#### Different pole structures in line shapes of the X(3872)

Eur. Phys. J. C
Different pole structures in line shapes of the X (3872)
Xian-Wei Kang 0
J. A. Oller 0
0 Departamento de Física, Universidad de Murcia , 30071 Murcia , Spain
We introduce a near-threshold parameterization that is more general than the effective-range expansion up to and including the effective range because it can also handle a near-threshold zero in the D0 D¯ ∗0 S-wave. In terms of it we analyze the CDF data on inclusive p p¯ scattering to J /ψ π +π −, and the Belle and BaBar data on B decays to K J /ψ π +π − and K D D¯ ∗0 around the D0 D¯ ∗0 threshold. It is shown that data can be reproduced with similar quality for X (3872) being a bound and/or a virtual state. We also find that X (3872) might be a higher-order virtual-state pole (double or triplet pole), in the limit in which the small D∗0 width vanishes. Once the latter is restored the corrections to the pole position are non-analytic and much bigger than the D∗0 width itself. The X (3872) compositeness coefficient in D0 D¯ ∗0 ranges from nearly 0 up to 1 in the different scenarios.
1 Introduction
The X (3872) has been analyzed in great
phenomenological detail by employing S-wave effective-range-expansion
(ERE) parameterizations in Refs. [1–3]. References [2,3]
includes only the D D¯ ∗ scattering length, a, while Ref. [1]
also includes the effective-range (r ) contribution.1 A detailed
comparison between both approaches is given in Sect. 6 of
Ref. [3]. Indeed, the use of the ERE up to and including
the effective range is more general than employing a Flatté
parameterization (also used in Refs. [4–6]), because only
negative effective ranges can be generated within the latter
[7].2
1 To shorten the presentation we actually refer by D D¯ ∗ to the C = +
combination (D D¯ ∗ + D¯ D∗)/√2.
2 As follows from Ref. [8], the Flatté parameterization and the ERE
including the effective-range contribution are equivalent if the former
is written in terms of the bare mass and coupling squared that need to
be tuned, taking a priori any sign, to reproduce the values of the residue
and pole position of the partial wave.
a e-mail:
However, the ERE convergence radius might be severely
limited due to the presence of near-threshold zeros of the
partial wave, in this case the D0 D¯ ∗0 S-wave. These zeros, also
called Castillejo–Dalitz–Dyson (CDD) poles [9], constitute
the major criticism to apply Weinberg’s compositeness
theorem to evaluate the actual compositeness of a near-threshold
bound state [10], because it is based on the ERE up to the
effective-range contribution.3 The same criticism is of course
applicable to the papers [1–3,5,6] referred in the previous
paragraph.
The issue about the possible presence of near-threshold
zero in the S-wave partial wave and the spoil of the
corresponding ERE was also discussed more recently in Ref. [11].
One of the main conclusions of this reference was that in
order to end with a near-threshold zero one needs also three
shallow poles. In this way this situation was qualified as
highly accidental by the authors of Ref. [11]. However, this
conclusion is not necessarily correct, that is, one can have a
near-threshold zero with only two nearby poles, without the
need of a third one. The reason for this misstep in the study of
Ref. [11] was a misuse of the relation between the position of
the zero and the location of the poles in the three-momentum
complex plane, as we discuss in detail in Sect. 6.3.
Twocoupled-channels effects were included in Ref. [12] along
the similar lines of mixing the exchange of a resonance with
direct interactions between the mesons, in the limit of
validity of the scattering length approximation for the latter ones.
In turn, the coupled-channel generalization of Ref. [11] was
derived in Ref. [13]. In the energy region around the D0 D¯ ∗0
threshold where X (3872) sits, the coupled-channel results
of Refs. [12,13] reduce to a partial wave whose structure
can be deduced from the elastic one-channel D0 D¯ ∗0
scatter
3 This serious criticism was originally due to R. Blankenbecler,
M. L. Goldberger, K. Johnson and S. B. Treiman, as explicitly stated
in the note added in proof in Ref. [10], warning about the possible
presence of CDD poles for E > −B in the Low equation used in this
reference. The only way to skip this problem is to ascertain the range
of convergence of the ERE, typically from data.
ing, because the D+ D∗− threshold is relatively much further
away. We also indicate here that Ref. [11] cannot reproduce
positive values for the D0 D¯ ∗0 S-wave effective range, while
our approach is more general in this respect and can also give
rise to positive values of this low-energy scattering
parameter. These two points are also shown explicitly below.
As in Refs. [1–3] we avoid any explicit dynamical model
for the D D¯ ∗ dynamics to study the X (3872) line shapes in
the BaBar [14,15] and Belle [16,17] data on the B decays to
K ± J /ψ π +π − and K J /ψ D0 D¯ ∗0. In addition, we also
consider the higher-statistics data from the inclusive p p¯
scattering to J /ψ π +π − measured by the CDF Collaboration [18]
and that gives rise to a more precise determination of the mass
of X (3872) [19]. However, we employ a more general
parameterization than the ERE expansion up to and including the
effective-range contribution by explicitly taking into account
the possibility of the presence of a CDD pole very close to the
D0 D¯ ∗0 threshold. Our formalism has as limiting cases those
of Refs. [1–3], but it can also consider other cases. In
particular, while in Refs. [1–3] X (3872) turns out to be either a
bound or a virtual-state pole, we also find other qualitatively
different scenarios that can reproduce data with similar
quality as well. In two of these new situations the X (3872) is
simultaneously a bound and a virtual state and for one of
them the D0 D¯ ∗0 compositeness coefficient is just of a few
per cent. This is also an interesting counterexample for the
conclusions of Ref. [11], because it has a CDD pole almost
on top of threshold with only two shallow poles.
Remarkably, we also find other cases with two/three virtual-states
poles, such that in the limit of vanishing width of D∗0 these
poles become degenerate and result in a second/third-order
S-matrix pole. Along the lines of the discussions, we also
match our resulting partial-wave amplitude from S-matrix
theory with the one deduced in Ref. [11] in terms of the
exchange of a bare state and direct interactions between the
D0 D¯ ∗0 mesons. Similarly, this is also done with the
onechannel reduction of Ref. [12] in the D0 D¯ ∗0 near-threshold
region.
The paper is organized as follows. After this Introduction
we present the formalism for the analysis of the line shapes of
the X (3872) in Sects. 2–5. The different scenarios and their
characteristics are the main subject of Sect. 6, where we also
give the numerical results of the fits in each case, the poles
obtained and their properties. After the concluding remarks
in Sect. 7, we give some more technical and detailed material
in Appendices A, B and C.
2 J/ψ π +π − partial-decay rate and differential cross
section
For the decay B → K F through X (3872) we have the decay
chain B → K X and then X → F . We can write the decay
Fig. 1 Skeleton Feynman
diagram for the B → K F decay
through the X (3872) resonance
amplitude TF , represented schematically by the Feynman
diagram in Fig. 1, as
V L VX
where VL and VX refer to the vertices from left to right in
Fig. 1, Q2 is the invariant mass squared of the subsystem
of final particles F (which also coincides with the invariant
mass squared of the X (3872) resonance) and the X (3872)
pole position is PX = MX − i X /2, with MX and X its
mass and width, respectively. The partial-decay width for this
process is
d3 pK
(2π )4δ( P − Q − pk ) (2π )32E K dF
In this equation, P is the total four-momentum of the system
(or that of the B meson), pK is the four-momentum of the
kaon and Q is the one of X (3872) (or F subsystem). Let us
denote by a subscript i (with i = 1, . . . , NF ) the particles in
F and denote the four-momentum of every particle as pi , so
that Q = iN=F1 pi . Then we define dF as the count of states
in the subsystem F ,
dF =
i=1
being Ei = mi2 + pi2 the energy of the i th particle with
mass mi and three-momentum pi . The phase space factor
for F , which we denote by df, can be obtained by extracting
from dF its total four-momentum contribution, so that
We take this into the expression for B→K F , Eq. (2), and
multiply and divide the integrand by Q0 = + Q2 + Q2,
which is the energy corresponding to a particle of mass Q2
and three-momentum squared Q2. Notice that Q0 > 0 and
Q2 > 0 because they are the total energy and invariant mass
squared, in order, of the asymptotic particles in F . We then
have
B→K F =
(2π )4δ( P − Q − pK )|VL |2
The decay width of a B meson into a kaon K and a resonance
X of mass Q2, B→K X (Q2), is the term on the right-hand
side of the second line in the previous equation:
1
B→K X (Q2) = 2MB
(2π )4δ( P − Q − pK )|VL |2
Similarly the decay width of X ( Q2) into F , X→F (Q2) is
given by
We also perform the change of variables from Q0 to Q2,
related by
Q2 = Q20 − Q2.
Then, in terms of Eqs. (6) and (7), we can rewrite Eq. (5) as
B→K F =
Q2 B→K X (Q2) X→F (Q2)
|Q2 − PX2 |2
One can formulate more conveniently the previous
expression by noticing that we are interested in event distributions
with invariant mass around the nominal mass of X (3872)
and X MX ( X < 1.2 MeV [19]). We then approximate
Q2 − PX2
Q2( Q2 − PX ),
in the propagator of X (3872) in Eq. (9). Measuring the
invariant mass of X (3872) with respect to the D0 D¯ ∗0 threshold,
we define the energy variable E as
E =
Q2 − MD0 − MD∗0 .
From Eqs. (10) and (11) we rewrite the differential decay rate
for Eq. (9) as
Q2 − PX |2
B→K X (Q2) X→F (Q2)
= 2π |E − E X + i X /2|2 ,
4 We follow a different sign convention for EX compared to Ref. [3],
so that here EX is negative for MX < MD0 + MD¯ ∗0 .
Next, let us assume that we describe the final-state
interactions of the D0 D¯ ∗0 system in terms of a function d(E )
that gives account of the X (3872) signal properly, which
is represented in Eq. (12) by the propagator factor squared
1/|E − E X + i X /2|2. This is strictly the case for a bound
state or for an isolated resonance such that | X /E X | 1.
For any other case (e.g. a pure virtual-state case) we make
use of the analytical continuation of the expressions obtained.
Then we can write d(E ) around this energy region as
α
d(E ) , (13)
(E − PX )
B→K X (Q2) X→F (Q2) |d(E )|2 .
2π |α|2
As in Refs. [1,3] it is convenient to introduce in Eq. (15) the
product of the branching ratios for the decays B → K X and
X → F , BF (Q2) = B→K X (Q2) X→F (Q2)/ B X , with
B the total decay width of a B meson. This equation then
reads
B BF (Q2)
However, for a final system F with a threshold relatively
far away from the D D¯ ∗0 threshold compared to |E X |, we
can neglect the Q2 dependence in BF . This criterion can
also be applied to a B → K J /ψ π +π − decay because
of the rather large width of the ρ around 150 MeV, which
washes out the sharp threshold effect for this state if we
neglected the ρ width [20]. However, this is not the case
for the B → K D0 D¯ ∗0 decay measured by the BaBar [15]
and Belle [17] Collaborations, which is discussed in the next
section.
We also consider here the J /ψ π +π − event distributions
from the inclusive p p¯ collisions at √s = 1.65 TeV measured
by the CDF Collaboration [18]. The basic Feynman diagram
now shown in Fig. 2. It is similar to Fig. 1 but changing
the kaon K by a set of undetected final particles that are
denoted collectively as F , with X (3872) decaying into a set
of particles denoted by F as above. Instead of Eq. (2) we
Fig. 2 Skeleton Feynman
diagram for the p p¯ → F F
scattering through the X (3872)
resonance
have now to calculate the cross section for p p¯ to F F that
reads
(2π )4δ( P − PF − Q)dF dF
3 D0 D¯ ∗0 partial-decay rate
The B → K D0 D¯ ∗0 decay rate measured be the
reconstruction of D¯ ∗0 from the decay channels D0 D¯ 0π 0 and
D0 D¯ 0γ decays [15,17] has a strong dependence on the
D0 D¯ ∗0 invariant mass in the energy region of the X (3872).
One obvious reason is that D0 D¯ ∗0 is almost at threshold.
Besides that, one also has the decay chain B → K X (3872),
(2π )4δ( P − PF − Q)|VL |2dF
Next, we perform the change of integration variable from
Q0 to Q2, cf. Eq. (8), and after integrating over F and Q
the factor on the right-hand side of the second line in the
previous equation is σ p p¯→XAll(Q2), namely,
(2π )4δ(P − PF − Q)|VL |2dF
Additionally, recalling the expression for
Eq. (7), the Eq. (19) becomes
dσ p p¯→F F = σ p p¯→XAll(Q2) X→F (Q2)
2 Q2
Approaching the inverse propagator as in Eq. (10) and
employing d(E ) to take into account the FSI, we finally
rewrite Eq. (21) as
dσ p dp¯ →EF F = σ p p¯→XAll(Q2)Br(X → F)(Q2) X2π|d|(αE|2)|2 .
dF =
i=1
where | p| is the CM thee-momentum of the initial p p¯ system
and
Splitting dF as in Eq. (4), we rewrite Eq. (17) as
X (3872) → D0 D¯ ∗0 and finally D¯ ∗0 → D¯ 0π 0 or D¯ 0γ , so
that the D¯ ∗0 Lorentzian has some overlapping with X (3872)
mass distribution that rapidly decreases for increasing energy
if the latter lies below threshold. As a result, the width of the
D¯ ∗0 has to be taken into account in the formalism from the
start to study the decays of the X (3872) through the D0 D¯ ∗0
intermediate state, particularly if this state manifests as a
D0 D¯ ∗0 bound state. This point was stressed originally in
Ref. [2].
A D0 D¯ 0π 0 event from the B decays to K X (3872) can be
generated by either B → K X (3872), X (3872) → D¯ 0 D∗0,
D∗0 → D0π 0 or X (3872) → D0 D¯ ∗0, D¯ ∗0 → D¯ 0π 0.
This is an interesting interference process for X (3872) being
mostly a D0 D¯ ∗0 molecule, as first discussed in Ref. [21]. This
latter reference shows that the interference effects vanish for
a zero binding energy while Ref. [3] elaborates that they
can be neglected for |E | 2(Mπ0 /MD0 )δ 1 MeV, with
δ = MD∗0 − MD0 − Mπ0 7.2 MeV, the energy delivered in
a D∗0 decay at rest. For the case of X (3872) with a nominal
mass E X = −0.12 ± 0.20 MeV [19] (adding in quadrature
the uncertainties in the masses of the X (3872), D0 and D∗0
given in the PDG [19]) this inequality is operative and one
might expect some suppression of these interference effects.
The latter were also worked out explicitly in Ref. [22] by
considering the three-body D0 D¯ 0π dynamics and it was shown
there that for a binding energy of 0.5 MeV, the interference
effects below the D0 D¯ ∗0 threshold at the peak of X (3872)
are sizable. This result is in agreement with the outcome of
Ref. [21] for the decay width of X (3872) to D0 D¯ 0π 0, which
found that they are substantial already for binding energies
|E X | 0.1 MeV. However, Ref. [22] derived that above the
D0 D¯ ∗0 threshold they are very modest and for the case of a
virtual state they are so in the whole energy range (both above
and below threshold). Additionally, these interference effects
are mostly proportional to the weight of the molecular D0 D¯ ∗0
weight of X (3872) or compositeness, as explicitly shown by
Voloshin in Ref. [21]. In turn, the interference contributions
in the decay channel D0 D¯ 0γ should be smaller because the
three-momentum of D0 from the decay D∗0 → D0γ is
significantly bigger than for D0π 0, so that the overlapping with
the wave function of D0 in the X (3872) is reduced, an
argument borrowed from Ref. [21]. Based on these facts
resulting from previous work [3,21,22] and because we are mostly
interested in our study in scenarios for the X (3872) in which
it is a double/triplet virtual state or it has a very small
molecular component, we neglect in the following any interference
effect in the D D¯ 0π 0 and D D¯ 0γ decays.5 Then we first
consider the diagonal processes and take for definiteness the
5 Mostly for comparison with less standard scenarios we also discuss
the case of a pure molecular X (3872) generated within the scattering
length approximation [2,3]. In this case it is true that interference effects
could be more important, around a 60% of the direct term at the
resonance mass according to Refs. [21,22]. Nonetheless, we are interested
chain of decays B → K X (3872), X (3872) → D¯ 0 D∗0 and
finally D∗0 → D0π 0. The resulting decay width is denoted
by γX→D¯ 0 D0π0 (Q2), which should be multiplied by 2 to have
the corresponding partial-decay width, X→D0 D¯ 0π0 (Q2), in
the limit in which we can neglect the aforementioned
interference. Analogous steps would apply to the decay B →
K D0 D¯ 0γ through X (3872).
Due to the closeness of MX and MD0 + MD∗0 one cannot
neglect the Q2 dependence of X→D0 D¯ 0π0 (Q2) in Eq. (15)
as noticed at the end of Sect. 2. All the factors on the
righthand side of Eq. (7) are Lorentz invariant and we evaluate
it in the X (3872) rest frame, where one finds the expression
(with F = D0 D¯ 0π 0)
× (2π )32E D¯ (2π )32Eπ (2MD∗0 )2|E + i 2∗ − 2μ |
p2D¯ 2
Several points need be discussed concerning this equation.
We have explicitly indicated the potentially most rapidly
varying kinematical facts in the decay X → F that
comprises the D∗0 propagator and the P-wave character of
D∗0 → Dπ0 , which implies the appearance of the
momentum squared of the pion.6 In Eq. (23) we have indicated by
βˆ2 a coupling constant squared, by μ the D¯ 0 D∗0 reduced
mass (μ = MD0 MD∗0 /(MD0 + MD∗0 ) and by ∗ the D∗0
width. We have used the non-relativistic reduction for the
energies of D0, D¯ 0 and π 0, as mass plus kinetic energy, in
the Dirac delta function for the conservation of energy. This
is also quite valid for the pion because δ Mπ0 . The
nonrelativistic expression is used for the D∗0 propagator as well.
Let us see how it emerges from its relativistic form:
Footnote 5 continued
in the D0 D¯ ∗0 production above its threshold for which the effect at
the peak of X (3872) is reduced by the remarkably narrow Lorentzian
associated with the D∗0 resonance for physical energies (E > 0), cf.
Eq. (25). In addition we also have the contribution from the D0 D¯ ∗0
production above threshold, which is of similar size as the former for
the X (3872) signal region in the pure molecular bound-state case, as
we have checked. For this case we then expect to do an error estimated
to be smaller than a 30%, already of similar size as the experimental
error, which can easily be accounted for by a renormalization about
the same amount of the normalization constant multiplying the signal
contribution.
6 At the end pπ 2Mπ0 δ because δ |EX |, and it could be
re-absorbed in βˆ2 of Eq. (23).
where we have employed the non-relativistic expression for
the energy of D¯ 0 and that in the rest frame of the X (3872),
pD¯ + pD∗ = 0. Neglecting quadratic terms in E , kinetic
energies and ∗ we are lead to the expression for the D∗0
propagator used in Eq. (23). Next, we insert in this equation
the integral identity
1 =
which corresponds to an intermediate D∗0 with
threemomentum pD∗ and energy E D∗ = MD∗0 +p2D∗ /2MD∗0 +E .
In this way we are explicitly extracting the phase space factor
corresponding to the final D0π 0 in the D∗0 decay, similarly
as done above in Eqs. (4) and (5) for the X (3872) resonance
and the subsystem F . We use this result to rewrite Eq. (23)
as
p2D¯ p2D p2π
δ δ + E + 2MD∗0 − 2MD0 − 2Mπ0
where we have split βˆ = βˆ1βˆ2, such that the term on the
right-hand side of the last line in the previous equation can
be identified with the partial-decay width D∗0 → D0π 0 at
rest, which we denote D∗0→D0π0 . As in Eq. (7) we see that
this partial-decay width should be strictly evaluated at the
corresponding D∗0 invariant mass. However, since X (3872)
is so close to the D0 D¯ ∗0 threshold and Q2 M X2 we can
simply take the invariant mass of D∗0 to be equal to MD∗0 ,
which furthermore has a tiny width.
Regarding the factor on the right-hand side of the first line
in Eq. (26) the integration over pD∗ and pD¯ are
straightforward, and then we are left with
2 ,
4π 2Q2 MD∗0 −∞ 2π E D¯ (E 2 + 4∗ )
E − E
where |pD¯ | = √2μ(E − E ). The integration in the previous
equation is a convergent one within a range that gives rise
to tiny kinetic energies compared to MD¯ in Eq. (27). Then
we can just keep the dominant |pD¯ | dependence that stems
from the factor √E − E in the numerator of the integrand and
replace E D by MD0 . In terms of the new integration variable
¯
E , defined as
E = E − E ,
the integral is now
Q2 MD∗0 MD0 ∗
X→D¯ 0 D∗0
E X +
E 2 + ∗2/4
E 2 + ∗2/4
X→D¯ 0 D∗0 =
Q2 MD∗0 MD0
To get the total partial-decay width of X (3872) into D0 D¯ 0π 0
we still have to multiply Eq. (29) by 2 because of the two
mechanisms involved, X (3872) → D¯ 0 D∗0 (the one
explicitly analyzed) and X (3872) → D0 D¯ ∗0. As argued above
we are neglecting interference effects. Then we end from
Eqs. (15), (26) and (29) with the following expression for
the partial-decay rate B → K D0 D¯ 0π 0:
E X +
E2X + ∗2/4
where BDπ = B→K X X→D0 D¯ ∗0 D∗0→D0π0 / B X ∗.
This equation was already derived in Ref. [3] and its main
characteristic energy dependence found before in Ref. [2].
However, what is measured experimentally is the D¯ 0 D∗0
invariant mass [15,17], which is given by |pD¯ |2/2μ when
measured with respect to the D0 D¯ ∗0 threshold in the
X (3872) rest frame. As indicated above because of the
energy conservation Dirac delta function in the first line of
Eq. (26) this quantity is equal to E . Thus, instead of the
differential rate d B→K D0 D¯ 0π0 /d E we should compare the
experimental data with d B→K D0 D¯ 0π0 /dE . The latter can
E , and replacing
be calculated from Eq. (29) by removing the integration in
βˆ12μ 23
8π√Q2 MD∗0 MD0 ∗
cf. Eq. (30). The result is multiplied by D∗→D0π0 , which
is present in Eq. (26), and by 2 because of the two ways of
decay involved. This is then placed in Eq. (15) instead of
X→F , which is then integrated with respect to E . We end
with,
X→D¯ 0 D∗0 ,
in terms of
E X +
This expression coincides with the one already deduced in
Ref. [3]. However, our derivation proceeds in a more
straightforward manner by having split the D0 D¯ 0π 0 phase space
factor in two terms of lower dimensionality [19], attached to
the decays X (3872) → D¯ 0 D∗0 and D∗0 → D0π 0,
employing Eq. (25). In this way the variable E enters directly into
the formulas.
Analogous steps can be followed to derive the
corresponding expression for d B+→K + D0 D¯ 0γ /dE and when summed
to Eq. (32) we have
E X +
BD =
B→K X X→D0 D¯ ∗0
The last equality follows by taking a 100% branching ratio
for the partial-decay width of a D∗0 into D0π 0 and D0γ
[19].
4 Final-state interactions
As discussed above in the Introduction the applicability of the
ERE (and hence of a Flatté parameterization as well) to study
near-threshold resonances, their properties and nature [1–
6,10], could be severely limited by the presence of a nearby
zero in the partial-wave amplitude.
This interplay between a resonance and a close zero indeed
recalls the situation in the presence of the Adler zero required
by chiral symmetry in the isoscalar scalar pion–pion (π π )
scattering and the associated σ or f0(500) resonance. The
presence of this zero distorts strongly the f0(500) resonance
signal in π π scattering while for several production
processes this zero is not required by any fundamental reason
and it does not show up. This is why the f0(500)
resonance could be clearly observed experimentally with high
statistics significance in D → π π π decays [23], where
the S-wave π π final-state interactions are mostly
sensitive to the pion scalar form factor which is free of any
low-energy zero; see e.g. Refs. [24–28] for related
discussions.
Regarding X (3872) there are data on event
distributions involving J /ψ [14,16,18,29] that show a clean
eventdistribution signal for this resonance without any distortion
caused by a zero. However, this does not exclude that a zero
could be relevant for the near-threshold D0 D¯ ∗0 scattering,
as it indeed happens for the f0(500) case. Of course, the
situation is not completely analogous because here X (3872) is
almost on top of the D0 D¯ ∗0 threshold and it has a very small
width, while f0(500) is wide and one does use the ERE to
study it because it is too far away from the π π threshold. This
implies that a CDD pole in the present problem on D0 D¯ ∗0
scattering must be really close to its threshold so as to spoil
the applicability of the ERE.
In this way, instead of using the ERE as in Refs. [1–6]
we employ another more general parameterization that
comprises the ERE up to the effective-range contribution (indeed
up to the next shape parameter) for some limiting case but at
the same time it is also valid even in the presence of a
nearthreshold CDD pole. This parameterization can be deduced
by making use of the N /D method as done in Ref. [30],
whose non-relativistic reduction is given in Ref. [31]. The
point is to perform a dispersion relation of the inverse of the
D0 D¯ ∗0 S-wave t (E ), which along the unitarity cut fulfills
the well-known unitarity relation
Imt (E )−1 = −k(E ), E ≥ 0,
where E is the center of mass (CM) energy of the system, cf.
Eq. (11), and k(E ) is the CM three-momentum given by its
non-relativistic reduction k(E ) = √2μE . Next, we neglect
crossed-channel dynamics based on the fact that the scale
associated with the massless one-pion exchange potential, as
worked out in Refs. [12,32], is = 4π fπ2/μg2 ∼ 350 MeV
( fπ = 92.4 MeV and g 0.6), which is much bigger than
the D0 D¯ ∗0 three-momentum ( 30 MeV) in the region of
the X (3872). In this estimate one takes into account that
the denominator in the exchange of a π 0 of momentum q
between D∗0 and D0 is q2 + Mπ20 − (MD∗0 − MD0 )2 and
that ((MD∗0 − MD0 )2 − Mπ20 )/Mπ20 0.1 1 because
MD∗0 − MD0 is larger than Mπ0 by only 7.2 MeV [33]. It
is then appropriate to write down a dispersion relation for
t (E )−1 with at least one necessary subtraction employing the
Fig. 3 Integration contour used for the dispersion relation of t (E)−1
giving rise to Eq. (36). The contour is closed by a circle centered at the
origin and of infinite radius
integration contour of Fig. 3. Then allowing for the presence
of a pole of t (E )−1 we then obtain
t (E ) =
E − MCDD
−1
with MCDD the position of the CDD pole measured with
respect to the D0 D¯ ∗0 threshold. Notice that this is a pole in
t (E )−1 and then a zero of t (E ) at E = MCDD.
Since the finite width effects of D∗0 could be important as
argued in Sect. 3, the CM three-momentum k(E ) is finally
calculated according to the expression
k(E ) =
For definiteness the three-momentum k(E ) is always defined
in the first Riemann sheet (RS), so that the phase of the
radicand is taken between 0 and 2π . Here an analytical
extrapolation in the mass of the D∗0 resonance until its
pole position MD∗0 − i ∗/2 has been performed, as also
done e.g. in Refs. [3,34,35]. By considering explicitly the
three-body channel D0 D¯ 0π 0 in a coupled-channel
formalism, Ref. [22] found that Eq. (37) is appropriate because
of the smallness of the P-wave D∗0 width into D0π 0,
which implies that ∗/2δ = 4.5 10−3 1. In Eq. (36)
the constant β for elastic D D¯ ∗0 scattering is real but it
becomes complex, with negative imaginary part, when
taking into account inelasticities from other channels, such as
J /ψ π +π −, J /ψ π +π −π 0, etc. [1–3]. We finally fix this
possible imaginary part in β to zero because, as already noticed in
Ref. [3], one can reproduce data equally well, as we have also
checked.
An ERE of t (E ) given in Eq. (36) is valid in the k2
complex plane with a radius of convergence coincident with
2μMCDD. Notice that a zero of t (E ) near threshold implies
that k cot δ = ∞ at this point and then it becomes
singular. As a result, its k2 expansion does not converge and the
ERE becomes meaningless for practical applications since
its radius of convergence is too small. In such a case, one
must consider the full expression for t (E ) in Eq. (36) and
not its ERE, which reads
λ
k cot δ = k2/2μ − MCDD + β
λ k2λ
= − MCDD + β − 2μMC2DD + · · ·
here the ellipsis indicate higher powers of k2. This expansion
can reproduce any values of the scattering length and
effective range (as well as of the next shape parameter v2) and we
obtain the expressions7
a = MCDD
It is then clear that in order to generate a large absolute value
for a, one needs a strong cancellation between λ/MCDD and
β unless both of them are separately small. But in order to
have a small magnitude of |a| and a large one for |r |, one
would naturally expect that MCDD → 0, though the explicit
value of λ plays also an important role. Equation (39) clearly
shows why the ERE could fail to converge even for very small
values of |k|2 as long as MCDD → 0.
In the limit MCDD → ∞ with λ/MCDD fixed, the
parameterization in Eq. (36) for t (E ) reduces to the function f (E )
used in Refs. [2,3]
t (E ) −−−−−−−−−→
MCDD → ∞
λ/MCDD = ct.
f (E ) =
where γ = 1/a is the inverse of the scattering length, using
the notation of Ref. [3]. The function f (E ) has a bound
(virtual) state pole for positive (negative) γ .
While the near-threshold energy dependence of f (E ) is
dominated by the threshold branch-point singularity and a
possible low-energy pole associated with a bound or virtual
state, this is not necessarily the case for t (E ) as long as MCDD
is small enough. In such a case one has to explicitly remove
the CDD pole from t (E ) by dividing it by E − MCDD. In this
way, we end with the new function d(E ), already introduced
in Sect. 2 just before Eq. (13), which is then defined as
d(E ) = 1 + E−MCDD (β − i k)
λ
7 Note the different sign convention for the scattering length here as
compared with Ref. [31].
such that its low-energy behavior is qualitatively driven by the
same facts mentioned for f (E ). This is also the function that
in general terms drives final-state interactions (FSI) when the
scattering partial wave is given by t (E ) in Eq. (36). A detailed
account of it can be found in Ref. [31], although Ref. [26,27]
could be more accessible depending on the reader’s taste and
education.
Next, we explicitly calculate the residue α for d(E ) needed
to work out the decay rates in Eqs. (16) and (33) and the
differential cross section of Eq. (22). This can be
straightforwardly determined by moving to the pole position as defined
in Eq. (13). Thus, it results
The three-momentum k P is evaluated at the pole position E P
in the energy plane,
k P =
E P + i ∗ ,
such that the phase of the radicand is between [0, 2π [ for
a bound-state pole in the first RS, while for a pole in the
second RS, a virtual-state one, the phase is between [2π, 4π [
and the sign of k P is reversed compared to its value in the
first RS.
The constant α, in the case of using the function f (E ) in
Eq. (40) for the decay rates in Eqs. (16) and (33), is defined
analogously as the residue of f (E ) at the pole position PX .
The function f (E ) has a different normalization compared
to d(E ), and α is then given by
Here we have taken into account that k P = i γ for the f (E )
parameterization.
The limit of decoupling a bare resonance from a
continuum channel, like D0 D¯ ∗0, requires the presence of a zero
to remove the pole of the resonance from t (E ). This simple
argument shows that CDD poles and weakly coupled bare
resonances are typically related. In this respect, we consider
the resulting t (E ) obtained in Ref. [11] by considering the
interplay between mesonic and quark degrees of freedom,
and that results by considering the exchange of a bare
resonance together with direct scattering terms in the mesonic
channel at the level of the scattering length approximation.
In the following discussions until the end of this section the
zero width limit of D∗0 should be understood in k(E ). The
resulting S-wave amplitude from Ref. [11] is8
Here, aV is the scattering length for the direct D0 D¯ ∗0
scattering (referred as potential scattering in Ref. [11]), γV = 1/aV ,
g f is the coupling squared between the bare resonance and
the mesonic channels, while E f is the mass of the bare
resonance in the decoupling limit g f → 0. By comparing t (E )
in Eq. (46) with our expression above Eq. (36) one has the
following relation between the parameters:
which shows that the results of Ref. [11] are a particular case
of ours, since it is always possible to adjust λ, MCDD and
β in terms of g f , E f and γV . However, the reverse is not
true because g f ≥ 0 [11], which implies that λ is restricted
to be positive as well, while the residue of the CDD pole
can have a priori any sign. This difference is also
important phenomenologically because, while our
parameterization for t (E ) can give rise to values of the effective range
with any sign, Ref. [11] generates only negative ones, cf.
Eq. (39). Equation (47) explicitly shows the above remark
that E f → MCDD in the decoupling limit, g f → 0, with
both g f and MCDD − E f being proportional to the residue
of the CDD pole. It is also interesting to notice that β
corresponds to the minus the inverse of the potential scattering
length aV . The language of the exchange of a bare resonance
and direct D0 D¯ ∗0 scattering could be more intuitive in some
aspects than the direct use of S-matrix theory, employed to
obtain Eq. (36), so that we will make contact with the former
when discussing our findings. The formalism of Ref. [11] was
extended to coupled channels in Ref. [13], and the inclusion
of inelastic channels was also addressed more recently in
Ref. [36,37].
The scattering length approximation for the D0 D¯ ∗0 S
wave of Refs. [2,38] was further generalized in Ref. [12] to
include as well the exchange of one bare resonance together
with the explicit coupling between the channels D0 D¯ ∗0 and
D+ D¯ ∗−. The expression obtained in Ref. [12] for the elastic
D0 D¯ ∗0 S-wave amplitude is
8 A minus sign is included due to the different convention in Ref. [11].
g2γ0(γ1 − 2κˆ2)
MZ = ν − γ0 + γ1 − 2κˆ2
The same comment as made above, concerning the
nonfully equivalence between our parameterization and the one
of Ref. [11], is also in order here regarding Eq. (48). The
point is that the latter implies again from Eq. (50) that λ ≥ 0
while the residue of a CDD pole can have any sign.
9 If higher orders are kept in the Taylor expansion of κ2 around E =
0 then the matching would require one to include more CDD poles
(with contributions suppressed by powers of E/ ); see Ref. [30] for
details. Another option is to follow the coupled-channel formalism there
developed as well, particularly when considering a wider energy range.
Here, κ2 = 2μ( − E − i 0+), where is the
difference between the thresholds of D− D∗+ and D¯ 0 D∗0.
Additionally, γ0,1 are the isoscalar and isovector scattering lengths
in the limit of decoupling the bare state with the continuum
channels and g is the coupling among them. The parameter
ν is the mass of the bare state measured with respect to the
lightest threshold. To match Eq. (48) in the near-threshold
region with the expression for t (E ) in Eq. (36) we rewrite
the former as
(E − ν + g2γ02)(2γ0γ1 − (γ0 + γ1)κ2) − g2γ02(2γ1 − κ2) −1
(E − ν + g2γ02)(−γ1 − γ0 + 2κ2) + g2γ02
which explicitly shows the correct form to fulfill elastic
unitarity below the D+ D∗− threshold, so that the term involving
the product κk in Eq. (48) has disappeared. Restricting
ourselves to our region of interest, |E | , we can perform
a Taylor expansion of κ2 around E = 0 and keep only its
leading term κ2 → κˆ2 = √2μ , so that all the energy
dependence of t (E )−1 is dominated by the CDD pole and
the right-hand cut for elastic scattering, as in our derivation
of Eq. (36).9 The explicit expressions of λ, β and MCDD as
a function of the parameters γ0,1, ν, g2 and ν in Eq. (49) are
N =
with the same expression replacing d(E ) by f (E ) if the latter
function is used [3]. The normalization integral is defined as
5 Formulas for the event distribution
The combination |d(E )|2 X /2π |α|2 in Eqs. (16), (22) and
(33) corresponds to the normalized standard non-relativistic
mass distribution for a narrow resonance or bound state
(taking in this last case X → 0). We then define this
combination as the spectral function involved in the energy-dependent
event distributions
which is equal to one for the cases mentioned before.
However, this is not the case when E P corresponds to a virtual
state or other situations for which the final-state interaction
function d(E ) has a shape that strongly departs from a
nonrelativistic Breit Wigner (which also includes a Dirac delta
function in the limit X → 0). When using f (E ) the
integration in Eq. (52) does not converge. Then we take as
integration interval [2E X , 0] as in Ref. [3], which embraces the
signal region and it is enough for a semiquantitative
understanding/picture based on the near value of N to 1 in the
bound-state case.
We consider data on event distributions for J /ψ π +π −
and D0 D¯ ∗0 from B → K X (3872) decays [14–17] and
inclusive p p¯ collisions [18]. In the B-decay cases the
number of B B¯ pairs produced at ϒ (4S) is given and we denote
it by NB B¯ , with the same amount of neutral and charged
B B¯ pairs produced. It is also the case that the experimental
papers [14–17] include the charge-conjugated decay mode
to the one explicitly indicated, a convention followed by
us too.
We perform fits to the data on the J /ψ π +π − event
distributions from charged B+ → K + J /ψ π +π − decays
measured by the Belle [16] and BaBar Collaborations [14]. The
predicted event number Ni at the i th bin, with the center
energy Ei and bin width , is given by the convolution of
the decay rate in Eq. (16) times NB B¯ / B+ with the
experimental energy-resolution function R(E , E ), and integrating
over the bin width. We divide Eq. (16) by B+ because all
the charged B+ B− pairs produced, NB B¯ /2, have decayed
(an integration over time of the rate of decay is implicit. The
latter is given by the product of the total width times the
number of B mesons decaying at a given time). In addition,
one has to multiply the signal function by the experimental
efficiency ε(J+). The resulting formula is
d E R(E , E )
The constant BJ attached to the signal contribution in Eq. (53)
can be interpreted as the product of the double branching
ratios Br(B+ → K + X )Br(X → J /ψ π +π −) when N =∼ 1,
cf. Eq. (52). In this case the product ε(J+) NB B¯ BJ is directly
the yield YJ . If this is not the case this interpretation is not
possible but we still call this product in the same way, though
its meaning is just that of a normalization constant. In this
way, we re-express Eq. (53) as
Ni = YJ
Ei − /2
On the other hand, the background contribution is specified
by the constant NB B¯ cbgJ , which can be determined by
simple eye inspection from the sidebands events around the
X (3872) signal region. The energy-resolution function is the
Gaussian function
R(E , E ) = √
exp −
Following Ref. [4], as also used in Ref. [3], we take σ = 3
MeV for both BaBar [14] and Belle [16] experiments on
J /ψ π +π − event distributions. We take both BJ and cbgJ to
be the same in the fits to BaBar and Belle data because once
we take into account the different NB B¯ for both experiments
(NB B¯ = 4.55 × 108 for BaBar [14], and NB B¯ = 6.57 ×
108 for Belle [16]; see also Table 1) the yields given in the
experimental papers [14,16] coincide. This means that the
ratio of the parameters YJ and cbgJ for BaBar and Belle is
the same as the quotient of their respective NB B¯ . Then, after
fitting the data, we will give only the values of the resulting
parameters for the former.
We also consider the CDF J /ψ π +π − event
distribution from inclusive p p¯ scattering [18]. We use Eq. (22)
times the integrated luminosity L, which for Ref. [18]
is 2.4 fb−1. In addition we neglect the Q2 dependence
except for d(E ) and after including the bin width,
experimental efficiency ε(Jp), energy resolution and background
we have
Ni = ε(Jp)Lσ p p¯→XAllBr(X → J /ψ π +π −)
Here the bin width is 1.25 MeV and the background in
the X (3872) region has been parameterized as a straight line
(which is easily determined from the sideband events),
following the outcome in Fig. 1 of the CDF Collaboration paper
D0 D¯ ∗0
NB B¯ = 3.83 × 108, =2 MeV
= 5 MeV
[18]. In this reference the experimental resolution function
is expressed as the sum of two Gaussians
R p p¯ (E , E ) = √
exp −
(E − E )2
exp −
(E − E )2
Ni = YJ( p)
Concerning the D0 D¯ ∗0 event distributions from charged
and neutral B → K X decays, D∗0 is fully reconstructed
from its decay products D0π 0 and D0γ in the data from
BaBar [15]. In the case of Belle data [17] we employ the one
in which D∗0 is reconstructed only from its decay product
D0π 0, because it has a much smaller background than for
D0γ . To reproduce the event distributions we employ the
decay rate of Eq. (33) and take into account the experimental
resolution, efficiency, bin width and background
contributions, similarly as done for Eq. (53) above. We end with the
expression
Ni =
YD
E2X + ∗2/4
Table 1 The B decays into K J/ψπ+π− and D0 D¯ ∗0 channels are
both measured by the BaBar [14,15] and Belle Collaborations [16,17].
The total number of B B¯ pairs (NB B¯ ), bin width ( ), and the Gaussian
width (σ ) used in the experimental resolution function are given. The
number of points included in the fits are also indicated. For the inclusive
p p¯ collision measured by the CDF Collaboration [18] we account for
similar parameters, but now the luminosity (L) is given instead of NB B¯ .
For more details see the text
NB B¯ = 6.57 × 108, =2 MeV
= 2.5 MeV
In this equation the background contribution is
parameterized as cbgD√E as in Ref. [3], giving rise after fitting to
similar background contributions as the ones in the
experimental papers Refs. [15,17] (though they are parameterized
in somewhat different form). The constant cbgD can easily be
determined from the events above the X (3872) signal region
which gives rise to a rather structureless pattern. The constant
YD can be interpreted again as a yield for N ≈ 1 because,
when integrating in E over all the energy range the signal
contribution in Eq. (59), the denominator below YD is
canceled because of Eq. (29). We again follow Refs. [3,4], as well
as the Belle experimental analysis [17], and take the
Gaussian width σ in the resolution function R(E , E ), Eq. (55),
to be energy dependent and given by the expression
(0.031 MeV)E ,
with E running through the values in Eq. (59). For the
D0 D¯ ∗0 event distributions the number of B B¯ pairs produced
is NB B¯ = 3.83 × 108 for BaBar [15], and NB B¯ = 6.57 × 108
for Belle [17], as also indicated in Table 1. For this case we
have to take different values for the yields and background
constants for fitting the BaBar and Belle data.
In all our formulas for the event distributions in Eqs. (54),
(58) and (59) the background contribution is added
incoherently because it is mostly combinatorial. This is the same
treatment as performed in the experimental papers [14–17]
as well as in the phenomenological analysis [1–5]. In a
Laurent expansion of the signal amplitude around the X (3872)
non-resonant terms appear that add coherently but they are
accounted for by the function d(E ) in the near-threshold
region, which, as discussed in Sect. 4, is assumed to have the
main dynamical features. Reference [12] attempts to unveil
further dynamical information from the B → K D0 D¯ ∗0
event distributions by considering them in a broader energy
region beyond the X (3872) signal and explicitly including
the D+ D∗− channel within the formalism. Nonetheless, the
present experimental uncertainty avoids extracting any
def)V12
e
M
/(2 8
s
t
n
ev 4
E
)V12
e
M
/(2 8
s
t
n
ev 4
E
)V60
e
M
/(540
s
t
n
e
v20
E
Fig. 4 Fit results for the different event distributions as a function of
energy. Panels a and b denote the mode D0 D¯ ∗0 from BaBar [15] and
Belle [17], respectively. Panels c–e denote the mode J/ψπ +π − from
BaBar [14], Belle [16], and CDF [18] Collaborations, respectively. The
black solid lines correspond to case 1 that employs f (E). The CDD
pole contribution is included in the expression of d(E), and the rest
inite conclusion beyond the smooth background out of the
X (3872) region.
For the fitting process it is advantageous to rewrite
Eqs. (54), (58) and (59) by using directly |d(E )|2 instead
of d Mˆ (E )/d E , and re-absorbing the factor 1/|α|2 in the
normalization constant. In this form one avoids working out the
dependence of α on the pole position, which numerically is
very convenient since it is not known a priori where the pole
lies when using d(E ). Once the fit is done one can actually
calculate α, cf. Eq. (42), and from its value and the fitted
constant the corresponding yield. This technicality is discussed
in more detail in Appendix A.
6 Combined fits
The data sets that we include in the fits were already
introduced in Sect. 5. A summary of their main characteristics
can be found in Table 1. Apart from the data on B → K X
decays we also include the high statistics J /ψ π +π − event
distribution from p p¯ collisions at √s = 1.96 TeV, measured
by the CDF Collaboration [18], which also has the
smallest bin width. In this way one can reach from these data a
better determination of E R . The value given by this
collaboration for the X (3872) mass is MX = 3871.61 ± 0.22 MeV,
of lines refer to cases making use of this parameterization. Case 2.I is
shown by the red dotted lines, and case 2.II by the brown dashed lines,
while cases 3.I and 3.II are given by the blue dash-double-dotted and
green dash-dotted lines, in order. Part of the lines overlap so much that
it is difficult to distinguish between them in the scale of the plots
from which we infer E R = −0.20 ± 0.22 MeV, that has a
smaller uncertainty than the one obtained in B decays from
Refs. [14–17]. From the point of view of mutual
compatibility between data sets [1] it is also interesting to perform
a simultaneous fit to all the data on B+ → K + J /ψ π +π −,
B+(B0) → K +(K 0)D0 D¯ ∗0 and J /ψ π +π − from inclusive
p p¯ collisions.
Experimental data points have typically asymmetric error
bars; see e.g. the data points in Fig. 4. Thus, as done in Ref. [3]
and also in other experimental analyses, the best values for
the free parameters are determined by using the binned
maximum likelihood method, which is also more appropriate in
statistics than the χ 2 method for bins with low statistics. At
each bin, the number of events is assumed to obey a Poisson
distribution, so that the predicted event numbers in Eqs. (54),
(58) and (59) are the corresponding mean value at the bin
(Ni ), while the experimentally measured number is called
Yi (experimental data). The Poisson distribution at each bin
reads Li (Ni , Yi ) = NiYi exp(−Ni )/Yi ! and the total
probability function for a data sample is given by their products.
One wants to maximize its value so that the function to be
minimized is defined as
− 2 log(L) = −2
(Yi log(Ni ) − Ni ).
Fig. 5 Detail of the fit results
for the near-threshold region in
the D0 D¯ ∗0 event distributions.
The left panel correspond to the
data from the BaBar
Collaboration [15] and the right
one to that of the Belle
Collaboration [17]. The types of
lines employed to plot the boxes
for every scenario are the same
as in Fig. 4
When including a CDD pole in the expression for t (E ),
Eq. (36), one has to fix three free parameters to characterize
the interaction, namely, λ, MCDD and β. However, this is a
too numerous set of free parameters to be fitted in terms of
the data taken (additionally we have the normalization and
background constants). This manifests in the fact that there
are many local minima when minimizing −2 log L, so that it
is not clear how to extract any useful information. Instead, we
have decided to consider five interesting and different
possible scenarios (cases 1, 2.I–II and 3.I–II), so that each of them
gives rise to an acceptable reproduction of the line shapes
but corresponds to quite a different picture for X (3872). In
addition, we also think that the pole arrangements that result
in every case are worth studying for general interest on
nearthreshold states. For every of these cases studied the number
of free parameters associated with d(E ) is one (only case 2.II
below has two free parameters), so that the interaction is well
constrained by data.
We gather together similar sets of information for each
scenario, so that the comparison between them is more
straightforward. The reproduction of the data fitted for all
the cases is shown in Fig. 4 by the black solid (case 1), red
dotted (case 2.I), brown dashed (case 2.II), blue
dash-doubledotted (case 3.I) and green dash-dotted (case 3.II) lines. A
detailed view of the more interesting near-threshold region
for the D0 D¯ ∗0 event distributions is given in the histogram
of Fig. 5. We also show separately the reproduction of the
J /ψ π +π − event distribution of the CDF Collaboration data
[18] in Fig. 9. In this figure we include the error bands too,
in order to show the typical size of the uncertainty in the
line shapes that stems from the systematic errors in our fits.
For all the figures we follow the same convention for the
meaning of the different lines. The spectroscopical
information is gathered in Table 2, where we give from left to right
the near-threshold pole positions, the compositeness for the
bound-state pole (if present), the residues to D0 D¯ ∗0 and the
yields. Finally, we show in Table 3 the scattering parameters
characterizing the partial wave t (E ) that result from the fits.
In the two rightmost columns we give the scattering length
and the effective range.
All fitted parameters are given in Eqs. (62)–(64), (66) for
cases 1, 2.I–II and 3.I–II, in order. The best values of the
parameters are obtained by the routine MINUIT [39]. The
error for a given parameter is defined as the change of that
parameter that makes the function value −2 log L less than
−2 log Lmin + 1 (one standard deviation), where −2 log Lmin
is the minimum value.
10 parameters YD1 = 7.49+−00..7411,
YD2 = 6.45+−00..3427,
YJ = 79.03+−56..6151,
10 parameters YD1 = 79.75+−2129..8416,
E R = −0.95+0.07 MeV,
−0.05
YD2 = 72.39+−98..1082,
YJ = 9.23+1.60
−1.57 × 103,
E R = −0.49+0.04 MeV,
−0.03
10 parameters YD1 = 25.45+−44..0155,
YD2 = 21.08+−11..3829,
YJ = 80.14−+55..1697,
E R = −0.49+0.02 MeV,
−0.01
10 parameters: YD1 = 22.90−+32..0924,
YD2 = 18.92−+01..7470,
YJ = 80.07+−55..1346,
= 6.23+0.71
−0.84 × 105, ζ = 1520.23+8.57 MeV−1,
−8.57
= 5.26+0.12
−0.08 × 103, ζ = 1524.70+7.91 MeV−1,
−7.91
= 5.23+0.07
−0.11 × 103, ζ = 1515.44+7.40 MeV−1,
−7.40
E R = −0.53+−00..0191 MeV, MCDD = −12.29−+11..1141 MeV,
11 parameters: YD1 = 83.13+−2126..1452,
YD2 = 68.84−+71.12.586,
YJ = 8.44+3.64
−2.59 × 103,
= 5.78+2.29
−1.64 × 105, ζ = 1498.57+9.73 MeV−1,
−9.73
N BaBar cbgD1 = 0.18+0.02 MeV−3/2,
B B¯ ;D −0.02
N BaBar cbgD2 = 0.11+0.01 MeV−3/2,
B B¯ ;D −0.01
N BaBar cbgJ = 3.59+0.13 MeV−1,
B B¯ ;J −0.13
N BaBar cbgD1 = 0.24+0.02 MeV−3/2,
B B¯ ;D −0.02
N BaBar cbgD2 = 0.14+0.01 MeV−3/2,
B B¯ ;D −0.01
N BaBar cbgJ = 3.46+0.14 MeV−1,
B B¯ ;J −0.14
N BaBar cbgD1 = 0.25+0.02 MeV−3/2,
B B¯ ;D −0.02
N BaBar cbgD2 = 0.15+0.01 MeV−3/2,
B B¯ ;D −0.01
N BaBar cbgJ = 3.65+0.13 MeV−1,
B B¯ ;J −0.13
N BaBar cbgD1 = 0.22+0.02 MeV−3/2,
B B¯ ;D −0.02
N BaBar cbgD2 = 0.13+0.01 MeV−3/2,
B B¯ ;D −0.01
N BaBar cbgJ = 3.66+0.13 MeV−1,
B B¯ ;J −0.13
N BaBar cbgD1 = 0.22+0.02 MeV−3/2,
B B¯ ;D −0.02
N BaBar cbgD2 = 0.13+0.01 MeV−3/2,
B B¯ ;D −0.01
N BaBar cbgJ = 3.66+0.13 MeV−1,
B B¯ ;J −0.13
= 5.28+0.05
−0.17 × 103, ζ = 1524.66+7.73 MeV−1,
−7.73
6.1 Case 1: Bound state
In this first case we fit the different data sets by using the
function f (E ) [2, 3], in order to take into account the FSI
sets separately. We also include this standard case as a
reference to compare with other less standard ones introduced
the expression of f (E ) can be taken complex (with negative
imaginary part) to mimic inelastic channels. Indeed complex
values were used in Ref. [3], though it was found that the
experimental data can be equally well described by taking
found). Physically, this indicates that an inelastic effect, as
the transition D0 D¯ ∗0 → J /ψ π +π −, has little impact on
FSI and we fix it always to zero in our fits, which are also
Table 2 Summary of the combined fits in Sect. 6 for all the cases. From
left to right, the pole positions, compositeness (X ), residue and yields
(Y ) are given. YD1, YD2 denote the yields corresponding to the BaBar
and Belle data on the D0 D¯ ∗0 mode, respectively; YJ denotes the one
Pole position [MeV]
for BaBar on the J/ψπ +π − mode and Y (p) applies to the CDF
Collaboration data with p p¯ collisions. The noJrmalization for the residues
is such that = kg2/(8π MX2 ) as in Ref. [19]
−0.19−+00..0011 − i 0.0325
−0.33−+00..0034 − i 0.31+0.02
−0.01
−0.84+0.07 −0.04
−0.05 + i 0.77+0.03
−1.67−+00..0180 − i 0.49+0.02
−0.02
The parameters corresponding to the yields and
background constants are YJ , YJ( p), YD, cbgJ, ξ , ρ and cbgD. The
most interesting free parameters are those fixing the
interaction t (E ), which for the present case just reduces to the
inverse scattering length γ . The background constants can be
determined rather straightforwardly, because the background
is very smooth in all cases and fixed by the sidebands events
around the signal region. The best fitted parameters that we
obtain for case 1 are given in Eq. (62). The reproduction of
the event distributions is shown in Figs. 4 and 5 by the black
solid lines. The different yields, which we denote globally as
YF in the following, can be interpreted properly in this way
because N = 0.98 1, as expected for a bound state.
With the values of the parameters at hand in Eq. (62),
X (3872) is a near-threshold bound-state pole in the
function f (E ) located at −0.19+−00..0011 − i ∗/2 MeV. Here the
imaginary part stems purely from the finite width of the
constituent D∗0. As a result the scattering length is large and
positive with the value a = 10.40+0.10 fm.
−0.26
X , of the resulting bound-state pole [40–42] can be written
as [8]
Residue [GeV2]
−47.48+−91.27.540 − i 66.06+10.87
−13.50
82.69+−1141..8884 + i 66.03+13.50
−10.87
−6.24−+22..2800 − i 1.41+0.14
−0.10 × 102
(2.32+0.16
−0.21 − i 1.77+−00..1018) × 102
(−3.26+−00..2126 + i 3.18+−00..1285) × 102
X = −i gk2,
with gk2 the residue of the amplitude in the momentum
variable k. For f (E ) = 1/(−1/a − i k) its residue at the pole
position in the variable k is i , so that X = 1. That is,
independently of which is the dynamical seed for binding (origin
of γ ) this is a bound state whose composition is exhausted
by the D0 D¯ ∗0 component [2,3,11]. This result is in
agreement with Ref. [6], which concludes that the scattering length
approximation is only valid for the bound-state case if its
compositeness is 1. We also give in the fourth column of
Table 2 the residue g2 of the S-wave scattering amplitude
for each near-threshold pole in a more standard
normalization, in which the partial decay width of a narrow resonance
is = kg2/(8π M X2 ) [19]. This residue for f (E ) reads
g2 = −i 16π k P PX2 /μ.
6.2 Case 2: Virtual state
In the previous section, as well as in Ref. [3], only the
scattering length is taken into account. However, considering the
Table 3 Parameters
characterizing the S-wave
interaction t (E ) for cases 1–3.II
of the combined fits in Sect. 6.
The ellipses indicate that it is
not appropriate to give the
corresponding magnitude in
such case. The elementariness Z
is calculated from the
knowledge of the bound-state
mass and the Weinberg’s
compositeness relations of
Eq. (84) either in terms of a (Za )
or r (Zr ). The error is not given
when its estimation is smaller
than the precision shown
analysis performed in Ref. [6], the effective range should
better be added into, as already done in the pioneering
analysis of Ref. [1], since the scattering length approximation is
only valid for pure molecular states. As discussed in Sect. 4,
one also has to face the problem of the possible presence of
zeros just around threshold. These two points can be better
handled by including a CDD pole, and the D¯ 0 D∗0 S-wave
scattering amplitude is given in Eq. (36). In this case FSI are
taken into account by the function d(E ), introduced above
and given in Eq. (41). We make use of this new formalism to
impose the presence of a virtual state when fitting data, so as
to distinguish the virtual-state scenario from the bound-state
one obtained above by using the function f (E ) in Sect. 6.1.
We also remark that proceeding in this way leads to quite
interesting situations in which X (3872) becomes a double or
triplet virtual-state pole in the zero width limit of D∗0.
Reference [11] already stressed the importance of taking care of
a possible near-threshold zero in scattering and production
processes.
The t -matrix for D0 D¯ ∗0 scattering, in the second RS sheet,
is obtained from its expression in the first RS, cf. Eq. (36),
but replacing k by −k, namely,
−1
tII(E ) =
E − MCDD
where k = √2μE is calculated such that Im k > 0. Notice
that here we are taking D∗0 without width, and then impose
a pure virtual-state situation, that is, a pole on the real axis
below threshold in the second RS. The presence of the virtual
state is granted by imposing the requirement that tII(E ) has a
pole at E P = E R −i G R /2, with E R < 0, G R > 0 and taking
at the end the limit G R → 0+. For an S-wave resonance it
is possible to have a non-zero width for a resonance mass
smaller than the two-particle threshold; see e.g. Refs. [43, 44]
for particular examples and Ref. [45] for general arguments.
The vanishing of the real and imaginary parts of tII(E P )−1
allows us to fix two parameters, e.g. for λ and β one has the
expressions
Thus, the function d(E ), Eq. (41), depends on E R and MCDD,
with a stable limit for G R → 0+. The latter can be performed
algebraically from Eq. (69) with the result
μ
with = √2μ|E R |. Notice that keeping G R finite and later
taking the limit G R → 0+ allows us to dispose of one more
constraint (one free parameter less) than if we had taken
directly G R = 0 and then imposed the requirement that
tII(E R ) = 0.
Next, let us consider the secular equation for the poles of
t (E ), Eq. (36), in the complex k plane:
λ + (E − MCDD)(−i k + β) = 0.
We substitute the expressions for λ and β of Eq. (70) in the
previous equation and obtain
= 0,
where the global factor −i /2μ has been dropped. This
equation explicitly shows that k = −i is a double virtual-state
pole.
It is also trivial from Eq. (72) to impose a triplet
virtualstate pole by choosing appropriately MCDD to
MCDD = −3E R .
In the following we denote by case 2.I the one with the double
virtual-state pole and by case 2.II that with the triplet pole.
To fit data we reinsert the finite width for D∗0 and use the
expressions for λ and β in Eq. (70). For case 2.I one has two
free parameters (E R and MCDD) to characterize the
interaction, while for case 2.II only one free parameter remains
(E R ) because of the extra Eq. (73). The fitted parameters in
each case are given in Eqs. (63) and (64).
The reproduction of the data for cases 2.I and 2.II are
shown by the red dotted and brown dashed lines in Figs. 4
and 5, respectively. There are visible differences between
case 1 and cases 2.I–II, in the peak region of the D0 D¯ ∗0 and
J /ψ π +π − event distributions. For the former, the
scenarios 2.I–II produce a signal higher in the peak that decreases
faster with energy, while for the latter there is a displacement
of the peak towards the threshold in the virtual-state cases.
This is more visible in Fig. 9 where we show only the
reproduction of the CDF data [18] including error bands as well.
The reason for this displacement is because the virtual-state
poles only manifests in the physical axis above threshold, so
that the peak of the event distribution happens almost on top
of it. Nonetheless, we have to say that the shift in the signal
peak for cases 2.I–II and the J /ψ π +π − data diminish
considerably if we exclude in the fit the D0 D¯ ∗0 data from the
BaBar Collaboration [15]. Furthermore, it is clear that these
data give rise to a line shape with a displaced peak towards
higher energies as compared with the analogous data from
the Belle Collaboration [17]; see Fig. 5. Thus, it is not fair
just to conclude that cases 2.I–II are not favorable because of
the shift of the signal shape in the J /ψ π +π − CDF data [18]
until one also disposes of better data for the D0 D¯ ∗0 event
distributions.
It is clear that when taking ∗ = 0 (so that standard ERE
is perfectly fine mathematically for |k2| < 2μ|MCDD|), all
the near-threshold poles are at E R . For case 2.I the CDD pole
lies relatively far away from the D0 D¯ ∗0 threshold. However,
if we kept only a = −11.82 fm in the ERE the pole
position in the second RS would be −0.14 MeV, if including
r = −5.64 fm it is −0.93 fm, with v2 then it still moves to
−0.58 MeV and with v3 one has −0.54 MeV. Thus, though
the CDD pole is around −12 MeV one needs several terms
in the ERE to reproduce adequately the S-wave amplitude.
In particular, it is not enough just to keep e.g. the
scattering length contribution as in case 1 or as in Ref. [2,3]. For
case 2.II the CDD pole is much closer, around 3 MeV, so that
the convergence of the ERE is much worse and many more
terms in the ERE should be kept to properly reproduce the
pole position.
At this point it is interesting to display the pole
trajectories as a function of g f , while keeping constants γV and E f ,
cf. Eq. (47). In this way, we have a quite intuitive
decoupling limit g f → 0 in which two poles at ±√2μE R
correspond to the bare state and an additional one at i γV = −iβ
stems from the direct coupling between the D0 D¯ ∗0 mesons.
As g f increases an interesting interplay between the pole
W =
This is a reasonable interval as explicitly shown in Fig. 7,
where several spectral density functions are shown for
increasing g f , from 0.1g f 0 up to 2g f 0, with γV and E F fixed.
The left panel is for case 2.I and the right one for case 2.II. In
the decoupling limit the spectral density is strongly peaked
and becomes more diluted as g f increases. The value of the
integral W is interpreted as the bare component in the
resonance composition, and we obtain the values of W = 0.38
for case 2.I and W = 0.75 for case 2.II. This result is in line
with our previous conclusion based on the value of MCDD,
movements arises reflecting the coupling between the bare
state and the continuum channel. For the fit of case 2.I one
has the central values g f 0 = 0.080, γV = 323 MeV and
E f = 0.63 → ± 2μE f = ±35 MeV. Its pole
trajectories, shown in the two top panels of Fig. 6, are obtained by
increasing g f from one tenth of the fitted value up to 10 times
it. In the left panel we show the global picture, including the
far away bound state, while in the right panel we show a finer
detail of the two near-threshold poles that stem from the bare
state, which for g f = g f 0 become degenerate. For case 2.II
we have the central values g f 0 = 0.039, γV = −128.6 MeV
and E f = 0.32 → ± 2μE f = ±24.7 MeV. The three
virtual-state poles, 2 from the bare state and another from
the direct interactions between the D0 D¯ ∗0 mesons, become
degenerate for g f = g f 0 and the triplet pole arises.
Compared with the pole trajectories explicitly shown in Ref. [11]
ours correspond to a much larger absolute value of γV than
those in Ref. [11] with |γV | between 20-55 MeV. There is no
pole trajectory with three poles merging neither in Ref. [11]
nor in Ref. [45].
Due to the relationship between a near-threshold CDD
pole and a bare state weakly coupled to the continuum (as
exemplified explicitly in the third expression of Eq. (47)) one
expects that for case 2.I the virtual-state pole has mostly a
dynamical origin while for case 2.II, with a much smaller
|MCDD|, one anticipates an important bare component. This
expectation can be put in a more quantitative basis by using
the spectral density function ω(E ) as introduced in Ref. [7],
which reflects the amount of the continuum spectrum in the
bare state. For the dynamical model of Ref. [11] the spectral
function can be calculated and reads
with θ (x ) the Heaviside function. We have used the
prescription argued in Ref. [7], so that the spectral density function
is integrated only along the X (3872) signal region, taken as
1 MeV above threshold, (74) (75)
since for the former case the D0 D¯ ∗0 component is dominant
(around a 60%) while for the latter is much smaller (around
a 25%). For different g f we also give in the legends of the
panels of Fig. 7 the resulting value of W , that increases as
g f decreases because the bare component is larger then. In
the limit g f → 1 W tends to 1, as it should.
We now discuss until the end of this section the situation
used to actually fit data with ∗ = 0. As already discussed
above for case 2.I we find two near-threshold virtual-state
poles in the second RS and one deep bound state in the first
RS. The latter is driven by the large negative value of β, so
that k ≈ −iβ.10 For the triplet case all poles lie close to
the threshold. Let us recall that the pole positions are given
in the second column of Table 2. Contrary to case 1 their
imaginary parts do not coincide with ∗/2 because of the
energy dependence of the CDD pole entering in d(E ). One
can observe that the imaginary parts of the pole positions for
case 2.I are much larger in absolute value than ∗/2, and
10 Of course, the deep bound state is out of the region of validity of
our approach and it is just referred for illustrative purposes.
Virtual-Resonance
Bound state
Virtual-Resonance
Fig. 6 Evolution of the poles as g f increases, with E f and γV fixed. The top panels are for case 2.I, the bottom-left panel for case 2.II and the
bottom-right one for case 3.II. In the right top panel a detail of the near-threshold region is given for case 2.I
that for case 2.II they are even larger than for case 2.I. This
noticeable fact is due to the dependence on
which shows a striking non-analytic behavior because of the
higher order of the virtual-state pole in the limit ∗ → 0,
and corrections to the pole positions are controlled by ρ1/n
with n = 2 and 3 for the double and triplet virtual-state
poles, respectively. This implies that these corrections are
significantly larger than expected as the order of the pole
increases. Of course, this is exemplified by the given splitting
in the pole positions for the double and triplet poles (being
correspondingly larger for the latter). The dependence of the
pole positions with ρ is worked out explicitly in Appendix B
and we give here the final results.
For case 2.I we can simplify formulas by taking into
account that |MCDD| |E R |. The poles are located at
k1,2 = − i
0 0
0 0
Fig. 7 Spectral density function for case 2.I (left) and case 2.II (right panel) for several values of g f , with E f and γV fixed. The value of W is
also indicated in the legends for each entry of g f
with = √2μ|E R |. Their positions in the energy plane,
E = k2/2μ − i ∗/2, are
1
E1,2 =E R 1 ± ρ 2 (1 − i ) ,
k2 = − i
k3 = − i
1 + 2
√
For the triplet virtual-state pole in case 2.II, we have the pole
positions
which imply the energies
1
E1 =E R 1 − i 2ρ 3 ,
E2 =E R 1 + 2
√
E3 =E R 1 − 2
1
Higher orders in ρ 3 have been neglected in Eqs. (79) and
(80)
As noticed above, because of this non-analytic behavior
in ρ, the imaginary parts for the pole positions in energy,
except for E3 in case 2.I which is just a simple pole in the
limit ∗ = 0, are much larger in absolute value than a naive
estimation from the width of the constituent D∗0. In
particular, one can immediately deduce from Eq. (78) that the
imaginary parts have opposite signs for the poles E1,2 of
case 2.I. For case 2.II it follows from Eq. (80) that the pole
at E1 has a positive imaginary part while the latter is
negative for both poles at E2 and E3. As far as we know this
is the first time that it is noticed such non-analytic behavior
of the pole positions in the width of one of its constituents
for higher degree poles. Of course, this might have important
phenomenological implications. In particular, for our present
analysis it favors extension of the virtual-state signal to
energies above the D0 D¯ ∗0 threshold, because it increases the
overlapping with the D∗0 Lorentzian in Eq. (59). Within other
context, non-analyticities of the pole positions as a function
of a strength parameter near a two-body threshold around the
point where the two conjugated poles meet have been derived
in Refs. [45,46]. Similar behavior has also been found as a
function of quark masses for chiral extrapolations [47–50].
The residues of the poles, given in the fourth column of
Table 2, are very large for case 2.I and huge for case 2.II. The
point is that they are affected by the extra singularity coming
from the other coalescing poles in the limit ∗ → 0+. For
the virtual-state cases one cannot interpret the normalization
constants YF as yields because the virtual-state pole is below
threshold in the second RS and then it is blocked by the
threshold branch-point singularity, so that it does not directly
influence the physical axis for E < 0. This also manifests in
that the normalization integral N , Eq. (52), is very different
from 1.
6.3 Cases 3: Simultaneous virtual and bound state
In this case we again use the more general
parameterization based on d(E ) and move towards a scenario in which
one finds simultaneously a bound-state pole in the first RS
and a virtual-state one in the second RS. To end with such
a situation we impose the requirement that in the isospin
limit there is a double virtual-state pole independently of the
common masses taken for the isospin multiplets (either the
masses of the neutral or charged isospin D(∗) members). This
is a way to enforce a weak coupling of the bare states with
the continuum, have poles in different RS’s and end with a
bound state with small compositeness (or large
elementariness). At this point we adapt, as an intermediate step to end
with our elastic D0 D¯ ∗0 S wave, the main ideas developed
in Ref. [31]. This reference takes into account the
coupledchannel structure of π + c0, π − c++ and π 0 c+ in relation
with the c(2595)+ resonance, where the symbol c actually
refers to the c(2455) [19]. However, the resulting
expression reduces to that of Eq. (36) for single coupled-scattering
since we focus on the X (3872) signal region around the
D0 D¯ ∗0 threshold, because of the same reasons as already
discussed when matching our results with those of Ref. [12]
in the last part of Sect. 4. Of course, these considerations
translate into a different dependence of λ and β on E R than
in cases 2.I–II analyzed in Sect. 6.2.
The basic strategy is the following: (i) We take the isospin
limit for D0 D¯ ∗0 and D+ D∗− coupled-channel scattering,
with masses equal to either those of the neutral particles for
each isospin doublet ( D0 and D∗0) or to the charged ones
( D+ and D∗+). At this early stage the zero width limit for
the D∗0 is taken. (ii) For every isospin limit defined in (i) we
impose having the same virtual-state pole position located at
E R − i G R /2, taking the limit G R → 0+ at the end, similarly
as done in Sect. 6.2 to end with less free parameters in a more
restricted situation. (iii) The previous point provides us with
four equations that are used to fix two of the three parameters
in d(E ), Eq. (41), namely β and λ.11 The remaining third
parameter, MCDD, is also fixed by imposing that d(E )−1
vanishes in the first RS below threshold at E R . (iv) In this
way all the parameters specifying d(E ) are given in terms
of E R , which is finally fitted to the data once the finite D∗0
width is restored in the definition of the three-momentum,
cf. Eq. (37).
The formulas derived to actually fix β, λ and MCDD are
given in Appendix C, Eqs. (C.3), (C.4), (C.7) and (C.8). There
it is shown that indeed one has two solutions, that we
indicate by case 3.I (first solution) and case 3.II (second solution).
The expressions simplify in the limit |E R |/ → 0, which is
relevant for X (3872) given its small energy, and the two
solutions coalesce in just one. In this case we show that there are
two poles in different RS’s, confirming the intuitive physical
reasons given at the beginning of this section.
The values for the fitted parameters are given in Eq. (65)
for case 3.I and in Eq. (66) for case 3.II. The resulting
event distributions are shown by the blue dash-double-dotted
(case 3.I) and green dash-dotted (case 3.II) lines in Fig. 4 and
in the histogram of Fig. 5. These lines are hardly
distinguish11 The other two equations give the values for the CDD pole positions
in the two isospin limits considered. This is information which is of no
use.
able among them and can only be differentiated with respect
to case 1 in the D0 D¯ ∗0 event distribution, as one can
appreciate clearly from Fig. 5. With respect to cases 2.I–II we have
the already commented shift of the peak in the J /ψ π +π −
event distributions, more clearly seen in Fig. 9. The global
reproduction of data is of similar quality as the one already
achieved by the pure bound-state and virtual-state cases. The
values for the pole positions and CDD parameters are given
in Tables 2 and 3, respectively.
For case 3.I the resulting pole position for the bound state
(the pole in the first RS) is −0.50−+00..0034 MeV, while the pole
position for the virtual state (the pole in the second RS) is
−0.68+−00..0053 MeV. In both cases there is a tiny imaginary part
of the order of 10−3 MeV which is beyond the precision
shown. The compositeness of the D0 D¯ ∗0 state in the bound
state, evaluated in the same way as explained at the end of
Sect. 6.1, is 0.06, i.e., the D0 D¯ ∗0 component only constitutes
around a 6% of the X (3872) due to the extreme proximity of
the CDD pole to the D0 D¯ ∗0 threshold. As shown in Table 3
the CDD pole is much closer to threshold than the bound-state
pole. As a result, other components are dominant, e.g. one
could think of the conventional χc1(2 P ) as cc¯, tetraquarks,
hybrids, etc. [4, 33, 51–56]. These facts about the smallness
of the imaginary part of the two near-threshold poles and the
small compositeness for the bound state can be understood
in algebraic terms in the limit |E R |/ → 0 as shown in
Appendix C; cf. Eqs. (C.18), (C.21) and (C.24). Indeed, they
are related because if the X (3872) has such a small value for
the compositeness, then it is fairly insensitive to the width
of D∗0. In addition, one also has a deep virtual state located
at E3 ≈ −β2/2μ − i ∗/2, that is quite insensitive to the
CDD pole contribution, which is strongly suppressed at those
energies as explained in more detail after Eq. (C.23).
Let us notice that the ERE for the present near-threshold
bound state fails because ERE is not applicable since the
zero is closer to threshold than the pole. Taking ∗ = 0 and
calculating a and r we obtain the central values (errors are
given in Table 3)
a = 0.27 fm → 0, r = −847.8 fm → ∞, (81)
while the binding momentum is κ = √2μ|E X | = 31.0 MeV
<< 1/|a| = 723 MeV. The pole position in the first (second)
RS that stems from the ERE up to the effective range is −0.17
(−0.18) MeV, which is indeed very different from the actual
pole position of the bound (virtual) state in the full amplitude
at −0.50 (−0.68) MeV.
For the second solution, i.e. case 3.II, we have much larger
values of λ and MCDD than for the first one, compare between
the last two rows in Table 3. This is a common characteristic
to any value E R < 0 as shown in Fig. 8, where the values of
λ (left panel) and MCDD (right panel) are given as a function
of E R for the first (black solid) and second (red dashed lines)
solutions.
The pole positions in the first and second RS’s are given
in the second column of the last line of Table 2. The fact that
for this second solution MCDD is further away from threshold
than for the solution case 3.I is an indication that
compositeness is larger for the former than for the latter. For case 3.II we
obtain now X = 0.16, while before it was around 0.06. This
is in agreement with our expectations, but still X is small
and the state is dominantly a bare (non-molecular) one.12
The residues for this case are also given in column four of
Table 2. They are larger by around a factor 3 compared to the
first solution, which is in line with the increase in the value of
compositeness. The bound states for cases 3.I–II have a
normalization integral N = 1 so that it is legitimate to interpret
the YF as yields.
The ERE expansion for case 3.II is better behaved because
MCDD is relatively further from threshold. We now have the
values ( ∗ = 0 should be understood in the following
discussions)
a = +1.57+0.05 fm,
−0.02
r = −43.75+0.24 fm,
−0.61
where r is still much larger than a typical range of strong
interactions and a is much smaller than 1/√2μ|E X |. These
facts just reflect the dominant bare nature of X (3872) in this
case. The ERE up to r gives rise to a bound state located at
−0.45 MeV, already very close to the full-solution result at
−0.51 MeV, which is much better than for case 3.I.
Regarding the virtual-state pole the ERE also produces a pole in
the second RS at −0.76 MeV, while the full result is at
−1.06 MeV, around a 25% of error. This worse behavior
12 Taking the same value for the residue of D+ D∗− as for D0 D¯ ∗0,
because of isospin symmetry, we can evaluate straightforwardly the
compositeness of the D+ D∗− channel and is a factor 3 smaller than for
D0 D¯ ∗0. Then, summing it to 0.15, we end with 0.20 as an estimated
value for the total compositeness of the D D¯ ∗ states, which is still small.
For case 3.I it would be around only 0.05.
of the ERE to determine the location of the virtual-state pole
is to be expected because the radius of convergence of the
ERE is 2μMCDD, and the virtual-state pole is closer to this
limit than the bound-state one. More contributions are
certainly needed as the ERE is applied to energies that are closer
to the radius of convergence of the expansion.
Related to this discussion we consider the Weinberg’s
compositeness theorem for a near-threshold bound state,
which reads [10]
where Z is the elementariness, or 1 − X . This criterion, as
discussed in the Introduction, cannot be applied if a CDD is
closer to threshold than the bound-state pole, as it happens
for case 3.I, because it relies on the applicability of the ERE
up to the effective range. For case 3.II this is not the case, but
still the CDD pole is quite close so that energy dependences
beyond the effective range play a role. The Weinberg’s
compositeness relation gives Za = 0.86 and Zr = 0.87, when
using a and r to calculate it from the first and second
expressions in Eq. (84), respectively. These numbers compare very
well with Z = 1 − X = 0.84, where X = 0.16 is determined
above and given in Table 2. These values of elementariness
so close to 1 for cases 3.I–II are also in agreement with the
expectation of having two poles close to threshold in adjacent
RS’s (virtual- and bound-state poles simultaneously), which
fits very well within Morgan’s criterion for a preexisting or
non-molecular state [57].
Our fit for case 3.II corresponds to the following central
values for the parameters characterizing the scattering model
of Ref. [11], in terms of the exchange of a bare state and direct
scattering between D0 D¯ ∗0: g f 0 = 0.014, γV = −561.7 and
E f = −0.71 MeV. Compared with cases 2.I–II one observes
a significant smaller value for g f 0 and |E f |. The resulting
pole trajectories as g f is increased from 0.1g f 0 up to 20g f 0,
with EV and γV held fixed, are shown in the bottom-right
panel of Fig. 6, with a similar behavior for case 3.I, which
is not shown. Notice that because E f < 0 one has in the
decoupling limit (g f → 0) a bound- and a virtual-state pole
at ±i 2μ|E f | = 33.2 MeV. This type of pole movement as
a strength parameter varies is different from those discussed
in Ref. [45], because the near-threshold poles do not belong
to the trajectory of two complex poles associated with the
same resonance. However, this is the case for the two virtual
state poles, the shallow and deep ones, as clearly seen in the
figure.
The poles trajectories in the last panel of Fig. 6 do not
belong either to the ones discussed explicitly in Ref. [11],
where much smaller values of |γV | are considered. The
reason behind is the misused performed in this reference of the
relationship between the pole positions ki , i = 1, 2, 3 and
the position of the CDD pole13
The point is that Ref. [11] concluded from this equation that
it is necessary that the three poles be shallow ones (|ki | )
in order to have a near-threshold CDD pole (|MCDD| ).
However, this conclusion is just sufficient but not necessary.
The other possibility is that k1 and k2 nearly cancel each other
(such that |k1 +k2| = O(|k1,2/ k3|2), without being necessary
that |k3| (which in our case is given by β ). This
is what happens particularly for case 3.I with k1 = i 31.0 and
k2 = −i 36.2 MeV, so that the CDD pole is almost on top
of threshold. Therefore, one does not really need that three
poles lie very close to threshold to end with a shallow CDD
pole.
It is also interesting to apply the spectral density
introduced in Eq. (74) to evaluate the compositeness and
elementariness of the bound states in cases 3.I–II. For such a purpose
one has to integrate the spectral density up to infinity with
W defined as
W =
and interpreted as the compositeness X [7]. The
normalization to 1 of the bare state then guarantees that Z = 1 − X ,
which provides us with the elementariness. Notice that this is
the third way that we have introduced to evaluate the
compositeness of a bound state. Namely, we can evaluate it in terms
of the residue of t (E ) at the pole position, Eq. (67),
Weinberg’s relations, Eq. (84), or in terms of the spectral density,
13 Similarly, one can also derive the equalities λ = −i (k32(k1 + k2) +
k3(k1 + k2)2 + k1k2(k1 + k2))/2μ and β = i i ki . These relations can
easily be worked out from the secular equation which is a third-order
polynomial.
Fig. 9 The curves of the different cases for the J /ψ π +π − event
distribution from the inclusive p p¯ scattering [18] are given with error
bands included. The black-, red-, brown-, blue-, and green-filled bands
correspond to cases 1, 2.I, 2.II, 3.I and 3.II, in order
Eq. (86). The latter also provides remarkably close values to
the previous ones so that for case 3.II one has W = 0.16,
while for case 3.I (in which case Weinberg’s result does not
apply) one obtains W = 0.06.
We have also checked that our results are stable if the
D+ D∗− channel is explicitly included in d(E ) by using
the same formalism as in Ref. [31]. At the practical level
this amounts to modifying the denominator of d(E ) such
that β → β − i [k(2)(E ) − k(2)(0)], with k(2)(E ) the
D+ D∗− three-momentum given by the expression k(2)(E ) =
2μ2(E + i ∗/2 − ), where μ2 is the reduced mass and
∗ the width of the D∗+ resonance [19], while k(2)(0) is
given by the same expression with E + i ∗/2 → 0. For
example, by redoing the fit in this case the fitted parameters
match very well the values in Eq. (65) within errors.
In order to have an extra perception on how the uncertainty
in the fitted parameters influences our results, we also show in
Fig. 9 the error bands of the curves obtained from the different
cases considered for the reproduction of the CDF data on
the inclusive p p¯ scattering to J /ψ π +π − [18]. We have not
shown the error bands for the other data, and just shown the
curves obtained from the central values of the fit parameters
in Fig. 4, because the typical width for every error band in
each line is of similar size as the one shown in Fig. 9. We
have chosen this data because it is the one with the smallest
relative errors, having the largest statistics and smallest bin
width. In addition, the curves are so close to each other that
in the scale of the Fig. 4 it would be nearly impossible to
distinguish between all the curves with additional error bands
included. Here we offer just one panel which also allows us
to use a larger size for it and be able to distinguish better
between lines with error bands. But even then one clearly
sees in Fig. 9 that the bands for cases 1 and 3.I–II mostly
overlap each other so that they are hardly distinguishable. The
cases 2.I–II can be differentiated from the rest because there
is a slight shift of the peak structure to the right. However, this
shift becomes smaller and all the bands overlap each other
if we had excluded in the fit the D0 D¯ ∗0 event distribution of
the BaBar Collaboration [15].
7 Conclusions
Since its exciting discovery [58] X (3872) has been
extensively studied, for a recent review see Ref. [59]. Among
the many theoretical approaches [1–6, 33, 51–56, 60–63], we
have paid special attention to the applicability of the popular
ERE approximation up to and including the effective-range
contribution to study near-threshold states like X (3872). We
have elaborated about the fact that the ERE convergence
radius might be severely limited due to the presence of
nearthreshold zeros of the partial wave, the so-called Castillejo–
Dalitz–Dyson poles. We have then derived a
parameterization that is more general than the ERE up to and including
effective range,14 but it can deal as well with the presence
of a CDD pole arbitrarily close to threshold. We have shown
too that other parameterizations based on the picture of the
exchange of a bare state plus direct interactions between
the D0 D¯ ∗0 can also be matched into our parameterization
[11, 12]. In particular, Ref. [11] already stressed the strong
impact that a possible near-threshold zero would have in the
D0 D¯ ∗0 S-wave amplitude. However, we have shown that the
conclusion there stated about the necessity of three
simultaneous shallow poles to end with a near-threshold zero is
sufficient but not necessary, because it is just enough having two
such poles. We have illustrated this conclusion with a
possible scenario for X (3872) in which there are only two
nearthreshold poles, a bound state in the first RS and a virtual-state
one in the second RS.
We have then reproduced several event distributions
around the D0 D¯ ∗0 threshold including those of D0 D¯ ∗0
and J /ψ π +π − from charged B decays measured by the
BaBar and Belle Collaborations and the higher-statistics
CDF J /ψ π +π − event distributions from inclusive p p¯
scattering at √s = 1.96 TeV. Our formalism has as limiting
cases those of Refs. [1, 3], but it can also include other cases
in which the presence of a CDD pole plays an important
role. In this respect we are able to find other interesting
scenarios beyond those found in Refs. [1, 3] that can reproduce
data fairly well, and without increasing the number of free
parameters. In two of these new situations X (3872) is
simultaneously a bound and a virtual state, while in others the
X (3872) is a double or a triplet virtual-state pole. In the limit
14 Indeed up to the next shape parameter, v2. Let us recall again that
the ERE is more general than a Flatté parameterization.
of vanishing width of D∗0 these poles become degenerate and
produce a higher-order pole (of second or third order). Thus,
our parameterization constitutes in the latter cases a simple
example for higher-order S-matrix poles that could have a
clear impact on particle physics phenomenology.
In this respect, we stress that the corrections to the pole
position when taking into account the finite width of the D∗0
resonance, ∗, are non-analytic for the higher-order poles
of order n > 1. In such situations one has that the
leading corrections are proportional to ρ1/n , with ρ = ∗/|E R |,
being E R the real part of the pole position with respect to the
D0 D¯ ∗0 threshold without the D∗0 width. This could be an
important source of D0 D¯ ∗0 partial width for the X (3872).
Indeed, with this mechanism the absolute value of twice the
imaginary part of the pole positions for the triplet-pole
scenario could be nearly as large as 1 MeV, despite ∗ being only
around 0.065 keV [2, 19]. Thus a measurement of the total
width of X (3872) might be useful to discriminate between
the discussed scenarios.
Further, while the compositeness is equal to 1 for the
bound-state case analyzed making use of the ERE
including only the scattering length [3], it is nearly zero for cases
3.I–II in which X (3872) is a simultaneous virtual and bound
state. Case 3.I has the closest CDD pole to threshold, even
closer than the pole positions. In this respect, we also estimate
that the X (3872) is mostly D0 D¯ ∗0 for the double
virtualstate case, because the CDD pole is relatively far away from
threshold, while in the triplet-pole case the elementariness is
dominant as indicated by the closeness of the CDD to
threshold. We have verified quantitative these conclusions as well
by employing the spectral density function.
From another perspective, we have shown that using a
more refined treatment of D0 D¯ ∗0 scattering X (3872) can be
a bound-state, a double/triplet virtual-state pole or two types
of simultaneous virtual and bound states with poles
occurring in both the physical and unphysical sheets, respectively.
All these scenarios can give a rather acceptable
reproduction of the experimentally measured event distributions. Up
to some extent this situation recalls the case of X (1835), for
which the energy-dependent J /ψ → γ p p¯ event
distribution is nicely reproduced by purely final-state interactions
of p p¯ [64]. However, this treatment fails to describe the
data when a more elaborate model is taken. Only the
generation of a p p¯ bound state in the scattering amplitude is able
to reproduce the data within this more sophisticated model
[65–67].
From our present results and this experience, more efforts
are still needed to finally unveil the nature of the acclaimed
X (3872), which is the first X Y Z state observed. In this
respect, we mention that there are visible differences between
the different scenarios analyzed in the D0 D¯ ∗0 invariant mass
distributions in the peak of X (3872), as shown in detail in
Fig. 5, in particular between the scenarios I, 2.II and the rest.
We have to indicate that the present data shows a clear
displacement towards higher masses of the X (3872) peak in the
BaBar Collaboration data [15] as compared with the Belle
Collaboration one [17]. Indeed, if the former data is excluded
in the fits the shift towards the right of the signal peak for
cases 2.I–II in the J /ψ π +π − CDF data [18], cf. Fig. 9,
diminishes considerably. Thus, a future high-statistic
experiment on B → K D0 D¯ ∗0 might be very helpful to
differentiate between different cases, if complemented with
highprecision data on J / π +π −. Another way to discriminate
between different possibilities might be the measurement of
the partial-decay width of X (3872) to D0 D¯ ∗0, as mentioned
above. Lattice QCD can also provide interesting information
from where one could deduce the D0 D¯ 0∗ near-threshold
scattering amplitude and then determine whether there is a CDD
pole or not. Indeed, present Lattice QCD results point towards
the importance of the interplay between quark and meson
degrees of freedom to generate X (3872) [68–70]. Another
interesting idea was put forward by Voloshin in Ref. [21],
indicating the convenience to measure the D0 D¯ ∗0π 0 Dalitz
plot to distinguish between the molecular and quarkonium
picture for X (3872).
Acknowledgements We would like to thank C. Hanhart for inspiring
and compelling discussions. Interesting discussions are also
acknowledged to Q. Zhao. This work is supported in part by the MINECO
(Spain) and ERDF (European Commission) Grant FPA2013-40483-P
and the Spanish Excellence Network on Hadronic Physics
FIS201457026-REDT.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: Obtaining properly normalized yields
In order to fit the event distributions when using d(E ) with
a CDD included, the evaluation of α entering in Eq. (51) for
calculating the event distributions requires one to work out
the pole position, cf. Eq. (43), which is not easily expressed
in terms of free parameters. This is not the case when using
the function f (E ) because in this case α, Eq. (45), is given
directly in terms of γ , a fit parameter. Thus, when d(E )
is employed we first perform the fits such that |α|2 is
reabsorbed in the normalization constants multiplying the
signal contribution. In this way one avoids having to calculate
the pole position for each iteration in the fit procedure. To be
specific, we use the following expression for the fits to the
D0 D¯ ∗0 event distributions:
d E R(E , E )√E
Ni (Ei ) =
Ni (Ei ) = ϑJ
where ϑJ,D = 1 for BaBar data and for Belle it corresponds
to the ratio of the number of B B¯ pairs produced in Belle and
BaBar for each type of B+ → J /ψ π +π − decays, namely,
N Belle/N BaBar. The number of B B¯ pairs is given in Table 1.
B B¯ B B¯
Once the fit is performed we can deduce the values of the
“yields” YD and YJ (here the quotation marks are introduced
because it is required that N 1, with the normalization
constant N introduced in Eq. (52), in order to interpret
meaningfully these constants as yields). The appropriate relations can
be deduced by comparing Eqs. (59) and (54) with Eqs. (A.1)
and (A.2), in order. They read
E X +
Ei − /2
YD =bD
cbgJ =NBBaB¯B;aJrcbgJ ,
cbgD =NBBaB¯B;aDrcbgD.
The pole position E X − i X /2 and associated momentum
k P are determined from the fitted values of the parameters.
We also have the trivial relations between the background
parameters
An analogous procedure is also applied when fitting the
J /ψ π +π − event distribution from the inclusive p p¯
scattering measured by the CDF Collaboration [18]. In this way, we
can extract YJ( p) by applying Eq. (A.3) too.
Appendix B: Pole positions with finite
∗ in cases 2.I, II
For ∗ = 0 the relation between energy and momentum is
E = k2/2μ − i ∗/2, cf. Eq. (37), and the secular equation
(71) becomes
After some simplifications it can be written as
(k + i )2 k − i
= 0,
with = √2μ|E R |.
For case 2.I |MCDD| |E R | and the previous equation
becomes in good approximation
(k + iβ)((k + i )2 − i μ ∗) = 0.
Its solution gives rise to the pole positions in the k plane given
in Eq. (77) and in the E plane are those of Eq. (78).
For case 2.II, MCDD = −3E R and Eq. (B.2) becomes
(k + i )3 − i μ ∗(k + i 3 ) = 0.
In terms of the dimensionless variable
t =
Eq. (B.4) reads
This equation can be solved in a power expansion of ρ with
the leading result
In the momentum and energy variables this solution gives
rise to Eqs. (79) and (80), in order.
Appendix C: Formalism for cases 3.I–II: simultaneous
virtual- and bound-state poles
Let us connect with the T -matrix derived in a previous paper
by one of the authors in which the resonance c(2535)+
was studied [31]. This is a resonance that also lies very
close to the π c thresholds. Namely, it has a small width
of 2.6 ± 0.6 MeV [19] and its mass is 4.37 MeV above the
π 0 c+ threshold, and 1.06 and 1.30 MeV below the nearly
degenerate thresholds of π + c0 and π − c++, respectively.
Reference [31] explored the viability of this resonance to be
a preexisting one, with Z 1, so that its actual pole position
is unaffected by taking as masses in the isospin limit those
in the π 0 c+ or in the π − c++ states.15
15 Contrarily to the more general expectations of Ref. [31] the
parameter λ could depend strongly on the isospin mass taken.
Here we have adopted a similar point of view and required
the invariance of the virtual-state pole position independently
of whether one takes in the isospin limit the masses of the
neutral mesons D0, D∗0 or of the charged ones D+, D∗+.
We denote the channel D0 D¯ ∗0 by 1 and the channel made by
the charged particles D+ D∗− by 2. In the isospin limit, we
replace λ by λ in Eq. (36) with
because of an isospin Clebsch–Gordan coefficient squared to
combine D D¯ ∗ in isospin 0. Specifically, in the isospin limit
we use the scattering amplitude
t j (√s) =
√s − M C(jD)D
−1
where s is the usual Mandelstam variable and the subscript
in t j (√s) refers to take the isospin limit with the masses of
the state j . For each case the CDD pole position is indicated
by M C(jD)D, and k( j)(√s) = 2μ j (√s − σ j ), with μ j the
reduced mass and σ j the threshold mass of the j th D D¯ ∗
state. In addition, following Ref. [31], we have
+ M D(j∗) log
M D(j∗)
The previous equation results by taking the non-relativistic
reduction of the unitarity loop function for the s-channel
intermediate state. We have also used quite an obvious
notation for the masses involved. The real and imaginary
parts of the momentum at the resonance in the second RS
are denoted by kr( j) and −ki( j) and can be calculated from
kr( j) − i ki( j) ≡ − 2μ j (MR − i R /2 − σ j ), with the
argument of the radicand taken between [0, 2π [. The
expressions for λ, α and M C(jD)D that result by imposing that each
t j (√s) has a pole in the second RS at MR − i R /2, with
MR = MD0 + MD∗0 + E R fixed can be found in Ref. [31].
We write them here as
R (ki(1) − ρ1) + 2kr(1)(M C(1D)D − MR ) ,
M C(1D)D =
χ21 = MR kr(1)(kr(1)σ 2
2 − kr(2)σ12) +
+ ρ1σ2 + ki(2)σ1 − ρ2σ1)/2,
are of O(δ M/MD), with δ M an isospin splitting mass in the
D(∗) multiplets. The resulting simplified expression is
E R ± 2α
√|E R |(
− E R ) + 3E R /2
The + applies to the first solution and the − to the second.
Substituting Eq. (C.9) in Eq. (C.5) we have the following
expressions for λ and β1:
E R + |E R |(
− E R ) ,
where R = 1 and 2 = √2μ( − E R ). From Eqs. (C.9)
and (C.11) we also have an explicit expression for MCDD,
−1
8μα2 E R + √|E R|( − E R) 2 .
+ 4μ(1 ± α) −E R ± α√|E R|( − E R) + R(1 ± α)2( R + 2)
(C.12)
It is interesting to consider the limit α → ∞ because it
is relevant for X (3872) given the fact that |E R | and
Eqs. (C.9), (C.11) and (C.12) largely simplify. In this limit
there is only one solution which is given by
− 2σ1σ2(ki(2) − ρ2)(ki(1) − ρ1)
+ σ12[kr(2)(kr(2) − kr(1)) + (ki(2) − ρ2)2]},
These equations provide us with two different solutions,
which stem from the ± sign in the expression for M C(1D)D.
We refer to them as the first and second solutions.
In the limit R → 0+ we end with similar expressions for
λ and β1 as Eq. (70),
β1 = μ1 (M C(1D)D − 3E R ) → α = 8π σ1 − ρ1,
β1
1
with 1 = √2μ1|E R |.
The coupled-channel S-wave amplitude for channels 1
and 2, using again the correspondingly adapted expression
of Ref. [31], reads
t (√s) =
+ β1 + β2 − ik(2)(√s) − ik(1)(√s)
In this formula one implicitly assumes that the main
isospin breaking corrections between the different
coupled channels are expected to arise from the dependence
of the three-momenta k(i) on their threshold because of
the associated branch point singularity at each nearby
threshold [31].
The parameters λ, β1 and β2, cf. Eq. (C.3), are fixed here
from Eq. (C.4) in terms of E R . We still have to determine
MCDD, which is fixed by requiring that t (√s) have a
boundstate pole (in the first RS) at √s = MR ,
MCDD = MR − 2λ/(β1 + β2 − i k(2)(MR ) − i k(1)(MR )).
In this way, the parameters to be employed in Eq. (36) for
case 3 introduced in Sect. 6.3 are:
β = β1 + β2 − i k(2)(σ1).
Notice that the three-momentum of the channel 2 has been
frozen at its value at the D0 D¯ ∗0 threshold because the
X (3872) signal happens around σ1 within an energy region
|E | . As commented above we have checked that our
results are stable if releasing it as in Eq. (C.6).
One can obtain an accurate numerical approximation to
the exact expression for M C(1D)D in Eq. (C.4) (in the limit of
R → 0+) if isospin breaking corrections in μi and βi are
neglected (that are set equal to μ1 and β1). These corrections
M C(1D)D =
MCDD = E R
We can also see that in this limit there is a virtual state
in the second RS, with similar energy as the bound state
imposed by construction. Since R is a root, and writing the
three-momentum of the new solution as i 2, we have from
the secular equation the still exact relation
2 = − R −
Now, implementing in this equation the values for the
constants obtained in Eq. (C.13) we simply have, for α → ∞,
Equation (C.14) considered for values of 22 much larger than
2μMCDD also implies that the third solution in this limit is
3 = −β.
To end this appendix let us discuss for α → ∞ how the
poles move when including the finite width of D∗0, that is,
with ∗ = 0. First, because of the condition imposed to
guarantee the presence of the bound state with ∗ = 0, one
can rewrite λ as λ = ( R2 /2μ + MCDD)(β + R ). The secular
equation to calculate its final pole position at i B is then
In the limit α → ∞ we can neglect R,B in front of β and
the previous equation takes us to the solution
B =
2 R
The corresponding energy E B is
E B = − 2μ − i 2∗ = − 2μ = E R ,
with quadratic terms in ∗ neglected both in Eqs. (C.17) and
(C.18).
Let us move to calculate the pole position of the
nearthreshold virtual state. Instead of Eq. (C.14) we now have
the exact relation,
The dominant contribution to the imaginary part stems from
the second term on the right-hand side of the previous
equation since β R . We then have
The associated energy E2 is
E R + i ∗ ,
where quadratic term in ∗ have been neglected. Notice how
Eqs. (C.18) and (C.21) imply a much smaller imaginary part
in absolute value for E B and E2 than the half of the width of
the constituent D∗0, in agreement with the numerical results
reported in Sect. 6.3.
For the deep virtual-state pole we consider again Eq. (C.19)
and neglect R and √2μ|MCDD| in front of and β (as
3 ≈ −β). We then have the following equation for the
solution 3 that gives rise to the leading contribution to its
imaginary part:
1
3 + β
= 0.
Neglecting quadratic terms in ∗ the imaginary part in 3
cancels and we obtain again the same result as above with
∗ = 0, 3 = −β. Its energy E3 is
and its width is just determined by that of its constituent D∗0.
This result is as expected because this pole is a deep one
that stems from the direct D0 D¯ ∗0 scattering, since at those
energies the CDD pole contributions is negligible compared
to β as λ/Eβ 2μλ/β3 ∝ √|E R |/ and tends to zero.
This is not the case for the lighter poles because they are
associated with the bare state with a small component of
D0 D¯ ∗0.
Indeed one can easily calculate the residue of t (E ) in the
variable k at the bound-state pole position for ∗ = 0 and
α → ∞ and apply Eq. (67). The following limit result is
obtained:
of similar size as those reported in Sect. 6.3. This finite small
0nubmutbtehreisqrueolatiteendtto|EthRe|/faMctCtDhDat =for3|/E4R√| E→R /0 also MCDD →
→ 0, so that
in relative terms the binding energy is much closer to zero
than MCDD for the limit α → ∞.
It is also worth remarking that for all the three poles the
corrections in their pole positions as a function of ∗ are
analytic because they are simple poles (isolated singularities).
1. C. Hanhart , Y.S. Kalashnikova , A.E. Kudryavtsev , A.V. Nefediev , Phys. Rev. D 76 , 034007 ( 2007 ). arXiv: 0704 .0605 [hep-ph]
2. E. Braaten , M. Lu , Phys. Rev. D 76 , 094028 ( 2007 ). arXiv: 0709 .2697 [hep-ph]
3. E. Braaten , J. Stapleton , Phys. Rev. D 81 , 014019 ( 2010 ). arXiv: 0907 .3167 [hep-ph]
4. O. Zhang , C. Meng , H.Q. Zheng , Phys. Lett . B 680 , 453 ( 2009 ). arXiv: 0901 .1553 [hep-ph]
5. Y.S. Kalashnikova , A.V. Nefediev , Phys. Rev. D 80 , 074004 ( 2009 ). arXiv: 0907 .4901 [hep-ph]
6. G.Y. Chen , W.S. Huo , Q. Zhao , Chin. Phys . C 39 ( 9 ), 093101 ( 2015 ). arXiv: 1309 .2859 [hep-ph]
7. V. Baru , J. Haidenbauer , C. Hanhart , Y. Kalashnikova , A.E. Kudryavtsev , Phys. Lett . B 586 , 53 ( 2004 ). arXiv:hep-ph/0308129
8. X.W. Kang , Z.H. Guo , J.A. Oller , Phys. Rev . D 94 ( 1 ), 014012 ( 2016 ). arXiv: 1603 .05546 [hep-ph]
9. L. Castillejo , R.H. Dalitz , F.J. Dyson , Phys. Rev . 101 , 453 ( 1956 )
10. S. Weinberg , Phys. Rev . 137 , B672 ( 1965 )
11. V. Baru , C. Hanhart , Y.S. Kalashnikova , A.E. Kudryavtsev , A.V. Nefediev , Eur. Phys. J. A 44 , 93 ( 2010 ). arXiv: 1001 .0369 [hep-ph]
12. P. Artoisenet , E. Braaten , D. Kang , Phys. Rev. D 82 , 014013 ( 2010 ). arXiv: 1005 .2167 [hep-ph]
13. C. Hanhart , Y.S. Kalashnikova , A.V. Nefediev , Eur. Phys. J. A 47 , 101 ( 2011 ). arXiv: 1106 .1185 [hep-ph]
14. B. Aubert et al. ( BaBar Collaboration), Phys. Rev. D 77 , 111101 ( 2008 ). arXiv:0803.2838 [hep-ex]
15. B. Aubert et al. ( BaBar Collaboration), Phys. Rev. D 77 , 011102 ( 2008 ). arXiv:0708.1565 [hep-ex]
16. I. Adachi et al . ( Belle Collaboration), arXiv:0809 .1224 [hep-ex]
17. T. Aushev et al. ( Belle Collaboration), Phys. Rev. D 81 , 031103 ( 2010 ). arXiv:0810.0358 [hep-ex]
18. T. Aaltonen et al. ( CDF Collaboration), Phys. Rev. Lett . 103 , 152001 ( 2009 ). arXiv:0906.5218 [hep-ex]
19. C. Patrignani et al. ( Particle Data Group), Chin. Phys. C 40 , 100001 ( 2016 )
20. E. Braaten , M. Kusunoki , Phys. Rev. D 72 , 054022 ( 2005 ). arXiv:hep-ph/0507163
21. M.B. Voloshin, Phys. Lett . B 579 , 316 ( 2004 ). arXiv:hep-ph/0309307
22. C. Hanhart , Y.S. Kalashnikova , A.V. Nefediev , Phys. Rev. D 81 , 094028 ( 2010 ). arXiv: 1002 .4097 [hep-ph]
23. E.M. Aitala et al. ( E791 Collaboration), Phys. Rev. Lett . 86 , 770 ( 2001 ). arXiv:hep-ex/0007028
24. J.A. Oller, Final state interactions in hadronic D decays . Phys. Rev. D 71 , 054030 ( 2005 ). doi:10.1103/PhysRevD.71.05403. arXiv:hep-ph/0411105
25. J.A. Oller, in Proceedings of the Second Workshop on the CKM Unitarity Triangle , eConf C 0304052, WG412 ( 2003 ). arXiv:hep-ph/0306294
26. D.V. Bugg , Phys. Lett . B 572 , 1 ( 2003 ) [(E) Phys. Lett . B 595 , 556 ( 2004 )]
27. J.A. Oller, Phys. Lett . B 477 , 187 ( 2000 ). arXiv:hep-ph/9908493
28. S. Gardner , U.G. Meißner , Phys. Rev. D 65 , 094004 ( 2002 ). arXiv:hep-ph/0112281
29. R. Aaij et al. ( LHCb Collaboration) , Eur. Phys. J. C 72 , 1972 ( 2012 ). arXiv:1112.5310 [hep-ex]
30. J.A. Oller , E. Oset , Phys. Rev. D 60 , 074023 ( 1999 ). arXiv:hep-ph/9809337
31. Z.H. Guo , J.A. Oller , Phys. Rev . D 93 ( 5 ), 054014 ( 2016 ). arXiv: 1601 .00862 [hep-ph]
32. M.P. Valderrama, Phys. Rev. D 85 , 114037 ( 2012 ). doi:10.1103/ PhysRevD.85.114037. arXiv: 1204 .2400 [hep-ph]
33. M. Suzuki , Phys. Rev. D 72 , 114013 ( 2005 ). arXiv:hep-ph/0508258
34. L. Alvarez-Ruso , J.A. Oller , J.M. Alarcon , Phys. Rev. D 80 , 054011 ( 2009 ). arXiv: 0906 .0222 [hep-ph]
35. L. Alvarez-Ruso , J.A. Oller , J.M. Alarcon , Phys. Rev. D 82 , 094028 ( 2010 ). arXiv: 1007 .4512 [hep-ph]
36. C. Hanhart , Y.S. Kalashnikova , P. Matuschek , R.V. Mizuk , A.V. Nefediev , Q. Wang , Phys. Rev. Lett . 115 (20), 202001 ( 2015 ). arXiv: 1507 .00382 [hep-ph]
37. F.-K. Guo , C. Hanhart , Y.S. Kalashnikova , P. Matuschek , R.V. Mizuk , A.V. Nefediev , Q. Wang , J.-L. Wynen, Phys. Rev. D 93 ( 7 ), 074031 ( 2016 ). arXiv: 1602 .00940 [hep-ph]
38. E. Braaten , M. Lu , Phys. Rev. D 77 , 014029 ( 2008 ). arXiv: 0710 .5482 [hep-ph]
39. F. James , MINUIT-Function Minimization and Error Analysis . CERN Program Library Long Writeup D506 , Version 94 .1
40. T. Hyodo , D. Jido , A. Hosaka , Phys. Rev. C 85 , 015201 ( 2012 ). arXiv:1108.5524 [nucl-th]
41. F. Aceti , E. Oset , Phys. Rev. D 86 , 014012 ( 2012 ). arXiv: 1202 .4607 [hep-ph]
42. T. Sekihara , T. Hyodo , D. Jido , PTEP 2015 , 063D04 ( 2015 ). arXiv: 1411 .2308 [hep-ph]
43. C. Hanhart , J.R. Pelaez , G. Rios , Phys. Rev. Lett . 100 , 152001 ( 2008 ). arXiv: 0801 .2871 [hep-ph]
44. M. Albaladejo , J.A. Oller , Phys. Rev. D 86 , 034003 ( 2012 ). arXiv: 1205 .6606 [hep-ph]
45. C. Hanhart , J.R. Pelaez , G. Rios , Phys. Lett . B 739 , 375 ( 2014 ). arXiv: 1407 .7452 [hep-ph]
46. T. Hyodo , Phys. Rev . C 90 ( 5 ), 055208 ( 2014 ). arXiv: 1407 .2372 [hep-ph]
47. V. Bernard , D. Hoja , U.-G. Meissner , A. Rusetsky , JHEP 0906 , 061 ( 2009 ). arXiv:0902.2346 [hep-lat]
48. T. Ledwig , V. Pascalutsa , M. Vanderhaeghen , Phys. Rev. D 82 , 091301 ( 2010 ). arXiv: 1004 .5055 [hep-ph]
49. F.-K. Guo , C. Hanhart , F.J. Llanes-Estrada , U.-G. Meissner , Phys. Lett . B 703 , 510 ( 2011 ). doi:10.1016/j.physletb. 2011 .08.022. arXiv: 1105 .3366 [hep-lat]
50. F.K. Guo, U.G. Meissner , Phys. Rev. Lett . 109 , 062001 ( 2012 ). arXiv: 1203 .1116 [hep-ph]
51. E. Cincioglu , J. Nieves , A. Ozpineci , A.U. Yilmazer, Eur. Phys. J. C 76(10) , 576 ( 2016 ). arXiv: 1606 .03239 [hep-ph]
52. P.G. Ortega , J. Segovia , D.R. Entem , F. Fernandez , Phys. Rev. D 81 , 054023 ( 2010 ). arXiv: 1001 .3948 [hep-ph]
53. D.R. Entem , P.G. Ortega , F. Fernandez , AIP Conf. Proc. 1735 , 060006 ( 2016 ). arXiv: 1601 .03901 [hep-ph]
54. C. Meng , J.J. Sanz-Cillero , M. Shi , D.L. Yao , H.Q. Zheng , Phys. Rev . D 92 ( 3 ), 034020 ( 2015 ). arXiv: 1411 .3106 [hep-ph]
55. C. Meng , K.T. Chao , Phys. Rev. D 75 , 114002 ( 2007 ). arXiv:hep-ph/0703205
56. J. Ferretti , G. Galat , E. Santopinto , Phys. Rev . C 88 ( 1 ), 015207 ( 2013 ). arXiv: 1302 .6857 [hep-ph]
57. D. Morgan, Nucl. Phys. A 543 , 632 ( 1992 )
58. S.K. Choi et al. ( Belle Collaboration), Phys. Rev. Lett . 91 , 262001 ( 2003 ). arXiv:hep-ex/0309032
59. R.F. Lebed , R.E. Mitchell , E.S. Swanson , Prog. Part. Nucl. Phys. 93 , 143 ( 2017 ). arXiv: 1610 .04528 [hep-ph]
60. D. Gamermann , E. Oset , Eur. Phys. J. A 33 , 119 ( 2007 ). arXiv: 0704 .2314 [hep-ph]
61. D. Gamermann , J. Nieves , E. Oset , E.R. Arriola , Phys. Rev. D 81 , 014029 ( 2010 ). arXiv: 0911 .4407 [hep-ph]
62. S. Fleming , M. Kusunoki , T. Mehen , U. van Kolck, Phys. Rev. D 76 , 034006 ( 2007 ). arXiv:hep-ph/0703168
63. Y. Liu , I. Zahed , Int. J. Mod . Phys . E 26 ( 01 , 02), 1740017 ( 2017 ). arXiv: 1610 .06543 [hep-ph]
64. A. Sibirtsev , J. Haidenbauer , S. Krewald , U.G. Meißner , A.W. Thomas , Phys. Rev. D 71 , 054010 ( 2005 ). arXiv:hep-ph/0411386
65. X.W. Kang , J. Haidenbauer , U.G. Meißner , JHEP 1402 , 113 ( 2014 ). arXiv: 1311 .1658 [hep-ph]
66. X.W. Kang , J. Haidenbauer , U.G. Meißner , Phys. Rev . D 91 ( 7 ), 074003 ( 2015 ). arXiv:1502.00880 [nucl-th]
67. Y.F. Liu , X.W. Kang , Symmetry 8 ( 3 ), 14 ( 2016 )
68. M. Padmanath , C.B. Lang , S. Prelovsek , Phys. Rev . D 92 ( 3 ), 034501 ( 2015 ). arXiv:1503.03257 [hep-lat]
69. S.H. Lee et al. ( Fermilab Lattice and MILC Collaborations) , arXiv:1411 .1389 [hep-lat]
70. S. Prelovsek , L. Leskovec , Phys. Rev. Lett . 111 , 192001 ( 2013 ). arXiv:1307.5172 [hep-lat]