendix B of Ref. [41].
However, this problem has already been carefully
analyzed in Ref. [41]. In fact, by comparing our expression for
f1(q2) in Eq.(30) with the expression for f (q2) in Eq. (B4) of
Ref. [41], we find that the integrand of f1(q2) is proportional
to the 1/wV term of the integrand of f (q2) in Ref. [41]. Thus,
we follow the same analysis in Ref. [41] of f (q2) to address
the possible zero-mode problem of f1(q2). The B((nm)) and
C((nm)) functions do not appear in the expression for f1(q2), so
we can still use the substitution in Eq. (28) because, first, we
utilize the relation C1(1) → 0 in the integrand, and second, the
amplitude associated with f1(q2) in A μν is contracted with
the transverse polarization vector of the photon. Explicitly,
our numerical calculation shows that all of the functions B((nm))
are numerically negligibly small, and hence we can neglect
all of the residual contributions to the form factors in the
present analysis.
3 Reduction to non-relativistic limit in application to
quarkonium systems
In this paper, we use the light-front formula discussed in
Sect. 2 to study the radiative decay (1). For this decay, the
non-relativistic electromagnetic dipole transition formulas
are widely adopted [
9
]. Thus it is interesting to investigate
the consistency between the LFQM and the non-relativistic
dipole transition formulas in the non-relativistic limit. In this
section we analyze the reduction of the light-front formula
for the decay width in the non-relativistic limit. This limit
is relevant here because (v/c)2 is substantially smaller than
unity for a heavy-quark Q Q¯ state. For a Coulombic
potential, αs ∼ v/c, and current data give αs = 0.21 at a scale of
mb = 4.5 GeV, yielding (v/c)2 ∼ 0.04 for the ϒ system.
There are several aspects of the non-relativistic limit for the
decay of a heavy-quarkonium system:
1. Masses of bound states. The masses of initial (M ) and
final state (M ) are close to the sum their constituents,
and the deviation is O (mQ−2) corrections:
M 2
4m2Q = 1 + O (mQ−2),
M 2
4m2Q = 1 + O (mQ−2).
(33)
Here and below, by O (mQ−2) we mean O (|p|2/m2Q),
where p is a generic three-momentum in the parent meson
rest frame.
2. No-recoil limit. In non-relativistic quantum mechanics,
the final state after the E1 radiative transition is assumed
to carry approximately the same three-momentum as the
initial state [47]. So the matrix element of this E1
transition is
r ∝ f (p )|r|i (p ) ,
p
= p .
In our analysis, we will adopt this no-recoil
approximation.
3. Normalization of wave function. In non-relativistic
quantum mechanics, the momentum-space wave function is
given by
p|n, lm
= Rnl ( p)Ylm (θ , φ) ,
with the normalization of the radial wave function
0
∞
d p p2 Rn∗l ( p) Rnl ( p) = 1,
where here p = |p|, and the normalization of the angular
wave function
d
Yl∗m (θ , φ)Yl m (θ , φ) = δll δmm .
(34)
(35)
(36)
(37)
(38)
In this paper we use harmonic oscillator wave functions
for the quarkonium 1P and 1S states. The general formula
for harmonic oscillator wave functions in momentum space
that satisfy the usual quantum mechanics normalization in
Eq. (36) is given by [
22, 48
]
Rnl ( p) =
1
3
β 2
×
2n
β
p l Ll+ 21
n−1
1
n + l + 2
p2
β2
exp
where Lln+−211( p2/β2) is an associated Laguerre polynomial.
Here, β is a parameter with dimensions of momentum that
enters in the light-front wave function (A.7) (and should not
be confused with the dimensionless ratio v/c, which serves
R1S( p) =
and
R1P ( p) =
2
3 1 exp
β 2 π 4
,
Notice that the normalization of these wave functions is
different from the normalization of the light-front wave
functions discussed in Appendix A. For example,
ψ ( p) = √
R1S( p).
1
4π
In non-relativistic quantum mechanics, the width of an E1
decay of the initial quarkonium state 1 P1 to the final
quarkonium state 1 S0 + γ is given by [
9
]:
1 P1 → 1 S0 + γ
4
= 9 αeQ2 Eγ3 |I3(1 P → 1S)|2
where Eγ = |q| is the energy of the emitted photon, and
I3(1 P → 1S) is the overlap integral in position space, which
represents the matrix element of the electric dipole operator:
I3(1 P → 1S) =
dr r 3 R1P (r )R1∗S(r ).
Similarly, we can define I5(1 P → 1S), which appears in
the relativistic correction to the electric dipole transition
width [
9
]:
I5(1 P → 1S) =
dr r 5 R1P (r )R1∗S(r )
For later use, we also list the analogous integrals in
momentum space:
0
0
∞
∞
0
0
∞
∞
I3p(1 P → 1S) =
d p p3 R1P ( p)R1∗S( p),
I5p(1 P → 1S) =
d p p5 R1P ( p)R1∗S( p).
We are now ready to reduce the light-front decay width in
Eq. (18) when applied to quarkonium systems to the standard
non-relativistic formula in Eq. (42). Using the explicit form
in Eq. (30) and taking the limit q2 → 0, the form factor in
Eq. (32), we can write
where we use the explicit form of light-front momentum
space wave function in 1. This expression can be further
simplified in the no-recoil limit, which is a valid approximation in
the study of an electric dipole transition in the non-relativistic
limit [47]. In this limit, we have
ψ ( p⊥2, pz 2) → ψ ( p⊥2, pz2). The corrections due to recoil
effect are suppressed by powers of (1/mQ):
ee11MM00 → 1, M0 → M0 and
e1 M0
e1 M0
=
2 p 2 + m2Q
p 2 + m2Q +
p 2 + m2Q
1 (p 2 − p 2)2
= 1 − 8
m4Q
+ O(mQ−6),
1 (p 2 − p 2)
M0 = M0 + 2 mQ
+ O(mQ−3).
(39)
(40)
(41)
(42)
(43)
(44)
(45)
(46)
(47)
(48)
(49)
⎞
(50)
× − p⊥2 − ( p⊥q·2q⊥)2
The last term in Eq. (47), −(q · P) p⊥q·2q⊥ , requires a more
careful treatment. It seems that linear p terms will not make
⊥
contributions after integrating over p , but the Taylor
expansion of the functions of p⊥ in the int⊥egrands will generate a
term that is proportional to ( p⊥ · q⊥), and this can combine
with −(q · P) p⊥q·2q⊥ term to produce a q2 independent term,
which is non-zero in the physical q2 → 0 and no-recoil limit.
Firstly we should expand p⊥ in powers of inverse of mQ:
⎛ 1
p⊥ = p⊥ − x2q⊥ = p⊥ − q⊥ ⎝ 2 +
pz
⊥ + pz2 ⎠
2 m2Q + p 2
= p⊥ − 21 q⊥ − 21 mpQz q⊥ + O(mQ−2).
We find in the physical limit q2 → 0, the dominant
contribution to the ( p⊥ · q⊥) term comes from the expansion
of ψ ( p⊥, pz ). Since ψ ( p⊥, pz ) is the wave function of
the 1S state, it is a function of p 2. Hence, we can write
ψ ( p⊥, pz ) = ψ ( p⊥2, pz 2) and expand it as follows:
ψ ( p⊥2, pz 2) ≈ ψ ( p⊥2, pz2) − p⊥ · q⊥
+ O(mQ−2)
1
= ψ ( p⊥2, pz2) + p⊥ · q⊥ 2β2 ψ ( p⊥2, pz2) + O(mQ−2),
where we use the explicit form of ψ ( p , pz ) ∝ exp[−p 2/
⊥
(2β2)] to calculate its derivative. Plugging the expansion of
ψ ( p⊥, pz ) into the integrands, we find the contribution of
the −(q · P) p⊥q·2q⊥ terms is
After this calculation, in the physical q2 = 0 and no-recoil
limit, the form factor f1(q2 → 0) is given by
f1(0) ≈ −eeQ
⊥ ψ p( p⊥, pz )ψ ( p⊥, pz ) p⊥2 ·
× (2x1 − 1)M 2 + M 2 + 2x1 M02 + 2(q · P)
d pz d2 p⊥ψ p( p⊥, pz )ψ ( p⊥, pz ) p⊥2 ·
+
+
This integral can be simplified by using symmetric property
of functions in the integrands. For functions F (p2) that have
spherical symmetry, the following relation is satisfied:
(51)
(52)
1
d3p F (p2) pi p j = 3 δi j
So Eq. (55) can be written as
4 √2
f1(0) = − 4π · β · eeQ
2 4 √2
= − 3 · 4π · β · eeQ
2 √2
= − 3 · 4 · β · eeQ
0
where we use the kinematic relation (q · P) = 2|q|M . In the
non-relativistic limit, it is more convenient to use notation of
wave functions in non-relativistic quantum mechanics. Using
Eq. (41), f1(q2 → 0) can be rewritten as
f1(q2 → 0) ≈ − 41π · √β2 ·
×eeQ
3 2M 2
d p R1S(p )R1S(p ) p⊥2 · 2 + 4(m2Q + p 2)
(54)
(55)
(56)
(57)
(58)
d3p F (p2)p2.
where p denotes the radial coordinate in the three
dimensional momentum space (and should not be confused with a
four-momentum). Using the definition of wave function in
Eq. (40),
1 2
R1P ( p ) = β 3 R1S( p ) p ,
we find that this integral is proportional to I3p(1 P → 1S):
2 √2 ∞
f1(0) = − 3 · 4 · β · eeQ d p p 4 R1S( p )R1S( p )
0
= −
= −
2(m2Q + p 2
⊥ + pz2) − 2 pz
m2Q + p 2
⊥ + pz2
In the non-relativistic limit, the mass can be interpreted as the
reduced mass of the Q¯ Q two-body system m = μ = mQ/2,
and in non-relativistic quantum mechanics the photon energy
is the difference of energy levels between initial and final
state, Ei − E f ≈ |q|, so we have
I3p(1 P → 1S) = |q|μ · I3(1 P → 1S).
Then f1(0) can be expressed as
f1(0) = −
Plugging this expression of f1(0) into the formula for the
decay width in Eq. (18), we get radiative decay width of
A(1 P1) → P(1 S0) + γ in the leading order non-relativistic
and no-recoil approximation:
|q|3μ 2 3 4
NR = 12π M 2 2 · 9 · 16 · 2 · e2eQ2|I3(1 P → 1S)|2
=
16μ 2
M 2
4
· 9 · αeQ2|q|3 · |I3(1 P → 1S)|2
4
= 9 · αeQ2|q|3 · |I3(1 P → 1S)|2 · (1 + O(mQ−2))
4
≈ 9 · αeQ2|q|3 · |I3(1 P → 1S)|2 ,
where we have made use of the approximate relations of
masses:
mQ
μ = 2
, M
2mQ , →
16μ 2
M 2
Equation (64) matches the non-relativistic electric dipole
transition formula for transition 1 P1 → 1 S0 in Eq. (42),
which proves the validity of light-front framework in the
non-relativistic limit in the application to heavy-quarkonium
systems.
3.2 Next-to-leading order correction
We next include the O(mQ−2) contributions in Eq. (54) with
the no-recoil approximation. In this case, f1(q2 → 0) is
given by
Now I3p(1 P → 1S) is proportional to I3(1 P → 1S), which
is evident in non-relativistic quantum mechanics, where we
have the operator relation:
(66)
(67)
(68)
(69)
(60)
(61)
(62)
(63)
(64)
(65)
1 √
f1(0) ≈ − 4π · β2 · eeQ
where we have made use of the symmetry property of the
integral for the function F (p2), which has spherical
symmetry:
d3p F (p2) pi p j pk pl
1
= 15 (δi j δkl + δik δ jl + δil δ jk )
and R1P,1S is given by
d3p F (p2)p4,
R1P,1S =
Combining Eqs. (66) and (18), we obtain the next-to-leading
order (O(mQ−2)) formula for the radiative decay width for
the heavy-quarkonium systems (1 P1 → 1 S0) in the
nonrelativistic and no-recoil approximation:
In this section we apply the radiative transition formulas for
the decay 1+−(1 P1) → 0−+(1 S0) + γ in the framework of
the light-front quark model, which we reviewed in Sect. 2, to
tial model (GI) [25], screened potential models with zeroth-order wave
functions (SNR0) and first-order relativistically corrected wave
functions (SNR1) [26]. For experimental data, we use the PDG value of
the total width hc(1P) = 700 ± 280 (stat.) ± 220 (syst.) keV and
B R(hc(1P) → ηc(1S) + γ ) = 51 ± 6% [27]
Fig. 2 Decay width for hc(1P) → ηc(1S) + γ (keV) as a function of
βhc(1P)(ηc)(1S) in LFQM, with mc = 1.5 GeV
Fig. 3 Decay width for hc(1P) → ηc(1S) + γ (keV) as a function of
mc in the LFQM, with βhc(1P)(ηc(1S)) = 0.63 GeV
study the radiative decay of the cc¯ state hc1(1 P ) via the
channel hc(1 P ) → ηc(1S) + γ and the bb¯ state hb(1 P ) via the
channel hb(1 P ) → ηb(1S) + γ . We present the results of our
numerical calculations of decay widths. Our results extend
those which we previously presented with Ke and Li in [45].
For the charmonium hc(1 P ) radiative decay, we compare our
result with experimental data on the width, as listed in the
Particle data group review of particle properties (RPP) [27].
We also list the theoretical calculations from other models,
including non-relativistic potential model (NR) [
9, 11, 25
],
relativistic quark model (R) [20], the Godfrey–Isgur
potential model (GI) [25], screened potential models with
zerothorder wave functions (SNR0) and first-order relativistically
corrected wave functions (SNR1) [26].
Although the PDG lists the width for the decay hc(1 P ) →
ηc(1S) + γ , it does not list the width for the hb(1 P ) →
ηb(1S)+γ decay, only the branching ratio. Since our
calculation yields the width itself, and a calculation of the branching
ratio requires division by the total width, we therefore
compare our results on the widths for these decays with
predictions from other models, including the non-relativistic
potential model (NR) [
9
], the relativistic quark model (R) [
20
], the
Godfrey–Isgur potential model (GI) [
22
], screened
potential models with zeroth-order wave functions (SNR0) and
first-order relativistically corrected wave functions(SNR1)
[
21
], as well as the non-relativistic constituent quark model
(CQM) [
23
].
First, we study the radiative decay hc(1 P ) → ηc(1S) + γ
in the LFQM, which depends on the corresponding harmonic
oscillator wave function (βhc(ηc)) and the effective charm
quark mass, mc. For the central values of mc and the wave
function parameters β, we use the central values of these
parameters suggested by previous study of LFQM [46]:
(70)
(71)
While Ref. [45] allowed a 10% variation in input
parameters, we investigate a somewhat larger variation, as indicated
in Eqs. (70) and (71). We present our numerical results in
Table 1, with the uncertainties arising from the
uncertainties in the β parameters and the value of mc. We also plot
the predicted width as a function of the input values for the
charm quark mass mc and wave function structure parameter
βhc(ηc) in Figs. 2 and 3. From these results, we find that the
main theoretical uncertainties come from variation of βhc(ηc).
With the same central value for βhc(ηc) as was used in [45],
we obtain a somewhat smaller central value for the width,
namely 398 keV as contrasted with 685 keV in [45]. As is
evident from Table 1, our current result for this width agrees
well with experimental data within the range of
experimental and theoretical uncertainties. The experimental data have
substantial uncertainties, and our result is relatively close
to the central experimental value, compared to other
nonrelativistic models. The reason that our current calculation of
the width (hc(1 P ) → ηc(1S) + γ ) yields a smaller result
Table 2 Decay width (in units of keV) of hb(1P) → ηb(1S) + γ in
the light-front quark model, denoted LFQM, as compared with
predictions from other theoretical models, including non-relativistic potential
model (NR) [
9
], relativistic quark model (R) [
20
], the Godfrey–Isgur
potential model (GI) [
22
], screened potential models with zeroth-order
wave functions (SNR0) and first-order relativistically corrected wave
functions (SNR1) [
21
] and the non-relativistic constituent quark model
(CQM) [
23
]
than that obtained in Ref. [46] may be due to the fact that
the numerical integration that is necessary in the calculation
of the amplitude involves significant cancellations between
different terms, and our current numerical integration routine
uses higher precision than was used in parts of the previous
calculation in Ref. [46].
Next we study radiative decay of hb(1 P ) → ηb(1S)+γ in
LFQM. For the central value of the effective bottom/beauty
quark mass mb, we use the value suggested by the
previous LFQM study [45] (see also [46]). For the wave function
parameter βhb(1P)(ηb(1S)), we estimate this to be in the range
β ∼ 0.9–1.3 GeV, which is suggested in [
22
], where β is
fitted by equating the rms radius of the harmonic oscillator
wave function for the specified states with the rms radius
of the wave functions calculated using the relativized quark
model. Our values for these input parameters are
We list the numerical results in the LFQM in Table 2. For
comparison, we also list other theoretical calculations from
various types of models, including the non-relativistic
potential model (NR) [
9
], the relativistic quark model (R) [
20
], the
Godfrey–Isgur potential model (GI) [
22
], screened
potential models with zeroth-order wave functions (SNR0) and
first-order relativistically corrected wave functions (SNR1)
[
21
] and the non-relativistic constituent quark model (CQM)
[
23
]. As can be seen from Table 2, with the given range
of uncertainties, our value agrees with predictions from the
non-relativistic potential model (NR) [
9
], the Godfrey–Isgur
potential model (GI) [
22
] and screened potential models
with relativistically corrected wave functions (SNR1) [
21
].
To show the theoretical uncertainties arising from
uncertainties in the βhb(1P)(ηb(1S)) parameter and the value of mb, we
also plot the decay width for hb(1 P ) → ηb(1S) + γ as a
function of these parameters in Figs. 4 and 5. We find that
the width is not very sensitive to the variation of mb and
the main uncertainties arise from the uncertainty in the wave
function parameter βhb(1P)(ηb(1S)).
These results show that the light-front quark model with
phenomenological meson wave functions (specifically,
harmonic oscillator wave functions) is suitable for the
calculation of quarkonium 1 P1 → 1 S0 + γ radiative decay
widths, since this model gives reasonable predictions for
Fig. 4 Decay width for hb(1P) → ηb(1S) + γ (keV) as a function of
βhb(1P)(ηb(1S)) in the LFQM, with mb = 4.8 GeV
Fig. 5 Decay width for hb(1P) → ηb(1S) + γ (keV) as a function of
mb in the LFQM, with βhb(1P)(ηb(1S)) = 1.0 GeV
these widths, as compared with experimental data and other
theoretical models.
5 Conclusion
In this paper we have revisited the calculation of the
radiative decay width of a 1+− axial vector meson A to a 0−+
pseudoscalar meson P via the channel 1+− → 0−+ + γ in
the LFQM approach, extending our previous work in Ref.
[45]. As part of our analysis, we have presented the
reduction of the LFQM results in the non-relativistic limit and
have shown the connection with the non-relativistic electric
dipole transition formula for heavy-quarkonium systems. We
have then applied the LFQM formula to the radiative decays
hc(1 P ) → ηc(1S) + γ and hb(1 P ) → ηb(1S) + γ . We have
performed numerical calculations and have compared our
results with experimental data and other model predictions.
We have shown that our results are in reasonable agreement
with data and other model calculations.
Acknowledgements We are grateful to Prof. Robert Shrock for his
illuminating suggestions and assistance. This research was partially
supported by the NSF Grant NSF-PHY-13-16617. We would like to
thank Profs. Hong-Wei Ke and Xue-Qian Li for collaboration on our
previous related work [45].
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: The wave functions
The normalization of the S-wave meson wave function in the
light-front framework is
1
2(2π )3
dx2d p⊥2 |ϕ(x2, p⊥)|2 = 1.
Here ϕ(x2, p⊥) is related to the wave function in normal
coordinates ψ ( p) by
3
ϕ(x2, p⊥) = 4π 2
d pz
dx2
ψ ( p),
d pz
dx2 = x1 x2 M0
e1e2
.
The normalization of ψ ( p) is given by
dp3 |ψ ( p)|2 = 4π
p2d p |ψ ( p)|2 = 1 .
The normalization for the P-wave meson wave function in
the light-front framework is [41]
1
2(2π )3
dx2d p⊥2 |ϕ p(x2, p⊥)|2 pi p j = δi j ,
where pi = ( px , py , pz ). In terms of the P-wave wave
function in normal coordinates,
3
ϕ p(x2, p⊥) = 4π 2
d pz
dx2
ψ p( p),
d pz
dx2 = x1 x2 M0
e1e2
, (A.5)
ψ p( p) =
ψ ( p).
2
β2
1
β2π
The explicit form of 1-S harmonic oscillator wave function
in the light-front approach is given by [41]
ψ ( p) =
we have the following normalization condition:
(A.6)
(A.7)
(A.8)
(B.1)
M02 = (e1 + e2)2 =
(A.1)
M0 2 = (e1 + e2)2 =
p 2
⊥ + m12
x1
p⊥2 + m12
x1
+
+
p 2
⊥ + m2
2 ,
x2
p 2 + m2
⊥ 2 ,
x2
(A.2)
(A.3)
(A.4)
M˜ 0 =
M˜ 0 =
pz =
pz =
M02 − (m1 − m2)2,
M0 2 − (m1 − m2)2,
x2 M0
x2 M0
2
2
−
−
m2
2 + p 2
⊥ ,
2x2 M0
The explicit expressions for A(ji)(i, j = 1 ∼ 4) and Z2 are
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