The nature of $Z_b$ states from a combined analysis of $\Upsilon(5S)\rightarrow h_b(mP) \pi^+ \pi^-$ and $\Upsilon(5S)\rightarrow B^{(\ast)}\bar B^{(\ast)}\pi$

0.25 MB·English

by  Wen-Sheng Huo

The nature of Zb states from a combined analysis of Υ(5S) hb(mP)π+π− and
Υ(5S) B(∗)B¯(∗)π →
→
Wen-Sheng Huo1, Guo-Ying Chen1,2
1) Department of Physics, Xinjiang University, Urumqi 830046, China and
2) State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China
(Dated: April6, 2016)
6 With a combined analysis of data on Υ(5S) → hb(1P,2P)π+π− and Υ(5S) → B(∗)B¯(∗)π in an
1 effectivefieldtheoryapproach,wedetermineresonanceparametersofZb statesintwoscenarios. In
0 one scenario we assume that Zb states are puremolecular states, while in theother one we assume
2
that Zb states contain compact components. Wefindthat thepresent data favor that thereshould
r besome compact componentsinsideZ(′) associated with themolecular components. By fittingthe
b
Ap invariantmass spectraofΥ(5S)→hb(1P,2P)π+π− andΥ(5S)→B(∗)B¯∗π,wedeterminethatthe
probability of findingthe compact components in Zb states may be as large as about 40%.

5
PACSnumbers:

]
h
p I. INTRODUCTION
-
p
he byTtwheocBheallregeCdoblloatbtoormaotinoinumin-ldikeecastyasteΥs(Z5Sb±)(1061Υ0()naSn)dπ+Zπb±(1f0o6r50n)—=d1e,n2o,toerd3asaZndbaΥn(d5ZSb′)—wehre(mdiPsc)oπv+erπed
[ → − → b −
for m=1 or 2 [1, 2]. The masses and decay widths averagedover the five channels are mZb =10607.2±2.0
4 MeV, ΓZb = 18.4±2.4 MeV, and mZb′ = 10652.2± 1.5 MeV, ΓZb′ = 11.5±2.2 MeV [3]. The average
v masses are about 2 MeV above the thresholds of both B B¯ and B B¯ . Recently, Belle [4] also reported
∗ ∗ ∗
9 the observation of these two Z states in Υ(5S) [BB¯ +c.c.]π, and Υ(5S) B B¯ π. The discovery of
8 b → ∗ → ∗ ∗
the Z states has inspired many interesting theoretical discussions. For example, it is suggested that these
1 b
2 states can be molecular states of the BB¯∗+c.c. or B∗B¯∗ meson pairs [5–12]. They are also proposed to be
0 candidatesoftetraquarkstates[13]. InRef.[14,15]thethresholdenhancementsareconsideredtobecaused
. by cusp effects.
1
AlthoughthemassesoftheseZ statesdeterminedfromtheexperimentalfitsareslightlyabovethethresh-
0 b
5 olds, one should note that the masses are extracted by the Breit-Wigner parametrization. As emphasized
1 in [9, 12], if an S-wave shallow bound state exists below the threshold, the amplitude should not be param-
: eterizedin the Breit-Wignerform. Using the line shape for a pure bound state, Ref. [9] shows that the data
v
Xi oennhΥan(5ceSm)e→ntshbin(mΥP()5πS+)π− a[BreB¯con+sics.tce.n]πt wanitdhΥth(e5Sb)oundBstBa¯teπnbaytuBreelolef Zarb(e′).veFruyrtchloesremtooret,htehethorbessehrovledds
∗ ∗ ∗
r of the B( )B¯( ) systems.→It is also found that the mas→ses of the Z states can be below the corresponding
a ∗ ∗ b
thresholds if these masses are extracted from data on Υ(5S) B( )B¯( )π [4].
∗ ∗
→
As a fact of observations, Z states and their analogues in the charmonium sector Z (3900) [16–18],
b c
Z (4020/4025) [19, 20] and also the famous X(3872) appear to be strongly correlated to the thresholds of
c
either B( ) or D( ) pairs. This feature makes it natural to interpret these states as molecules. However, as
∗ ∗
was pointed out in Ref. [21, 22], it is difficult to understand the large production rates of these states in
B-factories, e.g. X(3872), if these states are assumed to be loosely bound molecular states. In particular,
the recent LHCb measurement of the ratio Rψγ = B(X(X(3(3887722)→) ψJ(/2ψSγ)γ)) = 2.46±0.64±0.29 [23] seems not
to support a pure D 0D¯0 molecular interpretation ofBX(3872→), since R is predicted to be rather small
∗ ψγ
for a pure D 0D¯0 molecule [24]. Meanwhile, a compact component inside such states can compromise both
∗
threshold phenomena and sizeable production rates. It is shown in Ref. [25, 26] that the radiative decays
of X(3872) are not only sensitive to long-range parts but also to short-range parts of the wave function.
The search for a hidden-beauty counterpart of X(3872), which is usually denoted as X , is important for
b
understandingthestructureofX(3872). AneffectivefieldtheorystudyshowsthatifX(3872)isamolecular
bound state of D 0 and D¯0 mesons, the heavy-quark symmetry requires the existence of molecular bound
∗
state X of B 0B¯0 with mass of 10604 MeV [27]. However, there is no significant signal of X near the
b ∗ b
threshold of B 0B¯0 in X π+π Υ(1S) [28] and in X ωΥ(1S) [29]. Ref. [30] suggests that X may
∗ b − b b
→ →
be close in mass to the bottomonium state χ (3P) and mixes with it. Therefore, the experiments which
b1
reported observing χ (3P) might have actually discoveredX .
b1 b
Obviously,moreexperimentaldataandtheoreticaldevelopmentarerequiredto clarifythe natureofthese
nearthresholdstates. InRef. [31] aneffective fieldtheory (EFT) approachis proposedfor the study ofnear
threshold states (see also an independent study in Ref. [32]). In this framework the compositeness theorem
can be incorporated with a determination of parameter Z which is the probability of finding an elementary
component in the bound state, and the nature of near threshold states can be described by the presence of
both molecular and compact components in their wavefunctions.
The main purpose of this work is to study structure of Z states by doing a combined analysis of data
b
on Υ(5S) h (mP)π+π and Υ(5S) B( )B¯( )π within EFT approach proposed in [31]. Our work is
b − ∗ ∗
→ →
organizedasfollows: inSec. II,werecalltheEFTapproachproposedinRef.[31]. InSec. III,wepresentthe
analysis of the Υ(5S) h (mP)π+π transitions and in Sec. IV, the Υ(5S) B( )B¯( )π. Our numerical
b − ∗ ∗
→ →
results are presented in Sec. V. Finally, a brief summary is given in Sec. VI.
+ + +···
FIG. 1: Feynmandiagrams for thetwo particle scattering. The double lines denote thebare state.
II. COMPOSITENESS THEOREM IN EFT
In Ref. [31], we have developed an EFT approach which incorporates Weinberg’s compositeness theo-
rem [33, 34]. Here we recall some of the main points; more details can be found in Ref. [31]. Consider a
barestate with baremass B and coupling g to the two-particlestate, where the baremass is defined
0 0
|Bi −
relative to the two-particle threshold. The two particles have masses m , m respectively. If is near
1 2
|Bi
the two-particle threshold, then the leading two-particle scattering amplitude can be obtained by summing
the Feynman diagrams in Fig. 1. Near threshold, the momenta of these two particles are non-relativistic.
Therefore,the loopintegralin Fig.1 canbe done inthe same wayas thatin Ref. [35,36]. The loopintegral
can be written as
dDℓ i i
= ,
I (2π)D ℓ0 ~ℓ2/(2m )+iǫ · E ℓ0 ~ℓ2/(2m )+iǫ
Z − 1 − − 2
dD 1ℓ i
−
= ,
Z (2π)D−1E−~ℓ2/(2µ)+iǫ
3 D 1−D D−3
= i2µΓ( − )(4π) 2 ( 2µE iǫ) 2 , (1)
− 2 − −
where µ is the reduced mass of the two particles, and E is the kinematic energy of the two-particle system.
Obviously, the above integral does not diverge in D =4. Using the minimal subtraction(MS) scheme which
subtracts the 1/(D 4) pole before taking the D 4 limit, one obtains
− →
µ
=i ( 2µE iǫ)1/2. (2)
I 2π − −
It is interesting to note that, with the MS scheme no counter term is needed in the renormalization. We
then have the two body elastic scattering amplitude for Fig. 1
g2
= 0 . (3)
A −E+B g2 µ √ 2µE iǫ
0− 02π − −
2
If a bound state exists, we can have the following relations
2 Z 2π√2µB
g2 =g2/Z, B = − B, g2 = (1 Z), (4)
0 0 Z µ2 −
where B is the binding energy,andZ is the probability offinding an elementarystate in the physicalbound
state. Note that for the bound state, we mean a below threshold pole in the physical sheet. With Eq. (4),
Eq. (3) can be re-expressedas
g2
= , (5)
A −E+B+Σ˜(E)
where
µ µ√2µB
Σ˜(E)= g2[ 2µE iǫ+ (E B)]. (6)
− 2π − − 4πB −
p
We can also express Eq. (5) in the form
i =ig G(E) ig , (7)
0 0
A · ·
where G(E) is the complete propagatorfor the S-wave near threshold state
iZ
G(E)= . (8)
E+B+Σ˜(E)+iΓ/2
We have added a constant width Γ in the propagator, which can simulate the decay channels other than
the bottom and anti-bottom mesons. From Eq. (7), one can find that the Feynman rule for the coupling
between the near threshold state and its two-particle component is ig . Treating the binding momentum
0
γ =(2µB)1/2 and the three-momentum of the two-particle state p as small scales, i.e., γ,p (p), one can
then find that the leading amplitude Eq. (5) is at the order of (p 1). ∼O
−
O
π π π π
Z h Z h
Υ b b Υ b b
(a) (b)
FIG. 2: Feynman diagrams for Υ(5S)→Zb(′)π→hb(mP)ππ,where theZb states are produced in direct production
processes. Solid lines in theloop represent bottom and anti-bottom mesons.
III. Υ(5S) DECAYS TO hb(1P,2P)π+π−
In this section, we study the decay Υ(5S) → Zb(′)π → hb(mP)ππ in the EFT approach. Generally, in
the decay Υ(5S)→ Zb(′)π, Zb states can be produced through both direct and indirect processes. In direct
productionprocesses,Z statesareproduceddirectlyviaitscompactcomponent,whileinindirectproduction
b
processes a bottom and anti-bottom meson pair is produced first in the Υ(5S) decay and then rescatters
to Zb(′). Similarly, the decay Zb(′) → hb(mP)π can proceed through both direct and indirect processes. In
direct decay, Zb state will decay to hb(mP)π directly. In indirect decay, Zb(′) will first decay into a bottom
and anti-bottom meson pair and then the meson pair rescatters into h (mP)π.
b
3
The three-momenta of heavy mesons in decay Υ(5S) → Zb(′)π → hb(mP)ππ are small compared with
their masses. Therefore, these heavy mesons can be treated as non-relativistic, and one can set up a power
counting in terms of the small three-momentum p [31, 37–39]. From the power counting, one can find that
if Z(′) contains a compact component, its production will be driven by this compact component [31] (see
b
also Ref. [21, 22]). In Fig.2, we show Feynman diagrams where the Z states are produced via compact
b
components and decay through both direct and indirect processes.
If Z(′) is a pure molecular state, its production should via indirect processes. The leading Feynman
b
diagrams for indirect production of Z(′) are shown in Fig. 3. Note that there are two kinds of indirect
b
productionmechanismsforZ states. InFig.3(a,b), Υ(5S)firstdecaysto abottom andanti-bottommeson
b
()
pair and pion in the same vertex, then the bottom and anti-bottom meson pair rescatters to Z ′ . While
b
in Fig. 3(c,d), Υ(5S) first decays to a bottom and anti-bottom meson pair, and after emitting one pion,
the bottom and anti-bottom meson pair rescatters to Z(′). It is shown in Ref. [12] that both mechanisms
b
contribute at leading order for the indirect production of Z states.
b
π π
Z h Z h
Υ b b Υ b b
π π
(a) (b)
π π
π π
Z h Z h
Υ b b Υ b b
(c) (d)
FIG.3: FeynmandiagramsforΥ(5S)→Zb(′)π→hb(mP)ππ,wheretheZb statesareproducedinindirectproduction
processes. Solid lines in theloops represent bottom and anti-bottom mesons.
Asweareonlyinterestedinlowenergyphysics,itisconvenienttocollectB mesonsina2 2matrix[40,41]
×
Ha =P~a∗·~σ+Pa, H¯a =−P~¯a∗·~σ+P¯a, Pa(∗) =(B(∗)−, B¯(∗)0), (9)
where σi are the Pauli matrices, and a is the light flavor index, P and P annihilate the vector and
a∗ a
pseudoscalar heavy mesons respectively, and P¯a(∗) annihilates the corresponding anti-particle. The leading
effective Lagrangian describing the coupling of Z states to the bottom and anti-bottom mesons can be
b
written as that in Ref. [9]
LZbHH = 2g√02Tr[Za†ibHaσiH¯b]+ 2g√02Tr[(ZT)ibaH¯b†σiHa†], Zab = √1Z2Z(′)b(+′)0 Z1b(′Z)−(′)0 ! , (10)
b −√2 b ab
where Zab annihilates Zab, Za†b createsZab, and g0 is defined in Eq. (4). The Lagrangianfor the coupling of
the P-wave quarkonia and the B mesons reads [37]
g
LhbHH = 2hTr[h†biHaσiH¯a]+H.c. (11)
4
The chiral Lagrangianfor the B mesons and the S-wave quarkonia can be written as [12]
1 1
LHHχPT =gπTr[H¯a†σiH¯b]Aiab−gπTr[Ha†Hbσi]Aiba+2g1Tr[ΥH¯a†Hb†]A0ab+2ig2Tr[ΥH¯a†σ·←→∂ Ha†]+H.c., (12)
where A←→∂ B A(∂B) (∂A)B, Υ is the 2 2 matrix field defined as Υ = Υ~(5S) ~σ+η (5S), and A is
b µ
≡ − × ·
the axial vector pion current which is given by
Aµ = 2i(ξ†∂µξ−ξ∂µξ†)=−∂µM/Fπ+··· , ξ =eiFMπ, M = √1π2−π0 −√π1+2π0 !, Fπ =132MeV. (13)
We set g = 0.25 as in [9, 42]. Note that our convention is different from that in [9], because a factor of
π
√2M has been absorbed into the field operator of the heavy meson in our convention [31], then our g is
π
halfofthevaluewhichisusedinRef.[9]. TheleadingeffectiveLagrangiandescribingtheZ h π interactions
b b
reads
LZbhbπ =gzεijkZaibh†bjAkab+H.c., (14)
which describes the direct decay of Zb(′) →hb(mP)π.
Finally, we come to the vertex describing decay of Υ(5S) into Z(′)π. The corresponding Lagrangian to
b
the leading order of the chiral expansion is given by [9]
LΥZbπ =gΥΥi(5S)Zb†aiA0ab+H.c.. (15)
Similar to Ref. [9], we use the same coupling g , g for Z and Z .
Υ z b b′
π π π π
h h h
Υ b Υ b Υ b
π π
a b c
( ) ( ) ( )
FIG.4: Feynmandiagrams fornon-resonantprocesses Υ(5S)→hb(mP)ππ.Solidlinesin theloop representbottom
and anti-bottom mesons.
With the above effective Lagrangians and Eq. (8) as the propagator of Z(′), one can then write out the
b
amplitudesforalltheFeynmandiagramsinFig.2and3. WetreattheloopintegralsaswasdoneinRef.[37].
We presentthe relevantoneloopthree-pointfunctions inAppendix A,andgivealltheamplitudes ofFig. 2
and 3 in Appendix B. In the following we address several points before ending this section.
As in Ref. [9], we assume that Z only couples to BB¯ while Z only couples to B B¯ . We then
• find that there is a relative minus sbignbetween i ∗ for Υ(b′5S) Z+π h (∗mP∗)π+π and
M3a,3b,3c,3d → b − → b −
sthinocseeifforonΥe(5aSss)u→meZsb′Z+π(−Z→) choub(pmlePs)tπo+Bπ−B¯. I(tBshB¯ou)ldwnitohttbheessuarmpreissintrgentogtfihndasththisarteolaftZive(Zmi)nucosuspiglens,
b b′ ∗ ∗ ∗ b b′
to BB¯ (B B¯ ), one would find that the meson loop amplitudes would be suppressed in heavy-quark
∗ ∗ ∗
spin symmetry world as noticed in [37].
5
Assuming that Z and Z are spin partners of each other, we can use the same Z for Z and Z . In
• b b′ b b′
this way, we can reduce the number of free parameters in our fitting.
We show the Feynman diagrams for non-resonant contributions to Υ(5S) h (mP)ππ in Fig. 4.
b
• →
Ref.[43]showsthatthenon-resonantdiagramsdonotsatisfythetwo-cutconditionnearΥ(5S)region.
Hence their contributions will not be enhanced by the kinematic singularity. We do not include their
contributions in the present work, since they are suppressed by the heavy-quark spin symmetry. The
experimental fits also find no significant non-resonant contributions [1, 2].
π π π
Z Z Z
Υ b Υ b Υ b
(a) (b) (c)
FIG. 5: Feynman diagrams for Υ(5S)→Z(′)π →B(∗)B¯(∗)π. Solid lines in the loop and in the final state represent
b
bottom and anti-bottom mesons.
IV. Υ(5S) DECAYS TO B(∗)B¯(∗)π
Inthissection,wewillstudythedecayΥ(5S) B( )B¯( )πinEFT.Forthepreviousstudyonemayreferto
∗ ∗
→
Ref.[12],wheretheZ statesareassumedtobemolecules. Insteadoffittingdatadirectly,Ref.[12]constrains
b
some parameters using data on Υ(5S) B( )B¯( ),and it then calculates the differential distribution for
∗ ∗
Υ(5S) B( )B¯( )πasafunctionofinvar→iantmassoftheB( )B¯( ) pair. Inthiswork,wegivetheamplitudes
∗ ∗ ∗ ∗
for Υ(5→S) B( )B¯( )π in EFT and constrain parameters by fitting the data directly.
∗ ∗
→
Similar to Υ(5S) → Zb(′)π → hb(mP)ππ, Zb states can be produced through both direct and indirect
processes. TheleadingorderFeynmandiagramsforthesetwodifferentproductionmechanismsarepresented
in Fig. 5. The Feynman diagrams for the non-resonant contributions are shown in Fig. 6. We give all the
amplitudes for Fig. 5 and 6 in Appendix C.
Υ Υ
π π
a b
( ) ( )
FIG. 6: Feynman diagrams of the non-resonant contribution to Υ(5S) → B(∗)B¯(∗)π. The solid lines represent the
bottom and anti-bottom mesons.
6
V. NUMERICAL RESULTS
With the amplitudes given in Appendix B and C, we do a combined fit to data on Υ(5S)
h (mP)π+π [1, 2] and Υ(5S) B( )B¯( )π [4]. Data on Υ(5S) B( )B¯ π are nonvanishing belo→w
b − ∗ ∗ ∗ ∗
the B( )B¯( ) thresholds, hence w→e have to convolve the invariant m→ass spectra with detector resolution
∗ ∗
function. Data on Υ(5S) h (mP)π+π are collected per 10 MeV, so the invariant mass spectra should
b −
→
be convolved with detector resolution function and integrated over 10 MeV histogram bin. The detector
resolution is parameterized by a Gaussian function with energy resolution parameter σ = 5.2 MeV for
Υ(5S) h (mP)π+π [1] and σ = 6 MeV for Υ(5S) B( )B¯( )π [4]. To compare different scenarios for
b − ∗ ∗
→ →
the structure of Z states, we do the fit with two alternative schemes:
b
a. We assume that Z states are pure molecular states, then we have to set Z = 0 in the fit. In this way,
b
only the diagrams in Fig. 3(a,c), 5(b,c) and 6 give nonvanishing amplitudes.
b. We assume that Z states contain substantial compact components, i.e., tetraquark component. It is
b
showninRef.[22]thattheproductionrateofamolecularstateisproportionaltoitswavefunctionsquare
at the origin Ψ(0)2. Because the wave function of the molecular component in a loosely bound state
| |
spreads far out in space, Ψ(0)2 is quite small, then the production rate of Z(′) through the molecular
| | b
component will be suppressed. Therefore, we further assume that Z(′) is mainly produced through the
b
compact component, and we set g = g = 0 in the fitting. It is worth mentioning that Ref. [21, 31]
1 2
demonstrate that the production of a near threshold state (by which we mean a mixture of the compact
component and molecular component) is driven by the compact component. On the other hand, the
hadronic decays of Zb(′) into hb(mP)π will mainly go through the molecular component. This can be
found from the power counting analysis. We treat the binding momentum γ, the three momentum of the
bottommesonp andthefourmomentumofpionp assmallscales,i.e.,theyareallattheorderof (p).
B π
O
Note that in the non-relativistic effective field theory, the propagator of the heavy meson is at the order
of (p 2), and the measure of the one loop integration is at the order of (p5). One can then find that
−
O O
Fig. 2(a) is at the order of (p 1/2), while Fig. 2(b) is at the order of (p0). Thus, as a leading order
−
O O
study, we set g = 0 and neglect the contribution from Fig. 2(b). Up to now, we have shown that while
z
the production of Z(′) is driven by the compact component, its hadronic decays mainly go through the
b
molecular component. It is interesting to note that similar features are adopted for X(3872) in Ref. [21].
By setting g = g = g = 0, one can find that only the diagrams Fig. 2(a) and 5(a) give nonvanishing
1 2 z
amplitudes, and the number of the relevant free parameters in this scheme is the same as that in scheme
(a) (see Table. I).
We then compare our fitting schemes with that used in Ref. [9]. Although scheme(b) and Ref. [9] use the
same decay mechanism for Υ(5S) Z π h ππ as shown in Fig. 2(a), there are some differences between
b b
→ →
them. The main difference is that Ref. [9] sets Z = 0, while in scheme(b) we let Z to be a free parameter
which satisfies 0<Z <1. As shownexplicitly in Appendix B, the amplitude for Fig. 2(a) is zero by setting
Z = 0. Physically, by setting Z = 0, one assumes the Z states as pure molecular states which do not
b
containcompactcomponents,hence they cannotbe producedthroughthe compactcomponents. Therefore,
if one uses Fig. 2(a) to describe the decay mechanism of Υ(5S) Z π h ππ, one cannot set Z = 0 as
b b
→ →
in Ref. [9]. The consistent treatment is to let Z to be a free parameter which satisfies 0 < Z < 1. On the
()
other hand,if one assumesZ ′ to be a pure molecular state,i.e., Z =0, one shouldnote that itcanonly be
b
produced through indirect process. Therefore, for the pure molecular scenario, one should use Fig. 3(a,c),
i.e., scheme(a), instead of Fig. 2(a) to describe the decay mechanism of Υ(5S) Z π h ππ.
b b
→ →
Now we come to discuss the applicability of EFT. In the decay Zb(′) → hb(2P)π, the momentum of the
pion is around 300 400 MeV in the energy region of our concern, hence the pion can be treated as soft,
∼
andonewouldexpectthe EFTexpansioncanconvergefastenough. ButinZb(′) →hb(1P)π,themomentum
of the pion is relatively large and around 600 700 MeV. Based on naive dimensional analysis, Ref. [10]
∼
warns that the EFT expansion may not be good enough for decay Zb(′) → hb(1P)π due to the relatively
large pion momentum. However,the results from the complete loop calculations can be more complex than
the naive dimensionalanalysis. One may refer to Ref. [44] for anexample. Generally, it is complex to study
7
the convergenceofthe effective fieldtheory,andreliableconclusionscanonlybe achievedoncethe complete
higher loop contributions are available. Since such a kind of study is beyond the scope of the this work, we
take a more pragmatic approach with two options in the fit.
1. We use data sets of Υ(5S) h (1P,2P)π+π , Υ(5S) BB¯ π and Υ(5S) B B¯ π in our fit.
b − ∗ ∗ ∗
→ → →
2. We use data sets of Υ(5S) h (2P)π+π , Υ(5S) BB¯ π and Υ(5S) B B¯ π in our fit.
b − ∗ ∗ ∗
→ → →
We choose an individual normalization factor for each final state in the fit. In this way, we need not to
fix values of g and g . We present all the fitted parameters in Table. I, and we show the fitting results of
Υ h
fit(1a),fit(2a)andfit(1b) inFig.7,8. Note thatthe widthΓinTable. Iis notthe totalwidth, butthe width
defined in Eq. (8).
TABLEI: Parameters for four fits.
Fit g2/g1 B B′ ΓZb ΓZb′ Z χ2/d.o.f.
1a 0.049(15) 0.11(12)eV 27(58)keV 2(1)keV 1.9(1.9)MeV 0 110/58
2a 0.0017(69) 12(21)keV 0.14(7)MeV 0.12(8)MeV 0.59(27)MeV 0 72/45
1b – 0.19(22)eV 1.6(1.8)eV 5.5(1.8)MeV 7.8(2.2)MeV 0.42(12) 81/58
2b – 0.38(65)eV 0.51(86)eV 6.1(2.8)MeV 4.6(2.2)MeV 0.42(18) 69/45
We give some brief discussions as regards our fitting results as follows
ItisfoundintheexperimentalfitsthattherelativephasebetweenZ andZ intheh (mP)ππ channel
• b b′ b
is 1800 [1,2]. Infitting scheme(a),the relativeminus signbetweeni forZ andZ canaccount
M3a,3c b b′
for this relative phase. However, one can not find such a relative phase in amplitudes which are used
in scheme (b). In our fitting, we find that scheme (b) gives a good fit only if such a relative phase is
included. This may be attributed to g (defined in Eq. (15)) which has a relative minus sign between
Υ
Z andZ . We note that a very recent paper Ref. [45] proposedan explanationfor this relative minus
b b′
sign.
From the fitting results of fit(1b) and fit(2b), one can find that the fitted parameters in fit(1b) and
•
fit(2b) are close to eachother. This indicates that the fitting results in scheme (b) are not sensitive to
data on Υ(5S) h (1P)π+π . Whether this means that the effective field theory can be successfully
b −
→
applied inΥ(5S) h (1P)π+π needs to be further investigated. Nevertheless,our numericalresults
b −
→
show that such a possibility exists. It is also interesting to find that in scheme (b) the fitted binding
energy and the width ofZ are close to those of Z . This seems to be consistent with the heavy-quark
b b′
spin symmetry.
With all data sets, scheme (1b) gives much better fitting quality than scheme (1a). Unfortunately,
• if data on Υ(5S) h (1P)π+π are dropped, the two schemes give almost equal fitting qualities.
b −
→
In this sense, it seems too early to claim conclusively that Z states contain substantial compact
b
components. However,a substantialcompactcomponentin Z(′) canexplain its largeproduction rates
b
in experiments. In contrast, a pure molecular state with the tiny binding energy as determined in
scheme (a) is not likely to have large production rates in Υ(5S) decays.
ThebindingenergiesoftheZ statesfromthefitaregenerallyverysmall. IfwefixB =0.1MeV,which
b
• isthecaseforX(3872),andZ =0.4infit(1b),wegetafittingqualityχ2 =90,whichisstillacceptable
and better than fit(1a). The other fitting parameters are B =0.23(14) MeV, Γ =6.5(9) MeV and
′ Zb
ΓZb′ =5.6(9) MeV. This result also seems to be consistent with the heavy-quark symmetry.
One can also analysis data on Υ(5S) Υ(nS)π+π in the EFT approach. However, different from
−
• h (mP)π+π and B( )B¯( )π, the non→-resonant contribution in Υ(nS)π+π is significant. It is im-
b − ∗ ∗ −
possible to consider the interference with the non-resonant contribution correctly in one-dimensional
analysis. To analysis data on Υ(5S) Υ(nS)π+π , one needs to fit the two-dimensional Dalitz
−
→
distribution, which is beyond the scope of the present manuscript.
8
VI. SUMMARY
We have done a combined analysis of data on Υ(5S) h (1P,2P)π+π , Υ(5S) BB¯ π and Υ(5S)
b − ∗
B B¯ π within EFT approach. With a combined analy→sis, we determine the reson→ance parameters of Z→
∗ ∗ b
states in two scenarios. In one scenario we assume that Z states are pure molecular states, while in the
b
otheroneweassumethatZ statescontaincompactcomponents. ItisfoundthatbyassumingthatZ states
b b
containsubstantialcompactcomponents,onecanhaveabetterdescriptionofalldatathanbypuremolecular
assumption. ByfittingtheinvariantmassspectraofΥ(5S) h (1P,2P)π+π andΥ(5S) B( )B¯( )π,we
b − ∗ ∗
→ →
determine that the probability of finding a compact component in Z(′) is about 40% . It is also interesting
b
to note thatthe probabilityoffinding a compactcomponentinZ(′) couldbe closeto thatin X(3872)which
b
are around 26% 44% [46, 47].
∼
20000
10000
2/c 2/c
V V
Me Me10000
0 5000 0
1 1
/ /
nts nts
ve ve
E E
0 0
10.56 10.58 10.60 10.62 10.64 10.66 10.68 10.70 10.58 10.60 10.62 10.64 10.66 10.68 10.70
M(hb(1P)π+)/GeV/c2 M(hb(2P)π+)/GeV/c2
FIG. 7: Comparison of the invariant mass spectra of hb(1P)π and hb(2P)π in fit(1a), fit(2a),fit(1b) and the experi-
ment. The dotted line is the result of fit(1a). The dashed line is the result of fit(2a). The solid line is the result of
fit(1b). Data are from [1].
30 20
2/cMeV20 2/cMeV15
N/5events10 N/5events10
5
0 0
10.60 10.62 10.64 10.66 10.68 10.70 10.64 10.65 10.66 10.67 10.68 10.69 10.70 10.71
MBB∗/GeV/c2 MB∗B∗/GeV/c2
FIG. 8: Comparison of the invariant mass spectra of BB¯∗ and B∗B¯∗ in fit(1a), fit(2a), fit(1b) and the experiment.
The dotted line is the result of fit(1a). The dashed line is the result of fit(2a). The solid line is the result of fit(1b).
Data are from [4] and have had background subtracted.
9
ACKNOWLEDGMENTS
We would like to thank Qiang Zhao for a careful reading of the manuscript and valuable comments and
Feng-KunGuoforveryusefuldiscussions. WewouldalsoliketothankGuang-YiTangforthehelpfuldiscus-
sions on the Belle result. This workis supported, in part, by National NaturalScience Foundation of China
(Grant Nos. 11147022and 11305137)and Doctoral Foundation of Xinjiang University (No. BS110104).
APPENDIX A: ONE LOOP THREE POINT FUNCTIONS
The three-point loop functions we will encounter are
I(m ,m ,m ,q)
1 2 3
ddℓ 1
= ( i)
− Z (2π)d(ℓ0− 2~ℓm21 +iǫ)(ℓ0+b12+ 2ℓm22 −iǫ)[ℓ0+b12−b23− (~ℓ2−mq~3)2 +iǫ]
µ µ 1 c c 2a+c c
= 12 23 tan−1( ′− )+tan−1( − ′ ) , (16)
2π √a" 2 a(c−iǫ) 2 a(c′−a−iǫ) #
p p
I(1)(m ,m ,m ,q)
1 2 3
µ m m (c c)
= 12 3(√c iǫ √c a iǫ)+ 3 ′− I(m ,m ,m ,q), (17)
2π~q2 − − ′− − 2µ ~q2 1 2 3
23
where µ =m m /(m +m ) are the reduced masses, b =m +m M, b =m +m +q0 M, and
ij i j i j 12 1 2 23 2 3
− −
2
µ µ
a= 23 ~q2, c=2µ b , c =2µ b + 23~q2. (18)
12 12 ′ 23 23
m m
(cid:18) 3(cid:19) 3
I(1)(m ,m ,m ,q) is defined as
1 2 3
ddℓ ℓi
qiI(1)(m ,m ,m ,q)=( i) .
1 2 3 − Z (2π)d(ℓ0− 2~ℓm21 +iǫ)(ℓ0+b12+ 2ℓm22 −iǫ)[ℓ0+b12−b23− (~ℓ2−mq~3)2 +iǫ]
For more details, one can refer to Ref. [37].
APPENDIX B: AMPLITUDES FOR Υ(5S)→hb(1P,2P)π+π−
The amplitudes for Υ(5S) Z+π h (mP)π+π in Fig. 2 and 3 read
→ b − → b −
2√2Zgg g g E 1
iM2a = FΥ2 π h π E+B+Σ˜(E)+iΓ /2εijkqiǫ∗j(hb)ǫk(Υ)
π Zb
[I(mB∗,mB,mB∗,q)+I(mB,mB∗,mB∗,q)]. (19)
×
g g E Z
i = i Υ z π εijkqiǫ j(h )ǫk(Υ). (20)
M2b − F2 E+B+Σ˜(E)+iΓ /2 ∗ b
π Zb
2g2g g g E µ 1
i = 1 π h π ( 2µE iǫ)1/2
M3a − F2 π − − E+B+Σ˜(E)+iΓ /2
π Zb
εijkqiǫ∗j(hb)ǫk(Υ)[I(mB∗,mB,mB∗,q)+I(mB,mB∗,mB∗,q)]. (21)
×
10