paperzb-rev2.dvi


Hidden-Beauty Charged Tetraquarks and
Heavy Quark Spin Conservation

A. Ali¶, L. Maiani∗, A.D. Polosa∗, V. Riquer∗

¶Deutsches Elektronen-Synchrotron DESY, D-22607 Hamburg, Germany
∗Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2, I-00185 Roma, Italy

and INFN Sezione di Roma, Piazzale Aldo Moro 2, I-00185 Roma, Italy

Abstract

Assuming the dominance of the spin-spin interaction in a diquark, we point out that the
mass differences in the beauty sector M(Z′

b
)± − M(Zb)± scale with quark masses as expected

in QCD, with respect to the corresponding mass difference M(Z′
c
)± − M(Zc)±.

Notably, we show that the decays Υ(10890) → (hb(1P),hb(2P))π+π− are compatible with
heavy-quark spin conservation once the contributions of Zb,Z

′
b

intermediate states are taken
into account, Υ(10890) being either a Υ(5S) or the beauty analog of Yc(4260).

We also consider the role of Zb,Z
′
b

in Υ(10890) → Υ(nS)ππ decays and of light quark spin
non-conservaton in Zb,Z

′
b

decays into BB∗ and B∗B∗. Indications on possible signatures of the
still missing Xb resonance are proposed.

Preprint # DESY 14-234



Tetraquark interpretation of the hidden charm and beauty exotic resonances has been ad-
vanced and studied in considerable detail (see Refs. [1] [2], and [3]). In a recent contribution [4],
a new scheme for the spin-spin quark interactions in the hidden charm resonances has been pro-
posed, which reproduces well the mass and decay pattern of X(3872), of the recently discovered [5]
Z

±,0
c (3900), Z

±,0
c (4020), and of the lowest lying J

P C = 1−− Y states.

Tetraquark states in the large Nc (color) limit of QCD have been considered in [6] and [7] (see
also the review [8] and references therein). Compact tetraquark mesons may have decay widths as
narrow as 1/Nc, contrary to previous beliefs, and therefore they are reasonable candidates for a
secondary spectroscopic meson series, in addition to the standard qq̄ one.

In this letter we consider the extension of the scheme presented in [4] for the hidden-charm to
the hidden-beauty resonances Z

±,0
b

(10610) = Zb and Z
±,0
b

(10650) = Z′
b
.

These resonances are interpreted as S−wave JP G = 1++ states with diquark spin distribution
(use the notation |s[bq],s[b̄q̄]〉 for diquark spins)

|Zb〉 =
|1bq, 0b̄q̄〉 − |0bq, 1b̄q̄〉√

2

|Z′b〉 = |1bq, 1b̄q̄〉J=1 (1)

The JP = 1+ multiplet is completed by Xb, which is given by the C = +1 combination

|Xb〉 =
|1bq, 0b̄q̄〉 + |0bq, 1b̄q̄〉√

2
(2)

Assuming the spin-spin interaction inside diquarks to dominate, we expect Xb and Zb to be degen-
erate, with Z′

b
heavier according to

M(Z′b) − M(Zb) = 2κb (3)

where κb is the strength of the spin-spin interaction inside the diquark. A similar analysis for the
hidden-charm resonances has produced the value [4]

2κc = M(Z
′
c) − M(Zc) ≃ 120 MeV (4)

The QCD expectation is κb : κc = Mc : Mb. The ratio can be estimated from the masses reported
in [9]

Mc
Mb

≃
1.27

4.18
= 0.30 (5)

giving 2κb ≃ 36 MeV, which fits nicely with the observed Z′b − Zb mass difference (≃ 45 MeV).

Next we consider another crucial prediction of QCD, namely conservation of the heavy quark
spin in hadronic decays.

We recall that Zb,Z
′
b

are observed in the decays of Υ(10890)

Υ(10890) → Zb/Z′b + π → hb(nP)ππ (6)

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The Υ(10890) is usually reported as the Υ(5S) since its mass is close to the mass of the 5S state
predicted by potential models. However, a different assignment was proposed in [10], namely
Υ(10890) = Yb, the latter state being a P−wave tetraquark analogous to the Y (4260). A reason for
this assignment is the analogy of Υ(10890) decay (6) with Y (4260) → Zc(3900) + π, with Y (4260)
being the the first discovered Y state [11]. Current experimental situation about Υ(10890) is still
in a state of flux. In our opinion, the possibility that Υ(10890) is an unresolved peak involving
both the Υ(5S) and Yb, reported by Belle some time ago [12], is plausible, providing a resolution of
the observed branching ratios measured at the Υ(10890) [13]. However, this identification is not a
requirement in the considerations presented below. In fact, following the assignment of Y (4260) as
a P−wave tetraquark with scc̄ = 1 [4], one sees that in both cases the initial state in (6) corresponds
to sbb̄ = 1. As is well known hb(nP) has sbb̄ = 0, pointing to a possible violation of the heavy-quark
spin conservation, as suggested in [13].

We show now that the contradiction is only apparent. Expressing the states in the the basis of
definite bb̄ and qq̄ spin, one finds

|Zb〉 =
|1qq̄, 0bb̄〉 − |0qq̄, 1bb̄〉√

2

|Z′b〉 =
|1qq̄, 0bb̄〉 + |0qq̄, 1bb̄〉√

2
(7)

Define

gZ = g(Υ → Zbπ)g(Zb → hbπ) ∝ 〈hb|Zb〉〈Zb|Υ〉
gZ′ = g(Υ → Z′bπ)g(Z′b → hbπ) ∝ 〈hb|Z′b〉〈Z′b|Υ〉 (8)

where g are the effective strong couplings at the vertices Υ Zb π and Zb hb π. Therefore, for both
assignments of Υ(10890), Eq. (7) and heavy quark spin conservation require

gZ = −gZ′ (9)

In Ref. [14] the amplitude for the decay (6) is fitted with two Breit-Wigners corresponding
to the Zb,Z

′
b

intermediate states. Table I therein, that we transcribe here in Table 1, shows the
relative normalizations and phases obtained by the fit, for decays into hb(1P) and hb(2P). Within
large errors, consistency with Eq. (8), that is with the heavy-quark spin conservation, is apparent.

It is interesting that the same conclusion was drawn using a picture in which Zb,Zb′ have a
“molecular” type structure [15]

Zb =
|B,B̄∗〉 − |B̄,B∗〉√

2
Z′b = |B∗,B̄∗〉J=1 (10)

It is conceivable that the subdominant spin-spin interactions may play a non negligible role in
the b-systems, as the spin-spin dominant interaction is suppressed by the large b quark mass. In

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hb(1P)π
+π− hb(2P)π

+π−

Relative Normalization 1.39 ± 0.37+0.05−0.15 1.6
+0.6+0.4
−0.4−0.6

Relative Phase 187+44+3−57−12 181
+65+74
−105−109

Table 1: Values of |gZ/gZ′| and of the relative phases (in degrees), for hb(1P),hb(2P), as reported
by [14].

this case the composition of Zb,Z
′
b

indicated in Eq. (7) would be more general:

|Zb〉 =
α|1qq̄, 0bb̄〉 − β|0qq̄, 1bb̄〉√

2

|Z′b〉 =
β|1qq̄, 0bb̄〉 + α|0qq̄, 1bb̄〉√

2
(11)

but the ratio gZ/gZ′ would still be unity. To determine α and β separately, one has to resort to
sbb̄ = 1 → sbb̄ = 1 transitions, such as Υ(10890) → Υ(nS)ππ where n = 1, 2, 3. The effective
couplings analogous to (8) would be

fZ = f(Υ → Zbπ)f(Zb → Υ(nS)π) =
|β|2
2

〈Υ(nS)|0qq̄, 1bb̄〉〈0qq̄, 1bb̄|Υ〉

fZ′ = f(Υ → Z′bπ)f(Z′b → Υ(nS)π) =
|α|2
2

〈Υ(nS)|0qq̄, 1bb̄〉〈0qq̄, 1bb̄|Υ〉

The Dalitz plot of these decays indicate indeed that a sizeable part of the transitions proceeds
through Zb and Z

′
b

[13, 14]. Parametrizing the amplitude in terms of two Breit-Wigner, one would
determine the ratio α/β.

As a side remark, we observe that a Fierz rearrangement similar to the one used in (7) puts
together bq̄ and qb̄ fields

|Zb〉 = |1bq̄, 1qb̄〉J=1

|Z′b〉 =
|1bq̄, 0qb̄〉 + |0bq̄, 1qb̄〉√

2
(12)

The labels 0bq̄ and 1bq̄ could be viewed as indicating B and B
∗ mesons, respectively, leading to the

prediction of the decay patterns Zb → B∗B̄∗ and Z′b → BB̄∗ [3]. This would not be in agreement
with the Belle data [13].

We remark however that this argument rests on conservation of the light quark spin which,
on the contrary, may change when the color octet pairs which appear in (12), evolve into pairs of

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color singlet mesons. Therefore predictions derived from (12) are not as reliable as those derived
from (7).

Finally we comment on the expected positive charge conjugation state, Xb. On the basis of the
assumed spin-spin interaction, one predicts M(Xb) ≃ M(Zb) ≃ 10600 MeV. Such a state has been
searched by ATLAS [16] in the region 10500 < M < 11000 MeV looking for the decay

Xb → Υ(1S)ππ (13)

so far with negative results.

In Ref. [2], it is noted that the near equality of the branching ratios for X(3872) → J/ψ 2π
and X(3872) → J/ψ 3π can be understood if X(3872) is predominantly isosinglet. The isospin
allowed decay in J/ψ ω is phase space forbidden and the decay in the J/ψ ρ mode, although isospin
forbidden, is phase space favoured, leading to similar rates.

In the Xb decay, both ω and ρ channels are allowed by phase space, so that, if Xb is isos-
inglet, the dominant mode would be into Υ(1S) ω. The suggestion therefore is to look at the
decay Xb(10600) → Υ(1S) 3π with the 3π in the ω mass band, in parallel with the search for the
Xb(10600) → Υ(1S) 2π channel with the 2π in the ρ band.

Acknowledgements

We thank Christoph Hanhart for interesting discussions.

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