X(3872), I^G(J^{PC})=0^+(1^{++}), as the \chi_{1c}(2P) charmonium

0.1 MB·English

by  N. N. Achasov

X(3872), IG(JPC) = 0+(1++), as the χ (2P) charmonium
c1
N.N. Achasova and E.V. Rogozinaa,b
aLaboratory of Theoretical Physics, Sobolev Institute
for Mathematics, 630090, Novosibirsk, Russia
bNovosibirsk State University, 630090, Novosibirsk, Russia
5
(Dated: October 16, 2015)
1
0
2 Abstract

t
c Contrary to almost standard opinion that the X(3872) resonance is the D∗0D¯0+c.c. molecule or
O
the qcq¯c¯four-quark state, we discussthe scenario wheretheX(3872) resonance is the cc¯= χ (2P)
5 c1
1
charmonium which ”sits on” the D∗0D¯0 threshold.

]
h We explain the shift of the mass of the X(3872) resonance with respect to the prediction of
p
-
p a potential model for the mass of the χc1(2P) charmonium by the contribution of the virtual
e
h D∗D¯ +c.c. intermediate states into the self energy of the X(3872) resonance. This allows us to
[
estimate the coupling constant of the X(7872) resonance with the D∗0D¯0 channel, the branching
3
v
3 ratio of the X(3872) → D∗0D¯0+c.c. decay, and the branching ratio of the X(3872) decay into all
8
5 non-D∗0D¯0 +c.c. states. We predict a significant number of unknown decays of X(3872) via two
3
0 gluon: X(3872) → gluon gluon → hadrons.
.
1
0 We suggest a physically clear program of experimental researches for verification of our assump-
5
1 tion.
:
v
i
X PACS numbers: 13.75.Lb, 11.15.Pg,11.80.Et,12.39.Fe
r
a
1
The X(3872) resonance became the first in discovery of the resonant structures XYZ
(X(3872), Y(4260), Z+(10610), Z+(10650), Z+(3900)), the interpretations of which as
b b c
hadron states assumes existence in them at least pair of heavy and pair of light quarks
in this or that form. Thousand articles on this subject already were published in spite of
the fact that many properties of new resonant structures are not defined yet and not all
possible mechanisms of dynamic generation of these structures are studied, in particular,
the role of the anomalous Landau thresholds is not studied. Anyway, this spectroscopy took
the central place in physics of hadrons.
Below we give reasons that X(3872), IG(JPC) = 0+(1++), is the χ (2P) charmonium
c1
and suggest a physically clear program of experimental researches for verification of our
assumption.
The two dramatic discoveries have generated a stream of the D∗0D¯0 +D0D¯∗0 molecular
interpretations of the X(3872) resonance.
The mass of the X(3872) resonance is 50 MeV lower than predictions of the most lucky
naive potential models for the mass of the χ (2P) resonance,
c1
m −m = −∆ ≈ −50MeV, (1)
X χc1(2P)
and the relation between the branching ratios
BR(X → π+π−π0J/ψ(1S)) ∼ BR(X → π+π−J/ψ(1S)), (2)
that is interpreted as a strong violation of isotopic symmetry.
<
But the bounding energy is small, ǫB ∼ (1÷3) MeV. That is, the radius of the molecule
is large, rX(3872) ∼> (3÷5) fm = (3÷5)·10−13 cm. As for the charmonium, its radius is less
one fermi, r ≈ 0.5 fm = 0.5 · 10−13 cm. That is, the molecule volume is 100 ÷1000
χc1(2P)
>
times as large as the charmonium volume, VX(3872)/Vχc1(2P) ∼ 100÷1000.
Howtoexplain sufficiently abundant inclusive productionoftherather extended molecule
X(3872) in a hard process pp → X(3872) + anything with rapidity in the range 2,5 - 4,5
and transverse momentum in the range 5-20 GeV [1]? Really,
σ(pp → X(3872)+anything)BR(X(3872) → π+π−J/ψ) = 5.4nb (3)
and
σ(pp → ψ(2S)+anything)BR(ψ(2S) → π+π−J/ψ) = 38nb. (4)
2
But, according to Ref. [2]
BR(ψ(2S) → π+π−J/ψ) = 0.34 (5)
while
0.023 < BR(X(3872) → π+π−J/ψ) < 0.066 (6)
according to Ref. [3]. So,
σ(pp → X(3872)+anything)
0.74 < < 2.1. (7)
σ(pp → ψ(2S)+anything)
The extended molecule is produced in the hard process as intensively as the compact char-
monium. It’s a miracle.
As for the problem of the mass shift, Eq. (1), the contribution of the D−D∗+ and D¯0D∗0
loops, see Fig. 1, into the self energy of the X(3872) resonance, Π (s), solves it easily.
X
D
X
¯∗
D
FIG. 1: The contribution of the D¯0D∗0 and D−D∗+ loops into the self energy of the X(3872)
resonance.
g2
Π (s) = ΠD¯0D∗0(s)+ΠD−D∗+(s) = A ID¯0D∗0(s)+ID−D∗+(s) , (8)
X X X 8π2
(cid:16) (cid:17)
where
IDD¯∗(s) = Λ2 (s′ −m2+)(s′ −m2−)ds′ ≈ 2ln 2Λ −2 m2+ −s arctan s , (9)
q s′(s′ −s) m s s m2 −s
mZ2 + s +
+
where
m+ = mD∗ +mD, m− = mD∗ −mD, s < m2+, Λ2 ≫ m2+. (10)
3
For the calculations we use the Lagrangian
L(x) = g Xµ D (x)D¯(x)+D¯ (x)D(x)
A µ µ
(cid:16) (cid:17)
= g Xµ D0(x)D¯0(x)+D¯0(x)D0(x)+D+(x)D−(x)+D+(x)D−(x) . (11)
A µ µ µ µ
(cid:16) (cid:17)
The width of the X → D∗0D¯0 +c.c. decay
Γ(X → D∗0D¯0 +c.c., s) ≈ (g2/8π)(2|~k|/s). (12)
A
The inverse propagator of the X(3872) resonance
D (s) = m2 −s−Π (s)−ım Γ, (13)
X χc1(2P) X X
where Γ = ΣΓ is the total width of the X(3872) decays into all {i} non-D∗0D¯0 + c.c.
i
channels. According to Refs. [4] and [5] Γ < 1.2 MeV!
The renormalization of mass [6]
m2 −m2 −Π (m2 ) = 0 (14)
χc1(2P) X X X
results in
∆(2m +∆) = Π (m2 ) ≈ g2/8π2 4ln(2Λ/m ). (15)
X X X A +
(cid:16) (cid:17)
The renormalized propagator has the form [7]
D (s) = m2 −s+Π (m2 )−Π (s)−ım Γ. (16)
X X X X X X
If ∆ = m −m ≈ 50 MeV, see Eq. (1), then g2/8π ≈ 0.2 GeV2 for Λ = 10 GeV.
χc1(2P) X A
According to Ref. [5] such g2/8π results in BR(X → D0D¯∗0 +D¯0D∗0) ≈ 0.3 [8].
A
Thus, we expect that a number of unknown mainly two-gluon decays of X(3872) into
non-D∗0D¯0+c.c. states are considerable [9]. For details see Ref. [5]. The discovery of these
decays would be the strong (if not decisive) confirmation of our scenario.
As for BR(X → ωJ/ψ) ∼ BR(X → ρJ/ψ), Eq. (2), this could be a result of dynamics.
In our scenario the ωJ/ψ state is produced via the three gluons, see Fig. 2.
As for the ρJ/ψ state, it is produced both via the one photon, see Fig. 3, and via the
three gluons (via the contribution ∼ m −m ), see Fig. 2.
u d
4
q¯
q
c c
c¯ c¯
FIG. 2: The three-gluon production of the ω and ρ mesons (via the contribution ∼ m −m ).
u d
All possible permutations of gluons are assumed.
ρ
c c
c¯ c¯
FIG. 3: The one-photon production of the ρ meson. All possible permutations of photon are
assumed.
′ ′
Close to our scenario is an example of the J/ψ → ρη and J/ψ → ωη decays. According
to Ref. [2]
BR(J/ψ → ρη′) = (1.05±0.18)·10−4 and BR(J/ψ → ωη′) = (1.82±0.21)·10−4. (17)
Note that in the X(3872) case the ω meson is produced on its tail (m − m = 775
X J/ψ
MeV), while the ρ meson is produced on a half.
¯
It is well known that the physics of charmonium (cc¯) and bottomonium (bb) is similar.
Let us compare the already known features of X(3872) with the ones of Υ (2P).
b1
Recently, the LHCb Collaboration published a landmark result [10]
3
BR(X → γψ(2S)) ω
ψ(2S)
= C = 2.46±0.7, (18)
X
BR(X → γJ/ψ) ωJ/ψ !
5
where ω and ω are the energies of the photons in the X → γψ(2S) and BR(X →
ψ(2S) J/ψ
γJ/ψ) decays, respectively.
On the other hand, it is known [2] that
3
BR(χ (2P) → γΥ(2S)) ω
b1 Υ(2S)
= C = 2.16±0.28, (19)
BR(χb1(2P) → γΥ(1S)) χb1(2P) ωΥ(1S)!
where ω and ω are the energies of the photons in the χ (2P) → γΥ(2S) and
Υ(2S) Υ(1S) b1
χ (2P) → γΥ(1S) decays, respectively.
b1
Consequently,
C = 136.78±38.89 (20)
X
and
C = 80±10.37 (21)
χb1(2P)
as all most lucky versions of the potential model predict for the quarkonia, C ≫ 1 and
χc1(2P)
C ≫ 1.
χb1(2P)
According to Ref. [2]
BR(χ (2P) → ωΥ(1S)) = 1.63±0.4 %. (22)
b1 0.34
(cid:16) (cid:17)
If the one photon mechanism dominates in the X(3872) → ρJ/ψ decay, see Fig.3, then
one should expect
BR(χ (2P) → ρΥ(1S)) ∼ (e /e )2 ·1.6% = (1/4)·1.6% = 0.4%, (23)
b1 b c
where e and e are the charges of the c and b quarks, respectively.
c b
If the three gluon mechanism (its part ∼ m −m ) dominates in the X(3872) → ρJ/ψ
u d
decay, see Fig.2, then one should expect
BR(χ (2P) → ρΥ(1S)) ∼ 1.6%. (24)
b1
We believe that discovery of a significant number of unknown decays of X(3872) into
non-D∗0D¯0+c.c. states and discovery of the χ (2P) → ρΥ(1S) decay could decide destiny
b1
of X(3872).
Once more, we discuss the scenario where the χ (2P) charmonium sits on the D∗0D¯0
c1
thresholdbutnotamixingofthegiantD∗D¯ moleculeandthecompactχ (2P)charmonium,
c1
see, for example, Refs. [11], [12], and references cited therein. Note that the mixing of such
6
states requests the special justification. That is, it is necessary to show that the transition of
the giant molecule into the compact charmonium is considerable at insignificant overlapping
of their wave functions. Such a transition ∼ V /V and a branching ratio of a
χc1(2P) X(3872)
q
decay via such a transition ∼ V /V .
χc1(2P) X(3872)
We are grateful to A.E. Bondar, M. Karliner, B.A. Kniehl, and J.L. Rosner for useful
discussions.
This work was supported in part by RFBR, Grant No 13-02-00039, and Interdisciplinary
project No 102 of Siberian division of RAS.
[1] R. Aaij et al. (LHCb Collaboration), Eur. Phys. J. C. 72, 1972 (2012).
[2] K.A. Olive et al. (Particle Data Group), Chin. Phys. C 38, 090001 (2014).
[3] C.-Z. Yuan (Belle Collaboration), arXiv: 0910.3138 [hep-ex], Proceedings of the XXIX
PHYSICS IN COLLISION, 2009, Kobe, Japan.
[4] S.K. Choi et al. (Belle Callaboration), Phys. Rev. D 84, 052004 (2011).
[5] N.N. Achasov and E.V. Rogozina, Pis’ma v ZhETF 100, 252 (2014) [JETP Lettrs 100, 227
(2014)].
[6] When (mD¯0 + mD∗0)2 < s < (mD+ + mD∗−)2, the renormalization of mass has the form
m2 −m2 −Re(ΠD¯0D∗0(m2 ))−ΠD−D∗+(m2 )= 0.
χc1(2P) X X X X X
[7] The exact formulae of Re(Π (m2 ))−Π (s) in all regions of s can be found in Ref. [5].
X X X
[8] The assumption of the determining role of the D∗D¯ +c.c. channels in the shift of the mass of
∗
the χ (P) meson is based on the following reasoning. Let us imagine that D and D mesons
c1
∗
are light, for example, as the K and K mesons. Then the width of X(3872) meson is equal
50 MeV for g2/8π = 0.2GeV2 that much more than the width of its decay into all non-
A
D∗0D¯0 channels, Γ < 1.2 MeV. That is, in our case the coupling of the X(383) meson with
the D∗D¯ +c.c. channels is rather strong.
[9] Note that in the χ (1P) case the width of the two-gluon decays equals 0.56 MeV [2] that
c1
agrees with Γ < 1.2 MeV satisfactory.
[10] R. Aaij et al. (LHCb Collaboration), Nucl. Phys. B 886, 665 (2014).
[11] M. Karliner and J.L. Rosner, Phys. Rev. D 91, 014014 (2015).
[12] M. Butenschoen, Z.-G. He, and B.A. Kniehl, Phys. Rev. D 88, 011501(R)(213).
7