INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS G: NUCLEAR AND PARTICLE PHYSICS
J. Phys. G: Nucl. Part. Phys. 28 (2002) 2771–2781 PII: S0954-3899(02)34868-0
Amassformula for light mesons from a potential
model
Fabian Brau and Claude Semay
Universit
´
edeMons-Hainaut, Place du Parc 20, B-7000 Mons, Belgium
E-mail: fabian.brau@umh.ac.be and claude.semay@umh.ac.be
Received 14 March 2002
Published 17 October 2002
Online at stacks.iop.org/JPhysG/28/2771
Abstract
The quark dynamics inside light mesons, except pseudoscalar ones, can be
quite well described by a spinless Salpeter equation supplemented by a Cornell
interaction (possibly partly vector, partly scalar). A mass formula for these
mesons can then be obtained by computing analytical approximations of the
eigenvalues of the equation. We show that such a formula can be derived by
combining the results of two methods: the dominantly orbital state description
and the Bohr–Sommerfeld quantization approach. The predictions of the mass
formula are compared with accurate solutions of the spinless Salpeter equation
computed with a Lagrange-mesh calculation method.
1. Introduction
Semirelativistic potential models have been proved extremely successful for the description
of light mesons (mesons containing u, d or s quarks). The main characteristics of the spectra
of these mesons, except pseudoscalar ones, can be obtained with a spinless Salpeter equation
supplemented with the Cornell interaction (a Coulomb-like potential plus a linear confinement)
[1, 2].
Numerous techniques have been developed in order to solve the semirelativistic equation
numerically with great accuracy. Nevertheless, it is always interesting to work with analytical
results. Several attempts to obtain some mass formulae for hadrons were already performed.
Some approaches rely on fundamental QCD properties [3, 4], but they are limited to the study
of ground states of hadrons. In other works, the hadron masses are given as a function of
some quantum numbers. They are based, for instance, on shifted large-N expansion (N is the
number of spatial dimensions) of the Schr
¨
odinger equation [5], a spectrum generating algebra
[6] or a completely phenomenological point of view [7]. We will adopt here a different point
of view by assuming that a semirelativistic potential model allows a good description of the
main features of meson spectra.
0954-3899/02/112771+11$30.00 © 2002 IOP Publishing Ltd Printed in the UK 2771
2772 FBrauandCSemay
Recently, a new method to tackle this problem wasdeveloped: the dominantly orbital
state (DOS) description, in which the orbitally excited states are obtained as a classical result
while the radially excited states are treated semiclassically [8–11]. A second method is
the Bohr–Sommerfeld quantization (BSQ) approach, with which precise information can be
obtained on the asymptotical behaviours of observables as a function of the quantum numbers
[12]. We show here that a quite well accurate mass formula for light mesons, as a function of
quantum numbers and parameters of a QCD-inspired potential, can be obtained by combining
the results of these two approaches. The idea is to calculate analytical approximate solutions
of the equation assumed to govern the quark dynamics inside a meson. There is yet some
uncertainties about the Lorentz structure of the interquark interaction. In this work, we will
assume that the confinement potential is partly scalar and partly vector. A related work using
aWKBapproach was performed in [13], but the Coulomb-like potential and the Lorentz
structure of the confinement interaction were not taken into account.
In order to test the validity of the mass formula, we compare its predictions with accurate
numerical solutions of the spinless Salpeter equation. These last ones are computed with a
Lagrange-mesh calculation method [14]. This technique is modified here in order to handle
semirelativistic equations with mixed scalar–vector potentials.
In section 2,the model Hamiltonian is presented with the two methods previously
developed to compute some analytical solutions. The mass formula is established in the
case of symmetric and asymmetric mesons, with or without a constant term in the potential.
In section 3,the mass formula is compared with accurate numerical solutions of the spinless
Salpeter equation. Some concluding remarks are given in section 4.
2. The model
2.1. Model Hamiltonian
Within the framework of a semirelativistic potential model, it is possible to describe the main
characteristics of the spectra of light mesons [1, 2]. If the spinless Salpeter equation is chosen,
instead of the Schr
¨
odinger equation, the quark–antiquark Hamiltonian is given by (we use the
natural units ¯h = c = 1)
H =
p
2
+
(
m
1
+ α
1
S(r)
)
2
+
p
2
+
(
m
2
+ α
2
S(r)
)
2
+ V(r), (1)
where V(r)and S(r) are, respectively, the vector and scalar interactions [8], and where p is
the relative momentum between the quark and the antiquark. The vector p is the conjugate
variable of the inter-distance r.Asusual,weassume that the isospin symmetry is not broken,
that is to say that the u and d quarks have the same mass (in the following, these two quarks
will be denoted by the symbol n). The parameters α
1
and α
2
indicate how the scalar potential
S(r) is shared among the two masses m
1
and m
2
.Anaturalchoice, used in this work, is to
take
α
1
=
m
2
m
1
+ m
2
and α
2
=
m
1
m
1
+ m
2
. (2)
It is generally admitted that the short-range part of the interquark potential is dominated
by the one-gluon exchange process, which gives rise to a Coulomb term of vector type. The
long-range part is dominated by a confinement that lattice calculations predict linear in the
interquark distance. As its Lorentz structure is not precisely known, we suppose here that
the confinement is partly scalar and partly vector, as in [9]. The importance of each one is
Amass formula for light mesons from a potential model 2773
reflected through a parameter f whose value is 0 for a pure vector and 1 for a pure scalar.
Consequently, the potentials considered here are given by
S(r) = far, (3)
in which a is theusual string tension, whose value should be around 0.2 GeV
2
,and
V(r) = (1 − f)ar −
κ
r
, (4)
in which κ is proportional to the strong coupling constant α
s
.Areasonable value of κ should
be in the range 0.1 to 0.6.
It is worth noting that it is not possible to describe the pseudoscalar mesons with such
asimplepotential. Spin contributions as well as flavour-mixing effects are very large in this
sector. An interaction stemming from instanton effects, which is not considered here, could
explain the properties of these mesons [2, 15].Consequently, the pseudoscalar mesons cannot
be described by our model.
2.2. Semiclassical method
Approximate analytical solutions of Hamiltonian (1)with potentials (3)and(4) can be obtained
within the DOS approach. The idea of the model is to make a classical approximation by
considering uniquely the classical circular orbits (lowest energy states with given total orbital
angular momentum J ), defined by r constant, and thus ˙r = 0. The radial excitations, numbered
with the quantum number ν,are calculated by making a harmonic approximation around the
previous classical orbits. A detailed description of this method is given in [8–11]. We just
recall here the main results. In the case of a symmetric meson, m
1
= m
2
= m,thesquare
meson mass M
2
is given by[10]
M
2
= aA(f )J + B(f )m
√
aJ + C(f )m
2
+ aD(f )κ + aE(f )(2ν +1) + O(J
−1/2
). (5)
The coefficients A, B, C, D and E are given by
A(f ) =
y
2
4
[
t +3(1 − f)
]
2
,
B(f ) =
y
f
[
(1+f)(3f − 1) + t(1 − f)
]
,
C(f ) =
1
f
2
t
[t(s + f
2
) + (1 −f)(2f − 1)], (6)
D(f ) =−
[
t +3(1 − f)
]
,
E(f ) = A(f )
t
t +1−f
,
where the auxiliary functions s, t and y are written as
s(f) = 1 −2f +3f
2
,t(f)=
s(f) +6f
2
,y(f)
4
=
8
s(f ) + (1 −f)t(f)
.
(7)
The coefficients A, B, C, D and E are monotonic functions of f and their ranges from f = 0
to f = 1are8 A(f ) 4, 0 B(f ) 4
√
2, 8 C(f) 3, −4 D(f ) −2
√
2and
4
√
2 E(f ) 4, respectively. Expression (5)isvalid for small values of m/
√
a and κ,
and/or large values of J .
As this method relies basically on a classical approximation, it is not possible to calculate
the zero-point energy of the orbital motion. Thus, a mass formula cannot be obtained.
2774 FBrauandCSemay
Moreover, the dependence of the energy as a function of the radial quantum number ν is
calculated by making a harmonic approximation around classical orbits with high values of
J .Wecannot expect a good ν behaviour for small values of the angular momentum. A more
serious flaw is that the method predicts a linear dependence of M
2
as a function of ν whatever
theform of the potential. So, we cannot be sure that the ν dependence found is the more
appropriate. A way to correct these drawbacks is to complete the previous analysis by a BSQ
method.
2.3. BSQ method
The basic quantities in the BSQ approach [16] are the action variables,
J
s
=
p
s
dq
s
, (8)
where s labels the degrees of freedom of the system, and where q
s
and p
s
are the coordinates
and conjugate momenta, respectively; the integral is performed over one cycle of motion. The
action variables are quantized according to the prescription
J
s
= ν
s
+1/2, (9)
where ν
s
( 0) is an integer quantum number. This corresponds to a WKB expansion limited
to the first order in h (see, for instance, [17–20]).
The calculations for the angular momentum J ,inthelimit of high values for this quantum
number, give simply the same J dependence for M
2
as in expression (5), butwith J replaced
by J +1/2. The calculations for the radial motion are more involved. A detailed description
of the procedure is given in [12] where the case of Hamiltonian (1)isstudied for m
1
= m
2
and f = 0. We use here the same technique and expand all expressions in powers of the
meson mass M.Assuming that M
2
/a is large and J finite and keeping only terms in M
2
,M
and 1/M,integral(8)with Hamiltonian (1) can be written, after tedious calculations, as
π(2ν +1) =
√
YZ
2a(1 − 2f)
−
X
2
2a(1 − 2f)
3/2
ln
Z +
√
(1 − 2f)Y
X
, (10)
with ν the radial quantum number, and with
X = Mf +2m(1 −f),
Y = M
2
− 4m
2
+2aκ(1 −f)+4amκf/M, (11)
Z = M(1 − f)+2mf + aκ(1 −2f)/M.
The above expression is valid only for f 1/2. A similar equation exists for f 1/2. It is
now necessary to extract M
2
as a function of ν in order to obtain an analytical result usable in a
mass formula. If we assume that quantities m/
√
a and κ are small, we can expand equation (10)
in powers of these small parameters. The first order gives (m/
√
a = κ = 0)
M
2
≈ aE
(f )(2ν +1) for ν 1, (12)
with
E
(f ) = 2π
1 − 2f
1 − f − f
2
H(f)
with
H(f) =
1
√
1−2f
ln
1+
√
1−2f
1−
√
1−2f
for f 1/2,
1
√
2f −1
arccos
1−f
f
for f 1/2.
(13)
Amass formula for light mesons from a potential model 2775
f
0.0 0.2 0.4 0.6 0.8 1.0
4.0
4.5
5.0
5.5
6.0
6.5
E'(f )
E(f )
Figure 1. Coefficients E(f) and E
(f ).
We h ave E
(0) = 2π, E
(1/2) = 3π/2andE
(1) = 4. This is in agreement with results
obtained in [12] for the case f = 0.
The ν square mass dependence obtained with this method is very similar to the one
obtained with the DOS approach. But, the coefficients E(f) and E
(f ) are different, as
shown infigure 1.Theapproximations used to calculate these coefficients are also very
different: E(f ) is expected to give good results when J ν,while E
(f ) is expected to give
good results when ν J .
By expanding the right-hand side of equation (10)inpowersofm/
√
a and κ,weobtain,
at the second order,
π(2ν +1) ≈
M
2
π
aE
(f )
+
Mmf
a(1 − 2f)
3/2
(f ) −
(f )f
(f ) − f
+2(f − 1) ln
(f )
f
− 1
+ κ, (14)
where (f ) = 1+
√
1 − 2f (note that this expression is well defined for f in the range
[0, 1]). We give these expressions for the sake of completeness. Nevertheless, as we can see
below, the first-order term is sufficient for our purpose.
2.4. Mass formula
Usingresults from the DOS approach and the BSQ method, we can write a square mass
formula for light mesons composed of two identical quarks with a mass m as a function of the
quantum numbers J and ν,andtheparameters of the potentials a,κ and f :
M
2
= aA(f )(J +1/2) + B(f )m
a(J +1/2) + C(f )m
2
+ aD(f )κ + aE
(
)
(f )(2ν +1).
(15)
![Figure 5. Square masses M2 of ns̄ mesons as a function of ν and J quantum numbers, with parameters of [1], and for g = 0.5 and f = 0.5. Solid lines join exact solutions of the spinless Salpeter equation. Approximate results from equation (18) with the coefficient E′(f ) are indicated by circles.](/figures/figure-5-square-masses-m2-of-ns-mesons-as-a-function-of-n-3b2mn2k7.webp)
![Figure 4. Square masses M2 of nn̄ mesons as a function of ν and J quantum numbers, with parameters of [1], and for g = 0 and f = 1. Solid lines join exact solutions of the spinless Salpeter equation. Approximate results from equation (15) are indicated by circles (E′(1) = E(1)).](/figures/figure-4-square-masses-m2-of-nn-mesons-as-a-function-of-n-15okw0r1.webp)
![Figure 2. Square masses M2 of nn̄ mesons as a function of ν and J quantum numbers, with parameters of [1], and for g = 0 and f = 0. Solid lines join exact solutions of the spinless Salpeter equation. Approximate results from equation (15) with the coefficient E(f ) are indicated by circles.](/figures/figure-2-square-masses-m2-of-nn-mesons-as-a-function-of-n-pwfnzygx.webp)