Reconciling the X(3872) with the near-threshold enhancement in the
  D^0\bar{D}^{*0} final state
FZJ–IKP(TH)–2007–14
Reconciling the X(3872) with the near-threshold enhancement in
the D0D̄∗0 final state.
C. Hanhart
Institut für Kernphysik, Forschungszentrum Jülich GmbH, D–52425 Jülich, Germany
Yu. S. Kalashnikova, A. E. Kudryavtsev, and A. V. Nefediev
Institute of Theoretical and Experimental Physics,
117218, B.Cheremushkinskaya 25, Moscow, Russia
We investigate the enhancement in the D0D̄0π0 final state with the mass M =
3875.2 ± 0.7+0.3
−1.6 ± 0.8 MeV found recently by the Belle Collaboration in the B →
KD0D̄0π0 decay and test the possibility that this is yet another manifestation of the
well–established resonance X(3872). We perform a combined Flattè analysis of the
data for the D0D̄0π0 mode, and for the π+π−J/ψ mode of the X(3872). Only if the
X(3872) is a virtual state in the D0D̄∗0 channel, the data on the new enhancement
comply with those on the X(3872). In our fits, the mass distribution in the D0D̄∗0
mode exhibits a peak at 2 ÷ 3 MeV above the D0D̄∗0 threshold, with a distinctive
non-Breit–Wigner shape.
PACS numbers: 14.40.Gx, 13.25.Gv, 12.39.Mk, 12.39.Pn
I. INTRODUCTION
The X(3872) state, discovered by Belle [1] in the B-meson decay, remains the most
prominent member of the family of “homeless” charmonia, that is those mesons which
definitely contain a c̄c pair but do not fit the standard charmonium assignment. The state
was confirmed then by CDF [2], D0 [3], and BaBar [4]. The charmonium option for the
X(3872) looks implausible as the state lies too high to be a 1D charmonium, and too low to
be a 2P one [5]. This could, in principle, mean that we simply do not understand the spectra
http://arxiv.org/abs/0704.0605v2
of higher charmonia. Indeed, most of the quark model predictions consider charmonia as cc̄
states in the quark potential model, with the potential parameters found from the description
of lower charmonia, with uncertainties coming from proper treatment of relativistic effects.
Another source of uncertainty is the role of open charm thresholds, the problem which is
far from being resolved, though the attempts in this direction can be found in the literature
— see, for example, Refs. [6, 7]. In any case, it looks premature to reject the cc̄ assignment
for the X(3872) on basis of the mass only. However, the further development has revealed
more surprises.
The discovery mode of the X(3872) is π+π−J/ψ. The observation of the X(3872) in
the γJ/ψ and π+π−π0J/ψ (ωJ/ψ) modes [8] implies that the X has positive C-parity, and
the dipion in the π+π−J/ψ mode is C-odd, that is it originates from the ρ. Coexistence of
the ρJ/ψ and ωJ/ψ modes points to a considerable isospin violation. Studies of the dipion
mass spectrum in X(3872) → π+π−J/ψ decay establish that only the 1++ or 2−+ quantum
number assignments are compatible with the data, while all other hypotheses are excluded
by more than 3σ [9].
Both 1++ or 2−+ quantum numbers options for the X(3872) require drastic revisions
of naive quark potential models, and no alternative explanation of the 2−+ state in this
mass region was suggested. On the other hand, it was pointed out in Refs. [10, 11] that
the DD̄∗ system with 1++ quantum numbers can be bound by pion exchange, forming a
mesonic molecule (see also Ref. [12])1. As confirmed by actual calculations [13], large isospin
mixing due to about 8 MeV difference between the D0D̄∗0 and D+D∗− thresholds can be
generated in the molecular model in quite a natural way. This model was supplied, in
Ref. [14], by quark–exchange kernels responsible for the transitions DD̄∗ → ρJ/ψ, ωJ/ψ,
predicting the ωJ/ψ decay mode of the X(3872). Note, however, that one-pion-exchange
as a binding mechanism in the DD̄∗ system should be taken with caution, as, in contrast
to NN case, here the pion can be on-shell, as pointed in [15], where the ability to provide
strong enough binding with one-pion exchange was questioned. For the most recent work
on the implications of the nearby pion threshold see Refs. [16, 17]. For recent work for the
X as quark state we refer to Ref. [18] and references therein.
The molecular model has received additional support with the new data on the mass of
1An obvious shorthand notation is used here and in what follows: DD̄∗ ≡ 1√
(DD̄∗ + D̄D∗).
the D0 meson [19] which yield a very weak binding,
MX −M(D0D̄∗0) = −0.6± 0.6 MeV. (1)
In the meantime, the Belle Collaboration has reported the first observation [20] of the
near–threshold enhancement in the D0D̄0π0 mode in the decay B → KD0D̄0π0, with the
branching fraction
Br(B → KD0D̄0π0) = (1.22± 0.31+0.23
−0.30) · 10−4. (2)
The peak mass of the enhancement is measured to be
Mpeak = 3875.2± 0.7+0.3−1.6 ± 0.8 MeV. (3)
Obviously it is tempting to relate this new state to the X(3872). However, the average
value of the X(3872) mass is [21]
MX = 3871.2± 0.5 MeV. (4)
The central value (3) of the D0D̄0π0 peak mass enhancement is about 4 MeV higher than
that, which obviously challenges attempts to relate this new state to the X(3872).
Quite recently, the indication appeared that the Belle result [20] is likely to be con-
firmed. Namely, the BaBar Collaboration has reported the preliminary data [22] on the
B → KD0D̄∗0 decay, where the enhancement with the mass of
M = 3875.6± 0.7+1.4
−1.5 ± 0.8 MeV, (5)
was found, in a very good agreement with (3). BaBar observes the enhancement in the
D0D̄0π0 and in the D0D̄0γ modes, which strongly supports the presence of the D0D̄∗0
intermediate state in the decay of the new X .
If the new BaBar data persist, and the enhancement at 3875 MeV is indeed seen in two
independent experiments, the possibility should be considered seriously of the presence of
two charmonium-like states, X(3872) and X(3875), surprisingly close to each other and to
the D0D̄∗0 threshold.
However, there exists another, less exotic possibility. Namely, if the X(3872) is indeed
strongly coupled to the D0D̄∗0 channel, and indeed has 1++ quantum numbers, one could
expect the existence of a near–threshold peak in the D0D̄∗0 mass distribution. In the present
paper we perform a phenomenological Flattè-like analysis of the data on the decay B →
KD0D̄0π0 in the near–threshold region under the assumption of theX → D0D̄∗0 → D0D̄0π0
decay chain and 1++ quantum numbers for the X . The data on the π+π−J/ψ decay modes
of the X(3872) are analyzed in the same framework, in order to investigate whether these
data can accommodate the X(3875) state as a manifestation of the X(3872).
II. FLATTÈ PARAMETRIZATION
In this Section we introduce the Flattè-like parametrization of the near–threshold ob-
servables. The relevant mass range is between the thresholds for the neutral and charged
D-mesons. A natural generalization of the standard Flattè parametrization for the near–
threshold resonance [23] of the D0D̄∗0 scattering amplitude reads
F (E) = − 1
, (6)
D(E) =
E −Ef −
− g2κ2
Γ (E)
, E < 0
E −Ef −
Γ (E)
, 0 < E < δ
E −Ef + i
Γ (E)
, E > δ
δ =M(D+D∗−)−M(D0D̄∗0) = 7.6 MeV,
2µ1E, κ1 =
−2µ1E, k2 =
2µ2(E − δ), κ2 =
2µ2(δ − E).
Here µ1 and µ2 are the reduced masses in the D
0D̄∗0 and D+D∗− channels, respectively, and
the energy E is defined relative to the D0D̄∗0 threshold. In what follows we assume isospin
conservation for the coupling constants, g1 = g2 = g.
The term iΓ/2 in Eq. (7) accounts for non-DD̄∗ modes. The X(3872) was observed in
the π+π−J/ψ, π+π−π0J/ψ, and γJ/ψ modes, with
Br(X → π+π−π0J/ψ)
Br(X → π+π−J/ψ)
= 1.0± 0.4± 0.3, (8)
Br(X → γJ/ψ)
Br(X → π+π−J/ψ)
= 0.14± 0.05, (9)
reported in Ref. [8]. Thus we assume that Γ (E) in Eq. (7) is saturated by the π+π−J/ψ and
π+π−π0J/ψ modes and, in accordance with findings of Ref. [8], the dipion in the π+π−J/ψ
mode comes from the ρ whereas the tripion in the π+π−π0J/ψ mode comes from the ω. The
γJ/ψ channel is neglected due to its small branching fraction (9).
The nominal thresholds for both ρJ/ψ and ωJ/ψ (3872 MeV and 3879 MeV, respectively)
are close to the mass range under consideration, but both the ω meson and, especially, the
ρ meson have finite widths, which are large in the scale under consideration. Thus Γ (E) is
calculated as
Γ (E) = Γπ+π−J/ψ(E) + Γπ+π−π0J/ψ(E), (10)
Γπ+π−J/ψ(E) = fρ
∫ M−mJ/ψ
q(m)Γρ
(m−mρ)2 + Γ2ρ/4
, (11)
Γπ+π−π0J/ψ(E) = fω
∫ M−mJ/ψ
q(m)Γω
(m−mω)2 + Γ2ω/4
, (12)
with fρ and fω being effective couplings and
q(m) =
(M2 − (m+mJ/ψ)2)(M2 − (m−mJ/ψ)2)
being the c.m. dipion/tripion momentum (M = E +M(D0D̄∗0)).
Now we are in a position to write down the differential rates in the Flattè approximation.
These are
dBr(B → KD0D̄∗0)
= B 1
|D(E)|2
, (14)
dBr(B → Kπ+π−J/ψ)
= B 1
Γπ+π−J/ψ(E)
|D(E)|2
, (15)
dBr(B → Kπ+π−π0J/ψ)
Γπ+π−π0J/ψ(E)
|D(E)|2
. (16)
We assume the short–ranged dynamics of the weak B → K transition to be absorbed into
the coefficient B. Obviously, the rate (14) is defined for E > 0 only, while the rates (15) and
(16) are defined both above and below the D0D̄∗0 threshold.
The formulae (14)–(16) are valid in the zero-width approximation for the D∗-mesons. In
principle, one could include the finite width of the D∗-mesons either analogous to Eqs. (11)
and (12) or in a more sophisticated way, as there are interference effects possible in the final
state, as described in Ref. [24]. However, the widths of the D∗ mesons are small. Indeed,
the total width of the D∗±-meson is measured to be 96 ± 22 keV [21]. There are no data
on the D∗0 width, but one can estimate the D0π0 width of the D∗0 from the data [21] on
charged D∗±, which gives Γ(D∗0 → D0π0) = 42 keV. The branching fractions of D∗0 are
known [21]:
Br(D∗0 → D0π0) = (61.9± 2.9)%, (17)
Br(D∗0 → D0γ) = (38.1± 2.9)%, (18)
so the total D∗0 width can be estimated to be only about 68 keV. The effect of such a small
width was checked to be negligible in our studies, and we assume the D0D̄0π0 differential
rate to be
dBr(B → KD0D̄0π0)
= 0.62B
|D(E)|2
, (19)
where the branching fraction (17) is taken into account.
Analogously we have for the D0D̄0γ differential rate
dBr(B → KD0D̄0γ)
= 0.38B 1
|D(E)|2
. (20)
Expressions (19) and (20) neglect final–state interactions; in particular, no DD̄ resonance
within a few MeV above D0D̄0 threshold is assumed to exist, and π-rescattering is neglected.
The latter is expected to be weak, as a consequence of chiral symmetry [17].
III. FLATTÈ ANALYSIS: PROCEDURE AND RESULTS
Let us first specify the data used in our analysis. For the π+π−J/ψ mode we use the
data from the B-meson decay. These are the ones reported by the Belle [1] and BaBar [25]
Collaborations. The X(3872) is seen by Belle in the charged B-meson decay, with 35.7±6.8
signal events, and with the branching fraction [1]
Br(B+ → K+X)Br(X → π+π−J/ψ) = (13.0± 2.9± 0.7) · 10−6. (21)
The BaBar Collaboration [25] has observed the X(3872) both in the charged and neutral
B-meson decays, with 61.2 ± 15.3 signal events for the charged mode, and only 8.3 ± 4.5
signal events for the neutral one. The branching fraction for the charged mode was found
to be
Br(B− → K−X)Br(X → π+π−J/ψ) = (10.1± 2.5± 1.0) · 10−6, (22)
while the result for the neutral mode is much less certain: a 90% confidence interval was
established as
1.34 · 10−6 < Br(B0 → K0X)Br(X → π+π−J/ψ) < 10.3 · 10−6. (23)
Due to large errors and much smaller number of events, the X(3872) peak in the neutral
mode looks much less convincing than the peak in the charged mode.
A similar situation takes place for the D0D̄0π0 final state. The Belle data [20] include
both B+ → K+D0D̄0π0 and B0 → K0D0D̄0π0 decays. There are 17.4± 5.2 signal events in
the charged mode, with the branching fraction
Br(B+ → K+D0D̄0π0) = (1.02± 0.31+0.21
−0.29) · 10−4, (24)
and 6.5± 2.6 signal events in the neutral mode, with
Br(B0 → K0D0D̄0π0) = (1.66± 0.70+0.32
−0.37) · 10−4. (25)
Data on the B+ and B0 decays separately are presented in Ref. [26]. The D0D̄0π0 enhance-
ment appears to be clearly seen in the data on charged B decays while, again, the neutral
mode displays, within the errors, a much less pronounced peak.
We conclude therefore that the data on charged and neutral B decays should be analyzed
separately. The present analysis is performed for the charged mode only. Namely, with the
Flattè formalism, we attempt to describe simultaneously the π+π−J/ψ mass spectrum from
the charged mode and the D0D̄0π0 spectrum from the B+ mode, taken from Ref. [26].
The branching fractions (21) and (22) differ but, within the errors, are consistent with
each other. In both sets of data, the fitted width of the signal is consistent with the resolu-
tion, so only the upper limits on the X(3872) width were established:
Γtot(Belle) < 2.3 MeV (26)
Γtot(BaBar) < 4.1 MeV, (27)
for the Belle and BaBar data, respectively. In view of this discrepancy we prefer to present
two sets of fits, based on the two aforementioned sets of the π+π−J/ψ data.
The π+π−J/ψ data are fitted in the interval −20 < E < 20 MeV (as before, E is the
energy relative to the D0D̄∗0 threshold), after subtraction of the full background found in
the corresponding analysis. The free parameters of the fit are the short–range factor B and
the Flattè parameters Ef , g, and fρ. The parameter fω is constrained, in accordance with
Eq. (8) through the condition
RρJ/ψ
RωJ/ψ
= 1, (28)
where
RρJ/ψ =
20MeV
−20MeV
dBr(B → Kπ+π−J/ψ)
dE, (29)
RωJ/ψ =
20MeV
−20MeV
dBr(B → Kπ+π−π0J/ψ)
dE. (30)
The limits of integration in Eqs. (29) and (30) are somehow arbitrary but, as most of the
support of the distributions (15) and (16) comes from within a few MeV around the D0D̄∗0
threshold, the uncertainty introduced by the limits of integration is much less than the
experimental errors in Eq. (8).
The D0D̄0π0 data are fitted in the energy region 0 < E < 20 MeV. Equation (19)
describes the production of the D0D̄0π0 mode via the X-resonance, while the DD̄∗ pairs are
known to be copiously produced in the B → K decay in a non-resonant way. Besides, the
D0D̄0π0 final state could come from non-D0D̄∗0 modes like, for example, B → K∗D0D̄0.
Therefore, we are to make assumptions on the background.
The background in Refs. [20] and [26] is mostly combinatorial, and this part, given explic-
itly in the publications, was subtracted prior to the analysis. For the rest of the background
it is not possible to separate the contributions of the D0D̄∗0 and the D0D̄0π0 due to a limited
phase space [20]. So we work under two extreme assumptions for the background. In Case
A we consider the D0D̄0π0 background as unrelated to the D0D̄∗0 channel, while in Case B
we assume that all the D0D̄0π0 events come from the D0D̄∗0 mode. The background was
evaluated by fitting the Belle data off–peak (25 < E < 50 MeV). In Case A the background
function is assumed to be proportional to the three–body D0D̄0π0 phase space R3 ∝ E2DDπ,
where EDDπ = E +mD∗0 −mD0 −mπ0 . Then the total B → KD0D̄0π0 differential rate is
dBrA(B → KD0D̄0π0)
= 0.62
|D(E)|2
+ cAE
DDπ, (31)
with cA as fitting constant. In Case B the background function is proportional to the
two–body D0D̄∗0 phase space R2 ∝ k1 (see the definition below Eq. (7)). Then the signal–
TABLE I: The set of the Flattè parameters for the best fits to the Belle data Ref. [1] and [20].
Fit g fρ fω Ef , MeV B · 104 φ
ABelle 0.3 0.0070 0.036 -11.0 11.0 —
BBelle 0.3 0.0086 0.046 -10.9 8.9 -144
background interference is to be taken into account:
dBrB(B → KD0D̄0π0)
= 0.62
+ cB cosφ
+ cB sin φ
with the relative phase φ and cB being fitting constants.
The differential rates are translated into number-of-events distributions as follows. There
are about 36 signal events in the Belle data, which corresponds to the branching fraction of
about 1.3 · 10−5 (see Eq. (21)). Thus the number-of-events per 5 MeV distribution for the
π+π−J/ψ mode is given by
ππJ/ψ
Belle
(E) = 5 [MeV]
1.3 · 10−5
dBr(B → Kπ+π−J/ψ)
. (33)
For the BaBar data, with 61 events and the branching fraction of about 1.02 · 10−5 (see
Eq. (22)), we have
ππJ/ψ
BaBar
(E) = 5 [MeV]
1.02 · 10−5
dBr(B → Kπ+π−J/ψ)
. (34)
As to the D0D̄0π0 mode, the Belle Collaboration states to have 17.4 signal events in the
charged mode [20], which corresponds to the branching fraction (24) of about 1.02 · 10−4.
The number-of-events distributions per 4.25 MeV for the D0D̄0π0 mode is calculated as
0D̄0π0
A,B (E) = 4.25[MeV]
1.02 · 10−4
dBrA,B(B → KD0D̄0π0)
. (35)
The best fit to the π+π−J/ψ data alone requires a vanishing value of the DD̄∗ coupling
constant, g = 0, so that such solution cannot accommodate the D0D̄0π0 enhancement as a
related phenomenon. To describe both π+π−J/ψ and D0D̄∗0 modes we are to compromise
on the π+π−J/ψ line–shape.
It appears that a decent combined fit can be achieved only if the π+π−J/ψ distribution
is peaked exactly at the D0D̄∗0 threshold, with the peak width (defined as the width at the
TABLE II: The set of the Flattè parameters for the best fits to the BaBar data of Ref. [25] and
the Belle data of Ref. [20].
Fit g fρ fω Ef , MeV B · 104 φ
ABaBar 0.3 0.0042 0.021 -8.8 11.4 —
BBaBar 0.3 0.0056 0.027 -8.8 8.9 -153
peak half–height) close to the upper limits given by Eq. (26) or (27). The values of the
coupling g were found to be of the order of magnitude or larger than 0.3. Finally, the fits
exhibit the scaling behaviour: they remain stable under the transformation
g → λg, Ef → λEf , fρ → λfρ, fω → λfω, B → λB, (36)
with tiny variations of the phase φ in the Case B.
In Tables I, II we present the sets of the best fitting parameters — for both Case A
and Case B and for g = 0.3 — for the Belle (Table I) and BaBar (Table II) data on the
π+π−J/ψ mode and for the Belle data for the D0D̄0π0 mode. To assess the quality of the
fits we calculate the π+π−J/ψ distributions integrated over the 5 MeV bins, as in Refs. [1]
and [25], and the D0D̄0π0 distributions integrated over the 4.25 MeV bins, as in Refs. [20]
and [26]. The results are shown at Fig. 1 together with the experimental data.
The above–mentioned scaling behaviour does not allow one to perform a proper fit with
the estimate of uncertainties in the parameters found. Indeed, the parameters of the best
fits found for the values of coupling constant g larger than 0.3 differ only by a few % from
the ones given by the scaling transformation (36), and the corresponding distributions are
very similar to those given at Fig. 1.
As seen from the figures, acceptable fits require the D0D̄∗0 differential rate to be peaked
at around 2 ÷ 3 MeV above the D0D̄∗0 threshold. The scattering length in the D0D̄∗0
channel which follows from the expression (6) of the D0D̄∗0 scattering amplitude, is given
by the expression
a = −
2µ2δ + 2Ef/g + iΓ (0)/g
2µ2δ + 2Ef/g)2 + Γ (0)2/g2
, (37)
and is calculated to be
(−3.98− i0.46) fm, Case ABelle
(−3.95− i0.55) fm, Case BBelle,
-20 -10 0 10 20
E [MeV]
-20 -10 0 10 20
E [MeV]
-20 -10 0 10 20
E [MeV]
-20 -10 0 10 20
E [MeV]
0 5 10 15 20
E [MeV]
0 5 10 15 20
E [MeV]
0 5 10 15 20
E [MeV]
0 5 10 15 20
E [MeV]
ABelle BBelle ABaBar BBaBar
FIG. 1: Upper plots: Our fits to the differential rates for the π+π−J/ψ channel measured by
Belle [1] and BaBar [25] using prescription A and B (see Eqs. (31) and (32)). Lower plots: Cor-
responding fits for the differential rates in the D0D̄0π0 channel measured by Belle [26]. The
distributions integrated over the bins are shown in each panel as filled dots, experimental data as
filled squares with error bars.
(−3.10− i0.16) fm, Case ABaBar
(−3.10− i0.22) fm, Case BBaBar.
The real part of the scattering length for all the fits appears to be large and negative, and the
imaginary part is much smaller. This, together with the beautiful cusp in the π+π−J/ψ mass
distribution, signals the presence of a virtual state in the D0D̄∗0 channel. The cusp scenario
for the π+π−J/ψ excitation curve in the X(3872) mass range was advocated in Ref. [27].
The X(3872) as a virtual DD̄∗ state was found in the coupled–channel microscopic quark
model [7].
A large scattering length explains naturally the scaling behaviour of the Flattè param-
eters. Such kind of scaling was described in Ref. [28] in the context of light scalar mesons
properties: the scaling behaviour occurs if the scattering length approximation is operative.
In the case of X the situation is more complicated, as there are two near–threshold channels,
neutral and charged. Nevertheless, if it is possible to neglect the energy E in the expression
(7) for the Flattè denominator D(E) then, as seen from the expression (6), scaling for the
D0D̄∗0 scattering amplitude indeed takes place. If the factor B obeys the scaling transfor-
mation, the differential rates (14)–(16) also exhibit the scaling behaviour. Note that, if the
energy dependence of the charged D+D∗− and non-DD̄∗ channel contributions is neglected
as well, this corresponds to the scattering length approximation, and neglect of the effective
radius term.
IV. DISCUSSION
Our analysis shows that the large branching fraction (2) implies the X to be a virtual
D0D̄∗0 state, and not a bound state. We illustrate this point by calculating the rates (14)
and (15) for the set of the Flattè parameters (fit C)
g = 0.3, Ef = −25.9 MeV, fρ = 0.007, fω = 0.036, B = 1.32 · 10−4. (40)
The values of the coupling constants coincide with those of the fit ABelle, while the parameter
Ef is chosen to yield the real part of the scattering length to be equal in magnitude to the
one evaluated for the given fit ABelle, but positive: ã = (+3.98− i0.46) fm. The parameter
B for this set yields the same value of the total branching fraction for the π+π−J/ψ mode as
the fit ABelle. The π
+π−J/ψ and D0D̄0π0 rates are shown in Fig. 2, together with the rates
obtained for the case ABelle (without background). The new curve (dashed line in Fig. 2)
displays a very narrow peak in the π+π−J/ψ distribution, corresponding to theD0D̄∗0 bound
state, with binding energy of about 1 MeV (there is no corresponding peak in the D0D̄0π0
distribution as the finite width of the D∗0 is not taken into account in our analysis). Note
that the π+π−J/ψ rates (Fig. 2) are normalized to give the branching ratio 1.3 · 10−5, which
requires the coefficient B to be much larger for the virtual state than for the bound state.
As a result, the D0D̄∗0 rate is much smaller for the bound state, as seen from Fig. 2.
Obviously, the difference between the bound–state and virtual-state cases for the ratio
Br(X → D0D̄0π0)
Br(X → π+π−J/ψ)
-3 -2 -1 0 1 2 3
E [MeV]
0 5 10 15 20
E [MeV]
FIG. 2: The differential rates for the π+π−J/ψ (first plot) and D0D̄∗0 (second plot) for the fits
ABelle (solid curves) and C (dashed curves).
is driven by the strength of the bound–state peak, as discussed in Ref. [29], where the
scattering length approximation was used to describe the X(3872). Following Ref. [29], let
us write down the scattering length in the D0D̄∗0 channel as
γre + iγim
. (42)
Then, in the scattering length approximation, the π+π−J/ψ differential rate is proportional
to the factor
γ2re + (k1 + γim)
, E > 0,
(γre − κ1)2 + γ2im
, E < 0,
while the D0D̄∗0 rate is proportional to
γ2re + (k1 + γim)
. (44)
The line–shape for the D0D̄∗0 channel does not depend on the sign of γre. The same is
true for the π+π−J/ψ line–shape above the D0D̄∗0 threshold while, below the threshold,
the line–shapes differ drastically: in the bound–state case there is a narrow peak below
threshold, and in the virtual-state case a threshold cusp appears.
For γre > 0 and γim → 0 the expression (43) becomes a δ-function (see Ref. [29]):
γreδ(E + γ
re/(2µ1)). (45)
Then the total rate does not depend on γim, if it is small enough. This simply means that,
for γim = 0, we have a real bound state, which is not coupled to inelastic channels. In
contrast to the bound–state case, for the virtual state, the rate (43) tends to zero with
γim → 0, while the D0D̄∗0 rate does not vanish in such a limit. So it is possible, adjusting
γim, to obtain large values of the ratio (41).
Exactly the same situation is encountered in our fit: we need g & 0.3 for the fit to
be reasonable and, in this scaling regime, as soon as we have a positive real part of the
scattering length, the ratio (41) becomes small while, with a negative real part, we get
a solution compatible with the data. The large branching fraction (2) was identified in
Ref. [30] as a disaster for the molecular model of the X(3872). Indeed, the bound–state
molecule decay into D0D̄0π0 is driven by the process D∗0 → D0π0 which gives the width
of order 2Γ(D∗0 → D0π0) (up to the interference effects calculated in Ref. [24] which, for
the bound–state case, cannot be neglected anymore and should be taken into account). The
main decay mode of the X is π+π−J/ψ because the phase space available is large. This is
confirmed by model calculations of Ref. [14] yielding
Br(X → D0D̄0π0)
Br(X → π+π−J/ψ)
≈ 0.08, (46)
in a strong contradiction with data.
The estimate (46) describes the decay of an isolated bound state. However, the suppres-
sion is more moderate as, in B-decay, the continuum contribution is also to be considered.
The bound–state contribution would be zero in the zero–width approximation for D∗0, while
the D0D̄∗0 continuum contribution remains finite if the D∗0 width is neglected. However, if
the X is a bound state, the continuum contribution is not large (see Fig. 2),
Br(X → D0D̄0π0)
Br(X → π+π−J/ψ)
≈ 0.62. (47)
Such a small rate would remain unnoticed against the background. So, in practice, the
bound–state X(3872) would reveal itself only as a narrow peak below threshold, with a very
small rate (see Eq. (46)). In contrast to this we get for the virtual state
Br(X → D0D̄0π0)
Br(X → π+π−J/ψ)
≈ 9.9. (48)
In our analysis, the X appears to be a virtual state in the D0D̄∗0 channel. This does not
contradict the assumption g1 = g2 = g employed in the analysis. The latter means that the
underlying strong interaction conserves isospin, and all the isospin violation comes from the
mass difference between charged and neutral DD̄∗ thresholds. No charged partners of the
X are observed, so it is reasonable to assume that the strong attractive interaction takes
place in the isosinglet DD̄∗ channel.
We do not specify the nature of this attractive force. It is known that in the one-pion-
exchange model for the X , the force is attractive in the isosinglet channel, and is repulsive
in the isotriplet one. However, as was already mentioned, the doubts were cast in [15] on the
role of one-pion-exchange in the DD̄∗ binding, and it was advocated there that the X may
fit the 23P1 charmonium assignment if the coupling to DD̄
∗ channel is taken into account.
In such a scenario the strong binding force obviously takes place in the isosinglet channel.
We note, however, that, with the Flattè parameters found, one can make a definite
statement: whatever the nature of the X(3872) is, the admixture of a compact cc̄ state in
its wavefunction is small. Both large scattering length and the scaling behaviour of the DD̄∗
amplitude are consequences of the large value of the coupling constant of the state to the
DD̄∗ channel. As shown in Ref. [31], this points to a large DD̄∗ component and a dynamical
origin of the X . Although formulated for quasi–bound states in Ref. [31] the argument can
also be generalized to virtual states. To clarify the connection between effective coupling
and the nature of the state observe that the two–point function g(s) for the resonance can
be written as
g(s) =
s−M2 − iΣ̄(s)
, (49)
whereM is the physical mass of the resonance and Σ̄(s) = Σ(s)−ReΣ(M2) is the self–energy
responsible for the dressing through the mesonic channels. In the near–threshold region the
momenta involved are much smaller than the inverse of the range of forces. As a result one
may neglect the s–dependence of the real part of Σ̄ and replace its imaginary part by the
leading terms
g(s) ≃ 1
s−M2 + iM
, (50)
where the sum is over near–threshold channels, and the contributions of distant thresholds
are absorbed into the renormalised mass M . Nonrelativistic reduction of Eq. (50) immedi-
ately yields the Flattè formula (6). Thus the Flattè parameter Ef acquires clear physical
meaning: the quantity M(D0D̄∗0) +Ef is the physical mass of the resonance, renormalised
by the coupling to the decay channels.
Now, if the couplings gi are small, the distribution for the resonance takes a standard
Breit–Wigner form, and the scattering length is small. Correspondingly, the state is mostly
cc̄, with a small admixture of the DD̄∗ component. If the couplings are large, the terms
proportional to giki control the denominator in Eq. (50), the Breit–Wigner shape is severely
distorted, the scattering length approximation is operative, and the mesonic component
dominates the near–threshold wavefunction.
Formulated differently: if the couplings are large, the properties of the resonance are
given mainly by the continuum contribution — which is equivalent to saying it is mostly
of molecular (dynamical) nature. It should be stressed that this kind of reasoning can only
be used, if the resonance mass is very close to a threshold, for then the contribution of
the continuum state is dominated by the unitarity cut piece which is unique and model
independent. This argument is put into more quantitative terms in Ref. [31]. It is also
important to note that our analysis does not allow for any conclusion on the mechanism that
leads to the molecular structure. On the level of the phenomenological parametrisations used
here a molecule formation due to t–channel exchanges and due to short–ranged s–channel
forces (cc̄–DD∗ mixing) would necessarily lead to the same properties of the state, once the
parameters are adjusted to the data.
V. SUMMARY
In this paper we present a Flattè analysis of the Belle data [20] on the near–threshold
enhancement in the D0D̄0π0 mode. We constrain the Flattè parametrization with the data
on the X(3872) seen in the π+π−J/ψ and π+π−π0J/ψ modes. With such constraints the
new state can be understood as a manifestation of the well–established X(3872) resonance.
We showed that the structure at 3875 MeV can only be related to the X(3872), if we
assume the X to be of a dynamical origin, however, not as a bound state but as a virtual
state. The situation is then similar to that of nucleon–nucleon scattering in the spin–
singlet channel near threshold: in contrast to the spin–triplet channel, where there exists
the deuteron as a bound state, the huge scattering length in the spin–singlet channel —
about 20 fm — comes from a near–threshold virtual state. The attractive interaction is just
not strong enough to form a bound state in this channel as well.
The line–shape in the D0D̄∗0 mode appears to differ substantially from the one extracted
previously from the Belle data directly. It peaks much closer to the D0D̄∗0 threshold, though
the overall description of the data looks quite reasonable within the experimental errors.
It is the π+π−J/ψ line–shape which, in our solutions, differ drastically from the one
described by a simple Breit–Wigner form. We found a threshold cusp, with a width close to
the limits imposed by the data analysis. While the data currently available allows for such
a line–shape, a considerable improvement in the experimental resolution could confirm or
rule out this possibility. In the meantime, we urge to perform an analysis of the data on the
D0D̄0π0 final state with Flattè formulae given in Eqs. (14)–(16).
Equally important is the Flattè analysis of the D0D̄0γ data [22]: if the structure in the
D0D̄0π0 is indeed due to D0D̄∗0 and is indeed related to the X(3872) as a virtual state, one
should observe an enhancement in D0D̄0γ similar to the one seen in the D0D̄0π0. The phase
space available in this final state is larger than that in D0D̄0π0, so it is easier to separate
the contributions of D0D̄∗0 and D0D̄0γ to the peak. The D0D̄0γ enhancement would be
described with the Flattè formula (20) and, up to background and possible FSI effects, the
ratio of branching fractions would be
Br(X → D0D̄0π0)
Br(X → D0D̄0γ)
≈ 1.6. (51)
The most interesting situation would happen if, due to an improved resolution in the
π+π−J/ψ mode, the combined Flattè analysis of the π+π−J/ψ, D0D̄0π0, and D0D̄0γ data
fails to deliver a self-consistent result. Such a situation would point to the new X(3875)
state being completely unrelated to the X(3872).
Acknowledgments
We would like to thank A. Dolgolenko for useful comments and suggestions and P.
Pakhlov for illuminating discussions on various aspects of the Belle experiment. This re-
search was supported by the Federal Agency for Atomic Energy of Russian Federation,
by the grants RFFI-05-02-04012-NNIOa, DFG-436 RUS 113/820/0-1(R), NSh-843.2006.2,
and NSh-5603.2006.2, and by the Federal Programme of the Russian Ministry of Industry,
Science, and Technology No. 40.052.1.1.1112. A.N. is also supported through the project
PTDC/FIS/70843/2006-Fisica.
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http://belle.kek.jp/belle/talks/ICHEP2006/Majumber.ppt
	Introduction
	Flattè parametrization
	Flattè analysis: procedure and results
	Discussion
	Summary
	Acknowledgments
	References
