William I. Fine Theoretical Physics Institute
University of Minnesota
FTPI-MINN-07/08
UMN-TH-2541/07
LPT-Orsay/07-19
March 2007
Isospin breaking in the yield of heavy meson pairs in
e+e− annihilation near threshold
S. Dubynskiya, A. Le Yaouancb, L. Oliverb, J.-C. Raynalb and M.B. Voloshinc
a School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA
b Laboratoire de Physique Théorique1, Université de Paris XI, Bâtiment 210, 91405 Orsay
Cedex, France
c William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN
55455, USA and Institute of Theoretical and Experimental Physics, Moscow, 117259,
Russia
Abstract
We revisit the problem of interplay between the strong and the Coulomb interac-
tion in the charged-to-neutral yield ratio for BB̄ and DD̄ pairs near their respective
thresholds in e+e− annihilation. We consider here a realistic situation with a resonant
interaction in the isospin I = 0 channel and a nonresonant strong scattering amplitude
in the I = 1 state. We find that the yield ratio has a smooth behavior depending on
the scattering phase in the I = 1 channel. The same approach is also applicable to the
KK̄ production at the φ(1020) resonance, where the Coulomb effect in the charged-to-
neutral yield ratio is generally sensitive to the scattering phases in both the isoscalar
and the isovector channels. Furthermore, we apply the same approach to the treatment
of the effect of the isotopic mass difference between the charged and neutral mesons
and argue that the strong-scattering effects generally result in a modification to the
pure kinematical effect of this mass difference.
1Unité Mixte de Recherche UMR 8627 - CNRS
http://arxiv.org/abs/0704.0293v1
1 Introduction
The JPC = 1−− resonances near the new flavor thresholds: Υ(4S), ψ(3770), and φ(1020) are
the well known sources in e+e− experiments of pairs of the new-flavor mesons: respectively
BB̄, DD̄, and KK̄. A number of experimental approaches depends on the knowledge of the
relative yield of pairs of charged and neutral mesons:
Rc/n =
σ(e+e− → P+P−)
σ(e+e− → P 0P̄ 0)
, (1)
where P stands for the pseudoscalar meson, i.e. B, D, or K, and dedicated measurements
of such ratio have been done at the Υ(4S) resonance [1] at ψ(3770) [2] and at φ(1020) [3].
The values of the ratio Rc/n at all three discussed resonances are close to one due to these
resonances being isotopic scalars, and it is the deviation of the discussed ratio from one that
presents phenomenological interest. This deviation is generally contributed by the following
factors: the isospin violation due to the Coulomb interaction between the charged mesons
and due to the isotopic mass difference between charged and neutral mesons, and, in the
case of the KK̄ production at the φ(1020) resonance, a non-negligible nonresonant isovector
production amplitude. The latter effect can be studied and described as the “tail of the ρ
resonance”, while the isospin breaking due to the mass difference is usually accounted for
as a kinematical effect in the P wave production cross section factor p3, where p is the the
c.m. momentum of each of the mesons. The Coulomb effect has attracted a considerable
theoretical attention. The expression for this effect in the ratio Rc/n in the limit, where the
resonance and the charged mesons are considered as point-like particles [4] has the simple
textbook form:
δRc/n =
, (2)
with α being the QED constant and v the velocity of each of the (charged) mesons in the
c.m. frame. However for the production of the real-life mesons the analysis is complicated
by the charge form factors of the mesons[5], by the form factor in the vertex of interaction of
the resonance with the meson pair [5, 6] and generally by the strong interaction between the
mesons [7, 8, 9]. In particular, it has been argued [8, 9] that the modification of the Coulomb
effect by the strong (resonant) interaction between the mesons is quite significant. The
previously considered picture of the strong interaction was however somewhat unrealistic.
Namely, it has been assumed [8, 9] that the wave function in the I = 1 state of the meson
pair is vanishing at short but finite distances, which would correspond to a singular behavior
of the strong interaction at finite distances. In this paper we derive the formulas for the
Coulomb effect in the ratio Rc/n under the standard assumption about the strong scattering
amplitude in the channels with I = 0 and I = 1. We find that in the case of the Υ(4S) and
ψ(3770) resonances, where the heavy meson pairs are produced by the isotopically singlet
electromagnetic current of the corresponding heavy quark, the strong-interaction effect in
the Coulomb correction depends on the scattering phase δ1 in the I = 1 channel and is a
smooth function of the energy across the resonance, while in the case of the Kaon production
at and near the φ(1020) there is also a smooth dependence on the nonresonant part of the
strong scattering phase δ0 in the isoscalar channel inasmuch as there is a contribution of the
isovector production amplitude at these energies. In either case we find that the behavior
of the Coulomb effect is smooth on the scale of the resonance width, unlike the behavior
previously found [8, 9] under less realistic assumptions.
We further notice that essentially the same calculation can be applied to considering the
effect on the ratio Rc/n of the isotopic mass difference ∆m between the charged and neutral
mesons, at least in the first order in ∆m, by considering the mass difference as a perturbation
by a (constant) potential. In this way we find that the result coincides with the linear in ∆m
term in the ratio of the kinematical factors p3 only in the limit of vanishing strong scattering
phase. Once the latter phase is taken into account, there arises a correction whose relative
contribution is determined by the parameter (p a) with a being the characteristic range of
the strong interaction. We therefore conclude that the conventionally used p3 approximation
for this effect may be somewhat applicable to the KK̄ production at the φ(1020) resonance,
where p ≈ 120MeV, but becomes quite questionable for the DD̄ production at the ψ(3770),
where p ≈ 280MeV.
The strong-scattering phase in the P -wave state of mesons produced in e+e− annihilation
near the threshold is proportional to p3. We therefore expect the discussed effects of the
strong interaction in the ratio Rc/n to exhibit a measurable variation with energy. A mea-
surement of this variation can thus provide an information on the strong scattering phases,
which is not readily available by other means.
The material in the paper is organized as follows. In Sec. 2 we consider the production of
meson pairs by an isosinglet source and derive the formula for the correction to Rc/n due to
a generic isospin-violating interaction potential V (r) viewed as a perturbation. In Sec. 3 we
generalize this treatment to the situation where the source is a coherent mixture of I = 0 and
I = 1. The specific expressions corresponding to the Coulomb interaction and the isotopic
mass difference are considered in Sec. 4. Sec. 5 contains phenomenological estimates of the
constraints on the parameters of the strong interaction between heavy mesons based on the
currently available data [1, 2] for BB̄ and DD̄ production. Finally, in Sec. 6 we summarize
our results.
2 General formulas for an isoscalar source
We start with considering the behavior of the scattering wave functions of a meson-antimeson
pair in the limit of exact isotopic symmetry, i.e. neglecting any Coulomb effects and the
isotopic mass difference. We adopt the standard picture (see e.g. in the textbook [10]),
where the strong interaction is confined within the range of distances r < a, so that beyond
that range, at r > a the motion of the mesons is free. The two relevant independent solutions
to the Schrödinger equation at r > a for the radial wave function in the P wave are the free
outgoing wave
f(pr) =
eipr (3)
and its complex-conjugate, f ∗(pr), describing the incoming wave. A general wave function
of a pair of neutral mesons, φn(r) as well as of a pair of charged mesons, φc(r), in this region
is a linear superposition of these two solutions.
In the region of strong interaction, i.e. at r < a, the isotopic symmetry selects as
independent channels the states with definite isospin, I = 0 and I = 1, corresponding to
the wave functions φ0 = φc + φn and φ1 = φc − φn. The detailed behavior of the I = 0 and
I = 1 wave functions inside the strong interaction region is not important for the present
treatment, and the important point is that the non-singular at r = 0 ‘inner’ wave functions
match at r = a particular linear superpositions of the incoming and outgoing waves (which
superpositions in fact correspond to standing waves):
χ0(r) = e
iδ0 f(pr) + e−iδ0 f ∗(pr) ,
χ1(r) = e
iδ1 f(pr) + e−iδ1 f ∗(pr) , (4)
where δ0 and δ1 are the strong scattering phases in respectively the isoscalar and isovector
states.
Consider now the production of meson pairs by a source localized inside the region of
strong interaction, i.e. r < a, such as e.g. the electromagnetic current. The wave function
of the produced meson pairs at r ≤ a is then determined by both the source and the
strong interaction, and the relevant solution to the Schrödinger equation is chosen by the
requirement that asymptotically at large distances, r → ∞, only an outgoing wave is present.
Let us first consider the simple case where the relevant electromagnetic current is a pure
isotopic singlet, which is the case for DD̄ and BB̄ pair production. Then in the limit of
exact isotopic symmetry the outgoing waves for the ‘n’ and the ‘c’ channels have exactly the
same amplitude, which for our present purpose can be chosen as one:
φ(0)c (r) = f(pr) and φ
n (r) = f(pr) at r → ∞ , (5)
where the superscript (0) stands for the approximation of exact isotopic symmetry. It can be
noted that the approximation of the free motion beyond the region of the strong interaction
in fact makes the expressions in Eq.(5) applicable at all r > a, i.e. all the way down to the
matching point r = a. It is helpful to notice for a later discussion that at the matching point
the I = 1 wave function is vanishing while the I = 0 function φ
0 contains only the outgoing
wave. When continued into the strong interaction region, i.e. at r < a, the function φ0
evolves into the solution determined by the strong interaction and the source.
The isospin-violating effects of the Coulomb interaction and of the mass difference ∆m
between the charged and neutral mesons can be generally described as being due to a presence
of an extra potential V (r) in the ‘c’ channel beyond the region of the strong interaction:
V = −α/r for the Coulomb interaction effect and a constant potential V = 2∆m describing
the mass difference. In other words the wave function φn of the ‘n’ channel is still determined
at r > a by the radial Schrödinger equation for free P -wave motion2, while the equation for
the ‘c’ channel function φc reads as
+ p2 −mV (r)− 2
φc(r) = 0 . (6)
It is assumed throughout the present consideration that the isospin-breaking potential
exists only at distances beyond the range of the strong interaction, i.e. that V (r) has support
only at r > a. The justification for such treatment is that in the region of the strong force
2Clearly, in the considered here first order in the isospin violation only the difference of the interaction
between the two channels is important, thus any such difference can be relegated to one channel, while
keeping the other one unperturbed. Also, any effect of the mass difference in the kinetic term p2/m is of
order v2/c2 as compared to the discussed here effect of ∆m in the overal energy difference between the two
channels, and is totally neglected in our treatment.
small isospin-violating effects are compared to the energy of the strong interaction, so that
the contribution of any such effects arising at r < a is very small, while in the region r > a
the relative contribution of the potential V (r) is determined by its ratio to the kinetic energy
of the mesons, which is small near the threshold.
It should be emphasized that although the interaction at distances r > a is present only
in the ‘c’ channel, the wave functions in both channels are modified in comparison with those
in Eq.(5), as a result of the coupling between channels imposed by the boundary conditions
at r = a. According to the setting of the problem of production of the meson pairs by a
localized source, the appropriate modified functions are those containing at r → ∞ only the
outgoing waves
φc → (1 + x) f(pr), φn → (1 + y) f(pr) , (7)
where the (complex) coefficients x and y arise due to the potential V , and are proportional to
V in the considered here first order of perturbation theory. These coefficients determine the
ratio of the production amplitudes: Ac/An = 1+ x− y, and the discussed here modification
of the yield ratio:
Rc/n = 1 + 2Rex− 2Re y . (8)
The modified wave function in both channels is subject to two conditions:
i: The channel with neutral mesons has only an outgoing wave at all r > a. In other words,
the expression for φn(r) in Eq.(7) is valid at all r down to r = a;
ii: The wave function of the channel with isospin I = 1 at r ≤ a should be proportional to
the standing-wave solution matching the function χ1 in Eq.(4), since there is no source for
the I = 1 state of the meson pairs.
These two conditions are sufficient to fully determine the modified functions at r > a and
thus to find the coefficients x and y.
The first order in V (r) perturbation of the wave function in the channel with charged
mesons is found in the standard way, using the P wave Green’s function G+(r, r
′) satisfying
the equation
+ p2 −
G+(r, r
′) = δ(r − r′) , (9)
and the condition that G+(r, r
′) contains only an outgoing wave when either of its arguments
goes to infinity. The Green’s function is constructed from two solutions of the homogeneous
equation, i.e. from the functions f(pr) and f ∗(pr), as
G+(r, r
2 i p
[f(pr) f ∗(pr′) θ(r − r′) + f(pr′) f ∗(pr) θ(r′ − r)] , (10)
where θ is the standard unit step function. The perturbation δφc is then found as
δφc(r) = m
G+(r, r
′) V (r′) f(pr′) dr′ . (11)
One readily finds from this explicit form of the solution that δφc contains only the outgoing
wave at asymptotic distances r → ∞:
δφc |r→∞ = −
f(pr)
V (r′) |f(pr′)|2 dr′ , (12)
so that the coefficient x is purely imaginary:
x = − i
V (r′) |f(pr′)|2 dr′ (13)
and gives no contribution to the ratio of the production rates Rc/n described by Eq.(8)3.
Consider now the matching of the wave functions at r = a. In this region of r one has
r < r′ in the integral in Eq.(11) so that the correction in the ‘c’ channel has only an incoming
wave:
δφc(r) |r→a = η f
∗(pr) (14)
η = −
V (r′) [f(pr′)]
dr′ . (15)
The wave functions φ0 = φc + φn and φ1 = φc − φn corresponding to the states with isospin
I = 0 and I = 1 are then found as
φ0 |r→a = 2f(pr) + y f(pr) + η f
∗(pr) and φ1 |r→a = η f
∗(pr)− y f(pr) . (16)
One can now apply the condition ii to determine the coefficient y. Indeed, the condition for
the wave function φ1 at r → a to be proportional to f ∗(pr) + e2iδ1 f(pr) requires y to be
given by
y = −η e2iδ1 . (17)
Upon substitution in Eq.(8) this yields
Rc/n = 1 +
e2iδ1
e2ipr
V (r) dr
 . (18)
3It can be noticed that the integral in Eq.(13) is divergent, which corresponds to the infrared-divergent
behavior of the perturbation for the phase of the wave function, logarithmic for the Coulomb interaction
and linear for a constant potential. This slight technical difficulty can be readily resolved, for our present
purposes, by introducing an infrared regularizing factor exp(−λ r) in the potential and setting λ → 0 in the
end result.
3 Mixed isoscalar and isovector source
The formula (18) gives the general expression for the isospin-breaking effect in the considered
yield ratio for the case where the mesons are produced by an isoscalar source. The presented
consideration can also be extended to a situation where the source is a general coherent mix-
ture of an isoscalar and isovector. The specific isotopic composition of the source determines
the ratio of the coefficients of the amplitudes of the running outgoing waves in the I = 1
and I = 0 channels at the matching point r = a, which ratio we denote as A1/A0, thus
defining A1 and A0 as the production amplitudes in the respective channels (in the limit of
exact isotopic symmetry). In this situation the generalization of the expressions in Eq.(5) for
radial wave functions in the ‘outer’ region r > a in the zeroth order in the isospin violation
can be written as
φ(0)c (r) = (A0 + A1) f(pr) and φ
n (r) = (A0 − A1) f(pr) . (19)
The isospin violation in the asymptotic form of these wave functions at r → ∞ can then be
parametrized, similarly to Eq.(7), by complex coefficients x and y as
φc → (A0 + A1) (1 + x) f(pr), φn → (A0 − A1) (1 + y) f(pr) , (20)
so that the yield ratio is found from
Rc/n =
A0 + A1
A0 − A1
(1 + 2Rex− 2Re y ) . (21)
The coefficient x, similarly to the previous discussion and the equation (13), is purely
imaginary and in fact does not contribute in Eq.(21), while the coefficient y is found from the
appropriately modified conditions on the wave functions. Namely, the previously discussed
condition i remains applicable, so that the asymptotic expression in Eq.(20) for the ‘n’
channel function remains valid in the entire ‘outer’ region r > a down to the matching point
r = a. In order to allow for the isovector component of the source the condition ii has to be
modified as will be described few lines below.
The perturbation by the potential V (r) of the ‘c’ channel wave function at the matching
point r = a is readily found, similarly to Eq.(14), as
δφc(r) | r→a = η (A0 + A1) f ∗(pr) (22)
with η given by Eq.(15).
One can now write the expressions for the resulting ‘outer’ wave functions in the isotopic
channels at the matching point:
φ0(r) | r→a = 2A0 f(pr) + η (A0 + A1) f ∗(pr) + y (A0 −A1)f(pr) =
2A0 + y (A0 −A1)− η (A0 + A1) e2iδ0
f(pr) + η (A0 + A1) e
iδ0 χ0(r) (23)
φ1(r) | r→a = 2A1 f(pr) + η (A0 + A1) f ∗(pr)− y (A0 −A1)f(pr) =
2A1 − y (A0 −A1)− η (A0 + A1) e2iδ1
f(pr) + η (A0 + A1) e
iδ1 χ1(r) , (24)
with χ0 and χ1 being the standing wave functions from Eq.(4) in the corresponding isotopic
channels, which when evolved in the region of strong interaction contain no singularity at
r = 0. The remaining parts in the latter expressions for the functions φ0 and φ1 describe
the proper running outgoing waves. These parts, when continued down in r into the strong
interaction region evolve to match the source at r < a. The ratio of the amplitudes of
the isovector and the isoscalar running waves is determined by the isotopic composition of
the source, and by the isotopically symmetric propagation through the strong-interaction
region. Thus the ratio of the amplitudes of these waves at r = a does not depend on the
isospin-breaking effects at r > a and should be equal to A1/A0. Applying this condition to
the isotopic wave functions given by the expressions (23) and (24), one finds the equation
for the coefficient y:
2A1 − y (A0 − A1)− η (A0 + A1) e2iδ1
2A0 + y (A0 − A1)− η (A0 + A1) e2iδ0
. (25)
This equation in fact replaces in this more general situation the previously discussed condi-
tion ii, which condition and the ensuing result in Eq.(17) are readily recovered in the limit
A1/A0 = 0 from Eq.(25).
Considering that both y and η are of the first order in the potential V , it is sufficient to
use the linear expansion of the equation (25) in y and η, finding in this way the solution for
y in the form
y = −ηA0 e
2iδ1 − A1 e2iδ0
A0 − A1
, (26)
and thus arriving at the final formula for the relative yield:
Rc/n =
A0 + A1
A0 −A1
2iδ1 − A1 e2iδ0
A0 − A1
e2ipr
V (r) dr
. (27)
Given that A0 = |A0| eiδ0 and A1 = |A1| eiδ1, the amplitude-dependent factor in this formula
can also be written in terms of the real ratio ρ = |A1/A0| as
2iδ1 −A1 e2iδ0
A0 −A1
= e2iδ1
1− ρ ei(δ0−δ1)
1− ρ e−i(δ0−δ1)
. (28)
4 The Coulomb and the mass-difference effects
The general formulas in Eq.(18) and (27) can now be applied to a discussion of the specific
isospin-breaking effects in the e+e− production of meson pairs at and near the threshold
resonances. We start with considering the effect of the Coulomb interaction. In a detailed
treatment of this correction one should include the realistic form factors of the mesons,
which cut off at short distances the difference in the electromagnetic interactions between
the charged and neutral mesons. In the present discussion we replace for simplicity the
gradual cutoff of the Coulomb interaction by an abrupt cutoff at an effective range r = ac,
where generally ac ≥ a 4. The master integral with the Coulomb potential V (r) = −α/r in
the equations (18) and (27) then takes the form
e2ipr
V (r) dr =
cos 2pac
2(pac)2
sin 2pac
− Ci(2pac)
− cos 2pac
sin 2pac
2(pac)2
− Si(2pac)
2 (pac)2
− ln(2 pac) + 1− γE
(pac)
, (29)
where the integral sine and cosine are defined in the standard way:
Si(z) =
sin t
and Ci(z) = −
cos t
and γE = 0.577 . . . is the Euler’s constant. The latter line in Eq.(29) shows few first terms of
the expansion of the integral in the parameter (pac). This expansion illustrates the behavior
of the correction toward the threshold. For the purpose of this illustration one can consider
first the simpler expression in Eq.(18). The imaginary part, which determines the discussed
Coulomb effect in Rc/n in the limit where there is no strong scattering, δ1 → 0, is not singular
4As previously mentioned, any extension of the isospin-breaking potential inside the strong interaction
region can result only in very small corrections.
at pac → 0, and the textbook formula (2) is recovered in this limit. The real part of the
integral in Eq.(29) is singular at small pac, but it multiplies in Eq.(18) the factor sin δ1. The
P -wave scattering phase in its turn is proportional at small momenta to p3: δ1 ∼ (pa)3, so
that the overall contribution of the real part of the integral is not singular at the threshold
either. Considering a more general expression for the Coulomb effect for the case of an
isotopically mixed source, following from the equation (27), one can readily arrive at the
same conclusion that the singular in (pac) real part of the integral (29) does not lead to
an actual singularity, since it only enters the ratio Rc/n multiplied by a combination of the
phases δ0 and δ1 (cf. Eq.(28)), each vanishing as p
3 toward the threshold.
As previously mentioned, the effect of the isotopic mass difference corresponds to that of
a constant potential V = 2∆m extending from the range of the strong interaction r = a to
infinity. The master integral with such potential has the form
e2ipr
V (r) dr =
2 cos 2 pa
+ sin 2 pa+ i
2 sin 2 pa
− cos 2 pa
− 2 pa+ 3 i+O
(pa)2
. (30)
In the limit of vanishing strong scattering phases the mass correction to Rc/n is determined
by only the imaginary part of the integral, which in the limit of small pa thus yields
Rc/n = 1− 3∆m
= 1− 3∆m
, (31)
where E is the total kinetic energy of the meson pair, and the found expression coincides
with the linear in ∆m term in the expansion of the usually assumed ratio of the kinematical
factors (p+/p0)
3. Clearly, in the more realistic case of presence of the strong scattering the
real part of the integral in Eq.(30) also contributes and the simple kinematical approximation
is generally invalidated.
5 Phenomenological estimates
In this section we discuss application of our formulas to interpreting the data on the charged
to neutral meson yield ratio Rc/n at the near-threshold resonances Υ(4S), ψ(3770) and
φ(1020). The purpose of this discussion is to illustrate the effect of the strong scatering on
the isospin breaking corrections, and we use here the simplified picture of a abrupt cutoff
of the Coulomb interaction and of the isotopic mass difference effects. Such simplification
generally can be used as long as the parameter (pa) is not large. A detailed analysis should
likely involve a model of a gradual cutoff, since the details of the transition become important
at lager momenta.
5.1 Υ(4S)
The simplest case for the study of the isospin breaking corrections in the relative production
of heavy mesons is offered by the BB̄ pair production near and at the Υ(4S) resonance.
Indeed, this process only is due to the purely isosinglet electromagnetic current of the b
quarks, and the isotopic mass difference between the B mesons is very small: ∆mB =
−0.33±0.28MeV [11], so that any deviation of the ratio Rc/n from one is essentially entirely
due to the Coulomb interaction. On the other hand, the parameter α/v for the Coulomb
effect in this case is the largest due to small velocity of the B mesons: at the energy of the
Υ(4S) peak vB/c ≈ 0.06. In particular, the numerical value in the expression (2) is 0.19.
The experimental data [1] however indicate a significantly smaller deviation of Rc/n from one.
The BaBar data with the smallest errors give Rc/n = 1.006±0.036±0.031. Such behavior is
likely a result of a combined effect of the meson and production vertex form factors [5, 6] and
of the discussed here modification of the Coulomb correction by the strong scattering phase.
These effects can in principle be separated and studied quantitatively by measuring the
energy dependence of the ratio Rc/n near the Υ(4S) resonance. With the presently available
data we can only use a simplified parametrization of the form factor effects by introducing
an abrupt cutoff for the Coulomb interaction at r = ac ≥ a and thereby estimate the likely
regions in the (ac, δ1) plane. Such estimate from the equations (18) and (29) is shown in Fig.1
as a one-sigma area, corresponding to the BaBar data with the statistical and systematic
errors added in quadrature: Rc/n = 1.006± .048. Clearly, more precise data from dedicated
measurements of the ratio Rc/n are needed for a better understanding of the parameters of
strong interaction between the B mesons.
0.2 0.4 0.6 0.8 1 1.2
Figure 1: The one sigma area (shaded) in the (ac, δ1) plane corresponding to the BaBar data
on the B+B−/B0B̄0 yield ratio at the Υ(4S) resonance.
5.2 ψ(3770)
The largest isospin-breaking effect in the DD̄ production at the ψ(3770) is that due to the
mass difference between the charged and the neutral D mesons: ∆mD = 4.78±0.10MeV [11].
The most precise measurements of this process have been done [2] at the energy
3773MeV. At this energy the momentum of each charged D meson is p+ = 254MeV and
that for a neutral D meson is p0 = 287MeV. Thus the ratio of the kinematical factors
(p+/p0)
3 ≈ 0.69 is significantly less than one. The Coulomb effect is somewhat smaller.
Indeed, the velocity of a charged meson at this energy is v+/c = 0.135 and the expression
(2) gives numerically 0.085. One can notice that if the kinematical and the Coulomb factors
are combined in a straightforward way to estimate Rc/n = (p+/p0)
3 [1 + πα/(2v+)] ≈ 0.75,
this would be in a very good agreement with the experimental number [2]: Rc/n = 0.776 ±
0.024+0.014
−0.006. Thus it is quite likely that at this particular energy there is a considerable
cancelation between the strong-interaction effects in the yield ratio, and such cancelation by
itself imposes constraints on the parameters of strong interaction between the D mesons,
which constraints is interesting to analyze.
An analysis of the strong-interaction effects along the lines discussed in the present paper
generally runs into two difficulties. One is that our approach is accurate only in the linear
in ∆m approximation, while the actual effect of the isotopic mass difference between the
D mesons is not very small. However, numerically, the first term in the expansion of the
kinematical factor (Eq.(31)) gives 0.67, which is quite close to the mentioned above value
0.69, and it looks like the linear term gives a reasonable approximation. The other point
is that the cutoff parameter ac for the Coulomb interaction at short distances does not
necessarily coincide with the range parameter a used for the short-distance cutoff of the
effect of the mass difference. However, as previously noted, the Coulomb effect is somewhat
small at the energy of the ψ(3770) resonance, and for the purpose of preliminary estimates
we set ac = a in our numerical analysis. In order to allow for possible errors introduced by
our approximations in comparing with the data, we linearly add a theoretical uncertainty
of 0.03 units to the combined in quadrature statistical and experimental errors. Proceeding
in this way we find that the only region in the (a, δ1) plane at a < 2 fm consistent with the
CLEO-c data at one sigma level is the one shown in Fig.2.
0.2 0.4 0.6 0.8 1
Figure 2: The area (shaded) in the (a, δ1) plane corresponding to the CLEO-c data on the
D+D−/D0D̄0 yield ratio at the ψ(3770) resonance. The uncertainty shown includes a one
sigma experimental error with our estimate of the theoretical uncertainty added linearly.
It is interesting to compare the plots in the Figures 1 and 2. In the heavy quark limit
applied to both b and c quarks the strong interaction between the heavy mesons should be
the same, corresponding to the same range parameters a and ac. The scattering phase δ1
for these two systems is generally different due to different masses. However, provided there
are no isovector ‘molecular’ bound states, the sign of the phase should be the same, with
the absolute value of the phase for heavier B mesons being larger than for the D mesons.
The comparison with the data for the D mesons favors small values of the range parameter,
as indicated by Fig.2. If one also assumes that ac ≈ a for the B mesons, the short range
of ac, according to Fig.1, is compatible with the B mesons data at a negative scattering
phase δ1, which sign of δ1 is also in agreement with the D meson data. A negative sign of δ1
corresponds to a repulsion, which for the I = 1 state of heavy meson pairs can be expected
on general grounds [12].
5.3 φ(1020)
We believe that the production of KK̄ pairs in e+e− annihilation at and near the φ(1020)
resonance merits a separate analysis along the lines discussed in the present paper and using
detailed data similar to those in Ref.[3]. As is known, this production receives a small but
measurable nonresonant contribution from the isovector part of the electromagnetic current
of the u and d quarks, which corresponds to an isotopically mixed source. Furthermore, it
has been pointed out [13] that a detailed theoretical analysis of the K+K−/K0K̄0 yield ratio
at the φ(1020) resonance produces a result which possibly is at a meaningful variance with
the data.
At present we limit ourselves to noticing that the formula in Eq.(27), applicable in this
situation, describes a smooth behavior of the considered isospin breaking effects across the
resonance in the I = 0 channel. Indeed, the I = 0 scattering phase at energy E near the
resonance energy E0 is given by the Breit-Wigner formula
e2iδ0 =
∆− i γ
∆+ i γ
e2iδ̃0 , (32)
where ∆ = E −E0, δ̃0 is the nonresonant scattering phase in the isoscalar channel, and γ is
the width parameter. Both δ̃0 and γ are smooth functions of the energy proportional to p
small momentum, and γ(E0) determines the resonance width Γ as γ = Γ/2. The ratio of the
isovector and isoscalar production amplitudes can then be parametrized near the resonance
∆+ i γ
ei (δ1−δ̃0) , (33)
where µ is a parameter with dimension of energy: µ ∼ mφ − mρ. The amplitude ratio
entering the correction factor in Eq.(27) can then be written in the form
2iδ1 − A1 e2iδ0
A0 − A1
= e2iδ1
µ− (∆− i γ) e−i(δ1−δ̃0)
µ− (∆ + i γ) e+i(δ1−δ̃0)
, (34)
which manifestly shows that this ratio is a pure phase factor of a complex quantity slowly
varying across the φ(1020) resonance.
6 Summary
We have considered the effects of the isospin breaking by the Coulomb interaction and by
the isotopic mass difference in the relative yield Rc/n of pairs of charged and neutral mesons
near threshold by a compact source, such as in the production of heavy mesons in e+e−
annihilation. These effects are modified by the strong interaction scattering phases. The
general formula for a situation where the source is an arbitrary coherent mixture of an
isoscalar and isovector is given by Eq.(27). In particular, for a purely isoscalar source, which
is the case for the e+e− annihilation into DD̄ and BB̄ pairs the strong-interaction effect is
determined by the scattering phase δ1 in the I = 1 channel (Eq.(18)). As a practical matter
we find that under the standard assumptions about the strong scattering amplitudes in the
near-threshold resonance region the ratio Rc/n has a smooth behavior with energy showing
no abnormal rapid variation on the scale of the resonance width. The energy dependence
of this ratio is rather determined by the non-resonant scattering scattering phase(s). In
the P -wave the phase δ1 is proportional to p
3, so that a measurement of the behavior ratio
Rc/n with energy can provide information on this phase, which is not readily accessible by
other means. The behavior of the ratio Rc/n at larger energies away from the threshold also
depends on the details of the onset of the strong interaction between the heavy mesons at
short distances and on the behavior of their electromagnetic form factors, and a study of
this behavior can provide an insight into these properties of the heavy-light hadrons.
Acknowledgements
The work of MBV is supported, in part, by the DOE grant DE-FG02-94ER40823.
References
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http://arxiv.org/abs/hep-ph/0208194
	Introduction
	General formulas for an isoscalar source
	Mixed isoscalar and isovector source
	The Coulomb and the mass-difference effects
	Phenomenological estimates
	(4S)
	(3770)
	(1020)
	Summary

  We revisit the problem of interplay between the strong and the Coulomb
interaction in the charged-to-neutral yield ratio for $B {\bar B}$ and $D {\bar
D}$ pairs near their respective thresholds in $e^+e^-$ annihilation. We
consider here a realistic situation with a resonant interaction in the isospin
I=0 channel and a nonresonant strong scattering amplitude in the I=1 state. We
find that the yield ratio has a smooth behavior depending on the scattering
phase in the I=1 channel. The same approach is also applicable to the $K {\bar
K}$ production at the $\phi(1020)$ resonance, where the Coulomb effect in the
charged-to-neutral yield ratio is generally sensitive to the scattering phases
in both the isoscalar and the isovector channels. Furthermore, we apply the
same approach to the treatment of the effect of the isotopic mass difference
between the charged and neutral mesons and argue that the strong-scattering
effects generally result in a modification to the pure kinematical effect of
this mass difference.

Introduction
The JPC = 1−− resonances near the new flavor thresholds: Υ(4S), ψ(3770), and φ(1020) are
the well known sources in e+e− experiments of pairs of the new-flavor mesons: respectively
BB̄, DD̄, and KK̄. A number of experimental approaches depends on the knowledge of the
relative yield of pairs of charged and neutral mesons:
Rc/n =
σ(e+e− → P+P−)
σ(e+e− → P 0P̄ 0)
, (1)
where P stands for the pseudoscalar meson, i.e. B, D, or K, and dedicated measurements
of such ratio have been done at the Υ(4S) resonance [1] at ψ(3770) [2] and at φ(1020) [3].
The values of the ratio Rc/n at all three discussed resonances are close to one due to these
resonances being isotopic scalars, and it is the deviation of the discussed ratio from one that
presents phenomenological interest. This deviation is generally contributed by the following
factors: the isospin violation due to the Coulomb interaction between the charged mesons
and due to the isotopic mass difference between charged and neutral mesons, and, in the
case of the KK̄ production at the φ(1020) resonance, a non-negligible nonresonant isovector
production amplitude. The latter effect can be studied and described as the “tail of the ρ
resonance”, while the isospin breaking due to the mass difference is usually accounted for
as a kinematical effect in the P wave production cross section factor p3, where p is the the
c.m. momentum of each of the mesons. The Coulomb effect has attracted a considerable
theoretical attention. The expression for this effect in the ratio Rc/n in the limit, where the
resonance and the charged mesons are considered as point-like particles [4] has the simple
textbook form:
δRc/n =
, (2)
with α being the QED constant and v the velocity of each of the (charged) mesons in the
c.m. frame. However for the production of the real-life mesons the analysis is complicated
by the charge form factors of the mesons[5], by the form factor in the vertex of interaction of
the resonance with the meson pair [5, 6] and generally by the strong interaction between the
mesons [7, 8, 9]. In particular, it has been argued [8, 9] that the modification of the Coulomb
effect by the strong (resonant) interaction between the mesons is quite significant. The
previously considered picture of the strong interaction was however somewhat unrealistic.
Namely, it has been assumed [8, 9] that the wave function in the I = 1 state of the meson
pair is vanishing at short but finite distances, which would correspond to a singular behavior
of the strong interaction at finite distances. In this paper we derive the formulas for the
Coulomb effect in the ratio Rc/n under the standard assumption about the strong scattering
amplitude in the channels with I = 0 and I = 1. We find that in the case of the Υ(4S) and
ψ(3770) resonances, where the heavy meson pairs are produced by the isotopically singlet
electromagnetic current of the corresponding heavy quark, the strong-interaction effect in
the Coulomb correction depends on the scattering phase δ1 in the I = 1 channel and is a
smooth function of the energy across the resonance, while in the case of the Kaon production
at and near the φ(1020) there is also a smooth dependence on the nonresonant part of the
strong scattering phase δ0 in the isoscalar channel inasmuch as there is a contribution of the
isovector production amplitude at these energies. In either case we find that the behavior
of the Coulomb effect is smooth on the scale of the resonance width, unlike the behavior
previously found [8, 9] under less realistic assumptions.
We further notice that essentially the same calculation can be applied to considering the
effect on the ratio Rc/n of the isotopic mass difference ∆m between the charged and neutral
mesons, at least in the first order in ∆m, by considering the mass difference as a perturbation
by a (constant) potential. In this way we find that the result coincides with the linear in ∆m
term in the ratio of the kinematical factors p3 only in the limit of vanishing strong scattering
phase. Once the latter phase is taken into account, there arises a correction whose relative
contribution is determined by the parameter (p a) with a being the characteristic range of
the strong interaction. We therefore conclude that the conventionally used p3 approximation
for this effect may be somewhat applicable to the KK̄ production at the φ(1020) resonance,
where p ≈ 120MeV, but becomes quite questionable for the DD̄ production at the ψ(3770),
where p ≈ 280MeV.
The strong-scattering phase in the P -wave state of mesons produced in e+e− annihilation
near the threshold is proportional to p3. We therefore expect the discussed effects of the
strong interaction in the ratio Rc/n to exhibit a measurable variation with energy. A mea-
surement of this variation can thus provide an information on the strong scattering phases,
which is not readily available by other means.
The material in the paper is organized as follows. In Sec. 2 we consider the production of
meson pairs by an isosinglet source and derive the formula for the correction to Rc/n due to
a generic isospin-violating interaction potential V (r) viewed as a perturbation. In Sec. 3 we
generalize this treatment to the situation where the source is a coherent mixture of I = 0 and
I = 1. The specific expressions corresponding to the Coulomb interaction and the isotopic
mass difference are considered in Sec. 4. Sec. 5 contains phenomenological estimates of the
constraints on the parameters of the strong interaction between heavy mesons based on the
currently available data [1, 2] for BB̄ and DD̄ production. Finally, in Sec. 6 we summarize
our results.
2 General formulas for an isoscalar source
We start with considering the behavior of the scattering wave functions of a meson-antimeson
pair in the limit of exact isotopic symmetry, i.e. neglecting any Coulomb effects and the
isotopic mass difference. We adopt the standard picture (see e.g. in the textbook [10]),
where the strong interaction is confined within the range of distances r < a, so that beyond
that range, at r > a the motion of the mesons is free. The two relevant independent solutions
to the Schrödinger equation at r > a for the radial wave function in the P wave are the free
outgoing wave
f(pr) =
eipr (3)
and its complex-conjugate, f ∗(pr), describing the incoming wave. A general wave function
of a pair of neutral mesons, φn(r) as well as of a pair of charged mesons, φc(r), in this region
is a linear superposition of these two solutions.
In the region of strong interaction, i.e. at r < a, the isotopic symmetry selects as
independent channels the states with definite isospin, I = 0 and I = 1, corresponding to
the wave functions φ0 = φc + φn and φ1 = φc − φn. The detailed behavior of the I = 0 and
I = 1 wave functions inside the strong interaction region is not important for the present
treatment, and the important point is that the non-singular at r = 0 ‘inner’ wave functions
match at r = a particular linear superpositions of the incoming and outgoing waves (which
superpositions in fact correspond to standing waves):
χ0(r) = e
iδ0 f(pr) + e−iδ0 f ∗(pr) ,
χ1(r) = e
iδ1 f(pr) + e−iδ1 f ∗(pr) , (4)
where δ0 and δ1 are the strong scattering phases in respectively the isoscalar and isovector
states.
Consider now the production of meson pairs by a source localized inside the region of
strong interaction, i.e. r < a, such as e.g. the electromagnetic current. The wave function
of the produced meson pairs at r ≤ a is then determined by both the source and the
strong interaction, and the relevant solution to the Schrödinger equation is chosen by the
requirement that asymptotically at large distances, r → ∞, only an outgoing wave is present.
Let us first consider the simple case where the relevant electromagnetic current is a pure
isotopic singlet, which is the case for DD̄ and BB̄ pair production. Then in the limit of
exact isotopic symmetry the outgoing waves for the ‘n’ and the ‘c’ channels have exactly the
same amplitude, which for our present purpose can be chosen as one:
φ(0)c (r) = f(pr) and φ
n (r) = f(pr) at r → ∞ , (5)
where the superscript (0) stands for the approximation of exact isotopic symmetry. It can be
noted that the approximation of the free motion beyond the region of the strong interaction
in fact makes the expressions in Eq.(5) applicable at all r > a, i.e. all the way down to the
matching point r = a. It is helpful to notice for a later discussion that at the matching point
the I = 1 wave function is vanishing while the I = 0 function φ
0 contains only the outgoing
wave. When continued into the strong interaction region, i.e. at r < a, the function φ0
evolves into the solution determined by the strong interaction and the source.
The isospin-violating effects of the Coulomb interaction and of the mass difference ∆m
between the charged and neutral mesons can be generally described as being due to a presence
of an extra potential V (r) in the ‘c’ channel beyond the region of the strong interaction:
V = −α/r for the Coulomb interaction effect and a constant potential V = 2∆m describing
the mass difference. In other words the wave function φn of the ‘n’ channel is still determined
at r > a by the radial Schrödinger equation for free P -wave motion2, while the equation for
the ‘c’ channel function φc reads as
+ p2 −mV (r)− 2
φc(r) = 0 . (6)
It is assumed throughout the present consideration that the isospin-breaking potential
exists only at distances beyond the range of the strong interaction, i.e. that V (r) has support
only at r > a. The justification for such treatment is that in the region of the strong force
2Clearly, in the considered here first order in the isospin violation only the difference of the interaction
between the two channels is important, thus any such difference can be relegated to one channel, while
keeping the other one unperturbed. Also, any effect of the mass difference in the kinetic term p2/m is of
order v2/c2 as compared to the discussed here effect of ∆m in the overal energy difference between the two
channels, and is totally neglected in our treatment.
small isospin-violating effects are compared to the energy of the strong interaction, so that
the contribution of any such effects arising at r < a is very small, while in the region r > a
the relative contribution of the potential V (r) is determined by its ratio to the kinetic energy
of the mesons, which is small near the threshold.
It should be emphasized that although the interaction at distances r > a is present only
in the ‘c’ channel, the wave functions in both channels are modified in comparison with those
in Eq.(5), as a result of the coupling between channels imposed by the boundary conditions
at r = a. According to the setting of the problem of production of the meson pairs by a
localized source, the appropriate modified functions are those containing at r → ∞ only the
outgoing waves
φc → (1 + x) f(pr), φn → (1 + y) f(pr) , (7)
where the (complex) coefficients x and y arise due to the potential V , and are proportional to
V in the considered here first order of perturbation theory. These coefficients determine the
ratio of the production amplitudes: Ac/An = 1+ x− y, and the discussed here modification
of the yield ratio:
Rc/n = 1 + 2Rex− 2Re y . (8)
The modified wave function in both channels is subject to two conditions:
i: The channel with neutral mesons has only an outgoing wave at all r > a. In other words,
the expression for φn(r) in Eq.(7) is valid at all r down to r = a;
ii: The wave function of the channel with isospin I = 1 at r ≤ a should be proportional to
the standing-wave solution matching the function χ1 in Eq.(4), since there is no source for
the I = 1 state of the meson pairs.
These two conditions are sufficient to fully determine the modified functions at r > a and
thus to find the coefficients x and y.
The first order in V (r) perturbation of the wave function in the channel with charged
mesons is found in the standard way, using the P wave Green’s function G+(r, r
′) satisfying
the equation
+ p2 −
G+(r, r
′) = δ(r − r′) , (9)
and the condition that G+(r, r
′) contains only an outgoing wave when either of its arguments
goes to infinity. The Green’s function is constructed from two solutions of the homogeneous
equation, i.e. from the functions f(pr) and f ∗(pr), as
G+(r, r
2 i p
[f(pr) f ∗(pr′) θ(r − r′) + f(pr′) f ∗(pr) θ(r′ − r)] , (10)
where θ is the standard unit step function. The perturbation δφc is then found as
δφc(r) = m
G+(r, r
′) V (r′) f(pr′) dr′ . (11)
One readily finds from this explicit form of the solution that δφc contains only the outgoing
wave at asymptotic distances r → ∞:
δφc |r→∞ = −
f(pr)
V (r′) |f(pr′)|2 dr′ , (12)
so that the coefficient x is purely imaginary:
x = − i
V (r′) |f(pr′)|2 dr′ (13)
and gives no contribution to the ratio of the production rates Rc/n described by Eq.(8)3.
Consider now the matching of the wave functions at r = a. In this region of r one has
r < r′ in the integral in Eq.(11) so that the correction in the ‘c’ channel has only an incoming
wave:
δφc(r) |r→a = η f
∗(pr) (14)
η = −
V (r′) [f(pr′)]
dr′ . (15)
The wave functions φ0 = φc + φn and φ1 = φc − φn corresponding to the states with isospin
I = 0 and I = 1 are then found as
φ0 |r→a = 2f(pr) + y f(pr) + η f
∗(pr) and φ1 |r→a = η f
∗(pr)− y f(pr) . (16)
One can now apply the condition ii to determine the coefficient y. Indeed, the condition for
the wave function φ1 at r → a to be proportional to f ∗(pr) + e2iδ1 f(pr) requires y to be
given by
y = −η e2iδ1 . (17)
Upon substitution in Eq.(8) this yields
Rc/n = 1 +
e2iδ1
e2ipr
V (r) dr
 . (18)
3It can be noticed that the integral in Eq.(13) is divergent, which corresponds to the infrared-divergent
behavior of the perturbation for the phase of the wave function, logarithmic for the Coulomb interaction
and linear for a constant potential. This slight technical difficulty can be readily resolved, for our present
purposes, by introducing an infrared regularizing factor exp(−λ r) in the potential and setting λ → 0 in the
end result.
3 Mixed isoscalar and isovector source
The formula (18) gives the general expression for the isospin-breaking effect in the considered
yield ratio for the case where the mesons are produced by an isoscalar source. The presented
consideration can also be extended to a situation where the source is a general coherent mix-
ture of an isoscalar and isovector. The specific isotopic composition of the source determines
the ratio of the coefficients of the amplitudes of the running outgoing waves in the I = 1
and I = 0 channels at the matching point r = a, which ratio we denote as A1/A0, thus
defining A1 and A0 as the production amplitudes in the respective channels (in the limit of
exact isotopic symmetry). In this situation the generalization of the expressions in Eq.(5) for
radial wave functions in the ‘outer’ region r > a in the zeroth order in the isospin violation
can be written as
φ(0)c (r) = (A0 + A1) f(pr) and φ
n (r) = (A0 − A1) f(pr) . (19)
The isospin violation in the asymptotic form of these wave functions at r → ∞ can then be
parametrized, similarly to Eq.(7), by complex coefficients x and y as
φc → (A0 + A1) (1 + x) f(pr), φn → (A0 − A1) (1 + y) f(pr) , (20)
so that the yield ratio is found from
Rc/n =
A0 + A1
A0 − A1
(1 + 2Rex− 2Re y ) . (21)
The coefficient x, similarly to the previous discussion and the equation (13), is purely
imaginary and in fact does not contribute in Eq.(21), while the coefficient y is found from the
appropriately modified conditions on the wave functions. Namely, the previously discussed
condition i remains applicable, so that the asymptotic expression in Eq.(20) for the ‘n’
channel function remains valid in the entire ‘outer’ region r > a down to the matching point
r = a. In order to allow for the isovector component of the source the condition ii has to be
modified as will be described few lines below.
The perturbation by the potential V (r) of the ‘c’ channel wave function at the matching
point r = a is readily found, similarly to Eq.(14), as
δφc(r) | r→a = η (A0 + A1) f ∗(pr) (22)
with η given by Eq.(15).
One can now write the expressions for the resulting ‘outer’ wave functions in the isotopic
channels at the matching point:
φ0(r) | r→a = 2A0 f(pr) + η (A0 + A1) f ∗(pr) + y (A0 −A1)f(pr) =
2A0 + y (A0 −A1)− η (A0 + A1) e2iδ0
f(pr) + η (A0 + A1) e
iδ0 χ0(r) (23)
φ1(r) | r→a = 2A1 f(pr) + η (A0 + A1) f ∗(pr)− y (A0 −A1)f(pr) =
2A1 − y (A0 −A1)− η (A0 + A1) e2iδ1
f(pr) + η (A0 + A1) e
iδ1 χ1(r) , (24)
with χ0 and χ1 being the standing wave functions from Eq.(4) in the corresponding isotopic
channels, which when evolved in the region of strong interaction contain no singularity at
r = 0. The remaining parts in the latter expressions for the functions φ0 and φ1 describe
the proper running outgoing waves. These parts, when continued down in r into the strong
interaction region evolve to match the source at r < a. The ratio of the amplitudes of
the isovector and the isoscalar running waves is determined by the isotopic composition of
the source, and by the isotopically symmetric propagation through the strong-interaction
region. Thus the ratio of the amplitudes of these waves at r = a does not depend on the
isospin-breaking effects at r > a and should be equal to A1/A0. Applying this condition to
the isotopic wave functions given by the expressions (23) and (24), one finds the equation
for the coefficient y:
2A1 − y (A0 − A1)− η (A0 + A1) e2iδ1
2A0 + y (A0 − A1)− η (A0 + A1) e2iδ0
. (25)
This equation in fact replaces in this more general situation the previously discussed condi-
tion ii, which condition and the ensuing result in Eq.(17) are readily recovered in the limit
A1/A0 = 0 from Eq.(25).
Considering that both y and η are of the first order in the potential V , it is sufficient to
use the linear expansion of the equation (25) in y and η, finding in this way the solution for
y in the form
y = −ηA0 e
2iδ1 − A1 e2iδ0
A0 − A1
, (26)
and thus arriving at the final formula for the relative yield:
Rc/n =
A0 + A1
A0 −A1
2iδ1 − A1 e2iδ0
A0 − A1
e2ipr
V (r) dr
. (27)
Given that A0 = |A0| eiδ0 and A1 = |A1| eiδ1, the amplitude-dependent factor in this formula
can also be written in terms of the real ratio ρ = |A1/A0| as
2iδ1 −A1 e2iδ0
A0 −A1
= e2iδ1
1− ρ ei(δ0−δ1)
1− ρ e−i(δ0−δ1)
. (28)
4 The Coulomb and the mass-difference effects
The general formulas in Eq.(18) and (27) can now be applied to a discussion of the specific
isospin-breaking effects in the e+e− production of meson pairs at and near the threshold
resonances. We start with considering the effect of the Coulomb interaction. In a detailed
treatment of this correction one should include the realistic form factors of the mesons,
which cut off at short distances the difference in the electromagnetic interactions between
the charged and neutral mesons. In the present discussion we replace for simplicity the
gradual cutoff of the Coulomb interaction by an abrupt cutoff at an effective range r = ac,
where generally ac ≥ a 4. The master integral with the Coulomb potential V (r) = −α/r in
the equations (18) and (27) then takes the form
e2ipr
V (r) dr =
cos 2pac
2(pac)2
sin 2pac
− Ci(2pac)
− cos 2pac
sin 2pac
2(pac)2
− Si(2pac)
2 (pac)2
− ln(2 pac) + 1− γE
(pac)
, (29)
where the integral sine and cosine are defined in the standard way:
Si(z) =
sin t
and Ci(z) = −
cos t
and γE = 0.577 . . . is the Euler’s constant. The latter line in Eq.(29) shows few first terms of
the expansion of the integral in the parameter (pac). This expansion illustrates the behavior
of the correction toward the threshold. For the purpose of this illustration one can consider
first the simpler expression in Eq.(18). The imaginary part, which determines the discussed
Coulomb effect in Rc/n in the limit where there is no strong scattering, δ1 → 0, is not singular
4As previously mentioned, any extension of the isospin-breaking potential inside the strong interaction
region can result only in very small corrections.
at pac → 0, and the textbook formula (2) is recovered in this limit. The real part of the
integral in Eq.(29) is singular at small pac, but it multiplies in Eq.(18) the factor sin δ1. The
P -wave scattering phase in its turn is proportional at small momenta to p3: δ1 ∼ (pa)3, so
that the overall contribution of the real part of the integral is not singular at the threshold
either. Considering a more general expression for the Coulomb effect for the case of an
isotopically mixed source, following from the equation (27), one can readily arrive at the
same conclusion that the singular in (pac) real part of the integral (29) does not lead to
an actual singularity, since it only enters the ratio Rc/n multiplied by a combination of the
phases δ0 and δ1 (cf. Eq.(28)), each vanishing as p
3 toward the threshold.
As previously mentioned, the effect of the isotopic mass difference corresponds to that of
a constant potential V = 2∆m extending from the range of the strong interaction r = a to
infinity. The master integral with such potential has the form
e2ipr
V (r) dr =
2 cos 2 pa
+ sin 2 pa+ i
2 sin 2 pa
− cos 2 pa
− 2 pa+ 3 i+O
(pa)2
. (30)
In the limit of vanishing strong scattering phases the mass correction to Rc/n is determined
by only the imaginary part of the integral, which in the limit of small pa thus yields
Rc/n = 1− 3∆m
= 1− 3∆m
, (31)
where E is the total kinetic energy of the meson pair, and the found expression coincides
with the linear in ∆m term in the expansion of the usually assumed ratio of the kinematical
factors (p+/p0)
3. Clearly, in the more realistic case of presence of the strong scattering the
real part of the integral in Eq.(30) also contributes and the simple kinematical approximation
is generally invalidated.
5 Phenomenological estimates
In this section we discuss application of our formulas to interpreting the data on the charged
to neutral meson yield ratio Rc/n at the near-threshold resonances Υ(4S), ψ(3770) and
φ(1020). The purpose of this discussion is to illustrate the effect of the strong scatering on
the isospin breaking corrections, and we use here the simplified picture of a abrupt cutoff
of the Coulomb interaction and of the isotopic mass difference effects. Such simplification
generally can be used as long as the parameter (pa) is not large. A detailed analysis should
likely involve a model of a gradual cutoff, since the details of the transition become important
at lager momenta.
5.1 Υ(4S)
The simplest case for the study of the isospin breaking corrections in the relative production
of heavy mesons is offered by the BB̄ pair production near and at the Υ(4S) resonance.
Indeed, this process only is due to the purely isosinglet electromagnetic current of the b
quarks, and the isotopic mass difference between the B mesons is very small: ∆mB =
−0.33±0.28MeV [11], so that any deviation of the ratio Rc/n from one is essentially entirely
due to the Coulomb interaction. On the other hand, the parameter α/v for the Coulomb
effect in this case is the largest due to small velocity of the B mesons: at the energy of the
Υ(4S) peak vB/c ≈ 0.06. In particular, the numerical value in the expression (2) is 0.19.
The experimental data [1] however indicate a significantly smaller deviation of Rc/n from one.
The BaBar data with the smallest errors give Rc/n = 1.006±0.036±0.031. Such behavior is
likely a result of a combined effect of the meson and production vertex form factors [5, 6] and
of the discussed here modification of the Coulomb correction by the strong scattering phase.
These effects can in principle be separated and studied quantitatively by measuring the
energy dependence of the ratio Rc/n near the Υ(4S) resonance. With the presently available
data we can only use a simplified parametrization of the form factor effects by introducing
an abrupt cutoff for the Coulomb interaction at r = ac ≥ a and thereby estimate the likely
regions in the (ac, δ1) plane. Such estimate from the equations (18) and (29) is shown in Fig.1
as a one-sigma area, corresponding to the BaBar data with the statistical and systematic
errors added in quadrature: Rc/n = 1.006± .048. Clearly, more precise data from dedicated
measurements of the ratio Rc/n are needed for a better understanding of the parameters of
strong interaction between the B mesons.
0.2 0.4 0.6 0.8 1 1.2
Figure 1: The one sigma area (shaded) in the (ac, δ1) plane corresponding to the BaBar data
on the B+B−/B0B̄0 yield ratio at the Υ(4S) resonance.
5.2 ψ(3770)
The largest isospin-breaking effect in the DD̄ production at the ψ(3770) is that due to the
mass difference between the charged and the neutral D mesons: ∆mD = 4.78±0.10MeV [11].
The most precise measurements of this process have been done [2] at the energy
3773MeV. At this energy the momentum of each charged D meson is p+ = 254MeV and
that for a neutral D meson is p0 = 287MeV. Thus the ratio of the kinematical factors
(p+/p0)
3 ≈ 0.69 is significantly less than one. The Coulomb effect is somewhat smaller.
Indeed, the velocity of a charged meson at this energy is v+/c = 0.135 and the expression
(2) gives numerically 0.085. One can notice that if the kinematical and the Coulomb factors
are combined in a straightforward way to estimate Rc/n = (p+/p0)
3 [1 + πα/(2v+)] ≈ 0.75,
this would be in a very good agreement with the experimental number [2]: Rc/n = 0.776 ±
0.024+0.014
−0.006. Thus it is quite likely that at this particular energy there is a considerable
cancelation between the strong-interaction effects in the yield ratio, and such cancelation by
itself imposes constraints on the parameters of strong interaction between the D mesons,
which constraints is interesting to analyze.
An analysis of the strong-interaction effects along the lines discussed in the present paper
generally runs into two difficulties. One is that our approach is accurate only in the linear
in ∆m approximation, while the actual effect of the isotopic mass difference between the
D mesons is not very small. However, numerically, the first term in the expansion of the
kinematical factor (Eq.(31)) gives 0.67, which is quite close to the mentioned above value
0.69, and it looks like the linear term gives a reasonable approximation. The other point
is that the cutoff parameter ac for the Coulomb interaction at short distances does not
necessarily coincide with the range parameter a used for the short-distance cutoff of the
effect of the mass difference. However, as previously noted, the Coulomb effect is somewhat
small at the energy of the ψ(3770) resonance, and for the purpose of preliminary estimates
we set ac = a in our numerical analysis. In order to allow for possible errors introduced by
our approximations in comparing with the data, we linearly add a theoretical uncertainty
of 0.03 units to the combined in quadrature statistical and experimental errors. Proceeding
in this way we find that the only region in the (a, δ1) plane at a < 2 fm consistent with the
CLEO-c data at one sigma level is the one shown in Fig.2.
0.2 0.4 0.6 0.8 1
Figure 2: The area (shaded) in the (a, δ1) plane corresponding to the CLEO-c data on the
D+D−/D0D̄0 yield ratio at the ψ(3770) resonance. The uncertainty shown includes a one
sigma experimental error with our estimate of the theoretical uncertainty added linearly.
It is interesting to compare the plots in the Figures 1 and 2. In the heavy quark limit
applied to both b and c quarks the strong interaction between the heavy mesons should be
the same, corresponding to the same range parameters a and ac. The scattering phase δ1
for these two systems is generally different due to different masses. However, provided there
are no isovector ‘molecular’ bound states, the sign of the phase should be the same, with
the absolute value of the phase for heavier B mesons being larger than for the D mesons.
The comparison with the data for the D mesons favors small values of the range parameter,
as indicated by Fig.2. If one also assumes that ac ≈ a for the B mesons, the short range
of ac, according to Fig.1, is compatible with the B mesons data at a negative scattering
phase δ1, which sign of δ1 is also in agreement with the D meson data. A negative sign of δ1
corresponds to a repulsion, which for the I = 1 state of heavy meson pairs can be expected
on general grounds [12].
5.3 φ(1020)
We believe that the production of KK̄ pairs in e+e− annihilation at and near the φ(1020)
resonance merits a separate analysis along the lines discussed in the present paper and using
detailed data similar to those in Ref.[3]. As is known, this production receives a small but
measurable nonresonant contribution from the isovector part of the electromagnetic current
of the u and d quarks, which corresponds to an isotopically mixed source. Furthermore, it
has been pointed out [13] that a detailed theoretical analysis of the K+K−/K0K̄0 yield ratio
at the φ(1020) resonance produces a result which possibly is at a meaningful variance with
the data.
At present we limit ourselves to noticing that the formula in Eq.(27), applicable in this
situation, describes a smooth behavior of the considered isospin breaking effects across the
resonance in the I = 0 channel. Indeed, the I = 0 scattering phase at energy E near the
resonance energy E0 is given by the Breit-Wigner formula
e2iδ0 =
∆− i γ
∆+ i γ
e2iδ̃0 , (32)
where ∆ = E −E0, δ̃0 is the nonresonant scattering phase in the isoscalar channel, and γ is
the width parameter. Both δ̃0 and γ are smooth functions of the energy proportional to p
small momentum, and γ(E0) determines the resonance width Γ as γ = Γ/2. The ratio of the
isovector and isoscalar production amplitudes can then be parametrized near the resonance
∆+ i γ
ei (δ1−δ̃0) , (33)
where µ is a parameter with dimension of energy: µ ∼ mφ − mρ. The amplitude ratio
entering the correction factor in Eq.(27) can then be written in the form
2iδ1 − A1 e2iδ0
A0 − A1
= e2iδ1
µ− (∆− i γ) e−i(δ1−δ̃0)
µ− (∆ + i γ) e+i(δ1−δ̃0)
, (34)
which manifestly shows that this ratio is a pure phase factor of a complex quantity slowly
varying across the φ(1020) resonance.
6 Summary
We have considered the effects of the isospin breaking by the Coulomb interaction and by
the isotopic mass difference in the relative yield Rc/n of pairs of charged and neutral mesons
near threshold by a compact source, such as in the production of heavy mesons in e+e−
annihilation. These effects are modified by the strong interaction scattering phases. The
general formula for a situation where the source is an arbitrary coherent mixture of an
isoscalar and isovector is given by Eq.(27). In particular, for a purely isoscalar source, which
is the case for the e+e− annihilation into DD̄ and BB̄ pairs the strong-interaction effect is
determined by the scattering phase δ1 in the I = 1 channel (Eq.(18)). As a practical matter
we find that under the standard assumptions about the strong scattering amplitudes in the
near-threshold resonance region the ratio Rc/n has a smooth behavior with energy showing
no abnormal rapid variation on the scale of the resonance width. The energy dependence
of this ratio is rather determined by the non-resonant scattering scattering phase(s). In
the P -wave the phase δ1 is proportional to p
3, so that a measurement of the behavior ratio
Rc/n with energy can provide information on this phase, which is not readily accessible by
other means. The behavior of the ratio Rc/n at larger energies away from the threshold also
depends on the details of the onset of the strong interaction between the heavy mesons at
short distances and on the behavior of their electromagnetic form factors, and a study of
this behavior can provide an insight into these properties of the heavy-light hadrons.
Acknowledgements
The work of MBV is supported, in part, by the DOE grant DE-FG02-94ER40823.
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http://arxiv.org/abs/hep-ph/0208194
	Introduction
	General formulas for an isoscalar source
	Mixed isoscalar and isovector source
	The Coulomb and the mass-difference effects
	Phenomenological estimates
	(4S)
	(3770)
	(1020)
	Summary
